ON THE PROBLEM OF THE RIGHT HAMILTONIAN UNDER SINGULAR FORM-SUM PERTURBATIONS SERGIO ALBEVERIO Institute f¨ ur Ang. Mathematik, Stochastik Universit¨ at Bonn, D 53155 Bonn, Germany Fakult¨ at f¨ ur Mathematik, Ruhr-Universit¨ at Bochum SFB 237, Essen-Bochum-D¨ usseldorf, Germany BiBoS Research Center, Bochum-Bielefeld, Germany CERFIM, Locarno and USI, Switzerland
VOLODYMYR KOSHMANENKO Institute of Mathematics, Kyiv, Ukraine E-mail :
[email protected] Received 30 January 1998 Revised 2 November 1998 1991 Mathematical Subject Classifications: 47A10, 47A55
Let a perturbation of the self-adjoint operator H0 > 0 in the Hilbert space H be given by an operator V (or by a quadratic form ν) which is possibly singular and in general nonpositive, so H0 + V on D(H0 ) ∩ D(V ) is only a symmetric operator with nontrivial deficiency indices. ˙ in the sense of quadratic forms is extended to cases The definition of the sum H = H0 +V which are not covered by the well-known KLMN-theorem and conditions are found which ensure the unique self-adjoint realization of H in H. It is also shown that H coincides with the strong resolvent limit of the approximating sequence Hn = H0 + Vn , where Vn are bounded self-adjoint operators such that Vn → V in a suitable sense. Essentially that operator V might be strongly singular and acts in the H0 -scale of spaces, V : H+ → H− . Keywords: Singular perturbations, generalized sum, self-adjoint extensions.
1. Preliminaries Let H0 > 0 be a self-adjoint operator in a separable complex Hilbert space H. One can consider H0 , resp. H, as the free Hamiltonian, resp. the state space of some physical system. Usually H0 = −∆ is the Laplace operator in an appropriate L2 space. Suppose the perturbation of H0 is given by a possibly nonpositive operator V (or by a quadratic form ν) which has a nontrivial singular part. We resrict our investigations to the situation where V is defined as a symmetric map in the H0 -scale 1/2 of spaces, more precisely, V acts from the positive Hilbert space H+ = D(H0 ) to −1/2 the negative one H− = D(H0 ), equipped with the corresponding graph-norms. Equivalently one can think that the perturbation of H0 is given by a symmetric 1 Reviews in Mathematical Physics, Vol. 12, No. 1 (2000) 1–24 c World Scientific Publishing Company
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S. ALBEVERIO and V. KOSHMANENKO
quadratic form ν(ψ, ϕ) = hV ψ, ϕi = hψ, V ϕi, ϕ, ψ ∈ D(V ) where h·, ·i stands for the dual inner product between H+ and H− . V might present the potential perturbation (multiplication operator) which is possibly nonpositive and contains δ-like singularities. The presence of singularities implies that the sum H0 + V , naturally defined on all ψ ∈ D = D(H0 ) ∩ D(V ) such that (H0 + V )ψ ∈ H, is only a symmetric operator in H with non-zero deficiency indices. Thus the following problems arise. * Which self-adjoint extension H of the restriction (H0 + V )|D of (H0 + V ) to D can be considered as the “right” Hamiltonian for the perturbed system? * When does the approximating sequence Hn = H0 + Vn have a unique nontrivial (6= H0 ) limit H∞ = limn→∞ Hn independent of Vn → V ? * Under what conditions do we have H = H∞ , assuming that the previous problems have been solved? The above problems have a rich history connected with quantum physics, see e.g. the publications [2–5, 7, 13–15, 17, 19, 24–27, 31–33]. Additional references can be found in [1, 12, 16, 18, 28–30, 32]. Recently H. Niedhardt and V. Zagrebnov in [25–27] have given an important momentum to these studies, by looking singular nonpositive perturbations in a single Hilbert space. These authors assume that a perturbation of H0 ≥ 0 is given by a self-adjoint operator W ≤ 0 such that there exists a “stability domain” D ⊂ D(H0 ) ∩ D(W ) dense in H with the following properties: (1) the restriction W |D satisfies the well-known KLMN-theorem’s condition: |(W f, f )| ≤ b(H0 f, f ) + akf k2 ,
0 ≤ b < 1,
a ≥ 0,
f ∈D
(1.1)
and therefore the restriction of the sum (H0 +W )|D generates a symmetric operator in H which is bounded from below; (2) the Friedrichs extension H0,F of the symmetric operator H0 |D is maximal with respect to the perturbation (see [26, 27]). They proved in particular that there exists an approximating sequence Hn = ˜ 0 + Wn , Wn → W , which has a uniquely defined limit H∞ independent of the H ˜ 0 of H0 |D and coinciding with the Friedrichs choice of the self-adjoint extension H extension of the symmetric operator (H0 +W )|D. These authors call H∞ the “right” Hamiltonian. Here we present an extension of the results of [26, 27]. First we replace W by a general nonpositive and singular (for the notion of singularity see [18, 20, 33]) operator V (resp. quadratic form ν) which acts in the H0 -scale of Hilbert spaces. We show that the above KLMN-theorem’s condition (1.1) may be replaced by the weaker requirement that −1 is regular for V |D. In this way we leave the framework of the KLMN-theorem approach. To define ˙ which generalizes the notion the perturbed operator we introduce the sum H0 +V of sum in the sense of quadratic forms. In fact we extend H0 and possibly V to operators H0 , V in the H0 -scale of Hilbert spaces. Then we take their operator
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
3
sum H0 + V which obviously extends the usual sum H0 + V in H. Finally we define ˙ of H0 and V as the “restriction” of H0 + V into the generalized sum H = H0 +V H. The obtained operator H will be self-adjoint in H if, for instance, the range R(H0 + V) contains the whole space H. This occurs if the point −1 is regular for V. Of course the above requirements are automatically fulfilled under condition (1.1). In our approach it is possible that V is purely singular with respect to H (in the sense that the range of V has zero intersection with H) and hence V may not have sense as an operator in a single space but only in the H0 -scale of spaces. In spite of that our conditions on V are weaker than those in [27], they ensure the construction of H as a self-adjoint operator in H in two different ways. H is the “restriction” into H of the operator sum H0 + V (for more details see [21]), and it is uniquely defined as the strong resolvent limit of the approximating sequence of the ˜ 0 + Vn . We show that the limit is independent of regularized perturbations Hn = H ˜ 0 of H˙ 0 = H0 |D and of the sequence Vn the choice of the self-adjoint extension H which converges to V in the appropriate sense. Our main results are given in the following: Theorem. (I) Let us consider in the Hilbert space H an unbounded self-adjoint 1/2 operator H0 > 0 with domain D(H0 ). Let H− ⊃ H ⊃ H+ , H+ = D(H0 ), be the rigged Hilbert space associated with H0 (shortly, the H0 -scale). Consider in the H0 -scale a symmetric (in general nonpositive and singular with respect to H) operator V : H+ → H− . Assume there exists a dense in H (but not necessarily dense in H+ ) linear subset D ⊂ H+ ∩ D(V )
(1.2)
such that the point −1 is regular for the operator V | := (V |D)cl , where cl stands for the closure. ˙ | which coincides Then there exists the self-adjoint in H operator H = H0,D +V with the generalized sum of H0,D and V | , where H0,D is associated with the closure of the quadratic form γ0 |D (γ0 is generated by H0 ). ˙ | where H0,F is the Friedrichs extension If the set D ⊂ D(H0 ) then H = H0,F +V of the symmetric operator H˙ 0 = H0 |D and then H0,D = H0,F . If H is bounded from below then it is associated with the closure in H of the quadratic form γ = γ0 + ν | , where ν | is generated by V | . (II) Assume in the condition (1.2) the set D is dense in H+ .
(1.3)
˙ is self-adjoint in H and moreover H0 +V ˙ = Then the generalized sum H = H0 +V | ˙ H0,D +V . Let Vn , n = 1, 2, . . . be a sequence of bounded self-adjoint operators in H regularizing of V in the sense of the strong convergence in the H0 -scale, i.e., lim kVn ϕ − V ϕkH− = 0 ,
n→∞
ϕ ∈ D.
(1.4)
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S. ALBEVERIO and V. KOSHMANENKO
Assume for some z0 ∈ C, Im z0 6= 0, the extended resolvents Rnz := (H0 + Vn ± zI) as maps from H− to H+ are uniformly bounded on n. Then Hn = H0 + Vn converge to H in the strong resolvent sense in H, −1
˙ = s.r.s. lim (H0 + Vn ) . H = H0 +V n→∞
(1.5)
˜n = ˜ 0 of H˙ 0 = H0 | D the sequence H Moreover for any self-adjoint extension H ˜ H0 + Vn also converges to H in the strong resolvent sense. ˙ is (III) Let as above the set D is dense in H+ and the operator H = H0 +V self-adjoint in H. Assume in addition that H is bounded from below in H. Let Dn ⊂ D, n = 1, 2, . . . be a sequence of subsets which are dense in H and which converge to D in the following sense: ∀ϕ ∈ D,
∃ ϕn ∈ Dn ,
ϕn →H+ ϕ .
Assume Vn , n = 1, 2, . . . be a sequence of self-adjoint bounded operators in H such that Vn |Dn = V |Dn and Vn converge to V in the operator norm sense in the H0 scale. Then the sequence of self-adjoint operators Hn = H0 + Vn converges to H in the uniform resolvent sense. The proof of this theorem is given below in Sec. 4 (Theorem 4.3) and in Sec. 7 (Theorems 7.1 and 7.2). Our methods differ from the ones in [27]. Essentially we use the conception of the generalized sum for operators as it was first introduced in [11, 23] (see also [21]) and base our arguments on the Theorem 2.3 from [5]. 2. The Rigged Hilbert Space Associated with a Self-Adjoint Operator Let H be a self-adjoint lower semibounded operator, with exact lower bound mH > −∞ and domain D(H) in the Hilbert space H. The rigged Hilbert space (2.1) H− ⊃ H ⊃ H+ is said to be associated with H if the positive Hilbert space H+ coincides with the completion of D(H) in any norm which is equivalent to kψk2+ = ((H + mI)ψ, ψ) ,
ψ ∈ D(H) ,
(2.2)
where I denotes the identity operator and m is such that H +mI is positive definite. We can take m > |mH | if mH is negative. We recall (for details see [9, 10]) that the negative Hilbert space H− in (2.1) is defined as the conjugate space to H+ with respect to H. It coincides with the completion of H in the norm kf k2− = ((H + mI)−1 f, f ) ,
f ∈ H.
We will always assume m is such that norms in the chain (2.1) satisfies: k · k− ≤ k · k ≤ k · k+ . In particular the inclusions of spaces in (2.1) are dense and continuous.
(2.3)
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
5
Let γH denote the closure of the quadratic form (Hψ, ψ). Then it is well known [16, 29] that its domain Q(γH ) is a Hilbert space in the norm (2.2), i.e., H+ = Q(γH ) = D((H + mI)1/2 ) .
(2.4)
Let D : H+ → H− denote the canonical unitary isomorphism in (2.1): (ϕ, ψ)+ = (Dϕ, Dψ)− = hDϕ, ψi = hϕ, Dψi ,
(2.5)
where h·, ·i stands for the dual inner product between H+ and H− . It is easy to see that the operator H from H+ to H− is closable. Let H denote its closure. Proposition 2.1. D = H + mI .
(2.6)
Proof. Due to (2.4) H+ coincides with D((H + mI)1/2 ) in the norm (2.2). Since H− is the completion of H in the norm kf k− := k(H + mI)−1/2 f k, f ∈ H we have Dψ = Hψ + mψ ,
ψ ∈ H+ .
(2.7)
Proposition 2.2. The operator H : H+ → H− ,
D(H) = H+
(2.8)
is bounded and self-adjoint as an operator in a pair of Hilbert spaces, i.e., H∗ = H : H+ → H− and (2.9) hHϕ, ψi = hϕ, Hψi , ϕ, ψ ∈ H+ . Proof. The assertions follow from (2.6) and (2.5) since k · k− ≤ k · k+ , and D is unitary. Let ρ(H) denote the resolvent set of H. Let us introduce the family of operators Rz = (H − zI)−1 : H− → H+ ,
z ∈ ρ(H) ,
as follows: for each ω ∈ H− , ω = Dϕ = (H + mI)ϕ, ϕ ∈ H+ we put Rz ω := ϕ + (z + m)Rz ϕ ,
Rz = (H − zI)−1 .
(2.10)
Thus for each z ∈ ρ(H) the domain D(Rz ) of Rz coincides with whole negative space: (2.11) D(Rz ) = H− and obviously the restriction of Rz onto the Hilbert space H coincides with the usual resolvent Rz of H, Rz |H = Rz = (H − zI)−1 .
(2.12)
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Proposition 2.3. For any z ∈ ρ(H) the map Rz : H− → H+ is a bounded bijection. Besides Rz has an analytic dependence on z. Proof. Due to (2.10) for any ω ∈ H− , ω = Dϕ, ϕ ∈ H+ , we have kRz ωk+ = k(H + mI)1/2 Rz ωk = k(H + mI)1/2 (ϕ + (z + m)Rz ϕ)k ≤ ck(H + mI)1/2 ϕk = ck(H + mI)−(1/2) ωk = ckωk− , where c = 1+|z+m| kRz k. This proves the boundedness of the map Rz : H− → H+ . Let us prove that the range R(Rz ) of Rz is the whole space H+ . Note that domain of H is only dense subset of H+ , i.e., D(H) ⊂ H+ . Let ϕ be an arbitrary element of H+ and let ϕn ∈ D(H), H+
ϕn → ϕ . Then ωn = Dϕn − zϕn converges to ω = Dϕ − zϕ in H− and therefore Rz ωn converges to ϕ in H+ since due to (2.11) Rz (H − zI)ϕn = Rz (H − zI)ϕn = Rz (H − zI)ϕn = ϕn . This means that ϕ ∈ R(Rz ) because Rz is bounded. Besides by (2.10) the map Rz has a trivial zero-set. The last assertion of the proposition also easy follows from (2.10). From (2.12) we observe in particular that Rz extends by closure the resolvent Rz = (H − zI)−1 as an operator from H− to H+ . Remark 2.4. The assertions of Proposition 2.3 are true also if spaces H+ , H− in the triplet (2.1) are replaced by any other ones, K− ⊃ H ⊃ K+ , with equivalent norms in positive and negative spaces respectively (not necessary introduced by operator H). Obviously also the map Rz : K− → K+ is always bounded if the following continuous inclusions hold: H− ⊃ K− ⊃ H ⊃ K+ ⊃ H+ (see also Proposition 4.5 below). 3. Symmetric and Self-Adjoint Operators in a Rigged Hilbert Space Let the triplet H− ⊃ H ⊃ H+
(3.1)
be an abstract rigged Hilber space. As usual [9] we assume dense and continuous inclusions in (3.1) and that the norms in (3.1) satisfy: k · k− ≤ k · k ≤ k · k+ .
(3.2)
Let D : H+ → H− be the canonical unitary isomorphism: kDψk− = kψk+ . Then we have hω, ϕi = hϕ, ωi = (ω, Dϕ)− = (Iω, ϕ)+ ,
ω ∈ H− ϕ ∈ H+ ,
(3.3)
where I = D−1 and h·, ·i denotes the dual inner product between H+ and H− .
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
7
We remark that hf, ϕi = (f, ϕ) ,
f ∈ H,
ϕ ∈ H+ .
A linear operator T from H+ to H− (write T : H+ → H− ) is said to be symmetric in the rigged Hilbert space (3.1) if T ⊆ T ∗ where T ∗ is defined as follows. A vector ψ from H+ belongs to D(T ∗ ) if the linear functional lψ (ϕ) = hT ϕ, ψi, ϕ ∈ D(T ), is continuous in H+ . Then lψ (ϕ) = hϕ, ψ ∗ i, ψ ∗ ∈ H− and by definition T ∗ ψ = ψ ∗ . If T = T ∗ then T is self-adjoint in (3.1). For any T : H+ → H− we can write T = DIT = DT+ where the operator T+ = IT acts in H+ . Obviously T is symmetric in (3.1) iff T+ is symmetric in H+ since it is easily seen that T ∗ = DT+∗ . Moreover we have the following analog of the well-known self-adjointness criterion (see for instance [8, 28]). Theorem 3.1. Let T : H+ → H− be a symmetric operator in the rigged Hilbert space (3.1). Then the following assertions are equivalent: (a) T = T ∗ , (b) T is closed and Ker(T ∗ ± iD) = {0}, (c) R(T ± iD) = H− . Proof. The proof is reduced to the case of a single Hilbert space. Indeed due to the unitary properties of D and I we look at the operator T+ = IT in H+ and use the well-known arguments as in [8, 28]. We remark that the operator D : H+ → H− is self-adjoint in (3.1) and its restriction D := D| into H defined as follows: D(D) = {ψ ∈ D(D)|Dψ ∈ H} ,
Dψ = Dψ .
(3.4)
is also self-adjoint in the single space H. Of course not every self-adjoint operator T : H+ → H− becomes again selfadjoint after analogical restriction into H. It may occur that T | defined in the same way as in (3.4) becomes trivial in H. This happens for instance if the bounded self-adjoint in the scale (3.1) operator T = Tω is fixed by an element ω ∈ H− \ H: T ψ = Tω ψ = hψ, ωiω ,
ψ ∈ H+ .
Indeed the above kind restriction of Tω into H gives the densely defined zero/ H (see [3, Theorem A.1]) operator since the set Ker Tω is dense in H due to ω ∈ / H if hψ, ωi = 6 0. and because Tω ψ ∈ Let now T : H+ → H− be a closed symmetric operator in the rigged Hilbert space (3.1). A point z ∈ C is said to be regular for T if the range of the operator T − zD fills the whole space H− and if this operator is boundedly invertible. Thus we write z ∈ ρ(T ) if f
(T − zD)−1 ∈ B(H− , H+ ) .
(3.5)
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Consider its restriction T | into H defined by D(T | ) = {ψ ∈ D(T )|T ψ ∈ H} ,
T |ψ = T ψ .
(3.6)
Theorem 3.2. Assume the point zero be regular for a closed symmetric operator T : H+ → H− in the rigged Hilbert space (3.1), 0 ∈ ρ(T ) .
(3.7)
Then the restriction T | of T into H defined by (3.6) is the self-adjoint operator. Proof. By the construction the operator T | is symmetric in H, (T | ϕ, ψ) = hTϕ, ψi = hϕ, Tψi = (ψ, T | ψ) ,
ϕ, ψ ∈ D(T | ) .
Further (3.7) implies that T | is densely defined and invertible. Moreover the domain of (T | )−1 is whole space H. Therefore T | is self-adjoint. It is clear that T | might be self-adjoint in H under a slightly weaker condition than (3.7). Proposition 3.3. Let T : H+ → H− be a symmetric operator in the rigged Hilbert space (3.1). Assume its range contains the whole space H: R(T ) ⊃ H .
(3.8)
Then the restriction T | of T into H defined by (3.6) is the self-adjoint operator. Proof. The operator T | is symmetric in H because T is symmetric in (3.1). Due to (3.8) it is densely defined and its range fills the whole space H. Theorem 3.4. Let T : H+ → H− be a symmetric operator in the rigged Hilbert space (3.1). Then the restriction T | of T into H defined by (3.6) is the self-adjoint operator iff for one and therefore for any point z ∈ C, Im z 6= 0 the range R(T ± zI) contains the whole space H, in particular iff R(T ± iI) ⊃ H ,
(3.9)
where I denotes the identical operator. Proof. If T | is self-adjoint in H then R(T | ± iI) = H and therefore (3.9) holds since T | ± I ⊂ T ± iI. Conversely (3.9) implies that R(T | ± iI) = H. This means, due to T | is symmetric, that in fact it is self-adjoint. 4. Generalized Sum of Operators Let us consider an unbounded self-adjoint operator H0 ≥ 0 with domain D(H0 ) in the Hilbert space H. Let H− ≡ H− (H0 ) ⊃ H ⊃ H+ (H0 ) ≡ H+
(4.1)
9
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
be the rigged Hilbert space associated with H0 (see Sec. 3). Shortly we call the chain (4.1) by the H0 -scale. Consider in (4.1) a symmetric in general nonpositive operator V with domain D(V ) ⊂ H+ and range R(V ) ⊂ H− . Our discussion also includes the case where the quadratic form ν[ψ] = hV ψ, ψi ,
ψ ∈ D(V )
is purely singular in H in the sense that the set Ker(ν) is dense in H. In this case the opertator V also is called singular in H and then usually H ∩ R(V ) = {0} . The problem is: to define in this generality the suitable version for sum of H0 and V in H. Here we will introduce the notion of generalized sum for operators H0 and V which coincides with the usual operator sum H0 + V in a single space and with the ˙ in the sense of quadratic forms if latter are valid. Our definition of the sum H0 +V generalized sum develops the ideas from [11, 23]. Definition 4.1. Let H0 be a positive self-adjoint operator in H and let an operator V acting from H+ to H− be symmetric in the rigged space (4.1) which assumed to be H0 -scale. Let H0 , V denote closures of H0 and V as operators from H+ to H− . Consider the sum H = H0 + V with domain D(H) = D(H0 ) ∩ D(V). The operator H in H defined as the following restriction of H into H, D(H) = {ψ ∈ D(H)|H0 ψ + Vψ ∈ H} ,
Hψ = Hψ
(4.2)
will be called the generalized sum of H0 and V in H. Such defined sum we denote ˙ . by H = H0 +V ˙ always is Since both H and V are symmetric in (4.1), the operator H = H0 +V symmetric (Hermitian) in H: (Hϕ, ψ) = h(H0 + V)ϕ, ψi = hϕ, (H0 + V)ψi = (ϕ, Hψ) ,
ϕ, ψ ∈ D(H) .
However it may even happen that H is not densely defined in H. Problem. Under what condition is H self-adjoint in H? ˙ coincides Clearly if V is a bounded self-adjoint operator in H, then H = H0 +V with the usual operator sum H0 + V which is self-adjoint on D(H0 ). ˙ gives the same operator as in the Moreover the generalized sum H = H0 +V form-sum method if the latter is valid (see Theorem 5.1 below). Besides it is clear that for any self-adjoint operator V in the H0 -scale the sum H0 + V is also self-adjoint in the H0 -scale since H0 is bounded. However in general one knows nothing about the set R(H0 + V) ∩ H. Therefore the restriction of H0 + V into H possibly is not even densely defined. In general case we have the following simple observation.
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Proposition 4.2. If the range of the sum H0 + V coincides with H− , or in particular it contains the whole space H, H ⊂ R(H0 + V) ,
(4.3)
˙ is a self-adjoint operator in H. then the generalized sum H = H0 +V Proof. (4.3) evidently implies H = R(H). Since H always is symmetric this means that it is densely defined and is in fact self-adjoint. We remark that for H in Proposition 4.2 the point 0 is regular and hence for H there is a spectral gap just above zero. However it may happens that H is unbounded from below and thus it is impossible to construct H by the form-sum method. The following theorem gives another simple sufficient condition for a positive answer to the above problem in the case where the usual form-sum method is not applicable. Theorem 4.3. Let an operator V : H+ ⊇ D(V ) → R(V ) ⊆ H− be symmetric and closed in the sense of a pair spaces. Assume the point −1 is regular for V in the sense that the range of V + D coincides with the whole space H− , H− = R(D + V )
(4.4)
and (V + D)−1 : H− → H+ is bounded. We write shortly −1 ∈ ρ(V ) .
(4.5)
˙ defined by (4.2) is the self-adjoint operator Then the generalized sum H = H0 +V in H. Proof. Let (D + V )| denote the restriction of D + V into H, D((D + V )| ) = {ψ ∈ D(V )|Dψ + V ψ ∈ H} ,
(D + V )| ψ = Dψ + V ψ .
Evidently (D + V )| is symmetric: ((D + V )| ψ, ϕ) = h(D + V )ψ, ϕi = hψ, (D + V )ϕi = (ψ, (D + V )| ϕ) . Further due to the theorem conditions the operator D + V is boundedly invertible. This ensures that the range of the restriction (D + V )| is the whole space H. Thus (D + V )| as well as (D + V )| − I are self-adjoint in H. Noting that D = H0 + I (see Proposition (2.1)) and that (D + V − I)| = (D + V )| − I, we conclude that ˙ , which obviously coincides with (D + V )| − I, is also the operator H = H0 +V self-adjoint in H. This theorem has the following immediate application.
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ON THE PROBLEM OF THE RIGHT HAMILTONIAN
Theorem 4.4. Let q(x), x ∈ Rn be a real-valued distribution which generates a bounded self-adjoint multiplication operator Vq in the scale of the Sobolev spaces, W 2,−1 (Rn ) ⊃ L2 (Rn ) ⊃ W 2,1 (Rn ), i.e., (Vq ψ)(x) = q(x)ψ(x) ,
ψ ∈ W 2,1 ,
qψ ∈ W 2,−1 .
Assume V = Vq satisfies condition (4.4), i.e., the equation ((−∆ + I)ψ)(x) + q(x)ψ(x) = ω(x) , has a solution ψ ∈ W 2,1 for any ω ∈ W 2,−1 , where the Laplacian −∆ : W 2,1 → W 2,1 is understood in the generalized sense. Then the differential expression −∆ + q(x) defines a self-adjoint operator in L2 (Rn ). Moreover the differential expression −∆ + q(x) generates a self-adjoint operator in L2 (Rn ) if instead (4.4) we have only H ⊂ R(−∆ + V + I) ,
(4.6)
i.e., if the above equation has solution only for any ω ∈ L2 . Let us consider the following abstract situation. Let V : H+ → H− be a bounded symmetric operator in the rigged Hilbert space (4.1) such that the restriction H := H|H of the operator H = H0 + V : H+ → H− into H (see (4.2)) is self-adjoint, i.e., ˙ gives the self-adjoint operator in H . the generalized sum H = H0 +V Assume now that H is bounded from below in H, H ≥ mH > − ∞ .
(4.7)
We want to derive the resolvent identity for H. Let us introduce in the H0 -scale the family of operators: Rz := (H0 + V − zI)−1 : H− → H+ ,
z ∈ ρ(H) ,
(4.8)
where ρ(H) = {z ∈ C | R(H0 + V − zI) = H− } ⊆ ρ(H) . For any pair z, ξ ∈ ρ(H) and each ω ∈ H− we obviously have Rz ω = ϕ + (z − ξ)Rz ϕ ,
Rz = (H − z)−1 ,
ω = (H − ξI)ϕ ,
ϕ ∈ H+ , (4.9)
where we used that Rz ϕ = Rz ϕ for each ϕ ∈ H+ . In particular if −1 ∈ ρ(V ) then −1 ∈ ρ(H) and from (4.9) with ξ = −1 we have Rz ω = ϕ + (z + 1)Rz ϕ ,
ω = (H0 + V + I)ϕ ,
We remark that now D = H0 + I. Proposition 4.5. Under conditions (4.7) the operators Rz : H− (H0 ) → H+ (H0 ) z ∈ ρ(H) defined by (4.9) are bounded.
ϕ ∈ H+ .
(4.10)
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S. ALBEVERIO and V. KOSHMANENKO
Proof. We emphasize that now the scale (4.1) is associated with H0 but not with ˙ . Let H = H0 +V H− (H) ⊃ H ⊃ H+ (H) denote the rigged Hilbert space associated with H. Due to D(H) ⊇ H+ (H0 ) and by condition (4.7) we have k · k+ ≤ k · kH+ (H) ,
k · kH− (H) ≤ k · k− .
(4.11)
Using (4.9) and (4.11) we obtain for any ω = (H+mI)ϕ, m ≥ |mH |, ϕ ∈ H+ (H): kRz ωk+ ≤ kRz ωkH+ (H) = k(H + mI)1/2 Rz ωk = k(H + mI)1/2 (ϕ + (z + m)Rz ϕ)k ≤ ck(H + mI)1/2 ϕk = ck(H + mI)−(1/2) ωk = ckωkH− (H) ≤ ckωk− , where c = 1 + |z + m| kRz k.
Surely by (4.11) we can say that above Proposition 4.5 follows from Remark 2.4. ˙ We are ready to derive the resolvent identity for the operator H = H0 +V. Theorem 4.6. Let H = H0 + V where V : H+ → H− be a bounded symmetric ˙ defined by (4.2) is self-adjoint in operator and let the generalized sum H = H0 +V H. Then for the perturbed resolvent Rz = (H − zI)−1 the following identity holds: Rz = Rz0 − Rz V Rz0 = Rz0 − (I + R0z V )−1 R0z V Rz0 ,
z ∈ ρ(H) ∩ ρ(H0 ) ,
(4.12)
where R0z = (H0 − zI)−1 , R0z = (H0 − zI)−1 . Proof. Evidently the space H1 contains the range of the bounded operator Rz −Rz0 . Therefore we can apply to this difference the operator H0 + V − zI which acts from the whole space H+ into H− : H0 + V − z : H+ → H− . Thus we have, (H0 + V − zI)(Rz − Rz0 ) = I − (H0 + V − zI)(H0 − zI)−1 = −V Rz0 . Noting that Rz = (H0 + V − z)−1 : H−1 → H1 is bounded due to the previous proposition we can rewrite the difference of the resolvents Rz − Rz0 as follows: Rz − Rz0 = −(H0 + V − zI)−1 V Rz = −Rz V Rz0
(4.13)
that proves the first equality in (4.12). Further again using that all operators are bounded in H0 -scale we have:
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ON THE PROBLEM OF THE RIGHT HAMILTONIAN
Rz − Rz0 = −Rz ((H0 − zI)R0z )V Rz0 = −(Rz (H0 − zI))R0z V Rz0 = −(R0z (H0 + V − zI))−1 R0z V Rz0 = −(I + R0z V )−1 R0z V Rz0
(4.14)
which finishes the proof. We can rewrite the latter relation in the form Rz = Rz0 − (I + Γz )−1 Γz Rz ,
Γz = R0z V ,
(4.15)
where (I + Γz )−1 is bounded operator in H+ . ˙ We emphasize that the resolvent identity (4.12) obtained above for H = H0 +V has the same form as in the case where V acts in the single space H. However now V acts from H+ to H− . This is the reason why Rz and R0z appear in (4.12) instead of Rz and Rz0 . Remark 4.7. In general, when H is not below bounded we have the generalized resolvent identity of the form: Rz ω = R0z ω − Rz V R0z ω ,
ω = (H0 + V − zI)ϕ ,
ϕ ∈ D(V ) .
(4.16)
However if V is bounded in H then we can rewrite (4.16) as Rz ω = R0z ω − Rz V R0z ω ,
ω ∈ H−
(4.17)
since the range of V belongs to H. We shall formulate now some variant of the Theorem 4.3 in a form which is more convenient for applications. Theorem 4.8. Let H0 > 0 in H and H− ⊃ H ⊃ H+ be the H0 -scale. Let the perturbation of H0 is given by a symmetric operator V acting from H+ to H− . Assume V has a self-adjoint extension V which admits the orthogonal decomposition into sum of three components: V = V− ⊕ V0 ⊕ V+ ,
D(V) ≡ D = D(V− ) ∩ D(V0 ) ∩ D(V+ ) ,
where all operators V− , V0 , V+ are self-adjoint in the H0 -scale. Assume the following conditions are fulfilled. The operator V+ is positive: hV+ ψ, ψi ≥ 0 . The operator V0 satisfies the KLMN-theorem’s condition: |hV0 ψ, ψi| ≤ b+ hH0 ψ, ψi + a+ kψk2 ,
0 ≤ b+ < 1, a+ > 0 ,
ψ ∈ D.
(4.18)
The operator V− (if it is nontrivial ) is purely negative and satisfies the inequality: hV− ψ, ψi ≤ −b− hH0 ψ, ψi − a− kψk2 ,
b− > 1, a− > 0 ,
˙ defined by (4.2) is self-adjoint in H. Then H = H0 +V
ψ ∈ D.
(4.19)
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S. ALBEVERIO and V. KOSHMANENKO
Proof. We note only that due to the theorem conditions the point −1 does not belong to the spectrum of the self-adjoint operator IV in H+ since b+ < 1 and b− > 1. Thus −1 is a regular point for V and therefore we have H− = R(H0 + I + V) = R(D + V). This ensures, due to Theorem 4.3, that the restriction (H0 +V)|H is the self-adjoint operator in the space H. Corollary 4.9. Let the self-adjoint operator V from H+ to H− be purely negative, hV ψ, ψi ≤ −bkψk , b > 0 , ψ ∈ D(V ) . Then for any coupling constant c such that cb > 1 the nonpositive operator H = ˙ is self-adjoint in H. H0 +cV 5. Connection with the Friedrichs Extension Let H0 = H0∗ > 0 and let H− ≡ H− (H0 ) ⊃ H ⊃ H+ ≡ H+ (H0 )
(5.1)
be the rigged Hilbert space associated with H0 . By γ0 we will denote the closed quadratic form associated with H0 . So we have H+ = Q(γ0 ) in the norm k · k2+ = γ0 [·] + k · k2 . Let ν be a densely defined (possibly nonpositive) symmetric quadratic form in H with domain Q(ν). Assume there exists a dense in H+ subset D ⊂ Q(γ0 ) ∩ Q(ν) such that: |ν[ψ]| ≤ bγ0 [ψ] + akψk2 ,
ψ ∈ D,
0 ≤ b < 1,
a ≥ 0.
(5.2)
Then by the KLMN-theorem’s arguments (see [16, 29]) the form γ[ψ] = γ0 [ψ] + ν[ψ] ,
ψ∈D
(5.3)
is closable and lower semibounded in H. It is easy to see that γ cl ≥ −a where cl stands for closure in H. Thus there exists the associated with γ cl operator Has ≥ −a which is self-adjoint in H. By the Friedrichs construction the operator Has arises in the following way. Let H+,ν be the completion of D in the norm k · k2+,ν = γ[·] + (a + 1)k · k2 and let H−,ν ⊃ H ⊃ H+,ν be the corrresponding rigged Hilbert space. Let Dν : H+,ν → H−,ν denote the canonical unitary isomorphism. Then the self-adjoint operator Has in H may be defined as follows: D(Has ) = {ψ ∈ H+,ν | Dν ψ ∈ H} ,
Has ψ = Dν ψ − (a + 1)ψ .
(5.4)
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
15
By this construction (Has ψ, ϕ) = γ cl (ψ, ϕ) ,
ψ, ϕ ∈ D(Has ) .
Now we will show that there exists another way to construct Has which uses the notion of generalized sum (see the previous section). The condition (5.2) implies that |ν[ψ]| ≤ ckψk2+ ,
c = max{a, b} .
So the quadratic form ν is bounded in H+ . Its closure in H+ will again be denoted by ν. Thus now Q(ν) = H+ . Let V be the self-adjoint operator in H+ associated with ν, D(V) = H+ . Put V = DV where we recall that D : H+ → H− is the closure of H0 + I. Obviously V : H+ → H− is self-adjoint as an operator in a pair of spaces and we have ν(ψ, ϕ) = (Vψ, ϕ)+ = hV ψ, ϕi = hψ, V ϕi ,
ψ, ϕ ∈ D(V ) = H+ .
Let H0 denote the closure of H0 : H+ → H− . Consider the sum H0 + V and ˙ of H0 and V (see Definition 4.1 and introduce the generalized sum H = H0 +V Theorem 4.3). Theorem 5.1. Under the condition (5.2) the operator Has associated with the form-sum γ = γ0 + ν coincides with the generalized sum of H0 and V, i.e., ˙ =H. Has = H0 +V
(5.5)
Proof. By above discussion we have (Has ψ, ϕ) = γ cl (ψ, ϕ) = γ(ψ, ϕ) + ν(ψ, ϕ) = hH0 ψ, ϕi + hV ψ, ϕi . Thus Has is an extension of H. Let us show that H is self-adjoint and therefore it coincides with Has . If V is positive then h(H0 + I + V )ψ, ψi ≥ kψk2+ , i.e., H0 + I + V is positive definite, H0 + I + V ≥ 1. So in this case we have R(H0 + I + V ) = H− . Therefore R(H + I) = H and thus due to Proposition 4.2, H is self-adjoint. In the general case we argue as follows. We note that H0 + V + c is positive definite for sufficiently large c. Indeed by (5.2) we have h(H0 + V + c)ψ, ψi = γ0 [ψ] + ν[ψ] + ckψk2 ≥ (1 − b)γ0 [ψ] + (c − a)kψk2 ≥ b0 kψk2+ ,
b0 = 1 − b ,
c = b0 + a .
˙ +cI) is self-adjoint in H since V +cI ≥ 0. Therefore H0 +(V ˙ +cI)−cI = Thus H0 +(V ˙ is also self-adjoint. H0 +V We remark that one can derive the above result using the fact that the norms in the positive Hilbert spaces, H+ , H+,ν , and resp. in negative ones, H− , H−,ν , are equivalent. Consider now the situation where the condition (5.2) does not hold.
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S. ALBEVERIO and V. KOSHMANENKO
Let ν be a symmetric quadratic form with domain Q(ν) in H. Assume there exists a linear subset D ⊂ D(H0 ) ∩ Q(ν) which is dense in H and such that: (a) the restriction ν | := ν|D
(5.6)
is bounded from below and closable as a form in H+,F , where H+,F denotes the positive Hilbert space constructed by the Friedrichs extension H0,F of the symmetric operator H˙ 0 = H0 |D, (b) the number −1 belongs to the resolvent set of the operator V | which is associated with the closure of ν | in H+,F : −1 ∈ ρ(V | ) .
(5.7)
Due to (5.6) we have ν cl,+ [ψ] = hV | ψ, ψi ≥ mν kψk2+,F ,
mν > −∞ ,
(5.8)
where cl, + stands for closure in H+,F . (5.7) means that the operator V | + DF : H+,F → H−,F is boundedly invertible and its range coincides with the whole space H−,F : H−,F = R(V | + DF ) = R(DF (I + V)) , V = IF V | , (5.9) where DF : H+,F → H−,F denotes the canonical isomorphism in the rigged Hilbert space associated with H0,F and IF = D−1 F . ˙ Theorem 5.2. Under the conditions (5.6) and (5.7) the generalized sum H0,F + | V is a self-adjoint operator in H. Proof. The proof uses the same arguments as in the proof of Theorem 4.2.
Remark 5.3. In spite of the fact that ν | is bounded from below in H+,F it is possible that the sum (γ0 + ν)|D is unbounded from below in H. Therefore in such a case it is impossible to define the sum of H0,F and V by the method of quadratic forms. Remark 5.4. If the set D is dense in H+ then H0,F = H0 and therefore in this ˙ is also the self-adjoint operator. However H case the generalized sum H = H0 +V might be unbounded from below. 6. H−1 -Bounded Perturbations Here we show that for any bounded self-adjoint operator V : H+ → H− the gen˙ eralized sum Hλ = H0 +λV is a self-adjoint operator in H for sufficiently small λ. We emphasize that in general the operator V does not exist as a map in H.
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
17
Let H0 = H0∗ > 0 and H− ⊃ H ⊃ H+ ⊇ H+,+ ,
(6.1)
be the extended H0 -scale of Hilbert spaces, where H+,+ = D(H0 ). Let V : H+ → H− be a bounded self-adjoint operator: kV ψk− ≤ M kψk+ ,
M = kV k =
sup kV ψk− .
(6.2)
kψk+ =1
Consider the sum Hλ = H0 + λV : H+ → H− ,
λ ∈ R1 ,
where H0 = H0cl : H+ → H− is the closure of H0 , and define ˙ Hλ = H0 +λV by D(Hλ ) = {ψ ∈ H+ |(H0 + λV )ψ ∈ H} ,
Hλ ψ = H0 ψ + λV ψ .
(6.3)
Theorem 6.1. The operator Hλ defined by (6.3) is self-adjoint in H if |λ| <
1 . M
(6.4)
Proof. Let us consider the operator Dλ = D + λV where D = H0 + I : H+ → H− is unitary. We have Dλ = D(I + λV) , V = D−1 V . Note that V is a bounded operator in H+ . Consider now the equation of the Fredholm type, ϕ + λVϕ = ψ (6.5) for any fixed ψ ∈ D(H0 ) ⊂ H+ . The usual iteration procedure beginning with ϕ(0) = ψ gives p X (p) ϕ = λk V k ψ , p = 0, 1, . . . . k=1
Evidently due to (6,2) kϕ(p) k+ ≤ |λ|k M k kψk+ . Therefore the Neumann series
∞ X
λk V k ψ
k=0
is majorized by the convergent numerical series ! ∞ X 1 k k kψk+ |λ| |M | kψk+ = 1 − |λ|M k=0
(6.6)
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S. ALBEVERIO and V. KOSHMANENKO
if |λ| < 1/M . Thus the Neumann series (6.6) is convergent in the norm of H+ under condition (6.4). So we can expect that a solution of the Eq. (6.5) has the form ϕ=
∞ X
λk V k ψ .
(6.7)
k=0
Let us check that in fact (6.7) satisfies (6.5). Indeed we have in H+ , (p) (p−1) + ψ = λVϕ + ψ . ϕ = s − lim ϕ = λV lim ϕ p→∞
p→∞
We assert that the solution (6.7) is unique. Indeed let ϕ0 be a solution of the homogeneous equation ϕ0 = λVϕ0 . Then kϕ0 k+ ≤ |λ|M kϕ0 k+ and (6.4) implies that ϕ0 ≡ 0. Thus the range of the operator I + λV contains the domain of H0 : D(H0 ) ⊂ R(I + λV) . It is known (see [3, Theorem A.1]) that DH+,+ = H since D(H0 ) = H+,+ . Therefore the range of the restriction of Dλ into H (de| | noted by Dλ ) contains the whole space H. This shows that operator Dλ which is symmetric in H, in fact it is self-adjoint. Using |
Dλ = Hλ + I we conclude that Hλ is also self-adjoint.
7. Convergence Theorems In this section we will prove the convergence theorems for singularly perturbed operators using the method of rigged Hilbert spaces. Theorem 7.1. Let H0 = H0∗ > 0 in H and H− ≡ H− (H0 ) ⊃ H ⊃ H+ ≡ H+ (H0 )
(7.1)
be the rigged Hilbert space associated with H0 . Let V : H+ ⊃ D(V ) → R(V ) ⊂ H− be a self-adjoint operator which satisfies the condition: (7.2) H ⊆ R(H0 + V ± zI) , Im z 6= 0 . ˙ exists as the self-adjoint operator in H. Then the generalized sum H = H0 +V Further let Vn , n = 1, 2, . . . be a sequence of bounded self-adjoint operators in H, so each Hn = H0 + Vn is self-adjoint and bounded from H+ to H− . Consider
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
19
Rnz = (H0 + Vn − zI)−1 , as operators from H− into H+ which is bounded due to Proposition 4.5. Assume for some z0 , Im z0 6= 0 kRn±z0 k+ ≤ c ,
(7.3)
uniformly of n, where k · k+ means the operator norm from H− to H+ . Then, (a) the sequence of self-adjoint in H operators Hn = H0 + Vn converges to H = ˙ in the strong resolvent sense, if H0 +V Vn ϕ → V ϕ ,
ϕ ∈ D(V ) ,
strongly in H− . ˙ in the uniform resolvent sense, if V is (b) Hn = H0 + Vn converges to H = H0 +V bounded, H is bounded from below : H ≥ mH > − ∞ ,
(7.4)
and Vn converges to V in the operator norm from H+ to H− . ˙ Proof. Under condition (7.2) the generalized sum H = H0 +V is a self-adjoint operator in H due to Theorem 3.4. Indeed the operator H = H0 + V : H+ → H− is self-adjoint and satisfies (3.9). (a) We will prove that k(Rzn − Rz )hk → 0 ,
h ∈ H,
(7.5)
ϕ ∈ D(V ) ,
(7.6)
if k(Vn − V )ϕk− → 0 ,
where Rz and Rzn denote resolvents of H and Hn . Let z = ±z0 ∈ C. For any h ∈ H, h = (H + z)ϕ, ϕ ∈ D(V ) we have (Rzn − Rz )h = (Rnz − Rz )h = Rnz (V − Vn )ϕ . Therefore k(Rzn − Rz )hk ≤ k(Rzn − Rz )hk+ ≤ kRnz k+ k(V − Vn )ϕk− → 0 due to condition (7.6) and because for z = ±z0 we have the uniform estimate (7.3). The assertion for arbitrary z is valued due to the analytic dependence of resolvent on z. (b) Take again z = ±z0 . We will prove kRz (Hn ) − Rz (H)k → 0 ,
(7.7)
k(Vn − V )k− → 0 ,
(7.8)
under assumption
20
S. ALBEVERIO and V. KOSHMANENKO
where we recall that k · k− means the operator norm from H+ to H− . We will use that now H is bounded from below: H > mH > − ∞ and that all operators H0 , V, Vn are bounded as maps from H+ to H− and besides that the operators Rnz = (H0 + Vn − zI)−1 , Rz = (H0 + V − zI)−1 are well defined and bounded from H− to H+ . In such case we have Rnz − Rz = Rnz (V − Vn )Rz . Therefore kRnz − Rz k+ ≤ kRnz k+ · k(V − Vn )k− · kRz k+ ≤ ck(V − Vn )k− · kRz k+ → 0 , due to (7.8), and (7.3) since kRz k+ ≤ (1 + |z + mH |/|Im z|) (see Remark 2.4 and Proposition 4.5). Theorem 7.2. Let H0 = H0∗ > 0 and H− ⊃ H ⊃ H+ be the H0 -scale of spaces. Let V : H+ → H− be a bounded self-adjoint operator. Assume there exists a linear dense subset in H+ D ⊂ H+ (7.9) such that the point −1 is regular for the operator V | = V |D, so that there exists the ˙ | which is a self-adjoint operator in H. generalized sum H = H0 +V Assume the operator H is bounded from below : H > mH ≥ − ∞ .
(7.10)
Let Dn ⊂ D, n = 1, 2, . . . be a sequence of linear subsets dense in H and convergent to D in the following sense: ∀ϕ ∈ D,
∃ ϕn ∈ Dn ,
ϕn →H+ ϕ .
(7.11)
Assume Vn , n = 1, 2, . . . is a sequence of self-adjoint bounded in H operators such that Vn → V in the operator norm from H+ to H− and such that Vn |Dn = V |Dn .
(7.12)
Then Hn = H0 + Vn converges to H in the uniform resolvent sense. ˜ 0 of H˙ 0 = H0 |D Moreover for any bounded from below self-adjoint extension H ˜ ˜ the sequence Hn = H0 + Vn also converges to H in the strong resolvent sense: ˜ n →s.r.s. H . H
(7.13)
Proof. Let Hn = H0 + Vn , H = H0 + V : H+ → H− . Since all operators H0 , Vn , V are bounded and Vn → V in the operator norm we conclude that Hn → H in the operator norm also. This implies that (Hn − zI)−1 → (H − zI)−1 in the operator
21
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
norm from H− to H+ . In particularthis is true for the resolvents of the operators ˙ | , i.e., Hn = H0 + Vn and H = H0 +V (Hn − zI)−1 → (H − zI)−1 ,
n→∞
in the operator norm in H. Consider now the sequence of symmetric operators H˙ n = H|Dn = (H0 + Vn )|Dn , where we used the condition (7.12). We will prove that the Friedrichs extensions Hn,F of H˙ n converge to H in the strong resolvent sense: n h → Rz h = (H − zI)−1 h , (Hn,F − zI)−1 h = Rz,F
h ∈ H,
z ∈ ρ(H) .
(7.14)
Note that thanks to (7.10) and (7.12) all Hn,F are uniformly bounded from below. Evidently also that the relation γH ⊃ γHn,F for quadratic forms generated by H and Hn,F holds. This implies that H−,n ≡ H− (Hn,F ) ⊃ H− (H) ⊃ H ⊃ H+ (H) ⊃ H+ (Hn,F ) ≡ H+,n
(7.15)
since each H+,n is a proper subspace of H+ (H) and by the definition of the negative norm in the rigged Hilbert space [9, 10], we have ϕ ≤ k · kH (H) . (7.16) k · k−,n := sup ·, − kϕkH+ (H) ϕ∈Dn Above we denote by H− (H) ⊃ H ⊃ H+ (H) the H-scale of rigged Hilbert spaces. Let D = H+mI : H+ (H) → H− (H) denote the canonical unitary isomorphism, m > kmH k, H = H cl and I = D−1 We will prove that In ω → Iω ,
∀ ω ∈ H− (H) ,
(7.17)
where In ≡ D−1 n denotes the canonical unitary isomorphisms in the rigged Hilbert cl + mI) : H+,n → H−,n . spaces associated with Hn,F , i.e., Dn := (Hn,F Let ω ∈ H− (H) be fixed. Then there exists ϕ ∈ H+ (H) such that ω = Dϕ. By the conditions (7.11) there exists a sequence ϕn ∈ Dn such that ϕn → ϕ in H+ (H). Hence ωn → ω in H− (H) where ωn = Dϕn ≡ Hn ϕn (we recall that D, H + mI, Hn + mI and Hn,F coincide on each subset Dn . So we have kIω − In ωkH+ (H) ≤ kIω − In ωn kH+ (H) + kIn ωn − In ωkH+ (H) = kϕ − ϕn kH+ (H) + kϕn − In ωkH+(H) , where we used ϕn = In Dn ϕn = In Dϕn = In ωn due to ϕn ∈ Dn and Dn ϕn = (H0 + Vn + mI)ϕn . Further by the same reasons we can write kϕn − In ωkH+(H) = kIn (ωn − ω)kH+ (H) = kIn (ωn − ω)kH+,n = kωn − ωkH−,n ≤ kωn − ωkH− (H) = kϕn − ϕkH+ (H) .
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S. ALBEVERIO and V. KOSHMANENKO
Thus kIω − In ωk+ ≤ kIω − In ωkH+ (H) ≤ 2kϕn − ϕkH+ (H) → 0 which proves (7.17). In particular this is true for each ω = h ∈ H. So we have the convergence of the resolvents Rzn h → Rz h ,
h∈H
for fixed real z = −m and therefore for any z ∈ ρ(H). Thus (7.14) is proved. Further for any n the operator Hn is a self-adjoint extension of the symmetric operator H˙ n . Therefore its resolvent has the representation by the Kreins formula [5, 6, 8, 22]: n −1 + Bn,z , (7.18) Rzn = Rz,F where the second term corresponds to the operator extension parameter Bn,z which acts from the defects subspaces Nz¯ to Nz . −1 → 0. In other words this means that Thanks to (7.14) we conclude that Bn,z Bn,z → ∞ .
(7.19)
Thus we proved that (Hn − zI)−1 → (H − zI)−1 (for more details see [5]). ˜ 0 + Vn . Each H ˜ n one can also ˜n = H Consider now the sequence of operators H consider as an self-adjoint extension of (H0 + Vn )|Dn = H|Dn . Their resolvents have the representation by the Krein formula: n −1 ˜ n − zI)−1 = Rz,F ˜n,z ˜ zn = (H +B , R
(7.20)
0 ˜n,z = Bn,z + Bn,z ˜n,z has two parts, B . where the operator parameter of extension B 0 One of them, Bn,z , corresponds to the Vn and the other, Bn,z , appears due to taking ˜ 0 . Evidently B 0 → B 0 , where B 0 denotes the operator extension parameter of H n,z z z ˜ 0 , i.e., H ˜ 0 − zI)−1 = (H0 − zI)−1 + (Bz0 )−1 . (H
Due to (7.19), ˜n,z = Bn,z + B 0 → ∞ . B n,z Therefore ˜ n → Rz , R z n → Rz . since Rz,F
Acknowledgment The second-named author gratefully acknowledges the financial support received from the SFB-237, INTAS, and DFG projects. References [1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer, Berlin, 1988.
ON THE PROBLEM OF THE RIGHT HAMILTONIAN
23
[2] S. Albeverio, J. F. Brasche and V. Koshmanenko, “Lippmann–Schwinger equation for singularly perturbed operators”, Methods of Funct. Anal. and Topol. 3(1) (1997) 1–27. [3] S. Albeverio, W. Karwowski and V. Koshmanenko, “Square power of singularly perturbed operators”, Math. Nachr. 173 (1995) 5–24. [4] S. Albeverio and V. Koshmanenko, “Singular rank one perturbations of self-adjoint operators and Krein theory of self-adjoint extensions”, Potential Anal. 11 (1999) 279–287. [5] S. Albeverio and V. Koshmanenko, “Form-sum approximation of singular perturbation of self-adjoint operators”, J. Funct. Anal. 169 (1999) 32–51. [6] S. Albeverio, V. Koshmanenko and K. Makarov, “Eigenfunction expansions under singular perturbations”, Methods of Funct. Anal. and Topol. 5(1) (1999) 13–27. [7] S. Albeverio and K. Makarov, “Attractors in a model related to the three body quantum problem”, C.R. Acad. Sci. Paris 323, Serie I (1996) 693–698. [8] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Moscow, 1966. [9] Yu. Berezanskii, Expansion in Eigenfunction of Self-Adjoint Operators, AMS Providence, Rhode Island, 1968. [10] Yu. Berezanskii, Self-adjoint Operators in Spaces of Function of Infinitely Many of Variables, AMS, Providence, Rhode Island, 1986. [11] Yu. Berezanskii, The bilinear forms and Hilbert equipments, in Spectral Analysis of Differential Operators, Inst. of Math., Kiev, 1980, 83–106. [12] Yu. Berezanskii and Yu. Kondratiev, Spectral Methods in Infinite-dimensional Analysis, Naukova Dumka, Kiev, 1988. [13] M. Combescure-Moulin and J. Ginibre, “Essential self-adjointness of many particle Schr¨ odinger Hamiltonians with singular two-body potentials”, Ann. Inst. Henri Poincar´e, Sect. A, 23(3) (1975) 211–234. [14] J. Glimm and A. Jaffe, “Singular perturbations of selfadjoint operators”, Commun. Pure Appl. Math. 22 (1969) 401–414. [15] C. N. Friedman, “Perturbations of the Schroedinger equation by potentials with small support”, J. Funct. Anal. 10 (1972) 346–360. [16] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1980. [17] A. Kiselev and B. Simon, “Rank one perturbations with infinitesimal coupling”, J. Funct. Anal. 130 (1995) 345–356. [18] V. D. Koshmanenko, Singular Bilinear Forms in Perturbation Theory of Self-Adjoint Operators, Naukova Dumka, Kiev, 1993. English translation: Singular Quadratic Forms in Perturbation Theory, Kluwer Academic Publ. Dordrecht, Boston, London, 1999. [19] V. D. Koshmanenko, “Towards the rank-one singular perturbations of self-adjoint operators”, Ukrainian Math. J. 43(11) (1991) 1559–1566. [20] V. D. Koshmanenko, “Perturbations of self-adjoint operators by singular bilinear forms”, Ukrainian Math. J. 41 (1989) 1–14. [21] V. D. Koshmanenko, “Singular perturbations at infinite coupling”, Funct. Anal. Appl. 33(2) (1999). [22] M. G. Krein, “Theory of self-adjoint extensions semibounded Hermitian operators and its applications”, I, Math. Trans. 20(3) (1947) 431–495. [23] M. G. Krein and V. A. Yavrian, “Spectral shift functions arising in perturbations of a positive operator”, J. Operator Theory 6 (1981) 155–191. [24] G. Nenciu, “Removing cut-offs form singular perturbations: An abstract result”, Lett. Math. Phys. 7 (1983) 301–306. [25] H. Neidhardt and V. Zagrebnov, “Regularization and convergence for singular perturbations”, Commun. Math. Phys. 149 (1992) 573–586.
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[26] H. Neidhardt and V. Zagrebnov, “Towards the right Hamiltonian for singular perturbations via regularization and extension theory”, Rev. Math. Phys. 8(5) (1996) 715–740. [27] H. Neidhardt and V. Zagrebnov, “On the right Hamiltonian for singular perturbations: general theory”, Rev. Math. Phys. 9(5) (1997) 609–633. [28] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, London, 1975. [29] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic Press, New York, San Francisco, London, 1975. [30] M. Reed and B. Simon, Methods of Modern Mathematical Physics. III. Scattering Theory, Academic Press, New York, San Francisco, London, 1979. [31] M. Schechter, “Cut-off potentials and form extensions”, Lett. Math. Phys. 1 (1976). [32] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton Univ. Press, Princeton, NJ, 1971. [33] B. Simon, “A canonical decomposition for quadratic forms with applications to monotone convergence theorems”, J. Funct. Anal. 28 (1978) 377–385.
ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE ¨ SCHRODINGER EQUATION FOR A TWO-LEVEL SYSTEM WITH A HAMILTONIAN DEPENDING QUASI PERIODICALLY ON TIME ˜ C. A. BARATA∗ JOAO Instituto de F´ısica Universidade de S˜ ao Paulo Caixa Postal 66 318 05315 970, S˜ ao Paulo, SP, Brazil E-mail:
[email protected] Received 10 April 1998 Revised 14 December 1998 We consider the Schr¨ odinger equation for a class of two-level atoms in a quasi-periodic external field for large coupling, i.e. for which the energy difference 2 between the unperturbed levels is sufficiently small. We show that this equation has a solution in terms of a formal power series in , with coefficients which are quasi-periodical functions of the time, in analogy to the Lindstedt–Poincar´ e series in classical mechanics.
1. Introduction and Summary The problem of existence of quasi-periodic solutions of the Schr¨ odinger equation with quasi-periodic coefficients is a classical problem in analysis, with recent applications to the theory of “quantum chaos” [1] (see also [2] for a review with recent results). In the scalar almost-periodic case, the solution is provided by a theorem of H. Bohr [3] (generalizing previous results by P. Bohl for the quasi-periodic case [4]). We now pass from the scalar case to the simplest non-commutative setting, i.e. the Schr¨ odinger equation for two-level systems (e.g. atoms) in a quasi-periodic (external) field, with Hamiltonian H(t) = σ3 − f (t)σ1 ,
(1.1)
where σ1 , σ2 and σ3 denote the Pauli matrices, which satisfy the SU(2)-commutation relations [σ1 , σ2 ] = 2iσ3 plus cyclic permutations. In (1.1), f (t) is a quasi-periodic function of time t X ·ω f t fm eim (1.2) f (t) = ˜˜ ˜ m∈ZB ˜ and is a “small” parameter (half of the energy difference between the unperturbed levels), which characterizes the large coupling domain [5]. Above ωf is a vector of
˜
∗ Partially
supported by CNPq. 25
Reviews in Mathematical Physics, Vol. 12, No. 1 (2000) 25–64 c World Scientific Publishing Company
26
J. C. A. BARATA
frequencies ω f = (ωf1 , . . . , ωfB ) for some B ≥ 1, where we will assume ωfj > 0 for all 1 ≤ j ≤ B. Moreover, m · ω f denotes m1 ωf1 + · · · + mB ωfB . We have chosen the regime where is small because the result which is equivalent [1] to the existence of quasi-periodic solutions in this case — the pure point spectrum of the so-called generalized quasi-energy operator [6] — has not yet been proven in full generality [5]. The analogous problem in classical mechanics has been very nicely sketched in [7] — albeit just as a pedagogical introduction to perturbative renormalization in quantum field theory. The main theorem of [7] — existence of formal quasi-periodic solutions of a class of quasi-periodic Hamiltonian systems as a series in — the Lindstedt–Poincar´e series — has a counterpart — for (1.1) — in Theorem 2.2 of the present paper. In classical mechanics there is, however, a method, due to L. H. Eliasson [8], to isolate the divergent contributions to the Lindstedt–Poincar´e series (see [7] for a simple but illuminating discussion on these terms) and introduce further renormalization counter-terms ensuring convergence. See [10] for a realization of this idea in a special model and [11] for a review of the models and method. Here we should perhaps remark that the resumation methods mentioned above may not generally lead to converging power expansions in . Situations of this kind have been analysed by L. H. Eliasson in [9] and by G. Benfatto, G. Gentile and V. Mastropietro in [12]. Typically such situations involve series solutions with denominators like (n · ω + 2 )−1 , i.e. with dense lying singularities in the imaginary complex axis that prevent analyticity in any open set containing = 0. The question now poses itself whether methods similar to those discussed above might yield the existence of a quasi-periodic solution to the Schr¨ odinger equation
˜
˜ ˜
i
∂ψ(t) = H(t)ψ(t) ∂t
(1.3)
with H(t) given by (1.1). The natural guide is the (convergent) Dyson perturbation series, which we describe now. By a rotation of π/2 around the 2-axis, we may replace H(t) by H1 (t) = e−iπσ2 /4 H(t)eiπσ2 /4 = σ1 + f (t)σ3
(1.4)
and the Schr¨ odinger equation becomes ∂Φ(t) = H1 (t)Φ(t) , ∂t
(1.5)
Φ(t) := e−iπσ2 /4 ψ(t) .
(1.6)
e Φ(t) ≡ exp(iα(t)σ3 )Φ(t) ,
(1.7)
i with Defining now where
Z α(t) ≡
t
dτ f (τ ) 0
(1.8)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
27
the Schr¨ odinger Eq. (1.5) becomes i
e ∂ Φ(t) e Φ(t) e , = H(t) ∂t
where e H(t) ≡
0
e2iα(t)
e−2iα(t)
0
(1.9) ! .
(1.10)
The Dyson series solution to (1.10) is e e Φ(t) = Φ(0) +
Z Z τn−1 ∞ X (−i)n t e n )Φ(0) e e 1 ), . . . , H(τ dτ1 , . . . , dτn H(τ . n! 0 0 n=1
(1.11)
Let, now, f be such that q(t) := e
iα(t)
Z = exp i
t
f (τ ) dτ
(1.12)
0
is quasi-periodic (see Appendix B), and write X gn ein·ωt , q(t)2 =
(1.13)
n∈ZA
where ω ≡ (ω1 , . . . , ωA ) is a frequency vector satisfying some Diophantine conditions to be stated later on, and assume that (for the purpose of this introduction) g0 ∈ R .
(1.14)
Relation (1.14) is seen to hold explicitly, for instance, if f (t) = f (ω1 t, . . . , ωB t) = PB i=1 cos(ωi t). Then, explicit summation of all orders in (1.11) of the n = 0 term in (1.10), which is, by (1.13) and (1.14) of the form g0 σ1 , we find e e Φ(t) = [cos(tg0 ) − i sin(tg0 )σ1 ]Φ(0) ,
(1.15)
which is quasi-periodic. This illustrates that non-uniform convergence of a sequence of non-quasi-periodic functions may yield a quasi-periodic function and, in fact, this does occur in the general solution (1.11). Coupled with the fact that the general nth term in (1.11) grows as tn , it is obvious that (1.11) is grossly unsuitable to show quasi-periodicity of the wave function. We therefore try to subtract (1.15) explicitly from the solution of (1.9) by writing e 1 (t) , e H(t) = M + H where M =
0 g0 g0 0
(1.16)
! = g0 σ1
(1.17)
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J. C. A. BARATA
under assumption (1.14). In correspondence to (1.16) we set e 1 (t) = eiM t Φ(t) e . Φ
(1.18)
e 1 (t) ∂Φ e (I) (t)Φ e 1 (t) , =H 1, ∂t
(1.19)
e 1 (t)e−iM t . e (I) (t) ≡ eiM t H H 1,
(1.20)
e 1 is The equation for Φ i where
The Dyson series (1.11) for (1.20) has the first order term Z t e (I) (τ )Φ e 1 (0) dτ H −i 1, 0 Z t dτ = −i 0
×
i sin(g0 τ ) cos(g0 τ )(λ(τ ) − λ(τ )) sin2 (g0 τ )λ(τ ) + cos2 (g0 τ )λ(τ )
!
sin2 (g0 τ )λ(τ ) + cos2 (g0 τ )λ(τ ) i sin(g0 τ ) cos(g0 τ )(λ(τ ) − λ(τ ))
e 1 (0) , ×Φ
(1.21)
where λ(t) := q(t)2 − g0 . Note, however, that, due to the -dependence of M in (1.17), this Dyson series is not a power series in : there has been a nonlinear in , non-perturbative “renormalization” embodied in (1.20). Looking at the first matrix element in (1.21) (the discussion of the others is similar), we find Z t sin(g0 τ ) cos(g0 τ )(q(τ )−2 − q(τ )2 )dτ 0
i = 2
Z
t
sin(2g0 τ ) 0
X
(g−n − gn )ein·ωτ dτ
n6=0
# " i X ei(n·ω−2g0 )t ei(n·ω+2g0 )t =− . (g−n − gn ) − 4 n · ω + 2g0 n · ω − 2g0
(1.22)
n6=0
In addition to the small denominators of the form n · ω ± 2g0 as found above, there occurs in (1.21) also small denominators of the form n · ω and it is necessary to require, for instance |n · ω ± 2g0 | ≥ γn = γ(1 + |n|2 )−α , 0
|n · ω| ≥ γn0 = γ 0 (1 + |n|2 )−α
(1.23) (1.24)
in order to ensure convergence in (1.22) with |gn | ≤ c exp(−d|n|), with d > 0, the exponential decay being a consequence of analyticity of f in a strip, (1.8) and (1.13). Above, |n| is the l1 norm of the vector n.
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
29
Assume now, for simplicity, that A = 2 in (1.13), and set α = ω1 /ω2 ∈ (0, 1). Suppose, now, that (1.25) g0 6= 0 and consider (without loss, because is supposed “small”) such that β ≡ 2g0 ∈ (0, 1). It is easy to show using Cassels theorem [13] that, if α, α0 > 1 in (1.23)–(1.24), the set of (α, β) in the open unit square such that (1.23)–(1.24) are satisfied for all γ, γ 0 > 0 is of full Lebesgue measure. The intersection of this set with the -axis is thus, however, not an open set and, therefore, no convergence may be expected for in any (small) open set around the origin. Hence, any power series in cannot be expected to be analytic, or to converge uniformly in some open set containing the origin. Other strong indications of this fact (not a proof!) will appear later in the context of (pure) power series in (see the heuristic discussion in Sec. 7). Our discussion above relied on (1.25) (otherwise does not appear in (1.23)). It is interesting to remark that (1.25) holds whenever f0 = 0, (see Appendix C), ˜ which was the case left open in [5]. Is it possible to go further and analyze the whole Dyson series for (1.20)? The problem is that higher order terms require further subtractions and we were not able to show that they can be “renormalized” as in (1.20) (with a different M , of course). It seems that the natural way to obtain a formal (asymptotic) series in with quasi-periodic coefficients is to write an Ansatz for (1.1) in exponential form and then to “renormalize” the exponential in an inductive way. This is the purpose of the present paper. As we will see in Sec. 2.1 and in Appendix A the solutions of the Schr¨ odinger Eq. (1.5) can be studied in terms of the solutions of a complex and quasi-periodic version of Hill’s equation. φ00 (t) + (2 + if 0 (t) + f (t)2 )φ(t) = 0 .
Rt
(1.26)
We attempt to solve this last equation using the Ansatz φ(t) = exp(−i 0 (g(τ )+ f (τ ))dτ ), where g satisfies a generalized Riccati equation, and we try to find soluP∞ tions for g in terms of a (formal) power expansion in like g(t) = q(t) n=1 n cn (t). As discussed above, perturbative solutions of quasi-periodically time-dependent systems are usually plagued by small denominators and by the presence of the socalled “secular terms”, i.e. polynomials in t, that spoil the analysis of convergence of the series and the proofs of quasi-periodicity of the perturbative terms. We discovered a particular way to eliminate completely the secular terms from the perturbative expansion of g and we were able, under some special assumptions, to show that the coefficients cn (t) are all quasi-periodic functions. We prove explicitly the convergence of our perturbative solution in the somewhat trivial case where f (t) is a non-zero constant function. Unfortunately, however, no conclusion could be drawn about the convergence of the perturbative expansion for g in the general case. We conjecture, however, that our expansion is uniformly convergent at least in the situation where f (t) − f0 is uniformly small. ˜ Remarks on the notation. In this paper R+ will denote the open interval (0, ∞). Given the Fourier representation (1.2) of the quasi-periodic function f , we will denote by ω the vector of frequencies defined by
30
J. C. A. BARATA
(
ω f ∈ RB ,
if f0 = 0
(1.27) ˜(ωf , f0) ∈ RB+1 , if f˜0 6= 0 , . ˜ ˜ ˜ , the definition above Since we are assuming that ωf ∈ RB says that all components + ˜ we will denote of ω are always non-zero. Moreover, ω :=
(
if f0 = 0 . (1.28) ˜ 6 0 if f0 = ˜ B B We will denote vectors in Z (or R ) by v and vectors in ZA (or RA ) by v. The symbol |n| will denote the l1 (ZA ) norm of a vector n = (n1 , . . . , nA ) ∈ ZA : |n| := |n1 | + · · · + |nA |. We will use the symbol 1l for the identity matrix. Mat(n, C) is the set of all n × n matrices with complex entries. For an almost-periodic (in particular, quasi-periodic) function h we denote by M (h) the mean value of h, defined as A :=
B, B +1,
˜
1 T →∞ 2T
Z
T
h(t)dt .
M (h) := lim
(1.29)
−T
2. The Main Results In this section we will state our two main results. The first one describes the solution of the Schr¨ odinger Eq. (1.5) in terms of a particular solution of a generalized Riccati equation. The reason for considering the Schr¨ odinger equation in the form (1.5) is that, as we will see below and in Appendix A, we will be able to decouple the two components of Φ(t) by increasing the order of the equation. Theorem 2.1. Let f : R → R, f ∈ C 1 (R) and ∈ R and let g : R → C, g ∈ C 1 (R) be a particular solution of the generalized Riccati equation G0 − iG2 − 2if G + i2 = 0 . Then, the function Φ : R → C2 given by Φ(t) =
φ+ (t)
! = U (t)Φ(0) ,
φ− (t)
where U (t) := with
(2.1)
R(t)(1 + ig(0)S(t))
−iR(t)S(t)
−iR(t)S(t)
R(t)(1 − ig(0)S(t))
Z t R(t) := exp −i (f (t0 ) + g(t0 ))dt0
(2.2) ! ,
(2.3)
(2.4)
0
and
Z S(t) := 0
t
R(t0 )−2 dt0
(2.5)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
31
is a solution of the Schr¨ odinger Eq. (1.5) with initial value φ+ (0) ∈ C2 . Φ(0) = φ− (0) In Sec. 2.1 we will present a proof of Theorem 2.1. The proof and the solution will be further discussed in Appendix A. Theorem 2.1 reduces the problem of solving (1.5) to the not necessarily easier question of finding solutions for (2.1). Somewhat surprisingly, however, some interesting results could be proven about the nature of some particular solutions of (2.1) for the case where f is a quasi-periodic function subjected to some additional restrictions. These results are described in Theorem 2.2. Theorem 2.2. Let f be quasi-periodic with X t fn eiω˜f ·n f (t) = ˜ , ˜ B n∈Z ˜ and such that the sum above contains only a finite number of terms. Assume that the vector ω (defined in (1.27)) satisfies Diophantine conditions, i.e. assume the existence of constants ∆ > 0 and σ > 0 such that, for all n ∈ ZA , n 6= 0, |n · ω| ≥ ∆−1 |n|−σ . (1) Assume that f satisfies the condition M (q 2 ) 6= 0. Then, there exists a formal power series ∞ X cn (t)n , g(t) = q(t) n=1
representing a particular solution of the generalized Riccati Eq. (2.1) such that all coefficients cn can be chosen to be quasi-periodic and can be represented as X (n) im·ωt cn (t) = Cm e , m∈ZA (n)
where, for the Fourier coefficients Cm , we have (n) Cm ≤ Kn e−χ0 |m| , where χ0 > 0 is a constant and Kn ≥ 0. (2) Assume that f satisfies the conditions M (q 2 ) = 0 and M (Q1 ) 6= 0, where Z t q −2 (t0 )dt0 . Q1 (t) := q(t)2 0
Then, there exists a formal power series g(t) = q(t)
∞ X n=1
en (t)2n ,
32
J. C. A. BARATA
representing a particular solution of the generalized Riccati Eq. (2.1) such that all coefficients en can be chosen to be quasi-periodic and can be represented as X (n) im·ωt Em e , en (t) = m∈ZA (n)
where, for the Fourier coefficients Em , we have (n) E ≤ Ln e−χ0 |m| , m
where χ0 > 0 is a constant and Ln ≥ 0. Sections 3 and 5 are dedicated to the proof of part 1 of this theorem and Secs. 4 and 6 to the proof of part 2. The proof of part 1 consists in establishing inductively the bound (n) Cm ≤ Kn e−(χ−δn )|m| , (2.6) for all n ∈ N, with positive Kn and χ and with an increasing positive sequence δn converging to some δ∞ < χ. We set χ0 := χ − δ∞ . Quasi-periodicity of the functions cn follows from the bound (2.6). The proof of part 2 is analogous. At this point we should stress that the conditions of this last theorem are not sufficient for proving the convergence of the power series expansions in for g. Uniform convergence of those power series in would imply quasi-periodicity of g. Unfortunately, as discussed at the end of Sec. 5, the behavior for large n of the constants Kn and Ln is apparently too bad to guarantee absolute convergence of the formal power series above. This bad behaviour may be explained if the Fourier coefficients of g are singular on a dense set of values of . This is what happens if they are, for instance, of the form (m · ω + )−1 , since the set {m · ω, m ∈ ZB } is dense on R, under the assumptions on the frequencies. In such cases there could be no open set of values of where g is analytic. In Sec. 7 this and other problems related to our expansions are further discussed. The hypothesis that the Fourier series of f contains only a finite number of terms is not really crucial and may be eliminated with more work. Actually this hypothesis is used only in Appendices C and D. This hypothesis is anyway closer to the physical reality, since f represents an external interaction coupled to the system. 2.1. Proof of Theorem 2.1 Let Φ(t) = V (t)Φ(0) be the solution of the Schr¨ odinger Eq. (1.5) with initial value Φ(0), where V : R → Mat(2, C). Then V is the solution of iV 0 (t) = H1 (t)V (t)
(2.7)
with V (0) = 1l. Hence, V also satisfies iV 0 (0) = H1 (0) and iV 00 (t) = D(t)V (t) ,
(2.8)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
33
where D(t) := H10 (t) − iH1 (t)2 . An explicit computation shows that D(t) is the diagonal matrix ! 0 f 0 (t) − i(2 + f (t)2 ) . (2.9) D(t) = 0 −f 0 (t) − i(2 + f (t)2 ) We will show that the matrix U (t) defined in (2.3) satisfies (a) U (0) = 1l, (b) iU 0 (0) = H1 (0) and (c) iU 00 (t) = D(t)U (t). By the uniqueness of solutions of systems of linear second order differential equations with continuous coefficients we conclude that U (t) = V (t). The proof of (a) follows from the fact that R(0) = 1 and S(0) = 0. To show (b) we note that, since R0 (t) = −i(f (t) + g(t))R(t) and S 0 (t) = R(t)−2 , an explicit computation gives ! ! ! R(t)−1 0 −g(0) f (t) + g(t) 0 0 U (t) + . iU (t) = R(t)−1 0 g(0) 0 −(f (t) + g(t)) (2.10) Since R(0) = 1 and U (0) = 1l, it follows from (2.10) that ! f (0) 0 iU (0) = = H1 (0) , −f (0)
(2.11)
proving (b). From (2.10) we have iU 00 (t) = E(t)U (t) where E(t) :=
f 0 (t) + g 0 (t) − i(f (t) + g(t))2
0
0
−f 0 (t) − g 0 (t) − i(f (t) + g(t))2
! . (2.12)
Now, under the assumption that g satisfies (2.1), we easily check that E(t) = D(t), completing the proof of Theorem 2.1. The solution (2.3) has not been guessed out of nothing. In Appendix A we present a more constructive proof of Theorem 2.1. There we also discuss some questions related to the solution (2.3), as its relation to the general solution of the generalized Riccati equation. Our starting point in Appendix A will be the fact that, translated back to the components φ± (t) of the wave function, equation iU 00 (t) = D(t)U (t) becomes a complex and quasi-periodic version of Hill’s equation: φ00± (t) + (±if 0 (t) + 2 + f (t)2 )φ± (t) = 0. 3. The Solution g and its Dependence on Now we will start our analysis of the formal particular solution of the generalized Riccati Eq. (2.1) and the proof of Theorem 2.2. The solution presented in (2.3) is still incomplete, since no particular solution g of the generalized Riccati Eq. (2.1) has been presented. First we note that for
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J. C. A. BARATA
the case ≡ 0, the solution for R tU (t) is well known and is a diagonal matrix, whose diagonal elements are exp(∓i 0 f (τ )dτ ). Comparing with (2.3), we see that for the case ≡ 0 we should consider the particular solution g(t) ≡ 0. For more about this, see the discussion of Appendix A. A natural proposal would be to express g as a formal power expansion on which vanishes at = 0. For convenience, we write this expansion as ∞ X cn (t)n , (3.1) g(t) = q(t) n=1
where q(t) was defined in (1.12). This would give the desired solution, provided the infinite sum converges for all values of and t contained in some open intervals. Inserting (3.1) into (2.1) leads to ! ∞ n−1 X X 0 2 (qcn ) − i q cp cn−p − 2if qcn n + i2 = 0 . (3.2) n=1
p=1
Under the assumptions above, we conclude (qc1 )0 − 2if qc1 = 0 ,
(3.3)
(qc2 )0 − iq 2 c21 − 2if qc2 + i = 0 ,
(3.4)
(qcn )0 − i
n−1 X
q 2 cp cn−p − 2if qcn = 0 ,
n ≥ 3.
(3.5)
c1 (t) = α1 q(t) , Z t 2 0 2 0 −2 0 (α1 q(t ) − q(t ) )dt + α2 , c2 (t) = q(t) i
(3.6)
p=1
The solutions of (3.3)–(3.4) are
" cn (t) = q(t) i
0 n−1 XZ t p=1
! 0
0
cp (t )cn−p (t )dt
0
(3.7)
# + αn ,
for n ≥ 3 ,
(3.8)
0
where the αn ’s above, n = 1, 2, . . . , are arbitrary integration constants. Since the differential Eq. (2.1) involves 2 , the reader may wonder why we try in (3.1) to find solutions given in terms of power series on and not on λ ≡ 2 . One should note that, in the case f ≡ 0, a particular solution of (2.1) is given simply by g(t) = , which leads to solutions of the form A cos(t) + B sin(t) for φ± , as expected. Therefore, solutions like (3.1) must also be considered, in principle. It is interesting to note, however, that we will meet in Sec. 4 a situation where the solution g involves a power expansion on λ. In the next two subsections we analyze two different choices of the constants αn . 3.1. First choice of the constants αn In order to illustrate some potential problems, we choose first α1 = 1 and αn = 0, ∀ n ≥ 2. We get
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
c1 (t) = q(t) , Z t (q(t0 )2 − q(t0 )−2 )dt0 , c2 (t) = iq(t)
35
(3.9) (3.10)
0
cn (t) = iq(t)
n−1 XZ t p=1
cp (t0 )cn−p (t0 )dt0 ,
for n ≥ 3 .
(3.11)
0
In this case we can easily find a condition guaranteeing the convergence of the formal power expansion (3.1). We have |c1 (t)| = 1 , Z |t| |c2 (t)| ≤ 2dt0 = 2|t| .
(3.12) (3.13)
0
Admitting that |ck (t)| ≤ (2|t|)k−1 for all k with 1 ≤ k < n, we have |cn (t)| ≤
n−1 X Z |t| p=1
(2|t0 |)p−1 (2|t0 |)n−p−1 dt0 = 2n−2 |t|n−1
0
≤ (2|t|)n−1 .
(3.14)
This proved by induction that |cn (t)| ≤ (2|t|)n−1 , for all n ≥ 1. Hence, |g(t)| ≤ 1.
P∞
n−1 n || , n=1 (2|t|)
(3.15)
which converges, provided 2|t||| <
3.2. Second choice of the constants αn . Elimination of polynomial terms on t Let us return to (3.6)–(3.8). We will show that there is a choice of the αn ’s for which all functions cn are quasi-periodic functions of t. If this holds, there is a chance that g be also quasi-periodic, provided the sum (3.1) converges uniformly on R. To fix some ideas, let us start recalling some well-known facts about almostperiodic functions. For an almost-periodic function h, let us denote by σ(h) the spectrum of h (see, e.g. [14]). The function h has the Fourier decomposition P iωi t that converges uniformly on R (see, e.g. [14] for a preh(t) = ωi ∈σ(h) hi e cise statement). According to a celebrated theorem by H. Bohr (see, e.g. [14]), a Rt necessary and sufficient condition for H(t) := 0 h(t0 )dt0 being also almost-periodic P |hi | is H ∈ L∞ (R). For this, it is sufficient that M (h) = 0 and ωi ∈σ(h) |ωi | < ∞. ωi 6=0
Bohr’s theorem generalizes a previous result by P. Bohl on the quasi-periodic case [4]. A quasi-periodic function h is an almost-periodic function with the following property. There exists N ∈ N, N > 0, and a N -tuple ω = (ω1 , . . . , ωN ) ∈ (R+ )N ,
36
J. C. A. BARATA
such that σ(h) ⊂ {n · ω, n ∈ ZN }. For a given quasi-periodic function h we will denote the associated N -tuple ω by β(h). For the analysis below we will admit that f is such that M (q 2 ) 6= 0. A sufficient condition for this will be discussed in Appendix C. The case M (q 2 ) = 0 will be considered in Sec. 4. Let us start with c2 . In order to have c2 quasi-periodic one needs at least to guarantee that the integrand in (3.7) contains no constant term, i.e., M (α21 q 2 − q −2 ) = 0 ,
(3.16)
that means, M (q −2 ) M (q 2 ) = . (3.17) 2 M (q ) M (q 2 ) More generally, in order to have cn quasi-periodic, n ≥ 3, one needs to guarantee at least that the integrand in (3.8) contains no constant term, i.e. α21 =
n−1 X
M (cp cn−p ) = 0 .
(3.18)
p=1
This means 2M (c1 c2 ) = 2α1 M (qc2 ) = 0 for n = 3 and 2α1 M (qcn−1 ) = −
n−2 X
M (cp cn−p ) .
(3.19)
(3.20)
p=2
for n ≥ 4. Defining for n ≥ 2
Z t 2 iq(t) (α21 q(t0 )2 − q(t0 )−2 )dt0 , n = 2, 0 ! dn (t) := q(t)(cn (t) − αn q(t)) = Z t n−1 X 0 0 2 cp (t )cn−p (t ) dt0 , n ≥ 3 , iq(t) 0
p=1
(3.21) we get from (3.19) M (d2 ) , (3.22) M (q 2 ) which fixes α2 . Note that the right-hand side of (3.22) makes sense provided d2 is quasi-periodic (and, hence, has a mean value M (d2 )). By (3.21) this is true provided c2 is quasi-periodic, what will be proven inductively in Sec. 5 for the value of α1 given in (3.17). For n ≥ 4 we have α2 = −
M (qcn−1 ) = M (dn−1 ) + αn−1 M (q 2 ) .
(3.23)
Condition (3.20), combined with (3.23), says that αn−1 for n ≥ 4.
1 =− M (q 2 )
! n−2 1 X M (dn−1 ) + M (cp cn−p ) 2α1 p=2
(3.24)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
37
Note again that the right-hand side of (3.24) makes sense provided dn−1 is quasiperiodic (and, hence, has a mean value M (dn−1 )). By (3.21) this is true provided cn−1 is quasi-periodic, what will be proven inductively in Sec. 5 for the values of α1 given in (3.17), α2 given in (3.22) and α3 , . . . , αn−2 given inductively in (3.24). Now, note that dn−1 depends only on {c1 , . . . , cn−2 } and, therefore, relation (3.24) fixes αn−1 for given {α1 , . . . , αn−2 }. This, together with (3.17) and (3.22), fixes recursively all constants α∗ and guarantees (3.16) and (3.18) for all n ∈ N. Fixing the constants αn in the way described above is an important step towards the proof of quasi-periodicity of the functions cn and, eventually, of g. With this choice for the αn ’s no polynomial terms on t will appear after performing the integrations found in (3.6)–(3.8). As a consequence, one should expect to get here better estimates for the behavior on t of the functions cn than that found in (3.15) for our first choice of the αn ’s. As a matter of fact we will be able to prove that all functions cn are quasi-periodic by analyzing recursively their Fourier coefficients. This will be performed in Sec. 5. Now we will consider the case M (q 2 ) = 0. 4. A Solution g in the Case M (q 2 ) = 0 In the case where M (q 2 ) = 0 there are two equivalent procedures to start with. In the first we adopt the Ansatz (3.1) and choose α1 = 0, what implies cn ≡ 0 for all odd n. In the second, which we follow here, we adopt directly the Ansatz ∞ X
g(t) = q(t)
en (t)λn ,
(4.1)
n=1
with λ := 2 . Inserting this into (2.1) leads to ∞ X
0
(qen ) − i
n=1
n−1 X
! q ep en−p − 2if qen λn + iλ = 0 , 2
(4.2)
p=1
or (qe1 )0 − 2if qe1 + i = 0 , (qen )0 − i
n−1 X
q 2 ep en−p − 2if qen = 0 ,
(4.3) n ≥ 2.
(4.4)
p=1
The solutions are
Z
e1 (t) = q(t) −i " en (t) = q(t) i
t
q
−2
0
0
p=1
0
,
(t )dt + β1
0 n−1 XZ t
! 0
ep (t )en−p (t )dt
0
(4.5)
# + βn ,
for n ≥ 2 ,
0
where the βn ’s above, n = 1, 2, . . . , are arbitrary integration constants.
(4.6)
38
J. C. A. BARATA
Since now M (q 2 ) = M (q −2 ) = 0, e1 is bounded and, therefore, quasi-periodic. For e2 we have Z t (4.7) e1 (t0 )2 dt0 + β2 . e2 (t) = q(t) i 0
Using the same strategy used in the case M (q 2 ) 6= 0 above, we require M (e21 ) = 0. Since ! Z t 2 Z t (4.8) e1 (t)2 = q(t)2 − q −2 (t0 )dt0 − 2iβ1 q −2 (t0 )dt0 + β12 , 0
0
we must have −2iβ1 M (Q1 ) = M (Q2 ) , with
Z
t
q −2 (t0 )dt0
Q1 (t) := q(t)
2
(4.9)
and
(4.10)
0
Z
t
Q2 (t) := q(t)2
q −2 (t0 )dt0
2 .
(4.11)
0
Hence, we can choose β1 = provided M (Q1 ) 6= 0, i.e. M (Q1 ) = i
i M (Q2 ) , 2 M (Q1 )
(4.12)
2 X Q(2) n n·ω
n∈ZA n6=0
6= 0 ,
(4.13)
what we will assume here. Note that (4.12) makes sense, since by the condition M (q 2 ) = 0 and by the bound (5.30) (which guarantee the L∞ condition for Q1 and Q2 ) both Q1 and Q2 are quasi-periodic and, hence, have a mean value. Pn−1 For all n ≥ 3 we require p=1 M (ep en−p ) = 0, that means, 2M (e1 en−1 ) +
n−2 X
M (ep en−p ) = 0 ,
(4.14)
p=2
with the convention that
Pn−2 p=2
M (ep en−p ) = 0 for n = 3. Define
jn (t) := q(t)(en (t) − βn q(t)) = iq(t)
2
n−1 XZ t p=1
ep (t0 )en−p (t0 )dt0 ,
(4.15)
0
for n ≥ 2. We get for n ≥ 3, M (e1 en−1 ) = M (q −1 e1 jn−1 ) + βn−1 M (e1 q) = M (q −1 e1 jn−1 ) + βn−1 (−iM (Q1 ) + β1 M (q 2 )) = M (q −1 e1 jn−1 ) − iβn−1 M (Q1 ) .
(4.16)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
39
Hence, condition (4.14) implies βn−1
n−2 X 1 M (ep en−p ) M (q −1 e1 jn−1 ) + 2 p=2 = −i , M (Q1 )
n ≥ 3.
(4.17)
Note again that this expression makes sense provided jn−1 is quasi-periodic (and, hence, q −1 e1 jn−1 has a mean value). By (4.15) this is true provided en−1 is quasi-periodic, what will be proven inductively in Sec. 6 for the values of β1 given in (4.12) and β2 , . . . , βn−2 given inductively in (4.17). With the choices (4.12) and (4.17) we guarantee that no polynomial terms on t will appear after performing the integrations in (4.5) and (4.6). In Sec. 6 we will establish that the functions en are all quasi-periodic. 4.1. The power series solutions for the case f (t) = constant Before we finish this section let us consider one particular but interesting situation. p For the case f (t) = f0 ≡ f˜0 , constant, the Riccati Eq. (2.1) has g = −f0 ± f02 + 2 as particular solutions and for both U (t) becomes ! ω0 cos(ω0 t) − if0 sin(ω0 t) − sin(ω0 t) 1 , (4.18) U (t) = ω0 − sin(ω0 t) ω0 cos(ω0 t) + if0 sin(ω0 t) p as expected, with ω0 := f02 + 2 . It is important to take notice of the fact that, in the situation where f (t) = f0 ≡ ˜ f0 , constant, with f0 6= 0, we have precisely the conditions described above, namely, we have M (q 2 ) = 0 and M (Q1 ) = i(2f0 )−1 6= 0. We should p therefore expect that the power series (4.1) reproduces the solution g = −f0 + f02 + 2 (we take f0 > 0 without loss). Let us prove that this is indeed the case, at least for || < |f0 |. We will show by induction, using (4.6) and (4.17), that em (t) = lm q(t)−1
(4.19)
βm = lm ,
(4.20)
and for all m ∈ N, where lm := (−1)m−1
(2m − 2)! 1 . m!(m − 1)! (2f0 )2m−1
(4.21)
An explicit computation using (4.12) and (4.5) shows that e1 (t) = (2f0 )−1 q(t)−1 = l1 q(t)−1 and that β1 = (2f0 )−1 = l1 . Let us assume (4.19) and (4.20) valid for 1 ≤ m ≤ n − 1. By (4.17) we have
40
J. C. A. BARATA
"
# n−1 X 1 βn = −(2f0 ) M (q −1 e1 jn ) + M (ep en+1−p ) , 2 p=2
(4.22)
but, by (4.19), M (ep en+1−p ) = lp ln+1−p M (q 2 ) = 0 and, hence, βn = −(2f0 )M (q
−1
e1 jn ) = −iM
n−1 XZ t p=1
=−
=
! 0
0
ep (t )en−p (t )dt
0
0
n−1 1 X lp ln−p 2f0 p=1
n−1 (−1)n−1 X (2p − 2)!(2n − 2p − 2)! . (2f0 )2n−1 p=1 p!(p − 1)!(n − p)!(n − p − 1)!
(4.23)
Using the identity n−1 X p=1
(2n − 2)! (2p − 2)!(2n − 2p − 2)! = , p!(p − 1)!(n − p)!(n − p − 1)! n!(n − 1)!
(4.24)
the proof of (4.20) for m = n is complete and from (4.20) we have ! # " Z n−1 t X 0 −2 0 en (t) = q(t) i q(t ) dt lp ln−p + ln 0
= ln q(t)−1 ,
p=1
(4.25)
using (4.6) and (4.24) again. This proves (4.19) and (4.20) by induction for all n ∈ N. Relation (4.24) can be obtained from the more general identity ! ! ! n X tn + r + s tk + r tn − tk + s s r+s r = tk + r tn − tk + s tn + r + s n k n−k k=0 (4.26) by taking t = 2, r = s = −1 and by moving the terms with k = 0 and k = n to the right-hand side of (4.26). Relation (4.26) is one of the so-called “convolution identities” for binomial coefficients and is valid for n ∈ N, t, r, s ∈ R. Its proof is indicated in [15, Chap. 5]. With (4.19) one can easily check that the series p (4.1) becomes exactly the Taylor series in (about = 0) of the function −f0 + f02 + 2 , which is a particular solution of (2.1) for the case f (t) = f0 . In this case the convergence of (4.1) is clearly restricted to || < |f0 |. In the general case we learn from this example that one should not expect convergence of (4.1) for all values of . Further, we see that, in the situation where M (f ) 6= 0 and f has small oscillations around its mean value, i.e. in the situation
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
41
where f (t) − M (f ) is small (in some sense to be precised), the series (4.1) should be expected to converge (at least for small values of ). Unfortunately our analysis below was insufficient to provide a proof of this conjecture. 5. Quasi Periodicity of the Functions cn In this section we will prove the quasi-periodicity of the functions cn for our second choice of the constants αn and for the case M (q 2 ) 6= 0. Let us briefly describe the general strategy we will follow and the problems we will face. The strategy and the problems will be the same in the case M (q 2 ) = 0, treated in Sec. 6. We first express formally the recursive relations (3.6)–(3.8) for the functions cn (with the choices (3.17), (3.22) and (3.24)) in terms of its Fourier (n) (n) coefficients Cm , m ∈ ZA . The recursive relations for the coefficients Cm involve convolutions (a consequence, lately, of the quadratic character of the generalized Riccati equation) and, inevitably, small denominators. Assuming an exponential decay of the form |Qm | ≤ Qe−χ|m| for the Fourier coefficients Qm of the function q, a fact that will be proven in Appendix D, we (n) prove inductively bounds of the form |Cm | ≤ Kn e−(χ−δn )|m| for the Fourier coefficients of the functions cn . This exponential decay justifies the formal expression of (n) the functions cn in terms of the coefficients Cm and the corresponding recursion relations and is enough to establish the quasi-periodicity of all the functions cn by induction. Due to the convolutions and to the small denominators appearing in the recursive (n) relations for the coefficients Cm , the sequence δn is positive and increasing. It is possible to keep the sequence δn bounded and smaller than χ, but this ruins the behavior in n of the constants Kn and obstructs the control of the absolute convergence of the formal power series (3.1). 5.1. The Fourier decomposition of the functions cn To prepare the proof that the functions cn are all quasi-periodic, we will first prove that c1 is quasi-periodic. To prove that q is quasi-periodic we note that Z t q(t) := eif˜0 t exp i (f (t0 ) − f0 )dt0 . 0 ˜ Assuming now that
X
2
n∈ZB n6=0
˜ ˜ ˜ one has
Z F (t) := 0
t
|fn | < ∞, |n · ˜ω f |
(5.1)
(5.2)
˜˜
(f (t0 ) − f0 ) dt0 ∈ L∞ (R) ˜
(5.3)
and we conclude, by Bohr’s theorem, that F is almost-periodic and, by the uniform convergence of the Fourier series (see, e.g. [14]), that
42
J. C. A. BARATA
F (t) = −i
X n∈ZB n6=0
fn ein˜·ω˜f t − 1 , ˜ n · ωf
(5.4)
˜˜
˜ ˜ ˜ which shows that F is quasi-periodic. A standard argument (see Appendix B) shows that exp(iF (t)) is also quasi-periodic with β(exp(iF (t))) = β(F ) = β(f ). Therefore, we have established that q is quasi-periodic with σ(q) = σ(exp(iF )) + f0 . We write ˜ X in·ωt Qn e , (5.5) q(t) = n∈ZA
where ω and A are defined in (1.27) and (1.28), respectively. See Appendix C for the relation between the Fourier coefficients of q and those of f . For n ≥ 1, let us write X (n) im·ωt Cm e . (5.6) cn (t) = m∈ZA
In the next section we will be concerned with the recursion relations for the (n) (2) Fourier coefficients Cm . For Cm we have from (3.7)
(2) X α21 Q(2) X Qm−n n − Q−n (2) (2) 2 (2) + Cm α = Qm − Q − Q α 2 1 n −n . n·ω n·ω A A n∈Z n6=0
n∈Z n6=0
(5.7) Above,
(2) Qn
2
are the Fourier coefficients of q , namely q(t)2 =
X
im·ωt Q(2) , m e
(5.8)
Qn Qm−n .
(5.9)
m∈ZA
with Q(2) m =
X n∈ZA
For n ≥ 3 we have from (3.8) (n) Cm
=
X
Qm−n1 −n2
n1 ,n2 ∈ZA
n−1 X
! Cn(p) Cn(n−p) 1 2
p=1
1 (n1 + n2 ) · ω
n +n 6=0 1 2
+ Qm αn −
X
n−1 X
n1 ,n2 ∈ZA
p=1
! Cn(p) Cn(n−p) 1 2
1 . (n1 + n2 ) · ω
(5.10)
n +n 6=0 1 2
For the constants αn we have, according to (3.17), (3.22) and (3.24), the following expressions:
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
(2)
(2)
α21 = Q0 /Q0 , α2 = −
1 M (q 2 )
X
(2)
(2)
α21 Qn − Q−n
(5.11)
n·ω
n∈ZA
43
(2)
(2)
(Q−n − Q0 ) ,
(5.12)
n6=0
n−1 X
(p) (n−p) X Cn1 Cn2 1 (2) (2) Q − Q 0 −(n1 +n2 ) M (q 2 ) (n1 + n2 ) · ω p=1 n1 ,n2 ∈ZA
αn = −
n +n 6=0 1 2
n−1 1 X X (p) (n+1−p) , Cn C−n 2α1 p=2 n∈ZA
+
n ≥ 3.
(5.13)
Inserting these expressions into (5.7) and (5.10) we get (1)
(2) Cm
Cm = α1 Qm , (2) (2) " (2) # X α21 Qn − Q−n Qm Q−n Qm−n − , = (2) n·ω Q0 n∈ZA
(5.14)
(5.15)
n6=0
X
(n) = Cm
n1 ,n2
1 (n1 + n2 ) · ω
∈ZA
n−1 X
! Cn(p) Cn(n−p) Qm−(n 1
1 +n2 )
2
p=1
(2)
−
Qm Q−n1 −n2 (2)
Q0
n1 +n2 6=0
−
Qm
X n−1 X
(2)
2α1 Q0
(n+1−p)
Cn(p) C−n
,
n ≥ 3.
(5.16)
n∈ZA p=2 (2)
Above we used M (q 2 ) = Q0 . (2)
5.2. An upper bound for |Qm | At this moment we have to introduce some restrictions, which ultimately reflect on restrictions on the function f . The first concerns the frequencies ω. We assume a so-called Diophantine condition, namely, we assume the existence of constants ∆ > 0 and σ > 0 such that, for all n ∈ ZA , n 6= 0, |n · ω| ≥ ∆−1 |n|−σ .
(5.17)
The second restriction concerns the decay properties of the Fourier coefficients Qm for |m| → ∞. We assume that there are constants Q > 0 and χ > 0 such that, for all m ∈ ZA ,
44
J. C. A. BARATA
|Qm | ≤ Qe−χ|m| .
(5.18)
This will be proven in Appendix D for the case where f has a Fourier decomposition given by a finite sum. As a consequence of (5.18), we have (2) Qm ≤ Q2 Ξ0 (χ, m) (5.19) where, for α > 0,
X
Ξ0 (α, m) :=
e−α(|n|+|m−n|) .
(5.20)
e−α(|n|+|ma −n|) .
(5.21)
n∈ZA
One has
A X Y
Ξ0 (α, m) =
a=1 n∈Z
A simple computation shows that X e−α(|n|+|ma −n|) = e−α|ma | (|ma | + (tanh(α))−1 ) .
(5.22)
n∈Z
Hence Ξ0 (α, m) ≤ (|m| + (tanh(α))−1 )A e−α|m| . Using now the inequality |x|σ ≤
σ σ
eδ|x| , eδ valid for all x ∈ C and all δ > 0, σ > 0, we arrive at # " A A eδ/ tanh(α) e−(α−δ)|m| . Ξ0 (α, m) ≤ eδ Defining
A A eδ/ tanh(χ) eδ for δ > 0, we conclude that, for some δ2 positive and small enough, (2) Qm ≤ Q(2) (δ2 /2)e−(χ−δ2 /2)|m| .
(5.23) (5.24)
(5.25)
Q(2) (δ) := Q2
(5.26)
(5.27)
Below we will frequently denote Q(2) ≡ Q(2) (δ2 /2). (2)
5.3. An upper bound for |Cm | Since c2 is expressed as the quasi-periodic function q times the integral of a quasiperiodic function, one has only to show, for proving that c2 is quasi-periodic, that Z t i (α21 q(t0 )2 − q(t0 )−2 )dt0 ∈ L∞ (R) . (5.28) 0
But, by the hypotheses above, and using the fact that |α1 | = 1, we have Z t X Q(2) X n 2 0 2 0 −2 0 i ≤ 2∆Q(2) (α1 q(t ) − q(t ) )dt ≤ 2 |n|σ e−(χ−δ2 /2)|n| . |n · ω| 0 A A n∈Z
n∈Z
n6=0
(5.29)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
45
The above term corresponding to n = 0 has been eliminated by the choice of α1 . Using (5.24) we get σ X X Q(2) 2σ n (2) ≤ 2∆Q e−(χ−δ2 )|n| ≤ ∞ (5.30) 2 |n · ω| eδ2 A A n∈Z
n∈Z
n6=0
by choosing δ2 < χ. This concludes the proof of (5.28) and shows that c2 is quasiperiodic. (2) The relation (5.15) gives us the following upper bound for |Cm |: " # (2) X (2) Q (2) σ −(χ−δ /2)|n| −χ|m−n| −χ|m|−(χ−δ /2)|n| 2 2 Cm ≤ 2∆Q Q e |n| e + (2) e Q A 0 n∈Z ≤ 2∆Q
(2)
Q
σ σ
L1 (m) + e
eδ
−χ|m|
! Q(2) (2) L2 (2χ − δ2 − δ) , Q
(5.31)
0
where we used (5.24) again and the Diophantine condition (5.17). Above L1 (m) = Ξ(χ − δ2 /2 − δ, χ, m) , where, for α, β > 0, Ξ(α, β, m) :=
X
e−α|n|−β|m−n|
X
and L2 (α) :=
n∈ZA
e−α|n| .
n∈ZA
Note that Ξ(α, β, m) = Ξ(β, α, m) and that Ξ(α, β, m) ≤ Ξ0 (min{α, β}, m) .
(5.32)
To proceed we have to find upper bounds for L1 and L2 . We have for L2 (α) simply L2 (α) =
1+2
∞ X
!A e
−αa
=
a=1
1 + e−α 1 − e−α
A (5.33)
for α > 0. For L1 we have, using (5.32), L1 (m) ≤ Ξ0 (χ − δ2 /2 − δ, m) ≤ e
−(χ−δ2 /2−2δ)|m| δ/ tanh(χ−δ2 /2−δ)
e
A eδ
A (5.34)
by (5.25), for any δ > 0, small enough. Choosing δ = δ2 /4 we get L1 (m) ≤ e
−(χ−δ2 )|m| δ2 /(4 tanh(χ−3δ2 /4))
e
4A eδ2
A .
(5.35)
Putting the estimates above together we have for all δ2 > 0 sufficiently small, (2) C ≤ K2 e−(χ−δ2 )|m| , (5.36) m
46
J. C. A. BARATA
where K2 := 2∆Q(2) Q
4σ eδ2
σ
eδ2 / tanh(χ−3δ2 /4)
4A eδ2
A
! Q(2) + (2) L2 (2χ − 5δ2 /4) . Q 0
(5.37) −(2A+σ)
Note that K2 ∼ δ2
for δ2 → 0. (n)
5.4. Recursive upper bounds for |Cm | with n ≥ 3 In order to show that all cn ’s, n ≥ 3, are quasi-periodic we note that the kind of reasoning used above can be applied inductively and indicates to be sufficient to show that cn ∈ L∞ (R). For this it is enough to establish that X C (n) ≤ ∞ . (5.38) m m∈ZA
Using the Diophantine condition (5.17) together with (5.18) and (5.27) we get from (5.16) ! n−1 X X (n−p) (n) σ (p) Cn Cn Cm ≤ ∆Q |n + n | 1
"
2
1
n1 ,n2 ∈ZA
× e
−χ|m−n1 −n2 |
2
p=1
Q(2) + (2) e−χ|m|−(χ−δ2 )|n1 +n2 | Q
#
0
X n−1 X Q Cn(p) C (n+1−p) , + e−χ|m| (2) −n 2 Q0 n∈ZA p=2
n ≥ 3.
(5.39)
The strategy we will follow is to use (5.39) for recursively proving a bound on (n) the absolute value of the Fourier coefficients Cm , from which (5.38) should follow. We will assume that for 1 ≤ a ≤ n − 1, there are positive constants Ka and δa , δa < χ, such that, for all m ∈ ZA , (a) Cm ≤ Ka e−(χ−δa )|m| . (5.40) We have already seen that we can adopt K1 := Q, δ1 := 0 and K2 as in (5.37) with δ2 arbitrary and small. Inserting (5.40) into (5.39) and using once more the inequality (5.24), for some δ > 0 small enough to be chosen conveniently, gives ! σ σ n−1 (2) X (n) Q −χ|m| Cm ≤ ∆Q Kp Kn−p L3 (p, m) + e (2) L4 (p) Q eδ p=1 0 n−1 Q X Kp Kn+1−p L5 (p) , + e−χ|m| (2) 2 Q0 p=2
(5.41)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
47
where L3 (p, m) := L3 (χ − δp , χ − δn−p , δ, χ, m) ,
(5.42)
L4 (p) := L4 (χ − δp , χ − δn−p , χ − δ2 − δ) ,
(5.43)
L5 (p) := L2 (2χ − δp − δn+1−p ) ,
(5.44)
with 1 ≤ p ≤ n − 1 for L3 and for L4 and 2 ≤ p ≤ n − 1 for L5 , where X
L3 (α, β, µ, ν, m) :=
n1 ,n2
exp(−α|n1 | − β|n2 | + µ|n1 + n2 | − ν|m − n1 − n2 |) ,
∈ZA
(5.45) X
L4 (α, β, γ) :=
exp(−α|n1 | − β|n2 | − γ|n1 + n2 |) ,
(5.46)
n1 ,n2 ∈ZA
with α, β, γ, µ, ν > 0 and µ small enough. We have to find convenient upper bounds for the sums above. For L5 (p) we have L5 (p) = L2 (2χ − δp − δn+1−p ) ≤ L2 (2χ − 2δ[n]) ,
(5.47)
where δ[n] := max δp0 . 0 p
For L4 (α, β, γ) we have L4 (α, β, γ) ≤ L4 (α, β, 0) = L2 (α)L2 (β) .
(5.48)
L4 (p) ≤ L2 (χ − δp )L2 (χ − δn−p ) ≤ L2 (χ − δ[n] )2 .
(5.49)
Hence,
Let us now consider L3 (p, m). For later use let us try to find bounds for L3 (α, β, µ, ν, m). We have, for any δ > 0 sufficiently small, L3 (α, β, µ, ν, m) ≤
X
exp(−(α − µ)|n1 | − (β − µ)|n2 | − ν|m − n1 − n2 |)
n1 ,n2 ∈ZA
=
X
exp(−(α − µ)|n1 |) Ξ(β − µ, ν, m − n1 )
n1 ∈ZA
≤
X n1 ∈ZA
exp(−(α − µ)|n1 |) Ξ0 (κ, m − n1 )
48
J. C. A. BARATA
≤ ≤
A eδ A eδ
A
exp
A
with
exp
S :=
A eδ
δ tanh(κ) δ tanh(κ)
2A
exp
Ξ(α − µ, κ − δ, m)
Ξ0 (ζ, m) ≤ S e−(ζ−δ)|m| ,
δ δ + tanh(κ) tanh(ζ)
(5.50)
,
κ := min{β − µ, ν} and ζ := min{α − µ, κ − δ} = min{α − µ, β − µ − δ, ν − δ} . Above, we made use of inequality (5.25) in the third and in the last inequalities, as well as of (5.32). After the corresponding replacements we conclude that " # 2A A 2δ exp (− (χ − δ[n] − 3δ) |m|) . exp L3 (p, m) ≤ eδ tanh(χ − δ[n] − 2δ) (5.51) Note that the bounds (5.47), (5.49) and (5.51) do not depend on p, but depend on n. We choose the sequence δa , a ≥ 1 as a strictly increasing bounded sequence converging to some δ∞ < χ. This makes δ[n] = δn−1 We also choose δ such that δn = δn−1 + 3δ. These choices give " 2A # 3A 2δ∞ exp (− (χ − δn ) |m|) , exp L3 (p, m) ≤ e(δn − δn−1 ) 3 tanh(χ − δ∞ ) (5.52) L4 (p) ≤ L2 (χ − δ∞ )2 ,
(5.53)
L5 (p) ≤ L2 (2χ − 2δ∞ ) ,
(5.54)
with 1 ≤ p ≤ n − 1 for L3 and for L4 and 2 ≤ p ≤ n − 1 for L5 . Returning to (5.41), we get σ n−1 X (n) 3σ C ≤ ∆Q Dn e−(χ−δn )|m| + e−χ|m| Kp Kn−p m e(δn − δn−1 ) p=1 n−1 X Q + e−χ|m| (2) L2 (2χ − 2δ∞ ) Kp Kn+1−p , 2 Q0 p=2
with Dn := max
(
3A e(δn − δn−1 )
2A
exp
2δ∞ 3 tanh(χ − δ∞ )
(5.55)
Q(2) , L2 (χ − δ∞ ) (2) Q 2
) .
0
(5.56)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
49
We finally got the bound (n) Cm ≤ Kn e−(χ−δn )|m| ,
(5.57)
with Kn := 2∆Q
3σ e(δn − δn−1 )
σ Dn
n−1 X
Kp Kn−p
p=1 n−1 X
Q
+ (2) L2 (2χ − 2δ∞ ) Kp Kn+1−p . 2 Q0 p=2 By induction, this establishes the bound (n) Cm ≤ Kn e−(χ−δ∞ )|m|
(5.58)
(5.59)
for all n ∈ N and this proves (5.38) for all n ∈ N. This, in turn, finally establishes that all functions cn are quasi-periodic on R since it implies that X (n) Cm < ∞. m∈ZA
5.5. The convergence of the power expansion on Now, we have to deal with the question of the absolute convergence of the formal P power series in (3.1). Since |cn (t)| ≤ Kn m∈ZA e−(χ−δ∞ )|m| we have to analyze the convergence of the formal infinite sum ∞ X
Kn ||n .
(5.60)
n=1
Since the increasing sequence δn , n ∈ N, introduced in the definition of Kn , has to converge to some δ∞ < χ, the difference δn −δn−1 occurring in (5.58) has to decrease to zero for n → ∞. Hence, the leading behavior for n large of the right-hand side of (5.58) is governed by Kn ≈
C
n−1 X
(δn − δn−1 )σ+2A
p=1
Kp Kn−p ,
(5.61)
for some positive constant C. Even choosing a slowly converging sequence δn , for instance such that δn −δn−1 ' n−(1+0 ) , with some small 0 > 0, we would probably find a large n behavior for Kn like Kn ≈ (n!)(1+0 )(σ+2A) cn for some constant c > 1. Unfortunately we conclude that the upper bounds on the Kn obtainable from the recursive definition (5.58) are too bad, for whatever choice of the converging sequence δn , to guarantee the convergence of the sum in (5.60) for any finite and, hence, analyticity on cannot be established through the above analysis. Perhaps, analyticity around = 0 should not be generally expected in the system we considered. See Sec. 7 for further discussion.
50
J. C. A. BARATA
6. Quasi Periodicity of the Functions en In this section we will prove the quasi-periodicity of the functions en . 6.1. The Fourier decomposition of the functions en For all n ≥ 1, we write
X
en (t) =
(n) im·ωt Em e .
(6.1)
m∈ZA (n)
We will study the Fourier coefficients Em . We have from (4.5) X
(1) =− Em
(2) Qm−n Q−n
n·ω
n∈ZA n6=0
X + Qm β1 +
n∈ZA n6=0
(2) Q−n
n · ω
,
(6.2)
and, for n ≥ 2, we have from (4.6) (n) Em
=
X
n−1 X
Qm−n1 −n2
!
1 (n1 + n2 ) · ω
En(p) En(n−p) 1 2
p=1
n1 ,n2 ∈ZA n +n 6=0 1 2
+ Qm βn −
X
n−1 X
n ,n ∈ZA 1 2 n +n 6=0 1 2
p=1
! En(p) En(n−p) 1 2
1 , (n1 + n2 ) · ω
n ≥ 2 . (6.3)
For the constants β∗ we have (see (4.12) and (4.17))
β1 =
1 2iM (Q1 )
X
(2)
X n−1 1 iM (Q1 ) p=1
×
+
n ,n ∈ZA 1 2 n +n 6=0 1 2
(n1 + n2 ) · ω n−1 X
X
p=1
n1 ,n2 ∈ZA n +n 6=0 1 2
+
+
X Q(2) n , n · ω A
1 2
(p)
(6.4)
n∈Z n6=0
X
(p) (n−p) En1 En2
(2)
(n1 · ω)(n2 · ω)
n ,n ∈ZA 1 2 n 6=0,n 6=0 1 2
βn =
(2)
Qn1 +n2 Qn1 Qn2
X (2) Q R + −n −n 1 2
(2) (2) Qn−n −n Qn 1 2
n·ω
n∈ZA
n6=0
n−1 X
X
(n+1−p)
En(p) E−n
p=2 n∈ZA
(n−p)
En1 En2 , (n1 + n2 ) · ω
for n ≥ 2 ,
(6.5)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
51
where R := β1 +
X Q(2) −n n∈ZA n6=0
n·ω
=
1 2iM (Q1 )
X
(2)
(2)
(2)
Qn1 +n2 Qn1 Qn2 (n1 · ω)(n2 · ω)
n1 ,n2 ∈ZA n 6=0,n 6=0 1 2
.
(6.6)
Finally, for the Fourier coefficients we get, (1) Em =
X Qm+n Q(2) Qm n + n · ω 2iM (Q1 ) A
n∈Z n6=0
=
X n∈ZA n6=0
X
(2)
n ,n ∈ZA 1 2 n 6=0,n 6=0 1 2
(n) = Em
p=1
×
!
n ∈Z n0 6=0
X n ,n ∈ZA 1 2 n1 +n2 6=0
(2)
(n1 · ω)(n2 · ω)
(2) X Q(2) Qm n+n0 Qn0 Qm+n + 2iM (Q1 ) 0 A n0 · ω
n−1 X
(2)
Qn1 +n2 Qn1 Qn2
(2)
Qn , n·ω
(6.7)
X Qm−n −n + Qm Q(2) −n1 −n2 R + 1 2 iM (Q1 ) A
(2) (2) Qn−n −n Qn 1 2
n·ω
n∈Z n6=0
(p) (n−p) n−1 X X En1 En2 Qm (n+1−p) + En(p) E−n , (n1 + n2 ) · ω 2iM (Q1 ) p=2 A
for n ≥ 2 .
(6.8)
n∈Z
In the next two subsections we will follow the same steps of the case M (q 2 ) 6= 0 and we will try to find convenient upper bounds for the Fourier coefficients given recursively in (6.7) and (6.8). (1)
6.2. An upper bound for |Em | (2)
We will again make use of (5.18) and of (5.27) in the form |Qm | ≤ Q(2) (δ) e−(χ−δ)|m| , for δ > 0 arbitrary but conveniently small. From (6.7) we have, using the same inequalities as in the previous case, X (1) Em ≤ QQ(2) (δ)
e−χ|m+n| + e−χ|m|
n∈ZA n6=0
X e−(χ−δ)(|n+n0 |+|n0 |) × |n0 · ω| 0 A n ∈Z n0 6=0
≤ W1
X
!
Q(2) (δ)2 2|M (Q1 )|
e−(χ−δ)|n| |n · ω|
e−χ|m+n| + e−χ|m| W2
n∈ZA
×
X n0 ∈ZA
0
0
e−(χ−δ)|n+n |−(χ−2δ)|n |) e−(χ−2δ)|n| ,
(6.9)
52
J. C. A. BARATA
Q (δ) σ σ with W1 := ∆QQ(2) (δ)( eδ ) and W2 := ∆ 2|M(Q ( σ )σ . Using the previously 1 )| eδ defined function Ξ0 , the last inequality in (6.9) leads to X (1) Em ≤ W1 Ξ0 (χ − 2δ, m) + e−χ|m| W2 Ξ0 (χ − 2δ, n)e−(χ−2δ)|n| (6.10) (2)
2
n∈ZA
and using (5.25) we get
(1) E ≤ L1 e−(χ−3δ)|m|
(6.11)
m
for some constant L1 (depending on χ, δ, σ etc.). Of course, one sees here that, choosing 0 < δ < χ/3, we have proven that e1 is a quasi-periodic function on R. (n)
6.3. Recursive upper bounds for |Em | with n ≥ 2 (n)
We start now from relation (6.8) and try to find recursive bounds for |Em |, with (n) n ≥ 2, as in the case of the |Cm |’s. For 1 ≤ p ≤ n − 1 we assume the bound (p) Em ≤ Lp e−(χ−δp )|m| (6.12) for Lp ≥ 0 and δp > 0, small enough. This assumption has been proven above for the case p = 1 (see (6.11)). From (6.8) we have n−1 −χ|m| (2) X X (n) Q (δ) E ≤ Q e−χ|m−n1 −n2 | + e Lp Ln−p m |M (Q1 )| p=1 n1 ,n2 ∈ZA n1 +n2 6=0
X e−(χ−δ)(|n+n1 +n2 |+|n|) −(χ−δ)|n1 +n2 | (2) × |R|e + Q (δ) |n · ω| A n∈Z n6=0
×
×
e−(χ−δp )|n1 |−(χ−δn−p )|n2 | |(n1 + n2 ) · ω| n−1 X
Lp Ln+1−p
p=2
X
+ e−χ|m|
Q 2|M (Q1 )|
e−(2χ−δp −δn+1−p )|n|
n∈ZA
,
for n ≥ 2. Using once more (5.17) we have
σ σ n−1 X (n) Q(2) (δ) Em ≤ Q∆ Lp Ln−p L3 (p, m) + e−χ|m| eδ p=1 |M (Q1 )| × |R|L4 (χ − δp , χ − δn−p , χ − 2δ) + ∆
σ σ eδ
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
53
X × e−(χ−2δ)|n| L3 (χ − δp , χ − δn−p , 0, χ − δ, n) A
n∈Z
+ e−χ|m|
n−1 X Q Lp Ln+1−p L2 (2χ − δp − δn+1−p ) . 2|M (Q1 )| p=2
Using L3 (χ − δp , χ − δn−p , 0, χ − δ, n) ≤ L2 (χ − δ[n])2 L2 (χ − δ) and L4 (χ − δp , χ − δn−p , χ − 2δ) ≤ L2 (χ − δ[n])2 (see (5.48) and (5.51)), we get (n) E ≤ Ln e−(χ−δ[n]−3δ)|m| ,
(6.13)
m
where Ln := Q∆
σ σ eδ
n−1 X p=1
! ( Lp Ln−p
A eδ
2A
exp
2δ tanh(χ − δ[n] − 2δ)
σ σ Q(2) (δ) |R|L2 (χ − δ[n])2 + ∆ L2 (χ − δ[n])2 L2 (χ − δ)2 + |M (Q1 )| eδ n−1 X Q L2 (2χ − 2δ[n]) Lp Ln+1−p + 2|M (Q1 )| p=2
)
! .
(6.14)
Choosing δn as an increasing sequence (converging to δ∞ < χ) one has δ[n] = δn−1 and, with the choice δ = (δn − δn−1 )/3, we get from (6.13) (n) Em ≤ Ln e−(χ−δn )|m| ,
(6.15)
thus proving inductively the bound (6.12) for Ln given by the recurrence (6.14). As in the case of the functions cn considered previously, this establishes that the functions en are, for all n ≥ 1, quasi-periodic on R since it implies that P (n) m∈ZA |Em | < ∞. Note that from (6.14) the leading behavior of Ln for large n is expected to be like n−1 X C Lp Ln−p , (6.16) Ln ≈ σ+2A (δn − δn−1 ) p=1 as in the case of Kn above. As in that case, we draw the same negative conclusion about the possibility of proving absolute convergence of the infinite sum in (4.1) with the methods used in this paper.
54
J. C. A. BARATA
7. Final Discussion To finish these notes let us now briefly discuss in a non-rigorous way some problems involving our analysis and our results. Our original intention was to prove quasi-periodicity of the solutions of the generalized Riccati Eq. (2.1) by analyzing its power series expansion in . A step in this direction was the method of elimination of the secular terms, another was the proof of quasi-periodicity of the coefficients cn and en . However, the series expansions for g in could not be proven to be uniformly convergent for all t ∈ R and in some open set containing the point = 0. The question is, when should we expect that g is quasi-periodic? As discussed in Rt Appendix A, the function g appeared through the Ansatz φ(t) = exp(−i 0 (f (t0 ) + g(t0 ))dt0 ) for one of the particular solutions of one of the components of the wave function Φ, or more precisely, for a particular solution of the quasi-periodic Hill’s Eq. (1.26). It is clear that g could not be expected to be quasi-periodic if φ(t) becomes zero at some finite t, because quasi-periodic functions are bounded. A theorem by H. Bohr [16] asserts, up to technicalities, that if φ(t) is bounded away from zero then g should be quasi-periodic. Hence, one of the questions would be to know whether there are particular solutions φ of the quasi-periodic Hill’s Eq. (1.26) such that |φ(t)| > δ for all t ∈ R and some δ > 0. It is hard to prove this condition a priori. However, in the situation where we know that there are particular solutions of (1.26) of the f (t) = f0 is constant, p form e±iω0 t with ω0 = 2 + f02 . These particular solutions do not vanish for t ∈ R. In this case we have been able to show that the series expansion for g is convergent for || < |f0 |. This leads to the conjecture that g should be quasi-periodic at least for small perturbations of the case f (t) = constant, for instance if f (t) − M (f ) is uniformly small. We have not been able to prove this conjecture. There are other problems which have to be faced for a better understanding of the whole picture. Some are related to the unitarity condition U (t)U (t)∗ = 1l, which reduces to |R(t)|2 (1 − 2 Im (g(0)S(t)) + (|g(0)|2 + 2 )|S(t)|2 ) = 1 .
(7.1)
Relation (7.1) has implications which are not easy to verify by looking at our expansions. First we note that, if we assume that a quasi-periodic solution g for the generalized Riccati Eq. (2.1) was found, then we must have M (g) ∈ R for real . For, if Im(M (g)) > 0, R(t) diverges exponentially for t → +∞ while S(t) stays bounded, violating (7.1) for large t. If, on the other hand, Im(M (g)) < 0, R(t) will decay like exp(−|Im(M (g))|t) for t → +∞ while S(t) will behave as exp(+2|Im(M (g))|t). Hence, the product |R(t)|2 |S(t)|2 will diverge for t → +∞ making (7.1) impossible again. It is difficult to verify the condition M (g) ∈ R for real directly from our perturbative expansions for g but, from the unitarity of the time evolution, this must be true. Moreover, the unitarity relation (7.1) imposes an additional condition which is hard to verify. Under the assumption that g is quasi-periodic with M (g) ∈ R for real it could still happen that M (R−2 ) 6= 0. In this case the function S(t) would
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
55
develop a term like M (R−2 )t, which is linearly increasing in t, again violating (7.1) for large t. Hence, unitarity imposes the condition M (R−2 ) = 0. If we assume that our perturbative expansions for g converge, the function R(t)−2 must presumably be of the form X hm ()e−im·ωt . R(t)−2 = e2i(M(f )+M(g))t m∈ZA
To ensure that M (R−2 ) = 0 one should expect that hm () is of the form hm () = (2M (f ) + 2M (g) − m · ω)h0m () , where h0m () is a regular function of in some eventually small neighborhood of = 0. Again, it was not possible to verify this directly from the expansions. Another possibility would be that the function g fails to be almost-periodic, due to some sort of resonance, for in some eventually dense set E0 ⊂ R. For such values of the linearly increasing term M (R−2 )t appearing in (7.1) would make no sense. Presumably, E0 should be the set of solutions in of 2M (f )+2M (g)−m·ω, m ∈ ZB . This could explain the apparent impossibility of proving uniform convergence of our perturbative expansions for g. If the Fourier coefficients of g are proportional to something like (m · ω + )−1 one would not find an open neighborhood of = 0 where g is analytic, since the set {m · ω, m ∈ ZB } is dense, under the assumptions on the frequencies. With the methods employed here it is not possible to rule out such cases. As we explained before, however, we conjecture that g is analytic on at least for the situation where f (t) − M (f ) is uniformly small. It would be very interesting to investigate further the problems and conjectures discussed here and the mechanisms responsible for eventually breaking the analyticity of the function g as a function of . This should involve a deeper analysis than that attempted here. Finally we note that one should expect to have much better estimates in the case where f is a periodic function, since there no small denominators should be found in the perturbative expansions. Results in this direction will be found in a forthcoming publication. Acknowledgments I am very indebted to Walter F. Wreszinski for his suggestion to study the system considered here, for many conversations, for encouraging me to write these notes and for contributions to this manuscript. I am also grateful to H. Jauslin, D. H. U. Marchetti, C. Moreira and T. Spencer for fruitful discussions and suggestions. Appendix A. The Wave Function, Hill’s Equation and a Generalized Riccati Equation
Let Φ:R→C , 2
R 3 t 7→
φ+ (t) φ− (t)
,
56
J. C. A. BARATA
with φ± ∈ C 2 (R), be a solution of the Schr¨ odinger Eq. (1.5) with Hamiltonian H1 above: (A.1) iΦ0 (t) = (σ1 + f (t)σ3 )Φ(t) , for f : R → R, f ∈ C 1 (R) and ∈ R, with initial data φ+ (0) ∈ C2 . φ− (0) Above σi are the Pauli matrices in their usual representations. Taking the time derivative of both sides of (A.1) we get iΦ00 (t) = f 0 (t)σ3 Φ(t) − i(σ1 + f (t)σ3 )2 Φ(t) = f 0 (t)σ3 Φ(t) − i[(2 + f (t)2 )1l + f (t)(σ1 σ3 + σ3 σ1 )]Φ(t) .
(A.2)
Hence, iΦ00 (t) = (f 0 (t)σ3 − i(2 + f (t)2 )1l)Φ(t) ,
(A.3)
which is a diagonal equation for Φ(t). For the components of Φ, (A.3) means φ00+ (t) + (+if 0 (t) + 2 + f (t)2 )φ+ (t) = 0 ,
(A.4)
φ00− (t) + (−if 0 (t) + 2 + f (t)2 )φ− (t) = 0 .
(A.5)
The solutions of (A.1) can be recovered from the solutions of the differential Eqs. (A.4) and (A.5) with initial data (φ+ (0), φ0+ (0)) and (φ− (0), φ0− (0)), respectively, by imposing (A.6) iΦ0 (0) = (σ1 + f (0)σ3 )Φ(0) as a restriction to the initial data, i.e. by imposing iφ0+ (0) = f (0)φ+ (0) + φ− (0) ,
(A.7)
iφ0− (0) = −f (0)φ− (0) + φ+ (0) .
(A.8)
Since (A.5) is obtained from (A.4) by the change f → −f , we will study only the equation (A.9) φ00 (t) + (if 0 (t) + 2 + f (t)2 )φ(t) = 0 , with initial data (φ(0), φ0 (0)). In order to motivate the Ansatz we are going to use, let us consider first a particular way of solving (A.9) for the case ≡ 0. Appendix A.1. The case = 0 In this case Eq. (A.9) reduces to φ00 (t) + (if 0 (t) + f (t)2 )φ(t) = 0 . A natural Ansatz is given by solutions of the form Z t 0 0 0 (G(t ) + f (t ))dt , φ(t) = φ0 exp −i 0
(A.10)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
57
where φ0 is an arbitrary constant. This leads to the following equation for G: G0 − iG2 − 2if G = 0 .
(A.11)
This is an equation of the Bernoulli type, and has one obvious solution: g ≡ 0. Another solution can be found defining v = 1/G, which, in face of (A.11), leads to the linear equation (A.12) v 0 + 2if v + i = 0 , whose general solution is v(t) =
1 p0 (t)
Z t a0 − i p0 (t0 )dt0 ,
(A.13)
0
where a0 is an arbitrary constant and Z t f (t0 )dt0 . p0 (t) := exp 2i
(A.14)
0
This leads to the following general expression for the solution of (A.11): G(t) =
p0 (t) . Z t a0 − i p0 (t0 )dt0
(A.15)
0
Since G ≡ 0 is also a solution of (A.11), we allow formally a0 ∈ C ∪ {+∞}. Hence, the general solution of (A.10) is Z t Z t 0 0 φ(t) = φ0 exp −i f (t )dt exp −i 0 0
p0 (t0 ) dt0 , Z t0 a0 − i p0 (t00 )dt00
(A.16)
0
for arbitrary constants φ0 and a0 . Since 0 Z t p0 (t) = i ln a0 − i p0 (t0 )dt0 Z t 0 a0 − i p0 (t0 )dt0 0
we get, finally, the general solution of (A.10), Z t Z t φ(t) = exp −i f (t0 )dt0 φ1 + φ2 p0 (t0 )dt0 , 0
(A.17)
0
with φ1 = φ0 a20 and φ2 = −iφ0 a0 , or in terms of the initial data, φ1 = φ(0) , φ2 = if (0)φ(0) + φ0 (0) .
(A.18)
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J. C. A. BARATA
After imposing (A.7) and (A.8) (with = 0, of course) we get, as expected, the solutions Z t (A.19) f (t0 )dt0 , φ+ (t) = φ+ (0) exp −i 0
Z t 0 0 f (t )dt . φ− (t) = φ− (0) exp +i
(A.20)
0
Appendix A.2. The case 6= 0 For 6= 0, we will follow steps analogous to the previous case, but the situation here is more complicated. For solving (A.9) we start with the same Ansatz, namely, Z t (G(t0 ) + f (t0 ))dt0 . φ(t) = φ0 exp −i 0
This leads to Eq. (2.1) for G. Equation (2.1) is a differential equation of the generalized Riccati type. According to the theory of the generalized Riccati equations, given a particular solution g of (2.1), a general solution is of the form G = g + u, where u satisfies the Bernoulli equation (A.21) u0 − 2i(f + g)u − iu2 = 0 . This, in turn, can be transformed into a linear equation by defining v := 1/u, which gives for v (A.22) v 0 + 2i(f + g)v + i = 0 . The general solution of (A.22) is v(t) =
1 p(t)
Z t a0 − i p(t0 )dt0 ,
(A.23)
0
where a0 is an arbitrary constant and Z t p(t) := exp 2i (f (t0 ) + g(t0 ))dt0 .
(A.24)
0
Hence, the general solution of (2.1) is G(t) = g(t) +
p(t) . Z t a0 − i p(t0 )dt0
(A.25)
0
Since u ≡ 0 is, in principle, also a solution of (A.21), we allow formally a0 ∈ C ∪ {+∞}. For (A.9) we get, in complete analogy with the = 0 case, the general solution Z t Z t 0 0 0 0 0 φ1 + φ2 (A.26) (f (t ) + g(t ))dt p(t )dt , φ(t) = exp −i 0
0
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
59
with φ1 , φ2 ∈ C, arbitrary constants. In terms of the initial data one easily checks that φ1 = φ(0) , (A.27) φ2 = i(f (0) + g(0))φ(0) + φ0 (0) . Expression (A.26) indicates the solution for φ+ . The solution for φ− can be obtained, as we pointed before, by interchanging f → −f . Since g depends on f , we must first discover what happens to g by changing the sign of f . The generalized Riccati Eq. (2.1) indicates that, if g is a solution for f , then −g is a solution for −f . Hence, using the constraints (A.7)–(A.8), we get finally Eqs. (2.2) and (2.3) which express φ± in terms of g and of the initial conditions. In the general case it is interesting to note that the whole dependence on in (A.21) is hidden in the still unknown function g. For = 0, Eq. (2.1) reduces to (A.11). Comparing (A.21) with (A.11), (A.14) with (A.24) and (A.15) with (A.25) we conclude that we should have g ≡ 0 for = 0. p In the case where f (t) = f0 is a constant, the solution g = −f0 + sign (f0 ) f02 + 2 for f0 6= 0 satisfies this condition. For f0 = 0, we have g = ±. Appendix B. The Quasi Periodicity of q(t) and exp(iF (t)) It is already clear that exp(iF (t)) is an almost-periodic function since F (defined in (5.3)) is quasi-periodic and since the set of all almost-periodic functions on R is a closed sub-algebra of L∞ (R) (see, e.g. [14]). Let P be the Taylor polynomial of degree n of the exponential function: Pn (x) = Pn xa n a=0 a! . We know that | exp(x) − Pn (x)| ≤
−1 |x| |x|n+1 , 1− n+2 (n + 1)!
(B.1)
for n + 2 > |x|, x ∈ C. Let Fn , n ∈ N, be given by X
Fn (t) := −i
n∈ZB , n6=0 |n|∞ ≤n
˜
with |n|∞
˜
fn ein˜·ω˜f t − 1 , ˜ n · ωf
(B.2)
˜˜
˜ ˜ ˜ := max{|n1 |, . . . , |nB |}, the l∞ norm on ZB , and define σn (t) ≡ σn [F ](t) :=
F1 (t) + · · · + Fn (t) . n
(B.3)
By the hypothesis, σn converges to F uniformly on R (see e.g. [14]). We also know that |Fn | ≤ D, uniformly on n and t, where D is the left-hand side of (5.2). Therefore |σn (t)| ≤ D uniformly on n and t. Hence, for n large enough, | exp(iF (t)) − Pn (iσn (t))| ≤ | exp(iF (t)) − exp(iσn (t))| + | exp(iσn (t)) − Pn (iσn (t))| −1 |σn (t)| |σn (t)|n+1 . ≤ | exp(iF (t)) − exp(iσn (t))| + 1 − n+2 (n + 1)!
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J. C. A. BARATA
From the fact that |σn (t)| is uniformly bounded on n and t and from the uniform continuity of the exponential function on compact sets we conclude that | exp(iF (t)) − Pn (iσn (t))| can be made smaller than any prescribed > 0, for all t ∈ R, by choosing n large enough. The Fourier coefficients W (ξ), ξ ∈ R, of the almost-periodic function exp(iF (t)) are given by (see, e.g. [14]) 1 T →∞ 2T
Z
T
W (ξ) = lim
exp(iF (τ ))e−iξτ dτ .
(B.4)
−T
According to our previous remark, we have Z 1 T −iξτ |W (ξ)| ≤ + lim Pn (iσn (τ ))e dτ , T →∞ 2T −T
(B.5)
RT for any prescribed > 0, for n large enough. Now, for each n fixed, −T Pn (iσn (τ )) e−iξτ dτ is bounded in T if ξ is not of the form n · ωf , with n ∈ ZB . Hence, for ξ not of the form n · w f , n ∈ ZB , we have W (ξ) = 0. This completes the proof that exp(iF (t)) (and hence q(t)) is quasi-periodic with β(exp(iF (t))) = β(F ) = β(f ).
˜˜
˜˜ ˜
˜
Appendix C. The Relation Between q and f Since f is real and quasi-periodic we write X fn ein˜·ω˜f t , f (t) = f0 + ˜ n∈ZB ˜n6=0 ˜ ˜ with f0 = M (f ) ∈ R and fn = f−n . ˜ consider here the case where the sum above is To simplify our analysis˜ we will a finite sum. This situation is physically more realistic anyway. To fix the notation we write f (t) = f0 +
2J X
fa ein˜a ·ω˜f t ,
(C.1)
a=1
with the convention na = −n2J−a+1 6= 0, 1 ≤ a ≤ J, with fa ≡ fna and where ˜ J ≥ 1. Clearly fa = f2J−a+1 , 1 ≤ a ≤ J. We get 2J Y fa iγf if0 t ina ·ω f t , (C.2) exp e q(t) = e e na · ω f ˜ ˜ a=1
˜
˜
˜
˜ ˜
with γf := i As one easily sees, γf ∈ R.
2J X
fa . n · ωf a=1 a
˜ ˜
(C.3)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
61
Expanding the exponential functions inside of the product we have ( 2J ! ! ) ∞ 2J X Y 1 fa pa X iγf exp i f0 + ω f · pb nb t . q(t) = e pa ! na · ω f p ,...,p =0 a=1 1
˜
˜ ˜
2J
b=1
˜
(C.4)
The function q 2 is obtained by replacing f → 2f , what gives ( 2J ) ∞ X Y 1 2fa pa 2 i2γf q(t) = e pa ! na · ω f p ,...,p =0 a=1 1
2J
× exp i 2f0 + ωf ·
2J X
˜
Hence, 2
M (q ) = e
b=1
. , p2J = 0 p1 , . . P
2f0 +ωf ·
2J
pb nb
˜
t
.
(C.5)
pa 2J Y 1 2fa . p ! na · ω f a=1 a
∞ X
i2γf
˜ !˜!
pb nb =0
(C.6)
˜ ˜
b=1 ˜ ˜ Depending on the values of f0 and of na · ω f , 1 ≤ a ≤ 2J, it may be impossible to find a solution for 2J X pb nb = 0 (C.7) 2f0 + ω f ·
˜ ˜
˜
˜
b=1
with p1 , . . . , p2J ≥ 0. If that happens we have M (q 2 ) = 0. Note that (C.7) has always solutions in the case f0 = 0. There we have ωf ·
˜
2J X
pb nb = ω f ·
˜ ˜
b=1
J X
(pb − p2J−b+1 ) nb
(C.8)
˜
b=1
and this can be made equal to zero by taking pb = p2J−b+1 for all b with 1 ≤ b ≤ J. From (C.4) we conclude that q has a Fourier decomposition X Qm eim·ωt q(t) = m∈ZA
with ω defined in (1.27) and with Qm = e
iγf
∞ X p1 ,...,p2J
pa 2J Y fa 1 , δ (P , m) pa ! na · ω f =0 a=1
(C.9)
˜ ˜
where
2J X pb nb ∈ ZB , b=1 P ≡ P (p1 , . . . , p2J , n1 , . . . , n2J ) := ! 2J X pb nb , 1 ∈ ZB+1 ,
˜
˜
˜
b=1
˜
if f0 = 0 , ˜ if f0 6= 0 , ˜
(C.10)
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J. C. A. BARATA
and where δ(P , m) is the Kr¨ onecker delta: ( 1, δ(P , m) := 0,
if P = m , else .
The presence of the factor δ(P , m) remembers the fact that Qm is trivially zero if P = m has no solutions for p1 , . . . , p2J ∈ N. We say that Qm is non-trivial if P = m has solutions for some pb ’s in N. (2) Since the Fourier coefficients Qm of q 2 can be obtained from those of q by replacing f → 2f we have i2γf Q(2) m = e
∞ X
δ(P (2) , m)
p1 ,...,p2J =0
pa 2J Y 2fa 1 . pa ! na · ω f a=1
(C.11)
˜ ˜
Appendix D. The Exponential Decay of Qm After the remarks of Appendix C we are now prepared for the proof of Ineq. (5.18). Of course, we have to concentrate on the non-trivial Qm ’s. In this case the vectors m are either of the form m = (m, 1) ∈ ZB+1 for f0 6= 0 or of the ˜ form m = m ∈ ZB for f0 = 0, where m ∈ ZB satisfies ˜ 2J X pb nb (D.1) m=
˜
˜
˜
˜
b=1
˜
for some p1 , . . . , p2J ∈ N. Let us denote by mk (respectively, by nkb ) the kth component of m (respectively, of nb ) with 1 ≤ k ≤ B. The condition (D.1) says that
˜
˜
mk =
2J X
pb nkb
b=1
with 1 ≤ k ≤ B. Hence, |mk | ≤
max pb
X 2J
1≤b≤2J
k nb
b=1
and, after summing over all k, we get max pb ≥ N −1 |m| ,
˜
1≤b≤2J
where |m| = kmk1 is the l1 norm of and where
˜
˜
N :=
2J X b=1
|nb | .
˜
Note that, since the N is non-zero and will not be changed.
(D.2)
¨ ON FORMAL QUASI-PERIODIC SOLUTIONS OF THE SCHRODINGER EQUATION
63
Let us now return to (C.9). It says that for the non-trivial Qm ’s we have ∞ X
|Qm | ≤
p1 ,...,p2J =0 max1≤b≤2J pb ≥N −1 |m|
ϕp1 +···+p2J p1 ! · · · p2J !
˜
∞ X
≤ 2J
∞ X
p1 ,...,p2J−1 =0 p2J ≥N −1 |m|
= 2Je(2J−1)ϕ
∞ X n≥dN −1 |m|e
ϕp1 +···+p2J p1 ! · · · p2J !
˜ ϕn , n!
(D.3)
˜
fa . ϕ := max 1≤a≤2J na · ω f
where
˜ ˜
We conclude that
|Qm | ≤ 2Je
(2J−1)ϕ
−1 ϕdN −1 |m|e ϕ ˜ 1− , dN −1 |m|e! dN −1 |m|e + 1
˜
(D.4)
˜
for dN −1 |m|e + 1 > ϕ. Above dxe is the lowest integer larger or equal to x. From this, and since |m| ≤ |m| ≤ 1 + |m| for the sort of vectors m ∈ ZA in consideration, Ineq. (5.18) follows immediately, for convenient choices of Q and χ > 0. Note that (D.4) implies an actually even stronger decay in |m| for Qm than that given in (5.18). Unfortunately, however, this does not seem to help to improve our estimates in a way to change the main results of this paper, especially those concerning the large n behavior of the constants Kn and Ln , important for the proof of quasi-periodicity of the solution g.
˜
˜
˜
References [1] H. Jauslin, “Stability and chaos in classical and quantum Hamiltonian systems” in II Granada Seminar on Computational Physics, eds. P. Garrido and J. Marro, World Scientific, Singapore, 1993. [2] W. F. Wreszinski, “Atoms and oscillators in quasi-periodic external fields”, Helv. Phys. Acta 70 (1997) 109–123. [3] H. Bohr, “Zur Theorie der fastperiodischen Funktionen, I”, Acta Math. 45 (1924) 29–127; “Zur Theorie der fastperiodischen Funktionen, II”, Acta Math. 46 (1925) 101–214; “Zur Theorie der fastperiodischen Funktionen, III”, Acta Math. 47 (1926) 237–281. ¨ [4] P. Bohl, “Uber eine Differentialgleichung der St¨ orungstheorie”, J. de Crelle 131 (1906) 268–321. [5] W. F. Wreszinski and S. Casmeridis, “Models of two level atoms in quasiperiodic external fields”, J. Stat. Phys. 90 (1998) 1061–1068. [6] H. Jauslin and J. L. Lebowitz, “Spectral and stability aspects of quantum chaos”, Chaos 1 (1991) 114. [7] J. Feldman and E. Trubowitz, “Renormalization in classical mechanics and many body quantum field theory”, J. D’Analyse Math. 58 (1992) 213.
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[8] L. H. Eliasson, “Absolutely convergent series expansions for quasi-periodic motions”, Math. Phys. Electronic J. 2(4) (1996) (see the URL http://www.ma.utexas.edu/ -mpej/MPEJ.html). [9] L. H. Eliasson, “Floquet solutions for the 1-dimensional quasi-periodic Schr¨ odinger equation”, Commun. Math. Phys. 146 (1992) 447–482. [10] G. Gallavotti, “Twistless KAM Tori”, Commun. Math. Phys. 164 (1994) 145–156. [11] G. Gentile and V. Mastropiero, “Methods for the analysis of the Lindstedt series for KAM Tori and renormalizability in classical mechanics. A review with some applications”, Rev. Math. Phys. 8 (1996) 393-444. [12] G. Benfatto, G. Gentile and V. Mastropietro, “Electrons in a lattice with an incomensurate potential”, J. Stat. Phys. 89 (1997) 655–708. [13] J. W. S. Cassels, “Some metrical theorems in Diophantine approximation. I”, Proc. Camb. Phil. Soc. 46 (1950) 209. [14] Yitzhak Katznelson, “An Introduction to Harmonic Analysis”, Dover Publ., 1978. [15] R. L. Graham, D. E. Knuth and Oren Patashnik, “Concrete Mathematics — A Foundation for Computer Science”, Addison-Wesley Publ. Co., 1994. ¨ [16] H. Bohr, “Uber fastperiodische ebene Bewegungen”, Comment. Math. Helv. 4 (1932) 51–64.
BUNDLE GERBES APPLIED TO QUANTUM FIELD THEORY ALAN L. CAREY Department of Mathematics University of Adelaide Adelaide SA 5005 Australia E-mail:
[email protected]
JOUKO MICKELSSON Department of Theoretical Physics Royal Institute of Technology S-10044 Stockholm Sweden E-mail:
[email protected]
MICHAEL K. MURRAY Department of Mathematics University of Adelaide Adelaide SA 5005 Australia E-mail:
[email protected] Received 19 November 1997 Revised 4 September 1998 This paper reviews recent work on a new geometric object called a bundle gerbe and discusses some new examples arising in quantum field theory. One application is to an Atiyah–Patodi– Singer index theory construction of the bundle of fermionic Fock spaces parameterized by vector potentials in odd space dimensions and a proof that this leads in a simple manner to the known Schwinger terms (Mickelsson–Faddeev cocycle) for the gauge group action. This gives an explicit computation of the Dixmier–Douady class of the associated bundle gerbe. The method also works in other cases of fermions in external fields (external gravitational field, for example) provided that the APS theorem can be applied; however, we have worked out the details only in the case of vector potentials. Another example, in which the bundle gerbe curvature plays a role, arises from the WZW model on Riemann surfaces. A further example is the “existence of string structures” question. We conclude by showing how global Hamiltonian anomalies fit within this framework.
Contents 1. Introduction 2. Bundle Gerbes 2.1. The definition and basic operations 2.2. The Diximer–Douady class and stable isomorphism 2.3 Local bundle gerbes 2.4. Bundle gerbe connections, curving and curvature 2.5. The lifting bundle gerbe 3. The Wess–Zumino–Witten Term 4. The Mickelsson–Faddeev Cocycle 65 Reviews in Mathematical Physics, Vol. 12, No. 1 (2000) 65–90 c World Scientific Publishing Company
66 68 68 69 70 70 71 72 75
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A. L. CAREY, J. MICKELSSON & M. K. MURRAY
5. Ures Bundles and String Structures 6. Global Anomalies 6.1. Bundle gerbes with other structure group 6.2. The framework for the examples 6.3. The case G = SU (2) 6.4. A general analysis 6.5. Algebraic considerations References
78 82 82 83 84 86 88 89
1. Introduction In [4] J.-L. Brylinski describes Giraud’s theory of gerbes and gives some applications particularly to geometric quantization. Loosely speaking a gerbe over a manifold M is a sheaf of groupoids over M . Gerbes, via their Dixmier–Douady class, provide a geometric realisation of the elements of H 3 (M, Z) analogous to the way that line bundles provide, via their Chern class, a geometric realisation of the elements of H 2 (M, Z). There is a simpler way of achieving this end which, somewhat surprisingly, is nicely adapted to the kind of geometry arising in quantum field theory applications. For want of a better name these objects are called bundle gerbes and are introduced in [24]. All this talk of sheaves and groupoids sounds very abstract. In this article we want to illustrate the importance and usefulness of bundle gerbes by describing five natural examples arising in different parts of quantum field theory. These are: • • • • •
string structures, Ures bundles, the Wess–Zumino–Witten action, local Hamiltonian anomalies (the Mickelsson–Faddeev cocycle), global Hamiltonian anomalies.
The value of the bundle gerbe picture can be seen from the fourth and fifth examples: they provide a geometric meaning to these anomalies which previously have been thought of as associated with cocycles on the gauge group. Some of the other examples are also considered in [4]. Just as a gerbe is a sheaf of groupoids a bundle gerbe is essentially a bundle of groupoids. A bundle gerbe has associated to it a three-class also called the Dixmier– Douady class. Every bundle gerbe gives rise to a gerbe with the same Dixmier– Douady class. Bundle gerbes behave in many ways like line bundles. There is a notion of a trivial bundle gerbe and a bundle gerbe is trivial if and only if its Dixmier–Douady class vanishes. One can form the dual and tensor products of bundle gerbes and the Dixmier–Douady class changes sign on the dual and is additive for tensor products. Every bundle gerbe admits a bundle gerbe connection which can be used to define a three form on M called the curvature of the connection and which is a de Rham representative for 2πi times the Dixmier–Douady class. A difference with the line bundle case is that one needs to choose not just the connection but an intermediate two-form called the curving to define the curvature.
BUNDLE GERBES APPLIED TO QUANTUM FIELD THEORY
67
There is a notion of holonomy for a connection and curving but now it is associated to a two-surface rather than a loop. We exploit this in our description of the Wess– Zumino–Witten action. There is a local description of bundle gerbes in terms of ˇ transition functions and a corresponding Cech definition of the Dixmier–Douady class. Finally bundle gerbes can be pulled back and there is a universal bundle gerbe and an associated classifying theory. The one significant difference between the two structures; lines bundles and bundle gerbes; is that two line bundles are isomorphic if and only if their Chern classes are equal, whereas two bundle gerbes which are isomorphic have the same Dixmier–Douady class but the converse is not necessarily true. For bundle gerbes there is a weaker notion of isomorphism called stable isomorphism and two bundle gerbes are stably isomorphic if and only if they have the same Dixmier–Douady class [25]. The reader with some knowledge of groupoid or category theory will recall that often the right concept of equal for categories is that of equivalence which is weaker than isomorphism. A similar situation arises for bundle gerbes essentially because they are bundles of groupoids. A common thread in the examples we consider is the relationship between bundle gerbes and central extensions. Because group actions in quantum field theory are only projectively defined one often needs to consider the so-called “lifting problem” for principal G bundles where G is the quotient in a central extension: ˆ → G. U (1) → G The lifting problem starts with a principal G bundle and seeks to find a lift of ˆ bundle. The obstruction to such a lift is well known to be a this to a principal G 3 class in H (M, Z). The connection with bundle gerbes arises because there is a socalled lifting bundle gerbe, which is trivial if and only if the principal G bundle lifts and its Dixmier–Douady class is the three class obstructing the lifting. In the first and third examples G is the loop group with its standard central extension (the Kac–Moody group) and in the second example G is the restricted unitary group Ures with its canonical central extension. It is important to note, and the central point of this paper, that the bundle gerbe description arises naturally and usefully in these examples and is not just a fancier way of describing the lifting problem for principal bundles. In summary form we present the basic theory of bundle gerbes in Sec 2. This is followed by the examples: the gerbes arising from global Hamiltonian anomalies are described in Sec. 6, the lifting problem for the restricted unitary bundles and string structures is in Sec. 5, the Mickelsson–Faddeev cocycle (local Hamiltonian anomalies) is in Sec. 4 and the geometric interpretation provided by bundle gerbes of the Wess–Zumino–Witten term is in Sec. 3. Section 4 is a summary of [6] and also forms part of a previous short review [7]. We include it because it is essential for the understanding of the later sections. The material in Secs. 3, 5 and 6 is new. Section 3 may be skipped on first reading (it is a bit technical). Section 5 depends a little on Secs. 4 and 6 on parts of both Secs. 4 and 5.
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We conclude this introduction by remarking that there are “bundle n-gerbes” associated with classes in H n+2 (M, Z). Examples are given in [11] and the general theory in [29] however it would take us too far afield to describe them here. 2. Bundle Gerbes This section is a review: we describe the basic theory of bundle gerbes. We will not prove any of the results but refer the reader to [6, 8, 24, 25] and the forthcoming thesis of Stevenson [29]. 2.1. The definition and basic operations Consider a submersion π:Y →M and define Ym = π −1 (m) to be the fibre of Y over m ∈ M . Recall that the fibre product Y [2] is a new submersion over M whose fibre at m is Ym × Ym . A bundle gerbe (P, Y ) over M is defined to be a choice of a submersion Y → M and a U (1) bundle P → Y [2] with a product, that is, a U (1) bundle isomorphism P(y1 ,y2 ) ⊗ P(y2 ,y3 ) → P(y1 ,y3 ) . The product is required to be associative whenever triple products are defined. Example 2.1. Let Q → Y be a principal U (1) bundle. Define P(x,y) = AutU (1) (Qx , Qy ) = Q∗x ⊗ Qy . A bundle gerbe of this form is called trivial. A morphism of bundle gerbes (P, Y ) over M and (Q, X) over N is a triple of maps (α, β, γ). The map β : Y → X is required to be a morphism of the submersions Y → M and X → N covering γ : M → N . It therefore induces a morphism β [2] of the submersions Y [2] → M and X [2] → N . The map α is required to be a morphism of U (1) bundles covering β [2] which commutes with the product. A morphism of bundle gerbes over M is a morphism of bundle gerbes for which M = N and γ is the identity on M . Various constructions are possible with bundle gerbes. We can define a pullback and product as follows. If (Q, X) is a bundle gerbe over N and f : M → N is a map then we can pull back the submersion X → N to obtain a submersion f ∗ (X) → M and a morphism of submersions f ∗ : f ∗ (X) → X covering f . This induces a morphism (f ∗ (X))[2] → X [2] and hence we can use this to pull back the U (1) bundle Q to a U (1) bundle f ∗ (Q) say on f ∗ (X). It is easy to check that (f ∗ (Q), f ∗ (X)) is a bundle gerbe, the pull-back by f of the gerbe (Q, X). If (P, Y ) and (Q, X) are bundle gerbes over M then we can form the fibre product Y ×M X → M and then form a U (1) bundle P ⊗ Q over (Y ×M X)[2] which we call the product of the bundle gerbes (P, Y ) and (Q, X). Notice that for any m ∈ M we can define a groupoid as follows. The objects of the groupoid are the points in Ym = π −1 (m). The morphisms between two objects x, y ∈ Pm are the elements of P(x,y) . The bundle gerbe product defines the groupoid
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product. The existence of identity and inverse morphisms is shown in [24]. Hence we can think of the bundle gerbe as a family of groupoids, parameterised by M . 2.2. The Dixmier Douady class and stable isomorphism Let {Uα } be an open cover of M such that over each Uα we can find sections sα : Uα → Y . Then over intersections Uα ∩ Uβ we can define a map (sα , sβ ) : Uα ∩ Uβ → Y [2] which sends m to (sα (m), sβ (m)). If we choose a sufficiently nice cover we can then find maps σαβ : Uα ∩ Uβ → P such that σαβ (m) ∈ P(sα (m),sβ (m)) . The σαβ are sections of the pull-back of P by (sα , sβ ). By using the bundle gerbe multiplication (written here as juxtaposition) we have that σαβ (m)σβγ (m) ∈ P(sα (m),sγ (m)) and hence can be compared to σαγ (m). The difference is an element of U (1) defined by σαβ (m)σβγ (m) = σαγ (m)gαβγ (m) and this defines a map gαβγ : Uα ∩ Uβ ∩ Uγ → U (1). It is straightforward to check that this is a cocycle and defines an element of H 2 (M, U (1)) independent of all the choices we have made. Here, if G is a Lie group we use the notation G for the sheaf of smooth maps into G. It is a standard result that the coboundary map δ : H 2 (M, U (1)) → H 3 (M, Z)
(2.1)
induced by the short exact sequence of sheaves 0 → Z → C → U (1) → 0 is an isomorphism. Either the class defined by gαβγ or its image under the coboundary map is called the Dixmier–Douady class of the bundle gerbe. We denote it by d(Q, Y ). The first important fact about the Dixmier–Douady class is: Proposition 2.1 [24]. A bundle gerbe is trivial if and only its Dixmier–Douady class is zero. Let (P, Y ) and (Q, X) be bundle gerbes over M and Z → M be a map admitting local sections with f : Z → Y a map commuting with projections to M . Then from [24] we have Theorem 2.1 [24]. If P and Q are bundle gerbes over M then
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(1) d(P ∗ , Y ) = −d(P, Y ), (2) d(P ⊗ Q, Y ×M X) = d(P, Y ) + d(Q, X), and (3) d(f ∗ (P ), X) = f ∗ (d(P, Y )). Because pulling back the submersion does not change the Dixmier–Douady class of a bundle gerbe it is clear there are many non-isomorphic bundle gerbes with the same Dixmier–Douady class. We define: Definition 2.1 [25]. Two bundle gerbes (P, Y ) and (Q, Z) are defined to be stably isomorphic if there are trivial bundle gerbes T1 and T2 such that P ⊗ T1 = Q ⊗ T2 . We have the following theorem: Theorem 2.2 [25]. For bundle gerbes (P, Y ) and (Q, Z) the following are equivalent : (1) P and Q are stably isomorphic, (2) P ⊗ Q∗ is trivial, (3) d(P ) = d(Q). 2.3. Local bundle gerbes The notion of stable isomorphism is useful in understanding the role of open covers in bundle gerbes. Let (P, Y ) be a bundle gerbe and assume we have an open cover and various maps sα , σαβ and gαβγ as defined in Subsec. 2.2. Let X be the disjoint union of all the open sets Uα . This can be thought of as all pairs (α, m) where m ∈ Uα . There is a projection X → M defined by (α, m) 7→ m which admits local sections. Moreover there is a map s : X → Y preserving projections defined by s(α, m) = sα (m). The pullback by s of the bundle gerbe (P, Y ) is stably isomorphic to (P, Y ) [25]. This pull-back consists of a collection of U (1) bundles Qαβ → Uα ∩ Uβ . On triple overlaps there is a bundle map Qαβ ⊗ Qβγ → Qαγ which on quadruple overlaps is associative in the appropriate sense. A completely local description of bundle gerbes can be given in terms of open covers, U (1) bundles on double overlaps and products on triple overlaps [29]. The results on stable isomorphism tell us that this is equivalent to working with bundle gerbes. 2.4. Bundle gerbe connections, curving and curvature Consider the space Ωq (Y [p] ) of all q forms on the iterated fibre product. Let πi : Y [p] → Y [p−1] be the projection map that omits the ith element for each i = 1, . . . , p. Then define δ : Ωq (Y [p−1] ) → Ωq (Y [p] )
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by δ(ω) =
p X
71
(−1)i πi∗ (ω) .
i=1
An important result from [24] is: Proposition 2.2. The complex π∗
δ
δ
δ
Ωq (M ) → Ωq (Y ) → Ωq (Y [2] ) → Ωq (Y [3] ) → · · · is exact. Because P → Y [2] is a U (1) bundle it has connections. It is shown in [24] that it admits bundle gerbe connections that is connections commuting with the bundle gerbe product. A bundle gerbe connection ∇ has curvature F∇ satisfying δ(F∇ ) = 0 and hence from Proposition 2.2 we see that there exists a two-form f on Y , satisfying the “descent equation” F∇ = π1∗ (f ) − π2∗ (f ) . Such an f is called a curving for the connection ∇. The choice of a curving is not unique, indeed Proposition 2.2 shows that the ambiguity in the choice is precisely the addition of the pull-back of a two-form from M . Given a choice of curving we then have that δ(df ) = dδ(f ) = dF∇ = 0 so that we can find some ω, a three-form on M , such that df = π ∗ (ω). Moreover ω is closed as π ∗ (dω) = ddf = 0. In [24] it is shown that ω/2πi is a de Rham representative for the Dixmier–Douady class. We call ω the three curvature of the connection and curving. A bundle gerbe connection and curving are called flat if their three curvature is zero, that is, if df = 0. Notice that it is not obvious that ω is an integral class, this follows, as for line bundles, from the formula (3.3) relating the integral of the three curvature to the holonomy of the connection and curving. 2.5. The lifting bundle gerbe Finally let us conclude this section with the motivating example of the so-called lifting bundle gerbe. That is the bundle gerbe arising from the lifting problem for principal bundles. Consider a central extension of groups: ι ˆ p →G→1 1 → U (1) → G
(2.2)
ˆ and a principal G bundle Y → M . Then it may happen that there is a principal G ˆ ˆ ˆ bundle Y and a bundle map Y → Y commuting with the homomorphism G → G. ˆ bundle. One way of answering the question In such a case Y is said to lift to a G ˆ bundle is to present Y with transition function gαβ relative of when Y lifts to a G to a cover {Uα } of M . If the cover is sufficiently nice we can lift the gαβ to maps ˆ and such that p(ˆ gαβ ) = gαβ . These are candidate transition gˆαβ taking values in G functions for a lifted bundle Yˆ . However they may not satisfy the cocycle condition gˆαβ gˆβγ gˆγα = 1 and indeed there is a U (1) valued map eαβγ : Uα ∩ Uβ ∩ Uγ → U (1)
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defined by ι(eαβγ ) = gˆαβ gˆβγ gˆγα . Because (2.2) is a central extension it follows that eαβγ is a cocycle and hence defines a class in H 2 (M, U(1)). Using the isomorphism (2.1) we define a class in H 3 (M, Z) which is the obstruction to solving the lifting problem for Y . There is a map g : Y [2] → G defined by y1 = y2 g(y1 , y2 ) and we can use this ˆ → G to define a U (1) bundle Q → Y [2] . More to pull back the U (1) bundle G concretely we have ˆ 1 p(g) = y2 } . Q(y1 ,y2 ) = {g ∈ G|y ˆ induces a bundle gerbe product on Q. It is shown in [24] The group product on G ˆ and moreover that the bundle gerbe Q is trivial if and only if the bundle P lifts to G the Dixmier–Douady class of (Q, Y ) is the class defined in the preceding paragraph. 3. The Wess Zumino Witten Term In quantum field theory the path integral can have contributions that are topological in nature. Often these arise as the holonomy of a connection. For example if L → M is a Hermitian line bundle we can consider the Hilbert space of all L2 sections of L as a space of states. A connection ∇ on L defines an operator on states by Z Z Pγ (∇)(ψ(y)) dy dγ , K∇ (ψ)(x) = M
γ∈Wx,y
where Wx,y is the set of all paths from x to y and Pγ (∇) : Ly → Lx is the operation of parallel transport along the curve γ from the fibre of L over y to the fibre of L over γ. If x = y then Pγ (∇) : Ly → Lx is an element of U (1) called hol(γ, ∇), the holonomy of the connection ∇ around the loop γ. Assuming that M is simply connected every loop γ bounds a disk D and we have the fact that Z (3.1) hol(γ, ∇) = exp F∇ , D
where F∇ , is the curvature two-form of ∇. The Wess–Zumino–Witten term is defined as follows. The space of states is replaced by the space of all maps (classical field configurations) ψ from a closed Riemann surface Σ into a compact Lie group G. Let X be a three-manifold whose boundary is Σ and ψ : Σ → G. Then ψ can be extended to a map ψˆ : X → G. Let ω be a closed three-form on G such that ω/2πi generates the integral cohomology of G. The Wess–Zumino–Witten action of ψ is then defined by Z ψˆ∗ (ω) . WZW(f ) = exp X
It follows from the integrality of ω/2πi that this is independent of the choice of ˆ Clearly this is analogous to defining the holonomy by using the rightextension ψ. hand side of Eq. (3.1) as if one knew nothing of connections, only that F was a two-form such that F/2πi was integral.
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It is natural to look for the analogous left-hand side of Eq. (3.1) in the case of the Wess–Zumino–Witten term. In [10] an interpretation of the Wess–Zumino– Witten term in terms of holonomy of a connection on a line bundle over the loop group of G is given. This essentially only worked for simple Riemann surfaces such as spheres or cylinders. In [16] Gawedski gave a construction that works for any Riemann surface. Gawedski starts by showing that isomorphism classes of line bundles with connection are classified by certain Deligne cohomology groups ˇ which can be realised in terms of Cech cohomology of an open cover of M . It is then shown that if M is a loop then this cohomology group is U (1) and the identification of isomorphism classes with elements of U (1) is just the holonomy of the connection around the loop. This is then generalised to the Wess–Zumino–Witten case. For such cohomology classes Gawedski shows that it is possible to define a holonomy associated to a closed Riemann surface in M . Bundle gerbes give a geometric interpretation of Gawedski’s results. The Deligne cohomology class in question defines a stable isomorphism class of a bundle gerbe with connection and curving over M and the element of U (1) is the holonomy of the connection and curving over the surface Σ. In the case that Y → M admits local sections a definition of the holonomy in terms of lifting Σ to Y is given in [24]. In the present work we are interested in more general Y , in particular a Y arising from an open cover, so we give an alternative definition of holonomy of a bundle gerbe connection and curving. To define the holonomy we need the notion of a stable isomorphism class of a bundle gerbe with connection and curving. To define this let P → Y be a line bundle with connection ∇ and curvature F . The trivial gerbe δ(P ) → Y [2] has a natural bundle gerbe connection π1∗ (∇) = π2∗ (∇) and curving π1∗ (F ) = π2∗ (F ). To extend the definition of stable isomorphism (Definition 2.1) to cover the case of bundle gerbes with connection and curving we assume that the trivial bundle gerbes T1 and T2 , in the definition, have connections arising in this manner and that the isomorphism in Definition 2.1 preserves connections. Then stable isomorphism classes of bundle gerbes with connection and curving are classified by the two dimensional real Deligne cohomology or the hyper-cohomology of the log complex of sheaves D
0 → U (1) → Ω1 → Ω2 → 0 , where D = d log/2πi and Ωp is the sheaf of p-forms on M . The hypercohomology of this sequence of sheaves can be calculated by taking a Leray cover U = {Uα } of M and considering the double complex: δ↑ C 2 (U, U (1))
δ↑ δ↑ D d → C 2 (U, Ω1 ) → C 2 (U, Ω2 )
δ↑
δ↑
→
0
δ↑
D
d
d
C 1 (U, U (1)) δ↑
→ C 1 (U, Ω1 ) → C 1 (U, Ω2 ) δ↑ δ↑
→
0
C 0 (U, U (1))
→ C 0 (U, Ω1 ) → C 0 (U, Ω2 )
D
→
0.
(3.2)
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The real Deligne cohomology is the cohomology of the double complex (3.2). This can be calculated by either of two spectral sequences which begin by taking the cohomology along the rows or columns respectively. If we assume that M is a surface Σ and take the cohomology of the columns we obtain the E 2 term of the spectral sequence: 0 0 0 H 1 (Σ, Ω× ) 0
×
H (Σ, Ω )
0
0
H (Σ, C) H (Σ, C) 1
2
where we use the fact that the sheaves Ωi have no cohomology and that H 2 (Σ, U (1)) = H 3 (Σ, Z) = 0 . The third cohomology is therefore the quotient of the image of the inclusion Z = H 2 (Σ, Z) = H 1 (Σ, Ω× ) → H 2 (Σ, C) = C induced by the second differential. It is straightforward to check that this map is just the natural inclusion Z → C and hence the quotient is just U (1). We call the resulting non-vanishing number attached to each connection and curving the holonomy, hol(Σ, ∇, f ), where ∇ is the bundle gerbe connection and f is the curving. If ψ : Σ → M is a map then we can pull-back the bundle gerbe, connection and curving to Σ and we define hol(ψ, ∇, f ) to be the holonomy of the pulled-back connection and curving over Σ. To calculate the holonomy we need to explain how to unravel these definitions. Let us begin with a bundle gerbe (Q, Y ) and choose a Leray cover U = {Uα } with sections sα : Uα → Y . Choose sections σαβ : Uα ∩ Uβ → Q as we did in Subsec. 2.2 and define gαβγ by σαβ σβγ = σαγ gαβγ . Let Aαβ be the pullback of the connection one form on Q by σαβ and fα the pullback of the curving by sα . These satisfy d log gαβγ = Aαβ − Aβγ + Aγα dAαβ = fα − fβ and hence (gαβγ , Aαβ /2πi, fα /2πi) is an element of the total cohomology of the complex (3.2). Because Σ is two dimensional the cocycle gαβγ is trivial and we can solve the equation gαβγ = hβγ h−1 αγ hαβ , where hαβ : Uα ∩ Uβ → U (1) and hence d log hαβ = Aαβ + kα − kβ , where the kα are one-forms on Uα . Hence the two-form fα − dkα on Uα agrees with the two-form fβ − dkβ on Uβ on the overlap Uα ∩ Uβ and hence defines a two-form
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on Σ. The exponential of the integral of this two-form over Σ is the holonomy of the connection and curving. It is straightforward to check that if we can extend the map of Σ into M to a map of a manifold X whose boundary is Σ then we obtain the analogue of (3.1) Z ∗ ψ (ω) , (3.3) hol(∇, f, ψ) = exp X
where ω is the curvature of the connection and curving (a 3-form). In the case that Σ has boundary one expects a result analogous to parallel transport along a curve γ. Gawedski shows that there is a naturally defined line bundle L over the space LM of loops in M . The boundary components b1 , . . . , br of Σ define points in LM and Gawedski shows that the holonomy can be interpreted as an element of Lb1 ⊗ Lb2 ⊗ · · · ⊗ Lbr for more details see [16]. 4. The Mickelsson Faddeev Cocycle Let M be a smooth compact connected manifold without boundary equipped with a spin structure. We assume that the dimension of M is odd and equal to 2n + 1. Let n S be the spin bundle over M , with fiber isomorphic to C2 . Let H be the space of square integrable sections of the complex vector bundle S ⊗ V , where V is a trivial vector bundle over M with fiber to be denoted by the same symbol V . The measure is defined by a fixed metric on M and V . We assume that a unitary representation ρ of a compact group G is given in the fiber. The set of smooth vector potentials on M with values in the Lie algebra g of G is denoted by A. The topology on A arises from an infinite family of Sobolev norms which via the Sobolev embedding theorem give a metric equivalent to that arising from the topology of uniform convergence of derivatives of all orders. For each A ∈ A there is a massless Hermitian Dirac operator DA . Fix a potential A0 such that DA does not have zero as an eigenvalue and let H+ be the closed subspace spanned by eigenvectors belonging to positive eigenvalues of DA0 and H− its orthogonal complement (with corresponding spectral projections P± ). More generally for any potential A and any real λ not belonging to the spectrum of DA we define the spectral decomposition H = H+ (A, λ) ⊕ H− (A, λ) with respect to the operator DA − λ. Let A0 denote the set of all pairs (A, λ) as above and let Uλ = {A ∈ A|(A, λ) ∈ A0 } . Over the set Uλλ0 = Uλ ∩ Uλ0 there is a canonical complex line bundle, which we denote by DETλλ0 . Its fiber at A ∈ Uλλ0 is the top exterior power DETλλ0 (A) = ∧top (H+ (A, λ) ∩ H− (A, λ0 )) , where we have assumed λ < λ0 . For completeness we put DETλλ0 = DET−1 λ0 λ . Since M is compact, the spectral subspace corresponding to the interval [λ, λ0 ] in the spectrum is finite-dimensional and the complex line above is well-defined.
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It is known [9, 20] that there exists a complex line bundle DETλ over each of the sets Uλ such that (4.1) DETλ0 = DETλ ⊗ DETλλ0 over the set Uλλ0 . In [8, 9] the structure of these line bundles was studied with the help of bundle gerbes. In particular, there is an obstruction for passing to the quotient by the group G of gauge transformations which is given by the Dixmier– Douady class of the bundle gerbe. (In [20] the structure of the bundles and their relation to anomalies was found by using certain embeddings to infinite-dimensional Grassmannians.) We shall describe the computation in [6] of the curvature of the (odd dimensional) determinant bundles from Atiyah–Patodi–Singer index theory and how to obtain the Schwinger terms in the Fock bundle directly from the local part of the index density. We may consider A0 as part of a bundle gerbe over A. The obvious map A0 → A is a submersion. For any λ ∈ R we have a section sλ : Uλ → A0 defined by sλ (A) = (A, λ). So we can apply the discussion in Subsec. 2.3 to obtain the disjoint union Y =
a
Uλ ⊂ A × R
as the set of all (A, λ) such that A ∈ Uλ . We topologize Y by giving R the discrete topology. Notice that as a set Y is just A0 but the topology is different. The identity map Y → A0 is continuous. It follows from the results of Subsec. 2.3 that using either topology on A0 gives rise to stably isomorphic bundle gerbes so we can work in either picture. An advantage of the open cover picture is that the map δ introduced in [24] is then just the coboundary map in the C´ech de-Rham double complex. In the next section A0 can be interpreted in either sense. If we restrict λ to be rational then the sets Uλ form a denumerable cover. It follows that the intersections Uλλ0 = Uλ ∩ Uλ0 also form a denumerable open cover. Similarly, we have an open cover by sets Vλλ0 = π(Uλλ0 ) on the quotient X = A/Ge , where Ge is the group of based gauge transformations g, g(p) = e the identity at some fixed base point p ∈ M . Here π : A → X is the canonical projection. We now describe the bundle gerbe J over A defined in [8] and extracted from the work of [28]. First recall that there is an equivalence between U (1) bundles and Hermitian line bundles, that is complex line bundles with Hermitian inner product on each fibre. In one direction the equivalence associates to any Hermitian line bundle the U (1) bundle of all vectors of unit norm. It is possible to cast the definition of bundle gerbes in terms of Hermitian line bundles and indeed this was done in [8]. So the bundle gerbe J is defined as a Hermitian line bundle over the [2] fibre product A0 . This fibre product can be identified with all triples (A, λ, λ0 ) where neither λ nor λ0 are in the spectrum of DA . The fibre of J over (A, λ, λ0 ) is just DETλλ0 . For this to be a bundle gerbe we need a product which in this case is a linear isomorphism DETλλ0 ⊗ DETλ0 λ00 = DETλλ00 .
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But such a linear isomorphism is a simple consequence of the definition of DETλλ0 and the fact that taking top exterior powers is multiplicative for direct sums. Let π : A0 → A be the projection and p : A → A/Ge be the quotient by the gauge action. It was shown in [9] that the line bundle DET on A0 satisfies [2] J = δ(DET). Here δ(DET) = π1∗ (DET)∗ ⊗ π2∗ (DET) where πi : A0 → A0 are the projections, π1 ((A, λ, λ0 )) = (A, λ) , π2 ((A, λ, λ0 )) = (A, λ0 ) . In other words J = δ(DET) is equivalent to DETλλ0 = DET∗λ ⊗ DETλ0 which is equivalent to Eq. (4.1). The fibering A0 → A has, over each open set Uλ a canonical section A 7→ (A, λ). These enable us to suppress the geometry of the submersion and the bundle gerbe J becomes the line bundle DETλλ0 over the intersection Uλλ0 and its triviality amounts to the fact that we have the line bundle DETλ over Uλ and over intersections we have the identifications DETλλ0 = DET∗λ ⊗ DETλ0 . [2]
We denote the Chern class of DETλλ0 by θ2 . Note that these bundles descend 0 0 to bundles over Vλλ0 = π(Uλλ0 ) ⊂ A/Ge . Therefore, the forms θ2λλ = θ2λ − θ2λ on Uλλ0 (where θ2λ is the 2-form giving the curvature of DETλ ) are equivalent (in 0 on Vλλ0 . The following cohomology) to forms which descend to closed 2-forms φλλ 2 result is established in [6]: 0
on Vλλ0 determines a repTheorem 4.1 [6]. The family of closed 2-forms φλλ 2 resentative for the Dixmier–Douady class ω of the bundle gerbe J/Ge . In addition, noting that δ(DET) = J, the connection with the Faddeev–Mickelsson cocycle on the Lie algebra of the gauge group is simply that it is cohomologous to the negative of the cocycle defined by the curvature FDET of the line bundle DET along gauge orbits. To obtain the Dixmier–Douady class as a characteristic class we recall that in the case of even dimensional manifolds, Atiyah and Singer [1] gave a construction of “anomalies” in terms of characteristic classes. In the present case of odd dimensional manifolds a similar procedure yields the Dixmier–Douady class. We begin with the observation that given a closed integral form Ω of degree p on a product manifold M × X (dim M = d = 2n + 1 and dim X = k) we obtain a closed integral form on X, of degree p − d, as Z Ω. ΩX = M
If now A is any Lie algebra valued connection on the product M × X and F is the corresponding curvature we can construct the Chern form c2n = c2n (F ) as a polynomial in F . Apply this to the connection A defined by Atiyah and Singer, [1, 13, p. 196], in the case when X = A/Ge .
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First pull back the forms to M × A. The Atiyah–Singer connection on M × X becomes a globally defined Lie algebra valued 1-form Aˆ on M × A. Let Fˆ be the ˆ We showed in [6] that curvature form determined by A. Z Z ΩX = c2n (Fˆ ) , S3
M×D3
where the disk D3 is the pullback to A of S 3 ⊂ A/Ge . But the integral of the Chern form over a manifold with a boundary (when the potential is globally defined) is equal to the integral of the Chern–Simons action: Z ˆ . CS2n−1 (A) M×∂D3
Along gauge directions the form Aˆ is particularly simple so for example when M = S 1 and 2n = 4 we get (here S 2 = ∂D3 ) Z Z Z 1 ˆ ΩX = CS3 (A) = tr(dgg −1 )3 , 24π 2 S 1 ×S 2 S3 where g = g(x, z), z ∈ S 2 , is a family of gauge transformations relating the vector potentials on the boundary S 2 = ∂D3 . Similar results hold in higher dimensions. Theorem 4.2 [6]. The class ΩX is a representative for the Dixmier–Douady class of the bundle gerbe J/Ge . 5. Ures Bundles and String Structures Let H = H+ ⊕ H− be a polarisation of a Hilbert space H into a pair of closed infinite-dimensional subspaces. We denote by Ures the restricted unitary group consisting of unitary operators in H such that the off-diagonal blocks are Hilbert– Schmidt operators. In a recent article [5] we described in some detail results about the Dixmier–Douady class arising from the problem of lifting principal Ures bundles to principal Uˆres bundles. Here Uˆres is a central extension of Ures , [27]. We now summarise the results in [5]. Theorem 5.1. There is an imbedding of the smooth loop group LG of a compact Lie group G in Ures which extends to give an imbedding of the canonical central d in Uˆres . Under this imbedding the obstruction to the existence of a extension LG d principal bundle) on the string structure (in the sense of Killingback [17]: a LG loop space of a manifold M may be identified with the Dixmier–Douady class of the lifting bundle gerbe of the corresponding principal Ures bundle. A different approach to the question of the existence of string structures is due to [18] and exploits Brylinski’s point of view whereas in [8] the problem is solved using the classifying map of the bundle over the loop space of M . There is also an imbedding of Ures in the projective unitary group of the skew symmetric Fock space (determined by the polarisation H = H+ ⊕ H− , the “Dirac
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sea” construction), [27]. Under conditions on the underlying manifold M this imbedding enables us to establish a relationship between the Dixmier–Douady class of a bundle gerbe over M determined by a Ures bundle and the second Chern class of an associated principal projective unitary group bundle over the suspension of M . In this section we describe the field theory examples which motivated the proving of the previous results. Let Gr be the space of all closed infinite-dimensional subspaces of H with the topology determined by operator norm topology for the associated projections. We may think of Gr as the homogeneous space U (H) . U (H+ ) × U (H− ) Here all the groups are contractible (in the operator norm topology) and therefore there is a continuous section Gr → U (H), that is, for W ∈ Gr we may choose a gW ∈ U (H) which depends continuously on W , such that W = gW · H+ . The example we shall study below comes from a quantization of a family of Dirac operators DA parameterized by smooth (static) vector potentials A. In the following we shall use the notations in Sec. 4. Choose a real number λ such that D = DA − λ is invertible. The set of bounded operators X such that kXk < k1/Dk−1 is an open set V containing 0 and the function X 7→ |D + X|−1 (D0 + X) is continuous in the operator norm of X; this is seen using the converging geometric series (D + X)−1 = D−1 − D−1 XD−1 + · · · . Since the operator norm of the interaction A depends continuously on the components Ai of the vector potential (with respect to the infinite family of Sobolev norms on A) we can conclude that A 7→ A,λ = (DA − λ)|DA − λ|−1 is continuous. Thus also the spectral projections P± (A, λ) = 12 (1 ± A,λ ) are continuous and H+ (A, λ) = P+ (A, λ)H ∈ Gr depends continuously on A ∈ Uλ . On the other hand, we know that there is a section Gr → U (H) and therefore we may choose a continuous function A 7→ gλ (A) ∈ U (H) such that H+ (A, λ) = gλ (A) · H+ . We shall show that these define transition functions, gλλ0 (A) = gλ (A)−1 gλ0 (A), for a principal Ures bundle P over A. By construction, these satisfy the cocycle property required for transition functions so the only thing which remains is to prove continuity with respect to the topology of Ures . The topology of Ures is defined by the operator norm topology on the diagonal blocks (with respect to the energy polarisation H+ ⊕ H− fixed by a “free” Dirac operator DA0 without zero modes) and by Hilbert–Schmidt norm topology on the off-diagonal blocks. As before, P± = P± (A0 , 0) and we set = P+ −P− . We already know that the gλλ0 ’s (assume e.g. that λ < λ0 ) are continuous with respect to the operator norm topology and we need only show that the off-diagonal blocks [, gλλ0 ] are continuous in the Hilbert–Schmidt topology. Let us concentrate on the upper right block K+− = P+ gλλ0 P− . and using the fact Multiplying from the left by gλ and from the right by gλ−1 0 that Hilbert–Schmidt operators form an operator ideal with kgKk2 ≤ kgk · kKk2 we conclude that K+− is continuous in the Hilbert–Schmidt norm if and only if gλ P+ gλ−1 gλ0 P− gλ−1 0
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is a continuous function of A in the Hilbert–Schmidt norm. Now the product of the first three factors in the above expression gives P+ (A, λ) whereas the product of the last three factors is P− (A, λ0 ). But P+ (A, λ)P− (A, λ0 ) is the spectral projection P (λ, λ0 ) to the finite-dimensional spectral subspace corresponding to the interval [λ, λ0 ]. On the other hand, the dimension of this subspace is fixed over Uλλ0 and therefore the Hilbert–Schmidt norm of the projection, which is the square root of its rank, is continuous. Furthermore, since P (λ, λ0 ) is continuous in the operator norm and it has a fixed finite rank it is also continuous in the Hilbert–Schmidt norm. We denote by Grres the restricted Grassmannian, defined as the orbit Ures · H+ in Gr. The fiber PA at A ∈ A can be thought of as the set of all unitary operators T : H → H such that T −1 (H+ (A, λ)) (for any λ) is in Grres . This is because gλ (A) provides such an operator for any A ∈ Uλ and any two such operators differ only by a right multiplication by an element of Ures . Being a principal bundle over a contractible parameter space, P → A is trivial. We choose a global trivialization A 7→ TA . On any Uλ the function A → P+ (A, λ) is continuous and TA−1 P+ (A, λ)TA ∈ Grres . Over Grres there is a canonical determinant bundle DETres . The action of Ures on Grres lifts to an action of Uˆres on DETres , [27]. Using the maps A → P+ (A, λ) we can pull back the determinant bundle DETres over Grres to form local determinant bundles DETλ over Uλ . This family is the right one for discussing the gerbes over A and A/Ge . The reason is that the class of the bundle gerbe is completely determined by the line bundles DETλλ0 over Uλλ0 . On the restricted Grassmannian we obtain an isomorphism between the fibers DETres (W ) and DETres (W 0 ), where W 0 ⊂ W are points in Grres ; the isomorphism ⊥ is determined by a choice of basis {v1 , . . . , vn } in W ∩W 0 as follows. Recalling from [27] that an element in DETres (W ) is represented by the so-called admissible basis {w1 , w2 , . . .}, modulo unitary rotations with determinant equal to one, the isomorphism is simply {w1 , w2 , . . . , } 7→ {v1 , . . . , vn , w1 , w2 , . . .}. In particular, we apply this when W, W 0 are the points obtained by mapping H+ (A, λ) and H+ (A, λ0 ) to Grres using TA −1 . Now the vectors TA −1 vi span a basis in the subspace corresponding to the interval [λ, λ0 ] in the spectrum of DA and thus they define an element in DETλλ0 in our earlier construction and the basis can be viewed as an isomorphism between DETλ and DETλ0 . Next we consider the trivial bundles A × Ures and A × Uˆres over A. The gauge −1 gTA This function takes group G acts in the former as follows. Define ω(g; A) = Tg·A values in Ures and is a 1-cocycle by construction, [22], ω(gg 0 ; A) = ω(g; g 0 · A)ω(g 0 ; A) . Thus the gauge group acts through g · (A, S) = (g · A, ω(g; A)S) in A × Ures . Since ω takes values in Ures the same construction which gives the lifting of the Ures action on Grres to a Uˆres action on DETres gives also an action of an extension Gˆ in A × Uˆres and in A × DETres . The pull-back with respect to the conjugation
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by TA ’s of the latter action defines an action of Gˆ on the local determinant bundles DETλ . Next we observe that the natural action (without center) vi 7→ gvi in the line DETλλ0 intertwines between the action of the group extension in the lines DETλ , DETλ0 parameterized by potentials g · A on the gauge orbit. This follows from the corresponding property of the determinant bundle over Grres (by pushing forward by TA ): An element gˆ ∈ Uˆres acts on w = {w1 , w2 , . . . , } ∈ DETres (W ) as wi 7→ Σj gwj = qji , where the basis rotation q is needed in order to recover a basis in the admissible set, [27]. The same element gˆ acts then on the basis w0 = w ∪ v extending the action on w by sending vi to gvi . The intertwining property of the natural action on DETλλ0 is exactly what was needed in the definition of the action of Gˆ in the Fock bundle over A. On the other hand, the obstruction to pushing the Fock bundle over A/Ge was precisely the class of the extension Gˆ → G. Thus we have: Theorem 5.2. The obstruction to pushing forward the trivial bundle A × Uˆres to a bundle over the quotient A/Ge , with the action of Gˆ coming from the Ures valued cocycle ω, is the Dixmier–Douady class of the Fock bundle. It is clear from the above discussion that we may view the Fock bundle over A as an associated bundle to the principal bundle A × Uˆres defined by the representation of Uˆres in the Fock space of free fermions. Example. Let us take a very concrete example for the discussion above. Let G = SU (2) and the physical space M = S 1 . Now A/Ge is simply equal to G since the gauge class of the connection in one dimension is uniquely given by the holonomy around the circle. Because topologically SU (2) is just the unit sphere S 3 any principal bundle over G is described by its transition function on the equator S 2 . In case of a Ures bundle we thus need a map φ : S 2 → Ures to fix the bundle and the equivalence class of the bundle is determined by the homotopy class of φ. The topology of Ures is known: it consists of connected components labelled by the Fredholm index of P+ gP+ , it is simply connected and so the second homotopy is given by H 2 (Ures , Z) = Z. Thus the equivalence class of a principal Ures bundle over S 3 ≡ A/Ge is given by the index of the map φ. The principal Ge bundle A → A/Ge is defined by a transition function ξ : S 2 → Ge . This is determined as follows. Since the total space is contractible, we actually have here a universal Ge bundle over S 3 . Thus the transition function ξ is the generator in π2 (Ge ). Such a map can be explicitly constructed. Any point Z on the equator S 2 ⊂ S 3 determines a unique half-circle connecting the antipodes ±1. We define gZ : S 1 → SU (2) by first following the great circle through a fixed reference point Z0 on the equator, as a smooth function of a parameter 0 ≤ x ≤ π−δ (where δ is a small positive constant), from the point +1 to the antipode −1. For parameters π − δ < x < π + δ we let gZ (x) to be constant, for π + δ ≤ x ≤ 2π − δ the loop continues from −1 to +1 through the point Z on the equator, and finally for 2π − δ ≤ x ≤ 2π it is constant. It is easy to see that the set of smooth loops so obtained covers S 3 exactly once and therefore gives a map g : S 2 → G of index one.
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Any element of G is represented as an element of Ures through pointwise multiplication on the fermion field in H. Thus by this embedding we get directly the transition function φ for the Ures bundle over A/Ge . The index of the map ξ can also be checked using the WZWN action, Z 1 ind ξ = tr(g −1 dg)3 24π 2 S 2 ×S 1 and in the fundamental representation of G = SU (2) this gives ind ξ = 1. For chiral fermions on the circle in the fundamental representation of G this is the same as the index of the map φ : S 2 → Ures . This latter index is evaluated by pulling back the curvature form c on Ures to S 2 and then integrating over S 2 . The curvature is defined by the same formula as the canonical central extension of the Lie algebra of Ures . Identifying left-invariant vector fields on the group manifold as elements in the Lie algebra we have 1 c(X, Y ) = tr [, X][, Y ] , 4 where is now defined by the polarisation to nonnegative and negative Fourier modes. Note that this curvature on Ures is the generator of H 2 (Ures , Z). The Dixmier–Douady class in our example, as a de Rham class in H 3 (A/Ge ), is simply the normalized volume form on S 3 . This is because the third cohomology group of S 3 is one-dimensional and the Dixmier–Douady class was constructed starting from the universal bundle A → S 3 . 6. Global Anomalies 6.1. Bundle gerbes with other structure group There is no particular reason to restrict attention to C× as the structure group for bundle gerbes. If Z is any Abelian topological group there is a theory of Z bundle gerbes obtained by replacing C× by Z throughout. We need Z Abelian in order to take tensor products of Z bundles — (this does not work if Z is not Abelian). Brylinski calls these: gerbes with “band” Z (where Z is the sheaf of smooth functions into Z). In such a theory the Dixmier–Douady class is in H 2 (M, Z) because the isomorphism H 2 (M, C× ) = H 3 (M, Z) is generally not available. In particular if Z is a subgroup of C× one may think of a Z bundle gerbe as a special ordinary bundle gerbe. It is one where the C× bundle P → Y [2] has a reduction to Z and that reduction is preserved by the bundle gerbe product. In such a case the Cech cocycle which is a priori in H 2 (M, C× ) naturally ends up in H 2 (M, Z). If there is a central extension ˆ→G Z→G and a G bundle P → M there is a lifting Z bundle gerbe whose Dixmier–Douady ˆ This may be seen by noting its construcclass is the obstruction to lifting G to G. tion. If P → M is a G bundle there is a map s : P [2] → G
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defined by ps(p, q) = q. Then the Z bundle gerbe, as a Z bundle over P [2] is just the ˆ → G under s. Hence any special properties of G ˆ → G are inherited pull-back of G by the bundle gerbe. If L → M is a flat line bundle with connection ∇ then it can be represented locally by transition functions gab that are locally constant. Hence its Chern class is in H 1 (M, C× ) rather than H 1 (M, C× ). In fact one can show that flat line bundles are classified by H 1 (M, C× ). Similarly stable isomorphism classes of flat bundle gerbes (Subsec. 2.4) with flat connection and curving are classified by H 2 (M, C× ). Particular examples of these can be obtained by looking at H 2 (M, Zn ) where Zn is the cyclic subgroup of U (1). These are flat line bundles whose Chern class is n torsion. One can also realise the Dixmier–Douady class as an element of H 3 (M, Z) when Z is say Zn . To see this consider the following commuting diagram of sheaves of groups: 0 → Z → C → C× → 0 ↑
↑ ↑ 1 Z → Zn → 0 . 0→ Z → n Here the first vertical arrow is an equality, and the second and third are inclusions. The coboundary map for the lower short exact sequence induces the so-called Bockstein map β ∗ : H 2 (M, Zn ) → H 3 (M, Z) whose image consists of n torsion classes in H 3 (M, Z). 6.2. The framework for the examples The notation is as in Sec. 4. Thus H is the tensor product of the Hilbert space of square-integrable spinor fields on a compact Riemannian spin manifold M and a finite-dimensional inner product space V . We assume that an action of a compact Lie group G on V is given. This gives a natural action of G = Map(M, G) on H. We have a polarisation H = H+ ⊕ H− corresponding to the splitting of the spectrum of the Dirac operator D = DA0 on M to nonnegative and negative parts. The Lie algebra of Ures has a central extension defined by the cocycle c(X, Y ) =
1 tr [, X][, Y ] 4
and the corresponding group extension Uˆres is a topologically nontrivial circle bundle over Ures . This bundle has a natural connection defined by the 1-form θ = prc g −1 dg, the central projection of the Maurer–Cartan form on Uˆres . The curvature Ω of this form is left-invariant and at g = 1 it is given by the 2-cocycle c. The curvature is integral, its integral over a closed surface is an integer. Starting from the Lie algebra central extension (or curvature form) one can construct Uˆres as follows. Consider the set P of smooth paths g(t) ∈ Ures , 0 ≤ t ≤ 1, in P × S 1 with g(0) = 1 and g(t) = g ∈ Ures . Define and equivalence relation R by (g1 (·), λ) ∼ (g2 (·), µ) if g1 (1) = g2 (1) and µ = λ · exp(2πi D Ω), where D is any surface in Ures such that the boundary of D is the union of the paths g1
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and g2 . Define a product in P × S 1 as (g1 (·), λ) · (g2 (·), µ) = (g3 (·), λµ), where g3 (t) = g1 (t)g2 (t). Then Uˆres = (P × S 1 )/ ∼. −1 gTA with values in Ures . As before, we construct the 1-cocycle ω(g; A) = Tg·A The obstruction to pushing forward the bundle A × F of Fock spaces over A to a bundle (A × F )/Ge over A/Ge is the obstruction to lifting the cocycle ω to a cocycle ω ˆ with values in Uˆres . We have earlier discussed the local part of this obstruction. The local obstruction is due to the fact that the pull-back with respect to the map g 7→ ω(g; A) of the circle bundle Uˆres over Ures might be nontrivial. (The gauge parameter A is irrelevant in this context because A is an affine space and so uninteresting for the problem of nontriviality of bundles over Ge × A.) The circle bundles are classified by the Chern class, given as a (cohomology class of) a 2-form. This was related, via families index theorem, to the Dixmier–Douady class on A/Ge . Now we shall assume that the local obstruction vanishes, i.e. the restriction of the curvature form Ω to the submanifold {ω(g; A)|g ∈ Ge } ⊂ Ures vanishes (for all A ∈ A.) 6.3. The case G = SU (2) This is the original case considered by Witten in even dimensions, [30]. In our situation the dimension of M = S 3 is three and then the curvature form of the local determinant bundles along gauge orbits is Z i tr A[dX, dY ] ≡ 0 . 24π 3 M This follows from tr X(Y Z + ZY ) ≡ 0 in the Lie algebra of SU (2). On the other hand, as we have seen, the curvature of the determinant bundles gives directly the 2-cocycle of the Lie algebra extension arising from the action in Fock spaces. Even if the local obstruction vanishes there can be a finite torsion obstruction for lifting the cocycle ω to ω ˆ . This obstruction can arise only if π1 (Ge ) 6= 0. The reason for this is understood using the construction in Subsec. 6.2. If g1 (t) and g2 (t) are two paths with the same end points then g1 is always homotopic to g2 (with end points fixed) if π1 Ge = 0. But now the curvature Ω vanishes along Ge and thus we have a lift ω(g; A) 7→ ω ˆ (g; A) = [(ω(g(·); A), 1)] where g(t) is any path joining g to 1 in Ge and the outer brackets denote equivalence classes modulo the relation defined in Subsec. 6.2. If π1 (Ge ) 6= 0 we have to examine further the existence of the obstruction. Note that in the case of G = SU (2) and dimension three, π1 (Ge ) = Z2 . R There is a homomorphism of π1 (Ge ) to S 1 defined by φ(g(·)) = exp(2πi D Ω), where D ⊂ Ures is any surface with boundary curve ω(g(t); A). Since A is connected the equivalence class of this discrete group representation cannot depend on the continuous parameter A and we can fix A = 0, for example. The torsion obstruction is then the potential nontriviality of this representation. R In order to determine the relevant representation we have to compute Di Ω for a set of generators gi (t) of π1 (Ge ) with ∂Di = ω(gi (·); 0) ⊂ Ures . In the case of G = SU (2) in the defining representation and dimension = 3 this is particularly
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simple. We use a trick due to Witten, [31]. Embed SU (2) ⊂ SU (3) and use the fact that π4 (SU (3)) = 0 and on the other hand, π1 (Map(M, SU (3))) = π4 (SU (3)). Correspondingly, we extend the number of (internal) Dirac field components (2) (3) from 2 to 3 and we have, in a self-explanatory notation, Ures ⊂ Ures . The re(3) striction of the curvature form on Ures to the subgroup gives the curvature on the former group. Since π1 (G3 ) = 0, we can choose D ⊂ Map(M, SU (3)) such that the boundary of D gives the generator of π1 (G2 ) = Z2 . For a given A ∈ A the pull-back of Ω with respect to the map g 7→ ω(g; A) is equal to the 2-form d d ω(g · etX ; A)|t=0 , ω(g · esY ; A)|s=0 ΦA (X, Y ; g) = Ω dt ds d d tX g sY g = Ω ω(g; A) ω(e ; A )|t=0 , ω(g; A) ω(e ; A )|s=0 dt ds d d tX g sY g ω(e ; A )|t=0 , ω(e ; A )|s=0 ≡ c(X, Y ; Ag ) , =Ω dt ds where we have used the left-invariance of the form Ω and c(X, Y ; A) is the Schwinger term induced by the cocycle ω and the central extension of Ures . For G = SU (2) and d = 3 the above formula gives Z tr Ag [dX, dY ] ≡ 0 . ΦA (X, Y ) ∼ M
If D ⊂ Map(M, G) is a disk (and the dimension of M is 3) parameterized by real parameters t, s then Z Z Z i ΦA = tr Ag [d(g −1 ∂t g), d(g −1 ∂s g)] . 3 24π D D M In particular, at A = 0 the result is Z ΦA = D
i 480π 3
Z
(g −1 dg)5
D×M
provided that we can ignore boundary terms in integrations by parts; this is the case if at the boundary of the disk g(t, s) ∈ Map(M, SU (2)). In this case the last integral has been computed in [31]; the result is 1/2 mod integers if the boundary circle represents the nonzero Relement in π4 (SU (2)), otherwise the integral is zero mod integers. Since exp(2πi D φ) is the factor appearing in the definition of the extension of the group of gauge transformations in the Fock spaces, this result shows that the double cover of Map(M, SU (2)) is represented nontrivially and therefore obstructing the lifting of the cocycle ω to a quantum extension ω ˆ. The global SU (2) anomaly in the bundle of Fock vacua can also be analyzed in terms of the spectral flow of a family of Dirac hamiltonians, [26]. The Z2 extension of the gauge group Map(M, SU (2)) has been used for deriving a boson-fermion correspondence in four space-time dimensions, [21].
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6.4. A general analysis The idea of the SU (2) example should work for any simple group G in any dimension. Given a complex k-dimensional representation of G acting on the spinor components we extend the number of components to a large value N and think of G as a subgroup of G∞ = SU (N ), [14]. If the dimension of M is d = 2n + 1 then πd+1 (SU (N )) = 0 for large enough N . So given a loop γ in Map(M, G) we can find a disk D in Map(M, G∞ ) such that ∂D = γ. Finally, checking the nontriviality of the obstruction, comes up to evaluating the integral
−i 2π
n+2
(n + 1)! (d + 2)!
Z
tr(g −1 dg)d+2
D×M
and checking whether it is zero mod integers. In order that there could be a non-trivial global anomaly we note that we need to be in a situation where π1 (Ge ) is non-trivial, say equal to Zn for some integer n, and there is no local anomaly. Then one has a central extension Zn → Gˆe → Ge .
(6.1)
The Cech 1-cocycle arising from this extension takes values in Zn ⊂ U (1). Using the usual exact sequence Z → R → U (1) to change coefficients one sees that this gives the Chern class as a torsion element of the Cech cohomology group H 2 (Ge , Z). We may consider the corresponding lifting bundle gerbe for the principle Ge bundle A Ge → A → . Ge This lifting bundle gerbe has Dixmier–Douady class in H 2 (A/Ge , U (1)) which, using the exact sequence Z → R → U (1) is represented by a torsion class in H 3 (A/Ge , Z). The argument of Theorem 4.1 of [5] shows that the Dixmier–Douady class is the transgression of the Chern class of the extension (6.1) as an element of H 2 (A/Ge , Z). In the SU (2) example we have a reduction of the (local) determinant bundles along gauge orbits in A to Z2 bundles. On A/Ge this corresponds to trying to lift the system of local Gˆe (where Gˆe is the Z2 extension of the group of gauge transformations) to a global Gˆe bundle. The obstruction is the Dixmier–Douady class: our torsion element in H 3 (A/Ge ). It is of interest to have a practical method for determining when the global Hamiltonian anomalies are non-trivial. There is a method for finding the extension Gˆe of Ge which acts on DETλ for each λ. The map which sends g ∈ Gˆe to g k is a homomorphism onto Ge for sufficiently large k. Choose the smallest such k. Then
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as Gˆe acts on DETλ so Ge acts on (DETλ )k . Thus the bundle gerbe given locally by (DETλλ0 )k = (DET∗λ )k ⊗ (DETλ0 )k admits an action of Ge on each factor in the tensor product and so descends to a trivial bundle gerbe over A/Ge . Finally, finding k may be done using the Witten method. As above we have a compact Lie group G with say πd+1 (G) torsion, say Zn , for d odd. Then the subgroup Ge of based gauge transformations in Map(S d , G) has π1 Ge = Zn . For large enough N we have πd+2 (G∞ ) = Z in addition to πd+1 (G∞ ) = 0. We assume that the local Hamiltonian anomaly for the pair S d−1 , G vanishes and we wish to know if there exists a global anomaly. The imbedding into G∞ enables us to exploit Witten’s trick. Consider part of the homotopy long exact sequence for the fibration G → G∞ → X , where X denotes the quotient space G∞ /G: · · · → πd+2 (G∞ ) → πd+2 (X) → πd+1 (G) → πd+1 (G∞ ) → · · · . Hence: · · · → Z → πd+2 (X) → Zn → 0 → · · · . When πd+2 (X) is known this is enough to give precise information on the map Z → πd+2 (X). In general we only know that πd+2 (X) = Zr ⊕ T where T is torsion and r is a positive integer. Assuming that the form θd+2 = tr(dgg −1 )d+2 vanishes on G (which is the case if πd+2 (G)R is torsion) we can determine exactly the extension Gˆe through a computation of θd+2 for each generator in πd+2 (X) which corresponds to an element in π1 (Ge ). An element in the latter group is represented by a map g : S 1 × M → G and this map extends to a map g : D × M → G which in turn defines (through canonical projection) a map from S 2 × M to X. By an integration α(g) =
−i 2π
n+2
(n + 1)! (d + 2)!
Z
tr(g −1 dg)d+2
D×M
we get a real number α(g) and the homotopy class [g] ∈ π1 (Ge ) is represented by exp(2πiα(g)) in U (1). Thus we are interested in the values α(g) modulo integers. If all these numbers are in Z then the kernel of the extension Gˆe is represented trivially and there is no global Hamiltonian gauge anomaly. In specific examples we can determine the existence of the global anomaly without doing any explicit computations. This occurs when πd+2 = Z and πd+1 (G) = Zn . In this case we conclude from the exact homotopy sequence above that the generator in πd+2 (G∞ ) is mapped to n times the generator in πd+2 (X) and therefore the value of the integral for the generator in πd+2 (X) (which is defined by a generator of π1 (Ge )) is equal to 1/n modulo integers. It follows that Gˆe is represented faithfully and there indeed is a global anomaly.
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For example the case of G = SU (3) in the fundamental representation and dim M = 5 works in this way. In this case the relevant homotopy is π6 (G) = Z6 and can R be represented using tr(g −1 dg)7 on the larger group SU (4) because π6 (SU (4)) = 0. Using the exact homotopy sequence SU (4) → π6 (SU (3)) → π6 (SU (4)) π7 (SU (4)) → π7 SU (3) gives the exact sequence Z → Z → Z6 → 0 since SU (4)/SU (3) = S 7 . This shows that the generator of π6 (SU (3)) gets mapped to 6 times the generator of π7 (S 7 ). Another nice example is the case of the exceptional simple group G = G2 in the real 7 dimensional representation. Here one uses the embedding G2 ⊂ SO(7) and the fact that SO(7)/G2 is also equal to S 7 . Since π6 (SO(7)) = 0, π6 (G2 ) = Z3 , and π7 (SO(7)) = π7 (S 7 ) = Z one obtains an exact sequence Z → Z → Z3 → 0 of homotopy groups. Thus that the generator of π7 (SO(7)) is mapped R to three 7 times the generator of π7 (S ) and therefore the (normalized) integral tr(dgg −1 )7 corresponding to the elements in π6 (G2 ) define the phase factors 1, exp(2πi/3), exp(4πi/3) and π6 (G2 ) is represented faithfully in U (1). 6.5. Algebraic considerations In the case of gauge group G = SU (2) and the dimension of the physical space is three there is a real structure which explains the appearance of Z2 determinant bundles (instead of U (1) bundles). For 2 × 2 complex matrices there is a real linear (but complex antilinear) automorphism J defined by ∗ −c∗ a b d J = −b∗ a∗ c d where star means complex conjugation. Because the spinor field has now 2 internal and 2 space-time components we can think of the Dirac field as a complex 2 × 2 matrix function and we can define J as a real linear operator acting on the Dirac field point-wise in space. The vector potential acts from the right by matrix multiplication on ψ whereas the gamma matrices (in this case the Pauli matrices) act from the left. The automorphism J has the properties J(g) = g for g ∈ SU (2) and J(g) = −g if g is Hermitian traceless. From this follows that if DA ψ = λψ is an eigenvector in the external potential A then also Jψ is an eigenvector corresponding to the same eigenvalue λ. For this reason we may choose a real basis of eigenvector ψ1 , . . . , ψn in any given energy range s < λ < t. Real means here that Jψk = ψk . Any other real basis is obtained by a real orthogonal transformation R from this basis. Thus the only ambiguity in choosing a representative in the determinant line is det R = ±1.
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This gives the required Z2 structure. Whether there is an algebraic structure which explains the global anomaly in other cases remains open. An interesting test case is the exceptional Lie group G = G2 . The homotopy group π6 of G2 is equal to Z3 . This would lead to Z3 torsion in the Fock bundle in 5 + 1 space-time dimensions for the G2 gauge group (maybe related to quarks in 5 + 1 dimensions. . .). There must be some Z3 structure in the local determinant bundles over open sets Uλλ0 in A, in the same way as there is a Z2 structure in 3+1 dimensions for SU (2). References [1] M. F. Atiyah and I. M. Singer, “Dirac operators coupled to vector potentials”, Natl. Acad. Sci. (USA) 81 (1984) 2597. [2] M. F. Atiyah, V. K. Patodi and I. M. Singer, “Spectral asymmetry and Riemannian Geometry”, I-III. Math. Proc. Camb. Phil. Soc. 77 (1975) 43; 78 (1975) 405; 79 (1976) 71. [3] E. J. Beggs, “The de Rham complex on infinite dimensional manifolds.” Quart. J. Math. Oxford 38(2) (1987) 131–154. [4] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkh¨ auser, Boston-Basel-Berlin, 1993. [5] A. L. Carey, D. Crowley and M. K. Murray, “Principal bundles and the Dixmier Douady class”, Commun. Math. Phys. 193 (1998) 171–196, hep-th/9702147. [6] A. L. Carey, J. Mickelsson and M. K. Murray, “Index theory, gerbes and Hamiltonian quantization”, Commun. Math. Phys., 183 (1997) 707–722. [7] A. L. Carey, J. Mickelsson and M. K. Murray, Bundle Gerbes and Field Theory, Proc. Int. Congress of Math. Phys. Brisbane, 1997. [8] A. L. Carey and M. K. Murray, Confronting the Infinite, Proc. Conference in Celebration of the 70th Years of H. S. Green and C. A. Hurst, World Scientific, 1995, hep-th/9408141. [9] A. L. Carey and M. K. Murray, “Faddeev’s anomaly and bundle gerbes”, Letts. in Math. Phys. 37 (1996) 29–36. [10] A. L. Carey and M. K. Murray, “Holonomy and the Wess–Zumino term”, Letts. in Math. Phys. 12 (1986) 323–328. [11] A. L. Carey, M. K. Murray and B. Wang, “Higher bundle gerbes, descent equations and 3-Cocycles”, to appear in J. Geometry and Phys., hep-th/9511169. [12] J. Dixmier and A. Douady, “Champs continus d’espaces hilbertiens et de C ∗ algebres”, Bull. Soc. Math. Fr. 91 (1963) 227. [13] S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Clarendon Press, Oxford, 1990. [14] S. Elitzur and V. P. Nair, “Nonperturbative anomalies in higher dimensions”, Nucl. Phys. B243 (1984) 205. [15] L. Faddeev and S. Shatasvili, “Algebraic and Hamiltonian methods in the theory of nonabelian anomalies”, Theoret. Math. Phys. 60 (1985) 770. [16] K. Gawedski, “Topological actions in two-dimensional quantum field theories”, in Nonperturbative Quantum Field Theory, eds. G. ’t Hooft et al., Cargese, 1987, 101–141, NATO Adv. Sci. Inst. Ser. B: Phys., 185, New York, Plenum Press, 1988. [17] T. Killingback, “World sheet anomalies and loop geometry”, Nucl. Phys. B288 (1987) 578–588. [18] D. A. McLaughlin, “Orientation and string structures on loop space”, Pac. J. Math. 155 (1992) 1–31. [19] J. Mickelsson, “Chiral anomalies in even and odd dimensions”, Commun. Math. Phys. 97 (1985) 361.
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[20] J. Mickelsson, “On the Hamiltonian approach to commutator anomalies in 3 + 1 dimensions”, Phys. Lett. B241 (1990) 70. [21] J. Mickelsson, Current Algebras and Groups, Plenum Press, London, New York, 1989. [22] J. Mickelsson, “Hilbert space cocycles as representations of (3 + 1) − D current algebras”, Lett. Math. Phys. 28 (1993) 97. [23] J. Milnor, “On infinite dimensional Lie groups”, preprint. [24] M. K. Murray, “Bundle gerbes”, J. London Math. Soc. 54(2) (1996), 403–416, dgga/9407015. [25] M. K. Murray and D. Stevenson, “Bundle gerbes: Stable isomorphism and classifying theory”, math.DG/9908135. [26] P. Nelson and L. Alvarez-Gaume, “Hamiltonian interpretation of anomalies”, Commun. Math. Phys. 99 (1985) 103. [27] A. Pressley and G. Segal, Loop Groups, Clarendon Press, Oxford, 1986. [28] G. Segal, “Faddeev’s anomaly in Gauss’s law”, preprint. [29] D. Stevenson, “Bundle gerbes”, PhD thesis in preparation. [30] E. Witten, An SU (2) anomaly, Phys. Lett. 117B (1982) 324. [31] E. Witten, “Current algebra, baryons, and quark confinement” Nucl. Phys. B223 (1983) 433.
COVER OF THE BROWNIAN BRIDGE AND STOCHASTIC SYMPLECTIC ACTION ´ R. LEANDRE Institut Elie Cartan D´ epartement de Math´ ematiques Universit´ e de Nancy I 54000, Vandoeuvre-les-Nancy, France Received 21 May 1996 Revised 16 November 1998 We define the universal cover of the Brownian bridge of a symplectic manifold. This allows us to define a non-trivial functional over it called the stochastic symplectic action, and to define local Sobolev spaces over the universal cover, such that the symplectic action belong to them. This is done in the purpose of an analytical Morse theory in the sense of Witten over the loop space associated to this symplectic action. In this purpose, a stochastic Witten complex is constructed over the universal cover of the loop space, and modulo some weights, it is shown that its cohomology is equal to the stochastic cohomology of the universal cover.
0. Introduction Floer homology is a Morse theory over the loop space (see [8, 41, 81] for introduction references). There are two models of symplectic Morse theory of the loop space, for a given symplectic manifold M with a given symplectic form ω. The first one considers two lagrangian submanifolds L and L0 of the symplectic manifold. We assume, that they are compact. We introduce the set of paths starting from L and arriving in time 1 in L0 denoted by P (L, L0 ) and we look the 1 closed form over it given by Z 1 ω(dγs , .) (0.1) τ (ω) = 0
τ (ω) is a particular case of a Chen form (see [19, 46]). The second model is involved with the based loop space Lx (M ) endowed with the 1 closed form: Z 1
τ (ω) =
ω(dγs , .) .
(0.2)
0
The purpose of the Morse theory is to exhibit a relation between the configuration space and the set of the critical points of a functional F over the configuration space. Let us assume for this purpose that there exists a functional, called the symplectic action, F such that dF = τ (ω) . (0.3) In the first model, the set of critical points is the intersection points of L and L0 . In the second model, the set of the critical points is the constant loop [41], but the situation becomes more complicated because an auxiliary functional appears. 91 Reviews in Mathematical Physics, Vol. 12, No. 1 (2000) 91–137 c World Scientific Publishing Company
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Namely, the authors consider for instance the space of contractible loops [41, p. 251]. They introduce the action functional: Z FH (γ) = −
γ∗ω +
D
Z
1
H(t, γt )dt,
(0.4)
0
where D is the closed disc, and γ is a map from D into M such that γ(exp[2πit]) = R γt . H is the Hamiltonian function and is periodic. They suppose that S 2 u∗ ω = 0 for any map u : S 2 → M such that the integral over D in (0.4) does not depend on the extension. The critical points of FH are the periodic solutions of the equation d/dtγt = XH (t, γt ), where XH is the Hamiltonian vector field associated to H. The authors introduce an almost complex structure J. J over M is compatible with ω. It is an endomorphism Jx of Tx (M ), the tangent space of M at x, depending smooothly on x such that Jx2 = −1 and such that g(X, Y ) = ω(X, Jx Y )
(0.5)
defines a Riemannian metric over M . In classical Morse theory, the authors study the gradient curves associated to the Morse functional. For FH , these gradient curves are given by the equation (see [41, p. 252]): ∂ ∂ u + Ju u + ∇H(t, u) = 0 . ∂s ∂t
(0.6)
They study the curves joining two critical loops γt1 and γt2 : lim u(s, t) = γt1 ;
s→−∞
lim
s→=+∞
u(s, t) = γt2 .
(0.7)
Starting from this set of gradient curves, they do algebraic computations over these, in order to define Z/2Z homology groups, and they show (see [41, p. 257]), that they are independents of H and J. A relation is performed between these cohomology groups and the singular homology groups of M with coefficients in Z/2Z. This allows to get estimates of the number of periodic solution of the Hamiltonian flow associated to H with respect of the Betti numbers of the manifold (see [41, p. 262]). The integration of a closed form is only possible locally, except when the configuration space is simply connected. For this reason, we have to integrate over the universal cover of the configuration space or of some smaller extension, called the minimum extension. In the case of the smooth loop space and of the symplectic action, these extensions are performed in [20, 40] and [75]. In this set up, no measure over the loop space is necessary. However, there is an analytical approach to the Morse theory over the loop space initiated by Guilarte [36]. It is an infinite dimensional analoguous of the finite dimensional considerations of Witten [86]. They use the complex d + dFH , compute the adjoint of it for the Lebesgue measure over the loop space, in order to introduce the supercharge Q = d + dFH + d∗ + idFH and the Wess–Zumino–Witten associated Laplacian Q2 . A careful analysis of exp[−tQ2 ] allows to get Morse inequalities. Guilarte [36] used the
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“Lebesgue measure” over the path space for the first model P (L, L0 ), in order to try to perform some analytical Morse theory over the path space. Unfortunately, the Lebesgue measure does not exist in infinite dimension. For that, we introduce the Brownian bridge measure. The difficulty is that τ (ω) is then almost surely defined, because there is a stochastic integral in its definition. We have to do a theory where we have to take derivative of almost surely defined functionals, a stochastic exterior derivative for almost surely defined forms: in the Gaussian case, it is the purpose of the Malliavin Calculus. The purpose of this paper is to analyse this symplectic action. For this purpose, we introduce a measure, that is the Brownian bridge measure. The loops are only continuous. The theory of stochastic forms, and in particular of stochastic Chen forms, is covered in [46, 58] and [59]. In particular, there is a stochastic cohomology of forms over the loop space, which is compatible with the Hochschild boundary for Chen forms. In [63], in the case of the first model, when P (L, L0 ) is supposed simply connected, we have found a stochastic functional such that its H-derivative satisfies to DF = τ (ω) . (0.8) In particular, this notion is involved with the fact that the tangent space of a path is a smaller set than the tangent space of a loop given by the differential geometry. Its choice is given by [10] and [46]. We are concerned with the second model in this paper. We construct a measure over the universal cover of (Lx (M ), µ) where µ is the law of the pinned Brownian ˜ x (M ), µ ˜), µ ˜ being a positive measure. motion. We get a space (L Moreover there exists a projection π ˜ x (M ), µ ˜) → (Lx (M ), µ) . (L
(0.9)
More precisely, we get, if we suppose the manifold is simply connected: Definition 0.1. Let Lx,fin(M ) be the space of finite energy loops γfin endowed with the reference loop γref = x. . The tangent space of a loop is endowed with the uniform norm. Let l(γfin ) be a C 1 path joining γref to γfin . We identify paths in the finite energy loop space if they have the same extremities and are homotopic. We introduce a metric over the quotient space, by considering the infimum of the 0 . We consider the infimun length for the uniform norm of a curve joining γfin to γfin of the length which are obtained with respect to the equivalence relation. We get a ˜ x (M ) of the space of based continuous loops distance called d˜x . The universal cover L Lx (M ) is the completion of the previous space for the metric d˜x . The projection π is the map which to a path in the loop space associates the loop which is the end of ˜ x (M ) has a small neighborhood which is isomorphic this path. Each element γ˜ of L ˜ x (M ) by lifting the to a small neighborhood of π(˜ γ ). We define a measure µ ˜ over L Brownian bridge measure µ by the local homeomorphism π. In other words, µ ˜ is ˜ be an open subset of L ˜ x (M ) such that π realized defined by this property: let O ˜ over its image. Let A˜ be a borelian subset of O. ˜ Then a homeomorphism from O ˜ = µ(π(A)). µ ˜ (A)
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The problem is that µ ˜ does not have a finite mass, except when the first homotopy group of Lx (M ), that is the second homotopy group of M is finite. An ˜ x (M ) is endowed with a tangent space and π is an isometry from γ˜ element γ˜ of L over π(˜ γ ) = γ. The tangent space of a loop γ. is given by Bismut [10] and in this shape precisely by Jones and L´eandre [46]. Let τs be the parallel transport over a loop for the Levi–Civita connection. A tangent vector is given by Xs = τs Hs ;
H0 = H1 = 0,
(0.10)
where Hs is of finite energy and is a loop in the tangent space at x. We endow it with the metric Z 1 2 kXkγ = kd/dsHs k2 ds . (0.11) 0
If H. is deterministic, we recall that for all cylindrical functional E[hdF, Xi] = E[F div X]
(0.12)
where div X belongs to all the Lp . ˜ x (M ) is locally isomorphic to Lx (M ): this works for the integration As a space, L by parts formulas. Without speaking of a partition of unity, (see [6]), we get local ˜ x (M ). In particular, if G is a functional which belongs to Sobolev spaces over L ˜ = Gπ(˜ γ ) belongs to all the all the Sobolev spaces over Lx (M ) (see [55, 56]), G ˜ local Sobolev spaces over Lx (M ). There is an Ornstein–Uhlenbeck operator over ˜ the universal cover such that ∆(F ◦ π) = (∆F ) ◦ π, where ∆ denotes the Ornstein– Uhlenbeck operator over Lx (M ). ˜ x (M ), A˜ which is a lift of the Dirichlet form A There is a Dirichlet form over L over Lx (M ) (see [26, 31, 67]). The process associated to the Dirichlet form A˜ is the lift of the process associated to A over Lx (M ). This construction of a process is an approach to the nonlinear σ-model, which gives topological invariants (see [36, 87]). The nonlinear σ-model should give a measure over the tori inside the compact manifold. Here, we consider random ˜ x (M ), but with a different action. cylinders, or more precisely their lift over L The form τ (ω) lifts into a form τ˜(ω) which can be integrated in a function F˜ , which belongs locally to all the Sobolev spaces. More precisely, we get: Theorem 0.2. There exists a distinguished path ˜l joining γref to γ and which ˜ x (M ), such that ˜ls is a semi-martingale represents γ˜ in a small neigborhood of L ∂ ˜ and such that ∂s ls is a semi-martingale over ˜ls . Moreover the expression Z F˜ (˜ γ ) = τ (ω) (0.13) ˜ l
is almost-surely defined and does not depend on the distinguished path ˜l when we ˜ x (M ) of definition of the work over the intersection of two small neighborhoods of L ∗ ˜ local primitive of τ (ω). This means that DF (γ) = π τ (ω). ˜ + kdF˜ k2 . We study the Schr¨ odinger semi-group associated to ∆
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˜ x (M ), µ In the second part, we study forms over (L ˜ ). We introduce a measure Hd˜ µ for some H > 0, such that the lift of all the forms which belong to all the Sobolev spaces over the loop space belongs to all the Sobolev spaces over the universal cover. More precisely, [58] and [59] study Sobolev structures with regularity assumption over its kernel in order to define the stochastic exterior derivative. We show that we can choose H well enough in order that the lift of stochastic forms belongs to all the Sobolev spaces with the regularity assumptions of [58, 59]; it is therefore possible to lift the stochastic Chen forms of [46] to the universal cover (see [41, 61, 81]). Then it becomes possible to speak of the stochastic exterior derivative ˜ x (M ), µ ˜). d˜ over (L Moreover it is possible to choose H > 0 such that for all λ > 0, ˜ exp[λ|F˜ |]] < ∞ . E[H
(0.14)
˜ Therefore, we can perform the stochastic gauge transform over d: d˜F = exp[−F˜ ]d˜exp[F˜ ]
(0.15)
˜ x (M ). Its cohomology is equal to and we get the stochastic Witten complex over L the cohomology of the original one because of (0.14). It is one of the key points of the analytical Morse theory. This introduction of a weight H allows to improve the result of [63], where the two complexes do not have have the same cohomology, because exp[F˜ ] does not belong to all the Sobolev spaces. We can precise these statements. We recall the definition of a form smooth in the Nualart–Pardoux sense. Over the tangent space, we introduce a connection ∇(τ. H. ) = τ. DH. .
(0.16)
An r-form σ is given by a random kernel k(s1 , . . . , sr ). Its covariant derivatives have kernels k(s1 , . . . , sr ; t1 , . . . , tr0 ). We define the Nualart–Pardoux Sobolev norms of 0 (σ): σ by the smallest constants Cp,r0 (σ) and Cp,r kk(s1 , . . . , sr ; t1 , . . . , tr0 ) − k(s01 , . . . , s0r ; t01 , . . . , t0r0 )kLp X q Xq ≤ Cp,r0 (σ) |si − s0i | + |ti − t0i |
(0.17)
over all simplices, s and t included and 0 kk(s1 , . . . , sr ; t1 , . . . , tr0 )kLp ≤ Cp,r 0 (σ) .
(0.18)
0 If all the constants Cp,r (σ) and Cp,r 0 (σ) are finite, we say that the form is smoooth in the Nualart–Pardoux sense. The main result of [58] is that the exterior stochastic derivative is defined and continuous over the forms smooth in the Nualart–Pardoux ˜ x (M ) is locally isometric to Lx (M ) and since the notion of differential sense. Since L form is purely local, we can define the notion of weighted Nualart–Pardoux Sobolev norms with respect to a weight H.
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˜ x (M ) σ Definition 0.3. The r-form over L ˜ belongs to all the Nualart–Pardoux spaces with respect to the weight H if over all simplices in s and t ˜ 0 , . . . , s0 ; t0 , . . . , t0 0 )kp ]1/p ≤ Cp,r0 ,H (˜ ˜ 1 , . . . , sr ; t0 , . . . , t0 0 ) − k(s ˜ σ) E[Hk k(s 1 r 1 r 1 r X q q X (0.19) |si − s0i | + |ti − t0i × σ ) and if for a finite constant Cp,r0 ,H (˜ ˜ 1 , . . . , sr ; t1 , . . . , tr0 )kp ]1/p ≤ C 0 0 (˜ ˜ E[Hk k(s p,r ,H σ ) < ∞ .
(0.20)
˜ ∞ (H) the space of forms smooth in the Nualart–Pardoux sense for the We call N.P weight H. Theorem 0.4. There exists a weight H > 0 such that the exterior derivative ˜ ∞ (H) and such that the Witten complex d˜ is defined and is continuous over N.P ˜ ˜ ˜ exp[−F ]d exp[F ] has the same cohomology groups as the original one. It is one of the basic ingredient of the analytical Morse theory of Witten. It was not checked for the stochastic Witten complex of [63] for P (L, L0 ). In the third part, we summarize the previous discussion in the case of the mini˜ x,min(M ) of Lx (M ), done in order to be able to integrate τ (ω) over mal extension L ˜ x,min(M ). L A lot of results are done by patching together local copies of small neighborhood of the based loop space. The reader can find in [23, 27] and [51] analoguous of these constructions in other stochastic contexts. 1. Loop Spaces and Non Scalar Infinite Dimensional Analysis The geometry of the free loop space of a manifold, that is the space of smooth applications from the circle S1 into M is closely related to the topology of the l-manifold by its natural circle action over it: namely, we can recover the manifold from the free loop space as the fixed point set of this circle action on the free loop space. In finite dimensions, there are two types of relations between the fixed point set and the whole manifold. • The Berline–Vergne localisation formulas [9]. Integrals of equivariant closed forms are written as integrals over the fixed point set. The relation between the equivariant cohomology of the manifold and the cohomology of the fixed point set should be seen from this point of view [48]. • The Lefschetz formulas. The equivariant index of an invariant operator is a finite Laurent series in the parameter q of the circle: each term can be written as an integral over the fixed point set. It was conjectured by Witten, based on observation on the “Dirac operator” on loop spaces, that only the constant term is not 0 for a large class of classical operators. Taubes has given an infinitesimal version of this results by studying
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infinitesimally small loops [83]. The main idea of the finite dimensional Lefschetz formula is to localize the operator on the normal bundle of the fixed point set (see [83]). The construction of the Dirac operator by Taubes for infinitesimally small loops is an infinite dimensional copy of these finite dimensional constructions. Moreover, there is a variant of the index theory over the loop space, which is Morse theory. Morse theory over the loop space is related to Floer homology. Guilarte pointed out the existence of an analytical Morse theory in infinite dimension (see works of Bismut [15], Hellfer–Sjostrand [37] which give in finite dimension a rigorous version to the considerations of Witten [86]). It is based upon a deformation of the exterior derivative and of a deformed Laplacian which operates over the bundle of forms. Guilarte’s statement [36] needs a measure: we will take the Brownian bridge measure. Section 1 is related to a survey of the rigorous works done for these three types of problems. We will begin by a review of the problems for the finite dimensional index theorem and the rigorous results which are performed for the finite dimensional index theory. In a second step, we will do a review of the problems and the results from the infinite index theory. Finally, we will give some review of the results which are performed for the analytical Morse theory with respect of the symplectic action over the path space of a symplectic manifold. 1.1. Finite dimensional index theory Let us introduce a compact orientable manifold M . Suppose that there is a periodic group of diffeomorphisms over M . We can suppose that these diffeomorphisms are isometries. The typical example is the sphere which rolls around one of her axis. We say that there is an action of the circle over the manifold. In the case of the sphere, we see two distinguished points: the north pole and the south pole. They are invariant under the circle action. We say that there are the fixed points under the action of the circle. We call a Killing vector field the vector field which generates this circle action, because we can suppose that it is an action by isometries, since the circle is compact. The set of fixed points is the set where the Killing vector field vanishes. Let us describe a little bit more the relation between the fixed point set and the global structure of the manifold. Let us introduce the set of invariant forms in the manifold. Infinitesimally, this means that the Lie derivative of an invariant form under the Killing vector field is equal to 0. (1.1.1) LX µ = (d + iX )2 µ = 0 . Over the set of invariant forms, d + iX is the differential of a complex and its cohomology is called the S1 equivariant cohomology. What is the big difference with the ordinary cohomology? If we consider an equivariantly closed form, its degree cannot be given, because d adds one degree to the form and iX removes one degree from the form. We can only speak of even and odd cohomology groups. Let us introduce an equivariantly closed form µ. We remark ([14] Eq. (1.25)): Z Z Z µ= µtop = exp[−t(d + ix)X] ∧ µ = cht (µ) (1.1.2) M
M
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where X is identified with the 1-form g(X, .). When t goes to the infinity, this constant integral localizes over the fixed point set, because iX X = |X|2 . This is a Berline–Vergne localisation formula. The typical example of a form which is equivariantly closed is the following: we consider a bundle where the circle action lifts. We define a characteristic class which is associated to this bundle [9]. Over the fixed point set, this characteristic class coincides with the Chern character of the restricted bundle. This leads to the conclusion that the equivariant cohomology is equal to the cohomology of the fixed point set. Let us consider the free loop space over the manifold, that is the space of C ∞ functions from S1 into M . We can rotate the loop: there is a natural circle action. The fixed point set is the target manifold, as we have already said. A form which is equivariantly closed is an infinite series of forms of finite degree. Let us recall that in the definition of the supersymmetric Fock space, we consider infinite sums of forms of finite degree which depends on a parameter. For the smooth loop space, there is no measure, and we don’t know what is a convergent series of forms. In this formal landscape, [48] have shown that the equivariant cohomology is equal to the cohomology of the manifold. One of the main tool is the Bismut Chern character chξ∞ : let us introduce a complex auxiliary bundle over the manifold. This induces a bundle ξ∞ over the loop space with fibers being C ∞ -sections over the loops. The circle action lifts clearly over ξ∞ . Bismut introduces an equivariant characteristic class [13, 14] which restricts over the constant loops to the Chern character of the bundle ξ over the manifold. It is related to the solution of a differential equations over the loop space as follows: let et be the evaluation map γ. → γt and let σ be a form over M . e∗t σ is a form over Tγ . We then consider the solution of the differential equation: dHt = Ht ∧ e∗t σ(dγt , .) which is solved by Picard’s method: XZ 1 H1 =
σ(dγs1 , .) ∧ · · · ∧ σ(dγsn , .) .
(1.1.3)
(1.1.4)
0<s1 <···<sn <1
The main remark of [33] is the following: the Chen iterated integral [19] Z σ(dγs1 , .) ∧ · · · ∧ σ(dγsn , .) Hn =
(1.1.5)
0<s1 <···<sn <1
can be interpreted as an element of Ω. (M )⊗n , where Ω. (M ) denotes the set of forms of strictly positive degree over M . It is the way used in [32] in order to try to localize the paths integrals over the loop space. Namely Atiyah–Witten [7] remark that modulo a system of well chosen formal infinite constants: (1.1.6) cht (1) = Ind D+ , where D+ denotes the Dirac operator over the manifold which is assumed to be spin. Bismut [13, 14] remarks that cht (chξ∞ ) = Ind D+,ξ ,
(1.1.7)
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where D+,ξ id the Dirac operator tensorized by ξ. We start with the following observations: • The forms in [33] are formal series. • The current cht defined in [32] is defined over the Chen forms, that is a small set of forms over the loop space. It should be nice to produce a theory which takes care of these two constraints. [46] tries to give a beginning of the analytical interpretation to these statements. It introduces a measure over the free loop space, which lives over continuous loops, and a Hilbert tangent space which follows closely the considerations of [10]. The main conclusion is the following: there is a relation between the entire cyclic cohomology of the manifolds and the theory of Lp forms over the loop space. In particular, the Bismut Chern character belongs to all the Lp . [58] shows that the definition of an exterior derivative over the free loop space is related to the definition of anticipative Stratonovitch integrals, because the tangent space of the loop space is not stable by Lie bracket. This leads to the introduction of new types of Sobolev spaces, which are involved with the continuity of the kernels of the forms and their covariant derivatives. In particular, we can introduce on the algebraic objects some Sobolev structures. The Hochschild boundary, which corresponds to the exterior derivative by using stochastic Chen integrals, is continuous over these Sobolev spaces. The map Chen iterated integrals goes from the intersection of Sobolev spaces at the algebraic level to the intersection of Sobolev spaces over the loop space. The exterior derivative is continuous over these Sobolev spaces. In particular, the Bismut Chern Character belongs to all the stochastic Sobolev spaces. A relation between the criterion of the entire cyclic cohomology and the criterion which says that a form belongs to these new Sobolev spaces is established. We can summarize these results as follow. The definition of the space of smooth forms in the Nualart–Pardoux sense can be done analogously for the free path space. This is the space of continuous maps from [0, 1] into M endowed with the measure dx ⊗ dP1x , where dP1x is the Brownian motion measure. The tangent space is the space of sections of the tangent bundle over the path γ. of the form Xs = τs Hs endowed with the Hilbert structure Z 1 kXk2γ = kX0 k2 + kd/dsHs k2 ds . (1.1.8) 0
We can define the space of smooth forms in the Nualart–Pardoux sense for the free loop space. It is the space of continuous applications from the circle into M endowed with the measure p1 (x, x)dx ⊗ dP1,x , where pt (x, y) is the heat kernel associated to the heat semi-group over the compact Riemannian manifold M , and dP1,x the law of the Brownian bridge. The tangent space of a loop is the space of Xs = τs Hs such that X1 = XR0 and such that Hs is of finite energy. If X1 = X0 = 0, we take as 1 Hilbert structure 0 kd/dsHs k2 ds. The orthogonal complement is the set of vector fields of the form τs ((1 − s)X0 + τ1−1 sX0 ), with Hilbert structure kX0 k2 . We can take the following rotation invariant Hilbert structure on tangent spaces:
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Z
Z
1
kXk2 =
1
hXs , Xs ids + 0
kd/dsHs k2 ds .
(1.1.9)
0
For an r-form σr over the free path space, we can define its Sobolev Nualart– P 0 (σr ). We can define a series of forms σ = σn , each Pardoux norms Cp,l (σr ) + Cp,l σn being of degree n, over the free path space. We define the Nualart–Pardoux Sobolev norms of σ as follows: X 2np 0 (Cp,l (σn ) + Cp,l (σn )) . (1.1.10) kσkp,r = (n − p)!n! n P 0 P σn belongs to all the Nualart– Theorem 1.1.1 [58]. If σ = σn , and if σ 0 = 0 Pardoux spaces, σ ∧ σ belongs to all the Nualart–Pardoux spaces. Let Ω(M ) be the space of forms smooth over M . Ω. (M ) is the space of forms smooth over M of degree larger than 1. Over Ω(M ) ⊗ Ω. (M )⊗n ⊗ Ω(M ), we introduce the Sobolev norm associated to the tensor products of Hilbert spaces associated to the operator (dd∗ + d∗ d + 2)k . ωk2,k its associated Sobolev norm. If ω ˜ ∈ Ω(M ) ⊗. (M )⊗n ⊗ Ω(M ), we define k˜ If ω ˜ n = ω0 ⊗ ω1 ⊗ · · · ⊗ ωn ⊗ ωn+1 , we associates the form over the path space: Z ω(dγs−1 , .) ∧ · · · ∧ ωn (dγsn , .) ∧ ωn+1 . (1.1.11) σ(˜ ωn ) = ω0 ∧ 0<s1 <···<sn <1
ω0 is the form which to X 1 , . . . , X r0 associates ω(γ0 )(X01 , . . . , X0r0 ) (ω0 is a r0 -form). We have an analoguous definition the form ωn+1 , but in time 1 now. P ˜ n ∈ Ω(M ) ⊗ Ω. (M )⊗n ⊗ Ω(M ). If for Theorem 1.1.2 [58]. Let ω ˜ = ω ˜ n and ω all k, the entire series X k˜ ωn k2,k φk,˜ω (z) = zn √ (1.1.12) n! P has an infinite radius of convergence, σ(˜ ωn ) is smooth in the Nualart–Pardoux sense over the free path space. The stochastic exterior derivative is continuous over the space of forms smooth in the Nualart–Pardoux sense. Over Ω(M ) ⊗ Ω. (M )⊗n ⊗ Ω(M ), the Hochschild boundary is defined as follows, for ω ˜ = ω0 ⊗ ω1 ⊗ · · · ⊗ ωn ⊗ ωn+1 : ˜ = b0,p ω ˜ + b1,p ω ˜, bp ω
(1.1.13)
b0,p ω ˜ = dω0 ⊗ ω1 · · · ⊗ ωn ⊗ wn+1 X − (−1)i−1 (ω0 ⊗ ω1 · · · ⊗ dωi · · · ⊗ ωn+1 ) ,
(1.1.14)
where i = deg ω0 +
P
1≤i≤n+1
deg ωj−1 , X ˜ = ω0 ∧ ω1 ⊗ ω2 · · · ⊗ ωn+1 − (−1)i ω0 ⊗ · · · ⊗ ωi ∧ ωi+1 ⊗ · · · ⊗ ωn+1 . b1,p ω 1<j
1≤i≤n
(1.1.15)
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Let us introduce the family of semi-norms
P
zn
101
k˜ ωn k2,k √ . n!
Theorem 1.1.3 [58]. b0,p and b1,p are continuous for the previous family of seminorms. bp is a complex. We can do the analoguous operations for the based loop space and the free loop space. We get 3 spaces Nent (Ω(M ), Ω(M ), Ω(M )), Nent (Ω(M )) and Nent (C, Ω(M ), C) endowed with 3 algebraic complexes (we consider forms with values in C, the set of complex numbers). We do not detail the problems which arise from the fact that the iterated integrals map has a kernel. We get some restriction maps from the free path space to the free loop space, and from the free loop space to the based loop space, which remain valid at the algebraic level. More precisely, we have the following theorem [58]: Theorem 1.1.4 [58]. The following diagram of complexes commute: Nent (Ω(M ), Ω(M ), Ω(M ))
→
↓ N P∞ (Path)
Nent (Ω(M ))
→
↓ →
Λrest,L
Nent (C, Ω(M ), C) ↓
→
(1.1.16)
Λrest,Lx
where N.P∞ (Path) denotes the space of forms smooth in the Nualart–Pardoux sense over the free path space. At this stage, we do not know what is the exact image of N.P∞ (Path) under the restriction map. It is a classical result (see [1, 33]) that the Hochschild cohomology in a given degree is equal to the cohomology of the based loop space or of the free loop space, if the manifold is simply connected. The proof is based upon three ingredients: • It is clearly true for the smooth path space. • The path space is a fibration whose fiber is the based loop space and whose basis is the product of the manifold by himself. • A spectral sequence argument. The theorem remains true in the stochastic context [59] if we suppose that the manifold is homogeneous. For that, we consider the stochastic differential equation: dγs = τs dBs ;
γ0 = x,
(1.1.17)
where Bs is a flat Brownian motion in Tx (M ) and τs is the parallel transport over the path γ. . This differential doesn’t realize an isomorphism between the flat model and the curved model, if we consider standard Sobolev space. But it realizes an isomorphism, if we consider Sobolev spaces with regularity conditions of the involved kernels, because the derivative of this map is closely related to some anticipative Stratonovitch integrals. The cohomology (with regularity condition over the kernels of the considered forms) of the flat model is equal to the cohomology of the manifold, because there are natural retract formulas given by the Clark–Ocone formula. The cohomology of the curved models (with regularity condition over the kernels of
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the forms) is equal to the cohomology of the manifold. This shows that the first point has a stochastic analoguous. (We do not need to assume that the manifold is homogeneous.) More precisely, we introduce the notion of limit model. It is the family of flat Brownian motion starting from 0 in Tx (M ), the tangent space of M at x, the starting point x being chosen with the law dx. AR tangent vector is given by a path 1 Xt in Tx (M ), endowed with the norm kX0 k2 + 0 kd/dsXs k2 ds. We can define as before the flat Nualart–Pardoux spaces of the limit model, the only difference is that the parallel transport does not appear. We get: Theorem 1.1.5 [58]. The Nualart–Pardoux cohomology of the flat limit model is equal to the cohomology of the manifold. This theorem reflects the fact that the limit model retracts over the manifold, by using the Clark–Ocone formulas (see [58]). We get: Theorem 1.1.6 [59]. The Nualart–Pardoux cohomology of the flat limit model is equal to the Nualart–Pardoux cohomology of the path space. The law of the Brownian motion (or the Brownian bridge) depends on the metric over the manifold and not only on the topology of the manifold. If there is a sufficiently big group of isometries, this shows that the law of a given Brownian bridge is modulo a transformation, absolutely continuous with respect to the law of any Brownian bridge: it is the purpose of the Albeverio–Hoegh–Krohn quasiinvariance formulas [4, 79]. This transformation is compatible with the regularity assumptions over the kernels of forms. This shows that the second point in the proof of this classical theorem has a stochastic counterpart. The third step comes automatically. Therefore the stochastic cohomology (with regularity conditions over the kernels of the forms) of the loop space is equal to the Hochschild cohomology of the manifold (with Sobolev structures). If we fix the degree of the form, this allows to get that the restriction map from the free path space to the free loop space is onto from N.P∞ (Path) into N.P∞ (Free loops) with some obvious notation. Moreover, with the same type of notations, the restriction map is onto from N.P∞ (Free loops) into N.P∞ (based loops). This allows to get: Theorem 1.1.7 [59]. The Nualart–Pardoux cohomology in degree r of the free loop space of a compact homogeneous manifold is equal to the Hochschild cohomology of the free loop space, if the manifold is simply connected. As a corollary, since the Hochschild cohomology of the free loop space is equal to the cohomology of the smooth free loop space, the Nualart–Pardoux cohomology of the free loop space is equal to the cohomology of the smooth free loop space, for a compact homogeneous manifold. Let us remark that in [59], we consider forms of given degree and not series of forms of any degree. There is, if we consider the based loop space, an additional structure which appears when we multiply loops together. There is a rescaling of
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the two loops which appear. If we consider the (deterministic) Moore loop space, that is the set of loop indexed by R+ which are constant after some time, there is no reparametrization of the loops which appear when we multiply them, such that the multiplication is associative. We say that the determinisitic Moore loop space is an H-space. Its cohomology is a Hopf algebra [72]. We give a beginning of stochastic interpretation of this Hopf algebra stucture for series of forms which belong to all the Sobolev spaces [62]. We especially show that the continuity of the coproduct is closely related to short time asymptotics and large time asymptotics of heat kernels (see [53, 54, 85] for surveys). 1.2. Infinite dimensional index theory A genus associated to a closed manifold is an element of a ring, such that the genus of a disjoint union is the sum of the genera, and such that the genus of a product is the product of the genera. Moreover we suppose that the genus of the boundary of a manifold is zero. Classical genera are the signature of the manifold, the Atiyah– Singer genus which gives spin manifold the index of the Dirac operator. The oriented cobordism ring tensorized by Q is spanned by the even dimensional complex proP 2n+1 jective spaces, so that a genus φ is determined by its logarithm φ(P 2n (C)) z2n+1 . Ochanine associated to each elliptic integral a genus Z z X z 2n+1 dt 2n √ = . (1.2.1) φ(P (C)) 2n + 1 1 − 2δt2 + t4 0 δ is φ(P 2 (C)) and is φ(P 2 (H)). P 2 (H) denotes the the quaternionic projective plane. Such genera play an important role in the so called elliptic homology. If we consider the inverse funtion of this elliptic integral, we get a doubly periodic function called an elliptic function. If we normalize its period lattice to be 2πiZ + 2πτ Z, the genus φq (M ) is a modular form when we have set q = exp[2iπτ ]. Let us recall what is the equivariant index of an operator D. Let D be an elliptic operator which transforms as a section from a bundle ξ− into a section of a bundle ξ+ over a compact manifold M . We suppose that M is endowed with a Riemannian metric and that the bundles ξ− and ξ+ are endowed with Hermitian structures such that we can define the adjoint D∗ of D. We suppose that there is a circle action compatible with the previous data, i.e. commutes with D. The circle leaves over Ker D and Ker D∗ invariant. We consider the virtual character of this representation: (1.2.2) Indq (D) = −TrKer Dq∗ + TrKer Dq . Let us make precise the meaning of this formula. Ker D∗ and Ker D are two representations of the circle (q is the typical element of the circle), because D and D∗ commute with the circle action. Moreover, by elliptic theory, Ker D and Ker D∗ are of finite dimension, and since the elements of the circle group commute, they can be simultaneously diagonalized. Over Ker Dn∗ , q acts by multiplication by q n . Over Ker Dn , q acts by multiplication by q n . We get X Indq (D) = q n (dim Ker Dn − dim Ker Dn∗ ) . (1.2.3)
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The index is given by a finite Laurent series whose coefficients depends on integrals over the fixed point set of the symbol of D and depend on the behaviour of the normal bundle of the fixed point set. Witten [87] put τ real such that q is an element of the circle. He assumes the existence of the following data: • There exists a spin bundle over the free loop space, which is compatible with the natural circle action. It is possible when the first Pontryaguin class of the manifold is equal to 0. • There exists a Dirac operator D and a signature operator which acts over the sections of the spin bundle over the free loop space or over the sections of the spin bundle tensorized by himself. These operators are invariant under rotations. • When q acts over Ker D− and Ker D+ by q n , the dimension of the corresponding subspaces is finite, although the global dimension of the kernels of these operators is infinite. The equivariant indices of these infinite dimensional operators is given by localization. Let us make precise these statements. Let X tk Λk (T (M ) ⊗ C), Λt (T (M )) =
(1.2.4)
k≥0
where Λk (E) is the kth exterior of the complexified tangent space and let X tk S k (T (M ) ⊗ C) St (T (M )) =
(1.2.5)
k≥0
where S k (E) is the kth symmetric power of the complexified tangent space. Let ˆ ) be the Atiyah–Singer genus of the manifold. A(M Let Z n − 24 ˆ )ch ⊗ Sqm (T (M )) ψ(q) = q (1.2.6) A(M M
and ψ 0 (q) = q − 24 n
Z ˆ )ch ⊗ Sqm (T (M ))ch ⊗ Λqm (T (M )) A(M
(1.2.7)
M
where n is the dimension of the manifold and the tensor product in the previous formula is taken over the positive integers m. Modulo a normalizing constant ψ 0 (q) is the universal elliptic genus and ψ(q) n is another genus called the Witten genus. q − 24 is called the anomaly term. ψ(q) is supposed to be the equivariant index of the Dirac operator over the loop space and ψ 0 (q) is supposed to be the equivariant signature of the free loop space. The equivariant Euler–Poincar´e number is equal by localization to the Euler–Poincar´e number of the manifold. Taubes [83] gives a construction of the Dirac operator over the normal bundle of the constant loops. The normal bundle of the constant loops is constituted of the family of based loop spaces in the family of tangent spaces. There is a natural polarisation which arises from the Fourier expansion in the tangent space.
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[83] keeps the Fourier mode which corresponds to n > 0 in the Fourier expansion. √ It introduces a Gaussian measure in the associated fiber with covariance −∆. In the fiber, he considers the supersymmetric Fock space associated to this measure, where we consider only positive modes for the Fermionic Fock space. He constructs ∗ in the fiber the operator ∂ ∞,x and its adjoint ∂ ∞,x . Under the assumption that the manifold is spin, he introduces the finite dimensional operator D with an infinite dimensional auxiliary bundle which is the supersymmetric Fock space bundle and he builds a limit Dirac operator: ∗
D∞ = D + ∂ ∞,x + ∂ ∞,x .
(1.2.8)
The equivariant index of the Dirac operator of Taubes is given by (1.2.6) modulo the anomaly term. This difference arises because we consider in the normal bundle a mixture of a spin structure (we work with the finite dimensional Dirac operator) and a SpinC structure (we work with ∂ ∞,x ). We can make the following observations: √ the Gaussian measure with covariance −∆ has as support in distributions and cannot be generalized to the curved setting. The goal is therefore the following: if we consider the Brownian bridge measure, there is a Hilbert tangent space and integration by parts associated to it [10, 25, 42, 55, 56, 65] such that the adjoint of operators are densely defined. [56] constructs an Ornstein–Uhlenbeck operator invariant under the circle action over the free loop space. So, it is expected that it is possible to extend these construction to non scalar operators, if we consider the Brownian bridge measure such that we can speak of the adjoint of an operator. Moreover, there is a deformation of the Hilbert space of the theory by considering the Brownian bridge in time 2 . When → 0, the fluctuations of the measure near the constant loops are given by a limit model in the √ sense of [83]: instead of using the Gaussian measure associated to the covariance −∆, we get the flat Brownian bridge measure starting from 0. Such limit theorems were already done in the level of integrals computations by Bismut for the finite dimensional index theory (see [11, 53, 54, 85]). So the programme is the following: • Compute a non scalar operator for the curved Brownian bridge in time 1. • Compute a corresponding operator over the limit model, whose index is known. • Show that the operator over the limit model is the limit in some sense of the curved non scalar operator when we take the Brownian bridge measure in time 2 . This programme is partially achieved in [47, 57] and [66]. [47] begins to consider the case of a non equivariant de Rham operator over the free loop space. The bundle is the full fermionic Fock space bundle. It determines an infinite dimensional bundle over the free loop space. It is split into a countable sum of finite dimensional bundle over the free loop space by considering the Fourier expansion. [47] introduces a connection which preserves this decomposition, whose adjoint is known, because the connection is compatible with the metric. It introduces an auxiliary operator, which multiplies each Fourier mode by a number A(n).
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He gets an operator dreg which should be homotopic to the exterior derivative (there is no Levi–Civita connection over the free loop space, because the tangent space is not stable by Lie bracket. In particular dreg is not a complex). [47] computes the adjoint d∗reg and the Laplacian ∆reg associated to dreg . He can formulate the following conjecture: Conjecture. dreg + d∗reg has a self adjoint extension. The heat semi-group has a trace if A(n) > |n|γ for a suitable positive γ. The index of dreg + d∗reg is the Euler–Poincar´e number of the manifold. Let us remark that these computations are not invariant by rotations because there is a full Euler–Poincar´e number of the free loop space (and not only an equivariant one). This conjecture is motivated by the following observation: we can compute the limit operators where the conjecture is satisfied. [47] used the finite dimensional Lichnerowicz formula in order to show that the heat semi-group as a trace. We used the explicit formulae of [3] and [80] for the Laplacian over a full supersymmetric Fock space in terms of the bosonic and the fermionic number operators. Moreover this limit operator is a deformation of the non limit operator. It remains to remark that the index is stable under deformation. Let us give some more details. We decompose the tangent Hilbert space of a loop in a countable orthogonal sum of finite dimensional bundles isomorphic to the pullback of the tangent bundle of the manifold M by the evaluation map γ. → γ0 . We have for n = 0, (1.2.9) X0 (ei )t = τt (ei + t(−1 + τ1−1 )ei ) . If n > 0,
√ Z t cos(2πns)dsei Xn (ei )t = 2τt
(1.2.10)
0
and if n < 0, Xn (ei )t =
√ Z t 2τt sin(2πns)dsei .
(1.2.11)
0
ei is a local section of orthonormal basis depending only of γ0 of Tγ0 (M ). We introduce a connection by deciding to take the derivative of ei only, for the pullback of the Levi–Civita connection of the pullback bundle of the tangent bundle by the evaluation map γ. → γ0 . We choose over each piece of the tangent bundle of the loop space constituted by the Xn (ei ) the metric kei k2 . The connection is metric compatible over the tangent space. The fiber is the fermionic Fock space. Since the connection is metric compatible, it passes to the fermionic Fock space and gives a metric compatible connection over the fermionic Fock space. Theorem-Definition 1.2.1 [47]. dreg (σ) =
X
A(n)Xn (ei ) ∧ ∇Xn (ei ) σ
(1.2.12)
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is defined over the dense core of sections of the Fock bundle defined as follows: we consider a finite sum of XJ = Xn1 (e1 ) ∧ · · · ∧ Xnr (er ) multiplied by cylindrical functionals. It is possible to do that if A(n) < C|n|γ for γ < 1/2. Over this core, d∗reg (σ) can be computed. ∆reg = (dreg + d∗reg )2 is defined over Γ if A(n) < C|n|1/4 , and has a self-adjoint extension. We define a limit model as before, by taking the family of flat Brownian loops. dl , d∗l and ∆l can be defined as before, with some obvious notations, but no parallel transports appear. Theorem 1.2.2 [47]. (i) If A(n) > C|n|γ for some γ < 1/4, the heat semi-group exp[−t∆l ] has a trace. (ii) If A(n) is not equal to 0 for all n, the index of dl + d∗l is equal to the Euler–Poincar´e characteristic of M. We define afterwards the B-H-K measure in time 2 : dµ =
p2 (x, x)dx ⊗ dP2 ,x R 2 M p (x, x)dx
(1.2.13)
dP2 ,x is the law of the Brownian bridge in time 2 starting from x. We can define the operators d,reg , d∗,reg and ∆,reg for this measure after using some rescaling. We get: Theorem 1.2.3 [47]. There exists an operation over a dense core of sections Γ0 called Bismut’s operation such that in law, when → 0: (i) B σ → σl , (ii) (d,reg + d∗,reg )B σ → (dl + d∗l )σl , (iii) ∆ B σ → ∆l σl . Moreover, the set of limit sections is dense in the set of sections of forms over the limit model. In the previous set up, we always work in L2 . The purpose of [66] is to study the Euler–Poincar´e characteristic of an orbifold. Let G be a finite group of isometries of a compact Riemannian manifold M . Let M/G be the associated orbifold. Let M g be the fixed point sets of g ∈ G. It is classical that the Euler–Poincar´e characteristic of the orbifold M/G satisfies to: 1 X χ(M g ) . (1.2.14) χ(M/G) = |G| g∈G
The physicists [24] have introduced another Euler–Poincar´e characteristic: 1 X χ1 (M/G) = χ(M g ∩ M h ) . (1.2.15) |G| gh=hg
It should be the Euler–Poincar´e number of the loop space of the orbifold. Let Lg the space of twisted loops starting from any x and arriving in gx. The loop space of the orbifold appears as the quotient space by the action of G of the union of the sectors constituted by a twisted loop space. It is an orbifold of twisted loops. We apply
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(1.2.14) to each sector Lg : we see that the condition gh = hg appears. [66] does the same computation as in [47], but for twisted loops, such that the measure over the loop going from x to gx concentrates over the measure over the fixed point set of g. A gaussian fluctuation appears, where the normal bundle of the fixed point set plays an important role. [66] performs at the level of operators the same computation as [12] for the finite dimensional Lefschetz formulas at the level of integrals. [38] has defined a new genus for complex manifolds, which is called the elliptic genus of level N. It considers over the free loop space of a complex manifold, the polarization which is deduced from the polarization of the manifold. It considers the ∂ ∞ operator associated to this polarization and He tensorized it by the series of the exterior bundle in the holomorphic fermions weighted by a suitable number. Later on [38] applies the localisation formula, and finds a divergent series: but if his weights are related to a root of the unit, [38] can renormalize this divergent series in a convergent series. [57] considers the case of a K¨ahler manifold such that the Hilbert space structure of the tangent space of a loop is compatible with the polarization over the loop space. It introduced the Brownian bridge measure in this set-up. It computes rigorously a stochastic regularized ∂ ∞ operator and its adjoint. Later on [57] performs the limit theorem which allows to get a limit ∂ operator over the limit model with respect to the polarization which is deduced from the complex structure of the manifold. He can compute formally his equivariant index. The series diverges, but may be renormalized in order to come back to the formula of [38]. [47] considers the polarization which arises from Fourier expansion. For that it needs to extend the Fourier expansion for curved loops in an equivariant way. It is only possible when the holonomy around a loop is close from the identity. [46] can split the complexified tangent Hilbert space into a countable sum of finite dimensional bundles which are invariant under the circle action. They are closely related to the eigenvalues of the covariant derivative over a loop. This allows [47] to extend for small loops the construction of [83] of the Dirac operator for the limit model. All the computations are done by hand. The limit theorems are performed. [47] find in the limit an operator in the manner of Taubes, modulo some coefficients which appears because of the rigorous choice of the Hilbert space structures used in [47] for the complexified tangent space. The formulas of [73] which give the square ∗ of ∂ + ∂ in terms of antiholomorphic number operators allow to conclude. The limit operator of [47] has an equivariant index given by the formula (1.2.6) modulo the anomaly term. More precisely, we operate over a small “neighborhood” of the fixed point set under the natural circle action of the loop space, which is defined by the fact that τ1 is close of Id, such that there exists a version of log τ1 which belongs to all the Sobolev spaces. We decompose a tangent vector over a “small” loop over the basis constituted by Xn (ei ): τt exp[2iπnt] exp[−t log τ1 ]ei Xn (ei )t = √ Cn2 + 1
(1.2.16)
where ei is a local orthonormal basis of the pullback of the tangent bundle of the manifold M by the evaluation map γ. → γ0 . For each n, we get a finite dimensional
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bundle Tn (γ) over γ. We have to take care, because, unlike the Euler–Poincar´e characteristic of the loop space, there is for the Dirac operator over the loop space only one equivariant index. So the construction we have to do in the present context has to be invariant by rotations. The decomposition of T (γ) into the orthogonal sum of Tn (γ) is invariant under rotation. We get a connection by taking only the derivative of the ei in Tn (γ) for the pullback of the Levi–Civita connection by the evaluation map. This connection is not invariant by rotation. We average it under the circle action. We get a connection ∇inv invariant under the circle action. We have an extension of the finite dimensional spin bundle over the fixed point set by looking the space of τt exp[−t log τ1 ]s where s is a spinor over γ0 . We get a bundle S over a small “neighborhood” of the fixed point set. We extend by hand the Clifford multiplication (see [47, p. 270]) of vector fields of the type X0 (ei ). . We P put Tγanti = n>0 Tn (γ). The fiber is S ⊗ Λ(Tγanti ) where Λ(Tγanti ) is the Fermionic Fock space associated to Tγanti . Theorem-Definition 1.2.4 [47]. There is a dense core of sections invariant by rotations of S ⊗ Λ(Tγanti ) such that D=
X n<0,i
A(n)X−n (ei ) ∧ ∇inv Xn (ei ) +
X
X0 (ei )∇inv X0 (ei )
(1.2.17)
i
is defined over it if A(n) < C|n|γ for some γ < 1/2. Moreover D has an adjoint. D + D∗ is called the regularized Dirac operator defined over an “small ” enough neighborhood of the constant loop. It is closable. Short time asymptotics can be defined as well as for the Euler–Poincar´e characteristic of the loop space, but we have to take care because we have to define Bismut’s dilatation over a core invariant by rotation and dense. In the limit, we find an operator a little bit different than the operator of Taubes, but with the same equivariant index. Let M be a compact oriented even dimensional manifold. A spin structure over M is a lifting of the frame bundle by the spinor group spin which is a Z/2Z extension of SO(n) where n is the dimension of M . We choose a Z/2Z cover of SO(n), because the spin representation in finite dimension is a projective representation of SO(n) with fiber Z/2Z [76]. There is a topological obstruction to constructing a spin structure over M [34], which is the second Stiefel–Whitney class of M . It cannot be seen generally at the level of differential forms of the de Rham theory over M . Unfortunately, [47] gives only an approach to the construction of the Dirac operator over the free loop space, because the fiber is not what it should be. Namely, the construction of the suitable fiber is related to the theory of representations of a loop group [16, 76]. Namely, the projective spin representation of a loop group is related to a central extension of this loop group by the circle and not by Z/2Z. If we consider the case of the smooth free loop space, the structure group is the loop group of SO(n). A spin structure is related to a lifting of the principal bundle in a loop group by a central extension of this loop group [49, 86]. It is called a string structure. The obstruction is related to the first Pontryaguin class of the original bundle [18, 22, 69, 78, 87], and more precisely to its transgression; but of
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course these considerations work only for the smooth loop space. On the other hand, the Brownian bridge measure is used in order to compute the adjoint of this type of operators [10, 39] and the Brownian bridge is only continuous. [18] has given a geometrical construction of such lifting, called a string structure, in the case of the based loop space of smooth loops Lfin M . The purpose of [60] is to try to unify the considerations of [18] over the smooth loop space with the stochastic considerations of [10, 46] over the continuous loop space. In order to construct a string structure, [18] introduce a commutative diagram. The holonomy over a loop plays an important role in this commutative diagram. Namely let us consider a G principal bundle P over M . [60] considers the based loop space LP of P of loop starting from (x, e) in P : it is a LG principal bundle. [18] construct a U1 ˜ which is a LG ˜ bundle for a given central extension of bundle over LP denoted LP LG. In order to give an explicit construction, they suppose that LP and LG are simply connected. Under this hypothesis, we can construct this bundle for a given representative of a integral cohomology class of LP of degree two. They avoid in this way all torsion phenomena: a circle bundle over a simply connected manifold is indeed determined by its curvature. So the construction of this lift is strongly related to a theory of forms and a theory of construction of bundle by starting from a given cohomology class. The treatment is simplified, because the loop space of P is supposed to be simply connected. In [10] or in [46] the continuous loop space is introduced. A suitable Hilbert space which is the tangent space of a continuous loop is introduced in order to get integration by parts formulas [55, 56, 65]. [46] developed a theory of forms over the loop space without differential operations. In [58, 59] a theory of stochastic cohomology of the Brownian loop is developed, where the stochastic Chen forms play a key role. The purpose of [60] is to go one step further: that is to construct bundles by starting from a stochastic 2 form. Generally, the law of the deformed stochastic loop is a singular law with respect to the Brownian bridge law. For this reason, we restrict our attention to the case where the form is a stochastic Chen form. This form is defined for a larger class of processes. [60] starts from a closed Chen form of degree 2 with integral values over Lfin M . An example is the transgression τ (ω) of a representative ω of an element of H 3 (M, Z). A surface in the loop space is constituted by a volume in the manifold. The main property is that the integral of the transgression over a surface in the smooth loop space is equal to the integral of the underlying form over the underlying volume in M . Let us recall [22] how we construct a line bundle over a simply connected manifold (of finite dimension or of infinite dimension) by starting from its curvature ω. ˜ It is the set of couples of the form (l. (x), α). (l. (x) is a path starting from a point of reference and arriving at x in the manifold and α is an element of the complex line. We say that the couple (l. (x), α) and (l.0 (x0 ), α0 ) are equivalent if the end points of the paths x and x0 are equal and if Z 0 α = α exp −2iπ ω ˜ . (1.2.18) S
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S is any surface bounded by the loop constituted of l. (x) and l.0 (x0 ) circled in the opposite sense. Such surface exists because R we suppose that the manifold is simply connected. In the stochastic context, S τ (ω) leads to the study of some stochastic integrals. [60] deals with some distinguished surfaces in order to be able to integrate this transgression over a given surface in the stochastic loop space. Instead of constructing the bundle globally as in (1.2.18), [60] constructs it by using a system of transition functions. But [60] cannot construct the stochastic line bundle associated to this Chen form over the stochastic based loop space. Namely the transition functions are only almost surely defined. We require that the transition functions belong to all the Sobolev spaces, in order to give some rigidity to them. Let us recall namely that all bundles over a finite dimensional manifold are measurably trivial. But [60] constructs the Hilbert space of sections of this formal line bundle, and a connection which operates over the space of sections. [60] hopes that there are some applications to the geometric quantization of the loop space [17]. In order to do this, we suppose that the based loop space of loops with finite energy Lfin (M ) is simply connected. [60] can determine explicitly the system of the measurable transition functions. A connection allows us to define the space of C 1 Sobolev sections of this formal line bundle. The bundle which is gotten is an extension in a certain sense of the underlying bundle over the finite energy loop space. Namely, if [60] takes the polygonal approximation of a continuous loop, the transition functions are surely defined. They are in fact the transition functions which defined the underlying line bundle over the smooth loop space. Moreover they tend almost surely to the transition functions which define the formal line bundle over the continuous loop space, when the length of the subdivision goes to infinity. Definition 1.2.5 [60]. Let Lx (M ) be the based loop space of the Riemannian manifold M endowed with the Brownian bridge measure. Let us suppose that M is two connected, and consider a 3-form ω which is closed and Z-valued. We can R1 ˜ where γ denotes the typical construct the 2-form over Lx (M ) 0 ω(dγs , ., .) = ω continuous loop in M . There exists a system of open subsets Oi for the uniform topology which is a cover of Lx (M ) and a set ρi,j of Brownian functionals with values in the unit circle of C, defined over Oi ∩ Oj , which are almost surely defined and which belong to all the Sobolev spaces. Over Oi ∩ Oj , they are such that ρi,j ρj,i = 1 almost surely and over a triple intersection Oi ∩ Oj ∩ Ok , they satisfy almost surely to the relation ρi,j ρj,k ρk,i = 1. Moreover, there exists a set of one forms over Oi denoted by Ai with values in the Lie algebra of thae unit circle of C, such that almost surely over Oi ∩ Oj we have: Ai = Aj + ρ−1 i,j dρi,j and such that √ ˜ . A section of the formal line bundle associated to the set of ρi,j is a dAi = −12π ω collection of random variables αi from Oi into C such that αj = αi ρi,j over Oi ∩ Oj . The Hilbert space of sections is the set of sections ψ such that E[|ψ|2 ] < ∞ where |αi | = |ψ| is intrinsically defined. [60] considers a G principal bundle P over M . We suppose that G is simple and simply connected. Moreover, [60] supposes that the based loop group of loops ˜ finG of finite energy Lfin G is simply connected. There exists a central extension L
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˜ fin G (the of Lfin G. Let us suppose that there is a unitary representation Spin of L reader can find references in this direction in [16] or [76]), which is a Hilbert space. ˜ fin G bundle which We construct over LM , the set of the Brownian bridge, an L extends the natural Lfin G bundle which exists over LM . [60] uses the parallel ˜ is only formal, in the sense that the transition transport over a loop. This bundle Q functions are only measurable, because they belong only to all the Sobolev spaces. This allows us to construct, by using the associated transition functions, a formal linear bundle whose fiber is Spin. But [60] can define rigorously the space of sections of this formal linear bundle, which belong to some Lp spaces. The Dirac operator over the based loop space should act over the Hilbert space of section of this formal bundle. [60] has the same topological obstruction which are involved with the first Pontryaguin class P of M as in the smooth case. Definition 1.2.6 [60]. Let Q → M be a principal bundle over a compact Riemannian manifold with compact structural group, which is supposed to be simply connected. We suppose that M is two connected. Let Lfin G be the based loop group of G of loops with finite energy. Let Le Q the space of loops in Q of the form τsG gs where gs is a path in G of finite energy and τsG is the parallel transport over a random loop in M for the Brownian bridge measure, if the principal bundle Q is endowed by a connection. Associated to Le Q, there exists a system of subsets Vi of Lx M such that ∪Vi = Lx M almost surely. Over Vi ∩ Vj , there exists a natural measurable application gi,j with values in Lfin G such that gi,j gj,i = e. almost surely over Vi ∩ Vj and such that almost surely over Vi ∩ Vj ∩ Vk , gi,j gj,k gk,i = e. . We suppose that the Pontryagin class of the bundle is equal to 0. There exists a refinement ˜ fin G, of Vi , still denoted by Vi , and a measurable application ρi,j from Vi ∩ Vj into L the basical central extension of Lfin G, which satisfies the following conditions: (i) Over Vi ∩ Vj , ρi,j ρj,i = e˜ almost surely. (ii) Over Vi ∩ Vj ∩ Vk , ρi,j ρj,k ρk,i = e˜ almost surely. ˜ fin G into Lfin G. (iii) πρi,j = gi,j almost surely, where π is the projection from L ˜ fin G. The space of measurable sections Let us suppose a unitary representation of L of the associated bundle to the system of transition functionals ρi,j is given by a set of measurable functionals in the representation space called ψi defined over Vi , satisfying ψj = ρj,i ψi almost surely over Vi ∩Vj . The Hilbert space of sections is the space of sections such that E[kψk2 ] < ∞, where the random variable kψk = kψi k over Vi is intrinsically defined. [61] considers the case of the free loop space. [22] has established that for the free loop space a string structure exists if the first Pontryagin class of the bundle vanishes (we suppose that the structural group is simply connected). For that [22] use the Chern–Simons theory. [61] shows that it is possible to follow closely the considerations of [22] in the stochastic context. In particular, [61] can define rigorously an Hilbert space of sections of spinor fields over the free loop space. The circle action lifts to this Hilbert space. More or less, [61] considers the associated bundle over the free loop space whose fiber is a central extension of a loop group.
