Open quantum systems I: the Hamiltonian approach

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1880 S. Attal · A. ...

Author: Stéphane Attal | Alain Joye | Claude-Alain Pillet

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1880

S. Attal · A. Joye · C.-A. Pillet (Eds.)

Open Quantum Systems I The Hamiltonian Approach

ABC

Editors Stéphane Attal Institut Camille Jordan Universit é Claude Bernard Lyon 1 21 av. Claude Bernard 69622 Villeurbanne Cedex France e-mail: [email protected]

Alain Joye Institut Fourier Universit é de Grenoble 1 BP 74 38402 Saint-Martin d'Hères Cedex France e-mail: [email protected]

Claude-Alain Pillet CPT-CNRS, UMR 6207 Université du Sud Toulon-Var BP 20132 83957 La Garde Cedex France e-mail: [email protected] Library of Congress Control Number: 2006923432 Mathematics Subject Classification (2000): 37A60, 37A30, 47A05, 47D06, 47L30, 47L90, 60H10, 60J25, 81Q10, 81S25, 82C10, 82C70 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-30991-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30991-8 Springer Berlin Heidelberg New York DOI 10.1007/b128449 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A EX package Typesetting: by the authors and SPI Publisher Services using a Springer LT Cover design: design & production GmbH, Heidelberg

Printed on acid-free paper

SPIN: 11602606

V A 41/3100/ SPI

543210

Preface

This is the first in a series of three volumes dedicated to the lecture notes of the Summer School ”Open Quantum Systems” which took place at the Institut Fourier in Grenoble from June 16th to July 4th 2003. The contributions presented in these volumes are revised and expanded versions of the notes provided to the students during the School. Closed vs. Open Systems By definition, the time evolution of a closed physical system S is deterministic. It is usually described by a differential equation x˙ t = X(xt ) on the manifold M of possible configurations of the system. If the initial configuration x0 ∈ M is known then the solution of the corresponding initial value problem yields the configuration xt at any future time t. This applies to classical as well as to quantum systems. In the classical case M is the phase space of the system and xt describes the positions and velocities of the various components (or degrees of freedom) of S at time t. In the quantum case, according to the orthodox interpretation of quantum mechanics, M is a Hilbert space and xt a unit vector – the wave function – describing the quantum state of the system at time t. In both cases the knowledge of the state xt allows to predict the result of any measurement made on S at time t. Of course, what we mean by the result of a measurement depends on whether the system is classical or quantum, but we should not be concerned with this distinction here. The only relevant point is that xt carries the maximal amount of information on the system S at time t which is compatible with the laws of physics. In principle any physical system S that is not closed can be considered as part of a larger but closed system. It suffices to consider with S the set R of all systems which interact, in a way or another, with S. The joint system S ∨ R is closed and from the knowledge of its state xt at time t we can retrieve all the information on its subsystem S. In this case we say that the system S is open and that R is its environment. There are however some practical problems with this simple picture. Since the joint system S ∨ R can be really big (e.g., the entire universe) it may be difficult, if not impossible, to write down its evolution equation. There is no solution to

VI

Preface

this problem. The pragmatic way to bypass it is to neglect parts of the environment R which, a priori, are supposed to be of negligible effect on the evolution of the subsystem S. For example, when dealing with the motion of a charged particle it is often reasonable to neglect all but the electromagnetic interactions and suppose that the environment consists merely in the electromagnetic field. Moreover, if the particle moves in a very sparse environment like intergalactic space then we can consider that it is the only source in the Maxwell equations which governs the evolution of R. Assuming that we can write down and solve the evolution equation of the joint system S ∨ R we nevertheless hit a second problem: how to choose the initial configuration of the environment ? If R has a very large (e.g., infinite) number of degrees of freedom then it is practically impossible to determine its configuration at some initial time t = 0. Moreover, the dynamics of the joint system is very likely to be chaotic, i.e., to display some sort of instability or sensitive dependence on the initial condition. The slightest error in the initial configuration will be rapidly amplified and ruin our hope to predict the state of the system at some later time. Thus, instead of specifying a single initial configuration of R we should provide a statistical ensemble of typical configurations. Accordingly, the best we can hope for is a statistical information on the state of our open system S at some later time t. The resulting theory of open systems is intrinsically probabilistic. It can be considered as a part of statistical mechanics at the interface with the ergodic theory of stochastic processes and dynamical systems. The paradigm of this statistical approach to open systems is the theory of Brownian motion initiated by Einstein in one of his celebrated 1905 papers [3] (see also [4] for further developments). An account on this theory can be found in almost any textbook on statistical mechanics (see for example [9]). Brownian motion had a deep impact not only on physics but also on mathematics, leading to the development of the theory of stochastic processes (see for example [12]). Open systems appeared quite early in the development of quantum mechanics. Indeed, to explain the finite lifetime of the excited states of an atom and to compute the width of the corresponding spectral lines it is necessary to take into account the interaction of the electrons with the electromagnetic field. Einstein’s seminal paper [5] on atomic radiation theory can be considered as the first attempt to use a Markov process – or more precisely a master equation – to describe the dynamics of a quantum open system. The theory of master equations and its application to radiation theory and quantum statistical mechanics was subsequently developed by Pauli [8], Wigner and Weisskopf [13], and van Hove [11]. The mathematical theory of the quantum Markov semigroups associated with these master equations started to develop more than 30 years later, after the works of Davies [2] and Lindblad [7]. It further led to the development of quantum stochastic processes. To illustrate the philosophy of the modern approach to open systems let us consider a simple, classical, microscopic model of Brownian motion. Even though this model is not realistic from a physical point of view it has the advantage of being exactly solvable. In fact such models are often used in the physics literature (see [10, 6, 1]).

Preface

VII

Brownian Motion: A Simple Microscopic Model In a cubic crystal denote by qx the deviation of an atom from its equilibrium position x ∈ ΛN = {−N, . . . , N }3 ⊂ Z3 and by px the corresponding momentum. Suppose that the inter-atomic forces are harmonic and only acts between nearest neighbors of the crystal lattice. In appropriate units the Hamiltonian of the crystal is then p2 κxy x + (qx − qy )2 , 2 4 3

x∈ΛN

x,y∈Z

where κxy =

1 if |x − y| = 1; 0 otherwise;

and Dirichlet boundary conditions are imposed by setting qx = 0 for x ∈ Z3 \ ΛN . If the atom at site x = 0 is replaced by a heavy impurity of mass M 1 then the Hamiltonian becomes H≡

x∈ΛN

κxy p2x (qx − qy )2 , + 2mx 4 3 x,y∈Z

where mx =

M if x = 0; 1 otherwise.

We shall consider the heavy impurity at x = 0 as an open system S whose environment R is made of the (2N +1)3 −1 remaining atoms of the crystal. To write down the equation of motion in a convenient form let us introduce some notation. We set Λ∗N = ΛN \ {0}, q = (qx )x∈Λ∗N , p = (px )x∈Λ∗N , Q = q0 , P = p0 . For x ∈ Z3 we denote by δx the Kronecker delta function at x and by |x| the Euclidean norm of x. We also set χ = |x|=1 δx . The motion of the joint system S ∨ R is governed by the following linear system q˙ = p, M Q˙ = P,

p˙ = −Ω02 q + Qχ, P˙ = −ω02 Q + (χ, q),

(1)

where −Ω02 is the discrete Dirichlet Laplacian on Λ∗N and ω02 = 6. According to the open system philosophy described in the previous paragraph we should supply some appropriate statistical ensemble of initial states of the environment. To motivate the choice of this ensemble suppose that in the remote past the impurity was pinned at some fixed position, say Q = P = 0, and that at time t = 0 the resulting system has reached thermal equilibrium at some temperature T > 0. The positions and momenta in the crystal will be distributed according to the Gibbs-Boltzmann canonical ensemble corresponding to the pinned Hamiltonian H0 = H|Q=P =0 , H0 =

1 (p, p) + (q, Ω02 q) . 2

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This ensemble is given by the Gaussian measure dµ = Z −1 e−βH0 (q,p) dqdp, where Z is a normalization factor and β = 1/kB T with kB the Boltzmann constant. At time t = 0 we release the impurity. The subsequent evolution of the system is determined by the Cauchy problem for Equ. (1). The evolution of the environment can be expressed by means of the Duhamel formula q(t) = cos(Ω0 t)q(0) +

sin(Ω0 t) p(0) + Ω0

sin(Ω0 (t − s)) χQ(s) ds. Ω0

t

0

Inserting this relation into the equation of motion for Q leads to t 2 ¨ K(t − s)Q(s) ds + ξ(t), M Q = −ω0 Q +

(2)

0

where the integral kernel K is given by K(t) = (χ,

and ξ(t) =

sin(Ω0 t) χ), Ω0

(3)

sin(Ω0 t) χ, cos(Ω0 t)q(0) + p(0) . Ω0

Since q(0), p(0) are jointly Gaussian random variables, ξ(t) is a Gaussian stochastic process. It is a simple exercise to compute its mean and covariance E(ξ(t)) = 0,

E(ξ(t)ξ(s)) = C(t − s) =

cos(Ω0 (t − s)) 1 (χ, χ). β Ω02

(4)

We note in particular that this process is stationary. The term ξ(t) in Equ. (2) is the noise generated by the fluctuations of the environment. It vanishes if the environment is initially at rest. The integral in Equ. (2) is the force exerted by the environment on the impurity in reaction to its motion. Note that this dissipative term is independent of the state of the environment. The dissipative and the fluctuating forces are related by the so called fluctuation-dissipation theorem K(t) = −β∂t C(t).

(5)

The solution z t = (Q(t), P (t)) of the random integro-differential equation (2) defines a family of stochastic processes indexed by the initial condition z 0 . These processes provide a statistical description of the motion of our open system. An invariant measure ρ for the process z t is a measure on R3 × R3 such that f (z t ) dρ(z 0 ) = f (z) dρ(z),

Preface

IX

holds for all reasonable functions f and all t ∈ R. Such a measure describes a steady state of the system. If one can show that for any initial distribution ρ0 which is absolutely continuous with respect to Lebesgue measure one has (6) f (z t ) dρ0 (z 0 ) = f (z) dρ(z), lim t→∞

then the steady state ρ provides a good statistical description of the dynamics on large time scales. One of the main problem in the theory of open systems is to show that such a natural steady state exists and to study its properties. The Hamiltonian Approach Remark that in our example, such a steady state fails to exist since the motion of the joint system is clearly quasi-periodic. However, in a real situation the number of atoms in the crystal is very large, of the order of Avogadro’s number NA 6 · 1023 . In this case the recurrence time of the system becomes so large that it makes sense to take the limit N → ∞. In this limit −Ω02 becomes the discrete Dirichlet Laplacian on the infinite lattice Z3 \ {0}. This is a well defined, bounded, negative operator on the Hilbert space 2 (Z3 ). Thus, Equ. (2),(3), (4) and (5) still make sense in this limit. In the sequel we only consider the resulting infinite system. We distinguish two main approaches to the study of open systems. The first one, the Hamiltonian approach, deals directly with the dynamics of the joint system S∨R. We briefly discuss the second one, the Markovian approach, in the next paragraph. In the Hamiltonian approach we rewrite the equation of motion (1) as ˜ Z˙ = −iΩZ, ˜ 2 = m−1/2 Ω 2 m−1/2 with m = I +(M −1)δ0 (δ0 , · ) the operator of multiwhere Ω plication by mx and −Ω 2 is the discrete Laplacian on Z3 . The complex variable Z is ˜ −1/2 m−1/2 p˜ and q˜ = (qx )x∈Z3 , p˜ = (px )x∈Z3 . It fol˜ 1/2 m1/2 q˜+iΩ given by Z = Ω ˜ is self-adjoint lows from elementary spectral analysis that for M > 1 the operator√Ω ˜ = [0, 2ω0 ] on 2 (Z3 ). ˜ = σac (Ω) with purely absolutely continuous spectrum σ(Ω) ˜ 2 shows A simple argument involving the scattering theory for the pair Ω02 ⊕ω02 /M , Ω that the system S has a unique steady state ρ such that (6) holds for all ρ0 which are absolutely continuous with respect to Lebesgue measure. Moreover, ρ is the marginal on S of the infinite dimensional Gaussian measure Z −1 e−βH dpdqdP dQ which describes the thermal equilibrium state of the joint system at temperature T = 1/kB β. This is easily computed to be the Gaussian measure ρ(dP, dQ) = N −1 e−β(P

2

/2M +ω 2 Q2 /2)

where N is a normalization factor and ω2 =

1 . (δ0 , Ω −2 δ0 )

dP dQ,

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Preface

The Markovian Approach A remarkable feature of Equ. (2) is the memory effect induced by the kernel K. As a result the process z t is non-Markovian, i.e., for s > 0, z t+s does not only depend on z t and {ξ(u) | u ∈ [t, t + s]} but also on the full history {z u | u ∈ [0, t]}. The only way to avoid this effect is to have K proportional to the derivative of a delta function. By Relation (5) this means that ξ should be a white noise. This is certainly not the case with our choice of initial conditions. However, as we shall see, it is possible to obtain a Markov process in some particular scaling limits. This is not a uniquely defined procedure: different scaling limits correspond to different physical regimes and lead to distinct Markov processes. As a simple illustration let us consider the particular scaling limit QM (t) ≡ M 1/4 Q(M 1/2 t),

M → ∞.

of our model. For finite M the equation of motion for QM reads t 2 ¨ QM (t) = −ω0 QM (t) + KM (t − s)QM (s) ds + ξM (t), 0

where KM (t) ≡ M 1/2 K(M 1/2 t), and the scaled process ξM (t) ≡ M 1/4 ξ(M 1/2 t) has covariance CM (t) ≡ M 1/2 C(M 1/2 t). One can show that C(t) is in L1 (R) and that σ = C(t) dt > 0. It follows that, in distributional sense, lim CM (t) = σδ(t),

M →∞

lim KM (t) = 0.

M →∞

We conclude that the limiting equation for Q is ¨ = −ω02 Q(t) + σ 1/2 η(t), Q(t) ˙ where η is white noise, i.e., E(η(t)η(s)) = δ(t − s). The solution (Q(t), Q(t)) is a Markov process on R3 × R3 with generator σ L = − ∆2P − P · ∇Q + ω02 Q · ∇P . 2 It is a simple exercise to show that the unique invariant measure of this process is the Lebesgue measure. Moreover, one can show that for any initial condition (Q0 , P0 ) and any function f ∈ L1 (R3 × R3 , dQdP ) one has ˙ lim E(f (Q(t), Q(t))) = f (Q, P ) dQdP, t→∞

a scaled version of return to equilibrium.

Preface

XI

It is worth pointing out that in many instances of classical or quantum open systems the dynamics of the joint system S ∨ R is too complicated to be controlled analytically or even numerically. Thus, the Hamiltonian approach is inefficient and the Markovian approximation becomes the only available option. The study of the Markovian dynamics of open systems is the main subject of the second volume in this series. The third volume is devoted to applications of the techniques introduced in the first two volumes. It aims at leading the reader to the front of the current research on open quantum systems. Organization of this Volume This first volume is devoted to the Hamiltonian approach. Its purpose is to develop the mathematical framework necessary to define and study the dynamics and thermodynamics of quantum systems with infinitely many degrees of freedom. The first two lectures by A. Joye provide a minimal background in operator theory and statistical mechanics. The third lecture by S. Attal is an introduction to the theory of operator algebras which is the natural framework for quantum mechanics of many degrees of freedom. Quantum dynamical systems and their ergodic theory are the main object of the fourth lecture by C.-A. Pillet. The fifth lecture by M. Merkli deals with the most common instances of environments in quantum physics, the ideal Bose and Fermi gases. Finally the last lecture by V. Jakˇsi´c introduces one of the main tool in the study of quantum dynamical systems: spectral analysis.

