Commun. Math. Phys. 279, 1–30 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0419-3
Communications in
Mathematical Physics
Extended Weak Coupling Limit for Pauli-Fierz Operators Jan Derezinski ´ 1 , Wojciech De Roeck2 1 Department of Mathematical Methods in Physics, Warsaw University, Ho˙za 74,
00-682, Warszawa, Poland. E-mail:
[email protected]
2 Instituut voor Theoretische Fysica, K.U.Leuven, Belgium.
E-mail:
[email protected] Received: 23 October 2006 / Accepted: 29 June 2007 Published online: 8 February 2008 – © Springer-Verlag 2008
Abstract: We consider the weak coupling limit for a quantum system consisting of a small subsystem and reservoirs. It is known rigorously since [10] that the Heisenberg evolution restricted to the small system converges in an appropriate sense to a Markovian semigroup. In the nineties, Accardi, Frigerio and Lu [1] initiated an investigation of the convergence of the unreduced unitary evolution to a singular unitary evolution generated by a Langevin-Schrödinger equation. We present a version of this convergence which is both simpler and stronger than the formulations which we know. Our main result says that in an appropriately understood weak coupling limit the interaction of the small system with environment can be expressed in terms of the so-called quantum white noise.
1. Introduction One of the main goals of mathematical physics is to justify various approximate effective models used by physicists by deriving them as limiting cases of more fundamental theories. This paper is devoted to a class of such models that one sometimes calls quantum Langevin dynamics. We show that quantum Langevin dynamics arise naturally as the limit of a dynamics of a small system weakly interacting with a reservoir where not only the small system, but also the reservoir is taken into account. We will call this version of a weak coupling limit the extended weak coupling limit, to differentiate it from the better known reduced weak coupling limit, which involves only the dynamics reduced to the small system. To our knowledge, the main idea of extended weak coupling limit first appeared in the literature in the work of Accardi, Frigerio and Lu in [1] under the name of stochastic limit. Our approach is inspired by their work, nevertheless we think that it is both simpler and more powerful.
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J. Derezi´nski, W. De Roeck
The reader may also find it useful to compare the present work with our previous paper [12], which describes the extended weak coupling limit on a relatively simple (and less physical) example of the Friedrichs model. [12], apart from presenting results, which we believe are mathematically interesting in their own right, can be viewed as a preparatory exercise for the present work. 1.1. Quantum Markov semigroups. Before we discuss quantum Langevin dynamics, we should recall a better known class of effective dynamics – that of quantum Markov semigroups (or, in other words, completely positive unity preserving time-continuous semigroups). They are often used as a phenomenological description of quantum systems. It is well known that every quantum Markov semigroup on B(K), where K is a finite dimensional Hilbert space, can be written as et L , where L can be written in the so-called Lindblad form [24] L(S) = −i(ϒ S − Sϒ ∗ ) + ν ∗ Sν,
S ∈ B(K),
(1.1)
ν is an operator from K to K ⊗ h for some auxiliary Hilbert space h and ϒ is an operator on K satisfying − iϒ + iϒ ∗ = −ν ∗ ν. (1.2) Note that given L, the operators ϒ and ν are not defined uniquely. 1.2. Reduced weak coupling limit. It is generally assumed that only reversible (unitary) dynamics appear in fundamental quantum physics. Nevertheless, in phenomenological approaches researchers often apply non-unitary quantum Markov semigroups to describe irreversible phenomena. A possible justification for their use is provided by the so-called weak coupling limit, an idea that goes back to Pauli and van Hove [22], and was made rigorous in an elegant work of E. B. Davies [10]. Davies proved that if a small quantum system is weakly coupled to the environment, then the reduced dynamics in the interaction picture, after rescaling the time as λ−2 t, converges to a quantum Markov semigroup defined on the observables of the small system. To be more specific, consider a system given by a Hilbert space H := K ⊗ s (HR ), where K is a finite dimensional Hilbert space, HR is the 1-particle space of the reservoir and s (HR ) is the corresponding bosonic Fock space. The composite system is described by the dynamics generated by the self-adjoint operator Hλ = K ⊗ 1 + 1 ⊗ d(HR ) + λ(a ∗ (V ) + a(V )).
(1.3)
Here K describes the Hamiltonian of the small system, d(HR ) describes the dynamics of the reservoir expressed by the second quantization of a self-adjoint operator HR on HR , and a ∗ (V )/a(V ) describe the interaction between the small system and the reservoir, which we assume to be given by the creation/annihilation operators of an operator V ∈ B(K, K ⊗ HR ). The notation that we use to define Hλ is explained only in Sect. 2 and may be unfamiliar to some of the readers. Therefore let us describe the operators appearing in (1.3) with perhaps a better known (although less compact) notation. To this end it is convenient to identify HR with L 2 (, dξ ), for some measure space (, dξ ), so that one can introduce aξ∗ /aξ – the usual creation/annihilation operators describing bosonic excitations of the reservoir. Let HR be the multiplication operator by a real function
Extended Weak Coupling Limit for Pauli-Fierz Operators
3
ξ → x(ξ ) and let ξ → v(ξ ) ∈ B(K) be the function describing the operator V . Then we have an alternative notation d(HR ) = x(ξ )aξ∗ aξ dξ, a ∗ (V ) = v(ξ )aξ∗ dξ, a(V ) = v ∗ (ξ )aξ dξ. Operators of the form (1.3) are often used in quantum physics in phenomenological descriptions of a small quantum system interacting with an environment. Some varieties of (1.3) are known under such names as the spin-boson, Fröhlich, Nelson and polaron Hamiltonian. Following [11], we will call operators of the form (1.3) Pauli-Fierz operators. (Note, however, that some authors use this name in a slightly different meaning.) The vacuum vector in s (HR ) will be denoted by . Let IK : K → H denote the ∗ :H→K isometric embedding, which maps a vector φ ∈ K on φ ⊗ ∈ H. Note that IK ∗ · I is the conditional expectation from B(H) onto B(K). equals 1K ⊗|, and IK K One version of the result of Davies says that under some mild assumptions the following limit exists: −2 t K
t (S) := lim eiλ λ0
−2 t H 0
∗ iλ = lim IK e λ0
−2 H λ
∗ −itλ IK e
−2 H λ
e−itλ
−2 H λ
S ⊗ 1 eitλ
−2 H λ
S ⊗ 1 eitλ
−2 t K
IK e−iλ
−2 t H 0
e−iλ
IK ,
(1.4)
and t is a quantum Markov semigroup. Thus we obtain a, possibly irreversible, quantum Markov semigroup as a limit of a family of reversible, physically realistic dynamics. We also obtain a concrete expression for the generator of t . More precisely, ϒ and ν appearing in (1.1) are uniquely defined in terms of K , HR and V . In the literature on both the reduced and the extended weak coupling limit, one usually considers a nontrivial reference state for the reservoir, whereas we reduce our treatment to a vector state. This is justified since one can always represent the reservoir state as a vector state via the GNS construction. In particular, in the case of a thermal bosonic state, we can use the Araki-Woods representations of the CCR, so that the reservoir state is given by the Fock vacuum. The free reservoir Hamiltonian and the interaction are modified appropriately. For this reason, it is not always appropriate to call (1.3) a “Hamiltonian”. In typical applications that we have in mind, the environment is a collection of heat baths at various positive temperatures, and then it is natural to take d(HR ) to be the sum of their Liouvilleans. In this case, Hλ is not bounded from below, and it probably should not be called a “Hamiltonian”. On the other hand, the name “Liouvillean” is not appropriate either, since on the small system K is actually the Hamiltonian, not the Liouvillean. Following the terminology introduced in [11], in such a case Hλ should be called a semi-Liouvillean.
1.3. Quantum Langevin dynamics. It is well known that a 1-parameter semigroup of contractions on a Hilbert space can be written as a compression of a unitary group. This unitary group is called a dilation of the semigroup.
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A similar fact is true in the case of a quantum Markov semigroup. It has been noticed that every such semigroup can be written as ∗ −it Z e S ⊗ 1 eit Z IK , S ∈ B(K). t (S) = IK
(1.5)
Here, Z is a self-adjoint operator on a Hilbert space Z = K⊗s (ZR ) for some 1-particle space ZR and IK : K → Z is defined analogously as before. Unfortunately, in the literature there seems to be no consistent and uniform terminology for this dilation. A possible name for the unitary dynamics eit Z seems to be a Langevin-Schrödinger dynamics or a stochastic Schrödinger dynamics for the semigroup t . The corresponding dynamics in the Heisenberg picture, that is e−it Z · eit Z , will be called a quantum Langevin dynamics or a quantum stochastic dynamics for the semigroup t . The first construction of a quantum Langevin dynamics was probably given by Hudson and Parthasaraty. In [23] they introduced the so-called quantum stochastic differential equation - a generalization of the usual stochastic differential equation known from the Ito calculus. The group e−it Z is then given by the solution to this equation. If the operators ϒ and ν that appear in the generator of t written in the Lindblad form (1.1) are given, then there exists a canonical construction of the space Z and of a Langevin-Schrödinger dynamics eit Z on Z, which apart from (1.5) satisfies the condition ∗ −it Z e IK . e−itϒ = IK
(1.6)
Thus e−it Z is a dilation of the contractive semigroup e−itϒ and e−it Z · eit Z is a dilation of the quantum Markov semigroup et L . In this construction, at least formally, Z can be written in the form of a Pauli-Fierz operator Z=
1 1 1 (ϒ + ϒ ∗ ) + d(Z R ) + √ a(|1 ⊗ ν) + √ a ∗ (|1 ⊗ ν). 2 2π 2π
(1.7)
The interaction that appears in (1.7) is quite singular and difficult mathematically. It is an example of a so-called quantum white noise [6]. Equation (1.5) suggests that quantum Langevin dynamics have perhaps more physical content than being just a mathematical device, and could be used as effective dynamics describing a small system interacting with environment. In fact, physicists (see e.g. [18]) often use such quantum Langevin dynamics to describe the interaction of a small system with an environment, e.g. with several heat baths. Quantum Langevin dynamics are also often used to describe processes involving “continuous quantum measurements” [7]. One can then introduce observables describing “measurements performed in a given interval of time”. Observables corresponding to measurements in non-overlapping time intervals commute, which can be a reasonable assumption in some idealized situations. Note that the generator of a Langevin-Schrödinger dynamics is necessarily unbounded from below. This is often put forward as an argument against physical relevance of quantum Langevin dynamics. This argument is actually not justified, since unbounded from below generators of dynamics appear naturally in physics, especially in positive temperatures. We have seen such a situation when we discussed (1.3), since semiLiouvilleans are typically unbounded from below. (See also a remark at the end of Subsect. 1.2).
Extended Weak Coupling Limit for Pauli-Fierz Operators
5
1.4. Extended weak coupling limit. In [1], it was proposed by Accardi et al. that one could extend the idea of the weak coupling limit from the reduced dynamics to the dynamics on the whole system, and as a result one can obtain a justification of using quantum Langevin dynamics to describe quantum systems. They called their version of the weak coupling limit the stochastic limit. In our opinion, this name is not the best chosen, since the reduced weak coupling limit is just as “stochastic” as the extended one. Therefore we will use the name extended weak coupling limit. The reduced weak coupling limit in the form considered by Davies has a rather clean mathematical formulation. Therefore, it was quickly appreciated by the mathematical physics community. The extended weak coupling limit is inevitably somewhat more complicated, in particular since it involves constructions that are, to a certain extent, arbitrary. Nevertheless, we believe that the idea of the extended weak coupling limit is valuable and sheds light on models used in physics, especially in quantum optics and quantum measurement theory. In our paper we would like to state and prove a new version of the extended weak coupling limit. We start again from a dynamics generated by a “Pauli-Fierz operator” (1.3). As we discussed above, the reduced weak coupling limit leads to a quantum Markov semigroup with the generator given in a Lindblad form involving the operators ϒ and ν. Given these data, we have a canonical construction of a quantum Langevin-Schrödinger dynamics e−it Z acting on the “asymptotic space” Z such that (1.5) and (1.6) are satisfied. We also construct an appropriate identification operator (Jλ ), which is a partial isometry mapping the physical space H into the asymptotic space Z. Its main role is to scale the physical energy. There is some arbitrariness in the construction of the identification operator, since the frequencies away from the Bohr frequencies (differences of eigenvalues of K ) do not matter in the limit λ 0. Finally, one needs what we call the “renormalizing operator” Z ren , which takes care of the trivial part of the dynamics involving the eigenvalues of K . The main result of our paper can be stated as −2 t Z
s∗ − lim eiλ λ0
ren
−2 t H λ
(Jλ )e−iλ
(Jλ )∗ = e−it Z ,
(1.8)
−2
where s∗ − lim denotes the strong* limit. Thus e−it (Z +λ Z ren ) can be viewed as the effective dynamics in the limit of λ 0. Note that in the Heisenberg picture we obtain for any B ∈ B(Z), −2 t H λ
s∗ − lim eit Z ren (Jλ∗ )e−iλ λ0
−2 H λ
(Jλ )B(Jλ∗ )eitλ
(J )∗ e−it Z ren
= e−it Z Beit Z . Replacing B with S ⊗ 1, pretending Jλ is unitary (which is justified, see e.g. Remark ∗ · I of both sides and 4.5 or expression (6.33)), taking the conditional expectation IK K using (1.5) we retrieve (1.4) – the reduced weak coupling limit. One can also choose B of the form 1 ⊗ A such that the strong limit (Jλ )B(Jλ∗ ) as λ 0, exists. In that case, one can study fluctuations of reservoir quantities, see Theorem 5.7.
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We can summarize the results of our paper in the following diagram (w.c.l. stands for weak coupling limit): physical dynamics ↓ reduction reduced physical dynamics
extended w.c.l. −→
quantum Langevin dynamics ↓ reduction ↑ dilation
quantum Markov semigroup reduced w.c.l. + specific decomposition of Lindblad −→ generator
1.5. Comparison with previous results. As mentioned already, we are surely not the first to come up with the concept of the extended weak coupling limit. Although the original idea is attributed to Spohn [29], the field was pioneered by Accardi et al. in [1] and a long list of works on the subject can be found in the book [3]. Recently, an interesting generalization has been made by [20]. On the heuristic level, the ideas of the extended weak coupling limit have been expressed by some physicists, e.g. by Gardiner and Collett, see [17] and Sect. 2.5 of [7]. The same idea was also applied to the low-density limit in [28] and [4], see also [5]. (The “reduced low density limit ”has been put on rigorous footing in [15].) Most previous results we are aware of have the following form: For a Hilbert space R, let ( f ) ∈ s (R) be the exponential vector for the 1-particle vector f ∈ R: (1.9) ( f ) = exp a ∗ ( f ) . −2
−2
Let u, v ∈ K, f, g ∈ HR , s1 < t1 , s2 < t2 ∈ R and put Wtλ := eiλ t H0 e−iλ t Hλ . Then, with all symbols having the same meaning as in the introduction above, ⎞ ⎛ ⎞ ⎛ t1/λ2 t2/λ2 ⎟ λ ∗ ⎜ ⎟ ⎜ −iu HR λ −iu HR e f du ⎠ (Wt ) (S ⊗ 1)Wt v ⊗ ⎝λ e gdu ⎠ u ⊗ ⎝λ s1 /λ2
s2 /λ2
→ u ⊗ (1[s1 ,t1 ] ⊗ f ) Wt∗ (S ⊗ 1)Wt v ⊗ (1[s2 ,t2 ] ⊗ g) ,
λ→0
(1.10)
where Wt is the solution of an appropriate Langevin Schrödinger differential equation on the space K ⊗ s (L 2 (R) ⊗ HR ) and 1[·,·] is the indicator function of the interval [·, ·]. Note that both our approach and (1.10) express essentially the same physical idea. The scaling that we use to define Jλ is implicit in (1.10). The main advantages of our approach with respect to the previous works are 1) The asymptotic space K ⊗ s (L 2 (R) ⊗ HR ) considered in (1.10) is much larger than the asymptotic space that we use (which is introduced in Subsect. 4.3). One can argue that our choice is more natural and “tailor-made” for the problem at hand – it closely resembles the original physical space without introducing unnecessary degrees of freedom. 2) We prove convergence in the ∗-strong sense, instead of (as outlined above) convergence of matrix elements of a class of rescaled coherent vectors. This is mathematically cleaner and more flexible.
Extended Weak Coupling Limit for Pauli-Fierz Operators
7
3) Our approach allows to consider also limits of certain reservoir observables, see in particular Theorem 5.7. 4) We highlight the clear connection between the work of Davies [10] and Dümcke [14], and extended weak coupling limits. The latter follows rather easily from the results in [10] and [14]. A less important point of difference is the following: In the early works on the weak coupling limit, quasifree reservoirs were fermionic. If one chooses bosonic reservoirs, as we do, one has to control the unboundedness of the interaction term (since the bosonic creation and annihilation operators are unbounded). Although this is not difficult, see Theorem 4.1, we know of no place in the literature on the weak coupling limit where this difficulty is addressed. Of course, it is possible (and easy) to describe a version of our result where the Hamiltonian HR is fermionic. From the physical point of view, our results justify a lot of the manipulations one does with quantum Langevin dynamics (this is discussed in detail in [13]). In particular, Theorem 5.7 allows to identify fluctuations of reservoir number operators with limits of reservoir observables. These reservoir number operators (more specifically: their fluctuations) are heavily studied objects, see e.g. [7,8,27].
1.6. Outline. In Sect. 3, we construct a Langevin-Schrödinger dynamics associated with a specific decomposition of a Lindblad generator. In the first subsection of Sect. 4 we introduce the class of our physical models considered in our paper — Pauli-Fierz operators. In the remaining subsections of Sect. 4 we describe how to connect the setup of the physical model with that of the corresponding quantum Langevin dynamics. Our results are listed in Sect. 5 and their proofs are postponed to Sect. 6.
2. Preliminaries and Notations We will use the formalism of second quantization, following the conventions adopted in [11]. For a Hilbert space R and n ∈ N, we recall the projector Symn , which projects elements of the tensor power ⊗n R onto symmetric tensors. Its range will be denoted sn (R) – it is the n-particle subspace of the bosonic Fock space over R. The symmetric (bosonic) second quantization of R is hence defined as ∞
s (R) = ⊕ sn (R).
(2.1)
n=0
Note that we use the convention that ⊗ and ⊕ denote the tensor product and the direct sum in the category of Hilbert spaces. Sometimes we will use their algebraic counterparts. If D1 is a subspace of a Hilbert space R, then al
n
al
n
s (D1 ) = (⊗ D1 ) ∩ sn (R),
(2.2)
al
where ⊗ denotes the algebraic tensor product. We will often need al n s (D1 ) = Span ψ | ψ ∈ s (D1 ), n ∈ N .
al
(2.3)
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J. Derezi´nski, W. De Roeck
For R ∈ B(K, K ⊗ R), we heavily use the generalized creation and annihilation operators a(R) and a ∗ (R), as defined in [11]. Actually, we need even a slightly more general definition which is given now. al Assume that D1 is a dense subspace of the Hilbert space R and R ∗ : K ⊗ D1 → K ∗ is an unbounded operator. Let R stand for the adjoint of R in the sense of quadratic forms. (Note that the adjoint in the sense of forms is different from the adjoint in the sense of operators.) Define for all n ∈ N, √ al al n a(R)ψ := n R ∗ ⊗ Symn−1 ψ, ψ ∈ K ⊗ s (D1 ); (2.4) a(R) is well defined as an unbounded operator and it defines a quadratic form on al al K ⊗ s (D1 ). Denote by a ∗ (R) its adjoint in the sense of quadratic forms. We write for the vacuum vector in s (R): = 1 ⊕ 0 ⊕ 0 ⊗ 0 ⊕ ... .
(2.5)
s−lim will denote the strong limit. We say that the operators Aλ∈R+ ∈ B(R) converge ∗-strongly to A ∈ B(R) (notation: s∗ − limλ↓0 Aλ = A) if s − lim Aλ = A λ↓0
and
s − lim A∗λ = A∗ . λ↓0
(2.6)
If A is an operator, we will write A :=
1 1 (A + A∗ ), A := (A − A∗ ). 2 2i
Our typical Hilbert space will be the tensor product of two Hilbert spaces. We will usually write A, B for A ⊗ 1 and 1 ⊗ B. 3. Dilations 3.1. Unitary dilation of a contractive semigroup. Let K be a Hilbert space and let the family t∈R+ be a contractive semigroup on K:
t s = t+s ,
t ≤ 1,
t, s ∈ R+ .
(3.1)
Definition 3.1. We say that (Z, IK , Ut∈R ) is a unitary dilation of t∈R+ if 1) Z is a Hilbert space and Ut∈R ∈ B(Z) is a unitary one-parameter group; 2) K ⊂ Z and IK is the embedding of K into Z; 3) for all t ∈ R+ , ∗ U t I K = t . IK
(3.2)
Assume that K is finite-dimensional and the semigroup t continuous. Then there exists a dissipative operator −iϒ ∈ B(K), − iϒ + iϒ ∗ ≤ 0, such that t = e−itϒ .
(3.3)
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3.2. Quantum Langevin dynamics. Let the family t∈R+ be a semigroup on B(K): t s = t+s ,
t, s ∈ R+ .
(3.4)
Definition 3.2. We say that (Z, IK , Ut∈R ) is a Langevin-Schrödinger dynamics for t∈R+ if 1) Z˜R is a Hilbert space and Ut∈R is a one-parameter unitary group on Z := K⊗Z˜R ; 2) is a normalized vector in Z˜R and IK (ψ) := ψ⊗ ∈ Z is the corresponding embedding of K into Z; 3) for all t ∈ R+ and all S ∈ B(K), ∗ IK U−t S⊗1Ut IK = t (S).
(3.5)
The Heisenberg dynamics eit Z · e−it Z corresponding to a Langevin-Schrödinger dynamics will be called a quantum Langevin dynamics. Definition 3.3. We say that t∈R+ is a quantum Markov semigroup iff it is a semigroup on B(K) such that for any t ∈ R+ the map t is completely positive and preserves the unity. Clearly, if a semigroup t admits a Langevin-Schrödinger dynamics in the sense of Definition 3.2, then it is a quantum Markov semigroup. Again, assume that K is finite dimensional. Assume that t is a continuous quantum Markov semigroup, so that we can define its generator L and we have t = et L . Recall that then there exists a dissipative operator ϒ on K, another finite dimensional Hilbert space h and an operator ν ∈ B(K, K ⊗ h), satisfying the condition − iϒ + iϒ ∗ = −ν ∗ ν, such that
L(S) = −i(ϒ S − Sϒ ∗ ) + ν ∗ Sν,
(3.6) S ∈ B(K).
(3.7)
Remark 3.1. If we choose an orthonormal basis b1 , . . . , bd in h, then ν can be represented by a family of operators ν1 , . . . , νd ∈ B(K), and then (3.7) can be rewritten as L(S) = −i(ϒ S − Sϒ ∗ ) +
d
ν ∗j Sν j ,
S ∈ B(K).
(3.8)
j=1
3.3. Construction of a Langevin-Schrödinger dynamics. Let K, h be finite dimensional Hilbert spaces, ϒ a self-adjoint operator on K and ν an operator from K to K ⊗ h. Setting ϒ := ν ∗ ν we obtain a dissipative operator ϒ := ϒ + iϒ on K. Given the data (K, ϒ, h, ν) as above, we will construct a dilation for eitϒ , which ¨ at the same time is a Langevin-Schrdinger dynamics for et L . Introduce the operator Z R on ZR := L 2 (R) ⊗ h ∼ = L 2 (R, h) as the operator of multiplication by the variable x ∈ R: (Z R f )(x) := x f (x). Put
Z = K ⊗ s (ZR ).
(3.9)
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We define an unbounded linear functional on L 2 (R) with domain L 1 (R) ∩ L 2 (R), denoted 1|, by the obvious prescription 1| f =
f.
R
By |1 , we denote the adjoint of 1| in the sense of forms. al We will also use the quadratic form from K to K ⊗ L 2 (R, h) ∩ L 1 (R, h) : |1 ⊗ ν.
(3.10)
Consider al al
D := K ⊗ s (Dom(Z R )) ,
(3.11)
which is a dense subspace of Z. As outlined in Sect. 2, using the fact that L 2 (R, h) ∩ L 1 (R, h) ⊂ Dom(Z R ), we can define the quadratic forms a(|1 ⊗ ν) and a ∗ (|1 ⊗ ν) on D. Hence, also the following expressions are quadratic forms on D: 1
1
− 21
− 21
Z + = ϒ + (2π )− 2 a(|1 ⊗ ν) + (2π )− 2 a ∗ (|1 ⊗ ν) + d(Z R ), Z − = ϒ ∗ + (2π )
a(|1 ⊗ ν) + (2π )
a ∗ (|1 ⊗ ν) + d(Z R ).
(3.12) (3.13)
It will be convenient to choose a family b j∈J ∈ h and C j∈J ∈ B(K) indexed by a finite index set J such that ν=
C j ⊗ |b j .
(3.14)
j∈J
This can always be done, of course in many ways. Define, analogously to (2.4), a ∗ (eit Z R |1 ⊗ b j ),
a(eit Z R |1 ⊗ b j ),
(3.15)
as quadratic forms on D. Note the equality a(|1 ⊗ ν) =
C ∗j ⊗ a(|1 ⊗ b j ).
(3.16)
j∈J
For a ≤ b, let n [a, b] ⊂ Rn be the simplex n [a, b] := {(t1 , . . . , tn ) : a < t1 < · · · < tn < b} .
(3.17)
Set C +j = C j , ∗ C− j = Cj.
(3.18)
Extended Weak Coupling Limit for Pauli-Fierz Operators
11
Now we combine these objects into something that is a priori a quadratic form, but turns out to be a bounded operator. For t ≥ 0 we define Ut := e
−itd(Z R )
∞
×(−i)n (2π ) ×
j1 ,..., jn ∈J 1 ,...,n ∈{+,−}
n=0 [0,t] n − n2
dtn · · · dt1
e−i(t−tn )ϒ C jnn e−i(tn −tn−1 )ϒ a ∗ (eit p Z R |1 ⊗ b j p )
· · · C j11 e−i(t1 −0)ϒ a(eit p Z R |1 ⊗ b j p );
p =1,...,n: p =−
p=1,...,n: p =+
U−t := Ut∗ . (In the above expression
(3.19) p=1,...,n: p =+
should be understood as the product over these
indices p = 1, . . . , n that in addition satisfy the condition p = +.) Finally, let IK be the embedding of K ∼ = K ⊗ into K ⊗ s (ZR ). Theorem 3.2. Let Z ± be as defined in (3.12) and Ut as defined in (3.19). 1) The one-parameter family of quadratic forms Ut extends to a strongly continuous unitary group on Z and does not depend on the decomposition (3.14). 2) For ψ, ψ ∈ D, the function R t → ψ|Ut ψ is differentiable away from t = 0, its derivative t → dtd ψ|Ut ψ is continuous away from 0 and at t = 0 it has the left and the right limit equal respectively to −iψ|Z + ψ = lim t −1 ψ|(Ut − 1)ψ ,
(3.20)
−iψ|Z − ψ = lim t −1 ψ|(Ut − 1)ψ .
(3.21)
t↓0
t↑0
3) The triple (Z, Ut , IK ) is a unitary dilation of the semigroup e−itϒ on K: ∗ Ut IK = e−itϒ . IK
(3.22)
4) The triple (Z, Ut , 1K ) is a Langevin-Schrödinger dynamics for the semigroup et L on B(K): ∗ IK U−t (S ⊗ 1)Ut IK = et L (S), S ∈ B(K). (3.23) We will say that Ut constructed in the above theorem is the Langevin-Schrödinger dynamics given by the data (K, ϒ, h, ν). Note that Ut can be written as e−it Z for a uniquely defined self-adjoint operator Z on Z. Clearly, D is not contained in the domain of Z and the quadratic forms Z + and Z − are not generated by the operator Z (in fact, they are even not self-adjoint). On an appropriate domain, Z has the formal expression 1
1
Z = ϒ + (2π )− 2 a(|1 ⊗ ν) + (2π )− 2 a ∗ (|1 ⊗ ν) + d(Z R ),
(3.24)
which is the obvious “self-adjoint compromise” between Z − and Z + . This expression is formal since one needs a suitable regularization to give it a precise meaning. Such a regularization, under an additional assumption on the commutativity of the small system operators, is discussed e.g. in [9]. See also [21,31].
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J. Derezi´nski, W. De Roeck
3.4. Alternative form of Langevin-Schrödinger equations. Proofs of Theorem 3.2 are contained in the literature, see e.g. [25]. In any case, this theorem involves well defined formulas and its proof follows by straightforward computations, which we leave to the reader. Nevertheless, we would like to mention a slightly different (though equivalent) form of Langevin-Schrödinger dynamics, which is closer to those usually appearing in the literature. Let G denote the normalized Fourier transform on L 2 (R): 1 G f (s) := (2π )− 2 f (x)e−isx dx. We can treat it as a unitary operator on ZR . We second quantize G, obtaining an operator (G), which can be treated as an operator on Z. Set Zˆ R := G Z R G ∗ , Uˆ t := (G)Ut (G)∗ .
(3.25)
Note that ˆ e−itd( Z R ) = exp −t dsd . Then for t ≥ 0 the formula (3.19) transforms into ∞ Uˆ t = exp −t dsd dtn · · · dt1 n=0 [0,t] n
(−i)n
j1 ,..., jn ∈J 1 ,...,n ∈{+,−}
×e−i(t−tn )ϒ C jnn e−i(tn −tn−1 )ϒ · · · C j11 e−i(t1 −0)ϒ × a ∗ (δtk ⊗ b jk ) a(δtk ⊗ b jk ),
(3.26)
k =1,...,n: k =−
k=1,...,n: k =+
where δt denotes the deltafunction at t ∈ R, and (3.26) should be understood as a quadratic form between appropriate dense spaces. Equation (3.26) is sometimes referred to in the literature as the representation by integral kernels. It was introduced by Maassen [25]. See also [30,31,6,19] and Sect. VI, 3.2 of [26]. Differentiating (3.26) with respect to time we obtain (at least formally) i
d itd( Zˆ R ) ˆ ˆ e Ut = ϒ + a ∗ (δt ⊗ ν) eitd( Z R ) Uˆ t dt ˆ + ν ∗j eitd( Z R ) Uˆ t a(δt ⊗ b j ),
(3.27)
j∈J
which essentially coincides with what is known in the literature under the name of the stochastic (or Langevin) Schrödinger equation. 4. The Pauli-Fierz Operator 4.1. Definitions and assumptions. Let H = K ⊗ s (HR ), where K, HR are Hilbert spaces. We assume that K is finite-dimensional. Fix a self-adjoint operator HR on HR and a self-adjoint operator K on K. The operator H0 on H given as H0 = K + d(HR )
Extended Weak Coupling Limit for Pauli-Fierz Operators
13
will be called the free Pauli-Fierz operator. We choose a V ∈ B(K, K ⊗ HR ) and we recall the generalized creation and annihilation operators a(V ) and a ∗ (V ) introduced in Sect. 2. Theorem 4.1. Set H I (t) := e−it H0 (a ∗ (V ) + a(V ))eit H0 . Then Wλ,t ψ =
∞
dtn · · · dt1 eit H0 (iλ)n H I (tn ) · · · H I (t1 )ψ
(4.1)
n=0 [0,t] n al al
is well defined for all ψ ∈ K ⊗ s (HR ). Wλ,t extends to a 1-parameter unitary group al al
on K ⊗ s (HR ) with a self-adjoint generator Hλ . The finite particle space K ⊗ (HR ) al al belongs to the domain of Hλ and on K ⊗ (HR ), Hλ = H0 + λ a(V ) + a ∗ (V ) . (4.2) Hλ will be called the full Pauli-Fierz operator. We write K = k1Kk ,
(4.3)
k∈sp(K )
where k, 1Kk , are the eigenvalues and the spectral projections of K . We collect all Bohr frequencies in a set F: F := ω ∈ R ω = k − k for some k, k ∈ spK . (4.4) We again denote by IK the embedding of K = K ⊗ into H, where ∈ s (HR ) is the vacuum vector. We now list the assumptions that we will need in our construction. Assumption 4.2. For any ω ∈ F there exists a Hilbert space hω and an open set Iω ⊂ R with ω ∈ Iω and an identification Ran1 Iω (HR ) L 2 (Iω ) ⊗ hω , such that HR is the multiplication by the variable x ∈ Iω . We assume that Iω are disjoint for distinct ω ∈ F and we set I := ∪ω∈F Iω . Thus if ⊕ f (x)dx ∈ Ran1 I (HR ),
f I
then (HR f )(x) = x f (x), for almost all x. Assumption 4.3. For any ω ∈ F, there exists a measurable function Iω x → v(x) ∈ B(K, K ⊗ hω )
(4.5)
14
J. Derezi´nski, W. De Roeck
such that for u ∈ K for almost all x ∈ I we have (V u)(x) = v(x)u. Moreover, we assume that v is continuous in F, so that for ω ∈ F we can unambiguously define v(ω) ∈ B(K, K ⊗ hω ). Assumption 4.4. For all S ∈ B(K), dt V ∗ S⊗1 e−it H0 V < ∞.
(4.6)
R+
4.2. Asymptotic reduced dynamics. Let h := ⊕ hω .
(4.7)
ω∈F
We define the map νω : K → K ⊗ hω , √ νω := 2π k, k ∈ spK , ω = k − k
1Kk v(ω)1Kk ,
where v(ω) is well-defined by Assumption 4.3. We also define ν : K → K ⊗ h ν := νω . ω∈F
Under Assumption 4.4, we can define ∞ ϒ := − i 1Kk V ∗ e−it (K +HR −k) V 1Kk dt k∈spK
= −i
(4.8)
0
∞
ω∈F k−k =ω 0
1Kk V ∗ 1Kk e−it (HR −ω) V 1Kk dt.
(4.9)
Remark that −iϒ is a dissipative operator and hence it generates a contractive semigroup on K. Note that ∞ iϒ − iϒ ∗ = 1Kk V ∗ 1Kk e−it (HR −ω) V 1Kk dt ω∈F k−k =ω −∞
= 2π
ω∈F k−k =ω
1Kk v ∗ (ω)1Kk v(ω) 1Kk
= ν ∗ ν,
and thus ϒ and ν satisfy the condition (3.6). Therefore, L(S) = −i(ϒ S − Sϒ ∗ ) + ν ∗ Sν, is the generator of a quantum Markov semigroup.
S ∈ B(K),
(4.10)
Extended Weak Coupling Limit for Pauli-Fierz Operators
15
4.3. Asymptotic space and dynamics. We introduce the asymptotic space and the asymptotic dynamics that we will use in our paper. The asymptotic reservoir one-particle spaces are ZRω := L 2 (R, hω ),
(4.11)
ZR := ⊕ ZRω = L (R, h). 2
ω∈F
(4.12)
For ω ∈ F, we have the orthogonal projections 1Rω : ZR → ZRω . Let Z R be the operator of multiplication by the variable in R on ZR . Clearly, we can construct from (Z, IK , ν, ϒ) the Langevin-Schrödinger dynamics of Theorem 3.2. We denote it by Ut and its generator by Z . Finally, we define a renormalizing Hamiltonian Z ren on Z: (4.13) Z ren := K + d ⊕ ω1Rω . ω∈F
4.4. Scaling. For λ > 0, we define the family of partial isometries Jλ,ω : ZRω = L 2 (R, hω ) → L 2 (Iω , hω ), which on gω ∈ ZRω act as 1 g ( y−ω ), if y ∈ Iω ; (Jλ,ω gω )(y) = λ ω λ2 (4.14) 0, if y ∈ R\Iω . Since L 2 (Iω , hω ) ⊂ HR , Jλ,ω can be viewed as a map from ZR,ω to HR . We have ∗ Jλ,ω Jλ,ω = 1λ−2 (Iω −ω) (Z R )1Rω −→ = 1Rω
∗ Jλ,ω Jλ,ω = 1 Iω (HR ).
strongly
λ↓0
(4.15)
We set Jλ : ZR → HR defined for g = (gω )ω∈F by Jλ,ω gω . Jλ g := ω∈F
Note that Jλ Jλ∗ = 1 I (HR ). In what follows, we will mainly need the second quantized (Jλ ), which will also be used to denote the operator 1 ⊗ (Jλ ) ∈ B(Z, H). Remark 4.5. In the definition of Jλ there is a lot of freedom. What matters is what happens near the Bohr frequencies. In fact, essentially the only requirement on Jλ is that Lemma 6.5 holds and that both Jλ∗ Jλ and Jλ Jλ∗ converge strongly to 1. The form of Jλ also reflects that different frequencies “do not see each other" in the weak coupling limit (see e.g. [16,2] for an explicit discussion). The following fact is immediate: Proposition 4.6. We have −2 t Z
s∗ − lim eiλ λ↓0
ren
−2 t H 0
(Jλ∗ )e−iλ
(Jλ ) = e−itd(Z R ) .
16
J. Derezi´nski, W. De Roeck
5. Results The full dynamics in the interaction picture will be denoted by Tλ (t, t0 ) = eit H0 e−i(t−t0 )Hλ e−it0 H0 .
(5.1)
We start with two versions of older results by Davies about the reduced weak coupling limit. However, in most presentations of this subject contained in the literature the perturbation is assumed to be bounded. This is not the case in Theorem 5.1. Theorem 5.1. Assume Assumptions 4.2, 4.3, 4.4. Let T ≤ ∞. 1) Let ϒ be as defined in (4.8). Then ∗ Tλ (λ−2 t, λ−2 t0 )IK = e−i(t−t0 )ϒ lim IK λ↓0
(5.2)
uniformly for T ≥ t ≥ t0 ≥ −T . 2) Let L be as defined in (4.10). Then ∗ Tλ (λ−2 t, λ−2 t0 ) S⊗1 Tλ (λ−2 t, λ−2 t0 )∗ IK = e(t−t0 )L (S) lim IK λ↓0
(5.3)
uniformly for T ≥ t ≥ t0 ≥ −T . We will prove Theorem 5.1 1) in Subsect. 6.3 – it is an important step of the proof of our main result. Theorem 5.1 2) can be proven by similar arguments, or, which is easier in our framework, it follows immediately from Theorem 5.7. The following result is a version of a result of Dümcke [14]. Apart from its intrinsic interest, we will need it as an important step in the proof of our main result. Theorem 5.2. Assume Assumptions 4.2, 4.3, 4.4 and let T < ∞, ∈ N and S1 , . . . , S ∈ B(K). Then ∗ lim IK Tλ (λ−2 t, λ−2 t )S · · · S2 Tλ (λ−2 t2 , λ−2 t1 )S1 Tλ (λ−2 t1 , λ−2 t0 )IK λ↓0
= e−i(t−t )ϒ S . . . S2 e−i(t2 −t1 )ϒ S1 e−i(t1 −t0 )ϒ
(5.4)
uniformly for ordered times T ≥ t ≥ t ≥ · · · ≥ t1 ≥ t0 ≥ −T . Clearly, Theorem 5.1 1 is a special case of Theorem 5.2, corresponding to = 0, or all Si = 1. Remark 5.3. Strictly speaking, Theorems 5.1 1) and 5.2 are somewhat different from the results in [10] and [14]. In our setup, the latter are consequences of Theorem 5.7 and Theorem 3.2. (see Remark 5.8). Note that the above results did not involve any dilations, nor the identification operator (Jλ ).
Extended Weak Coupling Limit for Pauli-Fierz Operators
17
Our main result describes the extended weak coupling limit for Pauli-Fierz operators and reads Theorem 5.4. Assume Assumptions 4.2, 4.3, 4.4. Let Ut be the Langevin-Schrödinger dynamics constructed from (Z, IK , ν, ϒ) with Z, ν, ϒ defined in Sect. 4.3. Let also Z ren be as defined in Sect. 4.3. Then, s∗ − lim (Jλ∗ )Tλ (λ−2 t, λ−2 t0 )(Jλ ) = eitd(Z R ) Ut−t0 e−it0 d(Z R ) , λ↓0
−2 t Z
s∗ − lim eiλ
ren
λ↓0
−2 t H λ
(Jλ∗ )e−iλ
(Jλ ) = Ut .
(5.5) (5.6)
Remark 5.5. Weaker versions of Theorem 5.1 and Theorem 5.2 follow immediately from Theorem 5.4. They are weaker because the uniformity in time is lacking. Remark that on Dom Z ren , [Z ren , Ut ] = 0,
(5.7)
as can be checked from the explicit expression for Ut . The generator Z ren could be considered as the free (i.e. K and R are decoupled) Hamiltonian in the weak coupling limit and hence (5.7) expresses the conservation of the ’decoupled’ energy. In the reduced weak coupling limit we have an analogous situation: the generator of the limiting quantum Markov semigroup L commutes with the generator of the free evolution i[K , ·]. A consequence of Theorem 5.4 is now given. Its advantage is that it does not involve explicitly the operators Jλ . Recall the notation in Assumption 4.2. Let I x → g(x) ∈ B(h(x)) be a measurable function such that supx∈I g(x) < 1 and x → g(x) is continuous in a neighbourhood of F. Remark that this requirement makes sense because of Assumption 4.3. Define the contractive multiplication operator G ∈ B(HR ) as, (G f )(x) = g(x) f (x),
(5.8)
and remark that (G) is also a contractive operator on s (HR ). Let C be the C ∗ -subalgebra of B(H), generated by S ⊗ 1, and 1 ⊗ (G),
(5.9)
with S ∈ B(K) and G as defined above. Let Cas be the C ∗ -subalgebra of B(Z) generated by S ⊗ 1 and 1 ⊗ (1 ⊗ p)
(5.10)
with S ∈ B(K) and p ∈ ⊕ B(hω ). ω∈F
Proposition 5.6. There exists a unique ∗-homomorphism : C → Cas such that , S ∈ B(K), (5.11)
(S ⊗ (G)) = S ⊗ 1 ⊗ ⊕ g(ω) ω∈F
18
J. Derezi´nski, W. De Roeck
where G and g are related by (5.8). We have
(A) = s∗ − lim (Jλ∗ )A(Jλ ),
A ∈ C.
λ0
(5.12)
Theorem 5.7. Assume Assumptions 4.2 4.3, 4.4 and let the family Ut be as in Theorem 5.4. For any ∈ N, any A1 , . . . , A ∈ C and (not necessarily ordered) times t0 , t1 , . . . , t , t ∈ R, lim I ∗ Tλ (λ−2 t, λ−2 t )An λ0 K
. . . A2 Tλ (λ−2 t2 , λ−2 t1 )A1 Tλ (λ−2 t1 , λ−2 t0 )IK
∗ = IK U(t−t ) (An ) . . . (A2 )U(t2 −t1 ) (A1 )U(t1 −t0 ) IK .
(5.13)
Remark 5.8. The results in [10] and [14] correspond to Theorem 5.7 where A1 . . . , A are elements of B(K) and hence (A1,..., ) = A1,..., .
6. Proofs 6.1. Proof of Theorem 4.1. Existence of the physical dynamics. We will prove a somewhat stronger theorem. Let Pn be the orthogonal projector on K ⊗ sn (HR ) and let the dense subspace D1 of H be defined as Cn ψ ∈ D1 ⇔ there exists C such that for n = 0, 1, 2 . . . we have Pn ψ ≤ √ . n! Theorem 6.1. For ψ ∈ D1 , the series (4.1) defining Wλ,t ψ is absolutely convergent, belongs to D1 , is continuous wrt t ∈ R and we have Wλ,t Wλ,s ψ = Wλ,t+s ψ, Wλ,t ψ2 = ψ2 . Therefore, Wλ,t extends uniquely to a strongly continuous unitary group on H. By Stone’s theorem, it has a self-adjoint generator Hλ , and by a theorem of Nelson, D1 is a core for Hλ . Proof. It is enough to assume that λ = 1, so that we will write Wt . We can also assume m that t ≥ 0. Let ψ ∈ D1 with Pm ψ ≤ √C . Note that we have m!
Pn Wt ψ =
∞
∞
dtm · · · dt1
q=0 [0,t] m=max{n−q,0} n
×eit H0 in Pn H I (tq ) · · · H I (t1 )Pm ψ. Note also that H I (tq ) · · · H I (t1 )Pm ≤
⎧ √ ⎨ 2q V q (m+q)! √ ,
m ≥ n − q,
⎩
m < n − q.
m!
0,
Extended Weak Coupling Limit for Pauli-Fierz Operators
19
Therefore, Pn Wt ψ ≤ ≤
∞
∞
q=0 m=max{n−q,0} p ∞
(2tV )q q!
√
(m + q)! C m √ √ m! m!
1 (2tV )q C p−q p! √ q!( p − q)! p! p=n q=0
≤
∞ 1 √ (2tV + C) p p! p=n
≤
∞ (2tV + C)n (2tV + C)r √ √ n! r! r =0
=
(2tV + C)n C1 . √ n!
This proves the absolute convergence of the series and the fact that it belongs to D1 . The rest of claims is now straightforward. 6.2. Decomposition of interaction. Lemma 6.2. There is a finite index set J and families D j∈J ∈ B(K) and φ j∈J ∈ HR such that D j ⊗ |φ j , (6.1) V = j∈J
and such that the function h(t) :=
|φ j |e−it HR φ j |
(6.2)
dt h(t) < +∞.
(6.3)
j, j ∈J
is integrable:
h1 :=
R
Moreover, we can choose this decomposition so that all φ j∈J are continuous in F and for all j ∈ J there is at most one ω ∈ F such that φ j (ω) = 0. Proof. Let {w p } p∈P be an orthonormal basis of eigenvectors of K , so that K w p = k p w p . For each p ∈ P, there exists a family {φ p,m }m∈M in HR such that wm ⊗φm, p . V wp = m∈M
We set S = |wm wm |. Now w p |V ∗ Seit H0 V w p = φ p,m |eit HR φ p,m eitkm is integrable by Assumption 4.4.
20
J. Derezi´nski, W. De Roeck
Then we choose a partition of unity χω ∈ Cc∞ (R) together with χ∞ ∈ C ∞ (R) such that χω = 1 on a neighborhood of ω, χω = 0 on a neighborhood of F\{ω} and χ∞ = 0 on a neighborhood of F and m χm +χ∞ = 1. We set φm, p,ω := χω (HR )φm, p , φm, p,∞ := χ∞ (HR )φm, p , Dm, p,ω = Dm, p,∞ := |wm w p |. Hence, the index set J is chosen as M×P ×(F ∪{∞}) and elementary properties of the Fourier transform imply the integrability of (6.2). If for a given j ∈ J and ω ∈ F, we have φ j (ω) = 0, then this ω will be referred to as ω( j). If for a given j, there is no ω ∈ F such that φ j (ω) = 0, then ω( j) is chosen arbitrarily. For further reference let us record the identity D j ⊗ |φ j (ω) , ω ∈ F. (6.4) v(ω) = j∈J : ω( j)=ω
6.3. Proof of Theorems 5.1. Reduced weak coupling limit. For operators A1 , . . . , A p we will write p
Ai := A p · · · A1 .
i=1
We will also write D(t) := e−it K Deit K . Define G λ (t, t0 ) ∞ −2n := (iλ) n=0 n
× =
(iλ)−2n
j1 ,..., j2n ∈J
n=0 n
dt1 . . . dt2n
∗ I −2 IK H (λ t2i )H I (λ−2 t2i−1 )IK
i=1 ∞
×
2n [t0 ,t]
2n [t0 ,t]
dt1 . . . dt2n −2 (t
D ∗j2 p (λ−2 t2 p )D j2 p−1 (λ−2 t2 p−1 )φ j2 p |eiλ
2 p −t2 p−1 )HR
φ j2 p−1 .
p=1
Lemma 6.3. For all T ≤ ∞, lim
sup
λ↓0 0≤t0 ≤t≤T
∗ IK Tλ (λ−2 t, λ−2 t0 )IK − G λ (t, t0 ) = 0.
(6.5)
Proof. Set D +j = D j , ∗ D− j = Dj.
(6.6)
For n = 0, 1, 2, . . . , let Pair(2n) denote the set of pairings of {1, . . . , 2n}. That means, σ ∈ Pair(2n) iff it is a permutation σ ∈ S2n satisfying σ (2 p−1) < σ (2 p), p = 1, . . . , n, and σ (2 p − 1) < σ (2 p + 1), p = 1, . . . , n − 1. We will write ( p) = + for even p
Extended Weak Coupling Limit for Pauli-Fierz Operators
21
and ( p) = − for odd p. One can visualize the above definitions as follows: σ (2 p − 1) corresponds to the p th creator in the order of increasing time and σ (2 p) corresponds to the annihilator paired with this creator by the Wick theorem. Using first the Dyson expansion and then the Wick theorem we obtain ∗ Tλ (λ−2 t, λ−2 t0 )IK IK ∞ = n=0 σ ∈Pair(2n) j1 ,..., j2n ∈I
× ×
2n i=1 n
(σ (i))
D ji
(iλ)−2n
dt2n · · · dt1 2n [t0 ,t]
(λ−2 ti ) −2 (t
φ jσ (2 p) |eiλ
σ (2 p) −tσ (2 p−1) )HR
φ jσ (2 p−1)
p=1
=:
+∞
Cn .
(6.7)
n=0
Assume for simplicity t0 = 0. Abbreviating D := sup j∈J D j , we obtain a uniform estimate Cn
≤ (Dλ−1 )2n
dt1 . . . dt2n
π ∈Pair(2n) [0,t] 2n
×
n
h(λ−2 (tπ(2 p) − tπ(2 p−1) ))
p=1
(Dλ−1 )2n = 2n n!
dt1 . . . dt2n
π ∈S2n [0,t] 2n
×
n
h(λ−2 |tπ(2 p) − tπ(2 p−1) |)
p=1
(Dλ−1 )2n = n 2 n!(2n)!
dt1 . . . dt2n
π ∈S2n
×
n
[0,t]2n
h(λ−2 |tπ(2 p) − tπ(2 p−1) |)
p=1
(Dλ−1 )2n t n ≤ 2n n!
t −t
dsh(λ
−2
n |s|)
(D)2n n (6.8) t hn1 . 2n n! First we used that each pairing can be represented by 2n n! permutations. Then we allowed to permute t1 , . . . , t2n . The last inequality has been obtained by a change of integration ≤
22
J. Derezi´nski, W. De Roeck
variables. The bound (6.8) shows that the series (6.7) is absolutely convergent. We will exploit this now since we estimate the series term by term. Given a pairing σ , the term in the sum (6.7) is estimated by λ
−2n
n
dt1 . . . dt2n
2n ([0,t])
h(λ−2 (tπ(2i) − tπ(2i−1) )).
(6.9)
i=1
We are going to show that (6.9) does not vanish only for the time consecutive pairing: that is for the pairing given by the identity permutation (also called “nonnested, noncrossing pairings” for obvious reasons). Assume there is i such that π(2i) − π(2i − 1) > 1 and let p be such that s1 := tπ(2i−1) < tπ( p) < tπ(2i) =: s2 . Then λ
−2n
2n ([0,t])
n
dt1 . . . dt2n
≤ λ−2 t n−2 (2h1 )n−1
i=1 t
0
h(λ−2 (s2 − s1 ))
t
≤ t n−2 (2h1 )n−1 0
dtπ( p)
dtπ( p)
tπ( p)
t
ds2 h(λ−2 (s2 − s1 ))
ds1 0
tπ( p)
ds1 0
tπ( p) ∞
λ−2 (tπ( p) −s1 )
du h(u).
(6.10)
The last line vanishes uniformly in 0 ≤ t ≤ T by the dominated convergence theorem, since the expression is dominated by t 2 h1 , and the du-integral vanishes as λ ↓ 0 whenever s1 < tπ( p) . This ends the proof since G λ (t, t0 ) is the sum of all terms with time-consecutive pairings. Now notice that G λ (t, t0 ) can be written in the form familiar from the weak coupling limit for Friedrichs Hamiltonians. In fact, if we consider the Hilbert space K ⊕ (K⊗HR ) with the Friedrichs-type Hamiltonian K λV , H˜ λ := λV ∗ K + HR then we can write −2 t K
G λ (t, t0 ) = eiλ
−2 t H ˜λ
∗ −iλ IK e
−2 t
IK e−iλ
0K
.
Therefore, we can apply Theorem 2.1 in [10]. More precisely, define s −2 ∗ I −2 Q λ,s := λ−2 du eiλ u K IK H (λ u)H I (0)IK =
0 λ−2 s
V ∗ e−iu(K +HR ) V du
(6.11)
0
and remark that by Assumption 4.4, 1) For all τ1 > 0, there is c > 0 such that 0 ≤ s ≤ τ1 , λ ≤ 1 ⇒ Q λ,s ≤ c.
(6.12)
Extended Weak Coupling Limit for Pauli-Fierz Operators
23
2) For all 0 < τ0 ≤ τ1 < ∞, lim sup Q λ,s − Q = 0,
(6.13)
λ↓0 τ0 ≤s≤τ1
with
+∞
Q :=
V ∗ e−iu(K +HR ) V du < ∞.
(6.14)
0
The aforementioned theorem by Davies allows us to conclude from 1) and 2) above, and the fact that ϒ = k∈spK 1Kk K 1Kk , that for all T < ∞, lim sup G λ (t, t0 ) − e−i(t−t0 )ϒ = 0. λ↓0 t0 ≤t≤T
(6.15)
6.4. Proof of Theorem 5.2. Weak coupling limit for correlations. We follow very closely the strategy of Dümcke in [14]. The case = 0 has been already proven. For notational reasons, we restrict ourselves to the case = 1. Higher are proven in exactly the same way. The theorem for the case = 1 follows immediately from Theorem 5.1 and the following lemma: Lemma 6.4. For all T ≤ ∞ and S ∈ B(K), lim
sup
λ↓0 −T ≤t0
∗ IK Tλ (λ−2 t, λ−2 t ) S⊗1 Tλ (λ−2 t , t0 )IK
∗ ∗ −IK Tλ (λ−2 t, λ−2 t )IK S IK Tλ (t , t0 )IK = 0.
(6.16)
Proof. Using first the Dyson expansion and then the Wick theorem we obtain ∗ IK Tλ (λ−2 t, λ−2 t ) S⊗1 Tλ (λ−2 t , t0 )IK ∗
−2
(6.17)
−2
−IK Tλ (λ t, λ t )IK ∞ = (iλ)−2n n=0 σ ∈Pair(2n) j1 ,..., j2n ∈J n
×
(6.18)
dt2n · · · dt1
p=0 [t ,t ]× p 0 2n− p [t ,t]
× ×
2n
(σ (i))
D ji
i= p+1 n
+∞ n=0
Cn (S).
p i =1
−2 (t
φ jσ (2 p) |eiλ
p=1
=:
(λ−2 ti )S
(σ (i ))
Dj i
σ (2 p) −tσ (2 p−1) )HR
(λ−2 ti )
φ jσ (2 p−1) .
24
J. Derezi´nski, W. De Roeck
Assume for simplicity t0 = 0. Exactly as in (6.8), we prove that (D)2n n t hn1 , 2n n!
Cn (S) ≤ S
(6.19)
so we can again estimate the series (6.18) term by term. The term in the sum (6.18) corresponding to the pairing σ is estimated by
D λ
2n
S
p
[0,t ]×
2n− p
[t ,t]
dt1 . . . dt2n
n
h(λ−2 (tπ(2i) − tπ(2i−1) )). (6.20)
i=1
We are going to show all such terms with a pairing crossing t vanish in the limit λ 0. Assume there is a i such that s1 := tπ(2i−1) < t < tπ(2i) =: s2 .
(6.21)
Then λ
−2n
dt1 . . . dt2n
p [0,t ]×n− p [t ,t]
≤ λ−2 (2th1 )n−1 ≤ (2th1 )
n−1
t
t
ds1 0
ds1
t +∞
n i=1
t
ds2 h(λ−2 (s2 − s1 ))
λ−2 (t −s1 )
0
h(λ−2 (s2 − s1 ))
du h(u).
To prove that this term vanishes uniformly in t , we have to show λ↓0 0≤t ≤t
t
lim sup
ds1 0
+∞
λ−2 (t −s1 )
du h(u) = 0.
(6.22)
This follows since for each s1 < t , the integral over u vanishes as λ ↓ 0 and the whole expression is bounded by t h1 . Since we have established that no pairing crosses the t point, the problem factorizes and (6.16) is true.
6.5. Convergence of annihilation operators. Lemma 6.5. For ψ ∈ D and j ∈ J , −2
lim λ−1 a(Jλ∗ eiλ t (HR −ω( j)) φ j )ψ λ↓0 = a eit Z R |1 ⊗ φ j (ω( j)) ψ uniformly in t ∈ R.
(6.23) (6.24)
Extended Weak Coupling Limit for Pauli-Fierz Operators
25
Proof. We use −2 t (H −ω( j)) R
λ−1 Jλ∗ eiλ = ⊕
ω∈F
φ j (x)
−2 ∗ λ−1 Jλ,ω eiλ t (HR −ω( j)) φ j (x).
(6.25)
Now −2 t (H −ω( j)) R
∗ eiλ λ−1 Jλ,ω
=e
it (x+λ−2 (ω−ω( j)))
−→ λ0
φ j (x)
1 Iω (HR )φ j (ω + λ2 x)
φ j (ω( j)), ω = ω( j); 0, ω = ω( j).
6.6. Resummation formula. The following lemma gives a convenient expression for the full dynamics in terms of the reduced dynamics. Lemma 6.6. Tλ (t, t0 ) ∞ =
m=0 m [t0 ,t]
dt1 . . . dtm
(−i)m λm
1 ,..., p ∈{+,−} j1 ,..., jm ∈J m ×IK Tλ (t, tm )D jm (tm )Tλ (tm , tm−1 ) · · · D j11 (t1 )Tλ (t1 , t0 )IK ∗
×
a ∗ (eiti HR φ ji )
i =1,...,m
i=1,...,m : i =+
a(eiti HR φ ji ).
⊗ 1s (HR ) (6.26)
: i =−
Proof. Pair(n) will denote the set of all pairings inside the set {1, . . . , n}. That means, σ ∈ Pair(n) iff there is p = 0, 1, . . . , [n/2] such that σ is an injection of {1, . . . , 2 p} into {1, . . . , n} satisfying σ (2i − 1) < σ (2i + 1), i = 1, . . . , p − 1 and σ (2i − 1) < σ (2i), i = 1, . . . , p. For σ ∈ Pair(n), let Ranσ denote the image of σ . We say that a sequence Pair(n) iff 1 , . . . n of {+, −} is compatible with σ ∈ σ (2i−1) = −, σ (2i) = +, i = 1, . . . , p. Applying the Dyson expansion and then the Wick theorem we obtain that the left-hand side of (6.26) equals ∞ n=0 n [t0 ,t]
×
n r =1
dt1 · · · dtn
j1 ,..., jn ∈J 1 ,...,n ∈{+,−}
(−i)λD jrr (tr )
×a n (eitn HR φ jn ) · · · a 1 (eit1 HR φ j1 )
26
J. Derezi´nski, W. De Roeck
=
∞ n=0 n
dt1 · · · dtn
j1 ,..., jn ∈J σ ∈ Pair(n) 1 ,...,n ∈{+,−} σ
compatible with
×
n r =1
(−i)λD jrr (tr )
×
i ∈{1,...,n}\Ranσ
i∈{1,...,n}\Ranσ : i =+
×
p
a ∗ (eiti HR φ ji )
a(eiti HR φ ji ) : i =−
φ jσ (2q) |ei(tσ (2q) −tσ (2q−1) )HR φ jσ (2q−1) .
(6.27)
q=1
Applying the Wick theorem and the Dyson expansion backwards we see that the right-hand side of (6.27) equals the right-hand side of (6.26). 6.7. Proof of Theorem 5.4. Set
D j,ω :=
e−e =ω
1Ke D j 1Ke .
Recall the operators νω and ν defined in Subsect. 4.2. They can be expressed in terms of D j,ω by νω =
√ 2π
D j,ω ⊗ |φ j (ω) ,
j∈J : ω( j)=ω
√ D j,ω( j) ⊗ |φ j (ω( j)) . ν = 2π j∈J
We use first the resummation formula (6.26), and then we replace D j (t) with
D j,ω e−iωt .
ω∈F
We compute in terms of a quadratic form on D: (Jλ∗ )Tλ (λ−2 t, λ−2 t0 )(Jλ ) ∞ dt1 . . . dtm =
(iλ)−m
j1 ,..., jm 1 ,...,m ∈{+,−}
m=0 [t ,t] m 0
×IK Tλ (λ t, λ tm )D jmm (λ−2 tm )Tλ (λ−2 tm , λ−2 tm−1 ) · · · × · · · D j11 (λ−2 t1 )Tλ (λ−2 t1 , λ−2 t0 )IK ⊗ 1s (HR ) ∗
×
−2
−2
i=1,...,m : i =+
−2 t H i R
a ∗ (Jλ∗ eiλ
φ ji )
Extended Weak Coupling Limit for Pauli-Fierz Operators
×(Jλ∗ Jλ ) =
∞
−2 t H i R
a(Jλ∗ eiλ
i =1,...,m : i =−
dt1 . . . dtm
φ ji )
j1 ,..., jm 1 ,...,m ∈{+,−}
m=0 [t ,t] m 0 m i(ω p −ω( j p ))λ−2 t
×
27
(iλ)−m
ω1 ,...,ωm ∈F
e
p=1 ∗ ×IK Tλ (λ−2 t, λ−2 tm )D jmm ,ωm Tλ (λ−2 tm , λ−2 tm−1 ) · · ·
× · · · D j11,ω1 Tλ (λ−2 t1 , λ−2 t0 )IK ⊗ 1s (HR ) −2 × a ∗ (Jλ∗ eiλ ti (HR −ω( ji )) φ ji ) i=1,...,m : i =+
×(Jλ∗ Jλ )
−2 t (H −ω( j )) R i i
a(Jλ∗ eiλ
i =1,...,m : i =−
φ ji ).
(6.28)
Now by Theorem 5.2, we have a uniform limit lim I ∗ Tλ (λ−2 t, λ−2 tm )D jmm ,ωm Tλ (λ−2 tm , λ−2 tm−1 ) · · · λ0 K × · · · D j11,ω1 Tλ (λ−2 t1 , λ−2 t0 )IK = e−i(t−tm )ϒ D jmm ,ωm e−i(tm −tm−1 )ϒ · · · × · · · D j11,ω1 ei(t1 −t0 )ϒ .
(6.29)
By Lemma 6.5, for ψ, ψ ∈ D we have the uniform limits −2 lim λ−1 a(Jλ∗ eiλ ti (HR −ω( ji )) φ ji )ψ λ0
i =1,...,m : i =−
=
i =1,...,m : i =−
lim
λ0
a eiti Z R |1 ⊗ φ ji (ω( ji )) ψ , −2 t (H −ω( j )) R i i
i=1,...,m : i =+
=
λ−1 a(Jλ∗ eiλ
φ ji )ψ
a eiti Z R |1 ⊗ φ ji (ω( ji )) ψ.
i=1,...,m : i =+
Clearly, s − limλ0 (Jλ∗ Jλ ) = 1. Thus, (6.28), as a quadratic form on D, up to an error of the order o(λ0 ) equals ∞
dt1 . . . dtm
m=0 [t ,t] 1 ,...,m ∈{+,−} m 0 m i(ω p −ω( j p ))λ−2 t
×
e
p=1
ω1 ,...,ωm ∈F
(−i)m
28
J. Derezi´nski, W. De Roeck
×e−i(t−tm )ϒ D jmm ,ωm e−i(tm −tm−1 )ϒ · · · × · · · D j11,ω1 e−i(t1 −t0 )ϒ ⊗ 1s (HR ) × a eiti Z R |1 ⊗ φ ji (ω( ji )) i=1,...,m : i =+
×
i =1,...,m : i =−
a eiti Z R |1 ⊗ φ ji (ω( ji )) .
By the Riemann-Lebesgue Lemma, in the limit λ 0, all the terms with ω( j p ) = ω p for some p disappear, and we obtain
∞
dt1 . . . dtm
(−i)m
1 ,...,m ∈{+,−}
m=0 [t ,t] m 0
×e−i(t−tm )ϒ D jmm ,ωm e−i(tm −tm−1 )ϒ · · · × · · · D j11,ω1 e−i(t1 −t0 )ϒ ⊗ 1s (HR ) × a eiti Z R |1 ⊗ φ ji (ω( ji )) i=1,...,m : i =+
×
i =1,...,m : i =−
=e
itd(Z R )
a eiti Z R |1 ⊗ φ ji (ω( ji ))
Ut−t0 e−it0 d(Z R ) .
Thus we obtained, for ψ, ψ ∈ D, limψ|(Jλ∗ )Tλ (λ−2 t, λ−2 t0 )(Jλ )ψ
(6.30)
= ψ|eitd(Z R ) Ut−t0 e−it0 d(Z R ) ψ .
(6.31)
λ↓0
By density, (6.30) can be extended to the weak limit on the whole space. But the weak convergence of contractions to a unitary operator implies the strong* convergence. This yields (5.5). Note that −2 t H 0
(Jλ∗ )e−itλ
(1 − Jλ Jλ∗ ) = 0.
Therefore, −2 t Z
eiλ
ren
−2 t H λ
(Jλ∗ )e−iλ
(Jλ )
iλ−2 t Z ren
−2 (Jλ∗ )e−iλ t H0 (Jλ ) =e −2 −2 ×(Jλ∗ )eiλ t H0 e−iλ t Hλ (Jλ ).
(6.32)
The equality (5.6) now follows from (4.16) and (5.5) since the strong limit of a product of uniformly bounded operators is the product of limits.
Extended Weak Coupling Limit for Pauli-Fierz Operators
29
6.8. Proof of Theorem 5.7. Remark that −2 t H 0
(1 − (Jλ Jλ∗ ))eiλ
−2 t H λ
e−iλ
strongly
(Jλ ) −→ 0. λ0
(6.33)
This follows since for ψ ∈ Z −2 t H 0
(1 − (Jλ Jλ∗ ))eiλ −2 t H 0
= eiλ
−2 t H λ
e−iλ
−2 t H λ
e−iλ
(Jλ )ψ2 −2 t H 0
(Jλ )ψ2 − (Jλ Jλ∗ )eiλ −2 t H 0
= (Jλ )ψ2 − (Jλ∗ )eiλ
−2 t H λ
e−iλ
−2 t H λ
e−iλ
(Jλ )ψ2
(Jλ )ψ2 −→ 0, λ0
where we used s− lim Jλ∗ Jλ = 1, Theorem 5.4 and the fact that (Jλ ) is a partial λ0
isometry. Theorem 5.7 is now proven by using (5.5), (5.12), (6.33) and the fact that for multiplication operators G as in the text preceding Theorem 5.7 we have [eitd(HR ) , (G)] = 0
[(Jλ Jλ∗ ), (G)] = 0.
Acknowledgements. The research of J. D. was partly supported by the Postdoctoral Training Program HPRNCT-2002-0277 and the Polish KBN grants SPUB127 and 2 P03A 027 25. Part of the work was done when both authors visited the Erwin Schrödinger Institute (J. D. as a Senior Research Fellow), as well as during a visit of J. D. at K. U. Leuven supported by a grant of the ESF. W. D. R. is a Postdoctoral Fellow supported by FWO-Flanders.
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14. Dümcke, R.: Convergence of multitime correlation functions in the weak and singular coupling limits. J. Math. Phys. 24(2), 311–315 (1983) 15. Dümcke, R.: The low density limit for an n-level system interacting with a free bose or fermi gas. Commun. Math. Phys. 97, 331–359 (1985) 16. Frigerio, A., Gorini, V.: Diffusion processes, quantum dynamical semigroups, and the classical kms condition. J. Math. Phys. 25(4), 1050–1065 (1984) 17. Gardiner, C.W., Collet, M.J.: Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. Phys. Rev. A 31, 3761–3774 (1985) 18. Gardiner, C.W., Zoller, P.: Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods With Applications to Quantum Optics. Berlin Heidelberg-New York: Springer Verlag, 2004 19. Gough, J.: Asymptotic stochastic transformations for non-linear quantum dynamical systems. Rep. Math. Phys. 44(3), 313–338 (1999) 20. Gough, J.: Quantum flows as markovian limit of emission, absorption and scattering interactions. Commun. Math. Phys. 254, 489–512 (2005) 21. Gregoratti, M.: The hamiltonian operator associated with some quantum stochastic evolutions. Commun. Math. Phys. 222, 181–200 (2001) 22. Van Hove, L.: Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21, 517–540 (1955) 23. Hudson, R.L., Parathasaraty, K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93(3), 301–323 (1984) 24. Lindblad, G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40, 147–151 (1975) 25. Maassen, H.: Quantum markov processes on fock spaces described by integral kernels. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications II, Volume 1136 of Lecture Notes in Mathematics. Berlin: Springer, 1984, pp. 361–374 26. Meyer, P.-A.: Quantum probability for probabilists. Volume 1538 of Lecture Notes in Mathematics. Berlin: Springer, 1995, pp. 384–411 27. De Roeck, W., Maes, C.: Fluctuations of the dissipated heat in a quantum stochastic model. Rev. Math. Phys. 18, 619–653 (2006) 28. Rudnicki, S., Alicki, R., Sadowski, S.: The low-density limit in terms of collective squeezed vectors. J. Math. Phys. 33(7), 2607–2617 (1992) 29. Spohn, H.: Kinetic equations from hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52, 569–616 (1980) 30. von Waldenfels, W.: Ito solution of the linear quantum stochastic differential equation describing light emission and absorption. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications IV, Volume 1055 of Lecture Notes in Mathematics, Berlin: Springer, 1986, pp. 384–411 31. von Waldenfels, W.: Symmetric differentiation and hamiltonian of a quantum stochastic process. Inf. Dim. Anal. & Quantum Prob. 8(1), 73–116 (2005) Communicated by H.-T. Yau
Commun. Math. Phys. 279, 31–75 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0415-7
Communications in
Mathematical Physics
Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property Valter Moretti Dipartimento di Matematica, Facoltà di Scienze M.F.N., Università di Trento, Trento, Italy Istituto Nazionale di Alta Matematica “F.Severi” - Unità Locale di Trento, Trento, Italy Istituto Nazionale di Fisica Nucleare - Gruppo Collegato di Trento, via Sommarive 14, I-38050 Povo (TN), Italy. E-mail:
[email protected] Received: 31 October 2006 / Accepted: 10 July 2007 Published online: 1 February 2008 – © Springer-Verlag 2008
Abstract: This paper continues the analysis of the quantum states introduced in previous works and determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state λ M , of a massless conformally coupled scalar field φ propagating in M. λ M is “holographically” induced in the bulk by the universal BMS-invariant state λ defined on the future null infinity + of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at future null infinity discussed in previous papers. This induction is possible when some requirements are fulfilled, in par˜ are globally ticular whenever the spacetime M and the associated unphysical one, M, hyperbolic and M admits future time infinity i + . λ M coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case of a curved spacetime M, the state λ M enjoys the following further remarkable properties: (i) λ M is invariant under the (unit component of the Lie) group of isometries of the bulk spacetime M. (ii) λ M fulfills a natural energy-positivity condition with respect to every notion of Killing time (if any) in the bulk spacetime M: If M admits a time-like Killing vector, the associated one-parameter group of isometries is represented by a strongly-continuous unitary group in the GNS representation of λ M . The unitary group has positive self-adjoint generator without zero modes in the one-particle space. In this case λ M is a so-called regular ground state. (iii) λ M is (globally) Hadamard in M and thus it can be used as the starting point for the perturbative renormalisation procedure of QFT of φ in M. 1. Introduction In this paper we continue the analysis of the states determined by the asymptotic structure of four-dimensional asymptotically flat spacetimes at null infinity started in [DMP06]
32
V. Moretti
and fully developed in [Mo06]. Part of those results will be summarised in Sect. 2. In [DMP06] and [Mo06] it has been established that the null boundary at future infinity + of an asymptotically flat spacetime admits a natural formulation of bosonic linear QFT living therein. A preferred quasifree pure state λ has been picked out in the plethora of algebraic states defined on the algebra of Weyl observables W(+ ) of the QFT on + . That state enjoys remarkable properties, in particular it is invariant under the action of the natural (infinite-dimensional) group of symmetries of + – the so-called BMS group – describing the asymptotic symmetries of the physical spacetime M. λ is the vacuum state for BMS-massless particles if one analyses the unitary representations of the BMS group within the Wigner-Mackey approach [DMP06]. λ is uniquely determined by a positive BMS-energy requirement in addition to the above-mentioned BMS invariance [Mo06] (actually the latter requirement can be weakened considerably). Finally, in the folium of λ there are no further pure BMS-invariant (not necessarily quasifree or positive energy) states. λ is universal: it does not depend on the particular asymptotically flat spacetime under consideration but it is defined in terms of the asymptotic extent which is the same for all asymptotically flat spacetimes. However, in every fixed asymptotically flat spacetime M and under suitable hypothe˜ λ it induces a preferred quasifree state λ M on the algebra W(M) of the ses on M and M, Weyl observables of a bosonic massless conformally-coupled linear field propagating in the bulk M. This is because, due to a sort of holographic correspondence discussed in [DMP06,Mo06], the algebra of bulk observables W(M) is one-to-one mapped to a subalgebra of boundary observables W(+ ) by means of a (isometric) ∗-algebra homomorphism ı : W(M) → W(+ ). In this way λ M is defined as λ M (a) := λ(ı(a)) for all a ∈ W(M). The existence of ı is assured if some further conditions defined in [DMP06] are fulfilled for the spacetime M and the associated unphysical spacetime M˜ (see (b) of Proposition 2.1). In particular both M and M˜ are required to be globally hyperbolic. Those requirements are valid when M is Minkowski spacetime and, in that case, λ M turns out to coincide with Minkowski vacuum [DMP06]. However, it has been established in [Mo06], that the conditions are verified in a wide class of spacetimes (supposed ˜ individuated by Friedrich to be globally hyperbolic with the unphysical spacetime M) [Fri86-88]: the asymptotically flat vacuum spacetimes admitting future time infinity i + . This paper is devoted to study the general features of the state λ M . In particular, in Sect. 3 we focus on isometry-invariance properties of λ M and on the energy positivity condition with respect to timelike Killing vectors in any bulk spacetime M (Theorem 3.1): we show that λ M is invariant under the unit component of the Lie group of isometries of the bulk spacetime M. This fact holds true also replacing λ M with any other state λM uniquely defined by assuming that λM (a) := λ (ı(a)) for all a ∈ W(M), where λ is any (not necessarily quasifree or pure) BMS invariant state defined on W(+ ). Furthermore λ M fulfills a natural energy-positivity condition with respect to every notion of Killing time in the bulk spacetime M: If M possesses a time-like Killing vector ξ , the associated one-parameter group of isometries is represented by a strongly continuous unitary group in the GNS representation of λ M , that group admits a positive self-adjoint generator H and H has no zero modes in the one-particle space. In this sense the quasifree state λ M is a regular ground state for H [KW91]. Actually these properties are proved to hold also if ξ is causal and future directed, but not necessarily timelike. Section 4 is devoted to discuss the validity of the Hadamard condition for the state λ M . First we show that the two-point function of λ M is a proper distribution of D (M × M) (Theorem 4.1) when the asymptotically flat spacetime M admits future time infinity i + . It is interesting to notice that the explicit expression of the two-point function of λ M we
Quantum Out-States Holographically Induced by Asymptotic Flatness
33
present strongly resembles that of Hadamard states in manifolds with bifurcate Killing horizons studied by Kay and Wald [KW91] when restricted to the algebra of observables localised on a Killing horizon. The last result we establish in this work is that λ M is Hadamard (Theorem 4.2). In this case the kernel of the two-point function of λ M satisfies the global Hadamard condition [KW91]. The proof – performed within the microlocal framework taking advantage of some well known result established by Radzikowski [Ra96a,Ra96b] – is based on a “from local to global” argument and the analysis of the wavefront sets of the involved distributions. Remark 1.1. After the submission of this paper one of the referees pointed out to the author that a result of a very similar nature as Proposition 4.3 was presented in Hollands’ unpublished Ph.D thesis [Ho00]. In that work, the definition of local vacuum states near a point x in spacetime is considered. The idea of the construction is to define the state by specifying the “initial data” of its 2-point function on the (say, past-) lightcone through x. The formula for this initial datum is the formula (46) when the lightcone in [Ho00] is replaced by − as in our work. In [Ho00] it is also established that the individuated state is of Hadamard form (in the sense of the microlocal condition (45)) employing a proof which is very analogous to our proof. The rest of this section is devoted to remind the reader of the basic geometric structures in asymptotically flat spacetimes. 1.1. Notations, mathematical conventions. Throughout R+ := [0, +∞), N := {0, 1, 2, . . .}. For smooth manifolds M, N , C ∞ (M; N ) (omitting N whenever N = R) is the space of smooth functions f : M → N . C0∞ (M; N ) ⊂ C ∞ (M; N ) is the subspace of compactly-supported functions. If χ : M → N is a diffeomorphism, χ ∗ is the natural extension to tensor bundles (counter-, co-variant and mixed) from M to N (Appendix C in [Wa84]). A spacetime is a smooth four-dimensional semi-Riemannian connected manifold (M, g), whose metric has signature − + ++, and it is assumed to be oriented and time oriented. We adopt definitions of causal structures of Chap. 8 in [Wa84]. If ˜ (M, g) and ( M, ˜ g) ˜ S ⊂ M ∩ M, ˜ being spacetimes, J ± (S; M) (I ± (S; M)) and J ± (S; M) ˜ indicate the causal (chronological) sets associated to S and are respectively (I ± (S; M)) ˜ Concerning distribution and wavefront-set theory referred to as the spacetime M or M. we essentially adopt standard definitions and notation used in [Hö89,Hör71] and also in [Ra96a,Ra96b]. 1.2. Asymptotic flatness at future null infinity and + . Following [AH78,As80,Wa84]1 a smooth spacetime (M, g) is called asymptotically flat vacuum spacetime at future ˜ g) null infinity if there is a second smooth spacetime ( M, ˜ such that M can be viewed as an ˜ + is an embedded submanopen embedded submanifold of M˜ with boundary + ⊂ M. ˜ = ∅. ( M, ˜ g) ˜ is required to be strongly causal in a ifold of M˜ satisfying + ∩ J − (M; M) + 2 ˜ is strictly ˜ M = M g M , where ∈ C ∞ ( M) neighbourhood of and it has to hold g positive on M. On + one has = 0 and d = 0. Moreover, defining n a := g˜ ab ∂b , there must be a smooth function, ω, defined in M˜ with ω > 0 on M ∪ + , such that ∇˜ a (ω4 n a ) = 0 on and the integral lines of ω−1 n are complete on + . The topology 1 Here, we use a stronger definition than that presented in the cited references where only the existence of the manifold-with-boundary M ∪ + is assumed.
34
V. Moretti
~ M
n
I+
M
Fig. 1. Asymptotically flat spacetime. (+ is indicated by I + in the figure.)
of + has to be that of S2 × R. Finally vacuum Einstein equations are assumed to be fulfilled for (M, g) in a neighbourhood of + or, more weakly, “approaching” + as discussed on p.278 of [Wa84]. It results that + is a 3-dimensional submanifold of M˜ which is the union of integral lines of the nonvanishing null field n µ := g˜ µν ∇˜ ν ; these lines are complete for a certain ˜ + regular rescaling of n, and + is equipped with a degenerate metric h˜ induced by g. is called future infinity of M. Remark 1.2. For brevity, from now on asymptotically flat spacetime means asymptotically flat vacuum spacetime at future null infinity. ˜ + is the conic surface indicated by I+ in the figure. The tip is not a point of M. Minkowski spacetime and Schwarzschild spacetime are well-known examples of asymptotically flat spacetimes. It is simply proved – for instance reducing to the Minkowski space case – that, with our conventions, the null vector n is always future directed with ˜ g) respect to the time-orientation of ( M, ˜ induced from that of (M, g). As far as the geometric structure on + is concerned, changes of the unphysical space˜ g), time ( M, ˜ associated with a fixed asymptotically flat spacetime (M, g), are completely encompassed by gauge transformations → ω valid in a neighbourhood of + , with ˜ n) ω smooth and strictly positive. Under these gauge transformations the triple (+ , h, transforms as + → + ,
h˜ → h˜ := ω2 h˜ ,
n → n := ω−1 n.
(1)
˜ n) transforming as in (1) for a fixed asymptotically If C is the class of the triples (+ , h, flat spacetime, there is no general physical principle to single out a preferred element in C. On the other hand, C is universal for all asymptotically flat spacetimes [Wa84]: If C1 and C2 are the classes of triples associated respectively to (M1 , g2 ) and (M2 , g2 ), there is a diffeomorphism γ : +1 → +2 such that for suitable (+1 , h˜ 1 , n 1 ) ∈ C1 and (+2 , h˜ 2 , n 2 ) ∈ C2 : γ (+1 ) = +2 , γ ∗ h˜ 1 = h˜ 2 , γ ∗ n 1 = n 2 . Choosing ω such that ∇˜ a (ω4 n a ) = 0 – this choice is permitted in view of the very definition of asymptotically flat spacetime – and using the fact that vacuum Einstein’s equations are fulfilled in a neighbourhood of + , the tangent vector n = ω−1 n turns out to be that of complete null geodesics with respect to the rescaled metric g˜ = ω2 g˜ (see Sect. 11.1 in [Wa84]). ω is completely fixed by requiring that, in addition, the non-degenerate metric on the transverse section of + is the standard metric of S2
Quantum Out-States Holographically Induced by Asymptotic Flatness
35
in R3 constantly along geodesics. We indicate by ω B and (+ , h˜ B , n B ) that value of ω and the associated rescaled triple (+ , h˜ , n ) respectively. For ω = ω B , a Bondi frame on + is a coordinate system (u, ζ, ζ ) on + , where u ∈ R is an affine parameter of the complete null g-geodesics ˜ whose union is + (n = ∂/∂u in these coordinates) and ζ, ζ are standard complex coordinates on the cross section S2 of + when S2 is identified with the Riemann complex sphere : ζ = eiϕ cot(θ/2) with θ, ϕ the usual spherical coordinates of S2 (see [KND77,MC72-75] for the use of these complex coordinates in the specific case of + ). Obviously ζ and ζ bring the same information as two real independent coordinates as in Wirtinger notation in the theory of complexvariable functions. With these choices, the metric on the transverse section of + reads 2(1 + ζ ζ )−2 (dζ ⊗ dζ + dζ ⊗ dζ ) = dθ ⊗ dθ + sin2 θ dϕ ⊗ dϕ. By definition χ : + → + belongs to the BMS group, G B M S [Pe63,Pe74,Ge77, AS81], if χ is a diffeomorphism and χ ∗ h˜ and χ ∗ n differ from h˜ and n by a rescaling (1) at most. Henceforth, whenever it is not explicitly stated otherwise, we consider as admissible realisation of the unphysical metric on + only those metrics h˜ which are accessible from a metric with associate triple (+ , h˜ B , n B ), by means of a transformation in G B M S . In coordinates of a fixed Bondi frame (u, ζ, ζ ), the group G B M S is realised as a semi-direct group product S O(3, 1)↑C ∞ (S2 ), where ( , f ) ∈ S O(3, 1)↑ ×C ∞ (S2 ) acts as u → u := K (ζ, ζ )(u + f (ζ, ζ )) , ζ → ζ := ζ :=
a ζ + b , c ζ + d
(2)
ζ → ζ := ζ :=
a ζ + b c ζ + d
.
(3)
K is the smooth positive function on S2 (1 + ζ ζ ) K (ζ, ζ ) := and (a ζ + b )(a ζ + b ) + (c ζ + d )(c ζ + d ) a b = −1 ( ). c d
(4)
Above is the well-known surjective covering homomorphism S L(2, C) → S O(3, 1)↑ (see [DMP06] for further details). Two Bondi frames are connected to each other through the transformations (2),(3) with ∈ SU (2). Conversely, any coordinate frame (u , ζ , ζ ) on + connected to a Bondi frame (u, ζ, ζ ) by means of an arbitrary BMS transformation (2),(3) is physically equivalent to the Bondi frame from the point of view of General Relativity, but it is not necessarily a Bondi frame in turn. A global refer ence frame (u , ζ , ζ ) on + related with a Bondi frame (u, ζ, ζ ) by means of a BMS transformation (2)-(3) will be called an admissible frame. By construction, the action of G B M S takes the form (2)-(3) in admissible fames too. The notion of Bondi frame is useful but conventional. Any physical object must be invariant under the whole BMS group and not only under the subgroup of G B M S connecting Bondi frames. The local one-parameter group of diffeomorphisms generated by a (smooth) vector field ξ defined in an asymptotically flat spacetime (M, g) is called an asymptotic Killing symmetry if (i) ξ extends smoothly to a field ξ˜ tangent to + and (ii) 2 £ξ g has a smooth extension to + which vanishes there. This is the best approximation of a Killing symmetry for a generic asymptotically flat spacetime which does not admit
36
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proper Killing symmetries (see e.g. [Wa84]). The following well-known result illustrates how G B M S describes asymptotic Killing symmetries valid for every asymptotically flat spacetime [Ge77,Wa84]. Proposition 1.1. Let (M, g) be asymptotically flat. The one-parameter group of diffeomorphisms generated by a vector field ξ˜ tangent to + is a subgroup of G B M S if and only if ξ˜ is the smooth extension to + of some vector field of (M, g) defining an asymptotic Killing symmetry of (M, g). 2. Summary of Some Previously Achieved Results 2.1. Quantum fields on + . Let us summarise how a natural linear QFT can be defined on + employing the algebraic approach and the GNS reconstruction theorem. Motivations for the following theoretical construction and more details can be found in [DMP06,Mo06]. Referring to a fixed Bondi frame on + , consider the real symplectic space (S(+ ), σ ), where S(+ ) := { ψ ∈ C ∞ (+ ) ψ , ∂u ψ ∈ L 2 (R × S2 , du ∧ S2 (ζ, ζ ))}, S2 (ζ, ζ ) :=
2dζ ∧ dζ i(1 + ζ ζ )2
,
(5)
S2 (ζ, ζ ) being the standard volume form of the unit 2-sphere, and the nondegenerate symplectic form σ is given by, if ψ1 , ψ2 ∈ S(+ ), ∂ψ1 ∂ψ2 (6) σ (ψ1 , ψ2 ) := ψ2 − ψ1 du ∧ S2 (ζ, ζ ). ∂u ∂u R×S2 There is a natural representation of G B M S acting on (S(+ ), σ ) discussed in [DMP06, Mo06]. Start from the representation A of G B M S made of transformations on functions ψ ∈ C ∞ (+ ), −1 ψ g −1 (u, ζ, ζ ) , where g = ( , f ). (7) (A g ψ)(u, ζ, ζ ) := K g −1 (u, ζ, ζ ) −1 It turns out that A g (S(+ )) ⊂ S(+ ). Moreover, due to the weight K , the G B M S representation A preserves the symplectic form σ . As a consequence the space (S(+ ), σ ) does not depend on the used Bondi frame. In this context it is convenient to assume that the elements of S(+ ) are densities which transform under the action of A when one changes admissible frame. In the following the restriction A g S(+ ) will be indicated by A g for the sake of simplicity. Naturalness and relevance of the representation A follow from the content of Proposition 3.4 below as discussed in [DMP06,Mo06]. As is well known [BR021,BR022], it possible to associate canonically any symplectic space, for instance (S(+ ), σ ), with a Weyl C ∗ -algebra, W(S(+ ), σ ). This is the, unique up to (isometric) ∗-isomorphisms, C ∗ -algebra with generators W (ψ) = 0, ψ ∈ S(+ ), satisfying Weyl commutation relations (we use here conventions adopted in [Wa94])
W (−ψ) = W (ψ)∗ ,
W (ψ)W (ψ ) = eiσ (ψ,ψ )/2 W (ψ + ψ ).
(8)
Quantum Out-States Holographically Induced by Asymptotic Flatness
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Here W(S(+ )) := W(S(+ ), σ ) has the natural interpretation of the algebra of observables for a linear bosonic QFT defined on + as discussed in [DMP06,Mo06] (see also Appendix A of [Mo06]). The representation A induces [BR022] a ∗-automorphism G B M S -representation α : W(S(+ )) → W(S(+ )), uniquely individuated (by linearity and continuity) by the requirement αg (W (ψ)) := W (A g ψ) for all ψ ∈ S(+ ) and g ∈ G B M S . Since we expect that physics is BMS-invariant we face the issue about the existence of α-invariant algebraic states on W(S(+ )). To this end it has been established in [DMP06] that there is at least one algebraic quasifree2 pure state λ defined on W(S(+ )) which is invariant under G B M S . It is that uniquely induced by linearity and continuity from: λ(W (ψ)) = e−µλ (ψ,ψ)/2 , µλ (ψ1 , ψ2 ) := −iσ (ψ1+ , ψ2+ ) , ψ ∈ S(+ );
(9)
the bar over ψ+ denotes the complex conjugation, ψ+ being the positive u-frequency part of ψ computed with respect to the Fourier-Plancherel transform discussed in Appendix C: −iku e (k, ζ, ζ )dk , (k, ζ, ζ ) ∈ R × S2 . ψ+ (u, ζ, ζ ) := √ (k)ψ 2π R Everything is referred to an arbitrarily fixed Bondi frame (u, ζ, ζ ) and (k) = 0 for k < 0 and (k) = 1 for k ≥ 0. Consider the GNS representation of λ, (H, , ϒ). Since λ is quasifree, H is a bosonic Fock space F+ (H) with cyclic vector ϒ given by the Fock vacuum and 1-particle Hilbert H space generated by the positive-frequency + := ψ . In other words one has that H ≡ parts of u-Fourier-Plancherel transforms ψ L 2 (R+ × S2 ; 2kdk ∧ S2 ). Indeed it arises from (9): (k, ζ, ζ )dk ∧ S2 (ζ, ζ ). (k, ζ, ζ )ψ ψ+ , ψ+ = 2k(k)ψ (10) R×S2
In [DMP06,Mo06] we used a different, but unitarily equivalent, definition of positive frequency part in Fourier variables (we parts defined, in Fou√ used, in fact, positive frequency
+ (E, ζ, ζ ) := 2E ψ + (E, ζ, ζ ) so that H ≡ L 2 (R+ ×S2 ; d E ∧S2 ).) rier variables, as ψ λ is a regular state, that is self-adjoint symplectically-smeared field operators σ (, ψ) are defined via Stone’s theorem: (W (tψ)) = e−itσ (,ψ) with t ∈ R and ψ ∈ S(+ ). As a remarkable result, it has been established in [DMP06] that, equipping G B M S with a suitable Fréchet topology, the unique unitary representation U of G B M S leaving ϒ invariant and implementing α is strongly continuous. Its restriction to H (which is invariant under U ) is an irreducible and strongly continuous Wigner-Mackey representation associated with a 0-elicity representation of the little group given by the double covering of 2D Euclidean group. The little group is the same as in the case of massless Poincaré particles. As a matter of fact, in the space of characters of G B M S , where a generalisation of Mackey machinery works [MC72-75,AD05,Da04,Da05,Da06] (notice that G B M S is not locally compact), there is a a notion of mass, m B M S , which is invariant under the action of G B M S . It turns out that the found G B M S representation is defined over an orbit in the space of characters with m B M S = 0. So we are dealing with BMS-invariant massless particles. 2 We adopt the definition of quasifree state given in [KW91], and also adopted in [DMP06,Mo06], summarised in Appendix A of [Mo06].
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V. Moretti
λ enjoys some further properties, in particular a uniqueness property, which will be re-visited later with a point of view different from that adopted in [Mo06]. As it should be clear, the existence of λ has no relation with the presence or the absence of Killing symmetries in M. This B M S-invariant state exists on + which admits the BMS group as symmetry group in any cases since we are dealing with asymptotically flat spacetimes. The interplay of λ and the symmetries of M (if any) will be discussed in Sect. 3. 2.2. Interplay with massless particles propagating in the bulk spacetime. We want now to summarise some achieved results in [DMP06,Mo06]) on the interplay of QFT defined on + and that defined in the bulk M, for a massless conformally coupled scalar field. Consider an asymptotically flat spacetime (M, g) with associated unphysical spacetime ˜ g˜ = 2 g). In addition to asymptotic flatness assume also that both M, M˜ are ( M, globally hyperbolic. Consider standard bosonic QFT in (M, g) based on the symplectic space (S(M), σ M ), where S(M) is the space of real smooth, compactly supported on Cauchy surfaces, solutions φ of the massless, conformally-coupled, Klein–Gordon equation in M: Pφ = 0, where P is the Klein-Gordon operator P = 2g − 16 R. The Cauchy-surface independent symplectic form σ M is: σ M (φ1 , φ2 ) := (φ2 ∇ N φ1 − φ1 ∇ N φ2 ) dµ(S) g ,
(11)
(12)
S
S being any Cauchy surface of M with normal unit future-directed vector N and 3-volume (S) measure dµg induced by g. Henceforth the Weyl algebra associated with the symplectic space (S(M), σ M ), whose Weyl generators are indicated by W M (φ), φ ∈ S(M), will be denoted by W(M). That C ∗ -algebra represents the basic set of quantum observables associated with the bosonic field φ propagating in the bulk spacetime (M, g). The generators W M (φ) are formally interpreted as the exponentials e−iσ M (,φ) , where σ M (, φ) = −σ M (φ, ) is the field operator symplectically smeared with a solution φ ∈ S(M) of field equations (concerning the sign of σ we employ conventions used in [Wa94] which differ from those adopted in [KW91]). The interpretation has a rigorous meaning referring to a GNS representation of W(M). If the considered state ω is regular, the unitary group R t → ω (W (tψ)) is strongly continuous and −iσ M (, φ) can be defined as the self-adjoint generator of it. The more usual field operator ( f ) smeared with functions f ∈ C0∞ (M) is related with σ M (, φ) by means of ( f ) := σ M (, E( f )), where the causal propagator E : C0∞ (M) → C ∞ (M) is the difference of the advanced and retarded fundamental solutions of a Klein-Gordon equation which exist in every globally hyperbolic spacetime [Le53,Di80,BGP96]. solves the KleinGordon equation in a distributional sense: (P f ) = 0 because E ◦ P = 0 by definition. The relation between QFT in M and that defined on + can be now illustrated as follows (a simplified form of Proposition 1.1 in [Mo06]) joined to Proposition 2.5 in [DMP06]. Proposition 2.1. Assume that both the asymptotically flat spacetime (M, g) and the ˜ g) unphysical spacetime ( M, ˜ are globally hyperbolic. The following holds. (a) Every φ ∈ S(M) vanishes approaching + , but (ω)−1 φ extends to a smooth field, ω being any (arbitrarily fixed) positive function defined in a neighbourhood of + associated with a gauge transformation of the geometry on + (see Sect. 1.1).
Quantum Out-States Holographically Induced by Asymptotic Flatness
39
i+ ~ M
I+
M
Fig. 2. Asymptotically flat spacetime with future time infinity i + . (+ is indicated by I + in the figure.)
For the special case ω = ω B we define the R-linear map M : S(M) φ → (ω B )−1 φ + . (b) If M fulfills both the following requirements: (i) M (S(M)) ⊂ S(+ ) and (ii) symplectic forms are preserved by M , that is, for all φ1 , φ2 ∈ S(M), it holds σ M (φ1 , φ2 ) = σ ( M φ1 , M φ2 ), then W(M) can be identified with a sub C ∗ -algebra of W(+ ) by means of a C ∗ -algebra isomorphism ı uniquely determined by the requirement ı(W M (φ)) = W ( M φ),
for all φ ∈ S(M).
(13)
In other words, if (i) and (ii) are valid, the field observables of the bulk M can be identified with certain observables of the boundary + . This is a sort of holographic correspondence. ˜ g) If (M, g) is Minkowski spacetime (so that ( M, ˜ is an Einstein closed universe), hypotheses (i) and (ii) are fulfilled so that ı exists [DMP06]. However there is a large class of asymptotically flat spacetimes which fulfill hypotheses (i) and (ii) as proved in Theorem 4.1 in [Mo06]. They are the asymptotically flat spacetimes, which are globally hyperbolic together with the associated unphysical spacetime and such that admit future time infinity i + in the sense of Friedirich [Fri86-88]. Roughly speaking we may define an asymptotically flat vacuum spacetime with future time infinity i + as an asymptotically flat vacuum spacetime at future null infinity (M, g) such that there is a point i + in ˜ i + ∈ + , such that the geometric extent of + ∪ {i + } the chronological future of M in M, + about i “is the same as that in a region about the tip i + of a light cone in a (curved) spacetime”. The precise definition is stated in Appendix B (see also the discussion in [Mo06]). 3. The State λ M : Invariance Under Isometries and Energy Positivity A straightforward but very important consequence of Proposition 2.1 is that, whenever (i) and (ii) are fulfilled, the G B M S -invariant quasifree pure state λ defined on + can be pulled back to a state λ M (quasifree by construction) acting on bulk observables defined by: λ M (a) := λ(ı(a)) for all a ∈ W(M).
(14)
40
V. Moretti
If (M, g) is Minkowski spacetime, it turns out that λ M coincides with Minkowski vacuum. The main goal of this paper is to study the general properties of λ M whenever it can be defined. Remark 3.1. The assignment of the state λ M has actually quite a little overlap with the difficult problem of finding the solution of a hyperbolic field equation in M˜ with general Cauchy data assigned on the null surface + (the so-called Goursat problem). Here we are instead dealing with fields on + obtained by extending to + solutions of field equations in M, determined by (compactly supported) Cauchy data assigned on Cauchy surfaces in M. The central point is that the symplectic form of the solutions in M is preserved passing from M to + (under suitable hypotheses in particular the presence of i + ). Only this fact individuates the ∗-homomorphism ı of field algebras which associates the observable algebra of the bulk M with a subalgebra of the observables on the boundary + as stated in Proposition 2.1. Since algebraic states are functionals on the corresponding algebras, a state on the algebra on + individuates automatically a state of the algebra in the bulk as in (14). In the rest of this section we prove, in particular, that the state λ M is invariant under the Killing symmetries of M if any and that the self-adjoint generator associated with any Killing time of the bulk must have positive energy (Theorem 3.1 below). The proof is partially based on the known nice interplay between bulk isometries and BMS symmetries, established by several authors in the past and discussed into details in the next section (Proposition 3.1 below). Summarising, the procedure is the following. If M admits a Killing symmetry g M (associated with a Killing field everywhere defined on M), this field extends to + and individuates an associated transformation, say g, of the BMS group. This correspondence is injective. Now one notices that the isometries (M) of the bulk have a natural action βg M on the algebra of observables of the bulk and, on the other hand, the associated BMS symmetries on + have a natural action βg on the algebra of observables defined on + . The remarkable fact, partially established in [DMP06], is that (Proposition 3.4 below) the actions of these corresponding transformations, respectively in the bulk M and on + , commute with the map ı which identifies the elements a of the bulk algebra with the elements ı(a) of the boundary algebra: (M) ı(βg M (a)) = βg (ı(a)). Since λ(ı(a)) = λ M (a) by definition and since λ is invariant under the whole B M S group, λ M must in turn be invariant under the bulk isometries: (M) (M) λ M (βg M (a)) = λ(ı(βg M (a))) = λ(βg ı(a)) = λ(ı(a)) = λ M (a). In this context, the energy positivity property of λ M with respect to timelike Killing vectors in the bulk descends from the fact that isometries generated by timelike Killing vectors of M corresponds to BMS symmetries of a particular type (timelike 4-translations studied in Sect. 3.1). It has been proved in [Mo06] that λ admits positive energy with respect to that type of transformations. Essentially by means of the previously mentioned machinery, it arises that the state λ M admits positive energy with respect to any notion of Killing time in the bulk. 3.1. The spaces of supertranslations, 4-translations and interplay with bulk symmetries. In this section we introduce some notions and results, missed in [DMP06,Mo06], which will play a central role in studying the properties of λ M mentioned in the previous section. We focus on the internal action of G B M S , G B M S α → g ◦ α ◦ g −1 , for any fixed g ∈ G B M S . The decomposition of h ∈ G B M S as a pair ( , f ) ∈ S O(3, 1)↑ ×C ∞ (S2 ) depends on the used admissible frame. However the factor := C ∞ (S2 )
Quantum Out-States Holographically Induced by Asymptotic Flatness
41
is B M S-invariant, i.e. invariant under the above-mentioned internal action for every fixed g ∈ G B M S 3 and thus it is well-defined independently from the used admissible frame, since admissible frames are connected to each other by BMS transformations: If h ∈ G B M S belongs to C ∞ (S2 ) (i.e. has the form (I, α) with α ∈ ) when referring to an admissible frame, the same result holds referring to any other admissible frame. is called the group of supertranslations. However there is another, more important normal subgroup of both G B M S and . That is the group of 4-translations: ⎧ ⎫ ⎨ ⎬ T 4 := α = c jm Y jm c jm ∈ C, α( p) ∈ R, ∀ p ∈ S2 , (15) ⎩ ⎭ j=0,1 |m|≤ j Y jm being the standard spherical harmonics normalized with respect to the measure of the unit sphere S2 . T 4 turns out to be –once-again – B M S-invariant and thus, like , it is well-defined independently from the used admissible frame. Notice that T 4 enjoys the structure of real vector space in addition to that of the additive group. By direct inspection one sees that the internal action of G B M S on T 4 defines in fact a representation of G B M S made of linear transformations with respect to the real-vector-space structure of T 4 . It is possible to pass from the complex basis of T 4 , {Y jm } j=0,1,|m|≤J to a real 4 ), with basis {Y √µ }m=0,1,2,3 (see √ [DMP06] and references√cited therein for more details √ Y0 := 2/π , Y1 = − 2/π sin θcos ϕ, Y2 = − 2/π sin θ sin ϕ, Y3 = − 2/π cos θ . Referring to that basis, if α := µ α µ Yµ and α˜ := µ α˜ µ Yµ , the Lorentzian scalar product < α, α˜ > B M S := −α 0 α˜ 0 + α 1 α˜ 1 + α 2 α˜ 2 + α 3 α˜ 3 , turns out to be B M S-invariant with respect to the above-mentioned internal (and linear) action of G B M S . As a consequence T 4 results to be equipped with a light cone structure: there is a B M Sinvariant decomposition of T 4 \{0} into spacelike, timelike and null 4-translations. Every fixed admissible frame (u, ζ, ζ ) individuates a time-orientation of T 4 . Indeed, consider the B M S diffeomorphism associated with positive rigid translations of + , ατ : R × S2 (u, ζ, ζ ) → (u√ + τ, ζ, ζ ), (τ > 0 fixed). Looking at (2)-(3) one finds that, trivially, ατ identifies with τ π/2Y0 ∈ T 4 . Since τ > 0, ατ picks out the same half of the light-cone not depending on τ . This choice for time-orientation is not affected by changes in the used admissible frame. This is because n = ∂/∂u is always future˜ g) directed with respect to the time-orientation of ( M, ˜ induced by that of (M, g) when working in a Bondi frame. The action of B M S group, to pass to a generic admissible frame, does not change the extent as a consequence of (2)-(3) as one can check by direct inspection. Therefore the light cone in T 4 has a natural preferred time-orientation. With our definition of time-orientation of T 4 , if α ∈ T 4 is causal and future-directed, its ˜ g) action on + displaces the points toward the very future defined in ( M, ˜ by the time orientation of (M, g). The G B M S -subgroup S O(3, 1)↑T 4 is isomorphic to the proper orthochronous Poincaré group. However, differently from T 4 , that group is not normal and different admissible frames select different copies of the G B M S -subgroup isomorphic to the proper orthochronous Poincaré group. We are now ready to state a key result concerning the interplay of the BMS group and symmetries. The following proposition is obtained by collecting together several known 3 In other words and the subsequent subgroup T 4 are normal subgroups of G BMS. 4 In Eq. (3.19) in [DMP06] the statement “if 1 < k ≤ l” has to be corrected to “if 1 ≤ k ≤ l”, whereas in
the right-hand side of the subsequent Eq. (3.20), Yl −k and Yl k have to be corrected to Yl −(k−l) and Yl (k−l) respectively.
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V. Moretti
results but spread out in the literature. In Appendix B there is a proof of statement (c). The results in (a)–(b) can be made much stronger as established in [AX78]. However we do not need here stronger statements than (a)–(b). Proposition 3.1. Let (M, g) be an asymptotically flat spacetime. Redefine → ω B , if necessary, so that + admits Bondi frames. The following facts hold. (a) [Ge77] If ξ is a Killing vector field of (M, g), then ξ smoothly extends to a vector field on the manifold-with-boundary M ∪ + . The restriction to + , ξ˜ , of such an extension is tangent to + , is uniquely determined by ξ , and generates a one-parameter subgroup of G B M S . (b) [AX78] The linear map ξ → ξ˜ defined in (a) fulfills the following properties: (i) it is injective (ξ˜ is the zero vector field on + only if ξ is the zero vector field in M); (iii) if, for a fixed ξ , the one-parameter G B M S -subgroup generated by ξ˜ lies in then, more strictly, it must be a subgroup of T 4 . (c) Consider an one-parameter subgroup of G B M S , {gt }t∈R ⊂ . Suppose that {gt }t∈R arises from the integral curves of a smooth vector ξ˜ tangent to + . Then, in any fixed Bondi frame one has: gt : R × S2 (u, ζ, ζ ) → u + t f (ζ, ζ ), ζ, ζ , where the function f ∈ C ∞ (S2 ) ≡ individuates completely the subgroup. Notice that the fields ξ˜ associated with the one-parameter subgroup of G B M S are always complete since the parameter of the generated one-parameter subgroup ranges in the whole real line by definition. This would be false in case of incompleteness of the field n. The following proposition can be established by direct inspection from (c) in Proposition 3.1 and (2)-(3). Proposition 3.2. Consider a nontrivial one-parameter subgroup of G B M S , {gt }t∈R ⊂ T 4 generated by a smooth complete vector ξ˜ tangent to + . The following facts hold true referring to the time-oriented light-cone structure of T 4 defined above. (a) {gt }t∈R is made of future-directed timelike 4-translations if and only if there is an admissible frame (u, ζ, ζ ) such that the action of {gt }t∈R reduces there to: gt : (u, ζ, ζ ) → (u + t, ζ, ζ ), ∀t ∈ R . (b) {gt }t∈R is made of future-directed causal 4-translations if and only if there is a Bondi frame (u, ζ, ζ ) and constants c > 0, a ∈ R with |a| ≤ 1, such that the action of {gt }t∈R reduces there to ζζ − 1 gt : (u, ζ, ζ ) → u + tc 1 − a (16) , ζ, ζ , ∀t ∈ R . ζζ + 1 These translations are null if and only if |a| = 1. They are timelike for |a| < 1. (c) A 4-translation of T 4 \ {0} viewed as a function f ∈ C ∞ (S2 ) in any arbitrarily fixed admissible frame: (i) is spacelike if and only if f attains both signs, (ii) is timelike and future-directed if and only if f is strictly positive,
Quantum Out-States Holographically Induced by Asymptotic Flatness
43
(iii) is null and future-directed if and only if f is positive and vanishes on a single point of S2 . Propositions 3.2 and 3.1 have the following technical consequence relevant for our goal. Proposition 3.3. Let ξ be a Killing vector of an asymptotically flat spacetime (M, g). Then: (a) ξ individuates an asymptotic Killing symmetry as expected; (b) If ξ is everywhere causal future-oriented, the associated one-parameter subgroup of G B M S is made of causal future-directed elements of T 4 . Proof. The proof is in Appendix B.
2
3.2. Isometry invariance and energy positivity of λ M . We prove that the state λ M is invariant under any isometry generated by the Killing vector ξ of the bulk spacetime M. Moreover we prove that the spectrum of the self-adjoint generator associated with ξ is positive whenever ξ is timelike and thus the generator may be interpreted as an Hamiltonian with positive energy as it is expected from physics. Positivity of the spectrum of the Hamiltonian is a stability requirement: it guarantees that, under small (external) perturbations, the system does not collapse to lower and lower energy states. The proof of invariance of λ M is based on the following remarkable result. Proposition 3.4. Assume that both the asymptotically flat spacetime (M, g) and the ˜ g) unphysical spacetime ( M, ˜ are globally hyperbolic and consider the linear map M : ∞ + S(M) → C ( ) in Proposition 2.1 and the B M S representation A defined in (7). If a complete vector field ξ on (M, g) smoothly extends to M˜ and defines the asymp(ξ ) totic Killing symmetry {gt }t∈R , then the action of that asymptotic symmetry on the field φ in M is equivalent to the action of a BMS-symmetry on the associated field ψ := M φ on + via the representation A: (ξ )
M (φ ◦ g−t ) = A
(ξ˜ )
gt
(ψ) for all t ∈ R if ψ = M φ with φ ∈ S(M),
(17)
(ξ˜ )
where {gt }t∈R is the one-parameter subgroup of G B M S generated by the smooth extension ξ˜ to + of ξ . Proof. The proof is in Appendix B.
2
(ξ )
Notice that, in general, φ ◦ g−t does not belong to S(M) if φ does. However it happens (ξ ) when gt is an isometry, since the Klein–Gordon equation and thus S(M) are invariant under isometries of (M, g). We now prove one of the main results of this work. As is known the identity component G1 of a Lie group G is the subgroup made of the connected component of G containing the unit element of G. Theorem 3.1. Assume that both the asymptotically flat spacetime (M, g) and the unphys˜ g) ical spacetime ( M, ˜ are globally hyperbolic and conditions (i) and (ii) in (b) of Proposition 2.1 are fulfilled. Consider the quasifree state λ M canonically induced on W(M) from the BMS-invariant quasifree pure state λ defined on + . The following facts are valid:
44
V. Moretti
(a) λ M coincides with (free) Minkowski vacuum if (M, g) is Minkowski spacetime. (b) λ M is invariant under the identity component G1 of the Lie group G of isometries of M: λ M (βg a) = λ M (a), for all a ∈ W(M) and every g ∈ G1 ,
(18)
where β is the (isometric) ∗-isomorphism representation of G uniquely induced (imposing linearity and continuity) by the requirement on Weyl generators βg (W (φ)) := W (φ ◦ g −1 ), for every φ ∈ S(M) and g ∈ G. Thus, in particular the Lie-subgroup G1 admits unitary implementation in the GNS representation of λ M . (b)’ (b) holds for any state λM with λM (a) := λ (ı(a)), ∀a ∈ W(M), where λ is any BMS invariant state (not necessarily quasifree or pure or satisfying some positivity-energy condition) defined on W(+ ). (c) Assume that (M, g) admits a complete causal future-directed Killing vector ξ . The unitary one-parameter group which leaves the cyclic vector fixed and implements the one-parameter group of isometries generated by ξ in the GNS (Fock) space of λ M satisfies the following properties: (i) it is strongly continuous, (ii) the associated self-adjoint generator, H (ξ ) , has nonnegative spectrum, (iii) the restriction of H (ξ ) to the one-particle space has no zero modes. Remark 3.2. (1) Concerning in particular the statement (c), when ξ is timelike and futuredirected, H (ξ ) provides a natural (positive) notion of energy, associated with ξ displacements. Since λ M is quasifree, its GNS representation is a Fock representation. When ξ is timelike, the collection of properties (i), (ii) are summarised [KW91] by saying that λ M is a ground state. Then property (iii) states that λ M is a regular ground state if adopting terminology in [KW91]. (2) The notion of “completeness” adopted in Proposition 2.1 and in (c) of Theorem 3.1 has the standard meaning: the vector field (which is defined everywhere on M, and satisfies Killing’s constraints in the second case) generates a global one-parameter group of isometries, i.e. the parameter of the group must range from −∞ to +∞ along every integral line. This requirement allows one to avoid problems with the associated one-parameter group of transformations acting on functions with domain in M: The functions are extended objects and one would prevent problems concerning the domains of the parameter of integral lines since, in principle they could depend on the considered integral curve. Proof of Theorem 3.1. (a) It was proved in Theorem 4.5 [DMP06]. (b) It is well-known [O’N83] that there is a unique way to assign a Lie-group structure to the group of isometries G of a (semi-)Riemannian manifold (M, g) such that the action of the oneparameter subgroup is jointly smooth when acting on the manifold. Moreover the Lie algebra of G is that of the complete (i.e. generating global group of isometries) Killing vectors of (M, g). Finally using the exponential map one sees that every element of the identity component G1 can be obtained as a finite product of elements which belong to one-parameter subgroups. As a consequence, to establish the validity of (b) it is sufficient to prove that λ M is invariant under the one-parameter subgroups generated by Killing vectors of (M, g). Let us prove it. Let ξ be the complete Killing vector of (M, g) and
Quantum Out-States Holographically Induced by Asymptotic Flatness
45
ξ˜ the associated generator of G B M S on + in view of Proposition 3.1. Employing the same notation as in Proposition 3.4 and using the definition (14), one achieves: (ξ ) λ M W M (φ ◦ g−t ) = λ W (A (ξ˜ ) (ψ)) . gt
The right-hand side is, by definition, λ α (ξ˜ ) (W (ψ)) = λ (W (ψ)) , gt
where, in the last step we have used the invariance of λ under the representation α of BMS-group defined in Sect. 2.1. Since ψ = M φ and using (14) again we finally obtain that (ξ ) λ M W M (φ ◦ g−t ) = λ M (W M (φ)) . By linearity and continuity this result extends the whole algebra W(M): λ M (βg(ξ ) (a)) = λ M (a), for every a ∈ W(M) . t
Since λ is invariant there is a unique unitary implementation of the representation β in the GNS representation of λ which leaves fixed the cyclic vector (e.g. see [Ar99]). The proof of (b)’ is the same as that given for (b), replacing λ M with λM . (c) As λ M is quasifree, its GNS representation is a Fock representation (e.g. see Appendix A of [Mo06] and references cited therein, especially [KW91]). As a consequence it is sufficient to prove the positivity property for the restriction of the unitary group which represents the group of isometries in the one-particle space H M . The GNS triple of λ M is obtained as follows. Consider the GNS triple of λ, (H, , ϒ), where H = F+ (H) is the bosonic Fock space with one-particle space H. As said above, that space is isomorphic to the space of (Fourier transforms of the) u-positive frequency parts L 2 (R+ × S2 , 2kdk ∧ S2 (ζ, ζ )) referring to a fixed Bondi frame (u, ζ, ζ ), k being the Fourier variable associated with u. Consider the Hilbert subspace H M of H obtained by taking the closure of the complex span of the u-positive-frequency parts of the wavefunctions M φ, for every φ ∈ S(M). Let H M = F+ (H M ) be the Fock space generated by H M which, in turn, is a Hilbert subspace of H. Notice that we are assuming that the vacuum vectors ϒ M and ϒ coincide. By construction H M is invariant under and M := HM is a ∗-representation of ı(W(M)). Moreover M (ı(W(M)))ϒ M = M (ı(W(M)))ϒ is dense in H M by construction. By the uniqueness (up to unitary maps) property of the GNS triple, we conclude that (H M , M , ϒ M ) is the GNS triple of λ M .5 Consider the unique unitary G B M S representation U which acts on H implementing α and leaving ϒ(= ϒ M ) fixed. It is the unitary B M S representation defined by linearity and continuity by the requirement: Ug (W (ψ))ϒ := (αg (W (ψ)))ϒ, for all ψ ∈ S(+ ) and g ∈ G B M S .
(19)
(ξ )
Since the space S(M) is invariant under the group of isometries gt and (17) holds true, it arises that α (ξ˜ ) (W (ψ)) ∈ ı(W(M)) if W (ψ) ∈ ı(W(M)) and thus (19) entails gt
U
(ξ˜ )
gt
M (W M (φ))ϒ M := (βg(ξ ) (W M (φ)))ϒ M , for all φ ∈ S(M) and t ∈ R. (20) t
5 Notice that may be reducible also if is irreducible: in other words λ may be a mixture also if λ M M
is pure.
46
V. Moretti
As a consequence of (20) we can conclude that the unique unitary representation U (ξ ) (ξ ) of {gt }t∈R on H M which leaves ϒ M invariant, is nothing but the restriction of U to (ξ˜ )
{gt }t∈R ⊂ G B M S and H M ⊂ H. This result allows us to compute explicitly the self(ξ ) adjoint generator of the unitary representation of {gt }t∈R . The representation U is obtained by tensorialization of a unitary representation (1) U of G B M S working in the one particle space [DMP06,Mo06] (notice that in those papers, as one-particle space, we used the unitarily isomorphic space L 2 (R+ × S2 ; dk ∧ S2 ) instead of L 2 (R+ × S2 ; 2kdk ∧ S2 ), therefore the expression below looks different, but it is equivalent to that given in [DMP06,Mo06]):
(1)
U ( , f ) ϕ (k, ζ, ζ )
= eik K (
−1 (ζ,ζ )) f ( −1 (ζ,ζ ))
ϕ k K −1 (ζ, ζ ) , −1 (ζ, ζ ) ,
(21)
for every ϕ ∈ L 2 (R+ ×S2 ; 2kdk ∧S2 ) and G B M S g ≡ ( , f ). The restriction of (1) U (ξ˜ )
(ξ )
to {gt }t∈R and H M defines a unitary representation of {gt }t∈R whose tensorialization (ξ˜ ) on F+ (H M ) is the very representation (1) U . Notice that (1) U restricted to {gt }t∈R leaves invariant H M by construction because H M is the closure of the span of vectors d/dt|t=0 (W (tψ))ϒ for ψ ∈ M (S(M)) (the derivative being computed using the Hilbert topology). To conclude the proof it is sufficient to prove that the self-adjoint (1) generator of U (ξ˜ ) H M exists and has positive spectrum. gt
t∈R
(ξ˜ )
In our hypotheses {gt } is a one-parameter group of causal future-directed 4-translations. As a consequence, selecting the Bondi frame as in (b) in Proposition 3.2, we have that there are a real a with |a| ≤ 1 and a real c > 0 such that (ξ˜ ) gt
: (u, ζ, ζ ) → u + tc 1 − a
ζζ − 1
ζζ + 1
, ζ, ζ , for every t ∈ R.
Therefore, if ϕ ∈ H ≡ L 2 (R+ × S2 ; 2kdk ∧ S2 ),
(1)
U
(ξ˜ ) gt
itkc 1−a ζ ζ −1 ζ ζ +1 ϕ k, ζ, ζ . ϕ (k, ζ, ζ ) = e
(22)
Strong continuity is obvious (also after restriction to H M ). Finally, using Lebesgue’s dominate convergence to evaluate the strong-operator topology derivative at t = 0 of (1) U (ξ ) , where the self-adjoint operator h (ξ ) (ξ˜ ) , one obtains that this derivative is i h gt
reads
(h
(ξ )
ϕ)(k, ζ, ζ ) := kc 1 − a
ζζ − 1 ζζ + 1
ϕ(k, ζ, ζ ),
(23)
with dense domain D(h (ξ ) ) made of the vectors ϕ ∈ L 2 (R+ × S2 ; 2kdk ∧ S2 ) such that the right-hand side of (23) belongs to L 2 (R+ × S2 ; 2kdk ∧ S2 ) again. In view of the
Quantum Out-States Holographically Induced by Asymptotic Flatness
Stone Theorem h (ξ ) is the self-adjoint generator of coordinates:
kc 1 − a
ζζ − 1 ζζ + 1
(1) U
47
(ξ˜ )
gt
. Passing to work in polar
= kc (1 − a cos θ ) ≥ 0
(24)
because k ∈ [0 + ∞), c > 0 and a ∈ R with |a| ≤ 1. Therefore interpreting the integral below as a Lebesgue integral in F := [0, +∞) × [0, π ] × [−π, π ]: ϕ, h (ξ ) ϕ = 2c |ϕ(k, θ, φ)|2 (1 − a cos θ ) k 2 sin2 θ dkdθ dφ ≥ 0, F
for all ϕ ∈ D(h (ξ ) ).
(25)
h (ξ )
is included in [0, +∞) via spectral theorem. This fact entails that the spectrum of The result remains unchanged when restricting (1) U (ξ˜ ) (and thus h (ξ ) ) to the invariant gt
Hilbert-subspace H M . Suppose that there is a zero mode of h (ξ ) , that is ϕ ∈ H M \{0} with h (ξ ) ϕ = 0. By (25), |ϕ(k, θ, φ)|2 (1 − a cos θ ) k 2 sin2 θ dkdθ dφ = 0. F
The integrand is nonnegative on F since (24) is valid, therefore the integrand must vanish almost everywhere in the Lebesgue measure of R3 . Since the function (k, θ, φ) → (1 − a cos θ ) k 2 sin2 θ is almost-everywhere strictly positive on F, it has to hold ϕ = 0 almost-everywhere. Thus ϕ = 0 as an element of L 2 (R+ × S2 ; 2kdk ∧ S2 ). In other words h (ξ ) has no zero modes. 2 3.3. Reformulation of the uniqueness theorem for λ. It is clear that there are asymptotically flat spacetimes which do not admit any isometry. In those cases the invariance property stated in (a) and the positivity energy condition (c) of Theorem 3.1 are meaningless. However those statements remain valid if referring to the asymptotic theory based on QFT on + and the universal state λ. Indeed λ is invariant under the whole G B M S group—which represents asymptotic symmetries of every asymptotically flat spacetime—and λ satisfies a positivity energy condition with respect to every smooth one-parameter subgroup of G B M S made of future-directed timelike or null 4-translations—which correspond to Killing-time evolutions whenever the spacetime admits a timelike Killing field, as established above. As proved in Theorem 3.1 in [Mo06], the energy positivity condition with respect to timelike 4-translations determines λ uniquely. We may restate the theorem in a more invariant form as follows. The possibility of such a re-formulation was already noticed in a comment in [Mo06]; here, using the introduced machinery, we are able to do it explicitly6 . Theorem 3.2. Consider a nontrivial one-parameter subgroup of G B M S , G := {gt }t∈R made of future-directed timelike 4-translations, associated with a smooth complete vector tangent to + and let α (G) be the one-parameter group of ∗-isomorphisms induced by G on W(+ ). 6 The author is grateful to A. Ashtekar for suggesting this improved formulation of the theorem.
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V. Moretti
(a) The BMS-invariant state λ is the unique pure quasifree state on W(+ ) satisfying both: (i) it is invariant under α (G) , (ii) the unitary group which implements α (G) leaving fixed the cyclic GNS vector is strongly continuous with nonnegative generator. (b) Let ω be a pure (not necessarily quasifree) state on W(+ ) which is B M S-invariant or, more weakly, α (G) -invariant. ω is the unique state on W(+ ) satisfying both: (i) it is invariant under α (G) , (ii) it belongs to the folium of ω. Proof. The proof is that given for Theorem 3.1 in [Mo06] working in the admissible frame, individuated in (a) of Proposition 3.2, where the action of G reduces to gt : (u, ζ, ζ ) → (u + t, ζ, ζ ), ∀t ∈ R . 2 4. The Hadamard Property 4.1. Hadamard states. It is well known that Hadamard states [KW91,Wa94] have particular physical interest in relation with the definition of physical quantities which, as the stress-energy tensor operator (e.g. see [Mo03,HW04]), cannot be represented in terms of elements of the Weyl algebra or the associated ∗-algebra of products of smeared field operators. In the last decade the deep and strong relevance of Hadamard states in local generally covariant QFT in curved spacetime has been emphasised from different points of view (e.g. see [BFK96,BF00,HW01,BFV03]). Consider a quantum scalar real bosonic field φ propagating in a globally hyperbolic spacetime (M, g) satisfying the Klein–Gordon equation with a Klein–Gordon operator P := 2 + V (x) (V being any fixed smooth real function) and consider the quantisation procedure based on Weyl algebra approach. The rigorous definition of Hadamard state ω for φ has been given in [KW91] in terms of a requirement on the behaviour of the singular part of the integral kernel of two-point function of ω. The two-point function of ω is the bi-linear functional: ∂2 ω( f, g) := − ω (W M (s E f + t Eg)) eistσ M (E f,Eg)/2 , ∂s∂t f, g ∈ C0∞ (M) × C0∞ (M).
C0∞ (M)
s=t=0
(26)
C ∞ (M)
Above E : → S(M) ⊂ is the previously mentioned causal propagator. The two-point function ω( f, g) exists if and only if the right-hand side of (26) makes sense for every pair f, g. This happens in particular whenever the GNS representation of ω is a Fock representation, that is when ω is quasifree [KW91] (see also Appendix A in [Mo06]). The proof is straightforward and it provides a heuristic motivation for the definition (26). For a quasifree state ω one finds ω( f, g) = ϒω , ( f )(g)ϒω ,
(27)
where ϒω is the cyclic GNS vector which, in this case, coincides with the Fock vacuum vector and ( f ) denotes the self-adjoint field operator smeared with the smooth function f defined in the GNS Hilbert space Hω . Notice that, since E ◦ P = P ◦ E = 0 if the two-point function exists one gets [Wa94]: ω(P f, g) = ω( f, Pg) = 0 (KG)
Quantum Out-States Holographically Induced by Asymptotic Flatness
and, directly from (26),
49
ω( f, g) − ω(g, f ) = −i
f (x) (E(g)) (x) dµg (x) (Com). M
If a two-point function ω( f, g) exists on C0∞ (M) × C0∞ (M), the integral kernel ω(x, y) is defined (if it exists at all) as the function, generally singular and affected by some → 0+ prescription, such that ω(x, y) f (x)g(y) dµg (x)dµg (y) = ω( f, g), for all f, g ∈ C0∞ (M) . (28) M×M
Referring to a quantum scalar real bosonic field φ propagating in a globally hyperbolic spacetime (M, g) satisfying the Klein–Gordon equation, a quasifree state ω which admits a two-point function is said to be Hadamard if its integral kernel ω(x, y) exists and satisfies the global Hadamard prescription. That prescription requires that ω(x, y) takes a certain—quite complicated—form in a neighbourhood of a Cauchy surface of the spacetime, as discussed in details in Sect. 3.3 of [KW91]. The global Hadamard condition implies the validity of the local Hadamard condition which states that, every point p ∈ M admits a (open geodesically convex normal) neighbourhood G p , such that U (x, y) ω(x, y) = w-lim+ + V (x, y) ln(σ (x, y) →0 σ (x, y) + 2i(T (x) − T (y)) + 2 + 2i(T (x) − T (y)) + 2 ) + ωr eg (x, y), if (x, y) ∈ G p × G p ,
(LH)
where σ (x, y) is the “squared geodesic distance” of x from y, T is any, arbitrarily fixed, time function increasing to the future and U and V are locally well-defined quantities depending on the local geometry only. Finally ωr eg is smooth and is, in fact, the only part of the two-point function determining the state. w-lim→0+ indicates that the limit as → 0+ is to be understood in the weak sense, i.e. after the integration of ω(x, y) with smooth compactly supported functions f and g. In a pair of very remarkable papers [Ra96a,Ra96b] Radzikowski established several important results about Hadamard states, in particular he found out a microlocal characterisation of Hadamard states (part of Theorem 5.1 in [Ra96a]; henceforth D (N ) is the space of distributions on C0∞ (N )): Proposition 4.1. In a globally hyperbolic spacetime (M, g), consider a quasifree state with two-point function ω (so that (KG) and (Com) are valid) defining a distribution of D (M × M). The state is Hadamard if and only if the wavefront set W F(ω) of the distribution is W F(ω) = ((x, k x ), (y, −k y )) ∈ T ∗ M\0 × T ∗ M\0 | (x, k x ) ∼ (y, k y ), k x 0 , (29) where (x, k x ) ∼ (y, k y ) means that there is a null geodesic joining x and y with co-tangent vectors at x and y given by k x and k y respectively, whereas k 0 means that k is causal and future directed. 0 is the zero section of the cotangent bundle. A second result by Radzikowski, which in fact proved a conjecture by Kay, establishes that (immediate consequence of Corollary 11.1 in [Ra96b]):
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V. Moretti
Proposition 4.2. In a globally hyperbolic spacetime (M, g), if the two-point function of a quasifree state is a distribution ω ∈ D (M × M) satisfying (LH) when restricted to G p × G p , for some open neighbourhood G P of every point p ∈ M, then the state is Hadamard. In the following we shall prove that, in the presence of i + , the following results hold true. (i) λ M is a distribution of D (M × M) and, making use of Radzikowski results, (ii) λ M is Hadamard. To tackle item (i) we have to introduce some notions concerning a straightforward extension of the Fourier–Plancherel transform theory for functions and distributions defined on + ≡ R × S2 . This is done in Appendix C. 4.2. The integral kernel of λ M is a distribution when (M, g) admits i + . Since the considered spacetimes are equipped, by definitions, with metrics and thus preferred volume measures, here we assume that distributions of D (M × M) work on smooth compactlysupported scalar fields of D(M) := C0∞ (M) as in [Fr75] instead of smooth compactlysupported scalar densities as in [Hör71]. As is well known this choice is purely a matter of convention since the two points of view are equivalent. First of all we prove that λ M individuates a distribution in D (M × M), i.e. it is continuous in the relevant weak topology [Fr75] whenever the spacetime (M, g) is a vacuum asymptotically flat at null infinity spacetime and admits future temporal infinity i + . We give also a useful explicit expression for the distribution. Theorem 4.1. Assume that the spacetime (M, g) is an asymptotically flat vacuum spacetime with future time infinity i + (Definition A.1) and that both (M, g) and the unphysical ˜ g) spacetime ( M, ˜ are globally hyperbolic. Let E : C0∞ (M) → S(M) be the causal propagator associated with the real massless conformally-coupled Klein–Gordon operator P defined by Eq. (11) on (M, g). Then the following facts are valid concerning the state λ M defined in Eq. (14). (a) Referring to a Bondi frame (u, ζ, ζ ) one has ψ f (u, ζ, ζ )ψg (u , ζ, ζ ) 1 λ M ( f, g) = lim+ − du ∧ du ∧ S2 (ζ, ζ ), (30) →0 π R2 ×S2 (u − u − i)2 where ψh := M (Eh) for all h ∈ C0∞ (M) with M : S(M) → S(+ ) defined in Proposition 2.1. (b) The two-point function of the state λ M individuates a distribution of D (M × M). Remark 4.1. It is intriguing noticing that the expression (30) is the same as that for two-point functions of quasifree Hadamard states obtained in [KW91] (Eq. (4.13)) in globally hyperbolic spacetimes with bifurcate Killing horizon. In that case the null 3-manifold + is replaced by a bifurcate Killing horizon, the 2-dimensional cross section S2 with spacelike metric corresponds to the bifurcation surface with spacelike metric and, finally, the null geodesics forming + , parametrised by the affine parameter u, correspond to the null geodesics forming the Killing horizon parametrised by the affine parameter U . Proof of Theorem 4.1. From now on we use the notations and the theory about the Fourier–Plancherel transform presented in Appendix C. In particular F : S (+ ) → S (+ ) denotes the extension to distributions of F+ as stated in (d) in Theorem C.1 whose inverse, F −1 , is the analogous extension of F− . We call F the Fourier–Plancherel transformation, also if, properly speaking this name should be reserved to its
Quantum Out-States Holographically Induced by Asymptotic Flatness
51
restriction to L 2 (R × S2 , du ∧ S2 (ζ, ζ )) defined in (e) in Theorem C.1. We also use the formal distributional notation for F (and the analog for F −1 ) eiku F [ψ](k, ζ, ζ ) := √ ψ(u, ζ, ζ )du, R 2π (k, ζ, ζ ) is also regardless if f is a function or a distribution. Throughout the notation ψ used for the Fourier(-Plancherel, extension to distributions) transform F [ψ](k, ζ, ζ ). We start with a useful lemma. Lemma 4.1. In the hypotheses of theorem above, if h ∈ C0∞ (M), the following holds. (a) ψh can be written in terms of the causal propagator E˜ for the massless conformally ˜ g˜ = 2 g) and the smooth function ω B > 0 coupled Klein–Gordon operator P˜ in ( M, defined on + (see Sect. 1.2): ˜ −3 h)+ (u, ζ, ζ ), for u ∈ R and (ζ, ζ ) ∈ S2 . (31) ψh (u, ζ, ζ ) = ω B (u, ζ, ζ )−1 E( (b) For any compact K ⊂ M there is u 0 ∈ R such that, if supp h ⊂ K , ψh (u, ζ, ζ ) = 0 for u < u 0 and all (ζ, ζ ) ∈ S2 . Proof. The proof is in Appendix B.
2
Let us pass to the main proof. (a) We start from the fact that, as found in the proof of Theorem 3.1, the Fock GNS triple of λ M , (H M , M , ϒ M ) is such that ϒ M = ϒ and M (W M (ψ)) = M (W ( M (ψ))). In our hypotheses, since λ M is quasifree, one has, referring to its GNS representation (H M , M , ϒ M ): λ M ( f, g) = ϒ M , ( f )(g)ϒ M = ϒ, σ (, M (E f ))σ (, M (Eg))ϒ = ψ f + , ψg+ , f is the Fouwhere ψh+ is the u-positive frequency part of M (Eh). Using (10), if ψ rier–Plancherel transformation of ψh one has finally: f (k, ζ, ζ )ψ g (k, ζ, ζ )dk ∧ S2 (ζ, ζ ). λ M ( f, g) = 2k ψ R+ ×S2
If (k) = 0 for k ≤ 0 and (k) = 1 for k > 0, the identity above can be rewritten as f (k, ζ, ζ )2k(k)ψ g (k, ζ, ζ )dk ∧ S2 (ζ, ζ ). λ M ( f, g) = ψ (32) R×S2
We remind the reader that, by definition of S(+ ), ψ f and ψg are real, smooth and ψ f , ψg , ∂u ψ f , ∂u ψg belong to L 2 (R × S2 , dk ∧ S2 (ζ, ζ )). Using the fact that the Fourier–Plancherel transformation defined on the real line is unitary one gets: f (k, ζ, ζ )2k(k)ψ g (k, ζ, ζ )dk ∧ S2 (ζ, ζ ) ψ R g ](u, ζ, ζ )du ∧ S2 (ζ, ζ ). = ψ f (u, ζ, ζ )F −1 [2k ψ (33) R
52
V. Moretti
Notice that the identity above makes sense because both ψ f , ∂u ψg ∈ L 2 (R × S2 , dk ∧ S2 (ζ, ζ )), by definition of the space S(+ ), so that the Fourier–Plancherel transform of g up to a constant factor, and the restriction to the latter to k ∈ R+ are ∂u ψg , which is k ψ 2 g (k, ζ, ζ ) converges, as → 0+ , to (k)ψ g (k, ζ, ζ ) L as well. Now, since (k)e−k ψ 2 2 in the sense of L (R × S , dk ∧ S2 (ζ, ζ )), and using the fact that the (inverse) Fourier– Plancherel transform is continuous, one has g ] → F −1 [k ψ g ], F −1 [e−k k ψ as → 0+ in the topology of L 2 (R × S2 , dk ∧ S2 (ζ, ζ )).
(34)
The left-hand side can be computed by means of a convolution theorem (the convo g (k, ζ, ζ ) and k → lution restricted to the variable u) since both functions k → ψ −k 2 2 (k)e belong to L (R × S , dk) by construction almost everywhere in (ζ, ζ ) fixed (for the former function it follows from Fubini-Tonelli’s theorem using the fact that g ∈ L 2 (R × S2 , dk ∧ S2 (ζ, ζ )) since ψg ∈ L 2 (R × S2 , du ∧ S2 (ζ, ζ ))). In this way, ψ by direct inspection one finds g ](u, ζ, ζ ) = F −1 [e−k k ψ
∂u ψg (u , ζ, ζ ) 1 du . 2π R u − u − i
Inserting it in (34) we have achieved that, as → 0+ in the topology of L 2 (R × S2 , dk ∧ S2 (ζ, ζ )), ∂u ψg (u , ζ, ζ ) 1 g ]. du → F −1 [k ψ 2π R u − u − i Inserting it in the right-hand side of (33) we have: λ M ( f, g) =
ψ f (u, ζ, ζ )∂u ψg (u , ζ, ζ ) 1 du , du ∧ S2 (ζ, ζ ) lim+ →0 R π R×S2 u − u − i
(35)
then, using the continuity of the scalar product of the Hilbert space L 2 (R × S2 , du ∧ S2 (ζ, ζ )) one obtains: λ M ( f, g) = lim+ →0
ψ f (u, ζ, ζ )∂u ψg (u , ζ, ζ ) 1 du . du ∧ S2 (ζ, ζ ) π R×S2 u − u − i R
(36)
Since, by hypotheses, both ψg , ∂u ψg belong to C ∞ (R) ∩ L 2 (R, du) almost everywhere in (ζ, ζ ), one has ψg (u, ζ, ζ ) → 0 for u → ±∞.7 Integrating by parts the last integral one obtains in that way ψ f (u, ζ, ζ )ψg (u , ζ, ζ ) 1 λ M ( f, g) = lim+ − du ∧ S2 (ζ, ζ ) du . →0 π R×S2 (u − u − i)2 R
(37)
7 Work at fixed (ζ, ζ ). Using elementary calculus, by continuity of ∂ ψ and Cauchy-Schwarz inequalu g ity, one has |ψg (u ) − ψg (u)| ≤ ∂u ψg || L 2 (R,du) |u − u | so that u → ψg (u) is uniformly continuous. If ψg → 0 as u → +∞ (the other case is analogous) one would find a sequence of intervals Ik centred on k = 1, 2, . . . with I du > and |ψg |Ik > M for some M > 0 and > 0. As a consequence it would be k
2 R |ψg (u)| du = +∞.
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To conclude the proof it is sufficient to show that, for > 0 the function (u, u ζ, ζ ) →
|ψ f (u, ζ, ζ )||ψg (u , ζ, ζ )| =: H (u, u , ζ, ζ ) (u − u )2 + 2
is integrable in the joint measure of R × R × S2 . Since the function is positive, it is equivalent to prove that the function is integrable under iterated integrations, first in du and then with respect to du ∧ S2 (ζ, ζ ). We decompose the iterated integration into four terms: du ∧ S2 (ζ, ζ ) du H (u, u , ζ, ζ ) 2 [0,u 1 )×S [0,u 1 ) + du ∧ S2 (ζ, ζ ) du H (u, u , ζ, ζ ) 2 [u ,+∞)×S [0,u 1 ) 1 + du ∧ S2 (ζ, ζ ) du H (u, u , ζ, ζ ) [0,u 1 )×S2 [u 1 ,+∞) + du ∧ S2 (ζ, ζ ) du H (u, u , ζ, ζ ). (38) [u 1 ,+∞)×S2
[u 1 ,+∞)
Above we have fixed the origin of u and u in the past of the support of ψ f and ψg on + . This is possible due to the last statement in Lemma 4.1. The point u 1 is taken as specified in the following lemma. Lemma 4.2. Assume that the spacetime (M, g) is an asymptotically flat vacuum spacetime with future time infinity i + (Definition A.1). Referring to a Bondi frame, for every β ∈ [1, 2) there are u 1 > 0, a compact ball B centred in i + defined with respect to a suitable coordinate patch x 1 , x 2 , x 3 , x 4 in M˜ centred on i + , and constants a, b > 0 such that if u ≥ u 1 , (ζ, ζ ) ∈ S2 : a M a M −1 + )(u, ζ, ζ ) ≤ , ∂u (ω−1 , ω B + (u, ζ, ζ ) ≤ B |u − b| |u − b|β ˜ and where: for every ∈ C ∞ ( M) M := max sup ||, sup |∂x 1 |, · · · , sup |∂x 4 | . B
Proof. The proof is in Appendix B.
B
(39)
(40)
B
2
If h ∈ C0∞ (M), the lemma above entails that (with β = 1), for some constants a, b > 0: |ψh (u, ζ, ζ )|, |∂u ψh (u, ζ, ζ )| ≤
a Mh , u−b
(41)
where
˜ ˜ ˜ Mh := max sup | E(h)|, sup |∂x 1 E(h)|, · · · , sup |∂x 4 E(h)| . B
B
B
(42)
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Enlarging u 1 if necessary, we can always assume that u 1 > u 0 , b. In the decomposition (38) we use that value for u 1 . Therein the first integral converges trivially. Concerning the last integral, due to Eq. (41), we have the estimation in its domain of integration H (u, u , ζ, ζ ) ≤
M f Mg a2 . (u − u )2 + 2 (u − b)(u − b)
Using that and the fact that the volume of S2 is finite, by direct computation one finds du ∧ S2 (ζ, ζ ) du H (u, u , ζ, ζ ) [u 1 ,+∞)×S2 +∞
[u 1 ,+∞)
4πa 2 M f Mg ≤ du (u − b)(1 + u − b) u1 ! " π −1 u−u 1 1 (u 1 − b)2 2 + tan − ln × < +∞. 2(u − b) (u 1 − u)2 + 2
By Fubini-Tonelli’s theorem (H is positive) the second iterated integral in (38) converges if the third does. Concerning the third we have the estimation in its domain of integration (notice that ψ f is smooth in [0, u 1 ] × S2 and thus bounded and u ≥ u 1 > b) H (u, u , ζ, ζ ) ≤
C C ≤ [(u − u )2 + 2 ](u − b) (u − u )2 + 2
for some constants C, C ≥ 0. Therefore, computing the integral in u and using the finite volume of S2 we find: u1 π 1 + tan−1 u−u < +∞. du ∧ S2 (ζ, ζ ) du H (u, u , ζ, ζ ) ≤ C du 2 [0,u 1 )×S2 [u 1 ,+∞) 0 We conclude that the function H is integrable in the joint measure of R × R × S2 so that (37) entails (30). (b) Due to the Schwartz kernel theorem [Hö89], statement (b) is equivalent to prove that (i) for every g ∈ C0∞ (M), C0∞ (M) f → λ M ( f, g) is continuous in the topology of C0∞ (M) and (ii) the linear map C0∞ (M) g → λ M (·, g) ∈ D (M) is weakly continuous. (ii) means that, for every fixed f ∈ C0∞ (M), if {gn }n∈N ⊂ C0∞ (M) converges to 0, as n → +∞, in the topology of C0∞ (M), then λ M ( f, gn ) → 0 as n → +∞. To prove that the couple of requirements is fulfilled notice that, by Cauchy-Schwarz inequality and (32) one finds kψ f 2 g 2 |λ M ( f, g)| ≤ ψ L (R×S2 ,dk∧S2 ) L (R×S2 ,dk∧S2 ) ≤ C g ψ f L 2 (R×S2 ,du∧ ) , (43) S2 ψ f 2 g 2 |λ M ( f, g)| ≤ kψ L (R×S2 ,dk∧S2 ) L (R×S2 ,dk∧S2 ) ≤ C f ψg L 2 (R×S2 ,du∧ ) , (44) S2
f || L 2 (R×S2 ) , C g := ||kψ g || L 2 (R×S2 ) and we where, in the last passages C f := ||kψ have used the fact that the Fourier–Plancherel transform is isometric. Thus, the statement (b) is true if ||ψgn || L 2 (R×S2 ) → 0 for gn → 0 in the topology of C0∞ (M). Let us prove
Quantum Out-States Holographically Induced by Asymptotic Flatness
55
this fact exploiting (31) and (41) for h = gn . It is known that the causal propagator ˜ → C ∞ ( M) ˜ is continuous in defined in a globally hyperbolic spacetime E˜ : C0∞ ( M) the standard compactly-supported test-function topology in the domain and the natural ˜ (see [Le53,Di80,BGP96]). Fix f ∈ C ∞ (M), a compact Fréchet topology in C ∞ ( M) 0 set K ⊂ M and a sequence {gn }n∈N ⊂ C0∞ (M) supported in K . From (b) in Lemma 4.1 there is u 0 ∈ R such that the support of every ψgn is included in the set u ≥ u 0 . Moreover from Lemma 4.2, we know that, if u 1 > 0 is sufficiently large, there is a compact ball B centred in i + defined with respect to a suitable coordinate patch centred on i + , and constants a, b > 0 such that if u ≥ u 1 , (ζ, ζ ) ∈ S2 (41) hold for u = gn (for every n), where ˜ n )|, sup |∂x 1 E(g ˜ n )|, · · · , sup |∂x 4 E(g ˜ n )| . Mgn := max sup | E(g B
B
B
Enlarging u 1 if necessary, we can always assume that u 1 > u 0 , b. Continuity of E˜ implies that Mgn → 0 as n → +∞. If B ⊂ M˜ is another compact set such that B ⊃ {(u, ζ, ζ ) ∈ + | u 1 > u > u 0 }, since ω−1 B is bounded therein, ˜ continuity of E entails by (31) that ψgn vanishes uniformly as n → +∞ in B . Now ψg (u, ζ, ζ )2 du ∧ S2 (ζ, ζ ). ||ψgn ||2L 2 (R×S2 ,du∧ ) = n S2
[u 0 ,+∞)×S2
Decompose the last integral into two terms, the former corresponding to the integration from u 0 to u 1 and the latter from u 1 to +∞. Both parts vanish as n → +∞. The former vanishes because ψgn vanishes uniformly on {(u, ζ, ζ ) ∈ + |u 1 > u > u 0 } as n → +∞, the latter vanishes as a consequence of (41) with h = gn , since Mgn → 0 as n → +∞ and +∞ 1 ψg (u, ζ, ζ )2 du ∧ S2 (ζ, ζ ) ≤ a 2 M 2 4π du. n gn (u − b)2 [u 1 ,+∞)×S2 u1 2 4.3. λ M is Hadamard when (M, g) admits i + . We are in place to state and prove the main result of this section. We prove that the two-point function of λ M —which we know to be a bi-distribution due to Theorem 4.1—has global Hadamard form. The idea of the proof is the following. Using the very definition of λ M in terms of λ one sees that the two-point function of λ M , restricted to any suitably-defined open neighbourhoods N ⊂ M, is obtained by means of a suitable convolution of the two-point function of λ and a couple of causal propagators in M˜ (this is nothing but a re-arranged version of (30)): ˜ −3 g)(u , ζ, ζ ) ˜ −3 f )(u, ζ, ζ ) E( E( 1 (N ) λ M ( f, g) = − π R2 ×S2 ω B (u, ζ, ζ )ω B (u , ζ , ζ ) (u − u − i0+ )2 ×du ∧ du ∧ S2 (ζ, ζ ), where f, g ∈ C0∞ (M) are supported in N . Next one evaluates the wavefront set of the bi-distribution defined by the right-hand side using Hörmander theorems of composition of wavefront sets (taking advantage also of the Hörmander theorem of propagation of
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singularities). This is the most complicated step of the whole proof and it is explicitly implemented in the last subsection before the section of final comments. The final result ) is stated in Proposition 4.3: The wave front set of λ(N M has Hadamard shape. Finally, varying the set N in M and “collecting together” all the wavefront sets by means of the “local-to-global” Radzikowski’s theorem (our Proposition 4.2), one achieves the global Hadamard form for λ M . Theorem 4.2. Assume that the spacetime (M, g) is an asymptotically flat vacuum spacetime with future time infinity i + (Definition A.1) and that both (M, g) and the unphysical ˜ g) spacetime ( M, ˜ are globally hyperbolic. Consider the quasifree state λ M —on the Weyl algebra W(M) of the massless conformally coupled real scalar field propagating in M—canonically induced by the BMS-invariant state λ on + . Under those hypotheses λ M is Hadamard. Proof. We start by considering properties of the restriction of λ M to sets I − ( p; M) ∩ I + (q; M), with p, q ∈ M. Since the class of all the sets I − (r ; M) ∩ I + (s; M) define a topological base of the strongly causal spacetime (M, g), and since the geodesically convex normal neighbourhoods define an analogous base, I − ( p ; M) ∩ I + (q ; M) must be contained in a geodesically convex normal neighbourhood U provided that p is sufficiently close to q . Taking p, q ∈ I − ( p ; M) ∩ I + (q ; M) with p ∈ I + (q; M) we have J − ( p; M) ∩ J + (q; M) ⊂ I − ( p ; M) ∩ I + (q ; M) ⊂ U . In the following, a set I − ( p; M) ∩ I + (q; M) ⊂ M with p, q ∈ M such that both − I ( p; M)∩ I + (q; M) and J − ( p; M)∩ J + (q; M) are contained in a geodesically convex normal neighbourhood of (M, g) will be called a standard domain. Standard domains form a base of the topology of every strongly causal spacetime. A strongly causal spacetime (M, g) is globally hyperbolic if and only if every set J − ( p; M) ∩ J + (q; M) is compact [Wa84]. In that case I − ( p; M) ∩ I + (q; M) = J − ( p; M) ∩ J + (q; M) holds as well [Wa84], the closure being referred to M. Since compactness is topologically invariant and compact sets are closed, the closure with respect to M coincides with that ˜ Summarizing, in the hypotheses of Theorem 4.2, the standard domains N referred to M. of (M, g) are open, form a base of the topology of M and N ⊂ M where N is compact and the closure can be interpreted indifferently as referred to M or M˜ since they coin˜ N ) are cide. Finally notice that, by construction, both spacetimes (N , g N ) and (N , g globally hyperbolic (the latter because the former is globally hyperbolic and g˜ = 2 g with 2 smooth and strictly positive on N ). In the hypotheses of Theorem 4.2, consider a standard domain N ⊂ M and the restriction of the two point function of λ M to C0∞ (N )×C0∞ (N ). Since we know that λ M is a distribution by Theorem 4.1, this is equivalent to restrict the distribution λ M ∈ D (M × M) to C0∞ (N × N ) producing a distribution of D (N × N ). We have the central result whose proof, given in Appendix B, relies on the known wavefront set of the causal propagator ˜ on several pieces of information on the wavefront set of λ M extracted from (30) and E, on standard results about composition of wavefront sets [Hö89]. Proposition 4.3. In the hypotheses of Theorem 4.2, consider a standard domain N ⊂ M (N ) equipped with the metric g N . If λ M is the restriction of the distribution λ M ∈ D (M × ∞ M) to C0 (N × N ), then ) ∗ W F(λ(N M ) = ((x, k x ), (y, −k y )) ∈ T N \ 0 (45) ×T ∗ N \ 0 | (x, k x ) ∼ (y, k y ), k x 0 .
Quantum Out-States Holographically Induced by Asymptotic Flatness
Proof. The proof is given in the next section.
57
2
We are now in place to take advantage of Radzikowski’s results illustrated in Sect. 4.1. (N ) Since λ M determines a quasifree state for the Klein–Gordon field confined inside the ) globally hyperbolic subspace N , the result established in (69) entails that λ(N M is Hadamard on N in view of Proposition 4.1. Therefore it verifies the local Hadamard condition (LH) in a open neighbourhood G p of every point p ∈ N . Since the sets N form a base (N ) of the topology of M and the restriction of λ M to G p × G p is nothing but the restriction of λ M to the same open neighbourhood, we conclude that λ M satisfies (LH) in G p × G p , for some open neighbourhood G p of every point p of M. By Proposition 4.2, λ M is Hadamard on (M, g). 2 Remark 4.2. It is worth specifying that the analysis of wavefront sets and their composition mentioned about the proof of Proposition 4.3 leads to the inclusion ⊂ only in (45). The other inclusion follows straightforwardly from Hörmander’s theorem about (N ) propagation of singularities [Hör71] using only the fact that λ M satisfies (Com) and (KG). This sort of argument and result can be found in the proof of Theorem 5.8 in [SV01] and in Proposition 6.1 in [SVW02]. 4.4. Proof of Proposition 4.3. Consider a fixed Bondi frame (u, ζ, ζ ) on + ≡ R × S2 and suppose that + is equipped with the measure du∧S2 , S2 being the standard volume form of the unit 2-sphere referred to the coordinates (θ, ϕ) with ζ = eiϕ cot(θ/2). In the following we view the measure du ∧ S2 as that induced by the Riemannian metric given by gS2 ⊕ gR , gS2 being the standard Riemannian metric on the unit 2-sphere represented in coordinates (θ, ϕ) and gR the usual Riemannian metric on R referred to the coordinate u. In this way we can exploit the definition of distribution on manifolds equipped with a nondegenerate metric as working on scalar fields. One may fix a different nonsingular smooth metric or define distributions as operating on scalar densities (see discussion on [Hö89]) and it does not affect the wavefront sets: Different choices change distributions u ∈ D (M) by means of smooth nonvanishing factors a, however from the definition of wavefront set, one has WF(au) ⊂ WF(u) ⊂ WF(au) since both a, 1/a ∈ C ∞ (M). General strategy. Let us pass to the main statement of Proposition 4.3. Fix a Bondi frame ) u, ζ, ζ , and the standard domain N ⊂ M. For f, g ∈ C0∞ (N ), λ(N M ( f, g) can be written via (30) as ˜ −3 g)(u , ζ, ζ ) ˜ −3 f )(u, ζ, ζ ) E( 1 E( (N ) λ M ( f, g) = − π R2 ×S2 ω B (u, ζ, ζ )ω B (u , ζ, ζ ) (u − u − i0+ )2 ×du ∧ du ∧ S2 (ζ, ζ ). (46) In the following we decompose the right-hand side of (46) into four terms (see (57)) and we study the wavefront set of each term separately. Each of those four distributions will ˜ with be viewed as the composition of parts of the two distributions E˜ ∈ D ( M˜ × M) one of the entries restricted to + , and the distribution T , T = F ⊗ D, with F(u, u ) :=
1 and D(ω, ω ) := δ(ω, ω ). (47) (u − u − i0+ )2
As a result we shall achieve the inclusion ⊂ in (45). The other inclusion will be proved by a known propagation of singularity argument.
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Preliminary constructions. We need a preliminary construction in order to extract the ˜ The construction is based on the following lemma. singular part of E. Lemma 4.3. Assume that the spacetime (M, g) is an asymptotically flat vacuum spacetime with future time infinity i + (Definition A.1) and that both (M, g) and the unphysical ˜ g) spacetime ( M, ˜ are globally hyperbolic. Consider a Bondi frame u, ζ, ζ on + . If ˜ g)) N ⊂ M is a standard domain, all the null geodesics (of ( M, ˜ joining points of N and points of + intersect + in a set contained in the compact [u 0 , u f ] × S2 for suitable numbers u 0 , u f ∈ R, where the former can be taken as the value u 0 determined in (b) of Lemma 4.1 for K := N . Proof. The proof is in Appendix B.
2
Consider two Cauchy surfaces: S1 in the past of N such that, referring to the compact set S1 ∩ + , max S1 ∩+ u ≤ u 0 , and S2 in the future of N but in the past of i + and such that, considering the compact set S2 ∩ + , it holds min S1 ∩+ u ≥ u 1 . u 0 and u f are those individuated in Lemma 4.3 for the fixed standard domain N . By construction no maximally extended future-directed null geodesics starting from N can meet S1 and S2 in ˜ (concerning S1 the proof is trivial, concerning S2 we observe that if a futureJ − (i + ; M) ˜ directed maximally extended null geodesics starting from N intersect S2 in J − (i + ; M) it must also meet + in a forbidden point with u > u 1 , since the geodesic cannot remain ˜ ∩ J − (i + ; M) ˜ as proved below (in the subsection confined in the compact D + (S2 ; M) entitled Analysis of the first term in the rhs of Eq. (57)). Let H be the compact subset ˜ bounded by S1 in the past and by S2 in the future and let χ ∈ C ∞ ( M) ˜ of J − (i + ; M) 0 with 1 ≥ χ ≥ 0, and χ = 1 constantly in a neighbourhood of H disjoint with i + and ˜ y) is the Schwartz kernel supp χ ∩ + ⊂ [U0 , U f ] × S2 . Define χ := 1 − χ . If E(x, ˜ ˜ of E, decompose E as ˜ ˜ ˜ y). E(x, y) = χ (x) E(x, y) + χ (x) E(x,
(48)
Restriction of E˜ to + and a bound for the W F set of the restriction. As we said we aim to interpret the right-hand side of (46) in terms of a composition of distributions. To this end we notice that an entry of the two copies of E˜ appearing in (46) is constrained to stay on + . Therefore we are first of all committed to focussing on the feasibility of such restrictions of distributions. ˜ ˜ By construction χ (x) E(x, y) has a nonempty singular support, whereas χ (x) E(x, y) ˜ Therefore χ E˜ can be restricted is a smooth kernel when y ∈ N and x ∈ J + (N ; M). to + × N without problems and it determines a smooth function. Let us consider the ˜ χ E˜ can in fact be restricted to + × N producing distribution of same issue for χ E. + D ( × N ). To show it define a local chart about + given by coordinates , u, ζ, ζ [Wa84]. In these coordinates, exactly for = 0, i.e. on + , the metric of M˜ reads − d ⊗ du − du ⊗ d + dS2 (ζ, ζ ),
(49)
dS2 (ζ, ζ ) being the standard metric on a 2-sphere. Let j : + × N → M˜ × M˜ be ˜ With the used coordinate patch one has the immersion map of + × N in M˜ × M. j : (, u, ζ, ζ , y) → (0, u, ζ, ζ , y) about + and for y ∈ N . Hence the set of normals of the map j in the sense of Theorem 8.2.4 in [Hö89]) is (using notations of Lemma 4.4) N j = {((x, k x ), (y, k y )) ∈ T ∗ M˜ × T ∗ M˜ | (y, k y ) ∈ T ∗ N , x ∈ + , k x = (k x ) d, (k x ) ∈ R}.
Quantum Out-States Holographically Induced by Asymptotic Flatness
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On the other hand [Ra96a]: ˜ 0 × T ∗ M\ ˜ 0 | (x, k x ) ∼ (y, k y )}. (50) ˜ = {((x, k x ), (y, −k y )) ∈ T ∗ M\ W F( E) ˜ = ∅ then E˜ can be restricted to + × N as stated in Theorem If N j ∩ W F( E) ˜ 8.2.4 in [Hö89]. Let us prove that this is the case. Suppose that N j ∩ W F( E) ((x, k x ), (y, −k y )). In this case there would be a null geodesic γ (with respect to ˜ g)) ( M, ˜ intersecting both y ∈ N and x ∈ + with co-tangent vector k x at x of the form k x = (k x ) d = 0. Using the expression of the metric on + (49) one sees that ∂ the tangent vector of γ at x would be proportional to ∂u and thus the remaining part of the geodesic after x would coincide (up to re-parametrisation) with one of integral lines ˜ g) ˜ of n emanating from i + and forming + , these are null geodesics with respect to ( M, as said in Sect. 1.2. In other words y ∈ M would be joined with i + by means of a null ˜ g). geodesic of ( M, ˜ This is impossible as established in the proof of Lemma 4.3 (see ˜ = ∅ and thus E˜ can safely be statement (A) therein). We conclude that N j ∩ W F( E) + restricted to × N . The presence of the smooth factor χ in front of E˜ does not affect the result by the very definition of wavefront set, so that χ E˜ can be restricted to + × N in the same way. The above-mentioned Theorem 8.2.4 in [Hö89] implies also that (since W F(χ E˜ + ×N ) ⊂ W F( E˜ + ×N ) which is known by (50)): W F(χ E˜ + ×N ) ⊂ ((x, k x ), (y, −k y )) ∈ T ∗ +\ 0 × T ∗ N\ 0 (k x , k y ) = t d j(x, y)(h x , h y ), (x, h x ) ∼ (y, h y ) .
(51)
˜ + ×N ). Using the basis Let us give an explicit expression of that bound for W F(χ E| ∗ ∗ + dx , du x , dζx , dζ x of Tx M˜ and du x , dζx , dζ x of Tx , it holds: t
d j(x, y) : ((h x ) d + (h x )u du + (h x )ζ dζ + (h x )ζ dζ , h y ) → ((h x )u du + (h x )ζ dζ + (h x )ζ dζ , h y ).
(52)
The requirement (implicit in (x, h x ) ∼ (y, h y ) in (51)) that h x is null, in view of the form (49) of the metric, means −2(h x )u (h x ) + |h x |2S2 = 0, ∂ ∂ + (h x )ζ ∂ζ with respect to the positive metric on where the |h x |2S2 is the norm of (h x )ζ ∂ζ
S2 . Notice that, if it were (h x )u = 0 (i.e. (h x ) = 0), one would have |h x |2 = 0 so that also the components (h x )ζ , (h x )ζ would vanish and h x would reduce to h x = (h x ) d. However this form for h x is not allowed when h x is, as in the considered case, the co-tangent vector of a null geodesic joining x ∈ + and a point y ∈ M as previously remarked. We conclude that (h x )u = 0 in (51). Summarising, taking (52) into account, we can say that: ˜ + ×N ) ⊂ ((x, k x ), (y, −k y )) ∈ T ∗ +\ 0 W F(χ E| ×T ∗ N\ 0 | (x, k x ) ∼ (y, k y ), (k x )u = 0 , (53)
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where, referring to the basis dx , du x , dζx , dζ x of Tx∗ M˜ and du x , dζx , dζ x of Tx∗ + , the covector k x ∈ Tx∗ M˜ is that uniquely determined by k x ∈ Tx∗ + \ {0} with (k x )u = 0 imposing the condition g( ˜ k x , k x ) = 0. k x is in fact the generic co-tangent vector in x of a null geodesic starting in N ⊂ M and reaching + in x. Remark 4.3. The bound (53) and the requirement (k x )u = 0 forbid the presence in ˜ + ×N ) of elements of the form ((x, 0), (y, k y )), the geodesic joining x ∈ + W F(χ E| and y ∈ N having dx as cotangent vector at x. (N )
Decomposition of λ M . Let us come back to (46) in order to study the wavefront set of the distribution in the right-hand side making use of the wavefront sets of the distributions E|+ ×N and T . It is convenient to introduce the rearranged distributions E ∈ D (+ × N ) and E ∈ D (+ × N ) ∩ C ∞ (+ × N ) individuated via Schwartz kernel theorem by the operators C0∞ (N ) → D (+ ), ˜ + ×N (−3 f )(u, ζ, ζ ), for f ∈ C ∞ (N ), (54) E( f )(u, ζ, ζ ) := ω B (u, ζ, ζ )−1 χ E 0 ˜ + ×N (−3 f )(u, ζ, ζ ), for f ∈ C ∞ (N ). (55) E( f )(u, ζ, ζ ) := ω B (u, ζ, ζ )−1 χ E 0 −3 being smooth, from The wavefront set of E is obviously empty, whereas as ω−1 B , (53), we get again W F(E) ⊂ ((x, k x ), (y, −k y )) ∈ T ∗ +\ 0 ×T ∗ N\ 0 | (x, k x ) ∼ (y, k y ), (k x )u = 0 . (56)
Indicating by ω the angular coordinates ζ, ζ on + , and with dudω the measure on + , du ∧ S2 (ζ, ζ ), by (46) one has the decomposition (for h ∈ C0∞ (N )): E( f )(u, ω)E(g)(u , ω) 1 (N ) λ M ( f, g) = − dudωdu 2 + π R2 ×S2 (u − u − i0 ) E( f )(u, ω)E(g)(u , ω) 1 − dudωdu π R2 ×S2 (u − u − i0+ )2 E( f )(u, ω)E(g)(u , ω) 1 − dudωdu π R2 ×S2 (u − u − i0+ )2 E( f )(u, ω)E(g)(u , ω) 1 − dudωdu . (57) π R2 ×S2 (u − u − i0+ )2 (N )
In the following we compute the various contributions to W F(λ M ) due to each of the terms in the right-hand side of (57). The wavefront set of T . Concerning WF(T ) for the distribution T in (47), we have the following result. Lemma 4.4. Consider the distribution T ∈ D (+ × + ) defined in (47). defined as T = F ⊗ D, with F(u, u ) :=
1 and D(ω, ω ) := δ(ω, ω ), (u − u − i0+ )2
where u ∈ R with covectors k ∈ Tu∗ R, ω ∈ S2 with covectors k ∈ Tω∗ S2 and similar notations are valid for primed variables. With those hypotheses it holds WF(T ) = A ∪ B,
(58)
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A := (u, ω, k, k), (u , ω , k , k ) ∈ T ∗ +\ 0T ∗ +\ 0 u = u , ω = ω , 0 < k = −k , k = −k , B := (u, ω, k, k), (u , ω , k , k ) ∈ T ∗ +\ 0T ∗ +\ 0 ω = ω , k = k = 0, k = −k . Proof. It is is a straightforward consequence of the discussion after Theorem 8.2.14 in [Hö89] and the known wavefront sets of the delta distribution and 1/(k ± i0+ ) (e.g. see [RS75]). 2 Analysis of the first term in the rhs of Eq. (57). Let us focus on the first term in the right-hand side of (57). First of all we notice that it is possible to replace, without affecting the final result, the kernel (u − u − i)−2 with the compactly supported kernel χ˜ (u, u ) (u − u − i)−2 , where χ˜ ∈ C0∞ (R2 ) attains the value constant 1 on the compact [U0 , U f ] × [U0 , U f ] mentioned above defining χ . The first term in the right-hand side of (57) can be re-written as, barring the factor −1/π : χ˜ T , E ⊗ E( f ⊗ g) ,
(59)
where the tensor product of Schwartz kernels E ⊗ E is used as a map C0∞ (N × N ) → D (+ × + ) and T ∈ D (+ × + ) has been introduced in Lemma 4.4. If {Tn } ⊂ C0∞ (+ × + ) is a sequence of functions which tends to χ˜ T in the topology of D (+ × + ), we have trivially # $ χ˜ T, E ⊗ E( f ⊗ g) = lim Tn , E ⊗ E( f ⊗ g) = lim t (E ⊗ E)(Tn ), f ⊗ g , n→+∞
n→+∞
(60) where t (E ⊗ E) : C0∞ (+ × + ) → D (N × N ) indicates the adjoint of E ⊗ E: t (E ⊗ E)(Tn ) (x, x ) := du du dω dω Tn (u, ω, u , ω )E(u, ω, x)E(u , ω , x ). R×S2
Since Tn → χ˜ T , one wonders if it is possible to re-write (60) as: # $ χ˜ T , E ⊗ E( f ⊗ g) = t (E ⊗ E)(χ˜ T ) , f ⊗ g ,
(61)
where t (E ⊗ E)(χ˜ T ) represents the action of t (E ⊗ E) on the compact support distribution χ˜ T . By Theorem 8.2.13 in [Hö89] it is possible and t (E ⊗ E)(χ˜ T ) exists as an element of D (N × N ), provided that (a) χ˜ T has compact support—and this is assured by the introduction of the function χ˜ which in turn may exist due to Lemma 4.3— and (b): % W F(χ˜ T ) = ∅, (62) W F (t (E ⊗ E))+ ×+ where, if K ∈ D (X × Y ), W F (K ) X := {(x, k y ) | ((x, −k x ), (y, 0)) ∈ W F(K ) for some x ∈ X } , and W F (K )Y is defined analogously. To achieve (61) from (60), the sequence of functions Tn → χ˜ T has to tend to χ˜ T in the sense of Hörmander pseudotopology in the domain specified in Theorem 8.2.13 in [Hö89]. Existence of such a sequence is however guaranteed by Theorem 8.2.3 in [Hö89]. (Notice that also if the theorems above concern distributions defined on Rn , we can reduce to this case since N
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is covered by a single normal Riemannian coordinate patch, whereas + is diffeomorphic to R3 \ {0}.) Let us prove that the condition (62) is fulfilled in our case. By Theorem 8.2.9 in [Hö89]: & W F(E ⊗ E) ⊂ (W F(E) ⊗ W F(E)) ((supp E × {0}) × W F(E)) & (W F(E) × (supp E × {0})). (63) Since there are no null geodesics with vanishing tangent vector in y ∈ N joining x ∈ + , we have W F (E ⊗ E)+ ×+ = ∅ and so W F (t (E ⊗ E))+ ×+ = ∅, therefore (62) is fulfilled. We have obtained that the first term in the right-hand side of (57) is nothing but the action of the distribution t (E ⊗ E)(χ˜ T ) ∈ D (N × N ) on f ⊗ g. Now Theorem 8.2.13 in [Hö89] gives the inclusion: W F(t (E ⊗ E)(χ˜ T )) ⊂ W F(t (E ⊗ E)) N ×N & ((x, k x ), (x , k x )) | ((x, k x ), (x , k x ), (u, ω, −ku , −k), (u , ω , −ku , −k )) ∈ W F(t (E ⊗ E)),
for some (u, ω, ku , k), (u , ω , ku , k ) ∈ W F(χ˜ T ) .
(64)
Similarly to W F (t (E ⊗ E))+ ×+ one finds (with the same argument) W F (t (E ⊗ E)) N ×N = ∅. Whereas the remaining part in the right hand side of (64), taking into account the inclusions (63), the inclusion W F(χ˜ T ) ⊂ W F(T ) and (56), exploiting (58), produces straightforwardly the result: W F(t (E ⊗ E)(χ˜ T )) is contained in the set G of pairs ((x, k x ), (x , −k x )) ∈ T ∗ N \ 0 × T ∗ N \ 0 such that: (a) (x, k x ) and (x , k x ) are points and associated cotangent vectors of the same maximal null geodesic γ intersecting + in some point p, and (b) k x is future directed. (Since the coordinate u is future directed and (49) holds, k x is future directed if and only if, considering the geodesic γ with initial conditions (x, k x ) ∈ Tx∗ N , the opposite of the covector tangent to γ , in the point p where γ meets + , has component (k p )u positive. This agrees with the condition k > 0 in the definition of the wavefront set of the distribution T (58) concerning the subset A, the subset B gives no contribution to G.) We can improve the obtained bound for W F(t (E ⊗ E)(χ˜ T )) as follows. Notice that, if (x, k x ), (x, −k y ) ∈ G, by construction it holds: (x, k x ) ∼ (x, k y ) with k x 0. Conversely, consider a pair (x, k x ) ∼ (y, k y ) with k x 0. Let us prove that ((x, k x ), (y, −k y )) ∈ G. The maximal null geodesic passing through x and y with respective cotangent vectors k x and k y must achieve + in some point, since it cannot ˜ ∩ D + (S; M), ˜ where S is a spacelike remain confined in the compact D := J − (i + ; M) Cauchy surface of M˜ which intersects x or y and lies in the past of the other point. (Indeed, if the maximally extended geodesic γ : (a, b) → M˜ were confined in D, there would be c ∈ D with γ (tk ) → c for some sequence of affine parameter points (a, b) tk → b. In normal coordinates centred on c, the geodesics would assume standard form t → t v µ for constants v µ ∈ R and t ∈ (a , b ) 0 being another parameter related to t by means of a nonsingular affine transformation. This would imply, on one hand, that b < +∞, and on the other hand that γ admits an extension beyond c, and
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˜ ∩ D + (S; M)) ˜ in this is not possible by hypotheses.) So γ gets intersecting ∂(J − (i + ; M) ˜ some point p. Since it cannot intersect S twice, the geodesic has to meet ∂ J − (i + ; M) somewhere. The point i + is forbidden as established in the proof of Lemma 4.3. We conclude that the geodesics must intercept some point of + . We have found that if, for x, y ∈ N , (x, k x ) ∼ (y, k y ) with k x 0, then it also holds (x, k x ), (y, −k y ) ∈ G. We have finally obtained that the contribution W F(t (E ⊗ E)(χ˜ T )) to the wavefront set of (N ) λ M due to the first term in the rhs of (57) fulfills the following bound: W F(t (E ⊗ E)(χ˜ T )) ⊂ ((x, k x ), (y, −k y )) ∈ T ∗ N \ 0 (65) ×T ∗ N \ 0 | (x, k x ) ∼ (y, k y ), k x 0 . Analysis of the last term in the rhs of Eq. (57). To go on, we remind the reader that E ∈ C ∞ (+ × N ) by construction. Furthermore E(u, ω, x) = 0 smoothly for u < u 0 . ˜ y) has smooth kernel when y ∈ N and Moreover by (55) recalling that χ (x, y) E(x, x ∈ J + (N ) so that it is smooth for x varying in a neighbourhood of i + when y ∈ N , we can control the behaviour as u → +∞ of ∂xα E(u, ω, x) and ∂xα ∂u E(u, ω, x) by Lemma 4.2 for every multi-index α and for any fixed β ∈ [1, 2): ∂xα E(u, ω, x) and ∂xα ∂u E(u, ω, x) are bounded, respectively, by functions of the form Mα (x)/|u − b| and Mαβ (x)/|u − b|β . The bounds Mα (x), Mαβ (x) can be made locally uniform in x taking the sup in (40) over B × B , B being a relatively compact neighbourhood of every fixed point x0 ∈ N . By integration by parts, the last term in the right-hand side of (57) can be re-written (omitting a constant overall factor): ∂u E(u, ω, x)E(u , ω, x ) f (x)g(x ). (66) du du dω dµg˜ (x) dµg˜ (x ) lim+ →0 R2 ×S2 ×N ×N u − u − i The functional in (66) can be rearranged using Fubini-Tonelli and Lebesgue’s dominate convergence and computing the limit under the symbol of dµg˜ (x) dµg˜ (x ) integration, obtaining that the last term in the right-hand side of (57) is, in fact, up to an overall factor: K (x, x ) f (x)g(x ) dµg˜ (x) dµg˜ (x ), N ×N
where the smooth kernel K (x, x ) reads: iπ du dω E(u, ω, x)∂u E(u, ω, x ) R×S2 E(u , ω, x ) − E(u, ω, x ) du du dω ρ(u − u ) ∂u E(u, ω, x) − u − u R×R×S2 E(u , ω, x ) du du dω ρ (u − u ) ∂u E(u, ω, x), − u − u R×R×S2 ρ := 1 − ρ and ρ ∈ C0∞ (R) being any, arbitrarily fixed, function which attains the value 1 constantly in a neighbourhood of 0. Absolute convergence of the integrals and smoothness of K (x, x ) can be checked by direct inspection taking derivatives under the symbol of integration by standard theorems based on dominate convergence theorem together with the uniform bounds on the behaviour as u → +∞ mentioned above.
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We conclude that the last term in right-hand side of (57) gives no contribution to (N ) the wavefront set of the two-point function of λ M since it is associated with a smooth kernel (K (x, x )). Analysis of the second and third term in the rhs of Eq. (57). Let us examine the third term in the right-hand side of (57); the second can be analysed with the same procedure obtaining the same result. As before this term can be re-arranged and the limit can be explicitly computed obtaining that the third term in the right-hand side of (57) equals (up to a constant overall factor) ∂u E( f )(u, ω) E(u , ω, x ) lim+ g(x ) du du dω dµg˜ (x ) →0 R2 ×S2 ×N u − u − i = dµg˜ (x ) H ( f, x )g(x ), N
with, for every fixed f ∈ C0∞ (N ), the smooth function H ( f, x ) given by: iπ du dω E(u, ω, x )∂u E( f )(u, ω) R×S2 E(u , ω, x ) − E(u, ω, x ) du du dω ρ(u − u ) ∂u E( f )(u, ω) − u − u R×R×S2 E(u , ω, x ) du du dω ρ (u − u ) ∂u E( f )(u, ω). − u − u R×R×S2 As before, the function ρ ∈ C0∞ (R) is any function with ρ = 1 in a neighbourhood of 0 and ρ := 1 − ρ. Each of the three integrals in the expression of H have the form, with a corresponding S ∈ C ∞ (+ × N ), F( f )(x ) := du dω S(u, ω, x )∂u E( f )(u, ω). R×S2 ×N
At least formally, one may think of F : C0∞ (N ) → D (N ) as individuated by the Schwartz kernel F(x, x ) composition of Schwartz kernels: F(x, x ) = du dω S(u, ω, x )∂u E(u, ω, x) i.e. t F = t S ◦ ∂u E. (67) R×S2
Similarly to the case of the first term in the right-hand side of (57), this interpretation makes rigorous sense in view of Theorem 8.2.14 of [Hö89] provided (a) the projection supp (∂u E) (u, ω, x) → x ∈ N is proper—and this can be straightforwardly verified true by the properties of the support of E—and (b) W F (t S)+ ∩ W F(∂u E)+ = ∅—and this is also true because W F (t S)+ is empty since t S is smooth, whereas W F(∂u E)+ ⊂ W F(E)+ which is empty as can be found by direct inspection using (56) (there are no null geodesics from N to + with zero tangent vector). The inclusion given in Theorem 8.2.14 in [Hö89] states that W F(t F) is a subset of the union of the following sets: (1) W F (t S) ◦ W F (∂u E), which is empty because W F (t S) is empty, (2) W F(t S) N × N × {0}, which is empty due to the same reason, and (3) N × {0} × W F (∂u E) N , which is empty because W F (∂u E) N ⊂ W F (E) N , and referring to (56), there are no null geodesics from N to + with zero tangent vector. Hence W F(t F) = ∅ and thus W F(F) = ∅.
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We conclude that the second and the third term in right-hand side of (57) give no (N ) contribution to the wavefront set of the two-point function of λ M . Collecting all the obtained results together, we realise that the only contribution to (N ) W F(λ M ) comes from the first term in right-hand side of (57) and thus, due to (65): ) ∗ ∗ W F(λ(N M ) ⊂ ((x, k x ), (y, −k y )) ∈ T N \ 0×T N \ 0 | (x, k x ) ∼ (y, k y ), k x 0 . (68) ∼ and in the right-hand side of (68) refer to the metric g˜ N . However rescaling by means of a smooth factor −2 > 0 does not modify the right-hand side of (68). One proves its immediately using the transformation rule of (co)tangent vectors of null geodesics under local rescaling of the metric (see Appendix D in [Wa84]). Therefore (68) holds true also employing ∼ and associated with the metric g N . Propagation of singularity argument. In view of (68), to conclude the proof of Proposition 4.3 it is now sufficient to establish the other inclusion: (N ) W F(λ M ) ⊃ ((x, k x ), (y, −k y )) ∈ T ∗ N \ 0×T ∗ N \ 0 | (x, k x ) ∼ (y, k y ), k x 0 . (69) Let us do it using an argument based on the theorem of singularity propagation. Let E N ∈ D (N × N ) be the causal propagator associated with the Klein–Gordon equation in the globally hyperbolic spacetime (N , g N ) and, in the following we denote by sing supp(S) the singular support of a distribution S. In this proof p ∼ q means that there is, in the considered spacetime, at least one null geodesic joining p and q. We prove (69) by means of a reductio ad absurdum. Our per absurdum claim is that ) there are p, q ∈ N with p ∼ q but (( p, k p ), (q, −kq )) ∈ W F(λ(N M ), where k p and kq are the cotangent vectors to a null geodesic joining p and q with k p 0. Actually that geodesic is uniquely determined —for both N and M—by p and q, from the very definition of standard domain N . (Notice also that the wavefront set is conic and thus the vectors k p , kq are determined up to a common, strictly positive, factor completely irrelevant in our discussion.) Since the singular support of a distribution of D (N × N ) is the projection on N × N of the wavefront set of the distribution, we must conclude that, (N ) in view of (68), ( p, q) ∈ sing supp(λ M ). However, as p ∼ q, ( p, q) must belong to sing supp(E N ) (this is because E N is the difference of the advanced and the retarded fundamental solutions whose known wavefronts and causal properties of supports [Ra96a] entails that sing supp(E N ) is made exactly by the pairs ( p, q) ∈ N × N with p ∼ q). ) Since (Com) holds true, we conclude that (q, p) ∈ sing supp(λ(N M ), and thus there are (N ) k p ∈ T p∗ N and kq ∈ Tq∗ N such that ((q, kq ), ( p, −k p )) ∈ W F(λ M ), and so, via Proposition 4.3, k p and kq are vectors cotangent to the null geodesic joining p and q (the same as before since it is unique) and, finally, kq 0. (N ) The distribution λ M satisfies the Klein–Gordon equation in both arguments (in other words (KG) holds), therefore the propagation of singularities theorem [Hör71] (see dis(N ) cussion after Theorem 4.6 in [Ra96a]) implies that the wavefront set of λ M is the union of sets of the form B(x, k x ) × B(y, k y ). Here B(z, h z ) is the unique null maximal geodesic (viewed as a curve in T ∗ N ) passing through z ∈ N with co-tangent vector h z ∈ Tz∗ N . (N ) As ((q, kq ), ( p, −k p )) ∈ W F(λ M ) and since q and p belong to the same null geo(N ) desic, we are committed to conclude that (( p, k p ), (q, −kq )) ∈ W F(λ M ), where the
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cotangent vector k p is cotangent to the geodesic at p and it has the same time orientation as kq , so that k p 0, and the vector kq is cotangent to the geodesic at q. In other words, ) changing the used names for cotangent vectors: (( p, k p ), (q, −kq )) ∈ W F(λ(N M ), where k p 0. This is in contradiction with our initial claim. 2 We have finally established that, on (N , g N ) ) ∗ ∗ W F(λ(N M ) = ((x, k x ), (y, −k y )) ∈ T N \ 0 × T N \ 0 | (x, k x ) ∼ (y, k y ), k x 0 .
2 5. Final Comments: Summary and Open Issues Let us summarise the main results achieved in this work. We started from the unique, positive B M S-energy, B M S-invariant, quasifree, pure state λ acting on a natural Weyl algebra defined on + . That state is completely defined using the universal structure of the class of (vacuum) asymptotic flat spacetimes at null infinity, no reference to any particular spacetime is necessary. In this sense λ is universal. It is the vacuum state for a representation of the BMS group with vanishing BMS mass. Afterwards, we have seen that λ induces in any fixed (globally hyperbolic) bulk spacetime M, a preferred state λ M for a conformally coupled massless real scalar field. This happens if M admits future time infinity i + (and the unphysical spacetime M˜ is globally hyperbolic as well). The induction of a state takes place by means of an injective isometric ∗ homomorphism ı : W(M) → W(+ ) which identifies Weyl observables of the field in the bulk with some Weyl observables of the boundary + . λ M (a) := λ(ı(a)) for all a ∈ W(M). Using a very inflated term, we may say that this is a holographic correspondence. The picked state λ M enjoys quite natural, as well as interesting, properties. These properties (barring the first one) have been established in this paper: (i) λ M coincides with Minkowski vacuum when M is Minkowski spacetime, (ii) λ M is invariant under every isometry of M (if any); (iii) λ M fulfills the requirement of energy positivity with respect to every timelike Killing field in M and, in the one-particle space, there are no zero modes for the self-adjoint generator of Killing-time displacements, (iv) λ M is Hadamard and therefore the state may be used as background for perturbative procedures (renormalisation in particular). The statement (ii) holds as it stands replacing λ M with any other state λM uniquely defined by assuming that λM (a) := λ (ı(a)) for all a ∈ W(M), provided that λ is a BMS-invariant state (not necessarily quasifree or pure or satisfying some positivityenergy condition) defined on W(+ ). The state λ M may have the natural interpretation of outgoing scattering vacuum, but also it provides a natural and preferred notion of massless particle in the absence of Poincaré symmetry. Indeed, all the construction works for massless conformally coupled scalar fields propagating in M. Notice that the two notions of mass arising in our picture, that in the bulk based on properties of the Klein–Gordon operator (and on Wigner analysis if M is Minkowski spacetime) and that referred to the extent on + relying upon Mackey-McCarthy analysis of BMS group unitary representations, are in perfect
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agreement: both vanish. We do not see any obstruction to generalise all the results for other massless conformally invariant field equations. However a natural question deserving future investigation is now: what about massive fields? How to connect, if possible, the massive particle defined in M to fields on + associated with known unitary BMS representations with positive BMS mass? For an interesting attempt in that direction, specialized to Minkowski space, see Dappiaggi’s recent paper [Da07]. Another technically interesting issue concerns the purity of the state λ M : λ is pure by definition, but purity of λ M is not evident in the general case. Finally, it would be nice to describe interactions in the bulk, at least at a perturbative level, by means of a theory on + . Acknowledgements. I would like to thank to R. M. Wald for technical suggestions (see the footnote in the proof of Lemma 4.3). I am grateful to A. Ashtekar for comments and suggestions after the appearance of [Mo06]. I am grateful to C. Dappiaggi, R. Brunetti and N. Pinamonti for useful comments. I would like to thank S. Hollands for having provided me with a copy of his unpublished Ph.D. thesis.
A. Asymptotically Flat Spacetime with Future Time Infinity Definition A.1. A time-oriented four-dimensional smooth spacetime (M, g) is called an asymptotically flat vacuum spacetime with future time infinity i + , if there is a smooth ˜ g) spacetime ( M, ˜ with a preferred point i + , a diffeomorphism ψ : M → ψ(M) ⊂ M˜ and a map : ψ(M) → (0, +∞) so that g˜ = 2 ψ ∗ g and the following facts hold. (We omit to write explicitly ψ and ψ ∗ in the following.) ˜ is closed and M = J − (i + ) \ ∂ J − (i + ; M). ˜ Moreover ∂ M = + ∪ {i + }, (1) J − (i + ; M) + − + + ˜ ˜ where := ∂ J (i ; M) \ {i } is the future null infinity. (Thus M = I − (i + ; M), + + i is in the future of and time-like related with all the points of M and ∩ ˜ = ∅.) J − (M; M) (2) M is strongly causal, and satisfies vacuum Einstein solutions in a neighbourhood of + at least. ˜ (3) can be extended to a smooth function on M. + , and d(i + ) = 0, but ∇ ˜ µ ∇˜ ν (i + ) = (4) ∂ J− (i + ; M) = 0, but d(x)
= 0 for x ∈ ˜ −2g˜ µν (i + ). (5) If n µ := g˜ µν ∇˜ ν , for a strictly positive smooth function ω, defined in a neighbourhood of + and satisfying ∇˜ µ (ω4 n µ ) = 0 on + , the integral curves of ω−1 n are complete on + . Remark A.1. Notice that ω in (5) can be fixed to be the factor ω B mentioned in Sect. 1.2. The original definition due to Friedrich actually concerned the existence of the past time infinity i − , our definition is the trivial adaptation to the case of the existence of i + . B. Proofs of Some Technical Propositions Proof of (c) in Proposition 3.1. Consider a one-parameter subgroup of G B M S , {gt }t∈R ⊂ . Suppose that {gt }t∈R arises from the integral curves of a complete smooth vector ξ˜ tangent to + . In every Bondi frame (u, ζ, ζ ) one finds: gt : R × S2 (u, ζ, ζ ) → u + f t (ζ, ζ ), ζ, ζ , where, due to smoothness of ξ˜ and because of standard theorems of ordinary differential equations, the function (t, u, ζ, ζ ) → u + f t (ζ, ζ ) is jointly smooth. In particular f is jointly smooth and thus continuous in the parameter t, satisfies f t ∈ C ∞ (S2 ) ≡ and verifies f t (ζ, ζ ) + f t (ζ, ζ ) = f t+t (ζ, ζ ) for
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all t, t ∈ R and (ζ, ζ ) ∈ S2 . The relation above entails f p t (ζ, ζ ) = q
p q f t (ζ, ζ )
for
S2 . Using continuity in t one finally gets: t, t ∈ R, a, b ∈ R and (ζ, ζ ) ∈ S2 . There-
all t ∈ R, p, q ∈ Z, q = 0 and (ζ, ζ ) ∈ a f t (ζ, ζ ) + b f t (ζ, ζ ) = f at+bt (ζ, ζ ) for all fore it holds: f t (ζ, ζ ) = t f 1 (ζ, ζ ) . We conclude that if the one-parameter sub-group {gt }t∈R ⊂ ⊂ G B M S arises from the complete integral curves of a smooth vector ξ˜ tangent to + , in any fixed Bondi frame: gt : R × S2 (u, ζ, ζ ) → u + t f 1 (ζ, ζ ), ζ, ζ , where the function f 1 ∈ C ∞ (S2 ) ≡ individuates completely the subgroup.
2
Proof of Proposition 3.3. (a) is an immediate consequence of (a) in Proposition 3.1 and the definition of asymptotic symmetry. (b) Since the extension of ξ to + , ξ˜ , has to be tangent to + , referring to a fixed Bondi frame, it must hold ξ˜ = α∂/∂u + β∂/∂ζ + β∂/∂ζ . Since the angular part of the degenerate metric on + is positive, whereas that on the space spanned by ∂/∂u (which is orthogonal to the angular part) vanishes, one has g( ˜ ξ˜ , ξ˜ ) ≥ 0—with g( ˜ ξ˜ , ξ˜ ) = 0 if and only if β = β = 0. On the other hand we know that g(ξ, ξ ) ≤ 0 in M by hypotheses and thus g(ξ, ˜ ξ ) ≤ 0 as well. Hence approaching + ˜ ˜ it must be g( ˜ ξ , ξ ) = 0 by continuity. We have found that: ξ˜ (u, ζ, ζ ) = α(u, ζ, ζ )∂/∂u. The (generally local) one-parameter group of transformations gt obtained by integration of ξ˜ acts only on the variable u: u → u t and so it has to hold du t (u, ζ, ζ ) |t=0 = α(u, ζ, ζ ). dt
(70)
On the other hand this one-parameter group must coincide with a suitable one-parameter subgroup of BMS group because ξ˜ is a one-parameter generator of such an action by (a) in Proposition 3.1. By comparison with the action (2)-(3) of BMS group on coordinates (u, ζ, ζ ), noticing that the subgroup leaves fixed the angular coordinates, the only possible action is u t = u + f (t, ζ, ζ ) for some smooth class of functions { f (t, ·, ·)}t∈R ⊂ C ∞ (S2 ). Therefore du t (u, ζ, ζ ) ∂ f (t, ζ, ζ ) |t=0 = |t=0 . dt ∂t Comparing with (70) we conclude that α cannot depend on u. (b) in Proposition 3.1 also entails that α cannot vanish identically on + . In other words, ξ˜ is a generator of a nontrivial subgroup of . Next, by (b) in Proposition 3.1 we conclude that ξ˜ is a generator of a nontrivial subgroup of T 4 . That is equivalent to say that α ∈ T 4 \ {0}. To conclude, as a consequence of (c) of Proposition 3.2, it is sufficient to prove that α cannot assume ˜ g), both signs. Since ξ is future directed with respect to (M, g) and ( M, ˜ the limit values + of ξ toward , α∂/∂u must either vanish or be future directed. Since ∂/∂u is future ˜ g) directed with respect to ( M, ˜ too, the factor given by the smooth function α cannot be negative anywhere. 2
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Proof of Proposition 3.4. In this proof B := ω B . Under our hypotheses on ξ and ξ˜ , consider a smooth vector field v defined on M˜ which reduces to ξ in M and reduces to ξ˜ on + . By construction the (jointly smooth in both arguments) one-parameter group of diffeomorphisms generated by v, g (v) , reduces to those generated by the relevant ˜ restrictions of v: g (ξ ) and g (ξ ) . The orbits of v in M ∪ + are complete by construction. Indeed, if an orbit starts in M it remains in M and it is complete by hypotheses, if it starts on + it must remain in + and must be complete anyway, since ξ˜ generates a (complete) one-parameter subgroup of G B M S . This fact entails, in turn, that the oneparameter group of diffeomorphisms generated by v in M ∪ + is global and thus its pull-back action on functions defined over M ∪ + is well defined. If y ∈ + and x ∈ M one has, by continuity of the flux of v: lim g (ξ ) (x) x→y τ
˜
= lim gτ(v) (x) = gτ(v) (y) = gτ(ξ ) (y). x→y
In the proof of Proposition 2.7 in [DMP06] (within a more generalised context) we have (ξ˜ )
found that, referring to a Bondi-frame where gτ
lim
(ξ ) B g−t (x)
x→(u,ζ,ζ )
B (x)
=K
(ξ˜ )
(ξ˜ )
= τ , f τ
(ξ˜ )
−t
(ξ˜ )
g−t (u, ζ, ζ )
and y ≡ (u, ζ, ζ ),
−1
.
(ξ )
Therefore one has trivially that (φ ◦ g−t )(y) coincides with lim
x→y
(ξ ) φ g−t (x) B (x)
(ξ ) (ξ ) φ g−t (x) B g−t (x) lim = lim (ξ ) x→y B (x) B g−t (x) x→y ˜ −1 ˜ (ξ ) (ξ ) = K (ξ˜ ) g−t (u, ζ, ζ ) ψ g−t (y) .
−t
(ξ )
Comparing with (7), we finally find that: (φ ◦ g−t ) = A proof.
2
(ξ˜ )
gt
(ψ) and this concludes the
Proof of Lemma 4.1. (a) Using the definition of M (see Proposition 2.1) and the fact that E maps compactly-supported smooth functions to smooth solutions of Klein–Gordon equation with compactly-supported Cauchy data, it arises: −1 −1 lim+ E(h) + (u, ζ, ζ ). (71) ψh (u, ζ, ζ ) = ω B (u, ζ, ζ ) →
˜ g) On the other hand since also ( M, ˜ is globally hyperbolic, the causal propagator E˜ for ˜ g) the massless conformally coupled Klein–Gordon operator P˜ in ( M, ˜ is well defined. ˜ Using the following facts: (1) that E and E are the difference of the advanced and ˜ g˜ = 2 g), retarded fundamental solutions in the corresponding spaces (M, g) and ( M, (2) that the following identity holds: ˜ −1 φ) = −3 Pφ P(
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and (3) that the causality relations are preserved under (positive) rescaling of the metric, one achieves the following identity valid on M: ˜ −3 h), if h ∈ C ∞ (M). −1 E(h) = E( 0 The right-hand side is anyhow smoothly defined also in the larger manifold M˜ and on + in particular. Therefore, exploiting Eq. (71), the expression of ψh (u, ζ, ζ ) found above can be re-written into a more suitable form given by (31). The singularity of −3 on + is harmless because the support of h does not intersect + by construction and > 0 in M. Notice that supp (−3 h) = supp h if h ∈ C0∞ (M). (b) To prove the thesis take h ∈ C0∞ (M) and a compact K ⊂ M with supp h ⊂ K . ˜ ˜ −3 h) (equivalently E(h) since We use Definition A.1 from now on. By definition E( −3 + − ˜ ˜ supp ( h) = supp h) is supported in J (supp h; M) ∪ J (supp h; M) and thus ˜ ∪ J − (K ; M). ˜ However J − (K ; M) ˜ has no intersection with + since in J + (K ; M) ˜ and + ⊂ ∂ J − (i + ; M), ˜ we conclude that the support of the soluK ⊂ I − (i + ; M) ˜ and thus in ˜ −3 h) intersects + in a set completely included in J + (K ; M) tion E( + + ˜ ∩ J (K ; M). As a consequence ˜ ˜ −3 h)+ is included in + ∩ J + (K ; M). the support of ψh = ω B (u, ζ, ζ )−1 E( (72) ˜ g) Now consider a spacelike Cauchy surface S of ( M, ˜ with K completely contained in the chronological future of S (such a Cauchy surface does exist due to global hyperbo˜ g) licity of ( M, ˜ and because K is compact, it is sufficient to use any Cauchy foliation of R × S ≡ M˜ taking the value of the smooth global time function t ∈ R far enough ˜ is compact in the past). Notice that the set C := S ∩ (+ ∪ {i + }) = S ∩ ∂ J − (i + ; M) − + + ˜ ˜ ˜ g) ˜ because it is a closed subset of J (i ; M) ∩ J (S; M) which is compact since ( M, + + + ˜ is globally hyperbolic (e.g. see [Wa84]). C cannot contain i because i ∈ I (K ; M), ˜ and S is achronal. Let u 0 = minC u, which is finite because the coorK ⊂ I + (S; M) + dinate u : → R is smooth and C ⊂ + is compact. By construction it arises that ˜ ⊂ I + (S; M) ˜ ⊂ J + (S; M) ˜ and so + ∩ J + (K ; M) ˜ ⊂ + ∩ J + (S; M) ˜ ⊂ J + (K ; M) + ˜ J (C; M). Since u increases toward the future, we have ˜ ⊂ [u 0 , +∞). u(+ ∩ J + (K ; M)) Therefore, by (72), we have that ψh vanishes for u < u 0 due to (73).
(73) 2
Proof of Lemma 4.2. The proof is essentially that given for Lemma 4.4 in [Mo06]. There ˜ was specialised to the case = M (φ) for some the smooth function ∈ C ∞ ( M) φ ∈ S(M), however such a restriction can be removed without affecting the proof as is evident from the proof of the cited lemma. The improvement concerning the exponent β is obtained by noticing that in the last estimation before Eq. (44) in [Mo06], e−λ(4+) can be replaced by the improved bound e−βλ(4+) for every β ∈ [1, 2) provided the free parameter > 0 fulfills < 4(2 − β)/(β + 4). 2 Proof of Lemma 4.3. We use here the geometric structure defined in Definition A.1. K := N ⊂ M (the closure being referred to M) is compact. Using the same procedure ˜ ∩ + is contained in a as in the proof of (b) in Lemma 4.1, we obtain that J + (K ; M) set of the form [u 0 , +∞) × S2 . Thus the null geodesics of M˜ joining N and + must intersect + in a set contained in [u 0 , +∞) × S2 . Let us prove that [u 0 , +∞) can actually
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be restricted to a compact [u 0 , u f ]. First of all we notice that the following statement holds: ˜ g)) (A) If p ∈ M, there is no null geodesic (with respect to ( M, ˜ joining p and i + . Indeed, suppose that there is such a geodesic γ for some p ∈ M. As is known from the general theory of causal sets in globally hyperbolic spacetimes and the structure of the boundary ˜ \ {i + } = + of J ± (x) (e.g [Wa84]), after starting from i + , γ must belong to ∂ J − (i + ; M) + − + ˜ till it encounters its cut locus c ∈ where ∂ J (i ; M) terminates along the direction of γ . We conclude, in particular, that c is the endpoint on + of one of the null geode˜ After c, γ leaves ∂ J − (i + ; M), ˜ enters M and reaches p. In sics forming ∂ J − (i + ; M). the portion of its trip which lies on + , with a corresponding subset of the domain for its affine parameter t ∈ (0, b], one has (γ (t)) = 0 for definition of + . Therefore γ˙ µ (t)∇µ (γ (t)) = γ˙ µ (t)n µ (γ (t)) = 0. Finally, since γ˙ is null as n (and both do not vanish anywhere), it has to be γ˙ (t) = f (t)n(γ (t)) for some non-vanishing smooth function f . In other words, the portion of γ contained in + is, up to a re-parameterisation, an integral line of n. Therefore c is the (past) endpoint on + of one of the integral lines of n forming + . This is in contradiction with the requirement (5) in Definition A.1 which implies that the integral lines of n cannot have endpoints on + . We proceed to conclude the proof of existence of u f . Suppose per absurdum that, for the compact set K := N ⊂ M, u f does not exist, so that the null geodesics starting from K can intersect + arbitrarily close to i + . In this case we can consider a sequence {γn } of null geodesics through K which intersect + in the corresponding points { pn } and pn → i + as n → +∞. However the following statement holds: (B) If the mentioned sequence of geodesics {γn } exists, there is a null geodesic γ from K ⊂ M to i + . Statement (B) is in contradiction with statement (A), hence there is no sequence {γn } with the claimed properties and thus u f must exist. To demonstrate statement (B)8 consider the sequence {γn } where the geodesics are ˜ g) ˜ spacelike Cauchy surface C extended maximally after i + and before K . Choose a ( M, through i + , and normalise the null-geodesic tangents so that they have unit inner product with the normal to C. Let xn denote the intersection point of the null geodesic with C and and let kn denote the normalized tangent at xn . Then {(xn , kn )} is a sequence in a compact subset of the tangent bundle, so there is a subsequence that converges to a point (x, k). Clearly x = i + . Let γ be the maximally extended null geodesic individuated by ( p, k) and we assume that all the used geodesics start from C with the value of the affine parameter s0 = 0. Moreover, since M˜ is globally hyperbolic, rescaling the metric g˜ with a strictly positive smooth factor, we can make complete every null geodesic ˜ In this way we (Theorem 6.5 in [BGP96]), without affecting the causal structure of M. ignore problems of domains of the parameters of the geodesics. Let C be a second Cauchy surface in the past of K . Since γ is causal, one has γ (s1 ) ∈ C for some s > 0. Consider an auxiliary Riemannian smooth metric defined on M˜ and denote by d the distance associated with that metric—whose metric balls, as is known, form a base of the pre-existent topology of M˜ –. Using the jointly continuous dependence of maximal ˜ from the parameter describing solutions of differential equations (in this case on T M) the curves and the initial data, and exploiting the fact that continuous functions defined on a compact set are uniformly continuous, we get easily the following statement: For every > 0, there is a natural N such that d(γ (s), γn (s)) < for all s ∈ [0, s1 ] if n > N . It is clear that, in this way, if γ does not intersect K , one can fix in order that 8 The kind of argument to prove statement (B) was suggested to the author by R. M. Wald.
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no γn meets K if n > N . This is in contradiction with the hypotheses on the curves γn . 2 C. Fourier–Plancherel Transform on R × S2 Define S (+ ) as the complex linear space of the smooth functions ψ : + → C such that, in a fixed Bondi frame, ψ with all derivatives vanish as |u| → +∞, uniformly in ζ, ζ , faster than |u|−k , ∀k ∈ N. The space S (+ ) generalises straightforwardly Schwartz’ function space on Rn , S (Rn ). S (+ ) can be equipped with the Hausdorff topology induced from the countable class of seminorms—they depend on the Bondi frame but the topology does not— p, q, m, n ∈ N, q ||ψ|| p,q,m,n := sup |u| p ∂u ∂ζm ∂ζn ψ(u, ζ, ζ ) . (u,ζ,ζ )∈+
S (+ ) is dense in both L 1 (R × S2 , du ∧ S2 (ζ, ζ )) and L 2 (R × S2 , du ∧ S2 (ζ, ζ )) (with the topologies of these spaces which are weaker than that of S (+ )), because it includes the dense space C0∞ (R × S2 ; C) of smooth compactly-supported complex-valued functions. We also define the space of distributions S (+ ) containing all the linear functionals from R × S2 to C which are weakly continuous with respect to the topology of S (+ ). Obviously S (+ ) ⊂ S (+ ) and L p (R × S2 , du ∧ S2 (ζ, ζ )) ⊂ S (+ ) for p = 1, 2. We introduce the Fourier transforms F± [ f ] of f ∈ S (+ ) ±iku e F± [ f ](k, ζ, ζ ) := f (u, ζ, ζ )du, (k, ζ, ζ ) ∈ R × S2 . √ 2π R F± enjoy the properties listed below which are straightforward extensions of the analogs for standard Fourier transform in Rn . The proof of the following theorem is in Appendix B.9 Theorem C.1. The maps F± satisfy the following properties: (a) for all p, m, n ∈ N and every ψ ∈ S (+ ) it holds ' ( p F± ∂u ∂ζm ∂ζn ψ (k, ζ, ζ ) = (±i) p k p ∂ζm ∂ζn ψF± [ψ](k, ζ, ζ ). (b) F± are continuous bijections onto S (+ ) and F− = (F+ )−1 . (c) If ψ, φ ∈ S (+ ) one has F± [ψ](k, ζ, ζ )F± [φ](k, ζ, ζ )dk R = ψ(u, ζ, ζ )φ(u, ζ, ζ )du, for all (ζ, ζ ) ∈ S2 , R F± [ψ](k, ζ, ζ )F± [φ](k, ζ, ζ )dk ∧ S2 (ζ, ζ ) R×S2 ψ(u, ζ, ζ )φ(u, ζ, ζ )du ∧ S2 (ζ, ζ ). = R×S2
(74)
(75)
9 The statement of (6) in Theorem C1 in Appendix C of [Mo06] is erroneous, but this fact by no means affects the results achieved in [Mo06] since that statement did not enter the paper anywhere.
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(d) If T ∈ S (+ ) the definition F± T [ f ] := T (F± [ f ]) for all f ∈ S (+ ), is wellposed, gives rise to the unique weakly continuous linear extension of F± to S (+ ) and one has, with the usual definition of derivative of a distribution, ' ( p F± ∂u ∂ζm ∂ζn T = (±i) p k p ∂ζm ∂ζn F± [T ], for all p, m, n ∈ N. (e) Plancherel theorem. F± extend uniquely to unitary transformations from L 2 (R × S2 , du ∧ S2 (ζ, ζ )) to L 2 (R × S2 , du ∧ S2 (ζ, ζ )) and the extension of F− is the inverse of that of F+ . These extensions coincide respectively with the restrictions to L 2 (R × S2 , du ∧ S2 (ζ, ζ )) of the action of F± on distributions. (f) If F˜± : L 2 (R, du) → L 2 (R, du) denotes the standard Fourier transform on the line, for every ψ ∈ L 2 (R × S2 , du ∧ S2 (ζ, ζ )) it holds: F± [ψ](k, ζ, ζ ) = F˜± (ψ(·, ζ, ζ ))(k), almost everywhere on R × S2 . (76) As a consequence, if ψ, φ ∈ L 2 (R × S2 , du ∧ S2 (ζ, ζ )), one may say that almost everywhere in (ζ, ζ ) ∈ S2 :
R
F± [ψ](k, ζ, ζ )F± [φ](k, ζ, ζ )dk =
R
ψ(u, ζ, ζ )φ(u, ζ, ζ )du.
(77)
(g) If m ∈ N and T ∈ S (+ ), F+ [T ] is a measurable function satisfying R×S2
(1 + |k|2 )m |F+ [T ]|2 dk ∧ S2 (ζ, ζ ) < +∞
if and only if the u-derivatives of T in the sense of distributions, are measurable functions with ∂un T ∈ L 2 (R × S2 , du ∧ S2 ), for n = 0, 1, . . . , m. Proof. For (a) and (b) the statements can be proved with the same procedure used in Rm in Theorem IX.1 in [RS75] with trivial changes, passing ζ, ζ -derivatives under the relevant symbols of integration in dk and du since it is allowed by compactness of S2 and fast ζ, ζ -uniform decaying for large |u|. Equation (74) is a trivial consequence of the analogous statement in R1 noticing that if f ∈ S (R × S2 ) then, from fixed ζ, ζ , the restriction u → f (u, ζ, ζ ) is a function of S (R). Hence (75) follows from (74) via Fubini-Tonelli’s theorem using the ζ, ζ -uniform decaying for large u of the integrands in both sides of (74) and the fact that S2 has finite measure. (d) has the same proof as the analog in Rn in Theorem IX.2 [RS75]. (e) Has the same proof as in the Rn case (Theorem IX.6 in [RS75]) noticing that (75) holds true and that S (+ ) is dense in the Hilbert space L 2 (R × S2 , du ∧ S2 (ζ, ζ )). The identity (76) in (f) is trivially fulfilled for ψ ∈ S (R × S2 ) by construction. Moreover, by Plancherel’s theorem on R, if ψ ∈ L 2 (R × S2 , du ∧ S2 (ζ, ζ )) (so that its restrictions at ζ, ζ fixed belong to L 2 (R, du) by Fubini-Tonelli’s theorem), one has R
|F˜± [ψ(·, ζ, ζ )](k)|2 dk =
R
|ψ(u, ζ, ζ )|2 du
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almost everywhere in ζ, ζ . By the Fubini-Tonelli theorem the right-hand side, and thus also the left-hand side is ζ, ζ integrable. By the Fubini-Tonelli theorem one finally has that the integrands are u, ζ, ζ jointly integrable so that: |F˜± (ψ(·, ζ, ζ ))(k)|2 dk ∧ S2 (ζ, ζ ) = |ψ(u, ζ, ζ )|2 du ∧ S2 (ζ, ζ ). R×S2
R×S2
We conclude that the map that associates every ψ ∈ L 2 (R × S2 , du ∧ S2 (ζ, ζ )) with the function (in the same space) (k, ζ, ζ ) → F˜± (ψ(·, ζ, ζ ))(k) is continuous and isometric and coincides with F± in the dense subspace S (R × S2 ), therefore it must coincide with F± extended to L 2 (R × S2 , du ∧ S2 (ζ, ζ )). In other words (76) holds true. Now (77) can be re-written replacing F± by F˜± and in this form is nothing but Plancherel’s theorem on the real line. The proof of (g) is immediate from (d) and (e). 2 References [AD05] [AH78] [AX78] [Ar99] [As80] [AS81] [BGP96] [BGP96] [BR021] [BR022] [BFK96] [BF00] [BFV03] [Da04] [Da05] [Da06] [Da07] [DMP06] [Di80] [Fr75]
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[Fri86-88]
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Friedrich, H.: On Purely Radiative Space-Times. Commun. Math. Phys. 103, 35 (1986); On the Existence of n-Geodesically Complete or Future Complete Solutions of Einstein’s Field Equations with smooth Asymptotic Structure. Commun. Math. Phys. 107, 585 (1986); On Static and Radiative Space-Times. Commun. Math. Phys. 119, 51 (1988) [Ge77] Geroch, R.: In: P. Esposito, L. Witten (eds.) Asymptotic Structure of Spacetime, New York: Plenum, 1977 [HW01] Hollands, S., Wald, R.M.: Local wick polynomials and time ordered products of quantum fields in curved space-time. Commun. Math. Phys. 223, 289 (2001) [HW04] Hollands, S., Wald, R.M.: Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005) [Ho00] Hollands, S.: Aspects of Quantum Field Theory in Curved Spacetime. Ph.D.thesis, University of York, 2000, advisor B.S. Kay, unpublished [Hö89] Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Second edition, Berlin: Springer-Verlag, 1989 [Hör71] H”89 Hörmander, L.: Fourier integral operators. I. Acta Math. 127, 79 (1971) [KW91] Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space-times with a bifurcate killing horizon. Phys. Rept. 207, 49 (1991) [KND77] Ko, M., Newmann, E.T., Tod, K.T.: In: P. Esposito, L. Witten (eds.) Asymptotic Structure of Spacetime, New York: Plenum, 1977 [Le53] Leray, J.: Hyperbolic Differential Equations. Unpublished Lecture Notes, Princeton (1953) [MC72-75] McCarthy, P.J.: Representations of the Bondi-Metzner-Sachs group I. Proc. R. Soc. London A330, 517 (1972); Representations of the Bondi-Metzner-Sachs group II. Proc. R. Soc. London A333, 317 (1973); The Bondi-Metzner-Sachs in the nuclear topology. Proc. R. Soc. London A343, 489 (1975) [Mo03] Moretti, V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189 (2003) [Mo06] Moretti, V. (2006) Uniqueness theorems for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence. Commun. Math. Phys. 268:727 [O’N83] O’Neill, B.: Semi-Riemannian Geometry with applications to Relativity. New York: Academic Press, USA, 1983 [Pe63] Penrose, R.: Asymptotic Properties of Space and Time. Phys. Rev. Lett. 10:66, 1963 [Pe74] Penrose, R.: In: A.O. Barut (ed.), Group Theory in Non-Linear Problems, Dordrecht: Reidel, 1974, p. 97, Chapter 1 [Ra96a] Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179:529, 1996 [Ra96b] Radzikowski, M.J.: A local to global singularity theorem for quantum field theory on curved space-time. Commun. Math. Phys. 180, 1 (1996) [RS75] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. II. Fourier Analysis, Self-Adjointness, New York: Academic Press, 1975 [SV01] Sahlmann, H., Verch, R.: Microlocal spectrum condition and hadamard form for vector valued quantum fields in curved space-time. Rev. Math. Phys. 13, 1203 (2001) [SVW02] Strohmaier, A., Verch, R., Wollenberg, M.: Microlocal analysis of quantum fields on curved space-times: analytic wavefront sets and reeh-schlieder theorems. J. Math. Phys. 43, 5514 (2002) [Wa84] Wald, R.M.: General Relativity. Chicago: University of Chicago Press, 1984 [Wa94] Wald, R.M.: Quantum field theory in curved space-time and black hole thermodynamics. Chicago: University of Chicago Press, 1994 Communicated by Y. Kawahigashi
Commun. Math. Phys. 279, 77–116 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0420-x
Communications in
Mathematical Physics
The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere Francesco D’Andrea1 , Ludwik D¸abrowski1 , Giovanni Landi2,3 1 Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014, Trieste, Italy.
E-mail:
[email protected]
2 Dipartimento di Matematica e Informatica, Università di Trieste, Via Valerio 12/1, I-34127, Trieste, Italy 3 INFN, Sezione di Trieste, Trieste, Italy
Received: 13 November 2006 / Accepted: 23 July 2007 Published online: 8 February 2008 – © Springer-Verlag 2008
Abstract: Equivariance under the action of Uq (so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the orthogonal quantum 4-sphere Sq4 . These representations are the constituents of a spectral triple on Sq4 with a Dirac operator which is isospectral to the canonical one on the round sphere S 4 and which then gives 4+ -summability. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an ‘instanton’ projection. We also introduce a real structure which satisfies all required properties modulo smoothing operators.
1. Introduction The recent constructions of spectral triples – with the consequent analysis of the corresponding spectral geometry – for the manifold of the quantum SU (2) group in [7,5, 11,12] and for its quantum homogeneous spaces (the Podle´s spheres) in [13,10,8,9], have provided a number of examples showing that a marriage between noncommutative geometry and quantum groups theory is indeed possible. A common feature of most of these examples is that the dimension spectrum is the same as in the commutative (q = 1) limit. Furthermore, with the only known exception of the 0+ -summable ‘exponential’ spectral triple on the standard Podle´s sphere given in [13], in order to have a real spectral triple one is forced to weaken the usual requirements that the real structure should satisfy. It is then only natural to try and construct additional explicit examples wondering in particular if these properties are common to all quantum spaces or are rather coincidences which happen for low dimensional examples (all related to the quantum group SUq (2)). In this paper we present an example in ‘dimension four’ given by a spectral triple on the orthogonal quantum sphere Sq4 which is isospectral to the canonical spectral triple on the classical sphere with the round metric. There exists also a real structure which satisfies all required properties modulo an ideal of smoothing operators.
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There are a few reasons why in dimension greater than or equal to four the orthogonal quantum sphere Sq4 is most interesting to study. Firstly, all the relevant irreducible representations of the symmetry algebra Uq (so(5)) are known [2] and both the algebra A(Sq4 ) of polynomial functions as well as the modules of chiral spinors carry representations of Uq (so(5)) which are multiplicity free. Secondly, the spectrum of the Dirac operator D / for the round metric on the undeformed sphere S 4 is known [20,1]. All this allows us to apply the already tested methods of isospectral deformations and indeed to construct an Uq (so(5))-equivariant spectral triple on Sq4 . The sphere Sq4 could also be relevant for noncommutative physical models. In particular, on Sq4 there is a canonical ‘instantonic vector bundle’ [16] and the study of the noncommutative geometry of Sq4 could be a first step for the construction of SUq (2) instantons on this space. In Sect. 2 we recall all generalities about spectral triples that we need. We give also some properties of finitely generated projective modules over algebras having quantum group symmetries. The rest of the paper is organized as follows. Sections 3 and 4 are devoted to the symmetry Hopf algebra Uq (so(5)) and its fundamental ∗-algebra module, the orthogonal quantum sphere Sq4 . In Sect. 5 we describe the A(Sq4 )-modules of chiral spinors over Sq4 . Section 6 is devoted to the left regular representation of the algebra A(Sq4 ) of polynomial functions over Sq4 and to the representations of A(Sq4 ) which in the q = 1 limit correspond to the modules of chiral spinors. These representations are Uq (so(5))-equivariant, that is they correspond to representations of the crossed product algebra A(Sq4 ) Uq (so(5)). In Sect. 7 we use the isospectral Dirac operator to construct a spectral triple on Sq4 ; it will be Uq (so(5))-equivariant, regular, even and of metric dimension 4. We also prove that it is non-trivial by pairing the Fredholm module canonically associated to the spectral triple to an ‘instanton’ projection e. It turns out that the projection e has charge 1, as in the classical case. In Sect. 8, we compute the part of the dimension spectrum contained in the right half plane {s ∈ C | Re s > 2}, as well as the top residue (which in the commutative case is proportional to the integral). This is done by quotienting by a suitable ideal of ‘infinitesimals’ I, which is larger than smoothing operators. At the moment we are unable to comment on the part of the dimension spectrum which is in the left half plane Re s ≤ 2. Finally, in Sect. 9 we produce an equivariant real structure for which both the ‘commutant property’ and the ‘first order condition’ are satisfied modulo the ideal of smoothing operators; this is consonant with the cases of the manifold of SUq (2) in [11] and of Podle´s spheres in [10,9]. In fact, we also show that these conditions are much easier to handle modulo the ideal I.
2. Some Useful Preliminaries In this section, we collect some basic notions concerning equivariant spectral triples. We also give some general properties of finitely generated projective modules over algebras having quantum group symmetries.
2.1. Generalities about spectral triples. We start with the notion of finite summable spectral triples [3].
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Definition 2.1. A spectral triple (A, H, D) is the datum of a complex associative unital ∗-algebra A, a ∗-representation π : A → B(H) by bounded operators on a (separable) Hilbert space H and a self-adjoint (unbounded) operator D = D ∗ such that, • •
(D + i)−1 is a compact operator; [D, π(a)] is a bounded operator for all a ∈ A .
We refer to D as the ‘generalized’ Dirac operator, or the Dirac operator ‘tout court’ and for simplicity we assume that it is invertible. Usually, the representation symbol π is removed when no risk of confusion arises. With n ∈ R+ , D is called n + -summable if the operator (D 2 + 1)−1/2 is in the Dixmier ideal Ln+ (H). We shall also call n the metric dimension of the spectral triple. A spectral triple is called even if there exists a grading γ , i.e. a bounded operator satisfying γ = γ ∗ and γ 2 = 1, such that the Dirac operator is odd and the algebra is even: γ D + Dγ = 0,
aγ = γ a, ∀ a ∈ A.
We recall from [6] a few analytic properties of spectral triples. To the unbounded operator D on H one associates an unbounded derivation δ on B(H) by, δ(a) = [|D|, a], for all a ∈ B(H). A spectral triple is called regular if the following inclusion holds, A ∪ [D, A] ⊂ dom δ j , j∈N
and we refer to OP0 := j∈N dom δ j as the ‘smooth domain’ of the operator δ. For a regular spectral triple, the class 0 of pseudodifferential operators of order less than or equal to zero is defined as the algebra generated by k∈N δ k (A ∪ [D, A]). If the triple has finite metric dimension n, the ‘zeta-type’ function ζa (s) := Tr H (a|D|−s ) associated to a ∈ 0 is defined (and holomorphic) for s ∈ C with Re s > n and the following definition makes sense. Definition 2.2. A spectral triple has dimension spectrum iff ⊂ C is a countable set, for all a ∈ 0 the function ζa (s) extends to a meromorphic function on C with poles as unique singularities, and the union of such singularities is the set . If is made only of simple poles, the Wodzicki-type residue functional − T := Ress=0 Tr(T |D|−s )
(2.1)
is tracial on 0 . We also recall the definition of ‘smoothing operators’ OP−∞ , OP−∞ := {T ∈ OP0 | |D|k T ∈ OP0 , ∀ k ∈ N}. The class OP−∞ is a two-sided ∗-ideal in the ∗-algebra OP0 , is δ-invariant and then in the smooth domain of δ. If T is a smoothing operator, ζT (s) is holomorphic on C and
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(2.1) vanishes. Thus, elements in OP−∞ can be neglected when computing the dimension spectrum and residue. Finally, we note that if the metric dimension is finite, rapid decay matrices – in a basis of eigenvectors for D with eigenvalues in increasing order – are smoothing operators. In analogy with the notion of spin manifold, one asks for the existence of a real structure J on a spectral triple (A, H, D). Motivated by the examples of real spectral triples on Podle´s spheres [10,9] and on SUq (2) [11], we use the following weakened definition of real structure. Definition 2.3. A real structure is an antilinear isometry J on H such that ∀ a, b ∈ A, J 2 = ±1,
[a, J b J −1 ] ⊂ I,
J D = ±D J,
[[D, a], J b J −1 ] ⊂ I.
If the spectral triple is even with grading γ , we impose the further relation J γ = ±γ J . The signs ‘±’ are determined by the dimension of the geometry [4]. A real spectral triple of dimension 4 corresponds to the choices J 2 = −1, J D = D J and J γ = γ J . The set I is a suitable two-sided ideal in the algebra OP0 of ‘order zero’ operators which is made of ‘infinitesimals’. The original definition [4] corresponds to I = 0; while in examples coming from quantum groups [10,11,9] one usually takes I = OP−∞ . Let F := D|D|−1 be the sign of D; if (A, H, D) is a regular even spectral triple, the datum (A, H, F, γ ) is an even Fredholm module. We say that the Fredholm module is p-summable if p ≥ 1 and, for all a ∈ A, [F, a] belongs to the p th Schatten-von Neumann ideal L p (H) of compact operators T such that |T | p is of trace class. Associated with a p-summable even Fredholm module there are cyclic cocycles defined by chnF (a0 , . . . , an ) =
1 2 n!
( n2 + 1) Tr(γ F[F, a0 ] . . . [F, an ]),
(2.2)
for all even integers n ≥ p − 1. By composing it with a matrix trace, chnF is canonically extended to matrices with entries in A. The pairing with elements [e] ∈ K 0 (A), given by chnF (e, e, . . . , e) build up to an integer-valued map ch F([e]) which depends only on the class [e] and which yields the index of the Dirac operator D twisted with the projection e (for further details see [3]). Finally, we turn now to symmetries; these will be implemented by an action of a Hopf ∗-algebra. Firstly, let V be a dense linear subspace of a Hilbert space H with inner product , , and let U be a ∗-algebra. An (unbounded) ∗-representation of U on V is a homomorphism λ : U → End(V) such that λ(h)v, w = v, λ(h ∗ )w for all v, w ∈ V and all h ∈ U. From now on, the symbol λ will be omitted. Next, let U = (U, , ε, S) be a Hopf ∗-algebra and let A be a left U-module ∗-algebra, i.e., there is a left action of U on A satisfying h ab = (h (1) a)(h (2) b),
h 1 = ε(h)1,
h a ∗ = {S(h)∗ a}∗ ,
for all h ∈ U and a, b ∈ A. As customary, (h) = h (1) ⊗ h (2) . A ∗-representation of A on V is called U-equivariant if there exists a ∗-representation of U on V such that, for all h ∈ U, a ∈ A and v ∈ V, it happens that hav = (h (1) a) h (2) v. Given U and A as above, the left crossed product ∗-algebra A U is defined as the ∗-algebra generated by the two ∗-subalgebras A and U with crossed commutation relations ha = (h (1) a)h (2) , ∀ h ∈ U, a ∈ A .
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Thus, U-equivariant ∗-representations of A correspond to ∗-representations of A U. A linear operator D defined on V is said to be equivariant if it commutes with U, i.e., Dhv = h Dv
(2.3)
for all h ∈ U and v ∈ V. On the other hand, an antilinear operator T defined on V is called equivariant if it satisfies the relation T hv = S(h)∗ T v ,
(2.4)
for all h ∈ U and v ∈ V, where S denotes the antipode of U. Notice that if T is an equivariant antilinear operator, its square T 2 is an equivariant linear operator, but T ∗ T is not an equivariant linear operator unless S 2 = 1. We use all these equivariance requirements in the following definition (see also [19]). Definition 2.4. Let U be a Hopf ∗-algebra and A a left U-module ∗-algebra. A (real, even) spectral triple (A, H, D, γ , J ) is called equivariant if U is represented on a dense subspace V of H, V ⊂ dom D, the representation of U commutes with the grading γ , the restriction of the representation of A on V is U-equivariant, the operator D is equivariant and J is the antiunitary part of the polar decomposition of an equivariant antilinear operator.
2.2. Projective module description of equivariant representations. In order to construct the analogues of the modules of chiral spinors on the sphere Sq4 we need some properties of finitely generated projective modules over algebras having quantum group symmetries. Let U be a Hopf ∗-algebra, A be an U-module ∗-algebra and ϕ : A → C be an invariant faithful state (i.e. ϕ is linear, ϕ(a ∗ a) > 0 for all nonzero a ∈ A, and ϕ(h a) = (h)ϕ(a) ∀ a ∈ A and h ∈ U). Suppose also that there exists κ ∈ Aut(A) such that the ‘twisted’ cyclicity ϕ(ab) = ϕ b κ(a) holds for all a, b ∈ A. Instances of this situation are provided by subalgebras of compact quantum group algebras with ϕ the Haar state and κ the modular involution. KMS states in Thermal Quantum Field Theory provide additional examples. In particular, for the case A = A(Sq4 ) and U = Uq (so(5)), ϕ comes from the Haar functional of A(S Oq 2 (5)) and the modular automorphism is κ(a) = K 18 K 26 a [15, Sect. 11.3.4]. For N ∈ N, let A N := A ⊗ C N be the linear space with elements v = (v1 , . . . , v N ), vi ∈ A, and C-valued inner product given by
v, w :=
N i=1
ϕ(vi∗ wi ).
(2.5)
Lemma 2.5. Let σ : U → Mat N (C) be a ∗-representation. The formulæ: (a.v)i := avi ,
(h.v)i :=
N j=1
(h (1) v j )σi j (h (2) ),
(2.6)
for all a, v ∈ A and h ∈ U (and i = 1, . . . , N ), define a ∗-representation of the crossed product algebra A U on the linear space A N .
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Proof. The inner product allows us to define the adjoint of an element of A U in the representation on A N . For x ∈ End(A N ), its adjoint denoted with x † , is defined
x † .v, w := v, x.w ,
∀, v, w ∈ A N .
Recall that being a ∗-representation means that x † .v = x ∗ .v for any operator x and any v ∈ AN . The nontrivial part of the proof consists in showing that h † .v = h ∗ .v for all h ∈ U and v ∈ A. For N > 1 we are considering the Hopf tensor product of the N = 1 representation with a matrix representation that is a ∗-representation by hypothesis. Thus it is enough to take N = 1. The U-invariance of ϕ implies: (h) v, w = ϕ h (v ∗ w) = ϕ (h (1) v ∗ )(h (2) w) . But h (1) v ∗ = {S(h (1) )∗ v}∗ by definition of module ∗-algebra. Then,
(h) v, w = S(h (1) )∗ .v, h (2) .w = v, S(h (1) )∗† h (2) .w . We deduce that for all h ∈ U one has that S(h (1) )∗† h (2) = (h).
(2.7)
Recall that the convolution product ‘’ for any F, G ∈ End(U) is defined by (F G)(h) := F(h (1) )G(h (2) )
∀h∈U ;
and (End(U), ) is an associative algebra with unity given by the endomorphism h → (h)1U , with S a left and right inverse for idU in (End(U), ), that is S idU = 1U = idU S. Let S ∈ End(U) be the composition S := † ◦ ∗ ◦ S. Equation (2.7) implies that S is a left inverse for idU : S idU = 1U . Applying S to the right of both members of this equation and using idU S = 1U we get S = S as endomorphisms of U, i.e. S(h)∗† = S(h) for all h ∈ U. Now, the antipode of a Hopf ∗-algebra is invertible, with S −1 = ∗ ◦ S ◦ ∗, thus we arrive at h ∗† = h for all h ∈ U. Replacing h with h ∗ we prove that h † = h ∗ for all h ∈ U, and this concludes the proof. Now, let e = (ei j ) ∈ Mat N (A) be an N × N matrix with entries ei j ∈ A. Let π : A N → A N be the (linear) endomorphism defined by: N vi ei j , (2.8) π(v) j := i=1
for all v ∈ A N and j = 1, . . . , N . Since A is associative, left and right multiplication commute and π(av) = aπ(v) for all a ∈ A and v ∈ A N . Thus we have the following lemma. Lemma 2.6. The map π defined by (2.8) is an A-module map.
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Recall that an endomorphism p of an inner product space V is a projection (not necessarily orthogonal) if p ◦ p = p. A projection p is orthogonal if the image of p and idV − p are orthogonal with respect to the inner product of V , and this happens exactly when p † = p. A simple computation shows that the map π in (2.8) is a projection iff e2 = e, that is the matrix e ∈ Mat N (A) is an idempotent. Now we use the twisted-cyclicity of ϕ to deduce:
v, π † (w) = π(v), w = ϕ(ei∗j vi∗ w j ) = ϕ vi∗ w j κ(ei∗j ) , ij
ij
for all v, w ∈ A N . Hence the adjoint π † of the endomorphism π is given by π † (w)i =
N j=1
w j κ(ei∗j ).
Let e∗ be the matrix with entries (e∗ ) jk := ek∗j . We have the following lemma. Lemma 2.7. The endomorphism π in (2.8) is an orthogonal projection iff e2 = e = κ(e∗ ). Next, we determine a sufficient condition for the endomorphism π to be not only an A-module map, but also an U-module map. Lemma 2.8. With ‘ t ’ denoting transposition, if h e = σ (h (1) )t e σ (S −1 (h (2) ))t ,
(2.9)
for all h ∈ U, the endomorphism π in (2.8) is an U-module map. Proof. Equation (2.9) can be rewritten as, σki (h (1) ) ekl σ jl (S −1 (h (2) )) ; h ei j = kl
by using it into the definition (2.6) one checks that π(h.v) = h.π(v) for all h ∈ U and v ∈ AN . When Lemma 2.7 and Eq. (2.9) are satisfied, the orthogonal projections π and π ⊥ = 1 − π split A N into the orthogonal sum of two sub ∗-representations π(A N ) and π ⊥ (A N ) of A U. The next lemma gives a (quite obvious) sufficient condition for π(A N ) and π ⊥ (A N ) to be not equivalent as representations of A. Recall that an isomorphism of A-modules is an invertible A-linear map, so isomorphic modules correspond to equivalent representations. Lemma 2.9. Let (A, H, F, γ ) be an even Fredholm module over A. If ch F([e]) = 0, the A-modules π(A N ) and π ⊥ (A N ) are not equivalent. Proof. The map K 0 (A) → Z, [e] → ch F([e]) is a homomorphism. Suppose π(A N ) and π ⊥ (A N ) are isomorphic A-modules, then [e] = [1 − e] and ch F([1 − e]) = ch F([e]). But from Eq. (2.2), ch F([1 − e]) = −ch F([e]) (since [F, 1 − e] = −[F, e] and n is even). Hence ch F([e]) = 0, and this concludes the proof by contradiction.
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3. The Symmetry Hopf Algebra Uq (so(5)) Let 0 < q < 1. We call Uq (so(5)) the real form of the Drinfeld-Jimbo deformation of soC (5), corresponding to the Euclidean signature (+, +, +, +, +); it is a real form of the Hopf algebra called U˘ q (so(5)) in [15, Sect. 6.1.2]. As a ∗-algebra, Uq (so(5)) is generated by {K i = K i∗ , K i−1 , E i , Fi := E i∗ }i=1,2 (i → 3 − i with respect to the notations of [15]), with relations: [K 1 , K 2 ] = 0,
K i K i−1 = K i−1 K i = 1,
[E i , F j ] = δi j K i E i K i−1 = q i E i ,
K 2j −K −2 j q j −q − j
,
K i E j K i−1 = q −1 E j if i = j,
together with the ones obtained by conjugation and Serre relations, explicitly, given by E 1 E 22 − (q 2 + q −2 )E 2 E 1 E 2 + E 22 E 1 = 0, E 13 E 2
− (q + 1 + q 2
−2
)(E 12 E 2 E 1
−
E 1 E 2 E 12 ) −
E 2 E 13
= 0,
(3.1a) (3.1b)
together with their adjoints. These relations can be written in a more compact form by defining [a, b]q := q 2 ab − ba. Then, (3.1) are equivalent to [E 2 , [E 1 , E 2 ]q ]q = 0,
[E 1 , [E 1 , [E 2 , E 1 ]q ]q ] = 0.
The Hopf algebra structure ( , , S) of Uq (so(5)) is given by: K i = K i ⊗ K i , E i = E i ⊗ K i + K i−1 ⊗ E i , (K i ) = 1, (E i ) = 0, S(K i ) = K i−1 , S(E i ) = −q i E i . For each non-negative n 1 , n 2 such that n 2 ∈ 21 N and n 2 − n 1 ∈ N there is an irreducible representation of Uq (so(5)) whose representation space we denote V(n 1 ,n 2 ) . We call it “the representation with highest weight (n 1 , n 2 )” since the highest weight vector is an eigenvector of K 1 and K 1 K 2 with eigenvalues q n 1 and q n 2 , respectively. Irreducible representations with highest weight (0, l) and ( 21 , l) (the ones that we need explicitly) can be found in [2] and are recalled presently. Let us use the shorthand notation Vl := V(0,l) if l ∈ N and Vl := V( 1 ,l) if l ∈ N + 21 . The vector space Vl , for all 2
l ∈ 21 N, has orthonormal basis |l, m 1 , m 2 ; j , where the labels ( j, m 1 , m 2 ) satisfy the following constraints. For l ∈ N: j = 0, 1, . . . , l,
j − |m 1 | ∈ N,
l − j − |m 2 | ∈ 2N,
while for l ∈ N + 21 : j = 21 , 23 , . . . , l − 1, l,
j − |m 1 | ∈ N,
l+
1 2
− j − |m 2 | ∈ N.
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Notice that for any admissible (l, m 1 , m 2 , j) there exists a unique ∈ {0, ± 21 } such that 1
l + − j − m 2 ∈ 2N (that is, = 0 if l ∈ N and = 21 (−1)l+ 2 − j−m 2 if l ∈ N + shall need the coefficients, 1 [l − j − m 2 + ][l + j + m 2 + 3 + ] , al ( j, m 2 ) = [2] [2( j + | |) + 1][2( j − | |) + 3] √ [l − (2 j + 1) − m 2 + 1][l − (2 j + 1) + m 2 + 2] bl ( j, m 2 ) = 2| | , [2 j][2 j + 2] (−1)2 [l − j + m 2 + 2 − ][l + j − m 2 + 1 − ] , cl ( j, m 2 ) = [2] [2( j + | |) − 1][2( j − | |) + 1]
1 2 ).
We
(3.2a) (3.2b) (3.2c)
where, as usual, [z] := (q z − q −z )/(q − q −1 ) denotes the q-analogue of z ∈ C. The ∗-representation σl : Uq (so(5)) → End(Vl ) is defined by the rules, σl (K 1 ) |l, m 1 , m 2 ; j = q m 1 |l, m 1 , m 2 ; j , σl (K 2 ) |l, m 1 , m 2 ; j = q m 2 −m 1 |l, m 1 , m 2 ; j ,
σl (E 1 ) |l, m 1 , m 2 ; j = [ j − m 1 ][ j + m 1 + 1] |l, m 1 + 1, m 2 ; j ,
σl (E 2 ) |l, m 1 , m 2 ; j = [ j − m 1 + 1][ j − m 1 + 2] al ( j, m 2 ) |l, m 1 −1, m 2 + 1; j + 1
+ [ j + m 1 ][ j − m 1 + 1] bl ( j, m 2 ) |l, m 1 − 1, m 2 + 1; j
+ [ j + m 1 ][ j + m 1 − 1] cl ( j, m 2 ) |l, m 1 − 1, m 2 + 1; j − 1 . When there is no risk of ambiguity the representation symbol σl will be suppressed. For l ∈ N the representation σl is real. That is, there is an antilinear map C : Vl → Vl , which satisfies C 2 = 1 and Cσl (h)C = σl (S(h)∗ ). This map is explicitly given by
The operator
C |l, m 1 , m 2 ; j := (−q)m 1 q 3m 2 |l, −m 1 , −m 2 ; j .
(3.3)
C1 := q −1 K 12 + q K 1−2 + (q − q −1 )2 E 1 F1 ,
(3.4)
is a Casimir for the subalgebra generated by (K 1 , K 1−1 , E 1 , we note the action of C1 on a vector of Vl , with l ∈ 21 N; it is
F1 ). For future reference,
C1 |l, m 1 , m 2 ; j = (q 2 j+1 + q −2 j−1 ) |l, m 1 , m 2 ; j .
(3.5)
4. The Orthogonal Quantum 4-Sphere Definition 4.1 ([18]). We call orthogonal quantum 4-sphere the virtual space underlying the algebra A(Sq4 ) generated by x0 = x0∗ , xi and xi∗ (with i = 1, 2), with commutation relations: xi x j = q 2 x j xi ,
∀ 0 ≤ i < j ≤ 2,
∀ i = xi∗ x j = q 2 x j xi∗ , ∗ 4 2 [x1 , x1 ] = (1 − q )x0 , [x2∗ , x2 ] = x1∗ x1 − q 4 x1 x1∗ , x02 + x1 x1∗ + x2 x2∗ = 1.
j,
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The original notations of Faddeev-Reshetikhin-Takhtadzhyan [18, Eq. (1.14)] can be
obtained by defining x1 := x2∗ , x2 := x1∗ , x3 := q(1 + q 2 ) x0 , x4 := x1 , x5 := x2 and q := q 2 . The notations in [16, Eq. (2.1)] can be obtained by the replacement xi → xi∗ and q 2 → q −1 . In the next propositions we summarize some well known facts. Proposition 4.2. The algebra A(Sq4 ) is an Uq (so(5))-module ∗-algebra for the action given by: K i xi = q xi , i = 1, 2, K 2 x1 = q −1 x1 , E 1 x0 = q −1/2 x1 ,
E 2 x1 = x2 ,
F1 x1 = q 1/2 [2]x0 , F1 x0 = −q −3/2 x1∗
F2 x2 = x1 ,
while K i x j = x j , E i x j = 0 and Fi x j = 0 in all other cases. Notice that the action on the xi∗ ’s is determined by compatibility with the involution: K i a ∗ = {K i−1 a}∗ ,
E 1 a ∗ = {−q F1 a}∗ ,
E 2 a ∗ = {−q 2 F2 a}∗ .
Proof. The bijective linear map from the linear span of {xi , xi∗ } to the representation space V1 defined (modulo a global proportionality constant) by x2 → |0, 1; 0 , x1 → |1, 0; 1 , x0 → (q[2])−1/2 |0, 0; 1 , x1∗ → −q |−1, 0; 1 , x2∗ → q 3 |0, −1; 0 , is a unitary equivalence of Uq (so(5))-modules (here unitary means that the real structure C on V1 is implemented by the ∗ operation on xi ’s). This guarantees that the free ∗-algebra C xi , xi∗ generated by {xi , xi∗ } is an Uq (so(5))-module ∗-algebra. The degree ≤ 2 polynomials generating the ideal which defines A(Sq4 ) span the real representations V0 and V(1,1) , inside the tensor product V1 ⊗ V1 . The quotient ∗-algebra of C xi , xi∗ by this ideal, A(Sq4 ), is then an Uq (so(5))-module ∗-algebra. Proposition 4.3. There is an isomorphism A(Sq4 ) ules.
l∈N
Vl of Uq (so(5)) left mod-
Proof. A linear basis for A(Sq4 ) is made of monomials x0n 0 x1n 1 (x1∗ )n 2 x2n 3 with n 0 , n 1 , n 2 ∈ N, n 3 ∈ Z and with the notation x2n 3 := (x2∗ )|n 3 | if n 3 < 0. Using this basis one proves that a weight vector of A(Sq4 ) is annihilated by both E 1 and E 2 if and only if it is of the form x2l , l ∈ N. Thus, highest weight vectors are proportional to x2l and the algebra decomposes as a multiplicity free direct sum of highest weight representations with weights (0, l). The algebra A(Sq4 ) has two inequivalent irreducible infinite dimensional representations. The representation space is the Hilbert space 2 (N2 ) and the representations are
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87
given by x0 |k1 , k2 ± := ±q 2(k1 +k2 ) |k1 , k2 ± , x1 |k1 , k2 ± := q 2k2 1 − q 4(k1 +1) |k1 + 1, k2 ± , x2 |k1 , k2 ± := 1 − q 4(k2 +1) |k1 , k2 + 1 ± .
(4.1)
The direct sum of these representations, with obvious grading γ and operator F given by F |k1 , k2 ± := |k1 , k2 ∓ , constitutes a 1-summable Fredholm module over A(Sq4 ). In the sequel we shall need both the quantum space SUq (2) as well as the equatorial Podle´s sphere, whose algebras are given in [21] and [17] respectively. Definition 4.4. The algebra A(SUq (2)) of polynomial functions on SUq (2) is the ∗-algebra generated by α, β and their adjoints, with relations: βα = qαβ, β ∗ α = qαβ ∗ , [β, β ∗ ] = 0, αα ∗ + ββ ∗ = 1, α ∗ α + q 2 β ∗ β = 1. We call equatorial Podle´s sphere the virtual space underlying the ∗-algebra A(Sq2 ) generated by A = A∗ , B and B ∗ with relations: AB = q 2 B A,
B B ∗ + A2 = 1,
B ∗ B + q 4 A2 = 1.
Proposition 4.5. There is a ∗-algebra morphism ϕ : A(Sq4 ) → A(SUq (2)) ⊗ A(Sq2 ) defined by: ϕ(x0 ) = −(αβ + β ∗ α ∗ ) ⊗ A, ϕ(x1 ) = −α 2 + q (β ∗ )2 ⊗ A,
(4.2)
ϕ(x2 ) = 1 ⊗ B. Proof. One proves by direct computation that the five elements ϕ(xi ), ϕ(xi )∗ satisfy all the defining relations of A(Sq4 ). 5. The Modules of Chiral Spinors We apply the general theory of Sect. 2.2, to the case A = A(Sq4 ) and U = Uq (so(5)). Recall that in this case κ(a) = K 18 K 26 a is the modular automorphism. We shall use the notations of Sect. 3 for the irreducible representations (Vl , σl ) of Uq (so(5)). By Proposition 4.3 we have the equivalence A(Sq4 ) l∈N Vl as left Uq (so(5)) modules. Using Lemma 2.5 for N = 1, we deduce that on the vector space l∈N Vl there exists at least one ∗-representation of the crossed product A(Sq4 ) Uq (so(5)) that extends the ∗-representation l∈N σl of Uq (so(5)). 4 Let e ∈ Mat 4 (A(Sq )) be the following idempotent: ⎛
⎞ 1 + x0 q 3 x2 −q x1 0 1 ⎜ q −3 x2∗ 1 − q 2 x0 0 q 3 x1 ⎟ ⎟. e := ⎜ ∗ −1 2 ⎝ 0 1 − q x0 q 3 x2 ⎠ 2 −q x1 0 q x1∗ q −3 x2∗ 1 + q 4 x0
(5.1)
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By direct computation one proves that K 18 K 26 e∗ = e = e2 and then, by Lemma 2.7, e defines an orthogonal projection π , by Eq. (2.8), on the linear space A(Sq4 )4 with inner product (2.5). Next, let σ : Uq (so(5)) → Mat 4 (C) be the ∗-representation defined by ⎞ ⎛ 1/2 ⎞ ⎛ q 0 0 0 1 0 00 ⎜ 0 q 1/2 0 0 ⎟ ⎜0 q −1 0 0⎟ ⎟, (5.2a) σ (K 2 ) = ⎝ σ (K 1 ) = ⎜ ⎠, −1/2 ⎠ ⎝ 0 0 0 q0 0 0 q 0 0 01 0 0 0 q −1/2 ⎛ 0 ⎜0 σ (E 1 ) = ⎝ 0 0
0 0 0 0
1 0 0 0
⎞ 0 1⎟ , 0⎠ 0
⎛ 0 ⎜0 σ (E 2 ) = ⎝ 0 0
0 0 1 0
0 0 0 0
⎞ 0 0⎟ . 0⎠ 0
(5.2b)
Again, by direct computation one proves that: K i e = σ (K i ) e σ (K i )−1 , E i e = σ (Fi ) e σ (K i )
−1
(5.3a) −i
− q σ (K i )
−1
e σ (Fi ).
(5.3b)
Since σ (K i ) = σ (K i )t and σ (Fi ) = σ (E i )t , we conclude that condition (2.9) is satisfied and that π and π ⊥ = 1 − π project A(Sq4 )4 onto sub ∗-representations of A(Sq4 ) Uq (so(5)). We state the main proposition of this section. Proposition 5.1. There exists two inequivalent representations of the crossed product algebra A(Sq4 ) Uq (so(5)) on l∈N+ 1 Vl that extend the representation l∈N+ 21 σl 2 of Uq (so(5)). The proof is in two steps. We first prove (in Lemma 5.2) that π(A(Sq4 )4 ) and π ⊥ (A(Sq4 )4 ) are not equivalent as representations of the algebra A(Sq4 ). Then we prove (in Lemma 5.3) that as Uq (so(5)) representations they are both equivalent to l∈N+ 1 Vl . 2
Lemma 5.2. The idempotent e in (5.1) splits A(Sq4 )4 into two inequivalent ∗-representations of the crossed product algebra A(Sq4 ) Uq (so(5)). Proof. To prove the statement we apply Lemma 2.9. We use the Fredholm module associated to the representation on 2 (N) ⊕ 2 (N) defined by Eq. (4.1). One has ch F([e]) = 21 Tr 2 (N)⊗C8 (γ F[F, e]) = 41 (1 − q 2 )2 Tr 2 (N)⊗C2 (γ F[F, x0 ]) q 2(k1 +k2 ) = 1. = (1 − q 2 )2 k1 ,k2 ∈N
The statement of Proposition 5.1 follows from the obvious observation that if the two representations of the crossed product algebra were equivalent, their restrictions to rep resentations of A(Sq4 ) would be equivalent too. Lemma 5.3. π(A(Sq4 )4 ) π ⊥ (A(Sq4 )4 )
l∈N+ 21
Vl as Uq (so(5)) representations.
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Proof. In this proof, ‘’ always means equivalence of representations of Uq (so(5)). Since σ in (5.2) is unitary equivalent to the spin representation V1/2 , the representation of Uq (so(5)) on A(Sq4 )4 is the Hopf tensor product of the representation over A(Sq4 ) with the representation V1/2 . From A(Sq4 ) l∈N Vl and from the decomposition Vl ⊗ V1/2 Vl− 1 ⊕ Vl+ 1 for all l ∈ {1, 2, 3, . . .}, we deduce that A(Sq4 )4 2 2 l∈N+ 1 (Vl ⊕ Vl ) and then, 2
π(A(Sq4 )4 )
l∈N+ 21
m l+ Vl ,
π ⊥ (A(Sq4 )4 )
l∈N+ 21
m l− Vl ,
with multiplicities m l± to be determined, such that m l+ + m l− = 2. For l ∈ N + 21 , the vectors l− 21
vl± := x2
(1 ± x0 , ±q 3 x2 , ∓q x1 , 0)
are highest weight vectors, being annihilated by both E 1 and E 2 , and have weight ( 21 , l). Furthermore, vl+ (1 − e) = vl− e = 0. Thus, vl+ ∈ π(A(Sq4 )4 ) and vl− ∈ π ⊥ (A(Sq4 )4 ). Then in both modules π(A(Sq4 )4 ) and π ⊥ (A(Sq4 )4 ) each representation Vl , l ∈ N + 21 , appears with multiplicity m l± ≥ 1. Since m l+ + m l− = 2, we deduce that m l± = 1 for all l ∈ N + 21 . 6. Equivariant Representations of A(Sq4 ) Next, we construct Uq (so(5))-equivariant representations of A(Sq4 ) which classically correspond to the left regular spinor representations. The representation spaces and chiral will be (the closure of) l∈N Vl and l∈N+ 1 Vl . 2 Equivariance of a representation means that it is a representation of the crossed product algebra A(Sq4 ) Uq (so(5)). The latter is defined by the crossed relations ha = (h (1) a)h (2) for all a ∈ A(Sq4 ) and h ∈ Uq (so(5)); explicitly, the relations between the generators read: [K 1 , x0 ] = 0,
K 1 x1 = q x1 K 1 ,
K 1 x2 = x2 K 1 ,
[K 2 , x0 ] = 0,
K 2 x1 = q −1 x1 K 2 ,
K 2 x2 = q x2 K 2 ,
[E 1 , x0 ] = q −1/2 x1 K 1 ,
E 1 x1 = q −1 x1 E 1 ,
E 1 x2 = x2 E 1 ,
[F1 , x0 ] = −q −1/2 K 1 x1∗ , F1 x1 = q −1 x1 F1 + q 1/2 [2]x0 K 1 , [E 2 , x0 ] = 0,
E 2 x1 = q x1 E 2 + x2 K 2 ,
F1 x2 = x2 F1 , E 2 x2 = q −1 x2 E 2 ,
F2 x1 = q x1 F2 , F2 x2 = q −1 x2 F2 +x1 K 2 . (6.1) In the previous section we proved that on V there is at least one equivariant repl l∈N resentation, the left regular one, and that on l∈N+ 1 Vl there are at least two equivariant [F2 , x0 ] = 0,
2
representations, corresponding to the projective modules A(Sq4 )4 e and A(Sq4 )4 (1−e). In this section we’ll prove that on such spaces there are no other equivariant representations besides the ones just mentioned.
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Let us denote with |l, m 1 , m 2 ; j the basis of the representation space Vl of Uq (so(5)) as discussed in Sect. 3. From the first two lines of (6.1) we deduce that m ,m x0 |l, m 1 , m 2 ; j = (6.2a) A j,1j ,l,l2 l , m 1 , m 2 ; j , l, j
x1 |l, m 1 , m 2 ; j =
l, j
x2 |l, m 1 , m 2 ; j =
l, j
2 B mj, 1j ,m ,l,l l , m 1 + 1, m 2 ; j ,
(6.2b)
2 C mj, 1j ,m ,l,l l , m 1 , m 2 + 1; j ,
(6.2c)
with coefficients to be determined. Notice that from the crossed relations x1 |l, m 1 , m 2 ; j = (F2 x2 − q −1 x2 F2 )K 2−1 |l, m 1 , m 2 ; j , x0 |l, m 1 , m 2 ; j = q −1/2 [2]−1 (F1 x1 − q −1 x1 F1 )K 1−1 |l, m 1 , m 2 ; j , the matrix coefficients of x0 and x1 can be expressed in terms of the coefficients of x2 . Lemma 6.1. Let k ∈ N. The following formulæ hold: = 0 if k k F1 |l, m 1 , m 2 ; j = = 0 if k = 0 if k k E 1 |l, m 1 , m 2 ; j = = 0 if k
> j + m1 , ≤ j + m1
(6.3a)
> j − m1 . ≤ j − m1
(6.3b)
Proof. By direct computation:
F1k |l, m 1 , m 2 ; j = [ j + m 1 ][ j + m 1 − 1] . . . [ j + m 1 − k + 1]
× [ j − m 1 + 1][ j − m 1 +2] . . . [ j − m 1 + k] |l, m 1 − k, m 2 ; j . The second square root is always different from zero since the q-analogues are in increasing order and j − m 1 + 1 ≥ 1. In the first square root q-analogues are in decreasing order and are all different from zero if and only if j + m 1 − k + 1 ≥ 1. This proves Eq. (6.3a). In the same way one establishes (6.3b) by computing that
E 1k |l, m 1 , m 2 ; j = [ j − m 1 ][ j − m 1 − 1] . . . [ j − m 1 − k + 1]
× [ j + m 1 + 1][ j + m 1 + 2] . . . [ j + m 1 + k] |l, m 1 + k, m 2 ; j . Lemma 6.2. The coefficients in (6.2) satisfy: m 1 ,m 2 2 Amj,1j,m ,l,l = B j, j ,l,l = 0 if | j − j | > 1,
2 C mj, 1j ,m ,l,l = 0 if j = j.
Proof. From (6.1), (6.3a) and (6.3b) we derive: x1 |l, m 1 , m 2 ; j = q − j+m 1 −1 x1 E 1 1 |l, m 1 , m 2 ; j = 0, j +m +2 j +m +2 F1 1 x1∗ l , m 1 + 1, m 2 ; j = q j +m 1 +2 x1∗ F1 1 l , m 1 + 1, m 2 ; j = 0. j−m 1 +1
E1
j−m +1
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91 j−m 1 +1
We expand the left-hand sides and use the independence of the vectors E 1 l , m 1 + 1, m 2 ; j and F j +m 1 +2 |l, m 1 , m 2 ; j to arrive at the conditions: 1
j−m 1 +1 2 l , m 1 + 1, m 2 ; j = 0, B mj, 1j ,m ,l,l E 1 j +m 1 +2 2 |l, m B¯ mj, 1j ,m F , m ; j = 0. ,l,l 1 2 1 By (6.3b) the graph parenthesis in the first line is different from zero if j − m 1 + 1 ≤ 2 j − m 1 − 1, i.e. B mj, 1j ,m ,l,l must be zero if j ≥ j + 2. By (6.3a) the graph parenthesis in 2 the second line is different from zero if j + m 1 + 2 ≤ j + m 1 , i.e. B¯ mj, 1j ,m ,l,l must be zero if j ≤ j − 2. This proves 1/3 of the statement 2 / { j − 1, j, j + 1}. B mj, 1j ,m ,l,l = 0 ∀ j ∈
A similar argument applies to x0 . From the coproduct of E 1n we deduce: E 1n x0 =
n n (E 1k x0 )E 1n−k K 1k = x0 E 1n − [n]q −1/2 x1 E 1n−1 K 1 . k k=0
j−m +2 j +m +2 This implies that E 1 1 x0 |l, m 1 , m 2 ; j = 0 and F1 1 x0 l , m 1 , m 2 ; j = 0. From these conditions we deduce that also x0 shift j by {0, ±1} only. Finally, let C1 be the Casimir element in Eq. (3.4). Then [C1 , x2 ] = 0 and from (3.5) we deduce that x2 is diagonal on the index j. Lemma 6.3. The coefficients in (6.2c) satisfy 2 C mj, 1j ,m ,l,l = 0 if |l − l | > 1 or if |l − l | = 0 and l ∈ N.
Proof. The elements {xi , xi∗ } are a basis of the irreducible representation V1 . Covariance of the action tells that xi |l, m 1 , m 2 ; j and xi∗ |l, m 1 , m 2 ; j are a basis of the tensor representation V1 ⊗ Vl . Equations (14–15) in Chapter 7 of [15] tell that V1 ⊗ Vl Vl−1 ⊕ Vl+1 if l ∈ N and that V1 ⊗ Vl Vl−1 ⊕ Vl ⊕ Vl+1 if l ∈ N+ 21 (with Vl−1 omitted if l −1 < 0). This Clebsh-Gordan decomposition tells that x2 |l, m 1 , m 2 ; j is in the linear span of the basis vectors l , m 1 , m 2 + 1; j with l − l = ±1 if l ∈ N or with l − l = 0, ±1 if l ∈ N + 21 . This concludes the proof of the lemma. 6.1. Computing the coefficients of x2 . By Lemma 6.3, we have to consider only the cases j = j, |l − l| ≤ 1 if l ∈ N + 21 or |l − l| = 1 if l ∈ N. The condition ,m 2 m2 [E 1 , x2 ] = 0 implies that C mj, 1j,l,l =: C j,l,l is independent on m 1 . Equations (E 2 x 2 − q −1 x2 E 2 ) |l, − j, m 2 ; j = 0 and (F2 x2∗ − q x2∗ F2 ) l , j, m 2 + 1; j = 0 imply, respectively:
m2 m 2 +1 −1 [l − j − m + ][l + j + m + 3 + ], C j,l,l 2 2 [l − j − m 2 − 1 + ][l + j + m 2 + 4 + ] = C j+1,l,l q m2 m 2 −1 C j,l,l [l + j − m 2 + 3 − ][l − j + m 2 − ] = C j+1,l,l q [l + j − m 2 + 2 − ][l − j + m 2 + 1 − ],
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with , ∈ {0, ± 21 } determined by the conditions l + − j − m 2 ∈ 2N and l − − j − m 2 ∈ 2N. Notice that if l − l ∈ 2N + 1 then = , while if l − l ∈ 2N then = − . Looking at the cases l − l = ±1, we deduce that 1
q − 2 ( j+m 2 ) C m2 √ [l + j + m 2 + 3 + ] j,l,l+1
1
√
and
q − 2 ( j+m 2 ) C m2 [l − j − m 2 + ] j,l,l−1
depend on j + m 2 only through their parity (i.e. they depend only on the value of ). Similarly, 1
√
q 2 ( j−m 2 ) C m2 [l − j + m 2 + 2 − ] j,l,l+1
1
√
and
q 2 ( j−m 2 ) C m2 [l + j − m 2 + 1 − ] j,l,l−1
depend on j − m 2 only through their parity. Combining these informations, we deduce that the following elements do not depend on the exact value of j, m 2 , but only on the value of : √
q −m 2 C m2 =: Cl,l+1 ( ), [l + j + m 2 + 3 + ][l − j + m 2 + 2 − ] j,l,l+1 q −m 2 C m2 =: Cl,l−1 ( ). √ [l − j − m 2 + ][l + j − m 2 + 1 − ] j,l,l−1
2 / N, we have to compute If l ∈ N there are no other coefficients C mj,l,l to compute. If l ∈ m2 also C j,l,l . In this case = − and we get:
m2 m 2 +1 −1 C j,l,l [l − j − m 2 − 1 − ][l + j + m 2 + 4 − ] = C j+1,l,l q [l − j − m 2 + ][l + j + m 2 + 3 + ],
m2 m 2 −1 [l + j − m 2 + 3 − ][l − j + m 2 − ] = C j+1,l,l q [l + j − m 2 + 2 + ][l − j + m 2 + 1 + ]. C j,l,l
Again, looking at the two cases = ± 21 we deduce that 1
q − 2 ( j+m 2 ) 2 C mj,l,l if = 1 [l + 2 − j − m 2 ]
1
1 2
and
q − 2 ( j+m 2 ) [l +
1 2
+ j + m 2 + 2]
2 C mj,l,l if = − 21
do not depend on j + m 2 (this time is fixed, so the parity of j + m 2 is fixed). Similarly, 1
q 2 ( j−m 2 ) [l +
1 2
− j + m 2 + 1]
1
2 C mj,l,l if = 21 and
q 2 ( j−m 2 ) [l +
1 2
+ j − m 2 + 1]
2 C mj,l,l if = − 21
do not depend on j − m 2 . Combining these informations, we deduce that the following element does not depend on the exact value of j, m 2 , but only on the value of : q −m 2 C m 2 =: Cl,l ( ). √ [l − 2 j − m 2 + 1 − ][l − 2 j + m 2 + 2 − ] j,l,l The denominator of the left-hand side is just [2 j][2 j +2]bl ( j, m 2 ) with bl the coefficient 2 in Eq. (3.2b). The formula C mj,l,l = q m 2 [2 j][2 j + 2]bl ( j, m 2 )Cl,l ( ) is valid for all l, since bl ( j, m 2 ) vanish for l integer.
Isospectral Dirac Operator on 4-Dimensional Quantum Euclidean Sphere
Summarizing, we find that
2 C mj,l,l+1 = q m 2 [l + j + m 2 + 3 + ][l − j + m 2 + 2 − ] Cl,l+1 ( ), 2 C mj,l,l 2 C mj,l,l−1
= q m 2 [2 j][2 j + 2]bl ( j, m 2 ) Cl,l ( ),
= q m 2 [l − j − m 2 + ][l + j − m 2 + 1 − ] Cl,l−1 ( ),
93
(6.4a) (6.4b) (6.4c)
with coefficients Cl,l ( ) to be determined. 6.2. Computing the coefficients of x1 . From Lemma 6.2, we have to consider only the three cases j = j, j ± 1. Using equation E 1 x1 = q −1 x1 E 1 we get, q −m 1 q −m 1 −1 ,m 2 B mj, 1j+1,l,l B m 1 +1,m 2 , √ = √ [ j + m 1 + 1][ j + m 1 + 2] [ j + m 1 + 2][ j + m 1 + 3] j, j+1,l,l q −m 1 q −m 1 −1 ,m 2 2 B mj, 1j,l,l B m 1 +1,m , √ = √ [ j − m 1 ][ j + m 1 + 1] [ j − m 1 − 1][ j + m 1 + 2] j, j,l,l q −m 1 q −m 1 −1 ,m 2 B mj, 1j−1,l,l B m 1 +1,m 2 . √ = √ [ j − m 1 ][ j − m 1 − 1] [ j − m 1 − 1][ j − m 1 − 2] j, j−1,l,l We see that the left hand sides of these three equations are independent of m 1 , and call:
,m 2 m1 B mj, 1j+1,l,l [ j + m 1 + 1][ j + m 1 + 2] B mj, 2j+1,l,l , (6.5a) =: q
m 1 ,m 2 m B j, j,l,l =: q m 1 [ j − m 1 ][ j + m 1 + 1] B j, 2j,l,l , (6.5b)
m 1 ,m 2 m2 m1 B j, j−1,l,l =: q [ j − m 1 ][ j − m 1 − 1] B j, j−1,l,l . (6.5c) Imposing the condition x1 K 2 = F2 x2 −q −1 x2 F2 on the subspace spanned by |l, j, m 2 ; j ,m 2 m 1 ,m 2 (so m 1 = j and B mj, 1j,l,l = B j, j−1,l,l = 0 on this subspace) we get: −1 2 2 −1 cl ( j + 1, m 2 − 1)C mj+1,l,l q m 2 B mj, 2j+1,l,l = cl ( j + 1, m 2 )C mj,l,l −q . 2 From this we deduce that, that since coefficients C mj,l,l vanish for |l − l | > 1, also m2 B j, j+1,l,l is zero in these cases. In the remaining three cases l = l, l ± 1, using Eq. (6.4) we get: [l + j + m 2 + 3 + ][l + j − m 2 + 3 − ] Cl,l+1 ( ), B mj, 2j+1,l,l+1 = (−1)2 q l− j+m 2 − [2( j + | |) + 1][2( j − | |) + 3] (6.6a) [l + 21 + j − 2 m 2 + 2][l + 21 − j − 2 m 2 ] Cl,l ( ), B mj, 2j+1,l,l = −2 q 2 l− j+m 2 −2+3 [2 j + 2] (6.6b) [l − j + m 2 − ][l − j − m 2 + ] B mj, 2j+1,l,l−1 = (−1)2 +1 q −l− j+m 2 −3+ Cl,l−1 ( ). [2( j + | |) + 1][2( j − | |) + 3] (6.6c)
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Imposing q x1∗ K 2 = x2∗ E 2 −q −1 E 2 x2∗ on the subspace spanned by l , − j + 1, m 2 ; j − 1 ,m 2 m 1 ,m 2 (so B mj, 1j,l,l = B j, j+1,l,l = 0 on this subspace) we get: −1 2 2 −1 q m 2 B mj, 2j−1,l,l = al ( j − 1, m 2 )C mj,l,l al ( j − 1, m 2 − 1)C mj−1,l,l −q .
We deduce that B mj, 2j−1,l,l vanishes if |l − l | > 1, while in the three remaining cases l = l, l ± 1 using Eq. (6.4) we get: m2 l+ j+m 2 +1+ [l − j − m 2 + 2 + ][l − j + m 2 + 2 − ] Cl,l+1 ( ), B j, j−1,l,l+1 = q [2( j + | |) − 1][2( j − | |) + 1] (6.7a) [l + 21 + j + 2 m 2 + 1][l + 21 − j + 2 m 2 + 1] m2 −2 l+ j+m 2 −1−3 B j, j−1,l,l = −2 q Cl,l ( ), [2 j] (6.7b) [l + j + m 2 + 1 + ][l + j − m 2 + 1 − ] Cl,l−1 ( ). B mj, 2j−1,l,l−1 = −q −l+ j+m 2 −2− [2( j + | |) − 1][2( j − | |) + 1] (6.7c) Moreover, the condition l , j, m 2 ; j|x1 K 2 + q −1 x2 F2 − F2 x2 |l, j − 1, m 2 ; j = 0 implies that −1 2 2 −1 bl ( j, m 2 − 1)C mj,l,l q m 2 B mj, 2j,l,l = bl ( j, m 2 )C mj,l,l −q .
(6.8)
A further elaboration on these coefficients is postponed to after the following section. 6.3. Computing the coefficients of x0 . The condition q 1/2 [2]x0 K 1 = F1 x1 − q −1 x1 F1 implies: 2 q m 1 + 2 [2]Amj,1j,m ,l,l = 1
2 [ j − m 1 ][ j + m 1 + 1] B mj, 1j ,m ,l,l
2 −q −1 [ j + m 1 ][ j − m 1 + 1] B mj, 1j −1,m ,l,l .
In the three non-trivial cases j − j = 1, 0, −1, using (6.5), we get: ,m 2 j+m 1 − 2 Amj,1j+1,l,l = q
1
,m 2 Amj,1j,l,l ,m 2 Amj,1j−1,l,l
[ j + m 1 + 1][ j − m 1 + 1] B mj, 2j+1,l,l , 1 = [2]−1 q −2− 2 q j+m 1 +1 [2][ j − m 1 ] − [2 j] B mj, 2j,l,l , 1 = −q − j+m 1 −1− 2 [ j + m 1 ][ j − m 1 ] B mj, 2j−1,l,l .
(6.9a) (6.9b) (6.9c)
,m 2 m 1 ,m 2 ¯ m 1 ,m 2 The hermiticity condition x0 = x0∗ means that Amj,1j+1,l,l = A j+1, j,l l and A j, j,l,l = A¯ mj,1j,l,m l2 . Thus, from (6.9) it follows that: 2 j+2 ¯ m 2 2 B mj+1, B j, j+1,l,l , j,l l = −q
B mj, 2j,l,l = B¯ mj, 2j,l l .
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Using (6.6), the first equation turns out to be equivalent to the following conditions: Cl+1,l ( ) = (−1)2 q 2l+4 C¯ l,l+1 ( ),
Cl,l ( ) = C¯ l,l (− ).
(6.10a)
The second of equation together with (6.8) implies: m2 −1 2 2 −1 bl ( j, m 2 − 1)C mj,l,l bl ( j, m 2 )C mj,l,l −q = bl ( j, m 2 )C j,l l
−q −1 bl ( j, m 2 − 1)C mj,l2−1 l . That is, using (6.4): Cl,l+1 ( ) = Cl,l+1 (− ),
Cl,l ( ) = C¯ l,l ( ).
(6.10b)
6.4. Again the coefficients of x1 . Now, using (6.10) together with (6.8) we are able to compute the last coefficients. Notice that from (3.2b) the coefficients bl vanish if = 0 (i.e. in the left regular representation), and then from (6.8) B mj, 2j,l,l vanish too if = 0. Moreover, from Lemma 6.2 B mj, 2j,l,l vanish also if |l − l | > 1. In the three cases l = l, l ± 1, using Eq. (6.4) we get: B mj, 2j,l,l+1
B mj, 2j,l,l
= 2| |[2]q
l+m 2 +1+ (2 j+1)
√ [l + 2 j − m 2 + 2 + ][l − 2 j + m 2 + 2 − ] [2 j][2 j + 2]
× Cl,l+1 ( ), 2| | [l − (2 j + 1) − m 2 + 1][l − (2 j + 1) + m 2 + 2] + = [2 j][2 j + 2] −q −2 [l + (2 j + 1) − m 2 + 2][l + (2 j + 1) + m 2 + 1] q −2 (2 j+1) [2l + 4] − q 2 (2 j+1) [2l + 2] − [2]q 2m 2 2| | Cl,l ( ), [2 j][2 j + 2] 1 − q2 √ [l + 2 j + m 2 + 1 + ][l − 2 j − m 2 + 1 − ] = −2| |[2]q −l+m 2 −2− (2 j+1) [2 j][2 j + 2] × Cl,l−1 ( ). =−
B mj, 2j,l,l−1
We have inserted the factor 2| |, so that the expressions remain valid also when = 0.
6.5. The condition on the radius. Orbits for S O(5) are spheres of arbitrary radius, equivariance alone not imposing constraints on the radius. Similarly, for the quantum spheres one has to impose a constraint on the radius to determine the coefficients of the representation. In fact, this will determine Cl,l+1 (0), Cl,l+1 ( 21 ) and Cl,l ( 21 ) only up to a phase. Different choices of the phases correspond to unitary equivalent representations and without losing generality we choose Cl,l ( ) ∈ R. A possible expression for the radius is q 8 x02 + q 4 x1∗ x1 + x2∗ x2 which we constrain to be equal to 1. Let then,
r (l, m 1 , m 2 ; j) := l, m 1 , m 2 ; j q 8 x02 + q 4 x1∗ x1 + x2∗ x2 l, m 1 , m 2 ; j .
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All these matrix coefficients must be 1. In particular, for l ∈ N the condition r (l, 0, l; 0) = 1 implies (up to a phase) that Cl,l+1 (0) = √
q −l−3/2 . [2l + 3][2l + 5]
(6.11)
For l ∈ N+ 21 we first require that r (l, 21 , l; 21 ) =r (l, − 21 , l; 21 ) obtaining two possibilities: Cl,l ( 21 ) = ±
[2]q l+2 Cl,l+1 ( 21 ). [2l + 2]
Then imposing r (l, 21 , l; 21 ) = 1 , yields (up to a phase) Cl,l+1 ( 21 ) =
q −l−3/2 , [2l + 4]
(6.12)
hence,
q 1/2 [2] . [2l + 2][2l + 4] With these, all the coefficients are completely determined. Cl,l ( 21 ) = ±
(6.13)
6.6. Explicit form of the representations. Let us recall what we know on the equivariant representations of the algebra A(Sq4 ). By the decomposition A(Sq4 ) l∈N Vl into irreducible representations of Uq (so(5)), there exists (at least) one representation of A(Sq4 ) Uq (so(5)) on the vector space l∈N Vl extending the representation l∈N σl of Uq (so(5)). As we computed above, the equivariance uniquely determines (for l ∈ N, up to unitary equivalence) the matrix coefficients of the representation, whose expression is characterized by (6.11). On the other hand, by Proposition 5.1 there are(at least) two inequivalent representations of A(Sq4 ) Uq (so(5)) on the vector space l∈N+ 1 Vl extending the representation 2 l∈N+ 1 σl of Uq (so(5)). These correspond, by Lemma 5.2, to the projective modules 2
A(Sq4 )4 e and A(Sq4 )4 (1 − e), with e the idempotent in Eq. (5.1). The computation above (for l ∈ N + 21 ), which culminates in Eq. (6.13), tells us that there are only two possibilities for the matrix coefficients (up to unitary equivalence). Therefore, the two possible choices in (6.13) must correspond to the inequivalent representations associated with the projective modules A(Sq4 )4 e and A(Sq4 )4 (1 − e). Let us summarize these results in the following two theorems, which correspond to the scalar (i.e. left regular) and chiral spinor representations, respectively. Theorem 6.4. The vector space A(Sq4 ) has orthonormal basis |l, m 1 , m 2 ; j with, l ∈ N, L 2 (Sq4 )
j = 0, 1, . . . , l,
j − |m 1 | ∈ N,
We call the Hilbert space completion of the left regular representation is given by
A(Sq4 ).
l − j − |m 2 | ∈ 2N. Modulo a unitary equivalence,
x0 |l, m 1 , m 2 ; j = A j,m 1 Cl,+ j,m 2 |l + 1, m 1 , m 2 ; j + 1 + A j,m 1 Cl,−j,m 2 |l − 1, m 1 , m 2 ; j + 1
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− + A j−1,m 1 Cl+1, j−1,m 2 |l + 1, m 1 , m 2 ; j − 1 + + A j−1,m 1 Cl−1, j−1,m 2 |l − 1, m 1 , m 2 ; j − 1 ,
x1 |l, m 1 , m 2 ; j = B +j,m 1 Cl,+ j,m 2 |l + 1, m 1 + 1, m 2 ; j + 1 + B +j,m 1 Cl,−j,m 2 |l − 1, m 1 + 1, m 2 ; j + 1 − + B− j,m 1 Cl+1, j−1,m 2 |l + 1, m 1 + 1, m 2 ; j − 1 + + B− j,m 1 Cl−1, j−1,m 2 |l − 1, m 1 + 1, m 2 ; j − 1 ,
x2 |l, m 1 , m 2 ; j = Dl,+ j,m 2 |l + 1, m 1 , m 2 + 1; j + Dl,−j,m 2 |l − 1, m 1 , m 2 + 1; j , with coefficients
[ j + m 1 + 1][ j − m 1 + 1] , [2 j + 1][2 j + 3] − j+m 1 −1/2 [ j + m 1 + 1][ j + m 1 + 2] , =q [2 j + 1][2 j + 3] j+m 1 +1/2 [ j − m 1 ][ j − m 1 − 1] , = −q [2 j − 1][2 j + 1]
A j,m 1 = q m 1 −1 B +j,m 1 B− j,m 1 and
Cl,+ j,m 2 Cl,−j,m 2 Dl,+ j,m 2 Dl,−j,m 2
[l + j + m 2 + 3][l + j − m 2 + 3] , [2l + 3][2l + 5] [l − j + m 2 ][l − j − m 2 ] , = −q m 2 −1 [2l + 1][2l + 3] −l+m 2 −3/2 [l + j + m 2 + 3][l − j + m 2 + 2] , =q [2l + 3][2l + 5] [l − j − m 2 ][l + j − m 2 + 1] . = q l+m 2 +3/2 [2l + 1][2l + 3] =q
m 2 −1
The two chiral spinorial representations (corresponding to the sign ± in Eq. (6.13)) are described in the following theorem. Theorem 6.5. Let H± be two Hilbert spaces with orthonormal basis |l, m 1 , m 2 ; j ± , where l ∈ N + 21 ,
j = 21 , 23 , . . . , l,
j − |m 1 | ∈ N,
l+
1 2
− j − |m 2 | ∈ N.
Let = ± 21 be defined by l + − j −m 2 ∈ 2N. On each space H± there is an equivariant ∗-representation of A(Sq4 ) defined by: x0 |l, m 1 , m 2 ; j ± = A+j,m 1 Cl,+ j,m 2 |l + 1, m 1 , m 2 ; j + 1 ±
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∓ A+j,m 1 Cl,0 j,m 2 |l, m 1 , m 2 ; j + 1 ± + A+j,m 1 Cl,−j,m 2 |l − 1, m 1 , m 2 ; j + 1 ± + A0j,m 1 Hl,+ j,m 2 |l + 1, m 1 , m 2 ; j ± ± A0j,m 1 Hl,0 j,m 2 |l, m 1 , m 2 ; j ± + + A0j,m 1 Hl−1, j,m 2 |l − 1, m 1 , m 2 ; j ± − + A+j−1,m 1 Cl+1, j−1,m 2 |l + 1, m 1 , m 2 ; j − 1 ±
∓ A+j−1,m 1 Cl,0 j−1,m 2 |l, m 1 , m 2 ; j − 1 ± + + A+j−1,m 1 Cl−1, j−1,m 2 |l − 1, m 1 , m 2 ; j − 1 ± ,
x1 |l, m 1 , m 2 ; j ± = B +j,m 1 Cl,+ j,m 2 |l + 1, m 1 + 1, m 2 ; j + 1 ± ∓ B +j,m 1 Cl,0 j,m 2 |l, m 1 + 1, m 2 ; j + 1 ± + B +j,m 1 Cl,−j,m 2 |l − 1, m 1 + 1, m 2 ; j + 1 ± + B 0j,m 1 Hl,+ j,m 2 |l + 1, m 1 + 1, m 2 ; j ± ± B 0j,m 1 Hl,0 j,m 2 |l, m 1 + 1, m 2 ; j ± + + B 0j,m 1 Hl−1, j,m 2 |l − 1, m 1 + 1, m 2 ; j ± − + B− j,m 1 Cl+1, j−1,m 2 |l + 1, m 1 + 1, m 2 ; j − 1 ± 0 ∓ B− j,m 1 Cl, j−1,m 2 |l, m 1 + 1, m 2 ; j − 1 ± + + B− j,m 1 Cl−1, j−1,m 2 |l − 1, m 1 + 1, m 2 ; j − 1 ± ,
x2 |l, m 1 , m 2 ; j ± = Dl,+ j,m 2 |l + 1, m 1 , m 2 + 1; j ± ± Dl,0 j,m 2 |l, m 1 , m 2 + 1; j ± + Dl,−j,m 2 |l − 1, m 1 , m 2 + 1; j ± , with coefficients A+j,m 1 A0j,m 1 B +j,m 1 B 0j,m 1 B− j,m 1
√
[ j + m 1 + 1][ j − m 1 + 1] , [2 j + 2] q j+m 1 +1 [2][ j − m 1 ] − [2 j] , = q −2 [2 j][2 j + 2] √ [ j + m 1 + 1][ j + m 1 + 2] , = q − j+m 1 −1/2 [2 j + 2] √ [ j − m 1 ][ j + m 1 + 1] , = (1 + q 2 )q m 1 −1/2 [2 j][2 j + 2] √ [ j − m 1 ][ j − m 1 − 1] , = −q j+m 1 +1/2 [2 j] =q
m 1 −1
and Cl,+ j,m 2 = −q m 2 −1−
√ [l + j + m 2 + 3 + ][l + j − m 2 + 3 − ] , [2l + 4]
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[l +
1 1 2 + j − 2 m 2 + 2][l + 2 − j − 2 m 2 ] Cl,0 j,m 2 = [4 ] q 2 l+m 2 −1+3 , [2l + 2][2l + 4] √ [l − j + m 2 − ][l − j − m 2 + ] , Cl,−j,m 2 = −q m 2 −1+ [2l + 2] √ [l + 2 j − m 2 + 2 + ][l − 2 j + m 2 + 2 − ] , Hl,+ j,m 2 = q m 2 −1+ (2 j+1) [2l + 4]
Hl,0 j,m 2 =
[l− (2 j+1)−m 2 +1][l− (2 j+1)+m 2 +2]−q −2 [l+ (2 j+1)−m 2 +2][l+ (2 j+1)+m 2 +1] , [2l+2][2l+4]
√ [l + j + m 2 + 3 + ][l − j + m 2 + 2 − ] , Dl,+ j,m 2 = q −l+m 2 −3/2 [2l + 4] √ [l − 2 j − m 2 + 1 − ][l − 2 j + m 2 + 2 − ] , Dl,0 j,m 2 = [2]q m 2 +1/2 [2l + 2][2l + 4] √ [l − j − m 2 + ][l + j − m 2 + 1 − ] . Dl,−j,m 2 = −q l+m 2 +3/2 [2l + 2] These two representations are inequivalent and correspond to the projective modules A(Sq4 )4 e and A(Sq4 )4 (1 − e), with e the idempotent in Eq. (5.1). 7. The Dirac Operator on the Orthogonal Quantum 4-Sphere We start by constructing a non-trivial Fredholm module on the orthogonal quantum sphere (with different representations a non-trivial Fredholm module was already constructed in [16]). Proposition 7.1. Consider the representations of A(Sq4 ) on H± given in Theorem 6.5. Then, the datum (A(Sq4 ), H, F, γ ) is a 1-summable even Fredholm module, where H := H+ ⊕ H− , γ is the natural grading and F ∈ B(H) is defined by F |l, m 1 , m 2 ; j ± := |l, m 1 , m 2 ; j ∓ . This Fredholm module is non-trivial. In particular, ch F([e]) := 21 Tr H⊗C2 (γ F[F, P]) = 1,
(7.1)
with e the idempotent defined by Eq. (5.1). Proof. That F = F ∗ , F 2 = 1 and γ F + Fγ = 0 is obvious. Then, it is enough to show that [F, xi ] ∈ L1 (H) for i = 0, 1, 2. From this and the Leibniz rule it follows that [F, a] is trace class, and then compact, for all a ∈ A(Sq4 ). Now, notice that [F, x0 ] |l, m 1 , m 2 ; j ± = ∓ 2 A+j,m 1 Cl,0 j,m 2 |l, m 1 , m 2 ; j + 1 ∓ ± 2 A0j,m 1 Hl,0 j,m 2 |l, m 1 , m 2 ; j ∓ ∓ 2 A+j−1,m 1 Cl,0 j−1,m 2 |l, m 1 , m 2 ; j − 1 ∓ , [F, x1 ] |l, m 1 , m 2 ; j ± = ∓ 2B +j,m 1 Cl,0 j,m 2 |l, m 1 + 1, m 2 ; j + 1 ∓
[F, x2 ] |l, m 1 , m 2 ; j ± =
± 2B 0j,m 1 Hl,0 j,m 2 |l, m 1 + 1, m 2 ; j ∓ 0 ∓ 2B − j,m 1 Cl, j−1,m 2 |l, m 1 + 1, m 2 ; j − 1 ∓ , ± 2Dl,0 j,m 2 |l, m 1 , m 2 + 1; j ∓ .
(7.2)
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All the coefficients appearing in these equations are bounded by q 2l . Thus the commutators are trace class and this concludes the first part of the proof. To prove non-triviality it is enough to prove (7.1). Substituting (5.1) into (7.1) yields ch F([e]) =
(1−q 2 )2 Tr H (γ 4
F[F, x0 ]),
and in turn, using Eq. (7.2), ch F([e]) = (1 − q 2 )2
l, j,m 1 ,m 2
A0j,m 1 Hl,0 j,m 2 .