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The theory of central extension of loop groups works well for finite energy loop groups. In order to study this representation, [61] introduced a new measure over a path group. It is given by the differential equation: dgs = gs (C + Bs )ds .
(1.2.19)
C follows an independent Gaussian law of B. of average 0 and coavariance Id in the Lie algebra of G. [61] introduces a Hilbert tangent space related to this equation, which allows one to get integration by parts formulas. These integration by parts formulas allow one to disintegrate the measure over the pinned path space going from e and arriving in g. The tangent space consists of vector fieldsRof the form gs Ks where Ks has two 1 derivatives. The Hilbert structure is kK00 k2 + 0 kKs00 k2 ds. If K. is deterministic, for a cylindrical functional: E[hdF (g. ), g. K. i] = E[F (g. )div(g. K. )],
(1.2.20)
where div(g. K. ) belongs to all the Lp . This allows to shows that the law of g1 has a smooth density and that the density of g1 in e is strictly positive. Let us consider the loop space of a compact manifold: that is the space of smooth applications from the circle into the manifold. The propagation of a loop is related to the conformal field theory, because we consider a path integral over the set of applications from a Riemann surface with boundary into the manifold with the conformal group as symmetry group. When there is no boundary, we consider random tori, and the integral over all the random tori for the given action gives the partition action of the theory. It is involved with a renormalization procedure and it is called the nonlinear σ-model. In the flat case, it corresponds to the free field, and the measure lives over distributions. If we add some fermionic part to the nonlinear sigma model, the partition function becomes the partition function of a supersymmetric nonlinear σ-model and gives the index of some relevant operator over the loop space. In particular, Witten [87] deduces the modularity properties of the universal elliptic genus from this path representation of the infinite dimensional version of the index of the signature operator over the free loop space. [5, 26] and [66] have constructed by using the theory of Dirichlet forms processes over the Brownian motion. These constructions require no renormalization procedure. In the case of [5], the process which is associated to the Dirichlet form is invariant under rotation. 1.3. Stochastic Wess–Zumino–Witten models The Wess–Zumino–Witten model is involved with the perturbation theory: we consider the exterior derivative over the loop space, its adjoint and we perturb it by multiplying by a one form. When this one form is dF for a suitable functional, we call this functional the Wess–Zumino–Witten functional. We consider the associated Laplacian and its semi-group which is given by a path integral. Witten [86] remarked that this perturbation of the free operator is related to the Morse theory, and this remark was fully exploited in [15] and [37] in finite
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dimension. The purpose of [63] is to try to give a rigorous formulation to the argument of Witten in infinite dimension. The purpose of [63] is to try to generalize this construction to the case of the symplectic Morse theory over a path space. The model used here is quite different from the model used in [47, 57] and [66], and is based upon the work of Guilarte [36]. [63] gives a stochastic interpretation of the Wess–Zumino–Witten model of [36], by using stochastic analysis. For this purpose, let us consider a compact symplectic manifold M and two compact lagrangian submanifolds in transverse positions L and L0 . Rabinowitz, Chaperon, Conley, Zehnder, have introduced the space P (L, L0 ) of paths going from L to L0 in order to relate the topology of the full manifold to the structure of the intersection points of L and L0 . This initial introduction was fully exploited by Floer later. How can it be seen? Over the configuration space, we consider the closed one form σ = τ (ω) which degenerates when we are over the constant loops. This form, at least when the path space is simply connected, can be integrated: σ = dF , for a suitable functional F , which is called the symplectic action. The global topology of the configuration space, which is involved with the topology of L, L0 and M is therefore related to the intersection points of L and L0 by means of the Morse theory. This Morse theory is the purpose of the Floer homology. Morse theory, as it was pointed out by Witten, is related to the Wess–Zumino–Witten model by considering the complex d + dF , its dual d∗ + idF and the associated Laplacian ∆F . Moreover the Morse theory arises when we work over a small neighborhood of the critical points: Guilarte pointed out that in a Morse system near the critical points over the path space, the Wess–Zumino–Witten laplacian is a supersymmetric infinite dimensional harmonic oscillator, whose structure is known. Near the constant paths, F is an iterated integral, and we meet the problem that the Hessian of F has an infinite number of negative eigenvalues as well as positive eigenvalues. A good understanding of this is needed to introduce the Dirac sea [36]. The purpose of [63] is to explain some parts of [36]. In order to compute the adjoint of an infinite dimensional operator, physicists use Lebesgue measure over the path space, which does not exist. [63] replaces it by the Brownian bridge measure, which allows to define Sobolev spaces and others functional spaces. [63] introduces after [10] and [46] a suitable tangent space chosen in order to get integration by parts formulas which depend on a parameter. This allows us to define the bundle of forms over the path space: it is a fermionic Fock bundle. Moreover, the form σ = τ (ω) which is a closed Chen form (see [46, 58]) can be integrated if we suppose that the path space is simply connected. [63] gets a stochastic functional which checks for λ > 0 small enough: E[exp[λ|F |]] < ∞ .
(1.3.1)
Let us explain how F is constructed. Theorem-Definition 1.3.1 [63]. Let Bn be a small tubular neighborhood for the uniform distance of a polygonal path γn of P (L, L0 ). There exists a curve ltn (γ)
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which joins over Bn the path γ to a reference path γref such that s → lt (γ)s is a ∂ n lt (γ)s is a semi-martingale. We put, over Bn : semi-martingale and s → ∂t Z Z ∂ n n Fn = (1.3.2) τ (ω) = ω ds lt (γ)s , lt (γ)s dt . ∂t ln [0,1]×[0,1] t (γ) Over Bn ∩ Bm , Fn = Fm , because P (L, L0 ) is supposed simply connected and τ (ω) is closed. Therefore the set of local functionals Fn glue in a global functional F. [63] introduces a connection which preserves the symmetry between L and L0 , the two lagrangian submanifolds, which arise when we change the sense of the time. This allows to define a regularized exterior derivative dr and a regularized Wess–Zumino– Witten operator dr.W.Z.W after performing the scalar gauge transform associated to exp[λF ] over the stochastic regularized exterior derivative. The interest to choose a connection is that [63] can compute the adjoint of dr.W.Z.W and the Wess–Zumino– Witten laplacian ∆r.W.Z.W . We put (1.3.3) Tγ = Tγ (L) ⊗ Tγ (based) ⊗ Tγ (L0 ), where Tγ (L) = {Xt = τt (1 − t)X0 ; X0 ∈ TL },
(1.3.4)
Tγ (L0 ) = {Xt = τt tτ1−1 X1 ; X1 ∈ TL0 },
(1.3.5)
Tγ (based) = {Xt = τt Ht ; H0 = H1 = 0} .
(1.3.6)
Over Tγ (L), we put kXk2 = kX0 k2 . Over Tγ (L0 ), we put kXk = kX1 k2R. 1 Over Tγ (based), we put kXk2 = 0 kd/dsHs k2 ds. We decide that these three pieces of a tangent space of a loop are orthogonal. We have an orthonormal basis of Tγ (based) given by Fourier expansion: if n > 0 Xn (ei )t = Cτt
sin(2πnt) −1 τ1/2 e1 , n
(1.3.7)
where ei is an orthonormal basis of Tγ1/2 (M ). In n < 0, Xn (ei )t = Cτt
cos(2πnt) − 1 −1 τ1/2 ei , n
(1.3.8)
where ei is an orthonormal basis of Tγ1/2 . We choose a connection by taking the pullback of the Levi–Civita connection of the tangent bundle of M by the evaluation map γ. → γ1/2 . In other words, we take only the derivative of the ei in order to get a Calculus where L and L0 play a symmetric role. Let (1.3.9) Y (ei ) = τt (1 − t)ei for a local orthonormal basis of Tγ0 (L) and let Y 0 (e0i ) = τt tτ1−1 e0i
(1.3.10)
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for a local orthonormal basis e0i of Tγ1 (L0 ). Over Tγ (L) we take the pullback of the Levi–Civita connection of the lagrangian submanifold by the evaluation map γ. → γ0 . We perform a similar construction for Tγ (L0 ). We take over Tγ the sum of the three connections: let us remark that L and L0 play a symmetric role in this construction. Definition-Theorem 1.3.2 [63]. There exists a dense core of sections of the fermionic Fock bundle such that X X ∇Xn (ei ) ∧ Xn (ei ) + ∇Y (ei ) ∧ Y (ei ) dr.W.Z.W = n6=0
+
X
∇Y 0 (e0i ) ∧ Y 0 (e0i ) + dF ∧
(1.3.11)
acts over it. Moreover, dr.W.Z.W has an adjoint d∗r.W.Z.W . The supercharge Qr.W.Z.W = dr.W.Z.W + d∗r.W.Z.W is symmetric closable and the Laplacian ∆r.W.Z.W = Q2r.W.Z.W is positive symmetric and admits a self-adjoint extension. In order to try to recover some topological information, [63] considers the Brownian bridge in small time. The probability law concentrates over the intersection points of the two lagrangians submanifolds. [63] performs the limit theorems, which allow us to study the fluctuations over the intersection point of the model. [54] finds Gaussian models, and the limit Wess–Zumino–Witten functional is strongly related to the area functional of Paul L´evy. These limit computations are very similar to those of [47, 57] and [66]. The reader can find in [44] computations which are similar in the domain of quantum field theory. But unlike these cases, the limit model is new and the limit probability measure is not related to finite dimensional index theory [11, 12]. In particular [63] chooses the couple (x, y) ∈ TL × TL0 with the probability law exp[−1/2kx − yk2 ]. This seems to be new in the probabilistic literature. At the limit, [63] finds a Gaussian supersymmetric harmonic oscillator. [2] has studied such operators in the domain of quantum field theory. [63] can study its behaviour by using a Morse system in infinite dimension in order to diagonalize it; it is strongly related to the study of the Ramer functional. The unique groundstate which is in L2 is exp[λFl ]. In particular, there is no cohomology, except for the dimension 0. As pointed out by [36], the good understanding of this limit harmonic oscillator needs the introduction of the Dirac sea, in order to get non trivial cohomology groups at the limit. This explains why the Floer homology is a middle homology theory of the path space [8]. [63] speaks of stochastics complexes: namely, the price to pay in order to be able to compute the adjoint of the exterior derivative is that we have only to consider an operator homotopic to the exterior derivative with its Wess–Zumino–Witten perturbation, by using suitable connections (see [55, 56]). The problem to define a complex is strongly related to define an anticipative Stratonovitch integral, because we have to take the covariant derivative of the parallel transport, which has a covariant derivative without finite variation.
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Following the lines of [58, 63] defines a stochastic Witten complex over the configuration space, which is continuous for forms which belong to all the Sobolev spaces in the Nualart–Pardoux sense: they are involved with the regularity of the kernels of the derivatives. In particular, [63] meets the problem that exp[λF ] belongs only to some Lp , and not to all the Sobolev spaces. Therefore, it seems that the cohomology of the Witten complex is not immediately related to the cohomology of the configuration space, because the gauge transform is not in all the Sobolev spaces, as can be checked on the limit model. On the other hand, dF is a Chen form: over Chen forms, dF acts as a shuffle product [62]. This allows to define a Witten–Hochschild complex; there is a map between this algebraic model [68] and the geometrical model by using Chen iterated integrals. Over the introduced Hochschild space, [58] defines Sobolev norms, such that the Witten–Hochschild complex is continuous and such that the map, to which elements of the Sobolev Hochschild space associates the corresponding stochastic Chen iterated integral, is continuous. Surveys about the Floer homology can be found in [8, 41] and [81]. 1.4. Other contributions The first person to study forms in infinite dimensional analysis is Ramer [77]. He studied the case of a Wiener manifold: it is defined by charts the transitions maps are of the type I + K. We consider a triple E ∗ ⊆ H ⊆ E where E is a Banach space and H is a Hilbert space. I is the identity of E and K takes its values in the Cameron–Martin space H and checks the following condition: the H-derivative of K is an Hilbert–Schmidt operator. We recall that in infinite dimensional analysis we always take the derivative of functionals over a smaller subset than the set where the functional is defined. He considers as set of form the set ΛE ∗ and as set of finite codimensional forms the set ΛE. These notions are compatible with the systems of charts. He can therefore define a bundle of forms and of finite codimensional forms. He constructs an exterior derivative which transforms a form of codimension n into a form of codimension n − 1. The second person to consider forms in infinite dimensional analysis is Shigekawa [79]. He introduces an abstract Wiener space. Over each element of this abstract Wiener space, he built the fermionic Fock space of the Hilbert space associated to the abstract Wiener space. He looks therefore at an infinite dimensional bundle over the abstract Wiener space (called the supersymmetric Fock space) which in this situation is trivial. He can construct by hand an exterior derivative, compute its adjoint and the associated Laplacian. It is the sum of the bosonic number operator (or the Ornstein–Uhlenbeck operator) and of the fermionic number operator. Therefore he can compute explicitly its kernels, which are identified to the stochastic cohomology groups of the model. Only forms of finite dimension or infinite sum of forms of finite dimension are considered. Simultaneously, Smolyanov [82] has considered a Hilbert space, and has introduced forms of finite codimension k as smooth measures with values in the exterior algebra of degree k of H. There is a natural pairing between forms of codimension
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k and forms of dimension k. By duality, we can compute the exterior derivative of such a form. If we consider a finite codimensional manifold of the Hilbert space, given locally by graphs, [82] can integrate a finite codimensional form over it and give a Stokes formula. This submanifold is surely defined. In [82] and in [77], the source of the definition of finite codimensional form comes from the following finite dimensional observation: over a finite dimensional manifold there is a natural pairing between form of degree k and forms of degree n − k, (n is the dimension of the manifold), given by the integration over the manifold of a form of degree n. Kusuoka [52] gives a notion of differentiable forms for a finite codimensional manifold of the Wiener space: instead of being defined surely as graphs, his submanifolds are defined globally by using an equation as it generally done in quasi-sure analysis [3]. His submanifolds are defined modulo a set of capacity 0. The tangent space is a subspace of the Cameron–Martin space. The associated bundle of forms is constructed by the set of Fermionic Fock spaces of the tangent set of a path. He can compute an exterior derivative, since the tangent space is stable by Lie brackets, and its adjoint, since the integration by parts over the Wiener space can be disintegrated. [2] considers the case of the Wess–Zumino model over the loop space of Rd . He considers the measure of quantum √ field theory over the loop space which is a Gaussian measure of covariance −∆. He considers a perturbation of the free exterior derivative (which is defined in [2] modulo a second quantization). The Wess–Zumino operator has an index. He considers the heat semi group, and shows that there is a trace modulo some condition about the second quantized operators. He deduced a path representation of this index. The computations of [2] are closely related to the considerations of [44]. 2. Universal Cover of the Brownian Bridge The purpose of this section is to construct the universal cover of the Brownian bridge. Let us start by recalling how we construct the universal cover of a finite dimensional manifold. Let M be a compact manifold. We consider the set of paths l starting from x and arriving at any point in the manifold. We identify two paths l and l0 if they have same end points and if the loop constituted of l and l0 circled in the opposite sense is contractible. Let π be the canonical projection from the universal cover to the manifold. To a path l, π associates its end point. The fundamental group Π1 of M clearly acts over this set of paths by right concatenation of the loop which represents an element of the Π1 and l which represents an element of the universal cover of M . This operation is clearly compatible with the different relations which give the definition of Π1 of M and of the universal cover of M . This action is free. By selecting locally a system of paths joining x to y in a small neighborhood of M , we see that π −1 O is equal to the product of O by ˜ in the universal cover is equivalent by π to a small Π1 . A small neighborhood O neighborhood O in M . Let us suppose that there is over M a measure µ. There exists a unique measure over the universal cover of M µ ˜ which is gotten from µ by
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the local diffeomorhism π. Moreover µ ˜ is invariant under the action of the Π1 . This local diffeomorphism allows us to transport to the universal cover a Riemannian structure over M . The action of the Π1 becomes an action by isometries. Let (M, ω) be a compact Riemannian symplectic manifold. Let Lx (M ) be the based loop space of M , that is the space of continuous map γ from the circle S 1 into M such that γ0 = γ1 = x . (2.1) We endow it with the Brownian bridge measure µ. Let us recall how we construct this measure. Let pt (x, y) be the heat kernel associated to the Laplace–Beltrami operator over the manifold M . Let F (γs1 , . . . , γsn ) be a cylindrical functional over the loop space. Its expectation is given by Z 1 F (x1 , . . . , xn )ps1 (x, x1 ) . . . psi+1 −si (xi , xl+1 ) E[F (γs1 , . . . , γsn )] = p1 (x, x) × . . . p1−sn (xn , 1)dx1 . . . dxn .
(2.2)
Let Lx,fin(M ) be the space of finite energy loop γfin : we consider the reference loop γref,s = x for all s in Lx,fin(M ) and we consider the tangent space of a loop γfin by the space of sections Xs over γfin in the tangent bundle. We endow it with the norm: (2.3) kXkB = sup kXs kγs . Let l(γfin ) be a C 1 path going from the reference loop γref to γfin . We say that 0 ) are equivalent if the loop in Lx,fin(M ) constituted of two paths l(γfin ) and l0 (γfin 0 ) run in the opposite sense is homotopic to l(γfin ) run in the direct sense and l0 (γfin 0 . Modulo this equivalence class, we call the set the constant loop, and if γfin = γfin ˜ x,fin (M ), which is the universal cover of Lx,fin(M ). We define over which we get: L 0 0 ˜ , called l(γfin , γfin ). We Lx,fin(M ) a metric: we choose a path going from γfin to γfin choose its length for (2.3). It is given by Z
1
0 kd/dtl(γfin , γfin )kB dt .
(2.4)
0 0 ) is equivalent We assume that the path formed by l(γfin ) and followed by l(γfin , γfin 0 0 0 0 to the path l (γfin ). The distance between l(γfin ) and l (γfin ) will be the minimum 0 ) of the minimum of the length of modulo the representative of l(γfin ) and of l0 (γfin 0 ˜ x,fin(M ). L ˜ x,fin(M ) is not l(γfin , γfin ). We get that by a distance called d˜x over L complete. We complete it.
˜ x,fin(M ). ˜ x (M ) is the completion for d˜x of L Definition 2.1. L ˜ x (M ) is the universal cover of the continuous based loop space endowed with L the uniform distance. This means that we can do the previous considerations by using continuous loops instead of using finite energy loops. ˜ x (M ) which is similar to this, but for S 1 bundle over loop For a definition of L group, the reader can see [70].
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˜ x (M ) → Lx (M ), we will In order to define a measure µ ˜ and a projection π : L ˜ choose a system of charts of Lx,fin(M ). Let li (γi ) be a countable set of finite energy curves in Lx,fin(M ) such that the end points γi constitute a dense set of finite energy curves over Lx (M ) for the uniform distance. Let B(γi , δ) the balls of radius δ and of center γi for the uniform distance. If δ is small enough, it is the same distance as the following distance: we choose the infimum of the length for the norm (2.3) of the curve joining γi to γ. If γ ∈ B(γi , δ), there is a path li (γ) joining γ to γi . It is defined by li,t (γ)(s) = expγi (s) t(γs − γi,s ).
(2.5)
δ is chosen small enough in order to define the exponential charts. ˜ x,fin(M ) represented by the paths going from γref ˜ i , δ) the subset of L We call B(l to γ by the composition li (γi ) for a given curve joining γref to γi and afterwards ˜ x,fin which by li (γ). If we work over finite energy loop, we get an open set of L ˜ i , δ) is therefore is isometric to B(γi , δ) for the uniform norm. The open set B(l isometric to B(γi , δ) where we choose γ only continuous in B(γi , δ) instead of being of finite energy. ˜ i , δ) ˜ x (M ) by balls B(l Therefore, we describe by this procedure an open cover of L all isometric for the uniform distance to the balls B(γi , δ) with li,1 = γi . If γ˜ ∈ ˜ i , δ), we can define its projection over Lx (M ) because γ˜ is represented by the B(l path going from γref to γ by li and after li (γ): π(˜ γ) = γ .
(2.6)
˜ i , δ) into B(γi , δ) which preserves the metric. Moreover π is a bijection from B(l ˜ ˜i by pulling back the measure µ over B(γi , δ). If A is Over B(li , δ), we define µ ˜ i , δ), we have by definition µ ˜i (A) = µ(π(A). µ ˜i checks the a Borelian subset of B(l ˜ j , δ) is a borelian set for the distance d˜x , ˜ ⊆ B(l ˜ i , δ) ∩ B(l consistency relations: in O we get ˜ =µ ˜ . ˜j (O) (2.7) µ ˜i (O) ˜ over Lx (M ). µ ˜ is Therefore the set of µ ˜i define by patching together a measure µ uniquely defined and is the unique lift of µ by π. µ ˜ is the lift of the measure µ to the universal cover of the Brownian bridge. µ ˜ generally has no finite mass except when Π2 (M ) is finite. Namely let us recall that Π2 (M ) is Π1 (Lx,finM ). Moreover this last group acts by isometries over Lx,fin(M ). The reciproque image of a loop ˜ is therefore invariant under the action of Π2 (M ) and is deduced by π is Π2 (M ). µ from µ by the local homeomorphism π. ˜ x (M ) with a continuous projection Theorem 2.2. There exists a metric space L ˜ ˜ map π : Lx (M ) → Lx (M ). Moreover Lx (M ) is simply connected. The reciproque image of a small neighborhood by π of a small open ball Oi of Lx (M ) for the uniform ˜ x (M ) which ˜ over L distance is isomorphic to Π2 (M ) × Oi . There exists a measure µ ˜ x (M ) such ˜ satisfies to the following property: let Oi be a small neighborhood of L ˜ ˜i . ˜ that π is a homeomorphic from Oi over its image. Let A a Borelian subset of O ˜ = µ(π(A)). ˜ Then µ ˜(A)
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˜ i , δ), the Hilbert tangent space of γ˜ is defined by pulling back the If γ˜ ∈ B(l Hilbert tangent space of γ = π˜ γ , which is defined by using the lines of [10, 25, 46], and of [55] as follows: (2.8) Xs = τs Hs ; H0 = H1 = 0. τs is the parallel transport over the loop; Hs is of finite energy and we choose as Hilbert norm of X: Z 1 kHs0 k2 ds = kXk2 . (2.9) 0
This tangent space is chosen in order to get integration by parts formula over the Brownian bridge. Let F be a cylindrical functional. If H. is deterministic there exists a functional div X which belongs to all the Lp such that E[hdF, Xi] = E[F div X] .
(2.10)
Let us recall how [47] (2.40) [57, p. 207] introduce smooth cutoff functionals with support in B(γi , δ). Let g be a smooth function from R into [1, ∞] which is equal to 1 if and only if x ≤ δ 0 for some δ 0 < δ and which behaves as |δ 00 − x|−r for a big r when x → δ 00 by lower values. g is infinite if x > δ 00 for some δ 00 between δ and δ 0 . g is smooth if x < δ 00 . Let h be a smooth function from [1, ∞[ over [0, 1] equal to 1 only in 1 and with compact support. We introduce the functional: Z 1 g(d(γs , γi,s ))ds . (2.11) Fi (γ) = h 0
Fi (γ) belongs to all the Sobolev spaces over Lx (M ). In order to see that, we remark by using the exponential inequality over the modulus of continuity of γ that if n is big enough, Z 1 1 > ; d(γs , γi,s )−n < C < C(p)p . (2.12) µ sup s d(γs , γi,s ) The same inequality holds if we replace d(γs , γi,s ) by (δ 00 − d(γs , γi,s ))+ . ˜ i , δ), we put If γ˜ ∈ B(l γ ) = Fi (π˜ γ) . F˜i (˜
(2.13)
Let F be a cylindrical functional over Lx (M ), which is smooth. Let us remark that ˜ i , δ). µ ˜ is the transport of the measure µ by π the support of F˜i is included in B(l over B(γi , δ), where we have integration by parts if we use the mollifer Fi . Clearly, we have the integration by parts formula, if F˜ = F (π˜ γ ), i i h h ˜ F˜i (˜ ˜ ˜ ˜ + E˜ F˜ F˜i div X1 ˜ ˜ ˜ 1˜ , hdF˜ , Xi γ ) = −E[1 γ ), Xi] F˜ hdF˜i (˜ E B(li ,δ)
B(1i ,δ)
B(li ,δ)
(2.14) ˜ is the vector field over γ˜ from τs Hs = Xs , Hs deterministic by pulling where X back along π. Moreover, ˜ = div X(π˜ div X γ) . (2.15) 0 γ ) is equal to 0 in (2.14) if sup d(γs , γi,s ) ≤ δ . The main remark is that dF˜i (˜
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Let Ξ(l. ) be the core of finite combination of F˜i (˜ γ )F (π˜ γ ) for any F˜i of the given 2 µ). The integration by shape and any cylindrical functional F . It is dense in L (˜ ˜ over parts formulas (2.14) allow us to define the H-derivative of a functional G ˜ x (M ) by starting from the core Ξ(l. ) (see [56, pp. 305–307] in a ˜ x (M ) over L L more complicated set-up). If we change of system of curves li into another system γ ) for the system li can be reached as a limit of of curves li0 , any functional F˜i (˜ γ ) for the system li00 , because the two systems of curves are dense. functionals F˜i00 (˜ We consider the Hilbert–Schmidt norm of the H-derivative of a functional F˜ . Let us denote it by kDF˜ k2 . ˜ 1,p of functionals F˜ is the space of functionDefinition 2.3. The Sobolev space W als limit of functionals of the core Ξl for the norm: ˜ ˜ p 1/p + E[kD ˜ kF˜ kW F˜ kp ]1/p . ˜ 1,p = E[|F |]
(2.16)
R1 ˜ ˜ DF˜ is given by a kernel k(s) with 0 k(s)ds = 0 with values in Tx (M ). We can take the H-derivative of the kernel, which allows to define higher derivative of F˜ , ˜ 1 , . . . , sr ). This means that over B(l ˜ i , δ), Dr F˜ which are given by kernels k(s Z 1 Z 1 ˜ 1 , s2 , . . . , sr )d/dsH 1 . . . d/dsH r ds1 . . . dsr ˜r i = ˜1, . . . , X ··· k(s hDr F˜ , X s1 sr 0
0
(2.17) ˜ j corresponds to the vector field τs H j over B(γi , δ). Moreover for each j, if X s R1 r ˜ ˜ 0 k(s1 , . . . , sr )dsj = 0. The Hilbert–Schmidt norm of D F is given by Z
Z
1
˜ 1 , . . . , sr )|2 ds1 . . . dsr |k(s
··· 0
1/2
1
= kDr F˜ k .
(2.18)
0
For more details, we refer to [56, pp. 305–307] where the computations are performed over the free loop space, and can be localized over each small ball centered in γi of radius δ for the uniform norm, because we have the cutoff functional Fi . Since the tangent space of the based loop space is trivial, we do not need any connection in the iteration of Sobolev spaces. The main difference lies in the choice of the core: here we consider finite sum of functionals whose support lies in a small open neighborhood of finite energy curves. The core is not globally defined. ˜ 1,p is the space of functionals F˜ limit of Definition 2.4. The Sobolev space W functionals of the core Ξl for the norm: kF˜ kW ˜ r,p =
r X
j ˜ p 1/p ˜ E[kD Fk ] .
(2.19)
j=0
This definition does not depend on the system of curves li chosen. ˜ r,p . We say that F˜ is a smooth functional if it belongs to all the W We can define over Lx (M ) Sobolev spaces Wr,p : the reader can see [55] for analoguous considerations over the free loop space, when we take the Hilbert space structure Z Z 1
1
hXs , Ys ids + 0
hDXs , DYs ids = kXk2 , 0
(2.20)
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R1 instead of 0 hDXs , DYs ids (here D denotes the operation of covariant derivative over a loop: Dτs Hs = τs Hs0 ). For the definition of the corresponding Sobolev spaces, we do not have to localize. In particular, if F belongs to all the Sobolev spaces Wr,p , we can consider the ˜ r,p functional F˜ = F ◦ π. In general, it does not belong to the Sobolev spaces W because µ ˜ has no finite mass. We will introduce a local notion of Sobolev spaces. ˜ r,p (loc) if for any F˜i (˜ γ ), Definition 2.5. F˜ belongs locally to the Sobolev spaces W ˜ r,p . γ )F˜ belongs to W F˜i (˜ It is the same to say that Fi = 1B(γi ,δ) F˜i (π −1 γ)F˜ (π −1 γ) belongs to all the Sobolev spaces Wr,p . Namely π is a local isometry from the support of F˜i over the support of Fi . We refer to [56, Theorems 3.6 and 3.7], for various examples of functionals which belong to all the Sobolev spaces Wr,p . Definition 2.4 implies immediately, since π realizes a local isomorphism of the Sobolev Calculi: Proposition 2.6. Suppose that F belongs to all the Sobolev spaces over Lx (M ). ˜ x (M ). Then F˜ = F ◦ π belongs locally to all the Sobolev spaces over L Let us remark that the notion of local Sobolev spaces avoids the mention of ˜ x (M ) (see [6]). partition of unity of L ˜ x (M ) which belongs locally Let us give the basical example of functional over L ˜ to all the Sobolev spaces over Lx (M ) and which is not the pull-back of a functional over Lx (M ). Let us introduce a 2 closed form over M called ω. ω can be a symplectic form. Let us introduce the stochastic Chen form τ (ω), called the transgression of ω, over Lx (M ). We refer to [46, 58, 59] for a theory of stochastic Chen forms. Z τ (ω)(X) =
1
ω(dγs , Xs ) .
(2.21)
0
The kernel of τ (ω) is given by Z
1
ω(dγs , τs. )
k(s) =
(2.22)
0
if we consider the path space, that is the space of applications starting from x from [0,1] into the manifold. If ω is closed, τ (ω) is closed over Lx (M ) (see [58] Theorem 2.7, where a relation between the stochastic cohomology of the loop space and the Hochschild cohomology is shown). Let us introduce τ˜(ω) = π ∗ τ (ω) .
(2.23)
This means that we consider the differential form π over each small open subset where the map π is an isometry. The basical theorem of this section is:
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Theorem 2.7. There exists a functional F˜ which belongs locally to all the Sobolev ˜ r,p such that spaces W DF˜ = τ˜(ω) . (2.24) ˜ i , δ). We integrate τ (ω) over the path li,t (π˜ Proof. Let γ˜ ∈ B(l γ ). In order to see ˜ that, we imbed the manifold into Rd . We extend the form ω in a form ωext over ˜ the total space Rd , and we extend the map (x, y) → expx [t(y − x)] = ft (x, y). The process ft (γi,s , γs ) is a semi martingale. The integral of τ (ω) over the stochastic small path joining γ. to γi,. is the restiction to B(γi , δ) of the stochastic integral given by Z 1Z 1 ωext (ds ft (γi,s , γs ), dt ft (γi,s , γs )) . (2.25) 0
0
It is a non anticipative Stratonovitch integral. It belongs therefore to all the Sobolev ˜ γ )F˜ i (˜ γ ) such that 1B(l γ ) belongs to all the spaces. We get a functional F˜ i (˜ ˜ i ,δ) Fi (˜ Sobolev spaces (see [56, Theorem 3.6], [61, pp. 247–249] for analoguous considerations), by using the theory of stochastic integrals. Moreover γ ) = τ˜(ω) DF˜ i (˜
(2.26)
˜ i , δ). over B(l γ ) patch together in a functional It remains to show that all the functionals F˜ i (˜ ˜ F (˜ γ ). ˜ j,. , δ). This means that the path li,. (π˜ ˜ i,. , δ) ∩ B(l γ ) and lj,. (π˜ γ ) have Let γ˜ ∈ B(l γ ) and of lj (π˜ γ ) run the same extremities π˜ γ and that the loop constituted by li (π˜ in the opposite sense is contractible. But γi and γj are not far. We can join γi to γj by a curve (2.27) li,j,t (s) = expγi (s) t(γj (s) − γi (s)) and we can find a surface Ci,j whose boundary is the loop constituted by the distinguished path going from γ to γi , the distinguished path going from γj to γ and li,j,t . Ci,j is parametrized by: li,j,t,u (s) = expli,t (s) u(lj,t (s) − li,t (s)) .
(2.28)
The integral of τ (ω) over this loop is a functional ξ(γ). It is the restriction to B(γi , δ) ∩ B(γj , δ) of a stochastic integral which is the sum of 3 terms of the type (2.25). If we look at the polygonal approximation γ n of γ which is defined if n is enough big, we get by using the theory of non anticipative Stratonovitch integral a random variable ξ(γ n ) such that ξ(γ n ) → ξ(γ)
(2.29)
almost-surely. Moreover τ (ω) is closed. Therefore ξ(γ n ) = 0, because it is the integral over the boundary of a surface, by using Stokes theorem. Therefore ξ(γ) = 0.
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Let us consider the integral of τ (ω) over the loop constituted of the path going from the constant loop x to γi , the path li,j,t and the path going from γj to the constant loop x. This loop is contractible in Lx (M ). It is contractible by definition of the equivalence relation in Lx (M ). We can regularize the surface whose boundary is this loop by a surface in lx,fin(M ) whose boundary is this loop, by using a convolution operation convolution after imbedding M into Rd . Therefore the integral of τ (ω) over this loop which is a classical integral is equal to 0. ˜ = D∗ D where ˜ x (M ) ∆ Let us introduce the Ornstein–Uhlenbeck operator over L ˜ x (M ). ∆ ˜ is defined over Ξ(l. ), and D is the H-derivative and D∗ its adjoint over L is symmetric positive. It has therefore a self-adjoint extension. We can speak of ˜ F˜i (˜ γ )F˜ for any F˜i belongs functionals which belong locally to the domain of ∆: locally to the domain of ∆ where ∆ is the Ornstein–Uhlenbeck operator over the based loop space (see [56, (4.50)] for the construction of the Ornstein–Uhlenbeck operator over the free loop space). Since Fi (γ) = 1 if sup d(γs , γi,s ) < δ 0 for small enough δ 0 , we get if d(γs , γi,s ) < δ 0 , DFi (γ) = 0 and ∆Fi = 0: ˜ F˜i (˜ ˜ . ∆ γ )F = ∆F
(2.30)
˜ is a vector field over L ˜ x (M ), we have the formula: Namely, if X ˜ γ ) = F˜i (˜ ˜ − hDF˜i (˜ ˜ γ )i . div F˜i (˜ γ )X(˜ γ )div X γ ), X(˜
(2.31)
˜ ∆F ˜ is intrinsically Therefore if F˜ belongs locally to the domain of the operator ∆, defined. In particular from (2.14) and (2.15), we get: Proposition 2.8. If F belongs to the domain of ∆, F˜ = F ◦ π belongs locally to ˜ and the domain of ∆ ˜ ∆(F ◦ π) = (∆F ) ◦ π . (2.32) Let us consider a solution of the equation DF˜ = τ˜(ω). We get: Proposition 2.9. Let F˜ be any solution of (2.24) (see Theorem 2.6). Then F˜ ˜ belongs locally to the domain of ∆. ˜ i , δ) and by transport to work over B(γi , δ). Proof. It is enough to work over B(l Let us compute the kernel of τ (ω): Z 1 Z 1 Z s Z 1 ω dγs , τs hu du = du ω(dγs , τs. hu ) . (2.33) τ (ω)(X) = 0
0
0
0
Therefore the kernel of τ (ω) is kω (u) given by Z
Z
1
ω(dγs , τs. ) − u
Z
1
0
Z
1
ω(dγs , τs. ) = −
du 0
u
ω(dγs , τs. ) + C
(2.34)
0
which is the sum of an adapted term with respect to the filtration spanned the process γs and a smooth constant. This constant arises because a kernel over the
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loop space has to check the average condition given after (2.17). div Fi (γ)kω (u) therefore exists and is in all the Lp (µ). the divergence of an adapted vector field is given by the following formula: (see [55, Theorem 1.1] for the free path space) Z Z 1 1 1 div X = hτs d/dsHs , δγs i + hSXs , δγs i, (2.35) 2 0 0 where Xs = τs Hs , S is the Ricci tensor and where δ denotes the Itˆo integral. Thus we get the result. Let us consider the Hilbert norm of τ˜(ω). It is given by k˜ τ k2 checking
2 Z 1 Z u
2
ω(d(π˜ γs ), τs (π˜ γ )) + C(π˜ γ ) k˜ τ (ω)k =
du . 0
(2.36)
0
˜ x (M ) Let us consider the Schrodinger operator over L ˜ =∆ ˜ + k˜ N τ (ω)k2 .
(2.37)
It is a positive symmetric operator and therefore admits a self-adjoint extension. ˜ ] associated to it. There is a semi-group exp[−tN Let us recall what is a set of capacity 0: we consider an open set and we consider ˜ 1,2 of the functional larger than 1 over 0. the infimum of the Sobolev norms in W We get a real called Cap(O). A set is of capacity 0 if there exists a sequence of open sets containing this set such their capacity tends to 0. In particular if A is of capacity 0 over Lx (M ), its reciproque image is of capacity 0, because Π2 (M ) is countable. Theorem 2.10. Outside a set of capacity 0, there exists a continuous proces γ ) starting from γ˜ such that t→w ˜t (˜ Z 1 ˜ F˜ (˜ ˜ exp − exp[−tN] γ) = E k˜ τ (ω)k2 (˜ ωs (˜ γs ))ds F˜ (˜ ωt (˜ γ )) (2.38) 0
for any F bounded. Proof. Let A be the Dirichlet form over Lx (M ): A(F, F ) = E[hDF, DF i]
(2.39)
defined by starting from the core of cylindrical functionals (see [5, 26, 31, 66] [67, Theorem 4.3.5.]). This core is dense, separated from the points and the Dirichlet form is tight. Associated to it, there exists a unique Markov process defined outside a set of capacity 0 wt (γ) whose generator is ∆ = D∗ D. Let us consider the γ ) of it starting from a given element of π −1 γ. This lift is defined outside lift w ˜t (˜ ˜ ˜ x (M ). It defines a Markov process whose generator is ∆, a set of capacity 0 over L this form the Proposition 1.7. (The proof is the same as in finite dimension.) Let ˜ is constructed: then π(w˜t (˜ γ) us suppose conversely that a process associated to ∆ is a Markov process with generator ∆ starting from π˜ γ . This shows the existence ˜ Moreover µ and the unicity of the Markov process associated to ∆. ˜ is an invariant measure for this process. This allows us to state (2.37) by using the Feynmann–Kac formula.
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Remark. This lifting property reflects the fact that ˜ ](F ◦ π) = (exp[−tN ]F ) ◦ π . exp[−tN
(2.40)
Remark. It is interesting to compare in the case where Π2 (M ) = 0, the global Sobolev spaces of [56] formula (3.59) with the local Sobolev spaces in this paper. First of all, there is no non trivial connection which appears here because the tangent space is trivial. That is we consider the connection ∇τ. H. = τ. DH. ,
(2.41)
where DHt is the H-derivative of the vector Ht . We work as in [63, Theorem 1.2], to produce a system of partition of unity which belongs to all the Sobolev spaces related to a special cover of the based loop space. We take as set of γi a set of polygonal paths γi,n such that B(γi,n , δ) constitute a cover of Lx (M ). They are associated with the subdivision k/n of the time interval. When n is fixed, they are at most C n such γi,n . Fi,n has Sobolev norms bounded by Cnα for some α independent of i and n. Namely (2.12) is true only for < C n1β for some suitable β. If sup d(γs , γi,n,s ) < δ 0 and sup d(γs , γi,k,s ) > δ 0 for k < n,
sup |s−t|<1/n
d(γs , γt ) > δ 00
for a suitable δ 00 . We regularize the functional sup|s−t|<1/n as in [47] or in [57, p. 207]. We put ψs,t = d(γs , γt ) for γs and γt close and ψs,t = 1 for γs and γt far. We put Z dsdt Hn = n (2.42) +,α |s−t|<1/n (−ψs,t + δ1 ) for some δ1 < δ 00 close from δ 00 . We choose a function F1 with support [0, C] smooth from [0, ∞[ into [0, ∞[, and we put Gn = 1 − F1 (Hn )
(2.43)
Gn belongs to all the Sobolev spaces (see [47, (2.14)] and sup|s−t|<1/n d(γs , γt ) > δ 00 implies Gn = 1. Moreover if C is big enough, Gn > 0 implies there exists s and t such that |s − t| < 1/n and d(γs , γt ) > δ2 for δ2 close of δ 00 . Therefore Gn is strictly positive with a probability exp[−C1 n]. If we choose the Brownian bridge in time 2 instead of time 1, C1 → ∞ when 2 → 0. Therefore we can introduce X Fi,n Gn . (2.44) Ψ= i,n
Ψ ≥ 1 and belongs, if is small enough, to all the Sobolev spaces because the Sobolev norms of Gn are smaller than C exp[−Cn]nβ for a big C if is small enough. Let us explain this statement: the exponential arise from the exponential inequality. The integral has bounded Sobolev norms. The nβ arises from the n before the
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integral and there is in this present situation a formula analoguous to (2.12) only 0 for < n−β . So if the time of the Brownian loop is small enough, the exponential decay which arises from the exponential inequality has a stonger influence than the number of polygonal curves γi,n . Let F be a cylindrical functional. We put X Fi,n Gn F . (2.45) Fk = Ψ i,n≤k
Fk belongs to all the local Sobolev spaces and tends to F in all the local Sobolev spaces. Therefore if the time of the Brownian loop is small enough, the local Sobolev spaces defined above and the global Sobolev spaces of [56, (3.59)] coincide. 3. Differential Forms Over the Cover One of the main problems we met in the first part is that µ ˜ has an infinite mass. We will overcome this problem by choosing a measure Hd˜ µ for a suitable H, which will allow to state the equivalence between the cohomology of the stochastic Witten ˜ x (M ) (see [58, 59]). complex (see [36, 63]) and the stochastic cohomology of L The main theorem in comparison of [63, Theorem 1.2] of this part is the following: Theorem 3.1. There exists a functional H > 0 such that ˜ exp[λ|F˜ |]] < ∞ E[H
(3.1)
for all λ > 0. ˜ i , δ), we choose Proof. Over B(l Gi = exp[−|F˜i |2 ]
(3.2)
sup exp[λx] exp[−x2 ] < C(λ) < ∞ .
(3.3)
˜ i , δ) such that over B(l x≥0
We choose as G: such that
X
X
αi 1B(1 ˜ i ,δ) Gi
(3.4)
˜ i , δ)] < ∞ ˜ [B(l αi µ
(3.5)
G=
for a given sequence αi > 0. It is possible to choose such αi because ˜ i , δ)) = µ(B(γi , δ)) < ∞ µ ˜ (B(l
(3.6)
and we normalize in order to get a probability measure. We will not mention this operation any more.
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Remark. It should be possible to choose H which belongs locally to all the Sobolev spaces, if the time of the Brownian bridge is small enough, by doing as in (2.43). Let us recall the definition of a Chen form. Let Ω(M ) be the space of forms over M and Ω. (M ) be the space of smooth forms of degree ≥ 1. Let us consider Ω. (M )⊗n . Let ω1 ⊗ · · · ⊗ ωn ∈ Ω. (M )⊗n , we put: Z Σ(ω)(γ) = ω1 (dγs1 , .) ∧ · · · ∧ ωn (dγsn , .) . (3.7) 0<s1 <···<sn <1
We give the definition of Σ(ω) over an example: for more general cases, see [58, Theorem 2.1]. Let us suppose that n = 2 and that ω1 and ω2 are two forms. In this case, Σ(ω)(γ)(X. , Y. ) is given by the following formula: Z ω1 (dγs , Xs )ω2 (dγt , Yt ) + antisymmetry . (3.8) 0<s
P If ωi is a form of degree ri , Σ(ω) is of degree (ri − 1). We consider a form Σ(ω) of degree r. It is a form over Lx (M ) (see [46, 58, Theorem 2.7], [59]]). Over Ω. (M ), we put the Sobolev Hilbert structure associated to the operator ∗ (dd + d∗ d + 2)k (see [58, Theorem 2.3 and (2.11)]. We deduce an Hilbert strucure ˜ is a sequence of elements of Ω. (M )⊗n , we put k.kk over Ω. (M )⊗n . If ωn = ω φk,˜ω (z) =
X
kωn kk zn √ . n!
(3.9)
(see [21, 58, Theorem 2.1] and [59]). We call Ar∞ the intersection of the Banach completion for all z and k which are P ω = Σωn is of total degree r. from the Banach norms φk. (z) when the form Σ˜ Let us recall what is a form smooth in the Nualart–Pardoux sense over Lx (M ) (see [58, Definition 1.1], [59, 62]). Let us consider an r form σ over Lx (M ). It is given by a kernel k(s1 , . . . , sr ). This means that: Z 1 Z 1 σ(X1 , . . . , Xr ) = ··· k(s1 , . . . , sr )d/dsHs11 . . . d/dsHsrr ds1 . . . dsr (3.10) 0
0
τs Hsi
and where k(s1 , . . . , sr ) is a cotensor over Tx (M ). Let us introduce if Xi,s = the connection over the tangent bundle of the stochastic loop space: ∇X τ. H. = τ. DX H. ,
(3.11)
where D is the traditional H derivative of Ht which lives in the fixed linear space Tx (M ). We deduce that σ can have covariant derivatives with respect to this connection ∇. Let us recall that the covariant derivative of a cotensor σ is given by the formula (1.14) in [58]. X σ(X1 , . . . , ∇Y Xi , . . . , Xr ) . (3.12) ∇Y σ(X1 , . . . , Xr ) = hDσ(X1 , . . . , Xr ), Y i − 0
Let us call such covariant derivative ∇r σ. It has kernels given by k(s1 , . . . , sr ; t1 , . . . , tr0 ) .
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The Nualart–Pardoux Sobolev norms of σ are given by the following: over each simplices in s and t included: kk(s1 , . . . , sr ; t1 , . . . , tr0 ) − k(s01 , . . . , s0r ; t01 , . . . , t0r0 )kLp X q Xq ≤ Cp,r0 (σ) |si − s0i | + |ti − t0i | (3.13) and 0 kk(s1 , . . . , sr ; t1 , . . . , tr0 )kLp ≤ Cp,r 0 (σ) < ∞ .
We put kσkp,r0 =
X
0 (Cp,l (σ) + Cp,r 0 (σ)) .
(3.14)
(3.15)
l≤r 0
We call N Pp,r0 the space which we get after completing and N P∞ = ∩Np,r0 . Let us recall the main theorem of [58, Theorem 2.7] the map Chen iterated integral is continuous from A∞ into N∞ In [58], this theorem is shown for the path space: it involves the fact that the kernels of the derivaite of the parallel transport check the Nualart–Pardoux conditions because it is the solution of a linear equation (see [56, Theorem 4.6]. There are only a few algebraic modifications in order to pass to the based loop space. They are treated in Theorem 4.5 of [62]). ˜ x (M ): after local trivialization, it can be seen Let us introduce a form σ ˜ over L ˜ 1 , . . . , sr ) is intrisically defined as a collection of forms σi over B(γi , δ). Its kernel k(s ˜ and by using the local diffeomorphism between B(li , δ) and B(γi , δ), we can define ˜ 1 , . . . , sr ; t1 , . . . , tr0 ). We define the Nualart–Pardoux its covariant derivative k(s Sobolev norms with respect to H as follows: ˜ 0 , . . . , s0 ; t0 , . . . , t0 0 )kp ]1/p ˜ 1 , . . . , sr ; t1 , . . . , tr0 ) − k(s ˜ E[Hk k(s 1 r 1 r X q Xq < Cp,r0 ,H (˜ σ) |si − s0i | + |ti − t0i | (3.16) and ˜ 1 , . . . , sr ; t1 , . . . , tr0 )kp ]1/p < C 0 0 (˜ ˜ E[Hk k(s p,r ,H σ ) . We put kHkp,r0 (H) =
X
0 (Cp,r0 ,H (σ) + Cp,r 0 ,H (σ)) ,
(3.17)
(3.18)
l≤r
˜ ∞ (H) ˜ p,r (H) the Sobolev spaces in the sense of Nualart–Pardoux and N.P We call N.P the intersection of these spaces. We get: Theorem 3.2. There exists a functional H > 0 such that the map π ∗ is continuous ˜ ∞ (H). from N.P∞ into N.P
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Proof. We choose H =
P
αi 1B(l ˜ i ,δ) . Therefore
∗ ˜ E[Hkπ h(s1 , . . . , sr ; t1 , . . . , tr ) − π ∗ h(s01 , . . . , s0r ; t01 , . . . , t0r )kp ] X = αi E[1B(γi ,δ) kk(s1 , . . . , sr ; t1 , . . . , tr ) − k(s01 , . . . , s0r ; t01 , . . . , t0r )kp ] .
(3.19) We deduce ∗ ˜ k(s1 , . . . , sr ; t1 , . . . , tr ) − π ∗ k(s01 , . . . , s0r ; t01 , . . . , t0r )kp ] ≤ E[Hkπ
X
αi kσkpp,r . (3.20)
Therefore the theorem is true if 0 ∗ true for Cp,r 0 ,H (π σ).
P
αi < ∞ and αi > 0. The analoguous of (3.18) is
We get: ˜ ∞ (H). Corollary 3.3. π ∗ Σ is a continuous map from A∞ into N.P Let us recall that the exterior derivative of an r form σ is given by X (−1)i−1 hd(σ(X1 , . . . , Xi−1 , Xi+1 , . . . , Xn )), Xi i dσ(X1 , . . . , Xr+1 ) = +
X (−1)i+j σ([Xi , Xj ], X1 , . . . , Xi−1 , Xi+1 , . . . , Xj−1 , Xj+1 , . . . , Xn ) . i<j
(3.21) Theorem 3.4. There exists a H > 0 such that the exterior derivative d˜ is defined ˜ ∞ (H). and continuous over N.P Proof. We choose Xs = τs Hs with hs deterministic. Let us recall [10] that Z s ∇X τs = τs τu−1 R(dγu , Xu )τu . (3.22) 0
Let us precise what we mean by this expression. Let et be the evaluation map γ. → γt . Let e∗t T (M ) be the pullback bundle over the based loop space of the tangent bundle by the evaluation map. Suppose ∇t the pullback of the Levi–Civita connection by the evaluation map over the pullback bundle e∗0 T (M ). Let X0 be a section of e∗0 T (M ) such that τt X0 is a section of e∗t Xt . By definition, we have ∇t (τt X0 ) = (∇τt )X0 + τt (∇0 X0 )
(3.23)
(see [56, 3.82]). The proof of the formula (3.22) is then elementary as shown in [56, ˜ i , δ). It is enough to consider the counp. 347]. It is enough to define d˜ over B(l ˜ i , δ) terpart of σ ˜ over B(γi , δ). We introduce a smooth cutoff functional F˜i over B(l and its counterpart Fi over B(γi , δ). Fi = 1 over the closed set sup d(γs , γi,s ) ≤ δ 0 for some δ 0 ≤ δ. δ 0 can be chosen as close as possible as δ. The tangent space is not stable by Lie bracket. Namely, the Levi–Civita connection is without torsion. We
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can compute [Xk , Xj ] by using the formulas (3.23) and (3.22). The only problem to understand d˜ is to treat the anticipative Stratonovitch integral which appears when we take the Lie–Bracket of [Xk , Xj ]. The only difficulty is Rto understand the be1 haviour over Fi = 1 of some stochastic integrals of the shape 0 ki (s; s1 , . . . , sn )dγs . It is a Stratonovitch integral. Lemma 3.5. Fi over Lx (M ) belongs to all the Nualart–Pardoux spaces. Proof. The derivative of Fi is given by Z 1 Z 1 0 g(d(γs , γi , s))ds hg 0 (d(γs , γi,s ) ◦ d0 (γs , γi,s ), Xs i . hDFi , Xi = h 0
(3.24)
0
We use the fact that the covariant derivative of τs satisfies the Nualart–Pardoux conditions (3.14) and (3.15), s included (see [56, Theorem 4.6]), and the fact that the function γs → d(γs , γi,s )2 is smooth if d(γs , γi,s ) < δ. We find that Z k hD Fi , X1 , . . . , Xk i = qi (s1 , . . . , sr )d/dsHs11 . . . d/dsHskk ds1 . . . dsk , (3.25) R1 where Xi = τ. H.i . Moreover qi is equal to 0 if 0 g(d(γs , γi,s )ds is big. Before qi R1 there is an element in h(k) ( 0 g(d(γs , γi,s ))ds. qi outside that is a sum of iterated integrals in polynomials expressions in the derivatives of τu and of g(d(γu , γi,s )) 0 which behaves as |d(γu , γi,u ) − δ 00 |−r for a big integer r0 when d(γu , γi,u ) → δ 00 by lower values. We use the fact 1 1 |d(γu , γi,u ) − δ 00 |r0 − |d(γu0 , γi,u0 ) − δ 00 |r0 < C
|d(γu , γi,u ) − d(γu0 , γi,u0 )| |d(γu , γi,u ) − δ 00 |r0 |d(γu0 , γi,u0 ) − δ 00 |r0
(3.26)
such that when h(k) 6= 0,
p 0 0 1 E[kd(γu , γi,u ) − δ 00 |−r − |d(γu0 , γi,u0 ) − δ 00 |−r |p ] p ≤ C(γi ) |u − u0 | .
(3.27)
The constant C(γi ) depends on γi . We deduce that the kernel of ∇k Fi when h 6= 0 are iterated integrals with possible frozen terms of expressions which satisfy the Nualart–Pardoux conditions. By Lemme A.2 of [58], these iterated integrals satisfy also the Nualart–Pardoux conditions. Let us come back to the proof of the original statement: Fi σi checks the Nualart– Pardoux conditions over Lx (M ) and therefore the Stratonovitch integral which appears in the definition of the exterior derivative of Fi σi still belongs to the Nualart–Pardoux conditions, and their Nualart–Pardoux norms can be estimated in terms of the Nualart–Pardoux norms C(i, p, r) of Fi and the local Nualart–Pardoux norms of σ ˜. This shows that X 1/4 ˜σ kp,r (H) < E[H 2 1B(l C(i, p00 , r00 )k˜ σ kp0 ,r0 (H) (3.28) kd˜ ˜ i ,δ) ] i
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for some p0 ≥ p, r0 ≥ r. and some p00 ≥ p and some r00 ≥ r. It is enough to choose P 00 00 H = αi 1B(l ˜ i ,δ) such that for all p and r , X
1
˜ i , δ)] 14 C(i, p00 , r00 ) < ∞ . αi4 P [B(l
(3.29)
Corollary 3.6. There exists a H > 0 such that the cohomology group for d˜ and for d˜F˜ are equal. Proof. Namely, exp[λF˜ ] belongs to all the Nualart–Pardoux spaces for some H ˜ ∞ (H) belongs to all the Nualart–Pardoux well chosen. Therefore, if σ ˜ ∈ N.P ˜ ∞ (H). It is the same for exp[−λ spaces for some H, exp[λF˜ ]˜ σ belongs to N.P ˜ F ]. 4. Minimum Extension In fact the universal cover of the Brownian bridge is too big a manifold. The authors [20, 40, 75] take an extension which is minimum for deterministic loop. Let us give a stochastic analoguous of this. ˜ x,fin,min (M ) be the set of finite energy paths l starting from the constant path L in M submitted to the equivalence relation: two paths l and l0 in Lx,fin(M ) are equivalent if they have the same extremities and if Z Z τ (ω) = τ (ω) . (4.1) l
l0
˜ x,fin,min(M ) as in the first We introduce the metric of uniform convergence over L ˜ x,min(M ) ˜ part. We get by completion a space called Lx,min(M ). We can describe L ˜ ˜ by a system of open sets Bmin (li , δ). Each Bmin (li , δ) is isometric naturally to ˜min,i B(γi , δ) (li,1 = γi ) for the distances involved. We get a system of measures µ ˜ ˜ over Bmin (li , δ) which are compatible and therefore a space (Lx,min (M ), µ ˜ min ) with a projection πmin : ˜ x,min(M ), µ (L ˜min ) → (Lx (M ), µ) . (4.2) ˜min (li , δ) are isomorphic to the balls B(γi , δ) for the differential The balls B ˜ r,p and local Sobolev calculus. We deduce as in the first part, Sobolev spaces W ˜ ˜ spaces Wr,p (loc) over Lx,min(M ). The form τ (ω) has a pull-back over the big space ˜ x,min(M ) which is called τ˜(ω). Let D be the operation of H-derivative: L ˜ x,min (M ) such that F˜ belongs Theorem 4.1. There exists a functional F˜ over L ˜ locally to all the Sobolev spaces over Lx,min(M ) and such that DF˜ = τ˜(ω) .
(4.3)
As in the second part, we can define forms which are smooth in the Nualart– ˜ ∞ (H). Pardoux sense with respect to a functional H > 0. We get a space called N.P ˜ ˜ d is continuous for a suitable H over N.P ∞ (H).
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MODULAR THEORY AND GEOMETRY B. SCHROER∗ and H.-W. WIESBROCK† Institut f¨ ur Theoretische Physik FU-Berlin, Arnimallee 14 Berlin 14195 Germany ∗ E-mail:
[email protected] † E-mail:
[email protected] Received 10 December 1998 In this communication we present some new results on modular theory in the context of quantum field theory. In doing this we develop some new proposals how to generalize concepts of finite dimensional geometrical actions to infinite dimensional “hidden” symmetries. The latter are of a purely modular origin and remain hidden in any quantization approach. The spirit of this work is more on a programmatic side, with many details remaining to be elaborated.
1. Introduction Modular theory is the basis of a new concepts and structures in QFT. Mathematically it is a vast generalization of the modular factor which accounts for the difference between right and left Haar measure in the case of non unimodular groups such as the group of upper triangle matrices. Tomita and later Takesaki succeeded in converting this idea into a powerful tool for the investigation of von Neumann algebras, see [32]. In fact, Alain Connes could not have carried out his pathbreaking work on the classification of factor algebras without this theory. Modular theory is also in the background of the subfactor theory of Vaughn Jones. The physics side is equally impressive. At the time when Tomita presented his theory, Haag Hugenholz and Winnink published their fundamental work on (heat bath) thermal aspects of QFT, see [16]. The KMS condition (a name which they introduced) was used before that time notably by Kubo, Martin and Schwinger as a clever mathematical trick in order to avoid computing cumbersome traces in evaluating thermal ensembles via the Gibbs trace formula. In the hands of Haag Hugenholz and Winnink this formula became the key conceptual tool for their formulation of equilibrium quantum statistical mechanics directly in the thermodynamic limit for which the Gibbs representation becomes meaningless. In fact many years later it was shown that this structure follows from the stability of states under local perturbations of the interaction and that it has close conceptual links with the second fundamental thermodynamical law (“passivity” of equilibrium states) [15]. The generator for the modular group operator turns out to be the “thermal Hamiltonian” which, different from the original ground state Hamiltonian, has two-sided spectrum and finite fluctuations in thermal infinite volume states. It acts 139 Reviews in Mathematical Physics, Vol. 12, No. 1 (2000) 139–158 c World Scientific Publishing Company
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on the original algebra as well as on its commutant, where the latter represents a kind of “shadow world”. In the above analogy to group theory it corresponds to the conjugate right-hand action [15].a The Tomita (antiunitary) involution J in the thermal case turned out to be precisely the antilinear “flip” operation which maps the algebra of the world into that of the “shadow world”. Later Bisognano and Wichmann [2] found another very nontrivial field theoretic illustration of the Tomita–Takesaki theory in their study of the algebra which is generated by fields restricted to the wedge region, i.e. the same region which already played an important role in Unruh’s discussion of the Hawking effect for the Rindler universe. In their case the modular group turned out to be the wedge-affiliated Lorentz boost and the Tomita J was (up to a 90 degree rotation around the boost direction) identified with the field theoretical TCP operator. It was Sewell who showed that this result of Bisognano and Wichmann might be interpreted within Unruh’s setting as a thermalization through a horizon as seen by an uniformly accelerated observer [29, 33]. In this note we will mention an algebraic criterion which implies this identification of the modular operator with the Lorentz boost in the setting of algebraic QFT, see also [5]. Different from the situation of a heat bath temperature, the temperature caused by localization of matter behind a classical (Killing) horizon or by quantum localization in LQP (local quantum physics) does not produce a “shadow-world” without geometrical interpretation, but rather remains part of the real world. In fact the commutant of the wedge algebra is the opposite wedge algebra in the causally disjoint region beyond the wedge horizon. Besides having a thermal meaning, the modular theory leads to profound geometric results which will be the subject of this work. The inverse question namely whether the heat bath shadow world also permits the invention of a (causally disconnected, behind a “horizon”) space-time interpretation will be discussed in a separate paper. Recent works (as well as some special prior observations on free fields [21]) strongly suggested that there was a deep relation between modular theory and relativistic causality and localization. In fact Borchers [3] and one of the present authors (H.-W. W.) [36] turned the geometrical action of modular theory around and showed that with two subalgebras in appropriate modular position one can build up space-time symmetries (d = 1 + 1 conformal [13] and, with somewhat stronger assumptions, also the higher dimensional cases [18, 38] ). Starting from Wigner’s representation theory of positive energy representations it is possible to construct the unique net associated with the (m, s = semiinteger) Wigner representation without using (nonunique and nonintrinsic) field coordinates as the generators of local algebras [10, 22] and illustrate these structural results by beautiful calculations within the Wigner one particle theory which are mapped directly via the Weylfunctor (or its CCA analogue) into the local nets. We believe that without taking notice of these deep relation between spacetime symmetry and pure local quantum physical properties one will not be able to understand the still evasive “Quantum Gravity”. a See [17] for recent results on the physical interpretation of the underlying thermal “shadow world”.
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The modular structure also promises to clarify some points concerning the physics of the Wightman domain properties [22]. In fact the modular groups act linearly on the “field space”, i.e. on the space generated by applying a local field on the vacuum. Therefore this space, which in the case of interacting theories with a particle interpretation is expected to be highly reducible under the Poincar´e group, may according to a conjecture based on the results in [9] in fact carry an irreducible representation of the union of all modular groups (conjecture of K. Fredenhagen, private communication). This would be an infinite dimensional group Gmod which contains in particular all local space-time symmetries. This role of fields as of irreducible representations of a universal infinite dimensional symmetry group Gmod would add a significant conceptual element to LQP and give the notion of quantum fields a deep role which goes much beyond that of being simply generators of local algebras.b Our arguments in Sec. 3 suggest that in chiral conformal QFT Gmod may include all diffeomorphism. A related group theoretical approach which aims at geometrical aspects of modular constructions, but this time starting from properties of modular involutions (“The Condition of Geometric Modular Action”), was proposed in [11]. One obtains directly transformation groups on the indexed sets of the net, which in general cannot be described by pointlike space-time transformations. All these QFT symmetry properties which go beyond pointlike diffeomorphisms remain invisible in any quantization approach, and to emphasize this fact we call them “hidden”. In particular they cannot be interpreted as originating as quantized Noether currents. The addition of new families of hidden symmetries is the main aim of this paper.c Combining modular theory with scattering theory, the actual Tomita involution J together with the incoming J in can be used to obtain a new framework for nonperturbative interactions [22]. This last topic will be treated in a separate paper by the present two authors [27]. Besides the old result of Bisognano/Wichmann, there are nowadays other interesting geometrical, but not pointlike, actions of modular theory in chiral theories [8]. We present some new examples in chiral and higher dimensional models, and give some outlook to future investigations of geometrical actions [28]. 2. The Bisognano Wichmann Property We will start with the famous result of Bisognano/Wichmann [2] and present a purely algebraic proof of that property.d Let us first introduce some notations. We assume given a Poincar´e covariant local net of observables which fulfills essential duality (wedge duality). Let U denote the representation of the Poincar´e’ group on the vacuum Hilbert space H, Ω the b It would also reveal that the traditional quantization approach to QFT is quite limited and in
no way captures the conceptual richness of LQP. c In a certain sense the “symmetry enhancement” mechanism on spacetime subsets of [29, 31] is the inverse of hidden symmetries. d After finishing this part we were informed by H.-J. Borchers of a recent result by himself, [5]. Below we compare both approaches.
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vacuum vector. Let l1 and l2 be linear independent forward lightlike vectors and W [l1 l2 ] the associated wedge: → → x ∈ R1,3 |− x = αl1 + βl2 + l⊥ , with: W [l1 , l2 ] = {− α > 0, β < 0, (l⊥ , li ) = (l⊥ , lj ) = 0} .
(1)
Λ12 (t) is the Lorentz boost to that wedge, i.e. the boosts which map the wedge W [l1 , l2 ] onto itself. Let A(W [l1 , l2 ]) be the observable algebra associated to the wedge. The Reeh– Schlieder property of the vacuum states that Ω is cyclic and separating for this algebra. We denote the associated modular objects by ∆12 , J12 . Before continuing with the results let us remind the reader on an observation by Borchers [6]. Assume wedge duality for the underlying net. Then we have ∆it 12 = F12 (t)U (Λ12 (−2πt))
(2)
with some unitary group F12 (t) which is defined by this equation. Moreover modular theory gives [F12 (t), J12 ] = [F12 (t), ∆is 12 ] = [F12 (t), U (Λ12 (−2πs))] = 0
(3)
(F12 leaves invariant A(W [l1 , l2 ]) and Ω). Due to Borchers’ result [3, 7] we immediately conclude (4) [F12 (t), U (translations)] = 0 , see [6]. Moreover there is a strongly dense set of elements A ∈ A(W [l1 , l2 ]) s.t. U (Λ1,2 (2πt))AΩ, A ∈ A(W [l1 , l2 ]) can analytically be continued to t ∈ 1 S( −1 2 , 0) = {z ∈ C/− 2 < Im z < 0}. So far Borchers’ results. Let us now come to an observation which one of the authors made in discussion with D. Guido: Theorem 1 (Guido/Wiesbrock). Additionally to the above wedge duality we make the technical assumption i A∗ Ω , A ∈ A(W [l1 , l2 ]) (5) AΩ −→ U Λ1,2 − 2 is uniformly bounded. Then the Bisognano–Wichmann Property holds for the net. Proof. Let F12 (t) = eitf12 be the unitary group above. From the assumption it follows that f12 is semibounded . Now F12 leaves invariant A(W [l1 , l2 ] ) which implies by a result of Borchers, [20], innerness of F12 , i.e. ∃ eitf ∈ A(W [l1 , l2 ] ) with eitf e−itf12 ∈ A(W [l1 , l2 ] )0 . J12 -invariance of F12 implies Ad eitf12 = Ad J12 eitf12 J12 and therefore J12 eitf e−itf12 J12 = eiλ eitf ,
λ∈R
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because A(W [l1 , l2 ]) is a factor. Putting together we conclude eitf12 = J12 eitf J12 eitf . eitf12 is ∆12 invariant and therefore eitf too. Similarly eitf is U (Λ12 ) invariant and translation invariant along the baseline of the wedge. However, due to the cluster property the later implies eitf = const. We now make a comparison with the work of Borchers [5]. As an algebraic criterion for the Bisognano–Wichmann property, Borchers proposed a so-called reality condition as follows: Let A ∈ W [l1 , l2 ], such that U (Λ1,2 (2πt))A∗ Ω can analytically be continued to Im t = − 21 . (One can find a ∗ -strongly dense subset of such elements, see [6]) Then there exists Aˆ and A˜ affiliated with A(W [l1 , l2 ])0 = A(W [l2 , l1 ]) are related by ˜ = U (Λ1,2 (−iπ))A∗ Ω . AΩ
ˆ = U (Λ1,2 (−iπ))AΩ , AΩ
(6)
ˆ∗
˜ i.e. The reality condition states now: For such A one has A Ω = AΩ, U (Λ1,2 (−iπ))A∗ Ω = Aˆ∗ Ω. Rewritten in terms of the modular data this means: U (Λ1,2 (−iπ))S AΩ = S ∗ U (Λ1,2 (−iπ))AΩ
(7)
∗
for a -strongly dense subset of A(W [l1 , l2 ]), where S denotes the Tomita conjugation to (A(W [l1 , l2 ]), Ω). Now due to commutativity T := U (Λ1,2 (−iπ))∆− 2 = ∆− 2 U (Λ1,2 (−iπ)) 1
1
(8)
is a self adjoint operator and the reality condition can be rephrased as JT J = T .
(9)
But modular theory tells us that JU (Λ1,2 (−iπ))J = U (Λ1,2 (iπ)) ,
J∆− 2 J = ∆ 2 1
1
(10)
which implies JT J = T −1 and therefore T T ∗ = T 2 = T JT J = T T −1 = 1 .
(11)
Therefore the reality condition gives boundedness of T by 1 which implies the assumption of our theorem. We do not see a natural physical motivation for our boundedness assumption. But the results suggest that one should look at a purely algebraic criterion which guarantees the Bisognano–Wichmann property. This would not rely on some generating fields which in the algebraic approach to quantum field theory play the role of a special choice of introducing “field-coordinates” into the physical system. 3. On the Modular Theory of Disjoint Intervals In this section we present a modular interpretation of conformal diffeomorphisms.
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3.1. The Virasoro-algebra We consider the U (1)- current algebra on the circle. Then we have an action of the Virasoro-algebra on the physical vacuum Hilbert space of our model. The vacuum is defined as the unique invariant vector L1 Ω = L−1 Ω = L0 Ω = 0 .
(12)
The usual commutation relations are: [Ln , Lm ] = (n + m)Ln+m +
1 (m2 − 1)mδm,n 12
so that: [L±2 , L0 ] = ±2L±2 ,
[L2 , l−2 ] = 4L0 +
1 2
(13)
(14)
holds. Then a simple computation shows L0 7→
1 1 L0 + 2 16
1 L1 7→ √ L2 2 2
(15)
1 L−1 7→ √ L−2 2 2 gives an isomorphism of sl(2R)-Lie algebras. This second representation belongs to a 3-dim. subgroup in Diff S 1 given as follows.e r a + bz 2 a b ∈ SU (1, 1) . (16) z 7→ c d c + dz 2 One notices the following (after discussion with M. Schmidt): • SU (1, 1) maps the circle into itself. 2 • the poles of a+bz c+dz 2 lie outside the circle, the zeros inside. • Choose the cuts by adjoining both zeroes and both poles. In that way we get well-defined maps. • We get a representation of a twofold covering of the M¨ obius group. Geometrically the associated dilations leave invariant disjoint intervals of the following type: Given an interval, look at the square root of its elements. We get a disjoint union of intervals as the inverse image. Both parts separately are left invariant under that group. Therefore, if the modular groups act like dilations as above, we do not have Reeh–Schlieder Property for an arbitrary local algebra because of Takesaki’s result, see [30]. This would either imply equality of the algebra of a small interval and the algebra associated to appropriate disjoint intervals or that the algebra is not cyclic. But notice that due to the geometry it never happens that one part of such a region covers both parts of another type of such disjoint intervals. We especially do e These diffeomorphisms are quasisymmetric transformations in the sense of [19]. Their distance from the M¨ obius transformation increases with increasing degree of the covering or number of disconnected components of intervals explained below.
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not get into conflict with additivity of the net and locality! This is the reason that such a modular theory is allowed. Next we show that this indeed happens. 3.2. Some formulas Let us collect some well-known formulas which help to understand the following computations. In the Sl(2R) group we have the generator D for the dilations, P and K for the translations, respectively conformal translations. They are related to the Virasoro group by 1 1 (P + K) , D = (L1 − L−1 ) , 2 2 1 1 (17) L1 = D + (P − K) , P = (L1 + L−1 ) + L0 , 2 2 1 1 L−1 = D − (P − K) , K = (L−1 + L1 ) − L0 . 2 2 It turns out to be helpful to move in both directions from the compact (circle) picture to the noncompact on the reals: L0 =
compact picture → noncompact picture:z 7→ −i
z−1 z+1
ix + 1 . −ix + 1 Under these transformation the relevant forms and the Ln are mapped to the following: noncompact picture → compact picture:x 7→
1 dx (−ix + 1)2 n+1 (−ix + 1)2 d ix + 1 n+1 d , Ln = . Ln = z dz 2i −ix + 1 dx √ A formal computation now shows: z 2 ◦ Ln ◦ z = 12 L2n , i.e.: dx = −i
2 dz , (z + i)2
dz = 2i
1 L±2 , 2 1 L0 , L0 → 7 2
(18)
L±1 7→
(19)
in the noncompact picture: z 7→ z 2 ' x 7→
2x , 1 − x2
s 2 √ 1 1 z 7→ z ' x 7→ − + sign(x) 1 + . x x
(20)
In the noncompact and compact picture it is easily seen that diffeomorphisms act symplectically: Z Z 1 1 g(x) df (x) , ω(g, f ) = g(z) df (z) . (21) ω(g, f ) = 2 2
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3.3. Transformation and invariant scalar product Let ( ac db ) ∈ Sl(2R), then the underlying symmetry action is: r x 7→ −A−1 (x) + signA(x)
with A(x) = Moreover
Z hf, gi =
1 + A−1 (x)
2
(22)
2ax + b(1 − x2 ) . 2cx + d(1 − x2 )
f (x)g(y) (1 + x2 )(1 + y 2 ) dx dy ((x − y)(1 + xy) + i0)2
(23)
gives an invariant scalar product.
q Proof. Denote Γ(x) = − x1 + sign(x) 1 + ( x1 )2 then we compute 2y 2x a + b + b a 2 1 − x2 g ◦ Γ 1 − y f ◦ Γ 2x 2y Z +d +d c c 2 2 1−x 1−y (1 + x2 )(1 + y 2 ) dx dy ((x − y)(1 + xy) + i0)2 2y 2x a a + b + b 2 1 − x2 g ◦ Γ 1 − y f ◦ Γ 2x 2y Z + d + d c c 2(1 + x2 )2(1 + y 2 ) 1 − x2 1 − y2 = dx dy . 2 (1 − x2 )2 (1 − y 2 )2 2y 2x − + i0 1 − x2 1 − y2 (24) 2
2(1+x ) 2x One notices d( 1−x 2 ) = (1−x2 )2 and thereby: 2y 2x a + b + b a 2 1 − x2 g ◦ Γ 1 − y f ◦ Γ 2x 2y Z + d + d c c 2y 2x 1 − x2 1 − y2 d = d 2 1 − x2 1 − y2 2y 2x − + i0 1 − x2 1 − y2
2x 1 − x2
2y Z f ◦Γ g◦Γ 1 − y2 = 2 2y 2x − + i0 1 − x2 1 − y2 Z =
d
2x 1 − x2
2y d 1 − y2
f (x)g(y) (1 + x2 )(1 + y 2 ) dx dy , ((x − y)(1 + xy) + i0)2
(25)
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2x where the last equality follows from f = f ◦Γ◦( 1−x 2 ). This establishes the invariance of the positive definite real scalar product.
Z
Now define I(f )(x) = with kI (x, y) =
f (y)KI (x, y) dy
1 −x(1 + y 2 ) −1 1 − = . (x − y) y (1 + xy) y(x − y)(1 + xy)
(26)
(27)
Then one computes (1 + x2 )(1 + y 2 ) −1 1 d kI (x, y) = = − . dx [(x − y)(1 + xy)]2 (x − y)2 (1 + xy)2
(28)
ω(If, g) = hf, gi = −ω(f, Ig) .
(29)
and get Next we show that this scalar product gives us a Fock state on the Weyl algebra: Denote ! r x 2 x Γ(f )(x) := f + sign(x) 1 + (30) 2 2 Z
and I0 (f )(x) =
−1 f (y) dy (x − y + i0)
(31)
which gives the usual vacuum Fock state of the theory [39]. Z f (x)g(y) hf, gi0 = ω(I0 f, g) = dx dy (x − y + i0)2 Then we compute Γ
−1
Z
f 1 − + x − y + i0 x
◦ I0 ◦ Γ(f )(x) = Z = Z = Z = Z =
−1
! r y 2 y + sign(x) 1 + dy 2 2
z2 + 1 −1 f (z) 2 dz 1 1 z − + x − − + z + i0 x z −x(z 2 + 1) f (z) dz x z 2 −1 + x2 + − zx + i0 z −x(z 2 + 1) f (z) dz z(−z + x2 z + x − z 2 x + i0) −x(z 2 + 1) f (z) dz = I(f )(x) , z(−z + x + i0)(1 + xz)
(32)
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so we get Γ−1 ◦ I0 ◦ Γ = I .
(33)
= −1, from which we immediately conclude I = −1, i.e. the As is well known invariant scalar product produces a Fock state on the Weyl algebra. I02
2
3.4. The KMS-Condition Let us show the KMS-Condition. For this we first make the following simple observation: f ∈ C0∞ ([0, 1]) ⇒ Γ(f ) ∈ C0∞ ([0, ∞[) , f ∈ C0∞ (] − ∞, −1]) ⇒ Γ(f ) ∈ C0∞ ([0, ∞[) .
(34)
Denote h , i0 the scalar product of the vacuum state, h , i1 the one above. Then we have hΓ(f ), Γ(g)i0 = hf, gi1 or explicity: s s 2 2 1 1 1 1 f − +sign(x) 1+ g − +sign(x) 1+ Z x x y y d xdy (x − y + i0)2 Z f (z)g(w) 1 + z2 1 + w2 = dz dw 2 2 2 (1 − z ) (1 − w2 )2 2w 2z − + i0 1 − z2 1 − w2 Z f (z)g(w) (1 + z 2 )(1 + w2 ) 1 dz dw = . (35) 2 (z − w + i0)2 (1 + zw)2 Let us also introduce the notion V (λ)(f )(x) = f (λx) , s 2 2 2 1 − x 1 − x U (λ)(f )(x) = f + sign(x) 1 + 2λ2 2λ2
(36)
then, an easy computation shows Γ−1 ◦ V (λ) ◦ Γ = U (λ) .
(37)
To prove the KMS-property for the disjoint interval ] − ∞, −1] ∪ [0, 1] w.r.t. U , we use the strategy as in [39]. Let f, g ∈ C0∞ ([0, 1]). Then hU (λ)f, gi1 = hV (λ) ◦ Γ(f ), Γ(g)i0 .
(38)
Due to (3.1), Γ(f ), Γ(g) ∈ C0∞ ([0, ∞[). J. Yngvason has shown in [39] that for such functions the expectation value with V (λ) can analytically be continued to Im λ = 2π with specific boundary values. This gives in our case: lim hU (λ)f, gi1 = lim hV (λ) ◦ Γ(f ), Γ(g)i = hΓ(g), Γ(f )i = hg, f i1 .
θ↑2π
θ↑2π
(39)
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This shows that U (λ) fulfils the KMS-condition for the Weyl-algebra to [0, 1]. By the invariance and relation (3.2) we immediately get the same result for f , g ∈ C0∞ (] − ∞, −1] ), respectively f ∈ C0∞ (] − ∞, −1] ), g ∈ C0∞ ([0, 1] ), and thereby learn that U gives the modular group to the disjoint intervals. The methods of [39] then also apply to give 1 . (40) J = Γ−1 ◦ J0 ◦ Γ ⇒ J(f )(x) = f ∗ x The same methods should also work for the L3 , L−3 case, where one has to use the third root. In this way we get an action of these other diffeomorphism groups as automorphisms on the local net. As is well known they can be implemented on the vacuum theory by Bogoliubov automorphisms. Our method shows that the diffeomorphisms beyond M¨ obius transformations which leave end points of multiintervals fixed, do have a modular origin. But it falls short to prove that this can be done in the vacuum net, i.e. that the diffeomorphism is modular on the subspace which the multi-interval algebra generates from a suitable vector in the vacuum space. We expect that the method of quasifree states carries over to all loop group models as they were constructed for example in [34]. As a result of the pointwise action of the diffeomorphisms we also expect that there exists a more direct proof using pointlike fields similar to the method used in [12]. The final upshot of this section is that diffeomorphism actions in chiral conformal QFT are expected to occur as the action of some modular groups of special multiinterval observable algebras w.r.t. properly chosen states. In higher dimensional QFT’s the modular groups for topologically nontrivial regions beyond those for wedge algebras in the massive case and beyond those for double cone algebras for conformal theories cannot be pointlike diffeomorphisms. Yet they should be viewed as the “hidden” analogues of the chiral diffeomorphisms beyond the Moebius transformations. We expect the non-locally acting infinite dimensional Lie-groups Gmod which they generate to be very important in the future development of QFT. 4. Hidden Symmetry and Modular Inclusions As was shown in [8] it might happen that the modular group acts only geometrically on a certain subregion. We show how this comes up very naturally in higher dimensional situations. We start with some purely group theoretical observations on an interesting subgroup of the Poincar´e group [24]. The standard wedge situation suggests decomposition of the Poincar´e group generators into longitudinal, transversal and mixed generators: 1 1 (±) P± = √ (Pt ± Pz ), Mtz ; Mxy , Pi ; Gi ≡ √ (Mit ± Miz ) , 2 2 (±)
i = x, y .
(41)
The generators Gi are precisely the “translational” pieces of the euclidean stability groups E (±) (2) of the two light vectors e(±) = (1, 0, 0, ±1) which appeared for the first time in Wigner’s representation theory of the “little group” for zero mass
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(+)
particles. As one reads off from the commutation relations, Pi , Gi , P± have the interpretation of a central extension of a transversal “Galilei group” with the two (+) “translations” Gi representing the Galilei generators, P+ the central “mass” and P− the “nonrelativistic Hamiltonian”. The longitudinal boost Mtz scales the Galilei (+) (+) generators Gi and the “mass” P+ . Geometrically the Gi change the standard wedge (it tilts the longitudinal plane) and the corresponding finite transformations generate a family of wedges whose envelope is the half-space x− ≥ 0. The Galilei group together with the boost Mtz generate an 8-parametric subgroup G(+) (8) inside the 10-parametric Poincar´e group: (+)
G(+) (8) : P± , Mtz ; Mxy , Pi ; Gi
.
(42)
The modular reflection J transforms this group into an isomorphic G(−) (8). All observation have interesting generalizations to the conformal group in massless theories in which case the associated natural space-time region is the double cone. This subgroup is intimately related to the notion of “modular intersection”. Let l1 , l2 and l3 be three linear independent light-like vectors and consider two wedges W (l1 , l2 ), W (l1 , l3 ) with Λ12 and Λ13 the associated Lorentz boosts. As a result of this common l1 the algebras N = A(W (l1 , l2 )), M = A(W (l1 , l3 )) have a modular intersection with respect to the vector Ω. Especially (N ∩M) ⊂ M, Ω) is a modular inclusion [1, 7, 35]. Identifying W (l1 , l2 ) with the above standard wedge, we notice that the longitudinal generators P± , Mtz are related to the inclusion of the standard (+) wedge algebra into the full algebra B(H), whereas the Galilei generators Gi are the “translational” part of the stability group of the common light vector l1 (i.e. of (+) the Wigner light-like little group). These generators Gi should be considered as being associated with a common light ray shared by two wedges whereas all the other generators in (42) are either longitudinal or transversal to one wedge (the standard wedge). To simplify the situation let us take d = 1 + 2 with G(+) (4), in which case there is only one Galilei generator G. In addition to the “visible” geometric subgroup of the Poincar´e group, the modular theory produces a “hidden” symmetry transformation which belongs to the intersection region of two wedges. Let us now consider the 3-dim. situation from the point of view of algebraic QFT. We assume the Bisognano–Wichmann property for the net and also assume that it fulfills additivity. Denote l1 , l2 , l3 three linear independent lightlike vectors and W [l1 , l2 ], W [l1 , l3 ] two wedges characterized by their two lightlike vectors. Denote Λ12 respectively Λ13 the related Lorentz boosts. Due to their common light-ray l1 , the associated observable algebras have a modular intersection w.r.t. Ω, see [4, 38]. To simplify the notation we use N , M for the algebras. Then ((N ∩ M) ⊂ M, Ω) is a modular inclusion, see [1, 4, 35, 38] and ia (ln ∆N ∩M − ln ∆M ) (43) UN ∩M,M (a) := exp 2π is a unitary group with positive generator. Moreover one has −it UN _M,M (1 − e−2πt ) = ∆it M ∆N _M ,
(44)
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−it UN ∩M,M (e−2πt a) = ∆it M UN ∩M,M (a)∆M ,
(45)
Ad UN ∩M,M (−1)(M) = N ∩ M ,
(46)
JM UN _M,M (a)JM = UN _M,M (−a) .
(47)
and Similar results hold for N replacing M, see [4, 37]. Due to the intersection property we finally have the relation [37] [UN _M,M (a), UN _M,N (b)] = 0 which enables one to define the unitary group UN _M (a) = UN _M,M (−a)UN _M,N (a) . This latter group can be rewritten as −it UN _M (1 − e−2πt ) = ∆it M ∆N
and thereby recognized to be in our physical application the 1-parameter Galilean subgroup G (42) in the above remarks. Now we notice that for a < 0, 1 −i( 2π ln −a)
Ad UN _M,M (a)(M) = Ad ∆M
1 −i( 2π ln −a)
= Ad ∆M
UN _M,M (−1)(M) (N ∩ M) .
(48)
Because ∆it M acts geometrically as Lorentz boosts, we fully know the geometrical action of UN _M,M (a) on M for a < 0. For a > 0 we notice Ad UN _M,M (1)(M) = Ad UN _M,M (2)(M ∩ N ) = Ad JM JN _M (M ∩ N ) = Ad JM (M0 ∪ N 0 )
(49)
and again, due to the geometrical action of JM we have a geometrical action on M for a > 0. −i( 1 ln a) JM (M0 ∪ N 0 ) . (50) Ad UN ∩M,M (a)(M) = Ad ∆M 2π −it From these observations together with UN ∩M,M (1 − e−2πt ) = ∆it M ∆M∩N we get for t < 0: i (− 2π ln(e−2πt −1)) JM (M0 ∪ N 0 ) (51) Ad ∆it N ∩M (M) = Ad ∆M
and in case of t > 0: (−
i
2π Ad ∆it N ∩M (M) = Ad ∆M
ln(1−e−2πt ))
(N ∩ M) .
(52)
Similar results hold for N replacing M. With the same methods we get: is it is −is Ad ∆it N ∩M ∆N (M) = Ad ∆N ∩M ∆N ∆M (M) −2πs − 1)(M) , = Ad ∆it N ∩M UM∩N (e
(53)
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where UN ∩M is the 1-parameter Lorentz subgroup (the Galilei subgroup G in (42) associated with the modular intersection, see [4, 38]. This gives is −2πt −2πs (e − 1))∆it Ad ∆it N ∩M ∆N (M) = Ad UM∩N (e N ∩M (M) −
1
= Ad UM∩N (e−2πt (e−2πs − 1))∆M2π
ln(1−e−2πt )
(M ∩ N ) , (54)
if t > 0 and similar for t < 0. Therefore we get a geometrical action of ∆it N ∩M on (M). Ad ∆is N A look at the proof shows that the essential ingredients are the special commutation relations. Due to it −2πt −2πt ) = ∆it − 1)JM ∆it M∩N = ∆M UN ∩M,M (1 − e M JM UN ∩M,M (e
and the well-established geometrical action of ∆it M and JM , it is enough to consider the action of UN ∩M,M or similarly UN ∩M,N . For these groups we easily get −it is it −2π(s+t) a)(N ) Ad UN ∩M,M (a)∆is N ∆M (N ) = Ad ∆N ∆M UN ∩M,M (e
and due to the above remarks the geometrical action of ∆it N ∩M on the algebras of −it the type Ad ∆is N ∆M (M). Now, the lightlike translations Utransl1 (a) in l1 direction fulfill the positive spectrum condition and map N ∩M into itself for a > 0. Therefore we have the Borchers commutator relations with ∆it M∩N and get −2πt a)∆it Ad ∆it N _M Utransl1 (a)(M) = Ad Utransl1 (e N _M (M) .
(55)
In order to visualize the above results we will assume additivity for the net: Definition 1. A local net A obeys additivity if for two space-time regions O1 and O2 with open intersection we have A(O1 ) ∨ A(O2 ) = A(O)
(56)
with O the causal completion of O1 ∪ O2 . The additivity of the net tells us that taking unions of the algebras corresponds to the causal unions of localization regions. The assumed duality allows us to pass to causal complements and thereby to intersections of the underlying localization regions. Therefore the algebraic properties above transfer to unions, causal complements and intersections of regions. We finally get: Theorem 2. Let G be the set of regions in R1,2 containing the wedges W [l1 , l2 ], W [l1 , l3 ] and which is closed under : (a) Lorentz boosting with Λ12 (t), Λ13 (s), (b) intersection, (c) (causal) union, (d) translation in l1 direction, (e) causal complement.
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Then ∆it W [l1 ,l2 ]∩W [l1 ,l3 ] maps sets in G onto sets in G in a well-computable way and extends the subgroup (42) by a “hidden symmetry”. Similarly we can look at a (1 + 3)-dim. quantum field theory. Then we get the same results as above for the modular theory to the region W [l1 , l2 ] ∩ W [l1 , l3 ] ∩ W [l1 , l4 ], where li are 4 linear independent lightlike vectors in R1,3 . Moreover in this case the set G contain W [l1 , l2 ], W [l1 , l3 ] and W [l1 , l4 ] and is closed under boosting with Λ12 (t), Λ13 (s), Λ14 (r). The arguments are based on the Borchers commutation relation and modular theory and also apply if we replace modular intersection by modular inclusion. One recovers in this way the results of Borchers and Yngvason, [8]. (Note that in thermal situations we have no simple geometrical interpretation for the commutants as the algebra to causal complements. Therefore in these cases we have to take care in (e) in the above theorem, see [28].) The final upshot of this section is to show that there can be sensible meanings of partially geometrical actions of modular groups by restricting on certain subsystems. For the interesting question whether massive higher dimensional QFT’s have hidden realizations of conformal SL(2, R) and even diffeomorphism lifts in the sense of Sec. 3, we refer to a forthcoming paper [28]. 5. 4-dim. Theories from a Finite Set of Algebras In the following we briefly mention some unpublished recent results of one of the present authors (H.-W. W) in collaboration with R. K¨ ahler [18]. It follows a line beginning with the work of Borchers, [3] and one of the authors, [36]. Starting with a finite set of algebras lying in a specified position w.r.t. their common modular theory one constructs a net of local observables. Theorem 3. Let Mij , 0 ≤ i < j ≤ 4, M0ij = Mji , be von-Neumann algebras acting on H, Ω ∈ H with: (a)
(Mij , Mik , Ω)
has modular intersection .
(57)
This part reflects the underlying wedge geometry. (b)
symmetric in indices.
This part means that there is no preferential “wedge” region. We can reformulate this as follows: −it0 it0 −it0 0 with: Let ΓP := ∆it 24 ∆34 ∆32 ∆42 (b1)
Ad ΓP (M14 ) = M12 ,
Ad ΓP (M13 ) = M14
(58)
Ad ΓP (M42 ) = M12
(59)
−it0 −it0 0 0 ∆−t and ΓP 0 := ∆it 13 ∆43 ∆14 13 with
(b2) (c)
Ad ΓP 0 (M12 ) = M32 ,
closed in some finite dim. Lie group.
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Essentially we use that the generators of the modular groups can be composed. We reformulate this as follows. it24 it34 14 Denote P 1,1 the group generated by {∆it 14 ∆24 ∆34 }, SO(1, 2) the one generit13 it12 it23 ated by {∆13 ∆12 ∆23 } Then let 1,1 (c0 ) Ad J12 (Pε1,1 ) ⊂ Pδ · SO(1, 2), Pε1,1 , ε − neighborhood of 1 ∈ P 1,1 .
(60)
Then we conclude: it24 it34 it12 it13 it23 14 {∆it 14 , ∆24 , ∆34 }, {∆12 , ∆13 , ∆23 } generate a reprs. of Sl(2C) .
(61)
Proof. See [18].
In order to get a representation of the Poincar´e group we use the following. First we again implement rudimentarily the wedge geometry by: Let N12 be a von-Neumann Algebra with (d)
(N12 ⊂ M12 , Ω) is hsm .
(62)
The resulting translations should Be reflected by the CPT-like conjugations: (e1)
Ad JM1j (JN12 JM12 ) = JM12 JN12 ,
(e2)
[Ad JMjk (JN12 JM12 ), JN12 JM12 ] = 0 ,
j = 3, 4
(63)
j, k = 2, 3, 4
(64)
and the symmetry in the indices (no preferential wedge) leads to: (f)
Ad ΓP (JN12 JM12 ) = JN12 JM12 .
(65)
−it0 0 Remark 1. Notice that Ad ∆it 12 ∆13 ΓP (M12 ) = M12 , so that (f) implies −it0 it0 −it0 0 Ad ∆it 12 ∆13 ΓP (JN12 ) = JN12 . Modular theory then shows Ad ∆12 ∆13 ΓP (N12 ) = N12 . Then we get a representation of R1,3 , the translations. The Lorentz group maps translations onto themselves which can be encoded in:
(g)
Ad Jjk (translations) ⊂ translations
(66)
Under these assumptions we get a representation of the full Poincar´e group with spectrum condition. Using this one easily constructs a net of observables in R1,3 to it. These results show that space-time can be encoded in a finite set of algebraic data. Moreover space-time is recovered by looking at the noncommutative structure measured w.r.t. the underlying physical state of the system (modular theory). This would not work in a classical setting. Saying in an overstretched manner, the underlying classical geometry of space-time results in our approach from the local quantum physics by drawing on the deviation from the commutative “classical” case. (Notice that by the “classical” case we do not refer to any underlying quantization
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procedure nor to a semiclassical approximation via some perturbation theory. The route we follow in our reconstruction of space-time grounds in the noncommutativity and is intrinsically non perturbative.) 6. Modular Group for Double Cones Consider a double cone algebra A(O) generated by a free massless field (for s = 0 take the infrared convergent derivative). Then the modular objects of (A(O), Ω)m=0 are well known [15]. In particular the modular group is a one parametric subgroup of the proper conformal group. The massive double cone algebra together with the (wrong) massless vacuum has the same modular group σt , however its action on smaller massive subalgebras inside the original one is not describable in terms of the previous conformal subgroup. In fact the geometrical aspect of the action is wrecked by the breakdown of Huygens principle, which leads to a nonlocal reshuffling of the net inside O, but is still local in the sense of keeping the inside and its causal complement apart. This mechanism can be shown to lead to a pseudo-differential operator for the infinitesimal generator of σt whose highest term still agrees with conformal zero mass differential operator. We are however interested in the modular group of (A(O), Ω)m with the massive vacuum which is different from the that of the wrong vacuum by a Connes cocycle. We believe that (following a conjecture of Fredenhagen) this modular cocycle will not wreck the pseudo-differential nature, however we were not able to show this. In this case the hidden (nonlocal) aspect of the modular symmetry has a somewhat different origin and manifestation from the thermal Borchers–Yngvason example or from our illustration using modular intersection of wedges. Namely it is expected to become asymptotically geometric near the horizon, i.e. the light cone boundary of the double cone region. This would fit in nicely with Sewell’s “axiomatics on the horizon” [29]. In fact in two space-time dimensions where light-cones and light-fronts coalesce, the conformal invariant nature of the restriction of a massive theory to the boundary becomes manifest. The algebra on the two lower light ray intervals I± which define the lower boundary of a double cone D is in fact the causal shadow region of I± A(D) = A(I− ) ∨ A(I+ ) .
(67)
The two light ray intervals are (apart from the apex in which they meet) causally disjoint and hence their algebras commute. The algebras A(I± ) would both be pieces of conformal QFT on the respective light rays. Whereas in the massless case the A(I± ) are mutual tensor factors and the cyclically generated Hilbert spaces H± = A(I± )Ω are tensor factor subspaces, in massive theories one could expect that those spaces are already complete, i.e. the validity of the Reeh–Schlieder theorem for light ray intervals (as the quantum counterpart of the classical distinction between the characteristic massive versus massless problem). In that case the modular objects ∆it I± , JI± , SI± for the chiral conformal interval theory are defined in the full “massive” Hilbert space H and furthermore ∆it I− , JI− , SI− becomes the parity transforms (without loss of generality we have chosen a symmetric double cone) of ∆it I+ , JI+ , SI+ , e.g.
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SI− = P SI+ P
(68)
so that the Tomita operator SI+ for (A(I+ ), Ω) is parity reflected into the Tomita operator SI− . The parity commutes with the geometric actions of the modular objects on the chiral pair (A(I+ ), Ω) and therefore it is consistent to define ∗ P · A∗+ Ω . SA+ · P B+ P Ω = S+ A+ · P B+ P Ω = P B+
(69)
In this way one would expect to construct the modular theory of the characteristic algebra and hence for the double cone algebra in its causal shadow. On the characteristic horizon the modular quantities act geometrically such that the restriction of the modular group to A(I± ) transform these chiral algebras into themselves. Inside the double cone, the modular action is “fuzzy”. This means that if we envisage a causal net within the double cone the modular action on a member of that net does not respect the causal propagation picture of that net. Only near the horizon does one recover the causal geometric situation. Thanks to the cyclicity assumption the modular objects are well-defined operators on the full space and hence define global transformations which however cannot be written in terms of diffeomorphisms of Minkowski space and whose generators have therefore no associated Noether currents. We hope that the above remarks as well as the modular view of geometric and hidden symmetries may not only open the path for the understanding of useful mathematical generalizations of the concept of Lie-groups to infinite dimensions (as it already happened in chiral conformal theories), but also prove helpful in a future investigation of matters of physical relevance. Acknowledgment We would like to thank H.-J. Borchers and M. Schmidt for several discussions and remarks.
Note Added As an update of the present work (especially the methods used herein) we have added Ref. [25]. The cyclicity of light ray restrictions for massive free theories has been recently shown by [14] and a “characteristic shadow property” for semiinfinite light rays in arbitrary massive theories (from which cyclicity would follow) was presented in [26]. References [1] H. Araki and L. Zsido, “An extension of the structure theorem of Borchers with an application to half-sided modular inclusions”, preliminary version (1995). [2] J. Bisognano and E. Wichmann, “On the duality condition for a Hermitian scalar field”, J. Math. Phys. 16 (1975) 985. [3] H.-J. Borchers, “The CPT-Theorem in two-dimensional theories of local observables”, Commun. Math. Phys. 143 (1992) 315.
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[4] H.-J. Borchers, “Half-sided modular inclusions and the construction of the Poincar´ e group”, Commun. Math. Phys. 176 (1996) 703. [5] H.-J. Borchers, “On Poincar´e transformations and the modular group of the algebra associated with a wedge”, preprint, 1998, to appear in Lett. Math. Phys. [6] H.-J. Borchers, “When does Lorentz invariance imply wedge duality”, Lett. Math. Phys. 35 (1995) 39. [7] H.-J. Borchers, “On the use of modular groups in quantum field theory”, Ann. Henri Poincar´e 63 (1995) 331. [8] H.-J. Borchers and J. Yngvason, “Modular groups of quantum fields in thermal states”, preprint, 1998. [9] K. Fredenhagen and Joerss, “Conformal Haag–Kastler nets, pointlike localized fields and the existence of operator product expansions”, Commun. Math. Phys. 176 (1996) 541. [10] R. Brunetti, D. Guido and R. Longo, “On the intrinsic construction of free theories via Tomita Takesaki theory”, private discussion with D. Guido, unpublished manuscript. [11] D. Buchholz, O. Dreyer, M. Florig and S. Summers, “Geometric modular action and spacetime symmetry groups”, hep 9805026. [12] D. Buchholz and H. Schulz-Mirbach, Rev. Math. Phys. 2(1) (1990) 105. [13] D. Guido, R. Longo and H.-W. Wiesbrock, “Extensions of conformal nets and superselection structures”, Commun. Math. Phys. 192 (1998) 217. [14] D. Guido, R. Longo, J. E. Roberts and R. Verch, “Charged sectors, spin and statistics in quantum field theory on curved spacetimes”, math-ph/9906019. [15] R. Haag, Local Quantum Physics, Springer, Verlag, 1992. [16] R. Haag, N. Hugenholtz and M. Winnink, “On the equilibrium state in quantum statistical mechanics”, Commun. Math. Phys. 5 (1967) 215. [17] Ch. Jaekel, “Cluster estimates for modular structure”, hep 9804017. [18] R. Kaehler and H.-W. Wiesbrock, “Modular theory and the reconstruction of 4-dim. quantum field theories”, in preparation. [19] O. Lehto, Univalent Functions and Teichmueller Spaces, Springer, Verlag, 1987. [20] G. Pedersen, C∗ -algebras and their Automorphism Groups, Academic Press, London, New York, San Francisco, 1979. [21] P. Leyland, J. Roberts and D. Testard, “Duality for quantum fields”, unpublished preprint, July 1978; J. P. Eckmann and K. Osterwalder, J. Funct. Anal. 13 (1973) 1. [22] B. Schroer, “Motivations and physical aims of algebraic quantum field theory”, Ann. Phys. 255 (1997) 270. [23] B. Schroer, “Modular wedge localization and the d = 1 + 1 formfactor program”, hep-th/9712124, to be published in AOP. [24] B. Schroer, “A course on localization and nonperturbative local quantum physics”, hep-th/9805093, Chaps. 3 and 6. [25] B. Schroer, “Modular theory and Eyvind Wichmann’s contributions to modern particle physics theory”, dedicated to Prof. E. Wichmann on the occasion of his seventieth birthday, hep-th/9906071. [26] B. Schroer, “New concepts in particle physics from solution of an old problem”, hepth/9908021. [27] B. Schroer and H.-W. Wiesbrock, “Modular constructions of quantum field theories with interactions”, hep-th/9812251, to be published in Rev. Math. Phys. [28] B. Schroer and H.-W. Wiesbrock, “Looking beyond the thermal horizon: Hidden symmetries in chiral models”, hep-th/9901031, to be published in Rev. Math. Phys. [29] G. Sewell, “Quantum fields on manifolds: PCT and gravitationally induced thermal states”, Ann. Phys. 141 (1982) 201. [30] S. Stratila, Modular Theory in Operator Algebras, Abacus Press, 1981. [31] S. Summers and R. Verch, Lett. Math. Phys. 37 (1996) 145.
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[32] M. Takesaki, “Tomita’s theory of modular Hilbert algebras and its applications”, Lect. Notes in Math. 128, Springer, Verlag, 1970. [33] W. Unruh, “Notes on black hole evaporation”, Phys. Rev. D29 (1976) 1047. [34] A. Wassermann, “Operator algebras and conformal field theory”, Univ. Cambridge, preprint, 1997. [35] H.-W. Wiesbrock, “Half-sided modular inclusions of von-Neumann algebras”, Commun. Math. Phys. 157 (1993) 83; Erratum Commun. Math. Phys. 184 (1997) 683. [36] H.-W. Wiesbrock, “Conformal quantum field theory and half-sided modular inclusions of von-Neumann algebras”, Commun. Math. Phys. 158 (1993) 537. [37] H.-W. Wiesbrock, “Symmetries and modular intersections of von-Neumann algebras”, Lett. Math. Phys. 39 (1997) 203. [38] H.-W. Wiesbrock, “Modular intersections of von-Neumann algebras in quantum field theory”, Commun. Math. Phys. 193 (1998) 269. [39] J. Yngvason, “A note on essential duality”, Lett. Math. Phys. 31 (1995) 127.
HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES YURI BEREST∗ and PAVEL WINTERNITZ Centre de recherches math´ ematiques, Universit´ e de Montr´ eal, C.P. 6128, succ. Centre-Ville, Montr´ eal (Qu´ ebec), H3C 3J7, Canada E-mail:
[email protected] E-mail:
[email protected] Mathematics Subject Classification: 35Q51, 35Q53, 35L05, 35L15, 35Q05. Received 16 September 1996 Revised 18 December 1997 We demonstrate a close relation between the algebraic structure of the (local) group of conformal transformations on a smooth Lorentzian manifold M and the existence of nontrivial hierarchies of wave-type hyperbolic operators satisfying Huygens’ principle on M. The mechanism of such a relation is provided through a local separation of variables for linear second order partial differential operators with a metric principal symbol. The case of flat (Minkowski) spaces is studied in detail. As a result, some new nontrivial classes of Huygens operators are constructed. Their relation to the classical Hadamard conjecture and its modifications is discussed.
1. Introduction. Huygens’ Principle, Decomposability and Separation of Variables 1.1. Let Mn+1 be a C ∞ -smooth time-oriented pseudo-Riemannian manifold with a metric form ds2 = (dx, dx) of Lorentzian signature (+, −, . . . , −), and let Ω ⊂ Mn+1 be a normal causal domain therein. We consider a linear scalar second order hyperbolic operator L := 2 + (a(x), ∇) + u(x)
(1)
with a metric principal part 2 and with C ∞ -smooth vector and scalar fields a(x) and u(x) defined in Ω as lower order coefficients of L. Here, 2 := (∇, ∇) stands for a Laplace–Beltrami operator associated with a standard metric connection ∇ on Mn+1 . Our main interest is in the structure of the solution of a general Cauchy problem for the equation L[ψ] = 0
∗ Current
address: Dept. of Mathematics, Cornell Univ., Ithaca, NY 14853-4201, USA. Current E-mail:
[email protected] 159
Reviews in Mathematical Physics, Vol. 12, No. 2 (2000) 159–180 c World Scientific Publishing Company
(2)
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Y. BEREST and P. WINTERNITZ
with initial data given on a smooth non-characteristic space-like submanifold S ⊂ Mn+1 of codimension one: ∂ψ = f1 (x) , (3) ψ|S = f0 (x) , ∂ν S where ∂/∂ν is a transversal derivative with respect to the hypersurface S. For a generic hyperbolic operator L of normal type (1) a (local ) domain of dependence D(ξ) ⊂ S of the solution ψ(ξ) of the Cauchy problem (2), (3) at a point ξ ∈ Ω \ S is determined by an intersection J± (ξ) ∩ S of the closure of the interior of the characteristic conoid J± (ξ) := {x ∈ Ω | γ(x, ξ) > 0} (with the vertex at ξ) and the initial value manifold S. The function γ = γ(x, ξ) ∈ C ∞ (Ω × Ω) stands above for a square of the geodesic distance between the points x and ξ in Ω, while the sign ± corresponds to the time orientation on Mn+1 . Definition 1.1. A hyperbolic operator L is said to satisfy Huygens’ principle in a domain Ω0 ⊆ Ω in Mn+1 , if for any proper Cauchy problem (2), (3) D(ξ) ⊆ ∂J± ∩ S at every point ξ ∈ Ω0 . Huygens’ principle is a rare analytic property which is intrinsic for a quite exceptional class of second order hyperbolic operators. The question of explicit determination of all such operators is nowadays a classical problem in mathematical physics. Originally posed by J. Hadamard [21], this problem has turned out to be very difficult and, at present, it is still far from its complete solution.a Hadamard’s problem can be split into two separate, although interrelated parts. Namely, (i) find and classify (modulo conformal equivalence) all local (smooth and/or analytic) pseudo-Riemannian structures of a given dimension compatible with Huygens’ principle; (ii) for each such a structure construct a complete hierarchy of Huygens’ hyperbolic operators with the metric principal part (1). The ordinary d’Alembertians
2n+1 =
∂ ∂x0
2
−
∂ ∂x1
2
− ··· −
∂ ∂xn
2 (4)
with an even number d := n+1 of independent variables are the simplest well-known examples of Huygens’ operators and, therefore, even-dimensional flat (Minkowski) spaces admit Huygens’ principle. The problem of type (ii) on such spaces has been studied most thoroughly. However, its exhaustive solution is available only for d = 4 ([2, 22, 32]), while for higher dimensions only partial results are known (see [3, 4, 6, 8, 9, 30, 31, 40]). 1.2. The purpose of this paper is to present a new result in the theory which, it is hoped, may be of interest because of its inherently algebraic nature. a Hadamard’s problem has received a good deal of attention and the literature is extensive (see, e.g. [7, 12, 14, 17, 19, 20, 23, 26, 33]). A comprehensive list of references can be found, e.g. in the monograph [20] and in a more recent review article [7]. An interesting historical account on Huygens’ principle is given in [14, 19].
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We will demonstrate a close relation between the inner algebraic structure of the (local) group Gc of conformal transformations on a Lorentzian manifold Mn+1 and the proper existence of non-trivial hierarchies of Huygens’ operators thereon. The mechanism of such a relation is provided through a (local) separation of variables. To make the idea more precise, we first consider a (locally) decomposable (pseudo-)Riemannian space Md . By definition (cf. [39]), there is a coordinate chart i j Ω = Ωp ×Ωq in M with local coordinates {x0 , x00 }, i = 1, . . . , p; j = 1, . . . , q; p+q = 2 d, in which the fundamental metric form ds restricted to the (local) tangent bundle T Ω is additively decomposed (5) ds2 T Ω = (ds0 )2 + (ds00 )2 . Here, the forms (ds0 )2 and (ds00 )2 are defined over T Ωp and T Ωq and, hence, depend only on the coordinates x0 ∈ Ωp and x00 ∈ Ωq , respectively. The operator (1) defined in Ω is properly called decomposable, if L = L0 + L00 ,
(6)
where L0 and L00 are second order partial differential operators with metric principal parts, depending on and acting with respect to the variables x0 and x00 only. There exists a natural (sometimes called functorial, cf. [20]) relation between fundamental solutions of a decomposable operator L and its decomposition summands L0 and L00 [29]. This relation is given in terms of expansion coefficients of the distributional fundamental solution in the vicinity of its singularity support. It enables one to reduce the question of Huygens’ principle for the operator L to a similar question concerning operators L0 and L00 . In a sense to be made precise later, the decomposable hyperbolic operator L is Huygensian, if both of its lower dimensional counterparts L0 and L00 are Huygensian. The decomposability property (5) can be viewed as a particular case of the (multiplicative) separation of variables for Eq. (2). More generally, we will say that the local coordinates {xi } in Ω: o i n i1 x = x , . . . , x im , 1
m
ik = 1, . . . , dk ,
m X
dk = d ,
k=1
are (partially) R-separable for a second order linear differential operator L, if there exist (locally) smooth nonzero functions {gk (x)|gk (x) ∈ C ∞ (Ω)} and R(x) ∈ C ∞ (Ω) such that m X R ◦ L ◦ R−1 = gk (x) ◦ Lk , (7) k=1
where each Lk is a second order differential operator depending n o non and acting o b x with respect to only an appropriate part of variables = x1 , . . . , xdk . k
k
k
If m = d, so that all L1 , . . . , Lm are one-dimensional (i.e. ordinary) differential b The operators L may also depend on some supplementary real or complex parameters (the k so-called separation constants).
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operators in the corresponding variables x, . . . , x , then the system {xi } is called m
1
totally (orthogonally) R-separablec relative to the operator L (cf. [10, 16, 27, 34–36, 43, 44] and references therein). Let F (Ω) be a ring of linear differential operators on M with coefficients smoothly defined in Ω, and let F L (Ω) ⊂ F (Ω) be its proper subring which consists of all differential operators Q such that [L, Q] = H ◦ L with some H ∈F(Ω), i.e. F L (Ω) := {Q ∈ F (Ω) | [L, Q] ≡ 0 (mod L)} . (0)
(1)
(8) (k)
The ring F L has a natural filtration F L ≡ F L ⊂ F L ⊂ · · · ⊂ F L ⊂ · · · induced (k) from F, such that F L consists of all operators in F L of order, at most, k. The (k) elements of the R-linear space F L are called kth order (local) symmetries of the (1) operator L. The space F L of the first order symmetries has a natural Lie algebra (1) (1) (1) structure since it is closed with respect to commutation in F , i.e. [F L , F L ] ⊆ F L . The corresponding (local) Lie group GL is called the group of point symmetries of L. The problem of a classification of all R-separable local coordinate systems for a given operator L modulo the action of its point symmetry group GL (or one of its proper subgroups G ⊂ GL ) admits an algebraic reformulation (see [10, 27, 34–36, 43, 44]). Namely, it is reduced to a classification of commutative subrings (abelian (2) R-algebras) in the space F L of all second order symmetry differential operators modulo the adjoint action AdG of the group G ⊆ GL . loc If M is a (locally) homogeneous (pseudo-)Riemannian space M ∼ G/K, and (2) L is a G-invariant second order differential operator on M, the set F L is known to be in the universal enveloping algebra U(g) of the Lie algebra g of the isometry group G. 1.3. Returning to our discussion of Huygens’ principle, the following natural question can be raised. Do there exist separable coordinate systems {xi } for second order differential operators with a metric principal part on a given (pseudo-) Riemannian space M such that (a) the separation of variables (7) in {xi } is not equivalent to the total decomposability (6) modulo the conformal group action on M; (b) there exists a universal (functorial) relation between the fundamental solutions of L and its lower dimensional decomposition ingredients Lk ? To make the problem more tractable, we will restrict our attention to separable coordinate systems allowing separation with ignorable variables. In such coordinates an operator L can be presented in the following form: L=
m X
gk (x) ◦ Lk ,
k=1 c If R(x) ≡ 1, the coordinates {xi } are simply called separable.
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HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES
163
where all functions gk (x) are independent of some (fixed) part of the separated variables, say x, . . . , x, p ≤ m, which are properly called ignorable. 1
p
From the algebraic point of view, the problem of classifying the separable coordinate systems for second order G-invariant differential operators on a homogeneous space M ∼ G/K with a maximal number of ignorable variables is reduced to a classification of maximal abelian subalgebras (MASAs) of a rank r ≤ dim M in the Lie algebra g of the isometry group G [35]. A series of articles has been devoted to a classification of MASAs of the classical Lie algebras sp(2n, R), su(p, q), o(n, C) and o(p, q) [13, 24, 25, 37]. More recently, MASAs of nonsimple complex Euclidean Lie algebras e(n, C) [28] and pseudo-Euclidean real Lie algebras e(p, q) [41] were classified. The results of the latter work [41] (restricted to the case p = n, q = 1) will allow us to study the problem of a functorial decomposability for wave-type operators on a Minkowski space Mn+1 . The answer to the question formulated above turns out to be affirmative. More precisely, we will show that there exist a class of separable coordinate systems in Mn+1 for second order (hyperbolic) differential operators with a metric principal part which admit a functorial relation between fundamental solutions (essentially, in the same sense as for totally decomposable operators) and which are conformally different from any (global) Minkowskian coordinate system in Mn+1 . Algebraically, these coordinates stem from (maximal) abelian nilpotent subalgebrasd in the Poincar´e algebra p(n, 1) which are not equivalent modulo the adjoint action of the pseudo-orthogonal groupe O(n + 1, 2). The existence of such a class of separable coordinate systems gives a possibility to extend (in a nontrivial way) the known hierarchies of generically indecomposable Huygens’ operators. 2. Functorial Properties of Hadamard’s Coefficients and Maximal Abelian Subalgebras of the Poincar´ e Algebra We start with a brief summary of some necessary facts from Hadamard’s theory [21] of Cauchy’s problem for second order hyperbolic operators. 2.1. Some results from Hadamard’s theory Let Mn+1 ∼ = R1,n be a Minkowski space, and let Ω be an open connected part therein. We consider a (formally) self-adjoint scalar second order hyperbolic operator (10) L = 2n+1 + u(x) , defined in Ω with u(x) ∈ C ∞ (Ω). For any ξ ∈ Ω, we fix a cone of isotropic (null) vectors in Mn+1 with its vertex at ξ: γ(x, ξ) := (x0 − ξ 0 )2 − (x1 − ξ 1 )2 − · · · − (xn − ξ n )2 = 0 ,
(11)
d An abelian subalgebra a of a Lie algebra g is called nilpotent if its adjoint representation in is nilpotent. e The group O(n + 1, 2) is locally isomorphic to the conformal group G in Mn+1 . c
g
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Y. BEREST and P. WINTERNITZ
and single out the following subsets in Mn+1 : C± (ξ) := x ∈ Mn+1 | γ(x, ξ) = 0, ξ 0 ≶ x0 , J± (ξ) := x ∈ Mn+1 | γ(x, ξ) > 0, ξ 0 ≶ x0 .
(12)
Let D(Ω) be the space of all complex-valued C ∞ -functions on Mn+1 with their supports compactly imbedded in Ω, and let D0 (Ω) be the corresponding space of distributions, i.e. a dual (topological) vector space consisting of all linear continuous complex functionals on D(Ω). Definition 2.1. A forward (backward) fundamental solution of the operator L is 0 a distribution ΦΩ ± (·, ξ) ∈ D (Ω) satisfying the conditions: (i) supp ΦΩ ± (x, ξ) ⊆ J± (ξ) , Ω (ii) L Φ± (·, ξ) (x) = δ(x − ξ) ,
(13)
for any fixed ξ ∈ Ω. The forward and backward fundamental solutions Φ± for normal hyperbolic differential operators with C ∞ -smooth coefficients are known to exist, and they are uniquely defined (see, e.g. [12, 17]). The question of Huygens’ principle is essentially a question on the structure of the support of the fundamental solution of a hyperbolic operator. It is proven that the operator L defined by (10) satisfies Huygens’ principle in a domain Ω0 ⊆ Ω in the Minkowski space Mn+1 if, and only if, supp ΦΩ ± (x, ξ) ⊆ C± (ξ) = ∂J± (ξ) for every point ξ ∈ Ω0 . The analytic description of singularities of fundamental solutions for second order scalar hyperbolic differential operators is provided in terms of their local asymptotic expansions in the vicinity of the null cone by a graded scale of distributions with weaker and weaker singularities. An appropriate scale of distributions in the Minkowski space Mn+1 is given by the classical Riesz’s convolution ring [38]. The elements of this ring are homogeneous distributions defined as values of a holomorphic D0 -valued mapping C → D0 (Mn+1 ), λ 7→ Rλ± (x, ξ), such that Rλ± (x, ξ) is an analytic continuation (in λ) of the following (regular) distribution: hRλ± (x, ·), g(x)i(ξ) =
Z
n+1
J± (ξ)
γ(x, ξ)λ− 2 g(x) dx , Hn+1 (λ)
Re λ >
n−1 , 2
(14)
where dx = dx0 ∧ dx1 ∧ . . . ∧ dxn is a volume form in Mn+1 , g(x) ∈ D(Mn+1 ), and Hn+1 (λ) is a constant given by Hn+1 (λ) = 2π
n−1 2
4λ−1 Γ(λ)Γ (λ − (n − 1)/2) .
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165
In terms of Riesz’s distributions Rλ± (·, ξ) the (local) asymptotic expansion for the fundamental solutions of L can be written in the following form:f Φ± (x, ξ) ∼
∞ X
± 4ν ν! Uν (x, ξ) Rν+1 (x, ξ) .
(16)
ν=0
The coefficients Uν (x, ξ) of expansion (16) are defined as two-point C ∞ -smooth functions Uν (x, ξ) ∈ C ∞ (Ω×Ω) , ν = 0, 1, 2, . . ., satisfying the following differentialrecurrence system (transport equations): 1 (x − ξ, ∂x ) Uν (x, ξ) + ν Uν (x, ξ) = − L [Uν−1 (·, ξ)] (x) , 4
ν ∈ Z≥0 ,
(17)
with an additional convention U−1 (x, ξ) ≡ 0. Moreover, it is required that each function Uν (x, ξ) be bounded in some neighborhood of the vertex of the characteristic cone (11): Uν (x, ξ) ∼ O(1) when
x→ ξ.
(18)
If we fix, in addition, the first (constant) coefficient U0 (x, ξ), e.g. U0 (x, ξ) ≡ 1 ,
(19)
then Eqs. (17) with the regularity and initial conditions (18) and (19) have a unique solution U = {Uν (x, ξ) | ν ∈ Z≥0 } which defines the asymptotics (16). Following the conventional terminology (cf. [20]), we will call the functions Uν (x, ξ) Hadamard’s coefficients of the operator L. It is important to note that for a hyperbolic differential operator with (locally) analytic coefficients u(x) ∈ C ω (Ω) the series in the right-hand side of (16) is locally uniformly convergent, and formula (16) gives in this case an explicit representation for the fundamental distributions Φ± (x, ξ). ± We also note that, if n = 2l is even, then supp Rν+1 (x, ξ) = J± (ξ) for all ν = 0, 1, 2, . . ., and, hence, Huygens’ principle never occurs in odd-dimensional Minkowski spaces M2l+1 . On the other hand, in the case of an odd number of space dimensions n ≥ 3, we have 1 (p−1) Φ± (x, ξ) = p V (x, ξ) δ± (γ) + W (x, ξ) η± (γ) , 2π (p−1)
where δ± (γ) stands for the proper derivative of Dirac’s delta-measure supported on the forward (respectively, backward) characteristic half-cone C± (γ), η± (γ) are regular distributions characteristic for the regions J± (ξ): Z hη± (γ), g(x)i =
g(x) dx ,
g(x) ∈ D(Mn+1 ) ,
J± (ξ) f From now on, we omit the superscript Ω in ΦΩ (x, ξ) for the sake of notational simplicity. ±
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and V (x, ξ), W (x, ξ) are regular functions in a neighborhood of the cone vertex x = ξ admiting the following expansions therein: V (x, ξ) =
W (x, ξ) =
p−1 X
1 Uν (x, ξ) γ ν , (1 − p) · · · (ν − p) ν=0 ∞ X
Uν (x, ξ)
ν=p
γ ν−p , (ν − p)!
p=
n−1 . 2
(20)
(21)
The function W (x, ξ) is usually called a logarithmic term (or a diffusion kernel) of the fundamental solution. It follows directly from the representation formula (20) that the operator L satisfies Huygens’ principle in an open neighborhood of the point x = ξ if, and only if, W (x, ξ) vanishes identically (in x) in this neighborhood. One can show that the latter condition is equivalent to vanishing of the pth Hadamard’s coefficient Up (x, ξ) on the cone surface C± (ξ), i.e. Up (x, ξ) , 0 ,
p=
n−1 , 2
(22)
where the symbol , means that an equality must hold only on γ = 0. Equation (22) is referred to as Hadamard’s criterion. It is of a particular importance for testing Huygens’ principle for second order hyperbolic operators. 2.2. Maximal abelian subalgebras of p(n, 1) A Lie algebra p(n, 1) of the Lie group GI of all continuous isometric motions in a Minkowski space Mn+1 is spanned by n + 1 (infinitesimal) translations Pµ ∈ tn+1 , n(n − 1)/2 rotations Lik ∈ o(n, 1), and n proper Lorentzian transformations L0k ∈ o(n, 1). The standard realization of p(n, 1) by first order differential operators on Mn+1 reads P0 =
∂ ∂ ∂ ∂ ∂ ∂ , Pi = , Lik = xi k − xk i , L0k = −x0 k − xk 0 , 0 i ∂x ∂x ∂x ∂x ∂x ∂x
(23)
where i < k, and i, k = 1, . . . , n. As discussed above, we will be interested in a separation of variables in the wave-type operators (10) in the space Mn+1 involving a maximal possible number of ignorable variables. The corresponding separable coordinate systems are contained among those which allow the required separation of variables for the proper Laplace–Beltrami (i.e. the standard wave) operator L0 = 2n+1 in Mn+1 . The construction of all such coordinate systems is based on a classification of maximal abelian subalgebras {a} of the Poincar´e algebra p(n, 1). A coordinate system allowing a maximal possible number of ignorable variables for the operator L0 is obtained by taking a maximal abelian subalgebra a = spanhX1 , . . . , Xm i of a rank Rk a ≤ m ≤ n + 1 in p(n, 1), and then complementing it by (n − m + 1) pairwise commuting second order elements in the centralizer of a in the universal enveloping algebra U(p(n, 1)). Representing the algebra p(n, 1) by operators (23), the ignorable coordinates {αk } can be determined
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HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES
by a local rectification of the vector fields Xk ∈ a: Xk =
∂ , ∂αk
k = 1, 2, . . . , m .
(24)
The procedure requires that the fields Xk not only commute, but that they be linearly independent at a generic point in the space Mn+1 . The latter condition can make it necessary to discard some MASAs or to eliminate some elements of a MASA. The canonical projection π : p(n, 1) ∼ = o(n, 1) n tn+1 → o(n, 1) can be restricted to any Lie subalgebra a of p(n, 1). Accordingly, we will call an abelian subalgebra a of p(n, 1) orthogonally decomposable (respectively, orthogonally indecomposable), depending on whether its projection π(a) onto o(n, 1) is orthogonally decomposable (respectively, is not so). In other words, the Lie algebra a ⊂ p(n, 1) is orthogonally decomposable if, and only if, it admits a reducible representation as an algebra of R-linear endomorphisms on the (affine) Minkowski space Mn+1 such that the corresponding invariant subspaces are mutually orthogonal to each other with respect to the metric on Mn+1 . According to classification results [41], the complete list of MASAs of the Poincar´e algebra p(n, 1), which are mutually non-equivalent under the adjoint action of the pseudo-orthogonal group O(n + 1, 2), consists of the following: (i) Orthogonal decomposition (1, 1) ⊕ l · (2, 0) ⊕ (n − 2l − 1) · (1, 0): hL01 i ⊕ hL23 , L45 , . . . , L2l,2l+1 i ⊕ hP2l+2 , . . . , Pn i ,
0 ≤ l ≤ [(n − 1)/2]
(ii) Orthogonal decomposition (0, 1) ⊕ l · (2, 0) ⊕ (n − 2l) · (1, 0): hP0 i ⊕ hL12 , L34 , . . . , L2l−1,2l i ⊕ hP2l+1 , . . . , Pn i ,
0 ≤ l ≤ [n/2]
(iii) Orthogonal decomposition (2, 1) ⊕ (l − 1) · (2, 0) ⊕ (n − 2l) · (1, 0): hP0 + P1 , L02 + L12 + P0 + P1 i ⊕ hL34 , L56 , . . . , L2l−1,2l i ⊕ hP2l+1 , . . . , Pn i , 0 ≤ l ≤ [n/2] (iv) Orthogonal decomposition (k, 1) ⊕ l · (2, 0) ⊕ (n − 2l − k) · (1, 0): hP0 + P1 , L02 + L12 − q2 P2 , . . . , L0k + L1k − qk Pk i ⊕hLk+1,k+2 , . . . , Lk+l,k+l+1 i ⊕hPk+l+2 , . . . , Pn i , Pk where qj are arbitrary reals such that j=2 qj = 0, 3 ≤ k ≤ n , 0 ≤ l ≤ [(n − k)/2], and at least two of qj ’s are different from zero.
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2.3. Functorial relations It follows from the above classification that when n ≥ 3, there exists precisely one family {a(q) } of orthogonally indecomposable O(n + 1, 2)-inequivalent MASAs in the Lie algebra p(n, 1). This family belongs to type IV with proper structural restrictions (l = 0, k = n): a(q) ∼ = spanhP0 + P1 , L02 + L12 − q2 P2 , . . . , L0n + L1n − qn Pn i .
(25)
The family {a(q) } is parameterized by (n−1)-tuples of real numbers q := (q2 , . . . , qn ) Pn ∈ Rn−1 such that k=2 qk = 0. It can be shown that a(q) is a maximal nilpotent subalgebra in p(n, 1), provided that, at least, two components of q are different from zero. The local separable coordinate system {α, β, τ } corresponding to the subalgebra a(q) ⊂ p(n, 1) is given by 1 1 x0 = β + τ + (α · αT ) τ + 2 2 1 1 x1 = β − τ + (α · αT ) τ + 2 2 k x = (τ + qk ) αk , 2 ≤ k ≤ n
1 α · Q · αT 2 1 α · Q · αT , 2
(26)
where Q := diag(q2 , . . . , qn ) , SpkQk = 0, and α := (α2 , . . . , αn ). The wave operator in Mn+1 is written in terms of coordinates (26) in the following form:
2n+1
∂2 + =2 ∂β∂τ
n X k=2
1 τ + qk
!
X 1 ∂ − ∂β (τ + qk )2 n
k=2
∂ ∂αk
2 .
(27)
From the representation (27) one can see directly that the variables αk , k = 2, . . . , n, and β are ignorable whereas τ is an essential (i.e. non-ignorable) one. We will consider a general (formally) self-adjoint wave-type operator (10) on Mn+1 with a metric principal part, for which the system {α, β, τ } is separable, the variables α, β being ignorable. More precisely, we take ∂2 + L := 2 ∂β∂τ
n X k=2
1 τ + qk
!
X 1 ∂ + ◦ Lk , ∂β (τ + qk )2 n
(28)
k=2
where Lk := −∂ 2 /∂α2k + vk (αk ), k = 2, . . . , n, are second order operators with (locally) smooth potentials vk = vk (αk ) depending only on the respective ignorable variables αk . Theorem 2.1. Let {UN |N ∈ Z≥0 } be a sequence of Hadamard’s coefficients of 2
n
the operator L, and let {U ν2 (α2 , α∗2 )|ν2 ∈ Z≥0 }, . . . , {U νn (αn , α∗n )|νn ∈ Z≥0 } be the corresponding functional sequences for operators L2 , . . . , Ln . Then the following
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HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES
k
(functorial) relation between {UN } and {U νk (αk , α∗k )} holds: X
UN (α, β, τ ; α∗ , β ∗ , τ ∗ ) =
2
π(ν) (τ, τ ∗ ) U ν2 (α2 , α∗2 )
(ν): |ν|=N 3
n
× U ν3 (α3 , α∗3 ) · · · U νn (αn , α∗n ) ,
(29)
where the summation is taken over all (n − 1)-tuples of nonnegative integers (νk ) := P (ν2 , . . . , νn ) ∈ Zn−1 of a weight |ν| := k νk = N . The corresponding weight functions π(ν) = π(ν) (τ, τ ∗ ) depend only on the non-ignorable variable τ and its conjugate τ ∗ : n Y π(ν) (τ, τ ∗ ) := [(τ + qk )(τ ∗ + qk )]−νk . (30) k=2 k
Proof. The Hadamard coefficients {U νk (αk , α∗k )} of an operator Lk , k = 2, . . . , n, satisfy the following differential-recurrence system (cf. formula (17)): k k ∂ 1 ∗ ∗ ∗ + νk U νk (αk , αk ) = − Lk U νk −1 (·, αk ) (αk ) , νk ≥ 1 , (31) (αk − αk ) ∂αk 4 with proper initial and regularity conditions: k
U 0 (αk , α∗k ) ≡ 1 ,
k
U νk (αk , α∗k ) ∼ O(1) when
αk → α∗k .
(32)
k
As discussed above, each function U νk (αk , α∗k ), νk ≥ 1, is uniquely determined k
by (31), (32) and depends on αk and its conjugate α∗k in a symmetric way, i.e. U νk k
(αk , α∗k ) = U νk (α∗k , αk ). Similarly, the coefficients UN are defined in a unique way by Eqs. (17)–(19). The regularity conditions (32) for all k = 2, 3, . . . , n automatically imply the validity of similar conditions (18) for all functions defined by the right-hand side of formula (29). Hence, in order to verify the latter formula, it remains to prove k
that the sequence {UN |N ∈ Z≥0 } satisfy Eq. (17), provided all U νk (αk , α∗k ) satisfy (31). For this, we rewrite the differential-recurrence relation (17) in terms of new variables (26): ! n X ∂ ∂ τ + q k + (αk − α∗k ) + N UN (τ − τ ∗ ) ∂τ τ ∗ + qk ∂αk k=2
=−
1 4
n X k=2
1 ◦ Lk [UN −1 ] . (τ + qk )2
(33)
Here, we have omitted the terms involving differentiation with respect to the nonignorable variable β, since, according to (29), all the functions UN are independent of this variable.
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3
2
n
For N = 0, we have U0 = U 0 U 0 · · · U 0 ≡ 1 by (29) and (32), so that formula (29) is compatible with the initial condition (19). Assuming now that the right-hand side of (29) satisfies Eq. (33) for all m = 0, 1, . . . , N − 1, we will verify its validity for m = N . We have ! n X ∂ ∂ τ + q k + (τ − τ ∗ ) (αk − α∗k ) + N UN ∂τ τ ∗ + qk ∂αk k=2
"
X
=
n X
∗
−(τ − τ )
k=2
(ν):|ν|=N
νk τ + qk
!
2
3
n
π(ν) (τ, τ ∗ )U ν2 U ν3 · · · U νn
k
+
n X 2 k−1 k+1 n ∂ U νk τ ∗ + qk (αk − α∗k ) π(ν) (τ, τ ∗ )U ν2 · · · U νk−1 U νk+1 · · · U νn τ + qk ∂αk k=2
2
!
#
n
+ N π(ν) (τ, τ ∗ ) U ν2 · · · U νn . By virtue of Eqs. (31), the right-hand side of the latter equation can be rewritten in the form: " n ! X X (τ ∗ − τ )νk − νk (τ ∗ + qk ) 2 3 n + N π(ν) (τ, τ ∗ )U ν2 U ν3 · · · U νn τ + qk (ν):|ν|=N
k=2
1X − 4 n
k=2
=
X (ν):|ν|=N
k 2 k−1 k+1 n τ ∗ + qk π(ν) (τ, τ ∗ ) Lk [U νk −1 ] U ν2 · · · U νk−1 U νk+1 · · · U νn τ + qk
" −
n X
! νk + N
2
3
!#
n
π(ν) (τ, τ ∗ )U ν2 U ν3 · · · U νn
k=2
# n 2 k n 1X 1 ∗ − Lk (τ + qk )(τ + qk )U ν2 · · · U νk −1 · · · U νn 4 (τ + qk )2 k=2
n 1 X 1 Lk =− 4 (τ + qk )2 k=2
"
X
∗
2
k
n
#
π(ν2 ,...,νk −1,...,νn ) (τ, τ )U ν2 · · · U νk −1 · · · U νn .
(ν):|ν|=N
k
Since, by convention, U −1 (αk , α∗k ) ≡ 0 for all k = 2, 3, . . . , n, the expression in the last brackets must be equal to UN −1 . Indeed, according to our induction assumption, we have X
2
k
n
π(ν2 ,...,νk −1,...,νn ) (τ, τ ∗ )U ν2 · · · U νk −1 · · · U νn
(ν):|ν|=N
=
X (µ):|µ|=N −1
2
n
π(µ) (τ, τ ∗ )U µ2 · · · U µn = UN −1 .
HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES
171
Thus, by induction Theorem 1 is proven. Remark 2.1. If all the components of q ∈ Rn−1 are chosen to be equal to each other and, hence, to zero, i.e. q2 = q3 = · · · = qn = 0, the associated abelian subalgebra a(q) can be reduced to the commutative ideal tn+1 spanned by infinitesimal translations in p(n, 1) by the adjoint action of the pseudo-orthogonal group O(n + 1, 2). This means that the corresponding separable coordinate system {α, β, τ } is conformally equivalent to a standard Minkowskian system {x0 , x1 , . . . , xn }. In this case, and only in this case, formula (29) is reduced (by a trivial transformation) to the functorial relation for totally decomposable operators found in [29]: UN (x, ξ) =
X
2
3
n
U ν2 (x2 , ξ 2 ) U ν3 (x3 , ξ 3 ) · · · U νn (xn , ξ n ) .
(34)
(ν):|ν|=N
In this way, modulo trivial transformations, the relation (34) can be viewed as a particular case of formula (29). Remark 2.2. Suppose that among the components of the vector q ∈ Rn−1 some (not all) are equal to each other. Then the diagonal matrix Q, maybe after a proper reordering of qj , is decomposed in the following way: Q := diag (q2 , . . . , qn ) = λ1 En1 + λ2 En2 + · · · + λm Enm , where λ1 = q2 = q3 = · · · = qj1 , λ2 = qj1 +1 = · · · = qj2 , . . . , λm = qjm−1 +1 = · · · = P nj = n − 1. In this case the class of admissible qn ; Enj is a unit (nj × nj )-matrix; operators L is wider than (28). Namely, we have m m 2 X X nj ∂ 1 ∂ + + ◦ Lj . (35) L=2 ∂β∂τ τ + λ ∂β (τ + λj )2 j j=1 j=1 Here, Lj := −∆nj + u(α(j) ), j = 1, 2, . . . , m, where α(j) is a part of ignorable variables α corresponding to the equal components of the parameter q, and ∆nj is a nj -dimensional Laplacian written in terms of these variables. It will be convenient to refer the numbers nj as multiplicities of respective ignorable variables α(j) . A straightforward verification shows that the assertion of Theorem 1 also remains valid in the case of arbitrary multiplicities (nj ≥ 1). 2.4. The question of Huygens’ principle Formula (29) can be used to clarify the question of Huygens’ principle for the class of hyperbolic operators (28) (or (35)) in Minkowski spaces Mn+1 . Following [29], we first adopt the following convenient definition. Let L be a linear second order differential operator of a non-parabolic type. We say that L is a terminating operator at some level p (or, simply, p-terminating), if the sequence of its Hadamard coefficients {Uν (x, ξ) | ν ∈ Z≥0 } contains only a finite number of nonzero elements, more precisely, there exists such a number p ∈ Z≥0 that (36) Up (x, ξ) 6= 0 and Uν (x, ξ) ≡ 0 for all ν > p .
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Now suppose that each operator Lk , k = 2, 3, . . . n, involved in (28) is terminating at some level pk ∈ Z≥0 , i.e. k
U pk (αk , α∗k ) 6= 0
k
and U pk +µ (αk , α∗k ) ≡ 0 for all µ > 1 ,
(37)
Then from formula (29) we have ∗
∗
∗
UNp (α, β, τ ; α , β , τ ) =
n Y
[(τ + qk ) (τ ∗ + qk )]
−pk
k=2 2
n
× U p2 (α2 , α∗2 ) · · · U pn (αn , α∗n ) , where Np :=
P
(38)
pk , and for all M ≥ 1 UNp +M (α, β, τ ; α∗ , β ∗ , τ ∗ ) ≡ 0 .
(39)
The latter equation means that the operator L is also terminating at some finite level, namely, equal to Np . According to Hadamard’s criterion (22), if Np ≤ (n − 3)/2, the operator L will satisfy Huygens’ principle in Mn+1 . In this way, we arrive at the following: Theorem 2.2. Let L be a (locally) smooth (formally) self-adjoint hyperbolic operator (10) defined in a domain Ω in the Minkowski space Mn+1 . Suppose that L allows a separation of variables in the form (28) (or (35)) in terms of local coordinates (26). Then, if each operator Lk in (28) (or (35)) is terminating at a level pk ∈ Z≥0 (k = 2, 3, . . . , n), the operator L is also terminating at the level Np equal to n X Np = pk . (40) k=2
Moreover, when n is odd, and n ≥ 3 + 2 Np , the operator under consideration satisfies Huygens’ principle (in the whole domain of its definition). 3. New Examples of Huygens’ Operators Suppose that all components of a (n − 1)-tuple q ∈ Rn−1 parameterizing the algebra a(q) are pairwise different. In this case the operator L can be presented in odinger-type) a separated form (28), and all Lk can be only one-dimensional (Schr¨ operators. A complete characterization of the class of all terminating one-dimensional Schr¨ odinger operators with (locally) smooth real potentials L=−
d dz
2 + V (z)
(41)
HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES
173
has been given by Lagnese in [30]. The corresponding functions V (z) turn out to be rationalg and expressed explicitly in terms of some polynomials Pl (z): 2 d [log Pl (z)] , l = 0, 1, 2, . . . . (42) V (z) = −2 dz The latter polynomials are determined by the differential-recurrence relation: 0 0 Pl−1 − Pl−1 Pl+1 = (2l + 1)Pl2 , Pl+1
l ≥ 1,
(43)
with initial conditions P0 ≡ 1, P1 = z. Using the result of Lagnese together with Theorems 1 and 2 we have the following: Corollary 3.1. Let L = 2n+1 +u be a (locally) smooth wave-type hyperbolic operator in Mn+1 which allows a separation of variables in the form (28) with parameters qk ∈ R such that qk 6= qj when k 6= j for all k, j = 2, . . . , n. Then L is finitely terminating if, and only if, the potential u has the form 2 n X −2 ∂ [log Plk (αk ) ] , (44) u= (τ + qk )2 ∂αk k=2
where {Plk (αk ) ∈ R[αk ]|lk ∈ Z≥0 , k = 2, . . . , n} is a subsequence of polynomials in proper ignorable variables defined via Eq. (43). The operator L satisfies Huygens’ principle in Mn+1 , if n is odd, and n≥3+2
n X
lk .
(45)
k=2
Examples. 1. Consider the simplest case (lk ) = (0, . . . , 0, 1, 0, . . . , 0). Then the operator L with potential (44) can be presented in Minkowskian coordinates {xi } as follows L = 2n+1 +
2 (xm )2
,
2 ≤ m ≤ n.
(46)
For odd n ≥ 5, the operator (46) is historically one of the first nontrivial examples of Huygens’ operators found by Stellmacher [40]. 2. Let (lk ) = (2, 2, . . . , 2, 0, 0, . . . , 0). Then we have in Minkowskian coordinates: L = 2n+1 +
m X 6xk ((xk )3 − 2γk (x0 − x1 + qk )3 ) , ((xk )3 + γk (x0 − x1 + qk )3 )2
3 ≤ m ≤ n,
(47)
k=2
where γk , k = 2, 3, . . . , m, are arbitrary (real) constants. Operator (47) is Huygensian, if n is odd and n ≥ 3 + 4 (m − 1) = 4 m − 1 . (48) g In fact, the potentials (42) can be identified with rational (decreasing) solutions of the Korteweg– de Vries equation (cf. [1]).
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Now we suppose that some components in q ∈ Rn−1 are allowed to be pairwise equal to each other. In this case we have a separation of variables in the form (35), odinger operators: where the corresponding Lk are two-dimensional Schr¨ L=−
∂ ∂z 1
2
−
∂ ∂z 2
2 + V (z 1 , z 2 ) .
(49)
The complete classification of all terminating Schr¨odinger operators (49) with (locally) smooth real potentials V (z 1 , z 2 ) homogeneous of degree (−2) has been given recently in [4]. The result of such a classification reads as follows. Let k := (kM , kM−1 , . . . , k1 ) ∈ ZM be an integer strictly monotonic partition of height M : (50) kM > kM−1 > · · · > k2 > k1 ≥ 0 , kj ∈ Z≥0 , and let ϕ := (ϕ1 , . . . , ϕM ) ∈ RM be an arbitrary real vector parameter. We associate to the pair (k, ϕ) a set of elementary 2π-periodic functions on the real line R1 : ϕ 7→ χj (ϕ), (51) χj (ϕ) := sin(kj ϕ + ϕj ) , and define the potential V = V (z 1 , z 2 ) in terms of polar coordinates in the plane (z 1 , z 2 ) ∈ R2 : 2 2 ∂ V =− 2 [log W [χ1 (ϕ), . . . , χM (ϕ)] ] , (52) r ∂ϕ where z 1 = r cos ϕ , z 2 = r sin ϕ, and W [χ1 , . . . , χM ] is a Wronskian of the set of functions (51). It is proven in [4] that operators (49) constitute the whole class of terminating two-dimensional Schr¨ odinger operators with (locally) smooth real homogeneous potentials.h Using this result together with Theorems 2.1 and 2.2 above we arrive at the following Corollary 3.2. The wave-type hyperbolic operator L = smooth potential u, given explicitly by the formula u=
m X j=1
−2 (τ + qj )2 rj2
∂ ∂ϕj
2n+1 + u with a (locally)
2 h i (j) (j) log W [χ1 (ϕj ), . . . , χMj (ϕj )] ,
is terminating at the level N(k) :=
m X
(53)
(j)
kMj .
j=1
Here, qj are arbitrary real parameters; (rj , ϕj ) are polar coordinates in Euclidean 2-planes corresponding to ignorable variables α(j) with double multiplicities nj = 2 (j) (j) (cf. (35)); W [χ1 , . . . , χMj ] is a Wronskian of a set of functions (51) related to an (j)
(j)
(j)
arbitrary monotonic partition k(j) = (kMj , kMj −1 , . . . , k1 ) ∈ ZMj and arbitrary real phases ϕ(j) ∈ RMj , j = 1, . . . , m. h It is well known that the “angular part” of (52) is precisely a periodic counterpart of a multisoliton solution of the hierarchy of KdV equations with amplitudes k and phases ϕ.
HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES
175
When n is odd, and n ≥ 3 + 2 N(k) , the operator L with potential (53) satisfies Huygens’ principle in the Minkowski space Mn+1 . The other known class of terminating generically indecomposable Schr¨ odinger operators in Rn has been found by Veselov and one of the authors in [8, 9] (see also [7]). These operators have the form L = −∆n + V (z)
(54)
with real potentials V = V (z), z ∈ Rn , associated to finite reflection groups (Coxeter groups) in Rn : X m(a)(m(a) + 1) (a, a) . (55) V (z) = (a, z)2 a∈<+
Here, < = {a} stands for the root system related to a Coxeter group W of a rank RkW ≤ n; <+ is its “positive” part with respect to a generic linear form on Rn ; m : < → Z≥0 , a 7→ m(a), is an arbitrary W-invariant integer-valued multiplicity function on the root system <. The Schr¨odinger operator (54), (55) is terminating at the level M :=
X
m(a)
(56)
a∈<+
for an arbitrary Coxeter group W. This result together with Theorems 2.1 and 2.2 above gives the following: Corollary 3.3. Let W be a (reducible) Coxeter group of a rank Rk W ≤ n − 1: W = W 1 × · · · × Wm ,
(57)
acting on a Euclidean space Rn−1 , and let < = <1 ⊕ · · · ⊕ <m be a root system of W decomposed into mutually orthogonal subsystems <j = {a(j) } with Rk <j ≡ P nj = n − 1. Rk Wj ≤ nj , We associate to W a wave-type hyperbolic operator in the Minkowski space Mn+1 of the form: (58) L = 2n+1 + u with the potential u given in terms of coordinates (26) by the formula u=
m X j=1
1 (τ + qj )2
X mj (a)(mj (a) + 1)(a, a) , (a, α(j) )2
(59)
a∈<j+
where α(j) ∈ Rnj is a set of ignorable variables corresponding to the same value of a real parameter qj (cf. (35)); mj : <j → Z≥0 , a 7→ mj (a), is a W-invariant multiplicity function defined on the subset of roots <j .
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Then L is a terminating operator at the level M=
X
m(a) =
m X X
mj (a) .
(60)
j=1 a∈<j+
a∈<+
Moreover, if n is odd and n ≥ 2M + 3, the operator under consideration satisfies Huygens’ principle in Mn+1 . In conclusion, we note that the examples of Huygens’ operators presented in this section are generically not reducible to their (totally) decomposable counterparts corresponding to the case q = 0. This follows directly from the fact that the proper separable coordinate systems are conformally non-equivalent to a Minkowskian system of coordinates in Mn+1 . 4. Concluding Remarks: A Modified Hadamard Conjecture In the present paper we have established a close relation between the functorial properties of the fundamental solutions of a class of second order hyperbolic operators on Lorentzian spaces and the algebraic structure of the corresponding (local) conformal group thereon. In the case of Minkowski spaces Mn+1 , n ≥ 3, these properties have been studied in detail. As a result, we have constructed some new hierarchies of wave-type Huygens’ operators in Mn+1 . However, the problem of complete determination of the whole class of such operators for arbitrary n still remains open. The famous Hadamard conjecture (cf. [12, 21]) claiming that any Huygens’ operator L can be reduced to the ordinary d’Alembertian 2n+1 with the help of trivial equivalence transformations (local changes of coordinates and conformalgauge transformations of the operator L) is proved to be valid only in M3+1 (see [2, 22, 32]). In Minkowski spaces of higher dimensions (n ≥ 5), it fails to be true. A relevant modification of Hadamard’s conjecture for Minkowski spaces of an arbitrary dimension has been proposed recently by one of the authors in [4]. Here, we discuss briefly this statement and show that it is quite consistent with the results of the present paper. Let F(Ω) be a ring of linear differential operators defined in an open set Ω in Mn+1 . For a fixed pair of operators L0 , L ∈ F(Ω) we define a map: adL,L0 : F (Ω) → F (Ω) ,
A 7→ adL,L0 [A]
(61)
in such a way that adL,L0 [A] := L ◦ A − A ◦ L0 .
(62)
Then, given a number N ∈ Z>0 , the N th iteration of adL,L0 -map is determined as follows: adN L,L0 [A] := adL,L0 [adL,L0 [. . . adL,L0 [A]] . . .] =
N X k=0
(−1)k
N k
LN −k ◦ A ◦ Lk0 .
(63)
HUYGENS’ PRINCIPLE AND SEPARATION OF VARIABLES
177
Definition 4.1. The operator L ∈ F(Ω) is called N -gauge related to the operator L0 ∈ F(Ω), if there exists a smooth nonzero function θ ∈ C ∞ (Ω) defining a multiplication operator in F (Ω) and an integer positive number N ∈ Z>0 , such that +1 adN (64) L,L0 [θ(x)] ≡ 0 . Remark 4.1. It is easy to see that Eq. (64) defines an equivalence relation on the set F ∗ (Ω) ⊂ F (Ω) of (formally) self-adjoint operators in F (Ω). It has been proven in [3] that if L0 is a second order partial differential operator with a metric principal part in Mn+1 terminating at a level N0 , and L is N -gauge related to L0 via (64), then L is also a differential operator of the second order with the same principal part, which is terminating, at least, at the level N + N0 . Moreover, it is shown that for all known classes of indecomposable terminating Schr¨ odinger operators (mentioned in Sec. 3), the identities of the form (64) do exist. This circumstance motivates the following modification of Hadamard’s conjecture put forward in [3]: Conjecture 4.1. A wave-type hyperbolic operator L of the general form L = 2n+1 + (a(x), ∂) + u(x) ,
(65)
in a Minkowski space Mn+1 (n is odd, n ≥ 3) satisfies Huygens’ principle if, and only if, it is N -gauge related to the ordinary wave operator 2n+1 in Mn+1 . We note that Eq. (64) is equivalent to a strongly overdeterminated system of nonlinear partial differential equations with respect to the lower order coefficients a(x) and u(x) of the operator L. To solve these equations directly for arbitrary N seems to be a hopeless task. However, relation (64) provides a very efficient analytical test for the validity of Huygens’ principle which appears to be quite adapted for computer symbolic manipulations. In this connection it is worthwhile to mention some related problems of current interest in the theory of integrable systems: the problem of a classification of supercomplete commutative rings of partial differential operators initiated in the work [11], and the bispectral problem posed and studied in the paper [15] (see also [18]). Various connections between Huygens’ principle and these problems, demonstrating their common algebraic nature, are discussed in [3, 5, 7, 42]. In conclusion of the paper we will show that our present results are well consistent with an algebraic modification of Hadamard’s conjecture formulated above. Proposition 4.1. Let L be a smooth wave-type hyperbolic operator in Mn+1 allowing a separation of variables in coordinates (26) in the form (28) (or (35)). Suppose that each Schr¨ odinger operator Lk is N -gauge related to its principal part Lk0 , i.e. k +1 adN Lk ,Lk0 [θk (αk )] ≡ 0
for some
Nk ∈ Z≥0 .
(66)
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Then L is also N -gauge related to the ordinary wave operator L0 = precisely, the following identity holds:
2n+1 . More
+1 adN L,L0 [θN (α, τ )] ≡ 0
with N :=
n X
(67)
Nk
k=2
and θN (α, τ ) :=
n Y
(τ + qk )Nk θk (αk ),
q := (qk ) ∈ Rn−1 ,
k=2
n X
qk = 0 .
k=2
Proof. Note that Eq. (64) is invariant under the similarity transformation: L 7→ L˜ := λ ◦ L ◦ λ−1 ,
θ 7→ θ˜ := λ θ
(68)
for any nonzero smooth function λ(·) ∈ C ∞ (Ω). Choosing in our case λ=
n Y
(τ + qk )−Nk
k=2
and applying transformation (68), we simplify Eq. (67). After that implication (66) ⇒ (67) is easily verified by induction in a way similar to that of the proof of Theorem 1. Acknowledgments One of the authors (Yu. B.) would like to thank Professor F. A. Gr¨ unbaum (University of California at Berkeley) for fruitful discussions of various aspects of Huygens’ principle and the bispectral problem during his stay at Berkeley (June, 1996). The work presented in this paper is partially supported by an ISM Fellowship (for Yu. B.) and by research grants from NSERC of Canada and FCAR du Qu´ebec (for P. W.). References [1] M. Adler and J. Moser, “On a class of polynomials connected with the Korteweg–de Vries Equation”, Commun. Math. Phys. 61 (1978) 1–30. [2] L. Asgeirsson, “Some hints on Huygens’ principle and Hadamard’s conjecture”, Commun. Pure Appl. Math. 9 (1956) 307–326. [3] Yu. Yu. Berest, “Hierarchies of Huygens’ operators and Hadamard’s conjecture”, Acta Appl. Math. 53 (1998) 125–185. [4] Yu. Yu. Berest, “Solution of a restricted Hadamard’s problem in Minkowski spaces”, Commun. Pure Appl. Math. 50 (1997) 1019–1052. [5] Yu. Yu. Berest, “Lacunae of hyperbolic Riesz kernels and commutative rings of partial differential operators”, Lett. Math. Phys. 41 (1997) 227–235. [6] Yu. Yu. Berest and I. M. Lutsenko, “Huygens’ principle in Minkowski spaces and soliton solutions of the Korteweg–de Vries equation”, Commun. Math. Phys. 190 (1997) 113–132.
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[7] Yu. Yu. Berest and A. P. Veselov, “Huygens’ principle and integrability”, Uspekhi Mat. Nauk 49(6) (1994) 3–77. [8] Yu. Yu. Berest and A. P. Veselov, “Huygens’ principle and Coxeter groups”, Russian Math. Surveys 48(2) (1993) 181–182. [9] Yu. Yu. Berest and A. P. Veselov, “Hadamard’s problem and Coxeter groups: new examples of Huygens’ equations”, Funct. Anal. Appl. 28(1) (1994) 1–15. [10] C. P. Boyer, E. G. Kalnins and P. Winternitz, “Separation of variables for the Hamilton–Jacobi equation on complex projective spaces”, SIAM J. Math. Anal. 16 (1985) 93–109. [11] O. A. Chalykh, and A. P. Veselov, “Commutative rings of partial differential operators and Lie algebras”, Commun. Math. Phys. 126 (1990) 597–611. [12] R. Courant and D. Hilbert, Methods of Mathematical Physics, V.II Interscience Publ., N.-Y., 1962. [13] M. A. del Olmo, M. A. Rodriguez, P. Winternitz and H. Zassenhaus, “Maximal Abelian subalgebras of pseudo-unitary Lie algebras”, Lin. Algebra Appl. 135 (1990) 79–151. [14] J. J. Duistermaat, “Huygens’ principle for linear partial differential equations”, in: Huygens’ Principle 1690-1990: Theory and Applications, eds. H. Blok, H. A. Ferwerda and H. K. Kuiken, Elsevier Publ. B.V., 1992, pp. 273–297. [15] J. J. Duistermaat and F. A. Gr¨ unbaum, “Differential equations in the spectral parameter”, Commun. Math. Phys. 103 (1986) 177–240. [16] L. P. Eisenhart, “Separable systems of St¨ ackel”, Ann. Math. 35 (1934) 284–305. [17] F. G. Friedlander, The Wave Equation on a Curved Space-time, Cambridge, Cambridge Univ. Press, 1975. [18] F. A. Gr¨ unbaum, “Time-band limiting and the bispectral problem”, Commun. Pure Appl. Math. 37 (1994) 307–328. [19] P. G¨ unther, “Huygens’ principle and Hadamard’s conjecture”, The Mathematical Intelligencer 13(2) (1991) 56–63. [20] P. G¨ unther, Huygens’ Principle and Hyperbolic Equations, Acad. Press, Boston, 1988. [21] J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, 1923. [22] J. Hadamard, “The problem of diffusion of waves”, Annals of Math. 43(3) (1942) 510–523. [23] S. Helgason, “Wave equations on homogeneous spaces”, Lect. Notes in Math. 1077, Springer-Verlag, Berlin, 1984, 252–287. [24] V. Hussin, P. Winternitz and H. Zassenhaus, “Maximal Abelian subalgebras of complex orthogonal Lie algebras”, Lin. Algebra Appl. 141 (1990) 183–220. [25] V. Hussin, P. Winternitz and H. Zassenhaus, “Maximal Abelian subalgebras of pseudo-orthogonal Lie algebras”, Lin. Algebra Appl. 173 (1992) 125–163. [26] N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht, 1985. [27] E. G. Kalnins, Separation of Variables for Riemannian Spaces of Constant Curvature, Longman Scientific & Technical, Harlow, John Wiley & Sons, N.-Y., 1986. [28] E. G. Kalnins and P. Winternitz, “Maximal Abelian subalgebras of complex Euclidean Lie algebras”, Canad. J. Phys. 72 (1994) 389–404. [29] J. E. Lagnese, “The fundamental solutions and Huygens’ Principle for decomposable differential operators”, Arch. Rat. Mech. Anal. 19(4) (1965) 299–307. [30] J. E. Lagnese, “A solution of Hadamard’s problem for a rectricted class of operators”, Proc. Amer. Math. Soc. 19 (1968) 981–988. [31] J. E. Lagnese and K. L. Stellmacher, “A method of generating classes of Huygens’ operators”, J. Math. Mech. 17(5) (1967) 461–472. [32] M. Mathisson, “Le probl`eme de M. Hadamard relatif a ` la diffusion des ondes”, Acta Math. 71 (1939) 249–282.
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[33] R. G. McLenaghan, “Huygens’ principle”, Ann. Inst. Henri Poincar´e 37(3) (1982) 211–236. [34] W. Miller Jr., Symmetry and Separation of Variables, Addison-Wesley, Reading, Mass., 1977. [35] W. Miller Jr., J. Patera and P. Winternitz, “Subgroups of Lie groups and separation of variables”, J. Math. Phys. 22 (1981) 251– 260. [36] J. Patera and P. Winternitz, “A new basis for representations of the rotation group: Lam´e and Heun polynomials”, J. Math. Phys. 14 (1973) 1130–1139. [37] J. Patera, P. Winternitz and H. Zassenhaus, “Maximal Abelian subalgebras of real and complex symplectic Lie algebras”, J. Math. Phys. 24 (1983) 1973–1985. [38] M. Riesz, “L’int´egrale de Riemann-Liouville et le probl` eme de Cauchy”, Acta Math. 81 (1949) 1–223. [39] H. Ruse, A. Walker and T. Wilmore, Harmonic Spaces, Ediziori Cremonese, Rome, 1961. [40] K. L. Stellmacher, “Ein Beispiel einer Huygensschen Differentialgleichung”, Nachr. Akad. Wiss. G¨ ottingen, Math.-Phys. K1, IIa, Bd. 10 (1953) 133–138. [41] Z. Thomova and P. Winternitz, “Maximal Abelian subgroups of the isometry and conformal groups of Euclidean and Minkowski spaces”, J. Phys. A 31(7) (1998) 1831–1858. [42] A. P. Veselov, “Huygens’ principle and algebraic Schr¨ odinger operators”, in Topics in Topology and Mathematical Physics, ed. S. P. Novikov Amer. Math. Soc. Transl. Ser. 2 170 (1995) 199–206. [43] P. Winternitz and I. Fris, “Invariant expansions of relativistic amplitudes and subgroups of the proper Lorentz group”, Sov. J. Nucl. Phys. 1 (1965) 636–643. [44] P. Winternitz, I. Lukac and Ya. A. Smorodinsky, “Quantum groups in the little groups of the Poincar´e group”, Sov. J. Nucl. Phys. 7 (1968) 139–145.
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS WITH MAGNETIC FIELDS KURT BRODERIX Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen Bunsenstr. 9, D-37073 G¨ ottingen, Germany E-mail:
[email protected]
DIRK HUNDERTMARK∗ Department of Physics, Jadwin Hall Princeton University P.O.Box 708, Princeton, New Jersey 08544, USA E-mail:
[email protected]
HAJO LESCHKE Institut f¨ ur Theoretische Physik Universit¨ at Erlangen-N¨ urnberg Staudtstr. 7, D-91058 Erlangen, Germany E-mail:
[email protected] Received 18 August 1998 Mathematical Physics Preprint Archive: math-ph/9808004 The objects of the present study are one-parameter semigroups generated by Schr¨ odinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multidimensional Euclidean space; in case that it is a proper subset, the Schr¨ odinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simon’s landmark paper [Bull. Amer. Math. Soc. 7 (1982) 447] to non-zero vector potentials and more general configuration spaces.
Contents 1. Introduction 2. Basic Definitions and Representations 3. Continuity of the Semigroup in its Parameter 4. Continuity of the Image Functions of the Semigroup 5. Continuity of the Semigroup in the Potentials 6. Continuity of the Integral Kernel of the Semigroup Appendix A. Approximability of Kato-type Potentials by Smooth Functions ∗ New
182 183 193 195 198 203 209
address: Department of Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail:
[email protected] 181
Reviews in Mathematical Physics, Vol. 12, No. 2 (2000) 181–225 c World Scientific Publishing Company
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Appendix B. Proof of the Feynman–Kac–Itˆ o Formula Appendix C. Brownian-Motion Estimates Acknowledgments References
211 217 223 223
1. Introduction In non-relativistic quantum physics Schr¨ odinger operators with magnetic fields play an important role [26, 27]. In suitable physical units they are given by differential expressions of the form HΛ (A, V ) =
1 (−i∇ − A(x))2 + V (x) 2
(1.1)
over the configuration space Λ ⊆ Rd . Here the scalar potential V is a real-valued function on Λ representing the potential energy. The vector potential A is an Rd valued function on Λ giving rise to the magnetic field ∇ × A. For the purposes of quantum physics HΛ (A, V ) has to be given a precise meaning as a self-adjoint operator acting on the Hilbert space L2 (Λ) of complex-valued wave-functions on Λ. Clearly, if Λ 6= Rd this requires to impose boundary conditions on the wave functions in the domain (of definition) of HΛ (A, V ). As impressively demonstrated in Simon’s landmark paper [58], many spectral odinger semiproperties of HΛ (A, V ) can efficiently be derived by studying the Schr¨ group {e−tHΛ (A,V ) }t≥0 . The latter can be done by probabilistic techniques via the Feynman–Kac–Itˆ o path-integral formula. In fact, the works [10, 58] make essential use of this approach for the case A = 0 and Λ = Rd , that is, when both the magnetic field vanishes and the configuration space is the whole Euclidean space. For A = 0 and Λ ⊆ Rd some results may be found in [64, Chap. 1]. Another recent monograph further elucidating the relation between “quantum potential theory” and Feynman–Kac processes is [11]. The main goal of the present work is to extend some of the key results in [10, 58] on continuity properties to rather general A and Λ. In doing so, we follow [58] in restricting ourselves to Kato decomposable [37] scalar potentials V . The class of vector potentials A considered will only be restricted by local Kato-like conditions. The configuration space will either be Λ = Rd or an arbitrary non-empty open subset Λ ⊂ Rd . Since vector potentials in one spatial dimension are of no physical interest, we will only consider dimensions d ≥ 2. As for the necessary boundary conditions for Λ 6= Rd , we are only able to handle Dirichlet conditions. Apart from that, in our opinion, the resulting class of Schr¨ odinger operators with magnetic fields is general enough to cover most systems of physical interest with finite ground-state energy. Continuity properties of the generated semigroups have turned out to be valuable in obtaining interesting results in that field of mathematical physics where magnetic fields play a major role. We mention so-called magnetic Lieb–Thirring inequalities [22], heat-kernel estimates [21, 23] and the existence and Lifshits tailing of the integrated density of states of random Schr¨ odinger operators [8, 9, 24, 67]. In proving the statements of the present paper we closely follow [10, 58, 64] using the probabilistic approach. Since we aim at a reasonably self-contained presentation
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which is intelligible for many readers, our arguments are perhaps more detailed than usual. The reader should note that Simon’s work [58] is followed by several other interesting developments in semigroup theory relating to Schr¨ odinger operators. While some of them rely on probabilistic techniques, others do not. In a setting closest to that of the present paper one of us [34, 35] has considerably relaxed the conditions imposed on the vector potential A. Related results with A 6= 0 but with weaker assumptions on the scalar potential V have been obtained in [42] building on the works [39, 68] for A = 0. Basically, in [39, 42, 68] the negative part of V no longer needs to be infinitesimally form-bounded relative to the unperturbed Schr¨ odinger operator HΛ (0, 0). There has also been a lot of activity [14–16, 43] in studying perturbations of positivity preserving semigroups with generators more general than HΛ (0, 0). Extensions into other directions investigate perturbations by measures instead of functions [3, 7, 28, 29, 61, 63]. Last but not least we mention the progress [62] for Schr¨ odinger-like semigroups with an underlying configuration space which is only locally Euclidean, see also [15, 16]. For a wealth of information on regularity and spectral properties of Schr¨ odinger operators (for Λ = Rd ) with or without magnetic field we recommend the recent works [31, 32, 45], see also [44] and references therein. In contrast to the present paper these works do not rely on semigroups. 2. Basic Definitions and Representations In this section we fix our basic notation and give a short compilation of the classes of scalar potentials V and vector potentials A which we will consider. Moreover, we will give a precise definition of the Schr¨ odinger operator (1.1) and recall the appropriate Feynman–Kac–Itˆ o formula for its semigroup. The open ball of radius % > 0 centered about the origin in the ν-dimensional Euclidean space Rν , ν ∈ N, is denoted by B% := {x ∈ Rν : |x| < %} .
(2.1)
Here |x| := (x · x)1/2 is the norm of x = (x1 , . . . , xν ) ∈ Rν derived from the Pν scalar product x · y := j=1 xj yj . Given a subset Ω of Rν we write ( ω 7→ χΩ (ω) :=
1
for ω ∈ Ω
0
otherwise
(2.2)
for its indicator function. The closure of Ω is written as Ω and its boundary is denoted by ∂Ω. ∂ , . . . , ∂x∂ ν ) in Rν as ∇ and the Lebesgue meaWe denote the nabla operator ( ∂x 1 ν sure on the Borel subsets of R as dx. All real-valued or complex-valued functions on Rν are assumed to be Borel measurable. If not stated otherwise, we identify functions which differ only on sets of Lebesgue measure zero. The Banach space Lp (Ω) of pth-power Lebesgue-integrable functions (1 ≤ p ≤ ∞) on a Borel set Ω ⊆ Rν of non-zero Lebesgue measure consists of complex-valued
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functions ψ : Ω → C such that the norm Z 1/p p Ω |ψ(x)| dx kψkp := ess sup |ψ(x)|
p<∞ (2.3)
for p=∞
x∈Ω
is finite. We recall that L2 (Ω) becomes a Hilbert space when equipped with the scalar product Z (2.4) hφ|ψi := (φ(x))∗ ψ(x) dx . Ω
Here the star denotes complex conjugation. The norm of an operator X : Lp (Ω) → Lq (Ω), 1 ≤ p, q ≤ ∞ is defined as kXkp,q := sup kXψkq .
(2.5)
kψkp =1
The space Lpunif,loc (Ω) of uniformly locally pth-power integrable functions (1 ≤ p ≤ ∞) consists of functions ψ : Ω → C such that the norm Z 1/p p sup |ψ(y)| χ (x − y) dy p<∞ B1 x∈Rν Ω (2.6) kψkLpunif,loc (Ω) := for p=∞ ess sup |ψ(x)| x∈Ω
Lploc (Ω)
of locally pth-power integrable functions (1 ≤ p ≤ ∞) is finite. The space consists of functions ψ : Ω → C such that ψχK ∈ Lp (Ω) for all compact K ⊆ Ω. The positive part f + and the negative part f − of a real-valued function f on Ω are defined by (2.7) f ± (x) := sup{±f (x), 0} , x ∈ Ω . In case that Ω ⊆ Rν is open, we write C(Ω) for the space of complex-valued continuous functions on Ω. The subspace of arbitrarily often differentiable functions with compact support inside Ω is written as C0∞ (Ω). In the following Λ denotes a fixed, non-empty, open, not necessarily proper subset of Rd , d ≥ 2. We stress that we will always assume d ≥ 2. In physical terms, Λ serves as the configuration space. Any complex-valued function f defined a priori on Λ will be understood to be trivially extended to Rd by defining f (x) := 0 for x 6∈ Λ without further notice. We use this extension convention to injectively embed Lp (Λ) into Lp (Rd ), 1 ≤ p ≤ ∞, and thus have Lp (Λ) ⊆ Lp (Rd ) and similarly C0∞ (Λ) ⊆ C0∞ (Rd ), etc. We write d=2 − ln |x| for (2.8) g% (x) := χB% (x) |x|2−d d≥3 for the repulsive Newton–Coulomb potential on Rd truncated outside the Ball B% .
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Definition 2.1. A scalar potential is a real-valued function V on Λ. The function V is said to be in the Kato class K(Rd ) if Z g% (x − y) |V (y)| dy = 0 . (2.9) lim sup %↓0 x∈Rd
The function V is said to be in the local Kato class Kloc (Rd ) if V χK ∈ K(Rd ) for all compact K ⊂ Rd . The function V is said to be Kato decomposable, in symbols V ∈ K± (Rd ), if V + ∈ Kloc (Rd ) and V − ∈ K(Rd ). The Kato norm is given by Z g1 (x − y)|V (y)| dy . (2.10) kV kK(Rd ) := sup x∈Rd
Note that due to the above extension convention it is reasonable to state V ∈ K(Rd ) even if V is a priori only defined on Λ ⊆ Rd . The following remarks are borrowed from [13, Chap. 1.2], [2, Sec. 4] and [11, Chap. 3]. Remark 2.1. (i) To find out whether or not a function belongs to the Kato class or local Kato class the following inclusions may be helpful. For p > d2 one has Lpunif,loc(Rd ) ⊂ K(Rd ) ⊂ L1unif,loc (Rd ) ,
(2.11)
Lploc (Rd ) ⊂ Kloc (Rd ) ⊂ L1loc (Rd ) .
(2.12)
(ii) The Kato class is complete with respect to the Kato norm [58, Erratum], [68, Sec. 5], that is, K(Rd ) is a Banach space. The inclusions (2.11) hold also with respect to convergence in norm, that is, convergence in k • kLpunif,loc (Rd ) implies convergence in k • kK(Rd ) which in turn implies convergence in k • kL1unif,loc (Rd ) , where p > d2 . (iii) Let 2HΛ (0, 0) denote the self-adjoint Friedrichs-extension on the Hilbert space L2 (Λ) of the negative Laplacian −∇ · ∇ on C0∞ (Λ), that is, −2HΛ (0, 0) is the usual Laplacian on Λ with Dirichlet boundary conditions. Then any V ∈ K(Rd ) is infinitesimally form-bounded [50, Definition p. 168] relative to HΛ (0, 0) [11, Corollary to Theorem 3.25]. Alternatively, this follows from [37, Proposition 2.1] or [2, Theorem 4.7] used with the fact that, by the extension convention, the form domain of HΛ (0, 0) is a subset of the form domain of HRd (0, 0). The class K(Rd ) is nearly maximal with respect to infinitesimal form-boundedness [13, Theorem 1.12]. Consequently, the set of Kato-decomposable functions should contain all physically relevant scalar potentials, which lead to Schr¨odinger operators (1.1) with a finite ground-state energy. (iv) An example illustrating the admissible local singularities of scalar potentials in K(Rd ) for d ≥ 3 is Θ1/2 (x) , µ > 0, (2.13) Vµ (x) := |x|2 | ln |x||µ
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where Θ1/2 is some real-valued function in C0∞ (Rd ) which vanishes outside the ball B3/4 and furthermore obeys Θ1/2 (x) = 1 for |x| < 1/2. According to [13, Example (a) p. 8], [2, Proposition 4.10] one has the equivalence Vµ ∈ K(Rd ) ⇔ µ > 1 .
(2.14)
Note that, for d = 3, Vµ obeys the Rollnik condition [53, Chap. 1] whenever µ > 12 , see [53, Example 1.6.3]. Finally, Vµ is infinitesimally form-bounded relative to HRd (0, 0) for all µ > 0. Useful for handling Kato-decomposable functions is the following possibility to approximate them by nice functions. Proposition 2.1. such that
Let V ∈ K± (Rd ). Then there is a sequence {Vn }n∈N ⊂ C0∞ (Rd ) lim k(V − Vn )χK kK(Rd ) = 0
n→∞
for all compact K ⊂ Rd and Z Z g% (x − y)Vn− (y) dy ≤ sup g% (x − y)V − (y) dy sup sup n∈N x∈Rd
(2.15)
(2.16)
x∈Rd
for all 0 < % ≤ 1. This approximability and the idea of its proof have been stated in [58, § B.10]. The approximating sequence {Vn } may be constructed from V in the following standard way. The potential V is smoothly truncated outside an increasingly large ball and then mollified by convolution with an approximate delta function in C0∞ (Rd ). Since V ∈ K± (Rd ) is locally integrable, the resulting approximations are in C0∞ (Rd ). Then patiently estimating shows that (2.15) and (2.16) can be fulfilled. The details can be found in Appendix A. We now define the classes of vector potentials to be dealt with in the sequel. Definition 2.2. A vector potential is an Rd -valued function A on Λ. A vector potential A is said to be in the class H(Rd ), if its squared norm A2 := A · A and its divergence ∇ · A, considered as a distribution on C0∞ (Rd ), are both in K(Rd ). It is said to be in the class Hloc (Rd ), if both A2 and ∇ · A are in Kloc (Rd ). Remark 2.2. (i) As it should be from the physical point of view, only local regularity conditions are required for a vector potential to lie in Hloc (Rd ). Moreover, since once continuously differentiable vector potentials are included, in symbols (C 1 (Rd ))d ⊂ Hloc (Rd ), most physically relevant vector potentials are covered. From this point of view the class H(Rd ) is of less interest, because, for example, any vector potential A ∈ (C 1 (R3 ))3 giving rise to a constant magnetic field ∇ × A lies in Hloc (R3 ) but not in H(R3 ). Therefore, we try to avoid global regularity assumptions as far as possible, that is, we aspire after results valid for vector potentials in Hloc (Rd ).
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(ii) With regard to the one-parameter family of vector potentials considered in [13, Theorem 6.2], we note that the spectrum of the associated Schr¨odinger operator dramatically changes its character precisely at that parameter value where the border between H(Rd ) and Hloc (Rd ) is reached. (iii) The local singularities admissible for vector potentials in H(Rd ), d ≥ 3, are illustrated by the example Aµ (x) :=
x (Vµ (x))1/2 , |x|
(2.17)
where Vµ is defined in (2.13). According to Remark 2.1(iv) one has A2µ ∈ K(Rd ) ⇔ µ > 1, but (2.18) Aµ ∈ H(Rd ) ⇔ µ > 2 , as can be seen by explicitly calculating ∇ · Aµ . Similar to Kato-decomposable scalar potentials, vector potentials in Hloc (Rd ) can be approximated by smooth functions. Proposition 2.2. Let A ∈ Hloc (Rd ). Then there is a sequence {Am }m∈N ⊂ (C0∞ (Rd ))d such that (2.19) lim k(A − Am )2 χK kK(Rd ) = 0 m→∞
and lim k(∇ · A − ∇ · Am )χK kK(Rd ) = 0
m→∞
(2.20)
for all compact K ⊂ Rd . The proof is similar to the one of Proposition 2.1 and is given in Appendix A. The next topic is the precise construction of the Schr¨ odinger operator (1.1) as 2 a self-adjoint operator acting on the Hilbert space L (Λ). Here we use the definition via forms [51, Sec. VIII.6], [50, Sec. X.3], [13, Sec. 1.1]. In a first step we consider A ∈ Hloc (Rd ) and V + ∈ Kloc (Rd ). The sesquilinear form + + : C0∞ (Λ) × C0∞ (Λ) → C , (φ, ψ) 7→ hA,V (φ, ψ) , (2.21) hA,V Λ Λ where +
1X h(−i∇ − A)j φ|(−i∇ − A)j ψi 2 j=1 d
1
1
(φ, ψ) := h(V + ) 2 φ|(V + ) 2 ψi + hA,V Λ
(2.22)
is densely defined in L2 (Λ) and non-negative. For Λ = Rd the closure of the form (2.21) has form domain +
1
) := {ψ ∈ L2 (Rd ) : (−i∇−A)ψ ∈ (L2 (Rd ))d , (V + ) 2 ψ ∈ L2 (Rd )} Q(hA,V Rd
(2.23) +
) of see [57], [13, Theorem 1.13]. For general open Λ ⊆ Rd the completion Q(hA,V Λ C0∞ (Λ) with respect to the form norm q + (ψ, ψ) (2.24) kψkhA,V + := kψk22 + hA,V Λ Λ
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cannot exceed L2 (Rd ) because the form is closable for Λ = Rd . Since L2 (Λ) is a closed subspace of L2 (Rd ) the completion is even a subspace of L2 (Λ). Thus the form + (2.21) extended to the domain Q(hA,V ) is non-negative and closed. According to Λ [51, Theorem VIII.15] it therefore defines a unique self-adjoint operator on L2 (Λ) denoted as HΛ (A, V + ). Before proceeding with the construction of the Schr¨ odinger operator we note that the just-defined Schr¨ odinger operator HΛ (A, 0), A ∈ Hloc (Rd ), obeys the socalled diamagnetic inequality |e−tHΛ (A,0) ψ| ≤ e−tHΛ (0,0) |ψ| ,
ψ ∈ L2 (Λ), t ≥ 0 .
(2.25)
For Λ = Rd the inequality may be found in [58, Proposition B.13.1], [5, Theorem 2.3]. For general open Λ ⊆ Rd the estimate (2.25) is given in [47, Corollary 3.6] and is a special case of [42, Corollary 2.3]. Moreover, we will get it in the version (2.25) from Lemma B.5 in Appendix B devoted to a proof of the Feynman–Kac–Itˆ o formula below. In the second step we consider A ∈ Hloc (Rd ) and V ∈ K± (Rd ). According to Remark 2.1(iii) V − ∈ K(Rd ) is infinitesimally form-bounded relative to HΛ (0, 0). This and the diamagnetic inequality (2.25) imply that V − is infinitesimally formbounded relative to HΛ (A, 0) ≤ HΛ (A, V + ). In the case Λ = Rd the last statement is proven in [5, Theorem 2.5], [56, Theorem 15.10]. For general open Λ ⊆ Rd it is proven in [47, Proposition 3.7]. All three proofs are virtually identical. Due to the established form-boundedness the KLMN theorem [50, Theorem X.17] is applicable and ensures that +
+
: Q(hA,V ) × Q(hA,V ) → C, hA,V Λ Λ Λ +
A,V (φ, ψ) 7→ hA,V (φ, ψ) − h(V − ) 2 φ|(V − ) 2 ψi Λ (φ, ψ) := hΛ 1
1
(2.26)
is a closed sesquilinear form bounded from below and with form core C0∞ (Λ). The associated semi-bounded self-adjoint operator is denoted as HΛ (A, V ). Remark 2.3. (i) Of course, the above construction of HΛ (A, V ) works for A2 , V + ∈ L1loc (Rd ) and V − form-bounded relative to HΛ (A, V + ) with bound strictly smaller than 1. If we even assume A2 , ∇ · A, V + ∈ L2loc (Rd ) the construction of HΛ (A, V ) is a bit easier, since then the form (2.21) comes directly from the nonnegative symmetric operator C0∞ (Λ) → L2 (Λ) ,
ψ 7→
1 (−i∇ − A)2 ψ + V + ψ 2
(2.27)
and is therefore closable [50, Theorem X.23]. (ii) C0∞ (Rd ) is not only a form core but even an operator core for HRd (A, V ), if in addition to A ∈ Hloc (Rd ), V ∈ K± (Rd ) one has A2 , ∇ · A, V ∈ L2loc (Rd ). This follows from [31, Theorem 2.5]. In [58, Theorem B.13.4] the statement without the restriction ∇ · A ∈ L2loc (Rd ) is incorrectly ascribed to [40]. Other criteria for the essential self-adjointness of HRd (A, V ) on C0∞ (Rd ) may be found for instance in [40, Theorem 3] and, more generally, in [41, Corollary 1.4].
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(iii) If Λ 6= Rd the above construction of HΛ (A, V ) corresponds, roughly speaking, to imposing Dirichlet boundary conditions on ∂Λ in order to render (1.1) for◦
+ 1 mally self-adjoint. In fact, for A = 0 one has Q(h0,V Λ ) = Q(V ) ∩ H (Λ) where ◦
H1 (Λ) is the Sobolev space of functions in L2 (Λ) vanishing on ∂Λ in distributional sense and having a square-integrable distributional gradient, see, for example, [20, Sec. 2.3]. We are going to recall the Feynman–Kac–Itˆ o formula for the semigroup odinger operator HΛ (A, V ). {e−tHΛ (A,V ) }t≥0 generated by the above-defined Schr¨ We denote by Px Wiener’s probability measure associated with standard Brownian motion w on Rd having diffusion constant 12 and almost surely continuous paths s 7→ w(s) starting from x ∈ Rd , that is, w(0) = x. The induced expectation is written as Ex . According to [2, Theorem 4.5] the Kato class is conveniently characterized in terms of Brownian motion by the following equivalence: Z t |f (w(s))| ds = 0 . (2.28) f ∈ K(Rd ) ⇔ lim sup Ex t↓0 x∈Rd
0
This is a consequence of the chain of inequalities Z t Z C1 (d) gtκ(d) (x − y) |f (y)| dy ≤ Ex |f (w(s))| ds 0
Z
≤ C2 (d)
g% (x − y) |f (y)| dy
+ C3 (d, %, t) kf kL1unif,loc (Rd ) ,
(2.29)
which hold for all d ≥ 2, 0 < t < 1, 0 < % < 12 . Here we have set κ(2) := 1 and κ(d) := 12 for d ≥ 3 and C1 (d), C2 (d) are strictly positive constants. Moreover, C3 is a function obeying limt↓0 C3 (d, %, t) = 0 for all d ≥ 2, 0 < % < 12 . This chain of inequalities is implicit in the proof of [2, Theorem 4.5]. Since, almost surely, Brownian-motion paths stay for finite times in bounded regions, the equivalence (2.28) proves the implication Z t |f (w(s))| ds < ∞ = 1 , x ∈ Rd , t ≥ 0 . (2.30) f ∈ Kloc (Rd ) ⇒ Px 0
Confer the proof of Lemma C.8 of Appendix C. Remark 2.4. (i) The implication (2.30) explains with the help of [25, Sec. 4.3] or [36, Definition 3.2.23] that the stochastic line integral in the sense of Itˆ o Z t A(w(s)) · dw(s) (2.31) t 7→ 0
of a vector potential A ∈ Hloc (Rd ) is for all x = w(0) ∈ Rd a well-defined stochastic process possessing a continuous version.
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(ii) For a vector potential A ∈ Hloc (Rd ) and a scalar potential V ∈ K± (Rd ) the potentials’ part of the Euclidean action t 7→ St (A, V |w) ,
t ≥ 0,
(2.32)
where Z
t
St (A, V |w) := i
A(w(s)) · dw(s) + 0
Z
i 2
Z
t
(∇ · A) (w(s)) ds 0
t
V (w(s)) ds
+
(2.33)
0
is for all x = w(0) ∈ Rd a well-defined complex-valued stochastic process possessing a continuous version. This follows from the preceding remark and (2.30), confer [10, Lemma 3.1]. o formula. Our basic tool will be the following variant of the Feynman Kac Itˆ Proposition 2.3. Let A ∈ Hloc (Rd ), V ∈ K± (Rd ) and Λ ⊆ Rd open. Then for any ψ ∈ L2 (Λ) and t ≥ 0 one has (e−tHΛ (A,V ) ψ)(x) = Ex [e−St (A,V |w) ΞΛ,t (w) ψ(w(t))]
(2.34)
for almost all x ∈ Λ, where ( ΞΛ,t (w) :=
1
if w(s) ∈ Λ f or all 0 < s ≤ t
0
otherwise .
(2.35)
For Λ = Rd , ∇ · A = 0 the proposition is a special case of [56, Theorem 15.5]. The proof given there can be extended easily to Λ = Rd , ∇ · A ∈ L1loc (Rd ). For Λ ⊆ Rd , A = 0, V = 0 the proposition is equivalent to [54, Theorem 3] and [56, Theorem 21.1]. Finally, for Λ ⊆ Rd , A = 0, V + ∈ Kloc (Rd ), V − = 0, the proposition follows from [64, Proposition 1.3.3, Theorem 1.4.11]. We have not found a proof for the above setting in the literature. Therefore we give a detailed proof in Appendix B. There we use Proposition 2.3 for Λ = Rd to get (2.34) for Λ ⊂ Rd , A ∈ Hloc (Rd ) and V ∈ L∞ (Rd ) in a way patterned after the proof of [54, Theorem 3], [56, Theorem 21.1]. Eventually we generalize the achieved result to the assumptions of the proposition by an approximation technique taken from the proof of [56, Theorem 6.2]. With considerably more effort it is possible to prove a Feynman–Kac–Itˆo formula for still more general vector potentials, see [34, 35]. There is even a Feynman–Kac– Itˆo formula under virtually minimal conditions [47], which suffers, however, from being less explicit than (2.34). From the triangle inequality and the proof of [10, Proposition 3.1] or [58, Theorem B.1.1] the right-hand side of (2.34) — considered as a function of x — is seen to lie in Lq (Λ) for any ψ ∈ Lp (Λ) where t > 0, 1 ≤ p ≤ q ≤ ∞. This statement is
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also a special case of Lemma C.1. Hence the right-hand side of (2.34) can be used t : Lp (Λ) → Lq (Λ). By the same argument this operator is to define an operator Tp,q bounded, that is, t kp,q < ∞ , kTp,q
t > 0,
1 ≤ p ≤ q ≤ ∞.
(2.36)
t }t≥0, 1≤p≤q≤∞ thus obtained constitutes a oneThe family of operators {Tp,q parameter semigroup in the sense that 0 = 1, Tp,p
s+t s t Tp,r = Tq,r ◦ Tp,q ,
s, t > 0 ,
1 ≤ p ≤ q ≤ r ≤ ∞.
(2.37)
This follows from the Markov properties of Brownian motion and of the Itˆ o integral in (2.33). Moreover, the family is self-adjoint in the sense that the dual mapping t is Tpt0 ,q0 where the indices p0 , q 0 dual to p, q are defined as usual through of Tp,q 1 1 1 1 p0 + p := 1 =: q0 + q . t }t≥0 is a strongly continuous semigroup In the next section we will see that {Tp,p for all finite 1 ≤ p < ∞. Nevertheless, for mnemonic reasons, we will always write t for all 1 ≤ p ≤ q ≤ ∞. e−tHΛ (A,V ) instead of Tp,q Within this notation the triangle inequality applied to the right-hand side of (2.34) generalizes the diamagnetic inequality (2.25) to |e−tHΛ (A,V ) ψ| ≤ e−tHΛ (0,V ) |ψ| ,
ψ ∈ Lp (Λ) ,
(2.38)
confer [58, Theorem B.13.2], [42, Eq. (2.13)]. For vanishing magnetic field A = 0 the semigroup is increasing in Λ in the sense that for Λ ⊆ Λ0 ⊆ Rd , Λ, Λ0 open, one has e−tHΛ (0,V ) χΛ ψ ≤ e−tHΛ0 (0,V ) ψ
(2.39)
for all ψ ≥ 0, ψ ∈ Lp (Λ0 ), 1 ≤ p ≤ ∞ and t ≥ 0. Because ΞΛ,t ≤ ΞΛ0 ,t , this follows from (2.34) by inspection. The above-mentioned boundedness of e−tHΛ (A,V ) can now be sharpened to the statement that for given V ∈ K± (Rd ) there are real constants C and E, independent of t, p and q, so that ke−tHΛ (A,V ) kp,q ≤ ke−tHΛ (0,V ) kp,q ≤ ke−tHRd (0,V ) kp,q −(q−p)d/2pq −tE e 1≤p
(2.40)
for all A ∈ Hloc (Rd ). The first inequality follows from (2.38), confer [58, Corollary B.13.3], the middle inequality is a consequence of (2.39) and the last inequality is given in [58, Eq. (B.11)]. For related estimates see also [2, Sec. 1] and references therein. Furthermore, C and E can be chosen [58, Theorem B.5.1] such that E is any number smaller than the infimum of the L2 (Rd )-spectrum of HRd (0, V ). Eventually, we turn to the notion of a configuration space with a regular boundary.
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Definition 2.3.
A set Λ ⊂ Rd is called regular, if it is open and Ex [ΞΛ,t (w)] = 0
(2.41)
for all x ∈ ∂Λ, t > 0. The set Λ ⊂ Rd is called uniformly regular, if it is open and lim sup Ex [ΞΛ,t−τ (w(• + τ ))] = 0 τ ↓0 x∈∂Λ
(2.42)
for all t > 0. Furthermore, it is understood that Λ = Rd is both regular and uniformly regular. Remark 2.5. (i) The above definition of regularity is equivalent to the standard one [6, Definition II.1.9], [48, Sec. 2.3], which employs first exit times. More precisely, Λ ⊂ Rd is regular if and only if it is open and Px {inf{s > 0 : w(s) 6∈ Λ} = 0} = 1
(2.43)
for all x ∈ ∂Λ. To prove this assertion we first mention that by the definition (2.35) of ΞΛ,t (w), (2.43) implies (2.41). To show the opposite direction, note that t 7→ {w(s) ∈ Λ for all 0 < s ≤ t}
(2.44)
is decreasing in the sense of set inclusion. Therefore, (2.43) is equivalent to [ {w(s) ∈ Λ for all 0 < s ≤ t} = 0 , (2.45) Px t>0,t∈Q
where Q denotes the set of rational numbers. Due to (2.35), (2.41) implies (2.45). (ii) Using dominated convergence, one checks that (2.41) is equivalent to lim Ex [ΞΛ,t−τ (w(• + τ ))] = 0 τ ↓0
(2.46)
for all x ∈ ∂Λ, t > 0. Therefore, regularity of Λ is implied by uniform regularity, as it should. There are several known conditions implying regularity, see, for example, [6, Sec. II.1], [48, Sec. 2.3]. Here we only recall Poincar´ e’s cone condition, because it may easily be adapted to uniform regularity. The open set Cr,β (x, u) := {y ∈ Rd : 0 < |y − x| < r, 0 < u · (y − x) < |y − x| cos β}
(2.47)
is called a finite cone with vertex at x ∈ Rd in direction u ∈ Rd , |u| = 1, with opening angle 0 < β < π2 and radius r > 0. Proposition 2.4.
Let Λ ⊂ Rd open.
(i) If for all x ∈ ∂Λ there is a finite cone Cr,β (x, u) ⊂ Rd \Λ, u ∈ Rd , |u| = 1, r > 0, 0 < β < π2 , then Λ is regular.
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
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(ii) If there are constants 0 < β < π2 , r > 0 such that for all x ∈ ∂Λ there is a finite cone Cr,β (x, u) ⊂ Rd \Λ, u ∈ Rd , |u| = 1, then Λ is uniformly regular. Proof. The first assertion is proven in [6] as Proposition 1.13. This may be seen, using Remark 2.5(i). To show the second part, we use for x ∈ ∂Λ the estimate Ex [ΞΛ,t−τ (w(• + τ ))] ≤ Ex [ΞRd \Cr,β (x,u),t−τ (w(• + τ ))] = E0 [ΞRd \Cr,β (0,u),t−τ (w(• + τ ))] ,
(2.48)
where the equality follows from the rotation and translation invariance of Brownian motion and u is any given unit vector. By the first part of the proposition, Rd \Cr,β (0, u) is regular. Thus the right-hand side of (2.48), which is independent of x ∈ Rd , tends to 0 as τ ↓ 0 due to Remark 2.5(ii). This proves (2.42), whence uniform regularity. 3. Continuity of the Semigroup in its Parameter The following result is a straightforward generalization of [10, Proposition 3.2] to non-zero magnetic fields and Λ ⊆ Rd . Theorem 3.1. Let A ∈ Hloc (Rd ), V ∈ K± (Rd ) and Λ ⊆ Rd open. Moreover, let 1 ≤ p < ∞ be finite. Then the semigroup {e−tHΛ (A,V ) : Lp (Λ) → Lp (Λ)}t≥0
(3.1)
as defined in Sec. 2 is strongly continuous, that is, 0
lim k(e−tHΛ (A,V ) − e−t HΛ (A,V ) )ψkp = 0
(3.2)
t0 →t
for all ψ ∈ Lp (Λ) and all t ≥ 0. Proof. Due to the semigroup property (2.37) for p = q = r we may assume 0 ≤ t, t0 ≤ 1. Then the norm in (3.2) is bounded from above by 0
−tHΛ (A,V ) sup ke kp,p e−|t−t |HΛ (A,V ) − 1 ψ . (3.3) p
0≤t≤1
Using (2.40) it is therefore sufficient to show (3.2) for t = 0. Since (3.2) holds in the free case A = 0, V = 0 and Λ = Rd , it is enough to establish lim kDt,t ψkp = 0 . t↓0
(3.4)
Here we have made use of the abbreviation Dt,τ := e−τ HRd (0,0) e−(t−τ )HΛ (A,V ) − e−tHΛ (A,V )
(3.5)
where 0 ≤ τ ≤ t. The Feynman–Kac–Itˆo formula (2.34) and the Jensen Ineq. |Ex [•]|p ≤ Ex [| • |p ] give |(Dt,t ψ)(x)|p ≤ Ex [|1 − e−St (A,V |w) ΞΛ,t (w)|p |ψ(w(t))|p ] .
(3.6)
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
Exploiting that St (A, V |w) turns into its complex conjugate under time reversal of Brownian motion, (3.6) leads upon integration over x ∈ Λ to Z p Ex [|1 − e−St (A,V |w) ΞΛ,t (w)|p ] |ψ(x)|p dx . (3.7) (kDt,t ψkp ) ≤ Λ 0 p
Using |z − z | ≤ 2 (|z| + |z 0 |p ) for z, z 0 ∈ C, we obtain Z Ex [|1 − e−St (A,V |w) |p ] |ψ(x)|p dx . (kDt,t ψkp )p ≤ 2p p
p
Λ
Z
+2
Ex [1 − ΞΛ,t (w)] |ψ(x)|p dx
p
(3.8)
Λ
and employing additionally −V ≤ V − and t ≤ 1 we get |1 − e−St (A,V |w) |p ≤ 2p (1 + e−S1 (0,−pV
−
|w)
).
(3.9)
The action St (A, V |w) vanishes for all x = w(0) almost surely as t ↓ 0 due to Remark 2.4(ii). Moreover, Z − − Ex [1 + e−S1 (0,−pV |w) ] |ψ(x)|p dx ≤ (1 + ke−HRd (0,−pV ) k∞,∞ )(kψkp )p < ∞ Λ
(3.10) by (2.34) and (2.40). Hence the dominated-convergence theorem implies that the first integral on the right-hand side of (3.8) vanishes as t ↓ 0. In order to show that the second integral vanishes too, we claim lim sup Ex [1 − ΞΛ,t (w)] = 0
(3.11)
Λr := {x ∈ Λ : |x − y| > r for all y ∈ ∂Λ}
(3.12)
t↓0 x∈Λr
for all r > 0. Here
denotes the set of points well inside Λ. In fact, one has sup Ex [1 − ΞΛ,t (w)] ≤ P0 sup |w(s)| ≥ r x∈Λr
(3.13)
0<s≤t
and the right-hand side vanishes as t ↓ 0 due to L´evy’s maximal inequality [56, Eq. (7.60 )]. This completes the proof of (3.4). Remark 3.1. (i) Even for A = 0 and Λ = Rd Theorem 3.1 is slightly different from [10, Proposition 3.2] as can be inferred from (2.11). For A = 0 and Λ ⊆ Rd with finite Lebesgue measure Theorem 3.1 is contained in [11, Theorem 3.17]. (ii) As we will see in the next section, the set e−tHΛ (A,V ) L∞ (Λ) contains only continuous functions if A ∈ Hloc (Rd ), V ∈ K± (Rd ) and t > 0. Therefore, the semigroup {e−tHΛ (A,V ) : L∞ (Λ) → L∞ (Λ)}t≥0 is not strongly continuous for all pairs (A, V ) ∈ Hloc (Rd ) × K± (Rd ), see also [10, Remark 3.4]. However, for Λ = Rd , consider the closed subspace C∞ (Rd ) of L∞ (Rd ) consisting of continuous functions
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vanishing at infinity. A slight modification of the proof of (4.6) below shows that the semigroup maps C∞ (Rd ) into itself. Moreover, with somewhat more effort it can be shown that this restriction yields a strongly continuous semigroup, confer [11, Theorem 3.17]. 4. Continuity of the Image Functions of the Semigroup In this section we prove that the operator e−tHΛ (A,V ) is smoothing in the sense that it maps Lp (Λ) into the set C(Λ) of complex-valued continuous functions on Λ. For the reader’s convenience we recall that a family F of functions f : Λ → C is called equicontinuous, if lim sup |f (x) − f (x0 )| = 0
(4.1)
x0 →x f ∈F
for all x ∈ Λ and it is called uniformly equicontinuous, if sup
lim
sup |f (x) − f (x0 )| = 0 ,
r↓0 x,x0 ∈Λ, |x−x0 |
(4.2)
see, for example, [51, Sec. I.6]. Theorem 4.1.
Let A ∈ Hloc (Rd ), V ∈ K± (Rd ) and Λ ⊆ Rd open. Then
e−tHΛ (A,V ) Lp (Λ) ⊆ Lq (Λ) ∩ C(Λ) ,
t > 0,
1 ≤ p ≤ q ≤ ∞,
(4.3)
and for fixed t > 0, 1 ≤ p ≤ ∞ the family {e−tHΛ (A,V ) ψ : ψ ∈ Lp (Λ), kψkp ≤ 1}
(4.4)
of functions on Λ is equicontinuous. Furthermore, the right-hand side of (2.34) gives the continuous representative of x 7→ (e−tHΛ (A,V ) ψ)(x) ,
x ∈ Λ.
(4.5)
Finally, lim (e−tHΛ (A,V ) ψ)(x) = 0
|x|→∞
(4.6)
for all ψ ∈ Lp (Λ) with finite 1 ≤ p < ∞ and all t > 0. Proof. Using the boundedness (2.40) and the semigroup property (2.37) with q = r = ∞ it is sufficient to prove that (4.4) is an equicontinuous family for p = ∞ in order to get both (4.3) and the equicontinuity for all 1 ≤ p ≤ ∞. Since this holds in the free case A = 0, V = 0 and Λ = Rd , one has due to (2.40) that {e−τ HRd (0,0) e−(t−τ )HΛ (A,V ) ψ : ψ ∈ L∞ (Λ), kψk∞ ≤ 1}
(4.7)
is an equicontinuous family for all 0 < τ ≤ t. Therefore, it is enough to show lim ess sup τ ↓0
x∈K
sup |(Dt,τ ψ)(x)| = 0
kψk∞ ≤1
(4.8)
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
for all compact K ⊂ Λ. To this end, we represent the image of ψ by the operator difference (3.5) as (Dt,τ ψ)(x) = Ex [eSτ (A,V |w)−St (A,V |w) × (ΞΛ,t−τ (w(τ + •)) − e−Sτ (A,V |w) ΞΛ,t (w)) ψ(w(t))]
(4.9)
where we have used the Feynman–Kac–Itˆo formula (2.34), the additivity of the integrals in the action (2.33) and the Markov property of the Brownian motion w. We use the right-hand side of (4.9) to give meaning to (Dt,τ ψ)(x) for all x ∈ Rd . Then, in order to show that the right-hand side of (2.34) defines the continuous representative of (4.5), it is sufficient to establish sup |(Dt,τ ψ)(x)| = 0
lim sup
τ ↓0 x∈K kψk∞ ≤1
(4.10)
for all compact K ⊂ Λ. The triangle inequality in combination with Sτ (0, V |w) − St (0, V |w) ≤ −St (0, −V − |w)
(4.11)
yields |(Dt,τ ψ)(x)| ≤ Ex [e−St (0,−V
−
|w)
|ΞΛ,t−τ (w(τ + •)) − e−Sτ (A,V |w) ΞΛ,t (w)|
× |ψ(w(t))|] .
(4.12)
By |ψ(w(t))| ≤ kψk∞ and the Cauchy–Schwarz Ineq. one arrives at |(Dt,τ ψ)(x)|2 ≤ (kψk∞ )2 Ex [e−St (0,−2V
−
|w)
](Ex [ΞΛ,t−τ (w(τ + •)) − ΞΛ,t (w)]
+ Ex [|1 − e−Sτ (A,V |w) |2 ]) .
(4.13)
Since Ex [ΞΛ,t−τ (w(τ + •)) − ΞΛ,t (w)] = Ex [ΞΛ,t−τ (w(τ + •))(1 − ΞΛ,τ (w))] ≤ Ex [1 − ΞΛ,τ (w)] ,
(4.14)
(4.10) follows from (3.11) and Lemmas C.2, C.5. This completes the proof of (4.3) and the equicontinuity and identifies the continuous representative. Finally, for the proof of (4.6) we may assume without loss of generality 1 < p < ∞ due to (2.40) and the semigroup property. Then (4.6) follows from H¨ older’s inequality 0
0
|(e−tHΛ (A,V ) ψ)(x)| ≤ (ke−tHRd (0,p V ) k∞,∞ )1/p ((e−tHRd (0,0) |ψ|p )(x))1/p where p0 := p/(p − 1) < ∞.
(4.15)
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Remark 4.1. (i) The assertion (4.3) reduces for A = 0, V ∈ K± (Rd ) and Λ = Rd to [58, Corollary B.3.2]. A related result, also for A = 0 and Λ = Rd , is [10, Propositions 3.1, 3.3]. Our proof is patterned after that of [10, Lemma 3.2] or [11, Propositions 3.11, 3.12] and, in contrast to the strategy in [58], does not use Propositions 2.1 and 2.2. (ii) For V = 0, ∇ · A = 0 and A2 ∈ Kloc (Rd ) [59, Theorem 3.1] asserts that e−tHRd (A,0) (L∞ (Rd ) ∩ L2 (Rd )) ⊆ L∞ (Rd ) ∩ C(Rd ) ,
t > 0.
(4.16)
The proof given there, however, applies for all A2 ∈ K(Rd ) but not for all A2 ∈ Kloc (Rd ), because it can happen that Z
t
sup Ex x∈K
(A(w(s)))2 ds = ∞
(4.17)
0
although A2 ∈ Kloc (Rd ) and K ⊂ Rd compact. Consider the example A(x) = (x2 , −x1 )e|x|
4
(4.18)
of a vector potential on R2 . Since HΛ (A, V ) is equipped with Dirichlet boundary conditions on ∂Λ one would expect that (4.19) lim (e−tHΛ (A,V ) ψ)(x) = 0 , y ∈ ∂Λ , x→y
for a sufficiently nice boundary of Λ. Theorem 4.2. Let A ∈ Hloc (Rd ), V ∈ K± (Rd ) and Λ ⊂ Rd regular. Then (4.19) holds for all ψ ∈ Lp (Λ), 1 ≤ p ≤ ∞ and t > 0. Furthermore, (4.4) is an equicontinuous family of functions on Λ for all 1 ≤ p ≤ ∞, t > 0. Proof. The semigroup property (2.37) and (2.40) ensure that it is sufficient to check the case p = ∞. The triangle inequality, the Cauchy–Schwarz Ineq. and (2.34) imply |(e−tHΛ (A,V ) ψ)(x)| ≤ kψk∞ (ke−tHRd (0,2V ) k∞,∞ )1/2 (Ex [ΞΛ,t (w)])1/2
(4.20)
for almost all x ∈ Rd . Now the assertions follow from Theorem 4.1, (2.40) and Lemma C.7. Not surprisingly, uniform continuity of the image functions can be achieved for sufficiently regular Λ ⊆ Rd by imposing global regularity conditions for the potentials, thereby, however, possibly excluding physically relevant cases, confer Remark 2.2(i). Theorem 4.3. Let A ∈ H(Rd ), V ∈ K(Rd ) and Λ ⊆ Rd uniformly regular. Then (4.4) is a uniformly equicontinuous family of functions on Λ for all 1 ≤ p ≤ ∞ and t > 0.
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Proof. Analogously to the reasoning at the beginning of the proof of Theorem 4.1 it suffices to check (4.21) lim ess sup sup |(Dt,τ ψ)(x)| = 0 τ ↓0
kψk∞ ≤1
x∈Λr
in order to get uniform equicontinuity on Λr , r > 0. This follows with the help of (4.13) and (4.14) from (3.11), Lemma C.2 and Lemma C.3. Using Lemma C.7 and the estimate (4.20) we conclude lim ess sup
sup
r↓0 x∈Λ\Λr kψk∞ ≤1
|(e−tHΛ (A,V ) ψ)(x)| = 0 ,
(4.22)
which is sufficient to extend the domain of uniform equicontinuity from Λr for r > 0 to Λ. Remark 4.2.
Proposition 3.1 in [60] is a special case of Theorem 4.3 and (4.6).
5. Continuity of the Semigroup in the Potentials As a motivation for this section consider the simple example Ah (x) := (0, x1 h) ,
h > 0,
(5.1)
of a vector potential on R2 . Clearly, Ah belongs to Hloc (R2 ) and gives rise to a constant magnetic field of strength h. The related semigroup {e−tHR2 (Ah ,0) : L2 (R2 ) → L2 (R2 )}t≥0 ,
(5.2)
considered as a function of h, is not norm-continuous because lim sup ke−tHR2 (Ah ,0) − e−tHR2 (0,0) k2,2 > 0
(5.3)
h↓0
for all t > 0. This can be deduced, for example, from [5, Theorem 6.3]. It reflects the fact that the character of the energy spectrum of an electrically charged pointmass in the Euclidean plane changes from purely continuous to pure point, when an arbitrarily low constant magnetic field, perpendicular to the plane, is turned on. The following theorem, however, shows that under suitable technical assumptions the weaker notion of local-norm-continuity holds. Theorem 5.1. Let A ∈ Hloc (Rd ), {Am }m∈N ⊂ Hloc (Rd ), V ∈ K± (Rd ) and {Vn }n∈N ⊂ K± (Rd ) such that for all compact K ⊂ Rd lim k(A − Am )2 χK kK(Rd ) = 0 ,
(5.4)
lim k(∇ · A − ∇ · Am ) χK kK(Rd ) = 0
(5.5)
lim k(V − Vn ) χK kK(Rd ) = 0 ,
(5.6)
m→∞ m→∞
and n→∞
lim sup sup
%↓0 n∈N x∈Rd
Z
g% (x − y)Vn− (y) dy = 0 .
(5.7)
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Moreover, let Λ ⊆ Rd open. Then lim
sup kχK (e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )kp,q = 0
(5.8)
lim
sup k(e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) ) χK kp,q = 0
(5.9)
m,n→∞ τ1 ≤t≤τ2
and m,n→∞ τ1 ≤t≤τ2
for all compact K ⊂ Rd , 0 < τ1 ≤ τ2 < ∞ and 1 ≤ p ≤ q ≤ ∞. Furthermore, for p = q one may allow τ1 = 0. Proof. According to the Riesz–Thorin interpolation theorem [50, Theorem IX.17] it is enough to prove the theorem for the three cases p = q = 1, p = q = ∞ and p = 1, q = ∞. Moreover, due to the self-adjointness of the semigroup, the assertions (5.8) and (5.9) are equivalent under the combined substitutions p 7→ (1 − p1 )−1 , q 7→ (1 − 1q )−1 . In consequence, it remains to show the following three partial assertions: (1) (5.8) for p = q = ∞, τ1 = 0 (2) (5.9) for p = q = ∞, τ1 = 0 (3) (5.8) for p = 1, q = ∞, τ1 > 0. We note that (5.8) and (5.9) in the case p = q, τ1 = 0 follow already from the assertions (1) and (2). As to assertion (1) Since the Feynman–Kac–Itˆo formula (2.34) and the triangle inequality give kχK (e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )k∞,∞ ≤ sup Ex [|e−St (A,V |w) − e−St (Am ,Vn |w) |] , x∈K
(5.10) the assertion follows from Lemma C.6. As to assertion (2) Let BR be the open ball of radius R > 0 centered about the origin in Rd , see (2.1). Then the diamagnetic inequality (2.38) in combination with (2.39) and the triangle inequality yields k(e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) ) χK k∞,∞ ≤ kχBR (e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )k∞,∞ + k(1 − χBR ) e−tHRd (0,V ) χK k∞,∞ + k(1 − χBR ) e−tHRd (0,Vn ) χK k∞,∞ . (5.11) Hence assertion (2) follows from assertion (1) provided that lim
sup sup k(1 − χBR ) e−tHRd (0,Vn ) χK k∞,∞ = 0 .
R→∞ 0≤t≤τ2 n∈N
(5.12)
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
To prove (5.12) we use the Cauchy–Schwarz inequality in the Feynman–Kac–Itˆo formula to obtain k(1 − χBR ) e−tHRd (0,Vn ) χK k∞,∞ = sup Ex [e−St (0,Vn |w) χK (w(t))] x6∈BR
≤ (k(1 − χBR ) e−tHRd (0,0) χK k∞,∞ )1/2 (ke−tHRd (0,2Vn ) k∞,∞ )1/2 .
(5.13)
By Lemma C.1 the second factor on the right-hand side is uniformly bounded with respect to n ∈ N and 0 ≤ t ≤ τ2 . The first factor is seen to vanish uniformly in 0 ≤ t ≤ τ2 as R → ∞ by an elementary calculation. As to assertion (3) In a first step we get with the help of the triangle inequality kχK (e−2tHΛ (A,V ) − e−2tHΛ (Am ,Vn ) )k1,∞ ≤ N1 + N2 + N3 ,
(5.14)
where N1 := kχK (e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) ) e−tHΛ (A,V ) k1,∞ ,
(5.15)
N2 := kχK e−tHΛ (Am ,Vn ) χBR (e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )k1,∞ ,
(5.16)
N3 := kχK e−tHΛ (Am ,Vn ) (1 − χBR )(e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )k1,∞ .
(5.17)
In a second step we will repeatedly use the inequality kXY kp,r ≤ kXkq,r kY kp,q
(5.18)
for bounded operators Y : Lp (Rd ) → Lq (Rd ) and X : Lq (Rd ) → Lr (Rd ). By (5.18) and (2.40) we get N1 ≤ kχK (e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )k∞,∞ ke−tHRd (0,V ) k1,∞ .
(5.19)
By employing additionally the self-adjointness of the semigroup we obtain N2 ≤ ke−tHRd (0,Vn ) k1,∞ k(e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) ) χBR k∞,∞
(5.20)
and similarly N3 ≤ kχK e−tHRd (0,Vn ) (1 − χBR )k∞,∞ (ke−tHRd (0,V ) k1,∞ + ke−tHRd (0,Vn ) k1,∞ ) . (5.21) Hence assertion 3 follows from assertions 1 and 2 together with Lemma C.1 and an asymptotic relation analogous to (5.12).
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Remark 5.1. (i) According to (2.29) the analytic condition (5.7) is equivalent to the probabilistic condition Z t Vn− (w(s)) ds = 0 . (5.22) lim sup sup Ex t↓0 n∈N x∈Rd
0
(ii) Theorem 5.1 is a generalization of [58, Theorem B.10.2] both to non-zero magnetic fields and Λ ⊆ Rd . Even for A = Am = 0 and Λ = Rd the result is slightly stronger than that of [58, Theorem B.10.2]. Nevertheless, we followed a similar strategy for the proof. (iii) Propositions 2.1 and 2.2 imply that for given A ∈ Hloc (Rd ) and V ∈ K± (Rd ) one can find sequences {Am }m∈N ⊂ (C0∞ (Rd ))d and {Vn }n∈N ⊂ C0∞ (Rd ) obeying the hypotheses of Theorem 5.1. Since the notion of local-norm-continuity occurring in Theorem 5.1 seems to us less common, it may be worth noting that it implies strong continuity of the semigroup in the potentials under the additional condition p < ∞. Corollary 5.1. Let A ∈ Hloc (Rd ), {Am }m∈N ⊂ Hloc (Rd ), V ∈ K± (Rd ) and {Vn }n∈N ⊂ K± (Rd ) obey (5.4)–(5.7) for all compact K ⊂ Rd . Moreover, let Λ ⊆ Rd open. Then lim
sup k(e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )ψkq = 0
m,n→∞ τ1 ≤t≤τ2
(5.23)
for all 0 < τ1 ≤ τ2 < ∞, ψ ∈ Lp (Λ), 1 ≤ p < ∞ being finite, and 1 ≤ p ≤ q ≤ ∞. Furthermore, for p = q < ∞ one may allow τ1 = 0. Proof. We repeatedly use the triangle inequality to achieve the estimate k(e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )ψkq ≤ k(e−tHΛ (A,V ) − e−tHΛ (Am ,Vn ) )χBR ψkq + (ke−tHRd (0,V ) kp,q + ke−tHRd (0,Vn ) kp,q )k(1 − χBR )ψkp
(5.24)
valid for any R > 0. The right-hand side of the estimate is uniformly bounded in n, m ∈ N, τ1 ≤ t ≤ τ2 , due to (2.40) and Lemma C.1, respectively. The proof is therefore accomplished with the help of Theorem 5.1 and the fact that lim k(1 − χBR )ψkp = 0 ,
R→∞
whenever ψ ∈ Lp (Λ), 1 ≤ p < ∞.
(5.25)
Remark 5.2. For p = 2 Corollary 5.1 is a special case of [42, Theorem 2.8] as can be seen from Remark 2.1(ii). If one defers norm-continuity in the potentials instead of local-norm-continuity, one has to make stronger assumptions as indicated in the beginning of this section.
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Let {Am }m∈N ⊂ H(Rd ) and {Vn }n∈N ⊂ K(Rd ) such that
Theorem 5.2.
lim k(Am )2 kK(Rd ) = 0 ,
(5.26)
lim k∇ · Am kK(Rd ) = 0
(5.27)
lim kVn kK(Rd ) = 0 .
(5.28)
m→∞
m→∞
and n→∞
Moreover, let A ∈ Hloc (Rd ), V ∈ K± (Rd ) and Λ ⊆ Rd open. Then lim
sup k(e−tHΛ (A,V ) − e−tHΛ (A+Am ,V +Vn ) )kp,q = 0
m,n→∞ τ1 ≤t≤τ2
(5.29)
for all 0 < τ1 ≤ τ2 < ∞ and 1 ≤ p ≤ q ≤ ∞. Furthermore, for p = q one may allow τ1 = 0. Proof. According to the Riesz–Thorin interpolation theorem [50, Theorem IX.17] and the self-adjointness of the semigroup it is enough to prove (5.29) for the two cases p = q = ∞, τ1 = 0 and p = 1, q = ∞, τ1 > 0. Since the Feynman–Kac–Itˆo formula (2.34), the triangle inequality and the Cauchy–Schwarz Ineq. give ke−tHΛ (A,V ) − e−tHΛ (A+Am ,V +Vn ) k∞,∞ ≤ (ke−tHRd (0,2V ) k∞,∞ )1/2
1/2 sup Ex [|1 − e−St (Am ,Vn |w) |2 ] ,
(5.30)
x∈Rd
the case p = q = ∞, τ1 = 0 follows from (2.40) and Lemma C.4. By reasoning in a similar way to the proof of assertion 3 in the proof of Theorem 5.1 we get the estimate ke−2tHΛ (A,V ) − e−2tHΛ (A+Am ,V +Vn ) k1,∞ ≤ ke−tHΛ (A,V ) − e−tHΛ (A+Am ,V +Vn ) k∞,∞ × (ke−tHRd (0,V ) k1,∞ + ke−tHRd (0,V +Vn ) k1,∞ ) .
(5.31)
The second factor on the right-hand side of (5.31) is uniformly bounded with respect to n ∈ N and τ1 ≤ t ≤ τ2 due to (2.40) and Lemma C.1, the latter being applicable because {Vn }n∈N ⊂ K(Rd ) and (5.28) together with the definition (2.10) imply (5.7). Thus the case p = 1, q = ∞, τ1 > 0 follows with the help of the preceding case. Remark 5.3. (i) Theorem 5.2 is a generalization of [58, Theorem B.10.1]. (ii) In analogy to Remark 5.1(iii) one can find for given A ∈ Hloc (Rd ) and V ∈ K± (Rd ) sequences {A + Am }m∈N ⊂ (C ∞ (Rd ))d and {V + Vn }n∈N ⊂ C ∞ (Rd ) obeying the hypotheses of Theorem 5.2. In contrast to Remark 5.1(iii) it is in general wrong to replace here the set C ∞ (Rd ) of arbitrarily often differentiable functions by C0∞ (Rd ), since the Kato-norm closure of C0∞ (Rd ) is a proper subspace of K(Rd ), see [68, Proposition 5.5].
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
203
6. Continuity of the Integral Kernel of the Semigroup From the Dunford–Pettis theorem [65, Theorem 46.1], [13, Corollary 2.14] it follows with the help of (2.40) that the operator e−tHΛ (A,V ) : Lp (Λ) → Lq (Λ), t > 0, 1 ≤ p ≤ q ≤ ∞, A ∈ Hloc (Rd ), V ∈ K± (Rd ), Λ ⊆ Rd open, has an integral kernel kt : Λ × Λ → C in the sense that Z −tHΛ (A,V ) ψ)(x) = kt (x, y) ψ(y) dy (6.1) (e Λ
for all ψ ∈ Lp (Λ) and almost all x ∈ Λ. Furthermore, the integral kernel is bounded according to ess sup |kt (x, y)| = ke−tHΛ (A,V ) k1,∞ . (6.2) x,y∈Λ
The existence of an integral kernel can also be inferred from the Feynman–Kac–Itˆ o formula (2.34) by conditioning the Brownian motion to arrive at y at time t. The resulting representation kt (x, y) = (2πt)−d/2 e−(x−y)
2
/2t
Ex [e−St (A,V |w) ΞΛ,t (w)|w(t) = y]
(6.3)
holds for almost all pairs (x, y) ∈ Λ × Λ and all t > 0. The purpose of this section is to show that there is a representative of the integral kernel, which is jointly continuous in (t, x, y), t > 0, x, y ∈ Λ. Moreover, this representative is given in terms of an expectation with respect to the Brownian bridge. To begin with, we collect some preparatory material concerning the Brownian bridge. We define a continuous version b of the Brownian bridge from x = b(0) to y = b(t) in terms of standard Brownian motion w starting from x = w(0) by Z s w(u) s du 0≤s
t t−s
d/2 e(x−y)
2
/2t
Ex [Z(s)e−(w(s)−y)
2
/2(t−s)
]
(6.5)
for all 0 ≤ s < t and all x, y ∈ Rd . This is a consequence of the Cameron–Martin– Girsanov theorem [36, Theorem 3.5.1], since the Brownian bridge is obtained from Brownian motion by adding a non-stationary drift, see (6.4). We note that one
204
K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
may alternatively use the right-hand side of (6.5) to construct the Brownian-bridge measure, see [64, Proposition A.1]. For a vector potential A obeying Z
t
Pt,y 0,x
(A(b(s))) ds < ∞ = 1 2
(6.6)
0
and
Z t y − b(s) ds < ∞ =1 A(b(s)) · Pt,y 0,x t−s 0
the stochastic line integral with respect to the Brownian bridge Z s Z s Z s y − b(u) du s 7→ A(b(u)) · db(u) := A(b(u)) · dw(u) + A(b(u)) · t−u 0 0 0
(6.7)
(6.8)
is a well-defined stochastic process for 0 ≤ s ≤ t having a continuous version, confer [52, Definition IV.2.9]. According to Lemma C.8, our usual assumption A ∈ Hloc (Rd ) suffices to ensure the conditions (6.6) and (6.7). Now we are able to state the main result of this section: Theorem 6.1. Let A ∈ Hloc (Rd ), V ∈ K± (Rd ), 1 ≤ p ≤ q ≤ ∞ and Λ ⊆ Rd open. Recall the definitions (2.33) and (2.35) of St and ΞΛ,t . Then kt (x, y) := (2πt)−d/2 e−(x−y)
2
/2t
−St (A,V |b) Et,y ΞΛ,t (b)] 0,x [e
(6.9)
defines pointwise that integral kernel for e−tHΛ (A,V ) : Lp (Λ) → Lq (Λ)
(6.10)
which is jointly continuous in (t, x, y), t > 0, x, y ∈ Λ. Proof. Since the Brownian-bridge expectation in (6.9) yields a regular version of the conditional expectation in (6.3), kt as defined by (6.9) is an integral kernel for (6.10). Moreover, recalling our preparations, the kernel kt is well defined for all t > 0, x, y ∈ Λ. Employing the time-reversal symmetry of the Brownian bridge we get from (6.9) the Hermiticity of the integral kernel, that is, kt (x, y) turns into its complex conjugate upon interchanging x and y. Therefore, it is sufficient to ensure lim
sup
sup
sup |kt (x, y) − kt0 (x, y 0 )| = 0
%↓0 τ1 ≤t≤t0 ≤τ2 ,|t−t0 |<% y,y 0 ∈K,|y−y 0 |<% x∈K
(6.11)
for all compact K ⊂ Λ and all 0 < τ1 ≤ τ2 < ∞. Note that we have assumed t ≤ t0 . For any 0 < s < τ1 we can rewrite the difference in (6.11) according to kt (x, y) − kt0 (x, y 0 ) = Υ(t0 , t0 − t + s, x, y 0 ) − Υ(t, s, x, y) + Γ(t, t0 , s, x, y, y 0 ) , (6.12)
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
205
where we have introduced the abbreviations Υ(t, s, x, y) := (2πt)−d/2 e−(x−y)
2
/2t
−St (A,V |b) Et,y ΞΛ,t (b) 0,x [e
− e−St−s (A,V |b) ΞΛ,t−s (b)]
(6.13)
and 0 2
Γ(t, t0 , s, x, y, y 0 ) := (2πt0 )−d/2 e−(x−y )
/2t0
− (2πt)−d/2 e−(x−y)
2
0
0
,y Et0,x [e−St−s (A,V |b) ΞΛ,t−s (b)]
/2t
−St−s (A,V |b) Et,y ΞΛ,t−s (b)] . 0,x [e
(6.14) Therefore, it is enough to show lim
sup
sup |Υ(t, s, x, y)| = 0
(6.15)
s↓0 τ1 ≤t≤τ2 x,y∈K
and lim
sup
sup |Γ(t, t0 , s, x, y, y 0 )| = 0
sup
%↓0 τ1 ≤t≤t0 ≤τ2 ,|t−t0 |<% y,y 0 ∈Rd ,|y−y 0 |<% x∈Rd
(6.16)
for all compact K ⊂ Λ and all 0 < s < τ1 ≤ τ2 < ∞. The assertion (6.16) is actually stronger than needed for the present purpose, but will turn out to be useful in the sequel. From the triangle inequality we get Υ(t, s, x, y) ≤ (2πt)−d/2 e−(x−y)
2
/2t
−St (A,V |b) × (Et,y − e−St−s (A,V |b) |] 0,x [|e −St−s (0,V |b) + Et,y |ΞΛ,t (b) − ΞΛ,t−s (b)|]) . 0,x [e
(6.17)
Now we make use of the Cauchy–Schwarz Ineq., the elementary estimate St−s (0, 2V |b) ≥ St (0, −2V − |b)
(6.18)
and the time-reversal symmetry of the Brownian bridge. We thus achieve Υ(t, s, x, y) ≤ N 1/2 (2πt)−d/4 e−(x−y)
2
/4t
−Ss (A,V |b) 2 1/2 1/2 × ((Et,x | ]) + (Et,x ) 0,y [|1 − e 0,y [1 − ΞΛ,s (b)])
≤ N 1/2 (2π(t − s))−d/4 × ((Ey [|1 − e−Ss (A,V |w) |2 ])1/2 + (Ey [1 − ΞΛ,s (w)])1/2 ) .
(6.19)
Here we have used (6.5) for the second step and have set N :=
sup
sup (2πt)−d/2 e−(x−y)
τ1 ≤t≤τ2 x,y∈Rd
2
/2t
−St (0,−2V Et,y 0,x [e
−
|b)
].
(6.20)
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
Employing the Cauchy–Schwarz Ineq., the time-reversal symmetry and (6.5) again, we get − (6.21) N ≤ (πτ1 )−d/2 sup Ex [e−Sτ2 /2 (0,−4V |w) ] , x∈Rd
which shows by virtue of Lemma C.2 that N is finite. Note that it is essential to take the supremum in (6.20) and not only the essential supremum as, for example, in (6.2). Having shown that N is finite, (6.15) follows from (6.19) with the help of (3.11) and Lemma C.5. The remaining task is to prove (6.16). In a first step we use (6.5) to rewrite Γ as given by (6.14) in terms of an expectation with respect to Brownian motion. Then the Cauchy–Schwarz Ineq. and (6.18) lead to 0 2
|Γ(t, t0 , s, x, y, y 0 )|2 ≤ (2π)−d M Ex [((t0 − t + s)−d/2 e−(w(t−s)−y ) − s−d/2 e−(w(t−s)−y)
2
/2(t0 −t+s)
/2s 2
) ],
where M := sup Ex [e−Sτ2 (0,−2V
−
(6.22) |w)
]
(6.23)
x∈Rd
is finite due to Lemma C.2. The expectation on the right-hand side of (6.22) can be calculated explicitly. The result implies (x − y 0 )2 0 0 2 −d 02 2 −d/2 |Γ(t, t , s, x, y, y )| ≤ (2π) M (t − (t − s) ) exp − 0 t +t−s (x − y)2 − 2(t t0 − (t − s)2 )−d/2 + s−d/2 (2t − s)−d/2 exp − 2t − s s(x − y 0 )2 + (t − s)(y − y 0 )2 + (t0 − t + s)(x − y)2 . × exp − 2(t t0 − (t − s)2 ) (6.24) Now we use the three elementary inequalities 1 1 1 1 1 1 , ≥ 0 , , ≤ ≤ t02 − (t − s)2 s(2t − s) 2t − s t + t − s t t0 − (t − s)2 s(t0 + t − s) (6.25) valid for all 0 < s < t ≤ t0 < ∞, and get 0
0
−d
|Γ(t, t , s, x, y, y )| ≤ 2(2π) 2
" × cosh
Ms
−d/2
−d/2
(2t − s)
1 (x − y 0 )2 + (x − y)2 exp − 2 t0 + t − s
(y − y 0 ) · (y + y 0 − 2x) 2(t + t0 − s)
t(t0 − t) − 1+ s(2t − s)
(t − s)(y − y 0 )2 + (t0 − t)(x − y)2 . × exp − 2s(t + t0 − s)
−d/2
(6.26)
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
207
In a final step we further estimate the right-hand side with the help of the elementary inequalities 1 2 a cosh(a) + 1 , a ∈ R , 2 d (1 + b)−d/2 ≥ 1 − b , b > 0 , e−c ≥ 1 − c , 2
cosh(a) ≤
(6.27) c ∈ R,
the first one of which may readily be derived from the Taylor expansion [1, Eq. 4.5.63] of the hyperbolic cosine. The assertion (6.16) then follows from the resulting bound by inspection. Remark 6.1. (i) Theorem 6.1 generalizes [58, Theorem B.7.1.(a000 )] to non-zero A ∈ Hloc (Rd ) and Λ ⊆ Rd . The proof given there relies on the local-norm-continuity of the semigroup in the scalar potential and the approximability Proposition 2.1 (see [58, Theorem B.10.2], but beware of a circularity in the proofs of [58, Proposition B.6.7, Theorem B.7.1.(a0 ), Lemma B.7.4]). By using Theorem 5.1 local-normcontinuity covers also the case A 6= 0. Results similar to Theorem 6.1 for A = 0 and Λ ⊆ Rd are contained in [11, Theorem 3.17] and [64, Proposition 1.3.5]. Like the last reference, our approach has the advantage of singling out the continuous representative of the integral kernel as a Brownian-bridge expectation, which is useful to know for further investigations and applications. The rather general Theorem 14.5 in [56] does not cover the above Theorem 6.1. A discriminating example corresponds to the hydrogen atom: Λ = R3 , A = 0, V (x) = −|x|−1 . In the notation of [56] this means: ν = 3, f = 0, F (a) = eia for Rt a ∈ R, g(ω) = exp{ 0 |ω(s)|−1 ds} for t > 0. Theorem 14.5 in [56] is not applicable, because the Wiener-essential supremum kgk∞ of g is infinite. (ii) The continuity of the integral kernel of the semigroup implies the continuity of the integral kernels for various functions of the Schr¨ odinger operator HΛ (A, V ). This can be shown along the same lines of reasoning as in the proof of [58, Theorem B.7.1], since the arguments given there do not use the fact that HRd (0, V ) is a particular Schr¨ odinger operator, but merely the continuity of kt (x, y) and the bound (2.40). For example, each spectral projection χΩ (HΛ (A, V )) : L2 (Λ) → L2 (Λ)
(6.28)
associated with a bounded Borel subset Ω ⊂ R is an integral operator with jointly continuous kernel. (iii) Since kt (x, y) is jointly continuous in (x, y) for t > 0, the trace of the non-negative operator e−tHΛ (A,V ) : L2 (Λ) → L2 (Λ) can be represented as Z −tHΛ (A,V ) = kt (x, x) dx, (6.29) tr e Λ
whenever one of the two sides of this equation is finite. (iv) Due to the continuity of the integral kernel, the diamagnetic inequality (2.38) takes the form (6.30) |kt (x, y)| ≤ kt (x, y)|A=0
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
and is valid pointwise for all x, y ∈ Λ, t > 0. Various upper bounds on the righthand side of (6.30) are available in the literature, see, for example, the estimate [58, Theorem B.7.1.(a0 )]. Using this estimate, Theorem 4.1 can be proven alternatively with the help of the continuity of the integral kernel and the dominated-convergence theorem. We have given a different proof of Theorem 4.1, mainly because we think it is nice to see how Carmona’s elegant argument for A = 0 can be extended to A 6= 0. Theorem 6.2. Let A ∈ Hloc (Rd ), V ∈ K± (Rd ) and Λ ⊆ Rd regular. Then kt (x, y), as defined by (6.9), is jointly continuous in (t, x, y), t > 0, x, y ∈ Λ and vanishes if x ∈ ∂Λ or y ∈ ∂Λ. Proof. From (6.9) we derive with the help of the Cauchy–Schwarz Ineq. and ΞΛ,t (b) ≤ ΞΛ,t/2 (b) the estimate |kt (x, y)| ≤ (2πt)−d/4 N 1/2 e−(x−y)
2
/4t
1/2 (Et,y , 0,x [ΞΛ,t/2 (b)])
(6.31)
where 0 ≤ N < ∞ is given by (6.20). With the help of (6.5) we achieve |kt (x, y)| ≤ (πt)−d/4 N 1/2 (Ex [ΞΛ,t/2 (b)])1/2 .
(6.32)
According to Lemma C.7 the right-hand side vanishes as x approaches any given point on the boundary ∂Λ. Due to the Hermiticity of kt and Theorem 6.1 the assertion follows. Remark 6.2. A 6= 0.
Theorem 6.2 is a partial generalization of [11, Theorem 3.17] to
We finally want to show that the same strategy as used in Sec. 4 allows to get uniform continuity for global regularity conditions. Theorem 6.3. Let A ∈ H(Rd ), V ∈ K(Rd ), 0 < τ1 < τ2 < ∞ and Λ ⊆ Rd uniformly regular. Then the function [τ1 , τ2 ] × Λ × Λ → C ,
(t, x, y) 7→ kt (x, y)
(6.33)
is uniformly continuous. Proof. Analogously to the reasoning at the beginning of the proof of Theorem 6.1, it is sufficient to ensure lim
sup
sup |kt (x, y) − kt0 (x, y 0 )| = 0
sup
%↓0 τ1 ≤t≤t0 ≤τ2 ,|t−t0 |<% y,y 0 ∈Λr ,|y−y 0 |<% x∈Λr
(6.34)
in order to get uniform continuity on [τ1 , τ2 ] × Λr × Λr , r > 0, where Λr is given by (3.12). Due to (6.12) and (6.16) we only have to check that lim
sup
sup |Υ(t, s, x, y)| = 0 ,
s↓0 τ1 ≤t≤τ2 x,y∈Λr
(6.35)
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
209
where Υ is given by (6.13). With the help of the estimate (6.19) this follows from Lemma C.3 and (3.11). On the other hand, Lemma C.7 and (6.32) imply sup
lim
sup
sup
r↓0 τ1 ≤t≤τ2 x∈Λr y∈Λ\Λr
|kt (x, y)| = 0 ,
(6.36)
which suffices to extend the domain of uniform continuity from [τ1 , τ2 ] × Λr × Λr , r > 0, to [τ1 , τ2 ] × Λ × Λ. Appendix A. Approximability of Kato-Type Potentials by Smooth Functions We want to prove Propositions 2.1 and 2.2. The idea of the proof has been given already in Sec. 2. It is similar to the one underlying the standard proof of the fact that C0∞ (Rd ) is dense in Lp (Rd ) for 1 ≤ p < ∞. To begin with, we choose δ1 ∈ C0∞ (Rd ) such that δ1 ≥ 0, δ1 (x) = 0 for |x| > 1 and kδ1 k1 = 1. Moreover, we set δr (x) := r−d δ1 (x/r) for 0 < r ≤ 1 and ΘR := δ1 ∗ χBR for R > 1, where Z (h ∗ F )(x) :=
h(x − y) F (y) dy
(A.1)
denotes the convolution of a function F : Rd → Rν , ν ∈ N, with a function h : Rd → R. We note that ΘR ∈ C0∞ (Rd ) and χBR−1 ≤ ΘR ≤ χBR+1 .
(A.2)
Now we are ready to formulate the following preparatory result. Lemma A.1.
Let F : Rd → Rν , |F |p ∈ Kloc (Rd ), 1 ≤ p < ∞. Then ν δr ∗ ΘR F ∈ C0∞ (Rd ) ,
kg% ∗ |δr ∗ ΘR F |p k∞ ≤ g% ∗ χBR+1 |F |p ∞
(A.3) (A.4)
and p
lim k|ΘR F − δr ∗ ΘR F | kK(Rd ) = 0 r↓0
(A.5)
for all 0 < r ≤ 1 < R < ∞ and 0 < % ≤ 1.
ν Proof. (A.3) holds, since ΘR F ∈ L1 (Rd ) by (2.12). From the Jensen Ineq. and (A.2) we have p
|δr ∗ ΘR F | ≤ δr ∗ χBR+1 |F |p .
(A.6)
Furthermore, by the commutativity and the associativity of the convolution and the monotonicity of integration we get
g% ∗ δr ∗ χBR+1 |F |p ≤ kδr k1 g% ∗ χBR+1 |F |p ∞ . (A.7) Now (A.4) follows from (A.6), (A.7) and kδr k1 = 1.
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
Since kf kK(Rd) = kg1 ∗ |f |k∞ , we observe for f ∈ K(Rd ) ∩ L1 (Rd ) that kf kK(Rd) ≤ kg% ∗ |f |k∞ + G% kf k1 ,
(A.8)
where G% :=
sup g1 (x) .
(A.9)
%≤|x|≤1
On the other hand, the inequality |u − v|p ≤ 2p (|u|p + |v|p ) for u, v ∈ Rν in combination with (A.2) and (A.6) yields p (A.10) |ΘR F − δr ∗ ΘR F | ≤ 2p χBR+1 |F |p + δr ∗ χBR+1 |F |p . With the help of (A.8), (A.10) and (A.4) we establish p
k|ΘR F − δr ∗ ΘR F | kK(Rd )
p ≤ 2p+1 g% ∗ χBR+1 |F |p ∞ + G% k|ΘR F − δr ∗ ΘR F | k1 .
(A.11)
ν Since ΘR F ∈ Lp (Rd ) , the second term on the right-hand side tends to zero as r ↓ 0, see, for example, [66, Lemma 1.3.6/2]. Moreover, the remaining first term vanishes as % ↓ 0 because χBR+1 |F |p ∈ K(Rd ). This proves (A.5). Proof of Proposition 2.1. By the triangle inequality |V − δr ∗ ΘR V | χK ≤ V (1 − ΘR ) χK + |ΘR V − δr ∗ ΘR V |
(A.12)
and the fact that (1 − ΘR ) χK = 0
for R > 1 + sup |x|
(A.13)
x∈K
the approximability (2.15) follows from (A.3) and (A.5). The Ineq. (2.16) follows from (A.4) for the choice p = 1 and F = V − . Proof of Proposition 2.2. We start from the pair of elementary inequalities (A − δr ∗ ΘR A)2 χK ≤ 2A2 (1 − ΘR ) χK + 2 (ΘR A − δr ∗ ΘR A)2
(A.14)
and |∇ · A − ∇ · (δr ∗ ΘR A)| χK ≤ |∇ · A| (1 − ΘR ) χK + |δr ∗ (A · ∇ΘR )| χK + |ΘR ∇ · A − δr ∗ (ΘR ∇ · A)| .
(A.15)
The second term on the right-hand side of (A.15) can be estimated further with the help of (A.2) according to |δr ∗ (A · ∇ΘR )| χK ≤ χK χBR+2 1 − χBR−2 (δr ∗ |A|) sup |(∇ΘR )(x)| (A.16) x∈Rd
and, hence, vanishes for R > 2 + supx∈K |x|. In consequence, the approximability (2.19), (2.20) follows from (A.3) with the help of (A.13) and (A.5).
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
211
Appendix B. Proof of the Feynman Kac Itˆ o Formula The purpose of this Appendix is to prove Proposition 2.3. The strategy of the proof has already been sketched in Sec. 2. Here we only recall that we take (2.34) in the case Λ = Rd , A ∈ Hloc (Rd ) and V ∈ K± (Rd ) for granted. Let {Kl }l∈N be a strictly increasing sequence of compact subsets of Λ that S exhausts Λ. That is, each Kl is contained in the interior of Kl+1 and l∈N Kl = Λ. Furthermore let {ϑl }l∈N ⊂ C0∞ (Rd ) be such that ϑl (x) = 1 for x ∈ Kl , ϑl (x) = 0 for Λ : Rd → R by x 6∈ Kl+1 , and 0 ≤ ϑl (x) ≤ 1 for all x ∈ Rd . Define U∞ X 2 −3 |(∇ϑl )(x)| + (inf{|x − y| : y 6∈ Λ}) for x ∈ Λ Λ U∞ (x) := l∈N (B.1) +∞ otherwise and UnΛ : Rd → R for n ∈ N by Λ UnΛ (x) := inf{U∞ (x), n} .
(B.2)
Λ A,µU∞ +V
, A ∈ Hloc (Rd ), V ∈ L∞ (Rd ), µ > 0, defined A,U Λ through (2.22), with domain Q = Q hRd ∞ given by (2.23) is closed A,U Λ but not densely defined. In fact C0∞ (Λ) ⊂ Q hRd ∞ ⊂ L2 (Λ) by the extension The sesquilinear form hRd
A,µU Λ +V h Rd ∞
convention of Sec. 2 and the form is densely defined on L2 (Λ). It gives rise to a Λ + V ). self-adjoint non-negative operator on L2 (Λ) which we denote by HRd (A, µU∞ For a function f : R → R,
f continuous ,
lim f (x) = 0
x→+∞
(B.3)
we define for ψ ∈ L2 (Rd )
Λ Λ f HRd (A, µU∞ + V ) ψ := f HRd (A, µU∞ + V ) (χΛ ψ) . (B.4) Λ Hence f HRd (A, µU∞ + V ) naturally extends to a bounded self-adjoint operator on L2 (Rd ). Our plan for the proof of Proposition 2.3 is summarized in the following six lemmas. Lemma B.1. Let A ∈ Hloc (Rd ), V ∈ L∞ (Rd ), V ≥ 0, ψ ∈ L2 (Rd ), µ > 0, t ≥ 0, Λ ⊆ Rd open. Then
Λ Λ
(B.5) lim e−tHRd (A,µUn +V ) − e−tHRd (A,µU∞ +V ) ψ = 0 . n→∞
e
2
Lemma B.1 serves for the proof of the Feynman–Kac–Itˆ o formula for .
Λ −tHRd (A,µU∞ +V )
Lemma B.2. Let A ∈ Hloc (Rd ), V ∈ L∞ (Rd ), V ≥ 0, ψ ∈ L2 (Rd ), µ > 0, t ≥ 0, Λ ⊆ Rd open. Then h i Λ Λ (B.6) e−tHRd (A,µU∞ +V ) ψ (x) = Ex e−St (A,µU∞ +V |w) ΞΛ,t (w) ψ(w(t)) for almost all x ∈ Rd .
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
The purpose of Lemma B.2 is twofold. inequality
First, it immediately implies the
|e−tHRd (A,µU∞ +V ) ψ| ≤ e−tHRd (0,0) |ψ| , Λ
ψ ∈ L2 (Λ), t ≥ 0 ,
(B.7)
which is crucial in the proof of the following lemma. d d Lemma B.3. Let A ∈ Hloc (R ), Λ ⊆ R open. Then the completion of the form Λ A,U∞ A,0 with respect to the form-norm k • khA,0 is Q hΛ . domain Q hRd Rd
This rather technical lemma singles out the difficult part of the proof of the next approximation. Lemma B.4. t ≥ 0. Then
Let A ∈ Hloc (Rd ), V ∈ L∞ (Rd ), V ≥ 0, ψ ∈ L2 (Rd ), Λ ⊆ Rd open,
Λ
(B.8) lim e−tHRd (A,µU∞ +V ) − e−tHΛ (A,V ) ψ = 0 . µ↓0
2
The second purpose of Lemma B.2 is, of course, to be used together with B.4 to get the next variant of the Feynman–Kac–Itˆo formula. Assume A ∈ Hloc (Rd ), V ∈ L∞ (Rd ), V ≥ 0. Then Lemma B.4 implies that the left-hand side of (B.6) tends to the left-hand side of (2.34) for almost all x ∈ Λ as µ ↓ 0 at least for a suitably chosen subsequence. Furthermore, the right-hand side of (B.6) converges to the right-hand side of (2.34) for almost all x ∈ Λ as µ ↓ 0 by the dominated-convergence theorem. One may relax the assumption V ∈ L∞ (Rd ), V ≥ 0 to V ∈ L∞ (Rd ) by multiplying both sides of (2.34) with et inf x∈Rd V (x) . We summarize these findings in the following lemma. Lemma B.5. Let A ∈ Hloc (Rd ), V ∈ L∞ (Rd ), ψ ∈ L2 (Λ), Λ ⊆ Rd open, t ≥ 0. Then (2.34) holds for almost all x ∈ Λ. With this statement at hand, the diamagnetic inequality (2.25) follows from the triangle inequality. This is just-in-time because we used (2.25) in the case Λ ⊂ Rd , Λ 6= Rd for the construction of HΛ (A, V ), A ∈ Hloc (Rd ), V ∈ K± (Rd ), in that case. Now we are ready to formulate the last approximation tool for the proof of Proposition 2.3. Lemma B.6. Let A ∈ Hloc (Rd ), V ∈ K± (Rd ), ψ ∈ L2 (Rd ), Λ ⊆ Rd open, t ≥ 0 and set for n, m ∈ N Vmn (x) := sup{−n, inf{m, V (x)}}
(B.9)
and consequently Vm∞ (x) = inf{m, V (x)}. Then lim k(e−tHΛ (A,Vn ) − e−tHΛ (A,V∞ ) )ψk2 = 0
(B.10)
lim k(e−tHΛ (A,V∞ ) − e−tHΛ (A,V ) )ψk2 = 0 .
(B.11)
m
m
n→∞
and m
m→∞
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
213
Proof of Proposition 2.3. From (B.10) and (B.11) we conclude that for a suitably chosen subsequence (e−tHΛ (A,Vn ) ψ)(x) → (e−tHΛ (A,V ) ψ)(x) m
(B.12)
as n, m → ∞ for almost all x ∈ Λ. From Lemma B.5 we know that the Feynman– Kac–Itˆ o formula holds for the left-hand side of (B.12) and it therefore remains to show lim Ex [e−St (A,Vn
m
|w)
n,m→∞
ΞΛ,t (w) ψ(w(t))] = Ex [e−St (A,V |w) ΞΛ,t (w) ψ(w(t))] (B.13)
for almost all x ∈ Λ. To this end note that |e−St (A,Vn
m
and Ex [e−St (0,V
−
|w)
|w)
ΞΛ,t (w)| ≤ |e−St (0,V
|ψ(w(t))|] ≤ ke−tHRd (0,V
−
−
)
|w)
|
k2,∞ < ∞
(B.14)
(B.15)
by (2.34) for Λ = Rd and [58, Theorem B.1.1], see also Lemma C.1. Employing the dominated-convergence theorem we conclude that it is enough to establish Z t Z t lim Vnm (w(s)) ds = V (w(s)) ds (B.16) n,m→∞
0
0
for all x = w(0) almost surely. Since |Vnm | ≤ |V | ∈ K± (Rd ) this is achieved by applying dominated convergence with the help of Remark 2.4(ii). Proof of Lemma B.1. Since UnΛ and V are bounded one has from (2.23) Λ Λ A,µU Λ +V A,µU∞ +V A,U∞ = Q hA,0 ⊃ Q h = Q h (B.17) Q h Rd n Rd Rd Rd Λ Λ A,U∞ A,Un A,0 = ψ ∈ Q hRd : sup hRd (ψ, ψ) < ∞ . Q h Rd
and
(B.18)
n∈N
Furthermore, we have by construction the monotonicity and boundedness Λ A,µUn +V
h Rd
Λ A,µUn+1 +V
≤ h Rd
Λ A,µU∞ +V
≤ h Rd
,
n ∈ N,
(B.19)
as well as the pointwise convergence Λ A,µUn +V
lim hRd
n→∞
Λ A,µU∞ +V
(φ, ψ) = hRd
(φ, ψ),
A,U Λ φ, ψ ∈ Q hRd ∞ .
(B.20)
These facts ensure that the monotone-convergence theorem for not necessarily densely defined sesquilinear forms [55, Theorem 4.1], [69, Theorem 3.1.b)] is applicable and yields the generalized strong resolvent convergence
1 1
− ψ (B.21) lim
=0 Λ +V) Λ +V) n→∞ 1 + HRd (A, µUn 1 + HRd (A, µU∞ 2
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
for all ψ ∈ L2 (Rd ). This implies [55, Theorem 4.2] the strong convergence of f HRd (A, µUnΛ + V ) n∈N for all functions f of type (B.3). In particular, (B.5) follows. Proof of Lemma B.2. The Feynman–Kac–Itˆ o formula (2.34) for Λ = Rd tells (e−tHRd (A,µUn +V ) ψ)(x) = Ex [e−St (A,µUn +V |w) ψ(w(t))] Λ
Λ
(B.22)
for almost all x ∈ Rd . From Lemma B.1 we learn that by passing to a suitable subsequence the left-hand side of (B.22) tends to the left-hand side of (B.6) as n → ∞ for almost all x ∈ Rd . For the proof of the convergence of the right-hand side of (B.22) to the desired limit we only have to check e−St (0,µUn |w) → e−St (0,µU∞ |w) ΞΛ,t (w) Λ
Λ
(B.23)
as n → ∞ for almost all w, because |e−St (A,µUn +V |w) | ≤ 1 Λ
(B.24)
allows the application of the dominated-convergence theorem. If x = w(0) 6∈ Λ the left-hand side of (B.23) tends to 0 for all continuous w. Therefore we may assume x ∈ Λ. In addition it is sufficient to consider a path w which is H¨older continuous of order 13 [56, Theorem 5.2]. Assume that w leaves Λ and set 0 ≤ τ ≤ t to be the first exit time of w from Λ, that is τ := inf{0 < s ≤ t : w(s) 6∈ Λ} .
(B.25)
Then by (B.1), (B.2) Z
t
St (0, µUnΛ |w) ≥ µ 0
Z ≥µ
0
t
inf |w(s) − w(τ )|−3 , n ds c , n ds → ∞ inf |s − τ |
(B.26)
as n → ∞, where the constant c > 0 depends on the path w. Thus (B.23) holds for almost all paths w that do not stay in Λ. But for a path w staying in Λ (B.23) is a consequence of the monotone-convergence theorem. Whence (B.23) is true for almost all w. A,0 is the completion of C0∞ (Λ) with Proof of Lemma B.3. By definition, Q hΛ A,U Λ respect to k • khA,0 . Moreover C0∞ (Λ) ⊂ Q hRd ∞ . Therefore, Q hA,0 is a Λ Rd Λ A,U∞ . To prove the converse inclusion it is enough subset of the completion of Q hRd A,U Λ ∞ to establish that C0 (Λ) is dense in Q hRd ∞ with respect to k • khA,0 . This Rd
in turn follows, confer the proof of [13, Theorem 1.13], from the following three assertions: A,U Λ A,U Λ (1) L∞ (Λ) ∩ Q hRd ∞ is dense in Q hRd ∞ .
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
215
Λ Λ A,U∞ A,U∞ ∞ is dense in L . (2) L∞ (Λ) ∩ Q h (Λ) ∩ Q h 0 Rd Rd Λ A,U∞ (3) C0∞ (Λ) is dense in L∞ . 0 (Λ) ∩ Q hRd Here denseness is always meant with respect to the form norm k • khA,0 and L∞ 0 (Λ) Rd
denotes the space of bounded complex-valued functions with compact support inside Λ. As to assertion (1) Λ ) by the Fix t > 0. Then e−tHRd (A,U∞ ) L2 (Λ) is an operator core for HRd (A, U∞ Λ
A,U Λ
spectral theorem [51, Sec. VIII.8]. Consequently it is also a form core for hRd ∞ . Λ From (B.7) we conclude that e−tHRd (A,U∞ ) L2 (Λ) ⊆ L∞ (Λ) and therefore Λ A,U Λ (B.27) L∞ (Λ) ∩ e−tHRd (A,U∞ ) L2 (Λ) ⊆ L∞ (Λ) ∩ Q hRd ∞ A,U Λ is dense in Q hRd ∞ with respect to k • k
h
assertion follows.
Λ A,U∞ Rd
. Since k • khA,0 ≤ k • k Rd
h
Λ A,U∞ Rd
the
As to assertion (2) Λ A,U Λ A,U∞ for all l ∈ N. We Pick ψ ∈ L∞ (Λ) ∩ Q hRd ∞ , then ϑl ψ ∈ L∞ 0 (Λ) ∩ Q hRd want to show kψ − ϑl ψkhA,0 → 0 as l → ∞. By the construction of ϑl we know that Rd
k(1 − ϑl )φk2 → 0 for all φ ∈ L2 (Λ). Therefore, it remains to show lim k(−i∇ − A)(ψ − ϑl ψ)k2 = 0 .
l→∞
(B.28)
The triangle inequality gives k(−i∇ − A)(ψ − ϑl ψ)k2 ≤ k(1 − ϑl )(−i∇ − A)ψk2 + kψ∇ϑl k2 .
(B.29)
The first term on the right-hand (−i∇ − A)ψ ∈ side tends to 0 as l → ∞, Λbecause d A,U∞ A,0 2 and therefore L (Λ) due to ψ ∈ Q hRd . We even know ψ ∈ Q hRd X
Λ A,U∞
kψ∇ϑl k22 ≤ hRd
(ψ, ψ) < ∞
(B.30)
l∈N
which implies kψ∇ϑl k2 → 0 as l → ∞. As to assertion (3) Λ A,U∞ , then δr ∗ψ ∈ C0∞ (Λ) for suitably small r > 0, where δr Pick ψ ∈ L∞ 0 (Λ)∩Q hRd
is the approximate δ-function defined in Appendix A. Since limr↓0 kφ − δr ∗ φk2 = 0 for all φ ∈ L2 (Λ) we aim to show k(−i∇ − A)(ψ − δr ∗ ψ)k2 → 0 as r ↓ 0 at least for a suitably chosen subsequence.
(B.31)
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
d We first claim that ∇ψ ∈ L2 (Λ) . This follows from the fact that on the one d d ∞ hand Aψ ∈ L2 (Λ) because A ∈ L2loc (Λ) ⊃ Hloc (Rd ) and ψ ∈ L0 (Λ) and on d . Therefore the other hand (−i∇ − A)ψ ∈ L2 (Λ) because ψ ∈ Q hA,0 Rd k(−i∇)(ψ − δr ∗ ψ)k2 = k∇ψ − δr ∗ (∇ψ)k2 → 0
(B.32)
as r ↓ 0. Let K ⊂ Λ denote the compact support of δR ∗ |ψ| for some small R > 0. Then for all 0 < r ≤ R |A(x)(ψ − δr ∗ ψ)(x)| ≤ 2kψk∞ χK (x) |A(x)|.
(B.33)
The right-hand side of this estimate — considered as a function of x — is in d L2 (Λ) because A ∈ L2loc (Λ) . Moreover, we may choose a subsequence such that (ψ−δr ∗ψ)(x) → 0 for almost all x ∈ Λ and then get from the dominated-convergence theorem (B.34) kA(ψ − δr ∗ ψ)k2 → 0 for this subsequence. The convergences (B.32) and (B.34) imply (B.31) and hence assertion 3.
Proof of Lemma B.4. Recall that A,µU Λ +V A,U Λ C0∞ (Λ) ⊂ Q hRd ∞ = Q hRd ∞ ⊆ Q hA,0 = Q hA,V ⊂ L2 (Λ) (B.35) Λ Λ independent of µ > 0, where the inclusion follows from Lemma B.3. Therefore the monotonicity statement Λ A,µ0 U∞ +V
h Rd
Λ A,µU∞ +V
(ψ, ψ) ≥ hRd
(ψ, ψ) ≥ hA,V Λ (ψ, ψ) ,
µ0 > µ ,
(B.36)
and the convergence Λ A,µU∞ +V
lim hRd µ↓0
(ψ, ψ) = hA,V Λ (ψ, ψ)
(B.37)
A,U Λ make sense for all ψ ∈ Q hRd ∞ and follow from the construction of the forms. We conclude with the help of Lemma B.3 that the monotone-convergence theorem for densely defined forms [55, Theorem 3.2], [51, Theorem S.16], see also [69, Theorem 3.1.a)], is applicable and yields the strong resolvent convergence of Λ + V ) to HΛ (A, V ) as µ ↓ 0. According to [51, Theorem VIII.20] this HRd (A, µU∞ implies (B.8). m Proof of Lemma B.6. First we note that Vnm ∈ L∞ (Rd ) and V∞ ∈ K(Rd ) implies m A,V m A,V∞ Q hΛ n = Q hA,0 = Q h . (B.38) Λ Λ m A,V∞
Moreover, we know that hΛ
is bounded from below.
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
217
By construction of Vnm we have the monotonicity A,Vnm
hΛ
m A,Vn+1
(ψ, ψ) ≥ hΛ
and
m A,V∞
hΛ
m A,V∞
(ψ, ψ) ≥ hΛ A,Vnm
(ψ, ψ) = lim hΛ n→∞
(ψ, ψ)
(ψ, ψ)
(B.39)
(B.40)
. These facts imply according to [55, Theorem 3.2], [51, Theofor all ψ ∈ Q hA,0 Λ m ) as n → ∞. rem S.16] the strong resolvent convergence of HΛ (A, Vnm ) to HΛ (A, V∞ By [51, Theorem VIII.20] Eq. (B.10) follows. To prove (B.11) we first note that m A,V∞ A,0 : sup hΛ (ψ, ψ) < ∞ (B.41) Υ := ψ ∈ Q hΛ m∈N
is dense in L2 (Λ) because C0∞ (Λ) ⊂ Υ. Furthermore we have the monotonicity m A,V∞
hΛ
m+1 A,V∞
(ψ, ψ) ≤ hΛ
(ψ, ψ) ≤ hA,V Λ (ψ, ψ)
and
m A,V∞
hA,V Λ (ψ, ψ) = lim hΛ m→∞
(ψ, ψ)
(B.42)
(B.43)
for all ψ ∈ Υ. With 3.1], [51, Theorem S.14] we con [55,+Theorem the help of A,V A,V = Q hΛ and the strong resolvent convergence of clude that Υ = Q hΛ m ) to HΛ (A, V ) as m → ∞. Thus (B.11) follows by using [51, TheoHΛ (A, V∞ rem VIII.20] again.
Appendix C. Brownian-Motion Estimates In this appendix we have gathered the probabilistic estimates which play a key role in the proofs of Theorems 4.1, 4.2, 4.3, 5.1, 5.2, 6.1, 6.2 and 6.3. Our proofs of these estimates are inspired by ideas for zero vector potentials in [10, Sec. III], [58, § B.1, § B.10] and the more recent monograph [11, Sec. 3.2]. Lemma C.1.
Let {Vn }n∈N ⊂ K± (Rd ) obey (5.7). Then sup sup ke−tHRd (0,Vn ) kp,q < ∞
τ1 ≤t≤τ2 n∈N
(C.1)
for all 0 < τ1 ≤ τ2 < ∞ and 1 ≤ p ≤ q ≤ ∞. Furthermore, for p = q one may allow τ1 = 0. Proof. The Riesz–Thorin interpolation theorem [50, Theorem IX.17] and the selfadjointness of the semigroup imply that it is enough to prove the Lemma for the two cases p = q = ∞, τ1 = 0 and p = 1, q = ∞, τ1 > 0. In order to show this we first observe that
2
(C.2) ke−2tHRd (0,Vn ) k1,∞ ≤ e−tHRd (0,Vn ) 2,∞
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
holds by the semigroup property, Ineq. (5.18) and by self-adjointness. Moreover, the inequality
1/2 −d/4
e−tHRd (0,2Vn ) (C.3) ke−tHRd (0,Vn ) k2,∞ ≤ (2πt) ∞,∞ follows from employing the Cauchy–Schwarz Ineq. in the Feynman–Kac–Itˆo formula and from the elementary estimate |(e−tHRd (0,0) |ψ|2 )(x)| ≤ (2πt)−d/2 (kψk2 )2 .
(C.4)
The combination of (C.2) and (C.3) shows that we are left to treat the case p = q = ∞, τ1 = 0. The semigroup property (2.37) and the inequality (5.18) yield
N (C.5) ke−tHRd (0,Vn ) k∞,∞ ≤ e−tHRd (0,Vn )/N ∞,∞ , N ∈ N . Therefore it suffices to prove the case p = q = ∞, τ1 = 0 for some small τ2 > 0. By (5.7) and Remark 5.1(i) we may assume τ2 small enough to ensure Z τ2 − Vn (w(s))ds < 1 . (C.6) α := sup sup Ex n∈N x∈Rd
0
Employing Khas’minskii’s lemma [38], [11, Lemma 3.7] we get Ex [e−St (0,Vn |w) ] ≤
1 1−α
(C.7)
for all 0 ≤ t ≤ τ2 . By the Feynman–Kac–Itˆo formula it follows that ke−tHRd (0,Vn ) k∞,∞ ≤
1 1−α
for all 0 ≤ t ≤ τ2 and n ∈ N.