Lyon, Grenoble and Toulon, September 2005

St´ephane Attal Alain Joye Claude-Alain Pillet

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References 1. Caldeira, A.O., Leggett, A.J.: Path integral approach to quantum Brownian motion. Physica A 121 (1983), 587. 2. Davies, E.B.: Markovian master equations. Commun. Math. Phys. 39 (1974), 91. 3. Einstein, A: Uber die von der molekularkinetischen Theorie der W¨arme geforderte Bewegung von in ruhenden Fl¨ussigkeiten suspendierten Teilchen. Ann. Phys. 17 (1905), 549. 4. Einstein, A: Investigations on the Theory of Brownian Movement. Dover, New York 1956. 5. Einstein, A: Zur Quantentheorie der Strahlung. Physik. Zeitschr. 18 (1917), 121. 6. Ford, G.W., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys. 6 (1965), 504. 7. Lindblad, G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40 (1975), 147. 8. Pauli, W.: Festschrift zum 60. Geb¨urtstage A. Sommerfeld. S. Hirzel, Leipzig 1928. 9. Reif, F.: Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York 1965. 10. Schwinger, J.: Brownian motion of a quantum oscillator. J. Math. Phys. 2 (1961), 407. 11. Van Hove, L.: Master equation and approach to equilibrium for quantum systems. In Fundamental Problems in Statistical Mechanics. E.G.D. Cohen ed., North Holland, Amsterdam 1962. 12. Wax, N. (Editor):Noise and Stochastic Processes. Dover, New York 1954. 13. Weisskopf, V., Wigner, E.: Berechnung der nat¨urlichen Linienbreite auf Grund der Diracschen Lichttheorie. Zeitschr. f¨ur Physik 63 (1930), 54.

Contents

Introduction to the Theory of Linear Operators Alain Joye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generalities about Unbounded Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adjoint, Symmetric and Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . 4 Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 L2 Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Stone’s Theorem, Mean Ergodic Theorem and Trotter Formula . . . . . . . . . 6 One-Parameter Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 13 15 22 29 35 40

Introduction to Quantum Statistical Mechanics Alain Joye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Fermions and Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Boltzmann Gibbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 42 46 53 54 54 57 67

Elements of Operator Algebras and Modular Theory St´ephane Attal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 First definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 70 70 71 71 71 73

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2.3 Representations and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.4 Commutative C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3 von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.1 Topologies on B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2 Commutant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.3 Predual, normal states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4 Modular theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.1 The modular operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 The modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Self-dual cone and standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Quantum Dynamical Systems Claude-Alain Pillet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2 The State Space of a C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.2 The GNS Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3 Classical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.1 Basics of Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.2 Classical Koopmanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4 Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.1 C ∗ -Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.2 W ∗ -Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.3 Invariant States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4 Quantum Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.5 Standard Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.6 Ergodic Properties of Quantum Dynamical Systems . . . . . . . . . . . . . . 153 4.7 Quantum Koopmanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.8 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5 KMS States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.2 Perturbation Theory of KMS States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 The Ideal Quantum Gas Marco Merkli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 2 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 2.1 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 2.2 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 2.3 Weyl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2.4 The C ∗ -algebras CARF (H), CCRF (H) . . . . . . . . . . . . . . . . . . . . . . . . 194 2.5 Leaving Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Contents

3

The CCR and CAR algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The algebra CAR(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The algebra CCR(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Schr¨odinger representation and Stone – von Neumann uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Q–space representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Equilibrium state and thermodynamic limit . . . . . . . . . . . . . . . . . . . . . 4 Araki-Woods representation of the infinite free Boson gas . . . . . . . . . . . . . . 4.1 Generating functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ground state (condensate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Equilibrium states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Dynamical stability of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XV

198 199 200 203 207 209 213 214 217 222 224 228 233

Topics in Spectral Theory Vojkan Jakˇsi´c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 2 Preliminaries: measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 2.2 Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 2.3 Riesz representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 2.4 Lebesgue-Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 2.5 Fourier transform of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 2.6 Differentiation of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 3 Preliminaries: harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 3.1 Poisson transforms and Radon-Nikodym derivatives . . . . . . . . . . . . . . 249 3.2 Local Lp norms, 0 < p < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.3 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.4 Local Lp -norms, p > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 3.5 Local version of the Wiener theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 3.6 Poisson representation of harmonic functions . . . . . . . . . . . . . . . . . . . 256 3.7 The Hardy class H ∞ (C+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 3.8 The Borel transform of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4 Self-adjoint operators, spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 4.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 4.2 Digression: The notions of analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . 269 4.3 Elementary properties of self-adjoint operators . . . . . . . . . . . . . . . . . . 269 4.4 Direct sums and invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 4.5 Cyclic spaces and the decomposition theorem . . . . . . . . . . . . . . . . . . . 273 4.6 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 4.7 Proof of the spectral theorem—the cyclic case . . . . . . . . . . . . . . . . . . . 274 4.8 Proof of the spectral theorem—the general case . . . . . . . . . . . . . . . . . 277

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4.9 Harmonic analysis and spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Spectral measure for A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 The essential support of the ac spectrum . . . . . . . . . . . . . . . . . . . . . . . . 4.12 The functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 The Weyl criteria and the RAGE theorem . . . . . . . . . . . . . . . . . . . . . . . 4.14 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Scattering theory and stability of ac spectra . . . . . . . . . . . . . . . . . . . . . 4.16 Notions of measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Non-relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Spectral theory of rank one perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Aronszajn-Donoghue theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Spectral averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simon-Wolff theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Some remarks on spectral instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Boole’s equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Poltoratskii’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 F. & M. Riesz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Problems and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 280 281 281 283 285 286 287 290 291 295 296 298 299 300 301 302 304 308 309 311

Index of Volume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Information about the other two volumes Contents of Volume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Volume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents of Volume III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Volume III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

318 321 323 327

List of Contributors

St´ephane Attal Institut Camille Jordan Universit´e Claude Bernard Lyon1 21 av. Claude Bernard 69622 Villeurbanne Cedex France email: [email protected] Vojkan Jakˇsi´c Department of Mathematics and Statistics McGill University 805 Sherbrooke Street West Montreal, QC, H3A 2K6 Canada e-mail: [email protected] Alain Joye Institut Fourier Universit´e de Grenoble 1 BP 74 38402 Saint-Martin d’H`eres Cedex France email: [email protected]

Marco Merkli Department of Mathematics and Statistics McGill University 805 Sherbrooke Street West Montreal, QC, H3A 2K6 Canada email: [email protected] Claude-Alain Pillet CPT-CNRS (UMR 6207) Universit´e du Sud Toulon-Var BP 20132 83957 La Garde Cedex France email: [email protected]

Introduction to the Theory of Linear Operators Alain Joye Institut Fourier, Universit´e de Grenoble 1, BP 74, 38402 Saint-Martin-d’H`eres Cedex, France e-mail: [email protected]

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Generalities about Unbounded Operators . . . . . . . . . . . . . . . . . . . . . . . .

2

3

Adjoint, Symmetric and Self-adjoint Operators . . . . . . . . . . . . . . . . . . .

5

4

Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

4.1 4.2

Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L2 Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 22

5

Stone’s Theorem, Mean Ergodic Theorem and Trotter Formula . . . .

29

6

One-Parameter Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

1 Introduction The purpose of this first set of lectures about Linear Operator Theory is to provide the basics regarding the mathematical key features of unbounded operators to readers that are not familiar with such technical aspects. It is a necessity to deal with such operators if one wishes to study Quantum Mechanics since such objects appear as soon as one wishes to consider, say, a free quantum particle in R. The topics covered by these lectures are quite basic and can be found in numerous classical textbooks, some of which are listed at the end of these notes. They have been selected in order to provide the reader with the minimal background allowing to proceed to the more advanced subjects that will be treated in subsequent lectures, and also according to their relevance regarding the main subject of this school on Open Quantum Systems. Obviously, there is no claim about originality in the presented material. The reader is assumed to be familiar with the theory of bounded operators on Banach spaces and with some of the classical abstract Theorems in Functional Analysis.

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Alain Joye

2 Generalities about Unbounded Operators Let us start by setting the stage, introducing the basic notions necessary to study linear operators. While we will mainly work in Hilbert spaces, we state the general definitions in Banach spaces. If B is a Banach space over C with norm · and T is a bounded linear operator on B, i.e. T : B → B, its norm is given by T = sup ϕ=0

T ϕ < ∞. ϕ

Now, consider the position operator of Quantum Mechanics q = mult x on L2 (R), acting as (qϕ)(x) = xϕ(x). It is readily seen to be unbounded since one can find a sequence of normalized functions ϕn ∈ L2 (R), n ∈ N, such that qϕn → ∞ as n → ∞, and, there are functions of L2 (R) which are no longer L2 (R) when multiplied by the independent variable. We shall adopt the following definition of (possibly unbounded) operators. Definition 2.1. A linear operator on B is a pair (A, D) where D ⊂ B is a dense linear subspace of B and A : D → B is linear. We will nevertheless often talk about the operator A and call the subspace D the domain of A. It will sometimes be denoted by Dom(A). ˜ D) ˜ is another linear operator such that D ˜ ⊃ D and Aϕ ˜ = Aϕ Definition 2.2. If (A, for all ϕ ∈ D, the operator A˜ defines an extension of A and one denotes this fact by ˜ A ⊂ A. That the precise definition of the domain of a linear operator is important for the study of its properties is shown by the following Example 2.3. : Let H be defined on L2 [a, b], a < b finite, as the differential operator Hϕ(x) = −ϕ (x), where the prime denotes differentiation. Introduce the dense sets DD and DN in L2 [a, b] by

DD = ϕ ∈ C 2 [a, b] | ϕ(a) = ϕ(b) = 0 (1)

2 DN = ϕ ∈ C [a, b] | ϕ (a) = ϕ (b) = 0 . (2) It is easily checked that 0 is an eigenvalue of (H, DN ) but not of (H, DD ). The boundary conditions appearing in (1), (2) respectively are called Dirichlet and Neumann boundary conditions respectively. The notion of continuity naturally associated with bounded linear operators is replaced for unbounded operators by that of closedness. Definition 2.4. Let (A, D) be an operator on B. It is said to be closed if for any sequence ϕn ∈ D such that ϕn → ϕ ∈ B and Aϕn → ψ ∈ B, it follows that ϕ ∈ D and Aϕ = ψ.

Introduction to the Theory of Linear Operators

Remark 2.5. by

3

i. In terms of the graph of the operator A, denoted by Γ (A) and given Γ (A) = {(ϕ, ψ) ∈ B × B | ϕ ∈ D, ψ = Aϕ} ,

we have the equivalence A closed ⇐⇒ Γ (A) closed (for the norm (ϕ, ψ) 2 = ϕ 2 + ψ 2 ). ii. If D = B, then A is closed if and only if A is bounded, by the Closed Graph Theorema . ¯ = B so that it is possible to extend A to the iii. If A is bounded and closed, then D whole of B as a bounded operator. iv. If A : D → D ⊂ B is one to one and onto, then A is closed is equivalent to A−1 : D → D is closed. This last property can be seen by introducing the inverse graph of A, Γ (A) = {(x, y) ∈ B × B | y ∈ D, x = Ay} and noticing that A closed iff Γ (A) is closed and Γ (A) = Γ (A−1 ). The notion of spectrum of operators is a key issue for applications in Quantum Mechanics. Here are the relevant definitions. Definition 2.6. The spectrum σ(A) of an operator (A, D) on B is defined by its complement σ(A)C = ρ(A), where the resolvent set of A is given by ρ(A) = {z ∈ C | (A − z) : D → B is one to one and onto, and (A − z)−1 : B → D is a bounded operator.} The operator R(z) = (A − z)−1 is called the resolvent of A. Actually, A − z is to be understood as A − z1l, where 1l denotes the identity operator. Here are a few of the basic properties related to these notions. Proposition 2.7. With the notations above, i. If σ(A) = C, then A is closed. ii. If z ∈ ρ(A) and u ∈ C is such that |u| < R(z) −1 , then z + u ∈ ρ(A). Thus, ρ(A) is open and σ(A) is closed. iii. The resolvent is an analytic map from ρ(A) to L(B), the set of bounded linear operators on B, and the following identities hold for any z, w ∈ ρ(A), R(z) − R(w) = (z − w)R(z)R(w) dn R(z) = n! Rn+1 (z). dz n

a

(3)

If T : X → Y, where X and Y are two Banach spaces, then T is bounded iff the graph of T is closed.

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Alain Joye

Remark 2.8. Identity (3) is called the first resolvent identity. As a consequence, we get that the resolvents at two different points of the resolvent set commute, i.e. [R(z), R(w)] = 0, ∀z, w ∈ ρ(A). Proof. i) If z ∈ ρ(A), then R(z) is one to one and bounded thus closed and remark iv) above applies. ii) We need to show that R(z + u) exists and is bounded from B to D. We have on D (A − z − u)ϕ = (1l − u(A − z)−1 )(A − z)ϕ = (1l − uR(z))(A − z)ϕ, where |u| R(z) < 1 by assumption. Hence, the Neumann series T n = (1l − T )−1 where T : B → B is such that T < 1,

(4)

n≥0

shows that the natural candidate for (A − z − u)−1 is R(z)(1l − uR(z))−1 : B → D. Then one checks that on B (A − z − u)R(z)(1l − uR(z))−1 = (1l − uR(z))(1l − uR(z))−1 = 1l and that on D R(z)(1l − uR(z))−1 (A − z − u) = (1l − uR(z))−1 R(z)(A − z − u) = (1l − uR(z))−1 (1l − uR(z)) = 1lD , where 1lD denotes the identity of D. iii) By (4) we can write R(z + u) =

un Rn+1 (z)

n≥0

so that we get the analyticity of the resolvent and the expression for its derivatives. Finally for ϕ ∈ D ((A − z) − (A − w))ϕ = (w − z)ϕ so that, for any ψ ∈ B, R(z)((A − z) − (A − w))R(w)ψ = R(w)ψ − R(z)ψ = R(z)R(w)(w − z)ψ, where R(w)ψ ∈ D.

Note that in the bounded case, the spectrum of an operator is never empty nor equal to C, whereas there exist closed unbounded operators with empty spectrum or d on L2 [0, 1] on the following empty resolvent set. Consider for example, T = i dx 2 dense sets. If AC [0, 1] denotes the set of absolutely continuous functions on [0, 1] whose derivatives are in L2 [0, 1], (hence in L1 [0, 1]), set

Introduction to the Theory of Linear Operators

5

D1 = {ϕ | ϕ ∈ AC 2 [0, 1]}, D0 = {ϕ | ϕ ∈ AC 2 [0, 1] and ϕ(0) = 0}. Then, one checks that (T, D1 ) and (T, D0 ) are closed and such that σ1 (T ) = C and σ0 (T ) = ∅ (with the obvious notations). To avoid potential problems related to the fact that certain operators can be a priori defined on dense sets on which they may not be closed, one introduces the following notions. Definition 2.9. An operator (A, D) is closable if it possesses a closed extension ˜ D). ˜ (A, ¯ D) ¯ Lemma 2.10. If (A, D) is closable, then there exists a unique extension (A, ¯ ˜ called the closure of (A, D) characterized by the fact that A ⊆ A for any closed ˜ D) ˜ of (A, D). extension (A, Proof. Let ¯ = {ϕ ∈ B | ∃ϕn ∈ D and ψ ∈ B with ϕn → ϕ and Aϕn → ψ}. D

(5)

¯ we have ϕ ∈ D ˜ and Aϕ ˜ = ψ is For any closed extension A˜ of A and any ϕ ∈ D, ¯ ¯ ¯ ¯ Then A¯ uniquely determined by ϕ. Let us define (A, D) by Aϕ = ψ, for all ϕ ∈ D. ˜ ¯ ˜ is an extension of A and any closed extension A ⊆ A is such that A ⊆ A. The graph ¯ of A¯ satisfies Γ (A) ¯ = Γ (A), so that A¯ is closed. Γ (A) Note also that the closure of a closed operator coincide with the operator itself. Also, before ending this section, note that there exist non closable operators. Fortunately enough, such operators do not play an essential role in Quantum Mechanics, as we will shortly see.