Summing over m 1 from − j to j we obtain that [l + (2 j + 1) − m 2 + 2][l + (2 j + 1) + m 2 + 1] [2l + 2][2l + 4][2 j][2 j + 2] l, j,m 2 × q 2 j+2 + q −2 j − [2]q 2m 1 +1
ch F([e]) = q −3 (1 − q 2 )2
m1
(2 j + 1)(q 2 j+1 + q −2 j−1 ) − [2][2 j + 1] = [2l + 2][2l + 4][2 j][2 j + 2] l, j × q 2l+2 (2 j+1)+3 + q −2l−2 (2 j+1)−3 − q 2m 2 −1 − q −2m 2 +1 . m2
The sum over m 2 requires additional care. For fixed, l − − j + m 2 = 0, 2, 4, . . . , 2(l − j). If we call 2i := l − − j + m 2 and sum first over i = 0, 1, . . . , l − j and then over = ±1/2 we get: ch F([e]) =
(2 j + 1)(q 2 j+1 + q −2 j−1 ) − [2][2 j + 1] [2l + 2][2l + 4][2 j][2 j + 2] l, j × (l − j + 1)(q 2l+2 (2 j+1)+3 + q −2l−2 (2 j+1)−3 ) 2 =±1
−(q 2 −1 + q −2 +1 )[2]−1 [2(l − j + 1)]
(2 j + 1)(q 2 j+1 + q −2 j−1 ) − [2][2 j + 1] [2l + 2][2l + 4][2 j][2 j + 2] l, j × (l − j + 1)(q 2l+3 + q −2l−3 )(q 2 j+1 + q −2 j−1 ) − [2][2(l − j + 1)] fl j (q) =: f (q). =:
=
l, j
We call fl j (q) the generic term of last series, explicitly written as fl j (q) = (1 − q )
2 4
1+q 2 (1 − q 4 j+2 ) 1−q 2 (1 − q 4l+4 )(1 − q 4l+8 )(1 − q 4 j )(1 − q 4 j+4 )
(2 j + 1)(1 + q 4 j+2 ) −
× q 2l−1 (l − j + 1)(1 + q 4l+6 )(1 + q 4 j+2 ) −
1+q 2 1−q 2
q 2 (q 4 j − q 4l+4 ) ,
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and consider it as a function of q ∈ [ 0, 1[ . Notice that each fl j (q) is a C ∞ function of q (they are rational functions whose denominators never vanish for 0 ≤ q < 1). From the inequality 0 ≤ fl j (q) ≤ 4(2 j + 1)q 2l−1 we deduce (using the Weierstrass M-test) that the series is absolutely (hence uniformly) convergent in each interval [0, q0 ] ⊂ [ 0, 1[ . Then, it converges to a function f (q) which is continuous in [ 0, 1[ . Being the index of a Fredholm operator, f (q) is integer valued in ] 0, 1[; by continuity it is constant and can be computed in the limit q → 0. In this limit we have fl j (q) = 2 j (l − j + 1)q 2l−1 + O(q 2l ). Thus, fl j (0) = δl,1/2 δ j,1/2 and ch F([e]) = f (0) = 1. The next step is to define a spectral triple whose Fredholm module is the one described in Proposition 7.1. Proposition 7.2. Let D be the (unbounded) operator on H := H+ ⊕ H− defined by D |l, m 1 , m 2 ; j ± := (l + 23 ) |l, m 1 , m 2 ; j ∓ . Then, the datum (A(Sq4 ), H, D, γ ) is a Uq (so(5))-equivariant regular even spectral triple of metric dimension 4. Remark. The operator D is isospectral to the classical Dirac operator on S 4 (whose spectrum has been computed in [20,1]). When q = 1, this spectral triple becomes the canonical one associated to the spin structure of S 4 . Proof. Clearly the representation of the algebra is even, D is odd, with compact resolvent and 4+ -summable (being isospectral to the classical Dirac operator on S 4 ). Let δ be the unbounded derivation on B(H) defined by δ(T ) := [|D|, T ]. Each generator of A(Sq4 ) is the sum of a finite number of weighted shifts; each of these weighted shifts is a bounded operator (the coefficients are all bounded by 1) and is an eigenvector of δ, i.e., if T shifts the index l by k, then δ(T ) = kT . Thus, such weighted shifts are not only bounded but also in the smooth domain of δ, which we denote by OP0 := j∈N dom δ j . As a consequence A(Sq4 ) ⊂ OP0 . Recall that [F, xi ] has coefficients decaying faster than q l ; thus |D|[F, xi ] is a matrix of rapid decay. In particular, |D|[F, xi ] ∈ OP−∞ ⊂ OP0 . The identity [D, xi ] = δ(xi )F + |D|[F, xi ],
(7.3)
tells us that [D, xi ] is not only bounded but even in OP0 – being the sum of two bounded operators contained in the ∗-algebra OP0 . Then, D defines a spectral triple and such a spectral triple is regular. Finally, since D is proportional to the identity in any irreducible subrepresentation Vl of Uq (so(5)), it commutes with all h ∈ Uq (so(5)) and it is equivariant. As a preparation for the study of the dimension spectrum in Sect. 8, let us explicitly verify the 4-summability of D. As one can easily check, the dimension of Vl is [1] dim Vl = 23 (l + 25 )(l + 23 )(l + 21 ).
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From this we get Tr(|D|−s ) =
l∈N+ 21
2(l + 23 )−s dim Vl =
4 3
∞ n=1
(n 2 − 1)n −s+1 ,
where n = l + 23 (and we added the term with n = 1 since it is identically zero). The above series is convergent in the right half-plane {s ∈ C | Re s > 4}, thus D has metric dimension 4. Notice that Tr(|D|−s ) has meromorphic extension on C given by Tr(|D / |−s ) =
4 3
{ζ (s − 3) − ζ (s − 1)} ,
(7.4)
where ζ (s) is the Riemann zeta-function. We recall that ζ (s) has a simple pole in s = 1 as unique singularity and that Ress=1 ζ (s) = 1.
8. The Dimension Spectrum and Residues To compute the dimension spectrum we shall use a very simple representation of the algebra which differs – in a sense which will be clear in Proposition 8.3 – from the chiral ones by a suitable ideal of operators. This is the class of operators, I := T ∈ OP0 T |D|− p ∈ L1 (H), ∀ p > 2 .
(8.1)
Lemma 8.1. The collection I is a two-sided ideal in OP0 . Proof. Clearly I is a vector space: if T1 , T2 ∈ I, that is T1 |D|− p ∈ L1 (H), T2 |D|− p ∈ L1 (H) for all p > 2, then T1 |D|− p + T2 |D|− p ∈ L1 (H) for all p > 2, which means T1 + T2 ∈ I. That I is a left ideal is straightforward. Since L1 (H) is a two-sided ideal in B(H), if T1 ∈ OP0 and T2 ∈ I, for all p > 2 we have that T1 · T2 |D|− p is the product of a bounded operator, T1 , with a trace class one, T2 |D|− p , thus it is of trace class, and T1 T2 ∈ I. From Appendix B of [6] for any p > 0, we know that the bounded operator |D|− p maps H to H p := dom |D| p , that T ∈ OP0 ⊂ op0 is a bounded operator H p → H p , and finally that |D| p is bounded from H p to H. Thus, for T ∈ OP0 , the product |D| p T |D|− p is a bounded operator on H. Now, if T1 ∈ OP0 and T2 ∈ I, for all p > 2 we can write T2 T1 |D|− p = T2 |D|− p · |D| p T1 |D|− p as the product of a bounded operator, |D| p T1 |D|− p , with a trace class one, T2 |D|− p ; thus T2 T1 |D|− p is of trace class so T2 T1 ∈ I and I is also a right ideal. Clearly, if T is of trace class, so is |D|− p T for any positive p, and L1 (H) ⊂ I. Since OP−∞ ⊂ L1 (H), smoothing operators belong to I as well. On the other hand, I is strictly bigger than L1 (H); indeed, the operator L q ∈ B(H), given by 1
L q |l, m 1 , m 2 ; j ± := q j+ 2 |l, m 1 , m 2 ; j ± , is not of trace class but belongs to I, by the following proposition.
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Proposition 8.2. For any s ∈ C with Re s > 2 one has that 1+q 4q 3 −s j+ 21 ζ L q (s) := ζ (s − 1) − (l + 2 ) q = ζ (s) 2 (1 − q) 1−q l, j,m 1 ,m 2
+ holomorphic function, where ζ (s) is the Riemann zeta-function. In particular, this means that L q belongs to the ideal I. Furthermore, since the series ζ L q (0) is divergent, L q is not of trace class. Proof. Calling n := l + 23 , k := j + 21 , we have ζ L q (s) = 4
∞
n −s
n=2
n−1
k(n − k)q k ,
k=1
We can sum starting from n = 1 and for k = 0, . . . , n (we simply add zero terms) to get ζ L q (s) = 4
∞ n=1
n ∞ 1 − q n+1 . n −s nq∂q − (q∂q )2 qk = 4 n −s nq∂q − (q∂q )2 1−q k=0
n=1
Terms decaying as q n give a holomorphic function of s, thus modulo holomorphic functions, ζ L q (s) ∼ 4
∞ n=1
n −s n
q (1−q)2
−
q(1+q) (1−q)3
.
The last series is summable for all s with Re s > 2, and its sum can be written in terms of the Riemann zeta-function as in the statement of the proposition. 8.1. An approximated representation. Let Hˆ be a Hilbert space with orthonormal basis ||l, m 1 , m 2 ; j ± labelled by, l ∈ 21 Z,
l + j ∈ Z,
j + m 1 ∈ N,
l+
1 2
− j + m 2 ∈ N.
Let I be the labelling set of the Hilbert space H± as in Theorem 6.5, and given by ! I := ( j, m 1 , m 2 , j) l ∈ N + 21 , j = 21 , 23 , . . . , l, " j − |m 1 | ∈ N, l + 21 − j − |m 2 | ∈ N . Notice that I is the subset of labels of Hˆ satisfying l ∈ N + 21 , m 1 ≤ j ≤ l and m 2 ≤ l + 21 − j. Define the inclusion Q : H → Hˆ and the adjoint projection P : Hˆ → H by, Q |l, m 1 , m 2 ; j ± := ||l, m 1 , m 2 ; j ± for all (l, m 1 , m 2 , j) ∈ I, |l, m 1 , m 2 ; j ± if (l, m 1 , m 2 , j) ∈ I, P ||l, m 1 , m 2 ; j ± := 0 otherwise.
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Clearly, P Q = idH . The Hilbert space Hˆ carries a bounded ∗-representation of the algebra A(SUq (2)) ⊗ A(Sq2 ) defined by, α ||l, m 1 , m 2 ; j ± = 1 − q 2( j+m 1 +1) ||l + 21 , m 1 + 21 , m 2 ; j + 21 ± , β ||l, m 1 , m 2 ; j ± = q j+m 1 ||l + 21 , m 1 − 21 , m 2 ; j + 21 ± , A ||l, m 1 , m 2 ; j ± = q l− j+m 2 − ||l, m 1 , m 2 ; j ± , B ||l, m 1 , m 2 ; j ± = 1 − q 2(l− j+m 2 +2− ) ||l + 1, m 1 , m 2 + 1; j ± , 1
where, as before, := 21 (−1)l+ 2 − j−m 2 . Composition of such a representation with the algebra embedding A(Sq4 ) → A(SUq (2)) ⊗ A(Sq2 ) given in Eq. (4.2) results in a ˆ The sandwich π˜ (a) := Pπ(a)Q defines a ∗-representation π : A(Sq4 ) → B(H). 4 ∗-linear map π˜ : A(Sq ) → B(H). Proposition 8.3. With I the class of operators defined in Eq. (8.1), one has that the difference a − π˜ (a) ∈ I for all a ∈ A(Sq4 ). Proof. Define Iˆ as the collection of bounded operators T : Hˆ → H such that |D|− p T is trace class for all p > 2. Since trace class operators are a two sided ideal in bounded operators, the space Iˆ is stable when multiplied from the right by bounded operators: ˆ ⇒ T1 T2 ∈ I. ˆ T1 ∈ Iˆ and T2 ∈ B(H) Next, suppose that a, b satisfy a − π˜ (a) ∈ I and b − π˜ (b) ∈ I and consider the following algebraic identity: ab − π˜ (ab) = a {b − π˜ (b)} + {a P − Pπ(a)} π(b)Q. Since I is a two-sided ideal in OP0 , the first summand is in I. The stability of Iˆ discussed ˆ but if T ∈ Iˆ clearly T Q ∈ I. Hence the above implies that {a P − Pπ(a)} π(b) ∈ I, second summand in I too. Thus, ab − π˜ (ab) ∈ I whenever this property holds for each of the operators a, b. We conclude that it is enough to show that a − π˜ (a) ∈ I when a is a generator of A(Sq4 ). By Proposition 8.2, this amounts to prove that the matrix elements 1
of a − π˜ (a) are bounded in modulus by q j+ 2 . Let us have a close look at the coefficients of a ∈ {xi , xi∗ } in Theorem 6.5. Firstly, + −2 j A0 −2 j B 0 A j,m 1 , B +j,m 1 , B − j,m 1 , q j,m 1 and q j,m 1 are uniformly bounded by a constant, as one can see by writing explicitly the q-analogues in their expressions, getting: + j+m 1 4 j+4 −1 A j,m 1 = q (1 − q ) (1 − q 2( j+m 1 +1) )(1 − q 2( j−m 1 +1) ), q −2 j A0j,m 1 = (1 − q 2 )(1 − q 4 j )−1 (1 − q 4 j+4 )−1 ([2]q 2( j+m 1 ) − q 4 j+1 − q −1 ), B +j,m 1 = (1 − q 4 j+4 )−1 (1 − q 2( j+m 1 +1) )(1 − q 2( j+m 1 +2) ), q −2 j B 0j,m 1 = (1 + q 2 )q j+m 1 +1 (1 − q 4 j )−1 (1 − q 4 j+4 )−1 × (1 − q 2( j−m 1 ) )(1 − q 2( j+m 1 +1) ), 2( j+m 1 )+1 4 j −1 B− = −q (1 − q ) (1 − q 2( j−m 1 ) )(1 − q 2( j−m 1 −1) ). j,m 1
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Analogously, the coefficients q 2 j Hl,0 j,m 2 , Cl,0 j,m 2 and Dl,0 j,m 2 are seen to be bounded by q l . Thus, modulo rapid decay matrices (i.e. smoothing operators), x0 |l, m 1 , m 2 ; j A+j,m 1 Cl,+ j,m 2 |l + 1, m 1 , m 2 ; j + 1 + A+j,m 1 Cl,−j,m 2 |l − 1, m 1 , m 2 ; j + 1 + A0j,m 1 Hl,+ j,m 2 |l + 1, m 1 , m 2 ; j + + A0j,m 1 Hl−1, j,m 2 |l − 1, m 1 , m 2 ; j − + A+j−1,m 1 Cl+1, j−1,m 2 |l + 1, m 1 , m 2 ; j − 1 + + A+j−1,m 1 Cl−1, j−1,m 2 |l − 1, m 1 , m 2 ; j − 1 ,
(8.2a)
x1 |l, m 1 , m 2 ; j B +j,m 1 Cl,+ j,m 2 |l + 1, m 1 + 1, m 2 ; j + 1 + B +j,m 1 Cl,−j,m 2 |l − 1, m 1 + 1, m 2 ; j + 1 + B 0j,m 1 Hl,+ j,m 2 |l + 1, m 1 + 1, m 2 ; j + + B 0j,m 1 Hl−1, j,m 2 |l − 1, m 1 + 1, m 2 ; j − − + B j,m 1 Cl+1, j−1,m 2 |l + 1, m 1 + 1, m 2 ; j − 1 + + B− j,m 1 Cl−1, j−1,m 2 |l − 1, m 1 + 1, m 2 ; j − 1 ,
x2 |l, m 1 , m 2 ; j Dl,+ j,m 2 |l + 1, m 1 , m 2 + 1; j + Dl,−j,m 2 |l − 1, m 1 , m 2 + 1; j .
(8.2b) (8.2c)
Since modulo smoothing operators the representations are the same we are omitting the label ‘±’ in the vector basis. Furthermore, using the inequalities 0 ≤ (1 − qu)−1 − 1 ≤ q(1 − q)−1 u,
1
0 ≤ 1 − (1 − u) 2 ≤ u,
(8.3)
which are valid when 0 ≤ u ≤ 1, we prove that modulo terms bounded by q l , one has Cl,+ j,m 2 −q l− j+m 2 − 1 − q 2(l+ j+m 2 +3+ ) , (8.4a) Cl,−j,m 2 −q l+ j+m 2 +1+ 1 − q 2(l− j+m 2 − ) , (8.4b) Hl,+ j,m 2 q l+m 2 +1 q 2 (2 j+1) − q 2(l+m 2 +2) , (8.4c) Dl,+ j,m 2 1 − q 2(l+ j+m 2 +3+ ) 1 − q 2(l− j+m 2 +2− ) , (8.4d) Dl,−j,m 2 −q 2(l+m 2 )+3 .
(8.4e)
Up to now, we neglected only smoothing contributions (the above approximation will be needed when dealing with the real structure later on). We use again (8.3) to get a rougher approximation by neglecting terms bounded by q j . This yields A+j,m 1 A˜ +j,m 1 := q j+m 1 1 − q 2( j+m 1 +1) , (8.5a) A0j,m 1 Hl,+ j,m 2 0, B +j,m 1 B˜ +j,m 1 :=
(8.5b) (1 − q 2( j+m 1 +1) )(1 − q 2( j+m 1 +2) ),
(8.5c)
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B 0j,m 1 Hl,+ j,m 2 0, B− j,m 1 + Cl, j,m 2 Cl,−j,m 2
B˜ − j,m 1
(8.5d) := −q
C˜ l, j,m 2 := −q 0,
Dl,+ j,m 2 D˜ l, j,m 2 := Dl,−j,m 2
2( j+m 1 )+1
,
l− j+m 2 −
(8.5e) ,
(8.5f) (8.5g)
1 − q 2(l− j+m 2 +2− ) ,
0.
(8.5h) (8.5i)
Plugging these coefficients in the equations for the xi ’s we see that, modulo operators in the ideal I, we get x0 |l, m 1 , m 2 ; j A˜ +j,m 1 C˜ l, j,m 2 |l + 1, m 1 , m 2 ; j + 1 + A˜ +j−1,m 1 C˜ l−1, j−1,m 2 |l − 1, m 1 , m 2 ; j − 1
= −P(αβ A)Q |l, m 1 , m 2 ; j − P(β ∗ α ∗ A)Q |l, m 1 , m 2 ; j , x1 |l, m 1 , m 2 ; j B˜ +j,m 1 C˜ l, j,m 2 |l + 1, m 1 + 1, m 2 ; j + 1 ˜ + B˜ − j,m 1 Cl−1, j−1,m 2 |l − 1, m 1 + 1, m 2 ; j − 1 = −P(α 2 A)Q |l, m 1 , m 2 ; j + P(q(β ∗ )2 A)Q |l, m 1 , m 2 ; j , x2 |l, m 1 , m 2 ; j D˜ l, j,m 2 |l + 1, m 1 , m 2 + 1; j = P B Q |l + 1, m 1 , m 2 + 1; j . The observation that −P(αβ + β ∗ α ∗ )AQ = π˜ (x0 ),
P(−α 2 + q(β ∗ )2 )AQ = π˜ (x1 ),
P B Q = π˜ (x2 ),
concludes the proof. 8.2. The dimension spectrum and the top residue. The approximation modulo I allows considerable simplifications when getting information on the part of the dimension spectrum contained in the half plane Re s > 2. To study the part of the dimension spectrum in the left half plane Re s ≤ 2 would require a less drastic approximation which we are lacking at the moment. Proposition 8.4. In the region Re s > 2 the dimension spectrum of the spectral triple (A(Sq4 ), H, D, γ ) given in Proposition 7.2 consists of the two points {3, 4}, which are simple poles of the zeta-functions. The top residue coincides with the integral on the subspace of classical points of Sq4 , that is 2π 2 −4 σ (a)(θ )dθ, − a|D| = 3π 0
(8.6)
with σ : A(Sq4 ) → A(S 1 ) the ∗-algebra morphism defined by σ (x0 ) = σ (x1 ) = 0 and σ (x2 ) = u, where u, given by u(θ ) := eiθ , is the unitary generator of A(S 1 ).
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Proof. Let 0 be the ∗-algebra generated by A(Sq4 ), by [D, A(Sq4 )] and by iterated applications of the derivation δ. Let A ⊂ A(SUq (2)) ⊗ A(Sq2 ) ⊗ Mat 2 (C) be the ∗-algebra generated by α, β, α ∗ , β ∗ , A, B, B ∗ and F. By Proposition 8.3 there is an inclusion A(Sq4 ) ⊂ PAQ + I. A linear basis for A is given by, T := α k1 β n 1 (β ∗ )n 2 An 3 B k2 F h , α k1
(8.7) (α ∗ )−k1
:= if k1 < 0 and where h ∈ {0, 1}, n i ∈ N, ki ∈ Z and with the notation B k2 := (B ∗ )−k2 if k2 < 0. For this operator, and [D, P T Q] = δ(P T Q)F. δ(P T Q) = 21 (k1 + n 1 − n 2 ) + k2 P T Q Thus, PAQ is invariant under application of δ and [D, .], and hence 0 ⊂ PAQ + I. For the part of the dimension spectrum in the right half plane Re s > 2, we can neglect I and consider only the singularities of zeta-functions associated with elements in PAQ. By linearity of the zeta-functions, it is enough to consider the generic basis element in Eq. (8.7). Such a T shifts l by 21 (k1 + n 1 − n 2 ) + k2 , m 1 by 21 (k1 − n 1 + n 2 ), m 2 by k2 , j by 1 2 (k1 + n 1 − n 2 ), and flips the chirality if h = 1. Thus it is off-diagonal unless h = ki = 0 and n 1 = n 2 . The zeta-function associated with a bounded off-diagonal operator is identically zero in the half-plane Re z > 4, and so is its holomorphic extension to the entire complex plane. It remains to consider the cases T = P(ββ ∗ )k An Q, with n, k ∈ N. If n and k are both different from zero, one finds ζT (s) = 2 (l + 23 )−s q n(l− j+m 2 − )+2k( j+m 1 ) l, j,m 1 ,m 2
=2
(l + 23 )−s q n(l− j+m 2 − )
l, j,m 2
1 − q 2k(2 j+1) . 1 − q 2k
For fixed, set 2i := l − − j + m 2 = ∈ {0, 2, . . . , 2(l − j)}. Then, l− j 2k(2 j+1) 3 −s 1 − q ζT (s) = 2 (l + 2 ) q 2ni 1 − q 2k l, j
=±1/2 i=0
1 − q 2k(2 j+1) 1 − q 2n(l− j+1) =4 (l + 23 )−s 1 − q 2k 1 − q 2n l, j
1 + (1 − q 4k )−1 + (1 − q 2n )−1 ζ (s) (1 − q 2k )(1 − q 2n ) + holomorphic function,
= 4ζ (s − 1) − 4
which has meromorphic extension on C with simple pole in s = {1, 2}. If n = 0 and k = 0, 1 − q 2k(2 j+1) ζT (s) = 4 (l + 23 )−s (l − j + 1) 1 − q 2k l, j q 4k 4 1 1 1 ζ (s − 1) + = 1−q ζ (s − 2) − + ζ (s) 2k 4k 4k 2 4k 2 2 1−q (1−q ) log q + holomorphic function,
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which has meromorphic extention on C with simple pole in s = {1, 2, 3}. If n = 0 and k = 0, 1 − q 2n(l− j+1) (l + 23 )−s (2 j + 1) 1 − q 2n l, j 2q 2n ζ (s − 1) + = 1−q4 2n ζ (s − 2) − 1 + 1−q 2n
ζT (s) = 4
2q 2n 1−q 2n
1+
q 2n (1−q 2n ) log q 2n
ζ (s)
+ holomorphic function, which has meromorphic extention on C with simple pole in s = {1, 2, 3}. Finally, if both n and k are zero we get (cf. Eq. (7.4)), ζT (s) =
4 3
{ζ (s − 3) − ζ (s − 1)} ,
and this is meromorphic with simple poles in {2, 4}. Thus, the part of the dimension spectrum in the region Re s > 2 consists at most of the two points {3, 4} and both are simple poles. Since we have considered the enlarged algebra PAQ + I, it suffices to prove that there exists an a ∈ 0 whose zeta-function is singular in both points s = 3 and s = 4. We take a = x2 x2∗ . From the definition π˜ (x2 x2∗ ) |l, m 1 , m 2 ; j ± = (1 − q 2(l− − j+m 2 ) ) |l, m 1 , m 2 ; j ± . Then, modulo functions that are holomorphic when Re s > 2, we have (l + 23 )−s q 2(l− − j+m 2 ) ζx2 x2∗ (s) ∼ ζπ(x ˜ 2 x2∗ ) (s) = ζ1 (s) − 2 l, j,m 1 ,m 2
∼
4 3
ζ (s − 3) −
ζ (s − 2).
4 1−q 4
This proves the first part of the proposition, that is ∩ {Re s > 2} = {3, 4}. The proof of Eq. (8.6) is based on the observation that the residue in s = 4 of ζT , for T a basis element of PAQ, is zero unless T = 1. That is, it depends only on the image of T under the map sending β, A and F to 0 while α → eiφ and B → eiθ . Composing this map with π˜ we get the morphism σ : A(Sq4 ) → A(S 1 ) of the proposition and that 2π − a|D|−4 ∝ σ (a)dθ. # The equality − |D|−4 =
0 4 3
fixes the proportionality constant.
9. Reality and First Order Conditions Classically, if (A(M), H, D, γ ) is the canonical spectral triple associated with a 4-dimensional spin manifold M, there exists an antilinear isometry J on H, named the real structure, satisfying the following compatibility condition: J 2 = −1,
J γ = γ J,
J D = D J.
(9.1)
There are also two additional conditions involving the coordinate algebra A(M): [a, J b J −1 ] = 0,
[[D, a], J b J −1 ] = 0,
∀ a, b ∈ A(M).
(9.2)
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The real structure on S 4 is equivariant and equivariance is sufficient to determine J . In the deformed situation one has to be careful on how to implement equivariance. Let us start with the working hypothesis that equivariance for J is the requirement that it satisfies J h = S(h)∗ J for all h ∈ Uq (so(5)). Then, consider the Casimir operator C1 given in Eq. (3.4). This operator commutes with J since S(C1 )∗ = C1 and from its expression, C1 |l, m 1 , m 2 ; j = (q 2 j+1 + q −2 j−1 ) |l, m 1 , m 2 ; j , we conclude that J leaves the index j invariant. Compatibility with γ and D in Eq. (9.1) and equivariance with respect to h = K 1 and h = K 2 yields J |l, m 1 , m 2 ; j ± = c± (l, m 1 , m 2 ; j) |l, −m 1 , −m 2 ; j ± , with some constants c± to be determined. Equivariance with respect to h = E 1 implies c± (l, m 1 , m 2 ; j) = (−1)m 1 +1/2 q m 1 c± (l, m 2 ; j). For h = E 2 , looking at the piece diagonal in j we deduce that the dependence on m 2 is through a factor q 3m 2 ; and looking at the piece shifting j by ±1 we conclude that c± (l, m 1 , m 2 ; j) = (−1) j+m 1 q m 1 +3m 2 c± (l). Such an operator J cannot be antiunitary unless q = 1. At q = 1 the antiunitarity condition requires that c± (l) ∈ U (1) and modulo a unitary equivalence we can choose c± (l) = i 2l+1 . In conclusion for q = 1 the operator J |l, m 1 , m 2 ; j ± = i 2l+1 (−1) j+m 1 |l, −m 1 , −m 2 ; j ± ,
(9.3)
is the real structure on S 4 (modulo a unitary equivalence). For q = 1 we keep (9.3) as the real structure and notice that conditions (9.1) are satisfied, but J no longer satisfies the requirement J h = S(h)∗ J for all h ∈ Uq (so(5)). Nevertheless, J is the antiunitary part of an antilinear operator T that has this property. The antilinear operator T defined by T |l, m 1 , m 2 ; j ± = i 2l+1 (−1) j+m 1 q m 1 +3m 2 |l, −m 1 , −m 2 ; j ± , has J in (9.3) as the antiunitary part and it is equivariant, i.e. such that T h = S(h)∗ T for all h ∈ Uq (so(5)). Next, we turn to the conditions (9.2). In parallel with the cases of the manifold of SUq (2) in [11] and of Podle´s spheres in [10,9], once again we need to modify them. For instance, the commutator [x2 , J x2 J ] is not zero, as one can see by computing the matrix element f (l, j, m 2 ) := ± l + 1, m 1 , m 2 ; j [x2 , J x2 J ]l, m 1 , m 2 ; j ± 0 + + 0 = Dl+1, j,m 2 −1 Dl, j,−m 2 − Dl, j,m 2 −1 Dl, j,−m 2 0 + + 0 + Dl+1, j,−m 2 −1 Dl, j,m 2 − Dl, j,−m 2 −1 Dl, j,m 2 ,
which for j = f (l,
1 2 , l)
1 2
= −q
and m 2 = l is −l−4
√ (q l−1 + q −l+1 ) [2l + 3] [l + 1][l + 2][l + 3] (1 − q ) [2] = 0. [2l + 2][2l + 4]2 [2l + 6] 2 2
It is relatively easy to prove that the two conditions are satisfied modulo the ideal I. It is much more cumbersome computationally to show that they are in fact satisfied modulo the smaller ideal of smoothing operators.
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Proposition 9.1. Let J be the antilinear isometry given by (9.3). Then, [a, J b J ] ∈ I, [[D, a], J b J ] ∈ I,
∀ a, b ∈ A(Sq4 ).
Proof. We lift J and D to the Hilbert space Hˆ defined in Sect. 8.1, as follows: Jˆ ||l, m 1 , m 2 ; j ± = i 2l+1 (−1) j+m 1 ||l, −m 1 , −m 2 ; j ± , Dˆ ||l, m 1 , m 2 ; j ± = (l + 23 ) ||l, m 1 , m 2 ; j ∓ . Notice that Jˆ2 = −1 on Hˆ (thanks to the phase i 2l+1 that is irrelevant when restricted to H). Let now {α, β, α ∗ , β ∗ , A, B, B ∗ } be the operators defined in Sect. 8.1, generators of the algebra A(SUq (2)) ⊗ A(Sq2 ). Due to Proposition 8.3 it is enough to prove that ˆ a], Jˆb Jˆ] are for all pairs (a, b) of such generators, the commutators [a, Jˆb Jˆ] and [[ D, weighted shifts with weight which are bounded by q 2 j . From ˆ α] = [ D,
1 2
ˆ [ D, ˆ β] = α F,
1 2
ˆ [ D, ˆ A] = 0, [ D, ˆ B] = B F, ˆ β F,
ˆ a], Jˆb Jˆ] follows from the same condition on [a, Jˆb Jˆ], and we the condition on [[ D, have to compute only the latter commutators. Since [a, Jˆb∗ Jˆ] = −[a ∗ , Jˆb Jˆ]∗ and [b, Jˆa Jˆ] = Jˆ[a, Jˆb Jˆ] Jˆ, we have to check the 16 combinations in the following table. b\a α β B A
α •
α∗ •
β × •
β∗ × ×
A • • • •
B • • •
B∗ • • •
By direct computations one shows that bullets in the table correspond to vanishing commutators. On the other hand, the commutators corresponding to the crosses in the table are given, on the subspace with j − |m 1 | ∈ N, by $ [β ∗ , Jˆα Jˆ] ||l, m 1 , m 2 ; j ± = q j+m 1 1 − q 2( j−m 1 +1) − 1 − q 2( j−m 1 ) × ||l, m 1 , m 2 ; j ± [β, Jˆα Jˆ] ||l, m 1 , m 2 ; j ± = −[β ∗ , Jˆα Jˆ] ||l + 1, m 1 − 1, m 2 ; j + 1 ± , [β ∗ , Jˆβ Jˆ] ||l, m 1 , m 2 ; j ± = −[2]q 2 j ||l, m 1 + 1, m 2 ; j ± . √ Since 1 − u ≤ 1 − u ≤ 1 for all u ∈ [0, 1], we have that $ 0 ≤ q j+m 1 1 − q 2( j−m 1 +1) − 1 − q 2( j−m 1 ) ≤ q j+m 1 (1 − 1 + q 2( j−m 1 ) ) ≤ q 2 j . Then, all three non-zero commutators are weighted shifts with weights bounded by q 2 j . Proposition 9.2. Let J be the antilinear isometry given by (9.3). Then, [a, J b J ] ∈ OP−∞ , [[D, a], J b J ] ∈ OP−∞ ,
∀ a, b ∈ A(Sq4 ).
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Proof. By Leibniz rule, it is sufficient to prove the statement when a and b are generators of the algebra. By (7.3), [D, a] − δ(a)F is a smoothing operator. Thus, it is enough to show that [a, J b J ] ∈ OP−∞ ,
[δ(a), J b J ] ∈ OP−∞ ,
(9.4)
for any pair (a, b) of generators. From [b, J a J ] = J [a, J b J ]J,
[δ(b), J a J ] = −J [δ(a), J b J ]J + δ([b, J a J ]),
it follows that if (9.4) is satisfied for a particular pair (a, b), then it is satisfied for (b, a) too. From [a ∗ , J b∗ J ] = −[a, J b J ]∗ ,
[δ(a ∗ ), b∗ ] = [δ(a), b]∗ ,
(9.5)
we see that if (9.4) is satisfied for a pair (a, b), then it is satisfied for (a ∗ , b∗ ) too. With these symmetries we need to check only the following 9 cases out of 25: b\a x0 x1 x2
x0 •
x1 • •
x1∗ •
x2 • • •
x2∗ • •
From Eqs. (8.2) and (8.4) we see that modulo smoothing operators 1 2
{x0 + δ(x0 )} |l, m 1 , m 2 ; j A+j,m 1 Cˆ l,+ j,m 2 |l + 1, m 1 , m 2 ; j + 1 + q −2 j A0j,m 1 Hˆ l,+ j,m 2 |l + 1, m 1 , m 2 ; j − + A+j−1,m 1 Cˆ l+1, j−1,m 2 |l + 1, m 1 , m 2 ; j − 1 ,
1 2
{x0 − δ(x0 )} |l, m 1 , m 2 ; j A+j,m 1 Cˆ l,−j,m 2 |l − 1, m 1 , m 2 ; j + 1 + + q −2 j A0j,m 1 Hˆ l−1, j,m 2 |l − 1, m 1 , m 2 ; j + + A+j−1,m 1 Cˆ l−1, j−1,m 2 |l − 1, m 1 , m 2 ; j − 1 ,
1 2
{x1 + δ(x1 )} |l, m 1 , m 2 ; j B +j,m 1 Cˆ l,+ j,m 2 |l + 1, m 1 + 1, m 2 ; j + 1 + q −2 j B 0j,m 1 Hˆ l,+ j,m 2 |l + 1, m 1 + 1, m 2 ; j ˆ− + B− j,m 1 Cl+1, j−1,m 2 |l + 1, m 1 + 1, m 2 ; j − 1 ,
1 2
{x1 − δ(x1 )} |l, m 1 , m 2 ; j B +j,m 1 Cˆ l,−j,m 2 |l − 1, m 1 + 1, m 2 ; j + 1 + + q −2 j B 0j,m 1 Hˆ l−1, j,m 2 |l − 1, m 1 + 1, m 2 ; j
ˆ+ + B− j,m 1 Cl−1, j−1,m 2 |l − 1, m 1 + 1, m 2 ; j − 1 , Dˆ l,+ j,m 2 |l + 1, m 1 , m 2 + 1; j ,
1 2
{x2 + δ(x2 )} |l, m 1 , m 2 ; j
1 2
{x2 − δ(x2 )} |l, m 1 , m 2 ; j Dˆ l,−j,m 2 |l − 1, m 1 , m 2 + 1; j ,
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where
Cˆ l,+ j,m 2 = −q l− j+m 2 − 1 − q 2(l+ j+m 2 +3+ ) , Cˆ l,−j,m 2 = −q l+ j+m 2 +1+ 1 − q 2(l− j+m 2 − ) , Hˆ l,+ j,m 2 = q 2 j q l+m 2 +1 q 2 (2 j+1) − q 2(l+m 2 +2) , Dˆ l,+ j,m 2 = 1 − q 2(l+ j+m 2 +3+ ) 1 − q 2(l− j+m 2 +2− ) ,
(9.6d)
Dˆ l,−j,m 2
(9.6e)
= −q 2(l+m 2 )+3 .
(9.6a) (9.6b) (9.6c)
We have divided the terms in three classes, which need to be analysed separately. All terms T which are not ‘boxed’ have coefficients which are uniformly bounded by q l+m 2 ; since the conjugation with J changes the sign of the labels m 1 , m 2 , for such T ’s, the coefficients of J T J are uniformly bounded by q l−m 2 . They give products (and so commutators) with coefficients bounded by q l+m 2 q l−m 2 = q 2l , and so (these products) are smoothing operators. Analogously, the coefficients of single-boxed terms are bounded by q l− j+m 2 , and become smoothing when multiplied by the J -conjugated of non-boxed terms (as q l− j+m 2 q l−m 2 ≤ q l ), and viceversa for the product of a non-boxed term with the J -conjugated of a single-boxed one (q l+m 2 q l− j−m 2 ≤ q l ). Next we consider pairs of single-boxed terms. A closer look at the single-boxed terms in x0 ± δ(x0 ) and x1 − δ(x1 ) (and then x1∗ + δ(x1∗ )) shows that they have coefficients bounded by q l+m 1 +m 2 , and become smoothing when multiplied by the J -conjugated of one of them (q l+m 1 +m 2 q l−m 1 −m 2 = q 2l ). Last single-boxed term is the one in x1 + δ(x1 ) (and x1∗ − δ(x1∗ )). The relevant terms for the commutators involving them are + δ(x1 ), J x0 J ] |l, m 1 , m 2 ; j + + ˆ+ A+j+1,−m 1 −1 Cˆ l+1, j+1,−m 2 B j,m 1 Cl, j,m 2 + −A+j,−m 1 Cˆ l,+ j,−m 2 B +j+1,m 1 Cˆ l+1, j+1,m 2 |l + 2, m 1 + 1, m 2 ; j + 2 + A+j,−m 1 −1 Cˆ l,+ j,−m 2 B +j,m 1 Cˆ l,+ j,m 2 + + ˆ+ −A+j−1,−m 1 Cˆ l−1, j−1,−m 2 B j−1,m 1 Cl−1, j−1,m 2 |l, m 1 + 1, m 2 ; j ,
1 2 [x 1
+ δ(x1 ), J x1 J ] |l, m 1 , m 2 ; j + + ˆ+ B +j+1,m 1 −1 Cˆ l+1, j+1,m 2 B j,−m 1 Cl, j,−m 2
1 2 [x 1
+ + + ˆ −B +j+1,−m 1 −1 Cˆ l+1, B C j+1,−m 2 j,m 1 l, j,m 2 |l + 2, m 1 , m 2 ; j + 2 − + ˆ+ + B +j−1,m 1 −1 Cˆ l−1, j−1,m 2 B j,−m 1 Cl−1, j−1,−m 2 ˆ l,+ j,−m B +j,m Cˆ l,+ j,m |l, m 1 , m 2 ; j , −B − C j+1,−m 1 −1 2 1 2
− δ(x1 ), J x1 J ] |l, m 1 , m 2 ; j + ˆ+ ˆ+ B− j+1,m 1 −1 Cl, j,m 2 B j,−m 1 Cl, j,−m 2
1 2 [x 1
+ + + ˆ l−1, ˆ −B − B C C j−1,m 2 j−1,−m 1 −1 l−1, j−1,−m 2 |l, m 1 , m 2 ; j , j,m 1
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1 ∗ 2 [x 1
+ δ(x1∗ ), J x1 J ] |l, m 1 , m 2 ; j + ˆ+ ˆ+ B− j+2,m 1 −2 Cl+1, j+1,m 2 B j,−m 1 Cl, j,−m 2
− + ˆ+ −B +j+1,−m 1 +1 Cˆ l+1, j+1,−m 2 B j+1,m 1 −1 Cl, j,m 2 |l + 2, m 1 − 2, m 2 ; j + 2 ,
1 ∗ 2 [x 1
− δ(x1∗ ), J x1 J ] |l, m 1 , m 2 ; j B +j,m 1 −2 Cˆ l,+ j,m 2 B +j,−m 1 Cˆ l,+ j,−m 2
+ + + ˆ −B +j−1,−m 1 +1 Cˆ l−1, B C j−1,−m 2 j−1,m 1 −1 l−1, j−1,m 2 |l, m 1 − 2, m 2 ; j − + ˆ+ + B +j−2,m 1 −2 Cˆ l−2, j−2,m 2 B j,−m 1 Cl−1, j−1,−m 2 + + + ˆ l−2, ˆ −B − B C C j−2,−m 2 j−1,m 1 −1 l−1, j−1,m 2 |l − 2, m 1 − 2, m 2 ; j −2 . j−1,−m 1 +1
+ We need to estimate products of the form Cˆ l,+ j,−m 2 Cˆ l+i, j+i,m 2 , for which, modulo smoothing operators, we find + 2(l− j) − q 2l q 2(l−m 2 +3− ) q 2(l− j) − q 2l q 2(l+m 2 +3+2i+ ) Cˆ l,+ j,−m 2 Cˆ l+i, q j+i,m 2 2(l− j) q q 2(l− j) = q 2(l− j) .
Using this we get + δ(x1 ), J x0 J ] |l, m 1 , m 2 ; j q 2(l− j) A+j+1,−m 1 −1 B +j,m 1 − A+j,−m 1 B +j+1,m 1 |l + 2, m 1 + 1, m 2 ; j + 2 + q 2(l− j) A+j,−m 1 −1 B +j,m 1 − A+j−1,−m 1 B +j−1,m 1 |l, m 1 + 1, m 2 ; j ,
1 2 [x 1
+ δ(x1 ), J x1 J ] |l, m 1 , m 2 ; j q 2(l− j) B +j+1,m 1 −1 B +j,−m 1 − B +j+1,−m 1 −1 B +j,m 1 |l + 2, m 1 , m 2 ; j + 2 − + − B B + q 2(l− j) B +j−1,m 1 −1 B − j,−m 1 j+1,−m 1 −1 j,m 1 |l, m 1 , m 2 ; j ,
1 2 [x 1
− δ(x1 ), J x1 J ] |l, m 1 , m 2 ; j − + + B − B B q 2(l− j) B − j+1,m 1 −1 j,−m 1 j,m 1 j−1,−m 1 −1 |l, m 1 , m 2 ; j ,
1 2 [x 1
1 ∗ 2 [x 1
+ δ(x1∗ ), J x1 J ] |l, m 1 , m 2 ; j − + + q 2(l− j) B − j+2,m 1 −2 B j,−m 1 − B j+1,−m 1 +1 B j+1,m 1 −1 |l + 2, m 1 − 2, m 2 ; j + 2 ,
1 ∗ 2 [x 1
− δ(x1∗ ), J x1 J ] |l, m 1 , m 2 ; j q 2(l− j) B +j,m 1 −2 B +j,−m 1 − B +j−1,−m 1 +1 B +j−1,m 1 −1 |l, m 1 − 2, m 2 ; j − + − B B + q 2(l− j) B +j−2,m 1 −2 B − j,−m 1 j−1,−m 1 +1 j−1,m 1 −1 |l − 2, m 1 −2, m 2 ; j −2 .
To prove that these commutators are smoothing we still need to check that the terms in braces are bounded by q j (since q 2(l− j) q j ≤ q l is of rapid decay). This is done by using
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Eqs. (8.5). For example the first two braces are identically zero, while the third one is B +j+1,m 1 −1 B +j,−m 1 − B +j+1,−m 1 −1 B +j,m 1 = B˜ +j+1,m 1 −1 B˜ +j,−m 1 − B˜ +j+1,−m 1 −1 B˜ +j,m 1 + O(q j ) = 0 + O(q j ). What remains to control are the commutators [x2 +δ(x2 ), J b J ] for b = x0 , x1 , x2 and the commutators [x2∗ − δ(x2∗ ), J b J ] for b = x1 , x2 (which involve the ‘doubly-boxed’ term). The operators x2 and δ(x2 ) do not shift m 1 , j and have coefficients independent on m 1 . Thus, any operator acting only on the label m 1 and with coefficients depending only on m 1 , j, commutes with x2 and δ(x2 ) and so can be neglected. In particular, x0 and x1 can be written as sums of products of operators of this kind (commuting with x2 and δ(x2 )) by operators yi ’s, y1 |l, m 1 , m 2 ; j := Cˆ l,+ j,m 2 |l + 1, m 1 , m 2 ; j + 1 , y2 |l, m 1 , m 2 ; j := Hˆ l,+ j,m 2 |l + 1, m 1 , m 2 ; j , y3 |l, m 1 , m 2 ; j := Cˆ l,−j,m 2 |l − 1, m 1 , m 2 ; j + 1 , and their adjoints. To prove that the commutators [x2 + δ(x2 ), J b J ], for b = x0 , x1 , x2 , and [x2∗ − δ(x2∗ ), J b J ], for b = x1 , x2 , are smoothing, is sufficient to establish the same for b = y1 , y2 , y3 . For these operators we have + δ(x2 ), J y1 J ] |l, m 1 , m 2 ; j + + + + ˆ ˆ ˆ − D D C Cˆ l+1, j,−m 2 −1 l, j,m 2 l+1, j+1,m 2 l, j,−m 2 |l + 2, m 1 , m 2 + 1; j + 1 + = Cˆ l,+ j,−m 2 Dˆ l,+ j,m 2 − Dˆ l+1, j+1,m 2 |l + 2, m 1 , m 2 + 1; j + 1 ,
1 2 [x 2
+ δ(x2 ), J y2 J ] |l, m 1 , m 2 ; j + + + + ˆ ˆ ˆ − D D H Hˆ l+1, j,−m 2 −1 l, j,m 2 l+1, j,m 2 l, j,−m 2 |l + 2, m 1 , m 2 + 1; j + |l + 2, m 1 , m 2 + 1; j , = Hˆ l,+ j,−m Dˆ l,+ j,m − Dˆ l+1, j,m
1 2 [x 2
2
2
2
+ δ(x2 ), J y3 J ] |l, m 1 , m 2 ; j − ˆ+ ˆ+ ˆ− Cˆ l+1, j,−m 2 −1 Dl, j,m 2 − Dl−1, j+1,m 2 Cl, j,−m 2 |l, m 1 , m 2 + 1; j + 1 + = Cˆ l,−j,−m 2 Dˆ l,+ j,m 2 − Dˆ l−1, j+1,m 2 |l, m 1 , m 2 + 1; j + 1 ,
1 2 [x 2
1 ∗ 2 [x 2
− δ(x2∗ ), J y1 J ] |l, m 1 , m 2 ; j + ˆ+ ˆ+ ˆ+ Cˆ l−1, j,−m 2 +1 Dl−1, j,m 2 −1 − Dl, j+1,m 2 −1 Cl, j,−m 2 |l, m 1 , m 2 − 1; j + 1 + + ˆ − D = Cˆ l,+ j,−m 2 Dˆ l−1, j,m 2 −1 l, j+1,m 2 −1 |l, m 1 , m 2 − 1; j + 1 ,
1 ∗ 2 [x 2
− δ(x2∗ ), J y2 J ] |l, m 1 , m 2 ; j + + + + ˆ ˆ ˆ − D D H Hˆ l−1, j,−m 2 +1 l−1, j,m 2 −1 l, j,m 2 −1 l, j,−m 2 |l, m 1 , m 2 − 1; j + ˆ+ = Hˆ l,+ j,−m 2 Dˆ l−1, j,m 2 −1 − Dl, j,m 2 −1 |l, m 1 , m 2 − 1; j ,
1 ∗ 2 [x 2
− δ(x2∗ ), J y3 J ] |l, m 1 , m 2 ; j
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− ˆ+ ˆ+ ˆ− Cˆ l−1, j,−m 2 +1 Dl−1, j,m 2 −1 − Dl−2, j+1,m 2 −1 Cl, j,−m 2 |l − 2, m 1 , m 2 + 1; j + 1 + + ˆ = Cˆ l,−j,−m 2 Dˆ l−1, − D j,m 2 −1 l−2, j+1,m 2 −1 |l − 2, m 1 , m 2 + 1; j + 1 . + l± j+m 2 , and q l± j+m 2 C ˆ+ ˆ± Now Dˆ l±1, l, j,−m 2 is bounded by j+1,m 2 − Dl, j,m 2 is bounded by q q 2l . Next, Hˆ + is bounded by q l+ j−m 2 , and q l+ j−m 2 Dˆ + 1. This proves that l, j,−m 2
l, j,m 2
all previous commutators are smoothing. For the yi∗ ’s the same statement follows from the symmetry (9.5). We have arrived at last two commutators. Modulo smoothing operators, the first one is 1 2 [x 2
+ δ(x2 ), J x2 J ] |l, m 1 , m 2 ; j + + + + ˆ ˆ ˆ − D D D Dˆ l+1, j,−m 2 −1 l, j,m 2 l+1, j,m 2 −1 l, j,−m 2 |l + 2, m 1 , m 2 ; j − ˆ+ ˆ+ ˆ− + Dˆ l+1, j,−m 2 −1 Dl, j,m 2 − Dl−1, j,m 2 −1 Dl, j,−m 2 |l, m 1 , m 2 ; j + = Dˆ l,−j,−m 2 Dˆ l,+ j,m 2 − Dˆ l−1, j,m 2 −1 |l, m 1 , m 2 ; j ,
where the second equality follows from the fact that both Dˆ l,+ j,m 2 and Dˆ l,−j,m 2 in (9.6) depend on l and m 2 only through their sum. For the same reason we have also that 1 ∗ 2 [x 2
− δ(x2∗ ), J x2 J ] |l, m 1 , m 2 ; j + ˆ+ ˆ+ ˆ+ Dˆ l−1, j,−m 2 +1 Dl−1, j,m 2 −1 − Dl, j,m 2 −2 Dl, j,−m 2 |l, m 1 , m 2 − 2; j − − + + ˆ ˆ l−1, ˆ − D D D + Dˆ l−1, j,m 2 −1 l−2, j,m 2 −2 l, j,−m 2 |l − 2, m 1 , m 2 − 2; j j,−m 2 +1 + ˆ+ = Dˆ l,−j,−m 2 Dˆ l−1, j,m 2 −1 − Dl−2, j,m 2 −2 |l − 2, m 1 , m 2 − 2; j .
+ ˆ− The final observation that Dˆ l,−j,−m 2 Dˆ l−i, j,m 2 −i Dl, j,−m 2 , for i = 0, 1, 2, gives that these commutators vanish modulo smoothing operators.
Acknowledgment. We are grateful to the referee whose remarks led to a much improved version of the paper. This work was partially supported by the ‘Italian project Cofin06 – Noncommutative geometry, quantum groups and applications’.
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7. Chakraborty, P.S., Pal, A.: Equivariant spectral triples on the quantum SU (2) group. K-Theory 28, 107–126 (2003) 8. D’Andrea, F., D¸abrowski, L.: Local Index Formula on the Equatorial Podle´s Sphere. Lett. Math. Phys. 75, 235–254 (2006) 9. D¸abrowski, L., D’Andrea, F., Landi, G., Wagner, E.: Dirac operators on all Podle´s spheres. J. Noncommut. Geom. 1, 213–239 (2007) 10. D¸abrowski, L., Landi, G., Paschke, M., Sitarz, A.: The spectral geometry of the equatorial Podles sphere. Comptes Rendus Acad. Sci. Paris, Ser. I 340, 819–822 (2005) 11. D¸abrowski, L., Landi, G., Sitarz, A., van Suijlekom, W., Várilly, J.C.: The Dirac operator on SUq (2). Commun. Math. Phys. 259, 729–759 (2005) 12. D¸abrowski, L., Landi, G., Sitarz, A., van Suijlekom, W., Várilly, J.C.: The local index formula for SUq (2). K-Theory 35, 375–394 (2005) 13. D¸abrowski, L., Sitarz, A.: Dirac operator on the standard Podle´s quantum sphere. In: Noncommutative geometry and quantum groups (Warsaw, 2001), Banach Center Publ. 61, Warsaw: Polish Acad. Sci., 2003, pp. 49–58 14. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry, Basel-Boston: Birkhäuser, 2001 15. Klimyk, A., Schmüdgen, K.: Quantum Groups and their Representations. Berlin-Heidelberg-New York: Springer, 1997 16. Hawkins, E., Landi, G.: Fredholm Modules for orthogonal quantum Spheres. J. Geom. Phys. 49, 272–293 (2004) 17. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) 18. Reshetikhin, N.Yu., Takhtadzhyan, L., Fadeev, L.D.: Quantization of Let groups and Lie algebras. Leningrad Math. J. 1, 193–225 (1990) 19. Sitarz, A.: Equivariant spectral triples. In: Noncommutative Geometry and Quantum Groups, Banach Centre Publications 61 Warszawa: Polisth Acad. Sci., 2003 pp. 231–263 20. Trautman, A.: Spin structures on hypersurfaces and the spectrum of the Dirac operator on spheres. In: “Spinors, twistors, Clifford algebras and quantum deformations”, Dordrecht: Kluwer Acad. Publishers, 1993, pp. 25–29 21. Woronowicz, S.L.: Twisted SU (2) group. An Example of a Non-Commutative Differential Calculus. Publ. Res. Inst. Math. Sci. 23, 117–181 (1987) Communicated by A. Connes
Commun. Math. Phys. 279, 117–146 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0425-5
Communications in
Mathematical Physics
The Effect of Disorder on Polymer Depinning Transitions Kenneth S. Alexander Department of Mathematics KAP108, University of Southern California, Los Angeles, CA 90089-2532, USA. E-mail:
[email protected] Received: 20 December 2006 / Accepted: 29 August 2007 Published online: 8 February 2008 – © Springer-Verlag 2008
Abstract: We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length n is given by n −c ϕ(n) for some 1 < c < 2 and slowly varying ϕ. Disorder is introduced by having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. To see the effect of disorder on the depinning transition, we compare the contact fraction and free energy (as functions of u) to the corresponding annealed system. We show that for c > 3/2, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point—the size of this neighborhood scales as β 1/(2c−3) , where β is the inverse temperature. For c < 3/2, given > 0, for sufficiently high temperature the quenched and annealed curves are within a factor of 1 − for all u near the critical point; in particular the quenched and annealed critical points are equal. For c = 3/2 the regime depends on the slowly varying function ϕ. We consider the following model for a polymer (or other one-dimensional object) in a higher-dimensional space, interacting with a potential located either at a single site or in a one-dimensional subspace. There is an underlying Markov chain {X i , i ≥ 0}, with state space Σ, representing the “free” trajectories of the object in the absence of the potential. We assume the chain is irreducible and aperiodic. There is a unique site in Σ which we call 0 where the potential can be nonzero; we consider trajectories of X length N starting from state 0 at time 0 and denote the corresponding measure P[0,N ]. The potential at 0 at time i has form u + Vi , where the Vi are i.i.d. with mean zero; we refer to {Vi : i ≥ 1} as the disorder. For N and a realization {Vi , i ≥ 1} fixed, we attach Research supported by NSF grant DMS-0405915.
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a Gibbs weight
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N X (u + Vi )δ{xi =0} P[0,N exp β ] (x [0,N ] )
(0.1)
i=1
to each trajectory x[0,N ] = {xi : 0 ≤ i ≤ N }, where β > 0 and u ∈ R. Here δ A denotes the indicator of an event A. This can be viewed as either a directed polymer which we view as existing in Z × Σ with configuration given by the space-time trajectory (0, 0), (1, X 1 ), .., (N , X N ), or an undirected polymer which exists in Σ with configuration given by the trajectory 0, X 1 , .., X N , with i merely an index. The undirected polymer is the appropriate model for a situation in which the potential is at a single site, and the polymer can configure itself in loops to place multiple monomers at this site. The directed polymer is appropriate when the potential is in a one-dimensional subspace, and the polymer cannot make loops. A number of disparate physical phenomena are subsumed in this formulation, which was studied in [1,11] and [13] in the mathematics literature; closely related models have been studied extensively in the physics literature, for example in [6,10,17,18] and [19]. Taking Σ = Zd we obtain a directed quenched random copolymer in d + 1 dimensions interacting, with strength that depends on the random monomer types, with a potential located in the first coordinate axis. Here the realization {Vi }, representing the random sequence of monomer types, is fixed. Alternatively, we may have a uniform polymer but spatial randomness in the potential (as when the “polymer” is actually a flux tube in a superconducting material with one-dimensional defects–see [14,20].) When d = 1 and X i ≥ 0, the state X i may represent the height at location i of an interface above a wall in two dimensions, which attracts or repels the interface with randomly spatially varying strength–see [6,9]. For consistency, here we always view the index i as time. It should be noted that mathematically, the full underlying Markov chain is unnecessary; all we need is the arrival process marking its returns to 0, which has independent interarrival times. The main questions of physical interest are (i) whether for given β, u the polymer is “pinned”, meaning roughly that it places a positive fraction of its monomers at 0 for large n with high probability; (ii) the location and nature of the depinning transition, if any, as we vary β and/or u; and (iii) the effect of the disorder, as seen by comparing the transition to the annealed case (which is effectively the same as Vi ≡ constant.) The partition function and finite-volume Gibbs distribution on length-N trajectories, β,u,{Vi } {Vi } corresponding to (0.1), are denoted Z [0,N ] (β, u) and µ[0,N ] , respectively; the correβ,u,{V }
sponding expectation is denoted ·[0,N ] i . We omit the {Vi } when Vi ≡ 0. We write V V for the distribution of (Va , .., Vb ) and P V for P[0,∞) . (Here and wherever we P[a,b] deal with indices, we take intervals to mean intervals of integers.) The corresponding V V (· | ·). Let expectation and conditional expectation are denoted ·[a,b] and E [a,b] L N = L N ({X i }) =
N
δ{X i =0} .
i=1
For fixed β, u, {Vi } we say the polymer is pinned at (β, u) if for some δ > 0, β,u,{V }
lim µ[0,N ] i (L N > δ N ) = 1. N
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There is a (possibly infinite) critical u c (β, {Vi }) such that the polymer is pinned P V -a.s. for u > u c (β, {Vi }), and not pinned, P V -a.s., for u < u c (β, {Vi }). In [1] it was established that self-averaging holds in the sense that there is a nonrandom quenched q q q critical point u c = u c (β) such that u c (β, {Vi }) = u c (β) with P V -probability one. d d The deterministic critical point u c = u c (β) is the critical point for the deterministic model, which is the case Vi ≡ 0. The annealed model is obtained (provided the moment generating function MV (β) = eβV1 V is finite) by averaging the Gibbs weight (0.1) over the disorder; the annealed model at (β, u) is thus the same as the deterministic model at (β, u + β −1 log MV (β)), and the corresponding annealed critical point is u ac = u ac (β) = u dc (β) − β −1 log MV (β). It is not hard to show that q
u ac ≤ u c ≤ u dc ;
(0.2)
see [1]. For the annealed case, we denote the partition function and finite-volume Gibbs β,u distribution by Z [0,N ] (β, u) and µ[0,N ] , respectively, and for the deterministic case we β,u,0
0 denote them by Z [0,N ] (β, u) and µ[0,N ] , respectively. The free energies for the deterministic, annealed and quenched models are given by
1 1 0 lim log Z [0,N ] (β, u), β N N 1 1 f a (β, u) = lim log Z [0,N ] (β, u) = f d (β, u + β −1 log MV (β)), β N N 1 1 {Vi } f q (β, u) = lim log Z [0,N ] (β, u). β N N
f d (β, u) =
The P V -a.s. existence and the nonrandomness of the last limit are proved in [13], for the cases we consider here. Clearly f d (β, u) depends only on βu, and f a (β, u) depends only on βu + log MV (β). The specific heat exponent αd in the deterministic case is given by log f d (β, u dc (β) + ∆) ; ∆0 log ∆
2 − αd = lim
clearly we get the same annealed exponent αa if we replace d with a in this definition. For the quenched case the same definition applies with d replaced by q, provided the limit exists. A related quantity of interest is the contact fraction, defined in the deterministic model to be the value C = C d (β, u) for which LN β,u ∈ (C − , C + ) = 1; (0.3) lim lim µ[0,N ] 0 N N the existence of such a C is established in [1]. In the annealed case the contact fraction is C a (β, u) = C d (β, u + β −1 log MV (β)). Now f d and f a are convex functions of u, and we have by standard methods that C d (β, u) =
∂ d ∂ a f (β, u), C a (β, u) = f (β, u) ∂u ∂u
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for all non-critical u; the necessary differentiability of f a (β, ·) follows readily from the differentiability and strict convexity established in the proof of Lemma 2.2 of [1] for the function F defined in (1.13) below. In the quenched case, then, we define C q (β, u) =
∂ q f (β, u); ∂u
differentiability of f q (β, ·) is proved in [12] when the underlying Markov chain satisfies (0.4) below. From convexity we have for fixed β that 1 L N β,u,{Vi } 1 ∂ ∂ q {Vi } log Z [0,N ] (β, u) → f (β, u) for all u, = N [0,N ] β ∂u N ∂u P V -a.s. The case most commonly considered in the literature has Σ = Zd and {X i } a symmetric simple random walk. In keeping with our requirement of aperiodicity we modify this by considering the case in which X i is the location of the walk at time 2i; we call this the symmetric simple random walk case in d+1 dimensions. Let τi denote the time of the i th return to 0 by the chain, with τ0 = 0, and let E i = τi − τi−1 denote the i th excursion length. Our interest here is in general Markov chains {X i } which satisfy P X (E 1 = n) = n −c ϕ(n)
(0.4)
for some c ∈ (1, 2) and slowly varying function ϕ on [1, ∞). This includes the symmetric simple random walk case in one and (by virtue of Theorem 5) three dimensions, where c = 3/2 and ϕ(n) ∼ K for some K > 0. We do not, however, include the cases of c = 1 and c ≥ 2, because the technical details, and many of the heuristics, are quite different in these cases, as we will discuss further in Sect. 4. We focus instead on our main purpose which is to understand the role of the tail exponent c in the effect of quenched disorder; as we will see, the main distinction is between c < 3/2 and c > 3/2. Among excluded examples are the symmetric simple random walk case in 2 + 1 dimensions (where c = 1 and ϕ(n) ∼ K (log n)−2 ), and in d + 1 dimensions for d ≥ 4 (where c = d/2 ≥ 2.) For recurrent chains satisfying (0.4) it is easily seen (see [1]) that u dc (β) = 0 for all β > 0, and hence u ac (β) = −β −1 log MV (β). (0.5) Further, again from [1], for the deterministic or annealed model the transition is first order if and only if E 1 X < ∞; in particular it is first order for c > 2 but not for c < 2. It is proved in [13] that (again as known nonrigorously from the physics literature–see e.g. [8]) the annealed specific heat exponent is (2c − 3)/(c − 1). Based on the physics literature ([6,8]), the following is believed for the non-first-order cases: for 3/2 < c < 2 (positive annealed specific heat exponent) the depinning transition is altered by the disorder; (ii) for c < 3/2 (negative annealed specific heat exponent) the depinning transition is not altered by the disorder.
(i)
Here physicists generally take “altered” to mean that the specific heat exponent is different, but a disorder-induced change in the critical point is also of interest ([5,6,15,18]). This produces the question, “change relative to what, u ac or u dc ?” It is most natural to ask whether the factor eβ(u+Vi ) in (0.1) gives the same critical u as if it were replaced by its q mean eβu eβVi V , which is equivalent to asking whether u c (β) = u ac (β). The question is
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of interest in part because it is intertwined with questions of just how the polymer depins as the quenched critical point is approached, or, put differently, of what “strategy” the polymer uses to stay pinned when near the critical point–see [5]. For example, do long stretches depin, leaving the polymer attached only where the disorder is exceptionally favorable, or is the depinning more uniform? Some of our results may be interpreted as saying that for 1 < c < 3/2 the depinning is quite uniform in the quenched system all the way to the critical point, and for 3/2 < c < 2 the depinning is quite uniform at least until very close to the critical point, at least for high temperatures. It was proved in [13] that disorder does alter the critical behavior when 3/2 < c < 2, in that the quenched specific heat exponent (assuming it exists) becomes non-positive, i.e. q the free energy increases no more than quadratically as u increases from u c . This brings us to one of the main questions we consider here: just how are the free energy and contact fraction curves (as functions of u) altered by the disorder? When the disorder changes the specific heat exponent from an annealed value αa > 0 to a quenched value αq ≤ 0, does a large section of the contact fraction curve ∆ → C a (β, u ac (β) + ∆) ∼ ∆1−αa change to approximate ∆ → ∆1−αq instead, or does significant change only occur very close to ∆ = 0? We will show that the latter is the case, at least for small β. Regarding the possible difference in critical points between quenched and annealed q systems, since u dc (β) = 0, it is reasonable to ask how close u c (β)/u ac (β) is to 1. As discussed in [1], this is related to the following “sampling” point of view. We may think of the Markov chain as choosing a sample from the realization {V1 , .., Vn } through the times of its returns to 0. This sample is of course not i.i.d., and we expect that the probabilities for large deviations of the average of the sampled Vi ’s will be smaller than the corresponding probabilities for an i.i.d. sample. Roughly, the more this sampling procedure is “efficient” in the sense that certain large-deviation probabilities are not too q different from the i.i.d. case, the closer u c (β)/u ac (β) is to 1. For small β, the size of the relevant large deviations is of order β (see [1].) The analogous question of ratios of quenched to annealed critical points has been considered for a related model, a polymer at a selective interface in 1 + 1 dimensions. Here the horizontal axis represents an interface separating two solvents which differentially attract or repel each monomer, with u + Vi representing roughly the preference of monomer i for the solvent above the interface. The Markov chain {X i } is a symmetric simple random walk on Z, so the excursion tail exponent is c = 3/2. The main mathematical difference from the model here is that the factor δ{xi =0} in (0.1) is replaced by −δ{xi <0} . It has been conjectured in the physics [15] and mathematics ([4,11]) literatures that for this model, q u c (β) < 1. (0.6) lim a β→0 u c (β) The limit being less than 1 says that the above-discussed sampling procedure does not become fully efficient even in the high-temperature limit, where the size of the relevant large deviation approaches 0. For technical convenience and clarity of exposition, we will restrict attention here to Gaussian disorder, but our results are easily extended to other distributions with a finite exponential moment. Our results imply that, in contrast to (0.6), for the model (0.1) under (0.4) with 1 < c < 2, the limit in (0.6) is 1; in fact for c < 3/2 the ratio is equal to 1 at least for sufficiently small β. We do not view this as casting any doubt on the conjecture (0.6) for the selective interface model, however, for the following reason: the factor δ{xi <0} in the Gibbs weight means that the sample selected by the Markov chain from {V1 , .., Vn } consists of blocks V j , V j+1 , .., V j+k of consecutive disorder values,
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corresponding to excursions of {X i } below the axis. This “block” aspect may make the sampling procedure inherently less efficient (for large deviations) in the selective interface model, compared to our model (0.1). From (0.5), in the Gaussian case we have u ac (β) = −β/2 for all β > 0. Associated to a slowly varying function η there is a conjugate slowly varying function η∗ characterized (up to asymptotic equivalence) by the property that η∗ (xη(x)) ∼
1 as x → ∞; η(x)
see [22]. Here “∼” means the ratio converges to 1. For many common slowly varying functions such as η(x) = (log x)γ with γ ∈ R, we have η∗ = 1/η, but this is not the case for some functions which are “barely slowly varying” such as η(x) = x 1/ log log x . Given ϕ as in (0.4) and a > 0, define 1 ∗ −1/a , ϕ ˆ ϕ a (x) = (x) = ϕ (x) , ϕ(x) ˜ = n −1 ϕ(n)−2 , a a ϕ(x 1/a ) n≤x which are all slowly varying. As an example, if ϕ(x) ∼ (log x)γ for some γ ∈ R, then these three functions are of order (log x)−γ , (log x)−γ /a and (log x)1−2γ , respectively, the last being valid only for γ < 1/2. Our first result is for the annealed case, included mainly so that the quenched case can be compared to it. Most of this result appeared in [13]; we have made only minor additions. Throughout the paper, K 1 , K 2 , ... are constants which depend only on the distributions of V1 and {X i }, unless otherwise specified. Theorem 1. Suppose that {Vi , i ≥ 1} are i.i.d. standard Gaussian random variables, and the Markov chain {X i } is recurrent, satisfying (0.4) with 1 < c < 2. Then u ac (β) = −β/2 for all β > 0, and there exist constants K i , depending only on c, ϕ, such that β 1 a 1/(c−1) β, − + ∆ ∼ K 1 (β∆) as β∆ → 0, βf ϕˆ c−1 2 β∆ and
β 1 (2−c)/(c−1) C β, − + ∆ ∼ K 2 (β∆) ϕˆc−1 2 β∆ a
as β∆ → 0.
In particular, the annealed specific heat exponent is (2c − 3)/(c − 1). Note that writing the parameter u as − β2 + ∆ in Theorem 1 makes the annealed free energy and contact fraction functions of β∆ only, as is reflected in the asymptotic approximations given there. The next theorem confirms the prediction that disorder does not alter the critical behavior when c < 3/2, at high temperatures; we can even include certain cases when c = 3/2, though not the symmetric simple random walk case in 1+1 dimensions. Theorem 2. Suppose that {Vi , i ≥ 1} are i.i.d. standard Gaussian random variables, and the Markov chain {X i } is recurrent, satisfying (0.4) with either 1 < c < 3/2, or c = 3/2 and n n −1 ϕ(n)−2 < ∞. Then given > 0, provided β > 0 and β∆ > 0 are sufficiently small we have
C q β, − β + ∆ f q β, − β2 + ∆ 2
≤ 1,
− 1 ≤ , (0.7) 1− ≤ C a β, − β2 + ∆ f a β, − β2 + ∆
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q
so that in particular u c (β) = u ac (β) = −β/2. As we will see, when c = 3/2, ϕ(n) ˜ is proportional to the mean overlap under P X between {X i } and an independent copy {Yi } of the chain over length n, that is, the number of i ≤ n with (X i , Yi ) = (0, 0). The condition in Theorem 2 that mean −1 ϕ(n)−2 < ∞, i.e. that ϕ˜ is bounded, is thus equivalent (for c = 3/2) to the n n condition that (X i , Yi ) is transient. The next theorem quantifies the change in the critical curve caused by the presence of disorder, for 3/2 < c < 2. It shows that at high temperatures, the significant alteration is confined to a very small interval above u ac , with length of order β 1/(2c−3) ϕˆc− 3 (β −1 )1/2 . 2
In particular, the critical points u c and u ac differ by no more than β 1/(2c−3) ϕˆc− 3 (β −1 )1/2 . 2 Note the exponent 1/(2c − 3) here is greater than 1, so the length of the interval where significant alteration occurs is o(β). q
Theorem 3. Suppose that {Vi , i ≥ 1} are i.i.d. standard Gaussian random variables, and the Markov chain {X i } is recurrent, satisfying (0.4) for some 3/2 < c < 2. Then there exists K 3 as follows. Let 1/2 1 1/(2c−3) ∆0 = ∆0 (β) = K 3 β ϕˆ c− 3 . 2 β Given > 0, there exists K 4 () such that for all sufficiently small β > 0 and β∆ > 0 we have ∆2 2 β β q q , C β, − + ∆ ≤ ∆ if ∆ ≤ ∆0 , βf (0.8) β, − + ∆ ≤ 2 2 2 β
C q β, − β + ∆ f q β, − β2 + ∆ 2 ≤
≤ 1,
if ∆ ≥ K 4 ∆0 . 1− ≤ − 1 C a β, − β2 + ∆ f a β, − β2 + ∆ (0.9) Consequently we have −
β β q ≤ u c (β) ≤ − + K 4 ∆0 (β), 2 2
and therefore
(0.10)
q
u c (β) = 1. β→0 u ac (β) lim
(0.11)
The free-energy bound in (0.8) is proved in [13] with an additional constant factor on the right side; it is shown in [11] that one can take this constant to be c. More importantly, the result is proved there with the annealed critical point −β/2 replaced by the quenched critical point, which means that the quenched specific heat exponent, if it exists, cannot take the annealed value (2c − 3)/(c − 1). Avoiding the factor of c in the free-energy bound is not important to us here, but we include a proof of the bound because the method is completely different from [13] and the ideas may be useful elsewhere. The free-energy and contact-fraction bounds in (0.8) are actually valid for all ∆ > 0, 2 but when ∆ > ∆0 (β) the free-energy upper bound ∆ /2 exceeds the trivial upper bound β β f a β, − 2 + ∆ , as we will obtain from Theorem 1, so it provides no useful information. ∆0 (β) is also (up to a constant) the magnitude of ∆ for which the upper bound
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2∆/β is equal to the annealed contact fraction, as we will show. We will also see later that a value M = M(β∆) of order 1/β f a (β, u) makes a natural (annealed) correlation length for the problem. For a block of length M we may view the average value of u + Vi in the block as a sort of “effective u” for that block; this effective u fluctuates by order M −1/2 from block to block. ∆0 , as we will show, can be characterized (up to a constant) as the value such that for ∆ ∆0 the fluctuations of the effective u are of order M −1/2 ∆, while for ∆ ∆0 we have M −1/2 ∆. Thus for ∆ ≤ ∆0 a substantial fraction of blocks have an effective u below u ac , but this does not occur for ∆ ∆0 . In the borderline case c = 3/2, where the annealed specific heat exponent is 0, outside of the case covered by Theorem 2 we are unable to say whether the critical behavior is altered by the disorder; there is disagreement on this question even in the physics literature ([6,9]). However, in this case the free energy changes significantly in at most an even smaller interval above u ac , with length of order o(β r ) for all r > 1, in 2 fact O(e−K /β ) for some constant K in the case of simple random walk in 1 + 1 or 3 + 1 dimensions. This may help explain why nonrigorous techniques such as renormalization group methods and numerical simulation do not provide consistent predictions for the c = 3/2 case. Theorem 4. Suppose that {Vi , i ≥ 1} are i.i.d. standard Gaussian random variables, and the Markov chain {X i } is recurrent, satisfying (0.4) with c = 3/2 and n n −1 ϕ(n)−2 = ∞. Then given 0 < < 1, there exist K 5 and K 6 = K 6 () as follows. Let
K 5 ϕ ϕ˜ −1 Kβ 26 ∆0 = ∆0 (β, ) = (0.12) 1/2 . β ϕ˜ −1 Kβ 26 Provided β and β∆ are sufficiently small and ∆ ≥ ∆0 , we have
C q β, − β + ∆ f q β, − β2 + ∆ 2 ≤ .
≤ 1,
1− ≤ − 1 C a β, − β2 + ∆ f a β, − β2 + ∆ In particular we have −
β β q ≤ u c (β) ≤ − + ∆0 (β, ), 2 2
and therefore
q
u c (β) = 1. β→0 u ac (β)
(0.13)
lim
As we will see, the condition ∆ ≥ ∆0 says that the mean overlap mentioned above is at most of order 1/β 2 , over one correlation length M. In the symmetric simple random walk case in 1 + 1 or 3 + 1 dimensions, we have c = 3/2 and ϕ asymptotically constant, so for some K i (), ϕ(x) ˜ ∼ log x and ∆0 ≤ K 7 e−K 8 /β . 2
For a transient chain satisfying (0.4), it is easily seen (see [1] or [11]) that u dc (β) = −β −1 log P X (E 1 < ∞).
(0.14)
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To make Theorems 2–4 useful in the transient case, we do the following. Given a measure P X which makes {X i } a transient Markov chain, we define a modified measure P X,R by P X,R (E 1 = n) = P X (E 1 = n | E 1 < ∞), keeping the independence of the E i ’s and the conditional distribution given the E i ’s. We q q denote the corresponding free energy and contact fraction by f R (β, u) and C R (β, u), respectively. Theorems 2 and 4 applied to P X,R , together with the following result, extend our results to the transient case. Theorem 5. Suppose that {Vi , i ≥ 1} are i.i.d. standard Gaussian random variables, and the Markov chain {X i } is transient, satisfying (0.4) for some 1 ≤ c < 2. Then
q q f q (β, u) = f R β, u − u dc (β) , C q (β, u) = C R β, u − u dc (β) for all β > 0, u ∈ R. 1. Preliminaries on Large Deviations and Asymptotics In this section we summarize various basic results on large deviations specialized to our context, and certain asymptotics including Theorem 1, for ready reference later. We assume throughout that (0.4) holds with 1 < c < 2. Let M E denote the moment generating function of E 1 and I E (t) = sup(t x − log M E (x))
(1.1)
x≤0
the large-deviation rate function of E 1 . Fix β > 0 and u > − β2 , and define ∆ by u = − β2 + ∆. From [1], we have 1 log P X (L N ≥ δ N ) = −δ I E (δ −1 ), N
lim
N →∞
and the free energy of the deterministic model is given by β f d (β, u) = lim N
1 log Z [0,N ] (β, u) = sup (βuδ − δ I E (δ −1 )). N δ∈(0,1)
Hence the annealed free energy is given by β f a (β, u) = β f d (β, u + β −1 log MV (β))
= sup δ (βu + log MV (β)) − δ I E (δ −1 ) δ∈(0,1)
= sup
δ∈(0,1)
δ∗
δ ∗ (β∆)
(1.2)
β∆δ − δ I E (δ −1 ) .
= be the unique value of δ which achieves the supremum in (1.2); it is Let easily shown that δ ∗ = C a (β, u). The uniqueness here follows from strict convexity of δ I E (δ −1 ); see [1] and the correction to [1] in [2]. Let δn = E X (L n /n). From Lemma 1 below we see that 1 n −(2−c) ϕ(n)−1 . (1.3) δn ∼ Γ (2 − c)Γ (c − 1)
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Let
M = M(β∆) = min{n : δn ≤ δ ∗ }, δ∗
(1.4)
M −1 )δ ∗ .
and observe that ≥ δ M ≥ (1 − M serves as a correlation length for the pinned polymer–we will see that the free energy gain from pinning is of order 1 over length M. It can also be shown that excursions much longer than M are rare, and on length scales shorter than M, the pinned Markov chain in many senses “looks like” the underlying “free” chain with law P X , but we will not formalize or prove these statements here. Since E 1 X = ∞, there exists α0 = α0 (β∆) > 0 such that (log M E ) (−α0 ) =
E 1 e−α0 E 1 X 1 = ∗. e−α0 E 1 X δ
(1.5)
From basic large-deviations theory, as shown in [13] we have α0 = β f a (β, u) and
β∆ = − log M E (−α0 ).
(1.6)
By (1.3), (log M E ) (−α0 ) =
1 1 ∼ ∼ Γ (2 − c)Γ (c − 1)M 2−c ϕ(M) ∗ δ δM
as β∆ → 0. (1.7)
A routine calculation shows that the derivatives of log M E satisfy (k)
(log M E )(k) (−x) ∼ M E (−x) ∼ Γ (k + 1 − c)x −(k+1−c) ϕ(x −1 ) as − x 0, k ≥ 1, (1.8) so for every fixed ν > 0, ν Γ (2 − c)M 2−c ϕ(M) (log M E ) − ∼ as β∆ → 0 (equivalently, as M → ∞). M ν 2−c (1.9) Taking ν to be K 9 given by 1 K 92−c
= Γ (c − 1),
we see from (1.7) and (1.9) that Mβ f a (β, u) = Mα0 → K 9 as β∆ → 0.
(1.10)
For δ > 0 let x0 = x0 (δ) = ((log M E ) )−1 (δ −1 ), that is, x0 < 0 is the point where the sup in (1.1) is achieved for t = δ −1 . Note that x0 (δ ∗ ) = −α0 . From (1.8) we have as δ → 0, 1 Γ (2 − c) 1 1 Γ (2 − c) 1 , (1.11) = = (log M E ) (x0 ) ∼ ϕ δ |x0 |2−c |x0 | |x0 |2−c ϕ 2−c |x0 |2−c so ∗ 1 Γ (2 − c) 1 1 ∼ , δ ϕ 2−c δ |x0 |2−c
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so x0 (δ) ∼ −K 10 δ Letting
1/(2−c)
1 1 . ϕ 2−c δ
F(δ) = β∆δ − δ I E (δ −1 )
(1.12)
(1.13)
we have −δ 2 F (δ) = δ −1 I E (δ −1 ), and by a standard computation and (1.8), |x0 |3−c 1 ∼ as δ → 0. (log M E ) (x0 ) Γ (3 − c)ϕ(|x0 |−1 ) (1.14) It follows from these and (1.11) that I E (δ −1 ) = x0 ,
I E (δ −1 ) =
δ 2 F (δ) ∼ K 11 x0 (δ).
(1.15)
Next observe that integrating (1.8) with k = 1 gives log M E (−x) ∼ −
Γ (2 − c) c−1 x ϕ(x −1 ) as − x 0. c−1
(1.16)
With (1.10) and (1.6) this shows that β∆ ∼ K 12 M −(c−1) ϕ(M) as β∆ → 0.
(1.17)
Observe that by (1.16) and (1.11) we have I E (δ −1 ) = δ −1 x0 − log M E (x0 ) ∼
Γ (3 − c) |x0 |c−1 ϕ(|x0 |−1 ) as δ → 0, c−1
(1.18)
so by (1.11) and (1.12), δ I E (δ −1 ) ∼
1 1 2−c |x0 | ∼ K 13 δ 1/(2−c) as δ → 0. c−1 ϕ 2−c δ
(1.19)
If we take δ = δ ∗ in (1.19), we have |x0 | = α0 and thus β∆δ ∗ −δ ∗ I E ((δ ∗ )−1 ) = α0 , β∆δ ∗ ∼
α0 2−c , δ ∗ I E ((δ ∗ )−1 ) ∼ α0 as β∆ → 0. c−1 c−1 (1.20)
From (1.10) and (1.20) we have both
and
β∆δ ∗ M → K 14 as β∆ → 0,
(1.21)
β f a (β, u) = α0 ∼ (c − 1)β∆δ ∗ as β∆ → 0.
(1.22)
From (1.17) we have 1 1 ∼ K 15 c−1 ϕ c−1 β∆ α0
1 α0c−1
as β∆ → 0.
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K. S. Alexander
By (1.20) and Lemma 1.10 of [22] this means that 1 ∗ K 16 1 ∼ c−1 ∼ K 17 ϕ ∗ c−1 (β∆δ ) β∆ c−1 α0
1 , β∆
or equivalently ∗
δ ∼ K 18 (β∆)
(2−c)/(c−1)
ϕˆc−1
1 , β∆
and then also, by (1.10) and (1.22), K9 ∼ α0 ∼ K 19 (β∆)1/(c−1) ϕˆc−1 M
1 . β∆
(1.23)
Thus Theorem 1 is proved. For the remainder of this section we consider c > 3/2. Define ∆1 = ∆1 (β) to be the unique positive ∆, where δ ∗ (β∆) = 2∆/β (i.e. where the linear upper bound intersects the contact fraction curve) and let M1 = M(β∆1 ); see the discussion after Theorem 3. Using (1.17) we get that for ∆ ∼ ∆1 (β), 2∆ ϕ(M) ∼ K 12 2 c−1 , β β M
δ∗ ∼ δM ∼
K 20 as β → 0, M 2−c ϕ(M)
so for ∆ ∼ ∆1 , or equivalently, 2∆/β ∼ δ ∗ , we have 1 M 2c−3 ∼ K as β → 0, 21 β2 ϕ(M)2
(1.24)
and therefore from (1.17) again, ∆21 =
(β∆1 )2 K 22 ∼ . β2 M1
(1.25)
A similar argument shows that for ∆ ∆1 we have ∆2 1/M, and for ∆ ∆1 we have ∆2 1/M, confirming comments after Theorem 3. Taking ∆ = K ∆1 for some K , we have from (1.6), (1.23) and (1.25) that 1
K 2 K 22 K 2− c−1 K 22 ∆2 ∼ ∼ as β → 0. 2 2M(β∆1 ) 2M(β∆) 2−
1
Letting K 23 be defined by K 23 c−1 K 22 = 2K 9 , and letting ∆2 = K 23 ∆1 , we see that ∆2 /2 ∼ rβ f a (β, u) with r < 1 if K < K 23 , r > 1 if K > K 23 , and r = 1 if K = K 23 , i.e. if ∆ ∼ ∆2 . Thus asymptotically, the annealed free energy curve intersects the upper bound ∆2 /2 at approximately ∆ = ∆2 . To complete verification of the comments after Theorem 3, we show that K 3 can be chosen (in the definition of ∆0 ) so that ∆0 ∼ ∆2 . From (1.23) and (1.25) we have 1 1/(c−1) ∗ 1 1/(c−1) 1 K9 ∼ ϕ c−1 as β → 0, K 22 K 19 β∆1 β∆1 ∆21
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or equivalently, 1 ∆2c−2 1
1 ∗ ∼ K 24 ϕ β∆1 c−1
or 1 ∆2c−2 1
ϕ c−1
1 ∆2c−2 1
∼
1 β∆1
,
K 24 , β∆1
or from (1.24) and (1.25), since c > 3/2, −1 1 1 K 24 1 1 , ϕ 3 ∼ = 2c−3 ϕ 2c−3 c− 2 2c−3 2 β ∆ ∆1 ∆1 ∆1 1 or 1 ∆2c−3 1
K 24 ∗ ∼ ϕ 3 β c− 2 −1/(2c−3)
Thus taking K 3 = K 23 K 24
−(c− 3 ) 2 1 1 K 24 = ϕˆc− 3 . 2 β β β
we get
∆2 = K 23 ∆1 ∼ K 3 β 1/(2c−3) ϕˆc− 3 2
1/2 1 = ∆0 as β → 0, β
as desired. 2. Proof of Theorem 3 We begin with the proof that the quenched free energy is close to the annealed when ∆ is not too small, which is the first part of (0.9). For a disorder realization {v j } and Markov chain trajectories {xn }, {yn } define R N ({vn }, {xn }) =
N (u + vn )δ{xn =0} , n=1
B N ({xn }, {yn }) =
N
δ{xn =yn =0} .
n=1
We write only R N , B N when confusion is unlikely. We first consider the free energy with a modified measure Pˆ X in place of P X in (0.1), which changes only the distribution of the length of the first excursion. Let M be the correlation length from (1.4). Pˆ X is defined by the specification that under Pˆ X , E 1 is uniform in {1, ..., l0 M} and E 2 , E 3 , ... are iid with distribution P X (E 1 ∈ ·), independent of E 1 . Here l0 is an integer to be specified. In a harmless abuse of notation, we denote ˆ ˆ the corresponding expectation and conditional expectation by · X and E X (·|·), respecβ,u,{V } {V } i i tively. We use Zˆ [0,N ] (β, u) and µˆ [0,N ] to denote the corresponding quenched partition
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K. S. Alexander
function and finite-volume Gibbs distribution, respectively. In the annealed case when β,u Pˆ X is in effect we use the notation Zˆ [0,N ] (β, u) and µˆ [0,N ] . The idea is that under Pˆ X , the return times of the chain are distributed approximately as if the chain, instead of starting at 0 at time 0, had undergone a long excursion which started at least an order-M number of time steps before 0 and ended in (0, l0 M]. For N ⊂ {0} × Σ N a set of length-N Markov chain trajectories with x0 = 0, let {Vi } {Vi } Z [0,N ] (β, u, N ) denote the contribution to Z [0,N ] (β, u) from trajectories in N , and ˆ similarly for Z in place of Z . Fix an integer k0 > 6l0 to be specified, and define a block length N = k0 M. The k th block is then ((k − 1)N , k N ] ∩ Z. We would like to choose events m N , m ≥ {Vi } {Vi } 1 such that Z [0,m N ] (β, u, m N ) is a “not too small” fraction of Z [0,m N ] (β, u), and {V }
i Z [0,m N ] (β, u, m N ) approximately factors into a product of contributions from each block. Such a product form can be made exactly true, as in [1] and [13], by conditioning on a return to 0 at the end of every block, but the fact that the entropy cost of such conditioning grows large as ∆ → 0 (i.e. as the values of N we need to consider approach infinity) makes it unworkable in our context. Here we modify the product requirement slightly, to allow trajectories to bypass “bad” blocks. To that end, in the k th block there is a landing zone ((k − 1)N , (k − 1)N + l0 M] ∩ Z at the beginning of the block, a prohibited zone (k N − l0 M, k N ] ∩ Z at the end of the block and a takeoff zone (k N − 5l0 M, k N − l0 M] ∩ Z just before the prohibited zone. Let m ≥ 1 and J ⊂ {1, .., m} with 1 ∈ J , and label the elements of J as j1 < .. < j|J | . For m ≥ 1 we define mJ N to be the set of trajectories x[0,m N ] satisfying the following criteria:
(a) (b) (c)
for each 2 ≤ k ≤ |J | there is an excursion which starts in the takeoff zone of block jk−1 and ends in the landing zone of block jk ; for each k ≤ |J | there is at least one return to 0 in the first half of the takeoff zone of block jk ; there are no returns to 0 after the takeoff zone of block j|J | . {1}
We write N for N . Let 0 < < 1/2. Suppose that we can choose k0 , l0 such that for constants K i (not depending on ) to be specified, we have the following properties: (P1) The partition function approximately factors, in that for all m ≥ 1 and J ⊂ {1, .., m} with 1 ∈ J , {V(k−1)N +i } {V } Zˆ i (β, u, J ) ≥ e−K 25 l0 m (β, u, N ). Zˆ [0,m N ]
[0,N ]
mN
k∈J
(P2) β,u
µˆ [0,N ] ( N ) ≥ e−K 26 l0 . (P3) For {X n } and {Yn } two independent realizations of the Markov chain, 2 β,u µˆ [0,N ] eβ B N ({X i },{Yi }) − 1 {X i }, {Yi } ∈ N ≤ . 8 Here {V(k−1)N +i } denotes the sequence {V(k−1)N +1 , V(k−1)N +2 , ...}. Note that (P3) says that two randomly chosen trajectories from N intersect each other at 0 only rarely. Also, there is some competition between (P3) and (P2), in that for (P3) we would like
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N not too big so that B N is not too big, while for (P2) to be a workable lower bound, we must have the number N /M of correlation lengths be much greater than l0 . Our basic technique for proving the first part of (0.9) from (P1)–(P3) is the second moment method. Note first that
{Vi } Zˆ [0,N ] (β, u, N )
V
Xˆ
= eβ∆L N δ N Xˆ
β,u = µˆ [0,N ] ( N ) eβ∆L N β,u
= µˆ [0,N ] ( N ) Zˆ [0,N ] (β, u).
(2.1)
We have for trajectories {xi }, {yi } that
eβ R N ({xi }) eβ R N ({yi })
V
= exp (β∆(L N ({xi }) − B N ({xi }, {yi }))) · exp (β∆(L N ({yi }) − B N ({xi }, {yi })))
· exp (2βu + 2β 2 )B N ({xi }, {yi }) = eβ∆L N ({xi }) eβ∆L N ({yi }) eβ
2B
N ({xi },{yi })
,
(2.2)
so by (P3), provided β is small,
{Vi } var V Zˆ [0,N ] (β, u, N ) eβ∆L N ({xi }) eβ∆L N ({yi }) = {xn }∈ N {yn }∈ N
2
· eβ B N ({xi },{yi }) − 1 Pˆ X ({xi }) Pˆ X ({yi }) 2 V 2 β,u {Vi } β B N ({X i },{Yi }) = Zˆ [0,N (β, u, ) µ ˆ − 1 | {X i }, {Yi } ∈ N ) N [0,N ] e ] V 2 ˆ {Vi } Z [0,N ] (β, u, N ) ≤ . 8
(2.3)
Also, by (2.1) and (P2),
{Vi } Zˆ [0,N ] (β, u, N )
V
≥ K 27 e−K 26 l0 Zˆ [0,N ] (β, u).
Therefore by Chebyshev’s inequality and (2.3), 1 {Vi } V −K 26 l0 ˆ ˆ P Z [0,N ] (β, u) ≤ . Z [0,N ] (β, u, N ) ≤ K 27 e 2 2 We say the k th block is good if k = 1 or 1 {V(k−1)N +i } Zˆ [0,N (β, u, N ) > K 27 e−K 26 l0 Zˆ [0,N ] (β, u), ] 2
(2.4)
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K. S. Alexander
and bad otherwise, and we let Jm = Jm ({Vi }) = {k ≤ m : block k is good}. By (P1), 1 1 {Vi } {Vi } Jm log Zˆ [0,m log Zˆ [0,m N ] (β, u) ≥ N ] (β, u, m N ) mN mN l0 |Jm | log(K 27 /2) ≥ − (K 26 + K 25 ) + log Zˆ [0,N ] (β, u), N N mN (2.5) so with P V -probability one, 1 {Vi } log Zˆ [0,m N ] (β, u) mN 1 l0 log(K 27 /2) − (K 26 + K 25 ) + 1 − log Zˆ [0,N ] (β, u). ≥ N N 2 N
β f q (β, u) = lim
m→∞
(2.6) To compare N −1 log Zˆ [0,N ] (β, u) to β f a (β, u) we use subadditivity and the represen Xˆ tation Zˆ [0,N ] (β, u) = eβ∆L N . It is easily seen that for all n, k,
eβ∆L n+k
X
X X
eβ∆(1+L k ) , ≤ eβ∆L n
(2.7)
X so an = β∆ + log eβ∆L n , n ≥ 1, defines a subadditive sequence. Therefore ak /k ≥ limn an /n = β f a (β, u) for each fixed k. Hence for each y ≤ l0 M we have
X a ˆ E X (eβ∆L N | E 1 = y) = eβ∆ eβ∆L N −y ≥ e(N −l0 M)β f (β,u) , and therefore Xˆ
a Zˆ [0,N ] (β, u) = eβ∆L N ≥ e(N −l0 M)β f (β,u) . It follows that
l0 1 β f a (β, u). log Zˆ [0,N ] (β, u) ≥ 1 − N k0
(2.8)
From (1.6) and (1.10) we have 1 l0 l0 ≤ ≤ K 29 β f a (β, u), N N k0 which with (2.6) and (2.8) shows that, provided k0 /l0 is sufficiently large (depending on ), we have β f q (β, u) ≥ (1 − )β f a (β, u), completing the proof of the first part of (0.9). Our remaining task to prove the first part of (0.9), then, is to establish (P1)–(P3). We begin with (P1). Fix m, J as is (P1), and assume |J | ≥ 2. We will refer to blocks with index in J as J -blocks. Let Uk and Tk denote the location of the first and last returns, respectively, in the k th block (when such exist), necessarily in the landing and takeoff zones, respectively, for trajectories in mJ N and J -blocks k. Suppose j|J | < m; we then
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first deal with the final excursion initiated before time m N , necessarily from the takeoff zone of the last J -block, as follows. We have {Vi } J Zˆ [0,m N ] (β, u, m N ) {V } P X (E 1 > m N − t) , = Zˆ [0,i j|J | N ] (β, u, Jj|J | N ∩ {T j|J | = t}) X P (E 1 > j|J | N − t) t
(2.9)
where the sum is over t in the takeoff zone of block j|J | . Since j|J | N − t ≥ l0 M and m N − t ≤ (m − j|J | )k0 M + 5l0 M, there exist constants K i such that, provided l0 ≥ K 30 , k0 ≤ l0 e K 31 l0 and β∆ is sufficiently small (depending on ) so that M is large, we have c−1 P X (E 1 > m N − t) 1 l0 ≥ ≥ e−K 32 (m− j|J | )l0 . P X (E 1 > j|J | N − t) 2 (m − j|J | )k0 + 5l0 Then by (2.9), {Vi } J −K 32 (m− j|J | )l0 ˆ {Vi } Z [0, j|J | N ] (β, u, Jj|J | N ). Zˆ [0,m N ] (β, u, m N ) ≥ e
(2.10)
Having effectively replaced m with j|J | , we next decompose according to the starting and ending points t, y of the excursion from block j|J |−1 to block j|J | , as follows: {V } Zˆ [0,i j|J | N ] (β, u, Jj|J | N ) {V } = Zˆ [0,i j|J |−1 N ] (β, u, Jj|J |−1 N ∩ {T j|J |−1 = t}) t,y
{V( j
· Zˆ [0,N|J] | ·
−1)N +i }
(β, u, N ∩ {U1 = y − ( j|J | − 1)N })
P X (E 1 = y − t) . P X (E 1 > j|J |−1 N − t) Pˆ X (E 1 = y − ( j|J | − 1)N )
(2.11)
Note that l0 + ( j|J | − j|J |−1 − 1)k0 M ≤ y − t ≤ 6l0 + ( j|J | − j|J |−1 − 1)k0 M, while j|J |−1 N − t ≥ l0 M and Pˆ X (E 1 = y − ( j|J | − 1)N ) = 1/l0 M. Assuming again that l0 ≥ K 30 , k0 ≤ l0 e K 31 l0 and β∆ is sufficiently small, it follows readily that P X (E 1 = y − t) P X (E 1 > j|J |−1 N − t) Pˆ X (E 1 = y − ( j|J | − 1)N ) k0 −c ≥ K 33 6 + ( j|J | − j|J |−1 − 1) l0 ≥ e−K 34 ( j|J | − j|J |−1 )l0 .
(2.12)
Note it is essential that the correlation length M cancels out in (2.12), so that the lower bound does not depend on β∆. From (2.11) and (2.12), {V } Zˆ [0,i j|J | N ] (β, u, Jj|J | N ) {V }
{V( j
≥ e−K 34 ( j|J | − j|J |−1 )l0 Zˆ [0,i j|J |−1 N ] (β, u, Jj|J |−1 N ) Zˆ [0,N|J] |
−1)N +i }
(β, u, N ).
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K. S. Alexander
Iterating this we obtain {V } Zˆ [0,i j|J | N ] (β, u, Jj|J | N ) ≥ e−K 34 j|J | l0
{V(k−1)N +i } (β, u, N ), Zˆ [0,N ]
k∈J
which with (2.10) completes the proof of (P1). We next prove (P2), for which we need only consider one length-N block. Let D N denote the event that there is at least one return to 0 in the first half of the takeoff zone, i.e. in (N − 5l0 M, N − 3l0 M], and let C N denote the event that there are no returns to 0 after the takeoff zone, i.e. after N − l0 M, so N = D N ∩ C N . We introduce the tilted measure Q = Q X , on trajectories of the Markov chain, given by Q X (E 1 = k1 , .., E m = km ) e−α0 (k1 +..+km ) X P (E 1 = k1 , .., E m = km ) = (e−α0 E 1 X )m
for all m, k1 , .., km , (2.13)
which by (1.5) satisfies E 1 Q =
1 , δ∗
(2.14)
where · Q denotes expected value under Q X . Equation (2.13) does not of course completely determine a distribution for trajectories, but the excursion lengths are all that we β,u need, as noted earlier. Observe that µˆ [0,N ] tilts P X so that the contact fraction becomes ∗ δ , i.e. so that the mean excursion length is approximately 1/δ ∗ for large N , so from β,u (2.14) we expect the measures µˆ [0,N ] and Q X to be similar, a relation we will now make precise. Given n ≤ k ≤ r we have from (1.6) that X X eβ∆n eβ∆L r −k eα0 k eβ∆L r −k X P (τ = k) = Q X (τn = k). n X X β∆L β∆L r r e e (2.15) Fix y ≤ l0 M and let v = N − 4l0 M − y. We now specify n = vδ ∗ . Then by (2.15), β,u
µ[0,r ] (τn = k) =
β,u
β,u
µˆ [0,N ] (D N | E 1 = y) ≥ µ[0,N −y] ((k0 − 5l0 )M − y < τn ≤ (k0 − 3l0 )M − y) eα0 (N −6l0 M) X ≥ X Q (v − l0 M < τn ≤ v + l0 M) . eβ∆L N
(2.16)
By (2.14) we have τn Q ∼ v as β∆ → 0, and the variance of E 1 under Q X is easily shown (using (1.10)) to satisfy var (E 1 ) ∼ Q
E 12 Q
=e
β∆
∞
e−α0 j j 2−c ϕ( j) ∼ K 35 M 3−c ϕ(M),
(2.17)
j=1
so using (1.7), var Q (τn ) ∼ K 35 δ ∗ v M 3−c ϕ(M) ≤ K 36 k0 M 2 ,
(2.18)
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135
also as β∆ → 0. Therefore provided k0 ≤ l02 /4K 36 , by Chebyshev’s inequality and (2.16) we have for β∆ small that β,u
µˆ [0,N ] (D N ) ≥
eα0 (N −6l0 M) X . 2 eβ∆L N
(2.19)
Let G N denote the time of the first return to 0 in the takeoff zone (necessarily in (N − 5l0 M, N − 3l0 M], if the event D N occurs), when such a return exists. For g ∈ (N − 5l0 M, N − 3l0 M] we have using (1.3) that β,u
µˆ [0,N ] (C N | G N = g) β,u
≥ µ[0,N −g] (τi = k, L N −g = i for some k ≤ N − g − l0 M and i ≥ 1)
=
k≤N −g−l0 M i≥1
eβ∆i
X eβ∆L N −g
P X (τi = k)P X (E 1 > N − g − k)
P X (E 1 > 5l0 M) P X (X k = 0)
X β∆L 5l0 M k≤2l0 M e X P X (E 1 > 5l0 M) L 2l0 M ≥
X eβ∆L 5l0 M
≥
K 37
≥
eβ∆L 5l0 M
X .
(2.20)
Combining this with (2.19) we get β,u
β,u
β,u
µˆ [0,N ] ( N ) = µˆ [0,N ] (D N )µˆ [0,N ] (C N | D N ) ≥
K 37 eα0 (N −6l0 M) X . X β∆L 5l0 M e 2 eβ∆L N
We claim that for some K 38 ,
X eβ∆L k M ≤ K 38 keα0 k M
for all k ≥ 1.
(2.21)
(2.22)
Presuming this is proved, we have from (2.21), presuming once more that k0 ≤ l0 e K 31 l0 and l0 is sufficiently large, β,u
µˆ [0,N ] ( N ) ≥
K 37 exp (α0 (N − 6l0 M) − α0 N − 5l0 α0 M) ≥ e−K 39 l0 , (2.23) 2 k l 10K 38 0 0
so (P2) is proved. It should be pointed out that the various conditions we have required on l0 and k0 are compatible and can be summarized as follows: there exist K 40 , K 41 , K 42 and K 43 = K 43 () > 1 such that k0 , l0 must satisfy K 40 ≤ K 43l0 ≤ k0 ≤ K 41 min(l02 , l0 e K 42 l0 ).
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K. S. Alexander
To prove (2.22), fix k ≥ 1 and observe that F(δ) = β∆δ − δ I E (δ −1 ) is concave with F(0) = 0 and maximum value F(δ ∗ ) = α0 > 0, so δ0 given by I E (δ0−1 ) = β∆ is the unique positive solution of F(δ) = 0. By (1.14) we have F (δ0 ) = δ0−1 x0 (δ0 ). By concavity F is below its tangent line at δ0 , so we have the bound α0 = |x0 (δ ∗ )|, δ ≤ δ0 , F(δ) ≤ x0 (δ0 ) (2.24) δ0 (δ − δ0 ), δ > δ0 . Let j0 = δ0 k M. Using the equivalence of L k M ≥ j and τ j ≤ k M we obtain from (2.24) that
eβ∆L k M
X
= 1 + β∆
kM
eβ∆j P X (τ j ≤ k M)
j=1
≤ 1 + β∆
kM
e F( j/k M)k M
j=1
≤ 1 + β∆j0 e
α0 k M
+ β∆
j> j0
≤ 1 + β∆δ0 k Meα0 k M + β∆
x0 (δ0 ) exp δ0
∞
j − δ0 k M kM
e(r −2)x0 (δ0 )k M
r =2 (r −1)δ0 k M< j≤r δ0 k M
≤ 1 + β∆δ0 k M(e
α0 k M
+ K 44 ),
(2.25)
where the last inequality follows from x0 (δ0 ) ≤ −α0 together with (1.10). Now we need to compare δ0 to δ ∗ . From (1.19) and (1.20) we have β∆δ0 δ0 = δ∗ β∆δ ∗
x0 (δ0 ) x0 (δ ∗ ) 1/(2−c) ψ(δ0−1 ) δ0 , ∼ (2 − c) ∗ δ ψ((δ ∗ )−1 ) ∼ (2 − c)
with ψ slowly varying, which implies that δ0 ∼ (2 − c)−(2−c)/(c−1) . δ∗ With (2.25) and (1.21) this proves the claim (2.22), completing the proof of (P2). Next we prove (P3). We use ·µ and µ(· | ·) to denote expectation and conditional expectation, respectively, under a measure µ. By (P2), β,u β 2 B N ({X i },{Yi }) − 1 {X i }, {Yi } ∈ N µˆ [0,N ] e ≤
1 β,u
eβ
2B
N ({X i },{Yi })
µˆ β,u [0,N ] −1
µˆ [0,N ] ( N )2
2 µβ,u [0,N ] ≤ e2K 26 l0 eβ (B N ({X i },{Yi })+1) − 1 .
(2.26)
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We can shift the problem from length scale N to the length scale M, on which the β,u measures µ[0,M] and P X are comparable, via the inequality
e
β 2 B N ({X i },{Yi })
µβ,u
[0,N ]
≤
e
β 2 (B M ({X i },{Yi })+1)
µβ,u
k0
[0,M]
.
(2.27)
To quantify the comparability, by (2.22) (with trivial modifications to deal with the 2 in the exponent) and (1.10), using the fact that (x − 1)2 ≤ x 2 − 1 for x ≥ 1, we have
2 µβ,u 2 [0,M] β (B M ({X n },{Yn })+1) e −1 ≤
e
β∆L M ({X n }) β∆L M ({Yn })
e
2
X 2 β (B M ({X n },{Yn })+1) e −1
X X 2
X e2β∆L M ({Yn }) e2β (B M ({X n },{Yn })+1) − 1 ≤ e2β∆L M ({X n })
2 X ≤ K 45 e2β (B M ({X n },{Yn })+1) − 1 .
(2.28)
Combining (2.27) and (2.28) we obtain
2 µβ,u [0,N ] eβ (B N ({X i },{Yi })+1) − 1
2 µβ,u k0 2 [0,M] eβ (B M ({X i },{Yi })+1) ≤ eβ −1 ≤e
β2
1+
1/2 K 45
e
2β 2 (B
X 1/2 M ({X n },{Yn })+1) − 1
k0 − 1.
(2.29)
To bound the expectation on the right side of (2.29) we will need some lemmas. The first is a result of Doney [7] on renewal processes, which we specialize here to our situation. Lemma 1 ([7]). Suppose that for some slowly varying function ϕ and constants K < ∞, 1 < c < 2, the excursion length distribution of an aperiodic Markov chain {X n } satisfies P(E 1 = n) ≤ K n −c ϕ(n) for all n and P(E 1 > n) ∼
1 n −(c−1) ϕ(n) as n → ∞. c−1
Then P(X n = 0) ∼
c−1 n −(2−c) ϕ(n)−1 as n → ∞. Γ (2 − c)Γ (c − 1)
Let E˜ i denote the length of the i th excursion from (0, 0), L˜ n the number of returns to 0 by time n, and τ˜n the time of the n th return to (0, 0), for the chain {(X i , Yi )}.
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Lemma 2. Let {X i } and {Yi } be independent copies of an aperiodic recurrent Markov starting at 0 and satisfying (0.4) with 3/2 < c < 2, or with c = 3/2 and ∞chain −1 −2 = ∞. Then there exist constants K such that for all sufficiently large i i=1 i ϕ(i) n, K 47 n −(2c−3) ϕ(n)2 if c > 23 , K 46 ˜ P( E 1 > n) ≥ X ≥ K 48 (2.30) if c = 23 . n −1 −2 i=1 i ϕ(i) L˜ n Proof. We use E 1 , E 2 , ... to denote excursion lengths for {X i }, as usual. Define σ1 = min{i ≥ 1 : E i > n},
σ2 = min{i ≥ 1 : E˜ i > n}.
Let Tn1 and Tn2 denote the starting times of excursions σ1 and σ2 for the chains {X i } and {(X i , Yi )}, respectively. Then
L˜ n
X
X
X ≥ L˜ n δ{Tn2 ≤n/2} = σ2 δ{Tn2 ≤n/2} .
Suppose we can show that for some K 49 > 0, n
≥ K 49 for all sufficiently large n. P X Tn2 ≤ 2
(2.31)
(2.32)
Since σ2 has a geometric distribution, this shows that
σ2 δ{Tn2 ≤n/2}
X
≥ K 50 σ2 X =
K 50 P X ( E˜
1
> n)
,
which with (2.31) completes the proof of the first inequality in (2.30). The second inequality in (2.30) is a consequence of Lemma 1 and the relation
L˜ n
X
=
n
P X (X i = 0)2 .
i=1
To prove (2.32) we observe that n
n
≥ P X Tn1 ≤ , P X Tn2 ≤ 2 2 and proceed analogously to (2.20). Using (1.3) we have n X n X P X Tn1 ≤ = P (E j+1 > n) P τj ≤ 2 2 j≥0 X ≥ P X (E 1 > n) L n/2 ≥ K 51 ,
(2.33)
so (2.32) is proved. Lemma 3. Let {X i } and {Yi } be independent copies of an aperiodic recurrent Markov chain starting at 0 and satisfying (0.4) with 1 < c < 2. Then
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(i) if c > 3/2 then there exists K 52 < ∞ such that k ϕ(N )2 for all N , k ≥ 1; (2.34) P(B N ≥ k) ≤ 1 − K 52 N 2c−3 (ii) if c = 3/2 and n n −1 ϕ(n)−2 = ∞, then there exists K 53 < ∞ such that k 1 P(B N ≥ k) ≤ 1 − for all N , k ≥ 1; (2.35) K 53 ϕ(N ˜ ) (iii) if c < 3/2, or if c = 3/2 and n n −1 ϕ(n)−2 < ∞, then there exists 1 > 0 such that P(B N ≥ k) ≤ (1 − 1 )k for all N , k ≥ 1. (2.36) Proof. By Lemma 1 if 1 < c < 2 we have for some K 54 , (2.37) P ((X i , Yi ) = (0, 0)) ≤ K 54 i −(4−2c) ϕ(i)−2 for all i ≥ 1. −1 −2 If 1 < c < 3/2, or if c = 3/2 and n n ϕ(n) < ∞, then by (2.37) we have i P ((X i , Yi ) = (0, 0)) < ∞ so the chain {(X i , Yi )} is transient and (2.36) follows. If c > 3/2, then by Lemma 2, we have P(B N > k) ≤ P max E˜ j ≤ N ≤ 1 − j≤k
which proves (2.34). Similarly, if c = 3/2 and gives (2.35).
n
ϕ(N )2 K 52 N 2c−3
k ,
n −1 ϕ(n)−2 = ∞ then Lemma 2
We can now continue with the bound on the expectation on the right side of (2.29). By Lemma 3(i) we have B M ({X n }, {Yn }) + 1 stochastically smaller (under P X ) than a −1 −(2c−3) M ϕ(M)2 . Let geometric random variable with parameter of form p M = K 52 a=
e−2K 26 l0 1/2
32k0 K 45
,
where K 26 is from (P2) and K 45 from (2.28), and suppose that, for K 55 to be specified, 1/2 1 , (2.38) ∆ ≥ K 55 a −(2c−2)/(2c−3) β 1/(2c−3) ϕˆc− 3 2 β which is a version of the assumption in (0.9). From (1.24) and the discussion following it, for each fixed K there exists g(K ), with g(K ) ∞ as K → ∞, such that if we let ∆ = ∆(β) → 0, then the statement that ∆ ∼ K ∆0 is equivalent to the statement that p M ∼ g(K )β 2 as β → 0. Thus if K 55 is large enough then
4β 2 1 2 1 − e−2β , (2.39) pM ≥ 2 ≥ 1 + 2 a a so from the bound by a geometric random variable,
2 X e2β (B M ({X n },{Yn })+1) − 1 ≤
2
e2β −1 2 1−(1− p M )e2β
≤ a2.
(2.40)
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Plugging this into (2.29) we obtain from (2.26) that provided β is small enough (depending on l0 ), 2
β,u µˆ [0,N ] eβ B N ({X i },{Yi }) − 1 {X i }, {Yi } ∈ N e−2K 26 l0 −1 ≤ e2K 26 l0 exp β 2 + 32 (2.41) < . 8 The proof of (P3), and thus of the free-energy inequality in (0.9), is now complete. We next consider the contact-fraction inequality in (0.9). Recall that for F from (1.13), we have F maximized at δ ∗ with x0 (δ ∗ ) = −α0 = −F(δ ∗ ) (see (1.6).) Hence from (1.12) and (1.15), for all |γ | < 21 , we have as β∆ → 0 that −(1 + γ )2 (δ ∗ )2 F ((1 + γ )δ ∗ ) ∼ (1 + γ )1/(2−c) K 56 F(δ ∗ ). This shows that −δ 2 F (δ)/F(δ ∗ ) is bounded away from 0 on [δ ∗ /2, 3δ ∗ /2], uniformly for small β∆. It follows that given 0 < λ < 1/2 there exists θ > 0 such that |δ − δ ∗ | ≥ λδ ∗ implies Fix 0 < λ <
1 2
F(δ) ≤ (1 − θ )F(δ ∗ ).
(2.42)
and define the events L N ({xn }) ≤ (1 − λ)δ ∗ , Φ1N = {xn } : N L N ({xn }) Φ2N = {xn } : ≥ (1 + λ)δ ∗ . N
From trivial modifications of Theorem 2.1 of [1] and from (2.42), we obtain that the contributions to the quenched and annealed free energy from trajectories in i N satisfy V 1 {Vi } lim sup log Z [0,N (β, u, ) i N ] N N
V 1 {Vi } ≤ lim sup log Z [0,N (β, u, ) i N ] N N ∗ ≤ sup{F(δ) : |δ − δ | > λδ ∗ } ≤ (1 − θ )F(δ ∗ ), (2.43) for i = 1, 2. From straightforward modifications of Theorem 3.1 of [1], the limits lim N
1 {Vi } log Z [0,N ] (β, u, i N ), i = 1, 2, N
both exist as nonrandom constants a.s. By (2.43) these constants are at most (1−θ )F(δ ∗ ), while by the free energy inequality in (0.9), provided β is sufficiently small (depending on k0 , l0 ) we have β f q (β, u) > (1 − θ )F(δ ∗ ). This means that 1 β,u,{Vi } L N ∗ ∗ lim sup log µ[0,N ] N − δ > λδ < 0, N N
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which establishes the contact-fraction inequality in (0.9). We turn next to (0.8). Let η, ν > 0 and let {k N } be a sequence of integers with χ N = k N /N ≥ η for all N . Define the events N vi ≤ νβηN . A N = {vi } : i=1
N Since the Vi are i.i.d. with standard normal distribution, we have that given i=1 Vi = k N 2 s N for some s, i=1 Vi is normal with mean sk N and variance N (χ N − χ N ). Therefore for |s| ≤ νβη, N k N β2 V β i=1 Vi Vi = N s = exp βsk N + E e (χ N − χ N2 )N 2 i=1 2 β (1 + 2νη − χ N )k N ≤ exp 2 2 β ≤ exp (1 − (1 − 2ν)χ N ) k N , 2 and hence also
kN E V eβ i=1 Vi
2 A N ≤ exp β (1 − (1 − 2ν)χ N ) k N . 2
(2.44)
This shows that conditioning on the event A N of a “typical disorder” reduces the exponent in the exponential moment by a factor of 1 − (1 − 2ν)χ N when there are k N returns. As discussed in [1], this reduction can be viewed as a consequence of the fact k N Vi must be compensated by large negthat under A N , large positive deviations of i=1 N ative deviations of i=k N +1 Vi , hence have a lower probability. Conditioning on A N is related to what is called the Morita approximation, in which moments of the disorder are effectively held fixed, or nearly fixed ([16,21].) It follows from (2.44) that
{Vi } E V Z [0,N ] (β, u, {L N ≥ ηN }) A N X β 2 (1 − (1 − 2ν) LNN ) ≤ exp βu + (2.45) L N δ{L N ≥ηN } . 2 Since ν is arbitrary it follows that
1 {Vi } lim sup E V log Z [0,N ] (β, u, {L N ≥ ηN }) A N N N
1 {Vi } ≤ lim sup log E V Z [0,N ] (β, u, {L N ≥ ηN }) A N N N β 2 δ2 β2 −1 δ− − δ I E (δ ) : δ ≥ η ≤ sup βu + 2 2 β 2 δ2 − δ I E (δ −1 ) : δ ≥ η . = sup β∆δ − 2
(2.46)
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One may view the two negative terms on the right side of (2.46) as two separate costs: the first, of order δ 2 , is the above-mentioned cost of the “compensating” large deviation. The second is the cost of lowering the average excursion length enough to get L N ≈ δ N . By (1.19), since c > 3/2, when δ is small the cost of the compensating large deviation (which does not exist in the annealed model) exceeds the cost of lowering the average excursion length. This is what underlies (0.8). For η > 2∆/β the right side of (2.46) is at most β 2 δ2 : δ ≥ η < 0. sup β∆δ − 2
(2.47)
As in the proof of Theorem 3.1 of [1], there exist constants β f q (β, u, η− ) ≥ 0 and β f q (β, u, η+ ) such that lim N
1 1 V {Vi } {Vi } (β, u, {L N ≥ ηN }) log Z [0,N E log Z [0,N ] (β, u, {L N ≥ ηN }) = lim ] N N N (2.48) = β f q (β, u, η+ ) a.s.
and lim N
1 1 V {Vi } {Vi } log Z [0,N E log Z [0,N (β, u, {L N ≤ ηN }) ] (β, u, {L N ≤ ηN }) = lim ] N N N = β f q (β, u, η− ) a.s. (2.49)
Further, again as in the proof of Theorem 3.1 of [1], by truncating the Vi at some large M to obtain random variables V˜i and applying Azuma’s inequality [3] to {V˜ }
i log Z [0,N ] (β, u, {L N ≥ ηN }),
since lim inf N P V (A N ) > 0 we obtain using (2.48) and (2.49) that V {Vi } E log Z [0,N ] (β, u, {L N ≥ ηN }) | A N )
{Vi } −E V log Z [0,N (β, u, {L ≥ ηN }) = o(N ). N ]
(2.50)
With (2.46) and (2.49) this shows that β f q (β, u, η+ ) < 0. We can then conclude from (2.48) and (2.49) that β,u,{V }
µ[0,N ] i (L N ≥ ηN ) → 0
as N → ∞,
proving the contact-fraction inequality in (0.8). It follows from (2.46) and (2.50) (with η = 0) that β β 2 δ2 ∆2 :δ≥0 = , β f q β, − + ∆ ≤ sup β∆δ − 2 2 2 which is the free-energy inequality in (0.8).
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3. Proof of Theorems 2, 4 and 5 Proof (Proof of Theorem 4). For the free-energy inequality, the only changes needed from the case 3/2 < c < 2 in Theorem 3 involve the definition of p M and the fact that for c = 3/2, (2.38) is not a sufficient condition for the first inequality in (2.39). From Lemma 3 the proper choice is now p M = 1/K 53 ϕ(M), ˜ and for some K 57 = K 57 () the first inequality in (2.39) then holds provided ϕ(M) ˜ ≤
K 57 , β2
for which, by (1.23), it suffices that 2 1 K 58 ϕ ∗1/2 β∆ K 58 = (β∆)2 2 (β∆) ϕˆ 1/2
1 β∆
≤ ϕ˜ −1
K 57 β2
,
or K 59 ≤ ϕ˜ −1 β∆
K 57 β2
1/2
ϕ 1/2 ϕ˜ −1
K 57 β2
1/2
1/2 ϕ˜ −1 Kβ57 2
, = ϕ ϕ˜ −1 Kβ57 2
which is equivalent to ∆ ≥ ∆0 , for an appropriate choice of K 5 in (0.12). Here all K i depend on . The proof of the contact-fraction inequality in (0.7) from the free-energy inequality remains unchanged from Theorem 3. Proof (Proof of Theorem 2). Most of the proof of the free-energy inequality in (0.7) is the same as that of the free-energy inequality in (0.9), but we have the following changes. As in the proof of (0.9), let {Yi } be an independent copy of the Markov chain {X i }, under the distribution P X . Under the hypotheses of the theorem, it follows from Lemma 1 that {(X i , Yi )} is transient. This means that B M ({X n }, {Yn }) + 1 is stochastically smaller than a geometric random variable with parameter p˜ = P X ( E˜ 1 = ∞), where we recall that E˜ 1 is the length of the first excursion for the chain {(X i , Yi )}. Thus the dependence of p M on M is effectively removed–we can achieve (2.39) (with p M replaced by p) ˜ merely by taking β sufficiently small, and (2.38) is not needed. The proof of the contact-fraction inequality in (0.7) from the free-energy inequality remains unchanged from Theorem 3. β,u,0
Proof (Proof of Theorem 5). Recall that µ[0,N ] denotes the Gibbs measure for the deterministic model. We have N
d X β,u d ,0 {Vi } βu c L N d Z [0,N ] (β, u) = e exp β (u − u c + Vi )δ{xi =0} µ[0,Nc ] (x[0,N ] ), {xi }
i=1
and by definition of u dc and continuity of the free energy, lim N
d X 1 log eβu c L N = 0. N
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Hence to show the free energies are equal, it suffices to show that there exist constants a N with log a N = o(N ) and β,u d ,0
dµ[0,Nc ] 1 ≤ ({xi }) ≤ a N for all {xi }. X,R aN d P[0,N ]
(3.1)
Let TN denote the time of the last return to 0 in [0, N ], and let F (x) = P X (x < E 1 < ∞), so F(x) ∼ (c − 1)−1 x −(c−1) ϕ(x). The main observation is that by (0.14), the difference β,u d ,0
X,R between µ[0,Nc ] and P[0,N ] involves only the final excursion in progress at time N , in the sense that β,u d ,0
dµ[0,Nc ]
X,R d P[0,N ]
where 1 = qN
= qN
F(N − TN ) + P X (E 1 = ∞) F(N − TN )
F(N − TN ) + P X (E 1 = ∞) F(N − TN )
,
X,R .
Here · X,R denotes expectation with respect to P X,R . We have
X,R 1 ≥ K 60 (N − TN )c−1 ϕ(N − TN )−1 qN N , ≥ K 61 N c−1 ϕ(N )−1 P X,R TN ≤ 2 and as in (2.20) and the calculations following it we have (since c < 2) N X,R P TN ≤ ≥ K 62 for all N . 2 Therefore β,u d ,0
dµ[0,Nc ]
X,R d P[0,N ]
≤
qN F(N )
≤ K 63 .
In the other direction, X,R X,R 1 (N − TN )c−1 N c−1 1 , ≤ ≤ K 64 ≤ K 65 qN ϕ(N − TN ) ϕ(N ) F(N − TN ) so β,u d ,0
dµ[0,Nc ]
X,R d P[0,N ]
≥ K 66
ϕ(N ) , N c−1
so (3.1) is proved. Equality of the contact fractions follows immediately from equality of the free energies, by definition of the contact fraction.
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4. The Excluded Cases c = 1 and c ≥ 2 We conclude with a few remarks about the cases c = 1 and c ≥ 2 not covered by our results. For c = 1, the left side of (2.12) is not bounded below uniformly in ∆, so a new proof of (P1) is needed. More importantly, in this case there are two disagreeing definitions of the correlation length. One, which we denote here by M ∗ , is given by (1.4) (i.e. M ∗ is the M we used above), and the other, which we denote by M0 , is suggested by (1.10): M0 = K 9 /α0 , the inverse of the free energy. For c > 1 these two values are (by (1.10)) asymptotically the same as ∆ → 0, but for c = 1 we have M0 /M ∗ → ∞ as β∆ → 0. Inequality (P2) can be interpreted as saying that the entropy cost of the event N is at most a small mulltiple of 1/M ∗ per unit length, but for c = 1 a small multiple of 1/M ∗ becomes an unacceptably large multiple of 1/M0 . Thus for c = 1 something else must substitute for the event N , which plays the role of creating an approximate renewal at the start of each block of length N , leading to the factoring of the partition function expressed in (P1). The case c ≥ 2 is really two cases: c = 2 with E 1 X = ∞, which means the transition in the annealed system is continuous, and E 1 X < ∞ which means the transition is discontinuous. For c < 2, the idea of Lemma 2 is that L˜ n = k typically means the k th excursion was one of the first few excursions of length at least of order n, so L˜ n is of the same order as a geometric random variable with parameter 1/P X ( E˜ 1 > n). This will not be valid for c ≥ 2, so a different approach is required. Overall, though, despite significant changes in the details, the core ideas of our proof should carry over well to the case of c = 2 with E 1 X = ∞. Recall that ∆0 (β) is (up to a constant) the magnitude of ∆ for which the upper bound 2∆/β is equal to the annealed contact fraction. When the transition is discontinuous (i.e. when E 1 X < ∞), we have C a (β, u) > 1/E 1 X for all u > u ac (β), and therefore ∆0 (β) is not o(β) (and hence not o(u ac (β)) as β → 0. Thus for c > 2 it is not accurate to describe the disorder-induced changes as being confined to a very small interval above u ac . References 1. Alexander, K.S., Sidoravicius, V.: Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16, 636–669 (2006) 2. Alexander, K.S.: Ivy on the ceiling: first-order polymer depinning transitions with quenched disorder. http://arxiv.org/list/:math.PR/0612625, (2006) 3. Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967) 4. Bodineau, T., Giacomin, G.: On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117, 801–818 (2004) 5. Caravenna, F., Giacomin, G., Gubinelli, M.: A numerical approach to copolymers at selective interfaces. J. Stat. Phys. 122, 799–832 (2006) 6. Derrida, B., Hakim, V., Vannimenus, J.: Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992) 7. Doney, R.A.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Rel. Fields 107, 451–465 (1997) 8. Fisher, M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–729 (1984) 9. Forgacs, G., Luck, J.M., Nieuwenhuizen Th., M., Orland, H.: Exact critical behavior of two-dimensional wetting problems with quenched disorder. J. Stat. Phys. 51, 29–56 (1988) 10. Galluccio, S., Graber, R.: Depinning transition of a directed polymer by a periodic potential: a d-dimensional solution. Phys. Rev. E 53, R5584–R5587 (1996) 11. Giacomin, G.: Random Polymer Models. Cambridge: Cambridge University Press, 2007 12. Giacomin, G., Toninelli, F.L.: The localized phase of disordered copolymers with adsorption. Alea 1, 149–180 (2006)
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13. Giacomin, G., Toninelli, F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006) 14. Gotcheva, V., Teitel, S.: Depinning transition of a two-dimensional vortex lattice in a commensurate periodic potential. Phys. Rev. Lett. 86, 2126–2129 (2001) 15. Monthus, C.: On the localization of random heteropolymers at the interface between two selective solvents. Eur. Phys. J. B 13, 111–130 (2000) 16. Morita, T.: Statistical mechanics of quenched solid solutions with applications to diluted alloys. J. Math. Phys. 5, 1401–1405 (1964) 17. Mukherji, S., Bhattacharjee, S.M.: Directed polymers with random interaction: An exactly solvable case. Phys. Rev. E 48, 3483–3496 (1993) 18. Naidenov, A., Nechaev, S.: Adsorption of a random heteropolymer at a potential well revisited: location of transition point and design of sequences. J. Phys. A: Math. Gen. 34, 5625–5634 (2001) 19. Nechaev, S., Zhang, Y.-C.: Exact solution of the 2D wetting problem in a periodic potential. Phys. Rev. Lett. 74, 1815–1818 (1995) 20. Nelson, D.R., Vinokur, V.M.: Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48, 13060–13097 (1993) 21. Orlandini, E., Rechnitzer, A., Whittington, S.G.: Random copolymers and the Morita aproximation: polymer adsorption and polymer localization. J. Phys. A: Math. Gen. 35, 7729–7751 (2002) 22. Seneta, E.: Regularly Varying Functions. Lecture Notes in Math. 508. Berlin: Springer-Verlag, 1976 Communicated by F. Toninelli
Commun. Math. Phys. 279, 147–168 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0411-y
Communications in
Mathematical Physics
Dividing Quantum Channels Michael M. Wolf, J. Ignacio Cirac Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany. E-mail:
[email protected] Received: 21 December 2006 / Accepted: 14 August 2007 Published online: 1 February 2008 – © Springer-Verlag 2008
Abstract: We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.
Contents I. II. III. IV. V. VI.
Introduction . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . Determinants . . . . . . . . . . . . . . . . . Divisible and Indivisible Maps . . . . . . . Infinitesimal Divisible Channels . . . . . . . Qubit Channels . . . . . . . . . . . . . . . . A Extremal qubit channels . . . . . . . . . B Divisible and indivisible qubit channels C Infinitesimal divisible qubit channels . . VII. Conclusion . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
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I. Introduction Completely positive linear maps describe the dynamics of a quantum system in all cases where the evolution is independent of the past of the system. In the realm of quantum information theory these maps are referred to as quantum channels [1] and, clearly, the concatenation of two quantum channels is again a quantum channel. In this paper we address the converse and investigate whether and how a channel can be expressed as a non-trivial concatenation of other channels. That is, we study the semigroup structure of the set of quantum channels whose input and output systems have equal finite dimension. Despite the fact that one-parameter semigroups of completely positive maps have been extensively studied since the late sixties, the semigroup structure of the set of quantum channels as a whole appears to be widely unexplored. The main purpose of our work is to classify the set of quantum channels with respect to (i) whether a division in terms of a concatenation is at all possible and (ii) whether a channel allows for a division into a large number of infinitesimal channels. This will lead us to the notions of divisibility and infinitesimal divisibility, where the latter property is equivalent to the existence of a continuous time-dependent completely positive evolution which has the given channel as endpoint. This classification will allow us to identify basic building blocks (generators) from which all channels can be obtained by concatenation. Furthermore, it helps us to identify those channels which are solutions of time-dependent master equations. A graphical depiction of different notions of divisibility and their relations is given in Fig. 1. The following gives an overview on the paper and a simplified summary of the obtained results: • Section II introduces basic results and provides a coarse but later on very useful Markovian approximation to any quantum channel. • In Sect. III we prove some properties of the determinant of quantum channels. In particular, its strict monotonicity under concatenation, continuity bounds and properties for Kraus rank-two channels and Markovian channels. • The notions of divisible and indivisible maps are introduced in Sect. IV. The existence of indivisible maps and generic divisibility is shown in any dimension, and it is proven that building equivalence classes under filtering operations preserves divisibility. • Section V shows that every infinitesimal divisible channel can be written as a product of Markovian channels and that infinitesimal divisibility is preserved under invertible filtering operations. Equivalence to the set of continuous completely positive evolutions is proven. • Section VI provides a complete characterization of divisible and indivisible qubit channels in terms of their Lorentz normal form. Solutions of time-dependent master equations are identified for both positive and completely positive evolutions. Channels with Kraus rank two are studied in greater detail separately. • It is shown that already in the qubit case the vicinity of the ideal channel contains all types of channels, in particular ones that are not infinitesimal divisible and even indivisible ones. Before going into detail we want to briefly mention some related fields and results. The notion infinite divisibility goes back to de Finetti and has thus its origin in classical probability theory where it means that for any n ∈ N a characteristic function χ is a power of another characteristic function χ = χnn . Examples are the normal and Poisson distribution. Similarly, the notion of indecomposable distributions exists for those that
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Fig. 1. Graphical depiction of the set of quantum channels regarding finer and coarser notions of divisibility, i.e, the possibility of expressing a channel in terms of a concatenation of other channels. Whereas the set of Markovian channels only contains elements of completely positive semigroups, the set of divisible channels only requires the existence of any non-trivial product decomposition. Definitions of the sets are given in the text (Sects. IV, V). Indivisible maps are discussed in Sects. IV, VI and in Sect. V it is shown that the sets of infinitesimal divisible and time-dependent Markovian channels coincide
cannot be represented as the distribution of the sum of two non-constant independent random variables. In the ‘non-commutative’ context, infinite divisibility of positive matrices with respect to the Hadamard product was studied by Horn [2] and the notion was extended to quantum measurements and quantum channels by Holevo [3] and Denisov [4]. In fact, the findings of Horn can also be translated to the quantum world when considering channels with diagonal Kraus operators, as those act on a density operator by a Hadamard product with a positive matrix. II. Preliminaries This section introduces the notation and recalls some basic results which we will need in the following. Throughout we will consider linear maps T : Md → Md from the space Md of d × d matrices into itself. It will be convenient to consider Md as a Hilbert space Hd equipped with the Hilbert-Schmidt scalar product A, BH = tr[A† B] and the 2 2-norm ||A||2 = A, AH.1 Since Hd is isomorphic to Cd the space of linear maps on Md (Liouville space) is in turn isomorphic to Md 2 . Eigenvalues and singular values of the map T are then understood as the respective quantities of the matrix representation Tˆ ∈ Md 2 of T . More explicit Tˆα,β := tr Fα† T (Fβ ) = Fα |T |Fβ H, where {Fα }α=1...d 2 is any orthonormal basis in Hd . Depending on convenience we will use three different bases: (i) matrix units {|i j|}i, j=1...d , (ii) generalized Gell-Mann matrices which are √ √ either diagonal or an embedding of the Pauli matrices σx / 2 and σ y / 2 in Md , or (iii) a normalized unitary operator basis given by Uα ,α Fα = √1 2 , Uα1 ,α2 = e2πir α2 /d |α1 + r r |, α1 , α2 = 0, . . . , d − 1. d r =0 d−1
1 In general the Schatten p-norms are denoted by ||A|| := tr[|A| p ] 1/ p . p
(1)
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When considering T as a linear map on Hd the natural norm is given by ||T || := sup ||T (A)||2 = ||Tˆ ||∞ . ||A||2 =1
(2)
We denote by P and P+ the sets of linear maps on Md which are positive and completely positive respectively. The corresponding subsets of trace preserving maps will be denoted by T, T+ and the elements of the latter are called channels (in the Schrödinger picture). Following Jamiolkowski’s state-channel duality [5,6] we can assign to every channel T a state (density operator) τ by acting with T on half of a maximally entangled state ω = d1 i,d j=1 |ii j j|: τ = (T ⊗ id)(ω).
(3)
The rank of this Jamiolkowski state τ (the un-normalized form of which is often called Choi matrix) is equal to the Kraus rank of T , i.e., the minimal number of terms in a Kraus representation [7] T (A) = α K α AK α† . Moreover, with the involution i j|τ |kl := ik|τ | jl the matrix τ leads to a matrix representation of T (with matrix units as chosen basis [8]) such that Tˆ = dτ = K α ⊗ K¯ α . (4) α
T∗
we will denote the dual of a map T defined by tr[T ∗ (A)B] = tr[AT (B)]. If By T is trace-preserving then T ∗ is unital, i.e., T ∗ (1) = 1 and the matrix representation corresponding to T ∗ is given by the adjoint Tˆ † . A channel will be called Markovian if it is an element of a completely positive continuous one-parameter semigroup. That is, there exists a generator L : Md → Md with L ∗ (1) = 0 such that Tt = et L ∈ T+ for all t ≥ 0. Two equivalent standard forms for such generators were derived in [9] and [10]:
1 † (5) Fβ Fα , ρ G α,β Fα ρ Fβ† − L(ρ) = i[ρ, H ] + + 2 α,β
1 = i[ρ, H ] + φ(ρ) − {φ ∗ (1), ρ}+ , 2
(6)
where G ≥ 0, H = H † and φ ∈ P+ . The decomposition of the generator L into a Hamiltonian part (i[·, H ]) and a dissipative part (L − i[·, H ]) becomes unique [10] if the sum in Eq. (5) runs only over traceless operators (tr[Fγ ] = 0) from an orthonormal basis in Hd (e.g., the one in Eq. (1)). We will in the following always understand Eq. (5) in this form and call L and the corresponding semigroup purely dissipative if H = 0 w.r.t. such a representation. Clearly, not every channel is Markovian (see [32] for necessary and sufficient conditions). However, the following lemma allows us to assign a semigroup to each channel: Lemma 1 (Markovian approximation). For every channel T ∈ T+ we have that et (T −id) , t ≥ 0 is a completely positive semigroup. Moreover, if U0 is the unitary conjugation2 for which the supremum supU tr H[T U ] is attained, then (T U0 − id) is the generator of a purely dissipative semigroup. 2 By unitary conjugation we mean a channel of the form ρ → VρV † with V being a unitary.
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Proof. We will first show that (T − id) is a valid generator by bringing it into the form of Eq. (6). Define φ(ρ) := α Aα ρ A†α with Kraus operators Aα = K α − xα 1, where {K α } are the Kraus operators of T and x is any unit vector. Then
(T − id)(ρ) = φ(ρ) + κρ + ρκ † , T ∗ (1)
(7) φ ∗ (1)
with κ = α x¯α Aα . The trace preserving property = 1 imposes that + κ + κ † = 0 so that the Hermitian part of κ is −φ ∗ (1)/2. If we denote by −i H with H = H † the anti-Hermitian part then κ = −φ ∗ (1)/2 − i H which leads to the form in Eq. (6), proving the first statement. Note that there is freedom in the choice of the anti-Hermitian part of κ as Eq. (7) is invariant under adding to κ a multiple of i1. In order to prove the second statement we have to exploit the freedom [11] in the decomposition into dissipative and Hamiltonian parts where the latter corresponds (up to multiples of i1) to the anti-Hermitian part of κ. Note that (T − id)(ρ) = (Aα − aα 1)ρ(Aα − aα 1)† + κa ρ + ρκa† , (8) α
κa = κ +
a¯ α Aα − 1|aα |2 /2
(9)
α
=
(x¯α + a¯ α )(K α − xα 1) − 1|aα |2 /2
(10)
α
gives other representations of the same generator for any complex vector a. The representation of the generator has traceless Kraus operators in Eq. (8) iff (xα + aα ) = tr[K α ]/d. For this choice of a the imaginary part of κa in Eq. (10) would thus indeed be a multiple of i1 if α tr[ K¯ α ]K α ≥ 0. Let us now show that exactly this is achieved by concatenating T with U0 . To this end note that by exploiting Eq. (4) we get (11) tr[K α V ]tr[K α V ] tr H[T U0 ] ≤ sup V,V
α
√ = sup φV |τ |φV ≤ sup || τ |φV ||2 V,V
(12)
V
= tr H[T U0 ], (13) √ d |ii. On the one hand the r.h.s. where V, V are unitaries and |φV = d(V ⊗ 1) i=1 of Eq. (11) is maximized if V and V are unitaries from the polar decomposition of the remaining parts, i.e., V for instance is the polar unitary of α tr[K α V ]K α . On the other hand it follows from equality to tr H[T U0 ] that the maximum is attained for V = V so † that for U0 (·) = V · V we get α tr[K α V ]K α V ≥ 0 concluding the proof. III. Determinants The multiplicativity property of determinants det(T1 T2 ) = (det T1 )(det T2 ) makes them an indispensable tool for the study of semigroup properties of sets of linear maps. The following theorem contains some of their basic properties. Though the results of this section are necessary for subsequent proofs they are not essential for understanding the parts on divisibility, so that this section might be skipped by the reader. Theorem 2 (Determinants). Let T : Md → Md be a linear positive and trace preserving map.
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1. det T is real and contained in the interval [−1, 1]. 2. | det T | = 1 iff T is either a unitary conjugation or unitarily equivalent to a matrix transposition. 3. If T is a unitary conjugation then det T = 1, and if det T = −1 then T is a matrix transposition up to unitary equivalence. In both cases the converse holds iff d2 is odd. Proof. First note that every positive linear map satisfies T (A† ) = T (A)† for all A ∈ Md . This becomes obvious by writing A as a linear combination of four positive matrices and using linearity of T . As a consequence all eigenvalues either come in complex conjugate pairs or are real so that det√ T is real. From the boundedness of the norm of any trace preserving T ∈ T (||T || ≤ d [12]) together with the fact [14] that the spectral radius equals limm→∞ ||T m ||1/m it follows that the spectral radius is one which implies det T ∈ [−1, 1]. Now consider the case det T = ±1 where all eigenvalues are phases. There is always a sequence n i such that the limit of powers limi→∞ T n i =: T∞ has eigenvalues which all converge to one.3 To see that this implies that T∞ = id consider a two-by-two block ˆ ni on thediagonal of the Schur decomposition of T . Up to a phase this block is of the 1 c . Thus by taking the p th power of T n i this is mapped to form 0 ei p−1 ik 1 k=0 e c . 0 ei p As → 0 for n i → ∞ the norm ||(T n i ) p || could be increased without limit (by increasing p with n i ) unless c → 0. However, all powers of T are trace preserving and positive and have therefore to have bounded norm. This rules out the survival of Jordan block-like off-diagonal elements so that T∞ = id. Hence, the inverse T −1 = T∞ T −1 = limi→∞ T n i −1 is a trace preserving positive map as well. Assume that the image of any pure state under T is mixed, i.e., T () = λρ1 + (1 − λ)ρ2 with ρ1 = ρ2 . Then by applying T −1 to this decomposition we would get a nontrivial convex decomposition for (due to positivity of T −1 ) leading to a contradiction. Hence, T and its inverse map pure states onto pure states. Furthermore, they are unital, which can again be seen by contradiction. So assume T (1) = 1. Then the smallest eigenvalue of T (1) satisfies λmin < 1 due to the trace preserving property. If we denote by |λ the corresponding eigenvector, then 1 − λmin2 +1 T −1 (|λλ|) is a positive operator, but its image under T would no longer be positive. Therefore we must have T (1) = 1. Every unital positive trace preserving map is contractive with respect to the Hilbert-Schmidt norm [12,15]. As this holds for both T and T −1 we have that ∀A ∈ Md : ||T (A)||2 = ||A||2 , i.e., T acts unitarily on the Hilbert space. In particu Schmidt Hilbert lar, it preserves the Hilbert Schmidt scalar product tr T (A)T (B)† = tr[AB † ]. Applying this to pure states A = |φφ| and B = |ψψ| shows that T gives rise to a mapping of the Hilbert space onto itself which preserves the value of |φ|ψ|. By Wigner’s theorem [16,17] this has to be either unitary or anti-unitary. If T is a unitary conjugation then det T = det(U ⊗ U¯ ) = 1. Since every anti-unitary is unitarily equivalent to complex conjugation, we get that T is in this case a matrix transposition T (A) = A T (up to unitary equivalence). The determinant of the matrix transposition is easily√ seen in √ the Gell-Mann basis of Md . That is, we take basis elements Fα of the form σx / 2, σ y / 2 3 This is an immediate consequence of Dirichlet’s theorem on Diophantine approximations [13].
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for α = 1, . . . , d 2 − d and diagonal for α = d 2 − d + 1, . . . , d 2 . In this basis matrix transposition is diagonal and has eigenvalues 1 and −1 where the latter appears with multiplicity d(d − 1)/2. This means that matrix transposition has determinant minus one iff d(d − 1)/2 is odd, which is equivalent to d2 being odd. From this we get the following important corollary: Corollary 3 (Monotonicity of the determinant). Consider the set T of positive trace preserving linear maps on Md . 1. T, T −1 ∈ T iff T is a unitary conjugation or matrix transposition. 2. The determinant of T ∈ T is decreasing in magnitude under composition, i.e., | det T | ≥ | det T T | for all T ∈ T where equality holds iff T is a unitary, a matrix transposition or det T = 0. Part 1 of this corollary is a simple consequence of Wigner’s theorem and was proven for completely positive maps for instance in [19].4 One might wonder whether completely positive maps can have negative determinants. The following simple example answers this question in the affirmative. It is built up on the map ρ → ρ Tc which transposes the corners of ρ ∈ Md , i.e., (ρ Tc )k,l is ρl,k for the entries (k, l) = (1, d), (d, 1) and remains ρk,l otherwise. Note that for d = 2 this is the ordinary matrix transposition. Example 4. The map T : Md → Md defined by T (ρ) =
ρ Tc + 1trρ 1+d
(14)
is trace preserving, completely positive with Kraus rank d 2 − 1 and has determinant 2 det T = − (d + 1)1−d . For d = 2 the channel is entanglement breaking and can be written as 1 |ξ¯ j ξ j |ρ|ξ j ξ¯ j |, 3 6
T (ρ) =
(15)
j=1
where the six ξ j are the normalized eigenvectors of the three Pauli matrices. Proof. A convenient matrix representation √ of the channel is given in the generalized Gell-Mann basis. Choose F1 as the σ y / 2 element corresponding to the corners and √ F2 = 1/ d the only element which is not traceless. Then Tˆ = diag[−1, 1+d, 1, . . . , 1]/ 2 (d + 1) leading to det T = −(d + 1)1−d . For complete positivity we have to check positivity of the Jamiolkowski state τ . The corner transposition applied to a maximally entangled state leads to one negative eigenvalue −1/d. This is, however, exactly compensated by the second part of the map such that τ ≥ 0 with rank d 2 − 1. The representation for d = 2 is obtained from tr[AT (B)] = tr[(A ⊗ B T )τ ]d by noting that in this case τ is proportional to the projector onto the symmetric subspace which in turn can be written as 21 j |ξ j ξ j |⊗2 in agreement with the given Kraus representation of the channel. 4 It is also a consequence of [18] and therefore sometimes called the Wigner-Kadison theorem.
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The above example has Kraus rank d 2 − 1. Channels of Kraus rank two and compositions thereof can never lead to negative determinants: Theorem 5 (Kraus rank two maps). All linear maps on Md which are completely positive with Kraus rank two have non-negative determinant. Hence every composition of such Kraus rank two maps has non-negative determinant. Proof. Let A and B be two Kraus operators of the map T and assume for the moment that det A = 0. Then, using matrix units as a basis for the Hilbert-Schmidt Hilbert space, we ¯ B⊗ B. ¯ If we use the singular value decomcan represent the channel by the matrix A⊗ A+ position of A = U SV we can write the determinant as det T = (det S)2d det(1+ B ⊗ B¯ ) with B = S −1/2 U † BV † S −1/2 . Denoting
the eigenvalues
of B by bk we obtain det T = (det S)2d k,l (bk b¯l + 1) = (det S)2d k
(17)
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and in the limit det T → 1 the distance between T and the unitary conjugation U0−1 appearing in Lemma 1 vanishes as √ ||T − U0 −1 || = O 1 − det T . (18) Remark. An explicit however lengthy bound for the norm distance in Eq. (18) can easily be deduced from the subsequent proof. Proof. To relate the purity to the determinant we exploit that by Eq. (4) Tˆ = dτ is a matrix representation of the channel, so that 2
d 1 1 2 tr[τ ] = 2 tr[Tˆ † Tˆ ] = 2 si , d d 2
(19)
i=1
where the si are the singular values of Tˆ (and thus T ). From this Eq. (17) is obtained
via the geometric–arithmetic mean inequality together with the fact that | det T | = i si . We now use the purity bound (17) to prove the scaling in Eq. (18). The aim of the following is to relate first the purity to the largest eigenvalue of τ and then the latter to the sought distance ||T − U0−1 || = ||T U0 − id||. The largest eigenvalue µ := ||τ ||∞ of the Jamiolkowski state is for tr[τ 2 ] ≥ 1/2 lower bounded by the purity via 1 1 + 2tr[τ 2 ] − 1 . (20) µ≥ 2 If we denote by = |ψψ| the projector onto the eigenstate corresponding to µ then ||1/d − tr A ||1 ≤ ||τ − ||1 ≤ 2(1 − µ),
(21)
where the first inequality follows from the monotonicity of the trace norm distance under the partial trace. We will now use the bounds between the trace norm distance and the fidelity [20] f (σ, ρ) := tr ρ 1/2 σρ 1/2 : 1 (22) 1 − f (ρ, σ ) ≤ ||ρ − σ ||1 ≤ 1 − f (ρ, σ )2 . 2 By Uhlmann’s theorem [21] we have that f (tr A , 1/d) = sup ||ψ|, where the supremum is taken over all maximallyentangled √ states . Denote by τ0 the Jamiolkowski state of T U0 and |0 = |ii/ d so that ω = |0 0 |. Then by i Eqs. (11–13) we have that 0 |τ0 |0 = sup |τ | which is in turn lower bounded by µf (tr A , 1/d)2 . Together with Eqs. (21,22) this gives 0 |τ0 |0 ≥ µ3 .
(23)
Finally, we have to relate the distance between the Jamiolkowski states τ0 and ω to that of the respective channels. This can be done by exploiting tr[AT (B)] = tr[(A ⊗ B T )τ ]d so that in general ||T1 − T2 || = d sup A,B |tr[(τ1 − τ2 )A ⊗ B]|, where the supremum is taken over all operators with ||A||2 = ||B||2 = 1 such that an upper bound is given by d||τ1 − τ2 ||1 . In this way we get ||T U0 − id|| ≤ d||τ0 − ω||1 ≤ 2d 1 − 0 |τ0 |0 ≤ 2d 1 − µ3 . (24) The scaling in Eq. (18) is then obtained by combining Eqs. (17, 20, 24) and expanding around det T = 1.
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IV. Divisible and Indivisible Maps In the following we will apply the above results and study decompositions of channels in terms of concatenations of other channels, i.e., the possibility of writing T ∈ T+ as T = T1 T2 , Ti ∈ T+ . As the notion decomposable is commonly used in the context of convex decompositions and often refers to a specific convex decomposition of positive maps [22], we will use the notion divisible instead. Clearly, every channel is divisible in a trivial way T = (T U −1 )U , where U is any unitary conjugation. In order to make the divisibility of a channel a non-trivial concept we thus define it up to unitary conjugation: Definition 8 (Divisibility). Consider the set T ∈ {T, T+ } of linear trace preserving positive or completely positive maps from Md into itself. We say that T ∈ T is indivisible if every decomposition of the form T = T1 T2 with Ti ∈ T is such that one of the Ti has to be a unitary conjugation. T is called divisible if it is not indivisible. That this concept is not empty, i.e., that indivisible maps indeed exist is now a simple consequence of Thm. 2. Corollary 9 (Indivisible positive maps). Consider the case where d2 is odd. Then the matrix transposition θ : ρ → ρ T , ρ ∈ Md is indivisible within the set T of positive trace preserving maps on Md . Proof. Assume that the matrix transposition θ has a decomposition θ = T1 T2 . Then by Thm. 2 det(T1 ) = − det T2 = ±1. Hence, one of the two maps has to be a transposition and the other a unitary. Clearly, this simple observation is reminiscent of the fact that a reflection can not be expressed in terms of rotations (with positive determinant). The following is a less trivial example based on the same idea: Corollary 10 (Indivisible completely positive maps). Consider the set T+ of completely positive trace preserving maps on Md . The channel T0 ∈ T+ with minimal determinant, i.e., for which det T0 = inf T ∈T+ det T , is indivisible. For d = 2 we have that T0 (ρ) = (ρ T + 1)/3 is the channel discussed in Example 4. Proof. By Example 4 there are always channels with det T < 0. As the set T+ of channels on Md is compact there is always a map T0 for which inf T ∈T+ det T is attained. Now consider a decomposition T0 = T1 T2 . Then by Thm. 2 and Cor. 3 either T1 or T2 has to be unitary. For the case d = 2 recall that T can be conveniently represented in terms of the real 4 × 4 matrix Tˆi j := tr[σi T (σ j )]/2, where the σi s are identity and Pauli matrices. In general 1 0 ˆ , (25) T = v where v ∈ R3 , is a 3 × 3 matrix and det = det T . To simplify matters we can diagonalize by special orthogonal matrices O1 O2 = diag{λ1 , λ2 , λ3 } corresponding to unitary operations before and after the channel. Obviously, this does neither change is conthe determinant, nor complete positivity. For the latter it is necessary that λ tained in a tetrahedron spanned by the four corners of the unit cube with λ1 λ2 λ3 = 1
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[23,24]. Fortunately, all these points can indeed be reached by unital channels (v = 0) for which this criterion becomes also sufficient for complete positivity. By symmetry we can restrict our attention to one octant and reduce the problem to maximizing p1 p2 p3 over all probability vectors p yielding p = ( 13 , 13 , 13 ) = (λ1 , λ2 , −λ3 ). Hence, the minimal determinant is −( 13 )3 and the corresponding channel can easily be constructed from Tˆ as T : ρ → 13 (ρ T + 1). The channels with minimal determinant lie at the border of T+ , i.e., they have reduced Kraus rank. In fact, for channels with full Kraus rank (d 2 ) one can easily see that they are all divisible: Theorem 11 (Divisibility of generic channels). If a channel T ∈ T+ on Md has Kraus rank d 2 , then T is divisible. Proof. Note that T has full Kraus rank iff the corresponding Jamiolkowski state τ = (T ⊗ id)(ω) has full rank. Let T1 ∈ T+ be any invertible non-unitary channel. Then (T1−1 T ⊗ id)(ω) is still positive if only T1 is sufficiently close to the identity. Therefore T2 := T1−1 T is an admissible channel so that T = T1 T2 . Clearly, one can always choose T1 such (e.g. of Kraus rank two, or det T1 = det T ) that neither T1 nor T2 are unitary.
We will now see that in searching for a decomposition of a trace preserving map T =T1 T2 we can essentially drop the trace preserving constraint on T1 and T2 . That is, if there exists a non-trivial decomposition into non-trace preserving maps, then there will be one in terms of trace preserving maps as well: Theorem 12. Let P ∈ {P, P+ } be either the set of positive or completely positive linear maps on Md , and T ∈ {T, T+ } the respective subset of trace preserving maps. Then for every concatenation T˜1 T˜2 = T , T˜i ∈ P, T ∈ T with det T˜1∗ (1) = 0 there exist T1 , T2 ∈ T with Kraus rank5 rank[Ti ] = rank[T˜i ] such that T1 T2 = T . Proof. We will explicitly construct T1 and T2 via their duals. Due to positivity and the absence of a kernel in T˜1∗ (1) we can find a positive definite matrix P > 0 which is the square root of T˜1∗ (1) = P 2 . Then T1∗ (X ) := P −1 T˜1∗ (X )P −1 fulfills T1∗ (1) = 1 and is thus the dual of a map T1 ∈ T . Defining T2∗ (X ) := T˜2∗ (P X P) we obtain T2∗ T1∗ = T ∗ so that indeed T1 T2 = T . Moreover, T2 ∈ T since T2∗ (1) = T2∗ T1∗ (1) = T ∗ (1) = 1. Equality for the Kraus ranks follows immediately from the fact that Ti and T˜i differ merely by concatenation with an invertible completely positive Kraus rank-one map.
For the classification of (in-)divisible maps this allows us to restrict to equivalence classes under invertible filtering operations. In Sect. VI this reduction will enable us to completely characterize the set of indivisible qubit channels. Corollary 13 (Reduction to normal form). Let T ∈ {T, T+ } and T, T˜ ∈ T be related via T = T A T˜ TB where T A , TB ∈ P+ are invertible completely positive maps with Kraus rank one. Then T is divisible iff T˜ is divisible. 5 For positive maps we define the Kraus rank as the rank of the corresponding Jamiolkowski operator (Choi matrix).
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V. Infinitesimal Divisible Channels In this section we will refine the somewhat coarse notion of divisibility by asking which channels can be broken down into infinitesimal pieces, i.e., into channels arbitrary close to the identity. This will lead us to a number of a priori different sets of channels, depending on the additional structure which we impose on the infinitesimal constituents. The main result will then be the equivalence of three of these sets, showing that the imposed structure is not an additional requirement but rather emerges naturally. Let us begin with the most structured and best investigated of these sets: the set of Markovian channels. Evidently, a Markovian channel, i.e., an element of a continuous completely positive one-parameter semigroup is divisible. Furthermore it can be divided into a large number of equal infinitesimal channels and it is the solution of a time-independent master equation ∂ρ = L(ρ), ∂t
(26)
with L of the form in Eqs. (5, 6). Following the terminology used in classical probability theory one calls a channel T infinitely divisible [1,4] if for all n ∈ N there is another channel Tn such that T = Tnn . It was shown in [4] that infinitely divisible channels are all of the form T = T0 e L , where L is a Lindblad generator of the form in Eq. (5) and T0 is an idempotent channel satisfying T0 L = T0 L T0 . Hence, an infinitely divisible channel becomes an element of a continuous completely positive one-parameter semigroup if T0 = id. Consider now a more general family, which one might refer to as continuous completely positive evolutions. That is, for some time interval [0, t] there exists a continuous mapping [0, t] × [0, t] → T+ onto a family of quantum channels {T (t2 , t1 )} such that 1. T (t3 , t2 )T (t2 , t1 ) = T (t3 , t1 ) for all 0 ≤ t1 ≤ t2 ≤ t3 ≤ t, 2. lim →0 ||T (τ + , τ ) − id|| = 0 for all τ ∈ [0, t). In other words there is a continuous path within T+ which connects the identity with each element of this family and along which we can move (one-way) by concatenation with quantum channels. Let us denote by J ⊂ T+ the set of all elements of such continuous completely positive evolutions. Clearly, this set is included in the following: Definition 14 (Infinitesimal divisibility). Define a set I of channels T ∈ T+ with the property that for all > 0 there exists a finite set of channels Ti ∈ T+ such that (i) ||Ti − id|| ≤ and (ii) i Ti = T . We say that a channel is infinitesimal divisible if it belongs to the closure I. Remark. Note that every infinitely divisible channel is also infinitesimal divisible. To see this note that for every idempotent channel T0 we have that [(1 − )id + T0 ]n is a product of channels which are -close to the identity with convergence to T0 for n → ∞. By continuity and multiplicativity of the determinant we obtain a simple necessary condition for a channel to be infinitesimal divisible: Proposition 15. If a channel T ∈ T+ is infinitesimal divisible, then det T ≥ 0. A similar notion of infinitesimal divisibility can be defined by introducing a set I analogous to the set I with the additional restriction that all the Ti ∈ T+ have to be Markovian, i.e., of the form Ti = e L i with L i a Lindblad generator. Clearly, I ⊆ I and
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intuitively the converse should also hold as every channel close to the identity should be ‘almost Markovian’. However, the closer the Ti are to the identity, the more terms we need in the product in Ti = T . Hence, n will be an increasing function of and the question whether or not one can safely replace each Ti by a Markovian channel amounts to the estimation of an accumulated error of the form “n ”. The following theorem shows that the scaling of the latter is benign so that indeed I = I . Moreover, since I ⊆ J ⊆ I both sets are equal to the set of continuous completely positive evolutions. Theorem 16 (Structure of infinitesimal divisible channels). With the above notation we have that I = I = J . In particular, every infinitesimal divisible channel can be arbitrary well approximated by a product of Markovian channels. Proof. We want to show that one can replace every channel Ti in the decomposition n T = i=1 Ti with ||Ti − id|| ≤ by a Markovian channel such that the error becomes negligible in the limit → 0 (and thus n → ∞). This is proven in two steps: (i) we calculate the error obtained from the Markovian approximation in Lem. 1 as a function of n and , and (ii) we relate n and by exploiting properties of the determinant shown in Thms. 6, 7. Strictly speaking, we will in both steps not use the distance to the identity but rather a distance δ ≤ to a nearby unitary.
First we write i Ti = i T˜i Ui , where T˜i = Ti Ui−1 is such that (T˜i − id) is a purely dissipative generator according to Lem. 1. The idea is then to approximate T˜i by exp(T˜i − id). The total error in this approximation is then given by ˜ (27) Ti − e Ti −id Ui = Ti − (Ti + i ) , i
i
i
i
where i = (exp[T˜i −id]− T˜i )Ui is an operator whose norm vanishes as O(||T˜i −id||2 ).
The product i (Ti + i ) contains ( nk ) terms of the form “Tin−k ik ” where the Ti s come √ in at most k + 1 groups for each of which we can bound the norm by d [12]. If we define δ := maxi ||T˜i − id|| we can therefore bound the error in Eq. (27) by √ n √ Ti − (Ti + i ) ≤ d 1 + d O(δ 2 ) − 1 . (28) i
i
This vanishes iff δ 2 n → 0 as n → ∞.6 To relate δ and n we use Thm. 6 from which we obtain δ ≤ − d2 mini ln det exp(T˜i − id). Exploiting continuity of the determinant7 and denoting by T˜δ the channel Ti giving rise to the maximum distance δ, this gives 2 (29) δ ≤ − ln det T˜δ − O(δ 2 ) . d
Since by assumption there are arbitrarily fine-grained decompositions T = i Ti we can w.l.o.g. assume that all Ti have equal determinant det Ti = (det T )1/n (or ones distributed within a sufficiently narrow interval). As det T˜i = det Ti Eq. (29) relates n and
n 6 An alternative way for obtaining this result is by defining C(l) := l i=1 Ti j=l+1 (T j + j ). Then n−1 Eq. (27) equals ||C(n) − C(0)|| = || k=0 C(k + 1) − C(k)|| ≤ (n + 1)dδ 2 , where the inequality follows from the triangle inequality. 7 | det A − det B| ≤ d||A − B|| max{||A||, ||B||}d−1 [14].
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δ—unfortunately in a way that we cannot yet conclude that δ = o(n −1/2 ). However, −q ) it enables us to lift any polynomial bound to higher order: assume that δ = O(n −2q for some q ∈ (0, 1). Then Eq. (29) gives rise to δ = O n − (ln det T )/n which leads recursively to δ = O(1/n) provided that det T > 0. Hence, any bound of the form δ = O(n −q ), q > 0 will suffice to show that the error given by Eq. (27) vanishes asymptotically. Such a bound is provided by Thm. 7 as we obtain from Eq. (18) that √ δ = O −(ln det T )/n . Note finally that it suffices to consider the case det T > 0 as singular channels are only included in Def. 8 by taking the closure of I and det T < 0 is excluded by Prop. 15. Similar to the notion of divisibility we may introduce infinitesimal divisible positive maps by replacing T+ in Def. 14 by T. In both cases we can again decide whether a map is infinitesimal divisible by considering its normal form under invertible filtering operations with Kraus rank one: Theorem 17 (Reduction to normal form). Let T ∈ {T, T+ } and T, T˜ ∈ T be related via T = T A T˜ TB , where T A , TB ∈ P+ are invertible completely positive maps with Kraus rank one. Then T is infinitesimal divisible iff T˜ is. Proof. As the statement is symmetric in T and T˜ (due to invertibility of T A , TB ) it is
n T˜i is infinitesimal sufficient to prove one direction. So let us assume that T˜ = i=1
n −1 Ri T˜i Ri+1 , where Ri ∈ P+ are invertible maps divisible. Then we can write T = i=1 −1 of Kraus rank one with R1 = T A and Rn+1 = TB . We will now show that the interme−1 diate Ri ’s can be chosen such that Ti := Ri T˜i Ri+1 ∈ T is such that ||Ti − id|| vanishes ˜ uniformly as ||Ti − id|| ≤ → 0. This is achieved by recursively constructing Ri+1 from Ri according to the proof of Thm. 12 and exploiting that −1 −1 − id|| + ||Ri || ||Ri+1 ||. ||Ti − id|| ≤ ||Ri Ri+1
(30)
Let us denote by K i = Ui Pi the polar decomposition of the Kraus operator of Ri (·) = −1∗ ˜ ∗ ∗ Ti Ri (1) = 1 which K i · K i† . The trace preserving requirement for Ti imposes that Ri+1 ∗ 2 is achieved by choosing Pi+1 = T˜i (Pi ). As any unital positive map is spectrumwidth decreasing [19] we have for the range of eigenvalues [λmin (Pi+1 ), λmax (Pi+1 )] ⊆ [λmin (Pi ), λmax (Pi )]. This allows us to bound the second term in Eq. (30) by ||Ri || −1 || ≤ λ2max (P1 )λ−2 ||Ri+1 min (P1 ). √ To bound the first term note that ||T˜i∗ (Pi2 ) − Pi2 || ≤ ||Pi2 ||2 ≤ λ2max (Pi ) d. By √ continuity of the square root8 this implies ||Pi+1 − Pi || ≤ d 1/4 λmax (Pi ). Hence, √ √ −1 ||Pi Pi+1 − 1|| ≤ d 1/4 λmax (Pi )λ−1 min (Pi+1 ) yielding a bound for the first term in Eq. (30) if we take Ui+1 = Ui . The latter choice might not be possible in the n th step (as the trace preserving requirement only fixes TB up to a unitary conjugation). However, we can always add an additional unitary without changing the property of being infinitesimal divisible. Note that the above reduction to normal form together with Thm. 16 preserves continuity in the sense that if T = T A e L TB with Markovian e L ∈ T+ , then we can write T =Te
1 0
L(τ )dτ
√ √ 8 || A − B|| ≤ ||A − B||1/2 for all positive A, B [14].
,
(31)
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where T is the time-ordering operator and τ → L(τ ) is a continuous mapping onto generators of the form in Eqs. (5, 6). In other words, T is then a solution of a timedependent master equation dρ/dt = L(t)ρ. The fact that every generic infinitesimal divisible channel can be written in this way is proven below for the case d = 2 of qubit channels. VI. Qubit Channels The simplicity of qubit channels (T : M2 → M2 ) often allows a more thorough analysis of their properties. An exhaustive investigation of the convex structure of the set of qubit channels and positive trace-preserving qubit maps was for instance given in [25] and [26] respectively. Similarly, their normal form under invertible filtering operations was determined in [27]. In the following we will make extensive use of these results in order to derive a complete characterization of the above discussed semigroup structure of this set. We begin by recalling some of the basic tools and treat the case of extremal qubit channels (two Kraus operators) first, as later argumentation will build on this. The main results—a complete characterization of divisible and infinitesimal divisible qubit channels—are then stated in Thm. 23 and Thm. 24. The representation we will mainly use in the following is a real 4 × 4 matrix Tˆi j := tr[σi T (σ j )]/2 (cf. [25]) which is in turn characterized by a 3 × 3 block and a vector v ∈ R3 encoding the correlations and the reduced density matrix of the Jamiolkowski state respectively: 1 0 Tˆ = . (32) v Since there is an epimorphism from SU (2) to the rotation group SU (3) we can always diagonalize by acting unitarily before and after T . More specifically, for any T ∈ T there exist unitary conjugations U1 , U2 such that U1 T U2 has = diag(λ1 , λ2 , λ3 ) with 1 ≥ λ1 ≥ λ2 ≥ |λ3 |. Expressing complete positivity in terms of v and λ is rather involved and discussed in detail in [23–25]. A necessary condition for complete positivity is that λ1 + λ2 ≤ 1 + λ3 ,
(33)
which becomes sufficient if the channel is unital, i.e., v = 0. A very useful standard form for qubit channels is obtained when building equivalence classes under filtering operations [27].9 Theorem 18 (Lorentz normal form). For every qubit channel T ∈ T+ there exist invertible T A , TB ∈ P+ , both of Kraus rank one, such that T A T TB = T˜ ∈ T+ is of one of the following three forms: 1. Diagonal: T˜ is unital (v = 0). This√is the generic case. √ 2. Non-diagonal: T˜ has = diag(x/ 3, x/ 3, 1/3), 0 ≤ x ≤ 1 and v = (0, 0, 2/3). These channels have Kraus rank 3 for x < 1 and Kraus rank 2 for x = 1. 3. Singular: T˜ has = 0 and v = (0, 0, 1). This channel has Kraus rank 2 and is singular in the sense that it maps everything onto the same output. 9 This standard form is referred to as Lorentz normal form as the mapping T → T T T corresponds to A B Tˆ → L A Tˆ L B , where L A,B are proper orthochronous Lorentz transformations [27].
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A concatenation of qubit channels T1 T2 = T corresponds to a multiplication of the respective matrices Tˆ1 Tˆ2 = Tˆ so that 1 2 = and 1 v2 + v1 = v. In this way we can for instance decompose every channel of the second form in Thm. 18 into ⎛ 1 √ x/ 3 ⎝
⎞
⎛ 1 √ 1/ 3 ⎠=⎝
√ x/ 3
2/3
1/3
⎞⎛
⎠⎝ x
√ 1/ 3
2/3
⎞
1
⎠,
x
1/3
(34)
1
which is a concatenation of two Kraus rank-two channels (unless x = 1 where the initial channel is already rank-two). Let us now have a closer look at qubit channels with Kraus rank two.
A. Extremal qubit channels. Channels with Kraus rank two play an important role regarding the convex structure of the set of qubit channels. It was shown in [25] that every extreme point of this set is either a unitary conjugation or a (non-unital) Kraus rank-two channel. In this context it has been shown that every Kraus rank-two channel can up to unitary conjugations be represented by ⎞ ⎛ 1
Tˆ = ⎝
⎠, cu = cos u, su = sin u.
cu su sv
cv
(35)
cu cv
For the remainder of this subsection we will, however, use a different representation which is very handy for our purposes albeit less explicit than the one in Eq. (35). This will allow us to prove the following: Theorem 19 (Infinitesimal divisibility of Kraus rank-two channels). Let T : M2 → M2 be a qubit channel with Kraus rank two. Then there exist unitary conjugations U1 , U2 , a continuous time-dependent Lindblad generator L and t > 0 such that U1 T U2 = T e
t 0
L(τ )dτ
.
(36)
In order to prove this result, we will first introduce the mentioned normal form and then explicitly construct the Lindblad generators. To this end consider the set of specific channels C ∈ T+ with Kraus operators A1 = |0a|, A2 = |0b| + x|11|.
(37a) (37b)
We will take x and the zero components of |a and |b real. The trace preserving condition gives |aa| + |bb| = 1 − x 2 |11|. (38) We will prove that all channels C are of the form on the r.h.s. of Eq. (36), which, together with the following lemma, will yield the proof of the theorem. Lemma 20. For any qubit channel T with Kraus rank two, there exist unitary conjugations U1 , U2 such that T = U1 CU2 .
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Proof. Given the Kraus operators K 1,2 of T , we can always find α1,2 such that α1 K 1 + α2 K 2 has rank 1 (i.e., zero determinant). Thus, a different set of Kraus operators can be chosen with Kˆ 1 = |e0 f 1 |, and Kˆ 2 = |e0 f 2 | + |e1 f 3 |, where e0,1 are orthonormal. Defining Ai = V1 Kˆ i V2 , with V1 , V2 unitaries, using the fact that we can multiply Kraus operators with complex numbers of unit modulus, and imposing that the channel is trace preserving, we easily reach the above form. Thus, from now on we concentrate on the specific channels C. Depending on the vectors a, b, we can have very different channels. We define: Definition 21. Given a channel of the above form C, we will call it: (i) class-1 if a|0 = b|1 = 0; (ii) class-2 if it is not in class-1 and x = 1; (iii) class-3 otherwise. The main difference between these channels lies on the number of pure states that are mapped into pure states. In fact, it can be easily checked that for all channels |0 → |0 and that for class-1 channels, either all pure states are mapped into |0 (for x = 0) or only |0 is mapped into a pure state (for x = 0), whereas for class-2 and 3, apart from |0, there is only one state |c ⊥ |a which is mapped into a pure state. In the following we will consider the different classes of channels independently. a. Class-1 channels. We can write |a = (1 − x 2 )1/2 |1 and |b = |0, so that all these channels are parametrized just by x, and therefore we will write C x . We have C x1 C x2 = C x1 x2 , C1 = 1.
(39)
Thus, this class forms a continuous 1-parameter semigroup. Using infinitesimal transformations one can easily show that C x = e− ln(x)L ,
L(ρ) = 2|01|ρ|10| − ρ|11| − |11|ρ.
(40)
b. Class-2 channels. In this case we can write |a = (1 − y 2 )1/2 |0 and |b = y|0, so that again we have a single parameter family C y . As before, we obtain a one-parameter semigroup C y = exp(− ln(y)L) but now with L(ρ) = 2σz ρσz − 2ρ. c. Class-3 channels. We show now that every channel C in this class is completely determined by the vector different from |0 which is mapped into a pure state. As mentioned above, this class is characterized by the fact that a normalized pure state |c ⊥ |a is mapped into another pure |c : |c = c0 eiϕ |0 + c1 |1, c0 , c1 ∈ R, |c = yc0 eiϕ |0 + xc1 |1,
(41a) (41b)
where y ≥ 1 ensures normalization. That is, since x < 1, the distance to the vector |0 decreases, whereas the azimuthal angle in the Bloch sphere remains constant. Now we will show the converse: Lemma 22. Given |c and |c as in Eq. (41) with x < 1, there exists a unique class-3 channel which maps |c → |c .
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Proof. The definition of c and c fixes the values of x and |a up to a normalization for the Kraus operators (37). Both ||a|| and |b are completely specified by the condition (38). Indeed, defining |a ˜ := |a/||a|| we have to fulfill that 1−x 2 |11|−||a||2 |a ˜ a| ˜ = |bb|, i.e., has rank 1, which automatically fixes ||a||2 =
1 − x2 1 − x 2 c12
(42)
and thereby |b through Eq. (38). The maps in this class are parametrized by x, c1 ∈ (0, 1) and ϕ ∈ [0, 2π ), and thus we will write Cc1 ,x,ϕ . They fulfill C xc1 ,y,ϕ Cc1 ,x,ϕ = Cc1 ,x y,ϕ .
(43)
Note that Cc1 ,x,ϕ → id for x → 1. Thus, we can determine the generator of an infinitesimal transformation as L c1 ,ϕ := lim →0 (id − Cc1 ,e− ,ϕ )/ . We obtain L c1 ,ϕ (ρ) = i[ρ, Hc1 ,ϕ ] + Dc1 ,ϕ (ρ), c1 iϕ e |01| − e−iϕ |10| , Hc1 ,ϕ = ic0
(44) (45)
single Kraus and Dc1 ,ϕ is a simple dissipative √ Lindblad generator characterized by 2a 1/2 operator of the form Ac1 ,ϕ = 2|0(c1 0| − c0 1|)/c0 , with c0 = (1 − c1 ) . Thus, we arrive at the result of Thm. 19 and can write − ln(x) Cc1 ,x,ϕ = T exp L c1 e−τ ,ϕ dτ . (46) 0
B. Divisible and indivisible qubit channels. We are now prepared to give a complete characterization of divisible/indivisible qubit channels. An indivisible example—the channel with minimal determinant—was already given in Corollary 10. Surprisingly, there are indivisible channels with positive determinant as well: Theorem 23 (Indivisible qubit channels). A non-unitary qubit channel is indivisible within T+ if and only if it has Kraus rank three and its Lorentz normal form (Thm. 18) is diagonal (i.e., unital). Proof. As all qubit channels with Kraus rank four are divisible due to Thm. 11 and all rank-two channels are divisible according to the previous subsection, the Kraus rank of indivisible qubit channels must be three (or one—trivially). Following Cor. 13 it suffices to consider the Lorentz normal form of Thm. 18. Since the non-diagonal case can be decomposed via Eq. (34) into divisible Kraus rank-two channels, it remains to show that all unital channels with Kraus rank three are indivisible. Suppose T is such a channel and we can write T = T1 T2 with non-unitary Ti ∈ T+ . Then there is also a decomposition T = T1 T2 into non-unitary unital channels Ti which can for instance be obtained by setting the v’s in Tˆi in Eq. (32) to zero and keeping i = i . This will still be a decomposition of T but neither change the determinant (and thus non-unitarity) nor complete positivity as Eq. (33) becomes a necessary and sufficient condition for unital channels.
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By assumption the Jamiolkowski state τ = (T1 T2 ⊗ id)(ω) has rank three. As unital qubit channels are convex combinations of unitary conjugations and (U ⊗ id)(ω) = (id ⊗ U T )(ω) we can write τ = (T1 ⊗ T2T )(ω),
(47)
where T2T is again a unital channel whose Kraus operators are related to those of T2 by transposition. It follows from Eq. (47) that the Kraus rank of T1 and T2 is at most three. Assume now T2 has Kraus rank three. Then τ ≥ µ(T1 ⊗ id)(1 − ) where is the projector onto a maximally entangled state √ and µ is the smallest non-zero eigenvalue of the Jamiolkowski state of T2 . Thus, if { pi Ui } are the Kraus operators of T1 with {Ui } orthogonal unitaries and { pi } probabilities, then † τ ≥µ 1− pi (Ui ⊗ 1)(Ui ⊗ 1) . (48) i
Since the projectors in the sum are orthogonal, τ can only be rank deficient if there is only a single term in the sum and T1 thus a unitary. The only remaining possibility is thus a decomposition into two unital channels each of Kraus rank two. In order to rule this out note that in this case the support of τ equals that of 1−
2
pi (Ui ⊗ 1)P(Ui ⊗ 1)† ,
(49)
i=1
where P is now some two-dimensional projector. Denoting by ψ the normalized and maximally entangled null vector of τ we have to have that P(Ui ⊗1)† |ψ = (Ui ⊗1)† |ψ so that P=
2 (U j ⊗ 1)† |ψψ|(U j ⊗ 1).
(50)
j=1
Now we exploit the fact that every basis of orthogonal unitaries {U j } in M2 is essentially equivalent to the Pauli basis in the sense that there are always unitaries V1 , V2 and phases eiϕ j such that U j = V1 σ j V2 eiϕ j [28]. It follows that U1 U2† equals U2 U1† up to a phase which in turn implies that the expression in Eq. (49) and thus τ have rank two—contradicting the assumption and therefore concluding the proof. C. Infinitesimal divisible qubit channels. We will now give a necessary and sufficient criterion for qubit channels to be infinitesimal divisible, formulated in terms of the matrix representation Eq. (32) of the channel’s Lorentz normal form (Thm. 18): Theorem 24 (Characterization of infinitesimal divisible channels). Consider a qubit channel and denote by smin the smallest singular value of the -block of its Lorentz normal form. The channel is infinitesimal divisible iff one of the following conditions is true 1. The Lorentz normal form is not diagonal. 2. The normal form is diagonal and rank() < 2.
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3. The normal form is diagonal and 2 smin ≥ det > 0.
(51)
Proof. We exploit the fact that by Thm. 17 a channel is infinitesimal divisible iff its Lorentz normal form is. If the normal form is not diagonal, then by Eqs. (34, 35) it has Kraus rank two or is a product of Kraus rank-two channels which are in turn infinitesimal divisible according to Thm. 19. Similarly, if the normal form is diagonal and = diag(λ, 0, 0) we can again factorize it into Kraus rank-two channels as = diag(1, 0, 0)diag(λ, 1, λ). To complete point 2 in the theorem note that the unital channel with = 0 is a limit of a Markovian unital channel as = limt→∞ e−t 1. Consider now the generic case where the Lorentz normal form is diagonal and det T = 0. Following Prop. 15 we have that det T ≥ 0 for every infinitesimal divisible channel. Moreover, by Thm. 16 we can express these channels in terms of products of Markovian channels, which can w.l.o.g. be chosen unital. The latter can in turn be decomposed into n even simpler pieces by exploiting the Lie-Trotter formula limn→∞ e L 1 /n e L 2 /n = e L 1 +L 2 . In this way every unital Markovian qubit channel can be written as a product of unitaries and unital Kraus rank-two channels with = diag(1, λ, λ) [29]. Note that 2 for these channels we have smin = det T . The inequality Eq. (51) follows then from concatenating these channels together with multiplicativity of the determinant and the fact that smin (1 )smin (2 ) ≤ smin (1 2 ). Let us now show the converse, i.e., that Eq. (51) together with a diagonal Lorentz normal form implies that the channel is infinitesimal divisible. To this end we introduce t := exp (t ln ), t ≥ 0 and show that it corresponds to a completely positive unital semigroup if (chosen positive definite and diagonal) satisfies Eq. (51). Following Eq. (33) we have to show that trt ≤ 1 + 2smin (t ) for complete positivity. Moreover, it suffices to prove this for infinitesimal t since larger times are obtained by concatenation which preserves complete positivity. In leading order we get trt = tr[1 + t ln ] + O(t 2 ) = 3 + t ln det T + O(t 2 ), 1 + 2smin (t ) = 1 + 2 (1 + t ln smin ()) + O(t 2 ),
(52) (53) (54)
from which we obtain 2 trt − [1 + 2smin (t )] = t ln det() − ln smin () ± O(t 2 ),
(55)
2 (). The case of equalwhich is indeed negative for infinitesimal t if det() < smin ity is covered by the fact that we can then express = diag(λ1 , λ2 , λ1 λ2 ) = diag (λ1 , 1, λ1 )diag(1, λ2 , λ2 ) as concatenation of two Kraus rank-two channels. What remains to discuss is the case of a diagonal normal form with = diag(λ1 , λ2 ,0), λi > 0. Note that channels of zero determinant can be infinitesimal divisible due to the fact that we took the closure in Def. 14. Hence, there must be an infinitesimal divisible channel T with non-zero determinant in every -neighborhood of T . If T is unital we can again w.l.o.g. choose T to be unital as well. In leading order the -block of T has singular values λ1 , λ2 and . For sufficiently small this can, however, never satisfy Eq. (51) so that there cannot be an infinitesimal divisible channel with non-zero determinant close to T and thus T itself cannot be infinitesimal divisible.
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Theorem 24 characterizes the set of qubit channels which are solutions of continuous time-dependent master equations for completely positive evolutions. As in the theory of open quantum systems complete positivity is often dropped in the context of timedependent master equations we provide the analogous statement for evolutions which are (locally) merely positivity preserving: Theorem 25 (Continuous positive evolutions). A qubit channel T ∈ T+ is infinitesimal divisible within the set T of positive trace preserving maps iff it has non-negative determinant. Proof. By multiplicativity and continuity of the determinant we know that det T ≥ 0 is indeed necessary for T to be infinitesimal divisible. In order to prove sufficiency we exploit once again the Lorentz normal form together with Thm. 17 and the fact that the sign of the determinant does not change upon concatenating with Kraus rank-one filtering operations. If the normal form is not diagonal, then T is infinitesimal divisible according to Thm. 24. If the normal form is diagonal and det T > 0, then the statement follows from the fact that the corresponding unital channel is an element of a positivity preserving semigroup given by t = exp[t ln ]. As t ≤ 1 for all t ≥ 0 the corresponding map is always positive. The remaining cases with det T = 0 are obtained by taking the closure. VII. Conclusion We have mainly addressed two questions: which quantum channels can be broken down into infinitesimal pieces, and which can be expressed as a non-trivial concatenation of other channels at all. This led us to the two notions of infinitesimal divisibility and divisibility respectively. Loosely speaking, the former class corresponds to the set of solutions of time-dependent master equations. However, to make this a strong correspondence continuity of the Liouville operator (at least piecewise) would clearly be desirable. This follows from our analysis only for qubit channels for which a rather exhaustive characterization was possible. For higher dimensions a similar complete classification might be hard to obtain unless one restricts to specific classes like diagonal or quasi-free channels [30]. We find it remarkable that in the vicinity of the ideal channel all types of channels can be found (i.e., indivisible, divisible, not infinitesimal divisible, Markovian, etc.). This is, in fact, what makes the proof of our main structure theorem non-trivial—if all channels close to the identity would be Markovian, it would follow immediately. Apart from the implications for the theory of open quantum systems and the abstract semigroup structure of the set of quantum channels we can think of applying the techniques and results presented in this work in various contexts. Renormalization-group transformations for quantum states on a spin chain [31] for instance use concatenations and—in the infrared limit—divisions of quantum channels. Moreover, when considering quantum channels with a classical output in the sense of the positive operator valued measure (POVM) formalism, then a similar train of thought leads to the notion of clean POVMs which cannot be expressed as a non-trivial concatenation of a quantum channel with a different POVM [19]. Finally, it would be interesting to know whether a concatenation of quantum channels allows for a quantitative estimate of the channel capacity based on the capacities of the constituents which goes beyond the trivial bottleneck-inequality. In this context also the stability of the above introduced notions under tensor products is an interesting problem.
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Acknowledgement. We thank T. Cubitt, J. Eisert and A. Holevo for valuable discussions.
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Commun. Math. Phys. 279, 169–185 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0426-4
Communications in
Mathematical Physics
On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy Sergiu Klainerman1, , Matei Machedon2 1 Department of Mathematics, Princeton University, Princeton, NJ 08540, USA 2 Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
E-mail:
[email protected] Received: 18 January 2007 / Accepted: 31 July 2007 Published online: 6 February 2008 – © Springer-Verlag 2008
Abstract: The purpose of this note is to give a new proof of uniqueness of the GrossPitaevskii hierarchy, first established in [1], in a different space, based on space-time estimates similar in spirit to those of [2]. 1. Introduction The Gross-Pitaevskii hierarchy refers to a sequence of functions γ (k) (t, xk , xk ), k = 1, 2, . . ., where t ∈ R, xk = (x1 , x2 , . . . , xk ) ∈ R3k , xk = (x1 , x2 , . . . , xk ) ∈ R3k which are symmetric, in the sense that γ (k) (t, xk , xk ) = γ (k) (t, xk , xk ) and γ (k) (t, xσ (1) , . . . xσ (k) , xσ (1) , . . . xσ (k) ) = γ (k) (t, x1 , . . . xk , x1 , . . . xk )
(1)
for any permutation σ , and satisfy the Gross-Pitaevskii infinite linear hierarchy of equations, k B j, k+1 (γ (k+1) ), i∂t + xk − xk γ (k) = j=1
with prescribed initial conditions (k)
γ (k) (0, xk , xk ) = γ0 (xk , xk ). The first author is supported by the NSF grant DMS-0601186.
(2)
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Here xk , xk refer to the standard Laplace operators with respect to the variables xk , xk ∈ R3k and the operators B j, k+1 = B 1j, k+1 − B 2j, k+1 are defined according to B 1j, k+1 (γ (k+1) )(t, xk , xk ) )γ (k+1) (t, xk+1 , xk+1 ) d xk+1 d xk+1 , = δ(x j − xk+1 )δ(x j − xk+1 B 2j, k+1 (γ (k+1) )(t, xk , xk ) = δ(x j − xk+1 )δ(x j − xk+1 )γ (k+1) (t, xk+1 , xk+1 ) d xk+1 d xk+1 . ) replacing both variables In other words B 1j, k+1 , resp. B 2j, k+1 , acts on γ (k+1) (t, xk+1 , xk+1 xk+1 and xk+1 by x j , resp x j . We shall also make use of the operators,
B k+1 =
B j, k+1 .
1≤ j≤k
One can easily verify that a particular solution to (2) is given by, γ (k) (t, xk , xk ) =
k
φ(t, x j )φ(t, x j ),
(3)
j=1
where each φ satisfies the non-linear Schrödinger equation in 3+1 dimensions (i∂t + ) φ = φ|φ|2 , φ(0, x) = φ(x).
(4)
In [1] L. Erdös, B. Schlein and H-T Yau provide a rigorous derivation of the cubic nonlinear Schrödinger equation (4) from the quantum dynamics of many body systems. An important step in their program is to prove uniqueness to solutions of (2) corresponding to the special initial conditions (k)
γ (k) (0, xk , xk ) = γ0 (xk , xk ) =
k
φ(x j )φ(x j )
(5)
j=1
with φ ∈ H 1 (R3 ). To state precisely the uniqueness result of [1], denote S j = (1 − x j )1/2 , S j = (1 − x j )1/2 and S (k) = kj=1 S j · kj=1 S j . If the operator given by the integral kernel γ (k) (xk , xk ) is positive (as an operator), then so is S (k) γ (k) (xk , xk ), and the trace norm of S (k) γ (k) is (k) γ Hk = S (k) γ (k) (xk , xk ) xk =xk dxk .
The authors of [1] prove uniqueness of solutions to (2) in the set of symmetric, positive operators γk satisfying, for some C > 0, sup γ (k) (t, ·, ·)Hk ≤ C k
0≤t≤T
(6)
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
171
In that work, Eqs. (2) are obtained as a limit of the BBGKY hierarchy (see [1]), and it is proved that solutions to BBGKY with initial conditions (5) converge, in a weak sense, to a solution of (2) in the space (6). The purpose of this note is to give a new proof of uniqueness of the Gross- Pitaevskii hierarchy (2), in a different space, motivated, in part, by space-time type estimates, similar in spirit to those of [2]. Our norms will be R (k) γ (k) (t, ·, ·) L 2 (R3k ×R3k ) . (7) Here, R j = (−x j )1/2 , R j = (−x j )1/2 and R (k) = kj=1 R j · kj=1 R j . Notice that for a symmetric, smooth kernel γ , for which the associated linear operator is positive we have R (k) γ (k) (t, ·, ·) L 2 (R3k ×R3k ) ≤ S (k) γ (k) (t, ·, ·) L 2 (R3k ×R3k ) ≤ γ (k) (t, ·, ·)Hk since |S (k) γ (k) (x, x )|2 ≤ S (k) γ (k) (x, x)S (k) γ (k) (x , x ). This is similar to the condition aii a j j − |ai j |2 ≥ 0 which is satisfied by all n × n positive semi-definite Hermitian matrices. Our main result is the following: Theorem 1.1. (Main Theorem). Consider solutions γ (k) (t, xk , xk ) of the GrossPitaevskii hierarchy (2), with zero initial conditions, which verify the estimates, T R (k) B j,k+1 γ (k+1) (t, ·, ·) L 2 (R3k ×R3k ) dt ≤ C k (8) 0
for some C > 0 and all 1 ≤ j < k. Then R (k) γ (k) (t, ·, ·) L 2 (R3k ×R3k ) = 0 for all k and all t. Therefore, solutions to (2) verifying the initial conditions (5), are unique in the spacetime norm (8). We plan to address the connection with solutions of BBGKY in a future paper. The following remark is however reassuring. Remark 1.2. The sequence γ (k) , given by (3) with φ an arbitrary solution of (4) with H 1 data, verifies (8) for every T > 0 sufficiently small. Moreover, if the H 1 norm of the initial data is sufficiently small then (8) is verified for all values of T > 0. Proof. Observe that R (k) B1,k+1 γ (k+1) (t, x1 , . . . , xk ; x1 . . . , xk ) can be written in the form, R (k) B1,k+1 γ (k+1) (t, ·, ·) = R1 |φ(t, x1 )|2 φ(t, x1 ) R2 (φ(t, x2 ) · · · Rk (φ(t, xk ) ·R1 (φ(t, x1 ) · · · Rk (φ(t, xk ). Therefore, in [0, T ] × R3k × R3k we derive R (k) B j,k+1 γ (k+1) L 1 L 2 ≤ |R1 (|φ|2 φ) L 1 L 2 · R2 φ L ∞ 2 · · · Rk φ L ∞ L 2 t Lx t x t
t
x
·R1 φ L ∞ 2 R2 φ L ∞ L 2 · · · Rk φ L ∞ L 2 t Lx t t x x
≤ C∇(|φ|2 φ) L 1 L 2 × ∇φ2k−1 , L∞ L2 t
x
t
x
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where the norm on the left is in [0, T ] × R3k × R3k and all norms on the right-hand side are taken relative to the space-time domain [0, T ] × R3 . In view of the standard energy identity for the nonlinear equation (4) we have apriori bounds for supt∈[0,T ] ∇φ(t) L 2 (R3 ) . Therefore we only need to provide a uniform bound for the norm ∇(|φ|2 φ) L 1 L 2 . We shall show below that this is possible for all values of t x T > 0 provided that the H 1 norm of φ(0) is sufficiently small. The case of arbitrary size for φ(0) H 1 and sufficiently small T is easier and can be proved in a similar manner. We shall rely on the following Strichartz estimate (see [3]) for the linear, inhomogeneous, Schrödinger equation i∂t φ + φ = f in [0, T ] × R3 , φ L 2 L 6 ≤ C f L 1 L 2 + φ L ∞ 2 . t Lx t
t
x
(9)
x
We start by using the Hölder inequality, in [0, T ] × R3 , ∇(|φ|2 φ) L 1 L 2 ≤ C∇φ L 2 L 6 φ 2 L 2 L 3 ≤ C∇φ L 2 L 6 φ2L 4 L 6 . t
t
x
t
x
x
t
x
t
x
Using (9) for f = |φ|2 φ we derive, ∇φ L 2 L 6 ≤ C ∇(|φ|2 φ) L 1 L 2 + ∇φ L ∞ 2 . t Lx t
t
x
x
Denoting A(T ) = |φ|2 φ L 1 L 2 ([0,T ]×R3 ) , t
x
B(T ) = ∇(|φ|2 φ) L 1 L 2 ([0,T ]×R3 ) , t
x
we derive B(T ) ≤ C B(T ) + ∇φ(0) L 2 φ2L 4 L 6 t x ≤ C B(T ) + ∇φ(0) L 2 φ L ∞ 6 φ L 2 L 6 t Lx t x ≤ C A(T ) + φ(0) L 2 B(T ) + ∇φ(0) L 2 ∇φ(0) L 2x . On the other hand, using (9) again, A(T ) ≤ C φ 3 L 1 L 2 + φ(0) L 2 t x ≤ C A(T )3 + φ(0) L 2 . Observe this last inequality implies that, for sufficiently small φ(0) L 2 , A(T ) remains uniformly bounded for all values of T . Thus, for all values of T , with another value of C, B(T ) ≤ C B(T ) + ∇φ(0) L 2 ∇φ(0) L 2x , from which we get a uniform bound for B(T ) provided that ∇φ(0) L 2 is also sufficiently small.
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
173
The proof of Theorem (1.1) is based on two ingredients. One is expressing γ (k) in terms of the future iterates γ (k+1) · · · , γ (k+n) using Duhamel’s formula. Since B (k+1) =
k j=1 B j, k+1 is a sum of k terms, the iterated Duhamel’s formula involves k(k+1) · · · (k+ n − 1) terms. These have to be grouped into much fewer O(C n ) sets of terms. This part of our paper follows in the spirit of the Feynman path combinatorial arguments of [1]. The second ingredient is the main novelty of our work. We derive a space-time estimate, reminiscent of the bilinear estimates of [2]. ) verify the homogeneous equation, Theorem 1.3. Let γ (k+1) (t, xk+1 , xk+1 (k+1) , γ (k+1) = 0, i∂t + k+1 ± = xk+1 − xk+1 ± (k+1)
γ (k+1) (0, xk+1 , xk+1 ) = γ0
(10)
(xk+1 , xk+1 ).
Then there exists a constant C, independent of j, k, such that R (k) B j,k+1 (γ (k+1) ) L 2 (R×R3k ×R3k ) (k+1)
≤ CR (k+1) γ0
L 2 (R3(k+1) ×R3(k+1) ) .
(11)
2. Proof of the Estimate Without loss of generality, we may take j = 1 in B j,k+1 . It also suffices to estimate the term in B 1j,k+1 , the term in B 2j,k+1 can be treated in the same manner. Let γ (k+1) be as in (10). Then the Fourier transform of γ (k+1) with respect to the variables (t, xk , xk ) is given by the formula, δ(τ + |ξ k |2 − |ξ k |2 )γˆ (ξ, ξ ), where τ corresponds to the time t and ξ k = (ξ1 , ξ2 , . . . , ξk ), ξ k = (ξ1 , ξ2 , . . . , ξk ) correspond to the space variables xk = (x1 , x2 , . . . , xk ) and xk = (x1 , x2 , . . . , xk ). We ) and, also write ξ k+1 = (ξ k , ξk+1 ), ξ k+1 = (ξ k , ξk+1 |ξ k+1 |2 = |ξ1 |2 + . . . + |ξk |2 + |ξk+1 |2 = |ξ k |2 + |ξk+1 |2 |ξ k+1 |2 = |ξ1 |2 + . . . + |ξk |2 + |ξk+1 |2 . 1 The Fourier transform of B1,k+1 (γ (k+1) ), with respect to the same variables (t, xk , xk ), is given by , ξ2 , · · · , ξk+1 , ξ k+1 )dξk+1 dξk+1 , (12) δ(· · · )γˆ (ξ1 − ξk+1 − ξk+1
where δ(· · · ) = δ(τ + |ξ1 − ξk+1 − ξk+1 |2 + |ξ k+1 |2 − |ξ1 |2 − |ξ k+1 |2 ) (k+1)
and γ denotes the initial condition γ0 equivalent to the following estimate:
. By Plancherel’s theorem, estimate (11) is
Ik [ f ] L 2 (R×Rk ×Rk ) ≤ C fˆ L 2 (Rk+1 ×Rk+1 ) ,
(13)
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applied to f = R (k+1) γ , where Ik [ f ](τ, ξ k , ξ k ) =
δ(. . .)
,ξ ,··· ,ξ |ξ1 | fˆ(ξ1 − ξk+1 − ξk+1 2 k+1 , ξ k+1 ) ||ξ |ξ1 − ξk+1 − ξk+1 k+1 ||ξk+1 |
. dξk+1 dξk+1
Applying the Cauchy-Schwarz inequality with measures, we easily check that |Ik [ f ]|2 ≤
δ(. . .) ·
|ξ1 |2 dξk+1 dξk+1 |2 |ξ 2 |ξ |2 |ξ1 − ξk+1 − ξk+1 | k+1 k+1
δ(. . .)| fˆ(ξ1 − ξk+1 − ξk+1 , ξ2 , · · · , ξk+1 , ξ k+1 )|2 dξk+1 dξk+1 .
If we can show that the supremum over τ , ξ1 · · · ξk , ξ1 · · · ξk of the first integral above is bounded by a constant C 2 , we infer that Ik [ f ]2L 2
≤C
2
, ξ2 , · · · , ξk+1 , ξ k+1 )|2 dξ k+1 dξ k+1 dτ δ(. . .)| fˆ(ξ1 −ξk+1 −ξk+1
≤ C 2 fˆ2L 2 (Rk+1 ×Rk+1 ) . Thus, Ik [ f ]2L 2 (R×Rk ×Rk ) ≤ C 2 fˆ2L 2 (Rk+1 ×Rk+1 ) as desired. Thus we have reduced matters to the following: Proposition 2.1. There exists a constant C such that
δ(τ + |ξ1 − ξk+1 − ξk+1 |2 + |ξk+1 |2 − |ξk+1 |2 )
|ξ1 |2 dξk+1 dξk+1 ≤C |2 |ξ 2 2 |ξ1 − ξk+1 − ξk+1 k+1 | |ξk+1 | uniformly in τ, ξ1 . The proof is based on the following lemmas: Lemma 2.2. Let P be a 2 dimensional plane or sphere in R3 with the usual induced surface measure d S. Let 0 < a < 2, 0 < b < 2, a + b > 2. Let ξ ∈ R3 . Then there exists C independent of ξ and P such that P
1 C d S(η) ≤ . |ξ − η|a |η|b |ξ |a+b−2
Proof. If P = R2 and ξ ∈ R2 this is well known. The same proof works in our case, by breaking up the integral I ≤ I1 + I2 + I3 over the overlapping regions:
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
Region 1.
|ξ | 2
175
< |η| < 2|ξ |. In this region |ξ − η| < 3|ξ | and 1 1 I1 ≤ C b d S(η) |ξ | P∩{|ξ −η|<3|ξ |} |ξ − η|a 1 1 1 ≤C b d S(η) a |ξ | P∩{3i−1 |ξ |<|ξ −η|<3i |ξ |} |ξ − η| i=−∞
≤C
1 1 C 1 (3i |ξ |)2 = , |ξ |b |3i ξ |a |ξ |a+b−2 i=−∞
where we have used the obvious fact that the area of P ∩ {3i−1 |ξ | < |ξ − η| < 3i |ξ |} is ≤ C(3i |ξ |)2 . Region 2.
|ξ | 2
< |ξ − η| < 2|ξ |. In this region |η| < 3|ξ | and 1 1 I2 ≤ C a d S(η) |ξ | P∩{|η|<3|ξ |} |η|b C ≤ |ξ |a+b−2
(14)
in complete analogy with Region 1. Region 3. |η| > 2|ξ | or |ξ − η| > 2|ξ |. In this region, |η| > |ξ | and 2|ξ − η| ≥ |ξ − η| + |ξ | ≥ |η|, thus 1 I3 ≤ C d S(η) a+b P∩{|η|>|ξ |} |η| ∞ 1 d S(η) ≤C a+b P∩{2i−1 |ξ |<|η|<2i |ξ |} |η| i=1
≤C
∞ i=1
1 C (2i |ξ |)2 = . |2i ξ |a+b |ξ |a+b−2 (15)
We also have 1 Lemma 2.3. Let P, ξ as in Lemma (2.2) and let = 10 . Then 1 C d S(η) ≤ . ξ 3−2 2− 2− |ξ | |ξ − η| | − η||η| P 2
Proof. Consider separately the regions |η| > previous lemma.
|ξ | 2 ,
and |ξ − η| >
We are ready to prove the main estimate of Proposition (2.1).
|ξ | 2 ,
and apply the
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S. Klainerman, M. Machedon
Proof. Changing k + 1 to 2, we have to show I =
δ(τ + |ξ1 − ξ2 − ξ2 |2 + |ξ2 |2 − |ξ2 |2 )
|ξ1 |2 dξ2 dξ2 ≤ C. |ξ1 − ξ2 − ξ2 |2 |ξ2 |2 |ξ2 |2
The integral is symmetric in ξ1 − ξ2 − ξ2 and ξ2 so we can integrate, without loss of generality, over |ξ1 − ξ2 − ξ2 | > |ξ2 |. Case 1. Consider the integral I1 restricted to the region |ξ2 | > |ξ2 | and integrate ξ2 first. Notice δ(τ + |ξ1 − ξ2 − ξ2 |2 + |ξ2 |2 − |ξ2 |2 )dξ2 = δ(τ + |ξ1 − ξ2 |2 − 2(ξ1 − ξ2 ) · ξ2 + |ξ2 |2 )dξ2 d S(ξ2 ) , = 2|ξ1 − ξ2 |
(16)
where d S is surface measure on a plane P in R3 , i.e. the plane ξ · ω = λ with ω ∈ S2 2 2 1 −ξ2 | +|ξ2 | and λ = τ +|ξ2|ξ . 1 −ξ2 | In this region
d S(ξ2 ) dξ2 2 2 2 R3 |ξ2 | |ξ1 − ξ2 | P |ξ1 − ξ2 − ξ2 | |ξ2 | d S(ξ2 ) dξ2 ≤ |ξ1 |2 2− |ξ |2− 2+2 |ξ − ξ | 1 2 R3 |ξ2 | P |ξ1 − ξ2 − ξ2 | 2 dξ2 2 ≤ C|ξ1 | 2+2 |ξ − ξ |3−2 1 2 R3 |ξ2 | ≤ C.
I1 ≤ |ξ1 |2
(17)
Case 2. Consider the integral I2 restricted to the region |ξ2 | < |ξ2 | and integrate ξ2 first. Notice δ(τ + |ξ1 − ξ2 − ξ2 |2 + |ξ2 |2 − |ξ2 |2 )dξ2 ξ1 − ξ2 2 ξ1 − ξ2 ξ1 − ξ2 2 ξ1 − ξ2 2 − (ξ2 − )| + |(ξ2 − )+ | − |ξ2 | dξ2 =δ τ +| 2 2 2 2
|ξ1 − ξ2 |2 ξ1 − ξ2 2 + 2|ξ2 − | − |ξ2 |2 dξ2 =δ τ+ 2 2 =
d S(ξ2 ) 4|ξ2 −
ξ1 −ξ2 2 |
,
(18)
where d S is a surface measure on a sphere P, i.e. the sphere in ξ2 centered at 21 (ξ1 − ξ2 ) |ξ −ξ |2 and radius 21 |ξ2 |2 − τ − 1 2 2 .
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
Thus
177
dξ2 d S(ξ2 ) I2 ≤ |ξ1 | |2 ξ −ξ2 1 3 |ξ 2 R P |ξ2 |2 |ξ2 − 2 2 ||ξ1 − ξ2 − ξ2 | dξ2 d S(ξ2 ) ≤ |ξ1 |2 2+2 ξ −ξ R3 |ξ2 | P |ξ2 |2− |ξ2 − 1 2 ||ξ1 − ξ2 − ξ |2− 2 2 dξ2 2 ≤ |ξ1 | 2+2 |ξ − ξ |3−2 ≤ C. 1 R3 |ξ2 | 2 2
3. Duhamel Expansions and Regrouping This part of our note is based on a somewhat shorter version of the combinatorial ideas of [1]. We are grateful to Schlein and Yau for explaining their arguments to us. (k) Recalling the notation ± = xk − xk and ±,x j = x j − x j , we write t1 (1) γ (1) (t1 , ·) = ei(t1 −t2 )± B2 (γ 2 (t2 )) dt2 0 t1 t2 (1) (2) = ei(t1 −t2 )± B2 ei(t2 −t3 )± (γ 3 (t3 )) dt2 dt3 0
0
= ············ tn t1 ··· J (t n+1 ) dt2 · · · dtn+1 , = 0
(19)
0
where, t n+1 = (t1 , . . . , tn+1 ) and (1)
(n)
J (t n+1 ) = ei(t1 −t2 )± B2 · · · ei(tn −tn+1 )± Bn+1 (γ (n+1) )(tn+1 , ·).
Expressing B (k+1) = kj=1 B j, k+1 , · · · , the integrand J (t n+1 ) = J (t1 , . . . , tn+1 ) in (19) can be written as J (t n+1 ; µ), (20) J (t n+1 ) = µ∈M
where (1)
(2)
J (t n+1 ; µ) = ei(t1 −t2 )± B1,2 ei(t2 −t3 )± Bµ(3),3 · · · (n)
ei(tn −tn+1 )± Bµ(n+1),n+1 (γ (n+1) )(tn+1 , ·). Here we have denoted by M the set of maps µ : {2, · · · , n + 1} → {1, · · · n} satisfying µ(2) = 1 and µ( j) < j for all j. Graphically, such a µ can be represented by selecting one B entry from each column of an n × n matrix such as, for example, (if µ(2) = 1, µ(3) = 2, µ(4) = 1, etc.), ⎛ ⎞ B1,2 B1,3 B1,4 · · · B1,n+1 B2,3 B2,4 · · · B2,n+1 ⎟ ⎜ 0 ⎜ ⎟ 0 B3,4 · · · B3,n+1 ⎟ . (21) ⎜ 0 ⎝··· ··· ··· ··· ··· ⎠ 0 0 0 · · · Bn,n+1
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To such a matrix one can associate a Feynman graph whose nodes are the selected entries, as in [1]. However, our exposition will be self-contained and will not rely explicitly on the Feynman graphs. We will consider I(µ, σ ) = J (t n+1 ; µ)dt2 · · · dtn+1 , (22) t1 ≥tσ (2) ≥tσ (3) ≥···≥tσ (n+1)
where σ is a permutation of {2, · · · , n+1}. The integral I(µ, σ ) is represented by (µ, σ ), or, graphically, by the matrix ⎛
tσ −1 (2) tσ −1 (3) tσ −1 (4) · · · tσ −1 (n+1) ⎜ B1,2 B1,3 B1,4 · · · B1,n+1 ⎜ ⎜ 0 B2,3 B2,4 · · · B2,n+1 ⎜ 0 0 B3,4 · · · B3,n+1 ⎜ ⎜ ··· · · · · · · · · · ··· ⎜ ⎝ 0 0 0 ··· Bn,n+1 column 2 column 3 column 4 · · · column n+1
⎞ row 1⎟ ⎟ row 2⎟ ⎟ row 3⎟ . ⎟ ⎟ row n⎠
(23)
Notice the columns of this matrix are labelled 2 to n + 1, while the rows are labelled 1 through n. We will define a set of “acceptable moves” on the set of such matrices. Imagine a board game where the names Bi, j are carved in, and one entry Bµ( j), j , µ( j) < j, in each column is highlighted. If µ( j + 1) < µ( j), the player is allowed to exchange the highlighted entries in columns j and j +1 and, at the same time, exchange the highlighted entries in rows j and j + 1. This changes µ to a new µ = ( j, j + 1) ◦ µ ◦ ( j, j + 1). Here ( j, j + 1) denotes the permutation which reverses j and j + 1. The rule for changing σ is σ −1 = σ −1 ◦ ( j, j + 1). In other words, σ −1 changes according to column exchanges. Thus going from ⎛
t2 ⎜B1,2 ⎜ ⎜ 0 ⎝ 0 0
t5 B1,3 B2,3 0 0
t4 B1,4 B2,4 B3,4 0
⎞ t3 B1,5 ⎟ ⎟ B2,5 ⎟ B3,5 ⎠ B4,5
(24)
t4 B1,3 B2,3 0 0
t5 B1,4 B2,4 B3,4 0
⎞ t3 B1,5 ⎟ ⎟ B2,5 ⎟ B3,5 ⎠ B4,5
(25)
to ⎛
t2 ⎜B1,2 ⎜ ⎜ 0 ⎝ 0 0
is an acceptable move. In the language of [1] the partial order of the graphs is preserved by an acceptable move. The relevance of this game to our situation is explained by the following, Lemma 3.1. Let (µ, σ ) be transformed into (µ , σ ) by an acceptable move. Then, for the corresponding integrals (22), I(µ, σ ) = I(µ , σ ).
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
179
Proof. We will first explain the strategy of the proof by focusing on an explicit example. Consider the integrals I1 corresponding to (24) and I2 corresponding to (25): I1 =
(1)
t1 ≥t2 ≥t5 ≥t4 ≥t3 (3)
i(t3 −t4 )±
e I2 =
(2)
ei(t1 −t2 )± B1,2 ei(t2 −t3 )± B2,3 (4)
B1,4 ei(t4 −t5 )± B4,5 (γ (5) (t5 , ·)dt2 · · · dt5 ,
t1 ≥t2 ≥t5 ≥t3 ≥t4
e
(1) i(t1 −t2 )±
(3)
B1,2 e
(2) i(t2 −t3 )±
(26)
B1,3
(4)
ei(t3 −t4 )± B2,4 ei(t4 −t5 )± B3,5 (γ (5) (t5 , ·)dt2 · · · dt5 .
(27)
We first observe the identity, (2)
(3)
(4)
ei(t2 −t3 )± B2,3 ei(t3 −t4 )± B1,4 ei(t4 −t5 )± (2)
(3)
= ei(t2 −t4 )± B1,4 e−i(t3 −t4 )(±
−±,x3 +±,x4 )
(4)
B2,3 ei(t3 −t5 )± .
In other words, (t3 , x3 , x3 ) and (t4 , x4 , x4 ) and the position of B2,3 and B1,4 have been exchanged. This is based on trivial commutations, and is proved below in general, see (37). Recalling the definition (3) we abbreviate the integral kernel of B j,k+1 , ) − δ(x j − xk+1 )δ(x j − xk+1 ). δ j,k+1 = δ(x j − xk+1 )δ(x j − xk+1
We also denote γ3,4 = γ (5) (t5 , x1 , x2 , x3 , x4 , x5 ; x1 , x2 , x3 , x4 , x5 ), γ4,3 = γ (5) (t5 , x1 , x2 , x4 , x3 , x5 ; x1 , x2 , x4 , x3 , x5 ). By the symmetry assumption (1), γ3,4 = γ4,3 . Thus the integral I1 of (26) equals
I1 =
t1 ≥t2 ≥t5 ≥t4 ≥t3 (2) i(t2 −t4 )±
(1)
R24
ei(t1 −t2 )± δ1,2 (3)
(4)
e δ1,4 e−i(t3 −t4 )(± −±,x3 +±,x4 ) δ2,3 ei(t3 −t5 )± δ4,5 γ4,3 dt2 · · · dt5 d x2 · · · d x5 d x2 · · · d x5 .
(28)
In the above integral we perform the change of variables which exchanges (t3 , x3 , x3 ) (2) with (t4 , x4 , x4 ). Thus, in particular, x3 becomes x4 , and (3) ± = ± +±,x3 becomes (2) (3) ± + ±,x4 . Thus, ± − ±,x3 + ±,x4 changes to (2)
(3)
± + ±,x4 − ±,x4 + ±,x3 = ± ,
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S. Klainerman, M. Machedon (4)
and ± stays unchanged. The integral (28) becomes (1) I1 = ei(t1 −t2 )± δ1,2 t1 ≥t2 ≥t5 ≥t3 ≥t4
R24
(2)
(3)
i(t2 −t3 )±
(4)
e δ1,3 ei(t3 −t4 )± δ2,4 ei(t4 −t5 )± δ3,5 γ3,4 dt2 · · · dt5 d x2 · · · d x5 d x2 · · · d x5 (1) (2) = ei(t1 −t2 )± B1,2 ei(t2 −t3 )± B1,3 t1 ≥t2 ≥t5 ≥t3 ≥t4 (3)
(4)
ei(t3 −t4 )± B2,4 ei(t4 −t5 )± B3,5 (γ (5) (t5 , ·)dt2 · · · dt5 = I2 Therefore, I1 = I2 as stated above. Notice the domain of integration corresponds to σ (2) = 2, σ (3) = 5, σ (4) = 3, σ (5) = 4, that is, σ = (3, 4) ◦ σ . Now we proceed to the general case. Consider a typical term, J (t n+1 ; µ)dt2 . . . dtn+1 (29) I(µ, σ ) = t1 ≥···tσ ( j) ≥tσ ( j+1) ≥···tσ (n+1) ≥0
=
t1 ≥···tσ ( j) ≥tσ ( j+1) ≥···tσ (n+1) ≥0 ( j−1)
( j)
( j+1)
· · · ei(t j−1 −t j )± Bl, j ei(t j −t j+1 )± Bi, j+1 ei(t j+1 −t j+2 )± (· · · )dt2 . . . dtn+1 with associated matrix of the form ⎛ · · · tσ −1 ( j) ⎜· · · Bi, j ⎜ ⎜· · · ··· ⎜ Bl,j ⎜· · · ⎜ ··· ⎜· · · ⎜· · · ··· ⎜ ⎝· · · ··· ··· ···
tσ −1 ( j+1) Bi,j+1 ··· Bl, j+1 ··· ··· ··· ···
⎞ ··· ··· ⎟ ⎟ ··· ⎟ ⎟ ··· ⎟ ⎟, ··· ⎟ row j ⎟ ⎟ row j+1⎠ ···
(30)
(31)
where µ( j) = l and µ( j + 1) = i and i < l < j < j + 1. It is understood that rows j or j+1 may in fact not have highlighted entries, as in the previous example. We plan to show I = I , where
I =
(32)
t1 ≥···tσ ( j) ≥tσ ( j+1) ≥···tσ (n+1) ≥0 ( j−1)
( j)
( j+1)
· · · ei(t j−1 −t j )± Bi, j ei(t j −t j+1 )± Bl, j+1 ei(t j+1 −t j+2 )± (· · · ) dt2 . . . dtn+1 .
(33)
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
181
The · · · at the beginning of (30) and (33) are the same. Any B j,α in (· · · ) in (30) become B j+1,α in (· · · ) in (33). Similarly, any B j+1,α in (· · · ) in (30) become B j,α in (· · · ) in (33), while the rest is unchanged. Thus I is represented by the matrix, ⎛ ⎞ · · · tσ −1 ( j) tσ −1 ( j+1) ··· ⎜· · · Bi, j+1 ··· ⎟ Bi,j ⎜ ⎟ ⎜· · · ··· ··· ··· ⎟ ⎜ ⎟ Bl, j Bl,j+1 ··· ⎟ ⎜· · · (34) ⎜ ⎟, ··· ··· ··· ⎟ ⎜· · · ⎜· · · ⎟ ··· ··· (row j)’ ⎟ ⎜ ⎝· · · ··· ··· (row j+1)’⎠ ··· ··· ··· ··· where the highlighted entries of (row j)’, respectively (row j+1)’ in (34) have the positions of the highligted entries of row j+1 , respectively row j in (31). ˜ (±j) = (±j) − ±,x j + ±,x j+1 . We consider the terms To prove (32) denote, ( j)
P = Bl, j ei(t j −t j+1 )± Bi, j+1
(35)
and ( j)
˜ P˜ = Bi, j+1 e−i(t j −t j+1 )± Bl, j .
(36)
We will show that ( j−1)
ei(t j−1 −t j )±
( j+1)
Pei(t j+1 −t j+2 )±
( j−1)
= ei(t j−1 −t j+1 )±
( j)
( j+1)
˜ i(t j −t j+2 )± Pe
.
(37)
( j)
Indeed in (35) we can write ± = ±,xi + (± − ±,xi ). Therefore, ( j)
( j)
ei(t j −t j+1 )± = ei(t j −t j+1 )±,xi · ei(t j −t j+1 )(±
−±,xi )
.
Observe that the first terms on the right can be commuted to the left of Bl, j , the second one to the right of Bi, j+1 in the expression for I . Thus, ( j)
P = ei(t j −t j+1 )(±,xi ) Bl, j Bi, j+1 ei(t j −t j+1 )(±
−±,xi )
and ( j−1)
ei(t j−1 −t j )±
( j+1)
Pei(t j+1 −t j+2 )± ( j−1)
= ei(t j−1 −t j )±
( j)
ei(t j −t j+1 )(± ·e
( j+1)
−±,xi ) i(t j+1 −t j+2 )±
( j−1)
= ei(t j−1 −t j )±
ei(t j −t j+1 )(±,xi ) Bl, j Bi, j+1 e
ei(t j −t j+1 )±,xi Bi, j+1 Bl, j
ˆ ±,x +···+±,x ) i(t j+1 −t j+2 )(±,xi +±,x j+1 ) i(t j −t j+2 )(±,x1 ···+ i j
where a hat denotes a missing term.
e
,
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˜ ± , we can write Similarly, in view of the definition of ˜ (±j) = (±j) − ±,x j + ±,x j+1 ( j−1)
= ±
( j−1) ±
=
+ ±,x j+1 − ±,xi + ±,xi + ±,x j+1 .
Hence, ˜ ( j)
( j−1)
e−i(t j −t j+1 )± = e−i(t j −t j+1 )(±
−± ,xi )
·e
−i(t j −t j+1 )(±,xi +±,x j+1 )
,
and consequently, ( j−1)
P˜ = e−i(t j −t j+1 )(±
−±,xi )
Bi, j+1 Bl, j e
−i(t j −t j+1 )(±,xi +±,x j+1 )
.
Now, ( j−1)
ei(t j−1 −t j+1 )±
( j+1)
˜ i(t j −t j+2 )± Pe ( j−1)
= ei(t j−1 −t j+1 )± ·e =e ·e
( j−1)
e−i(t j −t j+1 )(±
−±,xi )
Bi, j+1 Bl, j
−i(t j −t j+1 )(±,xi +±,x j+1 ) i(t j −t j+2 )(±j+1)
e
( j−1)
i(t j−1 −t j )±
e
i(t j −t j+1 )±,xi
Bi, j+1 Bl, j
ˆ ±,x +···+±,x ) i(t j+1 −t j+2 )(±,xi +±,x j+1 ) i(t j −t j+2 )(±,x1 ···+ i j
e
and (37) is proved. Now the argument proceeds as in the example. In the integral (29) use the symmetry (1) to exchange x j , x j with x j+1 , x j+1 in the arguments of γ (n+1) (only). Then use (37) in the integrand and also replace the B s by their corresponding integral kernels δ. Then we make the change of variables which exchanges t j , x j , x j with t j+1 , x j+1 , x j+1 in the whole integral. To see the change in the domain of integration, say σ (a) = j and σ (b) = j + 1, and say b < a. Then the domain t1 ≥ · · · σ (b) ≥ · · · ≥ σ (a) · · · changes to t1 ≥ · · · σ (a) ≥ · · · ≥ σ (b) · · · . In other words, a = σ −1 ( j) and b = σ −1 ( j + 1) have been reversed. This proves (32). Next, we consider the subset {µs } ⊂ M of special, upper echelon matrices in which each highlighted element of a higher row is to the left of each highlighted element of a lower row. Thus (25) is in upper echelon form, and (24) is not. According to our definition, the matrix ⎛ ⎞ B1,2 B1,3 B1,4 B1,5 B2,3 B2,4 B2,5 ⎟ ⎜ 0 (38) ⎝ 0 0 B3,4 B3,5 ⎠ 0 0 0 B4,5 is also in upper echelon form. Lemma 3.2. For each element of M there is a finite set of acceptable moves which brings it to upper echelon form.
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
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Proof. The strategy is to start with the first row and do acceptable moves to bring all marked entries in the first row in consecutive order, B1,2 through B1,k . If there are any highlighted elements on the second row, bring them to B2,k+1 , B2,l . This will not affect the marked entries of the first row. If no entries are highlighted on the second row, leave it blank and move to the third row. Continue to lower rows. In the end, the matrix is reduced to an upper echelon form. Lemma 3.3. Let Cn be the number of n × n special, upper echelon matrices of the type discussed above. Then Cn ≤ 4n . Proof. The proof consists of 2 steps. First dis-assemble the original special matrix by “lifting” all marked entries to the first row. This partitions the first row into subsets {1, 2, · · · k1 }, {k1 + 1, · · · , k2 }, etc. Let Pn be the number of such partitions. Look at the last subset of the partition. It can have 0 elements, in which case there is no last partition. This case contributes precisely one partition to the total number Pn . If the last subset has k elements then the remaining n − k, can contribute exactly Pn−k partitions. Thus Pn = 1 + P1 + · · · Pn−1 , and therefore Pn ≤ 2n by induction. In the second step we will re-assemble the upper echelon matrix by lowering {1, 2, · · · k1 } to the first used row (we give up the requirement that only the upper triangle is used, thus maybe counting more matrices) {k1 + 1, · · · , k2 } to the second used row, etc. Now suppose that we have exactly i subsets in a given partition of the first row, which will be lowered in an orderpreserving way to the available n rows. This can be done in exactly ni ways. Thus
Cn ≤ Pn i ni ≤ 4n . This is in agreement with the combinatorial arguments of [1]. Theorem 3.4. Let µs be a special, upper echelon matrix, and write µ ∼ µs if µ can be reduced to µs in finitely many acceptable moves. There exists D a subset of [0, t1 ]n such that tn t1 ··· J (t n+1 ; µ)dt2 · · · dtn+1 = J (t n+1 ; µs )dt2 · · · dtn+1 . (39) µ∼µs
0
0
Proof. Start with the integral
D
I(µ, id) =
t1
···
0
with its corresponding matrix ⎛ t2 t3 ⎜Bµ(2),2 B1,3 ⎜ Bµ(3),3 ⎜ 0 ⎜ 0 0 ⎜ ⎝ ··· ··· 0 0
0
tn
J (t n+1 ; µ)dt2 · · · dtn+1 ,
t4 Bµ(4),4 B2,4 B3,4 ··· 0
··· ··· ··· ··· ··· ···
⎞ tn+1 B1,n+1 ⎟ ⎟ B2,n+1 ⎟ . B3,n+1 ⎟ ⎟ ⎠ ··· Bn,n+1
(40)
As in Lemma (3.2) perform finitely many acceptable moves on it, transforming the matrix determined by the pair (µ, id) to the special upper echelon form matrix corresponding to a pair (µs , σ ), ⎞ ⎛ tσ −1 (2) tσ −1 (3) tσ −1 (4) · · · tσ −1 (n+1) ⎜ B1,2 B1,3 B1,4 ··· B1,n+1 ⎟ ⎟ ⎜ B2,3 B2,4 ··· B2,n+1 ⎟ . (41) ⎜ 0 ⎝ 0 0 B3,4 ··· B3,n+1 ⎠ ··· ··· ··· ··· ···
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By Lemma (3.1), I(µ, id) = I(µs , σ ). Now observe that if (µ1 , id) and (µ2 , id), with µ1 = µ2 lead to the same echelon form µs the corresponding permutations σ1 and σ2 must be different. The lemma is thus proved with D the union of all {t1 ≥ tσ (2) ≥ tσ (3) ≥ · · · tσ (n 1 ) } for all permutations σ which occur in a given class of equivalence of a given µs . Proof of Main Theorem (1.1). We start by fixing t1 . Express γ
(1)
(t1 , ·) =
t1
···
0
µ
tn
0
J (t n+1 , µ),
(42)
where, we recall, (1)
(2)
J (t n+1 , µ) = ei(t1 −t2 )± B1,2 ei(t2 −t3 )± Bµ(3),3 · · · (n)
ei(tn −tn+1 )± Bµ(n+1),n+1 (γ (n+1) )(tn+1 , ·). Using Theorem (3.4) we can write γ (1) (t1 , ·) as a sum of at most 4n terms of the form D
J (t n+1 , µs ).
(43)
Let C n = [0, t1 ] × [0, t1 ] × · · · × [0, t1 ] (product of n terms). Also, let Dt2 = {(t3 , · · · , tn+1 )|(t2 , t3 , · · · , tn+1 ) ∈ D}. We have R (1) γ (1) (t1 , ·) L 2 (R3 ×R3 ) (1) (2) = R (1) ei(t1 −t2 )± B1,2 ei(t2 −t3 )± Bµs (3),3 · · · dt2 · · · dtn+1 L 2 (R3 ×R3 ) D
t1
= 0
t1
≤
(1)
ei(t1 −t2 )±
0
t1
=
0
≤
t1
0
≤
(1)
Cn
(2)
ei(t1 −t2 )±
D t2
(2)
D t2
R (1) B1,2 ei(t2 −t3 )± Bµs (3),3 · · · dt3 · · · dtn+1 L 2 (R3 ×R3 ) dt2 (2)
R (1) B1,2 ei(t2−t3 )± Bµs (3),3 · · · dt3 · · ·dtn+1 dt2 L 2 (R3×R3 )
D t2
R (1) B1,2 ei(t2 −t3 )± Bµs (3),3 · · · dt3 · · · dtn+1 L 2 (R3 ×R3 ) dt2 R
(1)
B1,2 e
(2)
i(t2 −t3 )±
D t2
Bµs (3),3 · · · L 2 (R3 ×R3 ) dt3 · · · dtn+1 dt2
(2)
R (1) B1,2 ei(t2 −t3 )± Bµs (3),3 · · · L 2 (R3 ×R3 ) dt2 dt3 · · · dtn+1 . (44)
On Uniqueness of Solutions to Gross-Pitaevskii Hierarchy
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Applying Cauchy-Schwarz in t and Theorem (1.3) n-1 times , we estimate (2) R (1) B1,2 ei(t2 −t3 )± Bµs (3),3 · · · L 2 (R3 ×R3 ) dt2 dt3 · · · dtn+1 Cn 1 (2) ≤ t12 R (1) B1,2 ei(t2 −t3 )± Bµs (3),3 · · · L 2 ((t2 ∈[0,t1 ])×R3 ×R3 ) dt3 · · · dtn+1 C n−1 1 (3) 2 ≤ Ct1 R (2) Bµs (3),3 ei(t3 −t4 )± Bµs (4),4 · · · L 2 (R6 ×R6 ) dt3 · · · dtn+1 C n−1
··· 1 2
≤ (Ct1 )
t1
n−1 0
R (n) Bµs (n+1),n+1 γ (n+1) (tn+1 , ·) L 2 (R 3n ×R 3n ) dtn+1
1
≤ C(Ct12 )n−1 . Consequently, 1 n−1 . R (1) γ (1) (t1 , ·) L 2 (R3 ×R3 ) ≤ C Ct12
(45)
If Ct1 < 1 and we let n → ∞ and infer that R (1) γ (1) (t1 , ·) L 2 (R3 ×R3 ) = 0. The proof for all γ (k) = 0 is similar. Clearly we can continue the argument to show that all γ (k) vanish for all t ≥ 0 as desired.
References 1. Erdos, L., Schlein, B., Yau, H.-T.: Derivation of the Cubic Non-linear Schródinger equation from Quantum dynamics of Many-Body Systems. Invent. Math. 167(3), 515–614 (2007) 2. Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math 46, 169–177 (1993) 3. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math 120(5), 955–980 (1998) Communicated by H.-T. Yau
Commun. Math. Phys. 279, 187–223 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0424-6
Communications in
Mathematical Physics
Diffusion Limited Aggregation on a Cylinder Itai Benjamini, Ariel Yadin Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail:
[email protected];
[email protected] Received: 21 January 2007 / Accepted: 23 July 2007 Published online: 6 February 2008 – © Springer-Verlag 2008
Abstract: We consider the DLA process on a cylinder G × N. It is shown that this process “grows arms”, provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most log(2−ε) |G|, the time it takes the cluster to reach the m th layer of the cylinder is at most of order m · log |G| log|G| . In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the “arms” grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3. 1. Introduction Diffusion Limited Aggregation (DLA), is a growth model introduced by Witten and Sander ([12]). The process starts with a particle at the origin of Zd . At each time step, a new particle starts a simple random walk on Zd from infinity (far away). The particle is conditioned to hit the existing cluster (when d ≥ 3). When the particle first hits the outer boundary of the cluster, it sticks and the next step starts, forming a growing family of clusters. We consider a variant of this model, where the underlying graph of the process is a cylinder with base G, G being some finite graph. A precise definition is given in Sect. 2. This paper contains three main results: The first, Theorem 2.1, states that if G has small enough mixing time, then the time it takes the cluster to reach the m th layer of the cylinder is o(m · |G|), where |G| is the size of G. In fact, for a graph G with mixing time at most log(2−ε) |G| (for any constant ε), the time to reach the m th layer is at most of order m · log |G| log|G| . This phenomenon is sometimes dubbed as “the aggregate grows
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arms”, i.e. grows faster than order |G| particles per layer. The analogous phenomenon in the original DLA model on Zd is considered a notoriously difficult open problem. In [6–8], Kesten provides upper bounds on the growth rate of the DLA aggregate in Zd . Eberz-Wagner [5] proved the existence of infinitely many holes in the two-dimensional DLA aggregate. The second result concerns the density of the limit cluster, the union of all clusters obtained at some finite time. Theorem 4.2 shows that the expected rate at which the cluster grows bounds this density. This has two implications: 1. Theorem 4.6 states that for any vertex transitive graph G, the DLA process on the G-cylinder has density bounded by 2/3. This includes the cases where G is a d-dimensional Torus. 2. Theorem 4.8 shows that for G with small enough mixing time, the density tends to 0 as the size of G tends to infinity. Finally, Theorem 5.1 is a lower bound on the expected time the cluster reaches the m th layer, complementing the upper bound in Theorem 2.1. This lower bound implies that the cluster cannot grow too fast, and in fact for many natural graphs it cannot grow faster than |G|c for some universal 0 < c < 1. The lower bound holds for a wider range of graphs at the base of the cylinder than the upper bound (including d-dimensional tori for d ≥ 3). We remark that our estimates for the upper bound are crude, and simulations indicate that there is much room for improvement. In fact we believe the truth to be closer to the lower bound, see Conjecture 2.2. Proving Conjecture 2.2 will imply that for any family of graphs {G n }, the density of the DLA process on the G n -cylinder tends to 0 as the size of G n tends to infinity (see Conjecture 2.2). For other very different variants of one dimensional DLA see [2,9]. Another paper dealing with random-walk related questions on cylinders with varying bases is [13]. The rest of this paper is organized as follows: First we introduce some notation. In Sect. 2 we define the process, and random variables associated with it. In Sect. 2.3 we state the first main result. Section 3 is devoted to proving Theorem 3.1, the main tool used to prove the main results. After the formulation of this theorem, a sketch of the key dichotomy idea is given, followed by a short discussion. In Sect. 4, we define the density of the DLA process on a cylinder. We also prove the theorems bounding the density in the above mentioned cases, Theorems 4.6 and 4.8. Finally, in Sect. 5 we prove the lower bound on the growth rate of the cluster, Theorem 5.1. Let us note that the set up of DLA on a cylinder suggests another natural problem we are now pursuing. That is, how long does it take until the cluster clogs the cylinder? (This problem may be related to [3].) Other possible directions for further research are presented in the last section, followed by an appendix which contains a few standard variants on some simple random walk results we need. 1.1. Notation. Let G be a graph. V (G) and E(G) denote the vertex set and edge set of G respectively. We use the notation v ∈ G to denote v ∈ V (G). For two vertices u, v in G we use the notation u ∼ v to denote that u and v are adjacent. For a graph G, define the cylinder with base G, denoted G × N, by: The vertex set of G × N, is the set V (G) × N. The edge set is defined by the following relations: For all u, v ∈ G and m, k ∈ N, (u, m) ∼ (v, k) if and only if: either m = k and u ∼ v,
Diffusion Limited Aggregation on a Cylinder
189
or |m − k| = 1 and u = v. The cylinder with base G is just placing infinitely many copies of G one over the other, and connecting each vertex in a copy to its corresponding vertices in the adjacent copies. By the simple random walk on a graph, we refer to the process where at each step the particle chooses a neighbor uniformly at random and moves to that neighbor. By the lazy random walk with holding probability α, we mean a walk that with probability α stays at its current vertex, and with probability 1 − α chooses a neighbor uniformly at random. By lazy random walk (without stating the holding probability) we refer to the walk that chooses uniformly at random from the set of neighbors and the current vertex. A lazy random walk is a simple random walk on the same graph with a self loop added at each vertex. For simplicity, this paper will only deal with regular graphs; i.e. graphs such that all vertices are of the same degree. We define the notion of the mixing time of a d-regular graph G: Let gt t≥0 be a lazy random walk on G. The mixing time of G, is defined by 1 def . m = m(G) = min t > 0 ∀ u, v ∈ G ∀ s ≥ t , Pr gs = u g0 = v ≥ 2 |G| (1.1) This is a valid definition, since for all v ∈ G, 1 lim max Pr gt = u g0 = v − = 0. t→∞ u∈G |G| (This can be seen via Lemma B.1.) For a probability event A, we denote by A the complement of A. 2. Cylinder DLA 2.1. Definition. Fix a graph G. We define the G-Cylinder-DLA process: Consider the graph G × N. Denote by G i the induced subgraph on the vertices V (G) × {i}, for all i ∈ N. We call G i the i th layer of G × N. The process is an increasing sequence, {At }∞ t=0 , of connected subsets of G × N. We start with A0 = G 0 . Given At , define the set At+1 as follows: Let ∂ At be the set of all vertices of G × N that are not in At , but are adjacent to some vertex of At . That is, ∂ At = u ∈ G × N u ∈ At , ∃v ∈ At : u ∼ v . Let a particle perform a simple random walk on G × N starting from infinity, and stop when the particle hits ∂ At . Let u be the vertex in ∂ At where the particle is stopped. Then, set At+1 = At ∪ {u}. We find it convenient alternative (but equivalent) definition: Let to use the following M(t) = min i ∈ N G i ∩ At = ∅ . That is, M(t) is the lowest layer of G × N that does not intersect the cluster At . Let (gt+1 (i), ζt+1 (i)) ∈ G × N, i = 0, 1, 2, . . ., be a simple random walk on G × N, such that gt+1 (0) is uniformly distributed in G, and ζt+1 (0) = M(t). Let κ(t + 1) be the first time at which the walk is in ∂ At . That is, κ(t + 1) = min r ≥ 0 (gt+1 (r ), ζt+1 (r )) ∈ ∂ At .
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200
400
600
800
1000
1200
1400
1600
1800
2000
Fig. 1. G-Cylinder-DLA, where G is the cycle on 500 vertices. The number of particles is approximately 64, 400.
Since the walk is recurrent, κ(t + 1) < ∞ with probability 1. Let κ = κ(t + 1). Then, (gt+1 (κ), ζt+1 (κ)) is distributed on the set ∂ At . Set At+1 = At ∪ {(gt+1 (κ), ζt+1 (κ))}. This construction is equivalent to “starting from infinity”; a simple random walk starting at higher and higher layers, will take more and more steps before reaching the layer M(t). Thus, as the starting layer tends to infinity, the distribution of the particle at the first time it hits the layer M(t) is tending to uniform. 2.2. Let {At }∞ t=0 be a G-Cylinder-DLA process. At is called the (G-Cylinder-DLA ) cluster at time t. Define the following random variables: For At , the G-Cylinder-DLA cluster at time t, define the load of the i th layer at time t by L t (i) = |At ∩ G i | . L t (i) is the number of particles in the cluster at time t, on the i th layer. Also define L t (≥ i) = L t ( j), and L t (> i) = L t ( j). j≥i
j>i
L t (≥ i) (respectively L t (> i)) is the total load on layers ≥ i (respectively > i). Note that L t (≥ i) =
M(t)−1
L t ( j),
and
L t (> i) =
j=i
M(t)−1
L t ( j).
j=i+1
When subscripts become too small, we write L(t, i) instead of L t (i) (and similarly for L(t, ≥ i) and L(t, > i)). Here are some properties of the Cylinder-DLA process, that we leave for the reader to verify. (This can help to get used to the notation.) 1. For all s > t, At As . 2. For any i ∈ N, and s ≥ t, L t (i) ≤ L s (i). 3. If L t (i) = 0, then L t ( j) = 0 for all j ≥ i. 4. For all i ≥ M(t), L t (i) = 0. For all i < M(t), L t (i) ≥ 1. 5. For all t, ∞ i=0
L t (i) =
M(t)−1
L t (i).
i=0
6. The following events are identical (for any t > 0): {L t (≥ i) > L t−1 (≥ i)} = {L t (≥ i) = 1 + L t−1 (≥ i)} = {ζt (κ(t)) ≥ i} .
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2.3. G-Cylinder-DLA grows arms, for quickly mixing G. Theorem 2.1. Let 2 ≤ d ∈ N. There exists n 0 = n 0 (d), such that the following holds for all n > n 0 : Let G be a d-regular graph of size n, and mixing time m(G) ≤
log2 (n) . (log log(n))5
Let {At } be a G-Cylinder-DLA process. For m ∈ N, define Tm = min t ≥ 0 At ∩ G m = ∅ . Tm is the time the cluster first reaches the layer m. Then, for all m, E [Tm ] <
4mn . log log n
The proof of Theorem 2.1 is via Theorem 3.1 below. Remark. One may suggest that the reason Theorem 2.1 can be proved, is that we use for the base graph G, graphs that are so highly connected that in some sense there is no geometry. We stress that the class of graphs that have log(2−ε) |G| mixing time, is much larger than what is known as “expander graphs”. This class includes many natural families of graphs, including lamplighter graphs on tori of dimension 2 and above (see [10]). We remark that Theorem 3.1 below is in some sense a “worst case” analysis. Thus, we believe that our results are not optimal. In fact, we conjecture that a stronger result than Theorem 2.1 should hold for any graph at the base of the cylinder: Conjecture 2.2. Let {G n } be a family of d-regular graphs such that limn→∞ |G n | = ∞. There exist 0 < γ < 1 and n 0 such that for all n > n 0 the following holds: Set G = G n and let {At } be a G-Cylinder-DLA process. For m ∈ N, define Tm = min t ≥ 0 At ∩ G m = ∅ . Tm is the time the cluster first reaches the layer m. Then, for all m, E [Tm ] ≤ m |G n |γ . 3. The Time to Stick to a New Layer The following theorem states that under the assumption that G has small enough mixing time, in the G-Cylinder-DLA process, the expected amount of particles until one sticks to the new layer, is substantially less than |G|. Note that since for all m, we can write the telescopic sum Tm =
m (T − T−1 ), =1
Theorem 2.1 follows from Theorem 3.1, by linearity of expectation.
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Theorem 3.1. Let 2 ≤ d ∈ N. There exists n 0 = n 0 (d), such that the following holds for all n > n 0 : Let G be a d-regular graph of size n, and mixing time m(G) ≤
log2 (n) . (log log(n))5
Let At be a G-Cylinder-DLA cluster at time t. Define T = min s > t M(s) > M(t) . T is the first time that a particle sticks to the empty layer, G M(t) . Then, E [T − t] <
4n . log log n
In order to prove Theorem 3.1, we need a few lemmas, stated and proved in this section. The proof of Theorem 3.1 is deferred to Sect. 3.4. The main idea of the proof is in the following proof sketch: Proof Sketch. The cluster At can be in two states: Either it is such that particles stick quickly to it; i.e. particles take few steps before sticking to the cluster. Or, the particles take many steps before sticking to the cluster. In the first case, the particles take few steps before sticking. Thus, the particles cannot stick many layers below M(t), so they build up a heavy load on the layers near M(t). Each time a layer has a heavy load, there is better
chance of the next particles to stick to n the layers above it. So, in less than O log log n particles, there is a heavy load on the layer M(t) − 1, and the probability of sticking to the layer M(t) is now substantially greater than 1/n. This case is dealt with in Lemma 3.5. In the second case, the particles take many steps before sticking. Thus, they also make many long excursions above the layer M(t). Because the base of the cylinder, G, has small enough mixing time, after each such excursion, there is a chance of at least 1/2n to stick to the layer M(t). This occurs many times, so the probability of sticking to the layer M(t) is much greater than 1/n. This case is dealt with in Corollary 3.14. The proof of Theorem 3.1 in Sect. 3.4 combines both cases, to show that in both cases, the expected time until a particle sticks to the new layer M(t), is substantially smaller than n. Remark. As stated above, the proof of Theorem 3.1 is in some sense a “worst-case” analysis. The first part, regarding the case where particles take few steps before sticking, is valid for any regular G (not only those with small mixing time). But in reality, simulations show that this is not what really happens. The particles do not build a series of higher and higher layers with large loads. On the other hand, the second part, (where particles take many steps and thus return to the layer M(t) many times, thus increasing the probability of sticking to M(t)) is probably what does actually occur. In fact, we suspect that this is true not only for graphs with small mixing time, but for any graph at the base of the cylinder (see Conjecture 2.2). Remark. It may be of use to note that Theorem 3.1 holds also if At is replaced with any subset of G × N intersecting all layers up to M(t). In particular, given any cluster, not necessarily grown by a G-Cylinder-DLA process, the expected time until a particle sticks to the new layer is bounded by order log |G| log|G| .
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3.1. A large load on a high layer. In this section, we show that if there is a high enough log n n layer (≥ M(t) − 4 log log n ) with large load (at least log n ), then the expected time until a particle sticks to the new layer, M(t), is o(n). Lemma 3.2. There exists n 0 , such that the following holds for all n > n 0 : Let G be a d-regular graph of size n. Set log(n) , and ν = ν(n) = log(n). µ = µ(n) = 4 log log(n) Let At be a G-Cylinder-DLA cluster at time t. Let T = min s > t M(s) > M(t) . T is the first time that the cluster reaches the new layer. Assume that there exists j ≥ M(t) − µ such that L t ( j) ≥ nν . Then, E [T − t] ≤
n . 4 log log(n)
The main idea of the proof is as follows: If a layer j has load m, then the probability m to stick above layer j is at least m/n. Thus, to get a layer i > j with load log(m) , we need o(n) particles. Thus, building higher and higher layers with high loads, we reach the empty layer in o(n) particles. Proof of Lemma 3.2. The following proposition states that if there is a layer with load m, then the probability of particles sticking above that layer is at least m/n. Proposition 3.3. Let G be a d-regular graph of size n. Let At be a G-Cylinder-DLA cluster at time t. Fix a layer j > 0. Assume that L t ( j) ≥ m. For s > t, let Is be the indicator function of the event that the s th particle sticks to a layer ≥ j + 1. That is, Is = 1{L(s,≥ j+1)>L(s−1,≥ j+1)} . Then, for all s > t, m Pr Is = 1 Ir , t < r < s ≥ , n for any values of Ir , t < r < s. Proof. Set s > t. Condition on the values of Ir , t < r < s. Let As−1 be the cluster at time s − 1. Let (g(·), ζ (·)) = ((gs (·), ζs (·)) be the walk of the s th particle. Note that for any r , if (g(r ), ζ (r )) ∈ ∂ As−1 ∪ As−1 , then κ(s) ≤ r . At time s − 1 the layer j has load L s−1 ( j) ≥ L t ( j) ≥ m. Thus, (∂ As−1 ∪ As−1 ) ∩ G j+1 ≥ m. Let k be the first time the walk ((g(·), ζ (·)) hits the layer j + 1. Then, since the uniform distribution on G is the stationary distribution, g(k) is uniformly distributed in G j+1 . Thus, m Pr κ(s) ≤ k As−1 ≥ Pr (g(k), ζ (k)) ∈ ∂ As−1 ∪ As−1 As−1 ≥ . n
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Since for all 0 ≤ r ≤ k we have that ζ (r ) ≥ j + 1, we get that m Pr Is = 1 As−1 ≥ Pr κ(s) ≤ k As−1 ≥ . n Let A be the set of all clusters A ⊆ G ×N such that Pr As−1 = A Ir , t < r < s > 0. Then we have, Pr Is = 1 Ir , t < r < s m = Pr Is = 1 As−1 = A Pr As−1 = A Ir , t < r < s ≥ . n A∈A
Assume there is a layer with load m. Since each particle sticks above this layer with probability at least m/n, the expected time until there are new particles above this layer should be at most · (n/m). This is captured in the following proposition: Proposition 3.4. Let G be a d-regular graph of size n. Let At be a G-Cylinder-DLA cluster at time t. Fix a layer j > 0. Assume that L t ( j) ≥ m. For ∈ N, define S = min s ≥ t L s (≥ j + 1) = + L t (≥ j + 1) . (3.1) That is, S is the first time that there are new particles in the layers ≥ j + 1 (so S0 = t). Then, E [S − t] = E [S − S0 ] ≤
n . m
Proof. By Proposition 3.3, for all k ≥ 1, Sk − Sk−1 is dominated by a geometric random variable with mean ≤ mn . Thus, E [S − t] =
k=1
m E Sk − Sk−1 ≤ . n
With these two propositions, we continue with the proof of Lemma 3.2. Set M = M(t). Set T0 = t. For r ≥ 0, define inductively the following stopping times:
Tr = min s ≥ Tr −1 ∃i ≥ j + r : L s (i) ≥ nν −(2r +1) . That is, Tr is the first time that there exists a “high enough” layer (higher than j + r ), such that the load on that layer is “large enough” (larger than nν −(2r +1) ). Consider time Tµ . At this time, we have that there exists a layer i ≥ j + µ ≥ M such that L Tµ (i) ≥ nν −(2µ+1) ≥ 1. So M(Tµ ) > M and T ≤ Tµ . Thus, we can write T −t =
µ
(min {T, Tr } − min {T, Tr −1 }) .
r =1
For all r ≥ 0, set τ (r ) = min {T, Tr }.
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Claim. For all r > 0, E [τ (r ) − τ (r − 1)] ≤
n . ν
Proof. Fix r > 0. For ∈ N, define S = min s ≥ Tr −1 L s (≥ j + r ) = + L Tr −1 (≥ j + r ) . That is, S is the first time that there are new particles in the layers ≥ j + r . So S0 = Tr −1 . Let a = µnν −(2r +1) . Case 1. T ≤ Tr −1 . Then τ (r ) − τ (r − 1) = 0 ≤ Sa − S0 . Case 2. T > Tr −1 and T ≤ Sa . Then τ (r ) − τ (r − 1) ≤ T − Tr −1 ≤ Sa − S0 . Case 3. T > Tr −1 and T > Sa . Note that if T > Sa , then M(Sa ) = M. At time Sa , there are at least a particles on the layers ≥ j + r . So, if T > Sa , then a≤
M(S a )−1
L Sa (i) =
i= j+r
M−1
L Sa (i).
i= j+r
So there exists some j + r ≤ i ≤ M − 1 such that L Sa (i) ≥ M−(aj+r ) . Since j ≥ M − µ, we have L Sa (i) ≥ µa ≥ nν −(2r +1) . So we conclude that if T > Sa then Tr ≤ Sa < T . So T > Sa implies that τ (r ) − τ (r − 1) = Tr − Tr −1 ≤ Sa − S0 . Thus, in all three cases, τ (r ) − τ (r − 1) ≤ Sa − S0 . At time S0 = Tr −1 , by the definition of Tr −1 , we have that for some i ≥ j + r − 1, there is a load L S0 (i) ≥ nν −(2r −1) (for r = 1 we can choose i = j, and since T0 = t we have by assumption that L t ( j) ≥ nν ). By Proposition 3.4, with j = i, t = Tr −1 = S0 and m = nν −(2r −1) , we have that for large enough n, E [τ (r ) − τ (r − 1)] ≤ E [Sa − S0 ] ≤ a
n n ≤ . ν nν −(2r −1)
Returning to the proof of Lemma 3.2, for all r > 0, E [τ (r ) − τ (r − 1)] ≤
n . ν
Thus, for large enough n, E [T − t] =
µ r =1
E [τ (r ) − τ (r − 1)] ≤ µ ·
n n ≤ . ν 4 log log(n)
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3.2. Particles take few steps. Recall that κ(s) is the number of steps the s th particle takes until it sticks (to ∂ As−1 ). In this section, we show that if κ(t + 1) is small, then all particles s > t, have a good chance of sticking at high layers. Thus, a small amount of particles is needed to get a high layer with large load. Lemma 3.5. There exists n 0 such that the following holds for all n > n 0 : Let G be a d-regular graph of size n. Set log(n) µ = µ(n) = , and ν = ν(n) = log(n). 4 log log(n) Let At be a G-Cylinder-DLA cluster at time t. Let T = min s > t M(s) > M(t) . T is the first time that the cluster reaches the new layer. Assume that Pr κ(t + 1) ≤ 1 4.
µ2 4
≥
Then, E [T − t] ≤
5n . 2 log log(n)
Proof. In the following two propositions, we use the fact that with probability at least 1/4, the particle takes a small amount of steps to stick. Proposition 3.6. Let G be a d-regular graph. Let At be a G-Cylinder-DLA cluster at time t, and consider the (t +1)th particle. Let y ∈ N and assume that Pr κ(t + 1) ≤ y 2 /4 ≥ 41 . Then, 1 Pr min ζt+1 (r ) ≥ M(t) − y ≥ . 0≤r ≤κ(t+1) 8 That is, with probability at least 1/8, the particle sticks without ever going below the layer M(t) − y. 2
Proof. Set x = y4 . Note that Pr [κ(t + 1) ≤ x] ≥
1 . 4
Let (g(·), ζ (·)) = (gt+1 (·), ζt+1 (·)) be the walk the (t + 1)th particle takes. That is, g(0) is uniformly distributed in G, and ζ (0) = M(t). Let κ = κ(t + 1) be the first time the walk hits ∂ At . Note that min ζ (r ) ≥ M(t) − y ∩ {κ ≤ x} implies min ζ (r ) ≥ M(t) − y . 0≤r ≤x
0≤r ≤κ
The walk ζ (0), . . . , ζ (x), is an x-step lazy random walk, with holding probability d . By Lemma A.6, we have that 1 − α = d+2 √ 1 Pr min ζ (r ) ≥ M(t) − y ≥ Pr max |ζ (r ) − M(t)| < 8αx ≥ 1 − . 0≤r ≤x 1≤r ≤x 8
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Thus, 1 Pr min ζ (r ) ≥ M(t) − y ≥ Pr min ζ (r ) ≥ M(t) − y − Pr [κ > x] ≥ , 0≤r ≤κ 0≤r ≤x 8 (where we have used the inequality Pr [A ∩ B] ≥ Pr [A] − Pr B , valid for any events A, B). Proposition 3.7. Let G be a d-regular graph. Let At be a G-Cylinder-DLA cluster at time t. Let y ∈ N and assume that 1 Pr κ(t + 1) ≤ y 2 /4 ≥ . 4 For s > t, define H (s) = ζs (κ(s)); i.e. H (s) is the height of the layer at which the s th particle sticks. Then, for all s > t, 1 Pr H (s) ≥ M(t) − y H (r ), t < r < s ≥ , 8 for any values of H (r ), t < r < s. Proof. Let s > t. Let (g(·), ζ (·)) = (gs (·), ζs (·)) be the walk the s th particle takes. That is, g(0) is uniformly distributed in G, and ζ (0) = M(s) ≥ M(t). Set k = min r ≥ 0 ζ (r ) = M(t) , and set k = min r ≥ k (g(r ), ζ (r )) ∈ ∂ At . k is the first time the s th particle is at the layer M(t) (this can be time 0, e.g. if M(s) = M(t)). k is the first time after k that the particle hits the outer boundary of the cluster At . Since At ⊆ As−1 , we have that κ(s) ≤ k . So, Pr H (s) ≥ M(t) − y As−1 ≥ Pr min ζ (r ) ≥ M(t) − y As−1 0≤r ≤κ(s) ≥ Pr min ζ (r ) ≥ M(t) − y As−1 0≤r ≤k ≥ Pr min ζ (k + r ) ≥ M(t) − y As−1 , 0≤r ≤k −k
the last inequality following from the fact that for all r < k, by definition, ζ (r ) ≥ M(t) ≥ M(t) − y. Since the uniform distribution is the stationary distribution on G, g(k) is uniformly distributed in G. Thus, the walk (g(k + r ), ζ (k + r )) has the same distribution as the walk (gt+1 (r ), ζt+1 (r )), and k − k has the same distribution as κ(t + 1). Using Proposition 3.6 we now conclude 1 Pr H (s) ≥ M(t) − y As−1 ≥ Pr min ζt+1 (r ) ≥ M(t) − y ≥ . 0≤r ≤κ(t+1) 8 Averaging over all A ⊂ G × N such that Pr As−1 = A H (r ), t < r < s > 0, we get that 1 Pr H (s) ≥ M(t) − y H (r ), t < r < s ≥ . 8
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Proposition 3.8. Let G be a d-regular graph. Let At be a G-Cylinder-DLA cluster at time t. Let y ∈ N and assume that 1 Pr κ(t + 1) ≤ y 2 /4 ≥ . 4 For ∈ N, define
S = min s ≥ t L s (≥ M(t) − y) = + L t (≥ M(t) − y) .
That is, S is the first time that there are new particles in the layers ≥ M(t) − y (so S0 = t). Then, E [S − t] ≤ 8. Proof. The proof is similar to the proof of Proposition 3.4. By Proposition 3.7, regardless of the previous particles, each particle s > t has probability at least 1/8 to stick to a layer ≥ M(t) − y. Thus, the expected time until there are particles above this layer is bounded by 8. We now put everything together to prove Lemma 3.5. We show that if κ(t + 1) is small, then after a small amount of particles there is a high layer with large load. Thus, after another small amount of particles, the cluster reaches the new layer M(t). Set M = M(t). Let
n . T = min s ≥ t ∃ j ≥ M − µ : L s ( j) ≥ ν For ∈ N, define
S = min s ≥ t L s (≥ M − µ) = + L t (≥ M − µ) .
Consider the time Sa for a = µ nν . Consider the case where T > Sa . Then M(Sa ) = M. At time Sa , there are at least a particles in the layers ≥ M − µ, so a≤
M(S a )−1 i=M−µ
L Sa (i) =
M−1
L Sa (i).
i=M−µ
Thus, there exists M − µ ≤ j ≤ M − 1 such that L Sa ( j) ≥ µa ≥ nν . So T ≤ Sa . We conclude that if T > Sa then T ≤ Sa . In other words, we have shown that min T, T ≤ Sa . Hence, because it was assumed that Pr κ(t + 1) ≤ µ2 /4 ≥ 41 , by Proposition 3.8, E min T, T − t ≤ E [Sa − t] ≤ 8a ≤ Define the event
2 log(n) 2n + . log log(n) log log(n)
n B = ∃ j ≥ M(T ) − µ : L T ( j) ≥ . ν
By Lemma 3.2, we have that for large enough n, n E 1{B} T − T ≤ . 4 log log n
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We have that T = min T, T + 1{T
M, and T ≤ T . So the event T < T implies the event B. Thus, for large enough n, E [T − t] ≤ E min T, T − t + E 1{B} T − T ≤
5n . 2 log log(n)
3.3. Particles take many steps. In the previous section, we analyzed what happens when κ(t + 1) is “small”. This section is concerned with the case where κ(t + 1) is “large”. The main goal of this section is proving Lemma 3.11 and Corollary 3.14. These are essential ingredients in the proof of Theorem 3.1. We begin with two technical lemmas: Lemma 3.9. Let G be a graph. Let G be the graph obtained from G by adding a self loop at each vertex. That is, V (G ) = V (G) , and E(G ) = E(G) ∪ {v, v} v ∈ G . Consider the and G -Cylinder-DLA Let V = V (G ×N) = V (G ×N). G-Cylinder-DLA } processes. Let Pt (A, v) = Pr At = A ∪ {v} At−1 = A , where {A t is a G-CylinderDLA process. Let Pt (A, v) = Pr At = A ∪ {v} At−1 = A , where {At } is a G -Cylinder-DLA process. Then, for all A ⊆ V , v ∈ V , and all t > 0, Pt (A, v) = Pt (A, v). Proof. Assume that At = A. We can couple the walk of the (t + 1)th particle in both processes to hit the same Denote by L the set of self loops vertex, as follows: added to G to form G . Let (g(r ), ζ (r )) r ≥ 0 be the walk of the (t + 1)th particle, in the G -Cylinder-DLA process. Define to be the set of all r > 0 such that the step from (g(r − 1), ζ (r − 1)) to (g(r ), ζ (r )) does not traverse one of the self loops in L. For process, let the (t + 1)th particle take the path the G-Cylinder-DLA (g(r ), ζ (r )) r ∈ ∪ {0} . This path has the correct marginal distribution, as it is a simple random walk on G × N. Note that both paths hit ∂ At at the same vertex, since traversing a self loop does not move the particle to a new vertex. Remark. If G already has self loops, then by adding a self loop at each vertex, we mean adding a new self loop, treated as different from the original loop. This only adds technical complications, so we will not go into this issue. The reader can treat all graphs as not having self loops, though the results carry out to graphs with self loops as well. The important consequence of Lemma 3.9 is that the Cylinder-DLA process does not change if we let the particles perform a lazy random walk on G × N. This is needed to avoid technical complications that arise from parity issues in bi-partite graphs. The following technical lemma is used to bypass this issue. Recall our definition of the mixing time of a d-regular graph G: Let gt t≥0 be a lazy symmetric random walk on G. The mixing time of G is defined by 1 def . m = m(G) = min t > 0 ∀ u, v ∈ G ∀ s ≥ t , Pr gs = u g0 = v ≥ 2 |G|
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Lemma 3.10. Let G be a d-regular graph, and let G be the graph obtained from G by adding a self loop at each vertex, as in Lemma 3.9. Let {gt }t≥0 be a simple random walk on G . Then, for all t ≥ m(G), and all u, v ∈ G, Pr gt = u g0 = v ≥
1 . 2 |G|
Proof. This is immediate from the definition of m(G), and the fact that {gt }t≥0 is distributed as a lazy symmetric random walk on G. This completes the two technical lemmas we require. Next we introduce some notation. Let G be a graph. Let (g(0), ζ (0)), (g(1), ζ (1)), . . . , be a simple random walk on G × N. For two times r1 < r2 denote r1 → r2 = {(g(r1 ), ζ (r1 )), (g(r1 + 1), ζ (r1 + 1)), . . . , (g(r2 ), ζ (r2 ))} . def
r1 → r2 is the path the walk takes between times r1 and r2 . Define L = r > 0 ζ (r ) = ζ (0) , and assume that L = {1 < 2 < · · · }. L is the set of times at which the walk visits the def original layer. For i ≥ 1 define ρi = i−1 → i , where 0 = 0. We call ρi an excursion. For i ≥ 1 and α ∈ R, we say that ρi = i−1 → i is a positive α-long excursion if the following conditions hold: 1. ζ (i−1 + 1) = ζ (0) + 1; i.e. the excursion is on the positive side of the origin of the walk. 2. The walk takes at least α steps in G during the excursion; that is, i
1{ζ (r )=ζ (r −1)} ≥ α.
r =i−1 +1
We stress that ‘α-long’ refers to the number of steps in G, not the total length of the excursion. Lemma 3.11. Let 2 ≤ d ∈ N. There exist c = c(d) > 0 and C = C(d) > 0 such that for any x ≥ 1 the following holds: Let G be a d-regular graph of size |G| = n and mixing time m(G). Let At be a G-Cylinder-DLA cluster at time t. Recall that M(t) is the lowest empty layer at time t, and that κ(t + 1) is the number of steps the (t + 1)th particle takes before it sticks. Then, 1 x 2 − Pr κ(t + 1) ≤ C x · . Pr M(t + 1) > M(t) At > c √ 2 n m(G) Proof. Let G be a d-regular ζ (0)), graph. Let (g(0), (g(1), ζ (1)), . . . , be a simple random walk on G × N. Let ρi = i−1 → i i ≥ 1 be the excursions of the walk. First, we need to calculate the probability of a positive α-long excursion. Proposition 3.12. For all i ≥ 1 and any 2 ≤ α ∈ R, the probability that ρi is a positive 1 √ α-long excursion is greater than 12(d+2) . α
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Proof. of the Markov property, and the fact that ζ (i ) = ζ (0) for all i, we get Because that ρi i ≥ 1 are independent and identically distributed. Thus, it suffices to prove the proposition for ρ = ρ1 . Fix 2 ≤ α ∈ R. Set m = ζ (0) + 1. So, the probability that ρ is a positive α-long excursion is equal to 1 Pr ζ (1) = m , 1{ζ (r )=ζ (r −1)} ≥ α r =1
= Pr [ζ (1) = m] Pr
1
1{ζ (r )=ζ (r −1)}
≥ α ζ (1) = m ,
(3.2)
r =2
(we use the fact that ζ (1) = m = ζ (0)). Define
= r > 1 ζ (r ) = ζ (r − 1) , Z = r > 1 ζ (r ) = ζ (r − 1) .
and
(respectively, Z ) is the set of times at which moves in G (respectively, N). Let the walk g1 = g(1) and let g2 , g3 , . . . , be the walk g(r ) r ∈ . So g1 , g2 . . . , is distributed as a simple randomwalk on G, starting at g(1). Let ζ1 = ζ (1) and let ζ2 , ζ3 , . . . , be the walk ζ (r ) r ∈ Z . So ζ1 , ζ2 . . . , is distributed as a simple random walk on N, starting at ζ (1). Set α = 8 α2 . For r > 1, let Ir = 1{ζ (r )=ζ (r −1)} . Set γ =
+1 α
Ir .
r =2
γ is the sum of α independent, identically distributed Bernoulli random variables, with d ≥ 21 . Using the Chernoff bound (see e.g. Appendix A in [1]), mean d+2 α 2 3 α ≤ 2 exp − ≤ < . Pr γ < 4 8 e 4 γ is independent of ζ (1), so
α 1 Pr γ ≥ α ζ (1) = m ≥ Pr γ ≥ ζ (1) = m > . 4 4
(3.3)
Consider the walk ζ1 , ζ2 , . . . , ζα +1 , conditioned on the event ζ (1) = m. Define the event B = {ζ2 ≥ m , ζ3 ≥ m , . . . , ζα +1 ≥ m} . Conditioned on ζ (1) = m, the walk ζ1 , ζ2 , . . . , ζα +1 is a simple random walk on N starting at ζ1 = m. Using Corollary A.2, α . Pr B ζ (1) = m ≥ Pr ζ2 ≥ m , . . . , ζα +1 ≥ m ζ (1) = m = 2−α α /2 A careful application of Stirling’s approximation gives 1 Pr B ζ (1) = m > √ , 3 α for all α ≥ 2.
(3.4)
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Since ζ (1 ) =ζ (0) = m − 1, we have that, conditioned on ζ (1) = m, the event B implies the event α + 1 ≤ 1 . Thus, 1
1{ζ (r )=ζ (r −1)} ≥ 1{B}
r =2
+1 α
1{ζ (r )=ζ (r −1)} = 1{B} γ .
r =2
Now, γ is independent of the event B, so, using (3.3) and (3.4), 1 Pr 1{ζ (r )=ζ (r −1)} ≥ α ζ (1) = m ≥ Pr B , γ ≥ α ζ (1) = m r =2
= Pr B ζ (1) = m Pr γ ≥ α ζ (1) = m 1 > √ . 12 α
Plugging this into (3.2), we have that the probability that ρ is a positive α-long excursion 1 √ is greater than 12(d+2) . α The next proposition bounds from below the probability of sticking to the layer M(t) at each excursion. Proposition 3.13. For all i ≥ 1, c , Pr κ(t + 1) = i κ(t + 1) > i−1 ≥ √ n m(G) where c = c(d) > 0 is a constant that depends only on d. Proof. It suffices to prove that for any u, v ∈ G, c Pr g(i ) = u g(i−1 ) = v ≥ √ , n m(G) for some constant c = c(d) > 0, depending only on d. Let G be the graph obtained from G by adding a self loop at each vertex. By Lemma 3.9 we can assume that (g(·), ζ (·)) is a walk on G × N. Let = i−1 < r ≤i ζ (r ) = ζ(r − 1) , and let γ = | |. Let h r 0 ≤ r ≤ γ be the walk g(i−1 + r ) r ∈ ∪ {0} . h 0 , h 1 , . . . , h γ is the walk measured only when moving in the G-coordinate. Note that conditioned on , the walk h 0 , h 1 , . . . , h γ has the distribution of a lazy random walk on G. Thus by Lemma 3.10, we have that for any u, v ∈ G, 1 . Pr h γ = u γ ≥ m(G) , h 0 = v ≥ 2n Note that if ρi is a m(G)-long excursion then γ ≥ m(G). Thus, for any u, v ∈ G, using Proposition 3.12, Pr g(i ) = u g(i−1 ) = v ≥ Pr h γ = u h 0 = v , γ ≥ m(G) Pr γ ≥ m(G) h 0 = v ≥
c 1 ·√ . 2n m(G)
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∞ Back to the proof of Lemma 3.11: Note that the events {κ(t + 1) = i }i=0 are pairwise disjoint, and that for every i ≥ 0, we have {κ(t + 1) = i } ⊂ {M(t + 1) > M(t)}. Thus, using Proposition 3.13 we now have for any x ≥ 1,
Pr [M(t + 1) > M(t)] ≥ =
∞ i=1 ∞
Pr [κ(t + 1) = i ] Pr κ(t + 1) = i κ(t + 1) > i−1 Pr κ(t + 1) > i−1
i=1
c , ≥ x Pr [κ(t + 1) > x ] · √ n m(G)
(3.5)
for a constant c = c(d) > 0 depending only on d. Since for any C > 0, Pr [κ(t + 1) > x ] ≥ Pr x ≤ C x 2 − Pr κ(t + 1) ≤ C x 2 , we are left with proving that there exists C > 0 such that Pr x ≤ C x 2 ≥ 21 for any x ≥ 1. Note that x > C x 2 implies that the number of times the walk ζ (·) visits the layer M(t) up to time C x 2 is less than x. Thus by Lemma A.5, there exists C = C(d) > 0 such that 1 Pr x > C x 2 ≤ . 2 Corollary 3.14. Let 2 ≤ d ∈ N. There exist n 0 = n 0 (d) such that the following holds for all n > n 0 : Let G be a d-regular graph of size n, and mixing time m(G) ≤
log2 (n) . (log log(n))5
Consider the G-Cylinder-DLA process. Let At be a G-Cylinder-DLA cluster at time t. Set log(n) . µ = µ(n) = 4 log log(n) Assume that Pr κ(t + 1) ≤ µ2 /4 < 41 . Then, Pr [M(t + 1) > M(t)] >
log log(n) . n
Proof. Let C and c be as in Lemma 3.11. We can choose x ≥ ≤ and 3.11, we get
Cx2
µ2 /4
√ cx m(G)
such that
≥ log log(n) for large enough n. Plugging this into Lemma
Pr [M(t + 1) > M(t)] ≥
log(n) (log log(n))(3/2)
log log(n) . n
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3.4. Proof of Theorem 3.1. For convenience, we restate the theorem: Theorem (3.1). Let 2 ≤ d ∈ N. There exists n 0 = n 0 (d), such that the following holds for all n > n 0 : Let G be a d-regular graph of size n, and mixing time m(G) ≤
log2 (n) . (log log(n))5
Let At be a G-Cylinder-DLA cluster at time t. Define T = min s > t M(s) > M(t) . T is the first time that a particle sticks to the empty layer, M(t). Then, E [T − t] ≤ Proof. Set M = M(t) and µ = µ(n) = For s ≥ t, define
log(n) , 4 log log(n)
4n . log log n
and
ν = ν(n) = log(n).
α(s) = Pr κ(s + 1) ≤ µ2 /4 As ,
(which is random variable that is a function of As ). Define 1 . τ = min s ≥ t α(s) ≥ 4 Fix s > t, and t + 1 ≤ r ≤ s. By Corollary 3.14, there exists n 0 = n 0 (d) such that for all n > n 0 , 1 1 Pr M(r ) = M(t) , α(r ) < ∀ t + 1 ≤ q ≤ r − 1 M(q) = M(t) , α(q) < 4 4 1 ≤ Pr M(r ) = M(t) ∀ t + 1 ≤ q ≤ r − 1 M(q) = M(t) , α(q) < 4 log log(n) . ≤1− n Thus, for all s > t, Pr [min {T, τ } > s] = Pr [T > s , τ > s] 1 ≤ Pr ∀ t + 1 ≤ r ≤ s M(r ) = M(t) , α(r ) < 4 s 1 1 ∀ t +1 ≤ q ≤r −1 M(q) = M(t) , α(q) < = Pr M(r )=M(t) , α(r ) < 4 4 r =t+1 s−t log log(n) . ≤ 1− n
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Since, Pr [min {T, τ } > t] ≤ 1, we get that E [min {T, τ } − t] ≤ Define
n . log log(n)
τ <∞ min s > 0 M(τ + s) > M(τ ) T = 0 τ = ∞. Then we have T ≤ min {T, τ } + T . If τ = ∞ then E T τ = ∞ = 0. Assume that τ < ∞. Then, at time τ , we have that Pr κ(τ + 1) ≤ µ2 /4 Aτ ≥ 41 . So, using Lemma 3.5,
E T τ < ∞ ≤
5n , 2 log log(n)
and consequently, E T <
3n . log log(n)
Thus, we conclude that E [T − t] ≤ E [min {T, τ } − t] + E T <
4n . log log(n)
4. Density 4.1. Definitions and notation. Definition 4.1. Fix a graph G, and let {At } be a G-Cylinder-DLA process. Define the cluster at infinity by A∞ =
∞
At .
t=0
For m ∈ N, define D(m) =
m 1 |A∞ ∩ G i | . mn i=1
D(m) is the fractional amount of particles in the finite cylinder G × {1, . . . , m}. Define the density at infinity by D = D∞ = lim D(m). (4.1) m→∞
Using standard arguments from ergodic theory it can be shown that the limit in (4.1) exists, and is constant almost surely. Since D(m) are bounded random variables, we get by dominated convergence (see e.g. Chapter 9 in [4]):
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D = E [D] = lim E [D(m)] . m→∞
Recall the random times:
Tm = min t ≥ 0 At ∩ G m = ∅ .
Tm is the time the cluster first reaches the layer m. Theorem 4.2. Let G be a d-regular graph of size n, and let {At } be a G-Cylinder-DLA process. Let D = D∞ be the density at infinity, and for all m let Tm = min t ≥ 0 At ∩ G m = ∅ . Then, 1 E [Tm ] . m→∞ mn
D = lim
Theorem 4.2 relates the density at infinity to the average growth rate. Theorem 4.2 is proved via the following propositions. The proof of the theorem is in Sect. 4.3. 4.2. The main objective of this section is Proposition 4.5. This proposition is the main observation in proving Theorem 4.2. First we require some notation: For a G-CylinderDLA process {At }, recall L t (i) = |At ∩ G i |, the load of the i th layer at time t. Define the load of the i th layer at infinity: L(i) = L ∞ (i) = |A∞ ∩ G i | . def
Define: L t (≤ i) = def
i
L t ( j).
j=1
L(≤ i) = L ∞ (≤ i) = def
i
L(i).
j=1
For 0 ≤ t ≤ ∞, L t (≤ i) is the total load of all layers below i, including i but not including the 0-layer. (When indices become too small we write L(t, ≤ i) instead of L t (≤ i).) Also define H (t) = ζt (κ(t)). That is, H (t) is the layer at which the t th particle sticks (the height of the t th particle). The following proposition bounds the probability that a particle sticks to a “low” layer. Proposition 4.3. Fix m < m ∈ N. Let G be a d-regular graph of size n, with spectral gap 1 − λ (i.e., λ is the second eigenvalue of the transition matrix of G). Consider the G-Cylinder-DLA process. Let t > Tm and let At−1 be the G-Cylinder-DLA cluster at time t − 1. Then, 1−λ (m − m) . Pr H (t) ≤ m At−1 < 3 exp − 8n
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Proof. Let t > Tm . Let M = M(t − 1) and ϕ = m − m. Note that M − 1 − m ≥ M(Tm ) − 1 − m = m − m = ϕ. Let (g(·), ζ (·)) = (gt (·), ζt (·)) be the walk of the t th particle. So ζ (0) = M and g(0) is uniformly distributed in G. Let k be the first step at which the walk is at the layer m. That is, k = min r > 0 ζ (r ) = m . Let κ = κ(t) be the step at which the particle sticks to the cluster. Note that the event {H (t) ≤ m} implies the event {κ ≥ k}. Moreover, {κ ≥ k} implies the event g(k − i) ∈ At−1 ∩ G ζ (k−i) , i = 1, 2, . . . , ϕ . Also, for all 1 ≤ i ≤ ϕ we have that At−1 ∩ G ζ (k−i) ≥ 1 (because ζ (k − i) ≤ m + i ≤ M − 1). Define
= k − ϕ ≤ r < k ζ (r ) = ζ (r − 1) , and assume that
= {r1 < r2 < · · · < rs } (note that s = | | is a random variable). For 1 ≤ i ≤ s let gi = g(ri ). So g1 , g2 , . . . , gs is distributed as an s-step simple random walk on G, starting from a uniformly chosen vertex. For all 1 ≤ i ≤ s define Ci = At−1 ∩ G ζ (ri ) . Thus, the event {H (t) ≤ m} implies the event {gi ∈ Ci , i = 1, 2, . . . , s}. By Lemma B.4 we have that 1 Pr gi ∈ Ci , i = 1, 2, . . . , s C1 , C2 , . . . , Cs , s ≥ ϕ 4 s 1−λ 1−λ |Ci | ≤ exp − ≤ exp − ϕ . 2n 8n i=1
Hence, Pr H (t) ≤ m At−1 ≤ Pr [gi ∈ Ci , i = 1, 2, . . . , s] 1−λ 1 ϕ . ≤ Pr s < ϕ + exp − 4 8n Note that s=
ϕ
1{ζ (k−i)=ζ (k−i−1)} .
i=1
That is, s is the sum of independent identically distributed Bernoulli random variables, d ≥ 21 . Thus, using the Chernoff bound (see e.g. Appendix A in [1]), with mean d+2 ϕ
1 . Pr s < ϕ < 2 exp − 4 8
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Thus, 1−λ Pr H (t) ≤ m At−1 < 3 exp − ϕ . 8n
Consider the following event in the G-Cylinder-DLA process: Given a cluster At , the next |G| particles appear in exactly the right order so that they completely fill up the layer M(t). (There is always such an order; e.g. consider a spanning tree of G rooted at a vertex in ∂ At .) Thus, an impassible “wall” is created. Specifically, we are interested in the event that L t+n (M(t)) = n. The following proposition bounds from below the probability of this event. Proposition 4.4. Let G be a d-regular graph of size n. Consider the G-Cylinder-DLA process. Let At be the G-Cylinder-DLA cluster at time t. Then, Pr L t+n (M(t)) = n At ≥ (d + 2)−(n−1) n −n . Proof. Consider the following event W : The (t + 1)th particle appears at a vertex in G M(t) that is in ∂ At . Since there is at least one such vertex, this happens with probability at least 1/n. For i = 2, . . . , n, the (t + i)th particle appears at the layer M(t) + 1, and moves to a vertex in G M(t) that is in ∂ At+i−1 . Since there is at least one such vertex, the probability of this is at least n −1 (d + 2)−1 , for each i = 2, . . . , n. Since the event W implies that L t+n (M(t)) = n, we have Pr L t+n (M(t)) = n At ≥ (d + 2)−(n−1) n −n . Proposition 4.5. Fix m ∈ N. Let ϕ = ϕ(m) be a positive integer, and let m = m + ϕ. Let G be a d-regular graph of size n, with spectral gap 1 − λ. Consider the G-Cylinder-DLA process. Let X (m) be the event that there exists t > Tm such that H (t) ≤ m. That is, X (m) is the event that a particle sticks to a layer ≤ m after the cluster has reached the layer m . Then, 1−λ ϕ . Pr [X (m)] ≤ (d + 2)n−1 (n + 1)n n · 3 exp − 8n Proof. Fix m ∈ N and let m = m + ϕ(m). Let t > Tm . For i ∈ N define the events W (t + i) = {L t+i+n (M(t + i)) = n} ,
and
B(t + i) = {H (t + i) ≤ m} .
Set F(t + i) = B(t + i) ∩ W (t + i). By Proposition 4.3 we have that for all i ≥ n + 1, 1−λ Pr B(t + i) ∀ 0 ≤ j ≤ i − (n + 1) F(t + j) ≤ 3 exp − ϕ(m) . 8n
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By Proposition 4.4 we have that for all i ≥ n + 1, Pr F(t + i) ∀ 0 ≤ j ≤ i − (n + 1) F(t + j) ≤ Pr W (t + i) ∀ 0 ≤ j ≤ i − (n + 1) F(t + j) ≤ 1 − (d + 2)−(n−1) n −n . Thus, for all i ≥ n + 1, Pr [B(t + i) , F(t + i − 1) , . . . , F(t)] ≤ Pr B(t + i) ∀ 0 ≤ j ≤ i − (n + 1) F(t + j) ×
i/(n+1)
Pr F(t + i − (n + 1)) ∀ 0 ≤ j ≤ i − ( + 1)(n + 1) F(t + j)
=1
≤ p(m)(1 − q)i/(n+1) , where
1−λ ϕ(m) p(m) = 3 exp − 8n
and
q = (d + 2)−(n−1) n −n .
Note that for all t > Tm , the event W (t) implies that B(t + i) for all i ≥ 0 (since the first n particles must stick to the layer M(t) > m, and after time t + n no particle can pass the layer M(t) > m). Thus, setting t = Tm + 1, the event X (m) implies that there exists i ≥ 0 such that B(t + i) ∩
i−1
F(t + j)
j=0
occurs (i.e. take the first i for which B(t + i) occurs). So, Pr [X (m)] ≤ ≤ =
∞ i=0 ∞ i=0 ∞ =0
Pr [B(t + i) , F(t + i − 1) , . . . , F(t)] p(m)(1 − q)i/(n+1) 1 (n + 1) p(m)(1 − q) = (n + 1) p(m) . q
4.3. Proof of Theorem 4.2. We restate the theorem: Theorem (4.2). Let G be a d-regular graph of size n, and let {At } be a G-Cylinder-DLA process. Let D = D∞ be the density at infinity, and for all m let Tm = min t ≥ 0 At ∩ G m = ∅ . Then, D = lim
m→∞
1 E [Tm ] . mn
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Proof. Let ϕ : N → N be any function such that lim ϕ(m) = ∞
m→∞
and
lim
m→∞
ϕ(m) = 0. m
For m ∈ N let m = m + ϕ(m). Recall that H (t) = ζt (κ(t)) is the height of the layer at which the t th particle sticks. For m ∈ N let X (m) be the event that there exists t > Tm such that H (t) ≤ m. Then, for all , we have that {L ∞ (≤ m) > }
implies
{L(Tm , ≤ m) > } ∪ X (m).
This is because if L(Tm , ≤ m) ≤ , then at least one more particle is needed to stick at a layer ≤ m after time Tm , in order for L ∞ (≤ m) > to hold. Thus, since L ∞ (≤ m) ≤ mn, using Proposition 4.5, E [L ∞ (≤ m)] = = ≤
∞
Pr [L ∞ (≤ m) > ]
=0 mn−1
Pr [L ∞ (≤ m) > ]
=0 ∞
mn−1
=0
=0
Pr [L(Tm , ≤ m) > ] +
= E [L(Tm , ≤ m)] + mn · p(m) for
1−λ p(m) = 3 exp − ϕ(m) 8n
and
Pr [X (m)]
n+1 , q
q = (d + 2)−(n−1) n −n .
Note that L t (≤ m) ≤ t for all t, so E [L(Tm , ≤ m)] ≤ E [Tm ] . Also, lim p(m)
m→∞
n+1 = 0. q
So, 1 E [L ∞ (≤ m)] mn 1 n+1 E [Tm ] + lim p(m) ≤ lim m→∞ mn m→∞ q 1 = lim E [Tm ] . m→∞ mn
E [D] = lim E [D(m)] = lim m→∞
m→∞
Since for all k > k, E [Tk − Tk ] ≤ n(k − k), we have that E [Tm ] = E [Tm ] + E [Tm − Tm ] ≤ E [Tm ] + ϕ(m)n.
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Thus,
211
1 ϕ(m) 1 E [Tm ] + lim = lim E [Tm ] . m→∞ mn m→∞ m m→∞ mn
D = E [D] ≤ lim
(4.2)
Note that for all m, Tm = L(Tm , ≤ m) ≤ L ∞ (≤ m), so E [Tm ] ≤ E [L ∞ (≤ m)]. Thus, lim
m→∞
1 1 E [Tm ] ≤ lim E [L ∞ (≤ m)] = E [D] = D. m→∞ mn mn
Equations (4.2) and (4.3) together give equality.