(C.8)
A glance at the preceding proof reveals that (C.7) not only holds for almost all x ∈ Rd but for all x ∈ Rd . We state a frequently used consequence of this separately, see, for example, [11, Proposition 3.8]. Lemma C.2.
Let V ∈ K± (Rd ). Then sup sup Ex [e−St (0,V |w) ] < ∞
(C.9)
0≤t≤τ x∈Rd
for all τ > 0. For our purposes it is essential to extend a previously known result for A = 0 [11, Proposition 3.9] to A = 6 0. Lemma C.3.
Let A ∈ H(Rd ) and V ∈ K(Rd ). Then lim sup Ex [|1 − e−St (A,V |w) |p ] = 0 t↓0 x∈Rd
for all 0 < p < ∞.
(C.10)
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
219
Lemma C.4. Let A ∈ H(Rd ), {Am }m∈N ⊂ H(Rd ), V ∈ K(Rd ) and {Vn }n∈N ⊂ K(Rd ) such that limm→∞ k(A − Am )2 kK(Rd ) = 0 ,
(C.11)
limm→∞ k∇ · A − ∇ · Am kK(Rd ) = 0
(C.12)
lim kV − Vn kK(Rd ) = 0 .
(C.13)
and n→∞
Then sup sup Ex [|e−St (A,V |w) − e−St (Am ,Vn |w) |p ] = 0
lim
m,n→∞ 0≤t≤τ x∈Rd
(C.14)
for all 0 ≤ τ < ∞ and 0 < p < ∞. Proof of Lemmas C.3 and C.4. We start from the inequality 0 0 |ez − ez | ≤ 21+1/q |z − z 0 |1/q esup{Rez,Rez }
(C.15)
valid for all z, z 0 ∈ C and 1 ≤ q ≤ ∞. It can be inferred from the elementary inequalities 1 − 2 |u|2/q ≤ cos u and |eu − 1| ≤ |u|1/q esup{u,0} for all u ∈ R. Choosing q such that γ := pq/(q − p) > 0 we get from (C.15) 2−2p Ex [|e−St (A,V |w) − e−St (Am ,Vn |w) |p ] ≤ Ex [|St (A − Am , V − Vn |w)|p/q e−pSt (0,−V
−
−Vn− |w)
≤ (Ex [|St (A − Am , V − Vn |w)|])p/q (Ex [e−γSt (0,−V
−
]
−Vn− |w) p/γ
])
≤ (Ex [|St (A − Am , V − Vn |w)|])p/q p/2γ − − × ke−τ HRd (0,−2γV ) k∞,∞ sup ke−τ HRd (0,−2γVn ) k∞,∞ .
(C.16)
n∈N
Here we have used the H¨ older inequality for the second step and the Cauchy– Schwarz Ineq. together with the Feynman–Kac–Itˆ o formula for the third step. As remarked similarly in the proof of Theorem 5.2, V ∈ K(Rd ), {Vn }n∈N ⊂ K(Rd ) and (C.13) imply (5.7). Hence Lemma C.1 can be applied to the right-hand side of (C.16). This guarantees together with (2.40) that it is sufficient to show lim
sup sup Ex [|St (A − Am , V − Vn |w)|] = 0
m,n→∞ 0≤t≤τ x∈Rd
(C.17)
in order to prove Lemma C.4. Similarly, to verify Lemma C.3 it is enough to show lim sup Ex [|St (A, V |w)|] = 0 , t↓0 x∈Rd
(C.18)
as can be inferred from (C.16) by putting there Am = 0 and Vn = 0 and appealing to (2.40).
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
To this end, the triangle inequality, the Jensen Ineq. and the isometry for stochastic integrals [25, Theorem 4.2.5] or [36, Proposition 3.2.10] yield Ex [|St (A − Am , V − Vn |w)|] ≤
Z t 1/2 2 Ex (Am (w(s)) − A(w(s))) ds 0
Z
t 1 + Ex |(∇ · Am )(w(s)) − (∇ · A)(w(s))| ds 2 0 Z t |Vn (w(s)) − V (w(s))| ds . + Ex
(C.19)
0
Now (C.17) is implied by further estimating the right-hand side of (C.19) with the help of Z t Z 1 1/2 f (w(s))ds ≤ τ Ex f τ w(s) ds , (C.20) Ex 0
0
valid for f ≥ 0 and 0 ≤ t ≤ τ , and (2.29) in combination with (2.10). Eventually (C.18) follows directly from (C.19) for Am = 0 and Vn = 0 by observing (2.28). Lemma C.5.
Let A ∈ Hloc (Rd ) and V ∈ K± (Rd ). Then lim sup Ex [|1 − e−St (A,V |w) |p ] = 0
(C.21)
t↓0 x∈K
for all compact K ⊂ Rd and 0 < p < ∞. Lemma C.6.
Under the hypotheses of Theorem 5.1 one has lim
sup sup Ex [|e−St (A,V |w) − e−St (Am ,Vn |w) |p ] = 0
(C.22)
m,n→∞ 0≤t≤τ x∈K
for all compact K ⊂ Rd , 0 ≤ τ < ∞ and 0 < p < ∞. Proof of Lemmas C.5 and C.6. By Lemma C.1 and a standard localization technique we will reduce Lemmas C.5 and C.6 to Lemmas C.3 and C.4, respectively. For this purpose it is convenient to define A0 := 0 and V0 := 0. Moreover, we introduce the abbreviation Υp,R (m, n, t, x) := Ex [|e−St (A,V |w) − e−St (Am ,Vn |w) |p ΞBR ,t (w)] .
(C.23)
By the Cauchy–Schwarz Ineq. we obtain 1/2
Υp,∞ (m, n, t, x) ≤ (Ex [1 − ΞBR ,t (w)]) + Υp,R (m, n, t, x) .
(Υ2p,∞ (m, n, t, x))
1/2
(C.24)
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
221
The use of |z − z 0 |2p ≤ 22p |z|2p + |z 0 |2p for z, z 0 ∈ C and of the Feynman–Kac–Itˆ o formula leads to − − Υ2p,∞ (m, n, t, x) ≤ 22p ke−τ HRd (0,−2pV ) k∞,∞ + sup ke−τ HRd (0,−2pVn ) k∞,∞ . n∈N
(C.25) L´evy’s maximal inequality [56, Eq. (7.60 )] gives lim sup Ex [1 − ΞBR ,t (w)] = 0 .
R→∞ x∈K
(C.26)
Now we insert the estimate (C.25) into (C.24) and use Lemma C.1 and (C.26) to observe that it is sufficient to show lim sup Υp,R (0, 0, t, x) = 0 t↓0 x∈Rd
(C.27)
and lim
sup sup Υp,R (m, n, t, x) = 0
m,n→∞ 0≤t≤τ x∈Rd
(C.28)
for all R > 0 in order to prove Lemmas C.5 and C.6, respectively. To this end, let ΘR be as in Appendix A. Then (A.2) implies that Υp,R (m, n, t, x) as defined in (C.23) does not change its value when one replaces A, Am , V and Vn by AΘR+1 , Am ΘR+1 , V ΘR+1 and Vn ΘR+1 , respectively. This shows that (C.27) follows from Lemma C.3 and (C.28) from Lemma C.4, since AΘR+1 ∈ H(Rd ), {Am ΘR+1 }m∈N ⊂ H(Rd ), V ΘR+1 ∈ K(Rd ) and {Vn ΘR+1 }n∈N ⊂ H(Rd ) by assumption. Lemma C.7.
If Λ ⊆ Rd is regular, then lim
sup Ey [ΞΛ,t (w)] = 0
r↓0 |y−x|
(C.29)
for all x ∈ ∂Λ, t > 0. If Λ is uniformly regular, then lim sup
sup Ey [ΞΛ,t (w)] = 0
r↓0 x∈∂Λ |y−x|
(C.30)
for all t > 0. Proof. The first assertion is not new. It is, for example, coded in [6, Corollary II.1.11]. Therefore, we only show the second assertion, but remark that the appropriate simplification of the following proof works for the first one, too. We start from the simple estimate Ey [ΞΛ,t (w)] ≤ Ey [ΞΛ,t−τ (w(• + τ ))]
(C.31)
valid for all 0 < τ < t. Using the Markov property of Brownian motion we get Z 2 Ey [ΞΛ,t−τ (w(• + τ ))] = (2πτ )−d/2 dz e−(y−z) /2τ Ez [ΞΛ,t−τ (w)] . (C.32)
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K. BRODERIX, D. HUNDERTMARK and H. LESCHKE
This shows that the right-hand side of (C.31), considered as a function of y ∈ Rd , is uniformly continuous for all 0 < τ < t. Therefore, lim sup
sup Ey [ΞΛ,t (w)] ≤ sup Ex [ΞΛ,t−τ (w(• + τ ))]
r↓0 x∈∂Λ |y−x|
(C.33)
x∈∂Λ
for all 0 < τ < t. Now (C.30) follows from the definition (2.42) of uniform regularity. Lemma C.8. x, y ∈ Rd .
Let A ∈ Hloc (Rd ). Then (6.6) and (6.7) hold true for all t > 0,
Proof. For given t > 0, x, y ∈ Rd we choose R > 0 such that x, y ∈ BR . To show (6.6) we use the estimate Z t Z t 2 2 t,y χ Pt,y (A(b(s))) ds = ∞ ≤ P (b(s)) A(b(s)) ds = ∞ 0,x 0,x BR 0
0
+ Pt,y 0,x
sup |b(s)| ≥ R
Since lim Pt,y 0,x
(C.34)
sup |b(s)| ≥ R
R→∞
.
0≤s≤t
= 0,
(C.35)
0≤s≤t
it is enough to show that Z Et,y 0,x
t
|f (b(s))| ds < ∞
(C.36)
0
for all f ∈ K(Rd ). Using (6.5) and the time-reversal symmetry of the Brownian bridge, one has "Z # Z t
Et,y 0,x
|f (b(s))| ds ≤ 21+d/2 e(x−y) 0
2
t/2
/2t
sup Ez z∈Rd
|f (w(s))| ds .
(C.37)
0
Due to (2.28) the right-hand side is finite and (6.6) is therefore established. By a reasoning analogous to the one leading to the condition (C.36) it is sufficient to check (6.7) for A ∈ H(Rd ) in order to prove it for A ∈ Hloc (Rd ). To this end, we observe Z t 2 y − b(s) ds 2−d/2 e−(x−y) /2t Et,y A(b(s)) · 0,x t−s 0 "Z # "Z # t/2 t/2 y − w(s) y − w(s) ≤ Ex ds + Ey ds . A(w(s)) · A(w(s)) · t − s s 0 0 (C.38) Here we have again used (6.5) and the time-reversal symmetry. The first expectation on the right-hand side is seen to be finite for A2 ∈ K(Rd ) by the Cauchy– Schwarz Ineq. The second expectation is finite for ∇ · A ∈ K(Rd ) due to the
¨ CONTINUITY PROPERTIES OF SCHRODINGER SEMIGROUPS
partial-integration identity y − w(s) = −Ey [(∇ · A)(w(s))] . Ey A(w(s)) · s Hence we are done.
223
(C.39)
Acknowledgments The authors are grateful to Karl-Theodor Sturm (Bonn, Germany) for clarifying discussions. D.H. is much indebted to Werner Kirsch (Bochum, Germany) for constant encouragement and stimulating discussions. Furthermore, K.B. and D.H. thank the Deutsche Forschungsgemeinschaft for financial support under grant-no. Br 1894/1-1 and Hu 773/1-1, respectively. References [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Ninth Printing, Dover, New York, 1972. [2] M. Aizenman and B. Simon, “Brownian motion and Harnack inequality for Schr¨ odinger operators”, Commun. Pure Appl. Math. 35 (1982) 209–273. [3] S. Albeverio, P. Blanchard and ZhiMing Ma, “Feynman-Kac semigroups in terms of signed smooth measures”, pp. 1–31 in [33]. [4] S. Albeverio, S. Casati, U. Cattaneo, D. Merlini and R. Moresi (eds.), Stochastic Processes – Physics and Geometry, World Scientific, Singapore, 1990. [5] J. Avron, I. Herbst and B. Simon, “Schr¨ odinger operators with magnetic fields. I. General interactions”, Duke Math. J. 45 (1978) 847–883. [6] R. F. Bass, Probabilistic Techniques in Analysis, Springer, New York, 1995. [7] P. Blanchard and ZhiMing Ma, “Semigroup of Schr¨ odinger operators with potentials given by Radon measures”, pp. 160–195 in [4]. [8] K. Broderix, D. Hundertmark and H. Leschke, “Self-averaging, decomposition and asymptotic properties of the density of states for random Schr¨ odinger operators with constant magnetic field”, pp. 98–107 in [30]. [9] K. Broderix, D. Hundertmark, W. Kirsch and H. Leschke, “The fate of Lifshits tails in magnetic fields”, J. Stat. Phys. 80 (1995) 1–22. [10] R. Carmona, “Regularity properties of Schr¨ odinger and Dirichlet semigroups”, J. Funct. Anal. 33 (1979) 259–296. [11] Kai Lai Chung and Zhongxin Zhao, From Brownian Motion to Schr¨ odinger’s Equation, Springer, Berlin, 1995. [12] E. C ¸ inlar, K. L. Chung and M. J. Shape (eds.), Seminar on Stochastic Processes 1992, Birkh¨ auser, Boston, 1993. [13] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators, with Application to Quantum Mechanics and Global Geometry, Springer, New York, 1987. [14] M. Demuth and J. A. van Casteren, “On the spectral theory of selfadjoint Feller generators”, Rev. Math. Phys. 1 (1989) 325–414. [15] M. Demuth and J. A. van Casteren, “Framework and results of stochastic spectral analysis”, pp. 123–132 in [17]. [16] M. Demuth and J. A. van Casteren, Stochastic Spectral Theory for Self-adjoint Feller Operators, Birkh¨ auser, Basel, 2000. [17] M. Demuth, P. Exner, H. Neidhardt and V. Zagrebnov (eds.), Mathematical Results in Quantum Mechanics, Birkh¨ auser, Basel, 1994. [18] M. Demuth, E. Schrohe and B.-W. Schulze (eds.), Pseudo-differential Calculus and Mathematical Physics, Akademie, Berlin, 1994.
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[19] M. Demuth, E. Schrohe, B.-W. Schulze and J. Sj¨ ostrand (Eds.), Schr¨ odinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Akademie, Berlin, 1996. [20] Yu. V. Egorov and M. A. Shubin, Foundations of the Classical Theory of Partial Differential Equations, Second Printing, Springer, Berlin, 1998. [21] L. Erd¨ os, “Estimates on stochastic oscillatory integrals and on the heat kernel of the magnetic Schr¨ odinger operator”, Duke Math. J. 76 (1994) 541–566. [22] L. Erd¨ os, “Magnetic Lieb–Thirring inequalities”, Commun. Math. Phys. 170 (1995) 629–668. [23] L. Erd¨ os, “Dia- and paramagnetism for nonhomogeneous magnetic fields”, J. Math. Phys. 38 (1997) 1289–1317. [24] L. Erd¨ os, “Lifschitz tail in a magnetic field: the nonclassical regime”, Probab. Theory Relat. Fields 112 (1998) 321–371. [25] A. Friedmann, Stochastic Differential Equations, Volume I, Academic, New York, 1975. [26] A. Galindo and P. Pascual, Quantum Mechanics I, Springer, Berlin, 1990. [27] A. Galindo and P. Pascual, Quantum Mechanics II, Springer, Berlin, 1991. [28] J. Glover, M. Rao and R. Song, “Generalized Schr¨ odinger semigroups”, pp. 143–172 in [12]. ˘ c and R. Song, “Quadratic forms corresponding to the [29] J. Glover, M. Rao, H. Siki´ generalized Schr¨ odinger semigroups”, J. Funct. Anal. 125 (1994) 358–378. [30] H. Grabert, A. Inomata, L. S. Schulman and U. Weiss (eds.), Path Integrals from meV to MeV: Tutzing ’92, World Scientific, Singapore, 1993. [31] A. M. Hinz and G. Stolz, “Polynomial boundedness of eigensolutions and the spectrum of Schr¨ odinger operators”, Math. Ann. 294 (1992) 195–211. [32] A. M. Hinz, “Regularity of solutions for singular Schr¨ odinger equations”, Rev. Math. Phys. 4 (1992) 95–161. [33] U. Hornung, P. Kotelenez and G. Papanicolaou (eds.), Random Partial Differential Equations, Birkh¨ auser, Basel, 1991. [34] D. Hundertmark, “Zur Theorie der magnetischen Schr¨ odingerhalbgruppe”, Ph.D. Thesis, Ruhr-Univ. Bochum, Bochum, 1996. [35] D. Hundertmark, “A second look at the Feynman–Kac–Itˆ o formula”, in preparation. [36] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer, New York, 1991. [37] T. Kato, “Schr¨ odinger operators with singular potentials”, Israel J. Math. 13 (1972) 135–148. [38] R. Z. Khas’minskii, Teor. Veroyatnost. i Primenen. 4 (1959) 332–341. English translation: “On positive solutions of the equation U u + V u = 0”, Theory Probab. Appl. 4 (1959) 309–318. [39] V. F. Kovalenko and Yu. A. Semenov, Ukrain. Mat. Zh. 41 (1989) 273–278. English translation: “Lp -theory of Schr¨ odinger semigroups I”, Ukrainian Math. J. 41 (1989) 246–249. [40] H. Leinfelder and C. G. Simader, “Schr¨ odinger operators with singular magnetic vector potentials”, Math. Z. 176 (1981) 1–19. [41] H. Leinfelder, “Gauge invariance of Schr¨ odinger operators and related spectral properties”, J. Operator Theory 9 (1983) 163–179. [42] V. Liskevich and A. Manavi, “Dominated semigroups with singular complex potentials”, J. Funct. Anal. 151 (1997) 281–305. [43] V. Liskevich and Yu. A. Semenov, “Some problems on Markov semigroups”, pp. 163– 217 in [19]. [44] H. Matsumoto and N. Ueki, “Spectral analysis of Schr¨ odinger operators with magnetic fields”, J. Funct. Anal. 140 (1996) 218–255.
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DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE AT A CUBIC ROOT OF UNITY R. COQUEREAUX Centre de Physique Th´ eorique, CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France E-mail:
[email protected]
A. O. GARC´IA and R. TRINCHERO Instituto Balseiro and Centro At´ omico Bariloche CC 439–8400, Bariloche, R´ıo Negro, Argentina E-mail:
[email protected] E-mail:
[email protected] Received 18 September 1998 Mathematics Subject Classification: 16W30, 81R50 We consider the algebra of N × N matrices as a reduced quantum plane on which a finitedimensional quantum group H acts. This quantum group is a quotient of Uq (sl(2, C)), q being an N th root of unity. Most of the time we shall take N = 3; in that case dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess–Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group. Keywords: geometry.
Quantum groups, differential calculus, gauge theories, non commutative
1. Introduction The algebra M of N × N complex matrices can be considered as a finite quantum space (indeed, a finite-dimensional vector space, as will be used in what follows). As such, it is a representation and corepresentation space for finite-dimensional (dual) quantum groups. If some physical problem involves such an algebra, it could therefore be tempting to think that this system has a (kind of) quantum symmetry. The role of fundamental object of the symmetry would now be played by a Hopf algebra, instead of a usual Lie group. In the case at hand the relevant quantum groups are particularly interesting, in view of the fact that, unlike the generic-q case, they have a non-trivial radical and their representation theory involves indecomposable nonirreducible representations. Being finite-dimensional, they are also much simpler to analyse explicitly. Moreover, our knowledge about them is still much less complete than that of the generic-q case. 227 Reviews in Mathematical Physics, Vol. 12, No. 2 (2000) 227–285 c World Scientific Publishing Company
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Although our paper discusses several topics that can already be found in the literature (we incorporate them to improve the reading, to set notations and for completeness sake) we develop several aspects that, up to our knowledge, cannot be found elsewhere. Mainly, these include the construction of the (finite-dimensional) differential algebra over M, the decomposition of this algebra and of its differential algebra as representations of the quantum group that acts on them, and the definition of a scalar product on M that is invariant under the quantum group action. The algebra M can indeed be generated by two Heisenberg generators x and y satisfying the commutation relation xy = qyx, where q is an N -root of unity. These basic facts are recalled in Sec. 2. As a consequence, there is on this algebra a coaction of a quotient F of the quantum group Fun(SLq (2, C)); this is reminded in Sec. 3. Consequently, the dual H of F , also a finite-dimensional quantum group, acts on the algebra of N × N matrices. Explicit formulae for the pairing between H and F and for the action of H on M have been given by [1, 2]. The structure of H had been studied before by [3] and its representation theory had been given by [4]. In Sec. 4 we recall these duality properties and examine this action from the point of view of the reducible indecomposable representations of the quantum group H, which is not a semisimple algebra. The unitary group of the semi-simple part of H turns out to be isomorphic with U (3)×U (2)×U (1); for this reason, this Hopf algebra was conjectured (see [12]) to encode a quantum group of “hidden symmetries” in the standard model of electroweak interactions. However, the task of implementing this observation at the level of the lagrangian describing the theory has not been completed, and it seems that this idea would require some non trivial modifications of the model itself; nevertheless this remark provided one of the motivations for the present work. In Sec. 5 we introduce a (unique) real structure on M, F and H and construct a compatible hermitian scalar product on the space of matrices (the star operation on M does not coincide with the usual hermitian conjugacy on matrices). Next, in Sec. 6 we introduce the Manin dual of our reduced quantum plane, and show how F coacts and H acts on it. In Sec. 7 we define differential forms on the algebra M and study their properties. The differential algebra ΩW Z (M) that we introduce is a quotient of the Wess–Zumino complex [6] constructed originally for the twodimensional quantum plane. When q is a third root of unity, M is the algebra of 3×3 matrices and H (or F) are of dimension 27. The differential algebra ΩW Z (M) is of dimension 36. Since H acts on M and on its Manin dual, it also acts on ΩW Z (M) and we study this algebra in terms of the representation theory of H. Given an associative algebra — not necessarily commutative — one can define connections and covariant derivatives, for any choice of a differential calculus over the algebra of interest. The definition and properties of such connections are studied in Sec. 8; we also consider, as a particular case, connections that are hermitian for the star operation introduced before. In Sec. 9, we show how to couple these differential forms to usual space-time by constructing a differential algebra equal to the tensor product of ΩW Z (M) with the usual differential forms (antisymmetric tensors on
DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE
229
space-time). Generalized connections can then be defined. They incorporate a usual one-form valued in the space of 3 × 3 matrices and two matrix-valued scalar fields. These connections transform covariantly under a simultaneous action of the usual Lorentz group and the finite-dimensional quantum group H. It would certainly be interesting to build a classical Lagrangian field theory along these lines, but this involves some deeper problems that are mentioned in the concluding section. The paper ends with a number of short appendices. In them we first describe a set of 3 × 3 generalized Gell–Mann matrices with entries in the quantum group F . Then we give the structure of the principal indecomposable modules and the most general covariant metrics on the representation spaces of the algebra H, we study the space of differential operators on M and finally write down the universal R-matrix of H. 2. The Space M of 3 × 3 Complex Matrices as a Reduced Quantum Plane 2.1. From elementary N × N matrices to the x,y generators It has been known for a long time [7] that the algebra of N × N matrices can be generated by two elements x and y with the relations xy = qyx ,
xN = y N = 1l ,
(1)
where q denotes an N th root of unity (q 6= 1) and 1l is the unit matrix. Let us make explicit, in the particular case N = 3, this well known (but sometimes forgotten. . .) property. Let q be a cubic root of unity (q 6= 1) and take 1 0 0 0 1 0 0 y = 0 0 1 . x = 0 q −1 0 0 q −2 1 0 0 It is then easy to check that the above relations between x and y are indeed satisfied (use q −1 = q 2 ), and that they generate the whole algebra. Many formulae that we shall write in the following can be easily generalized when N is an arbitrary integer, but we shall stick to the case N = 3. Any 3 × 3 matrix can obviously be expanded on the base made of the nine elementary matrices Eij (such a matrix has a single non-zero entry 1 in position (i, j) and is filled with zeros elsewhere). One can express the elementary matrices themselves in terms of x and y. Calculations are straightforward, and give E11 = (1l + x + x2 )/3
E12 = (y + xy + x2 y)/3
E13 = (y 2 + xy 2 + x2 y 2 )/3 E22 = (1l + qx + q 2 x2 )/3
E21 = (y 2 + qxy 2 + q 2 x2 y 2 )/3 E23 = (y + qxy + q 2 x2 y)/3 . E32 = (y 2 + q 2 xy 2 + qx2 y 2 )/3
E31 = (y + q 2 xy + qx2 y)/3 E33 = (1l + q 2 x + qx2 )/3
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The unit matrix (1l) itself can be written in terms of the generators x and y (since x3 = y 3 = 1l). Therefore, one can also express the usual Gell–Mann matrices that generate Lie(SU (3)) in terms of the generators x and y. These expressions are shown in Appendix A. Warning: the set of 3 × 3 matrices is endowed with a usual star operation (that we denote † ): hermitian conjugacy. It is clear that x and y are unitary elements (with respect to † ): x† = x−1 = x2 and y † = y −1 = y 2 . As a matter of fact, this star operation does not have good properties with respect to a quantum group action that we shall introduce later. We shall return to this important problem in a forthcoming section. C) as a reduced quantum plane 2.2. M = M3 (C The associative algebra generated, over the complex numbers, by x and y with the single (quadratic) relation yx = q −1 xy is known as “the algebra of polynomials over the quantum plane” and is often denoted by Funq (C2 ) or by Cq [x, y]. We shall just call it Cq . When q = 1, this algebra is commutative and can be considered as the algebra of polynomials C[x, y] over the usual plane, x and y being the two coordinate functions. The dimension of Cq — as a vector space — is infinite, since powers of the generators do not satisfy any particular new relation. On the contrary, in the algebra M3 (C) of 3 × 3 matrices over complex numbers, the generators x and y, on top of the above quadratic relation, satisfy also the cubic relations x3 = 1l and y 3 = 1l. The dimension is then clearly equal to 9, as it should, since one can choose the following base of generators: {1l, x, y, x2 , y 2 , xy, x2 y, xy 2 , x2 y 2 }. One can ˆ q , when q 3 = 1, therefore consider M3 (C) as the quotient of the associative algebra C 3 3 ˆq by the bilateral ideal generated by the relations x − 1l = 0 and y − 1l = 0. Here C denotes the unital extension of Cq (the former is obtained by adding a unit, namely 1l to the later). For this reason we can consider the space of 3 × 3 matrices over C as a reduced quantum plane. . Warning: M = M3 (C) is not a quantum group. 3. The Finite-Dimensional Quantum Group F 3.1. Construction of F as an algebra Let us start with a classical analogy and call x and y the coordinate functions on the plane. One can make a linear change of coordinates and call x0 and y 0 the new coordinate functions: ! ! ! x0 a b x = ⊗ . (2) y0 c d y We can assume the transformation to be unimodular (ad − bc = 1). Rather than considering x and y as numbers, we think of them as coordinate functions. In the same way, we do not take the matrix elements a, b, c and d as numbers but as functions on the group of coordinate transformations: when g denotes such a transformation then a(g), b(g), c(g) and d(g) are numbers, namely, the matrix elements of g. This change in the perspective explains why we write a tensor product
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231
sign in the previous formula. . . evaluation on points of the group and of the space gives the transformed coordinate functions. One could also introduce line vectors, with coordinate functions x ˜, y˜. The same change of coordinates would read ! a b (˜ x y˜) = (x y) ⊗ . (3) c d From now on, we shall no longer assume that symbols x and y commute but that they should satisfy the relations discussed in the previous section, namely xy = qyx, x3 = y 3 = 1l. Symbols x and y can therefore be represented by the 3 × 3 matrices already given. As before, M shall denote the algebra generated by x and y. One then introduces non-commuting symbols a, b, c and d and imposes that ˜, y˜) obtained by the previous matrix equalities should satisfy quantities x0 , y 0 (and x the same relations as x and y. We call F 0 the algebra generated by the elements . a, b, c and d. The product in F 0 ⊗ A is defined by (f ⊗ z)(g ⊗ w) = f g ⊗ zw, in other words, a, b, c, d commute with x, y. We have a similar definition for the ˜y˜ = q y˜x ˜ lead to multiplication in A ⊗ F 0 . The two constraints x0 y 0 = qy 0 x0 and x the six quadratic relations [8] ac = qca
bd = qdb
ab = qba bc = cb
cd = qdc ad − da = (q − q −1 )bc .
(4)
The algebra generated by a, b, c and d (take products and sums) together with the six above relations is usually denoted Fun(GLq (2, C)) and is the algebra of would-be functions over the quantum group GLq (2, C). Calling such elements “functions” is, of course, a misnomer, since they are not valued in any field of numbers and. . . do not commute. . The element D = da − q −1 bc = ad − qbc is central (it commutes with all the elements of Fun(GLq (2))); it is called the q-determinant and we set it equal toa 1l. Adding this extra relation defines the algebra Fun(SLq (2, C)). Now, we should remember that x3 = y 3 = 1l. Imposing x03 = y 03 = 1l (and also x ˜3 = y˜3 = 1l) again implies new relations. For instance, x03 = (a ⊗ x + b ⊗ y)3 = a3 ⊗ x3 + a2 b ⊗ x2 y + aba ⊗ xyx + ba2 ⊗ yx2 + ab2 ⊗ xy 2 + bab ⊗ yxy + b2 a ⊗ y 2 x + b3 ⊗ y 3 = a3 ⊗ x3 + (1 + q + q 2 )a2 b ⊗ x2 y + (1 + q + q 2 )ab2 ⊗ xy 2 + b3 ⊗ y 3 = a3 ⊗ x3 + b3 ⊗ y 3 where we used 1 + q + q 2 = 0 since q is a third root of unity. a We will use 1l to denote the unit element of M, F and the — to be defined — quantum group H, indistinctly. Which one this symbol refers to, should be easily understood from context. In case this is not obvious we will use 1lM , 1lF , 1lH .
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This, together with the analogous constraints on y 0 , x˜, y˜, imply: a3 = 1l , 3
c = 0,
b3 = 0 , d3 = 1l .
(5)
Imposing these cubic relations on Fun(SLq (2, C)) defines a new algebra that we denote with F . We call it the reduced quantum unimodular group associated with a cubic root of unity. Actually one should better call it the algebra of wouldbe functions over the reduced quantum unimodular group SLq (2, C) but we shall shorten the terminology. Since a3 = 1l, multiplying the relation ad = 1l + qbc from the left by a2 leads to d = a2 (1l + qbc)
(6)
so d is not needed and can be eliminated. The algebra F can therefore be linearly generated — as a vector space — by the elements aα bβ cγ where indices α, β, γ run in the set {0, 1, 2}. We see that F is a finite-dimensional associative algebra, whose dimension is dim(F ) = 27 . Let us stress the fact that this very particular algebra F (which turns out to be a quantum group) emerges naturally in relation with the study of the algebra M of 3 × 3 matrices. 3.2. F as a quantum group F is not only an algebra but also a quantum group, in other words, a Hopf algebra. This amounts to say that besides its algebra structure (a product), it has a coalgebra structure (a coproduct) and that the two structures are compatible (hence it is a bialgebra); moreover one can also define maps called antipode and counit obeying the appropriate axioms. There are now several more or less elementary textbooks on the subject of quantum groups and the interested reader should refer to them for general properties. We shall simply give the definitions for the coproduct, antipode and counit, in the present case. The reader will easily show that all required properties are indeed satisfied. These three maps being algebra morphisms (or antimorphisms) it is actually enough to define them on the generators. Coproduct: It is an algebra morphism ∆ from the algebra F to the algebra F ⊗ F, i.e. ∆(uv) = ∆u ∆v. It is given by ∆a = a ⊗ a + b ⊗ c, ∆b = a ⊗ b + b ⊗ d, ∆c = c ⊗ a + d ⊗ c, ∆d = c ⊗ b + d ⊗ d. Antipode: It is a (linearb ) antimorphism S from F to F , i.e. S(uv) = Sv Su. It is given by Sa = d, Sb = −q −1 b, Sc = −qc, Sd = a. Counit: It is a morphism from F to the complex numbers C and is given by (a) = 1, (b) = 0, (c) = 0, (d) = 1. b We stress that the antipode is a linear antimorphism, in contrast with the star operation that will be introduced in Sec. 5 which is an antilinear antimorphism.
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Actually, one can give a rather short proof of the fact that F is a Hopf algebra. It is enough to prove that the two-sided ideal I defining the quotient F = Fun(SLq (2, C))/I is a Hopf ideal. I is the ideal generated by the relations a3 − 1l = 0, d3 − 1l = 0, b3 = 0 and c3 = 0. Being I a Hopf ideal means: ∆I ⊂ I ⊗ Fun(SLq (2, C)) ⊕ Fun(SLq (2, C)) ⊗ I, (I) = 0 and S(I) ⊂ I. This is easy to show. For instance, we see that ∆a3 = (∆a)3 = (a ⊗ a + b ⊗ c)3 = a3 ⊗ a3 + (1 + q + q 2 )a2 b ⊗ a2 c + (1 + q + q 2 )ab2 ⊗ ac2 + b3 ⊗ c3 = a 3 ⊗ a 3 + b 3 ⊗ c3 but ∆1l = 1l ⊗ 1l, so that 2∆(a3 − 1l) = a3 ⊗ (a3 − 1l) + (a3 − 1l) ⊗ a3 + (a3 − 1l) ⊗ 1l + 1l ⊗ (a3 − 1l) + 2b3 ⊗ c3 which is indeed in I ⊗ Fun(SLq (2, C)) ⊕ Fun(SLq (2, C)) ⊗ I. F is, by construction, an associative algebra. However, it is not semisimple. This can be seen easily from the faithful realization given in Appendix C (this realization, in terms of matrices with entries in an algebra generated by two commuting symbols ξ1 , ξ2 whose cube power vanishes is due to Ogievetsky, [9]). F is therefore not a matrix quantum group in the sense of Woronowicz [10]: whatever the involution and the scalar product we choose, it will never be a C ∗ -algebra. 3.3. F coacting on M What is called coaction of F on M is precisely what was described, in simple terms, at the beginning of this section. Actually, we have two coactions: a left coaction and a right one. We have . x0 = ∆L x = a ⊗ x + b ⊗ y . y 0 = ∆L y = c ⊗ x + d ⊗ y
(7)
. x˜ = ∆R x = x ⊗ a + y ⊗ c . y˜ = ∆R y = x ⊗ b + y ⊗ d .
(8)
and
Both ∆L and ∆R can be extended to arbitrary elements of M by imposing the propertyc ∆L,R (zw) = ∆L,R (z) ∆L,R (w) for any z, w ∈ M. c Taking into account this condition, we see that the equations
∆R,L (xy − qyx) = 0 are completely equivalent to the ones we used in Sec. 3.1 to obtain the relations (4) for the quantum group F . Therefore they also imply (4).
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M itself, endowed with the two coactions ∆L and ∆R is not a quantum group but a left and right comodule, i.e. a corepresentation space of the quantum group F. This means that the right coaction (for instance) maps M 7→ M ⊗ F , in such a way that (9) (∆R ⊗ id )∆R (z) = (id ⊗ ∆)∆R (z) and (id ⊗ )∆R (z) = z ,
(10)
for any z ∈ M. Here one should not confuse ∆ (the coproduct on the quantum group) with ∆R,L (the R, L-coaction on M)! These conditions are indeed satisfied. . . the interested reader may check that. Moreover, M is a left and right comodule algebra over F . There are two extra axioms that have to be satisfied in order to have a comodule algebra structure, in particular a compatibility axiom between the coaction and the product in M. These extra conditions read: ∆L,R (zw) = ∆L,R (z) ∆L,R (w)
(11)
for z, w any elements of the algebra. This is just the equation we have used to extend the coaction to the whole M, thus is trivially satisfied. If the algebra also has a unit, the coaction should verify ∆L,R (1l) = 1l⊗ .
(12)
Using this coaction, and recalling that M is the algebra of 3×3 matrices, one can build a set of generalized Gell–Mann matrices with entries in the quantum group F, as it is done in Appendix B. Notice that although F coacts on M, it does not act on it. This is exactly what happens in the usual (commutative) situation: the algebra of functions on the group of unimodular transformations of the plane coacts on the algebra of functions on the plane (there are indeed tensor product signs in the formula x → a ⊗ x + b ⊗ y, even if it is sometimes convenient not to write them. . .). In the present situation, we have no “points” and it makes a priori no sense to speak about the action of a would-be group on would-be points. The coaction of F on M is however well defined and allows such a terminological abuse. As a matter of fact, there is an algebra that acts on M, it is not F but its dual, that we call H. We shall return to this in the next section. 4. The Dual H of F 4.1. Classical analogies One should think of F as an analogue of the space of functions Fun(G) over a Lie group G (in our case, G itself does not exist) and H — that we shall introduce now — as a finite-dimensional analogue of the universal enveloping algebra of the Lie algebra of G. Another fruitful analogy is to take F as the analogue of the space of functions over a finite group G and H as an analogue of the group algebra CG.
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As in the classical case, arbitrary elements X of H can be understood in several ways: • As distributions on F: we evaluate X on the “function” u ∈ F. The result is a number called hX, ui. • As invariant differential operators acting on the “functions” u. The result is another element of F , that we call X[u]. Actually, as in the classical case, one can define two kinds of invariant operators: left-invariant ones (coming classically from the right action of a group on itself) and right-invariant ones (coming from the left action). • As a differential operator acting on the space of “functions” M (classically functions on a manifold on which the group G acts). Let z be such a function. We call X[z] the result. This is another element of M. • Abstractly, as an element of the associative algebra H, i.e. classically, an element of the enveloping algebra U(Lie(G)). The remaining part of this section is devoted to a short study of these various aspects. 4.2. The finite-dimensional quantum group H Being the dual of F, it is clear that H is a vector space of dimension 27. It can be generated, as a complex algebra, by elements X± , K and K −1 obeying a number of relations. Multiplication and comultiplication in H can be obtained from the corresponding ones in its dual F , but here we gather most of the relevant formulae abstractly defining H as a Hopf algebra, with generators X± and K, without using its duality with F. Product: KX± = q ±2 X± K [X+ , X− ] =
1 (K − K −1 ) (q − q −1 )
K 3 = 1l
(13)
3 3 = X− =0 X+
Coproduct: The comultiplication is an algebra morphism, i.e. ∆(XY ) = ∆X ∆Y . It is given by . ∆X+ = X+ ⊗ 1l + K ⊗ X+ . ∆X− = X− ⊗ K −1 + 1l ⊗ X− (14) . ∆K = K ⊗ K . ∆K −1 = K −1 ⊗ K −1 Antipode: The antipode S is an anti-automorphism, i.e. S(XY ) = SY SX. It acts as follows: S1l = 1l, SK = K −1 , SK −1 = K, SX+ = −K −1 X+ , SX− =
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−X− K. As usual, the square of the antipode is an automorphism (and it is a conjugacy given — up to multiplication by a central element — by K −1 , hence we can write S 2 u = K −1 uK). Counit: The counit is defined by 1l = 1, K = 1, K −1 = 1, X+ = 0, X− = 0. The previous multiplication relations allow to order any monomial of the algebra α β γ 3 freely generated by X± and K as X+ K X− . Moreover, as X± = 0 and K 3 = 1l, it 3 α β γ is evident that the 27 = 3 elements {(X+ K X− )}α,β,γ∈{0,1,2} span H as a vector space over C. Note that the element X+ X− − (q −1 K + qK −1 )/3 commutes with all elements of H. Therefore its plays the role of a usual Casimir operator. Remark. In [1] the authors study the pairing between a (reduced) universal enveloping algebra and the algebra of functions on quantum SL(2, C), in the case q N = 1. However, they factorise the universal algebra over the relations k 6 = 1l, 3 3 = 0, rather than K 3 = 1l, X± = 0. Their choice for generators I± and k differs I± from our X± and K (actually, k = K 1/2 , I+ = qX+ K and I− = qX− K 2 ). The ˜ is twice as big as ours (k 3 is a central element but is not equal obtained algebra H to 1l). 4.3. H as the dual of F (distributions) . Being F a quantum group (a Hopf algebra), its dual H = F ∗ is a quantum group as well. Let u ∈ F and X ∈ H. We call hX, ui the evaluation of X on u (a complex number). • Using the coproduct ∆ in F , one defines a product in H (this is the so-called convolution product of distributions and it is often denoted by ?, but we shall omit this symbol in the sequel): . hX1 X2 , ui = hX1 ⊗ X2 , ∆ui . • Using the product in F, one defines a coproduct (that we again denote ∆) in H:d . h∆X, u1 ⊗ u2 i = hX, u1 u2 i . • The interplay between unit and counit is described by the two relations: h1lH , ui = F (u) and hX, 1lF i = H (X). The two structures of algebra and coalgebra are clearly interchanged by duality. H is the linear dual of F. In other words, if F were a space of smooth (resp. continuous) functions over a compact space, H would be the corresponding space of distributions (resp. measures). . d Some authors define h∆X, u ⊗ u i = hX, u2 u1 i, in order to increase the correspondence with 1 2
the classical case. This alternative definition corresponds to use our ∆op = τ ◦ ∆ as a coproduct, where τ produces a permutation of factors in the tensor product.
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All possible pairings can be computed from those between the generators K, X± and a, b, c. They are given by: hK, ai = q hX+ , ai = 0
hK, bi = 0 hX+ , bi = 1
hK, ci = 0 hX+ , ci = 0
hK, di = q 2 hX+ , di = 0
hX− , ai = 0
hX− , bi = 0
hX− , ci = 1
hX− , di = 0 .
(15)
The fourth column of this table can be obtained from the first three (remember that d = a2 (1l + qbc)). Remark. The previously given multiplication and comultiplication equalities for the generators of H (in particular the appearance of q rather than its inverse) cannot be given arbitrarily; indeed they have to be compatible with those given for the dual, F. In other words, once the duality formulae defining the generators X± have been chosen, one cannot choose independently “the q of F ” and “the q of H”. Conversely, if we start with the formulae defining multiplication and comultiplication in F and H, the pairing formulae are essentially unique. To see this, let us suppose given, for instance, the comultiplication relations for H and determine the constraints for both the pairings and the multiplication in H. Choose the basis ai bj ck in the vector space F . At the same time we choose a basis j k X− in the dual vector space F , with pairings K i X+ hK, ai = α hX+ , bi = β
hK, bi = hK, ci = 0 hX+ , ai = hX+ , ci = 0
hX− , ci = γ
hX− , ai = hX− , bi = 0
etc. where α, β, γ are numbers that we are going to determine. First of all, let us calculate hX+ , abi and hX+ , bai. Known commutation relations in F (or comultiplication in H) will imply that α = q. Indeed, hX+ , abi = h∆X+ , a ⊗ bi = hX+ ⊗ 1l + K ⊗ X+ , a ⊗ bi = 0 + αβ hX+ , bai = h∆X+ , b ⊗ ai = hX+ ⊗ 1l + K ⊗ X+ , b ⊗ ai = β + 0 . But ab = qba, therefore α = q. Then, we shall calculate hKX+ , bi and hX+ K, bi. This will imply KX+ = q 2 X+ K. To do that, we need to find hK, di. We know that d = a2 (1l + qbc), therefore hK, di = h∆K, a2 ⊗ (1l + qbc)i = hK, a2 ihK, (1l + qbc)i = h∆K, a ⊗ ai(hK, 1li + qhK, bci) = q 2 (1 + 0) = q 2 . Now, hKX+ , bi = hK ⊗ X+ , ∆bi = hK ⊗ X+ , a ⊗ b + b ⊗ di = hK, aihX+ , bi + hK, bihX+ , di = qβ + 0 .
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But hX+ K, bi = hX+ ⊗ K, ∆bi = hX+ ⊗ K, a ⊗ b + b ⊗ di = hX+ , aihK, bi + hX+ , bihK, di = 0 + βq 2 . Therefore hX+ K, bi = qhKX+ , bi
−→
KX+ = q 2 X+ K .
Finally, there are no conditions on β, γ and we can take β = 1 and γ = 1. We conclude that the following conditions are compatible: ab = qba, hK, ai = q, X+ K = q 2 X+ K. Such considerations justify the defining relations given a priori, in Sec. 4.2. 4.4. Actions of H One can define several actions of H on H, of H on F and of H on M. We briefly mention them in the present subsection. ♦
H acting on H
There is a natural action of H on itself given by the multiplication, thus we can define: • • • •
the the the the
right action R[X]Y = Y X, right action R0 [X]Y = S(X)Y , left action L[X]Y = XY , left action L0 [X]Y = Y S(X).
However, none of these actions is compatible with the algebra structure in H, i.e. none of these make H an H-module algebra. ♦
H acting on F
Let X, Y be elements of H, i.e. linear forms on F , or even distributions on the would-be group G. Take u ∈ F . Using the pairing h, i between these dual Hopfalgebras, one can define actions of H on F , that are dual to the ones on H we have just defined: • the left action defined by hY, X[u]i = hY X, ui . It comes from the right action of H on itself given by multiplication, R[X]Y = Y X. This is a left action of H on F , since (XY )[u] = X[Y [u]]. It is also compatible with the algebra structure of F because X[uv] = X1 [u]X2 [v], where ∆X = X1 ⊗X2 (implicit summation). Thus we also say that F is a left H-module algebra. Notice that hY, X[u]i = hY X, ui = hY ⊗ X, ∆ui = hY ⊗ X, u1 ⊗ u2 i = hY, u1 ihX, u2 i. Therefore, we have explicitly X[u] = u1 hX, u2 i .
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• the left action defined by hY, X[u]i = hS(X)Y, ui, which is not compatible with the multiplication of F : more precisely, it involves a twist because X[uv] = X2 [u]X1 [v]. • the right action defined by hY, X[u]i = hXY, ui . It is obviously a right action on F , since (XY )[u] = Y [X[u]]. • the right action defined by hY, X[u]i = hY S(X), ui. Again, the first right action is compatible with the algebra structure of F whereas the second one involves a twist. From now on, we shall choose consistently the left action of H on F defined by hY, X[u]i = hY X, ui. Explicitly, we find for the left action of the generators X± and K on F K[a] = qa K[b] = q 2 b
X+ [a] = 0 X+ [b] = a X− [a] = b X− [b] = 0 X+ [c] = 0
X+ [d] = c X− [c] = d X− [d] = 0 K[c] = qc
K[d] = q 2 d .
(16)
For every X in H, one can associate a linear operator X[ ] from F to F . This operator is sometimes called a “left invariant operator for the coproduct ∆” because ∆ ◦ X[ ] = (id ⊗ X[ ] ) ◦ ∆, where ◦ denotes the composition of maps and id is the identity map. Indeed, ∆(X[u]) = ∆(u1 )hX, u2 i = (u11 ⊗ u12 )hX, u2 i = (id ⊗ id ⊗ hX, ·i)(u11 ⊗ u12 ⊗ u2 ). But (u11 ⊗ u12 ⊗ u2 ) = (u1 ⊗ u21 ⊗ u22 ), because of the coassociativity of ∆ (recall that (∆ ⊗ id ) ◦ ∆ = (id ⊗ ∆) ◦ ∆). So ∆(X[u]) = (id ⊗ id ⊗ hX, ·i)(u1 ⊗ u21 ⊗ u22 ) = u1 ⊗ u21 hX, u22 i = (id ⊗ X[ ] )(u1 ⊗ u2 ) = (id ⊗ X[ ] )∆(u). The space of left invariant operators on F — the dual of H — is isomorphic X[ ] and with H itself since (XY )[u] = X[Y [u]]. Explicitly, the isomorphism X X is given by X[u] = (id ⊗ hX, ·i ) ◦ ∆ and hX, ui = ◦ X[u] where is the X[ ] counit of F. In a classical (i.e. group-like) situation, it would have been equivalent to evaluate X[u] at the identity of the group to get hX, ui. On the contrary, the space of right invariant operators (for the coproduct ∆) is anti-isomorphic with H, since with a right action (XY )[u] = Y [X[u]].
;
;
Remark about the classical case An easy way to remember the previous results is the following. First we introduce classical generators of Lie(SL(2)) in the fundamental representation, i.e. the usual matrices X ± , X 3 and define K = q X 3 : X+ =
X3 =
!
0 0
1 0
1
0
0
−1
X− =
,
0 1
0 0
! ,
K=q
X3
=
! ,
q
0
0
q −1
! .
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240
Now we have a left action on F class , which for a generic Y ∈ Lie (SL(2)) readse " !# ! a b . a b L Y = Y, c d c d and a right one which is defined analogously. For the generators, this action reads explicitly a b 0 a a b b 0 X+ = , X− = , c d 0 c c d d 0 a −b a b . X3 = c −d c d Observe that X ± and X 3 act by derivation on F class , and K as an automorphism. Indeed, the above left classical action (and the corresponding right one) of Lie(SL(2)) on F class may also be written in terms of operators (classical derivations) ∂ ∂ +c ∂b ∂d ∂ ∂ +d XL − = b ∂a ∂c ∂ ∂ ∂ ∂ L −b +c −d X3 = a ∂a ∂b ∂c ∂d XL + =a
∂ ∂ +d ∂a ∂b ∂ ∂ =a +b ∂c ∂d ∂ ∂ ∂ ∂ +b −c −d . =a ∂a ∂b ∂c ∂d
XR + = c XR − XR 3
It is easy to check that, for instance (and as it should!) [X 3 , X + ] = +2X +
R R [X R 3 , X + ] = −2X + .
L L [X L 3 , X + ] = +2X +
We recover the fact that left fundamental vector fields build up, in the classical case, the Lie algebra itself. As we see, it turns out that the formulae (16) expressing the action of H on the quantum generators a, b, c, d are the same as the classical ones. However, the e Note that the following equations should be interpreted as equalities amongst entries of the matrices, i.e. L X + [a] X L a b a b . + [b] = XL ≡ X+ , + c d c d XL XL + [c] + [d]
etc. This explains why this action is a left one, in spite of being given by a right multiplication; for example:
L XL + X−
a
b
c
d
= XL +
=
b
0
d
0
a
b
c
d
=
XL + [b]
0
XL + [d]
0
L
X + X − = (X + X − )
=
a
0
c
0
a
b
c
d
.
This action is nothing more than the left action of the classical enveloping algebra of a Lie algebra on functions U ∈ C(G) of the corresponding classical group G. Being dual to the right-product . action R[X]Y = Y X, it is given infinitesimally for the Lie generators x by x[U ]|g = U |g(1l+x) −U |g .
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quantum action cannot be extended by derivations, when the same quantities act on arbitrary elements of F: the action of invariant linear operators is then carried out by twisted derivations (derivations twisted by automorphisms). This can be easily seen remembering that one should use the coproduct of H to act on products of F. ♦
H acting on M
Using the fact that F coacts on M (an algebra of matrices) in two possible ways, and that elements of H can be interpreted as distributions on F , we obtain two actions of H on the quantum space M. We shall describe them for arbitrary elements and give explicit results for the generators. Let z ∈ M. We know that F coacts on M from the left and from the right. For the left coaction, we have ∆L z = uα ⊗ zα with uα ∈ F and zα ∈ M (implied summation). For instance ∆L x = a ⊗ x + b ⊗ y. Let X be an element of H. Using the pairing between H and F, we set . X R [z] = (hX, ·i ⊗ id)∆L z = hX, uα izα ,
(17)
which makes it into a right action of H on M. Moreover, it makes M a right-HR [x] = hX+ , aix + hX+ , biy = 0x + 1y = y. module algebra. For instance, X+ F also coacts on M from the right, so that ∆R z = zα ⊗ vα with zα ∈ M and vα ∈ F (implied summation). For instance ∆R x = x ⊗ a + y ⊗ c. Taking X ∈ H, and using the H-F pairing we can define the left action . X L [z] = (id ⊗ hX, ·i)∆R z = hX, vα izα .
(18)
With this L-action we can check that M a left-H-module algebra. In particular, L [x] = xhX+ , ai + yhX+ , ci = 0x + 0y = 0. X+ We stress again that we have denoted with X L the operators on M corresponding to the left-action of an element X ∈ H on M, but this action is indeed dual to the right-coaction of F on M. The same happens for the right-action. One should therefore use the notation involving upper indices L or R with some care. To obtain the action of an arbitrary element X ∈ H on a product zw of elements of M, one may use the definition (above) or the following property (notation should Table 1. Right and left actions of H on M. Right
KR
R X+
R X−
Left
KL
L X+
L X−
1l x2 y xy 2 x y x2 y 2 x2 xy y2
1l qx2 y q 2 xy 2 qx q2 y x2 y 2 q 2 x2 xy qy 2
0 −xy 2 1l y 0 −x −xy y2 0
0 1l −x2 y 0 x −y 0 x2 −xy
1l x2 y xy 2 x y x2 y 2 x2 xy y2
1l qx2 y q 2 xy 2 qx q2 y x2 y 2 q 2 x2 xy qy 2
0 q 2 1l −x2 y 0 x −qy 0 qx2 −q 2 xy
0 −xy 2 q 2 1l y 0 −qx −q 2 xy qy 2 0
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242
be clear):
X L [zw] = m (∆X)L [z ⊗ w] .
It is therefore enough to know the action of the generators X± , K on M (complete 27 × 27 tables, with other conventions, can be found in [2]). One finds: Notice that, up to multiplicative factors,
-
y2
0 .. .. .. R ... X− .. .. .. .. xy .. .. .. R ... X− .. .. .. . 2 ........... 0 x
6 R X+
6 R X+
?
6 I ,
?
y2 .. .. .. L ... L X− ..X+ .. .. .. xy .. .. .. L ... L X− ..X+ .. .. . x2
6 6
?
,
-0
xy 2
and
?
-0 ........... ?
R X+
-
0 .... .. .. .. . .. .. .. .. .. .. .. .. .. . .. . R. x2 y 2 X− .. .. .. ... .. .. .. .. .. . x ..................... 0 y
-0 y ..................... .. ..6 I @@ .. .. .. @@ ... .. @ .. ..
L. L x2 y 2 X− ..X+ .. . .. .. .. .. .. . .. .. .. .. .. . . .. .. .. 0 x
?
-
R X+
6 .......@ .. @ .. @ .. @@R .. .. .. X ... 1l (19) .. . .. .. .... . .. . .. .. .. .. .. .. .. .... .. .. .. .. .. . .. ..? ?? R −
x2 y
0
xy 2 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. L .. L X− ..X+ .. .. .. .. .. .. .. .. . x2 y
6
and
?
R 1l .. .. .....
(20)
.. .. .. ..
?? 0
We see clearly on these diagrams that (up to numerical factors) the right action is obtained from the left one by interchanging arrows for X+ and X− . Differential operators on M associated with the H action Here, we are only concerned with those differential operators on M associated with the quantum group action of H. A more complete analysis can be found in L,R , K L,R Appendix F. They are obtained by considering products and powers of X± acting from the left or from the right as before. From now on we will stick with the left action. We also consider elements of M acting by multiplication as differential operators . of order zero: let z ∈ M, then, for example, x[z] = xz. Let X ∈ H; the reader . should distinguish between Xx — which is a differential operator on M (Xx[z] = X[x[z]] = X[xz]) — from X[x] which is an element of M. Therefore it makes perfect sense to study the commutation relations, say, between X and x. We shall use such relations later. Here, we establish the commutation relations between x, y — taken as differential operators of degree 0 — and X± , K.
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Let z ∈ M. Consider, for instance, L L L L X+ [xz] = X+ [x] 1l[z] + K L [x] X+ [z] = 0 + qxX+ [z] . L L Therefore X+ x = qx X+ . Similarly, we obtain: L L X+ x = q x X+ L X− x = y KL L
−1
K x = qxK
L + x X−
L
L L X+ y = x + q 2 y X+ L L X− y = y X−
(21)
K L y = q2 y K L . Any element of H can also be written explicitly in terms of differential operators on M; see Appendix F. It may be useful to notice that it is possible to introduce “conformal weights” so that previous formulae are homogeneous with the following weight assignments: L L , K L , X− ) 7→ (1, 0, −1), and (x, y) 7→ (1/2, −1/2). (X+ 4.5. The structure of the non-semisimple algebra H The algebra H can be explicitly defined — and many of its properties understood — without using anything more sophisticated than basic multiplication or tensor products of matrices as well as elementary calculus involving anti-commuting numbers (Grassmann numbers). This construction, inspired from [3], was explicitly performed in [4] for N = 3 and used to analyse the representation theory of H. The analysis for other values of N can be found in [9]. We just recall here the results and establish the relations with the previous notations. . The finite-dimensional quantum group H can be defined as the algebra H = M3 ⊕ (M2|1 (Λ2 ))0 , where M3 is the set of 3 × 3 complex matrices and (M2|1 (Λ2 ))0 is the Grassmann envelope of the associative Z2 -graded algebra M2|1 (C2 ), i.e. the even part of its graded tensor product with a Grassmann algebra Λ2 of two generators. To see this explicitly, let Λ2 be the Grassmann algebra over C with two generators, i.e. the linear span of {1, θ1 , θ2 , θ1 θ2 } with arbitrary complex coefficients, where the relations θ12 = θ22 = 0 and θ1 θ2 = −θ2 θ1 hold. This algebra has an even part, generated by 1 and θ1 θ2 , and an odd part, generated by θ1 and θ2 . We call M3 the algebra of 3 × 3 matrices over the complex numbers and M2|1 another copy of this algebra that we grade as follows: a matrix V ∈ M2|1 is called even if it is of the type V11 V12 0 0 , V = V21 V22 0 0 V33 and odd if it is of the type
0 V = 0 V31
0 0 V32
V13 V23 . 0
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244
We call (M2|1 (Λ2 ))0 the Grassmann envelope of M2|1 which is defined as the even part of the tensor product of M2|1 and Λ2 , i.e. the space of 3 × 3 matrices V with entries V11 , V12 , V21 , V22 , V33 that are even Grassmann elements (of the kind α + βθ1 θ2 ) and entries V13 , V23 , V31 , V32 that are odd Grassmann elements (of the kind γθ1 + δθ2 ). We define H as . H = M3 ⊕ (M2|1 (Λ2 ))0 . Explicitly,
∗ H = ∗ ∗
∗ ∗ ∗
α11 + β11 θ1 θ2 ∗ ∗ ⊕ α21 + β21 θ1 θ2 ∗ γ31 θ1 + δ31 θ2
α12 + β12 θ1 θ2 α22 + β22 θ1 θ2 γ32 θ1 + δ32 θ2
γ13 θ1 + δ13 θ2 γ23 θ1 + δ23 θ2 . α33 + β33 θ1 θ2
All entries besides the θ’s are complex numbers (the above ⊕ sign is a direct sum sign: these matrices are 6 × 6 matrices written as a direct sum of two blocks of size 3 × 3). It is obvious that this is an associative algebra, with usual matrix multiplication, of dimension 27 (just count the number of arbitrary parameters). H is not semisimple, because of the appearance of Grassmann numbers in the entries of the matrices. Its semisimple part H, given by the direct sum of its block-diagonal θ-independent parts is equal to the 9 + 4 + 1 = 14-dimensional algebra H = M3 (C) ⊕ M2 (C) ⊕ C. The radical (more precisely the Jacobson radical) J of H is the left-over piece that contains all the Grassmann entries, and only the Grassmann entries, so H = H/J. The radical has therefore dimension 13. In order to write generators for H, we need to consider 6×6 matrices that have a (3 × 3) ⊕ ((2|1) × (2|1)) block diagonal structure. We introduce elementary matrices Eij for the M3 (C) part and elementary matrices Fij for the M2|1 (C) part. The associative algebra H defined previously can be generated by the following three matrices: X+ = E12 + E23 + (1 − θ1 θ2 /2)F12 + θ1 (F23 + F31 ) X− = −E21 − E32 + (1 − θ1 θ2 /2)F21 + θ2 (F13 − F32 ) K = q 2 E11 + E22 + q −2 E33 + qF11 + q −1 F22 + F33 . Explicitly, one gets
0 1 0 0 0 1 0 0 0 . X+ = ()
() 0
0 θ1
1−
θ1 θ2 2 0 0
0 θ1 0
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0 0 0 −1 0 −1 . X− = ()
q2
0 0 . K=
0 1 0 ()
0 0 0
0 0 −θ2
0
0 q −2
θ2 0
() 0 1 − θ1 θ 2 2 0
0
245
()
q 0 0
0 q
−1
0
. 0 0 1
Performing explicit matrix multiplications or using the relations Eij Ejk = Eik , Fij Fjk = Fik and Eij Fjk = Fij Ejk = 0, it is easy to see that the defining relations of H are indeed satisfied. As mentioned in Sec. 4.2, the center of H is generated by a Casimir operator C. Its explicit expression reads −2/3 0 0 −2/3 0 () 0 0 0 −2/3 . C= 0 0 1/3 − θ1 θ2 0 0 1/3 − θ1 θ2 () 0
0
1/3 + θ1 θ2
The problem of studying representation theory for H is solved by considering separately all the columns defining H = M3 ⊕ (M2|1 (Λ2 ))0 as a matrix algebra over a ring. We just “read” the following three indecomposable representations from the explicit definition of H. First of all we havef a three-dimensional irreducible rep. resentation 3irr = (c1 , c2 , c3 ) (where ci are complex numbers) coming from the M3 block. Notice that the three columns of the M3 factor give equivalent irreducible representations. Next we have two reducible indecomposable representations (also called “PIM’s” for “Projective Indecomposable Modules”) coming from the columns of (M2|1 (Λ2 ))0 . Notice that the first two columns give equivalent representations — that we call Pe ≡ 6eve — and the last one gives the representation Po ≡ 6odd . Each of these two PIM’s is of dimension 6. The PIM’s are also called “principal modules”. Po and Pe , although indecomposable, are not irreducible: submodules (subrepresentations) are immediately found by requiring stability of the representation f The following should be read as “column vectors”.
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spaces under the left multiplication by elements of H. The lattice of submodules of H is given and discussed in [4]. We recall these results in Appendix D. Here we just want to note (because it will be useful shortly) that 6o and 6e both contain indecomposable (but not irreducible) sub-representations of dimension 3. Indeed, . take λ1 , λ2 ∈ C, set λ = λλ12 ∈ CP 1 , and define θλ = λ1 θ1 + λ2 θ2 . We obtain, for each λ, a representation 3odd spanned by (γθλ , γ 0 θλ , αθ1 θ2 ) and a representation 3eve spanned by (βθ1 θ2 , β 0 θ1 θ2 , δθλ ). Here the coefficients β, β 0 , γ, γ 0 , α, δ are arbitrary complex numbers. 4.6. Decomposition of M = M3 (C) into representations of H Since there is an action of H on M, it is a priori clear that M, as a vector space, can be analysed in terms of representations of H. It turns out, as it is easy to see from (19) or (20), that M ∼ 3irr ⊕ 3eve ⊕ 3odd . More precisely, • 3irr is spanned by x2 , xy, y 2 , • 3eve is spanned by x, y, x2 y 2 , • 3odd is spanned by 1l, x2 y, xy 2 . If we attribute a Z3 -grading 1 to x and y, these three spaces carry gradings respectively equal to 2, 1, 0. This same analysis can be done in a more general N case, the interested reader can find it in [11]. The algebra M, as a vector space, therefore carries a nine-dimensional reducible representation of H and splits into the direct sum of an irreducible representation of dimension 3 and two inequivalent representations of dimension 3 that are reducible but not fully reducible. As described previously, these last two representations 3eve and 3odd appear as subrepresentations of two projective indecomposable modules + 1 where 2 of dimension 6. It is easy to see (cf. diagrams below) that 3eve = 2⊂ +2 denotes a two-dimensional invariant subspace (but 1 is not) whereas 3odd = 1⊂ where 1 denotes a one-dimensional invariant subspace (but 2 is not). One possible identification between H acting from the left on M and 3irr ⊕3eve ⊕ 3odd as elements of the left regular representation of H, is as follows: x2 ∼ E12 xy ∼ E22
x ∼ θ1 θ2 F11 y ∼ θ1 θ2 F21
x2 y ∼ (θ1 − θ2 )F13 xy 2 ∼ (θ1 − θ2 )F23
y 2 ∼ E32
x2 y 2 ∼ (θ1 − θ2 )F31
1l ∼ θ1 θ2 F33 .
In order to establish this identification, it is enough to compare the values of X L [z], for the generators X of H and z ∈ M, with those obtained by left matrix multiplication using the above realisation of H as M3 ⊕ (M2|1 (Λ2 ))0 (of course, we could use right matrix multiplication and compare with the values of X R [z] given before). For instance, in the case of the generator X+ we obtain X+ E12 = 0
X+ θ1 θ2 F11 = 0
X+ (θ1 − θ2 )F13 = −θ1 θ2 F33
X+ E22 = E12 X+ E32 = E22
X+ θ1 θ2 F21 = θ1 θ2 F11 X+ (θ1 − θ2 )F31 = −θ1 θ2 F21
X+ (θ1 − θ2 )F23 = (θ1 − θ2 )F13 X+ θ1 θ2 F33 = 0
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to be compared with L 2 [x ] = 0 X+ L X+ [xy] = q x2
L X+ [x] = 0 L X+ [y] = x
L X+ [1l] = 0 L 2 X+ [x y] = q 2 1l
L 2 [y ] = −q 2 xy X+
L 2 2 X+ [x y ] = −q y
L X+ [xy 2 ] = −x2 y .
One obtains in this way two identical 3 × 9 tables. This can also be seen by comparing the following diagrams with those obtained previously:
-
6 X+
6 X+
-
0 0 θ1 θ2 F11 (θ1 − θ2 )F13 .. .. . . .. .. .. .. .. .. .. .. . .. .. / X− ... .. . .. .. .. .. .. . . .. .. . . .. .. (θ1 − θ2 )F31 and X− ....X+ E22 θ1 θ2 F33 , X− ....X+ .. .. .. . .. . . .. .. .. .. ... . .. . . .. .. .. . .. .. .. .. .. X− .... .. . . . .. . .. .. .. . . . . .. .. .. .. . .. . . . . θ1 θ2 F21 .................. 0 (θ1 − θ2 )F23 0 E32 ................... 0 E12
?
I 6
6@@ @@ @R
?
-
?
-
?
??
5. Reality Structures We now want to obtain a reality structure (a star) and a scalar product on the space of matrices M, with covariance properties under the quantum group transformations. 5.1. Real forms and stars on quantum groups A ∗-Hopf algebra F 0 is an associative algebra that satisfies the following properties (for all elements a, b in F 0 ): • F 0 is a Hopf algebra (a quantum group), with coproduct ∆, antipode S and counit . • F 0 is an involutive algebra, i.e. it has an involution ∗ (a “star” operation). This ¯ ∗ a, where λ is a complex operation is involutive (∗ ∗ a = a), antilinear (∗(λa) = λ number), and anti-multiplicative (∗(ab) = (∗b)(∗a)). • The involution is compatible with the coproduct, in the following sense: if ∆a = a1 ⊗ a2 , then ∆ ∗ a = ∗a1 ⊗ ∗a2 . • The involution is also compatible with the counit: (∗a) = (a). • The following relation with the antipode holds: S ∗ S ∗ a = a. Actually, the last relation is a consequence of the others. It can also be written S ∗ = ∗ S −1 . It may happen that S 2 = id, in which case S ∗ = ∗ S, but it is not generally so (usually the square of the antipode is only a conjugacy).
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If one wishes, using the ∗ on F 0 , one can define a star operation on the tensor product F 0 ⊗ F 0 , by ∗(a ⊗ b) = ∗a ⊗ ∗b. The third condition then reads: ∆∗ = ∗∆, so one can say ∆ is a ∗-homomorphism between F 0 and F 0 ⊗ F 0 (each with its respective star). It can also be said that is a ∗-homomorphism between F 0 and C with the star given by complex conjugation. A star operation as above, making the Hopf-algebra a ∗-Hopf algebra, is also . called a real form for the given algebra. Very often, it is convenient to write a∗ = ∗a. A real element is, by definition, an element such that a∗ = a. Notice that a product of real elements a and b is not real in general ((ab)∗ = b∗ a∗ = ba), but the anticommutator of two real elements, i.e. the Jordan product, is real ((ab + ba)∗ = b∗ a∗ + a∗ b∗ = ab + ba). 5.1.1. Twisted stars It may happen that one finds an involution ∗ on a Hopf algebra for which the third axiom fails in a special way, namely, the case where ∆a = a1 ⊗ a2 but ∆ ∗ a = ∗a2 ⊗ ∗a1 . In this case S ∗ = ∗ S. Such an involution is called a twisted star operation. Remember that, whenever one has a coproduct ∆ on an algebra, it is possible to construct another coproduct ∆op by composing the first one with the . tensorial flip: if ∆a = a1 ⊗ a2 , then ∆op a = a2 ⊗ a1 . If one defines a star operation . on the tensor product (as before) by ∗(a ⊗ b) = ∗a ⊗ ∗b, the property defining a twisted star reads ∆ ∗ = ∗ ∆op . One should be cautious: some authors introduce a different star operation on . the tensor product, namely ∗0 (a⊗b) = ∗b⊗∗a. In that case, a twisted star operation obeys ∆ ∗ = ∗0 ∆ ! ∗-Hopf algebras seem to be “better” than Hopf algebras endowed with twisted stars (their representation theory is more interesting). In any case we shall only be interested in genuine “true” ∗-Hopf algebras and the only reason why we mention these twisted stars here is that we want to rule out some of the possibilities that we shall meet later. 5.2. General results concerning real forms for the classical group SL(2,C) and the quantum group Fun(SLq (2,C)) The following results are known and can be found easily in the literature. Classically, one can define several real forms for SL(2, C), by selecting the elements which are unitary with respect to a given scalar product G, which defines a conjugacy on the group. This last one, in turn, may be thought as coming from a real formon α β the universal enveloping algebra of the Lie algebra of the group. If u = γ δ ∈ SL(2, C), one may impose ut Gu = G. There are three cases: (1) Here G = 10 01 . The corresponding real form is called SU (2). The matrix elements obey α∗ = δ, β ∗ = −γ, γ ∗ = −β and δ ∗ = α.
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1 0 (2) Here G = 0 −1 . The corresponding real form is called SU (1, 1). The matrix elements obey α∗ = δ, β ∗ = γ, γ ∗ = β and δ ∗ = α. (3) Here G = 01 10 . The corresponding real form is called SL(2, R). The matrix elements obey α∗ = α, β ∗ = β, γ ∗ = γ and δ ∗ = δ.
In the classical case, SU (2) and SL(2, R) have the same finite-dimensional representations (they do not have the same infinite-dimensional ones). For this reason there is a one to one correspondence between the Hopf algebras of these two groups (indeed, the Hopf algebra of a group can be considered as the space of polynomials on matrix elements of finite-dimensional representations). The star operation allows one to distinguish between the two groups. Moreover, in the classical case, the two groups SU (1, 1) and SL(2, R) are isomorphic. One could also consider the complex group SL(2, C) itself as a real group (the spin covering of the Lorentz group) but the dimensionality is twice bigger and we are not going to follow this route in the quantum case — at least not in the present paper. In the quantum case, one also has three possibilities for the star operations on Fun(SLq (2, C)) (up to Hopf automorphisms): (1) The real form Fun(SUq (2)). The matrix elements obey a∗ = d, b∗ = −qc, c∗ = −q −1 b and d∗ = a. Moreover, q should be real. (2) The real form Fun(SUq (1, 1)). The matrix elements obey a∗ = d, b∗ = qc, c∗ = q −1 b and d∗ = a. Moreover, q should be real. (3) The real form Fun(SLq (2, R)). The matrix elements obey a∗ = a, b∗ = b, c∗ = b and d∗ = d. Here q can be complex but it should be a phase. In the quantum case, it is already clear from these results that taking q a root of unity is incompatible with the SUq real forms, and that the only possibility if we assume q N = 1 is to choose the star corresponding to Fun(SLq (2, R)). Notice also that, in the quantum case, the two real forms Fun(SUq (1, 1)) and Fun(SLq (2, R)) are different (already the range of values where q can be chosen do not coincide). 5.3. Real forms on F Having only one real form compatible with a complex q on Fun(SLq (2, C)) already tells us that there is at most one real form on its quotient F . We only have to check that the star operation preserves the ideal and coideal defined by a3 = d3 = 1l, b3 = c3 = 0. This is trivial because a∗ = a, b∗ = b, c∗ = c and d∗ = d. Hence, this star operation can be restricted to F . This real form can be considered as a reduced quantum group associated with the real form Fun(SLq (2, R)) of Fun(SLq (2, C)). 5.4. Real structures and star operations on M,H Now we would like to introduce an involution (a star operation) on the comodule algebra M. This involution should be compatible with the coaction of F . That is, we are asking for covariance of the star under the (right, left) F -coaction, (∆R,L z)∗ = ∆R,L (z ∗ ) ,
for any z ∈ M .
(22)
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R. COQUEREAUX, A. O. GARC´ IA and R. TRINCHERO
Here we have also used the notation ∗ for the star on the tensor products, which 0 is defined as (A ⊗ B)∗ = A∗ ⊗ B ∗ . Using, for instance, the left coaction xy0 = x a b ⊗ y in (22), we see immediately that Fun(SUq (2)) corresponds to the choice c d x∗ = y, y ∗ = −q −1 x. Writing xy = qyx, one also recovers the fact that, in this case, q should be real. In the same way, we find that the real form Fun(SUq (1, 1)) corresponds to the choice x∗ = y, y ∗ = q −1 x. Finally, the real form Fun(SLq (2, R)) corresponds to x∗ = x and y ∗ = y. As q 3 = 1, we stick to the Fun(SLq (2, R)) case since it is the only one compatible with a complex q. We therefore define the star on M = M3 (C) by x∗ = x ,
y∗ = y .
(23)
We now want to find a compatible ∗ on the algebra H. Of course, one could proceed by using duality with F , but it is both handy and instructive to proceed otherwise. We let M act on itself by multiplication and we want to promote this star operation to an involution defined for all operators acting on M. In particular, on the operators determined by the action of H, given in Sec. 4.4. To this end, we apply the ∗ operation to the commutation relations between x, y and the differential L L y = x + q 2 y X+ operators X L (or X R ), and read off the results. For instance, X+ L ∗ L ∗ L ∗ 2 L 2 implies y (X+ ) = x + q (X+ ) y and this suggests (X+ ) = −q (X+ ). In this way we obtain: L ∗ L ) = −q 2 X+ (X+ L ∗ L (X− ) = −qX−
(K L )∗ = K L and R ∗ R ) = −qX+ (X+ R ∗ R (X− ) = −q 2 X−
(K R )∗ = K R . Notice the change of q into q −1 when going from left to right. As the space of operators coming from a left action is isomorphic with H itself, the star operation L , K L , namely in H is the same as the one obtained for the operators X± ∗ = −q 2 X+ , X+
∗ X− = −qX− ,
K∗ = K .
(24)
Now, of course, one has to check that this involution satisfies all the axioms given at the beginning of this section. This is rather straightforward. For illustration, let ∗ ) and S ∗ S ∗ X− : us compute ∆(X− ∗ ) = −q∆X− = −q(X− ⊗ K −1 + 1l ⊗ X− ) ∆(X− ∗ ∗ ⊗ (K −1 )∗ + 1l∗ ⊗ X− = X−
= (∆X− )∗
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and ∗ ) S ∗ S ∗ X− = S ∗ S(−qX− ) = S ∗ (−qSX− ) = S(∗(qX− K)) = S(q 2 K ∗ X−
= q 2 S(K(−qX− ) = −S(X− )S(K) = −(−X− KK −1 ) = X− . Adding the cubic relations x3 = 1l, y 3 = 1l in M, and the corresponding ones in the algebra H, does not change anything to the determination of the star structures. This is because the (co)ideals are preserved by the involutions, and thus the quotients can still be done. Anyway, as H is dual to F (or Uq (sl(2)) dual to Fun(SLq (2, C))), we should have dual ∗-structures. This means the relation hh∗ , ui = hh, (Su)∗ i ,
h ∈ H, u ∈ F
(25)
holds. Moreover, the covariance condition for the star, Eq. (22), may also be written dually as a condition for ∗ under the action of H. This can be done pairing the F component of Eq. (22) with some h ∈ H. One should next recall the definition of the left action (18) (the right action if we are using the left coaction in (22)) and the duality (25) that relates both ∗-Hopf stars. One gets finally the constraint on ∗M to be H covariant, h(z ∗ ) = [(Sh)∗ z]∗ ,
h ∈ H, z ∈ M .
(26)
Note that the ∗-operation that we obtain on the algebra M = M3 (C) does not coincide with the usual hermitian conjugacy († -operation) on matrices, at least when x and y are represented with the 3 × 3 matrices given in Sec. 2 (they were such that x† = x−1 = x2 and y † = y −1 = y 2 ). Moreover, it can be seen that it is not possible to find a star operation on the quantum group F that would respect the † involution on M. The situation here is reminiscent of the case where one introduces a “bar”operation (Dirac conjugacy) on Dirac matrices (Clifford algebra for usual spacetime) that is distinct from usual hermitian conjugacy. One remembers that, using this involution, one can build a scalar product on the space of spinors which is spin-invariant (the scalar product built using the usual hermitian conjugacy being positive but not invariant). The situation here is somewhat similar. Remark. Note finally that one could be tempted to choose the involution defined ? ? = ±X− and X− = ±X+ . However, as it is easy to check (look for by K ? = K −1 , X+ instance at the compatibility with the coproduct), this is a twisted star operation. This last star operation is the one needed to interpret the unitary group of H as U (3) × U (2) × U (1), which could be related to the gauge group of the Standard Model [12]. Instead of using the standard Hopf-structure on Uq (sl(2, C)) with this twisted star, a better sounding mathematical alternative would be to look for a non-standard Hopf-structure for this quantum group (i.e. a different coproduct, but preserving the multiplication), such that ? becomes a normal star operation.
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5.4.1. Matrix representation of the ∗-operation on M The star operation for M introduced above (∗x = x, ∗y = y) can be written explicitly in terms of 3 × 3 matrices. Let us define the charge conjugation matrix C: 1 0 0 . C = 0 0 1 . (27) 0 1 0 For every element m of the algebra M we set . ∗ m = (C m C −1 )† ,
(28)
where † denotes the usual hermitian conjugation (the transpose of the complex conjugate matrix). This is clearly a star operation: it is conjugate-linear, involutive (∗(∗m) = m), anti-multiplicative (∗(mn) = (∗n)(∗m)), and unital (∗1l = 1l). Remark also that C = C −1 = C † . 1
0
0
and Let us represent the generators x and y, as before, by x = 0 q−1 0 0 0 q−2 0 1 0 y = 0 0 1 . With (28) it is then easy to check that ∗x = x, ∗y = y. Therefore 1 0 0
the star operation defined explicitly as before in terms of the charge conjugation matrix C implements the abstract star operation introduced above. We already know that this star structure on M gives rise to star structures on the quantum group F and its dual H that are compatible with their Hopf algebra structure. The most general self-adjoint element should satisfy ∗m = m, hence the vector space generators 1l, x, y, x2 , q xy, y 2 , q 2 x2 y, q 2 xy 2 , q x2 y 2 are “real” in the sense that they are self-adjoint for ∗. The same is obviously true for any linear combination of these generators with real coefficients. Using this, it is easy to write the most general 3 × 3 matrix that is self-adjoint for the star ∗. It can be written as: α = αirr + αeve + αodd , with
αirr
a20 = a02 q 2 a11
αeve
a10 2 = q a22 a01
a00
αodd = q a12 q a21
a02
a02 a11 = a20 x2 + a11 qxy + a02 y 2 , q 2 a20
a01 q 2 a10
q a22 a01 = a10 x + a01 y + a22 qx2 y 2 ,
q a11 q a20
a22
q a10
q 2 a21
q 2 a12
a00 a12
a21 = a00 1l + a21 q 2 x2 y + a12 q 2 xy 2 , a00
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where all the ars are arbitrary real numbers (warning: indices r, s of ars refer to the coefficients of xr y s and not to the line or column indices of the corresponding matrix elements). Notice that starting from the defining relations for the algebra M in terms of x and y, and imposing x∗ = x, y ∗ = y, it is not possible to find a matrix representation of this algebra for which these matrices would be († -)hermitian. Indeed such matrices would be in such a case diagonalisable with real eigenvalues, but the condition x3 = y 3 = 1l would then imply x = y = 1l. 5.4.2. Inverses and unitary elements in M To study the inverses of elements of M, it is useful to define 1 (exp(t) + exp(qt) + exp(q 2 t)) 3 d C1 (t) = C0 (t) dt d C2 (t) = C1 (t) . dt C0 (t) =
These functions appear as generalizations of sine and cosine functions when one replaces Z2 by Z3 . They seem to have been invented several times in the literature (an old reference is [13]). Call D(a, b, c) = a3 + b3 + c3 − 3abc T (a, b, c) = a2 − bc . In particular, it is easy to see that D(C0 , C1 , C2 ) = 1 (a generalization of sin2 + cos2 = 1). Writing M = 3irr ⊕ 3eve ⊕ 3odd , it can be shown that (with obvious notations) Inverse (3eve ) ⊂ 3irr ,
Inverse (3irr ) ⊂ 3eve ,
but Inverse (3odd ) ⊂ 3odd .
More precisely, given an arbitrary element aodd ∈ 3odd , we have aodd = a00 1l + a12 q 2 xy 2 + a21 q 2 x2 y a−1 odd =
1 [T (a00 , a12 , a21 ) 1l + T (a21 , a00 , a12 ) q 2 xy 2 D(a00 , a12 , a21 ) + T (a12 , a21 , a00 ) q 2 x2 y] .
Taking aeve ∈ 3eve , aeve = a10 x + a01 y + a22 qx2 y 2 a−1 eve =
1 [T (a10 , a01 , a22 ) x2 + T (a22 , a10 , a01 ) qxy D(a10 , a01 , a22 ) + T (a01 , a22 , a10 ) y 2 ] .
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254
Finally, for a general airr ∈ 3irr , we see that airr = a20 x2 + a11 qxy + a02 y 2 a−1 irr =
1 [T (a20 , a11 , a02 ) x + T (a02 , a20 , a11 ) y D(a20 , a11 , a02 ) + T (a11 , a02 , a20 ) qx2 y 2 ] .
The result Inverse(3odd) ⊂ 3odd should not be surprising, since 3odd is an abelian subalgebra of M (the two other subspaces are not subalgebras); this algebra is actually isomorphic with the algebra C[Z3 ] of the abelian group Z3 . Moreover, using this last isomorphism we can write M = C[Z3 ] ⊕ x C[Z3 ] ⊕ x2 C[Z3 ] = 3odd ⊕ 3eve ⊕ 3irr . Remark that when the coefficients aij are real, the generic elements aodd , aeve and airr written above are also real. Since 3odd is a subalgebra, it is interesting to consider the unitary elements within 3odd (with respect to our star operation). A unitary element (u∗ = u−1 ) can be written u = exp(ia), with a∗ = a (a is “real”). Therefore, working on 3odd , u can be expanded as u = exp[i(a00 1l + a12 q 2 xy 2 + a21 q 2 x2 y)], where aij ∈ R. Since x2 y commutes with xy 2 , the previous exponential can be written as a product of three exponentials, each with its own real parameter: u = u0 u1 u2 , with u0 = exp(ia00 1l) = exp(ia00 ) 1l u1 = exp(ia12 q 2 xy 2 ) = C0 (ia12 ) 1l + C1 (ia12 ) q 2 x2 y + C2 (ia12 ) q 2 xy 2 u2 = exp(ia21 q 2 x2 y) = C0 (ia21 ) 1l + C1 (ia21 ) q 2 x2 y + C2 (ia21 ) q 2 xy 2
5.5. A quantum group invariant scalar product on the space of 3 × 3 matrices The usual scalar product on the space M = M3 (C) of 3 × 3 matrices is m1 , m2 → † (m1 , m2 ) = Tr(m1 m2 ). For every linear operator ` acting on M, we can define the usual adjoint `† ; however, this adjoint does not coincide with the star operation introduced previously. Our aim in this section is to find another scalar product better suited for our purpose. We call z, z 0 two generic elements of M (linear combinations of xr y s ), and h a α β γ K X− ). We know that the first generic element of H (a linear combination of X+ ones act on M like multiplication operators and that the others act on M by twisted derivations or automorphisms. We also know the action of our star operation ∗ on these linear operators. We shall now obtain a unique scalar product by imposing that ∗ coincides with the adjoint associated with this scalar product. That is, we are asking for an inner product on M such that the (left) actions of M and H (each with its respective star) on that vector space may be thought as ∗-representations.
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Hence, for every linear operator ` acting on the nine-dimensional vector space M, we impose: (29) (z, `.z 0 ) = (`∗ .z, z 0) . Moreover, (·, ·) should be hermitian, so that (z 0 , z) = (z, z 0 ) . As usual, (·, ·) is taken as antilinear with respect to the first argument and linear with respect to the second: (αz, βz 0 ) = αβ(z, z 0 ), with α, β ∈ C. Due to the fact that (z, z 0 ) = (1l, z ∗ z 0 ), it is enough to compute (1l, z) for all the z belonging to M. We now use the action of H in (29) and obtain (h∗ .1l, z) = (1l, h.z) . Note that we are now calling h.z the left action that was previously denoted with hL [z]. Taking h = K and z = xr y s we see that (use the results of Sec. 4.4, Table 1): (1l, K.xr y s ) = q (r−s) (1l, xr y s ) but (1l, K.xr y s ) = (K ∗ .1l, xr y s ) = (K.1l, xr y s ) = (1l, xr y s ) , and therefore we conclude that (1l, xr y s ) = 0 unless r = s. We have then to find (1l, 1l), (1l, xy) and (1l, x2 y 2 ). Taking h = X+ we find ∗ .1l, z) = (−q 2 X+ .1l, z) (1l, X+ .z) = (X+
= −q (0, z) = 0 . With z = x2 y we have X+ .x2 y = q 2 1l, so (1l, 1l) = 0. Selecting z = y 2 , we have X+ .y 2 = −q 2 xy and thus (1l, xy) = 0. Hence, the only non vanishing scalar product of the type (1l, z) is (1l, x2 y 2 ). From this quantity we deduce eight other non-zero scalar products (z, z 0 ), where z and z 0 are basis elements xr y s . For instance, (x, xy 2 ) = (1l, x∗ .xy 2 ) = (1l, x2 y 2 ). Hermiticity with respect to ∗ implies that (xy, xy) should be real, indeed (xy, xy) = (xy, xy). We set (xy, xy) = 1 . We should however remember that this normalization condition is arbitrary and could be set to any other positive real number (this comment may become physically relevant when one introduces a “coupling constant” playing a role in some physical model). One can summarise the results in the formula: (xp y t , xr y s ) = q 2 δp+r,2 δt+s,2 [q 2 δs,0 + qδs,1 + δs,2 ] , where δi,j is the Kroneker delta. In the basis {{x2 y, xy 2 , x2 y 2 , y 2 }, {y, x, 1l, x2 }, {xy}}
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R. COQUEREAUX, A. O. GARC´ IA and R. TRINCHERO
the 9 × 9 matrix of scalar products (z, z 0 ) reads 0 U 0 † U 0 0 , 0
0
1
where U is the 4 × 4 diagonal matrix Diag (1, q, q, q). Notice that the trace and determinant of this matrix are equal to 1. The total space splits (Witt’s decomposition) into the sum of two complementary isotropic spaces of dimension 4 and a supplementary space of dimension 1. The real part of this non degenerate hermitian scalar product is a bilinear form of signature (5, 4). More precisely, if one uses the basis: u1 = −y + x2 y
u2 = y + x2 y
v1 = −1 + x y w1 = −x + xy 2
v2 = 1 + x2 y 2 w2 = x + xy 2
t1 = −x2 + y 2 s = xy
t2 = x2 + y 2
2 2
our hermitian form can be written −2 0 0 0 0 0 2 0 0 0 0 2 0 1 q − q 0 0 0 −q + q 2 −1 0 0 0 0 0 1 0 0 0 0 −q + q 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 q − q2 −1 0 0 0
0 0 0 0 0 0 1 −q + q 2 0
0 0 0 0 0 0 q − q2 −1 0
0 0 0 0 0 . 0 0 0 1
Its real part is diagonal in this base (since q¯ = q 2 ), and reads Diag (−2, 2, 1, −1, 1, −1, 1, −1, 1) . The signature (5, 4) can be read immediately from the above. As we saw in Sec. 4.6, the algebra M can be written as the sum of three vector spaces of dimensionality 3 corresponding to the decomposition of the underlying vector space as the sum of three indecomposable representations of the algebra H: M = 3irr ⊕ 3eve ⊕ 3odd , where 3irr = Span (x2 , xy, y 2 ) 3odd = Span (1l, x2 y, xy 2 ) 3eve = Span (x, y, x2 y 2 ) . It is therefore interesting to write the scalar product in a basis adapted to this decomposition.
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In the basis {{x2 , xy, y 2 }, {x, y, x2 y 2 }, {1l, x2 y, xy 2 }}, the scalar product can be written as B 0 0 0 B , G=0 0 B 0 where B is the 3 × 3 block
0 B = 0 q
0 1 0
q2 0. 0
Using the charge conjugation matrix (27) to write the star as in (28), the scalar product on M can be written as 1 1 † Tr(B t C m∗1 m2 ) = Tr(B t m1 C m2 ) , 3 3 where the trace is a usual trace on the matrices. (m1 , m2 ) =
(31)
5.5.1. Quantum group invariance of the scalar product We should now justify why the above scalar product was called a quantum group invariant one. Remember we only said the scalar product was such that the stars coincide with the adjoint-operators, or such that the actions are given by ∗-representations. We refer the reader to [14], where it is shown that the ∗-representation condition on the scalar product, (h.z, w) = (z, h∗ .w) ,
h ∈ H,
(32)
automatically fulfils one of the two alternative invariance conditions that can be imposed on the scalar product, namely ((Sh1 )∗ .z, h2 .w) = (h)(z, w) ,
with ∆h = h1 ⊗ h2 .
(33)
As a side note, we can check the classical limit: let H be the universal enveloping algebra of some Lie algebra, and x any antihermitian generator of the Lie algebra (such that exp(x) is unitary). As the natural Hopf structure on this particular cocommutative H is given by (x) = 0, S(x) = −x, and ∆x = x ⊗ 1 + 1 ⊗ x, (33) reduces in this case to ((∗Sx).z, 1.w) + ((∗S1).z, x.w) = (x.z, w) + (z, x.w) = 0. Note that this last equation is just what is needed to get the scalar product invariance in the usual sense: . (z 0 , w0 ) = (exp(x).z, exp(x).w) = ((1 + x).z, (1 + x).w) + O(x2 ) = (z, w) + (x.z, w) + (z, x.w) + O(x2 ) = (z, w)
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The relations dual to (33) and (32) are those that apply to the coaction of F instead of the action of H. These are (∆R z, ∆R w) = (z, w)1lF ,
(34)
where (∆R z, ∆R w) should be understood as (zi , wj )Ti∗ Tj if ∆R z = zi ⊗ Ti , etc. and (35) (z, ∆R w) = ((1 ⊗ S)∆R z, w) , respectively. We have used here the right-coaction, but the formulas for the left coaction can be trivially deduced from the above ones. It is worth noting that these equations for the coaction of F imply the previous ones for the action of H, and are completely equivalent assuming non-degeneracy of the pairing h·, ·i. Moreover, (34) is a requirement analogous to the condition of classical invariance by a group element action. If we select an orthonormal basis {zi } of M such that ∆R zi = zj ⊗ Tij with ∆Tij = Tkj ⊗ Tik , (34) and (35) reduce to (Tik )∗ Tjk = δij 1lF and STij = (Tji )∗ , respectively. This last equation is exactly what we use in the classical (Lie algebra) case: the unitarity condition for matrices (the antipode is the correct quantum generalization of the group inverse), that warrants the unitarity of the representation (in the sense of invariance of the scalar product). Applying the above discussion to the invariant subspaces of the Hopf-algebra H we obtain on each one the most general invariant scalar products that we list in Appendix E. 6. The Manin Dual M! of M In order to find the Manin dual [8] of a given quadratic algebra A given by the relations Eij xi xj = 0 amongst its generators {xi }, one has first to determine all the matrices E such that Eij Eij = Tr(E t E) = 0. The Manin dual A! of A is defined as a quadratic algebra with relations Eij ξ i ξ j = 0. This is equivalent to say that the quadratic relations in A! are orthogonal to the relations in A, with the pairing (or “scalar product”) hξ i , xj i = δ ij . The quadratic relations defining M (xy = qyx) can be written Eij xi xj = 0, . . 0 1 with x1 = x, x2 = y and Eij = −q 0 . Hence we have, in the present case, −qE E11 . Thus E t E = −qE21 E12 22 Tr(E t E) = −qE21 + E12 ≡ 0
;
E12 = qE21 .
The matrices solution of our problem span therefore a vector space of dimension 3: ! E11 qE21 = E11 E (1) + E21 E (2) + E22 E (3) E= E21 E22
DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE
with E
(1)
=
1 0
0 0
!
,
E
(2)
=
0 1
q 0
E
,
(3)
=
0 0
0 1
259
! .
Our algebra M is not quadratic since we impose the relations x3 = y 3 = 1l. Nevertheless, forgetting momentarily those constraints, we define its Manin dual M! as the algebra generated over the complex numbers by ξ 1 , ξ 2 , satisfying the relations (σ)
Eij ξ i ξ j = 0 ,
σ ∈ {1, 2, 3}
i.e. (ξ 1 )2 = 0 ,
q ξ1ξ2 + ξ2ξ1 = 0 ,
(ξ 2 )2 = 0 .
This means, in particular, that any cubic polynomial in the ξ’s is identically zero. There is no need to introduce additional constraints for ξ 1 , ξ 2 orthogonal to the cubic relations in M. . . We shall write dx = ξ 1 and dy = ξ 2 , so M! is defined by these two generators and the relations dx2 = 0 ,
dy 2 = 0 ,
q dx dy + dy dx = 0 .
(36)
6.1. F coacting on M! Once the coaction of F on M has been defined as in Sec. 3, and once the quantum group product relations have been obtained, it is easy to check that the coaction of F on M! is given by the same formulae as for M itself. Namely, writing 0 a b dx dx = ⊗ dy 0 c d dy
and ˜ dy) ˜ = (dx (dx
dy) ⊗
a c
b d
˜ dy ˜ + dy ˜ dx ˜ = 0, given that the relation ensures that q dx0 dy 0 + dy 0 dx0 = 0 and q dx q dx dy + dy dx = 0 is satisfied. Actually, one can also recover Fun(SLq (2, C)) imposing just the invariance of relations (1) and (36) under the quantum group left-coactions. The coaction can be read from those formulae, for instance ∆R (dx) = dx ⊗ a + dy ⊗ c. 6.2. H acting on M! Since the formulae for the (left or right) coactions are the same, the formulae for the L [x] = y we actions of H on M and M! must also coincide. For instance, using X− L find immediately X− [dx] = dy. This corresponds to an irreducible two-dimensional representation of H. Anyway, we shall return to this problem in the next section, since we are going to introduce a differential algebra ΩW Z (M) built over M and investigate its contents in terms of representations of H.
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7. Covariant Differential Calculus on M 7.1. Differential algebras associated with a given algebra Given an algebra A, there is a universal construction that allows one to build the P∞ so-called algebra of universal differential forms Ω(A) = p=0 Ωp (A) over A. This differential algebra is universal, in the sense that any other differential algebra with Ω0 (A) = A will be a quotient of the universal one. Schematically, the construction . of Ω(A) goes as follows. Let m be the multiplication map: m(a ⊗ b) = ab, for a and b in A and let ker(m) be the kernel of this map (a ⊗ b ∈ ker(m) implies that ab = 0). One builds the algebra of universal forms as the graded vector space: . Ω0 (A) = A . Ω1 (A) = ker(m) ⊂ A ⊗ A . Ωp (A) = Ω1 (A) ⊗A Ω1 (A) ⊗A · · · ⊗A Ω1 (A)
(p + 1 times) .
The above tensor products are taken over the algebra A. Multiplication is defined as follows: . (a0 ⊗ a1 ⊗ · · · ⊗ an )(b0 ⊗ b1 ⊗ · · · ⊗ bn ) = (a0 ⊗ a1 ⊗ · · · ⊗ an b0 ⊗ b1 ⊗ · · · bn ) . . One also writes a0 da1 = a0 ⊗ a1 − a0 a1 ⊗ 1l, etc. Notice that d1l = 0. More details about this construction can be found, for instance, in [15]. In the present situation, A = M. Since Ω1 (M) = ker(m), we find that dim(Ω1 (M)) = dim(M ⊗ M) − dim(M) = 92 − 9 = 72. The dimension of Ω(M) itself is infinite (notice that there are no particular relations between dx and dy so Ωp (M) never vanishes). For practical purposes, it is often not very convenient to work with the algebra of universal forms. First of all, it is very “big”. Next, it does not remember anything of the coaction of F on the algebra M (the 0-forms). Starting from a given algebra, there are several constructions that allow one to build “smaller” differential calculi. As already mentioned, they will all be quotients of the algebra of universal forms by some (differentiable) ideal. One possibility for such a construction was described by [16], another one by [17], and yet another one by [18]. In the present case, however, we shall use something else, namely the differential calculus ΩW Z introduced by Wess and Zumino [6] (see also [5]) in the case of the quantum 2-plane. Its main interest, as we shall see below, is that it is covariant with respect to the action (or coaction) of a quantum group. This differential algebra is, as usual, a graded vector space, and our first step will be to define the differentials dx and dy together with their commutation relations. 7.2. The Wess Zumino complex The differential algebra ΩW Z that we are going to use was introduced historically by [6], in the case of the quantum plane. It is only one of the many possible differential calculi that one can associate with this algebra, but it is the only one (up to trivial
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261
modifications) that is both quadratic and compatible with the coaction of the group Fun(SLq (2, C)). Since it will play an important role in what follows, we shall briefly recall how it can be obtained. First of all ΩW Z = Ω0W Z ⊕ Ω1W Z ⊕ Ω2W Z is a graded vector space. • Forms of grade 0 just coincide with the functions on the quantum plane, i.e. they are polynomials in x and y. • Forms of grade 1 are of the type ars xr y s dx + brs xr y s dy, where dx and dy are those introduced in Sec. 6, devoted to the construction of the Manin dual M! . The Manin dual is the subalgebra {λ00 + λ10 dx + λ01 dy + λ11 dx dy} of ΩW Z , where the coefficients λij belong to the field of scalars. • Forms of grade 2 are of the type crs xr y s dx dy. Next, ΩW Z is an algebra. We need therefore to find the relations between x, y, dx and dy. Moreover, we want ΩW Z to be a differential algebra, so we set d(x) = dx, d(y) = dy and the Leibniz rule (for d) should hold. Also, we set d1l = 0 and d2 = 0. Assuming quadraticity of the algebra, we expand a priori x dx, x dy, y dx and y dy in terms of dx x, dy x, dx y and dy y. This involves sixteen unknown coefficients. Applying d to the above, we get four relations between these coefficients. We are left with twelve independent parameters. Differentiating the relation xy − qyx = 0 and replacing x dx, x dy, y dx and y dy by what was postulated before gives one identity that fixes three of the unknown coefficients (actually one finds four constraints but one of them is not independent of the others). We are left with 12 − 3 = 9 independent parameters. Next, one uses compatibility with the (left or right) coaction of Fun(SLq (2, C)), i.e. one writes x0 = a ⊗ x + b ⊗ y, y 0 = c ⊗ x + d ⊗ y, dx0 = a ⊗ dx + b ⊗ dy, dy 0 = c ⊗ dx + d ⊗ dy and imposes that the relationsg between x0 , y 0 and dx0 , dy 0 are the same as the relations between the corresponding untransformed elements (the unprimed ones). In this way one obtains four identities that fix 8 additional parameters. One is left with just one (= 9 − 8) free parameter. The last parameter is fixed by checking associativity of the cubics, for instance (x dy) dx should be equal to x (dy dx). One finds that this last parameter should be either equal to q or to 1/q. The final result is as follows: xy = qyx x dx = q 2 dx x y dx = q dx y
x dy = q dy x + (q 2 − 1)dx y y dy = q 2 dy y .
dx2 = 0
dy 2 = 0
(37)
dx dy + q 2 dy dx = 0 It may be useful to notice that dy x = (q − 1)y dx + q 2 x dy. g Observe that asking for xy = qyx and dx dy = −q −1 dy dx to be preserved under the coaction provides an alternative way of defining the quantum group. Here we are just refering to relations that mix one of {x, y} with one of {dx, dy}.
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262
7.3. A reduced Wess Zumino complex In the case q 3 = 1, we add the following to the defining relation of the quantum plane: x3 = 1l, y 3 = 1l. As we know, this defines the algebra M (the space of 3 × 3 complex matrices) as a quotient of the quantum plane. Adding the same two cubic relations to the differential algebra ΩW Z defines a differential algebra that we shall denote ΩW Z (M). The fact that it is well defined as a differential algebra is not totally obvious and requires some checking. Technically, one has to verify that we are taking the quotient by a differential ideal. In plain terms, one has to check that d(x3 ) = d1l = 0 and that d(y 3 ) = d1l = 0. This is indeed so: d(x3 ) = d(x2 )x + x2 dx = (dx)x2 + x(dx)x + x2 dx = (1 + q + q 2 ) x2 dx = 0 d(y 3 ) = d(y 2 )y + y 2 dy = (dy)y 2 + y(dy)y + y 2 dy = (1 + q + q 2 ) y 2 dy = 0 . Note that dim(Ω0W Z ) = 9, dim(Ω1W Z ) = 9 + 9 = 18 and dim(Ω2W Z ) = 9. 7.4. The differential of an element of M Let m be an arbitrary 3 × 3 matrix. Let us call mij , i, j ∈ {1, 2, 3} the matrix elements of m. By using elementary matrices, we know how to decompose m on the basis xr y s (see Sec. 2). This way we can write m = mrs xr y s (where, of course, the coefficients mrs do not coincide at all with the mij !). It is then straightforward to compute the one-form dm belonging to the reduced Wess–Zumino differential algebra ΩW Z (M) by using the Leibniz rule together with the commutation relations given previously. If we write dm = (dm)x dx + (dm)y dy , we obtain (here the indices are of type i, j, i.e. refer to the matrix elements of m): (m11 − m33 )(1 − q 2 ) (m12 − m31 )(q 2 − q) (m32 − m13 )(1 − q) 1 (m23 − m12 )(1 − q 2 ) dx dm = (m21 − m13 )(q 2 − q) (m11 − m22 )(1 − q) 3 (m23 − m31 )(1 − q) (m32 − m21 )(1 − q 2 ) (m33 − m22 )(q 2 − q) m12 −m13 q 2 0 0 m23 −m21 q 2 + dy . −m32 q 2
0
m31
Notice that differences of cubic roots of 1 appear in the matrix elements of (dm)x . 7.5. The action of H on ΩWZ (M) 7.5.1. The action of H on Ω0W Z (M) = M We already saw that the nine-dimensional space M can be decomposed into three indecomposable representations of H called 3irr , 3eve and 3odd . The 3irr is irreducible and spanned by x2 , xy, y 2 . The 3eve is reducible indecomposable and spanned by x, y, x2 y 2 ; it contains an invariant two-dimensional subspace spanned by x, y. The 3odd is also reducible indecomposable and spanned by 1l, x2 y, xy 2 ; it contains an invariant one-dimensional subspace spanned by 1l.
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7.5.2. The action of H on Ω1W Z (M) Since Ω1W Z (M) = M dx ⊕ M dy, and as {dx, dy} span the two-dimensional irreducible representation of H, it is a priori clear that we should decompose 3irr ⊗ 2, 3eve ⊗2 and 3odd ⊗2 in indecomposable representations of H. The action of elements X ∈ H on one-forms is obtained, as usual, from the coproduct rule; for instance, L L L [y dy] = X+ [y]1l[dy] + K L [y]X+ [dy] = x dy + q 2 y dx . X+
In this way we obtain the following results (we only give the tables associated with the left action): • The case 3odd ⊗ 2 = 3irr ⊕ 3eve .
1l dx x2 y dx xy 2 dx 1l dy x2 y dy xy 2 dy
KL
L X+
L X−
q dx q 2 x2 y dx xy 2 dx q 2 dy x2 y dy qxy 2 dy
0 q 2 dx −x2 y dx dx q 2 dy + qx2 y dx −x2 y dy + q 2 xy 2 dx
dy −q 2 xy 2 dx + x2 y dy qdx + xy 2 dy 0 −qxy 2 dy dy
This gives 3irr ⊕ 3eve , since, up to multiplicative factors (dashed arrows stand for L L and continuous ones for X+ ) the action of X−
-
0 q x2 y dx − dy .. .. .. .. .. .. .. .. 2 −x y dy + q 2 xy 2 dx .. .. .. .. .. .. .. .. −q dx + q xy 2 dy ....... 0
6
?
6
?
-
dy ............................. 0 .. .. .. .. .. .. .. .. .. .. x2 y dy + q xy 2 dx .. .. . .. .. .. .. .. . .. .. .. .. .. . . .. .. . 0 dx
6 I@ @@ @@
⊕
?
-
-
Note that 3eve contains an irreducible two-dimensional representation spanned by {dx, dy}. • The case 3eve ⊗ 2 = 3irr ⊕ 3odd .
x dx y dx x2 y 2 dx x dy y dy x2 y 2 dy
KL
L X+
L X−
q 2 x dx y dx qx2 y 2 dx x dy qy dy q 2 x2 y 2 dy
0 x dx −qy dx qx dx x dy + q 2 y dx −qy dy + x2 y 2 dx
q 2 y dx + x dy y dy −x dx + x2 y 2 dy qy dy 0 −q 2 x dy
This corresponds to 3irr ⊕ 3odd , since, up to multiplicative factors,
R. COQUEREAUX, A. O. GARC´ IA and R. TRINCHERO
264
-
y dy ............. 0 x2 y 2 dy − x dx .. .. .. .. .. ... .. .. .. . .. .. .. .. .. .. . .. .. . .. .. .. .. . .. .. q x dy + y dx x dy − q y dx ⊕ .. .. .. .. .. .. ... . .. . .. .. .. .. . .. .. .. .. . .. .. .. .. . .. . . x dx 0 y dy − q 2 x2 y 2 dx 0 Remark that 3odd contains an irreducible one-dimensional representation spanned by x dy − q y dx. • The case 3irr ⊗ 2 = 6eve .
6
6
R
?
6
?
? -
x2 dx xy dx y 2 dx x2 dy xy dy y 2 dy
??
KL
L X+
L X−
x2 dx qxy dx q 2 y 2 dx qx2 dy q 2 xy dy y 2 dy
0 qx2 dx −q 2 xy dx q 2 x2 dx qx2 dy + xy dx −q 2 xy dy + qy 2 dx
−qxy dx + x2 dy y 2 dx + xy dy y 2 dy −xy dy q 2 y 2 dy 0
This actually gives the six-dimensional indecomposable representation 6eve (which is projective, cf. Appendix D.). It contains the four-dimensional indecomposable representation 4eve, a family of indecomposables of the type 3λe , and one irreducible of dimension two, spanned by {−q 2 x2 dy + xy dx, −qxy dy + y 2 dx}. 0
6 ....
.. .. ..
−q 2 x2 dy + xy dx .. .. .. .. .. .. .. .. −qxy dy + y 2 dx .. .. .. .. .. .. .. .. 0
6
?
..
x2 dx . .
@I@ @@ @
@I@ ....... . ..@ .. @ . . @
@I@ @@ @ ?
-0
xy dx + x2 dy .. ... .. .. .. .. .. . xy dy + y 2 dx . .. .. . .
6
?
. .. .. . . y 2 dy ...................... 0
-
7.5.3. The action of H on Ω2W Z (M) The differential 2-form dx dy that generates (over M) Ω2W Z (M) is H-invariant, since L L [dx dy] = X− [dx dy] = 0 and K L [dx dy] = dx dy. Hence Ω2W Z (M) = M dx dy X+ has exactly the same decomposition in representations of H as M.
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7.6. Cohomology of d on ΩWZ (M) and H-representations We will now analyse the action of the differential operator d on each level of the differential complex, as given by a basis adapted to the action of the quantum group . H. As usual, we shall call Z p = {ω ∈ Ωp with dω = 0}, the space of p-cocycles . and B p = {ω ∈ Ωp such that ω = dφ, for φ ∈ Ωp−1 }, the space of p-coboundaries. . Of course, B p ⊂ Z p , and we set H p = Z p /B p . 7.6.1. Action of d on Ω0W Z (M) The differential operator d acts on a basis of M as shown in the following table. The range and image vectors are classified according to the H-representation to which they belong. d 3odd
1l x2 y xy 2
→ → →
0 −qxy dx + x2 dy qy 2 dx − q 2 xy dy
2
3eve
x y x2 y 2
→ → →
dx dy −xy 2 dx − q 2 x2 y dy
3eve
3irr
x2 xy y2
→ → →
−q 2 x dx q 2 y dx + x dy −q 2 y dy
3irr
As it should (d was built as a covariant differential operator), d preserves the representations, mapping 3eve 7→ 3eve , 3irr 7→ 3irr , and 3odd 7→ the quotient 3odd/1 = 2. 7.6.2. Action of d on Ω1W Z (M) Dividing the space according to representations of H, d acts as follows: d 3odd ⊗ 2 3irr
3eve
x2 y dx − q 2 dy −x2 y dy + q 2 xy 2 dx −q dx + qxy 2 dy dx dy x2 y dy + qxy 2 dx
3eve ⊗ 2 3irr
3odd
y dy x dy + q 2 y dx x dx x dy − qy dx y dy − q 2 x2 y 2 dx x2 y 2 dy − x dx
∈ B1 ⊂ Z 1 ∈ B1 ⊂ Z 1 ∈ B1 ⊂ Z 1 ∈ B1 ⊂ Z 1 ∈ B1 ⊂ Z 1 ∈ B1 ⊂ Z 1
3irr ⊗ 2
6eve
xy dx + x2 dy xy dy + y 2 dx x2 dx y 2 dy x2 dy − qxy dx qy 2 dx − q 2 xy dy
∈ Z1 ∈ Z1 ∈ B1 ⊂ Z 1 ∈ B1 ⊂ Z 1
→ → → → → →
−qx2 dx dy −xy dx dy q 2 y 2 dx dy 0 0 0
→ → → → → →
0 0 0 −q dx dy −q 2 x2 y dx dy −xy 2 dx dy
→ → → → → →
x dx dy −qy dx dy 0 0 0 0
3irr
3odd
2
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7.6.3. Action of d on Ω2W Z (M) Here d gives trivially 0 on all the elements, because there is no Ω3 subspace in our differential complex; however, one should distinguish between exact and closed forms: 3odd : {dx dy, x2 y dx dy, xy 2 dx dy} ∈ B 2 ⊂ Z 2 3eve : {x dx dy, y dx dy} ∈ B 2 ⊂ Z 2 {x2 y 2 dx dy} ∈ Z 2 3irr : {x2 dx dy, xy dx dy, y 2 dx dy} ∈ B 2 ⊂ Z 2
7.6.4. Cohomology of d From the previous tables, we see that dim(Z 0 ) = 1 ,
dim(B 0 ) = 0 ,
hence dim(H 0 ) = 1 ,
dim(Z 1 ) = 10 , dim(Z 2 ) = 9 ,
dim(B 1 ) = 8 , dim(B 2 ) = 8 ,
hence dim(H 1 ) = 2 , hence dim(H 2 ) = 1 .
Remark that χ = dim(H 0 ) − dim(H 1 ) + dim(H 2 ) = 1 − 2 + 1 = 0. 7.7. Star operations on the differential calculus ΩWZ (M) Given the ∗ operation on the algebra M, we want to extend it to the differential algebra ΩW Z (M). As usual, it has to be involutive, anti-multiplicative for the algebra structure in ΩW Z (M), and complex sesquilinear. Moreover, it should be compatible with the coaction of F . However, there is no reason a priori to impose that ∗ should commute with d. The quantum group covariance condition is, again, just the commutativity of the ∗, ∆R,L diagram, or, algebraically, (∆R,L ω)∗ = ∆R,L (ω ∗ ) .
(38)
In any case, it is enough to determine the action of ∗ on the generators dx and dy, since we already determined the ∗ operation on M (∗x = x, ∗y = y). Taking ∆L dx = a ⊗ dx + b ⊗ dy, we get (∆L dx)∗ = a ⊗ dx∗ + b ⊗ dy ∗ , to be compared with ∆L (dx∗ ). It is trivial that the solution dx∗ = dx is a permissible one. But it is also the only one, up to complex phases. To see this one should expand dx∗ as a generic element of Ω1W Z (M) (we want a grade-preserving ∗) in the previous condition. Solving for the coefficients, and adding the requirement x∗ dx∗ = qdx∗ x∗ (for instance, it is the ∗ of dx x = q 2 x dx), we see that the result is: (39) dx∗ = dx , dy ∗ = dy . Remark. On ΩW Z , d is a graded derivation. First, ∗ is defined on Ω0W Z = M. It is also defined on the d of the generators of M (in our case ∗dx = dx and ∗dy = dy).
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Then ∗ is extended to the whole of the differential algebra ΩW Z by imposing the anti-multiplicative property ∗(ω1 ω2 ) = (∗ω2 )(∗ω1 ). Let φ = ai dxi be an arbitrary element of Ω1W Z . Then d(ai dxi ) = d(ai )dxi , so ∗d(ai dxi ) = (dai dxi )∗ = (dxi )∗ (dai )∗ = dxi (dai )∗ . But, at the same time, d ∗ (ai dxi ) = d((dxi )∗ (ai )∗ ) = d(dxi (ai )∗ ) = −dxi d(a∗i ) . In the case of a (real) manifold, we have an almost trivial star, with ∗ai = ai and ∗dai = dai for ai ∈ Ω0W Z , so d ∗ ai = dai and one findsh ∗dφ = −d ∗ φ, when φ ∈ Ω1W Z . One recovers the fact that ∗ is almost trivial, “almost” since it is still antimultiplicative: ∗(dx dy) = dy dx. In our case the conclusion is the same, indeed, it can be checked that, for all elements ai in M, one has ∗(dai ) = d(∗ai ). Therefore (with φ = ai dxi ∈ Ω1W Z ), d ∗ φ = − ∗ dφ. More generally, d ∗ ω = (−1)p ∗ dω
when
ω ∈ ΩpW Z .
(40)
A general expression for the structure of the most general hermitian one-forms will be given later. Warning. We are not saying that the above involution is the only one that one can define on the Wess–Zumino complex. For instance, one could very well try to extend the involution † (the one for which x† = x2 ) to this differential algebra. Such an extension may be possible, but it would not be compatible with the coaction of F (it would only be compatible with dilations of x and y by numerical scaling factors). Losing the compatibility with the coaction of F is however clearly inacceptable since the main interest of the differential complex of Wess–Zumino rests on the fact that it is compatible with the coaction (indeed, one can construct many other differential algebras over M that are not compatible with the coaction of F !). 7.8. Multiplicative properties of ΩWZ (M) and H-representations We now analyze the behaviour — as representations of H — of the product in ΩW Z (M). Some facts are trivially obtained: Ω2W Z (M) · Ω2W Z (M) = 0 Ω1W Z (M) · Ω2W Z (M) = Ω2W Z (M) · Ω1W Z (M) = 0 . Moreover, as dx dy is H-invariant we know that Ω2W Z (M) is isomorphic to M. Using this, and the fact that dx dy is M-central, we see that the products Ω2 · Ω0 and Ω0 · Ω2 can be understood in terms of products in M. h This shows in particular that usual differential forms on a real manifold (forms that are real in the naive sense) cannot be “real” for the star operation. Indeed, for a one-form, ∗φ = φ implies ∗dφ = −dφ. This peculiar aspect of usual differential forms is a general feature of any differential algebra and can only be circumvented at the expense of introducing graded star operations. We do not use such graded star operations in the present paper.
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Anyway, the multiplication table for representations in M is very easy to write down (for instance using the grading of the generators): M·M
3odd
3eve
3irr
3odd 3eve 3irr
3odd 3eve 3irr
3eve 3irr 3odd
3irr 3odd 3eve
Next we summarize the product relations amongst elements of Ω1W Z (M): Ω1 (M) · Ω1 (M)
3odd ⊗ 2
3eve ⊗ 2
3irr ⊗ 2 = 6eve
3odd 3eve 3irr
3eve 3irr 3odd
3irr 3odd 3eve
3odd ⊗ 2 3eve ⊗ 2 3irr ⊗ 2 = 6eve
Finally, here is the corresponding table for products between elements of M in a given representation and elements of Ω1W Z (M), also with fixed transformation properties: M · Ω1 (M) or Ω1 (M) · M
3odd ⊗ 2
3eve ⊗ 2
3irr ⊗ 2 = 6eve
3odd 3eve 3irr
3odd ⊗ 2 3eve ⊗ 2 3irr ⊗ 2
3eve ⊗ 2 3irr ⊗ 2 3odd ⊗ 2
3irr ⊗ 2 3odd ⊗ 2 3eve ⊗ 2
8. Noncommutative Generalized Connections on M and their Curvature 8.1. Generalized connections in non commutative geometry Let Ω be a differential calculus over a unital associative algebra A, i.e. a graded differential algebra with Ω0 = A. Let A0 be a right module over A. A covariant differential ∇ on A0 is a map A0 ⊗A Ωp 7→ A0 ⊗A Ωp+1 , such that ∇(ψλ) = (∇ψ)λ + (−1)s ψ dλ whenever ψ ∈ A0 ⊗A Ωs and λ ∈ Ωt . ∇ is clearly not linear with respect to the algebra A but it is easy to check that the curvature ∇2 is a linear operator with respect to A. In the particular case where the module A0 is taken as the algebra A itself, any one-form ω (any element of Ω1 ) defines a covariant differential. One sets simply ∇1l = ω, where 1l is the unit of the algebra A. When f ∈ A, one obtains ∇f = ∇1lf = (∇1l)f + 1l df = df + ωf . Moreover, ∇2 f = ∇(df + ωf ) = d2 f + ωdf + (∇ω)f − ωdf = (∇ω)f . The curvature, in that case, is . ρ = ∇ω = ∇1l ω = (∇1l) ω + 1l dω = dω + ω 2 . Take u as an invertible element of A, and act with d on the equation u−1 u = 1l. Using the property d1l = 0 one obtains du−1 = −u−1 du u−1 . Define ω 0 = u−1 ωu + u−1 du
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and compute the new curvature ρ0 = dω 0 + ω 02 . One obtains immediately ρ0 = u−1 (dω + ω 2 )u = u−1 ρu. This shows that the usual formulae hold without having to assume commutativity of the algebra A. Remark that here we take the module A0 (a necessary ingredient in the construction of any gauge theory) as the algebra A itself. More generally, we could have chosen a free module Ap , or even a projective module over A. We shall not consider here this more general situation. Therefore, in some sense, our connections are an analogue of usual “abelian Yang Mills fields”, but the algebra A, contrarily to what happens in conventional gauge field theories, will be non-commutative. 8.2. Connections on M and their curvature We now return to the specific case where A = M is the algebra of functions over the quantum plane at a cubic root of unity. The most general connection is defined by an element φ of Ω1W Z (M). Since we have a quantum group action of H on ΩW Z , it is convenient to decompose φ into representations of this algebra as obtained in Sec. 7.5. We set φ = φ3i + φ03i + φ3e + φ3o + φ6e , where φ3i = ai1 (qx2 y dx − dy) + ai2 (q 2 xy 2 dx − x2 y dy) + ai3 q(dx − xy 2 dy) φ03i = bi1 q 2 y dy + bi2 (y dx + qx dy) + bi3 q 2 x dx φ3e = ae1 dy + ae2 (qxy 2 dx + x2 y dy) + ae3 dx φ3o = bo1 q(x dx − x2 y 2 dy) + bo2 q(qy dx − x dy) + bo3 q(x2 y 2 dx − qy dy) φ6e = c1 (xy dx − q 2 x2 dy) + c2 (y 2 dx − qxy dy) + c3 qx2 dx + c4 qy 2 dy + c5 q 2 (xy dx + x2 dy) + c6 q(xy dy + y 2 dx) . The coefficients ai and bi refer to the three-dimensional irreducible representations, ae and bo to the even and odd three-dimensional indecomposable representations, and c to the six-dimensional indecomposable representation 6eve. The exact expression of the curvature ρ = dφ + φ2 is not very illuminating, but, thanks to the knowledge of the general features of the multiplication in ΩW Z (Sec. 7.8) and the properties of d (Sec. 7.6) we can make the following observations: • φ ∈ 3irr ⊂ 3odd ⊗ 2 ⇒ dφ ∈ 3irr , φ2 ∈ 3odd . Actually, direct calculation shows that φ2 = 0, so ρ = dφ ∈ 3irr . • φ ∈ 3eve ⊂ 3odd ⊗ 2 ⇒ dφ = 0, and ρ = φ2 ∈ 3odd . • φ ∈ 3irr ⊂ 3eve ⊗ 2 ⇒ dφ = 0, and ρ = φ2 ∈ 3irr . • φ ∈ 3odd ⊂ 3eve ⊗ 2 ⇒ dφ ∈ 3odd , φ2 ∈ 3irr . In fact, it can be shown that here also φ2 = 0, so ρ = dφ ∈ 3odd . • φ ∈ 6eve = 3irr ⊗ 2 ⇒ dφ ∈ 2, φ2 ∈ 3eve , and ρ ∈ 3eve.
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Hermitian connections As we know, the only star operation compatible with the quantum group action of H on the differential algebra ΩW Z , when q 3 = 1, is the one obtained in Sec. 7.7 (dx∗ = dx, dy ∗ = dy). Imposing the hermiticity property φ = φ∗ on the connection implies that all the coefficients ai , bi , ae , bo , c should be real. Consider, for instance, the most general hermitian connection with coefficients in the representation 3eve , namely φ = φ3e = ae1 dy + ae3 dx + ae2 (x2 y dy + qxy 2 dx) , with real coefficients ae1 , ae2 , ae3 . In this case, dφ = 0, automatically. The curvature is then equal to φ2 and its expression is quite simple, it reads ρ = (ae1 ae3 − a2e2 )(1 − q) dx dy . Notice that it is a singlet, the one-dimensional representation obtained as a subrepresentation of the 3odd of Ω2W Z . 9. Incorporation of Space-Time 9.1. Algebras of differential forms over C ∞ (M ) ⊗ M Let Λ be the algebra of usual differential forms over a space-time manifold M . (the De Rham complex) and ΩW Z = ΩW Z (M), the differential algebra over the reduced quantum plane introduced in Sec. 7. Remember that Ω0W Z = M, Ω1W Z = Mdx + Mdy, and that Ω2W Z = Mdx dy. Calling Ξ the graded tensor product of these two differential algebras, . Ξ = Λ ⊗ ΩW Z , we see that: • An element of Ξ0 = Λ0 ⊗ Ω0W Z is a 3 × 3 matrix with elements in C ∞ (M ). It can be thought as an M3 (C)-valued scalar field. • A generic element of Ξ1 = (Λ0 ⊗ Ω1W Z ) ⊕ (Λ1 ⊗ Ω0W Z ) is given by a triplet ω = (Aµ , φx , φy ), where Aµ determines a one-form (a vector field) on the manifold M with values in M3 (C) (which can be considered as the Lie algebra of the Lie group GL(3, C)), and where φx and φy are M3 (C)-valued scalar fields. Indeed φx (xµ )dx + φy (xµ )dy ∈ Λ0 ⊗ Ω1W Z . • An arbitrary element of Ξ2 = (Λ0 ⊗ Ω2W Z ) ⊕ (Λ1 ⊗ Ω1W Z ) ⊕ (Λ2 ⊗ Ω0W Z ) consists of — Fµν dxµ dxν ∈ Λ2 ⊗ Ω0W Z , a matrix-valued 2-form on the manifold M , — a matrix-valued scalar field on M , i.e. an element of Λ0 ⊗ Ω2W Z , — two matrix-valued vector fields on M , given by an element of Λ1 ⊗ Ω1W Z . The algebra Ξ is endowed with a differential (of square zero, of course, and . obeying the Leibniz rule) defined by d = d ⊗ id ± id ⊗ d. Here ± is the (differential)
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parity of the first factor of the tensor product upon which d is applied, and the two d’s appearing on the right-hand side are the usual De Rham differential on antisymmetric tensor fields and the differential of the reduced Wess–Zumino complex, respectively. If G is a Lie group acting on the manifold M , it acts also (by pull-back) on the functions on M and, more generally, on the differential algebra Λ. For instance, we may assume that M is Minkowski space and G is the Lorentz group. The Lie algebra of G and its enveloping algebra U also act on Λ, by differential operators. Intuitively, elements of Ξ have an “external” part (i.e. functions on M ) on which U act, and an “internal” part (i.e. elements belonging to M) on which H acts. We saw that H is a Hopf algebra (neither commutative nor cocommutative) whereas U, as it is well known, is a non-commutative but cocommutative Hopf algebra. To conclude, we have an action of the Hopf algebra U ⊗ H on the differential algebra Ξ. 9.2. Generalized gauge fields Since we have a differential algebra Ξ on the associative algebra C ∞ (M ) ⊗ M we can define, as usual, “abelian”-like connections by choosing a module which is equal to the associative algebra itself. A Yang–Mills potential ω is an arbitrary element of Ξ1 and the corresponding curvature, dω + ω 2 , is an element of Ξ2 . As we have said above, ω = (Aµ , φx , φy ) consists of a usual Yang–Mills field Aµ and a pair φx , φy of scalar fields, all of them valued in M3 (C). We have ω = A + Φ, where A = Aµ dxµ and Φ = φx dx + φy dy ∈ Λ0 ⊗ Ω1W Z ⊂ Ξ1 . We can also decompose A = Aα λα , with λα denoting the usual Gell–Mann matrices (together with the unit matrix) and Aα a set of complex valued one-forms on the manifold M . To make the distinction clearer, let us call δ the differential on Ξ, d the differential on Λ and d the differential on ΩW Z (as before). The curvature is then δω + ω 2 . Explicitly, δA = (dAα )λα − Aα dλα and δΦ = (dφx )dx + (dφy )dy + (dφx )dx + (dφy )dy . It is therefore clear that the corresponding curvature will have several pieces: • The Yang–Mills strength F of A, . F = (dAα )λα + A2
∈ Λ2 ⊗ Ω0W Z .
• A kinetic term DΦ for the scalars, consisting of a purely derivative term, a covariant coupling to the gauge field and a mass term for the Yang–Mills field (linear in the Aµ ’s), . DΦ = (dφx )dx + (dφy )dy + AΦ + ΦA − Aα dλα
∈ Λ1 ⊗ Ω1W Z .
• Finally, a self interaction term for the scalars (dφx )dx + (dφy )dy + Φ2
∈ Λ0 ⊗ Ω2W Z .
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We recover the usual ingredients of a Yang–Mills–Higgs system (the mass term for the gauge field, linear in A, is usually obtained from the “AΦ interaction” after shifting Φ by a constant). By choosing an appropriate scalar product on the space Ξ2 , one obtains therefore a quantity that is quadratic in the curvatures (quartic in A and Φ) and could be a candidate for the Lagrangian of a theory of Yang–Mills–Higgs type. However, if we do not make specific choices for the connection (for instance by imposing reality constraints or by selecting one or another representation of H), the results are a bit too general and, in any case, difficult to interpret physically. Regarding this construction, there is also another caveat that will be explained in the next section. 10. Discussion: Quantum Group Symmetry and Noncommuting Fields As mentioned in Sec. 9.2, the building of physical models of gauge type will involve the consideration of one-forms. If we restrict ourselves to the “internal space” part of these one-forms, we have to consider objects of the form X ϕi ωi . Φ= i
Here {ωi } is a basis of some non-trivial indecomposable representation of H (or any other non-cocommutative quantum group) on the space of 1-forms, and ϕi are functions over some space-time manifold. What about the transformation properties of the fields ϕi ? This is a question of central importance, since, ultimately, we will integrate out the internal space (whatever this means), and the only relic of the quantum group action on the theory will be the transformations of the ϕi ’s. There are several possibilities: one of them, as suggested from the results of Sec. 9 is to consider H as a discrete analogue of the Lorentz group (actually, of the enveloping algebra U of its Lie algebra). In such a case, “geometrical quantities”, like Φ should be H-invariant (and U-invariant). This requirement obviously forces the ϕi to transform. Another possibility would be to assume that Φ itself transforms according to some representation of this quantum group (in the same spirit one can study, classically, the invariance of particularly chosen connections under the action of a group acting also on the base manifold). In any case, the ϕi are going to span some non-trivial representation space of H. Usually, the components φi of fields are real (or complex) numbers and are, therefore, commuting quantities. However, this observation leads to the following problem: If the components of the fields commute, so that ϕi ϕj = ϕj ϕi , then we get h.(ϕi ϕj ) = h.(ϕj ϕi ), for any h ∈ H. This would imply (here ∆h = h1 ⊗ h2 ) (h1 .ϕi )(h2 .ϕj ) = (h1 .ϕj )(h2 .ϕi ) = (h2 .ϕi )(h1 .ϕj ) . This equality cannot be true in general unless we have a cocommutative coproduct. Hence we should generally have a noncommutative product for the fields. In our specific case, there is only one abelian H-module algebra, the 3odd one (which is
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nothing else than the group algebra of the group Z3 ). Only fields transforming according to this representation could have an abelian product. However, covariance strongly restricts the allowable scalar products on each of the representation spaces (for instance, the results for the quantum group H are shown in Appendix E, where we get both indefinite and degenerate metrics). This fact is particularly important as one should have a positive definite metric on the physical degrees of freedom. To this end one should disregard the non-physical (gauge) ones, and look for representations such that only positive definite states survive. Thus we see that the selection of the representation space upon which to build the physical model is not easy. The fact of having noncommuting fields has a certain resemblance with the case of supersymmetry. As the superspace algebra is noncommutative, the scalar superfield must have noncommutative component fields in order to match its transformation properties. In this last situation, everything can be casted in terms of Grassmann variables and fields. As a consequence, instead of having — on each space-time point — just the Grassmann algebra over the complex numbers, we see the appearance of an enlarged algebra generated by both the variables and fields. Introducing a q-deformed superspace leads to a more complicated algebraic structure [19]. Therefore, it is reasonable to expect that the addition of a non-trivial quantum group as a symmetry of the space forces an even more constrained algebra. We should point out that the reasoning followed above is very general, and is independent of the details of the fields. That is, the arguing leading to such a conclusion relies only in the existence of a non-cocommutative Hopf algebra acting in a nontrivial way on the fields. For this reason, we did not plan, in this paper, to make specific choices and discuss Lagrangian models. Actually, trying to write down such a definite physical model would involve the making of very particular choices; many of them are possible and we do not know, at the present level of our analysis, which kind of constraint could give rise to interesting physics. Our purpose was rather to investigate the structures involved and work out (or present) the mathematical tools that would allow such physical applications. We hope that the present exposition of our recent work will trigger some interesting ideas in that direction. Appendices Appendix A. Expression of Gell Mann Matrices in Terms of x and y Usual Gell–Mann matrices {λi }i=1...8 (a base for the Lie algebra of hermitian traceless 3 × 3 matrices) can be written in terms of elementary matrices, and therefore also in terms of the generators x and y. Remember that λ1 = E12 + E21 λ4 = E13 + E31
λ2 = i(−E12 + E21 ) λ5 = i(−E13 + E31 )
λ6 = E23 + E32
λ7 = i(−E23 + E32 )
λ3 = E11 − E22 1 λ8 = √ (E11 + E22 − 2E33 ) 3
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hence we obtain λ1 = (y + xy + y 2 + x2 y + qxy 2 + q 2 x2 y 2 )/3 λ2 = −i(y + xy − y 2 + x2 y − qxy 2 − q 2 x2 y 2 )/3 λ3 = ((1 − q)x + (1 − q 2 )x2 )/3 λ4 = (y + q 2 xy + y 2 + qx2 y + xy 2 + x2 y 2 )/3 λ5 = i(y + q 2 xy − y 2 + qx2 y − xy 2 − x2 y 2 )/3 λ6 = (y + qxy + y 2 + q 2 x2 y + q 2 xy 2 + qx2 y 2 )/3 λ7 = −i(y + qxy − y 2 + q 2 x2 y − q 2 xy 2 − qx2 y 2 )/3 √ λ8 = −(q 2 x + qx2 )/ 3 . Appendix B. A Set of Gell Mann Matrices with Coefficients in F Here we replace x and y by ∆L x and ∆L y in the expression of the usual Gell–Mann matrices given in Appendix 1. We obtain in this way a new set of matrices with entries in the quantum group F , denoted by λ0i . We only show the results for λ3 and λ8 since we do not need these matrices explicitly in our work. (1 − q) a + (1 − q 2 ) a2 (1 − q) b + (q − q 2 ) ab (1 − q 2 ) b2 . 1 (1 − q 2 ) b2 (q 2 − 1) a + (q − 1) a2 (1 − q) b + (1 − q) ab λ03 = 3 (1 − q) b + (q 2 − 1) ab (1 − q 2 ) b2 (q − q 2 ) a + (q 2 − q) a2 −q a − q a2 −q 2 b + ab −q b2 . 1 −a − q 2 a2 −q 2 b + q 2 ab . λ08 = √ −q b2 3 −q 2 b + q ab −q b2 −q 2 a − a2 Using the commutation relations for the reduced quantum group F , one can check that the usual commutation relations of SU (3) are satisfied. One can also check that Tr(λi ) = 0 and that Tr(λi λj ) = 2δij (when i 6= j the trace turns out to be proportional to 1 + q + q 2 = 0). Appendix C. A Faithful Representation of F Let ξ1 and ξ2 be two commuting symbols (ξ13 = ξ23 = 0). Let us write 0 1 0 0 b = ξ1 0 0 1 c = ξ2 0 1 0 0 1 . and, given that a = (1 + qbc)d2 ,
1 a = ξ1 ξ2 q 0
(ξ1 ξ2 = ξ2 ξ1 ) whose cube power vanishes 1 0 0
0 q ξ1 ξ2 q 2
0 1 0
1 d = 0 0
ξ1 ξ2 0 . q2
0 q −1 0
0 0
q −2
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It can be checked that this is a faithful representation of the algebra F. It is a representation in terms of 3 × 3 matrices with entries in the ring generated over C by 1, ξ1 , ξ2 . This representation is due to [9]. Appendix D. The Lattice of Submodules of H The theory of complex representations of quantum groups of type Uq (sl(2, C)) at roots of unity has been investigated by a number of people, but it is not yet in a satisfactory state. Here we are interested in representation theory of the finitedimensional quotient H. Notice that these representations can be considered as par3 3 ) = ρ(X− )=0 ticular representations ρ of Uq (sl(2, C)), namely those for which ρ(X+ 3 and ρ(K ) = 1. The structure of the regular representation was studied in [3] and the representation theory — in particular the lattice of submodules — was given in [4]; the general study, for arbitrary N , is to be found in [9]. For the reduced algebra H there are three principal modules (i.e. projective indecomposable modules). One is three-dimensional and irreducible. The two others, 6eve and 6odd are six-dimensional. When H is written in terms of matrices with entries in the Grassmann algebra with two generators, these two representations can be written as . . 6odd = (γθ1 +δθ2 , γ 0 θ1 +δ 0 θ2 , α+βθ1 θ2 ) and 6eve = (α+βθ1 θ2 , α0 +β 0 θ1 θ2 , γθ1 +δθ2 ). Their lattices of submodules are obtained by requiring stability under the H action. They are given in Fig. 1. The notation Po ≡ 6odd and Pe ≡ 6eve refers to the fact
Fig. 1. The lattices of submodules for the principal modules of H.
that, when quotiented out by their respective radicals, those two indecomposable modules give irreducible representations of odd and even dimensions. Remember that there exists a one-to-one correspondence between irreducible representations of the algebra H and the principal modules 3irr , 6eve and 6odd . Irreducible representations are obtained from these principal modules by factorizing
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their radical, which amounts to kill the Grassmann “θ” variables. As seen on the figure, the radical of 6eve is of dimension 4 and the radical of 6odd , of dimension 5. This gives us three irreducible representations, of dimensions 3, 2 = 6 − 4 and 1 = 6 − 5. These are the three irreducible representations corresponding to the quotient H of H by its Jacobson radical: namely H = C ⊕ M2 (C) ⊕ M3 (C). The dashed lines in Fig. 1. refer to the projective covers of the various representations. The explicit definition given for H allows one to compute any tensor product of representations and reduce them. Here we consider only the tensor products of projective indecomposable representations (6odd , 6eve and 3irr ) and the nontrivial irreducible ones (2 and 3irr ). 2×2
≡ 1 + 3irr
6eve × 3irr ≡ 2(6eve) + 2(3irr)
2 × 3irr
≡ 6eve
6odd × 3irr ≡ 2(6eve) + 2(3irr)
3irr × 3irr ≡ 6odd + 3irr 6eve × 2 ≡ 6odd + 2(3irr)
6eve × 6eve ≡ 2(6eve) + 2(6odd) + 4(3irr) 6eve × 6odd ≡ 2(6eve) + 2(6odd) + 4(3irr)
6odd × 2 ≡ 6eve + 2(3irr)
6odd × 6odd ≡ 2(6eve) + 2(6odd) + 4(3irr)
Appendix E. Metrics on Indecomposable Representations of H Using the unique Hopf compatible star operation ∗ on H, we can calculate the most general metric on the vector spaces of each of the indecomposable representations of H. Obviously, as we did in Sec. 5.5, we restrict the inner product to be a quantum group invariant one. On each representation space we use the basis obtained from appropriate restrictions of the natural basis of the column representations of H given in Sec. 4.5. For each indecomposable representation, we first write down the matrices of X+ , X− and K in the selected base, then we give an explicit expression of the most general covariant metric, and finally we calculate its signature. • 3irr
0
X+ = 0 0
1 0 0
0
1 0
0
0
X− = −1 0
0
0 0
0 −1
q2
K=0 0
0 1 0
0
0 q
We get for the metric, up to a real global normalization,
0
G= 0 −q
0 1 0
−q 2
0 . 0
Diagonalizing this matrix we get G ∼ Diag (1, 1, −1), so the signature is σ = (+ + −) .
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• 6odd
0 0 0 X+ = 0 0 0
q 0 0 K= 0 0 0
0 0 0 0 0
0 0
1 0
0 1
0 0
0 0
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0 q
0 0
0 0
0 0
0 0
q2 0
0 q2
0 0
0 0
0 0
0 0
1 0
0 0 1 X− = 0 0 0
0
0 0
0 0
0 0
0 1
0 1
0 0
0 0
0 0
0 0
0 1
0 0
0 0
0 0 0 0 0 0
0 0 0 . 0 0 1
Up to a normalization, the metric should be (β ∈ R)
0
0 0 G= q2 0
0
q
0
0 −q 2
−q 0
0 0
0 0
0 0
0 0
0 0
0 β
0 0 . 0 1
0
0
0
1
0
0
0
0
A change of basis tells us that G ∼ Diag (1, 1, −1, −1, λ+, λ− ), with λ+ > 0, λ− < 0. Thus, in this case, the signature is σ = (+ + + − − −) . • 5odd
0 0 0 0 0
0 0 X+ = 0 0
0 0
1 0
0 1
0 0
0 0
0 0
0
1
0
0
q
0
0
0
q 0
0 q2
0 0
0 0
0 0
q2 0
0 K= 0 0 0
0
0 0 X− = 1 0
0 0
0 0
0 0
0 1
0 0
0 0
0
0
1
0
0 0 . 0 1
0 0 0 0 0
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Up to a real factor, the metric we obtain is (β, γ ∈ R, g ∈ C)
0 0
2 G= −iq γ g¯ 0
0 0
iqγ −q 2 g¯
g iqβ
−qg −iq 2 β
0 0
0 0
0
0
0
0 0 0 . 0 0
Its signature is σ = (+ + − − 0) , because G ∼ Diag (λ+ , −λ+ , λ− , −λ− , 0). • 3odd
0 X+ = 0 λ2
1 0
0 0
0
0
0 X− = 1 0
Up to a real factor,
0
G = −iq 2 0
0 0
0 0
λ1
0
iq
0
q K = 0 0
0 q2 0
0 0 . 1
0
0
0
0
thus σ = (+ − 0) . • 6eve 0 0 0 X+ = 0 1 0
q 0 0 K= 0 0 0
0
1
0
0
0
−1/2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 q
0 0
0 0
0 0
0 0
q2 0
0 q2
0 0
0 0
0 0
0 0
1 0
0 0 0 1 0 X− = −1/2 1 0 0 0 0 0
0 0 0 . 0 0 1
0
0
0
0
0 0
0 0
0 0
−1 0
1 0
0 0
0 0
0 0
0 0 0 0 0
0
−1
0
0
0
DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE
Up to a normalization, the metric should be (β ∈ R) 0 0 iqβ −iq 0 0 0 −iq 0 0 −iq 2 β iq 2 0 0 0 G= iq 2 0 0 0 0 0 0 0 0 0 0 0 0 0 −i
279
0 0 0 . 0 i 0
The signature is σ = (+ + + − − −) , because diagonalizing G 0, λ− < 0. • 4eve 0 0 X+ = 0
we find G ∼ Diag (1, −1, λ+ , −λ+ , λ− , −λ− ) with λ+ >
0
0 0
0 0
1 0
0
0
0
0
q
0 K= 0
0
0
q2 0
0 1
0 . 0
0
0
0
1
0
1
0
0
1 X− = 0 0
0
−1
0 0
0 0
0 0
0
0
0
0
We obtain the metric (α, β ∈ R, g ∈ C) 0 0 0 0 G= 0 0 0
0 0 . g
0 0 α g¯
0
β
The signature is now dependant upon the parameters, because the non-null block is an arbitrary hermitian matrix. • 3eve 0 1 0 0 0 −λ1 q 0 0 0 K = 0 q2 0 . X+ = 0 0 λ2 X− = 1 0 0 0 0 0 0 0 0 0 1 In this case, we find simply
0
G = 0 0
0 0 0
0
0 . 1
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• 2eve X+ =
0
1
0
0
! X− =
Therefore G=
0
iq
−iq 2
0
0
0
1
0
! K=
q
0
0
q2
! .
! ∼ Diag (1, −1) .
Appendix F. The Space of Differential Operators on M We now look at the structure of differential operators on the reduced quantum plane M, i.e. the algebra of 3 × 3 matrices. The operators ∂x and ∂y We already know what the operator d : Ω0W Z = M −→ Ω1W Z is. A priori we can set df = dx ∂x (f ) + dy ∂y (f ), where ∂x (f ), ∂y (f ) ∈ M. This equation defines ∂x and ∂y as (linear) operators on M. We shall see later that they are twisted derivations on the algebra M. This definition implies in particular (take f = x or f = y): ∂x (x) = 1 ∂y (x) = 0 ∂x (y) = 0
∂y (y) = 1 .
The space D of differential operators on M Generally speaking, operators of the type f (x, y)∂x or f (x, y)∂y are called differential operators of order 1. Composition of such operators gives rise to differential operators of order higher than 1. Multiplication by an element of M is considered as a differential operator of degree 0. The space of all these operators is a vector space D = ⊕4i=0 Di , graded by operators of Order 0: xr y s . dim(D0 ) = 9. Order 1: xr y s ∂x , xr y s ∂y . dim(D1 ) = 9 + 9 = 18. Order 2: xr y s ∂x ∂x , xr y s ∂x ∂y , xr y s ∂y ∂y . dim(D2 ) = 9 + 9 + 9 = 27. Order 3: xr y s ∂x ∂x ∂y , xr y s ∂x ∂y ∂y . dim(D3 ) = 9 + 9 = 18. Order 4: xr y s ∂x ∂x ∂y ∂y . dim(D4 ) = 9 All these operators are linearly independent. Moreover, since dim(D) = 9 + 18 + 27 + 18 + 9 = 81 = 92 = dim(End (M)), we can identify D with End (M). Warning. We have here a problem of notations: the reader should distinguish, for instance, ∂x f which is a differential operator from M to M (namely, when acting on something, it multiplies what follows by f , and then acts with ∂x on the result)
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from ∂x (f ) which is the evaluation of ∂x on f , hence an element of M (which can, in turn, be considered as a differential operator of order 0). The presence, or absence, of parenthesis should be enough to make this clear. The twisting automorphisms σ and τ i Since we know how to commute x, y with dx, dy, we can write, for any element f ∈M f dx = dx σxx (f ) + dy σyx (f ) f dy = dx σxy (f ) + dy σyy (f ) where each matrix element of σxx σyx
σxy σyy
!
is an element of End(M) to be determined. In particular, taking f = x and f = y in the above equations leads to σxx (x) = q 2 x
σyx (x) = 0
σxx (y) = qy
σyx (y) = 0
σxy (x) = (q 2 − 1)y
σyy (x) = qx
σxy (y) = 0
σyy (y) = q 2 y .
Moreover, let f and g be elements of M. Using the associativity property (f g)dx = f (gdx), and the same with dy, we find σij (f g) = σik (f ) σkj (g) , with a summation over the repeated index k ∈ {x, y}. Thus the map σ : f ∈ M → σ(f ) ∈ M2 (M) is an algebra homomorphism (σ(f g) = σ(f )σ(g)) from M to the algebra M2 (M) of 2 × 2 matrices with elements in M. In the same way, we could have written for any element f ∈ M (note the transposed index convention) dx f = τxx (f ) dx + τxy (f ) dy dy f = τyx (f ) dx + τyy (f ) dy hence defining another M → M2 (M) homomorphism τ , since τij (f g) = τik (f )τkj (g). ∂x and ∂y are twisted derivations The usual Leibniz rule for d, namely d(f g) = d(f )g + f d(g), implies ! ! ! ! σxx (f ) σxy (f ) ∂x (f )g ∂x (g) ∂x (f g) = + σyx (f ) σyy (f ) ∂y (f g) ∂y (f )g ∂y (g) i Some properties of these automorphisms are discussed in [20].
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or even ∂i (f g) = ∂i (f ) g + σij (f ) ∂j (g) . This shows that ∂x and ∂y are twisted derivations (derivations twisted by an homomorphism). Actually, it is conceptually interesting to notice that one can get rid of the (explicit) twisting by introducing two distinct module structures on Ω1W Z . First of all, elements of Ω1W Z are of the kind dx g1 + dy g2 and can be written as a g1 column vector g2 . We can then consider Ω1W Z ' M2 as an M-bimodule, taking as right module structure the standard one: ! ! g1 . g1 f ·f = , g2 g2 f and making use of the automorphism σ for the left module structure: ! ! ! x y g1 g1 . σx (f ) σx (f ) f· = . σyx (f ) σyy (f ) g2 g2 . ∂ The operator d = ∂xy is a derivation on the algebra M with values in the bimodule Ω1W Z ' M2 . Indeed, by using the two (distinct) left and right module structures just given, the twisted derivation properties of the operators ∂x and ∂y , when expressed in terms of d, read simply d(f g) = d(f ) · g + f · d(g) . Relations in D For calculational purposes, it is useful to know the commutation relations between x, y and ∂x , ∂y , those between ∂x and ∂y , and the relations between the σij . Calculations are straightforward but slightly cumbersome. . . here are the results (see also [6, 8]). Calculating ∂x (xf ), ∂x (yf ), ∂y (xf ), ∂y (yf ) gives the following relations (here we do not suppose that q N = 1): ∂x x = 1 + q 2 x ∂x + (q 2 − 1)y ∂y ∂x y = qy ∂x ∂y x = qx ∂y ∂y y = 1 + q 2 y ∂y . One can check that all these relations are compatible with the defining relations of M. We then calculate ∂x ∂y (x), ∂x ∂y (y), ∂y ∂x (x), ∂y ∂x (y). Compatibility of the results implies the commutation relation ∂y ∂x = q ∂x ∂y .
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3 Moreover, the fact that ∂x,y are central elements of D if q 3 = 1 leads to
∂x3 = ∂y3 = 0 . The commutation relations between the σ’s can be obtained from the values of the σji (x). Taking into account that σyx ≡ 0, the remaining non-trivial relations are σxx σxy = q 2 σxy σxx σxx σyy = σyy σxx σxy σyy = q 2 σyy σxy . Scaling operators It is useful to introduce the operators [21] µx = 1 + (q 2 − 1)(x ∂x + y ∂y ) µy = 1 + (q 2 − 1) y ∂y . Indeed, µx x = q 2 x µx
µy x = x µy
µx y = q 2 y µx
µy y = q 2 y µy .
Observe that µx rescales x and y in the same way. This is not so for µy . Their product satisfies µx µy = µy µx and also µ3x = µ3y = 1l . It is sometimes handy to rewrite the commutation relations between x, y and the first order operators ∂x , ∂y in terms of the µx , µy . For instance, ∂x x = µx + x ∂x ∂y y = µy + y ∂y . The action of H in terms of differential operators The twisted derivations ∂x , ∂y considered previously constitute a q-analogue of the notion of vector fields. Their powers (including zero) build up arbitrary differential operators. Elements of H act also like powers of generalized vector fields (consider L , K L ), but, of course, they are differential for instance the left action generated by X± operators of a special kind: remember that dim(H) = 27 whereas dim(D) = 81. One can say that elements of H act on M as fundamental differential operators since they are associated with the action of a (quantum) group on a (quantum) space.
R. COQUEREAUX, A. O. GARC´ IA and R. TRINCHERO
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L A priori, the generators X± , K L can be written in terms of x, y, ∂x , ∂y . In order to find these expressions, it helps to notice that ∂x and ∂y are respectively of weight −1/2, 1/2 in Z/3Z (see also the discussion at the end of Sec. 4.4). Writing the generators as arbitrary differential operators of a fixed weight, one can determine the coefficients by imposing that Eqs. (14), (22) are satisfied. A rather cumbersome calculation leads to the following unique solution: L = x ∂y + (q − 1) xy ∂y2 X+ L X− = y ∂x + (q − q 2 ) xy ∂x2 + (1 − q) y 2 ∂x ∂y
K L = 1l + (q − 1) x ∂x + (q 2 − 1) y ∂y − 3q x2 ∂x2 + 3(1 − q) x2 y ∂x2 ∂y + 9 x2 y 2 ∂x2 ∂y2 L K− ≡ (K L )2 = 1l + (q 2 − 1) x ∂x + (∗/q − 1) y ∂y − 3 xy ∂x ∂y − 3q y 2 ∂y2 .
An alternative way of writing these, is to make use of the scaling operators, L K− = µx µy
K L = µ2x µ2y L X+ = µ2y x ∂y L X− = q µx y ∂x .
Appendix G. The Universal R Matrix of H We did not discuss the R-matrix aspects of H in the main body of this paper, however it can be seen that H is actually a braided and quasi-triangular finitedimensional Hopf algebra — as it is well known, the quantum enveloping algebra of SL(2) does not posess these properties when q is a root of unity. The universal R matrix can be obtained directly from a general formula given in [22] but one also obtain in a pedestrian way. In any case, the answer is simply the following: R=
1 [1l ⊗ 1l + (1l ⊗ K + K ⊗ 1l) + (1l ⊗ K 2 + K 2 ⊗ 1l) 3q + q 2 (K ⊗ K 2 + K 2 ⊗ K) + qK ⊗ K + qK 2 ⊗ K 2 ] 2 2 ⊗ X+ ]. × [1l ⊗ 1l + (q − q −1 )X− ⊗ X+ + 3qX−
By using the explicit formulae for the generators X+ , X− and K given in Appendix E, one can obtain the expression of R in any representation (including reducible indecomposable ones). One can then check that the defining quadratic equations for the quantum plane and its Manin dual are respectively recovered by writing A xx = 0 (respectively S xx = 0). Here S and A are the two projectors ˆ = (f lip).R matrix entering the spectral decomposition of the numerical R ˆ = q S − q −1 A R S + A = 1l (Tr S = 3, TrA = 1) , R being this time the numerical R-matrix in the fundamental representation.
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Acknowledgement We are greatly indebted to Oleg Ogievetsky for his comments and discussions. References [1] D. V. Gluschenkov and A. V. Lyakhovskaya, “Regular representation of the quantum Heisenberg double (q is a root of unity)”, Zapiski LOMI 215 (1994). [2] L. D¸abrowski, F. Nesti and P. Siniscalco, “A finite quantum symmetry of M (3, C)”, Int. J. Mod. Phys. A13 (1998) 4147. [3] A. Alekseev, D. Gluschenkov and A. Lyakhovskaya, “Regular representation of the quantum group SLq (2) (q is a root of unity)”, St. Petersburg Math. J. 6 (1994) 88. [4] R. Coquereaux, “On the finite dimensional quantum group M3 ⊕ (M2|1 (Λ2 ))0 ”, Lett. Math. Phys. 42 (1997) 309, hep-th/9610114. [5] W. Pusz and S. L. Woronowicz, “Twisted second quantization”, Rep. Math. Phys. 27 (1989), 231. [6] J. Wess and B. Zumino, “Covariant differential calculus on the quantum hyperplane”, Nucl. Phys. B (Proc. Suppl.) 18B (1990) 302. [7] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publ., 1931. [8] Yu. I. Manin, “Quantum groups and non commutative geometry”, preprint Montreal Univ. CRM-1561, 1988. [9] O. Ogievetsky, “Matrix structure of SLq (2) when q is a root of unity”, preprint CPT96/P.3390, to appear. [10] S. L. Woronowicz, “Compact matrix pseudogroups”, Commun. Math. Phys. 111 (1987) 613. [11] R. Coquereaux and G. Schieber, “Action of a finite quantum group on the algebra of N × N matrices”, Proc. Lodz Conf., 1998. [12] A. Connes, “Gravity coupled with matter and the foundation of noncommutative geometry”, Commun. Math. Phys. 182 (1996) 155. [13] P. Humbert, Sur les Fonctions de Bessel de Troisi`eme Ordre, C. R. Acad. Sci. Paris, 1930. [14] R. Coquereaux, A. O. Garc´ıa and R. Trinchero, “Finite dimensional quantum group covariant differential calculus on a complex matrix algebra”, Phys. Lett. B443 (1998) 221. [15] R. Coquereaux, “Noncommutative geometry and theoretical physics”, J. Geom. Phys. 6 (1989) 425. [16] A. Connes, “Noncommutative geometry and reality”, preprint IHES M/95/52. [17] M. Dubois-Violette, K. Kerner and J. Madore, “Classical bosons in a non-commutative geometry”, Class. Quant. Grav. 6 (1989) 1709. [18] R. Coquereaux, R. Haussling and F. Scheck, Int. J. Mod. Phys. A7 (1992) 6555. [19] H. Montani and R. Trinchero, “Quantum mechanics over a q-deformed (0 + 1)dimensional superspace”, Int. J. Mod. Phys. A13 (1998) 4173. [20] Yu. I. Manin, “Notes on quantum groups and quantum De Rahm complexes”, Teor. i Matem. Fiz. Tom 92 N3 (1992) 425. [21] O. Ogievetsky, “Differential operators on quantum spaces for GLq (N ) and SOq (N )”, Lett. Math. Phys. 24 (1992) 245. [22] M. Rosso, “Quantum groups at root of unity and tangle invariants”, in Topological and Geometrical Methods in Field Theory, eds. J. Mickelsson and O. Pekonen, World Scientific, 347 (1992).
ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF GINZBURG LANDAU AND RELATED EQUATIONS∗ YU. N. OVCHINNIKOV Landau Institute, Moscow, Russia
I. M. SIGAL Department of Mathematics University of Toronto Toronto, Canada M5S 3G3 E-mail:
[email protected] Received 24 February 1997 Revised 7 January 1999 In this paper we find asymptotic behaviour of solutions of the Ginzburg–Landau equation at the spatial infinity. Our method uses very little of a particular structure of the non-linearity and can be extended to a larger class of related equations.
1. Introduction In this paper we study asymptotic behaviour as |x| → ∞ of solutions of the Ginzburg–Landau equation −∆ψ + (|ψ|2 − 1)ψ = 0
(1.1)
for ψ: R2 → C subject to the boundary condition |ψ| → 1 as |x| → ∞
(1.2)
(uniformly in x ˆ ≡ x|x|−1 ). This boundary condition allows one to introduce the notion of degree of ψ, deg ψ, as an index (winding number) at ∞ of ψ, considered as a vector field on R2 , i.e. Z 1 d(arg ψ) (1.3) deg ψ = 2π |x|=R for R sufficiently large. We look, to begin with, for weak bounded solutions to (1.1), but elliptic regularity implies that such solutions are, in fact, smooth. Let (r, θ) be the polar coordinates of x (w.r. to the origin). The main result of this paper is the following.
∗ Supported
by NSERC under Grant NA7901. 287
Reviews in Mathematical Physics, Vol. 12, No. 2 (2000) 287–299 c World Scientific Publishing Company
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Theorem 1.1. Assume there is R0 > 0 s.t. the phase ϕ = arg ψ obeys the estimate |∇ϕ| ≤ Cr−1 for r ≥ R0 and for some C > 0. Then there is γ > 1 s.t. for any m ≥ 0 there are k = k(m) ≥ 1, Am 6= 0 and γm s.t. ϕ is of the form ϕ = nθ + Am
sin(kθ − γm ) + O(¯ r−k r−1 ) + O(r0−m r−1 ) , r¯k
(1.4)
where n = deg ψ and r¯ ≡ rr0 ≥ 1 for any r0 ≥ max(R0 , γ). Here the coefficients Am obey |Am | ≤ Ck m with C depending on R0 only and the symbols O(. . .) stand for estimates uniform in r and r0 . If n 6= 0, then by shifting the origin if necessary one can achieve that k ≥ 2. Observe that the statement implies that either maxm k(m) < ∞ or the phase ϕ is of the form ϕ = nθ + O(r−N ) ∀ N . (1.5) √ Indeed, if k(m) → ∞ as m → ∞, then for given r ≥ max(R0 , γ 2 ) take r0 = r to derive from (1.4) that ϕ = nθ + O(r−`(m) ) with `(m) = 12 min(k(m), m). Since `(m) → ∞ as m → ∞, (1.5) follows. In order to keep this paper relatively short we do not investigate carefully the case of ϕ’s of the form nθ+O(r−m )∀ m, which we cannot rule out on general grounds and which, in fact, corresponds to exponential corrections to nθ at ∞. It will be considered R 2π in a separate publication. A simple argument given in Sec. 3 also shows that if 0 (ϕ − nθ)(r, θ)eisθ dθ = 0 for s = 0, 1 and r ≥ r0 for some sufficiently large r0 , then the latter, exceptional, case does not occur. In order to keep the arguments as simple as possible we develop them for Eq. (1.1) only. In fact, these arguments are rather general and applicable to a large class of non-linear equations. For instance they hold for the equations of the form (1.6) −∆ψ + p(|ψ|2 )ψ = 0 , where p(v) is a smooth real function on R+ , has a single zero at v = 1 and satisfies (v − 1)p0 (v) > 0
(1.7)
for v ≥ 0, v 6= 1. The leading terms in asymptotics of |ψ| as |x| → ∞ (see Sec. 2) were derived in [3, 7]. For a mathematical review of Ginzburg–Landau equations, see [2, 5]. 2. Proof of Theorem 1.1 We divide the proof into seven steps. Step 1. We derive equations for f = |ψ| and ϕ = arg ψ. Writing ψ = f eiϕ and substituting this into Eq. (1.1), multiplying the result by e−iϕ and taking the real and imaginary parts of the resulting equation, we obtain the equations for f and ϕ: −∆f + |∇ϕ|2 f + (f 2 − 1)f = 0
(2.0)
ASYMPTOTICS OF SOLUTIONS OF GINZBURG–LANDAU EQUATIONS
289
and −∆ϕ · f − 2∇ϕ · ∇f = 0 . We rewrite these equations as u = |∇ϕ|2 −
∆f f
(2.1)
and
∇ϕ · ∇u , f2 where u = 1 − f 2 . We have in addition the restriction I 1 dϕ = n ∀ R sufficiently large . 2π |x|=R ∆ϕ =
(2.2)
(2.3)
Thus to begin with ϕ is a multivalued function on R2 . We return to this point later. Step 2. Instead of solving Eqs. (2.1)–(2.3) in R2 we solve them in a neighbourhood of infinity, say, in Brc0 := {x ∈ R2 | |x| ≥ r0 }, for some r0 1, with arbitrary boundary conditions on |x| = r0 : ϕ||x|=r0 = α ,
(2.4)
1 where α is a multivalued, if n 6= 0, real function on the unit H circle S , s.t. dα 2 is a one-form on {x ∈ R | |x| = r0 } satisfying the relation |x|=r0 dα = 2πn, and similarly for n. For n 6= 0, the phase ϕ can be considered as a function on the 2 \{0}, satisfying (2.3). neighbourhood |x| ≥ r0 of the universal covering R^
Step 3. We consider an auxiliary linear problem
1 2π
Z
∆ϕ0 = 0 |x| ≥ r0
|x|=R
dϕ0 = n ∀ R ≥ r0
(2.5) (2.6)
ϕ0 ||x|=r0 = α
(2.7)
ϕ0 is bounded .
(2.8)
This problem has a unique solution. Indeed, if it had two solutions, then their difference would satisfy (2.5)–(2.8) with n = 0 and α = 0. The condition n = 0 would imply that this difference is a univalued function on Brc0 and therefore due to the condition α = 0 it would be the identical zero in Brc0 . The solution of (2.5)–(2.8) is given in terms of a Poisson integral: Z 2π 2 1 r − r02 α1 (θ0 )dθ0 , (2.9) ϕ0 (x) = nθ(x) + 2π 0 |x − r0 yˆ|2 where, recall, θ(x) is the polar angle of x, yˆ = (cos θ0 , sin θ0 ) and α1 = α − nθ||x|=r0 . Recall r¯ = rr0 . Expanding integral in (2.9) in r¯−1 , we obtain for r¯ > 1 ϕ0 = nθ +
X s≥k
as
sin(sθ − γs ) , r¯s
(2.10)
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Y. N. OVCHINNIKOV and I. M. SIGAL
where k ≥ 1, as and γs are constants determined by the boundary condition and where we have used the polar coordinates (r, θ) for x. Since ϕ0 (r0 , θ) ∈ L2 (S 1 ), |as | ≤ kϕ0 (r0,· )kL2 (S 1 ) . So the sum in (2.10) converges absolutely for r¯ > 1. In fact, since ϕ(r0 , θ) is smooth in θ, we have that |as | ≤ Cm (1 + s)−m for any m. Step 4. Now we return to problem (2.1)–(2.4). Our task is to show that this problem can be set up as a fixed point one. Writing ϕ = ϕ0 + ϕ1 and observing that ∆(f 2 ) |∇(f 2 )|2 ∆f = − , f 2f 2 4f 4
(2.11)
we obtain the following equations for ϕ1 and u. In |x| ≥ r0 u = |∇ϕ|2 +
|∇u|2 ∆u + , 2(1 − u) 4(1 − u)2
∆ϕ1 =
∇ϕ · ∇u , 1−u
(2.13)
ϕ1 |x|=r0 = 0 , ϕ1 → 0 as
(2.12)
(2.14)
|x| → ∞ ,
(2.15)
ϕ = ϕ0 + ϕ1 ,
(2.16)
where ϕ0 is given by (2.9)–(2.10). Let ∆D be the Dirichlet Laplacian in the domain Brc0 . Moving the second term on the r.h.s. of Eq. (2.12) to the left side, multiplying the resulting equation by 1 − u and then applying the operator g = (1 − 12 ∆D )−1 , we obtain the equation u = v0 + g(1 − u)|∇ϕ|2 + gu2 + g
|∇u|2 , 4(1 − u)
(2.17)
where v0 = O(r−N ) for N sufficiently large and v0 |r=r0 = u|r=r0 (The term v0 accounts for the values of u on the boundary r = r0 , see Sec. 3). Now starting with ϕ(0) = ϕ0 and u(0) = v0 + g|∇ϕ0 |2 , we set up the iterations of Eqs. (2.13) and (2.17) as (m)
∆ϕ1
=
∇ϕ(m−1) · ∇u(m−1) 1 − u(m−1)
(2.18)
and u(m) = v0 + g|∇ϕ(m) |2 − gu(m−1) |∇ϕ(m−1) |2 + g(u(m−1) )2 + g
|∇u(m−1) |2 . 4(1 − u(m−1) )
(2.19)
It is shown in Sec. 3 that for r0 sufficiently large Eqs. (2.13)–(2.17) have a unique solution for ϕ and u twice differentiable and satisfying |∇ϕ| ≤ Cr−1 for r ≥ r0 . Moreover, for any m ≥ 1, this solution satisfies the estimates ϕ = ϕ(m) + O(r0−m r−1 )
(2.20)
ASYMPTOTICS OF SOLUTIONS OF GINZBURG–LANDAU EQUATIONS
291
and u = u(m) + O(r0−m r−3 ) .
(2.21)
This reduces our original problem to the linear problem of finding the iterants ϕ(m) . (m)
Step 5. Now we tackle the linear problem of solving Eqs. (2.18) and (2.19) for ϕ1 (m−1) and u(m) , given ϕ1 and u(m−1) . We introduce a class of functions X sin(sθ − αs` ) (m) 4m−N |a . as` | ≤ C (1 + `) ∀ N An,k = f (r, θ) f = s` N r¯` `−k≥s≥n `−s even
(2.22) (m)
(m)
We write f = 0 mod An,k iff f ∈ An,k . Using (2.10) and using that for any M the operator g = (1 − 12 ∆D )−1 can be written as g=
k M+1 M X 1 1 ∆ + ∆ g 2 2 k=0
and that g maps r−α L∞ (Brc0 ) for any α ≥ 0 into itself, uniformly in r0 , we conclude easily that the solutions to (2.18) and (2.19) are of the form = O(r−M ) mod
1 (m) A r02 1,0
(2.23)
u(m) = O(r−M ) mod
1 (m) A r2 0,0
(2.24)
(m)
ϕ1 and
for any M and for r¯ > 1. Combining Eqs. (2.20) and (2.21) with Eqs. (2.23) and (2.24), we conclude that the unique solution to Eqs. (2.13) and (2.17) is of the form ϕ = nθ + +O(r0−m r−1 ) mod A1,0
(2.25)
u = O(r0−m r−3 ) mod A0,0
(2.26)
(m)
and (m)
for arbitrary m > 0. Thus we obtained: Lemma 2.2. Under the assumptions of Theorem 1.1, the phase ϕ = arg ψ and the modulus u = 1 − |ψ|2 are of the form (2.25) and (2.26), respectively. In particular, ϕ = nθ + O(r−1 )
as r → ∞ .
Step 6. We now shift the origin in such a way that in the new coordinates ϕ can be written as X sin(sθ − αs ) as + ϕ2 + O(r0−m r−1 ) (2.27) ϕ = nθ + r¯s s≥k
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Y. N. OVCHINNIKOV and I. M. SIGAL
(m)
with k ≥ 2, |as | ≤ CN (1 + s)4m−N ∀ N and ϕ2 ∈ A1,2 . The latter is possible since θ(x0 + h) − θ(x0 ) (where x0 is the new variable) is a harmonic function in R2 with bounded values on r0 = r0 , where r0 = |x0 |. Therefore it can be written, in r0 ≥ r0 , as X sin(sθ0 − γs ) θ(x0 + h) = θ(x0 ) + cs , (2.28) r¯0 s s≥0
where θ = θ(x ), the polar angle for x , r¯0 = r0 r0−1 , cs = O(s−N ) for N sufficiently large, γ1 is the angle between h and the x1 -axis and 0
0
0
c1 = −
|h| . r0
(2.29)
Now we pick up h so that γ1 = α11 and c1 + a11 = 0. Hence |h| = r0 a11 and since, by Lemma 2.1, (ϕ − θ)|r=r0 → 0 as r0 → ∞, we see that h = o(r0 ). From now on we omit the primes at the new coordinates. Step 7. Now using that u and ϕ, given by (2.26) and (2.27), must satisfy Eqs. (2.12) and (2.13), we show that their asymptotic behaviour as r → ∞ must be of a particular form. In particular, we show that ϕ is of the form (2.27) but with the sum defining the term ϕ2 (see Eq. (2.22)) restricted further as ` ≥ k + 2. To prove this, let `0 = `0 (m) be the smallest ` in the second sum in (2.27). We assume now that maxm min(k, `0 ) < ∞, otherwise ϕ is of the form (1.5) and the proof is completed. If `0 ≥ k + 1, then our claim follows. Let now `0 ≤ k. In this case X sin(sθ − α ¯s) a ¯s + · · · + O(r0−m r−1 ) , (2.30) ϕ = nθ + r¯`0 1≤s≤`0
¯ s = αs`0 if s ≤ `0 − 2, a ¯ ` 0 = ak , α ¯`0 = αk if k = `0 and a ¯ `0 = 0 where a ¯s = as`0 , α if k > `0 , and the dots stand for the terms in the two sums on the r.h.s. of (2.27) with s > `0 and ` > `0 , respectively. Substituting (2.26) and (2.30) into (2.12), we conclude that u must be of the form X ¯ ¯bs sin(sθ − βs ) + · · · + O(r−m r−3 ) , (2.31) u = u0 + 0 r¯`0 r2 1≤s≤`0 P where u0 = `≥0 b0` r−2 r¯−` and the dots, as before, stand for the terms in the sum on the r.h.s. of (2.26) with s ≥ 1 and ` > `0 . The last two equations imply (1 − u)−1 ∇ϕ · ∇u =
X 1≤s≤`0
c¯s
sin(sθ − γ¯s ) + · · · + O(r0−m r−4 ) . r¯`0 r4
On the other hand, we have X sin(sθ − α ¯s) a ¯s + · · · + O(r0−m r−2 ) . ∆ϕ = r¯`0 r2
(2.32)
(2.33)
1≤s≤`0
m/2`
Remembering now Eq. (2.13) and taking r = r0 0 , we conclude that a ¯s = −(m/`0 ) sin(sθ−α ¯s ) O(r0 ). Thus the terms a ¯s can be absorbed into the remainder r¯`0 O(r0−m1 ) with m1 = m/`0 and so forth. This proves our claim.
ASYMPTOTICS OF SOLUTIONS OF GINZBURG–LANDAU EQUATIONS
The verified claim shows that there is k ≥ 2 s.t. ϕ is of the form (1.4).
293
3. Convergence of Iterations The main result of this section is the following: Proposition 3.1. Let the assumptions of Theorem 1.1 hold and let r0 be sufficiently large. Then Eqs. (2.12)–(2.16) (or (2.1)–(2.4)) have a unique solution for every ϕ0 (i.e. for every α). Moreover this solution obeys, for every m ≥ 1, estimates (2.20)–(2.21), where (ϕ(m) , u(m) ) is the mth iteration defined in (2.18)–(2.19). Proof. To begin with we rewrite the basic Eqs. (2.12)–(2.16) in a form which allows us to use a contraction mapping argument. First multiplying Eq. (2.12) by 1 − u and moving the laplacian term to the l.h.s., we obtain 1 |∇u|2 . (3.1) 1 − ∆ u = (1 − u)|∇ϕ|2 + u2 + 2 4(1 − u) Now let u0 be an arbitrary function satisfying u0 |r=r0 = u|r=r0
and u0 = O(r−N )
(3.2)
for r ≥ r0 and N sufficiently large. Define u1 = u − u0 .
(3.3)
u1 |r=r0 = 0 .
(3.4)
Then Now, using u = v0 + g(1 − 12 ∆)u, Eqs. (2.13) and (3.1) can be rewritten as ∆ϕ1 = a∇ϕ · ∇u ,
(3.5)
w = −u|∇ϕ|2 + u2 + 4−1 a|∇u|2 ,
(3.6)
u = v0 + g|∇ϕ|2 + gw ,
(3.7)
ϕ = ϕ0 + ϕ1 ,
(3.8)
where a = (1 − u)−1 , v0 = u0 − g(1 − 12 ∆)u0 and g = (1 − 12 ∆D )−1 . Here, recall, ∆D is the Dirichlet Laplacian on r ≥ r0 . Observe that due to (3.2), v0 satisfies v0 = O(r−N )
for r ≥ r0
(3.9)
and N sufficiently large. −s ∞ c ∞ L (Brc0 ), where BR = {x ∈ R2 | |x| ≥ R}. On the space L∞ Let L∞ s =r θ ⊕ Lθ with θ > 2 we introduce the map ! a∇ϕ · ∇u G(∆ϕ1 , w) = , (3.10) −u|∇ϕ|2 + u2 + 4−1 a|∇u|2
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Y. N. OVCHINNIKOV and I. M. SIGAL
where, on the r.h.s., ϕ and u are expressed in terms of ∆ϕ1 and w through relations (3.7), (3.8) and (3.11) ϕ1 = (∆D )−1 ∆ϕ1 . In the last relation we used that ϕ1 |r=r0 = 0. This equation is derived by solving the identity ∆ϕ1 = ∆ϕ1 . Now Eqs. (3.5)–(3.6) can be rewritten as α = G(α) ,
(3.12)
. Denote by k · ks the norm in the space L ∞ the operator norm on the spaces 2i=1 L∞ 4 and L4 .
where α =
∆ϕ1 w
L2 i=1
L∞ s , and by k · k
Lemma 3.2. Assume ϕ0 entering the definition of G satisfies |∇s (ϕ0 − nθ)| ≤ L2 ∞ Cr−1−s for s = 0, 1 and r ≥ r0 . Let θ > 2. The function G maps i=1 Lθ into L 2 −λ ∞ r0 i=1 Lµ , provided λ + µ = min(4, θ + 1), λ, µ ≥ 1 and (θ, µ) 6= (3, 4), with the norm uniformly bounded in r0 , and its Fr´echet derivative verifies the estimate k∂α G(α)k ≤ M r0−1 ,
(3.13)
where M is an r0 -independent constant. Proof. The basic ingredient in estimating the norms to follow is the following Lemma 3.3. Let θ ≥ 0. We have θ ∞ ∞ k∇∆−1 D kLθ →Lθ0 ≤ Cr0
0
+1−θ
,
(3.14)
where θ0 = min(θ − 1, 2), provided θ > 2 and θ 6= 3, ≤C, k∇n gkL∞ →L∞ θ θ
(3.15)
for θ ≥ 0 and for n = 0, 1. Proof. To establish Ineq. (3.15) we first scale r0 out by rescaling the problem as x → x0 = rx0 . Thus in proving this estimate we can set r0 = 1. We use an explicit formula for the Green function of ∆D : Z 1 −1 (ln |x − y| − ln(|y| |x − y/|y|2 |))f (y)d2 y (3.16) ∆D f (x) = 2π |y|≥1 (the Kelvin reflection principle). This yields Z 1 x − y/|y|2 x−y f (y)d2 y f (x) = − ∇∆−1 D 2π |y|≥1 |x − y|2 |x − y/|y|2 |2 = I1 + I2 , where I1 is given by the integral over |y| ≥ 1, |y| ≥ |y| ≤ 12 |x|.
(3.17) 1 2 |x|,
while I2 , over |y| ≥ 1,
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ASYMPTOTICS OF SOLUTIONS OF GINZBURG–LANDAU EQUATIONS
We note to begin with that |I1 | and |I2 | are bounded by some constant. Thus it suffices to estimate these integrals, say, for |x| ≥ 4. In this case |x − y/|y|2 | ≥ 12 |x| and therefore ! Z Z 1 d2 y d2 y kf kL∞ |I1 | ≤ C + θ θ |x| |y|≥ 12 |x| |y|θ |y|≥ 12 |x| |x − y||y| ≤ C|x|−θ+1 kf kL∞ , θ provided θ > 2. To estimate I2 we write the expression in parentheses in (3.17) as y 1 1 y/|y|2 − y , + x − − |x − y|2 |y|2 |x − y|2 |x − y/|y|2 |2 which leads to the estimate Z ∞ |I2 | ≤ Ckf kLθ
1≤|y|≤ 12 |x|
( = Ckf kL∞ θ
C d2 y ≤ kf kL∞ θ |x − y|2 |y|θ−1 |x|2
Z 1≤|y|≤ 12 |x|
d2 y |y|θ−1
|x|− min(θ−1,2) , θ 6= 3 |x|−2 ln |x| ,
θ = 3.
This implies the estimate |∇∆−1 D f (x)|
( ≤ Ckf kL∞ θ
|x|− min(θ−1,2) , θ 6= 3 |x|−2 ln |x| ,
θ=3
,
which completes the proof of (3.14). Now we proceed to the proof of (3.15). The cases n = 0 and n = 1 are treated similarly, so we consider here only the case n = 0. By the Green function method we can write Z G(x, y)ψ(y)d2 y , (gψ)(x) = |y|≥r0
where the integral kernel G(x, y) is of the form 2 |y| x − r02 y , G(x, y) = G0 (x − y) − G0 r02 |y|2 where G0 (x − y) is the Green function of −∆ + 1 in R2 . It satisfies the estimates |G0 (x)| ≤ Ce−|x| | ln |x|| . Now we estimate
Z |gψ(x)| ≤ kψk
L∞ θ
|y|≥r0
|G(x, y)| |y|−θ d2 y .
By breaking the integral on the r.h.s. into the integrals over the sets {y ∈ Brc0 | |x − y| ≤ 12 |x|} and {y ∈ Brc0 | |x − y| ≥ 12 |x|}, we conclude that it is bounded by const |x|−θ . Thus we have |gψ(x)| ≤ CkψkL∞ |x|−θ , θ which implies (3.16) for n = 0.
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Y. N. OVCHINNIKOV and I. M. SIGAL
∞ Now we prove bounds on G(α). Let α ∈ L∞ θ ⊕ Lθ , θ > 2 and θ 6= 3. Denote β = ∆ϕ1 . We write ∇ϕ = ∇ϕ0 + ∇∆−1 D β.
The assumptions on ϕ0 and Eq. (3.14) imply that |∇(ϕ − nθ)| ≤ Cr− min(θ−1,2) . This and Eq. (3.7) yield that |u − n2 r−2 | ≤ Cr− min(θ,3) ,
(3.18)
which together with (3.7) and (3.15) implies that |∇u| ≤ Cr− min(θ,3) .
(3.19)
The last three inequalities together with the definition of G(α) and Ineq. (3.15) L2 −λ L2 ∞ imply that G(α) maps 1 L∞ θ , θ > 2, into r0 1 Lµ with λ and µ satisfied in Lemma 3.2. Observe for future references that the previous two inequalities imply that |∇n u| ≤ Cr−2
for n = 0, 1 .
(3.20)
Now we estimate ∂α G(α). Compute the Jacobian matrix of G: ! A B , ∂α G(α) = C D
(3.21)
where A = (a∇u + 2a∇ϕ · ∇g∇ϕ) · ∇∆−1 D ,
(3.22)
B = a∇ϕ∇g + a2 ∇ϕ · ∇ug ,
(3.23)
C = (−2u∇ϕ − 2|∇ϕ|2 g∇ϕ + 4ug∇ϕ + a∇u · ∇g∇ϕ + 2−1 a2 |∇u|2 g∇ϕ) · ∇∆−1 D
(3.24)
and D = −|∇ϕ|2 g + 2ug + 2−1 a∇u · ∇g + 4−1 a2 |∇u|2 g .
(3.25)
Our task now is to estimate norms of A, B, C and D as linear operators on L∞ 4 . First of all, (3.20) with n = 0 implies that ≤ 2, kakL∞ 0 provided r0 is sufficiently large. This and estimates (3.14) and (3.15) imply + k∇ϕk2L∞ ), kAk ≤ Cr0−1 (k∇ukL∞ 2 1 + k∇ϕkL∞ k∇ukL∞ ), kBk ≤ C(k∇ϕkL∞ 0 0 0 k∇ϕkL∞ + k∇ϕk3L∞ kCk ≤ Cr0−1 (kukL∞ 1 1
2/3
+ k∇ukL∞ k∇ϕkL∞ + 1 1
k∇uk2L∞ k∇ϕkL∞ ) 1 1
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ASYMPTOTICS OF SOLUTIONS OF GINZBURG–LANDAU EQUATIONS
and kDk ≤ C(k∇ϕk2L∞ + kukL∞ + k∇ukL∞ + k∇uk2L∞ ). 0 0 0 0 Therefore, due to estimates |∇ϕ| ≤ Cr−1 for r ≥ r0 and (3.20) for n = 0, 1, we conclude that kF k ≤ Cr0−1 , where F stands for either of the operators A, B, C or D. This implies (3.13).
Now we prove a priori estimates on ∆ϕ1 and w. First we notice that it is shown in [3] (see also [6]) that |ψ| ≤ 1. Next we show that |∇n v| ≤ Cr−2
for r ≥ r0
(3.26)
and n = 0, 1, 2, where v = 1 − |ψ|. To this end we rewrite Eq. (2.0) as (−∆ + f (f + 1))v = |∇ϕ|2 f .
(3.27)
Using that ∆r−2 = 4r−4 , we rewrite this equation as (−∆ + f (f + 1))(C1 r−2 − v) = b , where b = (f +1)f C1 r−2 +4C1 r−4 −|∇ϕ|2 f . Since |∇ϕ|2 ≤ Cr−2 , we conclude that b ≥ 0 for r ≥ r0 , provided C ≤ C1 . Furthermore, choose C1 s.t. |v| |r=r0 ≤ C1 r0−2 . Then the maximum principle (see e.g. [4, 8]) implies that |v| ≤ C1 r−2 for r ≥ r0 . This and Eq. (2.0) implies that |∆f | ≤ Cr−2
for r ≥ r0 .
Interpolating between the last two inequalities (see Lemma 3.1 of [1]), one arrives at |∇f | ≤ Cr−2 for r ≥ r0 . Now using (2.11), we rewrite Ineq. (3.26) as |∇n u| ≤ Cr−2
for r ≥ r0 and n = 0, 1, 2 .
(3.28)
This and Eq. (3.5) yield that |∆ϕ1 | ≤ Cr−3 , while estimate (3.28) with Eq. (3.6) and the condition |∇ϕ| ≤ Cr−1 implies that |w| ≤ Cr−4 . Thus we know that to L2 begin with α ∈ i=1 L∞ 3 . Then Lemma 3.2 implies the existence and uniqueness part of Proposition 3.1. The estimate of iterations is done in a standard way. First we notice that estimate (3.13) implies, in particular, the Lipschitz estimate ˜ − αk4 . kG(˜ α) − G(α)k4 ≤ M r0−1 kα Now we iterate Eq. (3.12) starting with α(0) =
(3.29)
0 0
:
α(m) = G(α(m−1) ) .
(3.30)
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Y. N. OVCHINNIKOV and I. M. SIGAL
Inequality (3.29) implies kα − α(m) k4 = kG(α) − G(α(m−1) )k4 ≤ M r0−1 kα − α(m−1) k4 ≤ · · · ≤ (M r0−1 )m kα − α(0) k4 .
(3.31)
Hence and
|w(x) − w(m) (x)| ≤ Cr0−m r−4
(3.32)
|∆ϕ1 (x) − ∆ϕ1 (x)| ≤ Cr0−m r−4
(3.33)
(m)
for r ≥ r0 . To obtain estimates for ∇ϕ1 we observe that ∇∆−1 D is a bounded operator from ∞ L4 to L∞ 2 and therefore |∇ϕ1 (x) − ∇ϕ1 (x)| ≤ Cr0−m r−2 (m)
(3.34) 1
for r ≥ r0 . A similar estimate and for the same reasons holds also for (−∆) 2 ϕ1 . Next, the estimate (m) (3.35) |ϕ1 (x) − ϕ1 (x)| ≤ Cr0−m r−1 operator (∆D )−1 analogous for r ≥ r0 follows from (3.33) and an estimate of the R 1 to (3.14). Here we use that the leading term, 2π |y|≥1 ln |y|f (x)dy, where f =
1 ∆ϕ1 − ∆ϕ1 , in the expansion of ϕ1 − ϕ1 = ∆−1 D f in |x| , is independent of x and can be absorbed into an overall constant phase factor. Furthermore using the equations ∇k+1 ϕ = ∇k+1 ϕ0 + ∆−1 ∇2 (∇k−1 α) , (m)
(m)
2 2 and standard estimates of the singular integrals ∆−1 D ∇ and g∇ , we conclude that
Z |∇ ∆(ϕ1 − s
p1
(m) ϕ1 )|p rpθ d2 x
≤ Cr0−m
r≥r0
for any 5 > θ ≥ 0 satisfying (5 − θ)p > 1, for any ∞ > p ≥ 1, and for any s ≥ 1. With a little more work (which we omit here) one can improve this estimate for s ≥ 2 to any 0 ≤ θ < 6 satisfying (6 − θ)p > 1. This implies in particular the estimate |∇s ∆(ϕ1 − ϕ(m) )| ≤ Cr0−m r−3−s for s = 1, 2, which together with (3.33)–(3.35) yields (2.20). Equation (2.21) follows from (2.19), (3.7), (3.30) and (3.32). Acknowledgments The second author is grateful to H. Brezis for useful discussions and for pointing out the results and references [2, 3, 7]. The authors are grateful to the referee for useful remarks.
ASYMPTOTICS OF SOLUTIONS OF GINZBURG–LANDAU EQUATIONS
299
References [1] F. Bethuel, H. Brezis and F. H´ elein, “Asymptotics for the minimization of a Ginzburg– Landau functional”, Calc. Variat. and PDE 1 (1993) 123–148. [2] F. Bethuel, H. Brezis and F. H´elein, Ginzburg–Landau Vortices, Birkh¨ auser, Basel, 1994. [3] H. Brezis, F. Merle and T. Rivi`ere, “Quantization effects for −∆u = u(1 − |u|2 ) in R2 ”, Arch. Rational Mech. Anal. 126 (1994) 35–58. [4] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer-Verlag, New York, 1983. [5] A. Jaffe and C. Taubes, Vortices and Monopoles, Birkh¨ auser, 1980. [6] Yu. N. Ovchinnikov and I. M. Sigal, “Ginzburg–Landau Equation I. Static vortices”, in PDE’s and their Applications (eds. L. Seco et al.), CRM Proceedings and Research Notes 12 (1997) 199–220. [7] I. Shafrir, “Remarks on solutions of −∆u = (1 − |u|2 )u in R2 ”, C.R. Acad. Sci. Paris, t.318, S´erie I (1994) 327–331. [8] M. Struwe, Variational Methods, Springer-Verlag, 1990.
MODULAR CONSTRUCTIONS OF QUANTUM FIELD THEORIES WITH INTERACTIONS Dedicated to the memory of Harry Lehmann B. SCHROER and H.-W. WIESBROCK Institut f¨ ur Theoretische Physik FU-Berlin, Arnimallee 14 Berlin, Germany E-mail:
[email protected] E-mail:
[email protected] Received 10 December 1998 We extend the previously introduced constructive modular method to nonperturbative QFT. In particular the relevance of the concept of “quantum localization” (via intersection of algebras) versus classical locality (via support properties of test functions) is explained in detail, the wedge algebras are constructed rigorously and the formal aspects of double cone algebras for d = 1+1 factorizing theories are determined. The well-known on-shell crossing symmetry of the S-Matrix and of formfactors (cyclicity relation) in such theories is intimately related to the KMS properties of new quantum-local PFG (one-particle polarization-free) generators of these wedge algebras. These generators are “on-shell” and their Fourier transforms turn out to fulfill the Zamolodchikov–Faddeev algebra. As the wedge algebras contain the crossing symmetry information, the double cone algebras reveal the particle content of fields. Modular theory associates with this double cone algebra two very useful chiral conformal quantum field theories which are the algebraic versions of the light ray algebras.
1. Introductory Remarks It is known that the local field algebras of interaction-free Fermions and Bosons can be directly obtained [1, 3] from the modular wedge-localized one-particle spaces, as a kind of inverse of the Bisognano–Wichmann [6] construction, without any reference to quantization. In fact the present paper should be viewed as one in a series of paper [4] which attempt a new access to QFT in the presence of interactions by avoiding quantization as well as references to (semi)classical approaches and use instead modular methods. The common new findings in all these papers is an enlargement of the symmetry concept which previously we have described in terms of a new kind of “hidden” (i.e. a non-Lagrangian non-Noetherian) symmetry structure [4]. This new concept also turns out to be useful in the present context (Sec. 4) which aims at an intrinsic modular understanding of nonperturbative interactions. In the following we will continue the investigations, which were started by one of the authors (B.S.), on the possibility of incorporating interacting theories into a constructive modular approach [1, 2]. One important step in this approach was the recognition of the close relation between the thermal Hawking–Unruh aspects of the wedge horizon on the one hand and the subject of crossing symmetry in particle physics on the other hand. Analogies between the cyclic properties of thermal KMS states and onshell crossing symmetry were also noticed in [5]. We have however 301 Reviews in Mathematical Physics, Vol. 12, No. 2 (2000) 301–326 c World Scientific Publishing Company
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some problems with the arguments proposed in that paper; in particular we do not believe that it is possible to give a proof of this relation without the construction of the “polarization free generators” (PFG’s) of wedge algebras which one of the present authors (B.S.) introduced previously in the context of factorizable models [2]. Since their use constitutes our new main tool in the present nonperturbative constructions, we will take some care to explain again their principal features in this paper. We assume some familiarity with the use of Tomita’s modular theory in QFT as it can be found in the cited work. Our first testing ground will therefore be the d = 1 + 1 factorizing models which are the simplest and presently the only interacting properly renormalizable field theoriesa which have a well-defined particle (and hence a scattering) content and allow for an explicit analytic understanding. We will show that the use of PFG’s indeed allows to understand that somewhat elusive correspondence between the offshell thermal KMS and the on-shell crossing symmetry relation. Here the thermal KMS behaviour is caused by wedge localization (the LQP version of the Hawking– Unruh thermal aspect of localization behind a classical Killing horizon), whereas the crossing symmetry is that somewhat mysterious generalization of the TCP property which allows to individually convert incoming particles into outgoing antiparticles at (unlike usual symmetries) on-shell analytically continued values of the momenta, and for which nonperturbative proofs of sufficient generality do not exist. The thermal behaviour via localization occurs when matter is localized behind an event horizon which may be either a classical curves spacetime Killing horizon leading to the Hawking/Unruh effect or simply the boundary of the causal completion of a localization region in QFT. In the first case the thermal aspects have a classical manifestation on the curved spacetime side (which led to its original discovery by Bekenstein), whereas the second mechanism is purely quantum and, apart from the Rindler–Unruh situation with its manifest geometric interpretation, needs abstract, modular theory for its form. Our new conceptual tools also reveal that the idea to interpret crossing symmetry as a kind of on-shell substitute for (off-shell) Einstein causality, favoured at the time of invention of the Veneziano dual model, is essentially correct. Although with these remarks we have come close to the dual model cradle of (the old version of) string theory, we will resist the temptation to draw parallels to the latter [24] and limit ourselves strictly to local quantum physics (LQP). The mentioned PFG’s, which constitute our main new (modular) tool, permit an explicit construction in the case of so-called factorizing models. The logic of the modular construction follows very closely the recipes of the bootstrap-formfactorprogram [13, 14, 10] and in fact provides a derivation of these recipes from the general spacetime-independent concepts of QFT. a In the quantization approach to interactions, it is the operator short distance dimension of the interaction density in Fock space which divides the models into superrenormalizable (for which the mathematical existence can be controlled), and the more interesting properly renormalizable ones, for which it was not possible to control their nonperturbative existence by the standard quantization methods (canonical or functional integration). The terminology refers to the short distance property of the distinguished Lagrangian field coordinate. The present constructive modular approach does not depend in any direct way on the short distance behaviour of particular field generators of local nets.
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For our present purpose it is very helpful to first picture the special position of these factorizable theories within the set of general d = 1 + 1 Wightman field theories as representing massive superselection classes in the following way. Imagine that we remove all particles by spatially separating their (centre of) wave packets “on shell” i.e. we take an extreme cluster limit in the scattering matrix. Using as a hindsight the analytic singularity structure of the multiparticle creation (annihilation) threshold, as well as the relation between momentum space singularities and fall-off behaviour in its Fourier transform, we are led to believe that these inelastic multiparticle matrix elements in this extreme cluster limit possess decreasing properties relative to the elastic ones. Furthermore, the direct multiparticle elastic processes in this limit is expected to become asymptotically negligible as compared with the two particle elastic contribution [2]. In this way we obtain factorizing S-matrices which fulfill the Yang–Baxter consistency equations without reference to quantization and infinitely many conserved Noether currents; in other words the Y–B structure is not imposed but rather derived from the general principles of QFT in the extreme cluster limit. Although, as a result of lack of detailed knowledge about the analytic structure of admissable S-matrices of QFT, this picture is presently mathematically not rigorous, it is very useful for the motivation behind our investigation and for an intrinsic local quantum physical understanding of the position of factorizable models within general QFT which removes some of their “freak appearance”. In fact it is one of the folklore theorems (but in this case rigorous under suitable analytic assumptions) that for d = 1 + 3 theories the limit S-matrix is trivial, i.e. S = 1. The corresponding field theories are then expected to be free fields which still carry the superselection charges (internal symmetries) of the original theory. However in d = 1 + 1 the simplification does not go all the way to free theories, since the cluster property is not capable of separating the elastic two-particle Tmatrix from the identity in the decomposition S = 1 + T (intuitively because two particles always meet in x-space or mathematically because the two particle energymomentum conservation delta function in d = 1 + 1 happens to be identical to the two-particle inner product delta function). All the multiparticle elastic scattering then takes place through two-particle scattering and the Yang–Baxter relation follows as a physical consistency requirement (and is not a mathematical imposition) and allows together with the unitarity and crossing symmetry (closely linked to the TCP symmetry) to separate and solve the construction of the S-matrix from the field theory. This is a peculiar feature of the so constructed simple but nontrivial “factorizing representative” of d = 1 + 1; in any other field theory including nonfactorizing d = 1 + 1 models, there is no possibility known to separate a pure S-matrix bootstrap from the off-shell field theory; rather the S-matrix is expected to be determined together with the fields, or in our algebraic setting, together with the net of local algebras. The principal purpose of this paper, which generalizes the previous findings of one of the present authors [2] concerning the modular understanding of crossing symmetry, is a more detailed explanation of the useful constructive role of so-called “PFG” (one-particle polarization-free wedge generators) i.e. on-shell but never-
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theless wedge localized operators F (f ) (we explain our notation in Sec. 2), which create one-particle states without those additional pair contributionsb known from the famous vacuum polarization mechanism in QFT. In fact they constitute a system of generators for the wedge algebra. These “fields”, unlike those pointlike local fields associated with Lagrangians or with the Wightman framework, have a quantum (but no classical) localization, i.e. they remain wedge localized even if smeared with test functions which have sharper localized supports inside the wedge region. With quantum localization, this sharper localization has to be obtained from the intersection of wedge algebras; the intersected algebras in turn have their own new generators. The PFG’s are more noncommutative than the pointlike localized fields and they are in fact not derivable from any known quantization approach, although in the next section we will try to make them appear at least plausible within the standard setting. On the other hand they are essential for our nonperturbative construction which uses (quantum) modular wedge localization; although they are themselves nonlocal, they have (unlike cut-off or regularized fields in the standard setting) a very precise conceptual and mathematical relation to local fields. In some intuitive sense one may say that the semilocal intermediary PFG’s have a milder short distance behaviour than pointlike local fields, whereas the use of ad hoc cutoff or regularized fields in the quantization approach (usually formally presented in the euclidean functional integral setting) is a mathematically and physically ill-defined concept, the PFG’s are valuable building blocks in agreement with physical principles, since they live in the same Hilbert space and create the same wedge algebra as local fields. Therefore the comparison with cutoffs in the standard approach requires caution. Their use in some sense detaches the problem of existence of the theory from the short distance behaviour of pointlike field coordinates. The borderline between renormalizable/nonrenormalizable theories based on the standard perturbative local coupling of free fields loses some of its significance, and instead the existence of local theories becomes linked with the nontriviality of intersections of certain algebras, as will be seen below. For the aforementioned factorizing models it turns out that although the PFG’s cannot be obtained by Lagrangian quantization methods, their mass shell Fourier transforms are not totally new objects in QFT since they obey the Zamolodchikov– Faddeev algebra [2]. In fact this algebraic structure which was introduced on purely formal grounds to facilitate calculations in the bootstrap formfactor approach for the first time acquires a physical spacetime interpretation via these PFG’s. The construction of PFG’s can be made somewhat suggestive within the standard LSZ setting of QFT using the retarded Yang–Feldman formalism. The PFG’s viewed in this way are on-shell analogues of the off-shell interacting fields, but unlike the incoming free fields they carry information about the interaction. b We have called such operators on previous occasions “vacuum-polarization free” and used the abbreviation FWG in [2], but a better terminology used in the following is to call them “oneparticle-polarization free generators” (PFG). Wheras the notion of vacuum-polarization which originated with conserved currents in free field theory in the work of Heisenberg and Weisskopf, the present concept of “one-particle polarization”, i.e. the impossibility of creating pure one particle states (without pair admixtures) from local operators is totally characteristic for interacting theories (and may be used as a good intrinsic definition of the latter).
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Whereas the Z-P algebraic structure is essential for the present analytic control of the bootstrap-formfactor-approach, the concept of quantum wedge localization points to a vastly more general algebraic structure of the commutator of the PFG wedge generators F together with their modular J-reflected expressions JF J. Namely the modular theory expresses the wedge localization of the F 0 s in terms of the KMS property of the F -correlation functions. Since their Fourier transforms are expressible in terms of S-matrix elements, it turns out that the crossing symmetry of the latter insures the KMS property of the F -correlations. In the case of factorizing models, the KMS property of the F 0 s and the crossing symmetry of the scattering matrix are in fact equivalent properties, and we believe that this is true in general. This is the content of Sec. 3. Different from the usual locality of local fields or nets of algebras which exists independently of the particular representation, the new quantum locality depends crucially on the presence of the vacuum representation and one-particle states (more precisely on the standard assumptions which led to the validity of rigorous timedependent scattering theory). It is only after the modular construction of a local net has been accomplished, that one may change the vacuum representation with another locally normal one. After the role of the PFG’s for quantum wedge localization has been presented, we describe the formal aspects of their intersections which leads to double cone algebras in Sec. 4. Via suitably defined relative commutants we construct two light ray theories whose conformal rotation is a “hidden” symmetry. The problem of nontriviality as well as the explicit construction of the double cone algebra is significantly facilitated in terms of these “smaller” chiral light ray algebras. In the next section we will continue our more physical intuitive presentation of motivations and concepts. The mathematical physics style of presentation starts only in the third section. 2. An Intuitive Argument for Existence of PFG’s In this section we will present some more motivating mathematically nonrigorous arguments on why we expect one-particle polarization free wedge generators (PFG’s) to exist, although we are presently only able to rigorously show this for d = 1 + 1 factorizing systems. Let us briefly recollect how the standard perturbative approach deals with interactions. The formulation which is most appropriate for the present purpose is that of Stueckelberg, Bogoliubov and Shirkov [7] combined with the (Wigner-)particlefree field relation as can be found in Weinberg [8]; here we do not need the more sophisticated version of Epstein and Glaser [9]. The first step goes from Wigner particles to Fockspace and free fields, and the second step implements interactions by forming invariant Wick-ordered local polynomials of composite free fields and defining perturbative transition operators Z S(g, h) = T exp i {g(x)W (x) + h(x)A(x)} dx (1) where for simplicity we used a symbolic notation and wrote the polynomial inter-
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action density as a monomial W and denotes by h the source function(s) of the basic free field coordinate(s) A in terms of which we specify the Fock space and the W . As usual T denotes time-ordering. These operator functionals of the testfunctions in Fock space fulfill the so-called Bogoliubov axiomatic and it is of no relevance to us whether this axiomatics has nonperturbative solutions or not; the reader is entitled to take the most pessimistic view concerning their existence. Formally these time-ordered exponentials would represent the scattering operator Ssc if the test functions approach the constant function on Minkowski space. If only this “on-shell” value of this time-ordered operator would be known, there would be no possibility of computing off-shell local fields. Therefore Bogoliubov et al. [7] assume (either by functional dependence or in some other way) that there exists an off-shell extrapolation of Ssc which can be related (by their functional derivative formalism which is not part of scattering theory) to local fields and is given by (1). This formalism then leads to an expression for the outgoing free field and to the Yang–Feldman equation Z ∗ Ain (x)Ssc Aout (x) = Ain (x) + ∆(x − x0 )j(x0 )dx0 ≡ Ssc Z A(x) = Ain (x) + j(x) = Kx S ∗ (h)
∆ret (x − x0 )j(x0 )dx0
(2)
δ S(h)|h=0 . δh(x)
Here ∆ is the mass-shell Pauli–Jordan commutator function, whereas ∆ret is the off-shell retarded function which is formally the on-shell commutator function multiplied with the step function in time. The alternative right-hand way of writing R in the first line indicates that the on-shell restriction of the ∆(x − x0 )j(x0 ) con∗ [Ain (x)Ssc ], i.e. the definition via tribution is determined by the on-shell object Ssc the off-shell S(g, h) is not needed. The crucial question now is if these formulas can be used in a suggestive manner for the construction of a semilocal wedge localized F (x) which like Ain (x) is still on-shell, but on the other hand like A(x) carries information about interactions. Assume for simplicity that we are in the situation of a factorizable model with a diagonal S-matrix. Such an elastic S-matrix in terms of the analytic phase shift δ, with δ(θ)ε(θ) = δphys (|θ|), would be represented by an exponential of a quadrilinear term in the incoming field ZZ 1 δ(θ − θ0 )ε(θ − θ0 ) : ρ(θ)ρ(θ0 ) : dθdθ0 (3) Ssc = exp i 2 where the ρ(θ) is the rapidity space charge density (resp. particle number density in our case of selfconjugate particles). From this we read off the relation between Ssc and aout ∗ aout (θ) = Ssc ain (θ)Ssc Z = ain (θ) exp i
∞
−∞
δ(θ − θ0 )ρ(θ0 )dθ0 .
(4)
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It is now suggested to try the following nonlocal field as a candidate for a wedge localized PFG field [2] Z 1 {Z(θ)e−ipx + Z ∗ (θ)eipx }dθ (5) F (x) = √ 2π Z Z(θ) = ain (θ) exp i
θ
−∞
δ(θ − θ0 )ρ(θ0 )dθ0 .
(6)
The analogue of the off-shell split of ∆ into ∆ret and ∆av is the split of the on-shell rapidity space integrals of the S-matrix. This is in the spirit of interpreting physical momenta as describing asymptotic x-space relations for large time-like distances. This analogy gains more credibility from the remark that the exponent is indeed some generalization of an integral over the current j(x) a fact which is particularly evident [2] in the case of the Federbush- and Thirring-model. There are also indications that this analogy between “off-shell space time and on-shell velocity” or rather its rapidity logarithm may be helpful in understanding certain “rapidity space cluster properties” [10] which are expected to select a subclass of local pointlike operators which recently were observed in formfactors of certain models. And last but not least, we expect this analogy to be helpful in a future unraveling of the structure of PFG’s in the nonperturbative analysis of non-factorizing QFT’s. Clearly the positive and negative frequency parts of the above F (x) obey a Zamolodchikov–Faddeev algebra [15] (see (17) of the next section). The strategy for extension to factorizing systems with more general nondiagonal S-matrices is obvious: generalize the structure on the algebraic side and prove that the so obtained PFG fields are indeed wedge localized in the sense of quantum localization. Modular theory offers a very neat way of doing this through the verification of the KMS property for the correlation functions of the operators Z F (f ) = F (x)f (x)dx supp f ⊆ W . By now the motivation for our notation should be obvious; whereas the symbol F denotes the general PFG operators which share with free fields that their Fourier transform is on-shell, the notation Z # for the positive and negative frequency components are meant to point to the fact that in the simplest case of d = 1 + 1 factorizing theories these generalized “creation and annihilation” operators fulfil the Zamolodchikov–Faddeev algebra. In fact the first success of our new concept of PFG operators is that the vacuum representation of the Z-F algebra obtains a spacetime interpretation in terms of wedge localization including its thermal (Hawking–Unruh) KMS aspect. For non-factorizing d = 1+1 theories and for higher dimensional QFT the KMS properties of the correlations of F (f )0 s do not allow such a simple algebraic characterization but acquire a much more involved structure in terms of the full scattering matrix. In order to achieve a more detailed understanding and proofs, we need to develop some more formalism.
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3. Generators of Quantum Wedge Localization In order to verify the modular wedge localization for the F (f ) operators for the case that the Z # (θ)0 s have the Z-F algebra structure, we first return to the d = 1 + 1 free n-particle wedge localization [2] in order to introduce some additional notation. Using free fields the wedge localization spaces may be written: Z (n) |ψ i = ψˆ(n) (xn , xn−1 , . . . x1 ) : A(x1 ) · · · A(xn ) : Ω ˆ(n)
supp ψ
⊆W
⊗n
1 , A(x) = √ 2π
Z
(7) (e
−ipx
a(θ) + e
ipx ∗
a (θ))dθ
where W denotes the right-hand wedge and the operator involves a Wick product of free fields A (for the sake of notation chosen scalar and selfconjugate). Instead of the vectors p and x we use their rapidity parametrization in the wedge region x1 > |x0 |: ! ! sinh χ cosh θ for x ∈ W . (8) , x=ρ p=m cosh χ sinh θ We also prefer the more intrinsic way of writing which avoids the use of field coordinates (which are not unique) and uses the momentum space creation and annihilation operators which are directly linked to the Wigner representation theory of irreducible particle representations and therefore are independent of the choice of covariant represention: |ψ (n) i = Aψ(n) Ω Z Z ... ψ (n) (θn , θn−1 , . . . θ1 ) : a(θ1 ) . . . a(θn ) : dθ1 . . . dθn . Aψ(n) = C
(9)
C
The used path notation C is a self-explanatory notation which generalizes the rapidity representation of the wedge localized fields [16]: Z Z Z A(fˆ) = f (θ)a(θ) + f (θ − iπ)a(θ − iπ) ≡ f (θ)a(θ) C
a(θ − iπ) ≡ a∗ (θ), f (θ − iπ) = f¯(θ)
(10)
C consists of the real θ-axis and the parallel path shifted down by −iπ and it is only the function f which is analytic in the strip −π < Im θ < 0 and conjugate-symmetric (i.e. fulfilling the Schwarz reflection principle) around the Im θ = 12 iπ line. For the operators this is only a notational convention and implies no analyticity.c The f analyticity is equivalent to the localization property of fˆ, and the analytic properties of the state vectors ψ and the path notation in (9) is an n-variable generalization of (10). In the application to the vacuum in (9) of course only the creation contribution c Operators in QFT, either in x-space or in momentum space, are never analytic, although some unfortunate notation and terminology especially in chiral conformal QFT suggests this. For more remarks see [2].
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from the lower rim of the strip survives. It is easy to see that these vectors fulfill the modular localization equation [2]: S|ψ (n) i = |ψ (n) i (11)
1
S = J∆ 2 , J = T CP, ∆it = U (Λ(2πt)) . The antilinear unbounded Tomita involution S (be aware to avoid confusions with the notation for the S-matrix!) consists (for Bosons) of the TCP reflection J and the analytically continued (by functional calculus) wedge affiliated Lorentz boost U (Λ(χ)) and the analytical strip properties guarantee that the localized vectors 1 1 |ψ (n) i are in the domain of ∆ 2 and hence of S. The action of ∆ 2 corresponds to the continuation to the lower rim and the action of J is just complex conjugation in momentum space (for selfconjugate situations): S
ψ (n) (θn , θn−1 , . . . θ1 ) → ψ (n) (θn − iπ, θn−1 − iπ, . . . θ1 − iπ)
(12)
so that modular localization Eq. (11) states that the value of the wave function on the −iπ-shifted boundary equals the complex conjugate of the upper boundary. Note that the hermiticity of Aψ(n) implies the reality condition ψ (n) (θn , θn−1 , . . . θ1 ) = ψ (n) (θ1 , . . . θn ) without analytic properties. For free Bosons the wave functions are symmetric. In case of Fermions it is well-known that the Tomita reflection J has a Klein twist in addition to the T CP which in the present modular approach manifests itself in a mismatch between the modular versus the geometrical “opposite” and which is easily repaired by a sign factor for the action on one-particle spaces. (n) The closure HR of the real subspace of solutions of (11) contains the spatial part of modular wedge localization. By applying the generators of the wedge algebra (10) to the vacuum n-times, we generate a dense set of localized state vectors in the complex space which turns out to be the n-particle component of the wellknown Reeh–Schlieder set of vectors. This dense space becomes the Hilbert space (n) (n) HR + iHR if one forms the closure in the graph norm: hψ2 , ψ1 i ≡ (ψ2 , ψ1 ) + (Sψ1 , Sψ2 ) .
(13)
Let us now pass to the case with interaction. For simplicity of notation we assume that the S-matrix of the factorizing interacting model describes the interaction of only one kind of particle (neutral, without bound states). Different from the free case, it follows from scattering theory [2] that J carries all the interaction whereas the ∆it remains unchanged. Whenever necessary we will add a suffix 0 for the free (incoming) objects. With this notation we have: J = Ssc J0 , ∆it = ∆it 0
(14)
where Ssc denotes the S-matrix. The realization that the scattering connection between asymptotic (e.g. incoming) free fields and interacting fields (resp. algebras) keeps the unitary representations of the connected part of the Poincar´e group unmodified and only changes those disconnected components which contain the antiunitary time reversal, is a well-known [17] piece of collective knowledge in QFT,
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albeit mostly not highlighted according to its importance. What is new is the realization that the interpretation within the modular framework attributes to the S-matrix the property of a relative modular invariant for wedges. To see this, one only has to remember that thanks to the Bisognano–Wichmann connection of the Lorentz boost and the TCP operation with modular theory, the relation J = Ssc J0 is just the TCP transformation law of the S-matrix with the TCP operator being expressed in terms of the Tomita conjugation. In d = 1 + 1 the J is identical with the TCP operator whereas in higher dimensions it is different by a π-rotation which commutes with the Ssc . (0) In analogy with the free n-particle Hilbert spaces HR we make the Ansatz: |ψ (n) i = Aψ(n) Ω Z ψ (n) (θn , θn−1 , . . . θ1 ) : Z(θ1 ) . . . Z(θn ) Aψ(n) =
(15)
C
Z(iπ − θ) := Z ∗ (θ) .
(16)
Here the wave functions are holomorphic in the multivariable [0, −iπ] θ-strip and the Z 0 s are also mass shell annihilation and creation operators but with more complicated nonlocal commutation relations: Z # (θ1 )Z # (θ2 ) = Ssc (θ1 − θ2 )Z # (θ2 )Z # (θ1 ), Z # ≡ Z or Z ∗ −1 Z(θ1 )Z ∗ (θ2 ) = Ssc (θ1 − θ2 )Z ∗ (θ2 )Z(θ1 ) + δ(θ1 − θ2 )
(17)
∗ (θ − iπ) . crossing : Ssc (θ) = Ssc (iπ − θ) = Ssc
Using again the previous notation in terms of the integration path C and the crossing property of S we may compress this as Z Z f1 (θ1 )f2 (θ2 )Z(θ1 )Z(θ2 )dθ1 dθ2 C
C
Z Z
Z f2 (θ1 )f1 (θ2 )S(θ1 − θ2 )Z(θ2 )Z(θ1 )dθ1 dθ2 +
= C
f¯2 (θ)f1 (θ)dθ . (18)
C
In a different and more formal context this algebraic structure (the “Z-F” algebra) was introduced by Zamolodchikov and later completed (by adding the δ-function term) by Faddeev. In the absence of bound state poles in S, a case to which we will restrict our considerations, the vector states generated by the Zs are connected with the in-states created by the a(θ) by Z ∗ (θ1 ) . . . Z ∗ (θn )Ω = a∗ (θ1 ) . . . a∗ (θn )Ω for θ1 > θ2 > · · · > θn
(19)
and the identification for permuted orders following from the Z-F algebra. For diagonal S-matrices it is easy to write down exponential realizations of the Z-F algebra [1]. It should be stressed that these consistent interpretations are only
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tentative, pending on their confirmation by (LSZ, Haag-Ruelle) time-dependent scattering theory applied to the local operators which will only be available after the modular construction of the local net of algebras. So at the end the modular approach will produce an operator formula for the Z’s in terms of incoming operators consistent with the above formula for vector states. If we treat the Z # (θ) in analogy to mass shell creation and annihilation operators, the candidate for the generator of the wedge algebra (following our intuitive discussion in the previous section) should be the “field”: Z 1 (e−ipx Z(θ) + eipx Z ∗ (θ))dθ F (x) = √ 2π (20) Z f (θ)Z(θ)dθ ,
F (f ) =
supp f ∈ W .
C
The fact that with the Z-F commutation relation this operator (contrary to what the notation may suggest) cannot be pointwise local does not bother us, since we only expect locality in the wedge sense. It is easy to see that the states it generates from the vacuum are localized in wedges. In fact a stronger statement is valid: the F’s are generators of a wedge localized algebra. In formula [JF (f )J, F (g)] = 0 ,
supp f, g ∈ W .
(21)
To prove this, one first notices that the Z(θ)# commutes with the JZ(θ)# J underneath the Wick-ordering. For models in which Z has the exponential form (6), the integrals over the particle number density extend over complementary rapidity regions. The numerical phase factors which originate from the commutation of these exponential factors with the preexponential a#0 s, mutually compensate. There remains the contraction between the pre-exponential a#0 s themselves which leads to Z Z ¯ (22) f(θ)g(θ − iπ) exp i δsc (θ − θ0 )n(θ0 )dθ0 . C
Shifting the integration by −iπ as required by the lower boundary of C, and using the crossing symmetry (particle-antiparticle Schwartz reflection symmetry around i π2 ) in the form: δsc (θ − iπ) = δsc (−θ) + 2πni, we see that this contraction is equal to that in the opposite order (which has the negative exponential). Again one easily verifies that this argument goes through if one only uses the commutation and vacuum annihilation properties of the Z #0 s. A more general elegant method consists in defining the von Neumann algebra which is generated by the F’s, checking its standardness (cyclicity and separability) on the vacuum and then showing that the Tomita relation (11) holds with the J given by (14). For more complicated (nondiagonal) Z-F algebras, their physical representation is most conveniently constructed with the help of the thermal KMS property of the vacuum on the F-correlation function (29) which is a consequence of the modular localization structure. The functional-analytic control over the Z # (θ) and F (f )-operators is not more difficult than that of the standard creation and annihilation operators a# (θ) and
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the smeared free fields A(f ). Since the vacuum is annihilated by the Z 0 s, the R ∗ n-particle vectors are generated by n-fold application of Z (θ)f (θ)dθ onto Ω. With S (2) being a crossing symmetric solution of the bootstrap program, the n-particle component of the action of this operator on a state vector ψ is then Z
(n)
∗
Z (θ)f (θ)dθψ
(θ1 , . . . θn )
Y 1 X f (θi ) S (2) (θk − θi )ψ (n−1) (θ1 , . . . θˇi , . . . θn ) = √ n i=1 n
i−1
(23)
k=1
where we used θˇ for the omission of a variable and where we have suppressed the indices on the wave functions on which the two-body S-matrix acts. As in the bosonic case, the norm of this wave function obeys the standard inequality involving the number operator N
Z
Z ∗ (θ)f (θ)dθψ 6 kf k kNψk (24)
and hence the closability of the Z #0 s and the selfadjointness of F (f ) for real test functions follows in a well-known manner [18], despite the fact that the total particle space (25) H = ⊕n Hn is not manifestly identical to a Boson/Fermion Fock space. In fact for none of the calculations we need a formula for Z # in terms of free incoming fields as in (5). The general algebraic relations (17) together with the vacuum annihilation property is all we use for the construction of wedge localized generators in the space (26). Whereas for the spatial aspects of the modular theory the unbounded nature of the generators F (f ) does not matter, the construction of the wedge-localized von Neumann algebras A(W ) ≡ alg{F (f ), supp ⊆ W } (26) requires a conversion of the closed unbounded operators into bounded ones. This can be done via selfadjointness and spectral decomposition of the F (f ) similar to free fields. For free Bose fields one also has the method of Weyl-generators, but for the PFG’s the associated Weyl-like bounded objects would not have a simple algebraic structure. This is a result of the more complicated commutation relations (18). In the case of absence of poles in the physical sheet of the S-matrix, an explicit analytic characterization can be obtained. We will stop short of presenting these technical aspects and refer to this and the generalization to S-matrices with bound state poles in a separate paper. Contrary to the previous free case, the sharpening of the support of the test function does not improve the localization within the wedge. This is equivalent to the statement that the reflection with J does not create an operator which is localized at the geometrically mirrored region in the opposite wedge W 0 . It only fulfills the commutation relation with respect to the full W 0 . In fact the breakdown of parity covariance is important for the existence of such nonlocal but wedge-localized
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fields since fields which are covariant under all transformations are expected to be either point local or completely delocalized (i.e. not even in a wedge). This breakdown of parity covariance for the Z-fields does not mean that the theory violates parity symmetry but only that these auxiliary fields only fulfill that symmetry and localization requirement which are expressible in terms of wedge algebras without reference to possible pointlike local field generators. We will later see that any sharper localization requires the operator to be an infinite power series in the Z 0 s: Z X 1 Z A= ··· an (θn , θn−1 , . . . θ1 ) : Z(θ1 ) · · · Z(θn ) : (27) n! C C where we again used the previously explained path notation. The sharper localization leads in fact to relations between the a0n s (see later) which, with one coefficient being different from zero, forces higher ones to be nonvanishing as well. In particular each double cone localized operator applied to the vacuum exhibits the full vacuum polarization structure. The absence of this polarization structure, i.e. the appearance of a kind of relativistic QM, only happens on the level of the wedge algebra as a result of the on-shell nature of the PFG’s. Therefore the PFG’s only serve as a natural basis for the wedge algebra. For smaller than wedge algebras A(O) with O ⊂ W , they are not generators of these algebras. These statements comply with the physical idea that, whereas the noncompact wedge region is big enough to allow the identification of particle states (it contains Lorentz boosts as automorphisms), on the other hand it is also small enough to contain no annihilators as required by having a unique relation between vector states and operators (the Reeh–Schlieder relation) which is the prerequisite for the modular theory. This delicate balance is broken if one passes to compact localized algebras which favour the field side, with the sharp incoming particle number (in particular one-particle states) losing its observable meaning and being replaced by one- or multi-charged states. Compactly localized algebras are simply too small to contain the projectors which project onto one-particle vector states. Note that since the commutations of Z 0 s produce unitary factors with the same two-particle S-matrix, the an must compensate this unitary factors upon commuting θ0 s. (2) (θi − θi+1 )an (θ1 , . . . θi+1 , θi , . . . θn ) . an (θ1 , . . . θi , θi+1 , . . . θn ) = Ssc
(28)
In this respect the phase factors are like statistics terms. However since the an have analytic properties in the multi-θ strip (actually for compact localization the analytic region is much bigger), these phase factors must be consistent with the univaluedness of the analytic continuation in the analytic domain. We now show that the KMS property relative to the boost is equivalent to the crossing symmetry of the S (2) -coefficients in the Z-F algebra. The first nontrivial correlation function which deviates from that of free fields is the 4-point function. The KMS property reads hF (f10 )F (f20 )F (f2 )F (f1 )itherm = hF (f20 )F (f2 )F (f1 )F (f12πi 0 )itherm
(29)
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where the superscript 2πi denotes the imaginary KMS shift in the boost parameter. Each side is the scalar product of vector states created by the two-fold application of F’s onto the vacuum which thanks to (19) can be written ZZ f2 (θ2 − iπ)f1 (θ1 − iπ)Z ∗ (θ1 )Z ∗ (θ2 )Ω + c − number · Ω F (f2 )F (f1 )Ω = ZZ =
f2 (θ2 − iπ)f1 (θ1 − iπ){χ12 a∗ (θ1 )a∗ (θ2 )Ω
+ χ21 S (2) (θ2 − θ1 )a∗ (θ2 )a∗ (θ1 )Ω + c − number · Ω
(30)
and the analogous formula for the bra-vector. The formula needs some explanation. The symbol χ with the permutation subscript denotes the characteristic function associated with the permuted rapidity order. The order for the free creation operators a∗ is governed by particle statistics. For each transposition starting from the natural order (19) one obtains an S (2) factor. The Yang–Baxter relation assures that the various ways of doing this are consistent. For the inner product the S (2) -dependent terms are ZZ f10 (θ2 )f20 (θ1 ){χ21 S (2) (θ2 − θ1 ) + χ12 S¯(2) (θ1 − θ2 )}f2 (θ2 − iπ)f1 (θ1 − iπ)dθ1 dθ2 ZZ = f10 (θ2 )f20 (θ1 )S (2) (θ2 − θ1 )f2 (θ2 − iπ)f1 (θ1 − iπ)dθ1 dθ2 . The analogous computation for KMS crossed term in (29) gives ZZ f20 (θ1 )f2 (θ2 )S (2) (θ1 − θ2 )f1 (θ1 − iπ)f10 (θ2 − iπ + 2πi)dθ1 dθ2 .
(31) (32)
(33)
This formula only makes sense if the F (f ) operators are restricted in such a way that the 2πi translation on them is well defined, i.e. for wave functions f which are analytic in a strip of size 2πi. It is well known that the KMS condition does not hold on all operators of the algebra but rather on a dense set of suitably defined analytic elements. The S-independent terms which we have not written down are identical to terms in the 4-point function of free fields. They separately satisfy the KMS property. What remains is to show the identity of (31) and (33). This is done by θ2 → θ2 − iπ contour-shift in (33). Using the denseness of the wave functions one finally obtains (34) S (2) (θ2 − θ1 ) = S (2) (θ1 − θ2 + iπ) which is the famous crossing symmetry or the z ←→ −z reflection symmetry around the point z0 = 12 iπ. The presence of bound state poles in the physical strip of S (2) would invalidate this argument as the result of the appearance of noncompensating pole contribution. In that case the formula (19) is invalid and is modified by the additional bound state contribution Z ZZ (35) f¯1 (θ1 )f¯2 (θ2 ){Z + (θ1 )Z + (θ2 )Ω}scat + f¯1 (θ + iθb )f¯2 (θ − iθb )Φb (θ)
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where the superscript denote the scattering component and Φb (θ) is the new composite particle vector at rapidity θ. Further details of modular method in the presence of bound states will be deferred to a separate paper. Higher inner products which appear for higher correlation functions involve products of S-matrices and it is easy to see that the KMS condition for the Z-F algebra is equivalent to the crossing property of the S-matrix. The presence of operators A ∈ A(O) ∈ A(W ) which are localized in e.g. a double cone, does not influence the validity of the KMS condition. hF (f10 )F (f20 ) · · · F (fm0 )AF (fn ) · · · F (f2 )F (f1 )itherm = hF (f20 ) · · · F (fm0 )AF (fn ) · · · F (f2 )F (f1 )F (f12πi 0 )itherm .
(36)
The rapidity space formulation of this KMS condition will turn out (see next section) to be the cyclicity relation for the formfactor of that local operator which hitherto [13] was a special consequence for factorizing systems of the formally derived crossing symmetry, which in turn follows from the LSZ scattering theory together with analytic assumptions. The connected part of these mixed correlation functions are the (with f’s integrated) coefficient functions in (27). A moment of thought reveals that everything in the above verification of the KMS-property goes through if we would only use the (generalized to matrix-valued coefficients S) algebraic structure (17) and the annihilation relation Z(θ)Ω = 0, together with the crossing relation for the matrix-valued structure functions S; in fact the consistency of the algebra (unitarity and Yang–Baxter relations) are equivalent to the KMS-property of the F (f )-correlation functions [24]. It is worthwhile to present this statement in the form of a proposition. Proposition 1. The KMS-property of F (f )-correlation functions with F being of the form (5) with (possibly) matrix-valued coefficients S is equivalent to the crossing symmetry of the latter. The algebraic Ansatz (17) corresponds physically to a kind of two-body interaction. Hence within the realm of this Ansatz, the defining structure of the d=1+1 factorizing S-matrix bootstrap approach corresponds precisely to the (existence and) KMS-property of our PFG wedge generators. However it needs to be emphasized that the quantum mechanical aspect of the PFG’sd for factorizing S-matrices, i.e. the absence of real particle creation on the level of the on-shell structure of A(W ) in no way prevents the smaller localized algebras A(O) to be fully “virtual-particlenonconserving”. In other words, with the exception of S = 1, which leads to free fields, the refinement of localization beyond wedges creates the virtual particle polarization clouds which are the characteristic hallmarks of interacting causal QFT. Therefore the quantum mechanical appearance of particles and their bound states for factorizing models which are associated with the PFG’s, disappears immediately after one passes from wedge localization to say double cone localization and d This includes the formation of bound states. In our derivation we have tacitly assumed that they are absent. We hope to return to this interesting point in a separate publication.
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the theory turns into a genuine interacting QFT to which concepts of (relativistic) QM are not any more applicable. This will be demonstrated explicitly in the next section. 4. The Construction of the Double-Cone Algebra Finally we indicate (again for the simplest one particle model) how one proceeds from the quantum localized F 0 s and the A(W ) they generate to the double cone algebras. Formally we have to look for solutions of JBJ = A ,
A, B ∈ A(Oa )
the equation for the elements of the algebraic intersection a a a a A(W )U ∩U A(W 0 )U − . A(Oa ) = U − 2 2 2 2
(37)
(38)
The wedge algebra is the von Neumann algebra associated with the ∗ -algebra Aformal (W ) ( =
) Z X 1 Z dθ1 · · · dθn an (θn , θn−1 · · · θ1 ) : Z(θ1 ) · · · Z(θn ) : A|A = n! C C n (39)
where the an are meromorphic in the lower strip. Here we call a would-be von Neumann operator algebra Aformal if we only study aspects of sesquilinear forms, (i.e. if we ignore convergence properties required by bona fide operators). These coefficient functions are related to the KMS property of the connected parts of the previously introduced mixed correlation functions hAF (f1 ) · · · F (fn )itherm A ∈ A(Oa ) ,
F (fi ) ∈ A(Wa ) .
(40)
Originally the KMS property gives the analyticity in the regions of ordered imagiQ nary parts for monomial vectors i Z(fi )Ω, but as a result of the denseness of the analytic fi0 s, one retains meromorphy (analyticity modulo poles) for the coefficient functions of general nth components an of wedge localized operators. In fact the expansion of A in terms of F 0 s may be directly written in x-space Z X 1 Z ··· a ˆ(x1 , . . . , xn ) : F (xn ) · · · F (x1 ) : A= n! Z X 1 Z ··· a(θ1 , . . . θn ) : Z(θn ) · · · Z(θ1 ) : = n! C C hAF (f1 ) · · · F (fn )itherm Z Z = ··· a ˆn (x1 , . . . , xn )f (x1 ) · · · f (xn ) + lower terms.
(41)
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The compact localization allows to extend the meromorphy to the product of the θ-planes. The content of the proposition in the previous section consisted in stating that the KMS property of the mixed functions involving one A (or in the standard setting a local field A(x) with x ∈ W ) is equivalent to a crossing symmetry relation, or the so-called cyclicity relation in the rapidity variables θ. an (θ2 , . . . , θn , θ1 − 2πi) = an (θ1 , . . . , θn ) .
(42)
The crossing symmetry of the S-matrix corresponds to the special case A = 1. To see this one uses the rapidity representation in (41). For the highest coefficient in the KMS relation one obtains the cyclicity relation by straightforward computation, again (as in the previous S-matrix case) using the density of the boundary values of the strip analytic test functions fi (θi ). The required double cone affiliation of A leads to a pole structure, which implies a recursive relation between the coefficient function, which will be derived in the sequel. For many considerations it is safer to argue directly in terms of the thermal correlators, in x-space than with the momentum rapidity a0n s. Even the following derivation of the pole structure should be transcribed into the thermal correlators, but we will present it in the more familiar momentum space rapidity form. In order to avoid questions about the necessity to introduce “fused” Z(θ)0 s and F 0 s for fused (bound states)e particles into the expansions (39) and other questions related to the completeness of Z 0 s, we will continue to assume the absence of such particles and defer a more general discussion to a future publication. Now B ∈ Aformal (W )0 has the analogue representation to (39) in terms of Z J and coefficient functions bn which are upper strip-meromorphic. Shifting the apex of the opposite wedge by a translation a into W amounts to multiply the bn -coefficients P (a) with exp i pi (θi )a. Let us call the shifted b’s simply bn . The formal double cone algebra Aformal (Oa ) is defined as Aformal ∩ U (a)Aformal (W )0 U ∗ (a). In order to compute matrix elements of Z J within vector states created by Z 0 s, we use the cumulant formula Z Z X ˙ ∗ (θ0 )a(θ0 )dθ0 =: exp (1 − Πi S(θi − θ0 ))a∗ (θ0 )a(θ0 )dθ0 . (43) δ(θi − θ0 )a exp i i
The computation is facilitated by using the following equivalent modular characterization of the intersection algebra in terms of either A ∈ A(W±a ) and A = As + iAs (44) JAs J = As where we decomposed a general operator of the double cone algebra into a Jselfconjugate and antiselfconjugate part. Here A belongs to the shifted algebra, but J is the Tomita involution of the original wedge algebra, i.e. our previous J. Since e The notion of bound states carries connotations of Q.M. which are somewhat misleading in relativistic field theories with (apart from free fields) a virtual particle polarization structure. The correct hierarchy is that of basic versus fused charges whose low-lying carries are (infra)particles.
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the J does not mix even and odd terms, we will assume that A has only even terms in its power series. Matching first the diagonal matrix elements for the A0s s and JAJ 0 s, we obtain the following recursion for the boundary values of the meromorphic functions: ¯0 a0 = a a2 (θ1 , θ2 − iπ) = a ¯2 (θ1 , θ2 + iπ) + δ(θ1 − θ2 )(1 − S(θ1 − θ2 − iπ))a0 a4 (θ1 , θ2 , θ3 − iπ, θ4 − iπ) = a ¯4 (θ1 , θ2 , θ3 + iπ, θ4 + iπ) + δ(θ1 − θ3 )(1 − S(θ1 − θ4 − iπ)S(θ2 − θ3 + iπ)S(θ1 − θ2 ))a2 (θ2 , θ4 − iπ) + 3 more such terms a6 (θ1 , . . . . . .) = etc .
(45)
with an analogous recursion for the odd coefficient functions. Taking into account the meromorphic properties of the coefficient functions, these δ-function terms in the boundary value mean that the meromorphic functions have poles if two rapidities coalesce modulo iπ. The matching conditions also contain the possibility of analytic extension into the tensor product of complex planes. In this way one obtains meromorphic functions in the multi-θ plane which have a Paley–Wiener type of increase in the imaginary direction which is related to the size a of the double cone. Again one easily notices that these consideration can be done solely on the basis of the algebraic properties of the Z # operators together with their annihilation property of the vacuum vector. Possible poles in the coefficient functions from other fused particle states appear in crossing symmetric pairs and compensate; only the above kinematical poles enter the formal determination of the double cone algebras A(Oa ). Hence our method applies to the general case of factorizable theories with admissable nondiagonal bootstrap S-matrices [19]. In addition to the above kinematical poles there are poles inside the physical strip associated with the crossing symmetric (pairwise) appearance of “bound states”.f As already mentioned, these interesting problems will be dealt with in a separate work. As a result, we obtain the well-known “kinematical pole structure” of formfactors (the coefficients in the series (27) known from the work of Smirnov [14, 16] (based on recipes) and later abstracted from LSZ scattering formalism in [13]. In our approach this structure (as the crossing symmetry) is equivalent to the J-invariance of the symmetrically placed double cone algebra. Proposition 2. Necessary and sufficient for the series (27) to fulfill (37) is that the coefficient functions are meromorphic in the multi θ-plane with certain (Payley– Wiener) fall-off behaviour in imaginary direction the and following pole structure f Strictly speaking the hierarchy of elementary versus bound states is limited to quantum mechanics and loses its meaning in the presence of (virtual) vacuum polarization. The only hierarchy which remains meaningful is that between basic and fused charges; on the level of particles QFT practices “Nuclear Democracy”, i.e. interactions couple all channels which are not separated by superselection rules.
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on the boundary of the old strip an+2 (θ + iπ + iε, θ, θ1 , . . . , θn ) |ε→0 " # n Y 1 (2) 1− = S (θ − θi ) an (θ1 , . . . , θn ) . ε i=1
(46)
Since each pair of θ0 s can always be transported into this canonical position by the application of (28), there is no loss of generality in this way of writing. The formfactors of pointlike fields A(0) at the origin which have appeared in the literature [14, 13] result by taking the limit a → 0, prescribing the Lorentz transformation property of the would be field and defining a convention which picks a particular (composite) field in the remaining infinite dimensional field space. Since fields at the origin are sesquilinear forms and not operators, they are improper members of Aformal (O). The operator problems show up in the computation of correlation functions of these fields. In fact there are two formidable tasks in the Karowski–Weisz–Smirnov approach. On the one hand one must construct a basis in field space (the analogue to the Wick-polynomials in the free field case), which even in the simplest nontrivial so-called Sinh–Gordon model led to presently difficult looking problems about symmetric polynomials in the xi ≡ exp θi [10]. But this is not enough; according to Wightman, one also has to take care of the operator problems of the smeared fields who’s prerequisite is the existence of their correlation functions. Apart from those special cases with a constant S-matrix as the Ising- and Federbush models [2], this has not been achieved. In the latter cases one notices immediately that there are quadratic expressions in the Z 0 s for which the iteration yields a vanishing factor from the factor in front of a2 on the right-hand side in (46). Since the operator aspects of such quadratic expressions as the energy-momentum tensor are easily controlled, it is clear that not only Aformal (O) but also A(O) is nonempty. For rapidity dependent S-matrices the iteration cannot stop and the construction of even a single operator causes serious mathematical problems. Using the algebraic approach, one can convert this rather heavy looking problems into a “light” version. The reason is that the modular construction liberates the existence proof of a model from the technical burden of field coordinates (and sometimes even insures the existence of local versions of symmetry generating operators). One only has to show that the double cone algebras are not trivial, i.e. contain operators which are not multiples of unity. This step would then allow to omit the subscript “formal” in the above space of sesquilinear forms and write A ∈ A(Oa ) = U (− a2 )A(W )U ( a2 ) ∩ U ( a2 )A(W 0 )U (− a2 ). Let us first note that a trivial double cone algebra (A(Oa ) = C · 1) would violate several properties of Local Quantum Physics. On the one hand the additivity property W = ∪d O(0, d) A(W ) = ∨A(O(0, d))
(47)
where O(0, d) is a family of double cones inside W which have a common left corner with W, would not hold. Furthermore the so-called split property cannot hold in
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such a case for it is one of the consequences [11] that a double cone algebra cannot be just a multiple of the identity, because it must contain at least a local version of the symmetries of the theory, e.g. an algebraic analog of the energy momentum tensor. But in our situation we need an argument which refers only to the wedge algebras A(W ), A(W 0 ) ≡ JA(W )J and the d=1+1 Poincar´e group. If we were to assume the split property for two dimensional wedges [12], we would immediately obtain the unitary equivalence of the double cone algebra to a tensor product of wedge algebras (48) A(Oa ) ' A(W a2 ) ⊗ A(W−0 a2 ) so that the nontriviality of the double cone algebra would be manifest. This elegant argument however has one drawback; all factorizable models are very different from superrenormalizable models which are locally normal with respect to free theories and therefore inherit the split property of the latter; as a result we have some doubts about the validity of the split property for wedges for factorizing models. Therefore we will try yet another argument based on the idea that the double cone algebra possesses a nontrivial “light ray reduction”. The mathematical formulation of this idea starts from the relative commutant associated with the following inclusion: A(Wa± ) ⊂ A(W )
(49)
i.e. A(Wa± )0 ∩ A(W ) .
(50)
Here W is the d=1+1 standard wedge and Wa± are the upper and lower wedges obtained by having W slide into itself along the upper/lower light ray (±horizon) with the distance a± . The geometric position of the associated relative commutant A(Wa± )0 ∩ A(W ) is the 1-dimensional interval on the upper/lower light ray which starts at the origin and ends at a± . It may be considered to be the limiting case of the 2-dim. relative commutant which is obtained by shifting the W into itself so that the upper/lower horizons of the shifted wedge do not yet touch the standard wedge. From these upper/lower relative commutants we may form the light ray algebras [ 0 Ad ∆it (51) M± ≡ W A(Wa± ) ∩ A(W ) ⊂ M ≡ A(W ) . t
A manifest way of exposing this structural property of this limiting theory on the light ray is to emphasize the improvement of symmetry on the light rayg [21] to full conformal symmetry. The cyclically generated space from the vacuum and the related conditional expectations are H± ≡ M± Ω ⊂ M Ω ≡ H = P± Ω E± (M) = P± MP± = M± .
(52)
The projector P± onto this possibly smaller space is associated with a conditional expectation E± , in agreement with the fact that the two theories share the same g The fact that this net on the light ray which is indexed by intervals is not only covariant under light ray translations and scale transformations but also under conformal rotations followsfrom [20].
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modular group. To recover the full algebra from its light cone projections in case that the latter are genuine, we have to combine M± . There are two ways of doing this M(+,−) = M+ ∨ M− M⊗ = M+ ⊗ M− .
(53)
The second way belongs to the construction of two-dimensional conformal QFT from their chiral components whereas the first one gives the equality of the combined light cones to reproduce the original massive algebra in case that the vacuum is cyclic since clos(M(+,−) Ω) = clos(MΩ) ↔ M(+,−) = M according to a theorem of Takesaki [24]. In order to see what is going on in the case of factorizing models in more detail, let us write down the commutation relation which characterize those operators A ∈ M (27) which commute with (the generators of) M+ Z 2 0 A, fa+ (x)Z(x)d x = 0, a0+ ≥ a+ . (54) Clearly the nth order contribution in the Z(θ) power series receives two contributions, one with a coefficient function of the form Z dθ(ΠS(θi − θ) − 1)an−1 (θ1 , . . . θn−1 )eimu exp θ (55) C
which results from the product of S-matrices from commuting the Z(θ) to the lefthand side, as well as contraction terms Z Z dθ dθn an+1 (θ1 , . . . θn−1 , θn , θ)eimu exp θ . (56) C
C
A pause of thought reveals that this relative commutator condition does not tell us anything on the pole structure of the coefficient functions which we do not know already from the previous proposition (46), however due to the presence of only one light cone coordinate, the vanishing of the commutator is identical to the much simpler problems in the algebraic formulation of chiral nets on the line with translation and scale covariance. The one-dimensional operator version of the wellknown Payley–Wiener–Schwartz theorem results in a much better control on the remaining meromorphy properties. The reduction to the light ray fields gives a powerful tool for the transition from the wedge algebras to the compactly localized double cone algebras. We will give a separate detailed account of its application to factorizable models. Here we will be content to present some more structural properties in the sequel. The light ray reductions are bona fide conformal nets since the translationdilation situation together with the modular inclusion M0a± ⊂ M is known to correspond to a SL(2,R)-covariant local conformal net [22]. It is very important not to equate this net with the generally divergent naive light cone restriction nor with
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the (Lagrangian) light cone quantizationh of fields. For factorizing theories whose local generating fields are more singular for short distances than free fields (they are not superrenormalizable as it would be required by local normality), the canonical structure is not defined and the light cone restriction is divergent since the field near to the + light cone does generally not become independent on the − light cone variable. It is only through the associated algebraic net that the light ray restriction is well defined. This algebraic reprocessing is even necessary in superrenormalizable theories since the light cone quantization in the presence of interactions seems to be nonlocal (similar to our PFG’s but with less conceptual and mathematical control) even when it is well defined. Only by reprocessing the nonlocal data via relative commutants one has a chance to reconstitute a local net on the light ray. In other words the modular method is essential for understanding and physically interpreting light ray physics. A simple illustration of this mechanism is obtained by looking at the free massive Fermion field {ψ(x), ψ ∗ (y)} = (γµ ∂ µ − m) i∆(x − y) {ψ1 (x), ψ2∗ (y)} → ±m
(57)
where the x, y in the last relation are on the upper/lower horizon so that the distance is ±time-like. In this case no algebraic reprocessing is needed, we obtain two conformal theories with different chirality which applied to the vacuum generate H± subspaces. The fact that they did not simply originate from a mass zero limit shows up in their nontrivial m-dependent relative commutation relation which is the memory of the relative time-like difference. The light cone restriction is a peculiar chiral conformal theory, a kind of hidden massive theory, which by the adjoint action of the opposite light cone translation U− can be cropped into a wedge-localized massive theory (58) alg {M+ , ad U− (a ≥ 0)M+ } = M . There are good arguments that in case one ignores the local net structure, massive theories lead to the equality M+ = M− = M of the holographically associated conformal algebras with the massive wedge algebra, a point to which we plan to return in a separate paper. As mentioned before, all modular constructions for factorizing theories can be based on the algebraic structure as well as the vacuum annihilation properties of the Z-operators. After having constructed the local net one would like to check however the interpretation of the n-particle Z-states in terms of incoming states by establishing an operator formula (which generalizes the well-known exponential formula for diagonal S-matrices) based on the (Haag-Ruelle, LSZ) time dependent scattering theory. The explicit knowledge of the local double cone net together with the h The formal light cone quantization procedure leads to a kind of quantum mechanics as a result of the surpression of vacuum polarization [23]. In none of the presentations of the light cone formalism an attempt is made to reconstruct the original local fields from the light ray data. We believe that this is not possible without the algebraic reprocessing by modular methods as used in this paper.
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isolated one particle states (the spectral gap property) allows for the construction of the incoming fields in terms of the Z 0 s a la Haag-Ruelle and by inversion also of (a priori unknown for the nondiagonal models) formulas for the Z 0 s in terms of the incoming free fields. At this point the correctness of the scattering interpretation of the coefficient functions (27) in terms of formfactors between scattering states can be confirmed by selfconsistency. 5. Outlook We have succeeded in reformulating the existence of “quantum localized” PFG’s for d = 1 + 1 factorizing models in terms of KMS properties of their correlation functions which in turn result from the representation theory of the Z-F algebra structure which in turn implements the unitarity, crossing symmetry and Yang– Baxter structure of the S-matrix bootstrap. Although in standard QFT terminology the pointlike fields in these models have controllable short-distance behaviour beyond that of the canonical formalism (they are strictly renormalizable, but not superrenormalizable), yet the short-distance behaviour of particular pointlike fields has played no role in our modular approach. In fact as far as the existence of local algebraic QFT is concerned, this is totally decided by the nontriviality of the double cone algebras. The omnipresent short distance problem of any standard QFT approach has given way to the verification of nontriviality in the process of “quantum localization” (with some stretch of imagination a kind of algebraic short-distance behaviour) which consists in forming algebraic intersections of wedge algebras. There are many other old questions in standard QFT which appear solvable in the modular approach. For example it is believed that although the generators of the wedge algebras F (f ) have a completely analytic behaviour in the coupling parameters (apart from changes required by the existence of bound states), the off-shell correlation functions of operators in double cone algebras are at best Borel resummable. Two questions immediately arise: can one use this gain of conceptual insight for the improvement of the Bootstrap-Formfactor program, and secondly could the new algebraic structure based on the general possible existence of PFG’s be useful for a kind of modular-based general new constructive approach, including a new intrinsic perturbation theory of nets which is free of field coordinations. As far as the first question is concerned, we remark that the outstanding problem is the physical understanding and the mathematical description of the “operator content” of local fields. By this we mean the understanding of their expansion coefficients in terms of an auxiliary basis (27) which leads to formfactors. As it turns out, this is a difficult task even for the simplest case of (diagonal Toda) factorizing field theories. The identification of the sequences of symmetric polynomials which characterize the formfactors of the list of (composite) fields up to this date has not been accomplished. Our algebraic viewpoint which shifts the emphasis from individual field basis dependent coordinates to double cone algebras, reveals that the traditional field construction carries a lot of nonintrinsic structure which adds little to the securing of the existence and the understanding of the important dynamical properties of these models. The construction of a useful basis of local fields (i.e. a basis in the local field space), which parallels the construction of Wick polynomials
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in the free case, is a highly technical problem with a large amount of arbitrariness. This is of course a phenomenon which had to be expected on the analogy with coordinates in differential geometry. In fact in (nonperturbative) LQP the technical difficulties in using field coordinates versus intrinsically defined local nets, if anything, become even greater than the use of coordinates in differential geometry. Perturbation theory which requires the selection of a set of field coordinates is no good guide for these aspects, but even there one realizes the clumsiness of field coordinates if one tries to understand, e.g. the insensitivity of the S-matrix (in the present setting its role as a relative modular invariant for wedge algebras) against local changes of field coordinates. Even formulations which are usually considered to be the quintessence of the nonperturbative approach as e.g. the Wightman approach are not completely intrinsic. In the present work we have used modular ideas which force us to understand the existence and physical content of QFT in terms of net of von Neumann algebras. The crucial step, the construction of the double cone algebras, is facilitated by the use of light ray algebras which turned out to be “hidden” chiral conformal theories. This approach, whose complexity increases with the distance from onshell properties (or distance to polarization free objects), is particularly suited for high energy particle physics where the main measurable quantities are not correlation functions of fields but rather the S-matrix and perhaps formfactors of some distinguished fields (Noether currents), both closely related to modular concepts associated with wedge localization. The situation is different in statical mechanics and condensed matter physics; in that case one is really interested in the calculation of correlation functions, which often can be measured (by neutron scattering). Our nonperturbative approach contains the clear message to separate the construction of a nontrivial field theory and the understanding of its physical content from the technically complicated construction of its field coordinates. The second question concerning the applicability of these ideas outside of factorizing models should be subdivided according to space-time dimension = 1 + 1, or > 1 + 1. In the first case the mass shell is still parametrized in terms of rapidities, but they do not furnish a uniformization variable for the S-matrix which retains its rich creation/annihilation threshold analytic structure. Even in d > 1 + 1 the KMS property as well as crossing symmetry are still meaningful concepts, the PFG’s lose their algebraic Z-F structure. The crucial question is whether one is able to define PFG’s through correlation functions which involve the full real particle creating S-matrix, such that the KMS property of wedge localization becomes related to crossing symmetry. It seems that one can, and the correlation functions are again related to S-matrix products where the concrete expression depends on the order of rapidities. But these expressions involve inclusive summations over particle states between these products. Since their presentation would surpass the more modest aims of this paper, we refer to future work. As a curious side remark we mention that the central idea of this paper, namely the existence of PFG operators as the best compromise between the off-shell localization of fields with the on-shell particle structure, has some vague resemblance with certain formal simplifications in the so-called light-cone or light-front quantiza-
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tion. Closing one’s eye to the fact that the canonical nature of the approach (as in all quantization approaches) is not compatible with realistic ideas about interactionsi (apart from superrenormalizable interactions), one realizes that formally such a quantization suppresses vacuum or one-particle polarization phenomena [23] which is of course precisely the characteristic property of the PFG’s. However whereas in the light-front approach the relation of the so obtained field operators to the original local fields seems to have been irrevocably lost, and with it the possibility of physically interpreting the theory (apart from the one particle spectral information), the PFG’s are auxiliary operators which have a conceptually clear and mathematically precise relation within LQP. The message from our modular approach is that via “quantum localization” (formation of algebras and intersections) one should be able to recover the local variables from the ones obtained by light cone quantization. In line with these analogies we may add that the Galilei group which appeared as a Poincar´e subgroup in light cone quantization [23] showed up as part of an 8-parametric subgroup in the modular approach in connection with so-called modular intersections (see the first paper in [4]). The modular construction program is extremely enigmatic in that its conceptual physical richness points towards fantastic new possibilities of understanding the particle content of LQP in terms of the representation theory of modular groups. This means, e.g. in the context of factorizing models, that the (incoming) particle content of local operators belonging to a region O (i.e. the shape of the particle/antiparticle polarization clouds) have a characteristic feature which depends only on O and the particular model and is determined by a new form of hidden (modular) group theory going beyond the local action of the infinite dimensional Lie-groups which appeared in chiral conformal QFT. In view of the popularity which ideas of noncommutative geometry have enjoyed within part of the physics community, the present message, that the nonperturbative construction of local fields requires very noncommutative intermediate steps based on modular theory, should have an attractive appeal. Acknowledgments The authors thank M. Karowski for explaining some of the more intricate features of the formfactor program and K.-H. Rehren for critical reading of an early draft version. Note Added The notion of holography which is used in this paper is that of a local relation of a massive d = 1 + 1 theory with a lower d = 1 dimensional chiral QFT. In order to recreate the original theory one needs in addition to the light ray theory the (nontrivial) action of the opposite light ray translation on the holographic image. This is quite different from the anti De Sitter space situation, in the case of which i In fact genuine 2-dim. conformal fields, which are sums over products of anomalous dimension (spin) chiral fields from the two light cones, already expose this short-distance problem in the light ray restriction of pointlike fields.
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was recently also understood in the algebraic framework [25]. For a more detailed recent presentation of modular constructions and holography in algebraic QFT, see [26, 27]. References [1] B. Schroer, Nucl. Phys. B499 (1997) 519, 547. [2] B. Schroer, Ann. Phys. 275(2) (1999) 190. [3] Private discussion with D. Guido and unpublished manuscript by R. Brunetti, D. Guido and R. Longo, “On the intrinsic construction of free theories via Tomita Takesaki Theory”. [4] B. Schroer and H.-W. Wiesbrock, “Modular theory and geometry” math-ph/9809003, to be published in Rev. Math. Phys.; B. Schroer and H.-W. Wiesbrock, “Looking beyond the Thermal Horizon: Hidden symmetries in chiral models”, hep-th/9901031, to be published in Rev. Math. Phys.; D. Buchholz, O. Dreyer, M. Florig and S. J. Summers, “Geometric modular action and spacetime symmetry groups”, mathph/9805026. [5] M. Niedermaier, Nucl. Phys. B519 (1998) 517, hep-th/9706172 to be published in CMP. [6] J. Bisognano and E. H. Wichmann, J. Math. Phys. 17 (1976) 303. [7] N. N. Bogoliubov, A. A. Logunov, A. I. Oksak and I. T. Todorov, General Principles of Quantum Field Theory, Kluwer Academic Publ. 1990, and references therein. [8] S. Weinberg, The Quantum Theory of Fields, I, Foundations, Cambridge Univ. Press, 1955. [9] H. Epstein and V. Glaser, Ann. Inst. H. Poincar´ e Sect. A19 (1973) 211. [10] A. Koubek and G. Mussardo, Phys. Rev. Lett. B316 (1993) 193; A. Fring, G. Mussardo and P. Simonetti, Nucl. Phys. B393 (1993) 413; M. Pillin, “The formfactors in the Sinh–Gordon model” hep-th/9712033. [11] R. Haag, Local Quantum Physics, Springer Verlag, 1992. [12] M. Mueger, Commun. Math. Phys. 191 (1998) 137. [13] H. M. Babujian, A. Fring, M. Karowski and A. Zappletal, Nucl. Phys. B538 [FS] (1999) 535. [14] F. A. Smirnov, Adv. Series in Math. Phys. 14, World Scientific, Singapore, 1992. [15] A. B. Zamolodchikov and Al. B. Zamolodchikov, Ann. Phys. 120 (1979) 253; L. D. Faddeev, Sov. Sci. Rev. Math. Phys. C1 (1980) 107. [16] M. Yu Lashkevich, “Sectors of mutually local fields in integrable models of Quantum Field Theory”, hep-th/9406118. [17] R. F. Streater and A. S. Wightman, Spin, Statistics and All That, Benjamin Inc., New York, 1964. [18] H.-J. Borchers and W. Zimmermann, Nuovo Cimento 31 (1963) 1047. [19] M. Karowski, H. J. Thun, T. T. Truong and P. Weisz, Phys. Lett. B67 (1977) 321. [20] H.-W. Wiesbrock, Lett. Math. Phys. 184 (1997) 683. [21] G. Sewell, Ann. Phys. 141 (1982) 201; S. Summers and R. Verch, Lett. Math. Phys. 37 (1996) 145. [22] D. Guido, R. Longo and H.-W. Wiesbrock, Commun. Math. Phys. 192 (1998) 217. [23] T. Heinzl, “Light-cone dynamics of particles and fields”, Chap. 4, hep-th/9812190. [24] M. Takesaki, J. Funct. Anal. 9 (1972) 306. [25] K.-H. Rehren, “Algebraic holography”, hep-th/9905179. [26] B. Schroer, “New concepts in particle physics from a solution of an old problem”, hep-th/9908021. [27] B. Schroer, “Local quantum theory beyond quantization”, hep-th/99120.
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP COVARIANT HEISENBERG AlGEBRAE GAETANO FIORE∗ Sektion Physik der Ludwig-Maximilians Universit¨ at M¨ unchen, Theoretische Physik Lehrstuhl Professor Wess, Theresienstraße 37 80333 M¨ unchen, Germany E-mail:
[email protected] Received 11 March 1998 Revised 26 October 1988 Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. q-deformations of Weyl or Clifford algebrae that were covariant under the action of a simple Lie algebra g are characterized by their being covariant under the action of the quantum group Uh g, q := eh . We present a systematic procedure for determining all possible corresponding changes of generators, together with the corresponding realizations of the Uh g-action. The intriguing relation between g-invariants and Uh g-invariants suggests that these changes of generators might be employed to simplify the dynamics of some g-covariant quantum physical systems.
1. Introduction Weyl and Clifford algebrae (respectively denoted by A+ , A− in the sequel, and collectively as “Heisenberg algebrae”) are at the hearth of quantum physics. One may ask if deforming them within the category of associative algebrae (i.e. deforming their defining commutation relations) yields new physics, or at least may be useful to better describe certain systems in conventional quantum physics. This question can be divided into an algebraic and a representation-theoretic subquestion. The first was addressed in the fundamental paper [1]. Essentially, it reads: Is there a formal realization of the elements of the deformed algebra in terms of elements of the undeformed algebra? The answer is affirmative, but in general the realization is not explicitly known. A general result [2] regarding the Hochschild cohomology of the universal enveloping algebra associated to a nilpotent Lie group states in particular that the first and second cohomology groups of any Weyl algebra A+ are trivial. This implies [3] that any deformation Ah+ (h denoting the deformation parameter) of the latter is trivial, in the sense that there exists an isomorphism of topological algebrae over C[[h]] (a “deforming map”, in the terminology of [4]), f : Ah+ → A+ [[h]], reducing to the identity in the limit h = 0 (a concise and effective presentation of these results can be found in Secs. 1 and 2 of h [5]). Practically this means that the generators A˜i , A˜+ i of A+ are mapped by f + + i i into power series in h A+ , A := f (A˜ ), Ai = f (A˜i ), fulfilling the same deformed ∗ EU-fellow,
TMR grant ERBFMBICT960921. 327
Reviews in Mathematical Physics, Vol. 12, No. 3 (2000) 327–359 c World Scientific Publishing Company
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with coefficients in commutation relations and going to the corresponding generators a i , a+ i of A+ in the limit h = 0. The second subquestion is: Do the deformed and undeformed representation theories coincide? One can already see in some simple model that the answer is negative, but in the general case, up to our knowledge, the relation between the two is an open question. Given any automorphism g : A+ [[h]] → A+ [[h]], g = id + O(h), then g ◦ f is a new deforming map; conversely, given two deforming maps f, f 0 , the map f 0 ◦ f −1 is an algebra automorphism. Now, by the vanishing of the first cohomology group of A+ , all automorphisms of A+ [[h]] are “inner”, i.e. of the form g(a) = α a α−1 . Hence, all deforming maps can be obtained from one through the formula fα (a) := αf (a)α−1
α = 1 + O(h) ∈ Ah+ .
(1.1)
These results apply [5] in particular to so-called “q-deformations” (q := eh ) of Weyl algebrae which are covariant under the action of some simple Lie algebra g; such deformations [6–8] are matched to the deformation of U g into the quantum group Uh g [10], in the sense that for all q the deformed algebrae are in fact Uh gmodule algebrae:a the commutation relations among A˜i , A˜+ i are compatible with the quantum group action ˜ .h : Uh g × Ah+ → Ah+ as the commutation relations among ai , a+ i were compatible with the classical action . : U g × A+ → A+ . Also quantum groups admit algebra isomorphisms ϕh : Uh g → U g[[h]] [11].b In spite of the existence of the algebra isomorphisms f, ϕh , the Uh g-module algebra .h ) is a non-trivial deformation of the one (U g, A+ , .), i.e. for structure (Uh g, Ah+ , ˜ .h = . ◦ (ϕh × f ) holds.c This is because Uh g itself as a no ϕh , f the equality f ◦ ˜ Hopf algebra is a non-trivial deformation of U g, in other words all ϕh ’s are algebra but not coalgebra (and therefore not Hopf algebra) isomorphisms (this is related to the non-triviality of the Gerstenhaber–Schack cohomology [12]). Given f we define .h as the map making the following diagram commutative: Uh g
Uh g
×
Ah±
l
id × f
×
A± [[h]]
˜ .
h −→
Ah± l f
.h
99K
(1.2)
A± [[h]]
.h ◦ (id ⊗ f −1 ) will realize ˜.h on A±,g,ρ [[h]]). In this (in other words, .h := f ◦ ˜ work we give a systematic procedure to construct all pairs (f, .h ). It is based on the properties of the “Drinfel’d twist” [11]. We first construct .h , then f . One particular .h can be naturally constructed in a “adjoint-like” way (Sec. 3). To determine a corresponding f it is sufficient to identify in A±,g,ρ [[h]] appropriate a They should not be confused with the celebrated Biedenharn–Macfarlane–Hayashi q-oscillator (super)algebrae [9], whose generators are not Uh g-covariant (in spite of the fact that they are usually used to construct a generalized Jordan–Schwinger realization of Uh g). b The existence of the latter and their being defined up to inner automorphisms of U g[[h]] again is a consequence of the triviality of the first and second Hochschild cohomology groups of U g. c By . we mean here actually its linear extension to . : U g[[h]] × A +,g,ρ [[h]] → A+,g,ρ [[h]], where both the domain and codomain have to be understood completed in the h-adic topology.
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˜i ˜+ realizations Ai , A+ i of A , Ai . With this aim in mind, we first show (formula (3.6)) how to construct candidates Ai , A+ i having the same transformation properties under .h as the generators A˜i , A˜+ of Ah±,g,ρ under ˜.h ; they are determined up to i
multiplication by some g-invariants. By requiring that the Ai , A+ i also have the i ˜+ ˜ same commutation rules as the A , Ai (Sec. 5) we find constraints (5.0.1)–(5.0.3) on these g-invariants. Requiring that the ∗-structures of A± realize the ∗-structures h,inv [[h]] of of Ah± yields further constraints (Sec. 6). The subalgebrae Ainv ± [[h]], A± A±,g,ρ [[h]] that are invariant respectively under ., .h coincide (Sec. 4), but invariants in the form of polynomials in Ai , A+ j are highly non-polynomial (analytic) functions of the classical invariants and hence of ai , a+ j , and conversely; we explicitly find these functions. Solving the system of Eqs. (5.0.1)–(5.0.3) depends on the particular g and on the particular U g-representation ρ to which the generators a+ i belong (the ai necessarily belong to the contragradient ρ∨ of ρ). We shall denote by A±,g,ρ the corresponding Heisenberg algebra and by Ah±,g,ρ its q-deformed version. We solve (Sec. 5) the above equations for the well-known Ah±,sl(N ),ρd , Ah+,so(N ),ρd , (ρd will denote the defining representations of either g). Finally, in Sec. 7 we extend the previous results to all other isomorphisms (1.1) while giving an outlook of the whole construction, we make some remarks on the representation theory, and we draw the conclusions. This serves also to better clarify the motivations for the present work. In [13] we started the program just sketched by sticking to the cases of triangular deformations of the Hopf algebra U g (in the present setting it is recovered by postulating a trivial coassociator) and of the q-deformation Ah±,sl(2),ρd . Examples of q-deforming maps for Heisenberg algebrae were explicitly constructed “by hand” in past works [5, 14–16].d Of course these deforming maps are related to ours by some automorphism (1.1); in Sec. 5.1 we determine the latter for Ah+,sl(2),ρd and the deforming map found in [15]. We underline that our construction is based instead on universal objects characterizing the quantum symmetry algebra. This allows the application of our method, e.g. to the physically relevant case that ρ be the direct sum of an arbitrary number of copies of ρd . 2. Preliminaries and Notation Some general remarks before starting. Although we will always denote the gen+ i erators of the Heisenberg algebrae by ai , a+ i , A , Ai , . . ., only the choice of a ∗-structure may give them the meaning of creators/annihilators, or coordinates/ derivatives, etc. (see, e.g. Sec. 6). Given an algebra B, B[[h]] will denote the algebra of formal power series in h ∈ C with coefficients belonging to finite-dimensional subspaces of B. Both B[[h]] and tensor products like B[[h]] ⊗ B[[h]] will be understood to be completeted in the h-adic topology. The symbol Uh g [10] will denote the algebra on the ring C[[h]] underlying the quantum group (completed in the h-adic topology). d For multidimensional Heisenberg algebrae the non-trivial task there was to find an intermediate transformation to a set of mutually commuting pairs {αi , α+ i } of deformed generators; these generators αi , α+ are covariant neither under the U g nor the U h g action. i
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2.1. Twisting groups into quantum groups Let H = (U g, m, ∆, ε, S) be the cocommutative Hopf algebra associated to the universal enveloping (UE) algebra U g of a Lie algebra g. The symbol m denotes the multiplication (in the sequel it will be dropped in the obvious way m(a⊗b) ≡ ab, unless explicitly required), whereas ∆, ε, S the comultiplication, counit and antipode, respectively. Assume that Hh = (Uh g, mh , ∆h , εh , Sh , R ) is a quasitriangular noncocommutative deformation of H [10]; h ∈ C denotes the deformation parameter, mh , ∆h , εh , Sh the deformed multiplication, comultiplication, counit and antipode, respectively, and R ∈ Uh g ⊗ Uh g the universal R-matrix. A well-known theorem by Drinfel’d, Proposition 3.16 in [11] (whose results are to a certain extent already implicit in preceding works by Kohno [17]), proves, for any simple finite-dimensional Lie algebra g, the following results. There exists: (1) an algebra isomorphism ϕh : Uh g → U g[[h]] and (2) a “twist”, i.e. an element F ∈ U g[[h]] ⊗ U g[[h]] satisfying the relations (ε ⊗ id)F = 1 = (id ⊗ ε)F
(2.1.1)
F = 1 ⊗ 1 + O(h)
(2.1.2)
(1 denotes the unit of U g; (2.1.2) implies that F is invertible as a formal power series in h) such that Hh can be obtained from H through the following equations. Let F = F (1) ⊗ F (2) , F −1 = F −1(1) ⊗ F −1(2) , in a Sweedler’s notation with upper indices; in P (1) (2) the RHS a sum of many terms is implicitly understood, e.g. i Fi ⊗ Fi . Then mh = ϕ−1 h ◦ m ◦ (ϕh ⊗ ϕh ) ,
(2.1.3)
εh := ε ◦ ϕh ,
(2.1.4)
−1 −1 ], ∆h (x) = (ϕ−1 h ⊗ ϕh )[F ∆[ϕh (x)]F
(2.1.5)
−1 S[ϕh (x)]γ] , Sh (x) = ϕ−1 h [γ
(2.1.6)
−1 −1 2 ). R = [ϕ−1 h ⊗ ϕh ](F21 q F
(2.1.7)
t
Here t := ∆(C) − 1 ⊗ C − C ⊗ 1 is the canonical invariant element in U g ⊗ U g (C is the quadratic Casimir), the maps m, ε, ∆, S have been linearly extended from U g to U g[[h]], and γ := SF −1(1) · F −1(2) ,
γ −1 = F (1) · SF (2) .