3 Adjoint, Symmetric and Self-adjoint Operators The arena of Quantum Mechanics is a complex Hilbert space H where the notion of scalar product · | · gives rise to a norm denoted by · . Operators that are self-adjoint with respect to that product play a particularly important role, as they correspond to the observables of the theory. We shall assume the following convention regarding the positive definite sesquilinear form · | · on H × H: it is linear in the right variable and thus anti-linear in the left variable. We shall also always assume that our Hilbert space is separable, i.e. it admits a countable basis, and we shall identify the dual H of H with H, since ∀l ∈ H , ∃! ψ ∈ H such that l(·) = ψ| ·. Let us make the first steps towards self-adjunction. Definition 3.1. An operator (H, D) in H is said to be symmetric if ∀ϕ, ψ ∈ D ⊆ H ϕ|Hψ = Hϕ|ψ. 2

2

d d For example, the operators (− dx 2 , DD ) and (− dx2 , DN ) introduced above are symmetric, as shown by integration by parts.

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Remark 3.2. If H is symmetric, its eigenvalues are real. The next property is related to an earlier remark concerning the role of non closable operators in Quantum Mechanics. Proposition 3.3. Any symmetric operator (H, D) is closable and its closure is symmetric. This Proposition will allow us to consider that any symmetric operator is closed from now on. ¯ ⊇ D as in (5) and χ ∈ D, ϕ ∈ D. ¯ We compute for any such χ, Proof. Let D ϕ|Hχ = limϕn |Hχ = limHϕn |χ = ψ|χ. n

n

(6)

As D is dense by assumption, the vector ψ is uniquely determined by the linear, bounded form lψ : D → C such that lψ (χ) = ψ|χ. In other words, ψ is charac¯ on D ¯ by Hϕ ¯ = ψ and linearity is easily terized by ϕ uniquely. One then defines H ¯ is a closed extension of ¯ = Γ (H) is closed, H checked. As, by construction, Γ (H) ¯ H. Let us finally check the symmetry property. If χn ∈ D is such that χn → χ ∈ D, ¯ (6) says with Hχn → η and ϕ ∈ D, ¯ ϕ|Hχn = Hϕ|χ n . Taking the limit n → ∞, we get from the above ¯ = limHϕ|χ ¯ ¯ limϕ|Hχn = ϕ|η = ϕ|Hχ n = Hϕ|χ. n

n

When dealing with bounded operators, symmetric and self-adjoint operators are identical. It is not necessarily true in the unbounded case. As one of the most powerful tools in linear operator theory, namely the Spectral Theorem, applies only to self-adjoint operators, we will develop some criteria to distinguish symmetric and self-adjoint operators. Definition 3.4. Let (A, D) be an operator on H. The adjoint of A, denoted by (A∗ , D∗ ), is determined as follows: D∗ is the set of ψ ∈ H such that there exists a χ ∈ H so that ψ|Aϕ = χ|ϕ, ∀ϕ ∈ D. As D is dense, χ is unique, so that one sets A∗ ψ = χ and checks easily the linearity. Therefore, ψ|Aϕ = A∗ ψ|ϕ, ∀ϕ ∈ D, ψ ∈ D∗ . In other words, ψ ∈ D∗ iff the linear form l(·) = ψ|A· : D → C is bounded. In that case, Riesz Lemma implies the existence of a unique χ such that ψ|A· = χ|·. Note also that D∗ is not necessarily dense. Let us proceed with some properties of the adjoint.

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Proposition 3.5. Let (A, D) be an operator on H. i. The adjoint (A∗ , D∗ ) of (A, D) is closed. If, moreover, A is closable, then D∗ is dense. ii. If A is closable, A¯ = A∗∗ . iii. If A ⊆ B, then B ∗ ⊆ A∗ . Proof. i) Let (ψ, χ) ∈ D∗ × H belong to Γ (A∗ ). This is equivalent to saying ψ|Aϕ = χ|ϕ, ∀ϕ ∈ D, which is equivalent to (ψ, χ) ∈ M ⊥ , where M = {(Aϕ, −ϕ) ∈ H × H, | ϕ ∈ D}, with the scalar product (ϕ1 , ϕ2 )|(ψ1 , ψ2 ) = ϕ1 |ψ1 + ϕ2 |ψ2 . As M ⊥ is closed, Γ (A∗ ) is closed too. Assume now A is closable and suppose there exists η ∈ H such that ψ|η = 0, for all ψ ∈ D∗ . This implies that (η, 0) is orthogonal to Γ (A∗ ). But, Γ (A∗ )⊥ = M ⊥⊥ = M . Therefore, there exists ϕn ∈ D, such that ϕn → 0 and Aϕn → η. As A is closable, ¯ = 0, i.e. (D∗ )⊥ = 0 and D∗ = (D∗ )⊥⊥ = H. η = A0 ii) Define a unitary operator V on H × H by V (ϕ, ψ) = (ψ, −ϕ). It has the property V (E ⊥ ) = (V (E))⊥ , for any linear subspace E ⊆ H × H. In particular, we have just seen Γ (A∗ ) = (V (Γ (A)))⊥ so that Γ (A) = (Γ (A)⊥ )⊥ = ((V 2 Γ (A))⊥ )⊥ = (V (V (Γ (A))⊥ ))⊥ = (V (Γ (A∗ )))⊥ = Γ (A∗∗ ), i.e. A¯ = A∗∗ . iii) Follows readily from the definition.

When H is symmetric, we get from the definition and properties above that H ∗ is a closed extension of H. This motivates the Definition 3.6. An operator (H, D) is self-adjoint whenever it coincides with its adjoint (H ∗ , D∗ ). It is therefore closed. An operator (H, D) is essentially self-adjoint if it is symmetric and its closure ¯ D) ¯ is self-adjoint. (H,

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Therefore, we have in general for a symmetric operator, ¯ = H ∗∗ ⊆ H ∗ , and H ∗ = H ∗ = H ∗∗∗ = H ¯ ∗. H⊆H In case H is essentially self-adjoint, ¯ = H ∗∗ = H ∗ . H⊆H We now head towards our general criterion for (essential) self-adjointness. We need a few more Definition 3.7. For (H, D) symmetric and denoting its adjoint by (H ∗ , D∗ ), the deficiency subspaces L± are defined by L± = {ϕ ∈ D∗ | H ∗ ϕ = ±iϕ} = {ϕ ∈ H | Hψ|ϕ = ±iψ|ϕ ∀ψ ∈ D} = Ran(H ± i)⊥ = Ker(H ∗ ∓ i). The deficiency indices are the dimensions of L± , which can be finite or infinite. To get an understanding of these names, recall that one can always write H = Ker(H ∗ ∓ i) ⊕ Ran(H ± i) ≡ L± ⊕ Ran(H ± i).

(7)

¯ Note that the definitions of L± is invariant if one replaces H by its closure H. For (H, D) symmetric and any ϕ ∈ D observe that (H + i)ϕ 2 = Hϕ 2 + ϕ 2 = (H − i)ϕ 2 = 0. This calls for the next Definition 3.8. Let (H, D) be symmetric. The Cayley transform of H is the isometric operator U = (H − i)(H + i)−1 : Ran(H + i) → Ran(H − i). It enjoys the following property. Lemma 3.9. The symmetric extensions of H are in one to one correspondence with the isometric extensions of U . ˜ D) ˜ be a symmetric extension of (H, D) and U ˜ be its Cayley transProof. Let (H, form. We have ˜ such that ϕ = (H ± i)ψ = (H ˜ ± i)ψ, ϕ ∈ Ran(H ± i) ⇐⇒ ∃ψ ∈ D ⊆ D ˜ ± i), and hence Ran(H ± i) ⊂ Ran(H ˜ ϕ = (H ˜ − i)(H ˜ + i)−1 ϕ = U ϕ, ∀ϕ ∈ Ran(H ± i). U

(8)

˜ : M + → M − , be a isometric extension of U , where Ran(H ±i) ⊆ Conversely, let U ± M . We need to construct a symmetric extension of H whose Cayley transform is ˜ . Algebraically this means, see (8), U

Introduction to the Theory of Linear Operators

˜ = (U ˜ − 1l)−1 2 − i. H i

9

(9)

˜ . If ϕ ∈ M + is a corresponding eigenLet us show that 1 is not an eigenvalue of U vector, and ψ = (H + i)χ, where χ ∈ D, then 2iϕ|χ = ϕ|(H + i)χ − (H − i)χ = ϕ|ψ − U ψ ˜ ϕ|U ˜ ψ = 0. = ϕ|ψ − U ˜ by (9) on D ˜ = (U ˜ − 1l)M + . It is By density of D, ϕ = 0, so that we can define H ˜ is a symmetric extension of H. not difficult to check that H We can now state the Theorem 3.10. If (H, D) is symmetric on H, there exist self-adjoint extensions of H if and only if the deficiency indices are equal. Moreover, the following statements are equivalent: 1. H is essentially self-adjoint. 2. The deficiency indices are both zero. 3. H possesses exactly one self-adjoint extension. Proof. 1)⇒ 3): Let J be a self-adjoint extension of H. Then H ⊆ J = J ∗ and ¯ ∗ = H, ¯ so that J = H. ¯ ¯ Hence J = J ∗ ⊆ H J ⊇ H. ¯ = H ∗ . For any ϕ ∈ L± = 1)⇒ 2): We can assume that H is closed so H = H ∗ Ker(H ∓ i), 0 = (H ∗ ∓ i)ϕ 2 = (H ∓ i)ϕ 2 = Hϕ 2 + ϕ 2 ≥ ϕ 2 , L± = {0}. (10) 2) ⇒ 1): Consider (H + i) : D → Ran(H + i). By (10) above, this operator is one to one, and we can define (H + i)−1 : Ran(H + i) → D. By the same estimate it satisfies (H + i)−1 ψ 2 ≤ (H + i)(H + i)−1 ψ 2 = ψ 2 . ¯ and L+ = {0}, we get that As H can be assumed to be closed (i.e. H = H) Ran(H +i) is closed so that H = Ran(H +i), due to (7). Therefore, for any ϕ ∈ D∗ , there exists a ψ ∈ D such that (H ∗ + i)ϕ = (H + i)ψ. As H ⊆ H ∗ , (H ∗ + i)(ϕ − ψ) = 0, i.e. ϕ − ψ ∈ Ker(H ∗ + i) = {0}, we get that ϕ ∈ D and H = H ∗ , which is what we set out to prove. 3) ⇒ 2): if K is a self-adjoint extension of H, its deficiency indices are zero (by 2)). Therefore, (see (7)), its Cayley transform V is a unitary extension of U , the Cayley transform of H. In particular, V |L+ : L+ → L− is one to one and onto, so that the deficiency indices of H are equal. That yields the first part of the Theorem. Now assume these indices are not zero. By the preceding Lemma, there exist an infinite number of symmetric extensions of H, parametrized by all isometries from L+ to L− . In particular, there exist extensions with zero deficiency indices, which by 2) and 1) are self-adjoint, contradicting the fact that K is the unique self-adjoint extension of H.

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Remark 3.11. It is a good exercise to prove that in case (H, D) is symmetric and H ≥ 0, i.e. ϕ|Hϕ ≥ 0 for any ϕ ∈ D, then H is essentially self-adjoint iff Ker(H ∗ + 1) = {0}. As a first application, we give a key property of self-adjoint operators for the role they play in the Quantum dogma concerning measure of observables. It is the following fact concerning their spectrum. Theorem 3.12. Let H = H ∗ . Then, σ(H) ⊆ R and, (H − z)−1 ≤

1 , if z ∈ R. |z|

(11)

Moreover, for any z in the resolvent set of H, ∗

(H − z¯)−1 = ((H − z)−1 ) .

(12)

Proof. Let ϕ ∈ D, D being the domain of H and z = x + iy, with y = 0. Then (H − x − iy)ϕ 2 = (H − x)ϕ 2 + y 2 ϕ 2 ≥ y 2 ϕ 2 .

(13)

This implies Ker(H − z) = Ker(H ∗ − z) = {0} i.e. Ran(H − z) = H, and H − z is invertible on Ran(H − z). (13) shows that (H − z) is bounded with the required bound, and as the resolvent is closed, it can be extended on H with the same bound. Equality (12) is readily checked. As an application of the first part of Theorem 3.10, consider a symmetric operator (H, D) which commutes with a conjugation C. More precisely: C is anti-linear, C 2 = 1l and Cϕ = ϕ . Hence ϕ|ψ = Cψ|Cϕ. Moreover, C : D → D and CH = HC on D. Under such circumstances, the deficiency indices of H are equal and there exist selfadjoint extensions of H. Indeed, one first deduces that C(D) = D. Then, for any ϕ+ ∈ L+ = Ker(H ∗ − i) and ψ ∈ D, we compute 0 = ϕ+ |(H + i)ψ = Cϕ+ |C(H + i)ψ = Cϕ+ |(H − i)Cψ, so that Cϕ+ ∈ Ran(H − i)⊥ = Ker(H ∗ + i) = L− . In other words, one has C : L+ → L− , and one shows similarly that C : L− → L+ . As C is isometric, the dimensions of L+ and L− are the same. A particular case where this happens is that of the complex conjugation and a differential operator on Rn , with real valued coefficients.

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An example of direct application of this criterion is the following. Consider the symmetric operator Hϕ = iϕ on the domain C0∞ (0, ∞) ⊂ L2 (0, ∞). A vector ψ ∈ D∗ iff there exists χ ∈ L2 (0, ∞) such that ψ|Hϕ = χ|ϕ, for all ϕ ∈ C0∞ (0, ∞). Expressing the scalar products this means χ(x)ϕ(x)dx ¯ = −i ψ(x)ϕ¯ (x)dx = iDx Tψ (ϕ), ¯ where Tψ denotes the distribution associated with ψ. In other words, we have ψ ∈ W 1,2 (0, ∞) = D∗ and H ∗ ψ = iψ in the weak sense. Elements of Ker(H ∗ ∓ i) satisfy 2 ∗ ±x ∈ L (0, ∞) H ψ = ±iψ ⇐⇒ ψ = ±ψ ⇐⇒ ψ(x) = ce ∈ L2 (0, ∞) Hence there is no self-adjoint extension of that operator. If it is considered on C0∞ (0, 1) ⊂ L2 (0, 1), the above shows that the deficiency indices are both 1 and there exist infinitely many self-adjoint extensions of it. Specializing a little, we get a criterion for operators whose spectrum consists of eigenvalues only. Corollary 3.13. Let (H, D) symmetric on H such that there exists an orthonormal basis {ϕn }n∈N of H of eigenvectors of H satisfying for any n ∈ N, ϕn ∈ D ¯ = and Hϕn = λn ϕn , with λn ∈ R. Then H is essentially self-adjoint and σ(H) {λn | n ∈ N}. Proof. Just note that any vector ϕ in L± satisfies in particular Hϕn |ϕ = ±iϕn |ϕ = λn ϕn |ϕ, so that ϕn |ϕ = 0 for any n. This means that L+ = {0}, hence that H is essentially self-adjoint. ¯ as an extension of H admits the ϕn ’s as eigenvectors with the same Then H, ¯ ⊃ {λn | n ∈ N}. If λ eigenvalues and as the spectrum is a closed set, we get σ(H) does not belong to the latter set, we define Rλ by Rλ ϕn = λn1−λ ϕn , for all n ∈ N. ¯ is closed, it is not difficult to see that Rλ is the resolvent of H ¯ Using the fact that H at λ, which yields the result. 2

d 2 ∞ As a first example of application we get that − dx (a, b)) 2 on C (a, b) (or C with Dirichlet boundary conditions is essentially self-adjoint with spectrum

2 2 n π ∗ n∈N , (b − a)2

as the corresponding eigenvectors ϕn (x) =

2 b−a

1/2 sin

nπ(x − a) b−a

, n ∈ N∗ ,

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are known to form a basis of L2 [a, b] by the theory of Fourier series. Another standard operator is the harmonic oscillator defined on L2 (R) by the differential operator 1 d2 x2 Hosc = − + 2 2 dx 2 with dense domain S the Schwartz functions. This operator is symmetric by integration by parts, and using creation and annihilation operators √ it is a standard exercise, √ b† = (x − ∂x )/ 2, b = (x + ∂x )/ 2 to show that the solutions of Hosc ϕn (x) = λn ϕn (x), n ∈ N, are given by λn = n + 1/2 with eigenvector ϕn (x) = cn Hn (x)e−x

2

/2

2

, with Hn (x) = (−1)n ex

dn −x2 e , dxn

√ −1/2 . These eigenvectors also form a basis of L2 (R), so that this and cn = (2n n! π) operator is essentially self-adjoint with spectrum N + 1/2. Note that we cannot work on C0∞ to apply this criterion here. Another popular way to prove that an operator is self-adjoint is to compare it to another operator known to be self-adjoint and use a perturbative argument to get self-adjointness of the former. Let (H, D) be a self-adjoint operator on H and let (A, D(A)) be symmetric with domain D(A) ⊇ D. Definition 3.14. The operator A has a relative bound α ≥ 0 with respect to H if there exists c < ∞ such that Aϕ ≤ α Hϕ + c ϕ , ∀ϕ ∈ D.