(4.3)
4.4. Density of Cylinder-DLA with transitive base. In this section we assume that G is vertex transitive; i.e. for any u, v ∈ G there exists an automorphism (of graphs) ϕuv : G → G such that ϕ(u) = v. Theorem 4.6. Let G be a vertex transitive graph. Let {At } be the G-Cylinder-DLA process. Let D = D∞ be the density at infinity. Then, D≤
2 . 3
The key to proving Theorem 4.6 is Lemma 4.7 below. The proof of the theorem follows the proof of the lemma. Lemma 4.7. Let G be a vertex transitive graph. Let At−1 be a G-Cylinder-DLA cluster at time t − 1. Then, 2d + 2 . Pr M(t) > M(t − 1) At−1 ≥ (d + 2)n Proof. Recall (gt (·), ζt (·)) is the walk of the t th particle, so gt (0) is uniformly distributed in G, and ζt (0) = M(t − 1). Define (t − 1) to be the newest particle in the top layer of the cluster At−1 . That is, if At−1 ∩ G (M(t−1)−1) = {v1 , . . . , v }, then (t − 1) is the vertex vi that is the last vertex to join the cluster. Note that ((t −1), M(t −1)) ∈ ∂ At−1 . Because the graph G is vertex transitive, we get that (t − 1) is uniformly distributed in G. Moreover, (t − 1) depends only on the clusters At−1 , . . . , A0 , and is independent of the walk (gt (·), ζt (·)). Define S(t) to be the set of vertices in G that the walk (gt (·), ζt (·)) visits before leaving the layer M(t − 1). That is: τ = min r > 0 ζt (r ) = M(t − 1) ; τ is the first step the t th particle is not in the layer M(t − 1), S(t) = v ∈ G ∃ 0 ≤ r ≤ τ − 1 : gt (r ) = v . Claim. For all t > 1, Pr [(t − 1) ∈ S(t)] =
1 E [|S(t)|] . n
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Proof. For any u ∈ G, 1 Pr (t − 1) = u S(t) = . n Consequently, Pr [(t − 1) ∈ S(t)] =
Pr (t − 1) ∈ S S(t) = S Pr [S(t) = S]
S
=
Pr (t − 1) = v S(t) = S Pr [S(t) = S]
S v∈S
1 1 |S| Pr [S(t) = S] = E [|S(t)|] . = n n S
Recall that d is the degree of G. Claim. For all t > 0, E [|S(t)|] ≥
2d + 2 . d +2
Proof. Let R(k) denote the range of a k-step random walk on G. Then, for s ≥ 2, k ∞ d 2 . Pr [|S(t)| = s] = Pr [R(k) = s] d +2 d +2 k=s−1
For s = 1, Pr [|S(t)| = 1] =
2 . d +2
Thus, E [|S(t)|] =
k 2 d 2 + s Pr [R(k) = s] d +2 d +2 d +2 s≥2 k≥s−1
k k+1 d 2 2 + · s Pr [R(k) = s] d +2 d +2 d +2 k≥1 s=2 k d 2 · = E [R(k)] . d +2 d +2 =
k≥0
Substitute in (4.4) the naive bound R(k) ≥ 2 for all k ≥ 1: k ∞ d 2 · 1+2 E [|S(t)|] ≥ d +2 d +2 k=1 2d + 2 2 d d +2 = = · 1+2 . d +2 d +2 2 d +2
(4.4)
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Thus, using the claim, we have that for all t > 1, 2d + 2 . (d + 2)n The lemma now follows from the fact that the event {(t − 1) ∈ S(t)} implies {S(t) ∩ ∂ At−1 = ∅}. So {(t − 1) ∈ S(t)} implies that the t th particle sticks to the layer M(t − 1). Pr [(t − 1) ∈ S(t)] ≥
Proof of Theorem 4.6. For m ∈ N recall Tm = min t ≥ 0 At ∩ G m = ∅ . By Lemma 4.7, for all m, E [Tm ] ≤ m
(d + 2)n . 2d + 2
Thus, 1 d +2 E [Tm ] ≤ . mn 2d + 2 Plugging this into Theorem 4.2, we have D≤
d +2 2 ≤ . 2d + 2 3
4.5. Density of Cylinder-DLA with quickly mixing base. In this section we combine two main results: For a family of graphs {G n } with small mixing time, we show that since the G-Cylinder-DLA process grows arms, the densities at infinity tend to 0 as n tends to infinity. Formally: Theorem 4.8. Let 2 ≤ d ∈ N. Let {G n } be a family of d-regular graphs such that limn→∞ |G n | = ∞, and for all n, m(G n ) ≤
log2 |G n | . (log log |G n |)5
For all n, let D(n) be the density at infinity of the G n -Cylinder-DLA process. Then, lim D(n) = 0.
n→∞
Proof. There exists n 0 = n 0 (d) such that the following holds for all n > n 0 : Set G = G n and consider {At }, a G-Cylinder-DLA process. By Theorem 2.1, for all m, E [Tm ] < mn
4 . log log n
Thus, using Theorem 4.2, D(n) ≤
4 , log log n
for all n > n 0 . Thus, lim D(n) = 0.
n→∞
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5. Lower Bound on the Growth Rate In this section we prove a lower bound on the expected growth rate of the G-CylinderDLA cluster, provided that the spectral gap is at least |G|−2/3 . This regime of the spectral gap includes graphs with small mixing time as in Theorem 2.1, and many more natural families of graphs such as discrete cubes and tori of dimension at least 3. Theorem 5.1. Let 2 ≤ d ∈ N. There exists n 0 = n 0 (d), such that the following holds for all n > n 0 : Let G be a d-regular graph such that |G| = n
and 1 − λ ≥ n −2/3 ,
where 1 − λ is the spectral gap of G. Consider {At }, a G-Cylinder-DLA process. For m ∈ N, define Tm = min t ≥ 0 At ∩ G m = ∅ . Tm is the time the cluster first reaches the layer m. Then, for all m, E [Tm ] > Cmn 1/20 , where C is some constant that depends only on d. Proof. Fix t > 0, and let At−1 be the G-Cylinder-DLA cluster at time t − 1. Claim. There exists a constant C = C(d) (that depends on d) such that for all t > 0, C ∂ At−1 ∩ G M(t−1) . Pr M(t) > M(t − 1) At−1 < n 1/10 Proof. Let (g(·), ζ (·)) = (gt (·), ζt (·)) be the walk of the t th particle. Set L = r > 0 ζ (r ) = ζ (0) = {1 < 2 < · · · } , and let ρi = i−1 → i be the excursions of the walk. For 2 ≤ α ∈ R, let p(α) be the probability that an excursion is a negative α-long excursion; that is p(α) is the probability that ζ (i−1 + 1) = ζ (0) − 1
and
i
1{ζ (r )=ζ (r −1)} ≥ α.
r =i−1 +1
(This is independent of i.) By symmetry and Proposition 3.12, we have that p(α) > (1/c(d))α −1/2 , where c(d) = 12(d + 2). Fix 2 ≤ α ∈ R, and set p = p(α). For an integer k ∈ N, let N (k) denote the number k of negative α-long excursions out of the first k excursions. So N (k) = i=1 Ii (α), where Ii (α) is the indicator of the event that ρi is a negative α-long excursion. Since {Ii (α)} are indpendent, we have by Chebychev’s inequality that √ 4 α 1 < 4c(d) . Pr N (k) ≤ pk ≤ 2 pk k
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Let Z = Z (k) be the number of times up to k the walk moves in G while on the negative side of ζ (0); i.e., Z (k) =
k
1{ζ (r )=ζ (r −1)} 1{ζ (r )<ζ (0)} .
r =1
We have that Z ≥ α · N (k) (since each negative α-long excursion contributes at least α to the sum). Thus, √ √ α αk α ≤ Pr Z ≤ pk < 4c(d) . Pr Z ≤ 2c(d) 2 k Set A = ∂ At−1 ∩ G M(t−1) . Set B = ∂ At−1 \ A. That is, B is the set ∂ At−1 with the highest layer removed. For all r ≥ 0 let Cr = B ∩ G ζ (r ) . Note that if ζ (r ) < ζ (0) then |Cr | ≥ 1 (because any layer below M(t − 1) contains at least one particle). Define a simple random walk on G by h 0 = g(0) and {h 1 , h 2 , . . . , h s } = g(r ) ζ (r ) = ζ (r − 1) , 1 ≤ r ≤ k . Let F = F(k ) be the event that the particle does not hit the set B up to time k . That is, F = {∀0 ≤ r ≤ k (g(r ), ζ (r )) ∈ B}. Conditioned on a specific path ζ (0), ζ (1), . . . , ζ (k ), and on At−1 , we have that h 0 , h 1 , . . . , h s is distributed as a simple random walk on G. Using Lemma B.4, we have that Pr F ζ (0), . . . , ζ (k ) , At−1 ≤ Pr ∀ 0 ≤ r ≤ s h r ∈ Cr ζ (0), . . . , ζ (k ) s 1−λ 1−λ ≤ exp − Z . Cr ≤ exp − 2n 2n r =1
Thus, averaging over all possible paths ζ (0), ζ (1), . . . , ζ (k ), we have that 1−λ α αpk Pr F At−1 < Pr Z ≤ pk + exp − 2 4n √ 1−λ √ α + exp − αk . < 4c(d) k 4c(d)n Note that the event {κ(t) > k } implies the event F, so we have that √ 1−λ √ α + exp − Pr κ(t) > k At−1 < 4c(d) αk . k 4c(d)n On the other hand, consider the times 0 , 1 , . . . , k . Since ∂ At−1 ∩ G ζ (0) = ∂ At−1 ∩ G M(t−1) = A, we have by a union bound, |A| (k + 1) Pr ∃ x ∈ A , ∃ 0 ≤ i ≤ k : (g(i ), ζ (i )) = x At−1 ≤ . n
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Now, the event {M(t) > M(t − 1)} implies that there exists i ≥ 0 such that the particle does not stick to ∂ At−1 before time i , and (g(i ), ζ (i )) = x for some x ∈ A. Thus, we have for all 2 ≤ α ∈ R and all k ∈ N, Pr M(t) > M(t − 1) At−1 ≤ Pr κ(t) > k At−1 + Pr ∃ x ∈ A , ∃ 0 ≤ i ≤ k : (g(i ), ζ (i )) = x At−1 √ |A| (k + 1) 1−λ √ α + exp − . αk + < 4c(d) k 4c(d)n n 1 Set ε = 1/10, k = n 1−ε , α = n 2−4ε . Then, if 1 − λ ≥ n 2/3 , we have that for large enough n (depending on d), C |A| , Pr M(t) > M(t − 1) At−1 < nε for some constant C = C(d).
Back to the proof of Theorem 5.1: Fix m > 0, and consider the time Tm . Note that for all 1 ≤ j ≤ n, ∂ A T + j−1 ∩ G M(T + j−1) ≤ j, m m (because at most j particles could have stuck to the layer M(Tm + j − 1) − 1 by time Tm + j − 1). Thus, for all 1 ≤ j ≤ n we have that for C and ε as above Cj Pr M(Tm + j) = M(Tm + j − 1) A Tm + j−1 > 1 − ε . n This implies that for λ <
nε C,
Pr [Tm+1 − Tm > λ] >
λ Cj Cλ λ 1− ε ≥ 1− ε , n n j=1
and so, there exists a constant
C
(depending on C) such that for λ = n ε/2 ,
E [Tm+1 − Tm ] > λ Pr [Tm+1 − Tm > λ] > C n ε/2 . Hence, we get that for all m ≥ 2, E [Tm ] >
C ε/2 mn . 2
For completeness, we state the immediate Corollary of Theorems 5.1 and 4.2. Corollary 5.2. Let 2 ≤ d ∈ N. There exists n 0 = n 0 (d), such that the following holds for all n > n 0 : Let G be a d-regular graph such that |G| = n
and 1 − λ ≥ n −2/3 ,
where 1 − λ is the spectral gap of G. Consider {At }, a G-Cylinder-DLA process. Let D∞ be the density at infinity. Then, for some constant C that depends only on d, D∞ ≥
C . n 19/20
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6. Further Research Directions The results and methods in this paper raise a few natural questions: 1. Let G be a d-regular graph. Let H be obtained from G by only adding edges to G, so that V (H ) = V (G) and H is (d + 1)-regular. Is there monotonicity in the expected speed of the cluster on the Cylinder-DLA processes with base G and with base H ? That is, let TmG , respectively TmH , be the first time the cluster reaches the layer m in the , respectively H -Cylinder-DLA, process. Is it true G-Cylinder-DLA that E TmG ≥ E TmH for all m? 2. Consider a G-Cylinder-DLA process, started with A0 = {x0 } for a specific vertex x0 ∈ G. Let τ be the mixing time of a simple random walk on G (i.e. the time it takes for a simple random walk to come close in total-variation distance to the stationary distribution). For m > 0, let xm ∈ G be the vertex in G that is the first vertex in the layer m that a particle sticks to. In our notation above xm = v such that A Tm ∩ G m = {(v, m)}. How long does it take for the distribution of xm to be close to the uniform distribution? Does there exist a constant c such that xcτ is close to being uniformly distributed on G? 3. Directed G-Cylinder-DLA : Consider a model of G-Cylinder-DLA where particles cannot move to layers above, only to layers below or in their current layer. Is the density of directed G-Cylinder-DLA always greater than undirected? Are there graphs G for which these quantities are of the same order? Are there graphs for which the ratio between the density of undirected G-Cylinder-DLA and directed G-Cylinder-DLA goes to 0 as the size of G goes to infinity? The model of directed G-Cylinder-DLA can also be generalized to a model α where particles move up with probability d+2 and down with probability 2−α d+2 (and 1 to a neighbor in the current layer with probability d+2 ), for some α < 1. Thus, there is a drift down. The same questions can be asked of this model. We remark that some of our results still hold in directed G-Cylinder-DLA . Mainly, Lemma 3.5 (that states that if the particle takes a small amount of steps to stick, then the expected time to reach the new layer is small,) still holds with the assumption that Pr [κ(t + 1) ≤ µ] ≥ 41 . 4. The G-Cylinder-DLA process, is of course not a stationary process (since At ⊂ At+1 for all t). But, each time a “wall” is built (i.e. L t+n (M(t)) = n, see Proposition 4.4), we start the cluster again, independently of the cluster below the wall. If we identify clusters that are the same above walls, we get a stationary Markov chain on clusters. Our analysis throughout this paper in some sense evades this stationary distribution. It would be interesting if some properties of the cluster generated under the stationary distribution could be worked out. Perhaps, calculating properties of the “typical cluster” could help improve the results of this paper (e.g., reduce the spectral gap required to grow arms). 5. As stated in the introduction, DLA on a cylinder suggests studying the problem of “clogging”. That is, run a G-Cylinder-DLA process for some graph G. Let T be the (random) time at which the cluster clogs the cylinder. That is, T is the first time at which there exists a layer such that no particle can pass this layer; i.e., T = min t > 0 ∃ m > 1 : Pr [H (t) ≤ m] = 0 . Provide bounds on E [T ]. How is T distributed?
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A. Random Walks on Z We collect some facts about a simple random walk on Z, S(n), starting at S(0) = 0. The following is Theorem 9.1 of [11]: Lemma A.1. Let ρ(1) = min i ≥ 0 S(i) = 1 . Then, for all n, Pr [ρ(1) > 2n] = 2
−2n
2n . n
Corollary A.2. For all n, Pr [∀ 1 ≤ i ≤ 2n , S(i) ≥ 0] = 2−2n
2n . n
Proof. Let τ = min i ≥ 0 S(i) = −1 . By symmetry, τ has the same distribution as ρ(1) above. Thus, for all n, Pr [∀ 1 ≤ i ≤ 2n , S(i) ≥ 0] = Pr [τ > 2n] = Pr [ρ(1) > 2n] = 2
−2n
2n . n
The following is Theorem 9.3 of [11]: Lemma A.3. Let L(n) be the number of times the walk has visited 0, i.e. L(n) = 1 ≤ i ≤ n S(i) = 0 . Then for m ≤ n, Pr [L(2n) < m] = 2−2n
m−1
2j
j=0
2n − j . n
Corollary A.4. For L(n) as above, and m ≤ n/2, Pr [L(n) < m] < √
m n − 2m
.
Proof. This is a careful application of Stirling’s approximation to (A.1).
(A.1)
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Lemma A.5. Let S(·) be a lazy random walk on Z, starting at S(0) = 0, with holding probability 1 − α. That is, S(n) =
n
xi ,
i=1
where xi are i.i.d., such that Pr [xi = 0] = 1 − α, and α . 2 Let L(n) be the number of times the walk visits 0 up to time n. That is, L(n) = 1 ≤ i ≤ n S(i) = 0 . Pr [xi = 1] = Pr [xi = −1] =
Then, for any ε > 0 there exists C = C(ε, α) > 0 such that for all n ≥ 1,
Pr L Cn 2 < n ≤ ε. Proof. Let m(n) be the number of times the walk moves in the first n steps. Then, n m(n) = i=1 ri , where ri are i.i.d. Bernoulli random variables of mean α. By the Chernoff bound (see e.g. Appendix A in [1]), α α2 α Pr m(n) ≤ n ≤ Pr |m(n) − αn| ≥ n < 2 exp − n . 2 2 2 Conditioned on m(n), the walk is a m(n)-step simple random walk. Thus, for 2k ≤ m, by Corollary A.4, k Pr L(n) < k m(n) = m ≤ √ . m − 2k and set j = Cn 2 . If i ≥ α2j then 2n ≤ i. Thus, α Pr [L( j) < n , m( j) = i] Pr [L( j) < n] ≤ Pr m( j) ≤ j + 2 i≥(α j)/2 √ α2 2n ≤ exp − Cn 2 + √ . 2 αCn 2 − 4n For large enough C this is less than ε. Let C >
4 α
Lemma A.6. Let S(·) be a lazy random walk on Z, starting at S(0) = 0, with holding probability 1 − α. That is, S(n) =
n
xi ,
i=1
where xi are i.i.d., such that Pr [xi = 0] = 1 − α, and Pr [xi = 1] = Pr [xi = −1] =
α . 2
Let m ≥ 1. Then, for all β > 0, ! 1 Pr max |S(i)| < βαm ≥ 1 − . 1≤i≤m β
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Proof. The assertion is equivalent to ! 1 Pr max |S(i)| ≥ βαm ≤ . 1≤i≤m β But this follows immediately from the Kolmogorov inequality, since S(i) is the sum of i.i.d. random variables, and Var [S(m)] = αm. B. Random Walks on Finite Graphs In this section we recall some properties of a simple random walk on a finite graph. Given a finite d-regular graph G we define two matrices, whose columns and rows are indexed by the vertices of the graph. The adjacency matrix of G is the matrix A(u, v) = 1{u∼v} for all u, v ∈ G. The transition matrix of G is the matrix P = d1 A. It is well known that the eigenvalues of P are all real. Further, if λ1 ≥ λ2 ≥ · · · ≥ λ|G| are the eigenvalues of P, then λ1 = 1, and if G is not bi-partite |λi | < 1 for 1 < i ≤ |G|. We denote by λ = maxi>1 |λi |. λ is called the second eigenvalue of G, and 1 − λ is called the spectral gap. The following lemma is standard in the theory of random walks on graphs, and in fact stronger statements can be proved. We omit the proof (see [1]). Lemma B.1. Let G be a non-bi-partite d-regular graph. Let λ be the second eigenvalue of G. Let µ(i), i ∈ G be any distribution on the vertices of G. Let x0 , x1 , . . . , xt be a random walk on G, such that x0 is distributed like µ. Then, for any j ∈ G, Pr [xt = j] − 1 ≤ λt . n We now prove that the spectral gap of a graph measures how close the random walk on the graph is to independent sampling of the vertices. This is a slight generalization of results from Chapter 9 of [1], and the proof is similar. In what follows G is a d-regular graph of size n. A is its adjacency matrix. λ is the second eigenvalue of the transition matrix. Thus, d is the largest eigenvalue of A, and all other eigenvalues are at most dλ. Let C ⊆ V (G) of size |C| = cn. Define the matrix A(i, j) if j ∈ C, Q C (i, j) = 0 otherwise. For two vectors we use the usual inner product x, y = x2 = x, x. Claim B.2. Q C ≤
!
cd 2 + (1 − c)d 2 λ2
Proof. Let x be any vector, and let x˜ be the vector defined by x(i) if i ∈ C, x(i) ˜ = 0 otherwise.
i
x(i)y(i), and norm
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Then Q C x = Q C x˜ = A x. ˜ Also, note that x2 = x(i)2 ≥ x(i)2 = x ˜ 2. i
i∈C
Thus, Q C 2 = max x=0
Q C x, Q C x A x, ˜ A x ˜ ≤ max . x, x x, ˜ x ˜ x˜ =0
So it is enough to prove that for all x such that x = 1 and such that x(i) = 0 for all i ∈ C, that Ax, Ax ≤ cd 2 + (1 − c)d 2 λ2 . Let x be a vector such that x(i) = 0 for all i ∈ C, and assume that x = 1. Let γ1 ≥ γ2 ≥ . . . ≥ γn be the eigenvalues of A, and let v1 , . . . , vn be the orthonormal basis of eigenvectors of A, corresponding to these eigenvalues. We have that v1 = n −1/2 e, where e is the all-ones vector. Decompose x, x=
n
αs vs .
s=1
So, by the Cauchy-Schwartz inequality, α1 = x, v1 =
i∈C
Note that
s
1 x(i) √ ≤ n
" i∈C
x(i)2
" 1 √ = c. · n i∈C
αs2 = x = 1. Thus,
Ax, Ax =
n
γs2 αs2 ≤ d 2 α12 + (1 − α12 )(dλ)2 ≤ cd 2 + (1 − c)(dλ)2 .
s=1
Claim B.3. Let C1 , C2 , . . . , Ct be subsets of V (G) such that |Cs | = cs n for all s. Let be the number of paths x0 , x1 , . . . , xt in G such that xs ∈ Cs for all s ≥ 1. Then, ≤n
t !
cs d 2 + (1 − cs )d 2 λ2 .
s=1
Proof. For 1 ≤ s ≤ t, let Q s = Q Cs . Let Q = Q 1 Q 2 · · · Q t . We claim that Q(i, j) is the number of paths i = x0 , x1 , . . . , xt = j such that xs ∈ Cs for all s ≥ 1. (B.1) This is proven by induction on t. For t = 1, Q = Q 1 . So Q(i, j) = 1 iff j ∈ C1 and i ∼ j, and Q(i, j) = 0 otherwise. Assume (B.1) for t − 1. Let Q = Q 1 Q 2 · · · Q t−1 . Then by the induction hypothesis, Q (i, k) is the number of paths i = x0 , x1 , . . . , xt−1 = k such that xs ∈ Cs for all 1 ≤ s ≤ t − 1. Thus, Q(i, j) = (Q Q t )(i, j) = Q (i, k)Q t (k, j) k
is the required quantity.
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Thus, if e is the all-ones vector, using (B.1) and Claim B.2, we get that = Q(i, j) = Qe, e ≤ e, e Q i, j
≤n
t s=1
Q s ≤ n
t !
cs d 2 + (1 − cs )d 2 λ2 .
s=1
Lemma B.4. Let x0 , x1 , . . . , xt be a random walk on G starting at a uniformly chosen vertex. Let C1 , C2 , . . . , Ct be subsets of V (G) such that |Cs | = cs n for all s. Let E be the event that xs ∈ Cs for all s ≥ 1. Set c = s (1 − cs ). Then, c
Pr[E] ≤ exp − (1 − λ) . 2 Proof. The total number of possible paths is nd t . Thus, by Claim B.3, Pr[E] =
t ! ≤ cs + (1 − cs )λ2 nd t s=1
=
t ! s=1
1 − (1 − cs ) + (1 − cs )λ2 ≤
= exp −
t s=1
t s=1
(1 − cs ) (1 − λ2 ) exp − 2
c (1 − cs ) (1 − λ2 ) < exp − (1 − λ) . 2 2
Acknowledgement. We wish to thank Amir Yehudayoff for many useful discussions, and remarks about a preliminary version of this note.
References 1. Alon, N., Spencer, J.H.: The Probabilistic Method. New York: John Wiley & Sons, 2000 2. Amir, G., Angel, O., Benjamini, I., Kozma, G.: One-Dimensional Long Range Diffusion Limited Aggregation (DLA). In preparation (2006) 3. Dembo, A., Sznitman, A.S.: A Lower Bound on the Disconnection Time of a Discrete Cylinder. Preprint. Available at: http://arxiv.org/list/math.PR/0701414, 2007 4. Doob, J.L.: Measure Theory. Berlin-Heidelberg-New York: Springer, 1994 5. Eberz-Wagner, D.: Discrete Growth Models. Ph.D thesis, Univ. of Washington, available at http://arxiv. org/list/math.PR/9908030, 1999 6. Kesten, H.: How long are the arms in DLA? J. Phys. A 20(1), L29–L33 (1987) 7. Kesten, H.: Hitting probabilities of random walks on Zd . Stochastic Processes and Their Applications 25, 165–184 (1987) 8. Kesten, H.: Upper bounds for the growth rate of DLA. Physica A 168(1), 529–535 (1990) 9. Kesten, H., Sidoravicius, V.: A problem in one-dimensional Diffusion Limited Aggregation (DLA) and positive recurrence of Markov chains. In a preparation (2006) 10. Peres, Y., Revelle, D.: Mixing times for random walks on finite lamplighter groups. Electronic J. Probab. 9, 825–845 (2004) 11. Révész, P.: Random Walk in Random And Non-Random Environments. River Edge, NJ: World Scientific Publishing Co., 2005
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12. Sander, L.M., Witten, T.A.: Diffusion-Limited Aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47, 1400–1403 (1981) 13. Sznitman, A.S.: How universal are asymptotics of disconnection times in discrete cylinders? Ann. Probab. 36(1), 1–53 (2008) Communicated by M. Aizenman
Commun. Math. Phys. 279, 225–250 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0394-0
Communications in
Mathematical Physics
Harmonic Bilocal Fields Generated by Globally Conformal Invariant Scalar Fields Nikolay M. Nikolov1,2 , Karl-Henning Rehren3 , Ivan Todorov1,2 1 Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, BG-1784 Sofia,
Bulgaria. E-mail: [email protected]; [email protected]
2 Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I–34014 Trieste, Italy 3 Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen,
Germany. E-mail: [email protected] Received: 11 May 2007 / Accepted: 5 June 2007 Published online: 8 February 2008 – © The Author(s) 2008
Abstract: The twist two contribution in the operator product expansion of φ1 (x1 ) φ2 (x2 ) for a pair of globally conformal invariant, scalar fields of equal scaling dimension d in four space–time dimensions is a field V1 (x1 , x2 ) which is harmonic in both variables. It is demonstrated that the Huygens bilocality of V1 can be equivalently characterized by a “single–pole property” concerning the pole structure of the (rational) correlation functions involving the product φ1 (x1 ) φ2 (x2 ). This property is established for the dimension d = 2 of φ1 , φ2 . As an application we prove that any system of GCI scalar fields of conformal dimension 2 (in four space–time dimensions) can be presented as a (possibly infinite) superposition of products of free massless fields. 1. Introduction Global Conformal Invariance (GCI) of Minkowski space Wightman fields yields rationality of correlation functions [14]. This result opens the way for a nonperturbative construction and analysis of GCI models for higher dimensional Quantum Field Theory (QFT), by exploring further implications of the Wightman axioms. By choosing the axiomatic approach, we avoid any bias about the possible origin of the model, because we aim at a broadest possible perspective. On the other hand, the assumption of GCI limits the analysis to a class of theories that can be parameterized by its (generating) field content and finitely many coefficients for each correlation function (see Sect. 2). As anomalous dimensions under the assumption of GCI are forced to be integral, there is no perturbative approach within this setting, but it is conceivable that a theory with a continuous coupling parameter may exhibit GCI at discrete values (that appear as renormalization group fixed points). An example of this type is provided by the Thirring model: it is locally conformal invariant for any value of the coupling constant g and becomes GCI for positive integer g 2 [5]. Previous axiomatic treatments of conformal QFT were focussed on the representation theory and harmonic analysis of the conformal group [6,10] as tools for the Operator
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Product Expansion (OPE). The general projective realization of conformal symmetry in QFT was already emphasized in [16,17] and found to constitute a (partial) organization of the OPE. GCI is complementary in that it assumes true representations (trivial covering projection). A necessary condition for this highly symmetric situation is the presence of infinitely many conserved tensor currents (as we shall see in Sect. 3.3). The first cases studied under the assumption of GCI were theories generated by a scalar field φ(x) of (low) integral dimension d > 1. (The case d = 1 corresponds to a free massless field with a vanishing truncated 4-point function w4tr .) The cases 2 d 4, which give rise to non-zero w4tr were considered in [12,13,11].1 The main purpose in these papers was to study the constraints for the 4-point correlation (= Wightman) functions coming from the Wightman (= Hilbert space) positivity. This was achieved by using the conformal partial wave expansion. An important technical tool in this expansion is the splitting of the OPE into different twist contributions (see (2.10)). Each partial wave gives a nonrational contribution to the complete rational 4point function. It is therefore remarkable that the sum of the leading, twist two, conformal partial waves (corresponding to the contributions of all conserved symmetric traceless tensors in the OPE of basic fields) can be proven in certain cases to be a rational function. This means that the twist two part in the OPE of two fields φ is convergent in such cases to a bilocal field, V1 (x1 , x2 ), which is our first main result in the present paper. Throughout, “bilocal” means Huygens (= space–like and time–like) locality with respect to both arguments. Proving bilocality exploits the bounds on the poles due to Wightman positivity, and the conservation laws for twist two tensors which imply that the bilocal fields are harmonic in both arguments. Trivial examples of harmonic bilocal fields are given by bilinear free field construc¯ 1 )γµ (x1 − x2 )µ ψ(x2 ) :, or (x1 − x2 )µ (x1 − x2 )ν tions of the form : ϕ(x1 )ϕ(x2 ) :, : ψ(x σ :Fµσ (x1 )Fν (x2 ) :. A major purpose of this paper is to explore whether harmonic twist two fields can exist which are not of this form, and whether they can be bilocal. Moreover, we show that the presence of a bilocal field V1 completely determines the structure of the theory in the case of a scaling dimension d = 2. The first step towards the classification of d = 2 GCI fields was made in [12] where the case of a unique scalar field was considered. Here we extend our study to the most general case of a theory generated by an arbitrary (countable) set of d = 2 scalar fields. Our second main result states that such fields are always combinations of Wick products of free fields (and generalized free fields). The paper is organized as follows. Section 2 contains a review of relevant results concerning the theory of GCI scalar fields. In Sect. 3 we study conditions for the existence of the harmonic bilocal field V1 (x1 , x2 ). We prove that Huygens bilocality of V1 (x1 , x2 ) is equivalent to the single pole property (SPP), Definition 3.1, which is a condition on the pole structure of the leading singularities of the truncated correlation functions of φ1 (x1 )φ2 (x2 ) whose twist expansion starts with V1 (x1 , x2 ). This nontrivial condition qualifies a premature announcement in [2] that Huygens bilocality is automatic. Indeed, the SPP is trivially satisfied for all correlations of free field constructions of harmonic fields with other (products of) free fields, due to the bilinear structure of V1 . Thus any violation of the SPP is a clear signal for a nontrivial field content of the model. 1 The last two references are chiefly concerned with the case d = 4 (in D = 4 space-time dimensions) which appears to be of particular interest as corresponding to a (gauge invariant) Lagrangian density. The intermediate case d = 3 is briefly surveyed in [19].
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Moreover, the SPP will be proven from general principles for an arbitrary system of d = 2 scalar fields (the case studied in [2]). Yet, although the pole structure of U (x1 , x2 ) turns out to be highly constrained in general by the conservation laws of twist two tensor currents, the SPP does not follow for fields of higher dimensions, as illustrated by a counter-example of a 6-point function of d = 4 scalar fields involving double poles (Sect. 3.5). The existence of V1 (x1 , x2 ) in a theory of dimension d = 2 fields allows to determine the truncated correlation functions up to a single parameter in each of them. This is exploited in Sect. 4, where an associative algebra structure of the OPE of d = 2 scalar fields and harmonic bilocal fields is revealed. The free-field representation of these fields is inferred by solving an associated moment problem. 2. Properties of GCI Scalar Fields 2.1. Structure of correlation functions and pole bounds. We assume throughout the validity of the Wightman axioms for a QFT on the D = 4 flat Minkowski space–time M (except for asymptotic completeness) – see [18]. Our results can be, in fact, generalized in a straightforward way to any even space–time dimension D. The condition of GCI in the Minkowski space is an additional symmetry condition on the correlation functions of the theory [14]. In the case of a scalar field φ(x), it asserts that the correlation functions of φ(x) are invariant under the substitution
∂g φ(x) → det ∂x
d
4
φ (g(x)) ,
(2.1)
where x → g(x) is any conformal transformation of the Minkowski space, ∂g ∂x is its Jacobi matrix and d > 0 is the scaling important point is that the dimension of φ. An invariance of Wightman functions 0 φ(x1 ) · · · φ(xn ) 0 under the transformation (2.1) should be valid for all xk ∈ M in the domain of definition of g (in the sense of distributions). It follows that d must be an integer in order to ensure the singlevaluedness of the prefactor in (2.1). Thus, GCI implies that only integral anomalous dimensions can occur. The most important consequences of GCI in the case of scalar fields φk (x) of dimensions dk are summarized as follows: (a) Huygens Locality ([14, Theorem 4.1]). Fields commute for non light–like separations. This has an algebraic version: N (x1 − x2 )2 [φ1 (x1 ), φ2 (x2 )] = 0 (2.2) for a sufficiently large integer N . (b) Rationality of Correlation Functions (cf. [14, Theorem 3.1]). The general form of Wightman functions is: 0 φ1 (x1 ) · · · φn (xn ) 0 = C{µ jk } (ρ jk )µ jk , (2.3) {µ jk }
j
where here and in what follows we set ρ jk := (x jk − i 0 e0 )2 = (x jk )2 + i 0 x0jk , x jk := x j − xk ;
(2.4)
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the sum in Eq. (2.3) is over all configurations of integral powers {µ jk = µk j } subject to the following conditions: µ jk = −dk , (2.5) j (=k)
d j + dk δd j dk − 1 + . Equation (2.5) follows from the and pole bounds µ jk − 2 2 conformal invariance under (2.1); the pole bounds express the absence of non-unitary representations in the OPE of two fields [14, Lemma 4.3]. Under these conditions the sum in (2.3) is always finite and there are a finite number of free parameters for every n-point correlation function. We shall refer to the form (2.3) as a Laurent polynomial in the variables ρ jk .2 tr (c) The truncated Wightman functions 0 φ1 (x1 ) · · · φn (xn ) 0 are of the same form like (2.3) but with pole degrees µtrjk bounded by
µtrjk > −
d j +dk 2
(2.6)
(cf. [14, Corollary 4.4]). The cluster condition, expressing the uniqueness of the vacuum, requires that if a non-empty proper subset of points xk among all xi (i = 1, . . . , n) is shifted by t · a (a2 = 0), then the truncated function must vanish in the limit t → ∞. For the two-point clusters {x j , xk }, this condition is ensured by (2.6) in combination with (2.5). For higher clusters, it puts further constraints on the admissible linear combinations of terms of the form (2.3). Note however, that because of possible cancellations the individual terms need not vanish in the cluster limit. The cluster condition will be used in establishing the single pole property for d = 2. 2.2. Twist expansion of the OPE and bi–harmonicity of twist two contribution. The most powerful tool provided by GCI is the explicit construction of the OPE of local fields in the general (axiomatic) framework. Let φ1 (x) and φ2 (x) be two GCI scalar fields of the same scaling dimension d and consider the operator distribution U (x1 , x2 ) = (ρ12 )d−1 (φ1 (x1 ) φ2 (x2 ) − 0|φ1 (x1 ) φ2 (x2 )|0) .
(2.7)
As a consequence of the pole bounds (2.6), U (x1 , x2 ) is smooth in the difference x12 . This is to be understood in a weak sense for matrix elements of U between bounded energy states. Obviously, U (x1 , x2 ) is a Huygens bilocal field in the sense that N (x1 − x)2 (x2 − x)2 [U (x1 , x2 ), ψ(x)] = 0 (2.8) for every field ψ(x) that is Huygens local with respect to φk (x). Then, one introduces the OPE of φ1 (x1 ) φ2 (x2 ) by the Taylor expansion of U in x12 , U (x1 , x2 ) =
∞
3
n=0 µ1 ,...,µn =0
µ
µ
x121 · · · x12n X µn 1 ...µn (x2 ) ,
(2.9)
2 Writing correlation functions in terms of the conformally invariant cross ratios is particularly useful to parameterize 4-point functions. A basis of cross ratios for an n-point function is used in the proof of Lemma 3.6. The general systematics of the pole structure, however, is more transparent in terms of the present variables.
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229
where X µn 1 ...µn (x2 ) are Huygens local fields. We can consider the series (2.9) as a formal power series, or as a convergent series in terms of the analytically continued correlation functions of U (x1 , x2 ). We will consider at this point the series (2.9) just as a formal series. (See also [1] for the general case of constructing OPE via multilocal fields in the context of vertex algebras in higher dimensions.) Since the prefactor in (2.7) transforms as a scalar density of conformal weight (1 − d, 1 − d) then U (x1 , x2 ) transforms as a conformal bilocal field of weight (1, 1). Hence, the local fields X µn 1 ...µn in (2.9) have scaling dimensions n + 2 but are not, in general, quasiprimary.3 One can pass to an expansion in quasiprimary fields by subtracting from X µn 1 ...µn derivatives of lower dimensional fields X µn 1 ...µn . The resulting quasiprimary fields Oµk 1 ...µ are traceless tensor fields of rank and dimension k. The difference k − (“dimension − rank”)
(2.10)
Oµk 1 ...µ . Unitarity implies that the twist is non-negative
is called twist of the tensor field [10], and by GCI, it should be an even integer. In this way one can reorganize the OPE (2.9) as follows: U (x1 , x2 ) = V1 (x1 , x2 ) + ρ12 V2 (x1 , x2 ) + (ρ12 )2 V3 (x1 , x2 ) + · · · ,
(2.11)
where Vκ (x1 , x2 ) is the part of the OPE (2.9) containing only twist 2κ contributions. Note that Eq. (2.11) contains also the information that the twist 2κ contributions contain a factor (ρ12 )κ−1 (i.e. Vκ are “regular” at x1 = x2 ), which is a nontrivial feature of this OPE (obtained by considering 3-point functions). Thus, the expansion in twists can be viewed as a light-cone expansion of the OPE. Since the twist decomposition of the fields is conformally invariant then each Vκ will behave, at least infinitesimally, as a scalar (κ, κ) density under conformal transformations. Every Vκ is a complicated (formal) series in twist 2κ fields and their derivatives: Vκ (x1 , x2 ) =
∞
=0
K κµ1 ...µ (x12 , ∂x2 ) Oµ +2κ (x2 ) , 1 ...µ
(2.12)
µ ...µ
where K κ 1 (x12 , ∂x2 ) are infinite formal power series in x12 with coefficients that are differential operators in x2 acting on the quasiprimary fields O. The important point µ ...µ here is that the series K κ 1 (x12 , ∂x2 ) can be fixed universally for any (even generally) conformal QFT. This is due to the universality of conformal 3-point functions. The µ ...µ explicit form of K κ 1 (x12 , ∂x2 ) can be found in [6,7] (see also [13]). Thus, we can at this point consider Vκ (x1 , x2 ) only as generating series for the twist 2κ contributions to the OPE of φ(x1 )φ(x2 ) but we still do not know whether these series would be convergent and even if they were, it would not be evident whether they would give bilocal fields. In the next section we will see that this is true for the leading, twist two part under certain conditions, which are automatically fulfilled for d = 2. The higher twist parts Vκ (κ > 1) are certainly not convergent to Huygens bilocal fields, since their 4-point functions, computed in [13], are not rational. The major difference between the twist two tensor fields and the higher twist fields is that the former satisfy conservation laws: ∂xµ1 Oµ +2 (x) = 0 1 ...µ
( 1).
3 Quasiprimary fields transform irreducibly under conformal transformations.
(2.13)
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N. M. Nikolov, K.-H. Rehren, I. Todorov
This is a well known consequence of the conformal invariance of the 2-point function and the Reeh–Schlieder theorem. It includes, in particular, the conservation laws of the currents and the stress–energy tensor. It turns out that V1 (x1 , x2 ) encodes in a simple way this infinite system of equations. Theorem 2.1. ([13]). The system of differential equations (2.13) is equivalent to the harmonicity of V1 (x1 , x2 ) in both arguments (bi–harmonicity) as a formal series, i.e., x1 V1 (x1 , x2 ) = 0 = x2 V1 (x1 , x2 ). The proof is based on the explicit knowledge of the K series in (2.12) and it is valid even if the theory is invariant under infinitesimal conformal transformations only. The separation of the twist two part in (2.11) amounts to a splitting of U of the form (x1 , x2 ). U (x1 , x2 ) = V1 (x1 , x2 ) + ρ12 U
(2.14)
This splitting can be thought of in terms of matrix elements of U (x1 , x2 ) expanded as a formal power series according to (2.9). It is unique by virtue of Theorem 2.1, due to the following classical lemma: Lemma 2.2. ([3,1]). Let u(x) be a formal power series in x ∈ C4 (or, C D ) with coefficients in a vector space V . Then there exist unique formal power series v(x) and u (x) with coefficients in V such that u(x) = v(x) + x2 u (x)
(2.15)
and v(x) is harmonic in x (i.e., x v(x) = 0). Equation (2.15) is called the harmonic decomposition of u(x) (in the variable x around x = 0), and the formal power series v(x) is said to be the harmonic part of u(x). 3. Bilocality of Twist Two Contribution to the OPE Let us sketch our strategy for studying bilocality of V1 (x1 , x2 ). The existence of the field V1 (x1 , x2 ) can be established by constructing its correlation functions. On the other hand, every correlation function4 · V1 (x1 , x2 )· of V1 is obtained (originally, as a formal power series in x12 ) under the splitting (2.14). It thus appears as a harmonic decomposition of the corresponding correlation function · U (x1 , x2 ) · of U :
(x1 , x2 ) · . · U (x1 , x2 ) · = · V1 (x1 , x2 ) · + ρ12 · U
(3.1)
Note that we should initially treat the left-hand side of (3.1) also as a formal power series in x12 in order to make the equality meaningful. It is important that this series is always convergent as a Taylor expansion of a rational function in a certain domain around x1 = x2 in MC×2 , for the complexified Minkowski space MC = M + i M, according to the standard analytic properties of Wightman functions. We shall show in Sect. 3.1 4 This short-hand notation stands for 0|φ (x ) · · · φ (x ) V (x , x ) φ 3 3 1 1 2 k+1 (xk+1 ) · · · φn (xn )|0, here and k k in the sequel.
Harmonic Bilocal Fields Generated by Globally Conformal Invariant Scalar Fields
231
that this implies the separate convergence of both terms in the right-hand side of (3.1). Hence, the key tool in constructing V1 are the harmonic decompositions 1 , x2 ) F(x1 , x2 ) = H (x1 , x2 ) + ρ12 F(x
(3.2)
of functions F(x1 , x2 ) that are analytic in certain neighbourhoods of the diagonal {x1 = x2 }. Recall that H in (3.2) is uniquely fixed as the harmonic part of F in x1 around x2 , due to Lemma 2.2. This is equivalent to the harmonicity x1 H (x1 , x2 ) = 0. On the other hand, according to Theorem 2.1 we have to consider also the second harmonicity condition on H , x2 H (x1 , x2 ) = 0, i.e., H is the harmonic part in x2 around x1 . This leads to some “integrability” conditions for the initial function F(x1 , x2 ), which we study in Sect. 3.2. Next, to characterize the Huygensbilocality of V1 , we should have rationality of its correlation functions · V1 (x1 , x2 ) · , which is due to a straightforward extension of the arguments of [14, Theorem 3.1]. But we have started with the correlation functions of U , which are certainly rational. Hence, we should study another condition on U , namely that its correlation functions have a rational harmonic decomposition. We show in Sect. 3.3 that this is equivalent to a simple condition on the correlation functions of U , which we call “Single Pole Property” (SPP). In this way we establish in Sect. 3.4 that V1 always exists as a Huygens bilocal field in the case of scalar fields of dimension d = 2. However, for higher scaling dimensions one cannot anymore expect that V1 is Huygens bilocal in general. This is illustrated by a counter-example, involving the 6-point function of a system of d = 4 fields, given at the end of Sect. 3.5.
3.1. Convergence of harmonic decompositions. To analyze the existence of the harmonic decomposition of a convergent Taylor series we use the complex integration techniques introduced in [1]. Let MC = M + i M be the complexification of Minkowski space, which in this sub
section is assumed to be D–dimensional, and E = x : (i x 0 , x 1 , . . . , x D−1 ) ∈ R D its Euclidean real submanifold, and S D−1 ⊂ E the unit sphere in E. We denote by · the Hilbert norm related to the fixed coordinates in MC : x 2 := |x0 |2 + · · · + |x D−1 |2 . Let us also introduce for any r > 0 a real compact submanifold Mr of MC : Mr = ζ ∈ MC : ζ = r eiθ w, ϑ ∈ [0, π ], w ∈ S D−1
(3.3)
(note that ϑ ∈ [π, 2π ] gives another parameterization of Mr ). Then there is an integral representation for the harmonic part of a convergent Taylor series. Lemma 3.1 (cf. [1, Sect. 3.3 and Appendix A]). Let u(x) be a complex formal power series that is absolutely convergent in the ball x < r , for some r > 0, to an analytic function U (x). Then the harmonic part v(x) of u(x) (around x = 0), which is provided by Lemma 2.2, is absolutely convergent for |x2 | + 2 r x < r 2 .
(3.4)
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N. M. Nikolov, K.-H. Rehren, I. Todorov
The analytic function V (x) that is the sum of the formal power series v(x) has the following integral representation: 2 d D z 1 − xz2 Mr V (x) = U (z) , V1 = d D z M = iπ |S D−1 |, D 1 V1 (z − x)2 2 M M1 r
(3.5) where r < r , |x2 | + 2 r x < r 2 , and the (complex) integration measure d D z M is r
obtained by the restriction of the complex volume form d D z (= dz 0 ∧ · · · ∧dz D−1 ) on M C (∼ = C D ) to the real D–dimensional submanifold Mr (3.3), r > 0. D 2 (z − x)2 2 and Proof. Consider the Taylor expansion in x of the function 1 − xz2 write it in the form (cf. [1, Sect. 3.3]) 1−
x2 z2
D (z − x)2 2
=
∞
D
(z2 )− 2 − H (z, x),
H (z, x) =
h µ (z) h µ (x), (3.6)
µ
= 0
where {h µ (u)} is an orthonormal basis of harmonic homogeneous polynomials of degree
on the sphere S D−1 . This expansion is convergent for 2 (3.7) x + 2 |z · x| < z2 since its left–hand side is related to the generating function for H : 1 − λ2 x 2 y 2 D
(1 − 2 λ x · y + λ2 x2 y2 ) 2
=
∞
λ H (x, y),
(3.8)
=0
the expansion (3.8) being convergent for λ 1 if |x2 y2 | + 2|x · y| < 1. Then if we fix r < r and z varies on Mr , a sufficient condition for (3.7) is |x2 | + 2 r x < r 2 (since sup |w · x| = x ). w ∈ S D−1 On the other hand, writing u(z) = ∞ k=0 u k (z), where u k are homogeneous polynomials of degree k, we get by the absolute convergence of u(z) the relation (valid for |x2 | + 2 r x < r 2 ) 2 d D z d D z ∞ 1 − xz2 D Mr Mr (z2 )− 2 − H (x, z) u k (z). (3.9) D U (z) = V1 V 1 (z − x)2 2 k, = 0 M M r
r
Noting next that in the parameterization (3.3) of Mr we have d D z M = i r D ei D ϑ r dϑ ∧ dσ (w), where dσ (w) is the volume form on the unit sphere, we obtain for the right-hand side of (3.9): ∞ k, = 0
π 0
dϑ iϑ(k− ) e iπ
S D−1
dσ (w) H (x, w) u k (w). |S D−1 |
Harmonic Bilocal Fields Generated by Globally Conformal Invariant Scalar Fields
Now if we write, according to Lemma 2.2, u k (z) = then we get by the orthonormality of h ,µ (w), ∞
k, = 0 2 j k
π δ ,k−2 j
µ
2 j k µ
233
ck, j,µ (z2 ) j h k−2 j,µ (z),
dϑ iϑ(k− ) e ck, j,µ h k−2 j,µ (x) iπ
0
=
∞ k =0
ck,0,µ h k,µ (x) = v(x) .
µ
The latter proves both the convergence of v(x) in the domain (3.4) (since r < r was arbitrary) and the integral representation (3.5). As an application of this result we will prove now Proposition 3.2. For all n and k, and for all local fields φ j ( j = 3, . . . , n) the Taylor series (3.10) 0 φ3 (x3 ) · · · φk (xk ) V1 (x1 , x2 ) φk+1 (xk+1 ) · · · φn (xn ) 0 in x12 converge absolutely in the domain 2 2 2 2
x12 + x12 + x12
x2 j + x2 j + x2 j < x22 j ∀ j
(3.11)
( j = 3, . . . , n). They all are real analytic and independent of k for mutually nonisotropic points. Proof. Let Fk (x12 , x23 , . . . , x2n ) = 0 φ3 (x3 ) · · · φk (xk ) U (x1 , x2 ) φk+1 (xk+1 ) · · · φn (xn ) 0
(3.12)
be the correlation functions, analytically continued in x12 . As Fk , which is a rational function, depends on x := x12 via a sum of products of −µ j it has a convergent expansion in x for powers (x − x2 j )2 2 (3.13) x + 2 x · x2 j < x22 j . If we want Fk to have a convergent Taylor expansion for x < r we get the following sufficient condition: r 2 < x22 j − 2 r x2 j . (3.14) By Lemma 3.1 we conclude that the series (3.10) is convergent for 2 |x12 | + 2 r x12 < r 2 .
(3.15)
Combining both (sufficient) conditions (3.14) and (3.15) for r we find that they are 2 compatible if x12 + x12 2 + x12 < x2 j 2 + x22 j − x2 j , which is equivalent to (3.11).
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Note that one can also prove a similar convergence property for the correlation functions of several V1 . Remark 3.1. The domain of convergence of (3.10) should be Lorentz invariant. Hence, (3.10) are convergent in the smallest Lorentz invariant set containing the domain (3.11). 2 , x2 and x · x and it turns Such a set is determined by the values of the invariants x12 12 2 j 2j out to be the set 1 2 2 2 2 21 x2 j − x12 1 1 2 2 2 2 x12 x2 j x12 · x2 j < 4 2 1 2 2 2 21 x − x 12 2j 2 2 x x + x12 · x2 j 2 < . (3.16) or equivalently 12 2j 4 Outside the domain of convergence (3.16), the correlations of V1 (x1 , x2 ) have to be defined by analytic continuation. When the correlations are rational, V1 is Huygens bilocal, but the counter-example presented in Sect. 3.5 shows that rationality is not automatic. Then, it is not even obvious that the continuations are single–valued within the tube of analyticity required by the spectrum condition, i.e., that V1 exists as a distribution in all of M × M. Nontrivial case studies, however, show that at least for xk space–like to both x1 and x2 , the continuation is single–valued and preserves the independence on the position k in (3.10) where V1 (x1 , x2 ) is inserted. This leads us to conjecture Conjecture 3.3. The twist two field V1 (x1 , x2 ), whose correlations are defined as the analytic continuations of the harmonic parts of those of U (x1 , x2 ), exists and is bilocal in the ordinary sense, i.e., it commutes with φ(x) and V1 (x, x ) if x and x are space–like to x1 and x2 . We hope to return to this conjecture elsewhere (see also the Note added in proof). Note that the argument that locality implies Huygens locality [14] does not pass to bilocal fields. 3.2. Consequences of bi–harmonicity. Now our objective is to find the harmonic decomposition of the rational functions F(x1 , x2 ) that depend on x1 and x2 through the intervals ρik = (xi − xk )2 , i = 1, 2, k = 3, . . . , n, for some additional points x3 , . . . , xn . The F’s, as correlation functions of U (x1 , x2 ), have the form F(x1 , x2 ) =
M
M
(ρ12 )q Fq (x1 , x2 ) ≡
q =0
Fq (x1 , x2 ) =
(ρ12 )q Fq {ρik }{i,k}={1,2} ,
(3.17)
q =0
Cq,{µ1 j },{µ2 j }
{µ1i },{µ2i }
n
(ρ1 j )µ1 j
j=3
n
(ρ2i )µ2 j ,
j=3
(3.18)
where M ∈ N and µ1 j , µ2 j ( j = 3, . . . , n) are integers > −d such that j 3 µ1 j = j 3 µ2 j = −1 − q, and the coefficients C q,{µ1 j },{µ2 j } may depend on ρ jk ( j, k 3). If H is the harmonic part of F in x12 , then the leading part F0 (of order (ρ12 )0 ) is also the leading part of H . We shall now proceed to show that bi–harmonicity of H (Theorem 2.1), together with the first principles of QFT including GCI, implies strong constraints on F0 .
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Proposition 3.4. Let F0 (x1 , x2 ) be as in (3.18), and let H (x1 , x2 ) be its harmonic part with respect to x1 around x2 . Then H is also harmonic with respect to x2 , if and only if F0 satisfies the differential equation (E 1 D2 − E 2 D1 )F0 = 0, n where E 1 = i=3 ρ2i ∂1i (with ∂ jk = ∂k j = similarly for E 2 and D2 , exchanging 1 ↔ 2.
∂ ∂ρ jk ),
(3.19)
D1 =
3 j
ρ jk ∂1 j ∂1k , and
Proof. By Proposition 3.2 (see also Remark 3.1) we can consider H as a function in the 2n − 3 variables ρ1i , ρ 2i (i 3) and ρ12 , analytic in some domain that includes ρ12 = 0. q Expanding H = q (ρ12 ) Hq /q!, the functions Hq are homogeneous of degree −1 − q in both sets of variables ρ1i and ρ2i , and H0 = F0 . To impose the harmonicity with respect to the variable x1 , we use the identity [11, App. C] ⎡ ⎤ x1 F = −4 ⎣ ρi j ∂1i ∂1 j F ⎦ , (3.20) 2 2i< j n
ρi j = (xi −x j )
valid for homogeneous functions of ρ1i of degree −1, to express the wave operator x1 as a differential operator with respect to the set of variables ρ1i (i 2). This yields the recursive system of differential equations E 1 Hq+1 = −D1 Hq .
(3.21)
Performing the same steps with respect to the variable x2 , one obtains E 2 Hq+1 = −D2 Hq .
(3.22)
Equation (3.19) then arises as the integrability condition for the pair of inhomogeneous differential equations for H1 (putting q = 0), observing that E 2 E 1 − E 1 E 2 = ρ1i ∂1i − ρ2i ∂2i vanishes on H1 by homogeneity. Conversely, if (3.19) is fulfilled, then H1 exists and satisfies (D1 E 2 − D2 E 1 )H1 = −(D1 D2 − D2 D1 )H0 = 0 because D1 and D2 commute. But this is equivalent to (D2 E 1 − D1 E 2 )H1 = 0, which is in turn the integrability condition for the existence of H2 , and so on. It follows that bi–harmonicity imposes no further conditions on the leading function H0 = F0 . The differential equation (3.19) imposes the following constraints on the leading part F0 of the rational correlation function F (3.17): Corollary 3.5. Assume that the function F0 as in (3.18) satisfies the differential equation (3.19). Then (i) If F0 contains a “double pole” of the form (ρ1i )µ1i (ρ1 j )µ1 j with i = j and µ1i and µ1 j both negative, then its coefficients must be regular in ρ2k (k = i, j). (ii) F0 cannot contain a “triple pole” of the form (ρ1i )µ1i (ρ1 j )µ1 j (ρ1k )µ1k with i, j, k all different and µ1i , µ1 j , µ1k all negative. The same hold true, exchanging 1 ↔ 2.
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Proof. Pick any variable, say ρ2k , and decompose F0 = r − p (ρ2k )r fr as a Laurent polynomial in ρ2k . The differential equation (3.19) turns into the recursive system ⎞ ⎛ ⎝ρ1k ρi j ∂1i ∂1 j − ρ2i ρk j ∂1i ∂2 j ⎠ r · fr = X r fr −1 + Y fr i, j=k
i< j
of differential equations for the functions fr which are Laurent polynomials in the remaining variables. The precise form of the polynomial differential operators X r and Y does not matter. Assume the lowest power − p of ρ2k to be negative. For r = − p, the right-hand-side vanishes. Because the term ρi j ∂1i ∂1 j on the left-hand-side would produce a singularity that cannot be cancelled by any other term, f − p cannot have a “double pole” in any pair of variables ρ1i , ρ1 j with i = j and i, j = k. This property passes recursively to all fr with r < 0, because also the right-hand-side never can contain such a pole. This implies that a double pole in a pair of variables ρ1i , ρ1 j with i = j cannot multiply a term that is singular in ρ2k unless k = i or k = j, proving (i). If the coefficient of the double pole were singular in ρ1k , k = i, j, then the resulting double pole in the pair ρ1i , ρ1k resp. ρ1 j , ρ1k would imply regularity also in ρ2 j resp. ρ2i . Hence the coefficient of a triple pole must be regular in all ρ2m , which contradicts the total homogeneity −1 of F0 in these variables. This proves the statement (ii). 3.3. A necessary and sufficient condition for Huygens bilocality. Definition 3.1. (“Single Pole Property”, SPP). Let f (x1 , . . . , xn ) be a Laurent polynomial in the variables ρi j , i.e., regarded as a function of x1 only, it is a finite linear combination of functions of the form µ1 j (x1 − x j )2 (ρ1 j )µ1 j ≡ , (3.23) j 2
j 2
where µ1 j ( j 2) are integers and the coefficients may depend on the parameters ρ jk ( j, k 2). Then f is said to satisfy the single pole property with respect to x1 if it contains no terms for which there are j = k ( j, k 2) such that both µ1, j and µ1,k are negative. The significance of SPP stems from the fact that the harmonic parts H of F0 , i.e., the correlation functions of V1 , are again Laurent polynomials if and only if F0 satisfies the SPP. Namely, if H is a harmonic Laurent polynomial, the same argument as in [11, Lemma C.1] (using the representation (3.20) of the wave operator) shows that H fulfills the SPP with respect to x1 , and so does F0 , because it is the leading part of order (ρ12 )0 of H . The converse is an immediate consequence of Lemma 3.6 (allowing for a relabelling and multiple counting of the points x3 , . . . , xn , which are not required to be distinct). Lemma 3.6. Let n 4. Every finite linear combination of monomials of the form &n &n 2 i=4 ρ1i i=4 (x1 − xi ) ≡ (3.24) gn (x1 ) = (ρ13 )n−2 [(x1 − x3 )2 ]n−2 has a rational harmonic decomposition in x1 around x2 , gn (x1 ) = h n (x1 ) + (x1 − x2 )2 · g˜ n (x1 ),
(3.25)
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237
i.e., h n is harmonic with respect to x1 and g˜ n is regular at x1 = x2 , and both h n and g˜ n are rational. More precisely, (ρ13 )n−2 (ρ23 )n−3 h n is a homogeneous polynomial of total degree 2(n − 3) in the variables {ρi j : 1 i < j}, which is separately homogenous of degree n − 3 in the variables {ρ1i : i 2} and in the variables {ρ12 , ρ2i : i 3}. Proof. It is convenient to introduce the variables ti =
ρ1i ρ23 , ρ13 ρ2i
si =
ρ12 ρ3i , ρ13 ρ2i
ui j =
ρ12 ρ23 ρi j ρ13 ρ2i ρ2 j
(4 i < j n).
We claim that h n (x1 ) is of the form n ρ2i f n (ti , si , u i j ) h n (x1 ) = , · ρ23 ρ13
(3.26)
(3.27)
i=4
&n where f n are polynomials of degree n − 3 such that f n (ti , si = 0, u i j = 0) = i=4 ti . Because all si and u i j contain a factor ρ12 , these properties ensure that g˜ n given by (gn − h n )/ρ12 is regular in ρ12 . Using again the identity (3.20) for the wave operator, and transforming this into a differential operator with respect to the set of variables (3.26), we find n ρ2i ρ23 x1 h n (x1 ) = −4 · D f n (ti , si , u i j ), (3.28) ρ23 (ρ13 )2 ρ12 i=4
where D is the differential operator D = (1 + t∂t + s∂s + u∂u )(s∂t + s∂s + u∂u ) − (s∂s + u∂u )∂t − u∂t ∂t
(3.29)
with shorthand notations for degree-preserving operators t∂t =
n
ti ∂ti , s∂t =
i=4
n
si ∂ti , s∂s =
i=4
n
si ∂si , u∂u =
u i j ∂u i j
4i< j n
i=4
and degree-lowering operators ∂t =
n
∂ti ,
u∂t ∂t =
u i j ∂ti ∂t j .
4i< j n
i=4
To solve the condition D f n = 0 for harmonicity, we make an ansatz (n) g K (sk , u kl ) · (ti − si ), f n (ti , si , u i j ) = K ⊂N
i∈N \K
(n)
where N ≡ {4, . . . , n}, g K are polynomials in the variables sk , u kl (k, l ∈ K ) only, and (n) g∅ = 1. Then the harmonicity condition D f n = 0 is equivalent to the recursive system (n) (n − 2 − |K | + ) g (n) g K \{k} + (u kl − sk − sl ) g (n) K = K \{k,l} , k∈K
k,l∈K ,k
where |K | is the number of elements of the set K and the differential operator = s∂s + u∂u measures the total polynomial degree r in sk and u kl . Since one can divide by
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(n − 2 − |K | + r )r if r > 0, there is a unique polynomial solution such that g K (sk = (n) 0, u kl = 0) = 0 (K = ∅), and g K is of order |K |. So f n is of order n − 3. (Explicitly, the first three functions are f 3 = 1, f 4 = t4 − s4 and f 5 = (t4 − s4 )(t5 − s5 ) + 21 (u 45 − s4 − s5 ).) An inspection of the recursion also shows that all possible factors ρ2i in the denominators of the arguments of f n cancel with the factors in the prefactor in (3.27), thus h n can have poles only in ρ13 and ρ23 of the specified maximal degree. This proves the lemma. The upshot of the previous discussion is a necessary and sufficient condition for the Huygens bilocality of V1 which directly refers to the local correlation functions of the theory: Theorem 3.7. The field V1 (x1 , x2 ) weakly converges on bounded energy states to a Huygens bilocal field which is conformal of weight (1, 1), if and only if the leading parts F0 of the Laurent polynomials F (3.17) satisfy the “single pole property” (Def. 3.1) with respect to both x1 and x2 . In this case, the formal series H converge to Laurent polynomials in (xi −x j )2 subject to the same pole bounds, specified in Theorem 2.1, as F. Proof. We know already that if V1 is a Huygens bilocal field, then its correlation functions H are Laurent polynomials of the form (2.3), and that this implies the SPP for F0 with respect to x1 and x2 . Conversely, if the SPP holds for F0 with respect to x1 and x2 , then H are Laurent polynomials by Lemma 3.6, and hence V1 is relatively Huygens bilocal with respect to the fields φi . Since the general argument [4] that relative locality implies local commutativity of a field with itself refers only to local fields, we want to give an explicit argument for the case at hand. All the previous remains true when in (3.10) or (3.17) a product of fields φk (xk )k+1 φ(xk+1 ) is replaced by U (xk , xk+1 ). By assumption, and because U is bilocal, the contributions of order (ρk,k+1 )0 to the correlation functions of U (xk , xk+1 ) fulfill the SPP with respect to xk and xk+1 . By Lemma 3.6, this property is preserved upon the passage to the harmonic parts with respect to x1 and x2 . One may therefore continue in the same way with xk , xk+1 , and eventually find that all mixed correlation functions of φ’s and V1 ’s converge to rational functions. By this convergence we conclude that all products of φ’s and V1 ’s converge on the vacuum, and this then defines V1 as a Huygens bilocal field, since its matrix elements will satisfy Huygens locality. The conformal properties of V1 follow from the preservation of the homogeneity and the pole degrees in the harmonic decomposition, as guaranteed by Lemma 3.6. For n = 4 points, is trivially satisfied because of homogeneity. Hence the the SPP 4-point function 0 V1∗ V1 0 is always rational. It follows that its expansion in (transcendental) partial waves [11] cannot terminate. This means that (unless V1 = 0 in which case there is not even a stress-energy tensor) a GCI QFT necessarily contains infinitely many conserved tensor fields of arbitrarily high spin. 3.4. The case of dimension 2. Let us consider now the case of scalar fields φk of dimension 2. We claim that in this case, Corollary 3.5 in combination with the cluster condition is sufficient to establish the SPP, Definition 3.1. Hence we conclude by Theorem 3.7 that the twist two harmonic fields V1 (x1 , x2 ) are indeed Huygens bilocal fields. To prove our claim, we use that by (2.6), µi j −1, hence the SPP is equivalent to the statement that there can be no term contributing to 0 φ1 (x1 ) · · · φn (xn ) 0 , for which
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239
there is i with more than two µi j negative ( j = i). Thus assume that there is a term with, say, µ12 = µ13 = µ14 = −1. It constitutes a double pole for each of the three harmonic fields V1 (x1 , x j ) ( j = 2, 3, 4). Then by homogeneity (2.5), there must be more poles in x j ( j = 2, 3, 4), but these cannot be of the form ρ jk with k > 4 by Corollary 3.5. Hence (up to permutations of 2, 3, 4) µ23 = µ24 = −1, µ34 = 0. Again by homogeneity (2.5), the dependence on x1 , . . . , x4 must be given by a linear combination of terms ρ1k ρ4
ρ12 ρ13 ρ14 ρ23 ρ24
(3.30)
with k, > 4. Applying the cluster limit (Sect. 2.1) to the points x1 , x2 , x3 , x4 in (3.30), the limit diverges ∼t 4 . This behavior is tamed to ∼t 2 by anti–symmetrization in k, , but it cannot be cancelled by any other terms. Hence the assumption leads to a contradiction. This proves the SPP if the generating scalar fields have dimension d = 2. 3.5. A d = 4 6-point function violating the SPP. We proceed with an example of a 6-point function violating the SPP in the case of two d = 4 GCI scalar fields L i (x) such that the bilocal field U (x1 , x2 ) obtained from L 1 (x1 )L 2 (x2 ) has a non-zero skew–symmetric part. Let L be any linear combination of L 1 and L 2 . The following admissible contribution to the truncated part of the 6-point function 0|U (x1 , x2 )L(x3 )L(x4 )U (x5 , x6 )|0 clearly violates the SPP:
F0 (x1 , x2 ) = A12 A56
ρ15 ρ26 ρ34 − 2ρ15 ρ23 ρ46 − 2ρ15 ρ24 ρ36 ρ13 ρ14 ρ23 ρ24 · ρ34 · ρ35 ρ45 ρ36 ρ46
,
(3.31)
where Ai j stands for the antisymmetrization in the arguments xi , x j . It is admissible as a truncated 6-point structure because (ρ12 ρ56 )−3 F0 obeys all the pole bounds of Sect. 2 for a correlation 0|L 1 (x1 )L 2 (x2 )L(x3 )L(x4 )L 1 (x5 ) L 2 (x6 )|0tr of six fields of dimension d = 4. On the other hand, F0 satisfies the differential equation (E 1 D2 − E 2 D1 )F0 (x1 , x2 ) = 0
(3.32)
(and similar in the variables x5 and x6 ), ensuring that F0 is the leading part of a bi–harmonic function, analytic in a neighborhood of x1 = x2 and x5 = x6 , representing a contribution to the twist two 6-point function 0|V1 (x1 , x2 )L(x3 )L(x4 ) V1 (x5 , x6 )|0, of which F0 is the leading part. This function cannot be a Laurent polynomial in the ρi j by our general argument that the leading part of a bi–harmonic Laurent polynomial cannot satisfy the SPP. Hence the twist two field V1 (x1 , x2 ) cannot be Huygens bilocal. The resulting contribution to the conserved local current 4-point function 0|Jµ (x1 )L(x3 )L(x4 )Jν (x5 )|0tr is obtained through Jµ (x) = i(∂xµ −∂yµ ) V1 (x, y)|x=y . It also satisfies the pertinent pole bounds. This structure is rational as it should be, because only the leading part F0 contributes. In fact, while the 6-point structure involving the harmonic field cannot be reproduced by free fields due to its double pole, the resulting 4-point structure does arise as one of the three independent connected structures contributing to 4-point functions involving two Dirac currents : ψ¯ a γ µ ψb : and two Yukawa scalars ϕ : ψ¯ c ψd : (allowing for internal flavours a, b, . . . ).
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4. The Theory of GCI Scalar Fields of Scaling Dimension d = 2 The scaling dimension d = 2 is the minimal dimension of a GCI scalar field for which one could expect the existence of nonfree models. It turns out however, that in this case the fields can be constructed as composite fields of free, or generalized free, fields. Namely, we will establish the following result. Theorem 4.1. Let {m (x)}∞ m = 1 be a system of real GCI scalar fields of scaling dimension d = 2. Then it can be realized by a system of generalized free fields {ψm (x)} and a system of independent real massless free fields {ϕm (x)}, acting on a possibly larger Hilbert space, as follows: m (x) =
∞
αm, j ψ j (x) +
j =1
∞ 1 βm, j,k : ϕ j (x)ϕk (x) : , 2
where αm, j and βm, j,k = βm,k, j are real constants such that ∞ j,k = 1
2 βm, j,k
(4.1)
j,k = 1
∞ j =1
2 αm, j < ∞ and
< ∞. Here, we assume the normalizations 0| ϕ j (x1 ) ϕk (x2 ) |0 =
δ jk (ρ12 )−1 , 0| ψ j (x1 ) ψk (x2 ) |0 = δ jk (ρ12 )−2 . The proof of Theorem 4.1 is given at the end of Sect. 4.2. The main reason for this result is the fact that in the d = 2 case the harmonic bilocal fields exist and furthermore, they are Lie fields. This was originally recognized in [12,2] under the assumption that there is a unique field φ of dimension 2. We are extending here the result to an arbitrary system of d = 2 GCI scalar fields. If we assume the existence of a stress-energy tensor as a Wightman field5 , the generalized free fields must be absent in (4.1), and the number of free fields must be finite. In this case, the iterated OPE generates in particular the bilocal field 21 i :ϕi (x)ϕi (y):. As this field has no other positive-energy representation than those occurring in the Fock space [2], nontrivial possibilities for correlations between non-free fields and the fields (4.1) are strongly limited. 4.1. Structure of the correlation functions. We consider a GCI QFT generated by a set of hermitian (real) scalar fields. We denote by F the real vector space of all GCI real scalar fields of scaling dimension 2 in the theory. (Note that the space F may be larger than the linear span of the original system of d = 2 fields of Theorem 4.1.) We shall find in this section the explicit form of the correlation functions of the fields from F. Theorem 4.2. Let φ1 (x), . . . , φn (x) ∈ F, then their truncated n-point functions have the form −1 1 (n) c (φσ1 , . . . , φσn ) ρσ1 σ2 · · · ρσn σ1 , (4.2) 0|φ1 (x1 ) · · · φn (xn )|0tr = 2n σ ∈ Sn
where c(n) are multilinear functionals c(n) : F ⊗n → R with the inversion and cyclic symmetries c(n) (φ1 , . . . , φn ) = c(n) (φn , . . . , φ1 ) = c(n) (φn , φ1 , . . . , φn−1 ). 5 A stress-energy tensor always exists as a quadratic form between states generated by the fields from m the vacuum [8].
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241
Before we prove the theorem, let us first illustrate it on the example of the free field realization (4.1). In this case one finds ∞ ∞ c(2) m 1 , m 2 = αm 1 , j αm 2 , j + βm 1 , j,k βm 2 , j,k
≡
j =1 ∞
j,k = 1
αm 1 , j αm 2 , j + Tr βm 1 βm 2 ,
j =1
c(n) m 1 , . . . , m n = Tr βm 1 · · · βm n for n > 2, where βm = βm, j,k j,k .
(4.3)
Proof of Theorem 4.2. We first recall the general form (2.3) of the truncated correlation function with pole bounds (2.6) that read in this case: µtrjk −1. The argument in Sect. 3.4 shows that the nonzero contributing terms in Eq. (2.3) have for every j = 1, . . . , n exactly two negative µtrjk or µtrk j for some k = k1 , k2 different from j. The nonzero terms are therefore products of “disjoint cyclic products of propagators” of the form 1/ρk1 k2 ρk2 k3 · · · ρkr −1 kr ρkr k1 . But cycles of length r < n are in conflict with the cluster condition (Sect. 2). We conclude that 0|φ1 (x1 ) · · · φn (xn )|0tr is a linear combination of terms like those in (4.2) with some coefficients cσ (φ1 , . . . , φn ) depending on the permutations σ ∈ Sn and on the fields φ j (multilinearly). Locality, i.e. 0|φ1 (x1 ) · · · φn (xn )|0tr = 0|φσ1 (xσ1 ) · · · φσn (xσn )|0tr , then implies cσ σ (φ1 , . . . , φn ) = cσ (φσ1 , . . . , φσn ) (σ, σ ∈ Sn ), so that cσ (φ1 , . . . , φn ) = c(n) (φσ1 , . . . , φσn ) for some c(n) : F ⊗n → R. The equalities c(n) (φ1 , . . . , φn ) = c(n) (φn , . . . , φ1 ) = c(n) (φn , φ1 , . . . , φn−1 ) are again due to locality. As we already know by the general results of the previous section, the harmonic bilocal field exists in the case of fields of dimension d = 2. Moreover, the knowledge of the correlation functions of the d = 2 fields allows us to find the form of the correlation functions of the resulting bilocal fields. This yields an algebraic structure in the space of real (local and bilocal) scalar fields, which we proceed to display. Let us introduce together with the space F of d = 2 fields also the real vector space V of all real harmonic bilocal fields. We shall consider F and V as built starting from our original system of d = 2 fields {m } of Theorem 4.1, by the following constructions: (a) If φ1 (x), φ2 (x) ∈ F then introducing the bilocal (1, 1)–field U (x1 , x2 ) = 2 [φ (x )φ (x ) − 0|φ (x )φ (x )|0] in accord with Eq. (2.7), we consider its harx12 1 1 2 2 1 1 2 2 (x, y). We denote V1 (x, y) by monic decomposition U (x, y) = V1 (x, y) + (x − y)2 U ∗ φ1 ∗ φ2 ; this defines a bilinear map F ⊗ F → V. (b) If now v(x, y) ∈ V then v t (x, y) := v(y, x) also belongs to V and γ (v) (x) := 21 v (x, x) is a field from F. (c) If v(x, y), v (x, y) ∈ V then there is a harmonic bilocal field 2 (v ∗ v ) (x, y) := w- lim x − y v x, x v y , y − 0|v x, x v y , y |0 . x →y
(4.4) The existence of the above weak limit (i.e., a limit within correlation functions) will be established below together with the independence of x = y and the regularity of the resulting field for (x − y)2 = 0.
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N. M. Nikolov, K.-H. Rehren, I. Todorov
(d) If v(x, y) ∈ V and φ(x) ∈ F then we can construct the following bilocal field belonging to V: 2 (v ∗ φ)(x, y) := w-lim v x, x φ − 0|v x, x φ |0 , (4.5) (x − y) (y) (y) x →y
where again the existence of the limit and the regularity for (x − y)2 = 0 will be established later. One can define similarly a product φ ∗ v ∈ V, but it would then be expressed as: (v t ∗ φ)t . ∗ ∗ ∗ To summarize, we have three bilinear maps: F ⊗ F → V, V ⊗ V → V, V ⊗ F → V, γ t and two linear ones: V → V, V → F. Applying these maps we construct F and V inductively, starting from our original system of d = 2 fields, given in Theorem 4.1, and at each step of this inductive procedure, we establish the existence of the above limits in (c) and (d). In fact, we shall establish this together with the structure of the truncated correlation functions for the fields in F and V.6 Before we state the inductive result it is convenient to introduce the vector space '= F ×V A and endow it with the following bilinear operation: (φ1 , v1 ) ∗ (φ2 , v2 ) := 0, φ1 ∗ φ2 + v1 ∗ v2 + v1 ∗ φ2 + (v2t ∗ φ1 )t ,
(4.6)
(4.7)
and with the transposition (φ, v)t := (φ, v t ).
(4.8)
' Thus, the new operation ∗ The spaces F and V will be considered as subspaces in A. ' combines the above listed three operations. We shall see later that A ' is actually an in A associative algebra under the product (4.7). We note that the transposition t (4.8) is an ' antiinvolution with respect to the product: (q1 ∗ q2 )t = q2t ∗ q1t , for every q1 , q2 ∈ A. Proposition 4.3. There exist multilinear functionals '⊗N → R c(N ) : A
(4.9) ' : qk := vk xk[0] , xk[1] ∈ V, where such that if we take elements q1 , . . . , qn+m ∈ A [ε] stands for a Z/2Z–value and k = 1, . . . , n, and qk := φk−n (xk ) ∈ F for k = n + 1, . . . , n + m, then the truncated correlation functions can be written in the following form: 0|v1 x1[0] , x1[1] · · · vn xn[0] , xn[1] φ1 (xn+1 ) · · · φm (xn+m ) |0tr −1 1 = K σ,ε Tσ,ε x1[0] , . . . , xn[1] , xn+1 , . . . , xn+m . (4.10) 2(n + m) σ ∈ Sn+m (ε1 ,...,εn ) ∈ (Z/2Z)n
6 Since we shall use the notion of truncated correlation functions also for bilocal fields, let us briefly recall it. If B1 , . . . , Bn are some smeared (multi)local fields functions are recursively & then their truncated correlation defined by: 0|B1 · · · Bn |0 = 0|B j1 · · · B jk |0tr (the sum being over all partitions
P of {1, . . . , n}).
˙ P = {1,...,n} { j1 ,..., jk } ∈ P ∪
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243
[ε ] [εσn+m ] σ , where we set Here: K σ,ε are coefficients given by K σ,ε := c(n+m) qσ1 1 , . . . , qσn+m [0] [1] t ' εn+1 = · · · = εn+m = 0, and q := q, q := q (for q ∈ A); the terms Tσ,ε are the following cyclic products of intervals: Tσ,ε
2 n−1 2 = xσn+m − xσ1 [ε1 ] xσk [1+εk ] − xσk+1 [εk+1 ] k =1
2 m−1 2 xσn+k − xσn+k+1 . × xσn [1+εn ] − xσn+1
(4.11)
k =1
It follows by Eq. (4.10) that the limits in the steps (c) and (d) above are well defined. Before the proof let us make some remarks. First, we used the same notation c(n) as in Theorem 4.2 since the above multilinear functionals are obviously an extension of the previous, i.e., Eq. (4.10) reduces to Eq. (4.2) for m = 0. Let us also give an example for Eq. (4.10) with n = m = 1: 1 (2) 0|v(x1 , x2 )φ(x3 )|0 = c (v, φ) (ρ23 ρ31 )−1 + c(2) (v t , φ) (ρ13 ρ32 )−1 4 (4.12) + c(2) (φ, v) (ρ31 ρ23 )−1 + c(2) (φ, v t ) (ρ32 ρ13 )−1 . As one can see, c(n) (as well as c(n) of Theorem 4.2) possess a cyclic and an inversion symmetry: c(n) (q1 , . . . , qn ) = c(n) (qn , q1 . . . , qn−1 ) = c(n) qnt , . . . , q1t . (4.13) This is the reason for choosing the prefactors in Eqs. (4.2) and (4.10) (the inverse of the orders of the symmetry groups). Proof of Proposition 4.3. According to our preliminary remarks it is enough to prove ∗ ∗ ∗ that Eq. (4.10) is consistent with the operations F ⊗ F → V, V ⊗ V → V, V ⊗ F → V γ and V → F. ∗ Starting with F ⊗ F → V one should prove that any truncated correlation functr tion · φ1 (x1 ) φ2 (x2 ) · given by Eq. (4.10) yields a harmonic decomposition: ρ12 tr tr · φ1 (x1 )φ2 (x2 ) · = · (φ1 ∗ φ2 ) (x1 , x2 ) · + ρ12 R(x1 , x2 ), with a correlation function tr · (φ1 ∗ φ2 ) (x1 , x2 ) · given by Eq. (4.10) and a rational function R regular at ρ12 = 0. This gives us relations of the type c(n+2) (q1 , . . . , φ1 , φ2 , . . . , qn ) = c(n+1) (q1 , . . . , φ1 ∗ φ2 , . . . , qn ). (4.14) tr or Next, having correlation functions of type · v1 (x1 , x2 )v2 (x3 , x4 ) · tr · v(x1 , x2 )φ(x3 ) · of the form (4.10), one verifies that the limits (4.4) and (4.5) exist tr within these correlation functions, and they yield expressions for · (v1 ∗ v2 ) (x1 , x4 ) · tr and · (v ∗ φ) (x1 , x3 ) · consistent with (4.10). As a result we obtain again relations between the c’s: c(n+2) (q1 , . . . , v1 , v2 , . . . , qn ) = c(n+1) (q1 , . . . , v1 ∗ v2 , . . . , qn ), c(n+2) (q1 , . . . , v, φ, . . . , qn ) = c(n+1) (q1 , . . . , v ∗ φ, . . . , qn ).
(4.15)
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tr Finally, one verifies that setting x1 = x2 in · v(x1 , x2 ) · we obtain the correlation tr functions · γ (v) (x1 ) · with the relation c(n+1) (q1 , . . . , (v + v t ), . . . , qn ) = 2 c(n+1) (q1 , . . . , γ (v), . . . , qn ).
(4.16)
This completes the proof of Proposition 4.3 as well as the proof that the products V ⊗ ∗ ∗ V → V and V ⊗ F → V are well defined. 4.2. Associative algebra structure of the OPE. Note that Eqs. (4.14), (4.15) read (under (4.7)) c(n) (q1 , . . . , qk , qk+1 , . . . , qn ) = c(n−1) (q1 , . . . , qk ∗ qk+1 , . . . , qn ) .
(4.17)
' is an associative product. This implies that the bilinear operation ∗ on A ' Indeed, consider the element q := (q1 ∗ q2 ) ∗ q3 − q1 ∗ (q2 ∗ q3 ) for q1 , q2 , q3 ∈ A. By (4.7) q is a bilocal field. Equation (4.17) implies that all c’s in which q enters vanish and hence, by Eq. (4.10) q has zero correlation functions with all other fields, including itself. But then this (bilocal) field is zero by the Reeh–Schlieder theorem, since its action on the vacuum will be identically zero. ' (4.6) was not only convenient for comThus, introducing the cartesian product A bining three types of bilinear operations in one but also as a compact expression for the ' carries a redundant information due to associativity (Eqs. (4.14), (4.15)). However, A the following relation: 1 1 t t −γ (v), (v + v ) ∗ q = 0 = q ∗ −γ (v), (v + v ) (4.18) 2 2 ' To prove (4.18) we point out first that it is equivalent to for every v ∈ V and q ∈ A. the identities v ∗ φ = γ (v) ∗ φ and v ∗ v = v ∗ γ (v) for v = v t ∈ V and any φ ∈ F, v ∈ V. These identities can be established again first for the c’s, and then proceeding by using the Reeh–Schlieder theorem, as in the above proof of associativity. ' is because we can identify symmetric bilocal fields Hence, the redundancy in A t v = v ∈ V with their restrictions to the diagonal, γ (v) ∈ F, and this is compatible with the product ∗. Let us point out that the restriction of the map γ to the t–invariant subspace Vs := {v ∈ V : v = v t } is an injection into F. The latter follows from a simple analysis of the 4-point functions of v and the Reeh–Schlieder theorem: if v(x, y) = v(y, x) and 0|v(x, x)v(y, y)|0 = 0 then 0|v(x, x )v(y, y )|0 = 0. In this way we ' the symmetric harmonic bilocal fields v = v t with their see that we can identify in A restriction on the diagonal γ (v) ∈ F. Formally, the above considerations can be summarized in the following abstract way. Let us introduce the quotient ) ( 1 t ' A := A −γ (v), (v + v ) : v ∈ V . (4.19) 2 '→ A ' can be It is an associative algebra according to Eq. (4.18). The involution t : A transferred to an involution on the quotient (4.19) and we denote it by t as well. The ' → A and spaces F and V are mapped into A by the natural compositions F → A ' → A. The injectivity of γ on Vs implies that the maps F → A and V → A V →A
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so defined are actually injections. Hence, we shall treat F and V also as subspaces of A. Furthermore, A becomes a direct sum of vector spaces with
A = F ⊕ Va ,
q ∈ A : q t = q = F ⊇ Vs := v ∈ V : v t = v ,
q ∈ A : q t = −q = Va := v ∈ V : v t = −v .
(4.20)
Hence, the t–symmetric elements of A are identified with the d = 2 local fields, while the t–antisymmetric elements of A, with the antisymmetric, harmonic bilocal (1, 1) fields. (Neither F nor Va are subalgebras of A.) ' by identifying the space To summarize, the associative algebra A is obtained from A Vs of symmetric bilocal fields with its image γ (Vs ) ⊆ F. ' again For simplicity we will denote the equivalence class in A of an element q ∈ A by q. Also note that the c’s can be transferred as well, to multilinear functionals on A, since the kernel of the quotient (4.19) is contained in the kernel of each c(n) by (4.16). We shall use the same notation c(n) also for the multilinear functional c(n) on A. Example 4.1. Let us illustrate the above algebraic structures on the simplest example of a QFT generated by a pair of d = 2 GCI fields 1 and 2 given by normal a pair of two mutually commuting free massless fields ϕ j : 1 (x) = 21 : ϕ12 (x) : − : ϕ22 (x) : and 2 (x) = ϕ1 (x) ϕ2 (x). Their OPE algebra involves a set of four independent harmonic ∗ bilocal fields V jk (x1 , x2 ) := : ϕ j (x1 ) ϕk (x2 ) : ( j, k = 1, 2), which satisfy V jk (x1 , x2 ) = Vk j (x1 , x2 ) = V jk (x2 , x1 ). For instance, we have 1 ∗ 2 = V12 − V21 .7 Also note that 1 = γ (V1 ) for V1 (x1 , x2 ) =: ϕ1 (x1 ) ϕ1 (x2 ) : − : ϕ2 (x1 ) ϕ2 (x2 ) :, etc. By the associativity and Eq. (4.17) we have c(n) (q1 , . . . , qn ) = c(2) (q1 ∗ · · · ∗ qn−1 , qn )
(4.21)
for q1 , . . . , qn ∈ A. Let us consider now c(2) and define the following symmetric bilinear form on A: q1 , q2 := c(2) q1t , q2 . (4.22) First note that F and Va are orthogonal with respect to this bilinear form: this is due to the fact that there is no nonzero three point conformally invariant scalar function of weights (2, 1, 1), which is antisymmetric in the second and third arguments. Next, we claim that (4.22) is strictly positive definite. This is a straightforward consequence of the Wightman positivity and the Reeh–Schlieder theorem (one should consider separately the positivity on F and Va ). In particular, (4.22) is nondegenerate. By Eqs. (4.13) and (4.17) we have: q1 ∗ q2 , q3 = q2 , q1t ∗ q3 (4.23) for all q1 , q2 , q3 ∈ A. Let us introduce now an additional splitting of F. Denote by F0 the kernel of the product, i.e., F0 := {ψ ∈ F : ψ ∗ q = 0 ∀q ∈ A} ≡ {ψ ∈ F : q ∗ ψ = 0 ∀q ∈ A}
(4.24)
7 I.e., in the OPE (x ) (x ) there appears the antisymmetric bilocal field V (x , x ) − V (x , x ) 1 1 2 2 12 1 2 21 1 2 that involves only odd rank conserved tensor currents in its expansion in local fields.
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(the second equality is due to the identity φ ∗ q = (q t ∗ φ)t ). Let F1 be the orthogonal complement in F of F0 with respect to the scalar product (4.22):
(4.25) F1 := φ ∈ F : φ, ψ = 0 ∀ψ ∈ F0 . The meaning of fields belonging to F0 becomes immediately clear if we note that c(n) for n 3 are zero if one of the arguments belongs to F0 (this is due to Eq. (4.21)). Hence, all their truncated functions higher than two point are zero, i.e., the fields belonging to F0 are generalized free d = 2 fields. Furthermore, these fields commute with all other fields from F1 and Va ≡ A(1) : this is because of the vanishing of c(2) (ψ, q) if ψ ∈ F0 and q ∈ F1 ⊕ Va , as well as of all c(n+1) (ψ, q1 , . . . , qn ) for n 2 if ψ ∈ F0 and q1 , . . . , qn ∈ A (by (4.21) and (4.24)). Clearly, F1 ⊕ Va is a subalgebra of A: this follows from Eq. (4.23) with q3 ∈ F0 along with the definitions (4.24) and (4.25). Let us denote it by B := F1 ⊕ Va .
(4.26)
We are now ready to state the main step towards the proof of Theorem 4.1. Proposition 4.4. There is a homomorphism ι from the associative algebra B into the algebra of Hilbert–Schmidt operators over some real separable Hilbert space, such that c(n) (q1 , . . . , qn ) = Tr (ι (q1 ) · · · ι (qn )) ,
(4.27)
and ι (F) are symmetric operators while ι (Va ) are antisymmetric. We shall give the proof of this proposition in the subsequent subsection. The main reason leading to it is that B becomes a real Hilbert algebra with an integral trace on it. Here we proceed to show how Theorem 4.1 can be proven by using the above results. Proof of Theorem 4.1. Let m = 0m + 1m be the decomposition of each field m according to the splitting F = F0 ⊕ F1 . Take an orthonormal basis ψm in F0 and let ∞ 0m = αm, j ψ j , and βm = βm, j,k j,k be the symmetric matrix corresponding to j =1 the Hilbert–Schmidt operator ι 1m (m = 1, 2, . . . ). Then Eqs. (4.3) and (4.27) show that the constants αm, j and βm, j,k so defined satisfy the conditions of Theorem 4.1. Remark 4.1. In general, we have F1 Vs . This is because the elements of F1 correspond, by Proposition 4.4, to Hilbert–Schmidt symmetric operators and on the other hand, the elements of V are obtained, according to the inductive construction of Sect. 4.1, as products of elements of F and will, hence, correspond to trace class operators. 4.3. Completion of the proofs. It remains to prove Proposition 4.4. We start with an inequality of Cauchy–Schwartz type. Lemma 4.5. Let q1 , q2 ∈ A be such that each of them belongs either to F or to Va . Then we have 2 q1 ∗ q2 , q1 ∗ q2 q1 ∗ q1 , q1 ∗ q1 q2 ∗ q2 , q2 ∗ q2 . (4.28) Proof. Consider q1 ∗q1 +λ q2 ∗q2 , q1 ∗q1 +λ q2 ∗q2 0 and use that q1 ∗q1 , q2 ∗q2 = ± q1 ∗ q2 , q1 ∗ q2 if each of q1 , q2 belongs either to F or to Va .
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The space B (4.26) is a real pre–Hilbert space with a scalar product given by (4.22). It is also invariant under the action of t (actually the eigenspaces of t are F1 and Va ). The left action of B on itself gives us an algebra homomorphism ι : B → LinR B
(4.29)
of B into the algebra of all operators over B. Moreover, the elements of F are mapped into symmetric operators and the elements of Va , into antisymmetric (this is due to (4.23)). Lemma 4.6. Every element of B is mapped into a Hilbert–Schmidt operator. Proof. Since B is generated by F1 (according to the inductive construction of F and V in Sect. 4.1) it is enough to show this for the elements of F1 . Let φ ∈ F1 and consider the commutative subalgebra Bφ of B generated by φ. The algebra Bφ is freely generated by φ, i.e., is isomorphic to the algebra λ R[λ] of polynomials in a single variable λ (↔ φ), since φ belongs to the orthogonal complement of F0 (4.24). For a p(λ) ∈ λ R[λ] we shall denote by φ [ p] the corresponding element of Bφ . In particular,
Setting
φ [ p1 ] ∗ φ [ p2 ] = φ [ p1 p2 ] .
(4.30)
φ ∗(n+1) := φ ∗n ∗ φ, c λn+1 := c(2) φ ∗n , φ ≡ φ ∗n , φ
(4.31)
∼ over the algebra λ2 R[λ] = (φ ∗1 := φ, n 1) we obtain a positive definite functional φ ∗ Bφ (due to Eq. (4.23) and the positivity of ·, · (4.22)). Then, by the Hamburger theorem about the classical moment problem ([9, Chap. 12, Sect. 8]) we conclude that there exists a bounded positive Borel measure dµ (λ) on R, such that 2 c λ p (λ) = p (λ) dµ(λ) (4.32) R
for every p(λ) ∈ R[λ]. Using this we can extend the fields φ [ p] (x) to φ [ f ] (x) for Borel measurable functions f having compact support with respect to µ in R\{0}. The latter can be done in the following way. Fix ε ∈ (0, 1) and let g1 , . . . , gn be Schwartz test functions on M. By 0|φ [ p1 ] [g1 ] · · · φ [ pn ] [gn ]|0 depend polynomially Theorem 4.2 the correlators [ pk j ] [ pk 1 ] (n) φ = c pk1 · · · pk j for all {k1 , . . . , k j } ⊆ {1, . . . , n}. But for ,...,φ on c every ε ∈ (0, 1) there exists a norm qk (λ) |qk (λ)| dµ(λ)
q ε = Aε sup 2 + Bε (4.33) λ |λ| ε R \ (−ε, ε)
on λ2 R[λ] q(λ), where Aε and Bε are some positive constants, such that for every q1 , . . . , qm ∈ λ2 R[λ], ⎫1 ⎧ m ⎨ m ⎬m |qk (λ)|m | c [q1 (λ) · · · qm (λ)]| dµ(λ)
qk ε . ⎭ ⎩ |λ|2 k =1
R
k =1
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n & Hence, 0|φ [ p1 ] [g1 ] · · · φ [ pn ] [gn ]|0 C
pk ε gk S for some constant C and k =1
Schwartz norm · S (not depending on pk and gk ). Since for every ε ∈ (0, 1) the Banach space L 1 (R\{(−ε, ε)}, µ) is contained in the completion of λ2 R[λ] with respect to the norms (4.33), we can extend the linear functional c[ p(λ)] as well as the correlators 0|φ [ p1 ] [g1 ] · · · φ [ pn ] [gn ]|0 to a functional c[ f (λ)] and correlators 0|φ [ f1 ] [g1 ] · · · φ [ fn ] [gn ]|0 defined for Borel functions f, f 1 , . . . , f n compactly supported with respect to µ in R\{0}. Thus, we can extend the fields φ [ p] by extending their correlators. By the continuity we also have for arbitrary Borel functions f, f k , compactly supported in R\{0}: φ [ f1 ] ∗ φ [ f2 ] = φ [ f1 f2 ] , c(n) φ [ f1 ] , . . . , φ [ fn ] = c [ f 1 · · · f n ] , f (λ) c[f] = dµ(λ) (4.34) λ2 R
(cp. (4.32)), and c(n) determine the correlation functions of φ [ fk ] as in Theorem 4.2. In particular, for every characteristic function χ S of a compact subset S ⊂ R\{0} we have φ [χ S ] ∗ φ [χ S ] = φ [χ S ] . Hence, for such a d = 2 field we will have that all its (n) truncated correlation functions are given (4.2) [χ ]by [χ with all normalization constants c (2) ] s s φ ,φ . Then, as shown in [12, Theorem 5.1], equal to one and the same value c Wightman positivity requires this value to be a non-negative integer, i.e., dµ(λ) c(2) φ [χ S ] , φ [χ S ] = c [χ S ] = ∈ {0, 1, 2, . . . } (4.35) λ2 S
(it is zero iff φ [χ S ] = 0). Hence, the restriction of the measure dµ(λ)/λ2 to R\{0} is a (possibly infinite) sum of atom measures of integral masses, each supported at some γk ∈ R\{0} for k = 1, . . . , N (and N could be infinity). In particular, the measure µ is supported in a bounded subset of R. By Lemma 4.5 we can define ι(φ [ f ] ) as a closable operator on B if f is a Borel measurable function with compact support in R\{0}. It follows then that the projectors ι(φ [χ S ] ), for a compact S ⊆ R\{0}, provide a spectral decomposition for ι(φ) (in fact, ι(φ [ f ] ) = f (ι(φ))). Thus, ι(φ) has discrete spectrum with eigenvalues γk (k ∈ N), each of a multiplicity given by the integer c(2) φ χ{γk } , φ χ{γk } . Then ι(φ) is a Hilbert–Schmidt operator since ∞ ∞ dµ(λ) γk2 c(2) φ χ{γk } , φ χ{γk } = γk2 = dµ(λ) < ∞ λ2 k =1
k =1
{γk }
R\{0}
(µ being a bounded measure). The completion of the proof of Proposition 4.4 is provided now by the following corollary. Corollary 4.7. For every q1 , q2 ∈ B one has c(2) (q1 , q2 ) = Tr (ι(q1 )ι(q2 )). Proof. If q1 = q2 ∈ F1 this follows from the proof of Lemma 4.6 and hence, by a polarization, for any q1 , q2 ∈ F1 . The general case can be obtained by using the facts that B is generated by F1 and c(2) has the symmetry c(2) (q1 ∗ q2 , q3 ) = c(2) (q1 , q2 ∗ q3 ).
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5. Discussion. Open Problems The main result of Sect. 4, the (generalized) free field representation of a system {φa } of GCI scalar fields of conformal dimension d = 2 (Theorem 4.1), is obtained by revealing and exploiting a rich algebraic structure in the space F × V of all d = 2 real scalar fields and of all harmonic bilocal fields of dimension (1, 1). However, this structure is mainly due to the fact that we are in the case of lower scaling dimension: there is only one possible singular structure in the OPE (after truncating the vacuum part). One try can to establish such a result in spaces of spin–tensor bilocal fields (of dimension 3 3 2 , 2 or (2, 2)) satisfying linear (first order) conformally invariant differential equations (that again imply harmonicity). If these equations together with the corresponding pole bounds imply such singularities in the OPE, which can be “split” one would be able to prove the validity of free field realizations in such more general theories, too. One may also attempt to study models, say in a theory of a system of scalar fields of dimension d = 4, without leaving the realm of scalar bilocal harmonic fields V1 (of dimension (1, 1)). In [11] there have been found examples of 6–point functions of harmonic bilocal fields, which do not have free field realizations. However, our experience with the d = 2 case shows that in order to complete the model (including the check of Wightman positivity for all correlation functions) it is crucial to describe the OPE in terms of some simple algebraic structure (e.g., associative, or Lie algebras). On the other hand going beyond bilocal V1 ’s is a true signal of nontriviality of a GCI model. Our analysis of Sect. 3 shows that this can be characterized by a simple property of the correlation functions: the violation of the single pole property (of Sect. 3.3). From this point of view a further exploration of the example of Sect. 3.5 within a QFT involving currents appears particularly attractive. Note added in proof: In [15], we have determined the biharmonic function whose leading part is given by Eq. (3.31). It involves dilogarithmic functions, whose arguments are algebraic functions of conformal cross ratios. This exemplifies the violation of Huygens bilocality for the biharmonic fields, Theorem 3.7. Yet, in support of Conjecture 3.3, it is shown that the structure of the cuts is in a nontrivial manner consistent with ordinary bilocality. Acknowledgement. We thank Yassen Stanev for an enlightening discussion. This work was started while N.N. and I.T. were visiting the Institut für Theoretische Physik der Universität Göttingen as an Alexander von Humboldt research fellow and an AvH awardee, respectively. It was continued during the stay of N.N. at the Albert Einstein Institute for Gravitational Physics in Potsdam and of I.T. at the Theory Group of the Physics Department of CERN. The paper was completed during the visit of N.N. and I.T. to the High Energy Section of the I.C.T.P. in Trieste, and of K.-H.R. at the Erwin Schrödinger Institute in Vienna. We thank all these institutions for their hospitality and support. N.N. and I.T. were partially supported by the Research Training Network of the European Commission under contract MRTN-CT-2004-00514 and by the Bulgarian National Council for Scientific Research under contract PH-1406. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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3. Bargmann, V., Todorov, I.T.: Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of S O(N ). J. Math. Phys. 18, 1141–1148 (1977) 4. Borchers, H.-J.: Über die Mannigfaltigkeit der interpolierenden Felder zu einer interpolierenden S-Matrix. N. Cim. 15, 784–794 (1960) 5. Buchholz, D., Mack, G., Todorov, I.T.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B (Proc. Suppl.) 5, 20–56 (1988) 6. Dobrev, V.K. Mack, G., Petkova, V.B., Petrova, S.G., Todorov, I.T.: Harmonic Analysis of the n-Dimensional Lorentz Group and Its Applications to Conformal Quantum Field Theory. Berlin et al.: Springer, 1977 7. Dolan, F.A., Osborn, H.: Conformal four point functions and operator product expansion. Nucl. Phys. B 599, 459–496 (2001) 8. Dütsch, M., Rehren, K.-H.: Generalized free fields and the AdS-CFT correspondence. Ann. H. Poincaré 4, 613–635 (2003) 9. Dunford, N., Schwartz, J.: Linear Operators, Part 2. Spectral Theory. Self Adjoint Operators in Hilbert Space. N.Y.-London: Interscience Publishers, 1963 10. Mack, G.: All unitary representations of the conformal group SU (2, 2) with positive energy. Commun. Math. Phys. 55, 1–28 (1977) 11. Nikolov, N.M., Rehren, K.-H., Todorov, I.T.: Partial wave expansion and Wightman positivity in conformal field theory. Nucl. Phys. B 722, 266–296 (2005) 12. Nikolov, N.M., Stanev, Ya.S., Todorov, I.T.: Four dimensional CFT models with rational correlation functions. J. Phys. A 35, 2985–3007 (2002) 13. Nikolov, N.M., Stanev, Ya.S., Todorov, I.T.: Globally conformal invariant gauge field theory with rational correlation functions. Nucl. Phys. B 670, 373–400 (2003) 14. Nikolov, N.M., Todorov, I.T.: Rationality of conformally invariant local correlation functions on compactified Minkowsi space. Commun. Math. Phys. 218, 417–436 (2001) 15. Nikolov, N.M., Rehren, K.-H., Todorov, I.: Pole structure and biharmonic fields in conformal QFT in four dimensions. In: Dobrev, V. (ed.) LT7: Lie Theory and its Applications in Physics, Proceedings Varna 2007, Sofia, Heron Press (2008). e-print arXiv:0711.0628, to appear 16. Schroer, B., Swieca, J.A.: Conformal transformations of quantized fields. Phys. Rev. D 10, 480–485 (1974) 17. Schroer, B., Swieca, J.A., Völkel, A.H.: Global operator expansions in conformally invariant relativistic quantum field theory. Phys. Rev. D 11, 1509–1520 (1975) 18. Streater R.F., Wightman A.S.: PCT, Spin and Statistics, and All That. Benjamin, 1964; Princeton, N.J.: Princeton Univ. Press, 2000 19. Todorov, I.: Vertex algebras and conformal field theory models in four dimensions. Fortschr. Phys. 54, 496–504 (2006) Communicated by Y. Kawahigashi
Commun. Math. Phys. 279, 251–283 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0417-5
Communications in
Mathematical Physics
Asymptotic Error Rates in Quantum Hypothesis Testing K. M. R. Audenaert1,2 , M. Nussbaum3 , A. Szkoła4 , F. Verstraete5 1 Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, London SW7 2PG, UK 2 Dept. of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK.
E-mail: [email protected]
3 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA.
E-mail: [email protected]
4 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany.
E-mail: [email protected]
5 Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria.
E-mail: [email protected] Received: 5 September 2007 / Accepted: 27 November 2007 Published online: 8 February 2008 – © Springer-Verlag 2008
Abstract: We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing. The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support. The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.
1. Introduction One of the basic tasks in information theory is discriminating between two different information sources, modelled by (time-discrete) stochastic processes. Given a source that generates independent, identically distributed (i.i.d.) random variables, according to one out of two possible probability distributions, the task is to determine which distribution is the true one, and to do so with minimal error, whatever error criterion one chooses. This basic decision problem has an equally basic quantum-informational incarnation. Given an information source that emits quantum systems (particles) independently and
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identically prepared in one out of two possible quantum states, figure out which state is the true one, with minimal error probability. In both settings, we’re dealing with two hypotheses, each one pertaining to one law represented by a probability distribution or a quantum state, respectively, and the discrimination problem is thus a particular instance of a hypothesis testing problem. In hypothesis testing, one considers a null hypothesis and an alternative hypothesis. The alternative hypothesis is the one of interest and states that “something significant is happening”, for example, a cell culture under investigation is coming from a malignant tumor, or some case of flu is the avian one, or an e-mail attachment is a computer virus. In contrast, the null hypothesis corresponds to this not being the case; the cells are normal ones, the flu can be treated with an aspirin, and the attachment is just a nice picture. This is inherently an asymmetric situation, and Neyman and Pearson introduced the idea of similarly making a distinction between type I and type II errors. – The type I error or “false positive”, denoted by α, is the error of accepting the alternative hypothesis when in reality the null hypothesis holds and the results can be attributed merely to chance. – The type II error or “false negative”, denoted by β, is the error of accepting the null hypothesis when the alternative hypothesis is the true state of nature. The costs associated to the two types of error can be widely different, or even incommensurate. For example, in medical diagnosis, the type I error corresponds to diagnosing a healthy patient with a certain affliction, which can be an expensive mistake, causing a lot of grievance. On the other hand, the type II error may correspond to declaring a patient healthy while in reality (s)he has a life-threatening condition, which can be a fatal mistake. To treat the state discrimination problem as a hypothesis test, we assign the null hypothesis to one of the two states and the alternative hypothesis to the other one. If all we want to know is which one of the two possible states we are observing, the mathematical treatment is completely symmetric under the interchange of these two states. It therefore fits most naturally in the setting of symmetric hypothesis testing, where no essential distinction is made between the two kinds of errors. To wit, in symmetric hypothesis testing, one considers the average, or Bayesian, error probability Pe , defined as the average of α and β weighted by the prior probabilities of the null and the alternative hypothesis, respectively. This paper will be concerned with symmetric as well as with asymmetric quantum hypothesis testing. Since we have developed the main techniques in the symmetric setting we will start with this case and address the asymmetric setting at the end. The optimal solution to the symmetric classical hypothesis test is given by the maximum-likelihood (ML) test. Starting from the outcomes of an experiment involving n independent draws from the unknown distribution, one calculates the conditional probabilities (likelihoods) that these outcomes can be obtained when the distribution is the one of the null hypothesis and the one of the alternative hypothesis, respectively. One decides then on the hypothesis for which the conditional probability is the highest. I.e. if the likelihood ratio is higher than 1, the null hypothesis is rejected, otherwise it is accepted. In the quantum setting, the experiment consists of preparing n independent copies of a quantum system in an unknown state, which is either ρ or σ , and performing an optimal measurement on them. We assume that the quantum systems are finite, implying that the states are associated to density operators on a finite-dimensional complex Hilbert space. Under the null hypothesis, the combined n copies correspond to an n-fold tensor
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product density operator ρ ⊗n , while under the alternative hypothesis, the associated density operator is σ ⊗n . The null hypothesis is then accepted or rejected according to the outcome of the measurement and the specified decision rule. The task of finding this optimal measurement is so fundamental that it was one of the first problems considered in the field of quantum information theory; it was solved in the one-copy case more than 30 years ago by Helstrom and Holevo [14,17]. We refer to the generalised ML-tests as Holevo-Helstrom tests. In the special case of equal priors, the associated minimal probability of error achieved by the optimal measurement can be calculated from the trace norm distance between the two states: 1 ∗ Pe,n (ρ, σ ) = (1 − ρ ⊗n − σ ⊗n 1 /2), (1) 2 where A1 := Tr |A| denotes the trace norm. Going back to the classical case again, in a seminal paper, H. Chernoff [8] investigated the so-called asymptotical efficiency of a class of statistical tests, which includes the likelihood ratio test mentioned before. The probability of error Pe,n in discriminating two probability distributions decreases exponentially in n, the number of draws from the distribution: Pe,n ∼ exp(−ξ n). For finite n this is a rather crude approximation. However, as n grows larger one finds better and better agreement, and the exponent ξ becomes meaningful in the asymptotic limit. The asymptotical efficiency is exactly the asymptotic limit of this exponent. Chernoff was able to derive an (almost) closed expression for this asymptotic efficiency, which was later named eponymously in his honour. For two discrete probability distributions p and q, this expression is given by ξC B ( p, q) := − log inf p(i)1−s q(i)s , (2) 0≤s≤1
i
which is of closed form but for a single variable minimisation. This quantity goes under the alternative names of Chernoff distance, Chernoff divergence and Chernoff information. While Chernoff’s main purpose was to use this asymptotic efficiency measure to compare the power of different tests – the mathematically optimal test need not always be the most practical one – it can also be used as a distinguishability measure between the distributions (states) of the two hypotheses. Indeed, fixing the test, its efficiency for a particular pair of distributions gives a meaningful indication of how well these two distributions can be distinguished by that test. This is especially meaningful if the applied test is the optimal one. A quantum generalisation of Chernoff’s result is highly desirable. Given the large amount of experimental effort in the context of quantum information processing to prepare and measure quantum states, it is of fundamental importance to have a theory that allows to discriminate different quantum states in a meaningful way. Despite considerable effort, however, the quantum generalisation of the Chernoff distance has until recently remained unsolved. In the previous papers, [21] and [1], this issue was finaly settled and the asymptotic error exponent was identified, when the optimal Holevo-Helstrom strategy for discriminating between the two states is used, by proving that the following version of the Chernoff distance ξ QC B (ρ, σ ) := − log inf Tr[ρ 1−s σ s ] , (3) 0≤s≤1
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has the same operational meaning as its classical counterpart: It specifies the asymptotic ∗ (recall definition (1)). Remarkably, rate exponent of the minimal error probability Pe,n it looks like an almost naïve generalisation of the classical expression (2). We remark that in the literature different extensions of the classical expression have been considered. Indeed, when insisting only on the compatibility with the classical Chernoff distance, there is in principle an infinitude of possiblities. Among those, three especially promising candidate expressions had been put forward by Ogawa and Hayashi [23], who studied their relations and found that there exists an increasing ordering between them. Incidentally, the second candidate coincides with (3) and thus turns out to be the correct one. Kargin [18] gave lower and upper bounds on the optimal error exponent ξ in terms of the fidelity between the two density operators and found that Ogawa and Hayashi’s third candidate (in their increasing arrangement) is a lower bound on the optimal error exponent for faithful states, i.e. it is an achievable rate. Hayashi [11] made progress regarding (3), by showing that for s = 1/2, − log Tr[ρ 1−s σ s ] is also an achievable error exponent. The proof of our main result consists of two parts. In the optimality part, which was first presented in [21], we show that for any test the (Bayesian) error rate − n1 log Pe,n cannot be made arbitrary large but is asymptotically bounded above by ξ QC B . In the achievability part, first put forward in [1], we prove that under the Holevo-Helstrom strategy the bound is actually attained in the asymptotic limit, i.e. 1 ∗ lim sup − log Pe,n ≥ ξ QC B . n n→∞ It is the purpose of this paper to give a complete, detailed, and unified account of these results. We will present the complete proof in Sect. 3. Moreover, we give an in-depth treatment of the properties of the quantum Chernoff distance in Sect. 4. More precisely, we show that it defines a distance measure between quantum states. Distinguishability measures between quantum states have been used in a wide variety of applications in quantum information theory. The most popular of such measures seems to be Uhlmann’s fidelity [28], which happens to coincide with the quantum Chernoff distance when one of the states is pure. The trace norm distance ρ − σ 1 = Tr |ρ − σ | has a more natural operational meaning than the fidelity, but lacks monotonicity under taking tensor powers of its arguments. The problem is that one can easily find states ρ, σ, ρ , σ such that ρ − σ 1 < ρ − σ 1 but ρ ⊗2 − σ ⊗2 1 > ρ ⊗2 − σ ⊗2 1 . This already happens in the classical setting: take the following 2-dimensional diagonal states 1/4 0 3/4 0 00 b 0 ρ= , σ = , ρ = , σ = , 0 3/4 0 1/4 01 0 1−b √ where 1 − 1/ 2 < b < 1/2. Then ρ − σ 1 = 1 > 2b = ρ − σ 1 , while ρ ⊗2 − σ ⊗2 1 = 1 < 2b(2 − b) = ρ ⊗2 − σ ⊗2 1 . The quantum Chernoff distance characterises the exponent arising in the asymptotic behaviour of the trace norm distance, in the case of many identical copies, and therefore by construction does not suffer from this problem. As such, the quantum Chernoff distance can be considered as a kind of regularisation of the trace norm distance. For the above-mentioned states, ξ QC B (ρ, σ ) = √ − log( 3/2) (optimal s = 1/2) and ξ QC B (ρ , σ ) = − log(1 − b) (optimal s = 1).
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A related problem that attracted a lot of attention in the field of quantum information theory was to identify the relative entropy between two quantum states. An informationtheoretical way of looking at the classical relative entropy between two probability distributions, or Kullback-Leibler distance, is that it characterises the inefficiency of compressing messages from a source p using an algorithm that is optimal for a source p (i.e. yields the Shannon information bound for that source). Phrased differently, it quantifies the way one could cheat by telling that the given probability distribution is p while the real one is p . By proving a quantum version of Stein’s lemma [15,24], it has been shown that the quantum relative entropy, as introduced by Umegaki, has exactly the same operational meaning. When using the relative entropy to distinguish between states, one faces the problem that it is not continuous and is asymmetric under exchange of its arguments, and therefore it does not represent a distance measure in a mathematically strict manner. Furthermore, for pure states, the quantum relative entropy is not very useful, since it is either 0 (when the two states are identical) or infinite (when they are not). In contrast, the quantum Chernoff distance seems to be much more natural in many situations. On the other hand, (quantum) relative entropy is a crucial notion in asymmetric hypothesis testing. There it obtains an operational meaning as the best achievable asymptotic rate of type II errors. Its properties, which are problematic for a candidate for a distance measure, reflect the asymmetry between the null and alternative hypothesis arising from treating the type-I and type-II errors in a different way. As exemplified by the medical diagnosis case mentioned above, the type II error is the one that should be avoided at all costs. Hence, one puts a constraint α < on the type I error, and minimises the β-rate. One obtains that the optimal β-rate is the relative entropy of the null hypothesis w.r.t. the alternative, independent of the constrained . The mathematical derivation of this statement goes under the name of Stein’s Lemma. When the constraint consists of a lower bound on the asymptotic exponential rate of the type II error, one obtains what is called the Hoeffding bound. Asymmetric hypothesis testing has been subject to a quantum theoretical treatment much earlier, although it is a much less natural setting for the basic state discrimination problem. The quantum generalisation of Stein’s Lemma was first obtained by Hiai and Petz [15]. Its optimality part was then strengthened by Ogawa and Nagaoka in [24]. In the last few years there has been a lot of progress extending the statement of the lemma in different directions. In [4] the minimal relative entropy distance from a set of quantum states, the null hypothesis, w.r.t. a reference quantum state, the alternative, has been fixed as the best achievable asymptotic rate of the type II errors, see also [13]. This may be seen as a quantum generalisation of Sanov’s theorem. In a recent paper [5] an extension of this result to the case where the hypotheses correspond to sources emitting correlated (not necessarily i.i.d.) classical or quantum data has been given. Additionally, an equivalence relation between the achievability part in (quantum) Stein’s Lemma and (quantum) Sanov’s Theorem has been derived. Just a few months after the appearance of [21,1], the techniques pioneered in those two papers were used to find a quantum generalisation of the Hoeffding bound under the implicit assumption of equivalent hypotheses, i.e. for states with coinciding supports, thereby (partially) solving another long-standing open problem in quantum hypothesis Just as in the case of the Chernoff distance, the Hoeffding bound contains testing. 1−s q(i)s as a sub-expression, and the quantum generalisation of the Hoeffding p(i) i bound is obtained by replacing this sub-expression by Tr[ρ 1−s σ s ]. The optimality of the bound (also called the “converse part”) was proven by Nagaoka [20], while its
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achievability (the “direct part”) was found by Hayashi [12]. Using the same techniques, Hayashi also gave a simple proof of the achievability part of the quantum Stein’s Lemma, in that same paper. In Sect. 5 we first formulate and prove an extended version of the classical Hoeffding bound, which allows nonequivalent hypotheses. Secondly, we present a complete proof of the quantum Hoeffding bound in a unified way. Moreover, we derive quantum Stein’s Lemma as well as quantum Sanov’s Theorem from the quantum Hoeffding bound combined with the mentioned equivalence relation proved in [5]. 2. Mathematical Setting and Problem Formulation We consider the two hypotheses H0 (null) and H1 (alternative) that a device prepares finite quantum systems either in the state ρ or in the state σ , respectively. Everywhere in this paper, we identify a state with a density operator, i.e. a positive trace 1 linear operator on a finite-dimensional Hilbert space H associated to the type of the finite quantum system in question. Since the (quantum) Chernoff distance arises naturally in a Bayesian setting, we supply the prior probabilities π0 and π1 , which are positive quantities summing up to 1; we exclude the degenerate cases π0 = 0 and π1 = 0 because these are trivial. Physically discriminating between the two hypotheses corresponds to performing a generalised (POVM) measurement on the quantum system. In analogy to the classical proceeding one accepts H0 or H1 according to a decision rule based on the outcome of the measurement. There is no loss of generality assuming that the POVM consists of only two elements, which we denote by {11 − Π, Π }, where Π may be any linear operator on H with 0 ≤ Π ≤ 11. We will mostly make reference to this POVM by its Π element, the one corresponding to the alternative hypothesis. The type-I and type-II error probabilities α and β are the probabilities of mistaking σ for ρ, and vice-versa, and are given by α := Tr[Πρ], β := Tr[(11 − Π )σ ]. The average error probability Pe is given by Pe = π0 α + π1 β = π0 Tr[Πρ] + π1 Tr[(11 − Π )σ ].
(4)
The Bayesian distinguishability problem consists in finding the Π that minimises Pe . A special case is the symmetric one where the prior probabilities π0 , π1 are equal. Before we proceed, let us first introduce some basic notations. Abusing terminology, we will use the term ‘positive’ for ‘positive semi-definite’ (denoted A ≥ 0). We employ the positive semi-definite ordering on the linear operators on H throughout, i.e. A ≥ B iff A − B ≥ 0. For each linear operator A ∈ B(H) the absolute value |A| is defined as |A| := (A∗ A)1/2 . The Jordan decomposition of a self-adjoint operator A is given by A = A+ − A− , where A+ := (|A| + A)/2,
A− := (|A| − A)/2
(5)
are the positive part and negative part of A, respectively. Both parts are positive by definition, and A+ A− = 0. There is a very useful variational characterisation of the trace of the positive part of a self-adjoint operator A: Tr[A+ ] = max{Tr[AX ] : 0 ≤ X ≤ 11}. X
(6)
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In other words, the maximum is taken over all positive contractive operators. Since the extremal points of the set of positive contractive operators are exactly the orthogonal projectors, we also have Tr[A+ ] = max{Tr[A P] : P ≥ 0, P = P 2 }. P
(7)
The maximiser on the right-hand side is the orthogonal projector onto the range of A+ . We can now easily prove the quantum version of the Neyman-Pearson Lemma. Lemma 1 (Quantum Neyman-Pearson). Let ρ and σ be density operators associated to hypotheses H0 and H1 , respectively. Let T be a fixed positive number. Consider the POVM with elements {11 − Π ∗ , Π ∗ }, where Π ∗ is the projector onto the range of (T σ − ρ)+ , and let α ∗ = Tr[Π ∗ ρ] and β ∗ = Tr[(11 − Π ∗ )σ ] be the associated errors. For any other POVM {11 − Π, Π }, with associated errors α = Tr[Πρ] and β = Tr[(11 − Π )σ ], we have α + Tβ ≥ α ∗ + Tβ ∗ = T − Tr[(T σ − ρ)+ ]. Thus if α ≤ α ∗ , then β ≥ β ∗ . Proof. By formulae (6) and (7), for all 0 ≤ Π ≤ 11 we have Tr[Π (T σ − ρ)] ≤ Tr(T σ − ρ)+ = Tr[Π ∗ (T σ − ρ)]. In terms of α, β, α ∗ , β ∗ , this reads T (1 − β) − α ≤ T (1 − β ∗ ) − α ∗ , which is equivalent to the statement of the lemma.
The upshot of this lemma is that the POVM {11 − Π ∗ , Π ∗ }, where Π ∗ is the projector on the range of (T σ − ρ)+ , is the optimal one when the goal is to minimise the quantity α + Tβ. In symmetric hypothesis testing the positive number T is taken to be the ratio π1 /π0 of the prior probabilities. We emphasize that we have started with the assumption that the physical systems in question are finite systems with an algebra of observables B(H), i.e. the algebra of linear operators on a finite-dimensional Hilbert space H. This is a purely quantum situation. In the general setting (of statistical mechanics) one associates to a finite physical system, classical or quantum, a finite-dimensional ∗-algebra A. Such an algebra has a block k k representation i=1 B(Hi ), i.e. it is a subalgebra of B(H), where H := i=1 Hi . If ∗ the Hilbert spaces Hi are one-dimensional for all i = 1, . . . , k, then A is -isomorphic to the commutative algebra of diagonal (k × k)-matrices. This covers the classical case. Now, in view of Lemma 1 it becomes clear that in the context of hypothesis testing there is no restriction assuming that the algebra of observables of the systems in question is B(H); indeed, the optimally discriminating projectors Π ∗ are always in the ∗-subalgebra generated by the two involved density operators ρ and σ . This implies that they are automatically elements of the algebra A characterising the physical systems. In particular, if the hypotheses correspond to mutually commuting density operators then the problem reduces to a classical one in the sense that the best test Π ∗ commutes with the density operators as well. Hence it coincides with the classical ML-test, although there are many more possible tests in B(H) than in the commutative subalgebra of observables of the classical subsystem. The basic problem we focus on in this paper is to identify how the error probability Pe behaves in the asymptotic limit, i.e. when one has to discriminate between the hypotheses H0 and H1 on the basis of a large number n of copies of the quantum systems. This means that we have to distinguish between the n-fold tensor product density operators ρ ⊗n and σ ⊗n by means of POVMs {11 − Πn , Πn } on H⊗n .
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We define the rate limit s R for any positive sequence (sn ) as 1 s R := lim − log sn , n→∞ n if the limit exists. Otherwise we have to deal with the lower and upper rate limits s R and s R , which are the limit inferior and the limit superior of the sequence (− n1 log sn ), respectively. In particular, we define the type-I error rate limit and the type-II error rate limit for a sequence Π := (Πn ) of quantum measurements (where, as mentioned, each orthogonal projection Πn corresponds to the alternative hypothesis) as 1 1 ⊗n α R (Π ) := lim − log αn = lim − log Tr[ρ Πn ] , (8) n→∞ n→∞ n n 1 1 (9) β R (Π ) := lim − log βn = lim − log Tr[σ ⊗n (11 − Πn )] , n→∞ n→∞ n n if the limits exist. Otherwise we consider the limit inferior and the limit superior α R (Π ) and α R (Π ), respectively. Similar definitions hold in the classical case. 3. Bayesian Quantum Hypothesis Testing: Quantum Chernoff Bound In this section we consider the Bayesian distinguishability problem. This means the goal is to minimise the average error probability Pe , which is defined in (4) and can be rewritten as Pe = π1 − Tr[Π (π1 σ − π0 ρ)]. By the Neyman-Pearson Lemma, the optimal test is given by the projector Π ∗ onto the range of (π1 σ − π0 ρ)+ , and the obtained minimal error probability is given by Pe∗ = π1 − Tr[(π1 σ − π0 ρ)+ ] = π1 − (π1 − π0 )/2 − Tr[|π1 σ − π0 ρ|/2] 1 = (1 − π1 σ − π0 ρ1 ) , 2 where A1 = Tr |A| is the trace norm. We will call Π ∗ the Holevo-Helstrom projector. Next, note that the optimal test to discriminate ρ and σ in the case of n copies enforces the use of joint measurements. However, the particular permutational symmetry of n-copy states guarantees that the optimal collective measurement can be implemented efficiently (with a polynomial-size circuit) [2], and hence that the minimum probability of error is achievable with a reasonable amount of resources. We need to consider the quantity ∗ Pe,n := (1 − π1 σ ⊗n − π0 ρ ⊗n 1 )/2.
(10)
∗ vanishes exponentially fast as n tends to infinity. The theorem below It turns out that Pe,n ∗ , i.e. the rate limit of P ∗ , provides the asymptotic value of the exponent − n1 log Pe,n e,n which turns out to be given by the quantum Chernoff distance. This is our main result.
Theorem 1. For any two states ρ and σ on a finite-dimensional Hilbert space, occurring ∗ , as defined by (10), with prior probabilities π0 and π1 , respectively, the rate limit of Pe,n exists and is equal to the quantum Chernoff distance ξ QC B , 1 ∗ 1−s s . (11) lim − log Pe,n = ξ QC B := − log inf Tr ρ σ n→∞ 0≤s≤1 n
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Because the product of two positive operators always has positive spectrum, the quantity Tr[ρ 1−s σ s ] is well defined (in the mathematical sense) and guaranteed to be real and non-negative for every 0 ≤ s ≤ 1. As should be, the expression for ξ QC B reduces to the classical Chernoff distance ξC B defined by (2) when ρ and σ commute.
3.1. Proof of Theorem 1: Optimality Part. In this section, we will show that the best discrimination is specified by the quantum Chernoff distance; that is, ξ QC B is an upper bound on 1 lim sup − log Pe,n n n→∞ for any sequence of tests (Πn ) and Pe,n := π1 − Tr [π1 σ ⊗n − π0 ρ ⊗n ]. The proof, which first appeared in [21], is essentially based on relating the quantum to the classical case by using a special mapping from a pair of d × d density matrices (ρ, σ ) to a pair of probability distributions ( p, q) on a set of cardinality d 2 . Let the spectral decompositions of ρ and σ be given by d
ρ=
λi |xi xi |, σ =
i=1
d
µ j |y j y j |,
j=1
where (|xi ) and (|y j ) are two orthonormal bases of eigenvectors and (λi ) and (µ j ) are the corresponding sets of eigenvalues of ρ and σ , respectively. Then we map these density operators to the d 2 -dimensional vectors pi, j = λi |xi |y j |2 , qi, j = µ j |xi |y j |2 ,
(12)
with 1 ≤ i, j ≤ d. This mapping preserves a number of important properties: Proposition 1. With pi, j and qi, j as defined in (12), and s ∈ R, Tr[ρ 1−s σ s ] =
s pi,1−s j qi, j ,
(13)
i, j
S(ρσ ) = H ( pq).
(14)
Here, S(ρσ ) is the quantum relative entropy defined as
S(ρσ ) :=
Tr[ρ(log ρ − log σ )], +∞,
if Supp ρ ≤ Supp σ otherwise,
(15)
where Supp ρ denotes the support projection of an operator ρ, and H ( pq) is the classical relative entropy, or Kullback-Leibler distance,
H ( pq) :=
i, j
+∞,
pi, j (log pi, j − log qi, j ),
if p q otherwise.
(16)
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Proof. The proof proceeds by direct calculation. For example: Tr[ρ 1−s σ s ] = λi1−s µsj |xi |y j |2 i, j
=
λi1−s µsj |xi |y j |2(1−s) |xi |y j |2s
i, j
=
s pi,1−s j qi, j .
i, j
A direct consequence of identity (13) is that p and q are normalised if ρ and σ are. Furthermore, tensor powers are preserved by the mapping; that is, if ρ and σ are mapped to p and q, then ρ ⊗n is mapped to p ⊗n and σ ⊗n to q ⊗n . Now define the classical and quantum average (Bayesian) error probabilities Pe,c and Pe,q as [π0 φ(i) pi + π1 (1 − φ(i))qi ], (17) Pe,c (φ, p, π0 , q, π1 ) := i
Pe,q (Π, ρ, π0 , σ, π1 ) := Tr[π0 Πρ + π1 (11 − Π )σ ],
(18)
where p, q are probability distributions, ρ, σ are density matrices, and π0 , π1 are the respective prior probabilities of the two hypotheses. Furthermore, φ is a non-negative test function 0 ≤ φ ≤ 1, and Π is a positive semi-definite contraction, 0 ≤ Π ≤ 11, so that {11 − Π, Π } forms a POVM. The main property of the mapping that allows to establish optimality of the quantum Chernoff distance is presented in the following proposition. Proposition 2. For all orthogonal projectors Π and all positive scalars η0 , η1 (not necessarily adding up to 1), and for p and q associated to ρ and σ by the mapping (12), Pe,q (Π, ρ, η0 , σ, η1 ) ≥
1 inf Pe,c (φ, p, η0 , q, η1 ), 2 φ
where the infimum is taken over all test functions 0 ≤ φ ≤ 1. Note that we have replaced the priors by general positive scalars; this will be useful later on, in proving the optimality of the Hoeffding bound. Proof. Since Π is a projector, one has Π = Π Π = j Π |y j y j |Π , where the second equality is obtained by inserting a resolution of the identity 11 = j |y j y j |. Likewise, 11 − Π is also a projector, and using another resolution of the identity, 11 = i |xi xi |, we similarly get 11 − Π = i (11 − Π )|xi xi |(11 − Π ). This yields Tr[Πρ] = λi Tr[Π |xi xi |] i
=
λi Tr[Π |y j y j |Π |xi xi |]
i, j
=
i, j
λi |xi |Π |y j |2 ,
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and, similarly, Tr[(11 − Π )σ ] =
µ j |xi |11 − Π |y j |2 .
i, j
Then the quantum error probability is given by Pe,q = η0 Tr[Πρ] + η1 Tr[(11 − Π )σ ] η0 λi |xi |Π |y j |2 + η1 µ j |xi |11 − Π |y j |2 . = i, j
The infimum of the classical error probability Pe,c is obtained when the test function φ equals the indicator function φ = χ{η1 q>η0 p} (corresponding to the maximum likelihood decision rule); hence, the value of this infimum is given by inf Pe,c = min(η0 pi, j , η1 qi, j ) φ
i, j
=
min(η0 λi , η1 µ j )|xi |y j |2 .
i, j
For a fixed choice of i, j, let a be the 2 × 2 non-negative diagonal matrix η0 λi 0 , a := 0 η1 µ j and let b be the 2-vector b := (xi |Π |y j , xi |11 − Π |y j ). The i, j-term in the sum for Pe,q can then be written as the inner product b|a|b. Similarly, the factor |xi |y j |2 occurring in the i, j-term in the sum for Pe,c can then be written as |b1 + b2 |2 . Now we note that b|b = b22 , while |b1 + b2 |2 ≤ b21 . For d-dimensional vectors, √ the inequality b2 ≥ b1 / d holds; in our case, d = 2. Together with the inequality a ≥ min(η0 λi , η1 µ j )112 this yields 1 b|a|b ≥ min(η0 λi , η1 µ j )b|b ≥ min(η0 λi , η1 µ j ) |b1 + b2 |2 . 2
(19)
Therefore, we obtain, for any i, j, η0 λi |xi |Π |y j |2 + η1 µ j |xi |11 − Π |y j |2 ≥
1 min(η0 λi , η1 µ j )|xi |y j |2 . 2
As this holds for any i, j, it holds for the sum over i, j, so that a lower bound for the quantum error probability is given by Pe,q ≥
1 1 min(η0 pi, j , η1 qi, j ) = inf Pe,c , 2 2 φ i, j
which proves the proposition.
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Using these properties of the mapping, the proof of optimality of the quantum Chernoff bound is easy. Proof of optimality of the quantum Chernoff bound. Let hypotheses H0 and H1 , with priors π0 and π1 , correspond to the product states ρ ⊗n and σ ⊗n . Using the mapping (12), these states are mapped to the probability distributions p ⊗n and q ⊗n . By Proposition 2, the quantum error probability is bounded from below as Pe,q (Πn , ρ ⊗n , π0 , σ ⊗n , π1 ) ≥
1 inf Pe,c (φn , p ⊗n , π0 , q ⊗n , π1 ). 2 φn
(20)
By the classical Chernoff bound, the rate limit of the right-hand side is given by s pi,1−s − log inf j qi, j 0≤s≤1
i, j
(provided the priors π0 , π1 are non-zero) and this is, therefore, an upper bound on the rate limit of the optimal quantum error probability. By Proposition 1 the latter expression is equal to − log inf 0≤s≤1 Tr[ρ 1−s σ s ], which is what we set out to prove.
In a similar way one can prove the converse part of the quantum Hoeffding bound by relating it to the classical problem in the sense of (12), as already noted by Nagaoka in [20]. This will be discussed in Sect. 5.4. 3.2. Proof of Theorem 1: Achievability Part. In this section, we prove the achievability of the bound, which is the statement that the error rate limit quantum∗ Chernoff limn→∞ − n1 log Pe,n is not only bounded above by, but is actually equal to the quantum Chernoff distance ξ QC B . This can directly be inferred from the following matrix inequality, which first made its appearance in [1]: Theorem 2. Let a and b be positive semi-definite operators, then for all 0 ≤ s ≤ 1, Tr[a s b1−s ] ≥ Tr[a + b − |a − b|]/2.
(21)
Note that inequality (21) is also interesting from a purely matrix analytic point of view, as it relates the trace norm to a multiplicative quantity in a highly nontrivial and very useful way. If we specialise this theorem to states, a = σ and b = ρ, with Tr ρ = Tr σ = 1, we obtain Q s + T ≥ 1,
0 ≤ s ≤ 1,
where Q s := Q s (ρ, σ ) := Tr[ρ 1−s σ s ] and T := T (ρ, σ ) := ρ − σ 1 /2 is the trace norm distance. Remark 1. Inequality (21) can be written in the form b1/2 | f s (∆a,b )b1/2 ≤ a − b1 , where ∆a,b is the relative modular operator acting on the matrix space endowed with the Hilbert-Schmidt inner product, and f s is the operator convex function fs (t) := 1+t −2t s , see [25]. The expression on the left-hand side is a quasi-entropy. This also implies some of the properties of Q s . For s = 1/2 inequality (21) becomes a 1/2 − b1/2 2 = Tr[(a 1/2 − b1/2 )2 ] ≤ a − b1 , which is known to hold also in infinite dimensions.
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Remark 2. The inequality Q s +T ≥ 1 is strongly sharp, which means that for any allowed value of T one can find ρ and σ that achieve equality. Indeed, take the commuting density operators ρ = |00| and σ = (1 − T )|00| + T |11|, then their trace norm distance is T , and Q s = 1 − T . Proof of achievability of the quantum Chernoff bound from Theorem 2. We will prove the inequality 1 ∗ lim inf − log Pe,n ≥ ξ QC B . (22) n→∞ n Put a = π1 σ ⊗n and b = π0 ρ ⊗n , so that the right-hand side of (21) turns into ∗ . (1 − π1 σ ⊗n − π0 ρ ⊗n 1 )/2 = Pe,n
The logarithm of the left-hand side of inequality (21) simplifies to log(π01−s π1s ) + n log Tr[ρ 1−s σ s ] . Upon dividing by n and taking the limit n → ∞, we obtain log Q s , independently of the priors π0 , π1 (as long as the priors are not degenerate, i.e. are different from 0 or 1). Then (22) follows from the fact that the inequality 1 ∗ lim inf − log Pe,n ≥ − log Q s n→∞ n holds for all s ∈ [0, 1] and we can replace the right-hand side by ξ QC B .
Proof of Theorem 2. The left-hand and right-hand sides of (21) look very disparate, but they can nevertheless be brought closer together by expressing a + b − |a − b| in terms of the positive part (a − b)+ . The inequality (21) is indeed equivalent to Tr[a − a s b1−s ] ≤ Tr[a − (a + b − |a − b|)/2] = Tr[(a − b + |a − b|)/2] = Tr[(a − b)+ ].
(23)
At this point we mention another equivalent formulation of this inequality, which will be used later in the proof of the achievability of the quantum Hoeffding bound. With Π the projector on the range of (a − b)+ , we can write: Tr[a s b1−s ] ≥ Tr[Π b + (11 − Π )a].
(24)
What we do next is strengthening the inequality (23) by replacing its left-hand side by an upper bound, and its right-hand side by a lower bound. Since, for any self-adjoint operator H , we have H ≤ H+ , we can write Tr[a − a s b1−s ] = Tr[a s (a 1−s − b1−s )] ≤ Tr[a s (a 1−s − b1−s )+ ] = Tr[a s Π (s) (a 1−s − b1−s )] = Tr[Π (s) (a − b1−s a s )], where Π (s) is the projector on the range of (a 1−s − b1−s )+ . Likewise, Tr[Π (s) (a − b)] ≤ Tr[(a − b)+ ],
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because Tr[(a − b)+ ] is the maximum of Tr[Π (a − b)] over all orthogonal projections Π . Inequality (21) would thus follow if, for that particular Π (s) , Tr[Π (s) (a − b1−s a s )] ≤ Tr Π (s) (a − b). The benefit of this reduction is obvious, as after simplification we get the much nicer statement Tr[Π (s) b1−s (a s − bs )] ≥ 0. Equally obvious, though, is the risk of this strengthening; it could very well be a false statement. Nevertheless, we show its correctness below. It is interesting to note the meaning here of this strengthening in the context of the optimal hypothesis test, i.e. when a = σ ⊗n and b = ρ ⊗n . While the Holevo-Helstrom projectors Πn∗ are optimal for every finite value of n, we can use other projectors that are ∗ suboptimal but reach optimality in the asymptotic sense. Here we are indeed using Π (s ) , ∗ ∗ the projector on the range of (a 1−s − b1−s )+ , where s ∗ is the minimiser of Tr[ρ 1−s σ s ] over [0, 1], if it exists. Otherwise we have to use the Holevo-Helstrom projector. In the next few steps we will further reduce the statement by reformulating the matrix powers in terms of simpler expressions. One can immediately absorb one of them into a and b via appropriate substitutions. As we certainly don’t want a power appearing in the definition of the projector Π (s) , we are led to apply the substitutions A = a 1−s ,
B = b1−s , t = s/(1 − s).
This yields a value of t between 0 and 1 only when 0 ≤ s ≤ 1/2. However, this is no restriction since the case 1/2 ≤ s ≤ 1 can be treated in a completely similar way after applying an additional substitution s → 1 − s. Inequality (21) is thus implied by the lemma below, which ends the proof of Theorem 2.
Lemma 2. For matrices A, B ≥ 0, a scalar 0 ≤ t ≤ 1, and denoting by P the projector on the range of (A − B)+ , the following inequality holds: Tr[P B(At − B t )] ≥ 0.
(25)
Proof. To deal with the t th matrix power, we use an integral representation (see, for example [3] (V.56)). For scalars a ≥ 0 and 0 ≤ t ≤ 1,
a sin(tπ ) +∞ t a = . d x x t−1 π a + x 0 For other values of t this integral does not converge. This integral can be extended to positive operators in the usual way:
sin(tπ ) +∞ At = d x x t−1 A(A + x 11)−1 . π 0 To deal with non-invertible A (arising when the states ρ and σ are not faithful), we define lim x→0 A(A + x 11)−1 = 11. The potential benefit of this integral representation is that statements about the integral might follow from statements about the integrand, which is a simpler quantity.
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Applying the integral representation to At and B t , we get
sin(tπ ) +∞ Tr[P B(At − B t )] = d x x t−1 Tr[P B(A(A + x)−1 − B(B + x)−1 )]. π 0 If the integrand is positive for all x > 0 (it is zero for x = 0), then the whole integral is positive. The lemma follows if indeed we have Tr[P B(A(A + x)−1 − B(B + x)−1 )] ≥ 0. As a further reduction, we note that a difference can be expressed as an integral of a derivative:
1 d f (b + (a − b)t). f (a) − f (b) = f (b + (a − b)) − f (b) = dt dt 0 Here, we will apply this to the expression A(A + x)−1 − B(B + x)−1 . Let ∆ = A − B. Then
1 d −1 −1 A(A + x) − B(B + x) = (B + t∆)(B + t∆ + x)−1 . dt dt 0 The potential benefit is again that the required statement might follow from a statement about the integrand, which is a simpler quantity provided one is able to calculate the derivative explicitly. In this case we are not dealing with a stronger statement, because the statement has to hold for the derivative anyway (when A is close to B). In the present case, we can indeed calculate the derivative: d (B + t∆)(B + t∆ + x)−1 = x (B + t∆ + x)−1 ∆ (B + t∆ + x)−1 . dt Therefore, Tr[P B(A(A + x)−1 − B(B + x)−1 )]
1 =x dt Tr[P B(B + t∆ + x)−1 ∆(B + t∆ + x)−1 ]. 0
Again, if the integrand is positive for 0 ≤ t ≤ 1, the whole integral is positive. Absorbing t in ∆ we need to show, with P the projector on ∆+ : Tr[P B V ∆ V ] ≥ 0,
where V := (B + ∆ + x)−1 ≥ 0.
After all these reductions, the statement is now in sufficiently simple form to allow the final attack. Since B = V −1 − x − ∆, we have BV ∆V = ∆(V − V ∆V ) − x V ∆V . Positivity of B implies V BV = V − V ∆V − x V 2 ≥ 0, thus V − V ∆V ≥ x V 2 . Furthermore, since P∆ = ∆+ ≥ 0, Tr[P BV ∆V ] = Tr[P(∆(V − V ∆V ) − x V ∆V )] = Tr[∆+ (V − V ∆V )] − x Tr[P V ∆V ] ≥ x(Tr[∆+ V 2 ] − Tr[P V ∆V ]). Because 11 ≥ P ≥ 0, ∆+ ≥ 0, and ∆+ ≥ ∆, Tr[∆+ V 2 ] = Tr[V ∆+ V ] ≥ Tr[P(V ∆+ V )] ≥ Tr[P(V ∆V )]. The conclusion is that, indeed, Tr[P BV ∆V ] ≥ 0, which proves the lemma.
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4. Properties of the Quantum Chernoff Distance In this section, we study the non-logarithmic variety Q of the quantum Chernoff distance ξ QC B , i.e. Q(ρ, σ ) := inf Tr[ρ 1−s σ s ], 0≤s≤1
(26)
where ρ, σ are density operators on a fixed finite-dimensional Hilbert space H. All properties of ξ QC B = − log Q can readily be derived from Q. It will turn out that ξ QC B is not a metric, since it violates the triangle inequality, but it has a lot of properties required of a distance measure on the set of density operators. 4.1. Relation to Fidelity and Trace Distance. The Uhlmann fidelity F between two states is defined as F(ρ, σ ) := ρ 1/2 σ 1/2 1 = Tr[(ρ 1/2 σρ 1/2 )1/2 ].
(27)
Here, the latter formula is best known, but the first one is easier and makes the symmetry under interchanging arguments readily apparent. The Uhlmann fidelity can be regarded as the quantum √ generalisation of the so-called Hellinger affinity [29] defined as B( p0 , p1 ) := i p0 (i) p1 (i), where p0 and p1 are classical distributions. It is an upper bound on Q, which can be shown as follows. By definition, for any fixed value of s ∈ [0, 1], Q s = Tr[ρ 1−s σ s ] is an upper bound on Q. In particular, this is true for s = 1/2. Furthermore, by replacing the trace with the trace norm · 1 , we get an even higher upper bound. Indeed, Q ≤ Tr[ρ 1/2 σ 1/2 ] = ρ 1/4 σ 1/2 ρ 1/4 1 ≤ ρ 1/2 σ 1/2 1 = F.
(28)
In the last inequality we have used the fact ([3], Prop. IX.1.1) that for any unitarily invariant norm |||AB||| ≤ |||B A||| if AB is normal. In particular, consider the trace norm, with A = ρ 1/4 σ 1/2 and B = ρ 1/4 . For a pair of density operators the trace distance T is defined by 1 T (ρ, σ ) := ρ − σ 1 . 2 Fuchs and van de Graaf [10] proved the following relation between F and T : (1 − F)2 ≤ T 2 ≤ 1 − F 2 .
(29)
Combining this with inequality (28) yields the upper bound Q 2 + T 2 ≤ 1.
(30)
Recall the relation 1 − T ≤ Q, following from Theorem 2. Then combining everything yields the chain of inequalities 1 − 1 − F 2 ≤ 1 − T ≤ Q ≤ F ≤ 1 − T 2. (31) There is a sharper lower bound on Q in terms of F, namely F 2 ≤ Q.
(32)
This bound is strongly sharp, as it becomes an equality when one of the states is pure [18]. Indeed, for ρ = |ψψ|, the minimum of the expression Tr[ρ 1−s σ s ] is obtained for s = 1 and reduces to ψ|σ |ψ, while F is given by the square root of this expression. We prove√(32) in Appendix A, where we also give an alternative proof of the upper bound Q ≤ 1 − T 2 . Both proofs go through in countably infinite dimensions.
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4.2. Range of Q. The maximum value Q can attain is 1, and this happens if and only if ρ = σ . This follows, for example, from the upper bound Q 2 + T 2 ≤ 1. The minimal value is 0, and this is only attained for pairs of orthogonal states, i.e. states such that Tr ρσ = 0. Consequently the range of the Chernoff distance is [0, ∞] and the infinite value is attained on orthogonal states; this has to be contrasted with the relative entropy, where infinite values are obtained whenever the states have a different support. 4.3. Triangle inequality. As already mentioned, on the set of pure states we have the identity Q = F 2 . The Uhlmann fidelity F does not obey the triangle inequality; however it can be transformed into a metric by going over to arccos F, while the Chernoff distance on pairs of pure states is equal to ξ QC B = − log Q = −2 log F. When considering the triangle inequality for ξ QC B , one should note first that in the classical case, the classical expression ξC B should be expected to behave like a squared metric, similarly to the relative entropy or Kullback-Leibler distance. Indeed consider two laws from the normal shift family N (µ, 1), µ ∈ R; then it is easy to see that ξC B = (µ1 − µ2 )2 /8. Thus ξC B defines a squared metric on the normal shift √ family, which will not satisfy the triangle inequality due to the square, but ξ will. CB √ However ξC B does not satisfy the triangle inequality in the general case. To see this, let Be(ε) be the Bernoulli law with parameter ε ∈ [0, 1]. Some computations show that ξC B (Be(1/2), Be(ε)) → log 2 and ξC B (Be(ε), Be(1 − ε)) → ∞ as ε → 0. As a consequence we have, for ε small enough, 1/2
1/2
1/2
ξC B (Be(ε), Be(1 − ε)) > ξC B (Be(ε), Be(1/2)) + ξC B (Be(1/2), Be(1 − ε)) contradicting the triangle inequality. 4.4. Convexity of Q s as a function of s. The target function s → Q s = Tr[ρ 1−s σ s ] in the variational formula defining Q has the useful property to be convex in s ∈ [0, 1] in the sense of Jensen’s inequality: Q ts1 +(1−t)s2 ≤ t Q s1 + (1 − t)Q s2 for all t ∈ [0, 1]. This implies that a local minimum is automatically the global one, which is an important benefit in actual calculations. Indeed, the function s → x 1−s y s is analytic for positive scalars x and y, and in this case its convexity may be easily confirmed by calculating the second derivative x 1−s y s (log y − log x)2 , which is non-negative. If one of the parameters, say x, happens to be 0, then s → x 1−s y s is a constant function equal to 0 for s ∈ [0, 1) and equal to 1 at s = 1. Hence, it is still convex, albeit discontinuous. Consider then a basis with respect to which the matrix representation of ρ is diagonal ρ = Diag(λ1 , λ2 , . . .). Let the matrix representation of σ (in that basis) be given by σ = U Diag(µ1 , µ2 , . . .)U ∗ , where U is a unitary matrix. Then Tr[ρ 1−s σ s ] =
λi1−s µsj |Ui j |2 .
i, j
As this is a sum with positive weights of convex terms λi1−s µsj , the sum itself is also convex in s.
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4.5. Joint concavity of Q in (ρ, σ ). By Lieb’s theorem [19], Tr[ρ 1−s σ s ] is jointly concave on pairs of density operators (ρ, σ ) for each fixed s ∈ R. Since Q is the point-wise minimum of Tr[ρ 1−s σ s ] over s ∈ [0, 1], it is itself jointly concave as well. Hence the related quantum Chernoff distance is jointly convex, just like the relative entropy. 4.6. Monotonicity under CPT maps. From the joint concavity one easily derives the following monotonicity property: for any completely positive trace preserving (CPT) map Φ on the C ∗ -algebra B(H) of linear operators, one has Q(Φ(ρ), Φ(σ )) ≥ Q(ρ, σ ).
(33)
We remark that this has been shown as a more general result in the framework of relative modular operators in [25]. Moreover, another proof appeared in [26]. We give an alternative proof omitting the notion of relative modular operators. First, we note that Q is invariant under unitary conjugations, i.e. Q(UρU ∗ , U σ U ∗ ) = Q(ρ, σ ). Secondly, Q is invariant under addition of an ancilla system: for any density operator τ on a finite-dimensional ancillary Hilbert space we have the identity Q(ρ ⊗ τ, σ ⊗ τ ) = Q(ρ, σ ). This is because Tr[(ρ ⊗ τ )1−s (σ ⊗ τ )s ] = Tr[ρ 1−s σ s ] Tr[τ ]. Exploiting the unitary representation of a CPT map, which is a special case of the Stinespring form, the monotonicity statement follows for general CPT maps if we can prove it for the partial trace map. As noted by Uhlmann [27,7], the partial trace map can be written as a convex combination of certain unitary conjugations. Monotonicity of Q under the partial trace then follows directly from its concavity and its unitary invariance. 4.7. Continuity. By the lower bound Q + T ≥ 1, the distance measures 1 − Q and ξ QC B are continuous in the sense that states that are close in trace distance are also close w.r.t. 1 − Q and w.r.t. ξ QC B . Indeed, we have 0 ≤ 1 − Q ≤ T and ξ QC B = − log Q ≤ − log(1 − T ) = T + O(T 2 ). 4.8. Relation of the Chernoff distance to the relative entropy. In the classical case there is a striking relation between the Chernoff distance ξC B and the relative entropy H (··). It takes its simplest version if the two involved discrete probability distributions p and q have coinciding supports since then s → log x p 1−s (x)q s (x) = log Q s is analytic over [0, 1] and its infimum, which defines the Chernoff distance, may be obtained simply by setting 0 = (log Q s ) = H ( ps p) − H ( ps q) (the prime denotes derivation w.r.t. s). Here ps := x
p 1−s q s p 1−s (x)q s (x)
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defines a parametric family of probability distributions interpolating between p and q as the parameter s varies between 0 and 1. In the literature, this family is called the Hellinger arc. It follows that the minimiser s ∗ ∈ [0, 1] is uniquely determined by the identity H ( ps ∗ q) = H ( ps ∗ p).
(34)
Furthermore, for any s ∈ [0, 1] we have: H ( ps p) = s(log Q s ) − log Q s ,
(35)
H ( ps q) = −(1 − s)(log Q s ) − log Q s ,
(36)
and similarly
This may be verified by direct calculation using essentially the identity log p 1−s q s = log p 1−s + log q s . For the minimiser s ∗ the formulas (35) and (36) reduce to H ( ps ∗ p) = H ( ps ∗ q) = ξC B ( p, q).
(37)
In the generic case of possibly different supports of p and q one has to modify (34) and (37) slightly, see [22]. It turns out that in the quantum setting the minimiser s ∗ ∈ [0, 1] of inf s∈[0,1] log Q s can be characterised by a generalized version of (34). However, the surely more remarkable relation (37) between the Chernoff distance ξC B and the relative entropy seems to have no quantum counterpart. We assume again that the involved density operators ρ and σ both have full support, i.e. are invertible. Then Q s = Tr(ρ 1−s σ s ) is an analytic function over [0, 1] and its local infimum over [0, 1], which is a global minimum due to convexity, can be found by differentiating Q s w.r.t. s: ∂ Tr[ρ 1−s σ s ] = − Tr[(log ρ) ρ 1−s σ s ] + Tr[ρ 1−s σ s log σ ] ∂s = − Tr[ρ 1−s σ s log ρ] + Tr[ρ 1−s σ s log σ ].
(38)
The infimum is therefore obtained for an s ∈ [0, 1] such that Tr[ρ 1−s σ s log ρ] = Tr[ρ 1−s σ s log σ ]. This is equivalent to the condition S(ρs ||ρ) = S(ρs ||σ ),
(39)
where S(ρ||σ ) denotes the quantum relative entropy defined by (15) and ρs is defined as ρ 1−s σ s . (40) ρs = Tr[ρ 1−s σ s ] Note that ρs , with s ∈ (0, 1), is not a density operator, because it is not even self-adjoint (except in the case of commuting ρ and σ ). Nevertheless, as it is basically the product of two positive operators, it has positive spectrum, and its entropy and the relative entropies used in (39) are well-defined. The value of s for which both relative entropies coincide is the minimiser in the variational expression (26) for Q.
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The family ρs , s ∈ [0, 1], can be considered as a quantum generalisation of the Hellinger arc interpolating between the quantum states ρ and σ , albeit out of the state space, in contrast to the classical case. When attempting to generalise relation (37) to the quantum setting one has to verify (35) or (36) with density operators ρ, σ replacing the probability distributions p, q. This would require the identity Tr ρs log ρ 1−s σ s = Tr ρs (log ρ 1−s + log σ s ) to be satisfied. However, this is not the case for arbitrary non-commutative density operators ρ, σ . Thus the second identity in (37) seems to be a classical special case only.
5. Asymmetric Quantum Hypothesis Testing: Quantum Hoeffding Bound In this section, we consider the applications of our techniques presented in Sect. 3 to the case of asymmetric quantum hypothesis testing. More precisely, we consider a quantum generalisation of the Hoeffding bound and of Stein’s Lemma.
5.1. The Classical Hoeffding Bound. The classical Hoeffding bound in information theory is due to Blahut [6] and Csiszár and Longo [9]. The corresponding ideas in statistics were first put forward in the paper [16] by W. Hoeffding, from which the bound got its name. Some authors prefer the more complete name of Hoeffding-BlahutCsiszár-Longo bound. In the following paragraph we review the basic results in Blahut’s terminology; at this point we have to mention that many different notational conventions are in use throughout the literature. Let p be the distribution associated with the null hypothesis, and q the one associated with the alternative hypothesis.1 Following [6], and for the purposes of this discussion, we initially assume that p and q are equivalent (mutually absolutely continuous) on a finite sample space. The Hoeffding bound gives the best exponential convergence rate of the type-I error under the constraint that the rate limit of the type-II error is bounded from below by a constant r , i.e. when the type-II error tends to 0 sufficiently fast. Blahut defines the error-exponent function e(r ), r ≥ 0, with respect to two probability densities p and q with coinciding supports, as a minimisation over probability densities x: e(r ) = inf {H (x p) : H (xq) ≤ r }, x
(41)
where H (··) is again the classical relative entropy defined in (16). This minimisation is a convex minimisation, since the target function is convex in x, and the feasible set, defined by the constraint H (xq) ≤ r , is a convex set. Pictorially speaking, the optimal x is the point in the feasible set that is closest (as measured by the relative entropy) to p. If p itself is in the feasible set (i.e. if H ( pq) ≤ r ), then the optimal x is p, and e(r ) = 0. Otherwise, the optimal x is on the boundary of the feasible set, in the sense that H (xq) = r , and e(r ) > 0. Obviously, if r = 0, the feasible set is the singleton {q}, and e(r ) = H (q p). The error-exponent function is thus a non-increasing, convex function of r ≥ 0, with the properties that e(0) = H (q p) and e(H ( pq)) = 0. It can be expressed in a 1 In [6], the null hypothesis corresponds to H , with distribution q , and the alternative hypothesis to H , 2 2 1 with distribution q1 .
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0.9
H(q||p)
0.8 0.7
e(r)
0.6 0.5 0.4 0.3 0.2
H(p||q)
0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
r Fig. 1. (Color online) Example plot of the error-exponent function e(r ), Eq. (42), for the distributions p = (0.95, 0.05) and q = (0.5, 0.5). The thick (red) line is the graph of e(r ), while the thin (blue) lines are instances of the linear function (−r s − log k qks pk1−s )/(1 − s) for various values of s, of which e(r ) is the point-wise maximum. For the chosen p and q, the value of H ( pq) = 0.49463 and the value of H (q p) = 0.83037
computationally more convenient format as −r s − log k qks pk1−s . 1−s 0≤s< 1
e(r ) = sup
(42)
An example is shown in Fig. 1. Let φ = (φn ) be a sequence of test functions. Recall the notations α R (φ) and β R (φ) introduced in Sect. 2 for the rate limits (if they exist) of the corresponding type-I and type-II errors, respectively: 1 1 α R (φ) = lim − log αn (φ), β R (φ) = lim − log βn (φ). n→∞ n n→∞ n Then the classical HBCL Theorem can be stated as follows. Theorem 3 (HBCL). Assume that p, q are mutually absolutely continuous. Then for each r > 0 there exists a sequence φ of test functions φn such that the rate limits of the type-II and type-I errors behave like β R (φ) ≥ r and α R (φ) = e(r ). Moreover, for any sequence φ such that α R (φ) and β R (φ) both exist, the relation β R (φ) > r implies α R (φ) ≤ e(r ). We remark that for sequences φ of test functions φn for which the rate limits α R (φ) or β R (φ) do not exist, the result still applies to subsequences (φn k ) along which both error rate limits exist. The is thus a statement about all second part of the HBCL theorem accumulation points of − n1 log αn (φ), − n1 log βn (φ) for an arbitrary test sequence φ. Referring to Fig. 1, the claim of this theorem is that for any sequence of test functions φ the point (β R (φ), α R (φ)) cannot be above the graph of e(r ) over r > 0 and for any point on the graph over r ≥ 0 one can find a sequence φ. Since β R (φ) = 0 may correspond to
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the case where β(φn ) vanishes subexponentially slowly as well as converges to a positive value, a rate limit of type-I error α R (φ) larger than e(0) = H (q p) is achievable. The case β R (φ) > r ≥ H ( pq), where e(r ) = 0, can be shown to correspond to α(φn ) converging to 1, rather than to 0. (This is basically the content of the so-called ‘Strong Converse’.) In the case β R (φ) = H ( pq) a convergence of α(φn ) to 0 is achievable, albeit only subexponantially slowly (this is due to Stein’s Lemma.) Note that in order to obtain a bound on β R under a constrained α R one just has to interchange p and q in the theorem. 5.2. Nonequivalent hypotheses. The Chernoff and Hoeffding bounds have typically been treated in the literature under a restrictive assumption that hypotheses p, q are mutually absolutely continuous (equivalent), cf., e.g., Blahut [6]. As a prerequisite for a quantum generalisation, unless one wants to limit oneself to faithful states, one has to understand the classical Hoeffding bound for nonequivalent hypotheses. For the Chernoff bound, a corresponding discussion can be found in [22] without restrictions on the underlying sample space. Here we limit ourselves to finite sample spaces, thereby excluding infinite relative entropies for equivalent measures p, q. For probability measures p, q on a finite sample space Ω, let D0 be the support of p, D1 be the support of q and B = D0 ∩ D1 . Let ψ0 = p (B), ψ1 = q (B) and note that ψ0 > 0, ψ1 > 0 unless the measures p, q are orthogonal (which we exclude for triviality). Define conditional measures given the set B: p˜ (·) = p (·|B), q˜ (·) = q (·|B). Note that p, ˜ q˜ are equivalent measures; we may have p˜ = q. ˜ We consider hypothesis testing for a pair of product measures p ⊗n , q ⊗n . Recall that a (nonrandomised) test is a mapping φn : Ω n → {0, 1}. In our setting, only observations in either D0n or D1n can occur, so we will modify the sample space to be D0n ∪ D1n . We will then establish the relation of tests φn in the original problem p ⊗n vs. q ⊗n to tests in the ‘conditional’ problem p˜ ⊗n vs. q˜ ⊗n , i.e. to tests φ˜ n : B n → {0, 1}. Call a test φn null admissible if it takes value 0 on D0n \B n and value 1 on D1n \B n . These tests correspond to the notion that if a point in the sample space Ω n is not in B n , then it identifies the hypothesis errorfree (either p or q). We need only consider null admissible tests; for any test there is a null admissible test with equal or smaller error probabilities αn , βn . The restriction φn |B n gives a test on B n , i.e. in the conditional problem. Lemma 3. There is a one-to-one correspondence between null admissible tests φn in the original problem p ⊗n vs. q ⊗n and tests φ˜ n in the conditional problem p˜ ⊗n vs. q˜ ⊗n , given by φ˜ n = φn |B n . The errror probabilities satisfy αn (φ) = ψ0n αn φ˜ , βn (φ) = ψ1n βn φ˜ , where ψ0 = p(B) and ψ1 = q(B). Proof. The first claim is obvious, if one takes into account that we took all tests in the original problem to be mappings φn : D0n ∪ D1n → {0, 1}. For the relation of error probabilities, note that p ⊗n (A) = ψ0n p˜ ⊗n (A ∩ B n ), A ⊂ D0n ∪ D1n and therefore
αn (φ) = φn dp ⊗n = φn dp ⊗n (by null admissibility) Bn
n ⊗n n φ˜ n d p˜ ⊗n = ψ0n αn φ˜ = ψ0 φn d p˜ = ψ0 Bn
and analogously for βn (φ) .
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This result already allows to state the general Hoeffding bound in terms of the error-exponent function for the conditional problem −r s − log k q˜ks p˜ k1−s . e(r ˜ ) = sup 1−s 0≤s< 1 Indeed, rate limits α R (φ) and β R (φ) for a null admissible test sequence φ exist if and ˜ and only if they exist for the corresponding test sequence φ, α R (φ) = − log ψ0 + α R φ˜ , β R (φ) = − log ψ1 + β R φ˜ . (43) Proposition 3. Let p, q be arbitrary probability measures on a finite sample space. (i) (achievability) For each r ≥ − log ψ1 there exists a sequence φ of test functions φn such that the rate limits of the type-II and type-I errors behave like β R (φ) ≥ r and ˜ + log ψ1 ). For the case 0 ≤ r ≤ − log ψ1 , there is a sequence α R (φ) = − log ψ0 + e(r φ of test functions φn obeying −n −1 log βn (φ) = − log ψ1 and αn (φ) = 0 for every n. (ii) (optimality) Consider any sequence φ such that α R (φ) and β R (φ) both exist. If r ≥ − log ψ1 then the relation β R (φ) > r implies α R (φ) ≤ − log ψ0 + e(r ˜ + log ψ1 ). Note that in (ii) the omission of the case 0 ≤ r ≤ − log ψ1 means that there is no upper bound on α R (φ), as shown by the achievability part (α R (φ) has to be set equal to ∞ for a test of vanishing error probability αn ). Proof. (i) Assume r ≥ − log ψ1 and take a test sequence φ˜ n in the conditional problem ˜ ≥ r +log ψ1 and α R (φ) ˜ = e(r p˜ ⊗n vs. q˜ ⊗n such that β R (φ) ˜ +log ψ1 ), which exists according to the HBCL theorem since p, ˜ q˜ are mutually absolutely continuous. According to Lemma 3, the corresponding null admissible test φn satisfies (43) and hence β R (φ) ≥ r and α R (φ) = − log ψ0 + e(r ˜ + log ψ1 ). Furthermore, consider the test φ˜ n ≡ 0 in p˜ ⊗n ⊗n vs. q˜ . This has αn (φ˜ n ) = 0 and βn (φ˜ n ) = 1, hence the corresponding null admissible test φn has αn (φn ) = 0 and βn (φn ) = ψ1n . (ii) Using a reduction to the conditional problem p˜ ⊗n vs. q˜ ⊗n similar to the one above, the optimality part also follows immediately from the HBCL theorem.
Remark. Consider the dual of the test used in the second part of (i), i.e. the null admissible extension of the test φ˜ n ≡ 1. This one obviously has αn (φ) = ψ0n and βn (φ) = 0. It can be used for achievability for large r , i.e. it has β R (φ) = ∞ and α R (φ) = − log ψ0 . It is possible to obtain a closed form expression for the Hoeffding bound, using the error-exponent function defined for r ≥ 0 exactly as in (42), for the case of nonequivalent p, q. The difference is that we now have to admit a value +∞ for certain arguments. Lemma 4. For general p, q, the error-exponent function e(r ) satisfies
− log ψ0 + e(r ˜ + log ψ1 ), for r ≥ − log ψ1 e(r ) = ∞, for 0 ≤ r < − log ψ1 . Remark. For two distinct p, q it is possible that p˜ = q. ˜ In that case e(r ˜ ) = 0 for r ≥ 0. It follows that e(r ) = ∞ for r < −logψ1 and e(r ) = − log ψ0 for r ≥ − log ψ1 . This case will be relevant in the quantum setting when the hypotheses will be represented by two non-orthogonal pure quantum states.
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Proof. Assume r ≥ − log ψ1 and set es (r ) = where Q s =
k
pk1−s qks . Let Q˜ s =
−r s − log Q s , 1−s
k
p˜ k1−s q˜ks and note Q s = ψ01−s ψ1s Q˜ s . Hence
−r s − (1 − s) log ψ0 − s log ψ1 − log Q˜ s 1−s −(r + log ψ1 )s − log Q˜ s = − log ψ0 + e˜s (r + log ψ1 ), = − log ψ0 + 1−s
es (r ) =
where e˜s is the analogue of the function es (r ) with Q s replaced by Q˜ s . Since e(r ) = sup0≤s< 1 es (r ) and the analogue is true for e˜s and e, ˜ the claim follows in the case r ≥ − log ψ1 . Assume now 0 ≤ r < − log ψ1 and ψ1 < 1, i.e. − log ψ1 > 0. Clearly we have Q s → ψ1 as s 1, hence −r s − log Q s → −r − log ψ1 > 0 as s 1. Hence lims1 es (r ) = ∞, and since e(r ) = sup0≤s< 1 es (r ), we also have e(r ) = ∞.
In conjunction with Proposition 3 we obtain a closed form description of the Hoeffding bound for possibly nonequivalent measures p, q, in terms of the original error-exponent function e(r ). Theorem 4. Let p, q be arbitrary probability measures on a finite sample space. Then the statement of the HBCL Theorem (Theorem 3) is true, where the error-exponent function defined in (42) obeys e(r ) = ∞ for 0 ≤ r < − log ψ1 if ψ1 < 1. We noted already that for e(r ) = ∞, the bound on α R (φ) is achievable in the sense that a test exists having exactly αn (φ) = 0 for all n. Using the properties of the rate function e˜ pertaining to equivalent measures p, ˜ q, ˜ as illustrated in Fig. 1, and the representation of Lemma 4 we obtain the following description of the general rate exponent function. In the interval [0, − log ψ1 ) it is infinity. At r = − log ψ1 it takes value e(r ) = − log ψ0 + H (q ˜ p) ˜ = H (q ˜ p). For r ≥ − log ψ1 it is convex and non-increasing. More precisely, over the interval [− log ψ1 , − log ψ1 + H ( p ˜ q) ˜ = H ( pq)] ˜ e(r ) is convex (even strictly convex) and monotone decreasing. Over the interval [H ( pq), ˜ ∞) it is constant with value − log ψ0 . A visual impression can be obtained by imagining the origin in Fig. 1 shifted to the point (− log ψ1 , − log ψ0 ). This picture will explicitly appear in Fig. 2 below, in a situation further generalized to two quantum states with different supports. 5.3. Quantum Hoeffding Bound. In the quantum setting the error-exponent function e(r ) has to be replaced by a function e Q : R+0 −→ [0, ∞] given by −r s − log Tr σ s ρ 1−s . 1−s 0≤s<1
e Q (r ) := sup
(44)
In view of Proposition 1, e Q (r ) coincides with the error-exponent function e(r ) for the pair of probability distributions ( p, q) associated with (ρ, σ ) via relation (12). Therefore, we can use Lemma 4 to describe properties of the function e Q (r ), or the remarks after Theorem 4.
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Recall that for a pair ( p, q), we defined a related pair of probability distributions ( p, ˜ q) ˜ by conditioning p and q, respectively, on the intersection B = D0 ∩ D1 of the two support sets D0 and D1 , and also ψ0 = p(B), ψ1 = q(B). In the present context, in accordance with (12) we have D0 = {(i, j) : 1 ≤ i, j ≤ d, λi > 0} , D1 = (i, j) : 1 ≤ i, j ≤ d, µ j > 0 . Let, as before, e(r ˜ ) be the error-exponent function pertaining to the pair ( p, ˜ q) ˜ according to (42). Then the quantum error-exponent function e Q (r ) for the hypotheses ρ, σ may be represented simply by
e Q (r ) = e(r ) =
− log ψ0 + e(r ˜ + log ψ1 ), ∞,
for r ≥ − log ψ1 for 0 ≤ r < − log ψ1 .
(45)
It obtains its characteristic properties function being convex and from the classical ˜ with e(− log ψ1 ) = H (q ˜ p), monotone decreasing in the interval − log ψ1 , H ( pq) and constant with value − log ψ0 in the interval [H ( pq), ˜ ∞). Lemma 5. Let supp ρ, supp σ be the support projections associated with ρ, σ . Then the critical points and extremal values of e Q (r ) may be expressed in a more direct way in terms of the density operators: ψ0 = Tr ρ supp σ , ψ1 = Tr σ supp ρ and H ( pq) ˜ = Sσ (ρσ ) H (q ˜ p) = Sρ (σ ρ), where the entropy type quantities on the right-hand side are defined as ρ ρ log − log σ supp σ , ψ0 ψ0 σ σ log − log ρ supp ρ . Sρ (σ ρ) := Tr ψ1 ψ1
Sσ (ρσ ) := Tr
Proof. Note that for B = D0 ∩ D1 we have ψ0 =
2 2 λi xi |y j = λi sgn(µ j ) xi |y j
(i, j)∈B
=
i, j
=
i, j
= Tr
i, j
2 2 λi xi |sgn(µ j )y j = λi xi | (supp σ ) y j i, j
2 λi (supp σ ) xi |y j = λi (supp σ ) xi 2
i
i
λi |(supp σ ) xi (supp σ ) xi | = Tr ρ supp σ
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and analogously for ψ1 . Furthermore 2 1 p˜ i, j λi H ( pq) ˜ = p˜ i, j log = λi xi |y j log qi, j ψ0 µ j ψ0 (i, j)∈B
(i, j)∈B
2 λi 2 λi λi = sgn(µ j ) xi |y j log − sgn(µ j ) xi |y j log µ j ψ0 ψ0 ψ0 i, j i, j ρ ρ ρ log supp σ − Tr = Tr (log σ ) supp σ , ψ0 ψ0 ψ0 where the third equality is analogous to the calculation in the proof of Proposition 1.
To shed some light on the entropy type quantity Sσ (ρσ ), note that it may be rewritten as a difference of usual (Umegaki’s) relative entropies: ρ ρ ρ . supp σ σ − S supp σ Sσ (ρσ ) = S ψ0 ψ0 ψ0 This may be verified by direct calculations similar to those in the proof of Lemma 5. The linear operator ψρ0 supp σ is a kind of conditional expectation of ρ. While it is not self-adjoint, the relative entropies on the right-hand side are well defined (in a mathematical sense) and real: first, the entropy of ψρ0 supp σ is defined in terms of its spectrum, which is positive and normalised to 1, hence giving a real, positive entropy, and second, Tr[ρ supp σ log(ρ)] can be written as Tr[supp σρ log ρ supp σ ], from which it is evident that this term is also real. It is easily seen from the above formula that Sσ (ρσ ) coincides with S(ρσ ) if σ is a faithful state, or more generally if supp ρ ≤ supp σ . Otherwise S(ρσ ) = ∞, while Sσ (ρσ ) is finite. Note also that Sρ (σ ρ) ≥ − log ψ0 and equality holds if and only if it holds in Sσ (ρσ ) ≥ − log ψ1 . This immediately follows from Sρ (σ ρ) + log ψ0 = H ( p ˜ q), ˜ which is seen from Lemma 5. This happens in particular if both ρ and σ are pure states. In this case there is only one pair (i, j) where both λi > 0 and µ j > 0, hence the set B consists of one element only. In this case we must have p˜ = q, ˜ hence H ( p ˜ q) ˜ = H (q ˜ p) ˜ = 0. The general shape of the quantum error-exponent function e Q (r ) is represented in Fig. 2. If both ρ and σ are pure states then the shape degenerates to ‘rectangular’ form (e Q (r ) = ∞ or e Q (r ) = − log ψ1 ). A quantum generalisation of the HBCL Theorem then reads as follows. Theorem 5 (Quantum HBCL). For each r > 0 there exists a sequence Π of test projections Πn on H⊗n for which the rate limits of type-I and type-II errors behave like α R (Π ) = e Q (r ) and β R (Π ) ≥ r , respectively. Moreover, for any sequence Π such that α R (Π ) and β R (Π ) both exist, the relation β R (Π ) > r implies α R (Π ) ≤ e Q (r ). The statement of the quantum HBCL Theorem is that for every sequence Π (for which both error rate limits exist) the point (β R (Π ), α R (Π )) lies on or below the curve e Q (r ) over (0, ∞], and for every point on the curve over the closed interval [0, ∞] there is a sequence Π achieving it. We remark that, just like (37), the relationship (41) seems to have no general quantum counterpart, even when both states are faithful. In other words, there is no known subset of linear operators τ with positive spectrum such that e Q (r ) = inf τ {S(τ ρ) : S(τ σ ) ≤ r }. To prove the quantum Hoeffding bound, the following lemmas are needed.
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Fig. 2. Example plot of the quantum error-exponent function e Q (r ) in the general case
Lemma 6. For scalars x, y > 0, bounds on log(x + y) are given by max(log x, log y) ≤ log(x + y) ≤ max(log x, log y) + log 2.
(46)
Proof. For the first inequality, put x = ea and y = eb , and note log(ea + eb ) = a + log(1 + eb−a ) ≥ a + max(0, b − a) = max(a, b). The second inequality follows directly from the fact that the logarithm increases monotonically, so that log((x + y)/2) ≤ log max(x, y).
A direct consequence of this lemma is Lemma 7. For two scalar sequences xn , yn > 0 with rate limits x R and y R , the rate limit of xn + yn is given by 1 lim − log(xn + yn ) = min(x R , y R ). n
n→∞
(47)
5.4. Proof of Optimality of the Quantum Hoeffding Bound. Again we use the mapping from the pair (ρ, σ ) to the pair ( p, q), so that, by Proposition 1, e(r ) = e Q (r ). From Proposition 2 we have that for any sequence Π of orthogonal projections Πn and for any real value of the scalar x, for all n ∈ N one as α(Πn ) + e−nx β(Πn ) ≥
1 α(φn ) + e−nx β(φn ) , 2
where φn are classical test functions corresponding to the maximum likelihood decision rule, cf. the proof of Proposition 2. Recall that the type-I and type-II errors are defined as α(φn ) = i pin φn (i) and β(φn ) = i qin (1 − φn (i)). On taking the rate limit on the left side, this gives 1 1 lim − log α(Πn ) + e−nx β(Πn ) ≤ lim inf − log α(φn ) + e−nx β(φn ) . n→∞ n n
n→∞
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By possibly taking a subsequence, we can ensure that the rate limits α R (φ), β(φn ) also exist. By Lemma 7, the above simplifies to min(α R (Π ), x + β R (Π )) ≤ min(α R (φ), x + β R (φ)).
(48)
Assume now that β R (φ) ≤ − log ψ1 . Then, by selecting x < 0 and |x| sufficiently large, we obtain x + β R (Π ) ≤ x + β R (φ), and hence β R (Π ) ≤ − log ψ1 . Since e Q (r ) = ∞ for r < β R (Π ) ≤ − log ψ1 according to the discussion above Lemma 5, the claim α R (Π ) ≤ e Q (r ) holds trivially. Henceforth we assume that β R (φ) > − log ψ1 . From the classical HBCL Theorem (more precisely, from Theorem 4), the right-hand side of (48) is bounded above by min(e(r ), x + β R (φ)), for any r with − log ψ1 ≤ r < β R (φ). Note that e(r ) is continuous for r ≥ − log ψ1 (since it is monotonely nonincreasing and convex). By letting r β R (φ) we obtain an upper bound min(e(r ), x + r ) with r ≥ − log ψ1 . We can now prove the optimality part of the quantum HBCL Theorem, using only this upper bound plus the fact that e(r ) is monotonously decreasing. The upper bound min(e(r ), x + r ) holds for some particular value r . We will find a further upper bound by maximizing over r ≥ − log ψ1 . For this we have to distinguish two cases, depending on the value of x. a) At r = − log ψ1 we have e(r ) > x +r . Since e(r ) is decreasing in r and continuous, and x + r is increasing, the maximum of min(e(r ), x + r ) is obtained when e(r ) = x + r . Let r ∗ (x) > − log ψ1 be the solution of x +r = e(r ). We now have that for any sequence of quantum measurements Π and for any real value of the scalar x, min(α R (Π ), x + β R (Π )) ≤ x + r ∗ (x) = e(r ∗ (x)). b) At r = − log ψ1 we have e(r ) ≤ x + r . Again by the properties of e(r ) and x + r , the maximum of min(e(r ), x +r ) is e(r ∗ ), attained for r ∗ (x) = − log ψ1 . We then obtain the upper bound min(α R (Π ), x + β R (Π )) ≤ e(r ∗ (x)). Now set x = α R (Π ) − β R (Π ), then both inequalities above yield α R (Π ) ≤ e(r ∗ ). Assume r < β R (Π ); we intend to show that this implies α R (Π ) ≤ e(r ). Indeed, in both cases a) and b) r ∗ is such that e(r ∗ ) ≤ x + r ∗ = α R (Π ) − β R (Π ) + r ∗ < α R (Π ) − r + r ∗ , hence r ∗ − r ≥ e(r ∗ ) − α R (Π ) ≥ 0. Therefore, from the monotonicity of the errorexponent function e(r ∗ ) ≤ e(r ) follows and we finally obtain α R (Π ) ≤ e(r ) = e Q (r ).
5.5. Proof of achievability of the quantum Hoeffding bound. The proof of achievability is mainly due to Hayashi [12], who used inequality (24), which is obtained as a byproduct of the proof of Theorem 2. However, we modify it avoiding any implicit assumption that the involved quantum states are faithful; hence we prove Theorem 5 in full generality, which includes for example the case of two non-orthogonal pure states. Let us fix an arbitrary s ∈ (0, 1), and set a = e−nx σ ⊗n , b = ρ ⊗n ,
(49) (50)
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where the value of x will be chosen in due course. Consider the sequence of POVMs {(11 − Πn , Πn )} with Πn the projector on the range of (a − b)+ ; again element 11 − Πn is assigned to the null hypothesis ρ ⊗n , and element Πn is assigned to the alternative hypothesis σ ⊗n . We will show that this POVM asymptotically attains the Hoeffding bound. Recall that inequality (24) states Tr[a s b1−s ] ≥ Tr[Π b + (11 − Π )a]. By positivity of Tr[Π b] and Tr[(11 − Π )a], this implies the two inequalities Tr[Π b], Tr[(11 − Π )a] ≤ Tr[a s b1−s ]. These yield the following upper bounds on the α and β errors of the chosen POVM (recall Q s = Tr[ρ 1−s σ s ]): βn (Πn ) = = ≤ = = αn (Πn ) = = ≤ = =
Tr[(11 − Πn )σ ⊗n ] enx Tr[(11 − Πn )a] enx Tr[a s b1−s ] enx(1−s) Q ns exp[n(x(1 − s) + log Q s )], Tr[Πn ρ ⊗n ] Tr[Πn b] Tr[a s b1−s ] e−nxs Q ns exp[n(−xs + log Q s )].
(51)
(52)
Choosing x such that x(1 − s) + log Q s = −r then yields, from (51), βn (Πn ) ≤ exp(−nr ), and from (52),
r + log Q s − log Q s αn (Πn ) ≤ exp −n −s 1−s −r s − log Q s = exp −n 1−s ≤ exp −ne Q (r ) ,
where in the last inequality we have used the fact that the parameter s was arbitrarily chosen from (0, 1). Thus, for the rate limits we get β R ≥ r, α R ≥ e Q (r ). The optimality, proven in the previous subsection, states that α R ≤ e Q (r ) if β R = r . Furthermore, since e Q (r ) is a non-increasing function, α R ≤ e Q (r ) if β R > r . This implies that for the chosen sequence of POVMs β R = r, α R = e Q (r ) must hold, which proves that the Hoeffding bound is indeed attained.
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5.6. Quantum Stein’s Lemma and quantum version of Sanov’s Theorem. The quantum generalisation of Stein’s Lemma deals with the asymptotics of the error quantity βn∗ () := inf {βn (Πn ) : αn (Πn ) ≤ }, Πn
(53)
for fixed 0 < < 1. Here, the infimum is taken over all positive semi-definite contractions Πn on H⊗n . Quantum Stein’s Lemma states that the rate limit β R∗ (ε) of the sequence (βn∗ ()) exists and is equal to S(ρσ ), independently of . It was first obtained by Hiai and Petz [15]. Its optimality part was then strengthened by Ogawa and Nagaoka in [24]. Here we use the quantum HBCL Theorem to prove that the relative entropy S(ρσ ) is an achievable error rate limit and deduce optimality of this bound from Proposition 1 in [5]. Proof of the quantum Stein’s Lemma. We need to show that there is a sequence Π with α(Πn ) ≤ achieving β R (Π ) = S(ρσ ). Let η > 0 be small and set r = S(ρ||σ ) − η. Achievability of the quantum Hoeffding bound means that a sequence Π exists for which β R ≥ r and α R = e Q (r ). Since e Q (r ) > 0 for all r < S(ρσ ) and η > 0, the sequence αn converges to 0. Thus, from a certain value of n onwards, αn will get lower than any value > 0 chosen beforehand. This means that Π is a feasible sequence in (53) for n large enough, exhibiting β R () ≥ r = S(ρσ ) − η. As this holds for any η > 0, we find that β R∗ () ≥ S(ρσ ). With β R∗ () ≥ S(ρσ ) the two hypotheses associated to the pair of density operators (ρ, σ ) satisfy the HP-condition in the terminology of the paper [5]. Thus Proposition 1 in [5] implies β R∗ () = S(ρσ ).
We remark that in [5] the HP-condition was introduced for (ordered) pairs (Ψ, Φ) of arbitrary correlated states on quantum spin chains, while in the present paper only density operators of the tensor-product form ρ ⊗n have been considered. These correspond to the special case of shift-invariant product states on the infinite spin chain (quantum i.i.d. states). A pair (Ψ, Φ) is said to satisfy the HP-condition if the relative entropy rate s(Ψ Φ) exists and is a lower bound on the lower rate limit β ∗R (ε) for all ε ∈ (0, 1). Specifically to our setting (the i.i.d. case), Theorem 1 in [5] states that the achievability part in quantum Stein’s Lemma (the HP-condition) is equivalent to a quantum version of Sanov’s Theorem, which has been presented in [4] and which is a priori a result extending quantum Stein’s Lemma in the following way: Let the null hypothesis H0 correspond to a family Γ of density operators on H instead of a single density operator ρ. Let the alternative hypothesis H1 be still represented by a fixed density operator σ . Then there exists a sequence Π of orthogonal projections Πn on H⊗n , respectively, such that for all ρ ∈ Γ the corresponding type-I error vanishes asymptotically, i.e. lim Tr[ρ ⊗n Πn ] = 0,
n→∞
(54)
while the type-II error rate limit β R (Π ) is equal to the relative entropy distance from Γ to σ : S(Γ σ ) := inf S(ρσ ). ρ∈Γ
Moreover S(Γ σ ) is the upper bound on type-II error (upper) rate limit, for any sequence Π of POVMs satisfying the constraint (54). With the above reasoning we obtain the statement of quantum Sanov’s Theorem from the quantum HBCL Theorem as well.
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Acknowledgements. We thank various institutions for their hospitality: the Max Planck Institute for Quantum Optics (FV, KA), the Erwin Schrödinger Institute in Vienna (FV, KA, AS), and the Physics Department of the National University of Singapore (KA). KA was supported by The Leverhulme Trust (grant F/07 058/U), by the QIP-IRC (www.qipirc.org) supported by EPSRC (GR/S82176/0), by EU Integrated Project QAP, and by the Institute of Mathematical Sciences, Imperial College London. MN has been supported by NSF under grant DMS-03-06497. The authors are also grateful to the anonymous referee regarding Remark 1 after Theorem 2.
A Proofs of Bounds on Q Inequality (32) stated in terms of general positive operators is Theorem 6. For positive operators A and B, and 0 ≤ s ≤ 1, A1/2 B 1/2 1 ≤ (Tr[As B (1−s) ])1/2 (Tr[A])(1−s)/2 (Tr[B])s/2 .
(55)
Specialising to states, A = σ and B = ρ, the left-hand side is just F(ρ, σ ), while the right-hand side is equal to Q s (ρ, σ )1/2 . Proof. We rewrite A1/2 B 1/2 as a product of three factors A1/2 B 1/2 = A(1−s)/2 (As/2 B (1−s)/2 )B s/2 , apply Hölder’s inequality on the 1-norm of this product, and exploit the relation p
X p q = X pq (for X ≥ 0) a number of times: A1/2 B 1/2 1 = A(1−s)/2 (As/2 B (1−s)/2 )B s/2 1 ≤ A(1−s)/2 2/(1−s) As/2 B (1−s)/2 2 B s/2 2/s = (Tr[A])(1−s)/2 As/2 B (1−s)/2 2 (Tr[B])s/2 = (Tr[As B (1−s) ])1/2 (Tr[A])(1−s)/2 (Tr[B])s/2 .
We now give a direct proof of inequality (30) that circumvents the proof of (29) and goes through in infinite dimensions. We state it in terms of general positive operators: Theorem 7. For positive operators A and B, A − B21 + 4(Tr[A1/2 B 1/2 ])2 ≤ (Tr(A + B))2 .
(56)
Proof. Consider two general operators P and Q, and define their sum and difference as S = P + Q and D = P − Q. We thus have P = (S + D)/2 and Q = (S − D)/2. Consider the quantity 1 (S + D)(S + D)∗ − (S − D)(S − D)∗ 4 1 = (S D ∗ + DS ∗ ). 2
P P ∗ − Q Q∗ =
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Its trace norm is bounded above as S D ∗ + DS ∗ 1 /2 ≤ (S D ∗ 1 + DS ∗ 1 )/2 = S D ∗ 1 ≤ S2 D2 . In the last line we have used a specific instance of Hölder’s inequality for the trace norm ([3] Cor. IV.2.6). Now put P = A1/2 and Q = B 1/2 , which exist by positivity of A and B, and which are themselves positive operators. We get S, D = A1/2 ± B 1/2 , hence A − B1 ≤ A1/2 + B 1/2 2 A1/2 − B 1/2 2 , which upon squaring becomes A − B21 ≤ Tr(A1/2 + B 1/2 )2 Tr(A1/2 − B 1/2 )2 = Tr(A + B + A1/2 B 1/2 + B 1/2 A1/2 ) × Tr(A + B − A1/2 B 1/2 − B 1/2 A1/2 ) = (Tr(A + B) + 2 Tr(A1/2 B 1/2 )) ×(Tr(A + B) − 2 Tr(A1/2 B 1/2 )) = (Tr(A + B))2 − 4(Tr(A1/2 B 1/2 ))2 .
References 1. Audenaert, K.M.R., Calsamiglia, J., Munoz-Tapia, R., Bagan, E., Masanes, Ll., Acin, A., Verstraete, F.: Discriminating States: The Quantum Chernoff Bound. Phys. Rev. Lett. 98, 160501 (2007) 2. Bacon, D., Chuang, I., Harrow, A.: Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms. Phys. Rev. Lett. 97, 170502 (2006) 3. Bhatia, R.: Matrix Analysis. Heidelberg: Springer, 1997 4. Bjelakovi´c, I., Deuschel, J.D., Krüger, T., Seiler, R., Siegmund-Schultze, Ra., Szkoła, A.: A quantum version of Sanov’s theorem. Commun. Math. Phys. 260, 659–671 (2005) 5. Bjelakovi´c, I., Deuschel, J.D., Krüger, T., Seiler, R., Siegmund-Schultze, Ra., Szkoła, A.: Typical support and Sanov large deviation of correlated states. http://arxiv.org/list/math/0703772, 2007 6. Blahut, R.E.: Hypothesis Testing and Information Theory. IEEE Trans. Inf. Theory 20, 405–417 (1974) 7. Carlen, E.A., Lieb, E.H.: Advances in Math. Sciences, AMS Transl. (2) 189, 59–62 (1999) 8. Chernoff, H.: A Measure of Asymptotic Efficiency for Tests of a Hypothesis based on the Sum of Observations. Ann. Math. Stat. 23, 493–507 (1952) 9. Csiszár, I., Longo, G.: Studia Sci. Math. Hungarica 6, 181–191 (1971) 10. Fuchs, C.A., van de Graaf, J.: Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory 45, 1216 (1999) 11. Hayashi, M.: Quantum Information, An Introduction. Berlin: Springer, 2006 12. Hayashi, M.: Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to ClassicalQuantum Channel coding. http://arxiv.org/list/quant-ph/0611013, 2006 13. Hayashi, M.: Asymptotics of quantum relative entropy from a representation theoretical viewpoint. J. Phys. A: Math. Gen. 34, 3413–3419 (2001) 14. Helstrom, C.W.: Quantum Detection and Estimation Theory. New York: Academic Press, 1976 15. Hiai, F., Petz, D.: The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143, 99–114 (1991) 16. Hoeffding, W.: Asymptotically Optimal Tests for Multinomial Distributions. Ann. Math. Statist. 36, 369–401 (1965) 17. Holevo, A.S.: On Asymptotically Optimal Hypothesis Testing in Quantum Statistics. Theor. Prob. Appl. 23, 411–415 (1978)
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18. Kargin, V.: On the Chernoff distance for efficiency of quantum hypothesis testing. Ann. Statist. 33, 959–976 (2005) 19. Lieb, E.H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math. 11, 267–288 (1973) 20. Nagaoka, H.: The Converse Part of The Theorem for Quantum Hoeffding Bound. http://arxiv.org/list/ quant-ph/0611289, 2006 21. Nussbaum, M., Szkoła, A.: A lower bound of Chernoff type in quantum hypothesis testing. http://arxiv. org/list/quant-ph/0607216, 2006 22. Nussbaum, M., Szkoła, A.: The Chernoff lower bound in quantum hypothesis testing. Preprint No. 69/2006, MPI MiS Leipzig 23. Ogawa, T., Hayashi, M.: On error exponents in quantum hypothesis testing. IEEE Trans. Inf. Theory 50, 1368–1372 (2004) 24. Ogawa, T., Nagaoka, H.: Strong converse and Stein’s lemma in quantum hypothesis testing. IEEE Trans. Inf. Theory 46, 2428 (2000) 25. Petz, D.: Quasi-entropies for finite quantum states. Rep. Math. Phys. 23, 57–65 (1986) 26. Ruskai, M.B., Lesniewski, A.: Monotone Riemannian metrics and relative entropy on noncommutative probability spaces. J. Math. Phys. 40, 5702–5742 (1999) 27. Uhlmann, A.: Sätze über Dichtematrizen. Wiss. Z. Karl-Marx Univ. Leipzig 20, 633–653 (1971) 28. Uhlmann, A.: The ‘transition probability’ in the state space of a *-algebra. Rep. Math. Phys. 9, 273 (1976) 29. van der Vaart, A.W.: Asymptotic Statistics. Cambridge: University Press, 1998 Communicated by M.B. Ruskai
Commun. Math. Phys. 279, 285–308 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0436-2
Communications in
Mathematical Physics
On the Spectrum of Certain Non-Commutative Harmonic Oscillators and Semiclassical Analysis Alberto Parmeggiani Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40126 Bologna, Italy. E-mail: [email protected] Received: 10 July 2006 / Accepted: 8 March 2007 Published online: 20 February 2008 – © Springer-Verlag 2008
Dedicated to Professor Cesare Parenti, friend and teacher, on the occasion of his sixty-fifth birthday. Abstract: A localization and “cardinality” property, along with a multiplicity result, of the spectrum of certain 2 × 2 globally elliptic systems of ordinary differential operators, a class of vector-valued deformations of the classical harmonic oscillator called noncommutative harmonic oscillators, will be described here. The basic tool is the study of a semiclassical reference system. 1. Introduction In this paper, we will consider the following system: 2 ∂x x 2 1 w w Q (x, D) = Q (α,β) (x, D) = A − + + J x∂x + , x ∈ R, 2 2 2 where
(1.1)
α 0 0 −1 A= , J= , α, β ∈ R+ , αβ > 1. 0 β 1 0
Put p0 (x, ξ ) = (x 2 + ξ 2 )/2. System (1.1) is the Weyl quantization of the matrix-valued quadratic form on T ∗ R = R × R, Q(x, ξ ) = Q (α,β) (x, ξ ) = Ap0 (x, ξ ) + i J xξ. Clearly, one has Q(x, ξ )∗ = Q(x, ξ ) for all (x, ξ ) ∈ R × R. The condition αβ > 1 ensures that det Q(x, ξ ) ≈ (x 2 +ξ 2 )2 , whence it follows that Q w is a (classical) globally elliptic self-adjoint operator in L 2 (R; C2 ) (see [7,29]) with domain ⎧ ⎫ ⎨ ⎬ B 2 (R; C2 ) = u ∈ L 2 (R; C2 ); ||x j ∂xk u|| L 2 (R;C2 ) < +∞ . ⎩ ⎭ j,k≥0, j+k≤2
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Since B 2 (R; C2 ) is compactly embedded into L 2 (R; C2 ), we have that the spectrum of Q w (x, D) is discrete, made of a diverging (to +∞) sequence of eigenvalues with finite multiplicities, and it turns out (see [21]) that its lowest eigenvalue is positive. Hence (with repetitions according to the multiplicity) Spec(Q w (x, D)) = {0 < λ1 ≤ λ2 ≤ . . . → +∞}. System (1.1) is called a non-commutative harmonic oscillator, in the terminology introduced by Wakayama and the author in [20] and [21], and (1.1) is actually a normal form of the class introduced there. In [20] and [21], Wakayama and the author gave a qualitative description of the spectrum of Q w , by using sl2 (R)-symmetries to construct suitable creation-annihilation operators and a “twisted” basis of vector-valued Hermite functions. The case α = β is completely understood: the system is unitarily equivalent to a scalar harmonic oscillator (see Corollary 4.1, p. 555, of [21]), and one has the following result. Theorem 1.1 ([21]). When α = β one has 1 Spec(Q w (x, D)) = ; N ∈ Z+ α2 − 1 N + 2 (where Z+ = {0, 1, . . .}), with eigenvalues of multiplicity 2.
(1.2)
√ 2 It is interesting √ to notice the appearance of the “symplectic” parameter α − 1 (denoted by = αβ − 1 in [20,21]). When α = β things are highly nontrivial. In this case, in [21] (see also [20]) to ± , were introduced. (The ± stands understand the spectrum two kinds of sets, 0± and ∞ in this case for the parity: the system preserves parity, whence it follows that one can study, separately, the even case, +, and the odd one, −, respectively.) The sets 0± are described as the sets of those eigenvalues that are roots of particular polynomials, whereas ± are described as the sets of those eigenvalues that are zeroes of particular the sets ∞ meromorphic functions (defined through continued fractions). These polynomials and meromorphic functions are related to certain three-term recurrence systems. While one + ∩ − has a very difficult description always has 0+ ∩ 0− = ∅, the intersection ∞ ∞ and is yet to be understood. Upon defining Vλ± = {u ∈ L 2 (R; C2 ); Q w (x, D)u = λu, u even/odd}, one has the following theorem. Theorem 1.2 ([21]; see also [20]). When α = β, one has + − , 0− ⊂ ∞ , 0+ ⊂ ∞ + − Spec(Q w (x, D)) = ∞ ∪ ∞ ,
and dim Vλ± =
⎧ ⎨ 2, whenever λ ∈ 0± ⎩
± \ ± 1, whenever λ ∈ ∞ 0
, ±−respectively.
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∓ \ ∓ ) = ∅, case that Notice that the theorem says nothing about whether 0± ∩ (∞ 0 would yield an eigenvalue of multiplicity 3. ± , although explicitly described, are complicated (see [20,21 and The sets 0± and ∞ also 22]), and it would be interesting to have other (hopefully simpler) descriptions. It is a remarkable fact, proved by Ochiai in [17], that the spectral problem for Q w is equivalent to a family of third-order Fuchsian differential equations with four regular singularities in the complex unit disk. Furthermore, important work on the spectral zetafunction ζ Q (s) (and its special values) associated with Q w (x, D), defined by (see [12 and 13]; see also [25])
ζ Q (s) =
λ∈Spec(Q w (x,D))
1 , s ∈ C, Re s > 1, λs
has been started by Ichinose and Wakayama in [12 and 13]. It is also worth mentioning that numerical study of the spectrum Q w (x, D) has been carried out by Nagatou, Nakao and Wakayama in [16], and that one can study the spectrum by Rellich’s perturbation theory in the limit αβ → +∞ with α/β a fixed constant = 1 (see Parmeggiani [23]). The trajectories associated with the Hamilton vector-fields of the eigenvalues λ± of Q(x, ξ ) are all periodic with least periods (see [24]) 2π µ2 + sin2 (2θ ) − α+β α±β , (1.3) T± = √ dθ, µ± := π∓ 2 (2θ ) 2 αβ(αβ − 1) αβ − sin 0 when α = β, and with periods 2π T+ = T− = T = √ , α2 − 1
(1.4)
when α = β, that in all cases do not depend on the energy. Hence it makes sense to see whether the spectrum of Q w (x,D) “clusters” near a sequence of reals tending to +∞. Notice that when α = β one has 0 < T+ < T− . When α = β one of course has that Spec(Q w (x, D)) is indeed the sequence { 2π T (k + 1 )} . 2 k∈Z+ Note that as functions of (α, β) ∈ H0 := {(α , β ) ∈ R+ × R+ ; α β > 1, α = β }, the periods T± are continuous. However, when (α, β) tends in H0 to (α0 , α0 ) with α0 > 1, we have that T± (α, β) −→ T± (α0 , α0 ) =
2π α02 − 1
∓
4 α02 − 1
arctan
1 α02 − 1
,
which are not the periods relative to Q (α0 ,α0 ) (x, ξ ), that are actually given by (1.4). In [24] we proved, along with a result on the singularities of the density of eigenvalues, the following theorem. Theorem 1.3. Suppose α = β and that 0<
T+ ∈ Q+ , T−
(1.5)
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that is, there exist m + , m − ∈ N, such that m + T+ = m − T− =: T , where we take the smallest of such m ± . (Notice that the integers m ± cannot be both even then.) Then there exists C0 > 0 such that π C0 π C0 w Spec(Q (x, D)) ⊂ . (1.6) k− , k+ T k T k k≥1
In particular, if (1.5) holds with m ± odd positive integers, one can then find C0 > 0 such that 2π 1 C0 2π 1 C0 (k + ) − , (k + ) + . (1.7) Spec(Q w (x, D)) ⊂ T 2 k T 2 k k≥1
The method of proof of Theorem 1.3 was based upon an adaptation (see [19]) of the classical decoupling argument due to Taylor [30] and the theory of Fourier integral operators as developed by Helffer and Robert (see [9], and also [7]). However, information about multiplicity of the spectrum, or even presence of spectrum, in each interval appearing in the right-hand side of (1.6) or (1.7) still remained difficult to detect. The limit of the argument was that the diagonalizers were not unitary, for they were so only up to smoothing operators. In this paper (it will be supposed throughout that α = β), we shall show how one can remedy this disadvantage by using semiclassical analysis to correct the diagonalizers into true unitary operators. The spectrum of Q w (x, D) and of its semiclassical analogue Q w (x, h D), h ∈ (0, 1] being the semiclassical parameter, are linked by the relation λ j ∈ Spec(Q w (x, D)) ⇐⇒ hλ j ∈ Spec(Q w (x, h D)).
(1.8)
This is due to the homogeneity of Q(x, ξ ). To be in a position to use semiclassical analysis in the study of large eigenvalues, we shall construct a reference system Q 0 (x, ξ ) (no longer homogeneous) which has (being α = β) distinct eigenvalues l± that coincide with those of Q(x, ξ ) when (x, ξ ) is sufficiently large. Upon h-Weyl quantization, one finds out that, for sufficiently large energy E, the eigenvalues in Spec(Q w (x, h D)) ∩ (E, +∞) are within O(h N ) distance to the large eigenvalues of Q w 0 (x, h D) (any fixed N ∈ Z+ ). Now, by an adaptation of Taylor’s decoupling argument, using the semiclassical calculus, one may find, for h 0 ∈ (0, 1] sufficiently small and all h ∈ (0, h 0 ], unitary h-dependent 2 4 operators, that that Spec(Q w 0 (x, h D)) is within O(h ) preservew B , so as to obtain w distance to ± Spec( ± (x, h D)), where ± (x, h D) are h-pseudodifferential operators whose respective principal symbols are the eigenvalues l± of Q 0 (x, ξ ). One can now rely on very precise information on spectra of scalar h-pseudodifferential operators with periodic bicharacteristics (see [1, 3, 4, 8, 10, 11, 14, 27 and 28]), such as Weylasymptotics and localization of the spectrum inside a given energy interval [E 1 , E 2 ] as a subset of the union of intervals Jk± (h) (±-respectively) of size O(h 2 ) centered at particular sequences (related to the period of the classical trajectories). In particular, by using Helffer-Robert’s results of [11] in the 1-dimensional case (see also [3 and 4]), one obtains that for h 0 ∈ (0, 1] sufficiently small and all h ∈ (0, h 0 ], the multiplicity of ± ± Spec( w ± (x, h D)) ∩ Jk (h) (with the center of Jk (h) belonging to [E 1 , E 2 ], 1 E 1 < E 2 ) is 1. At this point, under particular assumptions on the relation between T+ and T− , we obtain information about Spec(Q w 0 (x, h D)) ∩ [E 1 , E 2 ], whence information, once h is fixed sufficiently small and upon using (1.8), on Spec(Q w (x, D)) ∩ [E 1 h −1 , E 2 h −1 ],
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where 1 E 1 < E 2 . For instance, we have that for h 0 ∈ (0, 1] sufficiently small and (all) h fixed in (0, h 0 ], Spec(Q w (x, D)) ∩ [E 1 h −1 , E 2 h −1 ] ⊂ h −1 Ik± (h), ±
k∈Z+
where Ik± (h) =
1 2π 1 (k + )h − C0 h 2 , (k + )h + C0 h 2 , T± 2 T± 2
2π
1 and that, as a corollary in case the sequences { 2π T± (k + 2 )}k∈Z+ are “far apart” (see condition (4.17) below), the intervals being disjoint then (in the first place h 0 is picked sufficiently small and then h is fixed in (0, h 0 ]), (1.9) multiplicity Spec(Q w (x, D)) ∩ h −1 Ik± (h) = 1, ∀k ∈ Z± E 1 ,E 2 (h),
±-respectively, where the Z± E 1 ,E 2 (h) are the sets of those k ∈ Z+ for which [E 1 , E 2 ], whence we may also conclude that
2π 1 T± (k + 2 )h
∈
+ − 0± ∩ h −1 Ik (h) = ∞ ∩ ∞ ∩ h −1 Ik (h) = ∅, ∀k ∈ Z E 1 ,E 2 (h), j = ±. j
j
j
This result in a sense complements Proposition 3.14 of [23] (see Remark 4.14 below), where it was proved by means of Rellich’s Theory that for constant α/β = 1, with α/β not a ratio of odd numbers, for any given E > 0 there is a constant C∗ = C∗ (α/β, E) > 1 such that for αβ > C∗ one has multiplicity Spec(Q w (x, D)) ∩ [0, E] = 1. We end this introduction by also noting that, since the fact that 0± = ∅ is related through Ochiai’s results (Theorems 2 and 3 of [18]) to the existence of certain holomorphic solutions (at the origin) in the kernel of the Heun operators H (z, ∂z ) and P e (z, ∂z ) introduced by Ochiai in [17] (see Eqs. (3.4) and (4.1) of [17]), from (1.9) we may conclude (see Corollary 4.13 below) that for such a range of eigenvalues these solutions cannot exist. 2. Background on the Semiclassical Calculus We recall here a few facts about the semiclassical calculus that will be needed in the sequel. We shall mainly follow Dimassi-Sjöstrand’s book [3] and Evans-Zworski’s notes [6] (see also Robert [27], Ivrii [14], Martinez [15] and Shubin [29]). We start by fixing some notation. Let m: Rn × Rn −→ (0, +∞) be an order function (see [3]), that is, there exist constants C0 , N0 > 0 such that m(x, ξ ) ≤ C0 (1 + |x − y|2 + |ξ − η|2 ) N0 /2 m(y, η) for all (x, ξ ) and (y, η). Given δ ∈ [0, 1/2] and µ ∈ R, µ consider the spaces of symbols S(m) and Sδ (m), µ ∈ R (see [3], Definition 7.5). Just recall that a symbol a ∈ C ∞ (Rnx × Rnξ ), possibly depending on the parameter h ∈ (0, 1], β
belongs to S(m) if |∂xα ∂ξ a| ≤ Cαβ m uniformly on R2n and h for all multiindices α and µ
β
β, and belongs to Sδ (m) if |∂xα ∂ξ a| ≤ Cαβ mh −µ−δ(|α|+|β|) uniformly on R2n and h for all multiindices α and β.
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Upon denoting by M N , N ∈ N, the set of complex-valued N × N matrices, we write µ µ S(m; M N ) = S(m) ⊗ M N and Sδ (m; M N ) = Sδ (m) ⊗ M N , etc. for the matrix-valued µ analogues of S(m) and Sδ (m) etc., respectively (and analogously S(m; C N ) etc. for the µ vector-valued versions). We shall write S µ (m; M N ) in place of S0 (m; M N ). µ Given a ∈ Sδ (m; M N ), for h ∈ (0, 1] one then defines the h-Weyl quantization (also semiclassical Weyl-quantization) x+y −1 a w (x, h D)u(x) := (2π h)−n ei h x−y,ξ a( , ξ ; h)u(y)dydξ, 2 where u ∈ S(Rn ; C N ). Of course, a w (x, h D) as a linear map a w (x, h D): S(Rn ; C N ) −→ S(Rn ; C N ) and as a linear map a w (x, h D): S (Rn ; C N ) −→ S (Rn ; C N ) is continuous. Moreover, if a ∈ Sδ0 (1; M N ), 0 ≤ δ ≤ 1/2, then (see [3], Theorem 7.11) a w (x, h D): L 2 (Rn ; C N ) −→ L 2 (Rn ; C N ) is bounded and there is a constant C > 0, independent of h, such that ||a w (x, h D)|| L 2 →L 2 ≤ C for all h ∈ (0, 1]. As regards the composition one has (see [3], Prop. 7.7 and Theorem 7.9) that given a ∈ Sδ0 (m 1 , M N ) and b ∈ Sδ0 (m 2 , M N ), then a w (x, h D)bw (x, h D) = (ah b)w (x, h D), where
ah b = ei hσ (Dx ,Dξ ;D y ,Dη )/2 (a(x, ξ ; h)b(y, η; h))x=y,ξ =η ∈ Sδ0 (m 1 m 2 ; M N ), (with σ = nj=1 dξ j ∧ d x j the canonical symplectic form in Rn × Rn ). Furthermore, when δ ∈ [0, 1/2) one has ∞ k 1 ih σ (Dx , Dξ ; D y , Dη ) a(x, ξ ; h)b(y, η; h)(x,ξ )=(y,η) . ah b ∼ k! 2
(2.1)
k=0
µ− j
µ
k (m; M N ), j0 = 0, jk < jk+1 +∞, Recall that for a ∈ Sδ (m; M N ) and ak ∈ Sδ a ∼ k≥0 ak means that for any given N0 ∈ Z+ ,
a−
N0
µ− j N0 +1
ak ∈ Sδ
(m; M N ).
k=0 0 (1; M ) with When δ = 1/2 one has (see [6]) that if a, b ∈ S1/2 N
supp a ⊂ K , and dist(supp a, supp b) ≥ γ > 0, where the compact K and the constant γ are independent of h, then ||a w (x, h D)bw (x, h D)|| L 2 →L 2 = O(h ∞ ).
(2.2)
Recall that f (h) = O(h N ) for some N ∈ Z+ if there exists C N > 0 such that | f (h)| ≤ C N h N , and that f (h) = O(h ∞ ) if for any given N ∈ Z+ one has f (h) = O(h N ). A very useful observation is the following.
On the Spectrum of Certain Non-Commutative Harmonic Oscillators
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√ Remark 2.1. Let E > 0. Define the L 2 -isometry U E : u(x) −→ E n/4 u( E x). Then, given any symbol a ∈ S µ (m; M N ) one has √ √ h U E a w (x, h D)U E−1 = a w ( E x, E h˜ D), where h˜ = . E In particular
(2.3)
√ √ Uh a w (x, h D)Uh−1 = a w ( h x, h D).
In the sequel we shall always use the order function m(x, ξ ) = (1 + |x|2 + |ξ |2 )1/2 and its powers. One has the following elementary lemma (whose proof is left to the reader). Lemma 2.2. Let φ j ∈ S(R; C2 ), j ≥ 1, be an eigenfunction of Q w (x, D) belonging to the eigenvalue λ j . Then x ϕ j (h; x) := (Uh−1 φ j )(x) = h −1/4 φ j ( √ ) h
(2.4)
belongs to the eigenvalue λ j (h) := hλ j of Q w (x, h D). In particular ϕj(
√ h ; x) = E 1/4 ϕ j (h; E x), j ≥ 1, E
belongs to the eigenvalue λ j ( Eh ) =
h E λj
of Q w (x,
Recall that the eigenvalues of Q(x, ξ ) are λ± (x, ξ ) = µ+ p0 (x, ξ ) ±
h E
(2.5)
D).
µ2− p0 (x, ξ )2 + x 2 ξ 2 ,
(2.6)
where α±β 1 , p0 (x, ξ ) = (x 2 + ξ 2 ), 2 2 and that one has, for α = β (as we are always assuming), µ± =
0 < λ− (x, ξ ) < λ+ (x, ξ ), ∀(x, ξ ) ∈ R × R \ {(0, 0)}. Furthermore, upon putting c1 := µ+ − µ2− + 1 =
αβ − 1 , c2 := µ+ + µ+ + µ2− + 1
µ2− + 1
(then 0 < c1 < c2 ), we have c1 p0 (x, ξ ) ≤ λ− (x, ξ ) ≤ λ+ (x, ξ ) ≤ c2 p0 (x, ξ ), ∀(x, ξ ) ∈ R × R.
(2.7)
Define (see [24]) for an energy E ≥ 0 the sets ± (E) := {(x, ξ ) ∈ R × R; λ± (x, ξ ) = E}. Then ± (E) are compact and connected for all E ≥ 0 (and they both reduce to {(0, 0)} when E = 0). We next prepare the ground (following [6]) for the localization of large eigenvalues. The first result is about uniform h ∞ -estimates.
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Proposition 2.3. Let E > 0. Let a ∈ Sδ0 (1; M2 ) with supp a ⊂ K , where K ⊂ R × R is a compact set independent of h such that K ∩ + (E) ∪ − (E) = ∅, and let γ := dist(K , + (E) ∪ − (E)). Let u(h) ∈ L 2 (R; C2 ) solve the eigenvalue equation Q w (x, h D)u(h) = λ(h)u(h). If |λ(h) − E| < , for some ∈ (0, 1/2] sufficiently small (compared to E and γ ), then ||a w (x, h D)u(h)|| = O(h ∞ )||u(h)|| (where the constants in O(h ∞ ) are allowed to depend on E and γ ). Proof. The set + (E) ∪ − (E) =: K is compact. We may therefore find χ ∈ C0∞ (R × R), with 0 ≤ χ ≤ 1, such that χ ≡ 1 near K and χ ≡ 0 near K . Define b(x, ξ ) = bh (x, ξ ) = Q(x, ξ ) − λ(h)I + iχ (x, ξ )I ∈ S 0 (m 2 ; M2 ). If is sufficiently small (with respect to E and γ ) we then have that |det b(x, ξ )|2 = |λ+ (x, ξ ) − λ(h)|2 + χ (x, ξ )2 |λ− (x, ξ ) − λ(h)|2 + χ (x, ξ )2 > 0, for all (x, ξ ) ∈ R × R, and C −1 ≤
|det b(x, ξ )|2 ≤ C, m(x, ξ )4
for some C = C E,γ > 0. Hence we may find c ∈ S 0 (m −2 ; M2 ) and r ∈ S 0 (1; M2 ) such that cw (x, h D)bw (x, h D) = I + r w (x, h D), and r w = O(h ∞ ), where by r w = O(h ∞ ) we mean that ||r w || L 2 →L 2 = O(h ∞ ). Then a w (x, h D)cw (x, h D)bw (x, h D) = a w (x, h D) + O(h ∞ ), where bw (x, h D) = Q w (x, h D) − λ(h) + iχ w (x, h D). We now claim that a w (x, h D)cw (x, h D)χ w (x, h D) = O(h ∞ ).
(2.8)
When δ ∈ [0, 1/2) the claim immediately follows, for we have supp a ∩ supp χ = ∅ uniformly in h. When δ = 1/2, we have that for any given N ∈ Z+ , cw (x, h D)χ w (x, h D) = (cχ )wN (x, h D) + O(h N +1 ), where (cχ ) N (x, ξ ) :=
N k 1 ih σ (Dx , Dξ ; D y , Dη ) c(x, ξ )χ (y, η)(x,ξ )=(y,η) k! 2 k=0
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has compact support (in fact supp(cχ ) N ⊂ supp χ ), uniformly disjoint from supp a. We are hence in a position to use (2.2) and obtain (by the continuity) the claim also when δ = 1/2. In all cases, on the other hand, from Q w (x, D)u(h) = λ(h)u(h) it follows a w (x, h D)u(h) = a w (x, h D)cw (x, h D)bw (x, h D)u(h) + O(h ∞ )u(h) = ia w (x, h D)cw (x, h D)χ w (x, h D)u(h) + O(h ∞ )u(h) = O(h ∞ )u(h), which concludes the proof of the proposition.
We next prove the following estimate. Proposition 2.4. Let u(h) ∈ L 2 (R; C2 ) be an eigenfunction of Q w (x, h D) belonging to λ(h). Let a ∈ S(1; M2 ) be independent of h. Suppose that, with R > 0, supp a ⊂ {(x, ξ ); x 2 + ξ 2 ≤ R/c2 } and λ(h) > R + 1. Then h ∞ w ||a (x, h D)u(h)|| = O ||u(h)||. λ(h) Proof. Let E > R + 1 be such that |λ(h) − E| ≤ E, where ∈ (0, 1/2] is sufficiently small, so as to have (1 − )E > R + 1. We have h U E Q w (x, h D)U E−1 = E Q w (x, h˜ D), where h˜ = , E √ √ ˜ since ˜ ξ ) := √ξ ) := Q( E x, E ξ ) = E Q(x, ξ ). Now, upon likewise writing a(x, √ Q(x, a( E x, E ξ ), we have β ˜ ξ )| ≤ Cαβ E (|α|+|β|)/2 ≤ Cαβ h˜ −(|α|+|β|)/2 , |∂xα ∂ξ a(x, 0 (1; M ) with respect to the parameter h, ˜ and that is a˜ ∈ S1/2 2
supp a˜ ⊂ {(x, ξ ); x 2 + ξ 2 ≤ R/(c2 E)} ⊂ {(x, ξ ); x 2 + ξ 2 ≤ (1 − )/c2 } =: K , ˜ − 1| < , for λ(h) = λ(h)E ˜ (see Lemma 2.2). Hence while |λ(h) K ∩ + (1) ∪ − (1) = ∅, for, by (2.7), we have that ± (1) ⊂ {(x, ξ ); x 2 + ξ 2 ≥ 2/c2 }. ˜ := U E u(h) we get that u( ˜ is an eigenfunction of Q w (x, h˜ D) Therefore, by putting u( ˜ h) ˜ h) ˜ belonging to the eigenvalue λ(h) = λ(h)/E, and that, by Proposition 2.3 applied to the ˜ h-Weyl quantization a˜ w (x, h˜ D) of a˜ (by shrinking if necessary), ˜ = O(h˜ ∞ )||u( ˜ ||a˜ w (x, h˜ D)u( ˜ h)|| ˜ h)||
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(now the constants in O(h˜ ∞ ) are absolute constants), which is, through the L 2 -isometry U E , equivalent to h ∞ ||u(h)||. ||a w (x, h D)u(h)|| = O E Since E/2 ≤ (1 − )E ≤ λ(h) ≤ (1 + )E ≤ 3E/2, we finally obtain h ∞ ||a w (x, h D)u(h)|| = O ||u(h)||, λ(h) which concludes the proof.
We hence obtain the following corollary, that will make it possible to study the large eigenvalues of Q w (x, h D) (and hence those of Q w (x, D)). Corollary 2.5. Let R > 0. Consider the orthogonal projection : L 2 (R; C2 ) −→ Span{u(h); Q w (x, h D)u(h) = λ(h)u(h), λ(h) ≤ R + 1}. Let a = a ∗ ∈ S(1; M2 ) be independent of h, with supp a ⊂ {(x, ξ ); x 2 + ξ 2 < R/c2 }. Then ||a w (x, h D)(I − )|| L 2 →L 2 = O(h ∞ ),
||(I − )a w (x, h D)|| L 2 →L 2 = O(h ∞ ).
Proof. Let Spec(Q w (x, h D)) = {−∞ < λ1 (h) ≤ λ2 (h) ≤ . . . → +∞} with repetitions according to multiplicity, and let {u j (h)} j≥1 be an orthonormal basis of L 2 (R; C2 ) made of eigenfunctions of Q w (x, h D), where u j (h) belongs to λ j (h), j ≥ 1. We can write I −= u j (h)∗ ⊗ u j (h), λ j (h)>R+1
where u j (h)∗ ⊗ u j (h) v = (v, u j (h))u j (h). Then a w (x, h D)(I − ) =
u j (h)∗ ⊗ a w (x, h D)u j (h) ,
λ j (h)>R+1
and
⎛ ||a w (x, h D)(I − )|| L 2 →L 2 ≤ ⎝
⎞1/2 ||a w (x, h D)u j (h)||2 ⎠
.
(2.9)
λ j (h)>R+1
By the Weyl law for the eigenvalues of Q w (x, D) (see [12], and [24]; see also [25] and [26]) we have that for some c > 0, λ j (h) ≥ cj h, so that by Proposition 2.4 (and recalling that we consider the eigenvalues λ j (h) for which λ j (h) > R + 1) we obtain, for each M < N , N N −M h h CN w M ≤ CN h ≤ N −M h M j −(N −M) . ||a (x, h D)u j (h)|| ≤ C N λ j (h) λ j (h) c
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Hence we choose N = M + 2, so that the sum in the right-hand side of (2.9) converges > 0 such that and get that for any given M ∈ Z+ there exists C M ||a w (x, h D)(I − )|| L 2 →L 2 ≤ C M hM.
Since a w (x, h D) = a w (x, h D)∗ , (I − )∗ = I − , and ||A|| L 2 →L 2 = ||A∗ || L 2 →L 2 (for an L 2 -bounded operator A), we also obtain the second desired inequality. 3. Construction of the Reference Operator To be able to use semiclassical analysis, we must first regularize the eigenvalues λ± of Q(x, ξ ), for they are not smooth at (0, 0). To this purpose the following lemma will be useful. Lemma 3.1. Without loss of generality we may suppose α > β.
(3.1)
Proof. Inequality (3.1) is no loss of generality for when β > α (recall that we suppose here α = β) we may then consider the system 0 1 . K Q (α,β) (x, ξ )K = K AK p0 (x, ξ ) − i J xξ, K = 1 0 β 0 Hence, being K AK = , by considering the symplectic transformation κ: T ∗ R 0α (x, ξ ) −→ (ξ, −x) ∈ T ∗ R, we have that Q (β,α) (x, ξ ) = K Q (α,β) K ◦ κ (x, ξ ). It thus follows, being λ± (x, ξ ) = λ± (ξ, −x), that Q (α,β) (x, ξ ) and Q (β,α) (x, ξ ) have exactly the same eigenvalues, and that, with Uκ the metaplectic operator (the Fourier transform) associated with κ, w −1 w Qw (β,α) (x, D) = (K Q (α,β) K ◦ κ) (x, D) = Uκ K Q (α,β) (x, D)K Uκ
are unitarily equivalent, for K Uκ : L 2 (R; C2 ) −→ L 2 (R; C2 ) is unitary. This concludes the proof of the lemma. Recall that from (2.7) we have λ± (x, ξ ) −→ +∞, as |(x, ξ )| → +∞, and λ+ (x, ξ ) ≤
c2 λ− (x, ξ ), ∀(x, ξ ) ∈ T ∗ R. c1
It hence follows that, for ε0 > 0 fixed, {(x, ξ ); λ+ (x, ξ ) ≤ ε0 } ⊂ {(x, ξ ); c1 p0 (x, ξ ) ≤ ε0 },
(3.2)
{(x, ξ ); c1 p0 (x, ξ ) ≤ ε0 } ∩ {(x, ξ ); c1 p0 (x, ξ ) ≥ 2ε0 } = ∅,
(3.3)
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and {(x, ξ ); λ− (x, ξ ) ≥ 2
c2 ε0 =: ε1 } ⊂ {(x, ξ ); c1 p0 (x, ξ ) ≥ 2ε0 }. c1
(3.4)
Because of (3.2), (3.3) and (3.4), we may therefore take χ ∈ C ∞ (R × R), 0 ≤ χ ≤ 1, with χ ≡ 0 when λ+ ≤ ε0 , χ ≡ 1 when λ− ≥ ε1 . Put χ1 = χ and χ2 = 1 − χ , so that χ1 + χ2 = 1 and χ2 ∈ C0∞ (R × R), with supp χ2 ⊂ {(x, ξ ); λ− (x, ξ ) ≤ ε1 }, and 0 ≤ χ2 ≤ 1.
(3.5)
Define hence (for α > β) the reference operator by
1 Q 0 (x, ξ ) = χ1 (x, ξ )Q(x, ξ ) + χ2 (x, ξ ) 0
0 . 1/2
(3.6)
We have the following lemma. Lemma 3.2. We have Q 0 = Q ∗0 ∈ S(m 2 ; M2 ) is positive elliptic,
(3.7)
hence det Q 0 (x, ξ ) ≈ m(x, ξ )4 , ∀(x, ξ ) ∈ R × R, and in particular Q 0 is a globally elliptic system (see [7,29]). Moreover, upon denoting by l± (x, ξ ) the eigenvalues of Q 0 (x, ξ ), we have l± ∈ C ∞ (R × R), 0 < l− (x, ξ ) < l+ (x, ξ ), ∀(x, ξ ) ∈ R × R, l± λ
− ≥ε1
= µ+ p0 (x, ξ ) ±
µ2− p0 (x, ξ )2 + x 2 ξ 2
l± λ
+ ≤ε0
=
λ− ≥ε1
= λ ± λ
3 1 ± , 4 4
l± (x, ξ ) ≈ p0 (x, ξ ), for |(x, ξ )| ≥ c > 0. Hence the l± are globally elliptic symbols and, in particular, l± (x, ξ ) −→ +∞, as |(x, ξ )| → +∞.
− ≥ε1
(3.8)
,
(3.9)
(3.10)
(3.11)
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Proof. We have, for all v ∈ C2 ,
1 0 v, vC2 . Q 0 (x, ξ )v, vC2 = χ1 (x, ξ )Q(x, ξ )v, vC2 + χ2 (x, ξ ) 0 1/2
Since 0 ≤ χ1 , χ2 ≤ 1, and χ1 +χ2 ≡ 1, and since λ± (x, ξ ) ≈ p0 (x, ξ ), we immediately obtain the existence of C > 0 such that on the one hand Q 0 (x, ξ )v, vC2 ≤ χ1 (x, ξ )λ+ (x, ξ ) + χ2 (x, ξ ) |v|2C2 ≤ Cm(x, ξ )2 |v|2C2 , (3.12) and on the other 1 Q 0 (x, ξ )v, vC2 ≥ χ1 (x, ξ )λ− (x, ξ ) + χ2 (x, ξ ) |v|2C2 ≥ C −1 m(x, ξ )2 |v|2C2 , 2 (3.13) for all (x, ξ ) ∈ R × R, and all v ∈ C2 . This proves (3.7). Now, it is easy to see that the eigenvalues of Q 0 (x, ξ ) are given by
3 l± (x, ξ ) = µ+ p0 (x, ξ )χ1 (x, ξ ) + χ2 (x, ξ ) ± δ(x, ξ ), 4 where the discriminant δ has the form 2 1 δ(x, ξ ) = µ− χ1 (x, ξ ) p0 (x, ξ ) + χ2 (x, ξ ) + χ1 (x, ξ )2 x 2 ξ 2 > 0 4
(3.14)
(3.15)
by construction (as it is easily seen, for we are supposing, by virtue of Lemma 3.1, that µ− > 0). Hence, from (3.14), (3.15), (3.12) and (3.13), we obtain (3.8), (3.9), (3.10) and (3.11). This concludes the proof of the lemma. Since w w Qw 0 (x, h D) = Q (x, h D) + R0 (x, h D), R0 = Q 0 − Q ∈ S(1; M2 ),
(3.16)
where supp R0 = supp χ2 is compact, we thus have that Q w 0 (x, h D) has the same domain B 2 as Q w (x, h D) and is self-adjoint, semi-bounded from below (i.e. Q w 0 (x, h D) ≥ 2 −C I on B for some C > 0), with a discrete spectrum, made of a diverging (to +∞) sequence of eigenvalues with finite multiplicities. We write (with repetitions according to multiplicity) Spec(Q w (x, h D)) = {−∞ < λ1 (h) ≤ λ2 (h) ≤ . . . → +∞}, Spec(Q w 0 (x, h D)) = {−∞ < µ1 (h) ≤ µ2 (h) ≤ . . . → +∞}. Moreover, from (3.9) and the previous considerations, we also obtain the following basic lemma. Lemma 3.3. Put
c2 E 0 := max 1, 2 ε1 , max l± . λ− ≤2ε1 c1
Then the set := T ∗ R ∩ {λ− ≥ E 0 } ⊂ {λ± ≥ E 0 } ⊂ {λ− ≥ 2ε1 }
(3.17)
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is foliated by periodic trajectories γ± (E) = {(x, ξ ); l± (x, ξ ) = E} = {(x, ξ ); λ± (x, ξ ) = E}, E ≥ E 0 , that have period T± , ±-respectively, where, recall (1.3), T± is the least period of the Hamilton flow exp(t Hλ± ) associated with λ± . We are now in a position to measure the distance of Spec(Q w (x, h D)) to Spec(Q w 0 (x, h D)). Fix the energy E 0 = E 0 + 10, and consider , the orthogonal projector of Lemma 2.5 with R + 1 = E 0 . Since E 0 − 1 > E 0 ≥ 2
c2 ε1 , c1
we get in particular that supp χ2 ⊂ {(x, ξ ); λ− (x, ξ ) ≤ ε1 } ⊂ {(x, ξ ); x 2 + ξ 2 < (E 0 − 1)/c2 }, whence the same is true for supp R0 . It follows that ||R0w (x, h D)|| L 2 →L 2 ≤ cχ , where cχ > 0 is independent of h.
(3.18)
Then Q w 0 (x, h D) is a compact operator (in fact, a finite-rank operator) with ||Q w 0 (x, h D)|| L 2 →L 2 ≤ E 0 + cχ .
(3.19)
Now Corollary 2.5 and (3.16) give w ∞ ||Q w 0 (x, h D)(I − )|| L 2 →L 2 = ||(I − )Q 0 (x, h D)|| L 2 →L 2 = O(h )
so that, writing w + w Q− 0 (h) := Q 0 (x, h D) and Q 0 (h) := (I − )Q 0 (x, h D)(I − ),
we have − + ∞ Qw 0 (x, h D) = Q 0 (h) + Q 0 (h) + O(h ),
(3.20)
where it is important to remark that (L 2 ) ⊂ B 2 , and u ∈ B 2 iff (I − )u ∈ B 2 . Put H− := (L 2 ), H+ = (I − )L 2 , so that L 2 = H− ⊕ H+ with orthogonal sum. Then ⎧ − ⎨ Q 0 (h) H+ = 0, Q +0 (h) H− = 0, (3.21) ⎩ − 2. Q 0 (h)Q +0 (h)u = Q +0 (h)Q − (h)u = 0, ∀u ∈ B 0 In addition, it is important to notice that Q +0 (h) is also self-adjoint and semi-bounded from below, with domain B 2 and a discrete spectrum, made of a diverging (to +∞) sequence of eigenvalues with finite multiplicities. We therefore have that − ) ∪ Spec(Q + (h) ) = O(h ∞ ), (3.22) dist Spec(Q w (x, h D)), Spec(Q (h) 0 0 0 H H −
+
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where − Spec(Q − 0 (h)) = Spec(Q 0 (h) H ) ∪ {0} ⊂ [−(E 0 + cχ ), E 0 + cχ ], −
where Q +0 (h) H : B 2 ∩ H+ ⊂ H+ −→ H+ , +
and where dist(Spec(P), Spec(Q)) ≤ c means that sup
ζ ∈Spec(P)
dist(ζ, Spec(Q)) ≤ c,
sup
ζ ∈Spec(Q)
dist(ζ, Spec(P)) ≤ c.
Using Corollary 2.5 once more yields ∞ 2 Q w (x, h D)(I − )u, (I − )u = Q w 0 (x, h D)(I − )u, (I − )u + O(h )||u|| , (3.23) for any given u ∈ S(R; C2 ). By the Minimax Theorem this proves, using (3.19), (3.20), (3.21), (3.22) and (3.23), the following proposition. Proposition 3.4. Let E, E > 0 with E > E > E 0 + 102 + cχ , where E 0 is given in (3.17) and cχ in (3.18). There is h 0 ∈ (0, 1] such that Spec(Q w (x, h D)) ∩ (E, +∞) ⊂
[µ(h) − h 10 , µ(h) + h 10 ],
E <µ(h)∈Spec(Q w 0 (x,h D))
for all h ∈ (0, h 0 ]. Moreover, for any given h ∈ (0, h 0 ], for each j ≥ 1 so large that λ j (h) > E, we have, with the same j, |λ j (h) − µ j (h)| ≤ h 10 ,
(3.24)
that is, for j ≥ 1 sufficiently large the j th eigenvalue of Q w (x, h D) is within h 10 -distance to the j th eigenvalue of Q w 0 (x, h D). By Proposition 3.4 we may hence get information about the large eigenvalues of Q w (x, h D), and hence about those of Q w (x, D), by studying the large eigenvalues of Q w 0 (x, h D), where “large” means that the eigenvalues belong to an energy-interval (E, +∞), for (classical) energy E # E 0 . 4. A Localization, Multiplicity and “Cardinality” Theorem Recall that α = β (and that we are considering, with no loss of generality, the case α > β). By virtue of (3.8), applying a variation of Taylor’s decoupling argument (as in [24]; see also [19] and [25]) yields (through Proposition 2.2 of [8] or Theorem A.2.2 of [29]) the following lemma.
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Lemma 4.1. For any given j ∈ Z+ we may find h-independent M2 –valued symbols E −2 j and real scalar symbols ± 2−2 j with, for all α, β ∈ Z+ , β
β
± 2−2 j−(|α|+|β|) |∂xα ∂ξ E −2 j | ≤ C j,αβ m −2 j−(|α|+|β|) , |∂xα ∂ξ ± , 2−2 j | ≤ C j,αβ m
(4.1)
such that ± ± 2 = l± , and 0 λ
− ≥2ε1
and such that for any fixed N ∈ Z+ , on setting E N (h) := N j ± j=0 h 2−2 j , Qw 0 (x, h D)
=
Ew N (h)
= 0, N
j=0 h
(4.2) jE
−2 j
and ±,N (h) :=
0 w ∗ N +1 +,N (h) Ew R N (h), N (h) + h 0 w −,N (h)
(4.3)
w ∗ N +1 ∗ w R N (h) = E w Ew N (h)E N (h) = I + h N (h) E N (h),
(4.4)
||R N (h)|| L 2 →L 2 ≤ C N , ||R N (h)|| L 2 →L 2 ≤ C N , ∀h ∈ (0, 1],
(4.5)
with
where R N (h) has symbol in S 0 (m −2(N +1) ; M2 ) and R N (h) L 2 (R; C2 ) ⊂ B 2 (R; C2 ). ± (Note that the ± 2 , resp. 0 , are the principal, resp. subprincipal, symbols of the ±,N (h).)
Now we fix N = 4, so that (4.4) gives the existence of h 0 ∈ (0, 1] such that h||R4 (h)|| L 2 →L 2 ≤ 1/2 for all h ∈ (0, h 0 ], whence I + h 5 R4 (h) > 0, and we may take the h–dependent operator in L 2 , −1/2 −1/2 5k k h R4 (h) = I + h 5 R (h), h ∈ (0, h 0 ]. = S(h) := I + h 5 R4 (h) k k≥0
˜ ˜ ˜ It is important to notice that R (h) = R4 (h) R(h) = R(h)R 4 (h), where || R(h)|| L 2 →L 2 ≤ 2 2 C, for all h ∈ (0, h 0 ]. Hence R (h)u ∈ B , for all u ∈ L . Setting E(h) = S(h)E 4w (h) gives the following lemma.
Lemma 4.2. There exists h 0 ∈ (0, 1] sufficiently small, and for all h ∈ (0, h 0 ] an L 2 – bounded operator E(h) and scalar (elliptic) symbols ± (h) ∈ S 0 (m 2 ) for which (4.2) holds, such that w + (h) 0 E(h)∗ + h 5 R(h), ∀h ∈ (0, h 0 ], (4.6) Qw (x, h D) = E(h) 0 0 w − (h)
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where ||R(h)|| L 2 →L 2 ≤ C for all h ∈ (0, h 0 ], and such that E(h)E(h)∗ = I = E(h)∗ E(h), ∀h ∈ (0, h 0 ],
(4.7)
E(h)∗ u, E(h)u ∈ B 2 (R; C2 ), ∀u ∈ B 2 (R; C2 ), ∀h ∈ (0, h 0 ].
(4.8)
and
In particular w 0 + (h) w E(h)∗ u, E(h)∗ u + O(h 5 )||u||2 , (4.9) Q 0 (x, h D)u, u = w 0 − (h) for all h ∈ (0, h 0 ]. From Lemma 4.2 and the Minimax Theorem we get the following result. Theorem 4.3. There exists h 0 ∈ (0, 1] and a constant C > 0 for which w dist Spec(Q w (x, h D)), Spec( (h)) ≤ Ch 5 , ± 0
(4.10)
±
for all h ∈ (0, h 0 ]. w w Moreover, with w (h) := diag( w + (h), − (h)) and Spec( (h)) = {−∞ < λ˜ 1 (h) ≤ λ˜ 2 (h) ≤ . . . → +∞}, with repetitions according to multiplicity, |µ j (h) − λ˜ j (h)| ≤ Ch 5 , ∀ j ≥ 1, ∀h ∈ (0, h 0 ].
(4.11)
Remark 4.4. From (3.9), Lemma 3.3 and Lemma 4.1 it follows that the subprincipal sym bols ± 0 λ− ≥2ε1 = 0 (see also [24] and [25]). Then, upon defining the (averaged) action integrals associated with the closed trajectories γ± (E) (where (x0 , ξ0 ) ∈ γ± (E)) by T± 1 1 A± (E) := ξdx − ± 2 exp(t Hl± )(x 0 , ξ0 ) dt, T± γ± (E) T± 0 by the homogeneity we have that for any given E ≥ E 0 , (x0 , ξ0 ) ∈ γ± (E) $⇒ A± (E) = 0,
since
± 2 exp(t Hl± )(x 0 , ξ0 )
≡ E for all t.
We now follow Helffer and Robert [11] (see also [3]). Take ∈ (0, 1) and energies E 1 , E 2 with E 0 < E 1 − < E 1 < E 2 , and define the set (E 1 , E 2 ) of ψ ∈ C0∞ (R), with supp ψ ⊂ (E 1 −, E 2 +), 0 ≤ ψ ≤ 1, and ψ [E ,E ] ≡ 1. Take hence ψ ∈ (E 1 , E 2 ). 1 2 By the Helffer-Robert theory we have that ± (h) := ψ( w ± (h)) is an h-admissible pseudodifferential operator with principal part ψ(λ± ), and bounded in L 2 , uniformly in h. −1 (E) = Remark 4.5. It is important to notice that whenever E > E 0 we have that l± −1 λ± (E) is a smooth closed curve. In other words, no such energy is critical for l± . −1 ([E 1 , E 2 ]) = λ−1 Moreover, for E 0 < E 1 ≤ E 2 one has that l± ± ([E 1 , E 2 ]) is compact and connected.
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Following Chazarain [1], Helffer and Robert [8–11] and Dozias [4] (see also Dimassi and Sjöstrand [3]), we have the following theorem about the spectra of the operators w w ± (h) and hence about that of Q 0 . Theorem 4.6. Let α = β. Let 4E 0 < E 1 < E 1 < E 2 < E 2 . There exists C0 > 0 and h 0 > 0 such that Jk± (h), ∀h ∈ (0, h 0 ], (4.12) Spec( w ± (h)) ∩ [E 1 , E 2 ] ⊂ k∈Z+
where Jk± (h) :=
C0 2 2π C0 2 1 1 h , h . (k + )h − (k + )h + T± 2 2 T± 2 2
2π
As a consequence, with Ik± (h) :=
1 2π 1 (k + )h − C0 h 2 , (k + )h + C0 h 2 , T± 2 T± 2
2π
one has, by possibly shrinking h 0 , Spec(Q w 0 (x, h D)) ∩ [E 1 , E 2 ] ⊂
± k∈Z+
Ik± (h), ∀h ∈ (0, h 0 ].
(4.13)
Proof. As already mentioned, the proof follows Chazarain’s approach used in [1] as developed by Helffer and Robert in [11] (see also [2–5,8,10,27,31]), and we recall it here for the sake of completeness. One uses the Schrödinger equation associated with ˜ ˜ w ± (h). Consider hence, for ψ, ψ ∈ (E 1 , E 2 ), with ψψ = ψ, Uψ± (t) := e−ith
−1 w (h) ±
± (h) = e−ith
−1 w (h) ˜ ± (h) ±
± (h).
(4.14)
(Notice in particular that as long as we are interested in the eigenvalues of w ± (h) in w (h) ˜ (h) by (h).) It is then well-known that for a time [E 1 , E 2 ], we may replace w ± ± ± ± T0 > 0 sufficiently small, one can construct h-Fourier integral operators F (t) approx−1 w ˜ imating the groups e−ith ± (h)± (h) for |t| ≤ T0 . The Schwartz kernels F ± (t; x, y) of ± the F (t) are written as −1 F ± (t; x, y) = (2π h)−1 ei h S± (t,x,η)−yη a± (t; x, y, η; h)dη, where the phase-functions S± are solutions (±-respectively) of the Hamilton-Jacobi equations ˜ ± ) (x, ∂x S± (t, x, η)) = 0, S± = xη, ∂t S± (t, x, η) + λ± ψ(λ t=0 and the amplitudes a± (t, x, y, η; h) ∼
j≥0
h j a± j (t, x, η),
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satisfy the transport equations with initial conditions a0± t=0 = 1, a ± j t=0 = 0, j ≥ 1. One then (see Helffer-Robert [11]) approximates the Uψ± (t) on the intervals [kT0 , (k + 1)T0 ] by using k ˜ ± (h) F ± (t − kT0 )± (h). Fk± (t)± (h) := F ± (T0 ) As in Chazarain [1], one has, for any given N ∈ Z+ , ||Fk± (t)± (h) − Uψ± (t)|| L 2 →L 2 = O(h N +1 ), uniformly for t ∈ [kT0 , (k + 1)T0 ]. Since the set {λ− ≥ E 0 } is invariant for the bicharacteristic flows exp(t Hλ± ), t ∈ R, one has the following lemma. Lemma 4.7. (Chazarain [1], Helffer and Robert [11]). The operators Uψ± (T± ) are hadmissible pseudodifferential operators with supports contained in the (bounded) sets λ−1 ± (E 1 − , E 2 + ), with principal symbols −2πiσ± (h) u± , ψ (x, ξ ) = ψ(λ± (x, ξ ))e
±-respectively, with α± T± β± + , 4 2π h
σ± (h) =
where α± = 2 is the Maslov index of the γ± (E) (the γ± (E) are homotopic to circles) and, with (x0 , ξ0 ) ∈ γ± (E), 1 β± = A± (E) + h T±
T±
0
exp(t H ± )(x , ξ ) dt = 0, λ 0 0 ± 0
for we have γ± (E) ⊂ {λ− ≥ E 0 }, E ≥ E 0 , and ± 0 λ
− ≥E 0
= 0 by Lemma 4.1.
Hence e−2πi h
−1 T± w (h)−hσ (h) ± 2π ±
± (h) = ± (h) + hW± (h),
where ||W± (h)|| L 2 →L 2 = O(1) uniformly in h ∈ (0, h 0 ]. Take now another φ ∈ (E 1 , E 2 ) such that ψφ = φ. Then, by composing with ± (h) := φ( w ± (h)), and by noting that [± (h), W± (h)] = 0, we obtain −1 T± w e−2πi h 2π ± (h)−hσ± (h) ± (h) = ± (h) I + hW± (h) . Hence, for h 0 sufficiently small, we have
I + hW± (h)
−1
= I + hW± (h),
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where the W± (h) are h–admissible of order 0, whence it follows that we may consider (by shrinking h 0 if needed) the logarithms 1 log I + hW± (h) , R± (h) = 2πi h where the R± (h) are h–admissible of order 0, with ˜ [R± (h), w ± (h)± (h)] = 0, and ||R± (h)|| L 2 →L 2 = O(1). Therefore (by (4.14)) e
−2πi h −1
T
± w 2 2π ± (h)−hσ± (h)−h R± (h)
± (h) = ± (h),
which, by the boundedness properties of the R± (h), gives (4.12) (see [11] and also [3]). Using (4.10) and (4.12) then yields, by possibly shrinking h 0 , (4.13) and concludes the proof of the theorem. Now, we may choose h 0 sufficiently small so as to have k = k $⇒ Ik (h) ∩ Ik (h) = ∅, ∀h ∈ (0, h 0 ], j = ±. j
j
Define, for 5E 0 ≤ E 1 < E 2 , the sets (±-respectively) 2π 1 Z± (h) := k ∈ Z+ ; (k + )h ∈ [E 1 , E 2 ] . E 1 ,E 2 T± 2
(4.15)
Since the T± are the least periods of exp(t Hλ± ), ±-respectively, by the Helffer-Robert result in [11] (see also [3] and [4]) we have, for h 0 sufficiently small and for all fixed h ∈ (0, h 0 ], N w± (h) (Jk± (h)) = 1, ∀k ∈ Z± E 1 ,E 2 (h), ±-respectively,
(4.16)
where N P (I ) denotes the number of eigenvalues of P, repeated according to multiplicity, which belong to I . One then gets the following theorem. Theorem 4.8. Let α = β. Suppose there exists C∗ > 0 such that T C∗ 2k + 1 − , ∀k, k ∈ Z+ . − ≥ T+ 2k + 1 2k + 1
(4.17)
Then there exists h 0 ∈ (0, 1], sufficiently small, such that for any fixed h ∈ (0, h 0 ], N Q w0 (x,h D) (Ik± (h)) = 1, ∀k ∈ Z± E 1 ,E 2 (h), ±-respectively. Proof. The proof follows immediately from (4.9), (4.10), (4.11), (4.16) and the Minimax Theorem, and the observation that (4.17) may be rephrased (for a new constant C∗ ) as 2π 2π 1 1 (k + ) ≥ C∗ , ∀k, k ∈ Z+ , (k + ) − T+ 2 T− 2 which yields, for h 0 sufficiently small and any given h ∈ (0, h 0 ], Ik+ (h) ∩ Ik− (h) = ∅, ∀k ∈ Z+E 1 ,E 2 (h), ∀k ∈ Z− E 1 ,E 2 (h), which concludes the proof.
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Define now the set T+ 2k + 1 = . Q E 1 ,E 2 (h) := (k, k ) ∈ Z+E 1 ,E 2 (h) × Z− E 1 ,E 2 (h); 2k + 1 T−
(4.18)
Hence (k, k ) ∈ Q E 1 ,E 2 (h) $⇒ Ik+ (h) = Ik− (h), k =
1 T− (2k + 1) − 1 . 2 T+
(Note, moreover, that (k, k ), (k, k˜ ) ∈ Q E 1 ,E 2 (h) yields k = k˜ , and likewise (k, k ), ˜ ˜ k ) ∈ Q E 1 ,E 2 (h) yields k = k.) (k, Remark 4.9. Notice that since α = β, one then has T− /T+ > 1. Hence it follows that if (k, k ) ∈ Q E 1 ,E 2 (h), then k = k and, being Ik+ (h) = Ik− (h), Ik+ (h) ∩ Ik+ (h) = Ik+ (h) ∩ Ik− (h) = Ik+ (h) ∩ Ik− (h) = ∅, ∀k = k, k . We therefore obtain (through (4.9), (4.10), (4.11), (4.16) and the Minimax) the following theorem. Theorem 4.10. Let α = β. There exists h 0 ∈ (0, 1] so small that for any given h ∈ (0, h 0 ], whenever Q E 1 ,E 2 (h) = ∅ one has N Q w0 (x,h D) (Ik+ (h)) = 2, ∀k ∈ proj1 Q E 1 ,E 2 (h) , where proj1 denotes the projection onto the first factor. By (1.8), Proposition 3.4 and, once more, the Minimax, we have the following consequences of Theorems 4.6, 4.8 and 4.10, that allow us to control the localization, the multiplicity and the “cardinality” of the large eigenvalues of Q w (x, D). Theorem 4.11. Let α = β. Let 10(E 0 + 102 + cχ ) ≤ E 1 < E 2 . With the notation of Theorems 4.6, 4.8 and 4.10, we have: • There exists h 0 ∈ (0, 1] so small that, for (all) h fixed in (0, h 0 ] one has h −1 Ik± (h), Spec(Q w (x, D)) ∩ [E 1 h −1 , E 2 h −1 ] ⊂ ±
(4.19)
k∈Z+
where Ik± (h) ∩ Ik± (h) = ∅, for all h ∈ (0, h 0 ] and all k = k , ±-respectively. • Suppose there exists C∗ > 0 such that (4.17) holds. Then there exists h 0 ∈ (0, 1] so small that, for (all) h fixed in (0, h 0 ] one has N Q w (x,D) (h −1 Ik± (h)) = 1, ∀k ∈ Z± E 1 ,E 2 (h), ±-respectively. Hence, ±-respectively, multiplicity Spec(Q w (x, D)) ∩ h −1 Ik± (h) = 1, ∀k ∈ Z± E 1 ,E 2 (h), (4.20) that is, the eigenvalues of Q w (x, D) belonging to the h −1 Ik± (h), k ∈ Z± E 1 ,E 2 (h), are all simple.
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• There exists h 0 ∈ (0, 1] so small that, for (all) h fixed in (0, h 0 ], whenever Q E 1 ,E 2 (h) = ∅ one has N Q w (x,D) (h −1 Ik+ (h)) = 2, ∀k ∈ proj1 Q E 1 ,E 2 (h) . Proof. The proof follows immediately, using the Minimax Theorem, through Proposition 3.4, Theorems 4.3, 4.6, 4.8 and 4.10 by picking h 0 so small that j
j
Ik (h) ∩ Ik (h) = ∅, ∀h ∈ (0, h 0 ], j = ±, and, whenever (4.17) holds, h 0 so small that Ik (h) ∩ Ik± (h) = ∅, ∀h ∈ (0, h 0 ], j = ±. j
Finally, using the relation Spec(Q w (x, h D)) = h Spec(Q w (x, D)) concludes the proof.
Corollary 4.12. Let α = β, and let 10(E 0 + 102 + cχ ) ≤ E 1 < E 2 . Suppose there exists C∗ > 0 such that (4.17) holds. Then there exists h 0 sufficiently small such that for (all) h fixed in (0, h 0 ] one has + − ∩ ∞ ∩ h −1 Ik (h) = ∅, ∀k ∈ Z E 1 ,E 2 (h), j = ±, 0± ∩ h −1 Ik (h) = ∞ j
j
j
± (recalled in the Introduction) are the sets introduced in [21] (see where 0± and ∞ also [20]).
Proof. This follows from the second point in Theorem 4.11 above, for in this case we have multiplicity Spec(Q w (x, D)) ∩ h −1 Ik± (h) = 1, ∀k ∈ Z± E 1 ,E 2 (h), ±-resp. + ∩ − (see [21]), the result Since higher multiplicity eigenvalues must lie in 0± or ∞ ∞ follows.
As a byproduct of the foregoing corollary, recalling Theorems 2 and 3 of [18] we immediately have the following result. Corollary 4.13. Following [17] and [18], for suitable real numbers a , b , c , d and complex numbers a, e with |a| > 1, let a b c d z − e Hλ (z, ∂z ) = ∂z2 + + + ∂z + , z z−1 z−a z(z − 1)(z − a) be the Heun operator associated with the eigenvalue λ of Q w (x, D) (with associated odd eigenfunction). Let α = β, and let 10 (E 0 + 102 + cχ ) ≤ E 1 < E 2 . Suppose there exists C∗ > 0 such that (4.17) holds. Then there exists h 0 sufficiently small such that for w (all) h fixed in (0, h 0 ], for all k ∈ Z± E 1 ,E 2 (h), ±-resp., and for λ ∈ Spec(Q (x, D)) ∩ ± h −1 Ik (h), one cannot find non-zero rational functions f 1 (z), f 2 (z) such that √ f 1 (z), f 2 (z) z − a ∈ Ker Hλ (z, ∂z ) at the origin.
On the Spectrum of Certain Non-Commutative Harmonic Oscillators
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Proof. By Corollary 4.12 we have that 0− ∩ h −1 Ik± (h) = ∅, for all k ∈ Z± E 1 ,E 2 (h), ±-resp. Hence the result follows at once from Theorems 2 and 3 of [18]. Remark 4.14. As already remarked in the Introduction, (4.20) and Corollary 4.12 complement Proposition 3.14 of [23]. Hence either when condition (4.17) is fulfilled, or when α/β = 1 is not a ratio of positive odd integers and αβ is sufficienly large, then the conclusions of Corollaries 4.12 and 4.13 hold. Notice that for α = β, upon putting !" " # !" " # α β α β m + := min , , , m − := max , β α β α one has 2π m ± (1 + o(1)), as T± = √ αβ
αβ → +∞, , α/β constant = 1
so that T+ /T− −→ m + /m − . Since, as shown in Proposition 3.14 of [23], the eigenvalues of Q w (x, D) smaller than or equal to any fixed E > 0 are still simple for all αβ sufficiently large with α/β fixed, even when m + /m − is a ratio of certain positive odd integers (that is, in the case α < β, say, when m + /m − = α/β = (2m 0 + 1)/(2n 0 + 1) with m 0 = n 0 − 2 − 4k, any given k ∈ Z+ ),√ it would be interesting to carry out a refined study of the periods T± as functions of 1/ αβ and α/β in order to get more precise spectral information also in the case Q± E 1 ,E 2 (h) = ∅ or, more generally, when condition (4.17) does not hold. Remark 4.15. We direct the reader also to [25], where it is shown (among other things) that Theorem 4.11 holds true for a suitable class of 2 × 2 positive globally elliptic homogeneous systems of order 2 that contains the non-commutative harmonic oscillators Qw (α,β) (x, D) (with α = β). Acknowledgement. We wish to thank the referee, L. Maniccia and F. Hérau for the useful remarks.
References 1. Chazarain, J.: Spectre d’un hamiltonien quantique et mécanique classique. Comm. P.D.E. 5(6), 595–644 (1980) 2. Colin de Verdière, Y.: Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques. Comment. Math. Helv. 54(3), 508–522 (1979) 3. Dimassi, M., Sjöstrand, J.: Spectral asymptotics and the semi-classical limit. London Math. Soc. Lect. Note Series 268, Cambridge: Cambridge University Press, 1999 4. Dozias, S.: Clustering for the spectrum of h-pseudodifferential operators with periodic flow on an energy Surface. J. Funct. Anal. 145, 296–311 (1997) 5. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975) 6. Evans, L.C., Zworski, M.: Lectures on Semiclassical Analysis. Notes of the course, UC Berkeley, http://math.berkeley.edu/~zworski/semiclassical.pdf 7. Helffer, B.: Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques. Astérisque 112, Paris: Soc. Math. de France, 1984 8. Helffer, B., Robert, D.: Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques. Ann. Inst. Fourier 31(3), 169–223 (1981) 9. Helffer, B., Robert, D.: Propriétés asymptotiques du spectre d’opérateurs pseudodifferentiels sur Rn . Commun. P. D. E. 7, 795–882 (1982)
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10. Helffer, B., Robert, D.: Comportement semi-classique du spectre des hamiltoniens quantiques hypoelliptiques. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 9(3), 405–431 (1982) 11. Helffer, B., Robert, D.: Puits de potentiel généralisés et asymptotique semi-classique. Ann. Inst. Henri Poincaré 41(3), 291–331 (1984) 12. Ichinose, T., Wakayama, M.: Zeta functions for the spectrum of the non-commutative harmonic oscillators. Commun. Math. Phys. 258, 697–739 (2005) 13. Ichinose, T., Wakayama, M.: Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations. Kyushu J. Math. 59(1), 39–100 (2005) 14. Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics. Springer Monographs in Mathematics. Berlin-Heidelberg-New York: Springer Verlag, 1998 15. Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Berlin-Heidelberg-New York: Universitext Springer-Verlag, 2002 16. Nagatou, K., Nakao, M.T., Wakayama, M.: Verified numerical computations for eigenvalues of noncommutative harmonic oscillators. Numer. Funct. Anal. Opt. 23, 633–650 (2002) 17. Ochiai, H.: Non-commutative harmonic oscillators and Fuchsian ordinary differential operators. Comm. Math. Phys. 217, 357–373 (2001) 18. Ochiai, H.: Non-commutative harmonic oscillators and the connection problem for the Heun differential equation. Lett. Math. Phys. 70, 133–139 (2004) 19. Parenti, C., Parmeggiani, A.: Lower Bounds for Systems with Double Characteristics. J. D’Analyse Math. 86, 49–91 (2002) 20. Parmeggiani, A., Wakayama, M.: Oscillator representations and systems of ordinary differential equations. Proc. Nat. Acad. Sci. U.S.A. 98(1), 26–30 (2001) 21. Parmeggiani, A., Wakayama, M.: Non-commutative harmonic oscillators-I, -II. Forum Math. 14, 539–604 (2002) ibid. 669–690 22. Parmeggiani, A., Wakayama, M.: Corrigenda and remarks to: Non-commutative harmonic oscillatorsI. Forum Math. 15, 955–963 (2003) 23. Parmeggiani, A.: On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators. Kyushu J. Math. 58(2), 277–322 (2004) 24. Parmeggiani, A.: On the spectrum of certain noncommutative harmonic oscillators. In: Proceedings of the Conference Around hyperbolic problems: in memory of Stefano, Annali dell’Università di Ferrara 52, 431–456 (2006) 25. Parmeggiani, A.: Introduction to the Spectral Theory of Non-Commutative Harmonic Oscillators. COE Lecture Note, vol. 8. Kyushu University, The 21st Century COE Program “DMHF”, Fukuoka, vi+233 (2008) 26. Robert, D.: Propriétés spectrales d’opérateurs pseudodifferentiels. Comm. Partial Differ. Eqs. 3(9), 755–826 (1978) 27. Robert, D.: Calcul fonctionnel sur les opérateurs admissibles et application. J. Func. Anal. 45, 74–94 (1982) 28. Robert, D.: Autour de l’Approximation Semi-Classique. Progress in Mathematics 68, Basel-Boston: Birkhäuser, 1987 29. Shubin, M.: Pseudodifferential Operators and Spectral Theory. Berlin-Heidelberg-New york: Springer Verlag, 1987 30. Taylor, M.: Pseudodifferential Operators. Princeton, NJ: Princeton University Press, 1981 31. Weinstein, A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44(4), 883–892 (1977) Communicated by P. Sarnak
Commun. Math. Phys. 279, 309–354 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0414-8
Communications in
Mathematical Physics
Linking and Causality in Globally Hyperbolic Space-times Vladimir V. Chernov (Tchernov)1 , Yuli B. Rudyak2 1 Department of Mathematics, 6188 Kemeny Hall, Dartmouth College,
Hanover, NH 03755, USA. E-mail: [email protected]
2 Department of Mathematics, University of Florida, 358 Little Hall, Gainesville,
FL 32611-8105, USA. E-mail: [email protected]
Received: 16 November 2006 / Accepted: 17 July 2007 Published online: 12 February 2008 – © Springer-Verlag 2008
Abstract: The classical linking number lk is defined when link components are zero homologous. In [15] we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the study of causality in Lorentzian manifolds. Let M m be a spacelike Cauchy surface in a globally hyperbolic space-time (X m+1 , g). The spherical cotangent bundle ST ∗ M is identified with the space N of all null geodesics in (X, g). Hence the set of null geodesics passing through a point x ∈ X gives an embedded (m − 1)-sphere Sx in N = ST ∗ M called the sky of x. Low observed that if the link (Sx , S y ) is nontrivial, then x, y ∈ X are causally related. This observation yielded a problem (communicated by R. Penrose) on the V. I. Arnold problem list [3,4] which is basically to study the relation between causality and linking. Our paper is motivated by this question. The spheres Sx are isotopic to the fibers of (ST ∗ M)2m−1 → M m . They are nonzero homologous and the classical linking number lk(Sx , S y ) is undefined when M is closed, while alk(Sx , S y ) is well defined. Moreover, alk(Sx , S y ) ∈ Z if M is not an odddimensional rational homology sphere. We give a formula for the increment of alk under passages through Arnold dangerous tangencies. If (X, g) is such that alk takes values in Z and g is conformal to g that has all the timelike sectional curvatures nonnegative, then x, y ∈ X are causally related if and only if alk(Sx , S y ) = 0. We prove that if alk takes values in Z and y is in the causal future of x, then alk(Sx , S y ) is the intersection number of any future directed past inextendible timelike curve to y and of the future null cone of x. We show that x, y in a nonrefocussing (X, g) are causally unrelated if and only if (Sx , S y ) can be deformed to a pair of S m−1 -fibers of ST ∗ M → M by an isotopy through skies. Low showed that if (X, g) is refocussing, then M is compact. We show that the universal cover of M is also compact.
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1. Preliminaries We work in the C ∞ -category, and the word “smooth” means C ∞ . An isotopy of a smooth embedding f : P → Q is a path in the space of smooth embeddings P → Q starting at f. Given an oriented manifold M m , consider its tangent bundle T M → M and put z : M → T M to be the zero section. Let R+ be the group of positive real numbers under multiplication that acts on T M as (r, µ) → r µ, r ∈ R+ , µ ∈ T M. We put ST M = (T M \ z(M)) /R+ and note that the tangent bundle T M → M yields the spherical tangent bundle pr : ST M → M of M. For the reasons discussed before Theorem 2, we will assume that dim M > 1. We denote by T ∗ M → M the cotangent bundle over M, and we construct the spherical cotangent bundle pr : ST ∗ M → M in a similar way. It is well known that ST ∗ M possesses the canonical contact structure and that the S m−1 -fibers of ST ∗ M are Legendrian submanifolds with respect to this contact structure, see [2] or Appendix A. Note also that the orientation of M yields canonical orientations on the fibers of spherical (co)tangent bundles. Namely, it is well known that every spherical (co)tangent bundle is canonically oriented, and we orient a fiber S m−1 via the convention that the orientation of ST ∗ M is given by the pair (orientation of the base M, orientation of the fiber S m−1 ). Given a path α : [a, b] → M, consider the bundle E → [a, b] induced by α from ST ∗ M → M. So, we have the commutative diagram E ⏐ ⏐
α
−−−−→ ST ∗ M ⏐ ⏐ α
[a, b] −−−−→
M.
Choose a trivialization ι : S m−1 × [a, b] → E of the bundle E → [a, b]. Define ι
α
εα : S m−1 × [a, b] −−−−→ E −−−−→ ST ∗ M.
(1.1)
Given a point v ∈ M, consider the constant path α : [0, 1] → M, α(0) = v and define εv : S m−1 −→ ST ∗ M, εv (s) = εα (s, 0). (1.2) Two such maps εv and εv are isotopic via an isotopy such that the projection of its trace lies in a small disk containing v. If the group Diff + (S m−1 ) of degree one autodiffeomorphisms of S m−1 is connected, then such an isotopy can be chosen so that its trace is inside pr −1 (v). For example this holds for S 3 , see [14], for S 2 , see [31,43] and for S 1 , for trivial reasons. So since dim M > 1, any two links (εu 1 , εv1 ) and (εu 2 , εv2 ), u 1 = v1 , u 2 = v2 , are isotopic. If m = 2, 3, 4 then π0 (Diff + (S m−1 )) = 0, and hence any two embeddings εv and εv are Legendrian isotopic via an isotopy whose trace is contained in pr −1 (v). One can show 1 that if m = 5, 6 then any εv and εv are Legendrian isotopic via an isotopy such that the projection of its trace is inside of a small disk containing v. For these cases any two links (εu 1 , εv1 ) and (εu 2 , εv2 ), u 1 = v1 , u 2 = v2 , are Legendrian isotopic. 1 Since the sequence π (Diff + (D m )) → π (Diff + (S m−1 )) → Γ → 0 is exact and the twisted sphere m 0 0 groups Γ5 , Γ6 are zero, every degree one autodiffeomorphism of S m−1 , m = 5, 6 extends to an autodifm m feomorphism of the unit disk D ⊂ R . By the results of Palais, Cerf, Milnor [33, Theorem 9.6] every orientation preserving embedding of D m to Rm is ambient isotopic to the identity map. Hence every degree one autodiffeomorphism of the standard unit S m−1 ⊂ Rm is isotopic to the identity map, cf. [33, Remark,
p. 122]. Now one uses the exponent map expv and the front projection description of Legendrian knots in ST ∗ M, see Example 8, to get the proof.
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Definition 1. Let f, g : S m−1 → ST ∗ M be two embeddings with disjoint images that are homotopic to a map εw for some w ∈ M m . We say that the pair ( f, g) is unlinked or trivially linked if there exists a path γ in the space of smooth embeddings S m−1 S m−1 → ST ∗ M that joins ( f, g) to a pair (εu , εv ), u, v ∈ M, u = v. If both embeddings f, g : S m−1 → ST ∗ M are Legendrian, we say that the pair ( f, g) is Legendrian unlinked or Legendrian trivially linked if there exists a path γ as above in the space of smooth Legendrian embeddings. (Any two trivial links are isotopic, but for m = 2, 3, 4, 5, 6 we do not know if it may happen that two Legendrian trivial links are not Legendrian isotopic.) Definition 2. A vector field on a manifold Y is a smooth section of the tangent bundle τY : T Y → Y , and a vector field along a (smooth) map φ : Y1 → Y2 of one manifold to another is a smooth map Φ : Y1 → T Y2 such that φ = τY2 ◦ Φ. Covector (direction, codirection, line, etc.) fields on a manifold and along a map φ are defined in a similar way. For brevity we will often write “φ is equipped with vector field” rather than “φ is equipped with a vector field along it”, etc. Now we recall some basic concepts of Lorentzian geometry. Definition 3. (a) Consider a smooth manifold X m+1 equipped with a Lorentz metric g. A nonzero vector ξ ∈ T X is said to be timelike, non-spacelike, null (lightlike), or spacelike if g(ξ, ξ ) is negative, non-positive, zero, or positive, respectively. A piecewise smooth curve is called timelike, non-spacelike, null, or spacelike if all of its velocity vectors are respectively timelike, non-spacelike, null, or spacelike. A smooth submanifold M m ⊂ X m+1 is spacelike if the restriction of g to M is a Riemannian metric. (b) For each x ∈ X the set of all non-spacelike vectors in Tx X consists of two connected components that are hemicones. A continuous (with respect to x ∈ X ) choice of a hemicone of non-spacelike vectors in Tx X is called the time orientation of (X, g). (c) The non-spacelike vectors from the chosen hemicones are called future pointing vectors. A piecewise smooth curve is said to be future directed if all of its velocity vectors are future pointing. Definition 4. (a) A space-time X = (X m+1 , g) is a smooth connected time-oriented Lorentz (m + 1)-manifold without boundary. An event is a point of the space-time X. (b) Two Lorentz metrics g and g on X m+1 are conformal if g = Ω 2 g for some nowhere zero smooth function Ω : X → R. If g and g are conformal, then a vector ξ ∈ T X is timelike, nonspacelike, null, or spacelike for g if and only if it is timelike, nonspacelike, null, or spacelike for g , respectively. (c) For two events x, y ∈ X we write x << y if there is a piecewise smooth future directed timelike curve from x to y. We write x ≤ y if x = y or if there is a piecewise smooth future directed non-spacelike curve from x to y. For x ∈ (X, g) we put the causal future of x to be J + (x) = {y ∈ X |x ≤ y} and we put the causal past of x to be J − (x) = {y ∈ X |y ≤ x}. We put the chronological future of x to be I + (x) = {y ∈ X |x << y} and we put the chronological past of x to be I − (x) = {y ∈ X |y << x.} It is easy to see that the causal and the chronological past and future of x depend only on the conformal class of the metric g on X. (d) Two events x, y are causally related if x ∈ J + (y) or y ∈ J + (x). (e) An open neighborhood is causally convex if there are no non-spacelike curves intersecting it in a disconnected set. A space-time is strongly causal if every point in it has arbitrarily small causally convex neighborhoods. A strongly causal space-time (X, g) is globally hyperbolic if J + (x) ∩ J − (y) is compact for all x, y ∈ X.
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(f) A Cauchy surface M is a subset of X such that for every inextendible non-spacelike curve γ (t) in X there exists exactly one value t0 of t with γ (t0 ) ∈ M. Clearly y ∈ J + (x) if and only if x ∈ J − (y); and y ∈ I + (x) if and only if x ∈ I − (y). The sets I ± (x) are always open, see [9, Lemma 3.5]; however, the sets J ± (x) are in general neither closed nor open, see [9, pp. 5–6]. A space-time can be shown to be globally hyperbolic if and only if it admits a Cauchy surface, see [22, pp. 211–212]. Example 1. Fix a connected oriented Riemannian manifold (M, g). Put · to be the standard Riemannian metric on R1 . Define the Lorentz metric g on M ×R via g((ξ1 , η1 ), (ξ2 , η2 )) = g(ξ1 , ξ2 ) − η1 · η2 , for (ξi , ηi ) ∈ T p M × Tt R = T( p,t) (M × R), i = 1, 2. We denote the space-time (M ×R, g) by (M ×R, g ⊕−dt 2 ) and call it a static space-time. If (M, g) is a complete Riemannian manifold, then the static space-time (M ×R, g⊕−dt 2 ) is globally hyperbolic, see [9, Theorem 3.66], and each M × t is a Cauchy surface. The pioneer result of Geroch [20] says that every globally hyperbolic space-time X is homeomorphic to Σ m × R, where every Σ × t ⊂ X is a (topological) Cauchy surface. The question of existence of smooth Cauchy surfaces was considered by Seifert [41] and Dieckmann [17], cf. [34, Remark 3.77]. Bernal and Sanchez [5, Theorem 1], [6, Theorem 1.1], [7, Theorem 1.2] proved the following strong result: Theorem 1. For every globally hyperbolic space-time X m+1 there is an isometry and in the same time a diffeomorphism h : (M m × R, −βdt 2 + g) → (X, g), where M is a smooth manifold, t : M ×R → R is the projection, β : M ×R → (0, +∞) is a smooth function and g is a smooth 2-covariant symmetric tensor field on M × R satisfying the following conditions: 1. for each q ∈ M × R the vector grad t ∈ Tq (M × R) is timelike and past pointing; 2. for each t the submanifold M ×t of M ×R is a smooth space-like Cauchy surface 2 (i.e. it is a Cauchy surface and the restriction of −βdt 2 + g to it is a Riemannian metric); 3. for each q ∈ M × R, the radical of g at q is equal to span{grad t} = span{∂/∂t} ⊂ Tq (M × R). Here the radical of g at q is the subspace of Tq (M × R) consisting of vectors that are g-orthogonal to all vectors in Tq (M × R). In particular, the vector ∂/∂t is time-like and future pointing, the function t is increasing along all future pointing non-spacelike curves, and the vector ∂/∂t is everywhere (−βdt 2 + g)-orthogonal to the smooth spacelike Cauchy surfaces M × t of M × R. Moreover, for every smooth spacelike Cauchy surface M ⊂ X there is an isometry h : (M × R, βdt 2 + g) → X as above such that h(m, 0) = m, for all m ∈ M. Also any two smooth spacelike Cauchy surfaces M1 , M2 of X are diffeomorphic. 2 The definition of the Cauchy surface Bernal and Sanchez use in [5] looks a bit weaker than the one we use. They define Cauchy surface to be a subset of X that is intersected exactly once by every inextendible timelike curve, rather than by every inextendible non-spacelike curve as we do. However as Sanchez explained to us, their spacelike Cauchy surface would be a Cauchy surface in our sense. This is since every non-spacelike curve intersects the Cauchy surface in their sense at least once, see [36, Sect. 14, Lemma 29]. Moreover since their Cauchy surface is spacelike, every non-spacelike curve would intersect it at most once, see [36, Sect. 14, Lemma 42].
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Convention. Throughout the paper the space-time (X, g) is assumed to be globally hyperbolic and oriented. The term “Cauchy surface” always means “smooth space-like Cauchy surface”. Since by our definition every space-time is connected, the Geroch Theorem [20] implies that a Cauchy surface is connected. Definition 5. Given a space-time (X, g) and a Cauchy surface M ⊂ X , let h : M ×R → X be an isometry as in Theorem 1. We say that the isometry h is M-proper if h(m, 0) = m for all m ∈ M. Given t ∈ R, we let Mt = h(M × t) ⊂ X and define h t : M → Mt , h t (m) = h(m, t) ∗ ∗ for all m ∈ M. Furthermore, we put gt = h −1 t : Mt → M. We put ST gt : ST Mt → ∗ ∗ ∗ ∗ ST M, ST gt : ST Mt → ST M, ST h t : ST M → ST Mt , ST h t : ST M → ST Mt to be the induced maps. Given a Cauchy surface M ⊂ X , choose an M-proper h : M × R → X and orient M × R by requiring h to be orientation-preserving. Now, we orient M so that the pair (orientation of M, orientation of R) gives the orientation of M ×R. Here the orientation of R is given by the time orientation of X . Definition 6. Let N denote the space of all future directed null-geodesics in (X, g) modulo orientation preserving affine reparameterizations. The sky Sx ⊂ N of an event x ∈ X is the subspace of all future directed null-geodesics passing through x. Example 2. Let g be the metric on S m induced by the identification of S m with the unit sphere in Rm+1 . Since (S m , g) is complete, the (m + 1)-dimensional Einstein cylinder (X, g) := (S m × R, g ⊕ −dt 2 ) is globally hyperbolic, see [9, Theorem 3.66]. Given s ∈ S m and t ∈ R, put x = (s, t) and x = (s, t + 2π ). Then, clearly, Sx = Sx although x = x . For our future goals (see Sect. 2) we need the concept of linking for skies. To this aim we relate skies in N with (lifted) wave fronts in the spherical cotangent bundle of a Cauchy surface. We explain this below. Fix a Cauchy surface M ⊂ X. An inextendible future directed null-geodesic γ intersects M in one point x = x(γ ). Since M is a spacelike surface, a g-orthogonal to M line field L y , y ∈ M, is not tangent to M. Since g is non-degenerate, the lines L y , y ∈ M, do not contain null vectors. Thus Tx X = Tx M ⊕ L x and γ˙ (x) = ξ + η ∈ Tx M ⊕ L x = Tx X, ξ ∈ Tx M, η ∈ L x ,
(1.3)
with ξ = 0, η = 0, and g(ξ, η) = 0. In this way we get a bijective map ϕ = ϕ M : N → ST M,
(1.4)
where ϕ(γ ) is the point of ST M corresponding to the nonzero vector ξ. Since M is a space-like surface, g| M is a Riemannian metric. This allows us to identify ST M with the total space ST ∗ M of the spherical cotangent bundle, that has the natural contact structure. Thus for a space-like Cauchy surface M we get a bijective map ψ = ψ M : N → ST ∗ M
(1.5)
that equips N with the structure of a smooth contact manifold. Low showed [28] that if M and M are two smooth Cauchy surfaces, then the map
−1 ∗ ∗ f MM = ψ M ◦ ψ M : ST M → N → ST M
is a contactomorphism. (Strictly speaking the work [28] deals only with 3 + 1-dimensional space-times. However this result holds for space-times of all the dimensions, see [35, pp. 252–253].)
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x,M = εx : Definition 7. Let M be a Cauchy surface in X and x ∈ M . We put W S m−1 → ST ∗ M . Now, for an arbitrary (smooth spacelike) Cauchy surface M ⊂ X , we x,M = f M ◦εx : S m−1 → ST ∗ M. We call this embedding a lifted wave front of x put W M x,M : S m−1 → M (with respect to the Cauchy surface M). A wave front Wx,M = pr ◦W x,M to a smooth spacelike Cauchy surface M. is the projection of the lifted front W x,M is a Legendrian embedding S m−1 → ST ∗ M. Indeed, if The lifted wave front W M is a smooth spacelike Cauchy surface passing through x that exists by Theorem 1, then x,M : S m−1 → ST ∗ M is εx that is Legendrian, see [2] or Appendix A. Since f M is W M x,M = f M ◦ εx is Legendrian. (This x,M = f M ◦ W a contactomorphism, we get that W M M explanation of why a lifted wave front is Legendrian was given to us by Jose Natario, and we are grateful to him for it.) Skies and lifted wave fronts are related as follows. Let ψ = ψ M . Since x ∈ M , we conclude that ψ(Sx ) is the unit cotangent sphere Sx that is the fiber of ST ∗ M → M over x. Identifying Sx with S m−1 , we conclude that ψ|Sx : S m−1 → ST ∗ M is the x,M . We see that for each Cauchy surface M the lifted wave front W x,M = map εx = W x,M is completely determined by the sky Sx . Moreover, if we know W x,M for f MM ◦ W some M, then we can restore the sky Sx . Note that there are examples of space-times where Sx = S y , for some x = y. Hence it is not generally possible to restore x from x,M . Sx or from W x,M to A front Wx,M is equipped with the natural codirection field defining the lifting W ∗ ∗ ST M. In view of the identification ST M = ST M (given by the Riemannian metric on x,M : S m−1 → ST M = ST ∗ M M), this codirection field yields a direction field. Since W is Legendrian, this direction field is everywhere orthogonal to the front with respect to the Riemannian metric g| M , cf. Example 8. In terms of skies, this (co)direction field can be described as follows. For every (equivalence class of a) null-geodesic γ ∈ Sx the direction ϕ M (γ ) ∈ ST M is orthogonal to Wx,M . So, the direction and codirection fields are {ϕ M (γ ) γ ∈ Sx } and {ψ M (γ ) γ ∈ Sx }, respectively. We see that there is no essential difference between the sky Sx and the lifted wave front Wx,M , for a Cauchy surface M. In fact, skies enable us to formulate the results in an elegant invariant way (without making a choice of a Cauchy surface), while the wave fronts play the role of technical tools that are useful for proofs. Definition 8. Assume that the events x and y do not lie on a common null geodesic in a space-time (X, g). Then the skies Sx and S y are disjoint and hence for every y,M ) is nonsingular. We say that the pair of x,M , W Cauchy surface M ⊂ X the link (W skies (Sx , S y ) is unlinked or trivially linked (respectively Legendrian unlinked or Legendrian trivially linked) if, for every Cauchy surface M ⊂ X , the pair of lifted x,M , W y,M ) is unlinked (respectively Legendrian unlinked), as defined wave fronts (W in Definition 1. Remark 1. (a) If the events x and y lie on a common null geodesic, then x and y are causally related for a trivial reason. In this case Sx ∩ S y = ∅ and hence for every x,M , W y,M ) is singular. So, the assumption that x and y Cauchy surface M the link (W do not lie on a common null geodesic does not lead to loss of generality. (b) In Theorem 9 we show that the skies are unlinked provided that the pair of lifted x,M0 , W y,M0 ) is unlinked for any particular Cauchy surface M0 . wave fronts (W (c) As we see from Example 2, it can happen that Sx = S y for some x = y. In this x,M = W y,M up to reparameterization. case, for each Cauchy surface M, we have W
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Construction 1. To make the picture more familiar for some of the readers, choose a Cauchy surface M and an M-proper isometry h : M × R → X . Given x ∈ X , define t xt = W x,M x,Mt : S m−1 → ST ∗ Mt → ST ∗ M. W = ST ∗ gt ◦ W
xt : S m−1 → M, t ∈ R can be regarded as the wave front Then the family Wxt := pr ◦W t = εgt (x) . propagating in M. Note that for x ∈ Mt we have W x,M Below we list notation and concepts that are used in the paper consistently. Glossary. The basic concepts of Lorentz geometry: space-time (assumed to be globally hyperbolic and oriented), conformal Lorentz metrics, causality, future and past directed curves, Cauchy surfaces (assumed to be smooth and space like), the sets J ± and I ± , etc, - are defined in Definitions 3 and 4. Differential-geometric notions for Lorentz manifolds are briefly reminded in Definition 27. Given a globally hyperbolic space-time X and a Cauchy surface M ⊂ X , put h : M ×R → X to be an isometry as in Theorem 1. We denote by π M = π M,h : X → M the composition of h −1 and of the projection M × R → M. Similarly πR = πR,h : X → R is the composition of h −1 and of the projection M × R → R. Given t ∈ R, we put Mt = h(M × t) ⊂ X , gt = π M | Mt : Mt → M and h t = gt−1 , cf. Definition 5. The maps εα and εv are described in (1.1) and (1.2), respectively. x,M and wave fronts Wx,M are described in Definition 7. For The lifted wave fronts W t and W t see Construction 1. the description of propagating wave fronts W x,M x,M Given a smooth curve γ : R → Y and t0 ∈ R, we denote by γ˙ (t0 ) the velocity vector of γ at t0 . 2. Introduction and Results Low [25–27] observed that two events (in a globally hyperbolic space-time) are causally related if their skies are linked in N . We explain this in greater detail in Sect. 3. The Low observation yielded a problem on V. I. Arnold’s 1998 Problem List [3, Problem 8], [4, Problem 1998-21] which is basically to study the relation between causality and the linking of skies. According to [3,4], the problem was communicated by R. Penrose. Our paper is motivated by the above and similar questions. Namely, we study relations between link theory and causality. Here we have the following three directions of research. 1. Detection of linking. Given two skies, how can we recognize whether they are linked or not? 2. Suitable space-times. Low conjectured that if the Cauchy surface is a 2-disk with holes, then two events x, y are causally related if and only if the skies Sx and S y are linked. Some special cases of this conjecture were proved by Natario and Tod [35]. For (m + 1)-dimensional space-times with m > 2 the obvious extension of the Low conjecture fails: Low [25] constructed an example of two causally related events x, y in a (3+1)-dimensional globally hyperbolic space-time with Cauchy surface diffeomorphic to R3 such that the pair (Sx , S y ) is unlinked. So an interesting question is: For which space-times the skies of every two causally related events are linked? One of results in this direction is Theorem 3.
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3. Suitable isotopies. It is not currently known whether the skies in the above mentioned example of Low [25] are Legendrian unlinked. The modified Low conjecture posed by Natario and Tod [35] says that for (3 + 1)-dimensional globally hyperbolic space-times, whose Cauchy surface M 3 is diffeomorphic to a submanifold of R3 , two events are causally related if and only if their skies are Legendrian linked. Nevertheless the following is an example of two causally related events in a globallyhyperbolic space-time whose skies are unlinked even in the Legendrian sense. Example 3. Let g be the metric on S m induced by the identification of S m with the unit sphere in Rm+1 . Since (S m , g) is complete, the (m + 1)-dimensional Einstein cylinder (S m × R, g ⊕ −dt 2 ) is globally hyperbolic, see [9, Theorem 3.66]. Given s ∈ S m , t ∈ R put x = (s, t), x = (s, t + 2π ) ∈ S m × R. Put n = −s ∈ S m ⊂ Rm+1 and put y = (n, t + 2π ) ∈ S m × R. It is easy to see that the events x and y are causally related but (Sx , S y ) = (Sx , S y ) are Legendrian unlinked. Since S m is not a submanifold of Rm , this example contrasts but does not contradict the modified Low conjecture of Natario and Tod [35]. We can, however, consider a link isotopy that is even finer than Legendrian. Namely, in Sect. 12 we prove that, for so-called nonrefocussing space-times, two events x, y are causally related if and only if the link (Sx , S y ) is isotopic through skies to the trivial link, see Theorem 6. Now we explain the results in greater detail. We start with detecting that the link (Sx , S y ) is nontrivial. One of the goals of the paper is to define a generalized linking number of the pair of skies (Sx , S y ) that vanishes (i.e. is equal to zero) for unlinked pairs. In particular, the events x and y are causally related if this invariant does not vanish for the pair (Sx , S y ). Note that for many space-times the vanishing of our invariant implies that (Sx , S y ) is unlinked, see Theorem 3. The (Gauss) linking number lk is the classical invariant that often allows one to detect that the link is nontrivial. It is defined as the intersection number of the singular chain whose boundary is one of the linked manifolds with the other linked manifold. In order for lk to be well-defined, the two linked submanifolds have to be zero homologous and the sum of their dimensions should be by one less than the dimension of the ambient x,M , W y,M ) consists of two copies of S m−1 in (ST ∗ M)2m−1 . Since space. The link (W y,M ) would be x,M , W (m − 1) + (m − 1) + 1 = (2m − 1), the linking number lk(W x,M , W y,M were zero homologous. Unfortunately, W x,M , W y,M are well-defined if W m−1 homotopic to a positively oriented S -fiber of pr : ST M → M which generally is x,M , W y,M ) is undefined. not zero homologous, and thus the linking number lk(W When M m = Int P m , for some manifold P with ∂ P = ∅, one can take two auxiliary m−1 ∗ negatively oriented fibers S m−1 points p1 , p2 ∈ p1 , S p2 of ST P → P over two distinct x,M , W y,M ) = lk (W x,M S m−1 ∂ P and define a (modified) linking number lk(W p1 ), y,M S m−1 (W p1 ) . This was exactly the trick used by Low [25–27] to define his linking numbers of the skies in ST ∗ Rm . Before Low this way of defining linking numbers for nonzero homologous circles in ST R2 was used by S. Tabachnikov [46]. The general theory of linking numbers when the linked objects are zero homologous in the homology group of the ambient manifold modulo boundary was developed by U. Kaiser [24]. When M is a closed manifold, the number lk defined using auxiliary negative fibers over some x,M , W y,M ), since it points is not an invariant of the equivalence class of the link (W changes when a link component passes through the auxiliary fiber corresponding to the other link component.