(2.1.8)
Equation (2.1.3) says that U g[[h]] and Uh g are isomorphic through ϕh as algebras over C[[h]]. Equation (2.1.4) says that, up to this isomorphism, εh and ε coincide. Equations (2.1.5) and (2.1.6) say that, up to the same isomorphism, ∆h , Sh differ from ∆, S by “similarity transformations”. Equation (2.1.5) is not
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in contradiction with the coassociativity of ∆ and ∆h ,e because the (nontrivial) coassociator φ := [(∆ ⊗ id)(F −1 )](F −1 ⊗ 1)(1 ⊗ F )[(id ⊗ ∆)(F )
(2.1.9)
commutes with ∆(2) (U g) (we denote by ∆(2) the two-fold coproduct), [φ, ∆(2) (U g)] = 0 .
(2.1.10)
From the properties of φ it follows also that γ −1 γ 0 ∈ Centre(U g), and Sγ = γ 0−1 . The above formulae can also be read in the other direction as giving a construction procedure of quasitriangular Hopf algebrae. They can be also applied to triangular deformations of H (one needs only to set t ≡ 0 in (2.1.7)), that are quasitriangular deformations with a trivial coassociator, φ = 1 ⊗ 1 ⊗ 1. In fact the theorem cited above generalizes an older theorem [18], also by Drinfel’d. The twist F is defined (and unique) up to the transformation F → FT ,
(2.1.11)
where T is a g-invariant [i.e. commuting with ∆(U g)] element of U g[[h]]⊗ such that T = 1 ⊗ 1 + O(h) , (ε ⊗ id)T = 1 = (id ⊗ ε)T . (2.1.12) 2
Under this transformation φ → [(∆ ⊗ id)(T −1 )](T −1 ⊗ 1)φ(1 ⊗ T )[(id ⊗ ∆)(T ) .
(2.1.13)
A function T = T (1 ⊗ Ci , Ci ⊗ 1, ∆(Ci ))
(2.1.14)
of the Casimirs Ci ∈ U g of U g and of their coproducts clearly is g-invariant. We find it plausible that any g-invariant T must be of this form; although we have found in the literature yet no proof of this conjecture, in the sequel we assume that this is true. We will often use a “tensor notation” for our formulae: F12,3 = (∆ ⊗ id)F12 , F123,4 = (∆(2) ⊗ id)F12 , and so on, and Definition 2.1.9 reads φ ≡ φ123 = −1 −1 F12 F23 F1,23 . φ satisfies the equations: F12,3 q
t13 +t23 2
q
t12 +t13 2
= φ−1 231 q = φ312 q
t13 2 t13 2
φ132 q φ−1 213 q
t23 2 t12 2
φ−1 123
(2.1.15)
φ123
(they follow from (∆h ⊗ id)R = R 13 R 23 , (id ⊗ ∆h )R = R 13 R 12 ). e To arrive at the above results Drinfel’d introduces the notion of quasitriangular quasi-Hopf algebra; the latter essentially involves the weakening of coassociativity of the coproduct into a property (“quasi-coassociativity”) valid only up to a similarity transformation through an ele3 ment φ ∈ U g[[h]]⊗ (the “coassociator”). This notion is useful because quasitriangular quasi-Hopf algebra are mapped into each other under twists (even if the latter is not trivial). As an intermediate result, he shows that U g[[h]], beside the trivial quasitriangular quasi-Hopf structure t 2 3 (U g[[h]], m, ∆, ε, S, R ≡ 1⊗ , φ ≡ 1⊗ ), has a non-trivial one (U g[[h]], m, ∆, ε, S, R = q 2 , φ 6= 3 1⊗ ).
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While for the twist F, apart from its existence, very little explicit knowledge is available, Kohno [17] and Drinfel’d [11] have proved that, up to the transformation (2.1.13), φ is given by g (x) , φm = gˆ−1 (x)ˇ
0 < x < 1,
(2.1.16)
where gˆ, gˇ(x) are U g[[h]]⊗ -valued “analytic”f solutions of the first order linear differential equation t23 t12 dg =~ + g, 0 < x < 1 (2.1.17) dx x x−1 3
(~ =
h 2πi )
with the following asymptotic behaviour near the poles: gˆ · x−~t12 −→ 1⊗ x→0
3
gˇ · (1 − x)−~t23 −→ 1⊗ . x→1
3
(2.1.18)
Using Eq. (2.1.17) it is straightforward to verify that the RHS of Eq. (2.1.16) is indeed independent of x.g Using Eq. (2.1.18) we can take the limit of Eq. (2.1.16): 12 gˇ(x0 ) . φm = lim + x−~t 0
(2.1.19)
x0 →0
We can formally solve Eqs. (2.1.17) and (2.1.18) for gˇ by a path ordered integral, Z 1−y0 t23 t12 ~t23 ~ + y0 P exp −~ (2.1.20) dx gˇ(x0 ) = lim x x−1 y0 →0+ x0 (P~ [A(x)B(y)] := A(x)B(y)ϑ(y − x) + B(y)A(x)ϑ(x − y)), whence Z 1−y0 t23 t12 −~t12 ~ ~t23 + y0 φm = lim + x0 . dx P exp −~ x x−1 x0 ,y0 →0 x0
(2.1.21)
Note that φm = 1⊗ + O(h2 ). We will say that the twist F is “minimal ” if the corresponding φ (2.1.9) is equal to φm or is trivial, respectively in the case of Hh = Uh g or Hh is a triangular deformation of U g. The algebra isomorphism ϕh : Uh g → U g[[h]] is defined up to an inner automorphism (a “similarity transformation”) of U g[[h]], 3
ϕh,v (x) := vϕh (x)v −1 ,
(2.1.22)
with v = 1 + O(h) ∈ U g[[h]]. It is easy to check that Drinfel’d theorem [11] remains true provided one replaces F by Fv := (v ⊗ v)F ∆(v −1 ) and all the objects derived f In the sense that the coefficients g (x) appearing in the expansion g(x) = P∞ g (x)hn of g in n n=0 n 3
h-powers are analytic functions of x with values in a finite-dimensional subspace of U g⊗ . g Kohno and Drinfel’d proved that φ can be obtained as the “monodromy” of a system of three first order linear partial differential equations in three complex variables zi (the socalled universal P 3 tij ∂f Knizhnik–Zamolodchikov [19] equations), with an U g⊗ -valued unknown f , ∂z = ~ j6=i z −z f. i
i
j
The system can be reduced to Eq. (2.1.17) exploiting its invariance under linear tranformations zi → azi + b. For a review of these results see for instance [20].
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
333
from F correspondingly; in particular, it is easy to check that the coassociator φ remains unchanged, because it is g-invariant φv = ∆(2) (v)φ∆(2) (v −1 ) = φ .
(2.1.23)
The freedom in choosing ϕh (and F ) is usually eliminated or reduced if one requires it to satisfy additional properties, such as to lead to a specific ∗-structure for Uh g. The Lie algebra g = sl(2) is the only g for which explicit ϕh ’s are known. In the sequel we shall often use Sweedler’s notations with lower indices for the coproducts: ∆(x) ≡ x(1) ⊗ x(2) for the cocommutative coproduct (in the RHS a sum P i i (n−1) (x) ≡ x(1) ⊗ · · · ⊗ x(n) i x(1) ⊗ x(2) of many terms is implicitly understood), ∆ for the (n − 1)-fold cocommutative coproduct and ∆h (x) ≡ x(¯1) ⊗ x(¯2) (with barred indices) for the non-cocommutative one. 2.2. Deforming group-covariant Heisenberg algebrae The generic undeformed Heisenberg algebra is generated by the unit 1A and elej ments a+ i and a satisfying the (anti)commutation relations: ai aj = ±aj ai + + + a+ i aj = ±aj ai
a
i
a+ j
=
δji 1A
±
(2.2.1) a+ j
a
i
j (the ± sign refers to Weyl and Clifford algebras, respectively). a+ i , a transform under the action of U g according to some law j + x . a+ i = ρ(x)i aj ,
x . ai = ρ(Sx)ij aj ;
(2.2.2)
here x ∈ U g and ρ denotes some matrix representation of g. We shall call the corresponding algebra A±,g,ρ . When x ∈ g, the antipode reduces to Sx = −x. Clearly ai belong to a representation of U g which is the contragradient ρ∨ = ρT ◦ S (T is the transpose) of the one of a+ i , ρ. Because of the linearity of the transfori mation (2.2.2) we shall also say that a+ i , a are “covariant”, or “tensors”, under .. The action . is extended to products of the generators using the standard rules of tensor product representations (technically speaking, using the coproduct ∆ of the universal enveloping algebra U g, see formula (2.2.8) below), and then linearly to all of A±,g,ρ , . : U g × A±,g,ρ → A±,g,ρ ; this is possible because the action of g is manifestly compatible with the commutation relations (2.2.1), and makes A±,g,ρ into a (left) module algebra of (H, .). In the sequel we shall denote by . also its linear extension to the corresponding algebrae of power series in h. For suitable ρ (specified below) A±,g,ρ admits a deformation Ah±,g,ρ with the same Poincar´e series and the following features. Ah±,g,ρ is generated by the unit ˜i 1Ah and elements A˜+ i , A fulfilling deformed commutation relations (DCR) which can be put in the form:
334
G. FIORE
ji A˜i A˜j = ±P F hk A˜k A˜h ,
(2.2.3)
F hk ˜+ ˜+ ˜+ A˜+ i Aj = ±P ij Ah Ak ,
(2.2.4)
i ˜F ˜+ ˜k A˜i A˜+ j = δj 1A ± P ihjk Ah A ,
(2.2.5)
(as before, the upper and lower sign refer to Weyl and Clifford algebras, respectively) and transforms under the action ˜.h of Uh g according to the law j + x˜ .h A˜+ i = ρhi (x)Aj ,
x˜.h A˜i = ρihj (Sh x)Aj .
(2.2.6)
Here x ∈ Uh g, ρh is the quantum group deformation of ρ, whereas P F and P˜ F are two suitable quantum-group-covariant deformations of the ordinary permutator ij = δki δhj ). By a redefinition (2.1.22) one can always matrix P (by definition Phk choose a ϕh such that (2.2.7) ρh = ρ ◦ ϕh . A˜i belong to a representation of Uh g which is the quantum group contragradient T ˜+ ρ∨ h = ρh ◦ Sh of the one of Ai , ρh . Because of the linearity of the transformation ˜i .h . The (2.2.6) we shall also say that A˜+ i , A are “covariant”, or “tensors”, under ˜ action ˜ .h is extended to products of the generators by the formula x˜ .h (ab) = (x(¯1) ˜.h a)(x(¯2) ˜.h b)
(2.2.8)
and then linearly to all of Ah±,g,ρ , ˜.h : Uh g × Ah±,g,ρ → Ah±,g,ρ ; this is possible because the action ˜ .h is compatible with the above commutation relations, and makes Ah±,g,ρ into a (left) module algebra of (Hh , ˜.h ). The latter also means that (xy)˜.h a = x˜.h (y˜.h a)
(2.2.9)
∀ x, y ∈ Uh g. In the undeformed setting the formulae corresponding to the previous two are obtained by just replacing ˜.h by ., and ∆h (x) ≡ x(¯1) ⊗ x(¯2) by the cocommutative coproduct ∆(x) ≡ x(1) ⊗ x(2) . To our knowledge, only the following deformed algebras with the same Poincar´e series as their classical counterpart have been constructed: Ah+,sl(N ),ρd , [6, 7] Ah−,sl(N ),ρd , [21] Ah+,so(N ),ρd , [8], Ah−,sp( N ),ρ [22] (for each of the above Lie algebras 2
d
ρd denotes the defining, N -dim representation) as well as those Ah±,g,ρ where ρ Lm is the direct sum of many copies (m, say) of ρd , ρ = µ=1 ρd,µ [22, 23].h ˆ (or to its inverse), Explicitly, in the case ρ = ρd the matrix P˜ F is equal to q R ˆ where R is the braid matrix [24] of Uh g, ( 1 qN if g = sl(N ) 2 ˆ := cg P [(ρd,h )⊗ (R )] cg := (2.2.10) R 1 otherwise h For a generic ρ, second degree relations of the type (2.2.3)–(2.2.5) may not be enough for a consistent definition of Ah ±,g,ρ ; more precisely, associativity may require the introduction of third or higher degree relations, which have no classical counterpart. In this case the Poincar´e series h of Ah ±,g,ρ will be smaller than its classical counterpart, and A±,g,ρ will be physically not so interesting.
335
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
(here ρd,h denotes the deformation of ρd and we have introduced the factor cg to match to the conventional normalization), whereas P F is a suitable first or second ˆ From formula (2.1.7) it then follows that degree polynomial in R. P F = F U F −1 ,
(2.2.11)
P˜F = F V F −1 ,
(2.2.12)
⊗2
⊗2
ρd (t/2) where F := ρ⊗ and U is also a polynomial in P q ρd (t/2) . d (F), V = cg P q One may actually choose U = P without affecting the commutation relations, since in this case one can easily show that (1 ∓ U ) ∝ (1 ∓ P ), the (anti)symmetric projector. For later use we recall that from the projector decomposition and the ˆ [24, 7, 8] it follows that the “q-number operator” N˜ := A˜+ A˜i of properties of R i h A±,sl(N )ρd satisfies the relations: 2
±2 ˜+ ˜ ˜+ N˜ A˜+ Ai N i = Ai + q
and the invariant elements A˜+ C A˜+ := satisfy the relations:
˜), N˜ A˜i = q ∓2 (−A˜i + A˜i N ij ˜+ A˜+ i C Aj ,
(2.2.13)
˜ A˜ := A˜i Cji A˜j of Ah AC +,so(N )ρd
˜ A) ˜ A˜i − A˜i (AC ˜ A) ˜ = 0, (AC
(2.2.14)
˜+ ˜+ ˜+ (A˜+ C A˜+ )A˜+ i − Ai (A C A ) = 0 ,
(2.2.15)
˜ A) ˜ = (1 + q 2−N )Cij A˜j , ˜ A) ˜ A˜+ − q 2 A˜+ (AC (AC i i
(2.2.16)
A˜i (A˜+ C A˜+ ) − q 2 (A˜+ C A˜+ )A˜i = (1 + q 2−N )C ij A˜+ j .
(2.2.17)
In the case that ρ is the direct sum of many copies (say m) of ρd the commutation relations between the different copies are not trivial. We refer the reader to [22] for the explicit form of P F , P˜F . The important point here is that one can show that also in this case these matrices can be put in the form (2.2.11) and (2.2.12), where F := ρ⊗ (F ) , 2
(2.2.18)
and U, V are suitable mN × mN matrices such that [U, ρ⊗ (∆(U g))] = [V, ρ⊗ (∆(U g))] = 0 . 2
2
(2.2.19)
3. Realization of the Quantum Group Action and of Candidates for the Deformed Generators It is immediate to check that one can define a Lie algebra homomorphism σ : g → A±,g,ρ by setting j (3.1) σ(x) := ρ(x)ij a+ i a for all X ∈ g, and therefore extend it to all of U g as an algebra homomorphism σ : U g → A±,g,ρ by setting on the unit element σ(1U g ) := 1A . σ can be seen as the generalization of the Jordan–Schwinger realization of g = su(2) [25] ↓ σ(j+ ) = a+ ↑a ,
↑ σ(j− ) = a+ ↓a ,
σ(j0 ) =
1 + ↑ ↓ (a a − a+ ↓ a ). 2 ↑
(3.2)
336
G. FIORE
Extending σ linearly to the corresponding algebrae of power series in h, we can also define an algebra homomorphism σϕh := σ ◦ ϕh : Uh g → A±,g,ρ [[h]] .
(3.3)
Since we know that a deforming map exists, although we cannot write it explicitly we can say that it must be possible to construct the map .h defined in the introduction. Our first step is to guess such a realization of ˜.h on Ah±,g,ρ. This requires fulfilling the conditions (2.2.8) and (2.2.9), which characterize a left module algebra. There is a simple way to find such a realization, namely: Proposition 1 [13]. The (left ) action .h : Uh g × A±,g,ρ [[h]] → A±,g,ρ [[h]] can be realized in an “adjoint-like” way: x .h a := σϕh (x(¯1) )aσϕh (Sh x(¯2) ) .
(3.4)
Using the basic axioms characterizing the coproduct, counit, antipode in a generic Hopf algebra it is easy to check that (2.2.8) and (2.2.9) are indeed fulfilled. The realization (3.4) is suggested by the cocommutative case, where it reduces to x . a = σ(x(1) )aσ(Sx(2) ) .
(3.5)
Our second step is to realize elements Ai , A+ j ∈ A±,g,ρ [[h]] that transform under i + (3.4) as A˜i , A˜+ in (2.2.6). Note that a , a do not transform in this way. We recall: j j Proposition 2 [13]. Let F be a twist associated to ϕh . For any choice of ginvariant elements u, u ˆ, v, vˆ = 1A + O(h) in A±,g,ρ [[h]] (in particular if they are trivial) the elements (1) A+ )v a+ ˆσ(SF (2) γ) i := σ(F i v
ˆ σ(F −1(1) ) Ai := σ(γ 0 SF −1(2) )uai u
(3.6)
+ i i transform under (3.4) as A˜i , A˜+ j in (2.2.6), and go to ai , a in the limit h → 0.
Looking for g-invariants making Ai , A+ i fulfil the DCR will be the third step of the construction (Sec. 5). Remark 1. Without loss of generality, we can assume F in Definitions 3.6 to be minimal. In fact, since any other twist can be written in the form F T with T ≡ T (1) ⊗ T (2) as in (2.1.14), one finds that ˆ)σ(ST (2) ) = v 0 a+ ˆ0 σ(T (1) )(va+ i v i v
(3.7)
ˆ)σ(T −1(2) ) = u0 ai u ˆ0 σ(ST −1(1) )(uai u
(3.8)
i The Ansatz (3.6) has some resemblance with the one in [26, Proposition 3.3], which defines an intertwiner α : U g[[h]] → Uh g of Uh g-modules.
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
337
with u0 , u ˆ0 , v 0 , vˆ0 = 1 + O(h) also g-invariant. This follows from the following observations: first, σ(Ci ), σ(SCi ) are g-invariant and central in σ(U g[[h]]), therefore the dependence of T on Ci ⊗ 1, 1 ⊗ Ci translates just into some replacements ˆ) transforms under . exactly as a+ u → u0 , v → v 0 , etc; second, (va+ i v i , whence (3.5)
σ(Ci(1) )(va+ ˆ)σ(SCi(2) ) = Ci . (va+ ˆ) = ci (va+ ˆ), where ci ∈ C is the value of i v i v i v the Casimir Ci in (the irreducible component of) the representation ρ to which a+ i belongs, and similarly for (uai u ˆ), in other words the dependence of T on Ci(1) ⊗ Ci(2) translates just into mutiplication of the generators by a constant. Remark 2. Note that if ρ is reducible the previous proposition holds also if we allow for different invariants u, u ˆ, v, vˆ within each irredicible component of ρ. In the sequel we shall often use the compact notation + ˆ) ai := (uai u ˆ) . a+ i := (vai v
(3.9)
In the appendix we prove the following: Lemma 1. If F is a “minimal ”, then −1(2)
F = γ −1 (SF −1(1) )F(1) −1(1)
= F(1)
−1(2)
⊗ F(2)
−1(2)
⊗ γ 0 (SF −1(2) )F(1)
F −1 = F(1) ⊗ F(2) (SF (2) )γ (1)
(1)
(3.11) (3.12)
= F(1) (SF (1) )γ 0−1 ⊗ F(2) . (2)
(3.10)
(2)
(3.13)
i We can find now useful alternative expressions for A+ i ,A .
Proposition 3. With a “minimal” F , Definitions 3.6 amount to + −1(2) )ρ(F −1(1) )li A+ i = al σ(F
(3.14)
(1) 0−1 l γ )i σ(F (2) )a+ A+ i = ρ(SF l
(3.15)
Ai = ρ(F (1) )il σ(F (2) )al
(3.16)
Ai = al σ(F −1(2) )ρ(γ −1 SF −1(1) )il .
(3.17)
Remark 3. In spite of its original definition (3.4), from the latter expressions we realize that only a “semiuniversal form” of the type (ρ ⊗ id)F ±1 for F is involved in the definition of Ai , A+ j . Proof of Proposition 3. Observing that σ(x)a = σ(x(1) )aσ(Sx(2) · x(3) )
(3.18)
aσ(x) = σ(x(3) Sx(2) )aσ(x(1) )
(3.19)
338
G. FIORE
for all x ∈ U g, a ∈ A±,g,ρ , we find −1(2)
A+ i
(3.5),(2.2.2)
(2) a+ )γ]ρ(F(1) )li = a+ l σ[F(2) (SF l σ(F1
Ai
(3.5),(2.2.2)
ρ F(1)
= =
(1)
(1)
(3.12)
(3.11) −1(1) i −1(1) σ[γ 0 (SF −1(2) )F(2) ]al = l
−1(1) l )i
)ρ(F1 (1)
(2)
,
ρil (F2 )σ(F2 )al .
(3.20) (3.21)
Similarly one proves the other relations. 4. Classical Versus Quantum Group Invariants
We have defined two actions ., .h on A±,g,ρ [[h]]. Their respective invariant subalgebras are respectively defined by Ainv ±,g,ρ [[h]] := {I ∈ A±,g,ρ [[h]]|x . I = ε(x)I
∀ x ∈ U g}
Ah,inv ±,g,ρ [[h]] := {I ∈ A±,g,ρ [[h]]|x .h I = εh (x)I
∀ x ∈ Uh g} .
(4.1) (4.2)
If f is a deforming map corresponding to .h , the second subalgebra clearly contains h,inv h f (Ah,inv ±,g,ρ ), where A±,g,ρ denotes the Uh g-invariant subalgebra of A±,g,ρ . What is h,inv inv the relation between A±,g,ρ [[h]] and A±,g,ρ [[h]]? h,inv Proposition 4. Ainv ±,g,ρ [[h]] = A±,g,ρ [[h]].
Proof. We show that both subalgebras coincide with the one {I ∈ A±,g,ρ [[h]]|[σ(U g), I] = 0} .
(4.3)
Given any I ∈ Ainv ±,g,ρ [[h]], y ∈ U g, we find (4.1)
Iσ(y) = Iε(y(1) )σ(y(2) ) = (y(1) . I)σ(y(2) ) (3.5)
= σ(y(1) )Iσ(Sy(2) · y(3) ) = σ(y(1) )Iε(y(2) ) = σ(y)I .
This proves that (4.3) contains Ainv ±,g,ρ [[h]]. Conversely, if [I, σ(U g)] = 0, then (3.5)
x . I = σ(x(1) )Iσ(Sx(2) ) = Iσ(x(1) · Sx(2) ) = Iε(x) for any x ∈ U g, proving that the set (4.3) is contained in Ainv ±,g,ρ [[h]]. Replacing in the previous arguments σ, ε, . and ∆(x) ≡ x(1) ⊗ x(2) by σϕh , εh , .h and ∆h (x) ≡ x(¯1) ⊗ x(¯2) , one proves that Ah,inv ±,g,ρ [[h]] also coincides with the algebra (4.3). In other words, the proposition states that invariants under the g-action . are also Uh g-invariants (under .h ), and conversely, although in general g-covariant objects (i.e. tensors) and Uh g-covariant ones do not coincide. Since . (resp. .h ) acts in a linear and homogeneous way on the generators ai , a+ j h,inv inv (resp. Ai , A+ ), in the vector space A [[h]] = A [[h]] we can choose a basis ±,g,ρ ±,g,ρ j {I n }n∈N (resp. {Ihn }n∈N ) consisting of normal ordered homogeneous polynomials
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
339
+ i in ai , a+ j (resp. A , Aj ). For all g the simplest invariant is the “number of particle + i 1 i 1 operator” n ≡ I := a+ i a and its deformed counterpart N ≡ Ih = Ai A . In general the invariants will take the form j ,...,j
(4.4)
j ,...,j
(4.5)
1 kn i1 + ihn , I n := a+ j1 · · · ajkn di1 ,...,ihn a · · · a 1 kn + i1 ihn , Ihn := A+ j1 · · · Ajkn Di1 ,...,ihn A · · · A
j ,...,j
j ,...,j
(kn , hn ∈ N ∪ {0}); the coefficients di11,...,ihknn (resp. Di11,...,ihknn ) depend on the particular g picked up and make up classical (resp. quantum) group isotropic tensors, i.e. satisfy iJn In 0 h kn hn J dIn0n = ε(x)dJInn (4.6) (ρ⊗ ⊗ ρ∨⊗ )(∆(bn −1) (x)) 0 I0 Jn n
h
ρ⊗ h
kn
⊗ ρ∨⊗ h
hn
iJn In J0 (b −1) ∆h n (y) DI 0n = εh (y)DIJnn 0 I0 Jn n
n
(4.7)
∀ x ∈ U g[[h]], y ∈ Uh g. Here and in the rest of the section we use the collectiveindex notation In ≡ (i1 · · · ihn ), Jn ≡ (j1 · · · jkn ) and the short-hand notation bn := hn + kn . Using formula (2.1.6) it is straightforward to verify that the d’s and D’s are related to each other by h iJn In 0 kn hn kn hn J DIJnn (ρ⊗ ⊗ ρ∨⊗ )((1⊗ ⊗ (γ 0−1 )⊗ )F12,...,bn ) dIn0n , (4.8) 0 I0 Jn n
and is given, where F12,...,b ∈ U g[[h]]⊗ is an intertwiner between ∆(b−1) and ∆h bn up to multiplication from the right by a g-invariant tensor Q ∈ U g[[h]]⊗ , by b
(b−1)
F12,...,b = F(b−1)b Fb−2,(b−1)b · · · F1,2,...,b .
(4.9)
The isotropic tensors corresponding to I 1 , Ih1 are dji = δij = Dji . The replacement 2 F → F · T , with T ∈ U g[[h]]⊗ and g-invariant, results also in multiplication from the right by a related Q. Relation (4.8) guarantees the existence of D’s in one-to-one correspondence with the d’s, but from the practical viewpoint is not of much help for finding the D’s (since the universal F is unknown and its matrix representations are known only for few representations); the latter can be found more easily from the knowledge of R and a direct study of ˜ .h . n can be The question whether Ainv ±,g,ρ is finitely generated, i.e. whether all I expressed as polynomials in a finite number of them, is part of an important problem originally raised by Hilbert.j This is in general not the case. However, they can be expressed as algebraic functions of a finite number of them [27]. The classification of the latter (for arbitrary g) is not completed yet [28]. We would like to ask here a different question. According to the above proposition, Ihn can be expressed as a power series in h with coefficients in Ainv ±,g,ρ , in other j The fourteenth problem proposed by Hilbert at the 1900 conference of mathematicians.
340
G. FIORE
words as a “function” Ihn = mn (h, {I n }). Since the latter are functions of ai , a+ j , n n ). How do we find m , f ? we can also express Ihn as a function Ihn = f n (h, ai , a+ j We give an explicit answer to the second question by proving (see the appendix): Proposition 5. Ihn = (a+ · · · a+ )Mn (a · · · a)Ln [(ρ⊗
kn
⊗ ρ∨⊗
hn
⊗ σ)
Mn L Jn −1 −1 × (φ−1 (bn −1)bn (bn +1) φ(bn −2),(bn −1)bn ,(bn +1) · · · φ1,2,...,bn ,(bn +1) )]Jn In dIn , n
where φ1,2,...,m,m+1 := (id ⊗ ∆(m−2) ⊗ id)φ123 and bn := hn + kn . Remark 4. Note that in these equations the whole dependence on the twist F is concentrated in the coassociator φ of g and in its coproducts. Consequently, use of formula (2.1.21) allows the explicit determination of the dependence of Ihn ’s on a i , a+ j ; clearly, the latter will be in general highly non-polynomial. Let us now address the first question. If Hh is triangular, then φ−1 and all its coproducts are trivial, the g-invariants u, v, u ˆ, vˆ may be chosen trivial as well [13], and from the previous proposition we find Ihn = I n . If Hh is a genuine quasitriangular Hopf algebra, such as Uh g, then in general Ihn 6= I n ; the Ihn will be some nontrivial function of the I m ’s, generally speaking highly non-polynomial, as well. This can already be verified for the simplest invariants. We will show in next section that e.g. Ih1 = (n)q2 ≡ (I 1 )q2 in the g = sl(N ) case. In general, Ihn = I n + O(h) . DJInn
dIJnn
i
(4.10) A+ i
i
a+ i
This follows from = + O(h), A = a + O(h), = + O(h). i In the g = so(N ), ρ = ρd case, beside δj , another basic isotropic tensor is the classical metric matrix cij = cji (with inverse cij = cji , to which there corresponds the deformed metric matrix [24] Cij , and its inverse C ij ): ij hk C ij = Fhk c = ρd (γ −1 )ih chj . Cij = cih ρd (γ)hj ;
(4.11)
the last two equalities follow from the so(N ) property li ρd (Sx)ij = ρd (x)m l c cmj
x ∈ Ug .
(4.12)
So one can build the invariants I 2,0 := ai cij aj ≡ aca
ij + + + I 0,2 := a+ i c aj ≡ a ca
Ih2,0 := Ai Cji Aj ≡ ACA
ij + + + Ih0,2 := A+ i C Aj ≡ A CA ;
(4.13)
we will see in next section that Ih2,0 6= I 2,0 , Ih0,2 6= I 0,2 . We leave the determination of the general dependence of Ihn on I m ’s as a subject for further investigation. 5. Realization of the Deformed Generators In Sec. 3 we have left some freedom in the definition of Ai , A+ i : the g-invariants u, u ˆ, v, vˆ appearing in the definitions (3.9) of ai , a+ i have not been specified. Can we choose them in such a way that Ai , A+ i fulfil the DCR (deformed commutation relations) of Ah±,g,ρ ? To explicitly study this question in the appendix we prove:
341
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
+ + i i Proposition 6. If we replace A˜i , A˜+ j → A , Aj [with A , Aj defined as in formulae (3.14) and (3.16), with a minimal F ], then Eqs. (2.2.3)–(2.2.5) become equivalent to m l ai aj = ±(M −1 U M )ji lm a a ,
(5.0.1)
+ + + −1 U M )lm a+ ij , i aj = ±al am (M
(5.0.2)
+ i −1 m V M )il a i a+ jm a , j = δj 1A ± al (M
(5.0.3)
ij ij k V ≡ kVhk k are the (numerical ) matrices introduced in where U ≡ kUhk ij k is the σ(U g[[h]])-valued matrix defined by Eqs. (2.2.19) and M ≡ kMhk
M := (ρ ⊗ ρ ⊗ σ)(φm ) .
(5.0.4) ⊗2
We recall that, if ρ = ρd , U is the permutation matrix P and V ∝ P q ρd
( 2t )
.
Remark 5. The above equations have to be understood as equations in the unknowns u, u ˆ, v, vˆ. They can be studied explicitly because the whole dependence on F is concentrated again in the coassociator φ of Uh g. Remark 6. If Hh is a triangular deformation, then U = V = P , φ = 1⊗ (and con3 sequently M = 1⊗ ), and Eqs. (5.0.1) and (5.0.2) are satisfied with trivial invariants + u, u ˆ, v, vˆ, i.e. with ai = ai , a+ i = ai . This was already shown in [13]. To look for solutions of Eqs. (5.0.1)–(5.0.3) for genuine quasitriangular deformations we have to treat the g’s belonging to different classical series separatly. We consider here Ah±,sl(N ),ρd , Ah+,so(N ),ρd . 3
5.1. The case of Ah ±,sl(N ),ρd
PN As a basis of g we choose {Eij }i,j=1,...,N with i=1 Eii ≡ 0 (so that there exist only N 2 − 1 linearly independent Eij ), satisfying [Eij , Ehk ] = Eik δjh − Ejh δik .
(5.1.1)
The quadratic Casimir reads C = Eij Eji ,
(5.1.2)
t = 2Eij ⊗ Eji .
(5.1.3)
implying The matrix representation of Eij in the fundamental representation ρ takes the form δij 1N , ρ(Eij ) = eij − (5.1.4) N where eij is the N ×N matrix with all vanishing entries but a 1 in the ith row and jth P column, and 1N = i eii is the N × N unit matrix; whereas the Jordan–Schwinger realization takes the form: δij j n. (5.1.5) σ(Eij ) = a+ i a − N
342
G. FIORE 2
As a consequence σ(C) = n(N ± n ∓ 1) − nN . From the previous equations one finds (ρ ⊗ ρ ⊗ σ) (ρ ⊗ ρ ⊗ σ) (ρ ⊗ ρ ⊗ σ)
t12 2 t23 2 t13 2
1 1⊗ 1N ⊗ 1N ⊗ 1A =: P − , N N 3
= eij ⊗ eji ⊗ 1A −
i = 1N ⊗ eij ⊗ a+ j a − 1N ⊗ 1N ⊗
2 n n =: A − 1⊗ N ⊗ N N
i = eij ⊗ 1N ⊗ a+ j a − 1N ⊗ 1N ⊗
2 n n =: B − 1⊗ ; N ⊗ N N
P denotes the permutation matrix on CN ⊗ CN , multiplied by 1A . i inv n := a+ i a is an element of A±,sl(N ),ρd [[h]]. We try to solve Eqs. (5.0.1)–(5.0.3) with invariants u, u ˆ, v, vˆ depending only on n. Using relations + [n, a+ i ] = ai
[n, ai ] = −ai ,
(5.1.6)
we can thus commute vˆ to the left of ai and u to the right of a+ i in formula (3.9), + +˜ i i directly in the form a := Ia , a := a I, with I = I(n) ∈ and look for ai , a+ i i i inv inv ˜ ˜ A±,g,ρ [[h]], I = I(n) ∈ A±,g,ρ [[h]]. From Eqs. (3.14) and (3.16) it follows N := + i + i i ˆ ˆ ˜ A+ i A = ai a = nI(n − 1), where I(n) := I(n)I(n). In order that N , Ai , A (n+1)q±2 ˆ satisfies the commutations relations (2.2.13), we therefore require I(n) = , with (x)a :=
ax −1 a−1 .
n+1
Summing up, we leave I(n) undetermined and we pick ai := Iai , +˜ a+ i := ai I
I ≡ I(n) , (n+1)q±2 ˜ . I(n) := I −1 (n) (n+1)
(5.1.7)
These ansatz can also be written in the equivalent form: ai = u(n)ai u−1 (n) ,
+ −1 a+ (n) , i = v(n)ai v
(5.1.8)
Γ(n + 1) Γq2 (n + 1)
(5.1.9)
where u, v are constrained by the relation uv −1 = y = ysl(N ) :=
and Γ, Γq2 are the Euler’s Γ-function and its deformation (A.4.9). We now have the right ansatz to show that the DCR of N -dimensional Uh sl(N )covariant Heisenberg algebra are fulfilled. In the appendix we prove: Theorem 1. When g = sl(N ), the objects Ai , A+ i (i = 1, 2, . . . , N ) defined in formulae (3.14), (3.16) and (5.1.7) satisfy the corresponding DCR (2.2.3)–(2.2.5). √ In [13] the case g = sl(2) was worked out explicitly. Choosing u = v −1 = ysl(2) , we found for the Ai , A+ i ∈ A+,sl(2),ρd [[h]] (i =↑, ↓), q ↑ q ↓ (n )q2 n↓ + (n )q2 + + + q a↑ A↓ = a↓ A↑ = n↑ n↓ (5.1.10) q ↑ q ↓ (n )q2 n↓ (n )q2 ↑ ↑ ↓ ↓ A =a A =a n↑ q n↓ ,
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
343
and for the Ai , A+ i ∈ A−,sl(2),ρd [[h]] ↓
−n + A+ a↑ ↑ = q ↓
A↑ = a↑ q −n
+ A+ ↓ = a↓
(5.1.11)
A↓ = a↓ .
i Here ni := a+ i a (no sum over i). Let us compare the generators (5.1.10) with the ones found in [15]. In our notation the latter would read ↓
n + A+ α↑ = q a↑
Aα↑ = a↑
(n↑ )q2 n↑
+ A+ α↓ = a↓ ↓
qn
Aα↓ = a↓
(n↓ )q2 n↓
(5.1.12) .
It is straightforward to check that the element α ∈ A±,g,ρ [[h]] such that Aαi = + −1 (formulae (1.1)) is αAi α−1 , A+ αi = αAi α s Γ(n↑ + 1)Γ(n↓ + 1) , (5.1.13) α := Γq2 (n↑ + 1)Γq2 (n↓ + 1) where Γ is the Euler Γ-function and Γq2 its q-deformation (A.4.9). 5.2. The case of Ah +,so(N ),ρd As a basis of g = so(N ) we choose {Lij }i,j=1,...,N with Lij = −Lji (so that there exist only N (N2−1) linearly independent Lij ), satisfying [Lij , Lhk ] = Lik cjh + Lkj cih − Lhj cik − Lih cjk ;
(5.2.1)
here cij denotes the (classical) metric matrix on the N -dimensional Euclidean space (cij = cji ), which in the special case we choose real Cartesian coordinates takes simply the form cij = δij . In the rest of this subsection classically covariant indices will be lowered and raised by means of multiplication by c : vi = cij v j v i = cij vj , etc. and v · w := v i wj cij = vi wi = v i wi . The quadratic Casimir reads 1 Lij Lji , 2
(5.2.2)
t = Lij ⊗ Lji .
(5.2.3)
C= implying
The matrix representation of Eij in the fundamental representation ρ takes the form: (5.2.4) ρ(Lij ) = eih chj − ejh chi , and the Jordan–Schwinger realization becomes + h h h + h + lij := σ(Lij ) = a+ i a chj − aj a chi = a ai chj − a aj chi .
It is easy to work out 2 2 N N − 1 − (a+ · a+ )(a · a) , = n+ l2 := σ(C) + 1 − 2 2
(5.2.5)
(5.2.6)
and to check that, as expected [l2 , a+ · a+ ] = 0 = [l2 , a · a] .
(5.2.7)
344
G. FIORE
A direct calculation also shows that + ji ji i [l2 , ai ] = −ai (2n + 1 + N ) + 2(a · a)a+ j c = −a (2n − 3 + N ) + 2aj c (a · a) , + + j + + + + j [l2 , a+ i ] = ai (2n + 3 + N ) − 2cij a (a · a ) = ai (2n − 1 + N ) − 2cij (a · a )a .
We look for “eigenvectors” of l2 l2 αi = αi λ
+ l2 α+ i = αi µ ,
+ + ji j + + in the form αi = ai γ + a+ j c (a · a)δ, αi = ai α + a cji (a · a )β with “eigenvalues” 2 second order equations λ, µ and “coefficients” α, β, γ, δ depending on n, l . We find √ for λ, µ with solutions λ, µ = (l ± 1)2 , where formally l = l2 . We can therefore consistently extend A+,so(N ),ρd by the introduction of a new generator l (whose square is constrained to give the l2 defined in Eq. (5.2.6)) such that N N i − 1 ± l − cij a+ + 1 ± l − (a · a)cij a+ n + (a · a) = a αi± := ai n + j j , 2 2 N + − 1 ± l − (a+ · a+ )cij aj n + α+ := a i,± i 2 N + 1 ± l − cij aj (a+ · a+ ) n + = a+ i 2
satisfy + lα+ i,± = αi,± (l ± 1) ,
lαi± = αi± (l ∓ 1) .
(5.2.8)
+ i After these preliminaries, let us determine the right ai , a+ i ’s for A , Ai to satisfy the DCR. To satisfy at once Eqs. (2.2.14) and (2.2.15) we make the ansatz:
ai = u(n, l)ai u−1 (n, l) ,
+ −1 a+ (n, l) . i = v(n, l)ai v
(5.2.9)
This implies A+ CA+ ACA
(5.2.9)
+ lm a+ = va+ · a+ v −1 , l am c
(5.2.10)
(5.2.9)
al am clm = ua · au−1 .
(5.2.11)
= =
The DCR determine only the product y := v −1 u; we are going to show now that Eqs. (2.2.16) and (2.2.17) completely determine the latter. It is immediate to check that the former implies + + i −1 (n + 2, l) − q 2 y(n − 2, l)(a+ · a+ )ai y −1 = (1 + q N −2 )cij a+ y[2cij a+ j (a · a )a ]y j . + + j Expressing a+ i , cij (a · a )a as combinations of αi± we easily move y past the c “eigenvectors” αi± of n, l; factoring out (from the right) 2lij we end up with a LHS being a combination of αi,+ , αi,− . Therefore Eq. (2.2.16) amounts to the condition that the corresponding coefficients vanish:
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
345
N N −2 + 1 − l y(n + 1, l + 1)y −1 (n + 2, l) (1 + q ) = n+ 2 N − 1 − l y(n − 1, l − 1)y −1 (n, l) . − q2 n + 2 N + 1 + l y(n + 1, l + 1)y −1 (n + 2, l) (1 + q N −2 ) = n + 2 N − 1 − l y(n − 1, l − 1)y −1 (n, l) . − q2 n + 2 Similarly, from Eq. (2.2.17) it follows that: N N −2 + 1 − l y(n, l)y −1 (n + 1, l − 1) ) = n+ (1 + q 2 N 2 − 1 − l y(n − 2, l)y −1 (n − 1, l − 1) −q n+ 2 N N −2 + 1 + l y(n, l)y −1 (n + 1, l + 1) (1 + q ) = n+ 2 N − 1 + l y(n − 2, l)y −1 (n − 1, l + 1) . − q2 n + 2 It is straightforward to check that the last four equations are solved by −n 1 + q N −2 −1 uv = y = yso(N ) := 2 N N 1 1 n+ +1−l Γ n+ +1+l Γ 2 2 2 2 , × N N 1 1 n+ + 1 − l Γq 2 n+ +1+l Γq 2 2 2 2 2
(5.2.12)
where Γ, Γq2 are the Euler’s Γ-function and its deformation (A.4.9). We now have the right ansatz to fulfill the DCR of N -dimensional Uh so(N )covariant Weyl algebra. We state without proof: Theorem 2. When g = so(N ), the objects Ai , A+ i (i = 1, 2, . . . , N ) defined in formulae (3.14), (3.16), (5.2.9) and (5.2.12) satisfy the corresponding DCR (2.2.3)– (2.2.5). 6. ∗-Structures Given the Hopf ∗-algebra Hh = (Uh g, m, ∆h , εh , Sh , R , ∗h ), we ask now whether the ∗-structures †h of Ah±,g,ρ compatible with the action .h of Uh g, i.e. such that (x˜ .h a)†h = Sh−1 (x∗h )˜.h a†h , can be naturally realized by the ones of A±,g,ρ .
(6.1)
346
G. FIORE
We stick to the case that Hh is the compact real section of Uh g. Then Uh g as an ˆ [[h]], where g ˆ is the compact section of g and h ∈ R, and algebra is isomorphic to U g the trivializing maps ϕh intertwine between ∗h and ∗, [ϕh (x)]∗ = ϕh (x∗h ) where ∗ is the classical ∗-structure in U g having the elements of gˆ as fixed points. If F is unitary then the corresponding γ, γ 0 , φ clearly satisfy γ0 = γ∗
φ∗⊗∗⊗∗ = φ−1 .
(6.2)
On the other hand, it is evident that the “minimal” coassociator φm (2.1.16) is also unitary (because h ∈ R); one could actually show that the unitary F is also minimal. If ρh is a ∗-representation of H, the ∗-structure (A˜i )†h = A˜+ i is clearly compatible with ˜ .h (condition (6.1)); the classical counterpart of ρh is also a ∗-representation ρ of H (i.e. ρ(x∗ ) = ρT (x)), and formula (ai )† = a+ i
(6.3)
defines in A±,g,ρ a ∗-structure (“hermitean conjugation”) † compatible with .. Correspondingly, it is immediate to check that σ, σϕh become ∗, ∗h -homomorphisms respectively, (6.4) σ ◦ ∗ = † ◦ σ σϕh ◦ ∗h = † ◦ σϕh , and .h as defined in formula (3.4) also satisfies (6.1). Under † the RHS of relations (3.14) and (3.15) are mapped into the RHS of relations (3.16) and (3.17), provided (ai )† = a+ i ;
(6.5)
(Ai )† = A+ i .
(6.6)
in this case we find, as requested
If g = sl(N ), so(N ) and ρ = ρd condition (6.5) is satisfied by choosing (√ ysl(N ) if g = sl(n) , −1 v =u= √ yso(N ) if g = so(n) .
(6.7)
A+,so(N ),ρd admits also an alternative ∗-structure compatible with .h , namely + † ji ˜ [24] together with a nonlinear equation for (A˜i )†h [30] which we (Ai ) = A˜+ j C omit here; in this case one usually denotes the generators by Xi , ∂ i instead of ˜i A˜+ i , A , because in the classical limit they become the Cartesian coordinates and partial derivatives of the N -dim Euclidean space respectively. The classical limit of this †h is + † (ai )† = −cij aj ; (6.8) (a+ i ) = aj cji using relations (6.8) and (4.12), tr(ρd ) = 0, Sx = −x if x ∈ g, one finds again relations (6.4). .h as defined in formula (3.4) also satisfies (6.1). Under † the RHS of relation (3.14) is mapped into the RHS of relations (3.15), provided that + † (a+ i ) = aj cji , i.e. (6.9) v = 1 u = yso(N ) ; in this case we find, as requested + † (A+ i ) = Aj Cji ,
(6.10)
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
347
and it is not difficult to show that (Ai )† is the (nonlinear) function of Ai , A+ i which was found in [30]. 7. Outlook, Final Remarks and Conclusions + i Given some solutions ai , a+ i (in the form (3.9)) of Eqs. (5.0.1)–(5.0.3), the A , Ai defined through formulae (3.6) (where we choose a minimal F) satisfy the deformed commutation relations of Ah±,g,ρ and are covariant under the Uh g action .h defined in formula (3.4). The corresponding basic algebra homomorphism f : Ah±,g,ρ → + A±,g,ρ [[h]] is defined iteratively starting from f (A˜i ) := Ai , f (A˜+ i ) := Ai . Explicit i + solutions a , ai of Eqs. (5.0.1)–(5.0.3) are given by
• formulae (5.1.8) and (5.1.9) for Ah±,sl(N ),ρd ; • formulae (5.2.9) and (5.2.12) for Ah+,so(N ),ρd . The main feature of such a realization (f, .h ) is that the g-invariant ground state |0i as well as the first excited states a+ i |0i of the classical Fock space representation are also respectively Uh g-invariant ground state |0h i and first excited states A+ i |0h i of the deformed Fock space representation. According to relation (1.1), all other elements of A±,g,ρ [[h]] satisfying the DCR of Ah±,g,ρ can be written in the form: Aαi = αAi α−1
+ −1 A+ , α i = αAi α
(7.1)
with α = 1A + O(h) ∈ A±,g,ρ [[h]]. They are manifestly covariant under the Uh gaction .h α defined by x .h,α a := ασϕh (x(¯1) )aσϕh (x(¯2) )α−1 .
(7.2)
For these realizations the deformed ground state in the Fock space representation reads |0h i = α|0i; thus in general, the g-invariant ground state and first excited states of the classical Fock space representation do not coincide with their deformed counterparts. In this way we have found all possible pairs (fα , .h,α ) making the diagram (1.2) in the introduction commutative. Note that the change ϕh → ϕh,v = vϕh (·)v −1 (formula (2.1.22)) of the algebra isomorphism Uh g → U g[[h]] amounts to the particular transformation (f, .h ) → (fα , .h,α ), with α = σ(v). In Sec. 4 we have shown (formula (4.5)) how to construct g-invariants Ihn ∈ A±,g,ρ [[h]] in the form of homogeneous polynomials in Ai , A+ i . It is immediate to verify that under a transformation (f, .h ) → (fα , .h,α ) these Ihn transform into Ihαn := αIhn α−1 . In Sec. 6 we have shown (sticking to the explicit case of Ah±,sl(N ),ρd and h A+,so(N ),ρd ) that, if Ah±,g,ρ is a module ∗-algebra (formula (6.1)) of the compact section of Uh g(q > 1), then one can choose (f, .h ) so that f is a ∗-homorphism, f (b†h ) = [f (b)]† , and .h also satisfies Eq. (6.1). It is straighforward to verify that (fα , .h,α ) satisfy the same constraints provided that α is “unitary”, α† = α−1 . Summing up, in the present work we have shown how to realize a deformed Uh gcovariant Weyl or Clifford algebra Ah±,g,ρ within the undeformed one A±,g,ρ [[h]].
348
G. FIORE
Given a deforming map f and a representation (π, V ) of A±,g,ρ on a vector space V , does (π ◦ f, V ) also provide a representation of Ah±,g,ρ ? In other words, can one interpret the elements of Ah±,g,ρ as operators acting on V , if the elements of A±,g,ρ are? If so, which specific role do the elements Ai , A+ i of A±,g,ρ [[h]] play? In view of the specific example we have examined in [13] the answer to the first question seems to be always positive, whereas the converse statement is wrong: e.g. for any h ∈ R, there are more (inequivalent) representations of the deformed ¯ algebra than representations of the undeformed one. This may seem a paradox, because in a h-formal-power sense f −1 can be defined and gives f −1 (ai ) = A˜i + O(h), ˜+ f −1 (a+ i ) = Ai + O(h), whence at least for small h one would expect the deformed and undeformed representation theories to coincide; but in fact there is no warranty that, also in some operatorial sense, for small h the would-be f −1 (ai ), f −1 (a+ i ) are . Of course, we are especially interested in Hilbert space represen“close” to A˜i , A˜+ i tations of ∗-algebras: in [13] we checked that in the operator-norm topology f −1 is ill-defined on all but exactly one deformed representation. Roughly speaking, the reason is that the “particle number” observables ni , which enter the transformation f (see e.g. (5.1.10)) are unbounded operators, therefore even for very small h the effect of the transformation on their large-eigenvalue eigenvectors can be so large to “push” the latter out of the domain of definition of the operators in f −1 (A±,g,ρ ). We are especially interested in the case of ∗-algebras admitting Fock space representations. The results presented in the previous paragraphs could in principle be applied to models in quantum field theory or condensed matter physics by choosing representations ρ which are the direct sum of many copies of the same fundamental representation ρd ; this is what we have addressed in [22]. The different copies would correspond respectively to different space(time)-points or crystal sites. One important physical issue is if Uh g-covariance necessarily implies exotic particle statistics. In view of what we have said the answer is no [31]. At least for compact g and Uh g (h is real), the undeformed Fock space representation, which allows a “Bosons & Fermions” particle interpretation, would carry also a representation of the deformed one. Next point is the role of the operators Ai , A+ j . Quadratic i , A act as creators commutation relations of the type (2.2.3)–(2.2.5) mean that A+ i and annihilators of some excitations; a glance at (3.6) and (7.1) shows that these are not the undeformed excitations, but some “composite” ones. The last point is: What could these operators be good for? As an Hamiltonian H of the system we may choose a simple combination of the Uh g-invariants Ihn of Sec. 4; thus the Hamiltonian is Uh g-invariant and has a simple polynomial structure in the composite operators Ai , A+ j . H is also g-invariant, but has a highly non-polynomial structure in the undeformed generators ai , a+ j (it would be tempting to understand what kind of physics it could describe!). This suggests that the use of the Ai , A+ j instead of the ai , a+ j should simplify the resolution of the corresponding dynamics (similar to what has been suggested in [32] for a 1-dim toy-model). Acknowledgments I thank J. Wess for his stimulating remarks in the early stages of this work, as well for the warm hospitality at his institute. I also thank Prof. B. Zumino for his
349
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
warm hospitality during a short visit at LBNL, where some involved computation of this work was completed. This work was supported in part through a TMR fellowship granted by the European Commission, Dir. Gen. XII for Science, Research and Development, under the contract ERBFMICT960921, and in part through the U.S. Dept. of Energy under contract No. DE-AC03-76SF00098 (during my visit at LBNL). Appendix A. Appendix A.1. Proof of Lemma 1 Lemma 2 [13]. If T ∈ U g[[h]]⊗ is g-invariant (i.e. [T , U g[[h]]⊗ ] = 0) then 2 mij Si T , mij Sj T (i, j = 1, 2, 3, i 6= j) are g-invariants belonging to U g[[h]]⊗ . 3
3
(Here Si denotes S acting on the ith tensor factor, and mij multiplication of the ith tensor factor by the jth from the right.) We may apply the previous lemma to T = φ, or T = φ−1 . Looking at Definition 2.1.9 one finds in particular the following g-invariants: (2)
(2)
T1 := m12 S1 φ = (SF (1) γ ⊗ 1)F (F(1) ⊗ F(2) ) , −1(1)
T2 := m23 S3 φ = (F(1)
−1(1)
⊗ F(2)
)F −1 (1 ⊗ γ −1 SF −1(2) ) ;
(A.1.1)
alternative expressions for these Ti can be obtained by applying the same operations to the identities φq q−
t12 +t13 2
t13 +t23 2
=q
t12 2
−1 −1 F23,1 F12 R 13 F32 F1,23 ,
−1 −1 − φ = F12,3 F21 R −1 13 F23 F3,12 q
t23 2
(A.1.2) ,
(A.1.3)
which directly follow from relations (2.1.15), (2.1.9) and (2.1.7) and the observation t12 +t13 +t23 2 that [φ, q ] = 0. Applying m12 S1 to (A.1.3), m23 S3 to (A.1.2) we get T1 = (SF (2) γ 0−1 ⊗ 1)F21 (F(1) ⊗ F(2) ) , (1)
−1(2)
T2 = (F(1)
−1(2)
⊗ F(2)
(1)
−1 )F21 (1 ⊗ γ 0 SF −1(1) ) .
(A.1.4)
From Eqs. (A.1.1) and (A.1.4) we easily find that the inverse of Ti take the form: −1(2)
T1−1 = F −1 [γ −1 (SF −1(1) )F(1)
−1(1)
−1 0 = F21 [γ (SF −1(2) )F(1)
−1(2)
⊗ F(2)
−1(1)
⊗ F(2)
T2−1 = [F(1) ⊗ F(2) (SF (2) )γ]F (1)
(1)
= [F(1) ⊗ F(2) (SF (1) )γ 0−1 ]F21 , (2)
±1(j)
±1(j)
(2)
],
]
(A.1.5) (A.1.6) (A.1.7) (A.1.8)
since [Ti , F(1) ⊗ F(2) ] = 0, with i, j = 1, 2. If F is minimal then it is easy to verify that, according to (2.1.16) and their 3 definitions (A.1.1), Ti ≡ 1⊗ . In the latter case the last four relations are equivalent to relations (3.10)–(3.13).
350
G. FIORE
Appendix A.2. Proof of Proposition 5 + We start by expressing (A · · · A)In ≡ Ai1 · · · Aihn , (A+ · · · A+ )Jn ≡ A+ j1 · · · Ajkn , respectively in the form (a · · · a)σ(·), (a+ . . . a+ )σ(·). First note that 0
(3.17)
Ai1 Ai2
0
=
ρ(γ −1 SF −1(1) )il11 ρ(γ −1 SF −1(1 ) )il22 al1 σ(F −1(2) )al2 σ(F −1(2 ) )
=
ρ∨ (F −1(1) γ 0 )li11 ρ∨ (F −1(1 ) γ 0 )li22 al1 σ(F −1(2) )al2 σ(F −1(2 ) )
0
(3.5),(2.2.2)
−1(2)
0
0
−1(2)
=
ρ∨ (F −1(1) γ 0 )li11 ρ∨ (F(1)
F −1(1 ) γ 0 )li22 al1 al2 σ(F(2)
=
∨l2 −1 −1 0⊗ 1 al1 al2 [(ρ∨l ⊗ 1))] i1 ⊗ ρi2 ⊗ σ)(F1,23 F23 (γ
0
F −1(2 ) )
2
whence, by repeated application, we find (4.9)
(A · · · A)In = (a · · · a)Ln [((ρ∨⊗
hn
−1 0⊗ n )L In ⊗ σ)(F12,...,(hn +1) (γ
hn
⊗ 1))] ;
similarly, starting from relation (3.14) we find −1 n (A+ · · · A+ )Jn = (a+ · · · a+ )Mn [((ρ⊗ )M Jn ⊗ σ)(F12,...,(kn +1) )] . kn
Putting these results together we find (A+ · · · A+ )Jn (A · · · A)In (2.2.2),(3.5)
=
(a+ · · · a+ )Mn (a · · · a)Ln ∨⊗ n × [((ρ⊗ )M Jn ⊗ (ρ kn
hn
−1 ⊗ n )L In ⊗ σ)(F12,...,(bn +1) (1
kn
⊗ γ 0⊗
hn
⊗ 1))] ,
whence, (4.8)
Ihn ∝ (a+ · · · a+ )Mn (a · · · a)Ln ∨⊗ n × [((ρ⊗ )M Jn ⊗ (ρ kn
hn
Ln −1 n )L In ⊗ σ)F12,...,(bn +1) F12,...,bn ,bn +1 ]dMn .
(A.2.1)
We prove now that −1 −1 −1 −1 F12,...,b = φ−1 F12,...,(b+1) (b−1)b(b+1) φ(b−2),(b−1)b,(b+1) · · · φ1,2,...,b,(b+1) F12,...,m,m+1 ;
(A.2.2) then the claim will follow from relation (A.2.1) and the observation that [(ρ⊗
kn
⊗ ρ∨⊗
hn
(4.6)
(1) n Ln Ln ⊗ id)(F12,...,bn ,bn +1 )]M )F (2) dIJnn Jn In dMn = ε(F
(2.1.1)
=
dIJnn . (A.2.3)
To prove relation (A.2.2) we start from −1 φ−1 123 F12,3
(2.1.9)
=
(4.9)
−1 −1 −1 F1,23 F23 F12 = F123 F12 ;
351
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
this is relation (A.2.2) for b = 2. Applying id ⊗ ∆ ⊗ id and multiplying the result from the left by φ−1 234 we find −1 −1 φ−1 234 φ1,23,4 F123,4
−1 −1 φ−1 234 F1,234 F2,34 F1,23
= (2.1.10)
−1 −1 F1,234 φ−1 234 F2,34 F1,23
(2.1.9)
−1 −1 −1 F1,234 F2,34 F34 F23 F1,23 ,
= =
(4.9)
=
−1 F1233 F123 ,
i.e. relation (A.2.2) for b = 3. Applying to the latter relation id ⊗ ∆(2) ⊗ id and multiplying the result from the left by φ−1 345 we find relation (A.2.2) for b = 4, and so on. Appendix A.3. Proof of Proposition 6 (2.2.3)
Ai Aj ∓ P F hk Ak Ah
(3.16)
(1) h (1 ∓ P F )ji )l ρ(F (1 ) )km σ(F (2) )am σ(F (2 ) )al hk ρ(F
ji
=
0
0
=
(3.5),(2.2.2)
=
0
(20 )
(20 )
(2)
(1) (1 ∓ P F )ji F(20 ) )km ρ(F (1) )hl σ(F(1) F(10 ) )am al ; hk ρ(F
−1 −1 F23 ) and noting that multiplying both sides from the left by (ρ ⊗ ρ ⊗ σ)(F1,23
P F 12
(2.2.11),(2.2.19)
=
−1 −1 [(ρ ⊗ ρ ⊗ σ)F12 F12,3 ]U12 [(ρ ⊗ ρ ⊗ σ)F12,3 F12 ],
we find −1 −1 −1 −1 m l F23 F12 F12,3 ]U12 [(ρ⊗ ⊗ σ)F12,3 F12 F23 F1,23 )]}hk {1⊗ ∓ [(ρ⊗ ⊗ σ)F1,23 lm a a = 0 , 3
2
2
i.e. relation (5.0.1), once we take definitions (5.0.4) and (2.1.9) into account. Using definition (3.14) one can prove in a similar way that relations (2.2.4) and (5.0.2) are equivalent. Similarly, 0
(2.2.5)
=
(3.16),(3.14)
=
i ˜ F ih + k Ai A+ j − δj 1A ∓ Pjk Ah A 0
0
−1(2 ) ρ(F (1) )il σ(F (2) )al a+ )ρ(F −1(1 ) )m m σ(F j 0
−1(2) (1 ) k ˜ F ih − δji 1A ∓ a+ )ρ(F −1(1) )m )l σ(F (2) )al ; m σ(F h Pjk ρ(F 0
0
multiplying both sides by ρ(F −1(1) )ii σ(F −1(2) ) from the left and by ρ(F (1 ) )jj 0 0 σ(F (2 ) ) from the right, and noting that F P˜12
(2.2.12),(2.2.19)
=
−1 −1 [(ρ⊗ ⊗ σ)F12 F12,3 ]V12 [(ρ⊗ ⊗ σ)F12,3 F12 ], 2
2
352
G. FIORE
we get 0
0
0
0
−1 i −1(1) i ⊗ )i σ(F −1(2) )a+ ⊗ σ)F13 F12 F12,3 ] ai a+ m {[(ρ j 0 − δj 0 1A ∓ ρ(F
=
2
0
0
−1 −1 l (1 ) j F12 F13 ]}im )j 0 σ(F (2 ) ) × V12 [(ρ⊗ ⊗ σ)F12,3 jl a ρ(F 2
(3.5),(2.2.2)
=
0
0
−1 −1 i + ⊗ a i a+ ⊗ σ)F1,23 F13 F12 F12,3 ] j 0 − δj 0 1A ∓ am {[(ρ 2
0
−1 −1 l F12 F13 F1,23 ]}ij 0m × V12 [(ρ⊗ ⊗ σ)F12,3 l a , 2
whence the equivalence between relations (2.2.5) and (5.0.3) follows, once one recalls Definition 2.1.9. Appendix A.4. Some properties of special and q-special functions The following results can be found in standard textbooks. If the parameters a, b, c ∈ C are such that none of the quantities c − 1, a − b, a + b − c is a positive integer, the general solution of the hypergeometric differential equation in the complex z-plane y 00 (1 − z)z + y 0 [c − (a + b + 1)z] − yab = 0
(A.4.1)
can be expressed as some combinations y(z) = αF (a, b, c; z) + βz 1−c F (1 + a − c, 1 + b − c, 2 − c; z) ,
(A.4.2)
= γF (a, b, a + b + 1 − c; 1 − z) + δ(1 − z)c−a−b F (c − a, c − b, c + 1 − a − b; 1 − z) ,
(A.4.3)
where α, β, γ, δ ∈ C and F (a, b, c; z) is the hypergeometric function. As known, F (a, b, c; 0) = 1 ,
(A.4.4)
ab d F (a, b, c; z) = F (a + 1, b + 1, c + 1; z) . dz c
(A.4.5)
The combinations (A.4.2) and (A.4.3) explicitly display the singular and nonsingular part of the solution respectively around the poles x = 0, 1. An essential identity to determine the asymptotic behaviour of a solution y around the pole x = 0 (resp. x = 1), known its asymptotic behaviour around the pole x = 1 (resp. x = 0), is F (a, b, c; z) =
B(c, a + b − c) B(c, c − a − b) F (a, b, a + b + 1 − c; 1 − z) + B(c − a, c − b) B(a, b) × (1 − z)c−a−b F (c − a, c − b, c + 1 − a − b; 1 − z) .
(A.4.6)
Here Γ(a) and B(a, b) are Euler’s Γ- and β-functions respectively; as known, Γ(a + 1) = aΓ(a) B(a, b) =
Γ(a)Γ(b) . Γ(a + b)
(A.4.7)
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
353
A less obvious property is Γ(a)Γ(−a) = −
π . a sin πa
(A.4.8)
The q-gamma function Γq can be defined when |q| < 1 by [33] Γq (a) := (1 − q 1−a )
∞ ∞ Y X (1 − q k+1 ) (q 1−a ; q)n na 1−a = (1 − q ) q , (1 − q a+k ) (q a ; q)n n=0
(A.4.9)
k=0
Qn−1 where (a; q)n := k=0 (1 − aq k ); it satisfies the following modified version of the property (A.4.7)1 : Γq (a + 1) = (a)q Γq (a) ,
(q a − 1) . (q − 1)
(a)q :=
(A.4.10)
We also introduce a different version of the q-gamma function by ˜ q (a) := Γq2 (a)q − Γ
a(a−3) 2
;
(A.4.11)
the latter satisfies ˜ q (a) , ˜ q (a + 1) = [a]q Γ Γ
[a]q :=
(q a − q −a ) . (q − q −1 )
(A.4.12)
Appendix A.5. Proof of Theorem 1 Proof. We need to show that Eqs. (5.0.1)–(5.0.3) are fulfilled. We get rid of indices by introducing the following vector notation: (aa)ij := ai aj
+ (a+ a+ )ij := a+ i aj
(av)ij := ai v j
(va)ij := v i aj
(a+ w)ij := a+ i wj
(wa+ )ij := wi a+ j
w · a := wi ai
i a+ · v := a+ i v ,
where v ≡ (v i ) ∈ CN , w ≡ (wi ) ∈ CN denote arbitrary covariant and controvariant vectors, respectively. If we plug (5.1.7) into (5.0.1)–(5.0.3), factor out of (5.0.1) and ˜ I(n ˜ + 1) respectively, multiply Eq. (5.0.3) by v j wi , (5.0.2) I(n)I(n + 1) and I(n) then we find the equivalent system (in vector notation): aa = ±(M −1 P M )aa
(A.5.1)
a+ a+ = ±a+ a+ (M −1 P M )
(A.5.2)
(n + 1)q±2 nq±2 + (w · a) · (a+ · v) = w · v1A ± q ±1 wa (M −1 V M )va . (n + 1) n
(A.5.3)
It is straightforward to show that Ava = ±(n − 1)va wa+ P = a+ w
a+ wA = −wa+ ∓ a+ w + (w · a)a+ a+ a+ wP = wa+ .
(A.5.4)
354
G. FIORE
As a consequence one finds, in particular, a+ w(A + P )k = (∓1)k a+ w ±
(±n)k − (∓1)k (w · a)a+ a+ ⇒ n+1
q ∓n − q ∓1 (w · a)a+ a+ . (A.5.5) n+1 Let us prove Eqs. (A.5.1) and (A.5.2). The matrix M (5.0.4) takes the form: Z 1−y0 A P (2.1.21) −2~P ~ 2~A + y0 x0 ; M = lim dx P exp −2~ x x−1 x0 ,y0 →0+ x0 a+ wq A+P = q ±1 a+ w ±
(A.5.6) ⊗3
the contributions of the central terms −2 1N , − N2 1⊗ ⊗ n to the integral are can12 , y0~t23 , in the limit x0 , y0 → celled by the corresponding contributions from x−~t 0 0+ . Since aa is an “eigenvector” both of A and P , the path-order P~ becomes redundant and we find that M acts trivially on M : Z 1−y0 1 n−2 ±2~(n−2) ∓2~ y0 dx ± ± M aa = aa lim + x0 exp −2~ x x−1 x0 ,y0 →0 x0 = aa
lim
x0 ,y0 →0+
3
(1 − y0 )∓2~ (1 − x0 )±2~(n−2) = aa .
(A.5.7)
Therefore M P M aa = ±aa. Similarly one proves Eq. (A.5.2).
In order to prove Eq. (A.5.3) it is convenient to recast M −1 V M in a more manageable form. Permuting the second and third tensor factor in Eq. (2.1.15)(1) , we find t12 t12 +t13 t23 2 φ 2 φ132 q − 2 , (A.5.8) φ−1 123 = q 213 q whence ⊗2
−1 ±1+ N +ρd q M −1 V M = P M21 1
( 2t )
M
(2.1.19),(5.1.6)
=
P q A+P lim+ x−2B f (x)q −A , x→0
(A.5.9) where f is the σ(U sl(N ))[[h]]-valued N 2 × N 2 matrix satisfying the differential equation and asymptotic conditions A B 0 + f lim f (x)(1 − x)−2A = 1 (A.5.10) f = 2~ x→1 x x−1 (the latter are obtained from Eq. (2.1.17) by permuting the second and third tensor factor and by getting rid of the central terms involved in (ρ ⊗ ρ ⊗ σ)(tij ) (formulae (5.1.6)) since, as in formula (A.5.6), the latter cancel with each other in the limit x → 0). It is convenient to introduce in A±,g,ρ [[h]] a grading g, by setting g(b) = l ∈ Z iff [n, b] = lb, b ∈ A±,g,ρ [[h]]. Since g(f va) = −1, and f v i aj is a doubly contravariant tensor, its most general expansion is f (x)va = avf1 (x) + vaf2 (x) + aa(a+ · v)f3 (x) , where fi are invariants with g(fi ) = 0; therefore fi = fi (n). Thus we find
(A.5.11)
355
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
wa+ · (M −1 V M )va (A.5.4),(A.5.9)
=
a+ wq A+P lim+ x−2~B f (x)q −A va x→0
(A.5.4),(A.5.5)
=
∓2~(n−1)
lim x
x→0+
q ±n − q ∓1 ∓1 + + + (w · a)a a q a w± n+1
× (f (x)va)q ∓(n−1)
q ±n − q ∓1 (w · a)a+ a+ q ∓(n−1) lim x∓2~(n−1) q ∓1 a+ w ± n+1 x→0+
(A.5.11)
=
× [avf1 (x) + vaf2 (x) + aa(a+ · v)f3 (x)] q ∓(n−1) lim+ x∓2~(n−1) q ∓1 (w · v)(nf1 ∓ f2 ) + (w · a)(a+ · v)
=
x→0
q ±n − q ∓1 (f1 ± f2 + (n + 1)f3 ) × q (nf3 ± f2 ) + n n+1 q ±(n+1) − 1 l3 , q ∓n (w · v)l1 + (w · a)(a+ · v) l2 + n n+1
=
∓1
(A.5.12) where we have defined l1 := lim+ x∓2~(n−1) (nf1 (x) ∓ f2 (x)) x→0
l2 := lim+ x∓2~(n−1) (nf3 (x) ± f2 (x)) x→0
l3 := lim+ x∓2~(n−1) [f1 (x) ± f2 (x) + (n + 1)f3 (x)] .
(A.5.13)
x→0
To evaluate the limits li let us consider the linear system of first order differential equations satisfied by fi . From (A.5.10) we find n−1 f2 1 0 + f1 − (A.5.14) f1 = ~ ± 1−x x x
n−1 1 + 1−x x
f20
f1 ∓ =~ 1−x
f30
f2 1 f1 1 + ∓ − (n − 1)f3 =~ ∓ 1−x x 1−x x
f2
(A.5.15)
(A.5.16)
and the asymptotic condition lim f1 (x) = 0 = lim f3 (x)
x→1
x→1
lim f2 (x)(1 − x)∓2~(n−1) = 1 .
x→1
(A.5.17)
The first two equations can be solved separately; then the third will yield f3 in terms of f1 , f2 just by an integration. Actually one of the combination we are interested in, [f1 (x)±f2 (x)+(n+1)f3 (x)], satisfies a completely decoupled equation,
356
G. FIORE
d [f1 (x) ± f2 (x) + (n + 1)f3 (x)] dx 1 1 + [f1 (x) ± f2 (x) + (n + 1)f3 (x)] , = ±2~(n − 1) x x−1 which (taking conditions (A.5.17) into account) is easily integrated to f1 (x) ± f2 (x) + (n + 1)f3 (x) = ±[x(1 − x)]±2~(n−1) .
(A.5.18)
This will yield therefore f3 in terms of f1 , f2 . Dividing (A.5.14) by f1 , (A.5.15) by f2 we find n−1 1 f2 1 f10 + − (A.5.19) = 2~ ± f1 1−x x x f1 f1 f20 1 n−1 1 + (A.5.20) = 2~ ∓ f2 (1 − x) f2 1−x x taking the difference of the two, one finds a Riccati equation in the unknown u := d u0 = ln u dx
f1 f2
=
f1 f2 :
1 u−1 u f10 f0 1 + − − ; − 2 = 2~ ±n f1 f2 x 1−x x 1−x (A.5.21) x→1
this should be supplemented with the condition u → 0. To get rid of its nonlinearity one can transform it into a (linear) second order equation in an unknown y(x) by a standard substitution, which in this case takes the form: u=
y 0 (1 − x) ; y 2~
(A.5.22)
the new equation will read y 00 (1 − x)x − y 0 (x ± n2~) + (2~)2 y = 0 .
(A.5.23)
We recognize the hypergeometric equation (formula (A.4.1)) with parameters a = ±2~ b = ∓2~ c = ∓2n~ .
(A.5.24)
Its general solution can be expressed in the form (A.4.3), in terms of the hyperge0 ometric function F . Imposing the condition limx→1 y (1−x) = 0 one finds that it 2~y must be δ = 0, implying 1−x d f1 ln[F (±2~, ∓2~, 1 ± 2~n; 1 − x)] . =u= f2 2~ dx We can now replace this result in the RHS in Eq. (A.5.20): d d n−1 1 ln(f2 ) = ln[F (±2~, ∓2~, 1 ± 2~n; 1 − x)] ∓ 2~ + ; dx dx 1−x x
(A.5.25)
(A.5.26)
357
DRINFEL’D TWIST AND q-DEFORMING MAPS FOR LIE GROUP
taking into account the condition (A.5.17), the latter is integrated into f2 (x) = x∓2~ (1 − x)±2~(n−1) F (±2~, ∓2~, 1 ± 2~n; 1 − x) .
(A.5.27)
Finally, we find f1 (x) = u(x)f2 (x) = −
1 0 F (±2~, ∓2~, 1 ± 2~n; 1 − x)x∓2~ (1 − x)1±2~(n−1) . 2~ (A.5.28)
From properties (A.4.4) and (A.4.5) we can easily read off the asymptotic behaviour of f2 , f1 for x → 0+ : f2 (x)
(A.4.6)
=
x∓2~ (1 − x)±2~(n−1)
B(1 ± 2~n, 1 ± 2~n) B(1 ± 2~(n + 1), 1 ± 2~(n − 1))
B(1 ± 2~n, −1 ∓ 2~n) B(±2~, ∓2~) × F (1 ± 2(n + 1)~, 1 ∓ 2~(n − 1), 2 ± 2~n; x)
× F (±2~, ∓2~, ∓2~n; x) + x1±2~n
(A.4.4),(A.4.7)
≈
f1 (x)
(A.4.5)
=
(A.4.6)
=
x∓2~
Γ(1 ± 2~n)Γ(1 ± 2~n) Γ(1 ± 2~(n + 1))Γ(1 ± 2~(n − 1))
2~ F (1 ± 2~, 1 ∓ 2~, 2 ± 2~n; 1 − x)x∓2~ (1 − x)1±2~(n−1) 1 ± 2~n 2~ B(2 ± 2~n, ±2~n) x∓2~ (1 − x)1±2~(n−1) 1 ± 2~n B(1 ± 2~(n + 1), 1 ± 2~(n − 1)) B(2 ± 2~n, ∓2~n) B(1 ± 2~, 1 ∓ 2~) × F (1 ± 2(n + 1)~, 1 ∓ 2~(n − 1), 1 ± 2~n; x)
× F (1 ± 2~, 1 ∓ 2~, 1 ∓ 2~n; x) + x±2~n
(A.4.4)
≈
2~ B(2 ± 2~n, ±2~n) ∓2~ x 1 ± 2~n B(1 ± 2~(n + 1), 1 ± 2~(n − 1)) ± 2~n, ∓2~n) B(1 ± 2~, 1 ∓ 2~)
±2~n B(2
+ x (A.4.7)
=
∓2~
±
x
1 Γ(1 ± 2~n)Γ(1 ± 2~n) n Γ(1 ± 2~(n + 1))Γ(1 ± 2~(n − 1)) ± 2~n)Γ(∓2~n) . Γ(1 ± 2~)Γ(1 ∓ 2~)
±2~n Γ(1
+ 2~x
358
G. FIORE
For the combination nf1 ∓ f2 we thus find nf1 ∓ f2
Γ(1 ± 2~n)Γ(∓2~n) Γ(1 ± 2~)Γ(1 ∓ 2~)
≈
2~x±2~(n−1)
(A.4.7)
∓nx±2~(n−1)
(A.4.8)
∓
x→0
= =
Γ(±2~n)Γ(∓2~n) Γ(±2~)Γ(∓2~)
q − q −1 ±2~(n−1) 1 ±2~(n−1) x =∓ x . n −n q −q [n]q
The limits li are thus given by l1 = ∓
1 [n]q
l3 = ±1
(A.5.29)
1 n 1 n l3 − l1 = ± 1+ , l2 = n+1 n+1 n+1 [n]q which plugged into Eq. (A.5.12) give (w · v) n 1 + −1 ∓n + ±(n+1) wa · (M V M )va = q − + (w · a)(a · v) +q [n]q n + 1 [n]q =∓
(n + 1)q±2 n q ∓1 ± (w · a)(a+ · v)q ∓1 ; nq±2 (n + 1) nq±2
Eq. (A.5.3) is manifestly satisfied once we replace the latter result in it.
(A.5.30)
References [1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Ann. Phys. 111(61) (1978) 111. [2] F. du Cloux, “Asterisque”, Soc. Math. France 124 125 (1985) 129. [3] M. Gerstenhaber, Ann. Math. 79 (1964) 59. [4] T. L. Curtright and C. K. Zachos, Phys. Lett. B243 (1990) 237; T. L. Curtright, G. I. Ghandour and C. K. Zachos, J. Math. Phys. 32 (1991) 676. [5] M. Pillin, Commun. Math. Phys. 180 (1996) 23. [6] W. Pusz and S. L. Woronowicz, Rep. Math. Phys. 27 (1989) 231. [7] J. Wess and B. Zumino, Nucl. Phys. Proc. Suppl. B18 (1991) 302. [8] U. Carow-Watamura, M. Schlieker and S. Watamura, Z. Phys. C49 (1991) 439. [9] L. C. Biedenharn, J. Phys. A22 (1989) L873; A. J. Macfarlane, J. Phys. A22 (1989) 4581; T. Hayashi, Commun. Math. Phys. 127 (1990) 129; M. Chaichian and P. Kulish, Phys. Lett. B234 (1990) 72. [10] V. G. Drinfeld, Proc. Int. Congress Math., Berkeley, 1986, Vol. 1, 798. [11] V. G. Drinfeld, Leningrad Math. J. 1 (1990) 1419. [12] M. Gerstenhaber and S. D. Schack, Proc. Natl. Acad. Sci. USA 87 (1990) 478. [13] G. Fiore, J. Math. Phys. 39 (1998) 3437–3452. [14] S. Vokos and C. Zachos, Mod. Phys. Lett. A9 (1994) 1; and references therein; see also: J. Seifert, Ph.D. Thesis, Munich Univ., 1996. [15] O. Ogievetsky, Lett. Math. Phys. 24 (1992) 245.
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[16] A. Lorek, A. Ruffing and J. Wess, Z. Phys. C74 (1997) 369. [17] T. Kohno, Ann. Inst. Fourier (Grenoble) 37 (1987) 139; Adv. Stud. in Pure. Math. 1 (1987) 189; Sugaku 41 (1989) 305; and references therein. [18] V. G. Drinfeld, Doklady AN SSSR 273 (1983) 531. [19] V. Knizhnik and A. Zamolodchikov, Nucl. Phys. B247 (1984) 83. [20] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, 1994. [21] W. Pusz, Rep. Math. Phys. 27 (1989) 349. [22] G. Fiore, J. Phys. A31 (1998) 5289. [23] C. Quesne, Phys. Lett. B322 (1994) 344; and references therein. [24] L. D. Faddeev, N. Y. Reshetikhin and L. A. Takhtajan, Algebra i Analysis 1 (1989) 178; translation: Leningrad Math. J. 1 (1990) 193. [25] See e.g.: L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and its Applications 8 (1981); Addison-Wesley; V. Manko, G. Marmo, P. Vitale and F. Zaccaria, Int. J. Mod. Phys. A9 (1994) 5541. [26] D. Gurevich and S. Majid, Pacific J. Math. 162 (1994) 27. [27] For a simple introduction to the subject see e.g.: J. Dieudonn´ e and J. Carrell, Invariant Theory, Old and New, Academic Press, 1971; and references therein. [28] See e.g.: J. Azc´ arraga, A. Macfarlane, A. Mountain and J. P´erez Bueno, Nucl. Phys. B510 (1998) 657; and references therein. [29] B. Jurco, Commun. Math. Phys. 166 (1994) 63. [30] O. Ogievetsky and B. Zumino, Lett. Math. Phys. 25 (1992) 121. [31] G. Fiore, in Proc. Quantum Group Symposium at ICGTMP 96, eds. H.-D. Doebner and V. K. Dobrev, Heron Press, Sofia, 1997; and hep-th/9611144. [32] J. Wess, Proc. of the Corfu Workshop on Elementary Particle Physics, Sept. 1995; and e-print q-alg/9607002. [33] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Math. and its Applications 35, Cambridge Univ. Press, 1990.
LONG RANGE SCATTERING AND MODIFIED WAVE OPERATORS FOR SOME HARTREE TYPE EQUATIONS I∗ J. GINIBRE Laboratoire de Physique Th´ eorique† Universit´ e de Paris XI, Bˆ atiment 210 F-91405 Orsay Cedex, France E-mail:
[email protected]
G. VELO Dipartimento di Fisica, Universit` a di Bologna and INFN Sezione di Bologna, Italy E-mail:
[email protected] AMS Classification: Primary 35P25. Secondary 35B40, 35Q40, 81U99 Received 8 July 1998 Revised 28 January 1999 We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n ≥ 3, including Hartree equations with potential V (x) = λ|x|−γ with γ < 1. For 1/2 < γ < 1 we prove the existence of modified wave operators with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. Keywords: Long range scattering, modified wave operators, Hartree equation.
1. Introduction In this paper, we study the theory of scattering and more precisely the existence of modified wave operators for a class of long range Hartree type equations 1 (1.1) i∂t u + ∆u = g˜(|u|2 )u 2 where u is a complex function defined in space time Rn+1 , ∆ is the Laplacian in Rn , and (1.2) g˜(|u|2 ) = λtµ−γ ω µ−n |u|2 with ω = (−∆)1/2 , λ ∈ R, 0 < γ < 1 and 0 < µ < n. The operator ω µ−n can also be represented by the convolution in x ω µ−n f = Cn,µ |x|−µ ∗ f
(1.3)
[27], so that (1.2) is a Hartree type interaction with potential V (x) = C|x|−µ . The more standard Hartree equation corresponds to the case γ = µ. In that case, the ∗ Supported
in part by NATO Collaborative Research Grant 972231. associ´e au Centre National de la Recherche Scientifique - URA D0063.
† Laboratoire
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nonlinearity g˜(|u|2 ) becomes g˜(|u|2 ) = V ∗ |u|2 = λ|x|−γ ∗ |u|2
(1.4)
with a suitable redefinition of λ. A large amount of work has been devoted to the theory of scattering for the Hartree Eq. (1.1) with nonlinearity (1.4) as well as with similar nonlinearities with more general potentials V [8, 9, 11–17, 19–21, 25]. As in the case of the linear Schr¨ odinger equation, one must distinguish the short range case, corresponding to γ > 1, from the long range case corresponding to γ ≤ 1. Fairly satisfactory results exist in the short range case. In particular it is known that the (ordinary) wave operators exist in suitable function spaces for γ > 1 [25]. Furthermore for repulsive interactions, namely for λ ≥ 0, it is known that all solutions in suitable spaces admit asymptotic states in L2 for γ > 1, and that asymptotic completeness holds in suitable spaces for γ > 4/3 [20]. In the long range case γ ≤ 1, the ordinary wave operators are known not to exist in any reasonable sense [20], and one expects that they should be replaced by modified wave operators including a suitable phase in their definition, as is the case for the linear Schr¨ odinger equation. A well developed theory of long range scattering exists for the latter. See for instance [3] for a recent and comprehensive treatment and for an extensive bibliography. In contrast with that situation, only preliminary results are available for the Hartree equation (and even less results for the more difficult nonlinear Schr¨ odinger (NLS) equation), all of them restricted to the case of small solutions. On the one hand, the existence of modified wave operators has been proved in the critical case γ = 1 for small solutions [8]. On the other hand, it has been shown recently, first in the critical case γ = 1 [12, 15] and then in the whole range γ < 1 [11, 13, 14] that the global solutions of the Hartree Eq. (1.1) with nonlinearity (1.4) with small initial data exhibit an asymptotic behaviour as t → ±∞ of the expected scattering type characterized by scattering states u± and including suitable phase factors that are typical of long range scattering. In particular, in the framework of scattering theory, the results of [11, 13, 14] are closely related to the property of asymptotic completeness for small data. In the simpler case of short range interactions, it is a fact of experience that for the Hartree equation as well as for other equations such that the NLS equation or the nonlinear wave (NLW) equation, asymptotic completeness for small data can be proved together with and by the same method as the existence of the wave operators with no size restriction on the data. The results of [11, 13, 14] therefore suggest that the same method that has been used there to prove asymptotic completeness for small data can be used to prove the existence of suitably modified wave operators, again with no size restriction on the data. It is the purpose of the present paper to explore that possibility in the case of the Hartree type Eq. (1.1) with nonlinearity (1.2). The special choice of nonlinearity (1.2) has been dictated by the following reasons. In one direction, at the present preliminary stage of long range nonlinear scattering theory, it seems more appropriate to look for the basic facts on a rather specific example rather than on a general class of nonlinearities of the type (1.4)
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with V replaced by some general function satisfying suitable smoothness and decay conditions. In the opposite direction, the strictly Hartree interaction (1.4) has the drawback that one single parameter γ serves two unrelated purposes. On the one hand, it characterizes the long distance behaviour of the interaction, leading in particular to the distinction between short range and long range cases. On the other hand, it characterizes the local regularity of the interaction, and as a consequence the local regularity that is required for the solutions. In order to avoid that confusion, we consider the time dependent interaction (1.2) with two independent parameters γ and µ. With that choice it turns out that γ again characterizes the long distance or equivalently long time behaviour, whereas µ characterizes the local regularity. Finally the nonlinearity (1.2) is covariant under dilations, which is an important simplifying property. The main result of the present paper is the existence of modified wave operators for Eq. (1.1) with nonlinearity (1.2), together with a description of the asymptotic behaviour in time of solutions in the ranges of those operators, with no size restriction on the data, in suitable spaces and for suitable values of γ and µ. The method is an extension of the energy method used in [11, 13, 14], and uses in particular the equations introduced in [13] to study the asymptotic behaviour of small solutions. The spaces of initial data, namely in the present case of asymptotic states, are Sobolev spaces of finite order similar to those used in [14]. The parameter µ characterizing the regularity of the interactions has to satisfy the condition µ ≤ n − 2 and µ < 2. The condition µ ≤ n − 2 is the really important one and is needed for the treatment of the problem in a neighborhood of infinity in time. It restricts the theory to space dimension n ≥ 3. That condition and consequently the restriction n ≥ 3 could most probably be relaxed at the cost of using more complicated spaces such as those used in [11, 13]. The condition µ < 2 is imposed in order to make contact with the available treatments of Eq. (1.1) at finite times [9, 18, 25]. It cannot be avoided in the attractive case λ ≤ 0, whereas it can most probably be relaxed to µ < Min(4, n) in the repulsive case λ ≥ 0. The parameter γ characterizing the long distance/long time behaviour of the interaction will have to satisfy 1/2 < γ < 1. The critical case γ = 1 can be treated easily by the same methods, but is excluded here in order to simplify the exposition, since it would require different formulas involving `n t instead of t1−γ . The case γ ≤ 1/2 can be treated by the methods of this paper, but it is more complicated and requires a more careful analysis of the asymptotic behaviour of the solutions of (1.1). It will be deferred to a subsequent paper. The construction of the modified wave operators is too complicated to allow for a more precise statement of the results at the present stage. That construction will be described in heuristic terms in Sec. 2 below. It involves in particular the study of an auxiliary system of equations involving a new function w and a phase ϕ instead of the original function u and the construction of local wave operators in a neighborhood of infinity for that system. After collecting some notation and a number of preliminary estimates in Sec. 3, we shall study the local Cauchy problem at finite times for the auxiliary system in Sec. 4. We shall then study the Cauchy problem at infinity and the asymptotic behaviour of solutions for the auxiliary
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system in Secs. 5 and 6. In particular we shall essentially construct local wave operators at infinity for that system. We shall then come back from the auxiliary system to the original Eq. (1.1) for u and construct the wave operators for the latter in Sec. 7, where the final result will be stated in Proposition 7.5. A more detailed description of the technical Secs. 3–7 will be given at the end of Sec. 2. The reader who wants to get quickly to the heart of the matter is invited to read Sec. 2, to skip most of Sec. 3 except for the notation at the beginning and for the definition of admissibility (Definition 3.1) to look at Proposition 4.1 and skip its proof, and to proceed to Sec. 5 where the main construction starts. We conclude this section with some general notation which will be used freely throughout this paper. We denote by k · kr the norm in Lr ≡ Lr (Rn ). For 1 ≤ r ≤ ∞, we define δ(r) = n/2 − n/r. For any interval I and any Banach space X, we denote by C(I, X) the space of strongly continuous functions from I to X and by L∞ (I, X) (resp. L∞ loc (I, X)) the space of measurable essentially bounded (resp. locally essentially bounded) functions from I to X. For real numbers a and b, we use the notation a ∨ b = Max(a, b), a ∧ b = Min(a, b) and [a] = integral part of a. Finally if (p · q) is the numbering of a double inequality, we denote by (p · qL) and (p · qR) the left-hand and right-hand inequality of (p · q) respectively. Additional notation will be given at the beginning of Sec. 3. 2. Heuristics In this section, we discuss in heuristic terms the construction of the modified wave operators for Eq. (1.1), as it will be performed below in this paper. The discussion applies to any Schr¨odinger like equation of the type (1.1) where g˜(|u|2 ) is real and depends on u only through |u|. In addition g˜ may also at this stage depend on x and t. For instance g˜ can be independent of u and depend only on (x, t), thereby leading for (1.1) to a linear Schr¨ odinger equation with time dependent potential; g˜ can be given by (1.4) or (1.2), thereby leading to the Hartree or Hartree-type equation considered here; g˜ can be a local function of |u|2 and possibly t such as g˜(|u|2 ) = λtµ−γ |u|2µ/n
(2.1)
thereby leading to an NLS equation with power nonlinearity, and with the parameters γ and µ playing the same role as in (1.2). Whatever the case, we assume that the Cauchy problem for (1.1) is globally well posed at finite time, namely that for any t0 ∈ R and any u0 in a suitable space, (1.1) has a unique solution u with u(t0 ) = u0 , defined for all t in R and depending continuously on u0 in suitable norms. The problem addressed by scattering theory is first of all that of classifying the possible asymptotic behaviour of the global solutions of (1.1) by relating them to a set of model functions with suitably chosen and preferably simple asymptotic behaviour. The first question to be considered is then the following one. For each function v of the previous set, construct a solution u of (1.1) such that u(t) behaves as v(t) when t → +∞, for instance in the sense that u(t) − v(t) tends to zero when t → +∞ in suitable norms. A similar question can be asked for t → −∞. From now on we restrict our attention to the case of positive times.
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A natural method to attack the previous question is the following one. Let v be a fixed model function. Take t0 > 0, t0 large. Using the wellposedness of (1.1) for finite initial time, define ut0 as the solution of the Cauchy problem for (1.1) with initial data ut0 (t0 ) = v(t0 ) at time t0 . For fixed v, take now the limit t0 → ∞. In favourable cases, ut0 will tend to a limiting solution u∞ of (1.1) answering the previous question. Of special interest is the case where the set of model functions v is the set of solutions of an evolution problem which is globally well posed (and preferably simpler than (1.1)). In that case, the set of model functions v can be characterized by its initial data at some prescribed time T , whereas the solutions u = u∞ as constructed above can also be characterized by their values u(T ) at time T . The map v → u∞ classifying (part of) the solutions of (1.1) through their asymptotic behaviour is then equivalent to the simpler map Ω+ : v(T ) → u(T ) relating the values of v and u at time T . That map is the wave operator (for positive time). At this stage there is some symmetry between the original evolution for u and the model evolution for v. In fact let u be a solution of (1.1) constructed by the previous limiting process. Then v can be recovered from u as follows. Take t0 > 0, t0 large. Define vt0 as the solution of the Cauchy problem for the model evolution with initial data vt0 (t0 ) = u(t0 ) at time t0 . For fixed u, take now the limit t0 → ∞. In general vt0 will then tend to the original function v from which u was constructed, and the limiting process will provide additional information on the asymptotic behaviour of u. In the short range case, the previous scheme is implemented by taking for v the solutions of the equation (2.2) i∂t v + (1/2)∆v = 0 hereafter referred to as the free equation. The generic solution of that equation is v(t) = U (t)u+ where U (t) is the unitary group U (t) = exp(i(t/2)∆) .
(2.3)
It is then natural to take T = 0. The initial data u+ for v is called the asymptotic state for the solution u = u∞ of (1.1) constructed by the method described above, and that method yields the ordinary wave operator Ω+ : u+ → u(0). Note also that the asymptotic closeness of u and v as t → ∞ can be better expressed in terms of the function u ˜(t) = U (−t)u(t) than in terms of u itself. In fact that function is expected to tend to u+ in suitable norms, whereas u(t) and v(t) separately tend weakly to zero in any reasonable sense. In the long range case, it is known that the previous ordinary wave operators fail to exist, which means that the previous set of v’s, namely the set of solutions of the free evolution, is badly chosen. A better set of model functions v is then obtained by modifying the previous ones by a suitable phase. That modification can be done in several ways and uses some additional structure of U (t). In fact U (t) can be written as U (t) = M (t)D(t)F M (t) (2.4)
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where M (t) denotes the operator of multiplication by the function, also denoted M (t), (2.5) M (t) = exp(ix2 /2t) , F is the Fourier transform and D(t) is the dilation operator defined by (D(t)f (x)) = (it)−n/2 f (x/t) .
(2.6)
Let now ϕ0 = ϕ0 (x, t) be a real function of space and time, to be chosen later, let z0 (x, t) = exp(−iϕ0 (x, t)), let ϕ0 (t) and z0 (t) be the operators of multiplication by the function ϕ0 (x, t) and z0 (x, t) respectively and let φ0 (t) and Z0 (t) be the operators φ0 (t) = ϕ0 (−i∇, t) = F ∗ ϕ0 (t)F , Z0 (t) = z0 (−i∇, t) = F ∗ z0 (t)F .
(2.7)
In what follows, we shall sometimes omit the time dependence of the various operators when no confusion is likely to arise. Instead of the free evolution v(t) = U (t)u+ one can now consider the following three modified free evolutions: v1 (t) = U (t)Z0 (t)u+ = U (t)F ∗ z0 (t)w+ ,
(2.8)
where w+ = F u+ , v2 (t) = U (t)M ∗ (t)Z0 (t)u+ = M (t)D(t)F Z0 (t)u+ = M (t)D(t)z0 (t)w+ = M (t)(D(t)z0 (t)D∗ (t))D(t)w+ ,
(2.9)
v3 (t) = U (t)M ∗ (t)Z0 (t)M (t)u+ = M (t)D(t)F Z0 (t)M (t)u+ = M (t)D(t)z0 (t)U ∗ (1/t)w+ = M (t)(D(t)z0 (t)D∗ (t))D(t)U ∗ (1/t)w+
(2.10)
where we have used the fact that F M (t)F ∗ = U ∗ (1/t) .
(2.11)
Note that D(t)z0 (t)D(t)∗ is the operator of multiplication by z0 (x/t, t) so that the modification due to ϕ0 appears only as an overall phase factor in v2 and v3 . Most of the literature on long range scattering for the linear Schr¨ odinger equation makes use of v1 . The function v2 has been introduced in [28] and further used in [4] in the linear case. It has been introduced independently in [26] and used in [8, 26] in the nonlinear case. The function v3 is mentioned but not really used in [8].
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In the short range case with v(t) = U (t)u+ , it was hinted that the function u ˜(t) = U (−t)u(t) was a better object of study than u(t). Similarly, in the long range case, it will be useful to introduce new functions which will be better suited than u for comparison with vi (t). Furthermore, it will be useful to express those functions as suitable combinations of a phase factor zi (t) = exp(−iϕi (t)) and of an amplitude u˜i (t) in such a way that the comparison of u with vi can be reduced ˜i (t) tends to u+ or to the facts that asymptotically ϕi (t) behaves as ϕ0 (t) and u ˜i (t) for i = 1, 2, 3. This is done equivalently wi (t) tends to w+ , where wi (t) = F u through the following definitions, to be compared with (2.8)–(2.10): u(t) = U (t)Z1 (t)˜ u1 (t) = U (t)F ∗ z1 (t)w1 (t) ,
(2.12)
u2 (t) = M (t)D(t)z2 (t)w2 (t) , u(t) = U (t)M ∗ (t)Z2 (t)˜
(2.13)
u3 (t) u(t) = U (t)M ∗ (t)Z3 (t)M (t)˜ u3 (t) = M (t)D(t)F Z3 (t)M (t)˜ = M (t)D(t)z3 (t)U ∗ (1/t)w3 (t) .
(2.14)
The study of the asymptotic behaviour of small solutions of Eq. (1.1) has been performed in [12, 15] by using (w1 , ϕ1 ) and in [11, 13, 14] by using essentially (w2 , ϕ2 ). We now explain the construction of the wave operators performed in the present paper. For technical reasons, that construction will use the variables (w3 , ϕ3 ), but we first explain it on the example of (w2 , ϕ2 ) because the algebra is slightly simpler. We shall then indicate the necessary modifications needed to switch to (w3 , ϕ3 ). Instead of trying to construct directly the wave operators for u, we first try to construct wave operators for (w2 , ϕ2 ) by using the general method given at the beginning of this section. Equation (1.1) is equivalent to the following equation for z2 w2 : (2.15) (i∂t + (2t2 )−1 ∆ − D∗ g˜D)(z2 w2 ) = 0 as can be seen by an elementary computation. Note also that |u(t)| = |D(t)w2 (t)|
(2.16)
g˜ ≡ g˜(|u|2 ) = g˜(|Dw2 |2 )
(2.17)
by (2.13), so that and g˜ depends only on w2 (actually only on |w2 |), but not on ϕ2 . Expanding the derivatives in (2.15), we obtain the equivalent form [13] {i∂t + (2t2 )−1 ∆ − i(2t2 )−1 (2∇ϕ2 · ∇ + (∆ϕ2 ))}w2 +(∂t ϕ2 − (2t2 )−1 |∇ϕ2 |2 − D∗ g˜D)w2 = 0 .
(2.18)
We are now in the situation of a gauge theory. The equation of evolution (2.15) and therefore also (2.18) is invariant under the transformation (w2 , ϕ2 ) →
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(w2 exp(iσ), ϕ2 + σ), where σ is an arbitrary function of space time, and the original gauge invariant Eq. (2.15) is not sufficient to provide evolution equations for the two gauge dependent quantities (w2 , ϕ2 ). At this point, we arbitrarily add a gauge condition, which will serve as a second evolution equation, and replace (2.18) by ( (2.19) {i∂t + (2t2 )−1 ∆ − i(2t2 )−1 (2∇ϕ2 · ∇ + (∆ϕ2 ))}w2 = 0 ∂t ϕ2 = (2t2 )−1 |∇ϕ2 |2 + D0∗ g˜
(2.20)
where D0∗ = D0∗ (t) is the modified dilation operator defined by (D0∗ f )(x) = f (tx) . The second equation, namely the gauge condition, is one of the Hamilton–Jacobi (HJ) equations for the classical system associated with (1.1) [3, 4]. The situation here is similar to that occurring for the Maxwell equations where for instance one can impose the Lorentz gauge condition ∂µ Aµ = 0 and use it as an evolution for A0 in order to reduce the gauge freedom in the study of the Cauchy problem. We have now replaced the original evolution (1.1) by the auxiliary system (2.19)– (2.20) and we try to study the asymptotic behaviour of its solutions and to construct wave operators for it by the same method that we intended to use for (1.1). In contrast with (1.1) however, the Cauchy problem for (2.19)–(2.20) cannot be expected to be globally well posed. It turns out however, and that is sufficient for our purposes, that this problem is locally well posed in a neighborhood of infinity in time. Roughly speaking for given initial data of arbitrary size, the Cauchy problem is well posed for initial time t0 in some interval [T, ∞) for some sufficiently large T depending on the size of the data. As a consequence, we shall be able to construct only local wave operators for (2.19)–(2.20) in a neighborhood of infinity. Wave operators for (1.1) will then be obtained from those by switching back to u and using the global wellposedness of (1.1) for finite times. In order to construct the local wave operators for the system (2.19)–(2.20), we need to choose a set of model functions playing the role of v, preferably defined through a model evolution. Keeping in mind that u is represented by (2.13) and should be asymptotic to v2 defined by (2.9), preferably with w2 (t) tending to w+ and ϕ0 (t) asymptotic to ϕ2 (t), we define the model evolution for a pair (w0 , ϕ0 ) corresponding to (w2 , ϕ2 ) by ( ∂t w0 = 0 (2.21) ∂t ϕ0 = (2t2 )−1 |∇ϕ0 |2 + D0∗ g˜(|Dw0 |2 ) . The first equation is immediately solved by w0 (t) = w+ , thereby leading to the form (2.9) of v2 where now the phase ϕ0 should be a solution of the equation ∂t ϕ0 = (2t2 )−1 |∇ϕ0 |2 + D0∗ g˜(|Dw+ |2 ) .
(2.22)
As in the case of the system (2.19)–(2.20), the Cauchy problem for (2.22) is well posed only in a neighborhood of infinity in time, but in general not globally in time. Although (2.22) is not the model evolution that we shall use later on, we use it to continue the heuristic discussion.
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The local wave operators at infinity for the system (2.19)–(2.20) as compared with (2.22) are now constructed by the general method described at the beginning of this section. Let Γ0 = (w+ , ϕ0 ) be a solution of (2.22), defined in some interval [T, ∞) with T sufficiently large, and depending on the initial data ϕ0 (T ). Let t0 > T and let Γt0 = (w2,t0 , ϕ2,t0 ) be the solution of (2.19)–(2.20) with initial data (w2,t0 (t0 ) = w+ , ϕ2,t0 (t0 ) = ϕ0 (t0 )) at time t0 . Under suitable assumptions, Γt0 will be defined in the interval [T, ∞) and will converge to a well defined limit Γ∞ = Γ when t0 → ∞ for fixed Γ0 . This could provide a basis for the definition of local wave operators at infinity for the system (2.19)–(2.20), although the fact that T depends on the size of the data would cause some difficulties, but we are actually interested in wave operators for u, and from the previous construction we keep only the map Γ0 → Γ. Reconstructing u from Γ = (w2 , ϕ2 ) by the use of (2.13), we obtain a map (w+ , ϕ0 (T )) → u where u is a local solution of Eq. (1.1) in a neighborhood of infinity, namely defined in [T, ∞), which behaves asymptotically as v2 when t → ∞ in a sense that is expressed by the relation between (w+ , ϕ0 ) = Γ0 and (w2 , ϕ2 ) = Γ at infinity, as it follows from the previous construction. We can finally complete the construction of u by solving the Cauchy problem for (1.1) with initial data u(T ) at time T obtained from the previous step down to time t = 1 and define accordingly a map (w+ , ϕ0 (T )) → u(1), which is a reasonable candidate for the wave operator for (1.1). The map (w+ , ϕ0 (T )) → u however is not yet satisfactory for two reasons. Firstly u depends on too many data. We want u to depend only on w+ and not in addition on an arbitrary initial condition for ϕ0 . That defect is easily remedied, as in linear long range scattering, by imposing arbitrarily some initial condition for ϕ0 . For instance, given w+ , one could choose in some preassigned way some T = T (w+ ) sufficiently large for all subsequent constructions to be possible, and choose for instance ϕ0 (T (w+ )) = 0. Secondly, because of gauge invariance, the map (w+ , ϕ0 (T )) → u has no chance of being injective, since different (w+ , ϕ0 (T )) can very well produce different but gauge equivalent (w2 , ϕ2 ), thereby leading to the same u. Fixing arbitrarily the initial condition for ϕ0 certainly will improve the injectivity, but at the risk of restricting the set of u obtained by the previous construction, namely the range of the wave operator, and making it dependent on that initial condition. We now show that in principle, this should not happen, and that fixing the initial condition for ϕ0 exactly removes the gauge freedom and ensures the injectivity of the map Γ0 → u without restricting its range. For that purpose we have to consider in more detail the gauge covariance of the map Γ0 → Γ. Now the HJ gauge condition (2.20) does not entirely remove the gauge freedom in Eq. (2.18) but only reduces it from that associated with an arbitrary function of space time to that associated with an arbitrary function of space only, for instance with some initial condition for (2.20). In fact let (w2 , ϕ2 ) and (w20 , ϕ02 ) be two gauge equivalent solutions of (2.19)–(2.20), namely such that w2 exp(−iϕ2 ) = w20 exp(−iϕ02 ). Then the difference ϕ− = ϕ02 − ϕ2 satisfies the equation ∂t ϕ− = (2t2 )−1 ∇ϕ− · ∇ϕ+
(2.23)
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where ϕ+ = ϕ02 + ϕ2 . Under suitable assumptions, it follows from (2.23) that ϕ− (t) has a well defined limit σ = limt→∞ ϕ− (t) as t → ∞, whereas both ϕ2 and ϕ02 grow indefinitely as t → ∞. Conversely, given a solution (w2 , ϕ2 ) of (2.19)–(2.20) and a suitable σ, the same equation written in the form ∂t ϕ− = (2t2 )−1 (2∇ϕ2 · ∇ϕ− + |∇ϕ− |2 )
(2.24)
with limiting condition limt→∞ ϕ− (t) = σ can be used to determine ϕ− in a neighborhood of infinity, and thereby determine ϕ02 = ϕ2 +ϕ− and w20 = w2 exp(iϕ− ) such that (w20 , ϕ02 ) be a solution of (2.19)–(2.20) which is gauge equivalent to (w2 , ϕ2 ). This argument shows that the gauge freedom left in (2.19)–(2.20) is the invariance under gauge transformations Gσ which can be parametrized by the change σ of ϕ2 at infinity. 0 , ϕ00 ) are two solutions of the model evolution In a similar way, if (w+ , ϕ0 ) and (w+ 0 (2.22) with |w+ | = |w+ |, then the difference ϕ00 − ϕ0 = ϕ− again satisfies Eq. (2.23) now with ϕ+ = ϕ00 + ϕ0 , and has a limit as t → ∞. Conversely the same equation rewritten in analogy with (2.24) determines ϕ00 from ϕ0 and from the value of ϕ− at some large initial time, possibly infinity, and possibly some preassigned time T . The map Γ0 → Γ as we shall construct it later will be such that ϕ0 (t)−ϕ2 (t) → 0 and w2 (t) → w+ as t → ∞. Consequently if we define the gauge transformation 0 , ϕ00 ) with Gσ on the solutions of the model evolution (2.22) by Gσ (w+ , ϕ0 ) = (w+ 0 = w+ eiσ , then the map Γ0 → Γ is gauge limt→∞ (ϕ00 (t) − ϕ0 (t)) = σ and w+ covariant in the sense that the image of Gσ Γ0 is Gσ Γ if Γ is the image of Γ0 . From the previous discussion it follows that imposing an initial condition on ϕ0 exactly fixes the gauge, thereby ensuring the injectivity of the map Γ0 → u without restricting its range. Actually in practice that picture is clouded by the fact that all the constructions involved produce a loss of regularity which prevents a complete proof of the previous statements. On the other hand the basic construction Γ0 → Γ also produces a similar loss of regularity, and it turns out that the former is hidden by the latter, so that an entirely satisfactory discussion of gauge invariance can be given at the level of regularity of the construction Γ0 → Γ. The previous heuristic discussion was based on the system (2.19)–(2.20) for (w2 , ϕ2 ) and the model evolution (2.22) for ϕ0 . For several reasons however, we shall use different equations. The first two reasons are rooted in the basic construction of the method, namely the construction of the solution Γt0 of the full evolution coinciding at t0 with a given solution Γ0 of the model evolution, and the third one is connected with gauge invariance. (1) We shall take w+ ∈ H k , where H k is the standard Sobolev space, and look 0 for w2 as a continuous function of time with values in H k for some k 0 ≤ k. In the construction Γ0 → Γt0 , the term ∇ϕ2 · ∇w2 in (2.19) produces a loss of one derivative, which seems difficult to avoid. The term ∆w2 on the other hand produces a loss of two derivatives but that loss can be avoided by switching from (w2 , ϕ2 ) to (w3 , ϕ3 ) and we shall therefore also use (w3 , ϕ3 ) in addition to (w2 , ϕ2 ). The equations for (w3 , ϕ3 ) could be obtained easily from the equation for u, but it is simpler to deduce them from the system (2.19)–(2.20). We impose ϕ2 = ϕ3 = ϕ and
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w3 (t) = U (1/t)w2 (t) = w(t), which is consistent with (2.13)–(2.14). The resulting system for (w, ϕ) is then ( (2.25) ∂t w = (2t2 )−1 U (1/t)(2∇ϕ · ∇ + (∆ϕ))U ∗ (1/t)w ∂t ϕ = (2t2 )−1 |∇ϕ|2 + D0∗ g˜(|DU ∗ (1/t)w|2 ) .
(2.26)
Correspondingly, the model evolution for (w0 , ϕ0 ) will replace (2.25) by ∂t w = 0, so as to produce model functions of the type (2.10), thereby leaving for ϕ0 the equation (2.27) ∂t ϕ0 = (2t2 )−1 |∇ϕ0 |2 + D0∗ g˜(|DU ∗ (1/t)w+ |2 ) . Since all the estimates on w will be made in spaces H k where U (1/t) is unitary, the explicit occurrence of that operator in (2.25)–(2.27) will not make any difference in those estimates. (2) In this paper we restrict our attention to the case γ > 1/2. It is well known in linear long range scattering theory that under that condition the correcting phase ϕ0 need not be a solution of the full HJ Eq. (2.27) and can be chosen simply according to the Dollard prescription, namely as a solution of the simpler equation ∂t ϕ0 = D0∗ g˜(|DU ∗ (1/t)w+ |2 ) .
(2.28)
We shall work with (2.28) instead of (2.27) in what follows. This produces a number of simplifications in all the questions involving only Γ0 . In particular the Cauchy problem for (2.28) is trivially solved globally by a simple integration, thereby allowing in particular for imposing an initial condition for ϕ0 at a fixed time independent of w+ . Whereas the term |∇ϕ0 |2 can be omitted from (2.27) for γ > 1/2, fiddling with the term |∇ϕ|2 in (2.26) may not be harmless. For instance shifting that term from (2.26) to (2.25) would produce additional restrictions on γ and require at least γ > 2/3 in the construction Γ0 → Γ. (3) The model evolution (2.27) and its simplified version (2.28) are best suited to study the asymptotic behaviour of the system (2.25)–(2.26), which minimizes the loss of regularity. On the other hand they are ill suited for a study of gauge invariance, which plays an important role in the reconstruction of u. In particular a change w+ → w+ eiσ produces a nontrivial change in the g˜ term in (2.27), whereas no such change occurs in (2.22). As a consequence, we cannot avoid also using (2.22), or rather its simplified version ∂t ϕ0 = D0∗ g˜(|Dw+ |2 )
(2.29)
obtained as previously by omitting the |∇ϕ0 |2 term. We shall therefore use both (2.29), which allows for a simple discussion of gauge invariance and for a cleaner construction of u at the cost of a loss of two derivatives, and (2.28), which produces only a loss of one derivative. In particular we shall use the construction Γ0 → Γ starting from (2.28) as an intermediate step for the corresponding construction starting from (2.29). The last technical modification is independent of the previous choices.
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(4) The right-hand sides of (2.25)–(2.29) contain ϕ and ϕ0 only through their gradients, and the construction of Γ can be discussed entirely in terms of those variables. Only for the discussion of gauge invariance and for the reconstruction of u are ϕ0 and ϕ themselves needed. We therefore introduce the Rn valued functions s = ∇ϕ and s0 = ∇ϕ0 and replace the basic Eqs. (2.26)–(2.29) by their gradients. Using the fact that X X (1/2)∂i |∇ϕ|2 = (∂i ∂j ϕ)(∂j ϕ) = (∂j ϕ)(∂j ∂i ϕ) = (∇ϕ · ∇)∂i ϕ j
we obtain
(
j
∂t w = (2t2 )−1 U (1/t)(2s · ∇ + (∇ · s))U ∗ (1/t)w ,
(2.30)
∂t s = t−2 (s · ∇)s + ∇D0∗ g˜(|DU ∗ (1/t)w|2 ) ,
(2.31)
∂t s0 = t−2 (s0 · ∇)s0 + ∇D0∗ g˜(|DU ∗ (1/t)w+ |2 ) ,
(2.32)
∂t s0 = ∇D0∗ g˜(|DU ∗ (1/t)w+ |2 ) ,
(2.33)
and either
or ∂t s0 = ∇D0∗ g˜(|Dw+ |2 ) .
(2.34)
The phase ϕ itself will then be recovered from s through the use of (2.26) as follows. Equation (2.31) is an Euler like equation for s and implies that the vorticity ω = ∇ × s remains zero for all time if it is zero for some initial time t0 , namely with an initial condition s(t0 ) = ∇ϕ(t0 ). In fact ω satisfies the linear equation ∂t ω = t−2 ((s · ∇)ω + Aω + ωAT )
(2.35)
where A is the matrix with entries Aij = ∂j si , and that equation implies the result just mentioned through the Gronwall inequality for sufficiently regular s. It follows then from (2.26) and (2.31) that ∂t (s − ∇ϕ) = t−2 (s × ω) = 0 and therefore s − ∇ϕ vanishes for all t if it vanishes for some t0 . We are now in a position to describe in more detail the contents of the technical part of this paper, Secs. 3–7. In Sec. 3, we introduce some notation, define the relevant function spaces needed to study the system (2.30)–(2.31), and we derive a number of estimates which are used throughout the paper. In Sec. 4, we study the Cauchy problem at finite times for the system (2.30)–(2.31), and we prove that that problem is locally well posed (Proposition 4.1). In Sec. 5, we study the local Cauchy problem at infinity for the same system, and construct local wave operators for it as compared with the model Eq. (2.33). We first solve the Cauchy problem in a neighborhood of infinity for finite but large t0 (Proposition 5.1) and derive a uniqueness result for given asymptotic behaviour (Proposition 5.2). We then prove the existence of asymptotic states for solutions Γ = (w, s) thereby obtained, in the
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373
following sense: firstly w(t) has a limit w+ when t → ∞ (Proposition 5.3). Secondly the solution Γ0,t0 of (2.33) which coincides with Γ at time t0 satisfies estimates uniform in t0 (Proposition 5.4) and has a limit Γ0 when t0 → ∞ which is asymptotic to Γ (Proposition 5.5). We then turn to the converse construction, which is that of the local wave operators at infinity. For a fixed solution Γ0 of (2.33), we construct a solution Γt0 of the system (2.30)–(2.31) which coincides with Γ0 at t0 and we estimate it uniformly in t0 (Proposition 5.6). We then prove that when t0 → ∞, Γt0 has a limit Γ which is asymptotic to Γ0 in the same sense as in Proposition 5.5 (Proposition 5.7). We conclude that section with some comments on the possible use of other equations, and in particular on the modifications required to use the more complicated Eq. (2.32) instead of (2.33). In Sec. 6, we perform exactly the same analysis of the system (2.30)–(2.31) at infinity, now however compared with the model Eq. (2.34) which is better suited for the study of gauge invariance and the reconstruction of u. Propositions 6.1–6.4 are the exact analogues of Propositions 5.4–5.7 with (2.33) replaced by (2.34) and their proofs rely to a large extent on those of the latter. Finally in Sec. 7 we exploit the results of Secs. 5 and 6, especially Propositions 5.7 and 6.4, to construct the wave operators for Eq. (1.1) and to describe the asymptotic behaviour of solutions in their range. We first supplement the constructions of Secs. 5 and 6 with the appropriate definitions in order to recover ϕ and ϕ0 from s and s0 , both at finite times and in their correspondence at infinity as it follows from Propositions 5.5, 5.7, 6.2 and 6.4. We then prove that the local wave operator at infinity for the system (2.30)–(2.31) as compared with (2.34) defined through Proposition 6.4 in Definition 7.1 is gauge covariant in the sense of Definitions 7.2 and 7.3 in the best form that can be expected with the available regularity (Propositions 7.2 and 7.3). With the help of some information on the Cauchy problem for (1.1) at finite time (Proposition 7.1), we then define the wave operator Ω : u+ → u (Definition 7.4), we prove that it is injective and has the expected range (Proposition 7.4). We then collect all the available information on Ω and on solutions of (1.1) in its range in Proposition 7.5, which contains the main results of this paper. Finally, by using some information on the global Cauchy problem at finite time for (1.1) (Proposition 7.6), we define the usual wave operator Ω1 : u+ → u(1) (Definition 7.5). A question that we leave unsettled in this paper is that of the intertwining property of the wave operator. That property can be stated in terms of Ω as the fact that for t sufficiently large and τ ≥ 0, (Ω(U (τ )u+ ))(t) = (Ω(u+ ))(t + τ ) .
(2.36)
That property is an asymptotic form of time translation invariance, and unfortunately that invariance has been severely broken by the change of variables from u to (w, ϕ) in (2.14). Therefore the method used in this paper is ill suited for a study of the intertwining property and we leave that question open here. We finally remark that the basic Eqs. (2.19)–(2.20) from [13], of which we used the modified form (2.25)–(2.26), are very similar to the equations used in [7, 10] to
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study the classical limit ~ → 0 of the nonlinear Schr¨ odinger equation 1 i~∂t u = − ~2 ∆u + g˜(|u|2 )u . 2 This comes as no surprise, since the latter are also obtained by separating u into an amplitude and a phase, u = w exp(−iϕ/~). Accordingly, the same energy methods can be applied to the small ~ problem [10] and to the large time problem. 3. Notation and Preliminary Estimates In this section we introduce some additional notation and we collect a number of estimates which will be used throughout this paper. We first define ¯2 , g0 (w1 , w2 ) = λ Re ω µ−n w1 w
(3.1)
g(w1 , w2 ) = g0 (U ∗ (1/t)w1 , U ∗ (1/t)w2 ) .
(3.2)
In particular g(0) (w1 , w1 ) − g(0) (w2 , w2 ) = g(0) (w− , w+ ) where w± = w1 ± w2 . By using the definitions (1.2) and (2.6) and that of D0∗ given after (2.20), we rewrite the nonlinearity in (2.26) as D0∗ g˜(|DU ∗ (1/t)w|2 ) = t−γ g(w, w) .
(3.3)
The nonlinearities in (2.31)–(2.34) can be rewritten in a similar way, so that the basic Eqs. (2.30)–(2.34) become respectively (
∂t w = (2t2 )−1 U (1/t)(2s · ∇ + (∇ · s))U ∗ (1/t)w
(2.30) ≡ (3.4)
∂t s = t−2 (s · ∇)s + t−γ ∇g(w, w)
(2.31) ∼ (3.5)
∂t s0 = t−2 (s0 · ∇)s0 + t−γ ∇g(w+ , w+ )
(2.32) ∼ (3.6)
∂t s0 = t−γ ∇g(w+ , w+ )
(2.33) ∼ (3.7)
∂t s0 = t−γ ∇g0 (w+ , w+ ) .
(2.34) ∼ (3.8)
We next introduce the function spaces where we shall solve the basic system (3.4)– (3.5). We denote multi-indices by greek letters α, β, . . ., their lengths by |α|, |β|, . . . , and nonnegative integers by j, k, `, . . . . For any function u and any function space norm k · k, we define X k∂ α uk . k∂ j uk = α:|α|=j
We shall use Sobolev spaces of integer order Hrk defined for 1 ≤ r ≤ ∞ by X k∂ j ukr < ∞ Hrk = u : ku; Hrk k ≡ 0≤j≤k
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375
and the associated homogeneous spaces H˙ rk with norm ku; H˙ rk k = k∂ k ukr . The subscript r will be omitted if r = 2. We first recall the well known Sobolev–Gagliardo–Nirenberg inequalities [6, 22, 23]. Lemma 3.1. Let 1 ≤ p, q, r ≤ ∞. Let j and k be nonnegative integers with j < k. If p = ∞, assume that k − j > n/r. Let σ satisfy j/k ≤ σ ≤ 1 and n/p − j = (1 − σ)n/q + σ(n/r − k) . Then the following inequality holds for any function u ∈ Lq : k∂ j ukp ≤ Ckuk1−σ k∂ k ukσr q except for p < ∞, j = 0 and q = ∞. In the latter case, for any function u ∈ L∞ there exists a constant c depending on u such that k σ ku − ckp ≤ Cku − ck1−σ ∞ k∂ ukr .
Remark 3.1. The statement as just given may differ from the usual ones (see for instance [6]) by unnecessarily excluding a few trivial cases. We shall use extensively the following spaces. Let `0 = [n/2] and define r0 by δ(r0 ) = `0 so that r0 = 2n for n odd and r0 = ∞ for n even. Let k and ` be nonnegative integers with ` ≥ `0 − 1. Let I ⊂ R+ be an interval. We shall look for k k w as a complex valued function in spaces L∞ loc (I, H ) or C(I, H ) and for s as an n n ∞ ` ` R or C vector valued function in spaces Lloc(I, X ) or C(I, X ) where X ` = Lr0 ∩ H˙ `0 ∩ H˙ `+1 .
(3.9)
For n odd, it follows from Lemma 3.1 that for s ∈ X ` , kskr0 ≡ ksk2n ≤ Cks; H˙ `0 k ≡ Cks; H˙ (n−1)/2 k so that the Lr0 norm will not need to be estimated separately. The space Lr0 has been included in the definition of X ` only in order to make Lemma 3.1 applicable by eliminating arbitrary polynomials of degree `0 − 1 which are not seen by the other norms. For ` ≥ `0 , the inclusion X ` ⊂ L∞ holds. In fact, by Lemma 3.1 again (3.10) ksk∞ ≤ C(kskr0 ks; H˙ `0 +1 k)1/2 ≤ C(ks; H˙ `0 k ks; H˙ `0+1 k)1/2 so that also the L∞ norm will not need to be estimated either. For n even, the Lr0 norm, namely the L∞ norm, is not controlled by the H˙ `0 norm, namely by the H˙ n/2 norm, and will require separate estimates. 0 The spaces X ` obviously satisfy the embedding X ` ⊂ X ` for `0 ≥ `.
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Because of the presence of Lr0 in the definition of X ` , one can replace H˙ `0 in that definition by and hxi−(n+1)/2 u ∈ L2 }
K `0 = {u : u ∈ H˙ `0
which is a Hilbert space, and similarly one can replace H˙ `+1 by K `+1 . As a consequence, X ` is the intersection of the duals of (compatible) Banach spaces and is therefore itself the dual of a Banach space. We shall use systematically the short hand notation |w|k = kw; H k k ,
|s|` = ks; X ` k
(3.11)
and the meaning of the symbol |a|b will always be made unambiguous by the fact that the pair (a, b) contains either the pair (w, k) or the pair (s, `). We shall need estimates of the solutions (in the sense of distributions) of transport diffusion equations of the form ∂t u = η∆u + ∇ · (vu) + h
(3.12a)
∂t u = η∆u + U (1/t)∇ · (vU ∗ (1/t)u) + h ,
(3.12b)
and where u and h are complex valued functions and v an Rn vector valued function defined in space time. Lemma 3.2. Let 2 ≤ p ≤ ∞, let I be an open interval. Let u, h ∈ Lploc (I, Lp ), let ∞ ∞ ∞ v ∈ L∞ loc (I, L ) with ∂v ∈ Lloc (I, L ) and let η ≥ 0. (1) Let u satisfy (3.12a) in I. Then for almost all t1 , t2 ∈ I, with t1 ≤ t2 , the following estimate holds: Z t2 dt0 {p−1 k∇ · v(t0 )k∞ ku(t0 )kp ku(t2 )kp ≤ ku(t1 )kp + t1
+ C0 k∂v(t0 )k∞ ku(t0 )kp + kh(t0 )kp }
(3.13)
for some absolute constant C0 . If η = 0, a similar estimate holds for t1 ≥ t2 . (2) Let p = 2 and let u satisfy (3.12b) in I. Then the same conclusion holds (with p = 2.) Proof. Part (1). The formal computation leading to (3.13) is given in Appendix A. The actual proof is obtained by following the methods in [5]. Part (2). This is a standard energy estimate, using in addition the unitarity of U (1/t) in L2 . Note that the estimate (3.13) is independent of η. In subsequent applications of Lemma 3.2, we shall for brevity state the results thereby obtained in the shorter differential form corresponding to ∂t kukp ≤ p−1 k∇ · v(t)k∞ ku(t)kp + C0 k∂v(t)k∞ ku(t)kp + kh(t)kp .
(3.14)
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377
We next give some preliminary estimates. They involve functions called s and w since that is suggestive of subsequent applications, but at the present stage it is irrelevant whether those functions are real or complex, scalar or vector valued. Lemma 3.3. Let α and β be multi-indices with β ≤ α and let ` = |α|. Then the following estimate holds: k∂ β s1 ∂ α−β s2 k2 ≤ Cks1 ; L∞ ∩ H˙ ` k ks2 ; L∞ ∩ H˙ ` k .
(3.15)
Proof. We estimate by the H¨older inequality k∂ β s1 ∂ α−β s2 k2 ≤ k∂ β s1 kr1 k∂ α−β s2 kr2 with 1/r1 + 1/r2 = 1/2. For β = α, we take r1 = 2, r2 = ∞. For β = 0, we take r1 = ∞ and r2 = 2. For 0 < β < α, we apply Lemma 3.1 and obtain 1 2 ks1 ; H˙ ` kσ1 ks2 k1−σ ks2 ; H˙ ` kσ2 · · · ≤ Cks1 k1−σ ∞ ∞
(3.16)
with σ1 (n/2 − `) = n/r1 − |β| , σ2 (n/2 − `) = n/r2 − ` + |β| , |β|/` ≤ σ1 ≤ 1 ,
1 − |β|/` ≤ σ2 ≤ 1 .
The equalities imply σ1 + σ2 = 1, while the inequalities imply σ1 + σ2 ≥ 1. As a consequence, all of them are satisfied for the unique choice 2/r1 = σ1 = |β|/` ,
2/r2 = σ2 = 1 − |β|/` .
Then (3.15) follows from (3.16).
Lemma 3.4. Let α and β be multi-indices with β ≤ α, |α| ≤ k, |β| ≤ ` and ` > n/2. Then the following estimate holds: k∂ β s∂ α−β wk2 ≤ Cks; L∞ ∩ H˙ ` k |w|k .
(3.17)
Proof. It is sufficient to consider the case α = β. The general case with α ≥ β will then follow therefrom by replacing w by ∂ α−β w. We estimate by the H¨ older inequality k∂ β swk2 ≤ k∂ β skr1 kwkr2 with 1/r1 + 1/r2 = 1/2. For β = 0, we take r1 = ∞, r2 = 2. For |β| = `, we take r1 = 2, r2 = ∞ and use the fact that in that case kwk∞ ≤ Ckw; H |β| k = Ckw; H ` k
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since ` > n/2. For 0 < |β| < `, we apply Lemma 3.1 and obtain ` σ · · · ≤ Cksk1−σ ∞ k∂ sk2 kwkr2
(3.18)
with σ(n/2 − `) = n/r1 − |β| and |β|/` ≤ σ ≤ 1. We then choose σ = |β|/` so that 2/r2 = 1 − |β|/` and δ2 = δ(r2 ) = n|β|/2` < |β| since ` > n/2, so that by Lemma 3.1 again kwkr2 ≤ kw; H |β| k, which together with (3.18), implies (3.17) in the special case α = β. Lemma 3.5. Let ϕ be a real function with s = ∇ϕ ∈ L∞ ∩ H˙ ` for some ` > n/2 and let k ≤ ` + 1. Then the following estimate holds: |e−iϕ w|k ≤ C(1 + k∇ϕ; L∞ ∩ H˙ ` k)k |w|k .
(3.19)
Let in addition ϕ ∈ L∞ . Then the following estimate holds: |(e−iϕ − 1)w|k ≤ C(kϕk∞ + k∇ϕ; L∞ ∩ H˙ ` k(1 + k∇ϕ; L∞ ∩ H˙ ` k)k−1 )|w|k . (3.20) Proof. For any multi-indices α, β with β ≤ α and 1 ≤ |β| ≤ |α| ≤ k, one has to estimate the L2 norm of X Y (−i)m C{βi } ∂ βi ϕ ∂ α−β w (3.21) ∂ β (e−iϕ )∂ α−β w = e−iϕ {βi }
1≤i≤m
where the sum runs over all possible decompositions β = β1 + · · · + βm of β as the sum of m ≥ 1 multi-indices. We have omitted the terms with β = 0 which trivially satisfy the required estimate with obvious assumptions on ϕ. We estimate each term in the RHS of (3.21) by Lemma 3.4 applied with s = ∇ϕ, β ≤ β1 , |β| = |β1 | − 1, α = β and w replaced by the product of the last m factors in (3.21). We obtain
Y X
β −iϕ α−β ∞ ` βi0 α0
˙ wk2 ≤ Ck∇ϕ; L ∩ H k ∂ ϕ ∂ w k∂ (e )∂
{β 0 ,α0 } 2≤i≤m i
βi0 (2
2
0
where ≤ i ≤ m) and α are multi-indices obtained by distributing at most |β1 | − 1 derivatives on β2 , . . . , βm , α − β. In particular, in the nontrivial case m ≥ 2, one has |βi0 | ≤ |βi | + |β1 | − 1 ≤ |β| − 1 ≤ |α| − 1 ≤ k − 1 ≤ `. One can then iterate the process, thereby extracting the mth power of k∇ϕ; L∞ ∩ H˙ ` k and obtaining for each m ≥ 1 a contribution k∇ϕ; L∞ ∩ H˙ ` km |w|k−m . Taking the sum over m and adding the contribution of the term m = 0 yields (3.19) and (3.20). The next estimate will be needed to estimate the nonlinearity g˜(|u|2 ).
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LONG RANGE SCATTERING FOR HARTREE TYPE EQUATIONS
Lemma 3.6. Let n ≥ 2, 0 < µ < n, let `, k1 and k2 be nonnegative integers with k1 ≤ k2 , and (3.22) n/2 < ` + µ ≤ (n/2 + k1 + k2 ) ∧ (n + k1 ) , and in addition k2 > n/2 if ` + µ = n + k1 . Then the following estimate holds : k∂ ` ω µ−n (w1 w2 )k2 ≤ C|w1 |k1 |w2 |k2 .
(3.23)
Proof. Let m = [n − µ] ∧ ` and define r by δ ≡ δ(r) = n − µ − m so that 0 ≤ δ(r) < n/2 if ` + µ ≤ n by (3.22L), while 0 ≤ δ(r) < 1 if ` + µ ≥ n. By the Hardy–Littlewood–Sobolev (HLS) inequality [22, p. 116] if m < n − µ and by inspection if m = n − µ, we estimate k∂ ` ω µ−n (w1 w2 )k2 = k∂ m ω µ−n ∂ `−m (w1 w2 )k2 ≤ Ckω −δ ∂ `−m (w1 w2 )k2 ≤ Ck∂ `−m (w1 w2 )kr¯
(3.24)
where 1/r + 1/¯ r = 1 and r¯ > 1. By the Leibnitz formula and the H¨ older inequality, we continue (3.24) as X k∂ j w1 kr1 k∂ `−m−j w2 kr2 (3.25) ··· ≤ C 0≤j≤`−m
where δi ≡ δ(ri ), i = 1, 2, have to satisfy δ1 + δ2 = n/2 − δ = µ + m − n/2 .
(3.26)
One can then continue (3.25) as · · · ≤ C|w1 |k1 |w2 |k2
(3.27)
provided ` − m ≤ k1 and provided for each j one can choose δ1 and δ2 satisfying (3.26) and ( 0 ≤ δ1 ≤ k1 − j (3.28) 0 ≤ δ2 ≤ k2 − (` − m − j) with the RHS inequality being strict if the corresponding δi is equal to n/2. The condition ` − m ≤ k1 is equivalent to ` ≤ k1 + [n − µ] and therefore to ` ≤ k1 + n − µ since ` and k1 are integers, and follows therefore from (3.22). The compatibility of (3.26) and (3.28) in δi reduces to µ + m − n/2 ≤ k1 + k2 − ` + m , again a consequence of (3.22). Finally the only possible exceptional case corresponds to δ = 0, j = ` − m = k1 , namely ` + µ = n + k1 , and requires k2 > n/2.
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We shall use Lemma 3.6 through the following corollary. We recall that `0 = [n/2]. Corollary 3.1. Let 0 < µ < n and let k, ` be nonnegative integers satisfying ` > n/2 and ` + 2 + µ ≤ (n/2 + 2k) ∧ (n + k) (3.29) and in addition k > n/2 if ` + 2 + µ = n + k. Then the following estimates hold : k∂ `0 +1 ω µ−n w1 w2 k2 + k∂ ` ω µ−n w1 w2 k2 ≤ C|w1 |k−2 |w2 |k ,
(3.30)
k∂ `+1 ω µ−n w1 w2 k2 ≤ C|w1 |k−1 |w2 |k ,
(3.31)
k∂ `+2 ω µ−n w1 w2 k2 ≤ C|w1 |k |w2 |k ,
(3.32)
k∂ `+3 ω µ−n |w|2 k2 ≤ C|w|k |w|k+1 ,
(3.33)
k∂ω µ−n w1 w2 k∞ ≤ C|w1 |k−1 |w2 |k ,
(3.34)
kω µ−n w1 w2 k∞ ≤ C|w1 |k−1 |w2 |k−1 .
(3.35)
If n is even, assume in addition that inequality (3.29) with ` replaced by n/2 + 1 (which is the lowest allowed value) is strict, namely n/2 + 3 + µ < (n/2 + 2k) ∧ (n + k) .
(3.36)
Then the following estimate holds: k∂ω µ−n w1 w2 k∞ ≤ C|w1 |k−2 |w2 |k .
(3.37)
Proof. The estimates (3.30)–(3.32) are direct applications of Lemma 3.6 with ` replaced respectively by `0 + 1 and `, by ` + 1 and by ` + 2, while (k1 , k2 ) are replaced respectively by (k − 2, k), by (k − 1, k) and by (k, k). The condition (3.22L) follows from n/2 < `0 + 1 ≤ `, while the condition (3.22R) follows from, actually reduces to (3.29) in all three cases. The estimate (3.33) follows from (3.32) applied to (w, ¯ ∂w) and to (w, ∂ w). ¯ In order to prove (3.34), (3.35) and (3.37), we note that by (1.3) ω µ−n w1 w2 and ∂ω µ−n w1 w2 belong to Lr+ + Lr− for w1 , w2 ∈ H 1 with n/r± = µ ± η. One can then estimate by Lemma 3.1 1/2
1/2
kω µ−n w1 w2 k∞ ≤ Ckω µ−n w1 w2 kn/ε k∂ω µ−n w1 w2 kn/(1−ε) which by the Hardy–Littlewood–Sobolev inequality implies 1/2
1/2
kω µ−n w1 w2 k∞ ≤ Ckω µ−n/2−ε w1 w2 k2 kω µ−n/2+ε w1 w2 k2
(3.38)
for any ε, 0 < ε ≤ 1. Similarly one can write the estimate 1/2
1/2
k∂ω µ−n w1 w2 k∞ ≤ Ck∂ω µ−n/2−ε w1 w2 k2 k∂ω µ−n/2+ε w1 w2 k2
.
(3.39)
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381
We rewrite each one of the four norms in the RHS of (3.38) and (3.39) as k∂ m1 ω µ−m0 −n/2±ε w1 w2 k2 with m1 = m0 in (3.38) and m1 = 1 + m0 in (3.39), the values of m0 to be chosen later. The proof is now achieved by repeated application of Lemma 3.6 after identification of the relevant variables, (3.35) being implied by (3.38) and (3.34) and (3.37) by (3.39). The quantity ε will always be taken sufficiently small. For n even we choose m0 = n/2. We apply (3.23) with (`, µ, k1 , k2 ) replaced by (n/2, µ ± ε, k − 1, k − 1) to (3.38) and by (n/2 + 1, µ ± ε, k − 1, k) to (3.39), thereby obtaining (3.35) and (3.34) respectively. Similarly we prove (3.37) by replacing (`, µ, k1 , k2 ) by (n/2 + 1, µ ± ε, k − 2, k). The condition (3.22L) is obviously satisfied while (3.22R) follows from (3.29) in the first two cases and from (3.36) in the last one. For n odd we choose m0 = (n + 1)/2 if 1/2 + ε < µ < n and m0 = (n − 1)/2 if 0 < µ ≤ 1/2 + ε. In the first case we apply (3.23) with (`, µ, k1 , k2 ) replaced by ((n + 1)/2, µ − 1/2 ± ε, k − 1, k − 1) to (3.38) and by ((n + 3)/2, µ − 1/2 ± ε, k − 1, k) to (3.39), thereby obtaining (3.35) and (3.34) respectively. In the second case we apply (3.23) with (`, µ, k1 , k2 ) replaced by ((n − 1)/2, µ + 1/2 ± ε, k − 1, k − 1) to (3.38) and by ((n + 1)/2, µ + 1/2 ± ε, k − 1, k) to (3.39), thereby obtaining (3.35) and (3.34) respectively. In both cases (3.22L) is obviously satisfied while (3.22R) follows from (3.29). We now introduce the following definition: Definition 3.1. A pair of nonnegative integers (k, `) will be called admissible if it satisfies k ≤ `, ` > n/2 and (3.29) and in addition k > n/2 if ` + 2 + µ = n + k, and (3.36) if n is even. Admissible pairs exist only if µ ≤ n − 2. For µ = n − 2, admissible pairs are pairs (k, `) such that k = ` > n/2. Admissible pairs always have k > 1 + µ/2 and therefore k ≥ 2. If (k, `) is an admissible pair, so is (k + j, ` + j) for any positive integer j. For n = 3, µ = 1, the pair (k, `) = (2, 2) is admissible. We next derive a number of a priori estimates for solutions of the basic system (3.4)–(3.5), to be understood in the distribution sense. In the rest of this section, we assume n ≥ 3, 0 < µ ≤ n − 2, and we denote by (k, `) an admissible pair. For solutions of (3.4) and (3.5), we shall derive a number of estimates similar to (3.13) in Lemma 3.2, and we shall state them in the shorter differential form corresponding to (3.14). Lemma 3.7. Let 0 < µ ≤ n − 2 and let (k, `) be an admissible pair. Let I ⊂ R+ k ` be an open interval, and let (w, s) be a solution of (3.4)–(3.5) in L∞ loc (I, H ⊕ X ). Then (w, s) satisfies the following estimates: |∂t |w|k | ≤ Ct−2 k∂s; L∞ ∩ H˙ ` k |w|k ≤ Ct−2 |s|` |w|k , |∂t ks; H˙ `+1 k | ≤ Ct−2 k∂s; L∞ ∩ H˙ ` k2 + Ct−γ |w|2k ,
(3.40) (3.41)
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|∂t ks; H˙ `0 k | ≤ Ct−2 ks; L∞ ∩ H˙ `0 k k∂s; L∞ ∩ H˙ `0 k + Ct−γ |w|k−2 |w|k , |∂t ksk∞ | ≤ Ct−2 ksk∞ k∂sk∞ + Ct−γ |w|k−1 |w|k , |∂t |s|` | ≤ Ct−2 |s|2` + Ct−γ |w|2k .
(3.42) (3.43) (3.44)
Proof. (3.40L). For any multi-index α with |α| ≤ k, we obtain from (3.4) X Cαβ (2∂ β s · ∇ + (∂ β ∇ · s))U ∗ (1/t)∂ α−β w , ∂t ∂ α w = (2t2 )−1 U (1/t)
(3.45)
β≤α
Q where Cαβ = i Cαβii and Cαβii are the binomial coefficients. We now use Lemma 3.2, Part (2) with u = ∂ α w, v = Ct−2 s, and h the sum of all terms not containing ∇∂ α w, thereby obtaining X |∂t k∂ α wk2 | ≤ Ct−2 k∂ j+1 s∂ k−j wk2 (3.46) 0≤j≤k
from which (3.40L) follows by Lemma 3.4 applied with s replaced by ∂s. (3.41). For any multi-index α with |α| = ` + 1, we obtain from (3.5) X Cαβ (∂ β s · ∇)∂ α−β s + t−γ ∂ α ∇g(w, w) . ∂t ∂ α s = t−2
(3.47)
β≤α
We now use Lemma 3.2, Part (1) with p = 2, u = ∂ α s, v = t−2 s, and h the sum of all terms with |β| ≥ 1 and of the contribution of g. The terms quadratic in s all have at least one derivative on each s and are estimated in L2 by Lemma 3.3 with s1 and s2 replaced by ∂s, while the term containing g is estimated by (3.32) of Corollary 3.1, thereby yielding (3.41). (3.42). For any multi-index α with |α| = `0 , we obtain again (3.47) from (3.5). We estimate the terms quadratic in s directly by Lemma 3.3 with s1 and s2 replaced by s and ∇s respectively, and with ` replaced by `0 . We estimate the contribution of g by (3.30) of Corollary 3.1. This yields (3.42). (3.43). The estimate of the terms quadratic in s is obvious and that of g follows from (3.34) of Corollary 3.1. (3.40R) and (3.44). The norms of s appearing in the LHS of (3.41)–(3.43) are precisely the norms defining X ` . The norms of s appearing in the middle member of (3.40) and in the RHS of (3.41)–(3.43) are again norms in the definition of X ` , with the exception of k∂sk∞ . However by Lemma 3.1 `+1 σ k∂sk∞ ≤ Cksk1−σ sk2 ≤ C|s|` ∞ k∂
(3.48)
with 1/(` + 1) < σ = 1/(` + 1 − n/2) < 1 since ` > n/2. This yields (3.40R) and (3.44) from (3.40L) and (3.41)–(3.43) respectively. The next result is a linear estimate of higher norms needed for regularity.
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Lemma 3.8. Let 0 < µ ≤ n − 2 and let (k, `) be an admissible pair. Let I ⊂ R+ be k+1 ⊕ X `+1 ). an open interval and let (w, s) be a solution of (3.4)–(3.5) in L∞ loc (I, H Then (w, s) satisfies the following estimates: |∂t kw; H˙ k+1 k | ≤ Ct−2 {k∂sk∞ kw; H˙ k+1 k + k∂ 2 s; L∞ ∩ H˙ ` k |w|k } , |∂t |w|k+1 | ≤ Ct−2 (|s|` |w|k+1 + |s|`+1 |w|k ) , |∂t ks; H˙ `+2 k | ≤ Ct−2 k∂s; L∞ ∩ H˙ ` k k∂ 2 s; L∞ ∩ H˙ ` k + Ct−γ |w|k |w|k+1 , |∂t |s|`+1 | ≤ Ct−2 |s|` |s|`+1 + Ct−γ |w|k |w|k+1 .
(3.49) (3.50) (3.51) (3.52)
Proof. The proof is very similar to that of Lemma 3.7 and will be sketched briefly. (3.49). We take the L2 norm of (3.45) now with |α| = k + 1, we apply Lemma 3.2, Part (2) and end up with (3.46) with k replaced by k + 1. We separate the term j = 0 and estimate the remaining terms by Lemma 3.4 with s replaced by ∂ 2 s. This yields (3.49). (3.51). We take the L2 norm of (3.47), now with |α| = ` + 2, we apply again Lemma 3.2, Part (1) with p = 2 followed by Lemma 3.3 with s1 and s2 replaced by ∂s and ∂ 2 s respectively, in order to estimate the terms quadratic in s, and we estimate the term containing g by (3.33) of Corollary 3.1. This yields (3.51). (3.50) and (3.52) follow immediately from (3.40) and (3.49) and from (3.44) and (3.51) respectively, from (3.48) and from the fact that by Lemma 3.1 `+2 σ sk2 ≤ C|s|`+1 k∂ 2 sk∞ ≤ Ck∂sk1−σ ∞ k∂
with the same σ as in (3.48).
(3.53)
We shall also need some estimates for the difference of two solutions of (3.4)– (3.5). Lemma 3.9. Let 0 < µ ≤ n − 2 and let (k, `) be an admissible pair. Let I ⊂ R+ be an open interval and let (w1 , s1 ) and (w2 , s2 ) be two solutions of (3.4)–(3.5) in k ` L∞ loc (I, H ⊕ X ). Let w± = w1 ± w2 and s± = s1 ± s2 . Then the following estimates hold : |∂t |w− |k−1 | ≤ Ct−2 {k∂s+ ; L∞ ∩ H˙ ` k |w− |k−1 + ks− ; L∞ ∩ H˙ ` k |w+ |k } ≤ Ct−2 {|s+ |` |w− |k−1 + |s− |`−1 |w+ |k } ,
(3.54)
|∂t ks− ; H˙ ` k | ≤ Ct−2 ks− ; L∞ ∩ H˙ ` k k∂s+ ; L∞ ∩ H˙ ` k + Ct−γ |w− |k−1 |w+ |k , (3.55) |∂t ks− k∞ | ≤ Ct−2 ks− k∞ k∂s+ k∞ + Ct−γ |w− |k−1 |w+ |k ,
(3.56)
|∂t |s− |`−1 | ≤ Ct−2 |s− |`−1 |s+ |` + Ct−γ |w− |k−1 |w+ |k .
(3.57)
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Proof. The proof is again very similar to that of Lemma 3.7 and will be sketched briefly. (3.54). Taking the difference of (3.4) for w1 and w2 and applying ∂ α with |α| ≤ k −1 yields X Cαβ {(2∂ β s+ · ∇ + (∂ β ∇ · s+ ))U ∗ (1/t)∂ α−β w− ∂t ∂ α w− = (4t2 )−1 U (1/t) β≤α
+ (2∂ β s− · ∇ + (∂ β ∇ · s− ))U ∗ (1/t)∂ α−β w+ } .
(3.58)
We apply Lemma 3.2, Part (2) with u = ∂ α w− and v = (2t2 )−1 s+ , followed by an application of Lemma 3.4 with (w, s, k, `) replaced by (w− , ∂s+ , k − 1, `) for the terms with (w− , s+ ) and by (w+ , s− , k, `) for the terms with (w+ , s− ). This yields (3.54L), while (3.54R) follows from the definition of X ` and from (3.48). (3.55). Taking the difference of (3.5) for s1 and s2 and applying ∂ α with |α| = ` yields X Cαβ {(∂ β s+ · ∇)∂ α−β s− + (∂ β s− · ∇)∂ α−β s+ } ∂t ∂ α s− = (2t2 )−1 β≤α
+ t−γ ∂ α ∇g(w− , w+ ) .
(3.59)
We apply Lemma 3.2, Part (1) with p = 2, u = ∂ α s− , v = (2t2 )−1 s+ , we estimate the terms quadratic in s by Lemma 3.3 with s1 and s2 replaced by s− and ∂s+ , and the contribution of g by (3.31) of Corollary 3.1. This yields (3.55). (3.56). We proceed in the same way, applying Lemma 3.2, Part (1) with p = ∞ to (3.59) with α = 0, u = s− and v = (2t2 )−1 s+ , we estimate the terms quadratic in s in the obvious way and the contribution of g by (3.34) of Corollary 3.1. This yields (3.56). Finally (3.57) follows from the definition of X ` , from (3.55) and its analogue with ` replaced by `0 , from (3.56) and from (3.48). Lemma 3.10. Let 0 < µ ≤ n − 2 and let (k, `) be an admissible pair. Let I ⊂ R+ be an open interval and let (w1 , s1 ) and (w2 , s2 ) be two solutions of (3.4)–(3.5) in k+1 k ` ⊕ X `+1 ) and L∞ L∞ loc (I, H loc (I, H ⊕ X ) respectively. Let w± = w1 ± w2 and s± = s1 ± s2 . Then the following estimates hold : |∂t |w− |k | ≤ Ct−2 {k∂s2 ; L∞ ∩ H˙ ` k |w− |k + k∂s− ; L∞ ∩ H˙ ` k |w1 |k + ks− k∞ kw1 ; H˙ k+1 k} (3.60) ≤ Ct−2 {|s2 |` |w− |k + |s− |` |w1 |k + ks− k∞ |w1 |k+1 } , ( ) X `+1 −2 ∞ ` ∞ ` `+2 k∂s− ; L ∩ H˙ k k | ≤ Ct k∂si ; L ∩ H˙ k+ks−k∞ ks1 ; H˙ k |∂t ks− ; H˙ i=1,2
+ Ct−γ |w− |k |w+ |k ,
(3.61)
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385
|∂t |s− |` | ≤ Ct−2 {|s− |` (|s1 |` + |s2 |` ) + ks− k∞ |s1 |`+1 } + Ct−γ |w− |k |w+ |k . (3.62) Proof. The proof is again very similar to that of Lemma 3.7 and will be sketched briefly. (3.60). We rewrite the difference of (3.4) for w1 and w2 as follows: ∂t w− = (2t2 )−1 U (1/t){(2s2 · ∇ + (∇ · s2 ))U ∗ (1/t)w− + (2s− · ∇ + (∇ · s− ))U ∗ (1/t)w1 } .
(3.63)
We apply ∂ α to (3.63) with |α| ≤ k, use Lemma 3.2, Part (2) with u = ∂ α w− and v = (2t2 )−1 s2 , estimate all the resulting terms by Lemma 3.4 with (w, s) replaced by (w− , ∂s2 ) or by (w1 , ∂s− ), except for the terms s− · ∇∂ α U ∗ (1/t)w1 which we estimate directly in an obvious way. This yields (3.60L), while (3.60R) follows from the definition of X ` and from (3.48). (3.61). We rewrite the difference of (3.5) for s1 and s2 similarly as ∂t s− = t−2 ((s2 · ∇)s− + (s− · ∇)s1 ) + t−γ g(w− , w+ ) .
(3.64)
We apply ∂ α to (3.64) with |α| = ` + 1, we use Lemma 3.2, Part (1) with p = 2, u = ∂ α s− and v = t−2 s2 , we estimate all the resulting terms quadratic in s by Lemma 3.3 with (s1 , s2 ) replaced by (∂s2 , ∂s− ) or by (∂s− , ∂s1 ), except for the terms (s− · ∇)∂ α s1 which we estimate directly in an obvious way. The contribution of g is estimated by (3.32) of Corollary 3.1. This yields (3.61). Finally (3.62) follows from the definition of X ` , from (3.57), (3.61) and from (3.48). Lemma 3.11. Let 0 < µ ≤ n − 2 and let (k, `) be an admissible pair. Let I ⊂ R+ be an open interval and let (w1 , s1 ) and (w2 , s2 ) be two solutions of (3.4)–(3.5) in k ` L∞ loc (I, H ⊕ X ). Let w± = w1 ± w2 and s± = s1 ± s2 . Then the following estimates hold : |∂t |w− |k−2 | ≤ Ct−2 {|s+ |` |w− |k−2 + |s− |`−2 |w+ |k } , |∂t |s− |`−2 | ≤ Ct−2 |s− |`−2 |s+ |` + Ct−γ |w− |k−2 |w+ |k .
(3.65) (3.66)
Proof. The proof is similar to that of Lemma 3.9. (3.65). We apply Lemma 3.2, Part (2) with u = ∂ α w− and v = (2t2 )−1 s+ to (3.58) with now |α| ≤ k − 2. Applying Lemma 3.4 to the terms containing (w− , s+ ) and omitting the irrelevant operator U ∗ (1/t) for brevity in the terms containing (w+ , s− ), we obtain |∂t |w− |k−2 | ≤ Ct−2 {k∂s+ ; L∞ ∩ H˙ ` k |w− |k−2 + ks− w+ ; H k−1 k} . We now distinguish two cases.
(3.67)
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If ` > n/2 + 1, we estimate the last norm in (3.67) by Lemma 3.4 with (w, s, k, `) replaced by (w+ , s− , k − 1, ` − 1) so that ks− w+ ; H k−1 k ≤ Cks− ; L∞ ∩ H˙ `−1 k |w+ |k−1 ≤ C|s− |`−2 |w+ |k .
(3.68)
If n/2 < ` ≤ n/2 + 1, namely for the lowest admissible value ` = `0 + 1, we estimate directly X 0 k∂ j s− ∂ j w+ k2 ks− w+ ; H k−1 k ≤ C j+j 0 ≤k−1
≤C
X
0
k∂ j s− kr1 k∂ j w+ kr2
(3.69)
j+j 0 ≤k−1
with 0 ≤ j ≤ k − 1 ≤ ` − 1 ≤ n/2 , (
0 ≤ δ(r1 ) = ` − 1 − j ≤ n/2 − j , 0 ≤ δ(r2 ) = n/2 − ` + 1 + j .
(3.70)
By Lemma 3.1, we then estimate k∂ j s− kr1 ≤ Ck∂ `−1 s− k2 = Cks− ; H˙ `0 k
(3.71)
with the only exception of the case of even n, ` = n/2 + 1 and j = 0 where r1 = ∞ and that norm reduces to ks− k∞ , so that in all cases k∂ j s− kr1 ≤ C|s− |`−2 .
(3.72)
On the other hand δ(r2 ) + j 0 ≤ n/2 − ` + 1 + k − 1 < k and therefore
0
k∂ j w+ kr2 ≤ C|w+ |k .
(3.73)
Substituting (3.72) and (3.73) into (3.69), substituting either the result thereof or (3.68) into (3.67) and using (3.48) yields (3.65). (3.66). We apply Lemma 3.2, Part (1) with p = 2, u = ∂ α s− , v = (2t2 )−1 s+ to (3.59) with |α| = ` − 1 and obtain |∂t k∂ `−1 s− k2 | ≤ Ct−2 k∂ `−1 ((∂s+ )s− )k2 + Ct−γ k∂ ` g(w− , w+ )k2 .
(3.74)
We distinguish again two cases. If ` > n/2 + 1, we apply Lemma 3.3 with (s1 , s2 , `) replaced by (∂s+ , s− , ` − 1) to estimate k∂ `−1 ((∂s+ )s− )k2 ≤ Ck∂s+ ; L∞ ∩ H˙ `−1 k ks− ; L∞ ∩ H˙ `−1 k ≤ C|s+ |`−1 |s− |`−2 .
(3.75)
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If n/2 < ` ≤ n/2 + 1, we estimate directly k∂ `−1 ((∂s+ )s− )k2 ≤ C
X
k∂ j s− kr1 k∂ `−j s+ kr2
(3.76)
j≤`−1
with r1 and r2 again given by (3.70), so that (3.72) holds as before, while δ(r2 ) + ` − j = n/2 + 1 so that k∂ `−j s+ kr2 ≤ C|s+ |` .
(3.77)
Substituting (3.72) and (3.77) into (3.76), and either the result thereof or (3.75) into (3.74) and estimating the contribution of g by (3.30) of Corollary 3.1 yields |∂t k∂ `−1 s− k2 | ≤ Ct−2 |s− |`−2 |s+ |` + Ct−γ |w− |k−2 |w+ |k .
(3.78)
In the case of even n, we estimate in addition |∂t ks− k∞ | ≤ Ct−2 ks− k∞ k∂s+ k∞ + Ct−γ |w− |k−2 |w+ |k
(3.79)
in the same way as in the proof of (3.56), but estimating now the contribution of g by (3.37) instead of (3.34) of Corollary 3.1. Collecting (3.78), its analogue with ` replaced by `0 + 1, and in addition (3.79) for even n yields (3.66). Remark 3.2. In Lemmas 3.7–3.11, not all the properties in the definition of admissibility for (k, `) are used in every single estimate. The condition ` > n/2 is used in many places. However the condition k ≤ ` is used only in the estimates of w or w− from (3.4), but not in the estimates of s or s− from (3.5). Conversely the condition (3.29) is used only in the estimates of s or s− from (3.5), but not in the estimates of w or w− from (3.4). Furthermore that condition is used only through the estimates (3.30)–(3.35) of Corollary 3.1, so that (3.29) could be replaced by (3.30)–(3.35) in the definition of admissibility, thereby opening the possibility of treating more general nonlinearities g˜ than simply (1.2). Finally the condition (3.36) for n even is used only through (3.37) in Lemma 3.11. 4. The Cauchy Problem at Finite Times In this section, we study the Cauchy problem for the basic system (3.4)–(3.5) at finite times. We use the basic spaces H k and X ` defined at the beginning of Sec. 3, as well as the notation (3.11) for the norm in those spaces. Admissible pairs (k, `) are defined in Definition 3.1. We prove that the Cauchy problem for the system (3.4)–(3.5) is locally well posed in R+ \{0} for positive initial time t0 and initial data in H k ⊕ X ` for admissible (k, `). A similar result holds in R− \ {0}. We make no effort to study the situation as t → 0, since that is of no interest for later purposes, and since the system (3.4)–(3.5) is singular at t = 0 anyway because of the choice (1.2) of g0 and of the change of variables (2.14) from u to (w, s). The main result can be stated as follows.
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Proposition 4.1. Let γ > 0, n ≥ 3 and 0 < µ ≤ n − 2. Let (k, `) be an admissible pair. Let t0 > 0. Then for any (w0 , s0 ) ∈ H k ⊕ X ` , there exist T± with 0 ≤ T− < t0 < T+ ≤ ∞ such that : (1) The system (3.4)–(3.5) has a unique solution (w, s) ∈ C(I, H k ⊕ X ` ) with (w, s)(t0 ) = (w0 , s0 ), where I = (T− , T+ ). If T− > 0 (resp. T+ < ∞), then |w(t)|k + |s(t)|` → ∞ when t decreases to T− (resp. increases to T+ ). 0 0 (2) If (w0 , s0 ) ∈ H k ⊕ X ` for some admissible pair (k 0 , `0 ) with k 0 ≥ k and `0 ≥ `, 0 0 then (w, s) ∈ C(I, H k ⊕ X ` ). (3) For any compact subinterval J ⊂⊂ I, the map (w0 , s0 ) → (w, s) is continuous from H k−1 ⊕ X `−1 to L∞ (J, H k−1 ⊕ X `−1 ) uniformly on the bounded sets of H k ⊕ X ` , and is pointwise continuous from H k ⊕ X ` to L∞ (J, H k ⊕ X ` ). Proof. Most of the proof proceeds by standard arguments, and we shall mainly concentrate on those which are not. We concentrate on the case of increasing time, namely t ≥ t0 . The case of decreasing time t ≤ t0 can be treated in the same way, possibly after changing t to 1/t and s to −s, thereby transforming (3.4) and (3.5) into the system ∂t w = (1/2)U (t)(2s · ∇ + (∇ · s))U (−t)w
(4.1)
∂t s = (s · ∇)s + tγ−2 ∇g(w, w)
(4.2)
and considering that system for increasing time. The (negative) powers of t in the coefficients of (3.4) and (3.5) are bounded on compact subintervals of [t0 , ∞), actually bounded on [t0 , ∞), and play no role in the present problem. We omit them for brevity. The proof proceeds in several steps. Step 1. We introduce a parabolic regularization and consider the system ∂t w = η∆w + U (1/t)(s · ∇ + (1/2)(∇ · s))U ∗ (1/t)w ≡ η∆w + F (w, s)
(4.3)
∂t s = η∆s + (s · ∇)s + ∇g(w, w) ≡ η∆s + G(w, s)
(4.4)
with η > 0. The Cauchy problem for the system (4.3)–(4.4) can be recast in the integral form ! Z ! ! t w w0 F (w, s) 0 0 + dt Vη (t − t ) (4.5) (t) = Vη (t − t0 ) (t0 ) s0 s G(w, s) t0 where Vη (t) = exp(ηt∆). The operator Vη (t) is a contraction in H k ⊕ X ` , while the operator ∇Vη (t) satisfies the bound k∇Vη (t); L(H k ⊕ X ` )k ≤ C(ηt)−1/2 .
(4.6)
From these facts and from estimates on F , G similar to, but simpler than, those in Lemma 3.7, it follows by a standard contraction argument that there exists T > 0
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depending only on η and on |w0 |k + |s0 |` such that the system (4.5) has a unique solution (wη , sη ) ∈ C([t0 , t0 + T ], H k ⊕ X ` ). Step 2. Estimates uniform in η. We estimate (wη , sη ) by Lemma 3.7, taking into account the fact that by Lemma 3.2, the terms η∆w in (4.3) and η∆s in (4.4) do not contribute to the estimates. Let y(t) = |wη (t)|k ,
z(t) = |sη (t)|` .
(4.7)
We obtain from Lemma 3.7 (with the powers of t omitted) ( ∂t y ≤ Cyz ∂t z ≤ C(y 2 + z 2 ) and by integration, with y0 = y(t0 ), z0 = z(t0 ), y(t) + z(t) ≤ (y0 + z0 )(1 − C(t − t0 )(y0 + z0 ))−1 ≤ 2(y0 + z0 ) for 2C(t − t0 )(y0 + z0 ) ≤ 1, so that for some T depending only on |w0 |k + |s0 |` , (wη (t), sη (t)) is estimated a priori in C([t0 , t0 + T ], H k ⊕ X ` ) uniformly in η. By a standard globalisation argument, the solution constructed in Step 1 can be extended to that new interval, now independent of η. Step 3. Limit η → 0. Let I0 = [t0 , t0 +T ] be the interval, independent of η, obtained in Step 2. We now prove that (wη , sη ) converges in norm in L∞ (I0 , H k−1 ⊕ X `−1 ). We already know that (wη , sη ) is estimated uniformly in η according to kwη ; L∞ (I0 , H k )k ≤ a ,
(4.8)
ksη : L∞ (I0 , X ` )k ≤ b .
(4.9)
Let now η1 , η2 > 0 and let wi = wηi , i = 1, 2. We estimate the difference (w1 − w2 , s1 − s2 ) by Lemma 3.9, except for the contribution of the terms coming from (η∆w, η∆s) in (4.3) and (4.4), which are estimated directly as follows. For |α| ≤ k − 1, we estimate ∂t k∂ α (w1 − w2 )k22 ≤ 2 Reh∂ α (w1 − w2 ), η1 ∂ α ∆w1 − η2 ∂ α ∆w2 i + other terms ≤ 4a2 (η1 + η2 ) + other terms where the other terms are those coming from Lemma 3.9. The contribution of the terms η∆s from (4.4) to the estimate of s1 −s2 are treated in the same way. Defining now (4.10) y(t) = |w1 (t) − w2 (t)|k−1 , z(t) = |s1 (t) − s2 (t)|`−1 , and combining the previous estimates with Lemma 3.9, we obtain ( ∂t y 2 ≤ C(a2 (η1 + η2 ) + by 2 + ayz) ∂t z 2 ≤ C(b2 (η1 + η2 ) + bz 2 + ayz)
(4.11)
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for t ∈ I0 , which together with y(t0 ) = 0, z(t0 ) = 0, implies that y and z tend to zero uniformly in I0 when η1 , η2 → 0 by Gronwall’s Lemma. As a consequence (wη , sη ) converges to a limit (w, s) in norm in (C ∩ L∞ )(I0 , H k−1 ⊕ X `−1 ). Clearly, (w, s) is a solution of the system (3.4)–(3.5) with the appropriate initial data. Furthermore, since (wη , sη ) is uniformly bounded in L∞ (I0 , H k ⊕ X ` ), it follows from a standard compactness argument that (w, s) belongs to that space with the same bound and that (wη , sη ) converges to (w, s) in that space in the weak-∗ sense. Step 4. Uniqueness. That step is independent of the previous ones and could equally well have been made at the very beginning. Actually uniqueness in L∞ (·, H k ⊕ X `) follows immediately from Lemma 3.9 and from Gronwall’s Lemma. Step 5. Regularity. It follows immediately from Lemmas 3.7 and 3.8 and from Gronwall’s Lemma that a solution (w, s) ∈ L∞ (I, H k ⊕ X ` ) with initial data (w, s)(t0 ) ∈ H k+1 ⊕ X `+1 belongs to L∞ (I, H k+1 ⊕ X `+1 ) for the same interval I. A similar regularity with general (k 0 , `0 ) follows by iteration. Using the previous five steps and standard arguments, one can then prove most of Proposition 4.1, with the only restriction that Parts (1) and (3) hold only with k ` C(·, H k ⊕ X ` ) replaced by C(·, H k−1 ⊕ X `−1 ) ∩ L∞ loc (·, H ⊕ X ), with continuity in Part (3) being in norm in the former space and in the weak-∗ sense in the latter, while Part (2) holds only with a similar restriction. We now turn to the proof of the missing continuities, which is more delicate. We follow a method used in [1]. We first derive an additional estimate for the difference of two solutions of (3.4)–(3.5) in L∞ (I, H k ⊕ X ` ) for some interval I. For brevity we introduce the short hand notation y = (w, s), y0 = (w0 , s0 ) and for two solutions yi = (wi , si ), y0i = (w0i , s0i ), i = 1, 2, y− = y1 − y2 and y0− = y01 − y02 . Furthermore for (k, `) an admissible pair and θ an integer, −2 ≤ θ ≤ 1, we denote j + θ = (k + θ, ` + θ) and Y j+θ = H k+θ ⊕ X `+θ . Let now yi ∈ L∞ (I, Y j ), i = 1, 2, be two solutions of (3.4)–(3.5) with initial data y0i at time t0 ∈ I for some compact interval I, satisfying the estimate kyi ; L∞ (I, Y j )k ≤ a < ∞ ,
i = 1, 2 .
(4.12)
Assume furthermore that y01 ∈ Y j+1 , so that by Step 5 y1 ∈ L∞ (I, Y j+1 ). We now estimate y− in Y j by Lemma 3.10. From (3.60) and (3.62) we obtain ∂t |y− |j ≤ C(a|y− |j + ks− k∞ |y1 |j+1 ) .
(4.13)
By Lemma 3.8, especially (3.50) and (3.52), we estimate ∂t |y1 |j+1 ≤ Ca|y1 |j+1 and therefore by Gronwall’s Lemma, for all t ∈ I, |y1 |j+1 ≤ C(a, |I|)|y01 |j+1 .
(4.14)
We next estimate for ` > n/2, possibly by using (3.10), ks− k∞ ≤ C|s− |`−2 for ` ≥ n/2 + 1 , ks k ≤ C|s |1/2 |s |1/2 − ∞ − `−2 − `−1
(4.15) for n odd, ` = (n + 1)/2 .
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We now estimate ∂t |y− |j−θ ≤ Ca|y− |j−θ with θ = 1, by Lemma 3.9, especially (3.54) and (3.57), and with θ = 2 by Lemma 3.11, especially (3.65) and (3.66), so that by Gronwall’s Lemma again, for θ = 1, 2 and for all t ∈ I |y− |j−θ ≤ C(a, |I|)|y0− |j−θ
(4.16)
and therefore by (4.15), for all t ∈ I, ks− k∞ ≤ C(a, |I|)ky0− kb where
ky0− kb = |y0− |j−2 ky
0− kb
1/2
(4.17)
for ` ≥ n/2 + 1 , 1/2
= |y0− |j−2 |y0− |j−1
(4.18) for n odd, ` = (n + 1)/2 .
Substituting (4.14) and (4.17) into (4.13) and applying Gronwall’s Lemma again, we obtain for all t ∈ I |y− |j ≤ C(a, |I|)(|y0− |j + ky0− kb |y01 |j+1 ) .
(4.19)
We now come back to the proof of the missing continuities, which will make an essential use of the estimate (4.19). Step 6. Continuity of the solutions in H k ⊕ X ` . Let R+ \ {0} be a compact interval and let (w, s) ∈ C(I, H k−1 ⊕X `−1 )∩L∞ (I, H k ⊕X ` ) be solution of the system (3.4)– (3.5) with initial data (w0 , s0 ) ∈ H k ⊕ X ` at some time t0 ∈ I. We shall prove that (w, s) ∈ C(I, H k ⊕ X ` ). We use the short hand notation y, y0 , Y j , etc. introduced above. We introduce aR regularisation defined as follows. We choose a function ψ1 ∈ S(Rn ) such that dxψ1 (x) = 1 and such that |ξ|−2 (ψb1 (ξ) − 1)|ξ=0 = 0, we define ψε (x) = ε−n ψ1 (x/ε) so that ψbε (ξ) = ψb1 (εξ) and we define the regularisation by f → fε = ψε ∗ f for all f ∈ S 0 . Clearly the regularisation is a bounded operator with norm at most kψ1 k1 and tends strongly to the unit operator when ε → 0 in Y j for all relevant j. We now regularize the initial data y0 to y0ε . By the previous steps, y0ε generates a solution yε of (3.4)–(3.5). For ε sufficiently small, and possibly after a small restriction of I, that solution can be assumed to be in L∞ (I, Y j ) for the same interval I as y and to be bounded there uniformly in ε, namely ky; L∞ (I, Y j )k ∨ kyε ; L∞ (I, Y j )k ≤ a < ∞ .
(4.20)
Furthermore, since y0ε ∈ Y j+1 , by regularity (Step 5), yε ∈ C(I, Y j ) ∩ L∞ (I, Y j+1 ). In order to prove that y ∈ C(I, Y j ) it is therefore sufficient to prove that yε converges to y in norm in L∞ (I, Y j ). For that purpose we apply the estimate (4.19) with y1 = yε , y2 = y, y− = yε − y. Now for all f k∂fε k2 = k∂(ψε ∗ f )k2 ≤ k∂ψε k1 kf k2 = ε−1 k∂ψ1 k1 kf k2 ,
(4.21)
kfε − f k2 = kψε ∗ f − f k2 = k(ψbε (ξ) − 1)fˆ(ξ)k2 ≤ εθ k |ξ|−θ (ψb1 (ξ) − 1)k∞ kf ; H˙ θ k
(4.22)
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for θ = 1, 2. Furthermore for n even and s ∈ L∞ ∩ H˙ n/2 ∩ H˙ n/2+θ so that |ξ|n/2 sb(ξ) ∈ L2 , and for θ = 1, 2, sk1 ≤ εθ k |ξ|−n/2−θ (ψb1 (ξ) − 1)k2 ks; H˙ n/2+θ k . kψε ∗ s − sk∞ ≤ k(ψbε − 1)b
(4.23)
It follows from (4.21)–(4.23) that for θ = 1, 2, (
|y0ε |j+1 ≤ Cε−1 |y0 |j , |y0− |j−θ ≤ Cεθ |y0 |j .
(4.24)
Substituting the estimate (4.24) into (4.18) and (4.19) and using the fact that |y0− |j → 0 when ε → 0 shows that |y− |j tends to zero when ε → 0 uniformly for t ∈ I, which completes the proof. Step 7. Continuity with respect to initial data in C(·, H k ⊕ X ` ). From Steps 1–5 and general arguments, it follows only that the solution (w, s) so far constructed is norm continuous in C(·, H k−1 ⊕ X `−1 ) and weak-∗ continuous in L∞ (·, H k ⊕ X ` ) as a function of the initial data (w0 , s0 ) ∈ H k ⊕ X ` . We now prove strong continuity in C(·, H k ⊕ X ` ). Again we use the short hand notation y, y0 , Y j , etc. introduced above. Let I ⊂ R+ \{0} be a compact interval and let y ∈ (C ∩L∞ )(I, Y j ) be a fixed solution of the system (3.4)–(3.5) with initial data y0 ∈ Y j at some time t0 ∈ I. Let y00 be an initial data in a small neighborhood of y0 in Y j and let y 0 be the solution of (3.4)–(3.5) thereby generated. We also consider the regularized initial data y0ε 0 and y0ε , and the solutions yε and yε0 thereby generated. By taking y00 sufficiently close to y0 and ε sufficiently small and possibly after a small restriction of I, we can assume that y 0 and yε , yε0 are in L∞ (I, Y j ) for the same interval I as y, and are bounded there uniformly in y00 and in ε so that both (4.20) and its analogue with y replaced by y 0 hold. For all t ∈ I, we estimate |y − y 0 |j ≤ |yε − y|j + |yε − yε0 |j + |yε0 − y 0 |j
(4.25)
and we estimate the three norms in the RHS by (4.19) with (y1 , y2 ) replaced by (yε , y), (yε , yε0 ) and (yε0 , y 0 ) respectively. Using in addition the first inequality in (4.24), we obtain 0 0 |j + |y0ε − y00 |j |y − y 0 |j ≤ C(a, |I|)(|y0ε − y0 |j + |y0ε − y0ε 0 0 + aε−1 (ky0ε − y0 kb + ky0ε − y0ε kb + ky0ε − y00 kb )) .
We now estimate 0 |j ≤ C|y0 − y00 |j |y0ε − y0ε 0 0 |y0ε − y00 |j ≤ |y0ε − y0ε |j + |y0ε − y0 |j + |y0 − y00 |j
≤ |y0ε − y0 |j + C|y00 − y0 |j
(4.26)
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and similarly 0 kb ≤ Cky0 − y00 kb ≤ C|y0 − y00 |j ky0ε − y0ε 0 ky0ε − y00 kb ≤ ky0ε − y0 kb + Cky00 − y0 kb
≤ ky0ε − y0 kb + C|y00 − y0 |j . Substituting those estimates into (4.26) yields |y − y 0 |j ≤ C(a, |I|)(|y0ε − y0 |j + aε− 1 ky0ε − y0 kb + (1 + aε−1 )|y0 − y00 |j )
(4.27)
which can be made to tend to zero uniformly for t ∈ I by letting y00 tend to y0 in Y j and letting ε tend to zero, in that order. Remark 4.1. Whereas the map y0 → y is uniformly continuous from Y j to (C ∩ L∞ )(·, Y j−1 ) on the bounded sets of Y j , it is only pointwise continuous from Y j to (C ∩ L∞ )(·, Y j ). In fact Step 7 is performed for fixed y0 , and does not yield an estimate of ky − y 0 ; L∞ (·, Y j )k in terms of |y0 − y00 |j . This is a standard situation in that kind of problems. 5. The Auxiliary System at Infinite Time. Existence and Asymptotics I In this section we study the existence of solutions in a neighborhood of infinity in time for the auxiliary system ( (2.30) ≡ (3.4) ≡ (5.1) ∂t w = (2t2 )−1 U (1/t)(2s · ∇ + (∇ · s))U ∗ (1/t)w (2.31) ∼ (3.5) ≡ (5.2)
∂t s = t−2 (s · ∇)s + t−γ ∇g(w, w)
where g is defined by (3.1) and (3.2), and we study the asymptotic behaviour in time of those solutions by essentially constructing local wave operators at infinity for the system (5.1)–(5.2) as compared with the auxiliary free equation ∂t s0 = t−γ ∇g(w+ , w+ ) . The general solution of (5.3) can be written as Z t dt0 t0−γ ∇g(w+ , w+ ) . s0 (t) = s0 (1) +
(2.33) ∼ (3.7) ≡ (5.3)
(5.4)
1
Since from (5.2) and (5.4) the functions s(t) and s0 (t) are expected to increase as t1−γ , we define the functions s˜(t) = tγ−1 s(t) ,
s˜0 (t) = tγ−1 s0 (t)
(5.5)
which are expected to be bounded in time. We shall use those functions throughout this section. It follows from Corollary 3.1 that for admissible (k, l) and for (w+ , s0 (1)) ∈ H k ⊕ X ` , s˜0 (t) ∈ (C ∩ L∞ )([1, ∞), X ` ) and that s0 (t) satisfies the estimate k˜ s0 ; L∞ ([1, ∞); X ` )k ≤ |s0 (1)|` + C(1 − γ)−1 |w+ |2k .
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We shall use the basic spaces H k and X ` defined at the beginning of Sec. 3, as well as the notation (3.11) for the norms in those spaces. We recall that admissible pairs (k, `) are defined in Definition 3.1. In all this section, we assume that n ≥ 3 and 0 < µ ≤ n − 2. The letter C in subsequent estimates will denote various constants depending on n, µ and possibly on an admissible pair (k, `). On the other hand we shall keep the dependence of the estimates on γ sufficiently explicit for the constants C to be uniform in γ in the range of γ where the estimates are stated. For instance a factor γ −1 will be kept explicitly in estimates valid for all γ > 0, but will be included in C for estimates valid for γ > 1/2. In the first three propositions, we study the existence of solutions of the system (5.1)–(5.2) defined in a neighborhood of infinity and some of their asymptotic properties. All those results hold for all γ > 0, but we restrict our attention to 0 < γ < 1 in order to simplify the exposition, as explained in the introduction. Most of those results are consequences of Proposition 4.1 and of a priori estimates where we now carefully keep track of the time dependence. We begin with the existence of solutions defined in a neighborhood of infinity in time. Proposition 5.1. Let 0 < γ < 1 and let (k, `) be an admissible pair. Let (w0 , s˜0 ) ∈ s0 |` . Then there exists T0 < ∞, depending H k ⊕ X ` and define y0 = |w0 |k and z˜0 = |˜ on y0 , z˜0 , such that for all t0 ≥ T0 , there exists T ≤ t0 , depending on y0 , z˜0 and t0 , , has such that the system (5.1)–(5.2) with initial data w(t0 ) = w0 , s(t0 ) = s˜0 t1−γ 0 a unique solution (w, s) in the interval [T, ∞) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ). One can take z0 + (1 − γ)−1 y02 ) , γT0γ = C(˜ , T = γT0γ t1−γ 0
(5.6) (5.7)
and the solution (w, s) is estimated by |w|k ≤ 2y0 z0 + C(1 − γ)−1 y02 )(t0 ∨ t)1−γ |s|` ≤ (2˜
(5.8) (5.9)
for all t ≥ T. Proof. The result follows from Proposition 4.1 and standard globalisation arguments, provided we can derive (5.8)–(5.9) as a priori estimates under the assumptions made on t0 and t. Let (w, s) be the maximal solution of (5.1)–(5.2) with the appropriate initial condition at t0 obtained by Proposition 4.1 and define y = |w|k and z = |s|` . By Lemma 3.7, y and z satisfy ( |∂t y| ≤ Ct−2 yz (5.10) |∂t z| ≤ Ct−2 z 2 + Ct−γ y 2 and we estimate y and z from those inequalities, taking C = 1 for the rest of the proof. We distinguish two cases.
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395
Case t ≥ t0 . Let t¯ > t0 and define Y = Y (t¯) = ky; L∞ ([t0 , t¯])k and Z = Z(t¯) = ktγ−1 z; L∞ ([t0 , t¯])k. Then for all t ∈ [t0 , t¯] ( ∂t y ≤ t−1−γ Y Z (5.11) ∂t z ≤ t−2γ Z 2 + t−γ Y 2 and therefore by integration with the appropriate initial condition at t0 ( Y ≤ y0 + γ −1 t−γ 0 YZ
(5.12) 2 −1 2 Z ≤ z˜0 + t−γ Y 0 Z + (1 − γ) Rt where we have used the fact that the function f (γ) = t0 dt0 t0−2γ is logarithmically convex in γ and therefore satisfies 1−γ . f (γ) ≤ f (0)1−γ f (1)γ = (t − t0 )(t0 t)−γ ≤ t−γ 0 t
(5.13)
Now (5.12) defines a closed subset R of R × R in the (Y, Z) variables, containing the point (y0 , z˜0 ), and (Y, Z) is a continuous function of t¯ starting from that point for t¯ = t0 . If we can find an open region R1 of R+ × R+ containing (y0 , z˜0 ) and such that R ∩ R1 ⊂ R1 , then (Y, Z) will remain in R ∩ R1 for all time, because R ∩ R1 = R ∩ R1 is both open and closed in R. We take for R1 the strip R1 = {(Y, Z) : Z < (1/2)γtγ0 }, so that in R ∩ R1 ( Y ≤ 2y0 (5.14) Z ≤ 2˜ z0 + 2(1 − γ)−1 Y 2 ≤ 2˜ z0 + 8(1 − γ)−1 y02 +
+
and the condition R ∩ R1 ⊂ R1 is ensured by z0 + 16(1 − γ)−1 y02 . γtγ0 > 4˜
(5.15)
Case t ≤ t0 . Let t¯ < t0 and define Y = Y (t¯) = ky; L∞ ([t¯, t0 ])k and Z = Z(t¯) = kz; L∞([t¯, t0 ])k. Then for all t ∈ [t¯, t0 ] ( −∂t y ≤ t−2 Y Z (5.16) −∂t z ≤ t−2 Z 2 + t−γ Y 2 and therefore by integration with the appropriate initial condition at t0 ( Y ≤ y0 + t−1 Y Z Z ≤ (˜ z0 + (1 − γ)−1 Y 2 )t1−γ + t−1 Z 2 0
(5.17)
so that for Z < t/2 ( Y < 2y0 Z < 2(˜ z0 + (1 − γ)−1 Y 2 )t1−γ < (2˜ z0 + 8(1 − γ)−1 y02 )t1−γ , 0 0
(5.18)
which ensures the condition Z < t/2 provided t > (4˜ z0 + 16(1 − γ)−1 y02 )t1−γ . 0
(5.19)
Putting back constants at appropriate places, we obtain (5.8)–(5.9) from (5.14)– (5.18), while (5.15)–(5.19) allow for the choice (5.6)–(5.7).
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Remark 5.1. The weakness of Proposition 5.1 is immediately apparent on (5.7) and (5.9). In fact, for the construction of wave operators, we want to solve the Cauchy problem for (5.1)–(5.2) for an initial time t0 → ∞. However when t0 → ∞, the interval [T, ∞) where the solution is defined disappears, while the estimate for |s|` blows up. This defect will be remedied in Proposition 5.6 below, but only for γ > 1/2. For the next results, we shall need the following estimate. Lemma 5.1. Let 0 < γ < 1, let a > 0, b > 0, t0 > 0 and let y, z be nonnegative continuous functions satisfying y(t0 ) = y0 , z(t0 ) = z0 , and ( |∂t y| ≤ t−1−γ by + t−2 az (5.20) |∂t z| ≤ t−1−γ bz + t−γ ay . Define y¯, z¯ by Then, for
γ(tγ0
(y, z) = (¯ y , z¯) exp[bγ −1 |t−γ − t−γ 0 |] . ∧ t ) ≥ 2a , the following estimates hold : ( y¯ ≤ 2(y0 + az0 t−1 0 ) γ
1−γ z¯ ≤ z0 + 2(1 − γ)−1 a(y0 + az0 t−1 0 )t
for t ≥ t0 , and
(5.21)
2
(
)t−1 y¯ ≤ y0 + 2a(z0 + (1 − γ)−1 ay0 t1−γ 0 z¯ ≤ 2(z0 + (1 − γ)−1 ay0 t1−γ ) 0
(5.22)
(5.23)
for t ≤ t0 . Proof. The inequalities (5.20) have been written in differential form, but should be understood in integrated form. Changing the variables from (y, z) to (¯ y , z¯) yields ( z ±∂t y¯ ≤ t−2 a¯ (5.24) y ±∂t z¯ ≤ t−γ a¯ for t > t0 , and in integrated form <
Z t 0 0−2 0 y ¯ ≤ y + a dt t z ¯ (t ) 0 t0
Z t z¯ ≤ z0 + a dt0 t0−γ y¯(t0 ) .
(5.25)
t0
Substituting the second component of (5.25) into the first one yields an inequality for y¯ alone: Z t 2 0 0−γ 0−1 −1 0 | + a dt t (t − t )¯ y (t ) (5.26) y¯ ≤ y0 + az0 |t−1 − t−1 0 . t0
We shall estimate y¯ by (5.26) and estimate z¯ by substituting the result into the second component of (5.25). (One could equally well obtain an equation for z¯
LONG RANGE SCATTERING FOR HARTREE TYPE EQUATIONS
397
alone similar to (5.26) and estimate z¯ directly therefrom.) We distinguish the cases t > t0 . <
Case t ≥ t0 . Let t¯ > t0 . We define Y = k¯ y; L∞ ([t0 , t¯])k. It then follows from (5.26) that 2 −1 −γ t0 Y Y ≤ y0 + az0 t−1 0 +a γ and therefore Y ≤ 2(y0 + az0 t−1 0 ) for γtγ0 ≥ 2a2 , while by (5.25) z¯ ≤ z0 + a(1 − γ)−1 t1−γ Y , which immediately yields (5.22). Case t ≤ t0 . Let t¯ < t0 . We define Y = kt(¯ y − y0 ); L∞ ([t¯, t0 ])k. It then follows from (5.26) that + a2 Y t−1 γ −1 t−γ y¯ ≤ y0 + az0 t−1 + a2 y0 t−1 (1 − γ)−1 t1−γ 0 so that + a2 γ −1 t−γ Y Y ≤ az0 + a2 y0 (1 − γ)−1 t1−γ 0 and therefore ) Y ≤ 2a(z0 + ay0 (1 − γ)−1 t1−γ 0 for γtγ ≥ 2a2 . Substituting that result into the second component of (5.25) yields z¯ ≤ z0 + ay0 (1 − γ)−1 t1−γ + aγ −1 t−γ Y 0 ) ≤ 2(z0 + ay0 (1 − γ)−1 t1−γ 0 by using again the condition γtγ ≥ 2a2 . The previous estimates immediately yield (5.23). Remark 5.2. Since (5.20) is a linear system, it is clear that one could obtain estimates of y and z valid for all t0 > 0, t > 0, but in view of Proposition 5.1, we are not interested in estimates for small t0 , t. The apparent lower restrictions on t0 , t come from the fact that the ansatz Y for y¯ is inadequate for small t. As a first consequence of Lemma 5.1, we obtain a uniqueness result at infinity for the system (5.1)–(5.2). Proposition 5.2. Let 0 < γ < 1 and let (k, `) be an admissible pair. Let (wi , si ), i = 1, 2 be two solutions of the system (5.1)–(5.2) such that (wi , s˜i ) ∈ L∞ ([T, ∞), H k ⊕ X ` ) for some T > 0 and such that |w1 − w2 |k−1 t1−γ and |s1 − s2 |`−1 tend to zero when t → ∞. Then (w1 , s1 ) = (w2 , s2 ).
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Proof. Define a = Maxkwi ; L∞ ([T, ∞), H k )k , i
y = |w1 − w2 |k−1 ,
b = Maxk˜ si : L∞ ([T, ∞), X ` )k , i
z = |s1 − s2 |`−1 .
Then by Lemma 3.9, y and z satisfy the inequalities ( |∂t y| ≤ Ct−1−γ by + Ct−2 az |∂t z| ≤ Ct−1−γ bz + Ct−γ ay
(5.27) (5.28)
(5.29)
and therefore (up to constants), the estimate (5.23) for all T ≤ t ≤ t0 , t and t0 sufficiently large. Taking the limit t0 → ∞ in (5.23) for fixed t shows that y(t) = 0, z(t) = 0. We now begin the study of the asymptotic behaviour of solutions of the system (5.1)–(5.2) by showing that for the solutions obtained in Proposition 5.1 (actually for slightly more general solutions, if any) w(t) tends to a limit when t → ∞. Proposition 5.3. Let 0 < γ < 1 and let (k, `) satisfy k ≤ ` + 1 and ` > n/2. Let (w, s) be a solution of the system (5.1)–(5.2) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ) for some T > 0. Let a = kw; L∞ ([T, ∞), H k )k ,
b = k˜ s; L∞ ([T, ∞), X ` )k .
(5.30)
Then there exists w+ ∈ H k such that w(t) tends to w+ strongly in H k−1 and weakly in H k when t → ∞. Furthermore the following estimates hold : |w+ |k ≤ a |w(t0 ) − w(t)|k−1 ≤ Cabγ −1 (t0 ∧ t)−γ |w(t) − w+ |k−1 ≤ Cabγ −1 t−γ
(5.31) (5.32) (5.33)
for t0 ∧ t ≥ T . Proof. Let T ≤ t0 ≤ t and w0 = w(t0 ). By exactly the same method as in Lemma 3.9 (see especially (3.54)) we obtain from (5.1) ∂t |w − w0 |k−1 ≤ Ct−2 |s|` |w|k ≤ Ct−1−γ ab and by integration between t0 and t |w − w0 |k−1 ≤ Cabγ −1 t−γ 0 . This proves (5.32), from which it follows that w(t) has a limit w+ ∈ H k−1 such that (5.33) holds. Since in addition w(t) is uniformly bounded in H k , it follows by a standard compactness argument that w+ ∈ H k , that w+ satisfies (5.31) and that w(t) tends weakly to w+ in H k .
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Remark 5.3. Note that the condition on (k, `) in Proposition 5.3 is weaker than admissibility, since in particular we do not use (5.2) and therefore do not require estimates of g. We now turn to the proof of existence of asymptotic states for the solutions of (5.1)–(5.2) constructed in Proposition 5.1. As explained in Sec. 2, we want to perform the following construction on such a solution. We take t0 ≥ T and we consider the solution (w0 , s0,t0 ) of the free Eq. (5.3) which coincides with (w, s) at time t0 . That solution is defined by w0 = w(t0 ) and Z t s0,t0 (t) = s(t0 ) + dt0 t0−γ ∇g(w0 , w0 ) (5.34) t0
or equivalently, by (5.2) Z t s0,t0 (t) = s(t) − dt0 {t0−2 (s · ∇)s + t0−γ (∇g(w, w) − ∇g(w0 , w0 ))} .
(5.35)
t0
We want to prove that (w0 , s0,t0 ) has a limit when t0 → ∞. Proposition 5.3 already provides us with the limit w+ = lim w(t0 ), and it remains only to be proved that also s0,t0 has a limit. Taking formally the limit t0 → ∞ in (5.35) leads us to expect that that limit should be defined by Z ∞ dt0 {t0−2 (s · ∇)s + t0−γ (∇g(w, w) − ∇g(w+ , w+ ))} . (5.36) s0 (t) = s(t) + t
From Corollary 3.1 and from (5.34), it follows that for (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H ⊕ X ` ), also (w0 , s˜0,t0 ) belongs to the same space, actually with constant w0 , and by (5.30), satisfies the estimate k
+ Ca2 (1 − γ)−1 (t0 ∨ t)1−γ . |s0,t0 (t)|` ≤ bt1−γ 0
(5.37)
That estimate however is not bounded uniformly in t0 , and is therefore useless to take the limit t0 → ∞. In order to proceed further, we shall need stronger assumptions, and in particular γ > 1/2. Under that condition and by using (5.35), we can indeed obtain estimates on s0,t0 that are uniform in t0 . Proposition 5.4. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let (w, s) be a solution of the system (5.1)–(5.2) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ) for some T ≥ 1 and define a and b by (5.30). Let t0 ≥ T, let w0 = w(t0 ) and define s0,t0 (t) by (5.34). Then s0,t0 (t) satisfies the estimates 1−γ |s0,t0 (t) − s(t)|`−1 ≤ C(b2 + (1 − γ)−1 a2 b)t−γ 0 t
for t ≥ t0 and
tγ0
(5.38)
≥ Cb, |s0,t0 (t) − s(t)|`−1 ≤ C(2γ − 1)−1 (b2 + a2 b)t1−2γ
(5.39)
for T ≤ t ≤ t0 and tγ ≥ Cb. Assume in addition that (1 − γ)tγ0 ≥ Ca2 and (2γ − 1)tγ ≥ C(a2 + b). Then s0,t0 satisfies the estimate (5.40) |s0,t0 (t)|`−1 ≤ Cbt1−γ .
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Proof. By the same method as in Lemma 3.9 (see especially (3.57)), we estimate the integrand in (5.35) by |{·}|`−1 ≤ Ct−2 |s|`−1 |s|` + Ct−γ |w − w0 |k−1 |w + w0 |k
(5.41)
which by (5.30) and Proposition 5.3, especially (5.32), can be continued as · · · ≤ Cb2 t−2γ + Ca2 bt−γ (t ∧ t0 )−γ
(5.42)
for (t ∧ t0 )γ ≥ Cb. Integrating (5.42) between t0 and t and using (5.13) if t ≥ t0 yields (5.38) for t ≥ t0 and (5.39) for t ≤ t0 . Finally (5.40) follows from (5.30), from (5.38) and (5.39) and from the additional conditions on t, t0 . We next prove that s0 (t) is actually well defined by (5.36) and is the limit of s0,t0 (t) when t0 → ∞. Proposition 5.5. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let (w, s) be a solution of the system (5.1)–(5.2) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ) for some T ≥ 1 and define a and b by (5.30). Let w+ be the limit of w(t) when t → ∞ obtained in Proposition 5.3. Then (1) The integral in (5.36) is absolutely convergent in X `−1 and defines a solution s0 of the Eq. (5.3) such that s˜0 ∈ (C ∩L∞ )([1, ∞), X `−1 ) and such that the following estimate holds for t ≥ T, tγ ≥ Cb : |s0 (t) − s(t)|`−1 ≤ C(2γ − 1)−1 (b2 + a2 b)t1−2γ .
(5.43)
If in addition (2γ − 1)tγ ≥ C(b + a2 ), the following estimate holds: |s0 (t)|`−1 ≤ Cbt1−γ .
(5.44)
(2) The function s0,t0 defined by (5.34) converges to s0 in norm in X `−1 when t0 → ∞ for t ≥ T, tγ ≥ Cb, uniformly in compact intervals, and the following estimate holds for t ∧ t0 ≥ T, (t ∧ t0 )γ ≥ Cb : 1−γ . |s0,t0 (t) − s0 (t)|`−1 ≤ C((1 − γ)−1 + (2γ − 1)−1 )(b2 + a2 b)t−γ 0 (t ∨ t0 )
(5.45)
Proof. Part (1). By the same method as in the proof of Proposition 5.4, we estimate the integrand in (5.36) by (5.41) with w0 replaced by w+ and therefore by (5.30) and (5.33) (5.46) |{·}|`−1 ≤ C(b2 + a2 b)t−2γ which proves the convergence of the integral in X `−1 and yields the estimate (5.43). Finally (5.44) follows from (5.30) and (5.43) and from the additional condition on t. Part (2). For t ≥ t0 , we estimate |s0,t0 (t) − s0 (t)|`−1 ≤ |s0,t0 (t) − s(t)|`−1 + |s0 (t) − s(t)|`−1 .
(5.47)
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We estimate the first norm in the RHS by (5.38) and the second norm by (5.43). This yields (5.45) for t ≥ t0 . For t ≤ t0 , we start from Z s0,t0 (t) − s0 (t) = s(t0 ) − s0 (t0 ) +
t0
dt0 t0−γ (∇g(w+ , w+ ) − ∇g(w(t0 ), w(t0 ))) .
t
(5.48) We estimate the first difference in the RHS by (5.43) with t = t0 and the integrand by Corollary 3.1 and (5.33) as |∇g(w+ , w+ ) − ∇g(w(t0 ), w(t0 ))|`−1 ≤ C|w(t0 ) − w+ |k−1 |w(t0 ) + w+ |k ≤ Ca2 bt−γ 0
(5.49)
which after integration yields (5.45) for t ≤ t0 . The convergence of s0,t0 to s0 in the norms indicated follows from (5.45).
Remark 5.4. Note that in Proposition 5.5 we are losing one degree of regularity in s, namely a solution (w, s) in H k ⊕ X ` has an asymptotic free solution (w+ , s0 ) in H k ⊕ X `−1 only. We have no uniform estimate in t0 of s0,t0 in X ` , the best we have being (5.37) and we are therefore unable to assert that the limiting s0 remains in X ` . We now turn to the main and more difficult question of existence of the wave operator. The construction is now the converse of that performed in Propositions 5.3–5.5. We consider a fixed solution (w+ , s0 ) of (5.3), we construct a solution (wt0 , st0 ) of (5.1)–(5.2) which coincides with (w+ , s0 ) at time t0 , and we take the limit of that solution when t0 → ∞. Here again we shall encounter a loss of derivative, now more severe than in the previous case. In particular we shall need to start with a free solution in H k+1 ⊕ X `+1 for admissible (k, `) and end up with a solution (w, s) which is only in H k ⊕ X ` . The crucial step is the construction of (wt0 , st0 ) and will be performed by an extension of the energy method used in Proposition 5.1. Proposition 5.6. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let (w+ , s0 (1)) ∈ H k+1 ⊕ X `+1 , define s0 by (5.4) and let a = |w+ |k+1 ,
b = k˜ s0 ; L∞ ([1, ∞), X `+1 )k .
(5.50)
Then, there exist T0 and T, 1 ≤ T0 , T < ∞, depending only on (γ, a, b), such that for all t0 ≥ T0 ∨ T, the system (5.1)–(5.2) with initial data w(t0 ) = w+ , s(t0 ) = s0 (t0 ), has a unique solution (wt0 , st0 ) in the interval [T, ∞) such that (wt0 , s˜t0 ) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ). One can take T0γ = C(b + (1 − γ)−1 a2 )
(5.51)
T γ = C(2γ − 1)−1 (b + a2 )
(5.52)
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and the solution satisfies the estimates ( |wt0 (t) − w+ |k ≤ Cabt−γ 0 1−γ |st0 (t) − s0 (t)|` ≤ C(b2 + (1 − γ)−1 a2 b)t−γ 0 t
for t ≥ t0 ,
(
|wt0 (t) − w+ |k ≤ Cabt−γ |st0 (t) − s0 (t)|` ≤ C(2γ − 1)−1 (b2 + a2 b)t1−2γ
(5.53)
(5.54)
for T ≤ t ≤ t0 , and |wt0 (t)|k ≤ Ca ,
|st0 (t)|` ≤ Cbt1−γ
(5.55)
for all t ≥ T. Proof. The result follows from Proposition 4.1 and standard globalisation arguments provided we can derive (5.53)–(5.54) as a priori estimates under the assumptions of the proposition. Let (wt0 , st0 ) be the maximal solution of (5.1)–(5.2) with the appropriate initial condition at t0 and define y = |wt0 − w+ |k and z = |st0 − s0 |` . We rewrite (5.1) and the difference between (5.2) and (5.3) as ( (5.56) ∂t (wt0 − w+ ) = (2t2 )−1 U (1/t)(2st0 · ∇ + (∇ · st0 ))U ∗ (1/t)wt0 ∂t (st0 − s0 ) = t−2 (st0 · ∇)st0 + t−γ (∇g(wt0 , wt0 ) − ∇g(w+ , w+ )) .
(5.57)
From (5.56) and (5.57), by exactly the same method as in Lemma 3.8, we obtain |∂t |wt0 − w+ |k | ≤ Ct−2 |st0 |` (|wt0 − w+ |k + |w+ |k+1 ) (5.58) |∂t |st0 − s0 |` | ≤ Ct−2 |st0 |` (|st0 − s0 |` + |s0 |`+1 ) −γ + Ct |wt0 − w+ |k |wt0 + w+ |k and therefore by (5.50) ( |∂t y| ≤ Ct−2 (z + bt1−γ )(y + a) |∂t z| ≤ Ct−2 (z + bt1−γ )2 + Ct−γ y(y + a) .
(5.59)
We estimate y and z from (5.59), taking C = 1 for the rest of the proof. We distinguish again two cases. Case t ≥ t0 . Let t¯ > t0 and define Y = Y (t¯) = ky; L∞ ([t0 , t¯])k and Z = Z(t¯) = ktγ−1 z; L∞ ([t0 , t¯])k. Then for all t ∈ [t0 , t¯] ( ∂t y ≤ t−1−γ (Z + b)(Y + a) (5.60) ∂t z ≤ t−2γ (Z + b)2 + t−γ Y (Y + a) and therefore by integration with the appropriate initial condition at t0 ( Y ≤ 2t−γ 0 (Z + b)(Y + a) 2 −1 Z ≤ t−γ Y (Y + a) 0 (Z(Z + 2b) + b ) + (1 − γ)
where we have used (5.13) again.
(5.61)
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403
As in the proof of Proposition 5.1, we impose an additional condition 2t−γ 0 (Z + 2b) < λ ≤ 1/2 and we obtain from (5.61) ( Y ≤ (1 − λ)−1 λa −3 Z ≤ (1 − λ)−1 b2 t−γ (1 − γ)−1 λa2 . 0 + (1 − λ)
(5.62)
(5.63)
We now choose λ = 8bt−γ so that (5.62 L) reduces to Z < 2b, and we obtain from 0 (5.63) ( Y ≤ 16abt−γ 0 (5.64) Z ≤ (2b2 + 64(1 − γ)−1 a2 b)t−γ 0 . The condition Z < 2b is ensured by (5.51) for t0 ≥ T0 , and the estimate (5.53) follows from (5.64). Case t ≤ t0 . Let t¯ ≤ t0 and define Y = Y (t¯) = ktγ y; L∞ ([t¯, t0 ])k and Z = Z(t¯) = kt2γ−1 z; L∞ ([t¯, t0 ])k. Then for all t ∈ [t¯, t0 ] ( −∂t y ≤ t−1−γ (Zt−γ + b)(Y t−γ + a) (5.65) −∂t z ≤ t−2γ (Zt−γ + b)2 + t−2γ Y (Y t−γ + a) and therefore by integration with the appropriate initial condition at t0 ( y ≤ 2abt−γ + (aZ + bY )t−2γ + Y Zt−3γ z ≤ (2γ − 1)−1 (b2 + aY )t1−2γ + 2(Y 2 + 2bZ)t1−3γ + Z 2 t1−4γ where we have used the condition γ > 1/2, so that ( Y ≤ 2ab + aZt−γ + (b + Zt−γ )Y t−γ Z ≤ (2γ − 1)−1 (b2 + aY ) + 2Y 2 t−γ + (4b + Zt−γ )Zt−γ .
(5.66)
(5.67)
We impose the additional conditions Zt−γ < 2b and tγ ≥ 12b and we obtain from (5.67) ( Y ≤ 8ab (5.68) Z ≤ 2((2γ − 1)−1 (b2 + aY ) + (6b)−1 Y 2 ) which yields (5.54), while the conditions Zt−γ < 2b and tγ ≥ 12b are ensured by (5.52). Finally (5.55) follows from (5.50)–(5.54). Remark 5.5. The major improvement of Proposition 5.6 over Proposition 5.1, which has been achieved by estimating the difference of (w, s) and (w+ , s0 ) instead of estimating (w, s) alone, is that now (w, s) is defined in a time interval [T, ∞) which is independent of t0 , and is estimated in that interval uniformly with respect to t0 . The condition γ > 1/2 plays a crucial role in obtaining that improvement (cf. Remark 5.1 above).
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Remark 5.6. One may wonder whether the loss of regularity from (k + 1, ` + 1) to (k, `) when constructing (wt0 , st0 ) from (w+ , s0 ) is unavoidable. Actually (wt0 (t0 ), st0 (t0 )) ∈ H k+1 ⊕ X `+1 and by regularity (Proposition 4.1, Part (2)), (wt0 , st0 ) ∈ C([T, ∞), H k+1 ⊕ X `+1 ). One can then estimate y = |wt0 |k+1 and z = |st0 |`+1 . By Lemma 3.8 and by (5.1)–(5.2) and (5.55), y and z satisfy the system (5.29) and therefore are estimated up to constants by (5.22)–(5.23). However under natural ), the estimate (5.23), which is the one assumptions y0 = O(1) and z0 = O(t1−γ 0 really relevant for large t0 , is not uniform in t0 , so that the estimate is lost when one takes the limit t0 → ∞, which is what we shall do next. As a consequence the regularity at the level of (k + 1, ` + 1) is also lost in that limit, and we have therefore made no effort to keep track of it at the stage of Proposition 5.6. We can now take the limit t0 → ∞ of the solution (wt0 , st0 ) constructed in Proposition 5.6 for fixed (w+ , s0 ). Proposition 5.7. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. (1) Let (w+ , s0 (1)) ∈ H k+1 ⊕ X `+1 and define s0 , a, b, by (5.4) and (5.50). Then there exists T, 1 ≤ T < ∞, depending only on (γ, a, b) and a unique solution (w, s) of the system (5.1)–(5.2) in the interval [T, ∞) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ), satisfying (5.36) and such that the following estimates hold for all t ≥ T : ( |w(t) − w+ |k ≤ Cabt−γ (5.69) |s(t) − s0 (t)|` ≤ C(2γ − 1)−1 (b2 + a2 b)t1−2γ , |w(t)|k ≤ Ca ,
|s(t)|` ≤ Cbt1−γ .
(5.70)
One can take T γ = C(2γ − 1)−1 (b + a2 ) .
(5.71)
(2) Let (wt0 , st0 ) be the solution of the system (5.1)–(5.2) constructed in Proposition 5.6 for t0 ≥ T0 ∨ T and in particular such that (wt0 , s˜t0 ) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ). Then (wt0 , st0 ) converges to (w, s) in norm in L∞ (J, H k−1 ⊕ X `−1 ) and in the weak-∗ sense in L∞ (J, H k ⊕ X ` ) for any compact interval J ⊂ [T, ∞), and in the weak-∗ sense in H k ⊕ X ` pointwise in t. (3) The map (w+ , s0 (1)) 7→ (w, s) defined in Part (1) is continuous on the bounded sets of H k+1 ⊕ X `+1 from the norm topology of (w+ , s0 (1)) in H k−1 ⊕ X `−1 to the norm topology of (w, s) in L∞ (J, H k−1 ⊕ X `−1 ) and to the weak-∗ topology in L∞ (J, H k ⊕ X `) for any compact interval J ⊂ [T, ∞), and to the weak-∗ topology in H k ⊕ X ` pointwise in t. Proof. Parts (1) and (2) will follow from the convergence of (wt0 , st0 ) when t0 → ∞ in the topologies stated in Part (2). We recall that (wt0 , st0 ) satisfies the estimates (5.53) and (5.54) which we rewrite more briefly as ( [wt0 − w+ |k ≤ M1 (t ∧ t0 )−γ (5.72) |st0 − s0 |` ≤ M2 (t ∧ t0 )−γ t1−γ
LONG RANGE SCATTERING FOR HARTREE TYPE EQUATIONS
405
where M1 , M2 depend on (γ, a, b) and can be read from (5.53) and (5.54), and satisfies the estimate (5.55). Let now T0 ∨ T ≤ t0 ≤ t1 . From (5.72) it follows that for all t ≥ t0 |wt0 − wt1 |k ≤ 2M1 t−γ 0 (5.73) |s − s | ≤ 2M t−γ t1−γ . t0 t1 ` 2 0 We now estimate (wt0 − wt1 , st0 − st1 ) in H k−1 ⊕ X `−1 for t ≤ t0 . Let y = |wt0 − wt1 |k−1 ,
z = |st0 − st1 |`−1 .
(5.74)
From Lemma 3.9 and from (5.55), it follows that y and z satisfy the system (5.29). Integrating that system for t ≤ t0 with initial data at t0 estimated by (5.73), we obtain from Lemma 5.1, especially (5.23) 1−2γ −1 y(t) ≤ M (t−γ t ) 0 + t0 (5.75) z(t) ≤ M t1−2γ 0 for some M depending on (γ, a, b) and for T ≤ t ≤ t0 . From (5.75) it follows that there exists (w, s) ∈ C([T, ∞), H k−1 ⊕ X `−1 ) such that (wt0 , st0 ) converges to (w, s) in L∞ (J, H k−1 ⊕X `−1 ) for all compact intervals J ⊂ [T, ∞). From that convergence, from (5.54) and (5.55) and from standard compactness arguments, it follows that (w, s˜) ∈ (L∞ ∩ Cw∗ )([T, ∞), H k ⊕ X ` ), that (w, s) satisfies the estimates (5.69) and (5.70) for all t ≥ T , and that (wt0 , st0 ) converges to (w, s) in the other topologies considered in Part (2). Furthermore, by the local result of Proposition 4.1, Part (1), (w, s) ∈ C([T, ∞), H k ⊕ X ` ). Obviously (w, s) is a solution of (5.1) and (5.2). We now prove that (w, s) satisfies (5.36). Let s00 (t) be the RHS of (5.36). By Proposition 5.5, Part (1), s00 (t) is well defined and satisfies the analogue of (5.43). Furthermore s00 (t) satisfies (5.3) so that s0 (t) − s00 (t) is constant in time. By (5.43) for s00 (t) and (5.69), that constant is zero, namely s00 (t) = s0 (t). This proves (5.36). From the uniqueness result of Proposition 5.2, it follows that (w, s) is unique under the condition (5.69). This completes the proof of Parts (1) and (2). 0 , s00 (1)) belong to H k+1 ⊕ X `+1 , define s0 and Part (3). Let (w+ , s0 (1)) and (w+ 0 s0 by (5.4) and its analogue, assume that 0 |k+1 ≤ a |w+ |k+1 ∨ |w+
ktγ−1 s0 ; L∞ ([1, ∞), X `+1 )k ∨ ktγ−1 s00 ; L∞ ([1, ∞), X `+1 )k ≤ b , let (w, s) and (w0 , s0 ) be the associated solutions of the system (5.1)–(5.2) constructed in Part (1), satisfying (5.69) and its analogue with the same (a, b) for 0 , s00 (1)) close to (w+ , s0 (1)) t ≥ T with the same T defined by (5.71). We take (w+ k−1 `−1 ⊕X in the sense that for some small ε, ε0 > 0 in H 0 |k−1 ≤ εa |w+ − w+
|s0 (1) − s00 (1)|`−1 ≤ ε0 b
(5.76) (5.77)
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and therefore by (5.4) and estimates from Corollary 3.1 |s0 − s00 |`−1 ≤ ε0 b + C(1 − γ)−1 a2 εt1−γ .
(5.78)
From (5.69) and its analogue for (w0 , s0 ) and from (5.76) and (5.78), it follows that for all t ≥ T ( |w − w0 |k−1 ≤ εa + Cabt−γ (5.79) |s − s0 |`−1 ≤ ε0 b + C(1 − γ)−1 a2 εt1−γ + C(2γ − 1)−1 (b2 + a2 b)t1−2γ . −1 We now define t0 by t−γ so that for t ≥ t0 , 0 = εb
|w − w0 |k−1 ≤ Cεa |s − s0 |`−1 ≤ ε0 b + C[(1 − γ)−1 a2 + (2γ − 1)−1 (b + a2 )]εt1−γ
(5.80)
and in particular |s(t0 ) − s0 (t0 )|`−1 ≤ ε0 b + M ε(2γ−1)/γ
(5.81)
for some M depending on (γ, a, b). We now estimate (w − w0 , s− s0 ) in H k−1 ⊕ X `−1 for t ≤ t0 . Let y = |w − w0 |k−1 , z = |s − s0 |`−1 . From Lemma 3.9, it follows that y and z satisfy the system (5.29). Integrating that system for t ≤ t0 with initial data at t0 estimated by (5.80) and (5.81), we obtain from Lemma 5.1, especially (5.23), (
y(t) ≤ Cεa + (Cε0 b + M ε(2γ−1)/γ )at−1 z(t) ≤ Cε0 b + M ε(2γ−1)/γ
(5.82)
for some M depending on (γ, a, b) and for all t, T ≤ t ≤ t0 . This implies the continuity of (w, s) as a function of (w+ , s0 (1)) in the norm topology of L∞ (J, H k−1 ⊕ X `−1 ) for all compact intervals J ⊂ [T, ∞). The other continuities follow therefrom and from the boundedness of (w, s˜) in L∞ ([T, ∞), H k ⊕ X ` ) by standard compactness arguments. Remark 5.7. By analogy with Proposition 4.1, Part (3), one expects the map (w+ , s0 (1)) → (w, s) to be also continuous from the norm topology in H k ⊕ X ` to the norm topology in L∞ (J, H k ⊕ X ` ) for compact J ⊂ [T, ∞). A proof of that fact would require a combination of Steps 6 and 7 in the proof of Proposition 4.1 with the proof just given of Proposition 5.7, Part (3) with however estimates at the level (k, `) instead of (k − 1, ` − 1). However, the coupling between t0 and ε in the latter has the effect that, when the time dependence is taken into account in the former, the resulting dependence of the estimates on ε or on t0 is not sufficiently good to establish the result without additional assumptions on γ, namely without assuming γ sufficiently close to 1. Since the argument is rather complicated for a result of restricted validity, we refrain from pushing it any further.
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407
In Propositions 5.5 and 5.7, we have defined two maps (w, s) → (w+ , s0 ) and (w+ , s0 ) → (w, s) between solutions (w, s) of the system (5.1)–(5.2) and solutions of the auxiliary free Eq. (5.3) defined in a neighborhood of infinity in time. Both of these maps suffer from the loss of one derivative. Nevertheless, they are inverse of each other (and in particular injective) whenever they can be applied successively. We state that fact in the following proposition. Proposition 5.8. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. (1) Let (w+ , s0 ) be a solution of (5.3) such that (w+ , s˜0 ) ∈ (C ∩ L∞ )([1, ∞), H k+1 ⊕ X `+1 ) or equivalently defined by (5.4) with (w+ , s0 (1)) ∈ H k+1 ⊕ X `+1 . Let (w, s) be the solution of (5.1)–(5.2) defined in Proposition 5.7, Part (1), and 0 , s00 ) be the solution of (5.3) defined from (w, s) in Propositions 5.3 and 5.5. let (w+ 0 = w+ and s00 = s0 . Then w+ (2) Let (w, s) be a solution of (5.1)–(5.2) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), k+2 ⊕ X `+2 ) for some T, 1 ≤ T < ∞. Let (w+ , s0 ) be the solution of (5.3) defined H in Propositions 5.3 and 5.5, so that (w+ , s˜0 ) ∈ (C ∩ L∞ )([1, ∞), H k+1 ⊕ X `+1 ) and let (w0 , s0 ) be the solution of (5.1)–(5.2) defined from (w+ , s0 ) in Proposition 5.7, Part (1), so that (w0 , s˜0 ) ∈ (C ∩ L∞ )([T 0 , ∞), H k ⊕ X ` ) for some T 0 , 1 ≤ T 0 < ∞. Then w0 = w and s0 = s for t ≥ T ∨ T 0 . Proof. Part (1). From Proposition 5.7, especially (5.69) we obtain |w − w+ |k ≤ M t−γ ,
|s − s0 |` ≤ M t1−2γ
for some M depending on (w+ , s0 ). From Propositions 5.3 and 5.5, especially (5.33) and (5.43), we obtain similarly 0 ]k−1 ≤ M t−γ , |w − w+
|s − s00 |`−1 ≤ M t1−2γ .
0 = w+ , so that by (5.4) s00 − s0 is constant Taking the limit t → ∞ shows that w+ in time, and therefore zero.
Part (2). From Propositions 5.3 and 5.5, we obtain in the same way |w − w+ |k+1 ≤ M t−γ ,
|s − s0 |`+1 ≤ M t1−2γ .
From Proposition 5.7, we then obtain |w0 − w+ |k ≤ M t−γ ,
|s0 − s0 |` ≤ M t1−2γ
so that |w − w0 |k ≤ M t−γ ,
|s − s0 |` ≤ M t1−2γ
for t ≥ T ∨ T 0 . The result then follows from Proposition 5.2.
We conclude this section with a brief discussion of the other systems of equations that can be used instead of (5.1)–(5.2) and (5.3) and on their drawbacks as compared with the latter. That discussion was briefly sketched at the end of Sec. 2 and can now be resumed at a more technical level.
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(i) Instead of the system (5.1)–(5.2) corresponding to (2.25)–(2.26) and to the choice w = w3 , we could have used a system corresponding to (2.19)–(2.20) and to the choice w = w2 , with an additional term (2t2 )−1 ∆w in the equation for w. That term would make no difference in the energy estimate (3.40) of Lemma 3.7 and therefore in Proposition 5.1. However it would produce an additional term Ct−2 |w+ |k+2 in the energy estimate (5.58) for |w − w+ |k , and therefore a loss of two derivatives instead of one on w in the crucial Proposition 5.6. (ii) Instead of the free auxiliary Eq. (5.3) corresponding to (3.7), we could have used the more accurate HJ equation corresponding to (3.6). However, since we need the assumption γ > 1/2 in a crucial way in order to obtain estimates uniform in t0 for t ≤ t0 , both in Proposition 5.4 (see especially (5.39)) and in Proposition 5.6 (see especially (5.54)), using (3.6) instead of (3.7) would not produce any improvement of the results. On the other hand it would make the treatment of s0 more complicated, since the Eq. (3.6) is hardly simpler than the system (3.4)–(3.5). In particular, instead of the explicit solution (5.4), we would need a proposition similar to Proposition 5.1 in order to solve (3.6) for s0 , with a result valid for large time only. Similarly, in order to estimate s0 (t) uniformly in t0 for t ≤ t0 and to take the limit t0 → ∞ so as to prove the existence of asymptotic states, we would have to replace the relatively simple Propositions 5.4 and 5.5 by more complicated ones of the same degree of complication as needed to prove the existence of wave operators, namely Propositions 5.6 and 5.7. Finally it would no longer be possible, in the definition of the wave operators, to characterize the asymptotic solution (w+ , s0 ) by an initial condition at a fixed time t = 1 independent of w+ . 6. The Auxiliary System at Infinite Time. Asymptotics II In this section, we perform a construction similar to that of Sec. 5, and we essentially construct local wave operators at infinity for the auxiliary system (5.1)–(5.2), now however compared with the free Eq. (3.8) which is better suited than (3.7) ≡ (5.3) for the study of gauge invariance, as explained in Sec. 2. We shall again encounter a loss of derivatives, now more severe than in Sec. 5 (see Remarks 5.4 and 5.6), namely we shall lose two derivatives when constructing (w, s) from w+ . The results of this section could be derived directly by the same method as in Sec. 5, but for simplicity we shall instead derive them from the results of that section. This has the additional advantage that the loss of derivatives could be slightly reduced, namely from 2 to 1 + 2(1 − γ) + 0, although we shall not exploit that possibility here. We recall that g0 and g are defined by (3.1) and (3.2). In order to make the comparison with Sec. 5 more transparent, we shall use exclusively g0 and refrain from using g in this and the next section. For brevity we shall also use the short hand notation g0 (w) = g0 (w, w) for the diagonal restriction of g0 . With that notation g(w, w) = g0 (U ∗ (1/t)w) . In order to distinguish solutions of (3.8) from those of (5.3) considered in the previous section, we shall use the notation s02 for the former, the additional subscript 2 referring to the fact that the nonlinearity in (3.8) is that of the free auxiliary
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Eq. (2.22) naturally associated with the system (2.19)–(2.20) for (w2 , ϕ2 ). With the previous notation, the Eq. (3.8) is rewritten as ∂t s02 = t−γ ∇g0 (w+ )
(6.1)
s02 (t) = s02 (1) + (1 − γ)−1 (t1−γ − 1)∇g0 (w+ )
(6.2)
and is trivially solved by
to be compared with the general solution (5.4) of (5.3). It follows from (6.2) and Corollary 3.1 that for admissible (k, `) and for (w+ , s02 (1)) ∈ H k ⊕ X ` , s˜02 ≡ tγ−1 s02 ∈ (C ∩ L∞ )([1, ∞), X ` ) and k˜ s02 ; L∞ ([1, ∞), X ` )k ≤ |s02 (1)|` + C(1 − γ)−1 |w+ |2k . We first compare solutions of (5.3) and (6.1) which coincide in a suitable sense at some time t0 . Lemma 6.1. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let w0 ∈ H k . Let 1 ≤ t0 ≤ ∞. Let s0 be a solution of (5.3) with w+ = w0 and let s02 be a solution of (6.1) with w+ = U ∗ (1/t0 )w0 such that s0 (t0 ) = s02 (t0 ), so that Z t dt0 t0−γ (∇g0 (U ∗ (1/t0 )w0 ) − ∇g0 (U ∗ (1/t0 )w0 )) . (6.3) s0 (t) − s02 (t) = t0
(For t0 = ∞, coincidence at t0 is defined by (6.3) and justified by the estimates to follow.) Then (1) The following estimates hold : −1/2 1−γ
|s0 (t) − s02 (t)|`−1 ≤ C(1 − γ)−1 |w0 |2k t0
(t ≥ t0 )
t
|s0 (t) − s02 (t)|`−1 ≤ C(2γ − 1)−1 |w0 |2k t1/2−γ
(t ≤ t0 ) .
(6.4) (6.5)
(2) Assume in addition that w0 ∈ H k+2 . Then the following estimates hold : −1/2 1−γ
|s0 (t) − s02 (t)|`+1 ≤ C(1 − γ)−1 |w0 |k+1 |w0 |k+2 t0
t
|s0 (t) − s02 (t)|`+1 ≤ C(2γ − 1)−1 |w0 |k+1 |w0 |k+2 t1/2−γ
(t ≥ t0 ) (t ≤ t0 ) .
(6.6) (6.7)
(3) Assume again that w0 ∈ H k+2 . Then the following estimates hold : 1−γ |s0 (t) − s02 (t)|` ≤ C(1 − γ)−1 |w0 |k |w0 |k+2 t−1 0 t
|s0 (t) − s02 (t)|` ≤ Cγ −1 |w0 |k |w0 |k+2 t−γ
(t ≥ t0 )
(t ≤ t0 ) .
(6.8) (6.9)
Proof. Part (1). From (6.3) and from Corollary 3.1, we obtain Z t dt0 t0−γ |(U ∗ (1/t0 ) − U ∗ (1/t0 ))w0 |k−1 |w0 |k |s0 (t) − s02 (t)|`−1 ≤ C t0
while |(U ∗ (1/t0 ) − U ∗ (1/t0 ))w0 |k−1 ≤ |1/t0 − 1/t0 |1/2 |w0 |k from which (6.4) and (6.5) follow immediately.
(6.10)
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Part (2). We estimate similarly by (6.3) and Corollary 3.1 Z t dt0 t0−γ |(U ∗ (1/t0 ) − U ∗ (1/t0 ))w0 |k+1 |w0 |k+1 |s0 (t) − s02 (t)|`+1 ≤ C t0
from which (6.6) and (6.7) follow as previously. Part (3). We estimate similarly by (6.3) and Corollary 3.1 Z t dt0 t0−γ |(U ∗ (1/t0 ) − U ∗ (1/t0 ))w0 |k |w0 |k |s0 (t) − s02 (t)|` ≤ C t0
from which (6.8) and (6.9) follow as previously.
We now follow step by step the constructions performed in Sec. 5 with s0 , leading to the existence of asymptotic states (Propositions 5.4 and 5.5) and to the existence of local wave operators at infinity (Propositions 5.6 and 5.7). We first consider a fixed solution (w, s) of the system (5.1)–(5.2) as constructed in Proposition 5.1 and we look for a solution of (6.1) which is asymptotic to s(t) at infinity. For that purpose we take some t0 large enough and we define the solution of (6.1) which coincides with s(t) at t0 by Z t dt0 t0−γ ∇g0 (U ∗ (1/t0 )w0 ) (6.11) s02,t0 (t) = s(t0 ) + t0
or equivalently
Z
t
s02,t0 (t) = s(t) −
dt0 {t0−2 (s · ∇)s + t0−γ (∇g0 (U ∗ (1/t0 )w) − ∇g0 (U ∗ (1/t0 )w0 ))}
t0
(6.12) with w0 = w(t0 ) (compare with (5.34) and (5.35)). As in Sec. 5, s02,t0 (t) satisfies the analogue of the estimate (5.37), which is not uniform in t0 , and the first task is to obtain an estimate uniform in t0 . This is done in the following proposition, which is the analogue of Proposition 5.4. Proposition 6.1. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let (w, s) be a solution of the system (5.1)–(5.2) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ) for some T > 0 and define a and b by (5.30). Let t0 ≥ T and w0 = w(t0 ). Then s02,t0 defined by (6.11) satisfies the estimates −1/2
−1 2 a t0 |s02,t0 (t) − s(t)|`−1 ≤ C((b2 + (1 − γ)−1 a2 b)t−γ 0 + (1 − γ)
)t1−γ
(6.13)
for t ≥ t0 and tγ0 ≥ Cb, |s02,t0 (t) − s(t)|`−1 ≤ C(2γ − 1)−1 ((b2 + a2 b)t1−2γ + a2 t1/2−γ )
(6.14)
for T ≤ t ≤ t0 and tγ ≥ Cb. 1/2 Assume in addition that (1 − γ)tγ0 ≥ Ca2 , (1 − γ)t0 ≥ Ca2 b−1 , (2γ − 1)tγ ≥ C(a2 + b) and (2γ − 1)t1/2 ≥ Ca2 b−1 . Then the following estimate holds: |s02,t0 (t)|`−1 ≤ Cbt1−γ .
(6.15)
LONG RANGE SCATTERING FOR HARTREE TYPE EQUATIONS
411
Proof. The result follows from Proposition 5.4 and Lemma 6.1, Part (1), applied to s02,t0 and s0,t0 defined by (6.11) and (5.34). We now want to prove that s02,t0 has a limit when t0 → ∞. Following the method of Sec. 5 would lead us to define the limiting function s02 by taking the formal limit t0 → ∞ in (6.12), namely Z ∞ s02 (t) = s(t) + dt0 {t0−2 (s · ∇)s + t0−γ (∇g0 (U ∗ (1/t0 )w) − ∇g0 (w+ ))} (6.16) t
where above s02 (t) define
w+ is the limit of w(t) as t → ∞ obtained in Proposition 5.3. As mentioned we use instead the results of Sec. 5, especially Proposition 5.5. We define in terms of s0 (t) obtained in the latter and defined by (5.36), namely we s02 (t) by Z ∞ dt0 t0−γ (∇g0 (U ∗ (1/t0 )w+ ) − ∇g0 (w+ )) . (6.17) s02 (t) = s0 (t) + t
The following proposition is the analogue for s02 of Proposition 5.5. Proposition 6.2. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let (w, s) be a solution of the system (5.1)–(5.2) such that (w, s˜) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ) for some T ≥ 1 and define a and b by (5.30). Let w+ be the limit of w(t) when t → ∞ obtained in Proposition 5.3. Then (1) The integral in (6.16) is absolutely convergent in X `−1 and defines a solution s02 of Eq. (6.1) such that s˜02 ∈ (C ∩ L∞ )([1, ∞), X `−1 ) and such that the following estimate holds for t ≥ T, tγ ≥ Cb : |s02 (t) − s(t)|`−1 ≤ C(2γ − 1)−1 ((b2 + a2 b)t1−2γ + a2 t1/2−γ ) .
(6.18)
If in addition (2γ − 1)tγ ≥ C(b + a2 ) and (2γ − 1)t1/2 ≥ Ca2 b−1 , the following estimate holds: (6.19) |s02 (t)|`−1 ≤ Cbt1−γ . (2) The function s02,t0 defined by (6.11) converges to s02 in norm in X `−1 when t0 → ∞ for t ≥ T, tγ ≥ Cb, uniformly in compact intervals, and the following estimate holds for t ∧ t0 ≥ T, (t ∧ t0 )γ ≥ Cb : |s02,t0 (t) − s02 (t)|`−1 −1/2
2 ≤ C((1 − γ)−1 + (2γ − 1)−1 )((b2 + a2 b)t−γ 0 + a t0
)(t ∨ t0 )1−γ .
(6.20)
Proof. Part (1). Let s0 (t) be defined by (5.36) supplemented by Proposition 5.5, Part (1) and define s02 (t) by (6.17). The result now follows from Proposition 5.5, Part (1) and Lemma 6.1, Part (1). In particular (6.18) follows from (5.43) and (6.5). Part (2). For t ≥ t0 , we estimate |s02,t0 (t) − s02 (t)|`−1 ≤ |s02,t0 (t) − s(t)|`−1 + |s02 (t) − s(t)|`−1
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and we estimate the first norm in the RHS by (6.13) and the second norm by (6.18). This yields (6.20) for t ≥ t0 . For t ≤ t0 , we obtain from (6.11) and (6.1) Z t0 dt0 t0−γ (∇g0 (w+ ) s02,t0 (t) − s02 (t) = s(t0 ) − s02 (t0 ) + t ∗
− ∇g0 (U (1/t0 )w(t0 ))) .
(6.21)
We estimate the first difference in the RHS by (6.18) with t = t0 , and the integral by the same method as in the proof of Lemma 6.1, Part (1), and by the use of (5.33) and (6.9) with t0 = ∞, so that Z t0 1/2−γ 0 dt · · · ≤ C(1 − γ)−1 (a2 bt1−2γ + a 2 t0 ) 0 t
`−1
which completes the proof of (6.20) for t ≤ t0 . The convergence stated in Part (2) follows from the estimate (6.20).
Remark 6.1. The approximation of s by s02 for solutions (w, s) of the system (5.1)–(5.2) is not as good as the approximation by s0 obtained in Sec. 5. This can be seen on (6.18) where the term a2 t1/2−γ is dominant for large t as compared with the term (b2 + a2 b)t1−2γ obtained from (5.43). We now turn to the converse construction, namely to the construction of a solution (w, s) of the system (5.1)–(5.2) which is asymptotic to a fixed solution (w+ , s02 ) of (6.1), defined by (6.2). The first step consists in constructing a solution (w, s) which coincides with (w+ , s0 ) at some time t0 . As mentioned above that construction produces a loss of two derivatives and requires w+ ∈ H k+2 . The next result is the analogue of Proposition 5.6. Proposition 6.3. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let (w+ , s02 (1)) ∈ H k+2 ⊕ X `+1 , define s02 by (6.2) and let a = |w+ |k+2 ,
b = k˜ s02 ; L∞ ([1, ∞), X `+1 )k .
(6.22)
Then, there exist T0 and T, 1 ≤ T0 , T < ∞, depending only on (γ, a, b), such that for all t0 ≥ T0 ∨ T, the system (5.1)–(5.2) with initial data w(t0 ) = U (1/t0 )w+ , s(t0 ) = s02 (t0 ), has a unique solution (wt0 , st0 ) in the interval [T, ∞) such that (wt0 , s˜t0 ) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ). One can take T0 = C{(b + (1 − γ)−1 a2 )1/γ ∨ (1 − γ)−2 a4 b−2 } T = C(2γ − 1)−2 ((b + a2 )1/γ ∨ a4 b−2 ) and the solution satisfies the estimates −1 |wt0 (t) − w+ |k ≤ C(abt−γ 0 + at0 ) |s (t) − s (t)| ≤ C((b2 + (1 − γ)−1 a2 b)t−γ + (1 − γ)−1 a2 t−1 )t1−γ t0 02 ` 0 0
(6.23) (6.24)
(6.25)
LONG RANGE SCATTERING FOR HARTREE TYPE EQUATIONS
for t ≥ t0 , (
|wt0 (t) − w+ |k ≤ C(abt−γ + at−1 0 ) |st0 (t) − s02 (t)|` ≤ C(2γ − 1)−1 ((b2 + a2 b)t1−2γ + a2 t−γ )
413
(6.26)
for T ≤ t ≤ t0 , and |wt0 (t)|k ≤ Ca ,
|st0 (t)|` ≤ Cbt1−γ
(6.27)
for all t ≥ T. Proof. Let w0 = U (1/t0 )w+ and define s0 (t) by (6.3) so that s0 (t) solves (5.3) and satisfies s0 (t0 ) = s02 (t0 ). By Lemma 6.1, Part (2), s0 ∈ C([1, ∞), X `+1 ) and s0 satisfies (6.28) k˜ s0 ; L∞ ([T, ∞), X `+1 )k ≤ Cb provided (1 − γ)t0 ≥ Ca2 b−1 and (2γ − 1)T 1/2 ≥ Ca2 b−1 which follow from (6.23) and (6.24) for t0 ≥ T0 . We now apply Proposition 5.6 with s0 just defined. Let (wt0 , st0 ) be the solution of the system (5.1)–(5.2) thereby obtained under the conditions (5.51) and (5.52) which also follow from (6.23) and (6.24). That solution satisfies the required initial condition wt0 (t0 ) = w0 and st0 (t0 ) = s0 (t0 ) = s02 (t0 ). Furthermore it satisfies the estimates (5.53)–(5.55) with however w+ replaced by w0 . The estimates (6.25)–(6.27) follow from the previous ones, from Lemma 6.1, Part (3) and from the estimate 1/2
|w0 − w+ |k = |(U (1/t0 ) − 1)w+ |k ≤ t−1 0 |w+ |k+2 .
(6.29)
We now take the limit t0 → ∞ of the solution (wt0 , st0 ) constructed in Proposition 6.3 for fixed (w+ , s02 ). The next result is the analogue of Proposition 5.7. Proposition 6.4. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. (1) Let (w+ , s02 (1)) ∈ H k+2 ⊕X `+1 and define s02 , a, b by (6.2) and (6.22). Then there exists T, 1 ≤ T < ∞, depending only on (γ, a, b) and a unique solution (w, s) of the system (5.1)–(5.2) in the interval [T, ∞) such that (w, s˜) ∈ (C∩L∞ )([T, ∞), H k ⊕ X ` ) and such that the following estimates hold for all t ≥ T : ( |w(t) − w+ |k ≤ Cabt−γ , (6.30) |s(t) − s02 (t)|` ≤ C(2γ − 1)−1 ((b2 + a2 b)t1−2γ + a2 t−γ ) , |w(t)|k ≤ Ca ,
|s(t)|` ≤ Cbt1−γ .
(6.31)
One can take T = C(2γ − 1)−2 ((b + a2 )1/γ ∨ a4 b−2 ) .
(6.32)
(2) Let (wt0 , st0 ) be the solution of the system (5.1)–(5.2) constructed in Proposition 6.3 for t0 ≥ T0 ∨ T and in particular such that (wt0 , s˜t0 ) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ X ` ). Then (wt0 , st0 ) converges to (w, s) in norm in L∞ (J, H k−1 ⊕ X `−1 ) and
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in the weak-∗ sense in L∞ (J, H k ⊕ X ` ) for any compact interval J ⊂ [T, ∞), and in the weak-∗ sense in H k ⊕ X ` pointwise in t. (3) The map (w+ , s02 (1)) 7→ (w, s) defined in Part (1) is continuous on the bounded sets of H k+2 ⊕X `+1 from the norm topology of (w+ , s02 (1)) in H k−1 ⊕X `−1 to the norm topology of (w, s) in L∞ (J, H k−1 ⊕ X `−1 ) and to the weak-∗ topology in L∞ (J, H k ⊕ X `) for any compact interval J ⊂ [T, ∞), and to the weak-∗ topology in H k ⊕ X ` pointwise in t. Proof. Part (1). Take now (6.17) as the definition of s0 in terms of s02 , so that s0 satisfies (6.28) with T given by (6.32) by Lemma 6.1, Part (2). Let (w, s) be the solution of the system (5.1)–(5.2) constructed in Proposition 5.7. Then (w, s) satisfies the properties stated in Part (1). In particular (6.30) follows from (5.69) and from (6.9). Uniqueness of (w, s) follows from Proposition 5.2. Part (2). We estimate y = |wt0 (t) − w(t)|k−1 and z = |st0 (t) − s(t)|`−1 in the same way as in the proof of Proposition 5.7, Part (2) by using Lemma 3.9 and Lemma 5.1, with initial conditions y(t0 ) and z(t0 ) estimated by (6.26) and (6.30) at time t0 . The rest of the proof is identical with that of Proposition 5.7, Part (2). Part (3). The proof follows from that of Proposition 5.7, Part (3) provided we can show that the function Z ∞ dt0 t0−γ (∇g0 (U ∗ (1/t0 )w+ ) − ∇g0 (w+ )) f (w+ ) = s02 (1) − s0 (1) = 1 k−1
`−1
is continuous from H to X uniformly on the bounded sets of H k+1 . By Corollary 3.1, especially (3.31) and suitable variants of (6.10), we estimate Z ∞ 0 0 0 dt0 t0−γ |(U ∗ (1/t0 ) − 1)(w+ + w+ )|k |f (w+ ) − f (w+ )|`−1 ≤ C|w+ − w+ |k−1 0 + C|w+ + w+ |k
Z
1 ∞
0 dt0 t0−γ |(U ∗ (1/t0 ) − 1)(w+ − w+ )|k−1
1 0 0 ≤ C|w+ − w+ |k−1 (2γ − 1)−1 |w+ + w+ |k+1 0 0 + C|w+ + w+ |k {(1 − γ)−1 t1−γ |w+ − w+ |k−1 0 + γ −1 t−γ (|w+ |k+1 + |w+ |k+1 )} 0 0 for any t ≥ 1. Taking t−1 = |w+ − w+ |k−1 (|w+ |k+1 + |w+ |k+1 )−1 yields the result.
Proposition 5.8 applies mutatis mutandis to the map (w+ , s02 (1)) → (w, s) because the map (w+ , s0 (1)) → (w+ , s02 (1)) is bijective since it is defined by the explicit formula (6.17) with suitable modifications in order to take into account the additional loss of derivative. 7. Wave Operators and Asymptotics for u In this section we complete the construction of the wave operators for Eq. (1.1) and we derive asymptotic properties of solutions in their range. The construction
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relies in an essential way on those of Secs. 5 and 6, especially Proposition 6.4, and will require a discussion of the gauge invariance of those constructions. The first task is to supplement them with the determination of the phase ϕ, which appears so far only through its gradient s (see (2.26) and (2.31)). Actually the treatment in Secs. 4–6 only involved the variable s, and did not even assume that s was a gradient. We recall that g0 , g are defined by (3.1) and (3.2) and we continue to use the short hand notation g0 (w, w) = g0 (w) so that g(w, w) = g0 (U ∗ (1/t)w) . We are interested in solving the system (2.25)–(2.26) which we rewrite as ∂t w = (2t2 )−1 U (1/t)(2∇ϕ · ∇ + (∆ϕ))U ∗ (1/t)w
(2.25) ≡ (7.1)
∂t ϕ = (2t2 )−1 |∇ϕ|2 + t−γ g0 (U ∗ (1/t)w) .
(2.26) ∼ (7.2)
In analogy with (5.5), we shall use the notation ϕ e = tγ−1 ϕ. The relevant spaces for the phase ϕ are the spaces Y ` defined by Y ` = {ϕ : ϕ ∈ L∞
and ∇ϕ ∈ X ` } .
(7.3)
We first consider the Cauchy problem with finite initial time t0 for the system (7.1)–(7.2) with initial data (w0 , ϕ0 ) ∈ H k ⊕ Y ` . That problem is solved by first solving the Cauchy problem for the system (5.1)–(5.2) with initial data (w0 , ∇ϕ0 ) at time t0 for (w, s) by Proposition 4.1 or 5.1 and then recovering ϕ from (w, s) by ϕ(t0 ) = ϕ0 and Z
t
ϕ(t) = ϕ(t0 ) +
dt0 {(2t02 )−1 |s|2 + t0−γ g0 (U ∗ (1/t0 )w)} .
(7.4)
t0
As mentioned in Sec. 2, it follows from (5.2) that the vorticity ω = ∇ × s satisfies Eq. (2.35). Furthermore ω(t0 ) = 0. Under the available regularity properties of s, it follows from Lemma 3.2 with p = ∞, u = ω, v = s and from Gronwall’s Lemma that ω(t) = 0 for all t. On the other hand, from (5.2) and (7.2), it follows that Z
t
(s − ∇ϕ)(t) =
dt0 t0−2 (s × ω)(t0 )
(7.5)
t0
P where (s × ω)i = j sj ωji , and therefore s(t) = ∇ϕ(t) for all t for which (w, s) is defined since ω = 0. If (w, ϕ) is a solution of the system (7.1)–(7.2) as obtained from Proposition 5.1 and from the previous argument, it follows from that proposition and from Corollary 3.1 that (w, ϕ) e ∈ (C ∩ L∞ )([T, ∞), H k ⊕ Y ` ). We next consider the Cauchy problem with infinite initial time covered by Propositions 5.5 and 5.7. There we have established a correspondence between solutions (w, s) of (5.1)–(5.2) and solutions (w+ , s0 ) of (5.3), and we now extend it to a
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correspondence between solutions (w, ϕ) of (7.1)–(7.2) and solutions (w+ , ϕ0 ) of the auxiliary free equation ∂t ϕ0 = t−γ g0 (U ∗ (1/t)w+ ) . The general solution of (7.6) can be written as Z t ϕ0 (t) = ϕ0 (1) + dt0 t0−γ g0 (U ∗ (1/t0 )w+ )
(2.28) ∼ (7.6)
(7.7)
1
and if s0 (t) and ϕ0 (t) are defined by (5.4) and (7.7) with s0 (t) = ∇ϕ0 (t) for one t (for instance t = 1), the same relation holds for all t. We supplement the correspondence established in Propositions 5.5 and 5.7 with the relation Z ∞ dt0 {(2t02 )−1 |s|2 + t0−γ (g0 (U ∗ (1/t0 )w) − g0 (U ∗ (1/t0 )w+ ))} ϕ(t) = ϕ0 (t) − t
(7.8) which will be used to define ϕ(t) and/or ϕ0 (t) in terms of each other. From (5.36) and (7.8) it follows that Z ∞ dt0 t0−2 (s × ω)(t0 ) . (7.9) (s − ∇ϕ)(t) − (s0 − ∇ϕ0 )(t) = − t
Corresponding to the situation of Proposition 5.5, let (w, ϕ) ∈ (C ∩ L∞ loc )([T, ∞), H ⊕ Y ` ) be a solution of (7.1)–(7.2) such that (w, s = ∇ϕ) satisfies the assumptions of Proposition 5.5. Define (w+ , s0 ) by Proposition 5.3, by (5.36) and by Proposition 5.5, Part (1), and define ϕ0 by (7.8). Then it follows from (7.9) that s0 (t) = ∇ϕ0 (t) for all t. Conversely, corresponding to the situation of Proposition 5.7, let (w+ , ϕ0 (1)) ∈ H k+1 ⊕ Y `+1 , define ϕ0 (t) by (7.7) and s0 (t) by s0 (t) = ∇ϕ0 (t), define (w, s) by Proposition 5.7, Part (1) and define ϕ(t) by (7.8). Let (wt0 , st0 ) be defined by Proposition 5.6. From the finite initial time results it follows that ωt0 (t) ≡ ∇ × st0 (t) = 0 for all t and t0 , while by Proposition 5.7, Part (2) ωt0 converges to ω in X `−2 uniformly in compact intervals, so that ω = 0. It then follows again from (7.9) that s(t) = ∇ϕ(t) for all t. This proves that the correspondence established in Propositions 5.5 and 5.7 extends to a correspondence between solutions (w, ϕ) of (7.1)–(7.2) and solutions (w+ , ϕ0 ) of (7.6) preserving the relations s = ∇ϕ, s0 = ∇ϕ0 . The same discussion applies mutatis mutandis to the situation of Propositions 6.2 and 6.4 and allows for an extension of the correspondence between solutions (w, s) of (5.1)–(5.2) and solutions (w+ , s02 ) of (6.1) established there to a correspondence between solutions (w, ϕ) of (7.1)–(7.2) and solutions (w+ , ϕ02 ) of the equation (2.29) ∼ (7.10) ∂t ϕ02 = t−γ g0 (w+ ) k
which preserves the relations s = ∇ϕ, s02 = ∇ϕ02 . The relation (7.8) between ϕ(t) and ϕ0 (t) is replaced by the relation Z ∞ dt0 {(2t02 )−1 |s|2 + t0−γ (g0 (U ∗ (1/t0 )w) − g0 (w+ ))} (7.11) ϕ(t) = ϕ02 (t) − t
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between ϕ(t) and ϕ02 (t), so that ϕ0 (t) and ϕ02 (t) are related by Z ∞ ϕ0 (t) = ϕ02 (t) − dt0 t0−γ (g0 (U ∗ (1/t0 )w+ ) − g0 (w+ ))
(7.12)
t
in agreement with (6.16) and (6.17). Equation (7.10) is trivially solved by ϕ02 (t) = ϕ02 (1) + (1 − γ)−1 (t1−γ − 1)g0 (w+ ) .
(7.13)
We can now embark on the explicit construction of wave operators, and we first construct a wave operator W for the auxiliary system (5.1)–(5.2). This could be based either on Proposition 5.7 or on Proposition 6.4. We choose the latter since as mentioned before it is better suited than the former for the discussion of gauge invariance to be performed next. Definition 7.1. We define W as the map W : (w+ , ϕ02 (1)) → (w, ϕ)
(7.14)
e ∈ (C ∩ L∞ )([T, ∞), H k ⊕ Y ` ) from H k+2 ⊕ Y `+1 to the set of (w, ϕ) such that (w, ϕ) for some T , 1 ≤ T < ∞, as follows. Define ϕ02 (t) by (7.13) and s02 (t) by s02 (t) = ∇ϕ02 (t). Define (w, s) by Proposition 6.4, Part (1) and finally define ϕ by (7.11), so that s = ∇ϕ by the previous discussion. Then W is well defined by (7.14) as a map between the spaces indicated. Before defining the wave operators for u, we now study the gauge invariance of W , which plays an important role in justifying that definition, as explained in Sec. 2. For that purpose we need some information on the Cauchy problem for Eq. (1.1) at finite times. In addition to the operators M = M (t) and D = D(t) defined by (2.5) and (2.6), we introduce the operator J = J(t) = x + it∇ ,
(7.15)
the generator of Galilei transformations. The operators M , D, J satisfy the commutation relation iM D∇ = JM D . (7.16) For any interval I ⊂ [1, ∞) and any nonnegative integer k, we define the space X k (I) = {u : D∗ M ∗ u ∈ C(I, H k )} = {u : hJ(t)ik u ∈ C(I, L2 )}
(7.17)
where hλi = (1 + λ2 )1/2 for any real number or self-adjoint operator λ and where the second equality follows from (7.16). Then Proposition 7.1. Let k be a positive integer and let 0 < µ < n ∧ 2k. Then the Cauchy problem for Eq. (1.1) with initial data u(t0 ) = u0 such that hJ(t0 )ik u0 ∈ L2 at some initial time t0 ≥ 1 is locally well posed in X k (·), namely
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(1) There exists T > 0 such that (1.1) has a unique solution with initial data u(t0 ) = u0 in X k ([1 ∨ (t0 − T ), t0 + T ]). (2) For any interval I, t0 ∈ I ⊂ [1, ∞), (1.1) with initial data u(t0 ) = u0 has at most one solution in X k (I). (3) The solution of Part (1) depends continuously on u0 in the norms considered there. Proof. The proof is obtained by minor variations of the corresponding results in [9]. The differences come from the factor tµ−γ in (1.2), which is irrelevant for the present problem, and from the replacement of C(I, H k ) by X k (I). That replacement is made possible by the properties of the operator J(t) and especially the commutation relation (7.16), which implies that J(t) behaves as a derivative on gauge invariant functions. See for instance [2]. In the study of gauge invariance for W we shall actually need only the uniqueness statement, Part (2) of Proposition 7.1. We recall that in the transition from the system (5.1)–(5.2) to Eq. (1.1), u should be defined by (2.14) with (w, ϕ) = (w3 , ϕ3 ) and accordingly we define the map Φ : (w, ϕ) → u = M D exp[−iϕ]U ∗ (1/t)w .
(7.18)
That map satisfies the following property. Lemma 7.1. The map Φ defined by (7.18) is bounded and continuous from C(I, H k ⊕ Y ` ) to X k (I) for any admissible pair (k, `) and any interval I ⊂ [1, ∞).
Proof. An immediate consequence of Lemma 3.5. We can now make the following definition.
Definition 7.2. Let (k, `) be an admissible pair and let (w, ϕ) and (w0 , ϕ0 ) be two solutions of the system (7.1)–(7.2) in C(I, H k ⊕ Y ` ) for some interval I ⊂ [1, ∞). We say that (w, ϕ) and (w0 , ϕ0 ) are gauge equivalent if they give rise to the same u, namely Φ((w, ϕ)) = Φ((w0 , ϕ0 )), or equivalently if exp[−iϕ(t)]U ∗ (1/t)w(t) = exp[−iϕ0 (t)]U ∗ (1/t)w0 (t)
(7.19)
for all t ∈ I. A sufficient condition for gauge equivalence is given by the following Lemma. Lemma 7.2. Let (k, `) be an admissible pair and let (w, ϕ) and (w0 , ϕ0 ) be two solutions of the system (7.1)–(7.2) in C(I, H k ⊕Y ` ). In order that (w, ϕ) and (w0 , ϕ0 ) be gauge equivalent, it is sufficient that (7.19) holds for one t ∈ I. Proof. An immediate consequence of Lemma 7.1, of Proposition 7.1, Part (2), and of the fact that (k, `) admissible implies k > 1 + µ/2. The gauge covariance properties of W will be expressed by the following two propositions.
LONG RANGE SCATTERING FOR HARTREE TYPE EQUATIONS
419
Proposition 7.2. Let 0 < γ < 1 and let (k, `) be an admissible pair. Let (w, ϕ) e (w0 , ϕ e0 ) ∈ and (w0 , ϕ0 ) be two solutions of the system (7.1)–(7.2) such that (w, ϕ), ∞ k ` 0 0 (C ∩ L )([T, ∞), H ⊕ Y ) for some T ≥ 1, and assume that (w, ϕ) and (w , ϕ ) are gauge equivalent. Then (1) There exists σ ∈ Y `−1 such that ϕ0 (t) − ϕ(t) converges to σ when t → ∞ strongly in Y `−2 and in the weak-∗ sense in Y `−1 . The following estimate holds: kϕ0 (t) − ϕ(t) − σ; Y `−2 k ≤ At−γ
(7.20)
for some constant A depending on T and on the norms of ϕ, e ϕ e0 in L∞ (·, Y ` ), with ∞ the exception of the case n even, ` = n/2 + 1 where the L norm of ∇ϕ satisfies only (7.21) k∇ϕ0 (t) − ∇ϕ(t) − ∇σk∞ ≤ At−γ/2 . (2) Assume in addition that γ > 1/2. Then ϕ0 (t) − ϕ(t) converges to σ in norm in Y `−1 and the following estimate holds : kϕ0 (t) − ϕ(t) − σ; Y `−1 k ≤ At1−2γ .
(7.22)
0 0 Furthermore w+ = w+ exp(iσ) where w+ , w+ are the limits of w(t), w0 (t) as t → ∞ obtained in Proposition 5.3. (3) Assume in addition that γ > 1/2 and that (w, ϕ), (w0 , ϕ0 ) ∈ R(W ). Then 0 ϕ02 (t) = ϕ02 (t) + σ for all t ≥ 1. In particular σ ∈ Y `+1 .
Proof. Part (1). Define ϕ− = ϕ0 − ϕ, s = ∇ϕ, s0 = ∇ϕ0 , s± = s0 ± s, and b = k˜ s; L∞ ([T, ∞), X ` )k ∨ k˜ s0 ; L∞ ([T, ∞), X ` )k .
(7.23)
From (5.2) and gauge equivalence it follows that ∂t s− = t−2 ((s− · ∇)s+ + (s+ · ∇)s− )
(7.24)
and therefore by Lemma 3.9 ∂t |s− |`−1 ≤ Ct−2 |s− |`−1 |s+ |` ≤ Cbt−1−γ |s− |`−1 so that by Gronwall’s Lemma, for all t ≥ T , |s− (t)|`−1 ≤ |s− (T )|`−1 exp(Cbγ −1 T −γ ) ≡ A0 , namely ks− ; L∞ ([T, ∞), X `−1 )k ≤ A0 .
(7.25)
From (7.2) and gauge equivalence, it follows that ∂t ϕ− = (2t2 )−1 (s− · s+ ) and therefore by (7.23) and (7.25) for any t ≥ t0 ≥ T Z t kϕ− (t) − ϕ− (t0 )k∞ ≤ dt0 bA0 t0−1−γ ≤ bA0 γ −1 t−γ 0 . t0
(7.26)
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This implies that ϕ− (t) converges in norm in L∞ to some σ ∈ L∞ and that kϕ− (t) − σk∞ ≤ bA0 γ −1 t−γ
(7.27)
which is the part of (7.20) involving ϕ− itself (and not s− only). From the uniform estimate (7.25) and standard compactness arguments, it follows that ∇σ ∈ X `−1 , that |∇σ|`−1 ≤ A0 and that s− converges to ∇σ in the weak-∗ sense in X `−1 , which together with (7.27) implies weak-∗ convergence of ϕ− to σ in Y `−1 . We finally prove the strong convergence of s− to ∇σ in X `−2 . From (7.24) which we rewrite as ∂t (s− − ∇σ) = t−2 {((s− − ∇σ) · ∇)s+ + (s+ · ∇)(s− − ∇σ) + (∇σ · ∇)s+ + (s+ · ∇)∇σ}
(7.28)
and by the same estimates as in the proof of Lemma 3.11, we obtain ∂t |s− − ∇σ|`−2 ≤ Ct−2 {|s− − ∇σ|`−2 |s+ |` + |∇σ|`−1 |s+ |` } ≤ Cbt−1−γ (|s− − ∇σ|`−2 + A0 )
(7.29)
with the only exception of the case n even, ` = n/2 + 1, where the L∞ norm of (s− − ∇σ) which occurs in the X `−2 norm, is not estimated as in (7.29) because k∂ 2 σk∞ is not controlled by |∇σ|`−1 . From (7.29) we obtain by Gronwall’s Lemma |s− (t) − ∇σ|`−2 ≤ A0 {exp(Cbγ −1 t−γ ) − 1}
(7.30)
which together with (7.27) completes the proof of (7.20). For n even, ` = n/2 + 1, we estimate simply ˙ n/2+1 k1/2 ks− (t) − ∇σk∞ ≤ Ckϕ− (t) − σk1/2 ∞ ks− (t) − ∇σ; H and (7.21) follows from (7.27) and from the H˙ n/2+1 part of (7.25). Part (2). Let t ≥ t0 ≥ T , s− (t) = s− and s− (t0 ) = s0 . We rewrite (7.24) as ∂t (s− − s0 ) = t−2 {((s− − s0 ) · ∇)s+ + (s+ · ∇)(s− − s0 ) + (s0 · ∇)s+ + (s+ · ∇)s0 } . By Lemma 3.9, we estimate ∂t |s− − s0 |`−1 ≤ Ct−2 {|s− − s0 |`−1 |s+ |` + |s− |`−1 |s+ |` + |s+ |`−1 |s0 |` } ≤ Ct−2 (|s− − s0 |`−1 + |s0 |` )|s+ |` ≤ Cbt−1−γ (|s− − s0 |`−1 + bt1−γ ) 0 and therefore by Gronwall’s Lemma |s− (t) − s− (t0 )|`−1 ≤ bt1−γ (exp(Cbγ −1 t−γ 0 0 ) − 1) . This yields a separate proof of the convergence of s− (t) in X `−1 , together with the estimate |s− (t) − ∇σ|`−1 ≤ bt1−γ (exp(Cbγ −1 t−γ ) − 1) , which together with (7.27) completes the proof of (7.22).
LONG RANGE SCATTERING FOR HARTREE TYPE EQUATIONS
421
0 We now prove that w+ = w+ exp(iσ). For that purpose we estimate 0 0 |w+ − w+ eiσ |k−1 ≤ |w+ − U ∗ (1/t)w0 (t)|k−1
+ |U ∗ (1/t)w0 (t) − exp[iϕ− (t)]U ∗ (1/t)w(t)|k−1 + |(exp[i(ϕ− (t) − σ)] − 1) exp(iσ)U ∗ (1/t)w(t)|k−1 + |exp(iσ)(U ∗ (1/t)w(t) − w+ )|k−1 .
(7.31)
We estimate the first norm in the RHS by Proposition 5.3, especially (5.33) and by (6.10) as 0 − U ∗ (1/t)w0 (t)|k−1 ≤ A(t−γ + t−1/2 ) . (7.32) |w+ The second norm in the RHS of (7.31) is zero by gauge equivalence. The third norm is estimated by Lemma 3.5 as | · | ≤ C(kϕ− − σk∞ + |∇(ϕ− − σ)|`−1 (1 + |∇(ϕ− − σ)|`−1 )k−2 ) × (1 + |∇σ|`−1 )k−1 |w|k−1
(7.33)
and the last norm is estimated by Lemma 3.5 again followed by the analogue of (7.32) for w as (7.34) | · | ≤ (1 + |∇σ|`−1 )k−1 A(t−γ + t−1/2 ) . Collecting (7.32)–(7.34) and using (7.25) and (7.22) shows that the RHS of (7.31) tends to zero when t → ∞ and therefore the LHS is zero since it is time independent. Part (3). We recall that in the situation of Proposition 6.4 and of the definition of W , ϕ(t) and ϕ02 (t) are related by (7.11), and by the estimates (3.35), (6.30), (6.31) and (7.35) kϕ(t) − ϕ02 (t)k∞ ≤ At1−2γ for some A depending only on a, b defined by (6.22) and for t sufficiently large. Similarly (7.36) kϕ0 (t) − ϕ002 (t)k∞ ≤ At1−2γ . It follows then from (7.35), (7.36) and (7.20) (or (7.22)) that kϕ002 (t) − ϕ02 (t) − σk∞ → 0 when t → ∞ . 0 On the other hand from (7.10) (or (7.13)) and from the condition w+ = w+ exp(iσ) 0 0 it follows that ϕ02 (t) − ϕ02 (t) is constant in time. Therefore ϕ02 (t) − ϕ02 (t) = σ for all t.
Proposition 7.2 prompts us to make the following definition of gauge equivalence for asymptotic states. 0 , ϕ002 (1)) are gauge equivalent if Definition 7.3. Two pairs (w+ , ϕ02 (1)) and (w+ 0 = w+ exp(iσ) and ϕ002 (1) − there exists a real function σ = σ(x) such that w+ ϕ02 (1) = σ.
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0 Two gauge equivalent pairs generate two solutions (w+ , ϕ02 ) and (w+ , ϕ002 ) of 0 (7.10) such that ϕ02 (t) − ϕ02 (t) = σ for all t ≥ 1. Those two solutions will also be said to be gauge equivalent. In Definition 7.3, we have not specified the regularity of (w+ , ϕ02 (1)) and 0 (w+ , ϕ002 (1)). This can be done easily, depending on the needs, and possibly with the help of Lemma 3.5. With the previous definition, Proposition 7.2 should be understood to mean that two gauge equivalent solutions of the system (7.1)–(7.2) in R(W ) are images of two gauge equivalent solutions of (7.10). The next proposition states that conversely two gauge equivalent solutions of (7.10) have gauge equivalent images under W .
Proposition 7.3. Let 1/2 < γ < 1 and let (k, `) be an admissible pair. Let 0 , ϕ002 (1)) ∈ H k+2 ⊕ Y `+1 be gauge equivalent, and let (w, ϕ), (w+ , ϕ02 (1)), (w+ 0 0 (w , ϕ ) be their images under W. Then (w, ϕ) and (w0 , ϕ0 ) are gauge equivalent, and limt→∞ ϕ0 (t) − ϕ(t) = σ ≡ ϕ002 (1) − ϕ02 (1). Proof. Let t0 be sufficiently large and let (wt0 , ϕt0 ) and (wt0 0 , ϕ0t0 ) be the solutions of the system (7.1)–(7.2) constructed by Proposition 6.3 supplemented with (7.4) with the appropriate initial conditions at t0 , namely ϕt0 (t0 ) = ϕ02 (t0 ) ,
ϕ0t0 (t0 ) = ϕ002 (t0 ) .
(7.37)
From (7.37) and from the initial conditions wt0 (t0 ) = U (1/t0 )w+ ,
0 wt0 0 (t0 ) = U (1/t0 )w+
imposed in Proposition 6.3 it follows that exp[−iϕt0 (t0 )]U ∗ (1/t0 )wt0 (t0 ) = exp[−iϕ0t0 (t0 )]U ∗ (1/t0 )wt0 0 (t0 ) and therefore by Lemma 7.2, (wt0 , ϕt0 ) and (wt0 0 , ϕ0t0 ) are gauge equivalent, namely satisfy (7.38) exp[−iϕt0 (t)]U ∗ (1/t)wt0 (t) = exp[−iϕ0t0 (t)]U ∗ (1/t)wt0 0 (t) for all t for which they are defined. We now take the limit t0 → ∞ for fixed t in (7.38). By Proposition 6.4, Part (2) supplemented with similar estimates on ϕt0 and ϕ0t0 , for fixed t, (wt0 , ϕt0 ) and (wt0 0 , ϕ0t0 ) converge respectively to (w, ϕ) and (w0 , ϕ0 ) in norm in H k−1 ⊕ Y `−1 . By an easy application of Lemma 3.5, it follows therefrom that one can take the limit t0 → ∞ in (7.38), thereby obtaining (7.19), so that (w, ϕ) and (w0 , ϕ0 ) are gauge equivalent. The last statement of Proposition 7.3 is a repetition of Proposition 7.2, Part (3). We can now define the wave operator for u. We recall from the heuristic discussion in Sec. 2 that we want to exploit the operator W defined in Definition 7.1, reconstruct u through the map Φ defined by (7.18), and eliminate the arbitrariness in ϕ02 by fixing some initial condition for it, namely ϕ02 (1) = 0, thereby purporting
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to ensure the injectivity of the wave operator without restricting its range. That program is implemented by the following definition and proposition. Definition 7.4. We define the wave operator Ω as the map Ω : u+ → u = (Φ ◦ W )(F u+ , 0)
(7.39)
from F H k+2 to X k ([T, ∞)) for some T , 1 ≤ T < ∞, where k is the first element of an admissible pair, and W , Φ are defined by Definition 7.1 and by (7.18). The fact that Ω acts between the spaces indicated follows from Proposition 6.4 and from Lemma 7.1. The value of T depends on u+ and can be taken according to (6.32) with b = C(1 − γ)−1 a2 , namely T = C((1 − γ)−2 + (2γ − 1)−2 )(|F u+ |k+2 ∨ 1)2/γ .