(14)

The infimum over such relative bounds is the relative bound of A w.r.t. H. Remark 3.15. The definition of the relative bound is unchanged if we replace (14) by the slightly stronger condition Aϕ 2 ≤ α2 Hϕ 2 + c2 ϕ 2 , ∀ϕ ∈ D. Lemma 3.16. Let K : D → H be such that Kϕ = Hϕ + Aϕ. If 0 ≤ α < 1, is the relative bound of A w.r.t. H, K is closed and symmetric. Moreover, A(H + iλ)−1 < 1, if λ ∈ R has large enough modulus. Proof. The symmetry of K is clear. Let us consider ϕn ∈ D such that ϕn → ϕ and Kϕn → ψ. Then, by assumption, Hϕn − Hϕm ≤ Kϕn − Kϕm + Aϕn − Aϕm ≤ Kϕn − Kϕm + α Hϕn − Hϕm + c ϕn − ϕm ,

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so that Hϕn −Hϕm ≤

1 c Kϕn −Kϕm + ϕn −ϕm → 0 as n, m → ∞. 1−α 1−α

H being closed, we deduce from the above that ϕ ∈ D and Hϕn → Hϕ. Then, from (14), we get Aϕn − Aϕ → 0 from which follows Kϕn → Kϕ = ψ. The proof of the statement concerning the resolvent reads as follows. Let ψ ∈ H, ϕ = (H + iλ)−1 ψ and 0 ≤ α < β < 1. Then, for |λ| > 0 large enough 2 Aϕ 2 ≤ (α Hϕ + c ϕ ) ≤ β 2 Hϕ 2 + λ2 ϕ 2 = β 2 (H + iλ)ϕ 2 = β 2 ψ 2 . Hence A(H + iλ)−1 ψ ≤ β ψ .

This leads to the Theorem 3.17. If H is self-adjoint and A is symmetric with relative bound α < 1 w.r.t. H, then K = H + A is self-adjoint on the same domain as that of H. Proof. Let |λ| be large enough. From the formal expressions (H + A + iλ)−1 − (H + iλ)−1 = −(H + A + iλ)−1 (A)(H + iλ)−1 ⇐⇒ (H + A + iλ)−1 = (H + iλ)−1 (1l + A(H + iλ)−1 )−1 we see that the natural candidate for the resolvent of K is Rλ = (H + iλ)−1 (−A(H + iλ)−1 )n . n∈N

By assumption on |λ|, this sum converges in norm and Ran(Rλ ) = D. Routine manipulations show that (H + A + iλ)Rλ = Rλ (H + A + iλ) = 1l so that Ran(H + A + iλ) = H. This implies that the deficiency indices of K = H + A are both zero, and since it is closed, K is self-adjoint. Note that one uses the fact that dim ker(K ∗ − iλ) is constant for λ > 0 and λ < 0.

4 Spectral Theorem Let us start this section by the presentation of another example of self-adjoint operator, which will play a key role in the Spectral Theorem, we set out to prove here. Before getting to work, let us specify right away that we shall not provide here a full proof of the version of the Spectral Theorem we chose. Some parts of it, of a purely analytical character, will be presented as facts whose detailed full proofs can be found in Davies’s book [1]. But we hope to convey the main ideas of the proof in these notes. Consider E ⊆ RN a Borel set and µ a Borel non-negative measure on E. Let H = L2 (E, dµ) be the usual set of measurable functions f : E → C such that

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f 2 = E |f (x)|2 dµ < ∞, with identification of functions that coincide almost everywhere. Let a : E → R be measurable and such that the restriction of a to any bounded set of E is bounded. We set D = {f ∈ H | (1 + a2 (x))|f (x)|2 dµ < ∞}, E

which is dense, and we define the multiplication operator (A, D) by (Af )(x) = a(x)f (x), ∀f ∈ D. Lemma 4.1. (A, D) is self-adjoint and if L2c denotes the set of functions of H which are zero outside a compact subset of E, then A is essentially self-adjoint on L2c . Proof. A is clearly symmetric. If z ∈ R, the bounded operator R(z) given by (R(z)f )(x) = (a(x) − z)−1 f (x) is easily seen to be the inverse of (A − z). Hence, σ(A) = C, so that A is closed. Moreover, the deficiency indices of A are both seen to be zero: ∗ A f = if ⇔ ∀h ∈ D Ahf dµ = i hf dµ ⇔

(a(x) − i)hf dµ = 0,

⇔ f = 0 µ a.e. So that A is closed and essentially self-adjoint, hence self-adjoint. Concerning the last statement, we need to show that A is the closure of its restriction to L2c . If f ∈ D and n ∈ N, we define f (x) if x ∈ E, |x| ≤ n, fn (x) = 0 otherwise. Hence |fn (x)| ≤ |f (x)| and fn ∈ L2c ∈ D. By Lebesgue dominated convergence Theorem, one checks that fn → f and Afn → Af as n → ∞. Lemma 4.2. The spectrum and resolvent of A are such that σ(A) = essential range of a = {λ ∈ R | µ({x | |a(x) − λ| < }) > 0, ∀ > 0}. If λ ∈ σ(A), then (A − λ)−1 =

1 . dist (λ, σ(A))

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Proof. If λ is not in the essential range of a, it is readily checked that the multiplication operator by (a(x) − λ)−1 is bounded (outside of a set of zero µ measure). Also one sees that this operator yields the inverse of a − λ for such λ’s, which, consequently, belong to ρ(A). Conversely, let us take λ in the essential range of a and show that λ ∈ σ(A). We define sets of positive µ measures by Sm = {x ∈ E | |λ − a(x)| < 2−m }. Let χm be the characteristic function of Sm , which is a non zero element of L2 (E, dµ). Then |χm |2 |a(x) − λ|2 dµ ≤ 2−2m |χm |2 dµ = 2−2m χm 2 , (A − λ)χm 2 = Sm

Sm

which shows that (A − λ)−1 cannot be bounded. Finally, if λ is not in the essential range of a, we set (a(·) − λ)−1 ∞ = essential supremum of (a(·) − λ)−1 , where we recall that for a measurable function f f ∞ = inf{K > 0 | |f (x)| ≤ K µ a.e.}. We immediately get that a(·) − λ ∞ is an upper bound for the norm of the resolvent, as, for any > 0 there exists a set S ⊂ E of positive measure such that |(a(x) − λ)−1 | ≥ K − , ∀x ∈ S. Considering the characteristic function of this set, one sees that the upper bound is actually reached and corresponds with the distance of λ to the spectrum of A. 4.1 Functional Calculus Let us now come to the steps leading to the Spectral Theorem. The general setting is as follows. One has a self-adjoint operator (H, D), D dense in a separable Hilbert space H. We first want to define a functional calculus, allowing us to take functions of self-adjoint operators. If H is a multiplication operator by a real valued function h, as in the above example, then f (H), for a reasonable function f : R → C, is easily conceivable as the multiplication by f ◦ h. We are going to define a function of an operator H in a quite general setting by means of an explicit formula due to Helffer and Sj¨ostrand and we will check that this formula has the properties we expect of such an operation. Finally, we will also see that any operator can be seen as a multiplication operator on some L2 (dµ) space. Let us introduce the notation < z >= (1 + |z|2 )1/2 and the set of functions we will work with. Let β ∈ R and S β be the set of complex valued C ∞ (R) functions such that there exists a cn so that n d (n) |f (x)| = n f (x) ≤ cn < x >β−n , ∀x ∈ R ∀n ∈ N. dx

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We set A = ∪β<0 S β and we define norms · n on A, for any n ≥ 1, by n ∞ |f (r) (x)| < x >r−1 dx. f n = r=0

−∞

This set of functions enjoys the following properties: A is an algebra for the multiplication of functions, it contains the rational functions which decay to zero at ∞ and have non-vanishing denominator on the real axis. Moreover, it is not difficult to see that f n < ∞ ⇒ f ∈ L1 (R), and f (x) → 0 as |x| → ∞ and that f − fk n → 0, as k → ∞ ⇒ sup |f (x) − fk (x)| → 0, as k → ∞. x∈R

Definition 4.3. A map which to any f ∈ E ⊂ L∞ (R) associates f (H) ∈ L(H) is a functional calculus if the following properties are true. 1. 2. 3. 4. 5.

f !→ f (H) is linear and multiplicative, (i.e. f g !→ f (H)g(H)) f¯(H) = (f (H))∗ , ∀f ∈ E f (H) ≤ f ∞ , ∀f ∈ E If w ∈ R and rw (x) = (x − w)−1 , then rw (H) = (H − w)−1 If f ∈ C0∞ (R) such that supp(f ) ∩ σ(H) = ∅ then f (H) = 0. For f ∈ C ∞ , we define its quasi-analytic extension f˜ : C → C by n (iy)r (r) ˜ σ(x, y) f (x) f (z) = r! r=0

with z = x + iy, n ≥ 1, σ(x, y) = τ (y/ < x >), where τ ∈ C0∞ is equal to one on [−1, 1], has support in [−2, 2]. We are naturally abusing notations as f˜ is not analytic in general, but it is C ∞ . Its support is confined to the set |y| ≤ 2 < x > due to the presence of τ . Also, the projection on the x axis of the support of f˜ is equal to the support of f . The choice of τ and n will turn out to have no importance for us. Explicit computations yield ∂ ∂ ˜ 1 ∂ +i f (z) = f˜(z) = (15) ∂ z¯ 2 ∂x ∂y n (iy)r (σx (x, y) + iσy (x, y)) (iy)n σ(x, y) + f (n+1) (x) . f (r) (x) r! 2 n! 2 r=0 As supp(σx (x, y)) and supp(σy (x, y)) are included in supp(τ (y/ < x >), i.e. in the set < x >≤ |y| ≤ 2 < x >, if x is fixed and y → 0, ∂ f˜(z) = O(|y|n ), ∂ z¯ which justifies the name quasi-analytic extension as y goes to zero.

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17

Definition 4.4. For any f ∈ A and any self-adjoint operator H on H the HelfferSj¨ostrand formula for f (H) reads ∂ ˜ 1 (16) f (H) = f (z)(H − z)−1 dxdy ∈ L(H). π C ∂ z¯ Remark 4.5. This formula allows to compute functions of operators by means of their resolvent only. Therefore it holds for bounded as well as unbounded operators. Moreover, being explicit, it can yield useful bounds in concrete cases. Note also that it is linear in f . We need to describe a little bit more in what sense this integral holds. Lemma 4.6. The expression (16) converges in norm and the following bound holds f (H) ≤ cn f n+1 , ∀f ∈ A and n ≥ 1.

(17)

Proof. The integrand is bounded and C ∞ on C \ R, therefore (16) converges in norm as a limit of Riemann sums on any compact of C \ R. It remains to deal with the limit when these sets tend to the whole of C. Let us introduce the sets U = {(x, y) | < x >≤ |y| ≤ 2 < x >} ⊇ supp τ (y/ < x >) V = {(x, y) | 0 ≤ |y| ≤ 2 < x >} ⊇ supp τ (y/ < x >). We easily get by explicit computations that |σx (x, y) + iσy (x, y)| ≤

cχU (x, y) , <x>

where χU is the characteristic function of the set U . Using the bound (11) on the resolvent, (15), and the fact that |y| < x > on U , we can bound the integrand of (16) by a constant times n

|f (r) (x)| < x >r−2 χU (x, y) + |f (n+1) (x)||y|n−1 χV (x, y).

r=0

After integration on y at fixed x, the integrand of the remaining integral in x is bounded by a constant times n

|f (r) (x)| < x >r−1 +|f (n+1) (x)| < x >n ,

r=0

hence the announced bound.

We need a few more properties regarding formula (16) before we can show it defines a functional calculus. It is sometimes easier to deal with C0∞ functions rather then with functions of A. The following Lemma shows this is harmless.

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Lemma 4.7. C0∞ (R) is dense in A for the norms · n . Proof. We use the classical technique of mollifiers. Let Φ ≥ 0 be smooth with the same conditions of support as τ . Set Φm (x) = Φ(x/m) for all x ∈ R and fm = Φm f . Hence, fm ∈ A and support considerations yield f − fm n+1 =

n+1 r=0

≤ cn

R

r d < x >r−1 dx (f (x)(1 − Φ (x))) m dxr

n+1 r=0

|x|>m

|f (r) (x)| < x >r−1 dx → 0, as m → ∞.

The next Lemma will be useful several times in the sequel. Lemma 4.8. If F ∈ C0∞ (C) and F (z) = O(y 2 ) as y → 0 at fixed real x, then ∂ 1 F (z)(H − z)−1 dxdy = 0. (18) π C ∂ z¯ Proof. Suppose suppF ⊂ {|x| < N, |y| < N } and let Ωδ , δ > 0 small such that Ωδ ⊂ {|x| < N, δ < |y| < N }. We want to apply Stokes Theorem to the above integral. Recall that ∂ 1 ∂ ∂ 1 ∂ ∂ ∂ = +i = −i , ∂ z¯ 2 ∂x ∂y ∂z 2 ∂x ∂y ⇐⇒ d¯ z = dx − idy, dz = dx + idy so that d¯ z ∧ dz = 2idx ∧ dy = 2idxdy. Moreover, since analyticity,

∂ ∂ z¯ (H

− z)−1 = 0 by

∂ (F (z)(H − z)−1 )dz ∧ dz ∂z ∂ (F (z)(H − z)−1 )d¯ z ∧ dz + ∂ z¯ ∂F (z)(H − z)−1 d¯ z ∧ dz. = ∂ z¯

d(F (z)(H − z)−1 dz) =

Therefore, if I denotes (18), we get by Stokes Theorem 1 1 I = lim d(F (z)(H − z)−1 dz) = lim F (z)(H − z)−1 dz δ→0 2πi Ω δ→0 2πi ∂Ω δ δ 1 −1 dz. = lim y=δ F (z)(H − z) δ→0 2πi y=−δ |x|
Hence the bound

Introduction to the Theory of Linear Operators

1 δ→0 2π

|I| ≤ lim

19

N

1 (|F (x + iδ| + |F (x − iδ|) dx = lim O(δ) = 0, δ→0 δ −N

where we used Taylor’s formula F (x, y) = (0, 1), so that |F (x, y)| ≤ c(x)y 2 .

y2 2 Fyy (x, θ(y, x)y)

with θ(x, y) ∈

Remark 4.9. It follows from the above proof that if f has compact support, we can write 1 f (H) = lim f˜(z)(H − z)−1 dz. δ→0 2πi ∂Ω δ Neglecting support considerations, if f˜ was analytic, this is the way we would naturally define f (H). We can now show a comforting fact about our definition (16) of f (H). Lemma 4.10. If f ∈ A and n ≥ 1, then f (H) is independent of σ and n. Proof. By density of C0∞ in A for the norms · n and Lemma 4.6, we can assume f ∈ C0∞ . Let f˜σ1 ,n and f˜σ2 ,n be associated with σ1 and σ2 . Then n r (iy) (r) f (x) f˜σ1 ,n − f˜σ2 ,n = (τ1 (y/ < x >) − τ2 (y/ < x >)), r! r=0 is identically zero for y small enough, so Lemma 4.8 applies. Similarly, if m > n ≥ 1, with similar notations, f˜σ,m − f˜σ,n =

m

f (r) (x)

r=n+1

and Lemma 4.8 applies again.