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In [15] we constructed the affine linking invariant that should be thought of as the generalization of the linking number lk to the case of linked oriented submanifolds realizing arbitrary homology classes. x,M , In this paper we use this theory to introduce the affine linking number alk(W y,M ). This alk invariant does not depend on the Cauchy surface M, see Theorem 9. W Hence it is an invariant of the two events x, y (that do not lie on a common null geodesic) and it can be interpreted as the affine linking number alk(Sx , S y ) of the skies Sx , S y . Actually, it was a very preliminary version [16] of this paper that motivated our work [15]. Since [15] was published before this work, we rewrote it to avoid reproving results proved in [15]. We also changed the setup of the work to be more familiar to people working in Lorentz geometry and included new results about space-times for which alk completely determines causality, about the relations between alk and the intersection index, and about space-times for which a weakened Low conjecture holds. The general alk invariant constructed in [15] takes values in a group that depends on the ambient manifold and on the homotopy classes of the linked submanifolds. The computation of the group is quite hard. In the rest of the paper we assume that space-times have dimension > 2, and hence that Cauchy surfaces have dimension > 1. The reason is the following. For a 2-dimensional globally hyperbolic space-time its Cauchy surface M is 1-dimensional and the lifted wave fronts are maps of S 0 . Since S 0 is not connected, [15, Theorem 7.4 and Corollary 7.5] that give a homotopy theoretical description of the range of values of the alk-invariant do not apply. Luckily in this case the Cauchy surface M is R or S 1 and all the links in ST M are easily classified by combinatorial methods. Combining Theorem 8, Theorem 9, Theorem 10, and Proposition 3 of this work we get the following result. Theorem 2. Let M m , m > 1, be a Cauchy surface in a globally hyperbolic space-time. Then the following holds: 1. If M is not an odd-dimensional rational homology sphere with finite π1 (M), then alk(Sx , S y ) is a well defined Z-valued invariant of the link (Sx , S y ). 2. If M is an odd-dimensional rational homology sphere, then the invariant alk(Sx , S y ) is well-defined if one regards it as having values in Z/(Im deg). Here deg : πm (M m ) → Z is the homomorphism that maps [α] ∈ πm (M m ) to the degree of α : Sm → M m . 3. The only odd-dimensional manifolds for which this quotient Z/ Im deg is the trivial group are odd-dimensional homotopy spheres. 4. If x, y are causally unrelated, then alk(Sx , S y ) = 0. Remark 2. In [15, Subsect. 3.2] we proved that the affine linking invariant alk is a Vassiliev-Goussarov [47,21] invariant of order ≤ 1. It is universal in the sense that it distinguishes all the link homotopy classes that can be distinguished using order ≤ 1 invariants with values in an abelian group. Since the invariant alk constructed in this paper is a particular case of the general construction from [15], we get that x,M , W y,M ) = alk(Sx , S y ) is a universal Vassiliev-Goussarov link homotopy alk(W invariant of order ≤ 1 of two linked S m−1 -spheres in ST ∗ M that are homotopic to a positively oriented fiber S m−1 of pr : ST ∗ M → M m , m > 1. A 2-plane E s ⊂ Ts X is called timelike if g| E s is nondegenerate and not positive definite. A timelike sectional curvature is a sectional curvature along a timelike 2-plane, see Definition 27. We prove the following result:
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Theorem 3 (see Theorem 11). Let (X, g), dim X > 2, be a globally hyperbolic space-time where g is conformal to g that has all the timelike sectional curvatures nonnegative. Assume moreover that a Cauchy surface M of (X, g) is such that alk is a Z-valued invariant (see Theorem 2). Then two events x, y ∈ X (that do not lie on the same null-geodesic) are causally related if and only if alk(Sx , S y ) = 0. In particular, they are causally unrelated if and only if (Sx , S y ) is a trivial link in N . Example 4. Take a complete connected oriented Riemannian manifold (M m , g), m > 1, of non-positive sectional curvature such that M is not an odd-dimensional rational homology sphere with finite π1 (M). Consider a globally hyperbolic static space-time (M × R, g ⊕ −dt 2 ) as in Example 1. Using [9, Eq. (3.21)] one immediately gets that (M × R, g ⊕ −dt 2 ) has nonnegative sectional curvature on every timelike two-plane. By Theorem 2 alk(Sx , S y ) is a Z-valued invariant. Thus (M × R, g ⊕ −dt 2 ) satisfies all the conditions of Theorem 3 and two events x, y ∈ (M × R, g ⊕ −dt 2 ) (that do not lie on the same null-geodesic) are causally related if and only if alk(Sx , S y ) = 0. The following theorem shows that for y ∈ J + (x) the invariant alk(Sx , S y ) gives an estimate from below on the number of times the light rays from x cross a generic past inextendible timelike curve to y. Theorem 4 (see Theorem 12). Let (X m+1 , g), m > 1 be a globally hyperbolic spacetime. Assume moreover that a Cauchy surface M ⊂ X is such that alk(Sx , S y ) is a Z-valued invariant. Let x, y ∈ X be events that do not belong to a common null geodesic and such that y ∈ J + (x). Then alk(Sx , S y ) is equal to the intersection index of the nullcone consisting of the future directed null geodesics from the point x and of a generic future directed past inextendible timelike curve to the point y. In Sect. 9 we develop a combinatorial method for computing alk(Sx , S y ). This is done from the shapes of (Wx,M , W y,M ) ⊂ M equipped with orthogonal to the fronts direction fields, defining their lifts to ST M. This method is motivated by Arnold’s [1] definition of the J + -invariant of planar wave fronts. (Please, do not confuse this J + with the causal future.) Arnold observed that generic double points of immersed Legendrian submanifolds in ST M = ST ∗ M correspond to the tangencies of their cooriented projections to M at which the coorienting normals to the two immersed tangent branches point to the same direction. These tangencies are called dangerous tangencies. Arnold defined his J + -invariant of a planar front by describing its increments under passages through the dangerous self-tangencies. Thus to compute J + one has to change the front to be “trivial” by a sequence of moves that are dangerous tangencies and the modifications corresponding to singularities of the front arising under a generic Legendrian isotopy. Then J + of the front is the value of J + on the trivial front plus the sum of the increments under the dangerous tangency moves that were used. We derive a formula for the increment of alk under the passage through the dangerous tangency between the two fronts. (Since alk is a link homotopy invariant, it does not change under the dangerous self-tangency move.) When fronts are one-dimensional, our alk changes similarly to Arnold’s J + . Now we explain the behavior of alk under a passage through dangerous tangency. Consider a positively oriented chart (x1 , · · · , xm ) such that the dangerous tangency happens at the origin where the common normal vector to the immersed branches of the two fronts that defines their lift to ST M is − ∂ x∂m . Locally the two fronts W1 , W2 can be expressed as graphs of some functions xm = f i (x1 , x2 , · · · , xm−1 ), i = 1, 2.
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Put σ to be the number of negative eigenvalues of the Hessian of f 2 − f 1 at the origin. Put ε to be +1 if the two oriented immersed tangent branches induce the same orientation on the common tangent (m − 1)-plane and put ε = −1 otherwise. Put α to be +1 (respectively α = −1) if the xm -coordinate of the point of W1 projecting to the origin in the (x1 , x2 , · · · , xm−1 )-hyperplane after the move is larger (respectively less) than the xm -coordinate of the corresponding point on W2 after the move. Theorem 5 (see Theorem 13). Under a passage through a dangerous tangency alk increases by εα(−1)σ . Recall that alk always takes values either in Z or in Zn , so this expression indeed makes sense. We use Theorem 5 to construct examples where we can conclude that the events are causally related from the shapes of their fronts, see Sect. 10. This conclusion can be made without the knowledge of the Lorentz metric on the space-time, of the event points, and in many cases even without the knowledge of topology of the globally hyperbolic space-time. In Sect. 11 we discuss the refocussing phenomena, see Definition 22. A good property of nonrefocussing globally hyperbolic space-times is that the map µ : X → {the space of skies}, x → Sx is a homeomorphism. Low [29] introduced the concept of refocussing spaces and noticed that a globally hyperbolic space-time with a noncompact Cauchy surface is nonrefocussing, see Proposition 6. We prove that a globally hyperbolic space-time (X, g) is nonrefocussing whenever any of its covering space-times is, see Theorem 14. In particular, if π1 (X ) is infinite, then (X, g) is nonrefocussing. As we discuss in Remark 7, the question on topology of a refocussing space-time is related to the problems similar to the Blaschke conjecture in Riemannian geometry, see Besse [11]. Low [29, Problem 7] asked: “Is there any construction intrinsic to the space N which will enable us to decide whether the points represented by two skies are causally related?” The following Theorem 6 gives an affirmative answer for all nonrefocussing globally hyperbolic (X, g). Also, Theorem 6 says that a weakened version of the Low conjecture holds for all globally hyperbolic nonrefocussing space-times. Theorem 6 (See Corollary 1, Definition 24). Let (X, g) be a nonrefocussing globally hyperbolic space-time of dimension > 2. Let (x1 , x2 ) be a pair of causally unrelated events and let (y1 , y2 ) be a pair of events that do not belong to a common null geodesic. Then the following two statements are equivalent: 1. y1 , y2 are causally related. 2. The link (S y1 , S y2 ) is not isotopic to (Sx1 , Sx2 ) via an isotopy through skies of events in (X, g). This theorem follows from the following more general fact that holds for all globally hyperbolic (X, g) and is also closely related to the above questions. Theorem 7 (See Theorem 15). Let (X m+1 , g), m > 1 be a globally hyperbolic spacetime. Let (x1 , x2 ) be a pair of causally unrelated events and let (y1 , y2 ) be a pair of events that do not belong to a common null geodesic. Then the following two statements are equivalent: 1. y1 , y2 are causally related. 2. For every pair of paths ρi : [0, 1] → X such that ρi (0) = xi and ρi (1) = yi , i = 1, 2, there exists t ∈ [0, 1] such that ρ1 (t) and ρ2 (t) belong to a common null geodesic.
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3. Linking and Causality The following theorem says that the skies of two causally unrelated events are Legendrian unlinked. In particular, we see that for every Cauchy surface M the lifted wave fronts x,M , W y,M of two causally unrelated events x, y are Legendrian unlinked. W Theorem 8. Let (X, g) be a globally hyperbolic space-time. If events x and y are causally unrelated, then the pair (Sx , S y ) is Legendrian unlinked. Proof. Choose a Cauchy surface M ⊂ X and an M-proper isometry h : M × R → X. x,M = W 0 and W y,M = W 0 are It suffices to prove that the lifted wave fronts W x,M y,M Legendrian unlinked in ST ∗ M. Take τ1 , τ2 ∈ R such that x ∈ Mτ1 and y ∈ Mτ2 . Thus h(x) = (m 1 , τ1 ) and h(y) = (m 2 , τ2 ), for some m 1 , m 2 ∈ M. Without loss of generality we assume that τ1 ≤ τ2 . There are three possible cases τ1 ≤ τ2 ≤ 0, τ1 ≤ 0 ≤ τ2 , and 0 ≤ τ1 ≤ τ2 . We prove the theorem only for the case τ1 ≤ 0 ≤ τ2 . The proof in the other two cases is similar and, in fact, even slightly easier. Let Si , i = 1, 2 be a copy of S m−1 . Consider I1 : (S1 S2 )×[0, τ2 ] → ST ∗ M defined t (s1 ) and I1 (s2 , t) = W t (s2 ), for s1 ∈ S1 , s2 ∈ S2 , t ∈ [0, τ2 ]. by I1 (s1 , t) = W x,M y,M Since x, y do not lie on a common null geodesic, we see that I1 is a Legendrian isotopy 0 , W 0 ) and (W τ2 , W τ2 ) = (W τ2 , εm 2 ). between (W x,M y,M x,M x,M y,M Consider a timelike curve ρ : [τ1 , τ2 ] → X given by ρ(t) = h(m 2 , t), for t ∈ [τ1 , τ2 ]. The future directed null geodesics of the sky Sx do not intersect ρ. Otherwise such a null geodesic followed by ρ after the intersection point is a piecewise smooth nonspacelike curve from x to y. This would contradict the assumption that x and y are causally unrelated. t (s1 ), Consider I2 : (S1 S2 ) × [τ1 , τ2 ] → ST ∗ M defined by I2 (s1 , t) = W x,M I2 (s2 , t) = εm 2 (s2 ), for s1 ∈ S1 , s2 ∈ S2 , t ∈ [τ1 , τ2 ]. Since ρ does not intersect the t null geodesics of the sky Sx , we get that m 2 ∈ Im Wx,M for all t ∈ [τ1 , τ2 ], and hence I2 is an isotopy. Since lifted wave fronts are Legendrian maps, we conclude that I2 is a τ1 , εm 2 ) and (W τ2 , εm 2 ). Legendrian isotopy between (εm 1 , εm 2 ) = (W x,M x,M x,M , W y,M ) is Legendrian isoCombining isotopies I1 and I2 , we conclude that (W topic to (εm 1 , εm 2 ). 4. Review of the alk Invariant In this section we adapt the general alk invariant constructed by us in [15] to the case of linked skies. Throughout the section M m is a smooth connected oriented manifold of dimension m > 1. Definition 9 (Bordism group). For a space Y, we put Ωn (Y ) to be the n-dimensional oriented bordism group of Y. Recall that Ωn (Y ) is the set of the equivalence classes of (continuous) maps g : V n → Y , where V is a smooth closed oriented manifold. Here two maps g1 : V1 → Y and g2 : V2 → Y are equivalent if there exists a map f : W n+1 → Y, where W is an oriented compact smooth manifold whose oriented boundary ∂ W is diffeomorphic to V1 (−V2 ) and f |∂ W = g1 g2 . Disjoint union operation turns Ωn (Y ) into an abelian group, and Ωn (Y ) is canonically isomorphic to Hn (Y ), for 0 ≤ n ≤ 3. See [39,44,45] for details.
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For a space Y , the group Ω0 (Y ) = H0 (Y ) is the free abelian group with the base π0 (Y ). So, every element of Ω0 (Y ) can be represented as a finite formal linear combination ak Pk with ak ∈ Z and Pk ∈ Y . Conversely every such linear combination gives us an element of Ω0 (Y ). Put S to be the connected component of the space of C ∞ -mappings S m−1 → (ST ∗ M)2m−1 that consists of the mappings homotopic to some (and hence to all) εv , v ∈ M m . (Note that the mappings in S are not assumed to be immersions or Legendrian mappings.) Let S • be the space of pointed maps (S m−1 , ) → (ST ∗ M, ) such that the corresponding maps S m−1 → ST ∗ M are in S. Lemma 1. For an oriented connected manifold M m , m > 1, the space S • is path connected. Proof. The standard π1 (M)-action on S • induces the bijection S = S • /π1 (M). Since S is a singleton by definition, we conclude that the above π1 (M)-action on S • is transitive. So it suffices to prove that the π1 (M)-action is trivial. Consider a loop γ : S 1 → ST ∗ M that realizes [γ ] ∈ π1 (ST ∗ M) and let [Sm−1 ] ∈ S • be the pointed homotopy class of the positively oriented fiber S m−1 of pr containing the base point . Consider the S m−1 -bundle over S 1 induced from pr : ST ∗ M → M by pr ◦γ : S 1 → M. This bundle is trivial, since pr is an oriented bundle. We choose its trivialization and obtain a bundle map S m−1 × S 1 −−−−→ ST ∗ M ⏐ ⏐ ⏐ ⏐ S1
pr ◦γ
−−−−→
M.
Now [γ ][Sm−1 ] = Sm−1 since π1 (S 1 × S m−1 ) acts trivially on πm−1 (S 1 × S m−1 ). Finally, [γ ]x = x for all x ∈ S • , since the π1 (M)-action on S • is transitive. Definition 10 (of B). Let B = BS ,S be the space of quadruples (φ1 , φ2 , ρ1 , ρ2 ), where φi : S m−1 → ST ∗ M, i = 1, 2, belong to S and ρi : pt → S m−1 are mappings of the one-point space pt such that φ1 ρ1 = φ2 ρ2 . Clearly, B can be regarded as a subset of S × S × S m−1 × S m−1 , and we equip B with the subspace topology. Lemma 2. For an oriented connected manifold M m , m > 1, the space B is path connected. Thus the augmentation aug : Ω0 (B) → Ω0 (pt) = Z induced by the map B → pt is an isomorphism. Proof. Our [15, Theorem 7.4] says that π0 (B) is the quotient of π0 (S • ) × π0 (S • ) by a certain right action of π1 (ST ∗ M) and a certain left action of π1 (S m−1 ) × π1 (S m−1 ). Now the result follows from Lemma 1. Definition 11 (of the µ-pairing). Let α1 : F1i → S be a map representing [α1 ] ∈ Ωi (S) j and let α2 : F2 → S be a map representing [α2 ] ∈ Ω j (S). Let αl : Fl ×S m−1 → ST ∗ M, l = 1, 2, be the adjoint maps i.e. maps such that αl ( f, s) = (αl ( f ))(s). Following standard arguments we can assume that α1 and α2 are transverse, see [12]. Consider the pullback diagram
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V ⏐ ⏐k 2
k1
−−−−→ F1 × S m−1 ⏐ ⏐ α1 α2
F2 × S m−1 −−−−→
(4.1)
ST ∗ M
of the maps αi , i = 1, 2. If α1 and α2 are transverse, then V = {( f 1 , s1 , f 2 , s2 )| α1 ( f 1 , s1 ) = α2 ( f 2 , s2 )} is a smooth closed (i + j + (m − 1) + (m − 1) − (2m − 1)) = (i + j − 1)-dimensional submanifold of F1 × S m−1 × F2 × S m−1 . It is identified with the transverse preimage of the diagonal in ST ∗ M × ST ∗ M under the map α1 × α2 : (F1 × S m−1 )×(F2 × S m−1 ) → ∗ ∗ ST M × ST M, and hence V is canonically oriented. Let µ( α1 , α2 ) : V → B be the map that sends ( f 1 , s1 , f 2 , s2 ) ∈ V to (α1 ( f 1 ), α2 ( f 2 ), ρs1 , ρs2 ), where ρsl (pt) = sl ∈ S m−1 , l = 1, 2. As we showed in [15, Theorem 2.2] the above construction yields a well-defined pairing µ = µi j : Ωi (S) ⊗ Ω j (S) → Ωi+ j−1 (B), µ ([α1 ], [α2 ]) = [V, µ( α1 , α2 )].
(4.2)
Definition 12. Let Σ be the discriminant in S × S, i.e. the subspace of S × S that consists of pairs ( f 1 , f 2 ) such that there exist s1 , s2 ∈ S m−1 with f 1 (s1 ) = f 2 (s2 ). (We do not include into Σ the maps that are singular in the common sense but do not involve double points between f 1 (S m−1 ) and f 2 (S m−1 ).) Put Σ0 to be the subset (stratum) of Σ consisting of all the pairs ( f 1 , f 2 ) for which there exists precisely one pair (s1 , s2 ) of points s1 , s2 ∈ S m−1 such that f 1 (s1 ) = f 2 (s2 ) and moreover 1. si is a regular point of f i , i = 1, 2; 2. (d f 1 )(Ts1 S m−1 ) ∩ (d f 2 )(Ts2 S m−1 ) = 0. Note that there is a canonical map of Σ0 into B. Namely, we assign the commutative diagram ( f 1 , f 2 , ρs1 , ρs2 ) with ρsi : pt → si ∈ S m−1 , i = 1, 2, to the pair ( f 1 , f 2 ) ∈ Σ0 with f 1 (s1 ) = f 2 (s2 ). Definition 13 (of the sign of the crossing of Σ0 and of a generic path in S × S). Consider a singular link ( f 1 , f 2 ) ∈ Σ0 . The double point z = f 1 (s1 ) = f 2 (s2 ) of it can be resolved in two (essentially different) ways. To a resolution ( f 1 , f 2 ) (that is a C ∞ -small deformation of ( f 1 , f 2 )) we associate the vector w ∈ Tz ST ∗ M that in a chart has the same direction as the vector from f 1 (s1 ) to f 2 (s2 ). We say that the resolution ( f 1 , f 2 ) is generic if span{(d f 1 )(Ts1 S m−1 ), w, (d f 2 )(Ts2 S m−1 )} = T f1 (s1 ) M. Let ri , i = 1, 2 be the positive (m − 1)-frames in Tsi S m−1 . Take a generic resolution of ( f 1 , f 1 ) and consider the (2m − 1)-frame {d f 1 (r1 ), w, d f 2 (r2 )} ⊂ Tz (ST ∗ M). We say that the resolution of the singular link is positive if this (2m − 1)-frame gives the canonical orientation of ST ∗ M, and we say that the resolution is negative, otherwise. One checks that the sign of the resolution does not depend on the choice of the chart used to define w. Let γ (t) be a path that intersects Σ at one point γ (t0 ) ∈ Σ0 . We say that γ intersects Σ transversally at γ (t0 ) if γ (t0 ) ∈ Σ0 and if the resolution ( f 1 , f 2 ) = γ (t) is generic for t close to t0 and different from t0 . We define the sign of the transverse intersection
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of Σ0 by γ to be the sign of the singular link resolution induced by γ , and denote this sign by σ (γ , t0 ) = ±1. Clearly if we traverse the path γ in the opposite direction, then the sign of the intersection changes. We say that a path γ in S × S is generic if it intersects Σ at a finite number of times and these intersections are transverse. We will also use the term “generic link homotopy” for a generic path. Definition 14 (of A (M) and of the alk invariant). Define the indeterminacy subgroup Indet of Ω0 (B) to be the subgroup generated by the images of µ0,1 and µ1,0 . Let A = A (M) = A (ST ∗ M) be the quotient group Ω0 (B)/ Indet and let q : Ω0 (B) → A be the quotient homomorphism. Our [15, Theorem 3.9] when applied to this work setup says that there exists a function alk : S × S \ Σ → A (M)
(4.3)
such that: a. alk is constant on path connected components of S × S \ Σ; b. if γ : [a, b] → S × S is a generic path such that γ (a), γ (b) ∈ Σ and ti , i ∈ I, are the moments when γ (ti ) ∈ Σ (and hence γ (ti ) ∈ Σ0 by the definition of the generic path), then
σ (γ , ti )γ (ti ) ∈ A (M). alk(γ (b)) − alk(γ (a)) = q i∈I
We showed that such alk is unique up to an additive constant. In this paper we normalize alk by the condition that alk(εu , εv ) = 0 for any two distinct u, v ∈ M. We proved [15, Corollary 7.5] that for every α ∈ A (M) there exists a nonsingular link ( f 1 , f 2 ) ∈ S × S \ Σ with alk( f 1 , f 2 ) = α. Thus A (M) is indeed the group of values of the alk-invariant. Definition 15 (of the affine linking number of a pair of skies). Let (X, g) be a globally hyperbolic space-time and let x, y ∈ X be events that do not lie on a common null geodesic. Choose a Cauchy surface M ⊂ X. Since x, y do not lie on a common null x,M , W y,M ) is a point in S × S \ Σ, and we geodesic, the pair of lifted wave fronts (W put x,M , W y,M ) ∈ A (M), alk M (Sx , S y ) = alk(W where the alk at the right-hand side means the function (4.3). Theorem 9 below states that the value alk M (Sx , S y ) ∈ A (M) does not depend on the Cauchy surface M. Thus we can and shall define alk(Sx , S y ) := alk M (Sx , S y ), for any choice of M. Theorem 9. Let x, y be two events in a globally hyperbolic space-time (X, g) that do not lie on a common null geodesic. Let M and N be two Cauchy surfaces in X , and let h : M × R → X and h : N × R → X be M-proper and N -proper isometries, respectively. Then for any t, τ ∈ R the following holds: t , W t ) is unlinked (respectively Legendrian unlinked) in ST ∗ M if and only 1. (W x,M y,M τ , W τ ) is unlinked (respectively Legendrian unlinked) in ST ∗ N . if (W x,N y,N t ) = alk(W τ , W τ ) ∈ A (M) = A (N ). t , W 2. alk(W x,M
y,M
x,N
y,N
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In particular, the value 0 0 x,M , W y,M ) = alk(W x,M y,M alk(Sx , S y ) = alk(W ,W ) ∈ A (M)
and the notion of the skies (Sx , S y ) being unlinked (respectively Legendrian unlinked) are well defined. t ) is Legendrian isotopic to (W 0 , W 0 ) in t , W Proof. Clearly, the link (W x,M y,M x,M y,M τ , W τ ) in ST ∗ N . Hence, to prove ST ∗ M, and a similar fact is true for the link (W x,N y,N 0 , W 0 ) is (Legendrian) unlinked if and only Statement 1, it suffices to show that (W x,M y,M 0 , W 0 ) is. if (W x,N y,N Since alk invariant does not change under link isotopy, to prove Statement 2 it suffices 0 , W 0 ) = alk(W 0 , W 0 ). to show that alk(W x,M y,M x,N y,N We prove the “Legendrian unlinked” part of statement 1. The proof of the “unlinked” part is obtained by omitting the word “Legendrian” everywhere in the proof. Assume 0 , W 0 ) is Legendrian unlinked in ST ∗ M. Let Si , i = 1, 2, be a that the link (W x,M y,M copy of S m−1 . Choose a Legendrian isotopy It : S1 S2 → ST ∗ M, t ∈ [0, 1], such 0 W 0 and I1 = εu εv for some u = v ∈ M. that I0 = W x,M y,M Put x = i(u) and y = i(v), where i : M → X is the inclusion. Then x and y belong to the same Cauchy surface M and hence are causally unrelated. Thus by Theorem 8 the 0 , W 0 ) is Legendrian unlinked in ST ∗ N . link (W x ,N y,N M Let f N : ST ∗ M → ST ∗ N be the contactomorphism (1.5). Now, f NM ◦ It , t ∈ 0 ) to the Legendrian trivial link 0 , W [0, 1], is a Legendrian isotopy that deforms (W x,N y,N 0 , W 0 ) in ST ∗ N . (W x ,N y,N Now we prove statement 2 of the theorem. Let γ : [0, 1] → S × S be a generic 0 , W 0 ) and γ (0) = (εu , εv ) for some smooth link homotopy such that γ (1) = (W x,M y,M u = v ∈ M. Let ti , i = 1, . . . , k, be the time moments when γ crosses Σ0 ⊂ Σ and let σ (γ , ti ) = ±1 be the signs of these crossings, see Definition 13. Lemma 2 says 0 , W 0 ) = q that B is connected and hence alk(W i∈I σ (γ , ti ) , for the map x,M y,M q : Ω0 (B) = Z → A (M). 0 , W 0 ) The smooth link homotopy f NM ◦γ (t), t ∈ [0, 1], deforms the trivial link (W x ,N y,N 0 0 M ∗ ∗ ,W ) in ST N . Clearly the link homotopy f ◦ γ (t), t ∈ in ST N to the link (W x,N y,N N [0, 1], is generic and it crosses Σ0 ⊂ Σ at the same time moments ti , i ∈ I, as γ (t). Moreover since f NM is orientation preserving, we conclude that σ ( f NM ◦γ , ti ) = σ (γ , ti ). Since M and N are diffeomorphic, they are homotopy equivalent. Hence A (M) = A (N ) and the maps q : Z → A (M) and q : Z → A (N ) are the same. Thus 0 0 0 x,N y,N x0,N , W y,N alk(W ,W ) − alk(W )=q
=q
i∈I
σ ( f NM
◦ γ , ti )
0 0 x,M y,M σ (γ , ti ) = alk(W ,W ).
i∈I
0 , W 0 ) is trivial, we have alk(W 0 , W 0 ) = 0 and hence Since the link (W x ,N y,N x ,N y,N 0 0 0 0 ,W ) = alk(W ,W ). alk(W x,N y,N x,M y,M
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5. Computation of the Group A (M). In this section M m , m > 1 is a smooth connected oriented manifold. α is special Definition 16. Given a map α : S 1 × S m−1 → ST ∗ M, we say that if α ×S m−1 has the form εv for some v ∈ M, see (1.2). Here ∈ S 1 is the base point. We define I ndet ⊂ Z to be the subgroup of Z generated by α∗ [(S m−1 × S 1 )] • [S m−1 ] ∈ Z = H0 (ST ∗ M) such that α is special . Here [S m−1 ] ∈ Hm−1 (ST ∗ M) is the homology class of a positively oriented fiber of pr : ST ∗ M → M and • is the intersection pairing of homology classes. ndet. Lemma 3. The isomorphism aug : Ω0 (B) → Z from Lemma 2 maps Indet onto I Proof. The subgroup Indet of Ω0 (B) was defined as the subgroup generated by the images of µ0,1 and µ1,0 where µi, j : Ωi (S) ⊗ Ω j (S) → Ωi+ j−1 (B), see Definition 11. In particular, the images of µ0,1 and of µ1,0 are subgroups of Ω0 (B) = Z. It is easy to see that µ0,1 (α0 , α1 ) = ±µ1,0 (α1 , α0 ), for any α0 ∈ Ω0 (S), α1 ∈ Ω1 (S), where the sign depends on the dimension of M. Thus Im(µ0,1 ) = Indet = Im(µ1,0 ). Take β : pt → εu ∈ S and α : S 1 → S. Without loss of generality we can assume (deforming α if necessary) that α() = εv , for some v ∈ M, and that the adjoint α : S 1 × S m−1 → ST ∗ M of α is transverse to β = εu . This homotopy does not change [S 1 , α] ∈ Ω1 (S). Now the adjoint α of α is a special map.
From Definition 11 one verifies that the bordism class µ1,0 [S 1 , α], [pt, β] ∈ Ω0 (B) is represented by the set of intersection points of the maps α : S 1 × S m−1 → ST ∗ M : pt ×S m−1 = S m−1 → ST ∗ M. The signs at these intersection points are equal and β to the signs obtained from the definition of the intersection number of two transverse oriented submanifolds of complimentary dimensions.
Since Ω0 (B) = Z[π0 (B)] = Z, we conclude that µ1,0 [S 1 , α], [pt, β] ∈ Z equals ∗ [S m−1 ] ∈ Z. the intersection number α∗ [S 1 × S m−1 ] • β Recall that Ωi (Y ) = Hi (Y ) for 0 ≤ i ≤ 3 and all spaces Y. In particular, every class in Ω0 (B) is parameterized by a collection of oriented points and every class in Ω1 (B) is parameterized by a collection of oriented circles. Thus Indet ⊂ I ndet ⊂ Z. On the other hand every smooth special α : S 1 × S m−1 → ST ∗ M is the adjoint of a certain map α : S 1 → S. Thus I ndet ⊂ Indet ⊂ Z. Remark 3. For future needs, it is convenient to regard the augmentation isomorphism as the identification Ω0 (B) = Z. For example, we can treat Lemma 3 as the equality I ndet = Indet. Definition 17. Given oriented m-dimensional manifolds N m , M m , and a continuous map β : N → ST ∗ M, we define d(β) to be the degree of the map pr ◦β : N m → M m . If one of N , M is not a closed manifold, then we put d(β) and the degree of pr ◦β to be zero.
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Lemma 4. A number i ∈ Z equals d( α ) for some special α : S 1 × S m−1 → ∗ m ∗ ST M if and only if i equals d(β) for some β : S → ST M. In particular, the set d( α ) such that α : S 1 × S m−1 → ST ∗ M is special is a subgroup of Z that is the image of the homomorphism πm (ST ∗ M) → Z sending the class of a map β : S m → ST ∗ M to d(β). Proof. We regard S m−1 and S m as pointed spaces. Assume that i = d( α ) for a special α . We show that i = d(β) for some β : S m → ST ∗ M. Consider a map α : S 1 × S m−1 → ST ∗ M such that: α , 1. α ×S m−1 = ×S m−1 2. α 1 = α 1 , S ×
S ×
3. α|t×S m−1 = εα(t,) , for all t ∈ S 1 . We regard S 1 × S m−1 as the C W -complex with four cells e0 , e1 , em−1 , em , dim ek = k. It is easy to see that the maps α and α coincide on the (m − 1)-skeleton. Thus, the maps α and α (restricted to the m-cell) together yield a map β : S m → ST ∗ M. Clearly d(α) = 0, and therefore i = d(β) = d( α ). Assume that i = d(β) for some β : S m → ST ∗ M. Let us show that i = d( α ) for some special α . Let π : S 1 × S m−1 → S m−1 be the projection. Choose v ∈ M and put α = εv ◦ π : S 1 × S m−1 → ST ∗ M. Consider the maps S 1 × S m−1 → ST ∗ M that coincide with α on the (m −1)-skeleton. Up to a homotopy fixed on the (m −1)-skeleton they are classified by πm (ST ∗ M). Consider such a map α : S 1 × S m−1 → ST ∗ M that ∗ corresponds to β ∈ πm (ST M). Since d( α ) = 0 we get that i = d(β) = d( α ). Proposition 1. aug(Indet) = I ndet ⊂ Z is the subgroup {d(β)|β : S m → ST ∗ M} ⊂ Z. Proof. By Lemma 3 aug(Indet) = I ndet ⊂ Z. Because of Lemma 4 it suffices to show that Indet ⊂ Z is the subgroup S = d( α ) such that α : S 1 × S m−1 → ST ∗ M is special ⊂ Z. Intersection number α∗ [(S m−1 × S 1 )] • [S m−1 ] ∈ Z = H0 (ST ∗ M) and d( α ) do not change if we substitute a special α : S 1 × S m−1 → ST ∗ M by a homotopic one. Thus in Definition 16 of the generating set of I ndet and in the description of S, it suffices to consider only special α that have some fixed v ∈ M as a regular value of pr ◦ α. Such α and εv are transverse and α −1 (Im( α )−1 (v). Comparing α ) ∩ Im(εv )) = (pr ◦ the orientations of the points in these two preimage sets we get that α∗ [(S m−1 × S 1 )] • m−1 [S ] = d( α ) ∈ Z. Thus, S is the subgroup Indet ⊂ Z. Lemma 5. We have A (M) = Z, unless M is a closed manifold that is an odd-dimensional rational homology sphere with finite π1 (M). In greater detail, if there exists a map β : S m → ST ∗ M with d(β) = 0, then M m is a closed manifold that is an odddimensional rational homology sphere with finite π1 (M). Proof. Set f = pr ◦β : S m → M and d = d(β). Since d(β) = 0, M is closed. Let f ! : H∗ (M) → H∗ (S m ) be the transfer map, see e.g. [39, V.2.12]. Since f ∗ ( f ∗ y ∩ x) = y ∩ f ∗ x, for all x ∈ H∗ (S m ) and y ∈ H ∗ (M), we conclude that f ∗ f ! (z) = dz, for all z ∈ H∗ (M). In particular, since Hi (S m ) = 0, for 0 < i < m, we conclude that d Hi (M) = 0, for 0 < i < m. Thus Hi (M; Q) = 0, for 0 < i < m, and M is a rational homology sphere. Let e ∈ H m (M) = Z be the Euler class of the bundle pr. Since f passes through ST ∗ M, we conclude that f ∗ (e) = 0. On the other hand, the map f ∗ : Z = H m (M m ) → H m (S m ) = Z
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is the multiplication by d with d = 0. Hence e = 0 and 0 = e = χ (M) = 1 + (−1)m , where χ (M) is the Euler characteristic of M. Thus m = dim M is odd. Finally since m > 1, every map f = pr ◦β : S m → M passes through the universal covering of M, and so d(β) = 0 whenever the fundamental group of M is infinite. Definition 18. We let deg : πm (M m ) → Z be the degree homomorphism, i.e. the homomorphism that assigns the degree deg f to the homotopy class of a map f : S m → M m . (In fact, it coincides with the Hurewicz homomorphism h : πm (M m ) → Hm (M m ) for M closed and is zero for M non-closed.) Proposition 2. If M is an odd dimensional closed manifold, then Indet = Im(deg). Proof. Proposition 1 implies that Indet ⊂ Im(deg). Since M is an oriented odd dimensional manifold, the projection pr : ST ∗ M → M has a section s : M → ST ∗ M, pr ◦s = 1 M . For closed M this follows, since the Euler characteristic of M is zero and it equals to the Euler class of T ∗ M → M. For non-closed oriented M the bundle T ∗ M → M has a section regardless of the dimension of M. So, every f : S m → M m can be written as f = pr ◦β with β = s f : S m → ST ∗ M. Since deg( f ) = d(β), we have Im(deg) ⊂ Indet . Hence Indet = Im(deg). Combining Definition 14, Lemma 2, Lemma 5, Proposition 2 and the results of the general theory of affine linking invariants reviewed in Sect. 4, we get the following result. Theorem 10. Let M m , m > 1, be a smooth connected oriented manifold. If M is not a closed manifold that is an odd dimensional rational homology sphere with finite π1 (M), then A (M) = Z and the homomorphism q : Ω0 (B) = Z → Z is the identity isomorphism. Otherwise A (M) = Z/(Im(deg : πm (M m ) → Z)) and q : Ω0 (B) = Z → A (M) is the quotient homomorphism. The affine linking number invariant alk of two component links in ST ∗ M with components homotopic to a positively oriented fiber εv : S m−1 → ST ∗ M of pr : ST ∗ M → M m is a link homotopy invariant such that 1. alk increases by q(1) ∈ A (M) (respectively by q(−1) ∈ A (M)) under a positive (respectively negative) transverse crossing of Σ0 , i.e. under homotopy of the link that involves exactly one positive (respectively negative) passage through a transverse double point between the two link components; 2. alk is invariant under Milnor’s [32] link homotopy that allows each link component to cross itself, but does not allow different components to cross. This alk is uniquely defined by the normalization that it is zero on links consisting of the positively oriented S m−1 -fibers over two different points of M. It is a universal order ≤ 1 Vassiliev-Goussarov link homotopy invariant of two such component links in ST ∗ M. Remark 4. One verifies that the sign of crossing of Σ0 does not depend on the order of link components if m = dim M is even. If m is odd, then the sign of the crossing of Σ0 gets reversed if one changes the order of the link components. Thus alk( f 1 , f 2 ) = (−1)m alk( f 2 , f 1 ), for all m. The group A (M) of the values of the alk-invariant appears to be quite nontrivial even in the case when M is an odd-dimensional rational homology with finite π1 (M).
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Proposition 3. Let M m , m > 1, be a smooth connected oriented manifold. Assume moreover that M is a closed manifold that is an odd-dimensional rational homology sphere with finite π1 (M). (This is the only case when Theorem 10 does not say that A (M) = Z.) Then the following statements hold: (i) If π1 (M) is a finite group of order k, then A (M) = Z/mZ, where k divides m (the case m = 0 i.e. A (M) = Z is also possible). (ii) If A (M) = 0, then M is homeomorphic to a sphere. Proof. (i) This follows because every map S m → M m passes through the universal → M which is of degree k. covering map p : M A (ii) If (M) = 0, then there exists a map S m → M m of degree 1. Since every map of degree 1 of connected closed oriented manifolds induces an epimorphism of fundamental groups and homology groups, we conclude that M is a homotopy sphere. The Poincaré conjecture proved in the works of Smale [42] for m ≥ 5, Freedman [19] for m = 4, and Perelman [37,38] for m = 3, implies that M is homeomorphic to a sphere. 6. Computing alk When One of the Linked Spheres is a Fiber of the Spherical Cotangent Bundle In this section M is a smooth connected oriented manifold of dimension m > 1. Definition 19. Let f : U k → V k be a smooth map of oriented manifolds, and let v be a regular value of f . A point u ∈ f −1 (v) is called positive (respectively negative) if a restriction of f to a small neighborhood of u is orientation preserving (respectively orientation reversing). Proposition 4. Suppose that the map f as in Definition 19 is an immersion and that U is connected. Then all the points of U have the same sign, i.e. either all of them are positive or all of them are negative. Proof. This follows since the set of all positive points is open, as well as the set of all negative points. Let F : S m−1 × [a, b] → ST ∗ M be a smooth map such that F(S m−1 × t) ∈ S, for some and then for all t ∈ [a, b]. Let v ∈ M be a regular value of G = pr ◦F : S m−1 × [a, b] → ST ∗ M → M m such that G −1 (v) ⊂ S m−1 × (a, b). Let n + (v, F) (respectively n − (v, F)) be the number of positive (respectively negative) points in G −1 (v). Recall that Ω0 (B) = Z by Lemma 2 and that q : Ω0 (B) = Z → Z/ Indet = A (M) is the quotient homomorphism. Lemma 6. We have the equality alk(F| S m−1 ×b , εv ) − alk(F| S m−1 ×a , εv ) = q(n − (v, G) − n + (v, G)). Proof. Using a C ∞ -small perturbation of F if necessary we can and shall assume that the [a, b]-coordinates of all the points in G −1 (v) are all different. Consider the link homotopy H : [a, b] → S × S such that H (t) = (F| S m−1 ×t , εv ). Let (si , ti ) ∈ S m−1 × [a, b], i = 1, . . . , k, be the points of G −1 (v). Clearly the crossings of Σ under the link homotopy H happen exactly at time moments ti . Since all the values ti , i = 1, . . . , k, are distinct and v is a regular value of G, we conclude
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that the crossings of Σ happen inside of Σ0 ⊂ Σ and these crossings are transverse. From the definition of the sign of the crossing of Σ0 we get that sign σ (H, ti ) is equal to +1 (respectively −1) exactly when (si , ti ) is a negative (respectively positive) point of G −1 (v). By Lemma 2, Ω0 (B) = Ω0 (pt) = Z. Thus by Definition 14, alk is a link homotopy invariant that increases by q(σ (H, ti )) under the crossing of Σ0 by the link homotopy H that happens at time ti . 7. alk-Invariant and Causality Let (X m+1 , g), m > 1 be a globally hyperbolic space-time with a Cauchy surface M. Let x, y ∈ X be two events that do not lie on a common null geodesic. If alk(Sx , S y ) = 0 ∈ A (M), then the events x, y are causally related by Theorem 8. The main result of this section is Theorem 11 saying that for many globally hyperbolic space-times the converse is also true, i.e. if alk(Sx , S y ) = 0, then the events x, y are causally unrelated. Lemma 7. Let (X m+1 , g), m > 1 be a globally hyperbolic space-time and let x, y ∈ X be events such that y ∈ J + (x). Let γ : (−∞, ∞) → X be an inextendible past directed nonspacelike curve with γ (0) = y. Then there exists a future directed null geodesic ν : [0, α) → X with ν(0) = x and τ1 ∈ [0, α), τ2 ∈ [0, ∞) such that ν(τ1 ) = γ (τ2 ). Proof. Let J + (x) and I + (x) be the causal and the chronological future of x. The set J + (x) is closed since (X, g) is globally hyperbolic, see [9, Proposition 3.16]. The set I + (x) is open, see [9, Lemma 3.5]. Let M ⊂ X be a Cauchy surface containing x. Then γ (t0 ) ∈ M for some t0 . We claim that t0 ≥ 0. Otherwise the curve γ |(−∞,0) followed by the past directed nonspacelike curve joining y to x is a past directed nonspacelike curve that intersects M twice. This is in contradiction with M being a Cauchy surface. Clearly γ (t) ∈ J + (x) for t > t0 ≥ 0. Since J + (x) is closed, I + (x) ⊂ J + (x) is open, γ (0) = y ∈ J + (x), and γ is continuous, we conclude that there exists τ2 > 0 with γ (τ2 ) ∈ J + (x) \ I + (x). Since (X, g) is globally hyperbolic, there exists a future directed null geodesic ν from x to γ (τ2 ), see [9, Corollary 4.14]. We reparameterize ν so that it is future directed and has ν(0) = x. Now put τ1 to be such that ν(τ1 ) = γ (τ2 ) and obtain the statement of the lemma. Take x ∈ X and let C = C + (x) ⊂ Tx X be the hemicone of all the future pointing null vectors. We have an obvious R+ -action on C. Clearly, C/R+ = S m−1 and in fact we have a diffeomorphism C∼ = S m−1 × R = S m−1 × (0, ∞). Similarly to Riemannian manifolds, one can use geodesics to define the exponential map exp = expx : Tx X → X , cf. Definition 27. Here the domain of exp is not the whole Tx X but rather a starshaped with respect to 0 ∈ Tx X subset V of it. We put U = V ∩ C. Lemma 8. Given a Cauchy surface M ⊂ X with x ∈ M and U ⊂ Tx X as above, take an M-proper isometry h : M × R → X . Consider the map F : S m−1 × (0, ∞) → M × t (s), t). Then there exists a diffeomorphism ω : U → S m−1 ×(0, ∞) R, F(s, t) = (Wx,M such that the diagram
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U ⏐ ⏐ ω
expx
−−−−→
X ⏐ ⏐ −1 h
(7.1)
F
S m−1 × (0, ∞) −−−−→ M × R commutes. Furthermore, U is an open subset of C. Proof. Since U ⊂ C ∼ = S m−1 × (0, ∞), every point of U can be written as (s, τ ) for m−1 some s ∈ S , τ ∈ (0, ∞). Now, given u = (s, τ ), there exists a unique t = t (u) such that expx (u) ∈ Mt . In other words, t (u) = πR (expx (u)).
(7.2)
We put ω(u) = (s, t). It is easy to see that the above diagram commutes and that ω is a bijection. Furthermore, ω is smooth because of (7.2). Moreover, for each s the velocity vectors of the curve γ = γs : τ → expx (s, τ ) are null, and hence dπR (γ˙ (τ )) = 0, for all τ in the domain of γ . Thus ∂t/∂τ = 0 everywhere. Now, since ω preserves the s-coordinate, we conclude that ω is a diffeomorphism. Now, the m-dimensional manifold U is a subset of the m-dimensional manifold C, and so U is open because of the Invariance of Domain Theorem. Definition 20. The timelike sectional curvatures in a space-time (X, g) are the sectional curvatures along the timelike 2-planes in T X, i.e. 2-planes E s ⊂ Ts X such that g| E s is a nondegenerate form that is not positive definite. (See Definition 27 for a more thorough exposition.) Proposition 5. Let (X m+1 , g), m > 1 be a globally hyperbolic space-time where g is conformal to g that has all the timelike sectional curvatures nonnegative. Take a point x ∈ X and Cauchy surface M ⊂ X with x ∈ M. Choose an M-proper isometry h. Define t G = G g : S m−1 × (−∞, ∞) → M, G(s, t) = Wx,M (s).
Then the restrictions of G onto S m−1 ×(−∞, 0) and onto S m−1 ×(0, ∞) are immersions. Proof. Let Ω : X → R be a nowhere zero smooth function such that g = Ω 2 g. A Cauchy surface M in (X, g) is a Cauchy surface in (X, g ). Moreover if h : (M × R, βdt 2 + g) → (X, g) is an M-proper isometry, then h : M × R, (Ω 2 ◦ h)(βdt 2 + g) → (X, g) also is an M-proper isometry. The null geodesics for g and g coincide up to reparameterization, see [9, Lemma 9.17], that is not in general an affine reparameterization. Hence m−1 × (−∞, ∞) → M are equal. Thus without loss of generality the maps G g , G g : S we assume that g has all the timelike sectional curvatures nonnegative. We prove that G : S m−1 × (0, ∞) → M is an immersion; the restriction G : t m−1 S × (−∞, 0) → M can be considered similarly. For brevity we denote Wx,M by t W and we denote expx by exp . First, we prove that the map F : S m−1 ×(0, ∞) → M ×R, F(s, t) = (G(s, t), t) is an immersion. Let V ⊂ Tx X be the maximal subset where exp is defined and let U ⊂ V be as in Lemma 8. Let ρ : [0, b] → X be a geodesic starting at x. The point ρ(b) is conjugate to x = ρ(0) along ρ if and only if the exponential map exp : Tx X → X is singular at bρ(0) ˙ ∈ Tx X, i.e. if and only if the differential (d exp)bρ(0) : Tbρ(0) ˙ ˙ (Tx X ) → Tρ(b) X is not of full rank, see [36, Proposition 10, Sect. 10]. All the non-spacelike geodesics in
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(X, g) do not have any conjugate points, since all the timelike sectional curvatures in (X m+1 , g) are nonnegative, see [9, Proposition 11.13]. Hence (d exp)u : Tu U → X is of full rank, for every u ∈ U. Therefore exp |U : U → X is an immersion. Since the diagram (7.1) is commutative, we get that F is an immersion. For all (s, t) ∈ S m−1 × (0, ∞), let V (s, t) be the image of the linear map (dW t )(s) : Ts S m−1 → Ta M, a = W t (s). Since F is the immersion, we get that W t : S m−1 → M is an immersion for all t > 0. So, in order to show that G : S m−1 × (0, ∞) → M is an immersion, it suffices to show that, for all (s, t) ∈ S m−1 × (0, ∞), dG(s, t)(∂/∂t) ∈ V (s, t).
(7.3)
So let us take a point (s0 , t0 ) ∈ S m−1 × (0, ∞) and prove that (7.3) holds for (s, t) = (s0 , t0 ). Let z = Wx,Mt0 (s0 ) ⊂ Mt0 ⊂ X and y = gt0 (z) ∈ M. We put
L = span dh(y, t0 )( ∂t∂ ) ⊂ Tz X. By definition of an M-proper isometry h we have a direct sum decomposition Tz X = Tz Mt0 ⊕ L ,
(7.4)
and L is the g-orthogonal compliment of Tz Mt0 in Tz X . Take a null curve γ (t) defined by γ (t) = Wx,Mt (s0 ) ∈ Mt ∈ X. Clearly up to reparameterization γ is a null geodesic through x. Put ξ = γ˙ (t0 ) ∈ Tz X and use (7.4) to decompose ξ as ξ = ξ1 + ξ2 , ξ1 ∈ Tz Mt0 , ξ2 ∈ L . Since L is g-orthogonal to Mt0 , the direction of ξ1 is the direction that defines the lifted x,Mt at s0 ∈ S m−1 , cf. (1.3). Since W x,Mt is Legendrian, ξ1 is a nonzero wave front W 0 0 vector that is g| Mt0 -orthogonal to Im(dWx,Mt0 )(s0 ). In particular, ξ1 ∈ Im(dWx,Mt0 )(s0 ). Since W t = gt (Wx,Mt ) = π M (Wx,Mt ) for all t and L = ker dπ M (z), we have dgt0 (ξ1 ) = dπ M (ξ ) = dG(s0 , t0 )( ∂t∂ ) and dgt0 Im(dWx,Mt0 )(s0 ) = Im dW t0 (s0 ) = V (s0 , t0 ). Since gt0 : Mt0 → M is a diffeomorphism, we get that dG(s0 , t0 )( ∂t∂ ) ∈ V (s0 , t0 ).
Recall that by Theorem 10 A (M) = Z for all smooth connected oriented M m , m > 1, unless M is a closed manifold that is an odd dimensional rational homology sphere with finite π1 (M). Theorem 11. Let (X m+1 , g), m > 1 be a globally hyperbolic space-time where g is conformal to g that has all the timelike sectional curvatures nonnegative. Furthermore, assume that A (M) = Z for a Cauchy surface M m of X. Let x, y ∈ X be two events that do not lie on a common null geodesic. Then the following statements (1), (2), and (3) are equivalent: 1. x and y are causally related; 2. alk(Sx , S y ) = 0 ∈ A (M) = Z; 3. the skies Sx , S y are nontrivially linked in N . Many space-times satisfying all the conditions of Theorem 11 are constructed in Example 4.
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Proof. By Theorem 10 alk is a link homotopy invariant that is normalized to be zero when the lifted wave fronts are unlinked. Thus (2) =⇒ (3). Furthermore, (3) =⇒ (1) by Theorem 8. Now we prove that (1) =⇒ (2). The null geodesics for g and g coincide up to reparameterization, see [9, Lemma 9.17], that is not in general an affine reparameterization. Clearly M is a Cauchy surface with respect to both g and g . Thus the spaces N = ST M for (X, g) and for (X, g ) are naturally diffeomorphic and the links (Sx , S y ) ⊂ ST M = N computed for the two metrics coincide. Moreover x, y are causally related in (X, g) if and only if they are causally related in (X, g ). Thus without loss of generality we assume that g has all the timelike sectional curvatures nonnegative. Remark 4 says that alk(Sx , S y ) = (−1)m alk(S y , Sx ). Thus it suffices to prove that alk(Sx , S y ) = 0 whenever y ∈ J + (x). So, we assume that y ∈ J + (x). Choose a Cauchy surface M x and an M-proper isometry h : M × R → X . For t brevity we denote Wx,M by W t . Let τ ∈ R be the unique value such that y ∈ Mτ . Clearly, τ > 0. Without loss of generality we can and shall assume that π M (x) = π M (y). Indeed, if π M (x) = π M (y), then we can construct an auxiliary event z ∈ Mτ such that π M (z) = π M (y), z ∈ J + (x), events x, z do not lie on a common null geodesic, and alk(Sx , S y ) = alk(Sx , Sz ). We construct the event z as follows. Put v = π M (y) = π M (x) so that y = h(v, τ ), x = h(v, 0). Since y ∈ J + (x) and x, y do not lie on a common null geodesic, [9, Corollary 4.14] says that y ∈ I + (x). ⊂M By [9, Lemma 3.5] I + (x) is open and hence there exists an open neighborhood U , τ ) ⊂ I + (x). Since x, y do not lie on a common null geodesic, containing v such that h(U y ∈ Im Wx,Mτ . Since Im Wx,Mτ is compact and y = h(v, τ ) ∈ Im Wx,Mτ , there exists containing v such that h(U, τ ) ∩ Im Wx,Mτ = ∅. Choose an open connected U ⊂ U u = v ∈ U and put z = h(u, τ ). Clearly z ∈ J + (x), π M (z) = u = v = π M (x), and since z ∈ Im Wx,Mτ , the events x, z do not lie on a common null geodesic. Let us prove that alk(Sx , S y ) = alk(Sx , Sz ). Take a path β : [0, 1] → U with β(0) = v and β(1) = u. Let S1 and S2 be two copies of S m−1 . Define I : (S1 S2 ) × τ (s1 ), I (s2 , t) = εβ (s2 , t), for s1 ∈ S1 , s2 ∈ [0, 1] → ST ∗ M by setting I (s1 , t) = W S2 , t ∈ [0, 1]. Since h(U, τ ) ∩ Im Wx,Mτ = ∅ we get that U ∩ Im W τ = ∅. Since τ , εv ) = alk(W τ , εu ). Im β ⊂ U, we conclude that I is a link isotopy and alk(W x,M x,M Using Theorem 9 and the above identity we have τ τ x,Mτ , W y,Mτ ) = alk(W x,M y,M alk(Sx , S y ) = alk(W ,W ) τ τ τ τ x,M x,M x,M z,M = alk(W , εv ) = alk(W , εu ) = alk(W ,W )
(7.5)
= alk(Sx , Sz ). Thus, we can and shall assume that π M (y) = π M (x). Let v = π M (y), so that h(v, τ ) = y, and let γ : (−∞, +∞) → X be an inextendible past directed timelike curve given by γ (t) = h(v, τ −t) ∈ Mτ −t ⊂ X. Lemma 7 applied to x, y and γ implies that there exists a future directed null geodesic ν : [0, α) → X and τ1 ∈ [0, α), τ2 ∈ [0, +∞) such that ν(0) = x and ν(τ1 ) = γ (τ2 ). Reparameterize ν as ν(t) = Wx,Mt (s), t ≥ 0, for some s ∈ S m−1 . Then there exists τ 1 ∈ [0, +∞) such that Mτ 1 ν(τ 1 ) = γ (τ2 ) ∈ Mτ −τ2 . Hence τ 1 = τ − τ2 and since τ 1 , τ2 ≥ 0, we have τ 1 ∈ [0, τ ] and v ∈ Im W τ 1 . Define G : S m−1 × [0, τ ] → M by setting G(s, t) = W t (s). Since v = π M (y) = π M (x), we conclude that v ∈ Im G| S m−1 ×0 = Im W 0 = x. Since x, y do not lie on a common null geodesic, y ∈ Im Wx,Mτ and therefore v ∈ Im W τ = Im G| S m−1 ×τ . So
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G −1 (v) ⊂ S m−1 × (0, τ ). By Proposition 5 G| S m−1 ×(0,τ ] is an immersion. Proposition 4 and the fact v ∈ Im W τ 1 = Im G| S m−1 ×τ 1 imply that all the points in G −1 (v) = ∅ have the same sign. Thus one of n + (v, G) and n − (v, G) is zero and the other is nonzero. By the assumption of the theorem A (M) = Z and hence q : Ω0 (B) = Z → Z has zero t (s), a = 0, b = τ and Theorem 9 imply that kernel. Now Lemma 6 for F(s, t) = W x,M τ τ ,W ) = alk(W τ , εv ) = alk(W 0 , εv ) + q(n − (v, G) − alk(Sx , S y ) = alk(W x,M y,M x,M x,M n + (v, G)) = alk(εx , εv ) + q(n − (v, G) − n + (v, G)) = q(n − (v, G) − n + (v, G)) = 0 ∈ A (M). 8. alk-Invariant and Intersection Numbers Recall the following definition. Definition 21 (intersection number f 1 • f 2 ). Let f 1 : N1 → L l and f 2 : N2 → L l be transverse mappings of oriented manifolds of complimentary dimensions into an oriented manifold L l . Assume that the preimages of Im f 1 ∩ Im f 2 under f 1 and f 2 are finite sets. Take (n 1 , n 2 ) ∈ N1 × N2 such that f 1 (n 1 ) = f 2 (n 2 ) and take positive orientation frames r1 ⊂ Tn 1 N1 and r2 ⊂ Tn 2 N2 . Put σ (n 1 , n 2 ) = +1 if {d f 1 (r1 ), d f 2 (r2 )} ⊂ T f1 (n 1 ) L = T f2 (n 2 ) L is a positive orientation frame of L , and put σ (n 1 , n 2 ) = −1 otherwise. Since f 1 and f 2 are transverse and dim N1 + dim N2 = dim L , f i is an immersion in a neighborhood of n i , i = 1, 2. Hence σ (n 1 , n 2 ) = ±1 is well defined and does not depend on the choices of r1 , r2 . The intersection number f 1 • f 2 ∈ Z is defined as
f1 • f2 = σ (n 1 , n 2 ). (8.1) {(n 1 ,n 2 )∈N1 ×N2 | f 1 (n 1 )= f 2 (n 2 )}
Let (X m+1 , g), m > 1 be a globally hyperbolic space-time, and let x, y be two events in X that do not lie on a common null geodesic and such that y ∈ J + (x). Let exp = expx and U be as in Lemma 8. Let γ be a future directed past inextendible timelike curve that ends at y and does not pass through x. We say that γ is generic (with respect to expx |U ) if it is transverse to expx |U and it does not pass through the self-intersection points of expx |U . Using the transversality results of Biasi and Saeki [12, Theorem 2.4, Remark 2.7] one can show that every γ as above can be made generic by a C ∞ -small deformation. Note that if γ is generic and exp(u) ∈ Im γ , for some u ∈ U, then exp |U is an immersion in a neighborhood of u, since otherwise γ and expx are not transverse for dimension reasons. Theorem 12. Let (X m+1 , g), m > 1 be a globally hyperbolic space-time. Let x, y, U, and γ be as above. Then alk(Sx , S y ) = q(exp |U • γ ) ∈ A (M), where q : Z = Ω0 (B) → A (M) is the homomorphism from Theorem 10 and M is any Cauchy surface. Proof. Take a Cauchy surface M and an M-proper isometry h : (M × R, g = −βdt 2 + g) → (X, g). Without loss of generality we assume that x ∈ M = M0 . Since y ∈ J + (x), we conclude that πR (y) > 0. So without loss of generality we assume that y ∈ M1 . Since the R-coordinate in M × R is strictly increasing along all future directed nonspacelike curves, we assume (reparameterizing γ if necessary) that πR (γ (t)) = t. For t t by W at , brevity we denote expx |U by exp, we denote Wa,M by Wat , and we denote W a,M for a ∈ M.
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Define F : S m−1 × [0, +∞) → M × R,
F(s, t) = (Wxt (s), t).
By Lemma 8 there exists an orientation preserving diffeomorphism ω : U → S m−1 × (0, +∞) such that h ◦ F ◦ ω = exp . Since γ does not pass through x = Im h F| S m−1 ×0 and ω, h are orientation preserving, we get that exp •γ = F • (h −1 γ ). Since πR (γ (t)) = t we get that h −1 γ (t) ∈ Im F, for t < 0. So in the computation of F • (h −1 γ ) = exp •γ we can substitute γ by its restriction to [0, 1]. For brevity we denote γ |[0,1] by γ . We define α : [0, 1] → M, α(t) = π M (γ (t)) and put u = α(0), v = α(1). Define γ : [0, 1] → M × R, γ (t) = h −1 γ (t). Clearly γ (t) = (α(t), t), t ∈ [0, 1]. Since x1 , W y1 ) by Lemma 9, it exp •γ = F • (h −1 γ ) = F • γ and alk(Sx , S y ) = alk(W 1 1 γ ) ∈ A (M). suffices to show that alk(Wx , W y ) = q(F • m−1 m−1 Let Si , i = 1, 2, be a copy of S . Let H : (S1m−1 S2m−1 ) × [0, 1] → ST ∗ M xt and H | m−1 = εα(t) , t ∈ [0, 1]. This be a link homotopy given by H | S m−1 ×t = W S2 ×t 1 x0 , εu ) = (εx , εu ) to (W x1 , W y1 ) = (W x1 , εv ). Thus homotopy deforms the trivial link (W to prove the theorem it suffices to show that the link homotopy H is generic and the sum of the signs of the crossings of Σ0 during H equals to F • γ ∈ Z. We do this below by showing that all the crossings of Σ under H are in the bijective correspondence with the points of Im F ∩ Im γ , they happen in Σ0 , and are transverse. After this we show that the sign of a crossing of Σ0 under H coincides with the sign of the corresponding intersection point of Im F ∩ Im γ. Clearly z ∈ Im F ∩ Im γ exactly when F(s1 , τ ) = z = γ (τ ), for some s1 ∈ S m−1 , τ ∈ [0, 1]. Since γ is generic, such s1 , τ are uniquely determined by z. Hence the points z ∈ Im F ∩ Im γ are in the bijective correspondence with the pairs (s1 , τ ), s1 ∈ S m−1 , τ ∈ [0, 1] such that Wxτ (s1 ) = α(τ ) ∈ M. xt , εα(t) ) also happen exactly at The crossings of Σ under the link homotopy Ht = (W τ time moments τ when Wx (s1 ) = α(τ ), for some s1 ∈ S1m−1 . Since γ is generic, this s1 is uniquely determined by τ. The preimages of the double point of the singular link are xτ (s1 ). So we established s1 ∈ S1m−1 and the unique s2 ∈ S2m−1 such that εα(τ ) (s2 ) = W a bijective correspondence between the points in Im F ∩ Im γ and the crossings of Σ under H. xτ , εα(τ ) are embeddings, Let us prove that all these crossings happen in Σ0 . Since W xτ and f 2 = εα(τ ) . we conclude that condition 1 from Definition 12 holds for f 1 = W Note that if exp(a) ∈ Im γ then exp is an immersion in a neighborhood of a, since γ is transverse to exp. Hence Lemma 8 implies that exp |ω−1 (S m−1 ×τ ) has image in Mτ and it is an immersion in a neighborhood of ω−1 (s1 , τ ). Thus Wxτ = π M exp |ω−1 (S m−1 ×τ ) = xτ is an immersion in a neighborhood of s1 . Since Im pr ◦εα(τ ) = α(τ ), we get pr ◦W xτ )(Ts1 S m−1 ) ∩ Im(dεα(τ ) )(Ts2 S m−1 ) = 0 and condition 2 of Definition 12 that Im(d W 1 2 holds. Thus all the crossings of Σ under link homotopy H happen in Σ0 . xτ at W xτ (s1 ), a tangent frame to εα(τ ) at If we show that a tangent frame to W τ εα(τ ) (s2 ) = Wx (s1 ), and the vector w from the definition of σ (H, τ ) form a linearly independent family, then the crossing of Σ0 is transverse. Consider the differential d pr : T ST ∗ M → T M of pr : ST ∗ M → M. Since εα(τ ) is the inclusion of an S m−1 fiber of pr : ST ∗ M → M, it suffices to show that the images under d pr of w and of a xτ at W xτ (s1 ) are linearly independent in TW τ (s1 ) M. As we remarked, tangent frame to W x
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Wxτ is an immersion in a neighborhood of s1 . So to prove that the crossing of Σ0 is transverse it suffices to show that d pr(w) ∈ / Im(dWxτ )(Ts1 S1m−1 ). For s ∈ S m−1 , we define βs : [0, 1] → M, βs (t) = Wxt (s). Clearly h(βs1 (t), t), t ∈ [0, 1], is (up to a reparameterization) an arc of the null geodesic whose velocity x,Mt (s1 ) ∈ ST ∗ Mt , t ∈ [0, 1]. For brevity we denote the vectors define the points W vector field ∂/∂t on R by ∂t . Put ξ = β˙s1 (τ ) ∈ Tα(τ ) M = TWxτ (s1 ) M and note that ξ := ξ + ∂t ∈ Tz (M × R) is the velocity vector of the curve (βs1 (t), t), t ∈ [0, 1]. Since h is an isometry, ξ is a future pointing null vector with respect to the Lorentz metric g. Put η = α(τ ˙ ) ∈ Tα(τ ) (M × R). Note that η := η + ∂t is the velocity vector of γ and hence it is a future pointing timelike vector with respect to the Lorentz metric g. It is easy to see that d pr(w) = ξ − η. Let g = h ∗τ (g) be the Riemannian metric on M. The direction of the vector ξ is τ xτ is Wx (s1 ), where we identify ST ∗ M and ST M via the metric g. Since W m−1 τ Legendrian, ξ is g-orthogonal to Im dWx (Ts1 S1 ). To show that d pr(w) = ξ − η ∈ Im dWxτ (Ts1 S1m−1 ) it suffices to show that g (ξ − η, ξ ) > 0. The Lorentz product of a future pointing timelike and of a future pointing null vector is negative. Hence 0 < 0 − g( η, ξ ) = g( ξ , ξ ) − g( η, ξ ) = g( ξ − η, ξ) = g (ξ − η, ξ ) = g(ξ − η, ξ ).
(8.2)
Thus g (ξ − η, ξ ) > 0 and all the crossings of Σ0 under H are transverse. To finish the proof of the theorem it suffices to show that the intersection point z ∈ Im F ∩ Im γ has the same sign as the corresponding crossing of Σ0 under homotopy H. Let r be a positive orientation frame in Ts1 S1m−1 = Ts1 S1m−1 ⊕ 0 ⊂ Ts1 S1m−1 ⊕ Tτ (0, +∞) = T(s1 ,τ ) (S1m−1 × [0, +∞)). Then the sign σ ((s1 , τ ), s2 ) of the intersection point z is the sign of the orientation of M × R given by the frame {d F(r), ξ, η}. The vectors of d F(r) are tangent to M × τ ⊂ M × R and are spacelike with respect to g. The vector ξ is null. The straight line homotopy (1 − λ) ξ + λξ, λ ∈ [0, 1], of ξ to the spacelike vector ξ is g-orthogonal to d F(r) and induces a homotopy of the frame {(d F)(r), ξ, η} to the frame {(d F)(r), ξ, η}. We claim that the frame stays nondegenerate during the homotopy, so that the orientations of M × R given by the initial frame and the final frame are equal. If this is false, then since the vectors in d F(r) are linearly independent and have zero R-coordinate, we get that there exist a (nonzero) spacelike vector ζ ∈ Tα(τ ) M ⊂ Tz (M × R), a value λ ∈ [0, 1], and a, b ∈ R (with at least one of a, b nonzero) such that ζ ∈ span(d F(r)) and such that a(η + ∂t ) + b ((1 − λ)(ξ + ∂t ) + λξ ) + ζ = 0. (8.3) Equating the coefficients at ∂t , we see that a = −b(1 − λ), and since at least one of a, b is non-zero, b = 0. Substitute a = −b(1 − λ) into (8.3) to get − b(1 − λ)η + bξ + ζ = 0.
(8.4)
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Since ξ is g-orthogonal to d F(r) and ζ ∈ span(d F(r)), we conclude that g (ξ, ζ ) = 0. 1 1 Thus from (8.4) we have that λ = 1, and hence η = (1−λ) ξ + b(1−λ) ζ. Thus 1 2 1 g(ξ, ξ ) + g(ξ, ζ ) + 2 g(ζ, ζ ) 2 2 (1 − λ) b(1 − λ) b (1 − λ)2 1 1 g(ξ, ξ ) + 0 + 2 g(ζ, ζ ) > g(ξ, ξ ), = (1 − λ)2 b (1 − λ)2
g(η, η) =
(8.5)
since λ ∈ [0, 1] and ζ is a spacelike vector. Since g (ξ, ∂t ) = 0 = g (η, ∂t ) , the vectors ξ is null, and (8.5) holds, we conclude that η and ∂t are timelike, the vector 0 > g( η, η) = g (η + ∂t , η + ∂t ) = g (η, η) + g(∂t , ∂t ) ξ , ξ ) = 0. > g (ξ, ξ ) + g(∂t , ∂t ) = g(
(8.6)
This is a contradiction. Since M × R is a product of oriented manifolds and the two frames above give equal orientations of it, we see that the sign σ ((s1 , τ ), τ ) of the intersection point z ∈ Im F ∩ Im γ is positive exactly when {d(Wxτ )(r), ξ } = {d(Wxτ )(r), β˙s1 (τ )} is a positive orientation frame of M. Recall that ST ∗ M was oriented in such a way that an m-frame projecting to a positive frame on M m followed by a positive orientation frame of the S m−1 -fiber is a positive orientation frame of ST ∗ M. Since εα(τ ) is an inclusion of the positively oriented fiber, we conclude that σ (H, τ ) = +1 exactly when {d(Wxτ )(r), d pr(w)} = {d(Wxτ )(r), ξ −η} is a positive orientation frame of M. Since ξ is g-orthogonal to the immersed branch of the front Wxτ , and since by (8.2) g(ξ −η, ξ ) > 0, we conclude that ξ and ξ −η point to the same half-space of TWxτ (s1 ) M \ Im d(Wxτ )(Ts1 S1m−1 ) . Thus the orientations of M given by the frames {d(Wxτ )(r), ξ } and {d(Wxτ )(r), d pr(w)} are equal. Hence the signs of the intersection points of F with γ and of the corresponding crossings of Σ0 under H coincide. 9. Computing the Increment of alk Under the Passage Through a Dangerous Tangency Let (X m+1 , g), m > 1 be a globally hyperbolic space-time. Definition 14 and Lemma 2 imply that alk(Sx , S y ) can be computed as follows. Take a Cauchy surface M m ⊂ X and t ∈ R and choose a generic path α : [a, b] → S × S that deforms a pair (εu , εv ) t , W t ) ⊂ ST ∗ M. Put ti , i ∈ I, to be the time moments when α crosses Σ. to (W x,M y,M Since α is generic these crossings happen in Σ0 , and we put σ (α, ti ) = ±1, i ∈ I, to be the signs of these crossings, see Definition 13. By Theorem 10,
alk(Sx , S y ) = q σ (α, ti ) ∈ A (M). i∈I
Such a path α : [a, b] → S × S can be described as a family of maps α τ : S1m−1 m−1 ∗ → ST M, τ ∈ [a, b]. It also can be described as a family of maps α τ = pr ◦ ατ : S2 m−1 m−1 ∗ τ S2 → M, τ ∈ [a, b], equipped with a covector field θs ∈ Tα τ (s) M, s ∈ S1
S1m−1 S2m−1 , that defines the lift of α τ to α τ . In terms of the last description the
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crossings of Σ by α correspond to the triples (τ, s1 , s2 ) ∈ [a, b] × S1m−1 × S2m−1 such that α τ (s1 ) = α τ (s2 ) ∈ M and the nonzero covectors θsτ1 , θsτ2 are positive multiples of each other in Tα∗τ (s1 ) M = Tα∗τ (s2 ) M. (The triples (τ, s1 , s2 ) at which α τ (s1 ) = α τ (s2 ) and the covectors θsτ1 , θsτ2 are negative multiples of each other do not correspond to the double α τ (s1 ) and α τ (s2 ) are the opposite points of the S m−1 -fiber of points of α τ , since then ∗ pr : ST M → M.) t , W t are Legendrian embeddings S m−1 → As we know, the lifted wave fronts W x,M y,M ST ∗ M that are each Legendrian isotopic to a Legendrian embedding εw : S m−1 → ST ∗ M, w ∈ M. Put L ⊂ S to be the connected component of the space of Legendrian immersions S m−1 → ST ∗ M that contains the Legendrian embeddings εw , w ∈ M. t , W t ) can be chosen so that Clearly the generic path α joining (εu , εv ) to (W x,M y,M Im(α) ⊂ L×L ⊂ S×S. Let λ : [a, b] → L×L be such a path. (We changed the notation from α to λ in order to emphasize that λ is a path in L×L rather than in the whole S ×S.) Let λτ : S1m−1 S2m−1 → ST ∗ M, τ ∈ [a, b], be the corresponding family of maps and λτ : S1m−1 S2m−1 → M, τ ∈ [a, b] be the family of maps equipped with a let λτ = pr ◦ λτ . Since λτ are Legendrian, the covectors covector field θsτ that defines the lift of λτ to m−1 m−1 ∗ τ τ θs ∈ Tλτ (s) M vanish on (λ )∗ (Ts Si ) for s ∈ Si , i = 1, 2. If λ : [a, b] → L × L is generic, then the crossings of Σ by λ happen in Σ0 and correspond to the triples (τ, s1 , s2 ) as above with an extra condition that λτ restricted to small neighborhoods of s1 , s2 is an immersion. Since θsτ vanishes on (λτ )∗ (Ts Sim−1 ) for s ∈ Sim−1 , i = 1, 2, we get that λτ | S m−1 and λτ | S m−1 are tangent at λτ (s1 ) = λτ (s2 ). Combining all this 1 2 together we see that the crossings of Σ0 by a generic λ : [a, b] → L × L correspond to the so called Arnold’s [1] dangerous tangencies of λτ | S m−1 and λτ | S m−1 . These are 1 2 the instances when the immersed branches of λτ | S m−1 and λτ | S m−1 are tangent at exactly 1
2
one point, this tangency point has exactly one preimage on each of S1m−1 , S2m−1 , and the covectors defining the Legendrian lifts of λτ | S m−1 and λτ | S m−1 at the tangency point 1 2 are positive multiples of each other. (We will see that the tangency point is of order one, since λ is a generic path and σ (λ, τ0 ) is well-defined.) Below we give a formula for computing the sign σ (λ, τ0 ) of the crossing of Σ0 that corresponds to the passage through Arnold’s dangerous tangency of λτ0 | S m−1 and 1 λτ0 | S m−1 . 2 Put λiτ = λτ | S m−1 , i = 1, 2, and equip them with the restrictions of the covector field i θsτ . Consider a positively oriented chart ϕ : U → Rm , U ⊂ M with local coordinates {x1 , . . . , xm } such that: – λ1τ0 (s1 ) = ϕ −1 (0) = λ2τ0 (s2 ) ∈ M is the dangerous tangency point; – the restriction of λiτ0 to the preimage Vi of U under λiτ0 is an embedding, i = 1, 2; – the common tangent hyperplane to ϕλiτ0 |Vi , i = 1, 2 at the point ϕλ1τ0 (s1 ) = 0 = ϕλ2τ0 (s2 ) is given by the equation xm = 0; ∗ τ0 ) is a positive multiple of −d x ; – (ϕ −1 m s1 ) (θ – Im ϕλiτ0 |Vi is given by an equation xm = f i (x1 , . . . , xm−1 ), for some smooth function f i , i = 1, 2. Since λτ10 and λτ20 are dangerously tangent at ϕ −1 (0), we conclude that (ϕ −1 )∗ (θsτ20 ) is a positive multiple of −d xm . We put ε to be +1 if λ1τ0 and λτ20 induce the same orientation on the common tangent (m − 1)-plane at ϕ −1 (0), and we put ε = −1 otherwise. Put g = f 2 − f 1 and let Hess g(0) denote the Hessian of g at 0 ∈ Rm−1 . Put α to be the sign
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of the m th coordinate of the difference ϕ(λ2τ (s2 )) − ϕ(λτ1 (s1 )) ∈ Rm , for τ slightly bigger than τ0 . Theorem 13. σ (λ, t0 ) = (−1)k αε = αε sign(det Hess g(0)), where k is the number of negative eigenvalues of Hess g(0). Remark 5. Since we consider the passage through a dangerous tangency point that corresponds to a transverse crossing of Σ0 , we know that σ (λ, τ0 ) is defined. In particular, by Theorem 13 Hess g(0) is nondegenerate and hence the tangency point is of order one. Similarly α is well-defined, i.e. the difference of the m-th coordinates that we used to define α is nonzero. A version of this theorem appeared in our preprint [16]. Also some ingredients of this formula appeared in the work of T. Ekholm, J. Etnyre, and M. Sullivan [18, Proposition 3.3 and Lemma 3.4] in a different situation, where the authors compute the ThurstonBennequin invariant of a Legendrian submanifold of R2n+1 . Proof. Since ϕ : U → Rm is a positively oriented chart, we get that the sign of the crossing of Σ0 under the lifts of λ1τ and of λτ2 to ST ∗ M is equal to the sign of the crossing of Σ0 under the lift to ST ∗ Rm of the branches of ϕλ1τ and of ϕλτ2 in Rm that are equipped with the covector field (ϕ −1 )∗ (θsτ ), s ∈ S1m−1 S2m−1 . We use the flat Riemannian metric on Rm to identify ST ∗ Rm and ST Rm . Under this identification the codirection of a covector θ ∈ T ∗ Rm corresponds to the direction of the vector θ + ∈ T Rm that is orthogonal to ker θ and satisfies θ (θ + ) > 0. Thus to prove Theorem 13, it suffices to show that the formula in its formulation indeed gives the sign of the crossing of Σ0 under the lifts of the branches of ϕλτ1 and of ϕλτ2 to ST Rm . If one changes orientation of one of the two S m−1 -spheres parameterizing λ1τ and τ λ2 , then both expressions in the statement of Theorem 13 change sign. Thus without loss of generality we can assume that the orientations induced by ϕλ1τ0 and by ϕλ2τ0 on a common tangent hyperplane {(x1 , . . . , xm )|xm = 0} ⊂ T0 Rm are equal to the standard orientation of the Rm−1 -plane, and hence ε = 1. Without loss of generality we identify Vi , i = 1, 2, with Rm−1 , and we put (v1 , . . . , vm−1 ) to be the coordinates on Rm−1 . We parameterize the branches of ϕλiτ0 |Vi , i = 1, 2, by the maps Rm−1 → Rm . After an orientation preserving reparameterization, the branch ϕλiτ0 |Vi , i = 1, 2 is given by the parametric equations xk = vk for k = 1, . . . , m − 1, xm = f i (v1 , . . . , vm−1 ). We consider the unit hemisphere S − = {(x1 , . . . , xm ) x12 +· · ·+ xm2 = 1, xm < 0} ⊂ Rm , and we equip S − with local coordinates {y1 , . . . , ym−1 } by setting yk ( p) = xk ( p) for all p ∈ S − and k = 1, . . . , m − 1. Put µiτ , i = 1, 2 to be the lift of the branch of ϕλiτ to ST Rm . It is obtained by mapping a point v ∈ Rm−1 to the direction of the unit vector normal to ϕλiτ at ϕλiτ (v) on which the corresponding covector (ϕ −1 )∗ (θvτ ) is positive. So at the dangerous tangency point 0 ∈ Rm the two unit length vector fields defining the lifts µiτ , i = 1, 2, are equal to −∂/∂ xm . Let b be the unique point in Im µτ10 ∩Im µτ20 . Clearly pr(b) = 0 ∈ Rm and the product m − R ×S can be considered as the codomain of the chart ψ = {x1 , . . . , xm , y1 , . . . , ym−1 }
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at b ∈ ST Rm . The parametric equations for the lifts µiτ0 : Rm−1 → ST Rm−1 , i = 1, 2 are xk = vk ,
yk =
where ri =
1+
1 ∂ fi for k = 1, . . . , m − 1; xm = f i (v1 , . . . , vm−1 ), ri ∂vk
m−1 ∂ fi 2 ∂vk
k=1
(9.1)
.
This holds since (y1 , . . . , ym−1 , −1/ri ) is the unit normal vector to Im ϕλiτ0 at and this normal vector for v = 0 ∈ Rm−1 coincides with −∂/∂ xm . Let w be the vector from Definition 13. Let w = (α1 , . . . , α2m−1 ) in the chart ψ. Clearly α from the statement of Theorem 13 is equal to the sign of αm . To make the notation simpler for a function h : Rm−1 → R we put ϕλiτ0 (v)
∂k h =
∂h ∂vk
and
∂k,l h =
∂ 2h . ∂vk ∂vl
For i = 1, 2, the positive tangent frame to µiτ0 is given by vectors ξk(i) = (∂k x1 , . . . , ∂k xm , ∂k y1 , . . . , ∂k ym−1 ), k = 1, . . . , m − 1, where xk and yk are from (9.1) with the corresponding value of i. So, according to Definition 13, the sign σ (λ, τ0 ) is equal to the sign of the polyvector (1)
(1)
(2)
(2)
ξ1 ∧ · · · ∧ ξm−1 ∧ w ∧ ξ1 ∧ · · · ∧ ξm−1 , i.e. to the sign of the determinant with column vectors ξ (1) ’s, w and ξ (2) ’s computed at v = 0. Clearly ∂k f i ri ∂k,l f i − ∂k f i ∂l ri ∂l = . ri ri2 Since ∂k f i (0) = 0, k = 1, · · · , m − 1, and r1 (0) = 1 = r2 (0), we get that ∂l (yk ) = ∂l ( ∂krifi )(0) = ∂k,l f i (0) for yk from (9.1) with the corresponding value of i. Thus σ (λ, τ0 ) equals the sign of the determinant 1 ··· 0 α1 1 ··· 0 0 ··· 0 α2 0 ··· 0 .. .. .. .. .. .. .. . . . . . . . 0 ··· 1 0 ··· 1 αm−1 αm ∂1 f 2 · · · ∂m−1 f 2 (9.2) ∂1 f 1 · · · ∂m−1 f 1 ∂ f ··· ∂ αm+1 ∂11 f 2 · · · ∂m−1,1 f 2 11 1 m−1,1 f 1 ∂ f ··· ∂ αm+2 ∂12 f 2 · · · ∂m−1,2 f 2 m−1,2 f 1 12 1 .. .. .. .. .. .. .. . . . . . . . ∂1,m−1 f 1 · · · ∂m−1,m−1 f 1 α2m−1 ∂1,m−1 f 2 · · · ∂m−1,m−1 f 2 evaluated at 0 ∈ Rm−1 = {(v1 , . . . , vm−1 )}. Here the up-left and up-right (m − 1) × (m − 1) blocks of the matrix are identity matrices.
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Fig. 1. Dangerous tangency
Subtract the k th column from the (m+k)th one, k = 1, . . . , m−1 to get the determinant 1 ··· 0 α1 0 ··· 0 0 ··· 0 α2 0 ··· 0 .. .. .. .. .. .. .. . . . . . . . 0 ··· 0 0 ··· 1 αm−1 αm ∂1 g · · · ∂m−1 g (9.3) ∂1 f 1 · · · ∂m−1 f 1 ∂ f ··· ∂ αm+1 ∂11 g · · · ∂m−1,1 g 11 1 m−1,1 f 1 ∂ f ··· ∂ αm+2 ∂12 g · · · ∂m−1,2 g m−1,2 f 1 12 1 .. .. .. .. .. .. .. . . . . . . . ∂1,m−1 f 1 · · · ∂m−1,m−1 f 1 α2m−1 ∂1,m−1 g · · · ∂m−1,m−1 g evaluated at 0. Since ∂k g(0) = 0 = ∂k f 1 (0), k = 1 · · · , m − 1, this determinant equals αm det Hess g(0) and we proved the theorem. Example 5. Let us show how Theorem 13 allows one to compute σ (λ, τ0 ). Consider the passage through a dangerous tangency point in a positively oriented chart (x1 , . . . , xm ) shown in Fig. 1. Assume that the tangency in the figure happens along the (x1 , . . . , xm−1 )hyperplane and that the xm -axis points to the right in the figure. Assume that λ1τ is the “left” surface in the figure and that λ2τ is “right” surface. The vector field in the Figure is the unit vector field normal to the branches of λτ1 and λτ2 on which the evaluation of the covector field θsτ , s ∈ S1m−1 S2m−1 , is positive. Then α = −1, sign det Hess f (0) = 1, and thus σ (λ, τ0 ) = −ε. That is σ (λ, τ0 ) = −1 if the two tangent branches induce the same orientation on the common tangent (m − 1)-hyperplane and σ (λ, τ0 ) = +1 otherwise. 10. Examples To illustrate the usage of the affine linking invariant consider the following examples. Example 6. Let us show how one can use alk to determine that two events are causally related. Let (X m+1 , g), m > 1 be a globally hyperbolic space-time and let M m be a a,M by Cauchy surface in X . For brevity we denote Wa,M by Wa and we denote W a , a ∈ X . Let x, y be two events that do not lie on a common null geodesic. From W
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This spherical front propagates outside
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This spherical front propagates inside
Fig. 2. Example of alk computation
Definition 8 and Remark 1 it follows that x, y do not lie on a common null geodesic if x , W y ) form a nonsingular link in ST ∗ M. and only if the lifted wave fronts (W To compute the value of alk(Wx , W y ) ∈ A (M) we take a generic homotopy deforx , W y ). Let p and n be respectively ming a trivial link (εu , εv ), u = v ∈ M, to (W the numbers of positive and negative crossings of Σ0 ⊂ Σ under the homotopy. xt , W yt ) = q( p − n) ∈ A (M), for the homomorphism Then alk(Sx , S y ) = alk(W x , W y ) = alk(Sx , S y ) = 0 ∈ A (M), then we q : Ω0 (B) = Z → A (M). If alk(W conclude that x and y are causally related, see Theorem 8 and Theorem 9. Observe that this computation and conclusion can be made just from the shape of the cooriented and oriented fronts Wx , W y on a Cauchy surface M, without the knowledge of the event points x, y and of the Lorentz metric g on X . Moreover, if M is not homeomorphic to an even dimensional sphere S 2k , then one does not have to equip the pictures of the fronts with orientations. This is since for such manifolds a positively oriented S m−1 -fiber of pr : ST ∗ M → M is not free homotopic to a negatively oriented fiber S m−1 , see Theorem 16. Thus if M is not homeomorphic to an even dimensional sphere, then the orientation of the cooriented wave front Wx on M is always the one such that the lifted wave front with this orientation is homotopic to a positively oriented fiber S m−1 of pr : ST ∗ M → M. As an example of the computation, consider a globally hyperbolic (X, g) such that its Cauchy surface M is not homeomorphic to a sphere. Thus in this case the orientation of the fronts does not have to be included in their description and A (M) = 0, see Proposition 3 and Theorem 10. Let (Wx , W y ) be two wave fronts located in a chart diffeomorphic to Rm . Assume that for some vector v ∈ Rm the straight line homotopy h τ,v = (Wx + τ v, W y ), τ ∈ [0, +∞) separates the fronts to be located in two different “half-spaces” of Rm . Assume moreover that this homotopy involves exactly one passage through a dangerous tangency point and this tangency point is nondegenerate, see for example Fig. 2. Then by Theorem 13 and x , W y ) = alk(Sx , S y ) = ±1 = 0 ∈ A (M). the discussion before it we have alk(W Here the sign ±1 depends on the sign of the determinant of the Hessian at the dangerous tangency point and on the coorientations and the actual orientations of the fronts. Hence the events x and y are causally related. Example 7. Let us show how one can use alk to estimate the number of times the exponential of the future directed null hemicone of a point x crosses a generic timelike
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z
y
a
b
Fig. 3. Example of using alk to estimate the number of crossing points of a timelike curve and the exponentiated future null hemicone of an event.
curve joining two points y, z. This number can be interpreted as the number of times that an observer traveling from y to z along a generic timelike curve sees the light from the event x. Let (X m+1 , g) be a globally hyperbolic space-time of dimension > 2 such that its Cauchy surface M m is not an odd-dimensional rational homology sphere with finite π1 (M). Theorem 10 says that A (M) = Z and q : Z → A (M) is the identity map. Assume moreover that M is not an even dimensional homotopy sphere, so that as we discussed in Example 6, we do not have to specify the orientations of the fronts Wx,M when depicting them. Let y, z ∈ X be two points that can be joined by a future directed generic timelike curve from y to z. Assume moreover that z ∈ J + (x). Let L y and N z be two Cauchy surfaces. Assume that Im Wx,L and y ∈ L are in the same chart of L and are shown in Fig. 3.a. Assume that Im Wx,N and z ∈ N are in the same chart of N and are shown in Fig. 3.b. (Figure 3.a depicts a trivially embedded sphere with y outside of it. Figure 3.b depicts a sphere that can be obtained from the trivially embedded sphere located far from z by passing three times through a point z and by creation of some singularities far away from z.) The normal vector fields to the fronts in Fig. 3.a and in Fig. 3.b are such that the evaluations of the covector fields defining the front lifts to ST ∗ L and to ST ∗ N on the vector fields are positive. That is, these are the vector fields defining the front lifts to ST L and to ST N that are identified with ST ∗ L and with ST ∗ N via the Riemannian metrics g| L and g| N . Using Lemma 6 we get that alk(Sx , S y ) = 0 and alk(Sx , Sz ) = 3. Let γ be a generic (as defined before Theorem 12) past inextendible future directed curve ending at y. Let U ⊂ Tx X be the part of the future pointing null hemicone where expx is defined. We have expx |U • γ = 0 ∈ Z, where • is the intersection number. Indeed if (Im expx |U ) ∩ (Im γ ) = ∅, then the statement is trivial. If (Im expx |U ) ∩ (Im γ ) = ∅, then y ∈ J + (x) and the statement follows from Theorem 12. Let β be a generic future directed timelike curve from y to z. Then β · γ is a generic future directed past inextendible curve ending at z. Since z ∈ J + (x), Theorem 12 says that expx |U • (β · γ ) = 3. Combining this with equality expx |U • γ = 0, we conclude that expx |U • β = 3. Thus an observer traveling from y to z along β sees the light from the event x at least 3 times regardless of which generic timelike curve s/he chooses to travel. (If β is not generic and the points of self-intersection of expx |U belong to Im β
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then, at a point of β, s/he might see the light coming from several different directions, and the total number of times s/he sees light may be less than 3.) 11. Refocussing and Nonrefocussing Space-times In this section we discuss (non)refocussing space-times. We use them in the next section. Let SKY denote the set of all skies in (X, g) with the following topology. For ∈ SKY the topology base at S is given by {S|S ⊂ W } for open W ⊂ N such that S ⊂ W. S Consider the map (11.1) µ : X → SKY, µ(x) = Sx . One verifies that if (X, g) is globally hyperbolic, then µ is continuous. Example 2 shows that µ is not even a bijection in general. Is µ a homeomorphism provided that it is a bijection? In order for µ to be open it suffices to show that for every x ∈ X and open U x there exists an open V x contained in U such that µ(V ) is open. This motivates the following definition, cf. [29]. Definition 22. A strongly causal space-time (X, g) (that is not necessarily globally hyperbolic) is called refocussing at x ∈ X if there exists a neighborhood O of x with the following property: For every open U with x ∈ U ⊂ O there exists y ∈ U such that all the null-geodesics through y enter U. A space-time (X, g) is called refocussing if it is refocussing at some x, and it is called nonrefocussing if it is not refocussing at every x ∈ X. Low [29] introduced the concept of nonrefocussing space-times and observed that if a globally hyperbolic (X, g) is nonrefocussing, then µ is bijective and open, i.e. µ : X → SKY is a homeomorphism. We note that in the original definition of Low [29] U was allowed to be any open neighborhood containing x that is not necessarily sufficiently small. This is clearly a typo, since for every (X, g) and U = X such a point y ∈ U does not exist. We need the following topological lemma. Lemma 9. Let M m be a (not necessarily orientable) non-compact manifold and let B be an open ball in M. Assume that the closure B of B is a smoothly embedded ball. Let V be an open subset in M such that its closure V is compact and V \ V ⊂ B. Then V ⊂ B. Proof. Without loss of generality we assume that V is connected. Since V \V is compact, there exists an open disk B0 ⊂ B such that V \ V ⊂ B0 ⊂ B 0 ⊂ B and the boundary B 0 \ B0 of B0 is a smoothly embedded (m − 1)-dimensional sphere S. We may assume that V ∩ S = ∅. Otherwise since V is connected, V ⊂ B0 ⊂ B and the proof is finished. Furthermore, the set V ∩ S = V ∩ S is open as well as closed in S. Since S is connected and V ∩ S = ∅, we conclude that S ⊂ V . Arguing by contradiction, suppose that V \ B = ∅. Since V \ V ⊂ B, we have that V \ B = ∅. Then V \ B 0 = ∅ is an open subset of M and Y := V \ B0 = V \ B0 is a compact connected smooth orientable manifold with the interior Int Y = V \ B 0 . Hence ∂Y = B 0 \ B0 = S.
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Take a point a ∈ Y \ ∂Y and consider the commutative diagram (Y, ∂Y ) ⏐ ⏐
−−−−→
(M, M \ a)
(M, B0 ) ⏐ ⏐ (M, M \ a)
of inclusions. This diagram induces the commutative diagram Z2
Hm (Y, ∂Y ; Z2 ) ⏐ ⏐
Z2
Hm (M, M \ a; Z2 )
−−−−→
Hm (M, B0 ; Z2 ) ⏐ ⏐
0
Hm (M, M \ a; Z2 )
Z2 .
Here Hm (M, B0 ; Z2 ) = Hm (M; Z2 ) = 0 since M is not compact. Since the left map is an isomorphism, we conclude that such a diagram cannot exist. Thus, V ⊂ B. Definition 23. A set A in a (not necessarily globally hyperbolic) space-time (X, g) is achronal if no timelike curve intersects A more than once. In particular every subset of a Cauchy surface is achronal. For an achronal set A its future Cauchy development D + (A) is the set of all the points x ∈ X such that every past inextendible non-spacelike curve through x meets A. Similarly, the past Cauchy development D − (A) is the set of all x ∈ X such that every future inextendible non-spacelike curve through x meets A. In particular A is a subset of both D + (A) and D − (A). The Cauchy development of A is D(A) = D + (A) ∪ D − (A). If M is a Cauchy surface in a globally hyperbolic space-time (X, g), then X = D + (M) ∪ D − (M). Proposition 6 (Low [29,30]). A globally hyperbolic space-time (X, g) with a non-compact Cauchy surface M, is nonrefocussing. Proof. A brief outline of the proof is contained in [30, Theorem 5]. We are grateful to Robert Low who explained to us the details of his proof. Assume that (X, g) is refocussing at a point x. Take an open neighborhood O of x such that for every open V with x ∈ V ⊂ O there exists y ∈ V such that all the null geodesics through y enter V. Take a Cauchy surface M through x and an open ball B in M with x ∈ B such that the closure B is a smoothly embedded ball. Put U = D(B). Then U is open, globally hyperbolic and contains B, see [36, Sect. 14, Lemma 42 and Lemma 43]. Clearly B is a Cauchy surface of U. Assume moreover that B is sufficiently small so that U ⊂ O. Take a point y ∈ X with y ∈ U = D(B) such that all the null-geodesics through y cross U . Without loss of generality we assume that y ∈ D + (M). By [9, Proposition 3.16 and Lemma 3.5], the set J − (y) is closed and the set I − (y) is open. Moreover, J − (y) is the closure of I − (y) by [36, Sect. 14, Lemma 6]. Put J − = J − (y) ∩ M and I − = I − (y) ∩ M. Because of what we said above, J − is the closure (in M) of the open subset I − of M. Since J − (y) ∩ D + (M) is compact by [36, Sect. 14, Lemma 40], we get that J − is compact in M. By [9, Corollary 4.14] if z ∈ J − (y) \ I − (y), then there is a null-geodesic from y to z. Thus if z ∈ J − \ I − , then z lies on a past directed null-geodesic from y. By our choice of y this null geodesic has to pass through U. Since B is a Cauchy surface of a
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globally hyperbolic U, this null geodesic crosses M in some point of B ⊂ M. Thus all the points of J − \ I − are in B, i.e. J − \ I − ⊂ B. By Lemma 9 applied to the case V = I − we get the inclusion J − (y) ∩ M ⊂ B. Thus y ∈ D + (B) ⊂ U and we get a contradiction. Clearly if p : (X 1 , g1 ) → (X, g) is a Lorentz cover of a globally hyperbolic spacetime and (X 1 , g1 ) is refocussing, then (X, g) is also refocussing. Below we prove the converse result. Theorem 14. Let (X m+1 , g) be a globally hyperbolic space-time that is refocussing, and let p : X 1 → X be a covering map. We equip X 1 with the induced Lorentz metric g1 . Then (X 1 , g1 ) is a refocussing globally hyperbolic space-time. In particular, if X has infinite fundamental group then X is nonrefocussing, see Proposition 6. Proof. First, we prove that (X 1 , g1 ) is globally hyperbolic. It suffices to prove that (X 1 , g1 ) admits a Cauchy surface. Choose a Cauchy surface M ⊂ X and put M1 = p −1 (M). We claim that M1 is a Cauchy surface. Indeed, if γ (t) is an inextendible nonspacelike curve in X 1 , then p ◦ γ (t) is an inextendible nonspacelike curve in X. Since M is a Cauchy surface, p ◦ γ (t) crosses M at exactly one value of t. Hence γ (t) also crosses M1 at exactly one value of t, and thus M1 is a Cauchy surface. Assume that X is refocussing at some x ∈ X. Take a Cauchy surface M in (X, g) with x ∈ M and consider the Cauchy surface M1 = p −1 (M) in (X 1 , g1 ). Choose x1 ∈ p −1 (x) ⊂ M1 . We will show that (X 1 , g1 ) is refocussing at x1 . Let grM1 and grM be the Riemannian metrics that are induced on M1 and on M by g1 and g, respectively. Choose an open ball B1 ⊂ M1 such that the closure B 1 is a smoothly embedded closed ball, p| B 1 is an embedding, and B1 is a normal neighborhood of x1 in (M1 , grM1 ). Put B = p(B1 ) to be the open ball containing x. Clearly B is a normal neighborhood of x in (M, grM ). Put V1 = D(B1 ) to be the Cauchy development of B1 . Then V1 is open globally hyperbolic and contains B1 , see [36, Sect. 14, Lemma 42 and Lemma 43]. Clearly B1 is a Cauchy surface for V1 . Put V = p(V1 ) x. Since p is a covering map, V is open. Clearly B is a Cauchy surface for V and hence V is globally hyperbolic. Since V is homeomorphic to B × R, it is simply connected and hence p : V1 → V is a diffeomorphism. Let O be a neighborhood of x described in Definition 22. It is not difficult to prove that the ball B1 can be chosen to be small enough so that V ⊂ O. Put O1 = V1 and let us show that (X 1 , g1 ) is refocussing at x1 . Choose any open U1 with x1 ∈ U1 ⊂ O1 . Put U = p(U1 ) ⊂ V ⊂ O. Since (X, g) is refocussing at x there exists y ∈ U such that all the null geodesics through y cross U. Without loss of generality y ∈ D + (M). Choose an M-proper isometry h : M × R → X and put (m y , t y ) ∈ M × R to be the point such that h(m y , t y ) = y. Define F : S m−1 × R → M × R via F(s, t) = t (W y,M (s), t). For s ∈ S m−1 put γs (t) = F(s, t). Clearly up to reparameterization the curves h ◦ γs (t), s ∈ S m−1 , are exactly all the null geodesics through y. Also h(γs (0)) ∈ B is exactly the intersection point of the corresponding null geodesic with B and h(γs (t y )) = y for all s ∈ S m−1 . For s ∈ S m−1 put ρs : [0, 1] → B to be uniform speed parameterization of the unique geodesic (with respect to grM ) arc in B from x ∈ B to h(γs (0)) ∈ B. For s ∈ S m−1 define the path δs : [0, 1 + t y ] → X from x to y via δs (t) = ρs (t) for t ∈ [0, 1] and δs (t) = h(γs (t − 1)) for t ∈ [1, 1 + t y ].
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For every s0 , s1 ∈ S m−1 the paths δs0 and δs1 are homotopic relative boundary. The homotopy is given by the family of paths δβ(τ ) constructed from a path β : [0, 1] → S m−1 with β(0) = s0 , β(1) = s1 . (Such β exists since S m−1 , m > 1 is connected.) For s ∈ S m−1 put δ1,s : [0, 1 + t y ] → X 1 to be the lift of δs starting at x1 . Since all the paths δs are homotopic relative boundary, we get that all the values δ1,s (1 + t y ) ∈ X 1 are equal and we put y1 = δ1,s (1 + t y ). Note that the parts of the lifted paths δs , s ∈ S m−1 that project to h ◦ γs are (up to reparameterization) the arcs of the null geodesics through y1 . Since y ∈ U and p : U1 → U is a diffeomorphism, we conclude that y1 ∈ U1 . Take a null geodesic β1,s (t), s ∈ S m−1 through y1 . By construction of y1 , we have that p(Im β1,s ∩ O1 ) ∩ U = ∅. Since p| O1 is a diffeomorphism, U1 ⊂ O1 , and p(U1 ) = U, we get that Im β1,s ∩ U1 = ∅. Thus, (X 1 , g1 ) is refocussing at x1 . Remark 6. Since the manifold M in Lemma 9 is not assumed to be orientable, Proposition 6 and Theorem 14 hold also for nonorientable time-oriented globally hyperbolic space-times. In this case the Cauchy surfaces are nonorientable. Remark 7. Refocussing space-times and problems related to the Blaschke conjecture. The following construction gives many examples of refocussing globally hyperbolic space-times. Let (M, g) be a complete oriented Riemannian manifold, such that for some x ∈ M and positive r ∈ R the exponential expx : Tx M → M maps the whole sphere of radius r centered at 0 ∈ Tx M to one point. A static Lorentz manifold (M ×R, g ⊕−dt 2 ) is globally hyperbolic, see [9, Theorem 3.66], and it is clearly refocussing at (x, r ). One can show that then x is the end point of all the length 2r geodesic arcs in M x -manifold in terms of Besse [11, Chapter 7.B]. The starting at x, i.e. (M, g) is a Y2r question on topology of such manifolds is closely related to the Blaschke conjecture type problems, see [11]. A weak form of the Bott-Samelson Theorem [13,40] says that x -manifold is a closed manifold with finite π whose rational cohomology ring every Y2r 1 is generated by one element, see Berard-Bergeri [10] and Besse [11, Theorem 7.37]. Clearly there are many examples of refocussing globally hyperbolic space-times that are not obtained by the above construction. However Proposition 6 due to Low and our Theorem 14 say that a Cauchy surface in all of them is a closed manifold with finite π1 . It would be interesting to know if its rational cohomology ring is necessarily generated by one element, i.e. if the Bott-Samelson type result holds for a Cauchy surface of a refocussing globally hyperbolic space-time. Since the only two-dimensional surfaces with finite π1 are S 2 and RP 2 , Proposition 6 and Theorem 14 imply that this is indeed so for (2 + 1)-dimensional globally hyperbolic refocussing space-times. 12. A Weakened Low Conjecture is True We show that a certain weakened version of the Low conjecture holds for a vast family of globally hyperbolic space-times (X m+1 , g), m > 1. Natario and Tod [35, Fig. 13, p. 18] considered (2 + 1)-dimensional space-times with a Cauchy surface diffeomorphic to R2 and presented several examples of causally related events whose skies are linked but have zero linking number. They also observed that since the skies of events are Legendrian submanifolds of N , it makes sense to ask if the skies of two causally related events are always nontrivially linked in the Legendrian sense. When a Cauchy surface M is diffeomorphic to an open subset of Rm , this is the modified Low conjecture due to Natario and Tod [35].
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However even for (2 + 1)-dimensional space-times not all of the Legendrian embeddings S m−1 → ST ∗ M = N that are Legendrian isotopic to εv , v ∈ M, correspond to skies, see [35, Theorem 4.5]. Thus one can weaken the Low conjecture even further and ask if it is always true that the skies of causally related events in (X, g) can not be unlinked by an isotopy through the skies of events in (X, g). Definition 24 (Isotopy through skies). Let (X m+1 , g), m+1 > 2, be a globally hyperbolic space-time. We say that two nonsingular links (S1 , S2 ) and (S1 , S2 ) are isotopic through skies if there exists a continuous map ρ : [0, 1] → SKY × SKY, ρ(t) = (ρ1 (t), ρ2 (t)) such that ρi (0) = Si , ρi (1) = Si , i = 1, 2 and for all t ∈ [0, 1] the intersection of the skies ρ1 (t) and ρ2 (t) in N is empty. Definition 25 (Sky-isotopy). Let x1 , x2 , y1 , y2 ∈ X be such that neither x1 , x2 nor y1 , y2 belong to a common null geodesic. We say that the pairs (x1 , x2 ) and (y1 , y2 ) are sky-isotopic if there exist paths p1 , p2 : [0, 1] → X such that pi (0) = xi , pi (1) = yi , i = 1, 2, and the skies S p1 (t) and S p2 (t) are disjoint, for all t ∈ [0, 1]. (The last condition is equivalent to requiring that for every t ∈ [0, 1] the points p1 (t), p2 (t) do not belong to a common null geodesic.) Remark 8 (Comparison of the “sky-isotopy” and of the “isotopy through skies” notions). If (x1 , x2 ) and (y1 , y2 ) are sky-isotopic, then clearly the links (Sx1 , Sx2 ) and (S y1 , S y2 ) are isotopic through skies. Indeed given the paths p1 (t), p2 (t) as in the definition of skyisotopy, put ρi (t) = S pi (t) , t ∈ [0, 1], i = 1, 2. The converse is not true in general, the pairs (x, y) and (x , y) of events in Example 3 yield a counterexample. For (X, g) nonrefocussing, (x1 , x2 ) and (y1 , y2 ) are sky-isotopic if and only if (Sx1 , Sx2 ) and (S y1 , S y2 ) are isotopic through skies. Indeed given ρi : [0, 1] → SKY, i = 1, 2, as in Definition 25, put pi (t) = µ−1 (ρi (t)), where µ : X → SKY is the homeomorphism from (11.1). The following Theorem 15 says that any two pairs of causally unrelated events in a globally hyperbolic (X, g) are sky-isotopic, and that no such pair is sky-isotopic to a pair of causally related events. Theorem 15. Let (X m+1 , g), m +1 > 2 be a globally hyperbolic space-time. Let (x1 , x2 ) be a pair of causally unrelated events, and let (y1 , y2 ) be two events that do not belong to a common null geodesic. Then the following two statements are equivalent: 1. The events y1 and y2 are not causally related. 2. The pairs (x1 , x2 ) and (y1 , y2 ) are sky-isotopic. Proof. Choose a Cauchy surface M ⊂ X and an M-proper isometry h : M × R → X. The proof of the implication 1 =⇒ 2 follows immediately from the following three claims that are proved below: Claim 1. For any causally unrelated v1 , v2 there exist t ∈ R and w1 , w2 ∈ Mt ⊂ X such that (v1 , v2 ) is sky-isotopic to (w1 , w2 ). Claim 2. If (v1 , v2 ) and (w1 , w2 ) are two pairs of distinct events in the same Cauchy surface Mτ ⊂ X, then (v1 , v2 ) is sky-isotopic to (w1 , w2 ). Claim 3. If t1 = t2 ∈ R, n 1 = n 2 ∈ M, then (h(n 1 , t1 ), h(n 2 , t1 )) and (h(n 1 , t2 ), h(n 2 , t2 )) are sky-isotopic pairs of events.
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We prove Claim 1. Let t1 , t2 ∈ R be such that vi ∈ Mti , i = 1, 2. Without loss of generality we assume that t1 ≤ t2 . Let γ be a future directed inextendible timelike curve through v2 . Reparameterize γ so that γ (t) ∈ Mt ⊂ X for all t ∈ R. Since v1 and v2 are causally unrelated, we conclude that Sv1 ∩ Sγ (t) = ∅ for all t ∈ [t1 , t2 ]. Indeed, if Sv1 ∩ Sγ (τ ) = ∅ for some τ ∈ [t1 , t2 ], then the arc of a null geodesic ν ∈ Sv1 ∩ Sγ (τ ) from v1 to γ (τ ) followed by γ |[τ,t2 ] is a future directed non-spacelike curve from v1 to v2 . Put w1 = v1 , w2 = γ (t1 ) ∈ Mt1 . Now, to see that (w1 , w2 ) = (v1 , γ (t1 )) is sky-isotopic to (v1 , γ (t2 )) = (v1 , v2 ), put p1 (t) = v1 , p2 (t) = γ (t), t ∈ [t1 , t2 ]. We prove Claim 2. Let (v1 , v2 ) and (w1 , w2 ) be two pairs of distinct events in the same Cauchy surface Mτ . Since dim(X ) > 2 and hence dim Mτ > 1, we can choose two paths p1 (t), p2 (t) in Mτ , t ∈ [0, 1] such that pi (0) = vi , pi (1) = wi , i = 1, 2 and p1 (t) = p2 (t) for all t ∈ [0, 1]. Since any two distinct points in the same Cauchy surface are causally unrelated, S p1 (t) ∩ S p2 (t) = ∅, for all t ∈ [0, 1]. Thus, (v1 , v2 ) is sky-isotopic to (w1 , w2 ). We prove Claim 3. Assume without loss of generality that t1 < t2 . Put p1 (t) = h(n 1 , t), p2 (t) = h(n 2 , t), t ∈ [t1 , t2 ]. Since p1 (t), p2 (t) ∈ Mt , the events p1 (t) and p2 (t) are causally unrelated, for t ∈ [t1 , t2 ]. Hence S p1 (t) ∩ S p2 (t) = ∅, for all t ∈ [t1 , t2 ], and (h(n 1 , t1 ), h(n 2 , t1 )) is sky-isotopic to (h(n 1 , t2 ), h(n 2 , t2 )). This completes the proof of Claim 3 and, hence, of the implication 1 =⇒ 2 of the theorem. To prove the implication 2 =⇒ 1, recall the notion of Lorentzian distance, see [9]. For points p, q in a (not necessarily globally hyperbolic) space-time (X, g) with q ∈ J + ( p) put Ω p,q to be the space of all piecewise smooth future directed non-spacelike curves δ : [0, 1] → X with γ (0) = p, γ (1) = q. For δ ∈ Ω p,q choose a partition 0 = t0 < t1 < t2 · · · < tn−1 < tn = 1 such that δ|(ti ,ti+1 ) is smooth for all i ∈ {0, 1, · · · (n − 1)}, and define the Lorentzian arc length L(δ) of δ by L(δ) = L g (δ) =
n−1
i=0
ti+1
˙ ), δ(τ ˙ ))dτ. −g(δ(τ
ti
For p, q ∈ (X, g) define the Lorentzian distance function d = dg : X × X → R ∞ as follows: set d( p, q) = 0 for q ∈ J + ( p); and set d( p, q) = sup{L g (δ)|δ ∈ Ω p,q }, for q ∈ J + ( p). By [36, Chap. 14, Corollary 1], if a << b and b ≤ c, or if a ≤ b and b << c, then a << c. Combining this with the definition of the Lorentzian distance d we get that d( p, q) > 0 if and only if q ∈ I + ( p). In general, the Lorentzian distance function is not continuous, and d( p, q) is not finite. (Also d( p, q) = d(q, p) and d( p, p) = 0 in many cases.) However, for (X, g) globally hyperbolic, d satisfies the finite distance condition and is a continuous function on X × X , see [9, Corollary 4.7]. We argue by contradiction. Assume that y1 , y2 are causally related, but the pairs (x1 , x2 ) and (y1 , y2 ) are sky-isotopic. Since y1 , y2 ∈ X are causally related, either y1 ∈ J + (y2 ) or y2 ∈ J + (y1 ). Without loss of generality assume that y2 ∈ J + (y1 ). If y2 ∈ J + (y1 ) \ I + (y1 ), then y1 and y2 lie on a common null geodesic, see [9, Corollary 4.14]. This contradicts the theorem assumptions. Hence y2 ∈ I + (y1 ) and d(y1 , y2 ) > 0. Since (x1 , x2 ) and (y1 , y2 ) are sky-isotopic, there exist p1 , p2 : [0, 1] → X such that pi (0) = yi , pi (1) = xi , i = 1, 2, and such that S p1 (t) ∩ S p2 (t) = ∅, for all t ∈ [0, 1]. Define a continuous function d : [0, 1] → R by d(t) = d( p1 (t), p2 (t)). We have d(0) = d( p1 (0), p2 (0)) = d(y1 , y2 ) > 0. Furthermore, d(1) = d( p1 (1), p2 (1)) =
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d(x1 , x2 ) = 0, since x1 , x2 are causally unrelated. Put τ = inf{t ∈ [0, 1]|d(t) = 0}, so that d(τ ) = 0 and d(t) > 0 for all t < τ. (12.1) Below we show that S p1 (τ ) ∩ S p2 (τ ) = ∅. This contradicts our assumptions about p1 , p2 . By [22, Proposition 6.6.1] (X, g) is causally simple, i.e. the sets J ± (K ) = ∪k∈K J ± (k) are closed for every compact K ⊂ X . By (12.1), d( p1 (t), p2 (t)) = d(t) > 0 for all t < τ. Hence p2 (t) ∈ I + ( p1 (t)) ⊂ J + ( p1 (t)) for all t < τ, and so Im( p2 |[t,τ ) ) ⊂ J + (Im( p1 |[t,τ ] )) for all t < τ. Since Im( p1 |[t,τ ] ) is compact and (X, g) is causally simple, we conclude that J + (Im( p1 |[t,τ ] )) is closed, and hence p2 (τ ) ∈ J + (Im( p1 |[t,τ ] )) for all t < τ. Choose an increasing sequence {ti ∈ [0, 1]}i∈N that converges to τ. Then for each i ∈ N there exists ti ∈ [ti , τ ] such that p2 (τ ) ∈ J + ( p1 ( ti )). Hence p1 ( ti ) ∈ J − ( p2 (τ )) for all i. Since (X, g) is causally simple, J − ( p2 (τ )) is closed and it contains the point p1 (τ ) = lim p1 ( ti ). Since p1 (τ ) ∈ J − ( p2 (τ )), we have p2 (τ ) ∈ J + ( p1 (τ )). i→∞
On the other hand, p2 (τ ) ∈ I + ( p1 (τ )) since d( p1 (τ ), p2 (τ )) = d(τ ) = 0 by (12.1). So, p2 (τ ) ∈ J + ( p1 (τ )) \ I + ( p1 (τ )), and therefore the points p1 (τ ), p2 (τ ) belong to a common null geodesic, see [9, Corollary 4.14]. Thus S p1 (τ ) ∩S p2 (τ ) = ∅. Contradiction. Remark 9. Looking carefully at the proof of the implication 2 =⇒ 1 of Theorem 15 one notices that in fact we proved the following stronger statement. Let (X m+1 , g) be a causally simple space-time such that the Lorentzian distance on it is a continuous function satisfying the finite distance condition. Let (x1 , x2 ) be a pair of causally unrelated events and let (y1 , y2 ) be a pair of causally related events. Then for every pair of continuous paths pi : [0, 1] → X such that pi (0) = xi , pi (1) = yi , i = 1, 2, there exists t ∈ [0, 1] for which p1 (t) and p2 (t) belong to the common null geodesic. The following Corollary 1 can be viewed as the proof of a weakened Low conjecture saying that two events y1 , y2 in a nonrefocussing globally hyperbolic (X m+1 , g), m +1 > 2, that do not belong to a common null geodesic, are causally unrelated if and only if the link (S y1 , S y2 ) is isotopic through skies to a trivial link. (Probably the best choice for the trivial link consists of skies of two events on the same Cauchy surface.) Corollary 1. Let (x1 , x2 ) be two causally unrelated events in a nonrefocussing globally hyperbolic space-time (X m+1 , g), m + 1 > 2. Let (y1 , y2 ) be two events that do not belong to a common null geodesic, then the following two statements are equivalent: 1. The nonsingular links (Sx1 , Sx2 ) and (S y1 , S y2 ) are isotopic through skies. 2. The events y1 , y2 are causally unrelated. Proof. Remark 8 says that for nonrefocussing globally hyperbolic (X, g) two events are sky-isotopic if and only if their skies are isotopic through skies. Now Corollary 1 follows from Theorem 15. Remark 10 (Isotopies that consist of skies at each time moment). Using Corollary 1 and the proof of the implication 1 =⇒ 2 of Theorem 15 one can show the following result. Let (X m+1 , g), m > 1 be a nonrefocussing globally hyperbolic space-time. Put Emb(S m−1 S m−1 , N ) to be the space of smooth embeddings S m−1 S m−1 → N . Let x1 , x2 ∈ X be two causally unrelated points and let y1 , y2 be two points that do not lie
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on a common null geodesic. Then y1 , y2 are causally unrelated if and only if there is an isotopy r = r (t) = (r1 (t), r2 (t)) : [0, 1] → Emb(S m−1 S m−1 , N ) such that Im ri (t) is a sky for all t ∈ [0, 1] and Im ri (0) = Sxi , Im ri (1) = S yi , i = 1, 2. Acknowledgement. The first author was partially supported by free-term research money from the Dartmouth College. The second author was supported by the by MCyT, projects BFM 2002-00788 and BFM 200302068/MATE, Spain, and by NSF grant 0406311. His visits to Dartmouth College were supported by funds donated by Edward Shapiro to the Mathematics Department of Dartmouth College. The authors are very thankful to Robert Caldwell, Paul Ehrlich, Robert Low, Jose Natario, Jacobo Pejsachowicz, Miguel Sanchez and Sergey Shabanov for useful discussions. We are grateful to the anonymous referee for the valuable comments and especially for the suggestion to work with conformal classes of metrics in Theorem 11.
A. A Brief Review of Contact and Lorentz Manifolds be a Definition 26 (Contact structures and Legendrian submanifolds) Let Q 2m−1 manifold equipped with a smooth hyperplane field η = {ηq2m−2 ⊂ Tq Q 2m−1 q ∈ Q}. This hyperplane field is called a contact structure, if it can be locally presented as the kernel of a 1-form α with α ∧ (dα)m−1 = 0. An immersion (respectively an embedding) f : Z m−1 → Q of an (m−1)-dimensional manifold Z m−1 into a (2m − 1)-dimensional contact manifold (Q 2m−1 , η) is called a Legendrian immersion (respectively a Legendrian embedding), if (d f )(Tz Z ) ⊂ η f (z) , for all z ∈ Z . Example 8 (The contact structure on ST ∗ M) For a smooth manifold M m a point p ∈ ST ∗ M can be regarded as a linear functional p on Tpr p M that is defined up to a multiplication by a positive number. Thus this point p is completely described by the hyperplane m−1 = ker p ⊂ Tpr( p) M and by the half-space Tpr( p) M \ m−1 , where p is positive. p p The natural contact structure η = {η2m−2 ⊂ T p (ST M)2m−1 , p ∈ ST ∗ M} p is given by η p = (d pr)−1 ( p ). If M is equipped with a Riemannian metric g, then we can identify the tangent and the cotangent bundles of M. Thus we can also identify the spherical tangent bundle with the spherical cotangent bundle. A smooth immersion ϕ : Z → ST M can be described as the map ψ := pr ◦ϕ : Z → M together with a smooth nowhere zero vector field ξz ∈ Tψ(z) M, z ∈ Z , where ξz points to the direction ϕ(z). It is easy to see that for an (m − 1)-dimensional manifold Z m−1 the immersion ∼ =
ϕ
f : Z −−−−→ ST M −−−−→ ST ∗ M is Legendrian exactly when ξz is g-orthogonal to dψ(Tz Z ), for all z ∈ Z . Definition 27 (Levi-Civita connection on Lorentz manifolds, geodesic, exponential map, curvature, etc.) Let (X, g) be a Lorentz manifold and let Ξ (X ) be the space of all smooth vector fields X → T X on X. A Levi-Civita connection on (X, g) is a connection ∇ g such that the following metric compatibility and torsion free conditions hold for every ξ1 , ξ2 , ξ3 ∈ Ξ (X ) : g
g
g
g
ξ1 g(ξ2 , ξ3 ) = g(∇ξ1 ξ2 , ξ3 ) + g(ξ2 , ∇ξ1 ξ3 ) and [ξ1 , ξ2 ] = ∇ξ1 ξ2 − ∇ξ2 ξ2 .
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Every Lorentz manifold (X, g) admits a unique Levi-Civita connection, see for example [9, p. 22]. When no confusion can arise we will often use ∇ rather than ∇ g . A geodesic c : (a, b) → (X, g) is a smooth curve such that ∇c c = 0 for all of its points. Similarly to Riemannian manifolds one can use geodesics to define the exponential map exp p : T p X → X. The map exp p is defined not on the whole T p X but rather on ⊂ Tp X a starshaped about 0 ∈ T p X subset of it. There is an open neighborhood U starshaped about 0 ∈ T p X and such that exp p |U is a diffeomorphism onto a neighbor) is called a normal neighborhood of p in (X, g). hood of p ∈ X. The image Im exp(U The curvature R of ∇ is a function that assigns to each pair ξ1 , ξ2 ∈ Ξ (X ) a map R(ξ1 , ξ2 ) : Ξ (X ) → Ξ (X ), R(ξ1 , ξ2 )ξ3 = ∇ξ1 ∇ξ2 ξ3 − ∇ξ2 ∇ξ1 ξ3 − ∇[ξ1 ,ξ2 ] ξ3 . It is well-known that for p ∈ X, R(ξ1 , ξ2 )ξ3 | p depends only on ∇ g and on ξ1 ( p), ξ2 ( p), ξ3 ( p), see for example [9, p. 20]. Moreover R(ξ1 , ξ2 )ξ3 | p depends linearly on ξ1 ( p), ξ2 ( p), ξ3 ( p). A two-dimensional plane E p ⊂ T p X is said to be spacelike if g| E p is positive definite, it is called timelike if g| E p is nondegenerate but it is not positive definite, and E p is called null or light-like if g| E p is degenerate. Let E p be timelike or spacelike and let v, w be a basis of E p , then one defines the sectional curvature K (E p ) =
g(R(w, v)v, w) , g(v, v)g(w, w) − (g(v, w))2
see [9, pp. 29–30]. (Note that for light-like E p the expression in the denominator is zero.) B. Manifolds for which the Positively and Negatively Oriented Sm−1 -Fibers of ST ∗ M → M m are Homotopic Definition 28 (Good manifolds) Let M m , m > 1 be a Cauchy surface in a globally hyperbolic (X m+1 , g). Let r : S m−1 → S m−1 be an autodiffeomorphism of degree −1. We call a manifold M “good” if the maps εv r, εv : S m−1 → ST ∗ M are not free homotopic. Since for any v1 , v2 ∈ M the maps εv1 and εv2 are free homotopic, this definition does not depend on the choice of v ∈ M. For a generic cooriented wave front Wx,M on M we can reconstruct the submanifold x,M ⊂ ST ∗ M from the cooriented Im Wx,M . The submanifold Im W x,M is diffeoIm W x,M : S m−1 → ST ∗ M can be reconstructed morphic to S m−1 and the lifted wave front W up to an autodiffeomorphism of S m−1 . x,M up to an orientation preserving autodifIf M is good, then we can reconstruct W m−1 x,M ⊂ M. feomorphism of S . Indeed, choose a diffeomorphism f : S m−1 → Im W Since M is good, exactly one of the maps f and f r is homotopic to εv and this map is x,M up to an orientation preserving autodiffeomorphism of S m−1 . equal to W Two links that are the same up to orientation preserving autodiffeomorphisms of the linked spheres are link homotopic. Since alk does not change under link homotopy, we see that for good M the methods of Examples 6 and 7 work even if the front orientations are not specified in the pictures of the cooriented fronts. The following theorem shows that almost all manifolds are good.
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Theorem 16 If a connected oriented manifold M m , m > 1 is not good, then M is homeomorphic to an even-dimensional sphere and Im{(εv )∗ : πm−1 (S m−1 ) → πm−1 (ST ∗ M)} ∼ = Z/2. Proof Since M is orientable, the π1 (ST ∗ M)-action on the class in πm−1 (ST ∗ M) of the positively oriented S m−1 -fiber of pr is trivial, see the proof of Lemma 2. Thus εv and εv r are homotopic if and only if the group G := Im{(εv )∗ : πm−1 (S m−1 ) → πm−1 (ST ∗ M)} is Z/2 or 0. Note that if M is not closed then it is good, since the bundle ST ∗ M → M has a section (the Euler class belongs to the trivial group), and therefore G = Z. So we assume that M is a closed oriented manifold and consider the following commutative diagram: ∂
(εv )∗
πm (M) −−−−→ πm−1 (S m−1 ) −−−−→ πm−1 (ST ∗ M) ⏐ ⏐ ⏐ ⏐ ∼ h =h χ (M)
Z
Hm (M) −−−−→ Hm−1 (S m−1 )
Z.
h
Here h and are the Hurewicz homomorphisms, the top sequence is a segment of the homotopy exact sequence of the spherical cotangent bundle ST ∗ M → M, and the bottom map is the multiplication by the Euler characteristic of M. (The commutativity follows since in the Leray–Serre spectral sequence of the spherical cotangent bundle the transgression τ : Hm (M) → Hm−1 (S m−1 ) is the multiplication by the Euler characteristic χ (M) of M.) Note that G ∼ = πm−1 (S m−1 )/ Im ∂. If G = Z/2, then Im ∂ = 2Z ⊂ Z = πm−1 (S m−1 ). Hence h is a non-zero homomorphism, i.e. there exists a map S m → M m of non-zero degree. Therefore M is a rational homology sphere, cf. Lemma 5. Hence χ (M) = 0 if m is odd and χ (M) = 2 if m is even. The case χ (M) = 0 is impossible, since h ∂ = 0. So χ (M) = 2 and therefore h must be surjective. Thus there exists a map S m → M m of degree 1. Similarly to the proof of Proposition 3.(ii), we get that M is homeomorphic to a sphere. Since χ (M) = 2, m is even. If G = 0, then ∂ is surjective. Hence h must be surjective and χ (M) = 1. Similarly to the case considered before, we get that M is a homotopy sphere. However this contradicts χ (M) = 1.
References 1. Arnold, V.I.: Invariants and perestroikas of fronts on the plane, Singularities of smooth maps with additional structures. Proc. Steklov Inst. Math. 209, 11–64 (1995) 2. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Second edition, Graduate Texts in Mathematics 60, New-York:Springer-Verlag, 1989 3. Arnold, V.I.: Problems, written down by S. Duzhin, September (1998), an electronic preprint at http:// www.pdmi.ras.ru/~arnsem/Arnold/prob9809.ps.gz, 1998 4. Arnold, V.I.: Arnold’s Problems. Translated and revised edition of the 2000 Russian original. With a preface by V. Philippov, A. Yakivchik, M. Peters. Berlin:Springer-Verlag, Moscow:PHASIS, 2004 5. Bernal, A., Sanchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243(3), 461–470 (2003) 6. Bernal, A., Sanchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic space-times. Commun. Math. Phys. 257(1), 43–50 (2005) 7. Bernal, A., Sanchez, M.: Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77(2), 183–197 (2006)
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8. Beem, J.K., Ehrlich, P.E.: Global Lorentzian Geometry. Monographs and Textbooks in Pure and Applied Math. 67, New York:Marcel Dekker, Inc., 1981 9. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry. Second edition. Monographs and Textbooks in Pure and Applied Mathematics 202, New York:Marcel Dekker, Inc., 1996 10. Bérard-Bergery, L.: Quelques exemples de variétés riemanniennes où toutes les géodésiques issues d’un point sont fermées et de même longueur, suivis de quelques résultats sur leur topologie. Ann. Inst. Fourier (Grenoble) 27(1), xi, 231–249 (1977) 11. Besse, A.L.: Manifolds All of Whose Geodesics are Closed, with appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Berard-Bergery, M. Berger J. L. Kazdan, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 93. Berlin-New York: Springer-Verlag, 1978 12. Biasi, C., Saeki, O.: On transversality with deficiency and a conjecture of Sard. Trans. Amer. Math. Soc. 350(12), 5111–5122 (1998) 13. Bott, R.: On manifolds all of whose geodesics are closed. Ann. of Math. (2) 60, 375–382 (1954) 14. Cerf, J.: Sur Les Difféomorphismes de la Sphère de Dimension Trios (Γ4 = 0). Lecture Notes in Mathematics, No. 53 Berlin-New York:Springer-Verlag, 1968 15. Chernov (Tchernov), V., Rudyak, Yu.B.: Toward a General Theory of Affine Linking Invariants. Geometry and Topology 9(42), 1881–1913 (2005) 16. Chernov (Tchernov), V., Rudyak, Yu.B.: Affine Linking Numbers and Causality Relations for Wave Fronts. http://arxiv.org/abs/math/0207219, 2002 17. Dieckmann, J.: Cauchy surfaces in a globally hyperbolic space-time. J. Math. Phys. 29(3), 578–579 (1988) 18. Ekholm, T., Etnyre, J., Sullivan, M.: Non-isotopic Legendrian submanifolds in R2n+1 . J. Differ. Geom. 71(1), 85–128 (2005) 19. Freedman, M.: The topology of four-dimensional manifolds. J. Differ. Geom. 17(3), 357–453 (1982) 20. Geroch, R.P.: Domain of dependence. J. Math. Phys. 11, 437–449 (1970) 21. Gusarov, M.: A new form of the Conway-Jones polynomial of oriented links. (Russian. English summary) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 193 Geom. i Topol. 1, 4–9, 161 (1991) 22. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-time. Cambridge Monographs on Mathematical Physics, No. 1, London-New York: Cambridge University Press, 1973 23. Hirsch, M.W.: Differential Topology. Corrected reprint of the 1976 original. Graduate Texts in Mathematics 33, New York: Springer-Verlag, 1994 24. Kaiser, U.: Link Theory in Manifolds. Lecture Notes in Mathematics 1669. Berlin: Springer-Verlag, 1997 25. Low, R.J.: Causal Relations and Spaces of Null Geodesics. PhD Thesis, Oxford University, 1988 26. Low, R.J.: Twistor linking and causal relations. Classical Quantum Gravity 7(2), 177–187 (1990) 27. Low, R.J.: Twistor linking and causal relations in exterior Schwarzschild space. Classical Quantum Gravity 11(2), 453–456 (1994) 28. Low, R.J.: Stable singularities of wave-fronts in general relativity. J. Math. Phys. 39(6), 3332–3335 (1998) 29. Low, R.J.: The space of null geodesics. In: Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000). Nonlinear Anal. 47(5), 3005–3017 (2001) 30. Low, R.J.: The Space of Null Geodesics (and a New Causal Boundary). Lecture Notes in Physics 692, Belin Heidelberg New York: Springer 2006, pp. 35–50 31. Munkres, J.: Differentiable isotopies on the 2-sphere. Michigan Math. J. 7, 193–197 (1960) 32. Milnor, J.: Link groups. Ann. of Math. (2) 59, 177–195 (1954) 33. Milnor, J.: Lectures on the H -Cobordism Theorem. Notes by L. Siebenmann and J. Sondow. Princeton N.J.: Princeton University Press, 1965 34. Minguzzi, E., Sanchez, M.: The Causal Hierarchy of Space-Times. http://arxiv.org/list/gr-qc/0609119, 2006 35. Natario, J., Tod, P.: Linking, Legendrian linking and causality. Proc. London Math. Soc. (3) 88(1), 251–272 (2004) 36. O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Pure and Applied Mathematics, 103. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], 1983 37. Perelman, G.: The Entropy Formula for the Ricci Flow and its Geometric Applications. http://arxiv.org/ list/math.DG/0211159, 2002 38. Perelman, G.: Ricci Flow with Surgery on Three-Manifolds. http://arxiv.org/list/math.DG/0303109, 2003 39. Rudyak, Yu.B.: On Thom Spectra, Orientability, and Cobordism. Berlin-Heidelberg New York: Springer, 1998 (to be republished in 2008) 40. Samelson, H.: On manifolds with many closed geodesics. Portugal. Math. 22, 193–196 (1963) 41. Seifert, H.G.: Smoothing and extending cosmic time functions. Gen. Relativ. Gravit. 8, 815–831 (1977) 42. Smale, S.: Generalized Poincare’s conjecture in dimensions greater than four. Ann. of Math. (2) 74, 391–406 (1961) 43. Smale, S.: Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc. 10, 621–626 (1959) 44. 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45. Switzer, R.: Algebraic Topology—Homotopy and Homology. Die Grundlehren der mathematischen Wissenschaften, Band 212. Berlin-Heidelberg-New York: Springer, 1975 46. Tabachnikov, S.L.: Calculation of the Bennequin invariant of a Legendre curve from the geometry of its wave front. Funct. Anal. Appl. 22(3), 246–248 (1988) 47. Vassiliev, V.A.: Cohomology of knot Spaces, Theory of singularities and its applications, Adv. Soviet Math. 1, Providence, RI: Amer. Math. Soc., 1990, pp. 23–69 Communicated by G.W. Gibbons
Commun. Math. Phys. 279, 355–379 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0437-1
Communications in
Mathematical Physics
Parametric Representation of “Covariant” Noncommutative QFT Models Vincent Rivasseau, Adrian Tanas˘a Laboratoire de Physique Théorique, bât. 210, CNRS UMR 8627, Université Paris XI, 91405, Orsay Cedex, France. E-mail: [email protected]; [email protected]; [email protected] Received: 11 January 2007 / Accepted: 30 August 2007 Published online: 19 February 2008 – © Springer-Verlag 2008
Abstract: We extend the parametric representation of renormalizable non commutative quantum field theories to a class of theories which we call “covariant”, because their power counting is definitely more difficult to obtain.This class of theories is important since it includes gauge theories, which should be relevant for the quantum Hall effect. I. Introduction Quantum field theories on a non-commutative space-time or NCQFT [1] deserve a systematic investigation. They are intermediate structures between ordinary quantum field theory on commutative space time and string theories [2,3]. They can also be better adapted than ordinary quantum field theory to the description of physical phenomena with non-local effective interactions, such as physics in the presence of a strong background field, for example the quantum Hall effect [4–6], and perhaps also the confinement. In the labyrinth of all possible Lagrangians and geometries, we propose to use renormalizability as an Ariane thread. Indeed renormalizable theories are the ones who survive under renormalization group flows, hence should be considered the generic building blocks of physics. Following the Grosse-Wulkenhaar breakthrough [7,8] and subsequent work [9–11], we have now a fairly good understanding of a first class of renormalizable NCQFT’s on the simplest non commutative geometry, the Moyal space. These models fall into two broad categories, depending on their propagators: • the ordinary models, such as the initial φ44 model [7][8] with harmonic potential, for which the propagator C(x, y) decays both as x − y and x + y tend to infinity, as shown in the ordinary Mehler kernel representation and • the so-called covariant models whose propagators involve covariant derivatives in a constant external field. The “orientable” Gross-Neveu model in two dimensions [10,11] and the LSZ model in 4 dimensions [12] fall into this category. The propagator
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C(x, y) in this case only decays as x − y tends to infinity, and simply oscillates when x + y tend to infinity. This second class of models is therefore harder to study but is the relevant one for the quantum Hall effect and for gauge theories. An important technical tool in ordinary QFT is the parametric representation. It is the most condensed form of perturbation theory, since both position and momenta have been integrated out. It leads to the correct basic objects at the core of QFT and renormalization theory, namely trees. It displays both explicit positivity and a kind of “democracy” between these trees: indeed the various trees all contribute to the topological polynomial of a graph with the same positive coefficient, as shown in (II.2). This is nothing but the old “tree matrix theorem” of XIXth century electric circuits adapted to Feynman graphs [13]. Finally parametric representation displays dimension of space time as an explicit parameter, hence it is the natural frame to define dimensional regularization and renormalization, which respects the symmetries of gauge theories. The parametric representation for ordinary renormalizable NCQFT’s was computed in [14]. It no longer involves ordinary polynomials in the Schwinger parameters but new hyperbolic polynomials. They contain richer information than in ordinary commutative field theory, since they are based on ribbon graphs, and distinguish topological invariants such as the genus of the surface on which these graphs live. The basic objects in these polynomials are “admissible subgraphs” which are more general than trees; among these subgraphs the leading terms which govern power counting are “hypertrees” which are the disjoint union of a tree in the direct graph and a tree in the dual graph. Again there is positivity and “democracy” between them. We think these new combinatorial objects will probably stand at the core of the (yet to be developed) non perturbative or “constructive” theory of NCQFT’s. In this paper we generalize the work of [14] to the more difficult second class of renormalizable NCQFTs, namely the covariant ones. The basic objects (the hypertrees) and the positivity theorems remain essentially the same, but the identification of the leading terms and the “democracy” theorem between them is much more involved. We rely partly on [11], in which the key difficulty was to check independence between the direct space oscillations coming from the vertices and from the covariant propagators. This independence implied renormalizability of the orientable Gross-Neveu model. Our more precise method uses a kind of “fourth Filk move” inspired by [11] and [14]. This paper is organized as follows. In the next section we briefly recall the parametric representation for commutative QFT and we present the noncommutative model as well as our conventions. The third section computes the first polynomial and its ultraviolet leading terms. We state here our main result, Theorem III.1, which sets an upper bound on the Feynman amplitudes. Moreover, exact power counting as a function of the graph genus follows directly from this theorem. This is an improvement with respect to [11], where only weaker bounds, sufficient just for renormalizability, were established. The fourth section analyses then the second polynomial, the noncommutative analog of the Symanzik polynomial (II.3). It allows us to recover also the proper power counting dependence in the number of broken faces. Finally, in the last section we present some explicit polynomials for different types of Feynman graphs.
II. Parametric Representation; the Noncommutative Model II.1. Parametric representation for commutative QFT. Let us give here the results of the parametric representation for commutative QFT (one can see for example [15] or
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[16] for further details). The amplitude of a Feynman graph is written A( p) = δ
p
∞ 0
L e−V ( p,α)/U (α) −m 2 α (e dα ), U (α)2
(II.1)
=1
where L is the number of internal lines of the graph and U and V are polynomials of the parameters α ( = 1, . . . , L) associated to each internal line. These so called “topological” or “Symanzik” polynomials have the explicit expressions: U = α , (II.2) T ∈T
V =
⎛ α ⎝
T2 ∈T2
⎞2 pi ⎠ ,
(II.3)
i∈E(T2 )
where T is a (spanning) tree of the graph and T2 is a 2−tree, i. e. a tree minus one of its lines.
II.2. The noncommutative model. For simplicity we treat in this paper the LSZ model in 4 dimensions, but the extension to the Gross-Neveu model is straightforwrd. We place ourselves in a Moyal space of dimension 4
where the matrix is
[x µ , x ν ] = iµν ,
(II.4)
2 0 0 θ . , 2 = = −θ 0 0 2
(II.5)
The Lagrangian is L=
1 ¯ − i x˜µ ) (∂ µ − i x˜ µ ) + ¯ ¯ , (∂µ 2
(II.6)
where the Euclidean metric is used, is the Moyal product and x˜ = 2−1 x. For such a model, the propagator between two points x and y was computed in [17] (see Corollary 3.1)
∞
˜
1
˜
˜
˜
˜
˜
e− 2 (cotanh t)(x +y )+ (cotanh t)x·y+i x∧y ˜ 2 (2π sinh t) 0 ∞ ˜ 1 ˜
2 ˜ ˜ = dt e− 2 (cotanh t)(x−y) +i x∧y , (II.7) 2 ˜ (2π sinh
t) 0
C(x, y) =
˜ = where
2
θ
dt
2
2
and x · y = (x 1 y 1 + x 2 y 2 ) + (x 3 y 3 + x 4 y 4 ), x ∧ y = (x 1 y 2 − x 2 y 1 ) + (x 3 y 4 − x 4 y 3 ).
(II.8)
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Let us now introduce the short and long variables: 1 1 u = √ (x − y), v = √ (x + y). 2 2
(II.9)
˜ and Moreover let α = t t = tanh
α . 2
(II.10)
The propagator (II.7) becomes C(x, y) = 2 0
1
dt
˜ − t 2 ) − 1 ˜ 1+t2 u 2 +i u∧v
(1 ˜ e 2 2t . (4π t )2
(II.11)
The vertex V is cyclically symmetric (note that this replaces the larger permutational symmetry of all the fields in the vertex which holds in ordinary commutative QFT). The vertex contribution is written, in position space, as ([10]) δ(x1V − x2V + x3V − x4V )e2i
i+ j+1 x V −1 x V 1≤i< j≤4 (−1) i j
,
(II.12)
where x1V , . . . , x4V are the 4−vectors of the positions of the 4 fields incident to the vertex V . For further use let us also define the antisymmetric matrix σ as
0 0 −i σ with σ2 = . (II.13) σ = 2 0 σ2 i 0 The δ−function appearing in the vertex contribution (II.12) is written as an integral over some new variables pV , called hypermomenta [14]. Note that one associates such a hypermomentum pV to any vertex V via the relation dpV i p (x V −x V +x V −x V ) δ(x1V − x2V + x3V − x4V ) = e V 1 2 3 4 (2π )4 dpV pV σ (x V −x V +x V −x V ) 1 2 3 4 , = e (II.14) (2π )4 where to pass from the first line to the second of the equation above one has used the change of variable i pV = pV σ , whose Jacobian is 1. II.3. Feynman graphs for NCQFT. In this subsection we give some useful conventions and definitions. Note that this subsection is a recall of [10,11] and [17]. Let us consider a graph with n vertices, L internal lines and F faces. One has 2 − 2g = n − L + F,
(II.15)
where g ∈ N is the genus of the graph. If g = 0 one has a planar graph, if g > 0 one has a non-planar graph. Furthermore, we call a planar graph a planar regular graph if it has no faces broken by external lines. Such a graph has 4n corners, 4 for each vertex. We denote by N the number of external positions and by I the set of 4n − N internal corners. The “orientable” form (II.12) of the vertex contribution of our model leads us to associate a sign “+” or “−” to each of the corners of each vertex. These signs alternate when turning around a vertex. The
Parametric Representation of “Covariant” Noncommutative QFT Models
359
Fig. 1. The first Filk Move: the line 2 is reduced; the two initial vertices are glued up into a “fatter” final one
model (II.6) has orientable lines in the sense of [10], that is any internal line joins a “−” corner to a “+” corner and this is the orientation we chose for the lines in our drawings. Consider T a tree of n − 1 lines. The remaining L − (n − 1) lines form the set L of loop lines. Let us now give some ordering relations. If one starts from the root and turns around the tree in trigonometrical sense, we can number each of the corners in the order they are met. Moreover, for each vertex V there is a unique tree line going towards the root. We denote it by V . This correspondence works both ways. The sign of a tree line ε(V ) is • −1 if the tree line is oriented towards the root and • 1 if not. Vertices V which are hooked to a single tree line (which is actually V ) are called leaves. We also define for any ∈ T a branch b() as the subgraph containing all the vertices “above” in the tree (see [11]). Moreover, we define the sign εk () of a loop line entering/exiting a branch associated to some vertex k to be • 1 if the loop line enters the branch, • −1 if the loop line exits the branch and • 0 if the loop line belongs to the branch. Let the lines l = (i, j), l = ( p, q) and the external position xk . We define: • • • • •
l ≺ l if i < p, q, and j < p, q, l ≺ k if i, j < k, l ⊂ l if p < i, j < q or q < i, j < p, k ⊂ l if i < k < j or j < k < i, l l if i < p < j < q and l l if i < q < j < p.
The first Filk move. In [18], T. Filk defined several contractions on a graph or its dual which we refer to as Filk moves. The first Filk move consists in reducing a tree line by gluing up together two vertices into a bigger one (see Fig. 1). Note that the number of faces or the genus of the graph do not change under this operation. Repeating this operation for the n − 1 tree lines, one obtains a single final vertex with all the loop lines hooked to it - a rosette. (See Fig. 2. If a rosette has only one face we refer to it as to a super-rosette (see [14].) Let us notice that the rosette can be considered as a vertex and one can write down its vertex factor, as done for any vertex entering some Feynman graph. We refer to it as to the rosette factor.
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Fig. 2. A rosette
Furthermore, let us remark that the order relations defined above do not change when performing this first Filk move. Thus, as observed in Sect. 3.1 of [10] one has Sign Alternation. Signs “+” and “−” alternate when turning around the rosette. We also define < if the starting point of precedes the end point of in the rosette. Finally, once we choose a root and an orientation around the rosette, we can define the sign of a loop line ε(w ) as +1 if the loop line goes in the same sense as the rosette orientation and −1 if it does not. II.4. Parametric representation for the noncommutative model. Note that, as pointed out in [14], the first polynomial in the noncommutative case is the determinant of the quadratic form integrated over all internal positions of the graph save one. One has thus to choose a particular “root” vertex whose position is not integrated. We denote this particular vertex by V¯ . Let us now generalize the notions (II.9) of short and long variables at the level of the whole Feynman graph. For this purpose we define the (L × 4)-dimensional incidence matrix ε V for each of the vertices V . Since the graph is orientable (in the sense defined in Subsect. II.3 above) we can choose V εi = (−1)i+1 , if the line hooks to the vertex V at corner i.
(II.16)
We also put V V η,i = |ε,i |, V = 1, . . . , n, = 1, . . . , L and i = 1, . . . , 4.
One now has 1 V V v = √ ηi xi , 2 V i 1 V V u = √ εi xi . 2 V i
(II.17)
Parametric Representation of “Covariant” Noncommutative QFT Models
361
Conversely, one has 1 V V xiV = √ ηi v + εi u . 2 Let us now express the amplitude A of such a noncommutative graph with the help of these long and short variables. One has to put together the expressions of all the propa√ gators (II.11) and vertices (II.12). Moreover, in order to avoid the 2 factors, we rescale the external positions xe to x¯e and the hypermomenta pV to p¯ V . One has: AG (xe ) = K e e
dt
(1 − t2 ) t2
du dv
d p¯ V e
2 ˜ 1+t 2 ˜ 2t u +i u ∧v
−
2
V
i+ j+1 (η V v +ε V u )−1 (η V v +ε V u ) 1≤i< j≤4 (−1) i i j j
e p¯ V σ
i+1 (η V v +ε V u ) i (−1) i i V V −1 −1 2i[ i=e ω(i,e)(ηi v +i u ) x¯e ]+4i e<e x¯e x¯e +2 e p¯ V σ (−1)e+1 x¯e
i
,
(II.18) with K some inessential normalization constant and ω(i, e) = 1 if i < e and −1 if i > e. Singling out the root vertex V¯ , we write AG (xe , p¯ V¯ ) = K e e
(1 − t2 )
dt
t2
du dv
d p¯ V e
2 ˜ 1+t 2 ˜ 2t u +i u ∧v
−
2
V =V¯
i+ j+1 (η V v +ε V u )−1 (η V v +ε V u ) 1≤i< j≤4 (−1) i i j j
e p¯ V σ
i+1 (η V v +ε V u ) i (−1) i i V V −1 −1 e+1 2i[ i=e ω(i,e)(ηi v +i u ) x¯e ]+4i e<e x¯e x¯e +2 e p¯ V σ (−1) x¯e
i
.
(II.19) From now on, in order to simplify notations, we forget the bar over the rescaled variables x¯e and p, ¯ but we keep the notation V¯ for the chosen root. One can write the amplitude (II.18) in the condensed way ˜
1 − t2 t dt du dv dp e− 2 X G X , (II.20) AG = K t2 where
X = xe pV¯ u v p , G =
M P . Pt Q
(II.21)
Furthermore, performing the Gaussian integration one obtains: AG = K
1 − t2 t2
1 dt √ e det Q
˜
−
2
−1 P t ] x e [M−P Q xe pV¯ pV¯ .
(II.22)
This form allows to define the polynomials HU and H V , the noncommutative analogs of U and V (see (II.2) and resp. (II.3)). One can write 1 H V (t,x ) − G e [dt (1 − t2 )]HUG (t)−2 e HUG (t) (II.23) AG (xe ) = K 0
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and resp. AG (xe , pV¯ ) = K
1
0
− [dt (1 − t2 )2 ]HUG,V¯ (t)−2 e
H V ¯ (t,xe , p ¯ ) G,V V HU ¯ (t) G,V
.
(II.24)
Note that we refer to the polynomials HU and H V as to hyperbolic polynomials, since they are polynomials in the set of variables t ( = 1, . . . , L), the hyperbolic tangent of the half-angle of the parameter α associated to each propagator line (see (II.10)). Using now (II.22) and (II.24) the polynomial HUG,V¯ is written HUG,V¯ = (det Q)
1 4
L
t .
(II.25)
=1
III. The First Hyperbolic Polynomial We now proceed with the analysis of the polynomial HUG,V¯ above, the study of the polynomial HUG being analogous. For this purpose we take a closer look at the matrix Q, a matrix which can be read out of the developed expression (II.19). Note that the (4(2L + n − 1))-dimensional matrix Q can be written as Q = A ⊗ I4 − B ⊗ σ,
(III.1)
where A is a (2L +n −1)-dimensional diagonal matrix and B is an antisymmetric matrix of the same dimension. In [14] it has been proven that for a matrix of the form (III.1) one has det Q = [det(A + B)]4 ,
(III.2)
which, using (II.25) leads to HUG,V¯ = det(A + B)
L
t .
(III.3)
=1
Let us now study both the diagonal A and the antisymmetric B parts of Q. One has ⎛ ⎞ S 0 0 A = ⎝ 0 0 0⎠ , 0 0 0 where S is a L−dimensional diagonal matrix with elements (1 + t2 ) \ (2t ). The antisymmetric part B is written
sE C B= −C t 0 with s=
2 ˜ θ
=
1
(III.4)
(III.5)
Parametric Representation of “Covariant” Noncommutative QFT Models
and
363
4 (−1)i+1 iV , CV = 4i=1 i+1 V i=1 (−1) ηi
uu E E uv E= , E vu E vv
(III.6) (III.7)
where vv E , =
4
V V (−1)i+ j+1 ω(i, j)ηi η j ,
V i, j=1 uu E ,
=
4
(−1)i+ j+1 ω(i, j)iV V j ,
V i, j=1 uv E , =
4
(−1)i+ j+1 ω(i, j)iV ηV j + 2 δ .
(III.8)
V i, j=1
Note that ω is the antisymmetric matrix for whom ω(i, j) = 1 if i < j. Finally in order to have the integer expression (III.6) of the matrix C coupling the u and v variables, we have rescaled by s the hypermomenta pV . We also define the integer entries matrix:
E C . (III.9) B = −C t 0 We denote by |I | the cardinal of the set I . Let us now state the following lemma: Lemma III.1. With A and B given by (III.4) and (III.5),
det(A + B) =
(s −1 )|I |+n−1−2L n 2I
I ⊂{1...L}, n+|I | odd
1 + t2
l∈I
2t
(III.10)
), the Pfaffian of the matrix B above with deleted lines and columns I with n I = Pf(B I among the first L indices (corresponding to short variables u).
Proof. The proof is straightforward, being just a particular case of Lemma III.3 of [14]. Let us now define the integer k I = |I | − L − F + 1.
(III.11)
Recalling that 2 − 2g = n − L + F one can use (II.25), (III.2) and Lemma III.1 above to obtain HUG,V¯ (t) =
I ⊂{1...L}, n+|I | odd
s 2g−k I n 2I
1 + t2
∈I
2t
∈{1,...,L}
t .
(III.12)
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Leading terms in the first polynomial. By leading terms we understand the terms of (III.12) which have the highest global degree in the t1 , . . . , t L variables. It is these terms that govern power counting. To obtain these leading terms one needs to express the I set in the development (III.12) of HUG,V¯ . If one takes I = {1, . . . , L} then the corresponding Pfaffian is 0 if F ≥ 2 and 2g iff F = 1 (by Lemma III.4 of [14]). Let us now consider F ≥ 2. We take I = {1, . . . , L} − J0 ,
(III.13)
where J0 is an admissible set in the sense defined in [14], i.e. • it contains a tree T˜ in the dual graph and • its complement contains a tree T in the direct graph. Then the rosette obtained by removing the lines of J0 and contracting the lines of T is a super-rosette, that is a rosette with exactly one face (see Subsect. II.3). In [14] it was proven that F − 1 ≤ |J0 | ≤ F − 1 + 2g.
(III.14)
Thus, the matrix B ˆ corresponding to (III.13) has an even size, which we denote by d. I We take the admissible set J0 such that |J0 | = F − 1. One then has: d = (n − 1) + L + (F − 1). From now on, amongst the long variables v we distinguish between the n − 1 ones corresponding to the tree lines (which we continue to call v) and the L − (n − 1) ones corresponding to the loop lines (which we refer to as to w variables). The determinant of B ˆ is written as a Grassmannian integral I u u w w det B Iˆ = d ψ¯ 1u dψ1u . . . d ψ¯ F−1 dψ F−1 d ψ¯ 1w dψ1w . . . d ψ¯ F−1+2g dψ F−1+2g p p p p v v d ψ¯ 1v dψ1v . . . d ψ¯ n−1 dψn−1 d ψ¯ 1 dψ1 . . . d ψ¯ n−1 dψn−1 e−
d
¯ i, j=1 ζi bi j ζ j
, (III.15)
where by ζ in the exponent we denote a generic Grassmannian in the set {ψ u , ψ w , p p ψ v , ψ p }. Integrating over the Grassmannian variables ψ¯ i , ψi (i = 1, . . . , n − 1) one gets u u w w dψ F−1 d ψ¯ 1w dψ1w . . . d ψ¯ F−1+2g dψ F−1+2g det B Iˆ = d ψ¯ 1u dψ1u . . . d ψ¯ F−1 v v v v d ψ¯ 1v dψ1v . . . d ψ¯ n−1 dψn−1 X 1v X¯ 1v . . . X n−1 e− X¯ n−1
(F−1)+L i, j=1
ψ¯ i bi j ψ j
(III.16)
with ˆ
ˆ
I I v X 1v = C1,1 ψ1u + . . . + C L+(F−1),1 ψn−1 , v Iˆ ¯ u Iˆ ψ1 + . . . + C L+(F−1),1 ψ¯ n−1 , X¯ 1v = C1,1 ... ˆ
ˆ
v v I I X n−1 = C1,n−1 ψ1u + . . . + C L+(F−1),n−1 ψn−1 , v v Iˆ Iˆ ψ¯ 1u + . . . + C L+(F−1),n−1 ψ¯ n−1 = C1,n−1 . X¯ n−1
(III.17)
Parametric Representation of “Covariant” Noncommutative QFT Models
365
ˆ
The matrix C I corresponds to the matrix (III.6), where only F − 1 of the L u−lines remain. Moreover, note that each pair X i , X¯ i (i = 1, . . . , n − 1) corresponds to some vertex i (which is of course different of the root vertex). Let us now put ε(k)X kv , ε j χ vj = X vj + ε(k) X¯ kv , (III.18) ε j χ¯ vj = X¯ vj + where the sign ε j is defined in Appendix A and the summation is performed on all the X k corresponding to the vertices entering or exiting the branch of the vertex corresponding to X vj . One can now prove (see Appendix A) that the change of variable (III.18) is equivalent to ψkv = χkv − ε(k) εk ( )ψw , ψu + ψ¯ kv = χ¯ kv − ε(k) (III.19) ψ¯ u + εk ( )ψ¯ w , where ε(k) is the sign of the tree line associated to the vertex k and εk ( ) is the sign of the loop line which enters or exits the branch associated to the vertex k (see Subsect. II.3). Note that the form (III.18) of this change of variable leads directly to v v v v X 1v X¯ 1v . . . X n−1 = χ1v χ¯ 1v . . . χn−1 χ¯ n−1 . X¯ n−1
(III.20)
Furthermore (III.19) shows that the Jacobian of this triangular change of variables is 1. Let us remark here that the antisymmetric character of the matrix is preserved. The determinant (III.16) is written as u u w w det B Iˆ = d ψ¯ 1u dψ1u . . . d ψ¯ F−1 dψ F−1 d ψ¯ 1w dψ1w . . . d ψ¯ F−1+2g dψ F−1+2g v v v v d χ¯ 1v dχ1v . . . d χ¯ n−1 dχn−1 χ1v χ¯ 1v . . . χn−1 χ¯ n−1 e−
(F−1)+L i, j=1
ζ¯i bij ζ j
,
(III.21)
where the general notation ζ denotes the Grassmannian variables belonging to the set {ψ u , ψ w , χ v } and the matrix elements bi j are obtained from the previous ones by the change of variables (III.19). v χ¯ v The presence of the factor χ1v χ¯ 1v . . . χn−1 n−1 in the Grassmann integral of (III.21) v v v v selects only the terms with no χ1 χ¯ 1 . . . χn−1 χ¯ n−1 in the development of the exponential. Thus we end up with an antisymmetric matrix of type
uu E uw E . (III.22) B Iˆ = E wu E ww As stated above, the elements of this matrix are obtained after performing the change of variable (III.19). This is nothing but the reduction of the graph via the first Filk move, reduction which has as a result the rosette vertex of the graph. One can thus just read the elements of the matrix (III.22) from the rosette factor of the corresponding graph (see Subsect. II.3) . Furthermore, let us recall here that this is the case since J0 contains no tree lines, so that B ˆ corresponds to the rosette graph. This remains true even for g = 0, since in both I cases one has reduced the graph via this first Filk move.
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Let us recall that L = (n − 1) + (F − 1) + 2g.
(III.23)
This leads to a distinction between the case g = 0 and the case g ≥ 1 (the non-planar case). We first treat the planar regular Feynman graphs. III.1. The planar regular case. Since g = 0, the matrix B in (III.22) is a 2(F − 1)−dimensional matrix. Let us now give the rosette factor of a planar regular graph, a result firstly obtained in [10], presented here under the form of Corollary 3.4 of [11]: N k+1 δ (−1) xk + u l exp ıϕ (III.24) l∈T ∪L
k=1
(III.25) where ϕ = ϕ E + ϕ X + ϕU , N
ϕE = 2
(−1)i+ j+1 xi −1 x j ,
i< j, i, j=1 N √ √ ϕX = 2 2 (−1)k+1 xk −1 u l + 2 2 u l −1 (−1)k+1 xk , k=1 ≺k
ϕU = 2
T
+4
(l)vl −1 u l + 2
⊂ , ∈L
()w −1 u
L
( )w
−1
(III.26)
k
ul + 4
u l −1 u l .
≺
Note the difference in the numerical factors with respect to [11] or [10], difference appearing from the definition (II.9) of the short and long variables. The rosette factor (III.24) leads to the following results: Lemma III.2. The block E ww in (III.22) is identically zero. Proof. It is straightforwrd from (III.24). This expresses the fact that in a rosette of a planar graph the loop lines never cross each other. Lemma III.3. The lower triangular block of E uw is identically zero. uw = 0 if ⊂ . Ordering now the loop lines one Proof. One sees from (III.24) that E obtains the result.
Lemma III.4. uw E = 2 ( − ()) .
Parametric Representation of “Covariant” Noncommutative QFT Models
367
uw = 2
Proof. Before the reduction of the matrix via the first Filk move, one has E (coming from the propagator). The contribution obtained from the first Filk move reduction of the tree is read from (III.24), thus completing the proof.
Let us also state: Lemma III.5. Let M be a 2d-dimensional antisymmetric matrix such that • m i, j = 0, i = 2, . . . , d, d + 1 ≤ j ≤ d + i − 1; • m i, j = 0, i = d + 2, . . . , 2d, d + 1 ≤ j ≤ i − 1. Then detM = (m 1,d+1 m 2,d+2 , . . . , m d,2d )2 .
(III.27)
Proof. We develop the determinant on its first line. The first contribution to be considered is the one of m 1,2 . We now develop the remaining determinant on its first line and choose, let’s say, the contribution of m 2,3 . We continue this procedure until we arrive to the contribution of the factor m d−1,d . From the first line of the remaining determinant, the only non-zero element is m d,2d . However, its corresponding determinant is zero (the determinant of a zero-matrix). Following the same type of arguments one can show that the only non-zero contribution in the determinant is the one in (III.27). We can now put all the pieces together. By Lemmas III.2, III.3 and III.4 one sees that the matrix B given in (III.22) is exactly of the form of Lemma III.5. We have thus proved: Proposition III.1. Pf B ˆ = I
2 ( − ()) .
(III.28)
∈L
Let us now deal with the case of non-planar Feynman graphs, keeping in mind that the interest for the planar non-regular case will be underlined in the next section. III.2. The non-planar case. Now since g ≥ 1 the matrix B ˆ of (III.22) is a I (2(F − 1) + 2g)-dimensional matrix (see (III.23)). We divide the (F − 1) + 2g loop variables w in two categories: • (F − 1) “face” variables, which we denote by w f and • g pairs of “genus” variables, which we denote by w g . Let us recall here the notion of “nice-crossing” [14] for a pair of lines of a rosette obtained from a Feynman graph which is not regular planar. Such a pair of lines 1 and 2 realize a nice-crossing if the start of 2 immediately precedes the end of 1 in the rosette. Choosing a tree in the dual graph, the respective F − 1 lines are our so-called facelines (see above). The remaining rosette has only one face (i.e. it is a super-rosette, see Subsect. II.3) and hence one can state that the g pairs of lines form the g nice-crossings of the rosette.
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After the reduction of the graph via the first Filk move (resp. the change of variable (III.19)) the entries of the matrix B (given again by the rosette factor, as above) are more complicated than in the planar regular case. The general form of the matrix B is ⎞ ⎛ f g E uw E uw E uu ⎜ f f f f g⎟ B Iˆ = ⎝ E w u E w w (III.29) E w w ⎠ . gu gw f g wg w w w E E E As before, we can read the entries of this matrix from the rosette factor, which now is written (see Corollary 3.3 of [11]) N δ (−1) jk +1 s jk + u l exp ıϕ, (III.30) l∈T ∪L
k=1
with ϕ = ϕ E + ϕ X + ϕU + ϕW , ϕE = 2
N
(−1) jk + jl +1 s jk −1 s jl ,
k
ϕU = 2
T
+2
(l)vl
−1
ul + 2
u l −1 u l + 2
≺
ϕW =
√ 2
()w
u
()w −1 u + ( )w −1 u + 4
, , ∈L
+4
jk −1
L
( )w −1 u l
⊂ , ∈L
u −1 u ,
, ∈L
(−1) jk s jk −1 ()w + 2
( )w −1 ()w .
, , ∈L
⊃ jk , ∈L
By a direct inspection of (III.30), one notices that the coupling between: • the u ’s and the w ’s for whom ⊂ and • the w f ’s themselves f
f
is trivial. Furthermore, the coupling between any u and its corresponding w is 2( − ()). Moreover (as also observed in [14]), the coupling between the w g variables has a Jordan-block form. Nevertheless, the situation is more complicated than in the planar regular case because the new elements w g couple in a non-trivial way with the rest of the matrix. In order to bring these new couplings to a trivial form, we perform a sort of “fourth Filk move”, which generalizes the third Filk move introduced in [14]. g g Let us first treat the case of just one pair of genus lines, say 1 2 . First notice that if some face line crosses such a pair of genus lines, then it must cross both of the lines
Parametric Representation of “Covariant” Noncommutative QFT Models
369
of the pair. Indeed, assuming it crosses only one of the genus lines, one can see on the rosette that an additional face is added, which cannot be the case. The fourth Filk move we propose here is the following change of Grassmannian variables: g g f f ψw − ψw η1w = ψ1w + g
g
g
<2 ; ∩2
+
g g <2 ; ∩2 g
ψu −
η2w = ψ2w − g
g
g g >2 ; ∩2
ψw + f
g
g g ⎜ −(1 )(2 ) ⎝−
g
g
+
g g <2 ; ∩2
g
f
g
⎞ g
g
⎟ ψu ⎠ ,
>1 ; ∩1 f ψ¯ w
g
g
>2 ; ∩2
ψ¯ u −
g
g
f ψ¯ w −
g g η¯ 2w = ψ¯ 2w −
ψw
ψu +
<1 ; ∩1
<2 ; ∩2
>1 ; ∩1
⎛
g g η¯ 1w = ψ¯ 1w +
ψu ,
g
<1 ; ∩1
g
>2 ; ∩2
g g >2 ; ∩2 f ψ¯ w +
g
ψ¯ u , g
<1 ; ∩1
f ψ¯ w
(III.31)
g
>1 ; ∩1
⎛
g g ⎜ −(1 )(2 ) ⎝−
g g <1 ; ∩1
ψ¯ u +
⎞
g g >1 ; ∩1
⎟ ψ¯ u ⎠ .
As proven in [14], the terms in ψ w above will produce a trivial coupling between all the w variables (except for the coupling between the two w variables of the lines of a nice crossing coupling, which are not affected by (III.31); these couplings will keep their Jordan-block form mentioned above). Furthermore, notice that the new ψ u terms present in the fourth-Filk move (III.31) do not affect the couplings between the ψ w ’s (they just affect the lines and columns corresponding to the u variables). Let us now investigate the change produced by (III.31) in the rest of the couplings of the matrix. We start this investigation with the coupling between an u variable and its f corresponding w variable. As mentioned above, before performing (III.31) this element g g has the value 2( − ()). If the line does not cross the pair 1 , 2 then the change f g g of variable (III.31) will not affect the element E uw . Suppose now that 1 < < 2 f uw is the (the other cases being analogous). The effect of the ψ w terms in (III.31) on E , following: uw uw uw uw E , → E , + E u,1 + E u,2 . f
f
g
g
uw and E uw can be read of (III.30) to be 2( ) and resp. 2( ). The values of E u,1 1 2 u,2 One can now distinguish several cases: g
g
g
g
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V. Rivasseau, A. Tanas˘a g
g
• (1 )(2 ) = −1, the contributions of the two lines cancel each other and thus the f entry (u , w ) does not change; g g • (1 )(2 ) = 1, one can see from the sign alternation property (see Subsect. II.3) g g that if (1 ) = 1 = (2 ) then () = −1 (and thus the resp. entry changes from 2( + 1) to 2( − 1)) or vice versa. Now that this part of the change of variables (III.31) (concerning the ψ w variables) is completed, the part concerning the ψ u ’s in (III.31) do not change anymore the value f f g of the (u , w ) entry. Indeed the new contribution is now given by the entries (w , w1 ) f g and (w , w2 ), which, as already stated above, are equal to 0 after the third Filk move. Using the same type of arguments one is able to prove that the coupling between the f u ’s and the w ’s for which ⊂ remains 0 and furthermore that the u’s now couple trivially with the w g ’s. Thus, the matrix (III.29) now has the form ⎞ ⎛ f E uw 0 E uu ⎟ ⎜ w f u (III.32) ⎝E 0 0 ⎠, g g 0 0 E w ,w f
where E w ,w has a Jordan-block form and E uw has a lower-diagonal zero block and the elements on its diagonal are of the form 2( ±1). Hence the Pfaffian is 2g 2( ±1). The case of several pairs of genus-lines which present nice-crossings is treated analogously. One performs a change of variables (III.31) for each of the g pairs of lines. The trickier cases when a face-line crosses two pairs of genus-lines is shown by the sign alternation property (see Subsect. II.3) not to lead to a different result than above. Using now (III.12) we can conclude this section with: g
g
f
Theorem III.1. HUG,V¯ (t) ≥
s 2[g+(F−1)]
2 2( ± 1) 2g
J0 admissible
1 + t2
∈I
2t
t .
(III.33)
∈{1,...,L}
Let us firstly remark that the RHS above never vanishes for ∈ [0, 1[. As anounced in Sect. I, this is the main result of our paper. Let us now argue on its meaning. Indeed, using now (II.24), one has an upper limit for the Feynman amplitude of any noncommutative graph G. Moreover, from this formula power counting follows easily, as exhibited in [14]. Through the method proposed in this section we obtain an exact power counting as a function of the graph genus, hence an improvement with respect to [11]. Furthermore we observe that the dimension D of space time would appear simply as a parameter in this representation. Therefore this work as well as [14] is the starting point to compute dimensional regularization and dimensional renormalization for all classes of scalar, LSZ or Gross-Neveu models. Moreover, via our approach, richer topological information is obtained, since we deal with a positivity theorem on some “hypertrees” related to the admissible sets J0 . The
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371
expressions deduced in this section allow the generalization of the notion of “democracy” between (hyper)trees, also present in the case of commutative QFT (see (II.2)). Let us end this section by another remark. A weaker result than Theorem III.1 can be obtained. This result does not require all the techniques used above and it just states that one has the same type of positivity and improved power counting but just for a smaller class of values for ∈ [0, 1[, namely for the transcendent values of . Indeed, the Pfaffians n I corresponding to the leading terms in (III.12) are integer coefficient polynomials in , of degree F − 1: nI =
F−1
ak k .
(III.34)
k=0
Developing the Pfaffian n I one can identify the coefficient a F−1 above as the Pfaffian n I,J0 of Sect. III of [14] (corresponding to the admissible set J0 ). Moreover, in Lemma III.5 of [14] it was proven that this coefficient is nonvanishing. Therefore, the polynomial (III.34) with integer coefficients can have only a finite number of algebraic roots. These roots could a priori vary when the graph varies, but none can be transcendent. We can thus conclude that, for a transcendent number, n I given by (III.34) never vanishes. Recall however that this is a weaker result than Theorem III.1, which holds for any ∈ [0, 1[, transcendent or not. IV. The Second Hyperbolic Polynomial We now proceed with the analysis of the second hyperbolic polynomial. As before, we focus on the study of H VG,V¯ , the polynomial H VG being similar. This analysis follows the same lines of the one of Sect. IV of [14]. Comparing (II.22) and (II.23) one has
˜ H VG,V¯
xe xe pV¯ P Q −1 P t . (IV.1) = pV¯ HUG,V¯ 2 Note that we left aside the matrix M appearing in (II.22) since it factorizes out of the integral. As in the previous section, the entries of the matrix P are read out of (II.19). Moreover P has the same form as in [14]. The expression (III.1) of the matrix Q implies that its inverse is: Q −1 ττ =
−1 −1 (A + B)−1 (A − B)−1 τ τ + (A − B)τ τ τ τ − (A + B)τ τ ⊗ I4 + ⊗ σ. (IV.2) 2 2
One thus notices that we deal with a real part given by the I4 terms and an imaginary part given by σ . As already mentioned in [14] we are interested here by the former, which R . leads to a hyperbolic polynomial that we denote by H VG, V¯ Furthermore, let Pf(B Iˆτˆ ) be the Pfaffian of the matrix obtained from B by deleting the lines and columns in the set I, τ , where τ ∈ / I . Moreover we define I,τ to be the signature of the permutation obtained from (1, . . . , d) by extracting the positions belonging to I and replacing them at the end in the order 1, . . . , d → 1, . . . , iˆ1 , . . . , iˆp , . . . , iˆτ , . . . , d, i τ , i p . . . , i 1 .
(IV.3)
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Note that by d we mean the dimension of the matrix (here 2L + n − 1) and by p we mean the cardinal of I . One has Lemma IV.1. 2 1 + t2 1 i (xe , pV¯ ) = t xe Peτ I τ Pf(B Iˆτˆ ) . HUG,V¯ HUG,V¯ 2ti e R H VG, V¯
I
τ ∈K /
i∈I
Proof. The proof is straightforward, being just a particular case of Lemma IV.1 of [14]. As in the previous section, we now proceed with the analysis of the leading terms in this sum. Leading terms. As we have seen, the leading terms are given by the sets I = {1, . . . , L} \ J0 , where J0 is an admissible set. The job to be done is the investigation of the conditions under which Pfaffians of type 1 I τ Pf(B Iˆτˆ ) = dχα χi χτ e− 2 χ Bχ , (IV.4) α=1...d
i∈I
are nonzero. As pointed out in [14], we first distinguish two types of graphs as • the graphs which do not have any vertex with two opposite external legs and • the graphs which have at least a vertex with two opposite external legs. Regarding the former case, we follow exactly the reasoning of [14], and for some external position xe we introduce a dummy Grassmann variable in the Pfaffian integral, which once exponentiated can be interpreted as if we add a new line between xe and the root. We denote this modified graph by G . One thus has a one-to-one correspondence between the leading terms in H VG,V¯ and the leading terms of HUG ,V¯ in which the dummy line is chosen to be a tree line (see again Subsect. IV.1 of [14] for further details). Note that these leading terms were computed in Sect. III here. To any tree in G which contains the above dummy line there corresponds a two-tree T2 in G (i.e. a tree without a line, the dummy one). Let us now recall the definition of a 2-admissible set J in G as a set which satisfies the conditions: • J is admissible in G and • the dummy line is a tree of G contained in the complement of J . Now, as argued in [14], the graph obtained from G by deleting the lines {1, . . . , L}\ J and contracting the two-tree T2 has two faces, the one broken by the root and another one, which we denote by FJ . The bound analogous to (III.33) is 2 R 2[g +(F −1)] g H VG, 2(
± 1) 2 (x ) ≥ (s v¯ e J 2−admissible in G
1 + t2
∈I
2t
t [
e∈FJ
(−1)e xe ]2 .
(IV.5)
Parametric Representation of “Covariant” Noncommutative QFT Models
373
Fig. 3. One loop, a first choice
The more complicated case of a graph containing at least a vertex with opposite external legs is more tricky because, as indicated in [14] one has a sum of two Pfaffians which could a priori cancel each other. These two Pfaffians correspond to two graphs G 1 of genus g1 (obtained by adding a dummy line to G, as described above) and G 2 (with one line erased with respect to G 1 ) of genus g2 = g1 − 1. However, the second one has an additional factor s in its weight in the sum of Lemma IV.1 (see again Sect. IV of [14] for details). In our case, checking the values of the s and the 2 factors (given by Theorem III.1 or (IV.5)) appearing in this sum one can easily see that no value of can cancel it. Note that these results do not hold for the particular case = 1 (to whom one may refer as to self-dual covariant models). For this case, the propagator depends only on one side of the ribbon of the graph. In the position space we used here, the propagator oscillates in a way that often exactly compensates the vertices oscillations. These models have definitely worse power counting than in the ordinary case. Nevertheless, one can presume that Ward identities may be used to show their renormalizability. V. Examples For the sake of completeness, we give here some examples of planar regular, non-regular and finally non-planar graphs. V.1. Planar regular graphs. We start with the simplest example, namely the 1−loop graph. One has here 1 vertex (n = 1), one internal line (L = 1) and two faces (F = 2). Applying the methods described in this article, one finds, for the loop of Fig. 3, HUG,V¯ = 4s 2 t ( + 1)2 ,
(V.1)
HUG,V¯ = 4s 2 t ( − 1)2
(V.2)
and resp.
for the loop of Fig. 4. Let us now go along to a more complicated case, the bubble graph (see Fig. 5) which has n = 2, L = 2 and F = 2. One obtains: HUG,V¯ = 2s 2 (t1 + t2 + t12 t2 + t1 t22 )( − 1)2 .
(V.3)
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Fig. 4. One loop, the second choice
Fig. 5. The bubble graph
Fig. 6. The sunshine graph
For the sunshine graph (see Fig. 6) one has n = 2, L = 3, F = 3 and HUG,V¯ = 8s 4 (t1 t2 + t1 t3 + t2 t3 + t12 t2 t3 + t1 t22 t3 + t1 t2 t32 )( − 1)2 ( + 1)2 .
(V.4)
For the half-eye graph (see Fig. 7) one has n = 3, L = 4, F = 3 and HUG,V¯ = 4s 4 ( − 1)2 ( + 1)2 t3 t4 + t22 t3 t4 + t2 (t3 + t4 + t32 t4 + t3 t42 ) +t12 t3 t4 + t22 t3 t4 + t2 (t3 + t4 + t32 t4 + t3 t42 ) +t1 (1 + t22 )(t4 + t32 t4 ) + t3 1 + 64s 2 t2 t4 + t42 + t22 (1 + t42 ) .
(V.5)
The most complicated example of a planar regular graph we consider is the eye-graph (see Fig. 8), with n = 4, L = 6, F = 4. We obtain: HUG,V¯ = 8 (−1 + )4 (1 + )2 s 6 (t3 t4 (t5 + t6 ) (1 + t5 t6 ) +t2 2 t3 t4 (t5 + t6 ) (1 + t5 t6 ) +t2 (t4 (t5 + t6 ) (1 + t5 t6 ) + t3 2 t4 (t5 + t6 ) (1 + t5 t6 ) +t3 (t5 + t6 + t5 t6 (t5 + t6 ) + t4 2 (t5 + t6 ) (1 + t5 t6 ) +t4 (1 + 64 2 s 2 t5 t6 + t6 2 + t5 2 (1 + t6 2 ))))
Parametric Representation of “Covariant” Noncommutative QFT Models
375
Fig. 7. The half-eye Graph
Fig. 8. The eye graph
+t1 ((1 + t2 2 ) t4 (t5 + t6 ) (1 + t5 t6 ) +(1 + t2 2 ) t3 2 t4 (t5 + t6 ) (1 + t5 t6 ) + t3 ((1 + t2 2 ) (t5 + t6 ) (1 + t5 t6 ) +(1 + t2 2 ) t4 2 (t5 + t6 ) (1 + t5 t6 ) + t4 (1 + t5 2 +64 2 s 2 t5 t6 + (1 + t5 2 ) t6 2 + 64 2 s 2 t2 (t5 + t6 ) (1 + t5 t6 ) +t2 2 (1 + t5 2 + 64 2 s 2 t5 t6 + (1 + t5 2 ) t6 2 )))) +t1 2 (t3 t4 (t5 + t6 ) (1 + t5 t6 ) + t2 2 t3 t4 (t5 + t6 ) (1 + t5 t6 ) +t2 (t4 (t5 + t6 ) (1 + t5 t6 ) + t3 2 t4 (t5 + t6 ) (1 + t5 t6 ) +t3 (t5 + t6 + t5 t6 (t5 + t6 ) + t4 2 (t5 + t6 ) (1 + t5 t6 ) +t4 (1 + 64 2 s 2 t5 t6 + t6 2 + t5 2 (1 + t6 2 )))))).
(V.6)
If one chooses the admissible set to be formed of the lines 3, 4 and 6 the leading term has n I = 8( + 1)( − 1)2 .
(V.7)
V.2. An example of a planar non-regular graph. Let us now show an example of a planar but not regular graph, the broken bubble graph (see Fig. 9). This graph has n = 2, L = 2, F = 2 (and thus g = 0) but both of the faces are broken by external lines (hence B = 2).
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V. Rivasseau, A. Tanas˘a
Fig. 9. The broken bubble graph
Fig. 10. The non-planar sunshine graph
Fig. 11. The twisted eye graph
One gets: HUG,V¯ = 2s 2 [(t2 + t12 t2 )( − 1)2 + (t1 + t1 t22 )( + 1)2 ].
(V.8)
V.3. Non-planar graphs. In order to investigate all possible cases, we conclude with some non-planar graphs. The simplest one we consider here is the non-planar sunshine graph (see Fig. 10) which has n = 2, L = 3, but F = 1, and hence g = 1. One gets: 1 HUG,V¯ = s 2 1 + t22 + t32 + 16s 2 t2 t3 ( + 1)2 + 16s 2 t1 (t2 + t3 + t22 t3 + t2 t32 )( + 1)2 2 +t12 (1 + t22 + t32 + t22 t32 + 16s 2 t2 t3 ( + 1)2 ) . (V.9) Let us consider the twisted-eye graph (see Fig. 11), which has n = 5, L = 6, F = 2 and hence g = 1. If one chooses as the admissible set the set formed of the line 4 or 6 then one gets n I = 22 ( − 1).
(V.10)
The most complicated example we have considered is the Feynman graph of Fig. 12.
Parametric Representation of “Covariant” Noncommutative QFT Models
377
Fig. 12. A more complicated non-planar graph
If one chooses the admissible set to be the set formed of the lines 7 and 8, one is able to calculate n I = 8( − 1)( + 1).
(V.11)
A. Proof of Formula (III.19) We now prove that the change of variables (III.18) is equivalent to the form (III.19). We first prove that the formula is true for a leaf vertex and then we proceed by induction to show that it is true for any vertex in the graph. For some vertex V one has (see (III.17)): ˆ
ˆ
v I I X Vv = C1,V ψ1u + . . . + C L+(F−1),V ψn−1 .
When dealing with a leaf, one has only one non-zero entry Ci,V corresponding to the set of variables ψ v . Note that this variable corresponds to the line V (see Subsect. II.3). We denote its corresponding ψ v variable by ψkv and we put, as in (III.18), εV χVv = X Vv . One sends the ψkv variable from the RHS on the LHS and forces it to have a “+” sign. We now have to investigate the signs of the remaining ψ u ’s and ψ w ’s in the RHS. Two cases are to be distinguished: • the tree line V exits its corresponding vertex V (i.e. is oriented towards the root, (V ) = −1) • the tree line V enters its corresponding vertex V (i.e. is (V ) = 1). Consider the first of these two cases. On the RHS one has Ck,V = −1 and all the ψ u variables will have +1 coefficient. Passing ψkv on the LHS, one has the appropriate signs for the ψ u of the loop lines entering the respective vertex V , as indicated by formula
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(III.19). The case of the signs of the ψ w variables of the second case above (when the tree line V enters the vertex V ) is analogous. Finally, let us notice that the sign εV is chosen such that the variable χ vj has a “+” sign on the LHS. One can remark that this sign is nothing but the sign ε(V ) of the tree line V . We have thus completed the proof of formula (III.19) for the case of a leaf vertex. We now prove by induction that our statement remains correct for any vertex in the graph. Let us take such a general vertex V , with its associated tree line going towards the root V . The vertex can then have a maximum of three other tree lines connecting it to three other vertices which belong to its branch b(V ). We analyze here just one of these possible other vertices, the other ones being analyzed in exactly the same manner. Let us call this vertex µ and thus µ is its associated line going towards the root and joining µ and V . We denote its corresponding ψ v variable by ψ vj . As indicated in (III.18), we put εV χVv = X Vv − (µ )X µv . Let us first remark that in the expression of X Vv one has ψkv and ψ vj . By an analysis of the signs following the different cases of orientations of the lines µ and V one sees that the contribution of ψ vj cancels, thus the only ψ v variable which remains is ψkv . Moreover, the loop present will be the loop lines entering the two vertices, that is the loop lines entering the branch b(V ). Finally, by the same type of analysis as above on the possibilities of orientations of the lines, one concludes that the signs are the ones indicated in (III.19). Acknowledgements. We thank R. Gur˘au and F. Vignes-Tourneret for useful discussions during the preparation of this work. We also thank our anonymous referee for his useful questions and suggestions.
References 1. Douglas, M., Nekrasov, N.: Noncommutative field theory. Rev. Mod. Phys. 73, 977–1029 (2001) 2. Connes, A., Douglas, M.R., Schwarz, A.: Noncommutative Geometry and Matrix Theory: Compactification on Tori. JHEP 9802, 3–43 (1998) 3. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 32–131 (1999) 4. Susskind, L.: The Quantum Hall Fluid and Non-Commutative Chern Simons Theory. http://arxiv.org/ list/htp-th/0101029.2001 5. Polychronakos, A.P.: Quantum Hall states on the cylinder as unitary matrix Chern-Simons theory. JHEP 06, 70–95 (2001) 6. Hellerman, S., Van Raamsdonk, M.: Quantum Hall physics equals noncommutative field theory. JHEP 10, 39–51 (2001) 7. Grosse, H., Wulkenhaar, R.: Power-counting theorem for non-local matrix models and renormalization. Commun. Math. Phys. 254, 91–127 (2005) 8. Grosse, H., Wulkenhaar, R.: Renormalizationof φ 4 -theory on noncommutative R4 in the matrix base. Commun. Math. Phys. 256, 305–374 (2005) 9. Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalization of noncommutative φ4 4 -theory by multi-scale analysis. Commun. Math. Phys. 262, 565–594 (2006) 10. Gur˘au, R., Magnen, J., Rivasseau, V., Vignes-Tourneret, F.: Renormalization of Non Commutative 44 Field Theory in Direct Space. Commun. Math. Phys. 267, 515–542 (2006) 11. Vignes-Tourneret, F.: Renormalization of the orientable non-commutative Gross-Neveu model. Ann. Henri Poincaré 8(3), 427–474 (2007) 12. Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of quantum field theory on noncommutative phase spaces. JHEP 0401, 17–87 (2004) 13. Abdelmalek, A.: Grasmann-Berezin Calculus and Theorems of the Matrix-Tree Type. Adv. in Appl. Math. 33, 51–70 (2004)
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14. Gur˘au, R., Rivasseau, V.: Parametric representation of noncommutative field theory. Commun. Math. Phys. 272(3), 811–835 (2007) 15. Itzkinson, C., Zuber, J.-B.: Quantum Field Theory. New York: McGraw-Hill, 1980 16. Rivasseau, V.: From perturbative to Constructive Field Theory: Princeton, NJ: Princeton University Press, 1991 17. Gur˘au, R., Rivasseau, V., Vignes-Tourneret, F.: Propagators for Noncommutative Field Theories. Ann. Henri Poincaré 7(7–8), 1601–1628 (2006) 18. Filk, T.: Divergencies in a field theory on quantum space. Phys. Lett. B 376, 53–58 (1996) Communicated by A. Connes
Commun. Math. Phys. 279, 381–399 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0422-8
Communications in
Mathematical Physics
Invariants of Spin Networks Embedded in Three-Manifolds João Faria Martins1,2 , Aleksandar Mikovi´c2,3 1 Departamento de Matemática, Instituto Superior Técnico (Universidade Técnica de Lisboa),
Av. Rovisco Pais, 1049-001 Lisboa, Portugal. E-mail: [email protected]
2 Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia,
Av do Campo Grande, 376, 1749-024, Lisboa, Portugal. E-mail: [email protected]
3 Grupo de Física Matemática da Universidade de Lisboa, Av. Prof. Gama Pinto, 2,
1649-003 Lisboa, Portugal Received: 15 January 2007 / Accepted: 2 July 2007 Published online: 8 February 2008 – © Springer-Verlag 2008
Abstract: We study the invariants of spin networks embedded in a three-dimensional manifold which are based on the path integral for SU(2) BF-Theory. These invariants appear naturally in Loop Quantum Gravity, and have been defined as spin-foam state sums. By using the Chain-Mail technique, we give a more general definition of these invariants, and show that the state-sum definition is a special case. This provides a rigorous proof that the state-sum invariants of spin networks are topological invariants. We derive various results about the BF-Theory spin network invariants, and we find a relation with the corresponding invariants defined from Chern-Simons Theory, i.e. the Witten-Reshetikhin-Turaev invariants. We also prove that the BF-Theory spin network invariants coincide with V. Turaev’s definition of invariants of coloured graphs embedded in 3-manifolds and thick surfaces, constructed by using shadow-world evaluations. Our framework therefore provides a unified view of these invariants.
Contents 1. 2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Chain-Mail . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Generalised Heegaard diagrams. . . . . . . . . . . 2.1.2 Skein theory and the -element. . . . . . . . . . . 2.1.3 The Chain-Mail invariant. . . . . . . . . . . . . . 2.2 The Witten-Reshetikhin-Turaev invariant . . . . . . . . . Spin Network Invariants . . . . . . . . . . . . . . . . . . . . 3.1 Definition of the invariants . . . . . . . . . . . . . . . . 3.1.1 Manifolds with boundary. . . . . . . . . . . . . . 3.1.2 Calculation of the invariants using triangulations-I. 3.2 Turaev’s description of the invariants . . . . . . . . . . .
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382 384 384 384 385 386 387 389 389 390 391 393
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J. F. Martins, A. Mikovi´c
3.2.1 Combinatorial expression of the observables in the case surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Invariants of graphs embedded in index-1 handlebodies. 3.2.3 Calculation using triangulations-II . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . .
of thick . . . . . . . . . . . . . . . . . . . .
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1. Introduction Spin networks, also called coloured graphs in this article, were invented by R. Penrose [P] in an attempt to find a combinatorial definition of quantum geometry. These are trivalent graphs whose edges are coloured with half-integers, i.e. spins. Since each spin corresponds to an irreducible representation of SU(2), one can consider graphs of arbitrary valence, by colouring the vertices with appropriate intertwiners, see [Ba]. Each vertex of valence higher than three can be decomposed into three-valent vertices; hence, without loss of generality, we can restrict our attention to three-valent spin networks. It was shown by C. Rovelli and L. Smolin that spin networks do arise in a quantum theory of gravity (see [RoS]), which is known as “Loop Quantum Gravity” (LQG). Each state | in the LQG Hilbert space can be expanded in the spin network basis {|}, which is in a one-to-one correspondence with the set of isotopy classes of closed spin networks embedded in a 3-manifold M.1 Explicitly we have | = I (M, )| ,
where I (M, ) is a certain invariant of spin networks embedded in M. The invariant I can be represented in the Euclidean gravity case as a functional integral: (1) I (M, ) = D A W [A][A] , called the loop transform of [A], where A is a real SU(2) connection on M, W [A] is the spin network wave-functional (the product of the holonomies along the edges of contracted by using the intertwiners assigned to the vertices of , see [Ba]) and [A] is a diffeomorphism and gauge invariant functional of A which satisfies the Hamiltonian constraint equation. In the Minkowski gravity case, the connection A is complex and the loop transform has a more complicated path-integral representation, see [Ko2]. Kodama [Ko1] has shown that an exponent of the Chern-Simons functional 1 2 CS [A] = exp (2) Tr A ∧ d A + A ∧ A ∧ A λ M 3 satisfies the Hamiltonian constraint equation, where λ is the cosmological constant. . The formula (1) then implies, see [MS,M1,FS], that Iλ (M, ) = ICS (M, ) can be related by an analytic continuation λ → iλ to the Witten-Reshetikhin-Turaev Invariant Z WRT (M, ) of the pair (M, ). Furthermore, it was conjectured that λ ∝ k −1 , where k/2 is the maximal spin used in the construction of the WRT Invariant; see [S,MS]. When λ = 0 then it is possible to show that the functional: 0 [A] = δ[F] = D B ei M T r (B∧F) , 1 In LQG the 3-manifold M represents the space, while the 4-manifold M × R represents the spacetime.
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satisfies the Hamiltonian constraint equation, where F = d A+ A∧ A and B is a one-form, see [M1]. Therefore the path integral: I0 (, M) = D A D B W [A] ei M T r (B∧F) provides us with a physical definition of quantum invariants of 3-dimensional oriented manifolds based on SU(2) BF-Theory. The path integral I0 also provides a set of observables for the theory of flat SU(2) connections on M. This theory can also be considered as a generalisation of three-dimensional Euclidean General Relativity. The invariant I0 can be defined by using the spin foam techniques [M1,M2], also see [O]. Let (M) be a triangulation of the oriented 3-manifold M. Let ∗ (M) be the dual cell complex; see [RS]. Let H− be the 3-dimensional index 1 handlebody obtained by thickening the 1-skeleton of ∗ (M). The spin network can be inserted into H− in a way such that each 0-handle of H− contains at most one vertex of , while each edge of goes along some 1-handle of H− , parallel to its core. The invariant I0 can be then expressed as dgl W (g) δ(gˆ f ) , (3) I0 (M, ) = l∈L
f ∈F
where L is the set of edges of∗ (M), F is the set of faces of ∗ (M), gl ’s are the dual edge holonomies and gˆ f = l∈∂ f gl . The spin network wavefunction W (g) is given by:
W (g) = D ( jl ) (gl )
ιv , l∈L()
v∈V ()
where L() is the subset of edges of the dual complex ∗ (M) which pass through the same 1-handles of H− as the edges of , jl is the spin assigned to the corresponding edge of , D ( jl ) (gl ) is the appropriate SU(2) representation matrix, V () is the subset of vertices of ∗ (M) which are contained in the same 0-handles of H− as the vertices of , ιv are the corresponding intertwiners and | denotes the contraction of appropriate indices. Performing the group integrations in (3) is equivalent to using the skein relations for the SU(2) group; see [Ba,O,M2]. Hence one obtains a state sum in terms of classical n j symbols where n ≥ 6 [M1,M2], which reduces to the Ponzano-Regge Model [PR] when the graph is empty. These infinite sums can be regularised by passing to the quantum SU(2) group at a root of unity, which then gives a Turaev-Viro type state sum containing quantum n j symbols [M1]. We will denote this invariant by Z CH (M, ), or Z CH (M, ; j1 , . . . , jn ) if we want to emphasise the colourings of the edges of . In this paper we are going to show that the state sum Z CH (M, ) can be obtained in the context of J. Roberts’ Chain-Mail formalism [R1,R2], which then provides a proof that Z CH (M, ) is a well-defined invariant of coloured framed graphs embedded in an oriented closed 3-manifold M. The Chain-Mail approach also provides a simple and natural definition of these spin network invariants, as well as an easier way to calculate them. We will also prove that Z CH (M, ) = Z WRT (M, )Z WRT (M),
(4)
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where M denotes the oriented 3-manifold M with the opposite orientation. Hence Eq. (4) gives a relation between BF-Theory spin network invariants and the corresponding Chern-Simons Theory invariants. The relation (4) also implies that the invariant Z CH (M, ) coincides (apart from normalisation factors) with the graph invariants Z TV (M, ) defined in a combinatorial way by V. Turaev in [T3,T4]; see also [BD,KS]. This is a surprising result, given the completely different ways of constructing these invariants. The definition of Turaev’s Invariant Z TV (M, ) is quite complicated. One has to excise from M a regular neighbourhood n() of , and what is left of M has to be triangulated. Then a Turaev-Viro type state sum is constructed, having some extra terms obtained by natural shadow-world evaluations at the boundary of n(). The construction of Z TV (M, ) will be outlined in this article, and we will also provide a direct proof that Z TV (M, ) coincides with Z CH (M, ), apart from normalising factors. The invariant Z CH (M, ) is also well defined for manifolds with boundary, a fact which we will prove carefully; compare with [BP]. This makes it possible to define Z CH (M, ) for the case of thick surfaces M = × I . In this case it is possible obtain a combinatorial definition of Z CH (M, ), making use of natural handle decomposition of × I , yielding exactly the shadow world definition in [T3,T4]. This framework also provides an alternative definition of the Turaev-Viro Invariant Z TV (M) for compact manifolds with boundary, see [KMS]. In fact, what we prove is that the equivalent J. Roberts’ Chain-Mail Invariant Z CH (M) extends directly to manifolds with boundary; cf. [BP]. This article should be compared with [BGM] where a different set of invariants of graphs embedded in oriented 3-manifolds is defined. In fact the techniques of proof in both articles are very similar, however adapted to different invariants. We will refer frequently to results obtained in [BGM] and [R1,R2] in the course of this article. See also [B1,GI,B2,B3]. 2. Preliminaries 2.1. Chain-Mail. We use the normalisations of [BGM]. The Turaev-Viro Invariant Z TV of 3-dimensional closed manifolds can be defined in the fashion shown in this section, due to J. Roberts see [R1,R2]. For an introduction to handle decomposition of manifolds, we refer the reader to [RS,GS] and [K]. One advantage of Roberts’ approach is that it makes the introduction of observables very natural, and all proofs can be obtained in a geometrical, rather than combinatorial way. The extension of the results to manifolds with boundary is immediate within this framework. 2.1.1. Generalised Heegaard diagrams. Let M be a closed oriented (piecewise-linear) 3-manifold. Choose a handle decomposition of M. Let H− be the union of the 0- and 1-handles of M. Let also H+ be the union of the 2- and 3-handles of M. Both H− and H+ have natural orientations induced by the orientation of M. There exist two non-intersecting naturally defined framed links m and in H− . The second one is given by the attaching regions of the 2-handles of M in ∂ H− = ∂ H+ , pushed inside H− , slightly. On the other hand, m is given by the meridional disks of the 1-handles, in other words by the belt-spheres of the 1-handles of M, living in ∂ H− . The sets of curves m and in H− have natural framings, parallel to ∂ H− . The triple (H− , m, ) will be called a generalised Heegaard diagram of the oriented closed 3-manifold M.
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Fig. 1. Evaluation of the Hopf Link with both strands coloured with . All strands are given the blackboard framing
Generalised Heegaard diagrams are also defined for compact 3-manifolds with a non-empty boundary. The only difference is that ∂ H+ is not equal to ∂ H− anymore. 2.1.2. Skein theory and the -element. Let L be a framed link in S 3 . Consider an integer 2 j+1 −q −2 j−1 iπ . parameter r ≥ 3, and let A = e 2r and q = A2 . Let also dimq j = q q−q −1 If the components of the framed link L are assigned spins j1 , j2 , . . . , jn ∈ {0, 1/2, . . . , (r − 2)/2}, then we can consider the value L; j1 , . . . , jn ∈ C (the skein space of S 3 ), obtained by evaluating the coloured Jones polynomial at q; see [KL]. For example, if 0 is the unlink with 0-framing, then 0 ; j = dimq j. Colourings by spins should not be confused with framing coefficients when drawing coloured links. It should be obvious from the context which colouring is meant in each case. We can in general consider the case in which the components of L are assigned linear combination of spins, with multilinear dependence on the colourings of each strand of L. The most important of these linear combinations is the “-element” given by: r −2
=
2
dimq ( j) j.
j=0
It is understood that: r −2
; =
2
dimq ( j) ; j ,
j=0
where “ ” represents some link component. For example for the evaluation of the Hopf Link with both strands coloured by the -element, one would proceed as in Fig. 1. By the “Killing an ” property (see Fig. 3) it actually follows that the end result is the element N ∈ R, defined below. A sample of the beautiful properties satisfied by the -element appears in [R1,R2, L,KL]. See Figs. 3, 4, 5 and 8 for different consequences of the important Lickorish Encircling Lemma satisfied by a 0-framed unknot coloured by the -element see [L]. Another extremely important property is the handleslide invariance property, shown in the references above. √ Define 0 ; = N and 1 ; = κ N , the evaluation of the 0- and 1-framed (r −2)/2 unknots coloured with the -element. Therefore we have that N = j=0 (dimq j)2 = √ −1
π 2 −3−r 2 e− iπ 4 , and −1 = N κ , see [R1,R2]. r sin r . On the other hand κ = A Note that the evaluation L; j1 , . . . , jn can be extended to 3-valent framed graphs whose edges are coloured with spins, or linear combinations of them. We use the normalisations of [KL] for the 3-valent vertex. By a framed graph we mean a finite graph
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Fig. 2. Weights associated with coloured simplices. All spin networks are given the blackboard framing
whose edges and vertices are extended to small 2-disks and bands. These are called fat graphs in [T3]. We will only consider framed graphs for which their associated surface with boundary is orientable. If and are non-intersecting framed graphs in S 3 , coloured with the spins j1 , . . . , jn and i 1 , . . . , i m , then there exists a colouring of their disjoint union ∪ ⊂ S 3, and its evaluation in the skein space of S 3 is denoted by ∪ ; j1 , . . . , jn , i 1 , . . . , i m . We will use this type of notation frequently in this article. 2.1.3. The Chain-Mail invariant. Let again M be a connected 3-dimensional closed oriented piecewise linear manifold. Consider a generalised Heegaard diagram (H− , m, ), associated with a handle decomposition of M. Give H− the orientation induced by the orientation of M. Let : H− → S 3 be an orientation preserving embedding. Then the image of the links m and under defines a link CH(H− , m, , ) in S 3 , called the “Chain-Mail Link”. J. Roberts proved that the evaluation CH(H− , m, , ); of the Chain-Mail Link coloured with the -element is actually independent of the orientation preserving embedding : H− → S 3 , see [R1], Proposition 3.3. The Chain-Mail Invariant of M (due to J. Roberts) is defined as: Z CH (M) = N g−K −1 CH(H− , m, , ); , where K is the number of components of the Chain-Mail Link and g is the genus of the Heegaard surface ∂ H− . Therefore K = n 1 + n 2 and g = n 1 − n 0 + 1, where n i is
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Fig. 3. Lickorish Encircling Lemma for the case of one strand: the “Killing an ” property. All networks are given the blackboard framing
Fig. 4. Lickorish Encircling Lemma for the case of two strands. All networks are given the blackboard framing
Fig. 5. Lickorish Encircling Lemma for the case of three strands. All networks are given the blackboard framing
the number of i-handles of M. It is proved in [R1,R2] that this Chain-Mail Invariant coincides with the Turaev-Viro Invariant Z TV (M) of M, see [TV]. We will go back to this later, see 3.1.2. Even though we need an orientation of M to define Z CH , it is possible to prove that the Chain-Mail Invariant is orientation independent. In fact, the equivalent Turaev-Viro Invariant is defined also for non-orientable closed 3-manifolds. 2.2. The Witten-Reshetikhin-Turaev invariant. The main references now are [L] and [RT]. Let M be an oriented connected closed 3-manifold. Then M can be presented by surgery on some framed link L ⊂ S 3 , up to an orientation preserving diffeomorphism. Any framed graph in M can be pushed away from the areas where the surgery is performed, and therefore any pair (M, ), where is a trivalent framed graph in the ˆ where ˆ is a framed oriented closed 3-manifold M can be presented as a pair (L , ), trivalent graph in S 3 , not intersecting L.
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The Witten-Reshetikhin-Turaev Invariant of a pair (M, ), where the framed graph is coloured with the spins j1 , . . . , jn , is defined as: Z WRT (M, ; j1 , . . . , jn ) = N −
m+1 2
ˆ , j1 , . . . , jn . κ −σ (L) L ∪ ;
Here σ (L) is the signature of the linking matrix of the framed link L. This is an invariant of the pair (M, ), up to orientation preserving diffeomorphism. Note that, in contrast with the Turaev-Viro Invariant, the Witten-Reshetikhin-Turaev Invariant is sensitive to the orientation of M. If M is an oriented 3-manifold, we represent the manifold with the reverse orientation by M. Some immediate properties of the Witten-Reshetikhin-Turaev Invariant are the following: Remark 2.1. We have: Z WRT (S 3 ) = N −1/2 , Z WRT (S 2 × S 1 ) = 1, Z WRT (M, ) = Z WRT (M, ),
1 Z WRT (P, )#(Q, ) = Z WRT (P, )Z WRT (Q, )N 2 . Here M, P and Q are oriented closed connected 3-manifolds. In addition, and are coloured graphs embedded in P and Q. It is understood that the connected sum P#Q is performed away from and . It is easy to see that this uniquely defines a framed graph in P#Q, up to an orientation preserving diffeomorphism. To prove the third equation we need to use the fact that N is a real number. The invariant Z WRT can be extended to manifolds with an arbitrary number of components by doing the product of the invariant Z WRT of each component, see [BGM]. Given oriented closed connected 3-manifolds P and Q, we define P#n Q in the following way, see [BGM]. Remove n 3-balls from P and Q, and glue the resulting manifolds P and Q along their boundary in the obvious way, so that the final result is an oriented manifold. We denote it by: P#n Q = P
Q .
∂ P =∂ Q
This definition extends to the non-closed case in the obvious way. It is easy to see that P#1 Q = P#Q, and P#n Q is diffeomorphic to (P#Q)#(S 1 × S 2 )#(n−1) , if n > 1. From Remark 2.1 it follows immediately that: Z WRT (P#n Q, ∪ ; j1 , . . . , j p , i 1 , . . . , i m ) n
= Z WRT (P, ; j1 , . . . , j p )Z WRT (Q, ; i 1 , . . . , i m )N 2 .
(5)
Here P and Q are closed oriented 3-manifolds. In addition, and are graphs in P and Q, coloured with the spins j1 , . . . , j p and i 1 , . . . , i m , respectively. As before, it is implicit that the multiple connected sum P#n Q is performed away from and . This defines a graph ∪ in P#n Q up to an orientation preserving diffeomorphism. This basically follows from the Disk Theorem (see [RS, 3.34]), and the fact that graph complements in connected 3-manifolds are connected.
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3. Spin Network Invariants 3.1. Definition of the invariants. Let M be a closed connected oriented piecewise linear 3-manifold. Let also be a framed trivalent graph embedded in M, whose edges are coloured with the spins j1 , . . . , jn ∈ {0, 1/2, . . . , (r − 2)/2}. There exists a handle decomposition of M such that is contained in H− , the union of the 0- and 1-handles of M. Let (H− , m, ) be the associated generalised Heegaard diagram of M. Note that H− is connected. Choose an orientation preserving embedding : H− → S 3 . The framed graph can be added to the Chain Mail Link CH(H− , m, , ), by using the map . Definition 3.1. The Chain-Mail Expectation Value is by definition: Z CH (M, ; j1 , . . . , jn ) = N g−K −1 CH(H− , m, , ) ∪ (); , j1 , . . . , jn , where, as before, K is the number of components of the Chain-Mail Link CH(H− , m, , ) and g is the genus of the Heegaard surface ∂ H− . Recall that g = n 1 − n 0 + 1 and K = n 1 + n 2 , where n i is the number of i-handles of M. The value of Z CH (M, ; j1 , . . . , jn ) actually does not depend on the orientation preserving embedding : H− → S 3 . This follows from the same argument of the proof of Proposition 3.3 of [R1]. In fact, Z CH (M, ; j1 , . . . , jn ) stays the same after births or deaths of 0 − 1 pairs of handles, now by the same argument of the proof of Theorem 3.4 of [R1]. In particular, we can alter the handle decomposition of M to a handle decomposition of M with a unique 0-handle, without changing Z CH (M, ; j1 , . . . , jn ). This fact will be of prime importance. The independence of the Chain-Mail Expectation Value on a general handle decomposition of M containing in the union of 0- and 1-handles of M can be proved as Theorem 1 of [BGM]. We can suppose that M has a unique 0-handle. In this case, the signature σ (CH(H, m,
, )) of the linking matrix of the Chain-Mail Link is zero (see [R1, proof of Theorem 3.7], [BGM, 4.2] or 3.1.1). Therefore it follows that Z CH (M, , j1 , . . . , jn ) is the Witten-Reshetikhin-Turaev Invariant of the pair graph-manifold defined, by using surgery, n 1 +n 2 +1
from (CH(H− , m, , ), ()), times N −n 0 −n 2 + 2 = N −n 3 /2 , where n i is the number of i-handles of M, see [BGM, proof of Theorem 1] and 3.1.1. Note that the Euler characteristic of any closed orientable 3-manifold is zero, thus n 0 − n 1 + n 2 − n 3 = 0. On the other hand (recall that we suppose that the chosen handle decomposition of M has a unique 0-handle), by the same argument appearing in the proof of Theorem 1 of [BGM], the pair (CH(H− , m, , ), ()) is a surgery presentation of the graph inside the manifold M#n 3 M, considering the inclusion of into M (not M), see Lemma 3.3 for an alternative proof of this important fact. Therefore: Z CH (M, ; j1 , . . . , jn ) = N −n 3 /2 Z WRT (M#n 3 M, ; j1 , . . . , jn ), thus, by Eq. (5), it follows that: Z CH (M, ; j1 , . . . , jn ) = Z WRT (M, ; j1 , . . . , jn )Z WRT (M). In particular:
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Theorem 3.2. Let M be a closed connected oriented piecewise linear manifold of dimension 3. Let also be a framed trivalent graph embedded in M, with its edges coloured with the spins j1 , . . . , jn ∈ {0, 1/2, . . . , (r − 2)/2}. The Chain-Mail Expectation Value Z CH (M, ; j1 , . . . , jn ) depends on neither the handle decomposition of M containing in the union of 0- and 1-handles, nor on the orientation preserving embedding : H− → S 3 , and it is therefore an invariant of the pair (M, ), up to an orientation preserving diffeomorphism. In fact: Z CH (M, ; j1 , . . . , jn ) = Z WRT (M, ; j1 , . . . , jn )Z WRT (M). This theorem is ultimately due to V. Turaev, though it did not appear exactly in this form. See [T1] and [T4, Theorem 7.3.1]. We will go back to this issue later, see 3.2.3. The construction given in [M1,M2] corresponds to the special case when each 0-handle of M contains at most one vertex of , while each edge of goes along some 1-handle of M. Notice that this restriction does not appear in this article. Note that the spin network invariant Z CH (M, ; j1 , . . . , jn ) is orientation dependent. In fact Z CH (M, ; j1 , . . . , jn ) = Z CH (M, ; j1 , . . . , jn ), see Remark 2.1. This is in sharp contrast with the observables defined in [B1,BGM]. 3.1.1. Manifolds with boundary. Let M be a compact connected oriented piecewiselinear 3-manifold, with a non-empty boundary. Consider a handle decomposition of M. As before, let H− be the union of the 0- and 1-handles of M. Let be a framed graph embedded in M, such that ⊂ H− . Suppose that the edges of are coloured with the spins j1 , . . . , jn . Consider an orientation preserving embedding : H− → S 3 . We define the ChainMail Expectation Value in an analogous way to the closed case: Z CH (M, ; j1 , . . . , jn ) = N g−K −1 CH(H− , m, , ) ∪ (); , j1 , . . . , jn . This is independent of the orientation preserving embedding : H− → S 3 , for the same reason why it is in the closed case. Recall that g is the genus of ∂(H− ), and K is the number of components of the Chain-Mail Link. Note that by the same argument as in the case of closed manifolds, we can suppose (and we will) that M has a handle decomposition with a unique 0-handle. Lemma 3.3. Suppose that M has a unique 0-handle. Then it follows that the pair (CH(H− , m, , ), ()) is a surgery presentation of the graph inside the boundary of the compact 4-manifold M (2) × I , where I = [0, 1]. Here M (2) is the union of the 0-,1and 2-handles of M, and is embedded as a copy × {0} in M (2) × {0} ⊂ ∂(M (2) × I ), in the obvious way. Proof (sketch). Turn the meridian curves m of the 1-handles of H− into dotted circles, as in [K], see also [GS]. Similarly to [R1, proof of Theorem 3.7], consider the 4-manifold W obtained from D 4 by looking at the Chain-Mail Link CH(H− , m, , ) ⊂ S 3 as being a Kirby diagram. In other words, we attach 4-dimensional 1-handles to the dotted circles of the Chain-Mail Link (after turning each of them into a disjoint union of 3-disks, paying attention to the remaining components of the Chain-Mail Link), and 2-handles to the remaining curves. Then W is diffeomorphic to M (2) × I , where M (2) is the union of the 0-, 1- and 2-handles of M. See for example [R1, proof of Theorem 3.7] or [GS, 4.6.8]. Under the natural diffeomorphism W → M (2) × I , the region in S 3 ⊂ D 4 inside the unique 0-handle of (H− ) ⊂ S 3 goes exactly to the corresponding
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area in H− ⊂ M (2) × {0} ⊂ ∂(M (2) × I ), see [GS, 4.6.8] and [BP]. This finishes the proof. Now, we will use the fact (see [R1,R2]) that the signature of the linking matrix of the Chain-Mail Link CH(H− , m, , ) is zero. This follows from the fact that the ChainMail Link defines a Kirby diagram for M (2) × I (a manifold of zero signature) and Lemma 3 of [BGM]; a result included in the discussion in [R1,R2]. This permits us to conclude: Z CH (M, ; j1 , . . . , jn ) = N g−K −1 CH(H− , m, , ) ∪ (); , j1 , . . . , jn n 1 +n 2 +1 = N n 1 −n 0 −n 1 −n 2 + 2 Z WRT ∂ M (2) × I , × {0}; j1 , . . . , jn χ (M)+n 3 = N − 2 Z WRT ∂ M (2) × I , × {0}; j1 , . . . , jn ;
here n i is the number of i-handles of M, note that M has a unique 0-handle, by assumption. Now, ∂(M (2) × I ) = ∂(M × I )#(S 1 × S 2 )#n 3 , since M has a non-empty boundary. The manifold ∂(M × I ) is exactly the double D(M) of M, in other words M ∪∂ M M. Thence, from Remark 2.1: Theorem 3.4. Let M be a connected compact oriented piecewise linear 3-manifold. Let also be a trivalent framed graph embedded in M, coloured with the spins j1 , . . . , jn ∈ {0, 1/2, . . . , (r − 2)/2}. Choose a handle decomposition of M such that is contained in the union of the 0- and 1-handles of M. The Chain-Mail Expectation Value: Z CH (M, ; j1 , . . . , jn ) = N g−K −1 CH(H− , m, , ) ∪ (); , j1 , . . . , jn does not depend on the handle decomposition of M containing in the union of the 0- and 1-handles, and moreover it is an invariant of the pair (M, ), up to an orientation preserving diffeomorphism. In fact: Z CH (M, ; j1 , . . . , jn ) = N −
χ (M) 2
Z WRT (D(M), ; j1 , . . . , jn ),
where is embedded in M ⊂ D(M) in the obvious way. Note that we have already proved this theorem for the case of closed manifolds. Considering the case when is the empty graph, it follows that the Chain-Mail Invariant Z CH (M) is also well defined for a manifold with boundary. See also [BP]. This gives a definition of the Turaev-Viro Invariant Z TV (M) for manifolds with boundary, cf. [KMS]. 3.1.2. Calculation of the invariants using triangulations-I. Let M be 3-dimensional closed connected oriented piecewise linear manifold. Consider a piecewise linear triangulation of M. We can consider a handle decomposition of M where each i-simplex of M generates a (3 − i)-handle of M. See for example [R1,R2]. Applying the ChainMail picture to this handle decomposition, yields the following combinatorial picture for the calculation of the Chain-Mail Invariant Z CH (M), which, in this form is called the Turaev-Viro Invariant Z TV .
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Fig. 6. Local configuration of the Chain-Mail Link at the vicinity of a tetrahedron
Fig. 7. Applying the Lickorish Encircling Lemma to the 0-framed link determined by a face of the triangulation of M
A colouring of M is an assignment of a spin j ∈ {0, 1/2, . . . , (r − 2)/2} to each edge of M. Each colouring of a simplex s gives rise to a weight W (s), in the way shown in Fig. 2. Using the identity shown in Fig. 5 and Fig. 6, it follows that: Z CH (M) = W (s) colourings
simplices s
= Z TV (M). Note that we apply the Lickorish Encircling Lemma to the 0-framed unknot defined from each face of the triangulation of M, see Fig. 7. The last expression for Z CH is the usual definition of the Turaev-Viro Invariant. For a complete proof of the fact that Z TV = Z CH , see [R1,R2]. Note that we need to use the fact that the Euler characteristic of any closed oriented 3-manifold is zero to obtain this. It is important to note that this argument only works if M has an empty boundary. If not, some extra terms will appear on the boundary of M, see 3.2.3. Suppose that is a coloured trivalent framed graph embedded in the closed triangulated 3-manifold M. Suppose also that is contained in the handlebody H− made from the 0- and 1-handles of M, where M is given the handle decomposition associated with
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Fig. 8. General Lickorish Encircling Lemma for the case of four strands. All networks are given the blackboard framing
a triangulation of M. Then the previous argument can be applied to obtain a combinatorial expression for the Chain-Mail Expectation Value Z CH (M, ; j1 , . . . , jn ). Anytime a strand of the graph passes though a face of the triangulation of M, then we must apply a general Lickorish Encircling Lemma. See Fig. 8 for the case of 4 strands. In this way one can express Z CH (M, ; j1 , . . . , jn ) as a state sum of a product of n j symbols. This state sum appeared originally in [M1], and as described in the introduction section, it was obtained by a quantum group regularisation of the group integral (3). In fact, the Lickorish Encircling Lemma can be understood as a property of the integral over the quantum group of a product of quantum group representations, see [O]. Later (see 3.2.3) we will present a different combinatorial expression for Z CH (M, ; j1 , . . . , jn ) if M is a piecewise-linear manifold with a piecewise-linear triangulation. 3.2. Turaev’s description of the invariants. 3.2.1. Combinatorial expression of the observables in the case of thick surfaces. Let be an oriented surface. Let also be a coloured trivalent framed graph embedded in × I . Let us find a combinatorial expression for the Chain-Mail Expectation Value Z CH ( × I, ; j1 , . . . , jn ). Consider a handle decomposition of constructed in the following way: First of all, project the graph onto a graph G in ∼ = × {1}, so that G is a regular projection of . Note that we need to keep track of under- and over-crossings of . By definition, the point (x, a) where x ∈ and a ∈ [0, 1] will be below (x, b) if a < b, here b ∈ [0, 1]. The graph G is in general not 3-valent, since it will be 4-valent at the crossing points of . Second, add some extra edges and vertices to G, and obtain a graph G ∗ , such that \ G ∗ is a disjoint union of disks, and, moreover, no edge of G ∗ closes up to a loop S 1 , see [R2, proof of Proposition 3.15]. Therefore, there exists a handle decomposition of × I for which the union of the 0- and 1-handles of it is a thickening n(G ∗ ) of G ∗ . In particular, each vertex of G ∗ induces a 0-handle of × I and each edge of G ∗ induces a 1-handle of × I . There exists also a 2-handle for each connected component of \ G ∗ . Note that is included in n(G ∗ ) in the obvious way. Each face f of \ G ∗ induces a 2-handle of × I , thus an element l f of the ChainMail Link. Let l F be the union of all these links, where F is the set of faces of \ G ∗ .
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A face colouring f → c f is a map F → {0, 1/2, . . . , (r − 2)/2}. Each face colouring induces a colouring c F of l F by spins in {0, 1/2, . . . , (r − 2)/2}. Let m be the meridian link of the 1-handles of × I . Therefore, each edge of G ∗ induces an element of m. Consider an orientation preserving map : × I → S 3 which factors through an unknotted embedding of a handlebody of index 1. We have: ∗
∗
Z CH ( × I, ; j1 , . . . , jn ) = N −#{vertices of G }−#{faces of \G } (m) ∪ (l F ) ∪ (); , c F , j1 , . . . , jn dimq (c f ). face colourings of \G ∗
f ∈F
G∗.
This will yield an addiNow, apply Lickorish Encircling Lemma to each edge of tional factor of N for each edge of G ∗ , in the previous formula. Applying it firstly to the edges of G ∗ which are not in G makes it clear that, in order that a face colouring of \ G ∗ does not evaluate to a trivial contribution for Z CH ( × I, ; j1 , . . . , jn ), then for each face of \ G, the colours of all faces of \ G ∗ contained in it must coincide. This follows from Fig. 4. Note that G ∗ is obtained from G by adding some extra edges, the thickening of each of them bearing a 0-framed unknot coloured with the -element. Therefore we define a colouring c of ( , G) as being a map from the set of connected components of \ G with spins j ∈ {0, 1/2, . . . , (r − 2)/2}. Note that the edges of G have colourings induced by the colourings of the edges of . Given a vertex v of G, we assign a weight W (v, c) to it in the fashion shown in Fig. 9. Note that even though we need to choose a square root for the evaluation of each θ -net, it is obvious from our construction that the final result will not depend on this choice. Similarly to the proof of the fact that Z TV = Z CH we obtain: ∗
∗
∗
Z CH ( × I, ; j1 , . . . , jn ) = N −#{vertices of G }+#{edges of G }−#{faces of \G } ⎞ ⎛
χ ( f ) ⎠, dimq (c f ) W (v, c) ⎝ colourings c of ( ,G)
vertices v of G
faces f of \G
where c f denotes the colouring of the face f , and χ ( f ) is the Euler characteristic of it. We refer to [R2, 3.5] for details, in the case when = S 2 . χ ( f )
is because after applying the Lickorish The reason for the factor dimq (c f ) Encircling Lemma in Fig. 4 to each edge of G ∗ not in G, there is still a contribution, which is, for each face f of \ G, the factor dimq (c f )k2 −k1 +k0 , where k1 is the number of edges of G ∗ which are not in G and are included in the face f , k2 is the number of faces of \ G ∗ contained in f , and finally k0 is the number of vertices of G ∗ not in G which are in the interior of f . Now k2 − k1 + k0 = χ ( f ). To see this note that f has a natural handle decomposition with k2 0-handles, k1 1-handles and k0 2-handles. Therefore if follows that: Z CH ( × I, ; j1 , . . . , jn ) = N −χ ( ) Z T ( × I, ; j1 , . . . , jn ), where Z T ( × I, ; j1 , . . . , jn ) = colourings c of ( ,G)
vertices v of G
⎛ W (v, c) ⎝
⎞ χ ( f )
⎠. dimq (c f )
faces f of \G
The invariant Z T of coloured graphs in thick surfaces × I was originally defined by V. Turaev, see [T3,T4]. Note that our normalisation is different.
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Fig. 9. Weights associated with vertices of a graph projection on a surface
3.2.2. Invariants of graphs embedded in index-1 handlebodies. Let H be a 3-dimensional oriented handlebody made from 0- and 1-handles. Consider the link m made from the meridian curves (the belt spheres) of the 1-handles of H . Suppose that is a coloured framed graph embedded in H . Consider an orientation preserving embedding : H → S 3 . Then, by definition: Z CH (H, ; j1 , . . . , jn ) = N g−n 1 −1 (m ) ∪ (); , j1 , . . . , jn = N −n 0 (m ) ∪ (); , j1 , . . . , jn , where n 0 and n 1 are the number of 0- and 1-handles of the chosen handle decomposition of H , and g is the genus of the boundary of H , in particular n 1 = #m . Another way to calculate Z CH (H, ; j1 , . . . , jn ) is to consider the combinatorial construction of the Chain-Mail Expectation Value for a graph embedded in ∂ H × I of 3.2.1. Let us explain this. First of all, choose an orientation of the surface of (like in [T3, 5.a]). If we are provided with such an orientation, then we can shift as close as we want to ∂ H , by using the normal vector field to the surface of determined from the orientation (by construction, the final result will not depend on the orientation chosen). We resume the notation and discussion of 3.2.1. Recall that we can define a handle decomposition of ∂ H × I for which the union of the 0- and 1-handles of it is a thickening of the graph G ∗ . To obtain a handle decomposition of H , we will need to add to G ∗ the projection of the meridians m of the 1-handles of H , which should stay below G ∗ . This defines a graph J living in ∂ H . If necessary, add some extra edges and vertices to J ,
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obtaining a graph J ∗ such that the complement of J ∗ in ∂ H is a disjoint union of disks, and, moreover, no edge of J ∗ closes up to a loop. Then, we attach a 2-handle along each circle of m , and complete the final result with 3-handles. This yields a handle decomposition of H , whose union of 0- and 1-handles is a thickening n(J ∗ ) of J ∗ in H . Of course, there is also defined a handle decomposition of ∂ H × I , and the only difference is that the handle decomposition of H has #m extra 2-handles, and also some 3-handles. Therefore, the Chain-Mail Link associated with the constructed handle decomposition of H has #m more components than the Chain-Mail Link of the handle decomposition of ∂ H × I . Let m be the meridian link of the handle decomposition of ∂ H × I , and the link defined from the attaching regions of the 2-handles of ∂ H × I . Consider an orientation preserving map : n(J ∗ ) → S 3 . We have: Z CH (H, ; j1 , . . . , jn ) = N g−1−K −#m (m ∪ m ∪ ∪ ); , , , j1 , . . . , jn
= N −#m Z CH (∂ H × I , ∪ m ; j1 , . . . , jn , ). Here g is the genus of ∂(n(J ∗ )) and K is the number of 1- and 2-handles of the handle decomposition of ∂ H × I . In particular:
Z CH (H, ; j1 , . . . , jn ) = N −χ (∂ H )−#m Z T (∂ H × I , ∪ m ; j1 , . . . , jn , ), by the discussion in 3.2.1. 3.2.3. Calculation using triangulations-II Let M be an oriented connected closed piecewise linear 3-manifold. Let also be a framed graph embedded in M. Suppose that is coloured with the spins j1 , . . . , jn . We present now a fully combinatorial way to evaluate the Chain-Mail Expectation Value Z CH (M, ; j1 , . . . , jn ), due to V. Turaev. This was in fact the original definition. See [T2,T3,T4]. Let M be M minus a regular neighbourhood n() of . This regular neighbourhood has a handle decomposition where each vertex of induces a 0-handle and each edge of a 1-handle of n(). If necessary, we still need to add up some extra bivalent vertices of , so that no edge of closes up to a loop. This yields a framed graph . Let m be the meridian link of this handle decomposition of n()= n( ). Recall the construction of the Turaev-Viro Invariant in 3.1.2. Consider a triangulation of M . This triangulation restricts to a triangulation of the boundary of the regular neighbourhood n() of . Consider the dual graph G to this triangulation of ∂(n()), pushed inside n(), slightly, with framing parallel to the surface of n(). Any colouring c of the edges of M by spins induces a colouring cG of this graph G in ∂(n()). Let H = n(). Define: Z TV (M, ; j1 , . . . , jn ) Z CH (H, ∪ G; j1 , . . . , jn , cG ) = colourings c of M
=
W (s)
simplices s of M
N −χ (∂(H ))−#m Z T (∂ H × I , ∪ G ∪ m ; j1 , . . . , jn , cG , )
colourings c of M
W (s).
simplices s of M
This coincides with Turaev’s definition in [T2,T3,T4], apart from normalisation.
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Theorem 3.5. We have:
Z TV (M, ; j1 , . . . , jn ) = N −χ (M ) Z CH (M, ; j1 , . . . , jn ). Proof. As above, choose a handle decomposition of n() for which each vertex of induces a 0-handle of n() = n( ) and each edge of a 1-handle of it. This handle decomposition of n() ⊂ M can be completed to a handle decomposition of M by using the triangulation of M , in the usual picture where an i-simplex of M generates an (3 − i)-handle of M. As usual, we denote the number of i-simplices of the triangulation of M by n i . Let m be the meridian link of the 1-handles of M, minus the meridian link m of n( ). Let also be the framed link obtained from the attaching regions of the 2-handles of M. Consider the Chain-Mail expression for Z CH (M, ; j1 , . . . , jn ) using this handle decomposition, thus: Z CH (M, ; j1 , . . . , jn )
= N n 2 −n 3 +#{edges of }−#{vertices of }−#{edges of }−n 1 −n 2
(m) ∪ (m ) ∪ ( ) ∪ (); , , , j1 , . . . , jn . Here is an orientation preserving embedding from the handlebody made from the 0- and 1-handles of M into S 3 . By applying the Lickorish Encircling Lemma to the circles determined by the triangles of M , in the same way as in the proof that Z CH = Z TV , this is the same as:
Z CH (M, ; j1 , . . . , jn ) = N −n 3 −n 1 −#{vertices of }+n 2 +n 0
() ∪ (G) ∪ (m ); j1 , . . . , jn , cG , colourings c of M
W (s),
simplices s of M
or Z CH (M, ; j1 , . . . , jn ) = N χ (M )
Z CH (H, ∪ G; j1 , . . . , jn , cG )
colourings c of M
W (s)
simplices s of M
= N χ (M ) Z TV (M, ; j1 , . . . , jn ). Considering empty graphs, this result also provides a new proof of the fact that the combinatorially defined Turaev-Viro Invariant can be extended to 3-manifolds with boundary; see also [BP]. Compare with [KS,KMS,BD,T3,T4], where similar invariants of graphs were considered. 4. Concluding Remarks Since Z CH (M, ) is based on the path integral (3), it is straightforward to define a state sum expression for Z CH (M, ) for higher-dimensional manifolds. It would be interesting to investigate whether there exists a higher-dimensional analog of relation (4). In the two dimensional case, the path integral (3) can also be regularised by using quantum groups, see [M3], and our results imply that the corresponding state sum is given by
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Z CH ( × I, ). These invariants will be relevant for three-dimensional Euclidean LQG as well as for topological string theory. Note that in two dimensions the path integral (3) is convergent for surfaces of non-zero genus and empty graphs (see [FK,Ba]), which then raises the possibility of defining the corresponding spin network invariants without using quantum groups. Such invariants will be observables of two-dimensional Quantum Gravity with zero cosmological constant. For the case of three-dimensional Quantum Gravity with zero cosmological constant see [FL]. As shown in [M2], in order to construct more realistic quantum gravity states one needs to consider more general wavefunctions. The associated spin-network coefficients will correspond to the chain-mail evaluations considered in this article, but with additional loops carrying insertions. Relation (4) should be helpful for understanding the loop transforms of both the Kodama state and the flat connection state. According to the proposal of [FS], the loop transform coefficients of the Kodama state should be given by Z WRT (M, )i·k , where iπ Z WRT (M, )k is the Witten-Reshetikhin-Turaev Invariant for the case when q = e k+2 , while the cosmological constant would be proportional to 1/k. On the other hand, according to the proposal of [M1], and the results in this article, the loop transform coefficients of the flat-connection state in the Euclidean case are given by Z WRT (M, )k Z WRT (M)k , where k is taken to be sufficiently large in order to obtain a vanishingly small, i.e. zero, cosmological constant. Note that these two loop transforms are essentially the same in the Euclidean case. However, extending them to the Minkowski case and obtaining a complete understanding of the role of the cosmological constant, are still open problems. Acknowledgements. J. Faria Martins was financed by Fundação para a Ciência e Tecnologia (Portugal), post-doctoral grant number SFRH/BPD/17552/2004, part of the research project POCTI/MAT/60352/2004 (“Quantum Topology”), also financed by the FCT. A. Mikovi´c was partially supported by the FCT grant PTDC/MAT/69635/2006. We would like to thank John W. Barrett for introducing us to some of the techniques used in this article and Vladimir Turaev for giving us the exact references for his work on invariants of coloured graphs.
References [B1] [B2] [B3] [Ba] [BD] [BGM] [BP] [FK] [FL] [FS] [GI]
Barrett, J.W.: Geometrical measurements in three-dimensional quantum gravity. Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics (2001). Internat. J. Modern Phys. A 18, October, suppl., 97–113 (2003) Barrett, J.W.: Feynman loops and three-dimensional quantum gravity. Mod. Phys. Lett. A 20 (17–18), 1271–1283 (2005) Barrett, J.W.: Feynman diagrams coupled to three-dimensional quantum gravity. Classical Quantum Gravity 23(1), 137–141 (2006) Baez, J.: An introduction to spin foam models of quantum gravity and BF theory. Lect. Notes Phys. 543, 25–94 (2000) Beliakova, A., Durhuus, B.: Topological quantum field theory and invariants of graphs for quantum groups. Commun. Math. Phys. 167(2), 395–429 (1995) Barrett, J.W., Garcia-Islas, J.M., Faria Martins, J.: Observables in the Turaev-Viro and Crane-Yetter models, math.QA/0411281. J. Math. Phys. 48(9), 093508 (2007) Benedetti, R., Petronio, C.: On Roberts’ proof of the Turaev-Walker theorem. J. Knot Theory Ramifications 5(4), 427–439 (1996) Freidel, L., Krasnov, K.: Spin foam models and the classical action principle. Adv. Theor. Math. Phys. 2, 1183–1247 (1999) Freidel, L., Louapre, D.: Ponzano-Regge model revisited II: Equivalence with Chern-Simons. http://arxiv.org/list/gr-qc/0410141 Freidel, L., Smolin, L.: The linearization of the Kodama state. Class. Quant. Grav. 21, 3831–3844 (2004) García-Islas, J.M.: Observables in three-dimensional quantum gravity and topological invariants. Classical Quantum Gravity 21(16), 3933–3951 (2004)
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Gompf, R.E., Stipsicz, A.I.: 4-manifolds and Kirby calculus. Graduate Studies in Mathematics, 20. Providence, RI: Amer. Math. Soc. 1999 Kirby, R.C.: The topology of 4-manifolds. Lecture Notes in Mathematics 1374. Berlin: SpringerVerlag, 1989 Kauffman, L.H., Lins, S.L.: Temperley-Lieb recoupling theory and invariants of 3-manifolds. Annals of Mathematics Studies 134. Princeton, NJ: Princeton University Press, 1994 Karowski, M, Schrader, R.: A combinatorial approach to topological quantum field theories and invariants of graphs. Commun. Math. Phys. 151(2), 355–402 (1993) Karowski, M., Müller, W., Schrader, R.: State sum invariants of compact 3-manifolds with boundary and 6 j-symbols. J. Phys. A 25(18), 4847–4860 (1992) Kodama, H.: Holomorphic wave function of the universe. Phys. Rev. D 42, 2548–2565 (1990) Kodama, H.: Quantum gravity in the complex canonical formulation. Int. J. Mod. Phys. D 1, 439–524 (1992) Lickorish, W.B.R.: The skein method for three-manifold invariants. J. Knot Theory Ramifications 2(2), 171–194 (1993) Major, S., Smolin, L.: Quantum deformation of quantum gravity. Nucl. Phys. B 473, 267–290 (1996) Mikovi´c, A.: Quantum gravity vacuum and invariants of embedded spin networks. Class. Quant. Grav. 20(15), 3483–3492 (2003) Mikovi´c, A.: Flat spacetime vacuum in loop quantum gravity. Class. Quant. Grav. 21(16), 3909– 3922 (2004) Mikovi´c, A.: String theory and quantum spin networks, http://arxiv.org/list/hepth/0307141, 2003 Oeckl, R.: Generalized lattice gauge theory, spin foams and state sum invariants. J. Geom. Phys. 46, 308–354 (2003) Penrose, R.: Angular momentum: an approach to combinatorial space-time. Quantum Theory and Beyond. T. Bastin, ed. Cambridge: Cambridge University Press, 1971 Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients, Spectroscopic and Group Theoretical Methods in Physics. F. Bloch et al, eds., Amsterdam: North-Holland, 1968 Rovelli, C., Smolin, L.: Spin networks and quantum gravity. Phys. Rev. D 52, 5743–5759 (1995) Rourke, C.P., Sanderson, B.J.: Introduction to piecewise-linear topology. Reprint. Springer Study Edition. Berlin-New York: Springer-Verlag, 1982 Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(3), 547–597 (1991) Roberts, J.: Skein theory and Turaev-Viro invariants. Topology 34(4), 771–787 (1995) Roberts, J.: Quantum invariants via skein theory. Phd. Thesis, University of Cambridge (April 1994). Available at http://math.ucsd.edu/~justin/Papers/phd.ps.gz Smolin, L.: Linking topological quantum field theory and nonperturbative quantum gravity. J. Math. Phys 36, 6417–6455 (1995) Turaev, V.G.: Quantum invariants of 3-manifolds and a glimpse of shadow topology. C. R. Acad. Sci. Paris SéR. I Math. 313(6), 395–398 (1991) Turaev V.G.: State sum models in low-dimensional topology. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Tokyo: Math. Soc. Japan, 1991, pp. 689–698 Turaev, V.G: Quantum invariants of links and 3-valent graphs in 3-manifolds. Inst. Hautes Études Sci. Publ. Math. No. 77, 121–171 (1993) Turaev V.G.: Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics 18. Berlin: Walter de Gruyter & Co., 1994 Turaev, V.G., Viro, O.Ya.: State sum invariants of 3-manifolds and quantum 6 j-symbols. Topology 31(4), 865–902 (1992)
Communicated by A. Connes
Commun. Math. Phys. 279, 401–427 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0431-7
Communications in
Mathematical Physics
Recursion and Growth Estimates in Renormalizable Quantum Field Theory Dirk Kreimer1,2, , Karen Yeats2 1 IHÉS, Le Bois-Marie, 35, route de Chartres, F-91440 Bures–sur–Yvette, France.
E-mail: [email protected]
2 Department of Mathematics and Statistics, Boston University, 111 Cummington Street,
Boston, MA 02215, USA. E-mail: [email protected] Received: 25 January 2007 / Accepted: 19 August 2007 Published online: 22 February 2008 – © Springer-Verlag 2008
Abstract: In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. In the nonnegative case the radius of convergence turns out to be min{ρ, 1/b1 }, where ρ is the bound on the convergent ones, the instanton radius, and b1 the first coefficient of the β-function, while in general it is bounded by the above. 1. Introduction We first put the paper into context, and then describe our approach.
1.1. Purpose and Motivation. In this paper we consider estimates for renormalizable quantum field theories. More specifically, we want to understand how the large order behaviour of superficially divergent Green functions relates to the behaviour of superficially convergent Green functions. We will answer this question in complete generality for any renormalizable field theory. We caution the reader though in immediately pointing out that we need an estimate on the large order behaviour of superficially convergent ones as an input. Such estimates are available at the moment either in the form of results in constructive field theory, with the work by Magnen, Nicolo, Rivasseau and Sénéor [22], a typical example, or by functional integral methods in combination with Lipatov’s work, see Zinn-Justin’s monograph [32] for a beautiful introduction and exhaustive overview. Also, we would like to stress that in some very special cases, notably φ44 theory, constructive field theory found its own way to turn an estimate for convergent amplitudes
Research supported by grant NSF-DMS/0603781. Supported by CNRS.
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into an estimate for divergent ones, see the paper by David, Feldman and Rivasseau [7] as a typical example. The constructive field theory derivation delivers a lower bound using a renormalon analysis, while for an upper bound it is affected by the assumption that higher kernels (6-point amplitudes) do not lead to “unexpected cancelations”. Below, in specializing our general result to φ44 and using [22] as input, we reconfirm the result of [7] without such an assumption, with no need to involve renormalons in particular. Let us compare this with our results below. In the case where all coefficients are nonnegative our final result is an exact expression for the radius for the divergent amplitudes in terms of the radius for the convergent ones. In cases, as above, where estimates are known for the convergent amplitudes, they are often only lower bounds, leading by our analysis to lower bounds for the divergent amplitudes too. However, even in these cases, the exact radius in terms of the unknown convergent radius tells us precisely how the convergent amplitudes affect the growth of the divergent ones, and shows that further analysis only of the convergent case is needed to get precise radii. In the case with possibly negative coefficients we can only bound the radius even if we were to know an exact radius for the convergent amplitudes. This leads, as with the constructivist analysis, to lower bounds for the convergent amplitudes giving lower bounds for the divergent amplitudes. Our method is very different in spirit from the constructive approach or the functional integral approach. It relies on a Hopf algebraic decomposition of terms in the perturbative expansion into primitive constituents, not unlike the decomposition of a ζ function into Euler factors. It promotes the early investigations in field theory which attempted to conclude the structure of Green functions from the structure of Dyson–Schwinger equations [21] to a powerful algebraic approach. We close this introduction by pointing out that the recursive structures which are typical for an underlying Hopf algebra structure are also visible in the constructive field theory approach: the use of rooted trees in the frequency decomposition in [22] (see Fig.3 in that reference) or the parquet decomposition in [7] (see Fig.2 in that reference) is tantamount to that. Utilizing the fact that rooted trees are the universal object in the Hochschild cohomology of graded commutative Hopf algebras, our set-up starts from the intimate connection between this fact and the fact that under the Feynman rules the generators of this cohomology map to the integral kernels underlying the Dyson–Schwinger equations. 1.2. The set-up. We now explore the recursive structure of the short-distance sector of a renormalizable quantum field theory. Such theories have a finite number of distinct amplitudes r ∈ R ⊂ A which need renormalization. We decompose each Green function in accordance with structure functions which need renormalization, the remaining structure functions and superficially convergent Green functions contribute to amplitudes a ∈ A R in the set of all amplitudes A [18]. For such a theory, the Dyson–Schwinger equations give any amplitude s ∈ A = A R ∪ R in terms of all those amplitudes r ∈ R in the theory which need renormalization, and in terms of an infinite series of integral kernels, the skeleton graphs of the theory. The very fact that we can renormalize by modifying the Lagrangian implies that we have a sub Hopf algebra at our disposal which has the graded elements ckr as generators which correspond to the k th order contribution to the amplitude r [19,18]. To understand our main results in Section Four, let us discuss some subtleties in these notions. All amplitudes r ∈ R necessarily belong to amplitudes which are superficially
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divergent by power counting, when expanded into graphs in perturbation theory. But this necessary condition is not sufficient. The four-photon graphs in quantum electrodynamics are divergent by power counting, but do not need renormalization. For non-vanishing external momenta they provide contributions free of superficially divergences, and hence their corresponding primitives have, at any loop order, vanishing residues. This is the crucial clue: a sufficient condition for amplitudes to belong to R is that their primitives have a non-vanishing residue. Our results in Section Four apply then to all Green functions for amplitudes precisely in this set R. We assume throughout that for an amplitude r ∈ R, its primitives have non-vanishing residues at any loop order n ≥ 1. In practice, it can happen that for example flavour symmetries postpone the appearance of residues to some higher finite order. For clarity of exposition we omit discussion of this case, which could be incorporated at the expense of extra notational burden. We can relate the integral kernels above to the primitives of the Hopf algebra structure underlying renormalization. The growth of these primitives is hence determined by the growth of integral kernels provided by overall convergent Green functions. A bound on such a growth is hence obtainable from a bound of amplitudes in A R, where results in constructive field theory are principally available. The typical example is quantum electrodynamics (QED), where the primitives for the vertex function are given by the superficially convergent four fermion e+ e− → e+ e− scattering kernel, two-particle irreducible in a suitable channel. To proceed from there to a bound for amplitudes in R, we need a handle on the behaviour of the singular integrations which encompass the short-distance singular sector. Here, we proceed by choosing a suitable set of primitives such that we can reduce the Dyson–Schwinger equations to recursive equations acting on one-variable Mellin transforms. A simple study of the conformal symmetries in the corresponding primitives suffice to determine the form of these Mellin transforms. For any r ∈ R, we can define a Green function G r (a, L) = 1 ± γkr (a)L k , (1) k
where L = ln −Q 2 /µ2 measures the scaling behaviour under the renormalization group flow with respect to a single Euclidean kinematical variable Q 2 < 0. Note that the renormalization group determines the coefficients γkr (a) of this next-to-· · · -to leading log expansion in terms of γ1r (a). This turns the Dyson–Schwinger equations into an equation for γ1r (a), upon using the scattering type formula [5] and the Dynkin operator S Y , as exemplified in [18,19] (also, see [10] for a thorough discussion of this operator) on the Mellin transforms of the primitives, as we will discuss shortly. Hence, if we have a recursive set of equations giving Green functions G r (a, L) in terms of themselves, inserted into integral kernels, we can apply Mellin transforms as defined below upon using 2 ρ k r 2 2 r , G (a, ln −k /µ ) = G (a, ∂ρ ) − 2 (2) µ ρ=0 to reduce the evaluation of all integral kernels above to a study of corresponding Mellin transforms. In such a situation, we hence show how the growth of superficially divergent amplitudes is related to the growth of the superficially convergent ones. To the extent the latter is under control by results in constructive field theory, we get results for the former.
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The representation of primitive integral kernels through their Mellin transform then allows us to turn the Dyson–Schwinger equations into a recursive system which determines the γk in terms of the γ1 , and determines γ1 (a) recursively through the Taylor coefficients of the relevant Mellin transforms. Modest knowledge of the structure of those transforms: res p M( p)(ρ) = , (3) ρ(1 − ρ) and a Lipatov bound for them, a | p| res p ∼ | p|!c| p| for some suitable c at large | p|, allows us to show that solutions to Dyson–Schwinger equations for Green functions G r (a, L) have a similar Lipatov bound. We emphasize that our construction of a basis of primitives with a given Mellin transform resolves overlapping divergences, thanks to the Hochschild cohomology of the relevant Hopf algebras [14]. We have thus a self-similar recursive system determining the formal sums X r , r ∈ R p;r in terms of themselves and the action of suitable maps B+k;r = | p|=k B+ [18]. For all r ∈ R, X r (a) = I ± a k B+k;r (X r Qk ), (4) k≥1
where aQ is the invariant charge of the theory, and Green functions are obtained as G(a, L) = φ R (X r (a))(L),
(5)
for renormalized Feynman rules φ R . The study of the Hochschild one-cocycles B+k,i;r is crucial for a QFT. The Hopf algebra elements B+k;r (I) provide the very integral kernels above underlying the DSEs and are the terms which drive the recursion. Their consistent construction automatically takes internal symmetries of the theory into account, [17], by dividing the Hopf algebra by suitable Hopf ideals [29,30,17]. A further ideal is defined by our desire to concentrate on amplitudes in R. 1.3. Ideals. Green functions in field theory decompose into structure functions which have logarithmic short-distance singularities. Locality of field theory guarantees that these short distance singularities are invariant under changes of dimensionful parameters in the theory, allowing for local counterterms. This is not the only invariance which we can observe in counterterms. A chosen subgraph needs the same counterterm wherever it appears in a larger graph, and in whatever orientation it is inserted, as long as we work in a symmetric renormalization scheme. Furthermore, its counterterm is invariant under modification of its external momenta. We can hence isolate short-distance singularities in subgraphs in a manner such that all subdivergences are functions of a chosen internal momentum of the cograph under consideration. In so doing we enlarge the Hopf algebra of graphs into a Hopf algebra of colored graphs or decorated rooted trees, which makes the underlying Hochschild cohomology obvious and resolves overlapping divergences [14,16]. We emphasize that the choices above depend on the theory, renormalization scheme and physical problem one wants to study. Here, we are only concerned with the general set-up, and will only sporadically restrict to a specific theory. Our final result applies
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to any renormalizable field theory. Specifics of any particular such theory are to be discussed elsewhere. The freedom we have in the above choices can be summarized by saying we work in an ideal given by ◦i End−1 = 0, (6) i
with (7) End(1 2 ) = End(1 )End(2 ). Here, we use Sweedler notation () = i ⊗ and the fact that Feynman graphs have an operadic structure, so that for each term in the sum we have an operadic composition [16] of graphs such that = ◦i .
(8)
Finally, End is a map which implements such a choice: it chooses a configuration of external momenta for the subgraph , permutes the orientation of external legs, or changes the insertion place: any modification which will not modify the required counterterm is allowed. By construction, elements in our ideal are superficially convergent, and hence contribute nothing to the short distance singular sector. The above ideal is determined by the choice of the map End, which is then determined by the physics one wants to study. We then decompose graphs as = ◦i End−1 ( )P( ) + ◦i End−1 P( )P( ). (9)
i
i
p()
By construction, the first term in the rhs is a primitive element p = p() in the Hopf algebra of decorated rooted trees, and End−1 = End ◦ S. Note that for a primitive graph γ and an endomorphism End which changes its external leg structure to say zero momentum transfer, the above ideal simply says that the short distance singular sector of γ and End(γ ) are the same. More instructive is the example of the case of say a graph with a single divergent subgraph = ◦ . We then have
(10) p() = ◦ − End( ) .
This is clearly primitive as − End( ) is in the chosen ideal, and p() in (9) is primitive in general because all its subgraphs are in the ideal, by construction. We emphasize that in all our applications we will never discard any terms on the rhs of (9), but will always calculate the full graph , but use (9) as a convenient tool to come to a manageable form for the primitives which drive the recursion, on the expense to have an enlarged set of such integral kernels. For our purposes, we choose an ideal which sets m = 0 in structure functions and evaluates external momenta at a symmetric point −Q 2 with regard to their external momenta, and iterates graphs into each other in accordance with the definition of the Mellin transform defined below. As a renormalization scheme we choose subtraction at a fixed −Q 2 = µ2 .
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1.4. Mellin transform. Any so constructed primitive p = p() is a degree homogeneous combination of Hopf algebra elements of degree | p|. We let −Q 2 be the above kinematical variable which we keep. We define the Mellin transform M( p) of p as ⎧ ⎫ | p| | p| ⎨ 1 ⎬ M( p)(ρ) = [−Q 2 ]ρ Int p (−Q 2 ) [ki2 ]−ρ d 4 ki , (11) ⎩ | p| ⎭ i=1
i=1
where Int p (−Q 2 ) is the integrand determined by p. We let 2 2 2 Int − p (−Q ) := Int p (−Q ) − Int p (µ ).
(12)
We define renormalized Feynman rules for a symmetric momentum scheme with subtractions at −Q 2 = µ2 by ⎧ ⎫ | p| | p| ⎨ 1 ⎬ p 2 2 − 2 2 2 φ R (B+ (X ))(−Q /µ ) = Int p (−Q ) φ R (X )(−ki /µ ) d 4 ki . (13) ⎩ | p| ⎭ i=1
i=1
We have
p φ R (B+ (X ))(−Q 2 /µ2 ) = lim φ R (X )(∂−ρ )M( p)(ρ) [−Q 2 /µ2 ]−ρ − 1 , ρ→0
(14)
∂ . where ∂−ρ = − ∂ρ A Green function is then defined as the image under such Feynman rules applied to a fixpoint of a combinatorial DSE. In the coming sections, we first study the asymptotics of a combinatorial DSE, then the growth after applying φ R . Let us now look at an example in φ44 theory. We define the vertex Green function by setting −Q 2 = s = t = u, and set m = 0 in all propagators in accordance with an ideal which isolates the short distance singularities in massless Green functions. Taking the symmetry in external legs into account, the Hopf algebra series reads to order g 2 ,
(15) We define a primitive
(16) We define a combinatorial DSE 3 p p X (g) = 1 + g B+ 1 (X (g)2 ) + 3g 2 B+ 2 (X (g)3 ), 2 where p1 = . It has a fixpoint which reproduces to order g 2 the above expansion.
(17)
Recursive Estimates in QFT
Let us turn this into an integral equation: G(g, ln −k 2 /µ2 )2 3 G(g, ln −Q 2 /µ2 ) = 1 + g d 4k 2 k 2 (k + Q)2 − 3 1 2 2 2 G(g, ln −k /µ ) + G(g, ln −l 2 /µ2 ) +3g 8 2k · l − 2(k + l) · Q − 2Q 2 × 22 d 4 kd 4 l, k l (k + Q)2 (l + Q)2 (l − k)2 −
407
(18)
where }− indicates subtraction at −Q 2 = µ2 . One verifies that the solution to this integral equation agrees to order g 2 with the perturbative renormalization in a symmetric momentum scheme for the vertex function in φ44 theory, as it should. Its solution also agrees to any order in g 2 with the leading order in L = ln −Q 2 /µ2 in that order of g 2 , and also with the next to leading order, as we have taken the two loop primitive into account. Addition of further primitives delivers the lower powers in L. The Mellin transform 1 M p1 = ρ(1−ρ) is trivial, the Mellin transform M p2 much less so and can be found in [3]. 1.5. Mellin transforms as a geometric series. The above Mellin transforms M p (ρ) have the form f p (ρ) , (19) M p (ρ) = ρ(1 − ρ) in a natural manner, where the pole at ρ = 0 reflects the short-distance singularity, and the pole at ρ = 1 reflects the fact that in our ideal we have massless internal propagation. The series f p (ρ) = res p + O(ρ) determines the residue res p of the transform. We keep the factor 1/(1 − ρ) explicit though, to maintain the form of the Mellin transform similar, in accordance with its conformal symmetries at −Q 2 = 1. By the above definition of the Feynman rules it is a purely algebraic exercise [3,15] p p p p to define a new series of primitives p1 = p, p2 = B+ (B+ (I)) − 21 B+ (I)B+ (I), and so on, such that res pn M p (ρ) = . (20) qn ρ(1 − ρ) n We henceforth assume a basis of primitives { p} such that res p M p (ρ) = , ρ(1 − ρ)
(21)
which is convenient in the following. This finishes the discussion of our analytic set-up and we next remind ourselves of some basic properties of recursive systems relevant to Dyson–Schwinger equations. 2. The Universal Law 2.1. The classical case. A classical result of combinatorics is Pólya’s asymptotic formula for the number, t (n), of rooted, unlabelled trees with n vertices [26] (translated in [27]): t (n) ∼ Cρ −n n −3/2 , (22)
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where C is an explicit constant and ρ is the radius of convergence of the generating function T(x) = n≥1 t (n)x n of the class of all rooted, unlabelled trees. Thus ρ is the reciprocal of Otter’s tree constant [25] (sequence A051491 in [28]). The key to Pólya’s proof is to convert the recursive equation ⎛ ⎞ T(x) = x exp ⎝ T(x m )/m ⎠ m≥1
for rooted, unlabelled trees to a bivariate function ⎛ ⎞ E(x, y) = xe y exp ⎝ T(x m )/m ⎠ m≥2
at which point the recursive equation becomes T(x) = E(x, T(x)). Since ρ < 1 this rewriting uses the new variable y to isolate the portion of the recursive equation which controls the radius of convergence. Consequently, y − E(x, y) is amenable to Weierstrass preparation, which tells us that around (ρ, T(ρ)), y − E(x, y) is a product of a holomorphic function which is nonzero around (ρ, T(ρ)) and a monic polynomial in y with coefficients which are analytic in x. The degree of the polynomial is the order of the first nonzero derivative. The failure of the implicit function theorem at ρ gives that T(ρ) − E(ρ, T(ρ)) = 0, 1 − (∂ y E)(ρ, T(ρ)) = 0, −(∂ yy E)(ρ, T(ρ)) = 0. So the Weierstrass polynomial is quadratic which gives that T has a square root singularity at ρ: √ T(x) = f (x) + g(x) ρ − x with f and g analytic at ρ. Then the Cauchy integral theorem gives the desired asymptotics along with a formula for C. Asymptotics of the form Cρ −n n −3/2 have since been widely found for classes of rooted trees with recursive definitions and hence (22) is sometimes known as the universal law. Some notable examples generalizing the reach of Pólya’s analysis include [4,12], and [23]. Surveys include [8] and [24]. See also chapter VII of Flajolet and Sedgewick’s forthcoming book Analytic Combinatorics currently available as an online draft [11].1 For all but the last of the above the focus is on improved analytic conditions which still must be checked for any particular case of interest. One approach for further generalization is to specify certain combinatorial constructions and then use these constructions to build recursive equations with solutions which automatically satisfy the universal law. This is the approach of [2]. Flajolet and Sedgewick also use a framework of combinatorial constructions to build their recursive equations, getting the universal law for various schemas, such as the exp-log schema [11, VII.1]. 1 References are drawn from the draft of October 23, 2006.
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2.2. Recursive systems. Another approach to generalizing Pólya’s analysis is to move to recursive systems of equations while restricting the complexity of each equation. This is of particular relevance to applications to counting Feynman diagrams since generally only polynomials and geometric series are needed, but systems are difficult to avoid. First note that any geometric series can be converted to a polynomial at the expense of a new variable. Namely replace 1/(1 − X ) with a new variable F and add the equation F = 1 + F · X. In view of this reduction we can focus on polynomial systems. Nonnegative polynomial recursive systems give the universal law under some reasonable conditions, as was shown independently by Drmota[9], Lalley[20], and Woods[31]. For our purposes the full generality of the above are not necessary and we’ll follow the presentation of Flajolet and Sedgewick [11, VII.6.3]. Suppose y1 = 1 (x, y1 , . . . , ym ) .. . ym = m (x, y1 , . . . , ym ) with the i polynomials with real coefficients. There are five conditions which together guarantee that each component solution to this system satisfies the universal law. First we say the system is nonlinear if at least one of the i is nonlinear in y1 , . . . , ym . The system is nonnegative if each i has nonnegative coefficients. The next condition guarantees that the system does in fact behave recursively. For y = (y1 , . . . , ym ) ∈ R[[x]]m define the x-valuation by val(y) = mini (val(yi )), where the valn uation of a series picks out the index of the first nonzero coefficient, i.e. val( ∞ n=k an x ) = k when ak = 0, and with the convention that val(0) = ∞. Also define d(y, y ) = 2−val(y−y ) . Then the system is proper if d( (y), (y )) < K d(y, y ) for some K < 1. To guarantee that the system behaves as one system rather than many we need to guarantee that all variables play a role in each equation. Specifically, define the dependency graph of the system to be the directed graph on {1, . . . , m} with an edge from k to j if y j figures in a monomial of k (x, y). The system is irreducible if the dependency graph is strongly connected, that is there is a path between any two vertices following edges only in their forward direction. Finally, to remove spurious zero coefficients in the solution series, and to avoid extra singularities on the circle of convergence, define a power series T(x) to be aperiodic if it is not the case that there is a power series U(x) and integers a ≥ 0 and d ≥ 2 such that T(x) = x a U(x d ). If the solutions for each yi of the system are aperiodic then the system itself is said to be aperiodic. Then Theorem 1. ([11] Theorem VII.6) Suppose y = (y) is a polynomial system that is nonlinear, proper, nonnegative, and irreducible. Then all component solutions y j have the same radius of convergence ρ < ∞ and have a square root singularity at ρ. If furthermore the system is aperiodic then all y j satisfy the universal law.
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An outline of the proof is as follows. For more details see [11, Theorem VII.6]. Properness gives that the component solutions are unique, because the solutions can be generated recursively from the zero vector, and along with nonnegativity we get further that the component solutions have nonnegative coefficients. Irreducibility, along with nonnegativity to prevent cancellations, gives that the radius of each yi is the same value ρ. Aperiodicity forces ρ to be the only singularity of each component solution on the circle of convergence. The key to the proof is the square root singularity at ρ. Each of the three independent proofs takes a different approach, though in all cases, as in the single equation situation, it comes down to using the failure of the implicit function theorem at ρ. Nonlinearity is necessary to avoid singularities which themselves are linear rather than of square root type. Drmota [9] proceeds by solving a subset of the equations for the remaining variables and then substituting back into the remaining equations to reduce the number of equations in the system. Iteratively he is able to reduce to one equation at which point the system can be treated classically. Lalley [20], summarized in [11, Theorem VII.6], considers the linearized Jacobian ∂ y j i (x, y1 , . . . , ym ) and uses Perron-Frobenius theory to show that the largest eigenvalue in absolute value at x = ρ of the Jacobian is precisely 1 and that there is an eigenvector v with positive coefficients. Then multiplying the system by v and expanding around ρ gives the desired asymptotics. Woods [31] also uses the Jacobian and the largest eigenvalue 1. He continues the analysis on block upper triangular matrices in order to deal with certain non-irreducible cases.
3. The Universal Law for Feynman Diagrams 3.1. QED with one primitive per loop order. Massless quantum electrodynamics provides three monomials in the Lagrangian which need renormalization, corresponding as one-particle irreducible Green functions to the inverse fermion and photon propagators, and the vertex. We do not yet impose the Ward identity and proceed by setting up the combinatorial structure as it rather typically holds for a theory with three-valent vertices and two types of edges (the degenerate case of a single type of edge is easily interfered). The smallest Hopf algebra which allows for a correct renormalization of the full theory treats the sum of all primitive graphs at a given loop order as a single Hochschild cocycle, and we start from there. This situation is described by the following system: X1 = 1 +
k≥1
xk
X 12k+1 , (1 − X 2 )2k (1 − X 3 )k
X1 , (1 − X 2 )(1 − X 3 ) X1 X3 = x . (1 − X 2 )2
X2 = x
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After converting the geometric series we get ⎧ X1 ⎪ ⎪ ⎪ ⎪ ⎪ F ⎪ ⎪ 1 ⎨ X2
= ⎪ F 2 ⎪ ⎪ ⎪ ⎪ X ⎪ 3 ⎪ ⎩ F3
= 1 + X 1 F1 = x X 12 F22 F3 + x X 12 F22 F3 F = x X 1 F2 F3 = 1 + F2 X 2 = x X 1 F22 = 1 + F3 X 3 .
is nonnegative, polynomial, nonlinear, and irreducible. can be seen to be aperiodic simply by calculating the first few terms. itself is not proper. However 2 describes the same solutions and the other properties, nonnegative, polynomial, nonlinear, irreducible, and aperiodic, remain true for powers. To see that 2 is proper suppose we have 2 vectors v = (x1 , f 1 , x2 , f 2 , x3 , f 3 ) and v = (x1 , f 1 , x2 , f 2 , x3 , f 3 ) at distance 2−n ; so write x1 = x1 + x1 with x1 having no term of degree less than n and similarly for the other coordinates. Consider the difference in X 1 coordinates after applying : x1 f 1 − x1 f 1 = −x1 f 1 − x1 f 1 − x1 f 1 , which has no terms of degree less than n. Further if f 1 has no constant term and f 1 has no term of degree n then the difference in the X 1 coordinate has no terms of degree less than n + 1. Argue similarly for F2 and F3 . For the X 2 coordinate after applying we get a difference of x x1 f 2 f 3 − x x1 f 2 f 3 = −x x1 f 2 f 3 − x x1 f 2 f 3 − x x1 f 2 f 3 − x x1 f 2 f 3 − x x1 f 2 f 3 − x x1 f 2 f 3 , which has no terms of degree less than n + 1. Notice also that the new X 2 coordinate has no constant term for both initial vectors. Argue similarly for X 3 and F1 . Now consider applying a second time. Apply the above arguments again but notice that we are in the “further” case for X 1 , F2 and F3 , so all coordinates now have a difference with no terms of degree less than n + 1, that is the distance has decreased by at least 1/2. Thus any 1 > K > 1/2 will give that 2 is proper. Consequently by Theorem 1 we know that all 6 solution series to 2 and hence all 3 solution series to the original system have the same radius of convergence ρ and have coefficients with the asymptotic form Cρ −n n −3/2 . In order to understand the asymptotic growth rate for the QED system with primitives summed at each loop order it remains to understand the radius ρ. Substituting X 2 and
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X 3 into F2 and F3 , respectively we get F2 = 1 + x X 1 F22 F3 = F3 . Thus the original system can be rewritten X1 = 1 +
x X 13 F23 1 − x X 12 F23
,
F2 = 1 + x X 1 F23 . Rearrange X1 = 1 +
X 12 (F2 − 1) , 1 − X 1 (F2 − 1)
then solve to get F2 =
1 − 2X 12 . X 1 (1 − 2X 1 )
Substitute back into the equation for F2 and expand to get −x + X 1 + (6x − 5)X 12 + 8X 13 + (−12x − 4)X 14 + 8x X 16 = 0. As a polynomial in X 1 this has discriminant 4096x 2 (32x 2 − 8x + 1)(−2 + 27x)2 . So the radius of the system is 2 . 27 In view of the fact that the radius is an important value associated to the system and that this system is canonically associated to QED we’re led to the following question as to what is the physical meaning of 2/27 in QED? The relevance of this number extends to any system which provides a recursive system similar to QED. We can proceed similarly for other theories; the results of some examples are summarized in Appendix 5. 3.2. Polynomially many primitives per loop order. Combining all primitives at a given loop order into one Hochschild cocycle which drives the Dyson–Schwinger equation defines the smallest sub Hopf algebra which still renormalizes the full theory correctly. It is often instructive to disentangle the primitives in different ways, for example in accordance with the transcendental nature of res p . This motivates to consider a slightly more general condition on the number of primitives, enough to apply the polynomial systems result. We next assume there to be p(k) primitives at k loops where p is a polynomial. To see that this circumstance reduces to a nonnegative polynomial system it suffices to show that p(k)B k k≥
can be written as a sum of powers of geometric series. This follows from two facts.
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First, the falling factorials {k(k − 1) · · · (k − n + 1) : n ≥ 0} form a basis for polynomials in k, and kn =
n
S2 (n, j)k(k − 1) · · · (k − j + 1),
j=1
where S2 (n, j) are the Stirling numbers of the second kind, A008277 in [28], so in particular are nonnegative, and hence for p(k) with nonnegative coefficients we only need nonnegative coefficients of the falling factorials. Second, notice that for ≥ n, k(k − 1) · · · (k − n + 1)B k k≥
dn k B d Bn k≥ n− j n n dj d 1 n =B B j dBj 1 − B d B n− j = Bn
j=0
=
n j=0
j+1 n 1 ( − 1) · · · ( − n + j + 1) B + j , 1− B j
where all the coefficients are nonnegative. One case where there is a natural interpretation is QED with a linear number of generators, namely X1 = 1 +
k≥1
p(k)x k
X 12k+1 (1 − X 2 )2k (1 − X 3 )k
with X 2 and X 3 as before and with p(k) linear, which corresponds to counting with Cvitanovi´c’s gauge invariant sectors [6]. 3.3. Other systems. Johnson, Baker, and Willey [13] use gauge invariance to reduce the QED system to x k x(1 − X ) . = X=x 1− X 1− X −x k≥0
While amenable to the universal law analysis, this recursive equation can be solved exactly by the quadratic formula. We get √ 1 + 1 − 4x X= , 2 giving the Catalan numbers, A000108 in [28], as coefficients. The radius is 1/4 which is considerably larger than 2/27, showing how powerful gauge invariance is. Note that it is only the inverse photon propagator 1 − X which needs renormalization, and that it appears in the denominator.
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4. The Growth of γ1 After these considerations of the combinatorial side, we discuss analytic aspects. 4.1. The recursions. Consider the Dyson–Schwinger equation X (x) = I −
tk
x k pi (k)B+k,i (X Q k ),
k≥1 i=0
where Q = X r with r < 0 an integer, and pi (k) coefficients, not necessarily polynomial. We’ll use the notation Fk,i (ρ) for the Mellin transform of the integral kernel. Starting with r < 0 is justified in light of the last example of the previous section; we will generalize this soon enough. Specializing [19, (26)] to this case we get the recursion γk (x) =
1 γ1 (x)(1 + r x∂x )γk−1 (x) k
(23)
independently of the pi (k). Using the dot notation, γ · U = γk U k , of [19] we have x k pi (k)(1 + γ · ∂−ρ )−r k+1 (1 − e−Lρ )Fk,i (ρ)|ρ=0 . (24) γ ·L= k
i
Taking one L derivative and setting L to 0 we get γ1 = x k pi (k)(1 + γ · ∂−ρ )−r k+1 ρ Fk,i (ρ)|ρ=0 . k
(25)
i
Restricting to ρ Fk,i (ρ) = rk,i /(1 − ρ) allows us to write a much tidier recursion for γ1 . Taking two L derivatives of (24) and setting L = 0 we get 2γ2 = − x k pi (k)(1 + γ · ∂−ρ )−r k+1 ρ 2 Fk,i (ρ)|ρ=0 k
=−
i
k
= −γ1 +
x k pi (k)(1 + γ · ∂−ρ )−r k+1 ρ Fk,i (ρ)|ρ=0 +
i
k≥1
x k rk,i pi (k)
k
rk,i pi (k)x
k
from (25).
i
Thus we can from now on ignore the sum over i and let p(k) = i rk,i pi (k). Then from (23), γ1 = p(k)x k − 2γ2 = p(k)x k − γ1 (1 + r x∂x )γ1 k≥1
k≥1
giving Proposition 2. γ1,n = p(n) +
n−1 (−r j − 1)γ1, j γ1,n− j . j=1
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4.2. the radius. We see from the proposition that if p(k)x k is Gevrey-n, that Finding k n is x p(k)/(k!) converges, but not Gevrey-(n − 1), then γ1 is at best Gevrey-n. Of most interest for our applications is the case where only finitely many p(k) are nonzero but all are nonnegative and the case where p(k) = ck k! giving the Lipatov bound; so in both cases p(k)x k is Gevrey-1. Assume that p(k) ≥ 0 and k≥1
xk
p(k) = f (x) k!
has radius 0 < ρ ≤ ∞ and f (x) > 0 for |x| ≤ ρ. The above two cases are included as f (x) a polynomial and f (x) = cx/(1 − cx) respectively. In such circumstances γ1 is also Gevrey-1 and the radius is the minimum of ρ and −1/(ra1 ) (where we view −1/(ra1 ) as +∞ in the case a1 = 0) which we can see as follows.2 Let an = γ1,n /n!. Then a1 = γ1,1 = p(1) and −1 n−1 p(n) n + an = (−r j − 1) a j an− j n! j j=1
=
−1 n−1 p(n) 1 n (−r j − 1 − r (n − j) − 1) a j an− j + n! 2 j j=1
n−1 −1 n p(n) n = a j an− j . + −r − 1 n! 2 j j=1
To achieve an upper bound on the radius of convergence of last terms of the sum to get an ≥
an x n take the first and
n−2 p(n) −r a1 an−1 n! n
for n ≥ 2. So the radius of an x n is no more than the radius of the recursively defined series with equality above, say bn =
n−2 p(n) −r b1 bn−1 n! n
for n ≥ 2 with b1 = a1 . Immediately we see that if a1 = 0 the radius of Otherwise consider n(n − 1)bn = Equivalently with B(x) =
p(n)n(n − 1) − r (n − 1)(n − 2)b1 bn−1 . n! b(n)x n we get
B (x) = f (x) − r b1 xB (x). Solving for B (x), B (x) =
f (x) 1 + ra1 x
2 Similarly p(k)x k Gevrey-n for n > 1 leads to γ Gevrey-n. n
b(n)x n is ρ.
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which, since differentiation does not change the radius of a series, has radius min{ρ, −1/(ra1 )}, and thus so does B(x). For the lower bound on the radius we need a few preliminary results. First a simple combinatorial fact. Lemma 3. Given 0 < θ < 1,
1 n θ − j+1 ≥ n j j
for 1 ≤ j ≤ θ n and n ≥ 2. Proof. Fix n. Write j = λn, 0 < λ ≤ θ . Then 1 n θ − j+1 λ−λn+1 1 n n λn−1 ≥ . = = ≥ n j n λn (λn)λn λn j Second we need to understand the behaviour of an x n at the radius of convergence. Lemma 4. Using notation as above and with A(x) = an x n with radius of convergence ρa we have that lim supx→ρa A(x)(1 + xra1 )/ f (x) ≤ 1. Proof. Take any 0 < θ < 1/2. Using the previous lemma, n−1 −1 n p(n) n an = + −r − 1 a j an− j j n! 2 j=1 −1 n −1 n p(n) n − rn ≤ a j an− j − r a j an− j j j n! 2 1≤ j≤θn θn≤ j≤n−θn −1 n p(n) n j−1 −r ≤ jθ a j an− j − r a j an− j n! 2 θn θn≤ j≤n−θn
1≤ j≤θn
p(n) −r ≤ n!
jθ j−1 a j an− j
1≤ j≤θn
r − nθ θn 2
a j an− j ,
θn≤ j≤n−θn
so the coefficients of A(x) are bounded above by the coefficients of d 2 r x A (θ θ x). f (x) − xr A (θ x)A(x) − 2 dx Since all coefficients are nonnegative, for any 0 < x < ρa we have d r x A2 (θ θ x), A(x) ≤ f (x) − xr A (θ x)A(x) − 2 dx which is continuous in θ , so for fixed 0 < x < ρa we can let θ → 0 giving A(x) ≤ f (x) − xra1 A(x) so A(x)(1 + xra1 ) ≤1 f (x) for 0 < x < ρa . The result follows.
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Let ρa be the radius of an x n . If ρa = ρ then we are done, so suppose ρa < ρ. To get the lower bound on ρa it remains then to prove that ρa ≥ −1/(ra1 ) when a1 = 0 and to prove a contradiction in the case that a1 = 0. Take any > 0. Then there exists an N > 0 such that for n > N , p(n) 1 − ra1 an−1 − r a j an− j n! n−1 n−2
an ≤
j=2
p(n) − ra1 an−1 + a j an− j n! n−2
≤
j=2
p(n) − ra1 an−1 + n!
≤ Define ! an cn = p(n) n!
− r c1 cn−1 +
n−1
j=1 c j cn− j
n−1
a j an− j .
j=1
n−1 if an > p(n) j=1 c j cn− j n! − r c1 cn−1 + . otherwise (in particular when n > N )
In c1 = a1 . The radius of an x n is at least as large as the radius of C(x) = particular cn x n . Rewriting with generating series, C(x) = f (x) − ra1 xC(x) + C2 (x) + P (x), where P (x) is some polynomial. This equation can be solved by the quadratic formula. The discriminant is (ra1 x + 1)2 − 4( f (x) + P (x)).
(26)
By assumption we are interested in |x| < ρ, where (26) has no singularities, so the radius of C(x) is the closest root to 0 of (26); call it ρ . Consider = 0 giving (ra1 x + 1)2 which has −1/(ra1 ) as its closest root to 0 when a 1 = 0; call this value ρ0 . Then to get the lower bound on the radius of an x n it remains only to prove the following lemma. Lemma 5. With notation and assumptions as above if a1 = 0, lim→0 |ρ | ≥ ρ0 , while if a1 = 0 we have a contradiction. Proof. In view of Lemma 4 this is a short exercise in analysis. By construction the coefficient of x n in P (x) is bounded by an + r c1 cn−1 ≤ an + ra1 an−1 , since cn ≥ an for all n ≥ 1. So P (x) has coefficients which are nonnegative and bounded by those of A(x)(1 + r xa1 ). Thus by Lemma 4, the continuity of P (x) at ρa , and the assumption that ρa < ρ, we see that f (ρa ) + P (ρa ) ≤ f (ρa ) + lim inf x→ρa (A(x)(1 +r xa1 )) < ∞. By the nonnegativity of the coefficients of f and P we can choose M > 0 such that | f (x) + P (x)| < M independently of for |x| ≤ ρa . Suppose a1 = 0. Take any η > 0. Consider |x| ≤ ρa . Choose δ > 0 such that (ra1 x + 1)2 < δ implies |x − ρ0 | < η. Pick < δ/(4M). Then (ra1 ρ + 1)2 = 4( f (ρ ) + P (ρ )) < δ so |ρ − ρ0 | < η. Suppose on the other hand that a1 = 0. Take 0 < δ < 1. Then, since |ρ | ≤ ρa , we get that for < δ/(4M), 1 = 4( f (ρ ) + P (ρ )) < δ which is a contradiction.
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Taking the two bounds together we get the final result. Theorem 6. Assume k≥1 x k p(k)/k! has radius ρ. Then x n γ1,n /n! converges with radius of convergence min{ρ, −1/(r γ1,1 )}, where −1/(r γ1,1 ) is interpreted to mean +∞ in the case γ1,1 = 0. 4.3. Nonnegative systems. Now suppose we have a system of Dyson–Schwinger equations r
X r (x) = I −
tk
x k pir (k)B+k,i;r (X r Q k )
k≥1 i=0
for r ∈ R with R a finite set, pir (k) ≥ 0, and where Q= X r (x)sr r ∈R
with integers sr < 0 for all r ∈ R. Then as before from [19, (26)] we have ⎞ ⎛ 1 j r γkr (x) = ⎝γ1r (x) + s j γ1 (x)x∂x ⎠ γk−1 (x) k
(27)
j∈R
again independent of the pr (k). Assume that there is one insertion place, and so one variable ρ, and that the Mellin r (ρ) = r transform of the integral kernel is a geometric series ρ Fk,i k,i;r /(1 − ρ). Rewriting the system of Dyson–Schwinger equations with dot notation we have r γr · L = x k pir (k) (1 + γ j · ∂−ρ )−s j k+1 (1 − e−Lρ )Fk,i (ρ)|ρ=0 . (28) k
j∈R
i
As before we can find tidier recursions for the γ1r by comparing the first and second L derivatives of (28). We get r x k pir (k) (1 + γ j · ∂−ρ )−s j k+1 ρ Fk,i (ρ)|ρ=0 γ1r = k
and 2γ2r = −
j∈R
i
k
= −γ1r +
x k pir (k)
k≥1
r (1 + γ j · ∂−ρ )−s j k+1 ρ 2 Fk,i (ρ)|ρ=0
j∈R
i
rk,i;r pir (k)x k since ρ Fk,i (ρ) =
i
rk,i;r . 1−ρ
Thus letting p(k) = i rk,i;r pi (k) and using (27), j p k (k)x k − 2γ2r = pr (k)x k − γ1r (x)2 − s j γ1 (x)x∂x γ1r (x), γ1r = k≥1
k≥1
j∈R
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419
giving r = pr (n) + γ1,n
n−1 n−1 j r r r (−sr i − 1)γ1,i γ1,n−i + (−s j i)γ1,n−i γ1,i . j∈R i=1 j =r
i=1
To attack the growth of the γ1r we will again assume that pr (k) = f r (x) xk k! k≥1
has radius 0 < ρr ≤ ∞ and as before. r /n!. Then Let anr = γ1,n anr
f r (x)
> 0 for |x| ≤ ρr . We will proceed by similar bounds
−1 −1 n−1 n−1 n pr (n) j r r r n + = (−sr i − 1)ai an−i + (−s j i)an−i ai . i i n! j∈R i=1 j =r
i=1
Taking the last term in each sum we have ⎛ ⎞ r (n) n−2 r p j −⎝ an−1 . anr ≥ s j a1 ⎠ n! n j∈R
Let bnr be the series defined by b1r = a1r and equality in the above recursion. Let Br (x) = r n j bn x . Then as before if j∈R s j a1 = 0 the radius of B(x) is ρr and otherwise consider
⎛ Br (x) = f r (x) − ⎝
⎞ s j a1 ⎠ xBr (x) . j
j∈R
Solving for Br (x) we get that the radius of !
anr x n is at most " −1 min ρr , , j j∈R s j a1 j again interpreting the second possibility to be ∞ when j∈R s j a1 = 0. In the other direction take any > 0, then there exists an N > 0 such that for n > N we get ⎛ ⎞ n−1 r (n) p j j r r ⎝ ⎠ an ≤ − s j a1 an−1 + air an−i . n! j∈R
i=1 j∈R
Taking Cr (x) to be the series whose coefficients satisfy the above recursion with equality for when this gives a result ≥ anr and equal to anr otherwise, we get ⎛
Cr (x) = f r (x) − ⎝
j∈R
where
Pr
is a polynomial.
⎞
s j a1 ⎠ xCr (x) + j
j∈R
Cr (x)C j (x) + Pr (x),
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Summing over r we get a recursive equation for r ∈R Cr (x) of the same form as in the single equation case. Note that since each Cr is a series with nonnegative coefficients, there can be no cancellation of singularities and hence the radius of convergence of each Cr is at least that of the sum. The equivalent of Lemma 4 for this case follows from −1 n−1 pr (n) n j j r anr ≤ s j a1 an−1 + (−s j i)an−i air − n! i r ∈R
r ∈R
j∈R
r ∈R
r, j∈R i=1
pr (n) j r − ≤ s j a1 an−1 n! r ∈R j∈R r ∈R # $# $ n−2 −1 n r r + max(−s j ) i an−i ai j i i=2
r ∈R
r ∈R
pr (n) j r − s j a1 an−1 = n! r ∈R j∈R r ∈R # $# $ n −1 r r + max(−s j )n an−i ai j i 2≤i≤θn r ∈R r ∈R $# $ −1 # n n r r + max(−s j ) an−i ai j i 2 θn≤i≤n−θn r ∈R r ∈R r A(x), where Ar (x) = a (n)x n . for θ as in Lemma 4 with r ∈R Ar (x) in place of r Then continue the argument as in Lemma 4 with r ∈R f (x) in place of f (x) and max j (−s j ) in place of −r , and using the second term to get the correct linear part as θ → 0. Thus by the analysis of the single equation case we get a lower bound on the radius r n j of an x of mins∈R {ρs , −1/ j∈R s j a1 }. In particular if r ∈ R is such that ρr is r n j minimal we see that the radius of an x is exactly min{ρr , −1/ j∈R s j a1 }. s n r n Suppose the radius of an x was strictly greater than that of an x . Then we can find β > δ > 0 such that anr > β n > δ n > ans for n sufficiently large. Pick a k ≥ 1 such that aks > 0. Then δ n > ans ≥
−sr k!aks −sr k!aks r an−k β n−k > n · · · (n − k + 1) n · · · (n − k + 1)
so δk −sr aks
n−k δ k! > β n · · · (n − k + 1)
which is false for n sufficiently large, giving a contradiction. Thus all the ans x n have j the same radius minr ∈R {ρr , −1/ j∈R s j a1 }. n r From this we can conclude that each x γ1,n /n! also converges with radius j minr ∈R {ρr , −1/ j∈R s j γ1,1 }, where the second possibility is interpreted as ∞ when j j∈R s j γ1,1 = 0.
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4.4. Systems with some sr > 0. Let us relax the restriction that sr < 0 and that pr (n) ≥ 0. It is now difficult to make general statements concerning the radius of convergence of the anr x n . For example consider the system an1 =
−1 −1 n−1 n−1 n n p 1 (n) 1 1 2 (2 j − 1)a 1j an− − ja a , + j j n− j j j n! j=1
an2 =
p 2 (n) − n!
j=1
−1 −1 n−1 n n 2 2 2 1 ( j + 1)a j an− j + 2 ja j an− j , j j
n−1 j=1
j=1
so s1 = −2 and s2 = 1. Suppose also that p 2 (2) = 0, a11 = a12 , 1 p 2 (n) = −2(n − 1)!a12 an−1 .
Then a22 = 0 and inductively an2 = 0 for n ≥ 2 so the system degenerates to −1 n−1 n p 1 (n) n−1 1 1 1 1 + a1 an−1 , = (2 j − 1)a j an− j − j n! n j=1 ! a 1 if n = 1 2 an = 1 . 0 otherwise an1
We still have a free choice of p 1 (n), and hence control of the radius of the a 1 series. On the other hand the a 2 series trivially has infinite radius of convergence. Generally, finding a lower bound on the radii of the solution series, remains approachable by the preceding methods while control of the radii from above is no longer apparent. Precisely, for any > 0, −1 n−2 r (n)| | p n j r r r + − |an | ≤ s j a1 |an−1 | + |(−sr i − 1)||air ||an−i | n! i j∈R
+
n−2
j∈R i=1 j =r
≤
| pr (n)| n!
i=1
−1 j r n |(−s j i)||an−i ||ai | i
n−1 j r j + − s j a1 |an−1 |+ |air ||an−i |. j∈R i=1 j∈R
So, for a lower bound on the radius we may proceed as in the nonnegative using case r x n , and the absolute value of the coefficients and achieving that the radius of the a n r /n!, is at least hence that of x n γ1,n " ! −1 , min ρr , j r ∈R sjγ j∈R
1,1
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j where the second possibility is interpreted as ∞ when j∈R s j γ1,1 = 0. Note that this gives the lower bound on the radius of convergence as the minimum of the first instanton singularity (which one expects to be the radius for p(k)/k!) and the inverse of the first term in the β function of the theory. Furthermore, we emphasize that Ward identities typically allow a restriction to systems where all sr < 0. A more detailed discussion will be given in future work where the general approach described here will be discussed with regard to the specific details of the relevant renormalizable theories of interest. Finally, we note that the appearance of the inverse of the first term in the β-function makes sense: in the conformal case of a vanishing β-function we would not expect a constraint on the minimum of the radius to come from perturbation theory. 5. Applications of the Growth of γ1 While expectations for the growth of p(k)/k! in terms of instanton singularities are routine in the context of path integral estimates, the path integral is merely a successful heuristic to parametrize our lack of understanding of quantum field theory. Rigorous estimates for the growth of superficially convergent Green functions, and hence the p(k), can sometimes be obtained using constructive field theory, at least as bounds for the radius [22]. We emphasize that such results can be turned by our methods into similarly rigorous results for superficially divergent Green functions. A more complete discussion, dedicated to the renormalizable quantum field theories in four dimensions, will be given elsewhere. Appendix A. Other Theories Combinatorially with One Primitive per Loop Order For each of the following systems the solution series have coefficients satisfying the universal law. In the mixed φ 3 , φ 4 case there is one primitive per vertex per loop order. Unfortunately the full power of symmetry factors is not available in this simple combinatorial set-up, leading to the different variants. Computations were done using GiNaC [1]. Theory
φ3
φ4
System X 1 = 1 + X 1 F1
Radius
F1 = x X 12 F23 (1 + F1 ) x X 1 F22 2 F2 = 1 + F2 X 2
X2 =
3 X 1 = 1 + x 2 X 13 F24 F1 + x X 1 F22 2 F1 = x X 1 F22 (1 + F1 ) X 2 = x F2 + x 2 X 1 F23 F2 = 1 + F2 X 2
Smallest positive real root of 3581577x 4 −4443984x 3 + 2332368x 2 − 539136x + 32768. Numerically 0.09061681898407704 . . .
Root of a degree 10 polynomial. Numerically 0.12968592295019730 . . .
Recursive Estimates in QFT
Theory
423
System X 1 = 1 + x 2 X 13 F24 F1 + x X 1 F22 rest as in previous case
φ 4 variant
X 1 =1 + x X 1 X 2 F32 + x X 12 F33
Radius Root of a degree 9 polynomial. Numerically 0.13856076790723086 . . .