(7.40)
Proposition 7.4. (1) The map Ω is injective. (2) R(Ω) = R(Φ ◦ W ). Proof. Part (1). Let u = Ω(u+ ) = Ω(u0+ ) and let (w, ϕ) = W (F u+ , 0), (w0 , ϕ0 ) = W (F u0+ , 0), so that u = Φ(w, ϕ) = Φ(w0 , ϕ0 ), namely (w, ϕ) and (w0 , ϕ0 ) are gauge equivalent. By Proposition 7.2, Part (3), also (F u+ , 0) and (F u0+ , 0) are gauge equivalent in the sense of Definition 7.3, and therefore u+ = u0+ . Part (2). Let u = (Φ ◦ W )(w+ , ϕ02 (1)) ∈ R(Φ ◦ W ). Then by Proposition 7.3, also 0 0 0 , 0) where w+ = w+ exp[−iϕ02 (1)], and w+ ∈ H k+1 by Lemma 3.5. u = (Φ ◦ W )(w+ ∗ 0 Therefore u = Ω(F w+ ) ∈ R(Ω). Note in particular that Proposition 7.4, Part (2) means that we have not restricted the range of the wave operators from W to Ω by arbitrarily imposing ϕ02 (1) = 0. We now collect all the available information on the solutions of the original Eq. (1.1) so far constructed, namely existence through the previous definition of Ω, some partial form of uniqueness coming from Proposition 5.2, and asymptotic decay estimates coming from Propositions 5.7 and 6.4. In order to state the result we need the phases ϕ02 (t) and ϕ0 (t) defined now by (7.13) and (7.12) with w+ = F u+ and ϕ02 (1) = 0, namely ϕ02 (t) = (1 − γ)−1 (t1−γ − 1)g0 (F u+ ) Z ∞ dt0 t0−γ (g0 (F M u+ ) − g0 (F u+ )) . ϕ0 (t) = ϕ02 (t) −
(7.41) (7.42)
t
(We recall that M = M (t) is defined by (2.5) and satisfies (2.11).) The main result of this paper can now be stated as follows. Proposition 7.5. Let n ≥ 3, 0 < µ ≤ n − 2, 1/2 < γ < 1 and let (k, `) be an admissible pair. Let u+ ∈ F H k+2 and define ϕ02 (t) and ϕ0 (t) by (7.41)–(7.42). Then
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(1) There exists a unique solution u ∈ X k ([T, ∞)) of Eq. (1.1) which can be represented as u = M D exp(−iϕ)U ∗ (1/t)w
(7.43)
where (w, ϕ) is a solution of the system (7.1)–(7.2) such that (w, ϕ) e ∈ (C ∩ L∞ ) k ` ([T, ∞), H ⊕ Y ) and such that |w(t) − F u+ |k−1 t1−γ → 0
(7.44)
kϕ(t) − ϕ02 (t); Y `−1 k → 0
(7.45)
when t → ∞. The time T depends on γ and u+ and can be taken in the form (7.40). (2) The solution u is obtained as u = Ω(u+ ) where the map Ω is defined in Definition 7.4. The map Ω is injective. (3) The map Ω is continuous on the bounded sets of M DH k+2 from the norm topology in F H k−1 for u+ to the norm topology in X k−1 (J) and to the weak-∗ topology in X k (J) for u for any compact interval J ⊂ [T, ∞), and to the weak topology in M DF H k pointwise in t. (4) The solution u satisfies the following estimates for t ≥ T : khJ(t)ik (exp[iϕ02 (t, x/t)]u(t) − M (t)D(t)F u+ )k2 ≤ A(2γ − 1)−1 t1−2γ ,
(7.46)
khJ(t)ik (exp[iϕ0 (t, x/t)]u(t) − U (t)u+ )k2 ≤ A(2γ − 1)−1 t1−2γ ,
(7.47)
for some constant A depending on γ and u+ and bounded in γ for fixed u+ and γ away from 1. Proof. Parts (1) and (2). All the results except uniqueness follow from Proposition 6.4 supplemented with the reconstruction of ϕ and from the subsequent definition of Ω. In particular (7.43) is essentially (7.18) and the injectivity of Ω is Proposition 7.4, Part (1). Uniqueness is an immediate consequence of Proposition 5.2, given the asymptotic behaviour of (w, ϕ) that follows from Proposition 6.4. Part (3) follows from Proposition 6.4, Part (3) and from Lemma 7.1. Part (4). From Proposition 5.7, Part (1), especially (5.69) and from Proposition 6.4, Part (1), especially (6.30), supplemented by similar estimates on ϕ − ϕ0 and on ϕ − ϕ02 easily obtained from (7.8) and (7.11), it follows that |w(t) − F u+ |k ≤ A0 t−γ
(7.48)
kϕ(t) − ϕ0 (t); Y ` k ≤ A0 (2γ − 1)−1 t1−2γ
(7.49)
kϕ(t) − ϕ02 (t); Y ` k ≤ A0 (2γ − 1)−1 t1−2γ
(7.50)
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for some constant A0 of the type stated for A. From the definition (7.15) of J, from the commutation relation (7.16), from (7.43) and from Lemma 3.5, we obtain khJ(t)ik (exp[iϕ02 (t, x/t)]u(t) − M DF u+ )k2 = | exp[i(ϕ02 − ϕ)]U ∗ (1/t)w − F u+ |k ≤ {kϕ02 − ϕk∞ + |∇(ϕ02 − ϕ)|`−1 (1 + |∇(ϕ02 − ϕ)|`−1 )k−1 }|w|k + |w − F u+ |k + |(U ∗ (1/t) − 1)F u+ |k
(7.51)
which yields immediately (7.46) by the use of (7.48), (7.50) and (6.9). Similarly khJ(t)ik (exp[iϕ0 (t, x/t)]u(t) − U (t)u+ )k2 = | exp[i(ϕ0 − ϕ)]U ∗ (1/t)w − U ∗ (1/t)F u+ |k ≤ {kϕ0 − ϕk∞ + |∇(ϕ0 − ϕ)|`−1 (1 + |∇(ϕ0 − ϕ)|`−1 )k−1 } × |w|k + |w − F u+ |k
(7.52)
from which (7.47) follows by the use of (7.48) and (7.49).
Remark 7.1. The uniqueness statement in Proposition 7.5 is rather restrictive because it requires the representation of u by (7.43). It would be more satisfactory to have uniquess under assumptions bearing directly on u, for instance under (7.46). However (7.46) seems insufficient to derive the asymptotic conditions on w and ϕ separately which are required in Proposition 5.2. Remark 7.2. The estimate (7.46) states that u behaves asymptotically as expected, namely u(t) ∼ exp[−iϕ02 (t, x/t) + ix2 (2t)−1 ](it)−n/2 (F u+ )(x/t) .
(7.53)
In order to have the strongest possible statement however, one has to shift the phase factor to u before taking derivatives as contained in hJ(t)ik . On the other hand, this is not necessary if one wants only asymptotic estimates in Lr . In fact one has the following corollary. Corollary 7.1. Let u be the solution of (1.1) obtained in Proposition 7.5. Let r satisfy 0 ≤ δ(r) ≤ k ∧ n/2, δ(r) < n/2 if k = n/2. Then u satisfies the following estimates: ku(t) − exp[−iϕ02 (t, x/t)]M DF u+ kr ≤ A(2γ − 1)−1 t−δ(r)+1−2γ ,
(7.54)
ku(t) − exp[−iϕ0 (t, x/t)]U (t)u+ kr ≤ A(2γ − 1)−1 t−δ(r)+1−2γ .
(7.55)
Proof. An immediate consequence of (7.46) and (7.47) and of the inequality kf kr = t−δ(r) kD∗ M ∗ f kr ≤ Ct−δ(r) kh∇ik D∗ M ∗ f k2 = Ct−δ(r) khJ(t)ik f k2 which follows from Lemma 3.1 and from the commutation relation (7.16).
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Remark 7.3. We have compared u with the solution U (t)u+ of the free Schr¨ odinger equation in (7.47), and with the standard asymptotic form M DF u+ thereof in (7.46), obtained in dropping the second M in U = M DF M . Those two choices correspond respectively to the parametrisation of u using w2 (see (2.9) and (2.13)) and w3 (see (2.10) and (2.14)). The difference between the asymptotic forms using M DF and U is estimated by |F (M (t) − 1)u+ |k ≤ t−1 |F u+ |k+2 and does not produce any difference in the time dependence of the estimates. So far we have constructed local solutions of Eq. (1.1) in a neighborhood of infinity associated with given asymptotic states u+ and defined the local wave operator at infinity Ω. In order to complete the construction of the standard wave operators, it remains only to extend the previous solutions from a neighborhood of infinity by using the results on the global Cauchy problem at finite times. This can be done with the help of the following result which is essentially contained in [25]. Proposition 7.6. Let k be a positive integer and let 0 < µ < 2. Then the Cauchy problem for Eq. (1.1) with initial data u(t0 ) = u0 such that hJ(t0 )ik u0 ∈ L2 at some initial time t0 ≥ 1 is globally well posed in X k ([1, ∞)), namely the local solutions of Proposition 7.1 can be extended to [1, ∞). Proof. A minor variation of Proposition 2.1, Part (1) in [25].
We can now define the standard wave operator Ω1 for Eq. (1.1). Definition 7.5. Under the assumptions of Proposition 7.5 supplemented with µ < 2, we define the wave operator Ω1 as the map Ω1 : u+ → u(1) where u is the solution of Eq. (1.1) obtained by continuing Ω(u+ ) down to t = 1 with the help of Proposition 7.6. From Propositions 7.5 and 7.6 it follows that Ω1 is an injective map from F H k+2 to the space K k = {u : exp(−ix2 /2)u ∈ H k } and that Ω1 satisfies continuity properties easily obtained from Proposition 7.5, Part (3). Since all the interesting information is already contained in that proposition, we refrain from a more formal statement. Appendix A Sketch of the Proof of Lemma 3.2 Part (1) We discuss only the case where p < ∞, the case p = ∞ being obtainable from the previous one by a limiting procedure. The proof proceeds by first writing a regularized equation for a suitable approximant of the function |u|p , then by proving an analogue of the estimate (3.13) for that function and finally by removing the regularisation. The method used is known and applied in [5] to Eq. (3.12a) with
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η = 0. For this reason, even though the assumptions on u and v used there are slightly different from those of Proposition 3.2, we refer to [5] for estimating the second and third term in the RHS of (3.12a). Here below we continue with the analysis of the first (η dependent) term. Let ϕ ∈ C0∞ (Rn ) denote a regularizing sequence of functions of the space variable, namely a sequence converging to the δ function. Let φ denote the operator of convolution with ϕ, i.e. φf = ϕ ∗ f , let ε > 0, let u1 ≡ φ ∗ u, and let w ≡ (|u1 |2 + ε2 )1/2 . The function w is uniformly bounded away from zero, is C ∞ in the space variable and satisfies Z lim lim w(t, x)p dx = ku(t)kpp . ϕ→δ ε→0
We compute ∂t wp by using (3.12a), thereby obtaining ∂t wp = 2wp−2 Re(¯ u1 ∂t u1 ) = p{ηJ1 + J2 + J3 } where J1 = wp−2 Re u ¯1 ∆u1 J2 = wp−2 Re u ¯1 φ∇(uv) ¯1 φh . J3 = wp−2 Re u The terms J2 and J3 are treated as in [5]. For the completion of the proof it is sufficient to show that the space integral of J1 is negative, so that J1 does not contribute to the estimates (3.13) and (3.14). Repeated application of the Leibnitz rule yields J1 = −wp−2 |∇u1 |2 + (1/2)wp−2 (∇ · ∇|u1 |2 ) = −wp−2 |∇u1 |2 − (1/4)(p − 2)wp−4 (∇|u1 |2 )2 + (1/2)∇ · {wp−2 ∇|u1 |2 } so that J1 ≤ (1/2)∇ · {wp−2 ∇|u1 |2 } which essentially implies that
Z J1 dx ≤ 0 .
Actually, in order to ensure integrability of J1 , the limit ε → 0 should be taken before performing the integration over the space variables up to infinity. References [1] L. Abdelouhab, J. L. Bona, M. Felland and J. C. Saut, “Nonlocal models for nonlinear dispersive waves”, Physica D40 (1989) 360–392. [2] T. Cazenave, “An introduction to nonlinear Schr¨ odinger equations”, Text. Met. Mat. 26, Inst. Mat., Rio de Janeiro, 1993.
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[3] J. Derezinski and C. G´erard, Scattering Theory of Classical and Quantum N -particle Systems, Springer, Berlin, 1997. [4] J. Derezinski and C. G´erard, “Long range scattering in the position representation”, J. Math. Phys. 38 (1997) 3925–3942. [5] R. Di Perna and P. L. Lions, “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math. 98 (1989) 511–547. [6] A. Friedman, Partial Differential Equations, Holt Rinehart & Winston, New York, 1969. [7] P. G´erard, “Remarques sur l’analyse semi-classique de l’´ equation de Schr¨ odinger non lin´eaire”, S´eminaire EDP, Ecole Polytechnique, Expos´e XIII, 1993. [8] J. Ginibre and T. Ozawa, “Long range scattering for nonlinear Schr¨ odinger and Hartree equations in space dimension n ≥ 2”, Commun. Math. Phys. 151 (1993) 619–645. [9] J. Ginibre and G. Velo, “On a class of nonlinear Schr¨ odinger equations with nonlocal interaction”, Math. Z. 170 (1980) 109–136. [10] E. Grenier, “Limite semi-classique de l’´equation de Schr¨ odinger non lin´eaire en temps petit”, C. R. Acad. Sci. Paris, Ser. I 320 (1995) 691–694; “Semiclassical limit of the nonlinear Schr¨ odinger equation in small time”, Proc. Am. Math. Soc. 464 (1998) 523–530. [11] N. Hayashi, E. I. Kaikina and P. I. Naumkin, “On the scattering theory for the cubic nonlinear Schr¨ odinger and Hartree type equations in one space dimension”, Hokka¨ıdo Math. J., to appear. [12] N. Hayashi and P. I. Naumkin, “Asymptotics for large time of solutions to the nonlinear Schr¨ odinger and Hartree equations”, Am. J. Math. 120 (1998) 369–389. [13] N. Hayashi and P. I. Naumkin, “Scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential”, preprint, 1997. [14] N. Hayashi and P. I. Naumkin, “Remarks on scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential”, SUT J. of Math. 34 (1998) 13–24. [15] N. Hayashi, P. I. Naumkin and T. Ozawa, “Scattering theory for the Hartree equation”, SIAM J. Math. Anal. 29 (1998) 1256–1267. [16] N. Hayashi and T. Ozawa, “Time decay of solutions to the Cauchy problem for time dependent Schr¨ odinger–Hartree equations”, Commun. Math. Phys. 110 (1987) 467–478. [17] N. Hayashi and T. Ozawa, “Scattering theory in the weighted L2 (Rn ) spaces for some Schr¨ odinger equations”, Ann. IHP (Phys. Th´ eor.) 48 (1988) 17–37. [18] N. Hayashi and T. Ozawa, “Smoothing effect for some Schr¨ odinger equations”, J. Funct. Anal. 85 (1989) 307–348. [19] N. Hayashi and T. Ozawa, “Time decay for some Schr¨ odinger equations”, Math. Z. 200 (1989) 467–483. [20] N. Hayashi and Y. Tsutsumi, “Scattering theory for Hartree type equations”, Ann. IHP (Phys. Th´ eor.) 46 (1987) 187–213. [21] H. Hirata, “Large time behaviour of solutions for Hartree equation with long range interaction”, Tokyo J. Math. 18 (1995) 167–177. [22] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, Vol. I, Springer, Berlin, 1983. [23] L. H¨ ormander, Lectures on Nonlinear Hyperbolic Equations, Math. & Appl. 26, Springer, Berlin, 1997. [24] A. Majda, “Compressible fluid flows and systems of conservation laws in several variables”, Appl. Math. Sci. 53, Springer, Berlin 1984. [25] H. Nawa and T. Ozawa, “Nonlinear scattering with nonlocal interaction”, Commun. Math. Phys. 146 (1992) 259–275.
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[26] T. Ozawa, “Long range scattering for nonlinear Schr¨ odinger equations in one space dimension”, Commun. Math. Phys. 139 (1991) 479–493. [27] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. [28] D. R. Yafaev, “Wave operators for the Schr¨ odinger equation”, Theor. Mat. Phys. 45 (1980) 992–998.
DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN R4 : A WAY TO FURTHER GENERALIZED WEIERSTRASS EQUATION SHIGEKI MATSUTANI 2-4-11 Sairenji, Niihama, Ehime 792 Japan Received 21 September 1998 In the previous report (J. Phys. A30 (1997) 4019–4029), I showed that the Dirac operator defined over a conformal surface immersed in R3 by means of confinement procedure is identified with the differential operator of the generalized Weierstrass equation and the Lax operator of the modified Novikov–Veselov (MNV) equation. In this article, using the same procedure, I determine the Dirac operator defined over a conformal surface immersed in R4 , which is for a Dirac field confined in the surface. Then it is reduced to the Lax operators of the nonlinear Schr¨ odinger and the MNV equations by taking appropriate limits. It means that the Dirac operator is related to the further generalized Weierstrass equation for a surface in R4 .
1. Introduction Investigation on an immersed geometry is current in various fields [1–6]. A certain class of the immersed surface in three dimensional space R3 is related to soliton theory. The surfaces, e.g. constant mean curvature surface, constant Gauss curvature surface, Willmore surface and so on, can be expressed by soliton equations and are sometimes called soliton surfaces. In the studies, Bobenko pointed out that an immersed surface in R3 should be studied through the spin structure over it [1, 2]. The auxiliary linear differential operator related to soliton surface should be regarded as an operator acting the spin bundle on the surface. Recently Konopelchenko discovered a Dirac-type operator defined over a conformal surface immersed in R3 which completely exhibits the symmetry of the immersed surface itself [3, 4]. The Dirac-type differential equation for a conformal surface immersed in R3 is given as ∂f1 = pf2 ,
¯ 2 = −pf1 , ∂f
(1.1)
√ where p := 12 ρH, H is the mean curvature of the surface parameterized by complex z and ρ is the conformal metric induced from R3 . Equation (1.1) is a generalization of the Weierstrass–Enneper equation for the minimal surface [3–6]. This equation is sometimes called generalized Weierstrass equation. As the geometrical interpretation of the modified Novikov–Veselov (MNV) equation [7], Konopelchenko and Taimanov studied the dynamics of the surface obeying the MNV equation [3–6]. Independently I proposed a construction scheme of a Dirac operator defined over an immersed object [8]. By adding a confinement mass potential to the ordinary 431 Reviews in Mathematical Physics, Vol. 12, No. 3 (2000) 431–444 c World Scientific Publishing Company
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Dirac equation for Dirac field in n-dimensional flat space-time Rn , one can confine the Dirac field in a quasi-low dimensional subspace along an embedded object in Rn . By taking a limit, a Dirac equation is effectively obtained for a Dirac field on the embedded object. By taking another limit, one can make the embedded object become an immersed object. Then the Dirac operator in the immersed object can be naturally defined. In [8–13], I showed that the Dirac operator in an immersed curve in Rn can be regarded as the Lax operator of a soliton equation while the extrinsic curvature of the immerse curve obeys the soliton equation itself. In other words, I obtained a natural spin structure over an immersed curve in Rn . In fact, I proved that the Dirac operator I obtained is a canonical object and its analytic index agrees with the topological index of the immersed curve [9, 11]. After I proposed the construction scheme of the Dirac operator on an immersed curve [8], Burgess and Jensen immediately applied my construction scheme to Dirac operator on a surface in R3 [14]. In previous work [15], I showed that if I make the surface conformal one, the Dirac operator on the surface turns out to be that of Konopelchenko (1.1) [3] and by investigation of its quantum symmetry, its functional space also exhibits the surface itself [16]. Hence it can be regarded that my construction scheme of the Dirac operator is canonical and gives the spin structure over the immersed objects as Bobenko pointed out [1, 2]. Recently using the Dirac operator in (1.1), Polyakov’s extrinsic string in R3 [17], whose world-sheet is an immersed surface in R3 and action consists of the Willmore functional and area of the surface, has been studied by several authors [16, 18–21]. As the Dirac operator in (1.1) completely exhibits the symmetry of an immersed surface in R3 , in terms of the operator, one can investigate its symmetry as the Frenet–Serret relation of a space curve gives us information of the curve itself. Since there is a natural relation between a surface and a space curve, the space curve also interests us from the point of view of string theory [17] although investigation of a space curve itself is of concern as a polymer physics and as a geometrical interpretation of soliton theory [22–25]. Using the Ferret–Serret relation (or the Dirac equation over the space curve) and isometry condition, I proposed another calculation procedure to obtain an exact partition function of an closed space curve (elastica) in two dimensional space R2 based on a soliton theory [25], beside the construction scheme of the Dirac operator [8–16]. There appear the modified Korteweg–de Vries (MKdV) hierarchy in the calculation of the partition function and the Painlev´e equation of the first kind at its critical point [25]. Thus the moduli of the partition function is closely related to string in 0-dimensional space and two-dimensional quantum gravity [26]. Further for the case of a space curve in R3 , its partition function is related to the complex MKdV equation and the nonlinear Schr¨ odinger (NLS) equation [22, 24, 27]. Using the NLS equation, Grinevich and Schmidt studied moduli of closed space curves from another viewpoint [28]; a complexfication of a space curve is related to world-sheet of string. In fact the MNV equation can be regarded as a
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kind of complexfication of the MKdV equation and corresponding surface (Willmore surface) in R3 can be interpreted as a generalization of an elastica in R2 [21]; both codimension are one. Using the correspondence and the Dirac operator in (1.1), I applied the calculation procedure of the partition function of [25] to the Willmore surface which has Polyakov extrinsic action [21]. It means that the surface (a kind of string in the string theory) in R3 is quantized [21]. In the calculation, I used the Dirac operator in (1.1) to look into the symmetry of the surface. Thus the Dirac operator in (1.1) plays the most important role in order to inspect the classical and quantized surface in R3 . However for example, a string in the string theory [17] might be immersed in ten or twenty-six dimensional space and should be quantized. Further, from mathematical point of view, investigation of a surface immersed in higher dimensional space is very interesting. However as long as I know, the differential operator which expresses a surface immersed even in four dimensional space R4 has not been determined. Thus we desire such an operator as a further generalization of the Dirac operator of Konopelchenko or the generalized Weierstrass operator (1.1). In this article, I will give a Dirac operator defined on a conformal surface immersed in R4 following the construction scheme I proposed in [8]. Using a confinement mass potential, I will confine a Dirac field in a quasi-two dimensional subspace along the surface and obtain an effective Dirac equation of the field by taking a limit. Then I will obtain the Dirac operator in an immersed surface in R4 . As I will show, it becomes the Lax operator of NLS equation [11] and that of the MNV equation [15] by taking appropriate limits. In other words, it naturally contains the differential operator in the Frenet–Serret relation and the Dirac operator in (1.1) as its special cases. Thus I conjecture that the Dirac operator which I will obtain exhibits the symmetry of the immersed object itself and is the operator of the further generalized Weierstrass equation for a surface in R4 . Organization of this article is as follows. Section 2 gives review of the extrinsic geometry of a conformal surface embedded in R4 and provides notations in this article. In Sec. 3, I will confine a Dirac particle in the surface and determine the Dirac equation over the surface. Using the complex representation, I will give more explicit form of the Dirac operator in Sec. 4. Section 5 gives confirmation whether obtained Dirac operator is a natural object by taking appropriate limits. I will discuss my result in Sec. 6. 2. Geometry of a Curved Surface Embedded in R3 In this section, I will set up a geometrical situation of the system which I will discuss, and give notations used in this article [11, 15, 16, 29]. I will deal with a conformal compact surface S embedded in R4 Σ → S ⊂ R4 ,
(2.1)
where Σ = H/Γ; H is the complex half plane and Γ is a Fuchsian group. Since the imaginary time and the euclidean quantum mechanics are very useful in the path integral method, I will deal with only the euclidean Dirac field in this article.
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Furthermore, for sake of simplicity, I assume that such a surface is embedded in R4 rather than immersed for a while. I assume that a position on the surface S is represented using the affine vector x(q 1 , q 2 ) = (xI ) = (x1 , x2 , x3 , x4 ) in R4 and normal vectors of S are denoted by e3 and e4 . Here q 1 and q 2 are natural parameters attached in the surface S. The surface S has the conformal flat metric which is induced from the natural metric of R4 , (2.2) gαβ dq α dq β = ρδα,β dq α dq β . I assume that the beginning parts of Greek indices (q α , q β , . . .) span from one to two and adopt Einstein convention. δ.,. means the Kronecker delta matrix. I will employ the complex parameterization of the surface as z := q 1 + iq 2 and ∂ :=
1 (∂q1 − i∂q2 ) , 2
1 ∂¯ := (∂q1 + i∂q2 ) , 2
d2 q := dq 1 dq 2 =:
1 2 1 d z := idzd¯ z. 2 2 (2.3)
I sometimes express it as f = f (q) for a real analytic function f over S but I will use the notation f = f (z) for a complex analytic function. The moving frame over S is denoted as eIα := ∂α xI and eIz := ∂xI , where z ∂α := ∂qα := ∂/∂q α . Their inverse matrices are denoted as eα I and eI . The induced metric is expressed as gαβ = δIJ eIα eIβ ,
1 ρ = hez , ez¯i := δI,J eIz eJz¯ , 4
(2.4)
where h, i denotes the canonical inner product in the euclidean space R4 . Since I will constrain a Dirac field in R4 to be on S by taking an appropriate limit, I can pay attention to only the vicinity of S or a tubular neighborhood T of the surface S even before squeezing limit. I will consider the properties of the tubular neighborhood T and its affine structure in R4 . I wish that geometry in the tubular neighborhood T is expressed using the variables of S. I will introduce a general coordinate system (q 1 , q 2 , q 3 , q 4 ) besides (q 1 , q 2 ) as a coordinate system of T . The surface S will be expressed as q 3 (x) = q 4 (x) = 0. Let the middle parts of the Greek indices (q µ , q ν , . . .) used as curved system, µ = 1, 2, 3, 4. Further I assume that the beginning parts of the Greek indices with ˙ dot (q α˙ , q β , . . .) run from three to four. I will assume that the normal structure is trivial; the metric of the normal plane is induced from R4 and vectors e3 and e4 are orthonormal: gα˙ β˙ = δα˙ β˙ . For later convenience, I also introduce the complex structure over the normal plane, ξ := q 3 + iq 4 ,
ξ¯ := q 3 − iq 4 .
(2.5)
The relation between the affine vector X := (X 1 , X 2 , X 3 , X 4 ) in T and natural coordinate systems q µ is given by X(q µ ) = x(q α ) + q α˙ eα˙ . This relation is uniquely determined if q α˙ is sufficiently small.
(2.6)
DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN R4
435
Next I will consider the extrinsic geometry of S itself. Using the moving frame eIα , I divide the ordinary derivative along S into the horizontal and vertical part; the horizontal part is written by ∇α defined as ∇α b := ∂α b − h∂α b, eα˙ ieα˙ ,
(2.7)
γ attached on S is for a vector field b. The two-dimensional Christoffel symbol γβα thus defined by γ eγ . (2.8) ∇α eβ = γβα
The second fundamental form is denoted by, ξ γβα =
α ˙ := heα˙ , ∂α eβ i , γβα
1 3 4 (γ + iγβα ). 2 βα
(2.9)
On the other hand, the Weingarten map, −γβαα˙ eα , is defined by γβαα˙ eα := ∇β eα˙ ,
γβαα˙ = heα , ∂β eα˙ i ,
(2.10)
γ α ˙ = −γαα which is associated with the second fundamental form as γβα ˙ gγβ . Here I will choose the normal vectors eα˙ satisfied with, β ∂α eα˙ = γαα ˙ eβ .
(2.11)
˜α˙ = ˜α˙ is given as ∂α e The derivative of a more general normal orthonormal base e β˙ β ˜ ˜ ˜ γ˜αα e e + γ ˜ instead of (2.11). From the orthonormal condition ∂ h˜ e , e i ˙ ˙ β α α ˙ αα ˙ ˙ β β = 0, ˙
β α ˙ α ˙ I have the relation, γ˜αα γβα ˜αα ˙ In other words, ˙ , and γ ˙ ≡ 0 (not summed over α). ˙ = −˜ 3 for α = 1, 2. Thus I will employ a SO(2) there are only two degrees of freedom; γ˜4α transformation so that I hold the relation (2.11);
e3 e4
=
cos θ sin θ
− sin θ cos θ
˜3 e ˜4 e
Z ,
θ :=
Z
q1 3 dq 1 γ˜41
+
q2 3 dq 2 γ˜42 .
(2.12)
This transformation is sometimes called as Hashimoto transformation [11, 22, 30]. As I prepared the languages to express the geometry of T , I will represent T . In terms of eIα and (2.6), the moving frame of T EµI := ∂µ X I is expressed by, β I EαI = eIα + q α˙ γαα ˙ eβ ,
EαI˙ = eIα˙ .
(2.13)
Its inverse matrix is denoted by (EIµ ). The induced metric of T is given as Gµν = δIJ EµI EνJ , and is explicitly expressed as ˙
γ γ γ α ˙ δ α˙ β Gαβ = gαβ + [γαα ˙ gδγ γββ ˙ gγβ + gαγ γαβ ˙ ]q q , ˙ ]q + [γαα
Gαα ˙ = Gαα˙ = 0 , Gα˙ β˙ = gα˙ β˙ = δα˙ β˙ .
(2.14)
The determinant of the metric G := det(Gµν ) becomes G = gζ ,
α α )q 3 + tr2 (γ4β )q 4 + K(q 3 , q 4 ))2 , ζ := (1 + tr2 (γ3β
(2.15)
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S. MATSUTANI
where g := det2 (gαβ ) and K(q 3 , q 4 ) = O((q 3 )2 , q 3 q 4 , (q 4 )2 ). Here tr2 is the twodimensional trace over α and β. I will denote them as 1 α ), H1 := − tr2 (γ3β 2
1 α H2 := − tr2 (γ4β ), 2
(2.16)
and introduce the “complex mean curvature”, Hξ = H1 + iH2 .
(2.17)
For an appropriate limit, it becomes ordinary mean curvature of one-codimensional case [15, 29] and for another limit, it becomes complex curvature of a space curve in R3 which is known as curvature given by the Hashimoto transformation [11, 22, 24, 27] as I will show in Sec. 5. Using (2.14), the Christoffel symbols associated with the coordinate system of the tubular neighborhood T is given as Γµνρ :=
1 µτ G (Gντ,ρ + Gρ,τ,ν − Gνρ,τ ) . 2
(2.18)
In terms of (2.18), the covariant derivative ∇µ in T is defined by, ∇µ Bν := ∂µ Bν − Γλµν Bλ ,
(2.19)
for a covariant vector Bµ . 3. Dirac Field on a Surface Embedded in R4 As I set up the geometrical situation of my system, in this section I will consider the Dirac field Ψ =: (Ψ Ψ1 , Ψ2 )T defined in the tubular neighborhood T ⊂ R4 . Following the construction scheme [8–16], I will confine it into the surface S by taking a limit. I will start with the original lagrangian given by, ¯ (x)i(ΓI ∂I − mconf (q α˙ ))Ψ Ψ(x)d4 x , Ld4 x = Ψ
(3.1)
4 ¯ = Ψ† Γ1 . I assume , ∂I := ∂/∂xI and Ψ where ΓI is the gamma matrix in the Rp α ˙ that mconf has the form, mconf (q ) := µ20 + ω 2 ((q 3 )2 + (q 4 )2 ) for a large ω and √ µ0 with µ0 ω [13, 15]. My confinement scenario is as follows. I will denote a thin tubular neighborhood by T0 . I can approximately regard T0 as a trivial disk bundle; T0 ≈ D1/ω × S, where D1/ω := {(y1 , y2 )| |(y1 , y2 )| < 1/ω}. As well as in [8–16, 30–32], by paying my attention only on the ground state, the Dirac field is approximately confined in the thin tubular neighborhood T0 . Even though there is a U(1)(≈ D1/ω − {0}) symmetry around S, I will choose a trivial rotation or constant function as a section of a sphere bundle U(1) × S because non-trivial rotational component of the energy is expected as order of ω. After taking squeezed limit, I can realize a quasi-twodimensional subspace in R4 and, by integrating the Dirac field along the normal direction, express the system using the two-dimensional parameters (q 1 , q 2 ). Then I will interpret the Dirac field as that over the surface S itself [8–16].
DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN R4
437
To make the above scenario meaningful, first I will express the lagrangian in terms of the curved coordinate system q µ . For the coordinate transformation (2.6), the Dirac operator becomes [8, 13, 15] iΓI ∂I = iΓµ ∂µ ,
(3.2)
and the spinor representation of the coordinate transformation is given as Ψ(q) = e−Σ
IJ
ΩIJ
Ψ(x) ,
(3.3)
where ΣIJ is the spin matrix, ΣIJ :=
1 I J [Γ , Γ ] , 2
(3.4)
and ΣIJ ΩIJ is a solution of the differential equation in terms of (2.19), ∂µ (ΣIJ ΩIJ ) = Ωµ ,
Ωµ :=
1 IJ ν Σ EI (∇µ EJν ) . 2
(3.5)
Hence the lagrangian density (3.1) becomes √ ¯ (q)i(Γµ Dµ − mconf (q α˙ ))Ψ Ψ(q) Gd4 q , Ld4 x = Ψ
(3.6)
where Γµ := ΓI EIµ and Dµ denotes the spin connection, Dµ := (∂µ + Ωµ ) .
(3.7)
Here I will note the relation for the normal direction, Dα˙ = ∂α˙
˙
module q β .
Since the measure on the curved system is given as, √ d4 x = G · d4 q ,
(3.8)
(3.9)
and −iDα˙ is not hermite nor a momentum operator, I redefine the field as [8–16, 30–33], (3.10) Ψ = ζ 1/2 Ψ . Then the lagrangian density (3.6) changes as, √ µ ¯ Dµ − mconf (q α˙ ))Ψ(q) gd4 q , Ld4 x = Ψ(q)i(Γ
(3.11)
where 1 Dα := Dα − ∂α log ζ , 4
Dα˙ := Dα˙ +
Hα−2 − ∂α˙ K/2 ˙ . 3 1 − 2H1 q − 2H2 q 4 + K(q 3 , q 4 )
(3.12)
Due to (3.10), in the deformed Hilbert space spanned by Ψ, −iDα˙ is a generator of the translation along the normal direction and thus is the momentum operator.
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S. MATSUTANI
After squeezing limit, the normal direction is independent space as several authors argued [8–16, 30–32]. The normal direction can be regarded as an inner space. Such an inner space comes from the symmetry of the euclidean space R4 and is determined by an embedded object. Hence at least, in the cases of a space curve in Rn , n > 1 and a surface in R3 , the Dirac operator completely exhibits the embedded symmetry [8–16]. After squeezing limit, due to the confinement potential mconf , the Dirac field for the normal direction is asymptotically factorized and can be expressed by the modes classified by a non-negative integer n [8–15]. Then there exists a mode with the lowest energy for normal direction and trivial rotation such that p ψ (q 1 , q 2 ) (3.13) Ψ(q 1 , q 2 , q 3 , q 4 ) ∼ δ(|q α˙ |)ψ and
p p ψ (q 1 , q 2 ) = m0 δ(|q α˙ |)ψ ψ (q 1 , q 2 ) . (Γα˙ ∂α˙ − mconf (q α˙ )) δ(q α˙ )ψ
(3.14)
By restricting the function space of the Dirac field to the ground state for the normal direction (n = 0), I will define the lagrangian density on a surface S [15], Z √ (0) √ 2 3 4 LS gd q := dq dq L G d2 q , (3.15) n=0
and then it has the form, √ ¯ (γ 1 D1 + γ 2 D2 + H1 γ 3 + H2 γ 4 + m0 )ψ gd2 q = iψ ψ gd2 q ,
(0) √
LS
(3.16)
where I rewrite the quantities as eIα ≡ EαI |qα˙ =0 ,
γ µ (q 1 , q 2 ) := Γµ (q 1 , q 2 , q α˙ ≡ 0) ,
ωα (q 1 , q 2 ) := Ωα (q 1 , q 2 , q α˙ ≡ 0) ,
Dα := ∂α + ωα .
(3.17)
Since a confined space is expressed by the two-dimensional parameter (q 1 , q 2 ) and can be regarded as a two-dimensional space, I can identify the confined space with the surface S itself and then the differential operator in (3.16) is interpreted as a Dirac operator in a surface S embedded in R4 . Hence the spin connection can be written as 1 ωα := Σij eβi (∇α ejβ ) , (3.18) 2 where the indices of i, j run from 1 to 2. It should be noted that the Dirac operator in (3.16) is not hermite in general as that in a space curve immersed in R4 is neither [8–13]. It is natural because the extra term in the corresponding Schr¨ odinger equation in the surface S behaves the negative potential [30–33]; roughly speaking the square root of the negative potential appears as a pure imaginary extra field in the Dirac equation [8–16]. As I argued in [15], an immersion can be approximately regarded as embedding an object in more higher dimensional space. For example, a loop soliton in R2 is mathematically realized as an immersed curve in R2 because it has crossings.
DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN R4
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However it is physically realized by using a thin rod on a plane in R3 whose overlap is negligible. The dimension of the rod can approximately be regarded as one; it can be considered as an immersed curve in R2 . Thus I can generalize this result of an embedding surface in R4 ⊂ Rn (n > 4) to that of immersed case. 4. Dirac Operator on a Complex Surface Immersed in R4 Generally m0 does not vanish, but I will neglect the mass m0 hereafter as in [8–16]. I am interested in the properties of the Dirac operator itself and I will investigate the high energy behavior of the field with m0 , i.e., the behavior of the high energy for surface direction and the lowest energy of the normal direction using the independency of the normal modes. Since the surface S is conformal, I will concretely express (3.16). Then the Christoffel symbol is calculated as [17], α = γβγ
1 −1 α ρ (δβ ∂γ ρ + δγα ∂β ρ − δβγ ∂α ρ) . 2
(4.1)
I will denote a natural euclidean inner space by y a , y b , . . ., (a, b = 1, 2). Then the moving frame is written as, (4.2) eaα = ρ1/2 δαa , and the gamma matrix is connected to the Pauli matrices σ a , 1 a γ α = eα aσ ⊗ σ .
(4.3)
Thus the spin connection becomes 1 ωα = − ρ−1 σ ab (∂a ρδαb − ∂b ρδαa ) , 4
(4.4)
where σ ab := 1 ⊗ [σ a , σ b ]/2. The Dirac operator in (3.16) can be expressed as 1 (4.5) γ α Dα = σ 1 ⊗ σ a δaα ρ−1/2 ∂α + ρ−3/2 (∂α ρ) . 2 Similar to (3.10), I will redefine the Dirac field in the surface S as ψ := ρ1/2ψ
(4.6)
and then the Dirac operator (4.5) becomes simpler [15, 17], γ α Dα ψ = ρ−1 σ 1 ⊗ σ a δaα ∂α ψ .
(4.7)
On the other hand, due to the properties of the Dirac matrix and noting the fact that the normal direction should be regarded as an inner space, γ α˙ is represented as, (4.8) γ 3 = σ1 ⊗ σ3 , γ 4 = σ2 ⊗ 1 . √ ¯ γ 1 Ψ gd2 q is the charge density, I expect the relation, Since Ψ ψ¯ = ψ † γ1 = ψ † σ 1 ⊗ σ 1 ρ1/2 .
(4.9)
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S. MATSUTANI
Accordingly the lagrangian density (3.16) is reduced to, (0) √
LS
¯ −1/2 (σ1 ⊗σ a δ α ∂α +ρ1/2 H1 σ1 ⊗σ3 +ρ1/2 H2 σ2 ⊗1)ψd2 q . gd2 q = iψρ a
(4.10)
Here I will denote the field as ψ1+ ψ1− ψ= ψ2+ . ψ2−
(4.11)
Using the complex parameterization of the surface z = q 1 + iq 2 and (2.3), the lagrangian density (4.10) can be concretely expressed. The equation of motion of the Dirac field in the conformal surface S immersed in R4 is derived as pc ∂ ψ1+ ∂ −pc ψ1− = 0, (4.12) p ψ2+ ∂ c
∂
ψ2−
−pc
where pc is the complex conjugate of pc and the “external” field pc is defined as, pc :=
1 1/2 ρ Hξ . 2
(4.13)
Equation (4.12) is the Dirac equation canonically defined over the conformal surface S immersed in R4 . This equation is a generalization of that of a space curve immersed in R3 [8–13] and that of a conformal surface immersed in R3 [14–16] following my procedure [8–16]. Physically speaking, this equation governs the Dirac particle confined in a surface immersed in R4 . 5. Some Limits In this section, I will check whether the obtained Dirac operator in (4.12) naturally behaves after taking some limits. 5.1. Reduction to a space curve First I will consider a space curve C in R3 , which is parameterized by q 1 ∈ S 1 . Its codimension is two. Due to the codimensionality, this situation resembles a surface immersed in R4 , which I dealt with until the previous section. I can extend C ⊂ R3 to S ⊂ R4 by multiplying a trivial line manifold R; C × R can be regarded as a surface S. This surface is not compact and contradict my assumption in (2.1). However it is not difficult to see that the assumption can be extended to this case. Hence I can regard this situation as a special case of the situation which I argued. I explicitly represent the “mean curvature” in (2.16) and metric in (2.4), 1 1 ), H1 = − tr2 (γ31 2
1 1 H2 = − tr2 (γ41 ), 2
ρ = 1.
(5.1)
DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN R4
441
They are a half of curvatures under the Hashimoto representation of the space curve and Hξ is a complex curvature. Vortex soliton obeys the NLS equation and whose dynamics is expressed in terms of Hξ [22–24], 1 i∂t Hξ + ∂12 Hξ + |Hξ |2 Hξ = 0 . 2
(5.2)
Now I will consider (4.12) in this case. Then there is a trivial solution ψ = ψ(q 1 ) and ∂2 ψ = 0. Noting ∂f (q 1 ) = ∂1 f (q 1 )/2, (4.12) reduces to ∂1 Hξ (5.3) L= ∂1 −H ξ which is one of its Lax operators of the NLS Eq. [34]. I obtained (5.3) by considering the Dirac operator over an immersed curve in R3 in [10, 11]. Further (5.3) can be regarded as the Frenet–Serret equation by inverse of the Hashimoto transformation [10, 11, 24]. It is a remarkable fact that the Dirac operator in (4.12) reproduces the Dirac operator in [10, 11] and is associated with the vortex soliton. 5.2. Reduction to a surface in R3 Next I will constrain the surface S to be immersed in R3 , which Konopelchenko and Taimanov dealt in [3–6] and I considered in the previous paper [15]. I will suppose that the direction e4 is identified with the direction of x4 and e4 geometry becomes trivial; (5.4) H2 = 0 . Then the external field is obtained as, pc ≡ p1 :=
1√ ρH1 2
(5.5)
and Eq. (4.12) is reduced to the Konopelchenko’s Dirac Eq. (1.1) [3–6], which represents the symmetry of the immersed surface in R3 ; p1 ∂ ψa+ = 0. (5.6) ψa− ∂ p1 This is one of the Lax operator of MNV equation [3–7]. Thus the Dirac operator in (4.12) contains my previous result in [15] as its special case. When I use the complex parameterization of the part of the euclidean space, Z := x1 + ix2 ,
(5.7)
we have special solutions of (5.6) [3–6], 2i(ψa+ )2 := −∂Z ,
2i(ψa− )2 := ∂Z ,
−2ψa+ ψa− = ∂x3 .
(5.8)
Hence the Dirac operator in (4.12) partially exhibits the symmetry of immersed surface.
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S. MATSUTANI
5.3. Reduction to a minimal surface in R3 Provided that Hξ = 0, the equation of motion (4.12) becomes the original Weierstrass–Enneper formula [1–6, 15] which was obtained as a scheme to obtain a minimal surface in last century and the lagrangian (4.10) becomes that in the ordinary (classical) conformal field theory [17]. Thus the Dirac operator in (4.12) contains the original Weierstrass–Enneper formula. 6. Discussion In this article, I obtained the Dirac operator defined over a conformal surface S immersed in R4 . As I argued in Sec. 5, the Dirac operator in (4.12) partially exhibits the symmetries of the immersed object. It recovers the Lax operators of the integrable surface and curve by taking appropriate limits. It also naturally contains differential operators of the Frenet–Serret relation for a curve in R3 and of the generalized Weierstrass relation for a conformal surface in R3 as its special cases. Thus it can be regarded as a generalization of the Dirac operator of Konopelchenko or that of the generalized Weierstrass equation for a conformal surface in R4 [3– 7, 15]. Consequently I conjectured that it exhibits the symmetry of an immersed surface in R4 [35]. After completing this article [35], the above conjecture was proved by Konopelchenko using more direct computation [36]; in his proof, he used only the words in the (classical) differential geometry. He showed that (4.12), which he did not call the Dirac operator, represents the surface in R4 and is associated with the (2+1) dimensional soliton of rank 2. Furthermore, Pedit and Pinkall also gave another proof, which does not directly derive (4.12), in terms of their quarternionic analysis [37, 38]. Even though the conjecture was proved, my approach definitely differs from theirs. If different approaches give the same result, they must be associated with each other and there is a (hidden) symmetry. Thus in order to lead us to deep insight into such symmetry, I believe that my computation and approach to the Dirac operator is still needed. At least, from physical point of view, it is surprising that Dirac particles defined over a surface know the surface itself and the Dirac Eq. (4.12) can reproduce the shape of the surface. Since it was proved that the Dirac operator in (4.12) can be Lax operator of the (2+1) dimensional soliton and represents the surface itself, it can be obvious that using the operator, we can quantize a surface in R4 as I did in [21]. This could be interpreted as a prototype of the recent approach of the quantization of gravity [39]. Finally I will comment upon higher dimensional case. As I did in [10], it is not so difficult that the argument in this article can be extended to a case of a surface in more higher dimensional space Rn , n > 3; using my procedure to construct the Dirac operator, it is easy to determine a proper Dirac operator for a conformal surface immersed in Rn , n > 3. It is expected that the Dirac operator is also related to the further generalized Weierstrass relation of a surface in Rn and using
DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN R4
443
the Dirac operator, we can inspect the immersion geometry and quantize a surface in R10 or R26 . Acknowledgments ˆ I would like to thank Y. Onishi and Prof. S. Saito for critical discussions and continual encouragements and Prof. B. G. Konopelchenko and Prof. I. A. Taimanov for helpful comment, sending me their interesting works and encouragement. I am grateful to Prof. P. G. Grinevich for sending me his interesting works. I acknowledge that the seminars on differential geometry, topology, knot theory and group theory with Prof. K. Tamano influenced this work. I also thank Prof. F. Pedit for kind discussions and sending me his recent works. Note Added in Proof In this article, I used only the ground state of the confinement mass potential but due to independence of the normal mode, we can deal with excited states with different mass spectrum as other particles. As I did in [16], the functional integral for these Dirac fields also reduce to the action of the Polyakov extrinsic strings [17]. Thus this system might be related to a phenomenological theory of confinement of quarks with different mass. References [1] A. I. Bobenko, Math. Ann. 290 (1991) 209–245. [2] A. I. Bobenko, “Surfaces in terms of 2 by 2 matrices: Old and new integrable cases”, in Harmonic Maps and Integrable Systems eds. A. P. Fordy and J. C.Wood, Vieweg, Wolfgang Nieger, 1994. [3] B. G. Konopelchenko, Studies in Appl. Math. 96 (1996) 9–51. [4] B. G. Konopelchenko and I. A. Taimanov, J. Phys. A: Math. & Gen. 29 (1996) 1261–1265. [5] I. A. Taimanov, Translations of the Amer. Math. Soc. 179 (1997) 133–151. [6] I. A. Taimanov, Ann. Global Analysis and Geometry 15 (1997). [7] A. P. Veselov and S. P. Novikov, Soviet Math. Dokl. 30 (1984) 588–591, 705–708. [8] S. Matsutani and H. Tsuru, Phys. Rev. A46 (1992) 1144–1147. [9] S. Matsutani, Prog. Theor. Phys. 91 (1994) 1005–1037. [10] S. Matsutani, Phys. Lett. A189 (1994) 27–31. [11] S. Matsutani, J. Phys. A: Math. & Gen. 28 (1995) 1399–1412. [12] S. Matsutani, Int. J. Mod. Phys. A10 (1995) 3091–3107. [13] S. Matsutani, Thesis in Tokyo Metropolitan Univ., 1995. [14] M. Burgess and B. Jensen, Phys. Rev. A48 (1993) 1861–1866. [15] S. Matsutani, J. Phys. A: Math. & Gen. 30 (1997) 4019–4029. [16] S. Matsutani, Rev. Math. Phys. 11 (1999) 171–186. [17] A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publ., London, 1987. [18] R. Carroll and B. Konopelchenko, Int. J. Mod. Phys. A11 (1996) 1183–1216. [19] P. G. Grinevich and M. U. Schmidt, “Conformal invariant functionals of immersions of tori into R3 ”, dg-ga/9702015. [20] B. G. Konopelchenko and G. Landolfi, “On classical string configurations”, solv-int9710008. [21] S. Matsutani, J. Phys. A31 (1998) 3595–3606. [22] H. Hashimoto, J. Fluid Mech. 51 (1972) 477–485.
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R. E. Goldstein and D. M. Petrich, Phys. Rev. Lett. 67 (1991) 3203–3206. A. Doliwa and P. M. Santini, Phys. Lett. A185 (1994) 373–384. S. Matsutani, J. Phys. A31 (1998) 2705–2725. M. Kaku, Strings, Conformal Fields, and Topology, Springer, New York, 1991. S. Matsutani, J. Geom. Phys. 29 (1999) 243–259. P. G. Grinevich and M. U. Schmidt, “Closed curves in R3 : A characterization in terms of curvature and torsion, the hasimoto map and periodic solutions of the filament equation”, dg-ga9703020. [29] D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles, Johon Wiley & Sons (republished in Dover 1989, New York), New York, 1975. [30] R. C. T. da Costa, Phys. Rev. A23 (1981) 1982–1987. [31] H. Jensen and H. Koppe, Ann. Phys. 63 (1971) 586–591. ``
`
[32] P. Duclos, P. Exner and P. S t ov´ıc ek, Ann. Inst. Henri Poincar´e 62 (1995) 81–101. [33] P. B. Gilkey, Invariance Theory, The Heat Equation and the Atiyah–Singer Index Theorem, Publish or Perish, Wilmington, 1984. [34] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Stud. Appl. Math. 53 (1973) 249–315. [35] S. Matsutani, “Dirac operator of a conformal surface immersed in R4 : Further generalized Weierstrass relation”, solv-int/9801006, 1998. [36] B. G. Konopelchenko, “Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy”, math-DG/9807129 (1998). [37] F. Pedit and U. Pinkall, Documenta Mathematica Extra Volume ICM 1998.II.389– 400. [38] U. Pinkall, Quaternionic Line Bundles and the Differential Geometry of Surfaces, ICM 1998 Berline, Abstracts of Plenary and Invited Lectures, 1998. [39] G. Landi and C. Rovelli, Phys. Rev. Lett. 78 (1997) 3051–3054.
EQUIVALENCE OF MODULAR K AND ANOSOV DYNAMICAL SYSTEMS H. NARNHOFER and W. THIRRING Institut f¨ ur Theoretische Physik, Universit¨ at Wien Boltzmanngasse 5, A-1090 Wien, Germany and Internationales Erwin Schr¨ odinger Institut f¨ ur Mathematische Physik Boltzmanngasse 9, A-1090 Wien, Germany Received 16 December 1998 We show that for modular dynamical systems the Anosov property and the existence of a refining subalgebra are equivalent. With some additional assumption they are also equivalent to the existence of an algebraic K-subsystem.
1. Introduction In quantum ergodic theory one studies the properties of the time evolution τ t given as a one-parameter group of automorphisms of an observable algebra M. We consider M to be a von Neumann algebra with a τ t invariant state ω represented via the GNS representation in a Hilbert space Hω . Actually the notion of ergodicity, i.e. the requirement that the only τ -invariant elements of M are multiples of 1 is too weak to be of much interest. For physics the relevant notions are refinements of mixing which specify how time correlations decay. Of these the most powerful are the Anosov and the K-properties [1–4]. The former postulates the existence of another −t transformation σ s (“transversal shift”) which is related to τ t by τ t ◦ σ s = σ se ◦ τ t . In general there will be a Lyapunov exponent e−λt but since t is continuous we may scale λ to one. What cannot be scaled away is the sad fact that if σ s is a group of automorphisms this excludes the possibility that there are τ -KMS-states over M. However the relevant consequences can also be derived by the mathematically weaker assumption that σ s , s ≥ 0, be a semigroup of (injective) endomorphisms of M. Definition 1.1. We say that the dynamical system (M, τ, ω) is Anosov if there exists a strongly continuous σs ∈ Aut B(Hω ) such that −t
(i) τ t ◦ σ s = σ se 6=
◦ τ t ∀ t ∈ R, s ∈ R, 6=
(ii) σ s M ⊂ M ∀ s > 0 (where ⊂ means proper inclusion). The Anosov property implies exponential sensitivity to initial conditions and exponential decay of time correlations [5]. The K-property ensures a uniformity in the observables of this decay. It is characterized by the existence of a subalgebra of M 445 Reviews in Mathematical Physics, Vol. 12, No. 3 (2000) 445–459 c World Scientific Publishing Company
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which by the time evolution is mapped isomorphically into a proper subalgebra of itself without having invariant subalgebras. Definition 1.2. We call a dynamical system (M, τ ) to be future (or past) refining if there exists a subalgebra N ⊂ M such that 6=
(i) τ t N ⊃ N for t > 0 (or t < 0). We call it a K system [6, 7], if in addition (ii) ∨t τ t N = M, (iii) ∩t τ t N = λ1. Remark 1.3. (ii) can be considered as definition of M since even if ∨t τ t N =: M0 is strictly contained in M it is τ -invariant. Then we can consider the dynamical system (M0 , τ ) and forget about M. So essential are (i) (“refining”) and (iii) (“triviality of the tail”). So far the considerations did not refer to a particular τ -invariant state over M. It turns out that if τ is the modular dynamics of a faithful state ω over M and all algebras are von Neumann algebras in the corresponding GNS-representation an unexpected relation between the two notions appears. What will be discussed in the present paper is: Theorem 1.4. If τ is the modular automorphism of a faithful state ω over M then (i) Anosov ⇐⇒ refining. (ii) If (M, N , τ ) is refining and ∨t (τ t N 0 ∩ M) 6= λ1, then (M, τ ) contains a Ksubsystem. (i) will be discussed in Sec. 2 and is essentially a reformulation of results in [8–10]. The K subsystem of (ii) will be defined in Sec. 3 and will be used to construct a time reflection map. 2. Some Standard Facts We shall always deal with von Neumann algebras M, N . . . N 0 is the commutant of N = N 00 , N1 ∩ N2 the (set-theoretic) intersection, N1 ∨ N2 the (algebraic) union with strong closure. ∩ and ∨ are associative, commutative, not distributive and commute with automorphisms. They are related to 0 via the Proposition 2.1. \ i
!0 Ni
=
_ i
Ni0 ⇔
\ i
Ni0 =
_
!0 Ni
.
i
Remark 2.2. The von Neumann property is essential for the equality in (2.1). For general C ∗ -algebras and the commutant relative to some embedding algebra only the inclusion (∩i Ni )0 ∩ M ⊃ ∨i (Ni0 ∩ M) holds.
EQUIVALENCE OF MODULAR K AND ANOSOV DYNAMICAL SYSTEMS
447
Now consider a faithful state ω over a von Neumann algebra M with modular automorphism τωt and a subalgebra N ⊂ M. Proposition 2.3. The following conditions are equivalent : (i) τωt N = N , t . (ii) τωt |N = τω|N (iii) There exists a unique conditional expectation ε(c.c.e.) : M → N with ω ◦ ε = ω, ε ◦ τωt = τωt ◦ ε. It is given by ε(m)P = P mP, P = P 2 = P ∗ ∈ N 0 , P M|Ωi = N |Ωi. For a proof see [11, 12]. t Corollary 2.4. If (a) τωt |N = τω| and (b) N |Ωi = M|Ωi, then N = M. N
Proof. (a) ⇔ (ii) ⇔ (iii) but ε(m)P = P mP where P is the projection M|Ωi → N |Ωi. Thus (b) ⇒ P = 1 ⇒ M = N . If N is not τ -invariant but mapped into itself we have Proposition 2.5. If M ⊃ N ⊃ τt N ∀ t ≥ 0 (or ≤ 0), then (i) N |Ωi = M|Ωi, (ii) ∨t∈R τωt N = M are equivalent. Proof. The proof is a variation of the Reeh–Schlieder Theorem [13]. (ii) ⇒ (i). We have to show that in M|Ωi only the zero vector is orthogonal to all n|Ωi, n ∈ N . Thus hΩ|mn|Ωi = 0 ∀ n ∈ N must imply m = 0. Define f (t) := hΩ|mτωt n|Ωi = 0 ∀ t > 0 (or < 0). We know that f (x + iy) is analytic in the strip 0 < y < β and for a dense set of n’s f and f 0 are continuous functions for y → 0. Since f (x) = 0 for x > 0 (or x < 0) it can be continued to some y < 0 by the Schwarz reflection principle. Then y = 0, x > 0 (or < 0) is in the domain of analyticity and f has to vanish altogether. By (ii) τ t N |Ωi, t ∈ R, is dense in M|Ωi so that hΩ|mτ t n|Ωi = 0 ∀ n, t ∈ R ⇒ m = 0. (i) ⇒ (ii). Denote ∨t∈R τ t N = M which is τ -invariant. Then from (2.3(iii)) we know ∃ ε : M → M, m → P mP with P M|Ωi = M|Ωi. But M|Ωi ⊂ N |Ωi = M|Ωi by (i). Thus P = 1 and M = M. If |Ωi is cyclic and separating for N and M it defines modular automorphisms t and τ t which are unitarily implemented by selfadjoint operators HN and HM : τN t and HN ). In terms of the modular operators ∆M τ t (·) = eiHM t · e−iHM t (resp. τN 1 1 and ∆N , HM = 2π ln ∆M and HN = 2π ln ∆N . If the situation of Proposition 2.5 is nontrivially realized (i.e. N 6= M) the HN and HM are distinct and satisfy: Proposition 2.6. [HM , HN ] = i(HN − HM ) .
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Proof. See [8–10] but note that we have rescaled the time to get rid of a factor 2π for the Lyapunov exponent. Remarks 2.7. 1. The domains of the modular operators ∆N and ∆M inherit an order relation from N ⊂ M. This can be converted in an operator inequality HN > HM or G := HN − HM > 0 if domain questions are observed. 2. The commutation relations [HM , G] = iG encode the Anosov properties of dynamical systems when eiHM t and eiGs are interpreted as time evolution and transversal shift. The multiplication laws of these unitaries are most conveniently deduced by a (non unitary) 2 × 2 representation of the group which they define: ! 0 et/2 iHM t , e = 0 e−t/2 ! 0 et/2 iHN t , = e et/2 − e−t/2 e−t/2 ! 1 0 eiGs = s 1 represents (2.5) and implies −t
eiHM t eiGs = eiGse eiHM t −t
eiHM t e−iHN t = e−iG(1−e
)
⇔ eiGs = e−iHN ln(1−s) eiHM ln(1−s)
= e−iHM ln(1+s) eiHN ln(1+s) . Compare also [14]. With these results we are equipped for: Proof of part (i) of (1.4). (a) Anosov ⇒ refining: Define N := σM ⇒ σ s N ⊂ N ∀ s ≥ 0 ⇒ τ t N = τ t σM = −t −t t→∞ σ e τ t M = σ e −1 N ⊂ N ∀ t ≤ 0 and τ t N −→ σ −1 N = M. Thus also (ii) of (1.2) holds. (b) Refining ⇒ Anosov: ω is faithful over N thus there exists HN the selfadjoint generator of τω|N . Define G = HN − HM and σ s (·) = eiGs · e−iGs . It satisfies the other conditions of (1.1). −t
t
t Corollary 2.8. (i) (or (ii)) imply that τN := σ 1−e τ t = τ t σ e parameter group of automorphism of N .
−1
is a one-
Proof. 0
−t0
−t
t t τN = σ 1−e τ t σ 1−e τN
0
−t
τ t = σ 1−e
0
+e−t −e−t−t
0
τ t+t ,
σ −1 τN σM = σ −1 τ t σ e M = τ t σ −e σ e M = M ∀ t ∈ R . t
t
t
EQUIVALENCE OF MODULAR K AND ANOSOV DYNAMICAL SYSTEMS
449
3. The Basic Construction Anosov subsystems are the chaotic parts of a modular dynamical system and one might wonder when they can occur. To formulate the simple answer we introduce another notion. Definition 3.1. We call a± := ∨t≥(≤)0 τ t a the future (or past) completion of the subalgebra a ⊂ M. (a may be generated by only one element a ∈ M and then we use the same letter for the subalgebra.) If a+ = a− for all subalgebras a ⊂ M we say (M, ω, τ ) is totally reversible. Proposition 3.2. (M, ω, τ ) is Anosov iff it is not totally reversible. Proof. (i) Anosov ⇒ not totally reversible. If there exists N ⊂ M, τ t N ⊂ N ∀ t > 0, then n+ ⊂ N ∀ n ∈ N ⇒ N + = ∨n∈N n+ ⊂ N and since N ⊂ N + ⇒ N + = N . If (M, ω, τ ) were totally reversible n+ = n− ∀ n ∈ N and thus N − = ∨n∈N n− = ∨n∈N n+ = N + . But if the system is Anosov ∃ N with M = N + ∨ N − 6= N + . (ii) Not totally reversible ⇒ Anosov. If there exists n ∈ M with n+ 6= n− , then n+ ∨ n− 6= n+ (or n− ). Take the one with 6= for N then M = ∨t τ t N = n+ ∨ n− 6= N . We shall first illustrate the notions introduced in (3.1) by: Example 3.3. Start with the CAR algebras generated by {af : [a∗f , ag ]+ = hf |gi, f, g ∈ L2 (R resp. S 1 )} . For τ we use the shift τ t af = aft , ft (x) = f (x − t) (x mod 1 in S 1 ) with associated KMS state Z X g (p) fe∗ (p)e dp resp. ω(a∗f , ag ) = in S 1 . 2π 1 + eβp p∈Z2π
(L2 (R) is isomorphic to L2 (S 1 ) and therefore also the C ∗ -CAR algebras are isomorphic. But the isomorphism does not carry over to the weak closures, so M(R) 6= M(S 1 ).) For M(S 1 ) we have ft = f−(1−t) , hence for all subalgebras N + = N − and the system is totally reversible. On the other hand, if N ⊂ M(R) is generated by {f, supp f ⊂ (a, b)} we get N + = M((a, ∞)), N − = M((−∞, b)) so M = N + ∨ N − 6= N + and (M, N + , ω, τ ) is refining. We have seen in Sec. 2 that Anosov implies refining and now we have to see when the third condition of (1.2), i.e. the triviality of the tail N := ∩t τ t N is satisfied. For modular systems the answer is simple. Proposition 3.4. With G defined in Remark 2.6.1, N = N ∩ G0 = M ∩ G0 .
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Remark 3.5. N is the biggest (setwise) τ -invariant subalgebra of N , in particular it contains the τ -invariant elements of M. Now we learn that it consists exactly of the σ-invariant elements of M. Proof of 3.4. We know from (2.7) N = σM thus MG := M ∩ G0 = N ∩ G0 . Since N is τ -invariant we know from (2.3(ii)) τ|N = τω|N . But N ⊂ N and hence ω|N = ω|N |N and we can continue τ|N = τω|N = τω|N |N = τω|N |N . From (2.7) −t −t . Thus on n ∈ N the actions of τ and τω|N are the we infer σ 1−e = τ t ◦ τω|N 0 same and we see n ∈ G ⇒ N ⊂ MG . Furthermore MG is also τ -invariant since t M is and τ preserves σ-invariance. σ s (m) = m ⇒ σ s τ t (m) = τ t σ se (m) = τ t (m). Since N contains all τ -invariant subalgebras of N we also get the converse inclusion N ⊃ MG . Remark 3.6. Proposition 3.4 is illustrated by the following observation. |Ωi is t0 cyclic for Nt := τ t N for all t thus there are modular automorphisms τN ∀ t and t 0 0 t −t t t = τ ◦ τ ◦ τ . From the representation (2.7.2) we infer they are conjugated by τN N t 0
−t
−t0
0
0
−t
t0
t = σ e (e −1) ◦ τ t = τ t ◦ σ e (1−e ) . Thus (for increasing N ) we see that for τN t t → ∞τNt → τ corresponding to limt→∞ τ t N → M. For t → −∞ τNt converges only on the σ-invariant elements corresponding to limt→−∞ τ t N → N . There it equals τ since σ-invariance implies τ = τω and since N is τ -invariant it also equals τω|N .
Proposition 3.7. In a modular K system M is a factor (i.e. M ∩ M0 = C1). Proof. Under the modular automorphism the center is elementwise invariant. N contains all τ -invariant elements. Thus M ∩ M0 ⊂ N = C1. Remark 3.8. If N 6= C1, then (3.5) tells us the first thing to do is to make the central decomposition of M. This does not disturb the dynamics so in the sequel we may assume that M is a factor. However this is not enough to get rid of the tail N . We know N acts in the zero space of G. But this can still be large. For instance, if we tensor a modular K system (M1 , N1 , τ1 , ω2 ) with the non-K system (M2 , N2 = M2 , τ2 , ω2 ) where M1 and M2 are factors. Then the product is refining, M1 ⊗ M2 a factor but the zero-space of G is |Ω1 i ⊗ Hω2 and N = 1 ⊗ N2 = G0 . However N 0 ∩ Nt = (B(H1 ) ⊗ M02 ) ∩ (N1,t ⊗ M2 ) = N1,t ⊗ 1 has a trivial tail. But this trick does not work generally, as we see from a counterexample: Example 3.9. Assume (M, N , τ, ω) is a modular K system and admits an additional free automorphism α, [τ, α] = 0, α2 = 1. Then the crossed product α c is a factor and again a c = M × Z (2) exists and allows an extension of τb. M M α cN b = N × Z (2) , τ, ω) (with the assumption αN = N , refining system with (M, b corresponds to the abelian c = SN bt ). The tail N which is necessary to have M c that implements α and is pointwise subalgebra generated by the element in M τb-invariant.
EQUIVALENCE OF MODULAR K AND ANOSOV DYNAMICAL SYSTEMS
451
Nevertheless there is another candidate for a subsystem with trivial tail: ¯ = ∪t Lt . Let ε be the c.c.e. from M Theorem 3.10. Let L := N 0 ∩ M and M ¯ ¯ ¯ ¯ ¯ L)) are an increasing into M. Let N = σ(M) = ε(N ). Then (M, N¯ ) (resp. (M, (resp. decreasing) K system. Proof. First we have to show that the c.c.e. ε (the canonical conditional expec¯ not only commutes tation uniquely defined by the condition ω = ω ◦ ε): M → M ¯ with τ but also with σ such that N = σ ◦ ε(M) = ε ◦ σ(M) = ε(N ). Since τ t (N 0 ) = (τ t N )0 and (2.7.2) we get σ s Nt0 = N−0 ln(s+e−t ) . Thus ! _ _ s ¯ s 0 s ¯ ∀s≥0 N ∩M = N0 σ M=σ −t ∩ σ M ⊂ M − ln(s+e
t
t
)
t
where we use that σ s M ⊂ M and Nt0 ⊃ Ns0 for t ≤ s. Therefore _ −t ¯. ¯ ⊃ N¯ ∀ t ≥ 0 and τ t N¯ = σ e M N¯t = M t
¯ N ¯ ) are refining, and by (2.4) Thus (M, ¯ = M|Ωi ¯ ¯ |Ωi . P H =: H =N This implies that eiGs as generator of an endomorphism σ s is not only normpre¯ N ¯ ), thus ¯ but also unitary: eiG mΩi ¯ =n ¯ |Ωi, (m, ¯ n ¯ ) ∈ (M, serving in H ¯=H ¯ ⇒ eiHt H ¯ = eiHt eiG H ¯ = eiGe−t H ¯ ∀ t ∈ R. eiG H ¯ N ¯ ), Therefore eiGs P = P eiGs . This verifies the properties (i), (ii) of (2.4) for (M, ¯ , L) they are trivial. Regarding the triviality of the tail, for L this is obvious: for (M \ \ Lt = Nt0 ∩ M = M0 ∩ M = λ1 . t
t
¯ we notice from L ⊂ N 0 that N ⊂ L0 . Since ε leaves L elementwise invariant For N ¯ Therefore ε(N ) ⊂ L0 ∩ M. ! \ \ \ ¯ =M ¯0 ∩M ¯ = λ1 . ¯t = ε Nt ⊂ L0 ∩ M N t
t
t
t
¯ is a factor according to (3.7) since we have already shown Here we used that M ¯ that (M, L) is a K system. ¯ N ¯ , L, ω) form a standard split inclusion as defined in [15], iff Notice that (M, L 6= λ1. Therefore some of the mixing properties can be found in [16]. ¯ ⊂M ¯ ¯ we have now two transversal shifts, G+ related to the imbedding N In M ¯ As was observed in [17] they satisfy a and G− related to the imbedding L ⊂ M. simple commutation relation: Lemma 3.11. With G+ = HN¯ − HM ¯ and G− = HL − HM ¯, [G+ , G− ] = 2iHM ¯ .
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H. NARNHOFER and W. THIRRING
¯ L⊂M ¯ and L ⊂ N ¯ 0 . First we have Proof. We use the three imbeddings N¯ ⊂ M, 0 to control that τN¯ 0 acts as endomorphism on L. In fact L = N ∩ M, therefore ¯ −iHN¯ t eiHN¯ t Le−iHN¯ t = eiHN¯ t N¯ 0 e−iHN¯ t ∩ eiHN¯ t Me −t ¯ −iG+ (1−e−t ) ⊂ L ∀ t > 0 . = N¯ 0 ∩ eiG+ (1−e ) Me
From the Anosov relations, [HM ¯ , HN ¯ ] = i(HN ¯ − HM ¯ ) =: iG+ , [HM ¯ , HL ] = −i(HL − HM ¯ ) = −iG− , since HN¯ 0 = −HN¯ ,
[−HN¯ , HL ] = i(HL + HN¯ ) := iG3 we infer
[G+ , G− ] = 2iHM . The corresponding M¨obius group can be represented as ! ! 0 et/2 1 0 iHM t iG+ s , eiG− s = , e e = = −t/2 s 1 0 e and e
iG3 t
=
1+t
−t
t
1−t
1
−s
0
1
! ,
!
.
Theorem 3.12. The map −1 γ = ad JM ¯ ◦ σ+ ◦ σ− ◦ σ+
¯ and reflects the time evolution acts as antiisomorphism on M γ ◦ τt = τ−t ◦ γ . ¯ 0. ¯ M Here JM ¯ is the modular conjugation, JM ¯ Ω = Ω and JM ¯ MJ ¯ = M Proof. The easiest way to see time reflection is the explicit calculation in the 2 × 2 representation (2.6.2). Notice that iH ¯ t iH ¯ t JM ¯ e M JM ¯ = e M .
Therefore there remains to calculate ! ! ! 1 0 1 −1 et 0 0
e−t
1
1
0
1
1
0
1
1
! =
=
!
et
0
0
e−t
0
−1
1
0
!
0 1
−1 0
e−t
0
0
et
! .
EQUIVALENCE OF MODULAR K AND ANOSOV DYNAMICAL SYSTEMS
453
It remains to argue −1 ¯ =M ¯0. σ+ σ− σ+ M
¯ 0 = σ −1 L0 . Observe that, if σ3 represents e−iG3 , then σ+ M = N = σ3 L0 and M − This reduces to verify −1 −1 σ+ σ− σ3 = σ− or explicitly to 1
0
1
1
!
1
−1
0
1
!
0
1
−1
2
! =
1
−1
0
1
! .
4. Cluster Properties In this section we shall study how the joint properties of Anosov and K allow to sharpen the estimates for the decay of time correlation functions. Once we got rid ¯ we conclude from (3.4) that in M ¯ only multiples of unity are of the tails in M σ-invariant. This entails: Proposition 4.1. Z ¯, ∀ m∈M
m ¯ := w − lim δ δ↓0
∞
e−δs σs (m)ds = ω(m) .
0
¯ since Proof. By weak compactness there are weak cluster points and they are in M it is weakly closed. They are σ-invariant and by (3.4) = 1C. Since ω is σ-invariant and weakly continuous C = ω(m). Remarks 4.2. 1. This shows that ω is the only σ-invariant state, if ω b were another ω b (m) = ω b (m) ¯ =ω b (ω(m)) = ω(m). 2. Weak convergence can be strengthened for means to strong convergence. Without mean this cannot hold, st-lims→∞ ms = ω(m) would imply st-lims→∞ m2s = ω(m2 ) = (ω(m))2 . This implies that ω((m − ω(m))2 ) = 0 but since ω is faithful this also implies m = ω(m). 3. Since |Ωi is separating for bounded sequences it is enough to prove strong convergence on |Ωi which we shall do now. ¯ Proposition 4.3. ∀ m ∈ M, Z m ¯ = st-lim δ δ→0
∞
e−δs σs (m)ds = ω(m)
0
or since Ω is separating ∀ ε > 0 ∃ δ0 such that
Z ∞
−sδ
δ dsσs (m)e − ω(m) |Ωi
<ε 0
∀ δ < δ0 .
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H. NARNHOFER and W. THIRRING
Proof. The above vector can be estimated by Z ∞ 0 2 2 dsds0 e−δ(s+s ) hΩ|(m∗ − ω(m∗ ))σs−s0 (m − ω(m))|Ωi k·k = δ 0
2
=
δ 2
δ = 2
Z
∞
−∞
Z
t
dthΩ|(m∗ − ω(m∗ ))σt (m − ω(m))|Ωi
Z
∞
due−δu
|t|
dte−δt hΩ|(m∗ − ω(m∗ ))σt (m − ω(m))
0
+ σt (m∗ − ω(m∗ ))(m − ω(m))|Ωi → 0 since the time average of (m − ω(m)) converges weakly to zero. Averaging with σs smoothens the action of this endomorphism and these smooth ¯ elements are strongly dense in M. Some cluster properties are discussed in [5]. They are not as strong as the classical multi-K clustering [3] due to the effects of noncommutativity. An improvement to control multi-clustering is based on the following observation: Proposition 4.4. The set of operators Z ∞ dsf (s)σs (m) with f ≥ 0 , RM ¯ := mf := 0
¯ , f = F 0 , F (0) = −1, kF k1, kF 00 k1 < ∞, m ∈ M ¯ In R σ is norm continuous and [kG, mf ]k < kF 00 k1 ∀ mf ∈ is strongly dense in M. ¯. RM ¯ . Similarly RN ¯ is dense in N ¯ ε > 0 ∃ s0 such that k(m − Proof. σs is strongly continuous thus ∀ m ∈ M, σs (m)|Ωik < ε ∀ 0 ≤ s ≤ s0 . Taking f ∈ C0∞ , supp f ⊂ [0, s0 ] we arrive at the first claim. Since Z ∞ ds0 |f (s0 ) − f (s0 + s)| → 0 for s → 0 kmf − σs mf k ≤ 0
for these f ’s and
Z
k[G, mf ]k =
0
also the other properties hold.
∞
0 dsf (s)σs (a)
≤ kf k1 0
R∞ Corollary 4.5. Elements of the form 0 dsh(s)σs (m) with h = H 0 , H(0) = 0, ¯ ω(m) = 0}. kHk1 , kH 00 k1 < ∞ are strongly dense in {m ∈ M,
455
EQUIVALENCE OF MODULAR K AND ANOSOV DYNAMICAL SYSTEMS
¯ we know from (4.2) that ∃ f = F 0 , F (0) = −1, kF k1 , Proof. ∀ ε > 0, m ∈ M 00 kF k1 < ∞ such that
Z ∞
m− dsf (s)σs (m) |Ωi
< ε/2 . 0
Furthermore (4.1) tells that ∃ δ such that
Z ∞
−sδ
ω(m) − dsδe σs (m) |Ωi
< ε/2 . 0
Adding we get with h(s) = f (s) − δe−sδ
Z ∞
m − ω(m) − ds(h(s)σs (m)) |Ωi
<ε 0
and H(s) = F (s) + e−sδ satisfies the specifications. Finally we appeal to the fact that for bounded sequences strong convergence on |Ωi implies strong convergence on all of Hω . ¯ τ ) clusters Using these results we can show how the dynamical system (M, exponentially fast. Theorem 4.6. For the dense set of elements m1 , m2 of the form (4.4) and m of the form (4.5) we have the bound |hΩ|m1 τt (m)m2 |Ωi − ω(m)ω(m1 m2 )| ≤ e−t kHk1 (kF100 k1 + kF200 k1 )
∀t>0
(where the m’s are taken of norm 1). Proof. Upon replacing m by m − ω(m) we may use (4.5) and proceed with ω(m) = 0, F (0) = 0. Then in view of (1.1(i)) the l.h.s. in (4.6) equals Z ∞ dsh(s)(σe−t s τt (m))m2 |Ωi hΩ|m1 0
and using h = H 0 and partial integration with H(0) = H(∞) = 0 it becomes Z ∞ −t e hΩ|m1 dsH(s)σe−t s τt (m), G m2 |Ωi . 0
Now kGmi |Ωik = k[G, mi ]|Ωi ≤ kFi00 k1 by (4.2) and
Z ∞
dsH(s)σe−t s τt (m)
≤ kHk1 .
0
5. Realizations We start with classical flows. The prototype of an Anosov system is given by repulsive harmonic forces
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H=
1 2 (p − x2 ) , 2
G = p − x,
(H corresponds to τM , G to σ)
τ t (x, p) = (x cosh t + p sinh t, p cosh t + x sinh t) =: (x(t), p(t))
(5.1)
σ s (x, p) = (x + s, p + s) . This situation can be looked at with many coordinate systems, for us the most convenient is to introduce coordinates in the stretching and shrinking directions X= t
x+p , 2
P = p − x, t
−t
τ (X, P ) = (Xe , P e ) ,
H = XP ,
G=P, (5.2)
s
σ (X, P ) = (X + s, P ) .
So here τ and σ become point transformations and do not mix X and P . The algebraic realization which appeals most to our geometrical intuition is by functions from L2 (R2 ) such that τ t f (X, P ) = f (X(t), P (t)). To construct the minimal Anosov subsystems according to (3.1) we start with small compact regions around (χ, K) in phase space. [χ, χ + ε] × [K, K + ε0 ] and a are the functions supported in this region. Since the only regions invariant under τ t , t ∈ R and σ s , s > 0 is R+ × R± we aim at M = L∞ (R+ × R+ ) = limt→∞ Nt hence we have to take K = 0. Furthermore N = σ 1 M = L∞ ([1, ∞] × R+ ) so we need χ = 1, ε0 = ∞. Then as,t = L∞ ([et + s, et (1 + ε) + s] × R+ ) , a+ = L∞ ([1, ∞] × R+ ) = N ,
a− = L∞ (R+ × R+ ) = M = a− ∨ a+ .
(5.3)
Note that N and M do not depend on ε and are in some sense minimal. For a non-abelian realization the Weyl algebra generated by W (α, β) = i(αX+βP ) e first comes to mind. Transfering the above relations one sees that it does not work, one ends up with N = M. The failure has the simple origin that the τ -dynamics on W does not admit KMS states [18]. Here we see another obstacle for the existence. σ s is inner (generated by W (0, s)) and thus strongly continuous in any representation where P exists. This implies that a τ -invariant state ω is also σ-invariant: t t→−∞ ω ◦ σ s = ω ◦ τ t ◦ σ s = ω ◦ σ e s τ t −→ ω . But a τ -KMS state can be only σ-invariant if [σ, τ ] = 0 which is not the case. After this attempt we turn to second quantization with fermi statistics. We shall denote with A(O) the CAR algebra formed with f, g ∈ L2 (O), [a∗f , ag ]+ = hf |gi ,
O a region in some Rn .
A has quasifree automorphisms, af → aeiHt f , H some selfadjoint operator in L2 (O). If H generates a point transformation x → x(t), then s ∂x(t) iHt f (x(t)) . (e f )(x) = ∂x
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If H generates a general canonical transformation we can use for H the corresponding Schr¨ odinger operator or can stay with our geometrical intuition and work in phase space. A canonical flow (x, p) → (x(t), p(t)) generates the unitary map f (x, p) → f (x(t), p(t)) with Jacobian 1 if it is canonical. In this way we realize the abelian picture by replacing L∞ by L2 . (a) Ordinary fermions With the transformations (5.2) in configuration space we start again with a local algebra a = A([1, 1 + ε]) to get at = A([et , et (1 + ε)] ,
N = a+ = A([1, ∞]) ,
M = A(R+ ) ,
a− = A([0, 1 + ε]) ,
N = z1 .
(b) “Classical” fermions The same with the replacement [1, 1 + ε] → [1, 1 + ε] × R+ . Remarks 5.1. 1. All our flows leave the region G > 0 invariant and we can introduce the canonical pair x = H, p = ln G since {H, G} = G ⇔ {x, p} = 1. The ensuing canonical flow is identical with the one in group space using t and s as coordinates: (t, s) ◦ (t0 , s0 ) = (t + t0 , s + e−t s0 ). The minimal realizations of the Anosov group by flows in a 2-dimensional symplectic manifold are essentially unique since they differ only by the choice of coordinates. The same is true for the “classical” fermions since they contain the same information. Quantum fermions depend on the choice of a one-dimensional submanifold as configuration space and they are not all isomorphic. Had we called p the coordinate of configuration space we had arrived at M = N and thus at the trivial case of Lyapunov exponent zero. 2. For A(R) there is the dilemma that σ acts even as automorphism and thus there should be no τ -KMS states since [τ, σ] 6= 0. On the other hand, we know that generally for quasifree automorphisms of CAR there are KMS states and here in particular 1 ∗ g , D = 1 (xp + px) , ω(af ag ) = f D 1+e 2 would do. The answer to this puzzle is that in πω σ s is not strongly continuous (and thus not Anosov in the sense of (1.1)). This follows since ω is obviously not σinvariant and according to the previous argument σ s cannot be strongly continuous. We shall soon see this more in detail. Finally we shall exhibit an expression for the τ -KMS state ω over A(R+ ). It is quasifree and thus determined by the 2-point function of the operator-valued R∞ distributions a(x) underlying af = 0 dxf ∗ (x)a(x), ω(a∗ (x)a(x0 )) = lim
1/2π , ε + i(x − x0 )
ω(a(x0 )a∗ (x)) = lim
1/2π . ε − i(x − x0 )
ε↓0
ε↓0
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We have ω(a∗ (x)τ −t a(x0 )) =
e−t/2 /2π et/2 /2π = ω(τ t a∗ (x)a(x0 )) = −t 0 ε + i(x − e x ) ε + i(et x − x0 )
since x, x0 , ε > 0 these expressions are analytic for −π < Im t < 0 and the KMS condition can be verified. We can also construct the τ -KMS state ω over A(R). Here we have only to notice that D commutes with the projection on R+ , Θ+ so that 1 − + Θ f Θ g = 0. D 1+e Therefore the KMS state is given by the operator valued distribution ω(a∗ (x)a(x0 )) = Θ+ (x)Θ+ (x0 ) lim ε↓0
1/2π 1/2π + Θ− (x)Θ− (x0 ) lim . ε↓0 ε + i(x0 − x) ε + i(x − x0 ) (5.4)
This demonstrates the discontinuity at x → 0 and shows that it is impossible to extend the shift automorphism on A(R) to the weak closure. At the same time we can explicitly realize the endomorphisms σ+ and σ− on π(A(R))00 : In the GNS representation the commutant of π(A(R))00 is again a representation of the CAR algebra. Therefore B(H) = π(A(R+ ))00 ∨ π(A(R+ ))0 ⊗ π(A(R− ))00 ∨ π(A(R− ))0
(5.5)
after a Bogolyubov transformation between π(A(R+ ))00 and π(A(R+ ))0 resp. between π(A(R− ))00 and π(A(R− ))0 can be naturally identified as the tensor product of two vacuum representations of the CAR algebra π0 (C(R)) or π0 (D(R), i.e. B(H) = π0 (C(R))00 ⊗ π0 (D(R))00 .
(5.6)
The endomorphisms σ+ and σ− act in π0 (C(R))00 and in π0 (DR))00 without mixing them. But as the shift automorphism they again act geometrically. Take N = π(A[1, ∞))00 ⊂ π(A[0, ∞))00 ⊂ π(C(R))00 , L = N 0 ∩ M = π(A(0, 1))00even . Then the automorphisms on B(H) (resp. endomorphisms on the subalgebras) are (we ignore the necessary normalization and concentrate on the geometry) τ t a(x) = a(et x) , s a(x) = a(x + s) , σ+ M = N , σ+ x 1 s =a , σ− a(x) = a 1/x + s 1 + sx
(5.7) σ− Meven = L .
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For t ∈ R and s ∈ R+ A(R+ ) is mapped into A(R+ ). For s < 0 we may be thrown into A(R+ )0 which we indicate by a negative argument. Now we can verify (3.12) −1 ◦ σ+ gives a bijection between π(A(R+ ))00 and explicitly by showing that σ+ ◦ σ− + 0 π(A(R )) : 1 x+1 −1 −1 =a − (5.8) σ+ σ− σ+ a(x) = σ+ σ− a(x + 1) = σ+ a − x x −1 so that σ+ σ− σ+ (C(R+ )) = C(R− ) = π(A(R+ ))0 .
¯ (defined Remark 5.2. In the realizations we constructed, the nontriviality of M in (3.10)) is satisfied. We did not find one where it is not. For us it is an open problem whether in Theorem 1.4(ii) this assumption has to be added. References [1] D. V. Anosov, Proc. Inst., Steklov 90 (1967) 1. [2] V. I. Arnold and A. Avez, Probl`emes Ergodiques de la M´ ecanique Classique, Gauthier– Villars, Paris, 1967. [3] I. P. Cornfeld, S. V. Fomin and Ta. G. Sinai, Ergodic Theory, Springer, New York, 1982. [4] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. [5] G. G. Emch, H. Narnhofer, G. L. Sewell and W. Thirring, J. Math. Phys. 35(11) (1994) 5582. [6] G. G. Emch, Commun. Math. Phys. 49 (1976) 191. [7] H. Narnhofer and W. Thirring, Lett. Math. Phys. 20 (1990) 231. [8] H. W. Wiesbrock, Commun. Math. Phys. 157 (1993) 83. [9] H. W. Wiesbrock, Commun. Math. Phys. 184 (1997) 683. [10] H. J. Borchers, Commun. Math. Phys. 143 (1992) 315. [11] M. Takesaki, Theory of Operator Algebras I, Springer, New York, 1979. [12] V. S. Sunder, An Invitation to Von Neumann Algebras, Springer, New York, 1986. [13] H. Reeh and S. Schlieder, Nuovo Cimento 22 (1961) 1051. [14] H. J. Borchers and J. Yngvason, “Modular groups of quantum fields in thermal states”, to be published in J. Math. Phys., 1998. [15] S. Doplicher and R. Longo, Invent. Math. 75 (1984) 493. [16] R. Longo, Adv. in Math. 63 (1987) 152. [17] H. W. Wiesbrock, Lett. Math. Phys. 39 (1997) 203. [18] H. Narnhofer and W. Thirring, Lett. Math. Phys. 27 (1993) 133.
LOOKING BEYOND THE THERMAL HORIZON: HIDDEN SYMMETRIES IN CHIRAL MODELS B. SCHROER and H.-W. WIESBROCK FB Physik der FU Berlin Arnimallee 14, 14193 Berlin, Germany E-mail:
[email protected] Received 4 January 1999 In thermal states of chiral theories, as recently investigated by H.-J. Borchers and J. Yngvason, there exists a rich group of hidden symmetries. Here we show that this leads to a radical converse of of the Hawking–Unruh observation in the following sense. The algebraic commutant of the algebra associated with a (heat bath) thermal chiral system can be used to reprocess the thermal system into a ground state system on a larger algebra with a larger localization space-time. This happens in such a way that the original system appears as a kind of generalized Unruh restriction of the ground state sytem and the thermal commutant as being transmutated into newly created “virgin space-time region” behind a horizon. The related concepts of a “chiral conformal core” and the possibility of a “blow-up” of the latter suggest interesting ideas on localization of degrees of freedom with possible repercussion on how to define quantum entropy of localized matter content in Local Quantum Physics.
1. Introduction In a previous paper we defined and illustrated the concept of a “hidden symmetry” [12]. We remind the reader that we use the terminology “hidden” for such symmetries of local quantum physics (LQP) which cannot be seen and understood by (Lagrangian-, functional integral-) quantization. They simply do not exist on a classical level nor in a euclidean functional integral representation, and hence cannot be transposed into the noncommutative realm e.g. by adapting (via quantizationa ) Noether’s theorem. Rather, one needs the power of the modular theory, which is totally characteristic of the noncommutative aspects of LQP (Local Quantum Physics), and which in addition to the classical space-time diffeomorphisms (Poincar´e- or Conformal-automorphisms) of Minkowski space-time also reveals a host of non-Noetherian “Hidden Symmetries” of modular origin. The latter escape the quantization approach to LQP (see the previous footnote), whereas the modular approach to LQP shows both kinds of symmetries; in fact the hidden symmetries may be considered in some sense as being the generalization of the diffeomorphisms of chiral conformal theory to higher dimensions [12]. In higher a If a quantization approach could lead to nonperturbative mathematically controllable real time local quantum physics, then these symmetries would show up in modular properties of the local algebras generated by the fields. But as everybody should know, we are (despite intense efforts for more than 40 years) still far away from such a goal, and it appears presently much more plausible that this will be achieved by modular methods [13] instead of quantization.
461 Reviews in Mathematical Physics, Vol. 12, No. 3 (2000) 461–473 c World Scientific Publishing Company
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dimensions the principles of LQP do not permit symmetries which represent diffeomorphism beyond the global symmetry of the (possibly curved) spacetime; rather there exists an infinite group of hidden symmetries of modular origin which extend the finite parametric diffeomorphism groups [Poincar´e, conformal, (anti) de Sitter, etc.] There are two mechanisms which reveal hidden structures: the modular theories of (possibly multiply) localized double cone algebras together with suitably chosen states, and the symmetries constructed from modular inclusions and modular intersections. The first mechanism might lead to a “fuzzy” action of the modular group for which the double cone situation for the free massive theory is the simplest illustration. It is believed that such modular actions become asymptotically pointlike near the horizon (boundary) of the localization region [12]. We will use the word hidden in this paper exclusively for the second inclusion/intersection mechanism which leads to (partially) geometric transformations of localization regions (i.e. of the net) which cannot be implemented as point transformations. For details see below and [12]. Modular inclusions/intersections of von Neumann algebras are significantly different from the well-studied V. Jones inclusions (subfactors). Whereas by definition the latter have conditional expectations onto the smaller algebra (“shallow” inclusions), the former, as a result of their modular structure, cannot have conditional expectations (“deep” inclusions). In passing we mention that there is a third type of (deep) inclusion which also has no conditional expectation but is not modular; it is the so-called split inclusion which is related to a strong form of statistical independence in LQP [15]. It is well known that the inclusions of the V. Jones subfactor theory lead to vast generalizations of compact group symmetry [7]. On the other hand modular inclusions/intersections have only been around for some years and very little is known about their consequences except that noncompact groups (as the two parametric translation/dilation group as well as SL(2, R)) emerge already from the simplest examples. Although modular (hsm) standard inclusions are known to be in one-toone correspondence to strongly additive chiral quantum fields (and thereby there are good reasons for expecting to be able to classify them), no results and even no efforts in this direction is known today. These groups act in a natural way on the two original algebras and their intersection and generate a space-time indexed net. The considerations in this paper lead to an increase in knowledge about modular inclusions/intersections. In our previous work we commented among other things on a very revealing chiral illustration of such a situation which was found by Borchers and Yngvason [2]. In this note we use their example as a starting point for a further going analysis of hidden symmetries in low-dimensional theories. As was shown by those authors, it might happen that the modular group acts only geometrically on a certain subregion. We show that in their case this group acts geometrically in a much larger region! Moreover there is a much richer “hidden symmetry” behind. In fact it turns out that in addition to the hidden symmetry
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generated by the two-parametric translation/dilation group [2] (with the dilation part being hidden), there is a hidden SL(2, R) M¨ obius symmetry with a differently localized net which is partially local relative to the old thermal net. In fact the thermal “shadow world” associated with a heat bath state becomes converted into a piece of real space-time which is to be localized behind the horizon! This is an inversion from the Hawking–Unruh situation in a two-fold sense: not only is the thermalization through localizing (Unruh) horizons undone and the thermal region returned as a piece of Minkowski space to the vacuum world (in analogy to the Unruh effect [14]), but even the “shadow world” (the apparently non-geometric thermal commutant) of a standard heat-bath thermal situation becomes converted into “virgin space-time” behind the horizon, thus giving additional weight to some speculative remarks of Jaekel [11]. Although we can presently illustrate this idea only in the chiral light ray setting, we consider the potential possibility of a space time interpretation of heat bath temperature, and a still unknown but expected ensuing conversion of a (still missing) notion of quantum entropy as sufficiently intriguing in order to justify publication. Using our recent ideas on the modular origin of the chiral diffeomorphism group which we illustrated with some computations on the Weyl-algebra [12], it could be interesting to extend also these ideas about a modular origin of chiral diffeomorphism groups to the thermal setting. Leaving this for future investigations, we make some remarks on the existence of hidden SL(2, R) symmetries in d = 1 + 1 massive theories and we conclude with some comments on higher dimensional theories. In all cases the modular method is used to extend the hidden action of translation/dilation subgroup [12] to a hidden SL(2, R) action. 2. Beyond the Borchers Yngvason Results Consider a chiral quantum field theory given by a local net of observables. The Bisognano–Wichmann property holds for the net, as it was in general shown by [3, 6]. For simplicity we also assume strong additivity or equivalently, Haag duality on the line, see [8]. This can always be achieved by passing to the dual net. With Borchers and Yngvason we take a thermal state on the net w.r.t. the lightlike translations. Let N denote the observable algebra to the half-line [0, ∞[, then Borchers and Yngvason showed that the modular group of N w.r.t. the thermal state transforms local algebras on the half-line geometrically (see below). The global thermal algebra is known to be of hyperfinite von Neumann type III1 and hence inequivalent to a ground state representation. This is the prerequisite for an Unruhlike interpretation in terms of a ground state representation thermalized by local restriction. We will show that this group also acts geometrically on the whole line and that even more, there is a “hidden” conformal group SU(2, R)/Z2 acting geometrically on intervals, but not pointwise in the sense of spatial automorphisms. For describing our observation, let us first recall the setting of Borchers and Yngvason. Denote by A the quasilocal algebra of the chiral model, by Ω+ the GNSvector to the thermal state w.r.t. the unitary group U (a) of lightlike translations
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e the C ∗ -algebra of observables localized in [0, ∞[. Let J0 be the CPTand by N symmetry of our model. Then we easily have by strong additivity [20] e = A. e J0 ∨ N J0 N
(1)
Let us pass to the weak closure of these algebras w.r.t. the thermal state, denoted by M and N . Under these circumstances U is the modular group associated with (M, Ω+ ). By the very definitions [19] we have (N ⊂ M, Ω+ ) is a -hsm inclusion. e J0 ⊂ N 00 ∩ M, and it is easy to see that therefore (N ⊂ M, Ω+ ) is a Now J0 N standard -hsm inclusion. But we even have M = N ∨ (N 0 ∩ M), see [8]. Now in general, we have due to Borchers result [1, 19] [J0 , U (a)] = 0 ,
(2)
which easily implies that A 7→ hΩ+ , J0 A∗ J0 Ω+ i ,
A∈A
(3)
is again a thermal state of the chiral theory w.r.t. the lightlike translations. The absence of phase transitions in chiral theories implies that we can anti-unitarily e J0 = N 0 ∩ A implement the CPT-operator. Then we immediately get with J0 N J0 N J0 = N 0 ∩ M .
(4)
This implies that the relative commutant of the observable algebra to a half-line is the algebra to the opposite half-line. In case of a -hsm standard inclusion we can always construct out of these data a chiral vacuum net [20, 8]. Therefore, within the above thermal situation we encounter a “hidden” vacuum theory. For this, let us sketch the general construction in the context of this particular case. One starts by associating the algebra M to the upper half-circle. Then exploiting the fact that the modular groups of M, obius group, N , and M ∩ N 0 generate a positive energy representation of the M¨ we implement the covariance of the algebra system by defining the various local algebras by mapping M according to the modular representation of the space-time symmetry. In this way we generate a M¨obius-covariant net. Let us now compare both theories from a more conceptual point of view. First, they have the weak closure M of the quasilocal algebra A in common. Further, the modular theory U (a) to that algebra w.r.t. Ω+ acts geometrically in both systems, namely as lightlike translations in the thermal case, and as half-line dilations in the vacuum case. Secondly, they have the observable algebras N and N 0 ∩ M in common, which in the thermal case are interpreted as the algebras to the two half-lines beginning at origin, whereas in the vacuum case they are associated to [1, ∞[, and [0, 1]. Now we notice that the physical symmetry in the thermal case is given by the lightlike translations, whereas in the vacuum theory we have a natural conformal symmetry SL(2, R)/Z2 behind, where the lightlike translations of the former theory are equal to the dilations in the latter.
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Let us exploit the common structures in order to compare the localizations in both theories. For this we denote the local algebras by Ath (a, b) if we refer to the thermal theory, A0 (a, b) in the vacuum case. So we can rephrase the above remark by (5) Ath (0, ∞) = N = A0 (1, ∞) , Ath (−∞, 0) = N 0 ∩ M = A0 (0, 1) and U (2πt) = ∆it M where we set the inverse temperature to be 2π. (The necessary modifications for the general case is left to the interested reader.) Then we have Ath (a, ∞) = Ad U (a/2π)(Ath (0, ∞)) = Ad U (a/2π)(N ) = A0 (e2πa , ∞)
(6)
and similarly Ath (−∞, a) = A0 (0, e2πa ) .
(7)
Now, by the very definition we have A0 (e2πa , e2πb ) = A0 (0, e2πb ) ∩ A0 (e2πa , ∞)
(8)
Ath (a, b) = Ath (−∞, b) ∩ Ath (a, ∞)
(9)
and similarly holds due to the strong additivity of the thermal net. Therefore we conclude Ath (a, b) = Ath (−∞, b) ∩ Ath (a, ∞) = A0 (0, e2πb ) ∩ A0 (e2πa , ∞) = A0 (e2πa , e2πb ) .
(10)
But now we know, by the very definition, that the conformal group Sl(2R)/Z2 acts geometrically on the vacuum theory. And due to the above relation we immediately get the action of that group on the thermal theory. Therefore, restricting on vacuum localization intervals in the half-line [0, ∞[ and to group actions mapping this interval into another one contained in the half-line, we can transfer this action onto a geometrical action in the thermal theory. Moreover we see, that there is a natural space-time structure also on the shadow world, i.e. on the thermal commutant to the quasilocal algebra on which this hidden symmetry naturally acts. In conclusion, we have encountered a rich hidden symmetry lying behind the tip of an iceberg, of which the tip was first seen by Borchers and Yngvason. Although we assumed the temperature to have the Hawking value β = 2π, the geometric modifications to the generic β values in [2] are easily done. By tensoring the two chiral theories one can obtain an isomorphic vacuum theory restricted to the forward light cone or the wedge. In fact using our previous result on the modular origin of the diffeomorphism for the special case of the Weyl-algebra [12], it seems plausible that there is an even much larger infinite dimensional symmetry of modular origin hidden in this thermal situation. Before we make some qualitative remarks about hidden SL(2, R) symmetries in the higher dimensional case, let us separately look at d = 1 + 1 massive theories.
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It is clear that in this case we should use the two modular inclusions which are obtained by sliding the (right-hand) wedge inside itself (M → Ma± ) by applying an upper/lower lightlike translation a± and forming the relative commutant (the size of a± is irrelevant) (11) M(I± ) ≡ M0a± ∩ M where the notation indicates that the localization of M(I± ) is thought of as the piece of the upper/lower light ray between the origin and the endpoint of the a± lightlike translation. By viewing this relative commutant as a lightlike limiting case of a spacetlike shift of W into itself (and using Haag duality), one obtains the interval I± as a limit of a double cone. The net obtained by applying the modular transformation ∆it M to the M(I± ) via its ad action is a chiral net with total algebra [ f± ≡ M ∆it (12) W (M(I± )) ⊂ M , t
f± , Ω) for the construction, where we used the hsm standard inclusion (M(I± ) ⊂ M see the Appendix below. Since this net is chiral, it cannot create the full space from the vacuum. Rather the cyclically generated space is a genuine subspace with f± , Ω) is obtained from restricting ∆it projector P± . Since the modular group of (M M to P± H, the projection is associated with a conditional expectation of the algebra f± . E± (M) = P± MP± = M
(13)
Although the two-dimensional conformal theory lives in the tensor-product space of the two chiral theories on the upper/lower light ray, the two chiral components constructed in the present way do not commute with each other unless one also performs the massless limit m → 0. Indeed since the upper/lower light ray have a relative ±time-like separation, one does not expect Huygens principle to become effective outside of m = 0. In fact a simple straightforward calculation on algebras generated by massive currents or energy-momentum tensor reveals that the relative commutator is proportional to a power of the mass but independent of the space-time coordinates. The mechanism can be illustrated in the simplest way by looking at the relative commutator of the 1-2 components of the Fermion field on the upper/lower light ray ¯ {ψ(x), ψ(y)} = (iγµ ∂ µ − m)i∆(x − y) i.e. {ψ1 (u)ψ2∗ (v)} ' ±m
(14)
where the last relation holds in the limit x, y → upper/lower light ray. The light ray reduction of the massive theory, although being a chiral conformal theory in its own right, can be transmuted into the original massive theory by adjoining the opposite light cone translation U− (a), i.e. by extending M (I+ ) to alg{M (I+ ), U− (a)} [13]. 3. Qualitative Remarks about Higher Dimensions In a previous paper we have shown how for d ≥ 1 + 2 one may use the situation of modular intersections to come to hidden symmetries [12]. In higher dimensions
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one expects that in general a conversion of thermal nets into spatially extended ground state nets, if possible at all, will involve SL(2, R)/Z2 symmetries of modular origin with positive energy translation operators, i.e. symmetries which (since a massive theory does not possess such pointwise acting symmetries) have hidden pieces already in their ground state realization. For this purpose we introduce the notion of a “conformal core” with the “blowup” property. Let us start with a d = 1 + 1 theory defined in terms of algebraic net data, i.e. a coherent map of space-time regions into von Neumann algebras. Such theories have two light-like translations U± (a) with positive generators. We use the right upper light ray translation U+ in order to construct a modular inclusion by sliding the standard wedge W at the origin into itself W → W+ ⊂ W using U+ (a = 1) [13]. Consider the modular inclusion defined in terms of the relative commutant of M+ ≡ U (1)A(W )U ∗ (1) in M = A(W ), (M0+ ∩ M ⊂ M, Ω) .
(15)
This inclusion leads to a full-fledged chiral conformal theory on the line (the light ray reduction) which generates from the vacuum its own Hilbert space which is a genuine subspace of the original space [ 0 ∆it A> ± ≡ W (M± ∩ M) ⊂ M ≡ A(W ) t
A> ± Ω = H± ⊂ H = MΩ
(16)
> A± = A> ± ∨ JA± J .
Note that the full conformal invariance is only realized on the reduced space H± . Since the modular theories of the combined upper/lower light ray reductions > A> + ∨ A− defines a 2-dim. net which lives on a possibly bigger Hilbert space than the reduced one and since this combined algebra has the same modular theory and the same light-like translations as M there exists according to Takesaki a conditional > expectation from M to A> + ∨ A− and the two algebras are either identical (iff the > cyclically generated Hilbert spaces are the same) or the A> + ∨ A− -algebra is obtain from M as a fixed point algebra under the action of an internal symmetry. A more efficient way to reconstruct the original massive net from its light ray reduction is the following “blow-up” of a light ray reduction, say A+ , by use of the opposite positive (a > 0) light-like translation, _ alg{A+ , U− (a)} . (17) B+ ≡ a>0 > Intuitively one expects that M, A> + ∨ A− and B+ , B− are all identical. Even more, based on analogies to the characteristic initial value problem in classical field theory one expects that apart from d = 1 + 1 conformal theory in all other cases > the algebra coalesce, i.e. M = A> + = A− = B+ = B− . This light ray reduction supplemented by the blow-up idea can be generalized to higher dimensions, where however it becomes more tricky. Let us explain the situation for d = 2 + 1, using the notation of [12]:
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U12,13 for the Galilean “translations” inside the Lorentz group, which in d = 2+1 exhausts the isotropy (“little”) group to l1 . Ul1 for the lightlike translations along l1 , A([l1 , l2 ]) for the wedge algebra to W [l1 , l2 ]. With this notational matter out of our way, we present now two interesting proposals for the definition of candidates for a “chiral conformal core” associated to a lightlike direction. The first one leads to a construction which by fiat is independent of the choice of the wedge and only depends on the light ray l1 . Start from Ad Ul1 (1)(A([l1 , l2 ]) ⊂ A([l1 , l2 ]) .
(18)
This defines a conformal theory, see Appendix. The resulting translations, namely U1 , commutes with the Galilean “translations”. We define: ∩λ∈R Ad Ul1 l2 ,l1 l3 (λ)(Ad Ul1 (1)(A([l1 , l2 ])) ⊂ A([l1 , l2 ]) .
(19)
This gives a modular standard inclusion in a canonical way, see Appendix. This proposal has the advantage to be covariant under the action of the isotropy: Ul1 l2 ,l1 l3 (µ) ∩λ∈R Ad Ul1 l2 ,l1 l3 (λ)(Ad Ul1 (1)(A([l1 , l2 ])) = Ad Ul1 (1)(∩λ∈R Ad Ul1 l2 ,l1 l3 (λ)Ul1 l2 ,l1 l3 (µ)((A([l1 , l2 ])))
(20)
⊂ Ul1 l2 ,l1 l3 (µ)(A([l1 , l2 ])) where the Galilean translation Ul1 l2 ,l1 l3 (µ) simply turns W [l1 , l2 ] into a W [l1 , l20 ]. The drawback of this construction could be that it is empty, as we will argue in the sequel. An equivalent construction based on the same intuition starts with the modular inclusion (21) A([l1 , l2 ]) ∩ A([l1 , l3 ])) ⊂ A([l1 , l2 ]) and suggests to represent the algebra (19) as a limiting intersection algebra: lim Ad Ul1 l2 ,l1 l3 (λ)(A([l1 , l2 ]) ∩ A([l1 , l3 ]))
λ→∞
(22)
and this seems to be C · 1, a multiple of the identity. Let us therefore mention another possible construction which starts with the modular inclusion: Ad Ul1 (1)(A([l1 , l2 ]) ∩ A([l1 , l3 ])) ⊂ A([l1 , l2 ]) .
(23)
Again intuitively this might be interpreted as a conformal core to associated the light ray l1 . This definition depends on the lightlike directions, but it does so in a covariant way since Ul1 l2 ,l1 l3 (λ) commutes with Ul1 and therefore the action of the isotropy group is computable. Moreover this modular inclusion is only invariant under the lightlike translations Ul1 and not under the transversal translations. This procedure for d ≥ 2 + 1 is more similar to a “conformal scanning” of the original theory than to its “conformal holography”. The reduction of higher dimensional
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QFT to a family of simpler chiral conformal field theories offers a great potential for future studies of nonperturbative QFT. In a generic curved space-time situation with a bifurcated Killing horizon and in the absence of additional symmetries, one can still apply the methods of LQP [14] and associate a chiral conformal theory as was shown recently in a remarkable paper by Guido, Longo, Roberts and Verch [9]. In that case the conformal theory is not associated to a particular light ray, but rather is induced via a restriction of the theory onto the horizon. Before we apply the achieved results to our main theme, namely how to convert heat bath temperature into Unruh temperature (or how to pass from the heat bath “shadow world” of the thermal commutant into new space-time behind a horizon), we cannot resist to make some comments about a fascinating but speculative connection of the blow-up picture of the chiral core with another speculative problem which presently is attracting the attention of many theoreticians: the problem of quantum entropy of algebras with a horizon or (in the context of LQP) a “quantum localization entropy”. The blow-up picture tell us that in the case of the wedge geometry we can represent the original wedge algebra M by the algebra generated by the chiral conformal algebra augmented with certain symmetry generators which do the blowing up (principally longitudinal and transverse translations). One would expect that for the counting of degrees of freedom the Poincar´e generators do not contribute. If we now use our physical imagination in an attempt to understand this picture beyond the wedge also for the rotationally symmetric double cone (in Minkowski space), and take notice that if the higher dimensional theory would be conformally invariant (zero mass), then there is a well-known transformation which carries the previous wedge situation into the double cone, then we obtain a very attractive picture. The more convincing part is the blow-up representation of the higher dimensional conformal algebra in terms of a chiral core algebra on the surface (plus symmetry generators of the higher dimensional theory which do not participate in the degree of freedom counting). The second, less rigorous step, is the idea that the unknown “fuzzy” modular theory of the massive double cone algebra is asymptotically (near the horizon) equal to the geometric modular situation of the conformal double cone theory. However to convert this “holographic degree of freedom picture” into a Bekenstein–Hawking-like formula for entropy expressed in terms of data of the chiral conformal core theory (e.g. the structure constants of W -algebras) more needs to be done. There is finally the problem of defining what we mean by localization entropy in the chiral theory. The local algebras, whether chiral or higher dimensional, are known to be of von Neumann hyperfinite type III1 which would lead to diverging (undefined) von Neumann entropies. As physicists we would try to regularize the situation. Indeed there is a natural way to do this, which is related to the phase space behavior of QFT. The latter is different from the behavior obtained by the box quantization at fixed time in the sense that the density based on the correct localization concept is bigger (“nuclear”) than for the box case (finite) [10], a fact overlooked in most relativistic entropy discussions. Closely related to the nuclearity property is the so-called split property which suggests to define a kind of fuzzy localized type I algebra. This algebra has its support in the
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double cone plus a “collar” around it, so that there is a bigger double cone containing both which the given double cone and the collar [10]. A type I algebra, unlike hyperfinite type III1 , has no principle obstruction against the existence of a von Neumann entropy. Such an entropy takes into account the quantum “entaglement” between the inside of the smaller and the outside of the bigger double cone across the collar. With vanishing size of the collar the entropy diverges in a way which depends on spacetime dimensions. Filling the double cone with different kind of matter, the strength of divergence is expected to be the same but the leading coefficients vary. Note that these arguments, if they withstand furthergoing detailed analysis, would interpret the holographic behavior as a generic property of nonperturbative local quantum physics and not as a particular behavior of special global properties (topological field theories, gauge theories, etc.). This yet speculative scenario build on modular theory has some resemblance to Wald’s recent more conservative ideas on localized entropy [18] and less so with string theory; although the results (but certainly not the physical concepts) may be similar. The reader may find an earlier account of this picture on holographic reduction of degrees of freedom from the viewpoint of LQP in section of a book manuscript draft by one of the authors [16]; the present blow-up mechanism of chiral light ray reduction lends considerably more credibility to those earlier remarks. There is a history and a long list of references on light cone holography in black hole physics by a variety of methods which seems to be based on quite different ideas than those of this work [5]. We hope to return with more results to this interesting and conceptually important entropy problem. Returning now to the thermal theme of this paper, the qualitative idea of generalizing our discussion of the Borchers–Yngvason model to higher dimensions is quite simple: Do the construction of the previous section on a conformal core and then use the blow-up construction using the longitudinal and transversal translations once in the ground state and once in the thermal representation. Whether this really works in higher dimensional model cases, still remains to be seen. Finally we comment on the problem of spontaneously broken symmetries. It is well known that the change of temperature is often accompanied by a transition of phase related to a change in global symmetry. The possibility of a modular transmutation of a thermal into a ground state theory (with spatial extension behind a horizon) creates all kinds of curious consistency problems. Unfortunately the well studied chiral conformal theories do not allow for spontaneous symmetry breaking. The only exceptional case is the thermal collapse of supersymmetry [4]. Since the “collapse mechanism” originates from the impossibility of annihilating a faithful state (any thermal state is faithful) on a C ∗ -algebra by an antiderivation, it is independent of space-time dimensions and holds in particular in chiral theories. Although it does not serve as an illustration of spontaneous symmetry breaking, its thermal aspects are interesting in their own right. Assuming that our transmutation mechanism also works for general chiral models including supersymmetric ones, and confronting it with the collapse mechanism one finds a somewhat curious situation whose only resolution seems to be that the (Unruh) thermal restriction wrecks the action of the antiderivative on only one side of the
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horizon and in this way “explains” the violent collapse. Whether this results allow for a deeper understanding of supersymmetry or only adds to the growing suspicion that this symmetry is of an accidental nature [17], remains to be seen. 4. Concluding Remarks Although the fully pointwise geometric symmetries which are well known from the quantization approach to relativistic QFT are, with the exception of chiral conformal QFT, restricted to finite dimensional automorphisms as Poincar´e symmetry and (only for zero mass) conformal symmetry, the modular structure of LQP opens the gates for a vast variety of infinite dimensional groups. One may either vary the states and consider the modular groups generated by wedge- and double conealgebras or one may investigate the modular inclusions/intersections generated from wedges with respect to natural reference states of curved space-time or the vacuum state in case of Minkowski space-time. As we argued in [12], we expect in the first case to obtain an (infinite dimensional) “hidden” analogue of the chiral diffeomorphism group. This is based on the fact that the chiral diffeomorphism group can be built up from infinitely many “M¨ obius layers” by “lifting” the vacuum M¨ obius group with a covering transformation and by realizing that each such lifted M¨ obius groupc is the modular group of a suitable multilocal pair (A(I1 ) ∨ · · · ∨ A(In ), Ωn ). The “lifted vacua” Ωn are suitably chosen states such that the endpoints of the obius transformation and Ωn Ik , k = 1 . . . n are fixed points of the nth layer M¨ is its unique invariant state [12]. The higher dimensional analogues of this construction beyond the Poincar´e, respectively conformal group are expected to be fuzzy (nonlocal) independent of the chosen state; in fact according to a conjecture of Fredenhagen their infinitesimal action on test functions should be described by pseudo-differential operators which are at most asymptotically geometric near the horizon, i.e. the space-time border of the localization region). In the present paper we have studied the partially hidden symmetry groups associated with thermal modular inclusions in chiral models. Even though in this case we do find geometrical aspects, these symmetries are not implemented by pointlike transformations in the underlying space-time and hence are hidden in the quantization approach. In addition the concepts of “chiral core” and its “blow-up property” which led us to a holographic degrees of freedom picture and gave some nice ideas about quantum entropy is obtained by the extremely noncommutative modular theory. It is a generic property of local quantum theory which probably cannot be seen by quantization procedures. The modular method based on real time noncommutative LQP seems to develop into a viable alternative to quantization methods [13]. Appendix. Canonical construction which underlies the calculation of the chiral core Let (N ⊂ M, Ω) hsm with non trivial relative commutant. Then look at the subspace (N 0 ∩ M)Ω ⊂ H. The modular groups to N and M leave invariant this c These lifted M¨ oebius transformations Moebn are called “quasisymmetric” of order n in the mathematical literature.
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0 subspace: ∆it M maps N ∩ M into itself by hsm for say positive t. Look at the orthogonal complement of (N 0 ∩ M)Ω ⊂ H. This orthogonal complement is mapped into itself by ∆it M for positive t. Let ψ be in that subspace, then 0 hψ, ∆it M (N ∩ M)Ωi = 0 for t > 0 .
(A.1)
Analyticity in t then gives the vanishing for all t. Due to Takesaki theorem we can restrict M to (N 0 ∩ M)Ω using a conditional expectation to this subspace. Then (N 0 ∩ M) ⊂ M|(N 0 ∩M)Ω is an hsm on the subspace defined above. N also restricts to that subspace and this restriction is obviously in the relative commutant of (N 0 ∩ M) ⊂ M|(N 0 ∩M)Ω . Moreover using arguments as above it is easy to see that the restriction is cyclic w.r.t. Ω on this subspace. Therefore we arrive at a hsm standard inclusion (N |(N 0 ∩M)Ω ⊂ M|(N 0 ∩M)Ω , Ω) .
(A.2)
Acknowledgment One of the authors (B.S.) thanks J. Yngvason, D. Buchholz and R. Verch for a helpful correspondence. Note Added For a more detailed discussion of holography from a modular viewpoint we refer to [21]. Meanwhile the subject of the present paper has been taken up and extended in a recent paper by Borchers [22]. References [1] H.-J. Borchers, “The CPT-Theorem in two-dimensional theories of local observables”, Commun. Math. Phys. 143 (1992) 315. [2] H.-J. Borchers and J. Yngvason, “Modular groups of quantum fields in thermal states”, preprint, 1998. [3] R. Brunetti, D. Guido and R. Longo, Commun. Math. Phys. 156 (1993) 201. [4] D. Buchholz and I. Ojima, Nucl. Phys. B498 (1997) 228; D. Buchholz, “On the implementation of supersymmetry”, hep-th/9812179. [5] S. Carlip, “Black hole entropy from Conformal Field Theory in any dimension”, hep-th/9812013, and references therein. [6] J. Fr¨ ohlich and F. Gabbiani, Commun. Math. Phys. 155 (1993) 569. [7] F. M. Goodman, P. de la Harpe and V. Jones, Coxeter Graphs and Towers of Algebras, Springer Verlag, 1989. [8] D. Guido, R. Longo and H.-W. Wiesbrock, “Extensions of conformal nets and superselection structures”, Commun. Math. Phys. 192 (1998) 217. [9] D. Guido, R. Longo, J. E. Roberts and R. Verch, “A general framework for charged sectors in Quantum Field Theory on curved spacetimes”, in preparation. [10] R. Haag, Local Quantum Physics, Springer Verlag, 1992. [11] Ch. Jaekel, “Cluster estimates for modular structure”, hep 9804017.
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[12] B. Schroer and H.-W. Wiesbrock, “Modular theory and geometry”, math-ph/9809003, Rev. Math. Phys. 12 (2000) 139. [13] B. Schroer and H.-W. Wiesbrock, “Modular theory and interactions”, hep-th/9812251, Rev. Math. Phys. 12 (2000) 301. [14] G. Sewell, Ann. Phys. 141 (1982) 201; S. Summers and R. Verch, Lett. Math. Phys. 37 (1996) 145. [15] S. Summers, “On the statistical independence of algebras of observables”, Rev. Math. Phys. 2(2) (1992) 201. [16] B. Schroer, “Localization and nonperturbative local quantum physics”, hep-th/9805093. [17] B. Schroer, “QFT at the turn of the century: old principles with new concepts” (an essay on local quantum physics), hep-th/9810080. [18] R. M. Wald, “Gravitation, thermodynamics and Quantum Theory”, gr-qc/9901033. [19] H.-W. Wiesbrock, Commun. Math. Phys. 157 (1993) 83. [20] H.-W. Wiesbrock, Lett. Math. Phys. 31 (1994) 303. [21] B. Schroer, “New concepts in Partial physics from solution of an old problem”, hepth/9908021. [22] H. J. Borchers, “On thermal states of (1 + 1)-dimensional quantum systems”, Univ. of G¨ ottingen preprint, October 1999.
GEOMETRIC MODULAR ACTION AND SPACETIME SYMMETRY GROUPS DETLEV BUCHHOLZ Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen, Bunsenstr. 9, D-37073 G¨ ottingen, Germany
OLAF DREYER Department of Physics, Pennsylvania State University University Park, PA 16802, USA
MARTIN FLORIG and STEPHEN J. SUMMERS Department of Mathematics, University of Florida Gainesville, FL 32611, USA Received 1 June 1998 Revised 12 February 1999 A condition of geometric modular action is proposed as a selection principle for physically interesting states on general space-times. This condition is naturally associated with transformation groups of partially ordered sets and provides these groups with projective representations. Under suitable additional conditions, these groups induce groups of point transformations on these space-times, which may be interpreted as symmetry groups. The consequences of this condition are studied in detail in application to two concrete spacetimes — four-dimensional Minkowski and three-dimensional de Sitter spaces — for which it is shown how this condition characterizes the states invariant under the respective isometry group. An intriguing new algebraic characterization of vacuum states is given. In addition, the logical relations between the condition proposed in this paper and the condition of modular covariance, widely used in the literature, are completely illuminated.
Contents 1. 2. 3. 4.
Introduction Nets of Operator Algebras and Modular Transformation Groups Geometric Modular Action in Quantum Field Theory Geometric Modular Action Associated with Wedges in R4 4.1. Wedge transformations are induced by elements of the Poincar´ e group 4.2. Wedge transformations generate the proper Poincar´ e group 4.3. From wedge transformations back to the net: Locality, covariance and continuity 5. Geometric Action of Modular Groups and the Spectrum Condition 5.1. The modular spectrum condition 5.2. Geometric action of modular groups 5.3. Modular involutions versus modular groups 6. Geometric Modular Action and de Sitter Space 6.1. Wedge transformations in de Sitter space 6.2. Geometric modular action in de Sitter space and the de Sitter group 7. Summary and Further Remarks Appendix. Cohomology and the Poincar´e Group Note References 475 Reviews in Mathematical Physics, Vol. 12, No. 4 (2000) 475–560 c World Scientific Publishing Company
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1. Introduction In [9, 10], Bisognano and Wichmann showed that for finite-component quantum field theories satisfying the Wightman axioms the modular objects associated by Tomita–Takesaki theory to the vacuum state and local algebras in wedgelike regions in Minkowski space have geometrical interpretation. This fundamental insight has opened up a number of fascinating lines of research for algebraic quantum field theory. To better appreciate the ramifications of their result, it is important to realize that the modular objects of the Tomita–Takesaki theory are completely determined by the choice of physical state and algebra of observables. That these modular objects can also have geometrical and dynamical significance thus allows the conceptually important possibility of deriving geometrical and dynamical information from the latter physical data. For example, it has become possible to characterize physically distinguished states by the geometric action of the modular objects associated with suitably chosen local algebras. This approach was taken in [24] (cf. also [13]), where it was shown how the vacuum state on Minkowski space can be characterized by the action of the modular objects associated with wedge algebras and how the dynamics of the theory can be derived from the modular involutions. The present paper is in several respects a refinement and generalization of [24]. Another program which has grown out of Bisognano and Wichmann’s insight is the construction of nets of local algebras and representations of a group acting covariantly upon the net, starting from a state, a small number of algebras, and a suitable “geometric” action of the associated modular objects upon these algebras. This line was first addressed in [12] and [58]. The most complete results in this direction have been, on the one hand, the construction of conformally covariant nets of local algebras in two spacetime dimensions in [71, 73] and, on the other, of Poincar´e covariant nets in three spacetime dimensions in [75] (see also [15, 74]). Yet another closely related research program is the generation of unitary representations of spacetime symmetry groups by modular objects which are assumed to implement the action of subgroups of these symmetry groups upon a given net of algebras. This course of study using the unitary modular groups was also opened up by Borchers [12] and followed in [21, 22, 35, 36, 70], whereas the derivation of such representations from the modular involutions was initiated in [24]. This aspect we also generalize in this paper. Moreover, we shall clarify the relations between these two different approaches to geometric action of modular objects. For a more detailed review of the prior literature, see [14]. As explained in our first paper on the subject [24], a further interesting step is the derivation of spacetime symmetry groups from the underlying algebraic structure and the given state. By “space-time” we here mean some smooth manifold without a priori given metric or conformal structure. From our point of view, if a given net of observable algebras happens to be covariant under the action of a unitary representation of some group of point transformations of the underlying manifold, then these point transformations should be regarded as the isometries of a metric structure to be imposed upon the space-time. We mention in this context the papers
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[43, 77], in which the causal (i.e. conformal) metric structure of the space-time is derived from the states and algebras of observables, under certain conditions. It is the essential lesson of the present paper that the various goals mentioned above — the derivation of spacetime symmetry groups, the generation of corresponding unitary representations, and the characterization of physically distinguished states from the algebraic data — can all be accomplished in physically interesting examples by a Condition of Geometric Modular Action proposed in [24]. This fact sheds new light on the results mentioned above and poses some new and intriguing questions. We shall present this condition somewhat imprecisely in this introduction — further details will be given in the main text. Let W be a suitable collection of open subsets of a space-time M and {A(W )}W ∈W be a net of C ∗ -algebras indexed by W, each of which is a subalgebra of the C ∗ -algebra A. A state on A will be denoted by ω and the corresponding GNS representation of A will be signified by (H, π, Ω). For each W ∈ W the von Neumann algebra π(A(W ))00 will be denoted by R(W ). The modular involution associated to the pair (R(W ), Ω) will be represented by JW , while the modular group associated to the same pair will be written as {∆it W }t∈R . Condition of Geometric Modular Action. Given the structures indicated above, then the pair ({R(W )}W ∈W , ω) satisfies the Condition of Geometric Modular Action if the collection of algebras {R(W )}W ∈W is stable under the adjoint action of the modular involution JW associated with the pair (R(W ), Ω), for all W ∈ W. In other words, for every pair of regions W1 , W2 ∈ W there is some region W1 ◦ W2 ∈ W such that JW1 R(W2 )JW1 = R(W1 ◦ W2 ) .
(1.1)
This condition was initially motivated by a number of examples in Minkowski space-time in which modular objects have a geometric action implying the above condition (see [9, 10, 23, 29, 39]). We emphasize that this condition does not assume that the adjoint action of the modular involutions upon the net acts in the detailed manner of the cited examples — indeed, it is not even assumed that this action can be realized as a point transformation on the space-time. In fact, we imagine that there will be situations of physical interest in which this geometric action is not implemented by point transformations, but where this condition will still serve as a useful selection criterion. Note that this condition can be stated sensibly for arbitrary space-time, indeed for arbitrary topological space M. This enables us to propose this Condition of Geometric Modular Action as a criterion for selecting physically interesting states on general space-times. We anticipate that in some applications this condition will have to be weakened in evident ways. In particular, there are circumstances where only (even) products of modular involutions will act “geometrically” in this manner — here we think, for example, of the Rindler wedge [41]. We expect that also these weakened versions should select states of notable physical interest.
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We emphasize that our selection criterion is one for a state and not an entire folium. In particular, previously suggested criteria, such as the Hadamard condition [42] and the microlocal spectrum condition [56], are valid for an entire folium of states. Though these criteria are valuable, they beg the question of which state (or states) of the respective folium is to be regarded as fundamental, i.e. as a reference state. In Sec. 2 we shall state and study our Condition of Geometric Modular Action in a very general form, which will enable us to explicate more clearly how it selects an intriguing class of transformation groups on the index sets of nets of von Neumann algebras and supplies them with projective representations. Returning to the original situation of nets indexed by open subsets of a space-time M in Sec. 3, we explain how to choose a suitable family W depending only on the space-time itself and present some results of conceptual importance for our framework. There we also outline the program opened up by our framework — a program we carry out explicitly in two examples in Secs. 4 and 6. In Sec. 4 we shall illustrate the power of our condition by choosing M to be topological R4 and W to be the set of wedgelike regions in R4 . It will be shown that with a few additional assumptions — all expressible in terms of the state, the net of algebras and the associated modular involutions — the transformations induced upon the index set W by (1.1) are implemented by point transformations — in fact, by the proper Poincar´e group P+ . We obtain after a series of steps a representation of P+ which acts covariantly upon the net. Therefore, we have an algebraic characterization of Poincar´e invariant states on nets of algebras indexed by open subsets of R4 , which induce Poincar´e covariant representations of these nets. A more detailed overview of Sec. 4 may be found at its beginning. Yet another example is worked out in Sec. 6, where it is shown how similar results for the de Sitter group in three dimensions may be obtained with suitable choices of M and W. Continuing the development presented in Sec. 4, Sec. 5 harbors a discussion of how the spectrum condition can also be characterized in terms of the modular objects, which then leads to how to derive algebraic PCT and Spin & Statistics Theorems in our setting. We present a striking new algebraic characterization of vacuum states on Minkowski space in terms of quantities which have meaning for arbitrary space-times. This condition may prove to be useful as a criterion for “stability” for quantum states on general space-times. Moreover, we show that if the adjoint action of the modular groups associated to the wedge algebras in R4 leaves the set {R(W )}W ∈W invariant, then these modular groups satisfy modular covariance, and all of the results in Sec. 4 hold once again, along with either the positive or negative spectrum condition. We provide further details which clarify the relation between our condition and the widely-used condition of modular covariance. Finally, in Sec. 7 we collect some further comments and speculations. An overview of an earlier version of the results of this paper has appeared in [63]. In addition to the detailed proofs, most of which were suppressed in [63], the present paper contains somewhat more transparent arguments, as well as many additional or strengthened results.
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2. Nets of Operator Algebras and Modular Transformation Groups We begin the main text of this paper with a more abstract setting of our Condition of Geometric Modular Action, since then its connection with transformation groups on partially ordered sets and projective representations of these groups emerges particularly clearly. We shall return to the original situation with further precisions in the next section. Let {Ai }i∈I be a collection of C ∗ -algebras labeled by the elements of some index set I. If (I, ≤) is a directed set and the property of isotony holds, i.e. if for any i1 , i2 ∈ I such that i1 ≤ i2 one has Ai1 ⊂ Ai2 , then {Ai }i∈I is said to be a net. However, for our purposes it will suffice that (I, ≤) be only a partially ordered set and that {Ai }i∈I satisfies isotony. We are therefore working with two partially ordered sets, (I, ≤) and ({Ai }i∈I , ⊆), and we require that the assignment i 7→ Ai be an order-preserving bijection (i.e. it is an isomorphism in the structure class of partially ordered sets). We note that any such assignment which is not an isomorphism in this sense would involve some kind of redundancy in the description. In algebraic quantum field theory the index set I is usually a collection of open causally closed subsets of an appropriate metric space-time (M, g). In such a case the algebra Ai is interpreted as the C ∗ -algebra generated by all the observables measurable in the space-time region i. Hence, to different spacetime regions should correspond different algebras. If {Ai }i∈I is a net, then the inductive limit A of {Ai }i∈I exists and may be used as a reference algebra. However, even if {Ai }i∈I is not a net, it is still possible [32] to naturally embed the algebras Ai in a C ∗ -algebra A in such a way that the inclusion relations are preserved. In the following we need therefore not distinguish these two cases and refer, somewhat loosely, to any collection {Ai }i∈I of algebras, as specified, as a net. Any state on A restricts to a state on Ai , for each i ∈ I. For that reason, we shall speak of a state on A as being a state on the net {Ai }i∈I . A net automorphism is an automorphism α of the global algebra A such that there exists an order-preserving bijection α ˆ on I satisfying α(Ai ) = Aα(i) ˆ . Symmetries, whether dynamical or otherwise, are generally expressed in terms of net automorphisms (or antiautomorphisms) [57]. An internal symmetry of the net is represented by an automorphism α such that α(Ai ) = Ai for every i ∈ I, i.e. the corresponding order-preserving bijection α ˆ is just the identity on I. Given a state ω on the algebra A, one can consider the corresponding GNS representation (Hω , πω , Ω) and the von Neumann algebras Ri ≡ πω (Ai )00 , i ∈ I. We shall assume that the representation space Hω is separable. We extend the assumption of nonredundancy of indexing to the net {Ri }i∈I , i.e. we assume that also the map i 7→ Ri is an order-preserving bijection.a If the GNS vector Ω is cyclic and separating for each algebra Ri , i ∈ I, then from the modular theory of Tomita– Takesaki, we are presented with a collection {Ji }i∈I of modular involutions (and a collection {∆i }i∈I of modular operators), directly derivable from the state and the a This is automatically the case if the algebras A are von Neumann algebras and ω induces a i S faithful representation of i∈I Ai .
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algebras. This collection {Ji }i∈I of operators on Hω generates a group J , which becomes a topological group in the strong operator topology on B(Hω ), the algebra of all bounded operators on Hω . Note that JΩ = Ω for J ∈ J . In the following we shall denote the adjoint action of Ji upon the elements of the net {Ri }i∈I by ad Ji , i.e. ad Ji (Rj ) ≡ Ji Rj Ji = {Ji AJi |A ∈ Rj }. Note that if R1 ⊂ R2 , then one necessarily has ad Ji (R1 ) ⊂ ad Ji (R2 ), in other words the map ad Ji is order-preserving. Hence, the content of the Condition of Geometric Modular Action in this abstract setting is that each ad Ji is a net automorphism. Thus, for each i ∈ I, there is an order-preserving bijection (an automorphism) τi on I ((I, ≤)) such that Ji Rj Ji = Rτi (j) , j ∈ I. The group generated by the τi , i ∈ I, is denoted by T and forms a subgroup of the transformations on the index set I. For the convenience of the reader, we summarize our standing assumptions. Standing Assumptions. For the net {Ai }i∈I and the state ω on A we assume (i) i 7→ Ri is an order-preserving bijection; (ii) Ω is cyclic and separating for each algebra Ri , i ∈ I; (iii) each ad Ji leaves the set {Ri }i∈I invariant.b We collect some basic properties of the group T in the following lemma. Lemma 2.1. The group T defined above has the following properties. For each i ∈ I, τi2 = ι, where ι is the identity map on I. For every τ ∈ T one has τ τi τ −1 = ττ (i) . If τ (i) = i for some τ ∈ T and some i ∈ I, then τ τi = τi τ . One has τi (i) = i, for some i ∈ I, if and only if the algebra Ri is maximally abelian. If T acts transitively on I, then τi (i) = i, for some i ∈ I, if and only if τi (i) = i, for all i ∈ I. Moreover, if τi (i) = i for some i ∈ I, then i is an atom in (I, ≤), i.e. if j ∈ I and j ≤ i, then j = i. (5) If i ≤ j ≤ k ≤ l, then τi (j) ≥ τl (k).
(1) (2) (3) (4)
Proof. (1) The first assertion is immediate since Ji2 = 1, the identity operator on Hω , hence for each j ∈ I one has Rj = Ji Ji Rj Ji Ji = Ji Rτi (j) Ji = Rτi (τi (j)) . Standing Assumption (i) then yields τi2 = ι. (2) Since every element of J leaves Ω invariant, standard arguments in modular theory show that the basic assumption Ji Rj Ji = Rτi (j) implies the relation Ji Jj Ji = Jτi (j) . Therefore one has the equalities R(τi τj τi )(k) = Ji Jj Ji Rk Ji Jj Ji = Jτi (j) Rk Jτi (j) = Rττi (j) (k) , for every k ∈ I. Once again, the nonredundancy assumption yields the assertion τi τj τi = ττi (j) , for each i, j ∈ I. Since T is generated by the set {τi |i ∈ I}, this entails assertion (2). (3) This is an immediate consequence of the preceding result. b and is a fortiori a net automorphism.
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(4) If τi (i) = i for some i ∈ I, then one has R0i = Ji Ri Ji = Rτi (i) = Ri , so that Ri is abelian. Moreover, since Ω is cyclic for this abelian von Neumann algebra, it must be maximally abelian. If T acts transitively on I, then since the modular involutions are (anti)unitary, every Ri must be maximally abelian. On the other hand, if Ri is maximally abelian, one has Ri = R0i = Ji Ri Ji = Rτi (i) . Hence, by the nonredundancy assumption, one has τi (i) = i. It follows that if every algebra Rk is maximally abelian, then τk (k) = k for every k ∈ I. As already pointed out, under Standing Assumption (ii), any abelian Ri must be maximally abelian. Hence, if there exist i1 < i2 with Ri1 and Ri2 both abelian, then Ri1 ⊂ Ri2 , which yields Ri1 = Ri2 , since both algebras are maximally abelian. This would violate Standing Assumption (i). (5) If i ≤ j ≤ k ≤ l, then one observes that Ji Rj Ji ⊃ Ji Ri Ji = R0i ⊃ R0l = Jl Rl Jl ⊃ Jl Rk Jl implies τi (j) ≥ τl (k).
For index sets without atoms, such as the index set W used as an example in Sec. 4 (however, not the example used in Sec. 6), Lemma 2.1 (4) implies that Ri must be nonabelian for every i ∈ I. Certain aspects of Lemma 2.1 may be interpreted as follows: Given the set I, we consider functions τ : I → T , where T is some subgroup of the symmetric group on I. There exist two natural automorphisms on these functions. The first one is given by the adjoint action on T — namely, ad τ0 (τ )(·) = τ0 τ (·)τ0−1 for each τ0 ∈ T , and the second one is induced by the action of T on I: (τ ◦τ0 )(·) = τ (τ0 (·)). If, for a given function τ , these two actions coincide for all τ0 ∈ T , we say that τ is T -covariant. Note that the T -covariant functions form a group under pointwise multiplication, the identity being the constant function on I with value ι. A particularly interesting case arises if the range of a function τ generates T ; we then say that τ is a generating function. The preceding proposition thus shows that the condition of geometric modular action provides us with subgroups T of the symmetric group on I which admit an idempotent, T -covariant generating function. This is a rather strong consistency condition on T . For example, the full symmetric groups of index sets do not in general admit such functions. What is of interest here is the fact that the structure is fixed once the index set I is given. We feel it is useful to elaborate further the relation between the groups J and T . Recall that an operator Z ∈ J is said to be an internal symmetry of the net {Ri }i∈I , if ZRk Z −1 = Rk for all k ∈ I. Proposition 2.1. The surjective map ξ : J → T given by ξ(Ji1 · · · Jim ) = τi1 · · · τim ,
i1 , . . . , im ∈ I , m ∈ N ,
is a group homomorphism. Its kernel is a group Z of internal symmetries of the net {Ri }i∈I which is contained in the center of J .
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Proof. If Ji1 · · · Jim = Jj1 · · · Jjn , then one has Rτi1 ···τim (k) = Ji1 · · · Jim Rk Jim · · · Ji1 = Jj1 · · · Jjn Rk Jjn · · · Jj1 = Rτj1 ···τjn (k) , for all k ∈ I. Thus the equality τi1 · · · τim = τj1 · · · τjn follows. It is therefore clear that the map ξ is well-defined. Moreover, ξ(Ji1 · · · Jim )ξ(Jj1 · · · Jjn ) = τi1 · · · τim τj1 · · · τjn = ξ(Ji1 · · · Jim Jj1 · · · Jjn ) , and by Lemma 2.1 (1), it follows that ξ(Ji1 · · · Jim )−1 = τi−1 · · · τi−1 = τim · · · τi1 = ξ(Jim · · · Ji1 ) = ξ((Ji1 · · · Jim )−1 ) . m 1 Hence ξ is a group homomorphism. If ξ(Ji1 · · · Jim ) = ι, then the operator Z = Ji1 · · · Jim is an internal symmetry, by definition. It remains to be shown that the set Z of internal symmetries is contained in the center of J . But as argued before, since ZΩ = Ω and ZRi Z −1 = Ri , for all i ∈ I, it follows from standard arguments in modular theory (see Theorem 3.2.18 in [18]) that Z commutes with the modular involutions Ji , i ∈ I. But J is generated by these operators and Z is an element of J , so the proof of the statement is complete. This proposition may be reformulated as the assertion that there exists a short exact sequence ξ ı {1} → Z → J → T → {ι} , where ı denotes the natural identification map. In other words, J is a central extension of the group T by Z, a situation for which the mathematics has reached a certain maturity. It is an immediate consequence of the preceding that there exists an (anti)unitary projective representation of the group T on Hω by operators in J . For an arbitrary τ ∈ T there may be many ways of writing τ as a product of the elementary {τi |i ∈ Qn(τ ) I}. For each τ ∈ T choose some product τ = j=1 τij ; which choice one makes is irrelevant for our immediate purposes. Having made such a choice for each τ ∈ T , Qn(τ ) define J(τ ) ≡ j=1 Jij . Corollary 2.1. The above construction provides an (anti) unitary projective representation of T on Hω with coefficients in an abelian group Z of internal symmetries in the center of J . Moreover, one has J(τ )Ω = Ω, for all τ ∈ T , as well as ZΩ = {Ω}. Proof. Consider τ , τ 0 ∈ T and the corresponding (anti)unitary operators J(τ ), J(τ 0 ) and J(τ τ 0 ). If ξ : J → T is the group homomorphism established in Proposition 2.1, one has ξ(J(τ τ 0 )−1 J(τ )J(τ 0 )) = ι, and the initial assertion thus follows from that proposition. The final assertions are trivial, since the modular conjugations Ji leave Ω invariant.
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It is an interesting mathematical question which groups and corresponding representations can arise in this manner. As we shall see, both finite and continuous groups can be obtained with appropriate choices of net and index set. Before dealing with infinite groups, let us briefly discuss the finite case and consider a family {Ri |i = 1, . . . , n} of von Neumann algebras with a common cyclic and separating vector Ω such that the corresponding modular conjugations Ji leave this family invariant, i.e. Ji Rk Ji = Rτi (k) for i, k = 1, . . . , n. The maps τi are in this case permutations on the set I ≡ {1, . . . , n} which are also involutions. Hence, the group T is a subgroup of the symmetric group Sn which is generated by involutions — a Coxeter group. Here we shall only consider the case where T acts transitively upon the set I. If the algebras Ri are nonabelian, then it is clear that T cannot be primitive. This is because Ji Ri Ji = R0i , so that if Ri is not maximally abelian, one must have, by hypothesis, R0i = Ri0 for some i0 6= i. But, since from J(τ )Ri J(τ )−1 = R0i follows J(τ )R0i J(τ )−1 = Ri , the index pair (i, i0 ) is either transformed by the elements of T onto itself or onto a disjoint pair. In other words, (i, i0 ) is a set of imprimitivity of T . Since T is not primitive, it also is not 2-transitive (Satz II.1.9 in [40]). Moreover, since the magnitude of every set of imprimitivity in I must be a divisor of the magnitude of I, (see, e.g. Satz II.1.2 in [40]), the magnitude n of I is then necessarily even. Hence, if n is odd, then all the algebras must be maximally abelian (the converse is false). It is easy to compute explicitly the possible groups T which arise in this manner for small values of n. In the case n = 2 one clearly obtains S2 ; for n = 3 one finds as the only possibility the symmetric group S3 . (And one can give corresponding examples of states and algebras which yield S3 .) The case n = 4 is not possible for a family of nonabelian algebras, since then the mentioned sets of imprimitivity are stable under the action of the group T ; in other words, T cannot act transitively on I when n = 4. This list can be continued without great effort, but a complete classification of the finite groups T which can be obtained in this manner is yet an open problem. 3. Geometric Modular Action in Quantum Field Theory We turn now to the physically interesting case of nets on a space-time manifold (M, g). The index set I appearing in the abstract formulation of our Condition of Geometric Modular Action in the previous section will be denoted henceforth by W and will consist of certain open subsets W ⊂ M. A natural question is: For the given manifold (M, g), how should the index set of the net {A(W )}W ∈W of algebras be chosen so that any state on that net satisfying the Standing Assumptions of Sec. 2 yields a group T which can be identified with a subgroup of isometries of the space-time? Evidently, not every choice of such regions will be appropriate. One purpose of this section is to explain which considerations should be made when choosing W, once the underlying space-time has been fixed. After this is done, we specify in detail the technical assumptions which constitute our
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Condition of Geometric Modular Action, which was heuristically presented in the introduction. We emphasize that in this section the starting point is a smooth manifold M and that some target space-time (M, g) has already been fixed. In other words, we have in mind a particular metric structure on M for which we are looking. If one does not have a specific target, that is to say if one just has a net {A(R)}R∈R indexed by open subregions R of the manifold without any further clue to the metric structure on the manifold M, then, in principle, one would have to test the Condition of Geometric Modular Action for various states and for various subnets {A(W )}W ∈W of {A(R)}R∈R . If the Condition of Geometric Modular Action would hold for one of these, then the program outlined below in this section would be applicable to that state and subnet. As the Condition of Geometric Modular Action is to be an a priori criterion for a characterization of elementary states on (M, g), the set W should depend only on the space-time manifold (M, g). Moreover, it should be sufficiently large to express all desired features of nets on (M, g) such as locality, covariance (in the presence of spacetime symmetries), etc. On the other hand, it should be as small as possible in order to subsume a large class of theories (on the target space-time). In light of these requirements, it is natural to assume that W has the following properties. (a) For each W ∈ W the causal (spacelike) complement W 0 of W (i.e. the interior of the set of all points in M which cannot be connected with any point in the closure W of W by a causal curve) is also contained in W. It is convenient to require each W ∈ W to be causally closed, that is to say W = (W 0 )0 ≡ W 00 . Moreover, the collection W should be large enough to separate spacelike separated points in M. (b) The set W is stable under the action of the group of isometries (spacetime symmetries) of (M, g). The latter constraint is consistent with the idea that the Condition of Geometric Modular Action should characterize the most elementary states on (M, g) with the highest symmetry properties. We append to the preceding conditions another constraint of a topological nature. In order to motivate it, let us assume for a moment that the transformations τW , W ∈ W, on the index set W arising from a given net and state satisfying the Condition of Geometric Modular Action are induced by diffeomorphisms (or even just homeomorphisms) of M and together act transitively on W. This is only possible if all regions in W belong to the same homotopy class. We therefore assume the following additional condition. (c) All regions W ∈ W are contractible. Condition (c) excludes, for example, the appearance of double cones in W when (M, g) is asymptotically flat (such as Minkowski space), since their causal complements, which are to be elements of W by condition (a), are not contractible. But double cones would be admissible in space-times such as the Einstein universe. We shall call families W of open regions W ⊂ M satisfying (a)–(c) admissible.
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Given an admissible family W of regions, it may contain proper subfamilies W0 ⊂ W which are also admissible. One could then base the Condition of Geometric Modular Action on the subnet indexed by W0 , instead. It should be noticed that there may exist nets which satisfy our condition with respect to W but not for W0 . In other words, the subgroup T0 ⊂ T induced by the underlying modular involutions corresponding to W0 ∈ W0 may not be a stability group of W0 in certain cases. However, it seems plausible that there exists a larger class of theories (nets and states) satisfying the condition based on W0 than that based on W, since there are fewer constraints imposed on the nets in the former case. So from this point of view, it appears to be natural to select sets W which, heuristically speaking, are small. It is of interest in this context that for certain space-times (M, g) with large isometry groups, there exist distinguished families W which are generated by applying the isometry group to a single region W , which itself has a maximal stability group, (i.e. a group which cannot be extended to the stability group of some other region which is still a member of the admissible family). Identifying W with the collection of corresponding coset spaces, it is then meaningful to say that these families are minimal and thus very natural candidates for a concrete formulation of the Condition of Geometric Modular Action. We shall consider certain examples of this type in the subsequent sections. As was explained in the introduction, it is one of the aims of the Condition of Geometric Modular Action to distinguish, for any given net {A(W )}W ∈W of C ∗ algebras on the manifold M, states ω on the net which can be attributed to the most symmetric physical systems in the space-time (M, g). Fix an admissible family W of regions in (M, g) and consider the von Neumann algebras {R(W )}W ∈W associated to ({A(W )}W ∈W , ω) as before. We state our Condition of Geometric Modular Action (henceforth, CGMA) for this structure. Condition of Geometric Modular Action. Let W be an admissible family of open regions in the space-time (M, g), let {A(W )}W ∈W be a net of C ∗ -algebras indexed by W, and let ω be a state on {A(W )}W ∈W . The CGMA is fulfilled if the corresponding net {R(W )}W ∈W satisfies: (i) W 7→ R(W ) is an order-preserving bijection, (ii) for W1 , W2 ∈ W, if W1 ∩ W2 6= ∅, then Ω is cyclic and separating for R(W1 ) ∩ R(W2 ), (iii) for W1 , W2 ∈ W, if Ω is cyclic and separating for R(W1 ) ∩ R(W2 ), then W1 ∩ W2 6= ∅, and (iv) for each W ∈ W, the adjoint action of JW leaves the set {R(W )}W ∈W invariant. The somewhat curious lack of symmetry in conditions (ii) and (iii) is introduced in order to admit theories for which W1 ∩ W2 = ∅, but nonetheless the vector Ω is cyclic and separating for the intersection R(W1 ) ∩ R(W2 ). This can occur, for example, in certain massless models in Minkowski space, when W1 and W2
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are disjoint wedgelike regions but where W1 ∩ W2 contains an unbounded lowerdimensional set [68]. We would like to emphasize that this condition is to be viewed as a selection criterion for states of particular physical interest. We do not assert that every state of physical interest will satisfy this condition. We observe that its formulation does not require any specific structure of the net {A(W )}W ∈W such as local commutativity, existence of spacetime symmetries, and so forth. As a matter of fact, {A(W )}W ∈W could be a free net on the manifold M satisfying no other relations but isotony. The above assumptions (i)–(iv) imply the Standing Assumptions of Sec. 2, so that all the results from that section will be available to us. In particular, we have a group T of bijections acting on W. The corresponding maps τW on W have additional convenient properties. Proposition 3.1. Let W be an admissible family of open regions in the spacetime (M, g), and let {A(W )}W ∈W be a net of C ∗ -algebras indexed by W. If ω is a state on {A(W )}W ∈W such that the CGMA is satisfied, then the involutions τW : W → W, W ∈ W, satisfy the following conditions: W1 ∩ W2 = ∅
implies τW (W1 ) ∩ τW (W2 ) = ∅ ,
(3.1)
τW (W1 ) ⊂ τW (W2 ) ,
(3.2)
and W1 ⊂ W2
if and only if
with W1 , W2 ∈ W. Proof. Since each JW is antiunitary and leaves Ω invariant, it is evident that the set (R(W1 ) ∩ R(W2 ))Ω is dense if and only if the set JW (R(W1 ) ∩ R(W2 ))Ω = (JW R(W1 )JW ∩ JW R(W2 )JW )Ω = (R(τW (W1 )) ∩ R(τW (W2 )))Ω is dense. Hence (3.1) follows from (ii) and (iii). The assertion (3.2) is a consequence of (i). The lack of symmetry in conditions (ii) and (iii) above entails the lack of symmetry in (3.1). If the map τW were continuous in the obvious sense, then (3.1) would imply τW (W1 ) ∩ τW (W2 ) = ∅
if and only if
W1 ∩ W2 = ∅ .
(3.3)
For the two examples worked out in the present paper, it will be seen that in Minkowski space the maps on the index sets W do indeed satisfy (3.3). In de Sitter space, condition (iii) is trivial and will be supplemented by an algebraic condition yielding (3.3). Having thus fixed the framework in detail, there arises the interesting question: Which transformation groups T are associated with states fulfilling this criterion
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and how do they act on the corresponding nets? In particular, are they implemented by point transformations on the manifold M, and are these isometries of the spacetime (M, g)? A comprehensive answer to this question does not seem to be an easy problem, but there are some engaging facts of a quite general nature which we wish to explain. Let us first consider the question of whether the elements of T could be implemented by point transformations on M. If we knew from the outset that the maps τW also leave stable a larger net {R(O)}O∈= containing {R(W )}W ∈W and indexed by a base for the topology on M, we could rely upon an approach initiated by Araki [4] (building upon [6, 7]) and further developed by Keyl [43] in order to prove under certain conditions that the maps τW are induced by point transformations of M which generate a group G. If these maps also preserved the causal structure on (M, g) in the sense of τW (W0 )0 = τW (W00 ) ,
for all W0 ∈ W ,
(3.4)
for each W ∈ W,c then since the regions in W separate spacelike separated points, we could, for a significant class of space-times, appeal to the well-known result of Alexandrov [2, 3], (see also [8, 11, 48, 78]) and conclude that the group G is a subgroup of the conformal group of (M, g).d Moreover, as was shown in the preceding section, there exists an (anti)unitary projective representation of T (and thus of G) on the Hilbert space Hω . Well-known examples which nicely illustrate this scenario are conformal quantum field theories on compactified Minkowski space (see [21]). However, in order to admit a priori a larger class of space-times, we avoid the strong initial assumption that the adjoint action of the modular involutions {JW |W ∈ W} leaves the net {R(O)}O∈= , i.e. a net indexed by a base for the topology of the space-time as in [4, 6, 7, 43], invariant. In particular, the CGMA can obtain without the maps τW being induced by point transformations of M. In order to indicate what can occur, let us consider any decreasing net T { i Wi,n }n∈N which converges to some point x ∈ M. Because τW is orderT preserving, the images { i τW (Wi,n )}n∈N also form a decreasing net, and if the limit set is nonempty, it is straightforward to show that it consists of a single point (see [4]). But the net may have no limit for certain points x ∈ M. Hence, loosely speaking, our CGMA admits the possibility of singular point transformations which are not contained in the conformal group of (M, g) but which nonetheless preserve the causal structure.e This flexibility is actually very advantageous for our purposes, since the conformal group is rather small for certain space-times and thus not suitable for the characterization of elementary physical states. Hence, the CGMA may still be a useful selection criterion for physically interesting states even c It is of interest to note that we shall derive, not assume, (3.4) in our examples, hence deduce, not postulate, locality and Haag duality for wedge algebras. d Some of the details of the argument which would be involved here may be gleaned from the proofs presented in Subsec. 4.1. The basic ideas are sketched in Sec. 3 of [63]. e See also the example discussed at the end of Subsec. 4.1.
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in these cases, where the point transformation group G has very little indeed to say about the underlying space-time. We reemphasize: We anticipate that the CGMA will serve to select physically distinguished states even on space-times with trivial isometry groups. In such cases the group T will not be implemented by point transformations. In the examples worked out in this paper, T will, in fact, be shown to be implemented by point transformations — indeed, the proof of that assertion is one of the primary technical difficulties which had to be overcome in the work we present here. But the reader should understand that if the CGMA holds for a state on a space-time with a small isometry group, the group T can still be quite large — only a subgroup of T would be implemented by point transformations on the space-time. We conclude this section with a list of mathematical problems which naturally arise if one wants to use our principle of geometric modular action for the determination of the possible symmetry groups T and their action on nets for a given space-time (M, g). The first step is to pick an admissible family W of regions W ⊂ M. We do not have a general algorithm for the choice of W, but, as previously mentioned, there do exist space-times for which the family W is uniquely fixed by our general requirements. One then has to solve, step by step, each of the following problems. (1) Are the transformations on W satisfying the conditions (3.1) and (3.2) induced by (singular) point transformations on (M, g) (forming a group G)? (2) Which subgroups T of the symmetric group on W can appear? More precisely, which groups are generated by families {τW }W ∈W of such automorphisms for which τW1 τW2 τW1 = ττW1 (W2 ) , for W1 , W2 ∈ W ? Of special interest are cases where T is large and acts transitively on W. (3) Do W and T (as an abstract group) determine the action of the automorphisms {τW }W ∈W ? (4) If the group G of point transformations is a continuous group or contains a continuous subgroup, (when) do the underlying modular involutions induce a continuous unitary projective representation of G, respectively of its continuous subgroup? (5) Can this projective representation be lifted to a continuous unitary representation of G? (6) If there exists a one-parameter subgroup in G which can be interpreted as time evolution on (M, g), what are the spectral properties of the generator of the corresponding unitary representation? In particular, when is the spectrum bounded from below (as one would expect in the case of elementary physical states such as the vacuum)? Whereas the latter three problems are standard in the representation theory of groups, the first three are problems in the theory of transformation groups of subsets of topological spaces, which apparently have not received the attention they seem to deserve. We discuss in the subsequent sections the physically interesting examples of Minkowski space and de Sitter space, for which the preceding program can be
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completely carried out. Our proofs are largely based on explicit calculations which do not yet provide the basis for a more general argument. But as our results are promising, we believe that a more systematic study of these mathematical problems would be worthwhile. 4. Geometric Modular Action Associated with Wedges in R4 We now carry out the program outlined at the end of the preceding section for the case of four-dimensional Minkowski space with the standard metric 1 0 0 0 0 −1 0 0 g = diag(1, −1, −1, −1) ≡ (4.1) 0 0 −1 0 0
0
0
−1
in proper coordinates as the target space. The isometry group of this space is the Poincar´e group P and an admissible family W of regions is obtained by applying the elements of P to a single wedge-shaped region of the form WR ≡ {x ∈ R4 |x1 > |x0 |} ,
(4.2)
i.e. W = {λWR |λ ∈ P}, where λWR = {λ(x)|x ∈ WR }. It is easy to show that W is an admissible family in four-dimensional Minkowski space. Because of the requirement that the admissible family be mapped onto itself by the isometry group of the space-time, an admissible family W in the case of Minkowski space must contain the orbit of each of its elements under the action of the Poincar´e group. Recall that an admissible family W is called minimal if it coincides with the orbit under the action of the isometry group of a single region with a maximal stability group. As the only open, causally closed regions which are invariant under the stability group InvP(WR ) of WR are WR itself, its causal complement WR0 and the entire space R4 , one concludes that R4 is the only open, causally closed region which is stable under the action of any proper extension of InvP(WR ). Hence, W is a minimal admissible family for four-dimensional Minkowski space. We therefore base the analysis in this section on this canonical choice of regions. We remark ↑ ↑ that, in fact, one has W = {λWR |λ ∈ P+ }, where P+ is the identity component of the Poincar´e group. Note that the metric is introduced because a specific target space is envisioned. The wedges in the smooth manifold R4 can be defined without reference to the Minkowski metric by introducing coordinates. Then the set W of wedges is determined only up to diffeomorphism, which is all we shall require. Nonetheless, it is clear that there is nothing intrinsic about such a definition of wedges. For a discussion of a possible means to determine an intrinsic algebraic characterization of “wedges” for our purpose, see Sec. 7. We commence with a state on an initial net {A(W )}W ∈W which satisfies the CGMA discussed in the previous section. In Subsec. 4.1 we consider the elements of the transformation group T associated with any such state and establish a considerable extension of the Alexandrov–Zeeman–Borchers–Hegerfeldt theorems by
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showing that these maps are induced by point transformations which form a subgroup G of the Poincar´e group. This section also contains a simple example of a space-time manifold and well-behaved transformations of a corresponding family of regions which are not induced by point transformations. In Subsec. 4.2 and the subsequent sections, we restrict attention to those cases where the transformation group T is large enough to act transitively upon the ↑ set W. It turns out that G then contains the full identity component P+ of the Poincar´e group P. The specific form of the Poincar´e elements corresponding to the generating involutions in T , which themselves arise from the adjoint action of the initial modular conjugations upon the net {R(W )}W ∈W , is also identified in Subsec. 4.2, and it is found that this form is uniquely fixed and agrees with the one first determined by Bisognano and Wichmann for the case of the vacuum state on Minkowski space and any net of von Neumann algebras locally associated with a quantum field satisfying Wightman’s axioms [9, 10]. It then follows from this explicit knowledge of the form of the implementing Poincar´e elements that G is exactly equal to the proper Poincar´e group P+ . Thus, starting with the CGMA, we find a unique and familiar solution for the possible symmetry groups and their respective actions. In the remaining portion of Sec. 6 we discuss the properties of the representations of T — and hence of G = P+ — which are induced by the modular conjugations. In Subsec. 4.3 we shall identify a natural continuity condition on the net {R(W )}W ∈W which implies that there exists a strongly continuous (anti)unitary projective representation of G = P+ . This requires a certain choice of product decomposition in the definition of the projective representation (cf. the discussion before Corollary 2.1). These results are used in Subsec. 4.3 for the proof, first of all, that one can always lift this projective representation to a continuous unitary representation of the covering ↑ . Our analysis, which is based on results in Borel measurable group group of P+ cohomology theory and is carried out in the Appendix, parallels to some extent the discussion in [22]; but our more global point of view and our explicit construction of the projective representation provide certain simplifications. In particular, we shall not need to argue via the Lie algebra, since the results of Subsec. 4.2 and modular theory give us sufficient control over our explicit representation. And then we show that, after all, this representation of the covering group provides a strongly continu↑ ous representation of P+ and coincides with the initially and explicitly constructed projective representation. It is worth emphasizing that, given a state satisfying the CGMA, we explicitly construct a strongly continuous unitary representation of the translation subgroup (using ideas of [24]), thereby determining the generator of the timelike translations, which has the physical interpretation of the Hamiltonian, or total energy operator, of the theory. In other words, we derive the dynamics of the theory from the physical data of the state and net of observable algebras. However, we do not here touch upon the problem of constructing such states. In this regard, the ideas expounded in Sec. 6 of [59] may be of relevance. We recall that it is the main purpose of this section to illustrate the steps which are necessary to apply the CGMA in our program. As already mentioned at the end
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of Sec. 3, the mathematics relevant to the first three group theoretical problems does not seem to be sufficiently well developed for our purposes, and we must therefore rely on explicit and sometimes tedious computations to carry out our program. But our results demonstrate that the CGMA, which at first glance appears very general and diaphanous, actually imposes strong constraints on the admissible states and allows one to characterize the vacuum states in the case of Minkowski space. 4.1. Wedge transformations are induced by elements of the Poincar´ e group The aim of this section is to show that the elements of the transformation group T acting upon the wedges W, which arises when one assumes the CGMA discussed in the previous section, are induced by point transformations on Minkowski space, indeed, by elements of the Poincar´e group. In other words, we wish to show that T can be identified with a subgroup of the Poincar´e group. Since one can define points as intersections of edges of suitable wedges, it is an intuitively appealing possibility that transformations of wedges could lead to point transformations. The assumptions made in this section are slightly more general than actually needed for our primary purpose, but these somewhat more general results have interest going beyond the immediate problem we are addressing. In particular, we shall also employ these results in Sec. 5, where we consider the consequences of the geometric action of modular groups. In the remainder of this section, we shall assume that we have a bijective map τ : W → W with the following properties: (A) If W1 , W2 ∈ W satisfy W1 ∩ W2 = ∅, then τ (W1 ) ∩ τ (W2 ) = ∅ and τ −1 (W1 ) ∩ τ −1 (W2 ) = ∅; (B) W1 , W2 ∈ W satisfy W1 ⊂ W2 if and only if τ (W1 ) ⊂ τ (W2 ). By Proposition 3.1, these are properties shared by the maps τW , W ∈ W, arising from states complying with the CGMA. We do not assume in this section that the map τ is an involution or that (3.3) holds. We shall show that conditions (A) and (B) imply (3.3). We introduce the following notation: ` ∈ R4 denotes a future-directed lightlike vector and p ∈ R a real parameter. For given `, p we define the characteristic half-spaces Hp [`]± ≡ {x ∈ R4 | ± (x · ` − p) > 0} .
(4.1.1)
Note that the boundary of such a half-space, Hp [`] = ∂Hp [`]± = {x ∈ R4 |x · ` = p}, is a characteristic hyperplane with the properties that all lightlike vectors parallel to this hyperplane are parallel to ` and all other vectors parallel to Hp [`] are spacelike. Given two such pairs, {`i, pi }, i = 1, 2, where `1 and `2 are not parallel, then W = Hp1 [`1 ]+ ∩ Hp2 [`2 ]− is a wedge. All wedges can be obtained in this manner. In particular, for any wedge W ∈ W there exist two future-directed lightlike vectors `± such that W ± `± ⊂ W . These vectors are unique up to a positive scaling factor. The half-spaces H ± generating W as above are given by
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H± =
[
(W + λ`∓ ) .
(4.1.2)
λ∈R
In the sequel, we shall denote by F ± the following family of wedges: F ± ≡ {W + λ`∓ |λ ∈ R} . We shall say that F ± generates H ± via (4.1.2). Note that every such family F ± has the following properties: (i) F ± is linearly ordered, i.e. if W1 , W2 ∈ F ± , then either W1 ⊂ W2 or W2 ⊂ W1 . (ii) F ± is maximal in the sense that if W1 , W2 ∈ F ± satisfy W1 ⊂ W2 and there exists a wedge W ∈ W such that W1 ⊂ W ⊂ W2 , then W ∈ F ± . (iii) F has no upper or lower bound in (W, ⊂), i.e. there exists no element W< ∈ W such that W< ⊂ W for all W ∈ F and also no element W> ∈ W such that W> ⊃ W for all W ∈ F. We shall call a collection of wedges F ⊂ W with the properties (i)–(iii) a characteristic family of wedges. Every characteristic family of wedges is, in fact, of the form of F ± . The proof of this assertion rests upon the following well-known properties of wedges. For wedges W , W0 ∈ W with W0 ⊂ W and W0 6= W , there exists a space- or lightlike translation a ∈ R4 such that W0 = W + a ⊂ W + λa ⊂ W
for all 0 ≤ λ ≤ 1 .
If the edge of W0 lies on the boundary of W , then the translation a can be chosen to be lightlike (and is therefore a multiple of one of the lightlike vectors `± determining W ). On the other hand, if the edge of W0 lies in the interior of W , then there exists an open set N ⊂ R4 such that W0 ⊂ W + a ⊂ W , for all a ∈ N . As in [24], we shall say that two wedges W1 , W2 ∈ W are coherent if one is obtained from the other by a translation, or, equivalently, if there exists another wedge W3 such that W1 ⊂ W3 and W2 ⊂ W3 . Hence, all wedges in a characteristic family are mutually coherent. We now prove the initial assertion. Lemma 4.1. Every characteristic family of wedges F has the form F = {W + λ`|λ ∈ R} , for some wedge W ∈ W and some future-directed lightlike vector ` with the property that W + ` ⊂ W or W − ` ⊂ W . Proof. Let W0 , W ∈ F. By the linear ordering of F, one may assume without loss of generality that W0 ⊂ W . If the edge of W0 would lie in the interior of W , then, as mentioned above, there exists an open set N in R4 such that W0 ⊂ W + a ⊂ W , for all a ∈ N . By the maximality of F in W, this would entail that W + a ∈ F, for all a ∈ N . However, the elements of {W + a|a ∈ N } clearly violate the linear ordering of F. Hence, the edge of W0 must lie on the boundary of W , so there exists a lightlike translation a ∈ R4 such that W0 = W + a ⊂ W . Let now W , W + a, W + b ∈ F be chosen such that a and b are lightlike and W + a ⊂ W ⊂ W + b. As in the preceding paragraph one shows that the edge of
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W + a lies on the boundary of W + b. The assumed inclusion then implies that the edge of W lies on the same characteristic hyperplane. This entails that a and b are proportional, i.e. the elements of F are all of the form W + λ` with real λ and future-directed lightlike vector ` ∈ R4 . That every λ ∈ R must occur follows at once from properties (ii) and (iii) of characteristic families. In the next lemma we show that order-preserving bijections τ : W → W map characteristic families onto characteristic families. Lemma 4.2. Let τ : W → W be a bijective map with the property (B). Then τ maps every characteristic family F of wedges onto a characteristic family τ (F) ≡ {τ (W )|W ∈ F}. In fact, if F1 = {W1 + λ`1 |λ ∈ R}, for some wedge W1 ∈ W and some future-directed lightlike vector `1 with the property that W1 + `1 ⊂ W1 or W1 − `1 ⊂ W1 , and if τ (W1 ) = W2 , then τ (W1 + λ`1 ) = W2 + f (λ)`2 , where f : R → R is a continuous monotonic bijection, f (0) = 0, and `2 is a future-directed lightlike vector with the property that W2 + `2 ⊂ W2 or W2 − `2 ⊂ W2 . Proof. Since τ is an order isomorphism, the linear ordering of τ (F), property (i), follows at once. If one has for some W ∈ W and W1 , W2 ∈ F the inclusions τ (W1 ) ⊂ W ⊂ τ (W2 ), one must also have the inclusions W1 ⊂ τ −1 (W ) ⊂ W2 , since τ −1 is also an order isomorphism. Hence, by the maximality of F it follows that τ −1 (W ) ∈ F, so that W ∈ τ (F), establishing the maximality of τ (F). Finally, if there were to exist a lower bound W< ∈ W to τ (F), then since τ is an order isomorphism, the wedge τ −1 (W< ) would be a lower bound for F, a contradiction. Similarly, one can exclude the existence of an upper bound in W for τ (F). Let F1 , W1 , `1 , and W2 be as indicated in the hypothesis. Since it has just been established that inclusion-preserving bijections on W map characteristic families of wedges onto characteristic families, one sees from Lemma 4.1 that there exist future-directed lightlike vectors k1 , k2 , such that W2 + k1 ⊂ W2 and W2 − k2 ⊂ W2 , and a function f : R → R such that for all λ ∈ R either τ (W1 + λ`1 ) = W2 + f (λ)k1
or τ (W1 + λ`1 ) = W2 − f (λ)k2 .
Since τ is an inclusion-preserving bijection, f is bijective and monotone; hence f is continuous. We wish now to show that the apparent asymmetry in condition (A) can be removed without loss of generality; in other words, condition (3.3) holds for the mappings considered in this section. Corollary 4.1. Let τ : W → W be a bijection which satisfies conditions (A) and (B). Then τ also satisfies W1 ∩ W2 = ∅
if and only if
τ (W1 ) ∩ τ (W2 ) = ∅ .
Relation (4.1.3) is also true for the mapping τ −1 .
(4.1.3)
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Proof. Let W1 , W2 ∈ W such that W1 ∩ W2 = ∅ but W1 ∩ W2 6= ∅. It suffices to show that in this case one has τ (W1 ) ∩ τ (W2 ) = ∅. First note that if N is a convex subset of the boundary W /W of the wedge W , it is contained in one of the two characteristic hyperplanes Hp [`± ] determined by W , and thus it is easy to see that either N ∩ W + λ`+ = ∅
or N ∩ W − λ`− = ∅ ,
for all λ > 0. Since both W1 and W2 are convex, so is their intersection W1 ∩ W2 ⊂ W1 /W1 ; hence, with `1 , `2 future-directed lightlike vectors with W1 + `1 ⊂ W1 and W1 − `2 ⊂ W1 , it follows that ∅ = W1 + λ`1 ∩ (W1 ∩ W2 ) = W1 + λ`1 ∩ W2 or ∅ = W1 − λ`2 ∩ (W1 ∩ W2 ) = W1 − λ`2 ∩ W2 , for all λ > 0. Consider the first case and note that Lemma 4.2 entails that τ (W1 + λ`1 ) = τ (W1 )+f (λ)`, with τ (W1 )+` ⊂ τ (W1 ) or τ (W1 )−` ⊂ τ (W1 ) and f : R → R a continuous bijection which is either monotone increasing or monotone decreasing. Consider the subcase where f is monotone increasing and τ (W1 )+` ⊂ τ (W1 ). Then by the continuity of f , one has τ (W1 ) ∩ τ (W2 ) = (τ (W1 ) + f (0)`) ∩ τ (W2 ) ! [ = (τ (W1 ) + f (λ)`) ∩ τ (W2 ) λ>0
=
[
(τ (W1 + λ`1 ) ∩ τ (W2 )) = ∅ ,
λ>0
using assumption (A). On the other hand, the subcase f monotone decreasing and τ (W1 ) + ` ⊂ τ (W1 ) cannot arise, since τ is inclusion-preserving. Similarly, the subcase f monotone increasing and τ (W1 ) − ` ⊂ τ (W1 ) cannot occur. Finally, in the subcase f monotone decreasing and τ (W1 ) − ` ⊂ τ (W1 ) one finds the same chain of equalities as above. In the second case, namely ∅ = W1 − λ`2 ∩ W2 , for all λ > 0, one similarly sees that the subcases τ (W1 ) + ` ⊂ τ (W1 ) with f increasing, and τ (W1 ) − ` ⊂ τ (W1 ) with f decreasing are excluded by the inclusion-preserving property of τ . In the other two subcases, one has from Lemma 4.2 in a like manner τ (W1 ) ∩ τ (W2 ) = (τ (W1 ) + f (0)`) ∩ τ (W2 ) ! [ = (τ (W1 ) + f (−λ)`) ∩ τ (W2 ) λ>0
=
[
(τ (W1 − λ`2 ) ∩ τ (W2 )) = ∅ ,
λ>0
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by assumption (A). Thus, one has proven that W1 ∩W2 = ∅ implies τ (W1 )∩τ (W2 ) = ∅. The argument for τ −1 is identical, completing the proof of the lemma. To proceed further, it is convenient to use the following notation for wedges. For any linearly independent future-directed lightlike vectors `1 , `2 ∈ R4 and any a ∈ R4 , we define the wedge W [`1 , `2 , a] ≡ {α`1 + β`2 + `⊥ + a | α > 0, β < 0, `⊥ ∈ R4 , `⊥ · `1 = `⊥ · `2 = 0} = W [`1 , `2 , 0] + a , where the dot product here represents the Minkowski scalar product. Then with `1± = (1, ±1, 0, 0) , `2± = (1, 0, ±1, 0) , `3± = (1, 0, 0, ±1) , one sees that WR = W [`1+ , `1− , 0]. Note that with this notation, one has W [`1 , `2 , a] + `1 ⊂ W [`1 , `2 , a] and W [`1 , `2 , a] − `2 ⊂ W [`1 , `2 , a], i.e. for this wedge `+ is a positive multiple of `1 and `− is a positive multiple of `2 . Moreover, the half-spaces H ± generating W [`1 , `2 , a] as above are given by H + = Ha·`2 [`2 ]+ and H − = Ha·`1 [`1 ]− , and the associated characteristic families are given by F + = {W [`1 , `2 , a + λ`2 ] |λ ∈ R} and F − = {W [`1 , `2 , a + λ`1 ] |λ ∈ R} . We next show a useful characterization of pairs of spacelike separated wedges. Lemma 4.3. Let W1 , W2 be wedges. W1 ⊂ W20 if and only if the two characteristic families F2+ and F2− containing W2 satisfy W1 ∩ W = ∅ for every W ∈ F2+ ∪ F2− . Proof. Let H2± be the characteristic half-spaces generated by the families F2± , so that one has W2 = H2+ ∩H2− and W20 = H2+c ∩H2−c , where the superscript c signifies that one takes the complementary half-space. From W1 ⊂ W20 follows therefore the containment W1 ⊂ H2±c and hence also W1 ∩ H2± = ∅. Conversely, the last equality follows from the disjointness of W1 from each member of the set F2+ ∪ F2− , so that one must have W1 ⊂ H2+c ∩ H2−c = W20 . It is next established that bijections on W satisfying conditions (A) and (B) preserve causal complements and thus causal structure. Corollary 4.2. A bijection τ : W → W which fulfills conditions (A) and (B) also satisfies the following condition: τ (W 0 ) = τ (W )0 ,
f or any
W ∈W.
(4.1.4)
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Proof. Consider an arbitrary wedge W ∈ W, and let F + and F − be the characteristic families of wedges containing W 0 . By Lemma 4.2, τ maps F + and F − onto two characteristic families τ (F + ) and τ (F − ) containing τ (W 0 ). Lemma 4.3 entails that W is disjoint from every element of F + ∪ F − , and hence Corollary 4.1 implies that τ (W ) is disjoint from every element of τ (F + ) ∪ τ (F − ). Thus, Lemma 4.3 yields the containment τ (W 0 ) ⊂ τ (W )0 . The reverse containment follows by applying the same argument to τ −1 . We continue now with our development of point transformations. A pair (W1 , W2 ) of disjoint wedges will be called maximal if there is no wedge W properly containing W1 , resp. W2 , such that W ∩ W2 = ∅, resp. W ∩ W1 = ∅. Note that a bijection τ : W → W fulfilling conditions (A) and (B) maps maximal pairs of wedges onto maximal pairs of wedges. We need a computational characterization of a maximal pair of wedges. To this end, we remark that given a pair (W1 , W2 ) such that W2 is not a translate of W1 or W10 , there exists a Poincar´e transformation (Λ, x) mapping W1 onto WR and W2 onto either the wedge W [`2+ , `, d] or its causal complement W [`2+ , `, d]0 , where ` is some positive lightlike vector which is not parallel to `2+ and d ∈ R4 . This follows from the observations that there always exists a Lorentz transformation Λ1 such that Λ1 W1 = WR and that every positive lightlike vector not parallel to `1± is mapped by some element of the invariance group of WR to `2+ . We shall therefore consider the pair (WR , W [`2+ , `, d]) — indeed, without loss of generality, the pair (WR , W [`2+ , `, d]), for suitable ` = (1, a, b, c) with a2 + b2 + c2 = 1, b 6= 1, and d ∈ R4 — and determine under which conditions this pair is maximal. In preparation, we prove the following simple lemma. Lemma 4.4. Let P : R4 → R2 be given by P (x0 , x1 , x2 , x3 ) = (x0 , x1 ) and let W = W [`2+ , `, d] with ` = (1, a, b, c), where a, b, c ∈ R satisfy a2 + b2 + c2 = 1, b 6= 1, and d ∈ R4 . Then P W = R2 for b < 0 or c 6= 0. On the other hand, if 0 ≤ b < 1 and c = 0, one has P W = {x ∈ R2 |(x − P d) · (1 − b, −a) > 0} , where here the dot product represents the Euclidean scalar product on R2 . Proof. Without loss of generality, one may assume d = 0. One has P W = P {α`2+ + β(1, a, b, c) + s(c, 0, c, 1 − b) + t(a, 1 − b, a, 0)|α > 0, β < 0, s, t ∈ R} = {α(1, 0) + β(1, a) + s(c, 0) + t(a, 1 − b)|α > 0, β < 0, s, t ∈ R} . And since 1 − b 6= 0, this shows that P W = R2 for c 6= 0. Hence, one may restrict one’s attention to c = 0. Since (1 − b, −a) is a normal vector for the line {t(a, 1 − b)|t ∈ R}, the remaining assertions readily follow from α(1, 0) · (1 − b, −a) = α(1 − b) > 0 ,
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for α > 0, and for β < 0 β(1, a) · (1 − b, −a) = β(1 − b − a2 ) = β(b2 − b) is nonnegative for 0 ≤ b < 1 and negative for b < 0.
This straightforward observation leads to the following characterization of maximal pairs of wedges. Lemma 4.5. The wedges WR = W [`1+ , `1− , 0] and W = W [`2+ , `, d], where ` = (1, a, b, c) and a, b, c ∈ R satisfy a2 + b2 + c2 = 1, b 6= 1, and d ∈ R4 , form a maximal pair of wedges if and only if 0 < a < 1, 0 < b < 1, c = 0, and the vector d is a linear combination of vectors whose associated translations leave either WR or W fixed. The statement is true if W is replaced by W 0 and the condition 0 < a < 1 is replaced by −1 < a < 0 or also if `2+ is replaced by `2− and 0 < b < 1 by −1 < b < 0. Proof. Using the projection P from Lemma 4.4, note that x ∈ P WR if and only if x = α(1, 1) − β(1, −1) for suitable α, β > 0 . (4.1.5) WR is invariant with respect to translations by vectors in the subspace generated by (0, 0, 1, 0) and (0, 0, 0, 1), so one has WR ∩ W = ∅ if and only if P WR ∩ P W = ∅. By Lemma 4.4, the condition P WR ∩ P W = ∅ is equivalent to c = 0, 0 ≤ b < 1 and (by (4.1.5)) 0 ≥ (α(1, 1) − β(1, −1) − P d) · (1 − b, −a) = (α − β)(1 − b) − (α + β)a − P d · (1 − b, −a) , for all α, β > 0. This clearly entails that a ≥ 0. Note also that a, b ≥ 0 and c = 0 imply a > 0, since b 6= 1. It is then easy to check that this implies 1 − b − a = (1, 1) · (1 − b, −a) ≤ 0
(4.1.6)
1 − b + a = (1, −1) · (1 − b, −a) > 0 .
(4.1.7)
and Hence, WR ∩ W = ∅ is equivalent to the conditions c = 0, 0 ≤ b < 1, a > 0, and −P d · (1 − b, −a) ≤ 0. Assume first the maximality of the pair (WR , W ). Then −P d · (1 − b, −a) ≤ 0 and the conditions just established entail (x − P d) · (1 − b, −a) ≤ −P d · (1 − b, −a) ≤ 0 ,
(4.1.8)
for all x ∈ P WR . The maximality then implies the equality −P d · (1 − b, −a) = 0 ,
(4.1.9)
since, if not, one could obtain a wedge which properly contains W and yet is still disjoint from WR by choosing a different d such that (4.1.8) is still satisfied. Thus,
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one concludes that P d is a multiple of (a, 1 − b) = P (a, 1 − b, a, 0). Therefore, d is a linear combination of the vectors (a, 1 − b, a, 0), (0, 0, 0, 1) and (0, 0, 1, 0), where translations by the former two leave W invariant and translations by the latter two leave WR fixed. The possibility that b = 0 still remains to be excluded. But b = 0 entails a = 1, so W , resp. P W , is invariant with respect to translations by multiples of (1, 1, 1, 0), resp. (1, 1). Translating the disjoint pair (W, WR ) by d = −(1, 1, 1, 0), one would therefore obtain another disjoint pair (W, W2 ) such that W2 = W [`1+ , `1− , d] properly contains WR , contradicting the assumed maximality of (W, WR ). For the converse, assume that W has the stated form. By the first part of this proof, one already knows that W and WR are then disjoint. Only the proof of maximality remains. By hypothesis, (4.1.9) holds in this direction, as well. Furthermore, (4.1.6) and (4.1.7) are fulfilled. Note that if b 6= 0, then (4.1.6) holds with strict inequality. A wedge W3 which contains WR must be coherent with WR and is thus obtained by translating WR by a vector of the form −α0 `1+ + β0 `1− , with α0 , β0 ≥ 0. For W3 6= WR , i.e. for α0 6= 0 or β0 6= 0, (4.1.6) and (4.1.7) imply P (−α0 `1+ + β0 `1− ) · (1 − b, −a) > 0 . The vertex of P W3 lies in P W (Lemma 4.4 and (4.1.9)), hence P W3 ∩ P W 6= ∅ and so W3 ∩ W 6= ∅. One can argue similarly to eliminate the possibility that there does not exist a wedge properly containing W and yet being disjoint from WR . To establish the final assertions of the lemma, one need but consider the wedges transformed by suitable reflections. Since the union of the elements of a characteristic family of wedges yields a characteristic half-space, it is natural to use Lemma 4.2 to extend the map τ to the set H of all characteristic half-spaces in R4 . In order to establish that this extension is well-defined, it is necessary to consider the possibility that two characteristic families generate the same half-space. According to Lemma 4.1, every characteristic family F can be represented in the form F = {W +λ`|λ ∈ R}. We define the complementary characteristic family F c ≡ {(W + λ`)0 |λ ∈ R}. The families F and F c generate complementary characteristic ¯ In order to simplify notation, half-spaces H and H c , respectively, i.e. H c = R4 /H. we shall write F1 ∩ F2 = ∅ for two characteristic families to mean W1 ∩ W2 = ∅ for all W1 ∈ F1 and all W2 ∈ F2 . Hence, one has F ∩ F c = ∅, for any characteristic family F. Lemma 4.6. Let τ : W → W be a bijection with properties (A) and (B). Moreover, let F1 and F2 be two characteristic families of wedges generating the S S S same half-space, i.e. W1 ∈F1 W1 = W2 ∈F2 W2 . Then one has W1 ∈F1 τ (W1 ) = S W2 ∈F2 τ (W2 ).
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Proof. Since F1 and F2 generate the same half-space, one must have F1 ∩ F2c = ∅. Hence, Corollary 4.1 entails τ (F1 ) ∩ τ (F2c ) = ∅. Similarly, one derives τ (F1c ) ∩ τ (F2 ) = ∅. From (4.1.4) it also follows that τ (F c ) = τ (F)c , so that one finds τ (F1 ) ∩ τ (F2 )c = ∅ and τ (F1 )c ∩ τ (F2 ) = ∅. By Lemma 4.2, τ (F1 ) and τ (F2 ) generate half-spaces H1 and H2 , respectively, for which the following relations must therefore hold: H1 ∩ H2c = ∅ and H1c ∩ H2 = ∅. It follows that H1 = H2 . Lemmas 4.2 and 4.6 ensure that the following map is well-defined: Definition 4.1. Let τ : W → W be a bijection satisfying the properties (A) and (B). Then an associated map τ : H → H is obtained by setting for H ∈ H [ τ (W ) , τ (H) ≡ W ∈F
where F is any characteristic family generating H. We permit ourselves this abuse of notation in order to keep the notation as simple as possible, and because there will be no possibility of confusion of context. We next collect some useful properties of this map. We let H± ⊂ H denote the set of all future-directed (resp. past-directed) characteristic half-spaces H ± . Lemma 4.7. Let τ : W → W be a bijection satisfying the properties (A) and (B), and let τ : H → H be the associated mapping of characteristic half-spaces. (1) τ is bijective on H; (2) τ (H c ) = τ (H)c , for all H ∈ H; (3) for H1 , H2 ∈ H, H1 ∩ H2 = ∅ if and only if τ (H1 ) ∩ τ (H2 ) = ∅; moreover, H1 ⊂ H2 if and only if τ (H1 ) ⊂ τ (H2 ); (4) for given H ∈ H and every element a ∈ R4 there exists an element b ∈ R4 (and vice versa) such that τ (H + a) = τ (H) + b; (5) for any W ∈ W, W = H+ ∩ H− if and only if τ (W ) = τ (H+ ) ∩ τ (H− ); (6) either τ (H± ) = H± or τ (H± ) = H∓ . Proof. (1) Let F1 , F2 be characteristic families such that [ [ τ (W1 ) = τ (W2 ) . W1 ∈F1
W2 ∈F2
S Since τ −1 has the same properties as τ does, Lemma 4.6 entails that W1 ∈F1 W1 = S τ is injective on H. Let now H ∈ H be generated by a characteristic W2 ∈F2 W2 , i.e.S S family F: H = W ∈F W . Then defining H0 = W ∈F τ −1 (W ), one has τ (H0 ) = H. i.e. τ is surjective on H. (2) Assertion (2) is an immediate consequence of the property (4.1.4) of the map τ on W. (3) Let F1 , F2 be characteristic families which generate the characteristic halfspaces H1 , H2 , respectively. If H1 ∩ H2 = ∅, then F1 ∩ F2 = ∅, which implies τ (F1 ) ∩ τ (F2 ) = ∅, by property (4.1.3) of the transformation τ . Hence one has
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τ (H1 ) ∩ τ (H2 ) = ∅. The converse is proven using the fact that the map τ −1 also has the stated properties. If one has instead the inclusion H1 ⊂ H2 , then by Lemma 4.1 there exist wedges W1 , W2 such that Hi = ∪{Wi + λ`|λ ∈ R}, i = 1, 2, for a fixed future-directed lightlike vector ` (one characteristic half-space is contained in another only if their boundaries are parallel hyperplanes). One can choose W1 , W2 such that W1 ⊂ W2 . From condition (B) it then follows that τ (W1 + λ`) ⊂ τ (W2 + λ`) for all λ ∈ R, so that one must have the inclusion τ (H1 ) ⊂ τ (H2 ). (4) One first notes some general properties of characteristic half-spaces: If H1 , H2 are half-spaces with H1 ⊂ H2 , then there exists a translation c ∈ R4 such that H2 = H1 + c. If, on the other hand, the latter relation holds, then one must have either H1 ⊂ H2 or H2 ⊂ H1 . Let now H ∈ H and a ∈ R4 be given. Then either H ⊂ H + a or H + a ⊂ H. In the former case, part (3) of this lemma entails the inclusion τ (H) ⊂ τ (H + a), so that τ (H + a) = τ (H) + b for some b ∈ R4 . The second case is handled analogously. Since the map τ −1 on H satisfies assertions (1)–(3) of this lemma, the assertion (4) also follows when the roles of a and b are exchanged. (5) Given a wedge W ∈ W there exist unique characteristic half-spaces H ± such that W = H + ∩ H − . They are determined by the characteristic families F ± = {W + λ`∓ |λ ∈ R}, where `± are future-directed lightlike vectors such that W ± `± ⊂ W . Clearly one has τ (W ) ∈ τ (F ± ). Since F ± are characteristic families, by Lemma 4.1 there exist future-directed lightlike vectors `± τ such that ± ± + − τ (F ) = {τ (W ) + λ`τ |λ ∈ R}. Since the set F ∪ F is not linearly ordered, condition (B) entails that also the set τ (F + ) ∪ τ (F − ) is not linearly ordered, in other − words, τ (F + ) 6= τ (F − ). Hence the vectors `+ τ and `τ are not parallel. Therefore, the intersection of the half-spaces τ (H ± ) generated by τ (F ± ) must coincide with τ (W ). (6) Let H ± ∈ H± . If the hyperplanes which form the boundaries of H ± are parallel, then one must have either H + ∩ H − = ∅ or H +c ∩ H −c = ∅. Parts (2) and (3) of this lemma then entail that either τ (H + ) ∩ τ (H − ) = ∅ or τ (H + )c ∩ τ (H − )c = ∅ must hold. Hence the boundary hyperplanes of the characteristic half-spaces τ (H ± ) are parallel, and the time-like orientations of these half-spaces are oppositely directed. On the other hand, if the boundary hyperplanes of H ± are not parallel, then their intersection H + ∩ H − = W is a wedge, and it follows from part (4) that τ (H + ) ∩ τ (H − ) = τ (W ) ∈ W. Hence, also in this situation the time-like orientations of the half-spaces τ (H ± ) are oppositely directed. Fixing H − and letting H + range through H+ , one concludes that either τ (H+ ) ⊂ + H and τ (H − ) ∈ H− or τ (H+ ) ⊂ H− and τ (H − ) ∈ H+ . Varying H − while holding H + fixed completes the proof of assertion (6), when one recalls the result of part (1). Each characteristic half-space Hp [`]± determines uniquely a characteristic hyperplane Hp [`] = Hp [`]+ ∩ Hp [`]− , and so the map τ on H naturally induces a map on the set of characteristic hyperplanes.
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Definition 4.2. Let τ : W → W be a bijection satisfying properties (A) and (B) and τ : H → H the associated mapping of characteristic half-spaces. Then τ (Hp [`]) ≡ τ (Hp [`]+ ) ∩ τ (Hp [`]− ) defines a mapping of characteristic hyperplanes onto characteristic hyperplanes. The following properties of this mapping of characteristic hyperplanes are an immediate consequence of Lemma 4.7. Corollary 4.3. Let τ : W → W be a bijection satisfying properties (A) and (B) and τ be the associated mapping of characteristic hyperplanes. (1) τ is bijective on the set of characteristic hyperplanes in R4 ; (2) for a given hyperplane Hp [`] and every element a ∈ R4 there exists an element b ∈ R4 (and vice versa) such that τ (Hp [`] + a) = τ (Hp [`]) + b; (3) τ maps distinct parallel characteristic hyperplanes onto distinct parallel characteristic hyperplanes. We next prove some further properties of this mapping τ which are not quite so obvious. Lemma 4.8. Let τ : W → W be a bijection satisfying properties (A) and (B) and τ be the associated mapping of characteristic hyperplanes. If `1 , `2 , `3 , `4 are linearly dependent future-directed lightlike vectors such that any two of them are linearly independent, then 4 \
\
τ (H0 [`i ]) =
i=1
τ (H0 [`i ])
f or k = 1, 2, 3, 4 .
i6=k
Proof. As pointed out earlier, an arbitrary maximal pair (W [`˜1 , `˜2 , d1 ], ˜ ˜ ˜ d]) W [`3 , `4 , d2 ]) with {`˜1 , `˜2 } 6= {`˜3 , `˜4 } can be brought into the form (WR , W [`2+ , `, 0 ˜ d] ) by a suitable Poincar´e transformation, and by Lemma 4.5 (or (WR , W [`2+ , `, it is no loss of generality to take d = 0. Hence, H0 [`1+ ], H0 [`1− ], H0 [`2+ ] and ˜ are the characteristic hyperplanes determined by these wedges. Since Lemma H0 [`] 4.5 entails that `˜ = (1, a, b, 0), with 0 < a < 1 and 0 < b < 1, one observes that any three of the four vectors `1+ , `1− , `2+ and `˜ are linearly independent. Hence, ˜ is the intersection of any three of the hyperplanes H0 [`1+ ], H0 [`1− ], H0 [`2+ ], H0 [`] one-dimensional. But, on the other hand, one evidently has ˜. {c(0, 0, 0, 1)|c ∈ R} ⊂ H0 [`1+ ] ∩ H0 [`1− ] ∩ H0 [`2+ ] ∩ H0 [`]
(4.1.10)
Therefore, one may conclude that the right-hand side of (4.1.10) is equal to the one-dimensional intersection of any three of the hyperplanes in that expression. Employing the suitable Poincar´e transformation, one sees that 4 \ i=1
Hci [`˜i ] =
\ i6=j
Hci [`˜i ] for j = 1, 2, 3, 4 ,
(4.1.11)
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where {Hci [`˜i ]}4i=1 are the hyperplanes determined by the maximal pair (W [`˜1 , `˜2 , d], W [`˜3 , `˜4 , d0 ]). Returning to the vectors {`1 , . . . , `4 } of the hypothesis, there exists a Lorentz transformation Λ with Λ`1 = a1 `1+ , Λ`2 = a2 `1− , Λ`3 = a3 `2+ , and Λ`4 = a4 `, where ` = (1, a, b, 0) , a, b ∈ R , a2 + b2 = 1 , and ai , i = 1, . . . , 4, are positive constants. Hence, one may once again consider the pair (ΛW [`1 , `2 , 0], ΛW [`3 , `4 , 0]) = (WR , W [`2+ , `, 0]) without loss of generality, since τ ◦ Λ−1 maps maximal pairs onto maximal pairs. If this pair is maximal, then (4.1.11) yields the desired assertion. If this pair and (WR0 , W [`2+ , `, 0]) are not maximal, then Lemma 4.5 entails b ≤ 0. But, in fact,b = 0 is excluded by the linear independence assumption. Set `0 = 1, √12 , √12 , 0 . Then using (4.1.11) for the maximal pairs (τ (WR ), τ (W [`2+ , `0 , 0])) and (τ (W [`2+ , `2− , 0]), τ (W [`1+ , `0 , 0])) as well as Lemma 4.5 and the fact that τ preserves the maximality of pairs of wedges, one finds 3 \
τ (ΛH0 [`i ]) = τ (H0 [`1+ ]) ∩ τ (H0 [`1− ]) ∩ τ (H0 [`2+ ])
i=1
= τ (H0 [`1+ ]) ∩ τ (H0 [`1− ]) ∩ τ (H0 [`2+ ]) ∩ τ (H0 [`]) = τ (H0 [`1+ ]) ∩ τ (H0 [`0 ]) ∩ τ (H0 [`2+ ]) = τ (H0 [`1+ ]) ∩ τ (H0 [`0 ]) ∩ τ (H0 [`2+ ]) ∩ τ (H0 [`2− ]) ⊂ τ (H0 [`2− ]) .
(4.1.12)
If a 6= 0, then either (WR , W [`2− , `, 0]) or (WR , W [`2− , `, 0]0 ) is maximal, by Lemma 4.5. Hence, (4.1.11) and (4.1.12) yield 3 \ i=1
τ (ΛH0 [`i ]) ⊂ τ (H0 [`1+ ]) ∩ τ (H0 [`1− ]) ∩ τ (H0 [`2− ]) = τ (H0 [`1+ ]) ∩ τ (H0 [`1− ]) ∩ τ (H0 [`2− ]) ∩ τ (H0 [`]) ⊂ τ (H0 [`]) ,
implying the desired assertion. If, on the other hand, a = 0, then ` is a positive multiple of `2− , so that H0 [`] = H0 [`2− ] and use of (4.1.12) completes the proof. It is evident that the intersection of four hyperplanes Hci [`i ] corresponding to a linearly independent set of four future-directed lightlike vectors `i and four real numbers ci is a set containing a single point. We now have established sufficient background to prove that the map τ preserves this property. Lemma 4.9. Let τ : W → W be a bijection satisfying properties (A) and (B) and τ be the associated mapping of characteristic hyperplanes. Then the intersection T ` τ (H0 [`]) taken over all future-directed lightlike vectors is a singleton set (a point) in R4 .
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Proof. Since, from Corollary 4.3, τ maps parallel characteristic hyperplanes onto parallel characteristic hyperplanes, there exist suitable pairwise linearly independent lightlike vectors `˜1 , `˜2 , `˜3 , `˜4 and c1 , c2 , c3 , c4 ∈ R such that τ (H0 [`1+ ]) = Hc1 [`˜1 ], τ (H0 [`1− ]) = Hc2 [`˜2 ], τ (H0 [`2+ ]) = Hc3 [`˜3 ], and τ (H0 [`3+ ]) = Hc4 [`˜4 ]. By part (2) of Corollary 4.3, there exist real numbers b1 , b2 , b3 , b4 ∈ R such that τ (Hb1 [`1+ ]) = H0 [`˜1 ], τ (Hb2 [`1− ]) = H0 [`˜2 ], τ (Hb3 [`2+ ]) = H0 [`˜3 ], and τ (Hb4 [`3+ ]) = H0 [`˜4 ]. If {`˜i}i=1,...,4 is a linearly dependent set, then Lemma 4.8 applied to τ −1 as a mapping on the set of characteristic hyperplanes would entail that {`1+ , `1− , `2+ , `3+ } is linearly dependent, a contradiction. Hence, {`˜i}i=1,...,4 is a linearly independent set T4 and so the intersection i=1 Hci [`˜i ] is a singleton set. An arbitrary lightlike vector ` 6= 0 is a linear combination of `3+ and two linearly independent lightlike vectors `1 , `2 with zero x3 -component. By Lemma 4.8, it follows that τ (H0 [`1 ]) ∩ τ (H0 [`2 ]) ∩ τ (H0 [`3+ ]) ⊂ τ (H0 [`]) (note that if `1 , `2 , `3+ , ` are not pairwise linearly independent, then ` is a positive multiple of one of the others and determines the same hyperplane as the latter) and also τ (H0 [`1+ ]) ∩ τ (H0 [`1− ]) ∩ τ (H0 [`2+ ]) ⊂ τ (H0 [`j ])
for j = 1, 2 .
This proves the claim, since 4 \
Hci [`˜i ] = τ (H0 [`1+ ]) ∩ τ (H0 [`1− ]) ∩ τ (H0 [`2+ ]) ∩ τ (H0 [`3+ ]) ⊂ τ (H0 [`]) ,
i=1
for arbitrary future-directed lightlike ` 6= 0.
This result entails that τ induces a point transformation on R4 . Definition 4.3. For each x ∈ R4 , W ∈ W, and each characteristic hyperplane H, let Tx (H) ≡ H + x and Tx (W ) ≡ W + x. Let τ : W → W be a bijection satisfying properties (A) and (B) and τ be the associated mapping of characteristic hyperplanes. Then define δ : R4 → R4 by \ {δ(x)} ≡ τ (Tx H0 [`]) for x ∈ R4 , `
where the intersection is taken over all non-zero future-directed lightlike vectors ` ∈ R4 . Note that the mapping τ ◦ Tx has the same properties as τ ; applying Lemma 4.9 to this mapping implies that δ is well-defined. We next need to show that this point transformation is consistent with the mapping τ . Proposition 4.1. Let τ : W → W be a bijection satisfying properties (A) and (B) and δ be the associated point transformation. Then δ is a bijection and τ (W ) = {δ(x)|x ∈ W }
f or all W ∈ W .
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Proof. Define a mapping γ : R4 → R4 by \ {γ(y)} ≡ τ −1 (Ty H0 [`]) . `
For a fixed x ∈ R , consider y ≡ δ(x), so that y ∈ τ (Tx H0 [`]) for all non-zero positive lightlike vectors `. But, by Corollary 4.3, for each such ` there exists a non-zero positive lightlike vector `0 such that τ (Tx H0 [`]) = Ty H0 [`0 ]. Since τ is bijective on the set of characteristic hyperplanes in R4 , it follows that \ \ {γ(δ(x))} = τ −1 (Ty H0 [`0 ]) = τ −1 (τ (Tx H0 [`]) = {x} ; 4
`0
`
hence, one has γ = δ −1 and δ is a bijection. For arbitrary W0 ∈ W and y ∈ W0 , there exists a wedge W1 ⊂ W0 such that y lies in the edge of W1 and such that the characteristic hyperplanes determined by W0 are different from (though parallel to) those determined by W1 . By Corollary 4.3, the same must be true of the hyperplanes determined by the wedges τ (W1 ) ⊂ τ (W0 ). Thus, one has τ (W1 ) ⊂ τ (W0 ). Let H1 and H2 be the characteristic hyperplanes determined by W1 . There are two characteristic families F1 and F2 , S S containing W1 , with H1 = ∂( W ∈F1 W ) and H2 = ∂( W ∈F2 W ). The wedge S τ (W1 ) is contained in both τ (F1 ) and τ (F2 ), so that τ (H1 ) = ∂( W ∈τ (F1 ) W ) S and τ (H2 ) = ∂( W ∈τ (F2 ) W ) are the characteristic hyperplanes determined by τ (W1 ). The characteristic hyperplanes containing the point y (H1 and H2 belong to this set) are mapped by τ into the set of characteristic hyperplanes containing δ(y), i.e. δ(y) ∈ τ (H1 ) and δ(y) ∈ τ (H2 ). This shows that δ(y) lies in the two characteristic hyperplanes determined by τ (W1 ). But this entails δ(y) ∈ τ (W1 ) ⊂ τ (W0 ), which yields {δ(x)|x ∈ W } ⊂ τ (W )
for every W ∈ W .
(4.1.13)
Since, by Corollary 4.1, τ −1 has the same properties as τ , one has similarly {δ −1 (x)|x ∈ W } ⊂ τ −1 (W )
for every W ∈ W .
Now let y ∈ τ (W ). Then one has x ≡ δ −1 (y) ∈ τ −1 (τ (W )) = W , and since δ(x) = y, it follows that τ (W ) ⊂ {δ(x)|x ∈ W }. The containment (4.1.13) completes the proof. We recall the well-known result of Alexandrov [2, 3] (see also Zeeman [78], Borchers and Hegerfeldt [11]) to the effect that bijections on R4 mapping light cones to light cones must be elements of the extended Poincar´e group, DP, generated by the Poincar´e group and the dilatation group. The above-established results can be used to show that the bijection δ : R4 → R4 constructed above does indeed map light cones onto light cones. However, a more concise argument can be obtained by appealing to a related result of Alexandrov [3], to wit: A bijection on R4 , who along with its inverse maps spacelike separated points onto spacelike separated points, is an element of the extended Poincar´e group.
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Lemma 4.10. Let τ : W → W be a bijection satisfying properties (A) and (B) and δ : R4 → R4 be the associated point transformation. Then δ is an element of DP. Proof. Note that two points x, y ∈ R4 are spacelike separated if and only if there exists a wedge W ∈ W such that x ∈ W and y ∈ W 0 . But by Proposition 4.1 and Corollary 4.2, one sees that x ∈ W and y ∈ W 0 if and only if δ(x) ∈ τ (W ) and δ(y) ∈ τ (W 0 ) = τ (W )0 , i.e. δ(x) and δ(y) are spacelike separated. It is therefore evident that both δ and δ −1 preserve spacelike separation. The desired assertion then follows from Theorem 1 of [3]. We have therefore established the following result, which we regard as a considerable extension of the theorems of Alexandrov et alia just cited. Theorem 4.1. Let τ : W → W be a bijection with the properties (A) and (B). Then there exists an element δ of the extended Poincar´e group DP such that for all W ∈ W one has τ (W ) = {δ(x)|x ∈ W } . Proof. This is an immediate consequence of Corollary 4.3, Proposition 4.1 and Lemma 4.10. Turning to the more special case of the transformations τW on W which arise when the CGMA holds, we know from Proposition 3.1 that they satisfy conditions (A) and (B) and are involutions. Each of those transformations will therefore fulfill the hypotheses of the next Corollary. Corollary 4.4. Let τ : W → W be an involutive bijection with the properties (A) and (B). Then there exists an element δ of the Poincar´e group such that for all W ∈W τ (W ) = {δ(x)|x ∈ W } . Proof. It follows from the preceding proposition that there exists an element δ of the extended Poincar´e group such that the stated equality of sets holds. Since τ is an involution, one sees that W = τ 2 (W ) = {δ 2 (x)|x ∈ W }, for each W ∈ W. Hence, by taking suitable intersections one may conclude that δ 2 (x) = x, for all x ∈ R4 . Since δ is an affine map, it is then clear that it cannot contain a nontrivial dilatation. Since the group T is generated by elements satisfying the hypothesis of Corollary 4.4, and since Poincar´e transformations are completely fixed by their action on the wedges W, we conclude that T is isomorphic to a subgroup G of the Poincar´e group. In order to indicate the strength of this result, we shall outline a closely related example, where the respective transformations are not induced by point transformations, even though properties (A) and (B) obtain.
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We consider the manifold M ≡ R4 \ V+ , which is the complement in Minkowski space of the closure of the forward light cone with apex at the origin, with the conformal structure inherited from Minkowski space, and we take as an admissible family W+ the set of all regions W+ = W \ V+ , where W ranges through the wedges in Minkowski space considered above. Note that W is uniquely determined once W+ is given and that (W1 )+ ∩ (W2 )+ = ∅ if and only if W1 ∩ W2 = ∅. However, this latter implication fails to be true in general for the intersection of more than two regions. Moreover, we also note that the equality (W+ )0 = (W 0 )+ holds for all wedges W . We pick now any Lorentz transformation which interchanges the forward and backward light cones in R4 , such as time reversal T , and define on W+ the mapping τ (W+ ) ≡ (T W )+ , W+ ∈ W+ . It follows from the preceding remarks that τ : W+ → W+ is well-defined and has properties (A) and (B). But if the intersection of three (or more) partial wedges W+ is contained in the backward light cone V− , their images under the map τ have empty intersection. This shows that τ cannot be induced by a point transformation on M. 4.2. Wedge transformations generate the proper Poincar´ e group In the preceding section we have seen that for any theory on R4 satisfying the CGMA for the wedge regions W, the corresponding transformation group T is isomorphic to a subgroup G of the Poincar´e group P. So the next question in our program is: Which subgroups of P can appear in this way? We do not aim here at a complete answer to this question and restrict attention to those cases where the group T is “large”. A natural way of expressing this mathematically is to assume that the group T acts transitively upon the set W. It would be interesting to consider situations where this transitive action fails.f However, as our intention in this paper is to illustrate the application of our approach to just a few, albeit physically important cases, we make this additional assumption and leave the other possibilities uninvestigated for the present. We remark that the condition that T acts transitively upon the set W is implied by the algebraic postulate that the adjoint action of the modular conjugations {JW |W ∈ W} acts transitively upon the set {R(W )}W ∈W . Hence, this condition also is expressible in terms of algebraically determined quantities. We have constructed a subgroup G of the Poincar´e group, which is isomorphic to T and related to the group T as follows: For each τ ∈ T there exists an element gτ ∈ G such that τ (W ) = gτ W ≡ {gτ (x)|x ∈ W }. To each of the defining involutions τW ∈ T , W ∈ W, there exists a unique corresponding involution gW ∈ G ⊂ P. The Poincar´e group has four connected components, and the transitivity of the action of G upon the set W, which implies that for every W1 , W2 ∈ W, there exists an element g ∈ G such that ggW1 g −1 = ggW1 = gW2 , entails the relation −1 −1 gW1 gW2 = gW1 ggW1 g −1 = gW1 ggW g , 1 f We shall return to this point in a subsequent publication.
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since gW1 is an involution. But the right-hand side is a group commutator, and in the Poincar´e group such commutators are always contained in the identity component ↑ P+ . Hence, for any wedges W1 , W2 ∈ W the product of the corresponding group ↑ elements gW1 gW2 must be contained in P+ , and the same is true for products of an even number of the generating involutions of G. Now pick a wedge W ∈ W and consider the corresponding involution gW ∈ G, which must lie in one of the four components of P. One then notes that if n ∈ N is odd, then it follows from gW1 · · · gWn = gW (gW gW1 · · · gWn ) that gW1 · · · gWn must lie in the same component of P as gW . But this implies the following lemma. Lemma 4.11. The group G has nonempty intersection with at most one connected ↑ component of the Poincar´e group P other than P+ . Thus we are dealing with a subgroup G of P which is generated by involutions, intersects at most two of the four connected components of P and acts transitively on W in the obvious sense. Which subgroups can such G be? Answering this question turned out to be a somewhat laborious task. We begin by discussing an analogous problem for the Lorentz group. Consider again the reference wedge WR = {x ∈ R4 |x1 > |x0 |}, whose edge contains the origin, and let InvL(WR ) ≡ {Λ ∈ L|ΛWR = WR } be its invariance subgroup in the full Lorentz group L. The involutions in InvL(WR ) given by the identity diag(1, 1, 1, 1) ∈ L↑+ , the temporal reflection T = diag(−1, 1, 1, 1) ∈ L↓− , the reflection through the 3-axis (in other words, about the x0 x1 x2 -hyperplane) P3 = diag(1, 1, 1, −1) ∈ L↑− , and their product P3 T = diag(−1, 1, 1, −1) ∈ L↓+ are distinguished, because all elements of InvL(WR ) can be obtained by multiplying elements of InvL↑+ (WR ) ≡ InvL(WR ) ∩ L↑+ by these involutions. It is important in what is to come that InvL↑+ (WR ) is an abelian group, since it is generated by rotations about the 1-axis and velocity transformations (boosts) in the 0-1 direction, whereas InvL(WR ) is not abelian, precisely because of the mentioned involutions. The fact that InvL↑+ (WR ) is abelian is heavily used in our arguments, and for that reason our proof does not function in higher-dimensional Minkowski spaces. We wish to prove the following proposition. Proposition 4.2. Any subgroup G of the identity component L↑+ of the Lorentz group, which acts transitively upon the set W0 of wedges whose edges contain the origin of R4 , must equal L↑+ . Furthermore, any subgroup G of the Lorentz group L, which is generated by a collection of involutions, has nontrivial intersection with at most two connected components of L and acts transitively upon the set W0 , must contain L↑+ . The proof will proceed in a number of steps, since we find it convenient to consider the following alternatives: (i) G ∩ InvL(WR ) is trivial, i.e. consists only of the identity 1, (ii) G ∩ InvL(WR ) is nontrivial but G ∩ InvL↑+ (WR ) is trivial, or (iii) G ∩ InvL↑+ (WR ) is nontrivial. We shall show that cases (i) and (ii) cannot obtain under our assumptions and that case (iii) implies the desired conclusion.
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We shall exclude case (i) by proving the following claim. Lemma 4.12. Let G be a subgroup of L which has nontrivial intersection with at most two connected components of L and which acts transitively upon the set W0 . Then one must have G ∩ InvL(WR ) 6= {1}. Before we prove this lemma, we explain how it immediately entails the first assertion of Proposition 4.2 and why the proof of the second assertion of Proposition 4.2 is more difficult. If we knew from the outset that the group G in the statement of Proposition 4.2 has the property that also G+ ≡ G ∩ InvL↑+ acts transitively on W0 (this is implied by the hypothesis of the first assertion in Proposition 4.2), Proposition 4.2 would follow directly from Lemma 4.12 and the fact that L↑+ is a simple group. For then the adjoint action of G+ applied to the nontrivial element in G+ ∩InvL(WR ), whose existence is assured by Lemma 4.12 would generate all of L↑+ . In detail: If G+ acts transitively upon W0 , then there exists for each Λ ∈ L↑+ a gΛ ∈ G+ and some ˜ Lemma 4.12 yields the existence of some ˜ ∈ InvL↑ (WR ) such that gΛ = ΛΛ. Λ + nontrivial element h0 ∈ G+ ∩ InvL(WR ). Since G+ ∩ InvL(WR ) ⊂ InvL↑+ (WR ) and ˜ 0Λ ˜ −1 Λ−1 = gΛ h0 g −1 ∈ G+ , the latter group is abelian, we conclude Λh0 Λ−1 = ΛΛh Λ ↑ ↑ for all Λ ∈ L+ . But L+ is simple (see, e.g. Sec. I.2.8 in [37]), so it follows in this case that G+ = L↑+ . This establishes the first assertion in Proposition 4.2, given Lemma 4.12. What makes the proofs of Lemma 4.12 and the second assertion of Proposition 4.2 somewhat cumbersome is the a priori possibility that for the transitivity of the action of G on W0 , the elements in G \ G+ are essential. As the proof of Lemma 4.12 is itself quite lengthy, we shall break it up into a series of sublemmas. We proceed by contradiction and make the assumption that the intersection G ∩ InvL(WR ) is trivial and that G acts transitively on W0 throughout the proof of Lemma 4.12. This assumption entails that for every Λ ∈ L↑+ ˜ ∈ InvL(WR ) such that gΛ = ΛΛ ˜ there exists exactly one gΛ ∈ G and a unique Λ −1 −1 ˜ ˜ ˜ ˜ (otherwise, one would have Λ = g1 Λ1 = g2 Λ2 , for g1 , g2 ∈ G and Λ1 , Λ2 ∈ ˜ −1 Λ ˜ 1 , yielding a contradiction unless both sides InvL(WR ), which entails g −1 g1 = Λ 2
2
are equal to the identity in L). Thus, under the given assumption we have a map ˜ = Λ−1 gΛ . Note that, in view of the assumpm : L↑+ → InvL(WR ) with m(Λ) = Λ tion G ∩ InvL(WR ) = {1}, the map m : L↑+ → InvL(WR ) is the identity map when restricted to InvL↑+ (WR ). Moreover, for any Λ ∈ L↑+ the elements m(Λ) and gΛ lie in the same component of the Lorentz group. Utilizing the fact that G is a group yields a strong condition on the map m. Consider any two elements Λ1 , Λ2 ∈ L↑+ and the corresponding gΛ1 , gΛ2 ∈ G. Then since G is a group, we must have f1 Λ2 Λ f2 = Λ1 (Λ f1 Λ2 Λ f1 gΛ1 gΛ2 = Λ1 Λ
−1
f1 Λ f2 ∈ G . )Λ
f1 Λ2 Λ f1 −1 ) we have on the other hand gΛ = ΛΛ ˜ with gΛ ∈ G Setting Λ = Λ1 (Λ −1 −1 f f ˜ and consequently gΛ gΛ1 gΛ2 = Λ Λ1 Λ2 ∈ G ∩ InvL(WR ) = {1}. This yields the equation m(Λ1 )m(Λ2 ) = m(Λ1 m(Λ1 )Λ2 m(Λ1 )−1 ) , (4.2.1)
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for all Λ1 , Λ2 ∈ L↑+ . For the solution of this equation it is convenient to proceed to the covering group SL(2, C) of L↑+ . One then has to consider the action of space and time reflections on SL(2, C). Adopting standard conventions (see, e.g. [61]), one obtains by a straightforward computation the following result, which we state without proof. Lemma 4.13. Space and time reflections (P and T ) acting on four-dimensional Minkowski space-time induce the same automorphic action upon SL(2, C), given by π(A) = A∗−1 , whereas the reflection of the 3-axis P3 induces the action π3 (A) = ¯ ∗ , where R = ( i 0 ) and the bar denotes complex conjugation. −RAR 0 −i With ρ : SL(2, C) → L↑+ the canonical homomorphism from the covering group, we proceed from m to the map M : SL(2, C) → InvL(W0 ) given by M ≡ m ◦ ρ. Note that according to our assumptions on G, the set M (SL(2, C)) is contained in at most two connected components of the Lorentz group. With Λ1 = ρ(A) and Λ2 = ρ(B), A, B ∈ SL(2, C), and the fact that ρ is a homomorphism, Eq. (4.2.1) yields the following functional equation for M : A, B ∈ SL(2, C) .
M (A)M (B) = M (AγA (B)) ,
(4.2.2)
γA is the unique automorphism of SL(2, C) satisfying ρ◦γA (·) = M (A)ρ(·)M (A)−1 . More concretely, for each A ∈ SL(2, C), M (A) can be written uniquely as a product of one of the reflections 1, T , P3 or T P3 and an element of InvL↑+ (WR ). The subgroup of SL(2, C) corresponding to InvL↑+ (WR ) (with the appropriate choice of coordinates) is the maximally abelian subgroup D of matrices in SL(2, C) of the form ! λ 0 , λ ∈ C/{0} . 0 λ−1 Hence, any choice of A ∈ SL(2, C) determines by the above decomposition of M (A) such an element Dλ ∈ D (up to a sign). With this in mind, the action of γA on SL(2, C) can be determined with the help of Lemma 4.13 and is given by
γA
α γ
β δ
!! =
α λ−2 γ
λ2 β δ
!
δ¯ −λ−2 β¯
−λ2 γ¯ α ¯
α ¯ −λ−2 γ¯
−λ2 β¯ δ¯ ! λ2 γ , α
δ −2
λ
β
if M (A) ∈ L↑+ ,
(a)
,
if M (A) ∈ L↓− ,
(b)
,
if M (A) ∈ L↑− ,
(c)
if M (A) ∈ L↓+ ,
(d)
, ! !
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where α, β, δ, γ ∈ C with αδ − βγ = 1. We shall refer to these four possibilities in the following as cases (a), (b), (c) and (d). After these preparations, we now turn to the solution of Eq. (4.2.2) and hence of Eq. (4.2.1). Let UC ≡ {( 10 z1 )|z ∈ C} be the subgroup of upper triangular matrices and LC ≡ {( z1 01 )|z ∈ C} be the subgroup of lower triangular matrices in SL(2, C). Note that in cases (a) and (c) γA leaves the sets UC and LC invariant, while in the other cases γA interchanges the two. Moreover, as long as A is in case (a) and γA is not the identity, one has for some λ2 6= 1 the equalities γA
1 z
0 1
γA
1 0
z 1
!!
1 z
0 1
1 0
z 1
!−1 =
1 −2 (λ − 1)z
=
1 0
0 1
!
and !!
!−1
(λ2 − 1)z 1
! ,
which entail {γA (X)X −1 |X ∈ LC } = LC , as well as {γA (X)X −1 |X ∈ UC } = UC . The following result is a simple consequence of the latter observation. Lemma 4.14. For any triangular matrix A in SL(2, C) such that M (A) ∈ L↑+ (i.e. case (a)), one has M (A) = 1. Proof. Let A be contained in UC or LC and satisfy M (A) ∈ L↑+ . If γA is not trivial, then from the above remarks there exists a matrix X ∈ SL(2, C) such that γA (X)X −1 = A−1 . Therewith one has the equality AγA (X) = X, and Eq. (4.2.2) implies M (A) = 1. This is a contradiction, since then γA is trivial. Therefore γA must act as the identity map on SL(2, C), and M (A) has to lie in the center of L↑+ , i.e. M (A) = 1. Some elementary properties of the elements of SL(2, C) which are mapped by M to the identity are collected in the following lemma. Lemma 4.15. Let E consist of all A ∈ SL(2, C) such that M (A) = 1. Then (1) E is a subgroup of SL(2, C), and (2) one has M (AB) = M (B) for all A ∈ E and B ∈ SL(2, C). Proof. If A ∈ E and B ∈ SL(2, C), then Eq. (4.2.2) and the triviality of γA entail that M (AB) = M (AγA (B)) = M (A)M (B) = M (B), proving assertion (2). Clearly the identity element of SL(2, C) is contained in E, and if A, B ∈ E, one has M (AB) = M (B) = 1. Thus, E is closed under products and taking inverses, hence assertion (1) follows. We exploit these results to show that, in fact, the image of any triangular matrix in SL(2, C) under M is the identity.
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Lemma 4.16. For any triangular matrix A in SL(2, C), one has M (A) = 1, i.e. UC ∪ LC ⊂ E. Proof. Since Lemma 4.14 has already established the claim for any triangular A in case (a) and since M (A) 6= 1 in the remaining cases, it is necessary to show that case (b), (c) and (d) cannot occur. Note that the set {M (A)|A 6∈ E} of Lorentz transformations lies in a single component of the Lorentz group L (unless, of course, it is empty), as a consequence of the assumption that the given group G intersects at most two components of L and here M (A) 6∈ L↑+ . Hence if A, B 6∈ E, it follows that M (A)M (B) ∈ L↑+ and consequently (4.2.2) yields M (AγA (B)) ∈ L↑+ . The details of the exclusion of cases (b)–(d) will be given for such A ∈ LC — the argument for A ∈ UC is similar. With A ∈ LC , one has AγA (B) ∈ LC whenever (if A is in case (c)) B ∈ LC , respectively (if A is in case (b) or (d)) B ∈ UC . Moreover, the equation AγA (X) = 1 −1 has, for any A ∈ LC , the solution X = γA (A−1 ) in LC (in case (c)), respectively −1 in UC (in cases (b) and (d)), since γA is an isomorphism between the respective groups. Given an element A0 ∈ LC which is not contained in E, one can choose X0 such that A0 γA0 (X0 ) = 1 holds and get M (A0 )M (X0 ) = M (A0 γA0 (X0 )) = 1. This shows that also X0 does not lie in E, hence, by the first paragraph, one finds that M (AγA (X0 )) ∈ L↑+ whenever A 6∈ E. If, in addition, A ∈ LC , then by the second paragraph one has AγA (X0 ) ∈ LC , and Lemma 4.14 implies that M (A)M (X0 ) = M (AγA (X0 )) = 1 . Thus one concludes that the Lorentz element M (A) does not depend upon the choice of the element A contained in LC but not contained in E. The same is therefore true for the corresponding automorphisms γA . Let E0 denote the subgroup E ∩ LC of E and choose now A, B ∈ LC \ E0 . By the −1 −1 preceding paragraph one also has γB (B −1 ) ∈ LC \ E0 (respectively γB (B −1 ) ∈ UC \ E0 ). Thus, taking into account the first paragraph and the fact that γA = γB and M (A) = M (B), one finds, using (4.2.2), −1 M (AB −1 ) = M (AγA (γB (B −1 ))) −1 = M (A)M (γB (B −1 )) −1 (B −1 )) = M (B)M (γB −1 = M (BγB (γB (B −1 )))
= M (1) = 1 . Therefore, AB −1 ∈ E0 . It has therefore been established that (1) E0 ⊂ LC is a group, (2) if A ∈ LC \ E0 , then E0 ·A ⊂ LC \E0 (this is the content of Lemma 4.15 (2)), and (3) if A, B ∈ LC \E0 , then AB −1 ∈ E0 . Hence, for each A ∈ LC \ E0 one has the disjoint decomposition
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LC = E0 ∪ (E0 · A). But for each A ∈ LC there exists an element X ∈ LC such that X 2 = A. If X ∈ E0 , then so is A, since E0 is a group. Thus for A ∈ LC \ E0 one must have X 6∈ E0 . But on the other hand, if X ∈ E0 · A, then XA−1 ∈ E0 , so that 1 = X · XA−1 ∈ X · E0 , which implies X −1 ∈ E0 . Then again one has X ∈ E0 . This is a contradiction unless the set LC \ E0 is empty. Hence, cases (b)–(d) cannot occur. We are now in the position to complete the proof of Lemma 4.12. Since UC ∪ LC generates all of SL(2, C), we may conclude from Lemmas 4.15 and 4.16 that M maps SL(2, C) onto {1}, and consequently m maps L↑+ onto {1}. But this contradicts the fact that m must be the identity map on InvL↑+ (WR ), so the assertion in Lemma 4.12 follows. Case (i) is therefore excluded. Next we turn to case (ii), which is easily eliminated by pointing out the following simple consequence of Lemma 4.12. Lemma 4.17. If the group G ⊂ L intersects at most one connected component of L other than L↑+ and acts transitively upon the set W0 , then the group G ∩ InvL↑+ (WR ) is nontrivial. Proof. From Lemma 4.12 there exists a nontrivial element g0 in the group G ∩ InvL(WR ). If one such g0 happens to lie in L↑+ , the proof is over. So assume that all such g0 are not contained in L↑+ . Since G intersects at most one other component of L besides L↑+ , one must have G = G+ ∪ G+ g0 , where G+ = G ∩ L↑+ . Thus, the transitivity of the action of G upon W0 implies W0 = G · WR = G+ · WR ∪ G+ · g0 WR = G+ · WR . In other words, also the group G+ acts transitively upon the set W0 , even though G+ ∩ InvL(WR ) = {1}. But this possibility has been excluded by Lemma 4.12. We are ready to show that in the only remaining case, case (iii), the identity component of the Lorentz group must be contained in G, which is the statement of Proposition 4.2. We begin by noting that for any involutive element j ∈ L, there exists some wedge W ∈ W0 which is mapped by j either onto itself or onto its causal complement W 0 = −W . This follows from the fact that either j maps every lightlike vector ` onto `, respectively −`, or there exists a lightlike vector `1 such that its (lightlike) image `2 = j`1 is not parallel to `1 . In the latter case, the set {`1 , `2 } is mapped onto itself by j, since j is an involution. As every wedge is determined by two lightlike vectors, the statement then follows after a moment’s reflection. Now, as above, let G+ = G ∩ L↑+ and let G− = G \ G+ . We first consider the case where G− is empty. Then G+ acts transitively upon W0 and we can conclude from the simplicity of L↑+ that G+ = L↑+ in this case (cf. the argument directly following the statement of Lemma 4.12).
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Note that the assumption that G is generated by involutions has not been used above. This assumption will be exploited now in the case where G− is nonempty. For then there must be some involution j ∈ G− and a wedge W ∈ W0 such that either jW = W or jW = −W . Without loss of generality, we may assume that W = WR . Since G = G+ ∪ G+ j, the relation jWR = WR lets us conclude, as in the proof of Lemma 4.17, that G+ acts transitively on W0 and hence G+ = L↑+ by the preceding argument. In the remaining case, where jWR = −WR , we have W0 = G · WR = G+ · WR ∪ −G+ WR . In other words, for each W ∈ W0 there exists an element g ∈ G+ such that either gWR = W or gWR = −W . Now consider the element R0 ∈ L↑+ which implements the rotation of angle π about the x2 -axis: R0 = diag(1, −1, 1, −1). This element maps WR to its causal complement: R0 WR = −WR . Moreover, conjugation by R0 takes the elements of ˜ ∈ InvL↑+ (WR ) we find InvL↑+ (WR ) into their inverses, i.e. for each Λ ˜ −1 = Λ ˜ −1 . R0 ΛR 0
(4.2.3)
Since W0 = L↑+ WR , we may conclude from the above arguments that for every Λ ∈ L↑+ there exists an element gΛ ∈ G+ such that ΛWR = gΛ WR or ΛWR = −gΛWR = gΛ R0 WR . Hence, for every Λ ∈ L↑+ there exist elements gΛ ∈ G+ and ˜ ∈ InvL↑ (WR ) so that either (1) gΛ = ΛΛ ˜ or (2) gΛ = ΛR0 Λ. ˜ Define therefore the Λ + subset L↑+ , resp. L↑+ , consisting of those elements Λ of L↑+ in case (1), resp. case (1)
(2)
(2). We have L↑+ = L↑+ ∪ L↑+ . According to Lemma 4.17 there exists a nontrivial element h0 ∈ G∩InvL↑+ (WR ). (1)
(2)
Since InvL↑+ (WR ) is abelian, we find as before for any Λ ∈ L↑+ the relation −1 Λh0 Λ−1 = gΛ h0 gΛ ∈ G+ . Moreover, since G ∩ InvL↑+ (WR ) is a group, it also (1)
↑ contains the element h−1 0 . It follows that for any Λ ∈ L+ , we have (2)
˜ −1 Λ ˜ −1 R−1 Λ−1 = gΛ h−1 g −1 ∈ G+ , Λh0 Λ−1 = ΛR0 Λh 0 0 0 Λ using (4.2.3). It has therefore been established that Λh0 Λ−1 ∈ G+ for any element Λ ∈ L↑+ . Once again, it then follows from the simplicity of L↑+ that G+ = L↑+ . The proof of Proposition 4.2 is therewith completed. The next step is to show that a similar statement holds also for the Poincar´ e group. ↑ Proposition 4.3. Any subgroup G of the identity component P+ of the Poin4 car´e group, which acts transitively upon the set W of wedges in R , must equal ↑ P+ . Moreover, any subgroup G of the Poincar´e group P, which is generated by involutions, intersects at most two of the four connected components of P and which ↑ acts transitively upon the set W of wedges in R4 , must contain P+ .
Proof. As a first step, consider the canonical homomorphism σ : G → L which acts as σ(Λ, a) = Λ for (Λ, a) ∈ G. Since G acts transitively on the set of wedges,
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it follows that σ(G) acts transitively on the subset W0 of wedges whose edges contain the origin. For if W ∈ W0 there exists an element (Λ, a) ∈ G such that W = ΛWR + a, and since ΛWR ∈ W0 , it follows that W = ΛWR . Since σ(G) ⊂ L is also generated by its involutions and intersects with at most two components of L, one may apply Proposition 4.2 and conclude that L↑+ ⊂ σ(G). Consider now the following alternatives. (1) There exist an element Λ ∈ L and a, b ∈ R4 with a 6= b such that both elements (Λ, a) and (Λ, b) are contained in G. Since G is a group, it follows that (Λ, a)(Λ, b)−1 = (1, a − b) ∈ G. As has already been seen, for every element Λ ∈ L↑+ there exists some element (Λ, c) ∈ G; hence it follows that (Λ, c)(1, a − b)(Λ, c)−1 = (1, Λ(a − b)) ∈ G ,
Λ ∈ L↑+ .
Since (1, c) ∈ G implies that (1, −c) ∈ G, one may conclude that G contains all translations (1, x) with x · x equal to some fixed constant κ. Since (1, x), (1, x0 ) ∈ G imply that (1, x + x0 ) ∈ G, and since every y ∈ R4 can be written in the form y = P4 1 xi with xi · xi = κ, i = 1, . . . , 4, it also follows that G contains all translations. Consider now for given Λ ∈ L↑+ an element c ∈ R4 for which (Λ, c) ∈ G. Then one has by the preceding result (Λ, c)(1, −Λ−1 c) = (Λ, 0) ∈ G ; in other words, G also contains all the pure Lorentz transformations, as well. Thus, ↑ in this case one has P+ ⊂ G. (2) For every element Λ ∈ σ(G) there exists exactly one a(Λ) ∈ R4 such that (Λ, a(Λ)) ∈ G. Since G is a group, this entails the following cocycle relation for the translations: a(ΛΛ0 ) = a(Λ) + Λa(Λ0 ) ,
Λ, Λ0 ∈ σ(G) .
(4.2.4)
Consider the subgroup G0 ⊂ G whose elements translate the wedge WR without rotating it. The elements of G0 have the form (Λ, a(Λ)) with Λ ∈ InvL(WR ). So it ↑ follows from the first paragraph of this proof that for G+ 0 ≡ G0 ∩ P+ the equality ↑ ↑ σ(G+ 0 ) = InvL(WR ) ∩ L+ holds. Since InvL(WR ) ∩ L+ is abelian, the cocycle Eq. (4.2.4) implies that a(Λ) + Λa(Λ0 ) = a(ΛΛ0 ) = a(Λ0 Λ) = a(Λ0 ) + Λ0 a(Λ) , for every Λ, Λ0 ∈ InvL(WR ) ∩ L↑+ , which itself entails that (1 − Λ0 )a(Λ) = (1 − Λ)a(Λ0 ) . Fixing an element Λ0 ∈ InvL(WR ) ∩ L↑+ such that the matrix (1 − Λ0 ) is invertible and setting a ≡ (1 − Λ0 )−1 a(Λ0 ), one obtains a(Λ) = (1 − Λ)a ,
Λ ∈ InvL(WR ) ∩ L↑+ .
(4.2.5)
↑ Hence G+ 0 is comprised of the elements {(Λ, (1−Λ)a)|Λ ∈ InvL(WR )∩L+ } for some 4 fixed a ∈ R .
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+ − Now, if G− 0 ≡ G0 \ G0 is nonempty, there exists some g0 = (Λ0 , a0 ) ∈ G0 − + such that G0 = G0 · g0 (recall that G intersects at most two of the connected components of P). Hence, without loss of generality, one may assume that (1 + Λ0 ) is invertible. Since g02 = (Λ20 , a0 + Λ0 a0 ) ∈ G+ 0 , it follows from Eqs. (4.2.4) and 2 2 (4.2.5) that (1−Λ0)a = a(Λ0 ) = a(Λ0 )+Λ0 a(Λ0 ) = (1+Λ0 )a(Λ0 ) and consequently a(Λ0 ) = (1 − Λ0 )a. Applying Eq. (4.2.4) another time yields
a(ΛΛ0 ) = a(Λ) + Λa(Λ0 ) = (1 − ΛΛ0 )a for arbitrary Λ ∈ InvL(WR ) ∩ L↑+ , which finally shows that G0 = {(Λ, (1 − Λ)a)|Λ ∈ σ(G0 )} . Hence, G0 induces solely translations of the edge of the wedge WR along some (two-sheeted) hyperbola or light ray, contradicting the assumption that G acts transitively on W. Therefore, only case (1) can arise and the proof of the proposition is complete. Summing up the results obtained so far in this section, we see that the symmetry groups G which arise by the CGMA in Minkowski space theories must contain the ↑ proper orthochronous Poincar´e group P+ if they act transitively on the set of wedges W. This result will enable us in the next step to determine G exactly, as well as the action of its generating involutions on Minkowski space. Proposition 4.4. Let the group T act transitively upon the set W of wedges in R4 , and let G be the corresponding subgroup of the Poincar´e group. Moreover, let gWR = (ΛWR , aWR ) be the involutive element of the Poincar´e group corresponding to the involution τWR ∈ T . Then aWR = 0 and ΛWR = P1 T = diag(−1, −1, 1, 1), where P1 is the reflection through the 1-axis and T is the time reflection. Since ↑ all wedges are transforms of WR under P+ , these assertions are also true, with the obvious modifications, for the involution gW corresponding to any wedge W ∈ W. In particular, one has gW W = W 0 , for every W ∈ W. In addition, G exactly equals ↑ the proper Poincar´e group P+ , and every element of P+ can be obtained as a product of an even number of involutions, gW , W ∈ W. Proof. If τ0 ∈ T leaves a given wedge W ∈ W0 fixed, then Lemma 2.1 (3) entails that τW τ0 = τ0 τW . Hence, if gW and g0 are the corresponding elements in the Poincar´e group, one must have g0 gW g0−1 = gW . In light of Lemma 4.11 and Proposition 4.3, this implies that gW must commute with every element of the invariance group InvP↑+ (W ). With gW = (ΛW , aW ), it follows that one must have −1 (Λ0 ΛW Λ−1 0 , a0 + Λ0 aW − Λ0 ΛW Λ0 a0 ) = (ΛW , aW ) ,
for arbitrary (Λ0 , a0 ) ∈ InvP↑+ (W ). By setting a0 = 0 and letting Λ0 vary freely through InvL↑+ (W ), this equation implies aW = 0, and therefore Λ0 ΛW Λ−1 0 = ΛW and (1 − ΛW )a0 = 0 , (4.2.6)
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for all (Λ0 , a0 ) ∈ InvP↑+ (W ). Furthermore, one has Λ2W = 1, since gW is an involution. Choosing W = WR , one concludes from (4.2.6) that ΛW must have the form ! X 0 ΛW = , Y 1 for suitable 2 × 2-matrices X, Y . Since ΛW is a Lorentz transformation, it is easy to see that Y = 0. The facts that ΛW must commute with the Lorentz boosts in the 1-direction (leaving WR invariant) and that Λ2W = 1 lead then, after some elementary computation, to X = ±1. But in the case where the positive sign is taken, one would have τWR (WR ) = WR , which is excluded by Lemma 2.1 (4) and the fact that there are no atoms in W. The remaining assertions are now easy to verify. 4.3. From wedge transformations back to the net: Locality, covariance and continuity Having established the geometrical features of the elements of the group T , we turn now to the discussion of its representations induced by the modular conjugations. Proposition 4.4 implies that there exists a projective representation J(P+ ) of the proper Poincar´e group with coefficients in the internal symmetry group of the net {R(W )}W ∈W . The next step is to verify that this projective representation acts geometrically correctly upon the net, in other words that the net is Poincar´e covariant under this projective representation. Proposition 4.5. Let the CGMA obtain with the choices M = R4 and W equal to the set of wedgelike regions in R4 , and let the adjoint action of J upon the set {R(W )}W ∈W be transitive. Then the projective representation J(P+ ) of the proper Poincar´e group whose existence is entailed by Corollary 2.1 and Proposition 4.4 acts geometrically correctly upon the net {R(W )}W ∈W , i.e. for each Λ ∈ P+ and each W ∈ W one has J(Λ)R(W )J(Λ)−1 = R(ΛW ) . Furthermore, Haag duality holds for {R(W )}W ∈W , hence the net {R(W )}W ∈W satisfies Einstein locality. Proof. By construction, for each g ∈ G = P+ there exists an element τg ∈ T such that τg (W ) = gW for all W ∈ W. Hence, for all W ∈ W, one has n(τg ) Y τij (W ) = R(τg (W )) = R(gW ) , J(g)R(W )J(g)−1 = R j=1
where the product indicated is taken over the chosen product for the element τg ∈ T implicit in the definition of the projective representation J(T ). By Proposition 4.4, one has R(W )0 = JW R(W )JW = R(gW W ) = R(W 0 ) ,
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for each W ∈ W. So Haag duality holds; thus, for each W1 ⊂ W 0 , one has R(W1 ) ⊂ R(W 0 ) = R(W )0 . Note that these results do not depend upon the choice of projective representation J(P+ ). We next provide conditions on the net {R(W )}W ∈W which imply ↑ that there exists a strongly continuous projective representation of P+ . These conditions essentially involve a continuity property of the map W 7→ R(W ). First ↑ note that since W = P+ WR , W is in 1-1 correspondence with the quotient space ↑ ↑ P+ /InvP+ (WR ); the latter’s topology induces thereby a topology on W. Consider then a continuous collection {W }>0 of wedges in W such that W → W as → 0, S T for some fixed W ∈ W. For δ > 0, let Aδ ≡ 0≤<δ W and Iδ ≡ 0≤<δ W , T S where W0 ≡ W . Define R(Iδ ) ≡ 0≤<δ R(W ) and R(Aδ ) ≡ ( 0≤<δ R(W ))00 to be the indicated intersection and union of wedge algebras. Note that {R(Aδ )}δ>0 , resp. {R(Iδ )}δ>0 , is a monotone decreasing, resp. increasing, family of von Neumann algebras. Our net continuity assumption is given next. Net Continuity Condition. For any W ∈ W and any continuous collection {W }>0 ⊂ W converging to W, the net {R(W )}W ∈W satisfies R(W ) =
[
!00 R(Iδ )
=
δ>0
\
R(Aδ ) .
δ>0
Moreover, there exists a δ0 > 0 such that Ω is cyclic for the algebras R(Iδ ), with 0 < δ < δ0 . We proceed with the following result, which establishes that the mentioned net continuity condition implies a certain continuity in nets of associated modular objects in the context of the CGMA. Condition (ii) of the CGMA entails that Ω is cyclic for R(Aδ ), 0 < δ < δ1 . And the Haag duality proven in Proposition 4.5 yields R(Aδ ) = 0
[
0≤<δ
0
R(W ) =
\
0≤<δ
R(W )0 =
\
R(W0 ) ,
0≤<δ
for which Ω is cyclic whenever δ is sufficiently small, by hypothesis. Hence, there exists a δ1 > 0 such that Ω is cyclic and separating for R(Aδ ), 0 < δ < δ1 . Moreover, since W 0 ⊂ Iδ0 , for all δ > 0, Ω is also separating for R(Iδ ). Hence, the CGMA and the net continuity condition imply that the modular objects JIδ , ∆Iδ , resp. JAδ , ∆Aδ , corresponding to the pair (R(Iδ ), Ω), resp. (R(Aδ ), Ω), exist for all 0 < δ < min{δ0 , δ1 }. Below we shall tacitly take 0 < δ < min{δ0 , δ1 } without further comment. Proposition 4.6. Assume the CGMA with the choices M described, as well as the mentioned net continuity condition. continuous net of wedges such that W → W as → 0. Then the verges strongly to JW as → 0. In addition, the net {∆it W }>0 to ∆it as → 0. W
= R4 and W as Let {W }>0 be a net {JW }>0 conconverges strongly
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Proof. By Corollary A.2 of [24], which is based upon a result of [25], it follows −1 from the hypotheses that ∆Iδ → ∆W and ∆−1 Aδ → ∆W in the strong resolvent sense, and JIδ → JW and JAδ → JW in the strong operator topology (note that T S R(W )0 = ( δ>0 R(Aδ ))0 = ( δ>0 R(Aδ )0 )00 ). On the other hand, from Eq. (2.6) in [31], one has the bounded operator inequality (1 + ∆Aδ )−1 ≤ (1 + ∆W )−1 ≤ (1 + ∆Iδ )−1 ,
(4.3.1)
for all 0 < < δ. Employing this inequality, the polarization identity, and the stated strong resolvent convergence, it follows easily that (1 + ∆W )−1 converges weakly to (1+∆W )−1 . By the positivity of the operators in (4.3.1) and the operator monotonicity of the operation of taking square roots, (4.3.1) also entails (1 + ∆Aδ )−1/2 ≤ (1 + ∆W )−1/2 ≤ (1 + ∆Iδ )−1/2 , for all 0 < < δ, so that by the same argument, also (1 + ∆W )−1/2 converges weakly to (1 + ∆W )−1/2 . In order to make the following computations somewhat more transparent, let RW ≡ (1 + ∆W )−1 and RW ≡ (1 + ∆W )−1 . One observes then that for any vector Φ ∈ H the expression 1/2
1/2
1/2
1/2
1/2
1/2
k(RW − RW )Φk2 = hΦ, (RW − RW RW − RW RW + RW )Φi 1/2
must converge to zero as → 0. Since {RW }>0 is uniformly bounded, RW converges also strongly to RW . Standard arguments then yield the strong convergence it of {∆it W }>0 to ∆W as → 0. 1/2
To proceed further, note that from the above it follows that ∆W is the strong 1/2 graph limit of the net {∆Iδ }. In particular, there exists a dense subset K in the 1/2
domain of definition of ∆W such that for each Φ ∈ K there exists a corresponding net {Φδ } with Φδ ∈ R(Iδ )Ω satisfying Φδ → Φ and 1/2
1/2
∆Iδ Φδ → ∆W Φ . Since the Tomita–Takesaki conjugations SIδ are restrictions of the corresponding conjugations SW to R(Iδ )Ω (for all 0 < < δ), one sees that this implies 1/2
1/2
1/2
JW ∆W Φδ = SW Φδ = SIδ Φδ = JIδ ∆Iδ Φδ → JW ∆W Φ , for all 0 < < δ, since JIδ converges strongly to JW . But this convergence of 1/2 JW ∆W Φδ entails the convergence of (δ → 0, 0 < < δ) 1 1+
1/2 ∆W
JW Φδ = →
As the nets {Φδ }δ>0 and
1 1+
1 1+
JW ∆W Φδ = 1/2
−1/2 ∆W
(
1
1/2
−1/2 ∆W
JW ∆W Φ = )
1 1/2
1 + ∆W
>0
1+
1/2
−1/2 ∆W
1 1/2
1 + ∆W
JIδ ∆Iδ Φδ
JW Φ .
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519
converge strongly, this proves the weak convergence 1 1+
1/2 ∆W
JW Φ →
1 1/2
1 + ∆W
JW Φ .
Hence, JW converges weakly (and thus also strongly, since the operators are antiunitary) to JW . The preceding proposition establishes that to every continuous net of wedges is associated a strongly continuous net of modular involutions. Using this fact and the explicit knowledge which Proposition 4.4 furnishes about the geometric action ↑ of the generators of the group G, we shall show that there exists a choice of J(P+ ) which is strongly continuous. In the following, U(H) denotes the group of unitary operators acting on the separable Hilbert space H. Proposition 4.7. Assume the CGMA with the choices M = R4 and W as described, along with the transitivity of the adjoint action of {JW |W ∈ W} on the net {R(W )}W ∈W , and the net continuity condition stated at the beginning of this sec↑ tion. Then there exists a strongly continuous projective representation V (P+ )⊂J ↑ which acts geometrically correctly upon the net {R(W )}W ∈W . of the group P+ ↑ As was already pointed out, any of the projective representations J(P+ ) furnished by Corollary 2.1 acts geometrically correctly upon the net {R(W )}W ∈W . Here, we merely make a particular choice amongst these in order to explicitly assure that the projective representation is continuous. The unitarity of the representation is already guaranteed by Proposition 4.4. We shall prove Proposition 4.7 in a series of steps. To begin, we shall define ↑ and show that they actually projective representations of certain subgroups of P+ yield continuous representations of their respective subgroups. Consider a wedge W (0) ∈ W0 containing the origin of R4 in its edge and denote by x(0) , y (0) , etc., any translation in the two-dimensional subspace R2W (0) generated by the two lightlike directions fixing the boundaries of W (0) . Denote by Jz(0) the modular involution associated with (R(W (0) + z (0) ), Ω). It follows from Propositions 4.4 and 4.5 that
Jz(0) R(W )Jz(0) = R(ΛW (0) W + 2z (0) ) , for all W ∈ W, where ΛW (0) ∈ L+ is the reflection which is equal to −1 on R2W (0) and equal to 1 on the two-dimensional subspace of R4 which forms the edge of W (0) . (This relation was Assumption (1) in [24].) One therefore sees that Jx(0) Jy(0) R(W )Jy(0) Jx(0) = R(W + 2x(0) − 2y (0) ) , for any W ∈ W, W (0) ∈ W0 , and x(0) , y (0) as described above. ↑ For x(0) ∈ R2W (0) ⊂ R4 ⊂ P+ , choose VW (0) (2x(0) ) ≡ Jx(0) JW (0) . Then Proposition 4.6 entails immediately that x(0) 7→ VW (0) (x(0) ) is a strongly continuous family of unitary operators implementing the action of the subgroup R2W (0) of the translation group on the net {R(W )}W ∈W . This is true for any choice of W (0) ∈ W0 .
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(0)
Next, consider the wedges Wi = {x ∈ R4 |xi > |x0 |}, i = 1, 2, 3, and the corresponding projective representations Vi (R2 (0) ), i = 1, 2, 3. These unitary Wi
operators will be used to build the desired representation of the translation group. We shall first show that they coincide on the subgroup of time translations. To this end we make use of the fact that the rotations in the time-zero plane are induced by unitary operators in J , cf. Proposition 4.5. Hence, if R is a rotation by π/2 about the 1-axis, we obtain from Lemma 2.1 (2), using the abbreviation x0 = (x0 , 0, 0, 0), the equalities J(R)V1 (2x0 )J(R)−1 = J(R)JW (0) +x0 JW (0) J(R)−1 1
1
= JRW (0) +x0 JRW (0) = V1 (2x0 ) , 1
(0) RW1
1
(0) W1 .
= Here we have made use of the important fact, a consequence since of Proposition 4.5 and the uniqueness of modular objects, that the modular conjugations associated with wedges transform covariantly under the adjoint action of the (anti)unitary operators in J , i.e. J(λ)JW J(λ)−1 = JλW ,
(4.3.2)
for any choice of wedge W ∈ W and Poincar´e transform λ ∈ P+ . Secondly, we know from Corollary 2.1 that V1 (x0 ) = Z(x0 )V2 (x0 ), where Z(x0 ) is an internal symmetry of the net {R(W )}W ∈W in the center of J . In the light of the equalities J(R)V2 (2x0 )J(R)−1 = J(R)JW (0) +x0 JW (0) J(R)−1 2
2
= JR(W (0) +x0 ) JRW (0) = V3 (2x0 ) , 2
using
(0) R(W2
+ x0 ) =
(0) W3
2
+ x0 , we arrive at the relation
Z(x0 )V2 (x0 ) = V1 (x0 ) = J(R)V1 (x0 )J(R)−1 = Z(x0 )V3 (x0 ) . Thus V2 (x0 ) = V3 (x0 ), and in a similar way one proves V1 (x0 ) = V3 (x0 ). We therefore write V ((x0 , 0, 0, 0)) for Vi (x0 ). This technique of establishing the equality of unitary implementers will also be used in the subsequent arguments in order to solve the cohomological problems involved in the discussion of the projective representation. Now, for any x = (x0 , x1 , x2 , x3 ) ∈ R4 , we define V (x) ≡ V ((x0 , 0, 0, 0))V1 ((0, x1 , 0, 0))V2 ((0, 0, x2 , 0))V3 ((0, 0, 0, x3 )) . As in the proof of Proposition 2.2 in [24], one verifies that x 7→ V (x) is a projective unitary representation of the translation subgroup of the Poincar´e group acting geometrically correctly on the net {R(W )}W ∈W . In order to prove that it is actually a representation, we must show that the various factors in the definition of V commute. Let us consider, for example, the operator V1 ((0, x1 , 0, 0)), which leaves Ω invariant and satisfies (0)
V1 ((0, x1 , 0, 0))R(W2
+ z (0) )V1 ((0, x1 , 0, 0))−1 = R(W2
(0)
+ z (0) ) ,
GEOMETRIC MODULAR ACTION AND SPACETIME SYMMETRY GROUPS
for every z (0) ∈ R2
(0)
W2
coherent family
521
. So it must commute with the modular involutions of the
(0) {R(W2 +z (0) )|z (0)
∈ R2
(0)
W2
} and therefore also with V ((x0 , 0, 0, 0))
= V2 ((x0 , 0, 0, 0)) and V2 ((0, 0, x2 , 0)). Similarly, V1 ((0, x1 , 0, 0)) commutes with V3 ((0, 0, 0, x3 )), and by the same argument one can establish the commutativity of the remaining unitaries. We next show that the unitaries in the definition of V define continuous representations of the respective one-dimensional subgroups. Lemma 4.18. Under the assumptions of Proposition 4.7, for any i = 1, 2, 3 and (0) x(0) ∈ R2 (Wi ), the mapping R 3 t 7→ Vi (tx(0) ) is a strongly continuous homomorphism. Proof. For convenience, set Jt ≡ JW (0) +tx(0) and V (t) = Jt/2 J0 , t ∈ R. It follows i from Proposition 4.6 that Jt/2 and hence V (t) is strongly continuous in t. Since V (t)J0 = J0 V (t)−1 and similarly V (t)Jt/2 = Jt/2 V (t)−1 , one obtains with the help of relation (4.3.2), for n ∈ N, V (t)2n J0 = V (t)n J0 V (t)−n = Jnt , and consequently one has V (t)2n = V (t)2n J02 = Jnt J0 = V (2nt) . Similarly one finds V (t)2n+1 = V (t)2n Jt/2 J0 = V (t)n Jt/2 V (t)−n J0 = J(n+1/2)t J0 = V ((2n + 1)t) . From these relations one sees in particular that for m1 , m2 ∈ N and 0 6= n ∈ Z, V (m1 /n)V (m2 /n) = V (1/n)m1 V (1/n)m2 = V (1/n)m1 +m2 = V ((m1 + m2 )/n) . Since m1 , m2 , n are arbitrary and V (t) is continuous, the remaining portion of the assertion follows. Combining this lemma with the preceding results, we have thus established the fact that the unitary operators V (x) introduced above define a continuous representation of the translations: Lemma 4.19. Under the assumptions of Proposition 4.7, there exists in J a strongly continuous unitary representation V (R4 ) of the translation subgroup which acts geometrically correctly upon the net {R(W )}W ∈W .g g It is possible to show the existence of a continuous representation of the translation group without the assumption of the net continuity condition. This argument will be presented in a subsequent publication.
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Let us now turn to the Lorentz transformations. As is well known, any Lorentz transformation Λ ∈ L↑+ can uniquely be decomposed in the chosen Lorentz system √ into a boost B and a rotation R, Λ = BR, where B = ΛΛT and R = B −1 Λ. It is apparent that the factors appearing in this decomposition are continuous in Λ. We first define unitary operators corresponding to the boosts and rotations individually. Given a nontrivial boost B there exists a unique two-dimensional subspace R2B in the time-zero plane {x ∈ R4 |x0 = 0} of the chosen Lorentz system which is perpendicular to the boost direction and therefore pointwise invariant under the action of B. We pick an arbitrary unit vector e ∈ R2B and consider the corresponding wedge (0) We = {x ∈ R4 |x · e > |x0 |}. An elementary computation using Proposition 4.4 shows that the Poincar´e transformations associated with the corresponding modular conjugations satisfy gBW (0) gW (0) = B 2 . This leads us to define e
e
Ve (B) ≡ JB 1/2 W (0) JW (0) , e
e
1/2
where B is the unique boost whose square is equal to B. If B = 1, we set Ve (1) = 1. This definition is consistent since J 2 (0) = 1 for any unit vector e. We In a similar manner we construct implementers of the rotations. Given any proper rotation R 6= 1 there is a unique two-dimensional subspace R2R which is perpendicular to the axis of revolution of R and therefore stable under the action of this rotation. As in the case of the boosts, we consider for e ∈ R2R the corresponding (0) wedge We and find gRW (0) gW (0) = R2 . Correspondingly, we set e
e
Ve (R) ≡ JR1/2 W (0) JW (0) , e
e
where R1/2 is defined as the rotation with the same axis of revolution as R but with half the rotation angle. This definition requires a consistency check because of the nonuniqueness of the square root of rotations. So let R1 , R2 be two different square roots of R (0) (0) (differing by a rotation by π). Then R2 We = −R1 We and, consequently, JR2 W (0) JW (0) = J−R1 W (0) JW (0) . Because of Haag duality, we have R(−W (0) ) = e
0
e
e
e
R(W (0) ) = R(W (0) )0 , for any wedge W (0) , and consequently J−W (0) = JW (0) . Hence, we have the equality J−R1 W (0) = JR1 W (0) , proving the consistency of the e e definition of Ve (R). We shall show in the next lemma that the implementers of boosts and rotations defined above do not depend on the choice of the vector e.
Lemma 4.20. Let Ve (B), Ve (R) be the unitary operators implementing the boost B and rotation R, respectively. These operators do not depend on the choice of the vector e within the above-stated limitations. Proof. Consider first the case of boosts. If B = 1, there is nothing to prove. So let B 6= 1, let R2B be the corresponding two-dimensional invariant subspace and let B1 be any other boost which leaves this subspace pointwise invariant. As in the case of the translations discussed in Lemma 4.18, it follows from relation (4.3.2) that for any e ∈ R2B one has Ve (B1 )n = Ve (B1n ), for n ∈ N.
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523
Now let Rφ be a rotation by φ about the axis established by the direction of the boost B and let J(Rφ ) be a corresponding implementer. Then one obtains from relation (4.3.2) J(Rφ )Ve (B1 )J(Rφ )−1 = JR
1/2 (0) We φ B1
JR
(0) φ We
= JB 1/2 W (0) JW (0) = VRφ e (B1 ) , 1
Rφ e
Rφ e
1/2
since Rφ and B1 commute. On the other hand, according to Corollary 2.1, there exists some element Zφ in the subgroup of internal symmetries Z of J such that VRφ e (B1 ) = Zφ Ve (B1 ) . Setting φ = 2mπ/n, for m, n ∈ N, one sees from the preceding two relations that n Ve (B1 ) = VRn2mπ/n e (B1 ) = J(R2mπ/n )n Ve (B1 )J(R2mπ/n )−n = Z2mπ/n Ve (B1 ) , n = 1. Hence, and consequently Z2mπ/n n VR2mπ/n e (B1n ) = VR2mπ/n e (B1 )n = Z2mπ/n Ve (B1 )n = Ve (B1n ) ,
and setting B1 = B 1/n one obtains VR2mπ/n e (B) = Ve (B) . According to Proposition 4.6, the operator JRφ W (0) depends continuously on φ for any wedge W (0) , and the same is thus also true of VRφ e (B). It therefore follows from the preceding relation that VRφ e (B) = Ve (B) for any rotation Rφ , proving the assertion for the case of the boosts. For the rotations R, one proceeds in exactly the same way as above. The role of Rφ is here played by the rotations about the axis of revolution fixed by R. In view of this result we may omit in the following the index e and set V (B) ≡ Ve (B) ,
V (R) ≡ Ve (R) .
We next discuss the continuity properties of these operators with respect to the boosts and rotations. Lemma 4.21. The unitary operators V (B) and V (R) depend (strongly) continuously on the boosts B and rotations R, respectively. Proof. Let Bn be a sequence of boosts which converges to B. If B 6= 1 it is clear that the distance between the unit disks in the corresponding invariant subspaces R2Bn and R2B converges to 0. In particular, there exists a sequence of unit vectors en ∈ R2Bn which converges to some e ∈ R2B and consequently the sequence of 1/2
(0)
(0)
wedges Bn Wen converges to B 1/2 We . Because of the continuity of the modular operators JW with respect to W , established in Proposition 4.6, one concludes that V (Bn ) = JB 1/2 W (0) JW (0) → JB 1/2 W (0) JW (0) = V (B) . n
en
en
e
e
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D. BUCHHOLZ et al.
If the sequence Bn converges to 1, the corresponding unit disks in R2Bn need not converge. But, because of the compactness of the unit ball in R3 , for any sequence of unit vectors en ∈ R2Bn , there exists a subsequence eσ(n) which converges to some 1/2
1/2
(0)
unit vector eσ . Since Bσ(n) → 1, the corresponding sequences of wedges Bσ(n) Weσ(n) (0)
(0)
and Weσ(n) converge to Weσ , and consequently one has V (Bσ(n) ) = JB 1/2
σ(n)
(0)
Weσ(n)
→ JW (0) JW (0) = 1 .
JW (0)
eσ(n)
eσ
eσ
Since the choice of the sequence en ∈ R2Bn was arbitrary, the proof of the continuity of the boost operators is complete. The argument for the rotations is analogous; the only difference being that the boost direction must be replaced by the axis of revolution. We are now in the position to prove Proposition 4.7. Given an element (Λ, x) ∈ ↑ P+ , we proceed to the unique and continuous decomposition (Λ, x) = (1, x)(B, 0)(R, 0) and set V ((Λ, x)) ≡ V (x)V (B)V (R) , where the unitary operators corresponding to the translations, boosts and rotations have been defined above. Since these operators depend continuously on their arguments, the assertion of Proposition 4.7 follows. As a matter of fact, we shall see ↑ . A that the unitary operators V ((Λ, x)) actually define a true representation of P+ first step in this direction is the following lemma. ↑ Lemma 4.22. Let V (·) be the continuous unitary projective representation of P+ introduced above. One has:
(1) V (R)V (B)V (R)−1 = V (RBR−1 ) and V (R)V (R0 )V (R)−1 = V (RR0 R−1 ), for all boosts B and rotations R, R0 ; (2) V (·) defines a true representation of every continuous one-parameter subgroup of boosts or rotations. Proof. The first statement in (1) follows from relation (4.3.2) and Lemma 4.20, which imply V (R)V (B)V (R)−1 = V (R)JB 1/2 W (0) JW (0) V (R)−1 e
e
= JRB 1/2 R−1 W (0) JW (0) = V (RBR−1 ) , Re
Re
where the last equality follows from the fact that RBR−1 is again a boost which leaves the subspace RR2B pointwise invariant. The argument for the rotations is analogous. Now let {G(u)|u ∈ R}, be a continuous one-parameter group of boosts or rotations. As in the proof of Lemma 4.18, one shows by an elementary computation on
525
GEOMETRIC MODULAR ACTION AND SPACETIME SYMMETRY GROUPS
the basis of relation (4.3.2) that V (G(u))n = V (G(u)n ) = V (G(nu)). Consequently, one finds that, for m1 , m2 ∈ N and 0 6= n ∈ Z, V (G(m1 /n))V (G(m2 /n)) = V (G(1/n))m1 V (G(1/n))m2 = V (G(1/n))m1 +m2 = V (G((m1 + m2 )/n)) . The stated assertion (2) thus follows once again from the continuity properties of V (·). Instead of proving by explicit but tedious computations that V (·) defines a ↑ true representation of P+ , we prefer to give a more abstract argument based on cohomology theory. In the appendix it is shown that the existence of a continuous ↑ unitary projective representation V (P+ ) with values in J implies that there is a ↑ . U continuous unitary representation U (·) of the covering group ISL(2, C) of P+ takes values in the closure J of J in the strong operator topology. Moreover, there exists a mapping Z : ISL(2, C) → Z, the closure of the internal symmetry group Z in the center of J , such that U (A) = Z(A)V (µ(A)), for all A ∈ ISL(2, C), where ↑ µ : ISL(2, C) → P+ is the canonical covering homomorphism whose kernel is a subgroup of order 2, the center of ISL(2, C). The preceding results enable us to show that U (·) acts trivially on the cen↑ ter of ISL(2, C) and therefore defines a representation of P+ . For let A1 , A2 ∈ ISL(2, C) be two elements corresponding to rotations by π about two orthogonal −1 axes, i.e. µ(Ai ) = Ri (π), i = 1, 2. It then follows that A1 A2 A−1 1 A2 = C, where C is the nontrivial element in the center of ISL(2, C). Consequently, we have U (C) = U (A1 )U (A2 )U (A1 )−1 U (A2 )−1 = Z(A1 )V (R1 (π))Z(A2 )V (R2 (π))V (R1 (π))−1 Z(A1 )−1 V (R2 (π))−1 Z(A2 )−1 = V (R1 (π))V (R2 (π))V (R1 (π))−1 V (R2 (π))−1 ,
(4.3.3)
where we made use of the fact that the operators Z(Ai ), i = 1, 2, are elements of Z and therefore commute through the product and cancel. Since R1 (π)R2 (π)R1 (π)−1 = R2 (π), we see from Lemma 4.22 (1) that V (R1 (π))V (R2 (π))V (R1 (π))−1 = V (R1 (π)R2 (π)R1 (π)−1 ) = V (R2 (π)) . So we conclude that U (C) = 1, as claimed. We can therefore set U (µ(A)) ≡ U (A) ,
A ∈ ISL(2, C) .
In the final step of our argument we make use of the fact that the Poincar´e group is perfect. (Recall that a group is perfect if it is equal to its commutator subgroup ↑ — see the appendix.) Given λi ∈ P+ , i = 1, 2, one can show in the same way as in relation (4.3.3) that −1 −1 U (λ1 λ2 λ−1 V (λ2 )−1 . 1 λ2 ) = V (λ1 )V (λ2 )V (λ1 )
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D. BUCHHOLZ et al.
Since the elements on the right-hand side of this equation are contained in J , we conclude that the representation U (·) also has values in J (so one does not need to proceed to the closure J¯). It then follows from Proposition 4.5 that the unitary ↑ operators U (λ), λ ∈ P+ , act geometrically correctly on the net {R(W )}W ∈W . Now, given a wedge W and the corresponding modular conjugation JW and reflection gW ∈ P+ — see Proposition 4.4 — it follows from relation (4.3.2) that −1 JW V (λ)JW = ZV (gW λgW ), where Z ∈ Z is some internal symmetry. As these central elements drop out in group theoretic commutators of the operators V (λ), ↑ we can compute the adjoint action of the modular conjugations JW on U (P+ ) by making use of the relation −1 −1 JW U (λ1 λ2 λ−1 V (λ2 )−1 JW 1 λ2 )JW = JW V (λ1 )V (λ2 )V (λ1 ) −1 −1 −1 −1 −1 −1 = V (gW λ1 gW )V (gW λ2 gW )V (gW λ1 gW ) V (gW λ2 gW ) −1 −1 = U (gW λ1 λ2 λ−1 1 λ2 gW ) . ↑ is perfect, this shows that Since P+ −1 JW U (λ)JW = U (gW λgW ),
↑ for λ ∈ P+ .