(iy)r σ(x, y) = O(y 2 ), as y → 0, x fixed, r!

We are now in a position to show that formula (16) possesses the properties of a functional calculus. Proposition 4.11. With the notations above, a. b. c. d.

If f ∈ C0∞ and supp(f ) ∩ σ(H) = ∅, then f (H) = 0. (f g)(H) = f (H)g(H), for all f, g ∈ A. f¯(H) = f (H)∗ and f (H) ≤ f ∞ . rw (H) = (H − w)−1 , w ∈ R.

Proof. a) In that case, since the compact set supp(f ) and the closed set σ(H) are disjoint, we can consider a finite number of contours γ1 , · · · , γr surrounding a region W disjoint from σ(H) containing the support of f˜. By Stokes Theorem again 1 f (H) = π

C

∂ ˜ 1 f (z)(H − z)−1 dxdy = ∂ z¯ 2πi j=1 r

γj

f˜(z)(H − z)−1 dz ≡ 0,

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by our choice of γj . b) Assume first f, g ∈ C0∞ , so that K =supp(f˜) and L =supp(˜ g ) are compact. ∂ ˜ 1 ∂ g˜(w)(H − z)−1 (H − w)−1 dxdydudv f (H)g(H) = 2 f (z) π K×L ∂ z¯ ∂w ¯ (H − w)−1 − (H − z)−1 ∂ ˜ 1 ∂ g˜(w) dxdydudv. = 2 f (z) π K×L ∂ z¯ ∂w ¯ w−z Then one uses the formula (easily proven using Stokes again) 1 ∂ ˜ dxdy f (z) = f˜(w), π K ∂ z¯ w−z the equivalent one for g˜ and one gets, changing variables to z, 1 ∂ ˜ ∂ ˜ f (H)g(H) = g˜(z) f (z) + f (z) g˜(z) (H − z)−1 dxdy π K∩L ∂ z¯ ∂ z¯ ∂ ˜ 1 (f (z)˜ g (z))(H − z)−1 dxdy. = π K∩L ∂ z¯ It remains to see that if k(z) = f˜(z)˜ g (z) − fg(z), the integral of ∂∂z¯ k(z) against the resolvent on C is zero. But this is again a consequence of Lemma 4.8, since k has compact support and explicit computations yield k(z) = O(y 2 ) as y → 0 with x fixed. The generalization to functions of A is proven along the same lines as Lemma 4.7 with Lemma 4.6. ∗ c) The first point follows from (H − z)−1 = (H − z¯)−1 , the convergence in norm of (16) and the fact that f¯(z) = f(¯ z ) if τ is even, which we can always assume. For the second point, take f ∈ A and c > 0 such that f ∞ ≤ c. Defining g(x) = c − (c2 − |f (x)|2 )1/2 , one checks that g ∈ A as well. The identity f f¯ − 2cg + g 2 = 0 in the algebra A implies with the above f (H)f (H)∗ − 2cg(H) + g(H)g(H)∗ = 0 ⇔ f (H)∗ f (H) + (c − g(H))∗ (c − g(H)) = c2 . Thus, for any ψ ∈ H, it follows f (H)ψ 2 ≤ f (H)ψ 2 + (c − g(H))ψ 2 ≤ c2 ψ 2 , where c ≥ f ∞ is arbitrary. d) Let us take n = 1 and assume w > 0. We further choose σ(x, y) = τ (λy/ < x >), where λ ≥ 1 will be chosen large enough so that w does not belong to the support of σ and then kept fixed in the rest of the argument. The sole effect of this manipulation is to change the support of τ , but everything we have done so far remains true for

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21

λ > 1 and fixed. Let us define, for m > 0 large, Ωm = {(x, y) | |x| < m and < x > /m < |y| < 2m}. Then, by definition and Stokes, 1 ∂ r˜w (z)(H − z)−1 dxdy rw (H) = lim m→∞ π Ω ∂ z ¯ m 1 r˜w (z)(H − z)−1 dz, = lim m→∞ 2πi ∂Ω m

(19)

where, since n = 1, (x)iy)σ(x, y). r˜w (z) = (rw (x) + rw

At this point, we want to replace r˜w (z) by rw (z) in (19). Indeed, it can be shown using the above explicit formula that lim (rw (z) − r˜w (z))(H − z)−1 dz = 0, m→∞

∂Ωm

by support considerations and elementary estimates on the different pieces of ∂Ωm . Admitting this fact we have 1 rw (z)(H − z)−1 dz rw (H) = lim m→∞ 2πi ∂Ω m = res(rw (z)(H − z)−1 )|z=w = (H − w)−1 , due to the analyticity of the resolvent inside Ωm .

We can now state the first Spectral Theorem for the set C∞ (R) of continuous functions that vanish at infinity C∞ (R) = {f ∈ C(R) | ∀ > 0, ∃K compact with |f (x)| < if x ∈ K}. Theorem 4.12. There exists a unique linear map f !→ f (H) from C∞ to L(H) which is a functional calculus. Proof. Replacing C∞ by A we have existence. Now, C0∞ ⊂ A ⊂ C∞ and it is a

· ∞

classical fact that C0∞ = C∞ , [4]. Hence A is dense in C∞ in the sup norm. As f (H) ≤ f ∞ ∀f ∈ A, a density argument yields an extension of the map to C∞ with convergence in norm. It is routine to check that all properties listed in Proposition 4.11 remain true for f ∈ C∞ . The uniqueness property is shown as follows. If there exists another functional calculus, then, by hypothesis, it must agree with ours on the set of functions R n R={ λi rwi , where λi ∈ C, wi ∈ R}. i=1

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But, it is a classical result also that the set R satisfies the hypothesis of the StoneWeierstrass Theoremb and R = C∞ , so that the two functional calculus must coincide everywhere. We shall pursue in two directions. We first want to show that any self-adjoint operator can be represented as a multiplication operator on some L2 space. Then we shall extend the functional calculus to bounded measurable functions. 4.2 L2 Spectral Representation Let (H, D) be self-adjoint on H. Definition 4.13. A closed linear subspace L of H is said invariant under H if (H − z)−1 L ⊆ L for any z ∈ R. Remark 4.14. It is an exercise to show that if ϕ ∈ L ∩ D, then Hϕ ∈ L, as expected. Also, for any λ ∈ R, Ker(H − λ) is invariant. If it is positive, the dimension of this subspace is called the multiplicity of the eigenvalue λ. Lemma 4.15. If L is invariant under H = H ∗ , then L⊥ is invariant also. Moreover, f (H)L ⊆ L, for all f ∈ C∞ (R). Proof. The first point is straightforward and the second follows from the approximation of the integral representation (16) of f (H) for f ∈ A by a norm convergent limit of Riemann sums and by a density argument for f ∈ C∞ (R). Definition 4.16. For (H, D) self-adjoint on H, the cyclic subspace generated by the vector v ∈ H is the subspace L = span {(H − z)−1 v, z ∈ R}. Remark 4.17. i. Cyclic subspaces are invariant under H, as easily checked. ii. If the vector v chosen to generate the cyclic subspace is an eigenvector, then, this subspace is Cv. iii. If the cyclic subspace corresponding to some vector v coincides with H, we say that v is a cyclic vector for H. iv. In the finite dimensional case, the matrix H has a cyclic vector v iff the spectrum of H is simple, i.e. all eigenvalues have multiplicity one. These subspaces allow to structure the Hilbert space with respect to the action of H. Lemma 4.18. For (H, D) self-adjoint on H, there exists a sequence of orthogonal cyclic subspaces Ln ⊂ H with cyclic vector vn such that H = ⊕N n=1 Ln , with N finite or not. b

Let X be locally compact and consider C∞ (X). If B is a subalgebra of C∞ (X) that separates points and satisfies f ∈ B ⇒ f¯ ∈ B, then B is dense in C∞ (X) for · ∞

Introduction to the Theory of Linear Operators

23

Proof. As H is assumed to be separable, there exists an orthonormal basis {fj }j∈N of H. Let L1 be the subspace corresponding to f1 . By induction, let us assume orthogonal cyclic subspaces L1 , L2 , · · · , Ln are given. Let m(n) be the smallest integer such that fm(n) ∈ L1 ⊕ · · · ⊕ Ln and let gm(n) be the component of fm(n) orthogonal to that subspace. We let Ln+1 be the cyclic subspace generated by the vector gm(n) . Then we have Ln+1 ⊥ Lr , for all r ≤ n and fm(n) ∈ L1 ⊕ · · · ⊕ Ln ⊕ Ln+1 . Then either the induction continues indefinitely and N = ∞, or at some point, such a m(n) does not exist and the sum is finite. The above allows us to consider each H|Ln , n = 1, 2 · · · , N separately. Note, however, that the decomposition is not canonical. Theorem 4.19. Let (H, D) be self-adjoint on H, separable. Let S = σ(H) ⊂ R. Then there exists a finite positive measure µ on S × N and a unitary operator U : H → L2 ≡ L2 (S × N, dµ) such that if h: S×N → R (x, n) !→ x, then ξ ∈ H belongs to D if and only if hU ξ ∈ L2 . Moreover, U HU −1 ψ = hψ, ∀ ψ ∈ U (D) ⊂ L2 (S × N, dµ) and U f (H)U −1 ψ = f (h)ψ, ∀ f ∈ C∞ (R), ψ ∈ L2 (S × N, dµ). This Theorem will be a Corollary of the Theorem 4.20. Let (H, D) be self-adjoint on H and S = σ(H) ⊂ R. Further assume that H admits a cyclic vector v. Then, there exists a finite positive measure µ on S and a unitary operator U : H → L2 (S, dµ) ≡ L2 such that if h:S→R x !→ x, then ξ ∈ H belongs to D if and only if hU ξ ∈ L2 and U HU −1 ψ = hψ ∀ ψ ∈ U (D) ⊂ L2 (S, dµ). Proof of Theorem 4.20. Let the linear functional Φ : C∞ (R) → C be defined by Φ(f ) = v|f (H)v. The functional calculus shows that Φ(f¯) = Φ(f ). And if 0 ≤ f ∈ C∞ (R), then, √ with g = f , we have Φ(f ) = g(H)v 2 ≥ 0, i.e. Φ is positive. Thus, by Riesz-Markov Theoremc , there exists a positive measure on R such that c

If X is a locally compact space, any positive linear functional l on C∞ (X) is of the form l(f ) = X f dµ, where µ is a (regular) Borel measure with finite total mass.

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v|f (H)v =

R

f (x)dµ(x), ∀ f ∈ C∞ (R).

Since in case supp (f ) ∩ σ(H) = ∅, f (H) is zero, we deduce that supp (µ) ⊆ S = σ(H). Also, note that f above belongs to L2 (S, dµ), since |f (x)|2 dµ(x) = v|f (H)∗ f (H)v ≤ f 2 ∞ v 2 < ∞. Consider the linear map T : C∞ (R) → L2 such that T f = f . It satisfies for any f, g ∈ C∞ (R) T g|T f L2 = g¯(x)f (x)dµ(x) = Φ(¯ gf ) S

= v|g ∗ (H)f (H)vH = g(H)v|f (H)vH . Defining M = {f (H)v ∈ H | f ∈ C∞ (R)}, we have existence of an onto isomorphism U such that U : M → C∞ (R) ⊆ L2 such that U f (H)v = f. Now, M is dense in H since v is cyclic and C∞ (R) is dense in L2 , so that U admits a unitary extension from H to L2 . Let f1 , f2 , f ∈ C∞ (R) and ψi = fi (H)v ∈ H. Then ψ2 |f (H)ψ1 H = f2 (H)v|(f f1 )(H)vH = f¯2 (x)f (x)f1 (x)dµ(x) = U ψ2 |f U ψ1 L2 , S

where f denotes the obvious multiplication operator. In particular, if f (x) = rw (x) = (x − w)−1 , we deduce that for all ξ ∈ L2 and all w ∈ R U rw (H)U −1 ξ = U (H − w)−1 U −1 ξ = rw ξ.

(20)

Thus, U maps Ran(H − w) to Ran(rw ), i.e. D and {ξ ∈ L2 | xξ(x) ∈ L2 } are in one to one correspondence. If ξ ∈ L2 and ψ = rw ξ, then ψ ∈ D(h), where D(h) is the domain of the multiplication operator by h : x !→ x. Then, with (20) U HU −1 ψ = U HU −1 rw ξ = U Hrw (H)U −1 ξ = wrw ξ + ξ = xrw ξ = hψ. Proof of Theorem 4.19. We know H = ⊕n Ln with cyclic subspaces Ln with vectors n vn . We n = 1/2 , ∀n ∈ N. By Theorem 4.19, there exist µn of will assume v 2 −2n and unitary operators Un : Ln → L2 (S, dµn ) such mass S dµn = vn = 2

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25

that Hn = H|Ln is unitarily equivalent to the multiplication by x on L2 (S, dµn ). Defining µ on S × N by imposing µ|S×{n} = µn and U by ⊕n Un , we get the result. eigenvalues λj with associated eigenIn case H = Cn and H = H ∗ has simple can be chosen as µ = j δ(x − λj ) and L2 = L2 (R, dµ) = vectors ψj , the measure ˜ = j aj δ(x − λj ), where aj > 0 is as good a measure as µ to Cn . Note also that µ n 2 ˜ represent C as L (R, dµ). Let us now extend our Spectral Theorem to B(R), the set of bounded Borel functions on R. Definition 4.21. We say that fn ∈ B(R) is monotonically increasing to f ∈ B(R), if fn (x) increases monotonically to f (x) for any x ∈ R. Thus, fn = supx∈R |fn (x)| is uniformly bounded in n. Theorem 4.22. There exists a unique functional calculus f → f (H) from B(R) to L(H) if one imposes s-limn→∞ fn (H) = f (H) if fn ∈ B(R) converges monotonically to f ∈ B(R). Recall that s-lim means limit in the strong sense, i.e. s-lim An = A in L(H) is equivalent to limn An ϕ = Aϕ, in H, ∀ϕ ∈ H. Proof. Consider existence first. By unitary equivalence, we identify H and L2 (S × N, dµ) and H by multiplication by h : (x, n) !→ x. We define f (H) for f ∈ B(R) by f (H)ψ(x, n) = f (h(x, n))ψ(x, n) on L2 (S × N, dµ), which is easily shown to satisfy the properties of a functional calculus. Then, by the dominated convergence Theorem, if fn converges monotonically to f : lim fn (H)ψ(x, m) = f (H)ψ(x, m).

n→∞

Uniqueness is shown as follows. Consider two functional calculus with the mentioned properties. Let C be the subset of B(R) on which they coincide. We know C∞ (R) ⊂ C and C is closed by taking monotone limits. But the smallest set of func tions containing C∞ (R) which is closed under monotone limits is B(R), see [4]. Remark 4.23. It all works the same if one considers functions fn that converge pointwise to f and such that supn fn ∞ < ∞ We have the following Corollary concerning the resolvent. Corollary 4.24. With the hypotheses and notations above, σ(H) is the essential range of h in L2 (S, dµ) and (H − z)−1 =

1 dist (z, σ(H))

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Proof. This follows directly from Theorem 4.20 and our study of multiplication operators. Another instance where our previous study of multiplication operators is useful is the case of constant coefficient differential operators on S(RN ), the set of Schwartz functions. Such an operator L is defined by a finite sum of the form L= aα Dα , (21) α

where α = (α1 , · · · , αN ) ∈ NN , aα ∈ C, Dαj =

∂ αj α α1 · · · DαN , α , D =D ∂xj j

and L acts on functions in S(RN ) ⊂ L2 (RN ). This set of functions being invariant under Fourier transformation F, this operator is unitarily equivalent to FLF −1 = aα (ik)α , on S(RN ) ⊂ L2 (RN ). α

The function α aα (ik)α is called the symbol of the differential operator. It is not difficult to get the following Proposition 4.25. Let L be the differential operator on Rn with constant coefficients defined in (21). Then L is symmetric iff its symbol is real valued in RN . In that case, ¯ is self-adjoint and L ¯ ={ σ(L) aα (ik)α | k ∈ RN }. α

Let us finally introduce spectral projectors in the general case of unbounded operators. Theorem 4.26. Let (H, D) be self-adjoint on H and (a, b) an open interval. Let fn be an increasing sequence of non-negative continuous functions on R with support in (a, b) that converges to χ(a,b) , the characteristic function of (a, b). Then s- lim fn (H) = P(a,b) , n

a canonical orthogonal projector, independent of {fn }, that satisfies P(a,b) H ⊂ HP(a,b) , and P(a,b) = 0 ⇐⇒ (a, b) ∩ σ(H) = ∅ Proof. The existence of the limit is ensured by Theorem 4.22 and and its properties are immediate.