+ 2x F32 X 1 + X 2 X 12 F3 (Fa Fb + Fa + Fb + Fa x X 22 F33 + Fb x X 2 F32 + x 2 X 12 X 2 F35 + x X 12 F33 + x X 2 F32 ) X 2 =1 + X 2 (Fa Fb + Fa + Fb + Fa x X 12 F33 + Fb x X 2 F32 mixed
φ3,
+ x 2 X 12 X 2 F35 )
φ4
+ X 13 F3 (Fa Fb + Fa + Fb
0.02145…
+ Fa x X 12 F33 + Fb x X 2 F32 + x 2 X 12 X 2 F35 + x X 12 F33 + x X 2 F32 ) 3 + x X 2 F32 2 Fa =x 2 X 22 F34 + x X 2 F32 Fa Fb =x 2 X 14 F36 + x X 12 F33 Fb X 3 =x X 1 F32 + x F3 + x 2 X 2 F33 F3 =1 + X 3 F3
Appendix B. Notation
Notation R A a L
G r (a, L) γkr
Explanation amplitudes which need renormalization, used as an index set all amplitudes the coupling constant as one of the two variables on which the Green functions depend ln(−Q 2 /µ2 ), the other variable on which the Green functions depend, where Q 2 is a Euclidean kinematical variable and µ2 is a subtraction point Green function indexed by the amplitude r k th leading log term of the Green function indexed by the amplitude r
First seen 1.2 1.2 1.2 1.2
1.2 1.2
424
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Notation Q p;r B+
Explanation First seen (combinatorial) invariant charge 1.2 insertion into the primitive p which has external 1.2 leg structure r p;r 1.2 B+k;r | p|=k B+ where | p| is the loop number of p, and p runs over primitive graphs B+k,i;r an alternate way of indexing over all primitives 1.2 at k loops X r (a) sum of all graphs with external leg structure r , 1.2 as a series in the coupling constant a I empty graph; unit of the Hopf algebra 1.2 ± (in a DSE) the positive sign is for the vertex case, and the 1.2 negative sign for the propagator case φR renormalized Feynman rules, used to convert 1.2 combinatorial DSEs to analytic DSEs End an endomorphism of the Hopf algebra; used to 1.3 construct Hopf ideals ◦i graph insertion at index i 1.3 , γ graphs 1.3 , Sweedler notation for the coproduct 1.3 S the antipode of the Hopf algebra 1.3 p a primitive, may be a single graph or a sum 1.4 ρ the argument of the Mellin transform which 1.4 marks the insertion place; in later sections radii of convergence M( p)(ρ) or M p (ρ) Mellin transform associated to the primitive p 1.4 defined in a symmetric renormalization scheme Intp integrand associated to p 1.4 g coupling constant in φ44 1.4 res p residue of the primitive p; constant term of 1.5 ρ(1 − ρ)M p (ρ) as a series in ρ O Landau big O notation 1.5 By combinatorial Dyson–Schwinger equations (combinatorial DSEs) we mean systems of equations of the form (4). Notation t (n) T(x) x E(x, y) F, T , y1 , . . . , ym
1 , . . . , m
Explanation the number of unlabelled rooted trees on n vertices ordinary generating series for t (n) indeterminant for forming generating series bivariate power series used to rewrite the recursive equation for T(x) series variables in recursive systems right-hand sides of a system of equations; later the whole system components of
First seen 2.1 2.1 2.1 2.1 2.2 2.2 2.2
Bold capital letters stand for generating series with the corresponding lowercase letters for coefficients. The universal law is the asymptotics Cρ n n −3/2 for the n th coefficient
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425
of a power series, where C is any constant and ρ is the radius of convergence of the series. Notation X1 X2 X3 F1 F2 , F3 x1 , f 1 , …, x1 , … x1 , f 1 , … p(k) B S2 X Notation X B+k,i
tk pi (k) Q r Fk,i (ρ) x γk γ ·U rk,i p(k) γ1,n f (x) ρ
Explanation generating function for the counting function for the vertex in QED with one primitive per loop order generating function for the counting function for the fermion propagator in QED with one primitive per loop order generating function for the counting function for the photon propagator in QED with one primitive per loop order variable for the geometric series appearing on the right-hand side of the expression for X 1 inverse propagators power series associated to the corresponding capital letters, generally thought of as partial solutions coming from successive iterations differences of the x1 , . . . and x1 , … polynomial in k counting the number of primitives we consider at k loops in the setup of Subsect. 3.2 notation for some expression in the variables of a recursive system Stirling numbers of the second kind photon propagator after the Baker, Johnson, and Willey analysis
Explanation X r (from Subsect. 1.2) in the case with only one r insertion into a primitive at k loops, with i an index running over primitives that is B+k,i;r (from Subsect. 1.2) in the case with only one r upper bound for the index over primitives at k loops coefficient giving the contribution of primitive i at k loops (combinatorial) invariant charge (written Q in 1.2) power of X in Q, assumed to be a negative integer Mellin transform associated with the primitive B+k,i (I) indeterminant for the series, now playing the role of the coupling constant k th leading log term of φ R (X )(x, L), that is γkr (from 1.2) in the case with only one r γk U k residue of ρ Fk,i (ρ), especially after reducing to geometric series, as permitted by Subsect. 1.5 i rk,i pi (k), the overall contribution of all primitives at k loops coefficient of x n in γ1 k k k≥1 x p(k)/k! when k≥1 x p(k) is Gevrey-1 the radius of convergence of f (x)
First seen 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.2 3.2 3.2 3.3 First seen 4.1 4.1
4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.2 4.2
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Notation a1 an A(x) bn B(x) ρa cn C(x) P ρ
Explanation γ1,1 γ1,n /n! generating function for an a particular lower bound for an generating function for bn radius of convergence of A(x) a particular upper bound for an implicitly depending on an > 0 generating function for cn implicitly depending on a polynomial appearing in the recursion for C(x) the radius of convergence of C(x)
First seen 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2
The notation in 4.3 is as in 1.2 with x in place of a and with the addition of coefficients pir (k) for the contribution of the i th primitive at k loops, and as in 4.1 and 4.2 with the additional index r . Notation tkr pir (k) r (ρ) Fk,i rk,i;r
pr (k) r γ1,n f r (x) ρr anr Ar (x) bnr Br (x) cnr
Cr (x) Pr an1 , an2
Explanation upper bound for the index over primitives at k loops with external leg structure r coefficient giving the contribution of primitive i at k loops with external leg structure r Mellin transform associated with the primitive B+k,i;r (I) r (ρ), especially after reducing to geometresidue of ρ Fk,i ric series, as permitted by Subsect. 1.5 r p (k), the overall contribution of all primitives i k,i;r i at k loops coefficient of x n in γ1r k r k r k≥1 x p (k)/k! when k≥1 x p (k) is Gevrey-1 r the radius of convergence of f (x) r /n! γ1,n generating function for anr a particular lower bound for anr generating function for bnr a particular upper bound for anr implicitly depending on an > 0 generating function for cnr implicitly depending on a polynomial appearing in the recursion for Cr (x) coefficients for an example system
First seen 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.4
Acknowledgement. It is a pleasure to thank Spencer Bloch and Ivan Todorov for useful discussions, and to thank the anonymous referees and the editor for helpful comments on the constructive approach.
References 1. Bauer, C., Frink, A., Kreckel, R.: Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language. J. Symb. Comp. 33, 1–12 (2002) 2. Bell, J., Burris, S., Yeats, K.: Counting Rooted Trees: The Universal Law t (n) ∼ Cρ −n n −3/2 . Elec. J. Combin. 13, R63 (2006)
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3. Bierenbaum, I., Kreimer, D., Weinzierl, S.: The next to leading ladder approximation. Phys. Lett. B 646, 129–133 (2007) 4. Canfield, E.R.: Remarks on an asymptotic method in combinatorics. J. Combin. Theory Ser. A 37(3), 348–352 (1984) 5. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The beta-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216, 215 (2001) 6. Cvitanovi´c, P.: Asymptotic estimates and gauge invariance. Nucl. Phys. B 127, 176–188 (1977) 7. David, F., Feldman, J., Rivasseau, V.: On the large order behaviour of φ44 . Commun. Math. Phys. 116, 215 (1988) 8. Drmota, M.: Combinatorics and asymptotics on trees. Cubo Journal 6, 2 (2004) 9. Drmota, M.: Systems of functional equations. Random Struct. Alg. 10, 103–124 (1997) 10. Ebrahimi-Fard, K., Gracia-Bondia, J.M., Patras, F.: A Lie theoretic approach to renormalization. Commun. Math. Phys. 276(2), 519–549 (2007) 11. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Online draft available at http://algo.inria.fr/flajolet/ Publications/books.html, to appear Camb. Univ. press, 2008 12. Harary, F., Robinson, R.W., Schwenk, A.J.: Twenty-step algorithm for determining the asymptotic number of trees on various species. J. Austral. Math. Soc. (Ser. A) 20(4), 483–503 (1975); Corrigendum: J. Austral. Math. Soc. (Ser. A) 41(3), 325 (1986) 13. Johnson, K., Baker, M., Willey, R.: Self-Energy of the Electron. Phys. Rev. 136(4B), B1111–B1119 (1964) 14. Kreimer, D.: On overlapping divergences. Commun. Math. Phys. 204, 669 (1999) 15. Kreimer, D.: Unique factorization in perturbative QFT. Nucl. Phys. Proc. Suppl. 116, 392 (2003); Étude for linear Dyson Schwinger Equations, IHES preprint 2006, http://www.ihes.fr/PREPRINTS/2006/P/ P-06-23.pdf, 2006 16. Kreimer, D.: Structures in Feynman graphs: Hopf algebras and symmetries. Proc. Symp. Pure Math. 73, 43 (2005) 17. Kreimer, D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757 (2006) 18. Kreimer, D.: Dyson Schwinger equations: From Hopf algebras to number theory. In: Renormalization and Universality in Math. Phys., Fields Inst. Commun. 50, I. Binder, D. Kreimer (eds.). Providence, RI: Amer. Math. Soc., 2007, pp. 225–248 19. Kreimer, D., Yeats, K.: An Étude in non-linear Dyson-Schwinger Equations. Nucl. Phys. B Proc. Suppl. 160, 116–121 (2006) 20. Lalley, S.P.: Finite range random walk on free groups and homogeneous trees. Ann. Probab. 21(4), 2087–2130 (1993) 21. Mack, G., Todorov, I.T.: Conformal-invariant Green functions without ultraviolet divergences. Phys. Rev. D 6, 1764 (1973) 22. Magnen, J., Nicolo, F., Rivasseau, V., Seneor, R.: A Lipatov bound for φ 4 in four-dimensions Euclidean field theory. Commun. Math. Phys. 108, 257 (1987) 23. Meir, A., Moon, J.W.: On an asymptotic method in enumeration. J. Combin. Theory Ser. A 51(1), 77–89 (1989) Erratum: J. Combin. Theory Ser. A 52(1), 163 (1989) 24. Odlyzko, A.M.: Asymptotic Enumeration Methods. Handbook of combinatorics, Vols. 1, 2. Amsterdam: Elsevier, 1995, pp. 1063–1229 25. Otter, R.: The number of trees. Ann. Math. 49, 583–599 (1948) 26. Pólya, G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math. 68, 145–254 (1937) 27. Pólya, G., Read, R.C.: Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. New York: Springer-Verlag, 1987 28. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, published on-line at www.research.att. com/~njas/sequences/, 2006 29. van Suijlekom, W.D.: The Hopf algebra of Feynman graphs in QED. Lett. Math. Phys. 77, 265 (2006) 30. van Suijlekom, W.D.: Renormalization of gauge fields: A Hopf algebra approach. Commun. Math. Phys. 276, 773–798 (2007) http://arxiv.org/list/hep-th/0610137 31. Woods, A.R.: Coloring rules for finite trees, probability of monadic second order sentences. Random Struct. Alg. 10, 453–485 (1997) 32. Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. 4th ed., Oxford: Oxford Univ. Press, 2003 Communicated by J.Z. Imbrie
Commun. Math. Phys. 279, 429–453 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0432-6
Communications in
Mathematical Physics
Asymptotic Lower Bounds for a Class of Schrödinger Equations Luis Vega1 , Nicola Visciglia2 1 Universidad del Pais Vasco, Apdo. 64, 48080 Bilbao, Spain. E-mail: [email protected] 2 Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy.
E-mail: [email protected] Received: 8 February 2007 / Accepted: 20 August 2007 Published online: 22 February 2008 – © Springer-Verlag 2008
Abstract: We shall study the following initial value problem: i∂t u − u + V (x)u = 0, (t, x) ∈ R × Rn ,
(0.1)
u(0) = f, where V (x) is a real short–range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to (0.1) that generalizes the ones proved in [12] and [21] in the free case. As a by–product of these identities we deduce some uniqueness results for solutions to (0.1), and a lower bound for the so-called local smoothing which becomes an identity in a precise asymptotic sense. 1. Introduction We shall study the following initial value problem: i∂t u − u + V (x)u = 0, (t, x) ∈ R × Rn ,
(1.1)
u(0) = f under suitable assumptions on V (x). Let us recall that if V (x) ∈ L ∞ (Rn ) then the operator L 2 (Rn ) ⊃ H 2 (Rn ) u → −u + V (x)u ∈ L 2 (Rn ) is self–adjoint (see [14] for the proof of this fact). In particular one can apply the classical Stone theorem in order to deduce the existence of a unique solution u(t, x) ∈ Ct (L 2 (Rn )) (here and in the sequel we shall denote by Ct (X ) the space of continuous functions of one variable valued in the Banach space X ) to the Cauchy problem (1.1), provided that f ∈ L 2 (Rn ).
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Hereafter we shall denote by eitV f the unique solution to (1.1) at time t ∈ R. Let us recall that the following conservation law is satisfied: eitV f L 2 (Rn ) ≡ f L 2 (Rn )
∀t ∈ R.
(1.2)
Notice that this identity implies that the operators eitV define a family of isometries on L 2 (Rn ). Moreover, as a by–product of the Stone theorem one can deduce the following implication: f ∈ H 2 (Rn ) ⇒ eit f ∈ Ct1 (H 2 (Rn )) (1.3) (here we have denoted by Ct1 (X ) the space of functions of one variable valued in the Banach space X with a continuous derivative). It is also well–known that the following conservation law holds: (|∇x u(t, x)|2 + V (x)|u(t, x)|2 ) d x Rn = (|∇x f (x)|2 + V (x)| f (x)|2 ) d x ∀t ∈ R, (1.4) Rn
and in particular |∇x u(t, x)|2 d x ≤ C (|∇x f (x)|2 + | f (x)|2 ) d x ∀t ∈ R, Rn
(1.5)
Rn
provided that V (x) ≥ 0 and V (x) ∈ L ∞ (Rn ). In the sequel we shall assume that V (x) satisfies the following decay assumption: 0 ≤ V (x) ≤
C ∀x ∈ Rn , (1 + |x|)1+
(1.6)
where , C > 0. We shall also assume either that V (x) is decreasing in the radial variable, i.e. ∂|x| V ≤ 0,
(1.7)
or that lim |x|∂|x| V (x) = 0.
|x|→∞
(1.8)
We shall specify in every theorem which kind of assumptions we assume on the derivative of V . In order to state our results let us introduce the perturbed Sobolev spaces H˙ Vs (Rn ), whose norm is defined as follows: √ s f H˙ s (Rn ) ≡ − + V u 2 n ∀s ≥ 0. (1.9) V
L (R )
Our first result contains a family of identities satisfied by solutions to (1.1). Let us underline that these identities represent a generalization of the identities proved in the free case, i.e. V (x) ≡ 0, in [12] and [21]. In fact in [22] a similar family of identities has been proved for the solutions to the conformally invariant nonlinear Schrödinger equation. Next we shall denote by D 2 ψ the Hessian matrix of the function ψ.
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Theorem 1.1. Let u(t, x) be the solution to (1.1), where n ≥ 1, f ∈ C0∞ (Rn ). Assume moreover that V (x) satisfies (1.6) and one of the conditions (1.7) or (1.8). Let ψ be a radially symmetric function such that the following limit exists: lim ∂|x| ψ = ψ (∞) ∈ [0, ∞)
(1.10)
|x|→∞
and moreover ∇ψ, D 2 ψ, 2 ψ ∈ L ∞ (Rn ). Then the following identity holds: T |u|2 2 2 lim ∇x u¯ D ψ∇x u − ( ψ + 4∂|x| V ∂|x| ψ) d xdt T →∞ −T Rn 4 = ψ (∞) f 2
1
H˙ V2 (Rn )
.
(1.11)
As a by–product of the argument involved in the proof of Theorem 1.1 we can 1 construct a natural Banach space 2 (whose definition will be given below) that is invariant along the flow associated to (1.1). Moreover we shall deduce one uniqueness 1 1 result for solutions to (1.1) provided that f ∈ 2 . In order to define the space 2 we first introduce the weighted Lebesgue space L 2|x| (Rn ) defined as the completion of C0∞ (Rn ) with respect to the following norm: 2 |x|| f (x)|2 d x. (1.12) f L 2 ≡ |x|
Rn
1 2
The Banach space is defined as follows: 1
2 ≡ H˙ V2 ∩ L 2|x| 1
(1.13)
and can be endowed with the norm f 2
1
2
≡ f 2
1
H˙ V2
+ f 2L 2 . |x|
We can state our second result. 1
Theorem 1.2. Let u(t, x) be the solution to (1.1), where n ≥ 2, f ∈ 2 and V (x) satisfies the same assumptions as in Theorem 1.1. Then we have the following a–priori estimate: u(t)2 1 ≤ f 2L 2 + C(1 + |t|) f 2 1 ∀t ∈ R, (1.14) 2
|x|
H˙ V2
1
for a suitable C > 0.In particular for every t ∈ R we have that eitV f ∈ 2 provided 1 that f ∈ 2 .Moreover |x| |u(t, x)|2 d x = 2 f 2 1 . (1.15) lim t→±∞ Rn |t| H˙ 2 V
In particular if
lim
t→±∞ Rn
then u ≡ 0.
|x| |u(t, x)|2 d x = 0 |t|
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Remark 1.1. From a technical point of view we assume n ≥ 2 in Theorem 1.2, since the proof of Lemma 4.1 (that in turn is needed in the proof of Theorem 1.2) does not work in dimension n = 1. Remark 1.2. Along the proof of Theorems 1.1 and 1.2, we shall make extensive use of the existence and completeness of the wave operator under the assumptions (1.6) and (1.8) on V (x) (see Sect. 2). Remark 1.3. In order to prove Theorems 1.1 and 1.2 we shall need some intermediate results, whose proof in some cases could be deduced by avoiding the use of the existence and completeness of the wave operator. For instance Lemma 2.1 in Sect. 2 follows from the general RAGE theory (see [15]). However we have proposed a proof that involves the existence and completeness of the wave operator in order to make the paper selfcontained as much as possible. In Appendix 7 we shall make some connections between the classical RAGE theorem and our results. Next we shall deduce some direct consequences from the identity (1.11). In particular we shall show how it allows us to prove a lower bound to the classical local smoothing estimate. For a proof of the local smoothing estimate in the free case see [5,18,20] and also their extensions in [3,7,17]. In particular in [17] the issue of the best constants involved in the local smoothing estimate is considered. First we shall present our results in dimension n ≥ 4. Theorem 1.3. Let u(t, x) be the solution to (1.1), where n ≥ 4 and V (x) satisfies (1.6) 1 and (1.7). Then the following a–priori estimate is satisfied for every f ∈ H˙ 2 (Rn ): V
f 2
1 H˙ V2 (Rn )
≤ sup R>0
1 R
∞
−∞ |x|
|∇x u|2 d xdt ≤ C f 2
1
H˙ V2 (Rn )
,
(1.16)
where C > 0 is a suitable constant independent of f . In the next result we give a better lower bound than the one in (1.16). Theorem 1.4. Let u(t, x) be the solution to (1.1) where n ≥ 4, f and V (x) are as in Theorem 1.3. Then we have: 1 ∞ |∂|x| u|2 d xdt = f 2 1 . (1.17) lim R→∞ R −∞ |x|
Therefore if lim
R→∞
1 R
∞
−∞ |x|
|∂|x| u|2 d xdt = 0
then u ≡ 0. Remark 1.4. Notice that if you choose formally ψ ≡ |x| in (1.11) and if you work in dimension n ≥ 4, then the general identity (1.11) becomes ∞ |∇τ u|2 (n − 1)(n − 3) |u|2 2 d xdt = f 2 1 + − ∂ V |u| (1.18) |x| 3 n |x| 4 |x| −∞ R H˙ 2 (Rn ) V
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433
(here ∇τ u denotes the angular part of the gradient of u). Let us underline that in the case V (x) ≡ 0 the previous identity has been proved in [12] with a different approach. Moreover (1.18) represents a precise version of the result in [9] and [10], where (1.18) is stated as an inequality and not as an identity. Notice also that the function ψ ≡ |x| does not satisfy all the assumptions required in Theorem 1.1. However in order to make precise the argument involved in the proof of (1.18), it is sufficient to choose in (1.11) the test function ψ to be equal to 2 + |x|2 and to get the limit in the corresponding identity as → 0. An alternative way to prove properly (1.18) it is to combine the proof of (1.11) with the argument used in [10] (in fact in [10] the integration by parts technique that we use in the proof of (1.11) is completely justified also when ψ ≡ |x|). Remark 1.5. Let us point out that Theorems 1.3 and 1.4 are stated in dimension n ≥ 4, while Theorem 1.1 is stated in any dimension n ≥ 1. The main reason is that in order to take advantage of the identity (1.11) we shall choose the test function ψ(x) in a suitable way. As it will be clear in the sequel, we shall be able to make such a good choice only in dimension n ≥ 4. In dimension n = 3 we are able to prove the following result. Theorem 1.5. Let u(t, x) be the solution to (1.1), where n = 3, V (x) satisfies (1.6) and 1 (1.7). Then the following a priori estimate is satisfied for every f ∈ H˙ V2 (R3 ): 1 ∞ 1 2 2 2 |∇x u| + 2 |u| d xdt ≤ C f 2 1 ≤ sup , (1.19) c f 1 R R>0 R −∞ |x|
V
where c, C > 0 are constants independent of f . Theorem 1.6. Let u(t, x) be the solution to (1.1) where n = 3, f and V (x) are as in Theorem 1.5. Then we have: 1 1 ∞ 1 2 2 lim inf |∂|x| u| d xdt + 2 |u| d xdt ≥ f 2 1 . (1.20) R→∞ R −∞ |x|
Therefore if 1 lim inf R→∞ R
∞ −∞
1 2 2 |∂|x| u| + 2 |u| d xdt = 0 R |x|
then u ≡ 0. Remark 1.6. Starting with (1.11) it is possible to show the following version of the identity (1.18) in dimension n = 3: ∞ ∞ |∇τ u|2 3 2 d xdt + π − ∂|x| V |u| |u(0, t)|2 dt = f 2 1 . (1.21) |x| 2 −∞ R3 −∞ H˙ 2 (R3 ) V
Exactly as for (1.18) the previous identity has been proved previously in [12] in the free case and it represents a precise version of a result proved in [9] and [10], where (1.21) is stated as an inequality. Indeed the proof of (1.21) follows formally by choosing the function ψ ≡ |x| in (1.11). However in dimension n = 3 the function ψ ≡ |x| is very singular since its bilaplacian is a multiple of the Dirac delta and hence in order to justify all the computations we have to argue as in dimension n ≥ 4 (see Remark 1.4).
434
L. Vega, N. Visciglia
The paper is organized as follows. In Sect. 2 we shall prove some asymptotic properties of solutions to (1.1). Sections 3 and 4 will be devoted to the proof of Theorems 1.1 and 1.2. The proof of Theorems 1.3 and 1.4 will be given in Sect. 5, while Theorems 1.5 and 1.6 will be proved in Sect. 6. Finally in Appendix 7 we shall discuss some connections between our results and the classical RAGE theorem. Next we shall fix some notations. Notations. For every potential V (x) ≥ 0 and for every real number s ≥ 0 we shall denote by H˙ Vs the perturbed Sobolev space whose norm is defined in (1.9). In particular when V ≡ 0 these spaces reduce to the standard Sobolev spaces H˙ s whose norm is defined as follows: 2 | fˆ(ξ )|2 |ξ |2s dξ, f H˙ s ≡ Rn
where fˆ(ξ ) ≡
Rn
e−2π ixξ f (x)d x.
In some cases we shall also write F( f ) ≡ fˆ. 1
The spaces L 2|x| and 2 are respectively the ones defined in (1.12) and (1.13),. For any 1 ≤ p, q ≤ ∞, p
p
q
L x and L t L x denote the Banach spaces
L p (Rn ) and L p (R; L q (Rn )). We shall also write p
p
p
L t L x ≡ L t,x . itV the group associated to (1.1) via the For every V (x) ∈ L ∞ x we shall denote by e Stone theorem. Given any couple of Banach spaces X and Y , we shall denote by L(X, Y ) the space of linear and continuous functionals between X and Y . Given a space–time dependent function w(t, x) we shall denote by w(t0 ) the trace of w at fixed time t ≡ t0 , in case that
it is well-defined.
We shall denote by ... d x, ... dt and ... d xdt the integral of suitable functions with respect to space, time, and space–time variables respectively. When it is not better specified we shall denote by ∇v the gradient of any time–dependent function v(t, x) with respect to the space variables. Moreover ∇τ and ∂|x| shall denote respectively the angular gradient and the radial derivative. If ψ ∈ C 2 (Rn ), then D 2 ψ will represent the hessian matrix of ψ. Given a set A ⊂ Rn we denote by χ A its characteristic function. We shall use the function x ≡ 1 + |x|2 .
Asymptotic Lower Bounds
435
2. Wave Operators and Asymptotic Behaviour of Solutions Let us recall that if V (x) satisfies (1.6), then the wave operators W± are well-defined and complete (see [1,6,15] and [16]). More precisely for every f ∈ L 2x there exist two functions W± ( f ) ∈ L 2x uniquely defined and such that lim u(t) − eit W± ( f ) L 2x = 0,
(2.1)
t→±∞
where u ∈ Ct (L 2x ) denotes the unique solution to (1.1) with initial data f and eit represents the propagator at time t associated to the free Schrödinger equation, i.e. (1.1) with V (x) ≡ 0 (for a proof of (2.1) see [1]). In the sequel we shall need the following asymptotic description of the free waves, whose proof can be found in [14]. Proposition 2.1. Assume f ∈ L 2x and n ≥ 1, then: 2 x ±i |x| it 4t e e f − e∓inπ/4 lim fˆ ± n/2 t→±∞ (4π t) 4π t
= 0.
(2.2)
L 2x
From now on we shall denote by W± the wave operators defined above and by F the Fourier transform. Next we shall state one of the basic results of this paper. Proposition 2.2. Let u(t, x) be the solution to (1.1), where n ≥ 1, f ∈ C0∞ (Rn ) and V (x) satisfies (1.6) and one of the two conditions (1.7) or (1.8). Assume that ψ is radially symmetric such that the following limit exists: lim ∂|x| ψ = ψ (∞) ∈ [0, ∞).
(2.3)
|x|→∞
Then
lim Im
t→±∞
u(t)∇u(t) ¯ · ∇ψ d x
= ∓2π ψ (∞)
|x||g± (x)|2 d x,
where g± = F[W± f ]. Moreover the following identity holds: f 2 1 = 2π |x||g± (x)|2 d x. H˙ V2
(2.4)
(2.5)
We shall need the following lemma which is a consequence of the RAGE theorem. In order to be self–contained, we have decided to include a proof of it which is based on the existence and completeness of the wave operators W± introduced at the beginning of the section. Lemma 2.1. Let u(t, x) be the solution to (1.1), where n ≥ 1, f ∈ L 2x and V (x) satisfies (1.6), then 1 t lim W (x)|u|2 d xds = 0, (2.6) t→±∞ t 0 where W ∈ L ∞ x is such that lim W (x) = 0.
|x|→∞
436
L. Vega, N. Visciglia
Proof. We shall prove the result only as t → ∞, since the case t → −∞ can be treated similarly. Notice the following identity: t (2.7) W (x)|u|2 d xds = I (t) + I I (t) ∀t > 0, 0
where I (t) =
t 0
1 x 2 W (x) |u| − d xds gˆ (4π s)n 4π s
2
and I I (t) =
t
x 2 d xds , W (x) gˆ 4π s (4π s)n
0
where g ≡ W+ ( f ). Due to (2.2) and to the assumption W ∈ L ∞ x we can deduce I (t) ≤ W L ∞ x where
t
h(s)ds and lim h(s) = 0,
|u(s)|2 −
h(s) =
(2.8)
s→∞
0
1 x 2 g ˆ d x. (4π s)n 4π s
On the other hand we have I I (t) =
t
2 W (4π sx)|g(x)| ˆ dsd x
0
that due to the dominated convergence theorem and to the decay assumption made on W (x) implies t I I (t) = H (s)ds and lim H (s) = 0, (2.9) s→0
0
where
H (s) =
2 W (4π sx)|g(x)| ˆ d x.
By combining (2.7) with (2.8) and (2.9) it is easy to deduce (2.6). Remark 2.1. Notice that in order to deduce (2.6) we have shown that lim W (x)|u(t)|2 d x = 0 t→±∞
(2.10)
which is stronger than (2.6). In fact (2.6) could be proved for a much larger class of potentials V (x) by using the general RAGE theorem. In Appendix 7 we shall show how to deduce (2.10) by using as a starting point (2.6).
Asymptotic Lower Bounds
437
Lemma 2.2. Let u(t, x) be the solution to (1.1) where n ≥ 1, f ∈ C0∞ (Rn ) and V (x) satisfies (1.6) and one of the two conditions (1.7) or (1.8). Then we have: x lim u(t) − 2i∇u(t) = 0. (2.11) t→±∞ t L 2x Proof. We prove (2.11) only for t → ∞, since the case t → −∞ is similar. First case. V (x) satisfies (1.7). The following identity is well–known (see [4]): xu(t) − 2it∇u(t)2L 2 + 4t 2 V (x)|u(t)|2 d x x
=
t
|x|2 | f (x)|2 d x +
sθ (s) ds ∀t ∈ R,
(2.12)
0
where
θ (s) = 8
1 V (x) + |x|∂|x| V (x) |u(s)|2 d x. 2
By combining the sign assumption done on V (x) and ∂|x| V with (2.12) we get:
t
2 2 x V (x)|u(s)|2 d x ds |x| | f (x)|2 d x 8 0 . (2.13) + u(t) − 2i∇u(t) 2 ≤ t t2 t Lx By combining this inequality with (2.6) we get (2.11). Second case. V (x) satisfies (1.8). In this case we can use (2.12) as above and we can deduce
t
2 2 x W (x)|u(s)|2 d x ds |x| | f (x)|2 d x 0 , (2.14) + u(t) − 2i∇u(t) 2 ≤ t t2 t Lx where
1 W (x) = 8 V (x) + |x|∂|x| V (x) . 2
Notice that due to (1.6) and (1.8) we have that lim|x|→∞ W (x) = 0, then we can use (2.6) in order to deduce (2.11). Remark 2.2. Let us underline that in [6] it is proved the existence of a sequence {tn }n∈N such that: x (2.15) lim tn = ∞ and lim u(tn ) − 2i∇u(tn ) 2 = 0. n→∞ n→∞ tn Lx Notice that (2.15) is a weaker version of (2.11), however in [6] it is a basic tool in order to prove the completeness of the wave operators, provided that V (x) satisfies the assumptions (1.6) and (1.8). Remark 2.3. Notice that we obtain (2.11) using Lemma 2.1 which we prove using the completeness of the wave operators. In Appendix 7 we shall give a proof of (2.11) that does not involve a–priori the completeness of the wave operator. Moreover we shall show that (2.11) is still satisfied for a class of potentials V (x) more general than the ones that satisfy the decay assumptions (1.6) and (1.8).
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Proof of Proposition 2.2. As usual we treat only the case t → ∞, the case t → −∞ can be treated similarly. Along the proof we shall use the function g(x) defined as follows: g ≡ F[W+ ( f )]. Let us introduce the following identity: Im u(t)∇u(t) ¯ · ∇ψ d x = I (t, R) + I I (t, R) where
∀t ∈ R, R > 0,
I (t, R) = Im
and
|x|>4π Rt
I I (t, R) = Im
(2.16)
|x|<4π Rt
u(t)∇u(t) ¯ · ∇ψd x u(t)∇u(t) ¯ · ∇ψ d x .
Estimate for I (t, R). Notice that the Cauchy–Schwartz inequality implies: 1 2 2 ≤ C∇ψ L ∞ f H 1 u(t)∇u(t) ¯ · ∇ψ d x |u(t)| d x , x x |x|>4π Rt |x|>4π Rt (2.17) where we have used (1.5). On the other hand due to (2.2) and due to the definition of g(x) we get: 1 x 2 g lim |u(t)|2 − dx = 0 t→∞ |x|>4π Rt (4π t)n 4π t
and then lim
t→∞ |x|>4π Rt
|u(t)|2 d x =
|x|>R
|g(x)|2 d x.
(2.18)
Since g ∈ L 2x we can combine (2.17) with (2.18) in order to deduce that ∀ > 0 ∃R() > 0 s. t. lim sup |I (t, R)| < ∀R > R().
(2.19)
t→∞
Estimate for I I (t, R). Notice that (1.2) and (2.11) imply: i 2 lim u(t)∇u(t) ¯ · ∇ψ d x + |x|∂|x| ψ|u(t)| d x = 0. (2.20) t→∞ 2t |x|<4π Rt |x|<4π Rt Moreover we have the following identities: dx |x|∂|x| ψ|u(t)|2 t |x|<4π Rt 1 x 2 d x 2 |x|∂|x| ψ |u(t)| − = g (4π t)n 4π t t |x|<4π Rt x 2 d x
1 |x| (∂|x| ψ − ψ (∞) g + n+1 (4π )n |x|<4π Rt 4π t t x 2 d x 1 + ψ (∞) |x| g n+1 . (4π )n 4π t t |x|<4π Rt
(2.21)
Asymptotic Lower Bounds
439
Notice that the following estimate is trivial: 1 x 2 d x 2 |x|∂|x| ψ |u(t)| − g (4π t)n 4π t t |x|<4π Rt x 2 1 2 ≤ 4π R∂|x| ψ L ∞ |u(t)| − (4π t)n g 4π t d x → 0 x
as t → ∞, (2.22)
where at the last step we have combined (2.1) with (2.2). Moreover the change of variable formula implies:
x 2 d x 1 g |x| (∂ ψ − ψ (∞) |x| (4π )n |x|<4π Rt 4π t t n+1 ≤ 4π R ∂|x| ψ(4π t x) − ψ (∞) |g(x)|2 d x, that in conjunction with the dominated convergence theorem and with assumption (2.3) implies:
x 2 d x 1 lim |x| (∂ ψ − ψ (∞) (2.23) n+1 = 0. g |x| t→∞ (4π )n |x|<4π Rt 4π t t Due again to the change of variable formula we get x 2 d x ψ (∞) |x| = 4π ψ (∞) |x||g(x)|2 d x, n+1 g (4π )n |x|<4π Rt 4π t t |x|
lim
t→∞
|x|
By combining (2.16) with (2.19) and (2.25) we get (2.4) at least in the case t → ∞. The other case is similar. Proof of (2.5). Recall that W+ : L 2x → L 2x is an isometry and moreover W+ ◦ f (−) = f (− + V ) ◦ W+ . By combining these facts with the definition of g, i.e. g ≡ F[W+ f ], we get: 1 1 (−) 4 ◦ W+ f 2L 2 |x||g(x)|2 d x = W+ f 2 1 = x 2π H˙ x2 1 1 1 1 1 = W+ ◦ (−V ) 4 f 2L 2 = (−V ) 4 f 2L 2 = f 2 1 . x x 2π 2π 2π H˙ 2 V
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3. Proof of Theorem 1.1 and Some Consequences Proof of Theorem 1.1. Following [2] we multiply (1.1) by the quantity 1 ∇ u¯ · ∇ψ + u¯ ψ, 2
(3.1)
and we integrate on the strip (−T, T ) × Rn . In this way we get the following family of identities: T 2 2 2 |u| 2 ∇ u¯ D ψ∇u − ψ − ∂|x| V ∂|x| ψ|u| d xdt 4 −T 1 = − Im u(±T ¯ )∇u(±T ) · ∇ψ d x, (3.2) 2 ± (for more details on this computation see [2] and [21]). Indeed all the integration by parts involved in the proof of (3.2) can be completely justified by a density argument due to (1.3). Notice that the identity (1.11) follows by combining (2.4), (2.5) and (3.2). Next we shall exploit (1.11) in order to deduce some a–priori estimates satisfied by the solutions to (1.1). 1
Lemma 3.1. Assume that u(t, x) solves (1.1), where n ≥ 4, f ∈ H˙ V2 , V (x) satisfies (1.6) and (1.7), then 1 |u|2 d xdt < ∞ (3.3) x3 and
|∂|x| V ||u|2 d xdt < ∞.
(3.4)
Proof. Choose in (1.11) the function ψ(x) ≡ |x| that is clearly a radially symmetric and convex function. Moreover we have ∂|x| ψ ≡ 1 and 2 (|x|) = −
(n − 1)(n − 3) ∀x ∈ Rn |x|3
where n ≥ 4.
(3.5)
Hence by choosing ψ(x) ≡ |x| in (1.11), it is easy to deduce that (n − 1)(n − 3) |u|2 d xdt − ∂|x| V |u|2 d xdt ≤ f 2 1 , 4 |x|3 H˙ 2
(3.6)
V
where C > 0 is a suitable constant, and hence we get easily (3.3) and (3.4).
Next we state a version of (3.4) in dimension n = 3. 1
Lemma 3.2. Assume that u(t, x) is solution to (1.1) with n = 3, f ∈ H˙ V2 , V (x) satisfies (1.6) and (1.7), then: |∂|x| V ||u|2 d xdt < ∞.
(3.7)
Asymptotic Lower Bounds
441
Proof. The proof of (3.7) is identical to the proof of (3.4). Notice that by choosing in (1.11) the test function ψ(x) ≡ |x| and arguing as in the proof of Lemma 3.1 we get 3 , π |u(0, t)|2 dt − ∂|x| V |u|2 d xdt ≤ f 2 1 2 H˙ 2 (R3 ) V
where we have used the property − 2 (|x|) = 6π δ0
on R3 .
(3.8)
We can now deduce the following 1
Proposition 3.1. Assume that u(t, x) is a solution to (1.1) with n ≥ 4, f ∈ H˙ V2 , V (x) satisfies (1.6) and (1.7), then: lim |2 φ R ||u|2 d xdt = 0, (3.9) R→∞
where φ is a radially symmetric function such that |2 φ| ≤ and φ R = Rφ
x R .
C ∀x ∈ Rn x3
Proof. Notice that (3.10) trivially implies |2 φ R ||u|2 d xdt ≤ C
(3.10)
|u|2 d xdt → 0 + |x|3
R3
as R → ∞,
where we have combined the dominated convergence theorem with (3.3).
1
Proposition 3.2. Assume that u(t, x) is solution to (1.1) with n ≥ 3, f ∈ H˙ V2 , V (x) satisfies (1.6) and (1.7), then: lim (3.11) |∂|x| V ||∂|x| φ R ||u|2 d xdt = 0, R→∞
where φ is a radially symmetric function such that
and φ R = Rφ
x R .
∂|x| φ(0) = 0, |∂|x| φ| ≤ C
∀x ∈ Rn
Proof. It follows from the following identity: |∂|x| V ||∂|x| φ R ||u|2 d xdt x 2 = |∂|x| V | ∂|x| φ |u| d xdt → 0 R
as R → ∞,
(3.12)
where at the last step we have combined the dominated convergence theorem with (3.4) (or with (3.7) in the specific case n = 3) and with the assumption ∂|x| φ(0) = 0.
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4. Proof of Theorem 1.2 We shall need the following lemma, whose proof in dimension n ≥ 3 follows an argument in [2]. Lemma 4.1. Assume that n ≥ 2 and h ∈ L 2x ∩ H˙ x1 , then we have the following inequality:
x ¯ h(x) ∇h(x) · d x ≤ Ch2 1 , |x| H˙ x2
(4.1)
where C > 0 is a constant that depends on n. Proof. We introduce the quadratic form x d x. a( f, g) = f¯(x) ∇g(x) · |x| Notice that the Cauchy–Schwartz inequality implies: |a( f, g)| ≤ f L 2x g H˙ 1 . x
(4.2)
Next we split the proof in two cases. First case: n ≥ 3. By assuming f, g regular enough we can use integration by parts in order to deduce: x 1 ¯ ¯ a( f, g) = − g(x)∇ f (x) · d x − (n − 1) f (x)g(x) d x. |x| |x| By combining the Hardy inequality and the Cauchy–Schwartz inequality with the previous identity we get: |a( f, g)| ≤ C f H˙ 1 g L 2x . (4.3) x
Notice that (4.1) will follow by interpolation from (4.2) and (4.3) and by choosing f = g = h. Second case: n = 2. In this case we are not allowed to use the Hardy inequality in order to deduce (4.3). Hence we shall look for a substitute of this inequality in dimension n = 2. Due to the Parseval identity we get
1 1 xj xj d x = |D| 2 f¯(x) |D|− 2 ∂ j g(x) d x, f¯(x)∂ j g(x) |x| |x| where |D| ≡ deduce
√
− and ∂ j ≡
∂ ∂x j
, and then due to the Cauchy–Schwartz inequality we
1 xj xj |D| 21 g L 2 ¯ d x ≤ |D| 2 f (x) x |x| |x| L 2x 1 xj ¯ = |D| 2 f (x) |x| 2 g H˙ 21 . f¯(x)∂ j g(x)
Lx
x
(4.4)
Asymptotic Lower Bounds
443
On the other hand we have the following chain of inequalities: |D| 21 f¯(x) x j ≤ |D| 21 f¯(x) x j − f¯|D| 21 x j |x| L 2x |x| |x| L 2x 1 1 xj xj ¯ ¯ + f |D| 2 |x| 2 ≤ f H˙ 21 + f |D| 2 |x| 2 , Lx
(4.5)
Lx
x
where we have used the inequality |D|s ( f g) − f (|D|s g) L 2x ≤ g L ∞ |D|s f L 2x x (for a proof see [8]). On the other hand the following inequality can be proved: |D| 21 x j ≤ C , |x| |x| 21 (for a proof see for example [13]) and due to the Sobolev embedding it implies 1 1 x j f¯|D| 2 ≤C f¯ ≤ C f 1 . |x| 21 2 |x| L 2x H˙ x2 Lx By combining this inequality with (4.5) we get: |D| 21 f¯(x) x j ≤ C f 1 , |x| L 2x H˙ x2 that in turn with (4.4) gives |a( f, g)| ≤ f The proof is complete.
1
H˙ x2
g
1
H˙ x2
.
Lemma 4.2. Let u(t, x) be the solution to (1.1), where n ≥ 1 and V (x) satisfies the same assumptions as in Theorem 1.1, then the following a –priori estimates are satisfied: u(t)2
1
H˙ x2
≤ u(t)2
1
H˙ V2
≤ f 2
1
H˙ V2
∀t ∈ R.
(4.6)
Moreover we have u(t)2L 2 + u(t)2H˙ 1 ≤ C( f 2L 2 + f 2H˙ 1 ) ∀t ∈ R. x
x
x
x
Proof. Due to (1.2) and (1.4) we get: u(t) L 2x = f L 2x
and
u(t) H˙ 1 = f H˙ 1 . V
V
Hence the r.h.s. in (4.6) follows by interpolation (see [19]). Next notice that by hypothesis V (x) ≥ 0 and then 2 h H˙ 1 ≤ (|∇h(x)|2 + V (x)|h(x)|2 ) d x = h2H˙ 1 . x
V
Hence the l.h.s. in (4.6) will follow again from an interpolation argument. The proof of (4.7)follows from (1.2) and (1.5).
(4.7)
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Proof of Theorem 1.2. Due to the r.h.s. in (4.6) it is easy to verify that (1.14) will follow from the following inequality: (4.8) |x||u(t)|2 d x ≤ |x|| f (x)|2 d x + C|t| f 2 1 . H˙ V2
In the sequel we shall prove (4.8) only for t > 0 (the proof is similar in the case t < 0). Proof of (4.8) for t > 0. Since u(t, x) solves (1.1) we have the following identity: d 2 ¯ + u(t)∂t u(t)) ¯ dx |x||u(t)| d x = |x|(∂t u(t)u(t) dt x = 2Im |x|u(t)u(t) ¯ d x = −2Im u(t)∇u(t) ¯ · d x, (4.9) |x| where we have used integration by parts. In order to simplify the notation we introduce the function x dx G(t) ≡ −2Im u(t)∇u(t) ¯ · |x|
(4.10)
that due to (4.1) and (4.6) satisfies: |G(t)| ≤ C f 2
1
H˙ V2
∀t ∈ R
(4.11)
(notice that in fact in order to justify this computation we have to work by density and first we assume f ∈ L 2x ∩ H˙ x1 . In this way due to (4.7) we have that u(t) ∈ L 2x ∩ H˙ x1 and hence we are in position to apply (4.1) with h = u(t)). By combining this inequality with (4.9) we get |x||u(t)|2 d x ≤ |x||u(0)|2 d x + Ct f 2 1 H˙ V2
=
|x|| f (x)|2 d x + Ct f 2
1
H˙ V2
∀t > 0.
Proof of (1.15). We shall prove (1.15) only in the case t → ∞ (the case t → −∞ can be treated in a similar way). Notice that (4.9) implies:
t G(s) ds |x| |x| 2 2 |u(t)| d x = | f (x)| d x + 0 ∀t > 0. (4.12) t t t On the other hand (2.4) and (2.5) (where we choose ψ ≡ |x|) imply lim G(t) = ±2 f 2 1 .
t→±∞
(4.13)
H˙ V2
By combining (4.11), (4.12) and (4.13) we finally get |x| |u(t)|2 d x = 2 f 2 1 . lim t→∞ t H˙ 2 V
(4.14)
Asymptotic Lower Bounds
445
5. Proof of Theorems 1.3 and 1.4 We split the proof in two steps. Proof of r.h.s. in (1.16). It is sufficient to consider the identity (1.11) where the generic function ψ is replaced with the family of rescaled functions Rφ Rx and φ(x) ≡ x. In fact notice that the function x is convex, increasing and moreover −2 (x) ≥ 0 ∀x ∈ Rn , n ≥ 3 as a direct computation shows. Let us point–out that the l.h.s. in (1.16) follows from Theorem 1.4. Proof of Theorem 1.4. Next we shall make use of the following identity: 2 ∇ u¯ D 2 ψ∇u = ∂|x| ψ|∂|x| u|2 +
∂|x| ψ |∇τ u|2 , |x|
(5.1)
where ψ is a radially symmetric function and u is a generic function. First of all let us notice that if we choose in the identity (1.11) the function ψ ≡ |x| then we get: |∇τ u|2 d xdt < ∞, |x|>1 |x| where we have used (3.5) and (5.1). In particular we get lim R→∞
|x|>R
|∇τ u|2 d x = 0. |x|
(5.2)
For any k ∈ N we fix a function h k (r ) ∈ C0∞ (R; [0, 1]) such that: h k (r ) = 1 ∀r ∈ R s.t. |r | < 1, h k (r ) = 0 ∀r ∈ R s.t. |r | > h k (r ) = h k (−r ) ∀r ∈ R.
k+1 , k (5.3)
C ∞ (R):
Let us introduce the functions ψk (r ), Hk (r ) ∈ r (r − s)h k (s)ds and Hk (r ) = ψk (r ) = 0
r
h k (s)ds.
Notice that ψk (r )
=
h k (r ), ψk (r )
(5.4)
0
∞
= Hk (r )∀r ∈ R and lim ∂r ψk (r ) = r →∞
h k (s)ds.
(5.5)
0
Moreover an elementary computation shows that: 2 ψk (x) =
C ∀x ∈ Rn s.t. |x| ≥ 2 and n ≥ 4, |x|3
(5.6)
where 2 is the bilaplacian operator. Thus the functions φ = ψk satisfy the assumptions of Propositions 3.1 and 3.2. In the sequel we shall need the rescaled functions x ∀x ∈ Rn , k ∈ N and R > 0, (5.7) ψk,R (x) = Rψk R
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where ψk is defined in (5.4). Notice that by combining the general identity (5.1) with (1.11), where we choose ψ = ψk,R defined in (5.7), and recalling (5.5) we get: ∂|x| ψk,R 2 ∂|x| |∇τ u|2 ψk,R |∂|x| u|2 + |x| 1 2 − ψk,R + ∂|x| V ∂|x| ψk,R |u|2 d xdt 4 ∞ = h k (s)ds f 2 1 ∀k ∈ N, R > 0.
(5.8)
H˙ V2
0
By using (3.9), (3.11) and (5.2) we get: 1 2 |∇τ u|2 ∂|x| ψk,R − ψk,R + ∂|x| V ∂|x| ψk,R |u|2 d xdt = 0 R→∞ |x| 4 lim
for every k ∈ N. We can combine this fact with (5.8) in order to deduce:
2 ψk,R |∂|x| u|2 d xdt = ∂|x|
lim
R→∞
∞
h k (s)ds f 2
1
H˙ V2
0
∀k ∈ N.
(5.9)
On the other hand, due to the properties of h k (see (5.3)), we get 2 |∂|x| u|2 d xdt ≤ ∂|x| ψk,R |∂|x| u|2 dtd x BR x 1 1 |∂|x| u|2 dtd x ≤ = |∂|x| u|2 d xdt hk R R R |x|< k+1 R k
1 R
that due to (5.9) implies: 1 lim sup R→∞ R ≤
|∂|x| u| d xdt ≤
∞
2
|x|
1 k+1 lim inf k R→∞ R
|x|
h k (s)ds f 2
|∂|x| u|2 d xdt
1
H˙ V2
0
∀k ∈ N.
Since k ∈ N is arbitrary and since the following identity is trivially satisfied: lim
k→∞ 0
∞
h k (s)ds = 1,
we can deduce easily (1.17) by using (5.10). The proof is complete.
(5.10)
Asymptotic Lower Bounds
447
6. Proof of Theorems 1.5 and 1.6 The proofs are similar in principle to the ones of Theorems 1.3 and 1.4, except that in dimension n = 3 we cannot use (3.3) and hence Lemma 3.1, that have been proved only in dimension n ≥ 4. However for completeness we shall give in the sequel the details of the proof in dimension n = 3. We split the proof in two steps. Proof of r.h.s. in (1.19). It is sufficient to consider the identity (1.11) where the generic function ψ is replaced with the family of rescaled functions Rφ Rx and φ(x) ≡ x. In fact notice that the function x is convex, increasing and moreover −2 (x) ≥ 0
∀x ∈ R3
as a direct computation shows. Let us point out that the l.h.s. in (1.19) follows from Theorem 1.6. Proof of Theorem 1.6. Following the proof of (5.2) we get: lim
R→∞
|x|>R
|∇τ u|2 d xdt = 0, |x|
(6.1)
(in this case of course we have to use (3.8) instead of (3.5)). We fix a function h(r ) ∈ C0∞ (R; [0, 1]) such that: h(r ) = 1 ∀r ∈ R s.t. |r | <
1 , h(r ) = 0 ∀r ∈ R s.t. |r | > 1, 2
h(r ) = h(−r )
∀r ∈ R.
Let us introduce the functions ψ(r ), H (r ) ∈ C ∞ (R): r ψ(r ) = (r − s)h(s)ds and H (r ) = 0
r
h(s)ds.
(6.2)
0
Notice that
∞
ψ (r ) = h(r ), ψ (r ) = H (r )∀r ∈ R and lim ∂r ψ(r ) = r →∞
h(s)ds.
(6.3)
0
Moreover an elementary computation shows that: 2 ψ(x) = 0 ∀x ∈ R3 s.t. |x| ≥ 1,
(6.4)
where 2 is the bilaplacian operator (recall that we are working in dimension n = 3) and ψ is defined in (6.2). Notice also that the function ψ given above satisfies the assumptions of Proposition 3.2. In the sequel we shall need the rescaled functions x ψ R (x) = Rψ ∀x ∈ R3 and R > 0. (6.5) R
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By combining the identity (5.1) with (1.11), where we choose ψ = ψ R , and recalling (6.3) we get: ∂|x| ψ R 2 ∂|x| |∇τ u|2 ψ R |∂|x| u|2 + |x| ∞ 1 2 ψ R + ∂|x| V ∂|x| ψ R |u|2 d xdt = − h(s)ds f 2 1 ∀R > 0. (6.6) 4 0 H˙ 2 V
By using (3.11) and (6.1) we get: |∇τ u|2 ∂|x| ψ R − ∂|x| V ∂|x| ψ R |u|2 d xdt = 0. lim R→∞ |x| We can combine this fact with (6.6) in order to deduce: ∞ 1 2 ∂|x| lim ψ R |∂|x| u|2 − 2 ψ R |u|2 d xdt = h(s)ds f 2 1 . (6.7) R→∞ 4 0 H˙ 2 V
On the other hand, due to the definition of the functions h(|x|) and ψ R (|x|) (see (6.2) and (6.5)), we get 1 2 2 2 2 ∂|x| ψ R |∂|x| u| − ψ R |u| d xdt 4 x 1 1 2 2 − x 2 h |∂|x| u| − |u| d xdt ( ψ) ≤ R R 4R 3 R (here (2 ψ)− represents the negative part of 2 ψ), that in turn due to (6.4) and again to the property of the support of h(|x|) implies: 1 2 2 2 2 ∂|x| ψ R |∂|x| u| − ψ R |u| d xdt 4 1 C 2 2 |∂|x| u| + 2 |u| d xdt. ≤ R R |x|
|x|→∞
and
lim |x||∂|x| V (x)| = 0.
|x|→∞
(7.1)
We point out that this appendix does not contain any essential novelty, since the content of Proposition 7.1 in part follows from the results in [11] (see Remark 7.1). However, our aim in this appendix is to show in a very simple way how to deduce a stronger version of the usual RAGE theorem by using as a starting point the classical RAGE theorem itself (see Proposition 7.1) and by avoiding the use of the general Mourre theorem in [11].
Asymptotic Lower Bounds
449
It is well–known that to every bounded potential V (x), we can associate a corresponding splitting of the Hilbert space L 2x as the direct sum of the projection onto the continuous spectrum and onto the pure point spectrum, that shall be denoted respectively as L 2c and L 2pp . The space L 2c can be splitted in turn as the direct sum of the projection onto the singular spectrum and onto the absolutely continuous spectrum, that shall be denoted 2 . respectively as L 2s and L ac Hereafter we shall make use of the following version of the RAGE theorem: if u ∈ Ct (L 2 ) solves (1.1) where f ∈ L 2c , 1 T |u(t)|2 d xdt = 0 ∀R > 0. then lim T →∞ T −T |x|
(7.2)
Actually the classical RAGE theorem is much more general than (7.2) (see [15]), however (7.2) will be enough for our purposes. One of the aims of this appendix is to show how the RAGE theorem implies (2.11), under the decay assumptions on V (x) given in (7.1). Let us recall that (2.11) is a stronger version of a result obtained in [6] (see Remark 2.2). Another point in this appendix is to show how it is possible to prove a decay of the solution u pointwisely in time, by using as a starting point the decay given in (7.2). Next we state the main result of the section. Proposition 7.1. Let V (x) be a function that satisfies (7.1). Assume that the point spectrum of − + V is the empty set. Then we have: x lim u(t) − 2i∇u(t) = 0, (7.3) t→±∞ t L 2x where u(t) ≡ e
itV
and
f with
|x|2 | f |2 d x
lim
t→±∞
W (x)|u(t)|2 d x = 0,
(7.4)
where u(t) ≡ eitV f with f ∈ L 2x and W (x) ∈ L ∞ x satisfies lim W (x) = 0.
|x|→∞
In particular we have: eitV f 0 as t → ±∞ ∀ f ∈ L 2x .
(7.5)
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Remark 7.1. Looking at the proof of the RAGE theorem, which is based on Wiener’s result about the decay of the Fourier transform of a measure, it is not difficult to show that (7.4) is trivially satisfied provided that the projection on the absolutely continuous 2 coincides with L 2 , or equivalently when the projection on the singular spectrum L ac spectrum L 2s is trivial. Actually this fact has been proved in [11], provided that the potential V (x) satisfies (7.1). However we have decided to present our own proof of Proposition 7.1 due to its simplicity. Proof of Proposition 7.1. For simplicity we shall treat only the limit as t → ∞ (the case t → −∞ can be studied in a similar way). Proof of (7.3). We have that L 2x ≡ L 2c . Hence we can use the RAGE theorem (see (7.2)) in order to deduce that 1 T |u(t)|2 d xdt = 0, (7.6) lim T →∞ T −T |x|
t
2 2 U (x)|u(s)|2 d x ds |x| | f (x)| d x 8 0 , ≤ + t2 t where
1 U (x) ≡ 8 V (x) + |x|∂|x| V (x) . 2
Due to (7.6) and to the decay assumption (7.1), the previous estimate can be rewritten as follows: 2 x u(t) − 2i∇u(t) 2 t Lx
2 2 |x| | f (x)| d x ≤ + ϕ(t), (7.7) t2 where lim ϕ(t) = 0.
t→∞
Hence the proof of (7.3) follows easily. Proof of (7.4). Due to (1.2) and to a density argument, it is sufficient to prove (7.4) under the assumption f ∈ C0∞ (Rn ).
Asymptotic Lower Bounds
451
Next notice that (7.7) can be written as
2 2 2 |x| | f (x)|2 d x i |x| + ϕ(t), 4 ∇(e 4t u(t)) ≤ t2 L 2x
(7.8)
where lim ϕ(t) = 0.
t→∞
Since we are assuming f ∈ C0∞ (Rn ) we have that the R.H.S. in (7.8) goes to zero as t → ∞. Assume n > 2. By the Sobolev embedding ∗ H˙ 1 (Rn ) ⊂ L 2 (Rn )
we get
where 2∗ ≡
2n , n−2
lim u(t) L 2∗ = 0.
(7.9)
t→∞
This estimate in conjunction with the Hölder inequality implies: |u(t)|2 d x ≤ C R 2 u(t)2L 2∗ ∀R > 0, |x|
that in turn can be combined with (7.9) in order to give |u(t)|2 d x = 0 ∀R > 0. lim t→∞ |x|
(7.10)
Notice that (7.4) follows by (7.10), (1.2) and the decay assumption of W (x) at infinity. Finally notice that (7.5) follows by combining (1.2) with (7.4). For n = 1 and n = 2 a similar argument works using Gagliardo–Nirenberg inequalities instead of the Sobolev embedding. The aim of the next proposition is to show that, despite (7.4), in general there is a–priori no rate of decay for the L 2 localized norm of the solutions u to (1.1). Proposition 7.2. Assume that V (x) ∈ L ∞ x is any bounded potential (possibly V (x) ≡ 0). Let R > 0 be a fixed positive number and γ ∈ C([0, ∞); R) be any function such that lim γ (t) = ∞.
t→∞
Then there exists g ∈ L 2x (that depends on R, V (x) and γ (t))such that n , |u(tn )|2 d x > γ (tn ) |x|
(7.11)
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Proof. We claim the following fact: U R,V (t)L(L 2x ,L 2x ) ≡ 1 ∀t ∈ R,
where
(7.12)
U R,V (t) : L 2x g → χ{|x|
and u(t) denotes the unique solution to the Cauchy problem (7.11). Notice that due to (7.12) we get: lim γ (t)U R,V (t)L(L 2x ,L 2x ) = ∞,
t→∞
and in particular due to the Banach-Steinhaus theorem the operators γ (t)U R,V (t) cannot be pointwisely bounded or in an equivalent way there exists at least one g ∈ L 2x such that sup γ (t)U R,V (t)g L 2x = ∞.
(7.13)
[0,∞)
On the other hand the function t → γ (t)U R,V (t)g L 2x is bounded on bounded sets of [0, ∞) and hence (7.13) implies that lim sup γ (t)U R,V (t)g L 2x = ∞, t→∞
which completes the proof. Next we shall prove (7.12). Let us fix any function f R ∈ L 2x such that f R L 2x ≡ 1
and
supp f R ⊂ {|x| < R}.
Notice that we have U R,V (t)e−itV f R ≡ χ{|x|
∀t ∈ R,
(7.14)
where we have used the group property of eitV and the assumption done on the support of f R . In particular due to (1.2) and (7.14) we get: U R,V (t)L(L 2x ,L 2x ) ≥ U R,V (t)(e−itV f R ) L 2x ≡ f R L 2x ≡ 1. On the other hand (1.2) implies trivially the opposite inequality U R,V (t)L(L 2x ,L 2x ) ≤ 1, and hence (7.12) is proved.
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References 1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2, 151–218 (1975) 2. Barcelo, J.A., Ruiz, A., Vega, L.: Some dispersive estimates for Schrödinger equations with repulsive potentials. J. Funct. Anal. 236(1), 1–24 (2006) 3. Ben–Artzi, M., Klainerman, S.: Decay and regularity for the Schrödinger equation. J. Anal. Math., 58, 25–37 (1992) 4. Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, New York: New York University Courant Institute of Mathematical Sciences, 2003 5. Constantin, P., Saut, J.C.: Local smoothing properties of Schrödinger equations. Indiana Univ. Math. J. 38(3), 791–810 (1989) 6. Enss, V.: Asymptotic completeness for quantum mechanical potential scattering, I short range potentials. Commun. Math. Phys., 61, 258–291 (1978) 7. Kato, T., Yajima, K.: Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1(4), 481–496 (1989) 8. Kenig, C., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math., 46(4), 527–620 (1993) 9. Lin, J.E., Strauss, W.A.: Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Funct. Anal. 30(2), 245–263 (1978) 10. Morawetz, C.: Time decay for the nonlinear Klein–Gordon equation. Proc. Roy. Soc. London A, 306, 291–296 (1968) 11. Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78, 391–408 (1981) 12. Perthame, B., Lions, P.L.: Lemmes de moments, de moyenne et de dispersion. C. R. Acad. Sci. Paris Sér. I Math. 314(11), 801–806 (1992) 13. Plamenevski, B.A.: Algebry Psevdodifferentsialnykh Operatorov, in Russian, “Algebras of Pseudodifferential Operators”, Moscow: Nauka, 1986 14. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier analysis, self-adjointness. New York: Academic Press [Harcourt Brace Jovanovich Publishers] 1975 15. Reed, M., Simon, B.: Methods of Modern Mathematical Physics III. Scattering theory. New York: Academic Press [Harcourt Brace Jovanovich Publishers] 1979 16. Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers](1978) 17. Simon, B.: Best constants in some operator smoothness estimates. J. Funct. Anal. 107(1), 66–71 (1992) 18. Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math J. 55(3), 699–715 (1987) 19. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Heidelberg: Johann Ambrosius Barth, 1995 20. Vega, L.: Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc., 102(4), 874–878 (1988) 21. Vega, L., Visciglia, N.: On the local smoothing for the Schrödinger equation. Proc. Amer. Math. Soc. 135, 119–128 (2007) 22. Vega, L., Visciglia, N.: On the local smoothing for a class of conformally invariant Schrödinger equations. Indiana Univ Math. J. 56(5), 2265–2304 (2007) Communicated by B. Simon
Commun. Math. Phys. 279, 455–486 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0418-4
Communications in
Mathematical Physics
Unbounded Violation of Tripartite Bell Inequalities D. Pérez-García1 , M. M. Wolf2 , C. Palazuelos1 , I. Villanueva1 , M. Junge3 1 Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040,
Madrid, Spain. E-mail: [email protected]
2 Max Planck Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, D-85748, Germany 3 Department of Mathematics, University of Illinois at Urbana-Champaign,
Illinois 61801-2975, USA Received: 22 February 2007 / Accepted: 29 November 2007 Published online: 12 February 2008 – © Springer-Verlag 2008
Abstract: We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck’s constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras. 1. Introduction Bell inequalities characterize the boundary of correlations achievable within classical probability theory under the assumption that Nature is local [76]. Originally, Bell [10] proposed the inequalities, which now bear his name, in order to put the intuition of Einstein, Podolski and Rosen [27] on logically firm ground, thus proving that an apparently metaphysical dispute could be resolved experimentally. Nowadays, the verification of the violation of Bell inequalities has become experimental routine [8,60,74](albeit there is a remaining desire for a unified loophole-free test). On the theoretical side—in the realm of quantum information theory—they became indispensable tools for understanding entanglement [6,68,72,77] and its applications in cryptography [2–4,28,65]
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and communication complexity [13]. In fact, the insight gained from the violation of Bell inequalities enables us even to consider theories beyond quantum mechanics [47,78] and allows us to replace quantum mechanics by the violation of some Bell inequality in the set of trusted assumptions for secure cryptographic protocols [2,3,9,48,65]. Most of our present knowledge on Bell inequalities and their violation within quantum mechanics is based on the paradigmatic Clauser-Horne-Shimony-Holt (CHSH) inequality [19]. It bounds the correlations obtained in a setup where two observers can measure two dichotomic observables each. In fact, it is the only non-trivial constraint on the polytope of classically reachable correlations in this case [29]. If we allow for more observables (measurement settings) per site or more sites (parties) the picture is much less complete. Whereas for two dichotomic observables per site the complete set of multipartite ‘full-correlation inequalities’ and their maximal violations within quantum mechanics is still known [79,83], the case of more than two settings is, despite considerable effort [41,52,82], largely unexplored. One reason is, naturally, that finding all possible Bell inequalities is a computationally hard task [7,59] and that in addition the violating quantum systems become vastly more complicated as the number of sites and dimensions increases. Another reason could be the lack of appropriate mathematics to tackle the problem. Thus far, researchers have primarily used algebraic and combinatorial techniques. In this work, following the lines already implicit in [70], we will relate tripartite Bell inequalities with two powerful theories of mathematical analysis: operator spaces and tensor norms. We will give new mathematical results inside these theories and show how to apply them to provide a deeper insight into the understanding of Bell inequalities, by proving some new and intriguing results on their maximal quantum violation. It is interesting to note here that operator spaces have recently also led to other applications in Quantum Information Theory [24]. We will start by outlining the main result and some of its implications within Quantum Information Theory. Sect. 3 will then recall basic notions from the theory of operator spaces and tensor norms and connect the language of Bell inequalities with the mathematical theories. In Sect. 4 we will prove that the violation remains bounded for GHZ states. Finally, Sect. 5 provides the proof for the main theorem. As a conclusion, in Sect. 6, we will present some questions that this work leaves open. Maybe the most interesting one is the search for an explicit form for the quantum states leading to unbounded violations. We will see how this question is intimately related to an old problem of Varopoulos [71] in the theory of Banach algebras. 2. Main Result and Implications We begin by specifying the framework. For the convenience of the non-specialist reader we will give first a brief introduction to Bell Inequalities. For further information we refer the reader to [76]. Bell inequalities can be dated back to the famous criticism of Quantum Mechanics due to Einstein, Podolski and Rosen [27]. This criticism was made under their belief that on a fundamental level Nature was described by a local hidden variable (LHV) model, i.e., that it is classical (realistic or deterministic) and local (or non-signaling). The latter essentially means that no information can travel faster than a maximal speed (e.g. of light) which implies in particular that the probability distribution for the outcomes of some experiment made by Alice cannot depend on what another (spatially separated) physicist Bob does in his lab. Otherwise, by choosing one or the other experiment, Bob
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could influence instantly Alice’s results and hence transmit information at any speed. On the other hand, saying that Nature is classical or deterministic means that the randomness in the outcomes that is observed in the experiments comes from our ignorance of Nature, instead of being an intrinsic property of it (as Quantum Mechanics postulates). That is, Nature can stay in different configurations s with some probability p(s) (s is usually called a hidden variable). But once it is in a fixed configuration s, then any experiment has deterministic outputs. We note that there are non-deterministic LHV models as well, but they can all be cast into deterministic models [76]. Let us formalize this a bit more. Consider correlation experiments where each of N spatially separated observers (Alice, Bob, Charlie,…) can measure M different observables with outcomes ±1: {Ai1 }iM1 =1 for Alice, {Bi2 }iM2 =1 for Bob and so on. By repeating the experiment several times, for each possible configuration of the observables (Alice measuring with the apparatus Ai1 , Bob with the apparatus Bi2 , …), they can obtain a good approximation of the expected value of the product of the outcomes of such configuration Ai1 Bi2 Ci3 · · · . If Nature is described by an LHV model, then Ai1 Bi2 Ci3 · · · = Ai1 Bi2 Ci3 · · · p =
p(s)Ai1 (s)Bi2 (s) · · · ,
(1)
s
where Ai1 (s) = ±1 is the deterministic outcome obtained by Alice if she does the experiment Ai1 and Nature is in state s (notice that we are including also the locality condition when assuming that Ai1 (s) is independent of i 2 , i 3 , . . .). For a quantum mechanical system in a state ρ we have to set Ai1 Bi2 Ci3 · · · = Ai1 Bi2 Ci3 · · · ρ = tr(ρ Ai1 ⊗ Bi2 ⊗ Ci3 · · · ),
(2)
where ρ is a density operator acting on a Hilbert space Cd1 ⊗· · ·⊗Cd N and the observables satisfy −11 ≤ Ai1 , Bi2 , Ci3 , . . . ≤ 11, describing measurements within the framework of positive operator valued measures (POVMs). Note the parallelism with (1). In fact the quantum mechanical expression coincides with the classical one if the matrices Ai1 ’s, B j2 ’s, …commute with each other (and therefore can be taken diagonal in some basis |s), and we take the state ρ to be the separable state given by ρ = s p(s)|ss|⊗|ss|⊗· · · . How then can one know if Nature allows for an LHV description or follows Quantum Mechanics? That is, how to discriminate between (1) and (2)? The key idea of Bell [10] was to realize that this can be done by taking linear combinations of the expectation values Ai1 Bi2 Ci3 · · · . So, given real coefficients Ti1 i2 ,... , if we maximize the expression M−1 (3) T A B C · · · i 1 ···i N i1 i2 i3 i1 ,...,i N =0 assuming (1) we get1 T := sup Ti1 ···i N ai1 bi2 ci3 · · · . ai 1 ,bi 2 ,ci 3 ,...=±1 N i∈Z M 1 We write T since the above expression is exactly the norm of T as an N -linear form from R D equipped with the sup-norm.
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Therefore, if all correlations predicted by quantum mechanics could be explained in a classical and local world, one would have the following Bell inequality: ≤ T . (4) T tr(ρ A ⊗ B ⊗ C · · · ) i 1 i 2 i 3 ... i1 i2 i3 i1 ,i2 ,i3 ,... However, Quantum Mechanics predicts examples for which we have a violation in (4). The largest possible violation of a given Bell inequality (specified by T ) within quantum mechanics is the smallest constant K for which (5) Ti1 i2 i3 ... tr(ρ Ai1 ⊗ Bi2 ⊗ Ci3 · · · ) ≤ K T i1 ,i2 ,i3 ,... holds independent of the state and the observables. For instance, for the √ CHSH inequality (M = 2, N = 2 and T the Hadamard matrix) we have K = 2 irrespective of the Hilbert space dimension. More generally, if we also allow for arbitrary M and T and just fix N = 2, there is (see Sect. 3) a universal constant (called Grothendieck’s constant) K G that works in (5) for all Bell inequalities, states and observables. This was firstly observed by Tsirelson [70] (see also [5]). As K G is known to lie in between 1.676.. ≤ K G ≤ 1.782.. the maximal Bell violation in Eq.(5) is bounded for bipartite quantum systems. This bound imposes some limitations to the use of Bell inequalities, where one usually desires as large violations as possible. Below, when talking about the implications of our main result, we will illustrate why having large violations can be useful in the contexts of communication complexity, quantum cryptography or noise robustness. Therefore, it would be very useful to know whether in the tripartite case we still have a uniform bound for the violations. To the best of our knowledge, this question was first considered in the review [70] of Tsirelson in 1993. Our main result will be to prove that this is not the case. (We will use , , in the following to denote ≥, =, ≤ up to some universal constant.) Theorem 1 (Maximal violation for tripartite Bell inequalities). 1. For every dimension d ∈ N, there exist D ∈ N, a pure state |ψ on Cd ⊗C D ⊗C D and a Bell inequality √ with dichotomic and traceless observables such that the violation by |ψ is d. 2. The (unnormalized) state can be taken |ψ = 1≤i≤d 1≤ j,k≤D j|Ui† |k |i jk, where Ui are unitary √ matrices. 3. The order d is optimal in the sense that, conversely, for every state acting on Cd ⊗ C D ⊗ C D and every Bell inequality√with dichotomic but not necessarily traceless observables the violation is also d. The existence of the desired unitaries Ui will be proven by probabilistic methods. Therefore to obtain them one needs to be able to sample with respect to the Haar measure. The exact conditions that we require for these random U ’s, and that are verified in average, are written down explicitly in Eq. (20) and (21) below. This theorem shows once more that random states exhibit unexpected extremal properties [1,14,35,36,50,80]. Unfortunately, though we know a class of these highly nonlocal states (see Fig. 1), there are some weaknesses in the above theorem, which mainly come from the techniques we use:
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Fig. 1. Quantum circuit that provides highly non-local states. Apart from using a maximally entangled state as an input, it requires the implementation of a controlled unitary with random unitaries (that is, if the control qubit is in state |i, the circuit applies the (random) unitary Ui )
• We do not have any control on the growth of D with respect to d. • We do not have an explicit form for the family of inequalities for which we have unbounded violation. As it will be shown in the proof, for both the choice of the observables and the choice of the coefficients of the Bell inequality we will use a lifting argument, which in our case goes back to some application of Hahn-Banach’s theorem and the clever use of approximate units C ∗ -algebraic ideals. This prevents us from having a constructive proof. It would be interesting to find this lifting in another way (even numerically or probabilistically). It is important to note here (see Sect. 4) that in contrast to what is known for the M = 2 case [79], GHZ states do not belong to this set of highly non-local states—they always lead to a bounded violation. Let us now discuss some of the implications of Theorem 1: Communication complexity. Using notions from [15] it was shown in [13] that for every quantum state that violates a Bell inequality there is a communication complexity problem for which a protocol assisted by that state is more efficient than any classical protocol. In fact, it turns out that there is a quantitative relation between the amount of violation and the superiority of the assisted protocol. Adapted to our case, the communication complexity problem discussed in [13,15,52] is the following: Each of the three parties (i = 1, 2, 3) obtains initially a random bit string encoding (xi , yi ), where each yi = ±1 is taken from a flat distribution and xi ∈ {0, . . . , M − 1} is distributed according to |Tx |/ x |Tx |, where Tx = Tx1 ,x2 ,x3 are the coefficients appearing in the violated Bell inequality. The goal is now that every party first broadcasts a single bit and then attempts to compute the function F(x, y) = Tx /|Tx | yi (6) i
upon the obtained information. The protocol was successful if all parties come to the right conclusion. If one compares the optimal classical protocol (assisted by shared randomness) with a protocol assisted by a quantum state violating the considered Bell inequality by a factor K , and denotes the respective probabilities of success by P and PK then PK − P−
1 2 1 2
= K.
(7)
Let us denote by H (P) the binary entropy and quantify the information I about the actual value of F(x, y) gained by a protocol with success probability P by I (P) = 1 − H (P).
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Taking the states and inequalities appearing in Theorem 1 and thus setting K leads to the ratio I (PK ) d. I (P)
√
d then (8)
Measuring the size of the Hilbert space. What do measured correlations tell us about a quantum system, if we do not have a priori knowledge about the observables or even the size of the underlying Hilbert space? This type of question naturally arises in cryptography, where one wants to avoid any kind of auxiliary assumption necessary for security [2,3,9,48,65]. For purposes of detecting entanglement, it is easy to see that the set of entanglement witnesses that remain meaningful when disregarding the Hilbert space dimension is exactly the set of Bell inequalities. In fact, if measured correlations do not violate any Bell inequality, then they can always be produced by a separable (i.e., unentangled) state in a sufficiently large Hilbert space [3]. Theorem 1 now shows that for multipartite systems the violation of a Bell inequality can in principle be used to give lower bounds to the Hilbert space dimension. It also answers a question posed by Masanes [46] in the negative: in contrast to the case M = 2 [46,79] the extreme points of the set of quantum correlations observable with dichotomic measurements are in general not attained for multi-qubit systems. Robustness against noise and detector inefficiencies. It is well known that for M = 2 the maximal quantum violation can increase exponentially in the number of sites N [49,79]. However, since the N parties have to measure in coincidence, in practice with imperfect detectors, this increase comes with the handicap that also the coincidence rates then decrease exponentially. This becomes clearly different if one increases the violation without increasing N as it is the case in Theorem 1. So, in spite of the opaqueness of our result it does not suffer from decreasing coincidence rates, although of course increasing dimensions and numbers of observables might cause similar problems. Similarly, Theorem 1 implies the existence of tripartite quantum states that can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. To √ see this let ρ belong to the family of states giving rise to a maximal violation K d and set ρ = pρ + (1 − p)
11 . tr(11)
(9)
As the violation K is attainable for√ traceless observables, ρ yields K = pK which is still a violation whenever p 1/ d (see [5] for a similar reasoning in the bipartite case). In this context, it is a natural question to ask which is the amount of noise needed to disentangle a quantum state. It happens that this is considerably bigger. In particular, it is shown in [63] (in a constructive way) that: Theorem 2 (Neighborhood of the maximally mixed state). Given d, there is an entangled state ρd in Cd ⊗ Cd ⊗ Cd such that ρd = pρd + (1 − p) d113 is still entangled whenever p d12 . We will give an independent proof in the Appendix. Up to now the optimal value of p is not known. The best bounds are given by d13 p d12 [34,63]. It is also known that ρd in Theorem 2 can be taken to be the generalized GHZ state [23] in contrast to what we will see for the maximal violation of multipartite Bell inequalities.
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3. Mathematical Tools We will use tools from the theory of Operator Spaces and Tensor Norms. The use of one or the other will depend on the point of view of our problem. If we put the focus on the Bell inequalities and ask for the largest possible violation within Quantum Mechanics, then we will work with Operator Spaces and the meta-theorem we have is the following (for a precise formulation see below): A Bell inequality for N observers and M dichotomic observables per site is given M × · · · × M −→ C with T = 1. The largest possible by an N -linear form T : ∞ ∞ violation within Quantum Mechanics is given by the completely bounded norm of T , T cb . If, however, we put the focus on the quantum states and ask, given a N -partite quantum state, which is the largest possible violation that this state gives to a Bell inequality, then we will work with the theory of Tensor Norms, and the meta-theorem now reads: The largest possible violation that a N -partite D × · · · × D state ρ gives to a Bell inequality (with an arbitrary number of dichotomic observables) is given by the extendible tensor norm ρ⊗ N
D j=1,αext S1
.
Operator spaces. The theory of operator spaces started with the work of Effros and Ruan in the 80’s (see e.g. [26,58]) where they characterized, in an abstract sense, the structure of a subspace of a C ∗ -algebra. Since then, this theory has found interesting applications in mathematical analysis. An operator space is a complex vector space E and a sequence of norms · n in the space of E-valued matrices Mn (E) = Mn ⊗ E, where Mn is the space of complex n × n matrices endowed with the operator norm, which verify the following compatibility properties: 1. For all n, x ∈ Mn (E) and a, b ∈ Mn we have that axbn ≤ a Mn xn b Mn . 2. For all n, m, x ∈ Mn (E), y ∈ Mm (E), we have that x 0 = max{xn , ym }. 0y n+m Any C ∗ -algebra has a natural operator space structure that is the result of embedding it inside the space B(H ) of bounded linear operators in a Hilbert space [26,58], where Mn (B(H )) = B(n2 ⊗ H ). In particular, k∞ (= Ck with the sup-norm), being a commutative C ∗ -algebra, has a natural operator space structure. To computeit we embed k∞ in the diagonal of Mk (with the operator norm) and then, given x = i Ai ⊗ |i ∈ Mn (k∞ ) = Mn ⊗ k∞ , we have xn = Ai ⊗ |ii| = max Ai Mn . (10) i i
Mnk
The morphisms in the category of operator spaces (that is, the operations that preserve the structure) are called completely bounded maps. They are linear maps u : E −→ F between operator spaces such that all the amplifications u n = 11n ⊗ u : Mn (E) −→ Mn (F) are bounded.2 The cb-norm of u is then defined as ucb = supn u n . We 2 By referring to the amplifications, one can define also the concepts of complete isometry, complete isomorphism, complete quotient or surjection, complete contraction, …[58, pp. 19, 43].
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will call C B(E, F) the resulting normed space, that is, in fact, an operator space by Mn (C B(E, F)) = C B(E, Mn (F)). Analogously one can define the cb-norm of a multilinear map T : E 1 × · · · × E N −→ F as T cb = sup Tn 1 ,...,n N , where now Tn 1 ,...,n N = T ⊗ 11n 1 ⊗ · · · ⊗ 11n N : Mn 1 (E 1 ) × · · · × Mn N (E N ) −→ Mn 1 ···n N (F). A multilinear map is called completely bounded if T cb < ∞. We will denote by C B N (E 1 , . . . , E N ; F) the resulting normed space, that is also an operator space by Mn (C B N (E 1 , . . . , E N ; F)) = C B N (E 1 , . . . , E N ; Mn (F)). M × · · · × M −→ C given With these definitions, if we have an N -linear form T : ∞ ∞ by T (|i 1 , |i 2 , . . .) = Ti1 i2 ... and we compute the usual norm and the cb-norm we obtain M−1 j 1 2 T = sup{ Ti1 i2 ... i1 i2 · · · ; |i j | ≤ 1}, i1 ,...,i N =0 ⎧ ⎨ M−1 T cb = sup Ti1 i2 ... tr(ρ Ai1 ⊗ Bi2 · · · ) ⎩ i 1 ,...,i N =0 :
Ai1 , Bi2 , . . . ∈ M D with operator norm ≤ 1 ρ ∈ M D N with trace norm ≤ 1
.
These expressions coincide respectively with the maximal value that one can achieve in the expression (3) M−1 Ti1 ···i N Ai1 Bi2 Ci3 · · · i1 ,...,i N =0 if we assume that Nature is deterministic and local, T , or if we assume Quantum Mechanics, T cb . This essentially proves the meta theorem stated at the beginning of the section. The only subtle point is that in the context of Bell inequalities everything is real while in this context of operator spaces we are in the complex case. Therefore, whenever we want to formally use this meta-theorem we will have to make some splits between real and imaginary parts. In [33], Grothendieck proved what he called the fundamental theorem of the metric theory of tensor products. This result, known as Grothendieck’s Theorem or Grothendieck’s Inequality reads as follows: There exists a universal constant K G such that no matter how we choose real coefficients Ti j and elements xi , y j in the unit ball of a real Hilbert space H with inner product ·, ·, we have that ≤ K T x , y sup T ν ij i j G ij i j . i ,ν j =±1 i j ij In particular, Ti j tr (ρ Ai ⊗ B j ) ≤ K G sup Ai sup B j sup Ti j i ν j . i ,ν j =±1 i j i j ij
(11)
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The second part tells us that Grothendieck’s Theorem provides a uniform bound K G for the violation of any bipartite Bell inequality with dichotomic observables. This was essentially Tsirelson’s observation [70]. But the above comments show how Grothendieck’s Theorem also implies that any bounded bilinear form from a commutative C ∗ -algebra has to be also completely bounded, which was first noticed in [40]. Since Grothendieck stated his theorem, a lot of effort has been devoted to find suitable multilinear generalizations (see for instance [11,12,17,53,56,69]). However, up until now, the validity of a trilinear Grothendieck’s Theorem in the context of operator spaces (and hence in the context of Bell Inequalities) has been open. Although it is conceivable that trilinear versions of Grothendieck’s inequality hold for operator spaces, our main theorem (Theorem 1) will show that the trilinear version of (11) fails. We will rewrite this now in the language of operator spaces which is instrumental in the proof. Then we will show how this theorem implies Theorem 1 and we will give the proof in Sect. 5. n2
N2
Theorem 3. For every n, there exist N , a state |ψ N , a trilinear form T : 2∞ × 2∞ × N2 n2 N2 ˆ
2∞ −→ C and elements b ∈ Mn (2∞ ), bˆ ∈ Mn (2∞ ), with |ψ N , T , b, b 1 and √ ˆ b)|ψ ˆ N n. ψ N |Tn,N ,N (b, b, Moreover √ 1. The order n is optimal. † 2. |ψ N can be taken √1 1≤i≤n 1≤ j,k≤N j|U N ,i |k|i jk, where U N ,i are unitary nN matrices. Again we use (resp. ) to denote ≥ (resp. =) up to some universal constant. In particular, we obtain that Corollary 4 (Bounded but not completely bounded trilinear forms). Given n, there √ exist N × N −→ C such that T ≥ T N and a trilinear map√T : n∞ ×∞ cb n,N ,N nT . ∞ Moreover, the order n is optimal. We will finish this section by showing how Theorem 3 implies Theorem 1. As we said before, it is simply a matter of splitting into real and imaginary parts. Theorem 3 tells us that there exist a complex matrix {Ti jk }i,N j,k=1 , n × n matrices bi and N × N matrices bˆ j (all of them with norm 1) such that √ ψ N | ˆ ˆ Ti jk bi ⊗ b j ⊗ bk |ψ N n. (12) i, j,k By splitting into real and imaginary parts it is not difficult to see that one can take i) in (12) T real and bi , bˆ j hermitian. Moreover, writing αi = tr(b n (so that |αi | ≤ 1), 1 0 1 bi = αi 11n , bi = bi − bi and applying the bipartite case to show that (ρ = |ψ N ψ N |) 1 tr(ρTi jk bi ⊗ bˆ j ⊗ bˆk ) = Ti jk αi tr(ρ2,3 bˆ j ⊗ bˆk ) ≤ K G , i, j,k i, j,k one can take the observables in (12) to be traceless.
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Tensor norms. The theory of tensor norms can be traced back to the work of Murray and von Neumann in the late 30’s, but it was definitely set by Grothendieck in his seminal paper [33]. Since then, several important contributions have been made (see [22] or [64] for a modern reference). N If X 1 , . . . , X N are normed spaces, by j=1,π X j we denote the algebraic tensor product Nj=1 X j endowed with the projective norm π(u) := inf
m
u i1 · · · u iN : u =
i=1
m
u i1 ⊗ · · · ⊗ u iN .
i=1
This tensor norm is both commutative and associative, in the sense that Nj=1,π X j = N N N j j j=1,π X σ ( j) for any permutation of the indices σ and that j=1,π i j =1,π X i j = N ,N j j j=1,i j =1,π X i j . The projective norm π is in duality with the injective norm , defined N on j=1 X j as
m 1 1 N N j ∗ j φ (u i ) · · · φ (u i ) : φ ∈ X j , φ ≤ 1 , (u) := sup i=1
where X ∗j denotes the topological dual of X j and u =
m
N 1 i=1 u i ⊗· · ·⊗u i . That is, if E j ∗ N 1, . . . , N , we have E = j j=1,π
is a finite dimensional normed space for every j = N ∗ j=1, E j . Moreover, the dual of the π tensor product can also be isometrically identified with the space of N -linear forms (with its usual operator norm). In fact, we have the natural isometric identification, ⎛ ⎛ ⎞∗ ⎞ N N N ∗ E j ⎠ = L ⎝ E1, Ej⎠. (13) L (E 1 , . . . , E N ; C) = ⎝ j=1,π
j=2,
Following [22] (or [30] for the multilinear version) we define a tensor norm β of order N as a way of assigning to every N -tuple of normed spaces (X 1 , . . . , X N ) a norm on Nj=1 X j (we call Nj=1,β X j the resulting normed space) such that • •
≤ β ≤ π . N N N N j=1 u j : j=1,β X j −→ j=1,β Y j ≤ j=1 u j , for every choice of linear bounded operators u j : X j −→ Y j . This is called the metric mapping property.
N Sometimes we will use the notation β (X 1 , . . . , X N ) instead of j=1,β X j to distinguish some space. We will say that β is finitely generated if, for every X j , j = 1, . . . , N , and z ∈ N j=1 X j we have β(z; X 1 , . . . , X N ) = inf
⎧ ⎨ ⎩
β(z; E 1 , . . . , E N ) : E j ∈ F I N (X j ), z ∈
N j=1
⎫ ⎬ Ej
⎭
,
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where we denote F I N (X ) = {E ⊂ X | dim E < ∞} (and C O F I N (X ) = {E ⊂ X |E is closed and dim(X/E) < ∞}). As one can find in [22, Sect. 17] and [30, Sect. 4] tensor norms are in one-to-one duality with ideals of multilinear operators. We explain this in what follows: A normed (Banach) ideal of N -linear continuous operators between Banach spaces is a pair (A, · A ) such that • A(X 1 , . . . , X N ; Y ) = A ∩ L N (X 1 , . . . , X N ; Y ) is a linear subspace of L N (X 1 , . . . , X N ; Y ) and the restriction · A |A(X 1 ,...,X N ;Y ) is a (complete) norm. • If u j ∈ L(Z j , X j ), T ∈ A(X 1 , . . . , X N ; Y ) and v ∈ L(Y, Z ), then the composition v ◦ T ◦ (u 1 , . . . , u N ) is in A, and v ◦ T ◦ (u 1 , . . . , u N )A ≤ vT A u 1 · · · u N . • The operator K N (x1 , . . . , x N ) → x1 · · · x N ∈ K is in A and it has · A -norm equal to one. An ideal (A, ·A ) is called maximal if T Amax := sup{q LY ◦T | E 1 ×···×E N A |E j ∈ F I N (X j ), L ∈ C O F I N (Y )} < ∞ implies T ∈ A and T A = T Amax . The following theorem shows the duality mentioned above Theorem 5. Let (A, ·A ) be a normed ideal of N-linear continuous mappings between Banach spaces. Then (A, ·A ) is maximal if and only if there exists a finitely generated tensor norm β of order N + 1 such that ⎛ ⎞∗ A(X 1 , . . . , X N ; Y ∗ ) = ⎝ (X 1 , . . . , X N , Y )⎠ , β
⎛ ⎞∗ (X 1 , . . . , X N , Y ∗ )⎠ ∩ L N (X 1 , . . . , X N ; Y ). A(X 1 , . . . , X N ; Y ) = ⎝ β
Here both identifications are isometric. For the purposes of this paper we will only need two of these ideals: the extendible and the (1; 2)-summing multilinear operators. Extendible multilinear operators. The lack of a multilinear Hahn-Banach extension theorem has motivated a considerable effort in the search of partial positive results (see [16,18,37,55] and the references therein). In this context, the natural space to work with is the space of extendible multilinear forms. That is, those continuous multilinear forms T : X 1 × · · · × X n −→ C (here X j are Banach spaces) such that for every choice of superspaces Y j ⊃ X j , there is a continuous and multilinear extension T˜ : Y1 × · · · × Yn −→ C. We define the extendible norm of T as T ext = sup inf T˜ , Yj
T˜
where the sup runs among all possible superspaces Y j and the inf among all possible extensions T˜ . As it can be found in [18], for infinite dimensional spaces X j , T ext can be ∞. We say that T is extendible if T ext < ∞.
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It can be easily seen that the extendible n-linear forms constitute a Banach ideal, which we denote by Lnext . Actually, it is trivial to check that ( nj=1,αext X j )∗ = Lnext (X 1 , . . . , X n ) isometrically, if we define the well known ([39], Sect. 3) finitely generated tensor norm αext (u; X 1 , . . . , X n ) = inf{π(u; Y1 , . . . , Yn ) : X j ⊂ Y j }, where the inf is taken among all superspaces Y j of X j . αext is called the extendible tensor norm (and it is, of course, the tensor norm associated to the ideal of extendible multilinear forms in the sense of Theorem 5). The next lemma will be a central result to connect this mathematical theory with the context of Bell inequalities: Lemma 6. Let X 1 , . . . , X n be n Banach spaces and u ∈ nj=1 X j . We have k 1 n αext (u) = sup Mi1 ···in Ai1 ⊗ · · · ⊗ Ain , u , M,Ai1 ,...,Ainn i 1 ,...,i n =1 1
where the sup is taken among (Ai11 )ik1 =1 ⊂ B X 1∗ , . . . , (Ainn )ikn =1 ⊂ k B X n∗ , (Mi1 ···in )i1 ,...,in =1 n k ≤ 1, k ∈ N, and the brackets denote as usual the j=1,ε 1 action by duality. Proof. By the injectivity of ∞ (see for instance [22, Chap I.1]), it follows that k αext (u) = sup a1 ⊗ · · · ⊗ an (u) n k∞ | a j : X j → ∞ , a j ≤ 1, j=1,π
j = 1, . . . , n; k ∈ N .
Now, we know that L(X j , k∞ ) is isometrically isomorphic to k∞ (X ∗j ) (see for instance [22, Chap I.3]). Thus, given a j ∈ L(X j , k∞ ) = k∞ (X ∗j ) by a j = ikj =1 |i j ⊗ Ai j , we have k 1 n a1 ⊗ · · · ⊗ an (u) n Ai1 ⊗ · · · ⊗ Ain , u|i 1 ⊗ · · · ⊗ |i n k = j=1,π ∞ i1 ,...,in =1 n k j=1,π ∞ ⎧ ⎫ k ⎨ ⎬ 1 n = sup Ai1 ⊗ · · · ⊗ Ain , uT (|i 1 , . . . , |i n ) : T ∈ B( n . k ∗ j=1,π ∞ ) ⎭ ⎩ i 1 ,...,i n =1 The statement follows now easily.
With this at hand we can now formalize the meta-theorem given in the introduction of Sect. 3: Theorem 7. Given a N -partite D1 × · · · × D N quantum state ρ, the largest possible violation that this state gives to a Bell inequality of an arbitrary number of dichotomic observables is bounded above by 2 N −1 ρ N j=1,αext
Dj
S1
.
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Proof. Given a Bell inequality with (real) coefficients Ti1 ,...,i N and observables −11 ≤ Ai1 , Bi2 , . . . ≤ 11, it is clear by the definition of αext that Ti1 ,i2 ,... tr(ρ Ai1 ⊗ Bi2 ⊗ · · · ) i1 ,i2 ,... 1 2 ≤ ρ N sup Ti1 ,i2 ,... i1 i2 · · · . Dj S j j j=1,αext 1 ∈C,| |=1 i 1 ,i 2 ,... ij
ij
To finish the proof of the theorem it is enough to notice that (see [51, Proposition 19])
j j
i
1 2 N −1 1 2 sup Ti1 ,i2 ,... i1 i2 · · · ≤ 2 sup Ti1 ,i2 ,... i1 i2 · · · . j j ∈C,| |=1 i 1 ,i 2 ,... =±1 i 1 ,i 2 ,... ij
ij
Summing operators. Since the work of Grothendieck [33], the class of absolutely summing linear operators plays a crucial role in the theory of tensor norms (see [25] for a reference). Motivated by that, A. Pietsch defined in [57] the following class of multilinear operators: A multilinear form T : X 1 × · · · × X N −→ C is called (s; r )-summing (1 ≤ s, r < j ∞) if there exists a constant K such that for any choice of finite sequences (xi )i ⊂ X j , we have that 1 N s s j ≤K (xi )i rω , T (xi1 , . . . , xiN ) i
(14)
j=1
where (xi )i rω denotes the supremum, among all elements x ∗j in the unit ball of the dual space X ∗j , of j
1 r r j . x ∗j (xi ) i
The smallest K valid in Eq. (14) is called the (s; r ) norm of T , and we write T (s;r ) . The key result is the following generalization of Grothendieck’s inequality, which appears explicitly in [53, Corollary 2.5] (see also [11,17,37,69]). Theorem 8. Every extendible N -linear form T is (1; 2)-summing and T (1;2) ≤ N −2 K G 2 2 T ext , where K G is Grothendieck’s constant.
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4. Bounded Violations for GHZ States The maximal violation of multipartite Bell inequalities with two dichotomic observables n−1 |iii (where per site [49,79,83] is known to be attained for GHZ states |ψ = √1n i=0 n = 2 is sufficient in this case). In contrast to that, we will show here that GHZ states do not give rise to the maximal violation in Theorem 1 but rather lead to a bounded violation. In other words, there is a fixed amount of noise (independent of the dimension) which makes the considered correlations of the GHZ state admit a description within a local hidden variable model. Before proving that we need a bit of work. We call ρ the unnormalized GHZ state 3 n n i j |i j| ⊗ |i j| ⊗ |i j| as a member of j=1 S1 (S1 the Banach space of trace class n operators on a n-dimensional Hilbert space 2 ). When we consider a tensor norm α on 3 3 n n ∗ ∗ j=1 S1 , ρ will be the same element as ρ, but considered in the dual ( j=1,α S1 ) . The key point is the following result: Proposition 9. For every tensor norm α, ρ 3
j=1,α
S1n
· ρ ∗ ( 3
j=1,α
S1n )∗
= n2.
We will follow [31, Theorem 2.5]. First we will need the next Lemma 10. Let α be any tensor norm and A = ⊗3j=1,α S1n . Let G be a topological compact group such that G ⊂ isom(A, A), the group of isometries of A. We suppose: (i) gρ = ρ for every g ∈ G. (ii) Given L ∈ A∗ , if L ◦ g = L for every g ∈ G, then L = λρ ∗ for some constant λ. Then we have that ρ A · ρ ∗ A∗ = n 2 . Proof. Let us take L ∈ A∗ such that L A∗ =1 and L(ρ) = ρ A . Let dg be the Haar measure on G. We define L 0 = G L ◦ gdg. It is easy to see that L 0 is well defined and ∗ ∗ ∗ belongs to A with L 0 A ≤ L A . Now, by (i), L 0 (ρ) = G L ◦ g(ρ)dg = L(ρ). On the other hand, for every g ∈ G we have L 0 ◦g = G L ◦g◦g dg = G L ◦gdg = L 0 , where we have used the translational invariance of the Haar measure. Using (ii) we conclude that L 0 = λρ ∗ . We have ρ A = L(ρ) = L 0 (ρ) = λρ ∗ (ρ) = λn 2 , and also λρ ∗ A∗ = λρ ∗ A∗ = L 0 A∗ ≤ L A∗ = 1. Then ρ ∗ A∗ ≤ λ1 , and thus ρ A · ρ ∗ A∗ ≥ n 2 . The other inequality is trivial.
Using the previous lemma we can easily prove Proposition 9: Proof. We only need to show that there exists a topological compact subgroup of isom(A, A) which verifies the hypothesis of Lemma 10. For every ε = (1 , . . . , n ), where i = ±1, we consider gε : Cn −→ Cn such that gε (|i) = i |i. For every σ permutation of {1, . . . , n} we consider h σ : Cn −→ Cn
Unbounded Violation of Tripartite Bell Inequalities
469
such that h σ (|i) = |σ (i). Now we take the group G generated by the elements of the form (gε∗ ⊗ gθ ) ⊗ (gε∗ ⊗ gθ ) ⊗ id , (gε∗ ⊗ gθ ) ⊗ id ⊗ (gε∗ ⊗ gθ ) and (h ∗σ ⊗ h τ ) ⊗(h ∗σ ⊗ h τ ) ⊗ (h ∗σ ⊗ h τ ). It is clear that G is a compact subgroup of isom(A, A) and that G verifies (i). Let us check (ii). Let λi, j,k,l,m,n |i j| ⊗ |kl| ⊗ |mn| L= i, j,k,l,m,n
be an arbitrary element of A∗ . We take g = (gε∗ ⊗ gθ ) ⊗ id ⊗ (gε∗ ⊗ gθ ) in G. If we have L = L ◦ g, we get, for every i, j, k, l, m, n, λi, j,k,l,m,n = λi, j,k,l,m,n i θ j m θn , for every choice of signs i , θ j . Therefore, λi, j,k,l,m,n = λi, j,k,l,m,n δi,m δ j,n . Then L=
λi, j,k,l |i j| ⊗ |kl| ⊗ |i j|.
i, j,k,l
We can repeat the step before (taking now g = (gε∗ ⊗ gθ ) ⊗ (gε∗ ⊗ gθ ) ⊗ id to see that, in fact, L= λi, j |i j| ⊗ |i j| ⊗ |i j|. i, j
Finally, taking g = (h ∗σ ⊗ h τ ) ⊗ (h ∗σ ⊗ h τ ) ⊗ (h ∗σ ⊗ h τ ) ∈ G, we see that λi, j = λσ (i),τ ( j) for every permutation τ, σ and every i, j. Then, we get that λi, j = λ, which finishes the proof. And finally we can get the desired bound for the GHZ violation: n−1 Theorem 11 (GHZ bound). Given the tripartite GHZ state |ψ = √1n i=0 |iii, the largest possible quantum violation for a Bell inequality with dichotomic observables is √ upper bounded by 4 2K G . Proof. By Theorem 7 it is enough to show that for the unnormalized GHZ state ρ = n−1 i, j=0 |i j| ⊗ |i j| ⊗ |i j| we have ρ 3 S n ≤ K · n. Due to Proposition j=1,αext
1
9 we only have to prove that ρ ∗ ext ≥ K1 n. To see this, we use that by Theorem 8 T (1;2) ≤ K T ext , and then it remains to be proven that ρ ∗ L3 (S n ) ≥ n. For (1;2)
1
that consider the sequence (|r 0|)rn=1 ⊂ S1n , which verifies (|r 0|)r w 2 ≤ 1, and we ∗ (|r 0|, |r 0|, |r 0|) = n. ρ r
Remark 12. Note √ that Theorem 11 holds also for N parties, where now the constant can be taken K G (2 2) N −1 . In [81] an explicit set of inequalities was derived for which GHZ states achieve a violation of the order (π/2) N .
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5. Proof of the Main Theorem More on operator spaces. Ruan’s Theorem [26,58,61] assures that any operator space can be considered as a closed subspace of B(H ) with the inherited sequence of matrix norms. Then we can define the minimal tensor product of two operator spaces E ⊂ B(H ) and F ⊂ B(K ) as the operator space given by E ⊗min F ⊂ B(H ⊗ K ). In particular, Mn (E) = Mn ⊗min E for every operator space E. The tensor norm min in the category of operator spaces will play the role of in the classical theory of tensor norms. In particular it is injective, in the sense that if E ⊂ X and F ⊂ Y , then E ⊗min F ⊂ X ⊗min Y as operator spaces. The analogue of the π tensor norm is the projective tensor norm, defined as u Mn (E⊗∧ F) = inf{α Mn,lm x Ml (E) y Mm (F) β Mlm,n : u = α(x ⊗ y)β}, where u = α(x ⊗ y)β means the matrix product u=
αr,i p β jq,s |r s| ⊗ xi j ⊗ y pq ∈ Mn ⊗ E ⊗ F.
r si j pq
Both tensor norms ∧ and min are associative and commutative and they share the duality relations of their classical counterparts π and . In fact, for finite dimensional operator spaces we have the natural completely isometric identifications (E ⊗∧ F)∗ = C B 2 (E, F; C) = C B(E, F ∗ ) = E ∗ ⊗min F ∗ ,
(15)
where, given an operator space E, we define its dual operator space E ∗ via the identification Mn (E ∗ ) = C B(E, Mn ). Depending on the way one embeds a Banach space inside B(H ), the same Banach space can have different operator space structures. This happens even in the simplest example: the case of a Hilbert space. A trivial way of embedding a finite dimensional Hilbert space n2 inside some B(H ) is to put it into the first column (resp. row) of Mn , that is, |i → |i0| (resp. |i → |0i|). This gives us the column operator space Cn (resp. the row operator space Rn ). It is trivial to verify Ai ⊗ |i i
Mm ⊗min Rn
1 2 † = Ai Ai , i
Ai ⊗ |i i
Mm ⊗min Cn
1 2 † = Ai Ai . i
We can also define the intersection of these two operator spaces RCn = Rn ∩ Cn , where given two operator spaces E, F [58, p. 55], · Mn ⊗min E∩F = max{ · Mn ⊗min E , · Mn ⊗min F }.
Unbounded Violation of Tripartite Bell Inequalities
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We will denote by RC2n to RCn ⊗min RCn . We have the following concrete expressions: ⎧ 1 1 ⎫ 2 2 ⎬ ⎨ † † Ai ⊗ |i = max Ai Ai , Ai Ai , ⎭ ⎩ i i i Mm ⊗min RCn ⎧ 1 1 2 2 ⎨ † † A ⊗ |i j = max A A , A A (16) , i j i i i i ⎩ ij
M ⊗
m min Ai j ⊗ |i j| ij
RC2n
Mm ⊗min Mn
i
, Ai j ⊗ | ji| ij
i
Mm ⊗min Mn
⎫ ⎬ ⎭
.
The first estimate is trivial and the second one can be easily derived by applying the following completely isometric identifications [26, p. 163]: Rn ⊗min Rn = Rn 2 ,
Cn ⊗min Cn = Cn 2 ,
Cn ⊗min Rn = Mn ,
and decomposing (Rn ∩Cn ) ⊗min (Rn ∩Cn ) = (Rn ⊗min Rn ) ∩ (Rn ⊗min Cn ) ∩ (Cn ⊗min Rn ) ∩ (Cn ⊗min Cn ) [58, p. 55]. With (16) it is trivial to verify that Lemma 13.
N |i j| ⊗ |i j i j=1
=
√
N.
M N (RC N 2 )
Moreover, we have the canonical completely isometric identifications Rn∗ = Cn , Cn∗ = Rn , and the formal identities Rn −→ RC∗n , Cn −→ RC∗n are completely contractive. The connection with Theorem 3 will be made by the following non-commutative Khintchine’s inequality, proved by Lust-Picard and Pisier in [44] (see also [58], Sect. 9.8). Before stating it, we need to recall the definition of the Rademacher functions. Given the group of signs Dn = {−1, 1}n and the normalized Haar measure on it µn , we define the i th Rademacher function i : Dn −→ R asn the i th coordinate function. If we call m E n = span{i : 1 ≤ i ≤ n} ⊂ L 1 (Dn , µn ) = 21 (where m 1 = (C , · 1 )) then Theorem 14 (Lust-Picard/Pisier). The canonical identity map id : RC∗n −→ E n given by |i → i verifies that idcb idn−1 cb ≤ C, where C nis somen universal constant, and the operator space structure on 21 is determined by 21 = (2∞ )∗ . Among all possible operator space structures for a finite dimensional Hilbert space m 2 , there is one that is the minimal in the sense that every bounded operator with range min(m 2 ) is always completely bounded. This is exactly the operator space structure inherm−1 ) given by |i → f where f (|φ) = φ|i ited from the embedding m i i 2 −→ ∞ (S m−1 for every |φ in the unit sphere S . There are some properties we will need about m min(m 2 ). The first one is that min(2 ) is a 1-exact operator space in the following sense ([58], Chap. 17 and Sect. 2.4):
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An operator space E is called λ-exact if, given any C ∗ -algebra A and any (closed A min E two-sided) ideal I ⊂ A, the completely contractive map Q : IA⊗ ⊗min E −→ I ⊗min E verifies that Q −1 ≤ λ. In particular, for min(m 2 ), Q is a complete isometry. m Moreover, for any operator space E, E ⊗min min(m 2 ) = E ⊗ 2 as Banach spaces. With this and (16) one can finally obtain Lemma 15.
|i j ⊗ |i jRC2 ⊗ n
ij
2
min
min(n2 )
≤ 1.
Random matrices and Wassermann’s construction. We start with the following application of Chevet’s inequality. Many of the ideas behind the proof come from the seminal work [45, Chapter V]. We essentially follow here [38] which is only available on a preprint server. Therefore we include a complete proof of the statement for the convenience of the reader. Lemma 16. Let n, N ∈ N and UnN be the n-fold product of the unitary group equipped with the normalized Haar measure. Then E sup
|λi |2 ≤1
n
λi Ui M N ≤ 32π(1 +
i=1
n ). 4N
Proof. We recall Chevet’s inequality. For Banach spaces E, F, [67, Theorem 43.1] E
gs,t xs ⊗ yt E⊗ε F ≤ b(xs )s ω2 E
t
gt yt F + b(yt )t ω2 E
s,t
s
gs xs E . (17)
Here gs,t are independent normalized real gaussian random variables, b = 1 if the spaces are real, whereas b = 4 if they are complex, and we recall that (xs )s ω2
= sup{
1 2
∗
|x (xs )|
2
| x ∗ ∈ E ∗ , x ∗ ≤ 1} .
s
Let us apply this twice to get
E
gikl |i ⊗ |k ⊗ |ln ⊗ε N ⊗ε N 2
i=1,...,n,k,l=1,...,N
≤ 4(|i)i ω2 E
kl
≤ 8E
N k=1
2
gkl |k ⊗ |l N ⊗ε N + 4E 2
2
√ √ √ gk |k2 + 4 n ≤ 8 N + 4 n .
2
n i=1
gi |i2 (|k ⊗ |l)kl ω2
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473
In order to transform this to unitaries we replace gi jk by complex gaussians g˜i jk = √ This gives an additional factor 2. Then, following [38, Lemma 3.2.1.5], we obtain (see below for the details) gi jk +igi jk √ . 2
n 1 E g˜ jk | jk| S N E |i ⊗ Ui n2 ⊗ M N 1 N jk i=1 √ √ ≤ E g˜i jk |i ⊗ | j ⊗ |kn ⊗ε N ⊗ε N ≤ 8 2N + 4 2n. 2
2
2
i=1,...,n; j,k=1,...,N
(18) Finally, we have E
jk
g˜ jk | jk| S N ≥ 1
real gaussians and real Hilbert spaces E
2N
3/2 N√ . π 2
To see this it is enough to show that for
we get
g jk | jk| N ⊗π N ≥ 2
jk
2
N 3/2 . π
Recall from above the notation Dn = {−1, 1}n , µn the Haar measure on Dn and i the i th Rademacher function. It is a simple exercise [22, Sect. 8.7] to verify that E[(
1
|g jk |2 ) 2 ] =
jk
≥
2 π
g jk (ω)| jk| N ⊗ 2
jk
DN 2
N 2 2
dP(ω)
jk (s) | jk| N ⊗ 2
jk
N 2 2
dµ N 2 (s) =
2 N. π
Using the duality (2N ⊗ε 2N )∗ = 2N ⊗π 2N and Hölder’s inequality this implies 1 1 2 N ≤ E[( |g jk |2 ) 2 ] = E[| g jk | jk|, gst |st|| 2 ] π st jk jk 1 1 g jk | jk| N ⊗ε N ) 2 (E g jk | jk| N ⊗π N ) 2 . ≤ (E 2
jk
2
Now, using Chevet’s inequality again, we know that (E √ 1 2N 4 . Thus we have E
jk
g jk | jk| N ⊗π N ≥ 2
2
jk
2
2
1
jk
g jk | jk| N ⊗ε N ) 2 ≤ 2
2
1 3 N 2. π
So it only remains to prove the first inequality in (18). We include the argument n ⊂ U N be a sequence of given in [38, Lemma 3.2.1.5] for completeness. Let (Ui )i=1 unitary matrices. We consider left multiplication L with respect to block diagonal of 2 2 Ui s, namely L : Cn N −→ Cn N defined by (x ijk ) jk −→ (Ui ◦ (x ijk ) jk ), ∀i
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D. Pérez-García, M. M. Wolf, C. Palazuelos, I. Villanueva, M. Junge 2
as well as the corresponding right multiplication. They are unitary operations on Cn N and therefore leave the complex gaussian density invariant. Given a sequence of random matrices G i (ω) with independent normalized complex gaussian entries, that is G ijk (ω) = g˜ ijk (ω), we denote by τ i (ω) the sequence of singular values of G i (ω), in the sense that there are unitaries U i (ω), V i (ω) with G i (ω) = U i (ω)Dτ i (ω) V i (ω). We denote by the Haar measure on the group G of sequences of permutations G = (Per m{1, . . . , N })n and Mπ the permutation matrix Mπ (|i) = |π(i). For i = 1, . . . , n and a diagonal operator Dτ i , we have
G
Mπ i Dτ i M(π i )−1 d = (
1 i τl )idC N . N l
2
Now, let C ⊂ Cn N be a finite set. Calling µ to the Haar measure in UnN we get |
sup
(xi jk )i jk ∈C
= =
xi jk g˜ ijk (ω)|dP(ω)
i jk
sup
UnN ×UnN
(xi jk )i jk ∈C
sup
UnN ×UnN (xi jk )i jk ∈C
|
xi jk (U i G i (ω)V i ) jk |dP(ω)dµdµ
i jk
|
xi jk (U i U i (ω)Dτ i (ω) V i (ω)V i ) jk |dµdµdP(ω).
i jk
By the invariance of the Haar measure we can write further = ≥ = =
G
sup
UnN ×UnN (xi jk )i jk ∈C
sup
UnN ×UnN (xi jk )i jk ∈C
|
|
xi jk (U i Mπ i Dτ i (ω) M(π i )−1 V i ) jk |dµdµddP(ω)
i jk
xi jk (U i (
i jk
G
Mπ i Dτ i (ω) M(π i )−1 ddP(ω))V k ) jk |dµdµ
1 1 τl (ω)dP(ω) sup | xi jk (U i V i ) jk |dµdµ n n N U N ×U N (xi jk )i jk ∈C i jk l 1 τl1 (ω)dP(ω) sup | xi jk u ijk |dµ. n N U N (xi jk )i jk ∈C
Now, since
l
i jk
1 N
ing the unit ball of
1 jk g˜ jk | jk| S1N , and taking N E n N N n ∗ (2 ⊗ 2 ⊗ 2 ) = (2 ⊗ M N )∗ , we get (18). j
τ 1j (ω)dP(ω) =
C approach-
We will use a theorem of Voiculescu in order to obtain a state of the form of Theorem 3 (defined by random unitary matrices). We will need to define some previous concepts. For a countable discrete group we recall that the left regular representation λ : G → B(2 (G)) is given by λ(g)δh = δgh . Here (δh ) stands for the unit vector basis in 2 (G). Then Cr ed (G), the norm closure of the linear span of λ(G), is called the reduced
Unbounded Violation of Tripartite Bell Inequalities
475
C ∗ -algebra of G. The reduced C ∗ -algebra sits in the von Neumann algebra V N (G) = λ(G)
. The normal trace τ on V N (G) is given by τ (x) = (δe , xδe ). For the free group Fn in n generators g1 , . . . , gn , the reduced C ∗ -algebra can be n realized by random unitaries in the following sense: Let (U N ,i )i=1 be random unitaries in N UnN , endowed with the Haar measure and τ N the normalized trace on U N (τ N (x) = N1 tr N (x)) for each N . According to [73, Theorem 4.3.3], we have that lim τ N (U Nε1,i1 · · · U Nεm,im ) = τ (λ(gi1 )ε1 · · · λ(gim )εm )) N
(19)
holds almost everywhere, for every string (i 1 , . . . , i m ) ∈ {1, . . . , n}m and ε j = ±1. Here τ is the normalized trace on the von Neumann algebra λ(Fn )
. This means that the right hand expression is 1 if and only if giε11 · · · giεmm is the trivial word e (after cancellation). In all the other cases, we obtain 0. We will use this result in a more quantitative way as follows: We define the set := {ω = {a1 , . . . , ak }|k ∈ N, 1 ∈ {a1 , . . . , ak } ⊂ Fn }. Given 1 = a ∈ Fn , Voiculescu’s theorem [73, Theorem 4.3.3] tells us that lim µ N ({(U1 , . . . , Un )|τ N (πU1 ,...,Un (a)) < N
1 }) = 1, k
where we call πU1 ,...,Un : Fn −→ M N to the representation of Fn uniquely determined by gi −→ Ui , i = 1, . . . , n. Given ω = {a1 , . . . , ak } ∈ of cardinality k, we deduce the existence of Nω such that µ Nω ({(U1 , . . . , Un )|τ Nω (πU1 ,...,Un (ai )) <
1 1 ∀i = 1, . . . , k}) > . k 2
(20)
Now, we know by Lemma 16 that E sup |λi
|2 ≤1
n
λi Ui M Nω ≤ 32π(1 +
i=1
n ). 4Nω
(21)
As a consequence of Chebychev’s inequality, for every ω ∈ there exists a sequence (U Nω , j )nj=1 ∈ UnNω which satisfies both (20) and (21) (multiplying by 2 the bound of (21)). These sequences of random unitary matrices will be crucial in our construction and will be fixed from now on. We simplify the notation a bit more: For every ω ∈ we call πω to πU Nω ,1 ,...,U Nω ,n , and τω : M Nω −→ C to the normalized trace τω (x) = τ Nω (x). We follow now a construction of Wassermann [75] to obtain a representation of the reduced C ∗ -algebra Cr ed (Fn ). We fix an ultrafilter U on refining the sets ω={a1 ,...,ak } := {{b1 , . . . , bn } ⊂ Fn |k ≤ n, ω ⊆ {b1 , . . . , bn }}. Then, we have that for a = 1, lim τω (πω (a)) = 0. U
We consider the space ∞ (, M Nω ), and we define the (closed two-side) ideal I = {(xω )ω ∈ ∞ (, M Nω ) : lim τω (xω† xω ) = 0}. U
(22)
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D. Pérez-García, M. M. Wolf, C. Palazuelos, I. Villanueva, M. Junge
We also consider the quotient MU = ∞ (, M Nω )/I, which is a finite von Neumann algebra. Finally, we consider the group representation π : Fn → MU , defined by π(a) = (πω (a))ω + IU . Remark 17. It is trivial to check that we can do the same construction taking π¯ ω : Fn → M Nω , defined by π¯ ω (gi ) = U Nω ,i . This construction was done in [75, Sect. 1]. Following this work, and using the crucial property (22), the next theorem follows directly Theorem 18 (Wassermann). π extends to an injective ∗ -homomorphism on λ(Fn )
, which we also call π . Following the same argument, we obtain a result for the product Fn2 = Fn × Fn of the free group. Here we use ∞ (, M Nω ⊗ M Nω ) and the ideal IU2 = {(xω )ω : limU τ Nω2 (xω† xω ) = 0}, and we write MU2 = ∞ (, M Nω ⊗ M Nω )/IU2 and τω2 for the corresponding trace τω2 (xω ) = τ Nω2 (xω† xω ). As before, we define π 2 : Fn × Fn −→ MU2 by π 2 (a1 , a2 ) = (π¯ ω (a1 ) ⊗ πω (a2 ))ω + IU2 , and again, using that lim τω2 ((π¯ ω (a1 ) ⊗ πω (a2 ))) = lim τω (π¯ ω (a1 ))τω (πω (a2 )) = δa1 ,1 δa2 ,1 , U
U
we can get the analogue of Wassermann’s result: Theorem 19. π 2 extends to an injective ∗ -homomorphism between λ(Fn × Fn )
and MU2 , which we also call π 2 . The next proposition will be crucial in the proof of the main theorem. Proposition 20. There exist matrices TiiN ω ∈ M Nω2 such that, if we define SiiN ω = U Nω ,i ⊗
U Nω ,i + TiiN ω , we have sup{| aii b j j ckk kk |SiiN ω | j j | ; |aii |2 ≤ 1, |b j j |2 ≤ 1, kk
ii j j kk
|ckk |2 ≤ 1} ≤ 5 and
ii
j j
Nω lim τω2 (U NTω ,i ⊗ U N† ω ,i Shh
) U
= δi h δi h .
Proof. We call id : RCn −→ Cr ed (Fn ) (resp. id 2 : RC2n −→ Cr ed (Fn × Fn )) to id(|i) = gi (resp. id 2 (|i j) = (gi , g j )). By [58, Theorem 9.7.1] idcb ≤ 2 and then id 2 cb ≤ 4 (just by tensoring with ⊗min , since Cr ed (Fn ) ⊗min Cr ed (Fn ) ⊂ Cr ed (Fn × Fn ) [58, Chapter 8]). We consider the map π 2 id 2 : RC2n → MU2 and the amplification 2
π 2 id 2 ⊗ 11n 2 : RC2n ⊗min min(n2 ) −→
∞ (M Nω ⊗M Nω ) 2 IU
2
⊗min min(n2 ).
Unbounded Violation of Tripartite Bell Inequalities
477
Using that any ∗-homomorphism (in particular π 2 ) is completely contractive, that min(m 2 ) is a 1-exact operator space and Lemma 15, there exists a lifting 2 U Nω ,i ⊗ U Nω ,i + TiiN ω ⊗ |ii ∈ M Nω2 ⊗min min(n2 ) Z Nω = ii
2
with TiiN ω ∈ IU2 and supω Z Nω ≤ 5. Now we use that M Nω2 ⊗min min(n2 ) = M Nω2 ⊗ 2 n2 to show that Z Nω = sup{| aii b j j ckk kk |SiiN | j j | ; |aii |2 ≤ 1, |b j j |2 ≤ 1,
ii j j kk
ii
j j
|ckk |2 ≤ 1} ≤ 5.
kk
To conclude it is enough to show that Nω lim τω2 ((U NTω ,i ⊗ U N† ω ,i )(U Nω ,h ⊗ U Nω ,h + Thh
)) = δi,h δi ,h .
U
(23)
Indeed, by (22) we have lim τω2 ((U NTω ,i ⊗ U N† ω ,i )(U Nω ,h ⊗ U Nω ,h )) = δi,h δi ,h . U
Nω 2 Moreover, since (Thh
) ∈ IU we deduce Nω † 2 T 1/2 lim |τω2 ((U NTω ,i ⊗ U N† ω ,i )Thh
)| ≤ lim τω ((U Nω ,i ⊗ U N ,i )(U Nω ,i ⊗ U Nω ,i )) ω
U
U
Nω † Nω 1/2 ×τω2 ((Thh
) (Thh )) Nω † Nω 1/2 = lim τω2 ((Thh = 0.
) (Thh ))
U
Remark 21. The operators TiiN are highly non-trivial. This can be seen by noticing that n n √ i=1 U N ,i ⊗ U N ,i M N 2 = n. This is by factor n larger than i=1 U N ,i ⊗ U N ,i + √ N Tii ≤ 5 n, guaranteed from the Wassermann lifting. Proof of the result. We define the (unnormalized) state |ψ Nω = j|U N† ω ,i |k|i jk.
√1 n Nω
1≤i≤n 1≤ j,k≤Nω
We know that these matrices verify the estimate from Lemma 16 and
hence
1 , |ψ Nω n2 ⊗ M Nω √ n Nω which means that ψ Nω |ψ Nω 1. 2
N2
N2
We define the trilinear form v Nω : n2 × 2 ω × 2 ω −→ C by v Nω (|ii , | j j , |kk ) = kk |Sii | j j .
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D. Pérez-García, M. M. Wolf, C. Palazuelos, I. Villanueva, M. Junge
Thanks to Proposition 20, v Nω ≤ 5. If we call q = id ∗ , where id is the map given in Theorem 14, we define T via the diagram n2
2 Nω
2 Nω
2∞ × 2∞ × 2∞ @
@ T @ @ ? R v Nω@ N2 N2 -C × 2 ω × 2 ω
q⊗q⊗q
n2
2
M −→ RC is a complete quotient It is clear that T 1. Moreover, since q : ∞ m 2 2 n Nω 2 2 ˆ (Theorem 14), there exist b ∈ Mn (∞ ) and b ∈ M Nω (∞ ) such that n 1 |ii | ⊗ |ii , (11n ⊗ q)(b) = √ n ii =1
Nω ˆ = √1 (11 Nω ⊗ q)(b) | j j | ⊗ | j j , Nω j j =1
ˆ 1 b, b
(by Lemma 13).
It remains to be proven that (for some Nω ) √ ˆ b)|ψ ˆ Nω n. ψ Nω |Tn,Nω ,Nω (b, b, To see this we notice ˆ b)|ψ ˆ Nω ψ Nω |Tn,Nω ,Nω (b, b, ⎛ ⎞ 1 = v Nω (|ii , | j j , |kk ) |ii | ⊗ | j j | ⊗ |kk |⎠ |ψ Nω √ ψ Nω | ⎝ Nω n ii j j kk 1 = 2 √ kk |SiiN ω | j j j|U N† ω ,i |k j |U N† ω ,i |k Nω n n ii j j kk 1 = 2 √ tr((U NTω ,i ⊗ U N† ω ,i )SiiN ω ) Nω n n ii √ 1 Nω † 2 T
= √ tr τω (U Nω ,i ⊗ U Nω ,i Shh )|ii hh | −→ n, U n n
ii hh
since, by Proposition 20, Nω
τ Nω2 (U NTω ,i ⊗ U N† ω ,i Shh lim
)|ii hh | = id n 2 . U
ii hh
The result follows trivially. The optimality part is a trivial consequence of the following
2
Unbounded Violation of Tripartite Bell Inequalities
479
N −→ N ⊗ N , if we call v the Proposition 22. For any N and any linear map v : ∞ 1 n 1 N N amplification vn = 11n ⊗ v : Mn (∞ ) −→ Mn (1 ⊗min 1N ), then
vn
√
nv.
Proof. We recall that E n is the linear span of the first n Rademacher functions in L 1 (Dn ). Fn will be E n ⊗ E n ⊂ L 1 (Dn × Dn ). By the classical Khintchine’s inequalities (see for instance [22], Sect. 8.5), we have that ⎞1 ⎛ 2 1⎝
|αi j |2 ⎠ α ij i j 2 ij ij
. L 1 (Dn ×Dn )
Hence, the norm of the identity id : Fn −→ S2n (i j → |i j|) is 1 and therefore √ (recall that · S2n ≤ n · Mn ) the norm of the adjoint map id : Mn −→ Fn∗ = √ L ∞ (Dn ×Dn ) is n. Using that the formal identities Rn −→ RC∗n , Cn −→ RC∗n are F⊥ n
completely contractive and Theorem 14, we get that the identity id : Rn ⊗∧ Cn −→ E n ⊗ E n ⊂ L 1 (Dn ) ⊗∧ L 1 (Dn ) = L 1 (Dn × Dn ) has completely bounded norm 1. Then the adjoint map id : Fn∗ −→ Mn cb 1. N ) = M ⊗ N with norm ≤ 1. There Let us take now x = i j |i j| ⊗ xi j ∈ Mn (∞ n ∞ √ N exists a function f ∈ L ∞ (Dn × Dn )(∞ ) such that f n and xi j =
Dn ×Dn
i j f (, )dµn ()dµn ( ).
For that we have used that if Q : X −→ Y is an isometric quotient, then Q ⊗ id : N −→ Y ⊗ N is also an isometric quotient (see for instance [22], Sect. 4.4). X ⊗ ∞ ∞ √ If we denote g = id ⊗ v( f ) ∈ L ∞ (Dn × Dn )(1N ⊗ 1N ), then g nv and v(xi j ) =
Dn ×Dn
If Q : L ∞ (Dn × Dn ) −→ Fn∗ =
i j g(, )dµn ()dµn ( ).
L ∞ (Dn ×Dn ) Fn⊥
is the canonical quotient map, the com i j h |i j|) has position id Q : L ∞ (Dn × Dn ) −→ Mn (given by id Q(h) = i j completely bounded norm 1 and then vn (x) Mn ( N ⊗min N ) = id Q ⊗ 11 N ⊗ N (g) Mn ( N ⊗min N ) 1
1
1
1
1
1
≤ id Qcb g L ∞ (Dn ×Dn )⊗ ( N ⊗min N )
1
1
√ nv,
since, by Grothendieck’s theorem, 1N ⊗min 1N 1N ⊗ 1N (see Sect. 3).
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D. Pérez-García, M. M. Wolf, C. Palazuelos, I. Villanueva, M. Junge
6. Conclusion We have shown that some tripartite quantum states, constructed in a random way, can lead to arbitrarily large violations of Bell inequalities with only two outcomes per observable. Moreover, and contrary to what happens in other cases, the GHZ state does not share this extreme behavior. Apart from the results (in particular we answer a long standing open question of Tsirelson) and from the applications that can be derived (see Sect. 2), we think that one of the main achievements in the paper is the use of completely new mathematical tools in this context. We hope that the techniques and connections we have established here will provide a better understanding of Bell inequalities in the near future. In this direction we would like to finish with some open problems.
Some open questions. We have proven that there are reasonably many states leading to large violations of Bell inequalities, since we have constructed them using random unitaries. However, if we focus on the inequalities (rather than on the states) the picture is much less clear. Apart from seeking for an explicit form (see the Remark after Theorem 1) one could ask the following: Question 1: How many Bell inequalities give large violation? Following the relations found in this paper, one can formulate this question in the following quantitative way: Question 1’: Are the volumes of the unit balls of n1 ⊗ n1 ⊗ n1 and n1 ⊗min n1 ⊗min n1 comparable? Once more Chevet’s inequality gives us the right estimate for the volume of the unit ball of n1 ⊗ n1 ⊗ n1 . So the question can be finally stated as Question 1”: Which is the (asymptotic) volume of the unit ball of n1 ⊗min n1 ⊗min n1 ? Unfortunately, the techniques used in this paper do not seem to help much to tackle this problem, and probably new ideas have to come into play. Another interesting question arising from the paper is the possibility of giving highly non-local states with a simpler structure than the ones given here. For instance, it would be nice to know if D Question 2: Can one find a diagonal state |ψ = i=1 αi |iii giving unbounded violation to a Bell inequality? We have proven that the GHZ (i.e. αi = √1 for every i) does not, but, interestingly D enough, Question 2 is equivalent to the following completely mathematical question Question 2’: Is S∞ (the space of compact operators in a Hilbert space) a Q-algebra with the Schur product? This question, that was formulated by Varopoulos in 1975 [71], is still open, though there has been some progress towards its solution [42,53]. A nice exposition about Q-algebras can be found in [25, Chapter 18]. We review here the basics to connect Questions 2 and 2’. A Q-algebra is defined as a commutative Banach algebra isomorphic to a quotient algebra of a uniform algebra, where a uniform algebra is simply a closed subalgebra of the algebra of continuous functions C(K ) for some compact Hausdorff space K . For a brief exposition of the history and importance of this kind of algebras we refer the reader to [25, Chapter 18, Notes and Remarks]. A very important step in the understanding of these algebras was made by Davie [20], by proving the following criterion:
Unbounded Violation of Tripartite Bell Inequalities
481
Theorem 23. A commutative Banach algebra X is a Q-algebra if and only if there is a universal constant K such that ti1 ...i N xi11 · · · xiNN X ≤ K N sup ti1 ...i N i11 · · · iNN . j | |=1 i 1 ,...,i N i 1 ,...,i N ij
j
j
For every choice of elements xi j ∈ X with xi j ≤ 1. To be precise this is not exactly the formulation made by Davie, but one can easily obtain it following the reasonings of [25, Prop. 18.6, Theorem 18.7]. Using Theorem 23, we can formalize the relation between Questions 2 and 2’: Theorem 24. S∞ is a Q-algebra if and only if there is a universal constant K such that D αi |ii · · · i, the largest violafor any N and any diagonal N -partite state |ψ = i=1 tion that |ψ can induce in a Bell inequality (with an arbitrary number of dichotomic observables) is bounded by K N . Proof. Let us assume first that S∞ is a Q-algebra. By Theorem 23, for real ti1 ...i N and j j hermitian Ai j ∈ M D ⊂ S∞ with Ai j M D ≤ 1 we have that ti1 ...i N Ai11 ∗ · · · ∗ AiNN S∞ ≤ K N sup ti1 ...i N i11 · · · iNN j |i |=1 i 1 ,...,i N i 1 ,...,i N j ≤ (2K ) N sup ti1 ...i N i11 · · · iNN , j =±1 i 1 ,...,i N ij
where ∗ means Schur (or Hadamard) product. We now notice that ti1 ...i N Ai11 ∗ · · · ∗ AiNN S∞ i 1 ,...,i N
= max r | |r
i 1 ,...,i N
ti1 ...i N Ai11 ∗ · · · ∗ AiNN
|r
= max ti1 ...i N a i a j ii · · · i| Ai11 ⊗ · · · ⊗ AiNN | j j · · · j 2 i |ai | =1 i ,...,i i, j 1 N 1 N = max (24) ti1 ...i N ψ| Ai1 ⊗ · · · ⊗ Ai N |ψ . |ψ diagonal i 1 ,...,i N
For the other implication we assume by hypothesis, and using (24), that 1 N N 1 N ti1 ...i N Ai1 ∗ · · · ∗ Ai N S∞ ≤ K sup ti1 ...i N i1 · · · i N j =±1 i 1 ,...,i N i 1 ,...,i N ij
(25)
482
D. Pérez-García, M. M. Wolf, C. Palazuelos, I. Villanueva, M. Junge j
j
for real ti1 ...i N and hermitian Ai j ∈ M N ⊂ S∞ with Ai j M N ≤ 1. By splitting into real and imaginary parts it is easy to obtain (25) for complex ti1 ...i N and arbitrary matrices j Ai j ∈ M D of norm 1 (maybe with a different constant K ). Since, given any > 0, we can approximate any element x ∈ S∞ of x ≤ 1 by a matrix A ∈ M D with A ≤ 1 and x − A S∞ ≤ , we obtain sup j xi ≤1 j
≤ K N
i 1 ,...,i N
ti1 ...i N xi11 ∗ · · · ∗ xiNN S∞
1 N
N 1 N sup ti1 ...i N i1 · · · i N ≤ K sup ti1 ...i N i1 · · · i N , j j =±1 i 1 ,...,i N | |=1 i 1 ,...,i N ij
which finishes the proof of the theorem.
ij
Finally, concerning the random procedure we use to get our main result, we would like to pose two additional questions. The first one is motivated by the recent solution [35,80], in the negative, to the maximal p-norm multiplicativity conjecture. In [80], the counterexample is given by the channel ρ → n1 i Vi ρVi† , where Vi are random unitaries. In fact, the state corresponding (via the Choi-Jamiolkowski isomorphism) to this channel is exactly our highly non-local state. To see that it is enough to look at Fig. 1. Question 3: Does our construction say something about the additivity problem [66]? The other question is motivated by the recent interest in unitary designs [32], as a way to derandomize many quantum protocols. Question 4: Is it possible to derandomize our construction to get an explicit form for our highly non-local state? If one wants to follow the path given in this paper, one should try to get a fixed finite set of unitaries satisfying at the same time the estimate of Lemma 16 and the conditions of Lemma 20. Acknowledgements. The authors are grateful to D. Kribs and M.B. Ruskai for the organization of the BIRS workshop Operator Structures in Quantum Information Theory, where part of this paper was made. We thank M. Zukowski, M. B. Ruskai and the referees for valuable comments and acknowledge financial support from Spanish grants MTM2005-00082 and Ramón y Cajal. Marius Junge is partially supported by the National Science Foundation Foundation DMS 05-56120.
Appendix: Proof of Theorem 2 The aim of this appendix is to show again the advantages of using the theory of tensor norms to tackle some problems on Quantum Information. Here we provide a new proof for Theorem 2. The key point of the proof is the following characterization of separability [62] (see also [54]): Theorem 25. A tripartite state ρ on Cd ⊗ Cd ⊗ Cd is separable if and only if it is in the closed unit ball of 3j=1,π S1d . The following lemma will be crucial.
Unbounded Violation of Tripartite Bell Inequalities
483
Lemma 26. The identity id : ⊗3j=1,2 d2 ⊗π ⊗3j=1,2 d2 −→ ⊗6j=1,π d2 has norm d 2 , where 2 is the usual (Hilbert-Schmidt) tensor norm that makes d2 ⊗2 2 d2 = d2 . We will need some concepts about unconditionality on Banach spaces. We refer the reader to [25, Chap. 17]. First we need to introduce some definitions. Given a linear operator between two finite dimensional Banach spaces u : X −→ Y , we define its 1-summing norm π1 (u) as the smallest constant K that makes the following inequality hold for arbitrarily chosen elements x j ∈ X : u(xi ) ≤ K sup |x ∗ (xi )|. i
x ∗ ∈X ∗ , x ∗ ≤1 i
We can also define the 1-factorable norm of u, namely γ1 (u), as inf ab, where a : X −→ 1N , b : 1N −→ Y and u = ba. Both norms define operator ideals [22] in the sense that they verify the inequalities π1 (uvw) ≤ uπ1 (v)w, γ1 (uvw) ≤ uγ1 (v)w. Now, given a finite dimensional Banach space X , we can define its Gordon-Lewis constant gl(X ) as the smallest constant k such that γ1 (u) ≤ kπ1 (u) for every linear operator u : X −→ 2N . Proof of Lemma 26. In [21] it is proven that the Gordon-Lewis constant gl(⊗3j=1, d2 ) d, which by duality [25, Prop. 17.9] implies gl(⊗3j=1,π d2 ) d. By the ideal property of γ1 and π1 , it can be easily deduced that if u : X −→ Y and v : Y −→ X are two operators such that id X = vu, then gl(X ) ≤ uvgl(Y ). Then, since gl(⊗3j=1,2 d2 ) 1 [25, Cor. 4.12], the norm of the identity id : ⊗3j=1,2 d2 −→ ⊗3j=1,π d2 has to be d. The lemma is then a consequence of the metric mapping property for the π tensor norm (see Section 3). Since the trace class S1d can be identified with d2 ⊗π d2 , Lemma 26 implies that there exists a d 3 × d 3 matrix ρ such that ρ S d 3 = 1 and ρπ d 2 . Using the Cartesian 1
decomposition ρ = Re ρ + i Im ρ one can assume ρ to be hermitian, and then, decomposing again into the positive and negative part, one obtains that ρ can indeed be taken positive, and hence with trace 1. But now, the state ρ = pρ + (1 − p) d113 verifies that 2 ρ π > 1 and is therefore entangled for every p > 1+ρ d12 . This proves Theorem π 2. References
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Commun. Math. Phys. 279, 487–496 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0434-4
Communications in
Mathematical Physics
Spreading of Lagrangian Regularity on Rational Invariant Tori Jared Wunsch Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208, USA. E-mail: [email protected] Received: 6 March 2007 / Accepted: 21 June 2007 Published online: 16 February 2008 – © Springer-Verlag 2008
Abstract: Let Ph be a self-adjoint semiclassical pseudodifferential operator on a manifold M such that the bicharacteristic flow of the principal symbol on T ∗ M is completely integrable and the subprincipal symbol of Ph vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes u h of Ph . We show that on a nondegenerate rational invariant torus, Lagrangian regularity of u h (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional “diffractive” manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics. 1. Introduction It is a well-known fact of semiclassical microlocal analysis, that the analogue of Hörmander’s theorem on propagation of singularities for operators of real principal type [7] holds for the semiclassical wavefront set (also known as “frequency set”): it propagates along null bicharacteristics of operators with real principal symbol [6,12]. Given a Lagrangian submanifold L of T ∗ M, we may introduce a finer notion of regularity, the local Lagrangian regularity along L. We show here that on rational invariant tori in integrable systems, local Lagrangian regularity not only propagates along bicharacteristics, but spreads in additional ways as well. Let Ph be a semiclassical pseudodifferential operator on a manifold M, with real principal symbol p (this is automatic if P is self-adjoint). Assume that the bicharacteristic flow of p is completely integrable. (In fact we only need to assume integrability locally, near one invariant torus.) Let u h be a family of quasimodes of Ph , i.e. assume that (Ph − λ)u h L 2 = O(h N ) for some N ∈ N, as h ↓ 0 either through a discrete sequence or continuously. (Note that this certainly includes the possibility of letting u h be a sequence of actual eigenfunctions). Let L be an invariant torus in the characteristic set { p = λ}. Then the bicharacteristic flow is by definition tangent to L, and we
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show (even in the absence of the integrability hypothesis) that Lagrangian regularity propagates along bicharacteristics—this is Theorem A below. If a single trajectory is dense in L, then this is the whole story for propagation, as the set on which Lagrangian regularity holds is open, hence the whole torus either enjoys Lagrangian regularity or none of it does. At the opposite extreme, if L is a torus on which all frequencis of the motion are rationally related, we may ask the finer question: what subsets of the space of all orbits may carry Lagrangian regularity? The answer (assuming a nondegeneracy condition holds) turns out to be somewhat constrained: given a single orbit, Lagrangian regularity along a small tube around it implies Lagrangian regularity along the orbit itself. This is our Theorem B. (In the special case of two-dimensional tori, we can go further: again, either the whole torus enjoys Lagrangian regularity or no points on it do.) The order of regularity up to which our result holds is constrained by the order of the quasimode. We speculate that a finer theorem may be obtainable by more authentically “second-microlocal” methods. Example 1. As a simple example of our main result, Theorem B, we consider the case M = Sx1 × S y1 , Ph = h 2 = −h 2 (∂ 2 /∂ x 2 +∂ 2 /∂ y 2 ); we consider Lagrangian regularity on the Lagrangian torus L = {ξ = 0, η = 1} for quasimodes satisfying (h 2 − 1)u h ∈ h k+1 L 2 (S 1 × S 1 ). Lagrangian regularity on this particular L is special in that we may test for it using powers of the differential operator Dx = i −1 (∂/∂ x). The theorem tells us the following in this case: let ϒ(x) be a smooth cutoff function supported on {|x| ∈ [, 3]} and nonzero at ±2. Let φ be another cutoff, nonzero at the origin and supported in [−2, 2]. If, for all k ≤ k, we have k Dx (ϒ(x)u h ) ≤ C < ∞, then for all k ≤ k,
k Dx (φ(x)u h ) ≤ C˜ < ∞,
i.e. the Dxk regularity fills in the “hole” in the support of ϒ. In this special case, the result can be proved directly by employing a positive commutator argument using only differential operators; the positive commutator will arise from the usual commutant h −1 x Dx . A less trivial example, that of the spherical pendulum, is discussed in §3 below. The methods of proof (and the idea of the paper) arose from work of Burq-Zworski [4,5] and a subsequent refinement by Burq-Hassell-Wunsch [3] on the spreading of L 2 mass for quasimodes on the Bunimovich stadium. The central argument here is a generalization of the methods used to prove that a quasimode cannot concentrate too heavily in the interior of the rectangular part of the stadium (which is essentially the example discussed above on M = S 1 × S 1 ). We remark that our hypotheses in this paper are quite far from those in the study of “quantum integrable systems” where one examines eigenfunctions of a system of n commuting operators on an n-manifold. For instance, if we take Ph = h 2 + h 2 V on the torus, with V a real valued, smooth bump-function, then the operator Ph satisfies the hypotheses of our Theorems A and B, and yet there does not exist a system of n −1 other
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operators commuting with Ph , with independent symbols. Moreover, even in the completely integrable case, given that we study eigenfunctions of a single operator, it may be possible to use the degeneracy of the system to construct non- or partially-Lagrangian quasimodes. Little seems to be known in this direction. 2. Lagrangian Regularity We begin by setting some notation and recalling some concepts of semiclassical analysis. For detailed background on this subject, we refer the reader to [6,12]. Let M n be a smooth manifold and fix L ⊂ T ∗ M a Lagrangian submanifold. Throughout the rest of the paper, we assume1 u h ∈ L 2 (M; 1/2 ), with h ∈ (0, h 0 ); here 1/2 denotes the bundle of half-densities on M, i.e. the square root of the density bundle | n M|. We will in future, however, suppress the half-density nature of u h as well as its h-dependence, writing simply u ∈ L 2 (M); similarly, all operators will tacitly be semiclassical families of operators, operating on half-densities. The hypothesis that our operators act on half-densities ensures that if A = Oph (a) with a(x, ξ ; h) ∼ a0 (x, ξ ) + ha1 (x, ξ ) + . . . , the terms a0 (principal symbol) and a1 (subprincipal symbol) are both invariantly defined as functions on T ∗ M (see [6]). Furthermore, we will deal with an operator P rather than P − λ, absorbing the constant term into the definition of the operator. We begin by defining a notion of Lagrangian regularity of a family of functions along L, following the treatment of the “homogeneous” case in [9]. Definition 2. Let M denote the module (over h (M)) of semiclassical pseudodifferential operators with symbols vanishing on L. Let q ∈ L, k ∈ N, and u ∈ L 2 (M). We say that u has Lagrangian regularity of k (u), if and only if there is a neighborhood U of q in order k at q, and write q ∈ SL ∗ T M such that for all k = 0, 1, . . . , k and all A1 , . . . , Ak ∈ M with WF A j ⊂ U, h −k A1 · · · Ak u ∈ L 2 (M). Proposition 3. Fix q ∈ L, and let Ai (i = 1, . . . , n) be a collection of elements of M with dσ (Ai ) spanning Nq∗ L. We have
q ∈ SL (u) ⇐⇒ h −k Ai1 · · · Aik u ∈ L 2 ∀(i 1 , . . . , i k ) ∈ {1, . . . , n}k , k ≤ k. Proof. We begin with the case k = 1. Given any B characteristic on L and microsup ported sufficiently close to q, we may factor σ (B) = ci σ (Ai ) by Taylor’s theorem. Thus, letting Ci be operators with symbol ci , we obtain h −1 Ci Ai u + Ru h −1 Bu = for some semiclassical operator R, hence we obtain the desired estimate on h −1 Bu since R is uniformly (in h) L 2 -bounded. More generally, if Bα1 , . . . , Bik is a k-tuple of operators characteristic on L, we have h −k Bi1 · · · Bik u = h −k
k
(Ci j Ai j + h Ri j )u;
j=1 1 We may just as well assume that h ↓ 0 through a discrete sequence; this will make no difference in what follows.
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we then obtain the desired estimate inductively, using the fact that each commutator of the form [C, A] or [R, A] produces a further factor of h. We note that it follows from the work of Alexandrova [1, Theorem 1] that SL = L if and only if we can actually write u in the form of an oscillatory integral a(x, θ, h)eiφ(x,θ)/ h dθ plus a term with semiclassical wavefront set away from L; here φ is a phase function parametrizing the Lagrangian L in the sense introduced by Hörmander. This is the semiclassical analog of a central result in the Hörmander-Melrose theory of conic Lagrangian distributions [9, Chap. 25]. We now observe that the analogue of Hörmander’s theorem on propagation of singularities for operators of real principal type is easy to prove in our setting. Theorem A. Let P ∈ h (M) have real principal symbol p. Let L ⊂ { p = 0} be a k (u) is invariLagrangian submanifold of T ∗ M. Then Pu ∈ h k+1 L 2 (M) implies that SL ant under the Hamilton flow of p. Proof (sketch). By [8, Theorem 21.1.6], there is a local symplectomorphism taking p to ξ1 and L to L0 ≡ {ξ = 0}. Following [6, Theorem 10.18] (or the development in [1]) we may quantize this to a semiclassical FIO that conjugates P to h Dx 1 modulo O(h ∞ ) (cf. [9, Theorem 26.1.3] in the non-semiclassical setting). Lagrangian regularity along L0 is iterated regularity under h −1 (h Dx i ), i.e. is just classical Sobolev regularity, uniform in h. The theorem thus reduces to the statement that Sobolev regularity for solutions to Dx 1 u ∈ h k L 2 (M) propagates along the lines (x 1 ∈ R, x = const), which is easily verified. 3. Integrable Flow We continue to assume that P ∈ h (M) has real principal symbol. We now further assume that p = σ (P) has completely integrable bicharacteristic flow, i.e. that there exist functions f 2 , . . . , f n on T ∗ M, Poisson commuting with p and with each other, and with dp, d f 2 , . . . , d f n pointwise linearly independent. We again emphasize that we in fact only require the f i ’s to exist in some open subset of interest in T ∗ M. Let denote the characteristic set in T ∗ M. Let (I1 , . . . , In , θ1 , . . . , θn ) be action-angle variables and let ωi = ∂ p/∂ Ii be the frequencies. We also let ωi j = ∂ 2 I /∂ Ii ∂ I j . (We refer the reader to [2] for an account of the theory of integrable systems, and in particular for a treatment of action-angle variables). Let L ⊂ be a rational invariant torus, i.e. one on which ωi /ω j ∈ Q for all i, j = 1, . . . , n. We further assume that L is nondegenerate in the following sense: we assume that the matrix ⎛ ⎞ ω11 . . . ω1n ω1 ⎜ .. . . .. .. ⎟ . . . ⎟ ⎜ . (1) ⎝ω . . . ω ω ⎠ n1
nn
n
ω1 . . . ωn 0
is invertible on L. This is precisely the condition of isoenergetic nondegeneracy often used in KAM theory (see [2], App. 8D). It is easy to verify that the condition is equivalent to the condition that the map from the energy surface to the projectivization of the
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frequencies { p = 0} I → [ω1 (I ) : · · · : ωn (I )] ∈ RPn be a local diffeomorphism. For later convenience, we introduce special notation for the frequencies and their derivatives on L : we let ωi = ω|L , ωi j = ωi j |L . k (u) is a union of orbits of On L, we of course only know from Theorem A that SL H p , which, being rational, are not dense in L. There are, however, further constraints k (u). on SL
Definition 4. An annular neighborhood of a closed orbit ρ is an open set U = V \K ⊂ L such that ρ ⊂ K ⊂ V with K compact and V open in L. We can now state our main result. Theorem B. Suppose Pu = f ∈ h k+1 L 2 . Let ρ be a null bicharacteristic for p on the k (u), rational invariant torus L. If a small enough annular neighborhood of ρ is in SL then so is ρ. The meaning of “small enough” depends only on the ωi ’s. If n = 2 then either SL (u) = L or SL (u) = ∅. Thus, conormal regularity propagates “diffusively” to fill in annular neighborhoods. Example 5. Horozov [10,11] has studied the spherical pendulum, i.e. the system on T ∗ S 2 with Hamiltonian h = (1/2)|ξ |2 + x3 on T ∗ S 2 (with x3 one of the Euclidean coordinates on S 2 ⊂ R3 ). Integrals of motion √ are h and pθ , the angular momentum. Horozov showed that when h ∈ (−1, 1] ∪ [7/ 17, ∞), all√values of pθ lead to isoenergetically nondegenerate invariant tori, while for h ∈ (1, 7/ 17), there are exactly two values of pθ for which isoenergetic nondegeneracy fails. Thus our results show that if we consider quasimodes for the operator Ph = (1/2)h 2 S 2 + x3 then for any torus L not associated to the one of the exceptional pairs of (h, pθ ) identified by Horozov, either SL = L or SL = ∅. Example 6. We now illustrate with an example the necessity of the isoenergetic nondegeneracy condition. As in the introduction, let M = S 1 × S 1 , but now let P = h Dx ; it is easy to verify that no Lagrangian torus is isoenergetically nondegenerate in this case. Let L = {ξ = η = 0}, the zero-section of T ∗ M. Lagrangian regularity in this setting is, as noted above, just Sobolev regularity, uniform in h. Let ψ(y) be a bump function supported near y = 0. Then √
u(x, y) = eiψ(y)/
h
has wavefront set only in L. It is manifestly Lagrangian on the complement of supp ψ, which forms an annular neighborhood of the orbit {x ∈ S 1 , y = 0, ξ = η = 0} ⊂ supp ψ. It is not Lagrangian, however, on supp ψ, as it lacks iterated regularity under h −1 (h D y ).
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4. Symbol Construction By shifting coordinates, we may assume that ρ is the orbit passing through {θ = 0}. For each i, j let γi j = mink,l∈Z ((θi + 2π k)ω j − (θ j + 2πl)ωi ), where min denotes the value with the smallest norm, i.e. may be positive or negative. Each γi j then takes values in an interval determined by ωi , ω j , and is smooth where it takes on values in the interior of the interval. (If ωi = p/q and ω j = p /q then γi j takes values in [−πa, πa] where a = gcd(qp , pq )/qq .) The “small enough” condition in the statement of Theorem B is just the following: each γi j should be smooth on the annular neighborhood of ρ where we assume Lagrangian regularity. Note that γi j (θ ) = 0 for all i, j exactly when there exists θ˜ ∈ Rn , equivalent to θ modulo 2π Zn , such that [θ˜1 : . . . θ˜n ] = [ω1 : · · · : ωn ]. Thus the functions γi j define ρ on L : we have {I = I , γi j = 0 ∀i, j} = ρ; indeed, the vanishing of each γi,i+1 and of γn,1 suffices to define ρ, and these n functions may be taken as coordinates on L in a neighborhood of ρ. The central point of our argument will be that the γi j are “propagating variables” with derivatives along the flow that, taken together, will suffice to give Lagrangian regularity. Since the γkl define ρ and are smooth on the annular neighborhood U where we have assumed regularity, there is a smooth cutoff function ψ := ψ(γ12, , γ23 , . . . , γn−1,n , γn,1 ) with ψ = 1 on ρ and ∇ψ having its support on L contained in U. We may also arrange for ψ to be the square of a smooth function. Let φ be a cutoff supported in [−, ], with smooth square root. Let ai j (x) = ψ · φ ( I − I ) · γi j · (ωi ω j − ω j ωi ). (2) We compute first that, where γi j ∈ C ∞ , { p, γi j } = (ωi ω j − ω j ωi )
(3)
(since γi j is locally given by expressions of the form ((θi + 2π k)ω j − (θ j + 2πl)ωi ) with k, l fixed) and hence that { p, ai j } = { p, ψ}φ ( I − I )γi j (ωi ω j − ω j ωi ) + ψφ ( I − I )(ωi ω j − ω j ωi )2 . We further note that as ψ is a function of the γi j ’s, by (3) the first term in this expression is a sum of terms divisible by (ωk1 ωl1 − ωl1 ωk1 )(ωk2 ωl2 − ωl2 ωk2 ) for various ki , li . Thus we may write ek fl + ψφ ( I − I )(ωi ω j − ω j ωi )2 , { p, ai j } =
(4)
where each ek and fl vanishes on L and with support intersecting L only in U. We will also employ a symbol that is invariant under the flow: for each j = 1, . . . , n, set w j = φ ( I − I )I j .
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5. Nondegeneracy Using a positive commutator argument, we will find that we can control operators whose symbols are multiples of (ωi ω j − ω j ωi ). These quantities vanish on L, but our nondegeneracy hypothesis permits us to use them to control Lagrangian regularity on L. To see this, rewrite (ωi ω j − ω j ωi ) = (ωi − ωi )ω j − (ω j − ω j )ωi and expand about L in the I variables, to rewrite this as (ωik ω j − ω jk ωi )(Ik − I k ) + O((I − I )2 ). k
We now prove a key algebraic lemma: Lemma 7. Let v1 , . . . , vn ,and vi j , i, j = 1, . . . , n be real numbers, with vi j = v ji . The functionals αi j (x) = k (vik v j − v jk vi )xk (for i, j = 1, . . . , n) together with the covector (v1 , . . . , vn ) span (R n )∗ if and only if the matrix ⎞ ⎛ v11 . . . v1n v1 ⎜ .. . . .. .. ⎟ . . .⎟ ⎜ . (5) ⎝v . . . v v ⎠ n1
nn
n
v1 . . . vn 0
is nondegenerate. Proof of Lemma. We may assume that not all of the vi ’s are zero, as the result is trivial in that case. Let ⎞ ⎛ v1i v j − v1 j vi ⎟ ⎜ .. ζi j = ⎝ ⎠. . vni v j − vn j vi
Letting A be the matrix with entries vi j and ui j = v j ei − vi e j , where ei is the standard basis for Rn , we have ζi j = Aui j . Let U denote the span of the ui j ’s. Thus,
U⊥ = ui⊥j = R · {w ∈ Rn |[wi : w j ] = [vi : v j ] ∀i, j} = R v, i, j
i, j
v ⊥ ). The where v = (v1 , . . . vn Thus, U = v⊥ . Hence the span of the ζi j is of A( assertion of the lemma is then that A( v ⊥ ) and v are complementary if and only if the matrix (5) is nondegenerate. This equivalence follows from the observation that Aw + z v A v w = , · v , w vt 0 z )t .
hence (5) has nontrivial nullspace if and only if there exists a nonzero w ∈ v⊥ with Aw ∈ R v.
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6. Proof of Theorem B We note, first of all, that in the special case when n = 2, a neighborhood of any closed orbit ρ = ρ is itself an annular neighborhood of ρ. Hence the special result for n = 2 follows directly from the general one. We now prove Theorem B by induction on k; we suppose it true for k ≤ K − 1 (and note that for k = 0 it is vacuous). Let Ai j ∈ h (M) be self-adjoint, with symbol ai j constructed above and vanishing subprincipal symbol. Then we have by (4), h −2 E k Fl + R, (6) i h −3 [P, Ai j ] = h −2 Bi2j + k,l
with Bi j self-adjoint with vanishing subprincipal symbols, and σ (Bi j ) = bi j = (ψφ ( I − I ))1/2 · (ωi ω j − ω j ωi ),
(7)
and with E k , Fl characteristic on L with the supports of σ (E k ), σ (Fl ) intersecting L only in U. (R, E k , and Fl of course depend on i, j but we suppress these extra indices.) Let W j have symbol w j constructed above, and be self-adjoint with vanishing subprincipal symbol. Then i h −3 [P, W j ] ∈ h (M). For a multi-index α with |α| = K − 1, set Q i j = Ai j W12α1 . . . Wn2αn . We will also need the operator denoted in multi-index notation α
W α = W1α1 . . . Wn K −1 . Now we examine i h −2K −1 (P ∗ Q i j − Q i j Pu, u = i h −2K −1 ( Q i j u, f − f, Q i∗j u ).
(8)
For any δ > 0, we may estimate the RHS by 2 2 2 Cδ h −K −1 f + δ(h −K Q i j u + h −K Q i∗j u ). Note that both Q i j and Q i∗j are (2K + 1)-fold products of operators vanishing on M, and that each contains the factors Ai j and W α . By (2) and (7), σ (Ai j ) is divisible by σ (Bi j ); thus, by elliptic regularity we may estimate the RHS by −2 2 2K −K −1 2 −K α C δ h f + Cδ h − j D j u, u , h Bi j W u + α
j=0
where C is independent of δ, and each D j is a sum of products of j elements of M, all microsupported on supp Ai j ; these arise from commutator terms in which we have reordered products of elements of M.
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Now we recall that P ∗ − P ∈ h 2 h (M), hence, by (6) we may write i h −2K −1 (P ∗ Q i j − Q i j P)u, u −2 2 2K h −K E k W α u, h −K Fl W α u + h − j D˜ j u, u = h −K Bi j W α u + k,l
(9)
j=0
with the D˜ j sharing the properties of the D j above. Putting together the information from our commutator, we now have, for all δ > 0, 2 (1 − Cδ)h −2K Bi j W α u −2 2 2 2 2K ≤ Cδ h −K −1 f + (h −K E k W α u + h −K Fl W α u ) + h − j D˜˜ j u, u , k,l
j=0
(10) with the D˜˜ j ’s satisfying the same properties as D j above. Each of the E k and Fl terms is ˜˜ terms are bounded by the inductive k u, while the D controlled by our hypothesis on SL j assumption. Now we use our nondegeneracy hypothesis as reflected in Lemma 7. Recallthat L ⊂ , hence the operator P is characteristic on L; moreover, we have dp|L = ω k d Ik , hence Lemma 7 tells us that P and Bi j , for i, j = 1, . . . , n, are a collection of operators fulfilling the hypotheses of Proposition 3. Thus, adding together Eqs. (10) for all possible values of i, j, and multi-index α, together with terms involving P rather than Bi j (which vanish up to commutators of P with W ’s), we obtain the desired estimate, by Proposition 3. Acknowledgements. The author is grateful to András Vasy for helpful discussions on Lagrangian regularity, and to Clark Robinson for introducing him to isoenergetic nondegeneracy. He has also benefitted greatly from comments on an earlier version of the manuscript by Maciej Zworski (who in particular suggested the brief proof of Theorem A given here), and by an anonymous referee. This work was supported in part by NSF grant DMS-0401323.
References 1. Alexandrova, I.: Semi-Classical Wavefront Set and Fourier Integral Operators. Can. J. Math, to appear 2. Arnol’d, V.I.: Mathematical methods of classical mechanics. Second edition, Graduate Texts in Mathematics, 60, New York: Springer-Verlag, 1989 3. Burq, N., Hassell, A., Wunsch, J.: Spreading of quasimodes in the Bunimovich stadium. Proc. AMS 135, 1029–1037 (2007) 4. Burq, N., Zworski, M.: Control in the presence of a black box. J. Amer. Math. Soc. 17, 443–471 (2004) 5. Burq, N., Zworski, M.: Bouncing Ball Modes and Quantum Chaos. SIAM Review 47(1), 43–49 (2005) 6. Evans, L.C., Zworski, M.: Lectures on semiclassical analysis. Preprint, available at www.math.berkeley. edu/~zworski, 2003 7. Hörmander, L.: On the existence and the regularity of solutions of linear pseudo differential equations. Enseign. Math. (2) 17, 99–163 (1971) 8. Hörmander, L.: The analysis of linear partial differential operators. III. Pseudodifferential operators. Grundlehren der Mathematischen Wissenschaften, 274. Berlin: Springer-Verlag, 1985 9. Hörmander, L.: The analysis of linear partial differential operators. IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften, 275. Berlin: Springer-Verlag, 1985
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10. Horozov, E.: Perturbations of the spherical pendulum and abelian integrals. J. Reine Angew. Math. 408, 114–135 (1990) 11. Horozov, E.: On the isoenergetical nondegeneracy of the spherical pendulum. Phys. Lett. A 173(3), 279–283 (1993) 12. Martinez, A.: An introduction to semiclassical and microlocal analysis. New York: Springer-Verlag, 2002 Communicated by P. Sarnak
Commun. Math. Phys. 279, 497–534 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0435-3
Communications in
Mathematical Physics
Stability in High Dimensional Steep Repelling Potentials A. Rapoport1 , V. Rom-Kedar2 , D. Turaev3 1 Weizmann Institute of Science, P. O. Box 26, Rehovot 76100, Israel.
E-mail: [email protected]
2 The Estrin Family Chair of Computer Science and Applied Mathematics, Weizmann Institute of Science,
P. O. Box 26, Rehovot 76100, Israel. E-mail: [email protected]
3 Ben Gurion University, Beer-Sheva 84105, Israel. E-mail: [email protected]
Received: 9 March 2007 / Accepted: 7 September 2007 Published online: 22 February 2008 – © Springer-Verlag 2008
Abstract: The appearance of elliptic periodic orbits in families of n-dimensional smooth repelling billiard-like potentials that are arbitrarily steep and limit to Sinai billiards is established for any finite n. For typical potentials, the stability regions in the parameter space scale as a power-law in 1/n and in the steepness parameter. Thus, it is shown that even though these systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity of this limit system is destroyed in the more realistic smooth setting. The considered example is highly symmetric and is not directly linked to the smooth many particle problem. Nonetheless, the possibility of explicitly constructing stable motion in smooth n degrees of freedom systems limiting to strictly dispersing billiards is now established. 1. Introduction At sufficiently high temperature, many-particle gas systems show fast decay of correlation, and, for most initial configurations, the time averages of this system and the appropriately defined ensemble averages coincide. This fundamental observation of Boltzmann lead to the development of the theory of statistical mechanics. It was further suggested by Boltzmann that at such temperatures the particles interaction resembles that of hard spheres, independent of the details of their effective potentials, hence, that a gas of hard spheres supplies an instructive universal model for studying statistical properties of gases. Notably, Boltzmann considered the many-particle case. Krylov explained that the fast decay of correlations of the hard sphere model is caused by the instability associated with the dispersive nature of the collision between the hard spheres, similar to the instabilities that appear in geodesic flows with negative curvature [15]. Sinai found that this instability appears in any dispersing billiard1 geometry (later on called 1 The behavior of a point particle travelling with a constant speed in a region D, undergoing elastic collisions at the region’s boundary, is known as the billiard problem. The billiard is dispersing if its boundary is piecewise strictly concave when looking from the billiards’s interior.
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Sinai billiards2 ) in any dimension, and set the mathematical foundation for rigorously studying such systems. Sinai proved, in his seminal works [30,32], that such systems are ergodic and hyperbolic in the two-dimensional billiard case. He further stated (the Sinai-Boltzmann conjecture3 ) that if one considers the motion of N hard spheres on a d-dimensional torus, this motion will be mixing4 for any d ≥ 2 and N ≥ 2. In particular, the Sinai-Boltzmann conjecture means that for any N , d ≥ 2, ergodicity is achieved independently of the number of particles because of the universal nature of the instability associated with the convex particles collision. We propose that the study of real particles, with smooth potentials, or, more generally, in studying Hamiltonians with smooth steep n-dimensional potentials, may shed light on the role of dimensionality in this problem. Thus, to formalize this notion, we consider a Hamiltonian n pi2 H= + W (x; ε), (1.1) 2 i=1
where W (x; ε) is a smooth potential that becomes a hard-wall potential5 in the limit ε → 0: 0 x ∈ D\∂ D, W (x; ε) → ε→0 c x ∈ ∂ D. In general, studying (1.1) for a finite ε value is a formidable task. Boltzmann’s insight and Sinai’s theory, in which the n-dimensional nonlinear system is replaced by the study of billiards, serve as a great simplification. To mimic the soft nature of the particles interactions and still obtain a tractable system, finite-range axis-symmetric potentials were introduced. It was established that these systems may be studied by a modified (non-smooth) billiard map, and thus that in two dimensions some configurations remain ergodic [30,31,16,6,3], while other configurations may possess stability islands [2,5]. More recently, some higher dimensional configurations were proved to be hyperbolic [4]. Yet, it was noticed in [35] that the behavior of any smooth approximation has to be fundamentally different from the discontinuous behavior of the billiards. Indeed, in mathematical terms the Krylov-Sinai instability translates to the existence of a universal hyperbolic structure in any dispersing billiard problem. More precisely, the family of cones d x · dp > 0 is forward invariant with respect to the billiard flow in the dispersing case independent of the details of the billiard’s shape. After each reflection from the billiard’s boundary, the cones are mapped into each other with flipped orientation (the normal component of the momentum p changes sign, while all other components are preserved), see [32,37,35]. In particular, nearby orbits experiencing a different number of reflections (i.e. near tangencies or near corners), have unstable manifolds with opposite orientability properties – one orientable and the other non-orientable [35]. Such a discontinuous dependence of the unstable manifold on initial conditions in smooth uniformly hyperbolic systems is impossible. 2 Strictly dispersing billiards for which the smooth boundary components intersect at positive angles (no cusps are allowed). 3 Proved initially for the N = d = 2 by Sinai [32], then for the N = 2, d = 3 by Sinai and Chernov [33] whereas the most general higher dimensional cases were studied by Krámli, Simányi, and Szász, see [11–14,26,25,29,27,28]. 4 On the reduced manifold, eliminating the total energy and momenta conservation laws. 5 Here c > 0 may be finite or infinite, and we always take the particle’s energy, h, to be positive and strictly smaller than c so that the particle cannot cross ∂ D.
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On the other hand, the hyperbolic structure near regular orbits of the billiard (e.g. periodic orbits that are bounded away from the singularity set) is typically inherited by steep billiard-like potentials [35,21]. It follows that the Krylov-Sinai instability mechanism indeed controls the smooth dynamics but only for some limited time scale, after which the non-hyperbolic behavior which stems from the billiard singularities will prevail. Therefore, we propose that the dependence of this instability time scale on the number of particles and other parameters is the most relevant question in the study of many-particle systems. One concludes then, that in order to study the dynamics of real particles, one needs to study (1.1) for smooth steep potentials, utilizing the theoretical advancements regarding the singular billiard limit as a tool in this study. This approach requires a well-defined limiting procedure that is well developed by now [17,22,21]. This formulation was first introduced in the most general two-dimensional setting of Sinai billiards (not necessarily axis-symmetric, nor of finite range) in [22]. After proving that regular hyperbolic orbits of the billiard persist in the smooth flow, two mechanisms by which the billiards ergodicity property is destroyed were identified [22,36]. One such mechanism is a tangency: periodic orbits or homoclinic orbits that are tangent to the billiard’s boundary produce islands of stability [22]. Another mechanism are corners–a sequence of regular reflections that begins and ends in a corner (termed a corner polygon) may, under some prescribed conditions, produce stable periodic orbits [36]. In both cases it was shown that a two-parameter family of potentials W (x; µ, ε) (ε is the steepness parameter and µ is responsible for a regular continuous change of the billiard’s geometry) possesses a wedge in the (µ, ε)-plane, at which the Hamiltonian flow has an elliptic periodic orbit. This orbit limits to the tangent billiard orbit or the corner polygon as ε → 0. These findings were shown to correctly describe the motion of cold atoms in atom-optics billiards in laboratory experiments [10]. What would one expect in the multi-dimensional case? Can there be other types of universal instabilities, besides the Krylov-Sinai one, that would make such systems ergodic for sufficiently steep potentials? Namely, would the billiard’s ergodicity be preserved for n-dimensional steep billiard-like potentials when n ≥ 3? While there are some conjectures regarding the generic appearance of islands in smooth n degrees of freedom systems, results of this nature appeared only in the case of C 1 -flows and assume the systems are not partially hyperbolic (see [34,19,1,24]), which is the heart of the problem here. Indeed, the above described mechanism of orientation flipping, which corresponds to a direct generalization of our previous two-dimensional results (e.g. [22]) to dispersing n-dimensional billiards, will produce orbits that have one pair of imaginary multipliers (ruining hyperbolicity), yet all the other (n − 2) pairs can still correspond to hyperbolic behavior. Thus, though destroying hyperbolicity, this mechanism is not necessarily going to kill ergodicity in the smooth case, as the existence of some uniform partially hyperbolic structure is not ruled out. This intuition might lead one to believe that the mechanisms described in [22,36] for ruining ergodicity are inherently two-dimensional. However, it was numerically demonstrated recently that regions of effective stability, hereafter called islands, are created in steep dispersing three-dimensional billiards for what appears to be arbitrarily small ε [20]. Before further describing this construction and its current generalization to the n degrees of freedom case, let us discuss the issue of islands in the multi-dimensional context. As opposed to the two-dimensional situation, due to the possible existence of Arnold diffusion, one cannot claim that in the vicinity of a non-degenerate non-resonant elliptic
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x2
21/2(L,L) R γ(ψ)
(0,l) r
1/2
2 r (l,0)
(ψc,ψc) x
1
Fig. 1. The billiard geometry in the two-dimensional and three-dimensional cases.
orbit there exists an invariant open neighborhood (on energy surfaces or on the full phase space). Nonetheless, by KAM theory, near such elliptic orbits there exists a positive measure set foliated by KAM-tori that corresponds to trajectories that remain forever near the elliptic trajectory. Furthermore, while other trajectories in this neighborhood may perhaps escape, this can take an exponentially long time [18,7] (namely, such islands may correspond to high-dimensional dynamical traps, generalizing the two-dimensional stickiness phenomena). Thus, hereafter, an island in the multi-dimensional context will be defined as the small neighborhood of the elliptic orbit which is effectively stable [7], bearing in mind that only in the two degrees of freedom case this neighborhood is known to correspond to an invariant set. The islands constructed in [20] are produced by a highly symmetric orbit of the smooth system which visits the vicinity of a symmetric 3-corner. The 3-corner is a point at which three smooth spheres of identical radius intersect in a symmetric fashion, so that only one characteristic parameter µ controls the angle of their intersection (µ = 0 corresponds to a cusp whereas µ = 1 corresponds to a complete overlap of the spheres, see Fig. 1). It is demonstrated numerically in [20] that for any value of ε there are intervals of µ values for which the symmetric orbit is elliptic. Here, we generalize this example to the n-dimensional case, for arbitrary large n, proving, that for certain classes of smooth repelling potentials (such as the power-law family) the smooth symmetric orbit which enters the vicinity of an n-corner has, for arbitrary small ε, intervals of µ values for which it is elliptic (all its 2n multipliers belong to the unit circle). Furthermore, these intervals converge to positive µ values and their length, for sufficiently small ε values, scales as a function of εn. In other words, we show that for arbitrarily large n, we can construct n-dimensional Sinai billiards and corresponding families of billiard-like smooth potentials, where, for arbitrary steepness the smooth flow possesses elliptic behavior. Our main result may be summarized by the following theorem: Theorem 1. There exist families of analytic billiard potentials that limit (in the sense of [36]), as the steepness parameter ε → 0, to Sinai billiards in n-dimensional compact domains6 , yet, for arbitrary small ε, the corresponding smooth Hamiltonian flows have stable (elliptic) periodic orbits. 6 In particular, for any finite n such billiards are hyperbolic, ergodic and mixing.
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Proof. We construct specific families of n-dimensional billiards depending on a parameter µ, such that the billiards are Sinai billiards for any µ > 0 depending smoothly on µ for µ ∈ (0, 1) (Sect. 2). We then consider families of potentials W (x; µ, ε) that limit as ε → 0, for any fixed µ, to these billiards. We establish that for sufficiently small ε these Hamiltonian flows have a periodic orbit γ (t, µ, ε) and we prove that the Floquet multipliers of this orbit may be found by solving a linear second order equation with a time-periodic coefficient (Sect. 3 ). This coefficient depends on µ, ε and n as parameters, and it approaches a sum of delta-like functions as ε → 0. For certain classes of W (x; µ, ε) (e.g. when W (x; µ, ε) decays as a power-law in the distance to the scatterers) we are able to analyze the asymptotic behavior of the emerging linear second order equation: we prove that for these potentials there are countable infinity values of µ, one of them given by √1n (i.e. bounded away from µ = 0, 1), from which a wedge of stability region in the (µ, ε) plane emerges. Namely, we prove that for any n, for arbitrary small ε, there exists an interval of µ values at which γ (t, µ, ε) is linearly stable (Lemma 1 in Sect. 3). In particular, this theorem proves that such systems are not partially hyperbolic. The paper is ordered as follows; we first construct the geometry of the limiting billiard domain. The construction of the billiards boundary, by intersecting several (n − 1)dimensional spheres in R n , is valid for any finite dimension n. Then, we establish that in the smooth case, for sufficiently small ε, there exists a symmetric periodic orbit γ (t) which corresponds to the one dimensional motion along the diagonal (in the n-dimensional space), and that this motion may be found by integrating a one-degree of freedom system which is independent of n. Next we show that the linear stability analysis about this motion is governed by a single second order linear differential equation with a time periodic coefficient in which n appears as a parameter. In the third section we construct asymptotic solutions to this equation showing that for small εn it has intervals of parameter values at which γ (t) is linearly stable, thus establishing the main theorem. Precise estimates of the length of these intervals are found for the power-law case. In the last section we integrate numerically these equations and compare the numerically found wedges of stability with the corresponding asymptotic estimates. Finally, we demonstrate the appearance of islands of effective stability by numerical integration of the symmetric n d.o.f. system and of a slight asymmetric perturbation of it for a few n values (n = 2, 3, 10) for two different types of potential families – the power-law family and the Gaussian family (e.g. we present islands of effective stability of dispersing, repelling, nonlinear 20 dimensional system). 2. Construction of the Billiard and the Limiting Smooth Flows 2.1. The billiard geometry. Define the n-dimensional billiard’s domain D as the region exterior to (n + 1) spheres S n−1 : one sphere n+1 of radius R which is centered on the diagonal at a distance L from the origin, i.e. at the point √1n (L , . . . , L), and n spheres 1 , . . . , n of radius r , each centered along a different principle axis at a distance n from the origin, i.e. the sphere k is centered at (0, . . . , l , . . . , 0) (Fig. 1). 0 ≤ l ≤ r n−1 k
To obtain a bounded domain, we enclose this construction by a large n-dimensional hyper-cube centered at the origin (we will look only at the local behavior near the diagonal connecting the radius-r spheres 1 , . . . , n to the radius-R sphere n+1 and thus we
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will not be concerned with the form of the outer boundary). The diagonal line (ξ, . . . , ξ ) intersects the radius-R sphere in the normal direction and the spheres 1 , . . . , n at their common intersection point Pc = (ξc , . . . , ξc ), where (Fig. 1):
l 1 1 ξc = + √ r 2 − l 2 1 − . (2.1) n n n √ Thus, for L > R + n ξc , it defines a corner ray
L−R γ = (ξ, . . . , ξ )| ξ ∈ ξc , √ n that starts at the corner Pc , gets reflected from the radius-R sphere and returns to Pc (and then gets stuck as there is no reflection rule at the corner). Notice that the dynamics in the billiard is unchanged when all the geometrical parameters are proportionally increased, hence, with no loss of generality, we may set r = 1 and regard all the other parameters as scaled by r . It is convenient for us to express the scaled l and L through √
1 l2 L − R − n ξc . (2.2) µ= 1− 1− and d = n r2 r The parameter d corresponds to the length of the diagonal ray γ whereas µ governs the angle created by the intersection of the n spheres at the corner point (both parameters have a finite limit as n → ∞). At µ = 0 the √ n spheres are tangent to each other, namely the corner becomes a cusp. The case µ = 1/ n corresponds to l = r, hence the spheres intersect at a right angle. The case µ = 1 corresponds to l = 0, namely the limit at which the n spheres collapse to a single sphere of radius r which is centered at the origin. In this case the diagonal becomes a hyperbolic periodic orbit of the billiard (note that the limit µ → 1 is singular: at µ = 1 the billiard’s boundary is smooth, whereas for all µ ∈ (0, 1) it has a corner). 2.2. Smooth motion – the diagonal periodic orbit. In this section we establish that for sufficiently small ε the diagonal corner ray γ of the billiard flow transforms into a periodic orbit of the smooth flow. Consider the smooth motion in the scaled billiard region, governed by the Hamiltonian (1.1), i.e. H=
n pi2 + W (x1 , . . . , xn ) 2
(2.3)
i=1
with
n Q n+1 1 Qk + V , W (x; ε) = V n ε ε
(2.4)
k=1
where Q k (x) (the pattern function of [23,21]) is the distance from x to k :
n
2 Q k (x) = xi − 2lxk + l 2 − 1 for k = 1, . . . , n, i=1
n
L Q n+1 (x) = (xi − √ )2 − R n i=1
(2.5)
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(recall that we scale r = 1). The potentials associated with the r −spheres i.e.V Qεk are multiplied by the 1/n factor so that for all n values the potential height near the corner is of the same magnitude as the potential near the R-sphere. The C k+1 (k ≥ 1) smooth function V satisfies at z > 0, V (z) > 0 and V (z) < 0,
(2.6)
so the potentials are repelling. We further assume that V (z) decays sufficiently rapidly for large z (in accordance to the assumptions in [23,21,36]), so there exists some α > 0 such that
1 (2.7) V (z) = O 2+α as z → +∞. z As a typical V , one can take the power-law potentials: α 1 V (z) = , α > 0, z the Gaussian potential
V (z) = exp −z 2 ,
(2.8)
(2.9)
or the exponential potential V (z) = exp (−z), that naturally appear in applications (e.g. the Gaussian form arises in the problem of cold atomic motion in optical traps [10], whereas the power-law and exponential potentials are abundant in various classical models of atomic interactions). The potential W (x; ε) given by (2.4), (2.5) is symmetric with respect to any permutation of the xi ’s (i = 1, . . . , n). This strong symmetry enables us to progress with the analysis for any n. Notice that it is easy to break this symmetry, by, for example, multiplying the terms V (Q k (x)/ε) in (2.4) by slightly different coefficients. Such a modification is studied numerically in Sect. 4.2. Now, consider the smooth motion along the diagonal x1 = · · · = xn = ξ . By the symmetry, ∂ ∂ W (ξ, . . . , ξ ) = W (ξ, . . . , ξ ) ∂ x1 ∂ xi
for i = 1, . . . , n,
so the plane {x1 = · · · = xn = ξ, p1 = · · · = pn = ξ˙ } is an invariant submanifold of the phase space. It follows from the conservation of energy that H =n
ξ˙ 2 + W (ξ, . . . , ξ ), 2
(2.10)
for the orbits in this manifold; by differentiating this identity we obtain the following equation of motion on the invariant plane: ξ¨ + Let
∂ W (ξ, . . . , ξ ) = 0. ∂ x1
(2.11)
√ n(ξ − ξc ),
(2.12)
ν=
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where ξc is defined by (2.1) with r = 1. The energy conservation law (2.10) at the fixed energy level h/2 reads as h ν˙ 2 (2.13) = + We f f (ν; ε, µ, d), 2 2 where the effective potential is as follows (see (2.4),(2.5) and (2.12)):
1 + 2µν + ν 2 − 1 d −ν We f f = V . (2.14) +V ε ε Equation (2.11) for the motion on the diagonal line transforms then into the equation (which is independent of n): ∂ We f f (ν; ε, µ, d) = 0. (2.15) ∂ν This is a Hamiltonian equation with the Hamiltonian given by the right-hand side of (2.13). Since V < 0, for any finite ε, the potential We f f (ν; ε, µ, d) has a minimal value for ν in the interval (0, d) and the potential is monotonically increasing as the boundaries of this interval are approached. Thus, by (2.13), it has periodic solutions for the non-critical values of h in the interval: ν¨ +
h > h min (ε, µ, d) := 2 min We f f (ν)
(2.16)
(at h = h min the periodic orbit degenerates into an equilibrium point). The critical values of h are those at which We f f has maxima, and then the periodic orbit is replaced by homoclinic or heteroclinic orbits. Summarizing, we have established the following lemma: Lemma 1. For every non-critical value of h > h min (ε, µ, d) the Hamiltonian flow (2.3) satisfying (2.4)–(2.6) possesses in the energy level H = h2 a periodic solution of the √ + ξc with v(t) ∈ (0, d) being diagonal form: γ (t) = (ξ(t), . . . , ξ(t)), where ξ(t) = v(t) n a periodic solution of (2.15) with energy h2 . Let T (ε, µ, d, h) denote the period of γ (t). To fix the notation, let us parameterize time along γ (t) so that t = 0 will correspond to the turning point near the corner whereas T /2 corresponds to the turning point near the large sphere, namely: We f f (ν(0)) = We f f (ν(T /2)) =
h 2
with ν(0) ≈ 0, ν(T /2) ≈ d. 3. Stability of the Periodic Orbit To study the stability of the periodic orbit γ (t), one needs to linearize the Hamiltonian equations of motion corresponding to (1.1) about this solution, solve the corresponding 2n-dimensional linear system with the time-periodic coefficients for a set of 2n orthonormal initial conditions and find the stability of the associated (2n × 2n)-dimensional monodromy matrix, leading finally to a set of 2n Floquet multipliers (2 of which are trivially one). The symmetric form of the potential allows to reduce this formidable task to a much simpler one – to solving a single second order homogeneous equation with a time periodic coefficient which depends on n as a parameter in a very simple form:
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Lemma 2. The Floquet multipliers of the T -periodic orbit γ (t) are (1, 1, λ, λ1 , . . . , λ, λ1 ), where λ is given by: Tr(A)2 1 − 1, (3.1) λ = Tr(A) + 2 4 and A is the monodromy matrix of the second order linear equation: y¨ + a(t)y = 0
(3.2)
with the T -periodic coefficient a(t) given by (see also (2.15)): V (ε−1 ( 1 + 2µν + ν 2 − 1)) V (ε−1 (d − ν)) a(t; ε, µ, d, R, n, h) = + ε(R + d − ν) ε 1 + 2µν + ν 2 1 − µ2 V (ε−1 ( 1 + 2µν + ν 2 − 1)) + n−1 ε2 (1 + 2µν + ν 2 ) V (ε−1 ( 1 + 2µν + ν 2 − 1)) − ε (1 + 2µν + ν 2 )3 = a − (ν(t); ε, µ, d, R, h) +
1 − µ2 + a (ν(t); ε, µ, d, h), (3.3) n−1
Proof. Consider the linearization about γ (t) of the system defined by (2.3). Let: ∂2 W (ξ(t), . . . , ξ(t)), ∂ x1 ∂ x2 ∂2 ∂2 W (ξ(t), . . . , ξ(t)) − W (ξ(t), . . . , ξ(t)). a(t) = 2 ∂ x1 ∂ x2 ∂ x1 b(t) =
(3.4)
By symmetry, ∂ x∂i ∂ x j W (ξ(t), . . . , ξ(t)) = b(t) for all i = j and ∂∂x 2 W (ξ(t), . . . , ξ(t)) = 2
2
a(t) + b(t) for all i. Hence, the linearization of (2.3) is given by x¨i + a(t)xi + b(t)
n
i
x j = 0, i = 1, . . . , n.
(3.5)
j=1
Let s = obtain
n
i=1 x i
and yi = xi −
s n
in (3.5). By summing the above equation on i we
s¨ + (a(t) + nb(t))s = 0, y¨i + a(t)yi = 0, i = 2, . . . , n.
(3.6)
Every equation in this system is decoupled from the others, therefore the spectrum of the Floquet multipliers of γ (t) is the union of the spectra of the monodromy matrices (i.e. the spectra of the time-T maps) corresponding to each of the equations. It is easy to check that the first equation is the linearization of (2.11) about ξ(t). Hence, both the eigenvalues of its monodromy matrix are equal to 1 (as (2.11) is a Hamiltonian equation). These correspond to trivial Floquet multipliers of γ (t). Since the rest of the equations in (3.6) are identical, the other Floquet multipliers of γ (t) correspond to the n − 1 identical pairs λ and λ−1 , the eigenvalues of the monodromy matrix of Eq. (3.2)
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with the T -periodic a(t) given by (3.4). By applying the above formulas to the system (2.3), (2.4), (2.5), and using the coordinate ν instead of ξ (see (2.12)), we obtain (3.3). To establish the main theorem, the spectral properties of the 2 ×2 monodromy matrix A of Eq. (3.2), that depend on n and the geometric parameters via a(t), need to be studied. For any finite n, when γ (t) is near the corner point (i.e. ν is close to zero) the third term of (3.3) is of order 1/ε2 and thus dominates a(t). This singular behavior leads to fast oscillations of the solutions of (3.2) at the corresponding time interval, so careful analysis of the resulting multipliers is needed. Thus, the rest of this section is dedicated to studying the dependence of the eigenvalues of A on the parameters. First, we show that in the limit of fixed ε and large n the periodic orbit γ (t) is unstable. Likewise, we show that in the limit of low energies (near h = h min (ε, µ, d), see (2.16)), the periodic orbit which oscillates near the fixed point is unstable for all n above some critical value. These observations show that the stable orbits we get do not correspond to a motion near the bottom of a potential well. Then, we prove the main result, that for any fixed n there exists a sequence of µ values, µk , such that the periodic orbit is stable in wedges in the (µ, ε) plane that are close to (µk , 0). The widths of these wedges is then found in two specific limits, with explicit formulae in the power-law potential case. In the limit n = +∞, Eq. (3.2) turns into y¨ + a − (t)y = 0. Since a − is always negative by (2.6), this equation cannot have non-trivial bounded solutions and the monodromy matrix A has multiplier λ > 1. Thus, at every fixed ε and h > h min (ε, µ, d), the diagonal solution γ (t) is linearly unstable for sufficiently large n. Therefore, it is not surprising that the stability zones that we find later on correspond to bounded values of εn, i.e. for higher dimension of the configuration space one should make the potential steeper in order to make the diagonal periodic orbit stable. The stability of the equilibrium state on the diagonal, at h = h min , is determined 1 − µ2 + by Eq. (3.2) of Lemma 2; the equilibrium is linearly stable if a − + a > 0, (n − 1) 2 1−µ + a < 0, where instead of ν(t) in a ± one should and linearly unstable if a − + (n − 1) substitute the value of ν = ν f that corresponds to the minimum of We f f (see (2.14)). Defining n c (µ, d, R, ε) = 1 +
a + (v f ) (1 − µ2 ), −a − (v f )
we see that the equilibrium (and small oscillations on the diagonal near it) are stable at n < n c and unstable at n > n c . In Fig. 2 we plot n c (µ, d, R, ε) for the power-law, exponential and Gaussian potentials, showing the dependencies of n c on µ, d and ε. In the case of power-law potential, n c does not depend on ε (see (2.8), (2.14) and (3.3)), thus, the stable periodic orbit that we find for small ε clearly does not inherit its stability from the equilibrium state, i.e. the effect has truly billiard origin. For the exponential and Gaussian cases n c diverges as ε → 0. In these cases the stable fixed point appears for exponentially small energies (see ((2.14)). Since the effective potential is essentially flat away from the scatterers, for energies that are not exponentially small, the amplitude of the oscillations becomes large and the linearization near v f is not applicable. Indeed, it is
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507
µ=0.5,d=2 60
n
c
40 20 0
0
0.1
0.2
0.3 ε
0.4
0.5
0.6
ε=0.2,d=2
20
n
c
15 10 5 0
0
0.1
0.2
0.3
0.4
0.5 µ
0.6
0.7
0.8
0.9
1
ε=0.2,µ=0.5
20
n
c
15 10 5 2
3
4
5
6 d
7
8
9
10
Fig. 2. The critical dimension, n c , beyond which the fixed point at the minimal energy level becomes unstable, for various (µ, d, ε) at R = 10. Results for three potentials are presented: power-law (solid), exponential (dotted) and Gaussian (dashed).
proved below that for such energies the periodic orbit changes its stability several times as ε → 0, so again, the stability regions we find do not correspond to small oscillations that inherit their stability from the equilibrium state. For any finite n, for sufficiently small ε, γ (t) has a finite positive period and a(t) changes sign7 as shown in Fig. 3, so the behavior of the monodromy matrix A in the limit ε → 0 becomes non-trivial. Our main result is that there are wedges in the (µ, ε) space at which the eigenvalues of A are on the unit circle: Theorem 2. Suppose the potential function V satisfies (2.6), (2.7). Then, given any h ∈ (0, 2V (0)), any natural n ≥ 2, and any positive d√and R, there exists a tending to zero countable infinite sequence 1 ≥ µ0 > µ1 = 1/ n > · · · > µk > · · · > 0 such that arbitrarily close to every point (µ = µk , ε = 0) there are wedges of (µ, ε) at which the orbit γ is linearly stable. Proof. Recall the definition of the monodromy matrix A: the linear second order differential equation (3.2) with the periodic coefficient a(t) defines the linear map: (y(t0 ), y (t0 )) → (y(t0 + T ), y (t0 + T )) = A(y(t0 ), y (t0 )). While A may depend on the choice of t0 , its eigenvalues, the Floquet multipliers of γ (t), do not. We choose t0 = − t, where t > 0 is slowly tending to zero as ε → 0, and express A as the product of two matrices: A = BC, where C corresponds to the map from t = − t to t = t (i.e. to the 7 While a − is always negative, for sufficiently small ε, there exists an interval of t values at which a + is positive (as V is negative, and V is bounded from below, it follows that V has to be positive somewhere). In fact, a + > 0 everywhere in the power-law potential case.
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A. Rapoport, V. Rom-Kedar, D. Turaev Coulomb potential
2 +
ε a (t)
0.4 0.2 0
−
εa (t)
0 −0.2 −0.4
0
1
2
3 t
4
5
6
Exponential potential
2 +
ε a (t)
0.5
0
−
εa (t)
0 −0.2 −0.4 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
t Gaussian potential
2 +
ε a (t)
1 0.5
−
εa (t)
0 0 −0.5 −1
0
0.5
1
1.5
2 t
2.5
3
3.5
4
Fig. 3. The rescaled ingredients of a(t) of Eq. (3.3). The peaks of a + (t) and a − (t) are shown to scale as 1/ε 2 and 1/ε respectively. Here µ = 0.5, and ε = 0.1, 0.01, 0.001 from widest to narrowest respectively.
linearized smooth motion in the neighborhood of the billiard corner, where the third term of (3.3) dominates and fast oscillations appear), and B corresponds to the time interval [ t, T − t] (i.e. to the linearization about the smooth regular motion along the diagonal and the regular reflection from the radius R sphere n+1 in the normal direction at t = T /2). Below, we find the form of B (Lemma 3) and C (Lemma 4) in the limit of small ε and fixed µ. In fact, we show in the proof of Lemma 4 that by rescaling time by δ = ε/µ and taking the appropriate limits of (2.15) and (3.3), the matrix C may be found by integrating a simplified scattering problem. There, only the rescaled third term of (3.3) appears with a rescaling parameter β: β=
1 − µ2 , (n − 1)µ2
(3.7)
and Eq. (2.15) is replaced by an equation which is independent of ε and µ (Eq. (3.11)). We then show that the trace of A is dominated by a term of the form s21δ(β) , where s21 (β) is a coefficient of the scattering matrix associated with the simplified scattering problem. We thus conclude (since δ → 0 as ε → 0) that the wedges in the (µ, ε) plane, where the trace of A varies between −2 to 2, emanate near the points at which s21 (β) changes sign. We then show (Lemma 5) that the zeroes of s21 (β) correspond to the “spectrum” of the
Stability in High Dimensional Steep Repelling Potentials
509
simplified scattering problem. Namely, bounded solutions appear if and only if s21 (β) vanishes, and the number of zeroes of a fundamental solution of the scattering problem is even when s21 > 0 and is odd when s21 < 0. We complete the proof by noticing that in this simplified scattering problem it is easy to establish that the number of zeroes of all solutions increases to infinity as β → ∞ (i.e. when µ → 0+ ), and to conclude that there is a countable number of µ values at which bounded solutions appear. These are the values at which s21 vanishes and wedges of stability are formed. The form of B is easily found by utilizing the billiard limit (using [21]): Lemma 3. For small t and sufficiently small ε, the linearized map about the diagonal orbit: (y( t), y ( t)) → (y(T − t), y (T − t)) = B(y( t), y ( t)) satisfies 2d √ 1 + 2d (1 + Rd ) R h √ B= + o(1). (3.8) 2 1 + 2d R h R Proof. Fixing t and letting ε → 0, the diagonal periodic orbit γ (t) on the interval [ t, T − t] approaches the boundary of the billiard domain only once, at t = T /2, hitting the radius-R sphere n+1 in the normal direction. This is a regular reflection, therefore, according to [21]8 , the flow map from any time moment before the reflection to any moment after the reflection is close to the corresponding map for the billiard flow. The closeness is along with k derivatives of the map (recall that V is C k+1 , k ≥ 1), i.e. the derivative of the flow map from t = t to t = T − t tends to the derivative of the billiard flow map as ε → 0. It is true for every fixed t, hence it remains true for a sufficiently slowly tending to zero t. Because of the symmetry of the diagonal orbit γ , the matrix of the derivative of the smooth flow has a block-diagonal structure with one idempotent block that corresponds to the variable s in (3.6) and the other blocks equal to B. The derivative matrix of the billiard flow has the same √ structure; to find this matrix, consider the billiard flow of a particle with a velocity h which starts at a distance d from the sphere of√radius R and reflects in the normal direction back to its original position at T = 2d/ h. Then, by direct computation, it can be shown that the ith block of the linearization of the billiard flow map is of the form: 2d d √ 1 + 2d (1 + ) ∂(yi (T − 0), y˙i (T − 0)) √R h 2d R = 2 ∂(yi (+0), y˙i (+0)) 1+ R R h and (3.8) follows from [21] as explained above (the same results can be achieved by asymptotic integration of Eq. (3.2), namely following a simplified version of the construction of C below). Finding the form of C is more complicated, and requires the integration of (3.2) in some asymptotic limits. In Appendix A, we prove the following: Lemma 4. For any fixed µ ∈ (0, 1), small t and sufficiently small ε, the linearized map about the diagonal orbit near the corner: (y(− t), y (− t)) → (y( t), y ( t)) = C(y(− t), y (− t)) satisfies
s11 + σ s21 (1 + o(1)) + o(1) s21 O(δσ 2 ) + O(δσ ) (3.9) C= 1 −1−α )) s + σ s (1 + o(1)) + o(1) , 22 21 δ (s21 (1 + o(1)) + O(σ 8 It is easy to verify that the conditions (2.6)–(2.7) on V suffice to guarantee that W (q, ε) satisfies the conditions in [21].
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where δ, σ −1 are small scaling parameters (tending to 0 as ε → 0) such that δσ = t, and S is a matrix which tends9 , as ε → 0, to a smooth limit S0 (µ). Let us explain the meaning of the matrix S and the parameters that appear in (3.9). It is shown in the Appendix that by rescaling time by δ = ε/µ, in the appropriate scaling limit, only the third term of Eq. (3.3) matters and so Eq. (3.2) near the corner reduces to d2 y + βV (z(τ )) y = 0, dτ 2
(3.10)
where βV corresponds to the limit of the third term of (3.3) (multiplied by δ 2 ), and −1 z(τ ) = ε ( 1 + 2µν(τ ) + ν(τ )2 −1) solves, in the asymptotic limit, an equation which is independent of ε and µ: h (z )2 = + V (z), z (0) = 0. 2 2
(3.11)
Notice that by (2.7) Eqs. (3.10)–(3.11) define a scattering matrix: it is shown in the Appendix that the solutions z(τ ) to (3.11) run from +∞ through some minimal positive value back to +∞ sufficiently rapidly and thus that (3.10) reduces, in the limit of τ → d2 ±∞ to dτ 2 y = 0. Then, as is usual in scattering theory, one may define two bases of solutions at the two asymptotic limits. Let y± (τ ) denote the uniquely defined solutions having the following asymptotic form as τ → ±∞ (respectively): y± (τ ) = 1 + O(|τ |−α ),
y± (τ ) = O(|τ |−1−α ).
(3.12)
Let y± (τ )) denote solutions10 with asymptotic: yˆ± = τ + O(|τ |1−α ),
y± (τ ) = 1 + O(|τ |−α )
(3.13)
so the Wronskians of (y− (τ ), y− (τ )) and of (y+ (τ ), y+ (τ )) are 1. Let S0 denote the scattering matrix which sends the coefficients of the solution in the basis (y− (τ ), y− (τ )) into the coefficients of the same solution in the basis (y+ (τ ), y+ (τ )). This matrix depends only on β and h – the only two parameters that appear in the above limit equations. In Appendix A, we derive the finite ε version of (3.10)–(3.11), the corresponding asymptotic bases and the scattering matrix S(µ, ε) which limits to S0 (µ) as ε → 0 for any fixed µ > 0. Notice that δ, the small time rescaling parameter, appears as a denominator in the C21 entry – this reflects the high sensitivity of y ( t) to changes in y(− t). Using the formulae for B and C ((3.8) and (3.9)) with σ δ = t tending to zero sufficiently slowly and δ → 0, one obtains that the trace of the monodromy matrix A = BC equals to
d 2d 2d 1+ Tr(A) = √ s21 (1 + o(1)) + (s11 + s22 ) 1 + + o(1). (3.14) R R δ h The periodic orbit is stable when | Tr(A)| < 2. Note that the main contribution to (3.14) is given by the term that includes s21 : since δ → 0 as ε → 0, if s21 (µ, ε) stays bounded away from zero, then for sufficiently small ε the trace of A is very large and positive for 9 Uniformly on any compact subset of µ > 0. 10 The functions y± (τ )) are defined in a unique way in Appendix A.
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511
positive s21 and very large negative for negative s21 . This means that if we fix h, n, d, R, choose ε sufficiently small and change µ, then Tr(A), as a function of µ, will change sign near the values of µ where s21 (µ, 0) changes sign. Then Tr(A) is necessarily small near these values of µ. Therefore, from these values of µ a wedge of parameter values for which the periodic orbit γ is linearly stable emerges. We need to establish that there is an infinite number of such values of µ. By definition, s21 is determined as follows (see the Appendix): take the solution y− (τ ) of (3.10) that tends to 1 as τ = −∞, then s21 =
dy− (+∞), dτ
(3.15)
namely, the asymptotic properties of y− (τ ) determine s21 . Next we establish a precise relation between the asymptotic form of y− (τ ) and the zeroes of s21 , and between the number of zeroes of y− (τ ) and the sign of s21 : Lemma 5. The limit system (3.10) has a non-trivial bounded solution y− (τ ; µ∗ ) for all τ ∈ (−∞, +∞) if and only if s21 (µ∗ , 0) = 0. Furthermore, √ ds21 β(µ∗ ) = 0, − h I, (3.16) = +∞ 1 (s)2 ds, β(µ∗ ) = 0, − βy− (+∞) −∞ y− dβ s21 =0 where 1 I =√ h
+∞ −∞
2 V (z(τ ))dτ = √ h
+∞ V −1 (h/2)
V (z) √
dz . h − 2V (z)
(3.17)
If s21 (µ, 0) = 0, then sign s21 = (−1)N (y− ) where N (y− ) denotes the number of zeroes of y− (τ ). Finally, if µ∗ < 1, or µ∗ = 1 and I > 0, then N (y− ) is decreased by one when µ changes from µ∗ − 0 to µ∗ + 0. Proof. Using the definition of the scattering matrix (see (3.12), (3.13)), y− has the following asymptotic as τ → +∞ (uniformly on any compact subset of positive values of µ): y− = s21 (τ + O(τ 1−α )) + s11 (1 + O(τ −α )),
y− = s21 (1 + O(τ −α )) + O(τ −1−α ),
(3.18)
It thus follows immediately that if s21 vanishes, then y− (τ ) is bounded. To prove the converse, notice that non-trivial bounded solutions must be proportional to y− (τ ) as τ → −∞, and therefore, if s21 = 0, these cannot remain bounded as τ → +∞. Next, we establish (3.16). Define u = dy− /dβ. By definition (see (3.15)) ds21 = u (+∞). dβ By differentiating (3.10) with respect to β we find that u is the solution of u + βV (z(τ ))u = −V (z(τ ))y− (τ ),
(3.19)
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which satisfies u(−∞) = u (−∞) = 0. By the variation of constants formula (recall that the Wronskian of y− (τ ) and yˆ− (τ ) is 1): τ τ V (z(s))y− (s) yˆ− (s)ds − yˆ− (τ ) V (z(s))y− (s)2 ds. (3.20) u(τ ) = y− (τ ) −∞
It follows that (+∞) u (+∞) = y−
−∞
+∞ −∞
V (z(s))y− (s) yˆ− (s)ds − yˆ− (+∞)
+∞ −∞
V (z(s))y− (s)2 ds.
−1 y− (+∞)
(+∞) = 0 and yˆ (+∞) = If s21 = 0, then we have y− (since the Wronskian − is 1). Thus, +∞ ds21 1 =− V (z(s))y− (s)2 ds at s21 = 0. (3.21) dβ y− (+∞) −∞ , hence, integrating by parts, we find At β = 0 we have V (z)y− = −β −1 y− ∞ 1 +∞ V (z(s))y− (s)2 ds = y (s)2 ds, β −∞ − −∞
(3.22)
which gives the second line of (3.16). At β = 0 the scattering matrix of system (3.10) is the identity so s21 (1, 0) = 0. In this case (3.10) has the bounded solution y(τ ) = 1 and by (3.21) the first line of (3.16) is obtained, or equivalently ds21 2 √ |µ=1 = h I. (3.23) dµ n−1 Finally, let us relate the number of zeroes of the fundamental solution y− , N (y− ), and the sign of s21 . By (3.15), if y− → +∞ as τ → +∞, then s21 > 0, and if y− → −∞ as τ → +∞, then s21 < 0. Recall that y− (−∞) = 1 is always positive. Clearly, if s21 > 0, then y− has an even number of zeros, and if s21 < 0, then the number of zeros of y− is odd so sign s21 = (−1)N (y− ) as claimed. Note that y− cannot have multiple zeros, as it is a non-trivial solution of a second order linear homogeneous equation. It follows that as µ varies, the number of zeros of y− can increase only when some zeros come out of +∞. 2 + s 2 is bounded away from zero by It follows from (3.18), and the fact that s21 12 preservation of the Wronskian, that y− may have only one zero at large τ . Therefore, if N (y− ) changes at some µ > 0, the increase/decrease in the value of N equals exactly to 1. It follows from (3.20) and (3.18) that at s21 = 0 (i.e. when y− (τ ) is bounded) d y− (τ ) = u(τ ) = τ u (+∞) + o(τ )τ →+∞ . dβ Hence, it follows from (3.19), and from (3.16) with µ∗ < 1, or µ∗ = 1 and I > 0, that for all τ sufficiently large sign
ds21 d y− (τ ) = sign = signy− (+∞) = (−1)N (y− ) , dµ dµ
so it follows that N (y− ) decreases when µ increases through µ∗ .
(3.24)
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513
It follows from the trace formula (3.14), that to complete the proof of Theorem 2, we need to show that the coefficient s21 of the scattering matrix for the limit equation (3.10), (3.11) changes its sign infinitely many times. By the above lemma, we need to examine the bounded solutions of (3.10) and their number of zeroes. Now, notice that independently of the choice of V and of the value of h there are two values of µ at which the bounded solutions are easily identified. At µ = 1 we have the bounded solution y− = 1 which has no zeroes. At µ = µ1 = n −1/2 there is a bounded solution with one zero: y(τ ) = z (τ ), where z is the solution of (3.11). It follows from (3.16) that when11 I > 0, s21 changes sign from negative to positive when µ increases through µ = 1 (recall that β (µ) < 0 for µ < 1), and when I < 0 (this is the case e.g. of Gaussian potential (2.9) at h close to 2) s21 changes sign from positive to negative. It follows from (3.24) that at µ = µ1 + 0 we have s21 < 0. Hence, using (3.23) we see that if I < 0, there exists µ = µ0 < 1 for which s21 = 0 (so there is a non-constant positive bounded solution at µ0 ). This is the tip of the 0th stability zone. Furthermore, since for µk < 1 the number of zeros of y− always decreases by one when µ changes from µk − 0 to µk + 0, it follows that for I < 0 there is only one such µ0 value in the interval (n −1/2 , 1), whereas for I > 0 we set µ0 = 1. We conclude that for k ≥ 1, the tip µ = µk of the k th stability zone corresponds to the existence of a bounded solution of (3.10), which has exactly k zeros. To establish that there is a countable infinity of values of such µk , recall that there is a non-empty interval of values of τ for which V (z(τ )) is strictly positive (by (2.6) and (2.7)). Since the coefficient β of V (z)y grows to +∞ as µ → +0, it follows that the number of zeros of every solution of (3.10) on this interval grows to infinity as µ → +0. In particular, the number of zeros of y− – hence the number of sign changes in s21 – grows to infinity as µ → +0, as required. This completes the proof of Theorem 2. Notice that the points µk where the stability zones touch the axis ε = 0 are determined by the behavior of the limit system (3.10)–(3.11) only. In particular, depending on the form of V and h there are the corresponding βk values at which the stability zones appear, and these are independent of n, d and R. Thus, we conclude from (3.7) that µk = (1 + βk (n − 1))−1/2 , where the numbers βk → +∞ depend only on h and on the potential function V . If I > 0 then β0 = 0. For all V and h we have β1 = 1. Note that in the proof of Theorem 2 the limit of fixed µ > 0 and ε → 0 was considered. It follows that for any finite k value a stability zone will appear near µk for sufficiently small ε (non-uniformly in k). In the Appendix we prove that an infinite number of these stability zones extend towards the ε axis: Lemma 6. Let L be a continuous curve in the region (µ ≥ 0, ε > 0) of the (µ, ε)-plane, which starts at (µ = 0, ε = 0). Then L intersects the region of stability of the diagonal periodic orbit γ in an infinite sequence of intervals converging to (µ = 0, ε = 0). 11 This is always the case if V > 0 for all z, e.g. for the power-law potentials (2.8), where the following explicit formula for I may be established:
I = 2(α + 1)(h/2)1/α
π/2 0
2
(cos θ ) α +1 dθ > 0.
(3.25)
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Proof. See Appendix A. After calculating the form of the matrix C in this limit of small (µ, ε), which involves deriving a rescaled system similar to (3.10), it is shown that the trace of A changes between ±2 whenever the number of zeroes of the bounded solutions of this rescaled system are changed. Then, we again argue that the number of zeros of this system tends to infinity as (µ, ε) → 0. 3.1. Estimates of the stability wedges width. We have thus established that for any finite dimension n there is an infinite number of wedges of linear stability zones emanating from µ values at (0, 1). Next we estimate their width in the (µ, ε)-plane at β values that are near β0 = 0 (corresponding to either µ close to 1 or to large n): Proposition 1. If I > 0 (see (3.17)), then the diagonal periodic orbit γ is stable for (µ, ε) values in the wedge enclosed by the two curves
−1 1 1 − µ2 1 − µ2 ε0+ = I 1 + (3.26) + o (n − 1)µ2 d+R (n − 1)µ2 and ε0−
1 − µ2 =I (n − 1)µ2
1 −1 1 − µ2 1+ . +o d (n − 1)µ2
(3.27)
Proof. See Appendix A, where formula (3.14) is expanded in β, ε near (0, 0) at which S limits the identity matrix. The other limit in which we are able to obtain analytical results regarding the stability wedges width corresponds to µ = 0, i.e. it is the limit of the zero angle between the spheres 1 , . . . , n at the corner point. We prove that for sufficiently large k, the stability zone emanating from (µk , ε = 0) extends towards the ε-axis as shown in Fig. 4: α Proposition 2. Consider the power-law potential V (Q, ε) = Qε . Then, for sufficiently small ε and µ, there exists an infinite number of disjoint stability tongues in the (µ, ε) plane at which γ (t; µ, ε, n) is linearly stable. For sufficiently large k the k th stability zone emanates from the µ axis near the bifurcation value: 1 2(α + 1) µk ≈ , (3.28) k α(n − 1) and extends up to the ε-axis, intersecting it near εk ≈ (h/2)
1/α
(α + 1) 4 α(n − 1) π 2 k 2
π/2
(sin θ )
2 dθ ,
(3.29)
1/2(α+1) ,
(3.30)
1/α
0
at a stability interval of length ( ε)k ≈
4εk π kG(0, α)d(1 +
d R)
4α(α + 1) (2εk )α n−1 h
where G(0, α) > 0 depends only on α and is defined by (B.12).
Stability in High Dimensional Steep Repelling Potentials n=3
nε
0.8
515
0.6
0.6
0.4
0.4
0.2
0.2
0
−1
0 Re( λ)
n=3
0.8
1
0
0
nε
0.8
0.6
0.6
0.4
0.4
0.2
0.2 −1
0 Re( λ)
1
0
0
nε
0.8
0.6
0.6
0.4
0.4
0.2
0.2 −1
0 Re( λ)
1
0
0
nε
n=1000 0.8
0.6
0.6
0.4
0.4
0.2
0.2 −1
0 Re( λ)
1
0.5 µ
1
n=1000
0.8
0
0.5 µ n=100
n=100 0.8
0
1
n=10
n=10 0.8
0
0.5 µ
1
0
0
0.5 µ
1
Fig. 4. Bifurcation diagram for the power-law potential. Left: real part of the eigenvalue λ(ε) at µ = 0; note that λ changes very fast as nε changes. Right: Wedges of stability in (µ, n) space (note that the ε-axis is scaled with n). The stability wedges lie between the saddle-center bifurcation curves (dotted lines) and the period doubling bifurcation curves (solid lines). The asymptotic predictions (thin lines) of formulae (3.27),(3.26) for the first wedge are shown.
Proof. See Appendix B for details. It is proved that any curve of the form L M = {(µ, ε) : 2εM = µ2 (1 − M)},
(3.31)
with M ∈ [0, 1] considered as a fixed parameter, intersects the stability wedges an infinite number of times. Moreover, the location and width of these intersections is evaluated
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along a parameterization of L M by an auxiliary parameter ρ: ρ=
2ε + µ2 .
(3.32)
Thus, formula (3.28) is established by applying formula (B.14) along L M=1 , whereas formulae (3.29), (3.30) are established by applying (B.14) and (B.15) along L M=0 (using (B.3) and (B.5)). As described next, the asymptotic formulae are in excellent agreement with our numerics. 4. Numerical Computations 4.1. Stability of the periodic orbit. In general, the numerical computation of a periodic trajectory and its stability in a steep n-dimensional potential is, for large n, a difficult problem; a high-dimensional scheme for locating the periodic trajectory is needed, and the search involves the integration of a nonlinear, stiff, high-dimensional system. Once the periodic orbit is found, the numerical computation of the linearized system and its Floquet multipliers for large n may be a formidable task. Here we use Lemmas 1 and 2 and some proper rescaling to reduce this problem to a simpler computational task. The search for the periodic orbit is unnecessary by Lemma 1 (using symmetry and proper parameters) and the need to compute eigenvalues of large matrices is demolished by Lemma 2: for all n we find the solutions of one second-order non-linear equation (2.15) and the monodromy matrix of one second-order linear equation (3.2), (3.3) which depends on n as a parameter. The steep limit is handled as in [20]: we fix ε and increase the size of the billiard domain (r in (2.2)) to get an effectively small ε = ε/r without running into stiffness problems (in the bulk of the domain the motion is essentially inertial and non-stiff). To find the stability regions, as shown in Fig. 4, we use the continuation scheme which was developed in [20]; first we compute the stability of γ (t) at µ = 0 (the case of a cusp created by n tangent spheres) along the ε-axis (see Fig. 4 left12 ). By symmetry (see Lemma 2), Re(|λn (µ = 0, ε))|) > 1 always corresponds to real eigenvalue (i.e. saddlefoci do not appear) and thus the values of ε = εk± (n) at which Re(λn (µ = 0, ε)) = ±1 correspond to degenerate saddle-center and degenerate period-doubling bifurcations respectively. Then, we use the values of ε = εk± (n) as the starting point for a continuation scheme in µ to locate the k th wedge of stability in the (µ, ε) plane (see Fig. 4 right). In accordance with a Theorem 2 and Propositions 1, 2, these calculations (performed for the Gaussian, exponential and power-law potentials, and shown here only for the power-law case) demonstrate that for any given n, at µ = 0, the stability of γε,n,µ = 0 (t) rapidly changes as ε → 0+ , whereas for any µ ∈ (0, 1), there is a finite number of intervals of ε in which γε,n,µ (t) is stable. Next, we demonstrate that the asymptotic formulae provided in these propositions are in good agreement with the numerics; in all the numerical simulations shown below we fix h = 1, R = 10 and d = 2 , consider the power-law case (2.8) with α = 1, and study, for each n, how the stability of γ (t) depends on µ and ε. 12 Note that the curves in this diagram represent the graph of Re(λ(µ = 0, ε)) so they are not horizontal. Their horizontal appearance reflects the rapid large oscillation of λ in the small ε limit.
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−2 +
log(ε0)≈ − 0.95log(n−1) − 0.58 − 0 + 1 log(ε+1)≈
log(ε )≈ − 0.92log(n−1) − 0.828
−3
log(ε )≈ − 0.99log(n−1)− 2.1 − 0.98log(n−1) − 2.1
−4
log(ε)
−5
−6
−7
−8
−9 2
2.5
3
3.5
4
4.5 log(n−1)
5
5.5
6
6.5
7
± Fig. 5. The intersection of the first and second wedges of stability with the ε-axis (ε0,1 (n)) is shown to scale like 1/n.
Figure 4 shows that the estimates (3.26), (3.27) of Proposition 1 for the stability boundary of the first wedge and their numerical calculation agree when either 1 − µ is 1−µ2 small or n is large (recall the o (n−1)µ2 correction term in (3.26), (3.27)). The origin at ε = 0 is found, by Theorem 2, to be given √ of the second stability zonen=10 by µ1 = 1/ n, so µn=3 ≈ 0.577 and µ ≈ 0.33, which agrees with the numerical 1 1 data at Fig. 4. ± The behavior near µ = 0 is examined next. In Fig. 5 we plot ε0,1 (n), the first and second ε value at which γε,n,µ=0 (t) becomes stable, as a function of n − 1. It shows β±
k that εk± (n) ≈ n−1 (k = 0, 1) in accordance with (3.29), even though k is not sufficiently large for the asymptotic estimates to hold. For larger k values, the oscillatory behavior in log ε of Re(λn (µ = 0, ε)) is shown in Figs. 6 and 7. Indeed, in the proof of Proposition 2, it is established that for the power-law potential, at α = 1 (see Appendix B):
1
h h(n − 1) 4 d sin 2 + 2ϕ(0, 1) + · · · (4.1) Tr(A) = G(0, 1) d 1+ ε R ε(n − 1)
with G(0, 1), ϕ(0, 1) some constants, and thus, using (3.1), λn (µ = 0, ε) may be estimated in this asymptotic limit; Fig. 7 shows the agreement between the numerical computation and the asymptotic form for sufficiently small ε; we fitted G(0, 1) = 1.85, ϕ(0, 1) = 1.1π for the n = 3 case and used these for the n = 10 case, suggesting that these constants are indeed independent of n as predicted by (4.1). Figure 6 shows the ε dependence of the envelope of Re(λn (µ = 0, ε)) for finite n. We observe an ε−0.29 envelope, whereas (4.1) suggests an ε−0.25 envelope for the power-law
518
A. Rapoport, V. Rom-Kedar, D. Turaev log|λ(µ=0)| vs. log((n−1)ε)
Re(λ(µ=0)) vs. log((n−1)ε)
y=−0.29x+1.7
50 n=3
4 0
−50
2
−6
−4
−2
0
2
4
0
−6
−4
−2
0
2
4
y=−0.29x+2.5
100 n=10
4 0
−100
2
−6
−4
−2
0
2
4
0
−6
−4
−2
0
2
100
4
y=−0.29x+3.7
n=100
4 0
−100
2
−6
−4
−2
0
2
4
100
0
−6
−4
−2
0
2
6
4 y=−0.5x+5.4
n=1000
4 0 2 −100
−6
−4
−2
0
2
4
0
−6
Re( λ(µ=0)) vs. log(ε) 200 n= ∞
−4
−2
0
2
4
log| λ(µ=0)| vs. log(ε) y=−0.5x+1.57
6 4
100
2 0 −8
−6
−4
−2
0
0 −8
−6
−4
−2
0
Fig. 6. The oscillatory nature of the Floquet multipliers at µ = 0 for several n values is shown.
potential13 . The discrepancy may be the result of finite ε effects. For the n = 1000 case we do not observe enough oscillations for (4.1) to be meaningful. This finding shows that for very small ε values, approaching the cusp limit, the orbit γε,n,µ=0 (t) has increasingly large multipliers that grow, on the appropriately defined subsequence of ε values, as a power law in (n/ε). 4.2. Non-linear stability – Phase space plots. To support the claim that for (µ, ε) values inside the stability wedges the linearly stable periodic orbit γε,µ (t) is surrounded by an island of effective stability (i.e. that KAM tori survive in its neighborhood), we 13 Similar fitting for the Gaussian case gives rise to an ε −0.61 envelope.
Stability in High Dimensional Steep Repelling Potentials G(0,1)=1.85, φ(0,1)=1.1π
n=3 0.02
0.02
0.018
0.018
0.016
0.016
0.014
0.014
0.012
0.012 nε
nε
519
0.01
0.01
0.008
0.008
0.006
0.006
0.004
0.004
0.002
0.002
0 −100
−50
0 Re(λ)
50
100
n=10
0 −100
−50
0 Re(λ)
50
100
Fig. 7. Oscillations of (λ(µ = 0; ε)) at n = 3, 10 . Thin line (blue) - analytical estimates (eq. (4.1)). Thick line (red) - numerical computations.
choose parameter values inside the wedges (using Fig. 4 right) and integrate the 2n equations of motion directly. The (x1 , p1 ) projection of the return map to the section 14 ξs = L−R 2 for the power-law potential with n = 10 is shown in Fig. 8 (left column). The islands of effective stability are clearly observed in this projection. To examine the non-degeneracy of these islands to asymmetric perturbations, we introduce the following family of potentials: per t
Vk
(x; ε) = Vk (x; ε) + δak Vk (x; ε),
(4.2)
where ak are uniformly distributed on the unit interval (i.e. we consider the case by which each sphere has a slightly different potential). The phase portraits of the perturbed motion with δ = 0.001 are shown in the right · column of Fig. 8 (we do verify that the projection plots of X = x − γ (0) , P = x − γ˙ remain bounded, namely that there is no instability in any direction of the 20-dimensional phase space).
5. Discussion We have constructed a set of examples that show that for an n-dimensional dispersing billiard, for any finite n, symmetric corners with n faces can produce islands of effective stability when the billiard is replaced by a more realistic model of a particle moving in a steep repelling potential, for arbitrarily high values of the steepness of the potential. In 14 Notice that the wedges emanating from µ < 1 or those corresponding to larger n values (where our k theoretical predictions for the wedges are in better agreement with the numerics) correspond to much smaller ε values, see 4 right. For such small ε’s, the computation of the phase portraits, in which long integrations that include many collisions are performed, becomes more prone to numerical errors.
520
A. Rapoport, V. Rom-Kedar, D. Turaev without perturbation
0.22
0.22
p
p1
0.222
1
0.222
0.218
0.218
0.216
0.216
0.36
0.365
0.37
0.375
0.36
0.37
0.375
1
µ=0.1
0.232
0.232
0.23
0.23
0.228
0.226
0.224 0.375
0.365
x
1
p1
1
x
p
with perturbation (δ=0.001)
µ=0
0.228
0.226
0.38
0.385
x
0.224 0.375
0.39
0.38
0.385
0.39
x
1
1
µ=0.2 0.238
0.238
0.236
p1
p
1
0.236
0.234
0.234
0.232
0.232
0.23 0.395
0.4
x
1
0.405
0.395
0.4
0.405
x1
Fig. 8. Islands in a 20 dimensional symmetric (left) and asymmetric (right) systems. Parameter values are chosen inside the first wedge of stability (see Fig. 4): ε = 0.0625, µ = 0, 0.1, 0.2. Return map projection to the (x1 , p1 ) plane is shown for the power-law potential with α = 1. On the right panel the potentials (4.2) with δ = 0.001 are used.
particular, for a certain symmetric geometry, we have found a specific (diagonal) periodic orbit for which we proved that for any n there is a countable set of wedges in the parameter plane where the periodic orbit is linearly stable. As the steepness parameter, ε−1 , tends to infinity, these stability zones do not disappear and remain in a finite region of the parameter plane up to ε = 0 that corresponds to the (dispersing) billiard limit. Moreover, we were able to estimate the width and location of these wedges for the powerlaw potentials. The qualitative results and the asymptotic formulae were supported by numerical computations for the power-law, the Gaussian and the exponential potentials. Finally, we conjecture that for most parameter values in the wedges, where the periodic orbit is linearly stable, a region of effective stability is created (namely, KAM-tori exist,
Stability in High Dimensional Steep Repelling Potentials
521
i.e. despite the symmetric form of the potential, the behavior near the elliptic points is similar to the behavior near generic elliptic points). This conjecture15 is supported by numerical simulations for several n values, for both the power-law and the Gaussian potentials: in these simulations islands of effective stability surviving small symmetry breaking perturbations of the potential are clearly seen (see Fig. 8, where projections of islands in 20-dimensional phase space are shown for the power-law potential). From the mathematical point of view, one generally expects that smooth Hamiltonian systems will have islands of stability. Here, we go beyond genericity type results – we identified specific mechanisms by which the ergodicity and hyperbolicity of the underlying dispersing billiard are destroyed, and a stable motion is created in the problem of a particle moving in a smooth, steep repelling n-dimensional potential. The proofs construction includes estimates for the scaling of the stability zones with the control parameters and a description of the bifurcation sequence associated with their creation– such explicit results may be of interest in specific applications. Admittedly, the presented construction has two limitations that we hope to abolish in future works; the first is the strong symmetry under which the example is constructed; it leads to a highly degenerate spectra – in fact all the non-trivial Floquet multipliers collapse onto only one pair (λ, 1/λ), (which is shown to belong to the unit circle in the intervals of stability). Thus, in the symmetric case resonance phenomena must be studied. When the symmetry is slightly broken, either all the eigenvalues remain on the unit circle, or some of them may bifurcate in quadruples to a Hamiltonian Hopf bifurcation. Such possibility may pose difficulties in proving that the periodic orbit remains stable (though one would expect that even in this case stable regions will be created, see [8,9]). The other limitation is that the constructed mechanism for the creation of islands requires an n-corner – it corresponds to the intersection of n truly n-dimensional strictly dispersing scatterers in an n-dimensional space. Currently, the most interesting applications of high-dimensional billiards (n > 3) are concerned with the problem of N particles in a d dimensional box. In this case the scatterers in the n = N d-dimensional configuration space are cylinders with only d − 1 dispersing directions [33,27], and the phase space structure may prohibit the appearance of the symmetric n-corners considered here. We believe that both of these issues may be resolved in future works. Indeed, the main ingredient in our construction is the concurrent singularity in n − 1 directions which is induced by the n-corner. We conjecture that it is possible to produce islands (non-degenerate elliptic orbits) in any smooth dispersing billiard family in which singular orbits are controlled by n − 1 independent parameters (here the angles between the n faces of the corner). The symmetric settings are simply convenient for collapsing the number of independent control parameters (here to one). Furthermore, we conjecture that the set of billiards having singular orbits that produce elliptic islands16 are dense in the family of Sinai billiards17 . Hence, while we did not prove yet that a system of N soft particles in a d-dimensional box is non-ergodic, we can now state that it is likely to be true – if strictly dispersive geometries give rise to elliptic islands, semi-dispersing 15 To establish these results analytically one may consider the symmetry breaking terms as perturbations that introduce small coupling to the linearized equations (3.5) and study under what conditions the degeneracy of the spectrum unfolds and remains on the unit circle. Proving that the orbit is non-linearly stable appears to be even more challenging. 16 Here other singularities such as multiple tangencies to the billiard boundary and multiple visits to k-corners with k < n need to be included. 17 The recent results of [4], in which hyperbolicity is proved for finite range potentials that have discontinuous derivatives at their outer perimeter, is consistent with these conjectures – we propose that in that work the hyperbolicity is linked to the lack of smoothness of the potentials.
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geometries should do so as well. The methodologies we develop might shed light on the scaling of the non-ergodic components with N and ε, supplying interesting insight on the Boltzmann ergodic hypothesis: while in the hard sphere case Sinai’s works show that there is no need to consider the large N limit (which is a major ideological cornerstone in Boltzmann’s argument), N does enter into the estimates of the non-ergodic component volume (and possibly their stickiness properties) in the smooth case. Acknowledgements. We thank Prof. Uzy Smilansky for stimulating discussions. We acknowledge support by the Israel Science Foundation (Grant 926/04), the Minerva Foundation and by a joint grant from the Ministry of Science, Culture and Sport, Israel and the Russian Foundation for basic research, the Russian Federation (MNTI-RFBR No. 06 - 01 - 72023).
Appendix A. Linearized Behavior near the Corner Here we construct the linearized reflection matrix near the corner in the finite µ case and establish Proposition 1 regarding the stability wedge width in this limit. Then we consider the limit of µ → 0+ : we construct the reflection matrix C in this limit and establish Lemma 6. First we present the proof of Lemma 4 regarding the form of C, the matrix corresponding to the linearized map near the corner: (y(− t), y (− t)) → (y( t), y ( t)) = C(y( t), y ( t)) in the limit of small ε and fixed µ > 0: Proof. On the time interval [− t, t] we scale time t → δ · τ , where δ = ε/µ.
(A.1)
Note that y˙ (t) then changes to y (τ )δ −1 , hence
1 0 1 0 ˆ , (A.2) C= C 0 δ 0 δ −1
y(σ ) y(−σ ) y(−σ ) ˆ ˆ → = C where C is the matrix of the linear map y (σ ) y (−σ ) y (−σ ) defined by the rescaled Eq. (3.2): y + δ 2 a(τ δ)y = 0,
(A.3)
on the interval τ ∈ [−σ, σ ], where we denote σ = t/δ.
(A.4)
Note that σ tends to +∞ as o(ε−1 ), because we assume that t = o(1)ε→0 . Let us introduce a new variable z by the rule 1 + 2µν + ν 2 = 1 + εz, (A.5) i.e. z is a rescaled distance to the corner. Recall that we choose our parameterization of time along γ in such a way that t = 0 corresponds to the point nearest to the corner. Hence, we have from (2.13 ), (2.14), (A.5), (2.7) that h = V (z) + O(εα ), 2
Stability in High Dimensional Steep Repelling Potentials
523
i.e. z(0) stays uniformly bounded for all ε. As the velocity ν˙ is bounded from above by virtue of (2.13), (2.6), it follows that ν(t) − ν(0) = O( t) at |t| ≤ t, so z(t) − z(0) = O( t/ε), i.e. z = o(ε−1 ) for all t from this interval. It is easy to see that Eq. (A.3) (see also (3.3)) takes the following form after the rescaling:
1 − µ2 y + V (z) + ε a(z, ˜ ε) y = 0, (A.6) (n − 1)µ2 where a˜ is uniformly bounded and
a˜ = O |z|−1−α ,
(A.7)
uniformly for all z such that εz is small. Equation (2.13) changes to
where
h (z )2 = (1 + φ(z, ε)) + V (z) + V˜ (z, ε), 2 2
(A.8)
V˜ = O(εα ), and φ = o(1)ε→0 .
(A.9)
As we mentioned, we consider Eqs. (A.6), (A.8) at z ≤ z ∗ with some z ∗ = o(ε−1 ). Therefore, at z > z ∗ we may define φ and V˜ in an arbitrary way, and we define there φ(z) = φ(z ∗ ) and V˜ (z) = V˜ (z ∗ )(z ∗ /z)α . Then, by virtue of (2.7), (A.9), the potential in the right-hand side of (A.8) uniformly (for all small ε) tends to zero as z → +∞. Hence, uniformly for all small ε, √ h + o(1) as τ → ±∞. z(τ ) = τ By plugging this into (A.6), and defining a(z) ˜ = a(z ˜ ∗ )(z ∗ /z)2+α we see from (2.7), (A.7) that Eq. (A.6) has the form y + Q(τ, ε)y = 0
where, uniformly for all ε, Q = O(|τ |−2−α ) as τ → +∞. (A.10) Moreover, Q is continuous with respect to ε and has a limit (uniformly for all z) as ε → 0: the limit system is y + βV (z) y = 0, 1 − µ2 β= , (n − 1)µ2
(A.11)
where z(τ ) solves
(z )2 h = + V (z). (A.12) 2 2 It is a routine fact that every solution y(τ ) of equation of type (A.10) grows at most linearly as τ → ±∞; and that there exists a limit for the derivative y : (A.13) y (τ ) = D1± + O |τ |−α ,
uniformly for any bounded set of initial conditions and for all small ε. Moreover, the solution is bounded as τ → +∞ if and only if D1+ = 0; and the solution stays bounded as τ → −∞ if and only if D1− = 0. Among the solutions bounded as τ → +∞, there
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exists exactly one solution y+ which tends to 1. Analogously, there exists exactly one solution y− which tends to 1 as τ → −∞: y± (τ ) = 1 + O(|τ |−α ),
y± (τ ) = O(|τ |−1−α ).
(A.14)
We also take a pair yˆ+ (τ ) and yˆ− (τ ) of solutions such that yˆ− (−∞) = 1,
hence
yˆ+ (+∞) = 1,
yˆ± = τ + O |τ |1−α .
(A.15) (A.16)
The solutions yˆ± are not uniquely defined, therefore we now fix a certain canonical choice of them, in order to ensure that they will depend continuously on ε and other parameters of the problem. To do that, let ϕ(τ ) denote the solution of (A.10) with initial conditions ϕ(0) = 1, ϕ (0) = 0, and let ψ(τ ) be the solution with initial conditions ψ(0) = 0, ψ (0) = 1 (we deal with time-reversible equations, and in this setting ϕ and ψ are, respectively, the even and odd solutions of (A.10); we do not use this in the proof of this theorem). Recall that
ϕ ψ =1 (A.17) det ϕ ψ for all τ , by Wronsky formula. As y+ is defined uniquely (by condition (A.14)), there exist uniquely defined constants K 1 and K 2 such that y+ = K 1 ϕ − K 2 ψ
(A.18)
(one can show that K 1 = ψ (+∞) and K 2 = ϕ (+∞), but we do not use this information). We will choose K2 K1 yˆ+ = 2 ϕ+ 2 ψ. (A.19) 2 K1 + K2 K 1 + K 22 Note that (y+ , yˆ+ ) are related to (ϕ, ψ) by a linear transformation with the determinant equal to 1. Therefore, by virtue of (A.17),
y+ yˆ+ =1 (A.20) det y+ yˆ+ for all τ . By taking a limit as τ → +∞, we obtain from this formula (see also (A.14), (A.13)) that yˆ+ (+∞) = 1, i.e. thus defined yˆ satisfies (A.15), (A.16), as required. Analogously one can fix the choice of yˆ − ; note that
y− yˆ− = 1. (A.21) det y− yˆ− As y+ and yˆ+ are linearly independent, every solution is a linear combination of them: y(τ ) = D0+ y+ + D1+ yˆ+ .
(A.22)
The same solution can be written as y(τ ) = D0− y− + D1− yˆ− .
(A.23)
Stability in High Dimensional Steep Repelling Potentials
525
It follows that the solutions of (A.3) define a continuously depending on ε scattering matrix S(ε): −
+
D0 D0 = S . (A.24) D1+ D1−
y(σ ) y(−σ ) → is given by Moreover, the matrix Cˆ of the map y (σ ) y (−σ )
y+ (σ ) yˆ+ (σ ) yˆ− (−σ ) − yˆ− (−σ ) ˆ C= · S(ε) · . (A.25) (−σ ) y+ (σ ) yˆ+ (σ ) −y− y− (−σ ) Recall that σ → +∞. By (A.2), (A.25 ), (A.14), (A.16), (A.13),
s21 O(δσ 2 ) + O(δσ ) s11 + o(1) + σ s21 (1 + o(1)) C= 1 −1−α )) s + o(1) + σ s (1 + o(1)) . 22 21 δ (s21 (1 + o(1)) + O(σ where si j (ε) are the entries of the scattering matrix.
(A.26)
The proof of Proposition 1, regarding the width of the stability wedges for small β values is established next: Proof. The stability zone corresponds to | Tr(A)| < 2. By (3.14), (A.1 ) the boundary Tr(A) = 2 is given by
2d 2 d ε(2 − (s11 + s22 ) 1 + + o(1)ε→0 ) = √ d 1 + s21 , (A.27) R R h and the boundary Tr(A) = −2 is given by
2d 2 d ε(−2 − (s11 + s22 ) 1 + + o(1)ε→0 ) = √ d 1 + s21 , R R h
(A.28) 2
1−µ where si j are the entries of the scattering matrix S(µ, ε) of Eq. (A.6). At β = (n−1)µ 2 = 0, ε = 0 Eq. (A.6) (the finite ε version of (3.10)) degenerates into y = 0, and the scattering matrix is equal to the identity. Thus, at β close to 0 and small ε, we find that
and
s11 + s22 = 2 + o(1)
(A.29)
s21 = q1 β + q2 ε + o (|ε| + |β|) ,
(A.30)
where q1 =
∂s21 | ∂β (β = 0,ε = 0)
q2 =
∂s21 | . ∂ε (β = 0,ε = 0)
and
(While S may be non differentiable in ε for general β, it can be shown, using (A.6), that at β = ε = 0 the expansion (A.30) is valid.) Thus, by plugging (A.29), (A.30), (3.23)
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A. Rapoport, V. Rom-Kedar, D. Turaev
into (A.27), (A.28), we find the following equations for the boundaries of the stability zone near (β = 0, ε = 0): √ √ 2 h ε + q2 + o(ε) = − h Iβ + o(β) d+R and
√ √ 2 h ε + q2 + o(ε) = − h Iβ + o(β). d
As we see, in order to prove the lemma, it remains to show that √ q2 = 2 h.
(A.31)
(+∞), where y (τ ) is the solution of (A.6) that satisfies By definition, s21 equals to y− − y− (−∞) = 1. Let us write (A.6) in the form (A.10). By differentiating (A.10) we find ∂ y− (τ ) satisfies that the derivative u(τ ) = ∂ε
u + Qu = −
∂Q y− . ∂ε
As Q = 0 and y− = 1 for all τ at β = 0, ε = 0, we obtain that +∞ ∂Q q2 = u (+∞)|(β = 0,ε = 0) = − dτ. −∞ ∂ε From (A.6), (A.3), (3.3), we find ∂Q | = a(z, ˜ 0)|β = 0 = V (z). ∂ε (β = 0,ε = 0) This gives us (see also (3.11)) +∞ q2 = − V (z(τ ))dτ = −∞
as required.
+∞
−∞
√ z (τ ))dτ = z (+∞) − z (−∞) = 2 h,
Next we find the form of C in the limit at which both µ and ε are small. Here, as in Lemma 4, on the time interval [− t, t] we scale time t → δ · τ, yet here we choose a different scaling coefficient δ (compare with (A.1)): δ=
ε 2ε + µ2
.
(A.32)
√ Obviously, δ → 0 (at least as O( ε)) as ε and µ tend to zero. Then, the matrix C is given by formula (A.2), where Cˆ is the corresponding matrix for system (A.3) obtained from (3.2) by the new time-scaling. With such scaling, system (A.3) gets the form
1 − µ2 1 y + V (z) + ε a(z, ˜ ε) y = 0, (A.33) 2ε + µ2 n − 1
Stability in High Dimensional Steep Repelling Potentials
527
where a˜ is uniformly bounded and satisfies (A.7) for all z such that εz is small. The equation for z(τ ) changes from (A.8) to
(z )2 h 2 ˜ (1 + εz) = − V (z) − V (z, ε) (M + (1 − M)z(1 + εz/2)) , (A.34) 2 2 where M(ε, µ) =
µ2 , µ2 + 2ε
(A.35)
and V˜ satisfies (A.9). Like in the proof of Lemma 4, we consider only the interval z ≤ z ∗ with z ∗ = o(ε−1 ), so outside this interval we may replace the terms εz with εz ∗ both in the right- and left-hand side of (A.34), and replace V˜ (z, ε) with V˜ (z ∗ , ε)(z ∗ /z)α . Then z(τ ) tends to +∞ linearly with τ or faster, with the velocity bounded away from zero. It follows that like in Lemma 4, the system (A.33), (A.34) belongs to the class (A.10), hence the matrix Cˆ is expressed by formula (A.25) via the scattering matrix S(ε, µ) defined by (A.24). Lemma 7. For small t and sufficiently small µ and ε, the linearized map about the diagonal orbit near the corner C : (y(− t), y (− t)) → (y( t), y ( t)) = C(y( t), y ( t)) is of the form
K 12 − K 22 1 + O((δ/ t)α ) O( t) ˜ + K 1 K 2 C, (A.36) C= 2 O(δ α ( t)1+α ) 1 + O((δ/ t)α ) K 1 + K 22 where δ → 0 as ε → 0, C˜ is a matrix whose exact form is irrelevant here and K 1 , K 2 are the coefficients of the even and odd components of the solution y− (τ ) of Eq. (A.33) with z solving (A.34). Proof. Here we will use the time-reversibility of Eq. (A.33), (A.34): if y(τ ) is its solution, then y(−τ ) is a solution as well. It follows that y− (τ ) = y+ (−τ ), hence, by (A.18),
y− (τ ) = K 1 ϕ(τ ) + K 2 ψ(τ ),
(A.37)
where ϕ and ψ are, respectively, the even and odd solutions of (A.33). Then, analogously to (A.19), K2 K1 ϕ+ 2 ψ. (A.38) yˆ+ = − 2 K 1 + K 22 K 1 + K 22 From (A.18), (A.19), (A.37), (A.38), (A.22), (A.23), (A.24) we obtain the following formula for the scattering matrix: ⎛ 2 ⎞ K 1 − K 22 2K 1 K 2 − ⎜ K2 + K2 (K 12 + K 22 )2 ⎟ ⎜ ⎟ 2 (A.39) S=⎜ 1 ⎟. 2 2 ⎝ K1 − K2 ⎠ 2K 1 K 2 K 12 + K 22 By (A.2), (A.25), (A.14), (A.16), (A.13), (A.4), (A.39) the required form of C, namely (A.36) is found.
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Finally we establish Lemma 6 regarding the stability wedges in this limit: Proof. As before, we represent the monodromy matrix A as the product of the two matrices B and C. Since B corresponds to the regular part of the diagonal orbit and is independent of µ, Lemma 3 applies in this small µ and ε limit as well and the matrix B is given by (3.8). As δ and t tend to zero, while t does this sufficiently slowly, we find from (A.36), (3.8) that −2(1 + 2d R ) + o(1)(ε,µ)→0 < −2 at K 1 = 0, Tr(A) = (A.40) 2(1 + 2d ) at K 2 = 0. R + o(1)(ε,µ)→0 > 2 The sought stability intervals on the curve L correspond to | Tr(A)| < 2. Therefore, by virtue of (A.40), we will prove the lemma if we show that there exists a converging to zero sequence of values of (ε, µ) ∈ L which corresponds to K 1 = 0 and a converging to zero sequence of values of (ε, µ) ∈ L for which K 2 = 0. By (A.39), vanishing of K 1 or K 2 corresponds to vanishing of s21 , i.e. to the boundedness of the solution y− of (A.33). At K 1 = 0 we have from (A.37) that y− = ψ, i.e. the bounded solution is odd, while at K 2 = 0 the bounded solution y− = ϕ is even. Thus, K 1 = 0 corresponds to the existence of a bounded solution y− with an odd number of zeros, and K 2 = 0 corresponds to the existence of a bounded solution with an even 1 − µ2 number of zeros. It remains to note that the coefficient of V (z)y in (n − 1)(2ε + µ2 ) (A.33) tends to +∞ as (ε, µ) → 0. From that, exactly like in the proof of Theorem 2, we obtain that the number of zeros of y− tends to infinity as (ε, µ) → 0. We also showed in the proof of Theorem 2 that each time the number of zeros changes, the increase is exactly 1. Now, the required existence of a converging to zero sequence of values of (ε, µ) ∈ L which corresponds to the existence of a bounded solution with odd number of zeros (i.e. K 1 = 0) and a converging to zero sequence of values of (ε, µ) ∈ L which corresponds to the existence of a bounded solution with even number of zeros (K 2 = 0) follows immediately. Appendix B. The Power-Law Potential To establish Proposition 2, we integrate Eq. (A.33) with the power-law potential in the asymptotic limit of small (ε, µ). In fact, we show below that by parameterizing the (µ, ε) plane by the parameters
µ2 2 (ρ, M) = (B.1) 2ε + µ , 2ε + µ2 we obtain estimates to the width of the wedge for all sufficiently small ρ uniformly in M. We first introduce some notations. Recall that the parabolas emanating from the origin L M = {(µ, ε) : 2εM = µ2 (1 − M)} were defined for a fixed parameter M ∈ [0, 1] and that ρ is used to parameterize these curves. Let +∞ dz . (B.2) J (M) = √ α 1/α z (hz − 2)(M + (1 − M)z) (2/ h) In particular, π J (1) = √ 2α
and
1/2α √ π/2 2 h J (0) = (sin θ )1/α dθ, 2 α 0
(B.3)
Stability in High Dimensional Steep Repelling Potentials
and at α = 1, J (0) =
√
529
h. Let
P(ρ, M) =
√
1 α+2 n−1 Mh κ α ρ 2 , α(α + 1)
where κ = κ(ρ, M) solves the equation
α+2 √ α 1 − M 1+ α α(α + 1) h κ α+2 + Mhκ α+2 . ρ2 = 4 n−1
(B.4)
(B.5)
Note that κ → +∞ as ρ → 0, while P(ρ, M) remains bounded: √ P ∈ [0, 1]. Moreover, one can rewrite (B.5) in the following form (recall that δ = ρε and M = µρ , see (A.32), (A.35)):
√ α(α + 1) 1/(α+2) h κδ + µ h = (ρ/κ)α/(α+2) , (B.6) 2 n−1 from which it follows immediately that ρ(1 − M) = o(1). 2
(B.7)
1 y(θ ) = 0 ((1 − P)θ 2 + Pθ )α+2
(B.8)
κδ = κ Consider an equation y (θ ) +
defined at θ > 0. In the limit θ → +0, the coefficient of y in (B.8) tends to +∞, which produces fast oscillations in y: every solution has the asymptotic given by
+∞ α+2 dθ y(θ ) ≈ E 1 ((1 − P)θ 2 + Pθ ) 4 cos − ((1 − P)θ 2 + Pθ )1+α/2 θ
+∞ α+2 dθ E 2 ((1 − P)θ 2 + Pθ ) 4 sin (B.9) ((1 − P)θ 2 + Pθ )1+α/2 θ with some constant E 1,2 . The asymptotic behavior as θ → +∞ is given by (A.22), (A.16), (A.14), (A.20) i.e. y(θ ) = F0 (1 + O(θ −α )) + F1 θ (1 + O(θ −α ))
(B.10)
ˆ with some constant F0,1 . Thus, solutions of (B.8) define the scattering matrix S(P, α):
F0 E1 = Sˆ . (B.11) F1 E2 ˆ For convenience of later computation we use the following general form for S: √
√ g cos ζ √ g sin ζ ˆ S(P, α) = √ . (B.12) G cos ϕ G sin ϕ Notice that det Sˆ = 1 by construction, hence Gg sin(ϕ − ζ ) = 1, where G, g, ϕ, ζ depend only on P and α.
(B.13)
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A. Rapoport, V. Rom-Kedar, D. Turaev
α Proposition 3. In the case of the power-law potential V (Q, ε) = Qε , every curve L M , M ∈ [0, 1], intersects infinitely many stability tongues; the intersections happen near (see (B.2)) √ 2J (α, M) α(α + 1) ρ = ρk = , (B.14) √ πk n − 1 and the length of the intervals is given by (see (B.4), (B.5) and (B.7)) √ h ρk ( ρ)k ≈ ρk κ(ρk , M)(1 − M). π k G (P(ρk , M), α) d 1 + Rd
(B.15)
Proof. As before, we need to estimate the scattering matrix S for the rescaled Eq. (A.33). For the power-law potential we have V > 0, so the coefficient of y is positive at small ε for all z. Thus, we may represent Eq. (A.33) in the form y + 2 (τ, ε, µ)y = 0,
(B.16)
We consider Eq. (B.16) separately on the interval |τ | ≤ R, and on the intervals |τ | > R, where R(ε, µ) tends sufficiently slowly to infinity as (ε, µ) → 0 (i.e. as ρ → 0). R is chosen so that for |τ | ≤ R the frequency is large, hence y is highly oscillatory, and so its envelope is found below by the method of averaging. Then, we show that on the intervals |τ | > R (B.16) limits, after some rescaling, to (B.8). Thus, the scattering matrix of (A.33) is found by composing the rescaled Sˆ with the oscillatory solution envelope and then with the rescaled Sˆ −1 . Once S is found, the stability regions are found from trace A. Let R(ε, µ) be chosen such that tends to +∞ uniformly on the interval |τ | ≤ R, as (ε, µ) → +0 (it tends to +∞ indeed on any finite interval of τ — hence it tends to +∞ on any sufficiently slowly growing interval as well). Then, there exists a limit of (A.33) and (A.34) by which α(α + 1) lim ρ 2 2 = , (B.17) ρ→0 (n − 1)z α+2 with z(τ ) solving (
1 h (z )2 − α )(M + (1 − M)z) = , z (0) = 0. 2 z 2
(B.18)
Let us apply an averaging procedure to (B.16) on the interval τ ∈ [−R, R]: define (r, φ) by √ √ y = r cos φ,
√ 1 √ y = r sin φ.
Then, Eq. (B.16) takes the form r =
(τ ) (τ ) r cos 2φ, φ = − − sin 2φ, 2
or, after we introduce the fast and slow phases η = (τ )dτ, = φ + η,
(B.19)
Stability in High Dimensional Steep Repelling Potentials
531
the following form dr = ωr cos 2( − η), dη
d ω = − sin 2( − η), dη 2
(B.20)
where ω := (τ )/ 2 (τ ); by (B.17) (and since z is bounded by (B.18)) ω = O( 2ε + µ2 ),
(B.21)
uniformly for |τ | ≤ R (provided R grows sufficiently slowly). Since ω in (B.20) is small, by virtue of the averaging principle, the solutions of (B.20) are close to the solutions of the averaged (with respect to η) system for every o(ω−2 )-long interval of values of η. +∞ In fact, the total change in η cannot exceed −∞ (τ )dτ = O(ρ −1 ) = o(ω−2 ) (see (B.19 ), (B.17), (B.21)). Hence, for all τ ∈ [−R, R], the solutions of (B.20) remain close to the solutions of the system averaged with respect to η, which is simply dr = 0, dη
d = 0. dη
Thus, the evolution from τ = −R to τ = R is, to the leading order, just a rotation R by the angle − −R (τ )dτ . Denote: ⎛
b
(τ )dτ ⎜ cos a b Sr ot (a, b) = ⎜ ⎝ − sin (τ )dτ a
sin
b
a b
cos
⎞ (τ )dτ ⎟ ⎟. ⎠ (τ )dτ
(B.22)
a
So the values of y and y at τ = ±R are related by: ⎛ ⎞ ⎛ (R) y(R) (R) ⎝ ⎠ ⎝ 1 1 (−R, R) · ≈ S r ot y (R) √ √ (R) (R)
y(−R) y (−R)
⎞ ⎠
(B.23)
(by time-reversibility, (R) = (−R)). Let us now consider the behavior of solutions of (A.33) on the interval τ > R. Here τ is large, and we estimate the solution of (A.34) as z(τ )(1 + εz(τ )/2) = h
√ 1−M 2 τ (1 + O(τ −α )) + Mhτ (1 + O(τ −α )). 4
Recall that we are interested only in the behavior for |τ | ≤ t/δ, which corresponds to z = o(ε−1 ) (see (A.32) and (A.35)), so we may write z(τ ) = h
√ 1− M 2 τ 1 + o(1)ρ→0 + Mhτ 1 + o(1)ρ→0 4
on the interval τ > R. After scaling the time τ = κθ , where κ is given by (B.5), we find (after some algebraic manipulations) that Eq. (A.33) on this interval transforms into 1 + o(1) y (θ ) + α+2 y(θ ) = 0, (1 − P)θ 2 + Pθ (1 + o(1))
(B.24)
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A. Rapoport, V. Rom-Kedar, D. Turaev
where P(ρ, M) is given by (B.4). Since Eq. (B.24) limits to (B.8) as ρ → 0, its scattering matrix is well approximated by the scattering matrix Sˆ of (B.8). Thus, returning to the time τ = κθ , we obtain from (B.11), (B.9), (B.10) that ⎛ ⎞ +
κ(τ )y(τ ) D0 E1 ⎠, 1 = Sˆ ≈ Sˆ · Sr ot (τ, ∞) · ⎝ (B.25) κ D1+ E2 κ y (τ ) √ κ(τ ) + are the coefficients of the expansion (A.22) for the solutions of (A.33). where D0,1 By time-reversibility, for the interval τ < −R we have ⎛ ⎞
(τ )y(τ ) − √ D0 ⎠, 1 (B.26) ≈ κ Sˆ · Sr ot (−∞, τ ) · ⎝ y (τ ) −√ −κ D1− (τ )
namely, ⎛
⎞ (τ )y(τ ) 1 1 ⎝ ⎠ 1 ≈ √ 0 y (τ ) √ κ (τ )
0 −1
·
1 = √ Sr ot (−∞, τ ) · κ
(−∞, τ ) · Sˆ −1
Sr−1 ot
1 0
0 −1
From (B.23), (B.25) and (B.27) we find D0+ κ D1+
≈ Sˆ · Sr ot (R, ∞) · Sr ot (−R, R) · Sr ot (−∞, R) ·
+∞ (τ )dτ cos −∞ +∞ = Sˆ (τ )dτ − sin −∞
+∞ sin −∞ (τ )dτ 1 +∞ 0 cos −∞ (τ )dτ
⎛
· Sˆ −1 ⎝
D0− −κ D1− D0− −κ D1−
⎞ ⎠ . (B.27)
D0− 1 0 · Sˆ −1 − 0 −1 −κ D1
−
D0 0 1 0 . Sˆ −1 −1 0 −1 κ D1−
By (B.12), this gives us the following formula for the scattering matrix S : (D0− , → (D0+ , D1+ ): √
Gg sin( + ϕ + ζ ) κg sin(ψ + 2ζ ) √ S≈ , (B.28) G Gg sin( + ϕ + ζ ) κ sin( + 2ϕ)
D1− )
where
#
α(α + 1) , (B.29) n−1 −∞ and G, g, ϕ, ζ are the coefficients of the scattering matrix Sˆ that depend only on P and α (see (B.17), (B.18), (B.2)). Now, like in the proof of Theorem 2, by virtue of (A.2), (A.25), (A.14), (A.16), (A.13), (3.8), (B.7), (B.28), we obtain the following formula18 for the trace of the monodromy matrix A = BC:
2G 2d d Tr(A) = √ sin(+2ϕ)(1+o(1))+2 Gg sin(+ϕ+ζ ) 1 + +o(1). d 1+ R R hδκ (B.30) =
+∞
2J (τ )dτ ≈ ρ
√ 18 In particular, setting µ = 0 and α = 1 in (B.30) (so M = P = 0, δ = √ε/2, ρ = 2ε, κ = 1 3
4 1 1 ε n−1
4 h
4
, J (0) =
√
h) gives formula (4.1).
Stability in High Dimensional Steep Repelling Potentials
533
Equating T r (A) to ±2 supply the stability intervals (B.14), (B.15); since δκ is small (see (B.7)) and G is non-zero (by (B.13)), only the first term is of importance, and the stability intervals are created when + 2ϕ ≈ π k, which gives (B.14) (see (B.29)). Formula (B.15) is found from:
d 2G d ρ ≈ 4. d 1+ √ dρ hδκ R
References 1. Arnaud, M.-C.: Difféomorphismes symplectiques de classe c1 en dimension 4. C. R. Acad. Sci. Paris Ser. I Math. 331(12), 1001–1004 (2000) In French 2. Baldwin, P.R.: Soft billiard systems. Phys. D 29(3), 321–342 (1988) 3. Bálint, P., Tóth, I.P.: Mixing and its rate in ‘soft’ and ‘hard’ billiards motivated by the Lorentz process. Phys. D 187(1–4), 128–135 (2004) 4. Bálint, P., Tóth, I.P.: Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards. Discrete Contin. Dyn. Syst. 15(1), 37–59 (2006) 5. Donnay, V.J.: Elliptic islands in generalized Sinai billiards. Ergod. Th. & Dynam. Sys. 16, 975–1010 (1996) 6. Donnay, V.J., Liverani, C.: Potentials on the two-torus for which the Hamiltonian flow is ergodic. Commun. Math. Phys. 135, 267–302 (1991) 7. Giorgilli, A., Delshams, A., Fontich, E., Galgani, L., Simó, C.: Effective stability for Hamiltonian systems near an elliptic point, with an application to the restricted three body problem. J. Diff. Eq. 77(1), 167 (1989) 8. Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.: Elliptic periodic orbits near a homoclinic tangency in four-dimensional symplectic maps and hamiltonian systems with three degrees of freedom. Regular and Chaotic Dynamics 3(4), 3–26 (1998) 9. Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.: Infinitely many elliptic periodic orbits in four dimensional symplectic diffeomorphism with a homoclinic tangency. Proc. Steklov Inst. Math. 244, 106–131 (2004) 10. Kaplan, A., Friedman, N., Andersen, M., Davidson, N.: Observation of islands of stability in soft wall atom-optics billiards. Phys. Rev. Let. 87(27), 274101–1–4 (2001) 11. Krámli, A., Simányi, N., Szász, D.: Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3D torus. Nonlinearity 2(2), 311–326 (1989) 12. Krámli, A., Simányi, N., Szász, D.: A “transversal” fundamental theorem for semi-dispersing billiards. Comm. Math. Phys. 129(3), 535–560 (1990) 13. Krámli, A., Simányi, N., Szász, D.: The K -property of three billiard balls. Ann. of Math. (2) 133(1), 37–72 (1991) 14. Krámli, A., Simányi, N., Szász, D.: The K -property of four billiard balls. Commun. Math. Phys. 144(1), 107–148 (1992) 15. Krylov, N.S.: Works on the foundations of statistical physics, Princeton, N.J.: Princeton University Press,1979, Translated from the Russian by Migdal, A.B., Sinai, Ya.G., Zeeman, Yu.L.: with a preface by Wightman, A.S., with a biography of Krylov by Fock, V.A., with an introductory article “The views of Krylov N. S. on the foundations of statistical physics” by Migdal and Fok, with a supplementary article “Development of Krylov’s ideas” by Sinai, Princeton Series in Physics 16. Kubo, I.: Perturbed billiard systems in the ergodicity of the motion of a particle in a compound central field. Nagoya Math. J. 61, 1–57 (1976) 17. Marsden, J.E.: Generalized Hamiltonian mechanics: A mathematical exposition of non-smooth dynamical systems and classical Hamiltonian mechanics. Arch. Rat. Mech. Anal. 28, 323–361 (1967) 18. Nekhoroshev, N.N.: An exponential estimate of the time of stability of near-integrable Hamiltonian systems. Russ. Math. Surveys 32(6), 1–65 (1977) 19. Newhouse, S.: Quasi-elliptic periodic points in conservative dynamical systems. Amer. J. Math. 99(5), 1061–1087 (1977) 20. Rapoport, A., Rom-Kedar, V.: Nonergodicity of the motion in three-dimensional steep repelling dispersing potentials. Chaos 16(4), 043108 (2006) 21. Rapoport, A., Rom-Kedar, V., Turaev, D.: Approximating multi-dimensional Hamiltonian flows by billiards. Commun. Math. Phys. 272(3), 567–600 (2007)
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22. Rom-Kedar, V., Turaev, D.: Big islands in dispersing billiard-like potentials. Physica D 130, 187–210 (1999) 23. Rom-Kedar, V., Turaev, D.: Big islands in dispersing billiard-like potentials. Physica D 130(3,4), 187– 210 (1999) 24. Saghin, R., Xia, Z.: Partial hyperbolicity of dense elliptic periodic points for c1 -generic symplectic diffeomorphisms. Trans. Amer. Math. Soc 358, 5119–5138 (2006) 25. Simányi, N.: The K -property of N billiard balls. I. Invent. Math. 108(3), 521–548 (1992) 26. Simányi, N.: The K -property of N billiard balls. II. Computation of neutral linear spaces. Invent. Math. 110(1), 151–172 (1992) 27. Simányi, N.: Proof of the ergodic hypothesis for typical hard ball systems. Ann. Henri Poincaré 5(2), 203–233 (2004) 28. Simányi, N.: The Boltzmann-Sinai Ergodic Hypothesis in Full Generality (Without Exceptional Models). http://arxiv.org/list/math/0510622, 2005 29. Simányi, N., Szász, D.: Hard ball systems are completely hyperbolic. Ann. of Math. (2) 149(1), 35–96 (1999) 30. Sinai, Ya.G.: On the foundations of the ergodic hypothesis for dynamical system of statistical mechanics. Dokl. Akad. Nauk. SSSR 153, 1261–1264 (1963) 31. Sinai, Ya.G.: On a “physical” system with positive “entropy”. Vestnik Moskov. Univ. Ser. I Mat. Meh., (5), 6–12 (1963) 32. Sinai, Ya.G.: Dynamical systems with elastic reflections: Ergodic properties of scattering billiards. Russian Math. Sur. 25(1), 137–189 (1970) 33. Sinai, Ya.G., Chernov, N.I.: Ergodic properties of some systems of two-dimensional disks and threedimensional balls. Usp. Mat. Nauk 42(3)(255), 153–174, 256 (1987) In Russian 34. Takens, F.: Homoclinic points in conservative systems. Invent. Math. 18(3–4), 267–292 (1972) 35. Turaev, D., Rom-Kedar, V.: Islands appearing in near-ergodic flows. Nonlinearity 11(3), 575–600 (1998) 36. Turaev, D., Rom-Kedar, V.: Soft billiards with corners. J. Stat. Phys. 112(3–4), 765–813 (2003) 37. Wojtkowski, M.: Principles for the design of billiards with nonvanishing lyapunov exponents. Commun. Math. Phys. 105(3), 391–414 (1986) Communicated by G. Gallavotti
Commun. Math. Phys. 279, 535–557 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0430-8
Communications in
Mathematical Physics
Stable Sets and -Stable Sets in Positive-Entropy Systems Wen Huang Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China. E-mail: [email protected] Received: 16 March 2007 / Accepted: 22 August 2007 Published online: 16 February 2008 – © Springer-Verlag 2008
Abstract: In this paper, the chaoticity appearing in the stable and unstable sets of a dynamical system with positive entropy is investigated. It is shown that in any positiveentropy system, there is a measure-theoretically “rather big” set such that the closure of the stable or unstable set of any point from the set contains a weak mixing set. Moreover, the Bowen entropy of these weak mixing sets are also estimated. At the same time, it is proved that the topological entropy of any topological system can be calculated in terms of the dispersion of the pre-images of -stable sets, which answers an open question posed by D. Fiebig, U.R. Fiebig and Z.H. Nitecki (Ergod. Th & Dynam. Sys. 23, 1785-1806 (2003)). 1. Introduction Throughout this paper, by a topologically dynamical system (X, T ) (TDS for short) we mean a compact metric space X with a homeomorphism T from X into itself; the metric on X is denoted by d. For x ∈ X , let W s (x, T ) = {y ∈ X : lim d(T n x, T n y) = 0} and n→+∞
W (x, T ) = {y ∈ X : lim d(T −n x, T −n y) = 0}. u
n→+∞
We call W s (x, T ) the stable set of x for T , and W u (x, T ) the unstable set of x for T . For Anosov diffeomorphisms on a compact manifold, pairs belonging to the stable set are asymptotic under T and tend to diverge under T −1 , while pairs belonging to the unstable set behave in the opposite way. Recently, Blanchard, Host and Ruette [1] showed that any positive-entropy system retains a faint flavor of this situation: the stable sets are not stable under T −1 . More precisely, if a T -invariant ergodic measure µ The author is supported by NSFC, 973 project and FANEDD (200520).
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has positive entropy, then there is δ > 0 such that for µ-a.e. x ∈ X , one can find an uncountable subset Fx of W s (x, T ) such that for any y ∈ Fx , one has lim inf d(T −n x, T −n y) = 0 and lim sup d(T −n x, T −n y) ≥ δ. n→+∞
n→+∞
In [15] N. Sumi got further information for stable and unstable sets of a C 2 diffeomorphism f of a closed C ∞ manifold M with positive entropy: if an f -invariant ergodic probability measure µ on M has positive metric entropy, i.e. h µ ( f ) > 0, then, for µ-a.e. x ∈ M, W s (x, T ) is a perfect ∗-chaotic set and W u (x, T ) contains a perfect ∗-chaotic set; and if µ is an SRB measure, W u (x, T ) is a perfect ∗-chaotic set. Note that f is said to be ∗-chaotic on F (and F is called a ∗-chaotic set) if the following two conditions are satisfied: (1) there is a constant τ > 0 (depending only on F) such that, for any non-empty open subsets U and V of F with U ∩ V = ∅ and any natural number N , there is an n ≥ N such that d( f n x, f n y) > τ for some x ∈ U , y ∈ V ; (2) for any non-empty open subsets U and V of F and any ε > 0, there is an n ≥ 0 such that d( f n x, f n y) < ε for some x ∈ U , y ∈ V . In [11] H. Kato proved that a homeomorphism T is ∗-chaotic on a perfect subset F of X iff there is δ > 0 and an uncountable subset S of F such that S is the union of a countable number of cantor sets, S is dense in F and for any x = y ∈ S, lim inf d(T n x, T n y) = 0 and lim sup d(T n x, T n y) ≥ δ. n→+∞
n→+∞
Using the above equivalent characterization, it is easy to check that the result in [2] implies that a TDS (X, T ) with positive entropy has a perfect ∗-chaotic set. It is a natural question whether the result of N. Sumi can be generalized to the general case. That is, for a general TDS with positive entropy, does the closure of the stable or unstable sets of “many” points contain a perfect ∗-chaotic set or a perfect set having stronger chaotic properties? Answering the question affirmatively, in this paper we show that in any TDS with positive entropy there is a measure-theoretically “rather big” set such that the closure of the stable or unstable sets of points in the set contains a weak mixing set, by means of ergodic theory. Note that the notion of a weak mixing set was first introduced in [3] and a weak mixing set must be a perfect ∗-chaotic set. At the same time, we also estimate the Bowen entropy of these weak mixing sets. More precisely, we show that if µ is an ergodic invariant measure of a TDS (X, T ) with h µ (T ) > 0, then for µ-a.e. x ∈ X , there exist closed subsets A(x) ⊆ W s (x, T ), B(x) ⊆ W u (x, T ) and E(x) ⊆ W s (x, T ) ∩ W u (x, T ) such that a) limn→+∞ diam(T n A(x)) = 0 and h(T −1 , A(x)) ≥ h µ (T ), where h(T −1 , A(x)) is the Bowen entropy of A(x) with respect to T −1 . b) limn→+∞ diam(T −n B(x)) = 0 and h(T, B(x)) ≥ h µ (T ), where h(T, B(x)) is the Bowen entropy of B(x) with respect to T . c) E(x) is weakly mixing for T, T −1 . d) h(T, E(x)) ≥ h µ (T ) and h(T −1 , E(x)) ≥ h µ (T ). A key tool in the proof of the above results a)-d) is the excellent partition constructed in Lemma 4 of [1]. Using the excellent partition, we also solve an open problem posed by D. Fiebig, U.R. Fiebig and Z.H. Nitecki in [6]. More precisely, let (X, T ) be a TDS
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with compatible metric d. Given > 0, the -stable set of x under T is the set of points whose forward orbit -shadows that of x: Ws (x, T ) = {y ∈ X : d(T n x, T n y) ≤ for all n = 0, 1, · · · }. The preimages of these sets can be nontrivial and hence can disperse at a nonzero exponential rate. Given x ∈ X and > 0, consider the dispersal rate h s (T, x, ) := lim lim sup δ→0 n→+∞
1 log sn (d, T, δ, T −n Ws (x, T )), n
where sn (d, T, δ, T −n Ws (x, T )) denotes the largest cardinality of any (n, δ)-separated subset of T −n Ws (x, T ). It was proved in Theorem 4.1 of [6] that when X has finite covering dimension, then sup h s (T, x, ) = h top (T ) for all > 0. x∈X
Our result in this paper shows that the finite-dimensionality hypothesis is redundant, that is, the above equality is true for any TDS (X, T ). The paper is organized as follows. In Sect. 2, some background in ergodic theory and TDS is introduced. In Sect. 3, using an ad hoc excellent partition, we show that the topological entropy of any TDS can be calculated in terms of the dispersion of the preimages of -stable set. In fact, we prove a stronger result: Given a T -ergodic measure µ on a TDS (X, T ), for µ-a.e. x ∈ X the dispersion of the preimages of the -stable set with respect to x is larger than the metric entropy h µ (T ). Moreover, we show that a TDS having a maximal entropy ergodic measure (in particular, every finite entropy TDS with an upper semi-continuous entropy map) has entropy points, where a point x for a TDS is an entropy point, if the dispersion of the preimages of -stable set with respect to x measures topological entropy. In the next section, we show by an ad hoc excellent partition that in any TDS with positive entropy there is a measure-theoretic “rather big” set such that the closure of the stable or unstable sets of points in the set contain a weak mixing set, and at the same time we also estimate the Bowen entropy of these weak mixing sets. More precisely, we obtain the above results a)-d). In the last section, we show that many results in Sects. 3 and 4 are also true for non-invertible TDS. 2. Preliminary Given a TDS (X, T ), denote by B X the σ -algebra of Borel subsets of X . A cover of X is a family of Borel subsets of X whose union is X . An open cover is one that consists of open sets. A partition of X is a cover of X by pairwise disjoint sets. Given a partition α of X and x ∈ X , denote by α(x) the atom of α containing x. We denote the set of finite partitions, finite covers and finite open covers, of X , respectively, by P X , C X and C oX , respectively. Given two covers U, V of X , U is said to be finer than V (denote by U V) if each element of U is contained in some element of V. Let U ∨ V = {U ∩ V : U ∈ U, V ∈ V}. It is clear that U ∨ V U and U ∨ V V. N to denote N −n U. Given integers M, N with M ≤ N and U ∈ C X , we use U M n=M T For U ∈ C X , we define N (U) as the minimum among the cardinalities of the subcovers of U. Then the topological entropy of U with respect to T is 1 1 h top (T, U) = lim log N U0N −1 = inf log N U0N −1 . N →∞ N N ∈N N
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The topological entropy of (X, T ) is defined by h top (T ) = sup h top (T, U). U ∈C oX
Let M(X ), M(X, T ), and Me (X, T ), respectively, be the set of all Borel probability measures, T -invariant Borel probability measures, and T -invariant ergodic measures on X , respectively. Then M(X ) and M(X, T ) are all convex, compact metric spaces when endowed with the weak∗ -topology. For any given α ∈ P X , µ ∈ M(X ) and any sub-σ -algebra C ⊆ Bµ , where Bµ is the completion of B X under µ, let −µ(A) log µ(A) and Hµ (α|C) = −E(1 A |C) log E(1 A |C)dµ, Hµ (α) = A∈α
X
A∈α
where E(1 A |C) is the conditional expectation of 1 A with respect to C. One standard fact states that Hµ (α|C) increases with respect to α and decreases with respect to C. For a given U ∈ C X , set Hµ (U) =
inf
β∈P X :β U
Hµ (β) and Hµ (U|C) =
inf
β∈P X :β U
Hµ (β|C).
Clearly, Hµ (U|C) ≤ Hµ (U). When µ ∈ M(X, T ) and C is T -invariant (i.e. T −1 C = C), it is not hard to see that Hµ (U0n−1 |C) is non-negative and a sub-additive sequence for a given U ∈ C X , so we can define h µ (T, U|C) = lim
n→+∞
1 1 Hµ (U0n−1 |C) = inf Hµ (U0n−1 |C). n≥1 n n
If C = {∅, X }(mod µ), we write Hµ (U|C) and h µ (T, U|C) by Hµ (U) and h µ (T, U) respectively. The measure-theoretic entropy of µ is defined by h µ (T ) = sup h µ (T, α). α∈P X
In [14] the author proved h µ (T ) = sup h µ (T, U). U ∈C oX
(2.1)
It is well known that for β ∈ P X , h µ (T, β) = h µ (T, β|Pµ (T )) ≤ Hµ (β|Pµ (T )), where Pµ (T ) is the Pinsker σ -algebra of (X, Bµ , µ, T ). Similarly one has Lemma 2.1. Let (X, T ) be a TDS, µ ∈ M(X, T ) and U ∈ C X . Then h µ (T, U) = h µ (T, U|Pµ (T )). Proof. The inequality h µ (T, U) ≥ h µ (T, U|Pµ (T )) is obvious. On the other hand, 1 Hµ (U0n−1 |Pµ (T )) n→+∞ n
h µ (T, U|Pµ (T )) = lim ≥ lim
= lim
1 inf n→+∞ n β∈P :β U n−1 X 0
1 inf h µ (T n , β) n→+∞ n β∈P :β U n−1 X 0
Note that last equality was proved in [14].
= h µ (T, U).
Hµ (β|Pµ (T ))
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Let (X, T ) be a TDS, µ ∈ M(X, T ) and Bµ be the completion of B X under µ. Then (X, Bµ , µ, T ) be a Lebesgue system. If {αi }i∈I is a countable family of finite partitions of X , the partition α = i∈I αi is called a measurable partition. The sets A ∈ Bµ , which are unions of atoms of α, form a sub-σ -algebra Bµ by α or α if there is no ambiguity. Every sub-σ -algebra of Bµ coincides with a σ -algebra constructed in this way (mod µ). +∞ −n α and α T = −n α. Given a measurable partition α, put α − = ∞ n=1 T n=−∞ T − T Define in the same way F and F if F is a sub-σ -algebra of Bµ . It is clear that for a − = ( T = ( measurable partition α of X , α α )− and α α )T (mod µ). Let F be a sub-σ -algebra of Bµ and α be the measurable partition of X with α=F (mod µ). µ can be disintegrated over F as µ = X µx dµ(x), where µx ∈ M(X ) and µx (α(x)) = 1 for µ-a.e. x ∈ X . The disintegration is characterized by the properties (2.2) and (2.3) below: for every f ∈ L 1 (X, B X , µ), f ∈ L 1 (X, B X , µx ) for µ-a.e. x ∈ X, f (y) dµx (y) is in L 1 (X, F, µ); and the map x → X 1 for every f ∈ L (X, B X , µ), Eµ ( f |F)(x) = f dµx for µ-a.e. x ∈ X.
(2.2)
(2.3)
X
Then, for any f ∈ L 1 (X, B X , µ), the following holds:
f dµx dµ(x) = f dµ. X
X
X
Lemma 2.2. Let (X, T ) be a TDS, µ ∈ M(X, T ) and F be a sub-σ -algebra of Bµ . If µ = X µx dµ(x) is the disintegration of µ over F, then (a) for V ∈ C X , Hµ (V|F) = X Hµx (V)dµ(x), (b) for U, V ∈ C X , Hµ (U ∨ V|F) ≤ Hµ (U|F) + Hµ (V|F). Proof. (b) is obvious. Now we are going to prove n (a). Let V = {V1 , V2 , . . . , Vn }. For any s = (s(1), . . . , s(n)) ∈ {0, 1}n , set Vs = i=1 Vi (s(i)), where Vi (0) = Vi and Vi (1) = X \ Vi . Let α = {Vs : s ∈ {0, 1}n }. Then α is the Borel partition of X generated by V and put P(V) = {β ∈ P X : α β V}, which is a finite family of partitions. It is well known that for every θ ∈ M(X ), Hθ (V) = min Hθ (β)
(2.4)
β∈P(V )
(for example see the proof of Proposition 6 in [14]). Now denote P(V) = {β1 , β2 , . . . , βl } and put Ai = {x ∈ X : Hµx (βi ) = min Hµx (β)}, i ∈ {1, 2, . . . , l}. β∈P(V )
Let B1 = A1 , B2 = A2 \ B1 , . . . , Bl = Al \ ( Eq. (2.4), µ(B0 ) = 0.
l−1 i=1
Bi ) and B0 = X \ (
l i=1
Al ). By
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Set β ∗ = {B0 ∩ β1 } ∪ {Bi ∩ βi : i = 1, . . . , l}. Then β ∗ ∈ P X (mod µ) and β ∗ V. It is clear that for i ∈ {1, 2, · · · , l} and µ-a.e. x ∈ Bi , Hµx (β ∗ ) = Hµx (βi ) = min Hµx (β) = Hµx (V), β∈P(V )
where the last equality follows from (2.4). Combining this fact with µ(B0 ) = 0 one gets Hµx (β ∗ ) = Hµx (V) for µ-a.e. x ∈ X . This implies Hµ (V|F) ≤ Hµ (β ∗ |F) = Hµx (β ∗ )dµ(x) (by (2.3)) X = Hµx (V)dµ(x) ≤ inf Hµx (β)dµ(x) β∈P X :β V
X
= That is, Hµ (V|F) =
X
inf
β∈P X :β V
X
Hµ (β|F) = Hµ (V|F).
Hµx (V)dµ(x).
Let K be a non-empty closed subset of X . For > 0, a subset F of X is called an (n, )-spanning set of K , if for any x ∈ K , there exists y ∈ F with dn (x, y) ≤ , where n−1 d(T i x, T i y); a subset E of K is called an (n, )-separated set of K dn (x, y) = maxi=0 if x, y ∈ E, x = y implies dn (x, y) > . Let rn (d, T, , K ) denote the smallest cardinality of any (n, )-spanning set for K and sn (d, T, , K ) denote the largest cardinality of any (n, )-separated subset of K . Put r (d, T, , K ) = lim sup n1 log rn (d, T, , K ), n→∞
s(d, T, , K ) = lim sup n1 log sn (d, T, , K ). n→∞
Obviously, r (d, T, , K ) and s(d, T, , K ) are monotone increasing when 0. Then set h ∗ (d, T, K ) = lim r (d, T, , K ) and h ∗ (d, T, K ) = lim s(T, d, , K ). →0+
→0+
h ∗ (d, T, K )
It is well known that h ∗ (d, T, K ) = is independent of the choice of any compatible metric d on X , so we write it simply as h(T, K ). When K = X , h(T, X ) = h top (T ). Given U ∈ C X , put
N (U|K ) = min{the cardinality of F|F ⊂ U, F ⊃ K }. F∈F
Then we define 1 h(T, U|K ) = lim sup log N n→+∞ n
n−1 i=0
It is easy to see that h(T, K ) = supU ∈C o h(T, U|K ). X
T −i U|K .
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3. -Stable Sets in Positive Entropy Systems Let (X, T ) be a TDS with a compatible metric d. Given > 0, the -stable set of x under T is the set of points whose forward orbit -shadows that of x: Ws (x, T ) = {y ∈ X : d(T n x, T n y) ≤ for all n = 0, 1, . . .}. The preimages of these sets can be nontrivial and hence can disperse at a nonzero exponential rate. Given x ∈ X and > 0, consider the dispersal rate h s (T, x, ) := lim lim sup δ→0 n→+∞
1 log sn (d, T, δ, T −n Ws (x, T )). n
It was proved in [6, Theorem 4.1] that when X has finite covering dimension, then sup h s (T, x, ) = h top (T ) for all > 0. x∈X
In this section, we will show that this is true for any compact metric space X . The main idea of the proof is that we obtain the result by proving that for any µ ∈ Me (X, T ) with positive entropy, lim→0 h s (T, x, ) ≥ h µ (T ) for µ-a.e. x ∈ X . For this purpose, we need several lemmas. The first lemma connects entropy using open covers and separated sets. Lemma 3.1. Let (X, T ) be a TDS and {K n } be a sequence of non-empty closed subsets of X . Then n−1 1 1 −i lim lim sup log sn (d, T, K n , δ) = sup lim sup log N T U|K n . δ→0 n→+∞ n U ∈C o n→+∞ n X
i=0
Proof. For a fixed δ > 0, choose V ∈ C oX with diam(V) < δ. For n ∈ N let A be an (n, δ)-separated set of K n with |A| = sn (d, T, K n , δ). Since B ∩ K n contains at n−1 −i T V, we get sn (d, T, K n , δ) ≤ most one element of A for each element B of i=0 n−1 −i n−1 −i N ( i=0 T V|K n ). That is, n1 log sn (d, T, K n , δ) ≤ n1 log N ( i=0 T V|K n ). Letting n → +∞, we have n−1 1 1 −i lim sup log sn (d, T, K n , δ) ≤ lim sup log N T V|K n n→+∞ n n→+∞ n i=0 n−1 1 −i ≤ sup lim sup log N T U|K n . U ∈C o n→+∞ n i=0
X
Letting δ → 0, we get 1 1 lim lim sup log sn (d, T, K n , δ) ≤ sup lim sup log N o δ→0 n→+∞ n n U ∈C n→+∞ X
n−1
T
−i
U|K n .
i=0
In the following, we show the converse inequality. For any fixed U ∈ C oX , let κ be the Lebesgue number of U. For n ∈ N, let E be an (n, κ)-separated set of K n with
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|E| = sn (d, T, K n , κ). Then E is also an (n, κ)-spanning set of K n . From the definition of spanning sets, we know
n−1
T −i Bκ (T i x) ⊃ K n , where Bκ (T i x) = {y ∈ X : d(T i x, y) ≤ κ}.
x∈E i=0
Now for each x ∈ E and 0 ≤ i ≤ n − 1, Bκ (T i x) is contained in the some element of U since κ is the Lebesgue number of the open cover U. Hence for each x ∈ n−1 −i n−1 −i E, i=0 T B (T i x) is contained in the some element of i=0 T U. Moreover, n−1 −i κ N ( i=0 T U|K n ) ≤ sn (d, T, K n , κ). This implies that n−1 1 1 −i T U|K n ≤ lim sup log sn (d, T, K n , κ) lim sup log N n n n→+∞ n→+∞ i=0
≤ lim lim sup δ→0 n→+∞
1 log sn (d, T, K n , δ). n
Since U is arbitrary, we get n−1 1 1 −i sup lim sup log N T U|K n ≤ lim lim sup log sn (d, T, K n , δ). δ→0 n→+∞ n U ∈C o n→+∞ n X
i=0
An immediate consequence of Lemma 3.1 is the following. Lemma 3.2. Let (X, T ) be a TDS. Then for each x ∈ X and > 0, n−1 1 −i −n s h s (T, x, ) = sup lim sup log N T U|T W (x, T ) . U ∈C o n→+∞ n X
i=0
To get Proposition 3.4 we need the following lemma. Lemma 3.3. Let (X, T ) be a TDS and µ ∈ M(X, T ). For every M ≥ 1, for every > 0, there exists δ = δ(M, ) > 0 such that if U = {U1 , U2 , . . . , U M }, V = M {V1 , V2 , . . . , VM } are two covers of cardinality M of X with µ(U V) := i=1 µ (Ui Vi ) < δ then n−1 n−1 1 1 −i −i T U|C − Hµ T V|C ≤ Hµ n n i=0
i=0
for each n ∈ N and sub-σ -algebra C of Bµ . Proof. Fix M ∈ N and > 0. Then there exists δ = δ (M, ε) > 0 such that for partitions α, β of cardinality M of X , if µ(α β) < δ then Hµ (β|α) < (see for example [16, Lemma 4.15]). Let U = {U1 , U2 , . . . , U M } and V = {V1 , V2 , . . . , VM } are any two δ covers of cardinality M of X with µ(U V) < M = δ. Claim. For every finite partition α U there exists a finite partition β V such that Hµ (β|α) < .
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Proof of Claim. Since α U, there exists a partition α = {A1 , . . . , A M } with Ai ⊂ Ui , i = 1, 2, . . . , M and α α , where Ai may be empty. Let
(Ak ∩ Vk ) , B1 = V1 k>1 Bi = Vi (Ak ∩ Vk ) ∪ B j , i ∈ {2, . . . , M}. k>i
j
Then β = {B1 , . . . , B M } ∈ P X which satisfies Bm ⊆ Vm and Am ∩ Vm ⊆ Bm for m ∈ {1, . . . , M}. It is clear that Am \ Bm ⊆ Um \ Vm and ⎞ ⎞ ⎛ ⎞ ⎛ ⎛
⎝ Am = ⎝ Bk ⎠ Aj⎠ Bk ⎠ Bm \ Am = ⎝ X \ ⊆
k=m
(Ak \ Bk ) ⊆
k=m
j=m
k=m
(Uk \ Vk ) .
k=m
Hence for every m ∈ {1, . . . , M}, Am Bm ⊆
M
(Uk Vk ) and µ(α β) ≤ M ·
k=1
µ(U V) < δ . This implies that Hµ (β|α ) < . Moreover, Hµ (β|α) ≤ Hµ (β|α ) < . This ends the proof of the claim. n−1 −i Fix n ∈ N and a sub-σ -algebra C of Bµ . For any α ∈ P X with α i=0 T U, we have T i α U for i ∈ {0, 1, · · · , n − 1}. By the above claim, there exists βi ∈ P X such n−1 −i T βi . Then that βi V and Hµ (βi |T i α) < , i.e., Hµ (T −i βi |α) < . Let β = i=0 n−1 −i β ∈ P X with β i=0 T V. Now n−1 −i Hµ T V|C ≤ Hµ (β|C) ≤ Hµ (β ∨ α|C) = Hµ (α|C) + Hµ (β|α ∨ C) i=0
≤ Hµ (α|C) +
n−1
Hµ (T −i βi |α) < Hµ (α|C) + n.
i=0
n−1 −i Since the above inequality is true for any α ∈ P X with α i=0 T U, we get n−1 −i n−1 −i 1 1 n Hµ ( i=0 T V|C) ≤ n Hµ ( i=0 T U|C) + . n−1 −i Exchanging the roles of U and V, we get n1 Hµ ( i=0 T U|C) ≤ n1 Hµ n−1 −i n−1 −i n−1 −i ( i=0 T V|C) + . This shows that | n1 Hµ ( i=0 T V|C) − n1 Hµ ( i=0 T U|C)| ≤ . With the help of Lemma 3.3, now we can prove the following. Proposition 3.4. Let (X, T ) be a TDS and µ ∈ M(X, T ). Then for any α ∈ P X and > 0, there exists U ∈ C oX such that n−1 n−1 1 1 −i −i T U|C − Hµ T α|C < Hµ n n i=0
for each n ∈ N and sub-σ -algebra C of Bµ .
i=0
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Proof. Let α = {A1 , A2 , . . . , A M }. By Lemma 3.3, there exists δ = δ(M, ) > 0 such that for every cover U = {U1 , U2 , . . . , U M } of X with µ(U α) < δ, one has n−1 n−1 1 1 −i −i T U|C − Hµ T α|C < Hµ n n i=0
i=0
for each n ∈ N and sub-σ -algebra C of Bµ . Now, it is sufficient to find an open cover U of cardinality M of X with µ(U α) < δ. Since µ is regular, there exist compact sets Bi ⊂ Ai , i = 1, 2, . . . , M such that M M δ δ µ(Ai \ Bi ) < 2M 2 . Let B0 = X \ ( i=1 Bi ). Then µ(B0 ) ≤ i=1 µ(Ai \ Bi ) < 2M . Now we set Ui = B0 ∪ Bi , i = 1, 2, . . . , M. Then µ(U α) =
M
µ(Ui Ai ) ≤
i=1
M (µ(B0 ) + µ(Ai \ Bi )) < δ, i=1
and U = {U1 , U2 , · · · , U M } ∈ C oX . Let (X, T ) be a TDS, µ ∈ M(X, T ) and Bµ be the completion of B X under µ. The Pinsker σ -algebra Pµ (T ) is defined as the smallest sub-σ -algebra of Bµ containing {ξ ∈ P X : h µ (T, ξ ) = 0}. It is well known that Pµ (T ) = Pµ (T −1 ) and Pµ (T ) is T -invariant, i.e. T −1 Pµ (T ) = Pµ (T ). We need the following lemma in the proof of the main result. ∞ ⊂ Lemma 3.5. Let (X, T ) be a TDS, µ ∈ M(X, T ) and δ > 0. Then there exist {Wi }i=1 P X and 0 = k1 < k2 < · · · such that
(1) diam(W1 ) < δ and lim diam(Wi ) = 0, i→+∞
(2) (3)
lim Hµ (Pk |P − ) = h µ (T ), where Pk =
k→+∞ ∞ −n P − T n=0
k i=1
T −ki Wi and P =
∞
k=1
Pk ,
= Pµ (T ).