(4.3.4)
Hence the involution JW induces the outer automorphism corresponding to gW on ↑ U (P+ ), so we may take U (gW ) ≡ JW . The fact that U (λ) ∈ J acts geometrically correctly on the net implies, according to relation (4.3.2), U (λ)JW U (λ)−1 = JλW ,
↑ for λ ∈ P+ .
(4.3.5)
↑ Let W1 , W2 ∈ W be arbitrary. There exists an element λ ∈ P+ such that W2 = λW1 . Hence the Poincar´e covariance of {R(W )}W ∈W and condition (i) of the CGMA entail the relation gW2 = λgW1 λ−1 . Using (4.3.4) and (4.3.5), we therefore have the equalities
U (gW1 )U (gW2 ) = JW1 JW2 = JW1 JλW1 = JW1 U (λ)JW1 U (λ)−1 −1 = U (gW1 λgW )U (λ)−1 1 −1 −1 = U (gW1 λgW λ ) 1
= U (gW1 gW2 ) ,
(4.3.6)
−1 since gW = gW1 . Hence, U (·) provides a representation for all of G = P+ . 1 ↑ Since J is generated by the conjugations JλW , λ ∈ P+ , we conclude that J = ↑ ↑ ↑ ) ∪ JW (0) U (P+ ), for any fixed wedge W (0) ∈ W0 . Moreover, J + = U (P+ ), U (P+ + where J is the subgroup of unitary operators in J which is generated by products ↑ of an even number of modular conjugations. As U is a faithful representation of P+ ↑ — cf. the standing assumptions in Sec. 2 — and P+ has trivial center, the center of J consists only of 1. Hence the representation U (·) must coincide with V (·). This ↑ shows finally that V (·) defines a representation of P+ , as claimed. We summarize these findings in the following theorem.
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Theorem 4.2. Assume the CGMA with the choices M = R4 and W the described set of wedges. If J acts transitively upon the set {R(W )}W ∈W and the net continuity condition mentioned at the beginning of Subsec. 4.3 holds, then there exists a strongly continuous (anti) unitary representation U (P+ ) of the proper Poincar´e group which acts geometrically correctly upon the net {R(W )}W ∈W and which sat↑ isfies U (gW ) = JW , for every W ∈ W. Moreover, U (P+ ) equals the subgroup of J ↑ ↑ consisting of all products of even numbers of JW ’s and J = U (P+ ) ∪ JWR U (P+ ). Furthermore, U (·) coincides with the representation V (·), which has been explicitly constructed above. 5. Geometric Action of Modular Groups and the Spectrum Condition A physically important property of a representation of the translation group on R4 is the spectrum condition, in other words, the condition that the generators of the given representation U (R4 ) have their joint sspectrum sp(U ) in the closed forward light cone V+ (for the positive spectrum condition) or in the closed backward light cone V− (for the negative spectrum condition). In Subsec. 5.1 we examine how to incorporate the spectrum condition into our setting, using only the modular objects. We shall show that the (positive or negative) spectrum condition holds whenever the group J generated by the initial modular involutions contains also the initial modular groups. Some further consequences of the spectrum condition in our setting, such as the PCT and Spin & Statistics Theorems, will also be discussed. We then turn our attention to the possible geometric action of the modular unitaries. In Subsec. 5.2 we shall reconsider the condition of modular covariance, which has been extensively discussed in the literature [22, 26, 35, 36]. If W0 ∈ W is a wedge, {∆it W0 }t∈R is the modular group corresponding to (R(W0 ), Ω), and {λ(t)}t∈R is the one-parameter subgroup of (suitably Poincar´e-transformed) boosts leaving W0 invariant, then modular covariance is said to hold if −it ∆it W0 R(W )∆W0 = R(λ(t)W ) ,
for all t ∈ R, W ∈ W ,
in other words, if the modular group associated to the algebra for the wedge W0 implements the mentioned boost subgroup. In fact, the subgroup {λ(t)}t∈R is usually more precisely specified: if `± are two positive lightlike translations such that ↑ W0 ± `± ⊂ W0 , one has in P+ the relation λ(t)(1, `± )λ(t)−1 = (1, e∓αt `± ) , with α = ±2π. The sign is a matter of convention fixing the direction of time. Bisognano and Wichmann [9, 10] (see also [29]) have shown that modular covariance holds for nets associated to finite-component Wightman fields in a Poincar´ e covariant vacuum representation. We shall show that if the adjoint action of the modular groups corresponding to the wedge algebras leaves the set {R(W )}W ∈W invariant, then modular covariance follows and either the positive or the negative spectrum condition holds. Moreover, under the same assumptions plus the locality of the net, the modular conjugations {JW }W ∈W will be seen to act geometrically
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as reflections about spacelike lines, i.e. as in Proposition 4.4. In Subsec. 5.3 we shall present some examples of nets satisfying all assumptions made in our program through Sec. 4, but violating the condition of modular covariance. In one of these examples the spectrum condition is violated, in the other the positive spectrum condition obtains. We then contrast the approaches to geometric modular action through the modular conjugations or through the modular groups in the light of the results of Subsec. 5.2 and the mentioned examples. 5.1. The modular spectrum condition Let V (R4 ) be any representation of the translation group acting covariantly on the net {R(W )}W ∈W and satisfying the relativistic spectrum condition with Ω as the ground state. Borchers [12] has isolated a condition on the modular group ∆ it W0 associated with the pair (R(W0 ), Ω) which is intimately connected to the spectrum condition. Borchers’ Relation. For every future-directed lightlike vector ` such that W0 +` ⊂ W0 , there holds the relation −it −2πt `) , ∆it W0 V (`)∆W0 = V (e
f or all t ∈ R .
(5.1.1)
Note that this is precisely the relationship which would result if ∆it W0 implemented the subgroup of boosts leaving the wedge W0 invariant. It has turned out that this condition is equivalent to the representation V (R4 ) satisfying the spectrum condition.h We cite the result as proven in [24]; the appearance of our theorem was preceded by that of an analogous result proven under slightly more restrictive conditions by Wiesbrock [69]. The proof of the deep result that the spectrum condition implies (5.1.1) is due to Borchers [12]. For a recent, considerably simplified proof of Borchers’ theorem, we recommend [30] to the reader’s attention. Proposition 5.1. Let V (R4 ) be a strongly continuous unitary representation of the translation group on R4 which acts geometrically correctly upon the net {R(W )}W ∈W and leaves Ω invariant. Then V (R4 ) satisfies the (positive) relativistic spectrum condition, i.e. sp(V ) ⊂ V¯+ , if and only if relation (5.1.1) holds for all wedges W0 , as described. We intend to utilize this proposition in a discussion of the spectral properties of the representation U (R4 ) of the translation group obtained in the previous section. ↑ Note that because we have a representation of P+ which acts geometrically correctly upon the net and which leaves the state invariant, if (5.1.1) holds (for U (·)) for one such wedge W0 , it must hold for all such wedges. In our approach, employing the modular involutions to derive symmetry groups and their representations, the only role played by the modular groups ∆it W is to characterize algebraically the spectrum condition as above. We next show that in h This connection has also been shown to be useful in applications to quantum fields defined on certain curved space-times associated with black holes [62].
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our framework, the Borchers relation (5.1.1) (with ±2π in the exponent on the right-hand side instead of −2π) already follows from the following assumption: Modular Stability Condition. The modular unitaries are contained in the group generated by the modular involutions, i.e. ∆it W ∈ J , for all t ∈ R and W ∈ W. In the situation described by this condition, the group generated by the modular unitaries and the modular conjugations associated to the net {R(W )}W ∈W by the vector Ω is minimal in a certain sense. The name of this condition is motivated by the use we envisage for it. We shall prove in Theorem 5.1 that the CGMA and the modular stability condition imply the spectrum condition, i.e. physical stability, in the special case of Minkowski space. Since both conditions are well-defined for nets based on arbitrary space-times, the modular stability condition, in the context of the CGMA, could perhaps serve as a substitute for the spectrum condition on spacetimes with no timelike Killing vector. In fact, as discussed below in Subsec. 6.2, recent results [17, 19] in de Sitter space support this picture. We remark that the Poincar´e covariance we have established entails that ∆it W ∈ J , for all t ∈ R and some W ∈ W, implies the modular stability condition. This ↑ follows from the transitive action of P+ upon the set W and the well-known fact it that if ∆ is the modular unitary for the pair (M, Ω) and if the unitary U leaves Ω invariant, then U ∆it U ∗ is the modular unitary for the pair (U MU ∗ , Ω). We can now show that, within our framework, the condition that the modular unitaries are contained in the group J implies the spectrum condition, up to a sign. We shall see in the examples in Subsec. 5.3 that each of the possible outcomes stated in this theorem can occur. Theorem 5.1. Assume the CGMA with the choices M = R4 and W the collection of wedgelike regions in R4 , the transitivity of the adjoint action of J on the net {R(W )}W ∈W , and the net continuity condition mentioned at the beginning of Subsec. 4.3. Let U (R4 ) be the representation of the translation group obtained in Subsec. 4.3. If ∆it W ∈ J , for all t ∈ R and some W ∈ W, i.e. if the modular stability condition obtains, then sp(U ) ⊂ V+ or sp(U ) ⊂ V− . Moreover, for every future-directed lightlike vector ` such that W + ` ⊂ W, there holds the relation −it −αt ∆it `) , W U (`)∆W = U (e
f or all t ∈ R ,
where α = ±2π. Proof. Recall that J + is the subgroup of J consisting of all products of even it/2 it/2 numbers of elements of {JW |W ∈ W}. Note that the relation ∆W ∆W = ∆it W ↑ it + and the assumption ∆it = U (P+ ) (using W ∈ J , for all t ∈ R, imply that ∆W ∈ J Theorem 4.2). Hence, for a fixed W ∈ W and each t ∈ R, there exists an element ↑ (Λt , at ) ∈ P+ such that −it R(W ) = ∆it W R(W )∆W
= U (Λt , at )R(W )U (Λt , at )−1 = R(Λt W + at ) .
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Therefore, one must have (Λt , at ) ∈ InvP↑+ (W ), t ∈ R, and the group GW = {(Λt , at )|t ∈ R} constituted by these transformations must be a one-parameter subgroup of InvP↑+ (W ) which is abelian, since the unitaries ∆it W , t ∈ R, mutually commute. Let W0 ∈ W be any wedge. One observes that if (`+ , `− ) is a pair of lightlike vectors such that W0 ± `± ⊂ W0 , the adjoint action of any element of InvL↑+ (W ) transforms the Poincar´e group element (1, `± ) to (1, c`± ), with c > 0. In particular, for each t ∈ R there must exist an element c± t > 0 such that −it ± ∆it W0 U (u`± )∆W0 = U (ct u`± ) ,
(5.1.2)
for all u ∈ R. Thus one has U (c± t+s `± ) = ∆W0
i(t+s)
−i(t+s)
U (`± )∆W0
−it −is it = ∆is W0 ∆W0 U (`± )∆W0 ∆W0
± = U (c± s ct `± ) , ± ± 4 which implies that c± s ct = ct+s , since U (R ) acts geometrically correctly upon the net {R(W )}W ∈W and since there exist wedges W1 such that W1 + s`± 6= W1 + t`± for s 6= t. From the left side of relation (5.1.2) one also sees that the map (t, u) 7→ 2 U (c± t u`± ) is strongly continuous, uniformly on compact subsets of R . As shall be ± shown, this implies that ct is continuous in t. Assume that c± t is discontinuous at t = 0. It then follows from the equation ± ± cs ct = c± that c± t+s t is unbounded in any neighborhood of t = 0. Thus, for any r 6= 0, there exist sequences tn → 0, un → 0 such that un ctn → r. Therefore, Eq. (5.1.2) and the mentioned strong continuity entail the equality 1 = U (r`± ), which is a contradiction. Thus, the function t 7→ c± t must be continuous at 0. The ± ± relation c± c = c then implies that there exist constants α± ∈ R such that s t t+s α± t c± = e . t It is important to notice that α± 6= 0. If, for example, one had c+ t = 1 for all t ∈ it it −1 R, then one would have [∆W0 , U (`+ )] = 0 and thus ∆W0 +`+ = U (`+ )∆it = W0 U (`+ ) it ∆W0 , for all t ∈ R. But since R(W0 + `+ ) ( R(W0 ), by the standing assumptions, this is in conflict with standard results in modular theory. For both algebras have Ω as a cyclic vector and the stability of the smaller algebra under the action of the modular group of the larger one would thus imply that these algebras must be equal (see [18]). One can now apply the arguments of Proposition 2.3 in [24]. There it was shown that relation (5.1.2) implies that for any two vectors Φ, Ψ ∈ H there exists a function f (z) which is continuous and bounded on the strip 0 ≤ Im(z) ≤ 1/2, analytic in the interior, satisfies the bound |f (z)| ≤ |ΦkkΨk, and on the real axis has the boundary value f (t) = hΦ, U (−eα+ t `+ )Ψi .
Since Φ and Ψ are arbitrary, one may conclude that the operator function z 7→ U (−eα+ z `+ ) is weakly continuous on the strip 0 ≤ Im(z) ≤ 1/2, analytic in the interior, and bounded in norm by 1. In particular, one has kU (−i sin(uα+ )`+ )k ≤ 1 ,
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for 0 ≤ u ≤ 1/2. Hence, it follows that either P · `+ ≥ 0, where P is the generator of the strongly continuous abelian unitary group U (R4 ), or P · `+ ≤ 0. By Lorentz covariance, these relations hold for arbitrary lightlike vector `+ , hence the spectrum of P must be contained either in the closed forward light cone or the closed backward light cone. The final assertion of the theorem then follows from Borchers’ theorem [12]. This observation reinforces our belief that the modular involutions are of primary interest in this context. Theorem 5.1 seems to leave open the possibility that the modular group associated to the wedge algebra R(W ) could conceivably act geometrically as some other subgroup of the invariance group InvP↑+ (W ) of W besides the boost subgroup. However, this is not the case, as we shall prove in the next section — cf. Proposition 5.3 and Theorem 5.4. We wish to emphasize the point that Borchers’ relation (5.1.1) is truly an additional assumption in our framework, as is the modular stability condition. In Subsec. 5.3 we present a simple example of a net satisfying our CGMA and all of the other assumptions made in this paper except the modular stability condition and (5.1.1). In this example the spectrum condition is therefore violated, and the action of the modular groups associated to wedge algebras does not coincide with the Lorentz boosts. Next, we wish to make a few comments about the uniqueness of the represen↑ tation of P+ which has been obtained above. There are uniqueness results for representations of the translation subgroup satisfying the spectrum condition in local quantum field theory — see [24] and references cited there. For the case of nets {R(W )}W ∈W based on wedges, the assertion can be derived easily from Borchers’ theorem. We state and prove this fact for completeness. Proposition 5.2. Let V (R4 ) be a continuous unitary representation of the translations on H which acts geometrically correctly on the net {R(W )}W ∈W , leaves Ω invariant, and satisfies the spectrum condition. Then there is no other representation on H with these properties. Proof. Let W be any wedge and let ` be any positive lightlike vector such that W + ` ⊂ W . Since V (·) acts geometrically correctly on the net, one has −1 V (`)∆it = ∆it W V (`) W +` , and because of the hypothesized spectral properties of −it −2πt V (·), Borchers’ relation holds: ∆it `). Combining these two W V (`)∆W = V (e −it −2πt it relations yields V (` − e `) = ∆W +` ∆W , and the operators appearing on the right-hand side of this equation are fixed by the net {R(W )}W ∈W and the vector Ω. Hence, V (·) is uniquely determined by these data for all lightlike vectors; the group property then yields the desired conclusion. For the representation of the entire Poincar´e group, the best result seems to be that of [22], which asserts that if the distal split property holds, then the representa↑ tion of P+ is also unique. (See also the results in the recent article by Borchers [16].)
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In Subsec. 5.3 we shall present an example of a well-behaved net covariant under two distinct representations of the Poincar´e group, only one of which is selected by the CGMA. With the additional condition (5.1.1) yielding the spectrum condition, algebraic PCT and Spin & Statistics theorems can be proven. A series of papers [35, 36, 45] (see also [26]) have demonstrated a purely algebraic version of the important relationship between spin and statistics, which was first pointed out by Fierz and Pauli and then proven rigorously in the context of Wightman quantum field theory by Burgoyne and L¨ uders and Zumino (see [61] for references). In the work [26, 35, 36] the assumption of modular covariance was made, which, as we shall see in Subsec. 5.3, does not necessarily hold in our more general setting. But if the conditions of Theorem 5.1 are satisfied, then the results established above do imply the hypotheses made in the approach by Kuckert [45] in order to derive the PCT and Spin & Statistics theorems. We shall not take further space to formulate the obvious theorem and refer the reader to [45] for details. 5.2. Geometric action of modular groups To obtain a deeper insight into the nature of the property of modular covariance on the one hand and the relation between the geometric action of modular involutions and that of the modular groups on the other, we shall assume in this section that the modular groups have a geometric action similar to that which we have heretofore assumed for the modular involutions. In particular, we shall assume that the adjoint action of the modular groups of the wedge algebras leaves the set {R(W )}W ∈W invariant. Throughout this section we shall assume that M = R4 and W is the set of wedges, as previously described. Condition of Geometric Action for the Modular Groups. The Condition of Geometric Action for the modular groups is fulfilled if the net {R(W )}W ∈W and vector Ω satisfy the first three conditions of the CGMA stated in Sec. 3 and the fourth condition is replaced by the following requirement : For each W0 ∈ W, the adjoint action of {∆it W0 }t∈R leaves the set {R(W )}W ∈W invariant, i.e. for any W ∈ W and any t ∈ R there exists a wedge Wt ∈ W such that −it ∆it W0 R(W )∆W0 = R(Wt ) .
This condition for the modular groups will be called CMG for short. We shall show that the analysis carried out in the preceding sections in the case of theories satisfying the CGMA can likewise be performed when one takes the CMG as the starting point. We denote by K the unitary group generated by the set {∆it W |t ∈ R, W ∈ W} . As in Sec. 2 one sees that the CMG entails that each ad ∆it W induces a bijection υW (t) on the set W of wedges. The group generated by these bijections will be denoted by U. We state the following counterpart to Lemma 2.1.
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Lemma 5.1. The group U defined above has the following properties. (1) For every υ ∈ U and W ∈ W, one has υυW (t)υ −1 = υυ(W ) (t), t ∈ R. (2) If υ(W ) = W for some υ ∈ U and W ∈ W, then υυW (t) = υW (t)υ, t ∈ R. (3) One has υW (t)(W ) = W, for all W ∈ W and t ∈ R. (4) If W1 ∈ W and υW (t)(W1 ) ⊂ W, for all t ∈ R, then W1 = W . Proof. The first two statements can be established in the same way as part (2) and (3) of Lemma 2.1. The third statement follows from the fact that each algebra R(W ) is stable under the adjoint action of the modular group {∆it W |t ∈ R}. Finally, the fourth assertion is a consequence of the basic result from Tomita–Takesaki theory that the only weakly closed subalgebra of a von Neumann algebra R which has Ω as a cyclic vector and is stable under the action of the modular group of (R, Ω) is R itself. Proposition 2.1 and Corollary 2.1, where J is replaced by K and T by U, also hold in the setting of the CMG, and it is still true that R(W ) is nonabelian for each W ∈ W. Moreover, an analogue of Proposition 3.1 obtains. We omit the straightforward proofs of these statements. For the set of wedgelike regions W in R4 , which we consider here, the elements υW (t) of the transformation group U satisfy the conditions (A) and (B) in Subsec. 4.1. We can thus apply Theorem 4.1 to conclude the following result. Lemma 5.2. Let the CMG hold as described. If {∆it W0 }t∈R is the modular group corresponding to an arbitrary wedge algebra R(W0 ) and the vector Ω, then for each t ∈ R there exists an element LW0 (t) of the extended (by the dilatations R+ ) Poincar´e group DP such that ad ∆it W0 (R(W )) = R(LW0 (t)W ) ,
f or all W ∈ W .
i(s+t)
it Because of the group law ∆is and the standing assumption that W0 ∆W0 = ∆W0 the relation between wedges and wedge algebras is a bijection, one has
LW0 (s)LW0 (t) = LW0 (s + t) ,
s, t ∈ R ,
(5.2.1)
for the corresponding transformations. In particular, LW0 (t) = LW0 (t/2)2 , so each ↑ LW0 (t) lies in the identity component DP+ of the extended Poincar´e group. We ↑ denote by G the subgroup of DP+ generated by the set {LW (t)|t ∈ R, W ∈ W}. In the next step of our analysis we shall determine this group. In order to abbreviate the argument, we shall make the additional simplifying assumption that G acts transitively on the set W of wedges (which follows from the assumption that the adjoint action of K upon {R(W )}W ∈W is transitive). However, this additional assumption can, in fact, be derived from the CMG as it stands; we shall present the proof in a subsequent publication. Lemma 5.3. If the CMG holds and K acts transitively upon {R(W )}W ∈W , then the group G of transformations coincides with the proper orthochronous Poincar´e ↑ group: G = P+ .
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Proof. Note, to begin, that the elements of the commutator G do not contain ↑ any nontrivial dilatations and therefore are contained in P+ . Moreover, they act transitively on W, as can be seen as follows: Let W1 be any wedge and let LW (t) ∈ G be any transformation associated with some wedge W . As G is assumed to act transitively on W, there exists, according to part (1) of Lemma 5.1, a transformation L ∈ G such that LLW1 (t)L−1 = LW (t). On the other hand, according to part (3) of that lemma, one has the relation LW1 (s)W1 = W1 , for all s ∈ R, and consequently LLW1 (t)L−1 LW1 (t)−1 W1 = LW (t)W1 . Since the wedge W1 and the transformation LW (t) were arbitrary, the transitive action of the commutator subgroup follows. The first part of Proposition 4.3 then ↑ implies that this subgroup of G coincides with P+ . Now let W be any given wedge, let LW (t) = (γW (t), ΛW (t), aW (t)) ∈ G be the corresponding transformation on Minkowski space, where γW (t) > 0 is a dilatation, ΛW (t) a Lorentz transformation and aW (t) a translation, and let (1, 1, a) ∈ G, a ∈ R4 , be any other nontrivial translation which leaves W invariant. Part (2) of Lemma 5.1 then implies that LW (t)(1, 1, a)LW (t)−1 = (1, 1, a), for all t ∈ R. On the other hand, one obtains by explicit computation LW (t)(1, 1, a)LW (t)−1 = (1, 1, γW (t)ΛW (t)a). Hence a is an eigenvector of ΛW (t) and thus would have to be lightlike if γW (t) 6= 1, in conflict with its choice. Therefore, one has γW (t) = 1 and ↑ G = P+ , as claimed. In the next step we want to determine the geometric action of the transformations LW (t) associated with the modular groups. The preceding results suffice to show that these transformations are Lorentz boosts. More detailed information will be obtained by making use of the continuity and analyticity properties of the modular groups. Proposition 5.3. Given the CMG and the transitive action of K upon {R(W )}W ∈W , the transformations LWR (t) ∈ G associated with the standard wedge WR are, for all t ∈ R, the boosts ! ! B(t) 0 cosh αt sinh αt LWR (t) = with B(t) = (5.2.2) 0 1 sinh αt cosh αt and α ∈ {±2π}. The form of LW (t) for arbitrary wedges W is obtained from LWR (t) by Poincar´e transformations — see the first part of Lemma 5.1. Proof. According to the second part of Lemmas 5.1 and 5.3, LWR (t) commutes ↑ with all elements of the stability group of WR in P+ . It thus must be a boost which leaves WR invariant and consequently has the block form given in (5.2.2). Moreover, because of relation (5.2.1), the matrix B(t) has the form given in (5.2.2), where the argument αt of the hyperbolic functions could, however, be a priori any additive function (homomorphism) β(t) on the reals. For the proof that β(t)
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has the asserted form, it suffices to show that β(t) is continuous — one then may apply standard results about continuous one-parameter subgroups of GL(n, C) (see, e.g. Theorem 2.6 and Corollary 1.5 in [38]). To this end one exploits the continuity properties of the group {∆it WR |t ∈ R}. According to the information about the action of LWR (t) accumulated up to this point, if ` is any positive lightlike vector such that WR + ` ⊂ WR , one has −it β(t) ∆it `) . WR R(WR + `)∆WR = R(WR + e
If β(t) is discontinuous at t = 0, one may assume without restriction (since β(·) is additive) that there exists a β0 > 0 and a sequence {tn }n∈N ⊂ R such that tn → 0 and βtn ≥ β0 > 0. By isotony and the preceding equality of algebras, one −itn β0 it n thus obtains ∆it WR R(WR + `)∆WR ⊂ R(WR + e `). As ∆WR is continuous in the β0 strong operator topology and R(WR + e `) is weakly closed, one can proceed on the left-hand side of this inclusion to the limit, yielding R(WR +`) ⊂ R(WR +eβ0 `). Since also R(WR + eβ0 `) ⊂ R(WR + `), by isotony, one concludes that these two algebras are equal, in conflict with the CMG. So β(·) is continuous at 0, and since it is a homomorphism it must be continuous everywhere. This shows that for some constant α, β(t) = αt, for all t ∈ R. In order to determine the value of this constant α, one can rely on results of Wiesbrock [72, 73], cf. also [14]. If ` is a lightlike vector as above, the specific form of the action of LWR (t), t ∈ R, on R(WR +`) implies that (R(WR +`) ⊂ R(WR ), Ω) is a ±-half-sided modular inclusion (where the ± depends on the sign of α). The claim α ∈ {±2π} then follows from the results in the quoted references. We have therefore derived modular covariance from our prima facie less restrictive Condition of Geometric Action for the modular groups. We next show that ↑ we have a strongly continuous unitary representation of P+ satisfying the spectrum condition with either negative or positive energy. Theorem 5.2. Assume that the CMG is satisfied and that the adjoint action of K upon {R(W )}W ∈W is transitive. Then there is a strongly continuous unitary rep↑ resentation U (·) of the covering group ISL(2, C) of P+ which generates K and acts geometrically correctly on the net. If , in addition, the net {R(W )}W ∈W satisfies locality, i.e. R(W ) ⊂ R(W 0 )0 for all W ∈ W, then U (·) yields a strongly continuous ↑ unitary representation U (·) of P+ satisfying either the positive or negative spectrum condition, depending on the sign of α in Proposition 5.3. Proof. This may be proven analogously to the arguments of Subsec. 4.3, but since Proposition 5.3 has already established that modular covariance holds, it suffices here simply to appeal to the results of [22, 36] — particularly Lemma 2.6 and Corollary 1.8 in [22] and Proposition 2.8 in [36]. In fact, the mentioned results of [22] imply that K provides a strongly continuous unitary representation of the ↑ covering group ISL(2, C).i For then the pair (K, ξ), where ξ : K → U ' P+ is the i Note that, given our hypotheses, the assumptions of locality and additivity in [22] are not required for the cited results.
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canonical homomorphism, is what those authors call a central weak Lie extension of ↑ the group P+ . With the additional assumption of locality, the results of [36] imply that the projective representation obtained above is actually a strongly continuous ↑ representation of P+ . The sign of α in Proposition 5.3 determines whether the inclusions (R(WR + `) ⊂ R(WR ), Ω), with WR , ` as in the proof of Proposition 5.3, are all +-half-sided modular inclusions or −-half-sided modular inclusions. That, together with Poincar´e covariance, then entails the spectrum condition with either positive or negative energy (see the argument of the proof of Theorem 5.1). It is of particular interest to note that the weak geometric action of the modular groups we have been studying in this section also entails the corresponding geometric action of the modular involutions, if and only if the net {R(W )}W ∈W is local. Theorem 5.3. If the CMG is satisfied and the group K generated by the modular unitaries of all wedge algebras acts transitively upon the net {R(W )}W ∈W , then K is equal to the group J + consisting of all products of even numbers of modular conjugations {JW |W ∈ W}. The adjoint action of the modular conjugations in {JW |W ∈ W} leaves the net {R(W )}W ∈W invariant (and so our CGMA holds) if and only if the net fulfills locality, i.e. R(W 0 ) ⊂ R(W )0 , for all W ∈ W. In that case, the modular conjugations {JW |W ∈ W} have the same geometric action upon the net as was found in Proposition 4.4 under different hypotheses. Furthermore, the net {R(W )}W ∈W satisfies wedge duality and the modular conjugations yield a representation of the proper Poincar´e group P+ which acts geometrically correctly upon the net. (A simple and well-known example of a net which complies with the CMG but where locality and hence also the CGMA fails is the net generated by a Fermi field [10]. It satisfies a twisted form of locality, however.) Proof. By the results of [22] appealed to in the proof of Theorem 5.2, K is iso↑ morphic to either P+ itself or to its covering group, ISL(2, C), and by Theorem 5.2 one knows that K = U (ISL(2, C)). Under the stated hypothesis, the conclusion of Corollary 2.7 in [36] still holds,j i.e. one has also here the relation for the modular (0) conjugations and groups associated with the wedges Wk , k = 1, 2, 3, based on the time-zero plane, −it JWR ∆it (0) JWR = ∆ (0) W k
Wk
k = 2, 3 ,
(0)
where WR = W1 is the standard wedge and JWR the corresponding modular involution. The corresponding relation for k = 1 is a basic result of Tomita–Takesaki theory. Furthermore, JWR commutes with those elements of K which act upon the j In the proof of Proposition 2.6 in [36], which is appealed to in the argument for Corollary 2.7, one should replace F(W1 ∩Λ2 (−t)W1 ) by R(WR )∩R(Λ2 (−t)WR ). Since WR ∩Λ2 (−t)WR is not empty, assumption (ii) in our CGMA entails that Ω is cyclic and separating for R(WR ) ∩ R(Λ2 (−t)WR ). The rest of the argument proceeds as before.
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net {R(W )}W ∈W as translations in the direction of the 2- or 3-axes, since their adjoint action leaves R(WR ) invariant and they leave Ω fixed. The adjoint action of JWR on those elements of K which act upon {R(W )}W ∈W as translations in the lightlike directions of `± fixed by WR inverts these elements, by [72]. Let θ1 denote the element diag(−1, −1, 1, 1) ∈ P+ . The above remarks imply the relations JWR U (µ−1 (λ))JWR = U (µ−1 (θ1 λθ1 )) ,
(5.2.3)
for any λ which is one of the translations or boosts just discussed, where µ is the ↑ canonical covering homomorphism from ISL(2, C) onto P+ . But since these boosts ↑ ↑ . and these translations generate P+ , it follows that (5.2.3) holds for any λ ∈ P+ Indeed, one has (5.2.3) for any wedge W , with θ1 replaced by the corresponding involution, and it follows that JW KJW = K, for any W ∈ W. Since the Poincar´e group acts transitively on W, for any pair of wedges Wa , ↑ Wb there exists some Poincar´e transformation λ ∈ P+ such that λWa = Wb . Consequently, one has JWb = U (A(λ))JWa U (A(λ))−1 for any A(λ) ∈ ISL(2, C) with µ(A) = λ, since U (·) acts geometrically correctly on the net and leaves Ω invariant. Hence, one has JWa JWb = (JWa U (A(λ))JWa )U (A(λ))−1 ∈ K , according to the preceding results, which shows that J + ⊂ K. On the other hand, ↑ it follows from relation (5.2.3) that for λ ∈ P+ JWR Jλ2 WR = U (A(θ1 λθ1 )2 A(λ)−2 ) . Hence the unitaries corresponding to the boosts in the 2- and 3-direction as well as to the lightlike translations in the direction of `1± are contained in J + . Similarly, (0) (0) (0) one can reproduce these arguments with WR = W1 replaced by W2 and W3 to show that the unitaries corresponding to the boosts in the 1-direction as well as the lightlike translations in the direction of `2± and `3± are contained in J + . Since these unitaries together generate U (ISL(2, C)), one concludes that K ⊂ J + , and therefore the two groups are equal. From the invariance of R(WR ) under the adjoint action of the unitaries implementing the stability group of WR , it follows that also the algebra R(WR )0 = JWR R(WR )JWR is invariant under this action. Hence, if R(WR )0 is a wedge algebra, then it must be equal to R(WR0 ) — it cannot coincide with R(WR ), since otherwise it would be abelian. Therefore, if the adjoint action of the elements of {JW |W ∈ W} leaves {R(W )}W ∈W invariant, the net must satisfy wedge duality and hence locality. Conversely, if the net satisfies locality, then R(W 0 ) ⊂ R(W )0 is 0 stable under the adjoint action of the modular group {∆−it W |t ∈ R}, of (R(W ) , Ω) according to Proposition 5.3. Since Ω is cyclic and separating for both algebras, Tomita–Takesaki theory then entails the equality R(W 0 ) = R(W )0 = JW R(W )JW . But this implies that, for any Wa , Wb ∈ W with corresponding modular involutions JWa , JWb , one has JWa R(Wb )JWa = JWa JWb R(Wb0 )JWb JWa ∈ {R(W )}W ∈W ,
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since JWa JWb ∈ J + = K and {R(W )}W ∈W is invariant under the adjoint action of K. The remaining assertions are therefore immediate consequences of the results of Sec. 4. To close the circle of implications relating the geometric action of the modular involutions to that of the modular groups, we conclude this section with the following result. Theorem 5.4. Assume the CGMA, with the choices M = R4 and W the collection of wedgelike regions in R4 , and the transitivity of the adjoint action of J on the net {R(W )}W ∈W . If ∆it W ∈ J , for all t ∈ R and some W ∈ W, i.e. if the modular stability condition obtains, and the adjoint action of K upon {R(W )}W ∈W is transitive, then modular covariance is satisfied. Proof. Since, by hypothesis, the adjoint action of any element of J leaves the set {R(W )}W ∈W invariant and since their transitive action on {R(W )}W ∈W implies ∆it W ∈ J , for all t ∈ R and W ∈ W, it is clear that the CMG is satisfied. Proposition 5.3 completes the proof. Hence, we derive modular covariance from our CGMA, whenever the modular stability condition also holds. We remark once again that in a later publication we shall show that the additional assumption of the transitive action of K is superfluous. 5.3. Modular involutions versus modular groups As explained in the introduction, there have been two distinctly different approaches to the study of the geometric action of modular objects and its consequences. In the one, initiated in [24], geometric action of the modular involutions was assumed, whereas in the other, initiated in [12], the starting point was the geometric action of the modular groups. However, even within each of these approaches, differing forms of concrete action have been studied. In most of the papers concerned with the consequences of geometric action of the modular groups, the action was assumed in the form of modular covariance (see [22, 36], among others). There are some variations of this condition in the literature [26, 35], but they all have in common that from the outset one is given an action of the Lorentz group on the spacetime. Certain exceptions are the papers by Kuckert [46] and Trebels [66], where the geometric action was assumed in the guise of requiring the adjoint action of the modular groups (or the modular involutions) to leave the set of local algebras in Minkowski space invariant. However, in both approaches the starting point is a vacuum representation of a net on Minkowski space which is covariant with respect to the translation group satisfying the spectrum condition. All of these approaches have in common that some a priori information about the geometric action of the modular groups or the spacetime symmetry group is required. But, as we have shown in the above analysis, this detailed information is derived if one starts from our CGMA. We also wish to emphasize that the condition
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of modular covariance and Borchers’ relation (5.1.1) are not implied in our framework. To illustrate these assertions, we present a simple example of a net satisfying our CGMA and all of the other assumptions made in this paper, except the modular stability condition. This example thus violates the spectrum condition and the modular groups associated to wedge algebras do not coincide with the representation of the Lorentz boosts, i.e. modular covariance fails in this example, though it is Poincar´e covariant. Subsequently, we give another example violating modular covariance but satisfying the spectrum condition and all of our assumptions. It is therefore clear that the assumption of modular covariance is more restrictive than the CGMA, even when the spectrum condition is posited. Turning to our first example, let {A(O)}O∈C be the standard net of von Neumann algebras generated by a (hermitian, scalar, massive) free field on the Fock space H. It is based on the set C of double cones in R4 and covariant under the ↑ standard action αλ , λ ∈ P+ , of the Poincar´e group. Let Θ be the PCT-operator on H and θ be the corresponding reflection in Minkowski space. For each double cone ˆ O define B(O) = A(θO) = ΘA(O)Θ. Let A(O) ≡ A(O) ⊗ B(O) act on H ⊗ H. The ˆ net {A(O)}O∈C is clearly local, since Θ is antiunitary and thus behaves properly under the taking of algebraic commutants. We observe that α ˆ λ ≡ αλ ⊗ βλ , with ↑ ˆ βλ ≡ αθλθ , λ ∈ P+ , defines an automorphic local action on {A(O)} O∈C , as can be ↑ seen as follows. With λ ∈ P+ , one has ˆ = αλ (A(O)) ⊗ βλ (B(O)) = A(λO) ⊗ (A((θλθ)θO)) α ˆλ (A(O)) ˆ = A(λO) ⊗ A(θλO) = A(λO) . With U (λ) the unitary implementation of αλ on H, one easily checks that V (λ) ≡ ΘU (λ)Θ implements the action of βλ . Setting U (x) = eixP , where P is the generator of the translations satisfying the positive spectrum condition, one has V (x) = ΘeixP Θ = e−ixP . Hence V (λ) satisfies the negative spectrum condition, ˆ (λ) ≡ U (λ) ⊗ V (λ) violates both the positive and the negative spectrum but U conditions. By the results of Bisognano and Wichmann [9], applicable to the free field, one knows that for the standard wedge WR the modular structure for the (weakly closed) wedge algebra A(WR ) and Ω is given by JWR = ΘR = ΘUπ , where Uπ implements the rotation by π about the 1-axis, and ∆it WR = U (λR (t)), t ∈ R, where the λR (t) are the Lorentz boosts in the 1-direction. The corresponding modular objects for (B(WR ), Ω) = (ΘA(WR )Θ, Ω) = (A(WR )0 , Ω) are given by B JWR = ΘR −1 and B ∆it = U (λR (−t)). It follows that the modular objects for WR = U (λR (t)) ˆ (A(WR ) = A(WR ) ⊗ B(WR ), Ω ⊗ Ω) are given by JˆWR = ΘR ⊗ ΘR ,
ˆ it = U (λR (t)) ⊗ U (λR (−t)) . ∆ WR
So, one has (with θR the transformation on Minkowski space corresponding to ΘR ) ˆ JˆWR A(O) JˆWR = ΘR A(O)ΘR ⊗ ΘR B(O)ΘR = A(θR O) ⊗ A(θR θO) ˆ R O) , = A(θR O) ⊗ A(θθR O) = A(θ
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and the modular conjugation JˆWR acts geometrically correctly on the net ˆ {A(O)} e covariance of the net, the same holds true for the modular O∈C . By Poincar´ involution JˆW , for any wedge W . Turning to the modular groups, one sees −1 ˆ it ˆ ˆ −it ∆ ⊗ U (λR (−t))B(O)U (λR (−t))−1 WR A(O)∆WR = U (λR (t))A(O)U (λR (t))
= A(λR (t)O) ⊗ A(λR (−t)θO) = A(λR (t)O) ⊗ A(θλR (−t)O) ˆ R (t)O) . = A(λR (t)O) ⊗ B(λR (−t)O) 6= A(λ ˆ it does not satisfy modular covariance. Note also that the modular Hence, ∆ WR ˆ (P+ ), so that the modular stability condition is groups are not contained in J = U violated, in accord with Theorem 5.4. We mention as an aside that in [36] Guido and Longo propose the split property, which yields the uniqueness of the representation of the Poincar´e group, as a natural candidate for the hypothesis needed in order to conclude that the modular group of a wedge algebra satisfies modular covariance. However, in the preceding example, the split property holds, though modular covariance does not. In our next example, we see that it is possible for all of our assumptions to hold, as well as the positive spectrum condition, but for modular covariance to be violated. For each W ∈ W, the set of wedgelike regions in four-dimensional Minkowski space, we denote by N (W ) the unique wedge in the coherent family of wedges determined by W which contains the origin in its edge. Once again taking {A(O)}O∈C to be the usual net for the free field on four-dimensional Minkowski space, we consider the net {A(W )}W ∈W indexed by the wedgelike regions and ˆ ) ≡ A(W ) ⊗ A(−N (W )). (Note that −N (W ) = define for this example A(W 0 N (W ) .) This net is local, since W1 ⊂ W20 entails N (W1 ) ⊂ N (W2 )0 and since ↑ A(W )0 = A(W 0 ) (Haag duality). Moreover, for each P+ 3 λ = (Λ, a), we set α ˆλ ≡ αλ ⊗ α(Λ,0) . Hence, the translation subgroup acts trivially upon the second factor of each local algebra. In this example, the unitary implementers of the action α ˆ λ are ˆ given by U(Λ, a) = U (Λ, a) ⊗ U (Λ, 0), and the translation subgroup is implemented ˆ (a) = U (a) ⊗ 1. Thus, the positive spectrum condition holds (though the by U vacuum is infinitely degenerate), whereas modular covariance is violated. In fact, ˆ ) there appears the ˆ it = U (λR (t))⊗U (λR (−t)), since in the second factor of A(W ∆ WR 0 0 algebra A(−N (W )) = A(N (W ) ) = A(N (W )) . On the other hand, the modular ˆ ), Ω ⊗ Ω) are given by JW ⊗ JN (W ) and hence conjugations corresponding to (A(W satisfy the CGMA and the assumption of transitive action on W. It is of interest to note that this example also violates the condition of modular ˆ it ∈ J , in spite of the validity of the spectrum condition. Furthermore, stability, ∆ W the local algebras associated with double cones O, \ ˆ ˆ ) A(O) ≡ A(W O⊂W ∈W
do not generate the wedge algebras. This resembles the situation which one expects to meet for the bosonic part of the field algebra in theories with topological or gauge charges.
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We sketch a final illustrative example, which makes a number of points about the interrelationship of the CGMA, uniqueness of representation of the Poincar´e group, and some further properties of interest. Consider an infinite component free hermitian Bose field with momentum space annihilation and creation operators satisfying the following canonical commutation relations [47]: [a(p0 , q 0 ), a∗ (p, q)] = 2ωp δ (3) (p − p0 )δ (4) (q − q 0 ) , p where p, p0 ∈ R3 , ωp = p2 + m2 , m > 0, and the variables q, q 0 ∈ R4 label the internal degrees of freedom. One unitary representation of the Poincar´e group on the corresponding Fock space of this field is determined by U (Λ, x)a(p, q)U (Λ, x)−1 ≡ eiΛp·x a(Λp, q) , where p = (ωp , p), while a second one is determined by ˜ U(Λ, x)a(p, q)U˜ (Λ, x)−1 ≡ eiΛp·x a(Λp, Λq) . It is evident that both representations satisfy the spectrum condition. Let {A(O)}O∈C be the net of von Neumann algebras generated by this free field. ↑ ↑ ˜ + Clearly, this net transforms covariantly under both U (P+ ) and U(P ). The work ↑ of Bisognano and Wichmann [9] shows that, using the representation U (P+ ), the net satisfies the special condition of duality, and hence it satisfies Haag duality for the wedge algebras, the CGMA, modular covariance and the modular stability condition, K ⊂ J . The arguments of Bisognano and Wichmann break down for ˜ ↑ ), because the extra action on the dummy variable would the representation U(P + destroy the analytic continuation crucial to their arguments. Applying the CGMA to the net {A(O)}O∈C in the Fock vacuum state would ↑ result in the construction of the representation U (P+ ) and not the representation ↑ ˜ U(P+ ). Note further that since both representations act geometrically correctly upon the net, we have ˜ (λ) = Z(λ)U ˜ U (λ) , ↑ for all λ ∈ P+ , with coefficients Z˜ which induce internal symmetries of the net and ↑ commute with U (P+ ). But they are not contained in J and, for this reason, this example escapes the uniqueness statement in Theorem 4.2. On the other hand, the net {A(O)}O∈C violates the distal split property and, for this reason, the example also escapes the uniqueness theorem of [22]. As we have shown, the CMG implies both modular covariance and the CGMA for the involutions (the latter in the presence of locality). The results in Subsec. 5.2 therefore generalize the results of both [22] and [46]. We have also seen that there exist Poincar´e covariant nets of local algebras on Minkowski space which do not satisfy the condition of modular covariance but which satisfy all of our assumptions, with or without the additional condition of positive spectrum. Though the CGMA (in application to the special case of Minkowski space) is weaker than the condition of modular covariance, it nonetheless allows one to systematically establish the same results which were proven under the assumption
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of modular covariance in the literature. Moreover, since the modular involutions depend only upon the characteristic cones of the pairs (A(W ), Ω), it would seem that they are more likely to encode some intrinsic information about the representation, as opposed to the modular unitaries, which are strongly state-dependent. 6. Geometric Modular Action and de Sitter Space As a further example of application of the program outlined in Sec. 3, we consider three-dimensional de Sitter space. The restriction on the dimension is made for simplicity, as it will allow us to apply some of the results obtained in the preceding analysis. It is well-known that three-dimensional de Sitter space dS 3 can conveniently be embedded into the ambient four-dimensional Minkowski space R4 . Choosing proper coordinates, it is described by dS 3 ≡ {x ∈ R4 |x20 − x21 − x22 − x23 = −1} , with the induced metric and causal structure from Minkowski space. Accordingly, the restriction of the Lorentz group L in the ambient space R4 to dS 3 is the isometry group of this space, simply called here the de Sitter group and commonly denoted by O(1, 3). As the elements of L are uniquely fixed by their action on dS 3 , we will identify the de Sitter group with L for later convenience. Similarly, the proper de Sitter group and its identity component are identified with L+ and L↑+ , respectively. In this section we shall assume the CGMA for a net {R(W )}W ∈W on M = dS 3 . Applying the reasoning advanced in Sec. 3, one is presented once again with a ˜ ∩ dS 3 |W ˜ ∈W ˜ 0 }, where W ˜ 0 is unique minimal admissible family, namely W ≡ {W the family of wedgelike regions in the ambient four-dimensional Minkowski space R4 containing the origin in their edges. Hence, we shall proceed with this choice of index set. Though there are clearly affinities between this setting and the Minkowski-space situation, there are nevertheless some nontrivial points to be worked out which do not automatically follow from the work in the previous sections. To begin, we shall prove in Subsec. 6.1 a general Alexandrov-like result in dS 3 along the lines of Theorem 4.1. In view of the different geometric structure of de Sitter space, the construction of the induced point transformations in dS 3 differs from the corresponding construction in Minkowski space. (An alternative construction made under stronger assumptions may be found in [28].) In Subsec. 6.2 it will be shown that the CGMA, with an additional technical postulate, implies that the bijections on W induced by the adjoint action of the modular involutions {JW |W ∈ W} upon the net {R(W )}W ∈W are obtained by elements of the de Sitter group. Then, under the assumption that the group generated by {ad JW |W ∈ W} acts transitively upon the set {R(W )}W ∈W , it will be shown that the group thereby generated is L+ . In contradistinction to the Minkowski space situation, all elements of the index set W are atoms; hence it is a priori possible for the wedge algebras to be abelian here. However, we shall see below that this possibility is once again excluded, albeit for different reasons.
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After this analysis, we shall proceed analogously to the development in Sec. 4 to obtain a strongly continuous unitary representation of L+ , resp. L↑+ , which acts geometrically correctly upon the net {R(W )}W ∈W . 6.1. Wedge transformations in de Sitter space In this section, we shall work with bijections τ : W → W satisfying the condition W1 ∩ W2 = W3 ∩ W4
⇔
τ (W1 ) ∩ τ (W2 ) = τ (W3 ) ∩ τ (W4 ) ,
(6.1.1)
for arbitrary pairs W1 , W2 and W3 , W4 in W. In the next section, we shall provide assumptions on the net {R(W )}W ∈W which entail condition (6.1.1). We shall use constantly without further mention the elementary fact that W ∈ ˜ in R4 such that W = W ˜ ∩ dS 3 and W determines uniquely a wedgelike region W ˜ . It will be vice versa. Hence we shall, where convenient for us, identify W with W 3 clear from the context whether W is regarded as a subset of dS or of the ambient space R4 . Adopting the notation of Sec. 4, we shall write ˜ [`1 , `2 , 0] ∩ dS 3 , W [`1 , `2 ] ≡ W
˜ [`1 , `2 , 0] ∈ W ˜0 where W
is the wedge in the ambient space fixed by the two positive lightlike vectors `1 , `2 and the translation 0. For the analysis of condition (6.1.1) we must make some elementary geometric points about pairs of wedges. Definition 6.1. Let W [`1 , `2 ], W [`3 , `4 ] ∈ W be wedges. If the positive lightlike vectors `1 , `4 , respectively `3 , `2 , are not parallel, then the pair of wedges (W [`1 , `4 ], W [`3 , `2 ]) will be called the pair of wedges dual to (W [`1 , `2 ], W [`3 , `4 ]) (or simply the dual pair). If (W3 , W4 ) is the pair dual to (W1 , W2 ), then W1 ∩ W2 = W3 ∩ W4 . If this intersection is nonempty, then (W1 , W2 ) and (W3 , W4 ) are the only pairs in W with this intersection. Hence, ∅ 6= W1 ∩ W2 = W3 ∩ W4 implies that the (unordered) pairs (W1 , W2 ) and (W3 , W4 ) are either the same or dual (for details, see [28]). We immediately have the following counterpart to Lemma 4.5. Lemma 6.1. Let `1± = (1, ±1, 0, 0), `2± = (1, 0, ±1, 0) and ` = (1, a, b, c) with a, b, c ∈ R, a2 + b2 + c2 = 1, b 6= 1. The wedges W1 = W [`1+ , `1− ] and W2 = W [`2+ , `] have empty intersection if and only if 0 < a ≤ 1, 0 ≤ b < 1 and c = 0. The statement is still true if W1 is replaced by W10 and the condition 0 < a ≤ 1 is replaced by −1 ≤ a < 0, or also if `2+ is replaced by `2− and 0 ≤ b < 1 by −1 < b ≤ 0. This result will be used in the proof of the next lemma. Lemma 6.2. Let τ : W → W be a bijection satisfying (6.1.1) and let `0 be a fixed future-directed lightlike vector. Then τ maps collections of wedges {W [`0 , `]|` lightlike, ` · `0 > 0}
and
{W [`, `0 ]|` lightlike, ` · `0 > 0}
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onto sets of the same form.k Furthermore, W1 ∩ W2 = ∅
⇔
τ (W1 ) ∩ τ (W2 ) = ∅ ,
(6.1.2)
f or any W ∈ W .
(6.1.3)
for any W1 , W2 ∈ W, and τ (W 0 ) = τ (W )0 ,
Therefore, if W1 ∩ W2 6= ∅ and the pair (W3 , W4 ) is dual to (W1 , W2 ), then (τ (W1 ), τ (W2 )) is dual to (τ (W3 ), τ (W4 )). Henceforth, we shall abbreviate {W [`0 , `]|` lightlike, ` · `0 > 0} by {W [`0 , `]|`}, etc. Proof. Let W1 ∩ W2 = ∅, with W1 , W2 ∈ W. There clearly exist infinitely many distinct pairs of disjoint wedges in W. Let (W1 , W2 ), (W3 , W4 ) and (W5 , W6 ) be any three of them. Then (6.1.1) implies τ (W1 ) ∩ τ (W2 ) = τ (W3 ) ∩ τ (W4 ) = τ (W5 ) ∩ τ (W6 ) . If this intersection is nonempty, (τ (W1 ), τ (W2 )), (τ (W3 ), τ (W4 )) and (τ (W5 ), τ (W6 )) are distinct (since τ is a bijection), mutually dual pairs of wedges, which is impossible. Hence, the assertion (6.1.2) is proven. The final assertion of the lemma follows at once. Let `1 , `2 , `3 , `4 be given. There exist corresponding lightlike vectors `01 , `02 , `03 , `04 such that τ (W [`1 , `2 ]) = W [`01 , `02 ] and τ (W [`3 , `4 ]) = W [`03 , `04 ] . Now (W [`1 , `2 ], W [`3 , `4 ]) equals its dual pair if and only if `1 is parallel to `3 or `2 is parallel to `4 , and since self-dual pairs of nondisjoint wedges are mapped by τ to self-dual pairs of nondisjoint wedges, this is equivalent to `01 is parallel to `03 or `02 is parallel to `04 , respectively. Thus, all pairs of images of the wedges W [`0 , `], `0 fixed but ` arbitrary, are self-dual. Therefore, one has {τ (W [`0 , `])|`} ⊂ {W [`00 , `]|`} or {τ (W [`0 , `])|`} ⊂ {W [`, `00 ]|`} for a suitable `00 . Since the same statement holds for τ −1 , the equality of these sets follows. It remains to prove (6.1.3). To this end assume W [`3 , `4 ] = W [`1 , `2 ]0 , i.e. `3 is parallel to `2 and `4 is parallel to `1 . The collection {τ (W [`1 , `])|`}, which contains the wedge W [`01 , `02 ], coincides with either {W [`01 , `]|`} or {W [`, `02]|`}. But by k Note that these collections of wedges are such that any two elements form a self-dual pair of wedges with nonempty intersection.
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relation (6.1.1) and a straightforward application of Lemma 6.1, each element of {τ (W [`1 , `])|`} is disjoint from τ (W [`1 , `2 ]0 ) = W [`03 , `04 ]. So `01 is a positive multiple of `04 in the first case (otherwise, one would have W [`01 , `04 ] ∈ {τ (W [`1 , `])|`} and W [`01 , `04 ] ∩ W [`03 , `04 ] = ∅, in contradiction to Lemma 6.1); in the second case one concludes that `02 is a positive multiple of `03 . On the other hand, by considering the collection {τ (W [`, `2 ])|`} instead, one can see that `02 is a positive multiple of `03 , resp. that `04 is a positive multiple of `01 . We next show that τ induces a map on the set of characteristic planes in the ambient space R4 , as in Subsec. 4.1. We use notation established there and recall ˜ 0. that we identify W with W Corollary 6.1. Let τ : W → W be a bijection satisfying (6.1.1). Then τ induces a bijection of characteristic planes, which we shall also denote by τ, such that τ (H0 [`1 ]) and τ (H0 [`2 ]) are the characteristic planes determined by τ (W [`1 , `2 ]) (with H0 [`1 ] 6= H0 [`2 ]). Proof. According to Lemma 6.2, one has for fixed `0 either {τ (W [`0 , `])|`} = {W [`00 , `]|`} or {τ (W [`0 , `])|`} = {W [`, `00 ]|`} , for a suitable `00 . Set τ (H0 [`0 ]) = H0 [`00 ]; the claim then follows easily.
By considering disjoint pairs of wedges instead of maximal pairs of wedges, one can follow the argument of Lemma 4.8 to prove the following. Lemma 6.3. Let τ : W → W be a bijection satisfying (6.1.1). If `1 , `2 , `3 , `4 are linearly dependent future-directed lightlike vectors such that any two of them are linearly independent, then 4 \ i=1
τ (H0 [`i ]) =
\
τ (H0 [`i ])
for k = 1, 2, 3, 4 .
i6=k
This leads to an induced map on spacelike lines through the origin. Lemma 6.4. Let τ : W → W be a bijection satisfying (6.1.1), and let x ∈ R4 be spacelike. Then the intersection \ τ (H0 [`]) {`|x∈H0 [`]}
is one-dimensional and spacelike. Hence, τ induces a bijection \ Rx 7→ τ (H0 [`]) {`|x∈H0 [`]}
on the set of spacelike one-dimensional subspaces of R4 . This map will again be denoted by τ .
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Proof. Let `1 , `2 , `3 , `4 be pairwise linearly independent lightlike vectors such that x ∈ H0 [`i ], for i = 1, 2, 3, 4. Then this quadruple of vectors is linearly dependent and consequently, by Lemma 6.3, one has \ {`|x∈H0 [`]}
τ (H0 [`]) =
3 \
τ (H0 [`i ]) .
i=1
We shall need the following geometric result about wedges. Lemma 6.5. Let x ∈ R4 be spacelike and let `k = (1, ak , bk , ck ), where ak , bk , ck ∈ R satisfy a2k + b2k + c2k = 1, k = 1, 2. Set W0 = W [`1 , `2 ]. Then Rx ∩ W0 6= ∅ if and only if W0 ∩W 6= ∅, for all W ∈ W whose edge contains Rx. For x = (0, 0, 1, 0), this is also equivalent to the statement that b1 b2 < 0. Moreover, (0, 0, 1, 0) ∈ W0 implies b1 > 0 and −(0, 0, 1, 0) ∈ W0 implies b1 < 0 (when b1 b2 < 0). Proof. One may assume without loss of generality that x = (0, 0, 1, 0). Since W0 is open, it is trivial that Rx ∩ W0 6= ∅ implies W0 ∩ W 6= ∅, for all W ∈ W whose edge contains x. For the converse, it will first be shown that W0 ∩ W 6= ∅, for all W ∈ W whose edge contains x, implies b1 b2 < 0. The case b1 = b2 = 0 is excluded, since it would imply that W0 is invariant under the translations Rx and consequently also W00 would be so invariant. Hence, it would follow that W0 ∩ W00 6= ∅, which is a contradiction. By considering W00 instead of W0 if b1 = 0, one may assume that b1 6= 0. By applying suitable Lorentz transformations leaving (0, 0, 1, 0) invariant, one may further assume that a1 = c1 = 0 and, after applying a suitable rotation, a2 > 0 and c2 = 0. Lemma 6.1 entails that if b1 > 0, a2 > 0 and b2 ≥ 0, or b1 < 0, a2 > 0 and b2 ≤ 0, then one has W0 ∩ W [`1+ , `1− ] = ∅, where `1± are as in the lemma. Since the wedge W [`1+ , `1− ] contains the line Rx in its edge, this is a contradiction. Hence, there holds b1 b2 < 0. Proceeding further, it may still be assumed that a1 = c1 = c2 = 0. The remaining assertion of the lemma follows for b1 > 0, b2 < 0 (and similarly for b1 < 0, b2 > 0), if one notices that the vector (0, 0, (1 − b2 )2 , 0) = −b2 (1 − b2 )(1, 0, 1, 0) − (1 − b2 )(1, a2 , b2 , 0) + a2 (a2 , 1 − b2 , a2 , 0)
is an element of W0 . This enables us to prove this final preparatory lemma.
Lemma 6.6. Let τ : W → W be a bijection satisfying (6.1.1), and let x ∈ R4 be spacelike. If x ∈ W1 ∩ W2 for W1 , W2 ∈ W, then ∅ 6= τ (Rx) ∩ τ (W1 ) = τ (Rx) ∩ τ (W2 ) .
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Proof. It has been seen that τ maps the set of wedges in W whose edges contain the line Rx onto the set of wedges in W whose edges contain the line τ (Rx). Lemmas 6.4 and 6.5 entail that both τ (Rx)∩τ (W1 ) and τ (Rx)∩τ (W2 ) are nonempty. There exist lightlike vectors `1 , `2 , `3 , `4 such that τ (W1 ) = W [`1 , `2 ] and τ (W2 ) = W [`3 , `4 ]. It is not possible for both the vectors `1 , `4 and the vectors `2 , `3 to be parallel, for otherwise one would have W [`3 , `4 ] = W [`1 , `2 ]0 , which implies that W1 ∩ W2 = ∅. Assuming that `1 and `4 are not parallel, Lemmas 6.2 and 6.5 then yield ∅ 6= τ (Rx) ∩ W [`1 , `2 ] = τ (Rx) ∩ W [`1 , `4 ] = τ (Rx) ∩ W [`3 , `4 ] , and a similar argument can be applied if `2 and `3 are not parallel.
We have seen above that every point x in the three-dimensional de Sitter space can be identified with a spacelike x ∈ R4 with x · x = −1. By Lemma 6.6, this then determines the nonempty intersection \ \ τ (Rx) ∩ τ (W0 ) = (τ (Rx) ∩ τ (W )) = τ (Rx) ∩ τ (W ) , W ∈W x∈W
W ∈W x∈W
where W0 ∈ W contains x. Since there exists a point y 6= 0 in this intersection, and τ (W0 ) ∈ W, while τ (Rx) is a spacelike line, the intersection τ (Rx) ∩ τ (W0 ) must contain the ray R+ y. Hence, there exists a unique point, call it δ(x), such that δ(x) ∈ τ (Rx) ∩ τ (W0 ) and δ(x) · δ(x) = −1. It thus represents a point in three-dimensional de Sitter space. We have therefore proven the following result. Proposition 6.1. Let τ : W → W be a bijection satisfying (6.1.1). Then there exists a bijection δ : dS 3 → dS 3 such that τ (W ) = {δ(x)|x ∈ W } , for all W ∈ W. The following Alexandrov-like theorem has been established for the case of de Sitter space by Lester [48]: Lemma 6.7. If φ : dS 3 → dS 3 is a bijection such that lightlike separated points are mapped to lightlike separated points, then there exists a Lorentz transformation Λ of the ambient Minkowski space R4 such that φ(x) = Λx, for all x ∈ dS 3 . We may therefore proceed to obtain the following extension of Lester’s theorem. Details may be found in Subsec. 1.5.2 of [28]. Theorem 6.1. Let τ : W → W be a bijection satisfying (6.1.1), and let δ : dS 3 → dS 3 be the associated bijection. Then there exists a Lorentz transformation Λ of the ambient Minkowski space R4 such that δ(x) = Λx, for all x ∈ dS 3 , and τ (W ) = ΛW, for all W ∈ W.
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6.2. Geometric modular action in de Sitter space and the de Sitter group We now turn to the discussion of nets on de Sitter space satisfying the Condition of Geometric Modular Action given in Sec. 3 with the choices M = dS 3 and the collection of wedges W specified in the previous section. In order to simplify the discussion, we work with the following somewhat more restrictive version of the CGMA. Strong CGMA. A theory complies with the strong form of the CGMA if the net {R(W )}W ∈W satisfies (i) W 7→ R(W ) is an order-preserving bijection, (ii) Ω is cyclic and separating for R(W1 ) ∩ R(W2 ) if and only if W1 ∩ W2 6= ∅, for W1 , W2 ∈ W, (iii) for any W0 , W1 , W2 ∈ W with W1 ∩ W2 6= ∅, there holds R(W1 ) ∩ R(W2 ) ⊂ R(W0 )
if and only if
W1 ∩ W2 ⊂ W0 ,
(6.2.1)
and (iv) for each W ∈ W, the adjoint action of JW leaves the set {R(W )}W ∈W invariant. The first and fourth conditions are the same as in Sec. 3 and entail the existence of an involution τW : W → W for each W ∈ W satisfying (3.1) and (3.2). The second condition is a strengthened version of the previous conditions (ii) and (iii). It directly implies relation (3.3). The third condition is an additional natural assumption which has no counterpart in the original CGMA. We note that the restriction to intersecting pairs of wedges is motivated by a curious fact pointed out to us by E. H. Wichmann. Already for the standard net of von Neumann algebras of the free field, there exists a counterexample to relation (6.2.1) if W1 , W2 are unrestricted wedges [68]. However, it has been shown that in a net satisfying the usual axioms as well as the condition of additivity of wedge algebras, the relation (6.2.1) holds for pairs satisfying W10 ∩ W20 6= ∅ [64, 65]. But for wedges W1 , W2 ∈ W0 , it is easy to see that W10 ∩ W20 6= ∅ if and only if W1 ∩ W2 6= ∅. Lemma 6.8. Let the strong CGMA with the choices M = dS 3 and the set of wedges W in dS 3 hold. Then for each W ∈ W the associated involution τW : W → W satisfies (6.1.1). Proof. As already pointed out, the strong CGMA entails relation (3.3). Therefore, in order to prove (6.1.1), it suffices to show that W1 ∩ W2 = W3 ∩ W4 6= ∅ implies τW (W1 ) ∩ τW (W2 ) = τW (W3 ) ∩ τW (W4 ). But W1 ∩ W2 = W3 ∩ W4 implies R(W1 ) ∩ R(W2 ) ⊂ R(W3 ), which itself entails R(τW (W1 )) ∩ R(τW (W2 )) ⊂ R(τW (W3 )). In the light of (3.3), one concludes that also τW (W1 ) ∩ τW (W2 ) 6= ∅, so by (6.2.1) one finds τW (W1 ) ∩ τW (W2 ) ⊂ τW (W3 ). By proving three similar inclusions, it follows that τW (W1 ) ∩ τW (W2 ) = τW (W3 ) ∩ τW (W4 ).
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Given the hypotheses of Lemma 6.8, we conclude from Theorem 6.1 that T is isomorphic to a subgroup G of the de Sitter group L. Since one has for W1 , W2 ∈ W the fact that the inclusion W1 ⊂ W2 entails the equality W1 = W2 , the index set W considered in this section consists exclusively of atoms, i.e. we cannot conclude from the argument of Sec. 2 that the algebras R(W ) are nonabelian. However, we shall see that the CGMA entails that they be nonabelian once again. Note that none of the arguments in Subsec. 4.2 relied upon the nonabelianness of the algebras R(W ). Hence, with the additional assumption that the adjoint action of J upon the net {R(W )}W ∈W is transitive, we conclude that the entire identity component of L↑+ of L is contained in G (Proposition 4.2). Lemma 6.9. Let the strong CGMA with the choices M = dS 3 and the set of wedges W in dS 3 hold. Moreover, let the adjoint action of J upon the set {R(W )}W ∈W be transitive. Then the algebra R(W ) is nonabelian for every W ∈ W, and the geometric action of the modular involutions is precisely that found in Proposition 4.4. Proof. Because of the transitive action of G upon W, it suffices to make the argument for the standard wedge WR and the corresponding involution gWR ∈ G. Since L↑+ ⊂ G, one sees from Lemma 2.1 that gWR commutes with the elements of the subgroup InvL↑+ (WR ) of L↑+ leaving WR invariant. But WR and WR0 are the only wedges which are stable under the action of InvL↑+ (WR ), so it follows that either gWR WR = WR0 or gWR WR = WR . In both cases one can proceed in a manner similar to the proof of Proposition 4.4. Making use of the fact that gWR is an involution which commutes with InvL↑+ (WR ), it is not hard to show that gWR has the block form ! X 0 gWR = , 0 Y where X, Y = ±1. In the first case, gWR WR = WR0 , one clearly has X = −1. If also Y = −1, then gWR commutes with all elements of L↑+ ⊂ G, which would be in conflict with the transitive action of J upon {R(W )}W ∈W . Hence Y = 1 and gWR has the form given in Proposition 4.4. The second case, gWR WR = WR , can be treated in the same manner. Finally, the relation R(WR )0 = JWR R(WR )JWR = R(gWR WR ) shows that if gWR WR = WR then R(WR ) is abelian. Conversely, if R(WR ) is abelian (and hence maximally abelian by the cyclicity of Ω), then one has R(WR ) = R(WR )0 , and the above relation together with the first part of the CGMA implies gWR WR = WR . Since L↑+ ⊂ G, the relations demonstrated above hold for all W ∈ W, not just for WR . Hence, for nonequal W1 , W2 ∈ W with nonempty intersection, condition (ii) of the strong CGMA entails that R(W1 ) ∩ R(W2 ) is maximally abelian. Condition (i) implies, however, that this maximally abelian algebra is strictly contained in the maximally abelian algebra R(W2 ). This is a contradiction. Hence the case gWR WR = WR is excluded, and R(W ) is nonabelian for all W ∈ W.
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It is now clear how to modify the arguments of Subsec. 4.2 to obtain the following result. Corollary 6.2. If the conditions of Lemma 6.9 are satisfied, then G coincides with L+ . The assumptions of Lemma 6.9 also directly yield an obvious counterpart to Proposition 4.5. The net continuity condition introduced in Subsec. 4.3 and the arguments presented there again entail that there exists a strongly continuous projective representation of L+ . Moreover, the reasoning presented in Subsec. 4.3 implies that this gives a true representation U (L+ ) of the proper de Sitter group. We summarize in the following theorem. Theorem 6.2. Let the strong CGMA with the choices M = dS 3 and wedges W in dS 3 hold, and let the adjoint action of J upon the set {R(W )}W ∈W be transitive. Moreover, assume that the net continuity condition stated at the beginning of Subsec. 4.3 holds. Then there exists a strongly continuous unitary representation of the proper de Sitter group L+ which acts geometrically correctly upon the net {R(W )}W ∈W . Moreover, the net satisfies Haag duality and is local. In light of the fact that the restricted Lorentz group L↑+ is also isomorphic to the group of motions of Lobaschewskian space, which can be modelled on a surface of transitivity of L↑+ in R4 (see, e.g. [34]), it is likely that the preceding arguments can be employed to handle that space-time, as well. To demonstrate that this theorem is not vacuous, we recall an example due to Fredenhagen [33]. Consider once again the standard net {A(O)}O∈C from Subsec. 5.3 associated with the free scalar field on R4 . We define for each region W ∈ W ˜ ), where W ˜ is the wedge fixed by W in the a corresponding algebra R(W ) ≡ A(W ˜ ) the corresponding algebra generated by the free field. ambient space R4 and A(W The results of Bisognano and Wichmann [9] and Thomas and Wichmann [65] entail that this net is covariant under the de Sitter group, and the assumptions in Theorem 6.2 are satisfied by this net in the vacuum state. Moreover, recent results in [17] and [19] concerning quantum field theory on de Sitter space-time are fully consistent with our findings, even though the starting point is quite different. These authors assume the existence of a preferred (vacuumlike) state vector Ω which is invariant under the de Sitter group L↑+ and satisfies a stability condition which can be expressed in terms of certain analyticity properties of the corresponding correlation functions. With this input they are then able to prove a Bisognano–Wichmann type theorem. In fact, they establish the Reeh– Schlieder property of Ω for wedge algebras (so the modular objects exist in their setting), and they also show that the modular conjugations associated with these algebras and Ω induce the geometric action upon the net found in the analysis presented here. Moreover, the modular groups comply with our proposal for a modular stability condition. These facts support our view of the relevance of our selection criterion for vacuum-like states in theories on curved space-times.
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7. Summary and Further Remarks As this paper is lengthy and involves many steps, it is perhaps not amiss to provide a final summary here. First of all, we showed that our Condition of Geometric Modular Action, CGMA, in the abstract form of the Standing Assumptions, yields special Coxeter groups T of automorphisms on the index set (I, ≤) of the net {Ai }i∈I and provides them with projective representations having coefficients in an abelian group Z of internal net symmetries. Some general properties of these groups, following from the modular theory, and a discussion of the finite case were given. In Sec. 3 it was explained how, starting from a smooth manifold M and with a target space-time (M, g) in mind, one would go about identifying the index set W before testing states on the net {R(W )}W ∈W for the CGMA. The resultant program using the CGMA for the determination of much of the geometrical structure of the space-time was then described. This program was then exemplified in application to the four-dimensional Minkowski space as target space. This involved a series of results of quite distinct natures. To begin, we showed that bijective inclusion-preserving mappings on the set of wedges which satisfy one additional condition are implemented by elements of the extended Poincar´e group, thus extending the Alexandrov-type theorems for Minkowski space. Then, it was shown that subgroups of the Poincar´e group which ↑ act transitively upon the set of wedges must contain the identity component P+ of the Poincar´e group. These results enabled us to show that the CGMA, applied to nets indexed by wedges in R4 and supplemented by the transitivity condition, implies that the induced isometry group G is equal to the proper Poincar´e group, and that the implementers for the generating involutions have exactly the geometric action found by Bisognano and Wichmann in their setting. This explicit knowledge of the geometric nature of the adjoint action of the modular involutions JW upon the net, along with the additional structure accompanying the modular theory, was used to construct a continuous projective representation ↑ of P+ , under the assumption of the net continuity condition. Using Moore’s Borel ↑ measurable cohomology theory, we showed that this projective representation of P+ lifts to a true representation of its universal covering group. The explicit geometric properties of the modular involutions already alluded to were then employed to prove that this representation of the covering group restricts to a strongly con↑ tinuous unitary representation of P+ and actually coincides with the constructed projective representation. In other words, the projective representation constructed in Subsec. 4.3 is actually a true representation. In Subsec. 5.1, we showed that if the modular unitaries are all contained in the group J generated by the modular involutions, i.e. if the modular stability condition holds, then the spectrum condition must hold. This is a purely algebraic stability condition which can be sensibly stated on any space-time. We next investigated the geometric action of the modular unitaries in detail. It was proven that, if the Condition of Geometric Action for modular groups is satisfied, then both modular covariance and the modular stability condition, K ⊂ J , hold and, if the net is local,
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↑ the group K yields a strongly continuous unitary representation of P+ satisfying the spectrum condition. Moreover, under the same assumptions, the CGMA holds if and only if the net is local. Furthermore, if the CGMA and the modular stability condition are satisfied, then again modular covariance follows. In Subsec. 5.3 a number of examples were given which make clear that modular covariance is, in fact, strictly stronger than the CGMA. Finally, in Sec. 6 we discussed the case of de Sitter space. In spite of its different geometric structure, results similar to the case of Minkowski space were recovered. Among other space-times, we expect our approach to function with little change in such examples as the (static) Robertson–Walker space-times. It is an interesting problem whether also in these cases the maps induced upon the index sets of the corresponding nets of algebras are implemented by point transformations. In this regard, it is relevant to note that Alexandrov-type theorems are available for many of the classical Lorentzian space-times (see [8]). But even if not every element of the group T of transformations is implemented by a point transformation on the space-time (and we have already presented such an example in Subsec. 4.1), we still anticipate that the CGMA could usefully select physically interesting states. Whatever the group of transformations which results, we would propose it as the symmetry group of the theory. We complete our comments in this final section by returning briefly to the conceptually interesting question of whether one can derive the space-time itself from our initial algebraic data. In this paper we began with a particular smooth manifold M and saw how the CGMA, for a certain choice of index set which was determined by the target space-time (M, g), enabled us to derive a metric-characterizing isometry subgroup. But is it possible to do without these initial data? We shall sketch here our program for meeting this question. We have shown that the abstract version of the CGMA in the form of the Standing Assumptions leads to a certain Coxeter group T of automorphisms on the index set (I, ≤) of the net {Ri }i∈I . There exists in the mathematical literature a branch of geometry known as absolute geometry, whose point of departure is precisely an abstract group T generated by involutions and whose aim is to investigate which algebraic relations in the group T entail the existence of a space-time (M, g) such that the group T can be realized as a metric-characterizing subgroup of the isometry group of (M, g). This has been carried out for all planar geometries [5, 76] and for three-dimensional Euclidean space [1]. So a first step in an attempt to characterize Minkowski space entirely in terms of the data ({Ri }i∈I , Ω) would be to find the algebraic relations in the group T which would enable one to derive in this manner four-dimensional Minkowski space. This has been accomplished in one form [44], but the particular geometric significance of the initial algebraic data in our setting entails that a different set of algebraic axioms be determined [67]. The second step would be to determine which additional structure on I, or equivalently, which relations among the algebras in the net {Ri }i∈I , imply via modular theory the requisite relations among the generating involutions Ji (equivalently, τi ) found in the first step. In application to Minkowski
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space, this would give an intrinsic characterization of “wedge algebras” (equivalently “wedges”). The results in this paper demonstrate that the CGMA is sufficiently strong to select physically interesting states and to actually determine metric-characterizing isometry groups in the examples of Minkowski and de Sitter space-times. We hope that the suggestive results and interesting perspectives of the present analysis will draw attention to the various mathematical problems opened up by our program. Appendix. Cohomology and the Poincar´ e Group In this appendix we shall prove the technical cohomological result used in the main ↑ text to the effect that the continuous projective representation V (P+ ) constructed in Subsec. 4.3 can be lifted to a true representation of the covering group. We include this appendix since we have not found in the literature the results in the form we need. Assume that G 3 g 7→ V (g) ∈ U(H) is a continuous projective representation of a semisimple Lie group by unitary operators on a separable Hilbert space H, which has coefficients in a closedl subgroup Z ⊂ U(H) left pointwise fixed by the adjoint action of the elements of {V (g)|g ∈ G}. Note that the group U(H) of unitary operators acting on the separable Hilbert space H is, when provided with the strong operator topology, a complete, metrizable, second countable topological group (cf. p. 33 in [27] and references cited there). It therefore follows that also Z is a complete, metrizable, second countable topological group; hence, it is a polonais (polish) group. In particular, Z is a trivial G-module. The first main theorem we want to prove is the following. (A related theorem with different assumptions and proof may be found in [22].) Theorem A.1. Let G, V (G) and Z be as described above.m Then there exists a strongly continuous unitary representation of the covering group E of the group G. The proof of this theorem will proceed in several steps, which we present in separate lemmata for the sake of clarity. For the reader’s convenience, we shall present some background information about the two-dimensional cohomology of groups, which can be found in textbooks on the subject (see, e.g. [20]). Since we are interested in the continuity of the representations, we shall need to work in the category of topological groups but find ourselves obliged to use the Borel cohomology on locally compact groups initiated by Mackey [49] and fully defined and extended by Moore [51–55], since the computational situation for continuous cohomologies seems to be exceedingly complicated. Fortunately, it can be shown that this will be sufficient for our purposes. For an overview of the various cohomologies for topological groups, see the review by Stasheff [60]. l It is no loss of generality to take Z closed. Though the subgroup Z ⊂ U (H) in the main text is
↑ not a priori closed, closing it in the strong operator topology still yields a trivial P+ -module, as used in this appendix. However, the restriction that Z be closed offers a technical problem in the main text which is dealt with there. m In fact, the arguments presented below are valid for a larger class of groups G than semisimple Lie groups, but we shall not tax the reader’s patience here with this generalization.
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Let G be a group and G0 ≡ [G, G] denote its derived subgroup, i.e. the group generated by the set {ghg −1 h−1 |g, h ∈ G} of commutators in G. If G0 = G, the group G is said to be perfect, and any connected semisimple Lie group has this property.n In particular, the group of interest to us in this paper, the proper ↑ orthochronous Poincar´e group P+ , is a perfect group. Let G be a group and A be an abelian group. A central extension of G by ˜ φ, ι) with G ˜ a group, ι an injective homomorphism from A to A is a triple (G, ˜ satisfying ι(A) ⊂ center(G) ˜ and φ a homomorphism from G ˜ onto G satisfying G kernel(φ) = ι(A). In other words, the sequence φ ι ˜ −→ {1} −→ A −→ G G −→ {1}
(A.1)
is exact, with {1} denoting the trivial group. Such a central extension is said to be equivalent to the central extension ι0
φ0
˜ 0 −→ G −→ {1} {1} −→ A −→ G ˜→G ˜ 0 such that the diagram if there exists an isomorphism ρ : G ι
{1} −−−−→ A −−−−→ yid. 0
φ ˜ −−− G −→ ρ y
G −−−−→ {1} yid.
0
φ ι ˜ 0 −−− {1} −−−−→ A −−−−→ G −→ G −−−−→ {1}
is commutative. The direct product G × A is an example of a central extension with the inclusion a 7→ (1, a) and the projection (g, a) 7→ g, where g ∈ G and a ∈ A. If the groups involved are topological groups and one wishes to keep track of continuity, as we do in this paper, then in the above the homomorphism ι is ˜ φ must also required also to be a homeomorphism onto a closed subgroup of G, ˜ ˜ be continuous and open (so that G/ι(A) ' G/kernel(φ) ' G), and ρ must be an isomorphism in the category of topological groups. If E is a topological group such that [E, E] is dense in E and p : E → G is a surjective continuous homomorphism, following Moore, we shall say that the pair (E, p) is a cover of G if the kernel of p is contained in the center of E. Then E is an extension of G by the trivial G-module kernel(p) (and, of course, [G, G] is necessarily dense in G). Moore showed that if G is locally compact and separable, then G has at most one simply connected covering group (in this sense) up to isomorphism of topological group extensions (see Lemma 2.2 in [53]). Moreover, if G is perfect, then there does exist such a (unique) simply connected covering group (called the universal covering group) E, which turns out to be perfect and a Lie group itself (Theorem 2.2 in [53] and Theorem 10 in [55]). What will be important for our arguments below is that if G is a semisimple Lie group, then this universal covering group coincides with the standard, topologically defined, universal covering group (cf. p. 49 in [55]). A central extension (U, ν, ) is called universal if for every central n Indeed, Moore [53] suggests the property G = G0 as the algebraic analogue of connectedness.
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˜ φ, ι) of G by A, there exists a (continuous, open) homomorphism h extension (G, ˜ from U to G such that φ ◦ h = ν. If such a universal central extension exists, then it is unique up to isomorphism over G. And it is known (cf. Theorem 5.7 in [50]) that a group G admits a universal central extension if and only if G is perfect. From the remarks above, it is now clear that for semisimple Lie groups, the (standard) universal covering group coincides with the universal covering group in the sense of Moore, which coincides with the universal central extension. ˜ is a Given a central extension (A.1) of G by A,o assume that σ : G → G section with σ(1) = 1, in other words it is a (Borel measurable) set map such ˜ defined that φ(σ(g)) = g for all g ∈ G. The function γ(σ) = γ : G × G → G −1 by γ(g, h) ≡ σ(g)σ(h)σ(gh) is a measure of the amount σ diverges from a ho˜ implies that γ is a 2-cocycle. momorphism, and, of course, the associativity in G Note that because φ(γ(g, h)) = 1, γ actually takes values in the subgroup A. Let Z 2 (G, A) denote the set of all such (Borel measurable) 2-cocycles (which turns out to be an abelian group). Let B 2 (G, A) denote the A-valued coboundaries, i.e. the subgroup of Z 2 (G, A) consisting of functions γ : G × G → A for which there exists a (Borel measurable) β : G → A such that γ(g, h) = β(g)β(h)β(gh)−1 for all g, h ∈ G. The quotient group Z 2 (G, A)/B 2 (G, A) is precisely the second cohomology group H 2 (G, A). One therefore sees that if H 2 (G, A) = {1}, then every ˜ determines a 2-cocycle γ which (A-valued) projective representation σ of G in G is actually a 2-coboundary. Thus, by defining σ ˜ ≡ β(g)−1 σ(g), a straightforward ˜ is a (Borel measurable) homomorphism,p i.e. a calculation shows that σ ˜:G→G representation, as desired. And if H 2 (G, A) is nontrivial, then it is possible to start with a section σ for which there exists no β for which β −1 σ yields a homomorphism. In this case, the question would have to be settled for a given section individually. In the setting of relevance to this paper, E 3 e 7→ V (p(e)) is a continuous projective representation of E with coefficients in Z. We prove the relevant cohomological result for the covering group E. Lemma A.1. Let G be a connected semisimple Lie group and E be its universal covering group. Then the second cohomology group H 2 (E, Z) is trivial. Proof. In the proof of Theorem 9 in [55], it is shown that for a perfect, almost connected group G, the second cohomology group H 2 (E, S 1 ) in Moore’s Borel measurable cohomology theory is trivial, where S 1 is the circle group. This result is thus applicable to the situation described by the hypothesis. Moreover, since G, and hence E, is perfect, it follows that also the first cohomology group H 1 (E, S 1 ) is trivial (see p. 48 in [55]). Thus Proposition 4 in [55] may be applied, yielding H 2 (E, A) is trivial for any unitary trivial G-module A, and, in particular, for A = Z. o For the purposes of his cohomology theory, in [51, 52] Moore took A to be an abelian, locally compact and second countable topological group. However, in [54] he extended his results to include second countable, Hausdorff polonais groups A. We may, therefore, take A = Z below. p The passage from Borel measurable to continuous will be addressed separately below.
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Hence, there exists a function Z : E → Z such that U (e) ≡ Z(e)V (p(e)), e ∈ E, is a true representation of E. One does indeed obtain a (unitary) representation of the group E. But in Moore’s cohomology, the cochains are only Borel measurable on the group; in other words, although the original section σ is continuous, the function β may only be Borel measurable, so that σ ˜ ≡ β −1 σ, i.e. U , may be only Borel measurable. However, the following result, attributed to Mackey in [79], closes this gap. Lemma A.2. If H1 is a locally compact second countable group, H2 is any second countable topological group, and h : H1 → H2 is a Borel measurable homomorphism, then h is continuous. Proof. This is Theorem B.3 in [79].
Hence, by taking H1 = E and H2 = U(H), it follows that E 3 e 7→ U (e) is, in fact, a strongly continuous unitary representation of E, completing the proof of Theorem A.1. In the more structured setting of the main text of this paper, G is the Poincar´e ↑ . There we get by an application of the preceding results: group P+ Corollary A.1. Let V (·) be the continuous unitary projective representation of ↑ P+ with values in J which has been constructed in Subsec 4.3, let J be the closure of J in the strong operator topology and let Z be the center of J . There exists a strongly continuous unitary representation U (·) of the covering group ISL(2, C) ↑ with values in J and a mapping Z : ISL(2, C) → Z of the Poincar´e group P+ ↑ with U (A) = Z(A)V (µ(A)), A ∈ ISL(2, C). Here, µ : ISL(2, C) → P+ is the canonical covering homomorphism whose kernel is a subgroup of order 2, the center of ISL(2, C). Acknowledgments As this paper has been simmering for many years, the authors have reason to thank many persons and institutions. DB thanks the Institute for Fundamental Theory at the University of Florida for an invitation in 1992, where this work was begun. He also acknowledges financial support from the Deutsche Forschungsgemeinschaft. SJS wishes to thank the Second Institute for Theoretical Physics at the University of Hamburg and DESY for invitations in the summers of 1993–95, as well as the University of Florida for travel support, which made the continuation of this collaboration possible. Part of this work was completed while SJS was the Gauss Professor at the University of G¨ ottingen in 1994. For that opportunity SJS wishes to thank Prof. H.-J. Borchers and the Akademie der Wissenschaften zu G¨ ottingen. Further progress was made while DB was a guest of the Department of Physics of the University of California at Berkeley in 1997, and he gratefully acknowledges the hospitality of E. H. Wichmann as well as a travel grant from the Alexander von Humboldt Foundation. Finally, DB and SJS express their gratitude to Prof. J. Yngvason and the Erwin Schr¨odinger International Institute for Mathematical
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Physics for providing the circumstances permitting the completion of this paper. All authors are grateful for useful comments by Profs. H.-J. Borchers, P. Ehrlich, C. Stark, H. V¨olklein and E. H. Wichmann, which helped bring this long-standing project to a successful end. Note In the nearly two years since this manuscript was completed for publication, we have made some developments and advances which may interest the reader of this paper. In [80] we have shown that the net continuity condition of Subsec. 4.3 is not necessary to derive a strongly continuous representation of the translation group from the remaining assumptions, and in [81] we do the same for the Lorentz, Poincar´e and de Sitter groups. Some of the arguments in Subsec. 4.3 are simplified and clarified. We also show that the assumption in Subsec. 5.2 that the group K acts transitively upon the net may be dropped and yet the conclusions in Subsecs. 5.2 and 5.3 are preserved. In [82] we provide a family of examples of nets and states on Robertson–Walker space-times which satisfy our CGMA, thereby significantly extending the range of known examples. And some of these nets provide physically interesting examples of the phenomenon briefly addressed at the end of Subsec. 4.1 and in Sec. 7 in which the modular objects satisfy the CGMA but the corresponding group T is not implemented by point transformations. These examples support the assertion made in Sec. 7 that the CGMA usefully selects physically significant states even when the group T is not implemented by point transformations. In addition, we mention that, though the main thrust of the paper is in another direction, in [83] we show that geometric modular action also manifests iteslf in any state on Anti-de Sitter space-time which satisfies the Second Law of Thermodynamics. Finally, in [84] may be found a number of technical improvements of the results and arguments presented in Sec. 6. References [1] J. Ahrens, “Begr¨ undung der absoluten Geometrie des Raumes aus dem Spiegelungsbegriff”, Math. Zeitschr. 71 (1959) 154–185. [2] A. D. Alexandrov, “On Lorentz transformations”, Uspehi Mat. Nauk. 5 (1950) 187. [3] A. D. Alexandrov, “Mappings of spaces with families of cones and space-time transformations”, Annali di Mat. Pura Appl. 103 (1975) 229–257. [4] H. Araki, “Symmetries in theory of local observables and the choice of the net of local algebras”, Rev. Math. Phys. (Special Issue) (1992) 1–14. [5] F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, second edition, Berlin, New York, Springer-Verlag, 1973. [6] U. Bannier, “Intrinsic algebraic characterization of space-time structure”, Int. J. Theor. Phys. 33 (1994) 1797–1809. [7] U. Bannier, R. Haag and K. Fredenhagen, “Structural definition of space-time in quantum field theory”, unpublished preprint, 1989. [8] W. Benz, Real Geometries, Mannheim, Leipzig, Vienna and Z¨ urich, B. I. Wissenschaftsverlag, 1994. [9] J. Bisognano and E. H. Wichmann, “On the duality condition for a hermitian scalar field”, J. Math. Phys. 16 (1975) 985–1007.
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ON THE SPECTRAL AND SCATTERING ¨ THEORY OF THE SCHRODINGER OPERATOR WITH SURFACE POTENTIAL AYHAM CHAHROUR and JAOUAD SAHBANI Institut de Math´ ematiques de Jussieu, CNRS UMR 7586 Physique math´ ematique et G´ eom´ etrie Universit´ e Paris 7-Denis Diderot U.F.R. de Math´ ematiques, case 7012 Tour 45–55, 5-` eme ´ etage, 2, place Jussieu 75251 Paris Cedex 05, France E-mail:
[email protected] E-mail:
[email protected] Received 20 December 1998 We consider a discrete Schr¨ odinger operator H = −∆+V acting in `2 (Zd+1 ), with potential V supported by the subspace Zd × {0}. We prove that σ(−∆) = [−2(d + 1), 2(d + 1)] is contained in the absolutely continuous spectrum of H. For this we develop a scattering theory for H. We emphasize the fact that this result applies to arbitrary potentials, so it depends on the structure of the problem rather than on a particular choice of the potential.
1. Introduction In this paper we will study the spectral properties and scattering theory of a discrete Schr¨ odinger operator in `2 (Zd+1 ) perturbed by a potential with support in the subspace Zd × {0}. More precisely, let E = Zd × Z(d ≥ 1) be the configuration space. Denote Pd X = (ξ, x) the elements of E and |X| = i=1 |ξi | + |x|. Let H = `2 (E) be the Hilbert space of square integrable sequences f : E → R equipped with the scalar product X hf, gi = f (X)g(X) . X∈E
Let us consider a discrete Schr¨ odinger operator H = H0 + V , acting in H, where H0 is the discrete Laplacian given by X (H0 ψ)(X) = − ψ(X 0 )
(1.1)
(1.2)
|X 0 −X|=1
and V is the multiplication operator by a potential V which we shall define now. Let us consider a function v(ξ) : Zd → R and let δ(x) be the Kronecker symbol, i.e. δ(x) = 1 if x = 0 and 0 elsewhere. The potential V that we consider in (1.1) is of the form V (ξ, x) = v(ξ)δ(x) 561 Reviews in Mathematical Physics, Vol. 12, No. 4 (2000) 561–573 c World Scientific Publishing Company
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acting by multiplication: ( (V ψ)(ξ, x) =
v(ξ)ψ(ξ, 0)
if x = 0
0
otherwise .
(1.3)
Let us remark that the support of V is contained in the subspace Zd × {0}, so V is called a surface potential. It is well known that H0 is a bounded self-adjoint operator on H, and σac (H0 ) = σ(H0 ) = [−2(d + 1), 2(d + 1)] , σpp (H0 ) = σsc (H0 ) = ∅ . For every real-valued potential v, it is clear that the operator H is self-adjoint in H, since H0 is bounded in H. Moreover, using Weyl’s criterion one can see that σ(H0 ) is contained in the spectrum of H. Our goal is to study the spectral and propagation properties of H in σ(H0 ). There is a large literature on the spectral properties of H (see [9, 5, 7, 8, 10] and references therein). For example in [5] the authors study the case where V belongs to a special class of unbounded quasiperiodic potentials. In that case they prove that away from σ(H0 ) the spectrum of H is pure point dense.a A typical example of potential considered in [5] is v(ξ) = λ tan(πα · ξ + θ) with α = (α1 , . . . , αd ) ∈ [0, 1]d and θ ∈ R. In that case H is the Maryland surface model. This model was also studied in [10] and [7] where the authors prove that if α has diophantine properties, i.e. if there exist constants C, k > 0 such that |ξ · α − n| > C|ξ|−k ,
∀ ξ ∈ Zd ,
∀n ∈ Z,
then the spectrum of H is dense and pure point outside σ(H0 ) for any λ 6= 0 and θ ∈ R. They also prove that if α1 , . . . , αd are Q-linearly independent, then for any λ 6= 0 and θ ∈ R, the spectrum of H is purely absolutely continuous on σ(H0 ). In [8] the authors deal with the case where v(ξ) = vω (ξ) are identically distributed random variables. Again the authors prove that the spectrum of H is almost surely pure point and dense outside σ(H0 ). Our goal here is to study the spectral and propagation properties of H inside σ(H0 ). Our main result is: Theorem 1.1. For every potential v we have σ(H0 ) ⊂ σac (H). For bounded v, the proof was sketched in [9] which is the only paper where the spectral properties of H in σ(H0 ) are obtained. This proof cannot work in our a We say that the spectrum of H is pure point dense on a real subset A if the set of eigenvalues of H belonging to A is dense in σ(H) ∩ A, and σc (H) ∩ A = ∅. The spectrum of H is pure point dense outside A if it is on R \ A.
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general case. In fact, we generalize the idea of [9] to the case where v is at most of polynomial growth (see the Appendix). This result remains valid for a Schr¨odinger operator in a half-space, more precisely for the self-adjoint operator H given by ( (H0 ψ)(ξ, x) if x > 0 (Hψ)(ξ, x) = (H0 ψ)(ξ, x) + v(ξ)ψ(ξ, 0) if x = 0 . Theorem 1.1 is based on the following theorem by which we establish the existence of the wave operators. Theorem 1.2. For every potential v the wave operators Ω± = s- lim eitH e−itH0 t→±∞
exist. It is well known that from the existence of these wave operators we get eitH Ω± = Ω± eitH0 . So using the fact that Z RH (z) =
∞
eit(H−z) dt ,
if Im z < 0 ,
eit(H−z) dt ,
if Im z > 0 ,
0
Z
0
−∞
we obtain R(z)Ω± = Ω± R0 (z) and by an obvious approximation argument E(J)Ω± = Ω± E0 (J) . Here E0 and E denote the spectral measures of H0 and H respectively. Theorem 1.1 follows: see [1] or [11] for more details. Remark 1.1. This brings us to the more subtle question of the existence of other components of the spectrum of H in σ(H0 ). In fact, this work shows that the existence of the absolutely continuous component of the spectrum of H in σ(H0 ) only depends on the structure of the problem. However the absence of each other component requires more knowledge on the potential V . Remark 1.2. Let us assume that V is an unbounded quasiperiodic potential as that considered in [5]. In this work the authors prove under some mild assumption the existence of a large number λ0 > 2d+1 such that outside [−λ0 , λ0 ] the spectrum of H is pure point, dense and of multiplicity one. Moreover the corresponding
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eigenfunctions are exponentially localized. Combining this result with ours, we obtain an example of Schr¨odinger operator with some mobility edges, i.e. whose spectrum is absolutely continuous in some region of the real axis and pure point sufficiently far. Remark 1.3. There is a similar result in the case where the potential is random as in [8]. More precisely, in their work Jaksic and Molchanov consider that if v = vω where the vω are independent and identically distributed random variables then the spectrum of H is almost surely pure point and dense outside [−4, 4] ([−4, 4] is the spectrum of the discrete Laplacian in dimension two, which is the case considered by these authors). This gives other examples of Schr¨odinger operators with different spectral structure in different region of the real axis. 2. Proof of Theorem 1.2 As explained in the introduction the proof of Theorem 1.1 is based on Theorem 1.2. In this section we give a detailed proof of Theorem 1.2. Let us start by proving the following lemma, which suffices also to prove Theorem 1.1. Lemma 2.1. For any potential v, the wave operators Ω± (H, H0 , 1 − δ) = s- lim eitH (1 − δ)e−itH0 t→±∞
exist. Proof. To prove the existence of the wave operators Ω± (H, H0 , 1 − δ), we use Cook’s method: it is sufficient to show that there is a dense set D in `2 (Zd+1 ), such that for all f ∈ D, we have Z ∞ k[H(1 − δ) − (1 − δ)H0 ]e−itH0 f kdt < ∞ . (2.1) 1
But we have A(t) = k[H(1 − δ) − (1 − δ)H0 ]e−itH0 f k = k[H0 , δ]e−itH0 f k . Here we have used the fact that V (1 − δ) = vδ(1 − δ) = 0. On the other hand we have in the representation H = `2 (Zd ) ⊗ `2 (Z): H0 = H0d ⊗ 1 + 1 ⊗ h0 δ = 1⊗δ, where H0d and h0 are the discrete laplacians in `2 (Zd ) and `2 (Z) respectively. It follows that d [H0 , δ]e−itH0 = e−itH0 ⊗ [h0 , δ]e−ith0 .
¨ SCHRODINGER OPERATOR WITH SURFACE POTENTIAL
Then Ineq. (2.1) follows from Z ∞ Z A(t)dt = 1
∞
k[h0 , δ]e−ith0 f kdt < ∞
565
(2.2)
1
for a dense set of vectors f ∈ `2 (Z). Let us calculate the commutator [h0 , δ] which is clearly trace class, since δ is trace class and h0 is bounded. Let ψ, ϕ be two vectors in `2 (Z). We have (ϕ, [h0 , δ]ψ) = (h0 ϕ, δψ) − (δϕ, h0 ψ) = (h0 ϕ)(0)ψ(0) − (h0 ψ)(0)ϕ(0) = (−ϕ(1) − ϕ(−1))ψ(0) − (h0 ψ)(0)ϕ(0) . Let us denote by δn the vector in `2 (Z) given by δn (x) = 0 if x 6= n and δn (x) = 1 if x = n. It is clear that {δn }n∈Z is an orthonormal basis of H. Then X k[h0 , δ]ψk2 = |(δn , [h0 , δ]ψ)|2 n∈Z
= 2|ψ(0)|2 + |h0 ψ(0)|2 . By taking ψ = e−ith0 f , it follows that Ineq. (2.2) is a consequence of Z ∞ [|(e−ith0 f )(0)| + |(h0 e−ith0 f )(0)|]dt < ∞
(2.3)
1
for a dense set of vectors f ∈ `2 (Z). We shall state this in the following lemma. Lemma 2.2. For all n > 0 there is D1 dense in `2 (Z) such that ∀ u ∈ D1 we have |(e−ith0 u)(0)| ≤ Cu,n |t|−n
(2.4)
|(h0 e−ith0 u)(0)| ≤ Cu,n |t|−n .
(2.5)
End of the proof of Lemma 2.1. Now it suffices to choose in Lemma 2.2 the integer n sufficiently large to obtain Ineq. (2.3), and so Lemma 2.1 follows too. Proof of Lemma 2.2. Let T = R/2πZ be the circle and dα its usual Lebesgue measure. Let us set φ(α) = −2 cos α. Let us consider the Fourier transformation F : `2 (Z) → L2 (T) given by 1 X (Fu)(α) = u ˆ(α) = √ u(n)e−inα . 2π n∈Z It is clear that h0 is unitary equivalent to the multiplication operator by φ(α) in L2 (T). Let us write Z 1 (e−ith0 u)(0) = √ e−itφ(α) u ˆ(α)dα . 2π T
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Let us assume that u is such that its Fourier transform is sufficiently smooth and has support included in (−π, 0) ∪ (0, π). Then by integration by parts we can easily deduce estimate (2.4). On the other hand, it suffices to take D1 = {u ∈ `2 (Z)|ˆ u ∈ C0∞ ((−π, 0) ∪ (0, π))} which is clearly dense in `2 (Z) and each of whose elements satisfies (2.4). For the remaining term, let us write Z −2 e−itφ(α) cos(α)ˆ u(α)dα . (h0 e−ith0 u)(0) = √ 2π T We obtain the desired result (2.5) similarly. This proves Lemma 2.2.
Proof of Theorem 1.2. (i) Using the obvious relation 1 = δ + (1 − δ), we get e−itH e−itH0 = e−itH δe−itH0 + e−itH (1 − δ)e−itH0 . By the preceding lemma the second term has a strong limit as t tends to ±∞ and this limit is Ω± (H, H0 , 1 − δ). Then it suffices to show that the first term is convergent. In fact, we shall prove that this limit exists and is equal to zero, that is Ω± (H, H0 ) = Ω± (H, H0 , 1 − δ) . (ii) Let f ∈ H. We have to prove that lim ke−itH δe−itH0 f k = lim kδe−itH0 f k = 0
t → ±∞
t → ±∞
for f belonging to a dense set. For this let us take f = f1 ⊗ f2 , where f1 ∈ `2 (Zd ) and f2 ∈ `2 (Z). We have kδe−itH0 f1 ⊗ f2 k = kf1 k · kδe−ith0 f2 k. But h0 is purely absolutely continuous and so e−ith0 f2 tends to zero weakly at infinity. Then δe−ith0 f2 tends to zero strongly, since δ is compact in `2 (Z), which finishes the proof. 3. Almost-Mathieu Surface Potential In this section we give an example of a surface potential where the spectrum of the associated Schr¨odinger operator is purely absolutely continuous. Let us consider the potential v given by v(ξ) = λ cos(πα · ξ + θ) ,
∀ ξ ∈ Zd
where λ ∈ R, α = (α1 , . . . , αd ) ∈ [0, 1]d and θ ∈ [0, π]. For simplicity we consider the Schr¨odinger operator defined on the half space Zd+1 = Zd × Z+ : + X − ψ(η, y) if x > 0 |x−y|+|ξ−η|=1 (Hψ)(ξ, x) = X −ψ(ξ, 1) − ψ(ξ, 0) + v(ξ)ψ(ξ, 0) if x = 0 , |ξ−η|=1
¨ SCHRODINGER OPERATOR WITH SURFACE POTENTIAL
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where (ξ, x) ∈ Zd+1 = Zd × Z+ . As explained in the introduction, in this case + Theorems 1.1 and 1.2 hold. In particular, [−2(d + 1), −2(d + 1] ⊂ σac (H). In this section we shall prove that for some coupling constants λ the spectrum of H is in fact purely absolutely continuous. Theorem 3.1. For each λ ∈ (−1, 1), for each α ∈ [0, 1]d and θ ∈ R, the spectrum of H is purely absolutely continuous. Proof. (i) Using Stone’s formula (see [11]), it suffices to show that lim sup |(f, (H − E − iε)−1 f )| < ∞
(3.1)
ε→0
for each spectral parameter E, and each vector f belonging to a dense subspace of H. Lemma 3.1. Let Hξ be the cyclic subspace generated by the vector δ(ξ,0) and H. Then the linear span of the Hξ ’s is dense in `2 (Zd+1 + ).
Proof. The proof of this lemma is easy and we omit it. See [8] for a proof. Then it suffices to prove that, for every ξ ∈ Zd , lim sup |(δ(ξ,0) , (H − E − iε)−1 δ(ξ,0) )| < ∞ .
(3.2)
ε→0
(ii) Let us consider the unitary operator Uξ acting by translation by the vector (ξ, 0). We shall use the dependence of the operator H on λ, α and θ, so let us denote H = H(λ, α, θ). We have Uξ H(λ, α, θ)Uξ = H(λ, α, θ0 ), with θ0 = θ −πα −ξ. Then (δ(ξ,0) , (H(λ, α, θ) − E − iε)−1 δ(ξ,0) ) = (δ(0,0) , (H(λ, α, θ0 ) − E − iε)−1 δ(0,0) ) . So to prove (3.2), it suffices to show that for each θ lim sup |(δ(0,0) , (H(λ, α, θ) − E − iε)−1 δ(0,0) )| < ∞ .
(3.3)
ε→0
(iii) For (ξ, x), (η, y) ∈ Zd × Z+ , let R((ξ, x), (η, y); z) = (δ(ξ,x) , (H − E − iε)−1 δ(η,y) ) be the kernel of the resolvent of H. Using (H − z)(H − z)−1 = 1 and the definition of V , we get that these matrix elements satisfy the equation X −R((ξ, x), (η, y + 1); z) − R((ξ, x), (η, y − 1); z) − R((ξ, x), (ζ, y); z) |η−ζ|=1
= δ(x − y)δ(ξ − η) + zR((ξ, x), (η, y); z)
(3.4)
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A. CHAHROUR and J. SAHBANI
if y > 0, and −R((ξ, x), (η, 1); z) −
X
R((ξ, x), (ζ, 0); z) + (v(η) − z)R((ξ, x), (η, 0); z)
|η−ζ|=1
= δ(x)δ(ξ − η) ,
(3.5)
if y = 0. Let T = R/2πZ. We consider the Fourier transformation 2 d 2 F : `2 (Zd+1 + ) → L (T ) ⊗ ` (Z+ )
defined by ˆ x) = (Fψ)(φ, x) = ψ(φ,
X 1 ψ(η, x)e−iξ·φ d/2 (2π) d Pd
where φ = (φ1 , . . . , φd ) ∈ Td . Let Φ(φ) = −2 two Eqs. (3.4) and (3.5) we find that for x = 0
η∈Z
k=1
cos φk . Then applying F to the
ˆ ˆ −R((ξ, 0), (φ, y + 1); z) − R((ξ, 0), (φ, y − 1); z) ˆ + (Φ(φ) − z)R((ξ, 0), (φ, y); z) = 0
(3.6)
if y > 0, and ˆ ˆ −R((ξ, 0), (φ, 1); z) + (Φ(φ) − z)R((ξ, 0), (φ, 0); z) c + vR((ξ, 0), (φ, 0); z) = eξ (φ)
(3.7)
if y = 0. Here eξ (φ) = (2π)−d/2 e−iξ·φ . It follows from the Eq. (3.6) that for y > 0, ˆ ˆ R((ξ, 0), (φ, y); z) = R((ξ, 0), (φ, 0); z)r(φ, z)y ,
(3.8)
where r(φ, z) is the root of the equation −X − X −1 + Φ(φ) = z ,
(3.9)
such that |r(φ, z)| < 1. Substituting (3.8) into (3.7) and using r + Φ − z = r˜ where r˜ = r−1 is the other root of (3.9), we get ˆ c R((ξ, 0), (φ, 0); z)˜ r(φ, z) + vR((ξ, 0), (φ, 0); z) = eξ (φ) .
(3.10)
Z
If we consider
eiξ·φ r˜(φ, z)dφ ,
j(ξ, z) = Td
applying F −1 to (3.10) we get X j(ξ − η, z)R((ξ, 0), (η, 0); z) + v(ξ)R((ξ, 0), (η, 0); z) = δ(ξ − η) . η
Replacing v by its value v(ξ) = λ cos(πα · ξ + θ) =
λ iπα·ξ+iθ (e + e−iπα·ξ−iθ ) , 2
(3.11)
¨ SCHRODINGER OPERATOR WITH SURFACE POTENTIAL
569
and applying F again, we get λ ˆ ˆ R((ξ, 0), (φ, 0); z)˜ r(φ, z) + eiθ R((ξ, 0), (φ + πα, 0); z) 2 λ −iθ ˆ e R((ξ, 0), (φ − πα, 0); z) = eξ (φ) . 2
+
(3.12)
(iv) Now we can establish (3.3). We have −1
(δ(0,0) , (H − E − iε)
−d/2
δ(0,0) ) = (2π)
Z ˆ R((0, 0), (φ, 0); E + iε)dφ . Td
So we need to prove that Z lim sup Td
ε→0
ˆ |R((0, 0), (φ, 0); E + iε)|dφ < ∞ .
It follows from (3.12) that ˆ |R((0, 0), (φ, 0); E + iε)˜ r(φ, E + iε)| ≤ C0 + +
|λ| ˆ R((0, 0), (φ + πα, 0); E + iε)| 2
|λ| ˆ R((0, 0), (φ − πα, 0); E + iε)| . 2
By integration on the torus we get Z ˆ |R((0, 0), (φ, 0); E + iε)˜ r(φ, E + iε)|dφ Td
Z ≤ C0 + |λ|
Td
ˆ |R((0, 0), (φ, 0); E + iε)|dφ ,
r(φ, E + ε)| > 1 we get where C0 = (2π)d/2 . Using that |˜ Z C0 ˆ |R((0, 0), (φ, 0); E + iε)|dφ, ≤ . 1 − |λ| Td Z
So lim sup ε→0
This finishes the proof.
Td
ˆ |R((0, 0), (φ, 0); E + iε)|dφ ≤
C0 . 1 − |λ|
Appendix As mentioned in the introduction, the only paper where the spectral properties of H in σ(H0 ) are studied is [9]. In fact, in this paper Theorems 1.1 and 1.2 are proved for any bounded potential. In this section we shall discuss ideas of this paper. The proof of Theorem 1.2 in [9] is different from our. Indeed the existence of the wave operators Ω± (H, H0 ) is proved directly. In this appendix we shall give another
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proof of Theorem 1.2 for any potential v but which is at most of polynomial growth. We shall also prove some lemma which is interesting for himself. Using Cook’s method, it suffices to show that there is a dense set D in `2 (Zd+1 ), such that for all f ∈ D, we have Z ∞ k(H − H0 )e−itH0 f kdt < ∞ . (A.1) 1
This is equivalent to
Z
∞
kV e−itH0 f kdt < ∞ .
(A.2)
1
Using the definition of V we get kV e−itH0 f k2 =
X
|v(ξ)(e−itH0 f )(ξ, 0)|2 .
(A.3)
ξ∈Zd
Since v is bounded, it is easy to see that (A.2) follows from Z 1
∞
X
12 |eitH0 f (ξ, 0)|2 dt < ∞ .
(A.4)
ξ∈Zd
In fact, it suffices to show that X
|e−itH0 f (ξ, 0)|2 ≤
ξ∈Zd
Cf , t4
(A.5)
for large t and for f in some dense set D in `2 (Zd+1 ). Let f = f2 ⊗ f1 where f1 is defined on Z and f2 is defined on Zd , then (eitH0 f )(ξ, 0) = (e−ith0 f1 )(0)(e−itH0 f2 )(ξ) . d
(A.6)
Hence X
|e−itH0 f (ξ, 0)|2 = |(e−ith0 f1 )(0)|2
ξ∈Z d
X
|(e−itH0 f2 )(ξ)|2 d
ξ∈Z d
= |(e−ith0 f1 )(0)|2 ke−itH0 f2 k , d
so
X
|e−itH0 f (ξ, 0)|2 ≤ |(e−ith0 f1 )(0)|2 kf2 k2 .
(A.7)
ξ∈Zd
Remark A.1. The proof of the theorem when v is bounded is now obvious. Indeed, it suffices to take n = 2 in Lemma 2.2, using (A.7) to show (A.5). Let us extends this proof to a class of unbounded potentials. In that case, by combining (A.3) and (A.6) one can see that we have to examine the behavior of d (e−itH0 f2 )(ξ) when |ξ| → ∞ in order to prove the suitable integrability property. While if v is bounded it is not necessary to do it.
¨ SCHRODINGER OPERATOR WITH SURFACE POTENTIAL
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Lemma A.1. Let m ∈ N, then there exists a dense set D2 in `2 (Z) such that for all f ∈ D2 we have (1 + |t|)m |(e−ith0 f )(ξ)| ≤ Cf . (1 + |ξ|)m In fact D2 = {f ∈ `2 (Z)|f is smooth and supp fˆ ∈ (−π, π)}, which is clearly dense. Proof. Let us write (e
ith0
1 f )(ξ) = 2π
Z
eiξα e−itφ(α) fˆ(α)dα.
T
It is clear that m integrations by parts give the result.
Remark A.2. Now we are able to prove Theorem 1.2 when d = 1 and the potential satisfies v(ξ) ≤ C(1 + |ξ|)s , for some constants C, s > 0 and every ξ ∈ Z. Indeed, we must prove X Cf kV e−itH0 f k = |v(ξ)(e−itH0 f )(ξ, 0)|2 ≤ 4 . t d ξ∈Z
But we have written that for f = f2 ⊗ f1 |v(ξ)(e−itH0 f )(ξ, 0)| = |v(ξ)k(e−ith0 f1 )(0)k(e−ith0 f2 )(ξ)| . By Lemma A.1 we get |v(ξ)(e−itH0 f )(ξ, 0)| ≤ |v(ξ)k(e−ith0 f1 )(0)|Cf2
(1 + |t|)m (1 + |ξ|)m
≤ Cf2 hξis−m |(e−ith0 f1 )(0)|(1 + |t|)m
where hξi =
p 1 + |ξ|2 . On the other hand, Lemma 2.2 implies that |v(ξ)(e−itH0 f )(ξ, 0)| ≤ Cf1 Cf2 hξis−m (1 + |t|)m−n .
It suffices to take m and n = m + 2 sufficiently large to have summability over ξ ∈ Z and integrability over t. Remark A.3. Let us mention that (A.1) is proved for any f ∈ D = D1 ⊗ D2 , which is clearly dense in our Hilbert space `2 (Z2 ). Now we have to prove the theorem for d ≥ 1, so we need a multidimensional version of Lemma A.1. Lemma A.2. Let H0d = −∆ acting in `2 (Zd ), then there exists a closed subset D2 in `2 (Zd ) such that for all f ∈ D2 we have d htim |(e−itH0 f )(ξ)| ≤ Cf Yd , hξi imi
i=1
where m = m1 + · · · + md .
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Proof. We have (e
−itH0d
f )(ξ) = (2π)
−d 2
Z
eiξα e−itφ(α) fˆ(α)dα .
Td
If we take f of the type f = f1 ⊗ f2 ⊗ · · · ⊗ fd , and using the fact that e−itφ(α) =
d Y
e−itφ(αi ) ,
i=1
we get d
(eitH0 f )(ξ) =
d Y (e−ith0 fi )(ξi ) . i=1
It follows from Lemma A.1 that d
|(eitH0 f )(ξ)| =
d Y
|(e−ith fi )(ξi )|
i=1
≤
d Y
htim Cfi htimi hξi i−mi ≤ Cf Yd . i=1 hξi imi i=1
This completes the proof of Lemma A.2. Combining Lemma 2.2 and Lemma A.1, one concludes easily that the wave operators Ω± (H, H0 ) exist for any potential v satisfying |v(ξ)| ≤ C(1 + |ξ|)s , for some real number s > 0. Acknowledgment The authors take this opportunity to express their gratitude to Professor Anne Boutet de Monvel for suggesting the subject of this work and for many helpful discussions.b References [1] A.-M. Berthier (Boutet de Monvel), “Spectral theory and wave operators for the Schr¨ odinger equation”, Research Notes in Math., 71, Pitman, Advanced Publ. Program, Boston, Mass.-London, 1982. [2] A. Boutet de Monvel and J. Sahbani, “On the spectral properties of discrete Schr¨ odinger operators”, C. R. Acad. Sci. Paris S´er. I Math. 326 (1998) 1145–1150. [3] A. Boutet de Monvel and J. Sahbani, “On the spectral properties of discrete Schr¨ odinger operators, the multidimensional case”, Rev. Math. Phys. 11 (1999) 1061–1078. [4] A. Boutet de Monvel and J. Sahbani, “Jacobi matrices with absolutely continuous spectrum”, C. R. Acad. Sci. Paris S´er. I Math. 328(5) (1999) 443–448. b In fact, Anne Boutet de Monvel announced the results of this paper in her conference at Bonn in September 1998 where she discussed the spectral properties of Schr¨ odinger operators with anisotropic potentials (see [2–4]).
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[5] A. Boutet de Monvel and A. Surkova, “Localisation des ´etats de surface pour une classe d’op´erateurs de Schr¨ odinger discrets ` a potentiels de surface quasi-p´eriodiques”, Helv. Phys. Acta 71 (1998) 459–490. [6] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer, Berlin, 1987. [7] V. Jaksic and S. Molchanov, “On the spectrum of the surface Maryland Model”, Lett. Math. Phys. 45 (1998) 185–193. [8] V. Jaksic and S. Molchanov, On the surface spectrum in dimension two, preprint. [9] V. Jaksic, S. Molchanov and L. Pastur, “On the propagation properties of surface waves”, in “Wave propagation in complex media”, G. Papanicolaou, IMA Vol. Math. Appl. 96 (1998) 145–154, Springer, New York, 1998. [10] B. Khoruzhenko and L. Pastur, “The localization of surface states: An exactly solvable model”, Phys. Reports 288 (1997) 109–126. [11] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 3, Scattering Theory, Academic Press, New York, 1979.
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY OF STATES OF AN UNBOUNDED RANDOM JACOBI MATRIX ´ ERIC ´ FRED KLOPP D´ epartement de Math´ ematique Institut Galil´ ee, U.R.A 742 C.N.R.S Universit´ e de Paris-Nord Avenue J.-B. Cl´ ement F-93430 Villetaneuse, France E-mail:
[email protected] Received 17 December 1997 Revised 22 February 1999 The purpose of this paper is to study the transition from the classical to the quantum asymptotics for the integrated density of states of an unbounded random Jacobi matrix. Therefore, we give precise results on the behavior of the tail of the integrated density of states near infinity. We study the evolution of these asymptotics when the decay of the tail of the distribution of the random potential increases. ´ ´ . Cet article est consacr´ e ` a l’´ etude de la transition entre le r´egime classique et E RESUM le r´ egime quantique pour la densit´e d’´ etats int´ egr´ee d’une matrice de Jacobi al´eatoire non born´ ee. Pour cela nous donnons des asymptotiques pr´ecises du comportement de la densit´ e d’´ etats int´ egr´ee au voisinage de l’infini. De plus, nous ´etudions le comportement de cette asymptotique lorsque la d´ ecroissance a ` l’infini de la distribution du potentiel al´ eatoire augmente.
Contents 0. Introduction 1. The Main Results 1.1. The classical regime 1.1.1. Asymptotic expansions in the classical regime 1.2. The quantum regime 2. Periodic Approximations 2.1. Some Floquet theory 3. The Classical Regime 3.1. The proofs of Theorems 1.1 and 1.2 3.2. The asymptotic expansion 4. The Quantum Regime 4.1. The proof of Lemma 4.1 4.1.1. The lower bound 4.1.2. The upper bound References
575 577 577 580 584 585 587 588 590 595 610 613 614 615 620
0. Introduction Let H be a translational invariant Jacobi matrix with exponential off-diagonal decay that is H = ((hk−k0 ))k,k0 ∈Zd such that: 575 Reviews in Mathematical Physics, Vol. 12, No. 4 (2000) 575–620 c World Scientific Publishing Company
576
F. KLOPP
• h−k = hk for k ∈ Z and for some k 6= 0, hk 6= 0. • there exists C0 > 0 such that, for k ∈ Zd , |hk | ≤ C0 e−|k|/C0 . 2
(0.1)
d
H defines a bounded self-adjoint operator on ` (Z ). By Fourier transformation, H ˆ is unitarily equivalent to the multiplication by the function θ 7→ H(θ) defined by P ikθ 2 d ˆ H(θ) = k∈Z hk e acting as an operator on L ([−π, π] ) (here and in the rest of the paper, we identify the square integrable functions over a cube C = [−c, c]d ⊂ Rd with the functions that are locally L2 and periodic of period the length of the cube C; moreover the L2 -norm is normalized so that the constant function 1 has norm ˆ ˆ −, H ˆ + ] = σ(H). 1). The spectrum of H is σ(H) = H([−π, π]d ). We define [H We denote by (δk )k∈Zd the vectors of the canonical basis of `2 (Zd ). More precisely, for k ∈ Zd , δk = (δjk )j∈Zd (where δjk is Kronecker’s symbol). Let us denote by Πk the orthogonal projection on δk in `2 (Zd ). Consider now the d-dimensional random Jacobi matrix Hω = H + Vω where Vω is a diagonal matrix with independent identically distributed real entries denoted by (ωk )k∈Zd that is X Vω = ωk Πk . k∈Zd
The tail of the probability distribution of the (ω)k∈Zd near +∞ will be denoted by F (λ) = P({ω0 ≥ λ}) (here P(Ω) denotes the probability of event Ω). We will assume that the random variables (ωk )k∈Zd are unbounded from above, i.e. ∀λ ∈ R
F (λ) > 0 ,
(0.2)
and that they have a finite expectation E(|ω0 |) < +∞ .
(0.3)
0 Moreover we assume R x 0 that F is absolutely continuous near +∞, i.e. F is locally integrable and a F (t)dt = F (x) − F (a) (see [8]). It is well known that the operator Hω has a density of states that we will denote by dν [1, 2, 13]. dν is a positive measure that essentially measures the number of states of the operator per units of volume. It can be defined by Z ϕdν = E(hδ0 , ϕ(Hω )δ0 i) , ϕ ∈ C0∞ (R) , R
where E(·) denotes the expectation with respect to the random variables (ωk )k∈Zd . Define Z +∞ N (λ) = dν . λ
So, in this paper, N denotes the complementary to 1 of the usual integrated density of states. Rough high energy asymptotics for N are classical: for any p ∈ N, we have ! ln(p) F (λ + a) ∗ (0.4) ∃a ∈ R , −−−−→ 1 ⇒ ln(p) N (λ) ∼ ln(p) F (λ) λ→+∞ ln(p) F (λ) λ→+∞
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
577
where ln(p) (·) = ln| ln | · · · k is the p times composed of ln| · | with itself. The equivalence (0.4) is just a simple consequence of the boundedness of H; one has F (λ + kHk) ≤ N (λ) ≤ F (λ − kHk) .
(0.5)
The asymptotic (0.4) tells us that, to first order, N is given by F . The purpose of this paper is to get more precise asymptotics for N (λ) when λ tends to +∞. As a result, we will be able to study the transition between the classical and the quantum regime for N , i.e. between the regime when N is essentially given by F (the classical regime) even at second (or third) order and the regime when the first order correction to the high energy behavior of N really results from tunneling (i.e. the quantum regime), the main term still being given by F . The parameter that will decide whether we are in one regime or the other is the decay rate of F at +∞. This may be understood heuristically in the following way. If one interprets N as the probability of Hω (restricted to some box of size L) having an eigenvalue above level λ, then, this eigenvalue will be created only if, at least one of the random variables (ωk )k is larger than λ. We know that, if we have just a single ωk that is larger than λ, then, to create an eigenvalue larger than λ, ˆ +. ωk will have to be larger than λ − h where h depends on H and satisfies h < H The probability of this event will roughly be of size F (λ − h). The other extreme ˆ + . The probability of this option is that sufficiently many ωk ’s are larger than λ − H M ˆ ˆ +. to happen is roughly F (λ − H+ ) if M is the number of ωk ’s larger than λ − H As M is assumed to be large, these two events are essentially disjoint; so that the probability we want to estimate will be the sum of the two probabilities, i.e. it will be the supremum of the two probabilities if one of the probabilities is much larger than the other one. We will be in the classical case if F (λ − h) is much larger ˆ + )M and, in the quantum case if F (λ − H ˆ + )M is much larger than than F (λ − H F (λ − h). One easily convinces oneself that it is the rate of decay of F at +∞ that will determine whether one is the classical or the quantum regime. That both cases may occur will be seen in examples. Here we do not say what M nor what L, the size of the box should be. All this will be made precise in the proofs. 1. The Main Results It will be convenient to introduce the function g = −ln F . It is increasing and tends to +∞ at +∞. 1.1. The classical regime We will say that we are in the classical regime if, g 0 (λ) −−−−→ 0 . g(λ) λ→+∞
(1.1)
The precision of our results will then depend on the rate of increase of g and of the precision with which g is known. Under no more assumptions than the one made above, we show:
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F. KLOPP
Theorem 1.1. Let Hω and N be defined as above. Then, for any δ > 0, we have N (λ) = F (λ + a0 + o(1)) + o(F (λ)2−δ ) where a0 = −h0 = −
1 Vol (T)
(1.2)
Z ˆ H(θ)dθ T
ˆ is the zeroth Fourier coefficient of H. (1.2) is not very precise but we will see that, under the sole assumption (1.1), one cannot do better. One can improve on the results of Theorem 1.1 if one assumes a minimal rate of decay for F at infinity. More precisely, if we assume that, for some η > 0, F (λ) = o(λ−d−η ) then we get: Theorem 1.2. For Hω and N defined as above. Let f : (0, +∞) → (0, +∞) a function tending to 0 at +∞ such that f (λ) · λ > 1. Then, there exists λδ > 0 and δ : (λδ , +∞) → (0, +∞), a positive function such that • δ(λ) ≥ f (λ), • limλ→+∞ δ(λ) = 0, and such that, for any ν > 0, there exists λs,ν > 0 such that, for λ > λs,ν , one has N (λ) = F (λ + a0 + a1 /λ + δ(λ)/λ) · (1 + o(1)) + E1 (λ) + E2 (λ)
(1.3)
where ˆ • a0 = −h0 is the zeroth Fourier coefficient of H, R R P 2 2 ˆ ˆ • a1 = − k∈(Zd )∗ |hk |2 = | Vol1(T) T H(θ)dθ| − Vol1(T) T |H(θ)| dθ, • E1 (λ) = O(λ2d F (λ)2−ν ), • E2 (λ) = O(F (λ + a0 + o(1))F ( lndδ(λ)λ ) lnd (λ/δ(λ))). (λ/δ(λ)) Let us first comment on formula (1.3). For ω0 > 0, consider the operator H + ω0 Π0 . Let E(ω0 ) denote the supremum of its spectrum. Then, for ω0 large, E(ω0 ) is an eigenvalue of H + ω0 Π0 . It is the unique solution of ω0 · I(E) = 1 where I(E) =
1 (2π)d
Z [−π,π]d
1 ˆ E − H(θ)
dθ .
Hence I(E) admits the following expansion: 1 = E + a0 + a1 /E + O(E −2 ) . I(E)
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1 So we see that the principal term in (1.3) is essentially F ( I(λ) ), i.e. the probability that E(ω0 ) ≥ λ. So, in this case, the density of states is essentially given by the probability that there is a single eigenvalue larger than λ. In this regime (at this level of precision), the density of states of Hω does not at all feel the tunneling between the different sites; or it looks like as if one would be dealing with infinitely many i.i.d. copies of H + ω0 Π0 each one located at a site of Zd . This also explains why one does not feel the rate of the off-diagonal decay of H.
Remark 1.1. The analysis used to prove Theorem 1.2 can also be performed in the continuous case, i.e. for a continuous Anderson model Hω = H + Vω acting on L2 (Rd ) where H is a periodic Schr¨odinger operator and Vω a potential of the form, e.g. X Vω = ωγ V (· − γ) γ∈Γ
• Γ is a non degenerate lattice in Rd , • V is a compactly supported potential, • (ωγ )γ∈Γ a family of i.i.d. random variables that are not lower semi-bounded. For such an operator the integrated density of states is well defined [13]; let us call it N (λ). Then the principal term of the behavior of N at −∞ should be given by P (E(ω0 ) ≤ −λ) where E(ω0 ) is the infimum of the spectrum of the operator H + ω0 V . This infimum will be an eigenvalue when ω0 is large (see [15]). In Theorem 1.2, we give a lower bound on the error of the above described approximation, i.e. the function δ(λ). We see that δ(λ) can not be taken in general of size O(1/λ). This will be verified in Theorem 1.3 where we give more precise results (under more restrictive assumptions on g). We will see that the size of this correction is growing with the increase of the rate of increase of g. The two error terms E1 and E2 can be analyzed in the following way: • The term E1 comes from the fact that we neglected the possibility that the spectrum above energy λ can be created by many sites at the same time. That is from the fact that we neglected the possibility that more than a single random variable takes a value larger than λ. • The second term E2 comes from the fact that we neglected the tunneling between two sites k and k 0 at which the random variables take values close to λ. We can now apply the asymptotic (1.3) to different examples. For example, if we α assume that F (λ) ∼ e−λ then, we get that +∞
N (λ)
∼
λ→+∞
e−(λ+a0 +a1 /λ+o(1/λ)) . α
(1.4)
This is immediately obtained from the fact that in this case δ(λ)λ d F (λ + a0 + o(1))F ln (λ/δ(λ)) + |λ2d F (λ)2−ν | d ln (λ/δ(λ)) = o(e−(λ+a1 +a2 /λ+o(1/λ)) ) . α
(1.5)
580
F. KLOPP
Let us now assume that F has a quicker decay, say F (λ) ∼ e− exp(λ
α
+∞
)
,
where 0 < α < 1
so that α
g(λ) ∼ eλ +∞
still satisfies (1.1). Then the remainder term in (1.3) is not smaller than the principal term any more. Indeed, as δ(λ) −→ 0, we have +∞
F (λ + a1 + o(1)) · F
δ(λ)λ d ln (λ/δ(λ))
lnd (λ/δ(λ)) F (λ + a1 + a2 /λ + δ(λ)/λ) .
In this case the tunneling between two sites at which the random variables take large values (close to λ) gives an important correction to the leading behavior. This may be understood in the following way: as g 0 /g tends to 0 in a slower way, the probability to have two random variables, say ωk and ωk0 of size λ becomes much larger than that of having a single random variable of size λ + C. Hence, the tunneling between the sites k and k 0 becomes more probable and its effect is felt in the asymptotics of N . This will be seen very clearly in the results presented in Theorems 1.3 and 1.4. 1.1.1. Asymptotic expansions in the classical regime ˆ is a trigonometric polynoLet us now assume that H is of finite range, i.e. that H mial. In this case, we will, under more restrictive assumption on H and F be able to get an asymptotic expansion of N . Therefore we need to introduce some geometric notions to control the tunneling induced by H (seen as a small perturbation of Vω ). Pick (k, k 0 ) ∈ Zd . An H-path from k to k 0 is defined to be an n-tuple γ(k, k 0 ) = (k1 , k2 , . . . , kn ) such that k1 = k, kn = k 0 and for j = 1, . . . , n − 1, hkj −kj+1 6= 0. For γ(0, k) = (k1 , k2 , . . . , kn ), an H-path from 0 to k, we define the length of γ(0, k) to be |γ(0, k)| = n. For k ∈ Zd , we define Γ(k) to be the set of all H-paths linking 0 to k. It may be empty. Let Λ be the set of k for which Γ(k) 6= ∅. Then, as H 6= k Id (k ∈ R), Λ 6= ∅. For example, in the case of the discrete Laplace operator, we have Λ = Zd . For k ∈ Λ, we define l(k) = inf{|γ|; γ ∈ Γ(k)}, the length of the shortest H-path leading from 0 to k, and Γ− (k) = {γ ∈ Γ(k); |γ| = l(k)} the set of H-path from 0 to k being of shortest possible length. As H is of finite range, Γ− (k) is finite. For l ∈ N, we also define Λl = {k ∈ Zd ; l(k) = l}. If h0 6= 0 then Λ0 = {0} if not Λ0 = ∅. For ` ∈ N, Λl is finite and Λ = ∪l Λl . Let k ∈ Zd such that l(k) = l. The tunneling between 0 and k through H will (l) be controlled by the coefficient hk defined as (l)
hk =
X γ=(k1 ,...,kl ) k1 =0, kl =k |γ|=l
hk1 −k2 · hk2 −k3 · · · hkl−2 −kl−1 · hkl−1 −kl .
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ˆ l , the lth power of the function This coefficient is also the kth Fourier coefficient of H ˆ ˆ H when the lth power of H is the first power where the kth Fourier coefficient may not vanish. We assume that, for any k ∈ Λ, we have (l(k))
hk
6= 0 .
(1.6)
This is for example the case if there is a unique path in Γ− (k) (e.g. if the dimension is 1). It is also the case is all the entries of H have the same sign, e.g. for the discrete Laplace operator. Assumption (1.6) is an assumption ensuring the non-degeneracy of the tunneling a given distance. To get an asymptotic expansion for N , we will, of course, also need a better knowledge of F (λ). We assume • all the moments of the i.i.d random variables (ωk )k∈Zd are finite (i.e. ∀ n ∈ N, E(|ω0 |n ) < +∞). • g admits a twice differentiable asymptotic expansion near +∞ of one of the following types: (1) for some α > 1 and g0 > 0, X g(λ) ∼ lnα λ gk λ−k . (1.7) λ→+∞
k≥0
(2) for some α > 0 and g00 > 0, g(λ)
∼
X
λ→+∞
gk0 λα(1−k) + ln λ
k≥0
X
gk1 λ−k .
(1.8)
k≥0
Then we have: Theorem 1.3. For Hω and N defined as above, ln N (λ) has a complete asymptotic expansion in terms of λ. Its precise form depends on the type of the asymptotic known for g. More precisely, • in the case of asymptotic of type (1.7), one has ln N (λ)
∼
λ→+∞
− lnα λ
X k≥0
• in the case of asymptotic of type (1.8), — if α ≤ 3 then, one has ln N (λ)
∼
λ→+∞
gk λ−k +
X k6=0
X i,j,m≥0 i+j+m≥1
where n3 = 0 if α < 3 (see (3.32)).
nj λ−j .
(1.9)
j≥1
− g00 λα + g00 αh0 λα−1 + g00 α
− g01 ln λ − g10 +
X
α − 1 |hk |2 − h20 λα−2 + n3 2
ni,j,m λ−i−αj
ln λ λ
m (1.10)
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F. KLOPP
— if α > 3 then, one has ln N (λ)
X
∼
2jα
nl,j λα−l− α−1 − g01 ln λ − g10
λ→+∞ j,l≥0 0≤2jα+l(α−1)≤α(α−1)
X
+
2j
ni,j,l,m,p λα(1−i)−l− α−1
i,j,l,m,p≥0 i+j+l+m+p≥1 α(α−1)<α(α−1)i+(α−1)l+2j
ln λ λα˜
m+pα
(1.11) where * α ˜= * α ˜=
2α α−1 2α α−1
α−1 2
−
α−1 2
if α 6∈ 2N + 1.
if α ∈ 2N + 1.
α−1 2 2 |h | − h k 0 . k6=0 2 P α 2 α−1 = g00 (α − 1) . k6=0 |hk |
* n0,0 = −g00 , n1,0 = g00 αh0 , n2,0 = g00 α * n0,j = n1,j = 0 for any j, n2,1
P
* all the coefficients (nj,l )j,l can be computed inductively. Remark 1.2. In the case when F (t) = e−t , the first terms of the asymptotics given in Theorem 1.3 were announced in [12]. β
Remark 1.3. For the sake of simplicity, in the case of asymptotics of type 1.7, we assumed β > 1. As the proofs show, we could also deal with β = 1 if the coefficient g01 is large enough, though the results we would get would not be as precise as the ones given below; indeed, we would not get a complete asymptotic expansion for N. One can also deal with much more complicated asymptotic expansion than (1.7) or (1.8). The price to pay is an increasingly complicated asymptotic expansion for N. In the case α < 2, we can relax the assumption on the finite range of H. Offdiagonal exponential decay is sufficient. One may also want to relax the assumption on the finiteness of the moments of the random variables (ωk )k∈Zd . This may be done at the expense of changing the type of asymptotics that one gets for N (λ). The tail at −∞ has a direct influence on the terms of negative order. We will not study this possibility as the asymptotics one gets are quite complicated already. Let us now comment on the results of Theorem 1.3. We can compare the different asymptotic expansions one gets when α is increasing. One sees that, with α increasing, new terms appear in the asymptotic expansion. These new terms carry the interaction (i.e. the tunneling) between the different sites in the lattice. The different terms of the principal part of the expansion may be analyzed in the following way:
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• the terms in λα−j (j ∈ N) come essentially from the interaction between the point 0 (that was privileged in our analysis) and itself through its H-neighbors, the nearest ones being considered first and the ordered by increasing order of distance, i.e. by decreasing order of importance for tunneling. It is essentially the asymptotic of the probability distribution of the highest eigenvalue of the operator H + ω0 Π0 . 2 • the first new term, the term of order λα−2− α−1 comes from the larger probability of tunneling between one site and its nearest H-neighbors when the random vari2j ables located at these sites both take large values. The other terms in λα−2− α−1 come from tunneling between one site and its more remote H-neighbors. The index j is counting how far away from each other the sites are. 2 As α → +∞, one notices that α − 2 − α−1 ∼ α − 2. Hence the correction terms become more and more important. This underlines the already noticed fact that, as g 0 /g tends to 0 in a slower way, the possible tunneling between neighbors affects more and more the behavior of N . Hence N behaves more and more in a quantum way. We can also compare the results of Theorems 1.3 and 1.2. We see that, if we rewrite (1.11) in the same form as (1.3), if α > 2, we get
ln N (λ) = ln F (λ + a0 + a1 /λ + b1 λ−(α+1)/(α−1) + o(λ−(α+1)/(α−1) )) . Comparing this with (1.3) and the analysis of (1.3) given after Theorem 1.2, we see that the principal correction to I −1 (λ) is of order λ−(α+1)/(α−1) . This order is increasing with α, i.e. the decay of F . To finish the discussion of this regime, let us just give a concrete example where assumption (1.8) is satisfied: suppose the (ωk )k∈Zd are i.i.d. random variables with α density Cα e−|t| (α > 0). In this case, g has the following asymptotic expansion: X g(λ) ∼ λα + (α − 1) ln λ + ln α + gk λ−kα . λ→+∞
k≥1
This is the case if the (ωk )k∈Zd are Gaussian (α = 2). In this case, we may use Theorem 1.3 to compute the asymptotics of N (λ). The quantum phenomenon will be even stronger if the growth of g at +∞ is faster than polynomial. In this case, we will see that the tunneling corrections already change the second term in the asymptotic. This is the object of our next result. To make the results simple, we will assume that, for λ large enough, we have α
g(λ) = g0 eλ
(1.12)
for some α ∈ (0, 1). Then, we have: Theorem 1.4. For g as above, we have
ln N (λ) = −eg0 λ 1 − αh0 g0 λα−1 + (αg0 )2 α
X
|hk |2 λ2(α−1) + o λ2(α−1) .
k6=0
(1.13)
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F. KLOPP
Let us rewrite (1.13) in the form (1.3). We get α−1
ln N (λ) = −eg0 (λ+h0 +aλ where
+o(λα−1 ))α
a = αg0
= ln F (λ + h0 + aλα−1 + o(λα−1 )) ,
X
|hk |2
k6=0
It is clear from Theorem 1.4 why the asymptotics given by Theorem 1.2 breaks down in the classical case when g 0 /g tends to 0 too slowly: the correction term we saw propagating upwards in the asymptotics in Theorem 1.3 has now overcome the correction computed in Theorem 1.2, i.e. it is the error terms in this expansion that now become principal. 1.2. The quantum regime We will say that we are in the quantum regime if g 0 (λ) −−−−→ + ∞ . g(λ) λ→+∞
(1.14)
It will be convenient to introduce h = ln g. Then h is increasing and tends to +∞ at +∞. (1.14) becomes h0 (λ) → +∞ as λ → +∞. To prove our result we will need to assume that: ˆ has a single maximum and this maximum is non-degenerate. (1) H (2) for λ large enough, h0 is increasing, differentiable and 0 1 h00 (λ) − (λ) = = −−−−→ 0 . h0 (h0 (λ))2 λ→+∞ ˆ + = supT H. ˆ Then we have the following: We recall that H ˆ we Theorem 1.5. In the quantum regime, under the above assumptions on H, have ! 0 ˆ +) d F (λ − H ˆ + )| + ln ln | ln N (λ)| = ln | ln F (λ − H (1 + o(1)) ˆ + ) ln F (λ − H ˆ+) 2 F (λ − H (1.15) when λ → +∞. To illustrate this result, one can take a simple example of F that is in the α quantum regime: ln F (λ) = −g(λ) = −eλ for α > 1. Remark 1.4. Let us comment on assumption (1). This assumption is clearly ˆ is the free discrete Laplacian (i.e. H(θ) ˆ fulfilled if H = cos θ1 + · · · + cos θd ). As in ˆ [6], we could have relaxed the assumption on H and have assumed that it reaches its maxima at isolated points. The technique we use here still works in that case. As in the quantum case h0 → +∞ at +∞, assumption (2) just means that h0 does not behave too wildly near +∞.
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
585
ˆ + . Notice The first correction to F in (1.15) is the shift of the energy λ by −H that, in the classical regime, there was also a shift in energy but only by an amount ˆ 0 . The new shift increases the value of N as −H ˆ + < −H ˆ 0. −H The second correction is due to the tunneling between many sites where the ˆ + . This effect is of purely quantum random variables take a value larger than λ − H nature: it is the same effect as the one discovered by I. Lifshitz at the edges of the spectrum when the support of the random variables (ωk )k∈Zd is compact (see [9]–[11]). The coefficient d/2 in (1.15) is nothing but the exponent giving the decay of the density of states of H at the upper edge of its spectrum. In case we replace ˆ by some analytic function, say H ˆ 0 , that has a degenerate critical point at its H maximum, we expect the coefficient d/2 to be replaced by the exponent giving the decay of the density of states of H0 at upper edge of its spectrum. This was proved in some simple cases for Lifshitz tails ([6]). Remark 1.5. As in the classical regime, in the quantum regime, one should be able to find a result similar to Theorem 1.5 for a continuous Anderson model that is not lower semi-bounded (cf. Remark 1.1). To prove such a result, one should be able to use the techniques developed here and in [7] (see [15]). 2. Periodic Approximations We will first state an approximation result for the density of states of Hω . Let (ωj )j∈Zd be a realization of the random variables defined above. Fix n ∈ N∗ . We define the following periodic operator acting on `2 (Zd ) X X Hωn = H + Vωn = H + ωk Πl+k k∈Zd 2n+1
l∈(2n+1)Zd
where Zd2n+1 = Zd /(2n + 1)Zd . In the sequel, we identify Zd2n+1 with the cube {−n, −n+1, . . . , n−1, n}d. Hωn is (2n+1)Zd-periodic. For Hωn , we define the density of states denoted by dNωn (e.g. [14]) to be the positive Borel measure satisfying Z X 1 n ϕ(x)dNωn (x) = hδk , ϕ(Hωn )δk i , ϕ ∈ C0∞ (R) . (ϕ, dNω ) = (2n + 1)d R d k∈Z2n+1
(2.1) Define
Z
+∞
Nωn (λ) =
dNωn . λ
As in the case of Hω , Nωn (λ) is the complementary to 1 of the usual density of states of Hωn . We have the Lemma 2.1. For any ν > 1, there exists Cν > 1 such that, for any ε > 0 and any n ∈ N∗ such that nε > 1, one has E(Nωn (λ + ε)) − δ(ν, ε, n) ≤ N (λ) ≤ E(Nωn (λ − ε)) + δ(ν, ε, n, λ)
(2.2)
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F. KLOPP
where 0 ≤ δ(ν, ε, n, λ) ≤ Cν
! 3 1/ν ! 1 nε 2 λ n+ + 2F (λ − kHk − 1) . exp − ε Cν 4
(2.3) Proof. Pick λ > 0. We write N (λ) = (N (λ) − N (λ2 )) + N (λ2 ) .
(2.4)
By (0.5), N (λ2 ) ≤ F (λ2 − |H|). Of course, we get the same kind of result for E(Nωn (λ2 )), i.e. E(Nωn (λ2 )) ≤ F (λ2 − |H|). A simple extension of Lemma 1.1 in [6] and of the discussion following the proof of Lemma 1.1 in [6] gives us ˜ ε, n, λ) E(Nωn (λ + ε) − Nωn (λ2 − 1)) − δ(ν, ≤ N (λ) − N (λ2 ) ˜ ε, n, λ) ≤ E(Nωn (λ − ε) − Nωn (λ2 + 1)) + δ(ν, where
(2.5)
1/ν ! 3 1 nε 4 ˜ 0 ≤ δ(ν, ε, n, λ) ≤ Cν n + λ exp − . ε Cν
Let us add that to extend the proof of Lemma 1.1 in [6] to the present case, in Eq. (7) of [6], one uses the estimate |hδ0 , (z − (H + Vω ))−1 δk i| ≤ Ce−|Im z|·|k|/C . that is uniform in Vω and the fact that E(|ωk |) < +∞ (i.e. assumption (0.3)). This ends the proof of Lemma 2.1. Let us now explain heuristically how we get the announced precise asymptotics for N (λ). To use Lemma 2.1, we need to choose ε and n in a suitable way depending on λ. ε will essentially be set by the precision of the asymptotics we want to get. This tells us immediately that ε should be chosen so that F (λ + ε)/F (λ) − 1 is of order the precision of the asymptotic we want to get. Nevertheless, in the case of (1.3), because of the precision we want to achieve in our asymptotics, we need to choose ε of order o(λ−1 ); this introduces a new length scale that is rather arbitrary in the sense that it is not directly related to F or g. This explains why a minimal decay is needed for F . The precision ε being chosen, we will want to take n as small as is possible while keeping the error δ(ν, ε, n, λ) smaller than the error prescribed by the precision wanted for the asymptotic. The main advantage of keeping n small is the following: by Lemma 2.2, the rough asymptotic (0.4) will also be valid for E(Nωn (λ)) (at least if δ(ν, ε, n, λ) is much smaller than F (λ)); hence, if n is much smaller than F (λ) then Nωn (λ) is well approximated by the probability that Hωn has some spectrum
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587
above energy λ. But this quantity is much easier to estimate than Nωn (λ) itself. This feature was already used (in a slightly different way) in [4, 5] or [6]. In terms of g, we get roughly that ε should be of order g 0 (λ)−1 (or o(λ−1 ) in the case of (1.3)). Then, as δ(ν, ε, n, λ) certainly will have to be of size less than F (λ), we get that n should be taken at least of size g(λ)ν · g 0 (λ)−1 . This is much smaller than F (λ) (only if F decreases fast enough in the case of (1.3)). Actually, in the quantum regime, in Theorems 1.1, 1.3 and 1.4, any power of n will be infinitesimal compared to F (λ). 2.1. Some Floquet theory To analyze Hωn , we will need some Floquet theory. It is just a d-dimensional version of the Floquet theory developed in [6]. Let us first introduce some notations. By π π Tn , we denote the d-dimensional torus [− 2n+1 , 2n+1 ]d . T0 will be denoted by d 2 2 d T = [−π, π] . We denote by F : L (T) → ` (Z ) the usual d-dimensional Fourier series transform. Then, we have, for u ∈ L2 (T), X ˆ ω u(θ) = (F ∗ Hω Fu)(θ) = H(θ)u(θ) ˆ H + ωj (Πj u)(θ) j∈Zd
where (Πj u)(θ) =
1 eijθ (2π)d
Z
e−ijθ u(θ)dθ .
T
For u ∈ L (T), we can decompose u in unique way X u(θ) = eikθ uk (θ) 2
(2.6)
k∈Zd 2n+1 2π -periodic. Define the operator U acting where the functions (uk )k∈Zd2n+1 are 2n+1 2 on L (T) by (U u)(θ) = (uk (θ))k∈Zd2n+1 . Then U is a unitary equivalence from
L2 (T) to L2 (Tn ) ⊗ `2 (Zd2n+1 ). For n > 1, we compute U F ∗ Hωn FU ∗ and get that ˆ j−j0 (θ) + ωj δjj0 ))(j,j0 )∈(Zd )2 acting it is the multiplication operator Mωn (θ) = ((H 2n+1 ˆ k )k∈Zd ˆ on L2 (Tn ) ⊗ `2 (Zd ); here the functions (H are the components of H 2n+1
2n+1
decomposed according to (2.6). The above discussion tells us that the Floquet eigenvalues and eigenvectors of Hωn with Floquet quasi-momentum θ (i.e. the solutions of the problem ( n Hω u = λu (where u = (uj )j∈Zd ) uj+k = e−ikθ uj
for j ∈ Zd , k ∈ (2n + 1)Zd )
are nothing but the eigenvalues and eigenvectors (continued periodically) of the (2n + 1)d × (2n + 1)d “matrix” Mωn (θ) acting on `2 (Zd2n+1 ). This gives us that, for ε > 0, Z 1 n Nω (λ) = ]{eigenvalues of Mωn (θ) larger than λ} dθ . (2.7) (2π)d Tn
588
F. KLOPP
Considering H as being (2n + 1)Zd -periodic on Zd , we get that the Floquet eigenˆ + 2πk ))k∈Zd values of H (for the quasi-momentum θ) are (H(θ each of them 2n+1 2n+1 being associated with the vector (uk (θ))k∈Zd2n+1 where uk (θ) =
2πk 1 (e−i(θ+ 2n+1 )j )j∈Zd2n+1 (2n + 1)d/2
in the canonical basis (δj )j∈Zd2n+1 . 3. The Classical Regime Our analysis of the classical regime will rely on Lemma 3.1. Assume that g satisfies assumption (1.1). Then, there exists C > 0 such that, for any n ≥ 1 and K > 0, we have Z 1 1 E(Nωn (λ)) − E F 1Ω0K dθ Vol (T) T I(λ, ω, θ) ˆ + )F (λ − K − H ˆ+) ≤ Cn2d F (λ − H where • I(λ, ω, θ) = hδ0 , (λ − Mωn0 (θ))−1 δ0 i, ˆ j−j0 (θ) + ω 0 δjj0 ))(j,j0 )∈(Zd • M n0 (θ) is the “matrix” ((H j
ω
2n+1
(3.1)
)2
acting on `2 (Zd2n+1 ),
• δ0 is the vector (δj0 )jınZd2n+1 and h·, ·i denotes the scalar product in `2 (Zd2n+1 ), • ωj0 = ωj if j 6= 0 and ω00 = 0,
ˆ + }. • Ω0K = {ω 0 ; ∀ j ∈ Zd2n+1 \ {0}, ωj0 < λ − K − H Remark 3.1. Though in (3.1), we integrate with respect to all the random variables (ωk )k6=0 , as they are i.i.d. and as Mωn0 only depends only on the random variables (ωk )k∈Zd \{0} , we could as well only have integrated with respect to 2n+1 these last random variables. Proof. For k ∈ Zd2n+1 , define Zn,k = Zd2n+1 \ {k}. We define the following sets: ˆ + }, • Ω− = {ω; ∀ k ∈ Zd2n+1 , ωj < λ − H ˆ + }, • Ωk = {ω; ωk ≥ λ; ∀ j ∈ Zn,k , ωj < λ − K − H d ˆ + , ωk ≥ λ − K − H ˆ + }. • Ω+ = {ω; ∃ j, k ∈ Z2n+1 , j 6= k, ωj ≥ λ − H These sets obviously form a partition of Ω. By (2.7), Nωn (λ) only depends on the random variables (ωk )k∈Zd2n+1 . Hence, we have E(Nωn (λ)) = E(Nωn (λ)1Ω− ) +
X
E(Nωn (λ)1Ωk ) + E(Nωn (λ)1Ω+ ).
(3.2)
k∈Zd 2n+1
ˆ + , on Ω− , Hωn (θ) < λ for any θ; so that As H ≤ H E(Nωn (λ)1Ω− ) = 0 .
(3.3)
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By definition, 0 ≤ Nωn (λ) ≤ 1 hence ˆ + )F (λ − K − H ˆ +) . |E(Nωn (λ)1Ω+ )| ≤ E(1Ω+ ) ≤ Cn2d F (λ − H
(3.4)
So that we only have to study the terms E(Nωn (λ)1Ωk ). For k ∈ Zd2n+1 , let us define the translation operator τk acting on `2 (Zd2n+1 ) by ∀ u ∈ `2 (Zd2n+1 ) (τk u)j = uj−k . On L2 (Tn ) ⊗ `2 (Zd2n+1 ), we can consider the translation 1 ⊗ τk that we will also denote by τk . In both cases τk is a unitary transformation. On Ω, we define the shift tk by ∀ ω ∈ Ω , (tk ω)j = ωj−k . {τk ; k ∈ Zd2n+1 } and {tk ; k ∈ Zd2n+1 } are Abelian groups. Moreover the product measure defined on Ω is invariant with respect to the action of the group {tk ; k ∈ Zd2n+1 }. One checks that τk∗ Mωn (θ)τk = Mtnk (ω) (θ) . This implies that Ntn−k (ω) (λ) = Nωn (λ) , hence that E(Nωn (λ)1Ωk ) = E(Nωn (λ)1tk (Ω0 ) ) = E(Ntn−k (ω) (λ)1Ω0 ) = E(Nωn (λ)1Ω0 ) .
(3.5)
We need only to study E(Nωn (λ)1Ω0 ). By (2.7), we need to analyze the number of eigenvalues of the “matrix” Mωn (θ) larger than λ. For ω ∈ Ω0 , there is only a single random variable ω0 larger than λ, so that, by the variational principle, the number of eigenvalues of Mωn (θ) larger than λ is bounded by 1. Hence, by (2.7), Z 1 E(Nωn (λ)1Ω0 ) = E(1{ω; Mωn (θ) has an eigenvalue larger than λ} 1Ω0 )dθ . (3.6) (2π)d Tn ˆ j−j0 (θ)))(j,j0 )∈(Zd )2 . We write Let M0n (θ) be the “matrix” ((H 2n+1 X
Mωn (θ) = M0n (θ) +
ωk Πk = ω0 Π0 + M0n (θ)
k∈Zd 2n+1
+
X
ωk Πk = ω0 Π0 + Mωn0 (θ)
k∈Zd \{0} 2n+1
where ω 0 is defined in Lemma 3.1 and Πk denotes the orthogonal projector (in `2 (Zd2n+1 )) on the vector δk = (δjk )j∈Zd . 2n+1 For ω ∈ Ω0 , Mωn0 (θ) is smaller than λ − K. Hence, by the Birman–Schwinger principle (see [14]), Mωn (θ) has an eigenvalue larger than λ if and only if ω0 hδ0 , (λ − Mωn0 (θ))−1 δ0 i ≥ 1
590
F. KLOPP
that is if and only if ω0 ≥
1 . hδ0 , (λ − Mωn0 (θ))−1 δ0 i
By (3.6), as the random variables (ωk )k are independent, we have Z 1 1 n E(Nω (λ)1Ω0 ) = E F 1Ω0K dθ (2π)d Tn hδ0 , (λ − Mωn0 (θ))−1 δ0 i
(3.7)
where Ω0K is defined in Lemma 3.1. Now, as Mωn0 (θ) is 2π/(2n + 1)-periodic, using (3.5) and (3.7), we may rewrite X
E(Nωn (λ)1Ωk )
k∈Zd 2n+1
1 = Vol T =
1 Vol T
X k∈Zd 2n+1
Z T
Z Tn
E F
E F
1 n hδ0 , (λ − Mω0 (θ +
1 hδ0 , (λ − Mωn0 (θ))−1 δ0 i
! 2kπ −1 δ i 0 2n+1 ))
! 1Ω0K
dθ
1Ω0K
dθ .
Plugging this into (3.2) and using (3.3) and (3.4), we get (3.1).
3.1. The proofs of Theorems 1.1 and 1.2 These theorems are now deduced quite easily from Lemma 3.1. Proof of Theorem 1.1. To choose ε for Lemma 2.1, we pick β > 0 small. We set ε = g(λ)−β . Let ν be as in Lemma 2.1 and choose it arbitrarily close to 1. Set n = g(λ)β+(2β+1)ν . Then the error estimate in Lemma 2.1 is of size δ(ν, ε, n, λ) ≤ Ce−g(λ)
1+β
.
Now we estimate E(Nωn (λ ± ε)) using Lemma 3.1. To do that we only need to estimate E(Nωn (λ)). For θ ∈ T, as ω00 = 0, we have I(λ, ω, θ) = hδ0 , (λ − Mωn0 (θ))−1 δ0 i 1 1 1 + hδ0 , Mωn0 (θ)δ0 i + 2 hMωn0 (θ)δ0 , (λ − Mωn0 (θ))−1 Mωn0 (θ)δ0 i λ λ2 λ 1 1 1 n 0 = 1 + hδ0 , M0 (θ)δ0 i + m0 (ω , θ, n) (3.8) λ λ λ
=
where m0 (ω 0 , θ, n) = hM0n (θ)δ0 , (λ − Mωn0 (θ))−1 M0n (θ)δ0 i . ˆ 0 (θ) Notice that m0 (ω 0 , θ, n) > 0 as Mωn0 (θ) < λ. By definition hδ0 , M0n (θ)δ0 i = H ˆ 0 (θ) is given by the decomposition (2.6) for the function H). ˆ As the Fourier (where H
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
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ˆ are exponentially decaying, writing the Fourier series for h0 (θ), coefficients on H we get hδ0 , M0n (θ)δ0 i = h0 + O(e−n/C0 ) . Fix K > 0 a large real number. For ω 0 ∈ Ω0K (see Lemma 3.1), we have, uniformly in θ, m0 (ω, θ, n) ≤ kHk2 /K . So that, by Neuman’s formula, for ω 0 ∈ Ω0K and all θ, 1 = λ − h0 + O(K −1 + e−n/C0 + 1/λ) . I(λ, ω, θ) Hence, as F is decreasing, we get that, for any η > 0, there exists λη > 0 and K > 0 large such that, for λ > λη , Z 1 1 F (λ − h0 + η) ≤ E F 1Ω0K dθ Vol (T) T I(λ ± ε, ω, θ) ≤ F (λ − h0 − η) .
(3.9)
As F is absolutely continuous near +∞, g too is absolutely continuous near +∞. In Rx particular, we know that, for x, y large enough, g(x) − g(y) = y g 0 (t)dt ([8]). We know that g satisfies assumption (1.1). That is for any η > 0, there exists λη > 0 such that, for λ > λη , g 0 (λ) ≤ ηg(λ). Hence, for λ > λη , Z λ+ √1 Z λ+ √1 η η 1 √ 1 0≤g λ+ √ g 0 (t)dt ≤ η g(t)dt ≤ ηg λ + √ − g(λ) = η η λ λ as g is non-decreasing. This shows that there exists a function K(λ) that tends to +∞ at +∞ such that, for any δ > 0, ˆ + ) ≤ F (λ − K(λ) − H ˆ + ) = o(F (λ)1−δ ) . F (λ − H This remark, (3.9) and Lemma 3.1 immediately imply Theorem 1.1 if we choose K = K(λ) in (3.9). Proof of Theorem 1.2. Theorem 1.2 is proved along the same lines as Theorem 1.1. We just need to choose ε somewhat differently and to get a more precise expansion for I(λ, θ, ω) to achieve the announced precision. We now have assumed F (λ) = o(λ−n−η ). Then pick ε = g(λ)−β/2 min(λ−1−η/(4d) , g(λ)−β/2 ) where β > 0 is small. Let ν as in Lemma 2.1 and choose it arbitrarily close to 1. Set n = g(λ)ν(1+β/2)+β/2 max(λ1+η/(4d) , g(λ)β/2 ). Then the error estimate in Lemma 2.1 is of size 1+β/2 δ(ν, ε, n, λ) ≤ Ce−g(λ) . We expand I(λ, ω, θ) a little further than in (3.8). We recall 1 I(λ, ω, θ) = λ
! ˆ 0 (θ) H 1 0 1+ + m0 (ω , θ, n) λ λ
592
F. KLOPP
where m0 (ω 0 , θ, n) = hM0n (θ)δ0 , (λ − Mωn0 (θ))−1 M0n (θ)δ0 i . P Let us define Πnω0 = k∈(Zd )∗ ωk0 Πk . For ω 0 ∈ Ω0K , we have λ − Πnω0 > K. Then 2n+1 we can rewrite m0 (ω 0 , θ, n) as m0 (ω 0 , θ, n) = hM0n (θ)δ0 , (λ − Πnω0 )−1 (1 − M0n (θ)(λ − Πnω0 )−1 )−1 M0n (θ)δ0 i = hM0n (θ)δ0 , (λ − Πnω0 )−1 M0n (θ)δ0 i + hM0n (θ)δ0 , (λ − Πnω0 )−1 (1 − M0n (θ)(λ − Πnω0 )−1 )−1 × (λ − Πnω0 )−1 M0n (θ)δ0 i = kv0 (ω 0 , θ, n)k2 + hv0 (ω 0 , θ, n), V (ω 0 , θ, n)v0 (ω 0 , θ, n)i
(3.10)
where v0 (ω 0 , θ, n) = (λ − Πnω0 )−1/2 M0n (θ)δ0 , V (ω 0 , θ, n) = (λ − Πnω0 )−1/2 (1 − M0n (θ)(λ − Πnω0 )−1 )−1 (λ − Πnω0 )−1/2 . Then, for K large, uniformly for ω 0 ∈ Ω0K , we have m0 (ω 0 , θ, n) = kv0 (ω 0 , θ, n)k2 (1 + O(K −1 )) .
(3.11)
On the other hand, we can expand kv0 (ω 0 , θ, n)k2 in kv0 (ω 0 , θ, n)k2 =
=
=
1 ˆ |H0 (θ)|2 + λ 1 λ
X
X k∈(Zd )∗ 2n+1
1 ˆ k (θ)|2 |H λ − ωk X
ˆ k (θ)|2 + |H
k∈(Zd )∗ 2n+1
k∈Zd 2n+1
1 X |hk |2 + λ d k∈Z
X ∗ k∈(Zd 2n+1 )
ωk ˆ k (θ)|2 |H λ − ωk
ωk ˆ k (θ)|2 + O(e−2n/C0 ) |H λ − ωk
(3.12)
ˆ In view of the asymptotic by the exponential decay of the Fourier coefficients of H. we announced, it is natural to estimate the probability that X ω k 2 ˆ |Hk (θ)| λ − ω k k∈(Zd )∗ 2n+1
is large. This is the purpose of: Lemma 3.2. For any δ > 0, uniformly in θ, one has that X ω k ˆ k (θ)|2 ≤ −δ −−−−→ 0 . Proba ω ∈ Ω0K ; |H λ→+∞ λ − ωk d ∗ k∈(Z2n+1 )
(3.13)
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and that, for some C > 0 and λ large enough, X ωk ˆ k (θ)|2 ≥ δ Proba ω ∈ Ω0K ; |H λ − ωk d ∗ k∈(Z2n+1 )
≤ C(ln(λ/δ)) F d
δλ C(ln(λ/δ))d + δ
.
(3.14)
Let us use this lemma to finish the proof of Theorem 1.2. Actually we will use: Corollary 3.1. Let f : (0, +∞) → (0, +∞) a function tending to 0 at +∞. Then, there exists δ, a positive function defined for λ ≥ λδ large enough such that • for λ ≥ λδ , δ(λ) ≥ f (λ) X ω k ˆ k (θ)|2 ≤ −δ(λ) −−−−→ 0 |H • Proba ω ∈ Ω0K ; λ→+∞ λ − ωk k∈(Zd )∗ 2n+1 X ω k ˆ k (θ)|2 ≥ δ(λ) • Proba ω ∈ Ω0K ; |H λ − ωk d ∗
(3.15)
k∈(Z2n+1 )
δ(λ)λ . (3.16) C(ln(λ/δ(λ)))d + δ(λ) P ωk ˆ k (θ)|2 ≥ δ(λ), then, Let λ > 0 be large. If ω ∈ Ω0K such that k∈(Zd )∗ λ−ω |H k 2n+1 by (3.8), (3.11) and (3.12), we get that 1 h0 1 −1 I(λ, ω, θ) = 1+ + O(K ) . λ λ λ P ωk ˆ k (θ)|2 ≤ −δ(λ), then, by (3.8), (3.11) and |H If ω ∈ Ω0K such that k∈(Zd )∗ λ−ω k 2n+1 (3.12), we get that 1 h0 1 X 1 I(λ, ω, θ) ≤ 1+ + 2 |hk |2 + 2 O(K −1 ) λ λ λ λ ≤ C(ln(λ/δ(λ)))d F
k6=0
ˆ 0 by h0 as, in this case, the last term in (3.12) is negative. Here we replaced H ˆ without damage because of the exponential decay of the Fourier coefficients of H. Now, we pick K(λ) as in the proof of Theorem 1.1; using the above estimates for I and plugging them into (3.1), we get that, for any ν > 0, there exists λδ,ν such that, for λ > λδ,ν , we have N (λ) = F (λ + a0 + (a1 + O(δ(λ)))/λ) · (1 + o(1)) + O(λ2d F (λ)2−ν ) + O(F (λ + a0 + o(1))F (δ(λ)λ/(C ln(λ/δ(λ)))d )(C ln(λ/δ(λ)))d ) . This ends the proof of Theorem 1.2.
(3.17)
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F. KLOPP
Proof of Lemma 3.2. Let us start with (3.13). Fix δ > 0. Then, for some kδ > 0, for any θ, X ˆ k (θ)|2 ≤ δ/2 |H |k|≥kδ k6=0
P P ωk 2 ˆ k (θ)|2 < +∞. Pick ω ∈ Ω0 such that ˆ as k supθ |H K k∈(Zd )∗ λ−ωk |Hk (θ)| ≥ δ. 2n+1 Then, as for ωk ≤ 0, ωk −1 ≤ ≤ 0, λ − ωk for λ large enough, we have X X ωk ωk ˆ k (θ)|2 ≤ 0 − δ − ˆ k (θ)|2 ≤ −δ/2 . |H |H λ − ωk λ − ωk k∈(Zd )∗ 2n+1 |k|≥kδ ωk ≤0
|k|
But
Proba ω ∈ Ω0K ;
X |k|
ωk ˆ k (θ)|2 ≤ −δ/2 |H −−−−→ 0 λ − ωk λ→+∞
as kδ does not depend on λ and, for any η > 0, ωk Proba ωk ; ≤ −η −−−−→ 0 . λ→+∞ λ − ωk ωk K This proves (3.13). For ω ∈ Ω0K , λ−ω ≤ λ−K . We pick C > 0 large enough k ˆ k (θ)| ≤ e−c|k| (independent of λ and δ) and set k(λ, δ) = C ln(λ/δ). Then, as supθ |H for some c > 0, we have X ωk ˆ k (θ)|2 ≤ δ/2 . |H λ − ωk |k|≥kδ k6=0
Hence, noticing that, for λ large enough, k(λ, δ) = o(n), we see that we just need to estimate X ωk 0 2 P0 = Proba ω ∈ ΩK ; |hk (θ)| ≥ δ . λ − ω k |k|≤k(λ,δ) k6=0
This probability is easily estimated by P0 ≤ Ck(λ, δ)d F
δλ Ck(λ, δ) + δ
for some C > 0 large enough. This ends the proof of (3.14) hence of Lemma 3.2. Proof of Corollary 3.1. By (3.14), (3.16) holds for any function δ. Obviously the two probabilities estimated in Lemma 3.2 are decreasing in δ. Hence, to prove
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Corollary 3.1, we just need to find some function δ(λ) that tends to 0 at +∞ such that (3.15) is satisfied. Let us define the function X ωk ˆ k (θ)|2 ≤ −δ . f (δ, λ) = Proba ω ∈ Ω0K ; |H λ − ωk d ∗ k∈(Z2n+1 )
By Lemma 3.2, f satisfies (1) for fixed δ, f (δ, λ) → 0 when λ → +∞, (2) for fixed λ, f (δ, λ) is a decreasing function of δ, (3) for fixed λ, f (δ, λ) is a right continuous function of δ. We define ∆λ = {δ ≥ 0; ∀ λ0 ≥ λ; 0 ≤ f (δ, λ) < δ} . Then, by (1), ∆λ 6= ∅. It is lower bounded by 0. We define δ(λ) = inf ∆λ . So that • for λ ≤ λ0 , δ(λ0 ) ⊂ δ(λ) (as ∆λ ⊂ ∆λ0 ), • by (1), δ(λ) → 0 when λ → +∞, • f (λ, δ(λ)) ≤ δ(λ) as f is right continuous and decreasing by (2) and (3). This ends the proof of Corollary 3.1. 3.2. The asymptotic expansion Under our assumptions on g, we know that the random variables (ωk )k∈Zd admit a continuous density near +∞, and that this density is non vanishing. R +∞ Let f be the absolute value of the logarithm of this density so that N (λ) = λ e−f (t) dt. We then know that f (t) = g(t) − ln(g 0 (t)) and that e−f (t) = o(t−∞ ) near +∞. Proof of (1.10). To keep things short, we will not prove the result in the case when g admits an asymptotic of type (1.7). This proof is very similar to the one in the case g admits an asymptotic of type (1.8) when α < 3 and the modification to be made to this proof are obvious. We now assume that g admits an asymptotic of type (1.8) with α < 3. Fix some arbitrary integer M ∈ N. This will be the order of the asymptotic expansion we want to get for N . Hence we need to take ε (of Lemma 2.1) of order λ−M so that it will be enough to take n in Lemma 2.1 of order λM+σ(α+1) . Now, in Lemma 3.1, set K = νλ (with 0 < ν < 1). In this case, let us consider the rest given by Lemma 3.1. For any η ∈ (0, 1), we have ˆ + )F (λ − K − H ˆ + ) = o(e−a1 (1+(1−ν)α )(1−η)λα ) ) . n2d F (λ − H
(3.18)
(Here o(·) depends on η). We now only need to get an asymptotic expansion of the principal term in (3.1). Therefore, for ω ∈ Ω0K , we compute an asymptotic expansion of I(λ, ω, θ); using the notations defined before (3.10), we have
596
F. KLOPP
I(λ, ω, θ) = hδ0 , (λ − Mωn0 (θ))−1 δ0 i −1 = hδ0 , (λ − Πnω )−1 1 − M0n (θ)(λ − Πnω )−1 δ0 i X p = hδ0 , (λ − Πnω )−1 M0n (θ)(λ − Πnω )−1 δ0 i p≥0
=
X p hδ0 , (λ − Πnω )−1 M0n (λ − Πnω )−1 δ0 i + O(e−n/C0 )
(3.19)
p≥0
where M0n = ((hj−j0 ))(j,j0 )∈(Zd2n+1 )2 acts on L2 (Tn ) ⊗ `2 (Zd2n+1 ). As K = νλ (0 < ν < 1), for ω ˜ ∈ Ω0K , we have kλ − Πnω k ≥ νλ. Hence the p-th ˜ ∈ Ω0K ). For summand in (3.19) is of size at most λ−p−1 (uniformly in θ ∈ T and ω p ≥ 0, we compute p Tp (λ, ω) = λ2 hδ0 , (λ − Πnω )−1 M0n (λ − Πnω )−1 δ0 i p−1 X Y 1 h−k1 hk1 −k2 · · · hkp−2 −kp−1 hkp−1 = λ − ω ˜ kj d p−1 j=1 (k1 ,...,kp−1 )∈(Z2n+1 )
(3.20) (we recall that ω ˜ 0 = 0 is constant). We have T0 (λ, ω) = λ and T1 (λ, ω) = h0 . Notice that Tp is of size λ−p+1 (uniformly in θ ∈ T and ω ˜ ∈ Ω0K ). −1 The function I(λ, ω, θ) admits an asymptotic expansion of the form j M+[α]+1 M+[α]+1 j X X 1 (−1) −M−[α]−1 T + O λ . (3.21) = λ1 + p I(λ, ω, θ) λj p=1 j=1 We define X
M+[α]+1
T =
j=1
j M+[α]+1 1 1 2 (−1)j X 1 Tp = − (T1 + T2 ) + 2 T1 + O 3 . j λ λ λ λ p=1
(3.22)
The quantity T is of size λ−1 . We compose expansion (3.21) with the expansion of g to obtain g
1 I(λ, ω, θ)
X
[M/α]+1
= λα (1 + T )α
gj0 λ−jα (1 + T )−jα
j=0
+ (ln λ + ln(1 + T ))
M+1 X
gj1 λ−j (1 + T )−j
j=0
+ O λ−M−1 = g00 λα (1 + T )α + g10 + g01 ln λ + R0 (λ, ω) + R1 (λ, ω) + R2 (λ, ω) + O(λ−M−1 )
(3.23)
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where X
[M/α]+1
R0 (λ, ω) =
gj0 λ(1−j)α (1 + T )(1−j)α = O(λ−α )
j=2
R1 (λ, ω) = ln(1 + T )
M+1 X
gj1 λ−j (1 + T )−j = O(λ−1 )
j=0
R2 (λ, ω) = ln λ
M+1 X
gj1 λ−j (1 + T )−j = O(λ−1 ln λ) .
j=1
Moreover, by (3.22), we have g00 λα (1 + T )α =
[α] X
g0,l λα T l + R4 (λ, ω) + O λ−M−1
(3.24)
l=0
where X
M+[α]+1
R4 (λ, ω) =
g0,l λα T l = O(λα−[α]−1 ) ,
l=[α]+1
g0,0 = g00 , g0,1 = αg00
and g0,2 =
α (α + 1)g00 . 2
Using (3.22), we get that [α] X
g0,l λα T l = g0,0 λα − g0,1 λα−1 T1 + g0,2 λα−2 T12 − g0,1 λα−1 T2
l=0
+ R5 (λ, ω) + O λ−M−1
(3.25)
where R5 (λ, ω) = O(λα−4 ) + O(λ−α ) . We compose (3.23), (3.24) and (3.25) with the asymptotic expansion of e−x to get exp(−g(I −1 (λ, ω, θ))) = e−g0,0 λ
α
−g01 ln λ−g10 +g0,1 λα−1 T1 −g0,2 λα−2 T12
α−1
· eg0,1 λ
T2
· (1 + R(λ, ω) + O(λ−M−1 ))
(3.26)
where R may be written R(λ, ω) =
M M X X j=0 l=0
+
X
λ−l
c0p1 ,...,pj ,l Tp1 · · · Tpj
p1 +···+pj ≤M
M M X X j=0 m,l=0
λ−l−m lnm λ
X p1 +···+pj ≤M
c1p1 ,...,pj ,l Tp1 · · · Tpj .
(3.27)
598
F. KLOPP
The coefficients (c0p1 ,...,pj ,l , c1p1 ,...,pj ,l ) only depend on the coefficients of the asymptotic expansion of g. Now, we want to integrate (3.26) in θ over the torus T and take its expectation in ω over Ω0K . The principal terms in (3.26) do not depend on θ: only the rest, i.e. the term O(λ−M−1 ) depends on θ. Hence, the θ-integration is performed trivially. Among the exponentials coming into (3.26), only the last one depends on the random variables; it depends on T2 where, by (3.20) X
T2 (λ, ω) =
k∈Zd 2n+1
1 |hk |2 = λ−ω ˜k λ
where
X k∈Zd 2n+1
X
T˜2 (λ, ω) =
|hk |2 +
1˜ T2 (λ, ω) λ
|hk |2 ωk . λ − ωk
k∈(Zd )∗ 2n+1
Hence we get that 1 Vol (T)
Z T
E F
= e−g0,0 λ
α
1 I(λ, ω, θ)
dθ
1Ω0K
+−g01 ln λ−g10 +g0,1 λα−1 T1
· exp −g0,2 λ
α−2
X
2
|hk |
· J(λ)
(3.28)
· (1 + R(λ, ω) + O(λ−M−1 ))1Ω0K ) .
(3.29)
T12
α−2
+ g0,1 λ
k∈Zd 2n+1
where α−2
J(λ) = E(eg0,1 λ
T˜2 (λ,ω)
We now distinguish three cases: (1) If α < 2, then λα−2 T˜2 (λ, ω) tends to 0 uniformly on Ω0K hence (3.29) reduces to [M/(α−2)] X 1 J(λ) = E1 + (g0,1 λα−2 T˜2 (λ, ω))l + R(λ, ω) + O(λ−M−1 ) 1Ω0K . l! l=0
1 P 2 0 (2) If α = 2, then T˜2 is bounded by 1−ν k∈Zd |hk | on ΩK . Hence the exponential in (3.29) is bounded. Using the fact that all the moments of the random variables (ωk )k are finite and the fact that n is polynomially bounded in λ, we get that ˜
J(λ) = E(eg0,1 T2 (λ,ω) · (1 + R(λ, ω) + O(λ−M−1 ))1Ω∗ 1/2 ) . λ
where Ω∗K = {ω; ∀ k ∈ (Zd2n+1 )∗ , −K ≤ ωk ≤ K}. But, on Ω∗λ1/2 , |T˜2 (λ, ω)| ≤ λ−1/2 . In this region, we again expand exp(g0,1 T˜2 (λ, ω)) according to its Taylor series as above and cut the expansion at order 2M getting thus a rest of size λ−M .
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
599
(3) If 2 < α < 3, we pick some exponent β such that 3 − α > β > 3−α α−1 . Then, as all the moments of the random variables (ωk )k are finite, we have Z νλ ω δ α−3 t − k e−f (t)+ λ−t dt = λe−λ(λ −λ /(1−ν)) = o(λ−∞ ) and E(e λ−ωk 1ωk ≤−λδ ) λδ
= o(λ−∞ ) . As n is polynomial in λ, we have α−2
J(λ) = E(eg0,1 λ
T˜2 (λ,ω)
· (1 + R(λ, ω) + O(λ−M−1 ))1Ω∗ δ ) .
(3.30)
λ
But, on Ω∗λδ , T˜2 = O(λδ+α−3 ), so that we can again expand the exponential inside Eq. (3.30) defining J using its Taylor series and cutting this series at order M/(δ + α − 3) getting thus a rest of size λ−M . The outcome of this discussion is that we will be finished if we show that expressions of the following form admit an infinite asymptotic expansion in λ: E((T˜2 )l Tp1 · · · Tpj 1Ω∗ δ ) or E((T˜2 )l Tp1 · · · Tpj 1Ω0K ) . λ
This follows immediately from the definition of the (Tp )p∈N and of T˜2 and the fact that all the moments of the random variables (ωk )k∈Zd are finite. When α = 3, we just have to change slightly the last part of the analysis. We write X X |hk |2 1 X 1 1 T2 (λ, ω) = = |hk |2 + 2 |hk |2 ωk + 2 T20 (λ, ω) λ − ω ˜ λ λ λ k d d d ∗ k∈Z2n+1
k∈Z2n+1
k∈(Z2n+1 )
(3.31) where T20 (λ, ω) =
X k∈(Zd )∗ 2n+1
|hk |2 ωk2 . λ − ωk
λx is increasing on any segment [λδ , νλ] (0 < δ < 1) As the function x 7→ x3 − λ−x and as n is polynomially bounded in λ, we see that (3.30) still holds. So that, if we choose δ ≤ η, we get P 2 g0,1 −M−1 d )∗ |hk | ωk k∈(Z J(λ) = E e · (1 + R(λ, ω) + O(λ )) · 1Ω∗ δ . λ
P
Notice that k∈(Zd )∗ |hk |2 ωk is a finite sum as H is of finite range. So we get the same result as in the case α < 3 (R however does not have the same value but only the same form as inthe case α < 3) except that we integrate P 2 the expansion against the measure exp g0,1 k∈(Zd )∗ |hk | ωk dP (ω) (instead of the measure dP (ω) in the case α < 3). This will just change the form of the obtained asymptotic expansion by a constant equal to X n3 = ln E exp αg00 |hk |2 ωk . (3.32) k∈(Zd )∗
600
F. KLOPP
Proof of (1.11). We now assume that g admits an asymptotic expansion of the form (1.8) with α > 3. In this case, our method to get a full asymptotic heavily relies on the finiteness of the range of H. Let us recall and introduce some new definitions that will be useful in this respect. Pick (k, k 0 ) ∈ Zd . We recall that an H-path from k to k 0 is an n-tuple γ(k, k 0 ) = (k1 , k2 , . . . , kn ) such that k1 = k, kn = k 0 and for j = 1, . . . , n−1, hkj −kj+1 6= 0. Of course, if γ(0, k) = (k1 , k2 , . . . , kn ) is an H-path from 0 to k, then k 0 + γ(0, k) = (k 0 + k1 , k 0 + k2 , . . . , k 0 + kn ) is an H-path from k 0 to k + k 0 . On the set of H-paths, one can naturally define an addition in the following way: if γ(k, k 0 ) = (k1 , k2 , . . . , kn ) and γ(k 0 , k 00 ) = (kn , kn+1 , . . . , kn+m ) then γ(k, k 0 ) + γ(k 0 , k 00 ) = (k1 , k2 , . . . , kn , kn+1 , . . . , kn+m ). Any H-path then admits an opposite. Let γ be an H-path. If γ = (k1 , k2 , . . . , kn ), then we define σ(γ), the support of γ to be the set {k1 , k2 , . . . , kn } and hγ to be the complex number hγ = hk1 −k2 hk2 −k3 . . . hkn−2 −kn−1 hkn−1 −kn . Let us now start with the proof of (1.11). Let M ∈ N∗ be the order of the expansion we want to prove. By (3.19) and (3.20), uniformly for ω ˜ ∈ Ω0K (here K = νλ for some ν ∈ (0, 1)), we have X
M+[α]+1 −2
I(λ, ω, θ) = λ
Tp (λ, ω) + O(λ−M−2−[α] )
(3.33)
j=0
where T0 (λ, ω) = λ, T1 (λ, ω) = h0 and we rewrite (3.20) p−1 X Y 1 Tp (λ, ω) = hγ . λ − ω ˜ kj j=2
(3.34)
γ=(k1 ,...,kp )∈Γ(0) |γ|=p
We now replace Tp by its expression (3.34) in (3.33) and group the thus obtained terms differently. Namely, we first sum over all the paths that never leave the point 0, then over all the paths with support contained in {0}∪Λ1 , then over all the paths with support contained in {0} ∪ Λ1 ∪ Λ2 , and so on (we recall that Λl was the set of points in Zd such that the shortest path from such a point to 0 has length exactly l). We get [M+α+1] X I(λ, ω, θ) = λ−2 Sp (λ, ω) + O(λ−M−2−[α] ) (3.35) j=0
where M+[α]+1
S0 (λ, ω) = S0 (λ) = λ + h0
X j=0
h0 λ
p =
λ2 + O(λ−M−2−[α] ) , λ − h0
and, for p ≥ 1, Sp (λ, ω) =
[(M+α)/2]
X
X
n=2p−1
γ=(k1 ,...,kn+1 )∈Γ(0) σ(γ)⊂{0}∪Λ1 ∪···∪Λp σ(γ)6⊂{0}∪Λ1 ∪···∪Λp−1
1 hγ . λ − ω ˜ k j j=2 n Y
(3.36)
(3.37)
601
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
Notice that each of the summands in Sp contains at least one factor (λ − ω ˜ k )−1 for −2p+1 some k ∈ Λp . Sp is of order λ and the term of smallest order in Sp (i.e. the term of order λ−2p+1 ) is the term p p X X Y Y 1 1 1 hγ h−γ 0 Sp0 (λ, ω) = 0 λ−ω ˜k λ − ω ˜ λ − ω ˜ k k j j j=2 j=2 k∈Λp
γ=(k1 ,...,kp+1 )∈Γ(k) γ 0 =(k0 ,...,k0 )∈Γ(k) 1 p+1 |γ|=|γ 0 |=p
2 p X X Y 1 1 hγ = λ−ω ˜k λ−ω ˜ kj j=2 k∈Λp p+1 )∈Γ(k) γ=(k1 ,...,k |γ|=p ∼
λ→+∞
X λ−2p+2 (p) 2 h λ−ω ˜k k
(3.38)
k∈Λp
ωk ) are all real. Notice that in (3.38), for k ∈ Λp , ω ˜k as h−k = hk and λ and (˜ never appears in the square in the right hand side as the H-paths over which one is summing are of length exactly p. Using the asymptotic expansion known for g, we write −1
e−g(I(λ,ω,θ)
)
λ2 = exp g S0 (λ)
1+
1 X[(M+α)/2] S p=1
p (λ,ω) S0 (λ)
−M−1 ) + O(λ
˜ ω) + O(λ−M−1 )) = e−g(J(λ,ω)) · (1 + R(λ, = e−g0 J 0
α
(λ,ω)−g10 −ln J(λ,ω)
where J(λ, ω) =
λ2 S0 (λ)
1+
· (1 + R(λ, ω) + O(λ−M−1 ))
1 X[(α−1)/2] S p=1
(3.39)
(3.40) p (λ,ω)
S0 (λ)
˜ and R have an expansion of the type (3.27) where the terms (Tp )p are and, both R replaced by the terms (Sp )p . −1 To compute E(e−g(I(λ,ω,θ) ) 1Ω0K ), we first note that J(λ, ω) only depends on S[(α−1)/2] the random variables (ωk )k∈Λα where we have defined Λα = j=0 Λj . So −1
that the principal term in e−g(I(λ,ω,θ) ) only depends on (ωk )k∈Λα . We will first integrate with respect to the random variables (ωk )k∈Λα . We recall that dP (ωk ) denotes the probability law of the random variable ωk and that f was defined by dP (ωk ) = e−f (ωk ) dωk . Hence the expectation we want to compute can be written as Z P Y −1 −1 E(e−g(I(λ,ω,θ) ) 1Ω0K ) = · · · e−g(I(λ,ω,θ) ) e k f (ωk ) dωk . (3.41) Ω0K
k
602
F. KLOPP
To perform this integration, we note that g(I(λ, ω, θ)−1 ) puts most of its weight on large ω’s. So that, if we cut the integral into two pieces one for ωk large and one for ωk small, the second piece should be negligible with respect to the first one. In the first piece, as we know an asymptotic expansion for g, we also know an asymptotic expansion for f . We will then be able to use Laplace’s method to get an asymptotic expansion for this piece. We shall follow this idea. However we will not be able to do this in such a direct way. Instead, we will, step by step inductively, integrate out, first all the random variables (ωk )k∈Λ1 , then all the random variables (ωk )k∈Λ2 , and so on, until we have integrated out all the random variables (ωk )k∈Λ[(α−1)/2] . Moreover, as in the case when α ≤ 3, we will have to distinguish between the case when α is an odd integer and the case when it is not. To implement this procedure, it will be convenient to enumerate the points in α Λ . We define the points {k0 , k1 , . . . , kM } = Λα in such a way that, if j ≤ j 0 then l(kj ) ≤ l(kj0 ). We also define the sets Λα n = {k0 , k1 , . . . , kn }. We will need some δ > 0 small (the size of which will be chosen later on) to control the cut off between large and small ωk . As the random variables are i.i.d, we write −1
E(e−g(I(λ,ω,θ)
)
−1
1Ω0K ) = E(E(e−g(I(λ,ω,θ)
)
1Ω0K |(ωk )k6=k0 )) .
We define −1
I1 (λ, (ωk )k6=k0 ) = E(e−g(I(λ,ω,θ) ) 1Ω0K |(ωk )k6=k0 ) Z −1 = e−g(I(λ,ω,θ) ) 1Ω0K dP (ωk0 ) .
(3.42)
− 0 δ 0 δ Let us define Ω+ K (k0 ) = {ω ∈ ΩK ; ωk0 ≥ λ } and ΩK (k0 ) = {ω ∈ ΩK ; ωk0 < λ }. We write −1
I1 (λ, (ωk )k6=k0 ) = E(e−g(I(λ,ω,θ)
)
1Ω+ (k0 ) |(ωk )k6=k0 ) K
−1
+ E(e−g(I(λ,ω,θ) =
I1+ (λ, (ωk )k6=k0 )
)
1Ω− (k0 ) |(ωk )k6=k0 ) K
+
I1− (λ, (ωk )k6=k0 ) .
(3.43)
−f (aωk0 ) For ω ∈ Ω− dωk0 where f (ωk0 ) = g(ωk0 ) + K (k0 ), we have dP (ωk0 ) = e 0 ln g (ωk0 ). Hence, f admits an asymptotic expansion of the same form (1.8) with different coefficients. More precisely
f (λ)
∼
λ→+∞
X k≥0
fk0 λα(1−k) +
X k≥0
fk3 λ−α−k + ln λ
X
fk1 λ−k +
k≥0
where, in particular, f00 = g00 , f10 = g10 + ln α, f01 = g01 + α − 1.
X k≥0
fk2 λ−α−k
603
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
Using the asymptotic expansion for f and (3.39), we write I1+ (λ, (ωk )k6=k0 ) Z = λδ ≤ωk0 ≤νλ
Z =
λδ ≤ω
k0 ≤νλ
exp(−g(I(λ, ω, θ)−1 ) − f (ωk0 ))dωk0 e−ψ((ωk )k∈Λα ) [(1 + Rf (ω))(1 + R(λ, ω) + O(λ−M )]dωk0
(3.44)
where ψ((ωk )k∈Λα ) = ψ0 ((ωk )k∈Λα ) + f0 (ωk0 ) , ψ0 ((ωk )k∈Λα ) = g00 (J(λ, ω))α + g10 + g01 ln J(λ, ω) f0 (λ) = f00 λα + f10 + f01 ln λ ,
(3.45)
and Rf (ω) has an expansion of the following form: Rf (ω) =
X
0 −αn3 cn0 ,n1 ,n2 ,n3 ωk−n 0
1≤n0 +n1 +α(n2 +n3 )+≤[M/δ] n0 ,n1 ,n2 ,n3 ∈N
ln ωk0 ωk0
n1 +αn2 .
(3.46)
The function ψ is analytic in (ωk )k∈Λα for ωk0 ∈ Ω+ K (k0 ). We are then interested in the minima of ψ with respect to the variable ωk0 , the (ωk )k∈Λα \{k0 } being considered as parameters. We compute ∂ψ ∂J g01 0 α−1 ((ωk )k∈Λα ) = g0 α(J(λ, ω)) + (λ, ω) ∂ωk0 J(λ, ω) ∂ωk0 + g00 α(ωk0 )α−1 + (g01 + α − 1)
1 ωk0
(3.47)
and ∂J 1 λ2 (λ, ω) = − 2 X[(α−1)/2] S (λ,ω) 2 ∂ωk0 (S0 (λ)) p 1+ S0 (λ)
X
[(α−1)/2]
p=1
∂Sp (λ, ω) ∂ωk0
p=1
λ2 1 =− 2 (S0 (λ))2 [(α−1)/2] X S (λ,ω) p 1 + S0 (λ)
X
[(α−1)/2]
p=1
p=1
×
! ∂Sp0 ∂(∆Sp ) (λ, ω) + (λ, ω) ∂ωk0 ∂ωk0
where ∆Sp (λ, ω) = (Sp − Sp0 )(λ, ω).
(3.48)
604
F. KLOPP
By (3.37) and (3.38), uniformly in Ω+ K (k0 ), for p ≥ 1, when λ → +∞, we have ∂(Sp − Sp0 ) (λ, ω) = O(λ−1−2p ) . ∂ωk0 Moreover, using (1.6), for k ∈ Λα , we have: • if p > 1 then, uniformly in Ω+ K (k0 ), when λ → +∞, ∂Sp0 (λ, ω) = O(λ−3 ) ∂ωk0 • if p = 1 then, as k0 ∈ Λ1 , 1 ∂S10 (λ, ω) = h1k0 ∂ωk0 (λ − ωk0 )2
∼
λ→+∞
h1k0 λ−2 .
So that the solution of ∂ωk0 ψ = 0 should roughly satisfy ωk0
1
α−3
∼ (h1k0 ) α−1 λ α−1 .
λ→+∞
Let us perform the following change of variables: set 1
α−3
• ωk0 = uk0 · (h1k0 ) α−1 · λ α−1 , • uk =
1 λ−ωk
for k ∈ Λα \ {k0 },
• v = λ1 , w = λ− α−1 and t1 = λ−α α−1 . α−3
2
Then, we have
1 λ−ωk0
=
v 1−w·uk0
and
∂ψ ((ωk )k∈Λα ) = h1k0 g00 αλα−3 Uk0 (uk0 , u0 , v, w, t) ∂ωk0 where Uk0 satisfies v t1 · w α−1 0 Uk (uk0 , u , v, w, t) = − · (1 + w · g(uk0 , u , v, w, t1 )) + uk0 1+ 1 − w · uk0 uk0 0
(3.49) where u0 = (uk )k∈Λα \{k0 } and g is analytic in some neighborhood of (1, 0, 0, 0, 0). We compute Uk0 (1, 0, 0, 0, 0) = 0 ,
∂Uk0 (1, 0, 0, 0, 0) = (α − 1) 6= 0 . ∂uk0
So we can apply the analytic version of the local inverse mapping theorem to the mapping (uk0 , u0 , v, w, t) → Uk0 near the point (1, 0, 0, 0, 0). This gives us that, for (u0 , t, v, w) in some neighborhood of 0, the equation Uk0 = 0 admits a unique solution u(u0 , t, v, w) that is analytic in (u0 , t, v, w). This solution satisfies u(0, 0, 0, 0) = 1. Hence, for λ large enough, ψ will have a unique critical point
605
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
0 in the variable ωk0 in Ω+ K (k0 ); we will denote this point by ωk0 ((ωk )k∈Λα \{k0 } , λ). Moreover, this critical point can be represented in following form:
ωk00 = ωk00 ((ωk )k∈Λα \{k0 } , λ) = (h1k0 )1/(α−1) λ(α−3)/(α−1) (1 + φ)
(3.50)
where φ admits a convergent expansion
X
φ= P
Y
a(nk )k ,l,m,n
k∈Λα \{k0 }
(nk )k∈Λα \{k } ,l,m,n 0
1 (λ − ωk )nk
nk +l+m+n≥1
× λ−l−2m/(α−1)−αn(α−3)/(α−1) .
(3.51)
Let us point out that, for k ∈ Λα , in (3.51), the term of lowest order in λ containing 1 0 λ−ωk contains at least the term Sl(k) /S0 . This in particular implies that, for k ∈ α Λ \ {k0 }, we have ! 0 ∂Sl(k) ∂ωk00 α−3 = λ α−1 O . (3.52) ∂ωk ∂ωk We compute the Hessian of ψ at ω 0 (λ) to get that ψ 00 (ωk00 , (ωk )k∈Λα \{k0 } ) = α(α − 1)g00 (h1k0 )(α−2)/(α−1) λ(α−3)(α−2)/(α−1) (1 + φ) where φ admits an asymptotic expansion of the type (3.51). So ω 0 (λ) is the unique critical of ψ in Ω+ K (k0 ) and it is a non degenerate minimum. The Hessian of ψ tends to +∞ with λ uniformly in Ω+ K (k0 ). Hence, to compute (3.42), as announced, we will use the Laplace method. Indeed, in Ω+ K (k0 ), outside an interval Iρ = α−3 α−3 0 0 α−1 α−1 [ωk0 − ρλ , ωk0 + ρλ ] (ρ > 0), we know that ψ(ω) ≥ ψ(ωk00 , (ωk )k∈Λα \{k0 } ) + ρ2 λ
(α−2)(α−1−2[(α−1]/2]) α−1
.
Inside the interval Iρ (if ρ is small enough), we write ψ(ω) = ψ(ωk00 , (ωk )k∈Λα \{k0 } ) + ψ 00 (ωk00 , (ωk )k∈Λα \{k0 } )(ωk0 − ωk00 )2 (1 + φ1 ) = ψ(ω 0 (λ), (ωk )k∈Λα \{k0 } ) + ψ 00 (ωk00 , (ωk )k∈Λα \{k0 } )[(ωk0 − ωk00 )(1 + φ)]2 (3.53) where φ1 and φ admit an asymptotic expansion of the type (3.51). We split the integral I1+ (given by (3.44)) into two parts I 1 + I 2 : in I 1 , we integrate over Iρ and, in I 2 , we integrate over the complementary of Iρ . Then, obviously I 2 ≤ λe−ψ(ωk0 ,(ωk )k∈Λα \{k0 } )−ρ 0
2
λ
(α−2)(α−1−2[(α−1)/2]) α−1
.
(3.54)
Hence, I 2 = 0(I 1 ) as λ → +∞. To compute I 1 , we use (3.53) and perform the change of variables ωk0 → (ωk0 − 0 ωk0 )(1 + φ) and compute the integral to get the following asymptotic expansion for I 1:
606
F. KLOPP
√ −ψ(ωk0 ,(ωk )k∈Λα \{k } ) 0 0 πe I = q · [(1 + RG (λ)) · (1 + Rf (ωk00 , (ωk )k∈Λα \{k0 } )) ψ 00 (ωk00 , (ωk )k∈Λα \{k0 } ) 1
· (1 + R(λ, ωk00 , (ωk )k6∈Λα \{k0 } )) + O(λ−M )] .
(3.55)
Hence, adding (3.54) and (3.55) and averaging with respect to the rest of the random variables, i.e. (ωk )k∈Λα \{k0 } , we get s π + E(I1 (λ, (ωk )k∈Λα \{k0 } )) = (1 + RG (λ)) · E (α−3)(α−2) α−2 (1) α(α − 1)g00 (hk0 )2 α−1 λ α−1 (3.56) where E = E[e−ψ(ωk0 ,(ωk )k∈Λα \{k0 } ,λ) · (1 + φ) · (1 + Rf (ωk00 , (ωk )k∈Λα \{k0 } )) 0
· (1 + R(λ, ωk00 , (ωk )k∈Λα \{k0 } )) + O(λ−M )] .
(3.57)
Here φ admits an asymptotic expansion of the type (3.51) and the asymptotic expansions of Rf and R are respectively given by (3.46) and (3.27). We now have to estimate I1− defined in (3.43) by −1
I1− (λ, (ωk )k6=k0 ) = E(e−g(I(λ,ω,θ)
)
1Ω− (k0 ) (ωk )k6=k0 ) . K
Therefore, let us notice that, by (3.47) and (3.48), g(I(λ, ω, θ)−1 ) is decreasing in −1 ωk0 on Ω− ) with respect K (k0 ). Using the estimate on the derivative of g(I(λ, ω, θ) − to ωk0 given by (3.47) and (3.48), we know that, uniformly in ΩK (k0 ), g(I(λ, ω, θ)−1 )ωk0 =λδ − g(I(λ, ω, θ)−1 )ωk0 =ωk0
0
∼
α−3
α
λ→+∞
αg00 (h1k0 ) α−1 λα−3+ α−1 .
Using (3.45), we estimate the difference ψ(ω)ωk0 =ωk0 − g(I(λ, ω, θ)−1 )ωk0 =ωk0 0
0
∼
λ→+∞
(1)
2α
α−3
g00 (hk0 ) α−1 λα α−1 .
+ This estimate is uniform in Ω− K (k0 ) (or ΩK (k0 ) as the functions used above only depend on (ωk )k∈Λα \{k0 } ). As α−1 > 0, we have that, for λ large enough, uniformly in Ω− K (k0 ), −1
e−g(I(λ,ω,θ)
)
≤e
−ψ(ω)ω
0 k0 =ωk 0
·e
(1)
2α
α
0 α−1 λ − α−1 2 g0 (hk ) 0
α−3 α−1
.
(3.58)
So that I1− (λ, (ωk )k6=k0 ) is negligible with respect to I1+ (λ, (ωk )k6=k0 ). To continue our scheme, we have to estimate the expectation defined in (3.57). This expectation does not contain the random variable ωk0 anymore. But it has the same form as (3.41) except for the fact that −g(I(λ, ω, θ)−1 ) has to be replaced with −ψ(ω)ωk0 =ωk0 . Let us denote this last function by ψ 0 ((ω)k6=k0 ). We want 0 to re-apply the same procedure as above to successively eliminate all the random variables (ωk )k∈Λα .
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
607
Let us point out the main characteristics of ψ 0 ((ω)k6=k0 ) that will enable us to continue the induction process, i.e. integrate with respect to k1 . By (3.45), (3.47) and (3.48), we have that ! 1 0 0 ∂ω 0 ∂ψ 0 g ∂S ∂S k 1 1 0 = g00 αJ α−1 + 0 (1 + o(1)) + (1 + o(1)) ∂ωk1 J ∂ωk1 ∂ωk0 ∂ωk1 ∂ωk00 1 0 α−1 1 + g0 α(ωk0 ) + (g0 + α − 1) ωk0 ∂ωk1 and, by (3.50), (3.51) and (3.52), we have that ωk00
= O(λ
α−3 α−1
)
∂ωk00 α−3 = λ α−1 O ∂ωk1
∂S10 ∂ωk1
.
Hence, as α − 1 ≥ α α−3 α−1 , by (3.38), we have
∂ψ 0 ∂S10 (l(k )) 2 = −g00 αJ α−1 (1 + o(1)) = −g00 αλα−1−2l(k1 ) hk1 1 (1 + o(1)) ∂ωk1 ∂ωk1
where o(1) → 0 as λ → +∞. So that, if we define ψ((ω)k6=k0 ) = ψ 0 ((ω)k6=k0 ) + f0 (ωk1 ), we may apply the same procedure as above to compute the expectation of 0 e−ψ with respect to the random variable ωk1 . Let us assume the procedure has been applied to all (kj )1≤j≤n−1 and that the minima (in the range [λδ , νλ]) that have been successively found applying the procedure are denoted by ωk0j ; our inductive assumption is that, for 1 ≤ j ≤ n − 1, ! 0 α−1−2l(kj ) α−1−2l(kj ) ∂Sl(k ∂ωk0j j) 0 α−1 α−1 ωkj = O(λ ) =λ O . (3.59) ∂ωk ∂ωk We also assume that these minima admit asymptotic expansions described by (3.50) and (3.51), and that the remark following (3.51) holds for any 1 ≤ j ≤ n−1. We set ψ((ωkj )j≥n ) = ψ n−1 ((ωk0j )1≤j≤n−1 , (ωkj )j≥n ) + f0 (ωkn ) ψ n−1 ((ωk0j )1≤j≤n−1 , (ωkj )j≥n ) = ψ0 ((ωk0j )1≤j≤n−1 , (ωkj )j≥n ) +
n−1 X
f0 (ωk0j ) .
j=0
We compute 0 0 n−1 0 1 X ∂Sl(k ∂S ∂ω ∂ψ n−1 g ) k l(k ) j j n 0 = g00 αJ α−1 + (1 + o(1)) + (1 + o(1)) ∂ωkn J ∂ωkn ∂ωkj ∂ωkn j=0 +
n−1 X
g00 α(ωkj )α−1 + (g01 + α − 1)
j=0
Then, as, for 1 ≤ j ≤ n − 1, we have α − 1 ≥ α ∂ψ n−1 = −g00 αJ ∂ωkn
0 ∂Sl(k n) α−1
∂ωkn
where o(1) → 0 as λ → +∞.
1 ωkj
∂ωk0j ∂ωkn
α−1−2l(kj ) , α−1
. (3.38) gives us
(l(k )) 2 (1 + o(1)) = −g00 αλα−1−2l(kn ) hkn n (1 + o(1))
608
F. KLOPP
We compute the minimum of ψ; it is unique, non-degenerate and satisfies (3.59) at the nth order (i.e. with j replaced by n). We also compute the Hessian of ψ at α−1−2l(kn )
the minimum to get that it is of order λ(α−2) α−1 . So that we can again use Laplace’s method to estimate the expectation with respect to ωkn at least in the interval [λδ , νλ]. Outside of this interval, we use the same method as above. ψ n−1 is a decreasing function of ωkn and we estimate n−1 n−1 ψω 0 δ − ψω k =λ k =ω n
n
kn
l(k )
∼
α
αg00 (hkn n ) α−1 λα
λ→+∞
α−2l(kn )−1 α−1
.
We also estimate the difference n−1 ψ(ω)ωkn =ωk0 − ψω 0 k =ω n
n
kn
∼
λ→+∞
l(k )
α
g00 (hkn n ) α−1 λα
α−2l(kn )−1 α−1
.
As above, as α−1 > 0, this proves that the expectation of e−ψ on the set {ωkn ≤ λδ } is exponentially small compared to the expectation of e−ψ on the set [λδ , νλ]. To end the proof of Theorem 1.3 at least in the case when α is not an odd integer, we just need to count what powers of λ come into the asymptotics that we have computed. Therefore, we notice that, for any 0 ≤ l ≤ [α − 1/2], there exists α−1−2[α−1/2] 2k k ∈ N such that α−1−2l + α−1 . α−1 = α−1 If α = 2n + 1, we can use the above procedure for all the sites k such that l(k) ≤ n − 1, i.e. k ∈ Λ1 ∪ · · · ∪ Λn−1 . To eliminate the random variables (ωk )k∈Λn , we will use the same idea as in the case α = 3. More precisely, by (3.40), we have
2n+1 2
1 λ (J(λ, ω))α = (J(λ, ω))2n+1 = X n−1 (λ,ω) S S0 (λ) 1 + p p=1 S0 (λ)
− (2n + 1)
λ2 S0 (λ)
2n+1
Sn (λ, ω) + Rn (λ, ω) S0 (λ)
˜ ω))2n+1 − (2n + 1)J ∗ (λ, ω) + Rn (λ, ω) = (J(λ, where Rn is a remainder term that admits an expansion of the form (3.27) and Rn = O( λ1 ) (uniformly in ω ∈ Ω0K ). ˜ ω))2n+1 is dealt with in the way described above. By (3.38), the The term (J(λ, ∗ new term J (λ, ω) satisfies J ∗ (λ, ω) =
X k∈Λn
|hnk |2
λ2 + Rn∗ (λ, ω) , λ − ωk
where Rn∗ is a remainder term that admits an expansion of the form (3.27) and Rn∗ = O( λ1 ) (uniformly in ω ∈ Ω0K ). Now, when we take the expectation of e−g in the random variables (ωk )ki nΛn , we may restrict the range where we take the expectation in ωk to [λδ , λβ ] (as the density is exponentially decaying). We then decompose J ∗ in the same way as in (3.31) and use the method described in the case α = 3 to conclude.
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PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
Proof of Theorem 1.4. The method used to prove this theorem is the same as the one used to prove the asymptotics (1.11). Hence we will not give all the details. First, because of the precision we ask for our asymptotic of N , we pick ε = λ−2 and n = e2(1+ν)λ in Lemma 2.1 (ν chosen as in Lemma 2.1). We fix K = λδ (where 0 < δ will be chosen later on). Hence ˆ + )F (λ − K − H ˆ + ) = o(e−eg0 λ ) . n2d F (λ − H We recall that by (3.8), (3.10) and (3.11), uniformly in Ω0K and T, we have 1 λ2 X = I(λ, ω, θ) λ + h0 +
|hk |2 k λ−ωk
+ δI(λ, ω, θ) ,
where δI is analytic in θ and ω, and, for some C > 0, it satisfies a uniform estimate |δI(λ, ω, θ)| ≤ Cλ−2δ . Hence, in Ω0K and T, we have g0 λ2α |hk |2 λ+h0 + k λ−ωk
P
g((I(λ, ω, θ))−1 ) = e
α + δg(λ, ω, θ) ,
(3.60)
where δg satisfies a uniform estimate α
|δg(λ, ω, θ)| ≤ C(λα−1−2δ + λ2(α−1)−δ ) · eg0 λ . So that ln E(e−g((I(λ,ω,θ)) where
−1
)
α
1Ω0K ) = ln E(λ) + O((λα−1−2δ + λ2(α−1)−δ )eg0 λ )
E(λ) = E exp −e
g0 λ2α |hk |2 k λ−ωk
λ+h0 +
P
α
1Ω0 . K
To get the asymptotic behavior of E, as above, we split the expectation into two parts: 0
(1) E1 where one takes the expectation over ωk ≤ λδ (where 0 < δ 0 is chosen small enough); 0 (2) E2 where one takes the expectation over λδ ≤ ωk ≤ λ − λδ . E2 may be rewritten as
Z
E2 (λ) = where ψ(ω) = e
Z ···
[λδ0 ,λ−λδ ]Λ1
g0 λ2α |hk |2 k λ−ωk
λ+h0 +
P
Y
e−ψ(ω)
dωk
k∈Λ1
α +
X k∈Λ0
α
eg0 ωk .
610
F. KLOPP 0
ψ admits a single minimum in [λδ , λ − λδ ]Λ1 that we denote by ω 0 . For k ∈ Λ1 , we have the following asymptotics 1−α 2λ1−α λ ωk0 = λ − ln (1 + o(1)) . αg0 αg0 |hk | Notice that this tells us that δ has to be chosen such that δ < 1 − α. The minimum ω 0 is non-degenerate as we have the following asymptotic for the Hessian of ψ α ∂2ψ 0 (ω ) = (αg0 )2 λ2(α−1) eg0 λ (1 + o(1)) 2 ∂ωk α ∂2ψ (ω 0 ) = o(λ2(α−1) eg0 λ ) ∂ωk ωj
where (k, j) ∈ Λ21 and k 6= j. We can then use the Laplace method to get the following asymptotics for E α
ln E(λ) = −ψ(ω 0 ) + o(λ2(α−1) eg0 λ )
= −eg0 λ 1 − αh0 g0 λα−1 + (αg0 )2 α
X
|hk |2 λ2(α−1) + o(λ2(α−1) ) .
k6=0
By what we have said above, ln E and ln N satisfy the same asymptotics at +∞. This ends the proof of Theorem 1.4. 4. The Quantum Regime We recall that we have defined h(λ) = ln g(λ). We have assumed that h is increasing, twice differentiable and tends to +∞ at +∞. In terms of h, assumption (1.14) becomes h0 (λ) −−−−→ + ∞ . (4.1) λ→+∞
ˆ is assumed to have a single maximum and that this maximum We also recall that H is non degenerate. Without restricting the generality of our discussion, we may ˆ assume that this maximum is attained at the point θ = 0 and that H(0) = 0. ˆ Hence, from now on, we assume that H ≤ 0 and that, near 0, we have 1 ˆ −C|θ|2 ≤ H(θ) ≤ − |θ|2 C
(4.2)
where C > 1. To prove (1.15), we will use Lemma 2.1. Hence we need to analyze E(Nωn (λ)). This is done in the 1 Lemma 4.1. For any η ∈ (0, 4d ), there exists λη > 0 and Cη > 0 such that, for ∗ −d/2 n ∈ N and δ > 0 such that n > δ , and for λ ≥ λη , we have
F (λ + Cη δ)δ
−d/2−η
≤ E(Nωn (λ)) .
(4.3)
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
611
1 For any η ∈ (0, 4d ), there exists λη > 0 and δη ∈ (0, 1) such that, for δ ∈ (0, δη ), −2d for m > δ (m ∈ N∗ ) and for λ ≥ λη , if we define
(2n + 1) = (2m + 1)[δ −η ]o [δ −1/2+2η ]o (where [·]o denotes the largest odd integer smaller than) then, we have E(Nωn (λ))
≤ (3n)
d+1
3d−2 δ−d/2(1+E(λ,δ,η))
F (λ − δ)
3d−2 δ−d/2 F (λ + δ 1−η/2 ) · 1+ 2F (λ − δ)4
+ 2(3n)d+1 F (λ + δ 1−η/2 )
(4.4)
where, uniformly in λ, δ and η, E(λ, δ, η) satisfies E(λ, δ, η) = O
ln δ F (λ + δ 1−η/2 ) + δ η/2 + +η ln F (λ − δ) F (λ − δ)
.
Lemma 4.1 will be proved later. We now finish the proof of Theorem 1.5. To do this we need to: • pick the precision ε(λ), • pick the scale n(λ), • pick the correction δ(λ) such that: • the ln ln of the upper and lower bounds given by Lemma 4.1 are asymptotically equivalent as λ → +∞, • the error estimate δ(ν, ε, n, λ) given by Lemma 2.1 becomes much smaller than the lower bound given by Lemma 4.1. We notice that, by assumption (1.14) and (2) of Sec. 4, we know that h0 is increasing and h00 /(h0 )2 → 0 at +∞. Integrating this last relation, we get, for λ large enough and δ > 0, Z
λ+δ
λ
h00 (t) 1 1 dt = 0 − 0 = δoλ (1) , 0 2 (h (t)) h (λ) h (λ + δ)
(uniformly in δ). This gives
oλ (1) −−−−→ 0 , λ→+∞
1 1 0 h λ+ 0 = 1 + oλ (1) . h0 (λ) h (λ)
We set δ(λ) =
1 . h0 (λ) ln h0 (λ)
As h0 → +∞, δ(λ) → 0 as λ → +∞. By (4.5), for any C > 0, we have 1 1 − oλ (1) = 0 0 δ(λ)h (λ) δ(λ)h (λ + Cδ(λ))
(4.5)
(4.6)
612
F. KLOPP
so that δ(λ)h0 (λ + Cδ(λ)) −−−−→ 0 . λ→+∞
As h0 is increasing, this implies that, for any C > 0, 0 ≤ h(λ + Cδ(λ)) − h(λ − Cδ(λ)) −−−−→ 0 . λ→+∞
(4.7)
Let η ∈ (0, 1). By (4.5) and (4.6), we have 1−η
h(λ + δ(λ)
) − h(λ) =
N X
h(λ + δ(λ)1−η − kδ(λ)) − h(λ + δ(λ)1−η − (k + 1)δ(λ))
k=0
+ h(λ + δ(λ)1−η − (N + 1)δ(λ)) − h(λ) ≥
N X
δ(λ)h0 (λ + δ(λ)1−η − (k + 1)δ(λ))
k=0
≥
N X
δ(λ)h0 (λ + δ(λ)) ≥
k=0
N 2
for N ≤ δ(λ) /2 and λ large enough. This proves that, for any η ∈ (0, 1), one has η
h(λ + δ(λ)1−η ) − h(λ) −−−−→ + ∞ . λ→+∞
(4.8)
This implies F (λ + δ(λ)1−η ) −−−−→ 0 . F (λ − δ(λ)) λ→+∞ Moreover, by (4.7), we know that
(4.9)
ln δ(λ) = −e−h(λ) ln h0 (λ)(1 + o(1)) . ln F (λ − δ(λ)) By assumption (2) of Subsec. 1.2, for any α > 0, (h0 e−αh )0 = −α(h0 )2 e−h (1 + o(1)). So h0 e−αh is decreasing and positive. Hence e−αh(λ) h0 (λ) −−−−→ 0 .
(4.10)
ln δ(λ) −−−−→ 0 . ln F (λ − δ(λ)) λ→+∞
(4.11)
λ→+∞
So that
We set ε(λ) = δ(λ) and n(λ) = Cν ε(λ)e(d+1)νh(λ) (where ν and Cν are fixed in Lemma 2.1). Then, (4.10) shows that, for any p > 0, δ(λ)p n(λ) −−−−→ + ∞ . λ→+∞
Hence, for λ large enough, n(λ) and δ(λ) satisfy n(λ) > δ(λ)−2d . We compute δ(ν, ε, n, λ) ≤ e−e
(d+1)h(λ)
.
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
613
Let us compare this estimate to the one we have for the lower bound in Theorem 4.1: for C > 0 and η ∈ (0, d/2) fixed, by (4.7), we have ! −d/2−η F (λ + Cδ(λ))δ(λ) ln ≥ eh(λ) (edh(λ) − δ(λ)−d/2−η )(1 + o(1)) δ(ν, ε, n, λ) ≥ 2e(d+1)h(λ) (1 − (δ(λ)eh(λ) )−d ) . Hence
−d/2−η
δ(ν, ε, n, λ) = o(F (λ + Cδ(λ))δ(λ)
).
(4.12)
At last, for λ → +∞, we have ln n(λ) = (d + 1)νh(λ)e−h(λ) (1 + o(1)) → 0 . ln F (λ) Now, by (4.3), (4.7), Lemma 2.1 and (4.12), we have ln|ln N (λ)| ≥ h(λ) +
d ln h0 (λ)(1 + o(1)) . 2
By (4.4), (4.7), (4.9), (4.11), Lemma 2.1 and (4.12), we have ln|ln N (λ)| ≤ h(λ) +
d ln h0 (λ)(1 + o(1)) . 2
As ln|ln F | = h and h0 = F 0 /(F ln F ), we get Theorem 1.5. 4.1. The proof of Lemma 4.1 To prove (1.15), we will prove a lower and an upper bound on ln|ln N (λ)|. Many of the ideas used to prove the lower and upper bound on N are similar to the ones developed in [6]. Therefore we will not give all the details and refer to [6] for additional material. We now want to analyze E(Nωn (λ)). We will use the Floquet theory described in Subsec. 2.1 to do this. In the sequel, it will sometimes be convenient to give vectors by their components in the basis of eigenvectors of M0n (θ) that we called (uk )k∈Zd . So that, if we denote the vectors of the canonical basis by (δl )l∈Zd , 2n+1 2n+1 their components in the basis (uk )k∈Zd2n+1 are δl =
2πk 1 (ei(θ+ 2n+1 )l )k∈Zd2n+1 . d/2 (2n + 1)
We also define the vectors (vl )l∈Zd
2n+1
vl = e−ilθ δl =
by
2πkl 1 (ei 2n+1 )k∈Zd2n+1 . (2n + 1)d/2
Remark 4.1. The canonical basis plays here a special role as it is the basis of eigenvectors for Vωn .
614
F. KLOPP
4.1.1. The lower bound By (2.7), we just need to prove the right lower bound for the probability that Mωn (θ) has an eigenvalue larger than λ. We will do this by explicitly constructing such an eigenvalue for a sufficiently large set of ω’s. Therefore we start with fixing θ. Let a ∈ `2 (Zd2n+1 ) be expressed by its coordinates in the basis (δl )l∈Zd2n+1 , i.e. a = P l∈Z2n+1 al δl . Then X X ˆ θ + 2kπ hMωn (θ)a, ai = H |Ak |2 + ωl |al |2 (4.13) 2n + 1 d d k∈Z2n+1
where Ak =
l∈Z2n+1
1 (2n + 1)d
X
al e−iθl e−2iπkl/(2n+1) .
l∈Zd 2n+1
Both bases being orthonormal, we have
P k∈Zd 2n+1
|Ak |2 = kak2 .
Let b ∈ C0∞ ((−1/4, 1/4)d) ⊂ L2 ([−1/2, 1/2]d), b non negative such that kbkL2([−1/2,1/2]) = 1. Pick η ∈ (0, 1/2) small. Set aδl = eilθ δ d/2+dη b(δ 1/2+η l) .
P Let aδ be the vector l∈Z2n+1 aδl δl . Notice that this vector has its support in the set {k ∈ Zd2n+1 ; |k| ≤ δ −1/2−η }. Using the smoothness of b and the fact that b is normalized, one obtains X kaδ k2 = |aδl |2 = 1 + O(δ 1/2+η ) . l∈Zd 2n+1
As in [6], one can use discrete integration by parts to estimate Aδk ; doing this in the variable kj (1 ≤ j ≤ d), one gets
∂ q b 2 δ q+2qη 1
δ 2 |Ak | ≤
. (2n + 1)d+2q |e−2iπkj /(2n+1) − 1|2q ∂xqj Hence, if |kj | ≥ δ 1/2 (2n + 1), we have |Aδk |2 ≤ C
δ 2qη−q δ 2qη ≤ C (2n + 1)d+2q (2n + 1)d
as n ≥ δ −d/2 . So that −Cδ
2qη
≤
X k∈Zd 2n+1 |k|≥δ1/2
and −Cδ ≤
X k∈Zd 2n+1 |k|≤δ1/2
2kπ ˆ H θ+ |Aδk |2 ≤ 0 2n + 1
2kπ ˆ H θ+ |Aδk |2 ≤ 0 2n + 1
ˆ + as, by (4.2), we know that, for |k| ≤ δ 1/2 , H(θ
2kπ 2n+1 )
≥ −Cδ.
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PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
Compiling all this, we see that, there exists C > 0 such that, if, for |l| ≤ δ −1/2−η , we have ωl ≥ λ + Cδ, then hMωn (θ)aδ , aδ i ≥ λ. This tells us that, for any 0 < η < d/2, there exists Cη > 0 such that E(Nωn (λ)) ≥ F (λ + Cη δ)Cδ
−d/2−η
.
This proves (4.3). 4.1.2. The upper bound We want to estimate E(Nωn (λ)) from above. By (2.7), we have E(Nωn (λ)) ≤ sup P ({Mωn (θ) has an eigenvalue larger than λ}) . θ
So we need to upper bound the probability that there exists a = (ak )k∈Zd2n+1 (the coordinates are given in the basis (uk )k∈Zd2n+1 ) normalized such that ha, Mωn (θ)ai ≥ λ, and we need to do this uniformly in θ. Pick 0 < η 0 < η <
1 2d
and write 0
{(ωk )k∈Zd2n+1 } = {(ωk )k∈Zd2n+1 ; ∀ k ∈ Zd2n+1 , ωk < λ + δ 1−η } 0
∪ {(ωk )k∈Zd2n+1 ; ∃ k ∈ Zd2n+1 , ωk ≥ λ + δ 1−η } = Ω− ∪ Ω+ . Then, one has
0
P (Ω+ ) ≤ (3n)d F (λ + δ 1−η ) . We define Ω− (δ, n, λ) = {(ωk )k∈Zd2n+1 ∈ Ω− ; Mωn (θ) has an eigenvalue larger than λ} , and the following auxiliary random −δ ω ˜ l = ωl − λ 1−η0 δ
variables: for l ∈ Zd , let if ωl ≤ λ − δ if λ − δ ≤ ωl ≤ λ + δ 1−η
0
0
if ωl ≥ λ + δ 1−η .
Then, for ω ∈ Ω− (δ, n, λ), we have ωl − λ ≤ ω ˜ l so that, if ha, Mωn (θ)ai ≥ λ then X ha, M0n (θ)ai + ω ˜ l |Al |2 ≥ 0 (4.14) l∈Zd 2n+1
where Al =
1 (2n + 1)d
X
ak e−2iπkl/(2n+1) .
k∈Zd 2n+1
We used the fact that the map a 7→ A = (Al )l∈Zd2n+1 is the discrete Fourier transP form; hence l∈Zd |Al |2 = kak2 = 1. As a conclusion, we see that we only need 2n+1 to upper bound the probability of the event
616
F. KLOPP
( ˜ − (δ, n, λ) = Ω
(ωk )k∈Zd2n+1 ∈ Ω− ; ∃ a ∈ `(Zd2n+1 ); × hM0n (θ)a, ai
)
X
+
ω ˜ l |Al | ≥ 0 . 2
l∈Zd 2n+1
Therefore, we define • 2L0δ + 1 = [δ −1/2+2η ]o , • 2Kδ0 + 1 = (2m + 1)[δ −η ]o . Then, by Lemma 4.1, we defined 2n + 1 = (2Kδ0 + 1)(2L0δ + 1). We have the 1 Lemma 4.2. For any 0 < η 0 < η < 4d , there exists δ0 > 0 such that, for 0 < δ < π −d δ0 and m ≥ δ , we have, for |θ| ≤ 2n+1 , [ ˜ − (δ, n, λ) ⊂ ˜ k (δ, n, λ) ∪ ∆Ω ˜ − (δ, n, λ) Ω Ω (4.15) − |k0 |≤Kδ0
where, for |k 0 | ≤ Kδ0 , we define the event X 1 3 0 0 0 Ωk− (δ, n, λ) = ω; ω ˜ δ ≥ 0 . + k (2Lδ +1)+l (2L0δ + 1)d 0 0 4 |l |≤Lδ
and we have the estimate ˜ − (δ, n, λ)) ≤ (3n)d P (ω0 ≥ λ + δ 1−η ) . P (∆Ω
(4.16)
Before proving Lemma 4.2, let us indicate how to derive the upper bound (4.4) from it. We want to estimate the probability of Ωk− (δ, n, λ). Therefore, we compute the expectation of ω ˜ 0 + 3δ/4 to get that E(˜ ω0 ) + 3δ/4 → −δ/4 as λ → +∞: we see that, in the limit λ → +∞, when we estimate P (Ωk− (δ, n, λ)), we are in a large deviation regime. Using Cr`amer’s ideas [3], we have P (Ωk− (δ, n, λ)) ≤ exp((2L0δ + 1)d ln E(et(˜ω0 +3δ/4) ))
(4.17)
for any t ≥ 0. Hence, we only have to estimate (2L0δ + 1)d ln E(et(˜ω0 +3δ/4) ) and minimize the thus obtained estimate in t over t ≥ 0. We compute E(et(˜ω0 +3δ/4) ) = e−tδ/4 P (ω0 ≤ λ − δ) + E(et(ω0 −λ+3δ/4) 1λ−δ≤ω0 ≤λ+δ1−η0 ) + etδ
1−η0
0
P (ω0 ≥ λ + δ 1−η )
≤ e−tδ/4 + etδ + (etδ
1−η0
1−η0
+3tδ/4
= f (t, λ, δ) .
0
P (ω0 ≥ λ + δ 1−η ) − 1)P (ω0 ≥ λ − δ)
617
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
Solving the equation ∂t f (t, λ, δ) = 0 gives 0
δ η e−tδ
1−η0
(1+δ
η0
0
)
3 0 P (ω0 ≥ λ + δ 1−η ) = 4P (ω0 ≥ λ − δ) 1 + δ η + e−t3δ/4 4 P (ω0 ≥ λ − δ)
! .
A short computation gives that the minimum of f (t, λ, δ) is attained in a point t0 given by 0 δ η −1 t0 = − ln P (ω0 ≥ λ − δ)(1 + E(λ, δ, η 0 )) 1 + δ η0 where, uniformly in λ large enough and δ and η small enough, we have ! 0 ln δ P (ω0 ≥ λ + δ 1−η ) 0 η0 E(λ, δ, η ) = O +δ + . ln P (ω0 ≥ λ − δ) P (ω0 ≥ λ − δ) We compute δη
0
E(et0 (˜ω0 +3δ/4) ) ≤ 2P (ω0 ≥ λ − δ) 4+4δη0
(1+E(λ,δ,η0 )) 0
×
1+
P (ω0 ≥ λ + δ 1−η ) 1
2P (ω0 ≥ λ − δ) 1+δη0
!
(1+E(λ,δ,η))
.
Using the fact that P (ω0 ≥ λ) is decreasing in λ, for δ small enough (depending on η0 ), we get ! 0 0 δη P (ω0 ≥ λ + δ 1−η ) t0 (˜ ω0 +3δ/4) (1+E(λ,δ,η0 )) 8 E(e ) ≤ 2P (ω0 ≥ λ − δ) 1+ . 2P (ω0 ≥ λ − δ)4 Plugging this into (4.17) and picking η 0 = η/2, we get that d−2 −d/2(1+E(λ,δ,η))
P (Ωk− (δ, n, λ)) ≤ 2P (ω0 ≥ λ − δ)3
δ
3d−2 δ−d/2 P (ω0 ≥ λ + δ 1−η/2 ) · 1+ 2P (ω0 ≥ λ − δ)4 where, uniformly in λ large enough and δ and η small enough, we have E(λ, δ, η) = O
ln δ P (ω0 ≥ λ + δ 1−η/2 ) + δ η/2 + +η ln P (ω0 ≥ λ − δ) P (ω0 ≥ λ − δ)
.
This ends the proof of Lemma 4.1.
Proof of Lemma 4.2. Let a ∈ `(Zd2n+1 ) normalized such that (4.14) holds. We define the vectors a1 and a2 by X X a1 = ak uk , a2 = ak uk . |k|≤ 14 δ1/2−η n
|k|> 14 δ1/2−η n
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F. KLOPP
So that a = a1 + a2 , a1 ⊥ a2 , and hM0n (θ)a, ai = hM0n (θ)a1 , a1 i + hM0n (θ)a2 , a2 i .
(4.18)
We notice that, by (4.2), hM0n (θ)a2 , a2 i ≤ −Cδ 1−2η ka2 k2 .
(4.19)
On the other hand, as A1 + A2 = A (here, as above, A1 (resp. A2 ) denotes the discrete Fourier transform of a1 (resp. a2 )), we have X X X ω ˜ l |Al |2 = ω ˜ l |A1l + A2l |2 + ω ˜ l |A1l + A2l |2 l∈Zd 2n+1
l∈Zd 2n+1 ω ˜ l ≥0
X
≤2
l∈Zd 2n+1 ω ˜ l ≤0
ω ˜ l (|A1l |2 + |A2l |2 ) +
l∈Zd 2n+1
X l∈Zd 2n+1 ω ˜ l ≤0
−ω ˜ l (2(|A1l |2 + |A2l |2 ) − |A1l + A2l |2 ) as |A1l + A2l |2 ≤ 2(|A1l |2 + |A2l |2 ). Hence, as −˜ ωl ≤ δ, we have X X X ω ˜ l |Al |2 ≤ 2 ω ˜ l (|A1l |2 + |A2l |2 ) + δ (2(|A1l |2 + |A2l |2 ) − |A1l + A2l |2 ) l∈Zd 2n+1
l∈Zd 2n+1
=2
X
l∈Zd 2n+1
ω ˜ l (|A1l |2 + |A2l |2 ) + δka1 k2 + δka2 k2 .
l∈Zd 2n+1
So that (4.14) and (4.18) give X
ω ˜ l |A1l |2 +
l∈Zd 2n+1
X l∈Zd 2n+1
1 1 ω ˜ l |A2l |2 ≥ − h(M0n (θ) + δ)a1 , a1 i − h(M0n (θ) + δ)a2 , a2 i . 2 2
Then, either of two things happen: (1) either
X l∈Zd 2n+1
(2) or
X l∈Zd 2n+1
1 ω ˜ l |A1l |2 ≥ − h(M0n (θ) + δ)a1 , a1 i , 2
1 ω ˜ l |A2l |2 ≥ − h(M0n (θ) + δ)a2 , a2 i . 2
In case (2), we know that that there exists some l ∈ Zd2n+1 such that ω ˜l ≥ (Cδ 1−2η − δ)/2. Indeed, if this were not the case then, by (4.19), we would have X (Cδ 1−2η + δ)/2ka2 k2 ≤ ω ˜ l |A2l |2 < (Cδ 1−2η − δ)/2ka2 k2 l∈Zd 2n+1
619
PRECISE HIGH ENERGY ASYMPTOTICS FOR THE INTEGRATED DENSITY
which is absurd. So that the probability that we are in case (2) is upper bounded by (3n)d P (ω0 ≥ λ + (Cδ 1−2η − δ)/2). ˜ − (δ, n, λ) as the event of all the ω for which we are in If we now define ∆Ω case (2), then the above analysis immediately gives us (4.16). We now turn to case (1). In this case, as M0n (θ) ≤ 0, we have X (˜ ωl + δ/2)|A1l |2 ≥ 0 . (4.20) l∈Zd 2n+1 0
A1 is the discrete Fourier transform of a1 , and a1 has its support in {|k| ≤ δ 1/2−η n}. We define • 2Lδ + 1 = [δ −1/2+2η ]o [δ −η ]o , • 2Kδ + 1 = (2m + 1). By an easy extension of Lemma 3.1 of [6], we know that there exists a ˜1 ∈ `2 (Zd2n+1 ) such that ˜1 k`2 (Zd2n+1 ) ≤ Cδ η ka1 k`2 (Zd2n+1 ) , (1) ka1 − a
(2) for l0 ∈ Zd2L0 +1 and k 0 ∈ Zd2K 0 +1 , we have δ
δ
A˜1l0 +k0 (2L0 +1) = A˜1k0 (2L0 +1) . δ
δ
Remark 4.2. Lemma 3.1 of [6] roughly says that if a is a vector in `2 (Zd2N +1 ) with coefficients concentrated near 0 in a region of width K < N , then up to a small error in `2 -norm, a can be considered to have Fourier coefficients that are constant over cubes of side length N/K. 0
Then, as ω ˜ l ≤ δ 1−η , X 1 2 1 2 ˜ (˜ ωl + δ/2)|Al | − |Al | ≤ l∈Zd 2n+1
X
|˜ ωl + δ/2|(2|A1l − A˜1l kA1l | + |A1l − A˜1l |2 )
l∈Zd 2n+1 0
˜1 kka1 k + ka1 − a ˜ 1 k2 ) ≤ (δ 1−η + δ/2)(2ka1 − a 0
≤ 8δ 1−η +η k˜ a1 k2 . As η 0 < η and as
P l∈Zd 2n+1
|A˜1l |2 = k˜ a1 k2 , for δ small enough, (4.20) gives X
(˜ ωl + 3δ/4)|A˜1l |2 ≥ 0 .
l∈Zd 2n+1
But by (2), we know that X (˜ ωl + 3δ/4)|A˜1l |2 = l∈Zd 2n+1
X k0 ∈Zd
2K 0 +1 δ
Bk0 (2L0δ + 1)d |A˜1k0 (2L0 +1) |2 δ
620
F. KLOPP
where Bk0 =
(2L0δ
X 1 + 1)d 0 d l ∈Z
(˜ ωl0 +k0 (2L0δ +1) + 3δ/4) .
2L0 +1 δ
Hence, for some k 0 ∈ Zd2K 0 +1 , we have δ
(2L0δ
1 + 1)d
X
(˜ ωl0 +k0 (2L0δ +1) + 3δ/4) ≥ 0 .
l0 ∈Zd 0 2L +1 δ
This ends the proof of Lemma 4.2.
Acknowledgments It is a pleasure to thank L. Pastur for suggesting that the problem studied here should be solvable using the methods developed in [6], and A. Trouv´e for enlightening discussions on Cr` amer’s Theorem. We also gratefully acknowledge support of the European TMR ERBFMRXCT960001. References [1] R. Carmona and J. Lacroix, Spectral Theory of Random Schr¨odinger Operators, Birkh¨ auser, Basel, 1990. [2] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators, Springer Verlag, Berlin, 1987. [3] J.-M. Deuschel and D. Stroock, Large Deviations, volume 137 of Pure and Applied Mathematics, Academic Press, 1989. [4] W. Kirsch and F. Martinelli, “On the density of states of Schr¨ odinger operators with a random potential”, J. Phys. A 15 (1982) 2139–2156. [5] W. Kirsch and B. Simon, “Lifshitz tails for the Anderson model”, J. Statistical Phys. 38 (1985) 65–76. [6] F. Klopp, “Band edge behaviour for the integrated density of states of random Jacobi matrices in dimension 1”, J. Statistical Phys. 90(3-4) (1998) 927–947. [7] F. Klopp, “Internal Lifshits tails for random perturbations of periodic Schr¨ odinger operators”, Duke Math. J. 98(2) (1999) 335–396. [8] A. Kolmogorov and S. Fomin, Introductory Real Analysis, Dover, 1975. [9] I. M. Lifshitz, “Structure of the energy spectrum of impurity bands in disordered solid solutions”, Soviet Phys. JETP 17 (1963) 1159–1170. [10] I. M. Lifshitz, “Energy spectrum structure and quantum states of disordered condensed systems”, Soviet Phys. Uspekhi 7 (1965) 549–573. [11] I. M. Lifshitz, S. A. Gredeskul and L. A. Pastur, Introduction to the Theory of Disordered Systems, Wiley, New York, 1988. [12] S. Molchanov, “Localization and intermittency”, In Proc. ICM 1990 Kyoto, The Mathematical Society of Japan, 1991, pp. 1091–1103. [13] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer Verlag, Berlin, 1992. [14] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol IV: Analysis of Operators, Academic Press, New York, 1978. [15] O. Saad, “Comportement en grandes ´energies de la densit´e d’´etats du mod`ele d’Anderson non born´e”, PhD thesis, Universit´e de Paris 13, 1999, en pr´eparation.
ON THE REPRESENTATION OF Tr(e(A − λB) ) AS A LAPLACE TRANSFORM PIERRE MOUSSA Service de Physique Th´ eorique, CEA-Saclay F-91191 Gif-Sur Yvette cedex, France Received 27 July 1998 Revised 15 October 1998 The conjecture made in 1976 by Bessis, et al. that Tr(e(A−λB) ) is the Laplace transform of a positive measure when A and B are Hermitean matrices and B semi positive definite is still not solved. We review some attempts made toward the solution of the question, and provide several unpublished partial results. We also quote some mathematical problems related to this question.
1. Introduction In 1976, Bessis, et al. [1] formulated Conjecture 1.1 below, which states that the function Z(λ) = Tr(e(A−λB) ) is the Laplace transform of a positive measure, when A and B are Hermitean matrices and B is semi positive definite (i.e. vanishing eigenvalues are permitted). In this paper, we report on various attempts to discuss this conjecture, and analyze some related questions. We will consider the following general situation: Let A and B be d × d complex matrices, and assume that there exists a basis B where B is diagonal and has eigenvalues, b1 , b2 , . . . , bd , which are not necessarily all different. Let U(B) be the closed convex hull (in the complex plane) of the set of eigenvalues, that is the smallest closed convex set which contains all eigenvalues of B. Indeed such a basis in which B is diagonal is not unique, and we will now make more precise which basis we want to select. When the eigenvalues bi all differ with one another, we select one of these bases arbitrarily (in this case, two bases differ only by a multiplicative factor on each basis vector). When some eigenvalues of B coincide, the degenerate eigenspaces have dimension more than one, and the restriction of B to each degenerate eigenspace is a multiple of the identity matrix. One can change the basis in such a way that the restriction of A to each degenerate eigenspace is in canonical Jordan form. By this canonical form, we mean first that all off diagonal elements (of the restriction) vanish except those of the first upper parallel to the diagonal (that is elements of the form ai,i+1 ), which are equal either to 0 or 1, and second, that ai,i+1 = 1 implies ai,i = ai+1,i+1 . We select one basis with all these properties and we call it BJ . We define the matrix M , which, in basis B, has vanishing diagonal elements and off-diagonal elements which coincide with the off-diagonal elements of A. Thus, in basis B, we have for 1 ≤ i ≤ d and 1 ≤ j ≤ d, 621 Reviews in Mathematical Physics, Vol. 12, No. 4 (2000) 621–655 c World Scientific Publishing Company
622
P. MOUSSA
Bi,j = bi δi,j ,
Ai,j = ai δi,j + Mi,j ,
and for
1 ≤ k ≤ d,
Mk,k = 0 .
(1.1)
For λ complex arbitrary, we define the matrix X, which in fact depends on λ, such that in basis B, Xi,j = Xi,j (λ) = (ai − λbi )δi,j ,
thus
A − λB = X + M .
(1.2)
We will consider the following functions of λ: E(λ) = e(A−λB) = e(X+M) ,
(1.3)
Z(λ) = Tr(E(λ)) = Tr(e(A−λB) ) = Tr(e(X+M) ) .
(1.4)
When A and B commute, we have E(λ) = eA e−λB , and therefore, E(λ)i,j = Pd (eA )i,j e−λbj , Z(λ) = i=1 (eA )i,i e−λbi . In this case, Z Z (1.5) (E(λ))i,j = e−λτ dµi,j (τ ) , Z(λ) = e−λτ dµ(τ ) , where µ is a complex measure made of a sum for i = 1 to d of Dirac masses situated at τ = bi with complex weight (eA )i,i . Note that the support of µ is contained in the closed set U(B) of the complex plane. The measures µi,j are also discrete, with support included in U(B), and we will have a similar representation for all combinations with constant coefficients (that is independent on λ), of matrix elements of E(λ). The matrix elements Ai,j vanish when bi 6= bj , and if now we work in basis BJ , which means that in case of degeneracy of the eigenvalues of B, the restriction of A to the corresponding eigenspace is diagonal or in Jordan form, we have (eA )i,i = eAi,i = eai . If in addition, the elements ai are real, the measure µ in (1.5) is real and positive. Such a situation occurs in the particular case where A and B are commuting Hermitean matrices, where moreover, the support of µ is on the real axis. We shall discuss possible generalizations of these results to non commuting matrices, and in particular the following conjecture, made in 1976 by Bessis, et al. [1, 2]: Conjecture 1.1. Let A and B be Hermitean matrices, Z Z(λ) = Tr(e(A−λB) ) = e−λτ dµ(τ ) ,
(1.6)
where µ is a positive measure with support in U(B), which is the smallest closed interval of the real line which contains all eigenvalues of B. When B is semi positive definite, its spectrum is on the closed positive real axis, in which case the conjecture (1.6) says that Z(λ) is the Laplace transform of a positive measure. The conjecture is proven in [1] for 2 × 2 matrices, and of course when A and B commute. Neither any proof, nor any counterexamples is available so far for 3 × 3 matrices. Nevertheless, a good reason to believe that the conjecture might be true, comes from the connection between the exponential function of an operator
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
623
R∞ and its resolvent, which reads in our case (z − (A − λB))−1 = 0 e−uz eu(A−λB) du. More precisely, if Conjecture 1.1 holds, similar positivity properties should hold for the trace of the resolvent. In fact, the property to be a Laplace transform of a positive measure is easy to get directly for the resolvent, as shown in the next proposition. Proposition 1.1. Let A and B be Hermitean matrices. Suppose B positive definite, and let β be its unique positive definite Hermitean square root: B = β 2 . Then for real z, not an eigenvalue of A − λB, we have Z Z ∞ 1 1 = dµR (u) = e−λτ dµz (τ ) , (1.7) Rz (λ) = Tr z − (A − λB) λ−u 0 where µR is a positive measure with support on the real axis, and where µz is a R∞ z (τ ) = 0 euτ dµR (u) with support on the positive positive measure defined as dµdτ real axis. The proof makes use of the following identity: 1 1 −1 −1 = Tr β . Rz (λ) = Tr β z − (A − λB) λ − β −1 (A − z)β −1
(1.8)
If we denote ui (z) the eigenvalues (which are real) of β −1 (A − z)β −1 , and ϕi (z) the P eigenvectors, so that β −1 (A−z)β −1 ϕi (z) = ui (z)ϕi (z), we get Rz (λ) = i,j Ci,j (λ− uj (z))−1 , where the coefficients Ci,j are the square of the modulus of the matrix element of β −1 between ϕi (z) and ϕj (z). Since the coefficients Ci,j are positive, with support on one gets the first half of (1.7) with µR a discrete positive measure, R the real axis. Then one get µz through the relation dµz (τ ) = exp(uτ )dµR (u). The conjecture appeared first in the research of rigorous upper and lower bounds for the partition function in quantum statistical mechanics [1]. In fact, the same question occurs for Schr¨ odinger operators, even if additional difficulties occur in the thermodynamic limit [1]. It happens that the question is already very hard in the case of finite matrices. Even in this apparently limited case, the conjecture remains mathematically interesting, due to its simplicity, and due to the numerous unsuccessful attempts which have been performed to its resolution. In Sec. 2, we use perturbation arguments in order to express the exponential of a matrix as a multidimensional Laplace transform in the variables of its diagonal elements, and show the analyticity properties of the inverse Laplace transform. In Sec. 3, we apply these results to the case of the matrix exp(A − λB), which we express as a Laplace transform in the variable λ. In Sec. 4, we consider the case of the diagonal elements, and show that they are Laplace transform of an object which is the sum of two contributions, the first being a positive discrete measure, and the second a bounded function, piecewise analytic, and vanishing outside the convex hull of the spectrum of B. We show in Sec. 5 that the positivity assertion of Conjecture 1.1 in the Hermitean case, cannot be extended from the trace to a single diagonal element. In Sec. 6, we consider the trace case, and we discuss the relation
624
P. MOUSSA
between Hermitean matrices of dimension d and real and symmetric matrices of dimension 2d. In Sec. 7, we consider the relation between the real and complex cases in the same dimension d, through an analysis of the inverse spectral problem for finite matrices, and we provide the best known result toward Conjecture 1.1 in dimension 3. In Sec. 8, we consider a variational approach to the Conjecture 1.1, in particular, an algebraic problem is set (Question 8.1), the solution of which would in turn resolve the Conjecture 1.1. We conclude in Sec. 9. 2. Perturbation Arguments: General Setting In this section, we consider a general complex d × d matrices, which we write in a given basis as X + M , where we have separated the diagonal part X, Xi,j = xi δi,j , and the off diagonal part M , Mk,k = 0. In this section, we analyze the dependence of eX+M , as a function of the diagonal elements xi of X, and express this dependence as a kind of Laplace transform [3]. For this purpose, we shall set up a perturbation scheme for eX+M as an expansion on powers of matrix elements of M [4]. We start from a classical formula: for real and positive u we have, Z u e(u−w)X M ew(X+M) dw , (2.1) eu(X+M) = euX + 0
Z eu0 X M eu1 (X+M) δ(u0 + u1 − u)du0 du1 ,
= euX +
(2.2)
+
R where + means that the integration is restricted to the positive values of the variables. By repetitive use of (2.2), one gets Z (X+M) X =e + eu0 X M eu1 X δ(u0 + u1 − 1)du0 du1 e +
+
n Z X m=2
e
u0 X
Me
u1 X
M · · · Me
um−1 X
Me
um X
δ
+
Z +
e
u0 X
Me
u1 X
M · · · Me
un X
Me
un+1 (M+X)
δ
+
m X
! ui − 1
m Y
i=0
i=0
n+1 X
! n+1 Y
ui − 1
i=0
dui
dui .
i=0
(2.3) Therefore we get the infinite power series Z (X+M) xj 0 0 0 )j,j = e δj,j + Mj,j exp(u0 xj + u1 xj 0 )δ(u0 + u1 − 1)du0 du1 (e +
+
∞ X Z X m=2 {i}m
+
exp
m X
! un xin
n=0
× Mi0 ,i1 Mi1 ,i2 · · · Mim−1 ,im δ
m X n=0
! un − 1
m Y n=0
dun ,
(2.4)
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
625
P where the short notation {i}m is used to mean the sum from 1 to d, over each in for 1 ≤ n ≤ m − 1, and the additional boundary conditions i0 = j and im = j 0 . The terms corresponding to m = 0 and m = 1 have been separated from the remainder of the sum. This series is absolutely convergent, as expected from the properties of the exponential function, but it is useful to exhibit an explicit bound for the term of order m in (2.4). For this one easily shows that ! m−1 ! m Z Z m m−1 X Y X Y um , (2.5) δ ui − u dui = θ u− ui dui = m! + + i=0 i=0 i=0 i=0 R where θ(x) = 1 when x ≥ 0, and θ(x) = 0 when x < 0 (we recall that + means that the integration is restricted on the positive values of the variables). Now, let kxk = sup |xi | and kM k = sup |Mi,j | . i
(2.6)
i,j
The series (2.4) is absolutely convergent, since for the term of order m (m ≥ 1) in (2.4), we immediately get the bound exp(kxk)kM km dm−1 (m!)−1 . We shall reorder this series according to the number of different values taken by the numbers i0 , i1 , . . . , im in the general term. We need a few notations: Let P(d,k) be the set of the non void subsets with k elements, 1 ≤ k ≤ d, of the sets of successive integers 1, 2 . . . , d. These subsets are labeled as ordered sequences of integers ` = {l1 ,l2 . . . lk }, with 1 ≤ l1 < l2 · · · < lk ≤ d. Their total number is N (k, d) = kd . Finally, the closed convex hull in the complex plane of the corresponding diagonal elements xl1 , xl2 , . . . , xlk is denoted Uk,` . We get d X X k,` Ej,j (2.7) (e(X+M) )j,j 0 = 0 , k=1 `∈P(k,d)
with k,` Ej,j 0
=
∞ X
X Z
m=k−1 {i,k,`}m
+
exp
m X
! un xin
n=0
× Mi0 ,i1 Mi1 ,i2 · · · Mim−1 ,im δ
m X n=0
! un − 1
m Y
dun ,
(2.8)
n=0
P where now the short notation {i,k,`}m means the sum over each in , for 1 ≤ n ≤ m − 1, (with i0 = j and im = j 0 ), but with the constraint that all in ∈ ` = {l1 , l2 , . . . , lk }, for all n, such that 0 ≤ n ≤ m. The diagonal elements of M vanish, therefore the only contribution to the case k = 1 comes from m = 0. We have N (1, d) = d, and we label the various possible terms by 1 ≤ l1 ≤ d, so that 1,l1 x l1 Ej,j δl1 ,j δl1 ,j 0 . 0 = e
(2.9)
In the case k = 2, using (2.5) and some easy changes n the integration variables, one gets contributions of the following kind, Z 2,l1 ,l2 1 ,l2 Ej,j 0 = L2,l (u1 , u2 ) exp(u1 xl1 + u2 xl2 )δ(u1 + u2 − 1)du1 du2 (2.10) j,j 0 +
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P. MOUSSA
where L2,l1 ,l2 (u1 , u2 ), is a matrix which an entire function of u1 and u2 . Indeed, we define the entire functions I˜e (z) =
∞ X √ zm = I0 (2 z) , 2 (m!) m=0
and, (2.11)
√ I1 (2 z) 1 zm ˜ − , = √ Id (z) = (m + 1)(m!)2 z 2 m=1 ∞ X
where the functions I0 (z) and I1 (z) are the usual Bessel functions. Then we get for the diagonal part 1 ,l2 (u1 , u2 ) = (u1 δj,l1 + u2 δj,l2 )I˜d (u1 u2 Ml1 ,l2 Ml2 ,l1 ) . L2,l j,j
(2.12)
The off diagonal part is given by 1 ,l2 (u1 , u2 ) = (δj,l1 δj 0 ,l2 Ml1 ,l2 + δj,l2 δj 0 ,l1 Ml2 ,l1 )I˜e (u1 u2 Ml2 ,l1 Ml2 ,l1 ) , L2,l j,j 0
(2.13)
for j 6= j 0 . More generally, we have the following proposition, which gives a precise meaning to the expression of e(X+M) as an integral matrix transform of the multivariable Laplace type, namely Z (X+M) = exp(x1 u1 + · · · + xd ud )L(u1 , . . . , ud )du1 · · · dud . (2.14) e Proposition 2.1. With the notation of this section, we have (e
(X+M)
)j,j 0 =
d X X
k,` Ej,j 0
=
k=1 `∈P(k,d)
× exp
k X p=1
d X
e
x l1
δl1 ,j δl1 ,j 0 +
up xlp
δ 1−
X
k=2 {l1 ,l2 ,...,lk }
l1 =1
!
d X
k X
Z +
k Y
! dup
p=1
! up
1 ,l2 ,...,lk Lk,l (u1 , u2 , . . . , uk ) , j,j 0
p=1
(2.15) where Lk,l1 ,l2 ,...,lk (u1 , u2 , . . . , uk ) is a d × d matrix which is an entire functions of the variables u1 , u2 , . . . , uk . The sum over ` = {l1 , l2 , . . . , lk } is subjected to the condition 1 ≤ l1 < l2 · · · < lk ≤ d. Note that, in the general term of order k, ` of (2.15) the integration over the Qk variables p=1 dup is in fact restricted to a (k − 1)-dimensional hyperplane. For the proof of the proposition, we just reorder the terms of the sum over the indices i0 = j, i1 , . . . , im = j 0 in (2.8), according to the number mp of occurrence of the value lp , for 1 ≤ p ≤ k. We then perform using (2.5) the integrations over sets of variables ui corresponding to the same xlp , and we get 1 L1,l j,j 0 = δl1 ,j δl1 ,j 0 ,
(2.16)
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
627
and for k ≥ 2, k,l ,l2 ,...,lk
Lj,j 01
=
k X
(u1 , u2 , . . . , uk )
!
k X
δj,lp
p=1
×
! δj 0 ,lp
p=1 ∞ X
up11 up21
p1 ,p2 ,...,pk =1
p1 ! p2 !
p
···
ukk pk !
Mi0 ,i1 Mi1 ,i2 · · · Mim−1 ,im {i,k,`}m
X
. m=p1 +p2 +···pk −1
(2.17) In the above formula, we have m = p1 + p2 + · · · pk − 1, and the notations used P {i,k,`}m means that we perform the sum of the products Mi0 ,i1 Mi1 ,i2 · · · Mim−1 ,im over all possible values of the indices i0 , i1 , . . . , im , with m+1 = p1 +p2 +· · ·+pk , i0 = j, im = j 0 , and with the additional conditions: for 0 ≤ p ≤ m, ip ∈ {l1 , l2 , . . . , lk }, and among the indices i0 , i1 , . . . , im , there are p1 times the value l1 , p2 times the values l2 , and so forth until pk times the value lk . We easily find an absolute bound for the series (2.17) by relaxing the above mentioned additional conditions, and we get for the left-hand side of (2.17) k,l1 ,l2 ,...,lk (u1 , u2 , . . . , uk ) Lj,j 0 ≤
1 kM kd2
∞ X p1 ,p2 ,...,pk
(|uk |dkM k)pk (|u1 |dkM k)p1 (|u2 |dkM k)p2 ··· , p1 ! p2 ! pk ! =1 (2.18)
1 ,l2 ,...,lk (u1 , u2 , . . . , uk ) are which is always convergent. Therefore, the functions Lk,l j,j 0 entire functions of the arguments u1 , u2 , . . . , uk . Notice that by the same argument, these functions are also entire functions of the matrix elements of M . These functions are multivariable generalizations of I˜e and I˜d given in (2.11).
Remark 2.1. A result similar to the above proposition could be obtained using Trotter’s formula [5], namely e(X+M) = limn→∞ (e(X/n) e(M/n) )n , which we rewrite more symmetrically for convenience e(X+M) = lim (E (n) ) , n→∞
E (n) = (e(X/2n) e(M/n) e(X/2n) )n .
with
(2.19)
More precisely, we have (E (n) )j,j 0 =
d X
exp
xj + 2
i1 ,i2 ,...,in−1 =1
M × en
M
en j,i1
i1 ,i2
Xn−1 p=1
xip + xj 0
2n M ··· e n
in−1 ,j 0
.
(2.20)
628
P. MOUSSA
Therefore, grouping together the indices which take the same value, we have expressions similar to (2.15) and we get (E (n) )j,j 0 d X
=
e
x l1
δl1 ,j δl1 ,j 0 +
d X
Z
X
k=2 {l1 ,l2 ,...,lk }
l1 =1
× exp
k X
! up xlp
δ 1−
p=1
k X
k Y
+
! dup
p=1
! up
k,l ,l2 ,...,lk
(n)j,j01
L
(u1 , u2 , . . . , uk ) ,
p=1
(2.21) k,l ,l2 ,...,lk
(n)j,j01
with the difference that now the functions L analytic, but in fact singular distributions. k,l ,l2 ,...,lk
(n)j,j01
L
(u1 , u2 , . . . , uk ) are no longer
=×
(u1 , u2 , . . . , uk ) ! k ! k X X δj,lp δj 0 ,lp p=1
p=1
p1 ,p2 ,...,pk =1
M M M en × en ··· e n j,i1 i1 ,i2 in−1 ,j 0
# k Y pp X δ up − 2n p=1
"
∞ X
{i,k,`}n
! .
(2.22)
2n=p1 +p2 +···+pk
In the above equation, pp is the number of occurrence of lp among the indices i0 , i1 . . . , in , but where we count twice the values taken by the indices i1 , . . . , in−1 , and once the values taken by i0 = j, and in = j 0 . What we have learned from the perturbative arguments, is that this sequence of discrete measures behaves as if it would reach a limit with a smooth density function, when applied to exponential functions. Notice that this does not necessarily mean that this limit exists in the sense of distributions. However in some cases, one can prove that the limit exists, for instance when the measure are positive, and supported by the real axis [1]. The main interest of the present argument using Trotter’s formula, is that it gives a better feeling for the support of the measures. In addition, this arguments displays some more or less precise connections with functional integration considerations. We now end this section with a comment made to us by G. A. Mezincescu. We P can rewrite the matrix X + M as M + di=1 xi Pi , where Pi is, for 1 ≤ i ≤ d, the projector such that (Pi )j,k = δj,i δk,i . Thus X + M also represents a perturbation of M by addition of a linear combination, with coefficients xi , of matrices Pi which commute with one another. It is natural to ask whether a representation of the type (2.14), as a multidimensional Laplace transform of a measure, holds for the Pd expression Tr exp(M + i=1 xi Bi ), where now the Hermitean matrices Bi do not all commute with each other (that is, there exist i and j such that [Bi , Bj ] 6= 0).