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27

Remark 4.27. The fact that P(a,b) is a projector follows from the identity χ(a,b) = χ2(a,b) , which makes χ(a,b) a projector, when viewed as a multiplication operator. These projectors are called spectral projectors and their range L(a,b) = P(a,b) H are called spectral subspaces. These spectral subspaces satisfy L(a,b) L2 (E(a,b) , dµ), where E(a,b) = {(x, n) | a < h(x, n) < b}, and denotes the unitary equivalence constructed in Theorem 4.19 It is also customary to represent a self-adjoint operator H by the Stieltjes integral ∞ λdE(λ), H= −∞

where E(λ) = P (−∞, λ) is projection operator valued. Let us justify this in case H has a cyclic vector, the general case following immediately. By polarization, it is enough to check that for ξ ∈ D ξ|HξH = λdξ|E(λ)ξ = λd E(λ)ξ 2 . By unitary equivalence to L2 (R, dµ), if ψ = U ξ ∞ 2 |χ(−∞,λ) (x)ψ(x)|2 dµ(x) d E(λ)ξ = d

−∞ λ

=d −∞

|ψ(x)|2 dµ(x) = |ψ(λ)|2 dµ(λ).

Hence ξ|HξH =

λ|ψ(λ)|2 dµ(λ) = ψ|hψL2 .

We close this Section about the Spectral Theorem by some results in perturbation theory of unbounded operators. Definition 4.28. Let (H, D) and (Hn , Dn ) be a sequence of self-adjoint operators on H. We say that Hn → H in the norm resolvent sense if lim (Hn + i)−1 − (H + i)−1 = 0.

n→∞

The point i ∈ C plays no particular role as the following Lemma shows. Lemma 4.29. If z = x + iy ∈ C \ R, and

h+i , g(x, y) = sup h∈R h − x − iy

then, there exists a constant c such that (Hn − z)−1 − (H − z)−1 ≤ cg(x, y) (Hn + i)−1 − (H + i)−1 .

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Proof. Once the identity (22) (Hn − z)−1 − (H − z)−1 = −1 −1 −1 −1 (Hn + i)(Hn − z) ((Hn + i) − (H + i) )(H + i)(H − z) is established, the Lemma is a consequence of the bound following from the Spectral Theorem (H + i)(H − z)−1 ≤ g(x, y). If one doesn’t take care of domain issues, (22) is straightforward. We refer to [1] for a careful proof of (22). Remark 4.30. If z ∈ ρ(H) ∩ ρ(Hn ), the result is similar. Theorem 4.31. If Hn → H in the norm resolvent sense, then lim f (Hn ) − f (H) = 0, ∀f ∈ C∞ .

n→∞

Remark 4.32. The result is not generally true if f ∈ B(R). Consider spectral projectors, for instance. See also Corollary 4.33 below. Proof. If f ∈ A, the definition yields, ˜ 1 ∂ f (z) f (H) − f (Hn ) ≤ (Hn − z)−1 − (H − z)−1 dxdy π C ∂ z¯ ˜ 4c ∂ f (z) ≤ g(x, y)dxdy (Hn + i)−1 − (H + i)−1 . π C ∂ z¯ ˜

It is not difficult to see that the last integral is finite, due to the properties of ∂ f∂(z) z¯ and of g(x, y). The convergence in norm is established for f ∈ A and the extension of the result to f ∈ C∞ (R) comes from the extension of the functional calculus to those f ’s and by density. We have the following spectral consequences. Corollary 4.33. If Hn → H in the norm resolvent sense, we have convergence of the spectrum of Hn to that of H in the following sense: λ ∈ R \ σ(H) ⇒ λ ∈ σ(Hn ), n large enough λ ∈ σ(H) ⇒ ∃λn ∈ σ(Hn ), such that lim λn = λ. n→∞

C0∞ (R)

Proof. If λ ∈ R \ σ(H), there exists f ∈ whose support is disjoint from σ(H) and which is equal to 1 in a neighborhood of λ. Then, Theorem 4.31 implies f (Hn ) → 0 and, in turn, the Spectral Theorem implies λ ∈ σ(Hn ) if f (Hn ) < 1. Conversely, if λ ∈ σ(H), pick an > 0 and a f ∈ C0∞ (R) such that f (λ) = 1 and supp (f ) ⊂ (λ − , λ + ). From limn f (Hn ) = 1 follows that σ(Hn ) ∩ (λ − , λ + ) = ∅, if n is as large as we wish.

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5 Stone’s Theorem, Mean Ergodic Theorem and Trotter Formula The Spectral Theorem allows to prove easily Stone’s Theorem, which characterizes one parameter evolution groups which we define below. Such groups are those giving the time evolution of a wave function ψ in Quantum Mechanics governed by the Schr¨odinger equation i

∂ ψ(t) = Hψ(t), with ψ(0) = ψ0 , ∂t

where H is the Hamiltonian. Definition 5.1. A one-parameter evolution group on a Hilbert space is a family {U (t)}t∈R of unitary operators satisfying U (t + s) = U (t)U (s) for all t, s ∈ R and U (t) is strongly continuous in t on R. Remark 5.2. It is easy to check that strong continuity at 0 is equivalent to strong continuity everywhere and that weak continuity is equivalent to strong continuity in that setting. Actually, we have equivalence between the following statements: the map t !→ ϕ|U (t)ψ is measurable for all ϕ, ψ and U (t) is strongly continuous, see [4] for a proof. Theorem 5.3. Let (A, D) be self-adjoint on H and U (t) = eitA given by functional calculus. Then a. {U (t)}t∈R forms a one parameter evolution group and U (t) : D → D for t ∈ R. b. For any ψ ∈ D, U (t)ψ − ψ → iAψ as t → 0. t c. Conversely, U (t)ψ − ψ lim exists ⇒ ψ ∈ D. t→0 t Proof. a) follows from the Functional Calculus and the properties of x !→ ft (x) = eitx . Similarly b) is a consequence of the functional calculus (see Theorem 4.22) applied to x !→ gt (x) = (eitx − 1)/t and of the estimate |eix − 1| ≤ |x|. c) Define

U (t)ψ − ψ U (t)ψ − ψ exists , and Bψ = lim on D(B). D(B) = ψ | lim t→0 t→0 t it (23) ¯ and One checks that B is symmetric and b) implies B ⊇ A. But A ⊆ B ⊆ B ∗ A = A is closed, thus A = B. Remark 5.4. The formula (23) defines the so-called infinitesimal generator of the evolution group U (t). The converse of that result is Stone’s Theorem.

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Theorem 5.5. If {U (t)}t∈R forms a one parameter evolution group on H, then there exists (A, D) self-adjoint on H such that U (t) = eiAt . Proof. The idea of the proof is to define A as the infinitesimal generator on a set of good vectors and show that A is essentially self-adjoint. Then one shows that ¯ U (t) = eiAt . Let f ∈ C0∞ (R) and define f (t)U (t)ϕ dt, ∀ϕ ∈ H. ϕf = R

Let D be the set of finite linear combinations of such ϕf , with different ϕ and f . 1) D is dense: Let j (x) = j(x/)/, where 0 ≤ j ∈ C0∞ with support in [−1, 1] and j(x) dx = 1. Then, for any ϕ, j (t)(U (t)ϕ − ϕ)dt ϕj − ϕ =

∞ ≤ j (t)dt sup U (t)ϕ − ϕ → 0 as → 0. −∞

|t|≤

2) Infinitesimal generator on D: Let ϕf ∈ D. (U (s) − 1l) U (t + s) − U (t) ϕf = ϕdt f (t) s s R f (τ − s) − f (τ ) = U (τ )ϕdτ s R s→0 − f (τ )U (τ )ϕdτ = ϕ−f . −→ R Hence, we set for ϕf ∈ D, Aϕf =

U (t) − 1l 1 ϕ−f = lim ϕf t→0 i it

and it is easily checked that U (t) : D → D, A : D → D, and U (t)Aϕf = AU (t)ϕf =

1 ϕ−f (·−t) . i

Moreover, for ϕf , ϕg ∈ D, U (−s) − 1l U (s) − 1l ϕg |Aϕf = lim ϕf | ϕg = lim ϕf |ϕg = Aϕf |ϕg , s→0 s→0 is −is so that A is symmetric. 3) A is essentially self-adjoint: Assume there exists ψ ∈ D∗ = D(A∗ ) such that A∗ ψ = iψ. Then, ∀ϕ ∈ D,

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31

d ψ|U (t)ϕ = ψ|iAU (t)ϕ = iA∗ ψ|U (t)ϕ = ψ|U (t)ϕ. dt Hence, solving the differential equation, ψ|U (t)ϕ = ψ|ϕet . As U (t) = 1, this implies ψ|ϕ = 0, so that ψ = 0 as D is dense. A similar reasoning holds for any χ ∈ Ker(A∗ + i), so that A is essentially self-adjoint and A¯ is self-adjoint. ¯ ¯ = D(A). ¯ On the one hand, 4) U (t) = eiAt : Let ϕ ∈ D ⊆ D ¯ ¯ and eiAt ϕ ∈ D

d iAt ¯ ¯ ¯ iAt e ϕ = iAe ϕ, dt

¯ for all t. Thus, introducing by b) Theorem 5.3. On the other hand, U (t)ϕ ∈ D ⊂ D ¯ iAt ψ(t) = U (t)ϕ − e ϕ; we compute ¯

¯ iAt ϕ = iAψ(t), ¯ ψ (t) = iAU (t)ϕ − iAe with ψ(0) = 0, so that

d 2 dt ψ(t)

¯

≡ 0, hence ψ(t) ≡ 0. As D is dense, eiAt ≡ U (t).

Examining the above proof, one deduces the following Corollary which can be useful in applications. Corollary 5.6. Let (A, D) be self-adjoint on H et E ⊂ D be dense. If, for all t ∈ R, eitA : E → E, then (A|E , E) is essentially self-adjoint. Remark 5.7. i. In the situation of the Corollary, one says that E is a core for A. ii. The solution to the following equation, in the strong sense, d ϕ(t) = iAϕ(t), ∀t ∈ R ϕ(0) = ϕ0 ∈ D dt is unique and is given by ϕ(t) = eiAt ϕ0 . In case A is bounded, the evolution group generated by A can be obtained from the power series of the exponential. This relation remains true in a certain sense when A is unbounded and self-adjoint. Indeed, if ϕ belongs to the dense set ∪M ≥0 P (−M, M )H, where P (−M, M ) denotes the spectral projectors of A, we get N (itA)k k=0

k!

ϕ → eitA ϕ, as N → ∞.

This formula makes sense due to the fact that ϕ ∈ ∩n≥0 D(An ), where D(An ) is the domain of An . Stone’s Theorem provides a link between evolution groups and self-adjoint generators, therefore one can expect a relation between essentially selfadjoint operators and the existence of sufficiently many vectors for which the above formula makes sense.

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Definition 5.8. Let A be an operator on a Hilbert space H. A vector ϕ is called an analytic vector if ϕ ∈ ∩n≥0 D(An ) and ∞ Ak ϕ k=0

k!

tk < ∞, for some t > 0.

The relation alluded to above is provided by the following criterion for essential self-adjointness. Theorem 5.9 (Nelson’s Analytic Vector Theorem). Let (A, D) be symmetric on a Hilbert space H. If D contains a total set of analytic vectors, then (A, D) is essentially self-adjoint. We refer the reader to [4] for a proof and we proceed by providing a link between the discrete spectrum of the self-adjoint operator H with the evolution operator eitH it generates. This is the so-called Theorem 5.10 (Mean Ergodic Theorem). Let Pλ be the spectral projector on an eigenvalue λ of a self-adjoint operator H of domain D ∈ H. Then t2 1 eitH e−itλ dt. Pλ = s − lim t2 −t1 →∞ t2 − t1 t 1 Proof. One can assume λ = 0 by considering H − λ if necessary. i) If ϕ ∈ P0 H, then Hϕ = 0 and eitH ϕ = ϕ, so that t2 1 eitH ϕdt = ϕ = P0 ϕ, for any t1 , t2 . t2 − t1 t1 ii) If ϕ ∈ Ran(H), i.e. ϕ = Hψ for some ψ ∈ D, then eitH ϕ = eitH Hψ = −i

d itH e ψ, dt

so that we can write t2 1 i eitH ϕdt = − (eit2 H − eit1 H )ψ → 0 = P0 ψ. t2 − t1 t1 t2 − t1 The result is thus proven for Ran(H) and Ran(P0 ). As the integral is uniformly bounded, the result is true on Ran(H) as well and we conclude by H = Ran(H) ⊕ Ker(H) = Ran(H) ⊕ P0 H, which follows from H = H ∗ . With a little more efforts, one can prove in the same vein

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33

Theorem 5.11 (von Neumann’s Mean Ergodic Theorem). If V is such that, uniformly in n, V n ≤ C and P1 projects on Ker(V − 1l), then N −1 1 n V ϕ → P1 ϕ, as N → ∞. N n=0

Remark 5.12. i. The projector P1 is not necessarily self-adjoint. ii. The projection on Ker(V − λ) where |λ| = 1 is obtained by replacing V by V /λ in the Theorem. iii. It follows from the assumption that σ(V ) ⊆ {z | |z| ≤ 1}, since the spectral radius spr(V ) = limn→∞ V n 1/n = 1. iv. A proof can be found in [5]. Another link between the spectrum of its generator and the behavior of an evolution group arises when a vector is transported away from its initial value at t = 0 by the evolution exponentially fast as |t| → ∞ . Proposition 5.13. Let (H, D) be self-adjoint on H and assume there exists a normalized vector ϕ ∈ H such that, for any t ∈ R and for some positive constants A, B, |ϕ|eitH ϕ| ≤ Ae−B|t| . Then σ(H) = R. Proof. Taking ϕ as first vector in the decomposition provided in Lemma 4.18, we have by the Spectral Theorem, that on L2 (dµ1 ), the restriction of L2 (dµ) unitary equivalent to that first cyclic subspace, ϕ 1, eitH ϕ eitx 1,

so that ϕ|eitH ϕ =

eitx dµ1 (x) ≡ f (t).