Proof. The lemma comes directly from the proof of Lemma 4 in [1]. For completeness, ∞ ⊂ P and 0 = k < k < · · · . we outline the construction of {Wi }i=1 X 1 2 ∞ Let {Wi }i=1 be an increasing sequence of finite partitions of X such that diam(W1 ) < δ and lim diam(Wi ) = 0. Take k1 = 0, then we find inductively k1 , k2 , . . . such that i→+∞
for each q ≥ 2 one has − ) − Hµ (Pk |Pq− ) < Hµ (Pk |Pq−1
1 1 , k = 1, 2, . . . , q − 1, k 2q−k
j where P j = i=1 T −ki Wi . It is not hard to check that (1)–(3) are true (see for example [13] or [8]).
Remark 3.6. Since limi→+∞ diam(Wi ) = 0, it is easy to see that (T −n P − )(x) ⊆ W s (x, T ) for each n ∈ N ∪ {0} and x ∈ X , where (T −n P − )(x) is the atom of T −n P − containing x. Now we are able to show the following theorem which clearly implies the main result we want to prove.
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Theorem 3.7. Let (X, T ) be a TDS and µ ∈ Me (X, T ) with h µ (T ) > 0. Then for µ-a.e. x ∈ X , lim→0 h s (T, x, ) ≥ h µ (T ). Proof. It suffices to prove that for a given > 0, h s (T, x, ) ≥ h µ (T ) for µ-a.e. x ∈ X . Fix > 0. Since T is a homeomorphism on X , there exists δ ∈ (0, ) such that ∞ ⊂ P satd(T −1 x, T −1 y) < when d(x, y) < δ. By Lemma 3.5, there exist {Pi }i=1 X ∞ − isfying that diam(P1 ) ≤ δ, n=0 T −n P − = Pµ (T ) and Hµ (Pk |P ) → h µ (T ) when ∞ k → +∞, where P = i=1 Pi . Since diam(P1 ) ≤ δ, it is clear that P − (x) ⊆ Ws (x, T ) for each x ∈X . Let µ = X µx dµ(x) be the disintegration of µ over P − . Then supp(µx ) ⊆ P − (x) ⊆ Ws (x, T ) for µ-a.e. x ∈ X . Let k ∈ N. By Proposition 3.4 there exists Uk ∈ C oX such that n−1 n−1 1 1 1 −i −n − −i −n − Hµ T Uk |T P T Pk |T P ≥ Hµ − for each n ∈ N. n n k i=0
i=0
(3.1) Since P −
n−1
T −i Pk ∨ T −n P − for each n ∈ N, we have n−1 1 −i −n − T Pk |T P lim sup Hµ n→+∞ n i=0 1 = lim sup Hµ (T −(n−1) Pk |T −n P − ) + Hµ (T −(n−2) Pk |T −(n−1) Pk ∨ T −n P − ) n→+∞ n n−1 −i −n − + · · · + Hµ Pk | T Pk ∨ T P i=1
i=1
1 = lim sup Hµ (Pk |T −1 P − ) + Hµ (Pk |T −1 Pk ∨ T −2 P − ) n→+∞ n n−1 + · · · + Hµ Pk | T −i Pk ∨ T −n P − i=1
≥ lim sup Hµ (Pk |P − ) = Hµ (Pk |P − ).
(3.2)
n→+∞
By (3.1) and (3.2), we get n−1 1 1 −i −n − T Uk |T P lim sup Hµ ≥ Hµ (Pk |P − ) − for each n ∈ N. k n→+∞ n
(3.3)
i=0
n For n ∈ N, let Fn : X → [0, log N (Uk )] with Fn (x) = n1 log N ( i=1 T i Uk |Ws s (x, T )). Note that W (x, T ) is a closed subset for each x ∈ X and the map x → Ws (x, T ) is an upper semi-continuous map from X to 2 X , where 2 X is the collection of all nonempty closed subsets of X endowed with the Hausdorff metric. It is not difficult to see that the function Fn is upper semi-continuous, that is, lim supx→x0 Fn (x) ≤ Fn (x0 ) for each x0 ∈ X . Hence Fn is a Borel measurable bounded function.
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Let F(x) = lim supn→+∞ Fn (x) for x ∈ X . Then F is a Borel measurable map from X to [0, log N (Uk )]. Since T Ws (x, T ) ⊆ Ws (T x, T ) for each x ∈ X , we have n 1 T i Uk |Ws (x, T ) F(x) = lim sup log N n→+∞ n i=1 n+1 1 ≤ lim sup log N T i Uk |T Ws (x, T ) n→+∞ n i=1 n+1 n+1 1 ≤ lim sup T i Uk |Ws (T x, T ) = F(T x). log N n n+1 n→+∞ i=1 That is, F(x) ≤ F(T x) for each x ∈ X . Since X (F(T x) − F(x))dµ(x) = 0, we have F(T x) = F(x) for µ-a.e. x ∈ X . Moreover, F(x) ≡ ak for µ-a.e. x ∈ X as µ is ergodic, where ak ≥ 0 is a constant. Now we are going to estimate ak . F(x)dµ(x) = lim sup Fn (x)dµ(x) ≥ lim sup Fn (x)dµ(x) ak = n→+∞ X X X n→+∞ n 1 ≥ lim sup Hµx T i Uk dµ(x) (Since supp(µx ) n→+∞ X n i=1
⊆ Ws (T, x) for µ-a.e. x ∈ X ) n 1 i − T Uk |P = lim sup Hµ (by Lemma 2.2 a)) n→+∞ n i=1 n−1 1 −i −n − T Uk |T P = lim sup Hµ n→+∞ n i=0
1 ≥ Hµ (Pk |P − ) − . (by (3.3)). k Since that F(x) ≤ h s (T, x, ) for each x ∈ X by Lemma 3.2, we have
1 = h µ (T ) h s (T, x, ) ≥ lim Hµ (Pk |P − ) − k→+∞ k for µ-a.e. x ∈ X . Now we investigate the -entropy point and entropy point which were introduced in [6]. Let (X, T ) be a TDS. For > 0 we call x ∈ X an -entropy point for T if h s (T, x, ) = h top (T ); it is simply an entropy point if lim→0 h s (T, x, ) = h top (T ). Since h s (T, x, ) is decreasing in , an entropy point is an -entropy point for each > 0. Note that while the notion of -entropy point depends on the choice of metric, that of entropy point does not. Denote by E(T ) the set of all entropy points of (X, T ). Theorem 3.8. Let (X, T ) be a TDS. If there exists µ ∈ Me (X, T ) such that h µ (T ) = h top (T ), then E(T ) = ∅. Particularly, if h top (T ) < ∞ and the entropy map ν : M(X, T ) → h ν (T ) ∈ R+ is upper semi-continuous, then E(T ) = ∅. Proof. By Theorem 3.7, for µ-a.e. x ∈ X , x is an entropy point for T . When h top (T ) < ∞ and the entropy map ν ∈ M(X, T ) → h ν (T ) is upper semi-continuous, we can always find µ ∈ Me (X, T ) with h µ (T ) = h top (T ).
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4. Stable Sets in Positive Entropy Systems In the section, we aim to investigate the rich dynamical properties contained in the stable and unstable sets in a positive-entropy system. Recall that for a TDS (X, T ) and x ∈ X , W s (x, T ) = {y ∈ X : lim d(T n x, T n y) = 0} and n→+∞
W u (x, T ) = {y ∈ X : lim d(T −n x, T −n y) = 0}. n→+∞
We call W s (x, T ) the T . Clearly, W s (x, T )
stable set of x for T , and call W u (x, T ) the unstable set of x for = W u (x, T −1 ) and W u (x, T ) = W s (x, T −1 ). The main results of the section are Theorem 4.2 and Theorem 4.6. To show Theorem 4.2 we need the following lemma. Lemma 4.1. Let (X, T ) be a TDS, U ∈ C oX and {Q i } ⊂ P X with Q 1 Q 2 Q 3 · · · . ∞ Then F : X → [0, N (U)] defined by F(x) = N (U| i=1 Q i (x)) for x ∈ X is a Borel measurable map. Proof. Let U = {U1 , U2 , . . . , U M }. First, we have Claim. For any non-empty open subset U of X , ∞
x∈X: Q i (x) ⊆ U = {A ∈ Q k : A ⊆ U }. i=1
n≥1 k≥n
Proof of Claim. Let x ∈ n≥1 k≥n {A ∈ Q k : A ⊆ U }. Then there exists N ∈ N such that x ∈ k≥N {A ∈ Q k : A ⊆ U }. Hence Q k (x) ⊆ U for k ≥ N . This implies ∞ Q i (x) ⊆ U . that i=1 ∞ Conversely, let x ∈ X with i=1 Q i (x) ⊆ U . Since Q 1 (x) ⊇ Q 2 (x) ⊇ · · · and U is an open set of X , there is N ∈ N such that Q k (x) ⊆ U for k ≥ N . This shows that x ∈ k≥N {A ∈ Q k : A ⊆ U }. This finishes the proof of claim. ∞ For any non-empty open subset U of X , put S(U ) = {x ∈ X : i=1 Q i (x) ⊆ U }. Then S(U ) is a Borel set of X by the claim. It is clear that for each x ∈ X , F(x) ∈ {1, 2, . . . , M}. Hence to show that F is a Borel map, it remains to prove that for any k ∈ {1, 2, . . . , M}, the set Ak := {x ∈ X : F(x) ≤ k} is a Borel subset of X . This follows by the following observation: for k ∈ {1, 2, . . . , M},
S(Ui1 ∪ Ui2 ∪ · · · ∪ Uik ). Ak = 1≤i 1
Using Lemma 4.1 and tools developed in the previous section, we now can show the following theorem. Theorem 4.2. Let (X, T ) be a TDS and µ ∈ Me (X, T ) with h µ (T ) > 0. Then for µ-a.e. x ∈ X , (1) there exist a closed subset A(x) ⊆ W s (x, T ) such that lim diam(T n A(x)) = 0 and h(T −1 , A(x)) ≥ h µ (T );
n→+∞
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(2) there exist a closed subset B(x) ⊆ W u (x, T ) such that lim diam(T −n B(x)) = 0 and h(T, B(x)) ≥ h µ (T ).
n→+∞
Proof. Since µ ∈ Me (X, T −1 ), h µ (T −1 ) = h µ (T ) and W s (x, T −1 ) = W u (x, T ), (1) implies (2). It remains to prove (1). ∞ ⊂ P and 0 = k < k < · · · satisfying that By Lemma 3.5, there exist {Wi }i=1 X 1 2 (1) limi→+∞ diam(Wi ) = 0, k (2) limk→+∞ Hµ (Pk |P − ) = h µ (T ), where Pk = i=1 T −ki Wi and P = ∞ k=1 Pk , ∞ −n − P = Pµ (T ). (3) n=0 T i Let Q i = j=1 T − j (P1 ∨ P2 ∨ · · · ∨ Pi ) for i ∈ N. Then Q i ∈ P X , Q 1 Q 2 · · · ∞ Qi = P −. and i=1 ∞ Q i (x). Then A(x) is a closed set and A(x) ⊇ P − (x). For x ∈ X , let A(x) = i=1 Now we show that for µ-a.e. x ∈ X , A(x) ⊆ W s (x, T ), lim diam(T n A(x)) = 0 and h(T −1 , A(x)) ≥ h µ (T ). n→+∞
Fix x ∈ X . For > 0, there exists r ∈ N such that diam(Wr ) ≤ . Then diam(T kr Pr ) ≤ . For j ≥ 0, since T kr + j Q r + j T kr Pr , diam(T kr + j Q r + j (x) ) = diam T kr + j (Q r + j (x)) = diam((T kr + j Q r + j )(T kr + j x)) ≤ diam((T kr Pr )(T kr + j x)) ≤ . ∞ Q i (x). This implies that Moreover diam(T kr + j A(x)) ≤ for j ≥ 0 as A(x) = i=1 lim diam(T n A(x)) = 0 and A(x) ⊆ W s (x, T ). n→+∞ Let µ = X µx dµ(x) be the disintegration of µ over P − . Then supp(µx ) ⊆ P − (x) ⊆ A(x) for µ-a.e. x ∈ X .
(4.1)
Next, we show that for µ-a.e. x ∈ X , h(T −1 , A(x)) ≥ h µ (T ). Since lim Hµ (Pk |P − ) = k→+∞
h(T −1 ,
h µ (T ), it is sufficient to prove that for each k ∈ N, A(x)) ≥ Hµ (Pk |P − ) − k1 for µ-a.e. x ∈ X . For a given k ∈ N, similar to the proof of Theorem 3.7, we know that there exists Uk ∈ C oX such that n−1 1 1 −i −n − lim sup Hµ T Uk |T P ≥ Hµ (Pk |P − ) − for each n ∈ N. k n→+∞ n
(4.2)
i=0
n For n ∈ N, let Fn : X → [0, log N (Uk )] with Fn (x) = n1 log N ( i=1 T i Uk |A(x)). By Lemma 4.1, Fn is a Borel measurable map. Let F(x) = lim supn→+∞ Fn (x) = n−1 i T Uk |A(x)) for x ∈ X . Then F is a Borel measurable map lim supn→+∞ n1 log N ( i=0 from X to [0, log N (Uk )].
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For any x ∈ X , T (A(x)) ∞ ∞ ∞ Q i (x) ⊆ T (Q i (x)) = T (Q i+1 (x)) =T i=1
=
∞
⎛
T⎝
i=1
=
∞
⎛ T⎝
i=1
=
i=1 i+1
i=1
⎞
(T − j (P1 ∨ · · · ∨ Pi+1 ))(x)⎠
j=1 i+1
⎞ T − j ((P1 ∨ · · · ∨ Pi+1 )(T j x))⎠
j=1
∞ i
T − j (P1 ∨ · · · ∨ Pi+1 )(T j (T x))
i=1 j=0
⊆ =
∞ i
T − j (P1 ∨ · · · ∨ Pi )(T j (T x))
i=1 j=1 ∞
Q i (T x) = A(T x).
i=1
That is, T (A(x)) ⊆ A(T x) for x ∈ X . Using this fact, we have n n+1 1 1 i i F(x) = lim sup log N T Uk |A(x) = lim sup log N T Uk |T (A(x)) n→+∞ n n→+∞ n i=1 i=1 n+1 n+1 1 i ≤ lim sup log N T Uk |A(T x) = F(T x). n n+1 n→+∞ i=1
That is, F(x) ≤ F(T x) for each x ∈ X . Since X (F(T x) − F(x))dµ(x) = 0, we have F(T x) = F(x) for µ-a.e. x ∈ X . This implies that F(x) ≡ ak for µ-a.e. x ∈ X as µ is ergodic, where ak ≥ 0 is a constant. Now we are going to estimate ak , F(x)dµ(x) = lim sup Fn (x)dµ(x) ≥ lim sup Fn (x)dµ(x) ak = n→+∞ X X X n→+∞ n 1 i ≥ lim sup Hµx T Uk dµ(x) (Since supp(µx ) ⊆ A(x) for µ-a.e. x ∈ X ) n→+∞ X n i=1 n 1 i − = lim sup Hµ T Uk |P (by Lemma 2.2 (a)) n→+∞ n i=1 n−1 1 −i −n − T Uk |T P = lim sup Hµ n→+∞ n i=0
1 ≥ Hµ (Pk |P − ) − . (by (4.2)). k
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Finally, for µ-a.e. x ∈ X , 1 h(T −1 , A(x)) ≥ h(T −1 , Uk |A(x)) = F(x) = ak ≥ Hµ (Pk |P − ) − . k This finishes the proof of theorem. A direct consequence of Theorem 4.2 is the following. Corollary 4.3. Let (X, T ) be a TDS. If there exists µ ∈ Me (X, T ) with h µ (T ) = h top (T ), then there exist x ∈ X , a closed subset A(x) ⊆ W s (x, T ) and a closed subset B(x) ⊆ W u (X, T ) such that lim diam(T n A(x)) = 0 and h(T −1 , A(x)) = h top (T ),
n→+∞
lim diam(T −n B(x)) = 0 and h(T, B(x)) = h top (T ).
n→+∞
In the remaining part of this section, we will investigate the chaotic properties happening in the stable and unstable sets for a positive-entropy system (see Theorem 4.6). For this purpose, we need the notions of weakly mixing sets and partially mixing systems which were introduced in [3]. Among properties of topological dynamical systems some are overall properties, like transitivity and weak mixing. Recall that a TDS (X, T ) is transitive if for each pair of non-empty open subsets U and V of X , there exist n ≥ 0 such that U ∩ T −n V = ∅; and it is weakly mixing, if (X × X, T × T ) is transitive. Does there exist a non-overall property corresponding to the overall property of weak mixing? In other words, is there a criterion that, when satisfied by some system (X, T ), allows one to say that (X, T ), without necessarily being weakly mixing, has ‘a certain amount of weak mixing’ in some consistent sense? The existence of a weakly mixing set inside a dynamical system is such a criterion. If X , Y are topological spaces, denote by C(X, Y ) the set of all continuous maps from X to Y . Definition 4.4. Let (X, T ) be a TDS and A ∈ 2 X . The set A is said to be weakly mixing (or to have weak mixing) for T if there exists B ⊂ A satisfying 1) B is the union of countably many Cantor sets; 2) the closure of B equals A; 3) for any C ⊂ B and g ∈ C(C, A) there exists an increasing sequence of natural numbers {n i } ⊂ N such that limi→+∞ T n i x = g(x) for any x ∈ C. It was proved in [17] that a TDS (X, T ) is weakly mixing if and only if the whole space X is a weakly mixing set according to Definition 4.4. Denote by W Ms (X, T ) the set of all subsets having weak mixing of (X, T ). The system (X, T ) itself is called partially mixing when it contains a weakly mixing set. While weakly mixing systems are partially mixing, the converse is not true in general; by Theorem 4.6 non-weakly mixing positive-entropy systems are obvious examples. A subset C ⊂ X is called perfect if it is a closed subset of X having no isolated points. Obviously, every E ∈ W Ms (X, T ) is a perfect ∗-chaotic set. The following result proved in [3, Prop. 4.2] shows that weakly mixing sets are characterized as those sets on which the transformation acts in a weakly mixing way.
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Proposition 4.5. Let (X, T ) be a TDS and A be a non-singleton closed subset of X . Then A is a weak mixing subset of X if and only if for any k ∈ N, any choice of nonempty open subsets V1 , . . . , Vk of A and nonempty open subsets U1 , . . . , Uk of X with A ∩ Ui = ∅, i = 1, 2, . . . , k, there exists m ∈ N such that T m Vi ∩ Ui = ∅ for each 1 ≤ i ≤ k. Now we are ready to prove the following. Theorem 4.6. Let (X, T ) be a TDS and µ ∈ Me (X, T ) with h µ (T ) > 0. Then for µ-a.e. x ∈ X , there exists a closed subset E(x) ⊆ W s (x, T ) ∩ W u (x, T ) such that (1) E(x) ∈ W Ms (X, T ) ∩ W Ms (X, T −1 ), that is, E(x) is weakly mixing for T, T −1 . (2) h(T, E(x)) ≥ h µ (T ) and h(T −1 , E(x)) ≥ h µ (T ). Proof. Let Bµ be the completion of B X under µ. Then (X, Bµ , µ, T ) is a Lebesgue system. Let Pµ (T ) be the Pinsker σ -algebra of (X, Bµ , µ, T ). Let µ = X µx dµ(x) be the disintegration µ over Pµ (T ). We divide the proof into the following steps. Step 1. For µ-a.e. x ∈ X , supp(µx ) ⊆ W s (x, T ) ∩ W u (x, T ). Since Pµ (T ) is also the Pinsker σ -algebra of (X, Bµ , µ, T −1 ) and W s (x, T −1 ) = u W (x, T ), by symmetry it remains to prove that for µ-a.e. x ∈ X , supp(µx ) ⊆ W s (x, T ). ∞ ⊂ P and 0 = k < k < · · · satisfying that By Lemma 3.5, there exist {Wi }i=1 X 1 2 (1) limi→+∞ diam(Wi ) = 0, k (2) limk→+∞ Hµ (Pk |P − ) = h µ (T ), where Pk = i=1 T −ki Wi and P = ∞ k=1 Pk , ∞ −n − P = Pµ (T ). (3) n=0 T It is clear that P − (x) ⊆ W s (x, T ) for x ∈ X . −n P − for n ∈ N. Then for Let µ = X µn,x dµ(x) be the disintegration µ over T −n − n ∈ N, µn,x ((T P )(x)) = 1 for µ-a.e. x ∈ X . Moreover, since (T −n P − )(x) ⊆ W s (x, T ) for each x ∈ X , supp(µn,x ) ⊆ W s (x, T ) for µ-a.e. x ∈ X . ∞ be a dense subset of C(X ; R) under the supremum norm. For each i ∈ N, Let { f i }i=1 by Martingale Theorem for µ-a.e. x ∈ X , −n P − )(x) = E( f |P (T ))(x) lim f i (y)dµn,x (y) = lim E( f i |T i µ n→+∞ X n→+∞ = f i (y)dµx (y). X
Hence there exists a measurable subset X 0 ⊆ X with µ(X 0 ) = 1 such that for each x ∈ X 0 and i ∈ N, lim f i (y)dµn,x (y) = f i (y)dµx (y). n→+∞ X
X
By a simple approximation discussion, we have that for f ∈ C(X ; R), lim f (y)dµn,x (y) = f (y)dµx (y) for each x ∈ X 0 . n→+∞ X
X
That is, limn→+∞ µn,x = µx for x ∈ X 0 under the weak∗ -topology. Moreover, since supp(µn,x ) ⊆ W s (x, T ) for x ∈ X 0 and n ∈ N, we have supp(µx ) ⊆ W s (x, T ) for µ-a.e. x ∈ X .
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Step 2. For µ-a.e. x ∈ X , supp(µx ) ∈ W Ms (X, T ) ∩ W Ms (X, T −1 ). Since Pµ (T ) = Pµ (T −1 ), we need only prove that for µ-a.e. x ∈ X , supp(µx ) ∈ W Ms (X, T ) by the symmetry of T and T −1 . Define a measure λn (µ) on (X (n) , T (n) ) by letting λn (µ) = µ(n) x dµ(x), X
X (n)
= X × X × · · · × X (n-times), T (n) = T × T × · · · × T (n-times), where (n) µx = µx × µx × · · · × µx (n-times). It is well known (see for example [2,10]) that µx is non-atomic for µ-a.e x ∈ X and λn (µ) is a T (n) -invariant ergodic measure on X (n) for every n ∈ N. Let W n = supp(λn (µ)) for n ∈ N. Since λn (µ) is an ergodic . n = {(x1 , x2 , . . . , xn ) ∈ W n : measure, (W n , T (n) ) is a transitive system, that is, WTrans {(T (n) )k (x1 , x2 , . . . , xn ) : k ∈ N} = W n } is a nonempty set. Let G n be the sets consisting of all generic points of λn (µ). Then λn (µ)(G n ) = 1 (n) n and G n ⊂ WTrans . Since 1 = λn (µ)(G n ) = X µx (G n )dµ(x) and µx is non-atomic for µ-a.e. x ∈ X , there exists a subset X n ⊂ X of X with µ(X n ) = 1 such that µ(n) x (G n ) = 1 and µx is non-atomic for x ∈ X n . For x ∈ X n , let Sx = supp(µx ) and (n) Sx = Sx × Sx × · · · × Sx (n-times). Then Sx is a perfect subset of X and . n G n ∩ Sx(n) ⊆ Wtrans ∩ Sx(n) = L nx . (n)
(n)
(n)
Since clearly µx (G n ∩ Sx ) = 1, it follows that G n ∩ Sx , and a fortiori also L nx is (n) dense subset of Sx . This shows that for x ∈ X n , Sx(n) = cl(L nx ) ⊂ W n .
(4.3)
Let X = ∩n∈N X n and fix x ∈ X . By the above discussion, Sx is a perfect subset of X . We claim that: for any k ∈ N, any nonempty open subsets V1 , . . . , Vk of Sx and nonempty open subsets U1 , . . . , Uk of X with Sx ∩ Ui = ∅, i = 1, 2, . . . , k, there exists m ∈ N such that T m Vi ∩ Ui = ∅ for 1 ≤ i ≤ k. k V = ∅, one can choose (x , x , . . . , x ) ∈ Let us show the claim. Since L kx ∩ i=1 i 1 2 k (k) k k k k L x ∩ i=1 Vi ⊂ Wtrans . As Sx ∩ Ui = ∅, i = 1, 2, . . . , k and Sx ⊂ W by (4.3), k U ∩ W k is an open non-empty subset of W k . Then there exists one knows that i=1 i m k U ∩ W k . This implies T m V ∩ U = ∅ for (k) m ∈ N with (T ) (x1 , x2 , . . . , xk ) ∈ i=1 i i i 1 ≤ i ≤ k, and ends the proof of the claim. Now, by Proposition 4.5 and the above claim, we know that the perfect subset Sx = supp(µx ) ∈ W Ms (X, T ) for any x ∈ X . Step 3. For µ-a.e. x ∈ X , h(T, supp(µx )) ≥ h µ (T ) and h(T −1 , supp(µx )) ≥ h µ (T ). Since h µ (T ) = h µ (T −1 ), by symmetry of T and T −1 it remains to prove that h(T, supp(µx )) ≥ h µ (T ) for µ-a.e. x ∈ X . Since Pµ (T ) is T -invariant, T µx = µT x for µ-a.e. x ∈ X , therefore, there exists a T -invariant measurable set X 0 ⊂ X with µ(X 0 ) = 1 and T µx = µT x for x ∈ X 0 . n−1 −i T U) for x ∈ X 0 . Then Fn (x) Fix U ∈ C oX and for n ∈ N, let Fn (x) = Hµx ( i=0 is a bounded non-negative function on X 0 and n+m−1 n−1 m−1 −i −i −n −i Fn+m (x) = Hµx T U ≤ Hµx T U + Hµx T T U i=0
= Fn (x) + Fm (T n x),
i=0
i=0
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that is, {Fn } is sub-additive. By Eq. (2.4), we know that Fn (x) is a measurable function on X 0 . Hence by the Kingman sub-additive ergodic theorem, we have limn→+∞ n1 Fn (x) ≡ aU for µ-a.e. x ∈ X , where aU is a non-negative constant. Since Fn (x) ≤ log N n−1 −i T U|supp(µx )) for any x ∈ X 0 and n ∈ N, for µ-a.e. x ∈ X one has ( i=0 h(T, U|supp(µx )) ≥ aU . Now we are going to compute aU . By Lemma 2.1, n−1 1 −i Hµ h µ (T, U) = h µ (T, U|Pµ (T )) = lim T U|Pµ (T ) n→+∞ n i=0 n−1 1 −i = lim Hµx T U dµ(x) (by Lemma 2.2 (a)) n→+∞ X n i=0 Fn (x) Fn (x) dµ(x) = dµ(x) = aU . lim = lim n→+∞ X n→+∞ n n X0 0 This implies that h(T, U|supp(µx )) ≥ h µ (T, U) for µ-a.e. x ∈ X . o Choose {Um }∞ m=1 ⊂ C X with lim m→+∞ diam(Um ) = 0. Then supm→+∞ h µ (T, Um ) = h µ (T ). Moreover, since for each m ∈ N, h(T, Um |supp(µx )) ≥ h µ (T, Um ) for µ-a.e. x ∈ X , we get that h(T, supp(µx )) = sup h(T, Um |supp(µx )) ≥ sup h µ (T, Um ) = h µ (T ) m∈N
m→+∞
for µ-a.e. x ∈ X . Finally we have the following corollary. Corollary 4.7. Let (X, T ) be a TDS. Then (1) supx∈X h(T, W s (x, T ) ∩ W u (x, T )) = h top (T ). (2) If there exists an ergodic T -measure µ with h µ (T ) = h top (T ), then for µ-a.e. x ∈ X there exists a closed subset E(x) ⊆ W s (x, T ) ∩ W u (x, T ) such that (a) E(x) ∈ W Ms (X, T ) ∩ W Ms (X, T −1 ) and (b) h(T, E(x)) = h(T −1 , E(x)) = h top (T ). 5. Non-Invertible Case In this section, we will generalize the results in Sects. 3 and 4 to the non-invertible case. Let (X, T ) be a non-invertible TDS, i.e. X is a compact metric space, and T : X → X is a surjective continuous map but not one-to-one. ) of a non-invertible TDS (X, T ) with a metric We now define the inverse limit ( X, T d. Let X = {(x1 , x2 , . . .) : T (xi+1 ) = xi , xi ∈ X, i ∈ N} be a subspace of the product ∞ X endowed with the compatible metric d : space X N = i=1 T dT ((x1 , x2 , · · · ), (y1 , y2 , · · · )) =
∞ d(xi , yi ) i=1
2i
.
: (x1 , x2 , · · · ) = (T (x1 ), x1 , x2 , · · · ), and T X→ X is the shift homeomorphism, i.e. T ) → (X, T ) is a πi : X → X is the projection to the i th coordinate. Clearly, πi : ( X, T factor map.
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) be the inverse limit of a non-invertible TDS (X, T ) and π1 : Lemma 5.1. Let ( X, T X → X be the projection to the 1th coordinate. Then for any sequence {K n } of non-empty closed subsets of X, lim lim sup
δ→0 n→+∞
1 , K n , δ) = lim lim sup 1 log sn (d, T, π1 (K n ), δ). log sn (dT , T δ→0 n→+∞ n n
Proof. It follows by Lemma 5.6 and Lemma 5.9 in [6]. For completeness we give a new proof. n−1 Let U ∈ C oX . Since N ((π −1 U)n−1 |π1 (K n )) for any n ∈ N, we have 0 |K n ) = N (U0 lim sup n→∞
1 1 n−1 N ((π1−1 U)n−1 |π1 (K n )). 0 |K n ) = lim sup N (U0 n n→+∞ n
Moreover by Lemma 3.1, we know lim lim sup
δ→0 n→+∞
1 , K n , δ) ≥ lim lim sup 1 log sn (d, T, π1 (K n ), δ). log sn (dT , T δ→0 n→+∞ n n
∈ C o . Then Conversely, let πi : X → X be the projection to the i th coordinate and U X Hence there exists U ∈ C 0 and i ∈ N such that π −1 (U) U. X
i
1 n−1 |K n ) ≤ lim sup 1 log N ((π −1 U)n−1 |K n ) log N (U 0 0 i n→+∞ n n→+∞ n 1 1 = lim sup log N (U0n−1 |πi (K n )) ≤ lim sup log N (U0n−1 |T −(i−1) π1 (K n )) n n n→+∞ n→+∞
lim sup
(as πi (K n ) ⊆ T −(i−1) π1 (K n )) 1 ≤ lim sup log N (T −(i−1) U0n−1 |T −(i−1) π1 (K n )) + N (U0i−2 |T −(i−1) π1 (K n )) n→∞ n 1 1 = lim sup log N (T −(i−1) U0n−1 |T −(i−1) π1 (K n )) = lim sup log N (U0n−1 |π1 (K n )) n→∞ n n→∞ n 1 ≤ lim lim sup log sn (d, T, π1 (K n ), δ). δ→0 n→+∞ n Moreover, by Lemma 3.1, we get lim lim sup
δ→0 n→+∞
1 , K n , δ) ≤ lim lim sup log sn (d, T, π1 (K n ), δ). log sn (dT , T δ→0 n→+∞ n
This finishes the proof of the lemma. Now we can show the following. Theorem 5.2. Let (X, T ) be a non-invertible TDS and µ ∈ Me (X, T ) with h µ (T ) > 0. Then for µ-a.e. x ∈ X , lim→0 h s (T, x, ) ≥ h µ (T ).
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) be the inverse limit of (X, T ). For Proof. Let ( X, T x ∈ X and > 0, let x = π1 ( x) −n W s ( ) for n ∈ N. Then π1 (K n ) ⊆ T −n Ws (x, T ) for any n ∈ N by and K n = T x, T 2 the definition of dT . Using Lemma 5.1, 1 log sn (d, T, T −n Ws (x, T ), δ) n 1 1 , K n , δ) ≥ lim lim sup log sn (d, T, π1 (K n ), δ) = lim lim sup log sn (dT , T δ→0 n→∞ n δ→0 n→+∞ n , ). x, T = h s ( 2 h s (x, T, ) = lim lim sup δ→0 n→∞
This implies that for any x∈ X, , ). x ), T, ) ≥ lim h s ( x, T lim h s (π1 ( →0 →0 2
(5.1)
) with π1 ( Choose µ ∈ Me ( X, T µ) = µ. Then by Theorem 3.7, there exists a Borel subset X0 ⊆ X with µ( X 0 ) = 1 such that for any x∈ X 0, , ≥ h ) ≥ h µ (T ). x, T (5.2) lim h s µ (T →0 2 X 0 ). Then X 0 ∈ Bµ and µ(X 0 ) = 1. For x0 ∈ X 0 , there exists x0 ∈ X Let X 0 = π1 ( such that π1 ( x0 ) = x0 . Moreover by (5.1) and (5.2), lim→0 h s (x0 , T, ) ≥ h µ (T ). This ends the proof of the theorem. Theorem 5.2 immediately leads to the following. Theorem 5.3. Let (X, T ) be a non-invertible TDS. If there exists µ ∈ Me (X, T ) such that h µ (T ) = h top (T ), then E(T ) = ∅. Particularly, if h top (T ) < ∞ and the entropy map ν : M(X, T ) → h ν (T ) ∈ R+ is upper semi-continuous, then E(T ) = ∅. Proof. By Theorem 5.2, for µ-a.e. x ∈ X , x is an entropy point for T . When h top (T ) < ∞ and the entropy map ν ∈ M(X, T ) → h ν (T ) is upper semi-continuous, we can always find µ ∈ Me (X, T ) with h µ (T ) = h top (T ). In Bowen [4] a TDS or a non-invertivble TDS (X, T ) is called h-expansive if there exists an > 0 such that supx∈X h top (T, Ws (x, T )) = 0 for some > 0, while in Misiurewicz [12] (X, T ) is called asymptotically h-expansive if lim sup h top (T, Ws (x, T )) = 0.
→0 x∈X
It is shown by Bowen [4] that expansive systems, expansive homeomorphisms, endomorphisms of a compact Lie group, and Axiom A diffeomorphisms are all h-expansive, by Misiurewicz [12] that every continuous endomorphism of a compact metric group is asymptotically h-expansive if its entropy is finite, and by Buzzi [5] that any C ∞ diffeomorphism on a compact manifold is asymptotically h-expansive. In [12], Misiurewicz showed that for an asymptotically h-expansive system (X, T ), the entropy map ν ∈ M(X, T ) → h ν (T ) ∈ R+ is upper semi-continuous. Hence there exist entropy points for an asymptotically h-expansive system (see also Theorem 6.4 in [6]). ) be the inverse limit of a non-invertible TDS (X, T ). If A ⊂ Lemma 5.4. Let ( X, T X , A) = h(T, π1 (A)). is weakly mixing, so is π1 (A) and h(T
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, A) = Proof. It is proved in Lemma 4.8 in [3] that π1 (A) is weakly mixing. Moreover, h(T h(T, π1 (A)) follows by Lemma 5.1. In fact Theorem 4.6 is also valid for non-invertible TDS as the following theorem shows. Theorem 5.5. Let (X, T ) be a non-invertible TDS and µ ∈ Me (X, T ) with h µ (T ) > 0. Then for µ-a.e. x ∈ X , there exists a closed subset E(x) ⊆ W s (x, T ) such that h(T, E(x)) ≥ h µ (T ) and E(x) ∈ W Ms (X, T ), that is, E(x) is weakly mixing for T . ) be the inverse limit of a non-invertible TDS (X, T ) and π1 : X→X Proof. Let ( X, T ) be the projection to the 1th coordinate. It is well known that there exists µ ∈ Me ( X, T such that π1 ( µ) = µ. Clearly, h ( T ) ≥ h (T ). By Theorem 4.6, there exists a Borel µ µ set X0 ⊆ X with µ( X 0 ) = 1 such that for x ∈ X 0 , there exists a closed subset s ) such that h(T , E( ) and E( ). x, T x )) ≥ h x ) ∈ W Ms ( X, T E( x ) ⊆ W ( µ (T Let X 0 = π1 ( X 0 ). Then X 0 is a µ-measurable set and µ(X 0 ) = 1. For any x ∈ X 0 , there exists x ∈ X 0 such that π1 ( x ) = x. Let E(x) = π1 (E( x )). Then E(x) ⊆ ) ) ⊆ W s (x, T ). By Lemma 5.4, h(T, E(x)) = h(T , E( ) ≥ π1 ( W s ( x, T x )) ≥ h µ (T h µ (T ) and E(x) ∈ W Ms (X, T ). The following corollary is immediate. Corollary 5.6. Let (X, T ) be a non-invertible TDS. Then (1) supx∈X h(T, W s (x, T )) = h top (T ). (2) If there exists an ergodic T -measure µ with h µ (T ) = h top (T ), then for µ-a.e. x ∈ X there exists a closed subset E(x) ⊆ W s (x, T ) such that E(x) ∈ W Ms (X, T ) and h(T, E(x)) = h top (T ). Acknowledgements. The author wishes to thank Prof. Xiangdong Ye for a careful reading of the manuscript.
References 1. Blanchard, F., Host, B., Ruette, S.: Asymptotic pairs in positive-entropy systems. Ergod. Th. and Dynam. Sys. 22(3), 671–686 (2002) 2. Blanchard, F., Glasner, E., Kolyada, S., Maass, A.: On Li-Yorke Pairs. J. Reine Angew. Math. 547, 51–68 (2002) 3. Blanchard, F., Huang, W.: Entropy sets, weakly mixing sets and entropy capacity. Discrete and Continuous Dynamical Systems 20(2), 275–311 (2008) 4. Bowen, R.: Entropy-expansive maps. Trans. Amer. Math. Soc. 164, 323–331 (1972) 5. Buzzi, J.: Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100, 125–161 (1997) 6. Fiebig, D., Fiebig, U.R., Nitecki, Z.H.: Entropy and preimage sets. Ergod. Th. and Dynam. Sys. 23, 1785– 1806 (2003) 7. Glasner, E.: A simple characterization of the set µ-entropy pairs and applications. Israel J. Math. 102, 13– 27 (1997) 8. Glasner, E.: Ergodic theory via joinings. Mathematical Surveys and Monographs 101. Providence, RI: Amer. Math. Soc. 2003 9. Huang, W., Ye, X.: Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topology Appl. 117(3), 259–272 (2002) 10. Huang, W., Ye, X.: A local variational relation and applications. Israel J. of Math. 151, 237–280 (2006) 11. Kato, H.: On scrambled sets and a theorem of Kuratowski on independent sets. Proc. Amer. Math. Soc. 126(7), 2151–2157 (1998) 12. Misiurewicz, M.: Topological conditional entropy. Studia Math. 55(2), 175–200 (1976)
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13. Parry, W.: Topics in ergodic theory. Cambridge Tracts in Mathematics 75. Cambridge-New York: Cambridge University Press, 1981 14. Romagnoli, P.P.: A local variational principle for the topological entropy. Ergod. Th. and Dynam. Sys. 23(5), 1601–1610 (2003) 15. Sumi, N.: Diffeomorphisms with positive entropy and chaos in the sense of Li-Yorke. Ergod. Th. and Dynam. Sys. 23(2), 621–635 (2003) 16. Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics 79. New York-Berlin: Springer-Verlag, 1982 17. Xiong, J., Yang, Z.: Chaos caused by a topologically mixing map. In: Dynamical systems and related topics, (Nagoya, 1990) Advanced Series in Dynamical Systems 9, River Edge, NJ: World Scientific (1990), pp. 550–572 Communicated by G. Gallavotti
Commun. Math. Phys. 279, 559–584 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0440-6
Communications in
Mathematical Physics
Typical Support and Sanov Large Deviations of Correlated States Igor Bjelakovi´c2,3 , Jean-Dominique Deuschel2 , Tyll Krüger1,2,4 , Ruedi Seiler2 , Rainer Siegmund-Schultze1,2 , Arleta Szkoła1,2 1 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany 2 Technische Universität Berlin, Fakultät II - Mathematik und Naturwissenschaften,
Institut für Mathematik MA 7-2, Straße des 17. Juni 136, 10623 Berlin, Germany. E-mail: [email protected] 3 Heinrich-Hertz-Chair for Mobile Communication, Technische Universität Berlin, Werner-von-Siemens-Bau (HFT 6), Einsteinufer 25, 10587 Berlin, Germany 4 Universität Bielefeld, Fakultät für Physik, Universitätsstr. 25, 33619 Bielefeld, Germany Received: 29 March 2007 / Accepted: 31 August 2007 Published online: 15 February 2008 – © Springer-Verlag 2008
Abstract: Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov’s theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states. 1. Introduction A relevant notion on the interface of classical discrete probability theory and information theory is that of typical subsets. For the quantum extensions of these fields there is a corresponding notion: typical subspaces. The general picture is that a stationary process (state in the case of quantum lattice systems) is asymptotically -i.e. observing a large finite interval-more and more confined to its typical support. The size of this support has an exponential growth rate (possibly zero) given by the essential supremum of the entropies of the ergodic components. In the classical situation this is the content of the Shannon-McMillan theorem. It clarifies the importance of Shannon entropy for several fields, from data transmission and compression to statistical mechanics or complexity theory. Under the much stronger condition of complete independence Sanov’s theorem (see [23 or 7]) specifies the exponential rate of this confinement of a classical iid process to its own typical set, or equivalently, the rate of avoidance of the supports of all other iid processes’ typical sets. This large deviations result is usually seen as a result on empirical distributions, as in its formulation a particular instance of typical set appears: typical for an iid process are realizations with an empirical distribution close to the probability distribution underlying this very process, see Ch. 3.2 in Deuschel and Stroock [9]. In the iid case Sanov’s theorem significantly extends the assertion of the ShannonMcMillan theorem. In fact, taking the equidistribution as reference measure, it follows
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from Sanov’s theorem that there is a universal typical set sequence of approximate size enh for all iid processes with (base e) entropy less than h. It is well-known in the classical situation that this extends to the general ergodic case (since there exist universal compression schemes like the Lempel-Ziv algorithm: the universal set of blocks of (fixed) length n can be defined as the set of those blocks, the Lempel-Ziv code of which has length less than hn/ log 2 ). This universality result was generalized to the quantum case by Kaltchenko and Yang [15], using a nice ’rotation technique’ and the quantum Shannon-McMillan theorem [1]. In the quantum iid case the universal coding result has been shown earlier by R. Jozsa, M. Horodecki, P. Horodecki and R. Horodecki [14] (1998). From the point of view of statistical hypothesis testing Sanov’s theorem asserts that there is a universally typical set sequence for any set of iid probability distributions (null hypothesis), separating it optimally from any other set of iid processes (alternative hypothesis) at a rate arbitrarily close to the infimum of the relative entropies between probability measures from the two hypotheses. So in the classical case Sanov’s theorem expresses a twofold universality in the choice of the typical sets. The special case of Sanov’s theorem with both hypotheses consisting of only one probability distribution each, is usually called Stein’s lemma. As already emphasized in [2], when passing from the classical to the quantum case, the universality mentioned above gets partially lost: there exists no longer a sequence of typical subspaces (of the underlying finite dimensional Hilbert spaces for the n-blocks of the system), which would work universally, whatever the reference states are. Consequently, speaking in the hypothesis testing terminology, for the alternative hypothesis only one process/state is admitted here. Universality with respect to the null hypothesis states is maintained, however. Also, in the quantum situation it is no longer possible to originate Sanov’s theorem on the concept of empirical distributions (states), see [2], Chap. 4. We mention here that the main techniques needed to generalize Sanov’s theorem to the iid quantum case were already presented in Hayashi [10] (1997), and in Hayashi [11] (2002) an equivalent result is shown. The authors of the present paper regrettably were not aware of this part of Hayashi’s work during the preparation of [2]. It is the aim of this paper as a continuation of [2] to extend the assertion of Sanov’s theorem in several directions. This concerns the classical case, too, but the main focus is on the quantum situation. First, the restriction to the uncorrelated case is substantially alleviated: No condition besides stationarity is imposed to the processes/states P of the null hypothesis. As for the alternative hypothesis (reference measure/state) Q, even stationarity is not assumed. The only requirements are the existence of relative entropy rates h(W, Q) ≤ +∞ for the ergodic components W occurring in the null hypothesis set, and the validity of the upper bound (achievability part) in Stein’s lemma concerning W and Q (see Theorems 11, resp. 13, in the classical situation). These are, in a sense, minimal requirements, since Stein’s lemma is a trivial consequence of Sanov’s theorem obtained by forgetting about universality. As an application of this general result we consider the case that a certain (admittedly very strong) mixing condition holds for the reference process Q . Observe that the very existence of the relative entropy rate for correlated processes can only be guaranteed in terms of mixing conditions, if the reference process is particulary strong mixing. Shields [25] gives an example where the reference process is even maximally mixing in the sense of Dynamical Systems theory (B-process, i.e. isomorphic to an iid process), but nonetheless there exists no asymptotic rate of the relative entropy. Though the mixing
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condition upon the reference processes is very strong (∗-mixing, cf. [4, or 5], where it is called ψ-mixing), the class of aperiodic irreducible Markov processes on a finite state space is covered. In this Markov case aperiodicity is necessary and sufficient for mixing, but not needed for Sanov’s theorem, showing that ∗-mixing is far from being a necessary condition for a Sanov type theorem. In fact, in the classical case a kind of average-mixing would yield the result, cf. condition ( U) on p. 86 of [9]. We also mention that in the classical case a usual mixing condition to derive large deviation results is hypermixing, cf. Chap. 5.4 in [9]. Secondly, we generalize the classical Sanov’s theorem to the (correlated) quantum situation (in Hayashi [11] and later in [2] the quantum iid case was considered). In fact, since the classical assertion is a special case of the quantum theorem, we only prove the latter. Again, the reference state only needs to fulfill the two minimal conditions mentioned above. We refer to those as the HP-condition. The states forming the null hypothesis have to be stationary only. It would be interesting to specify the set of all states which fulfill the HP-condition with respect to any ergodic (null hypothesis) state. We call these states HP-states. As already said, this set comprises all ∗-mixing states, but can be expected to be much larger. In the classical situation we remind the reader of an interesting example by Xu [26]: There exists a B-process (i.e. again maximally mixing in the sense of Dynamical Systems) Q which has the property that the relative entropy rate h(P, Q) exists and is zero for any stationary process P. So this process cannot be separated at exponential speed from an arbitrary other stationary process. It would be interesting to find conditions weaker than ∗-mixing ensuring exponential separability in the case that the relative entropy rate is positive. In the presented form, the quantum Sanov theorem comprises and extends several earlier results on typical subspaces and their connection with the von Neumann entropy and relative entropy. In particular, the result [2] of the present authors (which was preceded by Hayashi [11]) is extended to the correlated case. The quantum Shannon-McMillan theorem [1] is covered and extended from the ergodic to the general stationary situation. The universality result of Kaltchenko and Yang [15] is covered, too, by using the tracial state as the reference state in Theorem 11. In fact this Kaltchenko-Yang universality is a main ingredient in our proof. The quantum Stein lemma (see [20,21], Chap. 1.1 for the iid case, [3] for the case of ergodic null hypothesis states) is covered and extended to the case of correlated reference states. Results of Hiai and Petz [12,13] are completed in the sense that their bound is shown to be sharp, which means that it is asymptotically optimal, and the condition of complete ergodicity concerning the null hypothesis is dropped. In particular, the case of irreducible aperiodic algebraic (reference) states on a quasi-local algebra over a finite-dimensional C ∗ -algebra A (also called finitely correlated states) considered in [13] is covered by ∗-mixing. We mention that Hiai and Petz emphasize in [13] that they derive almost all assertions using ∗ -mixing, only. As already emphasized, the quantum Sanov theorem is a special type of quantum large deviations result. We refer the reader to some other work in this direction, see Lebowitz, Lenci and Spohn [16], Lenci and Rey-Bellet [17], Netoˇcný and Redig [19], De Roeck, Maes and Netoˇcný [8]. We give a short account of the principal steps to prove the main result. In Chapter 3 we show that ’one half’ of Stein’s lemma, namely the assumed achievability of the relative entropy rate as separation rate, already implies Sanov’s theorem.
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First it is shown that the optimality of the relative entropy rate (as separation rate) is a consequence of its achievability. In fact, for two states , such that s(, ) and s() exist, the quantity −n(s(, )+s()) is the asymptotic average of the logarithmic eigenvalues of D(n) , which denotes the density operator of the local state (n) on the disrete interval of length n, with respect to the probability measure generated on the corresponding eigenvectors by the operator D (n) . On the other hand, the achievability part of Stein’s lemma implies that −n(s(, ) + s()) is also an essential upper bound for these logarithmic eigenvalues. The key tool to show the latter is Lemma 8. Now, roughly speaking, with the asymptotic average being the asymptotic upper bound, it must be an asymptotic lower bound, too. This observation yields a relative AEP (asymptotic equipartition property) for the logarithmic eigenvalues of D(n) : the vast majority of them (with respect to the considered probability distribution) is close to −n(s(, ) + s()). Because for ergodic by the quantum Shannon-McMillan theorem the relevant dimension of the corresponding subspace of eigenvectors of D(n) is close to ens() , it easily follows now (applying Lemma 8 once again) that the optimally separating subspaces can essentially be described as those which are close to the span of the mentioned eigenvectors of D(n) fulfilling the relative AEP: the (n) -expectation is close to ens() · e−n(s(,)+s()) = e−ns(,) . Next we make use of the proven relative AEP, combined with Kaltchenko and Yang’s universality result to show Sanov’s theorem: We subdivide the null hypothesis set into small slices of almost constant value of the ’mixed’ term (aka cross-entropy) smix := s(, ) + s() = − lim n1 TrD (n) log D(n) , and within these slices the entropy rate is bounded from above by smix − inf s(, ). Then, by Kaltchenko-Yang universality, there exists a common support of dimension ≈ en(smix −inf s(,)) which by the relative AEP can be chosen to consist of eigenvectors of D(n) with eigenvalues close to e−nsmix . So this common support has an asymptotic (n) -expectation close to en(smix −inf s(,)) · e−nsmix = e−n inf s(,) . This essentially proves Sanov’s theorem under the HP-condition. In Chapter 4 we prove that ∗-mixing implies the HP-condition, hence the quantum Sanov theorem. The idea is borrowed from [13]: under ∗-mixing, the reference state is sufficiently close to some block-iid state, so that we may apply the techniques developed in [12 and 3] in order to prove the achievability part of Stein’s lemma. In Chapter 5 we use the ergodic decomposition of stationary states to extend our results to the case where the null hypothesis states are only assumed stationary. 2. Basic Settings and Notations As announced in the introduction, we address both the classical and the quantum situation. Let us first consider the classical case. Let a finite set A of symbols be given. We deal with processes P on [AZ , AZ ], where AZ denotes the σ -field over AZ which is generated by finite dimensional cylinders. We denote the set of all processes by P(AZ ). Let P (n) denote the marginal of a process P, restricted to the positive (time) indices {0, 1, ..., n − 1} ⊂ Z . The relative entropy rate between two processes P, Q is defined as h(P, Q) := lim
n→∞
1 H (P (n) , Q (n) ) n
whenever this limit exists in R+ = R+ ∪ {+∞}. Here H (·, ·) denotes the relative entropy of two probability measures given on a finite set.
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If Q ∈ P(AZ ), ⊆ P(AZ ) and h(P, Q) exists for each P ∈ we write h(, Q) for inf P∈ h(P, Q). The following very strong mixing property of Q was introduced by Blum, Hanson and Koopmans [4] (referred to as ψ-mixing in the survey paper [5]), which implies the existence of the relative entropy rate h(P, Q) for any stationary P (see [13], where the more general quantum case is treated): Definition 1. A stationary process Q on [AZ , AZ ] will be called ∗-mixing if for each 0 < α < 1 there exists an l ∈ N such that α Q(B)Q(C) ≤ Q(B ∩ C) ≤ α −1 Q(B)Q(C)
(1)
whenever B ∈ A{...,−2,−1,0} , C ∈ A{l,l+1,...} . Here AT , T ⊂ Z, denotes the sub-σ -field of AZ concerning only times t ∈ T . Observe that irreducible and aperiodic (i.e. weakly mixing) Markov chains are automatically ∗-mixing, even with α = α(l) tending to 1 exponentially fast as l → ∞. In the general situation, even strong mixing (α-mixing in the terminology of [5]) does not imply ∗-mixing, because rare events may still deviate much from independence. We note that in the following we use the seemingly weaker condition, that (1) is fulfilled for some α > 0 and some l. (The same was emphasized for most of the results in [13].) But, in fact, in the stationary classical situation this is already equivalent to full ∗-mixing, see [5], Theorem 4.1. Let Pstat (AZ ), Perg (AZ ), resp. P∗ (AZ ), denote the set of stationary, of ergodic, resp. stationary, ∗-mixing processes with state space A. We briefly introduce now the corresponding quantum set-up. Consider a finite-dimensional C ∗ -algebra A. The classical case is covered choosing A to be abelian. It is well-known that A can always be represented as a finite direct sum of matrix algebras m A∼ Mki , (2) = i=1
where Mk is the algebra of complex k × k matrices. The abelian case is covered if all ki are 1, meaning that A is simply the commutative algebra of complex functions over a finite set A = {1, 2, ..., m}. A state ψ on A is a positive functional on A with the property ψ(1) = 1, where 1 is unity. The set of all states on A is denoted by S(A). This is the set of probability measures on A in the abelian case. Any state ψ on the finite-dimensional algebra A is uniquely given by its density operator Dψ ∈ A, which is a positive trace-one operator fulfilling ψ(X ) = TrA (Dψ X ) for each X ∈ A. Here TrA denotes the canonical trace in A which is nothing but the sum of the matrix traces in the above representation (2). The quantum generalization of a stochastic process is usually constructed as follows (and in correspondence to the definition of a process by its compatible finitedimensional distributions via Kolmogorov’s extension theorem): For each finite subset T ⊂ Z consider the C ∗-algebra AT := t∈T A. Then for any T ⊂ T ⊂ Z there is a
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canonical embedding of AT into AT as a C ∗-subalgebra. With respect to this identification consider the algebra := A
AT =
T ⊂Z T finite
A{−n,...,n} .
n∈N
is not norm-complete. We denote the completion by AZ . It is a C ∗ -algebra and is A called the quasilocal algebra constructed from A. Again, a state on AZ is a positive functional on AZ with the property (1) = 1. If A is abelian, there is a one-to-one correspondence between states on AZ and stochastic processes with alphabet A: The restrictions (T ) := AT of to the local algebras AT correspond to the marginals P (T ) of the stochastic process P on the cylinder σ -algebras AT . This comes from the fact that any compatible family of local states (T ) has a unique extension to AZ just as any compatible family of marginals can be extended to a stochastic process. There is a canonically defined shift operator τ on AZ (mapping in particular A{0} ⊂ Z A onto A{1} ⊂ AZ ). The set of stationary states Sstat (AZ ) is the subset of states in S(AZ ) which are invariant with respect to τ . This is a Choquet simplex; the extremal points are called ergodic states Serg (AZ ). The notions coincide with the classical ones in the abelian case. We complete the picture by defining a mixing property (cf. [13] ) as above: Definition 2. A stationary state in S stat (AZ ) will be called ∗-mixing if for each 0 < α < 1 there exists an l ∈ N such that for each k ∈ N, α({−k,−k+1...,0}) ⊗ ({l,l+1,...,l+k}) ≤ ({−k,−k+1...,0}∪{l,l+1,...,l+k}) ≤ α −1 ({−k,−k+1...,0}) ⊗ ({l,l+1,...,l+k}) . We denote the set of stationary ∗-mixing states by S∗ (AZ ). Next we introduce the quantum version of the relative entropy rate. Let ψ, ϕ ∈ S(A). The relative entropy is defined as S(ψ, ϕ) :=
TrA Dψ (log Dψ − log Dϕ ), if supp(ψ) ≤ supp(ϕ) ∞, otherwise.
Here supp(D) is the smallest projection p ∈ A fulfilling p Dp = D (with D ∈ A self-adjoint). Now, for , ∈ S(AZ ), we define the relative entropy rate 1 S( (n) , (n) ), n→∞ n
s(, ) := lim
whenever this limit exists in R+ (we write for short (n) instead of ({0,1,...,n−1}) and A(n) instead of A{0,1,...,n−1} ). Again, if ∈ S(AZ ), ⊆ S(AZ ) and s(, ) exists for each ∈ we write s(, ) for inf ∈ s(, ).
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3. Equivalence of Sanov’s Theorem and Stein’s Lemma The maximally separating exponents for two states , on AZ are defined by βε,n (, ) := min{log (q) : q ∈ A(n) projection, (q) ≥ 1 − ε}, for ε ∈ (0, 1). By β¯ε (, ) we denote limsupn→∞ n1 βε,n (, ), and if the limit exists in −R+ := −[0, ∞], we denote it by βε (, ). Definition 3. We say that the pair (, ) satisfies the HP-condition if the relative entropy rate s(, ) exists and β ε (, ) ≤ −s(, ) for all ε ∈ (0, 1). This condition was first proved to be fulfilled by Hiai and Petz in [12] for the special case that is completely ergodic and is a stationary product state (i.e. an iid state) and later in [13] for completely ergodic and ∗-mixing states . Definition 4. We say that ∈ S(AZ ) is a HP-state if, for any ergodic state ∈ Serg (AZ ), the pair (, ) satisfies the HP-condition. As it turns out, the statement in Sanov’s theorem is equivalent to the HP-condition: Theorem 5. Let be a state on AZ and ⊆ Serg (AZ ). Then following statements are equivalent: 1. For each ∈ the pair (, ) satisfies the HP-condition. 2. The quantity s(, ) ≤ +∞ exists for each ∈ , and to each subset ⊆ and any η > 0 there exists a sequence { pn }n∈N of projections pn ∈ A(n) with lim (n) ( pn ) = 1,
n→∞
for all ∈
(3)
such that if s(, ) < ∞, 1 limsup log (n) ( pn ) ≤ −s(, ) + η, n→∞ n
(4)
1 1 limsup log (n) ( pn ) ≤ − . n η n→∞
(5)
otherwise
Moreover, for each sequence of projections { pn } fulfilling (3) we have 1 liminf log (n) ( pn ) ≥ −s(, ). n→∞ n Hence −s(, ) is the lower limit of all achievable separation exponents. Remark 6. 1. There are examples showing that in general one cannot choose η = 0, meaning that the exact value −s(, ) is not necessarily achievable. 2. If is stationary and, moreover, ∗-mixing, statement 1 of the theorem is fulfilled with = Serg (AZ ). This will be seen in Sect. 4.
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The implication 2 ⇒ 1 is trivial. The proof of the converse implication is carried out in Subsect. 3.2. As an immediate consequence, we have the following assertion for the classical case: Let the maximally separating exponents for two processes P, Q ∈ P(AZ ) be defined by βε,n (P, Q) := min{log Q (n) (M) : M ⊆ An , P (n) (M) ≥ 1 − ε}, for ε ∈ (0, 1). By β¯ε (P, Q) we denote limsupn→∞ n1 βε,n (P, Q), and if the limit exists in −R+ , we denote it by βε (P, Q). Theorem 7. Let Q ∈ P(AZ ), ⊆ Perg (AZ ) and suppose that the relative entropy rate h(P, Q) exists for all P ∈ . Then the following statements are equivalent: 1. β¯ε (P, Q) ≤ −h(P, Q) for all P ∈ and all ε ∈ (0, 1). 2. For each set ⊆ and each η > 0 there is a sequence of subsets {Mn }, Mn ⊆ An , such that lim P(Mn ) = 1, for all P ∈ , (6) n→∞
and 1 limsup log Q (n) (Mn ) ≤ −h(, Q) + η n→∞ n if h(, Q) < ∞, otherwise if h(, Q) = ∞, 1 1 limsup log Q (n) (Mn ) ≤ − . η n→∞ n n } fulfilling ( 6) we have Moreover, for each sequence of subsets { M 1 n ) ≥ −h(, Q). liminf log Q (n) ( M n→∞ n Hence −h(, Q) is the lower limit of all achievable separation exponents.
3.1. A quantum relative AEP and achievability in Stein’s lemma. We start with a useful lemma which allows to translate some standard techniques and estimates used in classical information theory into the quantum setting. Lemma 8. Let p, q be arbitrary projections and τ be a state on A. Suppose that u is a projection commuting with Dτ . Then we have τ (qpq) ≥ τ (qpqu) ≥ τ ( p) − 2 (τ (1 − q))1/2 − τ (1 − u).
(7)
Let c > 0. If Dτ u ≤ cu then Tr( pq) ≥
1 τ ( p) − 2 (τ (1 − q))1/2 − τ (1 − u) . c
(8)
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Proof. The first inequality in (7) is trivial. The second follows applying the CauchySchwarz inequality for the Hilbert-Schmidt inner product: τ ( p) = τ ( pq) + τ ( p(1 − q)) 1
≤ |τ ( pq)| + (Tr(Dτ (1 − q))) 2 ≤ τ (qpq) + |τ ((1 − q) pq)| + (τ (1 − q))1/2 ≤ τ (qpq) + 2 (τ (1 − q))1/2 = τ (qpqu) + τ (qpq(1 − u)) + 2 (τ (1 − q))1/2 ≤ τ (qpqu) + τ (1 − u) + 2 (τ (1 − q))1/2 . In the last inequality the assumption [u, Dτ ] = 0 has been used. Finally observe that u ≥ 1c Dτ u and 1 Tr(qpq Dτ u). c Inequality (8) follows immediately inserting (7) into Eq. (9). Tr( pq) = Tr(qpq) = Tr(qpqu) + Tr(qpq(1 − u)) ≥
(9)
For 0 ≤ s < ∞, write u ε(n) (s) for the finite direct sum spece−ns ((n) ), u ε(n) (s) := s−ε<s <s+ε
where specλ (·) denotes the eigen-projection of its argument’s density operator (here D(n) ) corresponding to the eigenvalue λ. We extend this definition to the case s = ∞ by setting u ε(n) (∞) := spec0 ((n) ) + spece−ns ((n) ). (10) ε−1 <s
Now we have Proposition 9. Let be ergodic and let be an arbitrary state on AZ . If the pair (, ) fulfills the HP-condition then: • βε (, ) = limn→∞ n1 βε,n (, ) exists and we have βε (, ) = −s(, ) for each ε ∈ (0, 1). • Moreover, for all ε > 0 it holds that lim (n) (u ε(n) (s() + s(, ))) = 1 (relative AEP)
n→∞
and for each sequence { pn } of projections fulfilling 1 log Tr pn → s() (11) n→∞ n there is a sequence εn 0 such that with u n := u εn(n) (s() + s(, )), the relations (n) ( pn ) → 1 and n→∞
(n) (supp(u n pn u n )) → 1 n→∞
and 1 log (n) (supp(u n pn u n )) → −s(, ) (max. separating projection) n→∞ n are fulfilled.
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Remark. The quantum Shannon-McMillan theorem [1] guarantees the existence of a sequence of projections { pn } with the properties assumed in (11). We refer to such sequences as entropy-typical w.r.t. . Roughly speaking, the above proposition shows that one obtains a sequence of maximally separating projections as an ’intersection’ of -entropy-typical projections with appropriate eigen-projections of the reference state . Proof. 1. First assume s(, ) < +∞. By the monotonicity of the relative entropy we may conclude that S( (n) , (n) ) < +∞ for each n. We have 1 1 1 S( (n) , (n) ) = − S( (n) ) − TrD (n) log D(n) . n n n (n) )
( Let {λi }ri=1
be the set of non-zero eigenvalues of D(n) . We get
1 1 1 S( (n) , (n) ) = − S( (n) ) − log λi TrD (n) specλi ((n) ). n n n i
Let ε > 0 and
pn,ε :=
specλi ((n) ).
λi >e−n(s()+s(,)−ε)
We claim that
lim ( pn,ε ) = 0
n→∞
for all ε > 0.
In fact, suppose on the contrary that for some ε > 0 we have limsup( pn,ε ) > 0. n→∞
We conclude the existence of some γ > 0 and some subsequence {n j } with (n j ) ( pn j ,ε ) > γ > 0. Fix some α ∈ (0, 1), δ > 0. Let pn j := pn j ,ε , qn j := arg min βα,n j (, ) and u n j := u δ (n j ) (s()). Then D (n j ) u n j ≤ cu n j for c = e−n j (s()−δ) and by Lemma 8 and the quantum Shannon-McMillan theorem we arrive at √ (n j ) (qn j pn j qn j u n j ) ≥ γ − 2 α − δ > 0, and
√ Tr(qn j pn j qn j ) ≥ en j (s()−δ) (γ − 2 α − δ), (12) √ if j is large enough and if 2 α + δ < γ . Now, observe that D(n j ) and pn j commute and that consequently we have D(n j ) ≥ e−n j (s()+s(,)−ε) pn j by definition of pn j . Thus we obtain qn j D(n j ) qn j ≥ e−n j (s()+s(,)−ε) qn j pn j qn j . (13)
After applying trace to both sides of this inequality, taking logarithms, dividing by n j , taking limit superior and using (12) we are led to β¯α (, ) ≥ −s(, ) + ε − δ > −s(, ), which contradicts the assumed HP-condition provided that δ < ε.
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In the case s(, ) = ∞ everything can be done in the same way, we just have to substitute the definition of pn,ε by pn,ε = specλi ((n) ) λi >e−n/ε
and obtain β¯α ≥ − 1ε + s() − δ, again in contradiction to β¯α (, ) = −∞, hence again the projectors pn,ε have asymptotically vanishing expectation with respect to for each positive ε. 2. Let first s(, ) < ∞. We have n1 S( (n) , (n) ) → s(, ) as n → ∞ by assumption, hence 1 − TrD (n) log D(n) −→ s() + s(, ) n→∞ n and the mixed term − n1 TrD (n) log D(n) is the expectation value of the random variable − n1 log λi with respect to the probability measure given by {TrD (n) specλi ((n) )}, where again {λi } runs through the non-zero eigenvalues of (n) . On the other hand, we have shown in 1 that the lower bounded random variable − n1 log λi ≥ 0 is bounded asymptotically in probability by the quantity s()+s(, ), being its asymptotic expectation value at the same time, i.e. limn→∞ ( pn,ε ) = 0 for all ε > 0. From this it easily follows that lim (tn,δ ) = 0 for all δ > 0, n→∞
where tn,δ :=
specλi ((n) ).
λi <e−n(s()+s(,)+δ)
This is the assertion lim (n) (u ε(n) (s() + s(, ))) = 1
n→∞
(14)
for all ε > 0. In the case s(, ) = ∞ the relative AEP follows immediately from 1. 3. First assume s(, ) < ∞. Fix some ε and some α ∈ (0, 1). Let {qn } be any sequence of projections fulfilling (n) (qn ) ≥ 1 − α for n large enough. Let pn := u ε(n) (s() + s(, )). We proved that (n) ( pn ) → 1. Now as in 1 we may conclude n→∞
(n) (qn ) ≥ e−n(s()+s(,)+ε) Tr pn qn . Using the quantum Shannon-McMillan theorem
and again Lemma 8, this time applied to the density operator of (n) and with u := λ<e−n(s()−δ) specλ ( (n) ), for arbitrary δ > 0 and n large enough we have Tr pn qn ≥ en(s()−δ) a for some 0 < a < 1 independent of n. Hence we get for any ε, δ > 0, 1 log (n) (qn ) > −s(, ) − ε − δ n
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for n large enough. Therefore, the quantity βα (, ) exists for any α ∈ (0, 1) and coincides with −s(, ), where we used again the HP-condition. In the case s(, ) = ∞ this assertion is a trivial consequence of β¯α (, ) = −∞. 4. Let { pn } be a sequence of projections pn ∈ A(n) with limn→∞ (n) ( pn ) = 1 and limn→∞ n1 log Tr( pn ) = s(). Fix some ε > 0. Let us write u n instead of u ε(n) (s() + s(, )) for short. From (14) and Lemma 8 we infer that (n) (u n pn u n ) → 1. n→∞ Now u n pn u n is a positive operator being upper bounded by its support projection supp(u n pn u n ) which proves (n) (supp(u n pn u n )) → 1. From this we easily conn→∞
clude that we may even substitute the ε in the definition of u n = u ε(n) (s() + s(, )) by a suitable sequence εn → 0 and still have (n) (supp(u n pn u n )) → 1. n→∞
On the other hand, we have supp(u n pn u n ) ≤ u n as well as Tr(supp(u n pn u n )) ≤ Tr( pn ). Hence we get in the case s(, ) < ∞, 1 log (n) (supp(u n pn u n )) n 1 ≤ (−n(s() + s(, ) − εn ) + log Tr ( pn )) −→ −s(, ), n→∞ n resp. for s(, ) = ∞, 1 log (n) (supp(u n pn u n )) n 1 ≤ (−n/εn + log Tr( pn )) −→ −s(, ) = −∞. n→∞ n This together with the fact we proved that no sequence of -typical projections has a better lower limit of the separation rate than −s(, ) shows now that 1 log (n) (supp(u n pn u n )) −→ −s(, ). n→∞ n We proved all assertions of the proposition. 3.2. Stein’s Lemma implies Sanov’s Theorem. With the preliminaries given in the last subsection, it is now easy to complete the proof of Theorem 5: Proof. Let smin := inf s() and smax := sup s(), ∈
∈
where s() denotes the von Neumann entropy rate of the ergodic state ∈ . Choose s1 , . . . , sm satisfying (for η := m −1 (smax − smin )) smin = s1 < s2 < . . . sm−1 < sm = smax and
si − si−1 = η, i ∈ {2, . . . , m}.
Define sm+1 = sm + η. Let first s(, ) < +∞. For i ∈ {1, . . . , m} we consider the collection of disjoint intervals η η Ii := si + s(, ) − , si + s(, ) + 2 2
Typical Support and Sanov Large Deviations of Correlated States
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and Im+1 := (smax + s(, ) +
η , ∞). 2
Moreover we define the following projections: u n,i := specλ ((n) ), − n1 log λ∈Ii
and u n,m+1 :=
specλ ((n) ),
− n1 log λ>smax +s(,)+η/2
where the summations extend over the eigenvalues of D(n) . Additionally we consider universally typical projections pn,i , i ∈ {1, . . . , m} (according to the Kaltchenko-Yang universality result [15]) to the levels si + η (i.e. lim
n→∞
and
1 log Tr( pn,i ) = si + η n
lim (n) ( pn,i ) = 1
n→∞
(15)
(16)
for each ergodic state with s() < si + η). In addition, set pn,m+1 := pn,m . We may choose the sequence of these projections to be ascending, i.e. pn,i ≤ pn,i+1 ,
(17)
since otherwise we may define pn,i :=
i
pn, j .
j=1
The pn,i fulfill (15) and (16) as well, so we may work with these instead of pn,i . Set rn,i =supp(u n,i pn,i u n,i ) for i = 1, 2, ..., m + 1 and define pn by pn :=
m+1
rn,i .
i=1
(Observe that the rn,i are mutually orthogonal.) For ∈ let i 0 ∈ {1, . . . , m + 1} be the index fulfilling s() + s(, ) ∈ Ii0 . This means that s() ≤ si0 + η/2 < si0 + η. Consequently, by (16) we obtain limn→∞ (n) ( pn,i0 ) = 1. Further, by the relative AEP (Proposition 9) limn→∞ (n) (u n,i0 + u n,i0 +1 ) = 1, for i 0 ∈ {1, . . . , m}, and limn→∞ (n) (u n,m+1 ) = 1 are satisfied. We add the projection u n,i0 +1 for i 0 ∈ {1, . . . , m} in order to cover the case where the mixed term is equal to the right end point of Ii0 . We conclude from (17) and Lemma 8 that (n) (rn,i0 +rn,i0 +1 ) → 1, for i 0 ∈ {1, . . . , m}, and (n) (rn,m+1 ) → 1. Therefore lim (n) ( pn ) = 1.
n→∞
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I. Bjelakovi´c et al.
On the other hand we have for n sufficiently large by (15) and by definition of η, ( pn ) = ≤
m+1 i=1 m+1
(rn,i ) ≤
m+1
Tr( pn,i )e−n(si +s(,)−η/2)
i=1
en(si +2η) e−n(si +s(,)−η/2)
i=1 5
= e−n(s(,)− 2 η− 5
= e−n(s(,)− 2 m
log(m+1) ) n
log(m+1) −1 (s ) max −smin )− n
.
So, by choosing m sufficiently large, we get statement (4). η The case s(, ) = +∞ easily follows by setting pn := u (n) (∞), see (10). By Proposition 9 the projection pn is asymptotically typical for all ∈ , and we have −1 −1 (n) ( pn ) ≤Tr(1A(n) )e−nη = e−n(η −log Tr(1A )) , so again we get statement (5). Finally, the fact that a better separation exponent than −s(, ) is not achievable immediately follows from Proposition 9. 4. ∗-Mixing Implies the HP-Condition We start with a proposition extending the result in [13] to the case of only ergodic (instead of completely ergodic) . Theorem 10. Let ∈ S∗ (AZ ). Then is an HP-state. Recall that S∗ (AZ ) denotes the set of stationary ∗-mixing states, see Definition 2. Proof. 1. Let ∈ Serg (AZ ). The relative entropy rate s(, ) ≤ +∞ exists in view of [13], Theorem 2.1, in connection with Remark 4.2, ibid. (even if only ∈ Sstat (AZ ) is assumed). 2. Fix some l ∈ N, another integer m (which in the sequel has to be chosen large enough) and represent the quasilocal C ∗ -algebra AZ as C ∗ -algebra (A⊗l ⊗ A⊗m )Z , i.e. partition the integers into blocks of length l + m, where each block consists of a starting part of length l and the remaining part of length m. Clearly, the entropy rate s(l+m) (, ) with respect to this new partitioning exists, and we have s(l+m) (, ) = (l + m)s(, ). With respect to the canonical shift operator τl,m := τ l+m acting in (A⊗l ⊗ A⊗m )Z , the state is still stationary, but may fail to be ergodic. Anyway, it has a finite ergodic decomposition l+m−1 1 (r,l+m) , = l +m r =0
where some of the ergodic components may coincide, and all (r,l+m) have the same entropy rate s(l+m) ((r,l+m) ) ≡ s(l+m) () = (l + m)s(), [1]. The ergodic components also have the same relative entropy rate s(l+m) ((r,l+m) , ) := s(l+m) (, ) = (l + m)s(, ). Observe that this was shown in [3] for the case of stationary product states . However as the proof only makes use of the existence of the relative entropy rates s(l+m) ((r,l+m) , ), which is guaranteed in our situation (see 1), the monotonicity
Typical Support and Sanov Large Deviations of Correlated States
573
of the relative entropy and the affinity of the relative entropy rate with respect to its first argument, the relation extends to ∗-mixing reference states. Next, denote by I the trivial subalgebra of A generated by the unit element 1A and consider the C ∗ -subalgebra I⊗l ⊗ A⊗m of A⊗l ⊗ A⊗m . Then (I⊗l ⊗ A⊗m )Z is a C ∗ subalgebra of the quasi-local algebra (A⊗l ⊗ A⊗m )Z , and by [6], Theorem 4.3.17, the (r,l,m) of the ergodic components (r,l+m) to (I⊗l ⊗ A⊗m )Z are ergodic, restrictions (l,m) := (I⊗l ⊗A⊗m )Z . too. They are the ergodic components of (l+m) on (A⊗l ⊗ A⊗m )Z which We introduce the (τl,m -) stationary product state is uniquely defined by its one-site marginals A⊗l ⊗A⊗m , and consider its restriction (l,m) to the C ∗ -algebra (I⊗l ⊗ A⊗m )Z , which is a (τl,m -) stationary product state, too. 3. In the following we have to take into account whether s(, ) is finite or infinite. Let us first treat the case s(, ) < +∞. Define for two states ψ, ϕ on a C ∗ -algebra A ,
ψ(qi ) Sco (ψ, ϕ) := sup ψ(qi ) log qi = I : qi projections with ϕ(qi ) i
(cf. [12]). (l+m) ), then we get by the superadditivity (l+m) , Consider the relative entropies S( (r,l,m) (l,m) of relative entropy and by the fact that the rates of the quantities S and Sco coincide (Hiai and Petz [12]) (l+m) ) (l+m) , S( (r,l,m) (l,m)
(18)
(r,l,m) , (l,m) ) = lim ≤ s(l+m) (
k→∞
1 (k(l+m)) , (k(l+m)) ). Sco ( (r,l,m) (l,m) k
From the definition of Sco and ∗-mixing we obtain now (l+m) , (l+m) ) S( (r,l,m) (l,m) 1 (k(l+m)) , (I⊗l ⊗A⊗m )⊗k ) − k log α . ≤ lim Sco ( (r,l,m) k→∞ k This is the same technique as used by Hiai and Petz in [13]. Again from the definition of Sco we get (l+m) , (l+m) ) S( (r,l,m) (l,m) ≤ − log α + lim
k→∞
1 (k(l+m)) Sco ((r,l+m) , (k(l+m)) ), k
and from the relation Sco ≤ S (see [12]), which is a consequence of the monotonicity of the relative entropy, we arrive at (l+m) , (l+m) ) S( (r,l,m) (l,m) 1 (k(l+m)) S((r,l+m) , (k(l+m)) ) k ≤ − log α + s(l+m) ((r,l+m) , ) = − log α + (l + m)s(, ).
≤ − log α + lim
k→∞
(19)
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I. Bjelakovi´c et al.
This upper bound may be utilized to derive an essential lower bound. For an arbitrarily chosen η > 0, define Al,m,η := {r : 0 ≤ r < l + m,
1 (l+m) , (m+l) ) < s(, ) − η}. S( (r,l,m) (m,l) l +m
The convexity of the relative entropy in its first argument together with ( 19) yields 1 1 (m+l) , (m+l) ) S( (m) , (m) ) = S( (m,l) (l,m) l +m l +m 1 (l+m) , (m+l) ) ≤ S( (r,l,m) (m,l) (l + m)2 r <
c # Al,m,η # Al,m,η (s(, ) − η) + l +m l +m
Fixing l and letting m → ∞, the expression immediately leads to the conclusion that
(20)
log α − + s(, ) . l +m
1 (m) , (m) ) l+m S(
tends to s(, ). This
# Al,m,η −→ 0 for each l, η. l + m m→∞
(21)
c , the relative entropy rate fulfills For each r ∈ Al,m,η
1 (r,l,m) , (l,m) ) ≥ s(, ) − η, s(l+m) ( l +m
(22)
(l,m) is a τl+m -stationary product state. too, since 4. Hence we are in the situation treated in [3]. The main assertion of [3] is the quantum Stein Lemma saying that for any given ε > 0 it is possible to construct (r,l,m) (i.e. projections pr,n,ε ∈ (I⊗l ⊗ A⊗m )⊗n which are ε-typical with respect to (n(l+m)) ((l+m)n) ( pr,n,ε ) ≥ 1 − ε for large n) and maximally separating: ( pr,n,ε ) ≤ (r,l,m)
(l,m)
e−n(s(l+m) ((r,l,m) ,(l,m) )−ε) for large n. Moreover, the quantum relative AEP (Theorem 2 in [3]) ensures, in particular, that if n is sufficiently large we have
Tr pr,n,ε ≤ en(s(l+m) ((r,l,m) )+ε) ((l+m)n) ( p) (l,m)
≤e
and
(r,l,m) )+s(l+m) ( (r,l,m) , (l,m) )−ε) −n(s(l+m) (
for each minimal projection p ≤ pr,n,ε . For our purpose we need a bit more information about the construction of these maximally separating projections. In the course of the proof of Theorem 2 in [3] the projections pr,n,ε are constructed in the following way: a) A super-block length L is chosen, where the only requirement about L is that it is large enough to ensure some appropriate entropy approximation (any larger L will do, too). b) The projections pr,n L ,ε are constructed as certain sub-projections of the projection (l+m) )⊗n L ). p (n L) := specλ (( (23) (l,m) (r,l,m) ) − n1L log λ≥s(l+m) ( (r,l,m) , (l,m) )−ε +s(l+m) (
Typical Support and Sanov Large Deviations of Correlated States
575
c) The remaining projections pr,n L+k,ε for 1 ≤ k < L are constructed as pr,n L ,ε ⊗ (IA⊗(l+m) )⊗k . For given l and m we may choose one and the same super-block length L for the different r ∈ {0, 1, ..., l + m − 1} and define our separating projections first for the multiples of L(l + m) by qn L(l+m),ε := pr,n L ,ε . c r ∈Al,m,ε
In view of (23) we get, using (22)
qn L(l+m),ε ≤ − n1L
(l+m) )⊗n L ) specλ (( (l,m)
(24)
(r,l,m) ) log λ≥ min s(l+m) ( c r ∈Al,m,ε
(r,l,m) , (l,m) )−ε + min s(l+m) ( c r ∈Al,m,ε
≤
(l+m)
⊗n L specλ (( ). (l,m) )
(r,l,m) ) − n1L log λ≥ min s(l+m) ( c r ∈Al,m,ε
+(l+m)s(,)−(l+m+1)ε
Next, observe that by the subadditivity of the entropy we have for each r , 1 1 (k(l+m)) ) = lim S((r,l+m) (I⊗l ⊗A⊗m )⊗k ) S( (r,l,m) k→∞ k k 1 (k(l+m)) ≥ lim (S((r,l+m) ) − klTr1A ) k→∞ k = s(l+m) ((r,l+m) ) − lTr1A = (l + m)s() − lTr1A ,
(r,l,m) ) = lim s(l+m) (
k→∞
and hence for sufficiently large m we may continue the chain of inequalities (24):
qn L(l+m),ε ≤
(l+m)
− log λ≥n L(l+m)(s()+s(,)−3ε)
⊗n L specλ (( ). (l,m) )
From this we derive the following upper bound, being valid for m large enough (using the Araki-Lieb inequality in the fifth line) n L(l+m) (qn L(l+m),ε ) (l,m) ≤ e−n L(l+m)(s()+s(,)−3ε) Trqn L(l+m),ε ≤ e−n L(l+m)(s()+s(,)−3ε) Tr pr,n L ,ε c r ∈Al,m,ε
≤ e−n L(l+m)(s()+s(,)−3ε)
en L(s(l+m) ((r,l,m) )+ε)
c r ∈Al,m,ε
≤ e−n L(l+m)(s()+s(,)−3ε) (l + m)en L(l+m)(s()+2ε) ≤ e−n L(l+m)(s(,)−6ε) .
576
I. Bjelakovi´c et al.
From ∗-mixing we get now the desired separation order (n L(l+m)) (qn L(l+m),ε ) ≤ e−n L(l+m)(s(,)−6ε) α −n L log α
= e−n L(l+m)(s(,)+ l+m −6ε) ≤ e−n L(l+m)(s(,)−7ε) (for m large enough). On the other hand, -typicality is guaranteed by ⎛ (n L(l+m)) (qn L(l+m),ε ) = (n L(l+m)) ⎝
⎞ pr,n L ,ε ⎠
c r ∈Al,m,ε
⎛ l+m−1 (n L(l+m)) 1 = (r ,l+m) ⎝ l +m c r =0
≥ ≥
1 l +m 1 l +m
⎞ pr,n L ,ε ⎠
r ∈Al,m,ε
(n L(l+m))
c r ∈Al,m,ε
(r ,l+m)
pr ,n L ,ε
(1 − ε),
c r ∈Al,m,ε
the last inequality being valid for large n. We may continue 1 (n L(l+m)) (qn L(l+m),ε ) ≥ (1 − ε) − (1 − ε) l +m r ∈Al,m,ε
≥ (1 − ε) −
# Al,m,ε ≥ 1 − 2ε l +m
for m large enough (by (21)). Now (in the usual way) we may interpolate the qn L(l+m),ε in order to define the projections qn,ε also for n ∈ N which are not multiples of L(l +m). We derived the existence of a sequence of projections being asymptotically ε-typical for and fulfilling (n) (qn,ε ) ≤ e−n(s(,)−ε) for large n. This proves that, for any α ∈ (0, 1) the separation exponent fulfills β α (, ) − s(, ) for finite s(, ). 5. Now assume s(, ) = +∞. Observe that in that case the estimates in (20) and hence (21) are not valid. But ( 21) becomes true if we replace the definition of Al,m,η most appropriately by Al,m,η := {r : 0 ≤ r < l + m,
1 (l+m) , (m+l) ) < η−1 }. S( (r,l,m) (m,l) l +m
In fact, choose M large enough to ensure S( (M) , (M) ) > η−1 M (we include the case S( (M) , (M) ) = +∞). Now we have the ergodic decomposition =
M−1 M−1 1 1 (r,M) = (0,M) ◦ τ −r M M r =0
r =0
(25)
Typical Support and Sanov Large Deviations of Correlated States
577
due to [1]. The states (r,M) = (0,M) ◦ τ −r are τ M -ergodic. In view of the (joint) (M) , convexity of the relative entropy we conclude that at least one of the r fulfills S((r,M) (M) ) > η−1 M. We may assume without any loss of generality that this is true for (M) , (M) ) > η−1 M. The τ M -ergodic state r = 0, i.e. S((0,M) (0,M) again has an ergodic decomposition with respect to τ 2M , (0,M) =
1 ( + ◦ τ −M ), 2
and, applying once again the convexity argument we find that we may assume S( (M) , (M) ) > η−1 M. is τ 2M -ergodic, and we obtain from (25) an ergodic decomposition of into τ 2M -ergodic states 2M−1 1 ◦ τ −r , 2M r =0
hence we may assume without loss of generality that (0,2M) = . So we have (M)
S((0,2M) , (M) ) > η−1 M for M large enough. This yields S(((0,2M) ◦ τ −r )({r,r +1,...,r +M−1}) , (M) ) > η−1 M for each r in view of the definition of τ , i.e. (using the stationarity of ) S(((r,2M) )({r,r +1,...,r +M−1}) ,
({r,r +1,...,r +M−1})
) > η−1 M.
In view of the monotonicity of the relative entropy we get now for r ≥ l, (r,l,2M−l) )(2M) , I⊗l ⊗A⊗m ) > η−1 M. S(( So again (21) is fulfilled for M sufficiently large. We conclude that asymptotically for the 1 (r,l,m) , (l,m) ) overwhelming part of the r in {0, 1, ..., l+m−1} the expression l+m s(l+m) ( 1 −1 is arbitrary large (i.e. > 2 η or even infinite) for large m. Now we may proceed essentially as in 4, employing the results of [3]. We find projections pr,n L ,η , separately for each c (r,l,m) and (l,m) exponentially well at a rate at , which distinguish between r in Al,m,η 1 −1 least 3 η , and we may join these projections to find -typical projections qn L(l+m),η . This is possible due to the properties a) and b) above, where now b) is modified to b’) The projections pr,n L ,η are constructed as certain sub-projections of the projection p (n L) :=
(l+m) )⊗n L ). specλ (( (l,m)
(26)
log λ≤−n L 13 η−1
But we have to take into account that [3] only treats the case of finite relative entropy (l+m) , (l+m) ) = +∞, are still not covered. For those rate; hence those r , for which S( (r,l,m) (l,m) r , simply choose (l+m) )⊗n L ). pr,n L ,η ≡ spec0 (( (l,m)
578
I. Bjelakovi´c et al. (l+m)
(l+m)
⊗n L (spec (( ⊗n L )) = 0, and it is a Obviously, this projection fulfills ( 0 (l,m) ) (l,m) ) (l+m) )⊗n L ) is asymptotsub-projection of p (n L) . We still have to show that spec0 (( (l,m)
(l+m) , (m+l) ) = +∞. In fact, represent the (r,l,m) with S( ically typical for each (r,l,m) (m,l) density operator of (m) as K D(m) = λjwj, j=1
where the w j are mutually orthogonal minimal projectors in A⊗m fulfilling j w j = 1A⊗m (and the λ j are the eigen-values of D(m) including 0). Let v j := 1A⊗l ⊗ w j . (l+m) , (l+m) ) = +∞ there is at least one j with λ = 0 but Observe that due to S( j (r,l,m) (l,m) ((l+m)) (r,l,m) v j > 0. Now we have (n(l+m)) (spec0 (( (l+m) )⊗n )) (r,l,m) (l,m)
=
( j1 ,..., jn )∈Nn
(n(l+m)) (r,l,m)
n
v jk ,
(27)
k=1
where Nn := {( j1 , ..., jn ) : nk=1 λ jk = 0} = ((N1c )n )c . Denote by B the abelian subalgebra of I⊗l ⊗ A⊗m generated by the set {v j }. Then the quasi-local algebra B Z is an (r,l,m) to this sub-algebra abelian sub-algebra of (I⊗l ⊗ A⊗m )Z and the restriction P of is a classical ergodic process with K symbols (Gelfand isomorphism and Riesz representation theorem). This process fulfills P (1) ({ j}) > 0. We may continue the left-hand side in (27) as follows: (n(l+m)) (spec0 (( (l+m) )⊗n )) = P (n) (Nn ) (r,l,m) (l,m) = 1 − P (n) ((N1c )n ) ≥ 1 − P (n) (({ j}c )n ). Now P (n) (({ j}c )n ) is the probability of all n -sequences of symbols where the symbol j does not appear at all. This tends to zero, since by the individual ergodic theorem the a.s. ((l+m)) v > 0 by assumption. asymptotic frequency of the symbol j is P (1) ({ j}) = j (r,l,m) Hence the conclusions of part 4 are valid in the case of infinite relative entropy, too. 5. The Stationary Case So far we formulated Theorems 5 and 7 for sets of ergodic states , resp. processes P to be optimally separated from a reference state or process. These results can be easily extended to the general stationary situation. Any stationary state ∈ Sstat (AZ ) can be represented as a mixture (ergodic decomposition) =
Serg (AZ )
γ (d)
of ergodic states (Sstat (AZ ) is a Choquet simplex, Serg (AZ ) is the corresponding set of extremal points, γ is a probability measure on the measurable space
Typical Support and Sanov Large Deviations of Correlated States
579
[Serg (AZ ), B(ϒAZ )], with ϒAZ denoting the weak-∗-topology and B(ϒAZ ) the corresponding Borel σ -field, cf. [22]). The measure γ is unique. Now let ∈ S(AZ ) be a state and ⊆ Sstat (AZ ) with the property that for any ∈ the relative entropy rate s(, ) exists for γ -almost all . We define the quantity s(, ) := essinfγ (d) s(, ), and for ⊆ the quantity s(, ) := inf s(, ). ∈
Theorem 11. Let be a state on AZ and ⊆ Sstat (AZ ) such that for each ∈ and γ -almost all the pair (, ) satisfies the HP-condition. Then the quantity s(, ) ≤ +∞ exists for each ∈ , and to each subset ⊆ and any η > 0 there exists a sequence { pn }n∈N of projections pn ∈ A(n) with lim (n) ( pn ) = 1,
n→∞
and
for all ∈
1 limsup log (n) ( pn ) ≤ −s(, ) + η. n→∞ n
(28)
(29)
If s(, ) < ∞, otherwise if s(, ) = ∞, 1 1 limsup log (n) ( pn ) ≤ − . η n→∞ n
(30)
Moreover, for each sequence of projections { pn } fulfilling (28) we have 1 liminf log (n) ( pn ) ≥ −s(, ). n→∞ n
(31)
Hence −s(, ) is the lower limit of all achievable separation exponents. Remark 12. If is stationary and, moreover, ∗-mixing, the assumption of the Theorem 11 is fulfilled with = S stat (AZ ), according to Sect. 4. ˜ := { ∈ Serg (AZ ) : (, ) satisfies the HP-condition and s(, ) ≥ Proof. Let ˜ is weak-∗ -measurable since it can be represented by a countable s(, )}. The set application of unions and intersections to local sets, defined via the measurable functions S(·, (n) ) and βε,n (·, ). . Then (29) or (30) are Let pn be chosen as in Theorem 5, with there specified as trivially fulfilled. For any ∈ we obtain by assumption (n) ( pn ) = (n) ( pn )γ (d) (32) =
Serg (AZ )
(n) ( pn )γ (d).
Now for each ∈ the expression (n) ( pn ) ∈ [0, 1] tends to 1 by the choice of the projections pn . Hence Lebesgue’s theorem on dominated convergence guarantees (28).
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I. Bjelakovi´c et al.
On the other hand, for each sequence of projections { pn } fulfilling (28) the identity (32) (with pn instead of pn ) proves that, for each ∈ , (n) ( pn ) tends to 1 in γ -probability as n → ∞. By the definition of s(, ) to any η > 0 we may choose in such a way that s(, ) ≤ s(, ) + η. We show that 1 liminf log (n) ( pn ) ≥ −s(, ), n→∞ n which implies (31) since η can be chosen arbitrarily small. In fact, assume the existence of a sub-sequence n such that lim n
1 log (n ) ( pn ) ≤ −s(, ) − δ, δ > 0. n
(33)
Along that sub-sequence there is still convergence in γ -probability of (n ) ( pn ) to 1. Since convergence in probability implies almost sure convergence of some sub pn ) = 1 holding γ sequence, we find another sub-sequence n of n with limn (n ) ( almost surely. Hence, in view of the definition of s(, ) there is some 0 ∈ Serg (AZ ) such that (0 , ) fulfills the HP-condition, s(0 , ) < s(, ) + δ, but (n ) pn ) = 1. Now Theorem 5, applied to the case = {0 } implies limn 0 ( liminf n
1 log (n ) ( pn ) > −s(, ) − δ, n
which contradicts (33). The classical case immediately follows (with γ P denoting the probability measure occurring in the ergodic decomposition of a stationary process P and h(P, Q) := essinf γ P (d W ) h(W, Q), supposing that h(W, Q) exists γ P -almost surely): Theorem 13. Let Q ∈ P(AZ ) be a process and ⊆ Pstat (AZ ). Assume that for each P ∈ and γ P -almost all W ∈ Pstat (AZ ) the relative entropy rate h(W, Q) exists and β¯ε (W, Q) ≤ −h(W, Q) for all ε ∈ (0, 1). Then the quantity h(P, Q) ≤ +∞ exists for all P ∈ , and to each subset ⊆ and any η > 0 there exists a sequence {Mn }n∈N of subsets Mn ⊆ An with lim P (n) (Mn ) = 1,
n→∞
for all P ∈
(34)
and
1 limsup log Q (n) (Mn ) ≤ −h(, Q) + η n→∞ n if h(, Q) < ∞, otherwise if h(, Q) = ∞, 1 1 limsup log Q (n) (Mn ) ≤ − . η n→∞ n n } fulfilling ( 34) we have Moreover, for each sequence of subsets { M 1 n ) ≥ −h(, Q). liminf log Q (n) ( M n→∞ n Hence −h(, Q) is the lower limit of all achievable separation exponents. Remark 14. If Q is stationary and, moreover, ∗-mixing, the assumption of the theorem is fulfilled with = Pstat (AZ ) , according to Sect. 4.
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6. The Quantum Shannon-McMillan Theorem for Stationary States and Other Corollaries As announced in the introduction, several earlier results on typical subspaces, resp. subsets, are contained in Theorem 11 in a version extended to the stationary case. We emphasize that the initial versions of the quantum Shannon-McMillan Theorem, Kaltchenko-Yang universality and the quantum Stein Lemma were important ingredients in our proof. Also, it should be mentioned that it is not difficult to prove the stationary case of the quantum Shannon-McMillan Theorem directly from the Kaltchenko-Yang result, without using the quantum Sanov Theorem. Corollary 15. (Quantum Shannon-McMillan Theorem for stationary states). Let ∈ Sstat (AZ ) be a stationary state and = Serg (AZ ) γ (d) be its ergodic decomposition. Then there exists a sequence { pn } of projections in A(n) , respectively such that • limn→∞ (n) ( pn ) = 1 (typicality) • limn→∞ n1 Tr pn =esssupγ (d) s() := s() (maximal ergodic entropy rate). For any sequence pn with limn→∞ (n) ( pn ) = 1 we have 1 pn ≥ s() (optimality). lim inf Tr n→∞ n Remark 16. We emphasize that the AEP does not hold in the stationary case. Also, observe that the relevant notion in the stationary case is not the von Neumann entropy rate s() of the state being the average of the entropy rates of the ergodic components of , but their essential supremum s(). Proof. Let be the tracial state in S(AZ ). It is ∗-mixing (even iid). Apply Theorem 11 η with = {}. This yields a sequence { pn } of -typical projections with 1 1 s() ≤ liminf Tr pnη ≤ limsup Tr pnη ≤ s() + η n→∞ n n n→∞ for any η > 0. Now the assertion of the theorem easily follows, since is a finite set. The next corollary extends the universality result of [15] to stationary states: Corollary 17. (Kaltchenko-Yang universality theorem for stationary states). Let s := { ∈ Sstat (AZ ) : s() < s}. Then there exists a sequence { pn } of projections in A(n) , respectively such that • limn→∞ (n) ( pn ) = 1 for each ∈ s (typicality) • limn→∞ n1 Tr pn = s (maximal ergodic entropy rate). For any sequence pn with limn→∞ (n) ( pn ) = 1, ∈ s , we have liminf n→∞
1 Tr pn ≥ s n
(optimality).
Proof. Let again be the tracial state in S(AZ ). Apply Theorem 11 in a similar way as in the proof of Corollary 15 to the sets s−η , η > 0.
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Remark 18. Observe that the condition s() < s defining s cannot be replaced by s() ≤ s. Finally, the quantum Stein Lemma [3] is extended to the case where the null hypothesis state is only assumed stationary, the reference state fulfills the HP-condition with respect to almost all ergodic components of (and the relative entropy rate s(, ) may be infinite): Corollary 19. (Stein’s Lemma for stationary states). Let ∈ S(AZ ) and ∈ Sstat (AZ ) such that for γ -almost all the HP-condition is fulfilled for (, ). Then there exists a sequence { pn } of projections with • limn→∞ (n) ( pn ) = 1 (typicality) • limn→∞ n1 log (n) ( pn ) = −s(, ) (achievability of the separation exponent −s(, )). For any sequence pn with limn→∞ (n) ( pn ) = 1 we have 1 pn ) ≥ −s(, ) (optimality) . liminf log (n) ( n→∞ n Remark 20. Note that the relative AEP does not hold in Stein’s Lemma in the stationary case. Again, the relevant quantity in the stationary situation is not the average relative entropy rate s(, ), but the essential infimum s(, ). Proof. With consisting of a single state only, we may proceed in the same way as in the proof of Corollary 15. 7. Conclusions The paper is devoted to a generalization of Sanov’s Theorem from the iid classical situation to the correlated case and, moreover, to the quantum setting. In the present form, the main result comprises and extends several earlier assertions including the (quantum and classical) Shannon-McMillan Theorem, Stein’s Lemma (with relative AEP), Kaltchenko and Yang’s universality and, of course, a version of Sanov’s Theorem itself. It is a continuation of [2], where the uncorrelated case is considered. It has to be pointed out again (see [2]), that any attempt to formulate a quantisized version of Sanov’s result has to face the problem that the very notion of a trajectory and its empirical distribution is problematic in quantum mechanics. Sanov’s classical theorem claims that for an iid process with marginal Q the probability to produce a trajectory with the empirical distribution belonging to some set of probability measures is (in general) exponentially small. The corresponding rate is specified as the minimal relative entropy between Q and the distributions in . In the interesting case the measure Q is of course not an element of or its topological closure. So it is a large deviation result: the typical behaviour of Q-trajectories is to have an empirical distribution close to Q. Whatever one tries to adopt as a quantum substitute for the empirical distribution, the natural choice in the case of a tensor product of vector states v ⊗ v ⊗ ... ⊗ v should be v itself. This leads into the problem that for a reference vector state w ⊗n the probability of measuring an ’empirical state’ v is at least TrPw⊗n Pv ⊗n = |w|v|2n , while the relative entropy of v wrt w is infinite, which would imply a super-exponential decay; for a more detailed
Typical Support and Sanov Large Deviations of Correlated States
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exposition see [2]. In this situation it proves useful to look at Sanov’s Theorem as an assertion about the likelihood of observing the classical iid process given by Q far from its original support in the vicinity of the supports of other iid processes. The most natural choice for ’typical support’ in the classical framework is the set of trajectories with empirical distribution close to the given probability distribution, since according to the individual ergodic theorem the empirical distribution tends to Q with probability one. So Sanov’s Theorem in its original form says that the probability of observing the trajectory in the typical support of other distributions, concretely specified by means of the corresponding empirical distributions, vanishes at a rate given by the minimum relative entropy. It is of course completely legitimate to insist on the point of view, that a quantum Sanov Theorem should be about empirical distributions, too (see [18], Remark 4, see also an attempt to formulate a quantum (iid) Sanov Theorem made in Segre’s Ph.D. thesis [24] (2004), Conjecture 7.3.1.). But, as explained, then one loses the relation to the established form of quantum relative entropy (Umegaki’s relative entropy). We chose to ’sacrifice’ empirical distributions in our approach but nonetheless calling it a version of Sanov’s Theorem: in the classical case the relative entropy is not only the rate of separation when empirical distributions as specifying typical sets are considered. It has a clear operational meaning as the optimal separation rate, whatever one considers as typical support in the sense that the probability goes to one. This perception of Sanov’s Theorem, closely connected with the statistical hypothesis testing aspect, appears to be natural. It allows useful generalizations to the correlated and quantum cases. Acknowledgements. This work was supported by DFG grants “Entropy and coding of large quantum systems” and by the Max-Planck Institute for Mathematics in the Sciences, Leipzig. Tyll Krüger, Rainer SiegmundSchultze and Arleta Szkoła are particularly grateful to Nihat Ay for his constant encouragement during the preparation of the manuscript.
References 1. Bjelakovi´c, I., Krüger, T., Siegmund-Schultze, Ra., Szkoła, A.: The Shannon-McMillan theorem for ergodic quantum lattice systems. Invent. Math. 155(1), 203–222 (2004) 2. Bjelakovi´c, I., Deuschel, J.-D., Krüger, T., Seiler, R., Siegmund-Schultze, Ra., Szkoła, A.: A Quantum Version of Sanov’s Theorem. Commun. Math. Phys. 260, 659–671 (2005) 3. Bjelakovi´c, I., Siegmund-Schultze, Ra.: An Ergodic Theorem for the Quantum Relative Entropy. Commun. Math. Phys. 247, 697–712 (2004) 4. Blum, J.R., Hanson, D.L., Koopmans, L.H.: On the strong law of large numbers for a class of stochastic processes. Z. Wahrsch. Verw. Gebiete 2, 1–11 (1963) 5. Bradley, R.C.: Basic Properties of Strong Mixing Conditions. Probability Surveys 2, 107–144 (2005) 6. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. New York: Springer, 1979 7. Cover, T.M., Thomas, J.A.: Elements of Information Theory. New York: John Wiley and Sons, 1991 8. De Roeck, W., Maes, C., Netoˇcný, K.: Quantum Macrostates, Equivalence of Ensembles and an H-Theorem. J. Math. Phys. 47, 073303 (2006) 9. Deuschel, J.-D., Stroock, D.W.: Large Deviations. Boston: Acad. Press, 2001 10. Hayashi, M.: Asymptotics of quantum relative entropy from a representation theoretical viewpoint. J. Phys. A: Math. Gen. 34, 3413–3419 (2001) 11. Hayashi, M.: Optimal sequence of quantum measurements in the sense of Stein’s lemma in quantum hypothesis testing. J. Phys. A: Math. Gen. 35, 10759–10773 (2002) 12. Hiai, F., Petz, D.: The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability. Commun. Math. Phys. 143, 99–114 (1991) 13. Hiai, F., Petz, D.: Entropy Densities for Algebraic States. J. Funct. Anal. 125, 287–308 (1994) 14. Jozsa, R., Horodecki, M., Horodecki, P., Horodecki, R.: Universal Quantum Information Compression. Phys. Rev. Lett. 81, 1714–1717 (1998)
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15. Kaltchenko, A., Yang, E.H.: Universal Compression of Ergodic Quantum Sources. Quant. Inf. and Comput. 3, 359–375 (2003) 16. Lebowitz, J.L., Lenci, M., Spohn, H.: Large deviations for ideal quantum systems. J. Math. Phys. 41(3), 1224–1243 (2000) 17. Lenci, M., Rey-Bellet, L.: Large Deviations in Quantum Lattice Systems: One-Phase Region. J. Stat. Phys. 119(3–4), 715–746 (2005) 18. Nagaoka, H., Hayashi, M.: An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses. IEEE Trans. Inf. Theo. 53(2), 534–549 (2007) 19. Netoˇcný, K., Redig, F.: Large deviations for quantum spin systems. J. Stat. Phys. 117, 521–547 (2004) 20. Ogawa, T., Nagaoka, H.: Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing. IEEE Trans. Inf. Th. 46(7), 2428–2433 (2000) 21. Ohya, M., Petz, D.: Quantum entropy and its use. Berlin-Heidelberg-New York: Springer-Verlag, 1993 22. Ruelle, D.: Statistical Mechanics. New York: W. A. Benjamin Publishers, 1969 23. Sanov, I.N.: On the probability of large deviations of random variables. Mat. Sbornik 42, 11–44 (1957) 24. Segre, G.: Algorithmic Information Theoretic Issues in Quantum Mechanics. Ph.D. Thesis, 2004 25. Shields, P.C.: Two divergence-rate counterexamples. J. Theor. Prob. 6, 521–545 (1993) 26. Xu, S.: An Ergodic Process of Zero Divergence-Distance from the Class of All Stationary Processes. J. Theor. Prob. 11(1), 181–195 (1998) Communicated by M.B. Ruskai
Commun. Math. Phys. 279, 585–594 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0448-y
Communications in
Mathematical Physics
A Dynamical Zeta Function for Pseudo Riemannian Foliations Bernd Mümken Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany. E-mail: [email protected] Received: 24 October 2006 / Accepted: 31 January 2008 Published online: 5 March 2008 – © Springer-Verlag 2008
Abstract: We investigate a generalization of geodesic random walks to pseudo Riemannian foliations. The main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity, quantum mechanics and dynamical systems. 1. Introduction Let (V, b) be a real vector space with a symmetric bilinear form. Let F denote the radical of b. For v, w ∈ V the value b(v, w) depends only on the classes of v and w in the quotient V /F. To globalize this linear model we start with a smooth, i.e. C ∞ , connected manifold M together with a symmetric 2-fold covariant tensor field b in the tangent bundle T M. Let the dimension of the radicals of b be constant and denote by F M the corresponding subbundle. We write g for the metric induced in the quotient QM = T M/F M and demand the index of g to be constant. To be able to talk about local quotient manifolds we require the bundle of radicals to be completely integrable. Now it is natural to assume that locally g is the pull back of a metric on the local quotient manifold. We end up with a pseudo Riemannian foliation. In the special case where the index of g is equal to (one less than resp.) the rank of QM the pair (M, b) is called a Riemannian (Lorentzian resp.) foliation. A map f : M → N between foliations is a smooth map of the underlying manifolds such that the derivative restricts to a map F f of the radicals. The induced map of quotients will be denoted by Q f . If not specified otherwise let (M, b) be a pseudo Riemannian foliation of a connected (n + D)-dimensional manifold with F M of rank n. Let M˜ be its tangential orientation ˜ Denote by Ω ∗ (Ωc∗ resp.) the real double cover with canonical involution ι : M˜ → M. tangential differential forms (with compact support resp.) and by H ∗ (Hc∗ resp.) the reduced real tangential cohomology (with compact support resp.). We choose Partially supported by DFG, SFB 478.
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− a bundle-like metric for the metric in QM, ˜ − a tangential orientation for M, ˜ − a representative Φ of the tangential Thom class of F M. Finally, let C : T M˜ ⊕ T M˜ → T T M˜ be the connection of the bundle-like metric in the ˜ tangent bundle π : T M˜ → M. Remark 1. To familiarize with foliations and their tangential cohomology confer [22,23] and the references therein. In particular, notice that the connection of the bundle-like metric is defined by a Levi-Civita type formula as in Sect. 4 of [23]. Moreover, recall the invariance of tangential cohomology under homotopies along the leaves and its applications in Sects. 7 to 10 of [23]. Generalities on pseudo Riemannian manifolds can be found in [24,30]. The structure of the work is as follows. In Sect. 2 we define the (N -step) partition function as a weighted integral over closed broken geodesics. We show that its semiclassical limit is computable from local data. In Sect. 3 we relate the partition function to transfer operators by a trace formula. Moreover, we indicate why differential operators appear in special cases by a variant of Pizzetti’s formula. In Sect. 4 we use a zeta function to prove that the spectra of the transfer operators determine the asymptotic behavior of the partition functions for large values of N . Finally, in Sect. 5 we give some examples. 2. Partition Function ˜ Consider the equation of the geodesic flow φh¯ on T M, d φh (ξ ) = C (φh¯ (ξ ), φh¯ (ξ )) , d h¯ ¯ and define eh¯ := π φh¯ . Differentiate with respect to h¯ at h¯ = 0 to find the fundamental Lemma 1. Choose coordinates (x 1 , . . . , ξ n+D ) on F M˜ ⊕ Q M˜ which are associated ˜ with a foliated chart on M˜ and use the bundle-like metric to identify F M˜ ⊕ Q M˜ = T M. Then for 1 ≤ i ≤ n + D and h¯ 0 there is an asymptotic expansion ehi¯ (x 1 , . . . , ξ n+D )
∼x + i
∞
h¯ L eij1 ... jL (x 1 , . . . , x n+D , L)ξ j1 . . . ξ jL ,
L=1
where the matrix eij (x 1 , . . . , x n+D , 1) has the form 01 repeated indices.
∗ 1
. As usual, we sum over
˜ h¯ > 0 This implies that there exist open neighborhoods Uh¯ of the zero section in T M, small, such that ˜ − Uh¯ is increasing with h¯ 0 and h¯ >0 Uh¯ = T M, ˜ − (π , eh¯ )(ξ ) := (π (ξ ), eh¯ (ξ )) defines a local isomorphism Uh¯ → M˜ × M. For N = 1, 2, . . . define a submanifold Z N by the embedding ˜ M˜ × M˜ × · · · × M˜ → M˜ × M˜ × M˜ × M˜ × · · · × M˜ × M, (a1 , a2 , . . . , a N ) → (a N , a1 , a1 , a2 . . . , a N −1 , a N ),
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and for κ ∈ {0, 1} write (ικ π , eh¯ )(ξ ) := (ικ π (ξ ), eh¯ (ξ )), where ι0 is the identity. Then the submanifold
−1 N −N κi ˜ Z h¯ := (ι π , eh¯ ) Z N ⊆ Uh¯ × · · · × Uh¯ κ∈{0,1} N
i=1
N times
is endowed with a tangential orientation and a transversal metric by the local isomorphism that maps (ξ1 , . . . , ξ N ) to (eh¯ (ξ1 ), . . . , eh¯ (ξ N )). Now fix ϕ1 , . . . , ϕ N ∈ Hc0 (QM), N ≥ 1, and let h¯ > 0 be small. We define the partition function 1 ϕ1 Φ ⊗ · · · ⊗ ϕ N Φ, Z ϕ1 ,...,ϕ N ,h¯ := (2h¯ D ) N Z˜ h−N ¯
where denotes the natural integral of top degree tangential differential forms with compact support against the transversal volume on a tangentially oriented pseudoRiemannian foliation. Basic properties of tangential cohomology imply that the germ at h¯ = 0 of Z ϕ1 ,...,ϕ N ,h¯ does not depend on the three choices we made above, cp. Prop. 9.2 in [23]. ˜ such that Theorem 1. Let h¯ > 0 be small. Then there exists Cϕ1 ,...,ϕ N ,h¯ ∈ Hc0 (Q M) 1 Z ϕ1 ,...,ϕ N ,h¯ = Cϕ1 ,...,ϕ N ,h¯ Φ, 2h¯ D M˜ where we embed M˜ → T M˜ as the zero section. Proof. Observe that for small values of h¯ only those components of Z˜ h−N with κi ¯ even contribute to the partition function. Now in order to construct Cϕ1 ,...,ϕ N ,h¯ use the map ˜ C N : M˜ × · · · × M˜ → M˜ × M˜ × · · · × M˜ × M, (a0 , . . . , a N ) → (a0 , a1 , a1 , a2 , . . . , a N −1 , a N ), to pull back the form 1 (2h¯ D ) N −1
κi even
N
κi
(ι π , eh¯ )
i=1
(ϕ1 Φ ⊗ · · · ⊗ ϕ N Φ). ∗
Then integrate over the fibers of (a0 , . . . , a N ) → (a0 , a N ) according to the rule ω0 ⊗ · · · ⊗ ω N = (−1)k(N −1)n ω0 ⊗ ω N ω1 ⊗ · · · ⊗ ω N −1 , ˜ M˜ M×···×
˜ M˜ M×···×
if ω0 is a k-form. Finally pull back with (π, eh¯ ) and integrate over the fibers of T M˜ → ˜ The theorem follows. Q M.
Denote by o : M˜ → F M˜ the zero section.
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Corollary 1. Let h¯ > 0 be small. Let X : M˜ → F M˜ be a section such that for each leaf the induced vector field is transversal to the zero field. Then 1 sgn(P) Cϕ1 ,...,ϕ N ,h¯ , Z ϕ1 ,...,ϕ N ,h¯ = 2h¯ D P P where the sum runs over the components of the zero locus of X and sgn(P) = ±1, depending on whether for a ∈ P the isomorphism ˜ im Fa o ⊕ im Fa X ∼ = Foa (F M) preserves or reverses orientation. In particular, Z ϕ1 ,...,ϕ N ,h¯ = 0 if the dimension of the leaves is odd. Proof. The equation follows from Theorem 1 by a tangential homotopy argument, cp. Theorem 8.1 in [23]. For the last assertion apply the formula to X and −X .
We have the following semiclassical limit. Theorem 2. For h¯ 0 there is an asymptotic expansion Cϕ1 ,...,ϕ N ,h¯ ∼
∞
h¯ L Cϕ1 ,...,ϕ N (L)
L=0
with Cϕ1 ,...,ϕ N (0)(ξ ) = ϕ1 ∗a · · · ∗a ϕ N (ξ ), where a = π (ξ ) and ∗a denotes the convolution in the fiber Qa M˜ containing ξ . Proof. The asymptotically relevant parts of the submanifold C N in the proof of Theorem 1 are isomorphic to an open neighborhood Vh¯ of the zero section in the N -fold ˜ For κi even, this is achieved by the maps Whitney sum T M˜ ⊕ · · · ⊕ T M. ˜ Vh¯ → M˜ × · · · × M, λ1 (ξ1 , . . . , ξ N ) → π (ξ1 ), ι eh¯ (ξ1 ), . . . , ιλ N −1 eh¯ (ξ N −1 ), eh¯ (ξ N ) , where λi ∈ {0, 1} is such that ιλi = ικi+1 . . . ικ N , 1 ≤ i ≤ N − 1. The result follows by a tangential homotopy argument and Lemma 1.
We can use this in Theorem 1 and Corollary 1 to obtain a semiclassical limit Z ϕ1 ,...,ϕ N ,h¯ ∼ h¯ −D
∞
h¯ L Z ϕ1 ,...,ϕ N (L),
L=0
h¯ 0, for the partition function. ˜ let eh¯ ,a be the restriction of eh¯ to the fiber Ta M˜ over a. Define Remark 2. For a ∈ M, Jh¯ ∈ H 0 (Uh¯ ) by Jh¯ (ξ ) := h¯ −D | det Qξ eh¯ ,π (ξ ) |. To apply the thermodynamic formalism [10,29] it will be of interest to replace ϕi by eβ A ϕi /Jh¯s , where β, s ∈ C and A ∈ H 0 (QM). Everything works similar in that case.
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Remark 3. The idea to interpret the parameter of the geodesic flow as the Planck constant is implicit in several approaches to quantize Riemannian manifolds, cf. e.g. [20], Sect. II.5 in [9,26]. The relevance of broken geodesics for the path integral on a Riemannian manifold is investigated in [2], cf. also [32,33]. This motivated our definition of the partition function. Asymptotic expansions appear frequently in Riemannian geometry and mathematical physics. Depending on the context, the asymptotic parameter stands for the Planck constant [19,25,8], the radius [18,16], the time [14] or the noise strength [11–13]. For an approach to a relativistic treatment confer [7,6]. 3. Transfer Operators Let ϕ ∈ Hc0 (QM) be given and let h¯ > 0 be small. Let αϕ,h¯ =
n
k k k,n−k ˜ ˜ αϕ, ( M × M), h¯ , αϕ,h¯ ∈ Ωc
k=0
be the decomposition with respect to bidegree of any representative of the class defined ˜ Integration over the second factor according to the by (π , eh¯ )∗ [ϕΦ] in Hcn( M˜ × M). rule M˜ ω1 ⊗ ω2 = ω1 M˜ ω2 defines the transfer operators k k ˜ k ˜ M˜ϕ, h¯ : Ωc ( M, C) → Ωc ( M, C), 1 + ι∗ k ω → (1 ⊗ ω) ∧ αϕ, h¯ , 2h¯ D M˜
˜ C) denotes complex valued tangential k-forms with compact support where Ωck ( M, ˜ on M, 0 ≤ k ≤ n. ∗ on the choice of α Remark 4. We suppress the dependence of M˜ϕ, ϕ,h¯ since we are h¯ interested in objects which are independent of it. Moreover, in the main applications there will be a canonical choice, cf. Sect. 5. Notice that M˜k induces a smoothing operator with compactly supported kernel on ϕ,h¯
˜ C), which is the completion of Ωck ( M, ˜ C) with respect to the Hilbert space L 2 Ωk ( M, the product ω, η := M˜ ω ∧ η. Here is the tangential star operator defined by the ˜ We have the trace formula metric on F M. Theorem 3. Let ϕ1 , . . . , ϕ N ∈ Hc0 (QM), N ≥ 1, be given and let h¯ > 0 be small. Then Z ϕ1 ,...,ϕ N ,h¯
n = (−1)k tr(M˜ϕk1 ,h¯ ◦ · · · ◦ M˜ϕkN ,h¯ ). k=0
Proof. We calculate Z ϕ1 ,...,ϕ N ,h¯ = = =
1 (2h¯ D ) N n k=0 n
N
κ
1 (2h¯ D ) N
ZN
κ
i=1
ZN
⎤ ⎡ n n k (ικi × 1) ⎣ αϕ11 ,h¯ ⊗ · · · ⊗ αϕk NN ,h¯ ⎦ ∗
k N =0
(ικ1 × 1)∗ αϕk 1 ,h¯ ⊗ · · · ⊗ (ικ N × 1)∗ αϕk N ,h¯
(−1)k tr(M˜ϕk1 ,h¯ ◦ · · · ◦ M˜ϕkN ,h¯ )
k=0
k1 =0
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and get the result. For the last equality use the fact that = (−1)k(n+1) on forms of degree n − k to rewrite integrals with the tangential star operator.
Let X : M˜ → F M˜ be a section such that for each leaf the induced vector field is transversal to the zero field. For the components P of the zero locus of X and for ϕ ∈ Hc0 (QM) define the smoothing operators Pϕ,h¯ : Cc∞ (P, C) → Cc∞ (P, C), 1 (1 ⊗ ω) · (π , eh¯ )∗ ϕ, ω → D h¯ P if h¯ > 0 is small. Note that by Lemma 1, for ω ∈ Cc∞ (P, C) and h¯ 0 there is an asymptotic expansion Pϕ,h¯ ω ∼
∞
h¯ L Pϕ (L)ω,
L=0
where Pϕ (L) is an L th order differential operator on P. Theorem 4. Let ϕ1 , . . . , ϕ N ∈ Hc0 (QM), N ≥ 1, be given and let h¯ > 0 be small. If X is a section as above then Z ϕ1 ,...,ϕ N ,h¯ =
1 sgn(P) tr(Pϕ1 ,h¯ ◦ · · · ◦ Pϕ N ,h¯ ). 2 P
Proof. Use Corollary 1 and find 1 Cϕ1 ,...,ϕ N ,h¯ = tr(Pϕ1 ,h¯ ◦ · · · ◦ Pϕ N ,h¯ ) h¯ D P by an argument similar to the proof of Theorem 3.
Remark 5. Close relatives of the transfer operators are the mean value operators of [17, 32] and the stochastic evolution operator in [11]. The literature on trace formulae is vast. A short introduction with focus on the semiclassical case is [34], many ideas can be found in [10]. 4. Zeta Function Fix ϕ ∈ Hc0 (QM) and let h¯ > 0 be small. If ϕ1 = · · · = ϕ N = ϕ, N ≥ 1, abbreviate N := Z Z ϕ, ϕ1 ,...,ϕ N ,h¯ and define the formal power series h¯ ζϕ,h¯ (z) := exp
∞ zN N Z . N ϕ,h¯
N =1
If there is a section X : M˜ → F M˜ such that for each leaf the induced vector field is transversal to the zero field, we get almost an Euler product for ζϕ,h¯ .
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Proposition 1. Let X : M˜ → F M˜ and the notation be as in Corollary 1. Let N0 ≥ 0 be given. Then for h¯ > 0 small, log ζϕ,h¯ (z) −
N0 zN 1 sgn(P) C N = O(z N0 +1 ) N P ϕ,h¯ 2h¯ D P N =1
N := C in the sense of formal power series, where Cϕ,
ϕ,...,ϕ,h¯ . h¯
Return to the general case and consider the formal power series ⎡ ⎤(−1)k n ⎢ ⎥ k , ∆ϕ,h¯ (z) := ⎣ (1 − zµϕ,h¯ )⎦ µkϕ,h¯
k=0
k counted with algebraic multiplicities. It is where µkϕ,h¯ runs over the spectrum of M˜ϕ, h¯ well known that ∆ϕ,h¯ (z) defines a function which is meromorphic on the whole complex plane, cf. [31,15]. We have a spectral representation for the zeta function.
Theorem 5. For small values h¯ > 0 we have the following equality of formal power series ζϕ,h¯ (z) · ∆ϕ,h¯ (z) = 1. In particular, the series ζϕ,h¯ defines a function which is meromorphic on the whole complex plane and Proposition 1 is valid as an asymptotic relation for z → 0. Proof. By definition − log ∆ϕ,h¯ (z) =
n ∞ zN k N µ (−1)k N ϕ,h¯ k µϕ,h¯ N =1
k=0
=
∞ N =1
The assertion follows from Theorem 3.
k N . (−1)k tr M˜ϕ, h¯
n zN
N
k=0
Write out the logarithmic derivative to see that the formula of Theorem 5 is equivalent to ∞ N =1
N z N Z ϕ, h¯ =
n k=0
(−1)k
µkϕ,h¯
zµkϕ,h¯ 1 − zµkϕ,h¯
.
N we conclude For the asymptotic behavior of the partition functions Z ϕ, h¯
Corollary 2. For all > 0 we have 1/N n N k k N (−1) µϕ,h¯ ≤ . lim sup Z ϕ,h¯ − N →∞ k k=0 |µϕ,h¯ |>
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Remark 6. The definition of ζϕ,h¯ is inspired by several examples of dynamical zeta functions, cf. [28,27,10]. Results similar to Theorem 5 go back to [21], cf. also [5] and recently [3,4]. 5. Some Examples Let M be compact. Let QM = Q+ M ⊕ Q− M be a foliated orthogonal decomposition, where the restriction of g to Q± M is positive/negative definite. Change the sign of the metric on Q− M to see that the tangentially harmonic representation of reduced ˜ If we define the transfer operators tangential cohomology, cf. [1], is valid for M and M. by tangentially harmonic representatives, they map to complex ι-invariant tangentially harmonic forms by the Künneth theorem, Theorem 2.1 of [23]. Example 1. Let ϕ ∈ Hc0 (QM) be given and let h¯ > 0 be small. If H ∗ (M) is finite dimensional then ζϕ,h¯ is rational. Example 2. Let n = 2, let M˜ be connected and assume that H 0 (M) = R. Use Poincaré duality, Theorem 2.3 of [23], on M˜ to conclude that if ϕ ∈ Hc0 (QM) is given and h¯ > 0 is small then ζϕ,h¯ is analytic except for at most one simple pole. Let π Q,∗ be integration ˜ Then if over the fibers of the projection π Q : Q M˜ → M. µ0ϕ,h¯ = π Q,∗ (ϕ Jh¯ ) = 0 the residuum of ζϕ,h¯ (z) at z = 1/µ0ϕ,h¯ is equal to −
1
µ0ϕ,h¯ 1 µϕ,h¯
(1 − µ1ϕ,h¯ /µ0ϕ,h¯ ).
Example 3. If M is Riemannian we have QM = Q− M. Obvious elements ϕ ∈ Hc0 (QM) are of the form ϕ(ξ ) = ψ (g(ξ, ξ )), ψ ∈ Cc∞ (R). Example 4. Let M be Lorentzian and let U be a foliated section of QM satisfying g(U, U ) ≡ 1. Then QM = R · U ⊕ Q− M, where Q− M is the orthogonal complement of U . Let c > 0, then obvious elements ϕc ∈ Hc0 (QM) are of the form ϕc (ξ ) = c · χ (c · g(ξ, U )) · ψ g(ξ − , ξ − ) , χ , ψ ∈ Cc∞ (R), where for ξ ∈ QM we write ξ − := ξ − g(ξ, U )U . In applications, the semiclassical limit h¯ 0 may be combined with the nonrelativistic limit c → ∞. Example 5. Let B be a compact manifold and let T be a compact pseudo Riemannian manifold. Let the pseudo-Riemannian foliation M be given by suspension of a homomorphism from the fundamental group of B to the isometry group of T . Let X be a vector field on B which is transversal to the zero field. Then X induces a section of F M˜ such that for each leaf the induced vector field is transversal to the zero field. Acknowledgement. The author thanks the referee for useful remarks to clarify the presentation.
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References 1. Álvarez López, J.A., Kordyukov, Y.A.: Long time behavior of leafwise heat flow for Riemannian foliations. Comp. Math. 125(2), 129–153 (2001) 2. Andersson, L., Driver, B.K.: Finite-dimensional approximations to Wiener measure and path integral formulas on manifolds. J. Funct. Anal. 165(2), 430–498 (1999) 3. Baillif, M.: Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions. Duke Math. J. 124(1), 145–175 (2004) 4. Baillif, M., Baladi, V.: Kneading determinants and spectra of transfer operators in higher dimensions: the isotropic case. Ergod. Th. Dynam. Syst. 25(5), 1437–1470 (2005) 5. Baladi, V., Ruelle, D.: Sharp determinants. Invent. Math. 123(3), 553–574 (1996) 6. Bolte, J.: Semiclassical expectation values for relativistic particles with spin 1/2. Found. Phys. 31(2), 423–444 (2001) 7. Bolte, J., Keppeler, S.: A semiclassical approach to the Dirac equation. Ann. Physics 274(1), 125–162 (1999) 8. Combescure, M., Ralston, J., Robert, D.: A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition. Commun Math. Phys. 202(2), 463–480 (1999) 9. Connes, A.: Noncommutative geometry. San Diego, CA: Academic Press Inc., 1994 10. Cvitanovi´c, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and Quantum. Copenhagen: Niels Bohr Institute, 2005. Available at http://ChaosBook.org 11. Cvitanovi´c, P.: Chaotic field theory: a sketch. Phys. A 288(1–4), 61–80 (2000) 12. Cvitanovi´c, P., Dettmann, C.P., Mainieri, R., Vattay, G.: Trace formulas for stochastic evolution operators: weak noise perturbation theory. J. Statist. Phys. 93(3–4), 981–999 (1998) 13. Cvitanovi´c, P., Dettmann, C.P., Mainieri, R., Vattay, G.: Trace formulae for stochastic evolution operators: smooth conjugation method. Nonlinearity 12(4), 939–953 (1999) 14. Gilkey, P.B.: Asymptotic formulae in spectral geometry. Studies in Advanced Mathematics. Boca Raton, FL: Chapman & Hall/CRC, 2004 15. Gohberg, I., Goldberg, S., Krupnik, N.: Traces and determinants of linear operators. Volume 116 of Operator Theory: Advances and Applications. Basel: Birkhäuser Verlag, 2000 16. Gray, A., Vanhecke, L.: Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142(3–4), 157–198 (1979) 17. Gray, A., Willmore, T.J.: Mean-value theorems for Riemannian manifolds. Proc. Roy. Soc. Edinburgh Sect. A 92(3–4), 343–364 (1982) 18. Gray, A.: The volume of a small geodesic ball of a Riemannian manifold. Michigan Math. J. 20, 329–344 (1974) 19. Gutzwiller, M.C.: Chaos in classical and quantum mechanics, volume 1 of Interdisciplinary Applied Mathematics. New York: Springer-Verlag, 1990 20. Liu, Z.J., Qian, M.: Gauge invariant quantization on Riemannian manifolds. Trans. Amer. Math. Soc. 331(1), 321–333 (1992) 21. Milnor, J., Thurston, W.: On iterated maps of the interval. In: Dynamical systems (College Park, MD, 1986–87), Volume 1342 of Lecture Notes in Math., Berlin: Springer, 1988, pp. 465–563 22. Mümken, B.: A coincidence formula for foliated manifolds. PhD thesis, Westfälische Wilhelms-Universität Münster, 2002 23. Mümken, B.: On tangential cohomology of Riemannian foliations. Amer. J. Math. 128(6), 1391– 1408 (2006) 24. O’Neill, B.: Semi-Riemannian geometry. Volume 103 of Pure and Applied Mathematics. New York: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1983 25. Paul, T., Uribe, A.: The semi-classical trace formula and propagation of wave packets. J. Funct. Anal. 132(1), 192–249 (1995) 26. Pflaum, M.J.: A deformation-theoretical approach to Weyl quantization on Riemannian manifolds. Lett. Math. Phys. 45(4), 277–294 (1998) 27. Pollicott, M.: Periodic orbits and zeta functions. In: Handbook of dynamical systems, Vol. 1A, Amsterdam: North-Holland, 2002, pp. 409–452 28. Ruelle, D.: Dynamical zeta functions and transfer operators. Notices Amer. Math. Soc. 49(8), 887–895 (2002) 29. Ruelle, D.: Thermodynamic formalism. Cambridge Mathematical Library. Cambridge: Cambridge University Press, second edition, 2004 30. Sachs, R.K., Wu, H.H.: General relativity for mathematicians. Graduate Texts in Mathematics, Vol. 48. New York: Springer-Verlag, 1977 31. Simon, B.: Trace ideals and their applications. Second ed., Volume 120 of Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc. 2005
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32. Sunada, T.: Spherical means and geodesic chains on a Riemannian manifold. Trans. Amer. Math. Soc. 267(2), 483–501 (1981) 33. Sunada, T.: Mean-value theorems and ergodicity of certain geodesic random walks. Comp. Math. 48(1), 129–137 (1983) 34. Uribe, A.: Trace formulae. In: First Summer School in Analysis and Mathematical Physics (Cuernavaca Morelos, 1998), Volume 260 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2000, pp. 61–90 Communicated by A. Connes
Commun. Math. Phys. 279, 595–636 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0428-2
Communications in
Mathematical Physics
Free Energy of a Dilute Bose Gas: Lower Bound Robert Seiringer Department of Physics, Jadwin Hall, Princeton University, P.O. Box 708, Princeton, NJ 08544, USA. E-mail: [email protected] Received: 26 November 2006 / Accepted: 20 August 2007 Published online: 22 February 2008 – © The Author 2008
Abstract: A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density and temperature T . In the dilute regime, i.e., when a 3 1, where a denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term 4πa(22 − [ − c ]2+ ). Here, c (T ) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and [ · ]+ = max{ · , 0} denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ∼ 2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one. 1. Introduction and Main Result The advance of experimental techniques for studying ultra-cold atomic gases has triggered numerous investigations on the properties of dilute quantum gases. From a mathematical point of view, several rigorous results have been obtained over the last few years. (See [8] for an overview.) The first of these, which has inspired much of the later work, was a study of the ground state energy of a Bose gas with repulsive interaction at low density . Per unit volume, it is given by e0 () = 4πa2 + o(2 )
for a 3 1
(1.1)
in three spatial dimensions. Here, a > 0 denotes the scattering length of the interparticle interaction, and units are chosen such that = 2m = 1, with m the mass of the particles. A lower bound on e0 () of the correct form (1.1) was proved by Lieb and Yngvason in Work partially supported by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship. © 2008 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes.
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[11]. Much earlier, Dyson [2] had already proved an upper bound of the desired form, at least in the special case of hard-sphere particles. An extension of his calculation to arbitrary repulsive interaction potentials was given in [9]. The methods introduced in [11] have been extended to treat the case of fermions as well, for the study of both the ground state energy [7] and the free energy at positive temperature [16]. We are concerned here with the extension of (1.1) to positive temperature, at least as far as a lower bound is concerned. That is, our goal is to derive a lower bound on the free energy of a dilute Bose gas at density and temperature T . Much of the complication in such an estimate is caused by the existence of a Bose-Einstein condensate for temperatures below some critical temperature. Although the existence of a condensate for interacting Bose gases has so far eluded a mathematical proof, its presence can easily be shown in the case of non-interacting particles. A short review of the Bose gas without interaction among the particles is given in Subsect. 1.2 below. One of the main ingredients in our estimate is a method to quantify correlations present in the state of the interacting system. This method has been introduced in [17]; it does not immediately apply below the critical temperature for Bose-Einstein condensation, however. We have been able to overcome this difficulty with the aid of coherent states.
1.1. Definition of the model. We consider a system of N bosons, confined to a threedimensional flat torus of side lengths L, which we denote by . The one-particle state space is thus L 2 (, d x), and the Hilbert space for the system is the symmetric N -fold tensor product H N = L 2sym ( N , d N x), i.e., the space of square integrable functions of N variables that are invariant under exchange of any pair of variables. The Hamiltonian is given as N HN = −i + v(d(xi , x j )). (1.2) i=1
1≤i< j≤N
Here, denotes the Laplacian on , and d(x, y) denotes the distance between points x and y on the torus . The particle interaction potential v : R+ → R+ ∪ {∞} is assumed to be a non-negative and measurable function. It is allowed to take the value +∞ on a set of positive measure, corresponding to hard sphere particles. In this case, the domain of the Hamiltonian has to be suitably restricted to functions that vanish on the set where the interaction potential is infinite. We assume that v has a finite range R0 , i.e., v(r ) = 0 for r > R0 . In particular, it has a finite scattering length, which we denote by a. We will recall the definition of a in Subsect. 1.3 below. We note that in a concrete realization of as the set [0, L]3 ⊂ R3 , is the Laplacian on [0, L]3 with periodic boundary conditions. Moreover, the distance d(x, y) is given as d(x, y) = mink∈Z3 |x − y − k L|. Note also that v(d(x, y)) = k∈Z3 v(|x − y − k L|) if L > 2R0 . The free energy (per unit volume) of the system at inverse temperature β = 1/T > 0 and density > 0 is given by f (β, ) = −
1 1 lim ln Tr H N exp (−β H N ) , β ||
(1.3)
where lim stands for the usual thermodynamic limit L → ∞, N → ∞ with = N /|| fixed. Here, we denote the volume of by || = L 3 . Existence of the thermodynamic limit in (1.3) can be shown by standard methods, see, e.g., [14,15].
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We are interested in a bound on f in the case of a dilute gas, meaning that a 3 is small. The dimensionless parameter β2/3 is of order one (or larger), however. Note that sometimes in the literature the case of small , but fixed a and β, is understood with the term “dilute”. This corresponds to a high-temperature (classical) limit and is not what we want to study here.
1.2. Ideal Bose gas. In the case of vanishing interaction potential (v ≡ 0), the free energy can be evaluated explicitly. It is given as
1 −β( p 2 −µ) . dp ln 1 − e f 0 (β, ) = sup µ + (2π )3 β R3 µ≤0
(1.4)
The supremum is uniquely attained at µ = µ0 (β, ) = d/d f (β, ) ≤ 0. If is bigger than the critical density c (β) ≡
1 1 = (4πβ)−3/2 dp −3/2 , 2 3 βp (2π ) R3 e −1 ≥1
(1.5)
the supremum is attained at µ0 = 0, whereas for < c (β), it is attained at some µ0 = µ0 (β, ) < 0. In particular, 1 1 = min{, c (β)}. dp 2 −µ ) 3 β( p 0 −1 (2π ) R3 e
(1.6)
Note also that the scaling relation f 0 (β, ) = 5/3 f 0 (β2/3 , 1) holds for an ideal Bose gas. In particular, the dimensionless quantity β2/3 is the only relevant parameter.
1.3. Scattering length. The scattering length of a potential v can be defined as follows (see Appendix A in [12], or Appendix C in [8]): For R ≥ R0 , 4πa = inf 1 − a/R
d x |∇φ(|x|)|2 + 21 v(|x|)φ(|x|)2 : |x|≤R φ : [0, R] → R+ , φ(R) = 1 .
(1.7)
For this definition to make sense, v need not necessarily be positive, one only has to assume that − + 21 v (as an operator on L 2 (R3 )) does not have any negative spectrum. We will restrict our attention to non-negative v, however. The infimum in (1.7) is attained uniquely. Moreover, the minimizer has a trivial dependence on R: for some function φv (independent of R) it can be written as φv (|x|)/φv (R). Note that a is independent of R, and also that φv (|x|) = 1 − a/|x| for |x| ≥ R0 .
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1.4. Main theorem. Our main result is the following lower bound on the free energy, defined in (1.3). It gives a bound on the leading order correction, compared with a non-interacting gas, in the case of small a 3 and fixed β2/3 . Theorem 1 (Lower bound on free energy of dilute Bose gas). There is a function C : R+ → R+ , uniformly bounded on bounded subsets of R+ , and an α > 0 such that f (β, ) ≥ f 0 (β, ) + 4πa 22 − [ − c (β)]2+ (1 − o(1)) , (1.8) with
o(1) ≤ C (β2/3 )−1 (a1/3 )α .
(1.9)
Here, [ · ]+ = max{ ·, 0} denotes the positive part. In the case of non-interacting particles, the expression [ − c (β)]+ is just the condensate density. Remarks. 1. Since C(t) is uniformly bounded for bounded t, our estimate is uniform in the parameter (β2/3 )−1 as long as it stays bounded. I.e., our result is uniform in the temperature for temperatures not much greater than the critical temperature (for the non-interacting gas). In particular, we recover the result (1.1) in the zero temperature limit. The error term is worse, however; in [11], it was shown that the exponent α can be taken to be α = 3/17 at T = 0, whereas our proof shows that α can be chosen slightly larger than 0.00087 (independent of T ). This value has no physical significance, however, it merely reflects the multitude of estimates needed to arrive at our result. 2. The error term, o(1), in our lower bound depends on the interaction potential v, besides its scattering length a, only through its range R0 . This dependence could in principle be displayed explicitly. By cutting off the potential in a suitable way, one can then extend the result to infinite range potentials (with finite scattering length). See Appendix B in [9] for details. 3. For ≤ c (β) (i.e., above the critical temperature), the leading order correction term is given by 8πa2 , compared with 4πa2 at zero temperature. The additional factor 2 is an exchange effect; heuristically speaking, it is a result of the symmetrization of the wave functions. This symmetrization only plays a role if the particles are in different one-particle states, which they are essentially always above the critical temperature. Below the critical temperature, however, a macroscopic number of particles occupies the zero-momentum state; there is no exchange effect among these particles, which explains the subtraction of the square of the condensate density in (1.8). 4. We note that f 0 (β, ) has a discontinuous third derivative with respect to at = c (β) or, equivalently, a discontinuous third derivative with respect to T = 1/β at the critical temperature. Since the specific heat cV (β, ) can be expressed in terms of the free energy as cV (β, ) = −T d 2 /(dT )2 f (β, ), it has a discontinuous derivative (with respect to T ) at the critical temperature. The first order correction term in (1.8) has a discontinuous second derivative at this value. Considering only this term and neglecting higher order corrections, this would mean that the specific heat is actually discontinuous at the critical temperature. 5. Our method applies also to particles with internal degrees of freedom, e.g., to particles with nonzero spin. For simplicity, we treat only the case of spinless particles here.
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6. Although we provide only a lower bound in this paper, one can expect that the second term in (1.8) gives the correct leading order correction to the free energy (see, e.g., [4, Chap. 12.4]). To prove this, one has to derive an appropriate upper bound on f (β, ), which has not yet been achieved, however. We note that a naive upper bound using first order perturbation theory yields (1.8) with 4πa replaced by 1 2 d x v(|x|), which need not be finite, however (and is always strictly greater than 4πa). The remainder of this paper is devoted to the proof of Theorem 1. The proof is quite lengthy and is split into several subsections. To guide the reader, we start every subsection with a short summary of what will be accomplished. 2. Proof of Theorem 1 In the following, we find it convenient to think of as the set [0, L]3 embedded in R3 . We will also assume L to be large. In particular, L > 2R0 , but L will also be assumed to be large compared with several other parameters (which are independent of L) appearing below. This is justified since we are only interested in quantities in the thermodynamic limit L → ∞. In many places in our proof, the Heaviside step function θ will appear. We point out that we use the convention that θ equals 1 at the origin, i.e., θ (t) = 0 for t < 0, and θ (t) = 1 for t ≥ 0. 2.1. Reduction to integrable potentials. Recall that we do not want to restrict our attention to interaction potentials that are integrable. For the Fock space treatment in the next subsection, it will be necessary that v has finite Fourier coefficients, however. As a first step, we will therefore replace the interaction potential v by a smaller potential v whose integral is bounded by some number 8π ϕ. The scattering length of the new potential will be smaller than a, however. In the following lemma, we show that as long as ϕ is much greater than a, the change in the scattering length remains small. Lemma 1. Let v : R+ → R+ ∪ {∞} have finite scattering ∞ length a. For any ε > 0, there exists a v , with 0 ≤ v (r ) ≤ v(r ) for all r , such that 0 dr r 2 v (r ) ≤ 2ϕ, and such that the scattering length of v , denoted by a , satisfies
(2.1.1) a ≥ a 1 − a/ϕ (1 − ε) . Without loss of generality, we may assume that ϕ > a. Let R = inf{s : Proof. ∞ 2 ) < ∞}. We note that R is finite; in fact R ≤ a. This follows from the fact s dr r v(r ∞ that 2a ≥ 0 dr r 2 v(r )|φv (r )|2 , where φv denotes the minimizer of (1.7) (for R = ∞), as introduced in Subsect. 1.3. Since it satisfies φv (r ) ≥ 1 − a/r (see Appendix B in ∞ [12]), s dr r 2 v(r ) is finite for s > a. ∞ ∞ Assume first that R dr r 2 v(r ) ≥ 2ϕ. The function s → s dr r 2 v(r ) is con∞ tinuous for s > R. We can thus choose s ≥ R such that s dr r 2 v(r ) = 2ϕ, and v (r ) = v(r )θ (r − s). To obtain an upper bound on a, we can use a trial function φ(r ) = (φ v (r ) − φ v (s)s/r )θ (r − s) in the variational principle (1.7). We note that φ is a non-negative function, since φ v (r ) is monotone increasing in r [12]. By partial integration, using
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1 the variational equation −φ v (|x|)φ v (|x|) + 2 v (|x|) = 0, we have d x |∇φ(|x|)|2 + 21 v(|x|)|φ(|x|)|2 4πa ≤ R3 ∞ s s = 4π ( a + sφ dr r 2 v(r ) φ v (s)) + 2π φ v (s) v (s) − φ v (r ) . (2.1.2) r r s The last term is negative and can be dropped for an upper bound. To obtain an upper a /s, and hence bound on sφ v (s), we note that φ v (s) ≥ 1 − a . (2.1.3) sφ v (s) ≤ 1/φ v (s) − 1 For an upper bound on φ v (s), we use again the monotonicity of φ v (r ), which allows us to estimate ∞ 2 2 a ≥ a ≥ 21 dr r 2 v(r )φ (2.1.4) v (r ) ≥ φ v (s) ϕ. s √ This yields φ v (s) ≤ a/ϕ. Altogether, we have thus shown that 1 . (2.1.5) a ≤ a + sφ a 1+ √ v (s) ≤ ϕ/a − 1 ∞ This proves (2.1.1) (with ε = 0) under the assumption that R dr r 2 v(r ) ≥ 2ϕ. ∞ Consider now the case when R dr r 2 v(r ) = 2ϕ − T for some T > 0. If R = 0, we can take v = v, and there is nothing to prove. Hence we can assume that R > 0. By R definition, we have that R(1−ε) dr r 2 v(r ) = ∞ for any ε > 0. Hence there exists a τ R (depending on T and ε) such that R(1−ε) dr r 2 min{v(r ), τ } = T . We can then take ⎧ for r ≥ R ⎨ v(r ) v (r ) = min{v(r ), τ } for (1 − ε)R ≤ r < R (2.1.6) ⎩0 otherwise.
Applying the same argument as a + Rφ v (R). Now √ in (2.1.2), with s = R, we have a ≤ R ≤ a, and φ a/ϕ using the same argument as in (2.1.4), noting that v (R(1 − ε)) ≤ v (r ) = 0 for r ≤ R(1 − ε). Moreover, since |∇φ a /|x|2 , as shown in [9, v (|x|)| ≤ −1 Eq. (3.33)], |φ a R /(1 − ε), and thus v (R(1 − ε)) − φ v (R)| ≤ ε a 1 +a . (2.1.7) a ≤ a 1−ε ϕ This finishes the proof of the lemma.
As an example, consider the case of a pure hard sphere interaction, i.e., v(r ) = ∞ for r ≤ a, and v(r ) = 0 for r > a. In this case, we √ can choose v (r√ ) = 6ϕa −3 θ (a − r ). The scattering length of v is given by a = a(1 − a/(6ϕ) tanh 6ϕ/a) in this case. √ Note that tanh t ≤ 1 for all t. In particular, a ≥ a(1 − a/(6ϕ)). N , with For a lower bound, we can simply replace v by v , i.e, we have H N ≥ H N = H
N i=1
−i +
1≤i< j≤N
v (d(xi , x j )).
(2.1.8)
√ √ a is of the order a/ϕ. We If we choose ε ≤ a/ϕ, the error in the scattering length will choose ϕ a below.
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2.2. Fock space. In the following, it will be convenient to give up the restriction on the particle number and work in Fock space instead. This has the advantage that the problem of condensation can be dealt with with the aid of coherent states, which will be introduced in the next subsection. Our treatment differs slightly from the usual grand canonical ensemble since we do not simply introduce a chemical potential as Lagrange multiplier to control the number of particles, but we add a quadratic expression in N to the Hamiltonian. This gives a bitter control on the particle number. Let µ0 ≤ 0 be the chemical potential of the ideal Bose gas, which is the quantity that maximizes the expression in (1.4). Let F = N H N be the bosonic Fock space over L 2 (). Let a †p and a p denote the usual creation and annihilation operators of plane waves in with wave functions L −3/2 e−i px . We define a Hamiltonian H on Fock space as H = T + V + K + µ0 N , (2.2.1) with T=
p 2 − µ0 a †p a p , V =
p
1 † † v ( p)ak+ p al− p ak al , 2||
(2.2.2)
p,k,l
and K = 4π a
C (N − N )2 . ||
(2.2.3)
3 Here and in the following, all sums are over p ∈ 2π L Z . The Fourier transform of −i px v (|x|)e . It is uniformly bounded; in fact, v is denoted by v , i.e., v ( p) = d x | v ( p)| ≤ v (0) ≤ 8π ϕ, where ϕ was introduced in the previous subsection. The number operator p a †p a p is denoted by N, whereas N is just a parameter. The parameter C is positive and will be chosen later on. The Hamiltonian H commutes with the number operator N, and can be thought of as a direct sum of its restrictions to definite particle number. Note that the restriction to N , i.e, H = H N on the sector of particle number N . This particle number N is just H implies, in particular, that N ≤ Tr F exp (−βH) . Tr H N exp −β H (2.2.4)
We will proceed deriving an upper bound on the latter expression.
2.3. Coherent states. To obtain an upper bound on the partition function Tr F exp(−βH), we use the method of coherent states [10]. Effectively, this replaces the operators a †p and a p by numbers. This can be viewed as a rigorous version of part of the Bogoliubov approximation, where one replaces the operators a0† and a0 by numbers. Such a replacement is particularly useful if the zero-mode is “macroscopically occupied”, i.e, if a0† a0 ∼ ||. We will use this method not only for p = 0, however, but for a whole range of momenta | p| < pc for some pc ≥ 0. Although not macroscopic, their occupation will be large enough to require this separate treatment. To be more precise, let us pick some pc ≥ 0 and write F = F< ⊗ F> , where F< and F> denote the Fock spaces corresponding to the modes | p| < pc and | p| ≥ pc ,
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respectively. Let M denote the number of p ∈ the Berezin-Lieb inequality [1,5] implies that Tr F exp(−βH) ≤
CM
2π 3 L Z
with | p| < pc . As shown in [10],
d Mz Tr F> exp −βHs (z ) .
(2.3.1)
M Here, z denotes the vector (z 1 , . . . , z M ) ∈ C M , d Mz = i=1 dz i and dz = π −1 d x d y s with x = (z), y = (z). Moreover, H (z ) is the upper symbol of the operator H. It is an operator on F> , parametrized by z , and can be written in the following way. Let |z ∈ F< denote the coherent state |z = exp
† | p|< pc z p a p
− z ∗p a p |0 ≡ U (z )|0,
(2.3.2)
with |0 the vacuum in the Fock space F< . Then the lower symbol of H is given by Hs (z ) = z |H|z . Since a p |z = z p |z , the lower symbol is obtained from the expression (2.2.1) by simply replacing all the a p by z p and the a †p by z ∗p for all | p| < pc . The upper symbol can be obtained from the lower symbol by replacing |z p |2 by |z p |2 − 1, for instance, and similarly with other polynomials in z p ; see, e.g., [10] for details. We can then write Hs (z ) in the following way. Denoting by Ns (z ) = |z |2 + | p|≥ pc a †p a p the lower symbol of the number operator, we have Hs (z ) = Hs (z ) − H(z ),
(2.3.3)
with H(z ) =
p 2 − µ0
| p|< pc
+2
|l|< pc , |k|≥ pc
⎡ 1 ⎣ + v (0) 2MNs (z ) − M 2 2||
v (l − k)ak† ak +
⎤ v (l − k) 2|z k |2 − 1 ⎦
|l|< pc , |k|< pc
4π aC 2|z |2 + M(2Ns (z ) − 2N − M) . + ||
(2.3.4)
Here, we have used that v ( p) = v (− p). Since v is a non-negative function, | v ( p)| ≤ v (0) ≤ 8π ϕ for all p. Hence we obtain the bound 16π ϕ 8π aC 2 MNs (z ) + |z | + M(Ns (z ) − N ) . H(z ) ≤ M pc2 − µ0 + || ||
(2.3.5)
(Here, we have used again the positivity of v .) Denoting by Ks (z ) = z |K|z the lower symbol of K (and, similarly, for T and V below), we have Ks (z ) =
4π aC 4π aC (Ns (z ) − N )2 + |z |2 ≥ (Ns (z ) − N )2 . || ||
(2.3.6)
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We can use part of Ks (z ) to estimate −H from below independently of z . More precisely, we have 1 z ) − H(z ) 2 Ks (
8π N 2ϕ 2 (M + 1)2 1+ ≥ −M pc2 − µ0 − (2ϕ M + a C) − 32π aC || || aC ≡ −Z (1) .
(2.3.7)
Note that M ∼ pc3 || in the thermodynamic limit. We will choose the parameters pc , ϕ and C such that Z (1) ||a2 for small a1/3 . With the definition 1 Fz (β) ≡ − ln Tr F> exp −β(Ts (z ) + Vs (z ) + 21 Ks (z )) , (2.3.8) β (2.3.1) and the estimates above imply that
1 1 ln Tr F exp(−βH) ≥ µ0 N − ln d Mz exp −β Fz (β) − Z (1) . (2.3.9) β β CM Hence it remains to derive a lower bound on Fz (β). Let z denote the Gibbs state of Ts (z ) + Vs (z ) + 21 Ks (z ) on F> , for inverse temperature β. Let 0 = |00| denote the vacuum state in F< . Denoting by ϒ z the state ϒ z ≡ U (z )0 U (z )† ⊗ z on the full Fock space F, we can write 1 Fz (β) = Tr F T + V + 21 K ϒ z − S(ϒ z ). (2.3.10) β −
Here, S() = −Tr F ln denotes the von-Neumann entropy. 2.4. Relative entropy and a priori bounds. In the following, we want to derive a lower bound on Fz (β). Although we do not have an upper bound available, we can assume an appropriate upper bound without loss of generality; if the assumption is not satisfied, there is nothing to prove (as far as a lower bound in concerned). This upper bound can be formulated as a bound on the relative entropy between the state ϒ z = U (z )0 U (z )† ⊗ z defined above and a simple reference state (describing noninteracting particles). Together with a bound on the total number of particles, this estimate on the relative entropy contains all the information we need in order to prove the desired properties of the state ϒ z that will allow us to derive a lower bound on (2.3.10). We note that, for any state of the form = U (z )0 U (z )† ⊗ > for some state > on F> , 1 1 Tr F [T] − S() = Tr F> Ts (z ) > − S( > ) β β 1 ≥ − ln Tr F> exp (−βTs (z )) . (2.4.1) β In fact, the difference between the right and left sides of (2.4.1) is given by β −1 S(, 0z ), where S denotes the relative entropy. For two general states and on Fock space, it is given by S(, ) = Tr F ln − ln . (2.4.2)
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R. Seiringer
Note that the relative entropy is a non-negative quantity. The state 0z is given by 0z = U (z )0 U (z )† ⊗ 0 , where 0 is the Gibbs state of Ts (z ) on F> (which is independent of z ). For = ϒ z , we have (2.4.3) S(ϒ z , 0z ) = Tr F> z ln z − ln 0 = S(z , 0 ). From these considerations, together with the positivity of V, we conclude that (2.3.10) is bounded from below by 1 1 Fz (β) ≥ − ln Tr F> exp (−βTs (z )) + 21 Tr F Kϒ z + S(z , 0 ). (2.4.4) β β Hence we can distinguish the following two cases: A) The following lower bound on Fz (β) holds: Fz (β) ≥ −
1 ln Tr F> exp (−βTs (z )) + 8π || a 2 . β
(2.4.5)
B) Inequality (2.4.5) is false, in which case and
a β2 S(z , 0 ) ≤ 8π ||
(2.4.6)
a 2 . Tr F Kϒ z ≤ 16π ||
(2.4.7)
From now on, we will consider Case B, i.e., we will assume (2.4.6) and (2.4.7) to hold. The lower bound we will derive on Fz (β) below will actually be worse than the bound (2.4.5) above; i.e., the bound in Case B holds in any case, irrespective of whether the assumptions (2.4.6) and (2.4.7) actually hold. Although the relative entropy does not define a metric, it measures the difference between two states in a certain sense. In particular, it dominates the trace norm [13, Thm. 1.15]: S(, ) ≥ 21 − 1 . (2.4.8) This inequality is a special case of the fact that the relative entropy decreases under completely positive trace-preserving (CPT) maps. In fact, inequality (2.4.8) can be obtained using monotonicity under the CPT map → Tr F [P] ⊕ Tr F [(1 − P)], where P is the projection onto the subspace where − ≥ 0. Although we have the upper bound (2.4.6) on the relative entropy, inequality (2.4.8) is of no use for us since the relative entropy is of the order of the volume of the system, while the right side of (2.4.8) never exceeds 2. To make use of (2.4.8), we must not look at the state on the full Fock space (over the whole volume) but rather on its restriction to a small subvolume. We do this in Subsect. 2.8 below. Again, the monotonicity of the relative entropy will be used in an essential way. We note that (2.4.7) implies the following simple upper bound on |z |2 . From (2.2.3) and (2.4.7), 1/2 2 |z |2 − N ≤ Tr F (N − N )ϒ z ≤ Tr F (N − N )2 ϒ z ≤ √ ||, (2.4.9) C and hence 2 |z |2 . (2.4.10) ≤ 1+ √ z ≡ || C We will choose C 1 below.
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2.5. Replacing vacuum. In the following, we want to derive a lower bound on the expectation value of the interaction energy V in the state ϒ z , i.e., on (2.5.1) Tr F Vϒ z = Tr F> Vs (z )z . For reasons that will be explained later (see Subsect. 2.13), we find it necessary to replace the vacuum 0 on F< in the definition of the state ϒ z = U (z )0 U (z )† ⊗ z by a more general quasi-free state. In this subsection, we show that such a replacement can be accomplished without significant errors. Let denote a (particle-number conserving) quasi free state on F< . It is completely determined by its one-particle density matrix, which we choose to be given as π= π p | p p|. (2.5.2) | p|< pc
Here, | p ∈ L 2 () denotes a plane wave of momentum p. We denote the trace of π by P = | p|< pc π p . Let ϒπz denote the state ϒπz ≡ U (z )U (z )† ⊗ z on F. We want to derive an upper bound on the difference (2.5.3) Tr F V ϒπz − ϒ z = Tr F V U (z ) ( − 0 ) U (z )† ⊗ z . A simple calculation yields 1 v (0) P 2 + 2P Tr F> Ns (z )z − 2 |k|< pc πk |z k |2 2|| 1 v (k − l) πk πl + 2|z k |2 πl + 2|| |k|< pc , |l|< pc 1 + v (k − l)πk Tr F> al† al z || |k|< pc , |l|≥ pc 8π ϕ P 2 + 2P Tr F Nϒ z . ≤ ||
(2.5.3) =
(2.5.4)
Here we have used again that | v (0) ≤ 8π √ ϕ. It follows easily from (2.4.7) v (k)| ≤ (compare with (2.4.9)) that Tr F Nϒ z ≤ N (1 + 2/ C). Hence we obtain from (2.5.4) that (2.5.5) Tr F Vϒ z ≥ Tr F Vϒπz − Z (2) , with Z
(2)
2 8π ϕ P 2 16π Pϕ . + N 1+ √ = || || C
(2.5.6)
Recall that C 1 and ϕ a. Hence Z (2) ||a2 as long as ϕ P a N . Note that the effect of the replacement of ϒ z by ϒπz on the kinetic energy is p 2 − µ0 π p . Tr F Tϒ z = Tr F Tϒπz − (2.5.7) | p|< pc
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R. Seiringer
We have thus obtained the lower bound 1 Fz (β) ≥ Tr F (T + V) ϒπz − S(ϒ z ) β 2 p − µ0 π p + 21 Tr F Kϒ z − Z (2) . −
(2.5.8)
| p|< pc
2.6. Dyson Lemma. Since the interaction potentials in V are very short range and strong (compared with the average kinetic energy per particle), we cannot directly obtain information on the expectation value of V in the state ϒπz . In fact, we cannot even expect that it yields the desired correction to the free energy, since part of the interaction energy leading to the second term in (1.8) is actually kinetic energy! Hence we will first derive a lower bound on V in terms of “softer” and longer ranged potentials, with the aid of part of the kinetic energy. More precisely, we will use only the high momentum part of the kinetic energy for this task, since this is the relevant part contributing to the interaction energy. The appropriate lemma to achieve this was derived in [7]; part of the idea for such an estimate is already contained in the paper by Dyson [2]. For this reason, we refer to this estimate as “Dyson Lemma”. Our goal is to derive an appropriate lower bound on the Hamiltonian T + V. Let χ : R3 → R be a radial function, 0 ≤ χ ( p) ≤ 1, and let h(x) =
1 (1 − χ ( p)) e−i px . || p
(2.6.1)
We assume that χ ( p) → 1 as | p| → ∞ sufficiently fast such that h ∈ L 1 () ∩ L ∞ (). For some L/2 > R > R0 , let f R (x) = sup |h(x − y) − h(x)|,
(2.6.2)
|y|≤R
and 2 w R (x) = 2 f R (x) π
dy f R (y).
(2.6.3)
Note that w R is a periodic function on R3 , with period L. Let U R : R+ → R+ be a non-negative function that is supported in the inter∞ val [R0 , R], and satisfies 0 dt t 2 U R (t) ≤ 1. The following is a simple extension of Lemma 4 (and Corollary 1) in [7]. The proof follows closely the one in [7]. For completeness, we present it in the Appendix. Lemma 2. Let y1 , . . . , yn denote n points in and, for x ∈ , let yNN (x) denote the nearest neighbor of x among the points y j , i.e., the yk minimizing d(x, y j ) among all y j . We then have, for any ε > 0, − ∇χ ( p)2 ∇ +
1 2
n
v (d(x, yi )) ≥ (1 − ε) a U R (d(x, yNN (x))) −
i=1
Here, the operator −∇χ ( p)2 ∇ stands for
n a i=1
p
p 2 χ ( p)2 | p p|.
ε
w R (x − yi ). (2.6.4)
Free Energy of a Dilute Bose Gas: Lower Bound
607
We note that yNN (x) is well defined except on a set of measure zero. Compared with Lemma 4 in [7], the main differences are the boundary conditions used, and the fact that we do not demand a minimal distance between the points yi . In [7], it was assumed that d(yi , y j ) ≥ 2R for i = j, inwhich case U R (d(x, yNN (x))) = i U R (d(x, y j )). Note also that only the inequality dt t 2 U (t) ≤ 1 is needed for the estimate, not equality, as stated in [7]. We will use Lemma 2 for a lower bound on the operator T + V on F. Note that the n , defined in (2.1.8). We restriction of this operator to the sector of n particles is just H write ⎤ ⎡ n n = ⎣− j + 1 H v (d(x j , xi ))⎦ , (2.6.5) 2 j=1
i = j
and apply the estimate (2.6.4) to each term in square brackets, for fixed j and fixed positions of the xi , i = j. We want to keep a part of the kinetic energy for later use, however. To this end, we pick some 0 < κ < 1, and write p 2 = p 2 (1 − (1 − κ)χ ( p)2 ) + (1 − κ) p 2 χ ( p)2 .
(2.6.6)
We split the kinetic energy in the Hamiltonian (2.6.5) accordingly, and apply (2.6.4) to the last part. Using also the positivity of the v , we thus obtain, for any subset J j ⊆ {1, . . . , j − 1, j + 1, . . . , n}, − j + 21 v (d(x j , xi )) i = j
≥ −∇ j (1 − (1 − κ)χ ( p j )2 )∇ j J
j +(1 − ε)(1 − κ) a U R (d(x j , xNN (x j ))) −
a w R (x j − xi ). ε
(2.6.7)
i∈J j
J
j (x j ) the nearest neighbor of x j among the points xi , i ∈ J j . Here we denoted by xNN Our choice of J j will depend on the positions xi , i = j. We want to choose it in such a way that d(xl , xk ) ≥ R/5 if l ∈ J j and k ∈ J j . Moreover, we want the set to be maximal, in the sense that if l ∈ J j , then there exists a k ∈ J j such that d(xl , xk ) < R/5. These properties of J j will be used in an essential way in Subsects. 2.9 and 2.10 below. There is no unique choice of J j satisfying these criteria. One way to construct it is the following. We first pick all i corresponding to those xi whose distance to the nearest neighbor (among all the other xk , k = i, j) is greater than or equal to R/5. Secondly, going through the list {x1 , . . . , x j−1 , x j+1 , . . . , xn } one by one, we add i to the list if d(xi , x j ) ≥ R/5 for all j already in the list. This last procedure depends on the ordering of the xi , and hence the resulting J j will depend on this ordering. The resulting interaction potential in (2.6.7) will thus be not symmetric in the particle coordinates. This is of no importance, however, since we will take the expectation value of the resulting operator only in symmetric (bosonic) states anyway. The set J j is chosen in order to satisfy the following requirements. On the one hand, we want the particles to keep a certain minimal distance, R/5; this is necessary in order to control the error terms coming from the potentials w R . We do not have sufficient control on the two-particle density to control these terms if all the particle configurations were taken into account. On the other hand, we want the balls of radius R centered at the particle coordinates to be able to overlap sufficiently much, such that the desired
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R. Seiringer
lower bound can be obtained. We note that we want to derive a lower bound which is independent of z ; for certain values of z , however, the system may be far from being homogeneous and particles may cluster in a relatively small volume. We want to ensure that there is still sufficient interaction among them. 2.7. Filling the holes. One defect of Lemma 2 above is that the resulting interaction potential U R is supported outside a ball of radius R0 , which is the range of v . For our estimates in Subsect. 2.9, it will be convenient to have a specific U R which is, in particular, positive definite and hence should not have a “hole” at the origin. We will show in this subsection that one can easily add the missing part to U R , at the expense of only a small amount of kinetic energy. We start with the description of our choice of U R . Let j : R+ → R+ denote the “hat function” 144 dy θ ( 21 − |y|)θ ( 21 − |y − et|) (2.7.1) j (t) = π R3 for some unit vector e ∈ R3 . Note that j is supported in the interval [0, 1], and 1 2 0 dt t j (t) = 1. An explicit computation yields j (t) = 12(t + 2)[1 − t]2+ .
(2.7.2)
R (t) = R −3 j (t/R). We will thus choose Our desired interaction potential will be U U R (t) = U R (t)θ (t − R0 ) in (2.6.4). R instead of U R . I.e., we would In the following, it will be convenient to work with U R ( · )θ (R0 − · ) to the interaction. In order to achieve this, like the add the missing part U we use the following lemma. It is an easy consequence of the definition of the scattering length, given in (1.7). Lemma 3. Let y1 , . . . , yn denote n points in , with d(yi , y j ) ≥ R/5 for i = j. Let 0 ≤ λ < π/2, and R0 < R/10. Then n λ2 θ (R0 − d(x, yi )) R02 i=1 n tan λ 3R0 −1 ≥− θ (R/10 − d(x, yi )). λ (R/10)3 − R03 i=1
− −
Proof. It suffices to prove that λ2 |∇φ(x)|2 − 2 |φ(x)|2 R0 |x|≤R0 |x|≤R/10 tan λ 3R0 − 1 ≥− |φ(x)|2 λ (R/10)3 − R03 |x|≤R/10
(2.7.3)
(2.7.4)
for any function φ ∈ H 1 . In fact, it is enough to prove (2.7.4) for radial functions. Note that the scattering length of the potential 2λ2 R0−2 θ (R0 − · ) is given by R0 (1−λ−1 tan λ). Hence, for any R0 ≤ s ≤ R/10, tan λ λ2 2 2 |∇φ(|x|)| − 2 |φ(|x|)| ≥ −4π R0 − 1 |φ(s)|2 . (2.7.5) λ R0 |x|≤R0 |x|≤s
Free Energy of a Dilute Bose Gas: Lower Bound
609
Equation (2.7.4) follows by multiplying this inequality by s 2 and integrating s between R0 and R/10. Let λ = π/4 for concreteness. Recall that d(xi , xk ) ≥ R/5 for i, k ∈ J j . Since R (t) ≥ j (1/10)/R 3 for t ≤ R/10, this lemma R (t) ≤ j (0)/R 3 = 24/R 3 , and U U implies, in particular, that 24 (4R0 )2 j π 2 R3 R03 1 18 R (d(x j , x J j (x j ))). + − π U (4 ) NN 3 3 3 (π/4) (R/10) − R0 j (1/10)
R − U R )(d(x j , x j (x j ))) ≤ − (U NN J
Define
R03 1 18 a ≡ a (1 − ε)(1 − κ) 1 − − π (4 ) 3 3 (π/4)3 j (1/10) (R/10) − R0 and κ ≡ κ −
(2.7.6)
24 a (4R0 )2 . 2 π R3
(2.7.7)
(2.7.8)
In the following, we will choose κ a R02 /R 3 and hence, in particular, κ > 0. Combining the estimates (2.6.7) and (2.7.6) and applying them in each sector of particle number n, we obtain the inequality T + V ≥ Tc + W, where Tc =
ε( p)a †p a p , ε( p) = κ p 2 + (1 − κ) p 2 (1 − χ ( p)2 ) − µ0 ,
(2.7.9)
(2.7.10)
p
and W is, in each sector with particle number n, given by the (symmetrization of the) multiplication operator ⎤ ⎡ n a J R (d(x j , x j (x j ))) − ⎣a U w R (x j − xi )⎦ . (2.7.11) NN ε j=1
i∈J j
Note again that the set J j depends on all the particle coordinates xi , i = j. We now describe our choice of the kinetic energy cutoff χ . Let ν : R3 → R+ be a smooth radial function with ν( p) = 0 for | p| ≤ 1, ν( p) = 1 for | p| ≥ 2, and 0 ≤ ν( p) ≤ 1 in-between. For some s ≥ R we choose χ ( p) = ν(sp).
(2.7.12)
We will choose pc ≤ 1/s below. This implies, in particular, that ε( p), defined in (2.7.10) above, is equal to (1 − κ + κ ) p 2 − µ0 for | p| ≤ pc . Hence (compare with (2.5.7)) (1 − κ + κ ) p 2 − µ0 π p . Tr F Tc ϒπz = Tr F Tc ϒ z + (2.7.13) | p|< pc
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R. Seiringer
Using the fact that 1 1 Tr F Tc ϒ z − S(ϒ z ) ≥ − ln Tr F> exp −βTcs (z ) , β β
(2.7.14)
we conclude from (2.5.8), (2.7.9) and (2.7.13) that 1 ln Tr F> exp −βTcs (z ) + Tr F Wϒπz + 21 Tr F Kϒ z β −(κ − κ ) p 2 π p − Z (2) . (2.7.15)
Fz (β) ≥ −
| p|< pc
Note that the first term on the right side of (2.7.15) can be computed explicitly. It is given by 1 (1 − κ + κ ) p 2 − µ0 |z p |2 − ln Tr F> exp −βTcs (z ) = β | p|< pc
+
1 ln 1 − exp (−βε( p)) . β
(2.7.16)
| p|≥ pc
In the following, we will derive a lower bound on the expectation value of W in the state ϒπz . 2.8. Localization of relative entropy. Our next task is to give a lower bound on the term Tr F Wϒπz . For that matter, we will show that we can replace the unknown state z in ϒπz = U (z )U (z )† ⊗ z by the quasi-free state 0 , which is the Gibbs state for the kinetic energy Ts (z ). The error in doing so will be controlled by the upper bound on the relative entropy, Eq. (2.4.6). In order to do this, we have to obtain a “local” version of this bound. Consider the quasi-free state π = ⊗ 0 . Its one particle density matrix is given by 1 | p p| (2.8.1) ωπ = ( p) − 1 e p with ( p) = β( p 2 − µ0 ) for | p| ≥ pc , and ( p) = ln(1 + 1/π p ) for | p| < pc . Let η : R3 → R be a function with the following properties: • η ∈ C ∞ (R3 ). • η(0) = 1, and η(x) = 0 for |x| ≥ 1. • η( p) = d x η(x)e−i px ≥ 0 for all p ∈ R3 . An appropriate η can, for instance, we obtained by convolving a smooth function supported in a ball of radius 21 with itself. Given such a function η, we define ηb (x) = η(x/b) for some b ≤ L/2. Moreover, with a slight abuse of notation, we define a one-particle density matrix ωb on H by the kernel ωb (x, y) = ωπ (x, y)ηb (d(x, y)).
(2.8.2)
Note that this defines a positive operator, with plane waves as eigenstates. Note also that |ωb (x, y)| ≤ |ωπ (x, y)| since |ηb | ≤ 1. We denote by b the corresponding (particle
Free Energy of a Dilute Bose Gas: Lower Bound
611
number conserving) quasi-free state on F, and bz = U (z )b U (z )† . Let also denote ω = ωb (x, x) = ωπ (x, x) the one-particle density of b (which is independent of x). Abusing the notation even more, we shall sometimes write ωb (x, y) = ωb (x − y) if no confusion can arise. For r < L/2, let χr,ξ ( · ) = θ (r − d( ·, ξ )) denote the characteristic function of a ball of radius r centered at ξ ∈ . The function χr,ξ defines a projection on the one-particle space H = L 2 (), and hence the Fock space F over H can be thought of as a tensor product of a Fock space over χr,ξ H and a Fock space over the complement. States on F can thus be restricted to the Fock space over χr,ξ H, simply by taking the partial trace over the other factor. We denote such a restriction of a state by χr,ξ . For d(ξ, ζ ) ≥ 2r , χr,ξ + χr,ζ defines a projection on H. Note that since ωb (x, y) vanishes if d(x, y) ≥ b, we have that b,χr,ξ +χr,ζ = b,χr,ξ ⊗ b,χr,ζ
(2.8.3)
if d(ξ, ζ ) ≥ 2r + b. This follows simply from the fact that the one particle density matrix of b,χr,ξ +χr,ζ is given by (χr,ξ +χr,ζ )ωb (χr,ξ +χr,ζ ) = χr,ξ ωb χr,ξ +χr,ζ ωb χr,ζ . The same factorization property (2.8.3) is obviously true with b replaced by bz = U (z )b U (z )† since the unitary U (z ) has the same product structure. As in [17, Sect. 5.1], we have the following superadditivity property of the relative entropy. Lemma 4. Let X i , 1 ≤ i ≤ k, denote k mutually orthogonal projections on H. Let be a state on F which factorizes under restrictions as i X i = ⊗i X i . Then, for any state , S(, ) ≥ S( X i , X i ). (2.8.4) i
We note that the lemma applies, in particular, to a (particle number conserving) quasifree state whose one-particle density matrix ω satisfies X i ωX j = 0 for i = j. We emphasize that the factorization property of is crucial; in general, the relative entropy need not be superadditive. This is the reason for introducing the cutoff b. Proof. Let X denote the projection X = i X i . The relative entropy decreases under restrictions [6,13], i.e., S(, ) ≥ S( X , X ) = S( X , ⊗i X i ) S( X i , X i ) + S( X i ) − S( X ). = i
(2.8.5)
i
The last two terms together are positive because of subadditivity of the von-Neumann entropy. We take the X i to be the multiplication operators by characteristic functions of balls of radius r , separated a distance 2b. By averaging over the position of the balls, Lemma 4 implies that, for any b ≥ 2r such that L/(2b) is a positive integer, and for any state , 1 z S(, bz ) ≥ dξ S(χr,ξ , b,χ ). (2.8.6) r,ξ (2b)3 We apply this inequality to the state = ϒπz = U (z )U (z )† ⊗ z .
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R. Seiringer
We remark that that restriction of L/(2b) being an integer will be of no concern to us, since we are interested in the thermodynamic limit L → ∞, with b independent of L. We can now apply inequality (2.4.8) to the right side of (2.8.6). Using the Schwarz inequality for the integration over ξ , we thus obtain 1/2 z z z 3 z dξ ϒπ,χ − ≤ 4 b ||S(ϒ , ) (2.8.7) 1 π b,χr,ξ b r,ξ
for any r ≤ b/2. Note that S(ϒπz , bz ) = S(ϒπ , b ) since the relative entropy is invariant under unitary transformations. Were it not for the cutoff b, we could use (2.4.6) to bound the right side of (2.8.7). We will estimate the effect of the cutoff in Subsect. 2.13. 2.9. Interaction energy, Part 1. The next step is to derive a lower bound on Tr F Wϒπz . The main input will be the bound (2.8.7) derived in the previous subsection. We split the estimate into three parts. First, we give a lower bound on the expectation value of the R in (2.7.11). In the next subsection, we bound the remaining energy terms containing U containing the terms w R . Finally, we combine the two estimates in Subsect. 2.11. One of the difficulties in our estimates results from the fact that z is rather arbitrary, and hence the system can be far from being homogeneous. From (2.7.11), we can write W = W1 − W2 , where
∞
W1 ≡
n
R (d(x j , x J j (x j ))) a U NN
(2.9.1)
(2.9.2)
n=0 j=1
and
∞
W2 =
n a
n=0 j=1 i∈J j
ε
w R (x j − xi ).
(2.9.3)
We start by giving a lower bound on the expectation value of W1 in the state ϒπz . Recall that ϒπz is defined after Eq. (2.5.2) as ϒπz = U (z )U (z )† ⊗ z . According to the R as decomposition (2.7.1), we can write U R (d(x, y)) = 144 dξ θ (R/2 − d(ξ, x))θ (R/2 − d(ξ, y)). (2.9.4) U π R6 This gives rise to a corresponding decomposition of W1 , which we write as 144a W1 = dξ w(ξ ). π R6
(2.9.5)
For r > 0, let nr,ξ denote the operator that counts the number of particles inside a ball of radius r centered at ξ ∈ . It is the second quantization of the multiplication operator χr,ξ ( · ) = θ (r − d(ξ, · )) on L 2 (). We claim that w(ξ ) ≥ n R/10,ξ θ (n R/10,ξ − 2). (2.9.6)
Free Energy of a Dilute Bose Gas: Lower Bound
613
This is just the second quantized version of the inequality J
j θ (R/2 − d(ξ, x j ))θ (R/2 − d(ξ, xNN (x j ))) ⎛ ⎞ # ≥ θ (R/10 − d(ξ, x j )) ⎝1 − θ (d(ξ, xi ) − R/10)⎠ .
(2.9.7)
i = j
To prove (2.9.7), we have to show that whenever x j and some xk , k = j, are in a ball of J
j radius R/10 centered at ξ , then xNN (x j ) is in a ball of radius R/2 (with the same center).
J
J
j j ) ≤ d(x j , xk ) ≤ R/5, whence d(ξ, xNN )≤ Assume first that k ∈ J j . Then d(x j , xNN 3R/10. If, on the other hand, k ∈ J j , then there exists an l ∈ J j such that d(xl , xk ) <
J
J
j j (x j )) ≤ d(x j , xl ) < 2R/5, and therefore d(ξ, xNN ) < R/2. This R/5. Hence d(x j , xNN proves (2.9.7). Hence, in particular, we have that
w(ξ ) ≥ w(ξ ) ≡ w(ξ ) θ (2−n 3R/2,ξ )+n R/10,ξ θ (n R/10,ξ −2)θ (n 3R/2,ξ −3).
(2.9.8)
We now claim that
w(ξ ) θ (2 − n 3R/2,ξ ) = n R/2,ξ n R/2,ξ − 1 θ (2 − n 3R/2,ξ ).
(2.9.9)
This implies, in particular, that the operator w(ξ ) acts non-trivially only on the Fock space restricted to a ball of radius 3R/2 centered at ξ . Equation (2.9.9) follows from the fact that if two particles with coordinates xi and x j are within a ball of radius R/2, and no other particle is in the bigger ball of radius 3R/2, then the two particles must be nearest neighbors. Moreover, j ∈ Ji and i ∈ J j by construction. Note that (2.9.9) is a bounded operator, bounded by 2. Moreover, since n R/10,ξ ≤ n 3R/2,ξ , we also see that |w(ξ ) − n R/10,ξ | ≤ 2. (2.9.10) Using (2.9.5), (2.9.8) and (2.9.10), we can estimate 144a z w(ξ )ϒ dξ Tr Tr F W1 ϒπz ≥ F π π R6 144a z z z ϒ dξ Tr w(ξ ) + n − ≥ R/10,ξ F π b b π R6 144a z z dξ ϒπ,χ − b,χ . (2.9.11) −2 3R/2,ξ 3R/2,ξ 1 π R6 Here we have also used that w(ξ ) acts non-trivially only on the Fock space over χ3R/2,ξ H. Note that the integral over the second term on the right side of (2.9.11) is equal to 4π R 3 z z dξ Tr F n R/10,ξ ϒπ − b = Tr F N ϒπz − bz . (2.9.12) 3 10 Moreover, for the last term in (2.9.11), we can use (2.8.7) to estimate 1/2 z z z 3 z dξ ϒπ,χ − ≤ 4 b ||S(ϒ , ) 1 π b,χ3R/2,ξ b 3R/2,ξ
as long as 3R ≤ b.
(2.9.13)
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R. Seiringer
We proceed with a lower bound on Tr F w(ξ )bz . In fact, we will derive two different lower bounds on this expression. First, neglecting the last term in (2.9.8) and using (2.9.9), Tr F w(ξ )bz ≥ Tr F n R/2,ξ n R/2,ξ − 1 bz −Tr F n 3R/2,ξ n 3R/2,ξ − 1 n 3R/2,ξ − 2 bz . (2.9.14) +
Since bz is a combination of a coherent and quasi-free state, the last expression in (2.9.14) is, in fact, easy to estimate. Let z denote the one-particle state |z = | p|< pc z p | p. We have Tr F n 3R/2,ξ n 3R/2,ξ − 1 n 3R/2,ξ − 2 bz 3 = Tr F n 3R/2,ξ bz + 2 tr (χ3R/2,ξ ωb )3 + 6z |(χ3R/2,ξ ωb χ3R/2,ξ )2 |z 2z |χ3R/2,ξ ωb χ3R/2,ξ |z + tr (χ3R/2,ξ ωb )2 +3 Tr F n 3R/2,ξ bz 3 ≤ 6 Tr F n 3R/2,ξ bz . (2.9.15) (Here, we use the symbol tr to denote the trace over the one-particle space L 2 (), while Tr is reserved for the trace over the Fock space.) A different lower bound can be obtained using (2.9.16) Tr F w(ξ )bz ≥ Tr F n R/10,ξ θ (n R/10,ξ − 2)bz . Equation (2.9.16) follows easily from (2.9.8) and (2.9.9). The latter trace is non-trivial only over the Fock space over χ R/10,ξ H. Denoting by F 0 the vacuum on F, we claim that 3 b,χ R/10,ξ ≥ e−4π(R/10) ω /3 F (2.9.17) 0,χ R/10,ξ , which implies that z b,χ ≥ e−4π(R/10) R/10,ξ
3 /3 ω
U (z )† F U ( z ) 0
χ R/10,ξ
.
(2.9.18)
Equation (2.9.17) follows from the fact that b,χ R/10,ξ is a particle-number conserving quasi-free state, whose vacuum part is given by exp −tr ln(1 + χ R/10,ξ ωb χ R/10,ξ ) ≥ exp −tr χ R/10,ξ ωb χ R/10,ξ ) & = exp(−4π(R/10)3 ω 3). (2.9.19) Equation (2.9.18) implies, in particular, that 3 U ( z ) . (2.9.16) ≥ e−4π(R/10) ω /3 Tr F n R/10,ξ θ (n R/10,ξ − 2)U (z )† F 0
(2.9.20)
The state U (z )† F z ) is a coherent state on F. Its restriction to the Fock space over 0 U ( χ R/10,ξ H is again a coherent state. In every sector of particle number n, it is given by the
Free Energy of a Dilute Bose Gas: Lower Bound
615
projection onto the n-fold tensor product of the wave function χ R/10,ξ z , appropriately normalized. Therefore Tr F n R/10,ξ θ (n R/10,ξ − 2)U (z )† F U ( z ) 0 = e−z |χ R/10,ξ |z
z |χ R/10,ξ |z n n n!
(2.9.21)
n≥2
z |χ R/10,ξ |z 2 = z |χ R/10,ξ |z 1 − e−z |χ R/10,ξ |z ≥ . 1 + z |χ R/10,ξ |z The last inequality follows from the elementary estimate x(1 − e−x ) ≥ x 2 /(1 + x) for x ≥ 0. Summarizing the results of this subsection, we have shown that, for any 0 ≤ λ ≤ 1, Tr F W1 ϒπz 1/2 24 a 8 a 3 z z z z ≥ N ϒ − 144 b Tr − ||S(ϒ , ) F π π b b 125 R 3 π R6 ' 3 ( 144 a z z +λ dξ Tr F n R/2,ξ (n R/2,ξ − 1)b − 6 Tr F n 3R/2,ξ b π R6 + z |χ R/10,ξ |z 2 144 a −4π(R/10)3 ω /3 . (2.9.22) +(1 − λ) e dξ π R6 1 + z |χ R/10,ξ |z The choice of λ will depend on the function |z |. If it is approximately a constant (in a sense to be made precise in Subsect. 2.11), we will take λ = 1, otherwise we choose λ = 0. 2.10. Interaction energy, Part 2. Next we are going to give an upper bound on the expectation value of W2 , defined in (2.9.3). To start, we claim that there exists a smooth function g of rapid decay (faster than any polynomial) such that w R (x − y) ≤
R2 g(d(x, y)/s). s5
(2.10.1)
Although w R depends on the box size L, g can be chosen independent of L for large L. This follows immediately from the following considerations. First of all, we have, from the definition (2.6.2) of f R and because of R ≤ s, f R (x) ≤ R
sup |∇h(y)|.
(2.10.2)
d(x,y)≤s
Recall that h(x) = ||−1 p (1 − ν(sp))e−i px , where 1 − ν is a smooth function supported in a ball of radius 2. We need the following elementary lemma. → C be a smooth function, supported in a cube of side length 4, Lemma 5. Let o : R3 and let u(x) = ||−1 p o(sp)e−i px . Then, for any non-negative integer n, |u(x)| ≤
s 16 d(x, 0)
2n (−) o∞ n
2 n+1 +2 πs L
3 .
(2.10.3)
616
R. Seiringer
Here, denotes the Laplacian on R3 , not on . Proof. Introducing coordinates x = (x1 , x2 , x3 ), we can write n 3 2 u(x) 2L (1 − cos(2π xi /L)) = ||−1 (−d )n o(sp)e−i px .
(2.10.4)
p
i=1
−2 Here, d denotes the d) discrete Laplacian in momentum space, which acts as L (− f ( p) = 8 f ( p) − |e|=1 f ( p + 2π e/L). It is easy to see that the function (−d )n f is bounded by (−)n f ∞ . Moreover, if f has support in a cube of side length , then (−d )n f is supported in a cube of length + 4π n/L. This implies that 2 n+1 3 +2 |(2.10.4)| ≤ s 2n (−)n o∞ . (2.10.5) πs L
On the other hand, note that 1 − cos(2π xi /L) ≥ 8L −2 mink∈Z |xi − k L|2 , and hence 2L 2
3
(1 − cos(2π xi /L)) ≥ 16 d(x, 0)2 .
(2.10.6)
i=1
This proves the lemma.
Applying the lemma to the function ∇h in (2.10.2), and using the definition (2.6.3) of w R , we immediately conclude (2.10.1). We now decompose the function g into an integral over characteristic functions of balls. Such decompositions have been studied in detail in [3]. Recall that j is defined in (2.7.1). According to [3, Thm. 1], we can write ∞ g(t) = dr m(r ) j (t/r ), (2.10.7) 0
where
1 r g (r ) − rg (r ) . (2.10.8) 72 Note that m is a smooth function of rapid decay. Since j is monotone decreasing, we can estimate 1 ∞ g(t) ≤ j (t) dr |m(r )| + dr |m(r )| j (t/r ). (2.10.9) m(r ) =
0
1
This estimate, together with (2.10.1), implies that 1 a R2 ∞ 144 dr δ(r − s) 0 dt |m(t)| + s −1 |m(r/s)| W2 ≤ 8 π εs s ∞ n dξ χr/2,ξ (x j )χr/2,ξ (xi ). ×
(2.10.10)
n=0 j=1 i∈J j
Let vr (ξ ) denote the integrand in the last line in (2.10.10). Because d(xi , xk ) ≥ R/5 for i, k ∈ J j , the number of xi inside a ball of radius r/2 is bounded from above by (1 + 5r/R)3 . Thus we have vr (ξ ) ≤ nr/2,ξ (1 + 5r/R)3 .
(2.10.11)
Free Energy of a Dilute Bose Gas: Lower Bound
617
Moreover, we trivially have that vr (ξ ) ≤ nr/2,ξ (nr/2,ξ − 1). By combining these two bounds, we obtain vr (ξ ) ≤ f (nr/2,ξ ), (2.10.12) where
f (n) = n min{n − 1, (1 + 5r/R)3 }.
(2.10.13)
Proceeding similarly to (2.9.11), using that | f (n)−n(1+5r/R)3 |
≤ (1+(1+5r/R)3 )/4, we can estimate Tr F vr (ξ )ϒπz ≤ Tr F f (nr/2,ξ )ϒπz ≤ Tr F f (nr/2,ξ )bz + (1 + 5r/R)3 Tr F nr/2,ξ ϒπz − bz 2 z z + 41 1 + (1 + 5r/R)3 ϒπ,χ − b,χ . (2.10.14) r/2,ξ r/2,ξ 1
When integrating over ξ , the last two terms can be handled in the same way as in the previous subsection, see Eqs. (2.9.12) and (2.9.13). We have to assume that r ≤ b, however. For the first term on the right side of (2.10.14), we estimate Tr F f (nr/2,ξ )bz ) * ≤ min Tr F nr/2,ξ (nr/2,ξ − 1)bz , (1 + 5r/R)3 Tr F nr/2,ξ bz . (2.10.15) Similarly to (2.9.15), 2 Tr F nr/2,ξ (nr/2,ξ − 1)bz ≤ 2 Tr F nr/2,ξ bz , and hence
Tr F f (nr/2,ξ )bz
2 4 Tr F nr/2,ξ bz ≤ . 1 + 2 Tr F nr/2,ξ bz / (1 + 5r/R)3
(2.10.16)
(2.10.17)
Moreover,
π Tr F nr/2,ξ bz = r 3 ω + z |χr/2,ξ |z . 6 2 Using convexity of the function x → x /(1 + x), we obtain the bound
(2.10.18)
π 2 8z |χr/2,ξ |z 2 r 3 ω + . (2.10.19) Tr F f (nr/2,ξ )bz ≤ 8 6 1 + 4z |χr/2,ξ |z /(1 + 5r/R)3 We use (2.10.19) in (2.10.14) and integrate over ξ . We obtain (assuming r ≤ b, as mentioned above) π 2 π r 3 ω dξ Tr F vr (ξ )ϒπz ≤ (1 + 5r/R)3 r 3 Tr F N ϒπz − bz + 8|| 6 6 2 1/2 z 3 3 z b ||S(ϒπ , b ) + 1 + (1 + 5r/R) 8z |χr/2,ξ |z 2 + dξ . (2.10.20) 1 + 4z |χr/2,ξ |z /(1 + 5r/R)3
618
R. Seiringer
In order to be able to compare the last term with the last term in (2.9.22), we note that
χr/2,ξ
5r ≤ 1+ R
3 −
|a|≤r/2+R/10
da χ R/10,ξ +a .
(2.10.21)
Here, we denote by − the normalized integral, i.e., we divide by the volume of the ball of radius r/2 + R/10. Using monotonicity and convexity of the map x → x 2 /(1 + x), we thus have the upper bound z |χr/2,ξ |z 2 1 + 4z |χr/2,ξ |z /(1 + 5r/R)3 z |χ R/10,ξ +a |z 2 5r 6 − . ≤ 1+ da R 1 + 4z |χ R/10,ξ +a |z |a|≤r/2+R/10
(2.10.22)
After integration over ξ , the right side of (2.10.22) simply becomes
6
5r R
dξ
z |χ R/10,ξ |z 2 ≤ 1 + 4z |χ R/10,ξ |z
6
z |χ R/10,ξ |z 2 . 1 + z |χ R/10,ξ |z (2.10.23) Here we have used the fact that r ≥ s ≥ R for the relevant values of r . As noted above, the estimates leading to (2.10.20) are only valid for r ≤ b. To bound the expectation value of W2 in (2.10.10) we have to consider all r ≥ s, however. For r ≥ b, we use (2.10.11) to obtain the simple estimate 1+
6r R
dξ
5r 3 dξ Tr F vr (ξ )ϒπz ≤ 1 + dξ Tr F nr/2,ξ ϒπz R 3 π 3 6r r Tr F Nϒπz . ≤ R 6
(2.10.24)
The contribution of r ≥ b to the integral in (2.10.10) is thus bounded from above by 1 s
∞
dr |m(r/s)|
b
π ≤ s3 6
6s R
3
dξ Tr F vr (ξ )ϒπz
Tr F
Nϒπz
∞
dr r 6 |m(r )|.
(2.10.25)
b/s
Since |m| is a function that decays faster than any polynomial, the last integral is bounded above by any power of the (small) parameter s/b. Let c denote the constant c= 0
1
dr |m(r )| + 1
∞
dr r 6 |m(r )|.
(2.10.26)
Free Energy of a Dilute Bose Gas: Lower Bound
619
To summarize, we have derived the upper bound 2 24 a z z 2 R Tr F W2 ϒπz ≤ 63 N ϒ + 32π || a c Tr − c F π ω b ε Rs 2 εs 2 1/2 a 144 (1 + 63 )2 4 2 c b3 ||S(ϒπz , bz ) + π εR s ∞ a 3 24 +6 Tr F Nbz dr r 6 |m(r )| 2 ε Rs b/s z |χ R/10,ξ |z 2 144 66 a . (2.10.27) +8 c dξ π εs 2 R 4 1 + z |χ R/10,ξ |z 2.11. Interaction energy, Part 3. We now put the bounds of the previous two subsections together in order to obtain our final lower bound on Tr F [Wϒπz ]. We will distinguish two cases, depending on the value of a certain function of |z |, given in (2.11.1) below. Assume first that z |χ R/10,ξ |z 2 π2 ≥ ||(R 3 )2 . dξ (2.11.1) 1 + |χ | 18 z R/10,ξ z This condition means, essentially, that |z | is far from being a constant. In this case, we choose λ = 0 in (2.9.22), and find that the contribution of the last terms in (2.9.22) and (2.10.27), respectively, is bounded from below by 8π || a
2
a 4π − a 3
R 10
3
66 R 2 ω − 8c 2 εs
.
(2.11.2)
Next, consider the case when (2.11.1) is false. In this case, using (2.10.21) for r = 3R, as well as convexity of x → x 2 /(1 + x), we find that
dξ
2 z |χ3R/2,ξ |z 2 6π ≤ 16 ||(R 3 )2 . 1 + 16−3 z |χ3R/2,ξ |z 18
Pick some D > 0, and let B ⊂ denote the set ) * B = ξ ∈ : z |χ3R/2,ξ |z ≥ 163 D R 3 .
(2.11.3)
(2.11.4)
Using (2.11.3), as well as monotonicity of x → x/(1 + x), we find that B
dξ z |χ3R/2,ξ |z ≤
163 π 2 ||R 3 1 + D R 3 . D 18
(2.11.5)
1 π2 3 1 + D R . D 2 18
(2.11.6)
Similarly, we have the estimate |B| ≤ ||
620
R. Seiringer
We choose λ = 1 in (2.9.22) and estimate the relevant term from below by ' 3 ( dξ Tr F n R/2,ξ (n R/2,ξ − 1)bz − 6 Tr F n 3R/2,ξ bz + 3 z . ≥ dξ Tr F n R/2,ξ (n R/2,ξ − 1)b − 6 Tr F n 3R/2,ξ bz \B
(2.11.7) Using Tr F n 3R/2,ξ bz = 9π R 3 ω /2+z |χ3R/2,ξ |z , the definition of B in (2.11.4) and convexity of x → x 3 , we can bound the last term as 3 3 2 9π 3 R ω + 18π |z |2 R 3 163 D R 3 . dξ Tr F n 3R/2,ξ bz ≤ 4|| 2 \B (2.11.8) We now investigate the first term on the right side of (2.11.7). A simple calculation shows that Tr F n R/2,ξ n R/2,ξ − 1 bz = Tr F n R/2,ξ n R/2,ξ − 1 b + 2z |χ R/2,ξ ωb χ R/2,ξ |z π (2.11.9) + R 3 ω z |χ R/2,ξ |z + z |χ R/2,ξ |z 2 . 3 Here, we have used again the translation invariance of b . Note that this invariance also implies that the first term on the right side of (2.11.9) is independent of ξ . Since b is a quasi free state, it can be rewritten in terms of the one-particle density matrix ωb as 2 Tr F n R/2,ξ n R/2,ξ − 1 b = tr χ R/2,ξ ωb + tr χ R/2,ξ ωb χ R/2,ξ ωb . (2.11.10) The first term is just (π R 3 ω /6)2 , and the second is bounded from above by this expression. Therefore, 2 π R 3 ω . dξ Tr F n R/2,ξ n R/2,ξ − 1 b ≤ 2|B| (2.11.11) 6 B Note also that z |χ R/2,ξ ωb χ R/2,ξ |z ≤ trχ R/2,ξ ωb z |χ R/2,ξ |z , and thus π dξ 2z |χ R/2,ξ ωb χ R/2,ξ |z + R 3 ω z |χ R/2,ξ |z 3 B 2π 3 R ω dξ z |χ R/2,ξ |z . (2.11.12) ≤ 3 B The last expression can be bounded from above using (2.11.5). For the last term in (2.11.9), we use Schwarz’s inequality, together with (2.11.5), to estimate 2 1 2 dξ z |χ R/2,ξ |z ≥ dξ z |χ R/2,ξ |z || \B \B π2 2π 163 z 1 + D R3 . ≥ || R 6 z2 − 36 3 D (2.11.13)
Free Energy of a Dilute Bose Gas: Lower Bound
621
Here we set again z = |z |2 /||. Putting all these estimates together, we have thus derived the lower bound \B
dξ Tr F n R/2,ξ (n R/2,ξ − 1)bz
1 π2 π 2 R6 2 ω 1 − 2 1 + D R3 + dξ tr χ R/2,ξ ωb χ R/2,ξ ωb 36 D 9 π 2 R6 2π 163 2 3 2ω z + z − z 1 + DR +|| 36 3 D 2π 3 R 6 163 1 + D R3 + 2 −|| dξ z |χ R/2,ξ ωb χ R/2,ξ |z . ω 3 18D (2.11.14)
≥ ||
The first integral on the right side of (2.11.14) can be rewritten as π R3 || 144
d x |ωb (x)|2 j (d(x, 0)/R) ≥ ||
π 2 R6 2 γ , 36 b
(2.11.15)
where we introduced the notation 1 γb = 4π R 3
d x ωb (x) j (d(x, 0)/R).
(2.11.16)
Equation (2.11.15) follows by applying Schwarz’s inequality to the integration over , noting that d x j (d(x, 0)/R) = 4π R 3 . It remains to integrate the last term in (2.11.14). We claim that
dξ z |χ R/2,ξ ωb χ R/2,ξ |z ≥ |z |2
π 2 R6 (γb − Rpc ω ) . 36
(2.11.17)
To see this, we write 144 2 dξ z |χ R/2,ξ ωb χ R/2,ξ |z − |z | d x ωb (x) j (d(x, 0)/R) π R3 = d x d y z (x + y)∗ − z (y)∗ z (y)ωb (x) j (d(x, 0)/R) × d x z (x + ·) − z (·)2 |ωb (x)| j (d(x, 0)/R). (2.11.18) ≥ −z 2
We can estimate |ωb (x)| ≤ ωb (0) = ω . Moreover, writing the 2-norm as a sum in momentum space, and using the fact that z has non-vanishing Fourier coefficients only for | p| < pc , it is easy to see that z (x+·)−z (·)2 ≤ z 2 pc d(x, 0). Since the range of j ( · /R) is R, the integral over can be estimated as d x j (d(x, 0)/R)d(x, 0) ≤ R d x j (d(x, 0)/R) = 4π R 4 . This yields (2.11.17).
622
R. Seiringer
Collecting all the estimates above, we conclude the following lower bound on the expectation value of W: 1 a 2 a z z z 3 R Tr F Wϒπ ≥ 24 3 Tr F N ϒπ − b −6 c 2 R 125 a εs 1/2 2 144 a 3 z z 3 2 R − 8 + (1 + 6 b ||S(ϒ , ) ) c π b π R6 εs 2 R 2 64 z + ω ∞ 6 −4π a || 8ω2 c 2 + dr r |m(r )| εs π ε Rs 2 b/s +4πa || min {A1 , A2 } . Here we have used the simple bound a ≤ a , and we have set a 66 R 2 4π R 3 2 A1 = 2 1 − ω − 8c 3 10 a εs 2 and
A2 = z2 + 2z γb + γb2 + 2ω z (1 − Rpc ) 1 π2 1 + D R 3 − 2π 38 R 3 ω +ω2 1 − 2 D 9 4π 163 a 66 R 2 −ω 1 + D R 3 − 2 16c 3 D a εs 2 3 4 3 2 2 3 2π 16 163 D R 3 + 1 + D R3 . −z π 3 D
(2.11.19)
(2.11.20)
(2.11.21)
We will choose D = (R 3 )−1/3 in order to minimize the error terms in A2 . Moreover, since a / a contains a factor (1 − ε) (see (2.7.7)), it is best to choose ε = R/s. We note that one can also use the simple bound (2.4.10) in order to estimate z in the error terms. is nonSince R0 R s, the term in round brackets in the first line of (2.11.19) negative and, therefore, we will need a lower bound on Tr F N ϒπz − bz . Moreover, we will need an upper bound on the relative entropy S(ϒπz , bz ). Appropriate bounds will be derived in the next two subsections.
2.12. A bound on the number of particles. Our lower bound onthe expectation value of z z W in the previous subsection contains the expression Tr F N ϒπ − b , multiplied by a positive parameter. Hence we need a lower bound on this expression in order to complete our bound. In fact, we will combine the first term on the right side of (2.11.19) with the last term 21 Tr F [Kϒ z ] in (2.7.15), which we have not used so far. I.e., we seek a lower bound on 1 a 2π aC a z z 3 R −6 c + Tr F (N − N )2 ϒ z . 24 3 Tr F N ϒπ − b R 125 a s || (2.12.1)
Free Energy of a Dilute Bose Gas: Lower Bound
623
(Here we have usedthat ε= R/s, as mentioned at the end of the previous subsection.) First, note that Tr F Nbz = |z |2 + Tr F [Nb ] = |z |2 + Tr F [Nπ ] and Tr F Nϒπz = |z |2 + Tr F [Nϒπ ]. Let N> = | p|≥ pc a †p a p denote the number operator on F> . Using that π = ⊗ 0 and that ϒπ = ⊗ z , we can thus write (2.12.2) Tr F N(ϒπz − bz ) = Tr F> N> (z − 0 ) . For the second term in (2.12.1), we use 2 (N − N )2 ≥ |z |2 + Tr F> N> 0 − N + 2 |z |2 +Tr F> N> 0 − N N−|z |2 −Tr F> N> 0 , (2.12.3) and hence 2 Tr F (N − N )2 ϒ z ≥ |z |2 + Tr F> N> 0 − N +2 |z |2 + Tr F> N> 0 − N Tr F> N> z − 0 . (2.12.4) Thus, we conclude that the expression (2.12.1) is bounded from below by 2 2π aC 2 |z | + Tr F> N> 0 − N + Tr F> (N> − N 0 )z || ' ( a 4π aC 2 aR 24 3 > 0 − 6 + | z | N − N . c + Tr × F > R 3 125 s ||
(2.12.5)
We will choose R, s and C below in such a way that R 3 1/C and R s. The last term in square brackets is thus positive, irrespective of the value of |z |. Hence we need to derive a lower bound on Tr F> N> z − 0 . To this end, we note that, for any µ ≤ 0, S(z , 0 ) − βµ Tr F> N> z ≥ β f (µ) − f (0) . (2.12.6) Here, we denoted
1 2 f (µ) = ln 1 − e−β( p −µ0 −µ) . β
(2.12.7)
| p|≥ pc
Equation (2.12.6) is simply the variational principle for the free energy f (µ). Since f is completely monotone (i.e., all derivatives are negative), we can estimate f (µ) ≥ f (0) + µ f (0) + 21 µ2 f (0) = −Tr F> N> 0 and hence, optimizing over f (0). But all (negative) µ, ⎛ Tr F> N> z − 0 ≥ − ⎝ S(z , 0 ) | p|≥ pc
⎞1/2 1 ⎠ . cosh(β( p 2 − µ0 )) − 1
(2.12.8) We can use (2.4.6) to estimate the relative entropy as S(z , 0 ) ≤ 8π || a β2 .
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R. Seiringer
For the sum over p, we use that cosh(x) − 1 ≥ x 2 /2. In the thermodynamic limit, we can replace the sum over p by an integral. We thus have to bound 2 || 1 dp 2 . (2.12.9) β 2 (2π )3 | p|≥ pc ( p − µ0 )2 We use two different bounds. On the one hand, 1 1 π2 dp 2 ≤ dp = . √ ( p − µ0 )2 ( p 2 − µ0 )2 −µ0 | p|≥ pc R3 On the other hand, since µ0 ≤ 0, 1 1 4π dp 2 ≤ dp 2 2 = . 2 ( p − µ0 ) (p ) pc | p|≥ pc | p|≥ pc In combination, these bounds imply that 1 || . (2.12.9) ≤ 2 2 √ π β max{ pc , 4π −1 −µ0 }
(2.12.10)
(2.12.11)
(2.12.12)
Using (2.12.12) in (2.12.8), we obtain the lower bound −1/2 √ a 1/2 Tr F> N> z − 0 ≥ −const. || 1/2 pc + −µ0 − o(||). β
(2.12.13)
We apply this bound in (2.12.5), noting again that the last term in square brackets is positive. We conclude that 2 2π aC 2 |z | + Tr F> N> 0 − N − Z (3) − o(||), (2.12.1) ≥ (2.12.14) || with ' ( √ −1/2 1 √ a 3/2 Z = const. || 1/2 pc + −µ0 + C 1 + 2/ C + ω . β R3 (2.12.15) Here we have used (2.4.10) to bound |z |2 from above, as well as the fact that Tr F> N> 0 ≤ Tr F [Nπ ] = ||ω . (3)
2.13. Relative entropy, effect of cutoff. Next, we are going to give an estimate on the relative entropy of the two states ϒπz and bz . This is needed for the lower bound on the expectation value of W obtained in (2.11.19). As already noted, the relative entropy is invariant under unitary transformations, and hence S(ϒπz , bz ) = S( ⊗ z , b ).
(2.13.1)
We want to bound this expression by the relative entropy of ⊗ z with respect to π = ⊗ 0 , which satisfies S( ⊗ z , π ) = S(z , 0 ) ≤ 8π a β||2
(2.13.2)
Free Energy of a Dilute Bose Gas: Lower Bound
625
according to (2.4.6). I.e., we want to estimate the effect of the cutoff b on the relative entropy S( ⊗ z , b ). Here it will be important that is not the vacuum state. The cutoff b corresponds to a mollifying of the one-particle density matrix in momentum space, and the error in doing so would not be small enough if this one-particle density matrix is strictly zero for | p| < pc . This is the reason for replacing the vacuum state 0 by a more general quasi-free state in Subsect. 2.5. If ω denotes a general (particle number conserving) quasi-free state with oneparticle density matrix ω, it is easy to see that S(, ω ) is convex in ω for an arbitrary state . The one-particle density matrix of b can be written as ωb =
1 1 ηb (q) 2 (ωπ ( p + q) + ωπ ( p − q))| p p|. || q p
Therefore, S( ⊗ z , b ) ≤
1 ηb (q)S( ⊗ z , q ), || q
(2.13.3)
(2.13.4)
where q is the quasi-free state corresponding to the one-particle density matrix with eigenvalues 21 (ωπ ( p + q) + ωπ ( p − q)). (This is the same estimate as in [17, Sect. 5.2].) Moreover, as in [17, Eq. (5.14)], simple convexity arguments yield S( ⊗ z , q ) ≤ 1 + t −1 S( ⊗ z , π ) 1 1 (2.13.5) + h q ( p) − h 0 ( p) − eh 0 ( p)+t (h 0 ( p)−h q ( p)) − 1 eh q ( p) − 1 p for any t > 0. Here h q ( p) = ln
2 + ωπ ( p + q) + ωπ ( p − q) . ωπ ( p + q) + ωπ ( p − q)
(2.13.6)
Note that h 0 ( p) = ( p), defined in (2.8.1). To estimate the expression (2.13.5) from above, we need the following lemma. Lemma 6. Let : R3 → R+ , and let L ± = ± sup p supq,q=1 ±(q∇)2 ( p) denote the supremum (infimum) of the largest (lowest) eigenvalue of the Hessian of . Let ωπ ( p) = [e( p) − 1]−1 , and let h q ( p) be given as in (2.13.6). Then h q ( p) − h 0 ( p) ≤ L + q 2 ,
(2.13.7)
and h q ( p) − h 0 ( p) ≥ q 2 L − + q 2 min{L − , 0} − 4q 2 sup[|∇( p)|2 ωπ ( p)] p
−2q (|q| + | p|) sup[|∇( p)| / p 2 ]. 2
2
2
(2.13.8)
p
Proof. By convexity of x → ln(1 + 1/x), h q ( p) ≤ 21 (( p + q) + ( p − q)) ≤ ( p) + L + q 2 ,
(2.13.9)
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R. Seiringer
proving (2.13.7). To obtain the lower bound in (2.13.8), we proceed similarly to [17, Lemma 5.2]. We can write 1 ∂2 h q ( p) − h 0 ( p) = dλ (1 − λ) 2 h λq ( p). (2.13.10) ∂λ 0 Denoting p± = p ± λq and ω± = ωπ ( p± ), we can write the second derivative of h λq ( p) as ( ' 2 ∂ 1 1 ∂2 (ω+ + ω− ) h λq ( p) = − ∂λ2 (ω+ + ω− )2 ∂λ (2 + ω+ + ω− )2 ∂2 2 − (ω+ + ω− ). (2.13.11) (ω+ + ω− )(2 + ω+ + ω− ) ∂λ2 The first term is positive and can thus be neglected for a lower bound. For the second term, we use ∂2 2 2 , ω = ω (1 + ω ) (1 + 2ω ) − (q∇) (q∇ ) + + + + + + ∂λ2
(2.13.12)
where we denoted + = ( p +λq). The last term in the square brackets is bounded above by −q 2 L − . Moreover, (1 + 2ω+ ) (q∇+ )2
≤ q 2 (| p| + |q|)2 sup |∇( p)|2 / p 2 + 2q 2 sup ωπ ( p)|∇( p)|2 . p
(2.13.13)
p
The same bounds hold with + replaced by −. Using in addition that 1 ω+ (1 + ω+ ) + ω− (1 + ω− ) ≤ ≤ 1, 2 (ω+ + ω− )(2 + ω+ + ω− ) we arrive at (2.13.8).
(2.13.14)
Let g : R3 → [0, 1] be a smooth radial function supported in the ball of radius 1. We assume that g( p) ≥ 21 for | p| ≤ 21 . We then choose ( p) = β( p 2 − µ0 ) + βpc2 g( p/ pc ).
(2.13.15)
Since ( p) = ln(1+1/π p ) for | p| < pc by definition, this corresponds to the choice π p = (exp(β( p 2 − µ0 ) + βpc2 g( p/ pc )) − 1)−1 . Note that, in particular, π p ≤ const. /(βpc2 ) and hence P ≤ const. M/(βpc2 ) ∼ pc ||/β. This bound is important for estimating the error term Z (2) in (2.5.6). For the given in (2.13.15), both β −1 L + and β −1 L − are bounded independent of all parameters. Moreover |∇( p)| ≤ const. β| p|. Using that ωπ ( p) ≤ ( p)−1 ≤ (βp 2 )−1 , the bounds in Lemma 6 imply that − Dβq 2 1 + β(| p| + |q|)2 ≤ h q ( p) − h 0 ( p) ≤ Dβq 2 (2.13.16) for some constant D > 0.
Free Energy of a Dilute Bose Gas: Lower Bound
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Using the fact that sinh(x)/x ≤ cosh(x) for all x ∈ R, we can estimate 1 1 h q ( p) − h 0 ( p) − eh 0 ( p)+t (h 0 ( p)−h q ( p)) − 1 eh q ( p) − 1 2 e−h q ( p) + e−h 0 ( p)+t (h q ( p)−h 0 ( p)) . ≤ 21 (1 + t) h q ( p) − h 0 ( p) 1 − e−h 0 ( p)+t (h q ( p)−h 0 ( p)) 1 − e−h q ( p)
(2.13.17)
The estimate (2.13.16) implies the bound
h q ( p) − h 0 ( p)
2
2 ≤ D 2 (βq 2 )2 1 + β(| p| + |q|)2 .
(2.13.18)
Moreover, using the upper bound in (2.13.16) to estimate h q ( p) − h 0 ( p) from above, we obtain e−h q ( p) + e−h 0 ( p)+tβ Dq 2 1 − e−h 0 ( p)+tβ Dq 1 − e−h q ( p) 2
= ωt ( p) +
1 2
(ωπ ( p + q) + ωπ ( p − q)) 1 + 2ωt ( p)
(2.13.19)
as an upper bound to the last fraction in (2.13.17). Here, we denoted ωt ( p) = 2 [eh 0 ( p)−Dβtq − 1]−1 , assuming t to be small enough such that h 0 ( p) − Dβtq 2 > 0 for all p. Recall that h 0 ( p) = ( p) is given in (2.13.15). In the thermodynamic limit, the sum over p in (2.13.5) converges to the corresponding integral and, hence, we are left with bounding 2 ωt ( p) + 21 (ωπ ( p + q) + ωπ ( p − q)) 1 + 2ωt ( p) dp 1 + β(| p| + |q|)2 R3
(2.13.20) from above. We can replace ωπ ( p − q) by ωπ ( p + q) in (2.13.20) without changing the value of the integral. Using Schwarz’s inequality, the fact that ωπ ( p) ≤ ωt ( p), and changing variables p → p − q, we see that (2.13.20) is bounded from above by 2 2 dp 1 + β(| p| + 2|q|)2 ωt ( p) 1 + 2ωt ( p) . (2.13.21) R3
It remains to bound ωt ( p). For this purpose, we need a bound on ( p) − Dβtq 2 for an appropriate choice of the parameter t. We will choose t = min{1, (b2 q 2 )−1 }. We then have tq 2 ≤ b−2 , and it is easy to see that ( ' D 2 2 1 1 2 ( p) − Dβtq ≥ β 2 p − µ0 + pc (2.13.22) − 8 b2 pc2 in this case. Since ( p) ≥ β( p 2 − µ0 ), this can be seen immediately in the case | p| ≥ 1 2 pc . For | p| ≤ pc /2 even a slightly better bound holds, this time using the fact that g( p) ≥ 1/2 for | p| ≤ 1/2. We will choose b and pc below in such that a way that bpc 1. Denoting by τ the (positive) number 1 D (2.13.23) − 2 2 , τ = −βµ0 + βpc2 8 b pc
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R. Seiringer
we thus have the bound ' (−1 1 1 t τ 2 βp 2 −τ − 2 βp 2 −1 ≤e e ω ( p) ≤ e e 1+
1 τ + 21 βp 2
.
(2.13.24)
The last bound follows from the elementary inequality (e x − 1)−1 ≤ e−x (1 + 1/x) for all x > 0. We insert this bound for ωt into (2.13.21). Simple estimates then yield e−τ (2.13.25) (2.13.21) ≤ const. 3/2 1 + τ −1/2 1 + (βq 2 )2 . β Combining (2.13.5), (2.13.17), (2.13.18) and (2.13.25), and using that t −1 ≤ 1+b2 q 2 , we have thus shown that S( ⊗ z , q ) ≤ 2 + b2 q 2 S( ⊗ z , π ) +const. ||β 1/2 q 4 τ −1/2 1 + (βq 2 )2 + o(||). (2.13.26) After inserting this estimate in (2.13.4) and summing over q, this yields the bound β 1/2 τ −1/2 S(ϒπz , bz ) ≤ const. || + o(||). (2.13.27) a β2 + b4 Here, we have used the assumed smoothness of η to estimate ηb (q)|q|n ≤ q −n const. ||b for integers n ≤ 8. We have also used the fact that we will choose b2 β and hence, in particular, βb−2 ≤ const. Moreover, the assumption (2.4.6) has been used to bound S( ⊗ z , π ) = S(z , 0 ). We have thus shown that the effect of the cutoff b on the relative entropy can be estimated by ||(β/τ )1/2 b−4 . We note that the exponent −4 of b is important, since the relative entropy has to be multiplied by b3 in the estimate (2.11.19). 2.14. Final lower bound on Fz (β). We have now obtained all the necessary estimates to complete our lower bound on Fz (β). For this purpose, we put all the bounds from Subsects. 2.7, 2.11, 2.12 and 2.13 together. In fact, from (2.7.15), (2.11.19), (2.12.14) and (2.13.27), we have the following lower bound on Fz (β): 1 Fz (β) ≥ − ln Tr F> exp −βTcs (z ) − Z (2) − Z (3) − Z (4) β −(κ − κ ) p 2 π p − o(||) | p|< pc
2 2π aC 2 |z | + Tr F> N> 0 − N + || * ) +4π a || min 22 , z2 + 2z (γb + ω ) + ω2 + γb2 , where we denoted Z
(4)
+
3 R0 R 3 1/3 = const. a || κ + + Rpc + (R ) + s R 1/2 , ∞ 1 β 1/2 τ −1/2 6 3 2 dr r |m(r )| + 6 b a β + + 2 . R s b/s R b
(2.14.1)
2
(2.14.2)
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629
Here, we have used the definition (2.7.7) of a , (2.4.10) to bound z in the error terms, together with γb ≤ ω and lim||→∞ ω ≤ . This last estimate follows from ( p) ≥ β( p 2 − µ0 ) and (1.6). The error terms Z (2) and Z (3) are defined in (2.5.6) and (2.12.15), respectively. Using (2.7.8), the term in the second line of (2.14.1) can be estimated by (κ − κ ) | p|< pc p 2 π p ≤ (κ − κ ) pc2 P ≤ const. ( a /R)3 pc3 ||/β. Here, we have also used the bound on P = | p|< pc π p derived after Eq. (2.13.15). The two terms in the third and forth line of (2.14.1) can be bounded from below independently of z , simply using Schwarz’s inequality. More precisely, introducing 0 ≡ ||−1 Tr F> N> 0 = ω − P/||, we obtain 2 2π aC 2 |z | + Tr F> N> 0 − N + 4π a || z2 + 2z (γb + ω ) + ω2 + γb2 || 4π a || 2 0 2 0 2 2 2 ( − ) + 2( − )(ω + γb ) + ω + γb − (ω + γb ) . ≥ 1 + 2/C C (2.14.3) We note that 0 =
1 (2π )3
| p|≥ pc
dp
1 2 eβ( p −µ0 )
−1
+ o(1)
in the thermodynamic limit. Hence, from (1.6), 1 1 0 = min { , c (T )} − + o(1) dp 2 −µ ) β( p 0 −1 (2π )3 | p|≤ pc e 1 pc ≥ min { , c (T )} − + o(1). 2π 2 β
(2.14.4)
(2.14.5)
This estimate is obtained by using eβ( p −µ0 ) − 1 ≥ βp 2 in the denominator of the integrand. Moreover, 0 ≤ ω ≤ min{, c (T )} + o(1). It remains to give a lower bound on γb . According to (2.11.16) and (2.8.2), 1 γb = d x ωπ (x)η(d(x, 0)/b) j (d(x, 0)/R). (2.14.6) 4π R 3 2
We note that, since ωπ (x) is real, ωπ (x) − ω =
1 p2 d(x, 0)2 1 px) − 1) ≥ − . (2.14.7) (cos( || p e( p) − 1 2|| e( p) − 1 p
2 We can estimate d(x, 0) ≤ R in 3the integrand in (2.14.6). Since ( p) ≥ β| p| , |η| ≤ 1 and d x j (d(x, 0)/R) = 4π R , the contribution of the last term to (2.14.6) is bounded by R2 1 p2 + o(1) (2.14.8) dp 2 2 (2π )3 β 5/2 R3 ep − 1 in the thermodynamic limit. Moreover, we can bound η from below as η(t) ≥ 1 − const. t 2 , and hence R2 (2.14.9) ω ≥ γb ≥ ω 1 − const. 2 − const. R 2 β −5/2 − o(1). b
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R. Seiringer
Using the bounds on 0 and γb just derived, we have 2 (2.14.3) ≥ 4π a || 22 − − c (T ) + 1 R2 pc R 2 −o(||). (2.14.10) + 2 + + 5/2 −const. a || 2 C b β β In particular, the terms in the third and forth line of (2.14.1) are bounded from below by the right side of (2.14.10).
2.15. The “free” free energy. We now insert the lower bound on Fz (β) derived in the previous subsection into (2.3.9). We note that the only z -dependence left is in the first term in (2.14.1), which is the “free” part of the free energy. Taking also the constant µ0 N in (2.3.9) into account, we are thus left with evaluating 1 ln d Mz Tr F> exp −βTcs (z ) β CM ⎞ ⎛ 1 ln (βε( p)) + ln 1 − e−βε( p) ⎠ . = µ0 N + ⎝ β
µ0 N −
| p|< pc
(2.15.1)
| p|≥ pc
Using x ≥ (1 − e−x ) for non-negative x, (2.15.1) becomes, in the thermodynamic limit, (2.15.1) ≥ N µ0 +
|| −βε( p) − o(||). dp ln 1 − e β(2π )3 R3
(2.15.2)
Recall that ε( p) is defined in (2.7.10). It is given by ε( p) = (1 − κ + κ ) p 2 − µ0 for | p| ≤ 1/s, and satisfies the bound ε( p) ≥ κ p 2 for | p| ≥ 1/s. Hence 2 dp ln 1 − e−βε( p) ≥ (1 − κ + κ )−3/2 dp ln 1 − e−β( p −µ0 ) R3 R3 1 2 dp ln 1 − e− p . (2.15.3) + 3/2 (κ β) | p|2 ≥κ β/s 2
We will choose s 2 κ β below, in which case the last integral is exponentially small in the parameter s 2 /(κ β). Inserting the definition (1.4) of f 0 (β, ), we have thus shown that (2.15.1) ≥ ||(1 − κ + κ )−3/2 f 0 (β, ) || − p2 − o(||). + 5/2 3/2 dp ln 1 − e β κ (2π )3 | p|2 ≥β κ /s 2 Here, we have also used that µ0 ≤ 0.
(2.15.4)
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631
2.16. Choice of parameters. We have now essentially finished our lower bound on f (β, ). It remains to collect all the error terms, and choose the various parameters in an appropriate way. All the error terms we have to take into account are given in (2.1.1), (2.3.9), (2.14.1), (2.14.10) and (2.15.4). We will choose the various parameters in our estimates as follows: 3/403 −121/403 R = −1/3 a2 β 5/2 , b = β 1/2 a2 β 5/2 , 1/3 1/403 a2 β 5/2 s = β−1/3 . Moreover,
−A ϕ = a a2 β 5/2
−B , C = a2 β 5/2
(2.16.1)
(2.16.2)
for 4/403 ≤ A ≤ 79/403 and 2/403 ≤ B ≤ 161/403. Depending on µ0 , we choose pc =
81/403 162/403 if β|µ0 | ≤ a2 β 5/2 β −1/2 a2 β 5/2 0 otherwise.
(2.16.3)
−δ Finally, we choose κ = s 2 β −1 a2 β 5/2 for some δ > 0. Our estimates then imply that (2.16.4) f (β, ) ≥ f 0 (β, ) + 4πa 22 − [ − c (β)]2+ (1 − o(1)) , with
2/403−δ o(1) ≤ Cδ (β2/3 )−1 a2 β 5/2
(2.16.5)
for some function Cδ , depending on δ, that is uniformly bounded on bounded intervals. The choice of the parameters pc , b, s, R and κ is determined by minimizing the sum of all the error terms. The main terms to consider are, in fact, the terms M pc2 ∼ || pc5 in Z (1) in (2.3.7), and || a 2 (κ+ R/s) as well as || a R −6 (b3 a β2 +( pc2 −µ0 )−1/2 b−1 )1/2 (4) in Z in (2.14.2). Moreover, we have to take the restriction s 2 κβ in (2.15.3) into account. This leads to the choice of parameters above. All other error terms are of lower order for small a2 β 5/2 . This is true, in particular, for all the terms containing ϕ and C, which explains the freedom in their choice above.
2.17. Uniformity in the temperature. Our final result, Eq. (2.16.4), does not have the desired uniformity in the temperature. It is only useful in case the dimensionless parameter a2 β 5/2 is small. In particular, one can not take the zero-temperature limit β2/3 → ∞. The reason for this restriction is that our argument was essentially perturbative, using that the correction term we want to prove is small compared to the main term, i.e., a2 f 0 (β, ). Below the critical temperature, f 0 (β, ) = const. β −5/2 , hence the assumption is only satisfied if a2 β 5/2 1. If the temperature is smaller, we can use a different argument for a lower bound on f , which uses in an essential way the result in [11]. There, a lower bound in the zero temperature case was derived.
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R. Seiringer
To obtain the desired bound for very low temperature, it is possible to skip Steps 1–5 entirely, and start immediately with the Dyson Lemma, Lemma 2, applied to the original potential v. Using this lemma, we have that N ' − ∇ j (1 − (1 − κ)χ ( p j )2 )∇ j HN ≥ (2.17.1) j=1 J
j (x j ))) − +(1 − ε)(1 − κ)aU R (d(x j , xNN
( a w R (x j − xi ) . ε i∈J j
Since d(xi , xk ) ≥ R/5 for i, k ∈ J j , we can estimate (using (2.10.1)) N a j=1
ε
i∈J j
w R (x j − xi ) ≤ const.
aN . ε Rs 2
(2.17.2)
Moreover, the calculation in [11] shows that, for the choice κ = (a 3 )1/17 and R = a(a 3 )−5/17 , N
κ Jj − j + (1 − ε)(1 − κ)aU R (d(x j , xNN (x j ))) 2 j=1 ≥ 4πa N 1 − ε − const. (a 3 )1/17 .
(2.17.3)
(Strictly speaking, this result was derived in [11] for Neumann boundary conditions, and with J j = {1, . . . , N } independent of all the particle coordinates. It is easy to see, however, that the same result applies to our Hamiltonian, being defined with periodic boundary conditions, and having a slightly smaller interaction.) For this choice of κ and R, we thus have N
1 3 1/17 HN ≥ l( − j ) + 4πa N 1 − ε − const. (a ) − const. , ε Rs 2 j=1
(2.17.4) with l(| p|) = p 2 (1 − κ/2 − (1 − κ)χ ( p)2 ). To obtain a lower bound on the free energy for this Hamiltonian, we can go to the grand-canonical ensemble, introducing a chemical potential in the usual way. Taking the thermodynamic limit, this yields 1 −β(l( p)−µ) f (β, ) ≥ sup µ + dp ln 1 − e (2π )3 β R3 µ≤0 +4πa2 1 − ε − const. (a 3 )1/17 − const. 2 (a 3 )5/17 . (2.17.5) εs Recall that χ ( p) = ν(sp), where ν is a function with 0 ≤ ν( p) ≤ 1 that is supported outside the ball of radius 1. This implies that l( p) = (1 − 21 κ) p 2 for | p| ≤ 1/s, and l( p) ≥ 21 κ p 2 for | p| ≥ 1/s. Hence 2 dp ln 1 − e−β(l( p)−µ) ≥ (1 − 21 κ)−3/2 dp ln 1 − e−β( p −µ) R3 R3 1 1 − 2 p2 . (2.17.6) + dp ln 1−e (κβ)3/2 | p|2 ≥κβ/s 2
Free Energy of a Dilute Bose Gas: Lower Bound
633
The last expression is exponentially small in the (small) parameter s 2 /(κβ). We choose, for some δ > 0, (a 3 )3/85 s 2 = β(a 3 )1/17+δ , ε2 = , (2.17.7) (a2 β 5/2 )2/5 and obtain f (β, ) ≥ f 0 (β, ) + 4πa2 (1 − o(1)) ,
(2.17.8)
with o(1) = const.
(a 3 )1/17 1 +
1 2 a β 5/2
+
(a 3 )3/170−2δ (a2 β 5/2 )1/5
.
(2.17.9)
Compared with our desired lower bound, we also have to take into account the missing term a(2 − [ − c (β)]2+ ), which can be bounded as a(2 − [ − c (β)]2+ ) ≤ const. aβ −3/2 .
(2.17.10)
In combination, the estimates (2.16.4) and (2.17.8) provide the desired uniform lower bound on the free energy. Depending on the value of a2 β 5/2 , one can apply either one of them. To minimize the error, one has to apply (2.16.4) for a2 β 5/2 ≤ (a 3 )403/6885 , and (2.17.8) otherwise. This yields our main result, Theorem 1, for α = 2/2295 − δ. A. Appendix: Proof of Lemma 2 For simplicity, we drop the on v and a in our notation in this Appendix. We start by dividing up into Voronoi cells, B j = {x ∈ : d(x, y j ) ≤ d(x, yk ) ∀k = j}.
(A.1)
For a given ψ ∈ H 1 (), let ξ be the function with Fourier coefficients ξ ( p) = ( p). We thus have to show that χ ( p)ψ Bj
d x |∇ξ(x)|2 + 21 v(d(x, y j ))|ψ(x)|2 ≥ (1 − ε)a −
a ε
Bj
d x U (d(x, y j ))|ψ(x)|2
d x w R (x − y j )|ψ(x)|2 .
(A.2)
The statement of the lemma then follows immediately by summing over j and using the positivity of v. We will actually show that (A.2) holds even when the integration region B j on the left side of the inequality is replaced by the smaller set B R ≡ B j ∩ {x ∈ : d(x, y j ) ≤ R}. Note that the first integral on the right side of (A.2) is also over this region, since the range of U is supposed to be less than R. As in Subsect. 1.3, let φv denote the solution to the zero-energy scattering equation − φv (x) + 21 v(|x|)φv (x) = 0
(A.3)
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subject to the boundary condition lim|x|→∞ φv (x) = 1. Let ν be a complex-valued function on the unit sphere S2 , with S2 |ν|2 = 1. We use the same symbol for the function on R3 taking values ν(x/|x|). For ψ and ξ as above, consider the expression d x ν(x − y j )∇ξ ∗ (x) · ∇φv (x − y j ) A≡ BR + 21 v(d(x, y j ))ψ(x)∗ φv (x − y j )ν(x − y j ). (A.4) BR
By using the Cauchy-Schwarz inequality, we can obtain the upper bound (A.5) d x |∇ξ(x)|2 + 21 v(d(x, y j ))|ψ(x)|2 |A|2 ≤ BR × d x |∇φv (x − y j )|2 + 21 v(d(x, y j ))|φv (x − y j )|2 |ν(x − y j )|2 . BR
For an upper bound, we can replace the integration region B R in the second integral by R3 . Since φv (x) is a radial function, the angular integration then can be performed by using S2 |ν|2 = 1. The remaining expression is then bounded by a because of 1 2 2 R3 d x |∇φv (x)| + 2 v(|x|)|φv (x)| = 4πa. Hence we arrive at
|A|2 d x |∇ξ(x)|2 + 21 v(d(x, y j ))|ψ(x)|2 ≥ a BR
for any choice of ν as above. It remains to derive a lower bound on |A|2 . By partial integration, d x ν(x − y j )∇ξ ∗ (x) · ∇φv (x − y j ) BR =− d x ξ ∗ (x)ν(x − y j )φv (x − y j ) BR + dω R ξ ∗ (x)ν(x − y j )n · ∇φv (x − y j ), ∂BR
(A.6)
(A.7)
where dω R denotes the surface measure of the boundary of B R , denoted by ∂B R , and n is the outward normal unit vector. Here we used the fact that ∇ν(x) · ∇φv (x) = 0. Now, by definition of h(x), ξ(x) = ψ(x) − (2π )−3/2 h ∗ ψ(x), where ∗ denotes convolution, i.e., h ∗ ψ(x) = dy h(x − y)ψ(y). Using the zero-energy scattering equation (A.3) for φv , we thus see that dω R ψ ∗ (x)ν(x − y j )n · ∇φv (x − y j ) A= ∂BR −(2π )−3/2 dω R (h ∗ ψ)∗ (x)ν(x − y j )n · ∇φv (x − y j ) ∂BR +(2π )−3/2 d x (h ∗ ψ)∗ (x)ν(x − y j )φv (x − y j ). (A.8) BR
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The last two terms on the right side of (A.8) can be written as ' ( −3/2 ∗ d x ψ (x) dµ(y) h(y − x) , (2π ) BR
(A.9)
where dµ is a (non-positive) measure supported in B R . Explicitly, dµ(x) = ν(x − y j )φv (x − y j )d x − n · ∇φv (x − y j )ν(x − y j )dω R , the second part being supported √ on the boundary ∂B R . Note that B R dµ = 0, and also B R d|µ| ≤ 2a S2 |ν| ≤ 2a 4π (by Schwarz’s inequality). Hence - √ dµ(y) h(y − x)-- ≤ 2a 4π f R (x − y j ), (A.10) BR
with f R defined in (2.6.2). The expression (A.9) is thus bounded from below by √ d x |ψ(x)| f R (x − y j ) (A.9) ≥ −(2π )−3/2 2a 4π
≥ −a
d x |ψ(x)| w R (x − y j ) 2
1/2 .
(A.11)
Here, we used Schwarz’s inequality as well as the definition of w R (2.6.3) in the last step. Note that this last expression is independent of ν. The only place where ν still enters is the first term on the right side of (A.8). By construction, ν depends only on the direction of the line originating from y j , which hits the boundary of B R at a distance not greater than R. We distinguish two cases. First, assume that the line hits the boundary at a distance R. In this case, we choose ν to be equal to the value of ψ at this boundary point. Secondly, if the length of the line is strictly less than R, we then choose ν to be zero. Of course we also have to normalize ν appropriately. The integrals are thus only over the part of the boundary of B j which is at a distance R from y j . Let us denote this part of ∂B R by ∂B R , assuming for the moment that it is not empty. We then have dω R |ψ(x)|2 n · ∇φv (x − y j ) ∂BR ∗ dω R ψ (x)ν(x − y j )n · ∇φv (x − y j ) = R . 1/2 ∂BR 2 dω |ψ(x)| R ∂BR (A.12) We note that n · ∇φv (x − y j ) = a/R 2 on ∂B R . We thus obtain from (A.8)–(A.12), 1/2 1/2 a A≥ dω R |ψ(x)|2 −a d x |ψ(x)|2 w R (x − y j ) . (A.13) R ∂BR With the aid of Schwarz’s inequality, we see that, for any ε > 0, a2 a2 |A|2 ≥ 2 (1 − ε) dω R |ψ(x)|2 − d x |ψ(x)|2 w R (x − y j ). R ε ∂BR
(A.14)
At this point we can also relax the condition that ∂B R be non-empty; in case it is empty, (A.14) holds trivially. In combination with (A.6), (A.14) proves the desired result (A.2) in the special case when U (|x|) is a radial δ-function sitting at a radius R, i.e., U (|x|) = R −2 δ(|x| − R).
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The case of a general potential U (|x|) follows simply by integrating this result (i.e., Ineq. (A.2) for this special U (|x|)) against U (R)R 2 d R, noting that d R U (R)R 2 ≤ 1 and that w R (x) is pointwise monotone increasing in R. Acknowledgements. It is a pleasure to thank Elliott Lieb and Jan Philip Solovej for many inspiring discussions.
References 1. Berezin, F.A.: Covariant and contravariant symbols of operators. Izv. Akad. Nauk, Ser. Mat. 36, 1134– 1167 (1972); English translation: USSR Izv. 6, 1117–1151 (1973); Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975) 2. Dyson, F.J.: Ground-State Energy of a Hard-Sphere Gas. Phys. Rev. 106, 20–26 (1957) 3. Hainzl, C., Seiringer, R.: General Decomposition of Radial Functions on Rn and Applications to N -Body Quantum Systems. Lett. Math. Phys. 61, 75–84 (2002) 4. Huang, K.: Statistical Mechanics. 2nd ed., New York: Wiley, 1987 5. Lieb, E.H.: The classical limit of quantum spin systems. Commun. Math. Phys. 31, 327–340 (1973) 6. Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973); Lieb, E.H., Ruskai, M.B.: A Fundamental Property of Quantum Mechanical Entropy. Phys. Rev. Lett. 30, 434–436 (1973) 7. Lieb, E.H., Seiringer, R., Solovej, J.P.: Ground State Energy of the Low Density Fermi Gas. Phys. Rev. A 71, 053605 (2005) 8. Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation. Oberwolfach Seminars, Vol. 34, Basel-Boston: Birkhäuser, 2005 9. Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional. Phys. Rev. A 61, 043602-1–13 (2000) 10. Lieb, E.H., Seiringer, R., Yngvason, J.: Justification of c-Number Substitutions in Bosonic Hamiltonians. Phys. Rev. Lett. 94, 080401 (2005) 11. Lieb, E.H., Yngvason, J.: Ground State Energy of the Low Density Bose Gas. Phys. Rev. Lett. 80, 2504–2507 (1998) 12. Lieb, E.H., Yngvason, J.: The Ground State Energy of a Dilute Two-Dimensional Bose Gas. J. Stat. Phys. 103, 509–526 (2001) 13. Ohya, M., Petz, D.: Quantum Entropy and Its Use, Texts and Monographs in Physics, Berlin-HeidelbergNew York: Springer, 2004 14. Robinson, D.W.: The Thermodynamic Pressure in Quantum Statistical Mechanics. Springer Lecture Notes in Physics, Vol. 9, Berlin-Heidelberg-New York: Springer, 1971 15. Ruelle, D.: Statistical Mechanics. Rigorous Results. OverEdge : World Scientific, 1999 16. Seiringer, R.: The Thermodynamic Pressure of a Dilute Fermi Gas. Commun. Math. Phys. 261, 729–758 (2006) 17. Seiringer, R.: A Correlation Estimate for Quantum Many-Body Systems at Positive Temperature. Rev. Math. Phys. 18, 233–253 (2006) Communicated by I.M. Sigal
Commun. Math. Phys. 279, 637–668 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0423-7
Communications in
Mathematical Physics
On the Renormalized Volume of Hyperbolic 3-Manifolds Kirill Krasnov1,2 , Jean-Marc Schlenker3 1 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK 2 Perimeter Institute for Theoretical Physics, Waterloo, N2L 2Y5, Canada.
E-mail: [email protected]
3 Institut de Mathématiques, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse Cedex 9, France
Received: 19 December 2006 / Accepted: 27 June 2007 Published online: 4 March 2008 – © Springer-Verlag 2008
Abstract: The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate its geometrical meaning. We use another regularization procedure based on surfaces equidistant to a given convex surface ∂ N . The renormalized volume computed via this procedure is equal to what we call the W -volume of the convex region N given by the usual volume of N minus the quarter of the integral of the mean curvature over ∂ N . The W -volume satisfies some remarkable properties. First, this quantity is self-dual in the sense explained in the paper. Second, it verifies some simple variational formulas analogous to the classical geometrical Schläfli identities. These variational formulas are invariant under a certain transformation that replaces the data at ∂ N by those at infinity of M. We use the variational formulas in terms of the data at infinity to give a simple geometrical proof of results of Takhtajan et al on the Kähler potential on various moduli spaces. 1. Introduction The renormalized volume. In this paper we study the so-called renormalized volume of hyperbolic 3-manifolds whose definition is motivated by the AdS/CFT correspondence of string theory [Wit98]. In the context of this correspondence one is interested in computing the gravity action 1 Sgr [g] = (R − 2)dv + H da (1) 2 M ∂M for an Einstein metric g on d-dimensional manifold M. Here we have set the dimensionful Newton constant typically present in the action (1) to 8π G = 1, and is the cosmological constant, which is assumed to be negative. The quantity R is the curvature scalar, H is the mean curvature of the boundary ∂ M, and dv, da are the volume and area forms correspondingly.
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In the context of AdS/CFT correspondence the manifold M is non-compact with a conformal boundary ∂ M. The metric g on M determines the conformal class of the boundary ∂ M. One would like to compute Sgr as a functional of this conformal class. However, because the manifold M is non-compact the functional Sgr [g] diverges. One notices, however, that this divergence is of a special type. Thus, let us introduce a compact sub-manifold N ⊂ M and compute the action Sgr [g, N ] (1) inside N . As one sends ∂ N towards the infinity of M one can show the divergent part of Sgr [g, N ] is given by an integral over ∂ N of a local quantity expressible in terms of the first and second fundamental forms of ∂ N . This suggests an idea of renormalization in which these divergent quantities are subtracted, after which the limit ∂ N → ∂ M can be taken. This idea works in any dimension, see [BK99] for the required subtraction procedure. 3-dimensional manifolds. In this paper we are interested in the simplest case of 3-dimensional spaces. The simplifications arising in this case are as follows. First, Einstein equations in 3 dimensions imply the metric g to be of constant curvature. Thus, in our case of a negative cosmological constant we are led to consider constant negative curvature manifolds. In 3 dimensions the √ radius of curvature l is related to the cosmological constant in a simple way l = 1/ −. We shall set the radius of curvature l = 1 in what follows, which in 3 dimensions is equivalent to = −1. Thus, the Riemannian manifolds (M, g) we are going to consider are hyperbolic. As is well known from Bers’ work [Ber60] on simultaneous uniformization and from its generalization to arbitrary Kleinian manifolds, the manifolds (M, g) are completely characterized by the conformal structures of all the boundary components of M. We would like to compute the gravity action Sgr as a functional of the conformal structure of ∂ M. The constant curvature condition gives R = −6. Therefore, the gravity action reduces to Sgr = −2 dv + H da. (2) M
∂M
It is now easy to show that the volume V (N ), N ⊂ M diverges as (1/2)A(∂ N ), where A(∂ N ) is the area of the boundary of N . There is also the so-called logarithmic divergence to be discussed below. The integrated mean curvature diverges as ∂ N H da ∼ 2 A(∂ N ). Altogether we see that the gravity action functional diverges as A(∂ N ). It seems natural, therefore, to introduce a renormalized gravity action given by 1 Sgr [g] = (R − 2)dv + (H − 1)da. (3) 2 M ∂M This action can then be computed for any compact sub-manifold N ⊂ M. However, because of the logarithmic divergence present in the volume, the limit N → M does not exist. It is this logarithmic divergence that causes all the difficulties that we are now to discuss. In an even number of dimensions, when a more involved but similar in principle subtraction procedure is used, there is no logarithmic divergence and the limit can indeed be shown to exist and to be independent of how exactly the surfaces ∂ N are taken to approach the conformal boundary ∂ M. However, when the dimension is odd, as is in the case of interest for us, the limit of (3) does not exist, except in the special situation when all the boundary components are of genus one. In general, the volume of N grows as the logarithm of the area of ∂ N times the Euler characteristic of ∂ N . As we shall see below, one can subtract this divergence as well, but the resulting renormalized gravity action (or volume) then turns out to depend on the limiting procedure. This is why in
On the Renormalized Volume of Hyperbolic 3-Manifolds
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odd dimensions the renormalized gravity action fails to be a true invariant of M. It is precisely for this reason that the concept of renormalized volume has not been developed by the geometry community. Indeed, the fact that the volume of a hyperbolic manifold grows as half of the area of its boundary has been known for a very long time and is mentioned, for example, in Thurston’s famous notes on the subject. It was also realized however that the limit V (N ) − (1/2)A(∂ N ) does not exist. The geometry community instead concentrated on e.g. a canonically defined volume of the convex core of M. The logarithmic divergence (referred to as the conformal anomaly in the physics literature) is studied in more detail in [HS98]. In [Kra00] one of us studied the renormalized volume of Schottky 3-manifolds. It was shown that there is a way to define the limit of the quantity V (N ) − (1/2)A(∂ N ) by choosing a foliation of M near its boundary by a family of surfaces Sρ [φ] parameterized by a “Liouville” (real-valued) field on the conformal boundary of M. Once such a family of surfaces is used, the limiting procedure becomes well-defined. Indeed, one takes the quantity Vρ − (1/2)Aρ and subtracts from it 2πρ(g − 1), where g is the genus of the boundary of M. It is then easy to see that the limit exists and defines the renormalized volume V R (M, φ), which in addition to being a function of the conformal structure of ∂ M also depends on the Liouville field φ on ∂ M. It was moreover shown by an explicit computation that V R (M, φ) is equal to the Liouville action SL (∂ M, φ) of Takhtajan and Zograf [TZ87]. To get a “canonical” quantity that depends only on the conformal structure of ∂ M one can evaluate V R (M, φ) = SL (∂ M, φ) on the canonical Liouville field corresponding to the metric of constant curvature −1 in the conformal class of ∂ M. This canonical φ can also be obtained by extremizing the functional V R (M, φ) keeping the area of ∂ M as determined by φ fixed. This variational principle leads to the unique canonically defined φ. One gets the “canonical” renormalized volume that depends only on the conformal structure of ∂ M. This way of defining the limit can be generalized to an arbitrary Kleinian manifold. This has been done in [TT03], where the limiting procedure of [Kra00] was also improved in the sense that an invariant family of Epstein surfaces [Eps84] was used for regularization, see also an earlier paper by one of the authors [Kra03], where the same Epstein family of surfaces was used for regularization, but of the CS formulation of gravity instead. In all cases, the renormalized volume obtained via the limiting procedure was shown to agree with the Liouville action on ∂ M. Variation of the renormalized volume = Liouville action under the changes of the conformal structure of ∂ M were studied in [TZ87,TT03]. It was shown that in all the cases the renormalized volume is equal to the Kähler potential for the Weil-Petersson metric on the moduli space of ∂ M. For quasi-Fuchsian spaces this implies, e.g. the quasi-Fuchsian reciprocity of McMullen [McM00]. The renormalized volume and equidistant foliations. In the present paper we undertake a further study of the renormalized volume of hyperbolic 3-manifolds. We show that the limiting procedure via which the volume is defined can be somewhat de-mystified by considering for regularization a family of surfaces equidistant to a given one, following an idea already used by C. Epstein [PP01] (and more recently put to use in [KS05]). Thus, the main idea of the present work is to obtain the renormalized volume by taking a convex domain N ⊂ M, and compute the renormalized volume of M with respect to N as V R (M, N ) = V (N ) + lim V (∂ N , ∂ Nρ ) −(1/2)A(∂ Nρ ) − 2πρ(gi − 1) , (4) ρ→∞
i
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where V (∂ N , ∂ Nρ ) is the volume between the boundary ∂ N of the domain N and the surface ∂ Nρ located a distance ρ from ∂ N . The quantity A(∂ Nρ ) is the area of the surface ∂ Nρ , the sum in the last term is taken over all boundary components of M and gi are the genera of these boundary components. The convexity of the domain N ensures that the equidistant surfaces ∂ Nρ exist all the way to infinity. This ensures that the limit ρ → ∞ can be taken. Similarly, using the combination (2) and subtracting the area of ∂ Nρ , as well as the term linear in ρ, one defines the renormalized gravity action. In both cases the limit exists and can be computed in terms of the volume or the gravity action for N , see below for the corresponding expressions. The limiting procedure used in [Kra03,TT03] is an example of the limiting procedure described above, for the Epstein surfaces [Eps84] are equidistant. Thus, the renormalized volume of references [Kra00,TT03] is an example of the renormalized volume (4) where N is a compact domain of M contained inside a particular Epstein surface. However, it is obvious that the renormalized volume (4) is more general as the domain N in (4) can be an arbitrary convex domain. It is also clear that the renormalized volume defined via (4) is “the most general one”. Indeed, the only constraint that enters into definition (4) is that equidistant surfaces are used for regularization. However, this seems to be a necessary requirement to be able to subtract the logarithmic divergence. Thus, there does not seem any other way to define the renormalized volume. Renormalized volume as the W -volume. The starting point is a simple formula for the renormalized volume (4): V R (M, N ) = W (N ) − π(gi − 1), (5) i
where the sum in the last term is taken over all the boundary components. Here the W-volume is defined as 1 W (N ) := V (N ) − H da. (6) 4 ∂N This formula for V R is a special case of a formula found by C. Epstein [PP01] for the renormalized volume of hyperbolic manifolds in any dimension. Thus, the renormalized volume of M with respect to N is, apart from an uninteresting term given by a multiple of the Euler characteristic of the boundary, just the W-volume of the domain N . All our other results concern this W-volume. Note already that W (N ) is not equal to the Hilbert-Einstein functional of N with its usual boundary term; it differs from it in the coefficient of the boundary term. Variation formula. We prove a formula for the first variation of W under changes of the metric inside N . This formula is a simple consequence of the formula obtained by Igor Rivin and one of us in [RS99]. It reads 1 H δW (N ) = δ II − δ I, I da. (7) 4 ∂N 2 Thus, this formula suggests that the W -volume of a domain N is a complicated functional of the shape of this domain, as well as of the manifold M. However, as we shall show, this functional depends on all these data in a very specific way, through a certain combination that we introduce below and refer to as the metric “at infinity”.
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Self-duality. One of the most interesting properties of the W-volume is that it is self-dual. Thus, we recall that the Einstein-Hilbert functional I E H (N ) := V (N ) −
1 2
∂N
H da
(8)
for a compact domain N ⊂ H 3 of the hyperbolic space (note a different numerical factor in front of the second term) is nothing but the dual volume. Thus, recall that there is a duality between objects in H 3 and objects in d S3 , the 2+1 dimensional de Sitter space. Under this duality geodesic planes in H 3 are dual to points in d S3 , etc. This duality between domains in the two spaces is easiest to visualize for convex polyhedra (see [RH93]), but the duality works for general domains as well. The fact that (8) is the volume of the dual domain then is a simple consequence of the Schläfli formula, analogous to (7), see the main text below. Thus, we can write: ∗
V (N ) = V (∗ N ) = V (N ) −
1 2
∂N
H da
(9)
for the volume of the dual domain. This immediately shows that 1 W (N ) = V (N ) − 4
∂N
H da =
V (N ) + ∗ V (N ) . 2
(10)
Thus, the W -volume is self-dual in that this quantity for N is equal to this quantity for the dual domain ∗ N : W (N ) = W (∗ N ). In the main text we shall also verify the self-duality more directly by applying the Legendre transform to W (N ). The W -volume and the Chern-Simons formulation. An interesting remark is that there is a very simple expression for the W -volume in terms of the so-called Chern-Simons formulation of 2+1 gravity [Wit89]. In this formulation the gravity action in the so-called first order formalism (in which the independent variables are the triads and the spin connection) is shown to be given by the difference of two Chern-Simons actions. This is easily shown for the “bulk” term of the gravity action, while to get the boundary term as in (1) one needs in addition a certain set of boundary terms in terms of the Chern-Simons connections. Remarkably, the combination that plays this role in the W -volume, namely 1 2
(R − 2) + M
1 2
∂M
H da,
(11)
which is different from the usual Einstein-Hilbert action, is exactly what appears naturally in the Chern-Simons formulation, without the need for any boundary terms. We refer the reader to e.g. [Kra03] for a demonstration of this fact, see formula (3.7) of this reference as well as the related discussion. It would be of interest to understand the relation, if any, between the self-duality of the W -volume and the fact that it has such a simple expression in the Chern-Simons formulation.
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Description “from infinity”. The other important property of the W-volume is that it can be interpreted as a quantity that depends on the metric “at infinity” only, instead of being a functional of the “shape” of the boundary ∂ N of N ⊂ M. To demonstrate this one introduces the first and second fundamental forms I ∗ , II ∗ obtained from those on the surfaces ∂ Nρ as ρ → ∞ by a simple rescaling, see the main text. The relation between the data at infinity and those on ∂ N are as follows: I∗ =
1 (I + II )I −1 (I + II ), 2
II ∗ =
1 (I + II )I −1 (I − II ). 2
(12)
These relations can be inverted, with the inverse relations looking exactly the same, with starred quantities replaced by non-starred everywhere, see (32) below. We note that the metric I ∗ is in the conformal class of the boundary ∂ M, but its precise form of course depends on the convex domain N used. The new variation formula. The formula (7) can then be re-written in terms of the variations δ I ∗ , δ II ∗ , the result being 1 H∗ ∗ ∗ δ II ∗ − δW (N ) = − (13) δ I , I da ∗ . 4 ∂N 2 This formula could be compared to a similar one given by Anderson in dimension 4 [Ando1]. Thus, the variational formula for the W -volume is essentially invariant under the transformation (12). One simple corollary of this formula is that the extremum of the W -volume under variations of the metric I ∗ that keep the area of ∂ M as defined by I ∗ fixed occurs for the hyperbolic (i.e. constant negative curvature) I ∗ . As there is always the unique canonical such I ∗ (of given area), the W -volume has a unique critical point for fixed conformal classes at infinity, so that the extremal W -volume becomes truly a functional of the conformal structure of the boundary components of M. By considering the second variation, we also prove that this critical point of W is a maximum. It is important to note that the second fundamental form II ∗ at infinity is completely determined by I ∗ on all the boundary components. This is essentially a consequence of Bers’ simultaneous uniformization. We give a direct proof of this fact in the main text. This fact implies that W (N ) is a functional of I ∗ only, a very important property of the renormalized volume. If one wishes, one can obtain a more canonical functional that depends on M only, by taking the metric I ∗ to be the canonical metric of curvature −1 in the conformal class of infinity of M. It is this “extremal” W -volume, called W M in the latter sections here that is of most interest due to the following. An important immediate corollary of (13) is the theorem by Takhtajan and Teo [TT03] that the extremal renormalized volume is equal to the Kähler potential on the moduli space of Kleinian manifolds. We refer to the main text for a proof of this. We note that our proof is entirely geometrical and avoids a reasonably complicated cohomology machinery that is necessary in [TT03]. For this reason our proof can be immediately extended even to situations where the methods of [TT03] are inapplicable, such as manifolds with cone singularities. See more remarks on this case below. Positivity. The (extremal, i.e the one for the hyperbolic I ∗ ) W-volume coincides with a multiple of the potential studied by Teo in [Teo05]. The result of this reference implies that the extremal W-volume is a non-negative function on the moduli space of manifolds, attaining the zero value only on the Fuchsian manifolds. Here we obtain a similar result
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for the W -volume of the convex core C M of M, see Sect. 4. These positivity results for the W -volume of two different convex domains in M lead us to suggest that the W -volume of any convex domain might be positive. We do not attempt to answer this question in the present work leaving it for future research. Manifolds with particles and the Teichmüller theory of surfaces with cone singularities. One key feature of the arguments presented in this work is that they are always local, in the sense that they depend on local quantities defined on the boundaries of compact subsets of quasi-Fuchsian manifolds. Thus, we make only a very limited use of the fact that the quasi-Fuchsian manifolds are actually quotients of hyperbolic 3-space by a group of isometries. One place where this is used is in the proof of the fact that II ∗ is determined by I ∗ (actually a direct consequence of the Bers double uniformization theorem). We expect that all the results should extend from quasi-Fuchsian (more generally geometrically finite) manifolds to the “quasi-Fuchsian manifolds with particles” which were studied e.g. in [KS05,MS06]. Those are actually cone-manifolds, with cone singularities along infinite lines running from one connected component of the boundary at infinity to the other, along which the cone angle is less than π . One problem towards such an extension is that although in the (non-singular) quasi-Fuchsian setting the Bers double uniformization theorem shows that everything is determined by the conformal structure at infinity, there is as yet no such result in the corresponding case “with particles”. It appears likely, however, that such a statement holds for “quasi-Fuchsian manifolds with particles”; a first step towards it is made in [MS06], while the second step is one of the objects of a work in progress between the second author and C. Lecuire. The result of [MS06] could actually already be used — even without a global Bers type theorem for hyperbolic manifolds with particles — to obtain results on the Teichmüllertype space of hyperbolic metrics with n cone singularities of prescribed angles on a closed surface of genus g. Note that this space, which can be denoted by Tg,n,θ (with θ = (θ1 , · · · , θn ) ∈ (0, π )n ) is topologically the same as the “usual” Teichmüller space Tg,n of hyperbolic metrics with n cusps (see [Tro91]) but it has a natural “WeilPetersson” metric which is different. It might follow from the considerations made here, extended to quasi-Fuchsian manifolds with particles, that this “Weil-Petersson” metric is still Kähler, with the renormalized volume playing the role of a Kähler potential. A global Bers-type theorem would not be necessary for this because, given any hyperbolic metric h ∈ Tg,n,θ on a surface , we can consider the “Fuchsian” hyperbolic manifold with particles defined as the warped product M := ( × R, dt 2 + cosh(t)2 h). Clearly the conformal structure at infinity on both connected components of the boundary at infinity of M are given by h. Moreover it follows from [MS06] that if h − := h and h + is in a small neighborhood U ⊂ Tg,n,θ of h then there exists a unique quasi-Fuchsian manifold with particles, close to M, with conformal structures at infinity given by h − and h + . The arguments developed here (extended to this singular context) should show that the renormalized volume is a Kähler potential for the natural Weil-Petersson metric on Tg,n,θ restricted to U . We leave such an extension to quasi-Fuchsian cone manifolds to future work. 2. Preliminaries In this section we collect various background information useful further on in the paper.
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Extrinsic invariants of surfaces in H 3 . Let S be a smooth surface in H 3 . Its Weingarten (or shape) operator is a bundle morphism B : T S → T S defined by: Bx := −∇x N , where N is the unit normal vector field to S and ∇ is the Levi-Cività connection of H 3 . B is then self-adjoint with respect to the induced metric on S, which we call I here. The second fundamental form of S is then defined by: II (x, y) := I (Bx, y) = I (x, By), for any two vectors x and y tangent to S at the same point. The third fundamental form of S is defined as: III (x, y) := I (Bx, By). When B has no zero eigenvalue, III is a Riemannian metric on S. The Gauss and Codazzi equations. The Weingarten operator satisfies two equations on S, the Codazzi equation: d ∇ B = 0, and the Gauss equation: det(B) = K + 1, where K is the curvature of the induced metric I on S. det(B) is called the extrinsic curvature of S, denoted by K e . Here d ∇ B makes sense if one considers B as a vectorvalued 1-form, so that, if x and y are two vector fields on S, (d ∇ B)(x, y) = (∇x B)(y) − (∇ y B)(x) = ∇x (By) − ∇ y (Bx) − B[x, y]. The Gauss and Codazzi equation are the only relations satisfied by the first and second fundamental forms of a surface. This can be expressed in a mildly complicated way as the “fundamental theorem of surface theory” stated below. Here and elsewhere we use a fairly natural convention: given two bilinear symmetric forms g and h on S, with g positive definite, we denote by g −1 h the unique bundle morphism b : T S → T S, self-adjoint for g, such that h(x, y) = g(bx, y) for any two vectors x, y tangent to S at the same point. For instance, I −1 II = B by definition. Theorem 2.1. Let g and h be two smooth symmetric bilinear forms on a simply connected surface S, with g positive definite at each point, and let ∇ be the Levi-Cività connection of g. Define B := g −1 h, and ke = det(B). If g, h satisfy the constraints: d ∇ B = 0 (Codazzi), ke = K g + 1 (Gauss),
(14) (15)
where K g is the Gauss curvature of g, then there exists a unique immersion of S into the hyperbolic space H 3 such that g, h are the induced metric and second fundamental forms of S respectively. The next lemma, which is elementary and well-known, describes the behavior of the surfaces at constant distance from a fixed surface.
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Lemma 2.2. Let S be a surface in H 3 , with bounded principal curvatures, and let I, B be the first fundamental form and the shape operator of S correspondingly. Let Sρ be the surface at distance ρ from Sρ . Then, for sufficiently small ρ the induced metric on Sρ is: Iρ (x, y) = I ((cosh(ρ)E + sinh(ρ)B)x, (cosh(ρ)E + sinh(ρ)B)y).
(16)
Here E is the identity operator. Note that this lemma also holds for a surface S in any hyperbolic 3-manifold M, not necessarily H 3 . We also note that when the surface S is convex, then the expression (16) gives the induced metric on any surface ρ > 0, where ρ increases in the convex direction. A proof of this lemma, and of the two corollaries which follow, can be found in [KS05]. Corollary 2.3. The area of the surfaces Sρ is given by: A(ρ) = det(cosh(ρ)E + sinh(ρ)B)da S cosh2 (ρ) + cosh(ρ) sinh(ρ)H + sinh2 (ρ)K e da. =
(17) (18)
S
Corollary 2.4. The integrated mean curvature of Sρ is given by: ∂ H da = A(ρ) = (sinh(2ρ)(1 + K e ) + cosh(2ρ)H )da. ∂ρ Sρ S
(19)
3. The New Volume Definition. We define the W -volume of a compact hyperbolic manifold with boundary as follows. Definition 3.1. Let M be a hyperbolic 3-manifold, and N be a compact subset of M with boundary ∂ N . Then: l2 W (N ) := V (N ) − H da, (20) 4 ∂N where V (N ) is the volume of N , l is the radius of curvature of the space, H is the mean curvature of the boundary, and da is the area element of the metric induced on ∂ N . Note that the volume defined above does not coincide with the usual Einstein-Hilbert action: √ S E H [g] = g(R − 2) + 2 H M
∂M
evaluated on a metric of constant curvature. Indeed, for such a metric R − 2 = 4. Thus, defining the radius of curvature as = −1/l 2 we have: l2 l2 I E H (N ) := − S E H (N ) = V (N ) − H, (21) 4 2 ∂N which is different from (20). We will set the radius of curvature l to one in what follows. The following property of W (N ) is obvious:
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Lemma 3.2. The W -volume is additive: if N1 and N2 are two compact sub-manifolds of M such that N1 ∩ N2 is a disjoint union of connected components of the boundary of both N1 and N2 then: W (N1 ∪ N2 ) = W (N1 ) + W (N2 ). Proof. This is obvious when N1 , N2 do not share any boundary components. For N1 , N2 such that a part of their respective boundaries is shared the additivity follows from the fact that the mean curvatures of that boundary component have the opposite sign as viewed from N1 and N2 . We note that the on-shell Einstein-Hilbert action (21) is also additive. Thus, at this stage there is no reason to prefer (20) to (21). However, as we shall see in the next section, it is the quantity (20) that behaves much more regularly for non-compact hyperbolic manifolds as well as for compact ones. Also, as we shall presently see, it is the W -volume that is the self-dual one. Self-duality. Here we prove self-duality of the new W -volume by considering its Legendre transform. We will need to use the variation formula (7) given in the introduction. An immediate consequence of this formula is that the variation of the W (N ) under the condition that the “conjugate” momentum
1 H π =− II − I (22) 4 2 is fixed is given by
δW (N ) =
∂N
π, δ I .
(23)
The quantity (22) is exactly the unique combination of I, II such that when it is kept fixed the variation of W (N ) produces exactly π . The dual W-volume can now be obtained by a Legendre transform: ∗ W (N ) := − π, I + W (N ). (24) ∂N
We see that, because π is traceless, the W -volume is self-dual: ∗ W (N ) = W (N ). Note that a similar argument applied to the usual volume V (N ) of a domain N shows that its Legendre transform is given by the Einstein-Hilbert functional I E H (N ). This demonstrates the fact that the Einstein-Hilbert functional is the dual volume I E H (N ) = ∗ V (N ) to which we referred to in the introduction. We have so far discussed the duality only in the context of a compact domain N ⊂ H 3 . The duality is, however, more general and holds also for domains in more general hyperbolic 3-manifolds. Note, however, that in this more general context one has to be careful about the geometrical meaning of the dual volume. Indeed, the 3-manifold dual to M is modeled on d S3 spacetime, and typically has two disconnected components, each having an internal boundary – a surface dual to the boundary of the convex core in M. Given a convex domain N that contains the convex core C M , one can meaningfully talk about the dual domain in d S3 as being a domain in the dual manifold located between the surfaces dual to ∂ N and the internal boundary of ∗ M. This discussion serves as a good introduction to the following section in which we discuss precisely those more general hyperbolic 3-manifolds for which the notion of the W -volume is of interest.
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4. Convex Co-compact Hyperbolic Manifolds Definitions and first properties. We first need to define convex co-compact hyperbolic manifolds. Definition 4.1. A complete hyperbolic 3-manifold M is convex co-compact if there is a compact subset N ⊂ M whose boundary ∂ N is convex and such that the normal exponential map from ∂ N to the conformal boundary ∂ M is a homeomorphism. Each connected component of the complement M\N is called a hyperbolic end of M. Note that the condition on N is equivalent to the fact that N is strongly convex in the sense that any geodesic segment in M with endpoints in N is actually contained in N . Simple examples of convex co-compact manifolds are: Schottky manifolds, each having one hyperbolic end; quasi-Fuchsian manifolds with two hyperbolic ends. The new volume W (N ) is especially interesting because it is (almost) defined not only for compact subsets N as in the above definition, but also for the hyperbolic ends. The following computation is central to motivate the definition that follows. Lemma 4.2. Consider a hyperbolic end of a convex co-compact manifold M, and let S be a connected component of the boundary ∂ N of the compact subset N of Definition 4.1. The W -volume of the sub-manifold contained between the surfaces S and Sρ is given by: ρ W [S, Sρ ] = − K da = 2πρ(g − 1), (25) 2 S where g is the genus of S. Proof. We have: ρ V (ρ) = A(r )dr 0 ρ = det(cosh(r )E + sinh(r )B)dadr 0 ρ S = cosh2 (r ) + H cosh(r ) sinh(r ) + K e sinh2 (r )dadr 0 ρ S = (cosh2 (r ) + sinh2 (r )) + K sinh2 (r ) + H cosh(r ) sinh(r )dadr S 0 ρ sinh(2r ) cosh(2r ) − 1 +H dadr, = cosh(2r ) + K 2 2 S 0 so that: 1 V (ρ) = 2
K H sinh(2ρ) + (sinh(2ρ) − 2ρ) + (cosh(2ρ) − 1) da. 2 2 S
Now the W -volume is given by: V (ρ) −
1 4
H da + Sρ
1 4
H da. S
The formula (25) follows by combining (26) with (19).
(26)
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The relative W -volume. Thus, the W -volume of a portion of a hyperbolic end is just the thickness of the portion considered times 2π(g − 1). This fact motivates the following definition: Definition 4.3. Let M be a convex co-compact hyperbolic 3-manifold with one or more hyperbolic ends. Let N be a compact convex subset of M as in Definition 4.1. The W -volume W (M, N ) of M relative to N is defined as the W -volume of N. Note that W (N ) is the same as the W -volume of M with the W -volumes (25) of the hyperbolic ends removed. Thus, one could also refer to the volume W (M, N ) as the renormalized volume. Indeed, it has a close relation to the renormalized volume that has appeared in the literature. Here we would like to give a comparison to the renormalized volume. Definition 4.4. Let M be a convex co-compact hyperbolic 3-manifold, and let Sri be a foliation by equidistant surfaces near each component i of the boundary. The renormalized volume of M relative to foliations Sρi is defined as V R (M, Sρ ) := lim V (ρ) − ρ→∞
1 i
2
Aiρ −
2πρi (gi − 1).
i
Here V (ρ) is the volume of the subset of M bounded by surfaces Sρi , Aiρ and gi are the areas and genera of the surfaces Sρi correspondingly. The limit is a multiple limit of all ρi → ∞. When M is convex co-compact there is a natural foliation of each end by surfaces equidistant to the (strongly) convex subset N . In this case we will talk about the renormalized volume V R (M, N ) of M relative to the (strongly) convex subset N . Lemma 4.5. Let M be convex co-compact, and let N be a (strongly) convex subset of M (as in Definition 4.1). The renormalized volume of M relative to N is the W -volume minus a multiple of the Euler characteristic of the boundary: V R (M, N ) = W (M, N ) − π(gi − 1). (27) i
The sum is taken over the boundary components. This formula is the 3-dimensional case of a formula given by C. Epstein for the renormalized volume of hyperbolic manifolds in [PP01]. We include a proof for the reader’s convenience. Proof. Consider one of the hyperbolic ends. Let S be the corresponding boundary component of ∂ N . The area (17) of Sρ can be rewritten as:
H K cosh(2ρ) + sinh(2ρ) + (cosh(2ρ) − 1) da. (28) A(ρ) = 2 2 S Subtracting half of this from (26) we get: 1 −2ρ 1 V (ρ)− A(ρ)/2−2πρ(g − 1) = − e H da − π(g − 1). (2 − H + K ) da − 4 4 S S The result now follows by taking the limit ρ → ∞ and adding the terms corresponding to all the different ends.
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Thus, the result (27) shows that the renormalized volume relative to a strongly convex subset N is basically the volume (20) we have defined, apart from a constant term proportional to the Euler characteristic of ∂ N . However, as it is clear from the proof, the two quantities agree only after the limit is taken. For a finite ρ the renormalized volume is a complicated functional, and it is only in the limit that a simplification occurs. The volume (20) we have defined is in contrast simple even for a finite ρ, as well as for any compact domain N . One could try to define an analog of the renormalized volume for a general subset N ⊂ M by taking the volume minus half of the area of ∂ N . This functional however fails to be additive and is thus of a very limited interest, apart from the limiting case when the surface ∂ N is sent to infinity. All this makes it clear that the functional W (N ) is much more natural to consider than the one that plays a role in the definition of the renormalized volume. Having motivated and defined the W -volume, the natural question to ask is what this quantity depends on. From its definition one may expect that it depends on the shape of the convex subset N in M in a complicated way. However, as we shall see, this dependence is actually rather simple in that the W -volume is just a certain functional of the so-called “asymptotic” metric constructed using the fundamental forms of the boundary of N . We deal with this in the next section. However, before we study this question, let us demonstrate certain positivity properties of the W -volume. Positivity estimates on W . We note that some of the quantities considered here are always positive on the convex core of a quasi-Fuchsian hyperbolic manifold. We actually prove this here under a technical hypothesis which is conjecturally always satisfied. Lemma 4.6. Let M be a quasi-Fuchsian manifold, and let C M be its convex core. Suppose that C M is the Gromov-Hausdorff limit of a sequence of convex cores of hyperbolic manifolds with bending laminations along closed curves. Then I E H (C M ) ≥ 0, with equality exactly when M is Fuchsian. A short explanation on the hypothesis is needed. Given M, let λ be the measured bending lamination on the boundary of its convex core. It is known (see [BO04,Lec06]) that λ is the limit of the measured bending laminations of a sequence of quasi-Fuchsian manifolds Mn for which the convex core converge, in the Gromov-Hausdorff distance, to the convex core of a quasi-Fuchsian manifold M , and that the measured bending lamination on the boundary of the convex core of M is λ. According to a conjecture of Thurston, this last point should imply that M = M, and then the hypothesis made in the lemma would be useless. Proof. We consider in the proof that M is not Fuchsian, since in that case it is quite obvious that I E H (C M ) = 0. Suppose first that the bending lamination of C M is along disjoint closed curves. Let li and λi be the lengths and bending angles at those closed curves. It is then known [BO04] that there exists a one-parameter family of quasi-Fuchsian manifolds, Mt , 0 ≤ t ≤ 1, with M0 Fuchsian, M1 = M, and such that, for all t ∈ [0, 1], the bending lamination of the convex core of Mt is t times the measured bending lamination of the convex core of M. In other terms, the bending angle of the curve i on the boundary of the convex core of Mt is tλi . Let li (t) be the length of this curve. Now a simple computation using the Schläfli formula (as seen in [Mil94], or more specifically in [Bon98]) shows that: 1 d I E H (C Mt ) dli (t) =− . tλi dt 2 dt i
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However Choi and Series [CS06] have recently proved that, in this context, the matrix of the differential of the lengths with respect to the angles is negative definite. Since the θi (t) = tλi here, it follows that:
λi
i
dli (t) < 0, dt
and therefore I E H (C Mt ) is a strictly increasing function of t. Since it vanishes for t = 0, it follows that I E H (M) > 0. If the bending lamination of M is general – i.e., it is not supported on closed curves – the result can be obtained, thanks to the technical hypothesis in the lemma, by approximating M by quasi-Fuchsian manifolds for which the bending lamination of the boundary of the convex core is along closed curves; the continuity of the I E H under GromovHausdorff limit follows from [KS95] (see also [Lec04] for related phenomena). Corollary 4.7. Under the same hypothesis, W (C M ) ≥ 0, with equality exactly when M is Fuchsian. Proof. This immediately follows from the previous lemma since 1 W (M) = I E H (M) + H. 4 ∂M
These results should be compared to a recent result by Teo [Teo05] that shows that the W -volume extremized in a certain way, to be explained below, is positive. This other extremized volume is simply the volume of a different convex domain in M. Taken together, these results suggest that the W -volume of any convex domain N might be positive. We will not attempt to prove this statement in the present work. 5. Description “From Infinity” The metric at infinity. In this section we switch from a description of the renormalized volume from the boundary of a convex subset to the boundary at infinity of M. This description from infinity is remarkably similar to the previous one from the boundary of a convex subset. Lemma 5.1. Let M be a convex co-compact hyperbolic 3-manifold, and let N ⊂ M be compact and “strongly” convex with smooth boundary. Let Sρ be the equidistant surfaces from ∂ N . The induced metric on Sρ is asymptotic, as ρ → ∞, to (1/2)e2ρ I ∗ , where I ∗ = (1/2)(I + 2II + III ) is defined on ∂ N . Proof. Follows from Lemma 2.2. It is the metric I ∗ that will play such a central role in what follows, so we would like to state some of its properties. Lemma 5.2. The curvature of I ∗ is K ∗ :=
2K . 1 + H + Ke
(29)
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Proof. The Levi-Cività connection of I ∗ is given, in terms of the Levi-Cività connection ∇ of I , by: ∇x∗ y = (E + B)−1 ∇x ((E + B)y). This follows from checking the 3 points in the definition of the Levi-Cività connection of a metric: • ∇ ∗ is a connection. • ∇ ∗ is compatible with I ∗ . • It is torsion-free (this follows from the fact that E + B verifies the Codazzi equation: (∇x (E + B))y = (∇ y (E + B))x). Let (e1 , e2 ) be an orthonormal moving frame on S for I , and let β be its connection 1-form, i.e.: ∇x e1 = β(x)e2 , ∇x e2 = −β(x)e1 . Then the curvature of I is √defined as: dβ = −K da. Now let (e1∗ , e2∗ ) := 2((E + B)−1 e1 , (E + B)−1 e2 ); clearly it is an orthonormal moving frame for I ∗ . Moreover the expression of ∇ ∗ above shows that its connection 1-form is also β. It follows that K da = −dβ = K ∗ da ∗ , so that: K∗ = K
2K da K = = . ∗ da (1/2) det(E + B) 1 + H + Ke
We note that the metric I ∗ is defined for any surface S ⊂ M. However, it might have singularities (even when the surface S is smooth) unless S is strictly horospherically convex, i.e., its principal curvatures are less than 1 (which implies that it remains on the concave side of the tangent horosphere at each point). If S is a strictly horospherically convex surface S embedded in a hyperbolic end of M then the metric I ∗ is guaranteed to be in the conformal class of the (conformal) boundary at infinity of M. For a general surface S the “asymptotic” metric has nothing to do with the conformal infinity, and in particular, does not have to be in the conformal class of the boundary. The W -volume as a functional of I ∗ . We claim that the W -volume of a convex co-compact manifold M relative to a convex subset N is a functional of only the metric I ∗ that is built using the fundamental forms of the boundary ∂ N . This claim can be substantiated in several ways. One way is to refer to the results about the renormalized volume. It is known from [Kra00,TT03] that the renormalized volume of M is given by the so-called Liouville functional for the asymptotic metric I ∗ . To prove this result one uses an explicit foliation of the covering space H 3 by certain equidistant surfaces. The easy part of the computation is then to identify the “bulk” part of the Liouville action. The hard part is to show that all the boundary terms that arise are exactly the boundary of the fundamental domain terms necessary to define the Liouville action. As we have said, this computation is done in the covering space and is not particularly illuminating as far as the geometry of the problem is concerned. In this paper we would like to give a more geometric perspective. We demonstrate the above assertion by proving an explicit variational formula for W (N ) in terms of I ∗ . However, before we do this, we would like to introduce some other quantities defined “at infinity”.
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Second fundamental form at infinity. We have already defined the metric “at infinity”. Let us now add to this a definition of what can be called the second fundamental form at infinity. Definition 5.3. Given a surface S with the first, second and third fundamental forms I, II and III , we define the first and second fundamental forms “at infinity” as: 1 1 1 (I + 2II + III ) = (I + II )I −1 (I + II ) = I ((E + B)·, (E + B)·), (30) 2 2 2 1 1 1 II ∗ = (I − III ) = (I + II )I −1 (I − II ) = I ((E + B)·, (E − B)·). 2 2 2 I∗ =
It is then natural to define: B ∗ := (I ∗ )−1 II ∗ = (E + B)−1 (E − B),
(31)
and III ∗ := I ∗ (B ∗ ·, B ∗ ·) = I ((E − B)·, (E − B)·). Note that, for a surface which has principal curvatures strictly bounded between −1 and 1, III ∗ is also a smooth metric and its conformal class corresponds to that on the other component of the boundary at infinity. This is a simple consequence of Lemma 2.2 and the fact that when the principal curvatures are strictly bounded between −1, 1 the foliation by surfaces equidistant to S extends all the way through the manifold M. Such manifolds were called almost-Fuchsian in our work [KS05]. As before, those definitions make sense for any surface, but it is only for a convex surface (or more generally for a horospherically convex surface) that the fundamental forms so introduced are guaranteed to have something to do with the actual conformal infinity of the space. The Gauss and Codazzi equations at infinity. We also define H ∗ := tr(B ∗ ). The Gauss equation for “usual” surfaces in H 3 is replaced by a slightly twisted version. Remark 5.4. H ∗ = −K ∗ : the mean curvature at infinity is equal to minus the curvature of I ∗ . Proof. By definition, H ∗ = tr((E + B)−1 (E − B)). An elementary computation (for instance based on the eigenvalues of B) shows that H∗ =
2 − 2 det(B) . 1 + tr(B) + det(B)
But we have seen (as Eq. (29)) that K ∗ = 2K /(1 + H + K e ), the result follows because, by the Gauss equation, K = −1 + det(B). However, the “usual” Codazzi equation holds at infinity. ∗
Remark 5.5. d ∇ B ∗ = 0.
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Proof. Let u, v be vector fields on ∂∞ M. Then it follows from the expression of ∇ ∗ found above that: ∗
(d ∇ B ∗ )(x, y) = ∇x∗ (B ∗ y) − ∇ y∗ (B ∗ x) − B ∗ [x, y] = (E + B)−1 ∇x ((E + B)B ∗ y) − (E + B)−1 ∇ y ((E + B)B ∗ x) −B ∗ [x, y] = (E + B)−1 ∇x ((E − B)y) − (E + B)−1 ∇ y ((E − B)x) −(E + B)−1 (E − B)[x, y] = (E + B)−1 (d ∇ (E − B))(x, y) = 0.
Inverse transformations. The transformation I, II → I ∗ , II ∗ is invertible. The inverse is given explicitly by: Lemma 5.6. Given I ∗ , II ∗ the fundamental forms I, II such that (30) holds are obtained as: 1 ∗ 1 (I + II ∗ )(I ∗ )−1 (I ∗ + II ∗ ) = I ∗ ((E + B ∗ )·, (E + B ∗ )·)), 2 2 1 1 II = (I ∗ + II ∗ )(I ∗ )−1 (I ∗ − II ∗ ) = I ∗ ((E + B ∗ )·, (E − B ∗ )·)). 2 2 I =
(32)
Moreover, B = (E + B ∗ )−1 (E − B ∗ ).
(33)
Having an expression for the fundamental forms of a surface in terms of the one at infinity, one can re-write the metric of Lemma 2.2 induced on surfaces equidistant to S in terms of I ∗ , II ∗ . Lemma 5.7. The metric (16) induced on the surfaces equidistant to S can be re-written in terms of the fundamental forms “at infinity” as: Iρ =
1 2ρ ∗ 1 e I + II ∗ + e−2ρ III ∗ . 2 2
(34)
This lemma shows the significance of II ∗ as being the constant term of the metric. This lemma also shows clearly that when the equidistant foliation extends all the way through M (i.e. when the principal curvatures on S are in (−1, 1)), the conformal structure at the second boundary component of M is that of III ∗ = II ∗ (I ∗ )−1 II ∗ . Thus, in this particular case of almost-Fuchsian manifolds, the knowledge of I ∗ on both boundary components of M is equivalent to the knowledge of I ∗ , II ∗ near either component. In other words, II ∗ is determined by I ∗ . This statement is more general and works for manifolds other than almost-Fuchsian.
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Fundamental Theorem of surface theory “from infinity”. Let us now recall that the Fundamental Theorem of surface theory, Theorem 2.1, states that given I, II on S there is a unique embedding of S into the hyperbolic space. Then (16) gives an expression for the metric on equidistant surfaces to S, and thus describes a hyperbolic manifold M in which S is embedded, in some neighborhood of S. It would be possible to state a similar result for hyperbolic ends, uniquely determined by I ∗ and II ∗ at infinity. But there is also an analogous theorem, based on a classical result of Bers [Ber60], in which the first (and only the first) form at infinity is used. This can be compared with arguments used in [Sch02]. Theorem 5.8. Given a convex co-compact 3-manifold M, and a metric I ∗ (on all the boundary components of M) in the conformal class of the boundary, there is a unique foliation of each end of M by convex equidistant surfaces Sρ ⊂ M (1/2)(Iρ + 2IIρ + IIIρ ) = e2ρ I ∗ , where Iρ , IIρ , IIIρ are the fundamental forms of Sρ . Remark 5.9. Note that one does not need to specify II ∗ . The first fundamental form I ∗ (but on all the boundary components) is sufficient. Proof. The surfaces in question can be given explicitly as an embedding of the universal cover S˜ of S into the hyperbolic space. Thus, let (ξ, y), ξ > 0, y ∈ C be the usual upper half-space model coordinates of H 3 . Let us write the metric at infinity as I ∗ = eφ |dz|2 ,
(35)
where φ is the Liouville field covariant under the action of the Kleinian group giving M on S 2 . The surfaces are given by the following set of maps: Epsρ : S 2 → H 3 , z → (ξ, y) (here E ps stands for Epstein, who described these surfaces in [Eps84]): √ −ρ −φ/2 2e e ξ = , (36) 1 + (1/2)e−2ρ e−φ |φz |2 e−2ρ e−φ y = z + φz¯ . 1 + (1/2)e−2ρ e−φ |φz |2 As is shown by an explicit computation, the metric induced on the surfaces Sρ is given by (34) with 1 ¯ z¯ 2 ) + φz z¯ dzd z¯ , (θ dz 2 + θd 2 1 θ = φzz − (φz )2 . 2
II ∗ =
(37) (38)
Thus, we see that II ∗ is determined by the conformal factor in (35). Remark 5.10. This theorem itself implies that the renormalized volume only depends on I ∗ . Indeed, the foliation Sρ of the ends does depend only on I ∗ , and this foliation can be used for regularization and a subtraction procedure. Then the fact that the W -volume is essentially the renormalized volume implies that the W -volume is a functional of I ∗ only. In the next section we will find a formula for the first variation of this functional. Corollary 5.11. If the principal curvatures at infinity (eigenvalues of B ∗ ) are positive the map Epsρ is a homeomorphism onto its image for any ρ.