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
629
The answer is in general no, and more precisely, it is easy to provide examples of P2 two-dimensional matrices for which Tr exp(M + i=1 xi Bi ) is a multidimensional Laplace transform of the type (2.14), not of a measure, but of a higher order distribution. 3. Perturbation Arguments: The exp(A − λB) Case Using (1.2), the results of the previous section apply in basis B, to the case considered in Sec. 1: exp(A − λB) = exp(X(λ) + M ). We can rewrite (2.15) as follows: (E(λ))j,j 0 = (e(A−λB) )j,j 0 =
d X X
k,` Ej,j 0 (λ)
k=1 `∈P(k,d)
≡
d X
X
k,l1 ,l2 ,...,lk Ej,j (λ) , 0
(3.1)
k=1 {l1 ,l2 ,...,lk }
which means explicitly for k = 1 ,
1,l1 −λbl1 al1 Ej,j e δl1 ,j δl1 ,j 0 , 0 = e
for k > 1 ,
k,l1 ,l2 ,...,lk (λ) Ej,j 0
Z
k Y
= +
(3.2) !
dup
1−
δ
p=1
× exp
k X
! up
exp −λ
p=1
k X
k X
! b lp u p
p=1
! 1 ,l2 ,...,lk alp up Lk,l (u1 , u2 , . . . , uk ) . j,j 0
p=1
(3.3) Therefore, we easily get the following representation for (E(λ))i,j : Proposition 3.1. We have Z (E(λ))j,j 0 = (e(A−λB) )j,j 0 = Z (E(λ))k,` j,j 0
=
e−λτ dµj,j 0 (τ ) ,
e−λτ dµk,` j,j 0 (τ ) ,
(3.4)
(3.5)
where the measure density associated to µj,j 0 (τ ) can be written as X d µj,j 0 (τ ) = dτ d
X
k=1 {l1 ,l2 ,...,lk }
with for k = 1,
d l1 ,l2 ,...,lk µ 0 (τ ) , dτ j,j
(3.6)
630
P. MOUSSA
d X d 1,l1 µj,j 0 (τ ) = δl1 ,j δl1 ,j 0 eal1 δ(τ − bl1 ) , dτ
(3.7)
l1 =1
and for k > 1, d l1 ,l2 ,...,lk µ 0 (τ ) = dτ j,j
Z +
!
k Y
dup
δ τ−
p=1
× exp
k X p=1
k X
! b lp u p δ
1−
k X
! up
p=1
! 1 ,l2 ,...,lk alp up Lk,l (u1 , u2 , . . . , uk ) , j,j 0
(3.8)
p=1
where Lk,l1 ,l2 ,...,lk (u1 , u2 , . . . , uk ), given by (2.17), is a d × d matrix which is an entire functions of the arguments u1 , . . . , uk , and of the matrix elements of M . When all bi differ, the decomposition of the measure dµj,j 0 as a sum over k and ` shows that the term k = 1 provides singular Dirac measures at the eigenvalues of B, and the terms k > 1 provide measures with support in the corresponding Uk,` (B), which is the closed convex hull of the eigenvalues bl1 , bl2 , . . . , blk of B. In the general situations, U2,l1 ,l2 (B) is an interval of a straight line joining the two eigenvalues bl1 , bl2 , and for k > 2, Uk,` (B) is a closed convex polygon with non empty interior, and the corresponding contribution to the measure has a two-dimensional support, and a piecewise analytic density. Of course, some of these polygons may collapse to intervals when more than two eigenvalues belong to the same straight line in the complex plane, a situation which happens for example when all eigenvalues of B are real. When some eigenvalues of B coincide, it is convenient to rewrite (3.6) and (3.8) so that the integrations to the variables ui which correspond to equal eigenvalues of B are performed. More precisely we shall reorder the sums over k and ` in (3.6) according to the number of different values taken by the numbers bl0 , bl1 , . . . , blk . The following proposition is an immediate consequence of Proposition 3.1, although it is rather cumbersome to write. Proposition 3.2. Let Sδ be the set of values taken by the eigenvalues of B, and let δ, be the number of its elements, that is the number of different values effectively taken by b1 , b2 , . . . , bd . We have 1 ≤ δ ≤ d, and we call eb1 , eb2 , . . . ebδ these values, Pδ 0 with respective degeneracies de1 , de2 , . . . deδ , so that i=1 dei = d. Let now Sk` 0 be all the possible non void subset of Sδ , labeled by the number of elements k 0 , 1 ≤ k 0 ≤ δ, and 0 `0 = {l1 , l2 , . . . , lk0 }, so that Sk` 0 = {ebl1 , ebl2 , . . . , eblk0 } with 1 ≤ l1 ≤ l2 · · · ≤ lk0 ≤ δ 0 0 (the number of possible Sk` 0 for a given k 0 is N (δ, k 0 ) = kδ0 ). We define Ik`0 as the 0 0 set of indices i, 1 ≤ i ≤ d, such that the eigenvalue bi ∈ Sk` 0 , therefore, if i ∈ Ik`0 , there exists one and only one r, 1 ≤ r ≤ k 0 , such that eblr = bi . Finally, we define 0 0 again Uk`0 (B) ⊂ U(B) as the convex hull of the eigenvalues contained in Sk` 0 . Then Eq. (3.4) holds with
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
δ X dµj,j 0 (τ ) = dτ 0
Z
X
+
k =1 {l1 ,l2 ,...,lk0 }
× δ τ −
dwp
p=1
0
k X
0
k Y
631
0
ebl wp δ 1 − p
p=1
k X
k0 ,l ,l2 ,...,lk0
˜ 01 wp L j,j
(w1 , w2 , . . . , wk0 ) ,
p=1
(3.9) 0
˜ k ,l1 ,l2 ,...,lk0 (w1 , w2 , . . . , wk0 ), is for 1 ≤ k 0 ≤ δ, an entire where the d × d matrix L functions of the matrix elements of M, and of the arguments w1 , . . . , wk0 . The support of the contribution of the term {k 0 , `0 } to the measure dµj,j 0 (τ ) is contained 0 0 0 0 ˜ k ,`0 ≡ L ˜ k ,l0 1 ,l2 ,...,lk0 (w1 , w2 , . . . , wk0 ) in the set Uk`0 (B) ⊂ U(B), and the functions L j,j j,j are given by Equation (3.10) below. 0
0
˜ k ,`0 for 1 ≤ k 0 ≤ δ, starting from (2.15), (2.16), One can obtain the functions L j,j k,l1 ,l2 ,...,lk (u1 , u2 , . . . , uk ), and (2.17), which make use of the functions Lk,` j,j 0 ≡ Lj,j 0 with 1 ≤ k ≤ d. We have to perform some integrations with respect to the group of variables up which corresponds to the same eigenvalues blp . For this purpose, it is convenient to define for 1 ≤ k 0 ≤ δ, the set σ(k 0 , `0 ) ≡ σ(k 0 ; l1 , l2 , . . . , lk0 ) as the set 0 of couples {k, `} ≡ {k; l1 , l2 , . . . , lk } with 1 ≤ k ≤ d, such that {l1 , l2 , . . . , lk } ⊂ Ik`0 , we get easily (3.9) with 0
˜ k ,l0 1 ,l2 ,...,lk0 (w1 , w2 , . . . , wk0 ) L j,j =
Z
X
+
{k,`}∈σ(k0 ,`0 )
× exp
k X
k Y p=1
! k0 X Y dup δ wr − ui {i;r}
r=1
! 1 ,l2 ,...,lk alp up Lk,l (u1 , u2 , . . . , uk ) , j,j 0
(3.10)
p=1
P
0 denotes the sum over i such that li ∈ Ik`0 , and bli = eblr . This equation 0 ˜ k ,l0 1 ,l2 ,...,lk0 have the requested analyticity properties, shows that the functions L j,j since the sum over {k, `} is over a finite set. It is convenient to give another form, directly derived from (2.4) and (2.8). We have δ X X k0 ,`0 (3.11) E˜j,j (e(A−λB) )j,j 0 = 0 (λ) ,
where
{i;r}
k0 =1 `0 ≡{l1 ,l2 ,...,lk0 }
with 1 ≤ k 0 ≤ δ, and 1 ≤ l1 ≤ l2 · · · ≤ lk0 ≤ δ. Note that in the case where B 0 0 ˜ k ,`0 (λ) differ from the set has some degenerate eigenvalues, the set of functions E j,j k,` Ej,j 0 (λ) of Eq. (3.1). Then we write 0
0
k ,` E˜j,j 0 (λ) =
Z
0
0
e−λτ d˜ µkj,j,`0 (τ ) ,
(3.12)
632
P. MOUSSA
with 0
0
d˜ µkj,j,`0 (τ ) dτ
=
∞ X
X
Z Y m
m=(k0 −1) {i,k0 ,`0 }m
×δ τ −
m X n=0
dun δ
+ n=0
! un bin
m X
exp
! un − 1
n=0 m X
! un ain
Mi0 ,i1 Mi1 ,i2 · · · Mim−1 ,im ,
n=0
(3.13) P where now the short notation {i,k0 ,`0 }m means the sum over each in for 1 ≤ n ≤ m−1 (with the condition i0 = j and im = j 0 ), but with the additional constraint that 0 0 all in for 0 ≤ n ≤ m are such that in ∈ Ik`0 , (which means bin ∈ Sk` 0 , as described in Proposition 3.2). One could have written directly Eqs. (3.12) and (3.13), but it helps to go through Propositions 2.1, 3.1 and 3.2, in order to show that (3.13) is 0 0 ˜ k ,`0 . equivalent to (3.9) with the required analyticity properties of L j,j We shall now give explicit expressions of (3.13) for k 0 = 1. In basis BJ , we easily obtain the diagonal elements, for 1 ≤ l ≤ δ, and 1 ≤ j ≤ d, X (τ ) d˜ µ1,l j,j = exp ai δ(τ − bl ) . (3.14) dτ l i∈I1
In basis BJ , the off diagonal part is easy to write only if the restrictions of A, (and also of M ), to the eigenspaces of B are all diagonal. In that case, there are no nontrivial Jordan blocks, and one just obtains X (τ ) d˜ µ1,l j,j 0 = δj,j 0 exp ai δ(τ − bl ) . (3.15) dτ l i∈I1
Remark 3.1. It is possible to write a closed form version of the {k 0 , `0 } decom0 position of Eq. (3.11). We define the characteristic function χ`k0 (i) of the set of 0 0 0 indices Ik`0 introduced in Proposition 3.2, namely χ`k0 (i) = 1 (resp. 0) when i ∈ Ik`0 , 0 0 0 0 0 (resp. i 6∈ Ik`0 ). Then we define the “restricted matrices” A{k ,` } and B {k ,` } , 0 0 {k0 ,`0 } = χ`k0 (j)χ`k0 (j 0 )Aj,j 0 and through their matrix elements in basis B, namely Aj,j 0 {k0 ,`0 }
0
0
Bj,j 0 = χ`k0 (j)χ`k0 (j 0 )Bj,j 0 . Indeed we have just replaced by zero all matrix elements for which one of the indices (at least) does not correspond to an eigenvalue of 0 0 B which belongs to the set selected by the label {k 0 , `0 }. As a consequence, A{k ,` } 0 0 and B {k ,` } , are still d × d matrices. By comparison of the expansions (3.13), using (3.12), one easily gets the d × d matrix equation X 0 0 0 0 00 00 ˜ k0 ,`0 (λ) + (3.16) E˜ k ,` (λ) , exp(A{k ,` } − λB {k ,` } ) = E k00 ,`00
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
633
P 00 0 where the sum k00 ,`00 is taken over the the values of k 00 and `00 such that Ik`00 ⊂ Ik`0 00 0 (but with Ik`00 6= Ik`0 ). With the same summation convention, we get ˜ k0 ,`0 (λ) = exp(A{k0 ,`0 } − λB {k0 ,`0 } ) + E
X
0
00
(−1)k −k exp(A{k
00
,`00 }
− λB {k
00
,`00 }
).
k00 ,`00
(3.17) From the above definitions, if we erase the lines and columns which contains only 0 0 0 0 vanishing elements, exp(A{k ,` } −λB {k ,` } ) can be reduced to a dk0 ,`0 ×dk0 ,`0 matrix, Pk 0 f 0 0 0 0 where dk0 ,`0 = i=1 dli . In some sense, E˜ k ,` (λ) is the contribution to exp(A{k ,` } − 0 0 λB {k ,` } ) which is irreducible, meaning here that it cannot be reduced to lower dimension. We finally observe that if we write (3.16) for k 0 = δ, we have only one choice for `0 = {1, 2, . . . , δ}, and (3.16) provides the decomposition of exp(A−λB) ≡ exp(A{δ,1,2,...,δ} − λB {δ,1,2,...,δ} ) into irreducible contributions. From the previous Propositions 3.1 and 3.2, one obtains using (2.17) the following immediate basic positivity result: Proposition 3.3. In basis B, let B be a complex diagonal matrix, and assume first, that all diagonal matrix elements of A are real, and second, that all off diagonal elements of A (and therefore of M ) are real and positive. Then the measures µj,j 0 , 0 µk,` j,j 0 of Proposition 3.1 are positive measures for any j, j ∈ {1, 2, . . . , d}, and for 0
0
any k, `, with 1 ≤ k ≤ d. In the degenerate case, the measures µ ˜kj,j,`0 introduced in (3.11) and (3.12), are also positive measures for any j, j 0 ∈ {1, 2, . . . , d}, and any k 0 , `0 , with 1 ≤ k 0 ≤ δ. As it has been stated, this result looks a bit anecdotic. However, it will later appear useful for the analysis of the trace Z(λ) defined in (1.4). Indeed, since the trace does not depend on the basis, it is sufficient to have the required positivity in one basis. We shall now consider in more details the case of diagonal elements, and the case of the trace. 4. Diagonal Matrix Elements: Cycle Description The diagonal elements of a matrix M are invariant under the change of basis of the kind M → D−1 M D, where D is an arbitrary diagonal invertible matrix with complex elements. Now we will define the k-cycles of matrix elements of M , as the non-vanishing products Mi1 ,i2 Mi2 ,i3 · · · Mik ,i1 , where the numbers i1 , i2 , . . . , ik are all differents. These cycles are invariant under the above mentioned transformation M → D−1 M D. Such a cycle may exist only for 2 ≤ k ≤ d (we recall that we have assumed that M has vanishing diagonal elements). We will denote the k-cycles as [i1 , i2 , . . . , ik ] = Mi1 ,i2 Mi2 ,i3 · · · Mik ,i1 . (4.1) Of course, two k-cycles which differ only by a cyclic permutation of the indices are identical, and therefore will be identified. The matrix M has d(d − 1) complex
634
P. MOUSSA
parameters (because it is diagonal free). If one count the numbers of matrices up to a transformation M → D−1 M D, it remains d(d − 1) − (d − 1) = (d − 1)2 complex parameters. The number of different 2-cycles is d(d − 1)/2, and the number of 3-cycles is d(d − 1)(d − 2)/3, since we have to distinguish cycles as [1, 2, 3] and [1, 3, 2] which are not cyclically equivalent. The total number of 2 and 3-cycles is therefore d(d − 1)(2d − 1)/6 and exceeds the number (d − 1)2 of independent complex parameters. There should therefore exist d(d − 1)(2d − 1)/6 − (d − 1)2 = (d − 1)(d − 2)(2d − 3)/6 relations between the 2 and 3-cycles, and it is possible to select one subset of the 3-cycles so that together with the 2-cycles, it forms a complete set of (d − 1)2 independent variables. However, when we express these variables in terms of the matrix elements Mi,j , singularities may occur for some Mi,j = 0. In the following lemma, we give a simple complete set of cycles with order not limited to three, which also will display some singularities for vanishing matrix elements. Lemma 4.1. For diagonal free matrices, a complete set of cycles is made of (1) the d(d − 1)/2 different 2-cycles, (2) for each k such that 3 ≤ k ≤ d, the k-cycles given for 1 ≤ q ≤ d − k + 1 by [q, q + 1, . . . , q + k − 1], together with the additional k-cycle [d − k + 2, d − k + 3, . . . , d, 1], that is (d − k + 2) different k-cycles for each k. We have therefore a number of (d − 1)(d − 2)/2 cycles of order greater or equal to 3, and a total number equal to (d − 1)2 . The statement is valid for complex matrices, up to a complex diagonal transformation, and leads to (d − 1)2 complex parameters. It holds for real matrices, up to a real diagonal transformation, in which case it leads to (d − 1)2 real parameters. In the case of Hermitean matrices, up to a diagonal unitary transformation, a complete set of real parameters is made of the d(d − 1)/2 different real 2-cycles and the arguments (d − 1)(d − 2)/2 of the complex cycles of order greater or equal to 3. For the proof, one observes first that the knowledge of the 2-cycles allows to deduce Mj,i once Mi,j is known. The arbitrariness of the basis allows to choose as wished the first upper parallel to the diagonal (that is Mi,i+1 ). The second upper parallel (that is Mj,j+2 ) is obtained from the known 3-cycles, and at each step, one obtains recursively the kth upper parallel from the known k + 1-cycles. If one want to obtain a set of parameters which contains only 2-cycles and 3-cycles, one can use recursively identities of the form [123 · · · k] × [1(k − 1)] = [12 · · · (k − 1)] × [1(k − 1)k] . Such an identity express k-cycles in terms of k − 1-cycles, 2-cycles, and 3-cycles. We will now rewrite Proposition 3.2 in the form (3.12) and (3.13) for the diagonal matrix elements, which leads to the following proposition, where the degeneracy of eigenvalues of B is permitted. Of course, one recovers the non-degenerate case by setting δ = d in the proposition.
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
635
Proposition 4.1. With the notations of Propositions 3.1 and 3.2, for 1 ≤ j ≤ d, in basis B, we have (e(A−λB) )j,j =
d X
X
δ X
k,` Ej,j (λ) =
X
0
0
˜ k ,` (λ) , E j,j
(4.2)
k0 =1 `0 ≡{l01 ,l02 ,...,l0 0 }
k=1 `≡{l1 ,l2 ,...,lk }
k
0
with 1 ≤ k ≤ d, 1 ≤ l1 ≤ l2 · · · ≤ lk ≤ d, and with 1 ≤ k ≤ δ, 1 ≤ l10 ≤ l20 · · · ≤ lk0 0 ≤ δ. Moreover, Z Z 0 0 k,` k,` k0 ,`0 −λτ ˜ dµj,j (τ ) Ej,j (λ) = e−λτ d˜ µkj,j,` (τ ) , (4.3) Ej,j (λ) = e with dµk,` j,j (τ ) = dτ
m X Z Y
∞ X
m=(k−1) {i,k,`}m
×δ τ −
m X
dun δ
+ n=0
! un − 1
n=0
! un bin
m X
m X
exp
n=0
! Mj,i1 Mi1 ,i2 · · · Mim−1 ,j ,
un ain
n=0
(4.4) and 0
0
d˜ µkj,j,` (τ ) = dτ
∞ X
X
Z Y m
m=(k0 −1) {i,k0 ,`0 }m
×δ τ −
m X n=0
dun δ
+ n=0
! un bin
exp
m X
! un − 1
n=0 m X
! un ain
Mj,i1 Mi1 ,i2 · · · Mim−1 ,j .
n=0
(4.5) P
In (4.4) the short notation {i,k,`}m means the sum over each in for 1 ≤ n ≤ m−1, but with the constraint that all in for 1 ≤ n ≤ m−1 are such that in ∈ {l1 , l2 , . . . , lk }, these contributions being included in the sum only if we also have j ∈ {l1 , l2 , . . . , lk }. P On the other hand, in (4.5) the short notation {i,k0 ,`0 }m means the sum over each in for 1 ≤ n ≤ m − 1, but with the constraint that all in for 1 ≤ n ≤ m − 1 are such 0 0 that in ∈ Ik`0 , that is bin ∈ Sk` 0 , these contributions being included in the sum only if 0 0 0 0 0 0 we also have j ∈ Ik`0 , that is bj ∈ Sk` 0 . The support of the measures µkj,j,` and µ ˜kj,j,` 0 are contained in the sets Uk` (B) and Uk`0 (B) respectively, both contained in U(B). A positivity property which reminds Proposition 3.3 is given now. Proposition 4.2. In basis B, let B be a complex diagonal matrix, and assume first, that all diagonal matrix elements of A are real, and second, that all 2-cycles of M 2,l1 ,l2 1 of Propositions 3.1 and are real and positive. Then the measures µ1,l j,j , and µj,j 4.1 are positive measures for any j ∈ {1, 2 . . . , d}, and for any l1 , l2 ∈ {1, 2, . . . , d}. If in addition for 2 < k ≤ d, all k-cycles are real and positive, then all measures ˜k,` µj,j , µk,` j,j , µ j,j are also positive.
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Notice here if some of the cycles vanish, the others remaining positive, the last sentence of the proposition (positivity of the mentioned measures) remains true. It is just a bit more intricate to describe their support more precisely than the statements of Propositions 3.1 and 3.2. We apply now the above proposition to the case where A (and therefore M ) 2,l1 ,l2 1 defined in (3.5), and are Hermitean. In dimension 2, all measures µ1,l j,j , µj,j therefore the measures µj,j and µ defined in (1.5) are also positive. If in addition, the eigenvalues bi are real (which happens when B is also Hermitean), all these measures have their support contained in U(B), which is an interval of the real axis. This solves Conjecture 1.1 in dimension 2. Moreover, the positivity result extends to the diagonal elements separately, and is not only true for the trace. 5. Diagonal Elements: Hermitean Case In the case where A is Hermitean, its diagonal elements are real, and moreover, all the 2-cycles of M are real and positive. However, we do not have a similar 1 ,l2 ,l3 is not controlled. If control on the 3-cycles. As a consequence, the sign of µ3,l j,j the eigenvalues of B are complex, and not in a special position (such as all on the same line), there is no chance to compensate the possible negative contributions 1 ,l2 ,l3 1 ,l2 by positive contributions coming from µ2,l , since the supports are to µ3,l j,j j,j different (and have different dimensions). When all the eigenvalues of B are on a same single straight line (which is the case when B is Hermitean), the negative contributions may be compensated. However, we will show an example where it is not the case. More precisely, following [6], we will give an example in dimension 3, where some of the measures µj,j are not positive definite. Note that the off diagonal elements µj,j 0 may be non positive and even complex already in dimension 2, as easily sean from (2.13) and (3.3): it is sufficient to consider the case where Mj,j 0 is negative (or even complex!), and small enough in modulus. From (4.4), or even directly from (2.4), we have d µj,j (τ ) = eaj δ(τ − bj ) dτ +
X
Z |Mij |2 +
i,i6=j
+
∞ X
u0 eu0 aj +u1 ai δ(u0 + u1 − 1)δ(u0 bj + u1 bi − τ )du0 du1 Z
X
δi0 ,j
m=3 {i0 ,i1 ,...,im−1 }
×δ
m−1 X n=0
! un − 1 δ
exp +
m−1 X n=0
m−1 X
! un ain
Mi0 ,i1 Mi1 ,i2 · · · Mim−1 ,i0
n=0
! un bin − τ
u0
m−1 Y
dun .
(5.1)
n=0
In the case where all eigenvalues of B are different, we get after some straightforward calculations, the expansion up to the the third order in the matrix elements of M ,
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
637
X d bi − τ µj,j (τ ) = eaj δ(τ − bj ) + Mji Mij dτ b i − bj i
bi aj − bj ai + τ (ai − aj ) bi − bj
χ[bi ,bj ] (τ ) |bi − bj | X a i − ak 2 Mji Mik Mkj (bi − bk ∆ijk ) exp aj + (τ − bj ) + bi − bk × exp
i,k
χ[b ,b ]∪[b ,b ] (τ ) × i j j k |bi − bk |
( 1+
∆ijk (1 − v+ ) bi − bk
exp
∆ijk v+ bi − bk
) ∆ijk (1 − v− ) ∆ijk v− exp + O(M 4 ) , − 1+ bi − bk bi − bk
(5.2)
where χ[bi ,bj ] (u) is the characteristic function of the closed interval [bi , bj ], ∆ is the following determinant: 1 1 1 (5.3) ∆ijk = ai aj ak , b bj bk i and the quantities v+ and v− which depends upon τ are the respective upper and lower limit of the remaining integration in the variable v = 1 − u0 , after elimination of the variables u1 and u2 by the mean of the delta functions. We must consider two different cases. If bj ∈ [bi , bk ], meaning either bi < bj < bk , or bk < bj < bi , we get a non vanishing contribution to the integral only if τ ∈ [bi , bk ] and we have τ − bj τ − bj . (5.4) , v+ = 1 , v− = sup bk − bj bi − bj / [bi , bk ], we get a non vanishing contribution only if τ ∈ If on the contrary, bj ∈ [bj , bk ] ∪ [bj , bi ], and τ − bj τ − bj τ − bj τ − bj , v− = inf . (5.5) , , v+ = inf 1, sup bk − bj bi − bj bk − bj bi − bj From (5.2), one sees that when the off diagonal part M is small, we get from the second order, a positive contribution on all intervals [bi , bj ]. So we would get the feeling that in the neighborhood of the set of diagonal matrices A, we would get always positive measures for the diagonal elements µj,j . We shall see now that this not true, as shown in the following example. We consider the case where A and B are three-dimensional Hermitean matrices. In order to simplify the calculations, we will assume more precisely, that the diagonal part of A vanishes, that is ai = 0, Mij = Mji , and b2 > b1 > b3 , 0 M12 M13 0 0 b1 0 M23 B = 0 b2 0. A = M21 (5.6) M31 M32 0 0 0 b3
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We will now assume all Mi,j small, but not of the same order, namely we will set M12 = ε2 m12 ,
M23 = εm23 ,
M31 = εm31 ,
(5.7)
with ε real and positive. We reorder the perturbation expansion according to powers d µ11 (τ ) comes only from the contribution M13 M31 , of ε. The ε2 contribution to dτ and therefore vanishes outside [b3 , b1 ]. As a consequence, the measure density on [b1 , b2 ] has an ε4 contribution as its lower term. This contribution is a sum of the 2cycle contribution |M12 |2 , the 3-cycles contributions M12 M23 M31 and M13 M32 M21 , and of the fourth-order contribution (in M ), coming from |M13 |2 |M23 |2 . In order to display a simple example, we will take b1 = 0 ,
so that
b2 = +1 , −λ
A − λB = ε2 m21 εm31
b3 = −1 ,
ε2 m12 1−λ εm32
εm13
(5.8)
εm23 . −1 − λ
(5.9)
After some calculations, we get the following ε-expansion: dµ11 = δ(τ ) + ε2 |m13 |2 (1 + τ )χ[−1,0] (τ ) + ε4 (|m12 |2 (1 − τ )χ[0,1] (τ )) dτ (1 − |τ |)2 + ε4 χ[−1,+1] (τ ) 2 Re(m12 m23 m31 ) 4 + ε4 |m13 |2 |m23 |2
(1 − |τ |)2 ((1 − |τ |)χ[−1,+1] (τ ) + 6|τ |χ[−1,0] (τ )) + O(ε5 ) . 24 (5.10)
So we have the following proposition: Proposition 5.1. For ε small enough and positive, one can find matrices of the d µ11 is negative on some subinterval of form (5.9) such that the measure density dτ the interval [0, 1]. For the proof, we first set (with Ψ real), Re (m12 m23 m31 ) = |m12 km23 km31 | cos Ψ, and we observe that on (0, 1), the density (5.10) is reduced to 1−τ dµ11 = ε4 (1 − τ ) |m12 |2 + |m12 km23 km31 | dτ 2 |m31 |2 |m23 |2 (1 − τ )2 × cos Ψ + 6 4
! + O(ε5 ) .
(5.11)
The above expression in between parentheses, is a second degree trinomial in the variable (1 − τ )/2, which can be negative only if the discriminant is positive, that is if (5.12) |m12 |2 |m23 |2 |m31 |2 (cos2 Ψ − (2/3)) > 0 .
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
639
In that case, the trinomial is negative between the two real roots, and the half sum of the roots is (1 − τ )/2 = −3 cos Ψ|m12 km31 |−1 |m23 |−1 ), which can take any real positive value, provided that cos Ψ < 0. For instance, take cos Ψ = −1, and 6|m12 | = |m31 km23 |, then in the vicinity of τ = 0, the measure is negative for τ > 0 and sufficiently small ε. An explicit example is m12 = 1, m23 = 2, m31 = −3, in√which case the trinomial becomes just (3τ 2 − 1)/2, which is negative for 0 < τ < 3/3. For each value τ in this interval, there exist an ε0 (τ ) such that for 0 < ε < ε0 (τ ), the density (5.11) is strictly negative. One could get effective values on this ε0 (τ ), using estimates of the same type as those of Sec. 2. Indeed, in addition to (2.5), one would have to bound integrals of the type ! ! m Z m m X X Y δ ui − u δ b mi u i − τ dui +
i=0
i=0
Z θ u−
= +
≤K
−1
m−1 X
i=0
! ui δ
i=0
Z θ u− +
m X
(bmi − bm1 )ui − τ
i=0 m−2 X i=0
ui
! m−2 Y
! m−1 Y
dui
i=0
dui = K −1
i=0
um−1 , (m − 1)!
(5.13)
where K = supi |bmi −bm1 |. By symmetry, one can extend this to K = supi6=j |bmi − bmj |. This bound is valid as long has there are at least two different values among the numbers bm1 , bm2 · · · bmm . Note that the estimates (5.13) also allow a direct proof of the piecewise analyticity properties of the measure density dµi,j /dτ of Eqs. (3.4), (3.6) and (3.8). Remark 5.1. It is unclear whether such examples, where the diagonal elements do not display a representation with a positive measure, do occur in the degenerate case even when δ = 2. A good example to treat would be the case b1 0 0 0 0 M13 M14 a1 0 0 0 0 b1 a2 M23 M24 . (5.14) , B = A= 0 0 0 b3 M31 M32 a3 0 0 0 0 b3 0 a4 M41 M42 In this case, all 3-cycles vanish, but there is a non trivial 4-cycle whose sign is not controlled, namely M13 M32 M24 M41 . 6. The Trace Case: Hermitean Versus Real Case For completeness, we begin this section by quoting the formula equivalent to (5.1) in the trace case. We have Z e−λτ dµ(τ ) , (6.1) Z(λ) = Tr(e(A−λB) ) = U(B)
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with X d µ(τ ) = eaj δ(τ − bj ) dτ j=1 d
+
1X |Mij |2 2 i,j
∞ X 1 + m m=3
×δ
m−1 X
Z eu0 aj +u1 ai δ(u0 + u1 − 1)δ(u0 bj + u1 bi − τ )du0 du1 +
Z
X
exp
{i0 ,i1 ,...,im−1 }
! un − 1 δ
n=0
+
m−1 X n=0
m−1 X
! un ain
Mi0 ,i1 Mi1 ,i2 · · · Mim−1 ,i0
n=0
un bin − τ
! m−1 Y
dun .
(6.2)
n=0
We will now show the following: Proposition 6.1. Assume Conjecture 1.1 holds with the further assumption that A and B are arbitrary real and symmetric matrices of real dimension 2d, then Conjecture 1.1 holds for arbitrary Hermitean matrices A and B of complex dimension d. This is an immediate consequence of the following lemma, which relates complex matrices with dimension d and real matrices with dimension 2d. Lemma 6.1. Let A = R + iI be the decomposition of the complex d × d matrix into its real part R and imaginary part I, and let v = φ + iψ the corresponding decomposition of the d-dimensional complex vector v. We have Av = (Rφ − Iψ) + i(Iφ + RΨ) = φ0 + iψ 0 . So that ! ! ! ! R −I φ0 φ φ = ≡A . (6.3) ψ0 I R ψ ψ Therefore, (6.3) induces an homomorphism A → A from the algebra of the complex d-dimensional matrices to the algebra of the real 2d-dimensional matrices. We have the following relation between the characteristic polynomials of A and A (A¯ is the complex conjugate of A): ¯ = | det(E − A)|2 (last equality for E real ) . det(E − A) = det(E − A) det(E − A) (6.4) This shows that the set eigenvalues of A is made of the real eigenvalues of A, with twice their multiplicity, and of the complex eigenvalues of A, together with their complex conjugates, and with the same multiplicity. When A is Hermitean, its real part R is symmetric, and its imaginary part I is antisymmetric, so that A is real and symmetric.
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
641
The correspondence A → A is an homomorphism between algebras over the real numbers, and the multiplication of A by a non zero complex number induces an automorphism of the real algebra of 2d-dimension real matrices. An obvious consequence of the previous lemma is: For A and B Hermitean, and λ real, we have Tr(exp(A − λB)) = 2 Tr(exp(A − λB)) , (6.5) where A and B are the 2d-correspondents of A and B, as a consequence, we get Proposition 6.1. The only not evident part of the lemma is Eq. (6.4) which we prove now. We apply to det(E − A) the bloc diagonal decomposition of the determinant (see Lemma 6.2 below), namely for det E 6= 0, ! C D (6.6) det = det(CE −1 F E − DE) , E F which gives when det I 6= 0, det(E − A) = det((E − R)I −1 (E − R)I + I 2 ) = det((E − R − iI)I −1 (E − R + iI)I) = det(E − R − iI) det(E − R + iI) ¯ , = det(E − A) det(E − A)
(6.7)
which is the expected result. The case det I = 0 is obtained by continuity. The missing lemma follows. Lemma 6.2. Let A=
C
D
E
F
!
be the bloc diagonal decomposition of the 2d-dimensional real matrix A into ddimensional real matrices C, D, E, F . If det E 6= 0, we have det A = det(CE −1 F E − DE) = det E det(CE −1 F − D) ,
(6.8)
and if in addition det C 6= 0, det A = det C det E det(E −1 F − C −1 D) . For the proof we first recall a classical result when E = 0, namely ! C D det = det(CF ) = det C det F , 0 F
(6.9)
(6.10)
which is easily checked by examination of the expansion of the determinants. Then, when E is invertible, we have the bloc product decomposition of the matrix ! ! ! C D C 0 1 C −1 D . (6.11) = 1 E −1 F E F 0 F
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Taking the determinant, we get det
C E
D F
! = det C det F det
= det C det F det
1 1
C −1 D E −1 F
!
1
C −1 D
0
E −1 F − C −1 D
! ,
(6.12)
which gives (6.9). Equations (6.8) follows immediately. Remark 6.1. The following situation happens: Let A be a complex d-dimensional Hermitean matrix having only real and positive cycles (see Sec. 4). For such a matrix, Proposition 4.2 states that Conjecture 1.1 holds. However, the corresponding real 2d-dimensional matrix A usually displays cycles with alternating signs (not all positive). Therefore we get examples for which Conjecture 1.1 holds, although the positivity property of cycles fails. Example are available even in dimension / {±1, ±i}. We get for the non vanishing off-diagonal eled = 2, with M12 = eiθ ∈ ments of A, M12 = M34 = cos θ, and M23 = −M14 = sin θ, so that the 4-cycle M12 M23 M34 M41 = −(cos θ sin θ)2 is negative. 7. The Trace Case: The Inverse Problem for Finite Matrices The previous section shows that we can get (up to a positive factor) the same function E(λ) for different sets of matrices A and B, of different dimensions (namely d and 2d). In this section, we will show that one can get the same result with different sets of matrices A and B, having the same dimension d. At first sight, the number of real parameters for pairs of complex Hermitean matrices A and B up to an equivalence is given by d2 + 1, counted as follows: B is diagonalized, so that we have d diagonal elements. The diagonal elements of A are real, and, by adjusting the relative phases of the basis vectors, one can get the first upper diagonal of A also real, that means 2d − 1 real parameters. It remains for the other off diagonal elements of A, (d − 1)(d − 2)/2 complex elements and the total amounts to d2 + 1 as mentioned. In the real symmetric case, we have only d2 + 1 − (d − 1)(d − 2)/2 = d(d + 3)/2 real parameters. However, the matrix A − λB is also characterized up to an equivalence by its characteristic polynomial det(E − (A − λB)), which is a polynomial in both the eigenvalue E and in λ. More precisely we have Ξ(E, λ) = det(E − (A − λB)) = Ed +
d X
Pk (λ)E d−k
k=1
= Ed +
d X k X k=1 n=0
n d−k p(k) . n λ E
(7.1)
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
643
In the case of complex Hermitean matrices, the polynomial Pk (λ) is real for real λ and of degree k, and therefore contains k + 1 real parameters. Therefore, the Pd number of real parameters is k=1 (k + 1) = d(d + 3)/2 even in the complex case. The discrepancy between the two ways of counting comes from the fact that in the first evaluation we classify pairs of Hermitean matrices up to an equivalence A → A0 = U AU −1 , and B → B 0 = U BU −1 , with U a constant unitary matrix, whereas in the second evaluation, we classify linear combinations A − λB, up to an equivalence A − λB → A0 − λB 0 = U (λ)(A − λB)U −1 (λ), where now the unitary matrix may depend on λ. In this case, we get larger equivalence classes, because now A → A0 = U (0)AU −1 (0), and B → B 0 = U (∞)BU −1 (∞). But these larger equivalent classes are that we need for our purpose, since the function E(λ) in (1.3) depends only on the eigenvalues of A − λB. In view of the counting of the parameters, we formulate the following conjecture. Conjecture 7.1. Let A and B be Hermitean matrices in dimension d, then there exist matrices A0 , B 0 , and U (λ) in the same dimension d, so that B 0 is diagonal real, A0 real symmetric, and U (λ) unitary such that for any real λ, we have A − λB = U (λ)(A0 − λB 0 )U −1 (λ) .
(7.2)
A weak version of the present conjecture is obtained by requiring only that A0 is real, and U (λ) invertible (that is by relaxing the conditions that A0 is symmetrical and U (λ) unitary). If the present conjecture holds, it will be sufficient to prove Conjecture 1.1 for real symmetric matrices in dimension d, to get the result that Conjecture 1.1 holds for complex Hermitean matrices in dimension d. The previous conjecture is interesting in view of Proposition 4.2: Among the possible complex values, the set of the cycles which are real and positive is certainly of measure zero (in any reasonable sense). However, in the real case (symmetric or not), selecting positive signs is a choice between a finite set of possibilities, and this means that Conjecture 1.1 holds certainly over a set of positive measure in the real case, and therefore also in the complex case if Conjecture 7.1 holds. From what we said in Sec. 4, Conjecture 7.1 (and Conjecture 1.1) hold for d = 2. If d = 3, it is easy to verify Conjecture 7.1 and Conjecture 1.1 when one of the off diagonal coefficients of A vanish, and also when the eigenvalues of B are degenerated, since in this case, one can take A in Jordan form. We shall now show that Conjecture 7.1 always holds in dimension d = 3. In fact, the problem can be formulated as follows: Given the characteristic polynomial in the form (7.1), is it possible to solve in term of the unknown real matrices A and B the following equation? Ξ(E, λ) = det(E − (A − λB)) = E d +
k d X X
n d−k p(k) . n λ E
(7.3)
k=1 n=0
This is exactly an inverse spectral problem: The spectral data are here the coefficients of the characteristic polynomial, and form a real manifold of dimension
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P. MOUSSA
Pk (k) d(d + 3)/2. Note that the sum σk (λ) = (−1)k n=0 pn λn is the symmetric function of degree k of the eigenvalues of A − λB. The problem is, given the coefficients of the characteristic polynomial, find the matrices A and B. The solutions are embedded in the space of pairs of complex Hermitean matrices which has real dimension 2d2 . Using constant unitary equivalence, which means here choosing the basis, one can reduce the embedding space to d2 + 1 real dimensions. The solutions corresponding to fixed spectral data form a manifold of real dimension d2 + 1 − d(d + 3)/2 = (d − 1)(d − 2)/2. The conjecture asserts that such a real manifold meets the d(d + 3)/2-dimensional real manifold which represents the real and symmetric matrices in the (d2 + 1)-dimensional space. In fact, we will not consider here the “existence” part of the inverse problem. We will merely consider the characterization of all possible solutions, assuming one solution (complex Hermitean) is given from start. As in the weak version of the conjecture, we will enlarge the problem in look(k) ing for solutions (given the pm ’s) of (7.3), such that: B is diagonal, and non degenerated, (which implies a choice of the basis), and A real (and not necessarily symmetric). For λ = small, we have λd Ξ(Eλ−1 , −λ−1 ) = E d +
k d X X
k−n d−k (−1)n p(k) E = det(E − (λA + B)) n λ
k=1 n=0
= det(E − B) + λ
=
d Y (E − bi ) + λ i=1
d det(E − B − λA) dλ ! d d Y (E − bi − λai ) dλ i=1
+ O(λ2 ) λ=0
+ O(λ2 ) ,
λ=0
(7.4) since up to first order in λ, only the diagonal elements contribute. Therefore we have det(E − B) =
d d Y X (k) (E − bi ) = E d + (−1)k pk E d−k , i=1
d Y
(E − bj )
j=1
d X i=1
(k)
(7.5)
k=1
X ai (k) = (−1)k pk−1 E d−k . E − bi d
(7.6)
k=1
Equation (7.5) shows that pk is for k = 1 to d the symmetric functions of degree k of (k) the eigenvalues of B: The knowledge of the coefficients pk is sufficient to determine b1 , b2 , . . . , bd . Furthermore, once the bi are known, and if they are all different (k) (no degeneracy), the knowledge of the coefficients pk−1 is sufficient to determine the diagonal elements a1 , a2 , . . . , ad of A. Indeed from (7.6), we get for the aj ’s Pd (k) the linear system i=1 Cik ai = −pk−1 , where Cik is the symmetric function of degree (k − 1) of the set of (d − 1) numbers obtained by erasing bi from the set
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
645
of eigenvalues of B. It is therefore easy to see that the determinant of this linear Q system is up to some constant the Vandermonde determinant i<j (bi − bj ) of the eigenvalues bj . These observations solve the diagonal part of the inverse problem: the ai ’s and bi ’s are uniquely fixed in the non degenerate case. Of course, the result is different in the degenerate case, because the basis is not fixed in the degenerate subspaces. However, there is little doubt that uniqueness (up to a permutation of the basis vectors) is recovered if A has diagonal restriction in degenerate space, although we have not checked this assertion throughout. Furthermore, we already know that Conjectures 7.1 and 1.1 hold for the degenerate case in dimension three. Now the inverse problem is restricted to the d(d−2)/2 real spectral data, namely (k) the pn ’s, with 2 ≤ k ≤ d, 0 ≤ n ≤ k−2. The diagonals ai and bi are known, and the unknown are either the off diagonal elements of A, or the cycles, which we extend in a straightforward way from the symmetric case of Sec. 4 to the non-symmetric case. We will now display the results in dimension 3. We get three equations for the six missing off diagonal coefficients, but these three equation really bind the four following expressions [12] = M12 M21 , [23] = M23 M32 , [13] = M31 M13 , and, [123] + [132] = M12 M23 M31 + M21 M32 M13 . Indeed we have (2)
[12] + [23] + [13] = C1 = a1 a2 + a2 a3 + a3 a1 − p0 ,
(7.7) (3)
(7.8)
(3)
(7.9)
b3 [12] + b1 [23] + b2 [13] = C2 = a1 a2 b3 + a2 a3 b1 + a3 a1 b2 − p1 , a3 [12] + a1 [23] + a2 [13] − ([123] + [132]) = C3 = a1 a2 a3 + p0 .
Now the question is: Assume that the right-hand sides of Eqs. (7.7), (7.8), (7.9) are obtained by writing these equations for a Hermitean matrix M , does it exist a solution of the same equations for M real, symmetric or not? From the previous equations, we get [12] =
b1 C1 − C2 b1 − b2 − [13] , b1 − b3 b1 − b3
(7.10)
[23] =
b3 − b2 b3 C1 − C2 − [13] , b3 − b1 b3 − b1
(7.11)
∆ a3 (b1 C1 − C2 ) − a1 (b3 C1 − C2 ) − C3 + [13] , b1 − b3 b3 − b1
(7.12)
[123] + [132] = where, as defined 1 ∆ = a1 b1
in (5.3), ∆ = ∆123 , that is 1 1 a2 a3 = (a2 − a1 )(b3 − b1 ) − (a3 − a1 )(b2 − b1 ) . b2 b3
(7.13)
Suppose now that the eigenvalues of B satisfy b1 < b2 < b3 , which we can always achieve by a suitable permutation of the eigenvectors and eigenvalues of B. In this case, the coefficients of [13] in (7.10) and (7.11) both have negative signs. Therefore,
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if we take [13] negative and large enough, we get positive signs for both [23], and [31], so that [12][23][31] < 0. Now, we have the identity [123] + [132] = [123] +
[12][23][13] . [123]
(7.14)
Once [12], [23], [13] are fixed, so that their product [12][23][31] < 0 is negative, (7.14) always allows to find a real solution for [123] (in fact we get two real solutions). Then, one can fix arbitrarily for instance M12 , and M13 and the other coefficients are easily obtained. This solves the weak form of Conjecture 7.1 in dimension three. We will now investigate the solutions of Eqs. (7.7) to (7.9), with all 2-cycles [12], [13], [23], real and non negative. We will assume that the coefficients C1 , C2 , C3 arise from the Hermitean matrix A and B, which for convenience, we parametrize as follows (with X0 , Y0 , and Z0 real and non negative, and φ0 an angle defined modulo 2π): a1 Z0 Y0 eiφ0 0 0 b1 a2 X0 B = 0 b 2 0, A = Z0 (7.15) Y0 e−iφ0
X0
a3
0
0
b3
and still b1 < b2 < b3 , so that we get, using notations analogous to (7.15), namely [23] = X 2 , [13] = Y 2 , [12] = Z 2 , and [123] + [132] = 2XY Zξ, C1 = X02 + Y02 + Z02 ,
(7.16)
C2 = b1 X02 + b2 Y02 + b3 Z02 ,
(7.17)
C3 = a1 X02 + a2 Y02 + a3 Z02 − 2X0 Y0 Z0 cos φ0 .
(7.18)
so that, using (7.10), (7.11) and (7.12), [12] = Z 2 = Z02 + [23] = X 2 = X02 +
b2 − b1 2 (Y − [13]) , b3 − b1 0 b3 − b2 2 (Y − [13]) , b3 − b1 0
[123] + [132] = 2XY Zξ = 2X0 Y0 Z0 cos φ0 +
∆ (Y 2 − [13]) . b3 − b1 0
We get [12], [23], and [13] > 0 if and only if b3 − b1 2 2 b3 − b1 2 2 . Z ,Y + X 0 < [13] < inf Y0 + b2 − b1 0 0 b3 − b2 0
(7.19) (7.20) (7.21)
(7.22)
When [13] = Y 2 increases from 0 to the upper bound (7.22), the quantity [123]+[132] in (7.21) increases if ∆ < 0, (resp. decreases if ∆ > 0), from V0 = 2X0 Y0 Z0 cos φ0 + ∆
Y02 b3 − b1
(7.23)
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
to V1 = 2X0 Y0 Z0 cos φ0 − ∆ inf
X02 Z02 , b2 − b1 b3 − b2
647
.
(7.24)
We let 0 = sign V0
and 1 = sign V1 .
(7.25)
It is immediate to see that a solution with −1 < χ < +1, corresponds to A complex Hermitean, and that a solution with χ = ±1 corresponds to A real and symmetric. Whereas a solution with |χ| > 1 corresponds to A real and non-symmetric (in this case, A can be taken of the form (7.15) with purely imaginary φ0 ). Let us know discuss the variation of X, Z and ξ when [13] varies in the interval given by (7.22) (this interval contains Y02 ). From Eqs. (7.19) to (7.21), we find that when Y = Y0 , we have X = X0 , Y = Y0 , and χ = cos φ0 . We also observe that on the edges of the interval given by (7.22), at least one of the quantities X, Y , Z vanish. Therefore at each end of this interval, either |χ| goes to infinity at both ends, or V0 (or V1 ) vanishes. If V0 (or V1 ) vanishes, it is easy to see that there is a solution where both members of a pair of symmetric off diagonal elements of A vanish (and also [123] and [132]). In this case, Conjectures 7.1 and 1.1 immediately hold. If |χ| goes to infinity, then χ must cross one of the values ±1, and this completes the proof of Conjecture 7.1 in dimension 3. More precisely, it is easy to see that if 0 = 1 = +1, then the value +1 is crossed exactly twice, (the value −1 being missed), and if 0 = 1 = −1, then the value −1 is crossed exactly twice, (the value +1 being missed), whereas, if 0 = −1 , then the values −1 and +1 are crossed exactly once. Each time it is possible to get χ = +1, Conjecture 1.1 holds, since one gets a matrix A with non negative off diagonal elements. We summarize the previous results in the following proposition in which we also include immediate corollaries. Proposition 7.1. (1) The Conjecture 7.1 holds in dimension 3. (2) If at least one of the signs ε0 and ε1 is positive, then Conjecture 1.1 holds for the pair of matrices A and B of (7.15) with b1 < b2 < b3 . (3) Conjecture 1.1 holds when some eigenvalues of B coincide. Conjecture 1.1 holds for the pair of matrices A and B of (7.15) with b1 < b2 < b3 , provided sup 2X0 Y0 Z0 cos φ0 + ∆
Y02 , b3 − b1
× 2X0 Y0 Z0 cos φ0 − ∆ inf
X02 Z02 , b2 − b1 b3 − b2
! ≥ 0.
(7.26)
(4) Assume X0 , Y0 , Z0 and the eigenvalues of B are given, then there exists a constant K > 0 such that Conjecture 1.1 holds when |∆| > K. (5) From (3) above, we see that Conjecture 1.1 holds in particular when cos φ0 ≥ 0. The item (5) of the proposition generalizes the result of Proposition 3.1 above, which assumes cos φ0 = +1. Note the following consequences: given X0 , Y0 , and
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P. MOUSSA
Z0 , the Conjecture 1 holds if ∆ is large enough in modulus. For instance, let A and B given, such that ∆ 6= 0: then for c real, denote Ac the matrix obtained from A by replacing the diagonal elements ai by cai , then there exists a constant K > 0 such that with |c| > K, then the modified matrix Ac (together with B) satisfies Conjecture 1.1. Clearly, the case where ∆ = 0 is the least favorable for a positive answer to Conjecture 1.1. The condition (7.27) is almost surely not optimal, since the conjecture holds on the boundary (when the left-hand side vanish), and the condition on A to satisfy Conjecture 1.1, is an open condition in any reasonable topology. Proposition 7.1 is the best result known so far toward Conjecture 1.1 in three dimension. Obviously, the method of this section can be generalized to dimension larger than three, but it becomes too intricate to give a nice form to a condition of the type (7.27) in higher dimensions, and the issue of Conjecture 1.1 remains mysterious. However, we guess that extension of the Conjecture 7.1 in higher dimension should be tractable, at least in its weak form. Indeed, a positive issue for Conjecture 7.1 is equivalent to finding an invertible matrix U (λ) solution of the linear system (A − λB)U (λ) = U (λ)(A0 − λB) ,
(7.27)
where A and B are given Hermitean matrices, B being diagonal, and A0 a real symmetric (or not symmetric) one. In this sense, Conjecture 7.1 is a linear problem. 8. Variational Arguments, or the Nicest Way not to Prove Conjecture 1.1 A classical theorem by Bernstein [7] states that a function Z(λ) is the Laplace R∞ transform of a positive measure, that is Z(λ) = 0 eλτ dµ(τ ), if and only if it is completely monotonic, which means that for all real λ and integer n ≥ 0, we have n (−1)n d n Z(λ) > 0. Defining I0 (A, B) = Tr(eA ), and for n > 0, dλ n dn n d Z(λ) = (−1) Tr exp(A − λB) , (8.1) In (A, B) = (−1)n n n dλ dλ λ=0 λ=0 it is easy from Bernstein’s theorem to deduce the following proposition (the condition for n = 0 being automatically satisfied). Proposition 8.1. The Conjecture 1.1 holds for the pair of Hermitean matrices A and B, with B semi positive definite, if and only if In (A − λ0 B, B) > 0 for all integer n > 0 and any λ0 real. The Conjecture 1.1 holds for every pair of matrices A and B, with B semi positive definite, if and only if In (A, B) > 0 for all integer n > 0 and all A and B with B semi positive definite. The variational approach was introduced by Le Couteur [8, 9]. The idea is to study the functions In (A, B) as function of A and B, and to investigate the stationary values under various kind of constraints. We first need an expression for
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
649
In (A, B). For n ≥ 1, we define n−1
Qn (A, B) = (−1)
dn−1 exp(A − λB) . dλn−1 λ=0
(8.2)
Using perturbative arguments similar to those which led us to Eq. (2.4), we have Z ex0 A Bex1 A B · · · exn−2 A Bexn−1 A Qn (A, B) = (n − 1)! +
× δ(x0 + x1 + · · · + xn−1 − 1)
n−1 Y
dxj ,
(8.3)
j=0
so that for n ≥ 1, In−1 (A, B) = Tr[Qn (A, B)] ,
In (A, B) = Tr[BQn (A, B)] .
(8.4)
The second equation in (8.4) is easily deduced from the first by performing an integration over one of the variables. For the lowest values of n, we have I0 (A, B) = Tr eA > 0 , and
I1 (A, B) = Tr(BeA ) = Tr(eA/2 BeA/2 ) > 0 ,
(8.5)
Z δ(x0 + x1 − 1)dx0 dx1 Tr[(ex0 A/2 Bex1 A/2 )(ex1 A/2 Bex0 A/2 )] > 0 ,
I2 (A, B) = +
(8.6) so that the conditions of Proposition 8.1 need to be verified only for n ≥ 3. Assume now that we perform a variation B into B + δB, at first order we get δIn = In (A, B + δB) − In (A, B) = n Tr[δBQn (A, B)] .
(8.7)
Suppose that A and B commute, then we can consider variations δB which are expressed in a basis where both A and B are diagonal. If δB has vanishing diagonal elements in this basis, then δIn = 0 at first order, and up to second order, we have P δIn = i,j Kij |δBi,j |2 where Kij is cumbersome to write, but it is immediate to see that Ki,j is a sum of integrals over positive quantities and therefore is positive. Therefore, with respect to variations δB which are purely off diagonal in the corresponding diagonalizing basis, In (A, B) reaches an absolute minimum on each commuting pair, and these minima are positive. Although In is not stationary with respect to diagonal variations, these diagonal variations link one pair of commuting matrices to another pair of commuting matrices, and along the variations path, all In remain positive. This observation is fascinating in view of Proposition 8.1, and it leads to discuss the occurrence of other stationary points, and the behavior on the boundary, (in case the variations of B are performed on a set with boundary). We now consider a variation A into A + δA, at first order we get δIn = In (A + δA, B) − In (A, B) = Tr[δAQn+1 (A, B)] .
(8.8)
650
P. MOUSSA
We get for the variation δA the same situation as for the variation δB. Suppose that A and B commute, then if δA has vanishing diagonal elements in the diagonalizing P 0 |δAi,j |2 basis, δIn = 0 at first order, and up to second order, we have δIn = i,j Kij 0 where Kij is positive. Therefore, with respect to variations δA which are purely off diagonal in the corresponding diagonalizing basis, In (A, B) reaches an absolute minimum on each commuting pair, and these minima are positive. For diagonal variations δA, we have the same situation as before for δB. We follow now more precisely Le Couteur [13], and we will parametrize the set of Hermitean positive matrices as follows. Let B = β2
(8.9)
be an Hermitian matrix in dimension d. We will restrict the variations of B by the following condition: X |βij |2 = T > 0 . (8.10) Tr B = ij
With these equations, one gets a parametrization of the set of positive (in fact semi positive) matrices B with fixed trace T in terms of the complex numbers βij (with ∗ ). βij = βji One could restrict the βij to the condition that β is itself a semi positive Hermitian matrix, in which case to a given B there corresponds a unique β, which commutes with B. However, with this choice the set of β’s do not form a manifold, or more precisely it forms a manifold with a boundary. As a consequence the search for extrema of some funtion on this set should include both stationary values and values taken on the boundary (whether they are stationary or not). The boundary is rather complicated, and the analysis of the values of the functions In on the boundary has not been performed. Therefore, it seems to be easier to relax the constraint that β is semi positive. In this case Eq. (8.10) above shows that the set of matrices B with constant trace T , is a true manifold (i.e. without boundary). So the analysis of the extremal values of In can be now restricted to the stationary values. However, to one matrix B, there correspond several matrices β, generically 2d , where d is the dimension. Moreover, in case of degeneracy of the eigenvalues there might exists a continuum of β’s as shown in the following example in dimension 2: B = 1, β = ux σx + uy σy + uz σz , where the σ’s are the usual Pauli matrices [14], and the reals ux , uy , uz are such that u2x + u2y + u2z = 1. The example shows in particular that two different β’s, both solutions of (8.9), might not commute with one another. Indeed it also shows that there might exist matrices commuting with B and not with β. Now, in order to find the extremal values of In for fixed A under the constraint (8.10) above, we apply the Lagrange multiplier method, and we have (writing Qn for Qn (A, B)) (8.11) δIn − µ Tr B = Tr(nQn δB − µδB) = 0 . Since δB = β(δβ) + (δβ)β, we get Qn β + βQn =
2µβ , n
(8.12)
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
651
and
2µ B − βQn β . n If the solution β is such that Qn commutes with β, we get Qn B = BQn =
Qn B = BQn = βQn β =
µ B. n
(8.13)
(8.14)
However, Qn commutes with B does not imply that Qn commutes with β. By chance, taking the trace in Eq. (8.13), one nevertheless gets In = Tr Qn B =
µT µ Tr B = . n n
(8.15)
2µ n
(8.16)
In addition if β is not singular, we have Qn + βQn β −1 = and
d µ. (8.17) n If all stationary β are not singular, then using (8.15) and (8.16), we see that In−1 > 0 implies µ > 0, which in turn implies In > 0, and through a recursion argument starting from I2 > 0, we get ∀n, In > 0 (in fact it would be enough that the minimizing β’s be non-singular). In principle, one should consider that in Eq. (8.12), µ is fixed by the implicit condition Tr β 2 = T . However, this is easily done due to the homogeneity properties of (8.12) and (8.3) in β (at fixed A). More precisely, suppose that we have for some µ a solution β of (8.12) and (8.3) with Tr β 2 = t. We then get for the same A another solution β 0 = (T /t)1/2 β such that Tr β 02 = T , corresponding to the rescaled value µ0 = (T /t)(n−1) µ. Note that this rescaling does not change the sign of µ. We can summarize these results by the following proposition: In−1 = Tr Qn =
Proposition 8.2. If Conjecture 1.1 fails, we have the following situation: For some A, and for some n ≥ 3, there is a Hermitean matrix β, such that we have B = β2 ,
Tr B = T > 0 ,
Tr Qn ≥ 0 ,
Qn β + βQn =
Qn as in (8.3) , 2µβ , n
(8.18) µ < 0.
We call such a β a pathological solution of (8.12). As already noticed, in (8.16) and (8.17), such a pathological matrix β should be singular if it exists. Let us add a few remarks on these pathological matrices. First, matrices β such that A and B = β 2 commute are never pathological. Indeed, in this case, one can diagonalize in the same basis A, B, and Qn , and we have Aij = ai δij , Bij = bi δij , and (Qn )ij = ≥ 0. Now from (8.12) we get (qi + qj )βij = (2µ/n)βij , and qi δij , with qi = eai bn−1 i therefore µ > 0, since we exclude the solution β = 0. Next, in the general case where A and B do not commute, we still have the commutation of B and Qn . In a
652
P. MOUSSA
basis where both B and Qn are diagonal, we have Bij = bi δij , and (Qn )ij = qi δij , Pd with i=1 qi > 0. We again have (qi + qj )βij = (2µ/n)βij ,
(8.19)
and also from (8.13), bi qi +
X
|βij |2 qj = (2µ/n)bi < 0 .
(8.20)
i
As a consequence, if βii 6= 0, we must have qi < 0, and if βij 6= 0 for i 6= j, we must have qi + qj < 0. This shows that pathological matrices β must be rather P sparse, if they exists, since one must have di=1 qi > 0. However, if all solutions B = β 2 of (8.12) commute with A, there does not exist pathological solutions of (8.12), and therefore Conjecture 1.1 would be proven. Therefore it is natural to ask the following question. Question 8.1. Let A and B be Hermitean matrices with dimension d, and let B be semi definite positive. Let Qn (A, B) be defined for n ≥ 1 by Eqs. (8.2) or (8.3). Assume that for some n, B commutes with Qn (A, B). Does it imply that B and A commute? A positive answer to Question 8.1 would exclude the existence of pathological solutions of (8.12), and therefore would imply the validity of Conjecture 1.1. Despite numerous efforts, the existence of pathological solutions has neither been excluded nor been proved, and therefore, the issue of Conjecture 1.1 is still not known. We have only few results on Question 8.1. The answer is yes for n = 1, since Q1 = eA . For n = 2, in a basis where A is diagonal, Aij = ai δij , we have (Q2 )i,j = Bij Kij , where Kii = eai , and for i 6= j, Kij = (eai − eaj )(ai − aj )−1 . Then P the condition [Q2 , B] = 0 implies k Bik Bkj (Kik − Kkj ) = 0. Suppose now that the dimension d = 2. If a1 = a2 , then B commutes with A. Now if a1 6= a2 , and [B, A] 6= 0, we have B12 6= 0, and therefore we get B11 (K11 −K12 ) = B22 (K22 −K12 ). Either B11 = B22 = 0, in which case B cannot be semi positive definite, or it is easy to see that the convexity property of the exponential function implies that B11 and B22 have opposite signs, and therefore again B cannot be semi positive. In other word, the answer to Question 8.1 is yes for d = n = 2, and shows explicitly that the positivity condition on B is essential: If this condition is removed, the answer is no. We will see now that the answer to Question 8.1 is yes for d = 2, any n. Indeed, the Question 8.1 might be closer to Conjecture 1.1 than one could think at first sight as shown by the complete solution of Question 8.1 for d = 2, obtained as follows. Using Pauli matrices, let A = a0 + aσ, and B = b0 + bσ. The matrix B is semi positive definite if and only if b0 ≥ |b|. Simple calculations show that ∞ X (−λ)n−1 sinh|a − λb| [Qn (A, B), B] = ea0 −λb0 i(a × b) · σ . (n − 1)! |a − λb| n=1
(8.21)
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
653
Therefore, [Qn , B] = 0 implies (a×b) = 0, that is [A, B] = 0, provided the derivative (−1)n−1
dn−1 a0 −λb0 sinh|a − λb| e dλn−1 |a − λb|
(8.22)
do not vanish. Using Bernstein’s Theorem for the limiting case b0 = |b|, one sees after some simple calculations and change of variables, that the derivatives (8.22) keep for all n a fixed sign (and do not vanish) if we have the following representation as a Laplace transform √ Z +1 sinh x2 + c2 √ = e−xτ dµc (τ ) , (8.23) 2 2 x +c −1 where µc is for any real c a positive measure with support on [−1, +1]. In order to prove (8.23), we consider the trace Tr(exp(A − λB)) = 2ea0 −λb0 cosh|a − λb| .
(8.24)
The expression of this trace as a Laplace transform in λ could easily be deduced from (2.11). It is sufficient for our purpose to notice that the same change of variable as above leads to √ Z Z +1 p c π cI1 (c 1 − τ 2 ) −xτ √ I1 (c sin θ)e−x cos θ dθ = e dτ , cosh x2 + c2 − cosh x = 2 0 2 1 − τ2 −1 (8.25) P∞ 1 ( z2 )2m+1 . Remark that (8.25) can be easily checked where I1 (z) = m=0 m! m+1! by direct computations using the expansion of the Bessel function I1 . Now, by derivation of (8.25) with respect to c, one gets (8.23) with ! √ ∂ cI1 (c 1 − τ 2 ) √ dτ , (8.26) dµc (τ ) = ∂c 2 1 − τ2 which shows that the measure dµc (τ ) is positive, since the expansion of the Bessel function has only positive coefficients. This ends the solution of Question 8.1 for d = 2. Notice that the argument used is similar to the solution of Conjecture 1.1 in this case. Curiously enough, the author does not know any other proof for Question 8.1. We do not know whether this strange connection extends to higher d. Other variational approaches have been also considered, leading to the same kind of open questions. We have mainly four kind of variations Hermitean variation of B, δB, A fixed: δIn = n Tr(Qn δB)
(8.27)
Hermitean variation of A, δA, B fixed: δIn = Tr(Qn+1 δA)
(8.28)
Unitary variation of B, δB = i[ε, B], A fixed: δIn = Tr(inε[B, Qn ])
(8.29)
Unitary variation of A, δA = i[ε, A], B fixed: δIn = Tr(iε[A, Qn+1 ]) .
(8.30)
654
P. MOUSSA
By unitary variation, we mean the infinitesimal version of the isospectral transformation B → eiε Be−iε , for ε an infinitesimal Hermitean matrix. If we perform the same unitary transformation on A and B, the function Z(λ) remains unchanged, and therefore the derivatives In are invariants. This leads to the identity n[B, Qn ] + [A, Qn+1 ] = 0 ,
(8.31)
which is easily checked directly from (8.3). An example of variations is to take A fixed, and to take unitary variations of B submitted to the constraint Tr(AB) = K. Using variation with Lagrange parameter δ(In − µ Tr(AB)), we get the equation: n[Qn , B] = µ[A, B] = [A, Qn+1 ] ,
(8.32)
and if we allow also arbitrary Hermitean variations of A, we get in addition Qn+1 = µB .
(8.33)
Now the question is, does Eqs. (8.32) and (8.33) have solutions with µ < 0? Indeed, µ = 0 implies In = 0. And there is also the question of the behavior around A infinity. Therefore this kind of variations looks much trickier than the first one. 9. Conclusion After more than twenty years, the present issue of Conjecture 1.1 remains rather confused, and for that reason many partial results remained unpublished. The present paper is an attempt to summarize all what is known on the subject. The analyticity arguments of Secs. 2 and 3 receive here a new presentation, although some results are already present in [3, 4]. Section 4 is given to popularize the cycle description which in my view, should receive a more formal treatment. Section 5 reproduces an unpublished result by Froissart [6]. Sections 6 and 7 are new, and if of course Sec. 6 is rather elementary, on the contrary, Sec. 7 is very intricate. I believe that there is something deep hidden behind the inverse problem for finite matrices. Finally, it remains very mysterious to me why the variational approach of Sec. 8 did not succeed, which explains why it has been extensively reported here. Acknowledgments Although the following reference list looks short, the problem has been considered by many colleagues, and I thank all them for discussions or for the correspondence exchanged with us. Among others, I especially thank R. Balian, G. A. Baker, J. Bellissard, D. Bessis, A. Bobylev, J. Bros, A. Connes, B. Dubrovin, M. Froissart, N. Gastinel, M. Gaudin, K. J. Le Couteur, E. Lieb, M. L. Mehta, G. A. Mezincescu, G. Roepstorff, M. Villani, R. Werner, A. Wulfsohn, J.-C. Yoccoz. For instance, Dyson, Lieb and Simon, found that Conjecture 1.1 would be a corollary of a more general statement [10, 12] which would have been useful for some problems of statistical mechanics models [11]. Unfortunately, this more general statement fails (see [12] for details), but no definitive conclusion can be drawn on Conjecture 1.1. We
ON THE REPRESENTATION OF Tr(e(A−λB) ) AS A LAPLACE TRANSFORM
655
also mention here that numerical calculations have been performed using various methods in order to find a counterexample to Conjecture 1.1, mainly by K. J. Le Couteur and G. A. Mezincescu: They did not find any numerical evidence in this direction. However, it should be noticed here that this numerical approach is rather difficult. We hope that this paper, by displaying all the unsuccessful, or more precisely uncomplete attempts toward the solution of Conjecture 1.1, will contribute to the solution of the problem, and help to develop the connected mathematical questions. Finally, I thank Anne-Marie Rocher for constant help during this work. References [1] D. Bessis, P. Moussa and M. Villani, “Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics”, J. Math. Phys. 16 (1975) 2318–2325. [2] P. Moussa, “Tr[exp(A − λB)] as a Laplace transform, in Polynˆ omes orthogonaux et applications, Problem Section”, Springer Lect. Notes in Math. 1171 (1980) 579–580. [3] M. Gaudin, “Sur la transform´ ee de Laplace de Tr e−A consid´er´ee comme fonction de la diagonale de A”, Ann. Inst. Henri Poincar´e, Sect. A, 28 (1978) 431–442. [4] M. L. Mehta and K. Kumar, “On an integral representation of the function Tr[exp(A− λB)]”, J. Phys. A: Math. Gen. 9 (1976) 197–206. [5] M. Suzuki, “Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems”, Commun. Math. Phys. 51 (1976) 183–190. [6] M. Froissart, private communication, 1976. [7] D. Widder, The Laplace Transform, Princeton Univ. Press, 1946, pp. 160–161. [8] K. J. Le Couteur, “Representation of the function Tr(exp(A − λB)) as a Laplace transform with positive weight and some matrix inequalities”, J. Phys. A: Math. Gen. 13 (1980) 3147–3159. [9] K. J. Le Couteur, “Some problems of statistical mechanics and exponential operators”, in Proc. Int. Conference and Winter School of Frontiers of Theoretical Physics, eds. F. C. Auluck, L. S. Kothari, V. S. Nanda, Indian National Academy, New Dehli, 1977, published by The Mac Millan Company of India, 1978, pp. 209–235. [10] E. Lieb, private communication, 1982. [11] F. Dyson, E. Lieb and B. Simon, “Phase transitions in quantum spin systems with isotropic and nonisotropic interactions”, J. Stat. Phys. 18 (1978) 335–383. [12] G. A. Mezincescu, “On a Conjecture by Dyson, Lieb, and Simon” in Topics in Theoretical Physics, a volume dedicated to Serban Titeica, ed. by Central Institute of Physics, Bucharest, Romania, 1978, pp. 1–6. [13] K. J. Le Couteur, unpublished, and private communication, 1993. [14] For a definition of the Pauli matrices, see any textbook on quantum mechanics, for instance, A. Messiah, Quantum Mechanics, North Holland, Chapter XIII, 1961.
THE LORENTZ DIRAC EQUATION, I BERNHARD RUF Dipartimento di Matematica “F. Enriques” Universit=E0 degli Studi di Milano Via Saldini 50 20133 Milano, Italy E-mail:
[email protected], http://www.mat.unimi.it/∼ruf
P. N. SRIKANTH TIFR Center, P.O.Box 1234 Bangalore, 560012, India E-mail:
[email protected] Received 21 January 1999 d The Lorentz–Dirac equation (LDE) τ x000 − x00 = dx V (x) models the point limit of the Maxwell–Lorentz equation describing the interaction of a charged extended particle with the electromagnetic field. Since (LDE) admits solutions which accelerate even if they are outside the zone of interaction, Dirac proposed to study so-called “non runaway” solutions satisfying the condition x00 (t) → 0 as t → +∞. We study the scattering of particles for a localized potential barrier V (x). We show, using global bifurcation techniques, that for every T > T0 there exists a reflection solution with “returning time” T , and for every T > 0 there exists a transmission solution with “transmission time” T . Furthermore, some qualitative properties of the solutions are proved; in particular, it is shown that for increasing T , these solutions spend more and more time near the maximum point s0 of V .
1. Introduction We consider the equation τ x000 (t) − x00 (t) =
d V (x(t)) , dx
τ > 0,
(1.1)
which was introduced by Abraham (1904) and Lorentz (1905) to study the interaction of a charged point particle with the electromagnetic field, taking into account the self-interaction of the accelerated particle with its own radiation via the modified exterior electromagnetic field. In 1938 P. Dirac [5] gave a classical relativistic theory of the point particle, based on Eq. (1.1). For information concerning interpretation and significance of this equation, see Lorentz [7], Jackson [6], Carati, et al. [3], Bambusi–Noja [2], Noja–Posilicano [9], R-S II [10], etc. The intention of this paper (and its sequel [10]) is to provide a mathematical analysis of the solution structure of Eq. (1.1), based on methods from nonlinear analysis and asymptotic estimates. For another mathematical treatment of (1.1) we refer to Hale–Stokes [4], where abstract existence theorems (based on generalized fixed-point arguments) are proved. With the change of variable y(t) = x(τ t) (and renaming τ 2 f to f ) one sees that one may set τ = 1 in (1.1), without restricting the generality. Furthermore, we 657 Reviews in Mathematical Physics, Vol. 12, No. 4 (2000) 657–686 c World Scientific Publishing Company
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assume that V is localized in the interval (0, 1). Thus, we consider the following equation, denoting f (s) = V 0 (s): ( 000 x = x00 + f (x) , 0 ≤ x ≤ 1 , (1.2) x000 = x00 otherwise . Equation (1.2) admits “non physical” solutions which accelerate for all times, even with no force present, as is easily seen if f (x) ≡ 0: the equation x000 = x00 admits the solutions x(t) = a et . To exclude such solutions, Dirac suggested to impose the so-called “non runaway” condition lim x00 (t) → 0 .
t→+∞
We make the following assumptions on the potential function V : / (0, 1). (V 1) V ∈ C 1 (R), with V (s) > 0 in (0, 1) and V (s) = 0, s ∈ 3 00 + (V 2) V ∈ C (0, 1) with V (0 ) > 0. (V 3) V has a unique maximum, say in s0 , with V 0 (s) > 0, s ∈ (0, s0 ), V 0 (s) < 0, s ∈ (s0 , 1). Example: V (s) = sin2 (πs), s ∈ (0, 1), V (s) = 0 otherwise. We are interested in the scattering of the point particle by the potential barrier V . Since outside of the barrier the motions are explicit, we may consider the following two sets of boundary conditions: Reflection :
x(0) = 0 ,
x(T ) = 0
x00 (T ) = 0 .
(1.3)
This corresponds to a particle which enters the obstacle at time t = 0, is reflected, and leaves the barrier at time t = T . The condition x00 (T ) = 0 represents the non runaway condition, imposed by Dirac. Transmission :
x(0) = 0 ,
x(T ) = 1 ,
x00 (T ) = 0 .
(1.4)
These conditions represent a particle which enters the obstacle at t = 0 and crosses the barrier to leave it at time t = T . We will study the solutions of (1.2) together with the reflection or transmission boundary conditions with methods from nonlinear analysis. In particular, we will use bifurcation theory and continuation techniques to obtain a global description of the solution structure. For this, we transform the equations by the change of variables z(t) = x(T t) into the equations ( 000 z = T z 00 + T 3 f (z) , t ∈ (0, 1) , (1.5) (Re) z(0) = 0 = z(1) , z 00 (1) = 0 respectively
( (T r)
z 000 = T z 00 + T 3 f (z) ,
t ∈ (0, 1)
z(0) = 0 ,
z 00 (1) = 0
z(1) = 1 ,
.
(1.6)
THE LORENTZ–DIRAC EQUATION, I
659
We can now use T as a parameter for the solutions of (Re) and (T r). Reflection: Note that (Re) admits for every T the trivial solution z ≡ 0. We will show the following global bifurcation result for problem (Re): Theorem 1.1. Suppose (V 1) and (V 2). Then there exists a T0 > 0 from which a global branch of positive solutions of (Re) bifurcates. Furthermore (i) the bifurcation branch extends to +∞ in T, i.e. for every T ∈ (T0 , +∞) there exists a positive solution zT ; (ii) 0 < zT (t) < 1, ∀ t ∈ (0, 1), ∀ T ∈ (T0 , +∞); (iii) zT0 (0) > 0 and zT00 (0) < 0, ∀ T ∈ (T0 , ∞). Transmission: Note that (T r) admits for T = 0 the “trivial” transmission solution z0 (t) = t. We show that this solution can be continued to a global branch of solutions. Theorem 1.2. Suppose (V 1) and (V 2). Then equation (T r) has a solution zT for all T > 0. Furthermore, one has (i) zT (t) → t in C 3 (0, 1) as T → 0; (ii) 0 < zT (t) < 1, ∀ t ∈ (0, 1), ∀ T > 0; (iii) zT0 (0) > 0 and zT00 (0) < 0, ∀ T > 0. The further qualitative properties depend on the “sharpness” of the maximum of the potential V (s), i.e. on V 00 (s0 ): Theorem 1.3. Assume (V 1)–(V 3) and (V 4) 0 > V 00 (s0 ) ≥ −4/27 Then we have for the solutions of (Re): • 0 < zT < s0 , for all t ∈ (0, 1), and zT is concave, for all T, and for the solutions of (T r): • there exists a unique point aT ∈ (0, 1) with zT00 (aT ) = 0; • zT is concave for 0 ≤ t < aT and convex for aT < t < 1; • zT (t) intersects the constant line s0 exactly once, in a value bT > aT . In both the reflection and the transmission case one has the following convergence theorem: Theorem 1.4. Assume (V 1)–(V 4). Let zT denote the solutions on the branch bifurcating from T0 for problem (Re), respectively the solutions for problem (Tr ). For any > 0 (sufficiently small ) given, denote with a,T the first point where zT (a,T ) = s0 − and with b,T the next point where |zT (b,T ) − s0 | = . Then, as T → ∞: c() c() , 1 − b,T ≤ . a,T ≤ T T
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We remark that the solutions of (Re) and (T r) (under conditions (V 1)–(V 4)) have the same qualitative properties as the solutions of Newton’s equation (formulated as boundary value problems for reflection and transmission solutions): (N-Re) 0 = z 00 + T 2 f (z) , (N-Tr) 0 = z 00 + T 2 f (x) ,
z(0) = z(1) = 0 ; z(0) = 0, z(1) = 1 .
In the following figures the solutions of the Lorentz–Dirac equation are plotted for 2 the parameters T = 20, 40 for the potential V (x) = 0.1 π 2 sin (πx), and the dotted curves represent the corresponding solutions of Newton’s equation. 1
1
1
Reflection solutions
1
Transmission solutions
The absolutely remarkable agreement of the solutions for Newton’s equation and the Lorentz–Dirac equation seems to fully justify the physically motivated “non runaway condition” x00 (1) = 0 imposed by Dirac. However, we would like to point out the following subtle but relevant difference for transmission solutions: in Newton’s equation, the transmitted particle reaches acceleration zero in the point s0 , that is “at the top” of the obstacle, while in the Lorentz–Dirac equation the particle reaches zero acceleration before s0 , and starts to reaccelerate before it reaches the top of the obstacle (see Proposition 3.1). Thus, the particle “feels” the accelerating force before it actually interacts with it! We emphasize that the condition (V 4) is necessary to have the qualitative agreement of the solutions of Newton’s and the Lorentz–Dirac equation. As first observed by Carati, et al. [3] (for piecewise linear potentials) the solutions of the Lorentz– Dirac equation show a strikingly different behaviour if V 00 (s0 ) < −4/27: for growing T the solutions oscillate around the value s0 an increasing number of times. In R-S [10] we will prove this phenomena under the general assumptions (V 1)–(V 3) and 4 . (V 5) V 00 (s0 ) < − 27
2. Reflection Solutions 2.1. Setup and linearization In this section we consider the problem (Re), which we set up as a bifurcation equation with bifurcation parameter T , on an appropriate function space. Define X = {z ∈ C 3 [0, 1] : z(0) = z(1) = z 00 (1) = 0} and Y = C(0, 1) .
(2.7)
Let F : X × R → Y denote the mapping F (z, T ) = z 000 − T z 00 − T 3 f (z) .
(2.8)
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THE LORENTZ–DIRAC EQUATION, I
Note that z(t) ≡ 0 is a solution of F (z, T ) = 0, for all T ∈ R. We will show that there exists T0 > 0 such that (0, T0 ) ∈ X × R is a bifurcation point from which emanates a continuous branch of solutions (zT (t), T ) of (2.23), with 0 < zT (t) < 1. Remark 2.1. Note that the last property allows us to use a C 2 (R) extension f˜ of the (only Lipschitz continuous) force f ; indeed, let f˜ ∈ C 2 (R) such that f˜(x) = f (x), x ∈ [0, 1]; then, having found the solution branch for F˜ (z, T ) = z 000 − T z 00 − T 3 f˜(z), we can modify the problem back to F (z, T ) without altering the solutions. We now consider the linearization of F (z, T ) with respect to z in z = 0 Fx (0, T ) = x000 − T x00 − T 3 f 0 (0)x = 0
(2.9)
and look for T ’s for which this equation has a nontrivial solution in the space X. This is equivalent to looking for T ’s for which the system · ˜(t) x ˜ (t) = AT x has a nontrivial solution in X, where 0 x(t) 0 ˙ , and AT = x˜ = x(t) 3 0 T f (0) x ¨(t)
(2.10)
1 0 0
0 1. T
(2.11)
Note that the eigenvalues for AT are precisely the eigenvalues of A1 multiplied by T . The matrix A1 has a real (positive) eigenvalue λ1 and two complex eigenvalues (with negative real part) µ ± iη, with corresponding eigenvectors v1 = (1, λ1 , λ21 ), v2 = (1, µ, µ2 − η 2 ), v3 = (0, η, 2µη), where v1 corresponds to λ1 , and v2 ± iv3 correspond to µ ± iη, respectively. A general solution of (2.10) (we are looking for a real solution) has the form: x˜(t) = ceλ1 T t v1 + (a + ib)e(µ+iη)T t(v2 + iv3 ) + (a − ib)e(µ−iη)T t(v2 − iv3 ) .
(2.12)
The boundary conditions force the following relations: (A1) c = −2a −eT µ cos ηT eµT sin T η
T λ1
(A2) b = −a[ e (A3)
(−λ21
]
+ µ − η ) sin T η + 2µη[cos T η − e(µ−λ1 )T ] = 0. 2
2
Note that there are values of T > 0 for which (A3) has a solution: taking T = and
(k+1)π , η
kπ η
k ∈ N sufficiently large, then the function G(T ) = (−λ21 + µ2 − η 2 ) sin T η + 2µη cos T η −
has opposite signs at T =
kπ η
and T =
(k+1)π . η
1 e(λ1 −µ)T
(2.13)
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Let T0 > 0 be the first positive T such that (A3) holds. The corresponding x(t) solving (2.14) x000 = T0 x00 + T03 f 0 (0)x , x ∈ X , is given by eT0 λ1 − eT0 µ cos T0 η T0 µt e cos T0 ηt + sin T0 ηt . eT0 µ sin T0 η
x(t) = −e
T0 λ1 t
+e
T0 µt
(2.15)
Lemma 2.1. Let x(t) denote the solution (2.15) of (2.14). Then x(t) has no zeros in (0, 1). Proof. The solution x(t) given by (2.15) can be rewritten as x(t) =
eT0 µ(1+t) {−eT0 (λ1 −µ)t sin T0 η + eT0 (λ1 −µ) sin T0 ηt + sin T0 η(1 − t)} . sin T0 η
eµT0
(2.16) It is therefore enough to show that L(t) = −e
T0 (λ1 −µ)t
sin T0 η + eT0 (λ1 −µ) sin T0 ηt + sin T0 η(1 − t)
(2.17)
does not change sign. We have L(0) = L(1) = 0, and it is easy to check that L00 (t) < 0, t ∈ (0, 1) (using that 3π 4 < T0 η < π, see Lemma A.2, Appendix). Hence L is concave and thus positive. 2.2. Local bifurcation We now apply the following result from Bifurcation Theory (see e.g. [1, 8]) to conclude that T0 is a bifurcation point. Theorem. Let φ(x, λ) be a C 2 -map of a neighbourhood of (0, λ0 ) in X × R into Y with φ(0, λ0 ) = 0. Suppose (i) (ii) (iii) (iv)
φλ (0, λ0 ) = 0 kerφx (0, λ0 ) is one dimensional, spanned by x0 Rφx (0, λ0 ) =: Y1 has codimension 1 φλ,λ (0, λ0 ) ∈ Y1 and φλ,x (0, λ0 )x0 6∈ Y1
Then (0, λ0 ) is a bifurcation point. Applying this theorem, we will prove: Theorem 2.1. The first positive zero T0 of G given by (2.13) is a bifurcation point of positive solutions for F (z, T ) = z 000 − T z 00 − T 3 f (z) = 0 ,
z∈X.
(2.18)
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THE LORENTZ–DIRAC EQUATION, I
Proof. We show that the map F (z, T ) defined in (3.45) satisfies the hypotheses of the above theorem. By Remark 2.1 we may assume that F is C 2 , and we have seen that kerFz (0, T0 ) is one dimensional, spanned by x given in (2.15). To prove (iii) of the above theorem, we consider the adjoint equation to x000 = T0 x00 + T03 f 0 (0)x given by
(
in X ,
−y 000 = T0 y 00 + T03 f 0 (0)y , y(0) = 0 = y 0 (0) ,
(2.19)
T0 y(1) = −y 0 (1) .
Converting this equation into a system y˜0 = BT0 y˜ 0 y 0 with y˜ = y˙ , BT0 = −T 3f 0 (0) y¨
1 0 0
0
1 −T
(2.20)
we see that the eigenvalues of BT0 are the negative of the eigenvalues of AT0 , since λ3 + λ2 + f 0 (0) = 0
⇐⇒
(−λ)3 − (−λ)2 − f 0 (0) = 0 .
Thus, a general solution of (2.19) turns out to be y˜(t) = ce−λ1 T0 t (1, −T0 λ1 , T02 λ21 ) + (a + ib)e−T0 (µ+iη)t (1, −T0 (µ + iη), T02 (µ + iη)2 ) + (a − ib)e−T0 (µ−iη)t (1, −T0 (µ − iη), T02 (µ − iη)2 ) . The boundary conditions imply (B1) c = −2a; (B2) b =
a(µ−λ1 ) ; η −T0 (µ−λ1 )
(B3) (λ1 − 1) + e
[cos ηT0 − µ cos ηT0 − η sin ηT0 ] −T0 (µ−λ1 ) (µ−λ1 ) [−µ sin ηT0 + η cos ηT0 + sin ηT0 ] = +e η
0.
Condition (B3) is equivalent to condition (A3): indeed, using 1 − λ1 = 2µ, we have 1 (B3) ⇔ (λ1 − 1)eT0 (µ−λ1 ) + (1 − λ1 ) cos T0 η = (η 2 + µ2 − λ1 µ − µ + λ1 ) sin T0 η η ⇔ 2ηµ[cos T0 η − e(µ−λ1 )T ] = (η 2 + λ21 − µ2 ) sin T0 η ⇔ (A3) Taking into consideration (B1) and (B2) we see that the solution of (2.20) is y(t) = −e−T0 λ1 t + e−T0 µt cos T0 ηt +
(µ − λ1 ) −T0 µt e sin T0 ηt , η
(2.21)
i.e. ker(Fz (0, T0 ))∗ = [y]. Since (R Fz (0, T0 ))⊥ = ker (Fz (0, T0 ))∗ , we see that condition (iii) of the previous theorem is satisfied.
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Lemma 2.2. The solution y(t) given in (2.21) satisfies y(t) < 0, t ∈ (0, 1). Proof. Note that y(0) = 0 = y 0 (0). Since by Lemma A.2 (Appendix) 3π 4 < T0 η < π, we have y(1) < 0, and it is easy to check that y 00 (0) < 0. Hence y(t) is negative in a neighborhood of the boundary points. We show now that there cannot exist s ∈ (0, 1) with y(s) = 0. Note that if y(s) = 0 for some s, then 0 < T0 ηs < π2 . Consider (µ − λ1 )µ 0 −λ1 tT0 −T0 µs −T0 µs . y (s) = T0 λ1 e − T0 λ1 cos T0 ηs e − T0 e sin T0 ηs η + η 1 )µ Note that η + (µ−λ is positive (see Appendix); furthermore, y(s) = 0 implies η cos T0 ηs e−T0 µs − e−λ1 tT0 > 0. Thus, we have y 0 (s) < 0, whenever y(s) = 0; therefore y(t) < 0, t ∈ (0, 1].
Note that FT,T (0, T0 ) = 6 T0 f (0) = 0 ∈ Y1 . Thus, in order to verify condition (iv), we need to show that FT,x (0, T0 )[x] = −x00 − 3T02 f 0 (0)x 6∈ Y1 , or equivalently, that
Z
1
(x00 + 3T02f 0 (0)x)y 6= 0 ,
(2.22)
0
where x is given by (2.15) and y by (2.21). The proof of Ineq. (2.22) uses some lenghty calculations; we give it in the appendix. 2.3. Global extension of the bifurcation branch We now proceed to extend the local branch obtained in Subsec. 2.2. By Theorem 2.1 we know that the equation z 000 = T z 00 + T 3 f (z) ,
z(0) = z(1) = z 00 (1) = 0
(2.23)
has a branch of solutions (z, T ) in some neighborhood of (0, T0 ) ∈ X × R. Note that Lemma 2.1 implies that the solutions zT bifurcating from (0, T0 ) do not change sign, since by the Bifurcation Theorem the solutions near (0, T0 ) can be represented ) 1 as zT = ρ(T )x + h(T ), ρ ∈ R and h ∈ X, with k h(T ρ(T ) kC → 0 as T → T0 . Furthermore, observe that the solutions (z, T ) of Eq. (2.23) are uniquely characterized by (z 0 (1), T ), since for fixed T the “initial” conditions z(1) = z 00 (1) = 0 and z 0 (1) determine the solutions uniquely. Thus, the bifurcation branch is characterized by a curve in some neighborhood of (0, T0 ) in R × R. To continue this branch, we consider now the initial value problem z 000 = T z 00 + T 3 f (z) z(1) = 0 ,
z 00 (1) = 0 , z 0 (1) = α ,
which we consider backward in time, i.e. for t < 1.
(2.24)
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THE LORENTZ–DIRAC EQUATION, I
We first choose some solution z¯ on the local branch, and keep the corresponding T¯ fixed. We now shoot (backwards), varying α; for α ¯ = z¯0 (1) the first zero of the solution z¯ is in t = 0. In the following Lemma 2.4 it is shown that this zero is always non degenerate, i.e. z¯T0 (0) > 0. Thus, by the continuous dependence of solutions on ¯ a (first) parameters for ODE’s the corresponding solution z = zα has for α near α zero in a = aα near 0; proceding similarly near any “starting point” α, we see that this procedure is open. Next, we show that this procedure is also closed, as long as aα is bounded from below. ¯: We first show that there exists a δ > 0 such that aα ≤ 1 − δ, for all α ≤ α ¯3 indeed, let δ > 0 such that T2 δ 2 m ≤ 12 |α|, where m = max |f (s)|. We show that y(s) ≥ (1−s)|α| > 0, for 1 − δ ≤ s < 1; if not, there exists some r ∈ (1 − δ, 1) with 2(1+T¯ ) y(r) <
(1−r)|α| . 2(1+T¯ ) 0
Integrating the equation twice we find Z
0
1
Z
y (t) = y (1)(1 + T¯ (1 − t)) + T¯y(t) + T¯ 3
f (y(τ ))dτ ds t
and hence 1 (1 − r) |α| + |α| < y (r) ≤ −|α| + T¯ 2 2(1 + T¯ ) 0
1
s
1 T¯ − + |α| , 2 2(1 + T¯)
which implies finally Z 1 |α| T¯ 1 y(r) = − |α| = (1 − r) , y 0 (s) > (1 − r) 2 2(1 + T¯) 2(1 + T¯ ) r contradicting the assumption. Let now αn → α ≤ α ¯ , with aαn ≥ −c. Since the first zero aαn satisfies −c ≤ aαn ≤ 1 − δ, we have (for a subsequence) aαn → a0 , and since on [a0 , 1] the equation is satisfied (with y(a0 ) = 0 = y(1), y 0 (1) = α), we see that a0 = aα . Thus, this extension procedure is open and closed, and we can extend it to the ¯ where α0 is such that aα → −∞ as α → α0 . interval (α0 , α], We now perform the following change of variables y(t) = z(aα (1 − t) + t), t ∈ (0, 1). Then y satisfies the equation y 000 = T¯(1 − aα )y 00 + T¯ 3 (1 − aα )3 f (y); y(0) = y(1) = y 00 (1) = 0 .
(2.25)
Denote Tα = T¯(1 − aα ); thus y satisfies y 000 = Tα y 00 + Tα3 f (y) ,
y(0) = y(1) = y 00 (1) = 0 ;
with y 0 (1) = z 0 (1)(1 − aα ) = α(1 − aα ) .
(2.26)
This yields a global branch of solutions, which is parametrized by α: indeed, we have the parametrization α → (yα0 (1), Tα ) = (α, T¯)(1 − aα ), where aα may vary in (−∞, 1 − δ).
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√ Proposition 2.4 below gives the estimate −T 2M ≤ y 0 (1) ≤ −dT √ as T → ∞, ¯ 2M : indeed, ≥ − T for some d > 0 and M√= max V (s); this implies that α 0 √ |α(1 − aα )| = |y 0 (1)| ≤ 2M Tα = 2M T¯ (1 − aα ). Furthermore, √ we observe that for T → ∞ (iff α → α0 ) we have yα0 (1)/Tα = α/T¯ → α0 /T¯ ≥ − 2M . Thus we have proved: Proposition 2.1. There exists a global extension Σ of the bifurcation branch, parametrized by (z 0 (1), T ) ∈ R2 . The first component covers (−∞, 0), while the T component√contains the semi-axis (T0 , +∞). Furthermore, there exists a constant v∞ with − 2M ≤ v∞ < 0 such that z 0 (1)/T → v∞ as T → +∞. 2.4. Properties of the solutions Next we show that along the branch given above the solutions stay always strictly positive and below 1. More precisely, we show Lemma 2.3. The solutions (zT , T ) ∈ X × R of (2.23) on the branch Σ constructed above satisfy 0 < zT < 1 on (0, 1). Proof. (i) zT (t) > 0, t ∈ (0, 1): suppose that zT (t0 ) = 0 for some t0 ∈ (0, 1); since zT (t) ≥ 0 we have zT0 (t0 ) = 0. Multiply the equation by zT0 (t) and integrate from t0 to 1. We have Z 1 Z 1 Z 1 000 0 00 0 3 zT · zT = T zT · zT + T f (zT ) · zT0 t0
t0
Z
1
−
|zT00 |2 =
t0
t0
T 02 T3 zT (1) + [V (zT (1)) − V (zT (t0 ))] 2 2
a contradiction. Hence zT (t) > 0. (ii) zT (t) < 1, t ∈ (0, 1): First note that for T = 0 Eq. (2.23) has only the trivial solution z ≡ 0. Thus, we may assume that T > 0. We show that zT (t) < 1, ∀ t ∈ (0, 1): since we are staying along a continuous branch bifurcating from the trivial solution, we conclude that if there exists a solution zT with zT (t) = 1 at some point t ∈ (0, 1) then there exists a T1 ≤ T such that zT1 has a maximum at some t0 with zT1 = 1, i.e. we have zT0001 (t) = T1 zT001 (t) + T13 f (zT1 (t))
t ∈ (0, 1)
(2.27)
and zT1 (t0 ) = 1, zT0 1 (t0 ) = 0. Multiply (2.27) by zT0 1 (t) and integrate from t0 to 1. We have Z 1 Z 1 Z 1 zT0001 zT0 1 = T1 zT001 zT0 1 + T13 f (zT1 )zT0 1 . t0
t0
t0
Using the boundary conditions and the facts zT1 (t0 ) = 1, zT0 1 (t0 ) = 0, we have Z 1 T1 0 2 |z | (1) , − |zT001 |2 = 2 T1 t0 which is a contradiction.
THE LORENTZ–DIRAC EQUATION, I
667
Next, we give some properties of the solutions in the boundary points. Lemma 2.4. If zT 6≡ 0 is a solution on the bifurcation branch Σ starting in (0, T0 ) of (2.23), then (i) zT0 (1) < 0, (ii) zT0 (0) > 0, and (iii) zT00 (0) < 0, for all (zT , T ) on the branch Σ. Proof. We have seen that near the bifurcation point the solutions zT are as affirmed in the lemma. Suppose now that T1 is the first T for which one of (i), (ii) or (iii) is not satisfied. We write z = zT1 (i) If z 0 (1) = 0, then by uniqueness of the solution for the initial value problem we would have z(t) ≡ 0 in [0, 1], contradicting the assumption. Since z(t) ≥ 0 we conclude that z 0 (1) < 0. (ii) Clearly z 0 (0) ≥ 0; if z 0 (0) = 0, then the equation yields (integrating against z 0 ) Z 1 T − |z 00 |2 = |z 0 |2 (1) , 2 0 which gives a contradiction of sign. (iii) (a) Suppose z 00 (0) > 0; since z(1) = 0, there exists a point t0 such that z 00 (t0 ) = 0 and z 00 (t) > 0, ∀ t < t0 . Integrating the equation (against z 0 from 0 to t0 yields Z t0 T T3 V (x(t0 )) |z 00 |2 = [(z 0 (t0 ))2 − (z 0 (0))2 ] + −z 00 (0)z 0 (0) − 2 2 0 Since the right-hand side is positive, while the second term on the left is negative, the first term on the left must be positive, and since z 0 (0) > 0, we must have z 00 (0) < 0. (4) (b) Suppose z 00 (0) = 0; then z 000 (0) = 0 by the equation, and zT (0) = 000 3 0 0 00 T zT (0) + T f (0)z (0) > 0. Thus zT (t) > 0 in some interval (0, δ), and thus the previous proof can be repeated. We now show that any solution of (2.23) which satisfies 0 < zT (t) ≤ s0 , where s0 ∈ (0, 1) is the unique point with f (s0 ) = 0, is concave: Proposition 2.2. Suppose that z(t) := zT (t) satisfies 0 < z(t) ≤ s0 ; then z 00 (t) < 0, for all t ∈ [0, 1). Proof. Consider the cases: (a) max z(t) < s0 : suppose that there exists a t0 ∈ (0, 1) with z 00 (t0 ) = 0; since f (z(t0 )) > 0 we conclude that z 000 (t0 ) > 0, which implies that z 00 (t) is increasing in [t0 , 1), and thus cannot satisfy the boundary condition z 00 (1) = 0: contradiction.
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(b) max z(t) = s0 : suppose again that z 00 (t0 ) = 0 for some t0 ∈ (0, 1); if t0 is not a maximum point for z, then z 000 (t0 ) > 0 and hence z 00 (t) > 0 for t in a right neighborhood of t0 , and hence for all t ∈ (t0 , 1]; thus, z 00 (1) 6= 0: contradiction. Finally, if t0 is a maximum point for z, then the initial conditions z(t0 ) = s0 , z 0 (t0 ) = 0, z 00 (t0 ) = 0 yield the unique (constant) solution z(t) = s0 , which does not satisfy the boundary conditions z(0) = z(1) = 0: contradiction.
Hence, the proposition is proved.
Next, we show that the solutions stay asymptotically away from the trivial solution. Proposition 2.3. There exists a constant d > 0 such that the solutions zT satisfy kzT kC 0 ≥ d
as T → +∞ .
Proof. (a) Suppose to the contrary that there is a sequence Tn → +∞ such that kzn k := kzTn kC 0 → 0. We show that then there exists a δ > 0 such that Z 1 f (zn (t)) sin πt ≥ δkzn kC 0 . (2.28) 0
Indeed, since kzn kC 0 → 0 we have for some 0 ≤ θn (t) ≤ kzn kC 0 and some c > 0 Z 1 Z 1 z2 0 00 sin πt f (0)zn + f (θn ) f (zn ) sin πt = 2 0 0 Z 1 ≥ f 0 (0) zn sin πt − ckzn k2C 0 . 0
We now show that
Z
1
zn sin πt ≥ 0
1 kzn kC 0 , π
(2.29)
which then yields (2.28). Let mn = maxt∈[0,1] zn (t) = zn (tn ); note that tn is unique, since zn00 (t) < 0 by Proposition 2.2. Using again the concavity of zn we can estimate Z 1 Z tn Z 1 mn mn zn sin πt ≥ t sin πt + (1 − t) sin πt . (2.30) t 1 − tn n 0 0 tn One then calculates Z 1 1 mT sin πtT mT sin πtT mT sin πtT ≥ mT , zn sin πt ≥ + = 2 2t 2 π 1 − t π π t (1 − t ) π T T T T 0 which proves (2.29). To prove the proposition, we multiply Eq. (2.23) by sin πt and integrate from 0 to 1: Z 1 Z 1 Z 1 zn cos πt + T π 2 zn sin πt = T 3 f (zn ) sin πt , π(zn0 (0) + zn0 (1)) + π 3 0
0
0
669
THE LORENTZ–DIRAC EQUATION, I
which implies by the above estimate π3 π(|zn0 (0)| + |zn0 (1)|) + kzn kT 2 T2
Z
1 0
π2 zn cos πt + kzn k T
Z
1
0
zn sin πt ≥ T δ . kzn k
Z
1
(2.31)
On the other hand, integrating (2.23) we have −zn00 (0) + T (zn0 (0) − zn0 (1)) = T 3
f (zn ) , 0
which yields, using that zn00 (0) < 0 and zn0 (0) > 0, zn0 (1) < 0 |zn (0)| + |zn0 (1)| ≤ c. T 2 kzn k Hence the terms on the left in (2.31) are bounded as T → ∞, while the right-hand side tends to +∞. This contradiction proves the proposition. 2.5. Asymptotic estimates In this section we derive some asymptotic properties of the solutions, assuming hypothesis (V 3) and (V 4). First, we show some asymptotic bounds: Proposition 2.4. Assume (V 1)–(V 4). Let zT denote the solutions on the branch Σ bifurcating from T0 . Then there exist positive constants 0 < d < c such that 00 |zT (t)| T2 0 |zT (t)| T
≤ c, for all (zT , T ) on Σ, ∀ t ∈ [0, 1]; √ |z 0 (1)| (b) ≤ c, ∀ t ∈ [0, 1], and TT ≤ 2M , for all (zT , T ) on Σ, where M = max V (s); R 1 00 2 (c) 0 |zT 3| ≤ c, for all (zT , T ) on Σ; R1 |z 0 (1)| |z 0 (0)| and d ≤ T13 0 |z 00 |2 dt, for T → ∞. (d) d ≤ TT , d ≤ TT (a)
Proof. (a) Let t0 ∈ (0, 1) denote an extremal point of zT00 (t) (since zT000 (0) = T 2 zT00 (0) < 0 and zT00 (1) = 0 there must exist an internal extremal point); then zT000 (t0 ) = 0, and hence the equation yields |zT00 (t0 )| = T 2 |f (z(t0 ))| ≤ T 2 c. (b) By Proposition 2.2 and Proposition 2.5 which we prove later, we have zT00 (t) < 0, for all t ∈ [0, 1) and for all T . Let a ∈ (0, 1) denote the maximum point of zT ; multiplying the equation by zT0 and integrating from 0 to a we get Z a T |zT00 |2 dt + T 3 V (z(a)) . −zT00 (0)zT0 (0) + |zT0 (0)|2 = 2 0 Noting that zT00 (0) < 0 and zT0 (0) > 0 we have 00 Z a Z a z 0 (0) 1 1 |zT0 (0)|2 −zT00 −zT 00 2 dt + c ≤ c T + c. · ≤ |z | dt + c ≤ max T 2 3 2 2 T T 0 T T T [0,a] 0
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B. RUF and P. N. SRIKANTH
z 0 (0)
Hence TT ≤ c. Multiplying again the equation by zT0 and integrating to an arbitrary point s ∈ (0, 1] we have Z s T |zT00 |2 = [zT0 (s)|2 − |zT0 (0)|2 ] + T 3 V (zT (s)) zT00 (s)zT0 (s) − zT00 (0)zT0 (0) − 2 0 and hence by (a) and the previous estimate |zT0 (s)| 1 |zT0 (0)|2 |zT00 (0)| |zT0 (0)| |zT00 (s)| |zT0 (s)| 1 |zT0 (s)|2 + ≤ c + c ≤ + 2 T2 2 T2 T2 T T2 T T which implies |zT0 (s)| ≤ cT . Furthermore, multiplying the equation by zT0 (t) and integrating from a to 1, where a ∈ (0, 1) denotes the first maximum point of zT , we obtain Z Z 1 T 1 1 00 |zT00 |2 (t)dt = |zT0 (1)|2 + T 3 (V (zT (1) − V (zT (a))|zT0 (1)|2 + |z (t)|2 dt − 2 T a T a ≤ 2T 2 V (z(a)) ≤ 2M T 2 .
(2.32)
(c) By (a) and (b) it now follows that 00 Z 1 Z 1 zT z 0 (0) − zT0 (1) 1 −zT00 00 2 dt ≤ c T ≤ c. |z | ≤ max T 3 2 T 0 [0,1] T T T 0 (d) first estimate: recall that by Proposition 2.2 any solution with max zT ≤ s0 is concave. In Proposition 2.5 below it is shown that under hypothesis (V 4) this is always the case, that is, one has for all T that max zT < s0 . Let now bT ∈ (0, T ) denote the last point where zT (bT ) = d/2, where d is the constant from Proposition 2.3, and cT ∈ (bT , 1) such that zT (cT ) = d/4. Integrating the equation from bT to cT we have cT 2 ≥ |zT00 (bT )| + |zT00 (cT )| + T |zT0 (bT )| + T |zT0 (cT )| Z cT f (zT ) ≥ T 3 (cT − bT )f (d/4) , ≥ T3 bT
and hence, using the concavity of zT , 1 − bT ≤ 2(cT − bT ) ≤ c/T . With this we conclude, again using the concavity of zT : |zT0 (1)| ≥
|zT (1) − zT (bT )| Td . ≥ 1 − bT c2
second estimate: from the energy equation (multiplying the equation by zT0 and integrating over (0, 1)) we have Z 1 T |zT00 (t)|2 = (|zT0 (1)|2 − |zT0 (0)|2 ) , −zT00 (0)zT0 (0) − 2 0 and hence cT 2 |zT0 (0)| + which implies
zT0 (0)
T 0 |z (0)|2 ≥ 2 T
≥ dT .
Z 0
1
|zT00 (t)|2 +
T 0 |z (1)|2 ≥ dT 3 , 2 T
THE LORENTZ–DIRAC EQUATION, I
671
third estimate: let aT denote the first point where T1 zT0 (aT ) = d2 , where d is the constant from the second then we have aT ≥ Tc as T → ∞, since R aT estimate; d 0 0 00 2 2 T = |zT (aT ) − zT (0)| = | 0 zT | ≤ aT cT . Further, there exists θ ∈ (0, aT ) such that z 0 (aT ) − zT0 (0) d2 ≤ − T2 . zT00 (θ) = T aT 2 Since the equation yields |zT000 (t)| ≤ cT 3 , ∀ t ∈ [0, 1], we obtain by integration that zT00 (t) R≤ −cT 2 , for R t t in some interval [t1 , t2 ] containing θ, and with t2 − t1 ≥ c/T . 1 Thus 0 |zT00 |2 ≥ t12 cT 4 ≥ cT 3 . Next we give an estimate for the approximation of the solution zT by the solution of the linearized equation in s0 . We write zT = s0 + yT . Lemma 2.5. Suppose that f (s0 ) = 0 and f 0 (s0 ) < 0; consider ( 000 yT = T yT00 + T 3 f (s0 + yT ) yT (a) = l1 ,
yT (b) = l2 ,
yT0 (b) = m
and the linearized equation in s0 ( 000 xT = T x00T + T 3 f 0 (s0 )xT xT (a) = l1
xT (b) = l2 ,
x0T (b) = m .
(2.33)
Suppose that |yT (t)| ≤ , t ∈ [a, b]. Then max[a,b] |yT − xT | ≤ cT 1/2 2 , and |yT0 (a) − x0T (a)| ≤ cT 3/2 2 . Proof. We have |yT (t)| ≤ , a ≤ t ≤ b, and yT satisfies for some |θ(t)| ≤ . y 000 = T y d00 + T 3 f (s + y ) = T y 00 + T 3 f 0 (s )y + T 3 f 00 (s + θ(t)) 1 (y )2 T 0 T 0 T 0 T T T 2 , yT (a) = l1 , yT (b) = l2 , yT0 (b) = x0T (b) = m (2.34) Note that wT := yT − xT satisfies w00 = T w00 + T 3 f 0 (s )w + T 3 f 00 (s + θ) 1 (y )2 0 T 0 T T T 2 (2.35) wT (a) = wT (b) = 0 , wT0 (b) = 0 Multiplying with wT and integrating over [a, b] yields b Z b Z b Z b 1 0 2 1 0 2 3 0 2 3 − |wT | + T |wT | + T |f (s0 )| wT = T f 00 (θ) yT2 wT , 2 2 a a a a and hence
Z
!1/2
b
|wT |
2
a
≤ c|yT |2∞ ≤ c2 ,
(2.36)
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which then yields 1 0 2 |w | (a) + T 2 T
Z
b
|wT0 |2 ≤ cT 3 4 ,
(2.37)
a
and hence Z
b
!1/2 |wT0 |2
and |yT0 (a) − x0T (a)| = |wT0 (a)| ≤ cT 3/2 2 .
≤ cT 2
a
Now, let σ ∈ (a, b) such that |wT (σ)| = |wT |∞ . Multiply Eq. (2.35) by wT0 and integrate from σ to b: Z b Z b 1 y2 − |wT00 |2 = T 3 |f 0 (s0 )| |wT (σ)|2 + T 3 f 00 (θ) T wT0 , (2.38) 2 2 σ σ which gives, using (2.37) 1 3 0 T |f (s0 )kwT (σ)|2 + 2
Z
b
|wT00 |2 ≤ T 3 c2
Z
σ
b
!1/2 |wT0 |2
≤ cT 4 4 ,
σ
which finally yields |yT − xT |∞ = |wT |∞ ≤ T 1/2 c2 .
(2.39)
We now show that if condition (V 4) holds, then the solutions zT remain below s0 , the maximum point of V (s), for all T , and thus they remain concave by Proposition 2.2. 4 . Then max zT (t) < s0 , for all Proposition 2.5. Suppose 0 > f 0 (s0 ) > − 27 (zT , T ) on Σ.
Proof. (1) Assume that there exists a T > 0 such that max zT (t) = s0 . Let > 0 be given. As zT is strictly concave, denote with a = a and b = b the unique points with z(a ) = z(b ) = s0 − . Again by the strict concavity (i.e. zT00 (t) ≤ −δ < 0, √ t ∈ [a , b]), we have b − a ≤ c . (2) The solution of the linearized equation is given by v(t) = αe−λ1 T t + βeλ2 T t + γeλ3 T t ,
with
λi > 0, i = 1, 2, 3 .
4 The λi are positive by the hypothesis 0 > f 0 (s0 ) > − 27 . The boundary conditions in a and b yield: (a) v(a) = − = αe−λ1 T a + βeλ2 T a + γeλ3 T a implies
α = −βe(λ2 +λ1 )T a − γe(λ3 +λ1 )T a − eλ1 T a .
(2.40)
Hence v(t) = (−βe(λ2 +λ1 )T a − γe(λ3 +λ1 )T a − eλ1 T a )e−λ1 T t + βeλ2 T t + γeλ3 T t .
THE LORENTZ–DIRAC EQUATION, I
673
(b) v(b) = (−βe(λ2 +λ1 )T a − γe(λ3 +λ1 )T a − eλ1 T a )e−λ1 T b + βeλ2 T b + γeλ3 T b = − and hence β=
−(1 − eλ1 T (a−b) ) − γeλ3 T b (1 − eT (λ3 +λ1 )(a−b)) ) . eλ2 T b (1 − eT (λ2 a+λ1 )(a−b) )
(2.41)
Developing β in terms of b − a we get (after some calculations and rearranging) β=
1 [−λ1 e−λ2 T b − γe(λ3 −λ2 )T b (λ3 + λ1 )] λ2 + λ1 +
b−a T [−λ1 e−λ2 T b λ2 + γe(λ3 −λ2 )T b (λ3 + λ1 )(λ3 − λ2 )] λ2 + λ1 2
+ ( + γ)O(b − a)2 .
(2.42)
(3) Claim: v 0 (a) > 0 =⇒ γ ≥ −c Indeed, developing again in b − a, we calculate v 0 (a) = βT eλ2 T a (λ2 + λ1 ) + γT eλ3 T a (λ3 + λ1 ) + λ1 T T2 (b − a) + γeλ3 T b T 2 (λ3 + λ1 ) 2 λ3 − λ2 (b − a) + ( + γ)O(b − a)2 × λ2 + 2
= λ1 λ2
which implies γ ≥ −c, for > 0 sufficiently small. (4) Claim: v 0 (b) < 0 =⇒ γ ≤ c Again developing in b − a yields v 0 (b) = βeλ2 T b T (λ2 + λ1 e(λ2 +λ1 )T (a−b) ) + γeλ3 T b × T (λ3 + λ1 e(λ3 +λ1 )T (a−b) ) + λ1 T eλ1 T (a−b) =
T2 [−λ1 λ2 + γeλ3 T b (λ3 + λ1 )(λ3 − λ2 )](b − a) + ( + γ)O(b − a)2 < 0 . 2
This implies γ ≤ c, for sufficiently small. (5) max[0,1] vT (t) ≤ − + c3/2 , for > 0 sufficiently small Again developing, we obtain now v(t) = βeλ2 T t (1 − e(λ2 +λ1 )T (a−t) ) + γeλ3 T t (1 − e(λ3 +λ1 )T (a−t) ) − eλ1 T (a−t) = [−λ1 [1 − λ2 T (b − t) + O(t − a)2 ] − γeλ3 T b [1 − λ2 T (b − t) + O(t − a)2 ](λ3 + λ1 )] · [T (t − a) − O(t − a)2 ] + γeλ3 T t (λ3 + λ1 )T (t − a) − + λ1 T (t − a) + ( + γ)O(t − a)2 = − + ( + γ)O(b − a) ≤ − + c3/2 (6) By the estimate of Lemma 2.5 we have for zT = s0 + yT |yT (t) − v(t)| ≤ c2 ,
for > 0 sufficintly small ,
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B. RUF and P. N. SRIKANTH
and hence for > 0 sufficiently small max zT (t) ≤ s0 + max v(t) + c2 ≤ s0 − + 2ca3/2 < s0 . [0,1]
[0,1]
3. Transmission Solutions 3.1. Continuation from the “trivial ” transmission solution We now consider the existence of transmission solutions, that is solutions of ( 000 z = T z 00 + T 3 f (z) , t ∈ (0, 1) , (3.43) (T r) z(0) = 0 , z(1) = 1 , z 00 (1) = 0 where we look for solutions satisfying 0 < z(t) < 1, for 0 < t < 1. We rewrite (3.43) setting z(t) = y(t) + t, t ∈ [0, 1]; then y satisfies y 000 (t) = T y 00 (t) + T 3 f (y(t) + t) ,
y(0) = y(1) = y 00 (1) = 0 .
(3.44)
Note that (3.44) has for T = 0 the trivial solution y(t) ≡ 0. The interpretation is that for T → 0 the solutions zT approximate the “trivial” transmission solution z0 (t) = t (which crosses the obstacle in zero time). We employ the Implicit Function Theorem to solve (3.44) near the trivial solution. Let X and Y denote the spaces as given in (2.7), and define the mapping F :R×X →Y
F (T, y) = y 000 − T y 00 − T 3 f (y + t)
(3.45)
Note that F is a C 1 mapping with F (0, 0) = 0; furthermore, Fy (0, 0)[v] = v 000 , and thus Fy (0, 0) : X → Y is an isomorphism. Hence by the Implicit Function Theorem we get a local branch of solutions (T, yT ) in a neighborhood of (0, 0) ∈ R × X. We will restrict attention to solutions (T, yT ) with T > 0. 3.2. Global extension of the branch and properties of the solutions The branch of transmission solutions can be extended to a global branch, proceeding as in the extension of the branch of reflection solutions, see Sec. 2.3. The main ingredient for this extension via shooting is the non degeneracy of the zeros of any solutions of (T r). This and other properties of transmission solutions are proved in the following: Lemma 3.1. (i) 0 < zT (t) < 1, for all t ∈ (0, 1) (ii) zT0 (1) > 0 and zT0 (0) > 0 (iii) zT00 (0) < 0 Proof. (i) NoteR that kyT kC 3 → 0 as T → 0. Thus we have for T > 0 sufficiently t small: yT + t = 0 (yT0 (s) + 1)ds > 0; furthermore, since (yT (t) + t)0 = yT0 (t) + 1 > 0 (for T small) and yT (1) + 1 = 1 we have yT (t) + t < 1. One now proceeds as in Lemmas 2.4 and 2.3 to extend the statement to all T > 0.
THE LORENTZ–DIRAC EQUATION, I
675
(ii) proceed as in Lemma 2.4, (i) and (ii) (iii) Assume zT00 (0) ≥ 0; then zT (t) > 0 to the right of 0 by the equation. If 00 zT (t) > 0 for all t ∈ (0, 1), we have 0 < zT0 (0) < zT0 (1), and hence we obtain from the equation the contradiction of sign Z 1 T −zT00 (0)zT0 (0) − |zT00 |2 = (|zT0 (1)|2 − |zT0 (0)|2 ) ; 2 0 thus, there must exist a first t0 ∈ (0, 1) with zT00 (t0 ) = 0. This yields 0 < zT0 (0) < zT0 (t0 ), which yields again a contradiction, integrating the equation (against zT0 ) from 0 to t0 . Next we show that if condition (V 4) holds, then any transmission solution has a unique turning point: Proposition 3.1. Suppose (V 4) is satisfied. Then (i) there exists a unique point aT ∈ [0, 1) such that zT00 (aT ) = 0, and zT is concave before aT and convex after aT . (ii) there exists a unique point bT ∈ [0, 1] such that zT (bT ) = s0 (iii) aT < bT , and zT0 (bT ) > 0, for all T > 0. Proof. (i) Note that in the proof of Lemma 3.1, (iii) we have seen that there exists a point t0 ∈ (0, 1) with zT00 (t0 ) = 0. Let now bT ∈ (0, 1) denote the last point where zT (t) = s0 . After this point the solution zT is convex: indeed, assume that there exists a point d > bT with zT00 (d) ≤ 0; since f (zT (d)) < 0 we get zT000 (d) < 0, and then the equation (and the assumption on bT ) implies that zT00 (t) is decreasing for t ∈ (bT , 1], thus contradicting the boundary condition zT00 (1) = 0. Note that also in bT we have zT00 (bT ) > 0: if zT00 (bT ) = 0, then zT0 (bT ) > 0 (otherwise zT would be the constant solution s0 ) and by the equation zT000 (bT ) = 0; then, taking the derivative of the equation, we get (4) zT (bT ) = T 3 f 0 (s0 )zT0 (bT ) < 0, which implies zT00 (t) < 0 for t > bT and close to bT , which yields as before a contradiction. Next, let cT ≤ bT denote the first point where zT (t) = s0 . Before this point there is at most one point, say aT with zT00 (aT ) = 0: indeed, in aT we have zT000 (aT ) = T 3 f (z(aT )) > 0, and hence zT00 (t) is increasing to the right of aT , and (by the equation) will continue to do so until zT reaches s0 . To complete the proof it suffices to show that cT = bT ; this is done in (ii). (ii) In the proof of Lemma 3.1 it was noted that for small T the solutions zT are increasing, and hence cT = bT for small T . The only way cT and bT may separate is in view of (i) that zT develops a local maximum before bT which then touches s0 for some T∗ ; but this is impossible by applying the proof of Proposition 2.5 to this local maximum. (iii) Note that in (i) we have shown that zT00 (t) > 0, for t ∈ [bT , 1); thus aT < bT . The second statment follows, since if zT is strictly monotone, then there is nothing to show, if not, then there exists either a unique horizontal turning point c with
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B. RUF and P. N. SRIKANTH
zT (c) < s0 (and hence zT0 (t) > 0 for t ∈ (c, 1]) or there exists a unique local maximum d with zT (d) < s0 , and hence we must again have zT0 (bT ) > 0. We now show the analogous estimates of Proposition 2.4 for transmission solutions. Proposition 3.2. Let zT , T > 0, denote the branch of transmited solutions. Then there exist constants 0 < d < c such that (a) (b) (c) (d)
00 |zT (t)| ≤ c, for all (zT , T ) on Σ, ∀ t T2 0 |zT (t)| ≤ c, for all (zT , T ) on Σ, ∀ t T R 1 |z00 |2 ≤ c, for all (zT , T ) on Σ 0 T3 0 |zT (1)| |z 0 (0)| for T → ∞, d ≤ T , d ≤ TT
∈ [0, 1] ∈ [0, 1] and d ≤
1 T3
R1 0
|z 00 |2 dt.
Proof. (a) as in Proposition 2.4. (b) zT0 (1) ≤ cT : Let yT (t) = zT (t) − t; then yT satisfies yT000 = T yT00 + T 3 f (yT (t) + t) ,
yT (0) = yT (1) = yT00 (1) = 0 .
(3.46)
Let a denote an extremal point of yT ; integrating (3.46) by yT0 and integrating from a to 1 yields Z 1 Z 1 Z 1 T |yT00 |2 = |yT0 (1)|2 + T 3 f (yT + t)(yT + t)0 − T 3 f (yT + t) − 2 a a a and hence T 0 |y (1)|2 + 2 T
Z
1
|yT00 |2 ≤ T 3 V (yT (a) + a) + T 3 c ≤ cT 3 .
a
|zT0 (0)| ≤ cT : suppose first that yT (t) = zT (t) − t starts negative, i.e. yT0 (0) ≤ 0; since zT0 (0) ≥ 0 we thus have yT0 (0) = zT0 (0) − 1 ≥ −1. If on the other hand yT starts positive, then let mT denote the first maximum point of yT , and let aT < s0 denote the unique point where yT00 (t) = zT00 (t) = 0 (cf. Proposition 3.1). Note that mT < aT , since otherwise yT would be increasing, contradicting yT (1) = 0. Now proceed as in Proposition 2.4, (b) to prove |zT0 (0)| ≤ cT . |zT0 (t)| ≤ cT , for all t ∈ [0, 1]: this follows now as in Proposition 2.4, (b). (c) as in Proposition 2.4. (d) first estimate: by Proposition 3.1 the solution zT is convex above s0 (i.e. for 0 t such that zT (t) ≥ s0 ). Let 0 < d < 1−s 2 , and let bT such that zT (bT ) = 1 − 2d and cT such that zT (cT ) = 1 − d. The claim now follows arguing now as in the proof of Proposition 2.4, (d), first estimate. second estimate: as in Proposition 2.4, (d), second estimate. third estimate: by the second estimate, we have zT0 (0) ≥ dT , for T large. Since yT (t) = zT (t) − t has an extremal in (0, 1), there exists a point t0 with zT0 (t0 ) = 1, and hence there exists a (first) value aT ∈ (0, 1) with zT0 (at ) = d2 T , for T large. Now argue as in Proposition 2.4, (d), third estimate, to complete the proof.
677
THE LORENTZ–DIRAC EQUATION, I
4. Convergence for Reflection and Transmission Solutions In this section we show that the reflection and transmission solutions converge pointwise on (0, 1) and uniformly on compact subsets of (0, 1) to s0 , the maximum point of the potential V (x). Proposition 4.1. Assume (V 1)–(V 4), and let > 0 be given. (a) Reflection: one has max zT → s0 , as T → ∞; thus denote with a,T the first and with b,T the next (and last ) value where a reflection solution zT assumes the value s0 − , i.e s0 − < zT (t) < s0 ,
for a,T < t < b,T .
(b) Transmission: denote with a,T the first value where the transmission solution zT assumes the value s0 − , and with b,T ∈ (a,T , 1) the value closest to a,T with |zT (b,T ) − s0 | = , i.e. s0 − < zT (t) < s0 + ,
for a,T < t < b,T .
Then there exists a constant c > 0 such that c c and 1 − b,T ≤ . a,T ≤ T T Proof. (a) We first show that max[0,1] zT → s0 ; indeed, suppose to the contrary that zT max zT ≤ s0 − δ, for some δ > 0 and all T . Integrate the equation from 0 to 1 Z 1
zT00 (1) − zT00 (0) = T (zT0 (1) − zT0 (0)) + T 3
f (zT ) ; 0
let aT , bT denote the first, resp. last value with zT (aT ) = d/2 = zT (bT ), where d is as in Proposition 2.3; then bT − aT ≥ 1/2 by the concavity of zT , and thus we have by Proposition 2.4 and the concavity of zT Z bT cT 2 ≥ T 3 min f (s)dt ≥ T 3 σ, ∀ T , aT
[d/2,s0 −δ]
for some σ > 0. This contradiction proves the claim. Let now aT , resp. bT denote the values where zT (aT ) = zT (bT ) = s0 /2; assuming that < s0 /4 one sees, using the concavity of zT , that aT ≤ 2(a,T − aT ) and 1 − bT ≤ 2(bT − b,T ). Integrating the equation from aT to a,T and estimating we get cT 2 ≥ |zT00 (a,T )| + |zT00 (aT )| + |T z 0 (a,T )| + |T z 0 (aT )| Z a,T f (zT (t))dt ≥ T 3 (a,T − aT ) min f (s) ≥ T 3 c(a,T − aT ) ≥ T3 {s0 /2,s0 −}
aT
Thus, a,T = a,T − aT + aT ≤ 3(a,T − aT ) ≤ One proceeds similarly for estimating 1 − b,T .
3c . T
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B. RUF and P. N. SRIKANTH
(b) Note that we do not know whether a transmission solution is monotone, that is, we do not exclude that a transmission solution zT may have an interior local minimum; however, in case it does, say in mT , then we can say that zT (mT ) ≥ δ > 0, for T large; indeed, multiplying the equation by zT0 and integrating from mT to 1 we have Z 1 T |zT00 |2 = |zT |2 (1) − T 3 V (zT (mT )) , − 2 mT i.e. V (zT (mT )) ≥ d2 /2, where d is the constant from Proposition 2.4, (d). We consider now three cases: (i) zT is increasing and concave up to zT (a,T ) = s0 −, and in b,T holds zT (b,T ) = s0 + (and thus zT is increasing and convex in (b,T , 1)). In this case, the proof proceeds as in a). (ii) the (unique) turning point τT of zT and the local maximum (if there is one) are below s0 − , and thus zT is increasing and convex in (a,T , 1). c c and 1 − a,T ≤ T , and hence this case cannot We show that a,T ≤ T occur for T sufficiently large; indeed, since zT is increasing and convex in (a,T , 1), there are unique points a,T < c,T < d,T < 1 with zT (c,T ) = s0 c and zT (d,T ) = s0 + . It follow as in i) that 1 − d,T ≤ T . Furthermore, since 000 3 3 zRT (t) ≥ T f (zT (t)) ≥ T c for t ∈ [a,T , c,T ], we have by zT00 (t) = zT00 (a,T ) + t z 000 (t)dt ≥ cT 3 (t − a,T ) the estimate cT 2 ≥ |zT00 (c,T )| ≥ cT 3 (c,T − a,T ), a,T T c . By the convexity, we now have also d,T − c,T ≤ and hence c,T − a,T ≤ T c c . c,T − a,T ≤ T . Thus, it follows that 1 − a,T ≤ T c On the other hand, we have also a,T ≤ T : from Proposition 2.4 we know that 2 d zT0 (0) ≥ dT and |zT00 (t)| ≤ cT 2 , from which we conclude that zT ( 2cT ) ≥ d4c =: p indeed, from Z t
zT0 (t) = zT0 (0) +
zT00 (s) ≥ dT − cT 2 t
0
we have zT0 (t) ≥
dT 2
i h d , and then , t ∈ 0, 2cT
zT
d 2cT
Z = zT (0) +
d/(2cT )
zT0 (s) ≥
0
d2 . 4c
d Let now aT = 2cT , and δ > 0 from the previous argument; then, on [aT , a,T ] we have 0 < m = min{p, δ} ≤ zT (t) ≤ s0 − . Integrating the equation from aT to a,T and estimating we get
cT 2 ≥ |zT00 (a,T )| + |zT00 (aT )| + |T zT0 (a,T )| + |T zT0 (aT )| Z a,T f (zT (t))dt ≥ T 3 (a,T − aT ) min f (s) ≥ T 3 c(a,T − aT ) ≥ T3 aT
{m,s0 −}
c d + 2cT . Thus, a,T = a,T − aT + aT ≤ T (iii) zT has a local maximum in (s0 − , s0 ) and a local minimum below s0 − ; then c as in (a). In b,T zT is increasing and concave up to a,T , and thus a,T ≤ T we have now again zT (b,T ) = s0 − , and furthermore there exist the values
679
THE LORENTZ–DIRAC EQUATION, I
b,T < c,T < d,T < 1 with zT (c,T ) = s0 − and zT (d,T ) = s0 + . In (c,T , 1) c . In zT is increasing and convex, and thus one obtains as in (ii) 1 − c,T ≤ T (b,T , c,T ) we have zT (t) ∈ [m, s0 − ), where m is as in (ii), and thus we can c . Thus proceed as there to prove c,T − b,T ≤ T 1 − b,T = 1 − c,T + c,T − b,T ≤
c . T
Appendix A.1. Relations between the eigenvalues of the matrix AT The characteristic polynomial of (2.9), setting f 0 (0) = d, is λ3 − λ2 − d = 0 ;
(A.47)
it has one real root and two complex roots. Denoting λ1 , µ ± iη these roots, one has 1 < λ1 < 1 + d, and the following identities hold: −µ3 + µ2 − η 2 + 3µη 2 + d = 0
(A.48)
2ηµ − 3µ2 η + η 3 = 0
(A.49)
λ1 (µ2 + η 2 ) = d
(A.50)
λ1 + 2µ = 1
(A.51)
µ2 + η 2 = −2µλ1 .
(A.52)
By (A.52) we have µ < 0, and also λ1 > η. Furthermore, by (A.49) we have, since µ<0 √ η 2 = 3µ2 − 2µ ⇒ η > 3|µ| . In addition we have Lemma A.1. The following relations hold : λ21 − µ2 + η 2 = λ1 + 2η 2 > λ1 − 3µη
(A.53)
3d + µ2 − η 2 > −2µ(λ1 − µ)
(A.54)
3d + µ2 − η 2 > 2η 2
(A.55)
3d + µ2 − η 2 > 4η 2 a
(A.56)
3d + µ2 − η 2 > λ21 − µ2 + η 2
(A.57)
and, provided that 3|µ| ≥ 1
and finally, if |µ| ≥ 1, then η≤
√ 5|µ| .
(A.58)
680
B. RUF and P. N. SRIKANTH
Proof. Relation (A.53): by (A.51) and (A.52) we have λ21 = (1 − 2µ)λ1 = λ1 + µ2 + η 2 .
(A.59)
√ √ Since η ≥ 3|µ| we get λ1 + 2η 2 > λ1 + 2η 3|µ| which yields the claim. Relation (A.54): by (A.50), (A.51) and (A.52) we have 3d + µ2 − η 2 = 3λ1 (µ2 + η 2 ) + µ2 − η 2 = 3(1 + 2|µ|)(µ2 + η 2 ) + µ2 − η 2 = 4µ2 + 2η 2 + 6|µ|3 + 6|µ|η 2 > 3µ2 + η 2 = −2µλ1 + 2µ2 relation (A.55) and (A.56): by (A.50) and (A.51) we have 3d + µ2 − η 2 = 3(1 − 2µ)(µ2 + η 2 ) + µ2 − η 2 ; thus (A.55) holds, and (A.56) holds if −3µ > 1. Relation (A.57): We show that 4η 2 ≥ λ21 − µ2 + η 2 holds if −3µ > 1. By (A.59) it is sufficient to show 2η 2 > λ1 ; this follows, by (A.49) and (A.52), from 4η 2 ≥ 3µ2 + 3η 2 = 6|µ|λ1 > 2λ1 . Relation (A.58): follows by (A.49). A.2. Estimate of T0 Consider the function G(T ) given by (2.13). We first show: Lemma A.2. The first positive zero T0 of G(T ) satisfies
3π 4η
< T0 <
π η.
Proof. We have G(0) = 0. We consider the following subintervals: 0 ≤ T ≤ we have G0 (T ) = −η(λ21 − µ2 + η 2 ) cos T η − 2µη 2 sin T η + 2µη(λ1 − µ)e−T (λ1 −µ) .
π 4η :
(A.60)
π The last term is negative, since µ < 0. Furthermore, in 0 ≤ T ≤ 4η we have 0 cos T η ≥ sin T η ≥ 0; hence it follows by (A.53) that G (T ) < 0 and hence G(T ) < 0. π π 4η < T ≤ 2η : since cos T η ≥ 0 and µ < 0 we have
G(T ) ≤ −(λ21 − µ2 + η 2 ) sin T η + 2|µ|ηe−T (λ1 −µ) . √ √ By λ1 − µ > η we have e−T (λ1 −µ) < e−T η < e−π/4 < 1/ 2; since sin T η > 1/ 2 we obtain G(T ) < 0 by (A.53). √ √ π 3π 2η < T ≤ 4η : in this√range we have | cos T η| ≤ 1/ 2, sin T η ≥ 1/ 2 and eT (λ1 −µ) ≥ eT η ≥ eπ/2 > 2 2. Hence, by (A.53) λ2 − µ2 + η 2 G(T ) < − 1 √ + 2|µ|η 2 Next we show the following lemma:
1 1 √ + √ 2 2 2
< 0.
681
THE LORENTZ–DIRAC EQUATION, I
Lemma A.3. Let T0 > 0 denote the first positive zero of G(T ); then sin T0 η <
1 2
if |µ| ≤ 1 ,
and |
sin T0 η <
sin2 T0 η 1 |< , T0 µ 4
1 1 + √ 2 20 3
if |µ| > 1 ,
∀µ < 0
Proof. From G(T0 ) = 0 we have by (2.13) and (A.53) | sin T0 η| =
2|µ|η −T0 (λ1 −µ) − cos T η + e . 0 λ1 + 2η 2
(A.61)
We consider separately the cases 1/3 < |µ| ≤ 1, |µ| > 1, and 0 < |µ| ≤ 1/3. √ Case 1: |µ| ≤ 1. Assume sin T0 η > 1/2; then | cos T0 η| ≤ 3/2. Since furthermore e−T0 (λ1 −µ) ≤ 1/4 we get the contradiction ! √ 1 2|µ|η 3 1 + ≤ | sin T0 η| ≤ 2 λ1 + 2η 2 4 2 √ where we have used η > 3|µ| and λ1 > |µ|η. From (A.61) we now get sin T0 η 2η 5η|µ| 1 T0 µ ≤ T0 (λ1 + 2η 2 ) (1 + 1/4) ≤ 2T0 (λ1 + 2η 2 ) < 2 ,
(A.62)
since T0 η > 2 and 2λ1 ≥ 2|µ|λ1 = µ2 + η 2 > η 2 . Case 2: |µ| √ > 1. Suppose that sin T0 η ≥ 1/2. Then we have the estimates √ > 1. Thus | cos T0 η| ≤ 3/2, T0 λ1 > T0 η > 3π/4 > 2.25 and T0 |µ| ≥ √15 T0 η ≥ 43π 5
by (A.61) and e−T0 (λ1 −µ) < e−3.25 < 1/20 ! √ 1 2η 2 2|µ|η 3 √ + ≤ | sin T0 η| < 2 λ1 + 2η 2 20 2η 2 3 while
! √ 1 1 1 3 + = + √ 2 20 2 20 3
sin T0 η 2η 2 1 1 ≤ T0 µ T0 η(λ1 + 2η 2 ) (1 + 1/20) < 2.25 1 + 20 ;
multiplying the two numbers yields the last estimate. A.3. Proof of the transversality condition A simple computation and rearrangement of terms shows that proving Z 0
1
[3dT02 x(t) + x00 (t)]y(t)dt 6= 0
682
B. RUF and P. N. SRIKANTH
is equivalent to showing Z
1
[(3d + µ2 − η 2 )L(t) + K(t)]H(t)dt 6= 0
(A.63)
0
where L(t) = [−eT0 (λ1 −µ)t sin T0 η + eT0 (λ1 −µ) sin ηtT0 + sin ηT0 (1 − t)] K(t) = −(λ21 − µ2 + η 2 )eT0 (λ1 −µ)t sin T0 η + 2µηeT0 (λ1 −µ) cos ηtT0 − 2µη cos ηT0 (1 − t) H(t) = −ηe−(λ1 −µ)tT0 + η cos T0 ηt + (µ − λ1 ) sin T0 ηt . An easy but tedious computation yields: Z
1
(3d + µ2 − η 2 )
L(t)H(t)dt 0
(
= (3d + µ − η ) 2
2
(λ1 − µ) 3 η sin T0 η + cos T0 η 2 2
sin2 T0 η T0 (λ1 −µ) (λ1 − µ) T0 (λ1 −µ) e e − T0 µ 2 1 (λ1 − µ)(λ21 − µ2 + η 2 ) − sin2 T0 ηeT0 (λ1 −µ) + 4η 2 T0 µ 2T0
− (λ1 − µ)
η2 2η 2 T0 (λ1 −µ) e cos T0 η + T0 [η 2 + (λ1 − µ)2 ] T0 [η 2 + (λ1 − µ)2 ] ) η2 e−T0 (λ1 −µ) (A.64) − T0 [η 2 + (λ1 − µ)2 ] −
and Z
1
K(t)H(t)dt 0
µ η sin T0 η = − µ + η ) + (λ1 − µ)µ − T0 4µη 2 (λ1 − µ) cos T0 η + −µη 2 + T0 [η 2 + (λ1 − µ)2 ] 2µη 2 (λ1 − µ) 1 cos T0 sin T0 η + − eT0 (λ1 −µ) + 2µη 2 eT0 (λ1 −µ) 2 2T0 η T0 [η 2 + (λ1 − µ)2 ]
(λ21
2
2
THE LORENTZ–DIRAC EQUATION, I
−
(λ1 − µ) T0 (λ1 −µ) 2 η2 e sin T0 η − sin2 T0 ηeT0 (λ1 −µ) T0 T0
−
(λ1 − µ)(λ21 − µ2 + η 2 ) sin2 T0 ηeT0 (λ1 −µ) T0 µ
−
2µη 2 (λ1 − µ) T0 [η 2 + (λ1 − µ)2 ]
e−T0 (λ1 −µ) .
683
(A.65)
We will compare the expressions from (A.64) and (A.65) to show that (A.63) < 0. Case 1: 1/3 < |µ| ≤ 1. We prove first that λ1 − µ 3 η sin T0 η + cos T0 η < 0 . 2 2 Multiplying (2.13) by
3 4η
(A.66)
and using (A.53) we have
3 λ1 3|µ| 3 η sin T0 η + sin T0 η + (cos T0 η − e−T0 (λ1 −µ) ) = 0 . 2 4 η 2 Since 3|µ| = λ1 − µ − 1 by (A.51) we get λ1 − µ 3 λ1 1 3 3 η sin T0 η + cos T0 η + sin T0 η − cos T0 η − |µ|e−T0 (λ1 −µ) = 0 . 2 2 4 η 2 2 (A.67) √
Note that − cos T0 η ≥ 23 (since sin T0 η ≤ 1/2), and that 32 |µ|e−T0 (λ1 −µ) < 38 (since T0 λ1 > 2 and |µ| ≤ 1); hence the sum of the last three terms is positive, which proves the claim (A.66). Thus, the first term in (A.64) is negative. Next, in view of Lemma A.3, the second line of (A.64) is ≤ −(3d + µ2 − η 2 )
λ1 − µ T0 (λ1 −µ) e , 4
(A.68)
while we conclude by ((λ1 − µ)T0 − 1) Tµ0 < 0 that the 1st line of (A.65) is ≤ (λ21 − µ2 + η 2 )η sin T0 η .
(A.69)
Since 3d + µ2 − η 2 > λ21 − µ2 + η 2 by Lemma A.1, and λ1 − µ > η and T0 (λ1 − µ) > 2 we have that the sum (A.68) + (A.69) < 0. Next, we compare the third line of (A.64) with the fifth line of (A.65): since by Lemma A.1 3d + µ2 − η 2 > 4η 2 we see that the sum of these two lines is negative. The fourth line of (A.64) is negative. The last line of (A.64) plus the last line of (A.65) is negative by Lemma A.1. The second line of (A.65) is negative since cos T0 η < 0 and the term in paren4(λ1 −µ) 1 −µ) theses is positive, as T0 (η4(λ 2 +(λ −µ)2 ) < 2T η(λ −µ) < 1 by T0 η > 2. 1 0 1 Using similar estimates we see that the third line of (A.65) is negative, and the fourth line of (A.65) is obviously negative.
684
B. RUF and P. N. SRIKANTH
Case 2: |µ| > 1: In this case (A.66) need not be valid. We show that the first plus the second line of (A.64) plus the first line of (A.65) is negative; indeed, since 2 T0 η 0 < sin T0 η ≤ 1/2 + 1/20, cos T0 η < 0, and sinT0 |µ| < 1/4 the sum of these lines is dominated by 1 1 3 1 2 2 T0 (λ1 −µ) + η − (λ1 − µ)e (3d + µ − η ) 2 2 20 4 1 1 + η. (A.70) + (λ21 − µ2 + η 2 ) 2 20 By T0 λ1 > 2, T0 |µ| > 1 (hence eT0 (λ1 −µ) > 20) and Lemma A.1 the expression (A.70) is 5 1 2 2 + η − 5(λ1 − µ) < 0 < (3d + µ − η ) 4 8 The other terms are treated like in case 1. Case 3: 0 < |µ| ≤ 1/3: first note that we can write (A.67) as λ1 − µ 3 λ1 1 3 η sin T0 η + cos T0 η = − sin T0 η + cos T0 η 2 2 4 η 3 3 1 cos T0 η + |µ|e−T0 (λ1 −µ) . (A.71) 6 2 Since 3|µ| < 1 and T0 λ1 > 2 the last term in (A.71) is bounded by 1/8. Thus √ we 1 3 −T0 (λ1 −µ) < 0, since sin T0 η < 1/2 and cos T0 η < − 3/2. see that 6 cos T0 η + 2 |µ|e Hence we can write λ1 − µ 3 λ1 1 3 η sin T0 η + cos T0 η < − sin T0 η + cos T0 η . (A.72) 2 2 4 η 3 +
We show that the first line + first term of second line + first term of third line of (A.64) plus the first term of first line + first term of third line + fifth line of (A.65) is negative. The remaining terms are easy to handle. Thus, taking into account (A.71) and Lemma A.1, we have to show that the follwoing expression is negative. 3 λ1 1 sin T0 η + cos T0 η − (3d + µ2 − η 2 ) 4 η 3 λ1 − µ T0 (λ1 −µ) (λ1 − µ)(λ21 − µ2 + η 2 ) e sin2 T0 ηeT0 (λ1 −µ) + − 4 4η 2 T0 µ + (λ21 −
2 T0 (λ1 −µ)
− µ + η )η sin T0 η + 2µη e 2
2
(λ1 − µ)(λ21 − µ2 + η 2 ) T0 (λ1 −µ) 2 e sin T0 η . T0 µ
!
1 cos To η sin T0 η + 2 2T0 η
(A.73)
685
THE LORENTZ–DIRAC EQUATION, I
We first show that 3 λ1 λ1 − µ T0 (λ1 −µ) sin T0 η − e + (λ21 − µ2 + η 2 )η sin T0 η < 0 . (3d + µ2 − η 2 ) − 4 η 4 In view of eT0 (λ1 −µ) > 8 and sin To η ≤ 1/2, this follows if 3 λ1 2 2 − 2(λ1 − µ) sin T0 η + (λ21 − µ2 + η 2 )η sin T0 η < 0 , (3d + µ − η ) − 4 η that is, by relation (A.53), if (3d + µ2 − η 2 )(3λ1 + 8(λ1 − µ)η) > 4(λ21 − µ2 + η 2 )η 2 = 4η 2 (λ1 + 2η 2 ) . Since 3d + µ2 − η 2 > 2η 2 , by (A.55), and 3λ1 + 8(λ1 − µ)η > 2(λ1 + 2η 2 ) (by λ1 − µ > η), this holds true. Next we show that also the sum of the remaining terms in (A.73) is negative: (λ1 − µ)(λ21 − µ2 + η 2 ) 1 2 T0 (λ1 −µ) cos T0 η + sin T ηe (3d + µ2 − η 2 ) 0 3 4η 2 T0 µ 1 cos To η sin T0 η + + 2µη 2 eT0 (λ1 −µ) 2 2T0 η −
(λ1 − µ)(λ21 − µ2 + η 2 ) T0 (λ1 −µ) e sin T02 η < 0 . T0 µ
Since 3d + µ2 − η 2 > 2η 2 , it is sufficient to show 1 (3d + µ2 − η 2 ) cos T0 η + 2µη 2 eT0 (λ1 −µ) 3 −
1 cos To η sin T0 η + 2 2T0 η
(λ1 − µ)(λ21 − µ2 + η 2 ) T0 (λ1 −µ) e sin T02 η < 0 . 2T0 µ
(A.74)
Note that since sin T0 η ≤ 1/2 we have T0 η ≥ 5π/6 > 2.5, and hence the second term in (A.74) can be estimated from above by 4 −2|µ|η 2 eT0 (λ1 −µ) (1/2 − 1/10) = − |µ|η 2 eT0 (λ1 −µ) . 5 Next, consider the third term in (A.74): multiplying (2.13) by this term is equal to
λ1 −µ 2T0 |µ|
we see that
η(λ1 − µ) −η(λ1 − µ) cos T0 η sin T0 ηeT0 (λ1 −µ) + sin T0 η . T0 T0 |µ| By (A.62) we have | sinTT00 η | ≤ 2(λ15η +2η2 )T0 η , and hence we can estimate, using T0 η > 2.5 η(λ1 − µ) 5η 3 |µ| T0 (λ1 −µ) (λ1 − µ)eT0 (λ1 −µ) < cos T0 η sin T0 ηe T0 2(λ1 + 2η 2 )T0 η 2
<
4 |µ|η 2 eT0 (λ1 −µ) 5
686
B. RUF and P. N. SRIKANTH
where the last inequality follows (since 3|µ| < 1 implies λ1 − µ < 2 by (A.51)) from 5η < 2(λ1 + 2η 2 ), i.e. 0 < 2λ1 + 4η 2 − 5η = 2λ1 + (2η − 5/4)2 − 25/16, which holds by λ1 > 1. Hence, to conclude that (A.74) is negative, it remains to show that η(λ1 − µ) sin T0 η 3d + µ2 − η 2 cos T0 η + < 0. 3 T0 √ Since cos T0 η ≤ − 3/2, | sinT0Tµ0 η | ≤ 1/2 and using relation (A.54), this holds if 1 1 −|µ|(λ1 − µ) √ + η(λ1 − µ)|µ| < 0 3 2
√ i.e. if η < 2/ 3 ;
this holds, since by (A.49): η 2 = 2|µ| + 3µ2 ≤ 1. References [1] A. Ambrosetti and G. Prodi, A Primer in Nonlinear Analysis, Cambridge Univ. Press, 1993. [2] D. Bambusi and D. Noja, “On classical electrodynamics of point particles and mass renormalization: some preliminary results”, Lett. Math. Phys. 37 (1996) 449–460. [3] A. Carati and L. Galgani, et al., “Nonuniqueness properties of the physical solutions of the Lorentz–Dirac equation”, Nonlinearity 8 (1995) 65–79. [4] J. Hale and A. P. Stokes, “Some physical solutions of Dirac–type equations”, J. Math. Phys. 3 (1962) 70–74. [5] P. A. M. Dirac, “Classical Theory of radiating electrons”, Proc. Roy. Soc. A167 (1938) 14. [6] J. D. Jackson, Classical Electrodynamics, John Wiley, New York, 1975. [7] H. A. Lorentz, The Theory of Electrons, 2nd ed., Dover, New York, 1952. [8] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Inst. of Math. Sciences Lecture Notes, 1974. [9] D. Noja and A. Posilicano, “The wave equation with one point interaction and the (linearized) classical electrodynamics of a point particle”, Ann. Inst. H. Poincar Phys. Thor. 68 (1998), 351–377. [10] B. Ruf and P. N. Srikanth, “The Lorentz–Dirac equation”, II, to appear.
ERRATA
COMMENTS ON “INTERACTING QUANTUM FIELDS”
[Rev. in Maths., Vol. 11, No. 7 (1999) 881–928 Glenn Eric Johnson Litton-TASC, 4801 Stonecroft Blvd. Chantilly, VA 20151, USA
The conclusions that Schwinger functions constructed from generalized random processes defined on sampling functions imply quantum fields with scattering is not supported by the arguments within [1]. The constructed Schwinger functions do analytically continue to define Wightman functions [1, 2]. These Wightman functions exhibit scattering with cross sections and equivalent potentials calculated in [1]. Nevertheless, there are two deficiencies in the arguments of [1] stating that there are quantum fields with the defined scattering amplitudes defined in spaces with positive metric. Development of scattering for quantum fields defined in spaces with indefinite metric is discussed in [2, 3] and their further references. The nonvanishing scattering amplitudes correspond to boundary values of the constructed analytic Wightman functions that are approached from beyond the algebra of functions E+ within which positivity is demonstrated. E+ is the algebra of sampling functions constructed in [1] for which the Schwinger functions define a nonnegative reflection (Osterwalder–Schrader [4]) positive form. Hence, the demonstrated nonnegativity results do not directly apply to the nontrivial scattering amplitudes. Indeed, the nonvanishing truncated four-point function corresponding to ha∗ (g1 , t1 )a∗ (g2 , t2 )Ω, a∗ (g3 , t3 )a∗ (g4 , t4 )ΩiT is a boundary value at vanishing Euclidean times approached from (just) beyond E+ . The boundary values approachable from within E+ exhibit no scattering. This nontriviality makes these models interesting, but requires a proof of positivity that extends beyond E+ , i.e. more than the “factor” positivity of [1]. Secondly, to get a representation in a Hilbert space of these field operators requires a demonstration that the null space is an ideal. There is a Hilbert space representation of states, but the accepted definition for a field operator within the 687 Reviews in Mathematical Physics, Vol. 12, No. 4 (2000) 687–689 c World Scientific Publishing Company
688
ERRATA
Hilbert space fails unless the null space is an ideal. The limited algebra of sampling functions E+ for which the Wightman functions define a nonnegative bilinear functional requires an explicit demonstration that the null space is an ideal since the algebra cannot be self-dual, i.e. a ∗-algebra. The issue of boundary values can be illustrated with the two-point function of the model constructed on E+ . T
S2 (ϑϕ∗1 , ϕ2 ) Z = dξ(R ∗ ϑϕ∗1 )(R ∗ ϕ2 ) Z =
Z dτ1 dτ2
dp
π 2ωm
−ωm |τ1 −τ2 | d e − dωm ωp
· h1 (p)∗ (ωm δ(−τ1 − τ10 ) − δ 0 (−τ1 − τ10 ))h2 (p)(ωm δ(τ2 − τ20 ) − δ 0 (τ2 − τ20 )) Z 0 0 d e−λ|τ1 +τ2 | d d π ∗ ωm + 0 − h1 (p) h2 (p) ωm + 0 = dp 2ωm dτ1 dτ2 dλ λ
Z =
dp Z
=
π d h1 (p)∗ h2 (p) − 2ωm dλ (
dp h1 (p)∗ h2 (p)
(ωm − λ sgn(τ10 + τ20 ))2
e
−λ|τ10 +τ20 |
λ
!
λ=ωm
λ=ωm
τ10 + τ20 > 0
0 0
0
2π|τ10 + τ20 |e−ωm |τ1 +τ2 |
τ10 + τ20 < 0 .
The boundary value approached from τ10 + τ20 < 0 is beyond E+ since τk0 ≥ 0 for ϕk ∈ E+ . Only the analytic extension of this boundary value is nontrivial. The desired field ordering corresponding to τ1 > τ2 also cannot be achieved while remaining within E+ . This two-point function is nonnegative for ϕ1 = ϕ2 , and its analytic extension diverges as |t1 − t2 | on the nontrivial boundary (i.e. at physical times). It is currently speculative whether the augmented models of [1]] define nontrivial quantum fields on spaces with positive metric. To encompass the non-trivial scattering amplitudes, the nonnegativity demonstrated for E+ needs to be extended to at least E0− , sampling functions with support for Euclidean times τ ≥ − for a real > 0. Acknowledgment I would like to thank Dr. H. Gottschalk for many insightful comments and correspondence on the problems with [1]. References [1] G. Johnson, “Interacting quantum fields”, Rev. Math. Phys. 11 (1999) 881–928. [2] S. Albeverio, H. Gottschalk and J.-L. Wu, “Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions”, Rev. Math. Phys. 8 (1996) 763–817.
ERRATA
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[3] S. Albeverio, H. Gottschalk and J.-L. Wu, “Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric”, Phys. Lett. B405 (1997) 243. [4] K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions”, Commun. Math. Phys. 31 (1973) 83–112.
CONSTRUCTION OF THE SPECTRAL MEASURE FOR DEFORMED OSCILLATOR POSITION OPERATOR IN THE CASE OF UNDETERMINED HAMBURGER MOMENT PROBLEM∗ V. V. BORZOV Department of Mathematics St. Petersburg University of Telecommunications Moika 61, 191065, St. Petersburg, Russia
E. V. DAMASKINSKY Department of Mathematics Defence Ingenering Technical University Zakharievskaya 22, 191185, St. Petersburg, Russia
P. P. KULISH St. Petersburg Department of Steklov Mathematics Institute Fontanka 27, 191011, St. Petersburg, Russia Received 2 March 1999 The spectral measure of the position (momentum) operator X for the q-deformed oscillator is calculated in the case of the undetermined Hamburger moment problem. The presentation being given for a particular choice of generators of the q-oscillator algebra, the developed technique can be applied to other cases of undetermined moment problem as well. The spectral measure of X and its Stieltjes transformation m(z) are expressed in terms of the Jacobi matrix entries only. Using the expression found for the Stieltjes transformation m(i), the connection between the parameters labeling the spectral measures σϕ and associated self-adjoint extensions Xϕ is established. The spectral measures σ0 and σπ are explicitly calculated.
Contents
1. Introduction 2. Background Material on q-Oscillator, Classical Moment Problem and q-Hermite Polynomials 2.1. Spectral theory of Jacobi matrices, classical moment problem and orthogonal polynomicals 2.2. Harmonic oscillator 2.3. The position operator for deformed oscillator and q-Hermite polynomials 3. Stieltjes Transformation m(z) of the Spectral Measure 3.1. Computation of the Nevanlinna matrix 3.2. Computation of m(i) 3.3. Computation of m(z) 4. Construction of the Spectral Measure σϕ 4.1. Support of the spectral measure σϕ ∗ This
research was supported by RFFI grants No. 97-01-01152 and No. 98-01-00310. 691
Reviews in Mathematical Physics, Vol. 12, No. 5 (2000) 691–710 c World Scientific Publishing Company
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4.2. Mass distribution of the spectral measure σϕ 5. Conclusion Acknowledgments References
708 708 709 709
1. Introduction The connection of the Hermite polynomials Hn (x) with the quantum harmonic oscillator is well known from the early days of quantum theory. This connection manifests itself in several aspects. First, these polynomials arise as the eigenvectors of the number operator (i.e. the Hamiltonian) of the quantum harmonic oscillator in coordinate representation. At the same time the Hermite polynomials are the so-called polynomials of the first kind for the Jacobi matrix X which represents the position operator in the number representation. These polynomials Hn (x) are 2 orthogonal with respect to the measure dσ(x) = e−x dx on the real line. This measure is the unique solution of the corresponding Hamburger moment problem. This last question of a measure reconstruction for a Jacobi matrix X(q) related to the deformed oscillator algebra Aq [1–6] is discussed in detail in the paper. One can choose different generators a(λ), a† (λ) of the deformed oscillator algebra Aq [6], and to connect with each choice the corresponding Jacobi matrix (X(q, λ))i,j = bi δi,j−1 + bi−1 δi,j+1 ,
i = 0, 1, . . . ,
and the appropriate set of the so-called q-Hermite polynomials Hn (x; q, λ) (see e.g. [7–13]). These polynomials are also orthogonal with respect to a measure which gives a solution of the associated Hamburger moment problem. However in this case the solution of the moment problem may be nonunique [14, 15, 17]. If such measure is unique the moment problem is determined. In the opposite case there is a family of such measures and the moment problem is known as undetermined. Some of these measures, which give the so-called extremal solutions of the moment problem, correspond to different self-adjoint extensions of the symmetric operator X(q, λ) (cf. [14–18]). Let us note that the undetermined moment problem appears even for the quantum harmonic oscillator with generators b, b† , [b, b† ] = 1. Indeed, it was pointed out in [19] that for the Jacobi matrices J (k) = bk + (b† )k , (which appear in the description of higher power squeezed states), the Hamburger moment problem is undetermined for k > 2. For those choices of q-oscillator generators [20, 21], when the corresponding q-Hermite polynomials and the spectral measures are known from the q-analysis [22–24], the moment problems are determined. In this paper we consider the opposite case of undetermined Hamburger moment problem. Although the construction proposed below can be applied to each choice of generators of the q-oscillator algebra Aq , all concrete expressions and proofs are given for the q-oscillator [4, 5] with nonzero entries of the Jacobi matrix equal to p a −q−a bn = [n + 1], with “symmetric” basic number [a] = qq−q −1 . Note also that [a] is exponentially growing for q > 1, as well as for 0 < q < 1 in contrast to the q-number a [a; q] = 1−q 1−q of the q-analysis [24] and basic hypergeometric functions [22, 23].
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Following the general theory of the moment problem and Jacobi matrices [14–16] we express the spectral measure and its Stieltjes transformation m(z) in terms of the entries bn of the Jacobi matrix only. Using the explicit expression for the Stieltjes transformation m(z) at z = i we establish the connection between parameters labeling the spectral measures and associated self-adjoint extensions of position operator X. This connection was missing in the general considerations. The paper is organized as follows. The relations between harmonic oscillator and Hermite polynomials, main formulas of the moment problem and definitions of the q-deformed oscillator algebra Aq and q-Hermite polynomials [9] are briefly reviewed in Sec. 2. In this section the Hamburger power moment problem for the deformed oscillator is also formulated. This moment problem is undetermined. In this case the extremal spectral measure is concentrated on the set of zeros of entire function expressed in terms of the related orthogonal polynomials [14–16]. Let us underline that such measures found in [17, 25, 26] were expressed in terms of the standard q-symbol [α; q] and the q-analogues of some classical special functions. At the same time the spectral measures we are interested in are expressed as infinite series containing the symmetric q-number [n]. These series have not yet been studied in the q-analysis and they could be of interest by themselves. Our main considerations and computations are given in Sec. 3. In this section we first calculate the value m(i) of the Stieltjes transformation m(z) of the spectral measure at the point i, and after that we find the value of m(z) at arbitrary complex z. Then in Sec. 4 we construct the spectral measure σϕ and give examples of the spectral measures σ0 and σπ . Moreover, in this section we establish connection between parameters labeling the spectral measures and associated self-adjoint extensions of position operator X. 2. Background Material on q-Oscillator, Classical Moment Problem and q-Hermite Polynomials 2.1. Spectral theory of Jacobi matrices, classical moment problem and orthogonal polynomials For the reader’s convenience we remind without proofs the relevant material from [14] (see also [15, 16]) thus making our exposition self-contained. Let {en |n ∈ Z+ } be the standard orthonormal basis in the Hilbert space `2 (Z+ ). We define the operator A which acts on the elements of this basis by Aen = bn en+1 + an en + bn−1 en−1 , Then the operator A can be represented as the a0 b0 0 0 b a b 0 1 1 0 0 b1 a2 b2 A= 0 0 b2 a3 . .. .. . . .. . . . .. .. .. .. . . . . which is known as a Jacobi matrix.
an ∈ R, bn > 0 .
matrix ··· ··· ··· .. . .. . .. .
··· ··· ···
(2.1)
, ··· .. . .. .
(2.2)
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There is an infinite sequence of polynomials Pn (x) of degree n related to such Jacobi matrix, and defined by the recurrence relation bn Pn+1 (x) + an Pn (x) + bn−1 Pn−1 (x) = xPn (x) ,
(2.3)
with “initial conditions” P0 (x) = 1 ,
P−1 (x) = 0 .
(2.4)
These polynomials Pn (x) are called the polynomials of the first kind. From now on we make the assumption that an = 0 and bn > 0. Under these conditions any polynomial Pn (x) has real coefficients and Pn (−x) = (−1)n Pn (x). Note that the recurrence relation (2.3) has two linearly independent solutions. The independent set of solutions to the relation (2.3) consists of polynomials Qn (x) satisfying another initial conditions Q0 (x) = 0 ,
Q1 (x) =
1 . b0
(2.5)
By definition we have deg Qn (x) = n − 1. Such polynomials are called the polynomials of the second kind for the Jacobi matrix A (2.2). The polynomials of the first and second kind are related Pn−1 (x)Qn (x) − Pn (x)Qn−1 (x) =
1 , bn−1
(n = 1, 2, 3, . . .) .
It is known that the polynomials {Pn (x)}∞ n=0 are orthonormal Z Pn (x)Pm (x)dσ(x) = δn,m
(2.6)
(2.7)
R
with respect to some positive Borel measure σ on R. In order to find the measure σ it is necessary to solve the following problem. Let sn denote the coefficient α0 at P0 (t) = 1 in the decomposition tn =
n X
αk Pk (t) .
k=0
For a given number sequence {sn }∞ n=0 a positive measure σ on the line has to be found such that Z ∞ tn dσ(t) , n = 0, 1, 2, . . . . (2.8) sn = −∞
This problem is known as the Hamburger (power) moment problem [14]. If such a measure is defined uniquely then the moment problem is called determined. Otherwise there is infinite family of such measures, and the moment problem is the undetermined one. Let D be a linear span of the basis {en }∞ n=0 . The matrix (2.2) defines the symmetric operator A on D. It is easy to prove that the deficiency indices of operator A are equal to (0, 0) or (1, 1).
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Proposition 2.1. Let A¯ denote the closure of A. Then the following conditions are equivalent: (1) The deficiency indices of operator A are equal to (0, 0). (2) A¯ is a self-adjoint operator in the space `2 (Z+ ). (3) The moment problem (2.8) for given numbers sn such that sn = (e0 , An e0 ) ,
(2.9)
is determined. P 2 (4) The series ∞ n=0 |Pn (z)| is divergent for all z ∈ C such that Im z 6= 0. From now on we will restrict our attention to the case when the deficiency indices of operator A are equal to (1, 1). Then the operator A¯ is not a self-adjoint operator, and there are infinitely many self-adjoint extensions. In this case the P∞ series n=0 |Pn (z)|2 < ∞ for all z ∈ C, and the moment problem (2.8) for numbers sn given by (2.9) is undetermined. The next proposition gives the sufficient condition for the deficiency indices of operator A to be equal (1, 1). Proposition 2.2. Under the following conditions: (1) There exist N such that the inequalities bn−1 bn+1 ≤ b2n
(2.10)
∞ X 1 < ∞; b n=0 n
(2.11)
hold for all n > N. (2)
the operator A has the deficiency indices (1, 1). Moreover, the following proposition is true. Proposition 2.3 (see [15]). Under the conditions (2.10) and (2.11), there are in¯ and the related moment problem finitely many self-adjoint extensions of operator A, (2.8) is undetermined for numbers sn given by (2.9). The spectral properties of self-adjoint extensions of operator A are closely related with the properties of measures, solving the associated moment problem. Let a parameter ϕ label different self-adjoint extensions Aϕ , ϕ ∈ R of a symmetric operator A with deficiency indices (1, 1) in the Hilbert space H, in the following way [27, 28]. By D(A), D(Aϕ ) (R(A), R(Aϕ )) we denote the domains (ranges) of closures A¯ and A¯ϕ of operators A and Aϕ , respectively. Let us fix vectors gi and g−i (with equal norms) from the subspaces Ti = H R(A + iI) ,
T−i = H R(A − iI) .
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For every g ∈ D(Aϕ ) there is [27] a unique representation g = f + αfϕ ,
α ∈ C,
(2.12)
where f ∈ D(A), and
i iϕ i (e 2 g−i − e− 2 ϕ gi ) . (2.13) 2 Note that for fixed values of norms of elements f , fϕ and g the parameter α also takes a fixed value. The action of the self-adjoint operator Aϕ on the vector fϕ is defined by fϕ =
Aϕ fϕ =
i 1 iϕ (e 2 g−i + e− 2 ϕ gi ) . 2
(2.14)
To obtain all possible self-adjoint extensions of the operator A it is sufficient to vary values of the parameter ϕ from 0 to 2π. It is known [27] that for complex number ω Z ∞ dσϕ (λ) , (2.15) ω= −∞ λ − i where the spectral measure σϕ is an “extremal” solution of the Hamburger moment problem (2.8) related to the self-adjoint extension Aϕ , one has ω = c∞ (i) − ie−iϕ r∞ (i) .
(2.16)
The center c∞ (i) and the radius r∞ (i) of the limit Weyl–Hamburger circle of the Jacobi matrix A are given by the formulas !−1 P∞ ∞ i X 2 2 − Pk=0 Qk (i)Pk (−i) c∞ (i) = , r∞ (i) = 2 |Pk (i)| . (2.17) ∞ 2 k=0 |Pk (i)| k=0
Denote by m(z)
Z
∞
m(z) = −∞
dσϕ (λ) λ−z
(2.18)
the Stieltjes transformation of the measure σϕ . Knowing m(z) one can reconstruct the spectral measure σϕ using the inverse Stieltjes transformation [28, 15] Z m(z) − m(z) σϕ (∆) = lim ψ(γ; τ )dγ ; z = γ + iτ , ψ(γ; τ ) = . (2.19) τ →0 ∆ 2πi Remark 2.1. It is not our purpose to study the domain D(Aϕ ) of a self-adjoint extension Aϕ of the operator A. But it is not difficult to describe D(Aϕ ) if the spectral measure σϕ of operator Aϕ on the space L2 (R; σϕ ) is obtained. The domain D(Aϕ ) consists [32] of all σϕ -measurable functions f (x) ∈ L2 (R; σϕ ) such that Z (1 + x2 )|f (x)|2 dσϕ (x) < ∞ . We now give few examples of the problem described above.
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2.2. Harmonic oscillator First we note that the quantum-mechanical position operator Xb can be represented on the space `2 (Z+ ) by infinite dimensional Jacobi matrix
0 b0 0 1 Xb = √ 2 0 . .. .. .
b0 0 b1
0 b1 0
0 0 b2
0 .. . .. .
b2 .. . .. .
0 .. .. .
.
··· ··· ··· .. . .. . .. .
··· ··· ···
, ··· .. . .. .
(2.20)
√ with bn = n + 1. ¯ b is self-adjoint, and the classical Hamburger It is well known that the operator X moment problem, associated with Jacobi matrix (2.20), is determined. The solution σ to this problem is given by 2 1 dσ(x) = √ e−x dx . π
(2.21)
In this case the polynomials Pn (x) of the first kind are given by Pn (x) = √
1 2n n!
Hn (x) ,
(2.22)
where Hn (x) is the usual Hermite polynomials Hn (x) = (−1)n ex
2
dn −x2 (e ), dxn
(2.23)
with recurrent relation 1 xHn (x) = nHn−1 (x) + Hn+1 (x) . 2 The Hermite polynomials are orthogonal polynomials Z ∞ Hm (x)Hn (x)dσ(x) = δmn d2n ,
(2.24)
(2.25)
−∞
with normalization d2n = 2n n! .
(2.26)
2.3. The position operator for deformed oscillator and q-Hermite polynomials Another important example of the problem described in Subsec. 2.1 is related to the so-called q-deformed oscillator. The deformed oscillator became a rather popular subject in the last decade due to its connection with quantum groups and algebras
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(see e.g. [10, 29]). Here we recall only main definitions and some of the properties of q-oscillator needed in the sequel. Let us remark that for the q-deformed oscillator the situation became significantly richer and more interesting than in the case of the usual harmonic oscillator mentioned above. Indeed, in this case besides a q-analogue of the Fock representation there are plenty of inequivalent representations for 0 < q < 1 [30]. Moreover, the related Hamburger power moment problem can be undetermined, i.e. there are different spectral measures. This means that the position operator X for the qoscillator may have deficiency indices (1, 1) and not only (0, 0). Recall that in the first case the position operator X has many different self-adjoint extensions. Below we consider the deformed oscillator algebra Aq , q ∈ R with fixed generators a± , N satisfying the commutation relations a− a+ − qa+ a− = q −N ,
[N, a± ] = ±a± ,
and usual hermiticity conditions (a± )† = a∓ , N † = N . In the oscillator-like representation the action of generators is p N |ni = n|ni ; a+ |ni = [n + 1]|n + 1i , n ≥ 0 ; p a− |ni = [n]|n − 1i , n ≥ 1 , a− |0i = 0 ; where
(2.27)
(2.28)
1 |ni = p (a+ )n |0i . [n]!
Here we use the notation [α] ≡ [α]q =
q α − q −α . q − q −1
(2.29)
+ − The p position operator X := a + a is described by Jacobi matrix (2.20) with bk = [k + 1], k = 0, 1, 2, . . . . P∞ From eigenvalue equation X|xi = x|xi, and the decomposition |xi = n=0 Pn (x; q)|ni one obtains the following recurrent relations for corresponding q-Hermite polynomials Pn (x; q) p p [n]Pn−1 (x; q) + [n + 1]Pn+1 (x; q) = xPn (x; q) , n ≥ 1 (2.30) P1 (x; q) = xP0 (x; q) , P0 (x; q) = 1 .
These q-Hermite polynomials Pn (x; q) are polynomials of the first kind for the Jacobi matrix X. Remind that polynomials of the second kind Qn (x; q) for the same Jacobi matrix satisfy the same recurrence relations (2.30) with initial conditions Q1 (x; q) = 1 ,
Q0 (x; q) = 0 .
The connection (2.6) of polynomials of the first and second kind looks in our case as 1 Pn−1 (x; q)Qn (x; q) − Pn (x; q)Qn−1 (x; q) = p . (2.31) [n]
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For q > 0, q 6= 1 the conditions (2.10) and (2.11) of the Proposition 2.2 hold p p [n] [n + 2] ≤ [n + 1] ,
n ≥ 0,
∞ X
[n + 1]− 2 < ∞ . 1
n=0
¯ of the position operator X is only closed but not self-adjoint Hence, the closure X ¯ are equal to (1.1), and there is one parameter operator. The deficiency indices of X ¯ family of self-adjoint extensions of the operator X. The related Hamburger moment problem is undetermined, and there exists a family of measures σ satisfying the relations (2.8). In the rest of the paper we construct the spectral measure σϕ which is the “extremal” solution of the Hamburger moment problem (2.8) related to the self-adjoint extension Xϕ . Remark 2.2. (1) The Stieltjes transformation m(z) of a spectral measure σϕ can be obtained according to [14] m(z) = −
A(z)t − C(z) , B(z)t − D(z)
(2.32)
where entire functions A(z), . . . , D(z) form a so-called Nevanlinna matrix (cf. (3.8)– (3.11)) A(z) C(z) , (2.33) B(z) D(z) ¯ Explicit and the real parameter t defines a self-adjoint extension of the operator X. connection of this parameter t with the parameter ϕ (2.18) is given in the next section (3.52). As mentioned above, the inverse Stieltjes transformation [28, 15] allows to reconstruct the spectral measure σϕ from given m(z) according to formula (2.19). (2) For the q-Hermite polynomials Hnq (x) such that 1 Pn (x) = p Hnq (x) , [n]! the following expression holds [9]: (n/2) n−1 X X Hnq (x) = xn + (−1)k xn−2k k=1
mk =2k−1
[mk ]
mX k −2
[mk−1 ] · . . . ·
mk−1 =2k−3
m 2 −2 X
[m1 ]
m1 =1
(2.34) where (x) = Ent(x) denotes the integer part of x. 3. Stieltjes Transformation m(z) of the Spectral Measure 3.1. Computation of the Nevanlinna matrix In this subsection we compute the elements of the Nevanlinna matrix (2.33).
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The polynomials Pn and Qn can be represented in the form ( n 2)
Pn (x; q) =
X (−1)m p α2m−1,n−1 xn−2m , [n]! m=0
α−1;n−1 ≡ 1 ;
n−1 X
α2m−1;n−1 =
n ≥ 0, kX 1 −2
[k1 ]
k1 =2m−1
(3.1) km−1 −2
X
[k2 ] · · ·
k2 =2m−3
[km ] ,
m≥1
km =1
(3.2) ( n 2)
Q0 (x; q) = 0 ,
β0;n ≡ 1 ;
Qn+1 (x; q) =
β2m;n =
n X
X
(−1)m p β2m,n xn−2m , [n + 1]! m=0 kX 1 −2
[k1 ]
k1 =2m
n ≥ 0,
(3.3)
km−1 −2
[k2 ] · · ·
k2 =2m−2
X
[km ] ,
m ≥ 1.
(3.4)
km =2
These relations (3.1)–(3.4) are some non-standard representations of the polynomials Pn and Qn . Here are some elementary properties of the coefficients αm;n and βm;n . From the recurrence relations for Pn and Qn (2.30) it follows that the coefficients αm;n and βm;n satisfy the relations α2m−1;n = [n]α2m−3;n−2 + α2m−1;n−1 ;
(3.5)
β2m;n = [n]β2m−2;n−2 + β2m;n−1 .
(3.6)
From definitions of the coefficients αk,n , βk,n (3.2) and (3.4) one gets α2k−1,2k−1 = [2k − 1]!! ,
β2k−2,2k−2 = [2k − 2]!! ,
(3.7)
where [2n]!! = [2n][2n − 2] · . . . · [2] ,
[2n − 1]!! = [2n − 1][2n − 3] · . . . · [1] .
The entries of the Nevanlinna matrix are given by [14] A(z) = z
∞ X
Qn (0)Qn (z) ,
(3.8)
0
B(z) = −1 + z
∞ X
Qn (0)Pn (z) ,
(3.9)
0
C(z) = 1 + z
∞ X
Pn (0)Qn (z) ,
(3.10)
0
D(z) = z
∞ X 0
Pn (0)Pn (z) .
(3.11)
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CONSTRUCTION OF THE SPECTRAL MEASURE
From the relations (3.1) and (3.3) one obtains if n = 2k : 0, k−1 Qn (0) = (−1) if n = 2k − 1 ; β2k−2,2k−2 , p [2k − 1]! if n = 2k − 1 : 0, k Pn (0) = (−1) α2k−1,2k−1 , if n = 2k . p [2k]!
(3.12)
(3.13)
Combining the relations (3.12), (3.13) and (3.7), one gets s [2k − 2]!! , k ≥ 1; Q2k−1 (0) = (−1)k−1 [2k − 1]!! 1, P2k (0) =
s
(−1)k
(3.14)
if k = 0 : [2k − 1]!! , [2k]!!
(3.15)
if k ≥ 1 .
From (3.3), (3.8) and (3.14) it follows that A(z) = z
∞ X k=1
k−1 X 1 (−1)k+m−1 β2m,2k−2 z 2(k−m−1) . [2k − 1]!! m=0
(3.16)
Similarly we obtain for other elements of the Nevanlinna matrix B(z) = −1 +
∞ X k=1
C(z) = 1 +
∞ X k=0
D(z) = z 1 +
k−1 X 1 (−1)k+m−1 α2m−1,2k−2 z 2(k−m) ; [2k − 1]!! m=0
k−1 1 X (−1)k+m β2m,2k−1 z 2(k−m) ; [2k]!! m=0 ∞ X k=1
k 1 X (−1)k+m α2m−1,2k−1 z 2(k−m) ; [2k]!! m=0
(3.17)
(3.18) ! ,
(3.19)
In particular, at z = i we get A(i) = i
∞ X k=1
k−1 X 1 β2m,2k−2 , [2k − 1]!! m=0
B(i) = −1 −
∞ X k=1
C(i) = 1 +
∞ X k=0
D(i) = i 1 +
(3.20)
k−1 X 1 α2m−1,2k−2 , [2k − 1]!! m=0
k−1 1 X β2m,2k−1 , [2k]!! m=0
∞ X k=1
k 1 X α2m−1,2k−1 [2k]!! m=0
(3.21)
(3.22) ! .
(3.23)
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3.2. Computation of m(i) In this subsection the explicit form of the Stieltjes transformation m(i) of the spectral measure σϕ will be calculated in terms of the elements of Jacobi matrix X. For this aim we have to consider some non-standard q-series, which probably are of a special interest by themselves. We stress that the explicit expression (2.32) for the spectral measure m(z) of some self-adjoint extension of the position operator X is known from the general theory. The parameter t in (2.32) labels these self-adjoint extensions. Unfortunately, it is not known from the general theory to which extension the concrete value of t corresponds. To clarify this point, it is convenient to compute first the value of √ m(z) at the point i = −1. To this end we take into account the known results from [27, 28]. If we compare the value m(i) computed below with the corresponding value of measure m(z) from the general formula (2.32) computed at the point z = i and take into account expressions for elements of the Nevanlinna matrix we obtain the exact connection between the values of the parameter t and the parameter ϕ which labels different self-adjoint extensions of X. From the relations (2.15), (2.16) and (2.18) we have [27]: Z ∞ dσϕ (λ) m(i) = (3.24) = c∞ (i) − ie−iϕ r∞ (i) . λ − i −∞ So to obtain m(i) it is sufficient to compute the center c∞ (i) and the radius r∞ (i) of the Weyl–Hamburger circle at z = i. To do this, some preliminary results are needed. Let us introduce the auxiliary functions Ψ(q) = lim Ψs (q) , s→∞
s X [2k − 1]!!
,
(3.25)
[2k]!! . [2k + 1]!!
(3.26)
[2k]!!
k=1
Φ(q) = lim Φs (q) , s→∞
Ψs (q) = 1 +
Φs (q) = 1 +
s X k=1
It is not difficult to check that for q > 0, q 6= 1 the functions Ψ(q) (3.25) and Φ(q) (3.26) are well defined as convergent q-series. In fact it follows from the inequality 1 [2s] ≥ ([2s − 1][2s + 1]) 2 that 1
[2k − 1]!! [1] 2 ≤ 1 . [2k]!! [2k + 1] 2
(3.27)
P∞ 1 In view of n=0 [n + 1]− 2 < ∞ for q > 0, q 6= 1 we have that Ψ(q) is well defined. The same is true for Φ(q). Lemma 3.1. The following relations are valid: p X α2m−1,2p−1 = −B(i) , p→∞ α2p−1,2p−1 m=0
lim
(3.28)
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CONSTRUCTION OF THE SPECTRAL MEASURE
p X D(i) α2m−1,2p = −i , p→∞ α Ψ(q) 2p−1,2p m=0
(3.29)
p X β2m,2p = C(i) , p→∞ β m=0 2p,2p
(3.30)
lim
lim
p X A(i) β2m,2p+1 lim = −i . p→∞ β Φ(q) m=0 2p,2p+1
(3.31)
This lemma is easily proved by means of the recurrent relations (3.5), (3.6). Now we can compute the radius r∞ (i) and the center c∞ (i) of the Weyl– Hamburger circle at point z = i. Theorem 3.1. The radius r∞ (i) of the Weyl–Hamburger circle at z = i is given by r∞ (i) =
1 . 2iB(i)D(i)
(3.32)
Proof. Using the formula (5) from [14, Chap. 1, Sec. 2] and (3.1) we have n−1 X
|Pk (i; q)|2 =
k=0
p [n] (Pn (i; q)Pn−1 (i; q) − Pn−1 (i; q)Pn (i; q)) 2i
=
( n 2)
X
1 α2m−1,n−1 [n − 1]! m=0
( n−1 2 )
X
α2m−1,n−2 .
(3.33)
m=0
It is convenient to separate the leading terms α2( n2 )−1,n−1 and α2( n−1 )−1,n−2 2 from the sums in the right-hand side of the expression (3.33). This gives us the common factor equal to (1) α2p−1,2p α2p−1,2p−1 = [2p]!Ψp (q) ,
if n = 2p + 1 ,
(2) α2p−1,2p−1 α2p−3,2p−2 = [2p − 1]!Ψp−1 (q) ,
if n = 2p ,
(3.34)
where we take into account that α2p−1,2p = [2p]!! Ψp (q) ,
α2p−1,2p−1 = [2p − 1]!! ,
(3.35)
which follows from (3.2) and (3.25). This allows us to rewrite (3.33) in the form (1)
2p X
|Pk (i; q)|2 = Ψp (q)
k=0
(2)
2p−1 X k=0
p p X α2m−1,2p X α2m−1,2p−1 , α2p−1,2p m=0 α2p−1,2p−1 m=0
p p−1 X α2m−1,2p−1 X α2p−1,2p−2 |Pk (i; q)| = Ψp−1 (q) . α2p−1,2p−1 m=0 α2p−3,2p−2 m=0 2
(3.36)
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V. V. BORZOV, E. V. DAMASKINSKY and P. P. KULISH
(1)
(2)
Introducing (Aα )p and (Aα )p and using the recurrent relations (3.5), one obtains (1) (Aα )p
p p k−1 X X X α2m−1,2p−1 1 := =1+ α2m−1,2k−2 , α2p−1,2p−1 [2k − 1]!! m=0 m=0 k=1
p X
p k−1 α2m−1,2p 1 X 1 X (2) (Aα )p := =1+ α2m−1,2k−1 . α2p−1,2p Ψp (q) [2k]!! m=0 m=0
(3.37)
k=1
From the Lemma 3.1 it is clear that lim (A(1) α )p = −B(i) ,
p→∞
lim (A(2) α )p = −i
p→∞
D(i) . Ψ(q)
(3.38) (3.39)
Finally the relation (3.32) follows from the relations (2.17), (3.36), (3.38), (3.39). Now we are going to consider the center c∞ (i) = limn→∞ cn (i), where (see [14]) (n−1)
D1 (−i; i) , Pn−1 2i k=0 |Pk (i)|2 p (n−1) D1 (−i, i) = [n]{Qn (i; q)Pn−1 (−i; q) − Qn−1 (i; q)Pn (−i; q)} . cn (i) = −
(3.40) (3.41)
Theorem 3.2. The center c∞ (i) of the Weyl–Hamburger circle at the point z = i is given by the relation c∞ (i) = −
B(i)C(i) + A(i)D(i) . 2B(i)D(i)
(3.42)
Proof. For the numerator of the expression (3.40) (n−1)
D1
(−i; i) = S(n) + T (n) ,
due to the relations (3.41), (3.1) and (3.3), one obtains p S(n) = − [n]Qn−1 (i; q)Pn (−i; q) n−2 n ( 2 ) ( 2 ) X X 1 = β2m,n−2 α2m−1,n−1 , [n − 1]! m=0 m=0 p [n]Qn (i; q)Pn−1 (−i; q) n−1 n−1 ( 2 ) ( 2 ) X X 1 = β2m,n−1 α2m−1,n−2 . [n − 1]! m=0 m=0
T (n) =
(3.43)
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CONSTRUCTION OF THE SPECTRAL MEASURE
From (3.4) and (3.26) one obtains also β2p,2p+1 = [2p + 1]!!Φp (q) ,
β2p,2p = [2p]!! .
(3.44)
Using (3.35) and (3.44) and repeating calculations given above to get the expression (3.32), one obtains limp→∞ S(2p) = limp→∞ T (2p − 1) = −B(i)C(i) ,
(3.45)
limp→∞ S(2p + 1) = limp→∞ T (2p) = −A(i)D(i) .
Finally from the relations (3.32), (3.43) and (3.45) one gets (3.42).
Now the final expression for the Stieltjes transformation m(z) at z = i can be given. From (2.18), (3.32) and (3.42) we have m(i) =
i sin ϕ cos ϕ + B(i)C(i) + A(i)D(i) − . 2B(i)D(i) 2B(i)D(i)
(3.46)
Remark 3.1. Note that from (3.1) and (3.3) it follows that Qn−1 (i; q)Pn (i; q) = (−1)n Qn−1 (i; q)Pn (−i; q) = (−1)n−1 S(n)[n]− 2 , 1
Qn (i; q)Pn−1 (i; q) = (−1)n−1 Qn (i; q)Pn−1 (−i; q) = (−1)n−1 T (n)[n]− 2 . 1
(3.47)
Then from (2.31), (3.45) and (3.47) we have A(i)D(i) − B(i)C(i) = 1 .
(3.48)
Remark 3.2. Let us emphasize that the arguments of the present section can be applied for any Jacobi matrix (2.20) with entries satisfying bn bn+2 ≤ (bn+1 )2 ,
n ≥ 0;
∞ X
(bn )−1 < ∞
n=0
so that the associated moment problem is undetermined. 3.3. Computation of m(z) Above we consider the Stieltjes transformation m(i) of the spectral measure σϕ connected with self-adjoint extension Xϕ of the operator X. Here we consider the Stieltjes transformation m(z) of the same spectral measure σϕ at arbitrary point z of complex plane. For this end we compare the rhs of (3.46) with that of (2.32) in which the entries A, B, C and D of the Nevanlinna matrix given at z = i by (3.20), (3.21), (3.22) and (3.23). This gives us the value of the parameter tϕ corresponding to a given value of the parameter ϕ thereby we find the related Stieltjes transformation m(z). We have −A(i)tϕ + C(i) = Uϕ 0 , (3.49) B(i)tϕ − D(i)
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where Uϕ = −
e−iϕ + (B(i)C(i) + A(i)D(i)) . 2B(i)D(i)
(3.50)
From the relation (3.49) we find the value of the parameter tϕ tϕ =
C(i) + D(i)Uϕ . A(i) + B(i)Uϕ
(3.51)
Substituting (3.50) into (3.51) we obtain finally the main result of this subsection tϕ = i
D(i) ϕ cot . B(i) 2
(3.52)
This relation allows us to obtain (using the (2.32)) the expression for m(z): m(z) = −
C(z)B(i) sin ϕ2 − iA(z)D(i) cos ϕ2 . D(z)B(i) sin ϕ2 − iB(z)D(i) cos ϕ2
(3.53)
Using the expressions for the Nevanlinna matrix entries one can obtain the explicit form of the rhs of this relation in terms of series in variable z. However, such expression in general case is very cumbersome and we omit it. Here only expressions for m(z) in two particular cases ϕ = 0 and ϕ = π are given: m0 (z) = −
A(z) ; B(z)
(for ϕ = 0)
(3.54)
mπ (z) = −
C(z) ; D(z)
(for ϕ = π) .
(3.55)
Using the expressions (3.16), (3.17), (3.18) and (3.19) for the Nevanlinna matrix entries we obtain P Pk−1 1 k+m−1 z ∞ β2m,2k−2 z 2(k−m−1) k=1 [2k−1]!! m=0 (−1) m0 (z) = − ; (3.56) P∞ Pk−1 1 k+m−1 α 2(k−m) −1 + k=1 [2k−1]!! 2m−1,2k−2 z m=0 (−1) P∞ Pk−1 1 k+m 1 + k=0 [2k]!! β2m,2k−1 z 2(k−m) m=0 (−1) mπ (z) = − . P∞ Pk 1 k+m α 2(k−m) ) z(1 + k=1 [2k]!! 2m−1,2k−1 z m=0 (−1)
(3.57)
4. Construction of the Spectral Measure σϕ In this section we construct the spectral measure σϕ for arbitrary self-adjoint extension of the position operator Xϕ . The spectral measure is reconstructed from the Stieltjes transformation m(z) by the relation (2.19). Since we have established already the connection between the parameters tϕ and ϕ we can avoid the direct cumbersome computations by using the results of the general theory of moments problem [14].
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CONSTRUCTION OF THE SPECTRAL MEASURE
4.1. Support of the spectral measure σϕ It is well known [15, Chap. VII, Sec. 1] that in the undetermined case the spectrum of Xϕ is discrete, real and can have accumulation points only at infinity. The support of the spectral measure σϕ coincides with the spectrum of Xϕ and with the set of zeros Πϕ of the function Λϕ (x) := D(x)B(i) sin
ϕ ϕ − iB(x)D(i) cos . 2 2
To find the set Πϕ = {xk }M≤∞ k=−N ≥−∞ ,
xN < · · · x1 < 0 ≤ x1 < x2 < · · · < xM ,
(4.1)
we must solve the equation [14] B(x) ϕ D(x) ϕ cos − i sin = 0 B(i) 2 D(i) 2
(4.2)
where the relation (3.52) is taken into account. For those particular cases ϕ = 0 and ϕ = π we are interested in, Eq. (4.2) takes the form B(x) = 0 ,
for
ϕ = 0,
(4.3)
D(x) = 0 ,
for ϕ = π .
(4.4)
If we substitute B(x) (3.19) and D(x) (3.17) into these equations we obtain 1+
∞ X k=1
x Ψ(q) +
k−1 X 1 (−1)k−m x2(k−m) α2m−1,2k−2 = 0 , [2k − 1]!! m=0
∞ X k=1
k−1 1 X (−1)k−m x2(k−m) α2m−1,2k−1 [2k]!! m=0
(4.5)
!
or, in terms of the polynomials of the first kind, s ∞ X [2k − 2]!! P0 (x) + x (−1)k P2k−1 (x) = 0 , [2k − 1]!!
= 0,
(4.6)
(4.7)
k=1
∞ X x P0 (x) + (−1)k k=1
s
! [2k − 1]!! P2k (x) = 0 . [2k]!!
(4.8)
Note that using the recurrent relation (2.30) we can rewrite the nth partial sum of the series in (4.7) as s s n X [2k − 2]!! [2n]!! P0 (x) + (−1)k xP2k−1 (x) = (−1)n P2n (x) . (4.9) [2k − 1]!! [2n − 1]!! k=1
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V. V. BORZOV, E. V. DAMASKINSKY and P. P. KULISH
4.2. Mass distribution of the spectral measure σϕ In this subsection we construct the mass distribution {σϕ (xk )} of the spectral measure σϕ . These quantities are [14] σϕ (xk ) = =
A(xk )tϕ − C(xk ) B 0 (xk )tϕ − D0 (xk ) 1 B 0 (xk )D(xk ) − B(xk )D0 (xk ) 1 = r∞ (xk ) . 2 n=0 Pn (xk )
= P∞
(4.10)
We omit the cumbersome expressions for σϕ (xk ) in case of general ϕ and restrict our attention to particular values ϕ = 0 and ϕ = π. In these cases we have σ0 (xk ) =
A(xk ) B 0 (xk )
= −xk
=
Q1 (xk ) + d dx (P0 (x)
xk
+
P∞
l−1 l=2 (−1)
q
P∞
l l=1 (−1)
q
[2l−2]!! [2l−1]!! Q2l−1 (xk )
[2l−2]!! [2l−1]!! xP2l−1 (x))(xk )
P∞
d dx (−1 +
Pj−1 1 j+m−1 β2m,2j−2 xk 2(j−m−1) j=1 [2j−1]!! m=0 (−1) P∞ P j−1 1 j+m−1 α 2(j−m) )(x ) 2m−1,2j−2 x k j=1 [2j−1]!! m=0 (−1)
,
(4.11) σπ (xk ) =
C(xk ) D0 (xk )
1 = xk 1 = xk
Q1 (xk ) + d dx (P0 (x)
+
P∞
q
P∞
q
l l=1 (−1)
l l=1 (−1)
[2l−1]!! [2l]!! xk Q2l (xk ) [2l−1]!! [2l]!! P2l (x))(xk )
P∞ Pj−1 2(j−m) 1 j−m 1 + j=1 [2j]!! β2m,2j−1 xk m=0 (−1) . P∞ Pj−1 d 1 j−m α 2(j−m)−1 )(x ) 2m−1,2j−1 x k j=1 [2j]!! m=0 (−1) dx (
(4.12)
5. Conclusion The classical Hamburger moment problem was applied to spectral analysis of coordinate operator X = a + a† of the q-deformed oscillator. Depending on the choice of the annihilation and creation operators a, a† (generators of the q-oscillator algebra Aq ), this Hamburger moment problem can be determined or undetermined. The most popular choice (2.27) results in dependence bn = ([n]q )1/2 of the Jacobi matrix entries, giving the undetermined moment problem. Using general theory, the corresponding q-Hermite polynomials are expressed in terms of these entries only, the Stieltjes transformation mt (z) of the spectral measure σϕ (x) and the measures
CONSTRUCTION OF THE SPECTRAL MEASURE
709
themselves are calculated. Explicit connection (3.52) between parameters t and ϕ is obtained. The self-adjoint “coordinate” operator Xϕ (q) has simple discrete spectrum (4.2). Let us mention once more, that the developed technique can be applied to any Jacobi matrix (2.20) entries of which satisfy the conditions bn bn+2 ≤ (bn+1 )2 , ∞ X
n ≥ 0,
(bn )−1 < ∞ ,
n=0
so that associated moment problem is undetermined (e.g. bn = ([n; q])1/2 , q > 1). Acknowledgments The authors thank the referee for careful reading of the manuscript and useful suggestions. References [1] D. D. Coon and M. Baker, Phys. Rev. D. D2 (1970) 2349; D. D. Coon, S. Yu and M. M. Baker, Phys. Rev. D. D5 (1972) 1429. [2] M. Arik and D. D. Coon, J. Math. Phys. 17(4) (1976) 524–527. [3] V. V. Kuryskin, Manuscript registered in VINITI, no. 3937–76, 1976 (in Russian); Ann. Found. L. de Broigle 5(2) (1980) 111–126. [4] L. C. Biedenharne, J. Phys. A. 22(18) (1989) L873–878. [5] A. J. Macfarlane, J. Phys. A. 22(21) (1989) 4581–4586. [6] P. P. Kulish and E. V. Damaskinsky, J. Phys. A. 23(9) (1990) L415–419. [7] R. Floreannini, Lett. Math. Phys. 22(1) (1991) 45–54; R. Floreanini, J. Le Teurneux and L. Vinet, J. Phys. A. 28(10) (1994) L287–294; R. Floreanini and L. Vinet, Ann. Phys. 221 (1993) 117–128. [8] N. M. Atakishiev and S. K. Suslov, Theor. Math. Phys. 85 (1990) 1055–1061; N. M. Atakishiev and Ph. Feinsilver, J. Phys. A. 29(8) (1996) 1659–1664; N. M. Atakishiev, A. Frank and K. B. Wolf, J. Math. Phys. 35(7) (1994) 3253–3260. [9] E. V. Damaskinsky and P. P. Kulish, Zap. Nauch. Sem. POMI 199 (1992) 81–90 (in Russian). [10] E. V. Damaskinsky and P. P. Kulish, Zap. Nauch. Sem. LOMI 189 (1991) 37–74 (in Russian), English transl: J. Soviet. Math. 62 (1992) 2963. [11] P. P. Kulish, “Irreducible representations of deformed oscillator and coherent states”, preprint KTH-96/21, 11pp., Stockholm, 1996. [12] E. V. Damaskinsky and P. P. Kulish, Int. J. Mod. Phys. A. 12(1) (1990) 153–158. [13] V. V. Borzov, E. V. Damaskinsky and S. B. Yegorov, Zap. Nauch. Sem. LOMI 245 (1997) 80–106 (in Russian), (q-alg/9509022). [14] N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner Publ. Co, New York, 1965. [15] Yu. M. Berezanski˘ı, Expansions in Eigenfunctions of Self-adjoint Operators, Transl. Math. Monographs 17, AMS, Providence, R.I., 1968. [16] J. A. Shohat and J. D. Tamarkin, The Problem of Moments, AMS, Providence, R.I., 1963. [17] T. S. Chihara, Pacif. J. Math. 27(3) (1968) 475–484; J. Math. Anal. Appl. 85 (1982) 331–346. [18] T. S. Chihara, An Introduction to Orthogonal Polynomials, Math. and Appl. 13, Gordon and Breach, New York, 1978.
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[19] B. Nagel, Higher power squeezed states, Jacobi matrices, and the Hamburger moment problem, Contribution to the 1997 Balatonfured conference on squeezed states. [20] I. M. Burban and A. U. Klimyk, Lett. Math. Phys. 29 (1993) 13–18. [21] W.-S. Chung and A. U. Klimyk, J. Math. Phys. 37(2) (1996) 917–932. [22] H. Exton, q-hypergeometric Functions and Applications, Chichester, Ellis Horwood, 1983. [23] G. Gasper and M. Rahman, Basic Hypergeometric Series, in Encyclopedia of Mathematics and its Applications 35, Cambridge Univ. Press, Cambridge, 1990. [24] G. A. Andrews, q-Series, Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Mathematics, AMS regional conference series 66, 1986. [25] D. S. Moak, J. Math. Anal. Appl. 81 (1981) 20–47. [26] C. Berg, J. Comp. Appl. Math. 65 (1965) 27–55; C. Berg and A. J. Duran, Math. Scand. 79 (1996) 209–223. [27] N. I. Akhiezer, Uspechi Matem. Nauk. (9) (1941) 126–156. [28] A. I. Plesner, Uspechi Matem. Nauk. (9) (1941) 3–125. [29] M. Chaichian and P. P. Kulish, Phys. Lett. B. 234(1/2) (1990) 72–80; Pr-t. CERN-TH 5969/90. [30] P. P. Kulish, Teor. Math. Phys. 85(1) (1991) 158–161. [31] H. Rampacher, H. Stumpf and F. Wagner, Fortschr. Phys. 13 (1965) 385. [32] M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Leningrad Univ. Press, 1980 (in Russian).
LAGRANGIAN AND HAMILTONIAN FORMALISM ON THE h-DEFORMED QUANTUM PLANE SUNGGOO CHO and SANG-JUN KANG Department of Physics, Semyung University Chechon, Chungbuk 390-711, Korea
KWANG SUNG PARK Department of Mathematics, Keimyung University Taegu 705-701, Korea Received 2 March 1999 It is known that there are only two quantum planes which are covariant under the quantum deformations of GL(2) admitting a central determinant. Contrary to the q-deformed quantum plane, the h-deformed quantum plane has a structure suitable for defining time derivatives and variations as closely as in the ordinary plane. From these we derive differential calculi including the skew-derivatives of Wess–Zumino as well as variational calculi on the quantum plane. These calculi enable us to generalize the Lagrangian and Hamiltonian formalism on the ordinary plane to the quantum plane. In particular, we construct commutation relations between noncommuting coordinates and momenta which do not depend on the initial choice of Lagrangian. We also discuss the symmetry of a Lagrangian and Noether’s theorem.
1. Introduction In the last few years, quantum spaces have attracted much attention as models for a microscopic structure of spacetime [7, 12]. Quantum spaces are often considered as spaces with noncommuting coordinates or as deformed spaces with deformation parameters [2, 3, 15]. It is known that there are only two quantum planes which are covariant under the quantum deformations of GL(2) admitting a centrtal determiant [10]. They are usually called q-deformed and h-deformed quantum planes. Some authors have examined the problem of defining Lagrangian for a particle moving on a q-deformed quantum space [2, 11, 14]. However, the q-deformed quantum plane seems to be too awkward for the natural generalization of the Lagrangian and Hamiltonian formulation on the ordinary plane. Contrary to the q-deformed quantum plane, the h-deformed quantum plane has rich properties enough to define time-derivatives and variations on it. From these we can derive commutation relations for the differential calculi including the skew-derivatives of Wess–Zumino. These calculi also enable us to generalize the Lagrangian and Hamiltonian formalism on the ordinary plane to the h-deformed quantum plane. In Sec. 2, we shall introduce commutation relations of the noncommuting coordinates at different times and define the time derivatives of the coordinates similarly to the classical mechanics. The time derivatives lead us naturally to differential calculi 711 Reviews in Mathematical Physics, Vol. 12, No. 5 (2000) 711–724 c World Scientific Publishing Company
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on the quantum plane and commutation relations between various quantities for the generalization of the Lagrangian and Hamiltonian formalism on the ordinary plane. In particular, we shall extend those to the extended h-deformed quantum plane that we are mostly interested in. In Sec. 3, we shall define the variations of noncommuting coordinates and derive their commutation relations. Using the variational method, we shall define partial derivatives as in [11]. We shall derive not only the commutation relations between partial derivatives obtained by the Wess–Zumino method [1, 17] but also other commutation relations between various quantities. In Sec. 4, we shall generalize the Lagrangian and Hamiltonian formalism on the ordinary plane to the h-deformed quantum plane. In particular, we shall construct commutation relations between coordinates and momenta which do not depend on the initial choice of Lagrangian contrary to the general belief as in [11, 14]. In Sec. 5, we shall also discuss the symmetry of a Lagrangian and Noether’s theorem. Some examples are given as an application. 2. The Kinematics on the h-Deformed Quantum Plane The h-deformed quantum plane is an associative algebra generated by noncommuting coordinates x, y such that xy − yx = hy 2 ,
(2.1)
where h is a deformation parameter. The quantum group GLh (2) is the symmetry group of the h-deformed quantum plane as is GLq (2) for the q-deformed quantum plane [8, 10]. R-matrix associated with the quantum group GLh (2) which solves the quantum Yang–Baxter equation [1]
is given by
ˆ 12 R ˆ 23 R ˆ 12 = R ˆ 23 R ˆ 12 R ˆ 23 R
(2.2)
1 −h h h2 0 0 1 h ˆ= R . 0 1 0 −h 0 0 0 1
(2.3)
Equation (2.1) can be written as for xa = (x, y) ˆ ab cd xc xd . xa xb = R
(2.4)
There are several methods to obtain the covariant differential calculi on the quantum plane [1, 17, 13]. In this work, however, we shall suggest a method similar to the classical one. Now let the time coordinate t be a real parameter of the system. We assume that the noncommuting coordinates xa be time-dependent generators satisfying, for any t and t0 , ˆ ab cd xc (t0 )xd (t) . xa (t)xb (t0 ) = R (2.5)
LAGRANGIAN AND HAMILTONIAN FORMALISM
713
Since the R-matrix is the inverse of itself, the above relation is consistent. This is not the case for the q-deformed quantum plane [11]. For the hermiticity of xa (t), h should be a pure imaginary, that is h∗ = −h. Now we define the time derivative of a function f (t) as in the classical case f (t + ∆t) − f (t) f˙(t) ≡ lim . ∆t→0 ∆t
(2.6)
Then it is straightforward to show the following: ˆ ab cd x˙ c (t0 )xd (t) . xa (t)x˙ b (t0 ) = R
(2.7)
If we define the 1-forms by dxa = x˙ a dt ,
(2.8)
the 1-forms satisfy the same relations as in [1] ˆ ab cd dxc xd . xa dxb = R
(2.9)
The definition of the 1-forms in this work seems to be more suitable for an interpretation of various quantities in the formulation of Lagrangian and Hamiltonian on the quantum plane. Equation (2.7) gives immediately the followings: ˆ ab cd x˙ c (t0 )x˙ d (t) , x˙ a (t)x˙ b (t0 ) = R ˆ ab cd x xa (t)¨ xb (t0 ) = R ¨c (t0 )xd (t) ,
(2.10)
ˆ ab cd x x˙ a (t)¨ xb (t0 ) = R ¨c (t0 )x˙ d (t) . In this work, we shall concern the extended h-deformed quantum plane. The extended h-deformed quantum plane is an associative algebra A generated by x, y, x−1 , y −1 satisfying Eq. (2.1). Now we introduce 1 u(t) = x(t)y(t)−1 + h , 2
v(t) = y(t)−2
(2.11)
for any time t. If x and y are hermitian, then so are u and v. This choice of generators is known to be useful in studying the commutative limit. The geometry of the quantum plane is well known already as a noncommutative version of the Poincar´e half-plane, a surface of constant negative Gaussian curvature [5]. From Eq. (2.5) we have the commutation relation [u(t), u(t0 )] = 2h(u(t) − u(t0 )) , [u(t), v(t0 )] = −2hv(t0 ) ,
[v(t), v(t0 )] = 0 .
(2.12)
A straightforward calculation also yields [u(t), u(t ˙ 0 )] = −2hu(t ˙ 0 ) , [u(t), v(t ˙ 0 )] = −2hv(t ˙ 0) , [v(t), u(t ˙ 0 )] = 0 ,
[v(t), v(t ˙ 0 )] = 0 .
(2.13)
Moreover, we have [u(t), ˙ v(t ˙ 0 )] = 0 ,
(2.14)
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S. CHO, S.-J. KANG and K. S. PARK
and [u(t), u ¨(t0 )] = −2h¨ u(t0 ) , [u(t), v¨(t0 )] = −2h¨ v(t0 ) , [v(t), u ¨(t0 )] = 0 ,
[v(t), v¨(t0 )] = 0 ,
[u(t), ˙ u ¨(t0 )] = 0 ,
[u(t), ˙ v¨(t0 )] = 0 ,
[v(t), ˙ u ¨(t0 )] = 0 ,
[v(t), ˙ v¨(t0 )] = 0 .
(2.15)
3. Variations on the Quantum Plane In order to make a Lagrangian formalism for particles on the extended h-deformed quantum plane, we need to define the variations δxa of the noncommuting coordinates xa . And from the variations we shall define partial derivatives by the rule of the right-derivatives [11]. In general, if we put x0a = xa + δxa and define δ x˙ a =
d a δx , dt
(3.1)
we have δ x˙ a = x˙ 0a − x˙ a .
(3.2)
Thus if we define the variation δf of a function f (x, y, x, ˙ y) ˙ as the first order term in δxa and δ x˙ a of the difference f (x0 , y 0 , x˙ 0 , y˙ 0 ) − f (x, y, x, ˙ y), ˙ it satisfies the Leibniz rule for any functions f , g as expected δ(f g) = (δf )g + f (δg) .
(3.3)
To initiate our formalism, we assume the existence of some noncommuting quantities x0 , y 0 at any time t and t0 satisfying ˆ ab cd x0c (t0 )xd (t) . xa (t)x0b (t0 ) = R
(3.4)
We define the variation of noncommuting coordinates x, y at time t by δx = x0 − x ,
δy = y 0 − y .
(3.5)
From the definition of δxa and commutation relation in Eq. (3.4), it is straightforward to see that ˆ ab cd δxc xd . xa δxb = R (3.6) In order to introduce the variation of a Lagrangian for a physical system on the quantum plane, we need to know the commutation relations such as [xa , δ x˙ b ], [x˙ a , δxb ] and [x˙ a , δ x˙ b ] other than [xa , δxb ] in Eq. (3.6). If we use Eq. (3.2) it is easy to show that ˆ ab cd δ x˙ c xd . xa δ x˙ b = R (3.7) Taking the time derivative of Eq. (3.6) as in the previous section gives ˆ ab cd δxc x˙ d x˙ a δxb = R
(3.8)
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and ˆ ab cd δ x˙ c x˙ d . x˙ a δ x˙ b = R
(3.9)
Now we define the partial derivatives of a function f (x, y, x, ˙ y) ˙ with respect to x, y, x, ˙ y˙ by the rule of the right derivatives [11] as δf (x, y, x, ˙ y) ˙ = δx
∂f ∂f ∂f ∂f + δy + δ x˙ + δ y˙ . ∂x ∂y ∂ x˙ ∂ y˙
(3.10)
A lengthy calculation yields not only the same commutation relations as in [1] [x, ∂x ] = −1 + hy∂x ,
[y, ∂x ] = 0 ,
[x, ∂y ] = −hx∂x − h2 y∂x − hy∂y , [y, ∂y ] = −1 + hy∂x , [∂x , ∂y ] =
h∂x2
(3.11)
,
but also [x, ˙ ∂x ] = hy∂ ˙ x,
[y, ˙ ∂x ] = 0
˙ x − h y∂ ˙ x − hy∂ ˙ y , [y, ˙ ∂y ] = hy∂ ˙ x, [x, ˙ ∂y ] = −hx∂ 2
[x, ∂x˙ ] = hy∂x˙ ,
[y, ∂x˙ ] = 0 ,
[x, ∂y˙ ] = −hx∂x˙ − h2 y∂x˙ − hy∂y˙ , [y, ∂y˙ ] = hy∂x˙ , [x, ˙ ∂x˙ ] = −1 + hy∂ ˙ x˙ ,
(3.12)
[y, ˙ ∂x˙ ] = 0 ,
[x, ˙ ∂y˙ ] = −hx∂ ˙ x˙ − h y∂ ˙ x˙ − hy∂ ˙ y˙ , [y, ˙ ∂y˙ ] = −1 + hy∂ ˙ x˙ . 2
Now we apply our methods to the (u, v)-quantum plane. Since u, v are functions of x and y, their variations satisfy the following relations: ∂u ∂u + δy = δx y −1 − δy(xy −2 + hy −1 ) , ∂x ∂y ∂v ∂v δv = δx + δy = δy(−2y −3) . ∂x ∂y
δu = δx
(3.13)
From these the corresponding formulae to Eq. (3.6) are given in the (u, v)-quantum plane as follows: [u(t), δu(t0 )] = −2hδu(t0 ) , [u(t), δv(t0 )] = −2hδv(t0 ) , [v(t), δu(t0 )] = 0 ,
[v(t), δv(t0 )] = 0 .
(3.14)
Taking the time derivatives yields the following relations for any time [u, ˙ δu] = 0 , [u, ˙ δv] = 0 , [v, ˙ δu] = 0 ,
[v, ˙ δv] = 0 .
(3.15)
If we also regard u˙ and v˙ as the functions of x, y, x, ˙ y, ˙ then it is easy to show that δ u˙ =
d δu , dt
δ v˙ =
d δv . dt
(3.16)
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S. CHO, S.-J. KANG and K. S. PARK
Hence if we take time derivative of Eq. (3.14), we obtain the following relations: [u, δ u] ˙ = −2hδ u˙ , [u, δ v] ˙ = −2hδ v˙ , [v, δ u] ˙ = 0,
[v, δ v] ˙ = 0,
(3.17)
as well as [u, ˙ δ u] ˙ = 0 , [u, ˙ δ v] ˙ = 0, [v, ˙ δ u] ˙ = 0,
[v, ˙ δ v] ˙ = 0.
(3.18)
Similarly, if we define partial derivatives with respect to u, v, u, ˙ v˙ by the right derivatives as before δf (u, v, u, ˙ v) ˙ = δu
∂f ∂f ∂f ∂f + δv + δ u˙ + δ v˙ , ∂u ∂v ∂ u˙ ∂ v˙
(3.19)
we obtain not only the same commutation relations as in [4] [u, ∂u ] = −1 + 2h∂u , [u, ∂v ] = 2h∂v , [v, ∂u ] = 0 ,
[v, ∂v ] = −1 ,
(3.20)
but also [u, ˙ ∂u ] = 0 ,
[u, ˙ ∂v ] = 0 ,
[v, ˙ ∂u ] = 0 ,
[v, ˙ ∂v ] = 0 ,
[u, ∂u˙ ] = 2h∂u˙ , [u, ∂v˙ ] = 2h∂v˙ , [v, ∂u˙ ] = 0 ,
[v, ∂v˙ ] = 0 ,
[u, ˙ ∂u˙ ] = −1 ,
[u, ˙ ∂v˙ ] = 0 ,
[v, ˙ ∂u˙ ] = 0 ,
[v, ˙ ∂v˙ ] = −1 .
(3.21)
Moreover, if we regard a function as a sum of polynomials, mathematical induction gives [∂u , ∂v ] = 0 . (3.22) Moreover, from an observation f δu = δu(v −1 f v) and the Leibniz rule it follows that for any functions f , g ∂u (f g) = (∂u f )g + (v −1 f v)∂u g .
(3.23)
Similary, we obtain ∂v (f g) = (∂v f )g + (v −1 f v)∂v g , ∂u˙ (f g) = (∂u˙ f )g + (v −1 f v)∂u˙ g ,
(3.24)
∂v˙ (f g) = (∂v˙ f )g + (v −1 f v)∂v˙ g . These relations are to be used in the next section. Now all preparations are made for the generalization of the Lagrangian and Hamiltonian formalism on the ordinary plane to the h-deformed quantum plane.
LAGRANGIAN AND HAMILTONIAN FORMALISM
717
4. Lagrangian and Hamiltonian Formalism on the Quantum Plane From now on, we shall consider only the (u, v)-quantum plane for simplicity. The variation of a Lagrangian L = L(u, v, u, ˙ v) ˙ is given by δL(u, v, u, ˙ v) ˙ = δu∂u L + δv∂v L + δ u∂ ˙ u˙ L + δ v∂ ˙ v˙ L d d d = δu ∂u L − ∂u˙ L + δv ∂v L − ∂v˙ L + (δu∂u˙ L + δv∂v˙ L) . dt dt dt (4.1) Thus we have the Euler–Lagrange equations of motion if we apply the action principle to the physical system. In fact,we have for ua = (u, v) d dt
∂L ∂ u˙ a
−
∂L = 0. ∂ua
(4.2)
Before continuing further, we note that some Lagrangians may not describe a physical system on the h-deformed quantum plane properly unless the equations of motion are consistent with the commutation relations. For instance if we choose a Lagragian as L(u, v, u, ˙ v) ˙ = 12 mu˙ 2 − 12 ku2 , the Euler–Lagrange equations of motion are given by u ¨ = −u + h which contradicts the commutation relation in Eq. (2.15). As in the classical mechanics, we define canonical momenta and Hamiltonian to be pu = ∂u˙ L , pv = ∂v˙ L (4.3) and H = up ˙ u + vp ˙ v −L.
(4.4)
Then the variation δH can be written as δH = −δu∂u L − δv∂v L + uδp ˙ u + vδp ˙ v.
(4.5)
On the other hand, we define the partial drivatives ∂pu and ∂pv also by the right derivatives ∂H ∂H ∂H ∂H + δpv . (4.6) δH = δu + δv + δpu ∂u ∂v ∂pu ∂pv The Hamilton’s equations of motion are obtained from each variation term δu, δv, δpu and δpv at the right-hand sides of Eqs. (4.5) and (4.6). It is usually believed that commutation relations between ua and pua and even Hamilton’s equations of motion and Poisson brackets are dependent on the initial choice of Lagrangian [11]. However, we can assume commutation relations between ua and pua and find the Hamilton’s equations of motion and Poisson brackets independent of the initial choice of Lagrangian as in classical mechanics. Now we shall construct commutation relations between various physical quantities to be used in the Hamiltonian formulation on the quantum plane. First we assume the following commutation relations between the noncommuting coordinates
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S. CHO, S.-J. KANG and K. S. PARK
and momenta for any time t and t0 : [u(t), pu (t0 )] = 2hpu (t0 ) ,
[v(t), pu (t0 )] = 0 ,
[u(t), pv (t0 )] = 2hpv (t0 ) ,
[v(t), pv (t0 )] = 0 ,
(4.7)
[pu (t), pv (t0 )] = [pu (t), pu (t0 )] = [pv (t), pv (t0 )] = 0 . The assumption can be introduced from Eqs. (3.20) and (3.22). In fact, Eqs. (3.20) and (3.22) will give a h-deformed Heisenberg algebra if one identifies Pˆu = −i~∂u and Pˆv = −i~(∂v − v −1 ) [9, 4]. By taking the classical limit ~ → 0 we can obtain the commutation relations between the noncommuting coordinates and momenta on the quantum plane. From Eq. (4.7) it follows then that [u, ˙ pu ] = 0 ,
[v, ˙ pu ] = 0 ,
[u, ˙ pv ] = 0 ,
[v, ˙ pv ] = 0 ,
[u, p˙ u ] = 2hp˙ u , [v, p˙u ] = 0 , [u, p˙ v ] = 2hp˙ v ,
[v, p˙v ] = 0 ,
[u, ˙ p˙ u ] = 0 ,
[v, ˙ p˙u ] = 0 ,
[u, ˙ p˙ v ] = 0 ,
[v, ˙ p˙v ] = 0 ,
[pu , p˙ u ] = 0 ,
[pv , p˙u ] = 0 ,
[pu , p˙ v ] = 0 ,
[pv , p˙v ] = 0 ,
[p˙ u , p˙ u ] = 0 ,
[p˙ v , p˙v ] = 0 ,
(4.8)
[p˙ u , p˙ v ] = 0 . In order to obtain commutation relations between physical quantities and their variations including the momenta, we note that Eqs. (3.20) and (4.7) give the following: [u, ∂u pu ] = 4h∂u pu ,
[u, ∂v pu ] = 4h∂v pu ,
[u, ∂u˙ pu ] = 4h∂u˙ pu ,
[u, ∂v˙ pu ] = 4h∂v˙ pu .
(4.9)
Similarly, we obtain [u, ∂u pv ] = 4h∂u pv ,
[u, ∂v pv ] = 4h∂v pv ,
[u, ∂u˙ pv ] = 4h∂u˙ pv ,
[u, ∂v˙ pv ] = 4h∂v˙ pv
(4.10)
and [v, ∂ua pub ] = [v, ∂u˙ a pub ] = 0 .
(4.11)
Now from the variations of the momenta pu and pv and Eqs. (4.9)–(4.11), it follows that [u(t), δpu (t0 )] = 2hδpu (t0 ) ,
[v, δpu (t0 )] = 0 ,
[u(t), δpv (t0 )] = 2hδpv (t0 ) ,
[v(t), δpv (t0 )] = 0 .
(4.12)
LAGRANGIAN AND HAMILTONIAN FORMALISM
719
Equations (4.7) and (4.12) give [pu , δu] = 0 ,
[pu , δv] = 0 ,
[pv , δu] = 0 ,
[pv , δv] = 0
[u, ˙ δpu ] = 0 ,
[v, ˙ δpu ] = 0 ,
[u, ˙ δpv ] = 0 ,
[v, ˙ δpv ] = 0 .
[pu , δ u] ˙ = 0,
[pu , δ v] ˙ = 0,
[pv , δ u] ˙ = 0,
[pv , δ v] ˙ = 0.
(4.13)
and (4.14)
Then it follows easily that (4.15)
Now Eqs. (4.8)–(4.11) yield [pu , δpu ] = 0 ,
[pv , δpu ] = 0 ,
[pu , δpv ] = 0 ,
[pv , δpv ] = 0 .
(4.16)
From the above relations and the definition of right derivatives, not only the relations in (3.20) but also the commutativity properties for ua and ∂pu , ∂pv , etc. are easily derived as follows: [pu , ∂u ] = 0 ,
[pv , ∂u ] = 0 ,
[pu , ∂v ] = 0 ,
[pv , ∂v ] = 0 ,
[u, ∂pu ] = −2h∂pu , [v, ∂pu ] = 0 , [u, ∂pv ] = −2h∂pv ,
[v, ∂pv ] = 0 ,
[pu , ∂pu ] = −1 ,
[pv , ∂pu ] = 0 ,
[pu , ∂pv ] = 0 ,
[pv , ∂pv ] = −1 .
(4.17)
Now from Eq. (4.6) and the above commutation relations we obtain the Hamilton’s equations of motion independent of the initial choice of Lagrangian u˙ = ∂pu H ,
v˙ = ∂pv H ,
p˙u = −∂u H , p˙v = −∂v H .
(4.18)
Before we apply the above Lagrangian and Hamiltonian formulation to the concrete examples, let us consider Poisson brackets on the quantum plane. For the time derivative of a function f = f (u, v, pu , pv , t), we put f (u + 4tu, ˙ v + 4tv, ˙ pu + 4tp˙u , pv + 4tp˙v , t + 4t) = f (u, v, pu , pv , t) + 4tuf ˙ 1 (u, v, pu , pv , t) + 4tvf ˙ 2 (u, v, pu , pv , t) + 4tp˙u f3 (u, v, pu , pv , t) + 4tp˙v f4 (u, v, pu , pv , t) + ∂t f + O((4t)2 ) .
(4.19)
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S. CHO, S.-J. KANG and K. S. PARK
If we consider the function as a sum of polynomials in u, v, pu , pv and t, a lengthy calculation using mathematical induction yields f1 = ∂u f ,
f2 = ∂v f ,
(4.20)
f3 = ∂pu f , f4 = ∂pv f . Hence we have df = u∂ ˙ u f + v∂ ˙ v f + p˙u ∂pu f + p˙v ∂pv f + ∂t f = {f, H} + ∂t f , dt
(4.21)
{f, H} ≡ ∂pu H∂u f + ∂pv H∂v f − ∂u H∂pu f − ∂v H∂pv f .
(4.22)
where
Since [u˙ a , p˙ u ] = [u˙ a , p˙ v ] = 0 in Eq. (4.8), {H, H} = 0 and hence the Hamiltonian whose momenta satisfy Eq. (4.7) is conserved unless the Hamiltonian is dependent explicitly on time t. Moreover, we have u˙ a = {ua , H} ,
p˙u = {pu , H} ,
p˙ v = {pv , H} .
(4.23)
On the h-deformed quantum plane, the Poisson bracket { , } does not satisfy the usual defining relations as expected: the skew-symmetry, the Leibniz rule of derivation and the Jacobi identity. Example 1: Free particles For the Lagrangian of a free particle in the h-deformed quantum plane, let us choose 1 L(u, v, u, ˙ v) ˙ = mv −2 (u˙ 2 + v˙ 2 ) . (4.24) 2 We choose this from the kinetic energy term on the Poincar´e half-plane [9]. Then from the Euler–Lagrange equations of motion we have the equations d −2 (v u) ˙ = 0, dt
v¨ − v −1 v˙ 2 + v −1 u˙ 2 = 0 .
In this case, the Hamiltonian becomes H = equations of motion are obtained as follows: u˙ =
1 2 v pu , m
v˙ =
1 2 2 2m v (pu
(4.25)
+ p2v ) and the Hamilton’s
1 2 v pv m
(4.26)
and p˙u = 0 ,
p˙v = −
1 v(pu 2 + pv 2 ) , m
(4.27)
which give the same equations as in (4.25). All relations are consistent with the previous commutation relations. The equations in (4.25) represent the geodesic equation on the Poincare half plane in the commutative limit h → 0.
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LAGRANGIAN AND HAMILTONIAN FORMALISM
Example 2: One-dimensional harmonic oscillators Let us consider a one-dimensional harmonic motion in the v-direction on the h-deformed quantum plane with a Lagrangian L(u, v, u, ˙ v) ˙ =
1 1 mv −2 v˙ 2 − k(ln v)2 . 2 2
(4.28)
Then we have Euler–Lagrange equation of motion m(¨ v v −1 − v˙ 2 v −2 ) = −k ln v .
(4.29)
The quantity ln v becomes the length from (0, 1) to (0, v) in the Poincar´e half-plane [16] in the commutative limit. Then the Hamiltonian is given by H=
1 2 2 1 v pv + k(ln v)2 2m 2
(4.30)
and the Hamilton’s equations of motion are v˙ =
v2 pv m
(4.31)
and
1 (4.32) vpv 2 − kv −1 ln v . m This is the same result as that of the Lagrangian method. Moreover, all relations are also consistent with the previous commutation relations. If we put lv = ln v, then the above equation can be rewritten as p˙v = −
¨lv = − k lv , m
(4.33)
which represents the harmonic oscillation obviously in the commutative limit. 5. Noether’s Theorem As in the classical mechanics, there is a connection between the existence of the first integral of a system of Euler–Lagrange equation and the invariance of the Lagrangian under certain transformation of the noncommuting coordinates u and v. Let us assume a Lagrangian L = L(u, v, u, ˙ v) ˙ is invariant under an infinitesimal transformation u → u + δ0 u v → v + δ0 v , (5.1) where δ 0 ua = λf (u, v) (λ 1) do not necessarily obey the commutation rule in Sec. 3 for δua in general. However, since the Leibniz rule for δ 0 is still valid as seen in Sec. 3, we can take variation δ 0 for the Lagrangian L. Let us put δ 0 L(u, v, u, ˙ v) ˙ = δ 0 u(∂u L + 4l1 ) + δ 0 u(∂ ˙ u˙ L + 4l2 ) = δ 0 v(∂v L + 4l3 ) + δ 0 v(∂ ˙ v˙ L + 4l4 ) .
(5.2)
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S. CHO, S.-J. KANG and K. S. PARK
Then the variation δ 0 L becomes the following form d d δ 0 L = δ 0 u ∂u L − (∂u˙ L) + δ 0 v ∂v L − (∂v˙ L) dt dt +
d 0 (δ u ∂u˙ L + δ 0 v ∂v˙ L) + 4l , dt
where 0
0
4l = δ u4l1 + δ v4l3 +
d 0 d 0 δ u 4l2 + δ v 4l4 . dt dt
(5.3)
(5.4)
Thus if the Euler–Lagrange equations are satisfied and 4l = 0, we have a conserved quantity δ 0 u ∂u˙ L + δ 0 v ∂v˙ L . (5.5) Example 3: A charged particle in a constant magnetic field Let us consider a charged particle in a constant magnetic field B on the h-deformed quantum plane with a Lagrangian 1 −2 2 v (u˙ + v˙ 2 ) − Bv −1 u˙ . 2m
L=
(5.6)
The commutative limit of this Lagrangian describes the motion of a charged particle in B on the Poincar´e half-plane [6]. The momenta pu = mv −2 u˙ − Bv −1 and pv = mv −2 v˙ satisfy the commutation relations in Eq. (4.7). The Hamiltonian is then given by H=
1 2 2 [v (pu + p2v ) + 2Bvpu + B 2 ] . 2m
(5.7)
In order to find a solution of the Euler–Lagrange equations, it is convenient to use conserved quantities. It is easy to show that the Lagrangian is invariant under the following infinitesimal transformations: dilation, translation along u and nonlinear rotation δ 0 u = λu ,
δ 0 v = λv ,
δ0u = λ ,
δ0 v = 0 ,
0
(5.8)
0
δ u = λ(v − u + 2hu) , δ v = −2λuv . 2
2
Now it is straightforward to see that 4l = 0 for each case. Thus we obtain the conserved quantities L1 = upu + vpv , L 2 = pu ,
(5.9)
L3 = (v 2 − u2 + 2hu)pu − 2uvpv + 2Bv . In terms of the conserved quantities, the Hamiltonian is given by H=
1 [L2 L3 + L21 + B 2 − h(L1 L2 − 2hL22 )] . 2m
(5.10)
LAGRANGIAN AND HAMILTONIAN FORMALISM
723
Moreover, we have a solution −1 2 −2 2 (u − L1 L−1 2 ) + (v + BL2 ) = 2mHL2 .
(5.11)
If we introduce a parameter K in the commutative limit to put L1 = 2K u ˜0 ,
L2 = 2K ,
L3 = −2K(˜ u20 + v˜02 ) ,
(5.12)
1 2 2 2 we have √ H = 2m (B − 4K v˜0 ). From this we can obtain the condition K < 0 and B > 2mH for a closed path as shown already in [6].
6. Conclusions In this work we have introduced the definition of the time derivatives of the noncommuting coordinates on the h-deformed quantum plane, which are not possible in the q-deformed quantum plane since the R-matrix is not the inverse of itself. The time derivatives enable us to obtain not only the relations for the differential calculi already known in the literature but also commutation relations between various quantities. We have also defined the variations of the noncommuting coordinates. By the rule of the right derivatives, we could derive not only the rules for skewderivatives of Wess–Zumino but also rules for partial drivatives with respective to various quantities. Using these, we have generalized the Lagrangian and Hamiltonian formulation on the ordinary plane to the h-deformed quantum plane and solved some physical systems such as free particles and one-dimensional harmonic oscillators. In particular, we have constructed commutation relations between noncommuting coordinates and momenta which do not depend on the initial choice of Lagrangian. We have also discussed the symmetry of a Lagrangian and Noether’s theorem. Contrary to the q-deformed quantum plane, the h-deformed quantum plane has a structure suitable for constructing Lagrangian and Hamiltonian formalism in a similar manner as in the classical mechanics. Acknowledgments This work was supported by Korean Ministry of Education, Project No. BSRI-972414 and a grant from TGRC-KOSEF 1997. References [1] A. Aghamohammadi, “The two-parametric extension of h deformation of GL(2), and the differential calculus on its quantum plane”, Mod. Phys. Lett. A8 (1993) 2607. [2] I. Ya. Aref’eva and I. V. Volovich, “Quantum group particles and non-Archimedean geometry”, Phys. Lett. B268 (1991) 179. [3] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1994. [4] S. Cho, “Quantum mechanics on the h-deformed quantum plane”, to appear in J. Phys. A: Math. and Gen. 32 (1999) 2091. [5] S. Cho, J. Madore and K. S. Park, “Noncommutative geometry of h-deformed quantum plane”, J. Phys. A: Math. and Gen. 31 (1998) 2639.
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[6] A. Comtet, “On the Landau levels on the hyperbolic plane”, Ann. Phys. 173 (1987) 185. [7] A. Connes, Non-Commutative Geometry, Academic, New York, 1994. [8] E. E. Demidov, Yu. I. Manin, E. E. Mukhin and D. V. Zhdanovich, “Non-standard quantum deformations of GL(n) and constant solutions of the Yang–Baxter equation”, Prog. Theor. Phys. (Suppl.) No. 102 (1990) 203. [9] C. Groche and F. Steiner, “The path integral on the Poincar´e upper half plane and for Liouville quantum mechanics”, Phys. Lett. A123 (1987) 319. [10] B. A. Kupershmit, “The quantum group GLh (2)”, J. Phys. A: Math. and Gen. 25 (1992) L1239. [11] M. Lukin, A. Stern and I. Yakushin, “Lagrangian and Hamiltonian formalism on a quantum plane”, J. Phys. A26 (1993) 5115. [12] J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge Univ. Press, Cambridge (2nd Ed.) 1999. [13] J. Madore and J. Mourad, “Quantum space-time and classical gravity”, J. Math. Phys. 39 (1998) 423. [14] R. P. Malik, “Lagrangian formulation of some q-deformed systems”, Phys. Lett. B316 (1993) 257. [15] Yu. I. Manin, Quantum groups and Noncommutative geometry, Centre de Recherches Math´ematiques, Montr´eal, 1988; Topics in Noncommutative Geometry, Princeton Univ. Press, Princeton, 1991. [16] S. Stahl, The Poincar´e Half-Plane, Jones and Bartlett Pub., London, 1993. [17] J. Wess and B. Zumino, “Covariant differential calculus on the quantum hyperplane”, Nucl. Phys. (Proc. Suppl.) B18 (1990) 302.
EXTENSION OF ANTI-AUTOMORPHISMS AND PCT-SYMMETRY∗ ROBERTO CONTI and CLAUDIO D’ANTONI Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica, I-00133 Roma, Italy E-mail:
[email protected] E-mail:
[email protected] Received 9 September 1998 Revised 21 December 1998 We study conditions for an anti-automorphism of a C ∗ -algebra to extend to the crossproduct by a compact group dual action. As a byproduct, we show that under a few natural assumptions any PCT transformation θ of A, the observable net of a local QFT, extends to the canonical field net F. Then we discuss the validity of the relation [θρθ−1 ] = [ρ] for every DHR morphism ρ of A with finite statistics.
1. Introduction Recently the TCP symmetry has been subject to several investigations that stemmed from H. J. Borchers result [4] (see [17, 18, 20] and also [26, 12, 10]). In [6] D. Buchholz and one of the authors suggested as a possible tool for extending a PCT-symmetry θ from the observables to the field algebra the use of BDLR results [8] on extension of automorphisms from a C ∗ -algebra to its cross product by the dual of a compact group. We discuss the generalization to the case of antiautomorphisms and prove that under very general assumptions any “reasonable” PCT transformation of the observable algebra A extends to the canonical field algebra F [16]. In the second section we prove an abstract extension theorem for antiautomorphisms much along the lines of [8]. Next we apply it to the PCT symmetry θ in QFT obtaining an anti-linear isomorphism θˆ on the field algebra that preserves the net structure and (under some cohomological restrictions) has the right commutation relations with the translations. There are now two natural questions that are closely related. The first one is whether the extension θˆ commutes with the action of G on F, and the second one is whether θ acting on the endomorphisms of A induces the conjugation. In the approach where geometrical modular action, in any of its variants, is assumed (see the references quoted above) these two problems are solved affirmatively exploiting the Tomita–Takesaki modular theory. More precisely, if Θ = V (R1 (π))J, where R1 (π) is a rotation and J = JA(W )00 ,Ω is the modular conjugation relative to a ˆ αg ] = 0, g ∈ G by the properties of J and the relation wedge algebra, then [θ, ∗ Research
supported by MURST, CNR-GNAFA, EU. 725
Reviews in Mathematical Physics, Vol. 12, No. 5 (2000) 725–738 c World Scientific Publishing Company
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ρ = j ◦ ρ ◦ j holds for every sector with finite statistics [17, 18] (this is true in any dimension, and in CFT where geometric modular action is automatic [19]). However at present it is not clear whether j := AdJ has always such a geometrical meaning (despite the results in [2, 9, 10]), or what the formula for the conjugate is in a possibly more general context. Perhaps a counterexample may be constructed arguing as in [34]. Our analysis originated with the desire to understand all of this from a different, and somehow complementary, point of view. In the third section we show that for any ρ with finite statistics θ ◦ ρ ◦ θ is a conjugate of ρ whenever θ admits an extension commuting with the gauge action. The opposite implication holds under the additional hypothesis on the group to be quasi-complete (see Appendix A and [11]). In the fourth section we shortly discuss the relation between our approach and the approaches based on the geometric modular action hypothesis. Finally we comment on lower dimensional theories where a braid symmetry appears. This is a first example of a wider range of applications of our techniques. In fact it seems likely that they could be used in a more general framework. In any case, we hope they can help to clarify the picture, and provide new insight to the known case. Some Notation. Throughout the paper H will designate a complex Hilbert space, ˆ its dual. By an automorphism A, B unital C ∗ -algebras, G a compact group, G (resp. an anti-automorphism) of a C ∗ -algebra A we always mean a ∗-isomorphism of A onto itself (resp. onto the opposite C ∗ -algebra Aop ). Aut(A) is the group of all the automorphisms of A. If A ⊂ B we denote by AutA (B) the group of all automorphisms of B leaving A pointwise fixed. End(A) is the semigroup of all unital endomorphisms of A. 2. Extension of Anti-Automorphisms In this section we prove an abstract extension result for automorphisms which will later be applied to Quantum Field Theory. In the following, if not otherwise stated, A is a C ∗ -algebra with trivial centre CI, (∆, ) a permutation symmetric, specially directed semigroup of unital endomorphisms of A with unit ι (see [15], Sec. 5). If T = T∆ denotes the full subcategory of End(A) with objects ∆, then (T , ) is a strict symmetric monoidal category with sufficiently many special objects (i.e. objects with determinant one) which is the abstract dual of a unique compact group G. Doplicher and Roberts define the cross product B := A × T of A by the (dual) action of T , and G appears as the stabilizer of A in B. A consequence of the uniqueness theorem for the cross product is an extension theorem for automorphisms of A which “preserve the category” [8]. Our first result is a natural generalization of Proposition 2.1 in [8]. Theorem 2.1. Let A, ∆, (T , ), and A1 , ∆1 , (T1 , 1 ) be as above, and let φ : A → A1 be an isomorphism of C ∗ -algebras. Consider the following conditions:
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(1) φ∆φ−1 = ∆1 , (10 ) A1 × T1 = A1 × T∆φ , where ∆φ := φ∆φ−1 (2) φ((ρ, σ)) = 1 (ρφ , σφ ), ρ, σ ∈ ∆, where ρφ := φρφ−1 . Then φ extends to an isomorphism φ˜ : A × T → A1 × T1 of the corresponding cross-products if (1), (2) hold, and (if and) only if (10 ) and (2) hold. Moreover if φ is implemented by a unitary operator from H0 onto H10 , where (π0 , H0 ), (π10 , H10 ) are faithful representations of A, A1 , respectively, then φ˜ is unitarily implemented in the corresponding representations π, π1 of B, B1 obtained by inducing up from π0 , π10 . Proof. Assuming that (1), (2) hold, it is straightforward to check that the quadruple {B1 , η1 ◦ φ, G1 , ρ ∈ ∆ → Hρφ } satisfies all the assumptions in [15], Theorem 5.1, thus by uniqueness there exists ˜ φ˜−1 = G1 , φ(H ˜ ρ ) = Hρ , an isomorhism φ˜ : B → B1 such that φ˜ ◦ η = η1 ◦ φ, φG φ 0 ρ ∈ ∆. If condition (1 ) holds, the same argument applies up to replacing ∆1 with ˜ then from Hρ = {ψ ∈ B|ψA = ρ(A)ψ, A ∈ A} we ∆φ . Conversely, if φ extends to φ, ˜ ρ ) = Hρ , ρ ∈ ∆. B is generated by A and {Hρ ; ρ ∈ ∆}, thus B1 = φ(B) ˜ have φ(H φ is generated by A1 = φ(A) and {Hρφ ; ρ ∈ ∆}; therefore, letting ∆φ := φ∆φ−1 , we have B1 = A1 × T1 = A1 × T∆φ . Now the condition (2) follows from (ρ, σ) = F (Hρ , Hσ ), where F (Hρ , Hσ ) is the canonical unitary operator in B implementing the flip symmetry on Hρ ⊗ Hσ . The last statement can be easily obtained from the results that follow. The next result is more abstract than needed here but we put it on record as a first step toward a group-free situation. Actually it is implicit in [1], Theorem 4.0.1. We report it for completeness, in a form which also makes explicit the connection with [8], (proof of) Proposition 2.1. We use Rieffel’s induction procedure, for which we refer to [29]. Proposition 2.1. Let A ⊂ B be an inclusion of C ∗ -algebras, E : B → A a conditional expectation, α ˜ an automorphism of B leaving A globally invariant (i.e. α := α ˜ |A ∈ Aut(A)), such that α ◦ E = E ◦ α ˜ . If α is covariant in a (faithful) represen∼ tation (π0 , H0 ) of A, namely π0 ◦ α = π0 , then α ˜ is covariant in the representation π induced from π0 up to B via E. Proof. Denote by H = B ⊗A H0 the Hilbert space obtained by separation and completion from the algebraic tensor product B ⊗ H0 with pre-inner product given by (B ⊗ η, B 0 ⊗ η 0 ) = (η, π0 (E(˜ α(B ∗ B 0 )))η 0 )H0 . H is equipped with the left B-module structure given by π(B)[C ⊗ η] = [BC ⊗ η]
(B, C ∈ B, η ∈ H0 ) .
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Let U0 be a unitary operator on H0 such that U0 π0 (A)U0∗ = π0 (α(A)), A ∈ A. We set U [B ⊗ η] = [˜ α(B) ⊗ U0 η], B ∈ B, η ∈ H0 . We compute hU [B ⊗ η], U [B 0 ⊗ η 0 ]iH = h[˜ α(B) ⊗ U0 η], [˜ α(B 0 ) ⊗ U0 η 0 ]iH = (U0 η, π0 (E(˜ α(B ∗ B 0 )))U0 η 0 )H0 = (η, π0 (E(B ∗ B 0 ))η 0 )H0 = h[B ⊗ η], [B 0 ⊗ η 0 ]iH
(B, B 0 ∈ B, η, η 0 ∈ H0 ) .
Thus we obtain a unitary operator with the required properties. In fact U π(B) [C ⊗ η] = π(˜ α(B))U [C ⊗ η](B, C ∈ B, η ∈ H0 ). Remark 2.1. Let B be a C ∗ -algebra, G a compact group of automorphisms of B, R A = B G , and E := G αg (·)dg : B → A the associated conditional expectation. If ˜ (A) = A implies G = AutA (B), i.e. G is a Galois-closed subgroup of Aut(B), then α ˆ (see below). α ˜ G˜ α−1 = G and thus α ˜E α ˜ −1 = E. This is the case when B = A × G Similarly to Proposition 2.1 we can prove: Proposition 2.2. Let (A, B, E), (A1 , B1 , E1 ) be two triples as in Proposition 2.1 above, and φ˜ : B → B1 an isomorphism extending the isomorphism φ : A → A1 and ˜ If φ is unitarily implemented in the faithful representasuch that φ ◦ E = E1 ◦ φ. tions π0 , π01 of A, A1 , i.e. U0 π0 (A) = π01 ◦ φ(A)U0 , A ∈ A, then φ˜ is unitarily implemented in the representations induced from π0 , π01 up to B, B1 via E, E1 , respectively. ˜ Proof. Define U : H = B ⊗A H0 → H1 = B1 ⊗A1 H01 by U [B ⊗ η] := [φ(B) ⊗ U0 η] (B ∈ B, η ∈ H0 ). Then U is unitary and satisfies the covariance relation (we omit details). ˆ B1 = A1 × G, ˆ then the condition φ ◦ E = E1 ◦ φ˜1 is Remark 2.2. If B = A × G, automatic so Proposition 2.2 applies. Now we specialize to the case where A1 is the opposite C ∗ -algebra Aop . In what follows, o = oA denotes the linear isomorphism A → Aop (as Banach spaces over C) such that o(AB) = o(B)o(A), A, B ∈ A, o(A∗ ) = o(A)∗ . We identify A and (Aop )op (and denote both oA , oAop by o), thus o2 = idA . For brevity we sometimes write Ao in place of o(A), A ∈ A. We also denote by ρo the endomorphism o◦ρ◦o of Aop , i.e. ρo (Ao ) = ρ(A)o , A ∈ A, and define T o as the full subcategory of End(Aop ) with objects ∆o := o ◦ ∆ ◦ o. Lemma 2.1. o : T → T o , defined on objects and arrows by the formulae above, is a unit preserving contravariant strict monoidal functor. Proof. ιo is a monoidal unit of End(Aop ). Furthermore we have (ρσ)o = ρo σo . T ∈ (ρ, σ) iff T o ∈ (σo , ρo ) i.e. (T ∗ )o ∈ (ρo , σo ). (T × S)o = T o × S o , where T ∈ (ρ1 , σ1 ), S ∈ (ρ2 , σ2 ) and T × S := T ρ1 (S) = σ1 (S)T ∈ (ρ1 ρ2 , σ1 σ2 ).
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Proposition 2.3. Let A, ∆, , T , o : A → Aop , T o be as above. Defining o (ρo , σo ) := ((ρ, σ)∗ )o , ρ, σ ∈ ∆, (T o , o ) becomes a unital strict symmetric monoidal categorya with sufficiently many special objects, and we have (A × T )op = Aop × T o . Proof. The first part of the statement follows by direct computation; given the system {A × T , η, g ∈ G → αg , ρ ∈ ∆ → Hρ } it is easy to verify that {(A × T )op , oA×T ηoAop , g ∈ G → αog , ρo ∈ ∆o → Hρo := oA×T (Hρ∗ )} is still such a system over Aop , thus the conclusion follows by the uniqueness of the cross-product. We recall that an antiautomorphism of a C ∗ -algebra A corresponds to a linear antimultiplicative map of A onto itself. By an antilinear isomorphism we mean a antilinear multiplicative bijection A → A. Proposition 2.4. Let θ : A → A be an antilinear isomorphism. Consider the following conditions: (1) θ∆θ−1 = ∆, (10 ) A × Tθ∆θ−1 = A × T∆ , (2) θ((ρ, σ)) = (ρθ , σθ ), ρ, σ ∈ ∆, where ρθ := θρθ−1 . Then θ extends to an antilinear isomorphism θ˜ of A × T if (1), (2) hold, and (if and) only if (10 ) and (2) hold. Proof. Since φ = φθ := o ◦ θ ◦ ∗ is an isomorphism A → Aop , by the previous theorem we obtain an isomorphism φ˜ : A × T → (A × T )op ; thus by the last proposition θ˜ := oA×T ◦ φ˜ ◦ ∗ is well defined. Note that from the proposition above it is immediate that Aop × Tφ∆φ−1 = Aop × T∆o ⇔ A × Tθ∆θ−1 = A × T∆ . Hopefully our method is general enough to deal with the extension of endomorphisms as well. 3. Application to PCT-Symmetry on Fields We refer to the book of R. Haag [21] for a detailed treatment of the Algebraic approach to Quantum Field Theory (AQFT). We present only a short survey of those aspects which are relevant for our purposes. Here, A is a net of von Neumann algebras over the set K of double cones in the 3 + 1-dimensional Minkowski spacetime, namely an inclusion-preserving map O → A(O) from double cones to von Neumann algebras acting on the vacuum Hilbert space H0 , satisfying the conditions of locality, irreducibility, Haag duality, a The (linear) functor of Lemma 2.1 is not symmetric according to the terminology of [14, Sec. 1].
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property B, covariance, et cetera. Then the Doplicher–Roberts reconstruction theorem applies [16]. The full subcategory T of End(A) with objects ∆f (see below) is a strict (permutation) symmetric monoidal C ∗ -category with conjugates, having direct sums and sub-objects, and also with (ι, ι) = C. By the abstract duality theory T is recognized to be a compact group dual, and the field algebra F is thus obtained as the cross product A × T . More precisely, we get another net O → F(O) of von Neumann algebras acting irreducibly on a bigger Hilbert space H, a canonical net injection πA of A in F containing all the physically relevant irreducible representations, and a (continuous) faithful unitary representation U of the compact groupb G = AutπA (A) (F) on H such that πA (A) = FG and πA (A)0 = U (G)00 . Any irreducible representation of G appears with infinite multiplicity, and there is a oneˆ the spectrum of G, and the “physical spectrum” to-one correspondence between G, of A. In this setting a sector is represented by the inner equivalence class of localized, transportable morphisms ρ of A. Given two such endomorphisms ρ, σ we set (ρ, σ) = {T ∈ A|T ρ(A) = σ(A)T, A ∈ A}. Attached to any endomorphism ρ there is a canonical unitary representation ρ of the group of finite permutations over the integers, and also an intrinsic notion of statistics d(ρ) ∈ N. We denote by ∆ (resp. ∆f ) the semigroup of localized transportable morphisms (resp. those with finite statistics). In particular any ρ ∈ ∆f becomes inner in F, i.e. there is a Hilbert space Hρ in F linearly spanned by isometries ψ1 , . . . , ψd(ρ) with mutually orthogonal ranges, summing up to IH , such that X
d(ρ)
πA (ρ(A)) =
ψi πA (A)ψi∗ , A ∈ A .
i=1 op
If A is a theory as above, then A is still such a theory as well. We remark that the definition of o given in Proposition 2.3 is consistent with the standard notion of the symmetry adopted in the AQFT setting, namely QFT (ρ, σ) := (V × U )∗ ◦ U × V , where U ∈ (ρ, ρ1 ), V ∈ (σ, σ1 ) and ρ1 , σ1 are localized in mutually space-like regions; in fact we have ((ρ, σ)∗ )o = (((V × U )∗ ◦ U × V )∗ )o = (U ∗ × V ∗ ◦ V × U )o = (V × U )o ◦ (U ∗ × V ∗ )o = (V o × U o ) ◦ (U ∗ )o × (V ∗ )o = ((V ∗ )o × (U ∗ )o )∗ ◦ (U ∗ )o × (V ∗ )o i.e. oQFT = QFTo (with the obvious meaning of symbols). We now introduce a “minimal” set of requirements for our main object of study. b We often write g, h, . . . for the elements of G considered as an abstract group, and α , α , . . . g h for the corresponding automorphisms Ad(U (g)), Ad(U (h)), . . . of F.
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Definition 3.1. A PCT transformation θ = θA for the theory determined by the net A, the automorphic action x → αx of the translation group R4 and the vacuum state ω0 , is an anti-linear isomorphism of A such that (a) θ(A(O)) = A(−O)(O ∈ K), (b) θ ◦ αx = α−x ◦ θ (x ∈ R4 ), (c) ω0 ◦ θ = ω0∗ (ω0∗ (A) := ω0 (A), A ∈ A) see also [30]. Later on at some stage we will also require θ to be involutive. Nevertheless, if this were not the case, θ−1 would be a PCT-symmetry for A as well. A PCT-operator for the pair (A, θ) is an anti-unitary operator Θ = ΘA on H0 such that θ = Ad(Θ), ΘV (x) = V (−x)Θ, V (x)Ω = Ω, x ∈ R4 . We always have such an operator from the “ω0 -invariance” of θ. ˆ for F. See [6, p. 541], for the notion of a PCT-operator Θ Remark 3.1. Assuming only θ(A(O)) = A(−O), O ∈ K, then for any double cone O, x ∈ R4 we have θαx (A(O)) = θ(A(O + x)) = A(−O − x) = α−x θ(A(O)) i.e. θ−1 αx θαx ∈ Autloc (A) := {α ∈ Aut(A)|α(A(O)) = A(O), O ∈ K}. For the moment we avoid considering explicitly the meaning of the action of θ on localized endomorphisms of A. Thus we postpone the question of whether ρθ := θ ◦ ρ ◦ θ−1 is actually a conjugate of ρ. Note that if ρ is localized in a double cone O, then ρθ is localized in −O. Applying our previous results, we can now state the following theorem. Theorem 3.1. Let A be a local net, and θ a PCT-symmetry for A as above. Then θ extends to an antiautomorphism of the field net F. Proof. In order to check that θ may be extended to an antiautomorphism θˆ of the cross product F = A × T we have to check the two conditions stated in Proposition 2.4. For this purpose we consider ∆f , the semigroup of localized transportable morphisms with finite statistics, and notice that (see [8, Sec. 3]): If ϕ is a left inverse of ρ, then θϕθ−1 is a left inverse of ρθ , and if ϕ is a standard left inverse then θϕθ−1 also is. Therefore if ρ has finite statistics, also ρθ has, and (1) holds true; condition (2) is satisfied, using the fact that θ preserves space-like separation.c We can also restrict our consideration to ∆c ⊂ ∆f , the semigroup of localized morphisms with finite statistics which are translation covariant (if we want F to be a covariant net): if αx ρα−x ' ρ, x ∈ R4 , i.e. αx ◦ ρ ◦ α−x = Ux ρ(·)Ux∗ , Ux ∈ A, then αx ρθ α−x = αx θρθ−1 α−x = θα−x ραx θ−1 ∗ = θ ◦ U−x ρ(·)U−x ◦ θ−1 = θ(U−x )ρθ (·)θ(U−x )∗ ' ρθ
which means that ρθ is still translation covariant if ρ is. c As in [8], the passage to the Bosonized symmetry ˆ does not cause any difficulties.
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ˆ It is clear from the discussion above that θ(F(O)) = F(−O). In fact this is an immediate consequence of the relation F(O) = F ∩ A(O0 )0 (see [7]). We digress for a while on the validity of the relation θˆα ˆx = α ˆ −x θˆ (x ∈ R4 ) in the covariant case. According to [15] there is a unitary cocycle Wρ (x) ∈ (ρ, αx ρα−1 x ) for every ρ ∈ ∆c , determined by α ˆx (ψ) = Wρ (x)ψ, ψ ∈ Hρ . The following conditions are easily seen to be equivalent: (a) θ(Wρ (x)) = Wθρθ (−x); ˆ ˆ −x θ. (b) θˆα ˆx = α Note that even if (a) is not satisfied (i.e. θ(Wρ (x)) ∈ (θρθ, α−x θρθαx ) but not ˆ x θˆα ˆ x extend θ. We associate to any exactly equal to Wθρθ (−x)), both θˆ and α ˆ x =: αq(x) . x an element q(x) in the gauge group through the relation θˆ−1 α ˆ x θˆα Then the map x 7→ q(x) is continuous. Straightforward computations also show that q ∈ Hom(R4 , G0 ) (G0 is the connected component of the identity in G). If, moreover, θˆ commutes with αg , g ∈ G (see below), then q ∈ Hom(R4 , Z(G)0 ) (Z(G) ˆ x θˆα is the center of G). In particular if G has discrete center then θˆ−1 α ˆ x = idF must hold. At this point it seems likely that one could proceed arguing as in [8, Sec. 3]. However, we will not elaborate on this in the sequel. We summarize the results obtained so far. θ extends to F; if θ is unitarily implemented, then the extension θˆ also is; if θ is ω0 -invariant, meaning that ω0 ◦θ = ω0 ◦∗, then θˆ is ω ˆ 0 -invariant, where ω ˆ 0 = ω0 ◦ E. Suppose now that we have an antiautomorphism θˆ of F extending the antiautomorphism θ of A. From now on we assume that θ2 = idA . It follows that θˆ2 , extending θ2 , is an automorphism of the field net acting identically on the observables, and is thus an element of the gauge group (possibly the neutral element). ˆ ρ ) = Hθρθ is a (non-degenerate) G-invariant Hilbert space in F. Lemma 3.1. θ(H Proof. If ψi ∈ F(O) is an ONB in the (non-degenerate) Hilbert space Hρ , then ˆ i ) ∈ F(−O) is an ONB for Hθρθ . In fact, given A ∈ A, we compute θ(ψ X X X ˆ i )Aθ(ψ ˆ i )∗ = ˆ i )θ(θ(A))θ(ψ ˆ i )∗ = θ(ψ θ(ψ θ(ψi θ(A)ψi∗ ) = θρθ(A) . i
i
i
ˆ ˆ ˆ It also follows that the operator Yρθ : Hρ → Hθρθ defined by θ(ψ) =: Yρθ (ψ)(ψ ∈ Hρ ) is anti-unitary since
ˆ ∗ θ(ψ) ˆ ˆ ∗ ψ) = (ϕ, ψ) θ(I) ˆ = (ψ, ϕ)I . θ(ϕ) = θ(ϕ Hρ Given an (antilinear) automorphism of A, namely θ, which can be extended to an (antilinear) automorphism of F according to the procedure outlined in [8] and generalized above, there are still two natural questions. On one hand it is not obvious that there is such an extension commuting with the gauge action, see [15, Sec. 8]. (This cannot be true at the same time for every extension: think about the
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case where the compact gauge group is not abelian.) On the other hand it is not clear that θ implements the conjugation on the sectors (in formulas θρθ = ρ). It turns out that these two properties are in fact (almost) equivalent. We remark that conditions under which the formula for the conjugate holds have been given in terms of the modular conjugation JW in [17]; we will make further comments on this in the next section. ˆ the class of the irreducible representation of G Notation: we denote by ξρ ∈ G corresponding to the irreducible ρ, ρ ∈ ∆f . Proposition 3.1. Assume the existence of an extension θˆ of θ commuting with the gauge action. Then for every localizable transportable sector with finite statistics ρ we have [ρθ ] = [ρ] . Proof. If ψi is an irreducible multiplet in Hρ ⊂ F(O) transforming according to P the representation u, [u] = ξρ , that is αg (ψi ) = U (g)ψi U (g)∗ = j uj,i (g)ψj , we have X ˆ g (ψi )) = ˆ j) . θ(α uj,i (g)θ(ψ j
ˆ g , then the left-hand side of the previous relation If in addition we have αg θˆ = θα ˆ ˆ i ) ∈ Hθρθ transforms according is also equal to αg (θ(ψi )), i.e. we deduce that θ(ψ to the conjugate representation class ξρ . Using the fact that the correspondence ˆ is bijective, ξθρθ = ξρ implies that θρθ is a conjugate of ρ. [ρ] ∈ Sect(A) → ξρ ∈ G The converse implication holds under some additional hypotheses. If θρθ is a conjugate of ρ for every irreducible ρ ∈ ∆f , then vectors in the G-Hilbert space ˆ ρ ) = Hθρθ transform according to the class ξρ . This information is (almost) θ(H ˆ g. sufficient to deduce that αg θˆ = θα ˆ Note that, for In the following we fix once and for all a reference extension θ. −1 ˆ ˆ ˆ g θˆ−1 = αη(g) . every g ∈ G, θαg θ ∈ AutπA (A) (F) = G. We define η : G → G via θα Lemma 3.2. η is a continuous automorphism of G such that η 2 is inner. Moreover, if θˆ2 = idF then we have η 2 = idG . Proof. Since θˆ2 = αz for some z ∈ G, we have θˆ2 αg θˆ−2 = αz αg α−1 z = αzgz −1 = αη2 (g) , g ∈ G. The rest is obvious. Lemma 3.3. If η ∈ Inn(G), then there is an extension θˆ1 of θ commuting with the gauge action. ˆ g θˆ−1 = αhgh−1 = Proof. Assuming that η = Ad(h) for some h ∈ G, we have θα αh αg αh−1 , therefore θˆ1 := αh−1 θˆ commutes with αg , g ∈ G. Proposition 3.2. Let us assume that the gauge group G is quasi complete (see the Appendix). If θρθ is a conjugate of ρ for every irreducible ρ ∈ ∆f , then there exists an extension θˆ of θ commuting with the gauge action.
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Proof. The notation is the same as in Proposition 3.1 above. By hypothesis and ˆ i ) transforms according to the class of the conjugate representation Lemma 3.1 θ(ψ ˆ η−1 (g) , we obtain u. From αg θˆ = θα X ˆ i )) = θ(α ˆ η−1 (g) (ψi )) = ˆ j) . αg (θ(ψ uj,i (η −1 (g))θ(ψ j
From this we infer that the unitary equivalence classes of u and u ◦ η coincide for every irreducible u. Since G is quasi-complete it follows that η is inner and the conclusion follows by the previous lemma. ˆ g , it follows ˆ on H. If αg θˆ = θα Let us now consider the anti-unitary operator Θ ˆ ˆ that U (g)Θ = ΘU (g), g ∈ G (the converse is clear). In the next section we will examine another basic example, namely the problem of the extension of Ad(J). 4. Connection with Geometric Modular Action A fundamental example of an antiunitary operator relevant for QFT is provided by the modular involution J = JW associated by the Tomita–Takesaki theory to the von Neumann algebra R(W ) = A(W )00 and to the cyclic and separating vacuum vector Ω (W = WR is the right wedge). Then j(·) := Ad(J)(·) is an antilinear isomorphism of R(W ) onto R(W )0 = R(W 0 ). j is a natural candidate to which we would apply our results; however we find some problems. An immediate obstruction is that j should induce an antilinear isomorphism of A. Other constraints follow by our analysis in Sec. 3 (see Theorem 3.1). Therefore we need an additional assumption linking j to the net structure. Following [9, 17] we require that the adjoint action j of the modular conjugation J of (A(WR )00 , Ω) on A is geometric, meaning that JA(O)J = A(jp O) holds for every double cone O, where jp is the pointwise transformation jp (x0 , x1 , x2 , . . . , xd ) := (−x0 , −x1 , x2 , . . . , xd ) . As a matter of fact this assumption automatically holds under much less restrictive conditions [26, 32, 10]. Furthermore if the group of local symmetries of A is compact (as we assume) it follows also that JV (x)J = V (jp x) [26]. Now we can apply our analysis to the present situation; in particular we consider the PCT symmetry θJ := Ad(V (R1 (π))J) (R1 (π) is the rotation by π around the x1 -axis). θJ verifies the hypothesis of the previous section, thus we obtain an antiautomorphism of A which can be extended to an antiautomorphism of F. However in this case we can say more, since we know the existence of an extension θˆJ ˆ H0 = J ([25, Lemma 3.3]), this commuting with the gauge action. Recalling that J| extension is implemented by Vˆ (R1 (π))Z Jˆ, where Z := (I +iU (k))/(i+1), k ∈ Z(G) ˆ ∗ . It follows by is the Bose–Fermi element such that k 2 = e, and we have Z Jˆ = JZ our previous analysis that jρj is a conjugate of ρ whenever ρ is a covariant DHR (i.e. transportable, localizable) morphism with finite statistics. ˆ ρ Jˆ = Hjρj , cf. [24]. Actually we have JH
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Just a word on the converse relation: given an antilinear isomorphism θ of the C ∗ -algebra with the properties listed in Sec. 2, when does it come from a modular involution? θ acts locally, that is, it preserves the net structure, hence it maps A(WR ) onto A(WL ) (as C ∗ -algebras). The assumption that it is spatially implemented by an antiunitary operator Θ amounts to saying that ΘA(WR )00 Θ = A(WL )00 = A(WR )0 (assuming wedge duality). Moreover ΘΩ = Ω. Using Araki’s characterization of the modular involution we may say that Θ = V (R1 (π))JW if and only if (Ω, AV (R1 (π))ΘAΩ) ≥ 0 (A ∈ A(WR )00 ) is satisfied, cf. [5]. 5. Comments on Low Dimensional Theories For theories in dimension larger than 2 we have (ρ, σ) = (σ, ρ)∗ , but in dimension less than or equal to 2 these operators are a priori different intertwiners between ρσ and σρ. This is related to the occurrence of braid group statistics. The monodromy operator M (ρ, σ) := (ρ, σ)(σ, ρ) measures the deviation from permutation statistics. An irreducible endomorphism ρ is said to be degenerate if M (ρ, σ) = 1 for every σ. For theories living in 1 + 1 dimensions (satisfying the split property) it is still possible to perform the DR construction provided we consider only semigroups of degenerate endomorphisms, see [28], cf. [3]. If we denote by ∆D the degenerate endomorphisms with finite statistical dimension, then (∆D , ) is a permutation symmetric, specially directed semigroup closed under direct sums and sub-objects. If F is the spatial cross product of A by (∆D , ), then O → F(O) is a normal field system with gauge symmetry satisfying twisted duality. An easy computation shows that for general morphisms ρ, σ, the relation θ((ρ, σ)) = (σθ , ρθ )∗ holds, and thus θ(M (ρ, σ)) = M (ρθ , σθ )∗ . Hence if ρ is degenerate ρθ also is, and we have θ((ρ, σ)) = (ρθ , σθ ), i.e. condition 2) of the previous section holds. Hence our results apply to this situation and give a PCT operation acting on the (“degenerate”) field algebra F starting from such an operation on A. Appendix A. Quasi-Complete Groups In this appendix we collect some results on automorphisms of groups that are relevant to our study in Sec. 3. A more detailed version will appear in a separate paper [11]. Definition A.1. If G is a (locally compact) group, we denote by Aut(G)Gˆ the set ˆ {α ∈ Aut(G)|π ◦ α ∼ = π, π ∈ G}. Note that Inn(G) ⊆ Aut(G)Gˆ . Lemma A.1. Aut(G)Gˆ is a normal subgroup of Aut(G). Definition A.2. A group G is quasi-complete if Inn(G) = Aut(G)Gˆ . Proposition A.1. If G is a compact Abelian group, then it is quasi-complete.
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ˆ Since the chaProof. The condition above becomes χ ◦ α = χ for every χ ∈ G. racters separate the points of G, we get α(g) = g, g ∈ G, i.e. α is the trivial automorphism. The case of compact (Lie) groups is enough understood. Compact connected groups are quasi-complete [27, 22, 11]. This can be seen in several ways. For instance, i ˆ → 1, is the existence of a short exact sequence 1 → Inn(G) → Aut(G) → Aut(G) ˆ shown in [27]; thus Out(G) ' Aut(G) (see also [22]). The proof in [11] is based on the highest weight theorem and some projective limit machinery. In fact any compact group which is a suitable inverse limit of quasi-complete groups is quasicomplete [11]. Remark A.1. (Unitary equivalence classes of) irreducible representations of SU (2) are classified by their dimension, thus they are invariant with respect to composition with an automorphism. Note that Out(SU (2)) is trivial. Arbitrary direct products of quasi-complete compact groups are quasi-complete. Finite groups are not necessarily quasi-complete. A counterexample is given in [23], and a general construction is described in [11]. Another example can be found in [22, Example 2.2]. Recall that a (finite) group G is called complete if Aut(G) = Inn(G). The permutation groups Sn (n ≥ 2), n 6= 6 are complete (see e.g. [31, p. 215]), thus they are quasi-complete. Although S6 is not complete, it is still quasi-complete. If p is odd prime, the dihedral group D2p is complete iff p = 3 [31, p. 218]. If H is finite simple then Aut(H) is complete. Defining by induction Gn = Aut(Gn−1 ), with G1 finite, then after a finite number of steps we find a complete group [33]. If a group G has an automorphism of order divisible by p, p prime, then |G| is divisible by p. It follows that if a theory has an involutive PCT transformation acting on the fields and a finite gauge group G of odd order, then the η defined in Lemma 3.2 has to be trivial. Acknowledgments This paper originated from a collaboration with D. Buchholz and C. D. would like to thank him for his continuing interest and encouragement. We thank S. Doplicher and J. E. Roberts for enlightening discussions. We are also grateful to M. Cowling, L. Geatti, T. Mach´ı, C. Moore and R. Schoof for useful comments about groups and their automorphisms. We thank the organizers of the program on Local Quantum Physics at the E. Schr¨odinger Institute for providing us hospitality and support in Wien, where part of this work has been written. R. C. was also supported by the EU Network in Non-Commutative Geometry and CNR. References [1] M. Aita, W. R. Bergmann and R. Conti, “Amenable groups and generalized Cuntz algebras”, J. Funct. Anal. 150 (1997) 48–64.
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[2] J. J. Bisognano and E. H. Wichmann, “On the duality condition for a hermitean scalar field”, J. Math. Phys. 16 (1975) 985–1007; “On the duality condition for quantum fields”, J. Math. Phys. 17 (1976) 303–321. [3] J. B¨ ockenhauer and J. Fuchs, “CFT fusion rules, DHR gauge groups, and CAR algebras”, DESY preprint 97-077. [4] H. J. Borchers, “The CPT theorem in two-dimensional theories of local observables”, Commun. Math. Phys. 143 (1992) 315–332. [5] R. Brunetti, D. Guido and R. Longo, “Modular structure and duality in conformal quantum field theory”, Commun. Math. Phys. 156 (1993) 201–219. [6] D. Buchholz and C. D’Antoni, “Phase space properties of charged fields in the theory of local observables”, Rev. Math. Phys. 7 (1995) 527–557. [7] D. Buchholz, S. Doplicher, R. Longo and J. E. Roberts, “A new look at Goldstone’s theorem”, Rev. Math. Phys. Special Issue (1992) 47–82. [8] D. Buchholz, S. Doplicher, R. Longo and J. E. Roberts, “Extensions of automorphisms and gauge symmetries”, Commun. Math. Phys. 155 (1993) 123–134. [9] D. Buchholz and S. J. Summers, “An algebraic characterization of vacuum states in Minkowski space”, Commun. Math. Phys. 155 (1993) 442–458. [10] D. Buchholz, O. Dreyer, M. Florig and S. J. Summers, “Geometric modular action and spacetime symmetry groups”, math-ph/9805026. [11] R. Conti, C. D’Antoni and L. Geatti, “Quasi-complete compact groups”, in preparation. [12] D. R. Davidson, “Modular covariance and the algebraic PCT/spin–statistics theorem”, preprint. [13] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics II”, Commun. Math. Phys. 35 (1974) 49–85. [14] S. Doplicher and J. E. Roberts, “A new duality theory for compact groups”, Invent. Math. 98 (1989) 157–218. [15] S. Doplicher and J. E. Roberts, “Endomorphisms of C ∗ -algebras, cross products and duality for compact groups”, Ann. Math. 130 (1989) 75–119. [16] S. Doplicher and J. E. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure of particle physics”, Commun. Math. Phys. 131 (1990) 51–107. [17] D. Guido and R. Longo, “Relativistic invariance and charge conjugation in Quantum Field Theory”, Commun. Math. Phys. 148 (1992) 521–551. [18] D. Guido and R. Longo, “An algebraic Spin and Statistics Theorem”, Commun. Math. Phys. 172 (1995) 517–533. [19] D. Guido and R. Longo, “The conformal Spin and Statistics Theorem”, Commun. Math. Phys. 181 (1996) 11–35. [20] D. Guido, “Modular covariance, PCT, spin and statistics”, Ann. Inst. Henri Poincar´e 63 (1995) 383–398. [21] R. Haag, Local Quantum Physics, Springer Verlag, Berlin, 1992. [22] D. Handelman, “Representation rings as invariants for compact groups and limit ratio theorems for them”, Int. J. Math. 4 (1993) 59–88. [23] B. Huppert, “Endliche gruppen I, Grundlehren der Matematischen Wissenschaft 134”, Springer-Verlag, Berlin Heidelberg New York, 1979. [24] T. Isola, “Modular structure of the crossed product by a compact group dual”, J. Operator Theory 33 (1995) 3–31. [25] B. Kuckert, “A new approach to Spin and Statistics”, Lett. Math. Phys. 35 (1995) 319–331. [26] B. Kuckert, “Borchers’ commutation relations and modular symmetries in quantum field theories”, Lett. Math. Phys. 41 (1997) 307–320. [27] J. R. McMullen, “On the dual object of a compact connected group”, Mat. Z. 33 (1984) 539–552.
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[28] M. M¨ uger, “On charged fields with group symmetry and degeneracies of Verlinde’s matrix S”, DESY preprint 97-080. [29] M. A. Rieffel, “Induced representations of C ∗ -algebras”, Adv. Math. 13 (1974) 176– 257. [30] C. Rigotti, “Remarks on the modular operator and local observables”, Commun. Math. Phys. 61 (1978) 267–273. [31] J. S. Rose, A Course on Group Theory, Cambridge Univ. Press, 1978. [32] S. J. Summers, “Geometric modular action and transformation groups”, Ann. Inst. Henri Poincar´e 64 (1996) 409–432, and private communication. [33] M. Suzuki, Group Theory I, Grundlehren der Matematischen Wissenschaft 247, Springer-Verlag, Berlin Heidelberg New York, 1980. [34] J. Yngvason, “A note on essential duality”, Lett. Math. Phys. 31 (1994) 127–141.
THE CAPPELLI ITZYKSON ZUBER A-D-E CLASSIFICATION TERRY GANNON Department of Mathematical Sciences, University of Alberta Edmonton, Alberta, Canada, T6G 2G1 E-mail:
[email protected] Received 19 February 1999 In 1986 Cappelli, Itzykson and Zuber classified all modular invariant partition functions for the conformal field theories associated to the affine A1 algebra; they found they fall into an A-D-E pattern. Their proof was difficult and attempts to generalise it to the other affine algebras failed — in hindsight the reason is that their argument ignored most of the rich mathematical structure present. We give here the “modern” proof of their result; it is an order of magnitude simpler and shorter, and much of it has already been extended to all other affine algebras. We conclude with some remarks on the A-D-E pattern appearing in this and other RCFT classifications.