This f admits a Fourier transform ω !→ fˆ(ω) which is analytic in a strip {ω | |ω| < B}. Therefore, we have dµ1 (x) = fˆ(x)dx and the support of dµ1 = R. Hence, σ(H) ⊃ R. Let us close this Section by a result concerning evolution groups generated by sums of self-adjoint operators. Theorem 5.14 (Trotter product formula). Let (A, DA ), (B, DB ) be self-adjoint and A + B be essentially self-adjoint on DA ∩ DB . Then n ei(A+B)t = s- lim eitA/n eitB/n , ∀ t ∈ R. n→∞

If, moreover, A and B are bounded from below, n e−(A+B)t = s- lim e−tA/n e−tB/n , ∀ t ≥ 0. n→∞

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Remark 5.15. The operator e−Ct , t ≥ 0 for C self-adjoint and bounded below can be defined via the Spectral Theorem applied to the function f defined as follows: f (x) = e−x , if x ≥ x0 and f (x) = 0 otherwise, with x0 small enough. Proof. (partial). We only consider the first assertion under the significantly simplifying hypothesis that A + B is self-adjoint on D = DA ∩ DB . The second assertion is proven along the same lines. Let ψ ∈ D and consider s−1 (eisA eisB − 1l)ψ = s−1 (eisA − 1l)ψ + s−1 eisA (eisB − 1l)ψ → iAψ + iBψ as s → 0 and s−1 (eis(A+B) − 1l)ψ → i(A + B)ψ as s → 0. Thus, setting K(s) = s−1 (eisA eisB − eis(A+B) ), the vector K(s)ψ → 0 as s → 0 or s → ∞, for any ψ ∈ D. Due to the assumed self-adjointness of A + B on D, A + B is closed so that D equipped with the norm ψ A+B = ψ + (A + B)ψ is a Banach space. As K(s) : D → H is bounded for each finite s and tends to zero strongly at 0 and ∞, we can apply the uniform boundedness principle or BanachSteinhaus Theorem d to deduce the existence of a constant C so that K(s)ψ ≤ C ψ A+B , ∀s ∈ R, ∀ψ ∈ D. Therefore, on any compact set of D in the · A+B norm, K(s)ψ → 0 uniformly as s → 0. We know from Theorem 5.3 that eis(A+B) : D → D and is strongly continuous, thus {eis(A+B) ψ | s ∈ [−1, 1]} is compact in D for · A+B , for ψ fixed. Hence t−1 (eitA eitB − eit(A+B) )eis(A+B) ψ → 0 uniformly in s ∈ [−1, 1] as t → 0. Therefore, writing n n eitA/n eitB/n − eit(A+B)/n ψ= n−1

eitA/n eitB/n

k

eitA/n eitB/n − eit(A+B)/n

eit(A+B)/n

n−1−k ψ,

k=0

we get that the RHS is bounded in norm by t −1 itA/n itB/n it(A+B)/n is(A+B) e e e −e ψ . |t| max |s|
If X and Y are two Banach spaces, and F is a family of bounded linear operators from X to Y such that for each x ∈ X , {T xY | T ∈ F} is bounded, then {T | T ∈ F} is bounded.

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Thus, if ψ ∈ D, we get n eitA/n eitB/n ψ → eit(A+B) ψ, as n → ∞. The operators being bounded and D being dense, this finishes the proof.

6 One-Parameter Semigroups This Section extends some results of the previous one in the sense that unitary groups discussed above are a particular case of semigroups. The setting used in this Section is that of a Banach space denoted by B. One parameter semigroups will be used in the study the time evolution of Open Quantum Systems. Definition 6.1. Let {S(t)}t≥0 be a family of bounded operators defined on B. We say that {S(t)}t≥0 is a strongly continuous semigroup or C0 semigroup if 1. S(0) = 1l. 2. S(t + s) = S(t)S(s) for any s, t ≥ 0. 3. S(t)ϕ is continuous as a function of t on [0, ∞), for all ϕ ∈ B. Remark 6.2. 3) is equivalent to requiring continuity at 0+ only. Definition 6.3. The infinitesimal generator of the semigroup {S(t)}t≥0 , is the linear operator (A, D) defined by D = {ϕ ∈ B | lim+ t−1 (S(t) − 1l)ϕ exists in B} t→0

(S(t) − 1l)ϕ , ϕ ∈ D. Aϕ = lim t t→0+ The main properties of semigroups and their generators are listed below. Proposition 6.4. Let {S(t)}t≥0 be a semigroup of generator A. Then a. There exist ω ∈ R and M ≥ 1 such that S(t) ≤ M eωt , for all t ≥ 0. b. The generator A is closed with dense t domain D. c. For any t ≥ 0, ϕ ∈ D, we have 0 S(τ )ϕdτ ∈ D and

t

S(τ )dτ

A

= S(t)ϕ − ϕ.

0

d. For any t ≥ 0, S(t) : D → D and if ϕ ∈ D, t !→ S(t)ϕ is in C 1 ([0, ∞)) and d S(t)ϕ = AS(t)ϕ = S(t)Aϕ, t ≥ 0. dt e. If {S1 (t)}t≥0 and {S2 (t)}t≥0 are two C0 semigroups with the same generator A, then S1 (t) ≡ S2 (t).

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Proof. a) By the right continuity at 0 and the Banach-Steinhaus Theorem, there exists > 0 and M ≥ 1 such that S(t) ≤ M if t ∈ [0, ]. For any given t ≥ 0, there exists n ∈ N and 0 ≤ δ < such that t = n + δ, hence S(t) = S(δ)S()n ≤ M n+1 ≤ M M t/ ≡ M eωt , with ω =

ln M ≥ 0.

c) For ϕ ∈ D, t ≥ 0 and any > 0, (S() − 1l)ϕ t 1 t S(τ )ϕ = (S(τ + ) − S(τ ))ϕdτ 0 0 t+ 1 1 = S(τ )ϕ − S(τ )ϕ t 0 which converges to S(t)ϕ − ϕ as → 0. d) For ϕ ∈ D, t ≥ 0 and any > 0, we have (S() − 1l)ϕ (S() − 1l)ϕ S(t)ϕ = S(t) ϕ → S(t)Aϕ, as → 0.

(24)

Thus S(t)ϕ ∈ D and AS(t) = S(t)A on D. As a consequence of (24) , the function t !→ S(t)ϕ has a right derivative given by S(t)Aϕ which is continuous on [0, ∞). Therefore, a classical result of analysis shows that the derivative at t ≥ 0 exists. b) Let ϕ and define ϕ for any > 0 by ϕ = 1 0 S(τ )ϕdτ . The vector ϕ ∈ D, by c) and ϕ → ϕ and → 0, so that D is dense. Closedness of A is shown as follows. Let {ϕn }n∈N be a sequence of vectors in D, such that ϕn → ϕ and Aϕn → ψ, for some ϕ and ψ. For any n ∈ N and t > 0, d) implies by integration t S(τ )Aϕn dτ. S(t)ϕn − ϕn = 0

Taking limits n → ∞ we get S(t)ϕ − ϕ =

t 0

S(τ )ψdτ , therefore

lim t−1 S(t)ϕ − ϕ = ψ.

t→0+

In other words, ψ ∈ D and ψ = Aϕ. e) Finally, for ϕ ∈ D, t > 0, we define, for τ ∈ [0, T ], ψ(τ ) = S1 (t − τ )S2 (τ )ϕ. In view of c) we are allowed to differentiate w.r.t. τ and we get d ψ(τ ) = −S1 (t − τ )AS2 (τ )ϕ + S1 (t − τ )AS2 (τ )ϕ = 0, dτ hence ψ(0) = ψ(t), i.e. S1 (t)ϕ = S2 (t)ϕ. The density of D concludes the proof.

Remark 6.5. If {S(t)}t≥0 is a semigroup that is continuous in norm, i.e. such that S(t) − 1l → 0 as t → 0+ , it is not difficult to show that (see e.g. [3]) that there exists A ∈ L(B) such that S(t) = eAt for any t ≥ 0.

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37

Definition 6.6. Semigroups characterized by the bound S(t) ≤ 1 for all t ≥ 0 are called contraction semigroups . Remark 6.7. There is no loss of generality in studying contraction semigroups in the sense that if the semigroup {S(t)}t≥0 satisfies the bound a) in the Proposition above, we can consider the new C0 semigroup S1 (t) = e−βt S(t) satisfying S1 (t) ≤ M . At the price of a change of the norm of the Banach space, we can turn it into a contraction semigroup. Let |||ϕ||| = sup S1 (t)ϕ , t≥0

such that ϕ ≤ |||ϕ||| ≤ M ϕ and |||S1 (τ )ϕ||| ≤ |||ϕ|||. The characterization of generators of C0 semigroups is given by the Theorem 6.8 (Hille Yosida). Let (A, D) be a closed operator with dense domain. The following statements are equivalent: 1. The operator A generates a C0 semigroup {S(t)}t≥0 satisfying S(t) ≤ M eωt , t ≥ 0 2. The resolvent and resolvent set of A are such that for all λ > ω and all n ∈ N∗ ρ(A) ⊃ (ω, ∞) and (A − λ)−n ≤

M (λ − ω)n

3. The resolvent and resolvent set of A are such that for all n ∈ N∗ ρ(A) ⊃ {λ ∈ C | %λ > ω} and (A − λ)−n ≤

M , %λ > ω. (%λ − ω)n

Remark 6.9. A proof of this Theorem can be found in [5] or [3]. We simply note here that A generates of a contraction semigroup, M = 1, ω = 0, if and only if 1 (A − λ)−1 ≤ λ , %λ > 0. This is trivially true if A = iH in a Hilbert space ∗ where H = H , as expected. Moreover, in that case, A and −A satisfy the bound, so that we can construct a group, as we already know, generated by iH. Let us specialize a little by considering contraction semigroups on a Hilbert space H. We can characterize their generator by roughly saying that their real part in nonpositive. Definition 6.10. An operator (A, D) on a Hilbert space H is called dissipative if for any ϕ ∈ D ϕ|Aϕ + Aϕ|ϕ = 2%(ϕ|Aϕ) ≤ 0. Proposition 6.11. Let A be the generator of a C0 semigroup {S(t)}t≥0 on H. Then {S(t)}t≥0 is a contraction semigroup iff A is dissipative.

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Proof. Assume S(t) is a contraction semigroup and ϕ ∈ D. Consider f (t) = S(t)ϕ|S(t)ϕ. As a function of t, it is differentiable and 0 ≥ f (0) = ϕ|Aϕ + Aϕ|ϕ. Conversely, if A is dissipative, as we have S(t) : D → D, we get for any t ≥ 0, f (t) = S(t)ϕ|S(t)Aϕ + AS(t)ϕ|S(t)ϕ ≤ 0, so f (t) is monotonically decreasing and S(t)ϕ ≤ ϕ . As D is dense, {S(t)}t≥0 is a contraction semigroup. Actually, the notion of dissipative operator can be generalized to the Banach space setting. Moreover, it is still true that dissipative operators and generators of C0 contraction semigroups are related. This is the content of the Lumer Phillips Theorem stated below. Let B be a Banach space and let B be its dual. The value of l ∈ B at ϕ ∈ B is denoted by l, ϕ ∈ C. Let us define for any ϕ ∈ B the duality set F (ϕ) ⊂ B by F (ϕ) = {l ∈ B | l, ϕ = ϕ 2 = l 2 }. By the Hahn-Banach Theorem, F (ϕ) = ∅ for any ϕ ∈ B. Definition 6.12. An operator (A, D) on a Banach space B is called dissipative if for any ϕ ∈ D, there exists l ∈ F (ϕ) such that %(l, Aϕ) ≤ 0. The following characterization of dissipative operators avoiding direct duality considerations can be found in [3]: Proposition 6.13. An operator (A, D) is dissipative if and only if (λ1l − A)ϕ ≥ λ ϕ , for all ϕ ∈ D and all λ > 0. The link between dissipativity and contraction semigroups is provided by the Theorem 6.14 (Lumer Phillips). Let (A, D) be an operator with dense domain in a Banach space B. a. If A is dissipative and there exists λ0 > 0 such that Ran(λ0 1l − A) = B, then A is the generator of a C0 contraction semigroup. b. If A is the generator of a C0 contraction semigroup on B, then one has Ran(λ1l − A) = B for all λ > 0 and A is dissipative. The proof of this result can be found in [3], for example. We close this Section by considerations on the perturbation of semigroups, or more precisely, of their generators. We stick to our Banach space setting for the end of the Section. We first show that the property of being a generator is stable under perturbation by bounded operators.

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Theorem 6.15. Let A be the generator of a C0 semigroup {S(t)}t≥0 on B which satisfies the bound 1) in Theorem 6.8. Then, if B is a bounded operator, the operator A + B generates a C0 semigroup {V (t)}t≥0 that satisfies the same bound with M !→ M and ω !→ ω + M B . Moreover, if B is replaced by xB, the semigroup generated by A + xB is an entire function of the variable x ∈ C. Proof. If A + B generates a contraction semigroup V (t), it must solve the following differential equation on D(A + B) = D(A) d V (t) = (A + B)V (t). dt Introducing S(t), the solution of that equation must solve

t

S(t − s)BV (s)ds.

V (t) = S(t) +

(25)

0

Strictly speaking the above integral equation is true on D(A) only, but as D(A) is dense and all operators are bounded, (25) is true on B, as a strong integral. We solve this equation by iteration V (t) = Sn (t), (26) n≥0

where

t

S(t − s)BSn (s)ds, n = 0, 1, . . . , and S0 (t) = S(t).

Sn+1 (t) = 0

All integrals are strongly continuous and we have the bounds Sn (t) ≤ M n+1 B n eωt tn /n! which are proven by an easy induction. The starting estimate is true by hypothesis on S(t). Thus we see that (26) is absolutely convergent and satisfies V (t) ≤ M e(ω+M B )t . To show that V (t) defined this way is actually generated by A + B, we multiply (25) by e−λt and integrate over [0, ∞], assuming %λ > ω + M B , to get ∞ e−λt V (t)dt r(λ) = 0 ∞ ∞ ∞ = e−λt S(t)dt + e−λt S(t)dt B e−λt V (t)dt 0

0 −1

= −(A − λ)

0 −1

− (A − λ)

Br(λ).

This yields (A + B − λ)r(λ) = −1l and, as λ ∈ ρ(A + B) by our choice of λ, it follows by computations we went through already that r(λ) = −(A + B − λ)−1 . We finally use Hille Yosida Theorem to conclude. For any k = 0, 1, 2, · · · ,

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Alain Joye

k ∞ 1 d r(λ) ≤ 1 (A + B − λ)−k−1 = tk |e−λt | V (t) dt k! k! dλk 0 M ∞ k −( λ−ω−M B )t ≤ t e dt = M (%λ − ω − M B )−k−1 , k! 0 which shows that the resolvent of A + B satisfies the estimate of the point 3) of Hille Yosida’s Theorem. Finally, we note that V (t) has the form of a converging series in powers of B, which proves the last statement. More general perturbations of generators of semigroups are allowed under the supplementary hypothesis that both the unperturbed generator and the perturbation generate contraction semigroups. Theorem 6.16. Let A and B be generators of contraction semigroups and assume B is relatively bounded w.r.t. A with relative bound smaller that 1/2. Then, A + B generates a contraction semigroup. For a proof, see [2], for example.

References 1. E. B. Davies, Spectral Theory and Differential operators, Cambridge University Press, 1995. 2. T. Kato, Perturbation Theory of Linear Operators, CIM, Springer 1981. 3. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. 4. M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol 1-4, Academic Press, 1971-1978. 5. K. Yosida, Functional Analysis, CIM, Springer 1980.

Introduction to Quantum Statistical Mechanics Alain Joye Institut Fourier, Universit´e de Grenoble 1, BP 74, 38402 Saint-Martin-d’H`eres Cedex, France e-mail: [email protected]

1

Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

1.1 1.2 1.3

Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermions and Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 46 53

Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.1

Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Boltzmann Gibbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 3

This set of lectures is intended to provide a flavor of the physical ideas underlying some of the concepts of Quantum Statistical Mechanics that will be studied in this school devoted to Open Quantum Systems. Although it is quite possible to start with the mathematical definitions of notions such as ”bosons”, ”states”, ”Gibbs prescription” or ”entropy” for example and prove theorems about them, we believe it can be useful to have in mind some of the heuristics that lead to their precise definitions in order to develop some intuition about their properties. Given the width and depth of the topic, we shall only be able to give a very partial account of some of the key notions of Quantum Statistical Mechanics. Moreover, we do not intend to provide proofs of the statements we make about them, nor even to be very precise about the conditions under which these statements hold. The mathematics concerning these notions will come later. We only aim at giving plausibility arguments, borrowed from physical considerations or based on the analysis of simple cases, in order to give substance to the dry definitions. Our only hope is that the mathematically oriented reader will benefit somehow from this informal introduction, and that, at worse, he will not be too confused by the many admittedly hand waving arguments provided.

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Some of the many general references regarding an aspect or the other of these lectures are provided at the end of these notes.