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Proof. We first note that the map E psρ is not always a homeomorphism, and the surfaces Sρ are not necessarily convex, but for sufficiently large ρ both things are true. A condition that guarantees that E psρ is a homeomorphism for any ρ is stated above. This condition can be obtained from the requirement that the principal curvatures of surfaces Sρ are in [−1, 1]. Let us consider the surface S := Sρ=0 the first and second fundamental forms of which are given by (32) (this immediately follows from (34)). The shape operator of this surface is then given by B = (E + B ∗ )−1 (E − B ∗ ). It is then clear that the principal curvatures of S are given by ki = (1 − ki∗ )/(1 + ki∗ ), where the ki∗ are the “principal curvatures” (eigenvalues) of B ∗ . The latter are easily shown to be given by ∗ = e−φ φz z¯ ± θ θ¯ . k1,2
(39)
It is now easy to see that the condition k1,2 ∈ (−1, 1) is equivalent to the condition ∗ > 0. This is a necessary and sufficient condition for the foliation by surfaces S to k1,2 ρ extend throughout M. If this condition is satisfied the map E psρ is a homeomorphism for any ρ. Interestingly, this condition makes sense not only in the quasi-Fuchsian situation but is more general. Thus, for example, it applies to the Schottky manifolds. But for the Schottky manifolds with their single boundary component the foliation by equidistant surfaces Sρ cannot be smooth for arbitrary ρ. It is clear that surfaces must develop singularities for some value of ρ. We therefore get a very interesting corollary: Corollary 5.12. There is no Liouville field φ on C invariant under a Schottky group such that φz z¯ is greater than |φzz − (1/2)φz2 | everywhere on C. Proof. Indeed, if such a Liouville field existed, we could have used it to construct a smooth equidistant foliation for arbitrary values of ρ, but this is impossible. A similar statement holds for a Kleinian group with more than two components of the domain of the discontinuity.
6. The Schläfli Formula “From Infinity” In this section we obtain a formula for the first variation of the renormalized volume. The computation of this section is a bit technical. Readers not interested in the details are advised to skip this section on the first reading. The result of the computation is given by the formula (45) below. The Schläfli formula. As we have seen in the previous sections, the renormalized volume of a convex co-compact hyperbolic 3-manifold M can be expressed as the W-volume of any convex domain N ⊂ M. The W-volume is equal to the volume of N minus the quarter of the integral of the mean curvature over the boundary of N . Let us consider what happens if one changes the metric in M. As was shown in [RS99], the following formula for the variation of the volume holds:
1 2δV (N ) = δ H + δ I, II da. (40) 2 ∂N
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Here H is the trace of the shape operator B = I −1 II , and the expression A, B stands for tr(I −1 AI −1 B). We can use this to get the following expression for the variation of the W-volume:
1 1 1 1 δ H + δ I, II da − δ H da − H δ(da), δW (N ) = 2 ∂N 2 4 ∂N 4 ∂N so that 1 δW (N ) = 4
∂N
H δ H + δ I, II − I da. 2
(41)
To get the last equality we have used the obvious equality da =
1 1 tr(I −1 δ I ) = δ I, I da. 2 2
(42)
The formula (41) can be further modified using δ H = δ(tr(I −1 II )) = −tr(I −1 (δ I )I −1 II ) + tr(I −1 δ II ) = − δ I, II + I, δ II .
(43)
We get 1 δW (N ) = 4
∂N
H δ II − δ I, I da. 2
(44)
It is this formula that will be our starting point for transformations to express the variation in terms of the data at infinity. Parameterization by the data at infinity. Let us now recall that, given the data I, II on the boundary of N one can introduce the first and second fundamental forms “at infinity” via (30). Conversely, knowing the fundamental forms I ∗ , II ∗ “at infinity” one can recover the fundamental forms on ∂ N via (32). Our aim is to rewrite the variation (44) of the W-volume in terms of the variations of the forms I ∗ , II ∗ . Lemma 6.1. The first-order variation of W can be expressed as 1 H∗ ∗ ∗ ∗ δW (N ) = − δ II − δ I , I da ∗ . 4 ∂N 2
(45)
Proof. Clearly the first-order variation of W contains two kinds of terms, those related to the first-order variation of I ∗ and those coming from the first-order variation of B ∗ , which we call δ B ∗ here. We define X := (I ∗ )−1 δ I ∗ , and consider separately the terms which are linear in X and those which are linear in δ B ∗ . To simplify notations we use the notation “O(X )” (resp. “O(δ B ∗ )”) to describe any term linear in X (resp. in δ B ∗ ). We will be using repeatedly a simple formula valid for any two 2 × 2 matrices A and B: det(A) tr(A−1 B) = tr(A) tr(B) − tr(AB). We consider first the terms linear in δ B ∗ . We already know that 2I = I ∗ ((E + B ∗ )·, (E + B ∗ )·),
(46)
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it follows that 2δ I = 2I ∗ ((E + B ∗ )·, δ B ∗ ) + O(X ), so that δ I, I = 2tr((E + B ∗ )−1 δ B ∗ ) + O(X ). Similarly, we know that 2II = I ∗ ((E + B ∗ )·, (E − B ∗ )·), it follows that 2δ II = I ∗ (δ B ∗ ·, (E − B ∗ )·) − I ∗ ((E + B ∗ )·, δ B ∗ ·) + O(X ), and therefore that δ II, I = tr((E + B ∗ )−1 (E − B ∗ )(E + B ∗ )−1 δ B ∗ ) − tr((E + B ∗ )−1 δ B ∗ ) + O(X ). Moreover H = tr((E + B ∗ )−1 (E − B ∗ )), and putting all terms together shows that H δ II − δ I, I = −tr((E + B ∗ )−1 δ B ∗ ) 2 +tr((E + B ∗ )−1 (E − B ∗ )(E + B ∗ )−1 δ B ∗ ) −tr((E + B ∗ )−1 (E − B ∗ ))tr((E + B ∗ )−1 δ B ∗ ) + O(X ). The last two terms can be treated as the right-hand side of (46), the equation becomes: H δ II − δ I, I = −tr((E + B ∗ )−1 δ B ∗ ) − det((E + B ∗ )−1 (E − B ∗ ))tr 2 ×((E − B ∗ )−1 δ B ∗ ) + O(X ). We now apply (46) to each of the two terms on the right-hand side and get for the above quantity: tr((E + B ∗ )δ B ∗ )−tr(E + B ∗ )tr(δ B ∗ )−tr(E − B ∗ )tr(δ B ∗ ) + tr((E − B ∗ )δ B ∗ ) + O(X ) det(E + B ∗ ) 2tr(δ B ∗ ) − 4tr(δ B ∗ ) −2tr(δ B ∗ ) = + O(X ) = + O(X ). ∗ det(E + B ) det(E + B ∗ ) The part which is linear in X can be computed in a similar way. First note that, by definition of X , δ I, I = tr(X ) + O(δ B ∗ ), while δ II = I ∗ ((E + B ∗ )·, X (E − B ∗ )·) + O(δ B ∗ ),
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so that δ II, I = tr((E + B ∗ )−1 X (E − B ∗ )) + O(δ B ∗ ). It follows, using (46) and forgetting all terms linear in δ B ∗ , that δ II −(H/2)δ I, I = tr((E + B ∗ )−1 X (E − B ∗ ))−(1/2)tr((E + B ∗ )−1 (E − B ∗ ))tr(X ) 1 = (tr(E + B ∗ )tr(X (E − B ∗ ))−tr((E + B ∗ )X (E − B ∗ )) det(E + B ∗ ) −(1/2)tr(E + B ∗ )tr(E − B ∗ )tr(X ) + (1/2)tr(E − (B ∗ )2 )tr(X )) 1 = (tr(E + B ∗ )tr(X (E − B ∗ )) − tr((E − (B ∗ )2 )X ) det(E + B ∗ ) −(1/2)(tr(E)2 − tr(B ∗ )2 )tr(X ) + (1/2)tr(E − (B ∗ )2 )tr(X )). The terms involving (B ∗ )2 can be replaced using the fact that (B ∗ )2 − tr(B ∗ )B ∗ + det(B ∗ )E = 0, so that tr((B ∗ )2 ) = tr(B ∗ )2 − 2 det(B ∗ ). It follows that, still forgetting all terms which are linear in δ B ∗ , we have 1 (tr(E + B ∗ )tr(X (E − B ∗ )) det(E + B ∗ ) −tr((1 + det(B ∗ ))X − tr(B ∗ )B ∗ X ) − (1/2)(4 − tr(B ∗ )2 )tr(X )+(1/2)(2 − tr(B ∗ )2 +2 det(B ∗ ))tr(X )) 1 = ((2 + tr(B ∗ ))(tr(X ) − tr(B ∗ X )) det(E + B ∗ ) − (1 + det(B ∗ ))tr(X ) + tr(B ∗ )tr(B ∗ X ) + (det(B ∗ ) − 1)tr(X )) tr(B ∗ )tr(X ) − 2tr(B ∗ X ) . = det(E + B ∗ )
δ II − (H/2)δ I, I =
Putting together the terms in X and the terms in δ B ∗ , and using the fact that da ∗ = (1/2) det(E + B ∗ )da by definition of I ∗ , we find that: δ II − (H/2)δ I, I = −(tr(δ B ∗ ) + tr(B ∗ X ) − (1/2)tr(B ∗ )tr(X ))(da ∗ /da) = − δ II ∗ − (H ∗ /2)δ I ∗ , I ∗ (da ∗ /da), and the result clearly follows. Formula (45) looks very much like the original formula (44), except for the minus sign and the fact that the quantities at infinity are used. The fact that we have got the same variational formula as in terms of the data on ∂ N is not too surprising. Indeed, the variational formula (45) was obtained from (44) by applying the transformation (32). As it is clear from (30), this transformation applied twice gives the identity map. In view of this, it is hard to think of any other possibility for the variational formula in terms of δ I ∗ , δ II ∗ except being given by the same expression (44), apart from maybe with a different sign. This is exactly what we see in (45). There is another expression of the first-order variation of W , dual to (41), which will be useful below.
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Corollary 6.2. The first-order variation of W can also be expressed as 1 δ H ∗ + δ I ∗ , II0∗ da ∗ , δW = − 4 ∂N where II0∗ is the traceless part (for I ∗ ) of II ∗ . 7. Conformal Variations of the Metric at Infinity In this section we use Corollary 6.2 to prove one simple corollary. Thus, we show that, when varying the W-volume with the area of the boundary defined by the I ∗ metric kept fixed, the variational principle implies the metric I ∗ to have constant negative curvature. The variations we consider in this section do not change the conformal structure of the metric I ∗ , and thus do not change the manifold M. Geometrically they correspond to small movements of the surface ∂ N inside the fixed manifold M. First variation. We consider in this section a conformal deformation of the metric I ∗ , i.e., δ I ∗ = 2u I ∗ , where u is some function on ∂ N . Clearly for such variations δ I ∗ , II0∗ = 0, precisely because II0∗ is traceless. Let us consider the following functional: λ F(N ) = W (N ) − da ∗ , (47) 4 ∂N appropriate for finding an extremum of the W-volume with the area computed using the metric I ∗ kept fixed. The first variation of this functional gives, using Corollary 6.2: 1 λ 1 λ ∗ ∗ ∗ ∗ ∗ δF = − (δ H )da − 4uda = (δ K )da − 4uda ∗ . 4 ∂N 4 ∂N 4 ∂N 4 ∂N But
δ
∂N
∗
∗
K da =
∂N
(δ K ∗ ) + 4u K ∗ da ∗ = 0
by the Gauss-Bonnet formula, so that δF = (−u K ∗ − uλ)da ∗ . ∂N
It follows that critical points of F are characterized by the fact that K ∗ = −λ. Second variation. In this paragraph we would like to verify whether the extremum found above is maximum or minimum. It is easy to compute the formula for the second variation, but for the sake of simplicity we only do it here for conformal deformations of I ∗ , corresponding to movements of ∂ N . According to the formula found right above, the gradient of F on the space of metrics conformal to I ∗ is simply D F = −(K ∗ + λ)/4. However a well-known formula on the conformal deformations of metrics (see e.g. Chapter 1 of [Bes87]) indicates that, under a conformal deformation δ I ∗ = 2v I ∗ , δ K ∗ = −2v K ∗ + v,
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where is the Laplacian for I ∗ . It follows directly that the Hessian of F at a critical point is given by (Hess(F))(2u I ∗ , 2v I ∗ ) =
∂N
2K ∗ uv − ( v)uda ∗ =
∂N
−2λuv − du, dv da ∗ .
This quantity is negative definite (because λ = −K ∗ > 0 by Gauss-Bonnet) so that the critical points of F are local maxima.
8. The Renormalized Volume as a Function on Teichmüller Space In this section we consider the renormalized volume as a function over the Teichmüller space of ∂ N ; in other terms, for each conformal class on ∂ N , we consider the extremum of W over metrics of given area within this conformal class. We have seen in the previous section that this extremum is obtained at the (unique) constant curvature metric. The main goal here is to recover by simple differential geometric methods important results of McMullen [McM00]—concerning his “quasi-Fuchsian reciprocity”—and Takhtajan and Zograf [TZ87], Takhtajan and Teo [TT03] – showing that the renormalized volume provides a Kähler potential for the Weil-Petersson metric. So the “volume” that we consider here is now defined as follows. Definition 8.1. Let g be a convex co-compact hyperbolic metric on M, and let c ∈ T∂ M be the conformal structure induced on ∂∞ M. We call W M (c) the value of W on the equidistant foliation of M near infinity for which I ∗ has constant curvature −1. In other terms, by the results obtained in the previous section, W M (c) is the maximum of W over the metrics at infinity which have the same area as the hyperbolic metric, for each boundary component of M. Throughout this section the metric at infinity I ∗ that we consider is the hyperbolic metric, while the second fundamental form at infinity, II ∗ , is uniquely determined by the choice that I ∗ is hyperbolic. Its traceless part is denoted by II0∗ . The second fundamental form at infinity as the real part of a HQD. It is interesting to remark that, in the context considered here – when I ∗ has constant curvature – the second fundamental form at infinity has a complex interpretation. This can be compared with the same phenomenon, discovered by Hopf [Hop51], for the second fundamental form of constant mean curvature surfaces in 3-dimensional constant curvature spaces. Lemma 8.2. When K ∗ is constant, II0∗ is the real part of a quadratic holomorphic differential (for the complex structure associated to I ∗ ) on ∂∞ M. This holomorphic quadratic differential is given explicitly by (38). Proof. By definition II0∗ is traceless, which means that it is at each point the real part of a quadratic differential: II0∗ = Re(h). Moreover, we have seen in Remark 5.5 that ∗ B ∗ satisfies the Codazzi equation, d ∇ B ∗ = 0. It follows as for constant mean curvature surfaces (see e.g. [KS05]) that h is holomorphic relative to the complex structure of I ∗ .
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The second fundamental form as a Schwarzian derivative. The next step is that, for each boundary component ∂i M of M, II0i∗ is actually the real part of the Schwarzian derivative of a natural equivariant map between the hyperbolic plane (with its canonical complex projective structure) to ∂i M with its complex projective structure induced by the hyperbolic metric on M. In the terminology used by McMullen [McM00], II0i∗ is the difference between the complex projective structure at infinity on ∂i M and the Fuchsian projective structure on ∂i M. A simple way to prove this assertion is to use the formula (38) for the holomorphic quadratic differential θ whose real part gives the traceless part of II ∗ . The Liouville field φ that enters into this formula can be simply expressed in terms of the conformal map from ∂i M to the hyperbolic plane. It is then a standard and simple computation to verify that θ is equal to the Schwarzian derivative of this map, see e.g. [TZ87]. To make this paper self-contained we decided to include yet a different, more geometric proof, which is spelled out in the Appendix. The proof we give is elementary, based on the conformal factor between the I ∗ metrics on corresponding surfaces in two foliations. It can be compared to the argument used in [McM00]. To state the result, let us call σ F the “Fuchsian” complex projective structure on ∂i M, obtained by applying the Poincaré uniformization theorem to the conformal metric at infinity on ∂i M. The universal cover of ∂i M, with the complex projective structure lifted from σ F , is projectively equivalent to a disk in CP 1 . We also call σ Q F the projective structure induced on ∂i M by the hyperbolic metric on M. Here “Q F” stands for quasiFuchsian (while M is only supposed to be convex co-compact), this notation is used to keep close to the notation in [McM00]. The map φ : (∂i M, σ F ) → (∂i M, σ Q F ) is conformal but not projective, so we can consider its Schwarzian derivative S(φ). Lemma 8.3. II0∗ = −Re(S(φ)). It is possible to reformulate this statement slightly by setting θi := S(φ) (this is analogous to the notations used in [McM00], the index i is useful to recall that this quantity is related to ∂i M). Then θi is a quadratic holomorphic differential (QHD) on ∂i M, and, still using the notations in [McM00], the definition of θi can be rephrased as: θi = σ Q F − σ F . The lemma can then be written as: II0i∗ = Re(θi ). A geometric proof of this lemma is given in the Appendix. Remark 8.4. Note that θi can also be considered as a complex-valued 1-form on the Teichmüller space of ∂i M. Indeed, it is well known that the cotangent vectors to T S , where S is a Riemann surface, can be described as holomorphic quadratic differentials q on S. The pairing with a tangent vector (Beltrami differential µ) is given by the integral of qµ over S. The complex structure on T S can then be described as follows: the image of the cotangent vector q under the action of the complex structure J is simply J (q) = iq. Another, more geometric way to state the action of J , is to note that it replaces the horizontal and vertical trajectories of q. Thus, holomorphic quadratic differentials q on S are actually holomorphic 1-forms on T S . The second fundamental form as the differential of W M . There is another simple interpretation of the traceless part of the second fundamental form at infinity. Lemma 8.5. The differential dW M of the renormalized volume W M , as a 1-form over the Teichmüller space of ∂ M, is equal to (−1/4)II0∗ .
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Proof. This is another direct consequence of Corollary 6.2 because, as one varies I ∗ among hyperbolic metrics, H ∗ (which is equal to K ∗ ) remains equal to −1, so that δ H ∗ = 0. Corollary 8.6. θi = −4∂ W M . Proof. This follows directly from the lemma, since we already know that θi is a holomorphic differential. Remark 8.7. We would like to emphasize how much simpler is the proof given above than that given in [TZ87,TT03]. Unlike in these references, which obtain the above result on the gradient of W M using an involved computation, Corollary 6.2 implies this result in one line. This demonstrates the strength of the geometric method used here. McMullen’s quasi-Fuchsian reciprocity. We can now recover McMullen’s quasiFuchsian reciprocity as a simple consequence of the proved above relation between W M and the second fundamental form at infinity. In the context considered here, it is just a consequence of the fact that the Hessian of a function is symmetric. From this point and until the end of this section we suppose that M is quasi-Fuchsian. It has two boundary components, which we call ∂− M and ∂+ M, which are homeomorphic. The renormalized volume W M is now a function on the space of quasi-Fuchsian metrics on M, which by the Bers theorem is T∂− M × T∂+ M . Let (c− , c+ ) ∈ T∂− M × T∂+ M ; they define a unique hyperbolic metric on M, and therefore a complex projective structure σ Q F,+ (c− , c+ ) (resp. σ Q F,− (c− , c+ )) on ∂+ M (resp. ∂− M). Using the Schwarzian derivative construction this can be used to define a QHD β+ (c− , c+ ) (resp. β− (c− , c+ )) on ∂+ M (resp. ∂− M) as β+ (c− , c+ ) = σ Q F,+ (c− , c+ ) − σ F (c+ ), and respectively β− (c− , c+ ) = σ Q F,− (c− , c+ ) − σ F (c− ). Then β+ (c− , c+ ) is a HQD on ∂+ M, and can therefore be identified with a complexvalued 1-form on T∂+ M . So c− determines a (complexified) cotangent vector field β+ (c− , ·) on T∂+ M , and similarly c+ determines a cotangent vector field β− (·, c+ ) on T∂− M . By Lemma 8.3 above, ∗ ∗ II0,+ = −Re(β+ (c− , c+ )), II0,− = −Re(β− (c− , c+ )).
Yet another way to state this relationship is that ∗ ∀v+ ∈ Tc+ T∂+ M , Re(β+ (c− , c+ )(v+ )) = − II0,+ , v+ ,
(48)
and similarly for β− . Using Lemma 8.5, β+ and β− can be expressed in terms of W M as follows: ∀v+ ∈ Tc+ T∂+ M , Re(β+ (c− , c+ )(v+ )) = 4dW M ((0, v+ )),
(49)
and similarly for β− . Here W M is considered as a function on T∂− M × T∂+ M , and (0, v+ ) is a vector tangent to T∂− M × T∂+ M .
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We can now consider the differential of the map β+ (·, c+ ) considered as a function of the conformal structure on ∂− M, Dβ+ (·, c+ ) : Tc− T∂− M → Tc∗+ T∂+ M , and of β− (c− , ·), considered as a function of the conformal structure on ∂+ M, Dβ− (c− , ·) : Tc+ T∂+ M → Tc∗− T∂− M . Theorem 8.8. (McMullen’s quasi-Fuchsian reciprocity [McM00]) The maps Dβ+ (·, c+ ) and Dβ− (c− , ·) are adjoint to each other. Proof. Let v− ∈ Tc− T∂− M and v+ ∈ Tc+ T∂+ M . Then Re( Dβ+ (c− , c+ )(v− , 0), v+ ) = 4(D(v− ,0) dW M )((0, v+ )) = 4(Hess(W M ))((v− , 0), (0, v+ )). (Note that the Hessian here can be considered without reference to a specific connection, because (v− , 0) and (0, v+ ) are tangent to T∂− M and T∂+ M respectively.) The symmetry in the right-hand side shows quite clearly that Re(Dβ+ (c− , c+ )(v− , 0)), v+ = Re(Dβ− (c− , c+ )(0, v+ )), v− , which directly yields the theorem. The renormalized volume as a Kähler potential. Finally we show here how to recover in this manner the result of Takhtajan and Teo [TT03] stating that the renormalized volume W M with c− fixed is a Kähler potential for the Weil-Petersson metric on T∂+ M . To simplify notations a little, we set θc− := β+ (c− , ·). Since we already know that θc− = 4∂ W M , we only need to prove that ∂(iθc− ) = −2ωW P , where ωW P is the Kähler form of the Weil-Petersson metric on T∂+ M . An important part of the argument is that dθc− , as a 2-form on T∂+ M , does not depend on c− . This appears as Theorem 7.2 in McMullen’s paper [McM00]. We include a proof for completeness, following the proof given in [McM00]. Proposition 8.9. The differential dθc− , considered as a complex-valued 2-form on T∂+ M , does not depend on c− . Proof. Let v− ∈ Tc− T∂− M ; we want to show that the corresponding first-order variation Dv− (dθc− ) of dθc− vanishes. This will follow from the fact that the first-order variation of θc− corresponding to v− , Dv− θc− , is the differential of a function defined on T∂+ M , namely the function f v− defined by f v− (c+ ) = β− (c− , c+ ), v− , where , is the WP pairing. The fact that Dv− θc− = d f v− can be proved by evaluating both sides on a vector v+ ∈ Tc+ T∂+ M and using the quasi-Fuchsian reciprocity. Dv− θc− , v+ = Dβ+ (c− , c+ )(v− , 0), v+ , = Dβ− (c− , c+ )(0, v+ ), v− = d f v− (v+ ). It clearly follows that dθc− , as a 2-form on T∂+ M , does not depend on c− .
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That W M is a Kähler potential is then reduced to a simple computation in the Fuchsian situation. Proposition 8.10. Suppose that M is a Fuchsian manifold, with c+ = c− . Let I ∗ be the hyperbolic metric in the conformal class c+ . Under a first-order deformation which does not change c− , the variation of I ∗ and II0∗ on ∂+ M are related by: δ II0∗ = −δ I ∗ . Proof. It follows from the constructions in the Appendix that B ∗ = (1/2)E. Under a first-order variation we have that δ II ∗ = (1/2)δ I ∗ + I ∗ (δ B ∗ ·, ·), δ III ∗ = δ I ∗ (B ∗ ·, B ∗ ·) + 2II ∗ (δ B ∗ ·, B ∗ ·) = (1/4)δ I ∗ + I ∗ (δ B ∗ ·, ·), so that δ III ∗ = 2δ ∗ II ∗ − δ I ∗ . But III ∗ is the hyperbolic metric in the conformal class c− and therefore it does not change under the deformation, so that 2δ II ∗ = δ I ∗ . Moreover, since I ∗ remains hyperbolic, H ∗ = tr I ∗ II ∗ = 1, so that II0∗ = II ∗ − (1/2)I ∗ , and so δ II0∗ = δ II ∗ − (1/2)δ I ∗ = −δ I ∗ .
We can reformulate this statement by calling θ R := Re(θc− ), so that, by Lemma 8.3, θ R (X ) = − X, II0∗ . Using the previous proposition, this can then be stated as (D X θ R )(Y ) = X, Y W P , where D is the Levi-Cività connection of the Weil-Petersson metric on T∂+ M . We can now compute explicitly an expression of ∂θc− , denoting by J the complex structure on T∂− M . ∂θc− (X, Y ) = (D X θc− )(Y ) + i(D J X θc− )(Y ) = (D X θ R )(Y )−i(D X θ R )(J Y )+i((D J X θ R )(Y )−i(D J X θ R )(J Y )) = X, Y − i X, J Y + i J X, Y + J X, J Y = 2( X, Y − i X, J Y ).
(50) (51) (52) (53)
This means precisely that ∂θc− (X, J X ) = 2iX 2W P , and we recover the result of Takhtajan and Teo [TT03] that W M is a Kähler potential for the Weil-Petersson metric. A. Appendix: A Geometric Proof of II0∗ = − Re(S(φ)). The proof is based on some preliminary but simple arguments on the conformal factor relating corresponding surfaces in the equidistant foliations of a quasi-Fuchsian and a Fuchsian manifold. We consider a point x0 ∈ ∂i M. For each point x in some neighborhood of x0 there is then a unique geodesic γx , orthogonal to the leaves of the foliation by the surfaces ρ already introduced above, with endpoint x. In addition to M, we also consider the “Fuchsian” manifold M with boundary at infinity equal to the disjoint union of two
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copies of ∂i M (with its conformal metric). So M is topologically ∂i M × R, with the hyperbolic metric dρ 2 + cosh2 (ρ)I ∗ ,
(54)
where I ∗ is the hyperbolic metric on ∂i M. Using this construction we can identify ∂i M also with the “upper” boundary component of M . To avoid ambiguities we call this upper boundary component ∂i M , and consider the map φ defined above as a map between ∂i M and ∂i M, and call x0 the point corresponding to x0 on ∂i M . There is a canonical foliation on M , given by the level surfaces ρ of ρ in (54), to which we can associate a metric I ∗ and a second fundamental form II ∗ at infinity on ∂i M . Taking the right choice of foliation near each boundary component (which involves an affine transformation on ρ) leads to 2I ∗ = II ∗ = 2φ ∗ I ∗ , so that B ∗ = (1/2)E. We now consider the geodesics γx0 (in M) and γx (in M ), with their respective para0 meterizations. There is a unique hyperbolic isometry 0 (defined from a neighborhood of γx to a neighborhood of γx0 ) sending γx to γx0 , preserving the parameterization by 0
0
ρ, and such that the differential at x0 of the boundary map φ0 : ∂∞ H 3 → ∂∞ H 3 is tangent to the isometry between I ∗ and I ∗ . Then φ0 is a complex projective map but it is not an isometry between I ∗ and I ∗ (although its differential at x0 is isometric). We will consider the map φ0−1 ◦ φ : ∂i M → ∂i M , and show that it is tangent to the identity at order 2 at x0 ; this shows that φ0 is the “best” approximation of φ at x0 by a projective map, and therefore that the third derivative of φ0−1 ◦ φ at x0 determines the Schwarzian derivative S(φ) at x0 (see e.g. [OS98], (1.3)). By construction, (φ0−1 ◦ φ)∗ I ∗ = (φ0−1 )∗ (φ∗ I ∗ ) = (φ0−1 )∗ I ∗ = φ0∗ I ∗ ,
so φ0−1 ◦φ is an isometry between I ∗ and φ0∗ I ∗ . Those two metrics on ∂i M are conformal, we will show that the Hessian at x0 of the conformal factor is φ ∗ II0∗ , and deduce from this the proof of Lemma 8.3. Proposition A.1. The shape operator of ρ is given by Bρ = (E + e−2ρ B ∗ )−1 (E − e−2ρ B ∗ ), so that, as ρ → ∞, Bρ − E −2e−2ρ B ∗ . Note that Bρ and B ∗ are defined at different points (Bρ on ρ and B ∗ on ∂i M) which are implicitly identified here through the Gauss flow. Proof. We have already computed above that Iρ = (1/2)(e2ρ I ∗ + 2II ∗ + e−2ρ III ∗ ) = (1/2)I ∗ ((eρ E + e−ρ B ∗ )·, (eρ E + e−ρ B ∗ )·). It follows that IIρ =
1 1 d Iρ 1 = (e2ρ I ∗ − e−2ρ III ∗ ) = I ∗ ((eρ E + e−ρ B ∗ )·, (eρ E − e−ρ B ∗ )·), 2 dρ 2 2
and the result is a direct consequence. Definition A.2. For each x in some neighborhood of x0 and all ρ large enough, we call dρ (x) the oriented distance, along γx , between its intersection with 0 (ρ ) and with ρ . Then we define d∞ (x) := limρ→∞ dρ (x).
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Proposition A.3. The limit distance d∞ vanishes at x0 along with its derivative, and its Hessian at x0 is equal to II0∗ . Proof. By construction, γx is orthogonal to ρ . But 0 (ρ ) is tangent to ρ at their common intersection with γx0 , so that dρ vanishes along with its derivative at x0 . Moreover, the second-order variation of dρ at x0 is determined by the difference between the shape operators at their intersection point with γx (for x close to x0 ) of 0 (ρ ) and of ρ . But, by Proposition A.1 applied to ρ and to ρ , the dominant term in the difference between those shape operators is Bρ − Bρ e−2ρ (2B ∗ − E). Integrating this second derivative of dρ shows that the Hessian of dρ is equal to Iρ (e−2ρ (2B ∗ − E)·, ·), which is equivalent to (1/2)(2II ∗ − I ∗ ), i.e., to II0∗ . The result then follows by taking the limit as ρ → ∞. Since φ0 is a projective map, it is conformal, and therefore the metrics I ∗ and φ0∗ I ∗ on ∂i M are conformal, which means that there exists a function u : ∂i M → R such that I ∗ = e2u φ0∗ I ∗ . Proposition A.4. u(x0 ) = 0 and du = 0 at x0 , while the Hessian of u at x0 is equal to II0∗ . Proof. First note that by definition of I ∗ it can be described in terms of the Gauss map, which we call G ρ , from ρ to ∂i M, sending a point x ∈ ρ to the endpoint at infinity of the geodesic ray starting from x orthogonal to ρ : I ∗ = (1/2)e−2ρ G ρ∗ (Iρ + 2IIρ + IIIρ ) 2e−2ρ G ∗ Iρ . The same can be said of 0 (ρ ), using the Gauss map G ρ of ρ : I ∗ (1/2)e−2ρ G ρ∗ (Iρ + 2IIρ + IIIρ ) 2e−2ρ G ρ∗ Iρ . Considering the image of ρ by 0 and the behavior of the geodesic rays orthogonal to 0 (ρ ) then shows that, in the neighborhood of x0 , φ0∗ (G ρ∗ Iρ ) e2dρ G ρ∗ Iρ , and the result follows by taking the limit as ρ → ∞. Proof of Lemma 8.3. Let f := φ0−1 ◦ φ. By construction the map f : ∂i M → ∂i M sends x0 to x0 , and it is tangent to the identity at x0 . We have seen above that f is an isometry between I ∗ and φ0∗ I ∗ , so it follows from Proposition A.4 that | f (z)|2 = e−2u , where u(x0 ) = 0, du = 0 at x0 , and the Hessian of u at x0 is II0∗ . Since (∂i M , I ∗ ) is hyperbolic, we can identify a neighborhood of x0 with a neighborhood of 0 in the unit disk. Since f is conformal it is then holomorphic, and can be written as: f (z) = z +
b 2 c 3 z + z + o(z 3 ), 2 6
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with b = f (0) and c = f (0), and clearly | f (z)| = e−u . But the fact that du = 0 at x0 clearly implies that b = 0, and it follows that:
2 c
| f (z)|2 = 1 + z 2 + o(z 2 ) = 1 + Re(cz 2 ) + o(z 2 ). 2 This means that: Re(cz 2 ) = −II0∗ (z, z), where II0∗ is considered as a real-valued symmetric bilinear form on C, identified with Tx0 ∂i M . So Re((φ0−1 ◦ φ) (x0 )) = −II0∗ , which means precisely that 2II0∗ = −Re(S(φ)), as claimed. Acknowledgements. We would like to thank Rafe Mazzeo for useful comments on a previous version of this text, and in particular for pointing out reference [PP01].
References Anderson, M.T.: L 2 curvature and volume renormalization of ahe metrics on 4-manifolds. Math. Res. Lett. 8(1-2), 171–188 (2001) [Ber60] Bers, L.: Simultaneous uniformization. Bull. Amer. Math. Soc. 66, 94–97 (1960) [Bes87] Besse, A.: Einstein Manifolds. Berlin-Heidelberg-NewYork: Springer, 1987 [BK99] Balasubramanian, V., Kraus, P.: A stress tensor for anti-de sitter gravity. Commun. Math. Phys. 208(2), 413–428 (1999) [BO04] Bonahon, F., Otal, J.-P.: Laminations mesurées de plissage des variétés hyperboliques de dimension 3. Ann. Math. 160, 1013–1055 (2004) [Bon98] Bonahon, F.: Variations of the boundary geometry of 3-dimensional hyperbolic convex cores. J. Differ. Geom. 50(1), 1–24 (1998) [CS06] Choi, Y.-E., Series, C.: Lengths are coordinates for convex structures. J. Differ. Geom. 73(1), 75–117 (2006) [Eps84] Epstein, C.L.: Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. Preprint, Princeton Univ, 1984 [Hop51] Hopf, H.: Über flächen mit einer relation zwischen den hauptkrümmungen. Math. Nachr. 4, 232–249 (1951) [HS98] Henningson, M., Skenderis, K.: The holographic weyl anomaly. JHEP 9807, 023 (1998) [Kra00] Krasnov, K.: Holography and riemann surfaces. Adv. Theor. Math. Phys. 4(4), 929–979 (2000) [Kra03] Krasnov, K.: On holomorphic factorization in asymptotically AdS 3-D gravity. Class. Quant. Grav. 20, 4015–4042 (2003) [KS95] Keen, L., Series, C.: Continuity of convex hull boundaries. Pacific J. Math. 168(1), 183–206 (1995) [KS05] Krasnov, K., Schlenker, J.-M.: Minimal surfaces and particles in 3-manifolds. Geometriae Dedicata 126, 187–254 (2007) [Lec04] Lecuire, C.: Continuity of the bending map, http://arxiv.org/list/math/0411412, 2004 [Lec06] Lecuire, C.: Plissage des variétés hyperboliques de dimension 3. Invent. Math. 164(1), 85–141 (2006) [McM00] McMullen, C.T.: The moduli space of riemann surfaces is kähler hyperbolic. Ann. of Math. (2) 151(1), 327–357 (2000) [Mil94] Milnor, J.: The Schläfli differential equality. In: Collected papers, Vol. 1. Berkeley, CA: Publish or Perish, 1994 [MS06] Moroianu, S., Schlenker, J.-M.: Quasi-Fuchsian manifolds with particles. http://arxiv.org/list/ math.DG/0603441, 2006 [OS98] Osgood, B., Stowe, D.: The schwarzian derivative, conformal connections, and möbius structures. J. Anal. Math. 76, 163–190 (1998) [PP01] Patterson, S.J., Perry, P.A.: The divisor of Selberg’s zeta function for Kleinian groups. Duke Math. J. 106(2), 321–390 (2001) (Appendix A by Charles Epstein) [Ando1]
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Rivin, I., Hodgson, C.D.: A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math. 111, 77–111 (1993) Rivin, I., Schlenker, J.-M.: The schläfli formula in einstein manifolds with boundary. Electronic Research Announcements of the A.M.S. 5, 18–23 (1999) Schlenker, J.-M.: Hypersurfaces in h n and the space of its horospheres. Geom. Funct. Anal. 12(2), 395–435 (2002) Teo, L.-P.: A different expression of the weil-petersson potential on the quasi-fuchsian deformation space. Lett. Math. Phys. 73(2), 91–107 (2005) Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324(2), 793–821 (1991) Takhtajan, L.A., Teo, L.-P.: Liouville action and weil-petersson metric on deformation spaces, global kleinian reciprocity and holography. Commun. Math. Phys. 239(1-2), 183–240 (2003) Takhtajan, L., Zograf, P.: On uniformization of Riemann surfaces and the Weil-Petersson metric on the Teichmüller and Schottky spaces. Mat. Sb. 132, 303–320 (1987); English translation in Math. USSR Sb. 60, 297–313 (1988) Witten, E.: Anti de sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998) Witten, E.: 2 + 1-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311(1), 46–78 (1988)
Communicated by L. Takhtajan
Commun. Math. Phys. 279, 669–703 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0416-6
Communications in
Mathematical Physics
Inverse Scattering Problem for a Two Dimensional Random Potential Matti Lassas1 , Lassi Päivärinta2 , Eero Saksman2 1 Department of Mathematics, Helsinki University of Technology, P.O. Box 1100,
02015 TKK, Finland. E-mail: [email protected]
2 Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68,
00014 Helsinki, Finland. E-mail: [email protected] Received: 30 January 2007 / Accepted: 21 June 2007 Published online: 4 March 2008 – © Springer-Verlag 2008
Abstract: We study an inverse problem for the two-dimensional random Schrödinger equation (∆ + q + k 2 )u = 0. The potential q(x) is assumed to be a Gaussian random function whose covariance operator is a classical pseudodifferential operator. We show that the backscattered field, obtained from a single realization of the random potential q, determines uniquely the principal symbol of the covariance operator of q. The analysis is carried out by combining harmonic and microlocal analysis with stochastic methods. Contents 1. 2. 3. 4. 5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the Problem and Results . . . . . . . . . . . . . . . . Regularity of the Potential . . . . . . . . . . . . . . . . . . . . . . . Direct Scattering Problem for a Distributional Potential . . . . . . . The First Order Term in the Born Series . . . . . . . . . . . . . . . . Higher Order Terms . . . . . . . . . . . . . . . . . . . . . . . . . . Existence of the Measurement: Convergence of the Ergodic Averages Conclusion: Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . .
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1. Introduction In inverse scattering theory the aim is to determine the scattering potential q from appropriate measurements. In many applications the scatterer is non-smooth and vastly complicated. For such scatterers, the inverse problem is not so much to recover the exact micro-structure of an object but merely to determine the parameters or functions describing the properties of the micro-structure. An example of such a parameter is the correlation length of the medium which is related to the typical size of “particles” inside the scatterer. In mathematical terms, one assumes that the potential has been created by a random process. This causes the scattered field to be random, as well.
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In applied literature the measured data is often assumed to coincide with the averaged data. This corresponds to the case when the measurements could be made from many independent samples of the scatterer and these measurements could be averaged. This appears not always to be a well justified assumption since often the scatterer does not change during the period of measurements. Also, in applications the multiple scattering is often omitted. This leads to a linearization of the inverse problem which approximates the original problem only when q is small. The main result of this paper (Theorem 1 below) shows that suitable mean values over the frequency k of the backscattered amplitude, obtained from a single realization of the random potential, almost surely determine the micro-structure of the random potential. More exactly, it determines the principal symbol of the covariance operator of the random function q. Below in Sect. 2 we describe these results in detail. Note that, after the model for the random potential is fixed, no approximations are made. This means that we study the full non-linear inverse problem. What is interesting, our stochastic setup leads to a new type of analytic problems. Our tools include basic stochastic analysis for generalized Gaussian fields, and, especially, we make use of harmonic and microlocal analysis, techniques that are also often used in the deterministic case, cf. [12,34,35,52]. An extensive review for this is given in [55]. In the stochastic settings the realizations of the potential are tempted to be rough, in many natural cases not even functions. For inverse problems involving non-smooth deterministic structures see e.g. [5,11,20,40,41]. A related approach to the scattering from a random medium is the study of the multiscale asymptotics of the scattered field. In this case the approximations is good when the frequency k and the spatial frequency of the scatterer have appropriate magnitudes. This type of asymptotic analysis has been studied by Papanicolaou and others in various cases ([6,9,43,44]). In many papers random Schrödinger operators have also been studied from the point of view of spectral theory ([10,17,18,29,30,45,48,49,54]). The paper is organized as follows: In Sect. 2 we set up the model for the random scattering problem. We also consider two important and natural examples of the processes that fit into our model; the two dimensional fractional brownian motion and the two dimensional Markov field. Next we formulate the main theorem of the paper. The regularity of the realizations of the random potential is considered in Sect. 3. In Sect. 4 we study the scattering problem where the emphasis is in the case were the potential is not a measure but true distribution. Section 5 considers oscillatory integrals in order to establish the asymptotic independence of the solutions u(x, y, k1 ) and u(x, y, k2 ) for large values of |k1 − k2 | in the Born approximation. The validity of this approximation in the context of our measurements is shown in Sect. 6. The results of the previous sections are combined in Sect. 7 to show that the measurements can be expressed as a deterministic weighted average over the unknown parameter µ. Sect. 8 verifies that this data allows us to recover µ almost surely. Part of the results of the paper have been announced without proofs in [31]. 2. Description of the Problem and Results 2.1. The model for the random potential. The setup for the stochastic scattering problem consists of the Schrödinger equation with outgoing radiation condition 2 (∆ in R2 , − q + k )u = δ y , ∂ − ik u(x) = o(|x|−1/2 ) as |x| → ∞, ∂r
(1)
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where the potential q is a random generalized function supported in a bounded and simply connected domain D in the plane. In the scattering problem the wave u is decomposed as u = u 0 (x, y, k) + u s (x, y, k), where u s (x, y, k) is the scattered field and i (1) u 0 (x, y, k) = Φk (x − y) = − H0 (k|x − y|) 4 is the incident field corresponding to a point source at y and H0(1) (· ) is the Hankel function of the first kind. We shall assume that the sources y are taken from a bounded and convex domain U ⊂ R2 \D. Since the measurements are done in the same set U , it is called the measurement domain. We assume that the potential q is a generalized Gaussian field supported in D. This means that q is a measurable map from the probability space Ω to the space of (realvalued) distributions D (R2 ) such that for all φ1 , . . . , φm ∈ C0∞ (R2 ) the mapping Ω ω → (q(ω), φ j )mj=1 is a Gaussian random variable. We will assume that the probability measure space (Ω, F, P) is complete. The distribution of q is determined by the expectation E q and the covariance operator Cq : C0∞ (R2 ) → D (R2 ) defined by ψ1 , Cq ψ2 = E (q − E q, ψ1 q − E q, ψ2 ).
(2)
Let kq (x, y) be the Schwartz kernel of the covariance operator Cq . We call kq (x, y) the covariance function of q. Then, in the sense of generalized functions, (2) reads as kq (x, y) = E ((q(x) − E q(x))(q(y) − E q(y))). We will assume that the potential is locally isotropic and moreover, that the average roughness or smoothness remains unchanged in spatial changes. However we allow the size of the (rough part of the) potential change from point to point in space. Eventually, it is this change, called the local strength of the potential that we would like to determine from our measurements. It is natural to assume that the covariance function kq (x, y) is singular only on the diagonal since the long range interactions depend often smoothly on the location. Also the basic stochastic processes like the Brownian bridge, Levy Brownian motion in the plane, or the free Gaussian field share this property. As the above properties are characteristic for Schwartz kernels of pseudodifferential operators, we introduce the following definition. Definition 1. Let µ ∈ C0∞ (D), µ(x) ≥ 0. A generalized Gaussian random field q on R2 is said to be microlocally isotropic (of order κ) in D, if the realizations of q are almost surely supported in the domain D and its covariance operator Cq is a classical pseudodifferential operator having the principal symbol µ(x)|ξ |−2κ−2 . In particular, we are interested in the case κ ∈ [0, 1/2), that corresponds to rough fields, cf. Subsects. 2.3 and 2.4 where natural examples of such fields are given. The case κ = 0 is especially interesting as in this case the potentials are proper distributions. Indeed with probability one the potentials in this case are not even measures. In Sect. 4 we will show that the Schrödinger equation has a.s. a unique solution for such potentials.
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We call µ the (local) strength of q. The role of µ and κ is better clarified as we now describe their effect on the covariance function – in this respect we refer also to the basic examples given in Subsects. 2.3 and 2.4 below. The covariance function kq (z 1 , z 2 ) is locally integrable for fixed z 2 . In the case κ = 0 it has the asymptotics kq (z 1 , z 2 ) = −cµ(z 2 ) log |z 1 − z 2 | + f (z 1 , z 2 ), where c is positive constant and f is a locally bounded function. Strength µ is closely related to the local correlation length of the random field. Namely, µ(z) determines approximately the radius of the set {z 1 : kq (z 1 , z 2 ) > M} with a given large bound M. In the case 0 < κ < 1/2 the asymptotic reads as kq (z 1 , z 2 ) = −cµ(z 2 )|z 1 − z 2 |2κ + f (z 1 , z 2 ), where f is smoother than the first term, as we shall see later. In this setting the parameter κ is tied to the Hölder continuity of the realizations of the potential. 2.2. The main result. We next formulate the measurement configuration. Recall that u s is the scattered field corresponding to problem (1). Definition 2. Given ω ∈ Ω and x, y ∈ U , the measurement m(x, y, ω) is the pointwise limit K 1 k 4+2κ |u s (x, y, k, ω)|2 dk. (3) m(x, y, ω) = lim K →∞ K − 1 1 An important special case is the backscattering measurement m(x, x, ω). Note that the measurement in the above definition is an average over frequencies whence it is not sensitive to measurement errors. For example the white noise error in the measurement is filtered out by frequency averaging. Note also that the measurement uses information only from the amplitude (not the phase) of the scattered field. It is truly a non-trivial fact that the above definition gives a well-defined, finite and non-zero quantity. That this is so, is part of Theorem 1 below, which is the main result of this paper. Theorem 1. Let D ⊂ R2 be a bounded simply connected domain, U ⊂ R2 \D be a bounded and convex domain, and let q be a microlocally isotropic Gaussian random field of order κ ∈ [0, 21 ) in D, as described in Definition 1. Then (i) For any x, y ∈ U the measurement m(x, y, ω) is well-defined (that is, the limit in (3) exists almost surely). (ii) There exists a continuous deterministic function m 0 (x, y) such that for any x, y ∈ U the equality m(x, y, ω) = m 0 (x, y) holds almost surely. In particular, the function n 0 (x) := m 0 (x, x) is almost surely determined by the backscattering data {m(x, x, ω) : x ∈ U }. (iii) The backscattering data i.e. n 0 (x), x ∈ U uniquely determines the microcorrelation strength µ in Ω through the linear relation 1 1 n 0 (x) = 8+2κ 2 µ(z) dz. 2 π D |x − z|2
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By the above result the principal structure of the covariance is determined by measurements from only one single realization of the potential. Observe that the needed data is the energy averages of the back-scattered field – no information on the phase is needed. We refer to Remark 4 at the end of Sect. 8 for a more thorough discussion of the relation of the above result to its deterministic counterparts. Property (ii) in Theorem 1 is called statistical stability, cf. [9]. For the simplicity of notations we will assume in the proof of Theorem 1 that E q = 0; one can easily dispense with this assumption. We refer the reader to the remarks at the end of the section where this fact and other generalizations are considered. Using the fact that the measurements m(x, y, ω) exist, our analysis could be generalized to other kinds of measurements. For instance, it would also be physically relevant to analyze measurements of fixed source point x → m(x, y0 , ω), where y0 ∈ U . We hope to come back to this and other related stochastic scattering problems in future work. 2.3. Example 1. Analogies of fractional Brownian fields. Let us recall that standard Brownian motion in the plane is a gaussian process on the real line with the covariance C(t, s) = max(t, s) for t, s ∈ R, and with a.s. continuous realizations. The Brownian paths are fairly regular: they are a.s. Hölder continuous with any exponent less than one half. In order to obtain a more rough stochastic model a natural analogue is fractional Brownian motion F B M H , where the Hurst index H takes values from the interval H ∈ (0, 1). The case H = 1/2 corresponds exactly to the Brownian motion, and rougher paths are obtained by considering Hurst indices with H ∈ (0, 1/2). Instead of recalling the definition of F B M H in one dimension we next give the definition in arbitrary dimension. The multidimensional fractional Brownian motion F B M H in Rn is easily obtained as follows: Let H ∈ (0, 1). One considers a centered Gaussian process X H (z) indexed by z ∈ Rn and with the following properties: E |X H (z 1 ) − X H (z 2 )|2 = |z 1 − z 2 |2H for all z 1 , z 2 ∈ Rn , X (z 0 ) = 0, the paths z → X H (z) are a.s. continuous. We refer e.g. to [26] for the proof of existence and basic properties of n-dimensional fractional Brownian motion. Especially, the obtained random functions R2 → R are almost surely Hölder-continuous with any exponent less than H . Observe that the differences of the process are completely invariant under rotation and translation, also there is a natural scaling in dilations. The deterministic zero-point z 0 with X (z 0 ) = 0 can of course be chosen arbitrarily, often one sets z 0 = 0. In the case H = 1/2 one calls F B M1/2 a Levy Brownian motion. There are other higher dimensional generalizations of Brownian motion (e.g. the so called Brownian sheets), but none other has the natural invariance properties just described. An important example of microlocally isotropic Gaussian fields is now obtained by considering the random functions q(z, ω) := a(z)X H (z, ω),
(4)
where X H (·) stands for a F B M H in the plane with Hurst index H ∈ (0, 1/2), and the deterministic function a ∈ C0∞ (D) is supported in the domain D. One observes that a modulates the size of the potential, (or, with another point of view, the size of the local
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correlations). We assume here that the zeropoint z 0 lies outside D. In order to verify that X H (z) really satisfies Definition 2 we observe that the covariance of the random field q can be computed as follows: Cq (z 1 , z 2 ) =
1 a(z 1 )a(z 2 )(|z 1 − z 0 |2H + |z 2 − z 0 |2H − |z 1 − z 2 |2H ). 2
The only singular term is a(z 1 )a(z 2 ) whence it is clear that in case the principal symbol has the form c H (a(z))2 |ξ |−2−2H , i.e. the potential q is microlocally isotropic of order κ = H. We may thus view (4) as a simplest type of natural examples of microlocally isotropic Gaussian potentials of positive order, for which our main result applies. More complicated examples can be easily constructed.
2.4. Example 2. Markov fields. We introduce the notion of Markov fields and briefly overview their basic properties (we refer to the monograph [47] for more information). These fields provide natural examples of microlocally isotropic fields. Let us assume in the present subsection that our random potential q is a localization of the generalized Gaussian Markov field Q, that is, q = χ Q, where χ ∈ C0∞ (D). The definition of Markov fields mimics the situation where physical particles in a lattice have no long-term interaction, i.e., only neighboring particles have direct interaction. Assume that S1 ⊂ D is an open set with S 1 ⊂ D. We set S2 = D\S 1 and S = {x ∈ D : d(x, ∂ S1 ) ≤ ε}, ε > 0, a collar neighborhood of the boundary ∂ S1 . Intuitively the Markov property means that the influence from the inside to the outside must pass through the collar. Definition 3. A generalized random field Q on R2 satisfies the Markov property if for any S1 , S2 and Sε as described above, and ε > 0 small enough, the conditional expectations satisfy E (h ◦ Q(ψ)|B(Sε )) = E (h ◦ Q(ψ)|B(Sε ∪ S1 )) for any complex polynomial h and for any test function ψ ∈ C0∞ (S2 ). Here B(S j ) is the σ -algebra generated by the random variables Q(φ), φ ∈ C0∞ (S j ), j = 1, 2, and B(Sε ) is defined respectively. The Markov property has dramatic implications to the structure of the field Q and especially to its covariance operator C Q . Under minor additional conditions (cf. [47, e.g. p. 112]), we may define the inverse operator (C Q )−1 which turns out to be a local operator: it cannot increase the support of a test function. By a well-known theorem of J. Peetre [46] (C Q )−1 must be a linear partial differential operator. As C Q is non-negative operator, (C Q )−1 has to be of even order. To obtain an isotropic situation we finally assume that (C Q )−1 is a non-degenerate elliptic operator, is of 2nd order, has smooth coefficients, and finally its principal part is positive and homogeneous. This implies that (C Q )−1 = P(z, Dz ) = −
2 j,k=1
∂ ∂ a(z) k + b(z), j ∂z ∂z
(5)
where a(z) > 0 and b(z) are smooth real functions in R2 . Then the field Q is microlocally isotropic of order zero as C Q is a pseudodifferential operator with an isotropic principal symbol.
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To motivate the assumption that the order of (C Q )−1 is two, let us consider the case where (C Q )−1 would be of fourth order or higher, with smooth coefficients. Then one could easily verify (cf. the proof of Theorem 2) that the realizations of q are in the s, p Sobolev class Hcomp (R2 ) for all s < 1 and 1 < p < ∞. As our aim is to consider the case of non-smooth potentials, the second order case is the most interesting in view of many applications. An important example of such random fields of this type is obtained by the free Gaussian fields, which appear in two dimension quantum field theory (cf. e.g. [21]). The free Gaussian field on the bounded domain D, corresponding to Dirichlet boundary values, has the (Dirichlet-)Green’s function as the kernel of its covariance operator. This corresponds to choices a(z) = 1, b(z) = 0. Examples with variable a(z) can be constructed easily. Finally, the covariance operator Cq of the potential q has the kernel kq (z 1 , z 2 ) = χ (z 1 )k Q (z 1 , z 2 )χ (z 2 ). This implies that q is microlocally isotropic of order zero in D and has the microcorrelation strength function µ(z) = χ (z)2 a(z)−1 . 3. Regularity of the Potential We will study what kind of regularity (or irregularity) is implied for the potential by Definition 1. In the case κ > 0 we will see that the realizations are almost Hölder continuous of exponent κ. In case κ = 0 it turns out that q(ω) is not a function (or even a measure); almost surely it is a proper distribution. This is not so surprising since similar phenomenon is well known in the case of a free Gaussian field. However, the potential just barely fails to be a function: almost every realization of the potential satisfies − , p
q(ω) ∈ H0
(D) for all ε > 0 and 1 < p < ∞.
(6)
Here, H s, p (R2 ) = F −1 ((1 + |ξ |2 )−s/2 F L p (R2 )) is the standard Sobolev space, defined s, p with the Fourier transform F and H0 (D) is the closure of C0∞ (D) in H s, p (R2 ). In this section we verify the stated Hölder continuity in case κ > 0, and for κ = 0 the fact (6), which is needed in the subsequent analysis of our problem. We start by recording a result which yields a criterion for realizations of a random field to lie in p>1 L p (D). Throughout the paper c denotes a generic constant the value of which may change even inside a formula. Lemma 1. Assume that the covariance operator K of a random field F on the open bounded set D ⊂ Rn has a locally integrable kernel (denoted also by K (x, y)) satisfying |K (x, y)| ≤ c < ∞ for every x, y ∈ D. Then the realizations of F belong almost surely to p>1 L p (D). Proof. This is an immediate consequence of [8, Prop. 3.11.15]. To sketch a direct proof p of this result, onep/2may first mollify F and observe that in the smooth case E (F p ) = c p D |K (x, x)| d x. Recall that Cq is the covariance operator of the random potential q. We next analyze the singularity of the Schwartz kernel kq (x, y) of Cq .
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Proposition 1. Let q be a microlocally isotropic Gaussian random field of order κ ∈ [0, 1/2). Then the Schwartz kernel of the covariance operator Cq has the form c (x, y) log |x − y| + r1 (x, y), κ = 0, Cq (x, y) = 0 c0 (x, y)|x − y|2κ + r1 (x, y), κ ∈ (0, 1/2), where c0 ∈ C0∞ (D × D) and r1 ∈ C0α (D × D) for any α < 1. Proof. By definition, Cq (x, y) is a kernel of a (compactly supported) classical pseudodifferential operator with symbol a(x, ξ ) = µ(x)(1 − ψ(ξ ))|ξ |−2−2κ + b(x, ξ ) in the −2 (R2 × R2 ) (cf. [24]), where the smooth cutoff ψ ∈ C0∞ (R2 ) equals 1 near the class S1,0 −3 (R2 × R2 ) is compactly supported in x-variable. We origin, and in any case b ∈ S1,0 2 obtain 2π C(x, y) = I (x, y) + r2 (x, y), where I (x, y) = µ(x) ei(x−y)·ξ (1 − ψ(ξ ))|ξ |−2−2κ dξ, r2 (x, y) = ei(x−y)·ξ b(x, ξ )dξ. R2
R2
Function I (x, y) may clearly be written in the form µ(x) log |x − y| + r0 (x, y) (resp. µ(x)|x − y|2κ + r0 (x, y)) with r0 ∈ C ∞ (R4 ) if κ = 0 (resp. κ ∈ (0, 1/2)). It remains to check that r2 is in C α for any α < 1. Let ∞ j=0 φ j (ξ ) = 1 be a smooth ∞ 2 partition of unity with φ0 , φ1 ∈ C (R ), supp(φ1 ) ⊂ {ξ : 1/2 < |ξ | < 2}, and φ j (ξ ) = φ1 (21− j ξ ) for j ≥ 2. By writing R2 (x, y) = r2 (x, x − y), we get k Dx φ j (D y )R2 (x, y) = ei y·ξ φ j (ξ )Dxk b(x, ξ )dξ, j, k ≥ 0. R2
Since ≤ Ck (1 + we see that ||Dxk φ j (D y )R2 || L ∞ (R4 ) ≤ Ck 2− j , where Ck 1 does not depend on j. This implies immediately that R2 in the Besov-space B∞,∞ (R4 ) 4 α 4 that coincides with the first Zygmund class 1 (R ) ⊂ C (R ) for all α < 1 (see [51, 5.3]. |Dxk b|
|ξ |)−3
The following immediate implication is needed for realizations of q Theorem 2. (i) Let κ = 0. Almost surely q(ω) ∈ H − , p (D) for all ε > 0 and 1 < p < ∞. (ii) Let κ ∈ (0, 1). Almost surely q(ω) ∈ C α for all α ∈ (0, κ). Proof. (i) Recall that for given s ∈ R the Bessel potential J s provides an isomorphism J s : H t, p (R2 ) → H t+s, p (R2 ) for all t ∈ R and 1 < p < ∞. Moreover, J s is a pseudodifferential operator, whence it preserves singular supports. Thus it is enough to verify that locally the covariance of J ε q has a uniformly bounded kernel for any small ε stand for a suitable localization of J ε we have to study ε > 0. That is, by letting Jloc ε C J ε . It is well known that for small ε > 0 the kernel has form the kernel of Jloc q loc J ε (x, y) =
c + S(x, y), |x − y|2−ε
where S has a lower order singularity. Now the claim follows by combining Proposition 1 and the fact | log |x|| dx < ∞ 2−ε B(0,R) |x| for any radius R > 0.
Inverse Scattering Problem for a Random Potential
677
(ii) One may reduce the situation to the one in case (i) by simply considering the field J −κ q. It follows that for any > 0 and p ∈ (1, ∞) we have almost surely that J −κ q(ω) ∈ H −ε, p . Equivalently, q(ω) ∈ H κ−ε, p and the claim now follows from the Sobolev imbedding theorem. 4. Direct Scattering Problem for a Distributional Potential 4.1. Unique continuation. We showed above that the random potential q(ω) belongs with probability one to the Sobolev space H − , p (D) for all 1 ≤ p < ∞ and > 0. Consequently, we need to study the existence and properties of the solution for the Schrödinger equation for such irregular potentials. In this section we accomplish this by considering scattering from a deterministic non-smooth potential q0 ∈ H − , p (D), and the obtained results have independent interest. The direct scattering theory from a potential that is in a weighted L 2 space is classical (cf. [7,3]). For the L p scattering theory the key tool is the unique continuation of the solution. Jerison and Kenig showed in [25] that the strong unique continuation principle for L p -potentials in Rn holds for p ≥ n/2 and fails for p < n/2 in dimensions n > 2. In dimension two the unique continuation holds in a space of functions that is close to L 1 [25]. For Sobolev space potentials, the selfadjointness of the operator has been studied in [37]. Below in Lemma 2 we show a positive result for negative index Sobolev spaces. More precisely, we study the scattering problem
(∆ − q0 + k 2 )u = δ y
∂ −1/2 ) , ∂r − ik u(x) = o(|x|
(7)
− , p
where the potential q0 ∈ Hcomp (R2 ), p −1 + ( p )−1 = 1, 1 < p < 2. We claim that the problem (7) is equivalent to the Lippmann-Schwinger equation u(x) = u 0 (x) − Φk (x − y)q0 (y)u(y)dy. (8) R2
In the proof we show that the pointwise product q0 u in the integrand of (8) is well defined and that the integral exists in the sense of distributions. We will then show that 2 p, (8) has a unique solution u ∈ Hloc (Rn ). The starting point is the unique continuation principle. Roughly speaking, it says that if u is a compactly supported solution of the Schrödinger equation with q0 ∈ H − ,r , r > n/2 and if is small then u must vanish identically. It appears to the authors that this result could also be obtained as a special case of D. Tataru’s and H. Koch’s recent unique continuation results based on L p Carleman estimates [28]. In our case, we present a direct and simple proof for unique continuation. We start by observing that known pointwise multiplication results allow us to define the product distribution q0 u. − , p
,2 p
(Rn ), 1 < p < ∞, > 0. Then Lemma 2. Assume that u ∈ Hloc (Rn ), q0 ∈ H0 − , p 2p p = 2 p−1 and the product q0 u is well-defined as an element of H0 (Rn ), where ||q0 u|| H − , p (Rn ) ≤ c||q0 || 0
− , p
H0
(R n )
||u|| H ,2 p (Rn ) .
(9)
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M. Lassas, L. Päivärinta, E. Saksman
Proof. Take φ ∈ C0∞ (Rn ) to be a test function. By duality, the product q0 u ∈ D (Rn ) is well defined through q0 u, φ = q0 , φu − , p
(10)
, p
when q0 ∈ Hcomp (Rn ) and u ∈ Hloc (Rn ). By using Bony’s paraproducts one can verify the following pointwise multiplier estimate in Sobolev spaces ([53, pp. 105])
φu H , p (Rn ) ≤ c φ L r1 (Rn ) u H ,r2 (Rn ) + u L r1 (Rn ) φ H ,r2 (Rn ) (11) for 1/ p = 1/r1 + 1/r2 . From (10) and (11) with r1 = r2 = 2 p it readily follows by − , p 2p duality that q0 u ∈ H0 (Rn ), where p = 2 p−1 . Proposition 2 (Unique continuation principle into an interior domain). Assume that − , p p ∈ (n/2, ∞), together with 0 < < n4 (2/n − 1/ p ). Let q0 ∈ Hcomp (Rn ). If
,2 p u ∈ Hloc (Rn ) is compactly supported and satisfies the Schrödinger equation (∆ − q0 + k 2 )u = 0 in the weak sense, then u = 0 identically. Proof. Assume that the support of q is contained in the bounded domain D ⊂ Rn . To prove the unique continuation we use the well-known techniques of exponentially growing solutions for the Schrödinger equation, cf. [22,27]. To this end we write the equation (∆ + k 2 )u = q0 u as (∆ + 2iζ · ∇)e−iζ ·x u = e−iζ ·x q0 u, where ζ ∈ Cn is such that ζ · ζ = k 2 . Since u is supposed to have compact support we
,2 p have v := e−iζ ·x u ∈ Hcomp (R2 ). For v we obtain the equation v = Gζ (q0 v),
(12)
where the Faddeev operator Gζ is defined as the Fourier multiplier Gζ ( f )(x) = F −1 (
ξ2
−1 fˆ)(x). + 2ζ · ξ
It is well known (see for example the proof of Theorem 4.1 in [39]) that for 0 ≤ s ≤ 21 . Gζ H −s (D)→H s (D) ≤ 0
c |ζ |1−2s
,
(13)
where H s (D) = H s,2 (D) and H0s (D) = H0s,2 (D) are L 2 -based Sobolev spaces. By [27], G ζ is a bounded operator
G ζ : L r (D) → L r (D),
(14)
2n if n ≥ 3 and for r > 1 for n = 2. We continue first in the case n ≥ 3. for r = n+2 Interpolation of (13) and (14) yields
||G ζ || H − , p (D)→H ,2 p (D) ≤ c|ζ |−(1−2s)θ , 0
(15)
Inverse Scattering Problem for a Random Potential
where = θ s and θ = 1 −
n 2 p .
679
Finally, (9), (12), and (15) show that
||v|| H ,2 p (D) ≤
c ||v|| H ,2 p (D) . |ζ |(1−2s)θ
(16)
Choosing 0 < s < 21 and ζ large enough, we conclude that v and hence u must vanish identically. Finally in the case n = 2 we interpolate (13) and (14) for r > 1 and by letting r → 1 the same conclusion follows. ε,r Remark. Note that for n = 2 the uniqueness follows for u ∈ Hloc (R2 ) when r > 2, −ε,r (R2 ). 0 < ε < r1 , and q0 ∈ Hcomp
4.2. Existence and uniqueness for solutions of the scattering problem. After having proven the unique continuation principle, the proofs of Theorems 3 and 4 below are relatively straightforward extensions of classical proofs for regular potentials. For the convenience of the reader, we include the details. − , p
Theorem 3. For q0 ∈ Hcomp (Rn ), with n ≥ 2, p ∈ (n/2, ∞), and 0 < < n4 ( n2 −
,2 p
1 p ),
the Lippmann-Schwinger equation (8) has a unique solution u ∈ Hloc (Rn ). Proof. Let D be a bounded domain such that supp (q0 ) ⊂ D. Consider Eq. (8) in H ,2 p (D). Since the operator Hk , Hk f = Φk ∗ f,
(17)
defines a bounded operator Hk : H0−s (D) → H s (D) for s ≤ 1 we see from the Sobolev − , p embedding theorem and Rellich’s compact embedding theorem that Hk : H0 (D) → H ,2 p (D) compactly. This and Proposition 2 give that the operator K k : H ,2 p (D) → H ,2 p (D), K k f = Hk q0 f is compact. Thus by Fredholm’s alternative it is enough to show that in H ,2 p (D) the homogeneous equation u = Hk q0 u
(18)
has only the trivial solution u = 0. If u ∈ H ,2 p (D) satisfies (18) then u belongs to the Schwartz class S and by taking the Fourier transform we obtain in the sense of distributions that (∆ + k 2 )u(x) = q0 u. In particular u must be smooth in Rn \D and satisfy (∆ + k 2 )u = 0 there. Note that by (18) the values of u in D define u in all of Rn . As the fundamental solution and its derivatives satisfy the radiation condition, we see from (18) that u also satisfies the radiation condition in (1). Thus, as u is a classical solution in Rn \D satisfying the radiation condition, it has a far field expansion (cf. [13, Thm. 2.14]). By Rellich’s lemma (cf. [13, Lem. 2.11]) and the unique continuation principle it is enough to show that the far field u ∞ of u, defined by
x eik|x| −(n−1)/2 + o |x| u u(x) = ∞ |x| 4π |x|(n−1)/2 as |x| → ∞, vanishes for u. Note that
,2 p
p ∆u = (q0 − k 2 )u ∈ H − , (R2 ) + Hloc (Rn ).
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2 and that u and ∆u belong locally to spaces that are dual to This implies that ∇u ∈ L loc each other. Take r > 0 so large that D ⊂ B(0, r ). Thus by approximating u by smooth functions we get from Green’s formula
∂ Im |∇u|2 + (q0 − k 2 )|u|2 d x = 0. u u ds = Im |x|=r ∂ν |x|≤r
Thus |x|=r
∂ 2 2 2 u + k |u| ds = ∂ν
|x|=r
2 ∂ u − iku ds → 0 ∂ν
as r → ∞. Especially, this implies that ||u|| L 2 ({|x|=r }) → 0 as r → ∞. This is possible only if u ∞ ≡ 0. Thus the assertion is proven. − , p
Theorem 4. For q0 ∈ Hcomp (Rn ), with n ≥ 2, p ∈ (n/2, ∞), and 0 < < n4 ( n2 − p1 ), the scattering problem (7) is equivalent to the Lippmann-Schwinger equation and thus
,2 p has a unique solution u ∈ Hloc (Rn ). Proof. As reasoned in the proof of the previous theorem a solution of the Lippmann ,2 p Schwinger equation satisfies (7). Suppose u ∈ Hloc (Rn ) ∩ S is a solution of (7). We need to show that (19) u s (x) = Φk (x − y)q0 (y)u(y) dy. − , p
,2 p
Since (∆+k 2 )u s = q0 u ∈ Hcomp (Rn ) and Φk (x −·) ∈ Hloc (Rn ) and both functions are real-analytic outside a large ball we have from (7) in the sense of distributions that Φk (x − y)(∆ + k 2 )u s (y) dy = Hk (q0 u). (20) |y|≤r
Denote the operator that operates to u s in the left-hand side of (20) by T . Now for φ ∈ C ∞ (Rn ), ∂ ∂ φ(y) − Φk (· − y)φ(y) ds(y). Φk (· − y) Tφ = φ + ∂r (y) ∂r (y) |y|=r Thus, approximating u s with smooth functions we obtain ∂ ∂ u s (y) − Φk (x − y)u s (y)) ds(y) = Hk (q0 u). u s (x) + (Φk (x − y) ∂r (y) ∂r (y) |y|=r From the radiation condition it follows that the boundary integral in the above formula approaches zero as r → ∞, cf. [13, Thm. 2.4]. This proves (19) and hence the theorem. Note that, in view of Theorem 2, Theorem 4 implies that the original stochastic scattering problem (1) has a unique solution almost surely.
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681
5. The First Order Term in the Born Series By iterating the Lippmann-Schwinger equation, one can formally represent u as the Born series, u(x, y, k) = u 0 (x, y, k) + u 1 (x, y, k) + u 2 (x, y, k) + . . . ,
(21)
where u 0 (x, y, k) = Φk (x − y) and u n+1 = (∆+k 2 +i0)−1 (qu n ). A considerable part of our work consists of analyzing the different terms in this expansion. We will later prove in Subsect. 6.2 that the series (21) converges for large enough values of k. In the proof of our main result we need to establish asymptotic independence for the first terms in the Born series, corresponding to different values of k. The verification of this fact leads to estimation of certain oscillatory integrals, and needs a fairly involved computation. As a useful tool we apply the calculus of conormal distributions. The results of this section will be applied later in Sect. 7. As the first term in the Born series is u 1 (x, y, k) = Φk (x − z)q(z)Φk (z − y) dz, D (1)
we start with the asymptotics of Φk (z) = − 4i H0 (k|z|), when k → ∞. These are given by ∞ 1 i(k|z|−π/4) 1 e , F(t) = Φk (z) = F d j t j , t > 0, (22) k|z| k|z| j=0
where d0 = − √i and d j are constants whose actual values are not important for us in 8π the sequel. The series (22) and its derivative have the property that for N > 1 (cf. [1, Formulae 9.1.27, 9.2.7–9.2.10]) ⎞ ⎛ N N d d j t j ≤ ct N +1 , ⎝ F(t) − d j t j ⎠ ≤ ct N , 0 < t < 1. (23) F(t) − dt j=0
j=0
Using first three terms in the asymptotics of Φk , we write u 1 (x, y, k) = u 1 (x, y, k) + b(x, y, k) where, for k ≥ 1,
(3)
u 1 (x, y, k) = D (3)
(24)
(3)
Φk (x − z)q(z)Φk (z − y) dz, 1
Φk (z) = (k|z|)− 2 ei(k|z|−π/4)
3
d j (k|z|)− j .
j=0
Let us denote by O(k1−n 1 k2−n 2 ) functions h(x, y, k1 , k2 ) which satisfy an estimate |h(x, y, k1 , k2 )| ≤ ck1−n 1 k2−n 2 for x, y ∈ U and k1 , k2 ≥ 1, where c is independent of x, y, k1 , and k2 . Next we compute the asymptotic expansion for the covariance of u 1 thus showing that the fields u 1 with different frequencies are asymptotically independent. We emphasize that formula (28) below is crucial for the construction of µ(z) in Sect. 8.
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Proposition 3. Assume that κ ∈ [0, 21 ). For k1 , k2 ≥ 1 the random variable u 1 satisfies uniformly for x, y ∈ U the estimates cn , (k1 + k2 )4+2κ (1 + |k1 − k2 |)n u 1 (x, y, k2 ))| ≤ cn (k1 + k2 )−n , | E ( u 1 (x, y, k1 )
| E ( u 1 (x, y, k1 ) u 1 (x, y, k2 ))| ≤
(25) (26)
where n is arbitrary. Moreover, for k1 = k2 = k we have the asymptotics u 1 (x, y, k)) = R(x, y)k −4−2κ + O(k −5 ), E ( u 1 (x, y, k) where R ∈ C ∞ (U × U ). Especially, it holds that µ(z) 1 dz for x ∈ U. R(x, x) = 8+2κ 2 2 π R2 |z − x|2
(27)
(28)
Proof. Denote φ(z, x, y) = |x − z|+|z − y|. As the covariance operator Cq has a weakly singular kernel C(z 1 , z 2 ) = kq (z 1 , z 2 ) with asymptotics given as in Proposition 1, we see that 3
E ( u 1 (x, y, k1 ) u 1 (x, y, k2 )) =
I j1 , j2 ,l1 ,l2 (k1 , k2 , x, y),
(29)
j1 , j2 ,l1 ,l2 =0
where I j1 , j2 ,l1 ,l2 (k1 , k2 , x, y) =
exp ik1 φ(z 1 , x, y) − ik2 φ(z 2 , x, y) E (q(z 1 )q(z 2 ))
×
d j1 d j2 d l1 d l2 1+ j1 + j2 1+l1 +l2 k1 k2
R4
|x − z 1 |
j1 + 21
|z 1 − y|
j2 + 21
l1 + 12
|x − z 2 |
l2 + 12
dz 1 dz 2 .
(30)
|z 2 − y|
Assumption 1 with κ ∈ [0, 21 ) states that kq (z 1 , z 2 ) = E (q(z 1 )q(z 2 )) is the Schwartz −2−2κ kernel of a pseudodifferential operator Cq with a classical symbol c(x, ξ ) ∈ S1,0 (R2 × R2 ), and the principal symbol of Cq is given by c p (z, ξ ) = µ(z)(1 + |ξ |2 )−1−κ . The support of Cq (z 1 , z 2 ) is contained in D × D. We may write (cf. [24]) −2 kq (z 1 , z 2 ) = (2π ) ei(z 1 −z 2 )·ξ c(z 1 , ξ ) dξ. (31) R2
All symbols appearing below will be classical symbols [24]. In order to obtain uniform estimates with respect to variables x and y we shall introduce them as variables in the covariance in the following way. Let us define the function C1 (z 1 , z 2 , x, y) = kq (z 1 , z 2 )θ (x)θ (y), where θ ∈ C0∞ (R2 ) equals one in the domain U and has its support outside D. The formula (31) now takes the form −2 ei(z 1 −z 2 )·ξ c1 (z 1 , x, y, ξ ) dξ, (32) C1 (z 1 , z 2 , x, y) = (2π ) R2
−2−2κ −2−2κ (R6 × R2 ). In fact, c1 ∈ S1,0 ((D × R4 ) × R2 ), where c1 (z 1 , x, y, ξ ) ∈ S1,0 but we consider it extended by zero to values z 1 ∈ D. By definition, (32) means that
Inverse Scattering Problem for a Random Potential
683
C1 (z 1 , z 2 , x, y) is a conormal distribution in R8 of Hörmander type having conormal singularity on the surface S1 = {(z 1 , z 2 , x, y) ∈ R8 : z 1 − z 2 = 0}. Using notations of [24], if X ⊂ Rn is an open set and S ⊂ X is a smooth submanifold of X , we denote by I (X ; S) the distributions in D (X ) that are smooth in X \S and have a conormal singularity at S. The set of distributions in I (X ; S) supported in a compact subset of X is denoted by Icomp (X ; S). Let D ⊂ R8 be an open set containing D × D × supp (θ ) × supp (θ ) so that C1 ∈ Icomp (D; S1 ∩ D). We employ the fact that conormal distributions are invariant under a change of coordinates. Actually, our plan is to consider several different coordinates systems. The first set of coordinates that we consider are (V, W, x, y), defined as V = z 1 − z 2 and W = z 1 + z 2 . Denote by η the change of coordinates η : (V, W, x, y) → (z 1 , z 2 , x, y) and consider the pull-back C2 = η∗ (C1 ). Then a direct substitution shows that −2 C2 (V, W, x, y) = (2π ) ei V ·ξ c2 (V, W, x, y, ξ ) dξ, R2
c2 (V, W, x, y, ξ ) = c1 (z 1 (V, W, x, y), x, y, ξ ) which means that C2 ∈ I (R8 ; S2 ), where S2 = {(V, W, x, y) : V = 0}. To find out how the symbol transforms in the change of coordinates, we have to represent C2 (V, W, x, y) with a symbol that does not depend on V . We can achieve this by way of the representation theorem for conormal distributions [24, Lemma 18.2.1] because of the special form of the surface S2 = {V = 0}. We have: ei V ·ξ c3 (W, x, y, ξ ) dξ, C2 (V, W, x, y) = (2π )−2 R2 (33) ∞ −2−2κ l 6 2 −i DV , Dξ c2 (V, W, x, y, ξ )|V =0 ∈ S1,0 (R × R ). c3 (W, x, y, ξ ) ∼ l=0
In particular, we see that c3 (W, x, y, ξ ) has the principal symbol
p c3 (W, x, y, ξ ) = µ(z 1 (V, W, x, y))(1 + |ξ |2 )−1−κ θ (x)θ (y)
V =0
.
(34)
The second set of coordinates that we consider are (v, w, x, y) defined below. For this, to consider the oscillatory integrals (30) we change the coordinates so that φ(z 1 , x, y) − φ(z 2 , x, y) will be a coordinate. We will do this change of coordinates in two steps. First we change the coordinates (z 1 , z 2 , x, y) to (Z 1 , Z 2 , x, y), where Z j = Z j (x, y, z j ) ∈ R2 , j = 1, 2 are related to ellipses having focal points in x and y. More precisely, we write Z j = (t j , s j ) ∈ R2 , ∇z j φ(z j , x, y) 1 1 ), t j = φ(z j , x, y), s j = φ(z j , x, y)· arcsin(e1 · 2 2 ||∇z j φ(z j , x, y)|| where e1 = (1, 0). In other words, here t j corresponds to the semi-major axis of the ellipse having focal points x and y and containing the point z j . The variable s j specifies a ’normalized’ angle of the normal vector of the ellipse with the x-axis at the point z j . Since the domain U is convex and D is simply connected, our definition of the new coordinates is well-posed in a neighborhood of the domain D.
684
M. Lassas, L. Päivärinta, E. Saksman
Secondly, we change from (Z 1 , Z 2 , x, y) to coordinates (v, w, x, y), where v = Z 1 − Z 2 , w = Z 1 + Z 2 . Together, the above steps define the coordinates (v, w, x, y) and the map τ : (v, w, x, y) → (z 1 , z 2 , x, y). Note that the first component of v(z 1 , z 2 , x, y) equals (φ(z 1 , x, y) − φ(z 2 , x, y))/2. To simplify the notation, we denote X 1 = D, X 2 = η−1 (D) and X 3 = τ −1 (D) so that τ : X 3 → X 1 and η : X 2 → X 1 . We are ready to represent the conormal distribution C1 (z 1 , z 2 , x, y) in coordinates (v, w, x, y) as the pull-back distribution C4 = τ ∗ (C1 ) ∈ I (X 3 ; S3 ∩ X 3 ), S3 = {(v, w, x, y) : v = 0}. By the invariance of conormal distributions under the change of variables we may write Icomp (X 1 ; S1 ∩ X 1 ) SSSS kk SSSSτ ∗ SSSS SSS) Icomp (X 3 ; S3 ∩ X 3 )
η∗ kkkkk
kkkk ukkkk Icomp (X 2 ; S2 ∩ X 2 )
To apply this diagram and the integral representation (33) of C2 ∈ Icomp (X 2 ; S2 ∩ X 2 ), consider the transformation κ = η−1 ◦ τ . We will below use [24, Theorem 18.2.9], to provide a representation for the pull-back C4 = κ ∗ C2 . Since surfaces S2 and S3 have the special form S2 = {V = 0} and S3 = {v = 0}, and κ maps S3 ∩ X 3 onto S2 ∩ X 2 , we obtain eiv·ξ c4 (w, x, y, ξ ) dξ, C4 (v, w, x, y) = (2π )−2 R2
−2−2κ where c4 (w, x, y, ξ ) ∈ S1,0 (R6 × R2 ) is a symbol satisfying c4 (w, x, y, ξ ) = c3 (κ2 (v, w, x, y), ((κ11 (v, w, x, y))−1 )t ξ ) × |det κ11 (v, w, x, y)|−1 + r (w, x, y, ξ ).
(35)
v=0
−3−2κ Here, r (w, x, y, ξ ) ∈ S1,0 (R6 × R2 ) and the coordinate transform κ is decomposed 2 into two parts, the R -valued function κ1 (v, w, x, y) = V (v, w, x, y) and the R6 -valued function κ2 (v, w, x, y) = (W (v, w, x, y), x, y). This yields for the differential κ of κ the corresponding representation κ11 κ12 . κ = κ21 κ22
We note that the transformation rule in κ ∗ in [24, Theorem 18.2.9] is presented for halfdensities. The proof of the analogous result for distributions, however, is immediate. Plugging the principal symbol of c3 (x, ξ ) given in (34) to formula (35), we see that the principal symbol of c4 (w, x, y, ξ ) is p c4 (w, x, y, ξ ) = µ(z 1 (v, w, x, y))(1 + |(κ11 (v, w, x, y))−1 )t ξ |2 )−1−κ · θ (x)θ (y)J (w, x, y), (0, w, x, y)|−1 . where J (w, x, y) = |det κ11
v=0
Inverse Scattering Problem for a Random Potential
685
We are ready to compute the asymptotics of I j1 , j2 ,l1 ,l2 (k1 , k2 , x, y). We denote j = ( j1 , j2 , l1 , l2 ). By writing the integral I j = I j1 , j2 ,l1 ,l2 (k1 , k2 , x, y) in coordinates (v, w, x, y) we obtain
−(1+ j1 + j2 ) −(1+l1 +l2 ) I j = k1 k2 exp(i (k1 + k2 )e1 · v + (k1 − k2 )e1 · w ) R4
j
· C4 (v, w, x, y)H (v, w, x, y) dvdw,
(36)
where e1 = (1, 0) is the unit vector and H j = H j (v, w, x, y) is d j1 d j2 d l1 d l2
Hj = |x − z 1 |
j1 + 21
|z 1 − y|
j2 + 12
l1 + 21
|x − z 2 |
l2 + 21
det (τ (v, w, x, y)),
|z 2 − y|
where z 1 = z 1 (v, w, x, y) and z 2 = z 2 (v, w, x, y). Since H j is smooth in X 3 in all variables and the class I (R8 ; S3 ) is closed under multiplication with a smooth function, we have C4 (v, w, x, y) H j (v, w, x, y) ∈ I (R8 ; S3 ). To evaluate the oscillatory integrals (36) in a convenient way, we need to represent this conormal distribution with a symbol that does not depend on v. Again, by using the representation theorem for conormal distributions [24, Lemma 18.2.1], we obtain j C4 (v, w, x, y) H j (v, w, x, y) = (2π )−2 eiv·ξ c5 (w, x, y, ξ ) dξ, 2 R (37) ∞ j l j −i Dv , Dξ (c4 (w, x, y, ξ )H (v, w, x, y))|v=0 . c5 (w, x, y, ξ ) ∼ l=0
j
In particular, we see that c5 (w, x, y, ξ ) has the principal symbol (v, w, x, y))−1 )t ξ |2 )−1−κ c5 (w, x, y, ξ ) = µ(z 1 (v, w, x, y))(1 + |((κ11 · θ (x)θ (y) J (w, x, y)H j (v, w, x, y) .
jp
v=0
(38)
By substituting (37) into (36) and using the Fourier inversion rule we obtain the important formula j −(1+ j1 + j2 ) −(1+l1 +l2 ) k2 (Fw c5 )((k2
I j = k1
− k1 )e1 , x, y, −(k1 + k2 )e1 ),
(39)
where Fw denotes the Fourier transform in the w variable, j j Fw c5 (η, x, y, ξ ) = e−iη·w c5 (w, x, y, ξ ) dw. R2
j
As the symbol c5 (w, x, y, ξ ) is C ∞ smooth and compactly supported in the (x, y, w) j
α c (w, x, y, ξ )| ≤ c (1 + |ξ |)−2−2κ for all |α| ≥ 0, where c variables, we see that |Dw α α 5 is independent of (w, x, y) ∈ R6 . This implies after n integrations by parts that
|I j (k1 , k2 , x, y)| ≤ cn
1 − j − j −1 k 1 2 k2−1 −2 −1 (1 + |k1 − k2 |)−n 1 + |k1 + k2 |2+2κ 1
686
M. Lassas, L. Päivärinta, E. Saksman
for all n ≥ 0. By considering separately the cases |k1 − k2 | ≤ |k1 + k2 |/2 and |k1 − k2 | ≥ |k1 + k2 |/2 we deduce that |I j (k1 , k2 , x, y)| ≤ cn (1 + |k1 − k2 |)−n (1 + |k1 + k2 |)−4−2κ− j1 − j2 −l1 −l2 , n > 0. (40) This verifies the estimate (25). Before proving (26) we consider the asymptotics when k1 = k2 = k. We denote 0 = (0, 0, 0, 0). For j = 0 we haveI j (k, k, x, y) = O(k −5−2κ ). Thus, in order to establish (27) it is enough to consider I0 (k, k, x, y). To obtain the leading order asymptotics of I0 , we consider the contributions of the principal symbol and the lower order remainder terms separately. Write 0p
c50 (w, x, y, ξ ) = c5 (w, x, y, ξ ) + cr (w, x, y, ξ ), −3−2κ where cr (w, x, y, ξ ) ∈ S1,0 (R6 × R2 ) is smooth and compactly supported in the α (w, x, y) variables. Thus |Dw cr (w, x, y, ξ )| ≤ cα (1 + |ξ |)−3−2κ for all multi-indices α and we infer as above that
|Fw cr (0, x, y, −2ke1 )| = O(k −3−2κ ), for all n > 0. Thus the contribution of cr to I0 is estimated by the right-hand side of (27). Hence it is enough to consider the principal part. To this end, we substitute the principal symbol (38) into formula (39) and obtain I0 (k, k, x, y) = k −2 θ (x)θ (y) µ(z 1 (0, w, x, y))H 0 (0, w, x, y)J (w, x, y) × dw + O(k −5−2κ ). 2 |((κ (0, w, x, y))−1 )t e |2 ]1+κ 2 [1 + 4k 1 R 11
(41)
(0, w, x, y))−1 )t e |2 = 0 we may apply for Since one may compute that a = 4|((κ11 1 1+κ
∞ large k the development (1 + k 2 a)−1−κ = a −1 k −2 j=0 k −2 j (−a)− j . We obtain the desired formula (27) with µ(z 1 (0, w, x, y))H 0 (0, w, x, y)J (w, x, y) 1 R(x, y) = 1+κ dw. (0, w, x, y))−1 )t e |2+2κ 4 |((κ11 1 R2 cos α + α sin α − sin α , where α = w2 /w1 . Moreover, for y = x we compute κ11 = sin α − α cos α cos α ) = 1 and |((κ )−1 )t e | = 1. Moreover, we also It follows that J (w, x, x) = det (κ11 1 11 1 −1 = 4. Put together, these 1 have det (τ (0, w, x, x)) = 4 , and (det( dz (v, w, x, x)| )) v=0 dw observations yield (28) for R(x, x). Finally we prove estimate (26). Observe that E ( u 1 (x, y, k1 ) u 1 (x, y, k2 )) is given by a linear combination of terms I j analogous to (30) where, in addition to changing constants d j , we only replace k2 with −k2 . Notice also that in the proof of formula (39) one may allow k2 to be negative, whence the estimate (26) follows immediately.
Lemma 3. In the decomposition (24) the random variable b(x, y, k) satisfies a.s. the condition |b(x, y, k)| ≤ c (1 + |k|)−3 , x, y ∈ U, k > 0, where the constant c depends only on H0−1,1 (D)-norm of q(z, ω).
Inverse Scattering Problem for a Random Potential
687 (3)
Proof. By (23), ||Φk (· −x)|| H 1,∞ (D) + ||Φk (· −y)|| H 1,∞ (D) ≤ ck 1/2 for k > 1, x, y ∈ U . This implies
(3) |b(x, y, k)| ≤ ||q|| H −1,1 (D) ||Φk (· −x) − Φk (· −x)|| H 1,∞ (D) ||Φk (· −y)|| H 1,∞ (D) 0 (3) (3) + ||Φk (· −x)|| H 1,∞ (D) ||Φk (· −y) − Φk (· −y)|| H 1,∞ (D) ≤ c||q|| H −1,1 (D) (1 + |k|)−1−m . 0
The above results have the following corollary that plays a crucial role in sequel. Corollary 1. Assume that k1 , k2 > 1 and x, y ∈ U . Then E Re (k12 u 1 (x, y, k1 ))Re (k22 u 1 (x, y, k2 )) ≤ cn (1 + |k1 − k2 |)−n , n > 0, where cn is independent of x, y ∈ U , and one may replace one or both of the real parts by imaginary parts. Proof. When k1 , k2 > 1 we have that |k1 + k2 |−n ≤ |k1 − k2 |−n . The claim now follows immediately from estimates (25) and (26) by simply observing that for any a, b, c, d ∈ R we may recover all the products ac, ad, bc and bd as linear combinations of real or imaginary parts of the numbers (a + ib)(c ± id) = (ac ∓ bd) + i(bc ± ad). 6. Higher Order Terms 6.1. The second order term. In this subsection we consider the second term u 2 of the Born series (21), given by Φk (x − z 1 )q(z 1 )Φk (z 1 − z 2 )q(z 2 )Φk (z 2 − y) dz 1 dz 2 . (42) u 2 (x, y, k) = D
D
It turns out that out of all terms in the formal Born series this one is the hardest one to analyze for our purposes. However, the following result yields exactly the estimate that will be used later to show that the contribution of u 2 can be ignored in the measurement (3). Theorem 5. Assume that κ ∈ [0, 1/2). For all x, y ∈ U it holds almost surely that K 1 lim k 4+2κ |u 2 (x, y, k, ω)|2 dk = 0. K →∞ K − 1 1 Proof. One may assume that x = 0, so we will abbreviate u 2 (k) = u 2 (0, y, k, ω) (the dependence on y and w is suppressed in the notation). A main reduction will be that we replace the Hankel functions, one by one, in (42) by the principal terms in the asymptotics (22). It will be useful to abbreviate f k (z) = Φk (z) − d0 (k|z|)−1/2 eik|z| and gk (z) = d0 (k|z|)−1/2 eik|z| ,
(43)
where the constant d0 comes from the asymptotics (22). We need two auxiliary results. The first one collects together useful knowledge on the behaviour of f k , gk , and Φk (·− y) with increasing k (in this section we always assume that k ≥ 1).
688
M. Lassas, L. Päivärinta, E. Saksman
Lemma 4. Let ε ∈ [0, 1]. Then (i) gk H ε, p (D) ≤ c p k ε−1/2 and Φk (·− y) H ε, p (D) ≤ c p k ε−1/2 for p > 1 and y ∈ U. (ii) f k (· − y) H ε, p (D) ≤ c p k 2ε−3/2 for p > 1 and y ∈ U. (iii) f k (z 1 − z 2 ) H ε, p (D×D) ≤ c p k 2ε−3/2 for p ∈ (1, 4/3). Proof. Assume y ∈ U and recall that dist (U, D) =: d > 0. Recall that d c c (1) (1) |H0 (t)| ≤ √ , H0 (t) ≤ √ , t ≥ d. dt t t
(44)
Then by denoting R = sup{|y − z| : y ∈ U, z ∈ D} we have p | f y (z)| dz ≤ Φk (u) p du ≤ c p k − p/2 (R 1− p/2 − d 1− p/2 ) = ck − p/2 , D
d≤|u|≤R
where c is independent of y, and by the same manner one may estimate the gradient ∇Φk (· − y). We thus have sup ||Φk (· − y)|| L p (D) ≤ ck −1/2 ,
y∈U
sup ||∇Φk (· − y)|| L p (D) ≤ ck 1/2 .
(45)
y∈U
These estimates interpolate for 0 ≤ s ≤ 1 to what was claimed for Φk . Next, the statement (i) for gk is obtained similarly by noting that direct computation of ∇gk shows that gk obeys bounds similar to (44). In order to prove (ii), observe that the asymptotics (23) yield the estimates f k L ∞ (D) ≤ ck −3/2 , ∇ f k L ∞ (D) ≤ ck 1/2 ,
(46)
from which the claim again follows by interpolation. To prove (iii), we again use the asymptotics (23) and observe that |∇Φk (z)| ≤ ck 1/2 max(|z|−1 , |z|−1/2 ) for all z. Moreover, by direct computation |∇gk (z)| ≤ ck 1/2 max(|z|−3/2 , |z|−1/2 ). These together yield that |∇ f k (z)| ≤ ck 1/2 max(|z|−1/2 , |z|−3/2 ), k ≥ 1. 3p
(47) 1
By direct computation, we obtain || f k || L p (B) ≤ c p,B (k −2 + k − 2 ) p for p < 4/3 in any bounded domain B. By combining this with (47) we obtain by interpolation the counterpart of (iii) for the function z → f k (z) on any bounded subdomain of R2 . The estimate for the map (z 1 , z 2 ) → f k (z 1 − z 2 ) follows since D is bounded. In order to state the second lemma, recall that the operator Hk was defined through (17) in Sect. 4. We also need to consider the operator K k which combines the multiplication operator with q to Hk , i.e. K k f = Hk (q f ). Lemma 5. For any p > 1, s ∈ (0, 1) and k ≥ 1 it holds that ||K k || H s,2 p →H s,2 p ≤ c1 k −1+2(s+(1−1/ p)) ,
(48)
||K k || H s,2 p →L ∞ ≤ c1 k
(49)
1+2s−1/ p
where the constant c1 = c1 (ω) is finite almost surely.
,
Inverse Scattering Problem for a Random Potential
689 −s, p
Proof. For 0 < s < 1 and 1 ≤ p ≤ 2 ≤ r ≤ ∞ one has that Hk : H0 H s,r (D) with the norm estimate
(D) →
||Hk || H −s, p (D)→H s,r (D) ≤ ck −1+2(s+(1/ p−1/r )) .
(50)
0
This estimate follows easily from the proof of Theorem 3.1 in [39]. An application of (50) together with Lemma 2 and Theorem 2 immediately yields the claim. Let us now replace the left-most Hankel-factor in the integral (42) defining u 2 (k) by the approximation gk and consider −1/2 u 2, (k) := d0 k eik|z 1 | q(z 1 )|z 1 |−1/2 Φk (z 1 − z 2 )q(z 2 )Φk (z 2 − y) dz 1 dz 2 . D
D
By the definition of the operator K k we have |u 2, (k) − u 2 (k)| = |q, f k K k (Φk (· − y)) |, where the brackets refer to distribution duality. According to the previous lemmata we may estimate the left hand side above by q H −ε,(1+ε) (D) f k K k (Φk (· − y)) H ε,1+ε (D) ≤ c2 f k H ε,2+2ε (D) K k H ε,2+2ε (D)→H ε,2+2ε (D) Φk (· − y) H ε,2+2ε (D) ≤ c2 k 2ε−3/2 k −1+2(ε+(1−(1+ε)
−1 ))
k ε−1/2 ,
where c2 = c2 (ω). Here Theorem 2 verifies that q H −ε,(1+ε) (D) < ∞ almost surely; we use here only the lower bound of smoothness obtained from Theorem 2 corresponding to the case κ = 0. Thus we obtain that
u 2 (k) = u 2, (k) + O(k ε −3 ) for all ε > 0.
(51)
We next apply the same asymptotics to the Φk -term on the right and consider eik(|z 1 |+|z 2 |) Φk (z 1 − z 2 )q(z 1 )q(z 2 )(|z 1 ||z 2 − y|)−1/2 dz 1 dz 2 . u 2,r (k) := d02 k −1 D
D
In this approximation the induced error term to u 2, (k) is given by |u 2,r (k) − u 2, (k)| = |q, h k K k ( f k (· − y) |, where h k (z 1 ) = d0 eik|z 1 | |kz 1 |−1/2 . Note that |z 1 |−1/2 is smooth on D. Clearly h k H ε,∞ (D) ≤ ck ε−1/2 , and hence we may apply again Lemma 4 and obtain analogously to the previous computation u 2, (k) = u 2,r (k) + O(k ε−1/2 k −1+2(ε+(1−(1+ε) = u 2,r (k) + O(k
ε −3
−1 ))
) for all ε > 0.
k −3/2+2ε ) (52)
To complete the reduction, we apply the same asymptotics to the remaining Φk -term in the integral defining u 2,r (k) and consider v(k) := d03 eik(|z 1 |+|z 2 |+|z 1 −z 2 |) q(z 1 )q(z 2 )(|z 1 − z 2 ||z 1 ||z 2 − y|)−1/2 dz 1 dz 2 . D
D
690
M. Lassas, L. Päivärinta, E. Saksman
Following our definitions, this integral is understood in the sense that one first does the integration (distributional duality) with respect to the z 1 variable. However, one verifies without difficulty that we also have u 2,r (k) = Φk (· − ·), sk , where sk (z 1 , z 2 ) = d02 k −1 q(z 1 )q(z 2 )eik(|z 1 |+|z 2 |) (|z 1 ||z 2 − y|)−1/2 , (z 1 , z 2 ) ∈ D × D. One easily verifies that a1 (z 1 )a2 (z 2 ) ∈ H −2ε,∞ (R4 ) whenever a1 , a2 ∈ H −ε,∞ (R2 ), and > 0. Thus, according to Theorem 2 and the Sobolev embedding theorem we have that q1 ⊗ q2 ∈ H0−ε,∞ (D × D) for all ε > 0. Observe that eik(|z 1 |+|z 2 |) H ε,∞ (D×D) ≤ ck ε . A simple duality argument using (11) shows that sk H −ε,∞ (D×D) ≤ ck ε−1 , for all ε > 0. 0
Combining this estimate with Lemma 4 we obtain |k −3/2 v(k) − u 2,r (k)| = | f k (· − ·), sk | ≤ f k (· −· ) H ε,5/4 (D×D) sk H −ε,5 (D×D) 0
≤ ck 3ε−5/2 . In conjunction with (51) and (52) this finally gives
u 2 (k) = k −3/2 v(k) + O(k ε −5/2 ) for all ε > 0.
(53)
We now enter the main difficulty of the proof, that is, the estimation of v(k). Our first observation is that it is possible to circumvent pointwise estimates with respect to k altogether. Namely, in order to prove Theorem 5 it will be enough to show that ∞ |v(k)|2 k 2κ dk < ∞ a.s. (54) 1
To see this, we notice that by (53) one may write K K 1 k 2+κ 2 |v(k)|2 k 2κ dk + O(K 2κ+2ε −1 ) |k u 2 (k)| dk ≤ 2 K 1 K 1 ∞ k min(1, )|v(k)|2 k 2κ dk + O(K 2κ+2ε −1 ). ≤2 K 1 This last integral converges a.s. to zero by the dominated convergence theorem as K → ∞, and the claim follows if ε is small enough. Towards (54) we will shortly express v as a one-dimensional Fourier transform and get rid of the variable k. Before that a couple of auxiliary considerations are needed to treat the case where κ = 0. First of all, in order not to have problems with interpreting the distribution dualities that will emerge, we introduce the modification vδ , δ > 0, of v that is obtained by replacing q by the standard mollification qδ := q ∗ ρδ , where ρδ (x) = δ −2 ρ(x/δ) and ρ ∈ C0∞ (R2 ) is a radially symmetric function satisfying ρ(x)d x = 1. We denote the mollification operator by Mδ : f → f ∗ ρδ . Observe for later use that the covariance operator of qδ equals Cδ = Mδ Cq Mδ . Clearly vδ (k) → v(k) as δ → 0. In order to verify (54) in case κ = 0 it is enough to show that ∞ sup E |vδ (k)|2 dk < ∞, (55) δ∈(0,1) 1
Inverse Scattering Problem for a Random Potential
691
since ∞an application of the Fubini theorem and Fatou’s lemma then yields that E ( 1 |v(k)|2 dk) < ∞, which immediately implies (54). We need to take a closer look at the phase function A(z 1 , z 2 ) := |z 1 |+|z 1 −z 2 |+|z 2 −y|. Observe first that A is smooth on D × D apart from the subset where z 1 = z 2 . Moreover, the gradient of A is bounded from below and above; 0 < c1 ≤ |∇ A(z 1 , z 2 )| ≤ c2 < ∞ for (z 1 , z 2 ) ∈ D × D, z 1 = z 2 .
(56)
The upper bound is evident. For the lower bound it is enough to apply the convexity of U , the fact that 0, y ∈ U and compute z2 − y ≥ c0 > 0 |z 2 − y| for z 1 = z 2 , z 1 , z 2 ∈ D.
(z 1 , z 2 ) · ∇ A(z 1 , z 2 ) = |z 1 | + |z 1 − z 2 | + z 2 ·
(57)
Here we are using the fact that the measurement domain U is convex. Moreover, it shows that the surfaces Γt := {(z 1 , z 2 ) ∈ D × D | A(z 1 , z 2 ) = t}, t > 0 are (if non-empty) locally boundaries of starshaped domains with respect to the origin. There are smallest and largest values 0 < T0 = T0 (y) and T1 = T1 (y) such that Γt is nonempty only for T0 ≤ t ≤ T1 , and we only need to consider these values of t. We now fix a t ∈ [T0 , T1 ]. Simple geometrical considerations verify that there is η = η( t) and an open cone K = K ( t) ⊂ R4 with center at the origin with the following properties: By writing t0 = t − η and t1 = t + η it holds that D × D ∩ {t0 < A(z 1 , z 2 ) < t1 } ⊂ K ∩ {t0 < A(z 1 , z 2 ) < t1 } =: Γ. By defining Γt = Γ ∩ {(z 1 , z 2 ) : A(z 1 , z 2 ) = t} for t ∈ (t0 , t1 ) we thus have Γ = t0 ≤t≤t1 Γt . Moreover, on the basis of (56) and (57) one deduces that there is a radial stretch Bt yielding a bi-Lipschitz chart Bt : F → Γt over a subdomain F of the unit ball. The bi-Lip constant of Bt is uniform over t0 < t < t1 and each Bt is actually a local diffeomorphism outside the subset where z 1 = z 2 . Moreover, since D has a positive distance to the origin one may also make the choice of η and K so that the condition |z 1 |, |z 2 | ≥ b > 0 for all (z 1 , z 2 ) ∈ Γ
(58)
holds true. In the preceding discussion the quantities Γ, Bt , F of course depend on fixed t, but we will suppress this in what follows. We hence see that the surfaces Γt , with varying t provide a fairly regular foliation of the domain Γ – actually (56) and (57) show that we may write Bt (w1 , w2 ) = (λ(t, w1 , w2 )w1 , λ(t, w1 , w2 )w2 ), where the dependence (w1 , w2 ) → λ(t, w1 , w2 ) is Lipschitz with respect to t with a uniform Lipschitz-constant with respect to w1 , w2 . The considerations in the preceding paragraph justify a generalized co-area formula for integrals over Γ : t2 1 3 dH (z 1 , z 2 ) dt, (59) g(z 1 , z 2 ) dz 1 dz 2 = g(z 1 , z 2 ) |∇ A(z 1 , z 2 )| Γ t1 Γt
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where the inner integral is with respect to the 3-dimensional Hausdorff measure on Γt , and g is any integrable Borel-function on Γ. In a similar vein, we may perform a change of variables and write for any fixed t, g(z 1 , z 2 ) dH3 (z 1 , z 2 ) = g(Bt (w1 , w2 ))Ht (w1 , w2 )dH3 (w1 , w2 ), (60) Γt
F
where the Jacobian Ht satisfies the uniform bound 0 < C1 ≤ Ht (z 1 , z 2 ) :=
|Bt (w1 , w2 )|3 |∇ A(Bt (w1 , w2 ))| ≤ C2 , |(w1 , w2 ) · ∇ A(Bt (w1 , w2 ))|
(61)
according to (56) and (57). Moreover, it is important to note for our later purposes that the dependence t → Ht (z 1 , z 2 ) is uniformly Lipschitz with respect to the variable t. Lemma 6. Given γ ∈ (0, 2) there is a finite constant c such that for every t ∈ [t0 , t1 ] we have |z 1 − z 2 |−γ dH3 (z 1 , z 2 ) ≤ c and (62) Γt
Γt ×Γt
|z j − z k |−γ dH3 (z 1 , z 2 )dH3 (z 1 , z 2 ) ≤ c for k, j = 1, 2.
(63)
Proof. Observe that for any (z 1 , z 2 ) ∈ Γt , in the change of variables (w1 , w2 ) = Bt−1 (z 1 , z 2 ) one has (w1 , w2 ) = (λ(z 1 , z 2 )z 1 , λ(z 1 , z 2 )z 2 ), where the scalar factor λ = λ(z 1 , z 2 ) ∈ R+ satisfies uniformly λ ≥ a > 0. Hence an application of (60) and (61) yields that |z 1 − z 2 |−γ dH3 (z 1 , z 2 ) ≤ c a −γ |w1 − w2 |−γ dH3 (w1 , w2 ). Bt−1 (Γt )
Γt
The last written integral can be estimated by the integral over the whole sphere S 3 , which is easily seen to be finite. In order to treat the integral (63) we first note that by elementary geometry one has |x − y| ≥ |x||x 0 − y 0 |/2, where x 0 = x/|x| (and similarly for y) stands for the corresponding unit vector. According to (58) we see that the integral (63) is bounded from above by γ −γ 2 b |(z j )0 − (z k )0 |−γ dH3 (z 1 , z 2 )dH3 (z 1 , z 2 ). (64) Γt ×Γt
Using again the change of variables as before with respect to both z and z we obtain that (64) is dominated by the expression |(w j )0 − (wk )0 |−γ dH3 (w1 , w2 )dH3 (w1 , w2 ), 2γ b−γ c2 H ×H
where H = {(w1 , w2 ) ∈ S 3 : |w1 |, |w2 | > b}. The last written integral is readily seen to be finite by an application of the Fubini theorem.
Inverse Scattering Problem for a Random Potential
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We return to our main theme and use (59) to write vδ (k) as a Fourier-transform vδ (k) = Sδ (k), k > 0,
(65)
where the function Sδ is compactly supported inside (T0 , T1 ), and for a fixed t ∈ [T0 , T1 ] and t ∈ [t0 ( t), t1 ( t)] one has Sδ (t) := qδ (z 1 )qδ (z 2 )(|z 1 − z 2 |)−1/2 L(z 1 , z 2 )|∇ A(z 1 , z 2 )|−1 dH3 (z 1 , z 2 ). Γt
Above L(z 1 , z 2 ) is a smooth cutoff of the function |z 1 |−1/2 |z 2 − y| that vanishes outside D × D. Hence L ∈ C0∞ (R4 ). Case 1. κ = 0. We claim that (55) (with κ = 0) follows as soon as we verify that for t) < ∞ such that each t ∈ [T0 , T1 ] there is a finite constant M = M( E |Sδ (t)|2 ≤ M for all δ ∈ (0, 1) and t ∈ [t0 ( t), t1 ( t)].
(66)
Namely, by compactness we may then cover [T0 , T1 ] by intervals associated to only finitely many values t ∈ [T0 , T1 ], and it follows that E |Sδ (t)|2 ≤ M for any t ∈ [T0 , T1 ], whence E Sδ 2L 2 (R) ≤ M (T2 − T1 ) < ∞. The desired inequality (55) will be a consequence of Parseval’s formula. t), t1 ( t)]. It remains to estimate E |Sδ (t)|2 . In Let us fix t ∈ [T0 , T1 ] and let t ∈ [t0 ( fact, by using the well-known Wick formulae for the expectation of n-fold products of centered Gaussian variables we obtain E (qδ (z 1 )qδ (z 2 )qδ (z 1 )qδ (z 2 )) = Cδ (z 1 , z 2 )Cδ (z 1 , z 2 ) + Cδ (z 1 , z 1 )Cδ (z 2 , z 2 ) + Cδ (z 1 , z 2 )Cδ (z 2 , z 1 ) and thus, in the new coordinates associated E |Sδ (t)|2 = (|z 1 − z 2 |)−1/2 |∇ A(z 1 , z 2 )|−1 L(z 1 , z 2 ) Γt ×Γt
·(|z 1 − z 2 |)−1/2 |L(z 1 , z 2 )∇ A(z 1 , z 2 )|−1 ·(Cδ (z 1 , z 2 )Cδ (z 1 , z 2 ) + Cδ (z 1 , z 1 )Cδ (z 2 , z 2 ) + Cδ (z 1 , z 2 )Cδ (z 2 , z 1 )) ·dH3 (z 1 , z 2 ) dH3 (z 1 , z 2 ). From Proposition 1 it is immediate that for any given a > 0 there is a finite constant ca such that |Cδ (z 1 , z 2 )| ≤ ca |z 1 − z 2 |−a for any δ ∈ (0, 1) and (z 1 , z 2 ) ∈ D × D. By (56) we obtain for any t ∈ [t1 , t2 ] the estimate 2 R(z 1 , z 2 , z 1 , z 2 )T (z 1 , z 2 , z 1 , z 2 ) dH3 (z 1 , z 2 )dH3 (z 1 , z 2 ), sup E |Sδ (t)| ≤ c δ∈(0,1)
Γt ×Γt
(67) where R(z 1 , z 2 , z 1 , z 2 ) = |z 1 − z 2 |−1/2 |z 1 − z 2 |−1/2
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and T (z 1 , z 2 , z 1 , z 2 ) =
(|r1 − r2 ||r3 − r4 |)−a .
{r1 ,r2 ,r3 ,r4 }={z 1 ,z 2 ,z 1 ,z 2 }
In this last formula we sum over all permutations of the four-element set. It is now clear by symmetry, Lemma 4, and an application of Hölder’s inequality on (67) that the integral (67) is finite for all t ∈ [t1 , t2 ] as soon as a is chosen small enough. Thus we have established (66). Case 2. κ ∈ (0, 1/2). In this case the realizations are Hölder continuous with probability one and the covariance operator also has a Hölder continuous kernel according to Proposition 1. Hence we denote S(t) = S0 (t), i.e. we leave out the mollification. For positive values of κ we claim that (54) follows if we establish for each t ∈ [T0 , T1 ] the estimate E |S(t) − S(t )|2 ≤ M|t − t |κ+1/2 for all t, t ∈ [t1 ( t), t2 ( t)].
(68)
Namely, then Fubini’s theorem yields that t1 t1 |S(t) − S(t )|2 dt dt < ∞ E |t − t |1+2s t0 t0 for s < (2κ + 1)/4. Especially, since κ < 1/2, this holds if s = κ. According to s,2 the Besov characterization of the homogeneous Sobolev norm Hhomog this means that κ,2 S|[t0 ,t1 ] ∈ Hhomog over any subinterval of ]t0 , t1 [. Again, as in the beginning of Case 1,
κ,2 we deduce by compactness that S ∈ Hhomog (R) almost surely, and by taking Fourier transforms this yields (54).
The verification of the estimate (68) leads to computations that are more cumbersome than in Case 1. Let us first fix t, t ∈ [t0 , t1 ]. By applying the change of variables (60) and introducing the abbreviation z j (u) = z j (u, w1 , w2 ) we obtain S(u) = q(z 1 (u))q(z 2 (u))Uu (w1 , w2 )Tu (w1 , w2 )dH3 (w1 , w2 ) for u ∈ [t0 , t1 ] F
with Uu (w1 , w2 ) := (|z 1 (u) − z 2 (u)|)−1/2 and Tu (w1 , w2 ) := L(z 1 (u), z 2 (u))Hu (w1 , w2 )(|∇ A(z 1 (u), z 2 (u))|)−1 . We next analyze the impact of the different factors in this integrand to the second moment of S(t)− S(t ). Let us observe first that Tt is uniformly bounded and satisfies the estimate |Tt (w1 , w2 ) − Tt (w1 , w2 )| ≤ c|t − t |. Hence, if we apply the Minkowski inequality after replacing in the definition of S(t ) the factor Tt (w1 , w2 ) by Tt (w1 , w2 ), it follows that S(t) − S(t ) L 2 (Ω) ≤ S1 (t) − S1 (t ) L 2 (Ω) +c|t − t | q(z 1 (t ))q(z 2 (t )) L 2 (Ω) |Ut (w1 , w2 )| dH3 (w1 , w2 ) F
≤ S1 (t) − S1 (t ) L 2 (Ω) + C|t − t |
(69)
Inverse Scattering Problem for a Random Potential
695
since the last written integral is obviously finite (cf. Case 1). Above S1 (u) := q(z 1 (u))q(z 2 (u))Uu (w1 , w2 )T (w1 , w2 )dH3 (z 1 , z 2 ), F
where T (w1 , w2 ) := Tt (w1 , w2 ) (remember that t, t are fixed). In order to perform a similar operation with respect to the term Uu (w1 , w2 ) we make use of homogeneity. What comes to |z 1 (u) − z 2 (u)|−1/2 we recall that (z 1 (u), z 2 (u)) = (λu w1 , λu w2 ), where the scalar factor λu depends on (w1 , w2 ), is uniformly bounded from above and below and stays uniformly Lipschitz in u. Accordingly, ||z 1 (t) − z 2 (t)|−1/2 − |z 1 (t) − z 2 (t)|−1/2 | = |(λt )−1/2 − (λt )−1/2 | |w1 − w2 |−1/2 ≤ C|t1 − t2 ||w1 − w2 |−1/2 . Since F q(z 1 (t ))q(z 2 (t ) L 2 (Ω) |w1 − w2 |−1/2 dH3 (w1 , w2 ) < ∞ we obtain as in (69) the estimate S1 (t) − S1 (t ) L 2 (Ω) ≤ S2 (t) − S2 (t ) L 2 (Ω) + C|t − t |, where
(70)
S2 (u) :=
q(z 1 (u))q(z 2 (u))U (w1 , w2 )T (w1 , w2 )dH3 (w1 , w2 ), F
and U (w1 , w2 ) := Ut (w1 , w2 ). Let us denote R(w1 , w2 ) := U (w1 , w2 )T (w1 , w2 ). In order to finally estimate S2 (t) − S2 (t ) L 2 (Ω) we write the difference S2 (t) − S2 (t ) as a double integral with the result S2 (t) − S2 (t )2L 2 (Ω) = G(w1 , w2 , v1 , v2 )R(w1 , w2 )R(v1 , v2 )dH3 (w1 , w2 )dH3 (v1 , v2 ), A×A
where G(w1 , w2 , v1 , v2 ) := Cq (z 1 , z 2 )Cq (u 1 , u 2 ) + Cq (z 1 , u 1 )Cq (u 2 , z 2 ) + Cq (z 1 , u 2 )Cq (u 1 , z 2 ) − 2Cq (z 1 , z 2 )Cq (u 1 , u 2 ) − 2Cq (z 1 , u 1 )Cq (u 2 , z 2 ) − 2Cq (z 1 , u 2 )Cq (u 1 , z 2 ) + Cq (z 1 , z 2 )Cq (u 1 , u 2 ) + Cq (z 1 , u 1 )Cq (u 2 , z 2 ) + Cq (z 1 , u 2 )Cq (u 1 , z 2 ). (71) Above we have denoted (u 1 , u 2 ) = Bt (v1 , v2 ) and (u 1 , u 2 ) = Bt (v1 , v2 ) for (v1 , v2 ) ∈ F, and similarly (z 1 , z 2 ) = Bt (w1 , w2 ) and (z 1 , z 2 ) = Bt (w1 , w2 ) for (w1 , w2 ) ∈ F. Recall that the covariance has the form Cq (z 1 , z 2 ) = a(z 1 , z 2 )|z 1 − z 2 |2κ + r (z 1 , z 2 ),
(72)
where a is smooth and r Hölder with exponent (1 − ε) for any ε > 0. Formula (71) yields immediately that |G(w1 , w2 , v1 , v2 )| ≤ c|t − t |2κ . Moreover, given δ > 0 it is easily checked that ||z 1 − z 2 |2κ − |z 1 − z 2 |2κ | ≤ c(κ)δ κ+1/2 for |z 1 − z 2 | ≥ 2δ and |(z 1 , z 2 ) − (z 1 , z 2 )| ≤ δ/2.
(73)
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A fortiori, by the bi-Lipschitz property of (t, w1 , w2 ) → Bt (w1 , w2 ) an analogous estimate follows for the covariance Cq : there is a constant c3 > 0 so that |Cq (z 1 , z 2 ) − Cq (z 1 , z 2 )| ≤ c (κ)δ κ+1/2 for |w1 − w2 | ≥ δ and |(w1 , w2 ) − (w1 , w2 )| ≤ c3 δ.
(74)
Consider the set 1 |t − t | for some i, j ∈ {1, 2}} 2 ∩{(w1 , w2 , v1 , v2 ) ∈ F × F : |w1 − w2 | ≤ |t − t | or |v1 − v2 | ≤ |t − t |}.
P = {(w1 , w2 , v1 , v2 ) ∈ F × F : |wi − v j | ≤
According to formulae (71) and (74) we have for |t − t | ≤ c4 that |G(w1 , w2 , v1 , v2 )| ≤ c |t − t |κ+1/2 if (w1 , w2 , v1 , v2 ) ∈ (F × F)\P.
(75)
Observe that |R(v1 , v2 )| ≤ c|v1 − v2 |−1/2 . By invoking crude estimates for the measure of the set P and applying Hölder inequality on the function (w1 , w2 , u 1 , u 2 ) → |w1 − w2 |−1/2 |u 1 − u 2 |−1/2 we easily obtain in view of Lemma 6 that R(w1 , w2 )R(v1 , v2 ) dH3 (w1 , w2 )dH3 (v1 , v2 ) P ≤c |w1 − w2 |−1/2 |v1 − v2 |−1/2 dH3 (w1 , w2 )dH3 (v1 , v2 ) P
≤ c|t − t |1/2 .
(76)
By dividing integration in (71) over the sets P ∩ (F × F) and (F × F)\P, the yield (together with the finiteness of the integral estimates (73), (75) and (76) 3 (w , w )dH3 (v , v )) that R(w , w )R(v , v ) dH 1 2 1 2 1 2 1 2 F×F
S2 (t) − S2 (t )2L 2 (Ω) ≤ c |t − t |2κ |t − t |1/2 + |t − t |κ+1/2 ≤ c |t − t |κ+1/2 . Together with the chain of our previous inequalities this yields (68) and hence finishes the proof of Theorem 5.
6.2. The convergence of the Born series. In this subsection we verify that the Bornseries converges to the solution (if k is large enough) and that the higher order terms decay in an appropriate way. Theorem 6. (i) There is a (random) index k0 = k0 (ω) such that k0 < ∞ almost surely and, if k ≥ k0 then the Born series (21) converges for any x, y ∈ U to the solution u(x, y, k). (ii) For any > 0 and k ≥ k0 there exist c = c( , ω), finite almost surely, such that ∞
sup |u n (x, y, k)| ≤ ck −5/2+ .
n=3 x,y∈U
Inverse Scattering Problem for a Random Potential
697
Proof. It is enough to consider the hardest case κ = 0, since for positive κ all the estimates below clearly hold true – actually many estimates become better in that case. We start from the expression u n (x, y, k) = (K kn Φk (· − y))(x). By Lemma 4 (i) and Lemma 5 we may estimate that ||u n (·, y, k)|| L ∞ (U ) ≤ ||K k || H s,2 p →L ∞ ||K k ||n−1 ||Φk (· − y)|| H s,2 p H s,2 p →H s,2 p ≤ cn k 1+2s−1/ p k (n−1)(−1+2(s+1−1/ p)) k −1/2+s .
(77)
Here the constant c = c(ω) is independent of y and thus the desired estimate follows. Let us denote s − 1 + 1p = 1 and 2(s + 1 − 1p ) = 2 , whence we can take 1 > 0 and 2 > 0 arbitrarily small. With these choices (77) yields that ||u n (·, ·, k)|| L ∞ (U ×U ) ≤ cn k 1/2+ 1 −n(1− 2 ) and consequently ∞
||u n (·, ·, k)|| L ∞ (U ×U ) ≤ c3 k −5/2+( 1 +3 2 )
n=3
1 . 1 − ck 2 −1
This proves (ii) as soon as we choose k0 large enough so that ck0ε2 −1 < 1/2. To obtain (i), observe that an iteration of the Lippmann-Schwinger equation yields the n th remainder term in the form (K k )n+1 u, which converges to zero by the operator norm estimate for K k used in (77). 7. Existence of the Measurement: Convergence of the Ergodic Averages Now we are ready to analyze the measurement m(x, y, ω). Theorem 7. For x, y ∈ U the limit (3) exist almost surely and equals K 1 k 4+2κ |u s (x, y, k, ω)|2 dk = R(x, y), lim K →∞ K − 1 1
(78)
where R(x, y) is the smooth function on U × U given in Proposition 3. Before giving the proof we first describe the philosophy behind Theorem 7. Let us write u s (x, y, k) = u 1 (x, y, k) + u r (x, y, k),
(79)
where u r = (b +u 2 +u 3 +u 4 +. . .) stands for the remainder term (recall that u 1 = u 1 +b). The results of the previous section will yield that the contribution of u r is negligible in the measurement, whence it remains to understand the mean behaviour of | u 1 |2 . The 4 2 analytic estimates of Sect. 5 show that the expectation E k | u 1 (x, y, k)| tends to a limit as k → ∞. In addition, the same estimates verify that the terms k12 u 1 (x, y, k1 ) and 2 k2 u 1 (x, y, k2 ) become asymptotically independent as k2 grows towards infinity (see the u 1 (x, y, k)|2 figure below). This makes it plausible that one could recover limk→∞ E |k 4 as a suitable ergodic average, in view of the strong law of large numbers, and this turns out to be true. We record an elementary lemma.
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Lemma 7. Let X and Y be zero-mean Gaussian random variables. Then E ((X 2 − E X 2 )(Y 2 − E Y 2 )) = 2( E X Y )2 . Proof. By scaling one may obviously assume that E X 2 = E Y 2 = 1. Denote E X Y = cos α ∈ [−1, 1]. Then (X, Y ) and (X, cos(α)X + sin(α)Y ) have the same distribution, where Y is an independent copy of X . The result follows now by a straightforward computation. Let us recall an ergodic theorem suitable for our purposes. The following is obtained e.g. as an immediate corollary of [14].
Theorem 8. Let X t , t ≥ 0 be a real valued stochastic process with continuous paths. Assume that for some positive constants c, ε > 0 the condition | E X t X t+r | ≤ c(1 + r )−ε holds for all t, r ≥ 0. Then almost surely lim
K →∞
1 K
K
X t dt = 0.
1
The ergodicity of the term u 1 (recall (24)) is verified in the following proposition. Proposition 4. For any x, y ∈ U we have almost surely lim
K →∞
1 K −1
1
K
k 4+2κ | u 1 (x, y, k)|2 dk = R(x, y).
(80)
Inverse Scattering Problem for a Random Potential
699
Proof. According to Lemma 3 we have limk→∞ E (k 4+2κ | u 1 (x, y, k)|2 ) = R(x, y). Hence it is clear that the claim follows as soon as we show that K 1 lim Y (x, y, k)dk = 0, (81) K →∞ K − 1 1 where Y (x, y, k) = k 4+2κ (| u 1 (x, y, k)|2 − E | u 1 (x, y, k)|2 ). Since
u 1 (x, y, k))2 − E (Re Y (x, y, k) = k 4 (Re u 1 (x, y, k))2 ) +(Im u 1 (x, y, k))2 − E (Im u 1 (x, y, k))2 ) , we may combine Corollary 1 together with Lemma 7 to obtain E|Y (x, y, k1 )Y (x, y, k2 )| ≤
c , 1 + |k1 − k2 |2
for any k1 , k2 ≥ 1. Statement (81) now follows immediately from Theorem 8.
We are ready for Proof of Theorem 7. By denoting u r (x, y, k) = b(x, y, k) + u 2 (x, y, k) + u R (x, y, k) we may decompose u s (x, y, k) = u 1 (x, y, k) + b(x, y, k) + u 2 (x, y, k) + u R (x, y, k). According to Lemma 3 and Theorem 6 we have a.s. lim k→∞ k 2+κ (b(x, y, k)+u R (x,y,k)) = 0. Together with Theorem 5 this yields that almost surely K 1 k 4+2κ |u r (x, y, k)|2 dk = 0. (82) lim K →∞ K − 1 1 The desired statements now follow directly by combining (82) and Proposition 4, as the obtained cross term may be estimated with the aid of the Cauchy-Schwartz inequality in the space [1, K ] equipped with the weight (K − 1)−1 dk. 8. Conclusion: Proof of Theorem 1 The results obtained so far (Theorem 7 from the previous section) prove directly parts (i) and (ii) of our main result, Theorem 1: the measurement (3) is almost surely well defined for any x, y ∈ U . It remains to prove part (iii) of the theorem, which deals with the recovery of µ from the measurements. Observe that in our case m 0 (x, x) = R(x, x) for any x ∈ U , and, by formula (28) in Sect. 5, we have that 1 1 R(x, x) = 8+2κ 2 µ(z) dz. (83) 2 π D |x − z|2 Especially, the function x → R(x, x) is continuous. Hence, by performing measurements in a dense set of points x ∈ U , Theorem 7 shows that almost surely we can recover R(x, x) for all x ∈ U.
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Thus, the relation (83) shows that we are left with a simple deconvolution problem: the values of the convolution H (x) := (h ∗ µ)(x), h(z) :=
1 28+2κ π 2 |z|2
are known in a open set U that has a positive distance to the support of µ ∈ C0∞ (R2 ), and we are to show that this knowledge is enough to recover µ. For that end, observe first that ∆z (|z|−2 p ) = 4 p 2 |z|−2 p−2 . Thus our data determines also the convolutions 1 p µ(z) dz c p ∆x H (x) = |x − z|2 p D for p > 1 and x ∈ U . Let us denote S(x, r ) =
|z−x|=r
µ(z) d|z|,
which corresponds to the Radon transform along circles. Fix any x ∈ U. It follows that we are able to recover the integrals S(x, r ) 1 Q( 2 ) dr, 2 r r R+ p where Q(t) = j=0 a j t j , p ≥ 0. The support of the continuous function r → S(x, r ) lies in a finite interval [a, b] with a, b > 0, and obviously the functions of the form Q(1/r 2 ) are dense in C([a, b]). Thus the function S(x, r ) is uniquely determined for all r > 0. The observation that we just made can be stated in another form: the data yields the knowledge of integrals of µ over all circles that are centered in the open set U . This is a classical problem of integral geometry, of the Radon type, which can be solved in a simple manner, cf. eg. [4] and the extensive list of references therein. Namely, let g(z) = exp(−|z|2 /2) for z ∈ R2 , and observe that knowing the integrals over the above mentioned circles we may compute the convolution g ∗ µ(z) for z ∈ U . However, g ∗ µ is clearly real analytic and the set U is open, whence we know g ∗ µ everywhere. As the Fourier transform of g is smooth and non-zero all over R2 , it follows that we can recover µ uniquely. This completes the proof of our main result. Remark 1. The proof of Theorem 1 goes through also without the assumption E q = 0. Namely, assume that E q = p ∈ C0∞ (D) and denote q0 = q − p. Then E (q(z 1 )q(z 2 )) = E (q0 (z 1 )q0 (z 2 )) + p(z 1 ) p(z 2 ).
(84)
We briefly analyze how the above proof should be modified for this case. We have again −ε, p that q ∈ H0 (D) a.s. Thus the results for the direct scattering problem given in Sect. 4 are valid without any change, and we see in particular that the higher order Born terms u 3 + u 4 + . . . do not contribute to the measurement (3). When the term p(z 1 ) p(z 2 ) in formula (84) is added to the covariance operator in −∞ formula (29), we see that this causes only a S1,0 perturbation for the symbol of the covariance operator Cq . Hence the proof of Proposition 3 remains unchanged. With small modifications the considerations in Subsect. 6.1 remain valid, too. Finally, as the stationary phase method yields E u 1 (x, y, k) = o(k −∞ ), we obtain Theorem 1 by finishing the proof as in Sects. 7 and 8.
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Remark 2. The unique solvability results of Sect. 4 allow us to extend the main result also to the case where κ < 0 with |κ| enough small. All the arguments remain essentially the same, only the treatment of the second term needs minor technical adjustment. Moreover, it should be pointed out that if κ were assumed to be unknown a priori, then Theorem 1 shows that (in principle) it would be possible to first determine κ from the above measurements. Remark 3. One may also consider as the measurement the average K 1 lim k 4+2κ |u s (x, x, k, ω)|2 φ(x)d xdk K →∞ K − 1 1 U with φ ∈ C0∞ (U ). The main result can also be stated in terms of this kind of ‘distributional measurements’. In this setup the proof of Theorem 1 remains essentially unchanged. One should also note that the function R(x, x) is uniquely determined from integrals R(x, x)φ(x)d x against a countable and dense set of smooth test functions φ. U Remark 4. It is interesting to compare the stability of the stochastical inverse problem with the deterministic one. In Theorem 1 the operator T is linear and thus the reconstruction of µ requires solving of a linear ill-posed inverse problem. More precisely, by the observations in the present section, T corresponds to a Radon transform over circles, which gives a pretty clear picture of the ill-posedness. This is markedly different from the corresponding deterministic problems. Remark 5. We mention that in the backscattering case y = x it is possible to avoid the use of the pseudodifferential calculus in Sect. 5, although the proof remains fairly technical. By this manner it is possible to relax somewhat the assumption of smoothness of µ. Remark 6. One might ask whether it is possible to formulate a more general ergodic theorem of the type considered in the present paper, perhaps trying to incorporate the heuristics that (working in the Fourier side) the main part of the scattering takes place where the potential corresponds to the wave length of the incoming wave. The quasi-local nature of the covariance operator makes room for the asymptotic independence. These kinds of ideas should be first studied in a simpler model, e.g. in the one-dimensional case. We mention in this connection that some of our results were announced without proofs in [31], where more heuristics can be found. Epecially, we sketch there a rough connection between the 2-dimensional case and an analogous one-dimensional problem. Acknowledgements. M.L. and L.P. were supported by the Finnish Centre of Excellence in Inverse Problems Research. E.S. was supported by the Academy of Finland, projects no. 113826 and 118765, and by the Finnish Center of Excellence Analysis and Dynamics.
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37. Neimam-Zade, M., Shkalikov, A.: The Schrödinger operators with singular potentials from the multiplicator’s spaces. (Russian) Matem. Zametki 66, 723–733 (1999) 38. Ola, P., Päivärinta, L., Serov, V.: Recovering singularities from backscattering in two dimensions. Comm. Partial Differ. Eqs. 26, 697–715 (2001) 39. Päivärinta, L.: Analytic methods for inverse scattering theory. In: New Analytic and Geometric Methods in Inverse Problems. Springer Lecture Notes Ed. Bingham, K., Kurylev, Y., Somersalo, E., Berline-Heidelberg New York: Springer, 2003, pp. 165–185 40. Päivärinta, L., Panchenko, A., Uhlmann, G.: Complex geometrical optics solutions for Lipschitz conductivities. Rev. Mat. Iberoamericana 19, 57–72 (2003) 41. Päivärinta, L., Serov, V.: An n-dimensional Borg-Levinson theorem for singular potentials. Adv. in Appl. Math. 29(4), 509–520 (2002) 42. Päivärinta, L., Somersalo, E. (eds.): Inverse Problems in Mathematical Physics. Lecture Notes in Physics, 422. Berlin: Springer-Verlag, 1993 43. Papanicolaou, G., Postel, M., White, B.: Frequency content of randomly scattered signals. Siam Review 33, 519–626 (1991) 44. Papanicolaou, G., Weinryb, S.: A functional limit theorem for waves reflected by a random medium. Appl. Math. Optim. 30, 307–334 (1994) 45. Pastur, L.: Spectral properties of disordered systems in the one-body approximation. Commun. Math. Phys. 75, 179–196 (1980) 46. Peetre, J.: Rectification à l’article Une charactérization abstraite des opérateurs differentiels. Math. Scand. 8, 116–120 (1960) 47. Rozanov, Y.: Markov Random Fields. Applications of Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 1982 48. Simon, B.: Localization in general one-dimensional random systems. I. Jacobi matrices. Commun. Math. Phys. 102, 327–336 (1985) 49. Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Comm. Pure Appl. Math. 39(1), 75–90 (1986) 50. Stefanov, P.: Generic uniqueness for two inverse problems in potential scattering. Comm. Partial Differ. Eqs. 17, 55–68 (1992) 51. Stein, E. M.: Harmonic Analysis. Princeton, NJ: Princeton University Press, 1993 52. Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125(1), 153–169 (1987) 53. Taylor, M. E.: Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. Mathematical Surveys and Monographs 81, Providence, RI: Amer. Math. Soc., 2000 54. Ueki, N.: Wegner estimates and localization for Gaussian random potentials. Publ. Res. Inst. Math. Sci. 40(1), 29–90 (2004) 55. Uhlmann, G.: Inverse boundary value problems for partial differential equations. Proceedings of the ICM. Vol. III (Berlin, 1998). Doc. Math. J. DMV Extra Vol. III, pp. 77–86 (1998) Communicated by B. Simon
Commun. Math. Phys. 279, 705–733 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0429-1
Communications in
Mathematical Physics
Canonical Sasakian Metrics Charles P. Boyer, Krzysztof Galicki, Santiago R. Simanca Department of Mathematics and Statistics, University of New Mexico, Albuquerque, N.M. 87131, USA. E-mail: [email protected]; [email protected]; [email protected] Received: 7 March 2007 / Accepted: 15 August 2007 Published online: 26 February 2008 – © Springer-Verlag 2008
Abstract: Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2 -norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric. 1. Introduction With the knowledge that the set of Kähler metrics representing a given Kähler class is an affine space modeled after the smooth functions, Calabi introduced [11,9] a natural Riemannian functional on this space with the hope of using it to find canonical representatives of the given class. In effect, his functional, or Calabi energy, is simply the squared L 2 -norm of the scalar curvature, and the critical point minimizing it would fix the affine parameter alluded to above, yielding the desired representative of the class. Calabi named these critical points extremal Kähler metrics. It was then determined that if the Futaki character [17] of the class vanishes, a plausible extremal Kähler representative must be a metric of constant scalar curvature, and if under that condition we look During the preparation of this work, the first two authors were partially supported by NSF grant DMS-0504367.
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at the case where the Kähler class in question is a multiple of the first Chern class, the extremal representative must then be Kähler-Einstein. One of the most important problems in Kähler geometry today involves the subtle questions regarding the existence of extremal Kähler metrics representing a given class. Over the years, starting with the formulation of the famous Calabi Conjecture and its proof by Yau in 1978, various tools have been used or developed to attack this problem. The continuity method, Tian’s α-invariant, the Calabi-Lichnerowicz-Matsushima obstruction, the Futaki invariant and its generalizations, the Mabuchi K-energy, and more recently the various notions of stability proposed and studied by Tian, Donaldson and others. Substantial progress has been made, but the general existence problem remains open. Sasakian geometry sits naturally in between two Kähler geometries. On the one hand, Sasakian manifolds are the bases of metric cones which are Kähler. On the other hand, any Sasakian manifold is contact, and the one dimensional foliation associated to the characteristic Reeb vector field is transversally Kähler. In many interesting situations, the orbits of the Reeb vector field are all closed, in which case the Sasakian structure is called quasi-regular. Compact quasi-regular Sasakian manifolds have the structure of an orbifold circle bundle over a compact Kähler orbifold, which must be algebraic and which has at most cyclic quotient singularities. Since much of the study of compact Kähler manifolds and extremal metrics can be extended to the orbifold case, extension often done in a fairly straightforward way, it is not surprising that we can then “translate” statements involving compact Kähler orbifolds to conclude parallel statements regarding quasi-regular Sasakian structures. This is an approach that has been spectacularly successful in constructing new quasi-regular Sasaki-Einstein metrics on various contact manifolds of odd dimension greater than 3 (cf. [5,3,4,20,21], and references therein). For some time now, it has been believed that the only interesting (canonical) metrics in Sasakian geometry occur precisely in this orbibundle setting. In 1994, Cheeger and Tian conjectured that any compact Sasaki-Einstein manifold must be quasi-regular [12]. Their conjecture was phrased in terms of the properties of the Calabi-Yau cone rather than its Sasaki-Einstein base,1 and until recently, compact Sasakian manifolds with non-closed leaves were certainly known, but there was no evidence to suspect that we could get such structures with Einstein metrics as well. Hence, it was reasonable to believe that all Sasaki-Einstein metrics could be understood well by simply studying the existence of Kähler-Einstein metrics on compact cyclic orbifolds. As it turns out, the conjecture of Cheeger and Tian mentioned above is not true, and the first examples of irregular Sasaki-Einstein manifolds, that is to say, Sasaki-Einstein manifolds that are not quasi-regular, came first from the physics surrounding the famous CFT/AdS Duality Conjecture [18,19,25–28,14]. It now appears that there are irregular structures of this type on many compact manifolds in any odd dimension greater than 3. These Sasaki-Einstein metrics represent canonical points in the space of metrics adapted to the underlying geometric setting. However, although their Calabi-Yau cones are smooth outside the tip of the cone, their space of leaves is not even Hausdorff, and so the whole “orbibundle over a Kähler-Einstein base” approach proves itself insufficient in the study of the problem. The discovery of these new metrics make a strong case in favor of a variational formulation of the study of these Sasakian metrics, in a way analogous to the notion of the Calabi energy and extremality. With the proper set-up, the quasi-regularity property 1 More precisely Cheeger and Tian used the term standard cone, and the conjecture states that all Calabi-Yau cones are standard [12]
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should no longer be a key factor, and all Sasaki-Einstein metrics should indeed appear as minima of a suitable Riemannian functional. This would put on equal footing the analysis of all Sasakian structures, the quasi-regular or the irregular ones. Thus, we should be able to study the existence and uniqueness of these canonical Sasakian metrics in ways parallel to those used in Kähler geometry. Until now, this approach for finding canonical Sasakian structures had not been pursued, perhaps due to the lack of evidence that the orbibundle approach would be insufficient. We do so here, and propose to look at the squared L 2 -norm of the scalar curvature functional, defined over a suitable space of Sasakian metrics that are determined by fixing the Reeb vector field, which we think of as polarizing the Sasakian manifold. Its critical points are, by definition, canonical Sasakian metrics representing the said polarization. Recently, Martelli, Sparks and Yau presented a similar point of view [28], opting to look at the Sasaki-Einstein metrics as minima of the Hilbert action instead, though under certain restrictions on M. Our point of view has important advantages, several of which are elucidated in the present article. In particular, we see that for certain manifolds of Sasaki type, the optimal Sasaki metric on it cannot possibly have constant scalar curvature, showing the need to enlarge the plausible set of metrics to be considered, if at least, from a mathematical point of view. In general, the minimization of the L 2 -norm of the scalar curvature over metrics of fixed volume is intimately related to the search for Einstein metrics. We use this functional here over a smaller space of metrics, thus laying the foundation for the study of canonical Sasakian metrics in a way that parallels what is done in Kähler geometry for the extremal metric problem. This point of view eliminates the need to make a distinction between the quasi-regular and the irregular case, discussing them both on an equal footing. For we introduce the notion of a polarized Sasakian manifold, polarized by a Reeb vector field, and analyze the variational problem for the L 2 -norm of the scalar curvature over the space of Sasakian metrics representing the said polarization. Given a CR structure of Sasaki type, we define the cone of Sasakian polarizations compatible with this underlying CR structure, and discuss the variational problem for this functional as we vary the polarization on this cone also. The quasi-regularity or not of the resulting critical Sasakian structures is just a property of the characteristic foliation defined by the Reeb vector field. This foliation must clearly sit well with the Sasakian metrics under consideration in our approach, but it stands in its own right. A canonical Sasakian metric, a critical point of the said functional, interacts with the underlying characteristic foliation, but neither one of them determines the other. We organize the paper as follows. In §2, we recall and review the necessary definitions of Sasakian manifolds and associated structures. In §3, we define the notion of a polarized Sasakian manifold, and describe the space of Sasakian metrics that represent a given polarization, a space consisting of metrics with the same transversal holomorphic structure. We then analyze the variational problem for the L 2 -norm of the scalar curvature with it as its domain of definition, and show that the resulting critical points are Sasakian metrics for which the basic vector field ∂g# sgT = ∂g# sg is transversally holomorphic, that is to say, metrics that are transversally extremal. In §4, we study various transformation groups of Sasakian structures and their Lie algebras, proving the Sasakian version of the Lichnerowicz-Matsushima theorem. In §5, we define and study the Sasaki-Futaki invariant, and prove that a canonical Sasakian metric is of constant scalar curvature if, and only if, this invariant vanishes for the polarization under consideration. In §6, we define and study the Sasaki cone, and end up in §7 by proving that the polarizations in
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the Sasaki cone that admit canonical representatives form an open set, proving that the openness theorem for the extremal cone in Kähler geometry [22] holds in the Sasakian context also. We illustrate the power of this result by providing a detailed analysis of the Sasaki cone for the standard CR-structure on the unit sphere S2n+1 , and use it to show that all of its elements admit canonical representatives. We describe explicitly those that are of constant scalar curvature, and show that the standard metric is the only one of these that is Sasaki-Einstein. 2. Sasakian Manifolds We recall that an almost contact structure on a differentiable manifold M is given by a triple (ξ, η, ), where ξ is a vector field, η is a one form, and is a tensor of type (1, 1), subject to the relations η(ξ ) = 1 , 2 = −1l + ξ ⊗ η . The vector field ξ defines the characteristic foliation Fξ with one-dimensional leaves, and the kernel of η defines the codimension one sub-bundle D. This yields a canonical splitting TM = D ⊕ L ξ , (1) where L ξ is the trivial line bundle generated by ξ . The sub-bundle D inherits an almost complex structure J by restriction of . Clearly, the dimension of M must be an odd integer 2n + 1. We refer to (M, ξ, η, ) as an almost contact manifold. If we disregard the tensor and characteristic foliation, that is to say, if we just look at the sub-bundle D forgetting altogether its almost complex structure, we then refer to the contact structure (M, D), or simply D when M is understood. Here, and further below, the reader can no doubt observe that the historical development of the terminology is somewhat unfortunate, and for instance, it is an almost contact structure that gives rise to a contact one, rather than the other way around. A Riemannian metric g on M is said to be compatible with the almost contact structure (ξ, η, ) if for any pair of vector fields X, Y , we have that g((X ), (Y )) = g(X, Y ) − η(X )η(Y ) . Any such g induces an almost Hermitian metric on the sub-bundle D. We say that (ξ, η, , g) is an almost contact metric structure. In the presence of a compatible Riemannian metric g on (M, ξ, η, ), the canonical decomposition (1) is orthogonal. Furthermore, requiring that the orbits of the field ξ be geodesics is equivalent to requiring that £ξ η = 0, a condition that in view of the relation ξ η = 1, can be re-expressed as ξ dη = 0. An almost contact metric structure (ξ, η, , g) is said to be a contact metric structure if for all pairs of vector fields X , Y , we have that g(X, Y ) = dη(X, Y ) .
(2)
We then say that (M, ξ, η, , g) is a contact metric manifold. Notice that in such a case, the volume element defined by g is given by dµg =
1 η ∧ (dη)n . n!
(3)
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It is convenient to reinterpret the latter structure in terms of the cone construction. Indeed, on C(M) = M × R+ , we introduce the metric gC = dr 2 + r 2 g . The radial vector field r ∂r satisfies £r ∂r gC = 2gC , and we may define an almost complex structure I on C(M) by I (Y ) = (Y ) − η(Y )r ∂r ,
I (r ∂r ) = ξ .
The almost contact manifold (M, ξ, η, ) is said to be normal if the pair (C(M), I ) is a complex manifold. In that case, the induced almost complex structure J on D is integrable. Definition 2.1. An contact metric structure (ξ, η, , g) on a manifold M is said to be a Sasakian structure if (ξ, η, ) is normal. A smooth manifold provided with one such structure is said to be a Sasakian manifold, or a manifold of Sasaki type. For a Sasakian structure (ξ, η, , g), the integrability of the almost complex structure I on the cone C(M) implies that the Reeb vector field ξ leaves both η and invariant [3]. We obtain a codimension one integrable strictly pseudo-convex CR structure (D, J ), where D = ker η is the contact bundle and J = |D, and the restriction of g to D defines a positive definite symmetric form on (D, J ) that we shall refer to as the transverse Kähler metric g T . By (2), the Kähler form of the transverse Kähler metric is given by the form dη. Therefore, the Sasakian metric g is determined fully in terms of (ξ, η, ) by the expression g = dη ◦ (1l ⊗ ) + η ⊗ η , (4) where the fact that dη is non-degenerate over D is already built in. Since ξ leaves invariant η and , it is a Killing field, its orbits are geodesics, and the decomposition (1) is orthogonal. Despite its dependence on the other elements of the structure, we insist on explicitly referring to g as part of the Sasakian structure (ξ, η, , g). The discussion may be turned around to produce an alternative definition of the notion of Sasakian structure. For the contact metric structure (ξ, η, , g) defines a Sasakian structure if (D, J ) is a complex sub-bundle of T M, and ξ generates a group of isometries. This alternative approach appears often in the literature. If we look at the Sasakian structure (ξ, η, , g) from the point of view of CR geometry, its underlying strictly pseudo-convex CR structure (D, J ), with associated contact bundle D, has Levi form dη. (In the sequel, when referring to any CR structure, we shall always mean one that is integrable and of codimension one.) Definition 2.2. Let (D, J ) be a strictly pseudo-convex CR structure on M. We say that (D, J ) is of Sasaki type if there exists a Sasakian structure S = (ξ, η, , g) such that D = ker η and |D = J . If (D, J ) is a CR structure of Sasaki type, the Sasakian structures S = (ξ, η, , g) with D = ker η and |D = J will be said to be Sasakian structures with underlying CR structure (D, J ). The following well known result will be needed later. Let us observe that since the fibers of the Riemannian foliation defined by a Sasaki structure (ξ, η, , g) are geodesics, their second fundamental forms are trivial.
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Proposition 2.3. Let (M, ξ, η, , g) be a Sasakian manifold. Then we have that a) Ricg (X, ξ ) = 2nη(X ) for any vector field X . b) Ricg (X, Y ) = RicT (X, Y ) − 2g(X, Y ) for any pair of sections X, Y of D. c) sg = sT − |A|2 = sT − 2n, where A is the O’Neill tensor of the “corresponding” Riemannian submersion. In these statements, the subscript T denotes the corresponding intrinsic geometric quantities of the transversal metric g T . Proof. The Riemannian submersion of the Sasakian structure has totally geodesic fibers. The vector field ξ spans the only vertical direction, and we have that A X Y = −(dη (X, Y ))ξ , and that A X ξ = −(X ) for arbitrary horizontal vector fields X, Y . The first two results follow easily from O’Neill’s formulae [32]. The computation of the L 2 -norm of A is a simple consequence of the J -invariance of the induced metric on D. 3. Canonical Representatives of Polarized Sasakian Manifolds Let us consider a Sasakian structure (ξ, η, , g) on M that shall remain fixed throughout this section. There are two naturally defined sets of deformations of this structure, those where the Reeb vector field remains fixed while the underlying CR structure changes, and those where the underlying CR structure stays put while the Reeb vector field varies. We study the first type of deformations in this section. They turn out to be the Sasakian analogues of the set of Kähler metrics representing a given polarization on a manifold of Kähler type. They have different underlying CR structures, but they all share the same transverse holomorphic structure. The other set of deformations that fix the underlying CR structure will lead to the Sasakian analogue of the Kähler cone of a manifold of Kähler type, and they will be analyzed in §6 below. We begin by recalling that a function ϕ ∈ C ∞ (M) is said to be basic if it is annihilated by the vector field ξ , that is to say, if ξ(ϕ) = 0. We denote by C B∞ (M) the space of all real valued basic functions on M. We observe that the notion of basic can be extended to covariant tensors of any order in the obvious manner. In particular, when looking at the transversal Kähler metric of (ξ, η, , g), its Kähler form is basic, and so must be all of its curvature tensors as well. This observation will play a crucial rôle in the sequel. We consider the set [3] ˜ g) S(ξ ) = {Sasakian structure (ξ˜ , η, ˜ , ˜ | ξ˜ = ξ } ,
(5)
and provide it with the C ∞ compact-open topology as sections of vector bundles. For ˜ g) any element (ξ˜ , η, ˜ , ˜ in this set, the 1-form ζ = η−η ˜ is basic, and so [d η] ˜ B = [dη] B . Here, [ · ] B stands for a cohomology class in the basic cohomology ring, a ring that is defined by the restriction d B of the exterior derivative d to the subcomplex of basic forms in the de Rham complex of M. Thus, all of the Sasakian structures in S(ξ ) correspond to the same basic cohomology class. We call S(ξ ) the space of Sasakian structures compatible with ξ , and say that the Reeb vector field ξ polarizes the Sasakian manifold M. Given the Reeb vector field ξ , we have its characteristic foliation Fξ , so we let ν(Fξ ) be the vector bundle whose fiber at a point p ∈ M is the quotient space T p M/L ξ , and let πν : T M → ν(Fξ ) be the natural projection. The background structure S = (ξ, η, , g) induces a complex structure J¯ on ν(Fξ ). This is defined by J¯ X¯ := (X ), where X is any vector field in M such that π(X ) = X¯ . Furthermore, the underlying CR structure
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(D, J ) of S is isomorphic to (ν(Fξ ), J¯) as a complex vector bundle. For this reason, we refer to (ν(Fξ ), J¯) as the complex normal bundle of the Reeb vector field ξ , although its identification with (D, J ) is not canonical. We shall say that (M, ξ, ν(Fξ ), J¯), or simply (M, ξ, J¯), is a polarized Sasakian manifold. ˜ g) We define S(ξ, J¯) to be the subset of all structures (ξ˜ , η, ˜ , ˜ in S(ξ ) such that the diagram ˜
TM → TM ↓πν ↓πν
(6)
J¯
ν(Fξ ) → ν(Fξ ), commutes. This set consists of elements of S(ξ ) with the same transverse holomorphic structure J¯, or more precisely, the same complex normal bundle (ν(Fξ ), J¯). We have [3], Lemma 3.1. The space S(ξ, J¯) of all Sasakian structures with Reeb vector field ξ and transverse holomorphic structure J¯ is an affine space modeled on (C B∞ (M)/R) × (C B∞ (M)/R)×H 1 (M, Z). Indeed, if (ξ, η, , g) is a given Sasakian structure in S(ξ, J¯), ˜ g) any other Sasakian structure (ξ, η, ˜ , ˜ in it is determined by real valued basic functions ϕ and ψ and integral closed 1-form α, such that 1 η˜ = η + d c ϕ + dψ + i(α) , 2 ˜ = − (ξ ⊗ (η˜ − η)) ◦ , ˜ + η˜ ⊗ η˜ , g˜ = d η˜ ◦ (1l ⊗ ) where d c = i(∂ −∂), and i : H 1 (M, Z) → H 1 (M, R) = H B1 (Fξ ) is the homomorphism induced by inclusion. In particular, d η˜ = dη + i∂∂ϕ. The complex structure defining the operators ∂ and ∂ in this lemma is J¯, as basic covariant tensors on M define multilinear maps on ν(Fξ ). We think of these as tensors on a transversal Kähler manifold that does not necessarily exist. Be that as it may, the cohomology class of the transverse Kähler metrics arising from elements in S(ξ, J¯) is fixed, and it is natural to ask if there is a way of fixing the affine parameters ϕ and ψ also, which would yield then a canonical representative of this set. We proceed to discuss and answer this question. We start by introducing a Riemannian functional whose critical point will fix a canonical choice of metric for structures in S(ξ, J¯). This, in effect, will fix the desired preferred representative that we seek. We denote by M(ξ, J¯) the set of all compatible Sasakian metrics arising from structures in S(ξ, J¯), and define the functional E M(ξ, J¯) → R , g → sg2 dµg ,
(7)
M
the squared L 2 -norm of the scalar curvature of g. The variation of a metric in M(ξ, J¯) depends upon the two affine parameters of freedom ϕ and ψ of Lemma 3.1. However, the transversal Kähler metric varies as a function of ϕ only, and does so within a fixed basic cohomology class. The critical point of (7),
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should it exist, will allow us to fix the parameter ϕ since, not surprisingly, it shall be determined by the condition that d η˜ = dη + i∂∂ϕ be an extremal Kähler metric [9] on D. The remaining gauge function parameter ψ represents nothing more than a change of coordinates in the representation of the form η˜ of the Sasakian structure in question. Thus, the finding of a critical point of E produces a canonical representative of S(ξ, J¯). 3.1. Variational formulae. In order to derive the Euler-Lagrange equation of (7), we describe the infinitesimal variations of the volume form, Ricci tensor, and scalar curvature, as a metric in M(ξ, J¯) is deformed within that space. We begin by recalling that the Ricci form ρ of a Sasaki structure (ξ, η, , g) is defined on D by the expression ρg (X, Y ) = Ricg (J X, Y ) . This is extended trivially on the characteristic foliation L ξ , and by Proposition 2.3, we easily see that ρg = ρgT − 2dη , (8) where ρ T denotes the form arising from the Ricci tensor of the transversal metric, a basic two form that we think of as a J¯-invariant two tensor in ν(Fξ ). Thus, ρg induces a well defined bilinear map on ν(Fξ ) that is J¯-invariant. Though the notation suggests so, it is not the case that the trace of the form (8) yields the scalar curvature of g. This form only encodes information concerning second covariant derivatives of g along directions in ν(Fξ ). Proposition 3.2. Let (ξ, ηt , t , gt ) be a path in S(ξ, J¯) that starts at (ξ, η, , g) when t = 0, and is such that dηt = dη + ti∂∂ϕ for a certain basic function ϕ, and for t sufficiently small. Then we have the expansions t dµt = 1 − B ϕ dµ + O(t 2 ) , 2 1
B ϕ + ϕ + O(t 2 ) , ρt = ρ − ti∂∂ 2 1 2
B ϕ + 2(ρ T , i∂∂ϕ) + O(t 2 ) , st = s T − 2n − t 2 for the volume form, Ricci form, and scalar curvature of gt , respectively. Here, the geometric terms without sub-index are those corresponding to the starting metric g, and
B is the Laplacian acting on basic functions. Proof. By Lemma 3.1, there exists a function ψ such that 1 c ηt = η + t d ϕ + dψ + O(t 2 ) . 2 Since ϕ, ψ and α are basic, we have that dµt = we obtain dµt =
1 n! ηt
∧ (dηt )n = η ∧ (dηt )n , and
1 t η ∧ (dη + ti∂∂ϕ)n = dµ + η ∧ (dη)n−1 ∧ i∂∂ϕ + O(t 2 ) . n! (n − 1)!
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Now, ω B = dη, the Kähler form of the induced metric on D, is basic. We then have that ∗ B (ω B )n−1 /(n − 1)! = ω B , and conclude that t dµt = 1 − B ϕ dµ + O(t 2 ) . 2 By (8), we may compute the variation of ρ by computing the variation of ρ T . This is well known to be [34] t ρtT = ρ T + i∂∂( B ϕ) + O(t 2 ) . 2 Since dηt = dη + ti∂∂ϕ, we obtain ρt = ρ + ti∂∂
1
B ϕ + ϕ + O(t 2 ) , 2
as stated. Finally, by Proposition 2.3 once again, we have that the variation of the scalar curvature arises purely from the variation of its transversal part. Since the transversal metric is Kähler, we obtain 1 2 T st = s − t
ϕ + 2(ρ , i∂∂ϕ) + O(t 2 ) 2 B as desired.
Remark 3.3. The forms ρ T and i∂∂ϕ are basic. Hence, the metric pairing of these forms that appears in the proposition above involves only the transversal metric. On the other hand, in view of the analogous variational formulae in the Kähler case [34], we might think that the expression for ρt above is a bit strange. We see that this is not so if we just keep in mind that the Sasakian ρg encodes ν(Fξ )-covariant derivatives information only. 3.2. Euler-Lagrange equations. Associated to any Sasakian structure (ξ, η, , g) in S(ξ, J¯), we introduce a basic differential operator L gB , of order 4, whose kernel consists of basic functions with transverse holomorphic gradient. Given a basic function ϕ : M → C, we consider the vector field ∂ # ϕ defined by the identity g(∂ # ϕ, · ) = ∂ϕ . (9) Thus, we obtain the (1,0) component of the gradient of ϕ, a vector field that, generally speaking, is not transversally holomorphic. In order to ensure that, we would need to ¯ # ϕ = 0, that is equivalent to the fourth-order equation impose the condition ∂∂ ¯ # )∗ ∂∂ ¯ #ϕ = 0 , (∂∂ ¯ # )∗ ∂∂ ¯ # ϕ2 2 . ¯ # ϕ L 2 = ∂∂ because ϕ, (∂∂ L We have that ¯ # )∗ ∂∂ ¯ # ϕ = 1 ( 2B ϕ + 4(ρ T , i∂∂ϕ) + 2(∂s T ) ∂ # ϕ) . L gB ϕ := (∂∂ 4
(10)
(11)
The functions on M that are transversally constant are always in the kernel of L gB . The only functions of this type that are basic are the constants. Thus, the kernel of L gB has dimension at least 1.
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Proposition 3.4. The first derivative of E at g ∈ M(ξ, J¯) in the direction of the deformation defined by (ϕ, ψ) is given by d ¯ # )∗ ∂∂ ¯ # ϕ dµ . E(gt ) |t=0 = −4 (s T − 2n) (∂∂ dt M Proof. This result follows readily from the fact that s = s T −2n, the variational formulae of Proposition 3.2, and identity (11). As a corollary to Proposition 3.4, we have: Theorem 3.5. A Sasakian metric g ∈ M(ξ, J¯) is a critical point of the energy functional E of (7) if the basic vector field ∂g# sgT = ∂g# sg is transversally holomorphic. We are thus led to our fundamental definition: Definition 3.6. We say that S = (ξ, η, , g) is a canonical representative of S(ξ, J¯) if the metric g satisfies the condition of Theorem 3.5, that is to say, if, and only if, g is transversally extremal. 4. Transformation Groups of Sasakian Structures In this section we discuss some important transformation groups associated to Sasakian structures, and their corresponding Lie algebras. Let us begin with a CR structure (D, J ) of Sasaki type on M. If η is a contact form, we have the group Con(M, D) of contact diffeomorphims, that is to say, the subgroup of the diffeomorphisms group Diff(M) consisting of those elements that leave the contact subbundle D invariant: Con(M, D) = {φ ∈ Diff(M) | φ ∗ η = f φ η , f φ ∈ C ∞ (M)∗ } .
(12)
Here, C ∞ (M)∗ denotes the subset of nowhere vanishing functions in C ∞ (M). The dimension of the group so defined is infinite. We may also consider the subgroup Con(M, η) of strict contact transformations, whose elements are those φ ∈ Con(M, D) such that f φ = 1: Con(M, η) = {φ ∈ Diff(M) | φ ∗ η = η} . This subgroup is also infinite dimensional. The Lie algebras of these two groups are quite important. The first of these is the Lie algebra of infinitesimal contact transformations, con(M, D) = {X ∈ − X (M) | £ X η = a(X )η , a(X ) ∈ C ∞ (M)} ,
(13)
while the second is the subalgebra of infinitesimal strict contact transformations con(M, η) = {X ∈ − X (M) | £ X η = 0} .
(14)
If we now look at the pair (D, J ), we have the group of CR automorphisms of (D, J ), defined by CR(M, D, J ) = {φ ∈ Con(M, D) | φ∗ J = J φ∗ } . (15)
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This is a Lie group [13]. Its Lie algebra cr(M, D, J ) can be characterized as cr(M, D, J ) = {X ∈ con(M, D) | £ X J = 0} .
(16)
Notice that in this defining expression, £ X J makes sense even though J is not a tensor field on M, the reason being that the vector field X leaves D invariant. Let us now consider a Sasakian structure S = (ξ, η, , g) with underlying CR structure (D, J ). We are interested in the subgroup of Diff(M) that leaves the tensor field invariant. So we define S = {φ ∈ Diff(M) : φ∗ ◦ = ◦ φ∗ } .
(17)
We also have Fol(M, Fξ ) = {φ ∈ Diff(M) : φ∗ Fξ ⊂ Fξ } , the subgroup of Diff(M) that preserves the characteristic foliation of the Sasakian structure S. In order to simplify the notation, we will often drop M from the notation when referring to these various groups and algebras. Lemma 4.1. Let (D, J ) be a strictly pseudo-convex CR structure of Sasaki type on M, and fix a Sasakian structure S = (ξ, η, , g) with underlying CR structure (D, J ). Then S = CR(D, J ) ∩ Fol(Fξ ) . Proof. If φ ∈ S , the identity η ◦ = 0 implies that φ preserves D. If X is a section of D, then we have that φ∗ J (X ) = φ∗ (X ) = (φ∗ X ) = J (φ∗ X ), which implies that φ ∈ CR(D, J ). But we have φ∗ (ξ ) = 0 = (φ∗ ξ ) also, which implies that φ ∈ Fol(Fξ ). Conversely, suppose that φ ∈ CR(D, J ) ∩ Fol(Fξ ). Then φ leaves all three D, Fξ , and J invariant, and therefore, it preserves the splitting (1). Since the relation between J and is given by J (X ) if X is a section of D , (X ) = (18) 0 if X = ξ , in order to conclude that φ ∈ S , it suffices to show that φ∗ (ξ ) = (φ∗ ξ ). But this is clear as both sides vanish. Since S is closed in the Lie group CR(D, J ), it is itself a Lie group. Generally speaking, the inclusion S ⊂ CR(D, J ) is strict, and the group Fol(M, Fξ ) is infinite dimensional. The automorphism group Aut(S) of the Sasakian structure S = (ξ, η, , g) is defined to be the subgroup of Diff(M) that leaves all the tensor fields in (ξ, η, , g) invariant. It is a Lie group, and one has natural group inclusions Aut(S) ⊂ S ⊂ CR(D, J )
(19)
whenever the CR structure (D, J ) is of Sasaki type, and S has it as its underlying CR structure. The Lie algebras of S and Fol(Fξ ) are given by X (M) | £ X = 0} , s = {X ∈ −
(20)
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and fol(Fξ ) = {X ∈ − X (M) | [X, ξ ] is tangent to the leaves of Fξ } ,
(21)
respectively. The latter is just the Lie algebra of foliate vector fields of the foliation Fξ . On the other hand, we may restate the defining condition for s as X ∈ s ⇐⇒ [X, (Y )] = ([X, Y ]) for all Y , and we can see easily that s = cr(D, J ) ∩ fol(Fξ ) .
(22)
We can characterize now CR structures of Sasaki type in terms of their relations to the Lie algebras above. Lemma 4.2. Let (D, J ) be a strictly pseudoconvex CR structure on M, and let η be a compatible contact form representing D, with Reeb vector field ξ . Define the tensor field by Eq. (18). Then (D, J ) is of Sasaki type if, and only if, ξ ∈ cr(D, J ). Proof. Given a strictly pseudoconvex CR structure (D, J ), with contact 1-form η that represents D, and Reeb vector field ξ , we consider the (1, 1) tensor field defined by (18). By Proposition 3.5 of [2], (ξ, η, ) defines a Sasakian structure if, and only if, the CR structure is integrable and £ξ = 0. Since ξ is a foliate vector field, the condition that ξ ∈ cr(D, J ) ensures that ξ ∈ s , and the result follows. For any strictly pseudoconvex CR structure with contact 1-form η, the Reeb vector field ξ belongs to con(M, η). Therefore, if ξ ∈ s , the structure S = (ξ, η, , g) is Sasakian, and ξ ∈ aut(S). The study of the group CR(D, J ) has a long and ample history [36,23,33]. We state the most general result given in [33]. This holds for a general CR manifold M, so we emphasize its consequence in the case where M is closed. Theorem 4.3. Let M be a 2n + 1 dimensional manifold with a strictly pseudoconvex CR structure (D, J ), and CR automorphism group G. If G does not act properly on M, then: (1) If M is a non-compact manifold, then it is CR diffeomorphic to the Heisenberg group with its standard CR structure. (2) If M is a compact manifold, then it is CR diffeomorphic to the sphere S2n+1 with its standard CR structure. In particular, if M is a closed manifold not CR diffeomorphic to the sphere, the automorphisms group of its CR structure is compact. We recall that the inclusion Aut(S) ⊂ CR(D, J ) (see (19)), generally speaking, is proper. However, we do have the following. Proposition 4.4. Let (D, J ) be a strictly pseudoconvex CR structure on a closed manifold M, and suppose that (D, J ) is of Sasaki type. Then there exists a Sasakian structure S = (ξ, η, , g) with underlying CR structure (D, J ), whose automorphism group Aut(S) is a maximal compact subgroup of CR(M, D, J ). In fact, except for the case when (M, D, J ) is CR diffeomorphic to the sphere S2n+1 with its standard CR structure, the automorphisms group Aut(S) of S is equal to CR(M, D, J ).
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Proof. Let G be a maximal compact subgroup of CR(D, J ). By Theorem 4.3, we have that G = CR(D, J ) except when (M, D, J ) is CR diffeomorphic to the sphere S2n+1 . ˜ g) Let S˜ = (ξ˜ , η, ˜ , ˜ be a Sasakian structure with underlying CR structure (D, J ). If φ ∈ G, then φ ∗ η˜ = f η˜ for some nowhere vanishing real-valued function f . By averaging η˜ over G, we obtain a G-invariant contact form η with associated contact structure D. Let ξ be its Reeb vector field. As ξ is uniquely determined by η, we conclude that φ∗ ξ = ξ . We then define a (1, 1)-tensor by the expression in (18). The conditions φ∗ J = J φ∗ and φ∗ ξ = ξ imply that φ∗ = φ∗ . The triple (ξ, η, ) defines a Sasakian structure S, and φ ∈ Aut(S). We now look at the case where the manifold M is polarized by (ξ, J¯). Then, (ν(Fξ ), J¯) is a complex vector bundle, and any φ ∈ Fol(M, Fξ ) induces a map φ¯ ∗ : ν(Fξ ) → ν(Fξ ). We define the group of transversely holomorphic transformations HT (ξ, J¯) by HT (ξ, J¯) = {φ ∈ Fol(M, Fξ ) | φ¯ ∗ ◦ J¯ = J¯ ◦ φ¯ ∗ }.
(23)
Since a 1-parameter subgroup of any smooth section of L ξ induces the identity on ν(Fξ ), this group is infinite dimensional. We are mainly interested in the infinitesimal version. Given a choice of S = (ξ, η, , g) in S(ξ, J¯), (ν(Fξ ), J¯) is identified with the underlying CR structure (D, J ). In this case, we may use the decomposition (1) to write any vector field as X = X D + c(X )ξ , (24) which defines the component function X → c(X ) := η(X ), and the class X¯ on ν(Fξ ) defined by the vector field X is represented by X D. The Lie bracket operation induces a bilinear mapping on ν(Fξ ) by [ X¯ , Y¯ ] := [X, Y ] . This operation allows us to generalize the notion of transversally holomorphic vector field already encountered in §3.2. Definition 4.5. Let (M, ξ, J¯) be a polarized Sasakian manifold. We say that a vector field X is transversally holomorphic if given any section Y¯ of ν(Fξ ), we have that [ X¯ , J¯ Y¯ ] = J¯ [X, Y ] . The set of all such vector fields will be denoted by hT (ξ, J¯). It is now desirable to express the defining condition for X to be in hT (ξ, J¯) in terms intrinsic to X itself. The reader may consult [7] for relevant discussions. Lemma 4.6. Let (M, ξ, J¯) be a polarized Sasakian manifold. If X ∈ hT (ξ, J¯) is a transversally holomorphic vector field, for any Sasakian structure (ξ, η, , g) ∈ S(ξ, J¯) we have that (£ X )(Y ) = η([X, (Y )])ξ . Proof. We have that (£ X )(Y ) = [X, (Y )] − ([X, Y ]) , which implies that c((£ X )(Y )) = η([X, (Y )]). Thus, (£ X )(Y ) = ((£ X )(Y ))D + η([X, (Y )])ξ . The result follows after simple considerations.
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The set hT (ξ, J¯) is a Lie algebra contained in fol(M, Fξ ). If we represent (ν(Fξ ), J¯) as (D, J ) for a choice of S = (ξ, η, , g) in S(ξ, J¯) with underlying CR structure (D, J ), by the decomposition (24) we see that for a transversally holomorphic vector field X we have that [X D, J (YD)]D = J ([X D, YD]D) , for any vector field Y . Thus, X D preserves the transverse complex structure J . This characterization can be reformulated by saying that if X ∈ hT (ξ, J¯), the vector field of type (1, 0) given by 1 (25) X = (X D − i J (X D)) 2 is in the kernel of the transverse Cauchy-Riemann equations. Thus, the mapping into the space of sections of ν(Fξ ) given by hT (ξ, J¯) → (ν(Fξ )) X → X¯
(26)
has an image that can be identified with the space of sections of (D, J ) satisfying the Cauchy Riemann equations; therefore, it is finite dimensional. We denote this image by hT (ξ, Fξ )/L ξ . Their one dimensional foliations made Sasakian manifolds a bit special. In particular, they carry no non-trivial parallel vector field (see [3]). For by a result of Tachibana, any harmonic one form must annihilate the Reeb vector field ξ , and so any parallel vector fields X must be orthogonal to ξ , that is to say, it must be a section of D. But then, since the metric is covariantly constant and ξ is Killing, we must have that 0 = ∇Y g(ξ, X ) = g((Y ), X ) for all Y , which forces X to be identically zero. Remark 4.7. Notice that for any Sasakian structure with underlying CR structure (D, J ), the Lie algebra cr(D, η) is reductive. For either CR(D, J ) is compact, or cr(D, η) = su(n +1, 1). In particular, let (N , ω) be a Kähler manifold, and consider the circle bundle π : M → N with integral Euler class [ω]. Then M has a natural Sasakian structure (ξ, η, , g) such that π ∗ ω = dη. Then the horizontal lifts of non-trivial parallel vector fields in (N , ω) are holomorphic but not parallel on M. On the other hand, if we take a holomorphic field that lies in the non-reductive part of the algebra of holomorphic vector fields of N , its horizontal lift is a transversally holomorphic vector field that does not lie in the reductive component of the algebra hT , and thus, it cannot possibly leave the contact subbundle D invariant. If X ∈ hT (ξ, J¯), given any real valued function f , X + f ξ ∈ hT (ξ, J¯) also, and so, T h (ξ, J¯) cannot have finite dimension. The remark above alludes to the special structure that hT (ξ, J¯)/L ξ has, and in fact, we are now ready to extend to the Sasakian context a result of Calabi [10] on the structure of the algebra of holomorphic vector fields of a Kähler manifold that carries an extremal metric. Calabi’s theorem is, in turn, an extension of work of Lichnerowicz [24] on constant scalar curvature metrics, and the latter is itself an extension of a result of Matsushima [29] in the Kähler-Einstein case. We also point the reader to the theorem for harmonic Kähler foliations in [31], which is relevant in this context. Consider a Sasakian structure (ξ, η, , g) in S(ξ, J¯). Let HgB be the space of basic functions in the kernel of the operator L gB in (11), and consider the mapping ∂g# : HgB → hT (ξ, J¯)/L ξ ,
(27)
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where ∂g# is the operator defined in (9). We use the Sasakian metric g to identify the quotient space in the right side above with the holomorphic vector fields that are sections of (D, J ), which we shall refer to from here on as h(ξ, D, J ). The notation for this Lie algebra is a bit non-standard in that h(ξ, D, J ) depends on ξ or rather on the foliation Fξ and are holomorphic sections of D with respect to J . It should be noted, however, that while elements in h(ξ, D, J ) leave both Fξ and J¯ invariant, they do not necessarily leave D invariant. We also define the operator L¯ gB on HgB by L¯ gB ϕ = L gB ϕ. ¯ It follows that ( L¯ gB − L gB )ϕ = ∂g# sg
∂ϕ − ∂g# ϕ
∂sg ,
(28)
where sg is the scalar curvature of g. The fact that sg is a basic function implies that ∂g# sg is a (1, 0) section of (D, J ). The identity above implies that L B and L¯ B coincide if sg is constant. If the metric g is canonical, then we have that ∂g# sg ∈ h(ξ, D, J ), and the operators L B and L¯ B commute. The image h0 ∼ = HgB /C of the mapping (27) is an ideal in h(ξ, D, J ), and can be identified with the space of holomorphic fields that have non-empty zero set. The quotient algebra h(ξ, D, J )/h0 is Abelian. We also denote by aut( J¯, g T ) the Lie subalgebra of hT (ξ, ν(Fξ )) that are holomorphic Killing vector fields of the transverse metric g T , that is (29) aut( J¯, g T ) = { X¯ ∈ h(ξ, D, J ) | £ X¯ g T = 0} . Suppose now that (ξ, η, , g) is a canonical representative of S(ξ, J¯), so that g is Sasaki extremal. Let z0 be the image under ∂g# of the set of purely imaginary functions in HgB . This is just the space of Killing fields for the transversal metric g T that are of the form J ∇g T ϕ, ϕ ∈ HgB . Furthermore, by (28) we see that the complexification z0 ⊕ J¯z0 coincides with the commutator of ∂g# sg : z0 ⊕ J¯z0 = {X ∈ h(ξ, D, J ) : [X, ∂g# sg ] = 0} . Theorem 4.8. Let (M, ξ, J¯) be a polarized Sasakian manifold. Suppose that there exists a canonical representative (ξ, η, , g) of S(ξ, J¯). Let HgB be the space of basic functions in the kernel of the operator L gB in (11), and let h0 be the image of the mapping (27). Then we have the orthogonal decomposition hT (ξ, J¯)/L ξ ∼ = h(ξ, D, J ) = a ⊕ h0 , where a is the algebra of parallel vector fields of the transversal metric g T . Furthermore, h0 = z0 ⊕ J¯z0 ⊕ (⊕λ>0 hλ ) , where z0 is the image of the purely imaginary elements of HgB under ∂g# , and hλ = { X¯ ∈ hT (ξ, J¯)/L ξ : [ X¯ , ∂g# sg ] = λ X¯ }. Moreover, z0 is isomorphic to the quotient algebra aut(ξ, η, , g)/{ξ }, so the Lie algebra aut( J¯, g T ) of Killing vector fields for the transversal metric g T is equal to aut( J¯, g T ) = a ⊕ z0 ∼ = a ⊕ aut(ξ, η, , g)/{ξ } .
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The presence of the algebra a above does not contradict the fact that there are no non-trivial parallel vector fields on a closed Sasakian manifold: a vector field can be parallel with respect to g T without being parallel with respect to g. Proof of Theorem 4.8. We prove the last statement first. In order to see this, we notice that there is an exact sequence [3] δ 0 → {ξ } → aut(ξ, η, , g) → aut( J¯, g T ) → H B1 (Fξ ) ,
(30)
where H B1 (Fξ ) denotes the basic cohomology associated to the characteristic foliation Fξ . Using the identification of aut( J¯, g T ) with elements in h(ξ, D, J ), let us describe the map δ. Since X¯ ∈ aut( J¯, g T ) it leaves dη invariant, so the 1-form X¯ dη is closed and basic. It, thus, defines an element in H B1 (Fξ ). So we can define δ( X¯ ) = [ X¯ dη] B . Now the section X¯ ∈ aut( J¯, g T ) can be extended to an element X = X¯ +aξ ∈ aut(ξ, η, , g) if, and only if, the basic cohomology class [ X¯ dη] B vanishes, and this determines a up to a constant. By Hodge theory and duality, the image of δ can be identified with the Lie algebra of parallel vector fields in aut( J¯, g T ). The splitting then follows as in the Kähler case [10]. For the first part of the theorem we sketch the main points, as the argument is an adaptation to our situation of that in [10]. Given a section X¯ in h(ξ, D, J ), we look at the Hodge decomposition of the (0, 1)-form that corresponds to it via the metric g T . It is ∂-closed, and both its harmonic and ∂ components are the dual of holomorphic fields. The vector field dual to the harmonic component is g T -parallel. Since g T is an extremal metric, the operators L gB and L¯ gB commute. We then restrict B L¯ g to the kernel of L gB , and use the resulting eigenspace decomposition together with the identity (28) to derive the remaining portion of the theorem. Remark 4.9. This result obstructs the existence of special canonical representatives of a polarized Sasakian manifold in the same way it does in the Kählerian case. For instance, let (N , ω) be the one-point or two-points blow-up of CP2 , and consider the circle bundle π : M → N with integral Euler class [ω]. If M is polarized by its natural Sasakian structure (ξ, η, , g), the one where π ∗ ω = dη, then S(ξ, J¯) cannot be represented ˜ g) by a Sasakian structure (ξ, η, ˜ , ˜ with g˜ a metric of constant scalar curvature. The structure of hT (ξ, J¯)/L ξ would obstruct it. 5. A Sasaki-Futaki Invariant Let (M, ξ, J¯) be a polarized Sasakian manifold. Given any structure (ξ, η, , g) ∈ S(ξ, J¯), we denote its underlying CR structure by (D, J ). The metric g is an element of M(ξ, J¯) whose transversal Ricci form ρ T is basic. We define the Ricci potential ψg as the function in the Hodge decomposition of ρ T given by ρ T = ρhT + i∂∂ψg , where ρhT is the harmonic representative of the foliated cohomology class represented by ρ T . Notice that if G Tg is the Green’s operator of the transversal metric, we have that ψg = −G Tg (sgT ) = −G Tg (sgT − 2n) = −G Tg (sg ) = −G(sg − sg,0 ) ,
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where sg and G g are the scalar curvature and Green’s operator of g, and sg,0 is the projection of sg onto the constants. The sequence of equalities above follows by (c) of Proposition 2.3, which implies that sg = sgT − 2n is a basic function. Thus, the Ricci potential ψg is itself a basic function. On hT (ξ, J¯), we define the function X → X (ψg )dµg . Since ψg is basic, the integrand in this expression can be fully written in terms of the transversally holomorphic realization X (see (25)) of X . Proposition 5.1. The mapping above only depends on the basic cohomology class represented by dη, and not on the particular transversal Kähler metric induced by g ∈ M(ξ, J¯) that is used to represent it. Proof. We take a path gt in M(ξ, J¯) starting at g for which the transversal Kähler form is of the form dηt = dη + ti∂∂ϕ , T − s T , we with the affine parameter ϕ a basic function. From the identity B ψg = sg,0 see that the variation ψ˙ g of ψg satisfies the relation
1 2(i∂∂ϕ, i∂∂ψg )g + B ψ˙ g = −s˙T = 2B ϕ + 2(ρ T , i∂∂ϕ)g . 2 Hence, ψ˙ g −
1 v
ψ˙ g dµg =
1
B ϕ + 2G Tg (ρhT , i∂∂ϕ)g , 2
where v is the volume of M in the metric. Since ρhT is harmonic, the last summand in the right side can be written as −2G Tg ∗ ∗ (∂ ∗ (∂ (ϕρhT ))). For convenience, let us set β = ∂ (ϕρhT ). Hence, 1 1 d T ∗
B ϕ − 2G g (∂ β)) − ψ B ϕ dµg . X X (ψt )dµgt = dt 2 2 By the Ricci identity for the transversal metric, we have that 1 ( B ϕ)α = −ϕ,γ αγ + ϕ,γ (ψ,αγ + (rhT )αγ ) = −ϕ,γ αγ + ϕ,γ ψ,αγ + βα , 2 and so, after minor simplifications, we conclude that γ d X (ψt )dµgt = αX ϕ,γ ψα − ϕ,γ α dµg dt + αX βα − 2(G g ∂ ∗ β),α dµg , where X is the (1,0)-component of X D (see (25)).
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The first summand on the right above is zero because X is holomorphic. This is just a consequence of Stokes’ theorem. The second summand is also zero since we have α ∗ X βα − 2(G g ∂ β),α dµg = (β − B G Tg β, X )dµg + (2∂ ∗ ∂G Tg β, X )dµg ,
and β − B G Tg η = 0 while ∂ X = 0. Here, of course, X is the (0, 1)-basic form corresponding the (1, 0)-vector field X . We may then define the transversal Futaki invariant F = F(ξ, J¯) of the polarized Sasakian manifold (M, ξ, J¯) to be the functional F : hT (ξ, J¯) −→ C
F(X ) =
X (ψg )dµg = − M
M
X (G Tg sgT )dµg ,
(31)
where g is any metric in M(ξ, J¯). The proposition above shows that F is well-defined, as this expression depends only on the basic class [dη] of a Sasakian structure (ξ, η, , g) in S(ξ, J¯), rather than the specific Sasakian structure chosen to represent it. It is rather obvious that F(X ) = F(X D), and the usual argument in the Kähler case implies also that F([X, Y ]) = 0 for any pair of vector fields X , Y in hT . Our next proposition extends to canonical Sasakian metrics a now well-known result in Kähler geometry originally due to Futaki [17]. The Sasaki version here is analogous to the expanded version of Futaki’s result presented by Calabi [10]. Proposition 5.2. Let (ξ, η, , g) be a canonical Sasakian representative of S(ξ, J ). Then, the metric g has constant scalar curvature if, and only if, F( · ) = 0. Proof. In one direction the statement is obvious: a constant scalar curvature Sasakian metric has trivial Ricci potential function, and the functional (31) vanishes on any X ∈ hT (ξ, J¯). In order to prove the converse, we first observe that if X ∈ hT (ξ, J¯) is a transversally holomorphic vector field of the form X = ∂g# f for some basic function f , then # F(X ) = − ∂g f (G g sg )dµg = −2 (∂ f, ∂ G Tg sg )g dµg ∗ = −2 f (∂ g ∂G Tg sg )dµg , ∗
because the scalar curvature sg is a basic function also. Since 2∂ ∂ = B , we conclude that # F(∂g f ) = − f (sg − sg,0 )dµg . Now, if the Sasakian metric g is a critical point of the energy function E in (7), then ∂g# s = ∂ # s T = ∂g# sg is a transversally holomorphic vector field, and we conclude that F(∂g# s T ) = − (sg − sg,0 )2 dµg . Thus, if F( · ) = 0, then sg must be constant.
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A particular case of constant scalar curvature Sasakian metrics is the case of Sasakian η-Einstein metrics. These are Sasakian metrics g that satisfy Ricg = λg + νη ⊗ η
(32)
for some constants λ and ν. The scalar curvature sg of these metrics is given by sg = 2n(1 + λ). We refer the reader to [6], and references therein, for further discussion of this type of metrics. Corollary 5.3. Let (ξ, η, , g) be a canonical representative of S(ξ, J¯), and suppose that the basic first Chern class c1 (Fξ ) is a constant multiple, say a, of [dη] B . Then (1) If a = 0, then (ξ, η, , g) is a null η-Einstein Sasakian structure with λ = −2, whose transverse metric is Calabi-Yau. (2) If a < 0, then (ξ, η, , g) is a negative η-Einstein Sasakian structure with λ < −2, whose transverse metric is Kähler-Einstein with negative scalar curvature. (3) If a > 0, then (ξ, η, , g) is a positive η-Einstein Sasakian structure with λ > −2, whose transverse metric is positive Kähler-Einstein if, and only if, the Futaki-Sasaki invariant Fξ, J¯ vanishes. Moreover, if Fξ, J¯ vanishes, g ∈ M(ξ, J¯) is Sasaki-Einstein if, and only if, λ = 2n. When an η-Einstein metric exists, the relation 2πa = λ + 2 holds. Proof. Parts (1) and (2) follow from Proposition 5.2 and Theorem 17 of [6]. For (3) we notice that if g is positive Sasakian η-Einstein, the result follows immediately from Proposition 5.2. Conversely, if g is a canonical representative S(ξ, J¯) and Fξ, J¯ vanishes, its scalar curvature is constant. It follows that the scalar curvature of the transversal metric is constant also, and this implies that ρg + 2dη = ρ T is transversally harmonic. As the latter form represents 2π c1 (Fξ ), which is also represented by a constant multiple of dη, the uniqueness of the harmonic representative of a class implies that ρg + 2dη = ρ T = 2πadη for some a > 0. It then follows from this that the transverse Ricci tensor RicT satisfies RicT = 2πag T . But then g is a positive Sasakian η-Einstein metric, and it follows from Eq. 32 that 2πa = λ + 2. This corollary applies whenever the first Chern class c1 (D) of the contact bundle is a torsion class. We mention also that one can always obtain a Sasaki-Einstein metric from a positive Sasakian η-Einstein metric by applying a transverse homothety. 6. The Sasaki Cone We have the following result for CR structures of Sasaki type on M, an immediate consequence of the argument in the proof of Lemma 4.2. Proposition 6.1. Let (D, J ) be a CR structure of Sasaki type on M, and let S = (ξ, η, , g) be a contact metric structure whose underlying CR structure is (D, J ). Then S is a Sasakian structure if and only if ξ ∈ cr(D, J ). We fix a strictly pseudoconvex CR structure (D, J ) on M, and define the set S = (ξ, η, , g) : S a Sasakian structure . S(D, J ) = (ker η, |ker η ) = (D, J )
(33)
We think of this as a subspace of sections of a vector bundle, and provide it with the C ∞ compact-open topology. This set is nonempty if, and only if, (D, J ) is of Sasaki type.
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Proposition 6.2. Let (D, J ) be a CR structure of Sasaki type, and let S0 = (ξ0 , η0 , 0 , g0 ) ∈ S(D, J ). If S = (ξ, η, , g) ∈ S(D, J ), we have that η0 (ξ ) > 0, and η = η0 . η0 (ξ ) Proof. Using the canonical splitting (1), we write any 1-form η as η = f η0 + α, with η0 and α orthogonal to each other. As the kernels of η and η0 equal D, we must have α = 0. Since η is a contact form, the function f is nowhere vanishing, and since |D= J = 0 |D, f must be positive. The result follows. We thus see that the underlying CR structure fixes both orientation and co-orientation of the contact structure. Definition 6.3. Let (D, J ) be a strictly pseudoconvex CR structure of Sasaki type. We say that a vector field X ∈ cr(D, J ) is positive if η(X ) > 0 for any S = (ξ, η, , g) ∈ S(D, J ). We denote by cr+ (D, J ) the subset of all positive elements of cr(D, J ). We consider the mapping ι defined by projection, ι
S(D, J ) → cr+ (D, J ) . S → ξ
(34)
By Proposition 6.2, we see that this mapping is injective. We have the following. Lemma 6.4. Let (D, J ) be a strictly pseudoconvex CR structure of Sasaki type. Then a) cr+ (D, J ) is naturally identified with S(D, J ), b) cr+ (D, J ) is an open convex cone in cr(D, J ), c) The subset cr+ (D, J ) is invariant under the adjoint action of the Lie group CR(D, J ). Proof. In order to prove (a), we show that the map ι in (34) is surjective. As in Proposition 6.2, we fix a Sasaki structure S0 = (ξ0 , η0 , 0 , g0 ) in S(D, J ). For ξ ∈ cr+ (D, J ), we define a 1-form η by η=
η0 . η0 (ξ )
Then, η(ξ ) = 1, and since ξ ∈ cr+ (D, J ), ξ leaves D invariant. This implies that ξ dη = £ξ η = 0. Thus, ξ is the Reeb vector field of η. We then define by = 0 − 0 (ξ ) ⊗ η, and a metric g by (4). The structure S = (ξ, η, , g) belongs to S(D, J ), and thus, ι is surjective. For the proof of (b), we observe that cr+ (D, J ) is open and convex, and that if ξ ∈ cr+ (D, J ), then so is aξ for any positive real number a. Indeed, all of these follow by the defining condition of positivity of a vector field X in cr(D, J ). For the final assertion, we observe that for groups of transformations, the adjoint action is that induced by the differential. Thus, given φ ∈ CR(D, J ) and ξ ∈ cr+ (D, J ), we have that η0 (φ∗ ξ ) = (φ ∗ η0 )(ξ ) = f φ η0 (ξ ) > 0 for some positive function f φ , which shows that φ∗ ξ ∈ cr+ (D, J ). Thus, cr+ (D, J ) is invariant. Hereafter, we shall identify the spaces S(D, J ) and cr+ (D, J ). We are interested in the action of the Lie group CR(D, J ) on S(D, J ) = cr+ (D, J ).
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Theorem 6.5. Let M be a closed manifold of dimension 2n + 1, and let (D, J ) be a CR structure of Sasaki type on it. Then the Lie algebra cr(D, J ) decomposes as cr(D, J ) = tk +p, where tk is the Lie algebra of a maximal torus Tk of dimension k, 1 ≤ k ≤ n +1, and p is a completely reducible Tk -module. Furthermore, every X ∈ cr+ (D, J ) is conjugate to a positive element in the Lie algebra tk . Proof. Let us assume first that M is not the sphere with its standard CR structure. By Proposition 4.4, there is a Sasaki structure S0 ∈ S(D, J ) such that CR(D, J ) = Aut(S0 ), which is a compact Lie group. A well known Lie theory result implies that every element in the Lie algebra aut(S0 ) is conjugate under the adjoint action of the group Aut(S0 ) to one on tk , and by (3) of Lemma 6.4, the positivity is preserved under this action. The possible restriction on the dimension of the maximal torus of cr(D, J ) is well-known in Sasakian geometry. In the case where (D, J ) is the standard CR structure on the sphere, we know [36] that CR(D, J ) = SU(n + 1, 1), and cr(D, J ) = su(n + 1, 1), which has several maximal Abelian subalgebras. A case by case analysis shows that the only Abelian subgroup where the positivity condition can be satisfied is in that of a maximal torus. (This can be ascertained, for instance, by looking at Theorem 6 of [15].) We wish to study further the action of the Lie group CR(D, J ) on the space S(D, J ). The isotropy subgroup of an element S ∈ S(D, J ) is, by definition, Aut(S), and this contains the torus Tk . More generally, we have Lemma 6.6. Let (D, J ) be a CR structure of Sasaki type on M. For each S ∈ S(D, J ), the isotropy subgroup of CR(D, J ) at S is precisely Aut(S). Furthermore,
Aut(S) = Tk . S∈S(D,J )
In particular, Tk is contained in the isotropy subgroup of every S ∈ S(D, J ). Proof. It suffices to show that for the generic Reeb vector field Aut(S) = Tk . So let ξ ∈ cr+ (D, J ) be such that the leaf closure of Fξ is a k-dimensional torus Tk . Since the Reeb field is in the center of Aut(S), continuity implies that all of Tk is in the center of Aut(S). But since the center is Abelian and Tk is maximal, the result follows. We are interested in the orbit space S(D, J )/CR(D, J ). We have Definition 6.7. Let (D, J ) be a CR structure of Sasaki type on M. We define the Sasaki cone κ(D, J ) to be the moduli space of Sasakian structures compatible with (D, J ), κ(D, J ) = S(D, J )/CR(D, J ) . Theorem 6.5 together with the mapping (34) says that each orbit can be represented by choosing a positive element in the Lie algebra tk of a maximal torus Tk . We denote the subset of positive elements by t+k , so we have an identification t+k /W = (tk /W)∩cr+ (D, J ) ≈ κ(D, J ), where W is the Weyl group of a maximal compact subgroup of CR(D, J ). Now the basic Chern class of a Sasakian structure S = (ξ, η, , g) is represented by the Ricci form ρ T /2π of the transverse metric g T (up to a factor of 2π ). Although the notion of basic changes with the Reeb vector field, the complex vector bundle D remains fixed. Hence, for any Sasakian structure S ∈ S(D, J ), the transverse 2-form
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ρ T /2π associated to S represents the first Chern class c1 (D) of the complex vector bundle D. It is of interest to consider the case where k = 1, that is to say, the case where the maximal torus of CR(D, J ) is one dimensional. Since the Reeb vector field is central, the hypothesis that k = 1 implies that dim aut(S) = dim cr(D, J ) = 1. Hence, we have that S(D, J ) = cr+ (D, J ) = t+1 = R+ , and S(D, J ) consists of the 1-parameter family of Sasaki structures given by Sa = (ξa , ηa , a , ga ), where ξa = a −1 ξ , ηa = aη , a = , ga = ag + (a 2 − a)η ⊗ η ,
(35)
and a ∈ R+ , the 1-parameter family of transverse homotheties. In effect, the homotheties described above are the only deformations (ξt , ηt , t , gt ) of a given structure S = (ξ, η, , g) in the Sasaki cone κ(D, J ) where the Reeb vector field varies in the form ξt = f t ξ , f t a scalar function. For we then have that the family of tensors t is constant, and since £ξt t = 0, we see that f t must be annihilated by any section of the sub-bundle D. But then (2) implies that d f t = (ξ f t )η, and we conclude that the function f t is constant. Thus, in describing fully the tangent space of S(D, J ) at S, it suffices to describe only those deformations (ξt , ηt , t , gt ) where ξ˙ = ∂t ξt |t=0 is g-orthogonal to ξ . These correspond to deformations where the volume of M in the metric gt remains constant in t, and are parametrized by elements of k(D, J ) that are g-orthogonal to ξ . The terminology we use here is chosen to emphasize the fact that the Sasaki cone is to a CR structure of Sasaki type what the Kähler cone is to a complex manifold of Kähler type. Indeed, for any point S = (ξ, η, , g) in κ(D, J ), the complex normal bundle (ν(Fξ ), J¯) is isomorphic to (D, J ), and so is the underlying CR structure of any element of S(ξ, J¯). In this sense, the complex structure J¯ is fixed with the fixing of (D, J ), the Reeb vector field ξ polarizes the manifold, and the Sasaki cone κ(D, J ) is the set of all possible polarizations. Definition 6.8. We say that (ξ, η, , g) ∈ κ(D, J ) is a canonical element of the Sasaki cone if the space S(ξ, J¯) admits a canonical representative. We denote by e(D, J ) the set of all canonical elements of the Sasaki cone, and refer to it as the canonical Sasaki set of the CR structure (D, J ). By the identification of κ(D, J ) with t+k , the canonical Sasaki cone singles out the subset of positive Reeb vector fields ξ in t+k for which the functional (7) admits a critical point. 7. Openness of the Canonical Sasaki Set Given a canonical Sasakian structure (ξ, η, , g) with underlying CR structure (D, J ), its isometry group will contain the torus Tk of Theorem 6.5. In fact by Lemma 6.6, for a generic element ξ ∈ t+k , Tk will be exactly the isometry group of g. Moreover, Theorem 4.8 says that the isometry group of the transversal metric g T is a maximal compact subgroup G of the identity component of the automorphism group of the transverse holomorphic structure, and the reductive part of the Lie algebra hT (ξ, J¯)/L ξ consists of the complexification of the Lie algebra of all Killing vector fields for g T that are Hamiltonian. If g is the Lie algebra of G and g0 is the ideal of Killing fields of g T that have zeroes, then z0 ⊂ g0 consists of sections of (D, J ) that are Hamiltonian gradients, and these can be lifted [3] to infinitesimal automorphisms of (ξ, η, , g). Hence,
Canonical Sasakian Metrics
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these vector fields are the transversal gradients of functions in M that are Tk -invariant, or to put it differently, they are generated by those elements of the Lie algebra tk that correspond to transversal holomorphic gradient sections of (D, J ). Thus, in searching for canonical representatives of elements of the Sasaki cone κ(D, J ), it will suffice to consider Sasakian structures that are invariant under Tk , and then seek the canonical representatives among them. We denote by S(ξ, J¯)Tk the collection of all Sasakian structures in S(ξ, J¯) that are Tk -invariant, and by M(ξ, J¯)Tk the space of all Tk -invariant metrics in M(ξ, J¯). The observation made above indicates that, in order to seek canonical representatives of S(ξ, J¯), it would suffice to do so among metrics in M(ξ, J¯)Tk . Given (ξ, η, , g) ∈ S(ξ, J¯)Tk , we let G g stand for the Green’s operator of g acting on functions. We consider a basis {X 1 , . . . , X k−1 } of z0 ∩ tk . Then the set of functions p0 (g) = 1, ∗ p j (g) = 2i G g ∂ g ((J X j + i X j )
dη) ,
j = 1, . . . , k − 1 ,
(36)
spans the space of Tk -invariant basic real-holomorphy potentials, real-valued functions solutions of Eq. (10) whose g T -gradients are holomorphic vector fields. Since the argument function on which G g acts in order to define p j (g) is basic, we could have used the Green’s operator of g T above instead of G g itself. Definition 7.1. We define πg to be the L 2 -projection onto the space of smooth real holomorphic potential functions in (36). By Theorem 3.5, (ξ, η, , g) is a canonical representative of S(ξ, J¯) if and only if (1 − πg )sg = 0, sg the scalar curvature of g. We denote by L 2B,l,Tk the Hilbert space of Tk -invariant basic real-valued functions of class L l2 . We consider deformations (ξα , ηα , α , gα ) of (ξ, η, , g) in the Sasaki cone κ(D, J ), where the Reeb vector field varies as ξα = ξ + α. We require that α be in a sufficiently small neighborhood of the origin in cr(D, J ) so that ξα remains positive. For ϕ in a sufficiently small neighborhood of the origin in L 2B,l+4,Tk , l > n, we then consider the deformations of (ξα , ηα , α , gα ) in S(ξα , J¯) to the Sasakian structure defined by ηα,ϕ = ηα + d c ϕ , α,ϕ = α − (ξα ⊗ (ηα,ϕ − ηα )) ◦ α , gα,ϕ = dηα,ϕ ◦ (1l ⊗ α,ϕ ) + ηα,ϕ ⊗ ηα,ϕ . Here, for (α, ϕ) = (0, 0), we have that (ξα , ηα,ϕ , α,ϕ , gα,ϕ ) = (ξ, η, , g). The restriction on l ensures that the curvature tensors of gα,ϕ are all well-defined because, under such a constraint, L 2B,l,Tk is a Banach algebra. We let U ⊂ cr(D, J ) × L 2B,l+4,Tk (M) be the open neighborhood of (0, 0) where the two-parameter family of deformations gα,ϕ of g is well-defined, and consider the scalar curvature map cr(D, J ) × L 2B,l+4,Tk (M) ⊃
S
U −→ L 2B,l,Tk (M) (α, ϕ) → sgα,ϕ ,
where sgα,ϕ is the scalar curvature of the metric gα,ϕ .
(37)
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Proposition 7.2. For l > n, the map (37) is well-defined and C 1 , with Fréchet derivative at the origin given by 1 2 T T DS(0,0) = −n B + s (η( · )) − ( B + 2r · ∇T ∇T ) , (38) 2 where the quantities in the right are associated to the transversal metric g T defined by g, and r T · ∇T ∇T denotes the full contraction of the Ricci tensor and two covariant derivatives of g T . ˜ g) Proof. Notice that when deforming (ξ, η, , g) to (ξ˜ , η, ˜ , ˜ while preserving the underlying CR structure (D, J ), the contact form η changes by the conformal factor f (ξ˜ ) = 1/η(ξ˜ ). The first component of the Fréchet derivative above follows via a simple calculation, after observing that the Ricci tensor of the transversal Kähler metric is computed in a holomorphic frame by −i∂∂ log det (giTk¯ ), a fact that we apply to the metric d η, ˜ and then differentiate. The second component of the Fréchet derivative follows by Proposition 3.2. For any integer l, we let Il ⊂ L 2B,l,Tk denote the orthogonal complement of the kernel ¯ g# )∗ ∂∂ ¯ g in (11), and set V = U ∩ (cr(D, J ) × Ik+4 ), where U is of the operator L gB = (∂∂ the neighborhood of (0, 0) in cr(D, J ) × L 2B,k+4,Tk in (37), shrunk if necessary so that ker(1 − πg )(1 − πgα,ϕ ) = ker(1 − πgα,ϕ ) whenever we have a Sasaki metric gα,ϕ of the type indicated above, parametrized by some (α, ϕ) ∈ U. Here, πg is the projection onto the finite dimensional space of functions (36) introduced in Definition 7.1. It is clear that the range of this projection changes smoothly with the metric. Since a Sasaki metric g is canonical if, and only if, its scalar curvature is annihilated by the projection operator 1 − πg , we introduce the map S
cr(D, J ) × Il+4 ⊃ V −→ cr(D, J ) × Ik , S(α, ϕ) := (α, (1 − πg )(1 − πgα,ϕ ) S(α, ϕ) )
(39)
where S(α, ϕ) is the map in (37). We have the following. Lemma 7.3. Suppose that g is a canonical Sasaki metric representing the polarization of M given by (ξ, J¯), and that gt = gt,α,ϕ is a curve of Sasakian metrics of the type above that starts at g when t = 0, and is parametrized by (α, ϕ) ∈ cr(D, J ) × L 2B,l+4,Tk . Then
d ∗ (1 − πg ) ( πgt )
sg = (1 − πg )[−2i G g ∂ (η(α)) ∂sg + (∂sg ∂ # ϕ)] , dt t=0
where G g is the Green’s operator of g. Proof. The result is clear if the basic scalar curvature function sg is constant. For the general case, we refer the reader to the original argument in [22] for the Kähler case (see also the significantly improved understanding, and related discussions, given in [35]). Its required extension follows by the observation that the transversal metric of gt is given by dηt , where the contact form ηt changes by the conformal factor 1/η(ξt,α,ϕ ), plus a deformation that preserves the basic class.
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Proposition 7.4. For l > n, the map (39) is C 1 with Fréchet derivative at the origin given by 1 0 1 0 ∗ , (40) DS(0,0) = 0 1−πg (−n B + s T ) (η( · ))+2i G g ∂ (η( · )) ∂sg −2L gB ¯ # )∗ ∂∂ ¯ #. where L gB = (∂∂ Proposition 7.5. Let M be a closed manifold, and (D, J ) be a CR structure of Sasaki type on it. Then the map S defined in (39) becomes a diffeomorphism when restricted to a sufficiently small neighborhood of the origin. Proof. We apply the inverse function theorem for Banach spaces. Hence, we just need to prove that DS(0,0) has trivial kernel and cokernel. Suppose that (α, ϕ) is in the kernel of DS(0,0) . By (40), we see that α = 0, and that (1 − πg )L gB ϕ = 0 . ¯ # )∗ ∂∂ ¯ # ϕ is a holomorphy potential, and consequently, it can It follows that L gB ϕ = (∂∂ be written as j ¯ #ϕ = ¯ # )∗ ∂∂ c j fg , (∂∂ j
in terms of an orthonormal basis of the space spanned by the functions in (36). If we take j ¯ # )∗ ∂∂ ¯ #, the inner product of this expression with f g , and dualize the symmetric map (∂∂ # ∗ # ¯ ¯ we see that c j = 0. Thus, (∂∂ ) ∂∂ ϕ = 0. But ϕ ∈ Il+4 , space orthogonal to the kernel ¯ # )∗ ∂∂ ¯ # . So ϕ must be zero, and the kernel of DS(0,0) consists of the point (0, 0). of (∂∂ Suppose now that (β, ψ) is orthogonal to every element in the image of DS(0,0) . Then, it must be orthogonal to the image of (0, ϕ) for any ϕ ∈ Il+4 , and therefore, ¯ # )∗ ∂∂ ¯ # )∗ ∂∂ ¯ # ϕ, ψ = ϕ, (∂∂ ¯ # (1 − πg )ψ = 0
(1 − πg )(∂∂ ¯ # )∗ ∂∂ ¯ # (1 − πg )ψ perpendicular to for all such ϕ. It follows that the component of (∂∂ j # ∗ # # ∗ # ¯ ) ∂∂ ¯ is zero, and thus, (∂∂ ¯ (1 − πg )ψ = ¯ ) ∂∂ c j f g . The same the kernel of (∂∂ ¯ # )∗ ∂∂ ¯ #. argument used above implies that c j = 0, and so, (1 − πg )ψ is the kernel of (∂∂ But the image of 1 − πg is orthogonal to this kernel. Hence, ψ = 0. Using this fact, we may now conclude that β must be such that β, α = 0 for all α ∈ cr(D, J ), and so β = 0. The cokernel of DS(0,0) is trivial. We now prove the following result, analogous to the openness of the extremal cone in Kähler geometry [22]. Theorem 7.6. Let (D, J ) be a CR structure of Sasaki type on M. Then the canonical Sasaki set e(D, J ) is an open subset of the Sasaki cone κ(D, J ). Proof. Let V0 ⊂ V be a neighborhood of (0, 0) ∈ cr(D, J ) × Ik+4 such that S |V0 is a diffeomorphism from V0 onto an open neighborhood of the origin in cr(D, J ) × Ik . For any point α in S(V0 ) ∩ (cr(D, J ) × {0}), we define ϕ(α) to be the projection onto Ik+4 of (S |V0 )−1 (α). Then we have (α, 0) = S(α, ϕ(α)) = (α, (1 − πg )(1 − πgα,ϕ(α) )sgα,ϕ(α) ) ,
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where gα,ϕ(α) is the deformation of the metric g associated to the parameters (α, ϕ(α)). Since the kernel of (1 − πg )(1 − πgα,ϕ(α) ) equals the kernel of 1 − πgα,ϕ(α) , it follows that (1 − πgα,ϕ(α) )sgα,ϕ(α) = 0. We then have that the scalar curvature of the Sasaki metric gα,ϕ(α) is a holomorphy potential, and so this metric is a canonical representative of S(ξ + α, J¯). This completes the proof. Example 7.7. Let us take coordinates z = (z 0 , . . . , z n ) in Cn+1 , and consider the unit sphere S2n+1 = {z ∈ Cn+1 : |z| = 1} . If z k = xk + i yk is the decomposition of z k into real and imaginary parts, then the vector fields Hk = (xk ∂ yk − yk ∂xk ), k = 0, . . . , n, form a basis for the Lie algebra tn+1 = Rn+1 of a maximal torus in the automorphism group U(n+1) of the standard Sasakian structure on S2n+1 . The latter is given by the contact form η = nk=0 (xk dyk − yk d xk ), Reeb vector field ξ = nk=0 Hk , and (1, 1)-tensor defined by the restriction J of the complex structure on Cn+1 to D = ker η, and the fact that (ξ ) = 0. The ensuing compatible metric (4) defined by (ξ, η, ) is the standard metric g on S2n+1 , which is Einstein with r g = 2ng. Thus, the metric g yields a canonical representative of S(ξ, J ). We have that the automorphism group of (ξ, η, , g) is U(n + 1). Its maximal torus Tn+1 has Lie algebra tn+1 with basis {H0 , . . . , Hn }. We now fix this CR structure (D, J ) on S2n+1 , and consider the set S(D, J ) of all Sasakian structures associated with it. A vector field X is positive if, and only if, η(X ) > 0, and such a vector field is conjugate to a positive vector in the Lie algebra tn+1 . We use the basis {H0 , . . . , Hn } to identify this Lie algebra with Rn+1 , so the point n+1 w = (w0 , . . . , wn ) in R yields the vector ξw = wk Hk . Then we have that η(ξw ) =
n
n wi xi2 + yi2 = wi |z i |2 ,
i=0
i=0
and so, the set of positive elements of tn+1 is just Rn+1 + . By Theorem 6.5, the Sasaki n+1 , this vector gives rise to the Sasakian struccone κ(D, J ) is equal to Rn+1 . If w ∈ R + + ture (ξw , η/η(ξw ), w , gw ), where w is defined by the conditions w |D= J and w (ξw ) = 0, respectively, and gw is determined by the expression (4) in terms of ξw , η/η(ξw ) and w . For any w ∈ Zn+1 + , the Sasakian structure (ξw , η/η(ξw ), w , gw ) is quasi-regular, and its transversal is a manifold with orbifold singularities, the weighted projective space CPnw . The space of metrics M(ξw , J¯) associated with the polarized Sasakian manifold T is Bochner (S2n+1 , ξw , J¯) has a representative gw whose transverse Kähler metric gw n flat [8] on CPw , and thus, extremal. Computing in an affine orbifold chart, it can be T is given by determined [16] that the scalar curvature of gw n n 2 j=0 w j (2( k=0 wk ) − (n + 2)w j )|z j | n , sgwT = 4(n + 1) 2 j=0 w j |z j | at z ∈ S2n+1 . Since the volume µgwT (CPnw ) = π n /(n! nj=0 w j ) [16], the volume of S2n+1 in the Sasakian metric gw is just µgw (S2n+1 ) = 2
1 π n+1 n . n! j=0 w j
Canonical Sasakian Metrics
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T is Similarly, since sgw = sgwT − 2n, and since the mean transverse scalar curvature of gw n 4n j=0 w j [16], we have that the projection sg0w of sgw onto the constants is given by ⎛ ⎞ n w j − 1⎠ . sg0w = 2n ⎝2 j=0
Thus, n sgw
− sg0w
= 4(n + 2)
j=0 w j
n
k=0 wk − (n + 1)w j n 2 j=0 w j |z j |
|z j |2
.
Notice that if the weight vector w is of the form w = l(1, . . . , 1), then sgwT = 4(n + 1)nl, and this yields the scalar curvature of the Fubini-Study metric when l = 1, as it should. For the Sasaki-Futaki character of the polarization (ξw , J¯), it suffices to determine its value on vector fields X that commute with ξw , and that are of the form X = ∂ # f for f a basic real holomorphy potential. In that case, we have F(ξw , J¯) (X ) = − f (sgw − sgw0 )dµgw . S2n+1
For convenience, let us set A j = nk=0 wk − (n + 1)w j . Working in an affine orbifold n chart for CPw , we then see that if f = i=0 bi |z i |2 we have that b0 + n b j x j (w0 A0 + n w j A j x j ) j=1 j=1 F(ξw , J¯) (X ) = −8(n+2)π n+1 d x1 . . . d xn n+3 Rn+ w0 + nj=1 w j x j ⎞ ⎛ n 1 π n+1 ⎝ bi 1 bi = −16 Ai + A j ⎠ n . (n + 1)! wi 2 wi j=0 w j i=0
i= j
Theorem 7.8. Let (D, J ) be the standard CR structure on the unit sphere S2n+1 , and κ(D, J ) and e(D, J ) be the associated Sasaki cone and canonical Sasaki set, respectively. Then e(D, J ) = κ(D, J ), and the only canonical points in e(D, J ) that yield metrics of constant scalar curvature are those representing transverse homotheties of the standard Riemannian Hopf fibration. Of these, the metric of constant sectional curvature one is the only Sasaki-Einstein metric whose underlying CR structure is (D, J ). n+1 Proof. We have identified above κ(D, J ) with points w in Rn+1 + . If w ∈ Z+ , the polarized Sasakian manifold (S2n+1 , ξw , J¯) admits a representative gw with transverse Bochner flat metric on the transverse space CPnw . Thus, these weights w belong to e(D, J ). Using the homotheties (35), we may obtain canonical representatives of the polarized Sasakian manifold (S2n+1 , ξw , J¯) for any weight w ∈ Qn+1 + . Applying Theorem 7.6, we obtain the same result for arbitrary weights w in Rn+1 + . Thus, e(D, J ) = κ(D, J ). The expressions computed above for the scalar curvature, volume, and Sasaki-Futaki character for w = (w0 , . . . , wn ) ∈ Zn+1 are rational functions of the weights w j , so + they also define the scalar curvature, volume, and Sasaki-Futaki character when w is
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an arbitrary vector in Rn+1 + , regardless of the fact that there may not be a transversal manifold to speak of in this general situation. The assertion about the scalar curvatures of the canonical representatives gw follows by the expression for F(ξw , J¯) given above, and Proposition 5.2. Indeed, the character F(ξw , J¯) is identically zero if, and only if, A j = Ak for all pairs of indices j, k. This only happens if the vector of weights w is of the form w = l(1, . . . , 1). The only one of these metrics that is Sasaki-Einstein is the standard metric. This follows by a simple analysis of the change of the Ricci curvature under homotheties of the metric. In this example, the first Chern class of the sub-bundle D is trivial. Thus, for any w in the Sasaki cone, the basic first Chern class of the resulting foliated manifold is proportional to the basic class defined by the transversal Kähler form dηw . However, only for the weight w = (1, . . . , 1) there exists a Sasaki-Einstein representative. In general, given a CR structure (D, J ) of Sasaki type on a closed manifold M, we do not expect the equality e(D, J ) = κ(D, J ) to hold, though this is likely to be so in the toric case. Acknowledgements. We would like to thank Vestislav Apostolov for spotting an incorrect statement in an earlier version of the paper.
References 1. Boyer, C.P., Galicki, K.: A note on toric contact geometry. J. Geom. Phys. 35, 288–298 (2000) 2. Boyer, C.P., Galicki, K.: Einstein manifolds and contact geometry. Proc. Amer. Math. 129, 2419–2430 (2001) 3. Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs, Oxford: Oxford University Press, 2008 4. Boyer, C.P., Galicki, K.: Sasakian geometry, hypersurface singularities, and Einstein metrics. Rend. Circ. Mat. Palermo (2) Suppl. 75, 57–87 (2005) 5. Boyer, C.P., Galicki, K., Kollár, J.: Einstein metrics on spheres. Ann. of Math. 162, 557–580 (2005) 6. Boyer, C.P., Galicki, K., Matzeu, P.: On Eta-Einstein Sasakian Geometry. Commun. Math. Phys. 262, 177–208 (2006) 7. Brinzanescu, V., Slobodeanu, R.: Holomorphicity and Walczak formula on Sasakian manifolds. J. Geom. and Phys. 57(1), 193–207 (2006) 8. Bryant, R.L.: Bochner-Kähler metrics. J. Amer. Math. Soc. 14, 623–715 (2001) 9. Calabi, E.: Extremal Kähler metrics. In: Seminar of Differerential Geometry, ed. S.T. Yau, Annals of Math. Studies, 102, Princeton, NJ: Princeton University Press, 1982, pp. 259–290 10. Calabi, E.: Extremal Kähler metrics II. In: Differential geometry and complex analysis I. Chavel, H.M. Farkas, eds. Berlin-Heidelberg-New York: Springer-Verlag, 1985, pp. 95–114 11. Calabi, E.: The Space of Kähler Metrics. Proc. Int. Cong. Math., Amsterdam, Vol. 2, 1954, pp. 206–207 12. Cheeger, J., Tian, G.: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118, 493–571 (1994) 13. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974) 14. Cvetic, M., Lü, H., Page, D.N., Pope, C.N.: New Einstein-Sasaki spaces in five and higher dimensions. Phys. Rev. Lett. 95, 4 (2005) 15. David, L.: The Bochner-flat cone of a CR manifold. http://arxiv.org/list/math.DG/0512604,2005 16. David, L., Gauduchon, P.: The Bochner-flat geometry of weighted projective spaces. C.R.M. Proceedings and Lecture Notes 40, Providence, RI: Amer. Math. Soc., 2006, pp. 104–156 17. Futaki, A.: An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73, 437–443 (1983) 18. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, W.: Sasaki-Einstein metrics on S 2 × S 3. Adv. Theor. Math. Phys. 8, 711–734 (2004) 19. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, W.: A new infinite class of Sasaki-Einstein manifolds. Adv. Theor. Math. Phys. 8, 987–1000 (2004) 20. Kollár, J.: Circle actions on simply connected 5-manifolds. Topology 45, 643–671 (2006) 21. Kollár, J.: Einstein metrics on five-dimensional Seifert bundles. J. Geom. Anal. 15, 445–476 (2005)
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22. LeBrun, C., Simanca, S.R.: On the Kähler Classes of Extremal Metrics. In: Geometry and Global Analysis, (First MSJ Intern. Res. Inst. Sendai, Japan) eds. T. Kotake, S. Nishikawa, R. Schoen, Sendai. Tohoku Univ. Press, 1993, pp. 255–271 23. Lee, J.M.: CR manifolds with noncompact connected automorphism groups. J. Geom. Anal. 6, 79–90 (1996) 24. Lichnerowicz, A.: Sur les transformations analytiques des variétés kählériennes compactes. C.R. Acad. Sci. Paris 244, 3011–3013 (1957) 25. Martelli, D., Sparks, J.: Toric Sasaki-Einstein metrics on S 2 × S 3. Phys. Lett. B 621, 208–212 (2005) 26. Martelli, D., Sparks, J.: Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals. Commun. Math. Phys. 262, 51–89 (2006) 27. Martelli, D., Sparks, J., Yau, S.-T.: The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds. Commun. Math. Phys. 268, 39–65 (2005) 28. Martelli, D., Sparks, J., Yau, S.-T.: Sasaki-Einstein manifolds and volume minimisation. http://arxiv.org/ list/hepth/0603021,2006 29. Matsushima, Y.: Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne. Nagoya Math. J. 11, 145–150 (1957) 30. Molino, P.: Riemannian Foliations. Progress in Mathematics 73, Boston, MA: Birkhäuser Boston Inc., 1988 31. Nishikawa, S., Tondeur, P.: Transversal infinitesimal automorphisms for harmonic Kähler foliations. Tôhuku Math. J. 40, 599–611 (1988) 32. O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966) 33. Schoen, R.: On the conformal and CR automorphism groups. Geom. Funct. Anal. 5, 464–481 (1995) 34. Simanca, S.R.: Canonical Metrics on Compact Almost Complex Manifolds. Publicações Matemáticas do IMPA, Rio de Janeiro: IMPA, 2004, 97 pp. 35. Simanca, S.R.: Heat Flows for Extremal Kähler Metrics. Ann. Scuola Norm. Sup. Pisa CL. Sci. 4, 187–217 (2005) 36. Webster, S.M.: On the transformation group of a real hypersurface. Trans. Amer. Math. Soc. 231, 179–190 (1977) Communicated by G.W. Gibbons
Commun. Math. Phys. 279, 735–768 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0438-0
Communications in
Mathematical Physics
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers Joseph Geraci1,2 , Daniel A. Lidar2 1 Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada.
E-mail: [email protected]
2 Departments of Chemistry, Electrical Engineering, and Physics, Center for Quantum Information Science
& Technology, University of Southern California, Los Angeles, CA 90089, USA Received: 16 March 2007 / Accepted: 5 September 2007 Published online: 1 March 2008 – © Springer-Verlag 2008
Abstract: We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC ) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date. 1. Introduction A wealth of results has been obtained since the dramatic early results [1,2] on quantum speedups relative to classical algorithms. A relatively unexplored field is quantum algorithms for problems in classical statistical mechanics. The earliest contribution to this subject [3] obtained a modest speedup in that it avoided critical slowing down [4] in the problem of sampling from the Gibbs distribution for Ising spin glass models. Subsequently Ref. [5] raised the question of providing a classification of classical statistical physics problems in terms of their quantum computational complexity. In this work we shed light on this classification by considering the problem of evaluating the Potts model partition function Z for classical spin systems on graphs. It is known that under particular conditions even certain approximations for Z are unlikely to be efficient, barring an N P = R P surprise [6]. Here we present a class of sparse graphs (which we call ICCC ) for which exact quantum evaluation of Z is possible with a polynomial speedup in the size of the graph and an exponential speedup in the number of per-spin states, over the best classical algorithms available to date.
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The Potts partition function of graphs in ICCC is equivalent to the weight enumerators of certain linear codes. The evaluation of weight enumerators (in this case) involves the evaluation of Gauss sums or Zeta functions. The evaluation of Gauss sums is in general hard and equivalent to the calculation of discrete log [7]. This suggests that ICCC includes cases that are unlikely to be solved as efficiently on classical computers. 1.1. The Potts model. Let Γ = (E, V ) be a weighted graph with edge set E and vertex set V . The q-state Potts model is a generalization of the Ising model where a q-state spin resides on each vertex. In the Ising model q = 2, whereas in the Potts model q ≥ 2. The edge connecting vertices i and j has weight Ji j , which is also the interaction strength between the corresponding spins. The Potts model Hamiltonian for a particular spin configuration σ = (σ1 , . . . , σ|V | ) is Ji j δσi σ j , (1) H (σ ) = −
where summation is over nearest neighbors, and where δσi σ j = 1 (0) if σi = σ j (σi = σ j ). Thus only nearest neighbor parallel spins contribute to the energy. The probability P(σ ), of finding the spin in the Potts model in some configuration σ at a given temperature T , is given by the Gibbs distribution P(σ ) =
e−β H (σ ) , Z (β)
(2)
where β = 1/(kB T ) is the inverse temperature in energy units, and kB is the Boltzmann constant. The normalization factor is the partition function Z (β) = e−β H (σ ) , (3) {σ }
which plays a central role in statistical physics, since many thermodynamic quantities can be derived from it [7]. When for all configurations β |H (σ )|, the probability distribution becomes flat: P(σ ) ≈ 1/Z (β), so that at high temperatures randomness dominates. The partition function can be rewritten as a polynomial: β βJ δ J δ Z (β) = e i j σi σ j = e i j σi σ j {σ }
=
{σ }
(1 + vi j (β)δσi σ j ),
(4)
{σ }
where vi j (β) = eβ Ji j − 1.
(5)
Now let us consider the case when the interactions Ji j are a constant J . Then the Hamiltonian (1) of this system can be written as H (σ ) = −J |U (σ )|,
(6)
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where U (σ ) is the subset of edges whose vertices have the same spin for a particular spin configuration σ , and |U (σ )| is the number of such subsets. If we let J
y = e− kT we can write the Potts partition function as y −|U (σ )| . Z (y) =
(7)
(8)
σ
1.2. Relation between the Potts partition function, knot invariants, and graph theory. There is a rich inter-relation between classical statistical mechanics and topology, in particular, the theory of the classification of knots. The first such connection was established by Jones [8], who discovered the second knot invariant (the Jones polynomial (a Laurent polynomial), after the Alexander polynomial) during his investigation of the topological properties of braids [9]. It is known that the classical evaluation of the Jones polynomial is P-hard [10]. A direct connection between knots and models of classical statistical mechanics was established by Kauffman [11]. Knot invariants are, in turn, also tightly related to graph theory; e.g., the graph coloring problem can be considered an instance of evaluation of the Kauffman bracket polynomial, via the Tutte polynomial [11,12]. The q-state Potts partition function on a graph Γ is connected to the Tutte polynomial TΓ for the same graph via q + v ,v + 1 , (9) Z Γ (v) = q n TΓ v where as in Eq. (5), v +1 = e−β . This means that the Potts partition function is equivalent to some, easily computed function times the Tutte polynomial along the hyperbola Hq = (x − 1)(y − 1) = q. But for planar graphs, when q > 2 the Tutte polynomial is P-hard to evaluate at points along Hq [6]. For a review of the connection between the Potts partition function and the various polynomials mentioned above, see [11] and also [5,13]. It immediately follows from Eq. (9) and complexity results concerning the Tutte polynomial, that the evaluation of the Potts partition function is also P-hard. It is not known whether there is an fpras (fully polynomial randomized approximation scheme) [6] for the q-state fully ferromagnetic Potts partition function, but it is known that if there is an fpras for the fully anti-ferromagnetic Potts partition function then N P = R P [6] and therefore it seems unlikely that an fpras will be found for this case. 1.3. Previous complexity results. The first connection between knots and quantum field theory was established by Witten, who showed that the Jones polynomial can be expressed in terms of a topological quantum field theory [14]. Recently this connection was extended to the possibility of efficient evaluation of the Jones polynomial by Freedman and co-workers, after showing that quantum computers can efficiently simulate topological quantum field theory [15]. More specifically, there are recent results demonstrating the efficacy of quantum computers in approximating the Jones polynomial at primitive roots of unity [16–18]. In Ref. [16] tools from topological quantum field theory [14] were utilized and it was shown that approximating the Jones polynomial at primitive roots of unity is BQP-complete, but no explicit algorithm was provided. More recently in
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[17], a combinatorial approach was taken which yielded an explicit quantum algorithm and which extended the results in [16] to all primitive roots of unity. This leads one to hypothesize that quantum computers will also be efficient at estimating partition functions. Indeed, an immediate corollary of the results in [16–19], is that the Potts partition function over any planar graph can be approximated efficiently on a quantum computer at certain imaginary temperatures (see also [5]). This follows by noting that in order to obtain an equality between the Potts partition function and the Jones polynomial (up to multiplication by an easily computed function), the Jones variable t and the temperature 2πi T must be related by t = −e±J± /kB T [11]. With t a root of unity (t = e r ) we then find: T =i
J±r , r ∈ N. kB π(2 + r )
This result is of interest mainly in light of quantum Monte Carlo simulations [20], where one retrieves real time dynamics from a simulation in terms of imaginary time, via analytic continuation. Perhaps a similar extrapolation can be achieved here between imaginary and real temperature dynamics. While this is interesting, here we are concerned with thermodynamics, and hence evaluations of the Potts partition function at physically relevant, real temperatures. Most closely related to our work is the very recent result due to Aharonov et al. [21] who – generalizing Temperley-Lieb algebra representations used in [17] – provided a quantum algorithm for the additive approximation of the Potts partition function (and other points of the Tutte plane) for any planar graph with any set of weights on the edges. These results are the most impressive to date in the context of approximate evaluations of the Potts partition function, but are also subject to certain caveats.1 In particular, Ref. [21] leaves as an open problem the complexity of physical instances (real temperature, positive partition function) under the restriction of an additive approximation. Nor is it clear whether the algorithm found in Ref. [21] provides a quantum speedup. The authors state: “We believe that the main achievement here is that we demonstrate how to handle non-unitary representations, and in particular, we are able to prove universality using non-unitary matrices.” Recently Ref. [22] gave a scheme for studying the partition function of classical spin systems including the Potts and Ising model. Their approach involves transforming the problem of evaluating the partition function into the evaluation of a probability amplitude of a quantum mechanical system and then using classical techniques to extract the pertinent information. In essence their method involves moving into a quantum mechanical formalism to obtain a classical result. The scheme is therefore classical and not a quantum algorithm. 1 To quote from the abstract of Ref. [21]: “Additive approximations are tricky; the range of the possible outcomes might be smaller than the size of the approximation window, in which case the outcome is meaningless. Unfortunately, ruling out this possibility is difficult: If we want to argue that our algorithms are meaningful, we have to provide an estimate of the scale of the problem, which is difficult here exactly because no efficient algorithm for the problem exists!”. And: “The case of the Potts model parameters deserves special attention. Unfortunately, despite being able to handle non-unitary representations, our methods of proving universality seem to be non-applicable for the physical Potts model parameters. We can provide only weak evidence that our algorithms are non-trivial in this case, by analyzing their performance for instances for which classical efficient algorithms exist. The characterization of the quality of the algorithm for the Potts parameters is thus left as an important open problem.” Finally, quoting from Sect. 1.5 of Ref. [21]: “Proving anything about the complexity of our algorithm for the Potts model remains a very important open problem. It is still possible that this case of the Tutte polynomial, with our additive approximation window, can be solved by an efficient classical algorithm.”
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In addition, two purely classical results should be mentioned here. One is a state of the art result by Hartmann [23], who provides an algorithm which is well suited to large ferromagnetic systems for either the Potts or Ising model. We do not know the exact complexity of this algorithm, however. The approach taken in our work is to utilize the connection between classical coding theory and the partition function. For this reason we mention the classical algorithm given in [24] for calculating the Zeta function of certain curves. This is also a state of the art algorithm and it can be used to find the Potts partition function via the scheme we present in this paper, though it is slower than using quantum resources. A quantum algorithm for finding the Zeta function of a curve is given in [25]. One could replace the role that the Gauss sum estimation [26] plays in our scheme with this quantum algorithm for the Zeta function. It seems that using Gauss sums is more efficient but further work is required to make this conclusive. Finally, we mention that it was recently shown that one can construct interesting classes of graphs for which the Potts model can be computed analytically [46]. These so called n-ladder graphs are recursively defined. 1.4. Structure of the paper. The structure of this paper is as follows. In Sect. 2 we define the class of graphs our quantum algorithm applies to, and present our main theorem. In Sect. 3 we compare the computational complexity of our algorithm with the state of the art in classical algorithms. In Sect. 4 we give a brief summary of the entire algorithm. In Sect. 5 we provide several illustrative examples of graphs and codes to which our algorithm applies. Finally, in Sect. 6 we conclude and discuss future directions. The Appendix provides pertinent background on matroids, irreducible cyclic codes, and Gauss sums. 2. A Theorem about Quantum Computation and Certain Instances of the Potts Model 2.1. Main Theorem. We present here a polynomial time quantum algorithm for the exact evaluation of the q-state (fully ferromagnetic or anti-ferromagnetic) Potts partition function Z for a certain class of graphs. This class of graphs, which we call “Irreducible Cyclic Cocycle Code” ICCC graphs, comprises graphs whose incidence matrices generate certain cyclic codes. This and other concepts used below are given precise definitions in Sect. 2.2. The key ingredients used are the connection of Z to the weight enumerators of codes [27] and a quantum algorithm for the approximation of Gauss sums [26]. The overall structure of the algorithm is the following: 1. Given a graph Γ = (E, V ), first determine if Γ belongs to the ICCC class. This decision problem can be solved efficiently using the quantum discrete log algorithm [1]. If Γ ∈ ICCC proceed to Step 2, otherwise the algorithm may not evaluate Z Γ efficiently. 2. Identify the linear code C(Γ ) for which we shall determine the weight spectrum. 3. Using the quantum Gauss sum estimation algorithm find the weight spectrum of the words in C. This step is believed to be classically hard but the exact complexity is unknown. It is known, however, that this step is at least as hard as determining discrete log [26]. This is the most expensive step of the algorithm due to the large number of words one has to deal with. This is because the number of possible spin configurations grows exponentially in the number of vertices.
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4. Take a tally of the weight spectrum obtained in the previous step. Grover’s search algorithm can be used to give an additional quadratic speed up but this does not help in reducing the overall complexity since the computational cost of Step 3 is greater than that of the current step. 5. Using the relation given by Eq. (15) between the weight spectrum of a code and Z , use the tally from the previous step to obtain Z (for graphs in ICCC ). We now give the main theorem, after the definition of the family of graphs for which the scheme applies. Definition 1. (ICCC ) Given a constant < 1, ICCC is the family of graphs whose cycle matroid matrix (CMM) representation generates a cyclic code whose dual is irreducible cyclic of dimension k and length n, such that n=
qk − 1 αk s(k)
(10)
(where α ∈ R is chosen so that n ∈ N and where s(k) is an arbitrary function whose role will be clarified below) and θn,k =
1 min S ( jn) q − 1 0< j≤αk s(k)
(11)
(where S (x) is the sum of the digits of x in base q) so that ≤
q θn,k −1 . 4 qk
(12)
Below we define the concepts entering this definition and clarify the role of θn,k and of the bound on . We work in units such that the Boltzmann constant kB = 1. Theorem 1 (Main Theorem). Let Γ = (E, V ) be a graph, n = |E| and k = |E| − |V | + c(Γ ), where c(Γ ) is the number of connected components of Γ . A quantum computer can return the exact q-state fully anti-ferromagnetic or ferromagnetic Potts partition function Z Γ for graphs in ICCC . For each family ( fixed), the overall running time is O( 1 k 2 max[1,s(k)] (log q)2 ) and the success probability is at least 1 − δ, where δ = [2((q k − 1)2 − 2)]−1 . Some remarks: 1. The function s(k) determines the complexity of the schemes. If s(k) = c ∈ R (constant) then we have a polynomial time algorithm for the exact evaluation of Z for each family ICCC . This restriction is reflected in the graphs by enforcing that n = O(q k /k s ), i.e., that the number of edges (n) and vertices (n − k) is close. We have numerically solved for the number of edges |E| as a function of the number of vertices |V |, given by the corresponding transcendental equation |E| = |V | − c(Γ ) + logq (|E|(|E| − |V | + c(Γ ))s + 1) [Eq. (10)]. A numerical fit reveals that to an excellent approximation |E| = |V | + a + b log |V |,
(13)
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741
b
q=2
1.6 1.4 1.2 2
4
6
8
s
10
0.8 0.6
35
q=9 q=2
a
30 25 20 15 10
q=9
5 2
4
6
8
10
s
Fig. 1. Coefficients a and b as a function of s, for different values of q. Here c(Γ ) = 1. See text for details
where the constants a and b depend on q and s, and both increase slowly with s, and decrease with q, as shown in Fig. 2.1. By direct substitution of Eq. (13) into the above transcendental equation it can be seen that the analytical solution will have a correction of order log log(|V |) to the right-hand side of Eq. (13). The fact that there are logarithmically more edges than vertices in the graphs that are members of ICCC is the reason we call these graphs sparse. The important point is that there are families of graphs for which there exist exact polynomial-time evaluation schemes via the methods presented in this paper. As we show below, in these cases we also obtain polynomial speed ups over the best classical algorithms available. 2. Note that if we have an efficient evaluation for ICCC then we also have an efficient evaluation for ICCC , provided > . 3. We provide a discussion of the computational complexity, both classical and quantum, in Subsect. 3. As argued there, we obtain a polynomial speed up in the difference between the number of edges and vertices and an exponential speed up in q over the best current classical algorithm for the ICCC class of graphs. Corollary 1. For a given graph Γ , whose CMM is the direct sum of the CMMs of two graphs Γ1 and Γ2 in ICCC , a quantum computer will be able to return Z Γ with a running time equal to the sum of the running times required to obtain Z Γ1 and Z Γ2 . Proofs of the main theorem and the corollary are provided in Sects. 2.4 and 2.5.
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2.2. Background. Theorem 1 connects the problem of estimating the Potts partition function to a quantum algorithm for Gauss sums, via weight enumerators for irreducible cyclic codes. In somewhat more detail, the connections we need are as follows. In [27], it was shown that the Potts partition function can be written as the weight enumerator of the cocycle code of the graph Γ , over which the Potts model is defined. Weight enumerators of irreducible cyclic codes are related to Gauss sums via the McEliece Theorem [28]. 2.2.1. Cycle matroid matrix representation of a graph A connected component of a graph is any subset of vertices which are all connected to each other via a path along the graph’s edges. We denote the number of connected components by c(Γ ). The incidence matrix of a finite graph Γ (E, V ) is a |V | × |E| binary matrix where column c represents edge c with non-zero entries in row i and j if and only if vertices i and j are the boundaries of edge c. Every finite graph Γ also gives rise to a cycle matroid matrix (CMM) [29], which essentially captures the presence and locations of cycles in the graph. Definition 2. The cycle matroid matrix of a graph Γ = (E, V ), CMM(Γ ), is formed as follows: write down the incidence matrix of Γ using 1 for the i th and −1 for the j th rows, where i < j. Then apply elementary row operations and Gaussian reduction to obtain a (|V | − c(Γ )) × |E| matrix of the form [I|V |−c(Γ ) |X ], where Ia is the a × a identity matrix and X is a (|V | − c(Γ )) × (|E| − |V | + c(Γ )) matrix. This is CMM(Γ ) (see Prop. 4.7.14 of [30]). We give more details on cycle matroids in Appendix 7. As an example consider the square [|V | = |E| = 4, c(Γ ) = 1] and its incidence matrix ⎛ ⎞ 1 0 0 −1 ⎜ −1 1 0 0 ⎟ ⎝ 0 −1 1 0 ⎠ . 0 0 −1 1 Applying elementary row operations and Gaussian reduction one obtains the CMM ⎛ ⎞ 1 0 0 −1 ⎝ 0 1 0 −1 ⎠ , 0 0 1 −1 which is indeed of the form [I|V |−c(Γ ) |X ] with dimensions as in the definition, i.e., X is (4−1)×(4−4+1). Over Z2 one would replace all the −1’s with +1’s. The column space of this matrix represents the cycle structure of the graph where a cycle (or circuit) is a path in the graph for which the first vertex of the path is the same as the last. Any set of columns that are linearly dependent indicate a cycle. The first three columns in the CMM of the square are linearly independent, but together with the fourth column they become linearly dependent, since there is a cycle in the graph involving the corresponding four edges. What is the equivalence class of graphs with the same CMM? This is answered by the following: Definition 3. Two graphs G and G are called 2-isomorphic if there exists a 1 − 1 correspondence between the edges of G and G such that the cycle (or circuit) relationships are preserved. Thus all 2-isomorphic graphs have the same CMM (up to elementary row and column operations).
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2.2.2. Irreducible cyclic codes We provide some background material in coding theory which allows us to exhibit the connection between Eq. (8) and the so-called cocycle code of the graph Γ . (A good reference is [31].) Definition 4. Let Fq be a finite field with q prime. A linear code C is a k dimensional subspace of the vector space Fqn and is referred to as an [n, k] code. The code is said to be of length n and of dimension k. In our case q is the number of possible states per spin. Definition 5. A k × n matrix whose rows are a basis for C is called a generator matrix for C. Recall from Definition 2 that CMM(Γ ) is a (|V | − c(Γ )) × |E| matrix. The |E| columns of CMM(Γ ) reflect the cycle structure of the given graph via linear independence in the vector space Fqn (see Appendix 7). We now view the |V | − c(Γ ) rows of the CMM as generating an [n = |V |, k = |E| − c(Γ )] “cocycle code” C: Definition 6. The cocycle code C(Γ ) of a graph Γ is the row space of CMM(Γ ) [27]. We focus our attention on a subclass known as cyclic codes and a smaller subclass known as irreducible cyclic codes. Definition 7. A linear code C is a cyclic code if for any word (c0 , c1 , . . . , cn−1 ) ∈ C, also (cn−1 , c0 , c1 , . . . , cn−2 ) ∈ C. If C contains no subspace (other than 0) which is closed under cyclic shifts then it is irreducible cyclic. Cyclic codes have an interesting underlying algebraic structure which we review in Appendix 8. In general the generator matrix of an [n, k] cyclic code can be written as ⎛
g0 ⎜0 ⎝0 0
g1 g0 0 0
⎞ · · · gn−k 0 0 ··· 0 · · · gn−k−1 gn−k 0 · · · 0 ⎟ . ··· ··· 0 ⎠ ··· g0 g1 · · · gn−k
(14)
The non-zero matrix elements can be used to construct the “generator polynomial” g(x) = g0 + g1 x + · · · gn−k x n−k . Both can be used to generate a cyclic code; the manner in which this is done via the generator polynomial is reviewed in Appendix 8. For an [n, k] non-degenerate irreducible cyclic code (no words are repeated) the relation between n and k is k = ordq n, i.e., k is the smallest integer such that q k = 1 mod n. Equivalently, q k − 1 = n N , where N counts the number of equivalence classes under cyclic permutations of words, which is an upper bound on the number of different weights. Non-degenerate irreducible cyclic codes have generator polynomials of the form gi (x) =
xn − 1 , wi (x)
where x n − 1 = w1 (x)w2 (x) · · · wt (x) is the decomposition of x n − 1 into irreducible factors. Here t is the number of q-cyclotomic cosets mod n (see Subsect. 2.6). Next we explain the connection to weight enumerators.
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Definition 8. Let C be a linear code of length n and let Ai be the number of vectors in C having i non-zero entries (Hamming weight of i). Then the weight enumerator of C is the bi-variate polynomial A(x, y) =
n
Ai x n−i y i .
i=0
The set {Ai } is called the weight spectrum of the code. In this paper our only concern for the weight spectrum is its connection to the Potts partition function, but in coding theory it can be used to reveal information about the effiency of a code [31]. The connection between Eq. (8) and the cocycle code of the graph Γ for the Potts model is given in the following theorem proved in [27]. Theorem 2. Let A(x, y) be the weight enumerator of the [n = |E|, k = |V | − c(Γ )] cocycle code C(Γ ) of the graph Γ = (E, V ), and let the number of states per spin (vertex) in the corresponding Potts model be a prime q. Then Z Γ (y) = y −n q c(Γ ) A(1, y).
(15)
We take q to be prime and not a power of a prime to simplify matters. In this manner the cocycle code has words whose entries are in Fq , as will the corresponding irreducible cyclic code in the trace representation over Fq r – see Appendix 8 for details. The connection between the Potts partition function and weight enumerators can also be understood via a previous result which shows that Z is equivalent to the Tutte polynomial (under certain restrictions) and that the weight polynomial of a linear code is also equivalent to the Tutte polynomial [32]. We also note that a relation similar to Eq. (15) was established in [5] for the Ising spin glass partition function and so-called quadratically signed weight enumerators, along with a discussion of computational complexity. 2.3. Testing the graph for membership in the ICCC class. We now have the tools to address the issue of whether a graph should be accepted as input into the main algorithm, i.e., whether a graph belongs to the ICCC class. This is handled as follows: - Input: A graph Γ with |E| edges and |V | vertices, the given Galois field of q k elements and . - Output: Accept or Reject. Let n = |E| and k = |E| − |V | + c(Γ ) as in the main theorem. - Overall Complexity: O(|E| · k 2 log k log log k) due the ability to take the discrete log |E| times efficiently with a quantum computer [1]. 1. Compute θn,k as given in Definition 1. 2. Find CMM(Γ ). It is a (n − k) × n matrix of the form [In−k |X ], where X is a (n − k) × k matrix. Form the k × n (transpose parity check) matrix H T = [−X T |Ik ]. H T generates an [n, k] code C ⊥ (Γ ) that is dual to the cocycle code C(Γ ). θn,k −1 3. Determine if ≤ q √ and if k is the multiplicative order of q mod n (i.e., k is the 4
qk
smallest integer such that q k = 1 mod n). If both are true then go to the next step. Otherwise skip the next step and continue.
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4. Main Loop: (a) Fix a basis of G F(q k ) over G F(q) and consider the columns of H T as coordinate vectors of some elements gi of G F(q k ). (b) Calculate the discrete logarithms log(gi ) of each gi with respect to a fixed primitive element g (every element in the field can be written as gl for some l) of G F(q k ) using Shor’s algorithm [1] on a quantum computer.2 (c) Accept or Reject Γ based on the fact that C ⊥ is (equivalent to) an irreducible cyclic code if and only if the numbers log(gi ) are some permuted list of consecutive integer multiples of N := (q k − 1)/n in some order. This is due to the fact that by definition the generator matrix of an irreducible cyclic code is equivalent to (1 g N j g 2N j . . . g (n−1)N j ) where gcd(n, j) = 1 [31]. 5. Step c failed. Using elementary row operations transform H T to a block diagonal matrix if possible. If not possible then Reject. If possible then go to Step c and input each sub-matrix and continue.
2.4. Proof of the Main Theorem. 2.4.1. Preliminaries We introduce Gauss sums as this is the vital link between the Potts partition function and quantum computation. Appendix 9 contains a more detailed exposition as well as an outline of a proof that there exists a polylog quantum algorithm for estimating Gauss sums. Here however it is essential only to understand the following. Given a field Fq k , there is a multiplicative and additive group associated with it. Namely, the multiplicative group is Fq∗ k = Fq k \ 0 and the additive group is Fq k itself. Associated with each group are canonical homomorphisms from the group to the complex numbers, named the additive and multiplicative characters. The multiplicative character χ is a function of the elements of Fq∗ k and the additive character is a function of Fq k and is parameterized by β ∈ Fq k . A Gauss sum is then a function of the field Fq k , the multiplicative character χ and the parameter β, and can always be written as G Fq k (χ , β) =
q k eiγ ,
(16)
where γ is a function of χ and β. It is in general quite difficult to find the angle γ . The complexity of estimating this quantity via classical computation is not known but there is evidence that it is hard [26]. We now introduce the trace function over finite fields. Definition 9. Let q be prime, k a positive integer, and let Fq k be the finite field with q k − 1 non-zero elements. The trace is a mapping Tr : Fq k → Fq and is defined as follows. Let ξ ∈ Fq k . Then Tr(ξ ) =
k−1
j
ξq .
(17)
j=0 2 For every non-zero x ∈ F r /{0} the discrete logarithm with respect to a primitive element (i.e., generator) q g of Fq r is given by logg (x) = logg (g j ) = j mod (q r − 1).
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Now let q k = 1 + n N , where n and N are both positive integers, and let γ generate the multiplicative (cyclic) group Fq∗ k = Fq k \{0}. Each of the q k words of an [n, k] irreducible cyclic code may then be uniquely associated with an element x ∈ Fq k and may be written as (Tr(x), Tr(xγ N ), Tr(xγ 2N ), . . . , Tr(xγ (n−1)N )),
(18)
where k is the smallest integer such that q k = 1 mod n. For a proof of this statement see [33] or [34]. As stated in Theorem 1, we are essentially interested in obtaining the weight spectrum of [n, k] irreducible cyclic codes. The number of words with different non-zero weight is at most N , where N = (q k − 1)/n (for a proof see Proposition 1 in Appendix 8). Now let w(x) be the Hamming weight of the code word associated with x ∈ Fq∗ k . The McEliece Theorem connects the weights of words of irreducible cyclic codes to Gauss sums. Theorem 3 (McEliece Formula). Let w(y) for y ∈ Fq∗ k be the weight of the code word given by Eq. (18), let q k = 1+n N , where q is prime and k, n and N are positive integers, let d = gcd(N , (q k − 1)/(q − 1)), and let the multiplicative character χ¯ be given by χ(γ ¯ ) = exp(2πi/d), where γ generates Fq∗ k . (χ¯ is called the character of order d.) Then the weight of each word in an irreducible cyclic code is given by q k (q − 1) q − 1 − χ¯ (y)−a G Fq k (χ¯ a , 1). qN qN d−1
w(y) =
(19)
a=1
For a proof of this see [33]. The important feature here is that if we had the ability to efficiently estimate G Fq k (χ , β), then we would be able to find the weights of the words in an irreducible cyclic code efficiently under the restrictions mentioned in Theorem 1. This would in turn allow us to find the weight spectrum {Ai } of the code. The following theorem reveals that a quantum computer can efficiently approximate Gauss sums. Theorem 4 (van Dam & Seroussi [26]). For any > 0, there is a quantum algorithm that estimates the phase γ in G Fq k (χ , β) = q k eiγ , with expected error E(|γ −γ˜ |) < . The time complexity of this algorithm is bounded by O( 1 · (log(q k ))2 ) [26]. (For details see Appendix 9.) The Gauss sum algorithm allows one to estimate γ in Eq. (16) to within any accuracy , i.e., the algorithm returns γ such that |γ − γ | < . The hope is that if one can approximate γ precisely enough then one would get an exact evaluation of the weight. In fact an essential step here is to use a quantum computer to obtain a list of approximate angles {γt } for t = 1, . . . , d − 1 for d given above. The next theorem gives some minimum distance between weights so that we can choose an appropriate error that will allow one to be able to distinguish between weights, which allows us to obtain accurate coefficients for A(1, y) and hence exact values for the exponents. Theorem 5 (McEliece [35]). All the weights of an [n, k] irreducible cyclic code are divisible by q θn,k −1 , where θn,k is given in Definition 1.
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2.4.2. The proof. We are now ready to prove Theorem 1. Proof. Assume that a given graph Γ = (E, V ) is a member of ICCC , where n = |E| θn,k −1 and k = |E| − |V | + c(Γ ). Hence it is given that ≤ q √ ≡ 0 . We want to obtain 4
qk
Z Γ for either the fully ferromagnetic or anti-ferromagnetic Potts model. It follows from Definition 1 that the dual of the cocycle code of Γ is an irreducible [n, k] cyclic code. We must demonstrate that we can obtain the weight enumerator A(1, y) of this dual code within the claimed number of steps. As mentioned above, since n N = q k − 1 there are then at most N different weights, with at least n words of each weight (see the Appendix). In order to find the spectrum {Ai }, we are faced with the computational task of finding the range of S(i) =
d−1
q k (q − 1) q − 1 − χ¯ (α i )−a q k ei γa qN qN
(20)
a=1
(where again d = gcd(N , q k − 1/q − 1) and i ∈ {0, . . . , N − 1}) and then performing a tally. The proof consists of five main parts: 1. Proof that with bounded by 0 as given, it is possible to distinguish between weights of the words of the code that corresponds to the given graph. This ability allows for an exact evaluation of Z Γ . 2. We need to justify our asymptotic approach and show that for a fixed error there are a countable number of graphs in ICCC . 3. Proof that the success probability δ is as stated in the theorem. 4. Proof that the running time is as stated in the theorem. 5. A transformation from the dual (irreducible cyclic) code to the cocycle code of the graph whose Potts partition function we are evaluating. Let us now prove each of these five parts. 1. The first question we must address is the following: how small do we need to make the error in the phases returned in the Gauss sum approximation algorithm so that we will be able to distinguish between weights? We now show that ≤ 0 is sufficient, and hence that for every member of the class ICCC it is possible to distinguish between weights. be the approximated weight returned by the quantum Gauss sum algorithm. Let w(y) It follows from Theorem 5 that two consecutive weights are separated by a distance that is an integer multiple of q θn,k −1 . Hence, a sufficient condition for being able to associate with the correct weight w(y) (and not another neighboring weight) is: w(y) θn,k −1
|γi − γi | < . Let us derive a bound on . Taking w(y) given in Theorem 3 and the necessary bound given in Eq. (21) we find that we need the inequality q N q θn,k −1 1 −a iγa −a i γa χ(y) ¯ e − χ(y) ¯ e < (22) (q − 1) 2 qk a a
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to be satisfied. Now, we have iγa −a iγa −a i γa χ(y) ¯ e − χ(y) ¯ e ≤ e − ei γa a a a ≤ (|cos(γa )−cos(γa ))|+|(sin(γa )−sin(γa ))|) a
≤ 2(d − 1) |γa − γa | < 2(d − 1), where the last inequality follows from the Mean Value Theorem of elementary calculus. Therefore, if we impose <
q θn,k −1 1 qN , (q − 1)(d − 1) 4 qk
then inequality (22) is satisfied. Consider the factor
qN (q−1)(d−1) .
(23) Noting from N ≤ αk s
qN that d = gcd(N , (q k − 1)/(q − 1)) ≤ αk s = N , it follows that 1 < (q−1)(d−1) ≤ q s(k) ). Thus we can replace the bound (23) by the tighter bound N = O(k q−1
<
q θn,k −1−k/2 = 0 , 4
(24)
and this is definitely small enough to satisfy the required bound given in Eq. (21).3 for different words y. This, in Hence, if < 0 it is possible to resolve the weights w(y) turn, gives us the ability to exactly reconstruct the weight enumerator A, and from there the partition function Z Γ . 2. We prove the following lemma. Lemma 1. Given a fixed < 1 there are countably many graphs in ICCC , i.e., there are infinitely many corresponding irreducible cyclic codes [n i , ki ] such that {θn i ,ki } satisfies 1 θn ,k −1− ki 2 > . q i i 4
(25)
What this means is that there is at least one family of graphs for a given fixed for which one will be able to obtain the exact Potts partition function. This also justifies the complexity arguments used herein. Proof. We shall construct one such family and show that it satisfies the required relations. For simplicity take = 4. We must construct one family of graphs for which the corresponding irreducible cyclic codes, [n i , ki ], satisfy θn i ,ki + logq ( −1 ) > 1 +
ki . 2
(26) θ
−1
3 Note, however, that when d = 2, we have in fact < k s(k) , where = q n,k , and in this 0 0 4 qk 1 2s(k) (log q)2 ) to case the computational cost of the algorithm (see Theorem 1) is scaled down from O( k 0 1 s(k) 2 O( k (log q) ), where the upper bound (24) still applies. This means that within the family ICCC0 some 0 instances can be solved faster than others by a factor of k s(k) , at fixed 0 .
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Take q fixed and consider the following countable set of irreducible cyclic codes: {[q m − 1, km ]}m=1,2,3,... . First we must note that θq m −1,km = m.
(27)
This follows from the properties of addition in base q: the k digits of q k − 1 in base q are all (q − 1), and adding integer multiples of q k − 1 will not decrease the digit sum. i.e., S (η(q m − 1)) ≥ m(q − 1) ∀η ∈ N. This is important to keep in mind when we consider extending this family later in this proof. Now we must demonstrate that there is at least one km that satisfies Eq. (26). Because we are dealing with irreducible cyclic codes we must have q km = 1modn = 1mod(q m − 1). This is trivially satisfied by km = m and indeed Eq. (26) becomes m − m2 > 1 + logq , which is clearly true for any < 1. This family is computationally trivial, however, being that N = 1. Let us now extend this family to include many interesting instances. Let us first consider a fixed code [q m − 1, km ] (i.e., N = 1, m fixed). Let us next generate a family of codes {[η j (q m − 1), km j ]} j=1,2,...,M by taking integer multiples η j of q m − 1, and picking km j ≥ km such that η j (q m − 1)N = q km j − 1 (this is just the irreducible cyclic code condition n N = q k − 1). We obtain a finite set of codes (M < ∞) because it follows from Eq. (25) that eventually the {km j } j will become too large for the fixed error , for each m. We then do this for every m ∈ N paying special attention to the integer multiples η j . The η j are selected in this construction so that two conditions are satisfied: (i) the corresponding {km j } j are sufficiently small to ensure that Eq. (25) is satisfied, (ii) s )} that N is bounded by {O(km j m, j . Regarding (i), the steps above are conveniently summarized as the following loop: Given : 1. For m = 1, 2, . . . 2. Repeat j = 1, 2, . . . n := j (q m − 1) calculate km j =ordq (n) θ −1−km j /2 if q n,km j < then reject km j , otherwise accept km j and let η j ≡ j. Until j = M Note that we are guaranteed to find such a non-empty finite set {km j } j due to the fact that if gcd(q, η j (q m − 1)) = 1, then ∃ km j ∈ N such that q km j = 1 mod η j (q m − 1) (see, e.g., Th. 7-1 of [36]). Regarding (ii), we still need to show that there exist solutions Nm j to q km j −1 = n Nm j s ). To see why such solutions exist consider solving q k − 1 = n N that scale as O(km j with N = αk s and n = η(q m − 1), where α ∈ R (we have dropped the subscripts for simplicity). The solution is m = logq [(q k − 1)/(αηk s ) + 1].
(28)
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In the loop above, only those m’s satisfying Eq. (28) are acceptable in terms of the scaling of our algorithm. However, note that asymptotically Eq. (28) yields m = k − s logq k − logq αη. This means that for every value of k and s it is possible to adjust α such that m is an integer by letting s logq k = logq αη. At this point we have constructed an infinite family of pairs [n, km j ] [where n = j (q m − 1) and where m satisfies Eq. (28)], each of which defines a graph which is a member of the set ICCC . Finally, we should mention without proof, that one can “fill” this family of graphs by considering the multitude of cases which do not conform to the restrictions in this construction, but which do obey relation (26) and obey the asymptotic conditions given in definition (1). Moreover, the graphs we have constructed are quite sparse but they are only a subset of ICCC . There are many more interesting graphs that can be handled by this fixed error bound. For example, graphs which are the direct sum of many copies of a smaller graph are excluded from this family. Further one may accept an error that decreases polynomially in k for example and define a family of graphs in that way. We do not pursue this here. 3. In the van Dam-Seroussi algorithm (Theorem 1 in [26]), a prepared state must must go through a phase estimation. In [37] it is demonstrated that if the number of qubits used in phase estimation is t = log 1/ + log(2 + 1/(2δ)) then the probability of success is at least 1 − δ. Ref. [38, p. 7] states that for the Gauss sum algorithm t = 2 log(q k − 1). After some elementary algebra we obtain δ = [2((q k − 1)2 − 2)]−1 . By the Chernoff bound, for fixed problem size k, we only need to pick such that the probability of failure δ is less than 1/2. 4. (a) We have that if α is a generator for Fq∗ k and if i = j mod n, then the code words associated with α i and α j are cyclic permutations of each other, and therefore are of the same weight. Let us denote by [α i ] the (equivalence) class of all words {α j } j with i = j mod n. In this step we wish to find the weight of [α i ]. This weight is given by S(i) =
d−1
q k (q − 1) q − 1 − χ¯ (α i )−a q k ei γa . qN qN
(29)
a=1
Hence (up to irrelevant classical computations) the computational cost of computing S(i) is d − 1 times the cost of computing γa . For any graph in ICCC , obtaining these d − 1 phases has a (quantum) cost of O(dk 2 (log q)2 ), where d is bounded above by N . This comes from the complexity of computing the Gauss sum d times. (Recall that one has to repeat this algorithm 1/ times in order to ensure that we obtain a sufficiently close approximation.) (b) How many times must we compute S(i)? The number of times is the number of different equivalence classes {[α i ]}. Each equivalence class [α i ] is clearly of size n, and there are q k − 1 words. Recall that n N = q k − 1 for non-degenerate irreducible cyclic codes, and hence N is the number of different equivalence classes. (Actually the answer to “How many times must we compute S(i)?” is that one must only do this for the number of cyclotomic cosets of N – see Subsect. 2.6). (c) For given S(i) we must compute a sum over d terms. The cost of computing each such term is constant once we have obtained the phases γa [which we have, in Step (a)]. Combining this with Step (b), we see that the total cost of computing all S(i)’s is (d − 1)N . At this point the total computational cost is therefore max[O(dk 2 (log q)2 ), O(d N )]. We choose N = O(k s(k) ) so if one takes s(k) to be a constant, then the algorithm is
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polynomial in k. Thus, the overall time complexity is O(d · k max[2,s(k)] (log q)2 ). Being that d ≤ N = O(k s(k) ), the complexity is ultimately O(k 2 max[1,s(k)] (log q)2 ). (d) We now have the list {S(i)}. Next, a tally of all the weights has to be done which has complexity O(k s(k)/2 ) using quantum counting [39]. The tally will return all the weights and counts of each weight (see Sect. 4) which are the exponents and coefficients respectively, of the polynomial A(1, y) which is the weight enumerator of the dual of the cocycle code. Note that this step does not effect the overall complexity of the algorithm as it has a smaller running time than the previous steps. 5. Note that so far we have dealt with the [n, k] irreducible cyclic code that is the dual of the cocycle code of Γ , i.e., we have used n = |E| and k = |E|−|V |+c(Γ ). However, recall that Γ = Γ (E, V ), and hence corresponds to the [n, n − k] = [|E|, |V | − c(Γ )] code, i.e., the cocycle code of the graph Γ as desired. (This correspondence means that we can obtain information about interesting graphs by considering codes of smaller dimension.) Thus, in order to complete the proof we need the weight enumerator of the [n, n − k] cocycle code itself, so that we can apply Theorem 2. The relation between the weight enumerator A of a code C over the field Fq k , and the weight enumerator A⊥ of the dual code C ⊥ is given by the MacWilliams Theorem [31]: n A⊥ (1, x) = q k(k−n) 1 + (q k − 1)x A(1, y), (30) where
x≡
1−y 1 + (q k − 1)y
.
(31)
Applying the MacWilliams theorem and Barg’s theorem [specifically Eq. (15) to A⊥ (1, x)], we arrive at the partition function Z (x) = x −n q c(Γ ) A⊥ (1, x) . e−β J
(32)
(where β = ); thus we have the following final expression for Recall that y = the partition function as a function of β: n Z (x(β)) = q c(Γ )+k(k−n) (q k − 1) + x(β)−1 A(1, y(β)). (33) 1 kB T
It is simple to verify that given any temperature T ≥ 0, and for both positive and negative J , Z (x(β)) is always positive, as it should be. 2.5. Proof of the corollary. We now give the proof of Corollary 1. Proof. Assume that we are given a graph Γ (E, V ) whose CMM is the direct sum of the CMMs of two graphs Γ1 and Γ2 in ICCC (we call such a graph Γ a “composite graph”). Let C be the code that corresponds to the graph Γ , i.e., C is the cocycle code of Γ . Let C1 and C2 be the corresponding cocycle codes of Γ1 and Γ2 . This means that we may apply our algorithm to each of these sub-graphs and obtain their weight enumerators. To do this we need to obtain the weight enumerators of C1 and C2 which we can do efficiently. By definition C = C1 ⊕ C2 . If the respective lengths and dimensions of C1 and C2 are [m, l] and [m , l ], then C is an [m + m , l + l ] linear code and its weight enumerator will be W = W1 W2 [31]. Thus, once one obtains the weight enumerators of the sub-graphs, one has the weight enumerator of Γ and by using the arguments already outlined one can see that we can efficiently compute Z Γ .
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The above corollary allows the scheme outlined in this paper to be efficiently applied to many graphs because if one knows the generator matrices for C1 and C2 then one can efficiently construct the generator matrix for C by just taking the direct sum of the matrices. This gives a way of constructing examples of graphs for which the Potts partition function can be efficiently approximated. On the other hand (recall Subsect. 2.3), we can efficiently check if a generator matrix decomposes into a direct sum of smaller matrices and we can efficiently check if these matrices generate codes whose duals are irreducible cyclic. 2.6. Reducing the computational cost of the algorithm via permutation symmetry. We now briefly review the concept of p-cyclotomic cosets. As an introduction see [31]. Consider the set of integers {0, 1, 2, . . . , N − 1} and take p a prime number such that p does not divide N . The p-cyclotomic cosets of this set are given by the collection of subsets {0}, {1, p, p 2 , . . . , pr (1) }, . . . , { j, j p, j p 2 , . . . , j pr ( j) }, . . . , where j is an integer and r ( j) is the smallest integer such that j ( pr ( j) − 1) = 0 mod N , i.e., r ( j) is the smallest integer before one begins to get repeats in the coset indexed by j. The number of cosets is finite so j is finite. As an example consider N = 16 and p = 3. One obtains {0}, {1, 3, 9, 11}, {2, 6}, {4, 12}, {5, 15, 13, 7}, {8}, {10, 14}. With regards to the scheme presented in this paper, we take q-cyclotomic cosets. We are guaranteed that gcd(N , q) = 1, which ensures that in our case the cyclotomic cosets are disjoint. That gcd(N , q) = 1 is due to the fact that there are solutions x, y ∈ Z k to N x + qy = 1 (Thm. 2-4 of [36]). For example, since N = q n−1 , one can take x = n(q − 1) and y = 1 + q k−1 − q k , which are both integers. The relevance of the q-cyclotomic cosets of {0, . . . , N − 1} is that each element in a given coset has the same value of S(i). This is because of the fact that the mapping x → x q is a permutation of Fq k and that the additive characters obey the identity exp(2πiTr(bq )/q) = exp(2πiTr(b)/q) for all b ∈ Fq k . Hence S(i) is invariant under the mapping x → x q . (See Appendix 9.1 for details on additive characters and the trace function Tr.) Therefore we only have to evaluate S(i) for one i in each coset. The computational cost of computing the coset representatives and the number of elements in each coset is linear in N [40]. This has the potential of significantly speeding up the algorithm, but how much will clearly depend on the number of cosets generated by each instance. The number of cosets is given by [41] φ( f ) NC = , (34) ordq f f |N
where φ( f ) is the Euler totient (the number of positive integers which are relatively prime to f and s = ordq f means that s is the smallest positive integer such that q s = 1mod f ). Note that NC replaces N in the overall computational cost of our algorithm and NC ≤ N . While this can lead to a significant speedup in some cases, for the sake of simplicity and of having uniform bounds we will not pursue this further here. As an illustration of the power of using cyclotomic cosets, consider the following numerical example. Let q = 2, 1/ ≥ 8192, and consider a binary [113, 85] code which
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is the dual to a binary [113, 28] irreducible cyclic code (i.e., 28 is the smallest integer such that 228 = 1mod113). This corresponds to either the fully ferromagnetic or fully anti-ferromagnetic Ising model on a graph with 113 edges and 86 vertices. Now note that n N = 228 − 1, which implies that N = 2375535. Without the use of cyclotomic cosets this value of N would set our computational cost in that it is the number of times that S(i) must be queried. However, it turns out that there are NC = 85439 cyclotomic cosets, and this is the actual number of queries to S(i). Note that there are instances where N n and cyclotomic cosets are not required. For example consider the binary [13981, 20] irreducible cyclic code. Here n = 13981 and N = 75. Physically this corresponds to either the fully anti-ferromagnetic or ferromagnetic Ising model over a connected graph with 13981 edges and 13962 vertices (considering the dual code).
3. Classical and Quantum Complexity of the Scheme Assuming one knew that a given graph was a member of ICCC , then classically one could proceed as follows using a state of the art algorithm ZETA for the computation of zeta functions of the family of curves Cα : y q − y = αx N [24]. Here N is as given in the relation n N = q k − 1 and the index α is in one-one correspondence with the code words in the given cocycle code (specifically α ∈ Fq k ). The connection between the weights of words of an irreducible cyclic code and the number of rational points on the curves Cα is well known, as is the connection between the zeta functions of such curves and Gauss sums [42]. The complexity of using ZETA to compute the N = αk s(k) different weights 5+ is O(k 6s(k)+3+ q2 ) [24] and a tally of these weights will take O(k s(k) ) operations ( is a small real number – unrelated to which parameterizes the class of graphs in question). The overall complexity of finding the range of S(i) will therefore be 5+ 6s(k)+3+ q classical cost = O k , 2
(35)
assuming that we know that a given graph is a member of ICCC . As far as we know this is the fastest classical algorithm for the problem we have considered here. For a quantum computer we do not need to assume that testing for membership is efficient: we know that this can be done efficiently using the discrete log algorithm [1]. Above we showed that the overall complexity of finding Z is bounded by O(k 2 max[1,s(k)] (log q)2 ). This should be contrasted with the best classical result available, (35). For example if we take s = 2, (both classical and quantum methods are polynomial when we take s(k) to be a constant) we obtain an O(k 11 ) improvement and an exponential speedup in q. One could imagine fixing a graph and calculating the partition function for increasing values of q. In this situation we have an exponential speedup over the best classical algorithm available. Note that there is a quantum algorithm for finding zeta functions of curves which is exponentially faster in q than the classical algorithm in [24] (as is ours). This is given in [25]. The use of this algorithm instead of the Gauss sum approximation algorithm is left for a future publication. On a final note, the classification ICCC we have chosen is meant to highlight the boundary between B Q P and P by fixing the acceptable error in the Gauss sum phases. One could opt for a perhaps more natural class of graphs by bounding the way that 1/
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grows instead. For example, one could restrict the class of graphs in such a way that k 1/ ∼ q 2 −θn,k +1 grows polynomially in k, in particular such that 1 < k 5s(k)+1 . For this class of graphs one would also have a speedup in the quantum case. 4. Detailed Summary For convenience we recollect our definitions and provide a diagram of our scheme. We are considering the q-state Potts model (fully ferromagnetic or fully anti-ferromagnetic) over a graph Γ = (E, V ), with q prime. This includes the Ising model (q = 2). Every graph Γ has a cycle matroid M(Γ ) associated with it and every cycle matroid has a (|V | − c(Γ )) × |E| matrix representation G (the CMM), where c(Γ ) is the number of connected components of Γ . The columns of G encode the cycle structure of the graph and the row space of G generates the cocycle code of length |E| and dimension |V | − c(Γ ). The length and dimension of the dual code are respectively n = |E| and k = |E| − |V | + c(Γ ). Following is a detailed synopsis of the algorithm for computing the partition function. 1. Given a graph, efficiently determine if it belongs to ICCC (Definition 1). This step appears to be hard on a classical computer in general, since it is equivalent to computing a discrete log. 2. If the CMM G = [I|V |−c(Γ ) |X ] is the matrix representation over Fq k of the cycle matroid of Γ , M(Γ ), then the row space of H = [−X T |I|E|−|V |+c(Γ ) ] will be the code C(Γ ). 3. Let N = O(k s ), where s is a constant integer that determines the complexity of k the algorithm. Take C(Γ ) as an irreducible cyclic code of length n = q N−1 and dimension k, i.e., we only consider graphs Γ , where C(Γ ) is an irreducible [n, k] cyclic code. 4. If we can evaluate the weight enumerator of C(Γ ) we will have successfully approximated the Potts partition function over the corresponding graph Γ . To do so: (a) Find the q-cyclotomic cosets of {0, 1, . . . , N − 1}. This step requires at most linear time in N . (b) Using the quantum algorithm for Gauss sums [26] we are be able to estimate the weights of the words. The error in the Gauss sum algorithm can be high in this setting, and therefore we have to restrict the class of graphs further in order to obtain exact evaluations. Use the Gauss sum algorithm to return the phases γ1 , . . . , γd−1 [Eq. (16)] and then input these values into the function S(i) [Eq. (20)]. According to the McEliece Theorem (Th. 3) we have to make k −1 d − 1 (where d = gcd(N , qq−1 )) calls to the quantum oracle and we can use these evaluations for each representative i of the q-cyclotomic cosets of {0, 1, . . . , N − 1}. This step has time complexity O(dk 2 (log q)2 ). (c) Let b1 , b2 , . . . , b NC be the coset representatives from the NC cosets. Now each coset has cardinality vi , i.e., bi belongs to coset i which has vi elements. We evaluate ωi = S(bi ) for each bi , remembering that each ωi occurs vi times. We end up with a list (ω1 , ω2 , . . . , ω NC ) as well as a list (v1 , v2 , . . . , v NC ) of multiplicities.
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Fig. 2. A diagrammatic overview of the algorithm. (Box shapes do not have a meaning.)
(d) Now perform a tally of repeats of the ωi for each i ∈ {1, . . . , NC }. This returns a set of indices Λi ≡ { ji } ⊆ {1, . . . , NC }. We add the corresponding v ji which yields ai = j∈Λi v j , the number of words of weight ωi up to cyclic permutations. To account for cyclic permutations due to the fact that we are working over cyclic codes, we have Ai = nai , which is the desired weight spectrum. 5. Now that we have determined the weight spectrum Ai in time O(k 2s (log q)2 ), we have the coefficients for A(1, y) and so via the MacWilliams identity (30) we finally obtain the partition function (33).
5. Examples and Discussion In this section we provide the reader with some simple examples for illustrative purposes. 5.1. Example. Consider the graph depicted in Fig. 3. This graph depicts three spins, one of which has a self-interaction. It can be verified that this graph corresponds to the dual of a [4, 2] irreducible cyclic code over G F(3), i.e., q = 3. The generator matrix for this code is given by 0111 . 1012 We see that the corresponding graph must have 4 edges and 3 vertices (if the graph is connected), and this is the reason for having the spin with the self-interaction. The second, third, and fourth columns correspond to a triangle (as they sum to zero modulo
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Fig. 3. A graph corresponding to a [4, 2] linear code over G F(3)
3) and the first column is the loop at one of the vertices. The self-interaction can be removed once the partition function has been obtained via a simple procedure described below. We need to find the weight spectrum for this code. After forming the weight enumerator using MacWilliams identity, we apply Barg’s theorem which will give the q = 3 Potts partition function for this graph. 1. Using a quantum computer we evaluate the necessary Gauss sums. From the identity q k − 1 = n N (necessarily satisfied by irreducible cyclic codes) we see that N = 2. This means that there can be at most two different weights (in fact the number of non-zero cyclotomic cosets is one). 2. Compute the number of times that one must repeat the quantum algorithm for Gauss sums in order to obtain an acceptable accuracy. We see that this number is given by √ 1 4 32 = θ −1 . 3 n,k Since 4 = 11 base 3 we have that θn,k = 21 [1 + 1] = 1 and so 1 = 12. This means that the algorithm must be repeated 13 times to ensure the desired accuracy. 3. After evaluating the Gauss sums and plugging them into Eq. (19), we obtain two weights: 0 and 3. 4. As only one word can have zero weight, the remaining 32 − 1 words have weight 3. This means that we have the weight enumerator A(1, y) = 1 + 8y 3 . 5. Using relation (33) derived earlier, we find that for this graph 4 1 Z (x(β)) = 8 + x(β)−1 [1 + 8y 3 (β)], 27 1−y(β) , y = e−β J , and β = k B1T . where x(β) = 1+8y(β) 6. At this point we can remove the self-interaction by dividing Z (x(β)) by y. This is due to the following theorems.
Theorem 6. Let T be the Tutte polynomial. If e is a loop then T (M; x, y) = yT (M − e; x, y), where M − e is the matroid (or graph) with the loop deleted [6].
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Theorem 7. A(1, y) = y
n−k
(1 − y) T k
1 + (q − 1)y 1 . M; , 1−y y
This is known as Greene’s identity and one can see either [27] or [6] for details. Putting these theorems together one finds that 1 + q − 1y 1 A M−e (1, y) = y (1 − y) T M − e; , 1−y y 1 1 + (q − 1)y n−k k , = y (1 − y) yT M; 1−y y = y A M (1, y),
n−k
k
(36) (37) (38)
and therefore we find that the partition function for the triangle is then given by Z (x(β)) =
4 1 8 + x(β)−1 27
1−x 1 + (q k − 1)x
[1 + 8y 3 (β)].
One should note that due to Corollary (1), we could form a string of these triangle graphs as shown in Fig. 4, and easily compute the partition function by multiplying the above partition function with itself three times (the number of copies of the triangle in the chain). This property is shared by all instances of the Tutte polynomial defined over direct sums of matroids. We can extend this to certain types of recursively defined graphs [46] by forming chains made of multiple copies of different graphs. We note however that recursively defined graphs [46] do not always fit into our construction because they may not be members of ICCC . For example, consider Fig. 5. This is known as a ladder graph and it is an example of a recursively defined graph. This graph corresponds to a [8,17] binary linear code which is not irreducible cyclic, nor dual to one [27].
Fig. 4. Chaining of the graph in Fig. 3
Fig. 5. A ladder graph illustrating a recursively defined graph. This graph corresponds to a [8,17] binary linear code
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5.2. Degenerate cyclic codes. Here we introduce an approach to construct examples that will help to classify the types of graphs that our scheme is tailored for. The motivation for this is to clarify the relationship between graphs and codes in the sense used in our scheme. The problem is the fact that many of the irreducible cyclic codes have duals that are not graphic in the sense of cycle matroids. We ask the following question: Given an irreducible cyclic code whose dual is not graphic (and hence does not correspond to a physical Potts model), can we find another code whose dual has a weight spectrum that is simply related to the original code, and which is graphic? We provide some arguments in favor of this idea. There exist codes whose words consist of several repetitions of a code of smaller length. Of particular interest to us is a class of degenerate codes related to irreducible cyclic codes in the following way. Lemma IV.2 in [45] states that a code of length n is degenerate if w(x)|x r − 1 [i.e, w(x) divides x r − 1] for some r |n, where w(x) is the check polynomial (see Appendix 8). In the case of irreducible cyclic codes the check polynomial is the denominator of the generator polynomial introduced earlier, given by r −1 g(x) = xw(x) . This means that if we have an [r, k] irreducible cyclic code with check polynomial w(x), we find some n such that w(x)|x n − 1 such that r |n. We then have n −1 a degenerate linear [n, k] code generated by xw(x) . The words in the degenerate code will look like (c , c , . . . , c ), where c is a word in the non-degenerate code. This means that once we know the weight distribution of the [r, k] code, we can easily construct the weight enumerator of the [n, k] code since the weights of the words of length n will be n/r times the weight of the corresponding word of length r . This construction allows one to loosen the constraints on the dimension and length and therefore on the number of vertices and edges of the corresponding graph. In other words, for many of the codes whose corresponding cycle matroids are not graphic we may use this construction to map these instances to graphic matroids. The definition for ICCC can be easily tailored to include these graphs as will be done in future work. As an example consider the [4, 5] irreducible cyclic code whose check polynomial is w(x) = 1 + x + x 2 + x 3 + x 4 . The dual of this code is non-graphic, because it requires forming a cycle of five edges with only two vertices. Now notice that w(x)|x 15 − 1 and 15 −1 . The dual of this code is a 5|15. In this way we form the [4, 15] code generated by xw(x) [11, 15] code and the corresponding graph is given by Fig. 6. The weight enumerator of the [4, 5] code is A(1, y) = 1 + 10y 2 + 5y 4 and the weight enumerator of the degenerate code is 1 + 10y 6 + 5y 12 . Note that the exponents are just multiplied by n/r = 3. The structure of this graph gives one a clue as to the structure of the types of graphs addressed by our approach. They will be graphs which consist of several repetitions of simple cycles of different lengths. In the example above all the simple cycles have length six, as can be seen in Fig. 6. As one explores codes with higher n, one finds that there will be multiple simple cycles of different lengths that will form the corresponding graph. The reason to believe this to be true in general comes from the fact that the weights of the code C correspond to the size of sets of linearly dependent columns of the generator matrix of the code dual to C. For example, the minimum weight of a code C is the size of the smallest set of linearly dependent columns of the code’s parity check matrix, which can be used as the generator matrix of the code dual to C. On the other hand, the length of the cycles (number of edges) are given by the weights or sums of the weights. The relation between codes and graphs is not yet well understood and future work in this regard based on our approach will hopefully reveal new results that will have applications to both statistical mechanics and knot theory.
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Fig. 6. Example of a graph corresponding to a [11, 15] code related to the [4, 5] irreducible cyclic code
6. Conclusions and Future Directions In this work we have given a quantum algorithm for the exact evaluation of the fully ferromagnetic or anti-ferromagnetic Potts partition function Z under the restriction to certain sparse graphs (with logarithmically more edges than vertices). The methods we used exploit the connection between coding theory and statistical physics. The motivation for this work is an ongoing effort to identify instances of classical statistical mechanics for which quantum computers will have an advantage over classical machines. The approach we described involves using the link between classical coding theory and the Potts model via the weight enumerator polynomial A. One should note that A is another instance of the Tutte polynomial and so this connection is not surprising. The weight enumerator encodes information about all the different Hamming weights of the code words in a linear code and the weight of a code word can be given by a formula involving a sum of Gauss sums when dealing with a specific type of linear code. Since there exists an efficient algorithm to approximate Gauss sums via quantum computation [26] we were able to efficiently calculate the weights of code words for certain codes. Much of this paper dealt with the necessary restrictions that one must impose in order to achieve this last step. For example, once an error in the Gauss sum algorithm is accepted, we demonstrated that there is a family of graphs for which one can find the exact partition function, and therefore the error does not scale within this family. Given a graph Γ , one can map the graph to a corresponding linear code via the incidence structure of Γ . The Potts partition function of Γ (with either fully ferromagnetic or anti-ferromagnetic interactions) is given by some easily computed function times the weight enumerator of the corresponding code. Due to the symmetries inherent in the mathematical structure of linear codes we were able to provide an efficient method to exactly determine Z for a class of graphs (ICCC ) which has a well defined correspondence to a subset of linear codes. In [43] it was shown that the exact evaluation of weight enumerators for binary linear codes is hard for the polynomial hierarchy. As our approach involved the exact evaluation of weight enumerators, it is not surprising that we had to make restrictions on the class of graphs so as to make our scheme efficient. The vantage that coding theory gives to this particular problem, however, allows one to utilize the fact that certain graphs have properties that a quantum computer can take advantage of to provide a speed up.
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Notice that the related results in [17,21] concern additive approximations; the methods used in this paper can be extended to a wider class of graphs if one relaxes the requirement of exact evaluation and instead similarly considers additive approximations of Z . An open question is what instances of the Potts partition function are amenable to an fpras (fully polynomial random approximation scheme). The methods used in [17,21] have proven to be quite powerful. There is hope to extend some of these methods to non-planar graphs. One idea is to extend the algorithm in [17] to the Jones polynomial for virtual knots and then use some correspondence between the virtual knots and nonplanar graphs. Another approach may involve seeing things in a new light. Note that the Jones polynomial is the Euler characteristic of a certain chain complex [44]. One can explore how effective quantum computers will be at approximating Euler characteristics in general. Perhaps there is a way of exploiting this in order to obtain knowledge about the Potts partition function. One may also consider strengthening the results given here by exploiting theorems about the minimal distance of cyclic codes. For example, there are theorems that guarantee a lower bound for the weight between any two words. By enforcing that the generator polynomial of the code be of a certain form, one would be guaranteed a certain distance between words and therefore the error in the Gauss sum approximation will be of little consequence for certain graphs [31]. As already mentioned in the Introduction, another potentially promising approach is to consider the scheme we have presented here but to replace the Gauss sum algorithm with the quantum algorithm for obtaining Zeta functions [25]. Work has to be done on understanding the exact cost of this algorithm when one is restricted to curves that are pertinent for the evaluation of the Potts model. Corollary 1 deals with the combination of graphs via a direct sum of codes and gives one a way of “tiling” graphs for which one knows the partition function. This gives a quick way of obtaining the partition function of certain graphs that are made of many repeats of a simpler graph. There are other ways of combining codes that may allow one to study the partition function of new graphs, for example the concatenation or direct product of two codes [31]. The coding theoretic approach does give us a way of evaluating the partition function of instances of the Potts model at arbitrary temperatures but precisely the kinds of graphs which are involved is a question for future research. Indeed, the identification of the physical instances represented by the graphs for which our algorithm is efficient will shed light on the question that motivated this work in the first place [5]: what is the quantum computational complexity of classical statistical mechanics? Acknowledgement. J.G. would like to thank Marko Moisio, Ravi Minhas, and Frank van Bussel for helpful discussions. D.A.L. gratefully acknowledges support under ARO grant W911NF-05-1-0440.
Appendix 7. Matroids Definition 10. A matroid M on a set E is the pair (E, I ), where I is a collection of subsets of E with the following properties: 1. The empty set is in I . 2. Hereditary Property: If A ∈ I and B ⊂ A, then B ∈ I . 3. Exchange Property: If A and B are in I and A has more elements than B, then ∃a ∈ A such that a ∈ / B but B ∪ {a} ∈ I .