1. The Problem One of the more important results in conformal field theory is surely the classification due to Cappelli, Itzykson, and Zuber [1; see also 2] of the genus 1 partition (1) functions for the theories associated to A1 (which in turn implies the classification of the minimal models). Their list was curious: the partition functions fall into the A-D-E pattern familiar from the simply-laced Lie algebras, finite subgroups of SU2 (C), simple singularities, subfactors with index < 4, representations of quivers, etc. See e.g. [3]. The problem can be phrased as follows. Fix any integer n ≥ 3. Let P+ = {1, 2, . . . , n − 1}, and let S and T be the (n − 1) × (n − 1) matrices with entries r Sab =
2 ab sin π , n n
a2 Tab = exp πi δa,b . 2n
Find all (n − 1) × (n − 1) matrices M such that • M commutes with S and T : M S = SM and M T = T M • M has nonnegative integer entries: Mab ∈ Z+ for all a, b ∈ P+ • M is normalised so that M11 = 1. Call any such M a physical invariant. Since most entries Mab are usually zero, it is more convenient to formally express M as the coefficient matrix for the combination n−1 X Z= Mab χa χ∗b . a,b=1
739 Reviews in Mathematical Physics, Vol. 12, No. 5 (2000) 739–748 c World Scientific Publishing Company
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def
Theorem 1.1. The complete list of physical invariants is (using Ja = n − a) An−1 =
n−1 X
|χa |2 ,
∀n ≥ 3
χa χ∗J a a ,
whenever
n is even 2
D n2 +1 = |χ1 + χJ1 |2 + |χ3 + χJ3 |2 + · · · + 2|χ n2 |2 ,
whenever
n is odd 2
E6 = |χ1 + χ7 |2 + |χ4 + χ8 |2 + |χ5 + χ11 |2 ,
for n = 12
a=1
D n2 +1 =
n−1 X a=1
E7 = |χ1 + χ17 |2 + |χ5 + χ13 |2 + |χ7 + χ11 |2 + χ9 (χ3 + χ15 )∗ + (χ3 + χ15 )χ∗9 + |χ9 |2 ,
for n = 18
E8 = |χ1 + χ11 + χ19 + χ29 | + |χ7 + χ13 + χ17 + χ23 | , for n = 30 . 2
2
These realise the A-D-E pattern, in the following sense. The Coxeter number h of the name X` equals the corresponding value of n, and the exponents of X` i (i.e. the mi in the eigenvalues 4 sin2 (π m 2n ) of its Cartan matrix) equal those a ∈ P+ for which Maa 6= 0. An interpretation of the nondiagonal entries of M has been provided by subfactor theory — by Ocneanu [4] and independently by B¨ockehauer and Evans [5]. Cappelli–Itzykson–Zuber proved their theorem by first finding an explicit basis for the space of all matrices commuting with S and T . Unfortunately their proof was long and formidable. Considering all of the structure implicit in the problem, we should anticipate a much more elementary argument. This is not merely of academic interest, because there is a natural generalisation of this problem to all other affine algebras. Several people had tried to extend the argument of [1] to these larger algebras, but with [6] it became clear that some other approach was necessary, or the generalisation would never be achieved. And of course another reason is that the more transparent the argument, the better the chance of ultimately understanding the connection with A-D-E. In this paper we provide a considerably shorter proof of the theorem, bearing no resemblance to the older arguments. Our proof is an example of the “modern” approach to physical invariant classifications. See [7] for a summary of the current status of these classifications for the other affine algebras. The argument which follows is completely elementary: no knowledge of e.g. CFT or Kac–Moody algebras is assumed. It is based on various talks I’ve given, most recently at the Schr¨odinger Institute in Vienna where I wrote up this paper and who I thank for generous hospitality. I also thank D. Evans, J. Fuchs, M. Flohr, J. McKay, V. Petkova, and J.-B. Zuber for correspondence. 2. The Combinatorial Background In this section we include some of the basic tools belonging to any classification of the sort, and we give a flavour of their proofs. We will state them for the specific problem given above, but everything generalises without effort [8].
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First note that commutation of M with T implies the selection rule Mab 6= 0 =⇒ a2 ≡ b2
(mod 4n) .
(2.1)
Next, let us write down some of the basic properties obeyed by S. S is symmetric and orthogonal (so M = SM S), and S1b ≥ S11 > 0 .
(2.2)
The permutation J of P+ , defined by Ja = n − a, corresponds to the order 2 (1) symmetry of the extended Dynkin diagram of A1 ; it satisfies SJa,b = (−1)b+1 Sab .
(2.3)
Note that the element 1 ∈ P+ is both physically and mathematically special; our strategy will be to find all possible first rows and columns of M , and then for each of these possibilities to find the remaining entries of M . The easiest result follows by evaluating M S = SM at (1, a) for any a ∈ P+ : S1a +
n−1 X
M1b Sba ≥ 0 ,
(2.4)
b=2
with equality iff the ath column of M is identically 0. Equation (2.4) has two uses: it severely constrains the values of M1b (similarly Mb1 ), and it says precisely which columns (and rows) are nonzero. Another simple observation is 1 = M11 =
n−1 X
2 S1a Mab S1b ≥ S11
a,b=1
n−1 X
Mab .
a,b=1
This tells us that each entry Mab is bounded above by S12 (we will use this below). 11 In particular, there can only be finitely many physical invariants for each n. (This same calculation shows more generally that there will only be finitely many physical (1) invariants for a given affine algebra Xr and level k.) Next, let’s apply the triangle inequality to sums involving (2.3). Choose any i, j ∈ {0, 1}. Then MJ i 1,J j 1 =
n−1 X
(−1)(a+1)i S1a Mab (−1)(b+1)j S1b .
a,b=1
Taking absolute values, we obtain MJ i 1,J j 1 ≤
n−1 X
S1a Mab S1b = M11 = 1 .
a,b=1
Thus MJ i 1,J j 1 can equal only 0 or 1. If it equals 1, then we obtain the selection rule: (a + 1)i ≡ (b + 1)j (mod 2) whenever Mab 6= 0; this implies the symmetry MJ i a,J j b = Mab for all a, b ∈ P+ .
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Whenever you have nonnegative matrices in your problem, and it makes sense to multiply those matrices, then you should seriously consider using Perron–Frobenius theory — a collection of results concerning the eigenvalues and eigenvectors of nonnegative matrices. Our M is nonnegative, and although multiplying M ’s may not give us back a physical invariant, at least it will give us a matrix commuting with S and T . In other words, the commutant is much more than merely a vector space, it is in fact an algebra. Important applications of this thought are the following two lemmas. Lemma 2.1. Let M be a physical invariant, and suppose Ma1 = δa,1 , i.e. the first column of M is all zeros except for M11 = 1. Then M is a permutation matrix, i.e. there is some permutation π of P+ such that Mab = δb,πa , and Sπa,πb = Sab . This is proved by studying the powers (M T M )L as L goes to infinity: its diagonal entries will grow exponentially with L, unless there is at most one nonzero entry on each column of M , and it equals 1. (Recall that the entries of (M T M )L must be bounded above.) Then show that also M1a = δ1,a (evaluate M S = SM at (1,1)), and look at (M M T )L . Lemma 2.1 was found independently by Schellekens and Gannon. That argument is elementary enough that it required no knowledge of Perron– Frobenius. But Perron–Frobenius is needed for generalisations. In this fancier language, what the preceding argument shows is: write M as the direct sum of indecomposable submatrices; then the largest eigenvalue of the submatrix containing (1, 1) bounds above that for each other submatrix. Arguing with a little more sophistication, we obtain much more. The special case we need is: Lemma 2.2 [8]. Let M be a physical invariant, and suppose Ma1 6= 0 only for a = 1 and a = J1, and similarly for M1a , i.e. the first row and column of M are all zeros except for MJ i 1,J j 1 = 1. Then the ath row (or column) of M will be identically 0 iff a is even. Moreover, let a, b ∈ P+ , both different from n2 , and suppose Mab 6= 0. Then ( 1 if c = b or c = Jb Mac = 0 otherwise and a similar formula holds for Mcb . This lemma says that the indecomposable submatrices of M which don’t involve (the fixed-point of J) will either be trivial (0) (for even places on the diagonal), or involve blocks 1 1 . You can check this for the Deven and E7 partition functions. 1 1 Our final ingredient is a Galois symmetry obeyed by S, and its consequence for M . Again, see e.g. [8] for a proof. Let L be the set of all ` coprime to 2n. For each ` ∈ L, there is a permutation a 7→ [`a] of P+ , and a choice of signs ` : P+ → {±1}, such that Mab = ` (a)` (b) M[`a],[`b] , (2.5) n 2
for all a, b ∈ P+ . In particular, write {x} for the unique number congruent to x (mod 2n) satisfying 0 ≤ {x} < 2n. Then if {`a} < n, put [`a] = {`a} and ` (a) = +1,
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while if {`a} > n, put [`a] = 2n − {`a} and ` (a) = −1. This “Galois symmetry” (2.5) comes from hitting M = SM S with the `th “Galois automorphism”. Any polynomial over Q with a 2nth root of unity ζ as a zero — and the entries of M = SM S can be interpreted in that way — also has ζ ` as a zero. We then use [`a]b sin(π ` ab n ) = ` (a) sin(π n ). From (2.5) and the positivity of M , we get for all ` ∈ L the Galois selection rule Mab 6= 0 =⇒ ` (a)` (b) .
(2.6)
(2.5) and (2.6), valid for any affine algebras, were first found independently by Gannon and Ruelle–Thiran–Weyers. The Galois interpretation, and extension to all RCFT, is due to Coste–Gannon. (1)
3. The “Modern” Proof of the A1
Classification
The last section reviewed the basic tools shared by all modular invariant partition (1) function classifications. In this section we specialise in A1 . The first step will be to find all possible values of a such that M1a 6= 0 or Ma1 6= 0. These a are severely constrained. We know two generic possibilities: a = 1 (good for all n), and a = J1 (good when n2 is odd). We now ask the question, what other possibilities for a are there? Our goal is to prove (3.4). Assume a 6= 1, J1. There are only two constraints on a which we will need. One is (2.1): (a − 1)(a + 1) ≡ 0
(mod 4n) .
(3.1)
More useful is the Galois selection rule (2.6), which we can write as sin(π` na ) sin(π` n1 ) > 0, for all ` ∈ L. But a product of sines can be rewritten as a difference of cosines, so we get a−1 a+1 cos π ` > cos π ` . (3.2) n n Since ` obeys (3.2) iff ` + n does, we can take ` in (3.2) to be coprime merely to n instead of 2n. Call L0 the set of these `. (3.2) is strong and easy to solve; here is my argument. Define d = gcd(a−1, 2n), d0 = gcd(a+1, 2n). Note from (3.1) that gcd(d, d0 ) = 2 and dd0 = 4n, so d, d0 ≥ 6. We can choose `0 , `0 ∈ L0 so that `0 (a + 1) ≡ d0 (mod 2n) and d n n − d if is odd and is even 2 2 d n `0 (a − 1) ≡ n − 2d if is odd and is odd (mod 2n) . 2 2 0 n − d otherwise, i.e. if d is odd 2 2 Now define `i = 2ni d + `0 . Then `i (a − 1) ≡ `0 (a − 1) (mod 2n) for all i, and for 0 ≤ i < d2 the numbers `i (a + 1) will all be distinct (mod 2n). For those i, precisely φ( d2 ) of the `i will be in L0 , where φ(x) is the Euler totient, i.e. the number of
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positive integers less than x coprime to x. (This count follows from the fact that for any prime p dividing d2 , p won’t divide 2n d and hence exactly one value of i (mod p) will be forbidden.) Now, the numbers `i (a + 1) are all multiples of d0 . So (3.2) with ` = `i gives us d n 2d if is odd and is even 2 2 d d n − 1 d0 < (3.3) φ 4d if is odd and is odd 2 2 2 d otherwise . Also, (3.2) with ` = `0 requires d0 > d(≥ 6). Combining this with (3.3), we get φ( d2 ) − 1 < 2, 4, or 1, which has the solutions d = 6 (for n some multiple of 4), and d = 6 or 10 (for n an odd multiple of 2). (3.3) now gives us exactly 3 possibilities: d = 6, d0 = 8, n = 12 (which yields E6 as we will see below); d = 6, d0 = 20, n = 30, and d = 10, d0 = 12, n = 30 (both which correspond to E8 ). So what we have shown is that, provided n 6= 12, 30, M obeys the strong condition Ma1 6= 0 or M1a 6= 0 =⇒ a ∈ {1, J1} . (3.4) Consider first Case 1: Ma1 = δa,1 . This is the condition in Lemma 2.1, and so we know Mab = δb,πa for some permutation π of P+ obeying Sab = Sπa,πb . We know π1 = 1; put m := π2. Then sin(π n2 ) = sin(π m n ), and so we get either m = 2 or m = J2. By T -invariance (2.1), the second possibility can only occur if 4 ≡ (n − 2)2 (mod 4n), i.e. 4 divides n. But for those n D n2 +1 is also a permutation matrix, so replacing M if necessary with the matrix product M D n2 +1 , we can always require m = 2, i.e. π2 = 2. Now take any a ∈ P+ and write b = πa: we have both sin(π na ) = sin(π nb ) and 2b a b sin(π 2a n ) = sin(π n ). Dividing these gives cos(π n ) = cos(π n ), and we read off that b = a, i.e. that M is the identity matrix An−1 . The other possibility, Case 2, is that both M1,J1 6= 0 and MJ1,1 6= 0. Then Lemma 2.2 applies. (2.1) says 1 ≡ (n − 1)2 (mod 4n), i.e. n2 is odd. n = 6 is trivial (the only unknown entry, M3,3 , is fixed by M S = SM at (1,3)), so consider n ≥ 10. The role of “2” in Case 1 will be played here by “3”. The only difference is the complication caused by the fixed-point n2 . Can M3, n2 6= 0? If so, then Lemma 2.2 would imply M3,a = 0 for all a 6= n2 . Evaluating M S = SM at (3, 1), we obtain M3, n2 = 2 sin(π n3 ), i.e. n = 18, which corresponds to E7 as we show later. Thus we can assume for now that both M3, n2 = M n2 ,3 = 0, and so by Lemma 2.2 there will be a unique m < n2 for which M3,m 6= 0. M S = SM at (3, 1) now gives m = 3. For any odd a ∈ P+ , a 6= n2 , can we have M n2 ,a 6= 0? If so then M S = SM at (1, a) and (3, a) would give us 2 sin(π na ) = M n2 ,a = 2 sin(π 3a n ), which is impossible. Therefore Lemma 2.2 again applies, and we get a unique b < n2 for which Mba 6= 0. The usual argument forces b = a, and we obtain the desired result: M = D n2 +1 . 3.1. The exceptional at n = 12 We know M1a ≥ 1 for some a ∈ P+ with gcd(a + 1, 24) = 8, i.e. a = 7. From (2.4) at a = 2, we get sin( π6 ) − M1,7 sin( π6 ) ≥ 0. Thus M1,7 = 1. Applying the
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Galois symmetry (2.5) for `5, 7, 11, we obtain the terms |χ1 + χ7 |2 + |χ5 + χ11 |2 in E6 . Now use (2.4) to show that among the remaining entries of M , only the 4th and 8th rows and columns will be nonzero. MJ1,J1 = 1 tells us M44 = M88 and M84 = M48 . These must be equal, by evaluating M S = SM at (4, 2), and now either Perron–Frobenius or M S = SM at (1, 4) forces that common value to be 1. We thus obtain M = E6 . 3.2. The exceptional at n = 18 We know M3,9 = 1 and that M3,a = 0 for all other a 6= 9. T -invariance (2.1) and Lemma 2.2 applied to the other odd a < 9, force Maa = 1. The only remaining entry is M9,9 , which is fixed by M S = SM at (9, 1). We get M = E7 . 3.3. The exceptional at n = 30 We know either M1,11 or M1,19 is nonzero; the only other (potentially) nonzero M1a are at a = 1, J1. Suppose first that M1,J1 = 1, so M1,11 = M1,19 . Then (2.4) at a = 3 forces M1,11 = 1; Galois (2.5) for ` = 7, 11, 13, 17, 19, 23, 29 gives us rows 7, 11, 13, 17, 19, 23, 29 of M , and (2.4) tells us all other rows must vanish. If instead M1,J1 = 0, then (2.4) at a = 2, 3, 4 gives our contradiction. 4. Closing Remarks There are two reasons to be optimistic about the possibilities of a classification of all modular invariant partition functions (= physical invariants) for all simple Xr . One is the main general result in the problem [8], which gives the analogue for any (1) Xr of the A1 physical invariants named A? , D? , and E7 . See [7] for a discussion. The other cause for optimism is the shortness and simplicity of the above proof (1) for A1 . (1) The reader should be warned though that A1 is an exceptionally gentle case — as we’ve seen, the proof quickly reduces essentially to combinatorics. Our argument here is a projection of the general argument onto this special case, and this projection loses most of the structure present in the general proofs. The general arguments are necessarily more subtle and sophisticated. Nevertheless this paper should help the interested reader understand the further literature on this fascinating problem, and make more accessible the proof of the important classification of Cappelli–Itzykson–Zuber. A big question is, does this new proof shed any light on the main mystery here: the A-D-E pattern to our theorem? It does not appear to. But it should be (1) remarked that it is entirely without foundation to argue that this A1 classification is “equivalent” to any other A-D-E one. There is a connection with the other A-D-E classifications which should be explained, and which has not yet been satisfactorily explained. But what we should look for is some critical combinatorial part of a proof which can be identified with critical parts in other A-D-E classifications. For instance, does an argument equivalent to that surrounding (3.2) appear elsewhere in the A-D-E literature?
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There has been some progress elsewhere at understanding our A-D-E. Nahm [9] constructed the invariant X` in terms of the compact simply-connected Lie group of type X` , and in this way could interpret the n = h and Mmi mi 6= 0 coincidences. A very general explanation for A-D-E has been suggested by Ocneanu [4] using a bimodule-conection approach on II1 subfactors and independently by B¨ ockenhauer and Evans [5] through a theory of induction and restriction of sectors on nets of III1 loop group subfactors. Also worth mentioning is the classification of boundary conditions in CFT (i.e. of partition functions on a finite cylinder rather than a torus). This has been claimed (though not everyone seems to agree) to be equivalent to a (1) classification of certain Z+ -valued representations of the fusion ring [10]; for A1 the problem is quickly reduced to considering symmetric Z+ -matrices with largest eigenvalue < 2, and from this we once again get an A-D-E pattern. Nevertheless, the A-D-E in CFT seems to remain almost as mysterious now as it did a dozen years ago . . . . An interesting question certainly is, what form if any does the A-D-E pattern (1) (1) take for A2 physical invariants? A3 ? etc. A step in this direction is provided by the work by Di Francesco, Petkova and Zuber on fusion graphs (see e.g. [11]), and in particular the physical invariant ↔ fusion graph ↔ subfactor theory developed in [5]. In particular this interprets and generalises the Mmi mi 6= 0 coincidence (at least for the so-called “block-diagonal” physical invariants, i.e. Z which can be expressed P as sums of squares: Z = |χ + χ0 + · · · |2 ). Related to this is the following. It is known that orbifolding a 4-dimensional N = 4 supersymmetric gauge theory by any finite subgroup G ⊂ SU2 (C) leads to a CFT with N = 2 supersymmetry, whose matter matrix (giving numbers of fermions and scalars) can be read off from the Dynkin diagram corresponding to G. For finite subgroups of SU3 (C) and SU4 (C), we would get N = 1 and N = 0 supersymmetry, respectively. The (directed) graphs corresponding to the matter matrices for G ⊂ SU3 (C) are given in [12] and closely (1) resemble the fusion graphs of [11] for A2 physical invariants. Indeed, [12] make the tantalising (though rather too vaguely stated) conjecture that there exists a McKaylike correspondence between certain singularities of type Cn /G (or corresponding (1) orbifold theories) for G ⊂ SUn (C), and the physical invariants of An−1 . Now, the finite subgroups of SLn (C), at least for n ≤ 7, are known (Blichfeldt 1917, Brauer 1967, Lindsey 1971, Wales 1968, for n = 4, 5, 6, 7 resp.), so presumably the work of [12] can with effort be extended and their conjecture more precisely stated and tested. It should be noted though that [12] also makes use of only those “block(1) (1) diagonal” A2 physical invariants (in analogy with the A1 classification, it is as if they would ignore Dodd and E7 — these graphs are also missing from the list of principal graphs of subfactors, although non-block diagonal invariants are related to nonflat connections [4] or non-local nets of subfactors [5]). What if anything should correspond to the remaining physical invariants is unknown? Incidently, there is a nice little curiousity contained within many modular invariants: another A-D-E! This A-D-E applies to any physical invariant (i.e. for any (1) RCFT, not necessarily related to A1 ) which looks like Z = |χ1 + χ10 |2 + stuff. 0 The label 1 can be anything in P+ , and “stuff” can be any sesquilinear combination of χi ’s, provided it doesn’t contain χ1 (the vacuum) or χ10 . In other words,
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the indecomposable submatrix of M containing (1, 1) is required to be 1 1 , 1 1 but otherwise M is unconstrained. Then to M we can associate several extended Dynkin diagrams of A-D-E type, as follows. Put a node on the left of the page for each a ∈ P+ whose row Ma? is nonzero, and put a node on the right of the page for each b ∈ P+ whose column M?b is nonzero. Connect a (on the left) and b (on the right) with precisely Mab edges. The result will be a set of extended Dynkin diagrams of A-D-E type! (For these purposes we will identify two nodes connected with 2 lines as the extended A1 diagram.) (1) For example, let’s apply this to our A1 classification. Any partition function D2` is of this kind, and its corresponding graph will consist of ` − 1 diagrams of (extended) A3 type, and one of extended A1 type. The exceptional E6 consists of three A3 ’s, and the exceptional E7 consists of three A3 ’s and one D5 . Again, this (1) fact (proved in [8]) is not restricted to the A1 physical invariants. This little curiousity is not as deep or mysterious as the Cappelli–Itzykson–Zuber A-D-E pattern, and has to do with the Z+ -matrices with largest eigenvalue 2. There are 4 other claims for A-D-E classifications of families of RCFT physical invariants, and all of them inherit their (approximate) A-D-E pattern from the more (1) fundamental A1 one. The two rigourously established ones are the c < 1 minimal models, also proven in [1], and the N = 1 superconformal minimal models, proved in [13]. In both cases the physical invariants are parametrised by pairs of A-D-E diagrams. The list of known c = 1 RCFTs [14] also looks like A-D-E (two series parametrised by Q+ , and three exceptionals), but the completeness of that list has never been successfully proved (or at least such a proof has never been published). The fourth classification often quoted as A-D-E, is the N = 2 superconformal minimal models. The only rigourous classification of these is accomplished in [15], assuming the generally believed but still unproven coset realisation (SU(2)k × U(1)4 )/U(1)2k+4 . The connection here with A-D-E turns out to be rather weak: e.g. 20, 30, and 24 distinct invariants would have an equal right to be called E6 , E7 , and E8 respectively. It appears to this author that the frequent claims that the N = 2 minimal models fall into an A-D-E pattern are without serious foundation, or at least require major reinterpretation. References [1] A. Cappelli, C. Itzykson and J.-B. Zuber, Commun. Math. Phys. 113 (1987) 1. [2] A. Cappelli, C. Itzykson and J.-B. Zuber, Nucl. Phys. B280 [FS18] (1987) 445; D. Gepner and Z. Qui, Nucl. Phys. B285 (1987) 423; A. Kato, Mod. Phys. Lett. A2 (1987) 585; P. Roberts, Ph.D. Dissertation, Univ. of G¨ oteborg, 1992; P. Slodowy, Bayreuther Math. Schr. 33 (1990) 197. [3] M. Hazewinkel, W. Hesselink, D. Siersma and F. D. Veldkamp, Nieuw Arch. Wisk. 25 (1977) 257; P. Slodowy, in: Algebraic Geometry, Lecture Notes in Math 1008, Springer, Berlin, 1983. [4] A. Ocneanu, Paths on Coxeter diagrams: “From platonic solids and singularities to minimal models and subfactors”, Lectures given at the Fields Institute (1995), notes recorded by S. Goto. [5] J. B¨ ockenhauer and D. E. Evans, Commun. Math. Phys. 200 (1999) 57; Commun. Math. Phys. 205 (1999) 183.
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[6] M. Bauer and C. Itzykson, Commun. Math. Phys. 127 (1990) 617. [7] T. Gannon, “The level 2 and 3 modular invariants for the orthogonal algebras”, preprint, math.QA/9809020, to appear in Canad. J. Math. [8] T. Gannon, “Kac–Peterson, Perron–Frobenius, and the classification of conformal field theories”, preprint, q-alg/9510026; T. Gannon, “The A? D? E7 -type invariants of affine algebras”, in preparation. [9] W. Nahm, Duke Math. J. 54 (1987) 579; W. Nahm, Commun. Math. Phys. 118 (1988) 171. [10] R. Behrend, P. Pierce, V. Petkova and J.-B. Zuber, Phys. Lett. B444 (1998) 163. [11] J.-B. Zuber, in Proc. of the XIth Int. Conference of Mathematical Physics, International Press, Boston, 1995. [12] A. Hanany and Y.-H. He, J. High Energy Phys. (2) (1999), p. 13. [13] A. Cappelli, Phys. Lett. B185 (1987) 82. [14] P. Ginsparg, Nucl. Phys. B295 [FS21] (1988) 153; E. Kiritsis, Phys. Lett. B217 (1989) 427. [15] T. Gannon, Nucl. Phys. B491 (1997) 659.
GLOBAL FOURIER INTEGRAL OPERATORS AND SEMICLASSICAL ASYMPTOTICS A. LAPTEV∗ Department of Mathematics, Royal Institute of Technology Lindstedtsv¨ agen 25, S-10044 Stockhold, Sweden
I. M. SIGAL† Department of Mathematics, University of Toronto 100 St. George Street, Room 4072, M5S 3G3 Toronto, Canada Received 13 June 1997 Revised 2 April 1999 In this paper we introduce a class of semiclassical Fourier integral operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schr¨ odinger equations. Our construction is elementary, it is inspired by the joint work of the first author with Yu. Safarov and D. Vasiliev. We consider several simple but basic examples. Keywords: Fourier integral operator, global phase, Schr¨ odinger equation, fundamental solution, parametrix, semiclassical asymptotics, Morse index, magnetic field.
1. Introduction The notion of Fourier integral operator (FIO) arises in the theory of partial differential equations in two contexts. Firstly, the FIO’s realize quantization of classical canonical transformations (Egorov’s theorem) (while pseudo-differential operators can be said to arise from quantization of classical observables). More precisely, a transformation of a pseudo-differential operator, P , induced by a change of its symbol under a canonical transformation can be realized, in the leading order, by the conjugation, F P F −1 , by an appropriate FIO F . Secondly, the FIO’s give a natural construction of approximate fundamental solutions (parametrices) for hyperbolic problems. These two applications are related. Indeed, on the level of symbols (i.e. the level of classical observables) a hyperbolic equation is reduced in the leading order to Hamilton equations. The solution of the latter equations gives a Hamiltonian flow. The latter is a family of canonical transformations (labelled by the time variable). Lifting this family to the level of operators yields a desired approximate solution. This lifting is valid only for sufficiently small times. In general, the construction of a FIO from a flow breaks down at some moment of time due to presence of focal points or caustics. To overcome this problem one either ∗ Supported † Supported
by the Swedish NSRC under Grant M-AA/MA 9364-320 and Magnussons fond. by NSERC under Grant No. NA7901.
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constructs a Maslov canonical operator or a global FIO. Both procedures are rather labor intensive and subtle (see [12, 13, 19, 21]). In this paper we propose a fairly elementary construction of global FIO’s giving, in particular, approximate fundamental solutions (or propagators) to the Schr¨ odinger equation ∂ψ iα = Hα ψ , (1.1) ∂t where α is a semiclassical parameter and Hα is an α-differential operator in Rn (see, e.g. [5, 7, 10, 20, 21, 26, 28]), in terms of semiclassical FIO’s. The simplicity of our construction is achieved by using, in our FIO’s, complex phases with quadratic imaginary parts. Our approach is inspired by works [17] and [18], where the global parameterization of homogeneous Langrangian distributions was introduced for construction of fundamental solutions of wave equations. Oscillatory integrals with complex phases having quadratic imaginary parts were first considered in [1, 2, 3, 8, 9, 16, 20, 25, 27] in connection with the problem of propagation of Gaussian wave packets. In another development, FIO’s with complex phases were studied in [22, 31] in relation to the problem of propagation of singularities for pseudodifferential operators with complex symbols. It seems that none of the works above have isolated the very useful class of FIO’s with complex phases having quadratic imaginary parts nor did they use FIO’s with complex phases in order to overcome the problem of caustics in deriving quasiclassical or short-wave asymptotics. Finally we note that in order to avoid the problem of infinite speed of propagation of singularities or oscillations, in the semiclassical context, we localize the construction of an approximate Schr¨ odinger propagator to a part of the phase space, T ∗ Rn , in which the momentum (i.e. the cotangent variable) is bounded (cf. [15]). This paper is organized as follows. In Sec. 2 we introduce our main construction and formulate our main theorem. This theorem is proven in Sec. 3 modulo an important classical statement of independent interest, whose proof is given in Sec. 4. We use our construction in Sec. 5 in order to obtain semiclassical asymptotics of solutions of Schr¨ odinger Eq. (1.1) with highly oscillating initial conditions. This is a classical problem treated in many texts (see [6, 19, 20, 21] and references therein) which allows a comparison of our method with standard treatments. In Sec. 6 we apply our construction to time-dependent Hamiltonians of a “quadratic type”. Our approach allows us to obtain a precise formula for the fundamental solution of the corresponding Schr¨ odinger equation via an oscillatory integral. In the last section we clarify the nature of the phase shift appearing in analysis of the motion of a particle in a singular magnetic potential concentrated at the origin. Some other applications will be presented elsewhere. In what follows we will be dealing with (vector) functions on Rn and on R2n = ∂h ∂h ∗ n T R . For a function h(x, ξ) from R2n to R, hx denotes the n-vector ( ∂x , . . . , ∂x ) 1 n 2
h (gradient of h in x) and hxξ the (n × n)-matrix ( ∂x∂i ∂ξ , i = 1, . . . , n, j = 1, . . . , n). j 2n n For a vector function x(y, η) from R to R , xη stands for the (n × n) matrix ∂x ( ∂ηji , i = 1, . . . , n, j = 1, . . . , n), etc. k · k will denote the norm in L2 (Rn ) as
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well as the norm of operators on this space, x · y = (x, y) will denote the real dot product of vectors in Rn . 2. Main Result In this section we present the main result of this paper. We begin with some basic definitions. Let h(t, x, ξ) be a real, smooth function on R1 × T ∗ Rn called a Hamiltonian function. We only assume that there are m > 0 and Cµν > 0, such that for every (t, x) ∈ R1 × Rn |∂xµ ∂ξν h(t, x, ξ)| < Cµν (1 + |ξ|)m .
(2.1)
∂ As usual we denote Dx = −i ∂x . Let Hα (t) be the α-pseudodifferential operator with the symbol h(t, x, ξ), i.e.
Hα (t)f (t, x) = h(t, x, αDx )f (t, x) Z Z = (2πα)−n h(t, x, ξ)ei(x−y)ξ/α f (y)dydξ , Rn
(2.2)
Rn
defined first on functions f ∈ C0∞ (Rn ). We assume that Hα (t), for every t, has a self-adjoint extention which we continue denoting by the same symbol Hα (t). Let U (t, s) be the (Schr¨ odinger) propagator for Hα (t), i.e. the family of operators solving the equation αi
∂ U (t, s) = Hα (t)U (t, s) ∂t
and U (s, s) = I ,
(2.3)
where I stands for the identity operator. Our task is to construct α-FIO’s, UN (t), which approximate U (t, 0) within O(αN +1−n ) for any N ≥ 1. In what follows (xt , ξ t ) denote the solutions of the Hamiltonian equations dxt ∂h dξ t ∂h = , =− , (2.4) dt ∂ξ dt ∂x subject to the initial conditions xt t=0 = y ,
ξ t t=0 = η .
Let gt denote the flow generated by h, i.e. gt (y, η) = (xt , ξ t ). Consider the action function S defined as Z t S(t, y, η) = (hξ (s, xs , ξ s ) · ξ s − h(s, xs , ξ s ))ds .
(2.5)
(2.6)
0
Definition 2.1. Given T > 0 and Ω ⊂ T ∗ Rn , denote by φ the class of functions (phase functions) ϕ = ϕ(t, x, y, η) ∈ C ∞ ([0, T ) × Rn × Ω) satisfying the following conditions: (1) ϕ(t, xt (y, η), y, η) = S(t, y, η), (2) ϕx (t, xt (y, η), y, η) = ξ t (t, y, η),
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(3) i−1 ϕxx (t, x, y, η) ≥ 0, and is independent of x, (4) det(ϕxη (t, xt (y, η), y, η)) 6= 0 for (t, y, η) ∈ [0, T ) × Ω. Condition (3) implies that phase functions ϕ ∈ φ are polynomials of the second degree with respect to x. Expanding ψ satisfying (1)–(3) in x around the point xt = xt (y, η), we find the following expression: ϕ(t, x, y, η) = S(t, y, η) + (x − xt ) · ξ t + i(x − xt ) · B(x − xt )/2 ,
(2.7)
where B = B(y, η, t) is a non-negative definite n × n matrix (= i−1 ϕxx (t, x, y, η)). Vice versa, functions of ψ of form (2.7) with B ≥ 0 and independent of x, satisfy conditions (1)–(3). We shall show in Lemma 4.1 below that if B > 0, then the condition (4) is fulfilled as well and therefore the class φ is not empty. Remark 2.1. Instead of the condition (3) we could have considered a more general condition <(i−1 ϕxx (t, x, y, η)|x=xt ) ≥ 0. This would lead to a wider class of phase function which is useful for analysis on manifolds. However, the class which is introduced in Definition 2.1 is sufficient for our purposes. In particular, for our purposes we can choose B > 0 and independent of y, η and t. The following matrix, appearing in (4), plays an important role in our analysis Z(t, y, η) = ϕxη (t, x, y, η) x=xt(y,η) . (2.8) Note that condition (4) implies that Z is always nonsingular. Differentiating condition (2) with respect to η, we obtain the following representation of Z: Z(t, y, η) = ξηt (y, η) − ixtη (y, η)B(t, y, η) , where we use our convention that (xη )ij = and, recalling condition (3), denote
∂xj ∂ηi
(2.9)
(see the end of the introduction)
B(t, y, η) = i−1 ϕxx (t, x, y, η) .
(2.10)
At t = 0 we obviously have that Z|t=0 = I. To be able to consider the square root of det Z we introduce the following agreement. Agreement 2.1. We choose the branch for the function argdet Z which is continuous in t for 0 ≤ t < T , and which yields zero for t = 0. This choice allows us to define the value of (det Z)1/2 uniquely for all the values of the parameter t, 0 ≤ t < T. We consider a bounded subset Ω of the phase-space T ∗ Rn and a number T > 0 satisfying the following condition: (H) Hamiltonian Eqs. (2.4) and (2.5) have solutions for every (y, η) ∈ Ω and for S 0 ≤ t < T , and the Hamiltonian function h is C ∞ on the set 0≤t
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Theorem 2.1. Assume condition (H) to be satisfied. Fix functions χ and θ as described above. For any function ϕ ∈ φ and integer N ≥ 0 there is an operator UN (t) such that (i) the Schwartz kernel of UN (t) is of the form Z −n UN (t, x, y) = (2πα) eiϕ/α uN (t, y, η, α)dη , (2.12) PN where uN = k=0 αk v(k) (det Z)1/2 with v(k) smooth functions supported in [0, T ) × Ω and, in particular, Rt −i sub h(s,xs ,ξs )ds 0 v0 (t, y, η) = e θ(y, η) (2.13) 1 Tr hxξ , the subprincipal symbol of h, and (ii) the operators UN (t) with sub h = − 2i approximate the propagator U (t, 0) on Ω in the sense that for any T0 < T
sup k(U (t, 0) − UN (t))χk ≤ CαN −2n ,
(2.14)
0≤t≤T0
where C = C(h, Ω, T0 ). The proof of this theorem is given in the next section. We leave out an important question of how the constant in (2.14) depends on h, Ω and T , which requires a rather involved analysis. Notice also that (2.12) gives a family of approximate fundamental solutions depending on a choice of the phase function ϕ (or the matrix B in (2.7)). 3. Proof of Theorem 2.1 We begin with deriving some simple equalities needed in the proof. Lemma 3.1. For all (t, y, η) ∈ [0, T ) × Ω the action function S defined in (2.6), satisfies the following identities: Sη = xtη ξ t
(3.1)
Sy = xty ξ t − η .
(3.2)
and
Proof. In the proof below we suppress the superindex t. Both sides of (3.1) are equal to zero when t = 0 since S|t=0 = 0 and xη · ξ|t=0 = yη · η = 0. Using (2.6), we find St = hξ (t, x, ξ) · ξ − h(t, x, ξ) . (3.3) Thus in view of (2.4) ∂ ∂ Sη = (hξ ξ − h) ∂t ∂η = xξ ˙ η + x˙ η ξ − xη hx − hξ ξη d = x˙ η ξ + xη ξ˙ = (xη ξ) . dt
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This leads to (3.1). Analogously both sides of (3.2) are equal to zero when t = 0. Using now (3.3) and (2.4) we have ∂ ∂ Sy = (hξ · ξ − h) = xξ ˙ y + x˙ y ξ − xy hx − hξ ξy ∂t ∂y d = x˙ y ξ + xy ξ˙ = (xy ξ) , dt
which completes the proof.
Now we proceed directly to the proof of Theorem 2.1. The proof is straightforward and standard except for one little twist. Since the integration in η is over bounded sets, all the estimates occuring below are elementary. We shall separate the proof into three parts. Step 1. Substitution. We look for a α-FIO UN (t) given by the integral kernel (2.12) which solves the equation ∂ αi − Hα (t) UN (t) = RN (t) , (3.4) ∂t where RN is a α-FIO with the phase ϕ and a symbol αN +2 rN , where rN = O(1) and rN (t, ·, ·) ∈ C0∞ (Ω). First we compute the integral kernel of the operator ∂ (αi ∂t − Hα (t)) UN (t). Taking into account (2.12), the fact that the phase function ϕ is a quadratic function with respect to x and applying the standard formula for the action of a pseudodifferential operator on an exponential function (see for example [13]), we obtain Z ∂ ∂ αi − Hα (t) UN (t, x, y) = (2πα)−n eiϕ/α αi uN − (g + O(α2 ))uN dη , ∂t ∂t (3.5) where g = ϕt + h(t, x, ϕx ) +
α (B∂ξ , ∂ξ )h (t, x, ϕx ) . 2
Since ϕ is a non-degenerated phase function and due to condition (2.1), the r.h.s. of (3.5) can be understood as an oscillatory integral. Using (2.4), (2.6) and (2.7), we find that ϕt = hξ (t, xt , ξ t ) · ξ t − h(t, xt , ξ t ) − x˙ t · ξ t + (x − xt ) · ξ˙t i ˙ − xt ) − ix˙ t B(x − xt ) + (x − xt ) · B(x 2 = −h(t, xt , ξ t ) − (x − xt ) · hx (t, xt , ξ t ) − ihξ (t, xt , ξ t )B(x − xt ) i ˙ − xt ) , + (x − xt ) · B(x 2
(3.6)
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where B˙ = ∂B ∂t . Applying the Taylor formula, we obtain that the third and the fourth terms on the right-hand side of (3.5) can be written as follows: h(t, x, ϕx ) = h(t, x, ξ t + iB(x − xt )) = h(t, xt , ξ t ) + hx (t, xt , ξ t ) · (x − xt ) + ihξ (t, xt , ξ t ) · B(x − xt ) +
1 2
X
(∂xν1 ∂ξν2 h)(x − xt )ν1 (iB(x − xt ))ν2 + O(|x − xt |3 )
|ν1 +ν2 |=2
(3.7) and α α (B∂ξ , ∂ξ )h (t, x, ϕx ) = (B∂ξ , ∂ξ )h (t, xt , ξ t ) + αO(|x − xt |) . 2 2
(3.8)
Summing up the expressions (3.6)–(3.8), we find that the terms with α0 and with (x − xt ) cancel. The rest can be rewritten as α (B∂ξ , ∂ξ )h (t, xt , ξ t ) + O(|x − xt |3 ) + αO(|x − xt |) , (3.9) g = g1 + 2 where g1 =
1 2
X
ν2 (∂xν1 ∂ξν2 h)(x − xt )ν1 (iB(x − xt )
|ν1 +ν2 |=2
i ˙ − xt ) . + (x − xt ) · B(x 2
(3.10)
Step 2. Integration by parts. In order to obtain the transport equations we integrate by parts in the integral involving g1 . Notice that ∂ iϕ/α i i e = eiϕ/α ϕη = eiϕ/α Sη − xtη ξ t + ξηt (x − xt ) − i(B(x − xt ), xtη ) . ∂η α α By Lemma 3.1 we find Sη −xtη ξ t = 0. Therefore taking into account (2.9), we obtain (x − xt )eiϕ/α = −iαZ −1
∂ iϕ/α . e ∂η
(3.11)
Using this equation and integrating twice by parts in order to convert all powers of (x − xt ) into powers of α, we obtain Z Z α eiϕ/α g1 uN dη = eiϕ/α Tr[Z −1 (hxx + iBhxξ 2i
˙ tη ]uN + + ihxξ B − Bhξξ B + iB)x
X |β|≤2
O(α)∂ β uN
dη . (3.12)
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Note that the leading order here is produced when the η-derivative is switched from eiϕ/α onto the remaining (first) power of x − xt . Differentiating Hamilton Eq. (2.4) with respect to η, we obtain x˙ tη = xtη hxξ + ξηt hξξ ,
ξ˙ηt = −xtη hxx − ξηt hξx .
(Remember that according to our convention established at the end of the introduction, we have, e.g., (xtη )ij =
∂xtj ∂ηi
and (hxξ )ij =
∂2 η ∂xi ∂ξj .)
This implies
˙ xtη (hxx + iBhxξ + ihxξ B − Bhξξ B + iB) = (xtη hxx + ξηt hxξ ) + iB(xtη hxξ + ξηt hξξ ) + ixtη B˙ − (ξηt − xtη iB)hxξ − (ξηt − iBxtη )iBhξξ = − Z˙ − Zhxξ − ZiBhξξ .
(3.13)
Finally, collecting together (3.5), (3.9), (3.12) and (3.13) and using (3.11) and integration by parts in order to convert the powers of x − xt in the remainder in (3.9) into powers of α, we derive ∂ αi − Hα (t) UN (t, x, y) ∂t Z 1 −n −n+1 iϕ/α ∂ −1 ˙ = i(2π) α e − Tr(Z Z + hxξ ) + O(α) uN dη , (3.14) ∂t 2 where O(α) is a differential operator in η with coefficients of order α. Consequently, Eq. (3.4) can be rewritten as ∂ 1 iα − Tr(Z −1 Z˙ + hxξ ) + O(α) uN = αN +2 rN . (3.15) ∂t 2 PN Now we write uN = k=0 αk u(k) and equate in Eq. (3.15) the terms containing the same power of the parameter α, starting with α and ending with αN +2 . As a result we obtain the recurrent system of transport equations. The terms with the multiplier αk+1 yield the equation ∂ 1 u(k) − Tr(Z −1 Z˙ + hxξ ))u(k) = O(1)u(k−1) , ∂t 2
(3.16)
where k = 0, 1, . . . , N , u(−1) = 0 and O(1) is a differential operator in η with coefficients of order 1 (O(1) = − α1 O(α), where O(α) is the same as in (3.15)). Setting rN = −O(1)u(N ) , we arrive at Eq. (3.4) for UN (t). Next, we require that UN (0) = θ(x, αDx ) + O(αN +1 ) ,
(3.17)
where θ is a smooth function supported in Ω and s.t. θ ≡ 1 on supp χ (see the paragraph containing Eq. (2.11)) and O(αN +1 ) stands for a bounded operator whose
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norm is bounded by CαN +1 . This leads to initial conditions for Eq. (3.16), to derivation of which we proceed. Using Eq. (2.7) for the phase ϕ, we obtain Z −n UN (0, x, y) = (2πα) ei(x−y)·η/α e−(B(0,y,η)(x−y),x−y)/2αuN (0, y, η)dη . (3.18) Expanding the function exp[−(B(0, y, η)(x − y), x − y)] in the Taylor series and integrating by parts, we obtain Z ∞ X α`
UN (0, x, y) = (2πα)−n
`=0
`!
ei(x−y)·η/α A` uN dη ,
(3.19)
where the operator A is defined by (Ag)(η) =
n X
∂η2k η` (bk` (0, y, η)g(η))
k,`=1
with bk` being the matrix elements of B. Comparing this with (3.17), using the standard definition of a pseudodifferential operator in terms of its symbol, and PN writting uN = k=0 αk u(k) , we obtain u(0) (0, y, η) = θ(y, η)
(3.20)
and, for k = 1, 2, . . . , N , u(k) (0, y, η) = −
k X 1 ` A u(k−`) (0, y, η) . `!
(3.21)
`=1
Linear differential Eq. (3.16) with initial conditions (3.20) and (3.21) can be solved explicitly. In particular, Rt s s 1/2 −i 0 sub h(s,x ,ξ )ds u(0) (t, y, η) = (det Z(t, y, η)) e θ(y, η) , (3.22) 1 where, recall sub h = − 2i Trhxξ , and similarly for u(k) , k ≥ 1. The explicit form of the solutions shows that u(k) (t, ·, ·) ∈ C0∞ (Ω) for k ≥ 0 and 0 ≤ t ≤ T .
Remark 3.1. Since every |x − xt |2 is traded for at least the first power of α, when integrating by parts, Eq. (3.16) involves no more than 2(N − k + 1) derivatives of the coefficients u(k) . Step 3. Error estimate (2.14). Since uN (t, ·, ·) ∈ C0∞ (Ω) and Ω is compact, we have that UN (t, x, ·) ∈ C0∞ (Rn ). Writing x as x − xt + xt and integrating by parts with a help of (3.11), it is easy to convince oneself that UN (t, ·, y) is in the Schwartz class S(Rn ). The last two properties can be combined into the formula UN (t, ·, ·) ∈ S(Rn , C0∞ (Ω)) .
(3.23)
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The definition of rN (see the paragraph after Eq. (3.16)) and an argument similar to one presented above show that sup kRN (t)k ≤ C(h, Ω, T )αN +1−n .
(3.24)
0≤t≤T
Now integrating Eq. (3.4), we obtain UN (t) − U (t, 0)UN (0) =
1 iα
Z
t
U (t, s)RN (s)ds .
(3.25)
0
Multiplying this by the operator χ from the right and using Eq. (3.24) and the fact that the norm of U (t, s) is uniformly bounded (in fact, is one), we obtain k(UN (t) − U (t, 0)UN (0))χk ≤ CαN −2n
(3.26)
with C = C(h, Ω, T ), Finally, due to (3.17) and the fact that θ ≡ 1 on supp χ, kUN (0)χ − χk ≤ CαN −2n
(3.27)
and therefore (2.14) follows. 4. Z-Matrix The main result of this section is the following:
Lemma 4.1. If B > 0, then the r.h.s. of (2.9) defines a non-singular matrixfunction. Consequently, the matrix-function Z introduced in (2.8) is non-degenerate for all values (t, y, η) ∈ [0, T ) × Ω. Proof. In this proof we write xt as x(t) and similarly for ξ t and often do not display this argument. First we show that ξη xTη = xη ξηT ,
(4.1)
where ξηT is the transpose of the matrix ξη . Indeed, xη |t=0 = 0 and (4.1) is obviously fulfilled for t = 0. Differentiating equations (2.4) with respect to η, we find d xη = xη hxξ + ξη hξξ , dt and
d ξη = −xη hxx − ξη hξx . dt
d d Using these equations it is easy to verify that dt (ξη xTη ) = ( dt xη ξηT ) and consequently (4.1) is true for all t. This proves (4.1) for all t, 0 ≤ t < T . Since B is a non-degenerate matrix, it suffices to show that Ker ZB −1 Z ∗ = ∅. Using (4.1), we obtain
ZB −1 Z ∗ = (ξη − ixη B)B −1 (ξηT + iBxTη ) = ξη B −1 ξηT + xη BxTη .
(4.2)
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Let us assume for a moment that ZB −1 Z ∗ c = 0 for some vector c ∈ Cn and s, 0 < s < T . Then, since B > 0, (4.2) implies that ξηT (s)c = xTη (s)c = 0. Therefore the system of ordinary differential equations x˙ Tη c = hξx xTη c + hξξ ξηT c , ξ˙ηT c = −hxx xTη c − hxξ ξηT c for xTη (t)c and ξηT (t)c, obtained by differentiating (2.4) w.r. to η, implies that xTη (t)c = 0 and ξηT (t)c = 0 for all t ∈ [0, T ]. This contradicts the relation ξη |t=0 = I. 5. Quasi-classical Asymptotics In this section we study asymptotic behaviour of solutions of the Schr¨ odinger equation ∂Ψ = Hα (t)Ψ (5.1) iα ∂t with initial conditions of the form Ψ|t=0 = Ψ0α = α−n/2 Ψ0 ((x−y0 )/α) for some y0 ∈ Rn and Ψ0 ∈ C0∞ (Rn ). Let ρ(x) be a smooth cut-off function, such that ρ ≡ 1 on R −n ˆ supp Ψ0α . We write Ψ0α as Ψ0α (x) = Ψ0α (x)ρ(x) = (2π) ρ(x) Ψ0 (ξ)ei(x−y0 )·ξ/α ˆ 0 (ξ) is the Fourier transform of Ψ0 (x). Thus it suffices to consider the dξ, where Ψ initial condition (5.2) Ψ t=0 = Ψ(0) := ρ(x)eix·η0 /α for a fixed vector η0 ∈ Rn (= Ty∗0 Rn ). For α → 0, such an initial condition is a fast oscillating function. 0 The point of taking the initial condition of the form α−n/2 Ψ0 ( x−y α ) is that in this case the average momentum Z ¯ 0α (−iα∇)Ψ0α dn x Ψ is bounded. Another type of the initial condition with bounded average momentum is a direct generalization of (5.2) as ρ(x)eiν(x)/α ,
(5.3)
where ν is a smooth function. Although Theorem 5.1 below deals with the initial condition (5.2) a generalization to initial condition (5.3) amounts to the changing just a new notation. Now observe that the initial value problem (5.1) and (5.2) has the unique solution Ψ = Uα (t, 0)Ψ(0) , where, recall, Uα (t, 0) is the propagator from 0 to t associated with (5.1). Due to Theorem 2.1 this propagator is approximated by a global Fourier integral operator
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UN (t) given in Theorem 2.1. Thus we are interested in asymptotic behaviour as α → 0 of the function Z (UN (t)Ψ(0) )(x) = UN (t, x, y)ρ(y)eiy·η0 /α dy . The latter is given in Theorem 5.1 below. Theorem 5.1. Let the conditions of Theorem 2.1 be satisfied and let UN be the approximate evolution operator (2.12). Let a point (t, y0 , η0 ) ∈ [0, T ) × Ω be s.t. det(xty (y0 , η0 )) 6= 0. Let Vxt0 and Vy0 be neighbourhoods of the points xt0 := xt (y0 , η0 ) for y in Vy0 for any x ∈ and y0 s.t. the equation x = xt (y, η0 ) has a unique solution R Vxt0 . Then for any smooth function ρ(y), supp ρ ⊂ Vy0 , UN (t, x, y)ρ(y)eiy·η0 /α dy = 0(α∞ ) unless x ∈ {xt (y, η0 ) | y ∈ Vy0 }. In the latter case we have Z UN (t, x, y)ρ(y)eiy·η0 /α dy = ρ(¯ y )ei¯y·η0 /α eiS(t,¯y,η0 )/α ei sub(t) ei 2 m(t,¯y,η0 ) |det xty (t, y¯, η0 )|−1/2 + O(α) , π
(5.4) where y¯ = y¯(t, x, η0 ) is the unique solution of the equation xt (y, η0 ) = x, sub(t) = Rt sub h(s, xs (¯ y , η0 ), ξ s (¯ y , y0 ))ds and m(t, y, η) ≡ m(γ t ) is the Morse index of the 0 t s trajectory γ = {x (y, η) | 0 ≤ s ≤ t}. Proof. We use representation (2.12) for UN (t, x, y) with B ≡ 1i ϕxx > 0. Substituting it into the l.h.s. of (5.4), we conclude that the latter is equal to ZZ eiψ/α uN (t, y, η, α)ρ(y)dηdy , (5.5) (2πα)−n where ψ(t, x, y, η) = ϕ(t, x, y, η) + y · η0 . We want to apply the stationary phase expansion to this integral. To this end we have to find stationary points of the phase ψ in η and y and the Hessian, Hess ψ, of ψ at those points. We begin with the former. Using (2.7), (3.1), (2.9) and the fact that B is symmetric, we obtain i ψη = ϕη = Sη − xtη ξ t + ξηt (x − xt ) − ixtη B(x − xt ) + (x − xt ) · Bη (x − xt ) 2 = Z(x − xt ) + O(|x − xt |2 ) .
(5.6)
Since det Z 6= 0, then in a sufficiently small neigbourhood Vxt0 of xt0 we have ψη = 0
⇒
x = xt (y, η) .
(5.7)
Furthermore, (2.7) and (3.2) imply i ϕy = Sy − xty ξ t + ξyt (x − xt ) − ixty B(x − xt ) + (x − xt ) · By (x − xt ) 2 = −η + Y (x − xt ) + O(|x − xt |2 ) ,
(5.8)
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where Y (t, y, η) = ξyt (y, η) − ixty (y, η)B . Hence ⇒
ψy = ϕy + η0 = 0 and (5.7)
η = η0 .
Thus on supp ρ × Rn , the phase ψ has a unique stationary point z¯ = (¯ y , η0 ) ,
where y¯ = y¯(t, x, η0 ) is the solution to xt (y, η0 ) = x ,
(5.9)
if x ∈ {xt (y, η0 )|y ∈ Vy0 } and no stationary points otherwise.
Now we compute the Hessian of the phase function ψ at the stationary point (5.9). First we notice that Hess ψ = Hess ϕ . Using expressions (5.7) and (5.9), we compute the matrix Hess ϕ at the point (t, x, y, η) s.t. xt (y, η) = x: ! Y xty I + Y xtη Hess ϕ = − . (5.10) Zxty Zxtη Note that the relation ϕηy = (ϕyη )T implies the constraint I + Y xtη = (Zxty )T . Matrix (5.10) can be factorized as follows: ! Y xty I Hess ϕ = − Zxty 0
(5.11)
I
(xty )−1 xtη
0
I
! ,
(5.12)
provided xt (y, η) = x, which implies det(Hess ϕ) = det Z · det xty ,
provided xt (y, η) = x .
(5.13)
The next lemma provides the last component needed to assemble the proof. Lemma 5.1. Let the matrix xty (y0 , η0 ) be nonsingular. Then, thinking of the sign as +1 or −1, we have sign det xty (y0 , η0 ) = (−1)m(γt )
(5.14)
where γ t is the trajectory {xs (y0 , η0 ) | 0 ≤ s ≤ t} and m(γt ), recall, its Morse index. Proof. By the definition of the Morse index (see [23, 24]) X m(γ t ) = multiplicity (conjugate point) where the sum is taken over all conjugate points along γ t . Recall that a conjugate point is a point xs0 (y0 , η0 ), 0 < s0 < t, at which det xsy0 (y0 , η0 ) = 0, and the multiplicity of a conjugate point xs0 (y0 , η0 ) is the multiplicity of the zero s = s0
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of det xsy (y0 , η0 ). Since det xsy (y0 , η0 )|s=0 = 1 and since passing through a conjugate point the sign of the determinant changes by (−1)multiplicity (conjugate point) , we conclude that at the end of the path we have (5.14). Now we are ready to apply the stationary phase formula (see e.g. [12, Sec. 7.7]) to oscillatory integral (5.5). We use Theorem 2.1 to expand the amplitude uN and Eqs. (5.9) and (5.10) for the critical points and Hessian of the phase. We make two 1 observations. Firstly we note that (det Z) 2 coming out of uN (see Theorem 2.1) 1 − 12 coming out of (det Hess ϕ)− 2 (see Eq. (5.13)) (remember that Z cancels (det Z) is a non-singular matrix). Secondly we see that the second factor on the r.h.s. of (5.13) and Lemma 5.1 leads to (det xty )− 2 = |det xty |− 2 e−i 2 m(γ ) , 1
1
π
t
(5.15)
where, recall, m(γ t ) is the Morse index of the trajectory γ t = {xs |0 ≤ s ≤ t}. This completes the proof of the theorem. 6. Quadratic Hamiltonians Let Q(t) be a smooth, Hermitian 2n × 2n matrix function and R(t), a smooth real 2n vector function. Let z = (x, ξ) ∈ R2n and introduce a quadratic Hamiltonian function by the formula h(t, z) =
1 z · Q(t)z + R(t) · z . 2
(6.1)
In this case Hamiltonian system (2.4) with the initial conditions (2.5) can be rewritten as the following system of linear equations (6.2) z˙ = J(Q(t) · z + R(t)) , z t=0 = z0 = (y, η) , 0 I where J = −I 0 . Let M (t) be the fundamental matrix of system (6.2), i.e. M (t) I 0 ˙ solves the initial value problem: M(t) = JQ(t) = M (t) and M (0) = 0 I . Due to the Liouville–Jacobi formula, Z t det M (t) = exp TrJQ(s)ds , 0
M (t) is a non-degenerate matrix. The solutions z t of (6.2) can be represented in terms of M (t) as Z t t 0 −1 M (s)JR(s)ds . (6.3) z = M (t) z + 0
Let mij (t), i, j = 1, 2, be the matrix elements of M (t). According to (2.7) and (2.8) we find Z(t, y, η) = m22 (t) − iB(t, y, η)m12 (t) . If we use a matrix B which is independent of (y, η), then the matrix Z is also independent of (y, η) and the first term of the approximation (2.12) to the fundamental solution of Schr¨ odinger Eq. (1.1) gives, in fact, the exact formula for this solution.
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Theorem 6.1. Let h be the Hamilton function defined in (6.1) and let B = B(t) from (2.7) be an arbitrary positive definite matrix independent of (y, η) for t ≥ 0. Then the Schwartz kernel of the Schr¨ odinger Eq. (1.1) is given by the following oscillatory integral Z −n 1/2 U (t, x, y) = (2πα) (det Z(t)) eiϕ/α dη , (6.4) where the phase function ϕ is defined in (2.7) and (2.6). Proof. The proof can be seen from (3.9). Since the Hamiltonian function is quadratic with respect to (x, ξ), we have that g = g1 . Moreover, if both B and Z are independent of (y, η), then the terms appearing in the sum in (3.12) are equal to 0. Notice that (6.4) gives us a family of solutions of the Schr¨ odinger equation depending on the choice of an axiliary matrix B. If for example h(x, ξ) =
1 (|ξ|2 + |ωx|2 ) 2
(6.5)
and ω is a matrix independent of t, then the matrix M (t) in (6.3) can be easiely found to be ! cos ωt ω −1 sin ωt . M (t) = −ω sin ωt cos ωt If now the matrix B is chosen so that B = ω, then we obtain that Z has a particularly simple form Z(t, y, η) = cos ωt − i sin ωt = e−itω and therefore (det Z)1/2 = e−it Trω/2 . Remark 6.1. Notice that xt (y, η) and ξ t (y, η), defined in (6.3), are linear functions with respect to the initial conditions (y, η). Therefore, the phase function in ϕ in (6.4) is quadratic and the integral (6.4) can be computed. In particular, if h is defined by (6.5), then we obtain the well-known Mehler formula for the propagator of the harmonic oscillator. 7. Schr¨ odinger Operator with Magnetic Potentials Now we consider the motion of a particle in a singular magnetic potential concentrated at the origin and having the total flux ψ0 . Then the corresponding vector potential is A(x) = (A1 (x), A2 (x)) = ψ0 (−x2 , x1 )/|x|2 . The Schr¨odinger operator for a particle moving in such a magnetic field is H0α =
1 (αDx − A(x))2 , 2
(7.1)
and the corresponding Hamiltonian function is h0 (x, ξ) = 12 (ξ − A(x))2 . It is a smooth function on T ∗ R2 outside the fiber, {(x, ξ) : (x, ξ) = (0, ξ), ξ ∈ R2 }, over 0. The Hamiltonian equations are
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(
x˙ tk = ξkt − Ak , ξ˙kt = (ξ t − A) · (A)xk ,
xtk |t=0 = yk , ξkt t=0 = ηk ,
k = 1, 2 .
(7.2)
This system obviously reduces to the system of ordinary differential equations of the second order x ¨tk =
2 X (ξjt − Aj )[(Aj )xk − (Ak )xj ] j=1
t = (−1)k+1 2π(ξ3−k − A3−k (xt ))ψ0 δ(xt ) ,
xt |t=0 = y ,
k = 1, 2 ,
x˙ t |t=0 = η − A(y) .
(7.3)
If we assume that xt stays away from zero, then Eq. (7.3) can be easily solved xt (y, η) = (η − A(y))t + y ,
ξ t (y, η) = η + A(xt ) − A(y) .
(7.4)
From (7.4) we see that xt (y, η) = 0 for some t if and only if η ∧ y := η1 y2 − η2 y1 = −ψ0 . Therefore we assume that in our construction Ω ⊂ {(y, η) : η ∧ y 6= −ψ0 } . Using (2.6) and (7.4), we obtain the expression for the action function S(t, y, η) = =
1 |η − A(y)|2 t + 2
Z
t
(η − A(y)) · A(xs )ds 0
1 |η − A(y)|2 t − (η ∧ y + ψ0 )ψ0 2
Z
t
|xs |−2 ds .
(7.5)
0
The integral on the right-hand side multiplying ψ0 is a correction term to the free motion due to the magnetic flux through the origin. Simple calculations show that ! ! 1 + 2xt1 xt2 |xt |−4 tψ0 ((xt2 )2 − (xt1 )2 )|xt |−4 tψ0 (ξ1t )η1 (ξ2t )η1 t ξη = = . (ξ1t )η2 (ξ2t )η2 ((xt1 )2 − (xt2 )2 )|xt |−4 tψ0 1 − 2xt1 xt2 |xt |−4 tψ0 Therefore the determinant, det ξη = 1 − t2 ψ02 |xt |−4 , p degenerates on the “sphere” |xt | = t|ψ0 |. At these points of degeneracy a nontrivial matrix B is necessary for our constructions of the corresponding fundamental solution. Now we consider a particle in R2 moving in the singular magnetic field with the vector potential A(x), given above, and in a quadratic “electric” potential: Hα =
1 1 (αDx − A(x))2 + |x|2 . 2 2
(7.6)
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The corresponding Hamiltonian function is h(x, ξ) =
1 1 (ξ − A(x))2 + |x|2 . 2 2
(7.7)
The solution of the Hamiltonian Eqs. (2.4) and (2.5) for y and η, s.t. xt avoids the origin, is xt = cos(t)y + sin(t)(η − A(y)) (7.8) and ξ t = − sin(t)y + cos(t)(η − A(y)) + A(xt ) .
(7.9)
The action function for this model is equal to Z t 1 S(t, y, η) = (hξ · ξ˙s − h)ds = sin(2t)(|η − A(y)|2 − |y|2 ) 4 0 Z t 1 + (cos(2t) − 1)y · η − ψ0 (η ∧ y + ψ0 ) |xs |2 ds . 2 0 By analogy with the previous example we see that the integral on the right-hand side of the last expression can be interpreted as a correction term to the motion generated by the harmonic oscillator. Acknowledgments The first author is grateful to L. Kapitanski and Yu. Safarov, while the second author, to V. Ivrii and A. Khovanskii, for many useful discussions. References [1] V. S. Buslaev, “Semi-classical approach for equations with periodic coefficients”, Soviet Math. Uspehy 62 (1987) 77–98. [2] V. M. Babich and V. V. Ulin, “Complex space-time ray method and ‘quasiphotons’ ” (in Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI ) 117 (1981) 5–11. [3] A. C´ ordoba and C. Fefferman, “Wave packets and fourier integral operators”, Comm. in Partial Diff. Eqs. 3(11) (1978) 979–1005. [4] M Combescure and D. Robert, “Propagation d’´equation de Schr¨ odinger et approximation semi-classique”, C. R. Acad. Sci. Paris t. 323, S´erie I (1996) 871–876. [5] J. Duistermaat, “Fourier integral operators”, in N.Y., Courant Inst. Lect. Notes, 1971. [6] Yu. V. Egorov and M. A. Shubin (eds.), Partial Diff. Equations I and IV, Encyclopaedia of Mathematical Sciences, Springer-Verlag 1991 and 1993. [7] V. Guillemin and S. Sternberg, “Geometric asymptotics”, Math. Surveys (1977). [8] G. A. Hagedorn, “Semiclassical quantum mechanics I: The ~ → 0 limit for coherent states”, Math. Phys. 71 (1980) 77–93. [9] K. Hepp, “The classical limit for quantum mechanical correlation functions”, Commun. Math. Phys. 35 (1974) 265–277. [10] B. Helffer, “Semiclassical analysis for the Schr¨ odinger operators and applications”, Lecture Notes in Mathematics 1336, Springer-Verlag. [11] B. Helffer and D. Robert, “Propri´et´es asymptotiques du spectre d’op´erateurs pseudodiff´erentieles sur Rn ”, Comm. Partial Diff. Eqs. 7 (1982) 795–882. [12] L. H¨ ormander, “Fourier integral operators I”, Acta Math. 127 (1971) 79–183.
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[13] L. H¨ ormander, The Analysis of Linear Partial Differential Operators. I–IV, SpringerVerlag, Berlin, Heidelberg, New York, Tokyo, 1984. [14] V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer, 1998. [15] V. Ivrii and I. M. Sigal, “Asymptotics of the ground state energy of large Coulomb systems”, Ann. Math. 138 (1993) 293–335. [16] V. V. Kucherenko, “Asymptotic solutions of equations with complex characteristics”, Math USSR Sb24 (1976). [17] A. Laptev and Yu. Safarov, “Global parametrization of Lagrangian manifolds and the Maslov index”, preprint, Link¨ oping Univ., 1989. [18] A. Laptev, Yu. Safarov and D. Vassiliev, “On global representation of Lagrangian manifolds and solutions of hyperbolic equations”, Comm. Pure Appl. Math. XLVII (1994) 1411–1456. [19] V. P. Maslov, Perturbation Theory and Asymptotic Methods, MGU, 1965 (in Russian); French translation: V. P. Maslov, Th´ eorie de Perturbations et M´ethodes Asymptotiques, Dunod, Gauthier-Villars, Paris, 1972. [20] V. P. Maslov, The Complex WKB Method in Nonlinear Equations. I. Linear Theory, Nauka, Moscow, 1977; English translation: Birkh¨ auser, 1994. [21] V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximation for Equations of Quantum Mechanics, Nauka, Moscow, 1976. [22] A. Melin and J. Sj¨ ostrand, “Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem”, Comm. PDE 1 (1975) 313–400. [23] J. Milnor, Morse Theory Ann. Math. Stud. No. 51, Princeton Univ. Press, 1963. [24] M. Morse, The Calculus of Variations in the Large AMS Col. Publ. 18, 1934. [25] T. Paul and A. Uribe, “The semiclassical trace formula and propagation of wave packets”, J. Funct. Anal. 132(1) (1995) 192–249. [26] V. Petkov and G. Popov, “Semi-classical formula and clastering of eigenvalues for Schr¨ odinger operators”, Ann. Inst. H. Poincar´e, Phys. Theor. 68 (1998) 17–83. [27] J. Ralston, “Gaussian beams and the propagation of Singularities”, MAA Studies in Mathematics 23 (1982). [28] D. Robert, Autour de l’Approximation Semi-Classique, Birkh¨ auser, Boston, 1987. [29] Yu. Safarov and D. Vassiliev, “The asymptotic distribution of eigenvalues of partial differential operators”, Transl. of Math. Monographs, ISSN : 0065-9282 155 (1996). [30] M. Shubin, Pseudodifferential Operators, Springer-Verlag, 1987. [31] J. Sj¨ ostrand, “Singularit´es analytiques microlocales”, Ast´erisque 95 (1982).
AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N-BODY SYSTEMS JACOB SCHACH MØLLER Department of Mathematical Sciences, Aarhus University, Denmark E-mail:
[email protected] Received 31 August 1998 In the setting of Mourre [18] we characterize the “outgoing” and “incoming” solutions, to the abstract inhomogeneous Schr¨ odinger equation (H − E)u = v, given by the Limiting Absorption Principle. The characterization is in terms of an abstract radiation condition and as an application we give a characterization, in the framework of weighted L2 -spaces, of the outgoing and incoming solutions for N -body Schr¨ odinger operators with and without Stark effect. The abstract radiation condition translates in the application into a radiation condition considered by Isozaki in [13].
1. Introduction Sommerfeld introduced in [24] a condition which characterizes the outgoing and incoming solutions to the reduced wave equation in R3 (−∆ − E)u = v ,
where E > 0 .
The solutions are, provided v has sufficient decay, Z u± (x) =
R3
√
e±i E|x−y| v(y)dy |x − y|
and they are determined uniquely by the asymptotic conditions √ u± = O(|x|−1 ) and (∂|x| ∓ i E)u± = o(|x|−1 ) , which is the radiation condition of Sommerfeld. These two solutions can be obtained from the Limiting Absorption Principle as the limits u± = lim (−∆ − E ∓ i)−1 v . ↓0
This characterization has been generalized to two-body scattering, see for example [17] and [22]. In the case of N -body systems we still have outgoing and incoming solutions by virtue of the Limiting Absorption Principle, but one should not expect them to be characterized by the radiation condition of Sommerfeld due to the more complex 767 Review of Mathematical Physics, Vol. 12, No. 5 (2000) 767–803 c World Scientific Publishing Company
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geometry of the problem. In [13] Isozaki gave a new type of radiation condition which does characterize the outgoing and incoming solutions to the N -body problem, provided √ the pair-potentials are non-singular. Instead of the differential operators ∂|x| ∓ i E one should apply localizations to certain classically forbidden regions of phase-space. In this paper we use Mourre theory to formulate an abstract radiation condition based on localizations in “forbidden regions” and we prove that it characterizes the “outgoing” and “incoming” solutions. We furthermore apply this to extend the result of Isozaki to systems with singular pair-potentials and we obtain new results for N -body systems in an external electric field. In order to elaborate we introduce some notation. We will use the abbreviation √ hti = 1 + t2 for real numbers, vectors and self-adjoint operators. Let H be a Hilbert space, T a self-adjoint operator on H and HTs the associated scale of rigged Hilbert spaces, i.e. HTs = D(hT is ) for s ≥ 0 and HT−s is the space dual to HTs . Assume H is a self-adjoint operator on H which has an interpretation as a closed operator on HTs (for example as distributional derivatives in the context of differential operators on weighted L2 -spaces). Let s > 0. For v ∈ HTs and E ∈ R we consider the inhomogeneous equation (H − E)u = v
(1.1)
and seek solutions in HT−s . Suppose the limits (H − E ∓ i0)−1 = s − lim (H − E ∓ i)−1 ↓0
(1.2)
exist in B(HTs ; HT−s ). Then u± = (H − E ∓ i0)−1 v is what we will understand as the “outgoing” (u+ ) and “incoming” (u− ) solutions to (1.1). The abstract theory of Mourre [18] and Perry, Sigal and Simon [20] provides a machinery to produce operators T for which the “outgoing” and “incoming” solutions to the associated Eq. (1.1) exist. Suppose one can find an operator which is conjugate to H, i.e. a self-adjoint operator A on H satisfying that the commutator form i[H, A] when localized in “energy” around E becomes strictly positive (as well as some technical requirements). Then the Limiting Absorption Principle holds (in −s s a neighbourhood of E). More precisely, the limits in (1.2) exist in B(HA ; HA ) 1 for s > 2 and provide the “outgoing” and “incoming” solutions u± to (1.1). It was proved by Mourre [19] and Jensen [16] that u± satisfy the abstract radiation condition s−1 F (A ≶ 0)u± ∈ HA , (1.3) and we prove in this paper, Theorem 3.1, that (1.3) is a characterization of u± . We illustrate this characterization by the following two explicit examples which fit into the abstract framework. Let H = L2 (R), T = x and H = p = 1i ∇x . The family Hxs is the usual family of weighted L2 -spaces. For s > 12 , E ∈ R and v ∈ Hxs the “outgoing” and “incoming” solutions to (1.1) exist u± = lim (p − E ∓ i)−1 v = ±eiEx {F (·\ ≶ x)v}(E) , ↓0
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where fˆ denotes the Fourier transform of f and F (S) denotes the characteristic function for a set S. An application of Cauchy–Schwartz inequality shows that u± respectively satisfy F (x ≶ 0)u± ∈ Hxs−1−
for all > 0
(1.4)
and since the homogeneous equation (p − E)u = 0 is solved uniquely by u = CeiEx , C ∈ C, which do not satisfy (1.4) we find that u± are characterized by (1.4). As a second example we consider the free one-dimensional Stark Hamiltonian 1 3 1 3 H = p2 −x with weight T = p. Since ei 3 p He−i 3 p = −x and the Fourier transform is an isometric isomorphism from Hps to Hxs we conclude from the previous example that the “outgoing” and “incoming” solutions to (1.1) exist for s > 12 and are characterized by the radiation condition F (p ≶ 0)u± ∈ Hps−1−
for all > 0 .
One should note that the abstract result implies that can be chosen equal to zero in both examples. In applications of the abstract theory to Schr¨ odinger operators one would like to go from L2 -spaces weighted with a conjugate operator to a framework of (position) weighted L2 -spaces. That is H = L2 (X), where X is a configuration space for the system and T is multiplication by some weight function r : X → R. We will consider two applications to Schr¨ odinger operators. The first application is to N -body systems. We will work in the center of mass frame and write X ⊂ RνN for the (ν − 1)N dimensional subspace of relative motion. Here ν is the dimension of the one-particle configuration space. Let A denote the dilation operator (suitably modified using Graf’s vector-field in order to handle singular pair-potentials, see [8] and [23]). Then A is conjugate to H. In particular the commutator form i[H, A] is strictly positive, locally in energy, for E ∈ σess (H)\{τ ∪ σpp (H)} where τ denotes the threshold set of H. For such energies we have the Limiting Absorption Principle −s s in weighted L2 -spaces. More precisely, the limits in (1.2) exist in B(H|x| ; H|x| ) for s > 12 , and thus provide the outgoing and incoming solutions u± to (1.1) (with T = |x|). It was proved by G´erard, Isozaki and Skibsted in [7] (see also [15, 30]) that u± becomes more regular when localized in the classically forbidden region ±A < ehxi, where e is smaller than the square root of the distance from E to the nearest threshold below E. These localizations were obtained using the operators Fs (±B < e) ,
where B = hxi− 2 Ahxi− 2 1
1
and Fs denotes a smoothed out indicator function. We show how to apply the abstract characterization result to prove that u± are characterized by the radiation condition −α Fs (±B < e)u± ∈ H|x| , for some e > 0 and 0 < α <
1 . 2
This is the characterization that was proven by Isozaki in [13] for non-singular N -body potentials.
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As a second application of the abstract result to Schr¨ odinger operators we consider N -body systems in an external electric field E (again in the center of mass frame). Let A denote the momentum in the direction of the field (modified using the vector-field constructed in [9] to handle singular potentials). Then A is conjugate to H. In particular the commutator form is strictly positive locally in energy for −s s any E ∈ R. The limits in (1.2) exist for any E ∈ R as operators in B(HE·x ; HE·x ) 1 for s > 4 and this gives the outgoing and incoming solutions u± to (1.1). For non-singular pair-potentials we prove that p u± become more regular when localized in the classically forbidden region ±A < e hE · xi (where A = E · p) and e < |E|. In analogy with the previous example we employ the operators Fs (±B < e) where B = hE · xi− 4 AhE · xi− 4 1
1
to obtain the localizations and prove that the canonical solutions are characterized by the radiation condition −α for some e > 0 and 0 < α < Fs (±B < e)u± ∈ HE·x
1 . 4
For singular pair-potentials we furthermore prove that each of the two criteria above are satisfied for at most one solution to (1.1). In [14] the characterization of outgoing and incoming solutions by localizations in classically forbidden regions of phase-space was applied to the two-body inverse scattering problem. It provides a formula connecting the limits of the resolvent in (1.2) with the Green’s function of Faddeev, see [14, Theorem 6.2], and thus creates a link between the physical scattering amplitude at energy E > 0 and the Faddeev scattering amplitude. The fact that the Green’s Function of Faddeev has good analyticity properties (with respect to energy) is useful in reconstructing the potential. See [14] for references on this subject. In the framework of geometric scattering theory, the uniqueness result of Isozaki and the resolvent estimate of [7] plays a central role in the study of the N -body scattering matrix. In [28] an N -body scattering matrix is constructed and in [29] new results on the free-free (3–3 cluster) 3-body scattering matrix is obtained. See [29] for references on geometric scattering theory. A central tool used in this paper is the application of almost analytic extensions to handle functions of self-adjoint operators, in particular to compute commutators of the form [B, fλ (A)] to leading order(s) and bound error terms uniformly in a (regularization) parameter λ. This idea goes back to [11] and it has been applied widely since. In [4] almost analytic extensions was used to construct a functional calculus and thus give a new proof of the spectral theorem. In [12] a construction similar to the one given in [4] was used to control commutator expansions in a manner close to the one presented in Sec. 4. See also [6] and [13]. In [25] and later in [23] the Fourier transform was used instead of almost analytic extensions to write functions of self-adjoint operators as an integral of the unitary group associated with the operator thus providing another tool to handle commutator expansions. In [2, Sec. 6.1] a third representation formula for functions of self-ajoint operators is
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introduced based on the observation that the limit to the spectrum of the imaginary part of the resolvent is a restriction to an energy shell. The formula is used to control commutator expansions here as well. The construction of almost analytic extensions given in this paper is inspired by [27, Chap. X.2] where such extensions are discussed in detail and later applied to prove a stationary phase Theorem for complex phase functions. In Sec. 2 we state the assumptions we need in order to prove the abstract characterization result and we collect the necessary tools from [18]. In Sec. 3 we interpret s H as a closed and densely defined operator on HA , where A is conjugate to H, and we formulate the abstract result. Section 4 is, as mentioned above, concerned with computing commutators of the form [B, fλ (A)] and controlling error terms uniformly in λ. In Sec. 5 we prove that the radiation condition (1.3) characterizes the “outgoing” and “incoming” solutions to (1.1) and in Sec. 6 we show how to apply this result to characterize outgoing and incoming solutions for a class of Scr¨odinger operators. In Sec. 7 we use the abstract characterization result and the result obtained in Sec. 6 to generalize the result of [13] to N -body systems with singular pair-potentials and as a new result we characterize the outgoing and incoming solutions for N -body Stark Hamiltonians with non-singular pair-potentials. In Appendix A we discuss the flows of self-adjoint first order partial differential operators which is important in the verification of our assumptions and in Appendix B we use the method developed in [7] to prove that N -body Stark Hamiltonians (with non-singular pair-potentials) do satisfy the radiation condition discussed in Sec. 7. An observation by E. Skibsted led to a reduction (from 6 to 2) in the number of commutators needed to prove Proposition 5.1. 2. Assumptions Let H be a seperable complex Hilbert-space with inner product h·, ·i conjugate linear in the first entry. Assumptions 2.1. Let n0 ≥ 1. Assume H and A are self-adjoint operators on H satisfying: (i) D = D(A) ∩ D(H) is dense in D(H). (ii) exp(itA) : D(H) → D(H) and sup|t|≤1 kH exp(itA)ψk < ∞ for all ψ ∈ D(H). (iii) The commutator form defined iteratively for n ≤ n0 by ad0A (H) = H and 0 in adnA (H) = i[in−1 adn−1 A (H) , A] as a form on D can be represented by a n n 0 symmetric operator i adA (H) with domain containing D(H). The choice of symmetric extension in Assumption 2.1(iii) is not significant since the operators in adnA (H)0 will only be applied to elements of D(H). We will usually write i[H, A]0 for i adA (H)0 . In applications where D is not explicitly known, Assumption 2.1(iii) may not be easy to verify. The following lemma comes in handy and the proof of [18, Proposition II.1] can be used in its verification. ˜ a symmetric operator Lemma 2.1. Let H and A be self-adjoint operators and H ˜ Assume there exist S ⊂ D = D(A) ∩ D(H) such that: with D(H) ⊂ D(H).
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S is a core for H. exp(itA)S ⊂ S and exp(itA)D(H) ⊂ D(H). sup|t|≤1 kH exp(itA)ψk < ∞ for all ψ ∈ D(H). ˜ A] defined on S can be represented by a symmetric The commutator form i[H, 0 ˜ operator i[H, A] with domain containing D(H). ˜ A] defined on D is also represented by i[H, ˜ A]0 . Then the commutator form i[H,
(i) (ii) (iii) (iv)
We will furthermore need some technical results the proofs of which can be found in [18, Sec. II]. Lemma 2.2. Suppose H and A satisfy Assumption 2.1 with n0 ≥ 1. Let z ∈ C, Im z 6= 0, 1 ≤ n ≤ n0 and write Aλ = iλ(A + iλ)−1 . There exist λ0 > 0 such that: (i) (H − z)−1 : D(A) → D(A) and kA(H − z)−1 (A + i)−1 k ≤ Chzi|Im z|−2 . (ii) There exist B(λ) ∈ B(H) with kB(λ)k ≤ 12 for |λ| ≥ λ0 such that Aλ (H − i)−1 = (H − i)−1 Aλ (I − B(λ))−1 . In particular, Aλ : D(H) → D(H) for |λ| ≥ λ0 . (iii) s − lim|λ|→∞ (H − i)Aλ (H − i)−1 = I. 0 (iv) in adnA (H)0 ψ = lim|λ|→∞ i[in−1 adn−1 A (H) , AAλ ]ψ for all ψ ∈ D(H). For a self-adjoint operator H, E ∈ R and δ > 0 we write PH (E, δ) for H’s spectral projection onto (E − δ, E + δ). For operators H and A satisfying Assumption 2.1(iii) with n0 = 1 we define M (A) to be the open set consisting of all E ∈ R for which there exist an e0 (E) > 0 and a map δE : (0, 1) → (0, ∞) such that the following Mourre estimate holds for any ∈ (0, 1) PH (E, δE ())i[H, A]0 PH (E, δE ()) ≥ e0 (E)PH (E, δE ()) .
(2.1)
Assumption 2.2. Assume H and A satisfy Assumption 2.1 for n0 = 2 and the set M (A) of “energies” for which the Mourre estimate (2.1) holds is non-empty. Since the assumptions imposed are stronger than those needed by Mourre [18] and [20] we find: Theorem 2.1 (Limiting Absorption Principle). Suppose H and A satisfy Assumption 2.2. For any E ∈ M (A) and s > 12 the limits limhAi−s (H − E ∓ i)−1 hAi−s ↓0
exist as bounded operators on H and they are attained locally uniformly in E. In particular M (A) contains neither eigenvalues nor singular continuous spectrum. 3. Weighted Spaces In this section we use Assumption 2.1 with n0 = 1 to interpret H as a closed and densely defined operator on some weighted spaces and within this framework we state the abstract characterization result.
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Definition 3.1. Let α ≥ 0 and suppose A is a self-adjoint operator on a Hilbertspace H. We define a scale of rigged Hilbert-spaces: α (i) HA = D(hAiα ), equipped with the graph-norm kvkα = khAiα vk. −α α (ii) HA is the dual space of HA , i.e. the completion of H in the norm
kuk−α = khAi−α uk =
sup v∈Hα ,kvkα =1 A
|hu, vi| .
Let α > 0. For a ψ ∈ H the map v → hψ, hAiα vi is a bounded linear functional −α α on HA . This induces a bounded operator ψ → hAiα ψ from H to HA and in fact −α α . for any α ∈ R the map ψ → hAi ψ is an isometric isomorphism from H to HA −1 α Suppose B ∈ B(H) satisfies hAiBhAi ∈ B(H). Then B ∈ B(HA ) for 0 ≤ α ≤ 1 −α ). If hAiB ∗ hAi−1 is also by interpolation, and by duality we find B ∗ ∈ B(HA α bounded then B ∈ B(HA ) for all −1 ≤ α ≤ 1. Lemma 3.1. For −1 ≤ α ≤ 1 and z ∈ C, Im z 6= 0 we have α (H − z)−1 ∈ B(HA )
and
k(H − z)−1 kB(HαA ) ≤ Chzi|Im z|−2 .
Proof. The result for α = 1 is contained in Lemma 2.2(i) and it extends by the remarks above to all −1 ≤ α ≤ 1. For |α| ≤ 1 we write
α α DA (H) = (H + i)−1 HA .
α We restrict attention to 0 ≤ α ≤ 1 and consider H as an operator on HA with α domain DA (H). Suppose for a moment that H as an operator on these spaces is closed and densely defined. This will by duality imply that H is a closed and densely defined −α operator on HA with domain −α D−α = {u ∈ HA : ∃C > 0 s.t. |hu, Hvi| ≤ Ckvkα ,
α for all v ∈ DA (H)} .
For u ∈ D−α the functional Hu is defined by the unique extension of the map −α α α α v → hu, Hvi from DA (H) to HA . Note that DA (H) ⊂ D−α . For v ∈ HA and −α u∈D we compute hu, vi = hu, (H − i)(H − i)−1 vi = h(H + i)u, (H − i)−1 vi = h(H + i)−1 (H + i)u, vi . −α (H). We furthermore see that the spectrum of H, as This shows that D−α = DA an operator on a weighted space, is a subset of the real axis and (H −z)−1 , Im z 6= 0 are resolvents. We thus only need to prove (i) of the following proposition in the case 0 ≤ α ≤ 1
Proposition 3.1. Let |α| ≤ 1.
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α α (i) The operator H on HA with domain DA (H) is closed and densely defined. −α (ii) The adjoint of H, as an operator on HA , equals H. (iii) The spectrum of H is a subset of the real axis and its resolvents away from the real axis are given by the usual resolvents. 1 α Proof. Let 0 ≤ α ≤ 1. Note that HA is dense in HA , with respect to k · kα , and 1 α DA (H) ⊂ DA (H). The identity 1 DA (H) = Aλ D(H) ,
for |λ| ≥ λ0 ,
(3.1)
1 1 (H) is dense in HA with which is a consequence of Lemma 2.2(ii), shows that DA respect to k · kα . We have thus proved the density part of (i). α α (H) and ψ, ϕ ∈ HA It remains to verify that H is closed. Assume {ψn } ⊂ DA satisfy lim Hψn = ϕ , lim ψn = ψ and n→∞ α in HA .
n→∞
α where the limits are Write ψn = (H + i)−1 fn for a sequence {fn } ⊂ HA . Then fn = Hψn + iψn which imply that fn → ϕ + iψ. This shows that ψ satisfies α ψ = (H + i)−1 (ϕ + iψ) and is thus in DA (H). Furthermore we find Hψ = ϕ. This α shows that H is closed as an operator on the domain DA (H).
Suppose Assumption 2.2 is satisfied. Let E ∈ M (A) and Theorem 2.1 and Proposition 3.1 we find that the limits
1 2
< s ≤ 1. By
(H − E ∓ i0)−1 = lim(H − E ∓ i)−1 ↓0
−s s B(HA ; HA ),
−s exist in Range(H−E∓i0) ⊂ DA (H) and (H−E)(H−E∓i0)−1 = I, −s s where I here denotes the inclusion of HA into HA . This shows that the “outgoing” and “incoming” solutions to (1.1) exist for 12 < s ≤ 1 and E ∈ M (A). We can now state the abstract characterization result. −1
Theorem 3.1. Suppose H and A satisfiy Assumption 2.2. Let E ∈ M (A) and −s u ∈ DA (H) for some 12 < s < 1. Assume u satisfies the equation (H − E)u = v ∈ s HA . Then u = (H − E ∓ i0)−1 v if and only if there exists 0 < α < 12 such that −α F (A ≶ 0)u ∈ HA . The fact that the limits of the resolvent to the real axis satisfies a better estimate when localized with respect to the sign of a conjugate operator was proven in [19] and [16], using the differential inequality technique introduced by Mourre in [18]. We will give an outline of an alternative proof which relies on the type of methods that will be applied throughout the paper. The particular proof is inspired by [14, Lemma 3.7] and [7, Lemma 2.6]. t In the following we write tˆ = hti for real numbers and self-adjoint operators. We introduce two classes of functions Definition 3.2. For σ > 0 we say F± ∈ F± (σ) if F± is smooth, σ 0 ≤ F± ≤ 1 , F± (t) = 1 , ±t < , F± (t) = 0, ±t > σ 2 q and 1 − F±2 = G± ∈ C ∞ (R).
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AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
Definition 3.3. Suppose Assumption 2.2 is satisfied. Let 0 < < 1. The function class P(E, ) ⊂ C0∞ (R)2 denotes pairs of functions (f, h) where 0 ≤ f ≤ 1 satisfies 1 1 supp(f ) ⊂ (E − δE (), E + δE ()) and f (t) = 1, t ∈ E − δE (), E + δE () . 2 2 There exists g ∈ C0∞ (R) such that gf = f and h(t) = tg(t). We furthermore write S P(E) = 0<<1 P(E, ). Outline of an existence proof. Let E ∈ M (A) and 0 < σ < 1. Write for F± ∈ F± (σ) s d 2 ˆ 00 2s−1 2 ˆ F (t)hti . T± (t) = (hti ∓ t) F± (t) and H± (t) = ∓ dt ± (We choose F± such that ∓F±0 ≥ 0 and H± is smooth.) Let 0 < e < e0 (E). Compute, to leading order, using the support properties of F± , the Mourre estimate (2.1) and (4.2) ˆ2 (1 − σ)2s−1 hAi2(s−1) F± (A) ˆ ±2 (A) ˆ ≤ (hAi ∓ A)2(s−1) (I ∓ A)F p p I ∓ Aˆ I ∓ Aˆ −1 ˆ ˆ ≤ e F± (A) i[H, A] F± (A) 1−s (hAi ∓ A) (hAi ∓ A)1−s = ∓{e(2s − 1)}−1 i[H, T± (A)] − {e(2s − 1)}−1 (hAi ∓ A)s− 2 H± (A)i[H, A]H± (A)(hAi ∓ A)s− 2 1
1
≤ ∓{e(2s − 1)}−1 i[H, T± (A)] . −1 v and compute Let > 0. We abbreviate u± = (H − E ∓ i) ± ± ± ± ∓hu∓ , i[H, T± (A)]u i = −2hu , T± (A)u i ∓ 2 ImhT± (A)u , vi .
Notice that the first term on the right-hand side is negative. We apply the Cauchy– Schwarz inequality to the last term and find for any δ > 0 C 2 ˆ ± kvk2s + δkhAis−1 F± (A)u k , δ from which the result follows. One can make this proof rigorous by using a pair from P(E) to localize H at the energy E. 2 ˆ ± khAis−1 F± (A)u k ≤
It is not always handy in applications to work with the weights hAi−α . For 0 < α < 1 and λ > 0 one can instead use the following representation formula: Z ∞ sin(απ) ∓i απ −α ± (A ± iλ) = Cα (A ± i(λ + y))−1 y −α dy , Cα± = ∓i e 2 , (3.2) π 0 to replace hAi−α with the weight (A + iλ)−α , which in some cases are easier to work with. We end this section with a result which can be viewed as an extension of Lemma 2.2(ii).
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Proposition 3.2. There exists λ0 > 0 such that for all λ ≥ λ0 , 0 ≤ α ≤ 1 and |s| ≤ 1 we have: (i) (A ± iλ)−α (H + i)−1 = (H + i)−1 (A ± iλ)−α (I − Bα± (λ))−1 , kBα± (λ)k ≤ 12 . α (ii) DA (H) = (A + iλ)−α D(H). s (iii) (A ± iλ)−α ∈ B(HH ) and k(A ± iλ)−α kB(HsH ) ≤ 2λ−α . Proof. We note that (ii) follows from (i) and the statement (iii) follows from the case s = 1 by interpolation and duality. We first consider (i) for α = 1, where we use the following step from Mourre’s proof of Lemma 2.2(ii) i[(A ± iλ)−1 , (H + i)−1 ] = −(A ± iλ)−1 (H + i)−1 i[H, A]0 (H + i)−1 (A ± iλ)−1 = (A ± iλ)−1 (H + i)−1 B1± (λ) , where kB1± (λ)k ≤ Cλ−1 . Choosing λ0 large enough proves (i) for α = 1 and (iii) for 1 α = s = 1. We now consider 0 < α < 1. By (3.2) we find that (A ± iλ)−α ∈ B(HH ) and Z sin(απ) ∞ −α k(A ± iλ) kB(H1H ) ≤ 2 (λ + y)−1 y −α dy π 0 Z sin(απ) −α ∞ =2 λ (1 + y)−1 y −α dy . π 0 Z
Since
∞
(1 + y)−1 y −α = Γ(α)Γ(1 − α) =
0
π sin(απ)
(3.3)
we conclude (iii). As for the remaining part of (i) we compute using (3.2) Z ∞ i[(A ± iλ)−α , (H + i)−1 ] = Cα± i[(A ± i(λ + y))−1 , (H + i)−1 ]y −α dy 0
= Cα±
Z
∞
(A ± i(λ + y))−1 (H + i)−1 B1± (λ + y)y −α dy
0
= (A ± iλ)−α (H + i)−1 Bα± (λ) , where Bα± (λ)
=
Cα±
Z
∞
(H + i)(A ± iλ)α (A ± i(λ + y))−1 (H + i)−1 B1± (λ + y)y −α dy .
0
We write (A ± iλ)α (A ± i(λ + y))−1 = (A + iλ)α−1 (I ∓ iy(A ± i(λ + y))−1 ) , which in conjunction with (iii) and (3.3) shows that kBα± (λ)k ≤ Cλ−1 , where C > 0 is independent of α and λ. This concludes the proof.
AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
777
One can extend Theorem 3.1 to higher weights (s ≥ 1) by assuming existence and regularity of higher order commutators between H and A. Such an extension is however not needed for characterization purposes. We note that one can relax the assumption on the second commutator as in [18]. 4. Almost Analytic Extension We start by giving a construction which is inspired by [27, Chap. X.2] and write ∂¯ for the differential operator ∂x + i∂y on C ∞ (C). Proposition 4.1. Consider a family of functions {fλ } ⊂ C ∞ (R) for which there (n) exists m ∈ R such that hxin−m fλ is uniformly bounded for all n ≥ 0. There exists a family of almost analytic extensions {f˜λ } ⊂ C ∞ (C) such that (i) supp(f˜λ ) ⊂ {z : Re z ∈ supp(fλ ) and |Im z| ≤ hRe zi}. (ii) |∂¯f˜λ (z)| ≤ CN hzim−N −1 |Im z|N for all N ≥ 0. Here CN > 0 does not depend on λ. Remark. The functions f˜λ are called almost analytic extensions of fλ because of ¯ = 0 characterizes the flatness of ∂¯f˜λ near the real axis. (Recall that the equation ∂ϕ analytic functions in open subsets of the complex plane.) Proof. The idea is to make a “Taylor expansion” of f˜λ around the real axis in such a way that the higher order terms in the expansion are localized increasingly close to the axis. Let (n)
Bn = sup khxin−m fλ k∞ ,
for n ≥ 0 .
λ
We can assume that Bn+1 ≥ Bn + 1. For n ≥ 0 we choose functions hn ∈ C0∞ (R) satisfying 0 ≤ hn ≤ 1, hn (y) = 1, 1 1 |y| ≤ Bn+2 , hn (y) = 0, |y| ≥ Bn+1 and kh0n k∞ ≤ C
1 Bn+1
−
1 Bn+2
−1 =C
Bn+1 Bn+2 , Bn+2 − Bn+1
for some C independent of n. We now define f˜λ (x + iy) =
∞ (n) X fλ (x)(iy)n x hn . n! hxi n=0
Notice that for all x and y this is a finite sum and the functions f˜λ are smooth. Computing ∂¯f˜λ we get ∂¯f˜λ = g1 + g2 , where
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J. S. MØLLER
∞ (n+1) X fλ (x)(iy)n y y g1 (x + iy) = hn − hn+1 n! hxi hxi n=0 g2 (x + iy) =
∞ (n) X fλ (x)(iy)n 0 y y hn i− x ˆ . hxin! hxi hxi n=0
First we notice that the sums in the expression for g1 and g2 only consist of one |y| 1 1 or two terms for each x and y. Consider x and y satisfying Bn+2 ≤ hxi ≤ Bn+1 . Let N ≥ 0. For n ≥ N + 3 we estimate N +3 |y| 1 n n |y| ≤ hxi ≤ 3 hxin−N |y|N . hxi Bn+1 For 0 < n < N + 3 we estimate n−N −2 N +2 B N +2−n n−N N |y| |y| n n |y| = hxi ≤ n+2 hxi |y| . 2 hxi hxi Bn+1 We obtain (i) and (ii) using the estimates above and the fact that hxis hzi−s is bounded uniformly in λ for all s ∈ R on the support of g1 and g2 . We write F0 = {f ∈ C ∞ (R) : ∃ > 0 such that khxi+n f (n) k∞ < ∞ for all n ≥ 0} . Let A be a self-adjoint operator on a Hilbert space H. For f ∈ F0 one can use any almost analytic extension to obtain the following representation of f (A) Z 1 f (A) = ∂¯f˜(z)(A − z)−1 dxdy , (4.1) π C where z = x + iy (see [12] and [4]). The existence of the integral follows from Proposition 4.1(ii). Assumption 4.1. Let n0 ∈ N be given. Assume A is a self-adjoint operator and B is a bounded operator on a Hilbert space H satisfying that the commutator form 0 adnA (B) defined iteratively by ad0A (B) = B and adnA (B) = [adn−1 A (B) , A] as a form n 0 on D(A) can be represented by a bounded operator adA (B) for all n ≤ n0 . Let A and B satisfy Assumption 4.1 for some n0 ≥ 1. Assume f ∈ F0 . Then Z nX 0 −1 1 ˜ [B, f (A)] = − adkA (B)0 ∂¯f(z)(A − z)−k−1 dxdy + Rn1 0 (A, B) π C k=1
=
nX 0 −1 k=1
(−1)k π
where Rn1 0 (A, B) = − Rn2 0 (A, B) =
1 π
Z C
(−1)n0 π
Z C
∂¯f˜(z)(A − z)−k−1 dxdy adkA (B)0 + Rn2 0 (A, B) ,
∂¯f˜(z)(A − z)−1 adnA0 (B)0 (A − z)−n0 dxdy Z C
˜ ∂¯f(z)(A − z)−n0 adnA0 (B)0 (A − z)−1 dxdy .
AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
Note that 1 π
779
Z
(−1)k (k) ˜ ∂¯f(z)(t − z)−k−1 dxdy = f (t) k! C
which implies [B, f (A)] =
nX 0 −1 k=1
=
nX 0 −1 k=1
(−1)k+1 k adA (B)0 f (k) (A) + Rn1 0 (A, B) k! 1 (k) f (A) adkA (B)0 + Rn2 0 (A, B) . k!
(4.2)
Proposition 4.2. Assume A and B satisfy Assumption 4.1 with n0 ≥ 1. Let 0 ≤ t1 ≤ n0 and 0 ≤ t2 ≤ 1, and suppose {fλ } satisfies the assumption of Proposition 4.1 for some m ∈ R. If m + t1 + t2 < n0 then (4.2) holds (as a form identity on −t1 t2 −t2 t1 D(hAim )); Rn1 0 (A, B) ∈ B(HA ; HA ), Rn2 0 (A, B) ∈ B(HA ; HA ) and their norms can be bounded independently of λ. Proof. First assume {fλ } ⊂ F0 . We only have to prove the stated bounds on the remainder terms Rn1 0 (A, B) and Rn2 0 (A, B). We restrict attention to the former. By Hadamards three-line interpolation Theorem we find khAit2 (A − z)−1 adnA0 (B)0 (A − z)−n0 hAit1 k ≤ Chzit |Im z|−n0 −1 , where t = t1 + t2 . We can now estimate the integrand in the expression for the remainder by Chzim+t−N −1 |Im z|N −n0 −1 (4.3) for all N ≥ 0. Pick N = n0 + 1 to avoid the singularity at Im z = 0 and note that m + t − n0 − 2 < −2 by assumption. As for the general result we pick χ ∈ C0∞ (R) with χ(0) = 1 and define gj,λ (x) = x χ( j )fλ (x). Then the family {gj,λ} also satisfies the assumption of Proposition 4.1 with the same m. Since gj,λ (A) → fλ (A) strongly on D(hAim ) as j → ∞ we can compute (as a form on D(hAim )) [B, fλ (A)] = lim [B, gj,λ (A)] . j→∞
We pick as analytic extensions the ones constructed in the proof of Proposition 4.1 (with the same choice of Bn and hn for {fλ } and {gλ,j }) and notice that we have the pointwise convergence as j → ∞ ¯gj,λ → ∂¯f˜λ . ∂˜ The estimate (4.3), for the integrand in the expression for the remainder, is now uniform in j and λ and we can use the Lebesgue Theorem on dominated convergence to conclude the result. After two applications of Proposition 4.2 (first with n0 = 2 and secondly on the leading term with n0 = 1) we get:
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J. S. MØLLER
Corollary 4.1. Let A and B satisfy Assumption 4.2 with n0 = 2 and suppose {fλ} satisfies the condition in Proposition 4.1 for some m ≤ 0. If 0 ≤ t1 ≤ 2, 0 ≤ t2 ≤ 1 −t1 t2 −t2 t1 ; HA ) ∩ B(HA ; HA ) and the two and m + t1 + t2 ≤ 1 then [B, fλ (A)] ∈ B(HA norms can be bounded independently of λ. 5. Uniqueness This section is devoted to proving the uniqueness part of the abstract characterization result Theorem 3.1. We begin by proving a technical lemma that enables us to do some basic operator manipulations in the abstract setting. Lemma 5.1. Assume H and A satisfy Assumption 2.1 for some n0 ≥ 1. Let f ∈ F0 . (i) We have the following expansion formula for multiple commutators: in adnA (f (H))0 =
n X j=1 β∈Nj ,|β|1 =n
Z Cβ
C
j i ∂¯f˜(z) Y h βi βi i adA (H)0 (H − z)−1 dxdy , H − z i=1
(5.1) where Cβ ∈ R are constants and 1 ≤ n ≤ n0 . (ii) Let 0 ≤ n < n0 . Assumption 4.1 is satisfied with B replaced by adnA (f (H))0 and n0 replaced by n0 − n. (iii) For any ψ ∈ H and 1 ≤ n ≤ n0 0 adnA (f (H))0 ψ = lim [adn−1 A (f (H)) , AAλ ]ψ . |λ|→∞
α (iv) For |α| ≤ 1 and 0 ≤ n < n0 we have adnA (f (H))0 ∈ B(HA ). (v) For f, g ∈ C0∞ (R) which satisfy gf = f we have
f (H)i[H, A]0 f (H) = f (H)i[g(H)H, A]0 f (H) . Proof. The expansion formula (i) follows from Lemma 3.1, (4.1), Assumption 2.1(iii) and the fact that in adnA (H)0 : D(H) ∩ D(A) → D(A) ,
for
0 ≤ n < n0 .
(5.2)
The statement (ii) is a consequence of (i) and (iii) follows from (ii) and a simple computation involving the fact that s − lim|λ|→∞ Aλ = I. The statement (iv) for α = 1 follows from (5.1) and (iii) and the general result can be verified by interpolation and duality. The last statement (v) follows from (iii) in conjunction with Lemma 2.2(iv) both applied with n = 1. We note that (iii) and (iv) are direct consequences of (ii) and does not rely on the particular operator B = f (H). Proposition 5.1. Assume H and A satisfy Assumption 2.2. Let E ∈ M (A), 12 < −s s < 1 and u ∈ DA (H). Suppose (H − E)u = 0 and there exists 0 < α < 12 and such −β −α that F (A ≶ 0)u ∈ HA . Then u ∈ DA (H), for any β > max{α, s − 12 }.
AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
781
Proof. We will repeatedly use Lemma 5.1(iv) without reference to justify the well definedness of inner products. Let max{α, s − 12 } < β < 12 , ρ ∈ C0∞ (R), ρ(t) = 1, |t| ≤ 1, and abbreviate fk (s) = s−2β ρ2 ( ks ). Consider the C0∞ (R) family of functions Z ∞
χk (t) =
hti
fk (s)ds .
R∞ R hti We write χk = Ck + φk , where Ck = 1 fk (s) and φk (t) = − 1 fk (s)ds. The family {φk } satisfies the assumption of Proposition 4.1 with m = 1 − 2β. Let (f, h) ∈ P(E). By Lemma 5.1(ii) applied with f replaced by h and n = 0 we find by (4.2) (as an identity in B(H)) hAi ˆ A + R21 (A, h(H)) . (5.3) −i[h(H), φk (A)] = i[h(H), A]0 hAi−2β ρ2 k From Proposition 4.2 applied with B replaced by h(H), n0 = 2 and the parameters −s s t1 = t2 = s we find R21 (A, h(H)) ∈ B(HA ; HA ) and its norm can be bounded uniformly in k. (Notice that m + t1 + t2 = 1 − 2β + 2s < 2.) −s s ; HA ) which are In the following we write Rk for operator families in B(HA bounded uniformly in k. Compute using u = f (H)u, (5.3) and Lemma 5.1(iv) with n ∈ {0, 1} (notice χk has compact support) 0 = ∓hi[H, χk (A)]u, ui = ∓hi[h(H), χk (A)]u, ui = ∓ lim hi[h(H), χk (A)]Aλ u, Aλ ui |λ|→∞
= ∓ lim hi[h(H), φk (A)]Aλ u, Aλ ui |λ|→∞
hAi ˆ 0 −2β 2 = ± i[h(H), A] hAi ρ Au, u ∓ hRk u, ui . k −α ˆ ∈ H−α for all F± ∈ The assumption F (A ≶ 0)u ∈ HA implies that F± (A)u A S 2 2 F(σ). For such a function we write 1 = F + G and ± ± 0<σ<1
hAi ˆ ± i[h(H), A]0 hAi−2β ρ2 Au, u . k 2 ˆ ui + hi[h(H), A]0 G2 (A)u, ui , = ±hi[h(H), A]0 Fk± (A)Au, k±
where −β
Fk± (t) = hti
ρ
hti k
F± (tˆ) and Gk± (t) = hti−β ρ
hti k
p ±tˆG± (tˆ)
are smooth functions both satisfying the conditions of Proposition 4.1 with m = −β. By Lemma 5.1(ii), applied with n = 1, and Proposition 4.2, applied with n0 = 1,
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J. S. MØLLER
t1 = s − β and t2 = s, we find [[h(H), A]0 , Fk± (A)] and [[h(H), A]0 , Gk± (A)] are −s+β s both in B(HA ; HA ) and their norms can be bounded uniformly in k. We get ˆ Fk± (A)ui 0 = ±hi[h(H), A]0 Fk± (A)Au, + hi[h(H), A]0 Gk± (A)u, Gk± (A)ui + hRk u, ui .
(5.4)
Similarly we have s−β β−s −s s [f (H), Gk± (A)] ∈ B(HA ; HA ) ∩ B(HA ; HA )
(5.5)
and both norms are bounded uniformly in k. In order to estimate the second term of (5.4) we write u = f (H)u and commute f (H) to the left and the right the of commutator. As for the leading term we apply Lemma 5.1(v) and we treat the error using (5.5) and Lemma 5.1(iv) with n = 1. We thus obtain for some 0 < e < e0 (E) hi[h(H), A]0 Gk± (A)u, Gk± (A)ui = hf (H)i[H, A]0 f (H)Gk± (A)u, Gk± (A)ui + hRk u, ui ≥ ekf (H)Gk± (A)uk2 + hRk u, ui , where we used the Mourre estimate in the final step. We insert this estimate into (5.4) and commute f (H) back again using (5.5) and Lemma 5.1(iv) with n = 0 to handle the error term
2
ˆ ≤ 2 kGk± (A)uk2
hAi−β ρ hAi G± (A)u
k σ ˆ 2 + hRk u, ui , ≤ CkhAi−β F± (A)uk −β where C > 0 is independent of k. That u ∈ HA now follows by assumption and the Lebesgue Theorem on monotone convergence since β > α, F±2 ≤ F± and G2± ≤ G± . β s Since DA (H) is a dense subset of DA (H) (in the graph-norm of H as an operator β on HA ), we conclude the result.
The uniqueness part of Theorem 3.1 follows from Proposition 5.1 in conjunction with the next result. Proposition 5.2. Let H and A satisfy Assumption 2.2. Let E ∈ M (A) and sup−α pose u ∈ DA (H) for some 0 < α < 12 . If (H − E)u = 0 then u = 0. Proof. Write u = hAi−α u. The aim is to prove sup>0 ku k < ∞ since this, in conjunction with the Lebesgue Theorem on monotone convergence, will imply that u ∈ H and thus u = 0 because E is not an eigenvalue. Note that for γ ≥ 0, the function family t → hti−γ satisfies the assumptions of Proposition 4.1 with m = 0. Let n ∈ {0, 1} and h ∈ C0∞ (R). By Lemma 5.1(ii), (iv) we can apply Corollary 4.1 (with t1 = 1 + α and t2 = α) in the case n = 0 and Proposition 4.2 (with n0 = 1 and t1 = t2 = α) in the case n = 1 to obtain −α α+1−n i[in adnA (h(H))0 , hAi−γ ] ∈ B(HA ; HA ),
(5.6)
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AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
and the norm of the commutator can be bounded uniformly in > 0. In the −α α following will be positive and R ∈ B(HA ; HA ) will denote operators which can be bounded uniformly in > 0. Let (f, h) ∈ P(E). We compute using (5.6) with n=0 hi[h(H), AAλ ]u , u i = −hi[h(H), hAi−α ]u, AA−λ u i − hAAλ u , i[h(H), hAi−α ]ui = αSλ (2 A2 hAi−2−α i[h(H), A]0 u, u) + Sλ (R u, u) , γ −γ where Sλ (u, v) = hu, A−λ vi + hu, Aλ vi for u ∈ HA and v ∈ HA for some γ ∈ R. Note that Sλ (u, v) → 2 Re {hu, vi} as |λ| → ∞. By Lemma 5.1(iii) we thus find
hi[h(H), A]0 u , u i = 2α Re {h2 A2 hAi−2−α i[h(H), A]0 u, u} + hR u, ui . We write 2 A2 hAi−2−α = hAi−α − hAi−2−α and use (5.6) twice with n = 1 to symmetrize the right-hand side. We move the leading order term to the left-hand side and obtain (1 − 2α)hi[h(H), A]0 u , u i = −2αhi[h(H), A]0 hAi−1 u , hAi−1 u i + hR u, ui . (5.7) We write u = f (H)u in the first term on the right-hand side of (5.7) and commute f (H) to the left and the right of the commutator. Then we apply (5.6) (with n = 0 and γ = 1 + α) and Lemma 5.1(iv) with n = 1 to handle the error term and finally we obtain from the Mourre estimate as well as the assumption that α < 12 the following inequality hi[h(H), A]0 u , u i ≤ hR u, ui . Using this estimate in conjunction with the Mourre estimate, Lemma 5.1(v), (5.6) (with n = 0) and Lemma 5.1(iv) with n = 1 gives for some 0 < e < e0 (E) kf (H)u k2 ≤
1 hf (H)i[H, A]0 f (H)u , u i e
=
1 hf (H)i[h(H), A]0 f (H)u , u i e
=
1 hi[h(H), A]0 u , u i + hR u, ui e
≤ hR u, ui . The result now follows after one last application of (5.6) and Lemma 5.1(iv) (both with n = 0) to move f (H) back onto u on the left-hand side. We end this section with an extension of Lemma 5.1 which will be useful in the following. Let Fb = {F ∈ C ∞ (R) : khxin F (n) k∞ < ∞ for any n ≥ 0} . Lemma 5.2. Assume H and A satisfy Assumption 2.1 with n0 ≥ 2. Let F ∈ Fb . Then the statements (ii), (iii) and (iv) in Lemma 5.1 hold with f replaced by F.
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J. S. MØLLER
Proof. The obstacle here compared to Lemma 5.1 is the first commutator, since the integral in (5.1) is not necessarily absolutely convergent. Note that (iii) and (iv) again follow from (ii). We apply the same idea which was used in the proof of Proposition 4.2. For χ ∈ C0∞ (R) with χ(0) = 1 we define the family fj (t) = χ( jt )F (t) which satisfy the assumptions of Proposition 4.1 with m = 0. We pick as almost analytic extension the one constructed in the proof of Proposition 4.1 and compute as a form on D(A) i[F (H), A] = lim i[fj (H), A] j→∞
= lim {i[H, A]0 fj0 (H) + R21 (A, H)} , j→∞
where we used (5.1) and the arguments that led up to (4.6) in the last step. The first term converges to i[H, A]0 F 0 (H) (as a form) which is bounded, and the last term converge to the remainder term corresponding to F instead of fj due to Proposition 4.1 and the Lebesgue theorem on dominated convergence. This shows that i adA (F (H))0 = i[H, A]0 F 0 (H) + R21 (A, H) . The remainder has (for the purpose of this lemma) the same structure as in (5.1) and thus cause no problems. One can now prove (by induction) that for 1 ≤ n ≤ n0 in adnA (F (H))0 =
X
1 0 cnl,k il adlA (H)0 ik adkA (F 0 (H))0 + in−1 adn−1 A (R2 (A, H)) ,
l+k=n l≥1,k≥0
where cnl.k ∈ Z. One can use (5.2) and the fact that Aik adkA (F 0 (H))0 : D(A) → D(H),
for
0 ≤ k < n0
which follows from Lemma 5.1(iv) and (5.1), to verify this identity.
6. Uniqueness in a Position-Weighted Space Let X be a real finite dimensional vector-space equipped with an inner product. As usual we write x for the position operator and p = −i∇x for the momentum operator on H = L2 (X). In this section we prove a result which can be used to obtain uniqueness statements in the framework of L2 (X) spaces, weighted with a function r. For such a weight-function and m ∈ R we define ∞ β max{−1,m−|β|} Qm }. r (X) = {q ∈ C (X) : |∂x q| ≤ Cβ hri
We write qm for functions in Qm r (X) and qm for vector functions from X to X with components in Qm (X). Let S = C0∞ (X). r
AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
785
Assumption 6.1. Let r ∈ C ∞ (X) be a real weight-function and V : X → R a measurable potential. There exist 12 ≤ θ ≤ 1 such that: (i) (ii) (iii) (iv)
r ∈ Q1r (X) and |r| ≤ Chxi for some C > 0. S ⊂ D(V ). H = p2 + V is essentially self-adjoint on S. The operator hri2(θ−1) p2 has an extension from S to an H-bounded operator.
Let A = 12 {qA · p + p · qA } and B = 12 {qB · p + p · qB }. (X) and (v) The components of qA and qB are real-valued elements of Q2θ−1 r Qθ−1 (X) respectively. r (vi) hri−θ qA = q˜02 qB for some q˜0 ∈ Q0r (X) satisfying inf x∈X q˜0 (x) > 0. (vii) Assumption 2.1(ii) is satisfied for the pairs {H, A} and {H, B}. n 0 n (viii) The commutator forms in adnA (H) = i[in−1 adn−1 A (H) , A] and i adB (H) = n−1 0 0 n−1 0 0 0 adB (H) , B], adA (H) = adB (H) = H, defined iteratively as forms i[i on S for 0 ≤ n ≤ 2 can be represented by symmetric operators in adnA (H)0 and in adnB (H)0 respectively. Furthermore for n ∈ {1, 2} X in adnA (H)0 = wβ hri|β|(θ−1) pβ |β|≤2
and in adnB (H)0 = hri−nθ
X
vβ hri|β|(θ−1) pβ ,
|β|≤2
where wβ and vβ are bounded C 2−|β| (X) functions with bounded derivatives. Note that A and B are essentially self-adjoint on C0∞ (X) by Proposition A.1. For θ = 1 these assumptions are satisfied by N -body Hamiltonians and for θ = 12 by N -body Stark effect Hamiltonians. See Sec. 7 for details. Proposition 6.1. Assume r, V, A and B satisfy Assumption 6.1. Then (i) The operator hri|β|(θ−1) pβ has an extension from S to an H-bounded operator for any multi-index β with |β| ≤ 2. (ii) Assumption 2.1 is satisfied with n0 = 2. (iii) Assumption 2.1 is satisfied with n0 = 2 and A replaced by B. (iv) D(H) ⊂ D(B). (v) D(A) ⊂ D(B) and hri−θ A has an extension from D(A) to D(B) and the extension equals q˜02 B + q−1 . Proof. The first statement follows from Assumption 6.1(i) and (iv). The statements (ii) and (iii) follow from (i) and Assumption 6.1(ii), (iii), (vii) and (viii) after an application of Lemma 2.1. The remaining statements follows from Assumption 6.1(v) and (vi) together with (i). Notice that H and hriθ satisfy Assumption 2.1 with n0 = 1 (apply Lemma 2.1 and Proposition 6.1(i)). We can thus apply Proposition 3.1 and interpret H as a
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s θs closed and densely defined operator on Hhri for |s| ≤ 1. It furthermore θ = Hr follows from Proposition 6.1(iv) and (v) that for E ∈ M (A) the limits
(H − E ∓ i0)−1 = lim (H − E ∓ i)−1 ↓0
exist in B(Hrθs ; Hr−θs ), for s > 12 , and Range(H − E ∓ i0)−1 ⊂ Dr−θs (H). In the s and Hrs respectively. following we write k · kA,s and k · kr,s for the norms on HA In order to reduce a uniqueness problem from an r-weighted space to an A-weighted space we need to discuss how we can interpret an element u of Hr−s , t , 0 < t ≤ 1. For this we 0 < s ≤ 1 as a bounded linear functional on HA note that {r, A} satisfies Assumption 2.1 (for any n0 ), see Corollary A.1. Since t t t DA (r) ⊂ HA ∩ Hrs is dense in HA , by Proposition 3.1(i), we find that u uniquely −t t determines a functional in HA if |hu, vi| ≤ CkvkA,t for v ∈ HA ∩ Hrs . We will denote this new functional by u as well. We will assume that M (A) is non-empty. The following theorem is the focus of this section. We will prove it through some lemmas, where Lemma 6.2 is the pivotal one. Note that the pair {B, r} also satisfies Assumption 2.1 (for any n0 ) which by Lemma 5.2(iv) assures that F (B) ∈ B(Hrs ) for F ∈ Fb and |s| ≤ 1. Theorem 6.1. Let 12 < s < 1, E ∈ M (A). Suppose u ∈ Dr−θs (H) solves (H − E)u = 0 and there exist σ > 0, F± ∈ F± (σ) and 0 < α < 12 such that F± (B)u ∈ −α0 −s Hr−θα . Then u ∈ DA (H) and there exists 0 < α0 < 12 such that F (A ≶ 0)u ∈ HA . Note that this result combines directly with Theorem 3.1 to produce uniqueness statements in position-weighted L2 (X) spaces. Several examples will be given in Sec. 7. By Lemma A.4 we find that Assumption 2.1(ii) is satisfied with H replaced by B. Since i[A, B] = q0 B + q−1 as an operator on C0∞ (X) we find by Proposition A.1 and Lemma 2.1 that Assumption 2.1 is satisfied with H replaced by B for any n0 , and furthermore in adnA (B)0 = q0 hri−θ A + q−1 = q0 B + q−1 .
(6.1)
Lemma 6.1. Let 0 ≤ n ≤ 2, −1 ≤ s ≤ 1, f ∈ F0 and F ∈ Fb . Then: (i) in adnB (f (H))0 ∈ B(Hr−t1 ; Hrt2 ) provided t1 + t2 ≤ θn. s (ii) For z ∈ C with Im z 6= 0 we have (B − z)−1 ∈ B(HA ) and k(B − z)−1 kB(HsA ) ≤ Chzi|Im z|−2 . s (iii) F (B) ∈ B(HA ). (iv) Suppose furthermore that F = 0 in a neighbourhood of 0 then F (B) ∈ −s s B(HA ; Hrθs ) ∩ B(Hr−θs ; HA ) for all 0 ≤ s ≤ 1.
Proof. By Proposition 6.1(iii) we can apply the commutator expansion in Lemma 5.1(i) with A replaced by B. The statement (i) now follows (after almost analytic extending f ) from Proposition 6.1(i), Assumption 6.1(viii) and the fact that hrit (H − z)−1 hri−t , is bounded by a constant times hzi|Im z|−2 for t ≥ 0.
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The statement (ii) follows from Lemma 3.1, with H replaced by B, and statement (iii) is a consequence of Lemma 5.2(iv) applied with H replaced by B and n = 0. In order to prove (iv) we compute using Proposition 6.1(v) hriθ F (B)hAi−1 = q˜0−2 {A + q0 }B −1 F (B)hAi−1 . This together with (iii) and Hadarmards three-line Theorem implies F (B) ∈ s B(HA ; Hrθs ) for all 0 ≤ s ≤ 1 and the result now follows by duality. For ρ > 0 we will in the following write Fρ for functions in C0∞ ([−ρ, ρ]) satisfying Fρ (t) = 1 for |t| < ρ2 . Lemma 6.2. Assume the conditions in Theorem 6.1 are satisfied. There exists −α ρ > 0 such that Fρ (B)u ∈ HA . Proof. Let 0 < ≤ 1, 0 < δ ≤ 1, 0 < ρ < uρ = Fρ (B)u ,
σ 4
uρ = hri−1 uρ
and (f, h) ∈ P(E). We abbreviate and uδρ = hδAi−α uρ
and note that uρ , uδρ ∈ D(B). We will write C(·) for positive constants and R(·) for bounded operators that can be bounded uniformly in those of the parameters , δ and ρ that are not supplied as arguments. For v ∈ H we abbreviate Sλ (v) = hv, AA−λ uδρ i + hv, AAλ uδρ i and compute hi[h(H), AAλ ]uδρ , uδρ i = −Sλ (i[h(H), hδAi−α ]uρ ) − Sλ (hδAi−α i[h(H), hri−1 ]uρ ) − Sλ (hδAi−α hri−1 i[h(H), Fρ (B)]u) .
(6.2)
By Proposition 4.2 and the formula for the remainder (4.2) we find that i[h(H), hδAi−α ] = −αi[h(H), A]0 δ 2 AhδAi−2−α + δhAi−1 RhδAi−α . We can treat the first term in (6.2) using the same computations and estimates that lead up to (5.7) to conclude lim hi[h(H), hδAi−α uρ , AA±λ uδρ i = −αhi[h(H), A]0 uδρ , uδρ i
|λ|→∞
+ αhi[h(H), A]0 hδAi−1 uδρ , hδAi−1 uδρ i + δhRuδρ , uδρ i .
(6.3)
As for the second term we will need some estimates. Notice that hrit (H − z) hri−t can be bounded by a constant times hzi|Im z|−2 for 0 ≤ t ≤ 1. Using almost analytic extension on h we get −1
i[h(H), hri−1 ] = hri−θ R0 hri−1 , where R0 =
α π
Z C
θ ˜ ∂¯h(z)hri (H − z)−1 hri−2 r∇r b · p(H − z)−1 hridxdy ,
(6.4)
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satisfy hAi−α R0 hAiα ∈ B(H) .
(6.5)
In order to prove (6.5) we note that it is sufficient to verify it with hAi−α replaced by (A + iλ)−α for some λ ≥ λ0 (see Proposition 3.2). We then write R0 as an integral of terms of the form (H − z)−1
k Y
˜ j ()(H − z)−1 } , {H
j=1
˜ j are H-bounded (uniformly in > 0) first-order differential operators where H (with coefficients in Qθ−1 (X)). We can now apply the representation formula (3.2) r in conjunction with Lemma 2.1 and Lemma 2.2(i) and (ii) to verify (6.5). In order to apply the decay obtained in (6.4) we pick an almost analytic extension of hti−α and apply Proposition 6.1(v) and (6.1) with n = 1 (twice) and obtain AhδAi−α hri−θ = hδAi−α {B q˜02 + q−1 } = {BR1 + hri−θ R2 + δR3 }hδAi−α .
(6.6)
Using the estimates (6.4–6) we write (see (6.2)) hhδAi−α i[h(H), hri−1 ]uρ , AA±λ uδρ i = hR1 uδρ , BA±λ uδρ i + hR2 uδρ , hri−θ A±λ uδρ i + δhR3 uδρ , A±λ uδρ i .
(6.7)
We take |λ| → ∞ and use the identity [hri−θ , hδAi−α ] = δRhδAi−α to rewrite the limit of (6.7) and obtain lim hhδAi−α i[h(H), hri−1 ]uρ , AA±λ uδρ i
|λ|→∞
= hR1 uδρ , Buδρ i + hR2 uδρ , hri−θ uρ i + δhR3 uδρ , uδρ i .
(6.8)
Lemma 2.2(iii) was used with H replaced by B to handle the first term on the right-hand side of (6.7). It remains to treat the third term in (6.2). We develop the commutator by almost analytic extending Fρ , see (4.2), to second order and use Lemma 6.1(i), with n = 2, and the fact that hriθ (B − z)−1 hri−θ is bounded by a constant times hzi|Im z|−2 to obtain i[h(H), Fρ (B)] = i[h(H), B]0 Fρ0 (B) + hri−θ R(ρ)hri−θ . Using (6.6) on the remainder term we get hhδAi−α hri−1 i[h(H), Fρ (B)]u, AA±λ uδρ i = hhδAi−α hri−1 i[h(H), B]0 Fρ0 (B)u2ρ , AA±λ uδρ i + hR1 (ρ)hri−θ u, BAλ uδρ i + hR2 (ρ)hri−θ u, A±λ uδρ i .
(6.9)
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AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
We wish to move hδAi−α onto Fρ0 (B) in order to utilize Lemma 6.1(iv) (Fρ was chosen equal to 1 in a neighbourhood of 0). For this we apply Lemma 6.1(i) and write hri−1 i[h(H), B]0 = hri−θ R00 hri−1 where R00 = hri−1 hriθ i[h(H), B]0 hri ∈ B(H) . By (6.6) and the estimate hAi−α R00 hAiα ∈ B(H) (which follows from an argument similar to the one that was used to prove (6.5)) we find AhδAi−α hri−θ R00 = {BR1 + R2 }hδAi−α .
(6.10)
−α Using that hδAi−α hri−1 = RhδAi−α (as an identity in B(HA ; H)) together with (6.9–10) and Lemma 6.1(iv) we find
lim hhδAi−α hri−1 i[h(H), Fρ (B)]u, AA±λ uδρ i
|λ|→∞
= hR1 (δ, ρ)hri−θα u2ρ , Buδρ i + hR2 (δ, ρ)hri−θα u2ρ , uδρ i + hR3 (ρ)hri−θ u, Buδρ i + hR4 (ρ)hri−θ u, uδρ i .
(6.11)
Combining (6.2–3), (6.8) and (6.11) we finally obtain for any κ > 0 (1 − 2α)hi[h(H), A]0 uδρ , uδρ i ≤ −2αhi[h(H), A]0 hδAi−1 uδρ , hδAi−1 uδρ i + C1 {(δ + κ)kuδρk2 + κ−1 kBuδρk2 } + C2 (δ, ρ){ku2ρk2r,−θα + kuk2r,−θ } . The next step is to apply the Mourre estimate on the left-hand side and on the first term on the right-hand side (as in the proof of Proposition 5.2). In order to do this we write u = f (H)u and commute f (H) to the left and the right of the commutator (and back again on the left-hand side of the equation). We treat the remainders using the following estimates which is similar to some of the estimates above. [f (H), Fρ (B)] = R(ρ)hri−θ ,
[f (H), hri−1 ] = Rhri−θ
[f (H), hδAi−γ ] = δRhδAi−α
and hriθ Fρ (B)hri−θ = R(ρ) ,
where γ ≥ 0 will be used as α and 1 + α. Lemma 6.1(i) was used to derive the first estimate. We obtain kuδρ k2 ≤ C1 {(δ + κ)kuδρ k2 + κ−1 kBuδρ k2 } + C2 (δ, ρ){ku2ρk2r,−θα + kuk2r,−θ } . The final step consists of writing Fρ (B) = F2ρ (B)Fρ (B), in the term involving kBuδρk, and commuting F2ρ (B) past hδAi−α hri−1 . For that we will use some estimates which follow from (6.1) and Lemma 6.1(ii) [hri−1 , F2ρ (B)] = (B + i)−1 R(ρ)hri−θ ,
[hδAi−α , (B + i)−1 ] = (B + i)−1 R
and [hδAi−α , F2ρ (B)] = δ(B + i)−1 R(ρ)hδAi−α ,
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J. S. MØLLER
where we used that [(δA − z)−1 , (B + i)−1 ] = (B + i)−1 R in order to verify the last of the above estimates. We thus obtain kuδρ k2 ≤ {κC1 + ρC2 (κ) + δC3 (ρ, κ)}kuδρk2 + C4 (δ, ρ, κ){ku2ρk2r,−θα + kuk2r,−θ } . One can now choose κ, ρ and lastly δ such that the constant in front of kuδρk2 on the right-hand side is less than 1. This implies sup khAi−α hri−1 Fρ (B)uk < ∞ .
(6.12)
0<≤1
In order to exchange the r- and the A-weights we use Proposition 3.2(iii) and (3.2) as in the proof of Proposition 3.2(i) to compute for δ ≥ λ0 (as operators in B(H; Hrα )) (A − iδ)−α hri−1 = hri−1 (A − iδ)−α (I − B,α (δ))−1 , where kB,α (δ)k ≤ 12 uniformly in > 0, 0 < α < 1 and large δ. Taking the adjoint we find ∗ (δ))−1 (A + iδ)−1 hri−1 hri−1 (A + iδ)−α = (I − B,α as an identity in B(Hr−α ; H). The result now follows from (6.12) and the Lebesgue Theorem on monotone convergence. Proof of Theorem 6.1. We will frequently use Lemma 6.1(iii) without reference to make sense of inner products. Pick ρ as in Lemma 6.2 and compute for F± ∈ F± (ρ) F± (B)u = F∓2 (B)F± (B)u + G2∓ (B)F± (B)u , −α which implies by assumption, Lemma 6.2 and Lemma 6.1(iv) that F± (B)u ∈ HA . −s 2 2 Since u = F± (B)u + G± (B)u we thus find by Lemma 6.1(iv) that u ∈ HA . Let
D = (H + i)−1 (A + iδ)−s (r + i)−1 H ,
for δ ≥ λ0 .
s By Lemma 3.1 and Proposition 3.2(iii) we find that D ⊂ DA (H) ∩ Drθs (H). −1 s Proposition 3.2(ii) shows on the other hand that D = (H + i) DA (r) which by −s s Proposition 3.1(i) is dense in DA (H). In order to verify that u ∈ DA (H) we theres fore only need to prove that v → hu, Hvi is bounded on DA (H) ∩ Drθs (H). This follows from the assumption that u as an element of Dr−θs (H) solves (H − E)u = 0. −α It remains to prove that F (A ≶ 0)u ∈ HA . Let F˜± ∈ F± (1) and α0 = 1 max{α, s − 2 }. For 0 < κ < 1 we define F˜κ± ∈ F± (κ) by F˜κ± (t) = F˜± ( κt ). It −α0 is sufficient to prove that F˜κ± (A)u ∈ HA for some 0 < κ < 1. We introduce a regularization. Consider for k > 0 and χ ∈ C0∞ (R) with 0 ≤ χ ≤ 1 and χ(0) = 1 t −α0 ˜ ˆ Fk± (t) = hti Fκ± (t)χ . k
We abbreviate u± k = Fk± (A)u and compute ± p ku± k k ≤ kF± (B)uk k + kG± (B)u mk k .
(6.13)
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AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
To treat the first term on the right-hand side of (6.13) we use Lemma 5.2(ii) with F = F± and apply Corollary 4.1 with m = −α0 , t1 = 0 and t2 = 1 (writing C for positive constants which can be bounded uniformly in k and locally uniformly in 0 < κ < 1) and obtain kF± (B)u± k k ≤ kF± (B)ukA,−α + CkukA,−1 .
(6.14)
As for the second term on the right-hand side of (6.13) we estimate using Proposition 6.1(v) ± 2 −1 hBG± (B)u± kG± (B)u± k k ≤ ±ρ k , G± (B)uk i ± ˆ 2q 2 ˜ θ G (B)u i = ±ρ−1 hAhAi ˜− θ G± (B)u± ± − k , hAi q k 1
1
2
2
± ± ρ−1 hq−1 G± (B)u± k , G± (B)uk i ,
(6.15)
where q˜− θ = q˜0−1 hri− 2 . We estimate the second term on the right-hand side as θ
2
± ρ−1 |hq−1 G± (B)u± k , G± (B)uk i| ≤
≤
1 ± 2 2 kG± (B)u± k k + Ckq−1 uk k 2 1 2 2 2 kG± (B)u± k k + C{kukr,−1 + kukA,−1 } , 2
where we applied Corollary 4.1 with B = q−1 in the last step. By Lemma 5.2(ii) (with F = G± ) we can apply Corollary 4.1 (twice) with m = −α0 , t1 = 12 and t2 = s, Lemma 6.1(iii) and estimate (6.15) as 1 κ 1 2 2 2 ˜ θ G (B)uk2 + C{kuk2 kG± (B)u± r,−1 + kukA,−s} . −2 ± k k ≤ 2 kFk± (A)hAi q 2 ρ
In the first term on the right-hand side we move Fk± (A) back onto u, again using Corollary 4.1 (as above), and obtain 1 κ ± 2 2 2 2 2 kG± (B)u± k k ≤ 8 khAi q− θ2 G± (B)uk k + C{kukr,−1 + kukA,−s } . ρ
(6.16)
Let f ∈ C0∞ (R) satisfy f = 1 in a neighbourhood of E. By Proposition 4.2 applied with n0 = 2, m = −α0 and t1 = t2 = 1 we estimate 0 i[f (H), Fk± (A)] = i[f (H), A]0 Fk± (A) + R , −1 −1 1 0 where R ∈ B(HA ; HA ). Using that Fk± (A) ∈ B(HA ; H), Proposition 4.1 and (5.1) we find
G± (B)[Fk± (A), f (H)] = {B −1 G± (B)}B{(H + i)−1 R1 + hAi−1 R2 }hAi−1 . (6.17) By Lemma 5.1(ii) (with A replaced by B), Lemma 6.1(i) and Proposition 4.2 applied with m = 0 we find [f (H), G± (B)] = G0± (B)[f (H), B]0 + (B + i)−1 Rhri−θ .
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J. S. MØLLER
This together with another application of Lemma 6.4(i) (to the first term on the right-hand side) and Corollary 4.1 applied with m = −α0 , t1 = 1, t2 = 0 and B replaced by hri−θ gives the estimate [G± (B), f (H)]Fk± (A) = (B + i)−1 {R3 hri−θ + R4 hAi−1 } .
(6.18)
The operators Rj , j ∈ {1, 2, 3, 4}, are all bounded uniformly in k and locally uniformly in 0 < κ < 1. We can now write u = f (H)u in the first term on the 1 right-hand side of (6.16) and apply (6.17–18) and the fact that hAi 2 q˜− θ is B- and 2 H-bounded to obtain 2 kG± (B)u± kk ≤
κ C1 ku± k2 + C2 {kuk2r,−θ + kuk2A,−s } , ρ
where C1 does not depend on κ. Inserting this estimate together with (6.14) into (6.13) we find κ 2 2 2 2 1 − C1 ku± k k ≤ kF± (B)ukA,−α + C2 {kukr,−θ + kukA,−s } ρ which implies the result after choosing κ smaller than k → ∞.
ρ C1
and taking the limit
We have in this section considered H in an abstract operator sense as an extension of −∆ + V from S. Note that V maps L2 (X) into the space of distributions. This follows from Assumption 6.1(ii) and [21, Theorem 6.8]. Then one can also view −∆ + V as an operator in a distributional sense. We denote this realization by h. Note that Hrα are distributions and h maps these spaces into the space of distributions. Define for |α| ≤ 1 the domains Dα (h) = {ψ ∈ Hrα : −∆ψ + V ψ ∈ Hrα } . On these domains h is well-defined as an operator in the distributional sense and h coincides with H on S, which is a core for H as an operator on Hrα . Let ψ ∈ Dα (h) and compute for ϕ ∈ S hψ, Hϕi = hψ, hϕi = hhψ, ϕi . This shows by assumption that ψ is in the domain of the operator dual to H considered as an operator on Hr−α , i.e. ψ ∈ Drα (H). On the other hand let ψ ∈ Drα (H) and compute for ϕ ∈ S hhψ, ϕi = hψ, hϕi = hψ, Hϕi = hHψ, ϕi . This shows that hψ ∈ Hrα and we have thus proved Dα (h) = Drα (H) and h = H. The above discussion shows in particular that if u ∈ Hr−α , α > 0, solves the distributional equation (H − E)u = v ∈ Hrα then Hu ∈ Hr−α , which implies that u ∈ Dr−α (H). This observation will be used in the next section to give a more attractive formulation of the examples.
AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
793
7. Applications to N-body Systems In this section we wish to consider Hamiltonians describing N ν-dimensional particles, possibly in an external electric field. That is operators of the form ( ) N X X p2j ˜ ˜ H= − qj E · xj + V on L2 (RN ν ) , V (x) = Vij (xk − xj ) , 2mj j=1 1≤j
where E˜ ∈ Rν is the external electric field, mj > 0 and qj ∈ R are the masses and the electric charges of the particles. We apply the standard procedure of removing the center of mass motion and obtain an operator on the space L2 (X), where X is the (N − 1)ν-dimensional vector-space given by N X mj xj = 0 , X = x ∈ RN ν : j=1
with the inner product x · y = mass frame is
PN j=1
2mj xj · yj . The Hamiltonian in the center of
H = p2 − E · x + V , where the effective electric field E ∈ X is Q q1 ˜ . . . , qN − Q E˜ − E= E, 2m1 2M 2mN 2M P PN and M = N j=1 mj and Q = j=1 qj are the total mass and charge of the system. We notice that E = 0 if and only if the charge to mass ratio of all particles are the same. See [1, 3] or [10] for the geometry of Stark systems. Let Π be an orthogonal projection on X. For any s ≥ 0 we write L2,s Π (X) for the measurable functions on X that are square-integrable with respect to the measure 2,−s hΠxi2s dx and for s ≤ 0 we write L2,s (X) (in the Π (X) for the space dual to LΠ 2,s 2,s sense of Definition 3.1). We abbreviate L (X) = LI (X). If E 6= 0 we write Π0 E·x for the projection Π0 x = |E| 2 E. The uniqueness part of all the examples below follow from the discussion at the end of Sec. 6 together with Theorems 6.1 and 3.1. Assumption 7.1. The external field E is zero. The functions Vij ∈ C ∞ (Rν ) are real-valued and satisfy the decay estimates |∂yβ Vij (y)| = O(hyi−|β|−µ ) , for any multi-index β and some µ > 0. It is well known, see [20], that under these assumptions the operator H and the generator of dilations A1 = 12 (x · p + p · x) satisfy Assumption 2.2. Employing the x·p+p·x ˆ} we find that Assumption 6.1 weight function r(x) = hxi and B1 = 12 {ˆ hxi 2 is satisfied with θ = 1 and q˜0 = hhxii . In this case we have M (A1 ) = [Σ, ∞)\{τ ∪ σpp (H)} ,
(7.1)
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where Σ ≤ 0 is the bottom of the essential spectrum of H and τ is the threshold set, i.e. the union of eigenvalues for all subsystem Hamiltonians. Furthermore e0 (E) = 2 inf{E − λ : λ ∈ τ and λ < E}. Theorem 7.1. Suppose Assumption 7.1. Let 12 < s < 1, E ∈ M (A1 ) and u ∈ L2,−s (X). Assume (H − E)u = v ∈ L2,s (X). Then u = (H − E ∓ i0)−1 v if and only if there exist σ > 0, 0 < α < 12 and F± ∈ F± (σ) such that F± (B1 )u ∈ L2,−α (X). That the outgoing and incoming solutions (H −E ∓i0)−1 v becomes more regular when localized with respect to B1 was proved in [7, Theorem 2.10] (for singular potentials using Graf’s vector-field, see [8]). More precisely for v ∈ L2,s (X), s > 12 , we have \ F± (B1 )(H − E ∓ i0)−1 v ∈ L2,−α (X) , (7.2) α>1−s
q 1 for any F± ∈ {F ∈ Fb : supp(F ) ⊂ {t ∈ R : ±t < 2 e0 (E)}}. A proof of the uniqueness part for this particular example was given in [13]. We note that the result can be formulated in terms of certain pseudo-differential operators instead of localizations in B1 . See [13, Theorem 1.3 and Lemma 4.4] and [7, Theorem 2.12]. Assumption 7.2. The external field E is zero. Let Vij = Wij + Vij,sing . The “pair” potentials Wij satisfy Assumption 7.1, Vij,sing has compact support and is relatively p2 -compact. For this example we can employ the vector-field constructed in [8], which is a gradient field. There exists (see [7]) a smooth function r satisfying C1 hxi ≤ r(x) ≤ C2 hxi for some C2 > C1 > 0 and such that Graf’s vector-field (up to a scaling) equals r∇r. The modified dilation operator becomes A2 = 12 {r∇r · p + p · r∇r} and Assumption 2.2 is satisfied, see [23, Appendix B] (and Lemma A.5(i)). We pick B2 = 12 {∇r · p + p · ∇r}. Consulting [7, Lemma 2.1] we find that Assumption 6.1 r is satisfied with θ = 1 and q˜02 = hri . The set M (A2 ) is as in (7.1) and the resolvent estimate (7.2) holds here as well. Theorem 7.2. Suppose Assumption 7.2. Let 12 < s < 1, E ∈ M (A2 ) and u ∈ L2,−s (X). Assume (H − E)u = v ∈ L2,s (X). Then u = (H − E ∓ i0)−1 v if and only if there exist σ > 0, 0 < α < 12 and F± ∈ F± (σ) such that F± (B2 )u ∈ L2,−α (X). This example includes the usual model for atoms and molecules, that is ν = 3 qi qj and Vij (y) = |y| . Assumption 7.3. The external field E is nonzero. The pair potentials Vij ∈ C 2 (Rν ) satisfy |∂yβ Vij (y)| = o(1) , f or |β| ≤ 1 , and |∂yβ Vij (y)| = O(1) ,
f or
|β| = 2 .
For this example we have the conjugate operator A3 = E · p for which Assumption 2.2 is satisfied (see [9, (5.1)]). Note that M (A3 ) = R and e0 (E) = |E|2 in this case.
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For any orthogonal projection Π on X with ΠE = E we define a weight function 1 1 r = hΠxi and choose B3 = 12 {r− 2 E · p + p · Er− 2 }. For these choices Assumption 6.1 q hΠxi is satisfied with θ = 12 and q˜02 = hhΠxii . (See [9, Lemma 3.1].) Theorem 7.3. Suppose Assumption 7.3. Let 14 < s < 12 , E ∈ R and u ∈ L2,−s (X). Π Assume (H − E)u = 0 and there exist σ > 0, 0 < α < 14 and F± ∈ F± (σ) such that (X). Then u = 0. F± (B3 )u ∈ L2,−α Π The operator B3 applied in this example (for Π = I) plays a significant role in the scattering theory for the N -body Stark problem. It was first used in [26], and later in [3] and [10]. For Π = Π0 we have an existence result as well. Let r(x) = E · x, B4 = 1 1 − 12 {hri E · p + p · Ehri− 2 }, For these choices Assumption 6.1 is satisfied with θ = 12 2 and q˜0 = 1. Theorem 7.4. Suppose Assumption 7.3. Let 14 < s < 12 , E ∈ R and u ∈ L2,−s Π0 (X). 2,s −1 Assume (H − E)u = v ∈ LΠ0 . Then u = (H − E ∓ i0) v if and only if there exist σ > 0, 0 < α < 14 and F± ∈ F± (σ) such that F± (B4 )u ∈ L2,−α Π0 (X). That the outgoing and incoming solutions satisfies the radiation condition is proved in Appendix B, see Theorem B.1. Assumption 7.4. The external field E is nonzero. The dimension of the particle space ν is greater than or equal to 3. The pair potentials are of the form Vij = Wij + Vij,sing where Wij satisfies Assumption 7.3. The singular part Vij,sing has compact support, and there exist l ≥ 1 and r1 , . . . , rl ∈ Rν such that |∂yβ Vij,sing (y)| ≤ C
l X
|rk − y|−1−|β| ,
k=1
for all multi-indices with |β| ≤ 1. For this example we can use the vector-field E(x) constructed in [9, Appendix B] which involves Derezi´ nski’s “distortion” of |x| (see [5]). By [9, Proposition 6.4(1), Lemma B.3, (B.5) and Proposition B.4] we find that Assumption 2.2 is satisfied with A5 = 12 {E(x) · p + p · E(x)} (see Lemma A.5(ii)). Notice that the derivatives of the vector field E(x) only decay in the direction of the electric field. We can thus 1 only obtain a uniqueness result for Π = Π0 . We pick B5 = 12 {hΠ0 xi− 2 E(x) · p + 1 p · E(x)hΠ0 xi− 2 }. Assumption 6.1 is therefore satisfied with the weight-function r(x) = E · x, θ = 12 and q˜02 = 1. Theorem 7.5. Suppose Assumption 7.4. Let 14 < s < 12 , E ∈ R and u ∈ L2,−s Π0 (X). 1 Assume (H − E)u = 0 and there exist σ > 0, 0 < α < 4 and F± ∈ F± (σ) such that F± (B5 )u ∈ L2,−α Π0 (X). Then u = 0. This example includes the usual model for atoms and molecules in an external electric field. The vector-field applied here is not a gradient field. The method used in Appendix B to prove an existence result for non-singular pair-potentials does therefore not apply in this case.
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Appendix A: Flows In this Appendix we analyse the group t → exp(itD) generated by a self-adjoint first order partial differential operator on H = L2 (X) (see Sec. 6). Most of the results are well-known. We consider operators of the form D=
1 i {q · p + p · q} = q · p − div q , 2 2
where q is a smooth function from X to X satisfying |q(x)| ≤ C0 hxi
and |∂xβ q(x)| ≤ Cβ ,
(A.1)
for any multi-index β with |β| ≥ 1. Under these assumptions q is globally Lipschitz and we can thus solve the ODE d y(t) = q(y(t)) , dt
y(0) = x ∈ X
(A.2)
globally in (t, x). Let φ(t, x) denote the solution to (A.2) (which is smooth in (t, x)). In the following R will denote a smooth function on X satisfying for some a > 0 a ≤ R(x) ≤ hxi ,
|∇R(x)| ≤ 1 and |∂xβ R(x)| ≤ Cβ ,
for any multi-index β with |β| ≥ 2. Lemma A.1. Let C0 be as in (A.1). Suppose |q(x)| ≤ C1 R(x)κ for some 0 ≤ κ ≤ 1. We have three estimates κ=0:
|φ(t, x) − x| ≤ C0 |t| .
κ=1:
R(φ(t, x)) ≤ R(x)eC1 |t| .
0≤κ<1:
1
1
R(φ(t, x)) ≤ 2 1−κ {R(x) + [(1 − κ)C1 |t|] 1−κ } .
Proof. We only consider the case t ≥ 0. The first estimate follows directly from (A.1–2). The last two estimates follow from the differential inequality d R(φ(t, x)) ≤ C1 R(φ(t, x))κ . dt
By differentiating φ0 (t, x) with respect to time we find that it solves the ODE d A(t) = q0 (φ(t, x))A(t) , dt
A(0) = I .
(A.3)
Lemma A.2. Let K be a compact subset of R. For any multi-index β with |β| ≥ 1 we find ∂xβ φ(t, x) is bounded uniformly in K × X.
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797
Proof. Let I = |∂xβ φ(t, x)|2 + 1 and compute using (A.1–2) d I ≤ CI + b , dt where the function b is the norm of a sum of products of the coordinate functions of ∂xβ1 q(φ(t, x)) and ∂xβ2 φ(t, x) with |β1 | ≥ 1 and β2 < β (b = 0 if |β| = 1). By (A.1) and an induction argument in β we find after solving the above differential inequality R I(t) ≤ I(0)e
t
0
C(s)ds
,
where s → C(s) is bounded locally uniformly in time. The result now follows since ∂xβ φ(0, x) is independent of x provided |β| ≥ 1. Lemma A.3. Let q be as in Lemma A.1 and assume furthermore that |∂xβ q(x)| = O(R(x)κ−1 ) for any multi-index β with |β| ≥ 1. Then (i) |φ0 (t, x) − I| ≤ C(t)R(x)κ−1 . (ii) |∂xβ φ(t, x)| ≤ Cβ (t)R(x)κ−1 for any multi-index β with |β| ≥ 2. Here C(t) and Cβ (t) are locally uniformly bounded in time. Proof. The first estimate follows from Lemmas A.1 and A.2, the group properties of the flow and the fact that φ0 (t, x) solves (A.3). The last estimate follows similarly noting that ∂xβ φ(0, x) = 0 for |β| ≥ 2. For ψ ∈ C0∞ (X) we define the family p ψt (x) = J(t, x)ψ(φ(t, x)) , where J(t, x) = det φ0 (t, x) . The Jacobian J is given by Rt J(t, x) = e
0
div q(φ(s,x))ds
.
This follows from the following identity which holds for matrix valued differentiable functions d det(A(t)) = det(A(t)) tr(A0 (t)A−1 (t)) dt as well as the fact that φ0 (t, x) solves (A.3). Proposition A.1. The operator D is essentially self-adjoint on C0∞ (X) and its associated group exp(itD) is given by the extension of the map t → ψt from C0∞ (X) to H. Furthermore exp(itD) : C0∞ (X) → C0∞ (X) for all t. Proof. By the formula for the Jacobian and the group properties of the flow we find that (ψt )s = ψt+s . Clearly kψt k = kψk so the invertible map ψ → ψt extends to a strongly continuous group of unitary operators U (t) on H. By Lemma A.1 (applied with R(x) = hxi) we have U (t)C0∞ (X) ⊂ C0∞ (X). It remains to check that −i
d ψt = Dψt dt
(A.4)
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J. S. MØLLER
holds at t = 0 for ψ ∈ C0∞ (X) since this will imply that D is essentially self-adjoint on C0∞ (X) and that its closure is the generator of U (t). We have the identity φ0 (t, x)q(x) = q(φ(t, x))
(A.5)
which holds by (A.2) since φ0 (t, x) solves (A.3). Using (A.5) and (A.2) one can now verify (A.4) by direct computations. This result can be applied in the verification of Assumption 2.1(ii). Corollary A.1. Assume R is as in Lemma A.1. Then Assumption 2.1(ii) is satisfied with H replaced by (multiplication with) R and A replaced by D. Lemma A.4. Let r, A and B be as in Assumption 6.1 for some Assumption 2.1(ii) is satisfied with H replaced by B.
1 2
≤ θ ≤ 1. Then
Proof. Let ψ ∈ C0∞ (X). Note that Proposition A.1 implies Aψt = (Aψ)t . We compute (B exp(itA)ψ)(x) = Q(x)−1 (Aψt )(x) + q−1 (x)ψt (x) Q(φ(t, x)) q−1 (φ(t, x)) = (Bψ)t (x) + q−1 (x) − ψt , Q(x) Q(x) where Q = q˜02 hriθ . The result now follows from Lemma A.1 applied with R = supx∈X {|∇r(x)|}−1 hri and κ = 2θ − 1. Lemma A.5. Let r, A and B be as in Assumption 6.1. We have (i) If θ = 1 then Assumption 2.1(ii) is satisfied for the pairs {p2 , A} and {p2 , B}. (ii) Let E ∈ X. Assume θ = 12 and hri−1 p2 (p2 − E · x + i)−1 ∈ B(H). Then Assumption 2.1(ii) is satisfied for the pairs {p2 − E · x, A} and {p2 − E · x, B}. Proof. It is sufficient to prove the result for A. Let ψ ∈ C0∞ (X). We compute ∇{ψ(φ)} = φ0t (∇ψ)(φ) = (∇ψ)(φ) + (φ0 − I)t (∇ψ)(φ) and ∇{(∇ψ)(φ)} = tr{φ0 (∇∇ψ)(φ)} = (∆ψ)(φ) + tr{(φ0 − I)(∇∇ψ)(φ)} . From these computations and Lemma A.3 applied with κ = 2θ − 1 we find X β p2 ψt = (p2 ψ)t + (vt pβ ψ)t , |β|≤2
where |vtβ (x)| ≤ C(t)r(x)2(θ−1) and C(t) is bounded locally uniformly in t. This proves (i) and the statement (ii) follows from the first estimate in Lemma A.1.
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Appendix B: Resolvent Estimates In this Appendix we prove, under Assumption 7.3, that the outgoing and incoming solutions for Stark Hamiltonians satisfies the radiation condition in Theorem 7.4. The proof is similar to the proof of [7, Lemmas 2.7 and 2.8] (for s close to 12 ) and a simpler version of the same idea was presented in connection with the outline given in Sec. 3. In fact one can generalize the result from [7] to models satisfying Assumption 6.1 provided the vector field qB is a certain gradient field. More precisely ∇hriθ ' qB , where the error should have sufficient decay in r. (This remark implies that the smoothness conditions in Assumption 7.1 can be replaced by a C 2 (X) condition.) The case of Stark effect with the usual choice of conjugate operator E · p 1 does not fit directly under this assumption since ∇hri 2 = zˆqB , where r = z = E · x, and zˆ is not close to 1 for large x. We will employ the following lemma to deal with this problem. We write H0 = p2 − E · x and H = H0 + V . √ Lemma B.1. The operator −zF (z < 0)(H − i)−1 is bounded. Proof. Let χ ∈ C ∞ (X) satisfy 0 ≤ χ ≤ 1, χ(t) = 1, t < −1 and χ(t) = 0, t > 0. It is sufficient to prove the result with F (· < 0) replaced by χ and H replaced by H0 . We estimate as a form on C0∞ (X) 0 ≤ −(H0 + i)−1 χ(z)zχ(z)(H0 − i)−1 = (H0 + i)−1 χ(z)(H0 − p2 )χ(z)(H0 − i)−1 ≤ (H0 + i)−1 χ(z)H0 χ(z)(H0 − i)−1 . This concludes the proof since χ(z) : D(H0 ) → D(H0 ).
This lemma (along with [9, Proposition 3.7]) imply that 1 1 i[H, hzi] = {ˆ z E · p + p · E zˆ} 2 2 is also a conjugate operator with e0 (E) = |E|2 . This choice of conjugate operator would fit under a generalization of [7, Lemmas 2.7 and 2.8]. It does however not carry as much information about the dynamics of the system as do 12 i[H, z] = E · p. Therefore we choose to consider the usual choice of conjugate operator and give an individual proof of the corresponding resolvent estimate. Definition B.1. Let e > 0 and 0 < δ < 13 . The set F˜± (e, δ) consists of C ∞ (R) functions, 0 ≤ F˜± ≤ 1, satisfying supp(F˜± ) ⊂ {t ∈ R : ±t ≤ (1 − δ)e} ,
supp(1 − F˜± ) ⊂ {t ∈ R : ±t ≥ (1 − 2δ)e} ,
d ˜ ∈ C0∞ (R) and ∓ dt F± ≥ 0. 1 1 In the following A = E · p and B = 12 {hzi− 2 E · p + p · Ehzi− 2 }. We note that Assumption 6.1 is satisfied with r = z and θ = 12 (as mentioned in Sec. 6). In this example M (A) = R and e0 (E) = |E|2 . d ˜ dt F±
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J. S. MØLLER
Theorem B.1. Let E ∈ R, s >
1 4
and v ∈ Hzs . There exist 0 < α <
1 4
such that
F± (B)(H − E ∓ i0)−1 v ∈ Hz−α for all F± ∈ {F ∈ Fb : supp(F ) ⊂ {t ∈ R : ±t < |E|}}. Proof. We can assume that s ≤ 38 since the spaces Hzs decreases as s increases. −1 Let α = 12 − s and u± v. It is sufficient to prove that = (H − E ∓ i) sup kF˜± (B)u± kz,α < ∞
0<≤1
for F˜± ∈ F˜± (|E|, δ), 0 < δ < 13 . Let 0 < δ < 13 and fix F˜± ∈ F˜± (|E|, δ). We abbreviate E0 = hzi−α F˜± (B)
1
and T± (t) = (|E| ∓ t)2s− 2 F˜±2 (t) .
We aim at proving the following form estimate on Dzs (H) σE0∗ E0 ≤ ∓i[H, hzi2s− 2 T± (B)] + 1
X
Ei∗ Ej ,
(B.1)
0≤i,j≤2 (i,j)6=(0,0)
for some σ > 0, where the error terms are of the form E1 = Rhzi−s
˜ s (1 − ηE (H)) . and E2 = Hhzi
˜ denotes H-bounded operators and η ∈ Here R denotes bounded operators, H ∞ C0 (R) denotes functions which are 1 in a neighbourhood of E. We will frequently in the following apply Proposition 4.2 with f = T± and n0 = 2 in order to justify the use of the expansion formula (4.2). The expression in (B.1) make sense since T± (B) is H-bounded and T± (B) : Dzs (H) → Hzs . We write T ' S provided T − S has the form of the error term in (B.1). Let (f, h) ∈ P(E, 1 − 12 δ) and compute 1
1
i[H, hzi2s− 2 T± (B)] ' f (H)i[h(H), hzi2s− 2 T± (B)]f (H) .
(B.2)
We compute using (4.6) (with n0 = 2) first as a form on Hzs i[h(H), hzi2s− 2 ] = h0 (H)i[H, hzi2s− 2 ] + E1∗ E1 1
1
= (2s − 12 )h0 (H)hzi2s−1 Re{ˆ z B} + E1∗ E1
(B.3)
AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N -BODY SYSTEMS
801
and secondly as a form on D(H) i[h(H), T± (B)] = i[h(H), B]0 T±0 (B) + hzis−1 Rhzi−s = h0 (H)i[H, B]0 T±0 (B) + hzis−1 Rhzi−s ,
(B.4)
where we used the assumption that s ≤ 38 in (B.3), and Lemma 6.1(i) as well as Assumption 6.2(viii) in (B.4) to estimate the remainder terms. We furthermore have 1 ˜ −s , i[H, B]0 = hzi− 2 {i[H, A]0 − Re {ˆ z B}B} + hzis−1 Hhzi (B.5) ˜ is H-bounded, and where H ( T±0 (B)
=∓
2s −
1 2
3
(|E| ∓ B)2s− 2 F˜±2 (B)
2s− 12
+ (|E| ∓ B)
) d ˜2 ∓ F± (B) . dt
(B.6)
Choose g ∈ C0∞ (R) such that g has the same support properties as f (i.e. (g, h) ∈ P(E, 1 − 12 δ)) and gf = f . Inserting (B.5–6) into (B.4) and subsequently (B.3–4) into the right-hand side of (B.2) yields (after rearrangement of the terms and an application of the assumption s ≤ 38 to the error terms in (B.4) and (B.5)) 1
∓i[H, hzi2s− 2 T± (B)] ' (2s − 12 )E0 f (H){∓|E| Re{ˆ z B} + g(H)i[H, A]0 g(H)}f (H)E0∗ ˜ ± (B){g(H)i[H, A]0 g(H) − B 2 }H ˜ ± (B)R1 + R1∗ H + R2∗ f (H)(ˆ z − 1)f (H)R2 , q ˜ ± (t) = ∓( d F˜ 2 )(t). The last term is an E ∗ E1 error term according to where H 1 dt ± Lemma B.1 and the second term is greater (modulo an error term of the desired 2 ˜± (B)R1 . As for the first term we write Re{ˆ z B} = B + form) than 13 δ|E|2 R1∗ H Re{(ˆ z − 1)B} and obtain by Lemma B.1 E0 f (H){∓|E| Re{ˆ z B} + g(H)i[H, A]0 g(H)}f (H)E0∗ ≥ E0 f (H){∓|E| Re{ˆ z B} + (1 − 12 δ)| E|2 }f (H)E0∗ ' E0 {∓|E|B + (1 − 12 δ)|E|2 }E0∗ ≥ E0 {(δ − 1)|E|2 + (1 − 12 δ)|E|2 }E0∗ ' 12 δ|E|2 E0∗ E0 . This proves (B.1) with σ = (2s − 12 ) 12 δ|E|2 .
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As in the outline in Sec. 3 we compute 2s− 2 ∓hu± T± (B)]u± , i[H, hzi i 1
2s− 2 = −2hu± T± (B)u± , hzi i 1
2s− 2 ± ∓ {hv, hzi2s− 2 T± (B)u± u , vi} i − hT± (B)hzi 1
1
s− 4 ± ' −2hhzis− 4 u± u i , T± (B)hzi 1
1
± Imhhzis v, (RE0 + E1 + E2 )u± i, where ' in this context means that the difference between the left- and the right-hand side is an expectation value of an error term of the type in (B.1) in the state u± . Inserting this estimate into (an expectation value of) (B.1) yields the result. Acknowledgments This work was part of the author’s Ph.D. thesis and I wish to thank the committee, which consisted of Henning Haahr Andersen, Jan Derezi´ nski and Arne Jensen, as well as the author’s supervisor, Erik Skibsted, for pointing out two errors and a number of misprints as well as making a suggestion which improved readability. References [1] T. Adachi, “Propagation estimates for N-body Stark Hamiltonians”, Ann. Inst. Poincar´e 62 (1995) 409–428. [2] W. Amrein, A. Boutet de Monvel and V.Georgescu, “C0 -groups, Commutator Methods and Spectral Theory of N-body Hamiltonians”, Progress in Mathematics, Birkh¨ auser Verlag 135 (1996). [3] T. Adachi and H. Tamura, “Asymptotic completeness for long-range many-particle systems with Stark effect”, Commun. Math. Phys. 174 (1996) 537–559. [4] E. B. Davies, Spectral Theory and Differential Operators, Cambridge Univ. Press, 1995. [5] J. Derezi´ nski, “Asymptotic completeness for N-particle long-range quantum systems”, Ann. Math. 138 (1993) 427–476. [6] C. G´erard, “Sharp propagation estimates for N-particle systems”, Duke Math. J. 67 (1992) 483–515. [7] C. G´erard, H. Isozaki and E. Skibsted, “N-body resolvent estimates”, J. Math. Soc. Japan 48 (1996) 135–160. [8] G. M. Graf, “Asymptotic completeness for N-body short-range quantum systems: a new proof”, Commun. Math. Phys. 132 (1990) 73–101. [9] I. Herbst, J. S. Møller and E. Skibsted, “Spectral analysis of N-body Stark Hamiltonians”, Commun. Math. Phys. 174 (1995) 261–294. [10] I. Herbst, J. S. Møller and E. Skibsted, “Asymptotic completeness for N-body Stark Hamiltonians”, Commun. Math. Phys. 174 (1996) 509–535. ´ [11] B. Helffer and J. Sj¨ ostrand, “Equation de Schr¨ odinger avec champ magnetique et ´equation de Harper”, Lecture Notes in Physics, Springer Verlag 345 (1989) 118–197. [12] W. Hunziker and I. M. Sigal, “Time-dependent scattering theory of N-body quantum systems”, Mathematical Physics Preprint Archive (1997) 97–406.
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[13] H. Isozaki, “A generalization of the radiation condition of Sommerfeld for N-body Schr¨ odinger operators”, Duke Math. J. 74 (1994) 557–584. [14] H. Isozaki, “Multi-dimensional inverse scattering theory for Schr¨ odinger operators”, Rev. Math. Phys. 8 (1996) 591–622. [15] H. Isozaki and H. Kitada, “A remark on the micro-local resolvent estimates for 2-body Schr¨ odinger operators”, Publ. RIMS Kyoto Univ. 21 (1985) 889–910. [16] A. Jensen, “Propagation estimates for Schr¨ odinger-type operators”, Trans. Am. Math. Soc. 291 (1985) 129–144. [17] W. J¨ ager, “Ein gew¨ ohnlicher differential-operator zweiter ordnung f¨ ur funktionen mit werte in einem Hilbertraum”, Math. Z. 113 (1970) 68–98. [18] E. Mourre, “Absence of singular continuous spectrum for certain self-adjoint operators”, Commun. Math. Phys. 78 (1981) 391–408. [19] E. Mourre, “Op´erateurs conjug´e et propri´et´es de propogations”, Commun. Math. Phys. 91 (1983) 279–300. [20] P. Perry, I. M. Sigal and B. Simon, “Spectral analysis of N-body Schr¨ odinger operators”, Ann. Math. 114 (1981) 519–567. [21] W. Rudin, Functional Analysis, McGraw-Hill, 1974. [22] Y. Sait¯ o, “The principle of Limiting Absorption for second-order differential equations with operator valued coefficients”, Publ. Res. Inst. Math. Sci. 7 (1972) 581–619. [23] E. Skibsted, “Propagation estimates for N-body Schr¨ odinger operators”, Commun. Math. Phys. 142 (1991) 67–98. [24] A. Sommerfeld, “Die Greenche funktionen der schwingungsgleichung”, Jahrsber. Deutch. Math. Verein. 21 (1912) 309–353. [25] I. M. Sigal and A. Soffer, “Local decay and propagation estimates for time-dependent and time-independent Hamiltonians”, preprint Princeton, 1988. [26] H. Tamura, “Spectral and scattering theory for 3-particle Hamiltonians with Stark effect: Asymptotic completeness”, Osaka J. Math. 29 (1992) 135–159. [27] F. Treves, Introduction to Pseudodifferential Operators and Fourier Integral Operators Volume 2, Plenum Press, 1980. [28] A. Vasy, “Asymptotic behaviour of generalized eigenfunctions in N-body scattering”, J. Func. Anal. 148 (1997) 170–184. [29] A. Vasy, “Propagation of singularities in three-body scattering”, Ast´erisque No. 262 (2000) vit151. [30] X. P. Wang, “Micro-local resolvent estimates for N-body Schr¨ odinger operators”, J. Fac. Sci. Univ. Tokyo 40 (1993) 337–385.
JAUCH PIRON STATES AND
σ -ADDITIVITY L. J. BUNCE Department of Mathematics Reading University Whiteknights, Reading RG6 AX UK
JAN HAMHALTER Czech Technical Univ. Faculty of Electrical Engineering Department of Mathematics Technicka 2, 166 27 Prague 6 Czech Republic E-mail:
[email protected] 1991 Mathematics Subject Classification: 46L30, 46L50, 46L60 Received 6 May 1998 Revised 14 June 1999 We study σ-additivity of the physically plausible Jauch–Piron states on a von Neumann algebra M . Amongst other consequences we extend our earlier results [5] by showing that geometric conditions much weaker than pureness imply σ-additivity for a Jauch–Piron state and, further, that if M is properly infinite and the continuum hypothesis is assumed to be true then all Jauch–Piron factor states are σ-additive.
1. Introduction A state % on a von Neumann algebra which, in the context of quantum mechanics, satisfies the physically plausible condition that % vanishes on e ∨ f whenever it vanishes on projections e and f , is said to be Jauch–Piron state. Thus, given that projections are interpreted as yes-no questions of the ambient physical system, it has been suggested that a state should of necessity satisfy the Jauch–Piron condition in order to qualify as a physical state. A convenient mathematical idealization of the Jauch–Piron condition is that of σ-additivity. However, whereas σ-additive states are always Jauch–Piron states, the converse is far from being true. So it seems both mathematically and physically desirable to determine conditions under which Jauch–Piron states are σ-additive. Initiated in [18], the study of mathematical implications of the Jauch–Piron condition has been pursued extensively for more general systems than are gone in to here (see [17]). For relevant discussion of quantum mechanical proposition system the reader is referred to [3, 11, 12, 17]. For Jauch–Piron states on von Neumann algebras key results were obtained by Amann [3]. In particular, for a von Neumann algebra with separable predual and without abelian part, Amann showed that all pure Jauch–Piron states are normal. 767 Reviews in Mathematical Physics, Vol. 12, No. 6 (2000) 767–777 c World Scientific Publishing Company
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Amongst further developments [5, 7, 10] was the extension [5] that Jauch–Piron pure states are σ-additive for all von Neumann algebras without abelian part. In the present article we extend this in two ways. We show that a geometric condition much weaker that pureness forces Jauch–Piron states to be σ-additive. For a properly infinite von Neumann algebra M we further show that if M is σ-finite or (without any restriction upon the “size” of M ) if the continuum hypothesis is assumed to be true, then all Jauch–Piron factor states are σ-additive. We also discuss Jauch–Piron factor states on finite von Neumann algebras (without appeal to the continuum hypothesis) giving mild conditions for σ-additivity if M is homogeneous Type I and exhibiting counter examples if M is otherwise. 2. Notation and Preliminaries In all that follows M denotes a von Neumann algebra and S(M ) denotes the set of all states of M . The G.N.S. data corresponding to a state % will be denoted by (π% , H% , x% ). We regard M as a C ∗ -subalgebra of M ∗∗ in the usual way. Accordingly, we identify S(M ) with the set of normal states of M ∗∗ . Definition 2.1. A state % ∈ S(M ) is said to be: (a) a Jauch–Piron state if %(e ∨ f ) = 0 whenever e and f are projections in M with %(e) = %(f ) = 0; P P (b) σ-additive if %( en ) = %(en ) whenever (en ) is a sequence of orthogonal projections in M ; (c) countably singular if there exists a sequence (en ) of orthogonal projections in P M with en = 1 and %(en ) = 0 for all n. The set of all Jauch–Piron states, all σ-additive states and all countably singular states on M , are, respectively, denoted by SJ (M ), Sσ (M ), Ssσ (M ). Every tracial state of M is a Jauch–Piron state as follows from the equivalence e∨f −f ∼ e−e∧f. In particular, if M is abelian then all states are Jauch–Piron states. Let R ⊂ S(M ) and let co(R) be the convex hull of R. The face of S(M ) generated by R is the set of all τ ∈ S(M ) such that, for some 0 < λ ≤ 1, λτ is majorized by an element of co(R). A face F of S(M ) is said to be a split face if there is a face F # of S(M ) for which S(M ) is the direct convex sum of F and F # . The face F # is uniquely determined by F and is called the complement of F . By [2, Theorem II.5], the split faces of S(M ) are precisely the sets Fz = {% ∈ S(M ) : %(z) = 1} where z is a central projection in M ∗∗ , and Fz# = F1−z . In particular, split faces are norm closed. Given % ∈ S(M ) we let E% denote the norm closure of the set of transformed states {%(a∗ a)−1 %a : a ∈ M, %(a∗ a) 6= 0} where %a is given by %a (x) = %(a∗ xa). Lemma 2.1. Let F be a norm closed face of S(M ). Then the following conditions are equivalent : (i) F is a split face ;
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(ii) E% ⊂ F for each % ∈ F ; (iii) %u ∈ F for each % ∈ F and each unitary u in M. Proof. By [2, Theorem II.5], F = {% ∈ S(M ) : %(e) = 1} for some unique projection e of M ∗∗ . If F is a split face of S(M ) then e is a central projection of M ∗∗ so that, for a in M , %a (1 − e) = %((a∗ a)(1 − e)) = 0 by the Cauchy–Schwarz inequality giving the condition (ii). On the other hand, if the condition (ii) holds so does the condition (iii) which, by the uniqueness of e, implies u∗ eu = e for all unitaries u of M . By the Kaplansky Density Theorem, this gives u∗ eu = e for all unitaries u of M ∗∗ so that e is central in M ∗∗ implying that F is a split face. The set of all normal states of M is a split face of S(M ) with complement equal to the set of all singular states of M [20, p. 127]. As the uniform limit of a sequence of σ-additive states of M is normal on the σ-finite von Neumann subalgebra generated by an orthogonal sequence of projections in M , it follows that Sσ (M ) is norm closed. It is readily seen that Sσ (M ) is a face of S(M ), and Sσ (M ) clearly satisfies condition (iii) of Lemma 2.1. Therefore Sσ (M ) is a split face. If e and f are projections in M and x = 12 (e + f ) there exists an increasing sequence (en ) of spectral projections of x with x ≥ n1 en for all n, and en % e ∨ f . So if % ∈ S(M ) is σ-additive with %(e) = %(f ) = 0, then %(e ∨ f ) = 0. Therefore, Sσ (M ) ⊂ SJ (M ). In the next section we show that Sσ (M ) is the largest split face of M contained in SJ (M ). 3. Jauch Piron Split Faces Throughout this section M continues to denote a von Neumann algebra, SJ (M ) its class of Jauch–Piron states and Sσ (M ) its class of σ-additive states. Proposition 3.1. The norm closure SJ (M ) of SJ (M ) is a split face of S(M ). Its complement #
SJ (M ) ⊂ {% ∈ S(M ) : %(e) = 0 f or all σ-f inite projections e of M } . Proof. It is easy to see that SJ (M ) is convex. Let τ lie in the face F of S(M ) generated by SJ (M ). Thus for some 0 < λ ≤ 1 and Jauch–Piron state % we have λτ ≤ %. If λ = 1 then τ = %. For λ 6= 1 we get % = λτ + (1 − λ)σ ,
where σ = (1 − λ)−1 (% − λτ ) ∈ S(M ) .
We note that every proper convex sum, ατ + (1 − α)σ, of τ and σ must be a ∗ Jauch–Piron state. Letting α → 1 we have τ ∈ SJ (M ). The order ideal of M+ , R+ F = {λ% : λ ≥ 0, % ∈ F }, has norm closure R+ F . By [9, Corollary 4.7], R+ F is an order ideal and so F is a face of S(M ). It follows that SJ (M ) = F . Further, as is clear, SJ (M ) is unitarily invariant. Therefore, SJ (M ) is unitarily invariant and so is a split face by Lemma 2.1.
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Let % ∈ S(M ) and suppose that %(e) 6= 0 for some σ-finite projection e of M . Let τ be a normal state with support equal to e and put ϕ = 1/2(% + τ ). If p and q are projections in M with ϕe (p) = ϕe (q) = 0, then ep = eq = 0 as τ is faithful on eM e so that ϕe (p ∨ q) = 0. Therefore, ϕ(e)−1 ϕe is a Jauch–Piron state. Hence, %(e)−1 %e ∈ SJ (M ), by the first part of the proof. Let J be the norm closed ideal of M generated by the σ-finite projections of M . As the set of all σ-finite projections is hereditary and closed under formation of finite suprema and von Neumann equivalence, it follows from [21, Theorem 19] that every projection in J is σ-finite. The state space, S(J), of J identifies with {% ∈ S(M ) : %(z) = 1} where z is the central projection in M ∗∗ with J ∗∗ = M ∗∗ z. It is enough to show that S(J) ⊂ SJ (M ). Let % ∈ S(J). By spectral theory, the σ-finite projections of M form an approximate unit (fλ ) for J. As %(z) = 1 we have π ˜% (z) = 1 where π ˜% : M ∗∗ → B(H% ) is the normal extension of π% . Hence, as fλ → z strongly in M ∗∗ , there exists a sequence (en ) of projections in J such that π% (en )x% → x% and %(en ) 6= 0 for all n. By the argument of [13, p. 309], or as repeated in [8, p. 41], k%(en )−1 %en − %k → 0. Hence, by the preceeding paragraph, % ∈ SJ (M ). We note that Proposition 3.1 implies that when M is σ-finite the set of Jauch–Piron states is norm dense in the state space of M . Corollary 3.1. The following statements are equivalent: (i) (ii) (iii) (iv)
SJ (M ) is a face of S(M ) ; SJ (M ) is norm closed ; SJ (M ) = S(M ) ; The non-abelian part of M is finite dimensional.
Proof. The implication (ii)⇒(i) follows from Proposition 3.1, whilst (iii)⇒(ii) is obvious. The equivalence, (iii)⇔(iv), was proved in [5, 2.2]. We shall show that (i)⇒(iv). Let SJ (M ) be a face in S(M ). We may suppose that M has no non-zero abelian part. Let e be a σ-finite projection of M and let % be a state of eM e. Put ϕ = 1/2(% + τ ), where % is the state extension of % to M and where τ is a normal state of M with support equal to e. As in the proof of Proposition 3.1, ϕe is a Jauch– Piron state. Therefore % ∈ SJ (M ), by assumption. Hence, % is a Jauch–Piron state of eM e. Thus, by the equivalence of (iii) and (iv), we have that for each σ-finite projection e of M , eM e has finite-dimensional non-abelian part. We note therefore that M must be Type I and that, in turn, by consideration of homogeneous decomposition [20, V. 1.27] it must be finite dimensional. In the following, for % ∈ S(M ), F% denotes the smallest split face of S(M ) containing %. The notation E% was defined in Sec. 2.
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Proposition 3.2. The following statements are equivalent for % ∈ S(M ), where M has no non-zero abelian part: (i) F% ⊂ SJ (M ) ; (ii) E% ⊂ SJ (M ) ; (iii) % is σ-additive. Proof. By Lemma 2.1, E% ⊂ F% , giving (i)⇒(ii). On the other hand if % is σ-additive, then F% ⊂ Sσ (M ) ⊂ SJ (M ) (F% is an intersection of all split faces containing %) and (iii)⇒(i) follows. It remains to prove (ii)⇒(iii). Let E% ⊂ SJ (M ). Then, by [13, Theorem C] ωh π% is a Jauch–Piron state of M for every unit vector h of H% (ωh denotes a vector state ωh (a) = (ah, h)). Let e and f be projections of M . We have π% (e) ∨ π% (f ) ≤ π% (e ∨ f ). Suppose that π% (e) ∨ π% (f ) 6= π% (e ∨ f ) and let h be a unit vector in (π% (e ∨ f ) − π% (e) ∨ π% (f ))H% . Then ωh π% (e) = ωh π% (f ) = 0 and ωh π% (e ∨ f ) = 1 which implies that ωh π% is not a Jauch–Piron state of M in contradiction to the previous remark. Therefore, π% (e ∨ f ) = π% (e) ∨ π% (f ). Hence, π% is σ-additive by [6]. Therefore, % is σ-additive. Corollary 3.2. If M is a von Neumann algebra with no non-zero abelian part then Sσ (M ) is the largest split face of S(M ) contained in SJ (M ). Proposition 3.2 both extends [5, 5.3] and at the same time provides a transparent alternative proof which we shall now give. Corollary 3.3. Let M have no non-zero abelian part. Then every Jauch–Piron pure state of M is σ-additive. Proof. Let % be a pure state of M . By [13, Theorem C] E% = {ωh π% : ωh is a vector state corresponding to a unit vector h ∈ H% } . Since π% is irreducible each element of E% is a pure state equivalent to % and therefore unitarily equivalent to % (see e.g. [15, Theorem 10.2.6, p. 730]). So E% = {%u : u is unitary in M }. Hence, E% ⊂ SJ (M ) if % ∈ SJ (M ). We remark that, the Jauch–Piron pure states of M are precisely the extreme points of the convex set SJ (M ). In fact, if % is an extreme point of SJ (M ) and % = 1 1 2 3 2 2 (%1 + %2 ) where %1 and %2 are states of M , then τ1 = 3 %1 + 3 %2 and τ2 = 5 %1 + 5 %2 3 5 are Jauch–Piron states with % = 8 τ1 + 8 τ2 . Therefore τ1 = τ2 so that %1 = %2 , and % is a pure state of M . 4. Countable Singularity Proposition 4.1, below, which is approached by a sequence of lemmas, shall be required in the final section. By [20, p. 127] when the von Neumann algebra M is
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σ-finite S(M ) is the direct convex sum of Sσ (M ) and Ssσ (M ), giving that Ssσ (M ), the countably singular states (see Sec. 2), is the split face complement of Sσ (M ). Without any restriction on the size of M we are able to show this with the aid of the continuum hypothesis. By the continuum hypothesis we mean that each subset of R is either countable or has the cardinality of the continuum. (We believe the continuum hypothesis to be purely technical mathematical condition acceptable from the physical standpoint because it cannot affect any real physical experiment.) The first result, however, is independent of such set theoretical considerations. Lemma 4.1. Sσ (M )# = Fsσ , the smallest split face containing all countably singular states of M. Proof. As Sσ (M ) is a split face of S(M ) and no countably singular state can majorize a positive multiple of a σ-additive state we have that Ssσ (M ) ⊂ Sσ (M )# . P P # . If % is not σ-additive then en = 1 and %(en ) < 1, for some Let % ∈ Fsσ P sequence (en ) of projections in M . In which case, f = π% (en ) 6= 1 in B(H% ) and there is a unit vector h ∈ (1 − f )H% with ωh π% (en ) = 0 for all n so that # ωh π% ∈ Ssσ (M ) ⊂ Fsσ . This leads to a contradiction because ωh π% ∈ E% ⊂ Fsσ , the first inclusion being given by [13, Theorem C] and the second by Lemma 2.1. We shall need certain results of [1, Chap. 6]. Lemma 4.2 [1, Lemma 6.2]. Assume that the continuum hypothesis is true. Let µ be finitely additive probability measure on the power set of R vanishing on all S singletons. Then there exists a sequence (An ) of disjoint subsets of R with R = An and µ(An ) = 0 for all n. M is said to be countably generated if it is generated as a von Neumann algebra by a sequence of projections. Lemma 4.3. Assume that the continuum hypothesis is true and let M be countably generated. Then all σ-additive states of M are normal and all singular states are countably singular. Proof. Let (eλ )λ∈Λ be a family of orthogonal projections in M . We claim that (without call upon the continuum hypothesis) countable generation of M implies card(Λ) ≤ card(R). Indeed, to borrow an idea from [19, Theorem 1], for each λ choose a normal state %λ of M with support projection majorized by eλ to obtain a mutually distinct family (%λ )λ∈Λ in M∗ . But M contains a norm separable weak∗ dense C ∗ -subalgebra, A. Since A∗ has cardinality at most that of R and M is isometric to a weak∗ closed ideal of A∗∗ , so that M∗ is isometric to a subspace of A∗ , the claim follows. Suppose that % is a state of M and that (eλ )λ∈R is a family of orthogonal P projections in M with eλ = 1 and %(eλ ) = 0 for all λ. Then, as observed in
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P [1, Lemma 6.4 (2)] with eA = λ∈A eλ for A ⊂ R, µ(A) = %(eA ) defines a finitely additive probability measure on the power set of R vanishing on all singletons. Therefore, by Lemma 4.2, % is countably singular. It follows that all singular states are countably singular giving immediately that all σ-additive states are normal. Proposition 4.1. If M is σ-finite or if the continuum hypothesis is assumed to be true then Ssσ (M ) = Sσ (M )# . Proof. Assume that the continuum hypothesis is true. Let σ and τ be countably singular states on M and let (en ) and (fn ) be sequences of projections in M with P P en = fn = 1 and σ(en ) = τ (fn ) = 0 for all n. Let N be the von Neumann subalgebra of M generated by all of the en and fn . As σ and τ are singular on N so is any convex combination, ϕ, of σ and τ . As N is countably generated ϕ restricts to a countably singular state of N , by Lemma 4.3. Therefore, ϕ is a countably singular state of M , proving that Ssσ (M ) is convex. It follows immediately that Ssσ (M ) is a face of S(M ). It remains to show that Ssσ (M ) is norm closed. Let % be the uniform limit of a sequence (%n ) of countably singular states of M . For each n we have a sequence (emn ) of orthogonal projections in M with sum 1 and %n (emn ) = 0 for all m. Passing to the countably generated von Neumann subalgebra generated by {emn : m, n ∈ N } and using Lemma 4.3 as before, we see that % must be countably singular. By this together with Lemma 2.1 and Lemma 4.1, Ssσ (M ) = Sσ (M )# . If M is abelian and % is a pure state of M then % restricts to a two-valued measure on projections of M . Therefore if % is not σ-additive then there exists a P en = 1 and %(en ) = 0 sequence (en ) of orthogonal projections in M such that for all n. So we have the following lemma needed later which makes no claim upon continuum hypothesis. Lemma 4.4. If M is abelian and % is a pure state of M, then % is either σ-additive or countably singular. We remark that Lemma 4.4 can be viewed as a special case of more general result of classical measure theory (see e.g. [4, Theorem 2.1]) saying that any state # % ∈ Sσ (M ) for M abelian is a σ-convex sum of countably singular states. The following is well known (one argument is given in [5, 2.5 (b)]). Lemma 4.5. If % is a state of M and (en ) is a sequence of orthogonal projections P in M with %(en ) = 0 for all n, then %( ∞ i=1 eni ) = 0 for some strictly increasing sequence (ni ). 5. Jauch Piron Factor States Throughout this section Z(M ) denotes the centre of the von Neumann algebra M . A state % is factor state if and only if %(z) = 1 for some minimal central projection
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z of M ∗∗ . Thus, by Proposition 4.1 (together with [20, p. 127]) if M is σ-finite or if the continuum hypothesis is assumed to be true, a factor state of M is either σ-additive or countably singular. We investigate Jauch–Piron factor states on a case by case basis. Every finite M with infinite dimensional centre possesses a Jauch–Piron factor state that is countably singular on Z(M ) (below). This tends to suggest that σ-additivity on the centre is key for Jauch–Piron factor states to be σ-additive. In the Type I finite case this has nothing to do with the Jauch–Piron condition, however, as the following result demonstrates. Proposition 5.1. Let M be a Type I finite von Neumann algebra and let % be a factor state of M such that % is σ-additive on Z(M ). Then % is σ-additive on M. Proof. As % is a σ-additive pure state of Z(M ) and M is a direct sum of countably many homogeneous summands, we may suppose that M is Type In for some n < ∞. Given a sequence (fj ) of orthogonal projections in M , let Ω be a maximal abelian von Neumann subalgebra of M containing all of the fj . By [14, 3.7], there exist orthogonal and equivalent abelian projections e1 , . . . , en of M contained in Ω and with sum 1. For each i = 1, . . . , n we have ei M ei = Z(M )ei and %(zei ) = %(z)%(ei ) for all z in Z(M ), the latter following from [20, IV, 4.11]. Further, if (zk ) is a sequence of projections in Z(M ) such that (zk ei ) is orthogonal for some i, then (zk ) is orthogonal as ei has central cover 1. Therefore, for each i = 1, . . . , n, % is P∞ P∞ Pn σ-additive on ei M ei so that j=1 %(ei fj ) = %( j=1 ei fj ). As i=1 ei = 1, we have P∞ P∞ j=1 %(fj ) = %( j=1 fj ). Corollary 5.1. A factor state on Type I finite von Neumann algebra is either σ-additive or countably singular. Proof. This follows from Lemma 4.4 together with Proposition 5.1.
A positive linear functional ϕ of M is said to be regular if whenever (xn ) is a sequence of self-adjoint elements in M with strong limit x and %(x2n ) = 0 for all n then %(x2 ) = 0 (see [5] for several equivalent conditions). Theorem 5.1. Let M = C(X) ⊗ Mn (C), where 2 ≤ n < ∞ and X is a compact hyperstonean space, and let % be a factor state of M. (a) If % is a Jauch–Piron state and is not faithful on 1⊗Mn(C), then % is σ-additive. (b) If % is faithful on 1 ⊗ Mn (C), then % is a Jauch–Piron state. Proof. Via [20, IV, 4.11] we see that % is a product state, % = %1 ⊗ %2 , where %1 is a pure state of C(X). (a) Let % be a Jauch–Piron state and suppose that %2 is not faithful. Choose minimal projections e and f of Mn (C) with %2 (e) = 0 and %2 (f ) 6= 0. Put N = (e ∨ f )Mn (C)(e ∨ f ), e = 1 ⊗ e, f = 1 ⊗ f and p = e ∨ f . As % is a Jauch–Piron
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state with %(e) = 0 and e ∼ f we have that τ = %(p)−1 % is a regular state of C(X) ⊗ N = pM p, by [5, 4.2. (i)]. Therefore, % is a regular pure state of Z(M ) and so is σ-additive on Z(M ). Hence, % is σ-additive on M , by Proposition 5.1. (b) Let %2 be faithful. By compactness there is an α > 0 such that %2 ≥ α on the positive part of the unit sphere of Mn (C). Let e be a projection of M . Via the identification C(X) ⊗ Mn (C) = C(X, Mn (C)), the set E = {t ∈ X : e(t) 6= 0} is clopen in X and the central cover, c(e), of e is given by c(e) = χE ⊗ 1. Given 0 < ε < 1, there exists (see [16, Lemma 8.3]) a projection f in M , Pk f = i=1 zi ⊗ fi , where z1 , z2 , . . . , zk are orthogonal projections of C(X) and f1 , . . . , fk are non-zero projections of Mn (C) such that ke − f k < ε. We have Pk i=1 zi ⊗ 1 = c(f ) = c(e). Therefore, %(e) > %(f ) − ε =
k X
%1 (zi )%2 (fi ) − ε ≥ α%(c(e)) − ε .
i=1
Hence, %(e) ≥ α%(c(e)). Therefore, if p and q are projections in M with %(p) = %(q) = 0, then %(c(p)) = %(c(q)) = 0 so that %(p ∨ q) ≤ %(c(q) ∨ c(q)) = 0, proving that % is a Jauch–Piron state of M . We note that the argument of Theorem 5.1(b) shows that every product state % = %1 ⊗ %2 on C(X) ⊗ Mn (C) for which %2 is faithful is a Jauch–Piron state of M . We give examples which indicate limits to possible improvements upon Proposition 5.1, Theorem 5.1 and Theorem 5.2 (below). P∞ Examples 5.1. (a) Let M = n=2 Mn where each Mn is homogeneous of Type In (n ≥ 2) or where each Mn is of Type II1 . Let J be a maximal ideal containing each Mn . Then [20, V, 5.2] M/J can be realized as a Type II1 factor and so has a unique tracial state ϕ. Now, ϕ is a factor state and so therefore is ϕ = ϕ ◦ π where π : M → M/J is the canonical map. As ϕ is a trace of M it is a Jauch–Piron state. Therefore, ϕ is a countably singular Jauch–Piron factor state of M . (b) Let M = Z ⊗ Mn (C), where Z is an infinite dimensional abelian von Neumann algebra. Let % = %1 ⊗ τ , where %1 is pure and countably singular and τ is the normalized trace. Then % is a countably singular factor Jauch–Piron state. Theorem 5.2. If M is Type II1 and % is a Jauch–Piron factor state such that % is normal on Z(M ), then % is normal on M. Proof. This follows from [5, 4.4].
Regarding Theorem 5.2 we have been unable to prove that, for a Jauch–Piron factor state of M , σ-additivity on the centre implies σ-additivity on M in the case when M is a direct summand of uncountably many Type II1 algebras.
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Lemma 5.1. Let % be a Jauch–Piron state of M and let e and f be projections of M with %(e) = 0 and e ∼ f. Let % be countably singular and suppose either that M is σ-finite or that the continuum hypothesis is true. Then %(f ) = 0. Proof. By [5, 4.2. (i)] % is regular on (e ∨ f )M (e ∨ f ) and hence is regular on f M f . If %(f ) 6= 0, then %(f )−1 %f is countably singular by Lemma 2.1 together with Proposition 4.1. In which cases, there exists a sequence of projections (en ) P Pn in M with en = 1 and %(f en f ) = 0, for all n. But then f ( i=1 ei )f converges strongly to f so that %(f ) = 0. Theorem 5.3. Let M be properly infinite. Suppose that M is σ-finite or that the continuum hypothesis is true. Then all Jauch–Piron factor states of M are σ-additive. Proof. Let % be a Jauch–Piron factor state of M . Suppose that % is countably P singular. Then en = 1 and %(en ) = 0 for all n, for some sequence (en ) of orthogonal projections in M . As M is properly infinite, there exists a sequence (pn ) of orthogonal projections Pn in M such that pn ∼ i=1 ei for each n. By Lemma 5.1, %(pn ) = 0 for all n. Consequently, by Lemma 4.5 there exists a strictly increasing sequence (ni ) of natural P∞ P∞ numbers such that %( i=1 pni ) = 0. We have, i=1 pni ∼ 1. Now Lemma 5.1 implies the contradiction that %(1) = 0. Therefore, % cannot be countably singular and so must be σ-additive. Acknowledgments The second author would like to express his gratitude to the Alexander von Humboldt Foundation, Bonn, and Prof. K˝olzow, Erlangen, for the support of his research and encouragement. He also acknowledges the support of the Grant Agency of the Czech Republic (Grant No. 201/96/0117) and the Grant J04/98/210000010 of the Czech Technical University. References [1] C. A. Akemann and J. Anderson, “Lyapunov theorems for operator algebras”, Memoirs of the Amer. Math. Soc. 458 (1991). [2] E. M. Alfsen and F. W. Shultz, “Non-commutative spectral theory for affine functions spaces on convex sets”, Memoirs of the Amer. Math. Soc. 172 (1976). [3] A. Amann, “Jauch–Piron states in W ∗ -algebraic quantum mechanics”, J. Math. Phys. 28(10) (1989) 2384–2389. [4] W. C. Bell, “A note on pure charges”, Bolletino U.M.I. 7 (1989) 147–154. [5] L. J. Bunce and J. Hamhalter, “Jauch–Piron states on von Neumann algebras”, Math. Zeitschrift 215 (1994) 491–502. [6] L. J. Bunce and J. Hamhalter, “Countably additive homomorphisms between von Neumann algebras”, Proc. Amer. Math. Soc. 123 (1995) 3437–3441. [7] L. J. Bunce and J. Hamhalter, “Extension of Jauch–Piron states on Jordan algebras”, Math. Proc. Camb. Phil. Soc. 189 (1996) 279–286.
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σ-ADDITIVITY
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[8] J. Dixmier, “C ∗ -Algebras”, North Holland, 1977. [9] E. G. Effros, “Order ideals in a C ∗ -algebra and its dual”, Duke Math. J. 30 (1963) 391–412. [10] J. Hamhalter, “Pure Jauch–Piron states on von Neumann algebras”, Ann. Inst. Henri Poincar´e 52 (1993) 173–187. [11] J. M. Jauch, Foundations of Quantum Mechanics, Addison-Wesley, 1986. [12] J. M. Jauch and C. Piron, “On the structure of quantum proposition systems”, Helv. Phys. Acta 42 (1969) 842–848. [13] R. V. Kadison, “States and representations”, Trans. Amer. Math. Soc. 103 (1962) 304–319. [14] R. V. Kadison, “Diagonalizing matrices”, Amer. J. Math. 106 (1984) 1451–1456. [15] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, Inc., 1986. [16] S. Maeda, “Probability measures on projections in von Neumann algebras”, Rev. Math. Phys. 1 (1990) 235–290. [17] P. Pt´ ak and S. Pulmannov´ a, Orthomodular Structures as Quantum Logics, Kluwer Academic Publ., 1991. [18] G. R˝ utimann, “Jauch–Piron states”, J. Math. Phys. 18(2) (1977) 189–193. [19] H. Takemoto, “On the homomorphisms of von Neumann algebra”, Tohoku Math. J. 21 (1969) 152–157. [20] M. Takesaki, Theory of Operator Algebras I, Springer Verlag, 1989. [21] D. Topping, “Jordan algebras of self-adjoint operators”, Memoirs of the Amer. Math. Soc. 53 (1965).
LOW TEMPERATURE PROPERTIES OF THE BLUME EMERY GRIFFITHS (BEG) MODEL IN THE REGION WITH AN INFINITE NUMBER OF GROUND STATE CONFIGURATIONS ˜ A. BRAGA and PAULO C. LIMA GASTAO Departamento de Matem´ atica-ICEX-UFMG Caixa Postal 1621 30161-970, Belo Horizonte MG, Brazil
MICHAEL L. O’CARROLL Departamento de F´ısica-ICEX-UFMG Caixa Postal 702 30161-970, Belo Horizonte MG, Brazil Received 14 October 1998 Revised 14 May 1999 For the low temperature Blume–Emery–Griffiths Z d , d ≥ 2, lattice model taking site spin values 0, +1, −1 we construct, using a polymer expansion, two pure states in the parameter region A where there are an infinite number of configurations with minimal energy. Each state is invariant under translation by two lattice spacings and the two states are related by a unit translation. Using analyticity techniques we show that the truncated n-point function decays exponentially with an n-independent lower bound on the decay rate. For the truncated two-point function, we find the exact exponential decay rate in the limit β → ∞. Keywords: Classical spin model, BEG model, pure phases, infinitely many ground state configurations, correlation functions.
1. Introduction and Results Here we obtain low temperature properties (β large) of the Z d , d ≥ 2, lattice discrete Blume–Emery–Griffiths (BEG) model with formal Hamiltonian: H(σ) = −J
X
[σi σj + yσi2 σj2 + x(σi2 + σj2 − 1)]
hi,ji
≡
X
Hi,j (σ)
(1)
hi,ji
and Gibbs factor exp(−βH). In this model, the spin variable σi , at the lattice site i ∈ Z d , d ≥ 2, takes values σi = 0, +1, −1, the parameters satisfy J > 0, −∞ < x, y < +∞ and the sum is over nearest neighbor pairs. Without loss of generality, we will assume J = 1. The BEG model [1] first appeared to describe mixtures of liquid Helium. Since then, it has been exhaustively studied by theoretical physicists. 779 Reviews in Mathematical Physics, Vol. 12, No. 6 (2000) 779–806 c World Scientific Publishing Company
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
In recent works, using Mean Field, Monte Carlo and approximate renormalization group techniques, phase diagrams have been found to have a very rich structure [2–6]. Rigorous results have been obtained in [7], where the number of pure phases for the 2-d model has been obtained using a generalized Pirogov–Sinai theory (for the standard theory, see [8]), and in [9, 10], where the two-point function is studied by correlation inequality techniques. This paper is devoted to the investigation of the low temperature states of the BEG model in a region of the phase diagram with a degeneracy of ground states. In its general part, it can be considered to be an illustration of the general theory extending Pirogov–Sinai theory to the case with degenerate ground states [11, 12]. Here we use the techniques developed in [13] to obtain low temperature properties of the model, in particular, a more precise decay rate for the correlation functions. The ground state degeneracy occuring here is rather simple — a similar situation in different model was discussed in [14]. Instead of using the notion of restricted ensembles from [11], we introduce the notion of equivalence classes introduced in [7]. The core of the paper is in the explicit evaluation of leading terms for the exponential decay of the two-point functions (Theorem 1.2). We develop a variant of the hyperplane separating technique [15] to obtain the rate of exponential decay. To understand the low temperature properties it is important to know the low energy configurations. We have written the Hamiltionian as a sum over nearest neighbors pairs. We denote the energy of a pair by Ess0 where s, s0 = 0, +1, −1 and decompose the parameter space according to the lowest pair energy. The regions and corresponding lowest pair energies are given by F: 1 + y + 2x > 0 and 1 + y + x > 0; E++ = E−− , D: 1 + y + 2x < 0 and x < 0: E00 , A: 1 + y + x < 0 and x > 0: E0+ = E0− = 0. In the region F (called ferromagnetic) there are two infinite lattice configurations with minimum energy, namely all sites have spin +1 or −1; for D (called disordered) there is only one configuration, namely spin zero at all sites. It is expected that states corresponding to the regions F and D can be constructed using low temperature expansion techniques as found in [16, 17]. A more difficult problem is the construction and properties of the states on the boundary of the regions as for instance in the ferrimagnetic phase region reported in [4] for d = 3, but not for d = 2, and non-zero but low temperature. In A the lowest pair excitation energies are E++ for 1 + y + 2x > 0 and E00 for 1 + y + 2x < 0; the excitation energies go to zero as the boundary of the region is approached. In this paper we consider the region A (called anti-quadrupolar). In this region there is an infinite number of configurations which have minimal energy which is zero, i.e., σi = 0 for i ∈ Le , the even sublattice, defined as Le ≡ {j ∈ Z d /j1 +· · ·+jd is even} and σi = +1, −1 for i ∈ Lo , the odd sublattice, defined as Lo ≡ {j ∈ Z d /j1 + · · · + jd is odd} as well as σi = 0 for i ∈ Lo and σi = +1, −1 for i ∈ Le . These ground state configurations will be split into the GA and GB equivalence classes (see [7]). GA (respectively GB ) is made of ground state configurations for which σi = 0 for i ∈ Le (respectively σi = 0 for i ∈ Lo ).
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
781
Starting from the finite lattice Λ = ([−L, L] ∩ Z)d, L even, and imposing boundary conditions σi = 0 for i on the boundary and i ∈ Le , i.e. the even boundary point, and σi = +1, −1 for the odd boundary point, we construct an infinite volume pure state α using a polymer expansion, where α is invariant under lattice translations by Le . From the state α another state is obtained by a unit lattice translation. We show that the n-point truncated correlation functions decay exponentially with an n-independent lower bound on the decay rate of order Eβ/2 where E ≡ min{E++ , E00 }. For the truncated two-point function we obtain, under certain conditions, the exact exponential decay rate as β → ∞. We now give a precise definition of the model we consider. For the finite volume partition function ZA (Λ) we write Z ZA (Λ) = exp(−βHΛ (σ))Dµ(σ) (2) where HΛ is H restricted to Λ and Dµ(σ) is the product measure Y dµi (σi ) i∈Λ
where dµi (σ) = δ(σ)dσ for i ∈ ∂Λ ∩ Le , dµi (σ) = [δ(σ − 1) + δ(σ + 1)]dσ for the remaining boundary points and dµi (σ) = [δ(σ) + δ(σ − 1) + δ(σ + 1)]dσ for i ∈ ([−(L − 1), (L − 1)] ∩ Z)d (∂Λ is the set of points in Λ which have at least one nearest neighbor outside Λ). Let # Z "Y hi σi ZA (Λ, h) = e (3) exp(−βHΛ (σ))Dµ(σ) i∈Λ
denote the generating function. In the same way, we define ZB (Λ) and ZB (Λ, h), except for the single spin measures on the boundary of Λ: dµi (σ) = δ(σ)dσ for i ∈ ∂Λ ∩ Lo , dµi (σ) = [δ(σ − 1) + δ(σ + 1)]dσ for the remaining (even) boundary points. ZA (respectively ZB ) can be thought as being a superposition of partition functions whose boundary conditions are taken from the GA equivalence class (respectively GB ). From now on we fix A boundary conditions, but the same results hold true for B boundary conditions. Let h·iΛ be the finite lattice expectation. The n-point truncated correlation function is ∂ ln ZA (Λ, h) T SΛ (i1 , . . . , in ) = hσi1 , . . . , σin iΛ = ∂hi1 , . . . , ∂hin h=0 where i1 , . . . , in ∈ Λ. We state our results on the existence of the free energy and decay of truncated correlation functions as: Theorem 1.1. Let x, y ∈ A. Then, there exists β0 = β0 (x, y) such that for any β > β0 and q = A, B (a) Fq ≡ limΛ→∞
−1 β|Λ|
ln Zq (Λ) exists.
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
(b) Sq (i1 , . . . , in ) ≡ limΛ→∞ SΛ,q (i1 , . . . , in ) exists and for a ∈ Z d there exists constants cn and m > 0 (independent of n) such that, for at least two distinct points |Sq (i1 , . . . , ir−1 , ir + a, . . . , in + a)| ≤ cn e−mda ({ik }) where da ({ik }) is the minimum distance between the groups {i1 , . . . , ir−1 } and {ir + a, . . . , in + a}. Also, limβ→∞ (m/β) = E/2, where E = min{E++ , E00 }. Moreover, Fq and Sq (i1 , . . . , in ) admit convergent expansions in the polymer system representation. To prove this result we obtain a representation of ZA (Λ, h) as a polymer system of outer contours as in [19] (see also [20]). The representation involves h-dependent polymer activities which in turn depend on ratios of partition functions with different boundary conditions in smaller volumes. We obtain bounds on the activities in terms of h = 0 quantities for which the results of [19] apply and allow us to control the ratio of partition functions. In this way we obtain a convergent expansion for (ln ZA (Λ))/|Λ| as well as analyticity in the {hi } for h summable and small. Cauchy estimates control the hi derivatives and the maximum modulus theorem is used to obtain decay of S. Due to possible h dependence of the ratios of partition functions throughout the volume enclosed by a contour in the polymer activities the usual proof of convergence of the polymer expansion (see for example [16, 17]) doesn’t go through. However, by a modified proof (see [13]) we obtain convergence of the expansion for SΛ . For the two-point function we can get finer results. Let m(x, y, d) be defined by the following limit: m(x, y, d) ≡ lim lim − β→∞ |i|→∞
ln|hS0 S(i,0,...,0) i| . β|i|
The next theorem gives the values of m(x, y, d) in terms of the pair energy Ess0 . These values are obtained by developing a perturbation theory around the configurations that give the low temperature main contributions to the two point function. Theorem 1.2. Let d ≥ 2 and x, y be such that E++ (x, y), E00 (x, y) > 0. (1) If 0 < E++ ≤ 2(d − 1)E00 , then m(x, y, d) = dE++ (x, y) . (2) If 0 < 4(d − 1)(2d − 1)E00 < E++ , then m(x, y, d) = E++ + (d − 1)(2d − 1)E00 . (3) If (2d − 1)E00 < E++ < 2(d − 1)E00 , then E++ + 2(d − 1)2 E00 ≤ m(x, y, d) ≤ dE++ (x, y) (4) If 4(d − 1)(2d − 1)E00 ≤ E++ ≤ (2d − 1)E00 , then E++ + 2(d − 1)2 E00 ≤ m(x, y, d) ≤ E++ + (d − 1)(2d − 1)E00 .
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
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Remark 1.1. In the above estimates in items (3) and (4), the upper bounds are the expected ones and they are proven analogously to the proof of Theorem 1.2. Remark 1.2. For the two-point function hS(−1,0,...,0) S(i−1,0,...,0) i we have similar bounds. To obtain a precise decay rate would require a more detailed analysis of the location and multiplicities of the excitation spectrum of the infinite lattice “transfer matrix” as carried out in [15, 21] for low temperature Ising Model. We now describe the organization of this paper. In Sec. 2 we show how the model’s partition function can be represented as a polymer system of outer contours as in [19] and also as a standard polymer system (see [16, 17]). In Sec. 3 we obtain bounds on the h-dependent activities which allow us to apply the results of [13] to obtain the existence and decay of correlation functions, hence proving Theorem 1.1. In Sec. 5 we make some concluding remarks and in Appendix A we prove some analytic properties of the complex correlation function which are used in Sec. 4, where we prove the upper bounds (lower bounds are proven in Appendix B) for the two-point function. 2. A Polymer Representation for ZA (Λ, h) Here we obtain (see Eq. (11) below) a low temperature polymer (see Definition 2.2 of polymer) representation as well as an outer contour representation (see Eq. (4) and [19]) of the Z d , d ≥ 2, lattice discrete Blume–Emery–Griffiths (BEG) model with formal Hamiltonian (1) for parameter values x and y in the anti-quadrupolar region A. In this region there is an infinite number of zero energy ground state configurations, constructed from the zero energy pairs E0+ = E0− = 0, which are split into the GA and GB equivalence classes. To prove our results, we first review some definitions and results from the theory of Pirogov–Sinai–Zahradnik [8, 20]. Let σ be a configuration which is equal to a ground state configuration s except for a finite number of points (then σ = s a.e.). Given such a configuration σ, we say that i ∈ Z d is correct if Hij (σ) = 0 ∀ j such that ki − jk = 1; otherwise, it is said to be non-correct. A correct point is classified as A-correct (if σi = 0 and i ∈ Le or if σi = ±1 and i ∈ Lo ) or as B-correct (if σi = 0 and i ∈ Lo or if σi = ±1 and i ∈ Le ). The union of all non-correct points of a given configuration σ is the boundary of σ and it is denoted by B(σ). A subset Y of Z d is connected if it contains only one point or if it is nearest neighbour bond connected. A contour will be a pair (Γ, suppΓ), where suppΓ is a finite connected subset of d Z and Γ is the restriction of some σ to suppΓ, satisfying the condition that any point i ∈ suppΓ is non-correct and if the bond hi, ji 6⊂ suppΓ but i or j ∈ suppΓ, then Hij (σ) = 0. B(σ) is uniquely written as the union of the contours of σ, where now suppΓ is a finite maximal connected component of B(σ). As usual (see, for instance, [8]), we will use the notation Γ to represent the contour (Γ, suppΓ). If Γ and Θ are two distinct contours, then dist(Γ, Θ) ≥ 2, by the very definition of a contour. Mudan¸ca de nota¸c˜ao para Z d \ suppΓ. The exterior of a contour Γ, written as ExtΓ, is the infinite connnected component of Z d \ suppΓ while the union of all
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
finite connected components of Z d \suppΓ is the interior, written as IntΓ (therefore, ExtΓ and IntΓ are sets of points). Proposition 2.1. Let σ = s a.e. be a given configuration and (Γ, suppΓ) one of its contours. For j ∈ ExtΓ or j in any of the connected components of IntΓ such that dist(j, suppΓ) = 1, j is correct (A-correct or B-correct for all j in the boundary of each connected component of Z d \ suppΓ). The restriction of σ to ∂(ExtΓ) or to the boundary of any of the connected components of IntΓ will be a boundary condition for Γ. Given a contour (Γ, suppΓ), the configuration Γ can be naturally extended to Z d by defining its extension to ExtΓ and IntΓ such that Proposition 2.1 is satisfied. Therefore, the contour Γ may have many distinct boundary conditions (although the equivalence class on the boundary of each connected component of Z d \ suppΓ remains the same for all possible boundary conditions). The union of all components of IntΓ whose boundary conditions are in the same equivalence class will be denoted by IntA Γ (for A-boundary conditions) or IntB Γ (for B-boundary conditions). Let {Γ, Γ1 , . . . , Γn } be the set of all contours of a configuration σ contributing to the sum (3) (therefore σ has A-boundary condition). If the contour Γ satisfies suppΓ ⊂ ExtΓi ∀ i, then Γ is said to be an external contour of σ. Any external contour of σ will be a ΓA contour, i.e. a contour whose external boundary condition is in GA . To be consistent with Borgs–Imbrie notation [19] (and to use their results), we define a Y -contour: Definition 2.1. A Y -contour is a pair (Y, q(.)) where Y is a finite connected set of Z d and q(.) is a function that assigns an equivalence class (A or B for GA or GB respectively) to the boundaries of Z d \ Y . As for Γ, we also define ExtY , IntY , IntA Y and IntB Y . Each contour Γ defines a Y -contour (Y, q(.)) by calling Y ≡ suppΓ and by observing, from Proposition 2.1, that on the boundary of each connected component of Z d \ suppΓ the equivalence class of the boundary condition remains the same, thereby defining the function q(.). Many distinct Γ contours may give rise to the same Y -contour (Y, q(.)) and the set of all of them will be denoted by [Y, q], where q stands for the function q(.). In what follows, Y A will denote a contour (Y, q(.)) for which q(∂(ExtY )) = A, while Y will denote its support. Next proposition is a representation formula for the generating function in terms of a sum over admissible Y A -contours: given a family of Y -contours {(Yi , qi (.))}, Yi ⊂ Λ ∀ i and dist(Yi , Yj ) ≥ 2 ∀ i 6= j, we say that this family is admissible if there is a spin configuration σ on Λ (with A-boundary conditions) whose external contours {(Γi , suppΓi )} satisfy Yi = suppΓi and qΓi (.) = qi (.) ∀ i.
785
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
Proposition 2.2. For x, y ∈ A, the partition function (3) satisfies ! X Y Y ZA (Λ, h) = 2 cosh hi K(YiA , h)
(4)
{(YiA ,qi (.))} i
i∈Λ∩Lo
where the sum is over all admissible families of Y A -contours. K(Y A , h) is defined as
The function
ZB (IntB Y A , h) ZA (IntB Y A , h)
(5)
Q [ i∈Y ehi Γi ]e−βH(Γ) Q . {i∈Y ∩Lo } 2 cosh hi
(6)
K(Y A , h) = ρ(Y A , h) where ρ(Y A , h) is X
A
ρ(Y , h) =
{Γ∈[Y A ,q]}
Remark 2.1. We point out that implicit in the sum (4) is a specification of the boundary conditions, through the function qi (.), on the internal boundaries of YiA . The number of internal boundaries and their boundary conditions must be taken into account when estimating the number of polymers of a given size containing a given point. Proof. Given a family of Y A -contours {(Yi , qi (.))}, dist(Yi , Yj ) ≥ 2 ∀ i 6= j, we say that Y˜ is a innermost contour if Yi ⊂ (V (Y˜ ))c for all Yi 6= Y˜ , where V (Y ) ≡ Y ∪ IntY is the volume of Y . We will prove by induction on |IntY˜ |, the size of IntY˜ , the following representation for the generating function: X Y ZA (Λ, h) = 2 cosh hj A A {Yi ,qi (.)}
˜ )∩Lo } {j∈(Λ/∪k IntY k
Y
×
˜ A ∀ k} {i/YiA 6=Y k
" # Y K(YiA , h) K(Y˜kA , h)ZA (IntY˜k , h)
(7)
k
where the sum is over all admissible families {(Yi , qi (.))} and where {Y˜k } is the subset of innermost Y -contours satisfying the condition |IntY˜ | ≤ |Λ| − n, where n is an integer. Therefore, after the first induction step, ZA (Λ, h) will be represented by a sum of products over innermost contours whose volumes are no bigger than |Λ| − 1. For each of these new innermost contours, we expand its partition function as in (7) to obtain innermost contours of size no bigger than |Λ| − 2, and so forth. The first induction step goes as follows: let (Γ, suppΓ) be a given contour and let Ω(Γ) be the set of all configurations on V (suppΓ) such that (Γ, suppΓ) is its unique external A-contour. Summing over these configurations: Y X ehi σi e−βHΛ (σ) = ρ˜(Γ, h)ZB (IntB Γ, h)ZA (IntA Γ, h) (8) {σ∈Ω(Γ)}
i∈suppΓ
786
G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
"
where ρ˜(Γ, h) ≡
Y
# e
hi Γi
e−βH(Γ) .
i∈Y
For any contour Θ ∈ [Y, qΓ (.)] (set of all contours with same support Y and same specification function qΓ (.)), we have that IntA Θ = IntA Y and IntB Θ = IntB Y . Therefore, summing (8) over contours Θ ∈ [Y, qΓ (.)], we obtain: ρ˜(Y, h)ZB (IntB Y, h)ZA (IntA Y, h) where ρ˜(Y, h) =
X
(9)
ρ˜(Θ, h) .
{Θ∈[(Y,qΓ (.)]}
Multiplying and dividing (9) by Y 2 cosh hj ZA (IntB Y, h)ZA (IntA Y, h) {j∈Y ∩Lo }
we finally obtain as a result of summing (8) over all contours Θ ∈ [Y, qΓ (.)]: Y 2 cosh hj K(Y, h)ZA (IntY, h)
(10)
{j∈Y ∩Lo }
where ZA (IntY, h) ≡ ZA (IntB Y, h)ZA (IntA Y, h) is a function of the set IntY and K(Y, h) has been defined by Eq. (5). To evaluate ZA (Λ, h), we fix a configuration µ contributing to the sum (3) and let {Y1 , . . . , Yn } be the support, respectively, of its external ΓA contours A {ΓA Then, the contour family {(Yi , qΓi (.))} is admissible (the 1 (µ), . . . , Γn (µ)}. definition of admissible contours is given right before Proposition 2.2) and we firstly sum over the set of all σ configurations whose external contours define the same family {(Yi , qΓi (.))}. Secondly, we sum over all admissible families. Using (10) to factorize over the volumes V (Yi ), the first sum is equal to Y 2 cosh hj A Y i
{j∈(Λ/∪i V (Yi ))∩Lo }
Y
{j∈YiA ∩Lo }
2 cosh hj K(YiA , h)ZA (IntYiA , h) .
Now, summing over all admissible families we obtain X Y Y ZA (Λ, h) = 2 cosh hj [K(YiA , h)ZA (Int Yi , h)] {YiA ,qi (.)}
{j∈(Λ/∪i Int Yi )∩Lo }
i
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
787
which is the representation (7) when all polymers are themselves the innermost ones. We iterate this process by applying the first induction step to ZA (Int Yi , h) whenever Int Yi 6= ∅ (if Int Yi = ∅, we define ZA (Int Yi , h) ≡ 1). When all innermost Y ’s satisfy Int Y = ∅ we stop the process and obtain the contour representation (4). Definition 2.2. A polymer Y is a finite connected subset of Z d . Proposition 2.3. Under the hypothesis of Proposition 2.2, the generating function (3) satisfies " # Y XY ZA (Λ, h) = 2 cosh hi z(Yi , h) (11) i∈Λ∩Lo
{Yi } i
where the sum is over all families of admissible polymers {Yi }. The polymer activity z(Y, h) is defined as 0 X z(Y, h) = (12) K(Y A , h) P0 where means a sum over all possible equivalence class assignments {q(.)} for the which {(Yi , qi (.))} is an admissible family of Y A -contours. 3. Proof of Theorem 1.1 In what follows we denote the number of sites of a polymer by |Y |. To establish bounds for ln ZA (Λ, h) and prove convergence for the polymer expansion for (ln ZA (Λ)/|Λ|), we follow [13] and find an -value, whose definition is the following: if z(Y, h) has a bound of the form |z(Y, h)| ≤ exp[−b|Y |] and if N (x, k), the number of polymers Y with |Y | = k and x ∈ Y , has a bound exp[d0 |Y |], then we call the product exp[d0 ] exp[−b] = , the -value. From [17] if < 1/6, then the polymer expansion for (ln ZA (Λ, h)/|Λ|) converges and (ln ZA (Λ, h)/|Λ|) has a bound O(). An upper bound for N (x, |Y | = k) is found by first fixing some function q(.) such that (Y, q(.)) is a Y A -contour and then bounding the number of Y A -contours of size |Y | = k intersecting x. This is at most (2d)2k (see for instance [17]). Since the function q(.) assumes at most two values A or B, the number of Y A -contours (Y, q(.)) of size |Y | = k is at most 2k and therefore N (x, k) ≤ 2k (2d)2k = exp{[ln 2+ 2 ln 2d]k}. A bound for |z(Y, h)| is found by first using the definition (12) and then taking the supremum of |K(Y A , h)| over all {q(.)} for the which (Y, q(.)) is a Y A -contour and |Y | = k: |z(Y, h)| ≤ 2k sup{Y A /|Y |=k} |K(Y A , h)| . An uniform bound on |K(Y A , h)| (uniform in all functions q(.) such that (Y, q(.)) is a Y A -contour and |Y | = k) is obtained using definition (5). We first find P an upper bound for |ρ(Y A , h)|. Since H(Γ) ≡ hi,ji Hij (Γ) ≥ EkY k, where kY k is the number of bonds hi, ji for which Hij (Γ) > 0 and E = E(x, y) ≡ min{E++ (x, y), E00 (x, y)} and since kY k ≥ |Y |/2 (for each point in Y there is
788
G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
associated at least one positive energy bond and for each of these bonds there corresponds at most two distinct points in Y ) we get |ρ(Y A , h)| ≤ 3k e3|h|1 e−
βEk 2
≤ e(ln 3+3|h|1 −
βE 2 )k
where we have used that for |hi | < 1/2, exp[−2|hi|] ≤ |2 cosh hi | ≤ exp[|hi |] and P A A where |h|1 = i∈Λ |hi |. Now, the ratio (ZB (IntB Y , h)/ZA (IntB Y , h)) can be rewritten as [ZB (IntB Y A , h)/ZB (IntB Y A , 0)] ZB (IntB Y A , 0) . (13) [ZA (IntB Y A , h)/ZA (IntB Y A , 0)] ZA (IntB Y A , 0) Both the numerator and denominator of the first factor in (13) can be bounded above and below using Lemma 2.1 of [13], which states that if |h|1 < 1/2 then e
−2|h|1
ZA (IntB Y A , h) ≤ e|h|1 , ≤ ZA (IntB Y A , 0)
the same inequality holding true for B boundary conditions also. Q The above bounds are also satisfied by {i∈Y ∩Lo } |2 cosh hi | under the condition |h|1 < 1/2 since for |hi | < 1/2, exp[−2|hi|] ≤ |2 cosh hi | ≤ exp[|hi |]. On the other hand, from the stability of A boundary conditions as in [19], we obtain an upper bound for the second factor of (13), which is exp[c|∂(IntB Y A )|] ≤ exp[ck], where c is β independent. Now consider the ratio of the h = 0 partition functions. The ratio will be controlled by using the results on stable contours of [19] to which we refer for notation. It will follow that A and B are stable boundary conditions. In (4)–(6) we have written ZA in the polymer system representation of [19] (up to the multiplicative factors of cosh hi ). We obtain a similar representation for ZB . We see that the infinite volume free energies hA and hB , associated with 0 0 the truncated partition functions ZA and ZB (at zero magnetic field), respectively, 0 are equal. This follows since for every polymer or stable contour occuring in ZA 0 there is a corresponding stable contour (with the same activity) occuring in ZB and obtained by a unit translation. Thus by Theorem 3.1(ii) of [19] for any finite Ω ⊂ Zd ZB (Ω) |∂Ω| ZA (Ω) ≤ e or
ZB (IntB Y A ) ck ZA (IntB Y A ) ≤ e .
Therefore, |ZB (IntB Y A , h)/ZA (IntB Y A , h)| ≤ exp[(3|h|1 + c)k]. Putting all this together, we get: |K(Y A , h)| ≤ exp{(ln 3 + 6|h|1 + c − βE 2 )k} and |z(Y, h)| ≤ exp{(ln 2 + ln 3 + 6|h|1 + c − βE )k} which gives an -value = 2 βE exp{(2 ln 2 + ln 3 + 2 ln 2d + 6|h|1 + c − 2 )}. It is clear that for β > β0 (h) ≡ {2(2 ln 2 + ln 3 + ln 6 + 2 ln 2d + 6|h|1 + c)/E} the condition < 1/6 is satisfied and we have convergence of the polymer expansion.
789
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
To prove Theorem 1.1(b) we proceed as in [13]. We write ZA and ln ZA in a multi-index notation (below A will occur as before and as a multi-index) as X ZA = M a(A)z(h, w)A {A}
ln ZA = ln M +
X
aT (A)z(h, w)A
{A}
where, as a device to obtain exponential decay, we have extended z(Y, h) to an analytic function in w by extending ρ(Y A , h) to ρ(Y A , h, w) =
w|Y
A
|
e−(βE/2−ln 3)|Y A |
ρ(Y A , h) .
The physical value of w is exp[−(βE/2 − ln 3)] ≡ wp . Using the Cauchy representation for the hi derivatives we have, with |h|1 < 1/2, Z 00 n Y X 1 dhj aT (A)z(h, w)A |w=wp SΛ (i1 , . . . , in ) = 2 2πi h |hj |=rj j j=1 A
P00 where if {Yi } is associated with the multi-index A, then means {(i1 , . . . , in } ⊂ ∪i V (Yi ) and we set hi = 0 for all i such that xi is not one of the points of the correlation function. We will show that the series converges for |w| < w0 ≡ exp[−(β0 E/2 − ln 3)] for β0 sufficiently large. Using Cauchy estimates for the hj integrals and, after multiplying and dividing by (w/w0 )da , the maximum modulus theorem in w we have, with |r|1 < 1/2 " # n 00 Y X w da 1 T 0 A suph,w0 |SΛ (i1 , . . . , in )| ≤ |a (A)||z(h, w )| rj w0 j=1
w=wp
A
where the sup is over {h : |hi | = ri , |r|1 < 1/2} and {w0 : |w0 | = w0 }. Thus we obtain the bound n Y 1 C 0 e−(β−β0 )da |SΛ | ≤ r j j=1 "
where 0
C ≡ suph,w0 and
P0
0 X
# 0
|a (A)||z(h, w )| T
A
A
means that only i1 ∈ ∪i V (Yi ). Now, concerning the -value |ρ(Y A , h, |w| = w0 )| ≤ e−(β0 E/2−ln 3−3|h|1 )k
which, upon proceeding as before, gives the same -value, i.e. e(2 ln 2+2 ln 2d+3|h|1 +c) e−(β0 E/2−ln 3−3|h|1 ) ≡ .
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
For β0 sufficiently large < 1/6 which implies by the results of Sec. 2 of [13] that C 0 is O(). Also a lower bound, m, for the exponential decay rate is m = (β − β0 )E/2 and limβ→∞ m/β = E/2. 4. Upper Bounds for the Spin Spin Correlation From the analyticity results of Appendix A and hyperplane considerations, in this section we prove Propositions 4.1 and 4.2 which imply the upper bounds for the two-point function and the lower bounds for m given in Theorem 1.2. At the end of this section we give Lemma 4.3, which contains explicit expressions for the two-point function for coincident points and for the points separated by two lattice units. This lemma will be needed in the lower bounds estimates of the two-point function in Proposition B.1. (2) We now give the upper bounds for GΛ (i) ≡ hσ0 σ(i,0,...,0) iΛ , which are uniform in Λ. Adjacent hyperplane analysis does not capture the dominant configurations for the correlation functions, but the use of three consecutive hyperplanes does. Without loss of generality we assume that i = 2k with k a positive integer. Consider the hyperplanes H0 , . . . , Hi , where Hp = {h ∈ Zd : h1 = p}. In order to do our analysis as local as possible, we group the 2k + 1 hyperplanes into k sets of three consecutive hyperplanes Hp−1 ∪ Hp ∪ Hp+1 , with p odd such that 0 < p < i. This grouping of three consecutive hyperplanes is suggested by expected dominant contribution for the two-point correlation function in the region A. Below we depict the expected dominant excitation for the two point function (in the case i = 2 and d = 2): (a) in the region 0 < E++ ≤ (2d − 1)E00 + +++ , + (b) in the region 0 < (2d − 1)E00 ≤ E++ 0 +0 0 0 +0 0 , 0 +0 where we have represented only those sites belonging to some bond hi, ji such that Eσi σj > 0. Notice that three hyperplanes is the minimum number of hyperplanes we need to see the dominant structure. Let B(Hl ) and B(Hl , Hl±1 ) be the set of nearest neighbors bonds in Hl and between Hl and Hl±1 , respectively. Define Bp = B(Hp−1 ) ∪ B(Hp ) ∪ B(Hp−1 , Hp ) ∪ B(Hp , Hp+1 ). In what follows, our estimates are local in the sense they depend only on the bonds of Bp for a fixed odd p, with 0 < p < i.
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
791
(2)
From Appendix A, the two point function GΛ (i, {zpˆ}) is shown to be analytic in the variables {zpˆ}, as long as |(zp )l | < r, l = 1, 2, 3. In particular, for each fixed p, with p odd and 0 < p < i, we have X (2) GΛ (i, {zpˆ}) = αnp ({zpˆ}p6ˆ=p ) zpnp , (14) np ∈Z+ 3
where zp ∈ C3 is taken such that |(zp )l | ≤ 2r (l = 1, 2, 3) and we use the multi-index notation zp np = (zp )1 (np )1 (zp )2 (np )2 (zp )3 (np )3 . In our applications (zp )1 = e−βE++ , (zp )2 = e−βE00 and (zp )1 = e−βE+− , which are the physical values. The quantities αnp do not depend on zp , namely, from Cauchy integral formula I 1 1 (2) dzp 0 G (i, {zpˆ}) , (15) αnp = (2πi)3 Cp (zp 0 )np +1 Λ where Cp = {zp 0 ∈ C3 : |(zp0 )l | = r, l = 1, 2, 3}. In formulas (14) and (15) we could have chosen |(zp )l | ≤ R where R < r. Q In (15), np + 1 ≡ ((np )1 + 1, (np )2 + 1, (np )3 + 1) and dzp0 ≡ 3l=1 d(zp0 )l . By spin symmetry, the hyperplanes H0 and Hi must be connected by bonds of (2) type 1, this already implies that exponential decay of GΛ (i) at a rate which is at least E++ β, which is far from optimal, since in the region where the first energy excitation is E++ the expected decay of is dE++ β +o(β) where o(β) β → 0 as β → ∞. Additional efforts are needed in order to optimize the upper bound of the decay rate for the two-point function and this is the goal of this section. Let β 0 (x, y, d) be the smallest β ≥ β0 such that max{e−βE++(x,y) , e−βE+−(x,y) , e−βE00 (x,y) } ≤
r . 2
Lemma 4.1. Let β ≥ β 0 and αnp be defined by (15), then αnp = 0
(16)
unless (np )1 +(np )3 ≥ 2. Moreover, if (np )1 +(np )3 = 2+l where l = 0, 1, . . . , 2d−2, then (16) holds, unless (np )2 ≥ (2d − 2)(2d − l − 2). In order to prove this lemma we should have in mind the relation between the zmn variables and the spin variables. The (np )l variable gives the multiplicity of the variable (zp )l and has the following relation with the spin variable: the variable (np )1 counts the number of nearest neighbor pairs hx, yi in Bp such that σx , σy = ±1, the variable (np )2 counts the number of nearest neighbor pairs hx, yi in Bp such that σx = σy = 0 and the variable (np )3 counts the number of nearest neighbor pairs hx, yi in Bp such that σx = −σy = ±1. In the proof of this lemma we will assume that the ground state boundary condition σ ˆ is such that its restriction to the even sublattice is 0. Proof of Lemma 4.1. By the spin flipping symmetry of the Gibbs measure of the BEG model, only those configurations whose boundary contains a contour of
792
G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
type 1 having the origin 0 and (i, 0, 0, . . . , 0) in its support will contribute to the two-point function. In particular, given the hyperplanes Hp−1 , Hp and Hp+1 they must be connected by bonds of type 1; therefore, (np )1 + (np )3 ≥ 2. Let b = hx, yi be a bond of type 1, with x ∈ Hp−1 and y ∈ Hp . Then either y or x belongs to the even sublattice A. Suppose that y ∈ A. Since σy = ±1, then y is incorrect with respect to the boundary condition σ ˆ , therefore, for each (±) k = 2, 3, . . . , d, we must have at least one incorrect bond, bk , on the half-line y ± tˆ ek , t ∈ Z+ , where (ˆ ek )j = δkj . Similarly, if x ∈ A is even, then for each k = 2, 3, . . . , d, there exists at least one incorrect bond, bk (±) , in the half-line x ± tˆ ek , t ∈ Z+ . Let b0 = hm1 , m2 i and define b0 ± eˆk ≡ hm1 ± eˆk , m2 ± eˆk i. Next we assume that y ∈ A. Given a bond b0 = hm1 , m2 i of type 0, with m1 , m2 ∈ Hp ( respectively, m1 , m2 ∈ Hp−1 , if x ∈ A) and m1 , m2 in the line y + tˆ eko , and t ∈ Z, ko ∈ {2, 3, . . . , d} (respectively, in the line x + tˆ eko , and t ∈ Z, ko ∈ {2, 3, . . . , d}, if x ∈ A), we say that b0 is free of adjacent bonds of type 1 (f ab1), if the following conditions hold: (i) none of the bonds b0 ± eˆk , k ∈ {2, 3, . . . , d}, k 6= ko , is of type 1 and (ii) the bond b0 − eˆ1 (respectively, the bond b0 + eˆ1 , if x ∈ A) is not of type 1. Notice that to each bond of type 0 which is f ab1 we can associate at least 2d − 3 additional bonds of type 0. In fact, from (i), for each k ∈ {2, 3, . . . , d}, k 6= ko , the configuration restricted to the bonds b0 ± eˆk must be either 0±, ±0 or 00, then at least one of the bonds hm1 , m1 + eˆk i, hm2 , m2 + eˆk i or hm1 + eˆk , m2 + eˆk i must be of type 0. Similarly, at least one of the bonds hm1 , m1 − eˆk i, hm2 , m2 − eˆk i or hm1 − eˆk , m2 − eˆk i must be of type 0. Hence from (i), we have at least 2(d − 2) bonds of type 0. Also, from (ii), similar arguments imply that at least one of the bonds hm1 , m1 − eˆ1 i, hm2 , m2 − eˆ1 i or hm1 − eˆ1 , m2 − eˆ1 i respectively, one of the bonds hm1 , m1 + eˆ1 i, hm2 , m2 + eˆ1 i or hm1 + eˆ1 , m2 + eˆ1 i, if x ∈ A must be of type 0. Let b and b0 be two bonds in Bp . We say that a bond b0 is adjacent to b if (i) b and b0 are in Hp (respectively, b and b0 are Hp−1 , if x ∈ A is even) and b0 = b ± eˆk for some k ∈ {2, 3, . . . , d} or (ii) b is in Hp (respectively, b is in Hp−1 , if x ∈ A is even) and b0 is in Hp−1 (respectively, in Hp , if x ∈ A is even) and b0 = b − eˆ1 (respectively, b0 = b + eˆ1 , if x ∈ A is even). We say that a bond b0 is adjacent to a half-line bk if it is adjacent to some (±) (±) bond b in bk . We say that a half-line bk is free with respect to a configuration (±) S, if there is no bond of type 1 in or adjacent to bk . (±) Given a configuration S, if a half-line bk is free with respect to σ, then we (±) have at least one bond of type 0 in bk which is free. In fact, from previous (±) considerations, by definition of the half-lines bk , they have at least one incorrect (±)
793
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
(±)
(±)
bond, so if some half-line bk is free, then all incorrect bonds in bk must be of type 0 and f ab1. Now suppose that (np )1 + (np )3 = 2 + l, with 0 ≤ l ≤ 2d − 2. In the spin variables language, we have 2 + l bonds of types 1 and 3 in Bp . From previous considerations, at least two of these bonds must be in B(Hp−1 , Hp ) ∪ B(Hp , Hp+1 ), therefore, we have at most l bonds of types 1 and 3 in B(Hp−1 ) ∪ B(Hp ), which we denote by b1 , . . . , bs , with s ≤ l. Let N ≤ s be the number of bonds in {b1 , . . . , bs } (±) which is adjacent to or is in one of the half-lines bk . Since a bond can be adjacent (±) (±) to or be in at most one bk , then the number of bk which is free is at least 2d − 2 − N ≥ 2d − 2 − l. Then we have at least 2d − l − 2 bonds of type 0 which is free and since to each such a bond we have at least 2d − 3 additional bonds of type zero, the total number of bonds of type zero in Bp , (np )2 , is at least 2d − l − 2 + (2d − l − 2)(2d − 3) = (2d − 2)(2d − l − 2). Therefore, (np )2 ≥ (2d − 2)(2d − l − 2), which proves Lemma 4.1. Corollary 4.1. Under the hypothesis of Lemma 4.1, we have the following representation: I 1 (2) dz 0 1 F (z 0 1 , z1 ) GΛ (i; {zpˆ}) = (2πi)3 C1 I 1 (2) ... dz 0 i−1 F (z 0 i−1 , zi−1 )GΛ (i; {zp0ˆ}) , (17) (2πi)3 Ci−1 where F (z, z 0 ) =
P7 k=1
F (k) (z, z 0 ) and
F (1) (z, z 0 ) =
(z1 )2d (z 0 1 )2d+1 z 0 2 z 0 3
F (2) (z, z 0 ) =
(z1 )2 (z2 )(2d−2) (z 0 1 )3 (z 0 2 )(2d−2)2 +1 z 0 3
2
F (3) (z, z 0 ) =
0 2d−2 XX n
l=0
F (4) (z, z 0 ) =
zn (z 0 )n+1
X
2
2≤n1 +n3 ≤2d
F (5) (z, z 0 ) =
X n1 ≤2d
F (6) (z, z 0 ) =
(z1 )n1 (z2 )(2d−2) +1 (z3 )n3 0 n +1 (z 1 ) 1 (z 0 2 )(2d−2)2 +1 (z 0 3 )n3 +1 (z 0 2 − z2 )
(z1 )n1 (z3 )2d+1 (z 0 1 )n1 +1 (z 0 3 )2d+1 (z 0 2 − z2 )(z 0 3 − z3 ) X
n1 +n3 ≥2d+1,n1 ,n3 ≤2d
F (7) (z, z 0 ) =
(z 0
(z1 )n1 (z3 )n3 n +1 (z 0 3 )n3 +1 (z 0 2 1) 1
(z1 )2d+1 , (z 0 1 )2d+1 (z 0 1 − z1 )(z 0 2 − z2 )(z 0 3 − z3 )
− z2 )
794
G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
P0 where, in the definition of F (3) , n means sum over all n = (n1 , n2 , n3 ) such that n 6= (2d, 0, 0), (2, (2d−2)2 , 0), n1 +n3 = 2+l, and (2d−2)(2d−2−l) ≤ n2 ≤ (2d−2)2 . Proof. Given any non-negative integer m and complex numbers z and z 0 , with |z| < |z 0 |, we use z m 1 X zn . (18) = (z 0 )n+1 z0 z0 − z n≥m
3 Let m = (m1 , m2 , m3 ) ∈ Z+ , z, z 0 ∈ C3 , with |zi | < |z 0 i | (i = 1, 2, 3), and define z 0 − z ≡ (z 0 1 − z1 )(z 0 2 − z2 )(z 0 3 − z3 ), then we have the following power series representation for z01−z : X zn 1 = (19) z0 − z (z 0 )n+1 3 n∈Z+
where n + 1 = (n1 + 1, n2 + 1, n3 + 1) and z n = (z1 )n1 (z2 )n2 (z3 )n3 . We can rewrite the sum (19) in the following way: X n
zn = (z 0 )n+1
X {n|n1 +n3 ≤2d, n2 ≤(2d−2)2 }
zn (z 0 )n+1
X
+
zn
{n|n1 +n3 ≤2d, n2 ≥(2d−2)2 +1}
X
+
{n|n1 +n3 ≥2d+1, n2 ≥0}
(z 0 )n+1
zn (z 0 )n+1
≡ G(1) (z, z 0 ) + G(2) (z, z 0 ) + G(3) (z, z 0 ) ,
(20)
where G(1) (z, z 0 ), G(2) (z, z 0 ) and G(3) (z, z 0 ) are the first, second and third terms, respectively, of the left-hand side of (20). Applying the resumation formula (19) to G(2) (z, z 0 ) and G(3) (z, z 0 ), G(2) (z, z 0 ) =
X
2
n1 +n3 ≤2d
(z1 )n1 (z2 )(2d−2) +1 (z3 )n3 (z 0 1 )n1 +1 (z 0 2 )(2d−2)2 +1 (z 0 3 )n3 +1 (z 0 2 − z2 )
and G(3) (z, z 0 ) =
=
1 z2 − z2
X n1 +n3 ≥2d+1
1 z 0 2 − z2 +
(z1 )n1 (z3 )n3 (z 0 1 )n1 +1 (z 0 3 )n3 +1
X n1 ≥2d+1,n3 ≥0
X n3 ≥2d+1,n1 ≤2d
(z1 )n1 (z3 )n3 (z 0 1 )n1 +1 (z 0 3 )n3 +1
(z1 )n1 (z3 )n3 (z 0 1 )n1 +1 (z 0 3 )n3 +1
(21)
795
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
X
+
n1 +n3 ≥2d+1,n1 ,n3 ≤2d
=
(z 0 +
1
)2d+1 (z 0
X n1 ≤2d
+
(z1 )n1 (z3 )n3 (z 0 1 )n1 +1 (z 0 3 )n3 +1
!
(z1 )2d+1 0 0 1 − z1 )(z 2 − z2 )(z 3 − z3 )
(z1 )n1 (z3 )2d+1 (z 0 1 )n1 +1 (z 0 3 )2d+1 (z 0 2 − z2 )(z 0 3 − z3 ) X
n1 +n3 ≥2d+1,n1 ,n3 ≤2d
(z 0
(z1 )n1 (z3 )n3 +1 n +1 (z 0 )n3 +1 (z 0 1 1) 3 2
− z2 )
= F (7) (z, z 0 ) + F (5) (z, z 0 ) + F (6) (z, z 0 ) . From Cauchy’s integral formula we have the following representation: I 1 1 (2) dz 0 1 0 GΛ (i; {zp }p ) = (2πi)3 C1 z 1 − z1 I 1 1 (2) ... dz 0 i−1 0 G (i; {zp0 }p ) , (22) (2πi)3 Ci−1 z i − zi Λ Q3 where z 0 p − zp = k=1 ((z 0 p )k − (zp )k ). In the representation given by (22), we replace each factor z0 p1−zp by its rep-
resentation given by G(1) (z, z 0 ) + G(2) (z, z 0 ) + G(3) (z, z 0 ) and from Lemma 4.1 we notice that the only terms in G(1) (z, z 0 ) which contribute are those having (n1 , n2 , n3 ) satisfying the conditions n1 + n3 = 2 + l, (2d − 2)(2d − 2 − l) ≤ n2 ≤ (2d − 2)2 , where l = 0, 1, . . . , 2d − 2. Therefore, in the Cauchy’s formula, we can replace each G(1) (z, z 0 ) by F (1) (z, z 0 ) + F (2) (z, z 0 ) + F (3) (z, z 0 ) and G(2) (z, z 0 ) by F (4) (z, z 0 ). Hence, we can identify F (z 0 p , zp ) with z0 p1−zp which concludes the proof of Corollary 4.1. Remark 4.1. In the region where 0 < E++ < 2(d−1)E00 , the dominant contribution to F comes from n = (2d, 0, 0), which corresponds to the dominant contribution for the two-point function. On the line E++ = 2(d − 1)E00 , the dominant contributions to F come from n = (2 + l, (2d − 2)(2d − l − 2), 0), l = 0, 1, . . . , 2d − 2. In the region 0 < 2(d − 1)E00 < E++ , the dominant contribution to F comes from n = (2, (2d − 2)2 , 0). Notice that this is not the truly dominant contribution to the two point function. In other words the local structure does not reflect the global one. Corollary 4.2. Let (zp )1 = e−βE++(x,y) = e−βE−− (x,y) , (zp )2 = e−βE00 (x,y) , (zp )3 = e−βE+− (x,y) and β ≥ β 0 . Then there is a β1 ≥ β 0 such that: (1) If 0 < E++ (x, y) < 2(d − 1)E00 (x, y), (np )1 + (np )3 ≤ 2d and np 6= (2d, 0, 0) (dominant contribution to F ) and β ≥ β1 , then (zp )np ≤ e−2dβE++ max{e−2β , e−β(2d−3)(2(d−1)E00−E++ ) } .
(23)
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
In particular, for all |z| ≤
r 2
and |z 0 | = r we have
|F (z, z 0 )| ≤
e−2dβE++ (1 + C1 (x, y, d)) r3
(24)
where 0 < C1 ≤ c1 (r, d) max{e−2β , e−βE++ , e−βE00 , e−β(2d−3)(2(d−1)E00−E++ ) }. (2) If E++ (x, y) = 2(d − 1)E00 (x, y), (np )1 + (np )3 ≤ 2d and np 6= (2 + l, (2d − 2)(2d − 2 − l), 0), l = 0, 1, . . . , 2d − 2 (dominant contributions to F ) and β ≥ β1 , then (zp )np ≤ e−2dβE++ max{e−2β , e−βE00 , e−βE++ } . (25) In particular, for all |z| ≤
r 2
and |z 0 | = r we have
|F (z, z 0 )| ≤ c3 (r, d)
e−2dβE++ (1 + C4 (x, y, d, β)) r3
(26)
where 0 < C4 ≤ c4 (r, d) max{e−2(np )3 β , e−βE00 , e−βE++ }. (3) If 4(d − 1)(2d − 1)E00 (x, y) ≤ E++ (x, y) ≤ 2(d − 1)E00 , (np )1 + (np )3 ≤ 2d and np 6= (2, (2d − 2)2 , 0) (dominant contribution to F ) and β ≥ β1 , then (zp )np ≤ e−β(2E+++4(d−1) In particular, for all |z| ≤
r 2
2
E00 ) −β(E++ −2(d−1)E00 )
e
.
(27)
and |z 0 | = r we have
e−β(2E+++4(d−1) |F (z, z )| ≤ r3 0
2
E00 )
(1 + C6 (x, y, d, β))
(28)
where 0 < C6 ≤ c6 (r, d) max{e−2β , e−βE++ , e−βE00 , e−β(E++−2(d−1)E00 ) }. Proof. We claim that (zp )np ≤ e−2βdE++ e−2β(np )3 e−β(2d−l−2)(2(d−1)E00 −E++ )
(29)
and (zp )np ≤ e−β(2E+++4(d−1)
2
E00 ) −2β(np )3 −βl(E++ −2(d−1)E00 )
e
e
.
(30)
Equation (29) implies (23) and (25), Eq. (30) implies (27). To prove (29) and (30), we notice by Lemma 4.1 that if (np )1 + (np )3 = 2 + l and (np )2 ≥ (2d − 2)(2d − 2 − l), where l = 0, 1, . . . , 2d − 2. Also, by definition E+− (x, y) = E++ (x, y) + 2. Therefore, (np )1 E++ + (np )2 E00 + (np )3 E+− = (np )1 E++ + (np )2 E00 + (np )3 (E++ + 2) = ((np )1 + (np )3 )E++ + 2(np )3 + (np )2 E00 = (2 + l)E++ + 2(np )3 + (np )2 E00 ≥ (2 + l)E++ + 2(np )3 + (2d − 2)(2d − 2 − l)E00 ( 2dE++ + 2(np )3 + (2d − 2 − l)((2(d − 1)E00 − E++ ) = 2E++ + 4(d − 1)2 E00 + 2(np )3 + l(E++ − 2(d − 1)E00 ) .
(31)
797
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
From (31) we have (29) and (30), respectively.
Proposition 4.1. Let d ≥ 2, suppose that i is even and let β1 (x, y, d) and r be defined as in Corollary 4.2. (1) Let 0 < E++ (x, y) ≤ 2(d − 1)E00 (x, y), then there is a positive number α11 (x, y, d) such that β ≥ β1 implies |hS0 S(i,0,...,0) i| ≤ K(α11 )
|i| 2
e−dβE++(x,y)|i| .
(32)
(2) Let 0 < 4(d − 1)(2d − 1)E00(x, y) < E++ (x, y) < 2(d − 1)E00 (x, y), then there is a positive number α21 (x, y, d) such that β ≥ β1 implies |hS0 S(i,0,...,0) i| ≤ K(α21 )
|i| 2
e−dβ(E++(x,y)+2(d−1)
2
E00 (x,y))|i|
.
(33)
where K = K(d, x, y). Proof. For each x and y such that E++ (x, y), E00 (x, y) > 0, there is a constant (2) K(d, x, y), such that |GΛ | ≤ K(d, x, y), independent of Λ. Then Corollaries 4.1 and 4.2, β ≥ βo imply I Y 1 (2) (2) |dz 0 p ||F (z 0 p , zp )||GΛ |∞ |GΛ (i; {zp })| ≤ (2π)3 Cp p odd,1≤p≤i−1
≤ K r3
max
z 0 ,z||(z 0 )l |=r,|(z)l |= r2
0
{|F (z , z)|}
|i| 2 .
(34)
Defining α11 = max{1 + C1 , c3 (1 + C4 )} and α21 = 1 + C6 , we have Proposition 4.1. The hyperplane and analyticity procedure described above can be applied to get an upper bound for the two-point function hσ(i,0...,0) σ(j,0,...,0) i for i, j ∈ Z and we get bounds similar to those given in Proposition 4.1. In Proposition 4.1, the line E++ (x, y) = 2(d − 1)E00 (x, y), does not play any special role — we expect that the bounds given in Eq. (32) hold in the whole region 0 < E++ (x, y) ≤ (2d − 1)E00 (x, y). |i| We expect an upper bound of the form c 2 e−β(E++ +(d−1)(2d−1)E00 )|i| , where c is β independent, in the whole region E++ (x, y) > (2d − 1)E00 (x, y) > 0. By doing a local analysis, using only three hyperplanes, the best we could get are the upper bounds given in item (2) of Proposition 4.1. To improve these estimates, we should in our hyperplanes analysis replace the sets of three hyperplanes by set of a larger number of hyperplanes, in other words, the local analysis using only three hyperplanes is not good enough to study this region. The reason why the local analysis fails in getting the right upper bounds in this region is the following: in our local analysis when we restrict to Bp we do not use the bonds which belong to B(Hp+1 ), then the dominant contribution we see is this region is the one below (depicted for d = 2) (we lose vertical bonds of type 0 in B(Hp , Hp+1 ))
798
G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
+ 0 0 +0 0 0 +0 instead of 0 +0 0 0 +0 0 . 0 +0 Notice that when the first picture above occurs, globally means that bonds of type zero have been replaced by bonds of type 1, and so the structure which locally looks like dominant, globally is not. In other words, even though the first local structure is dominant, it belongs to a global structure which is not. Therefore, the dominant character of the local structure does not reflect the global one to which it belongs. In the region 2(d − 1)E00 > E++ > 0, the local structure below + +++ + dominates the two structures above, and the local analysis reflects the global one. Next our goal is to obtain the expected upper bounds for the two-point function in the region 2(d−1)E00 > E++ > 4(d−1)(2d−1)E00 > 0. From the above remarks, we must quit our local analysis in which we have restricted only to three consecutive hyperplanes and we think globally. In this case, we replace (zij )k by zk for all i, j, therefore we have only three complex variables z1 , z2 and z3 . Nevertheless, in order to count their multiplicities, we may consider, locally, the contribution of each set of three consecutive hyperplanes. The following analysis is much simpler than the one we have used in the previous estimates; however, in order to extend the upper bounds obtained for the two point function for a larger region (the expected region for which the upper bounds should be true is 0 < (2d − 1)E00 < E++ ) we would need a more detailed analysis which is not our present goal. As before, we conclude that GΛ (i, z) is analytic, in the region |zk | ≤ r2 (k = 1, 2, 3). In particular, X GΛ (i, z) = αn z n n∈Z+ 3
where αn =
1 (2πi)3
I C
Q
GΛ (z 0 ) 0 0 0 0 − z ) dz1 dz2 dz3 , (z k k k
0
C = {z ∈ C | |zk | = r, k = 1, 2, 3} and I 1 GΛ (z 0 ) αn = dz 0 . n +1 3 (2πi) C z10 1 z20 n2 +1 z30 n3 +1 3
(35)
799
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
Lemma 4.2. Let β ≥ β o , lo =
|i| 4
and αn be defined by (35), then αn = 0 ,
(36)
unless, (i) n1 + n3 = |i| + l, 0 ≤ l ≤ lo and n2 ≥ (d − 1)(2d − 1)(|i| − 4l), or (ii) n1 + n3 = |i| + l, l ≥ lo + 1 and n2 ≥ 0. Proof. By symmetry, n1 + n3 ≥ |i| (the sites 0 and i must be connected by bonds of type 1), hence, we can write n1 + n3 = |i| + l, where l ≥ 0. On the other hand, if n1 + n3 = |i| + l and 0 ≤ l ≤ lo , then, since each one of the l bonds of type 1 can be “shared” by at most two sets of three consecutive hyperplanes, we have at least |i| 2 −2l sets of three consecutive hyperplanes which are free, i.e. the number of bonds of type 1 on them is exactly two (i.e. among all “horizontal” bonds in their three hyperplanes and all “vertical” ones connecting two adjacent hyperplanes in these sets, we have precisely the vertical ones, making vertical connection between them, as required). From each one of these sets of three hyperplanes, we can associate at least (2d − 2)(2d − 1) bonds of type 0. Therefore, |i| − 2l (2d − 2)(2d − 1) = (d − 1)(2d − 1)(|i| − 4l) . (37) n2 ≥ 2 Corollary 4.3. Let E++ (x, y) > 4(d − 1)(2d − 1)E00 (x, y), β ≥ β o and d ≥ 2, then GΛ (z, i) = z1 |i| z2 (d−1)(2d−1)|i| 1 × (2πi)3
I
1 + R1 (z 0 , z) + R2 (z 0 , z) + R3 (z 0 , z) (z20
C
−
! GΛ (z 0 , i)dz 0 ,
z2 )z10 |i|+1
where X
z1 n1 −|i| z3 n3
n1 +n3 =|i|,n3 6=0
z10 n1 −|i| z30 n3 +1
R1 (z 0 , z) =
R2 (z 0 , z) =
X
z2 −4l(d−1)(2d−1)
1≤l≤lo
R3 (z 0 , z) =
X
X
,
(38)
X
z1 n1 −|i| z3 n3
n1 +n3 =|i|+l
z10 n1 −|i| z30 n3 +1
z1 n1 −|i| z2 −(d−1)(2d−1)|i| z3 n3
l>lo n1 +n3 =|i|+l
z10 n1 −|i| z30 n3 +1
.
,
(39)
(40)
In particular, there exists a β2 ≥ βo , such that β ≥ β2 , implies |R1 (z 0 , z)| ≤ 2e−β ,
(41)
|R2 (z 0 , z)| ≤ 4e−βΘ ,
(42)
|R3 (z 0 , z)| ≤ e
βE − 2++
.
(43)
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
Proposition 4.2. Suppose that E++ > 4(d − 1)(2d − 1)E00 > 0 and d ≥ 2. Then lim lim −
β→∞ |i|→∞
ln |GΛ | ≥ E++ + (d − 1)(2d − 1)E00 . β|i|
By explicit calculation and analyticity we have the following lemma. Lemma 4.3. There is a β3 > β o , such that (1) If 0 < E++ (x, y) < (2d − 1)E00 (x, y), then for β ≥ β3 hS0 2 i =
e−2dβE++ (1 + R1 ) , 22d−1
hS1 2 i = 1 − R2 ,
(44) (45)
hS−1 S1 i =
e−2dβE++ (1 + R3 ) , 22d−1
(46)
hS0 S2 i =
e−4dβE++ (1 + R4 ) . 24d−2
(47)
(2) If 0 < (2d − 1)E00 (x, y) < E++ (x, y), then for β ≥ β3 hS0 2 i =
e−βd(2d−1)E00 (1 + R5 ) , 22d−1
hS1 2 i = 1 − R6 hS−1 S1 i = hS0 S2 i =
(48) (49)
e−β(2E+++(2d−2)(2d−1)E00 ) (1 + R7 ) , 22d−1
(50)
e−β(2E+++2(2d−1) 24d−2
(51)
2
E00 )
(1 + R8 ) .
(3) If (2d − 1)E00 (x, y) = E++ (x, y), then for β ≥ βo hS0 2 i = 2e−2dβE++ (1 + R9 ) ,
(52)
hS1 2 i = 1 − R10
(53)
e−2dβE++ (1 + R11 ) , 2
(54)
hS0 S2 i = e−4dβE++ (1 + R12 ) .
(55)
hS−1 S1 i =
where Ri (d, e−βE++ , e−βE+− , e−βE00 ) < ri , with ri independent of β. 5. Concluding Remarks For low temperature we have constructed two pure states of the BEG model in the anti-quadrupolar region. The question of the existence of other states, i.e. with surfaces as in the d = 3 Ising model, is open. Also, an analysis of the precise location and multiplicity of excitations is an interesting problem. Since we have shown in the
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
801
local Boltzman factors for a finite number of hyperplanes such an analysis could be carried out as for the low temperature Ising model in [15, 21]. There are questions of reentrance into phases at low temperature suggested by the phase diagrams in [2, 4], which possibly could be answered by a construction of the states for regions adjacent to the anti-quadrupolar region. There is also the question of the existence and construction of the ferrimagnetic phase. Acknowledgments This work has been partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico CNPq-Brazil and Funda¸ca˜o de Amparo a` Pesquisa do Estado de Minas Gerais. G. Braga and P. Lima thank CAPES for the financial support while visiting the Institute for Advanced Study and Rutgers University, respectively. Appendix A. In this appendix we prove the analiticity properties needed in Sec. 4 in order to obtain an optimal exponential upper bound for the two point function. It is inspired in the paper of Borgs and Imbrie [19]. In what follows, we take h = 0 and work with a modified version of the original model (2) by introducing complex variables in the following way: in Eq. (6) defining ρ(Y ), for each spin configuration σ and for each bond hi, ji belonging to some Bp , 0 < p < i (see Sec. 4), we replace the local Gibbs factor e−βEσi σj by (zij )k , where k = 1 if (σi , σj ) = (+, +) or (−, −), k = 2 if (σi , σj ) = (0, 0) and k = 3 if (σi , σj ) = (+, −). These redefined ρ functions are polynomials (and therefore analytic functions) in the z variables and satisfy the following Peierls estimate: |ρ(Y )| ≤ e−τ |Y | for large τ and for |(zij )k | < e− 2 ≡ r, where E = min{E++ , E00 } and τ is proportional to β. A contour Yα q is stable if Zm (Intm Yα q ) 4|∂Yα q | Zq (Intm Yα q ) ≤ e E
for m in {A, B}. The truncated partition function Z 0 (V ) is defined by leaving out the unstable contours in (11), up to a factor of 2|Λ| : Zq0 (V ) =
0 Y X {Yαq }
z(Yα q )
(56)
α
P0 where the sum goes over sets {Yαq } of nonoverlapping, stable contours with outer q boundary condition q. Notice that for the truncated model |z(Yαq )| ≤ e−(τ −8d)|Y | , which implies the convergence of the polymer expansion (see [16, 17] for notations), namely, X n(X) z 0 X log Z 0 (Λ) = , (57) |X|! X∈C(Λ)
802
G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
where z 0 (γ) = z(γ) if γ is stable and 0, otherwise. In particular, since the series in (57) is absolutely summable, it follows that Z 0 (Λ) 6= 0. We next state a theorem about the existence of the thermodynamic limit for the truncated non-translation invariant model. Theorem A.1. Consider the BEG model. For τ large enough (depending on dx, y) X log Zq0 (V ) n(X)ˆ zX hq = lim = = h, (58) |Λ|→∞ |Λ| |X|!|suppX| X∈Cx
here zˆ(γ) is the activity of the original model, q ∈ {A, B}. The independence of the above limit on q follows from the fact that for the original model these two boundary conditions are related by symmetry. Proof. Let Λ+ = {(x1 , . . . , xd ) ∈ Z d | x1 > i} and Λ− = {(x1 , . . . , xd ) ∈ Z d | x1 < 0} and define M = Λ − (Λ− ∪ Λ+ ). Our starting point is the representation (57). We break the sum over C(Λ) into three pieces: C(Λ+ ), C(Λ− ) and CM , where CM is the subset of C(Λ) consisting of all contours, X, such that supp X ∩ (M ∪ ∂M ) 6= ∅. Notice that the polymer model restricted to C(Λ− ) and C(Λ+ ) is translation invariant. Using this property we obtain X X∈C(Λs )
X n(X)z X n(X)ˆ zX = |Λs | + Rs (Λ, i, q) |X|! |X|!|suppX|
(59)
X∈Cx
where s ∈ {+, −} and where Rs (Λ, i, q) is equal to Rs (Λ, i, q) =
X
X
x∈Λs X∈Cx (Λs )
X X n(X)ˆ zX n(X)ˆ zX − . |X|!|suppX| |X|!|suppX| x∈Λs X∈Cx
It follows from our hypothesis that RM (Λ, i, q) ≡
0 X |n(X)|e−(τ −8d)|X| X n(X)z X ≤ |∂M | . |X|! |X|!
X∈CM
X∈Cx
Therefore, log Zq0 (Λ) = |Λ|
X n(X)ˆ X n(X)z X zX − |M | |X|!|X| |X|!|X|
X∈Cx
X∈Cx
+ R− (Λ, i, q) + R+ (Λ, i, q) + RM (Λ, i, q) . To conclude our proof we notice that as |Λ| → ∞.
|M| |∂M| |R− (Λ,i,q)| |Λ| , |Λ| , |λ|
and
|R+ (Λ,i,q)| |Λ|
(60) go to zero
In order to state and prove our next theorem, we impose some restriction on k k the variables z. We assume that zij = zmn for all bonds hi, ji and hm, ni in M
LOW TEMPERATURE PROPERTIES OF THE (BEG) MODEL
803
which are related by “horizontal” shifts, namely, hm, ni = hi + tˆ el , j + tˆ el i, t ∈ Z, l = 2, 3, . . . , d. Under this assumption, z(γ) = z(γ 0 ), if supp γ 0 = supp γ +tˆ el . Notice that we can go from the boundary condition A to B and vice-versa, by performing 0 a unit shift in any of the directions eˆlo , lo 6= 1. Hence, the contours in ZA (Λ) and 0 ZB (Λ) whose distance to ∂Λ is bigger than 1, are related by a shift of eˆlo . This means that these two partition functions can differ only from contributions which come from those polymers X ∈ C(Λ) such that supp X ∩ ∂Λ 6= ∅. Therefore, from the polymer representation for the truncated model, we have 0 0 (Λ) − log ZB (Λ)| ≤ 2 | log ZA
X
X |n(X)|e−(τ −8d)|X| |X|!
x∈∂− Λ X∈Cx
≤ 2|∂− Λ|
X |n(X)|e−(τ −8d)|X| |X|!
X∈Cx
≤ O(e
−τ
)|∂− Λ|
≤ 4 |∂Λ|
(61)
for τ large enough. Theorem A.2. Assume that |ρ(Y )| ≤ e−τ |Y | , ρ(Y ) is (jointly) analytic in the z variables, for |(zp )k | < r, and that hA = hB . Then there is a τo (depending on d, x, y) such that for τ ≥ τo , (V ) 4|∂V | (i) Zq (V ) 6= 0 and | ZZmq (V . )|≤e (ii) Zq (V ) is (jointly) analytic in the z variables, for |(zp )k | < r.
Remark A.1. From the polymer representation [16], the two-point function is expressed as a uniformly convergent sum of products of polymer activities, which, from Theorem A.2, are analytic functions of (zp )k , for |(zp )k | < r. This implies that GΛ ({zp }) is an analytic function in |(zp )k | < r. Proof. By induction on |V |, suppose that (i) and (ii) hold for all V such that |V | ≤ k − 1. Then for V such that |V | = k, all contours Y in V satisfy |Y | ≤ k − 1 and, therefore, are stable. Hence, by the induction hypothesis, Zq0 (V ) = Zq (V ) 6= 0, q = A, B, and, by (61): Zq (V ) Zq0 (V ) 4|∂V | = , Zm (V ) Z 0 (V ) ≤ e m which proves (i) for V = k. Notice that since |Intm Yαq | ≤ k − 1, for all contours in V , by the induction hypothesis Zm (Intm Yαq ) and Zq (Intm Yαq ) are analytic functions in the given polidisks and Zq (Intm Yαq ) 6= 0. Therefore, Y Zm (Intm Y q ) α
m
Zq (Intm Yαq )
804
G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
is an analytic function. Since ρ(Yαq ) is a polynomial in the z variables (see Eq. (6) with h = 0), it follows from our expression for K (5) that z as defined by (12), is analytic, implying that Zq (V ) is also analytic because its polymer representation converges uniformly, which proves (ii). Appendix B. In this section, following [16, 18], we construct a quantum Hilbert space and an infinite lattice “transfer matrix” and obtain a Feynman–Kac formula. Using H¨ older inequality in the spectral representation for the two-point function gives lower bounds for large distances in terms of the two-point function for coincident points and for the points separated by two lattice units. For small distance behavior we use the results obtained in Lemma 4.3, Sec. 4. The construction of the quantum Hilbert space and positive self-adjoint transfer “matrix” is standard. We single out the first component of Z d as the imaginary time direction and write x = (x0 , ~x) ∈ Z d . We use time translations of two lattice units since the spatial boundary conditions are invariant with a translation of two rather than one unit. The discrete time transfer “matrix”, denoted by Tˆ(x0 ), x0 ∈ 2Z, has the spectral representation Z 1 Tˆ(xo ) = λ dE(λ) . 0 ˆ We remark that if the null space of Tˆ is zero then Tˆ can be written as Tˆ = e−2H , ˆ is the quantum mechanical Hamiltonian. where H These considerations extend from Λs to Z d−1 and hence to the thermodynamic limit. The corresponding Hilbert space and i