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The collection of sets in I are called the independent sets and E is referred to as the ground set. Definition 11. A cycle matroid of a graph Γ is the set of all edges of Γ as the ground set E together with I as the subsets of E which do not contain a cycle. So the independent sets are collections of edges which do not have cycles. Recall that in graph theory one refers to such an edge set (the above independent set) as a forest. In matroid theory a matrix representation is a matrix whose column vectors have the same dependence relations as the matroid it is representing. More clearly, the column vectors represent the matroid elements and the usual notion of linear dependency determines the dependent sets and therefore the independent sets as well. Thus, the matrix can be said to generate the matroid. As an example, imagine the triangle graph of three nodes with three edges A, B, and C. The cycle matroid consists of each of the edges individually and any collection of two edges. All three edges form a cycle so it cannot be included. We require our matrix representation to encode this independence structure of the edges. One may work over any field here because we are only concerned with graphic matroids, i.e., matroids which can be represented as a cycle matroid of some graph. (Graphic matroids are representable over any field [29].) Now, if we think of column 1,2 and 3 as edges A, B and C respectively we can take the following matrix as a representation in F2 : 101 . 011 Since addition is mod 2 here, a cycle is any collection of columns that sum to the 0-vector. We can take all collections where this does not happen and these collections will form I . In this way, this matrix is a representation of the cycle matroid for the triangle graph. In matroid theory one has the familiar notion of a base. Definition 12. A base of a matroid M = (E, I ) is a maximal independent subset of E. It is not a coincidence that the left part of the matrix is the 2 × 2 identity matrix. In general one can form a representation (known as the standard matrix representation) where one begins with an identity matrix which is r × r , where r is the size of the base of M and append to it columns that capture the dependence structure of the matroid in question. In this way, the columns of the identity matrix represent the chosen basis of M. So M is isomorphic to the matroid induced on the columns of the matrix by linear dependence. A more precise explanation can be found in [29]. What is important for us is that such a matrix representation is possible. 8. Algebraic Approach to (Irreducible) Cyclic Codes 8.1. Irreducible cyclic codes as minimal ideals. Let us recall some definitions from algebra. Take q to be prime or a power of a prime. Definition 13. A ring is a set R which is an abelian group (R, +) with 0 as the identity, together with (R, ×), which has an identity element with respect to × where × is associative.
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Definition 14. An ideal I is a subset of a ring R which is itself an additive subgroup of (R, +) and has the property that when x ∈ R and a ∈ I then xa and ax are also in I . Definition 15. A principle ideal is an ideal where every element is of the form ar , where r ∈ R. Thus, a principle ideal is generated by the one element a and a principal ideal ring is a ring in which every ideal is principle. There is an important isomorphism between powers of finite fields Fqn and a certain ring of polynomials. Recall that the multiples of x n − 1 form a principal ideal in the polynomial ring Fq [x]. Therefore the residue class ring Fq [x]/(x n − 1) is isomorphic to Fqn since it consists of the polynomials {a0 + a1 x + · · · + an−1 x n−1 |ai ∈ Fq , 0 ≤ i < n}. Therefore, taking multiplication modulo x n − 1 we can make the following identification: (a0 , a1 , . . . , an−1 ) ∈ Fqn ←→ a0 + a1 x + · · · + an−1 x n−1 ∈ Fq [x]/(x n − 1). (39) This implies the following theorem. Theorem 8. A linear code C in Fqn is cylic ⇐⇒ C is an ideal in Fq [x]/(x n − 1) [31]. Proof. In one direction this is easy since if C is an ideal in Fq [x]/(x n − 1) and c(x) = a0 + a1 x + · · · + an−1 x n−1 is a codeword, then by definition xc(x) ∈ C as well and so (an−1 , a0 , a1 , . . . , an−2 ) ∈ C. In the other direction, one just has to note that since C is cyclic, xc(x) is in C for every c(x) ∈ C which means that x k c(x) is in C for every k. But C is linear by assumption so if h(x) is any polynomial then h(x)c(x) is in C and thus C is an ideal. Note that Fq [x]/(x n − 1) is a principal ideal ring and therefore the elements of every cyclic code C are just multiples of g(x), the monic polynomial of lowest degree in C; g(x) is called the generator polynomial of C. Because of the correspondence (39) above we know that given g(x) = g0 + g1 x + · · · gn−k x n−k [g(x) divides x n − 1 since otherwise g(x) could not be the monic polynomial of lowest degree in C], we have the vector (g0 , g1 , . . . , gn−k ). We then can write the k × n generator matrix of the code as ⎛ ⎞ g0 g1 · · · gn−k 0 0 ··· 0 ⎜ 0 g0 · · · gn−k−1 gn−k 0 · · · 0 ⎟ . ⎝ 0 0 ··· ··· 0 ⎠ 0 0 ··· g0 g1 · · · gn−k In this way, the row space generates C. If we can write x n −1 = w1 (x)w2 (x) · · · wt (x) as the decomposition of x n − 1 into irreducible factors, then the code generated by x n −1 wi (x) is called an irreducible cyclic code. In algebraic terms what this means is that the code C is a minimal ideal of Fq k [x]/(x n − 1), i.e., C contains no subspace (other than 0) which is closed under the cyclic shift operator [34]. The reason we are interested in irreducible cyclic codes is that there is an established connection between the weights of the code words and Gauss sums. For convenience, we also introduce the check polynomial and parity check matrix.
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Definition 16. The polynomial h(x) = polynomial.
x n −1 g(x)
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in an [n, k] cyclic code is called the check
It has earned this name due to the following fact. If a word (v0 , v1 , . . . , vn−1 ) ∈ C then (v0 + v1 x + · · · + vn−1 x n − 1)h(x) = 0
mod x n − 1.
This follows from the observation that every word in C is equal to a polynomial p(x) multiplied by the generator polynomial g(x) and thus we have that (v0 + v1 x + · · · + vn−1 x n − 1)h(x) = p(x)g(x)h(x) = p(x)(x n − 1) = 0
mod x n − 1.
Definition 17. The parity check matrix H of a code C is the generator matrix for the code dual to C. If c ∈ C then H c = 0. We now turn to the representation of irreducible cyclic codes, specifically 1) the form that the generator matrix can take, 2) a description of the codewords in terms of the trace function. Issue 1) relates back to the matrix representation of the cycle matroid of graphs and issue 2) will allow us to make the connection to Gauss sums. 8.2. Generator matrix of a cyclic code and the cycle matroid matrix. There is an alternative (but equivalent) way of constructing the generator matrix of a cyclic code which will immediately show its usefulness in its relationship with the cycle matroid matrix representation. Let C be an [n, k] cyclic code and let g(x) be the generator polynomial. Now, divide x n−k+i by g(x) for 0 ≤ i ≤ k − 1. We have x n−k+i = qi (x)g(x) + ri (x), where deg ri (x) < deg g(x) = n − k or ri (x) = 0. What this means is that we have a set of linearly independent code words. Namely, we have the k code words given by x n−k+i − ri (x) = qi (x)g(x) in C. More explicitly, take the remainder polynomials ri (x) after applying the division algorithm and using the correspondence (39) above, form the k × (n − k) matrix R and append the k × k identity matrix to it. The rows of R are the coefficients of the ri (x) and one then has the k × n generator matrix [Ik |R]. This is precisely the form of the matrix representation for matroids discussed above. Thus, we have a correspondence between the generator matrix for an irreducible cyclic code and the matrix representation for the cycle matroid of a graph. Proposition 1. In an [n, k] irreducible cyclic code there are at most N words of different non-zero weight where N = (q k − 1)/n. Proof. For any irreducible cyclic code we have the relation q k − 1 = n N over the field Fq . The length of each word is n and any cyclic permutation of a word preserves the Hamming weight. Therefore, for each word there are n − 1 other words of equal weight. As there are q k − 1 words of non-zero weight, if we assume that every word that does not arise from the cyclic permutation of another word is of a different weight, then there are (q k − 1)/n words of different weight. Being however that there is the possibility of repeats in weight among words which are not cyclic permutations of each other, there are at most N different weights.
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9. Gauss Sums and a Quantum Algorithm for the Estimation of Gauss Sums Gauss sums are sums of products of group characters. 9.1. Characters. A character of a finite group (G, ∗) is a homomorphism Φ from G to the group of the non-zero complex numbers C. We are interested in two types of characters, namely the multiplicative and additive characters. Let F ≡ Fq k (where k is a positive integer) be a finite field as defined previously, and let F∗ be the multiplicative group of F. Let g be a primitive element of F (i.e., g generates F). Let ζq = e2πi/q denote the q th root of unity. Let x = g k ∈ F∗ . A multiplicative character χ j (x) is a mapping from the set of powers {m} in x = g m to powers of roots of unity. Specifically, the group of multiplicative characters χ = {χ j } j consists of the elements jm
χ j (x) = χ j (g m ) = ζq k −1 , m = 0, . . . , q − 2 ∈ Fq ;
j = 0, . . . , q k − 2 ∈ F.
Let a ∈ F. An additive character e j (a) is a mapping from F to powers of roots of unity via the trace function. Specifically, the group of additive characters e = {eβ }β consists of the elements eβ (a) = ζqTr(βa) ∀a, β = 0, . . . , q k − 1 ∈ F, where the trace is defined in Eq. (17). 9.2. Discrete log. For every non-zero x ∈ F∗ the discrete logarithm with respect to a primitive element g ∈ F is given by logg (x) = logg (g m ) = m mod (q k − 1). This means that every multiplicative character can be written j log (x)
χ j (x) = χ j (g m ) = ζq k −1g
(40)
for x = 0 and χ (0) = 0. 9.3. Gauss sums. Let eβ and χ j be an additive and multiplicative character respectively. Then the Gauss sum G(χ j , eβ ) is defined as: G(χ j , eβ ) =
χ j (x)eβ (x).
(41)
x∈F∗
Gauss sums are used extensively in number theory, e.g., in the study of quadratic residues or Dirichlet L-functions.
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To compute a Gauss sum we need to specify the field F and the indices β ∈ F and j ∈ F of the additive and multiplicative characters respectively. Thus the input size to a Gauss sum computation is O(k log q) bits. We can now define the Gauss sum over F as G F (χ j , β) =
χ j (x)ζqTr(βx) .
x∈F∗
It is well known that if χ j = 1 then [33]: G F (χ j , β) =
iγ qr e ,
(42)
where γ = γF (χ j , β). This means that all we need to do is approximate the angle γ mod (2π ) in order to approximate the Gauss sum. This is precisely the Gauss sum approximation problem for finite fields.
9.4. Quantum algorithm for Gauss sums. Van Dam and Seroussi devised an efficient quantum algorithm to estimate Gauss sums [26]. The following is an outline of the essentials of the proof; we refer the reader to [26] for a complete description as well as a discussion of the complexity of estimating Gauss sums. Theorem 9 {Quantum Amplitude Amplification}. Let f : S → {0, 1} be a function for which we know the total weight f l1 but not those values x ∈ S for which f (x) = 1. Then the corresponding state |f =
1 f (x)|x f l2 x∈S
can be efficiently and exactly prepared on a quantum computer where we have to make |S| a number of queries to f of the order O f l . 1
This is an essential ingredient in Grover’s quantum search algorithm. For a proof and details see [2]. It follows from Eq. (40) and Shor’s discrete log algorithm [1] that given g, q k and j, we can efficiently create the state |χ j . The following lemma is essential in this regard. First note that for any set S we define 1 |S ≡ √ |x . |S| x∈S Lemma 2. For a finite field Fq k and the triplet (q k , g, r ) (the specification of a multiplicative character χr ), the state 1 |χr = χr (x)|x q k − 1 x∈F k q
and its Fourier transform |χˆr can be created in polylog(q k ) time steps on a quantum computer.
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Proof. We first create the state ˆ |Fq∗ k |1
1
= q k (q k − 1)
|x
k −2 q
x∈F∗ k
j
ζq k −1 | j
j=0
q
by using Grover’s amplitude amplification on Fq k and the Fourier transform. Next, in superposition over all x ∈ Fq∗ k , we calculate logg (x) and subtract r logg (x). ˆ |Fq∗ k |1
1
−→ q k (q k − 1)
x∈F∗ k q
j=0
k −2 q
1
= q k (q k − 1) 1 = k q −1
|x
k −2 q
|x
x∈F∗ k
x∈F∗ k q
j
ζq k −1 | j − r logg (x) r log (x)
j
ζq k −1 ζq k −1g
|k
(43)
(44)
j=0
q
r log (x)
ζq k −1g
ˆ |x |1
(45)
ˆ = |χr |1 .
(46)
To get |χˆr we just need to apply the Fourier transform.
The technique used in the above proof is known as the phase kickback trick. Now we are ready for the following. Theorem 10. Algorithm for approximating Gauss sums. Consider Fq k , a nontrivial multiplicative character χr and β ∈ Fq∗ k . If we apply the quantum Fourier transform over this field to |χr , followed by a phase change |y −→ χr2 (y)|y ,
(47)
then we generate an overall phase change given by G Fq k (χr , β) 1 χr (x)|x −→ |χr . |χr = q k − 1 x∈F k qk q
Proof. After a Fourier transform we have
⎛
⎞
⎜ 1 Tr(βx y) ⎟ |χˆr = χr (x)ζq ⎝ ⎠ |y q k (q k − 1) y∈F∗ x∈F k qk
1 = k q (q k − 1)
y∈F∗ k q
q
G Fq k (χr , βy)|y
1 = χr (y −1 )G Fq k (χr , β)|y . q k (q k − 1) y∈F∗ qk
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Then G Fq k (χr , β) χr (y −1 )|y . |χr = k k q (q − 1) y∈F∗ qk
Now we know that we can efficiently (and exactly) create the phase change given by (47). Doing so gives us G Fq k (χr , β) G Fq k (χr , β) |χ ˆ −→ χr (y −1 )χr2 (y)|y = |χr , q k (q k − 1) y∈F∗ qk qk
since |χr = √ 1k
q −1
y∈F∗ k q
χr (y)|y and χr (y −1 )χr (y) = 1. Thus, the coefficient of
|χr is just eiγ . It is well known that one can efficiently estimate the phase of such a function to within an expected error of O(1/n), where n is the number of copies of eiγ |χr we sample. Therefore we arrive at an estimate of γ and hence of the Gauss sum in question. This gives way to the following theorem about the time complexity of the algorithm and is the culmination of the first part of the paper [26]. Theorem 11. For any > 0, there is a quantum algorithm that estimates the phase γ in G Fq k (χr , β) = q k eiγ , with expected error E(|γ − γ˜ |) < . The time complexity of this algorithm is bounded by O( 1 · polylog(q k )) [26]. Note that the “poly” in polylog refers to a quadratic polynomial. References 1. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. on Comp. 26, 1484 (1997) 2. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, New York: ACM, 1996, p. 212 3. Lidar, D.A., Biham, O.: Simulating Ising spin glasses on a quantum computer. Phys. Rev. E 56, 3661 (1997) 4. Swendsen, R.H., Wang, J.-S.: Phys. Rev. Lett. 58, 86 (1987) 5. Lidar, D.A.: On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants. New J. Phys. 6, 167 (2004) 6. Welsh, D.J.A.: Complexity: Knots, Colourings and Counting. Volume 1 of London Mathematical Society Lecture Note Series 186. London: Cambridge University Press, 1993 7. Reichl, L.E.: A Modern Course in Statistical Physics. New York: John Wiley & Sons, 1998 8. Jones, V.F.R.: On Knot Invariants Related to Some Statistical Mechanical Models. Pacific J. Math. 137, 311 (1989) 9. Jones, V.F.R.: A Polynomial Invariant for Knots via von Neumann Algebras. Bull. Amer. Math. Soc. 12, 103 (1985) 10. Jaeger, F., Vertigen, D., Welsh, D.: On the Computational Complexity of the Jones’ and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108, 35 (1990) 11. Kauffman, L.H.: Knots and Physics. Volume 1 of Knots and Everything. Singapore: World Scientific, 2001 12. Alon, N., Frieze, A.M., Welsh, D.: Polynomial Time Randomised Approximation Schemes for TutteGröthendieck Invariants: The Dense Case. Electronic Colloquium on Computational Complexity, 1(5) (1994), available at http://eccc.hpi-web.de/eccc-reports/1994/TR94-005/index.html, 1994 13. Nechaev, S.: Statistics of knots and entangled random walks. http://arxiv.org/list/cond-mat/9812205, 1998
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14. Witten, E.: Topological quantum field theory. Commun. Math. Phys. 117, 353 (1988) 15. Freedman, M.H., Kitaev, A., Wang, Z.: Simulation of topological field theories by quantum computers. Commun. Math. Phys. 227, 587 (2002) 16. Freedman, M.H., Kitaev, A., Larsen, M.J., Wang, Z.: Topological Quantum Computation. http://arxiv. org/list/quant-ph/0101025, 2001 17. Aharonov, D., Jones, V., Landau, Z.: A Polynomial Quantum Algorithm for Approximating the Jones Polynomial. http://arxiv.org/list/quant-ph/0511096, 2005 18. Wocjan, P., Yard, J.: The Jones polynomial: quantum algorithms and applications in quantum complexity theory. http://arxiv.org/list/quant-ph/0603069, 2006 19. Kauffman, L.H., Lomonaco, S.J.: q-Deformed spin networks, knot polynomials and anyonic topological computation. J. of Knot Theory and Its Ramifications 16, 267 (2007) 20. Bonca, J., Gubernatis, J.E.: Real-Time Dynamics from Imaginary-Time Quantum Monte Carlo Simulations: Test on Oscillator Chains. http://arxiv.org/list/cond-mat/9510098, 1995 21. Aharonov, D., Arad, I., Eban, E., Landau, Z.: Polynomial Quantum Algorithms for Additive approximations of the Potts model and other Points of the Tutte Plane. http://arxiv.org/list/quant-ph/0702008, 2007 22. Van den Nest, M., Dur, W., Briegel, H.J.: Classical spin models and the quantum stabilizer formalism. Phys. Rev. Lett. 98, 117207 (2007) 23. Hartmann, A.K.: Calculation of partition functions by measuring component distributions. Phys. Rev. Lett. 94, 050601 (2005) 24. Denef, J., Vercauteren, F.: Counting Points on Cab Curves using Monsky-Washnitzer Cohomology. http:// citeseer.ist.psu.edu/denef04counting.html, 2004 25. Kedlaya, K.: Quantum Computation of zeta functions of curves. Comput. Complex. 15, 1–19 (2006) 26. van Dam, W., Seroussi, G.: Efficient Quantum Algorithms for Estimating Gauss Sums. http://arxiv.org/ list/quant-ph/0207131, 2002 27. Barg, A.: On some polynomials related to Weight Enumerators of Linear Codes. SIAM J. Discrete Math. 15, 155 (2002) 28. Baumert, L., McEliece, R.: Weights of Irreducible Cyclic Codes. Inform. and Control 20, 158 (1972) 29. Welsh, D.J.A.: Matroid Theory. London: Academic Press Inc, 1976 30. Gross, J., Yellen, J.: Graph theory and its applications. Discrete mathematics and its applications. Boca Raton, FL: CRC Press, 1999 31. van Lint, J.H.: Introduction to Coding Theory. Berlin-Heidelberg-New York: Springer-Verlag, 1982 32. Jaeger, F.: The Tutte Polynomial and Link Polynomials. Proc. Amer. Math. Soc. 103, 647 (1998) 33. Evans, J., Berndt, B.C., Williams, K.S.: Gauss and Jacobi Sums. New York: Wiley-Interscience, 1998 34. Moisio, M.: Exponential Sums, Gauss Sums and Cyclic Codes. 1997. Available at http://lipas.uwasa.fi/ ~mamo/vaitos.pdf, 1997 35. Aubry, Y., Langevin, P.: On the weights of irreducible cyclic codes. 2005. Available at http://iml.univ-mrs. fr/~aubry/LNCS.pdf, 2005 36. Andrews, G.E.: Number Theory. New York: Dover Publications Inc., 1994 37. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000 38. van Dam, W., Seroussi, G.: Quantum algorithms for estimating Gauss sums and calculating discrete logarithms. Available at http://www.cs.ucsb.edu/~vandam/gausssumdlog.pdf 39. Brassard, G., Hoyer, P., Tapp, A.: Quantum Counting. http://arxiv.org/list/quant-ph/9805082, 1998 40. van Bussel, F., Geraci, J.: A Note on Cyclotomic Cosets and an Algorithm for finding Coset Representatives and Size and a theorem on the quantum evaluation of weight enumerators for a certain class of cycliccades. http://arxiv.org/list/cs.0703129, 2007 41. Lidl, R., Niederreiter, H.: Finite Fields, Volume 20 of Encyclopedia of Mathematics. Cambridge: Cambridge University Press, 1997 42. Van Der Glugt, M.: Hasse-Davenport Curves, Gauss Sums and Weight Distributions of Irreducible Cyclic Codes. J. Number Theory 55, 145 (1995) 43. Vyalyi, M.N.: Hardness of approximating the weight enumerator of a binary linear code. http://arxiv. org/list/cs.CC/0304044, 2003 44. Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101, 359 (2000) 45. Martinez-Perez, C., Willems, W.: Is the Class of Cyclic Codes Asymptotically Good?. IEEE Trans. Inf. Theory 52(2), 696 (2006) 46. Shrock, R.: Exact Potts model partition functions on ladder graphs. Physica A 283, 388 (2000) Communicated by M.B. Ruskai
Commun. Math. Phys. 279, 769–787 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0444-2
Communications in
Mathematical Physics
Absence of Ground States for a Class of Translation Invariant Models of Non-relativistic QED D. Hasler, I. Herbst Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA. E-mail: [email protected]; [email protected] Received: 23 March 2007 / Accepted: 9 September 2007 Published online: 5 March 2008 – © Springer-Verlag 2008
Abstract: We consider a class of translation invariant models of non-relativistic QED with net charge. Under certain natural assumptions we prove that ground states do not exist in the Fock space.
1. Introduction Over the years there has been much interest in trying to develop an appropriate mathematical framework to describe the interaction of charged particles with the quantized electromagnetic field. Here we only cite [21] and references given therein but later we briefly mention other work. Of course relativistic quantum electrodynamics (QED) is a very successful theory but has not been shown to provide a Hilbert space framework for describing the states of charged particles interacting with photons. In spite of this there are certainly prescriptions for getting correct answers to the “right” questions [14]. One of the first questions which arises is perhaps the most elementary: Are there “dressed one-electron states” of fixed momentum which are eigenstates of the appropriate Hamiltonian. These states should of course have an adhering photon cloud. In [15] Faddeev and Kulish gave a suggestion as to what form such states should take. The Fadeev-Kulish states do not live in Fock space because of the nature of the photon cloud. At this time, however, we are far from understanding the mathematics of relativistic QED. In order to understand the infrared problem in a simpler model, Fröhlich [9,10], studied the massless Nelson model. This is a model of a non-relativistic particle interacting with a scalar massless bose field (“photon” field). Among other results, in [9] he outlined a construction of asymptotic dressed one particle states (with a low energy photon cloud). Recently, Pizzo [18,19] has taken Fröhlich’s outline, added some important ingredients, and rigourously constructed a Hilbert space of asymptotic dressed one-particle states (with certain smallness assumptions on particle velocity and on various parameter values).
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In recent years the more realistic model of non-relativistic QED has been studied by many authors, see for example [21] and references given therein. This model suffers from various difficulties but it is hoped that it may serve as a reasonably realistic model for low energies, and a testing ground for understanding the infrared problem. One of the main difficulties is that this model is neither Galilean nor Poincaré covariant. The charged particles are treated non-relativistically while the photons are relativistic. There remains an ultraviolet cutoff in the photon field to produce a well defined theory, but the theory is well defined without an infrared cutoff. More recently, Chen and Fröhlich [7] have also outlined the construction of asymptotic dressed one-particle states in non-relativistic QED, partly relying on some of the ideas in [9,19]. In this work we define our Hamiltonians on the Hilbert space consisting of the Fock space for photons tensored with the usual Hilbert space for the non-relativistic charged particles. We consider a class of translation invariant models of non-relativistic QED having a total net charge. The generator of translations defines the operator of total momentum. Translation invariance implies that the Hamiltonian commutes with this operator. We can thus restrict the Hamiltonian to any subspace of fixed total momentum ξ . This restricted Hamiltonian is denoted by H (ξ ). For any momentum ξ , H (ξ ) is bounded from below. We denote the infimum of its spectrum by E(ξ ). One can easily show the function E(·) is almost everywhere differentiable. In this paper we show that for momenta ξ at which E(·) has a non-vanishing derivative, H (ξ ) does not admit a ground state. We do not impose an infrared cutoff. The coupling constant is arbitrary, but nonzero. First we consider an electron (with spin 1/2) coupled to the quantized electromagnetic field. We show that for any value of the coupling constant H (·) does not admit a ground state at points where E(·) has a non-vanishing derivative. This model has been previously investigated in [6,3,7]. There it was shown that for small values of the coupling constant, E(·) has a non-vanishing derivative for all nonzero ξ with |ξ | < ξ0 , where ξ0 is some explicit positive number. Furthermore, for small coupling it was shown that H (0) does have a ground state. Moreover, for small coupling and nonzero ξ , with |ξ | < ξ0 , it was shown that an infrared regularized Hamiltonian does have a ground state. As the infrared regularization is removed this ground state does not converge in Fock space, however it can be shown that it does converge as a linear functional on some operator algebra, [9,7]. Absence of ground states has been previously proved for the massless Nelson model [17,8,13], under various conditions, and the spin-boson model, for example in [2]. The model is introduced and the result is stated in Sect. 3. The proof of the result is presented in Sect. 4. Although on the basis of the work cited above, our result is expected, we have not found a proof in the literature. We then generalize the above result to a positive ion. More specifically, we consider a spinless nucleus with nuclear charge Z e and N electrons each with charge −e where the allowed interaction between the particles includes the Coulomb potential. If Z = N , we show that H (·) does not admit a ground state at points where E(·) has a non-vanishing derivative. This model has been recently investigated in [1,16], where it was shown that under natural assumptions H (ξ ) does have a ground state provided N = Z . It was known previously that if the nucleus has infinite mass, then the relevant Hamiltonian does have a ground state if Z ≥ N , [4,11]. In contrast to our result, Coulomb systems without coupling to the quantized electromagnetic field do have positive ions, with fixed nonzero total momentum. In Sect. 3 we introduce the model describing an ion and state the result. Its proof is presented in Sect. 4. Although perhaps surprising, the intuition for
Absence of Ground States for Translation Invariant Non-relativistic QED
771
our result comes from the fact that from a distance, a charged bound state looks like a point particle. A basic ingredient in the proof of our results is the pull-through formula (see Eqs. (4) and (11)). For an early application of this formula see [20], p.345 and references therein. The pull-through formula has been used in rigorous work by many others including the pioneering work of Fröhlich on the infrared problem in the Nelson model [10]. We prove that the infrared behavior of this formula gives the projection onto the space of putative ground states. This is the first key step in our proof. Using this we prove an infrared divergent lower bound which rules out the existence of a ground state. Here, we only need ∇ E(ξ ) = 0. We perform our analysis for multi-particle systems (ions) for arbitrary nonzero coupling constant. To obtain our result we have to restrict to a suitable subset of momentum space. 2. Fock Space of Photons The degrees of freedom of the photons are described by a symmetric Fock space, introduced as follows. Let h := L 2 (Z2 × R3 ) ∼ = L 2 (R3 ; C2 ) denote the Hilbert space of a transversally polarized photon. The variable k = (λ, k) ∈ Z2 × R3 consists of the wave vector k or momentum of the particle and λ describing the polarization. The symmetric Fock space, F, over h is defined by F =C⊕
∞
Sn (h⊗n ),
n=1
where Sn denotes the orthogonal projection onto the subspace of totally symmetric tensors. The vacuum is the vector := (1, 0, 0, ...) ∈ F. The vector ψ ∈ F can be identified with sequences (ψn )∞ n=0 of n-photon wave functions, ψn (k 1 , ..., k n ) ∈ L 2 ((Z2 × R3 )n ), which for n ≥ 1 are totally symmetric in their n arguments. The Fock space inherits a scalar product from h, explicitly (ψ, ϕ)F = ψ 0 ϕ0 +
∞
ψ n (k 1 , ..., k n )ϕn (k 1 , ..., k n )dk 1 ...dk n ,
n=1
where we used the abbreviation dk = λ=1,2 dk. The number operator N is defined by (N ψ)n = nψn . It is self-adjoint on the domain D(N ) := {ψ ∈ F|N ψ ∈ F}. For each function f ∈ h one associates an annihilation operator a( f ) as follows. For a vector ψ ∈ F we define 1/2 (a( f )ψ)n (k 1 , ..., k n ) = (n + 1) f (k)ψn+1 (k, k 1 , ..., k n )dk, ∀ n ≥ 0. The domain of a( f ) is the set of all ψ such that a( f )ψ ∈ F. Note that a( f ) = 0. The creation operator a ∗ ( f ) is defined to be the adjoint of a( f ). Note that a( f ) is anti-linear, and a ∗ ( f ) is linear in f . They are well known to satisfy the canonical commutation relations [a ∗ ( f ), a ∗ (g)] ⊂ 0, [a( f ), a(g)] ⊂ 0, [a( f ), a ∗ (g)] ⊂ ( f, g),
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where f, g ∈ L 2 (Z2 × R3 ) and ( f, g) denotes the inner product of L 2 (Z2 × R3 ). Since a( f ) is anti-linear, and a ∗ ( f ) is linear in f , we will write ∗ f (k)ak dk, a ( f ) = f (k)ak∗ dk, a( f ) = where the right hand side is merely a different notation for the expression on the left. For a function f ∈ L 2 (R3 ) and λ = 1, 2, we will write aλ ( f ) := a( f λ ) and aλ∗ ( f ) := a ∗ ( f λ ), where f λ ∈ h is the function defined by f λ (µ, k) := f (k)δλ,µ . The field energy operator denoted by H f is given by n |ki | ψn (k 1 , ...k n ). (H f ψ)n (k 1 , ...k n ) = i=1
It is self-adjoint on its natural domain D(H f ) := {ψ ∈ F|H f ψ ∈ F}. The operator of momentum P f is given by n ki ψn (k 1 , ...k n ). (P f ψ)n (k 1 , ...k n ) = i=1
Its components (P f ) j are each self-adjoint on the domain D((P f ) j ) := {ψ ∈ F|(P f ) j ψ ∈ F}. In this paper we will adopt the notation that | · | denotes the standard norm in R, R3 , C, or C2 . 3. The Electron: Model and Statement of Result At first we consider a single free electron interacting with the quantized electromagnetic field. The Hilbert space describing the system composed of an electron and the quantized field is H = L 2 (R3 ; C2 ) ⊗ F. The Hamiltonian is H = {σ · ( p + e A(x))}2 + H f , where A(x) =
λ=1,2
ρ(k) ∗ aλ,k eik·x ελ,k + aλ,k e−ik·x ελ,k dk, √ 2|k|
(1)
k = k/|k|, such that where the ελ,k ∈ R3 are vectors, depending measurably on
(k/|k|, ε1,k , ε2,k ) forms an orthonormal basis; and σ = (σ1 , σ2 , σ3 ), where σi denotes the i th Pauli matrix: 0 1 0 −i 1 0 , σ2 = , σ3 = . σ1 = 1 0 i 0 0 −1 By x we denote the position of the electron and its canonically conjugate momentum by p = −i∇x . We have introduced the function ρ(k) =
1 χ (|k|), (2π )3/2
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where χ is the characteristic function of the set [0, ]. Since we are interested in the infrared problem we fix the ultraviolet cutoff 0 < < ∞. The Pauli matrices satisfy the commutation relations [σ1 , σ2 ] = 2iσ3 and cyclic permutations thereof. Using these commutation relations, we can write the Hamiltonian as H = ( p + e A(x))2 + eσ · B(x) + H f , where
ρ(k)(ik ∧ ελ,k ) ∗ B(x) = (∇ ∧ A)(x) = e−ik·x dk. aλ,k eik·x − aλ,k √ 2|k| λ=1,2
(2)
The Hamiltonian is translation invariant and commutes with the generator of translations, i.e., the operator of total momentum Ptot = p + P f . Let F be the Fourier transform in the electron variable x, i.e., on L 2 (R3 ), 1 e−iξ ·x ψ(x)d x. (Fψ)(ξ ) = (2π )3/2 R3 Set W = exp(i x · P f ). Note W Ptot W ∗ = p so that in the new representation p is the total momentum. We compute
2 W H W ∗ = σ · ( p − P f + e A) + H f , where A := A(0). Then the composition U = F W yields the fiber decomposition of the Hamiltonian and the Hilbert space ⊕ ⊕ H (ξ )dξ, U : H → L 2 (R3 ; C2 ) ⊗ F ∼ C2 ⊗ Fdξ U HU ∗ = = R3
with
R3
2
H (ξ ) = σ · (ξ − P f + e A) + H f
:= C2 ⊗ F. Note that H (ξ ) can also be written as an operator on F H (ξ ) = (ξ − P f + e A)2 + eσ · B + H f , where B := B(0). The explicit self-adjoint realization of H (ξ ) is given by the following lemma. 2 + H f )ψ ∈ Lemma 1. The operator H (ξ ) is self-adjoint on D(P 2f + H f ) = {ψ ∈ F|(P f 2 and essentially self-adjoint on any core of P + H f . F} f For a proof of Lemma 1 see [12,16]. The operator H (ξ ) is bounded from below and we write E(ξ ) := inf σ (H (ξ )).
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Proposition 2. The function E(·) is almost everywhere differentiable. By spherical symmetry E(·) is invariant under rotations. We want to point out that differentiability properties stated in [6] imply that for small e and |ξ | < 16 , E(·) is C 1 and has non-vanishing derivative for ξ = 0. In [9] it is shown that for large ξ , E(ξ ) = |ξ | + O(1). It seems probable that for all e and ξ = 0, E(·) is differentiable with non-vanishing derivative. Theorem 3. Let e = 0. If E(·) is differentiable at ξ and has a nonzero derivative, then H (ξ ) does not have a ground state We want to relate this to results obtained in [5,3,7,6], where A = A(0) in (1) −1/2 (a ∗ ε is replaced by an infrared regularized Aσ (0) = λ σ ≤|k| ρ(k)(2|k|) λ,k λ,k + 1 aλ,k ελ,k )dk. It is shown that if e is small and |ξ | < 6 then for any σ > 0, there exists a normalized ground state ψσ (ξ ). For ξ = 0, ψσ (0) converges weakly as σ → 0 to a nonzero vector. However for nonzero ξ , with |ξ | < 16 , it was shown that ψσ (ξ ) converges weakly to zero. We want to note that in principle this does not rule out the possibility that there could suddenly appear a ground state in Fock space at σ = 0. 4. The Electron: Proof of Results First we give a well known proof of Proposition 2, see [9]. Proof of Theorem 2. We set T (ξ ) := H (ξ ) − ξ 2 = −2ξ · (P f − e A) + (P f − e A)2 + eσ · B + H f . Since for each ψ ∈ D(P 2f + H f ) = D(H (ξ )), the function ξ → (ψ, T (ξ )ψ) is linear, it follows that the function ξ → t (ξ ) := inf {(ψ, T (ξ )ψ)|ψ ∈ D(H (ξ )), ψ = 1} is concave. From concavity it follows that t (·) is a.e. differentiable and hence also the function ξ → E(ξ ) = ξ 2 + t (ξ ). For notational convenience we write v(ξ ) = (ξ − P f + e A). Before we present the proof of Theorem 3, we need a few lemmas. For E(·) differentiable at ξ and > 0, we fix ξ and consider the following subset of the unit sphere: S := {ω ∈ S 2 | ω · ∇ E(ξ ) ≤ 1 − }. We denote normalized vectors by
k = k/|k|. Lemma 4. Assume that E(·) is differentiable at ξ . Uniformly for
k ∈ S , we have H (ξ − k) + |k| − E(ξ ) ≥ |k| + o(|k|), as |k| → 0.
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Proof. Using that E(ξ − k) is a lower bound for H (ξ − k) and the differentiability of E(·) at ξ , we have H (ξ − k) + |k| − E(ξ ) ≥ E(ξ − k) − E(ξ ) + |k| = −k · ∇ E(ξ ) + |k| + o(|k|) ≥ |k| + o(|k|). Let P0 = P0 (ξ ) denote the orthogonal projection onto the kernel of H (ξ ) − E(ξ ). we set For ϕ ∈ F, (ak ϕ)n (k 1 , ..., k n ) = (n + 1)1/2 ϕn+1 (k, k 1 , ..., k n ).
(3)
For λ = 1, 2, a.e. k, and all n, (ak ϕ)n ∈ Sn (h⊗n ) ⊗ C2 . The relation to a( f ) is outlined in the following lemma. and suppose the function k → ak ϕ is in L 2 (Z2 × Lemma 5. Let ⊂ R3 and ϕ ∈ F ). Then for all f ∈ h, with f vanishing outside of Z2 × , and η ∈ F, ; F (η, a( f )ϕ) =
f (k)(η, ak ϕ)dk.
Proof. We have (η, a( f )ϕ) =
∞ ηn (k1 , ..., k n ), (n + 1)1/2 f (k)ϕn+1 (k, k 1 , ..., k n+1 ) dkdk 1 ...dk n n=0
=
f (k)
∞
ηn (k 1 , ..., k n ), (n + 1)1/2 ϕn+1 (k, k 1 , ..., k n+1 )
n=0
× dk 1 ...dk n dk = f (k)(η, ak ϕ)dk, where the interchange of the order of integration and summation is justified since ∞ ηn (k1 , ..., k n ), (n + 1)1/2 f (k)ϕn+1 (k, k 1 , ..., k n+1 ) dkdk 1 ...dk n n=0
1/2
≤ f η
ak ϕ dk 2
< ∞.
1/2 with Lemma 6. For each ϕ ∈ D(H f ), the function k → ak ϕ is in L 2loc (Z2 × R3× ; F), 3 3 R× = R \{0}.
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Proof. Since ϕ ∈ D(H f ), we conclude that ∞ n=0
=
|k||(ak ϕ)(k 2 , ..., k n+1 )|2 dk 2 ...dk n+1 dk
∞ n+1 n=0
=
|k j ||ϕn+1 (k 1 , k 2 , ..., k n+1 )|2 dk1 dk 2 ...dk n+1
j=1
1/2 H f ϕ2
< ∞.
This implies that the function k → ak ϕ2 is integrable over any compact subset of Z2 × R3× . The next result uses the pull-through formula (see for example [20,9]). Lemma 7. Suppose E(·) is differentiable at ξ and that ψ is a ground state of H (ξ ). Let
> 0. Then there exists a δ > 0 such that for all η ∈ F,
eρ(k) η, ak ψ = √ H (ξ, k)−1 η, −2 k · v(ξ ) + i(k ∧ k ) · σ ψ , 2|k|
(4)
for a.e. k, with 0 < |k| < δ and
k ∈ S , where H (ξ, k) := H (ξ − k) + |k| − E(ξ ). Proof. Let f ∈ C0∞ (R3 \{0}) . Let ϕ ∈ Ran(P[0,ν] (N )) be a state having less than or equal to ν photons, for some finite ν, and assume each ϕn has compact support. By a calculation using the canonical commutation relations, we find for real f , ∗ (aλ ( f )H (ξ, k) − (H (ξ ) − E(ξ ))aλ∗ ( f ))ϕ, ψ = (A∗ ( f ) + R0∗ ( f ) + R1∗ ( f ))ϕ, ψ , with
R0 ( f ) :=
+
f (y)2(y − k) · v(ξ )aλ,y dy +
f (y)(k 2 − y 2 )aλ,y dy
f (y)(|k| − |y|)aλ,y dy,
eρ(y) (k · ελ,y ) dy, f (y) √ 2|y| eρ(y) −2ελ,y · v(ξ ) + i(y ∧ ελ,y ) · σ dy. A( f ) := f (y) √ 2|y|
R1 ( f ) :=
Since ψ ∈ D(H f + P 2f ) ⊂ D(aλ ( f )), (H (ξ, k)ϕ, aλ ( f )ψ) = (ϕ, (A( f ) + R0 ( f ) + R1 ( f ))ψ).
(5)
Note that this holds for all ϕ in an operator core for H (ξ, k). For any > 0, there exists by Lemma 4 a δ > 0 such that for all k with 0 < |k| < δ and
k ∈ S , H (ξ, k) has a bounded inverse. This and Eq. (5) imply that in fact for all such k and all η ∈ F, (η, aλ ( f )ψ) = (H (ξ, k)−1 η, (A( f ) + R0 ( f ) + R1 ( f ))ψ).
(6)
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For k, with 0 < |k| < δ and
Now fix η ∈ F. k ∈ S , we choose a δ–sequence centered at k. Explicitly, we choose a nonnegative function g ∈ C0∞ (R3 ) with g(y)dy = 1 and support in the unit ball. We set f k,m (y) := m 3 g(m(y −k)). By Lemmas 5 and 6 it follows that the left-hand side of (6) yields limm→∞ (η, aλ ( f m,k )ψ) = (η, aλ,k ψ) a.e. k. The term (H (ξ, k)−1 η, A( f m,k )ψ) converges to the right hand side of (4). Below we will show that the terms (H (ξ, k)−1 η, R0 ( f m,k )ψ) and (H (ξ, k)−1 η, R1 ( f m,k )ψ) vanish as m tends to infinity for a.e. k. The expression containing R1 vanishes since k · λ,k = 0. To show that the expression involving R0 vanishes we will only consider one term. The other terms will follow similarly. We set φl := vl (ξ )H (k, ξ )−1 η and estimate −1 f k,m (y)2(y − k) · v(ξ )aλ,y ψ dy R0,1 ( f m,k ) := H (k, ξ ) η, 3 −1 ≤ f k,m (y)2(y − k)l aλ,y ψ dy vl (ξ )H (k, ξ ) η, l=1 3 ∞
(φl )n (k , ..., k ), ≤ f k,m (y)2(y − k)l (n + 1)1/2 1 n l=1 n=0 ×ψn+1 (λ, y, k 1 , ..., k n )dy dk 1 ...dk n ≤ φ f k,m (y)2|y − k|h λ (y)dy, where h λ (y) =
∞
2 (n + 1) ψn+1 (λ, y, k 1 , ..., k n ) dk 1 ...dk n
1/2
n=0
and φ2 =
3
φl 2 .
l=1 1/2
Since ψ is in D(H f ), ∞ 2 2 1/2 (n + 1)|y| ψn+1 (µ, y, k 1 , ..., k n ) dydk 1 ...dk n |y| h λ (y) dy ≤ n=0 µ=1,2
= (ψ, H f ψ). Thus h λ ∈ L 1loc (R3 \{0}). Therefore a.e. point is a Lebesgue point of h λ . At such points k, f k,m (y)h λ (y)dy → h λ (k), by Lebesgue’s differentiation theorem, see for example [22], Theorem 1.25. Thus R0,1 ( f k,m ) tends to zero as m → ∞, a.e. k.
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Lemma 8. If E(·) is differentiable at ξ , then P0 2v(ξ )P0 = ∇ E(ξ )P0 . Proof. Suppose ψ ∈ Ran P0 , with ψ = 1, then E(ξ + k) − E(ξ ) ≤ (ψ, (H (ξ + k) − H (ξ ))ψ) = 2k · (ψ, v(ξ )ψ) + |k|2 . This implies k · ∇ E(ξ ) ≤ 2k · (ψ, v(ξ )ψ) + o(|k|),
as |k| → 0.
Since k can have any direction we conclude that ∇ E(ξ ) = 2(ψ, v(ξ )ψ). Since this holds for any ψ ∈ Ran P0 the claim follows by polarization. We set Q(k) = |k|(H (ξ − k) + |k| − E(ξ ))−1 , whenever this exists. And for |k| > 0, we set Q 0 (k) = |k|(H (ξ ) + |k| − E(ξ ))−1 . By the spectral theorem P0 = P0 (ξ ) = s − lim Q 0 (k). |k|→0
(7)
Lemma 9. Let E(·) be differentiable at ξ . Given > 0, then w − lim Q(k) − (1 −
k · ∇ E(ξ ))−1 P0 = 0.
k∈S ,|k|→0
Proof. Fix ξ Step 1. v(ξ )Q 0 (k) is uniformly bounded for small |k|. Since B := B(0), introduced in 1/2 Eq. (2), is H f operator bounded, we see that there exists a finite constant C0 such that v(ξ )2 ≤ H (ξ ) + C0 ≤ (H (ξ ) + |k| − E(ξ )) + (E(ξ ) + C0 ).
(8)
On the other hand 3 (v(ξ )Q 0 (k))l∗ (v(ξ )Q 0 (k))l = l=1
|k| |k| v(ξ )2 . H (ξ ) − E(ξ ) + |k| H (ξ ) − E(ξ ) + |k|
By inequality (8) we see that the right hand side is uniformly bounded for small |k|. This shows Step 1. Step 2. We have s − lim|k|→0 v(ξ )Q 0 (k) = v(ξ )P0 . By the resolvent identity |k| H (ξ ) − E(ξ ) + |k| 1 |k| |k| − v(ξ ) . (9) = v(ξ ) H (ξ ) − E(ξ ) + |k| + 1 H (ξ ) − E(ξ ) + |k| + 1 H (ξ ) − E(ξ ) + |k|
v(ξ )
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Again using the resolvent identity and an argument similar to the one in Step 1, 1 1 v(ξ ) − v(ξ ) H (ξ ) − E(ξ ) + |k| + 1 H (ξ ) − E(ξ ) + 1 1 1 = v(ξ ) |k| H (ξ ) − E(ξ ) + 1 H (ξ ) − E(ξ ) + |k| + 1 |k|→0
−→ 0.
This implies that the first term on the right hand side in (9) converges in norm to zero and the second term converges strongly to v(ξ )P0 . Step 3. Uniformly for
k ∈ S as |k| → 0, w w k · v(ξ )P0 (P0 Q(k)P0 ) −→ P0 and Q(k) − P0 Q(k)P0 −→ 0. P0 Q(k)P0 − P0 2
Using the second resolvent identity twice we obtain for small |k| and
k ∈ S , Q(k) = Q 0 (k) + Q 0 (k)(2
k · v(ξ ) − |k|)Q(k) = Q 0 (k) + Q 0 (k)(2
k · v(ξ ) − |k|)Q 0 (k)
+Q 0 (k)(2k · v(ξ ) − |k|)Q(k)(2
k · v(ξ ) − |k|)Q 0 (k)
(10) (11)
Now using (11) and the results of Step 1 and Step 2, we find w
w
Q(k)(1 − P0 ) −→ 0, (1 − P0 )Q(k) −→ 0, where the limit is uniform for
k ∈ S . It follows that w
Q(k) − P0 Q(k)P0 −→ 0, uniformly for
k ∈ S . Now this, (7), and (10) show Step 3. The claim of the lemma is now an immediate consequence of Lemma 8 and Step 3. Now we are ready to prove Theorem 3. Proof of Theorem 3. Suppose H (ξ ) has a ground state ψ with ψ = 1. We want to show this leads to a contradiction. We choose an η ∈ D((N +1)1/2 ) such that (η, ψ) = 0. Choose with 0 < < 1 and δ > 0 sufficiently small. Then by Lemma 7 for a.e. k with
k ∈ S and |k| < δ, (η, aλ,k ψ) = η, H (ξ, k)−1 (2|k|)−1/2 eρ(k) −2ελ,k · v(ξ ) + i(k ∧ ελ,k ) · σ ψ eρ(k) =√ η, Q(k) −2ελ,k · v(ξ ) + i(k ∧ ελ,k ) · σ ψ . 2|k|3/2 Now uniformly for
k ∈ S ,
|k|→0 k · ∇ E)−1 ελ,k · (P0 η, 2v(ξ )ψ) η, Q(k) −2ελ,k · v(ξ ) + i(k ∧ ελ,k ) · σ ψ −→ −(1 −
= −(ελ,k · ∇ E)(1 −
k · ∇ E)−1 (η, ψ),
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where in the last step we used Lemma 8. We introduce the set 1 K := ω ∈ S 2 | − |∇ E| ≤ ω · ∇ E ≤ 0 ⊂ S . 2
k ∈ K, Then there exists a positive constant c0 such that for all
|(ελ,k · ∇ E)|2 ≥ c0 > 0. λ=1,2
k ∈ K, By the above, there exists a nonzero δ2 such that for a.e. k with |k| < δ2 and
|(η, aλ,k ψ)|2 ≥
λ=1,2
1 |eρ(k)|2 (1 −
k · ∇ E)−2 |(η, ψ)|2 c0 . 2 2|k|3
k ∈ K , we have Therefore, there exists a c1 > 0 such that for a.e. small k with
|eρ(k)c1 |2 ≤ |(η, aλ,k ψ)|2 |k|3 λ=1,2 ⎛ ≤ (1 + N )1/2 η2 ⎝
∞
⎞ |ψn+1 (λ, k, k 1 , ..., k n )|2 dk 1 ...dk n ⎠ .
λ=1,2 n=0
Integrating over the set of all k with
k ∈ K and |k| ≤ δ2 , we see this is inconsistent with Thus H (ξ ) does not have a ground state. ψ being in F. 5. Positive Ion: Model and Statement of Results We consider an ion consisting of a spinless nucleus of mass m 0 and charge Z e and N spin 1/2 electrons having charge −e and mass 1. The energy of this system is described by the operator 1 1 {σ j · ( p j + e A(x j ))}2 + H f + V (x0 , ..., x N ), ( p0 − Z e A(x0 ))2 + 2m 0 2 N
H=
j=1
acting on the Hilbert space ⎛ H = L 2 (R3 ) ⊗ ⎝
N
⎞ L 2 (R3 ; C2 )⎠ ⊗ F,
j=1
where p0 = −i∇0 acts on the first factor and p j = −i∇ j and σ j , the three-vector of Pauli matrices, act on the j th factor of the antisymmetric tensor product. We take the spin of the nucleus to be zero only to simplify notation. We will make the following assumptions about the potential V : V (x0 , ..., x N ) = Vi j (xi − x j ). 0≤i< j≤N
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Each Vi j is infinitesimally bounded with respect to the Laplacian in three dimensions, which we denote by −, i.e., there exists for any a > 0 a finite constant b such that for all f in the domain of −, Vi j f ≤ a − f + b f . The Hamiltonian is translation invariant and therefore commutes with the generator of translations, i.e., the operator of total momentum Ptot =
N
pj + Pf .
j=0
Let F be the Fourier transform in the variable x0 , i.e., on L 2 (R3 ), 1 (Fψ)(ξ ) = e−iξ ·x0 ψ(x0 )d x0 . (2π )3/2 R3 Let
⎛
⎛
W = exp ⎝i x0 · ⎝ P f +
N
⎞⎞ p j ⎠⎠ .
j=1
Note that WPtot W ∗ = p0 so that in a new representation, p0 is the total momentum. Then the composition U = FW yields the decomposition of the Hamiltonian ⊕ UHU ∗ = H (ξ )dξ, R3
with
⎞2 ⎛ N N 2 1 ⎝ 1 H (ξ ) = σ j · ( p j + e A(x j )) + H f + V p j − P f − Z e A(0)⎠ + ξ− 2m 0 2 j=1
N
j=1
= V |x0 =0 . Let us cite the L 2 (R3 ; C2 ) ⊗ F, and where we have set V acting on following Theorem [12,16]. Theorem 10. The operator H (ξ ) is self-adjoint on N
D( p 2j ) ∩ D(P 2f + H f )
j=1
and essentially self-adjoint on any core of
N j=1
p 2j + P 2f + H f .
It is easy to show that for every ξ the operator H (ξ ) is bounded below. Let E(ξ ) = inf σ (H (ξ )) be the infimum of the spectrum. By a simple argument as in the proof of Proposition 2 we see that E(·) is almost everywhere differentiable. The following theorem is the main result. Its proof is given in the next section. Theorem 11. Suppose N = Z and e = 0. If E(·) is differentiable at ξ with non-vanishing derivative then H (ξ ) does not have a ground state.
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6. Positive Ion: Proof of Result First we show the following lemma. | is infinitesimally form bounded with respect to H (ξ ). Lemma 12. |V Proof. By Theorem 10, we know that H (ξ ) is self-adjoint on the domain of P 2f + N 2 j=1 p j + H f . Therefore there exist finite constants c1 and c2 such that P 2f +
N
p 2j + H f ≤ c1 H (ξ ) + c2 .
j=1
| is is infinitesimally small with respect to Nj=1 p 2 . Therefore, |V By assumption V j N 2 infinitesimally form bounded with respect to j=1 p j . Hence for any a > 0 there exists a finite b such that ⎛ ⎞ N N | ≤ a p 2j + b ≤ a ⎝ P 2f + p 2j + H f ⎠ + b ≤ ac1 H (ξ ) + ac2 + b. |V j=1
j=1
We will prove Theorem 11 using a sequence of lemmas. For notational convenience we set v(ξ ) = ξ −
N
p j − P f − Z e A(0).
j=1
Recall the definitions S := {ω ∈ S 2 | ω · ∇ E(ξ ) ≤ 1 − } and
k := k/|k|, which are the same as in Sect. 4. Lemma 13. Assume that E(·) is differentiable at ξ . Given > 0, then for
k ∈ S , we have H (ξ − k) + |k| − E(ξ ) ≥ |k| + o(|k|). The proof of Lemma 13 is the same as the proof of Lemma 4. Lemma 14. Let H0 be any Hilbert space. Let ⊂ R3 and ϕ ∈ H0 ⊗ F, and suppose the function k → ak ϕ is in L 2 (Z2 × ; H0 ⊗ F). Then for all f ∈ h, with f vanishing outside of Z2 × , and η ∈ H0 ⊗ F, (η, a( f )ϕ) = f (k)(η, ak ϕ)dk. The proof of this lemma is analogous to the proof of Lemma 5. We merely have to replace the inner product of C2 by the inner product of H0 . Likewise, one generalizes the proof of Lemma 6 to prove the next lemma. Anticipating our application we set N 2 henceforth H0 := L (R3 ; C2 ) .
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1/2
Lemma 15. Let ϕ ∈ D(H f ). Then the function k → ak ϕ is in L 2loc (Z2 ×R3× ; H0 ⊗F), with R3× = R3 \{0}. Lemma 16. Suppose E(·) is differentiable at ξ and that ψ is a ground state of H (ξ ). Let > 0. Then there exists a δ > 0 such that for all η ∈ H0 ⊗ F, ⎛ ⎛ N Z eρ(k) ⎝ η, aλ,k ψ = √ e−ik·x j H (ξ, k)−1 η, ⎝ v(ξ ) − m0 2|k| j=1 1 ik ∧ σ j + p j + e A(x j ) × · ελ,k ψ , (12) 2 for a.e. k, with 0 < |k| < δ and
k ∈ S , where H (ξ, k) := H (ξ − k) + |k| − E(ξ ). Proof. Let f ∈ C0∞ (R3 \{0}). Let ϕ ∈ Ran(P[0,ν] (N )) be a state having less than or equal to ν photons, for some finite ν, and assume ϕn is smooth and has compact support. Then a straightforward calculation using the canonical commutation relations yields for f real, ∗ (aλ ( f )H (ξ, k) − (H (ξ ) − E(ξ ))aλ∗ ( f ))ϕ, ψ = (A∗ ( f ) + R0∗ ( f ) + R1∗ ( f ))ϕ, ψ , with
(|k| − |y|) f (y)aλ,y dy + m −1 f (y)(y − k) · v(ξ )aλ,y dy 0 +(2m 0 )−1 f (y)(k 2 − y 2 )aλ,y dy, Z ρ(y) f (y)k · ελ,y dy, := − e √ R1 ( f ) 2m 0 2|y| N ρ(y) −i y·x j A( f ) := − e e √ f (y)ελ,y · ( p j + e A(x j )) dy 2|y| j=1 Z ρ(y) f (y)ελ,y · v(ξ ) dy + e √ m0 2|y| N 1 ρ(y) −i y·x j e + e √ f (y)(ik ∧ ελ,y ) · σ j dy. 2 2|y| j=1
R0 ( f ) :=
Since ψ ∈
N j=1
D( p 2j ) ∩ D(P 2f + H f ) ⊂ D(aλ ( f )),
(H (ξ, k)ϕ, aλ ( f )ψ) = (ϕ, (A( f ) + R0 ( f ) + R1 ( f ))ψ). Note that this holds for all ϕ in an operator core for H (ξ, k). For > 0, there exists by Lemma 13 a δ > 0 such that for all k with 0 < |k| < δ and
k ∈ S , H (ξ, k) has a bounded inverse. Thus we conclude by density that for all such k and all η ∈ H0 ⊗ F, (η, aλ ( f )ψ) = (H (ξ, k)−1 η, (A( f ) + R0 ( f ) + R1 ( f ))ψ).
(13)
Now fix η ∈ H0 ⊗ F. For k, with 0 < |k| < δ and
k ∈ S , we choose a δ-sequence, f m,k , centered at k as in the proof of Lemma 7. We insert f m,k for f in Eq. (13). As m → ∞,
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it follows by Lemmas 14 and 15 that the left hand side of (13) converges to the left hand side of (12) for a.e. k. In the same limit the term involving A converges to the right hand side of (12). As demonstrated in the proof of Lemma 7 the terms involving R0 and R1 vanish as m tends to infinity for a.e. k. This implies the assertion of the lemma. The next lemma would follow easily from the formal commutation relation [H (ξ ), i x j ] = −
1 v(ξ ) + p j + e A(x j ) m0
if we ignored domain considerations. Lemma 17. Let P0 be the projection onto the kernel of H (ξ ) − E(ξ ). Then for all j with 1 ≤ j ≤ N , P0
1 v(ξ )P0 = P0 ( p j + e A(x j ))P0 . m0
Proof. Fix a j ∈ {1, 2, ..., N }. Let χ ∈ C ∞ (R+ ; [0, 1]) with χ |`[0, 1] = 1 and χ |`[2, ∞) = 0. We set χn (x j ) = χ (|x j |/n). Let ψ ∈ Ran P0 , then for all n, 0 = ψ, H (ξ )iχn (x j )x j ψ − ψ, iχn (x j )x j H (ξ )ψ 1 = χn (x j )ψ, − v(ξ ) + p j + e A(x j ) ψ m0 1 1 +Re ψ, (∇χ )(|x j |/n)x j · − v(ξ ) + p j + e A(x j ) ψ n m0 1 1 1 (∇χ )(|x j |/n)ψ + −i ψ, 2m 0 2 n 1 n→∞ −→ ψ, − v(ξ ) + p j + e A(x j ) ψ . m0 The limit as n tends to infinity follows from dominated convergence. By polarization this yields the claim. The proof of the next lemma is the same as the proof of Lemma 8. Lemma 18. Let P0 be the projection onto the the kernel of H (ξ ) − E(ξ ). If E(·) is differentiable at ξ , then P0
1 v(ξ )P0 = ∇ E(ξ )P0 . m0
We set Q(k) = |k|(H (ξ − k) + |k| − E(ξ ))−1 , whenever this exists. And for |k| > 0, we set Q 0 (k) = |k|(H (ξ ) + |k| − E(ξ ))−1 . Let P0 be the orthogonal projection onto the kernel of H (ξ ) − E(ξ ). By the spectral theorem P0 = P0 (ξ ) = s − lim Q 0 (k). |k|→0
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Lemma 19. Let E(·) be differentiable at ξ . Given > 0. Then for
k = k/|k|, w − lim Q(k) − (1 −
k · ∇ E(ξ ))−1 P0 = 0.
k∈S ,|k|→0
The proof follows the steps of Lemma 9, where Step 1 uses Lemma 12. We now present the proof of Theorem 11. Proof of Theorem 11. Suppose H (ξ ) has a ground state ψ with ψ = 1. We want to show that this leads to a contradiction. Choose with 0 < < 1, and choose η ∈ D((N + 1)1/2 ) with (η, ψ) = 0. By Lemma 16 there exists a δ > 0 such that for a.e. k, with 0 < |k| < δ and
k ∈ S , ⎞ ⎡⎛ N eρ(k) ⎣⎝ 1 −ik·x j (η, aλ,k ψ) = √ e (ik ∧ ελ,k ) · σ j ψ ⎠ η, Q(k) 2 2|k|3/2 j=1 " Z +ελ,k · η, Q(k) v(ξ )ψ m0 ⎛ ⎞⎫⎤ N ⎬ + ⎝η, Q(k) (−e−ik·x j )( p j + e A(x j ))ψ ⎠ ⎦ . ⎭ j=1
Since Q(k) is uniformly bounded on S for small |k|, ⎛ ⎝η, Q(k)
N
⎞ |k|→0
e−ik·x j i(k ∧ ελ,k ) · σ j ψ ⎠ −→ 0,
j=1
uniformly for k ∈ S . Using Lemma 9, we find uniformly for
k ∈ S as |k| → 0, Z Z η, Q(k) v(ξ )ψ −→ (1 −
k · ∇ E)−1 P0 η, v(ξ )ψ m0 m0 −1
= Z (∇ E)(1 − k · ∇ E) (η, ψ), where we used Lemma 18. Again by Lemma 9 and using that e−ik·x j converges in the strong operator topology to 1, we find uniformly for k ∈ S as |k| → 0, ⎛
⎞ N ⎝η, Q(k) (−e−ik·x j )( p j + e A(x j ))ψ ⎠ −→ (1 −
k · ∇ E)−1 j=1
⎛
⎞ N × ⎝ P0 η, − ( p j + e A(x j ))ψ ⎠ j=1
−N −1
P0 η, = (1 − k · ∇ E) v(ξ )ψ m0 = −N (∇ E)(1 −
k · ∇ E)−1 (η, ψ),
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where in the second line we used Lemma 17 and in the last again Lemma 18. We introduce the set 1 K := ω ∈ S 2 | − |∇ E| ≤ ω · ∇ E ≤ 0 ⊂ S . 2
Then, since by assumption ∇ E = 0, there exists a positive constant c0 such that for all
k ∈ K, |ελ,k · ∇ E|2 ≥ c0 > 0. λ=1,2
Collecting the above estimates we conclude that for small |k| uniformly for
k ∈ K, λ=1,2
|(η, aλ,k ψ)|2 ≥
1 |eρ(k)|2 (1 −
k · ∇ E)−2 |Z − N |2 |(η, ψ)|2 c0 . 2 2|k|3
k ∈ K , we find By this and N = Z , there exists a c1 > 0 such that for all small k with
|ρ(k)c1 |2 ≤ |(η, ak,λ ψ)|2 3 |k| λ=1,2 ⎛ ≤ (1 + N )1/2 η2 ⎝
∞
⎞ ψn+1 (λ, k, k 1 , ..., k n )2 dk 1 ...dk n ⎠,
λ=1,2 n=0
where in the last inequality we used Cauchy-Schwarz. This is inconsistent with ψ being in H0 ⊗ F. Thus H (ξ ) does not have a ground state. Acknowledgements. D.H. wants to thank Marcel Griesemer, Volker Bach, and Michael Loss for interesting discussions. I.H. would like to acknowledge an interesting conversation with Benoit Grébert.
References 1. Amour, L., Grébert, B., Guillot, J.: The dressed mobile atoms and ions. J. Math. Pures Appl. (9) 86(3), 177–200 (2006) 2. Arai, A., Hirokawa, M., Hiroshima, F.: On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff. J. Funct. Anal. 168(2), 470–497 (1999) 3. Bach, V., Chen, T., Fröhlich, J., Sigal, I.M.: The renormalized electron mass in non-relativistic quantum electrodynamics. J. Funct. Anal. 243(2), 426–535 (2007) 4. Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999) 5. Chen, T.: Operator-theoretic infrared renormalization and construction of dressed one-particle states in non-relativistic QED. preprint mp_arc 01-310 6. Chen, T.: Infrared renormalization in non-relativistic QED and scaling criticality. http://arxiv.org/list/ math-ph/0601010, 2006 7. Chen, T., Fröhlich, J.: Coherent infrared representations in non-relativistic QED. In: Proceedings of “Spectral Theory and Mathematical Physics”, a Festscrift in honor of Barey Simon’s 60th birthday, Vol. 1, Proc. Symp. Pure Math. 76, Providence, RI: Amer. Math. Soc., 2007, pp. 25–46 8. Derezi´nski, J., Gérard, C.: Scattering theory of infrared divergent Pauli-Fierz Hamiltonians. Ann. Henri Poincaré 5(3), 523–577 (2004) 9. Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré Sect. A (N.S.) 19, 1–103 (1973)
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10. Fröhlich, J.: Existence of dressed one electron states in a class of persistent models. Forts. Der Phy. 22, 159–198 (1974) 11. Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3), 557–595 (2001) 12. Hiroshima, F.: Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of the coupling constants. Ann. Henri Poincaré 3, 171–201 (2002) 13. Hirokawa, M.: Infrared catastrophe for Nelson’s model—non-existence of ground state and soft-boson divergence. Publ. Res. Inst. Math. Sci. 42(4), 897–922 (2006) 14. Itzykson, C., Zuber, J.-B.: Quantum Field Theory. New York: McGraw-Hill, 1980 15. Kulish, P., Fadeev, L.D.: Theor. and Math. Phys. 4, 247 (1970) 16. Loss, M., Miyao, T., Spohn, H.: Lowest energy states in nonrelativistic QED: atoms and ions in motion. J. Funct. Anal. 243(2), 353–393 (2007) 17. Lörinczi, J., Minlos, R.A., Spohn, H.: The infrared behaviour in Nelson’s model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincaré 3(2), 269–295 (2002) 18. Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré 4(3), 439–486 (2003) 19. Pizzo, A.: Scattering of an infraparticle: The one particle sector in Nelson’s massless model. Ann. Henri Poncaré 6(3), 553–606 (2005) 20. Schweber, S.: An introduction to relativistic quantum field theory. Evanston, IL/Elmsford, NY: Row, Peterson and Company, 1961 21. Spohn, H.: Dynamics of charged particles and their radiation field. Cambridge: Cambridge University Press, 2004 22. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton, NJ: Princeton Press, 1971 Communicated by I.M. Sigal
Commun. Math. Phys. 279, 789–813 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0439-z
Communications in
Mathematical Physics
Unstable and Stable Galaxy Models Yan Guo1 , Zhiwu Lin2 1 Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University,
Providence, RI 02912, USA. E-mail: [email protected]
2 Mathematics Department, University of Missouri, Columbia, MO 65211, USA.
E-mail: [email protected] Received: 9 April 2007 / Accepted: 14 August 2007 Published online: 29 February 2008 – © Springer-Verlag 2008
Abstract: To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models f 0 (E), for which the distribution function f 0 depends on the particle energy E only. In the first part of the article, we derive the first sufficient criterion for linear instability of f 0 (E) : f 0 (E) is linearly unstable if the second-order operator A0 ≡ − + 4π f 0 (E){I − P}dv has a negative direction, where P is the projection onto the function space {g(E, L)}, L being the angular momentum [see the explicit formulae (29) and (28)]. In the second part of the article, we prove that for the important King model, the corresponding A0 is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric perturbations. 1. Introduction A galaxy is an ensemble of billions of stars, which interact by the gravitational field they create collectively. For galaxies, the collisional relaxation time is much longer than the age of the universe ([8]). The collisions can therefore be ignored and the galactic dynamics is well described by the Vlasov - Poisson system (collisionless Boltzmann equation) ∂t f + v · ∇x f − ∇x U f · ∇v f = 0, U f = 4π f (t, x, v)dv, (1) R3
where (x, v) ∈ R3 × R3 , f (t, x, v) is the distribution function and U f (t, x) is its gravitational potential. The Vlasov-Poisson system can also be used to describe the
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dynamics of globular clusters over their period of orbital revolutions ([11]). One of the central questions in such galactic problems, which has attracted considerable attention in the astrophysics literature, of [7,8,11,31] and the references there, is to determine dynamical stability of steady galaxy models. Stability study can be used to test a proposed configuration as a model for a real stellar system. On the other hand, instabilities of steady galaxy models can be used to explain some of the striking irregularities of galaxies, such as spiral arms as arising from the instability of an initially featureless galaxy disk ([7]), ([32]). In this article, we consider stability of spherical galaxies, which are the simplest elliptical galaxy models. Though most elliptical galaxies are known to be non-spherical, the study of instability and dynamical evolution of spherical galaxies could be useful to understand more complicated and practical galaxy models . By Jeans’s Theorem, a steady spherical galaxy is of the form f 0 (x, v) ≡ f 0 (E, L 2 ), where the particle energy and total momentum are E=
1 2 |v| + U0 (x), L 2 = |x × v|2 , 2
and U0 (x) = U0 (|x|) satisfies the self-consistent Poisson equation. The isotropic models take the form f 0 (x, v) ≡ f 0 (E). The cases when f 0 (E) < 0 (on the support of f 0 (E)) have been widely studied and these models are known to be linearly stable to both radial ([9]) and non-radial perturbations ([2]). The well-known Casimir-Energy functional (as a Liapunov functional) 1 1 2 H( f ) ≡ Q( f ) + (2) |v| f − |∇x U f |2 , 2 8π is constant along the time evolution. If f 0 (E) < 0, we can choose the Casimir function Q 0 such that Q 0 ( f 0 (E)) ≡ −E for all E. By a Taylor expansion of H( f ) − H( f 0 ), it follows that formally the first variation at f 0 is zero, that is, H(1) ( f 0 (E)) = 0 (on the support of f 0 (E)), and the second order variation of H at f 0 is 1 g2 1 (2) d xdv − (3) H f0 [g] ≡ |∇x Ug |2 d x, 2 8π { f 0 >0} − f 0 (E) where Q ( f 0 ) = − f 1 (E) , g = f − f 0 and Ug = gdv. In the 1960s, Antonov ([1,2]) 0 proved that |Dh|2 1 1 |∇ψh |2 d x d xdv − [Dh] = (4) H(2) f0 f (E) 2 8π 0 is positive definite for a large class of monotone models with f 0 < 0. Here D = v · ∇x − ∇x U0 · ∇v ,
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h(x, v) is odd in v and −ψ = Dhdv. He showed that such a positivity is equivalent to the linear stability of f 0 (E). In [9], Doremus, Baumann and Feix proved the radial stability of any monotone spherical models. Their proof was further clarified and simplified in [10,22,37], and more recently in [21,33]. In particular, this implies that any monotone isotropic models are at least linearly stable. Unfortunately, despite its importance and a lot of research (e.g., [5,6,13,20]), to our knowledge, no rigorous and explicit instability criterion of non-monotone models (2) has been derived. When f 0 (E) changes sign, functional H f0 is indefinite and it gives no stability information, although it seems to suggest that these models are not energy minimizers under symplectic perturbations. In this paper, we first obtain the following instability criterion for general spherical galaxies. We define the f 0 (E) −weighted L 2 R3 × R3 space L 2| f | with the norm ·| f | as 0 0 ||h||2| f | ≡ | f 0 (E)|h 2 d xdv. (5) 0
Theorem 1.1. Assume that f 0 (E) has a compact support in x and v, and f 0 is bounded. For φ ∈ H 1 , define the quadratic form 2 (A0 φ, φ) = |∇φ| d x + 4π f 0 (E) (φ − Pφ)2 d xdv, (6) where P is the projector of L 2| f | to 0
ker D = g E, L 2 ,
and more explicitly Pφ is given by (20) for radial functions and (28) for general functions. If there exists φ0 ∈ H 1 such that (A0 φ0 , φ0 ) < 0,
(7)
then there exists λ0 > 0 and φ ∈ H 2 , f (x, v) ∈ L ∞ given by (14), such that eλ0 t [ f, φ] is a growing mode to the Vlasov-Poisson system (1) linearized around [ f 0 (E), U0 ] . A similar instability criterion canbe obtained for symmetry preserving perturbations of anisotropic spherical models f 0 E, L 2 , see Remark 2. We note that the term Pφ in the instability criterion is highly non-local and this reflects the collective nature of stellar instability. The proof of Theorem 1.1 is by extending an approach developed in [25] for 1D Vlasov-Poisson, which has recently been generalized to Vlasov-Maxwell systems ([26,28]). There are two elements in this approach. One is to formulate a family of dispersion operators Aλ in (15) for the potential, depending on a positive parameter λ. The existence of a purely growing mode is reduced to find a parameter λ0 such that the Aλ0 has a kernel. The key observation is that these dispersion operators are selfadjoint due to the reversibility of the particle trajectories. Then a continuation argument is applied to find the parameter λ0 corresponding to a growing mode, by comparing the spectra of Aλ for very small and large values of λ. There are two new complications in the stellar case. First, the essential spectrum of Aλ is [0, +∞) and thus we need to make sure that the continuation does not end in the essential spectrum.This is achieved by using some compactness property due to the compact support of the stellar model. Secondly, it is more tricky to find the limit of Aλ when λ tends to zero. For that, we need
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an ergodic lemma (Lemma 2.4) and use the integrable nature of the particle dynamics in a central field to derive an expression for the projection Pφ which appeared in the limit. In the second part of the article, we further study the nonlinear (dynamical) stability of the normalized King model: f 0 = [e E 0 −E − 1]+
(8)
with a negative constant E 0 < 0, motivated by the study of the operator A0 . The famous King model describes isothermal galaxies and the core of most globular clusters [24]. Such a model provides a canonical form for many galaxy models widely used in astronomy. Even though f 0 < 0 for the King model, it is important to realize that, because of the Hamiltonian nature of the Vlasov-Poisson system (1), linear stability fails to imply nonlinear stability (even in the finite dimensional case). The Liapunov functional is usually required to prove nonlinear stability. In the Casimir-energy functional (2), it is natural to expect that the positivity of such a quadratic form H(2) f 0 [g] should imply stability for f 0 (E). However, there are at least two serious mathematical difficulties. First of all, it is very challenging to use the positivity of H(2) f 0 [g] to control the higher order remainder in H( f ) − H( f 0 ) to conclude stability [38]. For example, one of the remainder terms is f 3 whose L 2 norm is difficult to be bounded by a power of the stability norm. The nonsmooth nature of f 0 (E) also causes trouble here. Second of all, even if one can succeed in (2) controlling the nonlinearity, the positivity of H f0 [g] is only valid for certain perturbation of the form g = Dh [22]. It is not clear at all if any arbitrary, general perturbation can be reduced to the form Dh. To overcome these two difficulties, a direct variational approach was initiated by Wolansky [39], then further developed systematically by Guo and Rein in [14,15,17–19]. Their method avoids entirely the delicate analysis of the (2) second order variation H f0 in (3), which has led to the first rigorous nonlinear stability proof for a large class of f 0 (E). The high point of such a program is the nonlinear stability proof for every polytrope [18] f 0 (E) = (E 0 − E)k+ . Their basic idea is to construct galaxy models by solving a variational problem of minimizing the energy under some constraints of Casimir invariants. A concentration-compactness argument is used to show the convergence of the minimizing sequence. All the models constructed in this way are automatically nonlinearly stable. Unfortunately, despite its success, the King model can not be studied by such a variational approach. The Casimir function for the normalized King model is Q 0 ( f ) = (1 + f ) ln(1 + f ) − 1 − f,
(9)
which has very slow growth for f → ∞. As a result, the direct variational method fails. Recently, Guo and Rein [21] proved nonlinear radial stability among a class of measure-preserving perturbations
Q( f, L) = Q( f 0 , L), for Q ∈ Cc∞ and Q(0, L) ≡ 0. . S f0 ≡ f (t, r, vr , L) ≥ 0 : (10) The basic idea is to observe that for perturbations in the class S f0 , one can write g = (2) f − f 0 as Dh = {h, E}. Therefore, H(2) f 0 [g] = H f 0 [Dh], for which the positivity was proved in [22] for radial perturbations. To avoid the difficulty of controlling the (2) remainder term by H f0 [g], an indirect contradiction argument was used in [21]. As our second main result of this article, we establish nonlinear stability of King’s model for general perturbations with spherical symmetry:
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Theorem 1.2. The King’s model f 0 = [e E 0 −E − 1]+ is nonlinearly stable under spherically symmetric perturbations in the following sense: given any ε > 0 there exists ε1 > 0 such that for any compact supported initial data f (0) ∈ Cc1 with spherical symmetry, if d ( f (0) , f 0 ) < ε1 then sup d ( f (t) , f 0 ) < ε, 0≤t<∞
where the distance functional d ( f, f 0 ) is defined by (38). For the proof, we extended the approach in [27] for the 1 21 D Vlasov-Maxwell model. To prove nonlinear stability, we study the Taylor expansion of H( f ) − H( f 0 ). Two difficulties as mentioned before are: to prove the positivity of the quadratic form and to control the remainder. We use two ideas in [27]. The first idea is to use any introduced finite number of Casimir functional Q i f, L 2 as constraints. The difference from [21] is that we do not impose Q i f, L 2 = Q i f 0 , L 2 in the perturbation class, but expand the invariance equation Q i f (t) , L 2 − Q i f 0 , L 2 = Q i f (0) , L 2 − Q i f 0 , L 2 to the first order. In this way, we get a constraint for g = f − f 0 in the form that the coefficient of its projection to ∂1 Q i f 0 , L 2 is small. Putting these constraints together, we deduce that a finite dimensional projection of g to the space spanned by ∂1 Q i f 0 , L 2 is small. To control the remainder term, we use a duality argument. Noting that it is much easier to control the potential φ, we use a Legendre transformation to reduce the nonlinear term in g to a new one in φ only. The key observation is that the constraints on g in the projection form are nicely suited to the Legendre transformation and yield a non-local nonlinear term in φ only with the projections kept. By performing a Taylor expansion of this non-local nonlinear term in φ, the quadratic form becomes a truncated version of (A0 φ, φ) defined by (6), whose positivity can be shown to be equivalent to that of an Antonov functional. The remainder term now is only in terms of φ and can be easily controlled by the quadratic form. The new complication in the stellar case is that the steady distribution f 0 (E) is non-smooth and compactly supported. Therefore, we split the perturbation g into inner and outer parts, according to the support of f 0 . For the inner part, we use the above constrained duality argument and the outer part is estimated separately. 2. An Instability Criterion We consider a given steady distribution f 0 (x, v) = f 0 (E) which has a bounded support in x and v and f 0 is bounded, where the particle energy E = 21 |v|2 +U0 (x). The steady gravitational potential U0 (x) satisfies a nonlinear Poisson equation U0 = 4π f 0 dv. The corresponding linearized Vlasov-Poisson system around f 0 (E) takes the form φ = 4π f (t, x, v)dv. (11) ∂t f + v · ∇x f − ∇x U0 · ∇v f = ∇x φ · ∇v f 0 ,
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A growing mode solution (eλt f (x, v), eλt φ(x)) to (11) with λ > 0 satisfies λ f + v · ∇x f − ∇x U0 · ∇v f = f 0 v · ∇x φ. We define [X (s; x, v), V (s; x, v)] as the trajectory of d X (s;x,v) = V (s; x, v) ds d V (s;x,v) = −∇x U0 ds
(12)
(13)
such that X (0; x, v) = x, and V (0; x, v) = v. Notice that the particle energy E is constant along the trajectory. Integrating along such a trajectory for −∞ < s ≤ 0, we have 0 f (x, v) = eλs f 0 (E)V (s; x, v) · ∇x φ(X (s; x, v))ds −∞
= f 0 (E)φ(x) − f 0 (E)
0 −∞
λeλs φ(X (s; x, v))ds.
(14)
Plugging it back into the Poisson equation, we obtain an equation for φ, 0 −φ + [4π f 0 (E)dv]φ − 4π f 0 (E) λeλs φ(X (s; x, v))dsdv = 0. −∞
We therefore define the operator Aλ as Aλ φ ≡ −φ + [4π f 0 (E)dv]φ − 4π f 0 (E)
0 −∞
λeλs φ(X (s; x, v))dsdv. (15)
Lemma 2.1. Assume that f 0 (E) has a bounded support in x and v and f 0 is bounded. For any λ > 0, the operator Aλ : H 2 → L 2 is self-adjoint with the essential spectrum [0, +∞) . Proof. We denote
K λ φ = −4π [
f 0 (E)dv]φ + 4π
f 0 (E)
0 −∞
λeλs φ(X (s; x, v))dsdv.
Recall that f 0 (x, v) = f 0 (E) has a compact support ⊂ S ⊂ R3x × R3v . We may assume S = Sx × Sv , both open balls in R3 . Let χ = χ (|x|) be a smooth cut-off function for the spatial support of f 0 in the physical space Sx ; that is, χ ≡ 1 on the spatial support of f 0 and has compact support inside Sx . Let Mχ be the operator of multiplication by χ . Then K λ = K λ Mχ = Mχ K λ = Mχ K λ Mχ . Indeed, f 0 (x, v) = f 0 (X (s; x, v), V (s; x, v))
because of the invariance of E under the flow. So the support of f 0 (E)dv is contained in Sx , and 0 f 0 (E) λeλs φ(X (s; x, v))dsdv (K λ φ) (x) = −4π [ f 0 (E)dv]φ + 4π = −4π [
f 0 (E)dv]φ + 4π
= (Mχ K λ Mχ φ)(x).
−∞
0
−∞
λeλs f 0 (E)φ (X (s; x, v))dsdv (16)
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First we claim that K λ L 2 →L 2 ≤ 8π f 0 (E) dv . ∞
Indeed, the L 2 norm for the first term in K λ is easily bounded by 4π f 0 (E)dv ∞ . For the second term, by the duality argument, we have for any ψ ∈ L 2 , |
0
−∞
≤ 4π
4π λeλs f 0 (E)φ(X (s; x, v))dsdvψ(x)d x| 0
−∞
× = 4π
0 −∞
λe
λs
| f 0 (E)|φ 2 (X (s; x, v))dvd x
| f 0 (E)|ψ 2 (x)dvd x λe
= 4π
λs
1 2
1 2
ds
| f 0 (E)|φ 2 (x)dvd x
| f 0 (E)|φ 2 (x)dvd x
1 2
f 0 (E) dv φ2 ψ2 . ≤ 4π
1 2
| f 0 (E)|ψ 2 (x)dvd x
| f 0 (E)|ψ 2 (x)dvd x
1 2
ds
1 2
(17)
∞
Moreover, we have that K λ is symmetric. Indeed, for fixed s, by making a change of variable (y, w) → (X (s; x, v), V (s; x, v)), so that (x, v) = (X (−s; y, w), V (−s; y, w)), we deduce that
4π f 0 (E)
=
0
−∞
=
=
λeλs
0 −∞
4π f 0 (E) 4π f 0 (E)
λeλs φ(X (s; x, v))dsdvψ(x)d x 4π f 0 (E)φ(y)ψ(X (−s; y, w))dydwds
0
−∞ 0 −∞
λeλs ψ(X (−s; y, −w))φ(y)dydwds λeλs ψ(X (s; x, v))φ(x)dvd xds.
Here we have used the fact [X (s; y, w), V (s; y, w)] = [X (−s; y, −w), −V (s; y, −w)] in the last line. Hence (K λ φ, ψ) = (φ, K λ ψ). Since K λ = K λ Mχ and Mχ is compact from H 2 into L 2 space with support in Sx , so K λ is relatively compact with respect to −. Thus by Kato-Relich and Weyl’s Theorems, Aλ : H 2 → L 2 is self-adjoint and σess (Aλ ) = σess (−).
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Lemma 2.2. Assume that f 0 (E) has a bounded support in x and v and f 0 is bounded. Let k(λ) =
inf
φ∈D(Aλ ),||φ||2 =1
(φ, Aλ φ),
then k(λ) is a continuous function of λ when λ > 0. Moreover, there exists 0 < < ∞ such that for λ > , k(λ) ≥ 0. (18) Proof. Fix λ0 > 0, φ ∈ D(Aλ ), and ||φ||2 = 1. Then k(λ0 ) ≤ (φ, Aλ0 φ) ≤ (φ, Aλ φ) + |(φ, Aλ0 φ) − (φ, Aλ φ)| 0 ≤ (φ, Aλ φ) + 4π | f 0 (E)| [λeλs − λ0 eλ0 s ]φ(X (s; x, v))φ(x)dsdvd x ≤ (φ, Aλ φ) + 4π ≤ (φ, Aλ φ) + C
0
| f 0 (E)|
λ
−∞ λ0
−∞ 0 λ
−∞ λ0 ˜
λ˜ s λ˜ s ˜ ˜ [λ|s|e +e ]d λφ(X (s; x, v))φ(x)dsdvd x
˜
λs ˜ ˜ [λ|s|e + eλs ]d λds
≤ (φ, Aλ φ) + C| ln λ − ln λ0 |. We therefore deduce that by taking the infimum over all φ, k(λ0 ) ≤ k(λ) + C| ln λ − ln λ0 |. The same argument also yields k(λ) ≤ k(λ0 ) + C| ln λ − ln λ0 |. Thus |k(λ0 ) − k(λ)| ≤ C| ln λ − ln λ0 | and k(λ) is continuous for λ > 0. To prove (18), by (14), we recall from Sobolev’s inequality in R3 , |(K λ φ, ψ)| λs 4π f 0 (E)e ∇φ(X (s; x, v))V (s)dsdvψ(x)d x = 1/2 0 ≤ eλs · |ψ|2 | f 0 (E)|dvd x −∞ ×[ |∇φ(X (s))|2 | f 0 (E)||V (s) |2 d xdv]1/2 ds =
0
−∞
eλs
|ψ|2 | f 0 (E)|dvd x
1/2
v 2 |∇φ(x)|2 | f 0 (E)|d xdv]1/2 ds
C C ≤ ||ψ||6 ||∇φ||2 ≤ ||∇ψ||2 ||∇φ||2 , λ λ since f 0 has compact support. Therefore, (Aλ φ, φ) = ||∇φ||2 − (K λ φ, φ) ≥ (1 − for λ large.
C )||∇φ||2 ≥ 0 λ
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We now compute limλ→0+ Aλ . We first consider the case when the test function φ is spherically symmetric. We recall the linearized Vlasov-Poisson system for spherically symmetric solutions in the r, vr , L coordinates takes the form 2 L − U0 ∂vr f = ∂r U f ∂vr f 0 , ∂t f + vr ∂r f + r3 2 ∂rr U f + ∂r U f = 4π f dv. r For points (x, v) with E < E 0 and L > 0, the trajectory of (X (s; x, v), V (s; x, v)) in the coordinate (r, E, L) is a periodic motion described by the ODE (see [8]) dr (s) = vr (s), ds dvr (s) L2 = 3 − U0 (r ), ds r with the period T (E, L) = 2
r2 (E,L)
r1 (E,L)
dr , 2(E − U0 − L 2 /2r 2 )
where 0 < r1 (E, L) ≤ r2 (E, L) < +∞ are zeros of E − U0 − L 2 /2r 2 . See [21] for more details. For any function g(r, E, L), we define its trajectory average as r2 (E,L) √ g(r,E,L)dr r1 (E,L) 2(E−U0 −L 2 /2r 2 ) . (19) g(E, ¯ L) ≡ r (E,L) dr 2 √ r (E,L) 2(E−U0 −L 2 /2r 2 )
1
Lemma 2.3. For a spherically symmetric function φ(x) = φ (|x|) , we have 2 lim (Aλ φ, φ) = (A0 φ, φ) ≡ |∇φ| d x + 4π f 0 (E)dvφ 2 d x λ→0+
− 32π 3
E
min U0
=
0
|∇φ|2 + 32π 3
∞
r2 (E,L) f 0 (E)
f 0 (E)
r1 (E,L)
r1 (E,L)
√
r2 (E,L)
φdr 2(E−U0 −L 2 /2r 2 )
√
r2 (E,L)
r1 (E,L)
2
dr 2(E−U0 −L 2 /2r 2 )
d Ld E
dr d Ed L ¯ 2 (φ − φ) . 2(E −U0 − L 2 /2r 2 ) (20)
Proof. Given the steady state f 0 (E), U0 (|x|) and any radial function φ (|x|) , to find the limit of 2 (Aλ φ, φ) = |∇φ| d x + 4π f 0 (E)dvφ 2 d x 0 λs λe φ(X (s; x, v))ds φ (x) d xdv, (21) − 4π f 0 (E) −∞
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we study the following
0
lim
λ→0+ −∞
λeλs φ(X (s; x, v))ds.
(22)
Note that we only need to study (22) for points (x, v) with E = 21 |v|2 +U0 | (x|) < E 0 and L = |x × v| > 0, because in the third integral of (21) f 0 (E) has support in {E < E 0 } and the set {L = 0} has a zero measure. So by Lin’s lemma in [25],
0
lim
λ→0 −∞
λeλs φ(X (s; x, v))ds =
1 T
T
φ(X (s; x, v))ds.
0
Since φ(X (s; x, v) = φ(r (s)), a change of variable from s → r (s) leads to
T
φ(X (s; x, v))ds = 2
0
φ(r )dr . 2(E − U0 − L 2 /2r 2 )
r2
r1
Then
0
λs
lim
λ→0+ −∞
r2
λe φ(X (s; x, v))ds = 2
r1
φ(r )dr 2(E − U0 − L 2 /2r 2 )
/T (E, L) = φ¯ (E, L) ,
¯ Thus by the and the integrand in third term of (21) converges pointwise to f 0 (E)φφ. dominated convergence theorem, we have lim (Aλ φ, φ) ¯ d xdv = |∇φ|2 d x + 4π f 0 (E)φ 2 d xdv − 4π f 0 (E)φφ = |∇φ|2 d x + 4π f 0 (E)φ 2 d xdv r2 (E,L) E ∞ dr d Ed L 3 φ¯ (E, L) φ (r ) f 0 (E) − 32π 2(E − U0 − L/2r 2 ) min U0 0 r1 (E,L) = |∇φ|2 d x + 4π f 0 (E)φ 2 d xdv
λ→0+
− 32π 3
E
min U0
∞ 0
=
|∇φ|2 + 32π 3
r2 (E,L) f 0 (E) f 0 (E)
r1 (E,L)
r1 (E,L)
√
r2 (E,L)
φdr 2(E−U0 −L/2r 2 )
√
r2 (E,L)
r1 (E,L)
2
dr 2(E−U0 −L/2r 2 )
¯ 2 (φ − φ)
d Ed L
dr d Ed L 2(E − U0 − L/2r 2 )
.
This finishes the proof of the lemma. To compute limλ→0+ (Aλ φ, φ) for a more general test function φ, we use the following ergodic lemma which is a direct generalization of the result in [26].
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Lemma 2.4. Consider the solution (P (s; p, q) , Q (s; p, q)) to be the solution of a Hamiltonian system P˙ = ∂q H (P, Q) , Q˙ = −∂ p H (P, Q) , with (P (0) , Q (0)) = ( p, q) ∈ Rn × Rn . Denote 0 Qλ m = λeλs m (P (s) , Q (s)) ds. −∞
Then for any m ( p, q) ∈ L 2 (Rn × Rn ), we have Qλ m → Pm strongly in L 2 (Rn × Rn ). Here P is the projection operator of L 2 (Rn × Rn ) to the kernel of the transport operator D = ∂q H ∂ p − ∂ p H ∂q and Pm is the phase space average of m in the set traced by the trajectory. Proof. Denote U (s) : L 2 (Rn × Rn ) → L 2 (Rn × Rn ) to be the unitary semigroup U (s) m = m (P (s) , Q (s)). By the Stone Theorem ([40]), U (s) is generated by i R = D, where R = −i D is self-adjoint and +∞ eiαs d Mα , U (s) =
−∞
1
where Mα ; α ∈ R is spectral measure of R. So 0 0 λs λs iαs λe m(P(s), Q(s))ds = λe e d Mα m ds = −∞
R
−∞
On the other hand, the projection is P = M{0} = and ξ(0) = 1. Therefore 0 2 λs = λe m(P(s), Q(s))ds − Pm −∞
L2
λ d Mα m. R λ + iα
R ξ d Mα ,
R
where ξ(α) = 0 for α = 0
2 λ dMα m2 2 − ξ(α) λ + iα L
by orthogonality of the spectral projections. By the dominated convergence theorem this expression tends to 0 as λ → 0+, as we wished to prove. The explanation of Pm as the phase space average of m is in our remark below. 0 Remark 1. Since −∞ λeλs ds = 1, the function
λ
Q m (x, v) =
0
−∞
λeλs m (P(s), Q(s)) ds
(23)
is a weighted time average of the observable m along the particle trajectory. By the same proof of Lemma 2.4, we have 1 T lim m (P(s), Q(s)) ds = Pm. (24) T →∞ T 0 But from the standard ergodic theory ([3])of Hamiltonian systems, the limit of the above time average in (24) equals the phase space average of m in the set traced by the trajectory. Thus Pm has the meaning of the phase space average of m and Lemma 2.4 states that
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the limit of the weighted time average (23) yields the same phase space average. In particular, if the particle motion is ergodic in the invariant set S I determined by the invariants E 1 , . . . , Ik , and if dσ I denotes the induced measure of Rn × Rn on S I , then 1 m ( p, q) dσ I ( p, q) . Pm = (25) σ I (S I ) S I For integrable systems, using action-angle variables (J1 , . . . , Jn ; ϕ1 , . . . , ϕn ) we have (Pm) (J1 , . . . , Jn ) = (2π )−n
2π
2π
···
0
m (J1 , . . . , Jn , ϕ1 , . . . , ϕn ) dϕ1 , . . . dϕn
0
(26)
for the generic case with independent frequencies (see [4]). Recall the weighted L 2 space L 2| f | in (5). Then U (s) : L 2| f | → L 2| f | defined by 0 0 0 U (s) m = m (X (s; x, v), V (s; x, v)) is an unitary group, where (X (s; x, v), V (s; x, v)) is the particle trajectory (13). The generator of U (s) is D = v · ∂x − ∇x U0 · ∇v and R = −i D is self-adjoint by the Stone Theorem. By the same proof, Lemma 2.4 is still valid in L 2| f | . In particular, for any φ (x) ∈ L 2 R3 we have 0
0
−∞
λeλs φ(X (s; x, v))ds → Pφ
(27)
in L 2| f | , where P is the projector of L 2| f | to ker D. 0 0 Now we derive an explicit formula for the above limit Pφ. Note that as in the proof of Lemma 2.3, we only need to derive the formula of Pφ for points (x, v) with E < E 0 and L > 0. Since U0 (x) = U0 (r ), the particle motion (13) in such a center field is integrable and has been well studied (see e.g. [8,4]). For particles with energy E < E 0 < 0, L > 0 and momentum L = x × v, the particle orbit is a rosette in the annulus A E,L = {r1 (E, L) ≤ r ≤ r2 (E, L)} = E − U0 − L 2 /2r 2 ≥ 0 , So we can consider the particle motion lying on the orbital plane perpendicular to L. to be planar. For such case, the action-angle variables are as follows (see e.g. [30]): the actions variables are 2π , T (E, L)
Jr =
Jθ = L ,
where T (E, L) = 2
r2 (E,L)
r1 (E,L)
dr 2(E − U0 − L 2 /2r 2 )
is the radial period, the angle variable ϕr is determined by dϕr =
dr 2π , T (E, L) 2(E − U0 − L 2 /2r 2 )
Unstable and Stable Galaxy Models
801
and ϕθ = θ − θ , where d (θ ) = and 1 θ (E, L) = T (E, L)
Lr −2 − θ 2(E − U0 − L 2 /2r 2 )
r2 (E,L)
dr
L
dr 2(E − U0 − L 2 /2r 2 ) is the average angular velocity. For any function φ (x) ∈ H 2 R3 , we denote φ L (r, θ ) Then by (26), for the to be the restriction of φ in the orbital plane perpendicular to L. generic case when the radial and angular frequencies are independent, we have 2π 2π −2 φ L dϕθ dϕr (Pφ) E, L = (2π ) r1 (E,L)
0
1 = π T (E, L)
r2
0 r2 (E,L) 2π
r1 (E,L)
0
φ L (r, θ ) dθ dr 2(E − U0 − L 2 /2r 2 )
In particular, for a spherically symmetric function φ = φ (r ), we recover r2 (E,L) φ(r )dr 2 . (Pφ) (E, L) = T (E, L) r1 (E,L) 2(E − U0 − L 2 /2r 2 )
.
(28)
(29)
We thus conclude the following Lemma 2.5. Assume that f 0 (E) has a bounded support in x and v and f 0 is bounded. For any φ ∈ H 1 R3 , we have lim (Aλ φ, φ) = (A0 φ, φ) 2 2 = |∇φ| d x + 4π f 0 (E)dvφ d x − 4π f 0 (E) (Pφ)2 d xdv = |∇φ|2 d x + 4π f 0 (E) (φ − Pφ)2 d xdv, (30)
λ→0+
where P is the projector of L 2| f | to ker D and more explicitly Pφ is given by (28). The 0 limiting operator A0 is A0 φ = −φ + [4π f 0 (E)dv]φ − 4π f 0 (E)Pφdv. (31) Now we give the proof of the instability criterion. Proof of Theorem 1.1. We define λ0 = sup λ. k(λ)<0
By assumption (7) and Lemmas 2.1 and 2.5, we deduce that 0 < λ0 ≤ < ∞.
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Therefore, by the continuity of k(λ), we have k(λ0 ) = 0. Hence, there exists an increasing sequence n ≥ 1 of 0 < λn < λn+1 < λ0 so that λn → λ0 , kn ≡ k(λn ) < 0, and kn → k(λ0 ) = 0. By Lemma 2.1, kn are negative eigenvalues of Aλn . So we get a sequence 0 = φn ∈ H 2 such that Aλn φn = kn φn (32) with kn < 0, kn → 0 and λn → λ0 > 0, as n → ∞. Recall χ (|x|) to be the cutoff function of the x−support of f 0 (E) such that χ ≡ 1 in { f 0 (E) > 0} . We claim that χ φn is a nonzero function for any n. Suppose otherwise, χ φn ≡ 0, then from Eq. (32) we have (− − kn ) φn = 0 which implies that φn = 0 since kn < 0, a contradiction.Thus we can normalize φn by χ φn 2 = 1. Taking inner product of (32 ) with φn and integrating by parts, we have φn 22 ≤ −4π f 0 (E)φn2 dvd x 0 λn eλn s φn (X (s; x, v))dsφn (x) d x + 4π f 0 (E) −∞ = −4π f 0 (E) (χ φn )2 dvd x 0 λn eλn s (χ φn ) (X (s; x, v))ds (χ φn ) (x) dvd x + 4π f 0 (E) −∞ ≤ 8π f 0 (E)dv χ φn 22 . ∞
Here in the second equality above, we have used the fact χ = 1 on the support of f 0 (E) and that (χ φn ) (X (s; x, v)) = φn (X (s; x, v)χ due to the invariance of the support under the trajectory flow, as in (16). In the last inequality, we use the same estimate as in (17). Thus, sup ||φn || L 6 ≤ C sup φn 2 < C , n
n
for some constant C independent of n. Then there exists φ ∈ L 6 and ∇φ ∈ L 2 such that φn → φ weakly in L 6 , and ∇φn → ∇φ weakly in L 2 . This implies that χ φn → χ φ strongly in L 2 . Therefore χ φ2 = 1 and thus φ = 0. It is easy to show that φ is a weak solution of Aλ0 φ = 0 or −φ = −[4π
f 0 (E)dv]φ +4π f 0 (E)
0 −∞
λ0 eλ0 s φ(X (s; x, v))dsdv ≡ 4πρ. (33)
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We have that ρd x = − f 0 (E)φ (x) d xdv +
f 0 (E)φ (x) d xdv
=−
0
−∞ 0
+
−∞
λ0 e
λ0 s
λ0 e
λ0 s
f 0 (E)φ(X (s; x, v))d xdvds f 0 (E)φ(x)d xdvds = 0,
and by (33) ρ has compact support in Sx , the x−support of f 0 (E). Therefore from the ρ(y) dy, we have formula φ (x) = |x−y| ρ (y) ρ (y) ρ (y) φ (x) = dy = dy − dy = O |x|−2 , |x − y| |x − y| |x| for x large, and thus φ ∈ L 2 . By elliptic regularity, φ ∈ H 2 . We define f (x, v) by (14), then f ∈ L ∞ with the compact support in S. Now we show that eλ0 t [ f, φ] is a weak solution to the linearized Vlasov-Poisson system. Since φ satisfies the Poisson equation (33), we only need to show that f satisfies the linearized Vlasov equation (12) weakly. For that, we take any g ∈ Cc1 R3 × R3 , and (Dg) f d xdv R 3 ×R 3 = (Dg) f 0 (E)φ(x) d xdv R 3 ×R 3
−
R 3 ×R 3
(Dg) f 0 (E)
0 −∞
λ0 eλ0 s φ(X (s; x, v))dsd xdv
= I + I I. Since D is skew-adjoint, the first term is I =− g D f 0 (E)φ d xdv = − R 3 ×R 3
For the second term, 0 λ0 s II = − λ0 e =−
−∞ 0 −∞
R 3 ×R 3
λ0 e
λ0 s
R 3 ×R 3 0
R 3 ×R 3
f 0 (E)g Dφd xdv.
f 0 (E) Dg(x, v) φ (X (s; x, v)) d xdvds f 0 (E) (Dg) (X (−s), V (−s)) φ (x) d xdvds
d λ0 eλ0 s − g (X (−s), V (−s)) ds φ (x) d xdv ds R 3 ×R 3 −∞ 0 = f 0 (E) λ0 g (x, v) − λ20 eλ0 s g (X (−s), V (−s)) ds φ (x) d xdv
f 0 (E)
=−
R 3 ×R 3
=
R 3 ×R 3
−∞
f 0 (E)λ0 φ (x) − f 0 (E)
= λ0 = λ0
3 3 R ×R
R 3 ×R 3
f 0 (E)φ (x) − f 0 (E)
f gd xdv.
0
−∞ 0 −∞
λ20 eλ0 s φ (X (s), V (s)) ds
g (x, v) d xdv
λ0 eλ0 s φ (X (s), V (s)) ds g d xdv
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Thus we have R 3 ×R 3
(Dg) f d xdv =
R 3 ×R 3
λ0 f − f 0 (E)Dφ gd xdv
which implies that f is a weak solution to the linearized Vlasov equation λ0 f + D f = f 0 (E) v · ∇x φ.
Remark 2. Consider an anisotropic spherical galaxy with f 0 (x, v) = f 0 E, L 2 . For a radial symmetric growing mode eλt (φ, f ) with φ = φ (|x|) and f = f |x| , E, L 2 . The linearized Vlasov equation (11) becomes λ f + v · ∇x f − ∇x U0 · ∇v f ∂ f0 ∂ f0 2 |x × v| ∇ = ∇x φ · ∇v f 0 = ∇x φ · v+ v ∂E ∂ L2 ∂ f0 ∂ f0 ∂ f0 x v + 2 2 [(x × v) × x] = v · ∇x φ, = φ (|x|) · |x| ∂E ∂L ∂E which is of the same form as in the isotropic case (22). So by the same proof of Theorem 1.1, we also get an instability criterion for radial perturbations of anisotropic galaxy, in terms of the quadratic form (20) with f 0 (E) being replaced by ∂∂ fE0 . 3. Nonlinear Stability of the King’s Model In the second half of the article, we investigate the nonlinear stability of the King model (8). We first establish: Lemma 3.1. Consider spherical models f 0 = f 0 (E) with f 0 < 0 on the support of f 0 . Then the operator A0 : Hr2 → L r2 , f 0 dv]φ − 4π f 0 Pφdv A0 φ = −φ + [4π is positive, where Hr2 and L r2 are spherically symmetric subspaces of H 2 and L 2 , and the projection Pφ is defined by (29). Moreover, for φ ∈ Hr2 we have (34) (A0 φ, φ) ≥ ε0 |∇φ|22 + |φ|22 for some constant ε0 > 0. Proof. Define k0 = inf (A0 φ, φ)/(φ, φ) . We want to show that k0 > 0. First, by the compact embedding of Hr2 → L r2 it is easy to show that the minimum can be obtained and k0 is the lowest eigenvalue. Let A0 φ0 = k0 φ0 with φ0 ∈ Hr2 and φ0 2 = 1. The fact that k0 ≥ 0 follows immediately from Theorem 1.1 and the nonexistence of radial modes ([9,22]) for monotone spherical models. The proof of k0 > 0 is more delicate. For that, we relate the quadratic form (A0 φ, φ) to the Antonov functional (4). We define D = 2,r v ·∂x −∇x U0 ·∇v to be the generator of the unitary group U (s):L 2,r | f | → L | f | defined by 0
0
Unstable and Stable Galaxy Models
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U (s) m = m (X (s; x, v), V (s; x, v)) . Here L 2,r | f | is the spherically symmetric subspace 0
of L 2| f | , which is preserved under the flow mapping U (s). By the definition of Pφ, we 0 have φ0 − Pφ0 ⊥ ker D. By the Stone Theorem i D is self-adjoint and in particular D is closed. Therefore by the closed range theorem ([40]), we have (ker D)⊥ = R (D) , where R (D) is the range of D. So there exists h ∈ L 2,r | f0 | such that Dh = φ0 − Pφ0 . Moreover, since φ0 − Pφ0 is even in v and the operator D reverses the parity in v, the function h is odd in v. Define f − = f 0 h. We have k0 = (A0 φ0 , φ0 ) = |∇φ0 |2 d x + 4π f 0 (φ0 − Pφ0 )2 d xdv f (φ0 − Pφ0 ) φ0 d xdv = |∇φ0 |2 d x − 8π 0 f (φ0 − Pφ0 )2 d xdv + 4π 0 D f − 2 1 d xdv + 2 φ0 D f − dvd x + |∇φ0 |2 d x = 4π f 4π 0 D f − 2 1 1 − 2 d xdv + |∇φ0 | d x = 4π φ0 φ d x + f 2π 4π 0 D f − 2 1 2 − d xdv + |∇φ0 | − 2∇φ0 · ∇φ d x = 4π f 4π 0 D f − 2 − 2 1 ∇φ d x , d xdv − ≥ 4π f 4π 0 where φ − = 4π D f − dv. Notice that the last expression above is the Antonov − − − functional H f , f . Since f is spherical symmetric and odd in v, we have − −4π > 0 by the proof in [22] which was further clarified in [33] and [21]. H f ,f Therefore we get k0 > 0 as desired and (A0 φ, φ) ≥ k0 |φ|22 . To get the estimate (34), we rewrite |∇φ|2 d x + 4π f 0 (φ − Pφ)2 d xdv + (1 − ε) (A0 φ, φ) (A0 φ, φ) = ε ≥ ε |∇φ|2 d x − 4π ε φ − Pφ2L 2 + (1 − ε) k0 |φ|22 | f0 | ≥ ε |∇φ|2 d x −8π ε φ2L 2 +(1 − ε) k0 |φ|22 (since P L 2 →L 2 ≤ 1) | f0 | | f0 | | f0 | k0 |φ|22 ≥ ε |∇φ|2 d x + ((1 − ε) k0 − Cε) |φ|22 ≥ ε |∇φ|2 d x + 2 if ε is small enough. The estimate (34) follows with ε0 = min ε, k20 . Next, we will approximate the ker D by a finite dimensional approximation. Let ∞ {ξi (E, L) = αi (E)βi (L)}i=1 be a smooth orthogonal basis for the subspace ker D =
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Y. Guo, Z. Lin
2,r {g(E, L)} ⊂ L 2,r | f0 | . Define the finite-dimensional projection operator P N : L | f0 | → L 2,r | f | by 0
PN h ≡
N (h, ξi )| f | ξi
(35)
0
i=1
and the operator A N : Hr2 → L r2 by A N φ = −φ + [4π
f 0 dv]φ − 4π
f 0 P N φdv.
Lemma 3.2. There exists K , δ0 > 0 such that when N > K we have A N φ, φ ≥ δ0 |∇φ|22
(36)
for any φ ∈ Hr2 . Proof. First we have A N → A0 strongly in L 2 . Indeed, for any φ ∈ Hr2 , N A φ − A0 φ = 4π f 0 (P N φ − Pφ) dv ≤ C P N φ − Pφ L 2| f | → 0 2 2 0 as N → ∞. We claim that for N sufficiently large, the lowest eigenvalue of A N is at least k0 /2, where k0 > 0 is the lowest eigenvalue of A0 , as in the proof of Lemma 3.1. Suppose otherwise, then there exists a sequence {λn } and {φn } ⊂ Hr2 with λn < k0 /2, φn 2 = 1 and An φn = λn φn . This implies that φn is uniformly bounded in L 2 , by elliptic estimate we have φn H 2 ≤ C for some constant C independent of n. Therefore there exists φ0 ∈ Hr2 such that φn → φ0 weakly in Hr2 . By the compact embedding of Hr2 → L r2 , we have φn → φ0 strongly in L r2 and φ0 2 = 1. The strong convergence of An φ0 → A0 φ0 implies that A n φn → A 0 φ0 weakly in L 2 . Let λn → λ0 ≤ k0 /2, then we have A0 φ0 = λ0 φ0 , soλ0 is an even smaller eigenvalue than k0 , a contradiction. Therefore we have A N φ, φ ≥ k20 |φ|22 for φ ∈ Hr2 , when N is large enough. The estimate (36) is established by the same proof of (34) in Lemma 3.1. Recalling (8) with f 0 = [e E 0 −E − 1]+ and Q 0 ( f ) = ( f + 1) ln( f + 1) − f, we further define functionals (related to the finite dimensional approximation of ker D) as
f
Ai ( f ) ≡
αi (− ln(s + 1) + E 0 )ds,
0
Q i ( f, L) ≡ Ai ( f )βi (L), for 1 ≤ i ≤ N for 1 ≤ i ≤ N . Clearly, ∂1 Q i ( f 0 , L) = αi (− ln( f 0 + 1) + E 0 )βi (L) = αi (E)βi (L) = ξi (E, L),
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807
N where {ξi (E, L)}i=1 are used to define P N in Lemma 3.2. Define the Casimir functional (E 0 < 0 ) 1 1 I ( f ) = [Q 0 ( f ) + |v|2 f − E 0 f ]d xdv − |∇φ|2 d x 2 8π which is invariant for spherically symmetric solution f to the nonlinear Vlasov-Poisson system. We introduce additional N invariants Ji ( f, L) ≡ Q i ( f, L)d xdv (37)
for 1 ≤ i ≤ N . We define to be the support of f 0 (E). We first consider 1 I ( f ) − I ( f 0 ) = [Q 0 ( f ) − Q 0 ( f 0 ) + |v|2 ( f − f 0 ) − E 0 ( f − f 0 )]d xdv 2 1 1 − ∇U0 · ∇(U − U0 ) − |∇(U − U0 )|2 d x 4π 8π 1 = [Q 0 ( f )− Q 0 ( f 0 )+(E − E 0 )( f − f 0 )]d xdv− |∇(U −U0 )|2 d x. 8π We define g = f − f0 ,
φ = U − U0
and
gin ≡ ( f − f 0 )1 ,
gout ≡ ( f − f 0 )1c , φin ≡
gin dv,
φout ≡
gout dv.
And we define the usual distance function for nonlinear stability and split it into 1 d( f, f 0 ) ≡ |∇φin |2 d x [Q 0 (gin + f 0 ) − Q 0 ( f 0 ) + (E − E 0 )gin ]d xdv + 8π (E − E 0 )gout d xdv + Q 0 (gout )d xdv + E≥E 0 1 (38) ≡ din + |∇φin |2 d x + dout , 8π for which each term is non-negative. We therefore can split: I ( f ) − I ( f0 ) 1 2 = |∇φin | d x [Q 0 ( f 0 + gin ) − Q 0 ( f 0 ) + (E − E 0 )gin ]d xdv − 8π (E − E 0 )gout d xdv + Q 0 (gout )d xdv + E≥E 0 1 1 − |∇φout |2 d x − ∇φout · ∇φin d x 8π 4π (39) = Iin + Iout . In the estimates below, we use C, C , C to denote general constants depending only on f 0 and quantities like f (t) L p ( p ∈ [1, +∞]) which equals f (0) L p and therefore always under control. We first estimate ∇φout 22 to be of higher order of d, which also implies that ∇φout · ∇φin d x is of higher order of d.
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Lemma 3.3. For ε > 0 sufficiently small, we have 1 |∇φout |2 d x ≤ C εd( f, f 0 ) + 5/3 [d( f, f 0 )]5/3 . ε Proof. In fact, since 2 |∇φout | d x ≤ C|| gout dv||2L 6/5 ≤ C|| gout 1 E 0 ≤E≤E 0 +ε dv||2L 6/5 + C|| gout 1 E>E 0 +ε dv||2L 6/5 . The first term is bounded by
5/3 2 3/5 3/5 [ gout dv] [ 1 E 0 ≤E≤E 0 +ε dv] d x
2/3 2 gout dvd x] × [ 1 E 0 ≤E≤E 0 +ε dv]3/2 d x 2 2 ≤ Cε[ gout dvd x] ≤ Cε[ gout dvd x]
≤[
≤ Cεd( f, f 0 ).
2 In the above estimates, we use that Q 0 (gout )dvd x ≥ c gout dvd x and 1 E 0 ≤E≤E 0 +ε dv ≤ Cε, which can be checked by an explicit computation when ε > 0 is sufficiently small such that E 0 + ε ≤ 0. On the other hand, since gout ≥ 0, and E 0 < 0, we have |v|2 gout = {2E − 2U0 }gout = 2{E − E 0 }gout + 2E 0 gout − 2U0 gout ≤ 2{E − E 0 }gout + 2 sup |U0 |gout . By the standard estimates (see [12, 120–121]) || gout 1 E>E 0 +ε dv||2L 6/5
≤C
9 7
gout 1 E>E 0 +ε d xdv
7 6
×
1 |v| gout 1 E>E 0 +ε d xdv 2
2
7 6 1 ≤ C||gout ||∞ (E − E 0 )gout 1 E>E 0 +ε d xdv ε
1 2 × gout 1 E>E 0 +ε d xdv 2(E − E 0 )gout 1 E>E 0 +ε d xdv + 2 sup |U0 | 1 3
≤
1 d ε
7 1 6 2 sup |U0 | 2 C d+ d ≤ 5/3 d 5/3 . ε ε
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By Lemma 3.3, and ∇φin 2 ≤ d 1/2 , we have ∇φout · ∇φin d x ≤ ∇φout 2 ∇φin 2 1 ≤ C ε1/2 d( f, f 0 ) + 5/6 [d( f, f 0 )]4/3 . ε By (39), for ε sufficiently small, 1 1 1/2 4/3 5/3 . Iout ≥ dout − C ε d( f, f 0 ) + 5/6 [d( f, f 0 )] + 5/3 [d( f, f 0 )] ε ε
(40)
To estimate Iin in (39), we split it into three parts: 1 Iin = τ |∇φin |2 d x [Q 0 ( f 0 + gin ) − Q 0 ( f 0 ) + (E − E 0 )gin + φin gin ]d xdv + 8π + (1 − τ ) [Q 0 ( f 0 + gin ) − Q 0 ( f 0 ) + (E − E 0 )gin 1 +(I − PN )φin gin ]d xdv + |∇φin |2 d x 8π + (1 − τ ) PN φin gin d xdv ≡ Iin1 + Iin2 + Iin3 , (41) 1 where φin = 4π gin dv, and |∇φin |2 d x. We estimate each term φin gin = − 4π in the following lemmas. Lemma 3.4. Iin1 ≥
τ din − Cτ 2
|∇φin |2 d x.
(42)
Proof. In fact, since the integration region is finite, we have
1 1 2 Iin = τ |∇φin | d x [Q 0 ( f 0 + gin )− Q 0 ( f 0 )+(E − E 0 )gin +φin gin ]d xdv+ 8π ≥τ [Q 0 ( f 0 + gin ) − Q 0 ( f 0 ) + (E − E 0 )gin ]d xdv − Cτ ||φin || L 6 ||gin || L 6/5 ≥τ [Q 0 ( f 0 + gin ) − Q 0 ( f 0 ) + (E − E 0 )gin ]d xdv − C τ ||∇φin || L 2 ||gin ||2 ≥
τ din − C τ ||∇φin ||22 , 2
since
din =
[Q 0 ( f 0 + gin ) − Q 0 ( f 0 ) + (E − E 0 )gin ]d xdv ≥ C||gin ||22 .
To estimate Iin2 , we need the following pointwise duality lemma from elementary calculus.
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Lemma 3.5. For any c, and any h, we have gc, f 0 (h) = Q 0 (h + f 0 ) − Q 0 ( f 0 ) − Q 0 ( f 0 )h − ch ≥ ( f 0 + 1)(1 + c − ec ). Proof. Direct computation yields that the minimizer f c of gc, f 0 (h) satisfies the EulerLagrange equation ln ( f c + f 0 + 1) − ln ( f 0 + 1) − c = 0, so f c = ( f 0 + 1) ec − 1 . Thus by using the Euler-Lagrange equation, we deduce min gc, f 0 (h) = gc,d ( f c ) = ( f c + f 0 + 1) ln(1 + f c + f 0 ) − ( f 0 + 1) ln(1 + f 0 ) − [1 + ln( f 0 + 1)] f c − c f c = ( f c + f 0 + 1)[ln(1 + f c + f 0 ) − ln( f 0 + 1) − c] + f c ln(1 + f 0 ) + c( f 0 + 1) − [1 + ln( f 0 + 1)] f c = ( f 0 + 1)(1 + c − ec ). Lemma 3.6. Iin2
(1 − τ ) δ0 ≥ 8π
1 2
3
|∇φin |2 d x − CeC d d 2 .
(43)
Proof. Recall (41). By using Lemma 3.5 for c = − (φin − PN φin ) and using the Taylor expansion, we have Iin2 = (1 − τ ) [Q 0 ( f 0 + gin ) − Q 0 ( f 0 ) + (E − E 0 )gin + (φin − PN φin ) f in ]d xdv 1 (1 − τ ) |∇φin |2 d x + 8π 1 (1 − τ ) |∇φin |2 d x + (1 − τ ) ≥ ( f 0 + 1)1 8π × (1 − φin + PN φin − e−φin +PN φin )d xdv 1−τ f (E) (φin − PN φin )2 d xdv ≥ |∇φin |2 d x − 4π 0 8π f (E) |φin − PN φin |3 d xdv − Ce|φin −PN φin |∞ 0 (1 − τ ) δ0 f (E) |φin − PN φin |3 d xdv. ≥ |∇φin |2 d x − Ce|φin −PN φin |∞ 0 8π (Note ( f 0 (E) + 1)1 = | f 0 (E)|). In the last line, we have used Lemma 3.2. To estimate the last term above and conclude our lemma, it suffices to show 1
|φin − PN φin |∞ ≤ C N d 2 .
Unstable and Stable Galaxy Models
811
This follows from the facts that for the fixed N smooth functions ξi , we have N |PN φin |∞ = (φin , ξi )| f | ξi ≤ C N |φin |∞ , 0 ∞
i=1
and since φin is spherically symmetric, r R 1 2 |φin | (r ) = u ρin (u) du + uρin (u) d∂ r 0 r √ 1 ≤ C R |ρin |2 ≤ C gin 2 ≤ C N d 2 , where ρin = gin dv and R is the support radius of ρin . We now estimate the term PN φin f in d xdv, for which we use the additional invariants (37). Lemma 3.7. For any ε > 0, we have 1 3 Iin ≤ C(d 1/2 (0) + ε1/2 d 1/2 + d)d 1/2 . ε
(44)
Proof. By the definition of Iin3 in (41), it suffices to estimate (gin , ξi ). From the construction of Q i and (37), we use Taylor expansion to get Ji ( f, L) − Ji ( f 0 , L) = Ji ( f 0 + gin , L) − Ji ( f 0 , L) + Ji (gout , L) = (gin , ξi ) + O(d) + Ji (gout , L), since the second derivative of Q i is bounded and ||gin ||2 ≤ Cd. Notice that |Ji (gout , L)| ≤ C||gout || L 1 ≤ C||1{E 0 ≤E≤E 0 +ε} gout || L 1 + C||1{E≥E 0 +ε} gout || L 1 C 1 ≤ ε1/2 ||gout || L 2 + ||1{E≥E 0 +ε} (E − E 0 )gout || L 1 ≤ C[ε1/2 d 1/2 + d]. ε ε It thus follows that 1 |(gin , ξi )| ≤ |Ji ( f (0), L) − Ji ( f 0 , L)| + C[ε1/2 d 1/2 + d] ε 1 ≤ Ci [d 1/2 (0) + ε1/2 d 1/2 + d]. ε Therefore
N 3 PN φin gin d xdv = (φin , ξi )| f | ξi gin d xdv Iin = (1 − τ ) 0 i=1
≤
N
N |φin |∞ |(ξi , gin )| (φin , ξi )| f0 | |(ξi , gin )| ≤ C
i=1
i=1
1 ≤ C N d 1/2 [d 1/2 (0) + ε1/2 d 1/2 + d]. ε
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Now we prove the nonlinear stability of King model. Proof of Theorem 1.2. The global existence of classical solutions of 3D Vlasov-Poisson system was shown in [34] for compactly supported initial data f (0) ∈ Cc1 . Let the unique global solution be ( f (t) , φ (t)). Let d (t) = d( f (t) , f 0 ). Combining estimates (40), (42), (43) and (44), we have I ( f (0)) − I ( f 0 ) = I ( f (t)) − I ( f 0 ) τ (1 − τ ) δ0 ≥ dout + din + − Cτ |∇φin |2 d x 2 8π 1 3 1 1 2 − C ε1/2 d (t) + 5/6 d (t)4/3 + 5/3 d (t)5/3 − CeC d(t) d (t) 2 ε ε 1 − Cd (t)1/2 [d 1/2 (0) + ε1/2 d (t)1/2 + d (t)]. ε Thus by choosing ε and τ sufficiently small, there exists δ > 0 such that 1 3 2 I ( f (0)) − I ( f 0 ) ≥ δ d(t) − C d (t)4/3 + d (t)5/3 + d (t)3/2 − CeC d(t) d (t) 2 − Cd (t)1/2 d 1/2 (0).
(45)
It is easy to show that I ( f (0)) − I ( f0 ) ≤ C d (0). Define the functions y1 (x) = 8/3 x 3 2 C δ x − Ce x − C x + x 10/3 + x 3 and y2 (x) = Cd (0)1/2 x + C d (0). Then the above estimates imply that y1 d (t)1/2 ≤ y2 d (t)1/2 . The function y1 is increasing in (0, x0 ), where x0 is the first maximum point. So if d (0) is sufficiently small, the line y = y2 (x) intersects the curve y = y1 (x) at points x1 , x2 , . . . , with x1 (d (0)) < x0 < x2 (d (0)) < · · · . Thus the inequality y1 (x) ≤ y2 (x) is valid in disjoint intervals [0, x1 (d (0))] and [x2 (d (0)) , x3 (d (0))], · · · . Because d (t) is continuous, we have that d (t)1/2 < x1 (d (0)) for all t < ∞, provided we choose d (0)1/2 < x0 . Since x1 (d (0)) → 0 as d (0) → 0, we deduce the nonlinear stability in terms of the distance functional d (t)1/2 . Acknowledgements. This research is supported partly by NSF grants DMS-0603815 and DMS-0505460. We thank the referees for many constructive comments to improve the presentation of the paper.
References 1. Antonov, V.A.: Remarks on the problem of stability in stellar dynamics. Soviet Astr, AJ., 4, 859–867 (1961) 2. Antonov, V.A.: Solution of the problem of stability of stellar system Emden’s density law and the spherical distribution of velocities, Vestnik Leningradskogo Universiteta, Leningrad University, 1962 3. Arnold, V.I., Avez, A.: Ergodic problems of classical mechanics. New York-Amsterdam: W. A. Benjamin, Inc., 1968 4. Arnold, V.I.: Mathematical methods of classical mechanics. New York-Heidelberg: Springer-Verlag, 1978 5. Barnes, J., Hut, P., Goodman, J.: Dynamical instabilities in spherical stellar systems. Astrophy. J. 300, 112–131 (1986) 6. Bartholomew, P.: On the theory of stability of galaxies. Monthly Notices of the Royal Astron. Soc. 151, 333 (1971) 7. Bertin, G.: Dynamics of Galaxies. Cambridge: Cambridge University Press, 2000 8. Binney, J., Tremaine, S.: Galactic Dynamics. Princeton, NJ: Princeton University Press, 1987
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9. Doremus, J.P., Baumann, G., Feix, M.R.: Stability of a Self Gravitating System with Phase Space Density Function of Energy and Angular Momentum. Astron. and Astrophys. 29, 401 (1973) 10. Gillon, D., Cantus, M., Doremus, J.P., Baumann, G.: Stability of self-gravitating spherical systems in which phase space density is a function of energy and angular momentum, for spherical perturbations. Astron. and Astrophys. 50(3), 467–470 (1976) 11. Fridman, A., Polyachenko, V.: Physics of Gravitating System. Vol I and II, Berlin-Heidelberg-New York: Springer-Verlag, 1984 12. Glassey, R.T.: The Cauchy problem in kinetic theory. Philadelphia, PA: SIAM, 1996 13. Goodman, J.: An instability test for nonrotating galaxies. Astrophys. J. 329, 612–617 (1988) 14. Guo, Y.: Variational method for stable polytropic galaxies. Arch. Rat. Mech. Anal. 147, 225–243 (1999) 15. Guo, Y.: On generalized Antonov stablility criterion for polytropic steady states. Contemp. Math. 263, 85–107 (1999) 16. Guo, Y., Rein, G.: Stable steady states in stellar dynamics. Arch. Rat. Mech. Anal. 147(3), 225–243 (1999) 17. Guo, Y., Rein, G.: Existence and stability of Camm type steady states in galactic dynamics. Indiana U. Math. J. 48, 1237–1255 (1999) 18. Guo, Y., Rein, G.: Isotropic steady states in galactic dynamics. Commun. Math. Phys. 219, 607–629 (2001) 19. Guo, Y., Rein, G.: Isotropic steady states in stellar dynamics revisited., Los Alamos Preprint, 2002 20. Henon, M.: Numerical Experiments on the Stability of Spherical Stellar Systems. Astron. and Astrophy. 24, 229 (1973) 21. Guo, Y., Rein, G.: A non-variational approach to nonlinear stability in stellar dynamics applied to the King model. Commun. Math. Phys. 271(2), 489–509 (2007) 22. Kandrup, H., Signet, J.F.: A simple proof of dynamical stability for a class of spherical clusters. The Astrophys. J. 298, 27–33 (1985) 23. Kandrup, H.E.: A stability criterion for any collisionless stellar equilibrium and some concrete applications thereof. Astrophys. J. 370, 312–317 (1991) 24. King, I.R.: The structure of star clusters. III. Some simple dynamical models. Astron. J. 71, 64 (1966) 25. Lin, Z.: Instability of periodicBG waves. Math. Res. Letts. 8, 521–534 (2001) 26. Lin, Z., Strauss, W.: Linear stability and instability of relativistic Vlasov-Maxwell systems. to appear in Comm. Pure Appl. Math. 27. Lin, Z., Strauss, W.: Nonlinear stability and instability of relativistic Vlasov-Maxwell systems. Comm. Pure Appl. Math. 60(6), 789–837 (2007) 28. Lin, Z., Strauss, W.: A sharp stability criterion for the Vlasov-Maxwell systems. submitted, http://arxiv. org/list/physics/0702023, 2007 29. Lynden-Bell, D.: The Hartree-Fock exchange operator and the stability of galaxies. Monthly Notices of the Royal Astron. Soc. 144, 189 (1969) 30. Lynden-Bell, D.: Lectures on stellar dynamics. Galactic dynamics and N-body simulations (Thessaloniki, 1993), Lecture Notes in Phys. 433, Berlin: Springer, 1994, pp. 3–31 31. Merritt, D.: Elliptical Galaxy Dynamics, The Publications of the Astronomical Society of the Pacific, Volume 111, Issue 756, 129–168 (1999) 32. Palmer, P.L.: Stability of collisionless stellar systems: mechanisms for the dynamical structure of galaxies. Dordrecht: Kluwer Academic Publishers, 1994 33. Perez, J., Aly, J.-J.: Stability of spherical stellar systems - I. Analytical results. Monthly Notices of the Royal Astron. Soc. 280(3), 689–699 (1996) 34. Pfaffelmoser, K: Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Differ. Eqs. 95(2), 281–303 (1992) 35. Rein, G.: Collisionless Kinetic Equations from Astrophysics - The Vlasov-Poisson system. In: Handbook of Differential Equations. Edited by C. M. Dafermos, E. Feireisl, Oxford: Elsevier B. V, 2007 36. Schaeffer, J.: Steady states in galactic dynamics. Arch. Rat. Mech. Anal. 172(1), 1–19 (2004) 37. Sygnet, J.F., des Forets, G., Lachieze-Rey, M., Pellat, R.: Stability of gravitational systems and gravothermal catastrophe in astrophysics. Astrophys. J. 276, 737–745 (1984) 38. Wan, Y-H.: On nonlinear stability of isotropic models in stellar dynamics. Arch. Rational. Mech. Anal. 147, 245–268 (1999) 39. Wolansky, G.: On nonlinear stability of polytropic galaxies. Ann. Inst. Henri Poincare 16, 15–48 (1999) 40. Yosida, K.: Functional analysis, Sixth edition. Grundlehren der Mathematischen Wissenschaften 123. Berlin-Heidelberg-New York: Springer-Verlag, 1980 Communicated by H. Spohn
Commun. Math. Phys. 279, 815–844 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0443-3
Communications in
Mathematical Physics
Integral Formulas for the Asymmetric Simple Exclusion Process Craig A. Tracy1 , Harold Widom2 1 Department of Mathematics, University of California, Davis, CA 95616, USA.
E-mail: [email protected]
2 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA.
E-mail: [email protected] Received: 26 April 2007 / Accepted: 3 August 2007 Published online: 5 March 2008 – © Springer-Verlag 2008
Abstract: In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice Z with nearest neighbor hopping rates p to the right and q = 1 − p to the left. For the most part we consider an N -particle system but for certain of these formulas we can take the N → ∞ limit. First we obtain, for the N -particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schütz for the case N = 2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m th left-most particle is at x at time t. In one of these formulas we can take the N → ∞ limit, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones. I Introduction Since its introduction nearly forty years ago [19], the asymmetric simple exclusion process (ASEP) has become the “default stochastic model for transport phenomena” [22]. Recall [8,9] that the ASEP on the integer lattice Z is a continuous time Markov process ηt , where ηt (x) = 1 if x ∈ Z is occupied at time t, and ηt (x) = 0 if x is vacant at time t. Particles move on Z according to two rules: (1) A particle at x waits an exponential time with parameter one, and then chooses y with probability p(x, y); (2) If y is vacant at that time it moves to y, while if y is occupied it remains at x. The adjective “simple” refers to the fact that the allowed jumps are only one step to the right, p(x, x +1) = p, or one step to the left, p(x, x −1) = q = 1− p. The totally asymmetric simple exclusion process (TASEP) allows jumps only to the right, so that p = 1. In a major breakthrough, Johansson [6], building on earlier work of Baik, Deift, and Johansson [3], related a probability in TASEP to a probability in random matrix
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theory. Specifically, if the initial configuration in TASEP is Z− then the probability that a particle initially at −m moves at least n steps to the right in time t equals the probability distribution of the largest eigenvalue in a (unitary) Laguerre random matrix ensemble. The realization [6,12] that the TASEP is a determinantal process [5,17] has led to considerable progress in our understanding of the one-dimensional TASEP. (See [18] for a recent review.) It is natural to ask to what extent these results for TASEP can be extended to ASEP. Since we no longer have the determinantal structure that is present in TASEP, random matrix theory methods, RSK-type bijections, or nonintersecting path techniques are not applicable (or at least not obviously so) to ASEP. However, it has been known for some time [1,4] that the generator of ASEP is a similarity transformation of the quantum spin chain Hamiltonian known as the XXZ model [20,21]. Since the XXZ Hamiltonian is diagonalizable by the Bethe Ansatz [21], it is reasonable to expect that these ideas are useful for ASEP. Indeed, Gwa and Spohn [4] applied Bethe Ansatz methods to the TASEP for a finite number of particles with periodic boundary conditions (i.e., for particles on a circle) to compute the dynamical scaling exponent. Subsequently for TASEP on Z for a finite number of particles, Schütz [16] showed that the probability that at time t the system is in configuration X = {x1 , . . . , x N }, given that its initial configuration was Y = {y1 , . . . , y N }, is expressible as an N × N determinant. From this determinant representation Rákos and Schütz [13] derived Johansson’s result relating TASEP to the Laguerre ensemble. The Rákos-Schütz derivation uses the crucial fact that for TASEP the probability for any particle depends only on the initial positions for that particle and those to its right, and so it is expressible in terms of probabilities for finite systems when the initial configuration is Z− .1 This is clearly no longer the case for ASEP. In this paper we obtain general integral formulas for probabilities in ASEP. For the most part we consider an N -particle system but for certain of these formulas we can take the N → ∞ limit, so that there are analogous formulas (involving infinite series) for infinite systems where the initial configuration is Y = {y1 , y2 , . . .},
y1 < y2 < · · · → +∞.
To specialize to infinite systems in TASEP with initial configuration Z− we would replace Z− by Y = Z+ and let q = 1. Denote by PY (X ; t) the probability that a system with initial configuration Y is in configuration X at time t. Then (Theorem 2.1) for an N -particle system PY (X ; t) is equal to a sum of N ! N -dimensional contour integrals. The integrand was suggested by the Bethe Ansatz (but there is no Ansatz!). The case N = 2 was established by Schütz [16] in different notation, and he also proposed that there was a general result such as this, with some contours. The main objective of the paper is to obtain probabilities for the individual particles at time t. To state the formulas we introduce some notation. We set ε(ξ ) = p ξ −1 + q ξ − 1, 1 The determinantal formula can be used to obtain scaling limit results for other initial conditions as well. See, e.g., [14].
Integral Formulas for the Asymmetric Simple Exclusion Process
817
and define an N -dimensional integrand I (x, Y, ξ ), with variable ξ = (ξ1 , . . . , ξ N ) and parameter x ∈ Z, by x−y −1 ξ j − ξi 1 − ξ1 · · · ξ N ξi i eε(ξi )t . (1.1) I (x, Y, ξ ) = p + qξi ξ j − ξi (1 − ξ1 ) · · · (1 − ξ N ) i< j
i
The parameter t appears in the last factor but not in the notation. For the probability for x1 (t), the position of the first particle at time t, the sum of N ! integrals given in Theorem 2.1 miraculously collapses into one integral. When p = 0 the first product in (1.1) is analytic when all the ξi lie inside some circle Cr with center zero and radius r . We also assume that r < 1. We show that for p = 0 and such r , we have (Theorem 3.1) N (N −1)/2 P(x1 (t) = x) = p ··· I (x, Y, ξ ) dξ1 · · · dξ N . (1.2) Cr
Cr
In order to take the N → ∞ limit we need integrals over large contours rather −y than small ones. This is because of the factors ξi i in the integrand (1.1). Given a set S ⊂ {1, . . . , N } we denote by I (x, Y S , ξ ) the integrand analogous to I (x, Y, ξ ), where only the variables ξi with i ∈ S occur. Then when q = 0 we have (Theorem 3.2) P(x1 (t) = x) = cS ··· I (x, Y S , ξ ) d |S| ξ, (1.3) S
CR
CR
where R > 1 is so large that all the poles of the first product in the integrand lie inside C R . The sum runs over all nonempty subsets S of {1, . . . , N } and c S are certain constants involving S and powers of p and q. Once we have this we are able to compute the expected value E(x1 (t)) and to take the N → ∞ limit in (1.3). If Y = {y1 , y2 , . . .} with y1 < y2 < · · · → +∞ then on the right side of (1.3) we simply take the sum over all finite sets S ⊂ Z+ , the resulting series being convergent. These results are contained in Sect. III. In Sect. IV we derive the analogue of (1.2) for the second-left particle. The main results of the paper are in Sect. V where we obtain the analogues of (1.2) and (1.3) for the general particle. The analogue of (1.2) has the form P(xm (t) = x) = c S,m,N ··· I (x, Y S , ξ ) d |S| ξ, (1.4) Cr
|S|>N −m
Cr
where we sum over all subsets S of {1, . . . , N } with cardinality at least N − m + 1, and c S,m,N is another explicitly given constant. (See Theorem 5.1) The analogue of (1.3) is c S,m ··· I (x, Y S , ξ ) d |S| ξ, (1.5) P(xm (t) = x) = |S|≥m
CR
CR
where we sum over all subsets S of {1, . . . , N } with cardinality at least m, and c S,m is yet another explicitly given constant. (See Theorem 5.2) Notice that in the latter representation the coefficients are independent of N , and this allows us to take the N → ∞ limit and so obtain probabilities for infinite systems. When the initial configuration is Z+ the sum over sets in (1.5) may be simplified to a sum over integers. (The corollary to Theorem 5.2.)
818
C. A. Tracy, H. Widom
Of course (1.2) and (1.3) and the second-particle formula are special cases of these, but we give their proofs first, so that we can introduce the new ingredients gently. The deduction of these formulas from Theorem 2.1 requires two algebraic identities which were discovered by computer computation of special cases. The first is ⎛ sgn σ ⎝ ( p + qξσ (i) ξσ ( j) − ξσ (i) ) σ ∈S N
i< j
⎞
−1 ξσ (2) ξσ2(3) ξσ3(4) · · · ξσN(N )
×
(1 − ξσ (1) ξσ (2) ξσ (3) · · · ξσ (N ) ) · · · (1 − ξσ (N −1) ξσ (N ) )(1 − ξσ (N ) )
i< j (ξ j − ξi ) N (N −1)/2
, =p j (1 − ξ j )
⎠
(1.6)
where the sum is over all permutations σ in the symmetric group S N . We also use an equivalent version of this identity, ⎛ p + qξσ (i) ξσ ( j) − ξσ (i) sgn σ ⎝ σ ∈S N
i< j
⎞ 1 ⎠ × (ξσ (1) − 1)(ξσ (1) ξσ (2) − 1) · · · (ξσ (1) ξσ (2) · · · ξσ (N ) − 1)
i< j (ξ j − ξi ) N (N −1)/2
, =q j (ξ j − 1)
(1.7)
obtained from (1.6) by interchanging p and q and letting ξi → 1/ξ N −i+1 . For the second identity we introduce the notation [N ] =
pN − q N , p−q
and then
[N ]! = [N ] [N − 1] · · · [1] ,
N m
=
[N ]! , [m]! [N − m]!
(1.8)
N is q m (N −m) times a q-binomial coefficient with where we set [0]! = 1. (Note that m q equal to our p/q. Hence the notation.) The identity is
N p + qξi ξ j − ξi N −1 1− · 1− ξ j = qm ξj m ξ j − ξi c i∈S
|S|=m
j∈S c
j∈S
(1.9)
j=1
for N ≥ m + 1. The sum runs over all subsets S of {1, . . . , N } with cardinality m, and S c denotes the complement of S in {1, . . . , N }. The proofs of these identities will be given in the last section.
Integral Formulas for the Asymmetric Simple Exclusion Process
819
II. Solution of the Master Equation We denote by Y = {y1 , . . . , y N } with y1 < · · · < y N the initial configuration of the process and write X = {x1 , . . . , x N } ∈ Z N . When x1 < · · · < x N then X represents a possible configuration of the system at a later time t. We denote by PY (X ; t) the probability that the system is in configuration X at time t, given that it was initially in configuration Y . Given X = {x1 , . . . , x N } ∈ Z N we set X i+ = {x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , x N }, X i− = {x1 , . . . , xi−1 , xi −1, xi+1 , . . . , x N }. The master equation for a function u on Z N × R+ is N d u(X ; t) = p u(X i− ; t) + q u(X i+ ; t) − u(X ; t) , dt
(2.1)
i=1
and the boundary conditions are, for i = 1, . . . , N − 1, u(x1 , . . . , xi , xi + 1, . . . , x N ; t) = p u(x1 , . . . , xi , xi , . . . , x N ; t) + q u(x1 , . . . , xi + 1, xi + 1, . . . , x N ; t).
(2.2)
The initial condition is u(X ; 0) = δY (X ) when x1 < · · · < x N . The basic fact is that if u(X ; t) satisfies the master equation, the boundary conditions, and the initial condition, then PY (X ; t) = u(X ; t) when x1 < · · · < x N .2 Recall that an inversion in a permutation σ is an ordered pair {σ (i), σ ( j)} in which i < j and σ (i) > σ ( j). We define Sαβ = −
p + qξα ξβ − ξα p + qξα ξβ − ξβ
and then Aσ =
{Sαβ : {α, β} is an inversion in σ }.
We also set ε(ξ ) = p ξ −1 + q ξ − 1. For now we shall assume p = 0, so the Aσ are analytic at zero in all the variables. Here and later all differentials dξ incorporate the factor (2πi)−1 , 2 The idea in Bethe Ansatz (see, e.g., [7,20,21]), applied to one-dimensional N -particle quantum mechanical problems, is to represent the wave function as a linear combination of free particle eigenstates and to incorporate the effect of the potential as a set of N − 1 boundary conditions. The remarkable feature of models amendable to Bethe Ansatz is that the boundary conditions for N ≥ 3 introduce no more new conditions, with the result that (2.2) involves only consecutive particles. The application of Bethe Ansatz to the evolution equation (master equation) describing ASEP begins with Gwa and Spohn [4] with subsequent developments by Schütz [16].
820
C. A. Tracy, H. Widom
Theorem 2.1. We have when p = 0, xi −yσ (i) −1 ··· Aσ ξσ (i) e i ε(ξi ) t dξ1 · · · dξ N , PY (X ; t) = σ ∈S N
Cr
Cr
(2.3)
i
where Cr is a circle centered at zero with radius r so small that all the poles of the Aσ lie outside Cr . Remark. For TASEP with p = 1 we have Sαβ = −(1 − ξα )/(1 − ξβ ). Fix α. In Aσ the factor 1 − ξα occurs for each inversion of the form {α, β} and (1 − ξα )−1 for each inversion of the form {β, α}. If α = σ (i) then the number of the former minus the number of the latter equals σ (i) − i. The number of inversions is the number of transpositions whose product is σ . These give (1 − ξσ (i) )σ (i)−i . Aσ = sgn σ Now the integrand in (2.3) factors and we obtain the sum of sgn σ times x −y −1 (1 − ξσ (i) )σ (i)−i ξσ i(i) σ (i) eε(ξσ (i) )t dξσ (i) Cr
i
=
i
Cr
(1 − ξ )σ (i)−i ξ xi −yσ (i) −1 eε(ξ )t dξ,
and so the sum over σ equals det (1 − ξ ) j−i ξ xi −y j −1 eε(ξ )t dξ . Cr
This is the determinant representation of PY (X ; t) obtained in [16]. To prove the theorem we shall show three things: (a) (b) (c)
The right side of (2.3) satisfies the master equation for all X ∈ Z N . The right side of (2.3) satisfies the boundary conditions for all X ∈ Z N . The right side of (2.3) satisfies the initial condition when x1 < · · · < x N .
Proof of (a). This is clear once the last factor in (2.3) is written as the exponential of i ε(ξσ (i) ) t. Proof of (b). We shall show that the boundary condition is satisfied pointwise by the integrand. Let Ti σ denote σ with the entries σ (i) and σ (i +1) interchanged. The boundary conditions will be satisfied provided that3 A Ti σ = Sσ (i+1),σ (i) Aσ 3 The Bethe Ansatz proposes an integrand as in (2.3) with some coefficients A . The fact that (2.2) only σ involves consecutive particles implies that satisfying it only requires relations between the coefficients As and A Ti σ . A straightforward computation shows that the relations are as stated.
Integral Formulas for the Asymmetric Simple Exclusion Process
821
for all σ . Let us see why this relation holds. Let α = σ (i), β = σ (i + 1), and suppose α > β. Then {α, β} is an inversion for σ but not for Ti σ , so Sαβ is a factor in Aσ but not in A Ti σ , and all other factors are the same. Therefore, using Sαβ Sβα = 1, we have A Ti σ = Sβα Aσ = Sσ (i+1),σ (i) Aσ . The same identity holds immediately if β > α, since {β, α} is an inversion for Ti σ but not for σ . Thus, (b) is established. Proof of (c). The initial condition is satisfied by the summand in (2.3) coming from the identity permutation id. So what we have to show is that xi −yσ (i) −1 ··· Aσ ξσ (i) dξ1 · · · dξ N = 0 σ =id Cr
Cr
i
when x1 < · · · < x N . This is the heart of the matter. We write I (σ ) for the integral corresponding to σ , and prove a series of lemmas. We think of σ also as the ordered N -tuple (σ (1), . . . , σ (N )). Lemma 2.1. Suppose that α appears in position α −1 in σ and that the entries preceding α are α − 2 of the numbers less than α. Then I (σ ) = 0. Proof. It follows from the assumption that there is a unique inversion of the form {α, β} and none of the form {δ, α}. Therefore the factor Sαβ appears in Aσ but no other Sαδ or Sδα . The variables ξα and ξβ appear in the integrand as x
Sαβ ξα α−1
−yα −1
x −yβ −1
ξβ α
.
We make the substitution ξα →
η
γ =α ξγ
,
so that η runs over a circle of radius r N , and we integrate with respect to ξβ . The power of ξβ that now appears is x −xα−1 +yα −yβ −1
ξβ α
.
(2.4)
The extra −1 in the exponent is due to the fact that dξα = dη/ξβ . Since α can appear in no other S factor, the only one that can introduce a pole inside Cr in the ξβ integration is Sαβ , which becomes
p + q η γ =α,β ξγ−1 − η γ =α ξγ−1 − .
p + q η γ =α,β ξγ−1 − ξβ The apparent simple pole at ξβ = 0 coming from the third summand in the numerator does not occur because the exponent in (2.4) is positive since yβ < yα and xα > xα−1 . The denominator is bounded away from zero when ξβ is inside Cr , since |ξβ | ≤ r and the second summand has absolute value O(r 2 ), so there is no pole inside Cr and the integral is zero. Lemma 2.2. Suppose that in the permutations σ and σ the entry α appears to the left of two adjacent smaller entries β and γ , and that the permutations differ only by an interchange of β and γ . Then I (σ ) + I (σ ) = 0.
822
C. A. Tracy, H. Widom
Proof. The pairs {α, β} and {α, γ } are inversions for both σ and σ , so Sαβ Sαγ is a factor in both Aσ and Aσ . Suppose for definiteness that β is to the left of γ in σ , so α = σ (i), β = σ ( j), γ = σ ( j + 1) with i < j. We make the substitutions η
ξα →
δ=α ξδ
.
Then the powers of ξβ and ξγ appear as x −xi +yα −yβ −1
ξβ j
x
ξγ j+1
−xi +yα −yγ −1
.
(2.5)
Both exponents are positive. We’ll integrate with respect to ξβ and ξγ . Any S other than Sαβ or Sαγ involving β or γ will not introduce poles, as in the proof of Lemma 2.1. The product Sαβ Sαγ becomes p+qη
−1 δ=α,β ξδ
p+qη
−η
−1 δ=α,β ξδ
−1 δ=α ξδ
p+qη
− ξβ
ξδ−1 − η
δ=α,γ
p+qη
δ=α,γ
−1 δ=α ξδ
ξδ−1 − ξγ
.
We do the ξβ integration first. The first factor has a pole at zero as before but, as before, does not introduce one in the integrand because of the power of ξβ in (2.5). Assume for the moment that q = 0. Then the second factor has a pole at ξβ =
qη ξγ − p
δ=α,β,γ
ξδ−1 .
(2.6)
The new second factor, its residue, has a pole of order 1 at ξγ = 0 (but again that does not contribute) and nowhere else. But the new first factor has a pole where ξγ =
qη ξβ − p
δ=α,β,γ
ξδ−1 ,
(2.7)
with ξβ satisfying (2.6). Then ξγ satisfies a quadratic equation with one of the roots inside Cr , in fact O(r 3 ), but all we will need is that (2.6) and (2.7) imply that ξβ = ξγ . Now we compare this with σ . The variables ξβ and ξγ occur in different positions in σ and σ and so their powers are different. After the variable change, for σ they are given by (2.5) whereas for σ they are x −xi +yα −yγ −1
ξγ j
x
ξβ j+1
−xi +yα −yβ −1
.
When we eventually have ξβ = ξγ these are the same. The only difference, then, is the factor Sβγ . It occurs for σ and not σ when β > γ , and for σ and not σ when β < γ . (Recall that β occurs to the left of γ in σ .) It equals −1 when ξβ = ξγ , so the sum of integrals equals zero. Recall that we assumed q = 0. If q = 0 then the integrals with respect to ξβ , after the substitution, for both σ and σ are zero, so I (σ ) = I (σ ) = 0 then. Lemma 2.3. For n > 1 the permutations in Sn can be grouped in pairs in such a way that the permutations in a pair differ only by one interchange of adjacent elements.
Integral Formulas for the Asymmetric Simple Exclusion Process
823
Proof. Let two permutations form a pair if they differ by an interchange of the first two entries. Lemma 2.4. For any N the set S N \{id} is the union of disjoint subsets, each of which consists of either of a single permutation satisfying Lemma 2.1 or a pair of permutations satisfying Lemma 2.2. Proof. We use induction, so we assume the result holds for N − 1. It clearly holds for N = 2, so we assume N > 2. For the set of permutations in which N appears in slot N we apply the induction hypothesis. Those permutations in which N appears in slot N −1 all satisfy Lemma 2.1 with α = N . Consider all those permutations that begin with a fixed (α1 α2 · · · α N −n−1 N ) with n ≥ 2, so N is in slot N − n. There are n! of them, corresponding to the permutations of the n remaining entries. Pair these permutations as in Lemma 2.3. The pairs of permutations in S N corresponding to these satisfy Lemma 2.2 with α = N . Putting all these together gives the desired decomposition of S N \{id}. Combining Lemmas 2.1, 2.2, and 2.4 completes the proof of (c), and so of Theorem 2.1. Remark. In case q = 0 the same formula (2.3) holds when the circle is sufficiently large instead of small. A similar argument to the one just given should hold, but there is another way to see this. If we set X − = {−x N , . . . , −x1 }, Y − = {−y N , . . . , −y1 } and denote by P˜ the probability density for the process with p and q interchanged, then PY (X ; t) = P˜Y − (X − ; t). If we apply (2.3) to this other process and then make the substitutions ξi → ξi−1 in the integrals we obtain (2.3) with a large C R . This duality will be use again in Sect. V, in the derivation of (1.5) from (1.4).
III. The Left-most Particle Here we determine the probability that the left-most particle x1 is at x at time t. Theorem 3.1. With Cr as before and I (x, Y, ξ ) given by (1.1), we have when p = 0 P(x1 (t) = x) = p N (N −1)/2
Cr
···
Cr
I (x, Y, ξ ) dξ1 · · · dξ N .
(3.1)
Proof. Since x1 < · · · < x N we may rewrite X = {x1 , x2 , . . . , x N } as x, x + z 1 , x + z 1 + z 2 , . . . , x + z 1 + · · · + z N −1 . Then P(x1 (t) = x) equals the sum of PY (X ; t) over all z i > 0. After summing, the integrand in (2.3) becomes
Aσ
−1 (1 − ξσ (1) · · · ξσ (N ) ) ξσ (2) ξσ2(3) · · · ξσN(N )
(1−ξσ (1) ξσ (2) · · · ξσ (N ) )(1−ξσ (2) · · · ξσ (N ) ) · · · (1 − ξσ (N ) )
x−y −1 ξi i eε(ξi )t . i
824
C. A. Tracy, H. Widom
If we observe that Aσ = sgn σ
i< j σ (i)>σ ( j)
p + qξσ (i) ξσ ( j) − ξσ (i) = sgn σ p + qξσ (i) ξσ ( j) − ξσ ( j)
( p + qξσ (i) ξσ ( j) − ξσ (i) )
i< j
( p + qξi ξ j − ξi )
,
i< j
(3.2) then we see that the theorem follows from (1.6). We derive here the alternative expression for P(x1 (t) = x). Given a set S ⊂ {1, . . . , N } there is a corresponding set Y S = {yi : i ∈ S} and corresponding |S|dimensional integrand I (x, Y S , ξ ). We define σ (S) = i∈S i, the sum of the indices in S. Theorem 3.2. When q = 0 we have P(x1 (t) = x) =
S
p σ (S)−|S| q σ (S)−|S|(|S|+1)/2
CR
···
CR
I (x, Y S , ξ ) d |S| ξ,
(3.3)
where R is so large that all the poles of the integrand lie inside C R . The sum runs over all nonempty subsets S of {1, . . . , N }. We begin with a lemma that replaces integrals such as appear in (3.1) by sums of integrals over large contours. Suppose f (ξ1 , . . . , ξ N ) is analytic for all ξi = 0 and that for i > k, f (ξ1 , . . . , ξ N ) = O(ξk ), ξi →(ξk − p)/qξk
as ξk → 0, uniformly when all ξ j with j = k are bounded and bounded away from zero. Define I f (ξ ) =
i< j
ξ j − ξi f (ξ1 , . . . , ξ N )
. p + qξi ξ j − ξi i (1 − ξi )
For a subset S of {1, . . . , N } let I f,S (ξ ) denote the analogous function where the variables are the ξi with i ∈ S, and in f (ξ1 , . . . , ξ N ) the ξi with i ∈ S c are replaced by 1. Lemma 3.1. Under the stated assumptions on f (ξ1 , . . . , ξ N ), we have when p, q = 0
Cr
···
Cr
I f (ξ ) d N ξ =
S
p |S
c |−σ (S c )
q σ (S)−|S|(|S|+1)/2
CR
···
CR
I f,S (ξ ) d |S| ξ ,
(3.4)
where r is so small that the poles of the integrand on the left lie outside Cr and R is so large that the poles of the integrand on the right lie inside C R . The sum runs over all subsets S of {1, . . . , N }. When S is empty the integral is interpreted as f (1, . . . , 1). Proof. We use induction. The result is easily seen to be true when N = 1, so we assume N > 1 and that the lemma holds for N − 1. We expand the ξ N -contour on the left side. In addition to the pole at ξ N = 1 we encounter poles at ξ N = (ξk − p)/qξk . We claim that the residue at this pole, when integrated over ξk , will give zero. The factor
Integral Formulas for the Asymmetric Simple Exclusion Process
825
f (ξ1 , . . . , ξ N ), after substituting for ξ N its value at the pole, is O(ξk ) as ξk → 0 by the assumption on f , while
i< j (ξi − ξ j )
, i (1 − ξi ) after substituting for ξ N its value at the pole, will be of the order ξk−N +2 at ξk = 0. The residue of 1/( p + qξk ξ N − ξk ) at the pole is 1/q ξk . The factor 1/( p + qξi ξ N − ξi ) with i = k equals ξk /( p(ξk − ξi )). The factor ξk − ξi is cancelled by the same factor in the numerator, so no new poles in the ξk variable are introduced. So the 1/( p + qξi ξ N − ξi ) combined, including the contribution of the residue, give the power ξkN −3 . Thus the product of all factors combined is O(1). Hence the ξk integral equals zero, as claimed. So after we expand the ξ N -contour we have an integral where ξ N is over an arbitrarily large contour C R and the other ξi over small contours Cr , and another integral (coming from the pole at ξ N = 1) in which ξ N does not appear and the other ξi are over Cr . Let us consider the latter. The integral we get is the left side of (3.4) with N replaced by N − 1 times ξ N − ξi 1 = N −1 . (3.5) p + qξi ξ N − ξi ξ N =1 p i
(Notice that ξ N appears in the denominator of I f (ξ ) as 1 − ξ N , and the pole at ξ N = 1 was outside the contour. The two minus signs cancel when we compute the contribution from the pole.) Our induction hypothesis tell us that this equals c c 1 p |S |−σ (S ) ··· I f,S (ξ ) d |S| ξ. p N −1 q σ (S)−|S|(|S|+1)/2 C R CR S⊂{1,...,N −1}
Now S c here indicates complement with respect to {1, . . . , N − 1}. If we want to express this in terms of the complement in {1, . . . , N } as in the statement of the lemma we have to make the substitutions |S c | → |S c | − 1, σ (S c ) → σ (S c ) − N , and therefore in the notation of the lemma the above equals c c p |S |−σ (S ) · · · I f,S (ξ ) d |S| ξ. q σ (S)−|S|(|S|+1)/2 C R CR S⊂{1,...,N −1}
This is the portion of the right side of (3.4) corresponding to those S not containing N . Now for the original integral, but where ξ N is taken over C R . If we integrate first with respect to ξ N , leaving us with an N − 1-dimensional integral with small Cr , we see that we are in the case N − 1 with f (ξ1 , . . . , ξ N ) replaced by ξ N − ξi f (ξ1 , . . . , ξ N ) ˜ f (ξ1 , . . . , ξ N −1 ) = dξ N . (3.6) p + qξ ξ − ξ 1 − ξN i N i CR i
We have to show that f˜ satisfies the required condition. For notational convenience we take k = 1 and i = 2, so we make the substitution ξ2 → (ξ1 − p)/qξ1 . All but the product are O(ξ1 ) as ξ1 → 0, by assumption on f . Write the product as the product over
826
C. A. Tracy, H. Widom
i = 1, 2, which is bounded if R was chosen large enough, uniformly for the ξi bounded away from zero, times ξ N − ξ1 ξ N − (ξ1 − p)/qξ1 p + qξ N ξ1 − ξ1 p + ξ N (ξ1 − p)/ξ1 − (ξ1 − p)/qξ1 ξ N − ξ1 ξ N − (ξ1 − p)/qξ1 = . p + qξ N ξ1 − ξ1 p + ξ N (ξ1 − p)/ξ1 − (ξ1 − p)/qξ1 The first factor equals 1/qξ1 while the second factor is O(ξ1 ). Thus (3.6) satisfies the required condition and we may again apply the induction hypothesis. If S˜ ⊂ {1, . . . , N − 1} to compute I f˜, S˜ we replace the ξi with i ∈ S˜ c by 1 in f˜(ξ1 , . . . , ξ N −1 ). We see that the product in the integrand in (3.6) is replaced by 1 c q | S˜ |
i∈ S˜
ξ N − ξi . p + qξi ξ N − ξi
If we set S = S˜ ∪ {N } then in terms of S the full integrand including the ξ N -variable is 1 c I f,S (ξ ). q |S | ˜ the power of p is For the coefficient on the right side of (3.4) with S replaced by S, unchanged while the power of q is ˜ − | S| ˜ (| S| ˜ + 1)/2 = σ (S) − N − (|S| − 1) |S|/2, σ ( S) and if we add to this |S c | = N − |S| we get σ (S) − |S| (|S| + 1)/2, which is the power of q in (3.4). Summing over these S gives the portion of the right side of (3.4) corresponding to those S containing N , and this completes the proof of Lemma 3.1. Proof of Theorem 3.2. First assume p = 0. Observe that I (x, Y, ξ ) is I f (ξ ) as defined above with x−y −1 ξi i eε(ξi ) t , (3.7) ξi s f (ξ1 , . . . , ξ N ) = 1 − i
i
and I (x, Y S , ξ ) is I f,S (ξ ). We show that f as defined this way satisfies the hypothesis of the lemma. The exponential summands ε(ξi ) and ε(ξk ) combine when ξi = (ξk − p)/qξk to give pqξk ξk − p p pqξk +q − 1, + + qξk − 2 = qξk + ξk − p qξk ξk ξk − p which is analytic at ξk = 0. The powers of ξi and ξk combine as
ξk − p qξk
x−yi −1
x−yk −1
ξk
y −yk
= O(ξk i
)
Integral Formulas for the Asymmetric Simple Exclusion Process
827
as ξk → 0, which is O(ξk ) since yi > yk . So the hypothesis on f is satisfied. Since f (1, . . . , 1) = 0 the sum may be taken over nonempty subsets S. Because of the factor p N (N −1)/2 in (3.1) we must multiply the factor in (3.4) by this, resulting in the factor in (3.3). We can remove the requirement p = 0 by taking the p → 0 limit since the power of p is nonnegative and no pole tends to infinity as p → 0. An immediate conclusion from Theorem 3.2 is that P(x1 (t) = x) tends exponentially to zero as x → −∞, and therefore so does P(x1 (t) < x). Thus P(x1 (t) ≥ x) tends exponentially to 1. We write ξi − ξ j x−y −1 x−y −1 ··· ξ1 1 · · · ξ N N Q(x) = p N (N −1)/2 p + qξi ξ j − ξi Cr Cr i< j
1 e i ε(ξi ) t dξ1 · · · dξ N , ×
i (1 − ξi )
where r is small. Clearly Q(x) → 0 as x → +∞ and P(x1 (t) = x) = Q(x) − Q(x + 1). It follows that Q(x) = P(x1 (t) ≥ x), and this tends exponentially to 1 as x → −∞. Therefore ∞ ∞ y [Q(y) − Q(y + 1)] = lim Q(y) + (x − 1) Q(x) E(x1 (t)) = lim x→−∞
= lim
x→−∞
x→−∞
y=x
∞
Q(y) + (x − 1) .
y=x
Now ∞
Q(y) = p N (N −1)/2
y=x
×
y=x
Cr
···
x−y1 −1
Cr
ξ1
1 1 e ·
1 − ξ1 · · · ξ N (1 − ξi ) i
x−y N −1
· · · ξN
i< j
i
ε(ξi ) t
ξ j − ξi p + qξi ξ j − ξi
dξ1 · · · dξ N .
(3.8)
If we apply the procedure of the last section to this integral we start by moving the ξ N -contour out. The resulting integral, with ξ N over a very large contour, is exponentially small as x → −∞, even though the other contours are small. As before the residues at the poles ξ N = (ξk − p)/qξk integrate out to zero. The contribution from the pole ξ N = 1 gives the same expression we started with but with N replaced by N − 1. But there is now also a pole at ξ N = 1/ξ1 · · · ξ N −1 whose contribution is − ··· ψ N −1 (ξ1 , . . . , ξ N −1 ; y N − y1 , . . . , y N − y N −1 ) dξ1 · · · dξ N −1 , Cr
Cr
where ψ N (ξ1 , . . . , ξ N ; z 1 , . . . , z N ) = p N (N +1)/2 ξ1z 1 · · · ξ Nz N
i< j
×
i
ξ j − ξi p + qξi ξ j − ξi
(ξ1 · · · ξ N )−1 −ξi 1 1 −1 e i ε(ξi )+ε((ξ1 ···ξ N ) ) t . · ·
−1 p + qξi (ξ1 · · · ξ N ) −ξi 1−ξ1 · · · ξ N i (1−ξi )
828
C. A. Tracy, H. Widom
For the right side of (3.8) when N = 1, which is what we are left with at the end, we expand the contour, encounter a double pole at ξ1 = 1, and find that it equals ( p − q) t + y1 − x + 1 + exponentially small term. Therefore E(x1 (t)) = ( p − q) t + y1 − ×
Cr
N −1 j=1
Cr
···
ψ j (ξ1 , . . . , ξ j ; y j+1 − y1 , . . . , y j+1 − y j ) dξ1 · · · dξ j .
The integral Cr ψ1 (ξ ; z) dξ has an explicit expression in terms of Bessel functions In (2t). Indeed, it equals 2 pt e−2t [Iz−1 (2t) + Iz (2t)] + (2z − 1) p
1 2
e−2t I0 (2t) −
1 + e−2t I j (2t) . 2 z−1
j=1
The integrals of the other ψ j (ξ ; z) are not so simple. We now show how to obtain the probability P(x1 (t) = x) for a system with infinitely many particles4 with initial configuration Y = {y1 , y2 , . . .},
y1 < y2 < · · · → +∞
(3.9)
when q = 0. In fact, it is very easy once we have Theorem 3.2 We just modify the right side so that the sum runs over all finite subsets of Z+ . Because C R may be taken arbitrarily large the resulting series converges, as we shall now show. Consider the various factors in the integrand in (3.1), where the ξi are replaced by the ξn with n ∈ S. The −1 parts of the exponents of the ξn may be removed if we replace dξn by dξn /ξn , which is O(1). Suppose that |S| = k as above. The analogue of the factor 1 − ξ1 · · · ξ N
i (1 − ξi )
is at most (R k + 1)/(R − 1)k = O(2k ) for R large. The product i< j (ξi − ξ j ) is at
most (2R)k(k−1)/2 . The denominator i< j ( p + qξi ξ j − ξi ) is at least (q R 2 /2)k(k−1)/2 . The product of the ξn is at most R kx− yn . The exponential factor is O(ea Rk ) for some a depending on t. So the integral is O (a/R)k(k−1)/2 R kx− yn ea Rk = R O(k)− yn a Rk can be incorporated into the R O(k) if we take R > a. (Since R is fixed the factor e term.) Since yn ≥ (y1 +n −1) = σ (S)+k (y1 −1) the above is at most R O(k)−σ (S) . 1/2 And since σ (S) ≥ k(k + 1)/2, this is at most R −σ (S)+O(σ (S) ) . The external factor in (3.3) is
p σ (S)−k q k(k+1)/2−σ (S) ≤ q −σ (S) . 4 It follows from the fact that ASEP is a Feller process [8] that the limit as N → ∞ equals the probability for the infinite system. We thank Thomas Liggett [10] for explaining this fact to us.
Integral Formulas for the Asymmetric Simple Exclusion Process
829
It follows that if we take R > 1/q 2 then the integral times the external factor is at most R −σ (S)/2 . Now consider all sets S with σ (S) = k. Since the largest i ∈ S is at most k, the number of such sets is at most 2k . Hence the sum of the absolute values of the terms of the infinite series is at most a constant times ∞
2k R −k/2 ,
k=1
which is finite when R > 4. Thus we have shown convergence for all t. IV. The Second-Left Particle In this section we compute the probability P(x2 (t) = x). It is somewhat more complicated than that given for P(x1 (t) = x) in Theorem 3.1 and the proof introduces some new elements. We use the notation (1.1), and for 1 ≤ k ≤ N we set Yk = Y \yk . Theorem 4.1. With the contours Cr as in Theorem 3.1 we have when p = 0, p N −1 − q N −1 (N −1)(N −2)/2 p ··· I (x, Y, ξ ) d N ξ P(x2 (t) = x) = −q p−q Cr Cr N k−1 q + p (N −1)(N −2)/2 ··· I (x, Yk , ξ ) d N −1 ξ. p Cr Cr
(4.1)
k=1
Proof. We rewrite X = {x1 , x2 , . . . , x N } as x − v, x, x + z 1 , . . . , x + z 1 + · · · + z N −2 . Then P(x2 (t) = x) equals the sum of PY (X ; t) over all v > 0 and z i > 0. If we sum
x −y −1
x−y −1 first over z 2 , . . . , z N −2 the product i ξσ i(i) σ (i) in (2.3) becomes, i ξi i times ξσ−v (1)
−2 ξσ (3) ξσ2(4) · · · ξσN(N )
(1 − ξσ (3) ξσ (4) · · · ξσ (N ) ) · · · (1 − ξσ (N ) ξσ (N −1) )(1 − ξσ (N ) )
.
We now move the ξσ (1) -contour out beyond the unit circle, and we do not encounter any poles. Here is the reason. From the first part of (3.2) we see that we get poles at ⎧ ⎨ (ξσ (k) − p)/qξσ (k) if k > i, ξσ (i) = (4.2) ⎩ p/(1 − qξ if k < i. σ (k) ) The poles in the ξσ (1) -variable are at ξσ (1) = (ξσ (k) − p)/qξσ (k) , and these are very large since ξσ (k) ∈ Cr . So we move the ξσ (1) -contour out beyond the unit circle and sum, giving 1
−2 ξσ (3) ξσ2(4) · · · ξσN(N )
ξσ (1) − 1 (1 − ξσ (3) ξσ (4) · · · ξσ (N ) ) · · · (1 − ξσ (N ) ξσ (N −1) )(1 − ξσ (N ) )
.
(4.3)
830
C. A. Tracy, H. Widom
If we move the contour back to Cr we pass a pole at ξσ (1) = 1 with residue −2 ξσ (3) ξσ2(4) · · · ξσN(N )
(1 − ξσ (3) ξσ (4) · · · ξσ (N ) ) · · · (1 − ξσ (N ) ξσ (N −1) )(1 − ξσ (N ) )
.
(4.4)
The factor Aσ when ξσ (1) = 1 equals q σ (1)−1 p + qξσ (i) ξσ ( j) − ξσ (i) sgn σ − : i < j, σ (i) > σ ( j), i, j > 1 , p p + qξσ (i) ξσ ( j) − ξσ ( j) (4.5) since there are σ (1) − 1 indices less than σ (1). Now we add all terms in which σ (1) = k. For the contribution from the pole at ξσ (1) = 1, when σ (1) = k (4.5) may be written (as in (3.2)) q k−1 1 sgn σ − ( p + qξσ (i) ξσ ( j) − ξσ (i) ), p ( p + qξi ξ j − ξi ) i< j i< j
where all indices are = k. This is to be multiplied by (4.4). If we use identity (1.6) to sum the product over those σ with σ (1) = k we get ⎛ ⎞ (ξ j − ξi ) k−1 q i< j , p (N −1)(N −2)/2 ⎝1 − ξj⎠ p (1 − ξ j ) ( p + qξi ξ j − ξi ) j=k j
i< j
where again all indices are = k. The exterior factor is now from these we obtain the sum on the right side of (4.1). Next we consider (4.3), which we rewrite as
i=k
x−y −1 ξi i eε(ξi )t , and
−2 ξσ (3) ξσ2(4) · · · ξσN(N (1 − ξσ (2) ξσ (3) · · · ξσ (N ) ) ) , ξk − 1 (1 − ξσ (2) ξσ (3) · · · ξσ (N ) ) · · · (1 − ξσ (N ) ξσ (N −1) )(1 − ξσ (N ) )
where σ is now a map from {2, . . . , N } to {1, . . . , N }\{k}. The factor sgn σ becomes (−1)k+1 sgn σ . We also rewrite Aσ as
1 ( p + qξi ξ j − ξi )
( p + qξk ξ j − ξk )
j=k
( p + qξσ (i) ξσ ( j) − ξσ (i) ).
i< j
i< j
If we use identity (1.6) to sum this over these σ we get (ξ j − ξi ) ( p + qξ ξ − ξ ) ⎛ ⎞ k j k i< j j = k i, j = k (−1)k p (N −1)(N −2)/2 ⎝1 − ξj⎠ . (1 − ξ j ) ( p + qξi ξ j − ξi ) j=k j
x−y −1 The other factors are still i ξi i e i ε(ξi )t .
i< j
(4.6)
Integral Formulas for the Asymmetric Simple Exclusion Process
831
To evaluate the sum of (4.6) over k we write it as (ξ j − ξi ) − p (N −1)(N −2)/2
i< j
(1 − ξ j )
j
×
⎛ ⎝1 −
j=k
k
⎞ ξj⎠
j=k
( p + qξi ξ j − ξi )
i< j
( p + qξ j ξk − ξk )
(ξ j − ξk )
.
j=k
(−1)k
The factor became −1 because of the way we rewrote the product of the ξ j − ξi in (4.6). Identity (1.9) with k = 1 tells us that the last sum equals ⎞ ⎛ N −1 N −1 p −q ⎝1 − q ξj⎠ . p−q j
If we recall the power of p in the first factor above and the ubiquitous factor x−y −1 ξi i eε(ξi )t we see that this gives the first term in (4.1).
i
V. The General Particle In this section we consider the m th particle from the left for general m. We prove, with the notation (1.8), Theorem 5.1. We have when p = 0, P(xm (t) = x) = p (N −m)(N −m+1)/2 q m(m−1)/2
c c |S| − 1 q σ (S )−m |S | m−1−|S c | × (−1) m − |S c | − 1 p σ (S c )−|S c |(|S c |+1)/2 |S c |<m × ··· I (x, Y S , ξ ) d |S| ξ, (5.1) Cr
Cr
where r is so small that the poles of the integrand lie outside Cr . The sum is taken over all subsets S of {1, . . . , N } with |S c | < m. We shall first establish a preliminary form of the result, and for that we use a lemma analogous to Lemma 3.1 and which follows from it. Now we have a function g(ξ1 , . . . , ξ N ) which is analytic for all ξi = 0 and satisfies, for i < k, = O(ξk−1 ) g(ξ1 , . . . , ξ N ) ξi → p/(1−qξk )
as ξk → ∞, uniformly when all ξ j with j = k are bounded and bounded away from zero. Define, as before, ξ j − ξi g(ξ1 , . . . , ξ N )
, Ig (ξ ) = p + qξi ξ j − ξi i (1 − ξi ) i< j
and similarly Ig,S (ξ ).
832
C. A. Tracy, H. Widom
Lemma 5.1. Under the stated assumptions on g(ξ1 , . . . , ξ N ), we have when p, q = 0, c c q σ (S )−N |S | N |S c | ··· Ig (ξ ) d ξ = (−1) ··· Ig,S (ξ ) d |S| ξ , c p |S|(N +1+|S |)/2−σ (S) Cr CR CR C r S (5.2) where r is so small that the poles of the integrand on the right lie outside Cr and R is so large that the poles of the integrand on the left lie inside C R . As before, S runs over all subsets of {1, . . . , N }. Proof. We apply Lemma 3.1 with p and q interchanged to the function f (ξ1 , . . . , ξ N ) = g(ξ N−1 , . . . , ξ1−1 ) ξi−1 . The required hypothesis on f follows from the hypothesis on g. The left side of (5.2), after the substitutions ξi → 1/ξ left side of (3.4) after interchanging p
N −i+1 , equals the
and q, times (−1) N , because (1 − ξi ) becomes (ξi − 1). So we apply Lemma 3.1 to f and then change variables again resulting in a factor (−1)|S| in each summand. The reason for the different coefficients is the reversal of the order of the variables. Thus the coefficient of the integral involving I f,S with p and q interchanged equals the coefficient of the integral involving Ig, S˜ , where S˜ = {N − i + 1 : i ∈ S}. This, together with the resulting power of −1, is what we see on the right side of (5.2). This proves Lemma 5.1. To state the preliminary form of the result we introduce more notation. For disjoint subsets T and U of {1, . . . , N }, we define
I (x, YT,U , ξ ) (ξ j − ξi )
i, j∈U i
( p + qξi ξ j − ξi )
i∈T, j∈U
( p + qξi ξ j − ξi )
x−y −1 ξi i eε(ξi ) t , i
i< j
(5.3) where indices with unspecified range run over T ∪ U . If T ⊂ U we define σ (T, U ) to be the sum of the positions of the elements of T in U . Thus, if U = {2, 3, 5} and T = {2, 5} then σ (T, U ) = 1 + 3 = 4. Finally, for a set U we define sgn U = (−1)#{(i, j) : i> j, i∈U, j∈U } . c
Lemma 5.2. With r small enough and p, q = 0 we have P(xm (t) = x) = p (N −m)(N −m+1)/2+m(m−1)/2 q (m−1)(m−2)/2 q σ (U \T )−(m−1) |U \T | × sgnU (−1)|U \T |+σ (U \T )−σ (U \T, T ) σ (U \T )+|T |(m+|U \T |)/2 p |U |=m−1 T ⊂U c × ··· I (x, YT,U c , ξ ) d |T ∪U | ξ, Cr
Cr
where U runs over all subsets of {1, . . . , N } with |U | = m − 1.
Integral Formulas for the Asymmetric Simple Exclusion Process
833
Proof. To begin we now write X as x − vm−1 − · · · − v1 , x − vm−2 − · · · − v1 , · · · , x − v1 , x, x + z 1 , . . . , x + z 1 + · · · + z N −m .
x−y −1
x −y −1 The product i ξσ i(i) σ (i) eε(ξi ) t in (2.3) is replaced by i ξi i eε(ξi )t times −v −···−vm−1
ξσ (1)1
z1 1 1 · · · ξσ−v (m−1) ξσ (m+1) · · · ξσ (N )
z +···+z N −m
,
and we have to sum over all vi > 0, z i > 0. As in the proof of Theorem 4.1 we can sum over the z i immediately and we get −v −···−vm−1
ξσ (1)1
1 · · · ξσ−v (m−1)
−m ξσ (m+1) ξσ2(m+2) · · · ξσN(N )
(1 − ξσ (m+1) · · · ξσ (N ) ) · · · (1 − ξσ (N ) )
.
Before we can sum over the vi we have to move the ξσ (i) -contours out, and we do them in the order i = 1, . . . , m − 1. As in the proof of Theorem 4.1 we see by referring to (4.2) that the poles obtained from moving the ξσ (1) -contour are very large and so we can move that contour out almost that far. Then if we want to move the ξσ (2) -contour out we encounter poles with k > 2 in (4.2), which are far out and so no problem, but also the pole with k = 1 when σ (2) < σ (1), which is at ξσ (2) = p/(1 − qξσ (1) ). We show that the residue at this pole, when integrated with respect to ξσ (1) , gives zero. Recall that the ξσ (1) -contour is large, and so ξσ (2) as a function of ξσ (1) is analytic outside the ξσ (1) -contour and is in fact O(ξσ−1 (1) ) at infinity. The part of the residue that comes from the factor p + qξσ (1) ξσ (2) − ξσ (1) p + qξσ (1) ξσ (2) − ξσ (2) is O(1) at infinity as are all the other factors in the first part of (3.2) because ξσ (2) , in terms of ξσ (1) , is small when ξσ (1) is large. The powers of the ξi involving ξσ (1) and ξσ (2) combine as −v1 −···−vm−2 −x−1 −yσ (2) p p −y −v −···−vm−1 −x−1 ξσ (1)1 ξσ (1)σ (1) , 1 − qξσ (1) 1 − qξσ (1) −v
−y
+y
which is analytic outside the ξσ (1) -contour and O(ξσ (1)m−1 σ (1) σ (2) ) at infinity. The exponent is ≤ −2 since vm−1 > 0 and yσ (2) < yσ (1) . Finally we have to check the exponential of ε(ξi ). The sum of those involving ξσ (1) and ξσ (2) is p ξσ (1)
+ qξσ (1) + (1 − qξσ (1) ) + q
p , 1 − qξσ (1)
which is bounded at infinity. Hence the ξσ (1) -integral is zero, as claimed. Continuing this way we move all the ξσ (i) -contours out for i < m. We then sum over the vi , obtaining 1 (ξσ (1) − 1) (ξσ (1) ξσ (2) − 1) · · · (ξσ (1) ξσ (2) · · · ξσ (m−1) − 1) ×
−m ξσ (m+1) ξσ2(m+2) · · · ξσN(N )
(1 − ξσ (m+1) · · · ξσ (N ) ) · · · (1 − ξσ (N ) )
834
C. A. Tracy, H. Widom
as replacement for what we had before. We also have, from the numerator in the second part of (3.2), a product which we write as ( p + qξσ (i) ξσ ( j) − ξσ (i) ) ( p + qξσ (i) ξσ ( j) − ξσ (i) ) i< j<m
×
m≤i< j
( p + qξσ (i) ξσ ( j) − ξσ (i) ).
i<m≤ j
Therefore we are to take the sum over all σ of sgn σ times the product ( p + qξσ (i) ξσ ( j) − ξσ (i) ) i< j<m
× ×
1 (ξσ (1) − 1) (ξσ (1) ξσ (2) − 1) · · · (ξσ (1) ξσ (2) · · · ξσ (m−1) − 1)
( p + qξσ (i) ξσ ( j) − ξσ (i) )
m≤i< j
×
−m ξσ (m+2) ξσ2(m+3) · · · ξσN(N )
(1 − ξσ (m+1) · · · ξσ (N ) ) · · · (1 − ξσ (N ) )
( p + qξσ (i) ξσ ( j) − ξσ (i) ).
i<m≤ j
Now we take a fixed U ⊂ {1, . . . , N } with |U | = m − 1, and sum over all permutations σ such that σ (i) ∈ U when i ≤ m − 1. Notice that the last product above is equal to ( p + qξi ξ j − ξi ), (5.4) i∈U j∈U c
and so is independent of the particular permutation. Therefore we may sum independently over bijective maps {1, . . . , m − 1} → U for the first factors and bijective maps {m, . . . , N } → U c for the second factors. To keep track of the signs of the permutations we observe that sgn σ equals the product of the signs of the two restrictions of σ times sgn U . So we eventually have to multiply by sgn U . For the second factor above we use (1.6) and find that the sum equals (ξ j − ξi ) i< j i, j∈U c p (N −m)(N −m+1)/2 1 − . (5.5) ξi (1 − ξi ) i∈U c i∈U c
For the first factor we use (1.6) and find that the sum equals (ξ j − ξi ) i< j i, j∈U
q (m−1)(m−2)/2 i∈U
(ξi − 1)
.
(5.6)
This is to be multiplied by (5.4), divided by i< j ( p +qξi ξ j −ξi ), and then integrated over large contours C R with respect to the ξi with i ∈ U . But we want all integrals to be
Integral Formulas for the Asymmetric Simple Exclusion Process
835
taken over the same contours Cr so we want to replace the integral with all contours C R with a sum of integrals with all contours Cr . In our application of Lemma 5.1 N will be replaced by m − 1, {1, . . . , N } will be replaced by U , and S will be replaced by T , which is the reason we defined σ (T, U ) as the sum of the positions of the elements of T in U . The coefficients become in this notation q σ (U \T, U )−(m−1) |U \T | . (5.7) p |T |(m+|U \T |)/2−σ (T,U ) If we put all integrands together the result is, aside from the powers of p and q in (5.6) and (5.5), ξi (ξ j − ξi ) 1− x−y −1 i i< j ξi i eε(ξi ) t (all indices in U c ) (5.8) ( p + qξi ξ j − ξi ) (1 − ξi ) i i< j
×
(ξ j − ξi )
i< j
( p + qξi ξ j − ξi )
i< j
×
i
(1 − ξi )
x−yi −1 ε(ξi ) t
ξi
e
(all indices in U )
(5.9)
i
i
i> j i∈U, j∈U c
p + qξi ξ j − ξi . p + qξi ξ j − ξ j
We apply Lemma 5.1 with x−y −1 ε(ξi ) t i ξi e ··· g(ξ ) = i∈U
Cr
Cr
i> j i∈U, j∈U c
p + qξi ξ j − ξi dξ j , p + qξi ξ j − ξ j c j∈U
where g(ξ ) = g({ξi }i∈U ). The poles of the integrand defining g are at ξi = (ξ j − p)/qξ j , where j ∈ U c . Since ξ j can be arbitrarily small the pole is outside C R for any R, so g is analytic for all ξi = 0. To check the main hypothesis on g we observe that after the substitution ξi → x−y −1 x−y −1 y −y p/(1 − qξk ) the product ξi i ξk k becomes O(ξk i k ) as ξk → ∞ which is O(ξk−1 ) since i < k. The sum of the exponents in the last factor which involve ξi and ξk is pq p 1 − qξk + + + qξk , 1 − qξk ξk which is bounded at infinity. As for the integral, we show that it is bounded when one ξk → ∞ while the others are bounded. If we take a fixed j > k with j ∈ U c , the pole at ξ j = p/(1 − qξk ) passes across the ξ j -contour when ξk → ∞. The residue of the factor in the product that contributes the pole is seen to be O(1) as are the other factors involving i = k. We do this with each j in turn and end up with a sum of integrals each
836
C. A. Tracy, H. Widom
of which is O(1). If we make the substitution ξi → p/(1 − qξk ) then ξk → ∞ while p/(1 − qξk ) remains bounded, so g satisfies the required hypothesis. To find the summand in (5.2), we must evaluate g(ξ ), where all the ξi with i ∈ U \T set equal to 1. Each factor in the product in the integrand with such an i is (−q/ p). For each such i the number of j in the product in the integrand satisfying i > j and j ∈ U c equals i minus the position of i in U . The sum of this over all i ∈ U \T equals σ (U \T ) − σ (U \T, U ). If we multiply (5.7) by (−q/ p) to this power the result may be written (−1)σ (U \T )−σ (U \T,U )
q σ (U \T )−(m−1) |U \T | p σ (U \T )+|T |(m+|U \T |)/2−m (m−1)/2
,
since σ (U \T, U ) + σ (T, U ) = m(m − 1)/2. |U | There are also the factors (−1)|U \T | coming from
(5.2) and (−1) coming from the fact that (ξi − 1) appears in (5.6) rather than (1 − ξi ). These factors combine as (−1)|T | . For the integrand we must combine (5.8), (5.9) with the ξi with i ∈ U \T set equal to 1, and the integrand in g with the ξi with i ∈ U \T set equal to 1. The result is (5.3) with U replaced by U c . If we take account of the original factors in (5.6) and (5.5) we have established Lemma 5.2. Proof of Theorem 5.1. Suppose temporarily that q = 0 also. We take a fixed S ⊂ {1, . . . , N } with |S c | ≤ m and in Lemma 5.2 sum over all T and U with T ⊂ U and T ∪ U c = S. Let us write everything in terms of T and U c . For any i ∈ U , the position of i in U equals #{ j : j ≤ i, j ∈ U }, so σ (T, U ) = #{(i, j) : i ≥ j, i ∈ T, j ∈ U }. In particular, σ (U \T, U ) = #{(i, j) : i ≥ j, i ∈ U \T, j ∈ U }, σ (U \T ) = #{(i, j) : i ≥ j, i ∈ U \T, j ∈ {1, . . . , N }}, and so σ (U \T ) − σ (U \T, U ) = #{(i, j) : i ≥ j, i ∈ U \T, j ∈ U c }. Also, sgn U = (−1)#{(i, j) : i> j, i∈U, j∈U } . c
Thus
(−1)|T |+#{(i, j) : i> j, i∈T, j∈U
c}
(5.10)
is the combined power of −1 that occurs. The powers of p and q that occur in the summation are, since U \T = S c , q σ (S )−(m−1) |S | , c c p σ (S )+|T |(m+|S |)/2 c
and only depends on |T |, given S.
c
(5.11)
Integral Formulas for the Asymmetric Simple Exclusion Process
837
In (5.3), we write
(ξ j − ξi ) =
i< j i, j∈U c or i, j∈T
(ξ j − ξi )
i< j i, j∈U c ∪T
(ξ j − ξi )
i< j i∈U c , j∈T
(ξ j − ξi )
i< j i∈T, j∈U c
The denominator may be written c (ξi − ξ j ) (ξ j − ξi ) = (−1)#{(i, j):i> j, i∈T, j∈U } i> j i∈T, j∈U c
.
(ξ j − ξi ).
i∈T, j∈U c
i< j i∈T, j∈U c
The power of −1 combined with (5.10) equals (−1)|T | and therefore our integrand, aside from this power of −1, may be written ( p + qξi ξ j − ξi ) (ξ j − ξi ) i∈T, j∈U c i< j 1− ξi (ξ j − ξi ) (1 − ξi ) ( p + qξi ξ j − ξi ) i∈U c i∈T, j∈U c
x−y −1 ξi i eε(ξi ) t ,
i
i< j
i
where indices not specified range over T ∪ U c . Now we take a fixed S ⊂ {1, . . . , N } with |S c | < m and in Lemma 5.2 first sum over all T and U such that T ∪ U c = S. The condition |U | = m − 1 translates to |T | = m − 1 − |S c |. Apply (1.9) with {1, . . . , N } replaced by S, with S replaced T , and with m replaced by m − 1. We obtain (5.1), and this completes the proof when q = 0. We can remove this condition by taking the q → 0 limit since no pole tends to zero when q → 0. We now obtain the expansion where all integrals are taken over large contours. Theorem 5.2. We have when q = 0, P(xm (t) = x) = (−1)m+1 ( pq)m(m−1)/2 |S| − 1 p σ (S)−m |S| × ··· I (x, Y S , ξ ) d |S| ξ, |S| − m q σ (S)−|S|(|S|+1)/2 C R CR |S|≥m
(5.12) where R is so large that the poles of the integrand lie inside C R . The sum is taken over all subsets S of {1, . . . , N } with |S| ≥ m. Proof. Denote by P˜ the probabilities for the process with p and q interchanged. As in the ˜ N −m+1 (t) = remark following the proof of Theorem 2.1, P(xm (t) = x) is equal to P(x −x) with initial configuration Y replaced by {−y N , . . . , −y1 }. In the integrals in (5.12) we make the replacements ξi → 1/ξ N −i+1 . The upshot is that in (5.1) we replace m by ˜ we multipy by (−1)|S|+1 (because of N − m + 1, in the coefficients we replace S by S,
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C. A. Tracy, H. Widom
the sign change in the integrand), and take the integrals over C R . A little algebra, using the general fact ˜ = σ ( S) (N − i + 1) = (N + 1) |S| − σ (S), i∈S
shows that the result is (5.12). This completes the proof of Theorem 5.2. As with the first particle when Y is infinite and bounded below, we can show that the sum (5.12) converges when it is taken over all finite subsets of Z+ . This gives the probability for infinitely many particles. In the special case Y = Z+ we can evaluate the sum over all sets of a given cardinality. We define the k-dimensional integrand ξ j − ξi 1 − ξ1 · · · ξk Jk (x, ξ ) = ξix−1 eε(ξi )t , p + qξi ξ j − ξi (1 − ξi ) (qξi − p) i i= j i
where all indices run over {1, . . . , k}. The result we obtain is
Corollary. When Y = Z+ we have when q = 0, P(xm (t) = x) = (−1)m+1 q m(m−1)/2 1 k−1 p (k−m)(k−m+1)/2 q k(k+1)/2 × k! k − m k≥m × ··· Jk (x, ξ ) dξ1 · · · dξk . CR
CR
(5.13)
Proof. We sum first over all S ⊂ Z+ with |S| = k. If S = {z 1 , . . . , z k } we make the variable changes ξz 1 → ξ1 , . . . , ξz k → ξk in the integral of I (x, Y S , ξ ), so all integration are over the variables ξi with i ∈ {1, . . . , k}. The integrand becomes ξ j − ξi 1 − ξ1 · · · ξk x−zi −1 ε(ξi )t ξi , e p + qξi ξ j − ξi (1 − ξi ) i i< j i
where all indices run over {1, . . . , k}. The only part of the coefficient in (5.12) involving more than |S| is ( p/q)σ (S) . When we multiply by this the product may be written (ξ j − ξi ) 1 − ξ1 · · · ξk x−1 ε(ξi )t i< j ξi e ( p + qξi ξ j − ξi ) (1 − ξi ) i i= j
×
i> j
i
( p + qξi ξ j − ξi )
q i
p
ξi
−zi
.
If we sum over all {z i } with 0 < z 1 < · · · < z k the last product becomes
1 . ( p + qξi ξ j −ξi ) ( qp ξ1 )( qp ξ2 ) · · · ( qp ξk ) − 1 ( qp ξ2 ) · · · ( qp ξk )−1 · · · ( qp ξk )−1 i> j
Integral Formulas for the Asymmetric Simple Exclusion Process
839
The integral is unchanged if we antisymmetrize this, because all other factors are symmetric except for the Vandermonde. We can bring this antisymmetrization to the form of identity (1.7) if we make the substitutions ξi =
p ηk−i+1 . q
The second factor becomes 1 , (η1 − 1) (η1 η2 − 1) · · · (η1 η2 · · · ηk − 1) while the first factor (after the index changes i → k − i + 1, j → k − j + 1) becomes k(k−1)/2 p (q + pηi η j − ηi ). q i< j
Now we can apply (1.7) with p and q interchanged and we see that the antisymmetrization of this is (η j − ηi ) (ξ j − ξi ) 1 p k(k−1) i< j 1 k(k+1)/2 i> j p = . k! q k(k−1)/2 k! (ηi − 1) (qξi − p) i
i
If we recall the remaining factor p −m |S| q −|S|(|S|+1)/2
=
q k(k+1)/2 p mk
in the coefficient in (5.12) we see that we obtain formula (5.13). Remark. The power of p on the right side of (5.13) is always nonnegative and is zero only when k = m. Hence when p = 0, in other words in the TASEP where particles move to the left, only one term survives and we obtain ξ1 · · · ξm − 1 (−1)m(m−1)/2 P(xm (t) = x) = ··· (ξ j − ξi )2 m! CR C R i< j (ξi − 1)m ξix−m−1 e(ξi −1)t dξ1 · · · dξm . ×
i
i
For P(xm (t) ≤ x) we sum P(xm (t) = y) over all y ≤ x (which we may, since R > 1), obtaining (−1)m(m−1)/2 P(xm (t) ≤ x) = ··· (ξ j − ξi )2 (ξi − 1)−m m! CR C R i< j i x−m (ξi −1)t ξi dξ1 · · · dξm . × e i
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C. A. Tracy, H. Widom
By a general identity [2] this equals (−1)m(m−1)/2 det ξ i+ j+x−m (ξ − 1)−m e(ξ −1)t dξ CR
.
i, j=0,...,m−1
After reversing the order of the columns this becomes a Toeplitz determinant equal to the determinant of Rákos-Schütz [13, (12)] which they used to obtain Johansson’s result [6]. VI. Proofs of the Identities Proof of identity (1.6).5 We use induction on N , and assume that the identity holds for N − 1. (It clearly holds for N = 1.) Call the left side ϕ N (ξ1 , . . . , ξ N ). We first sum over all permutations such that σ (1) = k, and then sum over k. If we observe that the inequality i < j becomes j = i when i = 1, we see that what we get for the left side is N 1 (−1)k+1 ( p + qξk ξ j − ξk ) · 1 − ξ1 ξ2 · · · ξ N k=1 j=k ξ j · ϕ N −1 (ξ1 , . . . , ξk−1 , ξk+1 , . . . , ξ N ), j=k
which may also be written N ξ1 ξ2 · · · ξ N (−1)k+1 ( p + qξk ξ j − ξk ) · ξk−1 ϕ N −1 (ξ1 , . . . , ξk−1 , ξk+1 , . . . , ξ N). 1−ξ1 ξ2 · · · ξ N k=1
j=k
We want to show that this equals the right side of (1.6), and the induction hypothesis gives (ξ j − ξi ) ϕ N −1 (ξ1 , . . . , ξk−1 , ξk+1 , . . . ξ N ) = p (N −1)(N −2)/2
i< j; i, j=k
(1 − ξ j )
.
j=k
After some multiplying, dividing, and computing powers of −1 we see that what we want to show is N k=1 j=k
( p + qξk ξ j − ξk ) ·
1 1 − ξ1 ξ2 · · · ξ N 1 − ξk
. = p N −1 ξk ξ1 ξ2 · · · ξ N j=k (ξ j − ξk )
(6.1)
If we change the first product on the left to run over all j we have to divide by p + qξk2 − ξk . Setting p = 1 − q shows that 1 − ξk 1 = . 2 p − qξk p + qξk − ξk 5 When we showed Doron Zeilberger the identity when it was still a conjecture he suggested [23] that problem VII.47 of [11], an identity of I. Schur, had a similar look about it and might be proved in a similar way. He was right.
Integral Formulas for the Asymmetric Simple Exclusion Process
841
So the left side of (6.1) equals N N
( p + qξk ξ j − ξk ) ·
k=1 j=1
1 1
. ξk ( p − qξk ) j=k (ξ j − ξk )
We evaluate this by integrating N
( p + qzξ j − z) ·
j=1
1 1 · N z ( p − qz) j=1 (ξ j − z)
over a large circle. Since the integrand is O(z −2 ) for large z the integral is zero. There are poles at 0 and the ξk , and the sum of the residues there is 1 p N −1 1
− ( p + qξk ξ j − ξk ) · . ξk ( p − qξk ) j=k (ξ j − ξk ) j ξj N
N
k=1 j=1
There is also a pole at z = p/q and for the residue there we compute p + pξ j − p/q q + qξ j − 1 =p = p, ξ j − p/q qξ j − p so the residue at p/q is − p N −1 . This gives N N
( p + qξk ξ j − ξk ) ·
k=1 j=1
1 p N −1 1
=
− p N −1 , ξk ( p − qξk ) j=k (ξ j − ξk ) ξ j j
as desired. This completes the proof of identity (1.6). Proof of identity (1.9). We use the easily established recursion formula
N N −1 N −1 m N −m =p +q m m m−1
(6.2)
and a preliminary simpler identity,
p + qξi ξ j − ξi N = , ξ j − ξi m i∈S
|S|=m
(6.3)
j∈S c
where S is as before. We prove this6 by induction on N , so assume (6.3) holds for N − 1. The left side is symmetric in the ξi and is O(1) as
any ξi → ∞ with the other ξ j fixed. If we multiply the left side by the Vandermonde i< j (ξi − ξ j ) we obtain an antisymmetric polynomial which is O(ξiN −1 ) as any ξi → ∞ with the others fixed, so it has degree at most N − 1 in each ξi . Being antisymmetric it is divisible by the Vandermonde, and having degree 6 The proof is a modification of one found by Anne Schilling [15] for an equivalent identity.
842
C. A. Tracy, H. Widom
at most N − 1 in each of the ξi separately it must be a constant times the Vandermonde. Thus the left side of (6.3) is a constant, say C N ,m . To evaluate the constant set ξ N = 1. For convenience we write p + qξi ξ j − ξi = U (ξi , ξ j ). ξj − ξj We have U (ξi , ξ j ) C N ,m = S
ξ1 =1
i∈S j∈S c
= q N −m
N ∈S
U (ξi , ξ j ) + p m
N ∈ S
i∈S\N j∈S c
U (ξi , ξ j ).
i∈S j∈S c \N
By the induction hypothesis the right side equals
N −1 N −1 N −m m +p , q m−1 m N and this equals m by (6.2). This establishes identity (6.3). The proof of (1.9) runs along the same lines. We interpret both sides to be zero when N = m, and do an induction on N ≥ m. So we assume N > m and that the formula holds for N − 1 ≥ m − 1. We quickly deduce that the left side is a polynomial of degree at most one in each ξi . If we call the left side C N ,m (ξ ) then U (ξi , ξ j ) · 1 − ξj C N ,m (ξ1 , . . . , ξ N −1 , 1) = q N −m N ∈S
+ pm
N ∈ S
i∈S j∈S c \N
i∈S\N j∈S c
j∈S c
U (ξi , ξ j ) · 1 − ξj . j∈S c \N
= q N −m C N −1,m−1 (ξ1 , . . . , ξ N −1 ) + p m C N −1,m (ξ1 , . . . , ξ N −1 ). Similar relations hold for the other ξi . Notice that when N = m the second sum above does not appear and Cm−1,m−1 (ξ ) = 0 so this is consistent with our initial condition. If we call the right side of (1.9) C N ,m (ξ ) we see from (6.2) that the same relations hold (ξ ) = 0) and so for the difference D N ,m (ξ ) = for C N ,m (ξ ) (with initial condition Cm,m C N ,m (ξ ) − C N ,m (ξ ). The induction hypothesis gives D N −1,m = D N −1,m−1 = 0, so wse have shown that for any i, D N ,m (ξ )|ξi =1 = 0. Any polynomial which
has degree at most one in each ξi and vanishes when any ξi = 1 is a constant times (ξi − 1),7 so D N ,m (ξ ) has this form. We have shown that C N ,m (ξ ) = C N ,m (ξ ) + c (ξi − 1) i
for some c. We show that c = 0 by computing asymptotics as ξ N → ∞. All terms are asymptotically a constant times ξ N . If in the sum in (1.9) N ∈ S then the corresponding 7 Such a polynomial must be of the form ξ −1 times a polynomial in ξ , . . . , ξ 1 N N −1 with the same property, so the statement follows by induction.
Integral Formulas for the Asymmetric Simple Exclusion Process
843
summand is O(1). So we need consider only those S for which N ∈ S. In the product
in the summand, if j = N then the corresponding product over i has the limit q m i∈S ξi since there are m factors with limit qξi . It follows that C N ,m (ξ ) = −q m ξ N →∞ ξN lim
= −q m
|S|=m i∈S S⊂{1,...,N −1} j∈S c , j
ξi
p + qξi ξ j − ξi · ξi · ξ j − ξi i∈S
|S|=m i∈S S⊂{1,...,N −1} j∈S c , j
i
ξi
j
p + qξi ξ j − ξi . ξ j − ξi
Identity (6.3) tells us that this equals −q
m
i
ξi
N −1 . m
Clearly C N ,m (ξ ) has the same asymptotics, so c = 0. Acknowledgements. The authors thank Thomas Liggett, Anne Schilling and Doron Zeilberger for their invaluable assistance. Helpful communications with Alex Gamburd and Gunter Schütz are gratefully acknowledged as are comments by Timo Seppäläinen and Pavel Bleher. This work was supported by the National Science Foundation under grants DMS–0553379 (first author) and DMS–0552388 (second author).
References 1. Alcaraz, F.C., Droz, M., Henkel, M., Rittenberg, V.: Reaction-diffusion processes, critical dynamics, and quantum chains. Ann. Phys. 230, 250–302 (1994) 2. Andréief, C.: Note sur une relation les intégrales définies des produits des fonctions, Mém. de la Soc. Sci., Bordeaux 2, 1–14 (1883) 3. Baik, J., Deift, P.A., Johansson, K.: On the distribution of the length of the longest increasing subsequence in a random permutation. J. Am. Math. Soc. 12, 1119–1178 (1999) 4. Gwa, L.-H., Spohn, H.: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844–854 (1992) 5. Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probability Surveys 3, 206–229 (2006) 6. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) 7. Lieb, E.H., Liniger, W.: Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605–1616 (1963) 8. Liggett, T.M.: Interacting Particle Systems. [Reprint of the 1985 original.] Berlin:Springer-Verlag, 2005 9. Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Berlin: SpringerVerlag, 1999 10. Liggett, T.M.: Private communication, April 17, 2007 11. Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis. Berlin, Springer-Verlag, 1964 12. Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In and Out of Equilibrium, Progress in Probability 51, 185–204 (2000) 13. Rákos, A., Schütz, G.M.: Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118, 511–530 (2005) 14. Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005) 15. Schilling, A.: Private communication, Apr. 6 2007 16. Schütz, G.M.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys 88, 427–445 (1997) 17. Soshnikov, A.: Determinantal random fields. Russ. Math. Surv. 55, 923–975 (2000) 18. Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Physica A 369, 71–99 (2006)
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19. Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970) 20. Sutherland, B.: Beautiful Models: 70 Years of Exactly Solvable Quantum Many-Body Problems, Singapore: World Scientific, 2004 21. Yang, C.N., Yang, C.P.: One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hypothesis for the ground state in a finite system. Phys. Rev. 150, 321–327 (1966) 22. Yau, H.-T.: (log t)2/3 law of the two dimensional asymmetric simple exclusion process. Ann. Math. 159, 377–405 (2004) 23. Zeilberger, D.: Private communication, Feb. 14 2007 Communicated by H. Spohn
Commun. Math. Phys. 279, 845–872 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0441-5
Communications in
Mathematical Physics
Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups Deepak Naidu, Dmitri Nikshych Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA. E-mail: [email protected]; [email protected] Received: 7 May 2007 / Accepted: 11 July 2007 Published online: 21 February 2008 – © Springer-Verlag 2008
Abstract: We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω (G), where G is a finite group and ω is a 3-cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of Rep(D ω (G)) and module categories over the category VecωG of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld’s characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra [D]. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories. 1. Introduction Throughout this paper we will work over an algebraically closed field k of characteristic zero. Unless otherwise stated all cocycles appearing in this work will have coefficients in the trivial module k × . All categories considered in this work are assumed to be k-linear and semisimple with finite-dimensional Hom-spaces and finitely many isomorphism classes of simple objects. All functors are assumed to be additive and k-linear. Let G be a finite group and ω be a 3-cocycle on G. In [DPR1,DPR2] R. Dijkgraaf, V. Pasquier, and P. Roche introduced a quasi-triangular quasi-Hopf algebra D ω (G). When ω = 1 this quasi-Hopf algebra coincides with the Drinfeld double D(G) of G and so D ω (G) is often called a twisted quantum double of G. It is well known that the representation category Rep(D ω (G)) of D ω (G) is a modular category [BK,T] and is braided equivalent to the center [K] of the tensor category VecωG of finite-dimensional G-graded vector spaces with the associativity constraint defined using ω. The category VecωG is a typical example of a pointed fusion category, i.e., a finite semisimple tensor category in which every simple object is invertible.
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In [DGNO] a criterion for a modular category C to be braided tensor equivalent to the center of a category of the form VecωG for some finite group G and ω ∈ Z 3 (G, k × ) is given. Namely, such a braided equivalence exists if and only if C contains √ a Lagrangian subcategory, i.e., a maximal isotropic subcategory of dimension dim(C). More precisely, Lagrangian subcategories of C parameterize the classes of braided equivalences between C and centers of pointed categories, see [DGNO, Sect. 4]. Note that any Lagrangian subcategory of C is equivalent (as a symmetric tensor category) to the representation category of some group by the result of P. Deligne [De]. This means that a description of Lagrangian subcategories of Rep(D ω (G)) for all groups G and 3-cocycles ω is equivalent to a description of all braided equivalences between representation categories of twisted group doubles. Such equivalences for elementary Abelian and extra special groups were studied in [MN] and [GMN]. A motivation for such study comes from a relation between holomorphic orbifolds in the Rational Conformal Field Theory and twisted group doubles observed in [DVVV,DPR2]. A complete classification of Lagrangian subcategories of Rep(D ω (G)) is the principal goal of this paper. 1.1. Main results. Let G be a finite group and let ω ∈ Z 3 (G, k × ) be a 3-cocycle on G. Theorem 1.1. Lagrangian subcategories of the representation category of the Drinfeld double D(G) are classified by pairs (H, B), where H is a normal Abelian subgroup of G and B is an alternating G-invariant bicharacter on H . The proof is based on the analysis of modular data (i.e., the S- and T -matrices) associated to D(G). Theorem 1.1 gives a simple classification of Lagrangian subcategories for the untwisted double D(G). In the twisted (ω = 1) case the notion of a G-invariant bicharacter needs to be twisted as well, cf. Definition 4.6. Theorem 1.2. Lagrangian subcategories of the representation category of the twisted double D ω (G) are classified by pairs (H, B), where H is a normal Abelian subgroup of G such that ω| H ×H ×H is cohomologically trivial and B : H × H → k × is a G-invariant alternating ω-bicharacter in the sense of Definition 4.6. Note that bicharacters in the statement of Theorem 1.2 are in bijection with equivalence classes of G-invariant cochains µ ∈ C 2 (H, k × ) such that δ 2 µ = ω| H ×H ×H , see (6). Let L(H,B) denote the Lagrangian subcategory of Rep(D ω (G)) corresponding to a pair (H, B) in Theorems 1.1 and 1.2. Then there is a group G , defined up to an isomorphism, such that L(H,B) is equivalent to Rep(G ) as a symmetric tensor category, where the braiding of Rep(G ) is the trivial one [De]. The group G can be described explicitly in terms of G, H , and B, see Remark 3.7. Note that G ∼ = G in general. There is a canonical subcategory L({1}, 1) ∼ Rep(G) corresponding to the forgetful = functor Rep(D ω (G)) ∼ = Z(VecωG ) → VecωG . We have L({1}, 1) ∩ L(H,B) ∼ = Rep(G/H ). In [Na] the first named author classified indecomposable VecωG -module categories M with the property that the dual fusion category (VecωG )∗M is pointed. Such module categories M can be thought of as categorical analogues of homogeneous spaces. The above property gives rise to an equivalence relation on the set of pairs (G, ω), where G is a finite group and ω ∈ Z 3 (G, k × ) with (G, ω) ≈ (G , ω ) if and only if (VecωG )∗M ∼ = VecωG for some M.
(1)
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In other words, VecωG and VecωG are weakly Morita equivalent in the sense of M. Müger [Mu2]. Theorem 1.3. There is a canonical bijection between equivalence classes of indecomposable VecωG -module categories M with respect to which the dual fusion category (VecωG )∗M is pointed and Lagrangian subcategories of Rep(D ω (G)). Lagrangian subalgebras of the double of a Lie bialgebra g were used by V. Drinfeld in [D] to describe Poisson homogeneous spaces of the Poisson-Lie group G corresponding to g. So Theorem 1.3 can perhaps be understood as a quantum version of the correspondence in [D]. Note that quantization of Poisson homogeneous spaces was studied in [EK]. Recall that a fusion category is called group-theoretical if it has a pointed dual. Theorem 1.4. Let C1 and C2 be group-theoretical fusion categories. Then C1 and C2 are weakly Morita equivalent if and only if their centers Z(C1 ) and Z(C2 ) are braided equivalent. If C1 and C2 are arbitrary (i.e., not necessarily group-theoretical) weakly Morita equivalent finite tensor categories then it was observed by M. Müger in [Mu2, Remark 3.18] that Z(C1 ) is braided tensor equivalent to Z(C2 ). Also, Z(C1 ) being equivalent to Z(C2 ) (even in a non-braided way) implies that C1 C1rev is weakly Morita equivalent to C2 C2rev , where C rev denotes the fusion category obtained by reversing the tensor product in C. At the moment of writing we do not know if braided equivalence of centers implies weak Morita equivalence of C1 and C2 in a general case. Note that it was shown by S. Natale [N] that a fusion category C is group-theoretical if and only if Z(C) is a braided equivalent to the representation category of some twisted group double D ω (G) (this also follows from [O2]). Combining the explicit description of weak Morita equivalence classes of pointed categories from [Na] and correspondence between braided equivalences of centers and Lagrangian subcategories from [DGNO] one obtains a complete description of braided equivalences of twisted quantum doubles. Recall that two finite groups G 1 and G 2 were called categorically Morita equivalent in [Na] if VecG 1 and VecG 2 are weakly Morita equivalent. Let us write G 1 ≈ G 2 for such groups. It follows from Theorem 1.4 that G 1 ≈ G 2 if and only if the corresponding Drinfeld doubles D(G 1 ) and D(G 2 ) have braided tensor equivalent representation categories. By Theorems 1.1 and 1.3 groups categorically Morita equivalent to a given group G correspond to pairs (H, B), where H is a normal Abelian subgroup of G and B is an alternating bicharacter on H . Note that such pairs (H, B) for which B is, in addition, a nondegenerate bicharacter were used by P. Etingof and S. Gelaki in [EG] to describe groups isocategorical to G, i.e., such groups G for which Rep(G ) ∼ = Rep(G) as tensor categories. This non-degeneracy condition in [EG] is the reason why the categorical Morita equivalence extends isocategorical equivalence. Indeed, since Rep(D(G)) is determined by the tensor structure of Rep(G), it is clear that isocategorical groups are categorically Morita equivalent. On the other hand, the first author constructed in [Na] examples of categorically Morita equivalent but non-isocategorical groups. denote the group of linear characters of H . For an Abelian group H let H Corollary 1.5. Let G, G be finite groups, ω ∈ Z 3 (G, k × ), and ω ∈ Z 3 (G , k × ). Then the representation categories of twisted doubles D ω (G) and D ω (G ) are equivalent as braided tensor categories if and only if G contains a normal Abelian subgroup H such that the following conditions are satisfied:
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(1) ω| H ×H ×H is cohomologically trivial, (2) there is a G-invariant (see (6))2-cochain µ ∈ C 2 (H, k × ) such that that δ 2 µ = ω| H ×H ×H , and ∼ ν (H \G) such that ◦ (a × a × a) and ω →H (3) there is an isomorphism a : G − are cohomologically equivalent. ) that comes from the G-invariance of Here ν is a certain 2-cocycle in Z 2 (H \G, H ν (H \G) that depends on ν and on the exact µ and is a certain 3-cocycle on H sequence 1 → H → G → H \G → 1 (see [Na, Theorem 5.8] for precise definitions). Note that in the special case when ω = 1 and µ is a non-degenerate G-invariant alternating 2-cocycle on H , our construction of the “dual” group G in Corollary 1.5 becomes the construction of a group isocategorical to G from [EG]. This can be seen by comparing [Na, 4.2] and [EG, Formula (2)].
1.2. Organization of the paper. Section 2 contains necessary preliminary information about fusion categories, module categories, and modular categories. We also recall definitions and results from [DGNO] concerning Lagrangian subcategories of modular categories. Section 3 (respectively, Sect. 4) is devoted to classification of Lagrangian categories of the representation category of the Drinfeld double (respectively, twisted double) of a finite group. The reason we prefer to treat untwisted and twisted cases separately is because our constructions in the former case do not involve rather technical cohomological computations present in the latter. We feel that the reader might get a better understanding of our results by exploring the untwisted case first. Of course when ω = 1 the results of Sect. 4 reduce to those of Sect. 3. Sections 3 and 4 contain proofs of our main results stated above. Theorems 1.1–1.4 and Corollary 1.5 correspond to Theorems 3.5, 4.12, 4.17, 4.19, and Corollary 4.20. Section 5 contains examples in which we compute Lagrangian subcategories of Drinfeld doubles of finite symmetry groups. Here we also show that four non-equivalent non-pointed fusion categories of dimension 8 with fusion rules of the dihedral group are pairwise weakly Morita non-equivalent, and hence their centers are pairwise non-equivalent as braided tensor categories. 2. Preliminaries 2.1. Fusion categories and their module categories. A fusion category over k is a k-linear semisimple rigid tensor category with finitely many isomorphism classes of simple objects, finite-dimensional Hom-spaces, and simple neutral object. In this paper we only consider fusion categories with integral Frobenius-Perron dimensions of simple objects. It was shown in [ENO, Props. 8.23, 8.24] that any such category is equivalent to the representation category of a semisimple quasi-Hopf algebra and has a canonical spherical structure with respect to which the categorical dimension of any object is equal to its Frobenius-Perron dimension. In particular, all categorical dimensions are positive integers. For any object X of a fusion category C let d(X ) denote its (Frobenius-Perron) dimension. By a fusion subcategory of a fusion category we will always mean a full fusion subcategory.
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A fusion category is said to be pointed if all its simple objects are invertible. A typical example of a pointed category is VecωG - the category of finite-dimensional vector spaces over k graded by the finite group G. The morphisms in this category are linear transformations that respect the grading and the associativity constraint is given by a 3-cocycle ω on G. Let C = (C, ⊗, 1C , α, λ, ρ) be a tensor category, where 1C , α, λ, and ρ are the unit object, the associativity constraint, the left unit constraint, and the right unit constraint, respectively. A right module category over C is a category M together with an exact bifunctor ⊗ : M × C → M and natural isomorphisms µ M, X, Y : M ⊗ (X ⊗ Y ) → (M ⊗ X ) ⊗ Y, τ M : M ⊗ 1C → M, for all M ∈ M, X, Y ∈ C such that the following two equations hold for all M ∈ M, X, Y, Z ∈ C: µ M⊗X, Y, Z ◦ µ M, X, Y ⊗Z ◦ (id M ⊗ α X,Y,Z ) = (µ M, X, Y ⊗ id Z ) ◦ µ M, X ⊗Y, Z , (τ M ⊗ idY ) ◦ µ M, 1C , Y = id M ⊗ λY . Note that having a C-module structure on a category M is the same as having a tensor functor from C to the (strict) tensor category of endofunctors of M. The coherence conditions on a module action follow automatically from those of a tensor functor. Let (M1 , µ1 , τ 1 ) and (M2 , µ2 , τ 2 ) be two right module categories over C. A C-module functor from M1 to M2 is a functor F : M1 → M2 together with natural isomorphisms γ M, X : F(M ⊗ X ) → F(M) ⊗ X , for all M ∈ M1 , X ∈ C such that the following two equations hold for all M ∈ M1 , X, Y ∈ C: (γ M, X ⊗ idY ) ◦ γ M⊗X, Y ◦ F(µ1M, X, Y ) = µ2F(M), X, Y ◦ γ M,X ⊗Y , 1 1 τ F(M) ◦ γ M, 1C = F(τ M ).
Two module categories M1 and M2 over C are equivalent if there exists a module functor from M1 to M2 which is an equivalence of categories. For two module categories M1 and M2 over a tensor category C their direct sum is the category M1 ⊕ M2 with the obvious module category structure. A module category is indecomposable if it is not equivalent to a direct sum of two non-trivial module categories. Example 2.1. Indecomposable module categories over pointed categories. Let G be a finite group and ω ∈ Z 3 (G, k × ). Indecomposable right module categories over VecωG correspond to pairs (H, µ), where H is a subgroup of G such that ω| H ×H ×H is cohomologically trivial and µ ∈ C 2 (H, k × ) is a 2-cochain satisfying δ 2 µ = ω| H ×H ×H , i.e., µ(h 2 , h 3 )µ(h 1 h 2 , h 3 )−1 µ(h 1 , h 2 h 3 )µ(h 1 , h 2 )−1 = ω(h 1 , h 2 , h 3 )
(2)
for all h 1 , h 2 , h 3 ∈ H (see [O1]). Let M := M(H, µ) denote the right module category constructed from the pair (H, µ). The simple objects of M are given by the set H \G of right cosets of H in G, the action of VecωG on M comes from the action of G on H \G, and the module category structure isomorphisms are induced from the 2-cochain µ. Let H, H be subgroups of G such that restrictions of ω are trivial in H 3 (H, k × ) and H 3 (H , k × ). Two pairs (H, µ) and (H , µ ), where δ 2 µ = ω| H ×H ×H and δ 2 µ = ω| H ×H ×H give rise to equivalent VecωG -module categories if and only if there is g ∈ G such that H = g H g −1 and µ and the g-conjugate of µ differ by a coboundary. We will say that two elements of {µ ∈ C 2 (H, k × ) | δ 2 µ = ω| H ×H ×H } are equivalent if they differ by a coboundary. Let (3)
H,ω := equivalence classes of µ ∈ C 2 (H, k × ) | δ 2 µ = ω| H ×H ×H .
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There is an (in general, non-canonical) bijection between H,ω and H 2 (H, k × ), i.e.,
H,ω is a (non-empty) torsor over H 2 (H, k × ). Note that H,1 = H 2 (H, k × ). Let M1 and M2 be two right module categories over a tensor category C. Let (F 1 , γ 1 ) and (F 2 , γ 2 ) be module functors from M1 to M2 . A natural module transformation from (F 1 , γ 1 ) to (F 2 , γ 2 ) is a natural transformation η : F 1 → F 2 such that the following equation holds for all M ∈ M1 , X ∈ C: 1 2 (η M ⊗ id X ) ◦ γ M, X = γ M, X ◦ η M⊗X .
Let C be a tensor category and let M be a right module category over C. The dual ∗ := Fun (M, M) whose objects category of C with respect to M is the category CM C are C-module functors from M to itself and morphisms are natural module transforma∗ is a tensor category with tensor product being composition of tions. The category CM module functors. It is known that if C is a fusion category and M is semisimple k-linear ∗ is a fusion category [ENO]. and indecomposable, then CM Let G be a finite group and ω ∈ Z 3 (G, k × ). For each x ∈ G, define ϒx : G×G → k × by ϒx (g1 , g2 ) :=
ω(xg1 x −1 , xg2 x −1 , x)ω(x, g1 , g2 ) , ω(xg1 x −1 , x, g2 )
It is straightforward to verify that δ 2 ϒx =
ω ωx ,
for all g1 , g2 ∈ G.
(4)
for all x ∈ G, where
ω x (g1 , g2 , g3 ) = ω(xg1 x −1 , xg2 x −1 , xg3 x −1 ), for all g1 , g2 , g3 ∈ G. For each x ∈ G, define νx : G × G → k × by νx (g1 , g2 ) :=
ω(g1 , g2 , x)ω(g1 g2 xg2−1 g1−1 , g1 , g2 ) ω(g1 , g2 xg2−1 , g2 )
,
for all g1 , g2 ∈ G.
It is easy to verify that the following relation holds: ϒx1 x2 (g1 , g2 ) −1 ϒx1 (x2 g1 x2 , x2 g2 x2−1 )ϒx2 (g1 ,
g2 )
=
νg1 (x1 , x2 )νg2 (x1 , x2 ) , νg1 g2 (x1 , x2 )
for all x1 , x2 , g1 , g2 ∈ G. (5)
Let H be a normal subgroup of G such that ω| H ×H ×H is cohomologically trivial. For any x ∈ G and µ ∈ C 2 (H, k × ) such that δ 2 µ = ω| H ×H ×H , define µ x := µx × ϒx | H ×H , where µx (h 1 , h 2 ) = µ(xh 1 x −1 , xh 2 x −1 ), for all h 1 , h 2 ∈ H . It is easy to verify that δ 2 (µ x) = ω| H ×H ×H . This induces an action of G on H,ω (defined in (3)). Indeed, that this is an action follows from (5). Let ( H,ω )G denote the set of G-invariant elements of H,ω , i.e., µx G 2 × ( H,ω ) := µ ∈ H,ω × ϒx | H ×H is trivial in H (H, k ), for all x ∈ G . µ (6)
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Example 2.2. Module categories over VecωG with pointed duals. Let G be a finite group and ω ∈ Z 3 (G, k × ). It is shown in [Na, Theorem 3.4] that the set of equivalence classes of indecomposable module categories over VecωG such that the dual is pointed is in bijection with the set of pairs (H, µ), where H is a normal Abelian subgroup of G such that ω| H ×H ×H is cohomologically trivial and µ ∈ ( H,ω )G (the description in [Na, Theorem 3.4] is given in somewhat different but equivalent terms). Two fusion categories C and D are said to be weakly Morita equivalent if there exists an indecomposable (semisimple k-linear) right module category M over C such that the ∗ and D are equivalent as fusion categories. It was shown by M. Müger categories CM [Mu1] that this is indeed an equivalence relation. A fusion category C is said to be group theoretical if it is weakly Morita equivalent to a pointed category. 2.2. Modular categories and centralizers. Let C be a modular fusion category with braiding c, twist θ , and S-matrix S (see [BK]). Let D bea full (not necessarily tensor) subcategory of C. Its dimension is defined by dim(D) := X ∈Irr(D) d(X )2 , where Irr(D) is the set of isomorphism classes of simple objects in D. In [Mu2], M. Müger introduced the notion of the centralizer of D in C as the fusion subcategory D := {X ∈ C | c(Y, X ) ◦ c(X, Y ) = id X ⊗Y , for all Y ∈ D} . It was also shown in [Mu2] that if D is a fusion subcategory then D = D and dim(D) · dim(D ) = dim(C).
(7)
Following M. Müger, we will say that two objects X, Y ∈ C centralize each other if c(Y, X ) ◦ c(X, Y ) = id X ⊗Y . For simple X and Y this condition is equivalent to S(X, Y ) = d(X )d(Y ) [Mu2, Corollary 2.14]. Remark 2.3. If D is a full subcategory of C such that all objects in D centralize each other, i.e., D ⊆ D then dim(D)2 ≤ dim(C). Indeed, we have dim(D) ≤ dim(D ) and so it follows from (7) that dim(D)2 ≤ dim(C). In particular, if D is a symmetric fusion subcategory of C, then dim(D)2 ≤ dim(C). Lemma 2.4. Let D be a full subcategory of C (which is not a priori assumed to be closed under the tensor product or duality) such that D ⊆ D . Then the fusion subcategory D˜ ⊆ C generated by D is symmetric. Proof. We may assume that D is closed under taking duals. Indeed, it follows from [ENO, Prop. 2.12] that X centralizes Y if and only if X centralizes Y ∗ for any two simple objects X, Y in C. ˜ There exist simple objects X 1 , X 2 , Y1 , Y2 in D Let Z 1 , Z 2 be simple objects in D. such that Z 1 is contained in X 1 ⊗Y1 and Z 2 is contained in X 2 ⊗Y2 . By [Mu2, Lemma 2.4 (i)], it follows that Z 1 centralizes X 2 ⊗ Y2 , and hence Z 1 , Z 2 centralize each other. Corollary 2.5. Let D be a full subcategory of C such that D ⊆ D and dim(D)2 = dim(C). Then D is a symmetric fusion subcategory.
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2.3. Lagrangian subcategories and braided equivalences of twisted group doubles. Let C be a modular category. Recall that we chose the canonical spherical twist for C with respect to which the categorical dimension of any object of C is equal to its Frobenius-Perron dimension. This is possible by [ENO, Props. 8.23, 8.24]. Let us recall some definitions and results from [DGNO]. Definition 2.6. A fusion subcategory D ⊆ C is said to be isotropic if the twist of C restricts to identity on D. An isotropic subcategory D ⊆ C is said to be Lagrangian if (dim(D))2 = dim(C). An isotropic subcategory D of C is necessarily symmetric and its objects have positive categorical dimensions. It follows from [De] that there is a (unique up to an isomorphism) group G such that D ∼ = Rep(G) as a symmetric fusion category, where Rep(G) is considered with its trivial braiding. ∼ → Z(P), where P is a Consider the set of all braided tensor equivalences F : C − pointed fusion category and Z(P) denotes its center. There is an equivalence relation ∼ ∼ → Z(P1 ) and F2 : C − → Z(P2 ) on this set defined as follows. We say that F1 : C − ∼ → P2 such that F2 ◦ F2 and are equivalent if there exists a tensor equivalence ι : P1 − ι ◦ F1 ◦ F1 are isomorphic tensor functors, where Fi : Z(Pi ) → Pi , i = 1, 2, are the canonical forgetful functors. Let E(C) be the collection of equivalence classes of such equivalences. Informally, E(C) is the set of all “different” braided equivalences between C and centers of pointed categories, i.e., representation categories of twisted group doubles. Let Lagr(C) be the set of all Lagrangian subcategories of C. In [DGNO, Theorem 4.5] it was proved that there is a bijection ∼
f : E(C) − → Lagr(C)
(8) ∼
defined as follows. Note that each braided tensor equivalence F : C − → Z(P) gives rise to the Lagrangian subcategory f (F) of C formed by all objects sent to multiples of the unit object 1 under the forgetful functor Z(P) → P. This subcategory is clearly the same for all equivalent choices of F. In particular, the center of a fusion category D contains a Lagrangian subcategory if and only if D is group-theoretical [DGNO]. 2.4. The Schur multiplier of an Abelian group. Let H be a normal Abelian subgroup of a finite group G. Let 2 H denote the Abelian group of alternating bicharacters on H , i.e., ⎧ ⎫ B(h 1 h 2 , h) = B(h 1 , h)B(h 2 , h), ⎨ ⎬ 2 H := B : H × H → k × B(h, h 1 h 2 ) = B(h, h 1 )B(h, h 2 ), and . ⎩ ⎭ B(h, h) = 1, for all h, h 1 , h 2 ∈ H Let Z 2 (H, k × ) be the group of 2-cocycles on H . Define a homomorphism alt : Z 2 (H, k × ) → 2 H : µ → alt (µ) by alt (µ)(h 1 , h 2 ) :=
µ(h 2 , h 1 ) , h 1 , h 2 ∈ H. µ(h 1 , h 2 )
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It is well known that alt induces an isomorphism between the Schur multiplier H 2 (H, k × ) of H and 2 H . By abuse of notation we denote this isomorphism also by alt: ∼
alt : H 2 (H, k × ) − → 2 H.
(9)
Note that both H 2 (H, k × ) and 2 H are right G-modules via the conjugation and that alt is G-linear. 3. Lagrangian subcategories in the untwisted case We fix notation for this section. Let G be a finite group. For any g ∈ G, let K g denote the conjugacy class of G containing g. Let R denote a complete set of representatives of conjugacy classes of G. Let C denote the representation category Rep(D(G)) of the Drinfeld double of the group G: C := Rep(D(G)). The category C is equivalent to Z(VecG ), the center of VecG . It is well known that C is a modular category. Let denote a complete set of representatives of simple objects of C. The set is in bijection with the set {(a, χ ) | a ∈ R and χ is an irreducible character of C G (a)}, where C G (a) is the centralizer of a in G (see [CGR]). In what follows we will identify with the previous set, := {(a, χ ) | a ∈ R and χ is an irreducible character of C G (a)}.
(10)
Let S and θ be (see, e.g. [BK,CGR]) the S-matrix and twist, respectively, of C. Recall that we take the canonical twist. It is known that the entries of the S-matrix lie in a cyclotomic field. Also, the values of characters of a finite group are sums of roots of unity. So we may assume that all scalars appearing herein are complex numbers; in particular, complex conjugation and absolute values make sense. We have the following formulas for the S-matrix, twist and dimensions: S((a, χ ), (b, χ )) =
|G| |C G (a)||C G (b)|
χ (a) θ (a, χ ) = , deg χ d((a, χ )) = |K a | deg χ =
χ (gbg −1 ) χ (g −1 ag),
g∈G(a, b)
|G| deg χ , |C G (a)|
for all (a, χ ), (b, χ ) ∈ , where G(a, b) = {g ∈ G | agbg −1 = gbg −1 a}. 3.1. Classification of Lagrangian subcategories of Rep(D(G)). Lemma 3.1. Two objects (a, χ ), (b, χ ) ∈ centralize each other if and only if the following conditions hold: (i) The conjugacy classes K a , K b commute element-wise, (ii) χ (gbg −1 ) χ (g −1 ag) = deg χ deg χ , for all g ∈ G.
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Proof. By [Mu2, Corollary 2.14] two objects (a, χ ), (b, χ ) ∈ centralize each other if and only if S((a, χ ), (b, χ )) = deg χ deg χ . This is equivalent to the equation
χ (gbg −1 ) χ (g −1 ag) = |G| deg χ deg χ ,
(11)
g∈G(a, b)
where G(a, b) = {g ∈ G | agbg −1 = gbg −1 a}. It is clear that if the two conditions of the lemma hold, then (11) holds since G(a, b) = G. Now suppose that (11) holds. We will show that this implies the two conditions in the statement of the lemma. We have
|G| deg χ deg χ = | χ (gbg −1 ) χ (g −1 ag)| g∈G(a, b)
≤
|χ (gbg −1 )| |χ (g −1 ag)|
g∈G(a, b)
≤ |G| deg χ deg χ . So
g∈G(a, b) |χ (gbg
−1 )| |χ (g −1 ag)|
|χ (gbg
−1
= |G| deg χ deg χ . Since
|G(a, b)| ≤ |G|, )| ≤ deg χ , and |χ (g −1 ag)| ≤ deg χ ,
we must have G(a, b) = G, |χ (gbg −1 )| = deg χ , and |χ (g −1 ag)| = deg χ . The equality G(a, b) = G implies that the conjugacy classes K a , K b commute elementwise, which is Condition (i) in the statement of the lemma. Since |χ (gbg −1 )| = deg χ , and |χ (g −1 ag)| = deg χ , there exist roots of unity αg and βg such that χ (gbg −1 ) = αg deg χ , and χ (g −1 ag) = βg deg χ , for all g ∈ G. Put this in (11) to get the equation
αg βg = |G|. (12) g∈G
Note that (12) holds if and only if αg βg = 1, for all g ∈ G. This is equivalent to saying that χ (gbg −1 ) χ (g −1 ag) = deg χ deg χ , for all g ∈ G and the lemma is proved. Lemma 3.2. Let E be a normal subgroup of a finite group K . Let Irr(K ) denote the set of irreducible characters of K . Let ρ be a K -invariant character of E of degree 1. Then
(deg χ )2 =
χ ∈Irr(K ):χ | E =(deg χ ) ρ
|K | . |E|
Proof. Suppose χ is any irreducible character of K . Since ρ is K -invariant, by Clifford’s Theorem, if ρ is an irreducible constituent of χ | E , then χ | E = (deg χ ) ρ.
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By Frobenius reciprocity, the multiplicity of any irreducible χ in Ind K E ρ is equal to the multiplicity of ρ in χ | E . The latter is equal to deg χ if χ satisfies the above condition and 0 otherwise. Therefore,
|K | , (deg χ )2 = deg Ind K Eρ = |E| χ ∈Irr(K ):χ | E =(deg χ ) ρ
as required.
Let H be a normal Abelian subgroup of G and let B be a G-invariant alternating bicharacter on H . Then H = a∈H ∩R K a . Let L(H, B) := full Abelian subcategory of C generated by a ∈ H ∩ R and χ is an irreducible character of C G (a) . (a, χ ) ∈ such that χ (h) = B(a, h) deg χ , for all h ∈ H (13) Proposition 3.3. The subcategory L(H, B) ⊆ Rep(D(G)) is Lagrangian. Proof. We have χ (gbg −1 ) χ (g −1 ag) = B(a, gbg −1 ) deg χ B(b, g −1 ag) deg χ = B(a, gbg −1 ) B(gbg −1 , a) deg χ deg χ = deg χ deg χ , for all (a, χ ), (b, χ ) ∈ L(H, B) ∩ , g ∈ G. The second equality above is due to G-invariance of B and the third equality holds since B is alternating. By Lemma 3.1, it follows that objects in L(H, B) centralize each other. χ (a) B(a, a) Also, we have θ(a, χ ) = deg χ = deg χ deg χ = 1, for all (a, χ ) ∈ L(H, B) ∩ . Therefore, θ |L(H, B) = id. The dimension of L(H, B) is equal to |G|. Indeed,
dim(L(H, B) ) = d(a, χ )2 (a, χ )∈L(H, B) ∩
=
|K a |2 (deg χ )2
(a, χ )∈L(H, B) ∩
=
|K a |2
=
|K a |2
a∈H ∩R
=
(deg χ )2
χ :(a, χ )∈L(H, B) ∩
a∈H ∩R
|C G (a)| |H |
|G| |K a | |H | a∈H ∩R
= |G|. The fourth equality above is explained as follows. Fix a ∈ H ∩ R. Define ρ : H → k × by ρ(h) := B(a, h). Observe that ρ is a C G (a)-invariant character of H of degree 1 and then apply Lemma 3.2.
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It follows from Lemma 2.4 that L(H, B) is a Lagrangian subcategory of Rep(D(G)) and the proposition is proved. Now, let L be a Lagrangian subcategory of C. So, in particular, the two conditions in Lemma 3.1 hold for all simple objects in L. Define Ka . (14) HL := a∈R:(a, χ )∈Lfor someχ
Note that HL is a normal Abelian subgroup of G. Indeed, that HL is a subgroup follows from the fact that L contains the unit object and is closed under tensor products. The subgroup HL is normal in G because it is a union of conjugacy classes of G. Finally, that HL is Abelian follows by Condition (i) of Lemma 3.1. For each a ∈ H ∩ R, define ξa : HL → k × by ξa (h) :=
χ (h) , deg χ
for h ∈ HL , where χ is any irreducible character of C G (a) such that (a, χ ) ∈ L ∩ . To see that this definition does not depend on the choice of χ , let (a, χ ), (a, χ ), (b, χ ) ∈ L∩ and apply Condition (ii) of Lemma 3.1 to pairs (a, χ ), (b, χ ) and (a, χ ), (b, χ ) to get −1 −1 −1 −1 χ (g ag) χ (gbg −1 ) χ (g ag) χ (gbg −1 ) = and = , deg χ deg χ deg χ deg χ χ |H
χ |H
for all g ∈ G. This implies that deg Lχ = deg χL , for any two pairs (a, χ ), (a, χ ) ∈ L∩. For any a, b ∈ HL ∩ R, by Condition (ii) of Lemma 3.1, ξa and ξb satisfy the equation: ξa (gbg −1 ) = ξb (g −1 ag)−1 ,
for all g ∈ G.
(15)
Define a map BL : HL × HL → k × by BL (h 1 , h 2 ) := ξa (g −1 h 2 g), where h 1 =
gag −1 , g
(16)
∈ G, a ∈ HL ∩ R.
Proposition 3.4. BL is a well-defined G-invariant alternating bicharacter on HL . Proof. First, let us show that BL is well-defined. Suppose gag −1 = kak −1 , where a ∈ HL ∩ R, g, k ∈ G. Then BL (gag −1 , lbl −1 ) = ξa ((g −1l)b(g −1l)−1 ) = ξb ((g −1l)−1 a(g −1l))−1 = ξb (l −1 (gag −1 )l)−1 = ξb (l −1 (kak −1 )l)−1 = ξa ((l −1 k)−1 b(l −1 k)) = ξa (k −1 (lbl −1 )k) = BL (kak −1 , lbl −1 ), for all b ∈ HL ∩ R, l ∈ G. The second and the fifth equalities above are due to (15).
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Let h 1 = kak −1 , h 2 ∈ HL , g ∈ G, where a ∈ HL ∩ R, k ∈ G. Then BL (gh 1 g −1 , gh 2 g −1 ) = BL (gkak −1 g −1 , gh 2 g −1 ) = ξa ((gk)−1 (gh 2 g −1 )(gk)) = ξa (k −1 h 2 k) = BL (kak −1 , h 2 ) = BL (h 1 , h 2 ). So, BL is G-invariant. Now, BL (gag −1 , gag −1 ) = BL (a, a) = ξa (a) χ (a) = deg χ = θ(a, χ ) = 1, for all a ∈ HL ∩ R, g ∈ G. The first equality above is due to the G-invariance of BL . So BL (h, h) = 1, for all h ∈ HL . Also, BL (g1 ag1−1 , g2 bg2−1 )BL (g2 bg2−1 , g1 ag1−1 ) = ξa (g1−1 g2 bg2−1 g1 ) ξb (g2−1 g1 ag1−1 g2 ) = 1, for all g1 , g2 ∈ G, a, b ∈ H ∩ R. We used (15) in the last equality. To see that BL is a bicharacter, observe first that ξa is a homomorphism, for all a ∈ HL ∩ R. We have BL (gag −1 , h 1 ) BL (gag −1 , h 2 ) = ξa (g −1 h 1 g) ξa (g −1 h 2 g) = ξa (g −1 h 1 h 2 g) = BL (gag −1 , h 1 h 2 ), for all a ∈ HL ∩ R, g ∈ G, h 1 , h 2 ∈ HL . We conclude that BL is a G-invariant alternating bicharacter on HL and the proposition is proved. C.
Recall that Lagr(C) denotes the set of Lagrangian subcategories of a modular category
Theorem 3.5. Lagrangian subcategories of the representation category of the Drinfeld double D(G) are classified by pairs (H, B), where H is a normal Abelian subgroup of G and B is an alternating G-invariant bicharacter on H . Proof. Let E := {(H, B) | H be a normal Abelian subgroup of G and B ∈ (2 H )G }. Define a map : E → Lagr(C) : (H, B) → L(H, B) , where C = Rep(D(G)) and L(H, B) is defined in (13). It was shown in Proposition 3.3 that L(H, B) is a Lagrangian subcategory. To see that is injective pick any (H, B), (H , B ) ∈ E and assume that ((H, B)) = ((H , B )). So in particular we will have L(H, B) ∩ = L(H , B ) ∩ . Note that H = ∪(a, χ )∈L(H, B) ∩ K a and H = ∪(a, χ )∈L(H , B ) ∩ K a . Since L(H, B) ∩ = L(H , B ) ∩ , it follows that H = H . Also note that for any (a, χ ) ∈ L(H, B) ∩ =
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L(H , B ) ∩ , we have χ (h) = B(a, h) deg χ = B (a, h) deg χ , for all h ∈ H = H . Since B, B are G-invariant, it follows that B = B . So is injective. To see that is surjective pick any L ∈ Lagr(C). Consider the pair (HL , BL ), where HL and BL are defined in (14) and (16), respectively. Proposition 3.4 showed that (HL , BL ) belongs to the set E. We contend that ((HL , BL )) = L. It suffices to show that L ∩ ⊆ L(HL , BL ) . But this holds by definition of L(HL , BL ) and the observation χ |H
χ |H
that deg Lχ = deg χL , for any two pairs (a, χ ), (a, χ ) ∈ L ∩ , a ∈ HL ∩ R. So is surjective and the theorem is proved. 3.2. Bijective correspondence between Lagrangian subcategories and module categories with pointed duals. Let D be a fusion category and let M be an indecomposable D-module category. There is a canonical braided tensor equivalence [Mu1,EO] ∼
∗ ιM : Z(D) − → Z(DM )
(17)
∗ )rev -module endodefined by identifying both centers with the category of D (DM functors of M. ∼ Let f : E(C) − → Lagr(C) be the bijection between the set of (equivalence classes of) braided tensor equivalences between C and centers of pointed fusion categories and the set of Lagrangian subcategories of C defined in [DGNO], see (8).
Theorem 3.6. The assignment M → ιM restricts to a bijection between the set of equivalence classes of indecomposable VecG -module categories M with respect to which the dual fusion category (VecG )∗M is pointed and E(Rep(D(G))). Proof. Comparing the result of [Na] (see Example 2.2) and Theorem 3.5 and taking into ∼ → (2 H ) is G-linear, we see that the account that the isomorphism alt : H 2 (H, k × ) − two sets in question have the same cardinality. Thus, to prove the theorem it suffices to check that for M := M(H, µ) one has f (ιM ) ⊆ L(H, alt (µ)) , where L(H, alt (µ)) is the Lagrangian subcategory defined in (13). By definition, f (ιM ) consists of all objects Z in C = Z(V ecG ) (identified with Rep(D(G))) such that the VecG -module endofunctor FZ : M → M : M → M ⊗ Z is isomorphic to a multiple of idM . Note that here we abuse notation and write Z for both the object of the center and its forgetful image. Let us recall the parameterization of simple objects of Z(VecG ) in (10). Suppose that a simple Z corresponds to the conjugacy class K a represented by a ∈ R and the character afforded by the irreducible representation π : C G (a) → G L(Vπ ). Then as a G-graded ∼ → Z ⊗g vector space Z = ⊕x∈K a Vπx and the permutation isomorphism cg,Z : g ⊗ Z − is induced from π , where we identify simple objects of VecG with the elements of the group G. It is clear that FZ is isomorphic to a multiple of idM as an ordinary functor if and only if K a ⊆ H . Note that this implies that H ⊆ C G (a). Note that for every VecG -module functor F : M → M the module functor structure on F is completely determined by ∼ → F(H 1) ⊗ h, h ∈ H , where H 1 denotes the collection of isomorphisms F(H 1 ⊗ h) − the trivial coset in H \G = Irr(M). For F = FZ the latter isomorphism is given by the composition ⊕x µ(h,x)−1 idVπx
id H 1 ⊗ch,Z
⊕x µ(x,h)idVπx
(H 1 ⊗ h) ⊗ Z −−−−−−−−−−→ H 1 ⊗ (h ⊗ Z ) −−−−−−→ H 1 ⊗ (Z ⊗ h) −−−−−−−−→ (H 1 ⊗ Z ) ⊗ h.
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The restriction of ch,Z to h ⊗ Vπa is given by π(h), for all h ∈ C G (a). If the above composition equals identity, then π(h) = alt (µ)(a, h) idVπ , for all h ∈ H . So Z ∈ L(H, alt (µ)) and, therefore, f (ιM ) ⊆ L(H, alt (µ)) , as required. Remark 3.7. Let us explicitly describe the subcategory L(H, B) . By [De] there is a unique up to an isomorphism group G such that L(H, B) ∼ = Rep(G ) as a symmetric category. This group G is precisely the group of invertible objects in the dual category (VecG )∗M(H,µ) , where µ ∈ Z 2 (H, k × ) is such that alt (µ) = B. It was shown in [Na] that G is an extension → G → H \G → 0, 0→H with the corresponding second cohomology class being the image of the cohomology ), see class of µ under the canonical homomorphism H 2 (H, k × )G → H 2 (H \G, H [Na] for details. Note that in general G ∼ = G , see Sect. 5 for examples. 4. Lagrangian subcategories in the twisted case In this section we extend the constructions of the previous section when the associativity is given by a 3-cocycle ω ∈ Z 3 (G, k × ). Note that the results of this section reduce to the results in Sect. 3 when ω ≡ 1. For this section we follow the notation fixed at the beginning of Sect. 3. Let ω be a normalized 3-cocycle on G, i.e., ω is a map from G × G × G to k × satisfying: ω(g2 , g3 , g4 )ω(g1 , g2 g3 , g4 )ω(g1 , g2 , g3 ) = ω(g1 g2 , g3 , g4 )ω(g1 , g2 , g3 g4 ), (18) ω(g, 1G , l) = 1, for all g, l, g1 , g2 , g3 , g4 ∈ G. Let C denote the representation category Rep(D ω (G)) of the twisted quantum double of the group G [DPR1,DPR2]: C := Rep(D ω (G)). The category C is equivalent to Z(VecωG ). It is well known that C is a modular category. Replacing ω by a cohomologous 3-cocycle we may assume that the values of ω are roots of unity. For all a, g, h ∈ G, define βa (h, g) := ω(a, h, g)ω(h, h −1 ah, g)−1 ω(h, g, (hg)−1 ahg).
(19)
The βa ’s satisfy the following equation: βa (x, y)βa (x y, z) = βa (x, yz)βx −1 ax (y, z),
for all x, y, z ∈ G.
(20)
Observe that the restriction of each βa to the centralizer C G (a) of a in G is a normalized 2-cocycle. Let denote a complete set of representatives of simple objects of C. The set is in bijection with the set {(a, χ ) | a ∈ R and χ is an irreducible βa -character of C G (a)}. In what follows we will identify with the previous set: := {(a, χ ) | a ∈ R and χ is an irreducible βa -character of C G (a)}.
(21)
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Let S and θ be the S-matrix and twist, respectively, of C. It is known that the entries of the S-matrix lie in a cyclotomic field. Also, the values of α-characters of a finite group are sums of roots of unity, so they are algebraic numbers, where α is any 2-cocycle whose values are roots of unity. So we may assume that all scalars appearing herein are complex numbers; in particular, complex conjugation and absolute values make sense. We have the following formulas for the S-matrix, twist, and dimensions (see [CGR]): S((a, χ ), (b, χ ))
=
g∈K a ,g ∈K b ∩C G (g)
θ (a, χ ) =
βa (x, g )βa (xg , x −1 )βb (y, g)βb (yg, y −1 ) χ(xg x −1 ) χ (ygy −1 ), βa (x, x −1 )βb (y, y −1 )
χ (a) , deg χ
d((a, χ )) = |K a | deg χ =
|G| deg χ , |C G (a)|
for all (a, χ ), (b, χ ) ∈ , where g = x −1 ax, g = y −1 by.
4.1. Classification of Lagrangian subcategories of Rep(Dω (G)). Remark 4.1. Let ρ : K → G L(V ) be a finite-dimensional projective representation with 2-cocycle α on the finite group K , i.e., ρ(x y) = α(x, y)ρ(x)ρ(y), for all x, y ∈ K . Let χ be the projective character afforded by ρ, i.e., χ (x) = Trace(ρ(x)), for all x ∈ K . Suppose that the values of α are roots of unity. Then |χ (x)| ≤ deg χ , for all x ∈ K and we have equality if and only if ρ(x) ∈ k × · idV . Lemma 4.2. Two objects (a, χ ), (b, χ ) ∈ centralize each other if and only if the following conditions hold: (i) The classes K a , K b commute element-wise, conjugacy βa (x,y −1 by)βa (x y −1 by,x −1 )βb (y,x −1 ax)βb (yx −1 ax,y −1 ) (ii) χ (x y −1 byx −1)χ (yx −1 ax y −1) = −1 −1 β (x,x )β (y,y ) a
b
deg χ deg χ , for all x, y ∈ G.
Proof. Two objects (a, χ ), (b, χ ) ∈ centralize each other if and only if S((a, χ ), (b, χ )) = deg χ deg χ . This is equivalent to the equation:
g∈K a ,g ∈K b ∩C G (g)
βa (x, g )βa (xg , x −1 )βb (y, g)βb (yg, y −1 ) χ (xg x −1 ) χ (ygy −1 ) βa (x, x −1 )βb (y, y −1 ) (22) = |K a ||K b | deg χ deg χ ,
where g = x −1 ax, g = y −1 by. It is clear that if the two conditions of the lemma hold, then (22) holds since the set over which the above sum is taken is equal to K a × K b .
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Now suppose that (22) holds. We will show that this implies the two conditions in the statement of the lemma. We have |K a ||K b | deg χ deg χ )β (xg , x −1 )β (y, g)β (yg, y −1 )
(x, g β a a b b −1 −1 χ(xg = x ) χ (ygy ) −1 )β (y, y −1 ) β (x, x a b g∈K a ,g ∈K b ∩C G (g) −1 −1
βa (x, g )βa (xg , x )βb (y, g)βb (yg, y ) χ(xg x −1 ) χ (ygy −1 ) ≤ −1 )β (y, y −1 ) β (x, x a b g∈K a ,g ∈K b ∩C G (g)
χ(xg x −1 ) χ (ygy −1 ) = g∈K a ,g ∈K b ∩C G (g)
≤ |K a ||K b | deg χ deg χ .
So
χ (xg x −1 ) |χ (ygy −1 )| = |K a ||K b | deg χ deg χ .
g∈K a ,g ∈K b ∩C G (g)
Since |{(g, g ) | g ∈ K a , g ∈ K b ∩ C G (g)}| ≤ |K a ||K b |, |χ (xg x −1 )| ≤ deg χ , and |χ (ygy −1 )| ≤ deg χ , we must have |{(g, g ) | g ∈ K a , g ∈ K b ∩ C G (g)}| = |K a ||K b |, i.e. {(g, g ) | g ∈ K a , g ∈ K b ∩ C G (g)} = K a × K b , |χ (xg x −1 )| = deg χ , and |χ (ygy −1 )| = deg χ . The equality {(g, g ) | g ∈ K a , g ∈ K b ∩ C G (g)} = K a × K b implies that K b ⊆ C G (g), for all g ∈ K a . This is equivalent to the condition that K a , K b commute element-wise which is Condition (i) in the statement of the lemma. Now, (22) becomes:
(g, g )∈K a ×K b
βa (x, g )βa (xg , x −1 )βb (y, g)βb (yg, y −1 ) βa (x, x −1 )βb (y, y −1 )
χ (xg x −1 ) χ (ygy −1 ) deg χ deg χ
= |K a ||K b |,
(23)
where g = x −1 ax, g = y −1 by. Since |χ (xg x −1 )| = deg χ , and |χ (ygy −1 )| = x −1 ) (ygy −1 ) and χ deg deg χ , by Remark 4.1, χ (xg deg χ χ are roots of unity. Note that (23) holds if and only if βa (x, g )βa (xg , x −1 )βb (y, g)βb (yg, y −1 ) χ (xg x −1 ) χ (ygy −1 ) = deg χ deg χ , βa (x, x −1 )βb (y, y −1 ) for all g ∈ K a , g ∈ K b , where g = x −1 ax, g = y −1 by. This is equivalent to Condition (ii) in the statement of the lemma. Note 4.3. Let E be a subgroup of a finite group K . Let α be a 2-cocycle on K . Let χ be a projective α-character of E. For any x ∈ K , define χ x by χ x (l) := α(lx, x −1 )−1 α(x, x −1 lx)−1 α(x, x −1 ) χ (x −1 lx), for all l ∈ E. Then χ x is a projective α-character of x E x −1 . Suppose E is normal in K . Then χ is said to be K -invariant if χ x = χ , for all x ∈ K .
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Lemma 4.4. Let E be a normal subgroup of a finite group K . Let α be a 2-cocycle on K . Let Irr(K ) denote the set of irreducible projective α-characters of K . Let ρ be a K -invariant projective α| E×E -character of E of degree 1. Then
(deg χ )2 =
χ ∈Irr(K ):χ | E =(deg χ ) ρ
|K | . |E|
Proof. The proof is completely similar to the one given in Lemma 3.2 except in this case we apply Clifford’s Theorem [Ka, Theorem 8.1] and Frobenius reciprocity [Ka, Proposition 4.8] for projective characters. Let H be a normal Abelian subgroup of G. Recall that ω ∈ Z 3 (G, k × ) gives rise to a collection (20) of 2-cochains βa , a ∈ G. Definition 4.5. We will say that a map B : H × H → k × is an alternating ω-bicharacter on H if it satisfies the following three conditions: B(h 1 , h 2 ) = B(h 2 , h 1 )−1 , B(h, h) = 1, 1 δ Bh = βh | H ×H ,
(24) (25) (26)
for all h, h 1 , h 2 ∈ H , where the map Bh : H → k × is defined by Bh (h 1 ) := B(h, h 1 ), for all h, h 1 ∈ H . Definition 4.6. We will say that an alternating ω-bicharacter B : H × H → k × on H is G-invariant if it satisfies the following condition: B(x −1 ax, h) =
βa (x, h)βa (xh, x −1 ) B(a, xhx −1 ), βa (x, x −1 )
for all x ∈ G, a ∈ H ∩ R, h ∈ H. (27)
Define 2ω H := {B : H × H → k × | B is an alternating ω − bicharacter on H },
(28)
(2ω H )G := {B ∈ 2ω H | B is G-invariant}.
(29)
and
Remark 4.7. If ω ≡ 1, then (2ω H )G is the Abelian group of G-invariant alternating bicharacters on H . Remark 4.8. If B is an alternating ω-bicharacter on H , then the restriction ω| H ×H ×H must be cohomologically trivial. Indeed, let ω H := ω| H ×H ×H . Then B defines a braid∼ → h 2 ⊗ h 1 is given by ing on the fusion category VecωHH . The isomorphism h 1 ⊗ h 2 − B(h 1 , h 2 ), for all h 1 , h 2 ∈ H , where we identify simple objects of VecωHH with elements of H . It is known (see, e.g., [Q,FRS]) that in this case ω H is an Abelian 3-cocycle on H . By a classical result of Eilenberg and MacLane [EM] the third Abelian cohomology group of H is isomorphic to the (multiplicative) group of quadratic forms on H . The value of the corresponding quadratic form q on h ∈ H is given by q(h) = B(h, h). Since B is alternating we have q ≡ 1 and so ω H must be cohomologically trivial.
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Let B ∈ (2ω H )G and define: L(H, B) := full Abelian subcategory of C generated by
(a, χ) ∈
a ∈ H ∩ R and χ is an irreducible βa -character of C G (a) such that χ(h) = B(a, h) deg χ, for all h ∈ H
. (30)
Proposition 4.9. The subcategory L(H, B) ⊆ Rep(D ω (G)) is Lagrangian. Proof. Pick any (a, χ ), (b, χ ) ∈ L(H, B) ∩ . We have
βa (x, y −1 by)βa (x y −1 by, x −1 )βb (y, x −1 ax)βb (yx −1 ax, y −1 ) χ(x y −1 byx −1 ) χ (yx −1 ax y −1 ) βa (x, x −1 )βb (y, y −1 ) =
βa (x, y −1 by)βa (x y −1 by, x −1 ) B(a, x y −1 byx −1 ) βa (x, x −1 ) ×
βb (y, x −1 ax)βb (yx −1 ax, y −1 ) B(b, yx −1 ax y −1 ) × deg χ deg χ βb (y, y −1 )
= B(x −1 ax, y −1 by) B(y −1 by, x −1 ax) deg χ deg χ = deg χ deg χ ,
for all x, y ∈ G. The second equality above is due to (27) while the third equality is due to (24). Note that K a , K b commute element-wise since H is Abelian. By Lemma 4.2, it follows that objects in L(H, B) centralize each other. Also, θ |L(H, B) = id. The proof of this assertion is exactly the one given in Proposition 3.3. Now, fix a ∈ H ∩ R and observe that Ba defines a C G (a)-invariant βa -character of H of degree 1. Indeed,
(Ba )x (h) =
βa (x, x −1 ) B(a, x −1 hx) βa (hx, x −1 )βa (x, x −1 hx)
= B(x −1 ax, x −1 hx)−1 B(a, h) B(a, x −1 hx) = B(a, h), for all x ∈ C G (a), h ∈ H . The second equality above is due to (27). The dimension of L(H, B) is equal to |G|. The proof of this assertion is exactly the one given in Proposition 3.3 except we appeal to Lemma 4.4 in this case. It follows from Lemma 2.4 that L(H, B) is a Lagrangian subcategory of Rep(D ω (G)) and the proposition is proved. Lemma 4.10. Let H be a normal Abelian subgroup of G. Let B : H × H → k × be a map satisfying (24),(25), and (27). Suppose δ 1 Ba = βa | H ×H , for all a ∈ H ∩ R. Then B ∈ (2ω H )G .
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Proof. We only need to verify that (26) holds. We have (δ 1 Bx −1 ax )(h 1 , h 2 ) B(x −1 ax, h 1 )B(x −1 ax, h 2 ) B(x −1 ax, h 1 h 2 ) βa (x, h 2 )βa (xh 2 , x −1 ) βa (x, h 1 )βa (xh 1 , x −1 ) −1 = B(a, xh 1 x ) × B(a, xh 2 x −1 ) βa (x, x −1 ) βa (x, x −1 ) βa (x, x −1 ) × B(a, xh 1 h 2 x −1 )−1 βa (x, h 1 h 2 )βa (xh 1 h 2 , x −1 ) =
=
βa (x, h 1 )βa (xh 1 , x −1 )βa (x, h 2 )βa (xh 2 , x −1 )βa (xh 1 x −1 , xh 2 x −1 ) βa (x, x −1 )βa (x, h 1 h 2 )βa (xh 1 h 2 , x −1 )
β −1 (h 1 , h 2 )βa (xh 1 , x −1 )βa (x, h 2 )βa (xh 2 , x −1 )βa (xh 1 x −1 , xh 2 x −1 ) = x ax βa (xh 1 , h 2 )βa (x, x −1 )βa (xh 1 h 2 , x −1 ) β −1 (h 1 , h 2 )βa (xh 1 , h 2 x −1 )βx −1 ax (x −1 , xh 2 x −1 )βa (x, h 2 x −1 )βx −1 ax (h 2 , x −1 ) = x ax βa (xh 1 , h 2 )βa (x, x −1 )βa (xh 1 h 2 , x −1 ) = βx −1 ax (h 1 , h 2 ),
for all x ∈ G, a ∈ H ∩ R, h 1 , h 2 ∈ H . In the second equality above, we used (27). In the third equality we used δ 1 Ba = βa | H ×H , and canceled some factors. In the fourth equality we used (20) with (x, y, z) = (x, h 1 , h 2 ). In the fifth equality we used (20) twice with (x, y, z) = (x, h 2 , x −1 ), (xh 1 , x −1 , xh 2 x −1 ). In the last equality we used (20) twice with (x, y, z) = (xh 1 , h 2 , x −1 ), (x, x −1 , xh 2 x −1 ). Now, let L be a Lagrangian subcategory of C. So, in particular, the two conditions in Lemma 4.2 hold for all objects in L ∩ . Define HL :=
Ka .
(31)
a∈R:(a, χ )∈L∩ for some χ
Note that HL is a normal Abelian subgroup of G. Define a map BL : HL × HL → k × by BL (h 1 , h 2 ) :=
βa (x, h 2 )βa (xh 2 , x −1 ) χ (xh 2 x −1 ) × , βa (x, x −1 ) deg χ
(32)
where h 1 = x −1 ax, x ∈ G, a ∈ HL ∩ R and χ is any βa -character of C G (a) such that (a, χ ) ∈ L ∩ . The above definition does not depend on the choice of χ . The proof of this assertion is similar to the proof given for the corresponding assertion in the untwisted case. Proposition 4.11. The map BL defined in (32) is an element of (2ω H )G .
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Proof. First, let us show that BL is well-defined. Suppose x −1 ax = z −1 az, where a ∈ HL ∩ R, x, z ∈ G. Then βa (x, y −1 by)βa (x y −1 by, x −1 ) χ (x y −1 byx −1 ) × βa (x, x −1 ) deg χ −1 −1 βb (y, x −1 ax)βb (yx −1 ax, y −1 ) χ (yx −1 ax y −1 ) = βb (y, y −1 ) deg χ −1 −1 −1 χ (yz azy −1 ) βb (y, z −1 az)βb (yz −1 az, y −1 ) = βb (y, y −1 ) deg χ
BL (x −1 ax, y −1 by) =
=
βa (z, y −1 by)βa (zy −1 by, z −1 ) χ (zy −1 byz −1 ) × βa (z, z −1 ) deg χ
= BL (z −1 az, y −1 by), for all b ∈ HL ∩ R, y ∈ G, where χ is any irreducible βb -character of C G (b) such that (b, χ ) ∈ L ∩ . The second and the fourth equalities above are due to Condition (ii) of Lemma 4.2. The map BL satisfies (24) because Condition (ii) of Lemma 4.2 holds. Let us show that (25) holds for BL : BL (x −1 ax, x −1 ax) =
βa (x, x −1 ax)βa (ax, x −1 ) χ (a) × deg χ βa (x, x −1 )
= ω(a, x, x −1 ax) ×
ω(a, ax, x −1 )ω(ax, x −1 , a) ω(x, x −1 ax, x −1 ) × × θ(a, χ ) −1 −1 ω(ax, x ax, x ) ω(a, x, x −1 )ω(x, x −1 , a)
ω(a, x, x −1 a)ω(ax, x −1 , a) ω(a, x, x −1 )ω(x, x −1 , a) = 1,
=
for all x ∈ G, a ∈ HL ∩ R. In the second equality we used the definition of βa . In the third equality we used (18) with (g1 , g2 , g3 , g4 ) = (a, x, x −1 ax, x −1 ) and used the fact that θ(a, χ ) = 1 . In the fourth equality we used (18) with (g1 , g2 , g3 , g4 ) = (a, x, x −1 , a). χ (xhx −1 ) deg χ , for χ (h 1 ) χ (h 2 ) deg χ deg χ = βa (h 1 ,
The map BL satisfies (27) because BL (a, xhx −1 ) =
all a ∈ H ∩
(h 1 h 2 ) R, x ∈ G, h ∈ H . We have BL (a, h 1 )BL (a, h 2 ) = h 2 ) χdeg χ = βa (h 1 , h 2 ) BL (a, h 1 h 2 ), for all a ∈ H ∩ R, h 1 , h 2 ∈ H . The second to last equality above is because H acts as scalars on the projective βa -representation of C G (a) whose projective character is χ . By Lemma 4.10 it follows that BL ∈ 2ω H and the proposition is proved.
Theorem 4.12. Lagrangian subcategories of the representation category of the twisted double D ω (G) are classified by pairs (H, B), where H is a normal Abelian subgroup of G such that ω| H ×H ×H is cohomologically trivial and B : H × H → k × is a G-invariant alternating ω-bicharacter in the sense of Definition 4.6. Proof. The proof is completely similar to the one given in Theorem 3.5.
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4.2. Bijective correspondence between Lagrangian subcategories and module categories with pointed duals. Recall [Na] that equivalence classes of indecomposable module categories over VecωG for which the dual is pointed are in bijection with pairs (H, µ), where H is a normal Abelian subgroup of G such that ω| H ×H ×H is cohomologically trivial and µ ∈ ( H,ω )G (defined in (6)). Theorem 4.12 showed that Lagrangian subcategories of Rep(D ω (G)) are in bijection with pairs (H, B), where H is a normal Abelian subgroup of G such that ω| H ×H ×H is cohomologically trivial and B ∈ (2ω H )G (the latter was defined in (28)). In this subsection we will first show that the set of equivalence classes of indecomposable module categories over VecωG such that the dual is pointed is in bijection with the set of Lagrangian subcategories of Rep(D ω (G)). Let H be a normal Abelian subgroup of G such that ω| H ×H ×H is cohomologically trivial. We will establish the aforementioned bijection by showing that there is a bijection between H,ω (defined in (3)) and 2ω H that restricts to a bijection between ( H,ω )G and (2ω H )G . Let µ ∈ C 2 (H, k × ) be a 2-cochain satisfying δ 2 µ = ω| H ×H ×H . Define alt (µ) by alt (µ)(h 1 , h 2 ) :=
µ(h 2 , h 1 ) , h 1 , h 2 ∈ H. µ(h 1 , h 2 )
Lemma 4.13. The map alt (µ) : H × H → k × defined above is an element of 2ω H . Proof. Clearly alt (µ)(h 1 , h 2 ) = alt (µ)(h 2 , h 1 )−1 and alt (µ)(h, h) = 1, for all h, h 1 , h 2 ∈ H . We have alt (µ)(h, h 1 ) alt (µ)(h, h 2 ) µ(h 1 , h) µ(h 2 , h) µ(h, h 1 h 2 ) = × × alt (µ)(h, h 1 h 2 ) µ(h, h 1 ) µ(h, h 2 ) µ(h 1 h 2 , h) µ(h 1 , h)µ(h 2 , h)µ(hh 1 , h 2 ) × ω(h, h 1 , h 2 ) = µ(h, h 2 )µ(h 1 h 2 , h)µ(h 1 , h 2 ) µ(h 1 , h)µ(hh 1 , h 2 ) ×ω(h, h 1 , h 2 )×ω(h 1 , h 2 , h) = µ(h, h 2 )µ(h 1 , hh 2 ) ω(h, h 1 , h 2 )ω(h 1 , h 2 , h) = ω(h 1 , h, h 2 ) = βh (h 1 , h 2 ), for all h, h 1 , h 2 ∈ H . In the second, third, and fourth equalities above we used (2) with (h 1 , h 2 , h 3 ) = (h, h 1 , h 2 ), (h 1 , h 2 , h), (h 1 , h, h 2 ), respectively. The map alt induces a map between H,ω and 2ω H . By abuse of notation we denote this map also by alt : alt : H,ω → 2ω H : µ → alt (µ).
(33)
Lemma 4.14. The map alt defined above is a bijection. Proof. First note that alt is well-defined. Fix µ0 ∈ C 2 (H, k × ) satisfying δ 2 µ0 = ∼ → 2 H : B → BB0 and ω| H ×H ×H . Let B0 := alt (µ0 ). Define bijections f 1 : 2ω H − ∼ f 2 : H,ω − → H 2 (H, k × ) : µ → µµ0 . Note that the cardinality of the two sets H,ω
and 2ω H are equal. Injectivity, and hence bijectivity, of alt follows from the equality f 1 ◦ alt = alt ◦ f 2 .
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Lemma 4.15. The following relation holds: βxh 1 x −1 (x, h 2 )βxh 1 x −1 (xh 2 , x −1 ) ϒx (h 2 , h 1 ) = , for all x ∈ G, h 1 , h 2 ∈ H. ϒx (h 1 , h 2 ) βxh 1 x −1 (x, x −1 ) Proof. We have βxh 1 x −1 (x, x −1 ) ϒx (h 2 , h 1 ) × ϒx (h 1 , h 2 ) βxh 1 x −1 (x, h 2 )βxh 1 x −1 (xh 2 , x −1 ) =
ω(xh 2 x −1 , xh 1 x −1 , x) ω(xh 1 x −1 , x, x −1 )ω(x, x −1 , xh 1 x −1 ) × ω(xh 2 x −1 , x, h 1 )ω(xh 1 x −1 , xh 2 x −1 , x) ω(x, h 1 , x −1 ) ×
= =
ω(xh 2 , h 1 , x −1 ) ω(xh 1 x −1 , xh 2 , x −1 )ω(xh 2 , x −1 , xh 1 x −1 )
ω(xh 2 x −1 , xh 1 x −1 , x)ω(xh 1 x −1 , x, x −1 )ω(x, x −1 , xh 1 x −1 )ω(xh 2 x −1 , xh 1 , x −1 ) ω(xh 1 x −1 , xh 2 x −1 , x)ω(xh 1 x −1 , xh 2 , x −1 )ω(xh 2 , x −1 , xh 1 x −1 )ω(xh 2 x −1 , x, h 1 x −1 ) ω(xh 1
x −1 ,
xh 2
x −1 ,
ω(x, x −1 , xh 1 x −1 )ω(xh 1 h 2 x −1 , x, x −1 ) x)ω(xh 1 x −1 , xh 2 , x −1 )ω(xh 2 , x −1 , xh 1 x −1 )ω(xh 2 x −1 , x, h 1 x −1 )
ω(x, x −1 , xh 1 x −1 )ω(xh 2 x −1 , x, x −1 ) = ω(xh 2 , x −1 , xh 1 x −1 )ω(xh 2 x −1 , x, h 1 x −1 ) = 1,
for all x ∈ G, h 1 , h 2 ∈ H . In the first equality above we used the definition of ϒ and β and canceled some factors. In the second, third, fourth, and fifth equalities we used (18) with (g1 , g2 , g3 , g4 ) = (xh 2 x −1 , x, h 1 , x −1 ), (xh 2 x −1 , xh 1 x −1 , x, x −1 ), (xh 1 x −1 , xh 2 x −1 , x, x −1 ), and (xh 2 x −1 , x, x −1 , xh 1 x −1 ), respectively. Lemma 4.16. The map alt defined in (33) restricts to a bijection between ( H,ω )G and (2ω H )G . Proof. Let us first show that alt (( H,ω )G ) ⊆ (2ω H )G . Pick any µ ∈ ( H,ω )G . So alt
µx µ
× ϒx | H ×H = 1, for all x ∈ G. We have
alt (µ)(x −1 ax, h) × alt (µ)(a, xhx −1 )−1 =
µ(h, x −1 ax) µ(a, xhx −1 ) × µ(x −1 ax, h) µ(xhx −1 , a)
µx (x −1 ax, h) µ(h, x −1 ax) × x −1 µ(x ax, h) µ (h, x −1 ax) x ϒx (h, x −1 ax) µ = alt × ϒx | H ×H (h, x −1 ax) × µ ϒx (x −1 ax, h)
=
=
ϒx (h, x −1 ax) ϒx (x −1 ax, h)
=
βa (x, h)βa (xh, x −1 ) , βa (x, x −1 )
for all x ∈ G, a ∈ H ∩ R, h ∈ H . In the fourth equality above we used the fact x that alt µµ × ϒx | H ×H = 1 and in the fifth equality we used Lemma 4.15. So
alt (( H,ω )G ) ⊆ (2ω H )G , as desired.
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Now let us show that (2ω H )G ⊆ alt (( H,ω )G). Pick any µ ∈ H,ω and suppose x that alt (µ) ∈ (2ω H )G . It suffices to show that alt µµ × ϒx | H ×H = 1, for all x ∈ G.
Let B := alt (µ). We have x µ ϒx (h 1 , h 2 ) alt × ϒx | H ×H (h 1 , h 2 ) × µ ϒx (h 2 , h 1 ) = B(xh 1 x −1 , xh 2 x −1 )B(h 1 , h 2 )−1
= B((yx −1 )−1 a(yx −1 ), xh 2 x −1 )B(y −1 ay, h 2 )−1
(where h 1 = y −1 ay)
βa (y, y −1 ) βa (yx −1 , xh 2 x −1 )βa (yh 2 x −1 , x y −1 ) × −1 −1 βa (yx , x y ) βa (y, h 2 )β(yh 2 , y −1 ) βa (yx −1 , xh 2 )βa (yh 2 , x −1 )βa (yh 2 x −1 , x y −1 ) = βxh 1 x −1 (xh 2 , x −1 )βa (y, y −1 )βa (yx −1 , x y −1 )βa (y, h 2 )βa (yh 2 , y −1 )
=
=
βa (yx −1 , x)βa (yh 2 , x −1 )βa (yh 2 x −1 , x y −1 )βa (y, y −1 ) βxh 1 x −1 (x, h 2 )βxh 1 x −1 (xh 2 , x −1 )βa (yx −1 , x y −1 )βa (yh 2 , y −1 )
=
βxh 1 x −1 (x, x −1 )βa (yh 2 , x −1 )βa (yh 2 x −1 , x y −1 )βa (y, y −1 ) βxh 1 x −1 (x, h 2 )βxh 1 x −1 (xh 2 , x −1 )βa (y, x −1 )βa (yx −1 , x y −1 )βa (yh 2 , y −1 )
ϒx (h 1 , ϒx (h 2 , ϒx (h 1 , = ϒx (h 2 , =
βa (y, y −1 )βh 1 (x −1 , x y −1 ) h2) × h 1 ) βa (y, x −1 )βa (yx −1 , x y −1 ) h2) , h1)
for all x ∈ G, h 1 , h 2 ∈ H . In the fourth through eighth equalities above we used (20) with (x, y, z) = (yx −1 , xh 2 , x −1 ), (yx −1 , x, h 2 ), (yx −1 , x, x −1 ), (yh 2 , x −1 , x y −1 ), and (y, x −1 , x y −1 ), respectively. It follows that (2ω H )G ⊆ alt (( H,ω )G ) and the lemma is proved. Recall that E(C) denotes the set of (equivalence classes of) braided tensor equivalences between a modular category C and the centers of pointed fusion categories. Theorem 4.17. The assignment M → ιM (defined in (17)) restricts to a bijection between equivalence classes of indecomposable VecωG -module categories M with respect to which the dual fusion category (VecωG )∗M is pointed and E(Rep(D ω (G))). Proof. The proof is completely similar to the one given in Theorem 3.6.
Remark 4.18. The equivalence type of the symmetric category L(H,B) of Rep(D ω (G)) can be explicitly described in a way similar to Remark 3.7, cf. [Na, Theorem 4.5]. Theorem 4.19. Let C1 , C2 be group-theoretical fusion categories. Then C1 , C2 are weakly Morita equivalent if and only if their centers Z(C1 ) and Z(C2 ) are equivalent as braided fusion categories. Proof. That the “if” part is true for all fusion categories was first observed by M. Müger in [Mu1, Remark 3.18]. This follows from the definition of weak Morita equivalence and a theorem of P. Schauenburg [S]. See also [N,O1,EO].
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For the “only if” part, let (G 1 , ω1 ), (G 2 , ω2 ) be two pairs of groups and 3-cocycles such that C1 is weakly Morita equivalent to VecωG11 and C2 is weakly Morita equivalent to VecωG22 . ω ω If Z(C1 ) ∼ = Z(C2 ) (as braided fusion categories) then Z(VecG11 ) ∼ = Z(VecG22 ) (as braided fusion categories) and therefore, VecωG11 and VecωG22 are weakly Morita equivalent by Theorem 4.17 and hence, C1 and C2 are weakly Morita equivalent. Corollary 4.20. Let G, G be finite groups, ω ∈ Z 3 (G, k × ), and ω ∈ Z 3 (G , k × ). Then the representation categories of twisted doubles D ω (G) and D ω (G ) are equivalent as braided tensor categories if and only if G contains a normal Abelian subgroup H such the following conditions are satisfied: (1) ω| H ×H ×H is cohomologically trivial, (2) there is a G-invariant (see (6))2-cochain µ ∈ C 2 (H, k × ) such that that δ 2 µ = ω| H ×H ×H , and ∼ ν (H \G) such that ◦ (a × a × a) and ω (3) there is an isomorphism a : G − →H are cohomologically equivalent. ) coming from the G-invariance of µ and Here ν is a certain 2-cocycle in Z 2 (H \G, H ν (H \G) depending on ν and on the exact sequence is a certain 3-cocycle on H 1 → H → G → H \G → 1 (see [Na, Theorem 5.8] for precise definitions). 5. Examples 5.1. Lagrangian subcategories of the Drinfeld doubles of finite symmetry groups. Example 5.1. Consider the group of symmetries of a regular n-gon, i.e., the dihedral group D2n = r, s | r n = s 2 = 1, r s = sr −1 , n ≥ 2. Let us describe the Lagrangian subcategories of Rep(D(D2n )) ∼ = Z(Vec D2n ). Let n = 2. Then D4 ∼ = Z/2Z × Z/2Z. There are six different Lagrangian subcategories of Rep(D(Z/2Z × Z/2Z)), all of them equivalent to Rep(Z/2Z × Z/2Z). Let n = 4. The group D8 has five normal Abelian subgroups which give rise to seven Lagrangian subcategories of Z(Vec D8 ). With an exception of the Lagrangian subcategory corresponding to the center r 2 of D8 , all Lagrangian subcategories are equivalent to Rep(D8 ). The one corresponding to r 2 is equivalent to Rep(Z/2Z×Z/2Z×Z/2Z). Applying [DGNO] we conclude that for some 3-cocycle ω on Z/2Z × Z/2Z × Z/2Z there is a braided tensor equivalence Rep(D(D8 )) ∼ = Rep(D ω (Z/2Z × Z/2Z × Z/2Z)) (existence of such an equivalence is already known to experts, see [CGR] and [GMN]). If n = 3 or n ≥ 5 then normal Abelian subgroups of D2n are precisely rotation subgroups r k , where k is a divisor of n. Each of these subgroups is cyclic and so has a trivial Schur multiplier. The Lagrangian subcategory of Z(Vec D2n ) corresponding to r k , k|n, is equivalent to Rep(Dih(Z/ nk Z × Z/kZ)), where for any Abelian group A we denote Dih(A) = A Z/2Z the generalized dihedral group (the action of Z/2Z on A is by inverting elements). Consequently, there is a 3-cocycle ω on Dih(Z/ nk Z × Z/kZ) such that Rep(D ω (Dih(Z/ nk Z × Z/kZ))) ∼ = Rep(D(D2n )) as braided tensor categories. Next, consider the symmetry groups of Platonic solids. Example 5.2. The tetrahedron group A4 has two normal Abelian subgroups: the trivial one and another isomorphic to Z/2Z × Z/2Z. The Schur multiplier of Z/2Z × Z/2Z
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is isomorphic to Z/2Z with the trivial A4 -action. So Rep(D(A4 )) has three Lagrangian subcategories. All three are equivalent to Rep(A4 ). The double of the cube/octahedron group S4 also has three Lagrangian subcategories (corresponding to the same data as in the case of A4 ). One can check that corresponding three Lagrangian subcategories of Rep(D(S4 )) are all isomorphic to Rep(S4 ). Finally, the group of symmetries of dodecahedron/icosahedron is a simple nonAbelian group A5 . It is clear that for any simple non-Abelian group G the category Rep(D(G)) contains a unique Lagrangian subcategory (corresponding to the trivial subgroup of G). Recall [Na] that a group G is called categorically Morita rigid if VecG being weakly Morita equivalent to VecωG implies that groups G and G are isomorphic. One can check that in this case ω must be cohomologically trivial. Our results imply that G is categorically Morita rigid if and only if all Lagrangian subcategories of Rep(D(G)) are equivalent to Rep(G) as symmetric categories. It follows from Examples 5.1 and 5.2 that groups A4 , S4 , A5 , as well as groups D2n , where n is a square-free integer, are categorically Morita rigid. It is clear that any group G without non-trivial normal Abelian subgroups is categorically Morita rigid. Finally, let us consider symmetries of vector spaces. Example 5.3. Let n be a positive integer and let Fq be the finite field with q elements. Let G = S L(n, q) denote the special linear group of n × n matrices with entries from Fq , i.e., matrices having determinant equal to 1. Let P S L(n, q) = S L(n, q)/Z (S L(n, q)), where Z (S L(n, q)) is the center of S L(n, q). Let us assume that (n, q) = (2, 2), (2, 3). It is known that in this case P S L(n, q) is simple. Let d = (n, q − 1) be the greatest common divisor of n and q − 1. The group Z (S L(n, q)) ∼ = Z/dZ is cyclic and any normal subgroup of S L(n, q) is contained in Z (S L(n, q)). Thus, Lagrangian subcategories of Rep(D(S L(n, q))) correspond to divisors of d. One can easily describe the equivalence types of these subcategories. For instance, the Lagrangian subcategory corresponding to Z (S L(n, q)) ∼ = Z/dZ is equivalent to Rep(Z/dZ × P S L(n, q)). Therefore, Rep(D(S L(n, q)) is equivalent (as a braided tensor category) to Rep(D ω (Z/dZ × P S L(n, q))) for some 3-cocycle ω.
5.2. Non-pointed categories of dimension 8. Example 5.4. It is known [TY] that there are exactly four non-equivalent fusion categories of dimension 8 with fusion rules of the dihedral group D8 : Rep(D8 ); Rep(Q 8 ), where Q 8 is the group of quaternions; K P, the representation category of the Kac-Paljutkin Hopf algebra [KacP]; and T Y , the category of representations of a unique 8-dimensional quasi-Hopf algebra which is not gauge equivalent to a Hopf algebra [TY] (equivalently, T Y is the unique non-pointed fusion category of dimension 8 with K 0 (T Y ) = K 0 (Rep(D8 )) which does not have a fiber functor). Let us show that these four categories belong to four different weak Morita equivalence classes and hence, in view of Theorem 4.19, their centers are not equivalent as braided tensor categories. All proper subgroups of Q 8 are normal Abelian and have trivial Schur multipliers. Hence all of them produce pointed duals. So Rep(Q 8 ) is the only non-pointed dual of Vec Q 8 .
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The only non-normal subgroups of D8 = r, s | r 4 = s 2 = 1, r s = sr 3 are reflection subgroups s, sr , and their conjugates. Such subgroups appear as factors in exact factorizations of D8 (one can take r as another factor). The corresponding dual categories (Vec D8 )∗M(s, 1) and (Vec D8 )∗M(sr , 1) admit fiber functors by [O2, Cor. 3.1] and so are representations of semisimple Hopf algebras. But Hopf algebras corresponding to exact factorizations of D8 are known to be either commutative or cocommutative. We conclude that (Vec D8 )∗M(s, 1) ∼ = (Vec D8 )∗M(sr , 1) ∼ = Rep(D8 ), and, hence, Rep(D8 ) is the unique non-pointed dual of Vec D8 . Thus, neither Rep(Q 8 ) nor Rep(D8 ) is weakly Morita equivalent to any other nonpointed category. It remains to check that the same is true for T Y . Let ω0 be a non-trivial 3-cocycle on D8 /r 2 , s ∼ = Z/2Z (corresponding to the non-zero element of H 3 (Z/2Z, k × ) ∼ = Z/2Z). Let π : D8 → D8 /r 2 , s be the canonical projection. Define a 3-cocycle ω on D8 by ω = ω0 ◦ (π × π × π ). Then ω ≡ 1 on r 2 , s and the restrictions of ω on each of the subgroups r and sr are cohomologically non-trivial. This means that the complete list of equivalence classes of indecomposable module categories over VecωD8 consists of M({1}, 1), M(r 2 , 1), M(s, 1), and M(r 2 , s, µ), where µ ∈ H 2 (Z/2Z × Z/2Z, k × ). Therefore, the only non-pointed dual category of VecωD8 corresponds to M(s, 1), see Example 2.2. It follows from the classification of fiber functors on group-theoretical categories obtained in [O2, Cor. 3.1] that the category (VecωD8 )∗M(s, 1) does not have a fiber functor and hence (VecωD8 )∗M(s, 1) ∼ = T Y . Since all other duals of VecωD8 are pointed, it follows that T Y is not weakly Morita equivalent to any other non-pointed fusion category. Hence, Rep(D8 ), Rep(Q 8 ), K P, and T Y are pairwise weakly Morita non-equivalent fusion categories. Our claim about their centers follows from Theorem 4.19. η ∼ Let us note that there is another 3-cocycle η on D8 such that (Vec )∗ = K P. D8 M(s, 1)
Up to a conjugation, such η must have a trivial restriction on r 2 , s and sr (this can be seen from the Kac exact sequence [Kac]). Remark 5.5. The braided tensor equivalence classes of twisted doubles of groups of order 8 were studied in detail in [GMN] using the higher Frobenius-Schur indicators. In particular, it was shown that there are precisely 20 equivalence classes of such nonpointed doubles. In view of results of the present paper this description can be interpreted in terms of weak Morita equivalence classes of pointed categories of dimension 8. Acknowledgements. The present paper would not be possible without [DGNO] and the second named author is happy to thank his collaborators, Vladimir Drinfeld, Shlomo Gelaki, and Victor Ostrik for many useful discussions. The authors thank Nicolas Andruskiewitsch, Pavel Etingof, and Leonid Vainerman for helpful comments. The second author thanks the Université de Caen for hospitality and excellent working conditions during his visit. The authors were supported by the NSF grant DMS-0200202. The research of Dmitri Nikshych was supported by the NSA grant H98230-07-1-0081.
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Communicated by Y. Kawahigashi