1 Quantum Mechanics We provide in this section an introduction to the quantum description of a physical system, starting from the Hamiltonian description of Classical Mechanics. The quantization procedure is illustrated for the standard kinetic plus potential Hamiltonian by means of the usual recipe. A set of postulates underlying the quantum description of systems is introduced and motivated by means of that special though important case. These aspects, and much more, are treated in particular in [2] and [4], for instance. 1.1 Classical Mechanics Let us recall the Hamiltonian version of Classical Mechanics in the following typical setting, neglecting the geometrical content of the formalism. Consider N particles in Rd of coordinates qk ∈ Rd , masses mk and momenta pk ∈ Rd , k = 1, · · · , N , interacting by means of a potential V : RdN → R q !→ V (q).

(1)

The space RdN of the coordinates (q1 , q2 , · · · , qN ), with qk,j ∈ R, j = 1 · · · , d which we shall sometimes denote collectively by q (and similarly for p), is called the configuration space and the space Γ = RdN × RdN = R2dN of the variables (q, p) is called the phase space. A point (q, p) in phase space characterizes the state of the system and the observables of the systems, which are the physical quantities one can measure on the system, are given by functions defined on the phase space. For example, the potential is an observable. The Hamiltonian H : Γ → R of the above system is defined by the observable N p2k + V (q1 , q2 , · · · , qN ), H(p, q) = 2mk k=1 Vij (|qi − qj |), where V (q1 , q2 , · · · , qN ) =

(2)

i<j

which coincides with the sum of the kinetic and potential energies. The equations of motion read for all k = 1, · · · , N as q˙k =

∂ ∂ H(q, p), p˙k = − H(q, p), with (q(0), p(0)) = (q 0 , p0 ), ∂pk ∂qk

(3)

where ∂q∂k denotes the gradient with respect to qk . The equations (3) are equivalent to Newton’s equations, with pk = mk q˙k ,

Introduction to Quantum Statistical Mechanics

mk q¨k = −

43

∂ V (q) with (q(0), q(0)) ˙ = (q 0 , q˙0 ), ∂qk

for all k = 1, · · · , N . In case the Hamiltonian is time independent, i.e. if the potential V is time independent, the total energy E of the system is conserved E = H(q(0), p(0)) ≡ H(q(t), p(t)), ∀t.

(4)

where (q(t), p(t)) are solutions to (3) with initial conditions (q(0), p(0)). More generally, a system is said to be Hamiltonian if its equations of motions read as (3) above. We shall essentially only deal with systems governed by Hamiltonians that are time-independent. These evolution equations are also called canonical equations of motion. Changes of coordinates pk !→ Pk , qk !→ Qk , such that H(q, p) !→ K(Q, P ) which conserve the form of the equations of motions, i.e. ∂ ∂ Q˙k = K(Q, P ), P˙k = − K(Q, P ), with (Q(0), P (0)) = (Q0 , P 0 ), ∂Pk ∂Qk (5) are called canonical transformations. The energy conservation property (4) is just a particular case of time dependence of a particular observable. Assuming the Hamiltonian is time independent, but not necessarily given by (2), the time evolution of any (smooth) observable B : Γ → R defined on phase space computed along a classical trajectory Bt (q, p) ≡ B(q(t), p(t)) is governed by the Liouville equation d Bt (q, p) = LH Bt (q, p), with B0 (q, p) = B(q, p), dt

(6)

where the linear operator LH acting on the vector space of observables is given by LH = ∇p H(q, p) · ∇q − ∇q H(q, p) · ∇p ,

(7)

with the obvious notation. Observables which are constant along the trajectories of the system are called constant of the motions. Introducing the Lebesgue measure on Γ = R2dN , N d d dµ = Πk=1 dqk dpk , with dqk = Πj=1 qk,j , and dpk = Πj=1 pk,j ,

and the Hilbert space L2 (Γ, dµ), one checks that LH is formally anti self-adjoint on L2 (Γ, dµ), (i.e. antisymmetric on the set of observables in C0∞ (Γ )). Therefore, the formal solution to (6) given by Bt (q, p) = etLH B0 (q, p) is such that etLH is unitary on L2 (Γ, dµ). Another expression of this fact is Liouville’s Theorem stating that

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∂(q(t), p(t)) ≡ 1, ∂(q 0 , p0 ) where the LHS above stands for the Jacobian of the transformation (q 0 , p0 ) !→ (q(t), p(t)). It is convenient for the quantization procedure to follow to introduce the Poisson bracket of observables B, C on L2 (Γ ) by the definition {B, C} = ∇q B · ∇p C − ∇p B · ∇q C.

(8)

{qk,m , pj,n } = δ(j,n),(k,m) , {pj,n , pk,m } = {qj,n , qk,m } = 0,

(9)

For example,

which are particular cases of {qk,m , F (q, p)} =

∂F (q, p) ∂G(q, p) , {G(q, p), pj,n } = . ∂pk,m ∂qj,n

These brackets fulfill Jacobi’s relations, {A, {B, C}} + circular permutations = 0

(10)

and, as {H, B} = −LH B, we can rewrite (6) by means of Poisson brackets as d Bt = −{H, Bt }. dt

(11)

Therefore, it follows that the Poisson bracket of two constants of the motion is a constant of the motion, though not necessarily independent from the previous ones. Before we proceed to the quantization procedure, let us introduce another Hamiltonian system we will be interested in later on. It concerns the evolution of N identical particles of mass m and charge e in R3 , interacting with each other and with an external electromagnetic field (E, B). Let us recall Maxwell’s equations for the electromagnetic field ∇B = 0,

∇∧E =−

∂B , ∂t

0 ∇E = ρe ,

∇ ∧ B = µ0 j +

1 ∂E , c2 ∂t

(12)

where ρe and j denote the density of charges and of current, respectively, the constant 0 and µ0 are characteristics of the vacuum in which the fields propagate, and c is the speed of light. A particle of mass m and charge e in presence of an electromagnetic field obeys the Newtonian equation of motion determined by the Lorentz force m¨ q = e(E + q˙ ∧ B).

(13)

When N charged particles interact with the electromagnetic field, the rule is that each of them becomes a source for the fields and obeys (13), with the densities given by N N ρe (x, t) = eδ(x − qj (t)), and j = eq˙j (t)δ(x − qj (t)). (14) j=1

j=1

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In order to have a Hamiltonian description of this dynamics later, we introduce the scalar potential V and the vector potential A associated to the electromagnetic field (E, B). They are defined so that E=−

∂A − ∇V, ∂t

B = ∇ ∧ A,

(15)

hence the first two equations (12) are satisfied, and ∂ (∇A) + ∆V = −−1 0 ρe ∂t 2 1 ∂ 1 ∂V ) = µ0 j A − ∆A + ∇(∇A + 2 2 2 c ∂t c ∂t

(16)

yield the last two equations of (12). There is some freedom in the choice of A and V in the sense that the physical fields E and B are unaffected by a change of the type V !→ V +

∂χ , A !→ A − ∇χ, ∂t

where χ is any scalar function of (x, t). A transformation of this kind is called a gauge transformation. This allows in particular to choose the potential vector A so that it satisfies ∇A = 0, (17) by picking a χ solution to ∆χ = ∇A, if (17) is not satisfied. This is the so called Coulomb gauge in which (16) reduces to 1 ρe (y, t) ∆V = −−1 ρ ⇐⇒ V (x, t) = dy e 0 4π0 |x − y| 2 1 ∂ 1 ∂∇V ≡ µ0 jT . A − ∆A = µ0 j − 2 (18) c2 ∂t2 c ∂t The subscript T stands here for transverse, since ∇jT ≡ 0. Assume we have N particles of identical masses m and identical charges e interacting with the electromagnetic field satisfying (18) with (14) so that V (x, t) =

N e 1 . 4π0 j=1 |x − qj (t)|

(19)

We want to write down a Hamiltonian function which will yield (13) back when we compute the equation of motions for the particles as (3). It is just a matter of computation to show that the following (time-dependent) Hamiltonian fulfills this requirement: H(q, p, t) =

1 e2 1 |pj − eA(q, t)|2 + . 2m 8π0 |qi − qj | j i=j

(20)

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The only thing to note is that when one computes the part of the electric field E that is produced by the ∇V part of (15) at the point qj (t), the double sum that appears due to the form (19) of V contains an infinite part when the indices take the same value, which is simply ignored . Remark: It follows also from the above that the momentum pj of the j’th particle isn’t proportional to its velocity, but is given instead by pj = mvj + eA(qj , t). This is an important feature of systems interacting with electromagnetic fields. As noted above, this Hamiltonian is time dependent. We can get a time independent Hamiltonian provided one takes also into account the energy of the field in the Hamiltonian. This new Hamiltonian H tot reads 1 2 H tot = H(q, p, t) + , dx 0 |∂A(x, t)/∂t|2 + µ−1 0 |∇ ∧ A(x, t)| 2 where the first term in the integral is the contribution to the electric field that is not provided by the Coulomb potential (19) (which is taken into account in (20)) and the second term is the magnetic energy. It can be shown also that the total energy H tot is conserved. 1.2 Quantization The Quantum description of a general classical system is given by a set of postulates we list here as P1 to P4. In order to motivate and/or illustrate their meaning, we consider in parallel the typical Hamiltonian (2) to make the link with its quantization by means of the traditional recipe. P1: The phase space Γ is replaced by a Hilbert space H whose scalar product is denoted by · | · , with anti-linearity on the left. The state of the system is characterized by a ray in this space, that is a unit vectors with an arbitrary phase. Actually, rays characterize the pure states of the system. When we consider Quantum Statistical Mechanics, we will make a distinction between pure states and mixed states that will be introduced then. However, in that section, we will go on talking about states. In case of our example, H = L2 (RdN ), RdN being the configuration space. The state of the system is characterized by a normalized complex valued function ψ(q) in L2 (RdN ), also called the wave function of the system. P2: The observables are given by (possibly unbounded) self-adjoint linear operators on H obtained from their classical counterparts by a quantization procedure. The quantization procedure is not always straightforward. In particular, if the phase space has a non trivial topological structure, sophisticated methods of quantizations have to be applied. The link with the corresponding classical observables should be achieved, at a formal level at least, by taking the limit → 0.

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For our example, the formal substitutions pk !→ −i∇k , qk !→ mult qk ,

k = 1, · · · , N

(21)

are used. Here mult qk denotes the operator multiplication by the variable qk , which we shall simply denote by qk below, and is Planck’s constant, whose numerical value is about 1.055 × 10−34 J.s. In particular, the classical Hamiltonian (2) gives rise to the (formally) self-adjoint operator H=

N k=1

−

2 ∆k + V (q1 , q2 , · · · , qN ) on H, 2mk

(22)

where ∆k denotes the Laplacian in the variables qk . This class of operators goes under the name Schr¨odinger operators and plays, for obvious reasons, an important role in Quantum Mechanics. Note that the quantization of observables by the formal rule (21) may need to be precised by a symmetrization procedure due to the noncommutativity of p and q, [pj,n , qk,m ] =

[∂qj,n , qk,m ] = 1l, i i

(23)

where 1l denotes the identity operator. The symmetrization can be performed by hand in some concrete cases. For example, the dilation operator (for N = 1) is the (selfadjoint) quantization of p · q given by p · q !→

−i (∇q q + q∇q ). 2

Note that in dimension d = 3, the angular momentum x ∧ p vector doesn’t require symmetrization and yields the operator x ∧ p !→ J := −iq ∧ ∇q ,

(24)

whose components satisfy the relations [Ji , Jj ] = iJk , for (i, j, k) a permutation of (1, 2, 3).

(25)

The (components of the) angular momentum are unbounded operators which are known to have discrete spectrum, see below. For a general classical observable B(q, p) (belonging to some reasonable class of smooth real valued functions on Γ R2d , say) the Weyl quantization procedure B !→ B W defined by q + q , p ei(q−q )·p/ ψ(q )dq dp B (B W ψ)(q) = (2π)−d 2 is a good prescription to obtain the corresponding self-adjoint observables. It maps functions of q ∈ Rd to the corresponding multiplication operators and polynomials

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in pj to the same polynomials in the differential operator i ∂qj . Note, however, that there exists other quantization prescriptions that have their own merits. Also, in other cases, if the geometry of the classical phase space of the system has more structure, the formal operator (22) needs to be supplemented by boundary conditions determined by physical considerations. For example, if the system is confined to a region Λ in configuration space, one customarily provides ∂Λ with Dirichlet boundary conditions. In particular, the Hamiltonian of a particle in Rd confined to a cube Λ whose sides have length L is given by HΛ = −

2 ∆ plus Dirichlet boundary condition at ∂Λ. 2m

(26)

P3: The result of the measure of an observable B on the quantum system characterized by ψ ∈ H is an element b ∈ R of the spectrum σ(B) of the self-adjoint operator B. Moreover, the probability to obtain an element in (b1 , b2 ] as the result of this measure on the state ψ is given by Pψ (B ∈ (b1 , b2 ]) = PB ((b1 , b2 ])ψ 2 ,

(27)

where PB (I) denotes the spectral projector of the operator B on the set I ⊂ R. Furthermore, once a measure of B is performed, and the result yields a value in a set (b1 , b2 ] ⊂ R, the wave function ψ is reduced, i.e. it undergoes the instantaneous change PB ((b1 , b2 ])ψ . (28) ψ !→ PB ((b1 , b2 ])ψ Another observable C is said to be compatible with B if B and C commute, i.e. if [PB (α), PC (β)] = 0, for any intervals α, β ⊆ R.

This postulate explains the importance of the efforts made by mathematical physicists in order to determine the spectral properties of operators related to Quantum Mechanics. As, in general, the spectrum of a self-adjoint operator is the union of its discrete and continuous components, the result of the measure of an observable may be quantized, even though its classical counterpart may take values in an interval. This justifies the adjective Quantum for the theory. Several examples of this fact will be discussed in the lectures on the spectral theory of unbounded operators, in particular for Schr¨odinger operators of the form −∆ + V . Note that due to the Spectral Theorem, the expectation value of an observable B in a state ψ can be written as b P (db)ψ 2 = bψ|P (db)ψ = ψ|Bψ. Eψ (B) = σ(B)

σ(B)

The reduction processes of the wave function (28) after the measurement of B insures that an immediate subsequent measure of B gives a result in the same set

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(b1 , b2 ] with probability one. The compatibility condition insures that the observables B and C can be simultaneously measured, in the sense that once B and C have been measured with results in the sets α and β respectively, further successive measurements of B and C, in any order, will give results in the same sets with probability one. Or in other words, B and C can be diagonalized simultaneously. This is not the case if B and C do not commute. Let us introduce some very classical examples as illustrations of the above. The interpretation of the wave function, in the setting where Γ = R2dN , is that |ψ(q)|2 dq ˆ is the probability that the system is at point q of configuration space and if ψ(p) 2 ˆ dp is the probability that it has momendenotes the Fourier transform of ψ, |ψ(p)| tum p. This is just a particular case of the above rule. Indeed, the operators qk,m , k = 1, · · · , N , m = 1, · · · , d all commute and they have continuous spectra R, as multiplication operators. Hence the interpretation of |ψ(q)|2 follows. That of |ψ(p)|2 is a consequence of the fact that the Fourier transform is unitary on L2 (RdN ) which transforms the derivative into a multiplication by the independent variable. Note that (23) shows that qj,n and pj,n cannot be simultaneously determined, an expression of Heisenberg’s uncertainty principle. Actually, Heisenberg’s principle can be put on more quantitative grounds as follows. Let A and B be two self-adjoint operators such that their commutator can be written as [A, B] = iC,

where C = C ∗ .

Then, denoting the variance of A in the state ψ by ∆ψ (A)2 = ψ|(A − Eψ (A))2 ψ, we get the inequality

Eψ (C) . 2 Applied to the operators p and q, we get the familiar relation ∆ψ (A)∆ψ (B) ≥

∆ψ (p)∆ψ (q) ≥

(29)

. 2

Similarly, the components of the angular momentum Jk , k = 1, 2, 3 are not compatible, however it follows from (24) that Jk and J 2 := J12 + J22 + J32 are compatible observables, for any k. Hence one can measure the third component of the angular momentum and its length simultaneously. The result of such measures belongs to the spectra of these operators which is discrete as we recall here. If we denote by Kj the eigenspace of J 2 associated with the quantum number j and consider the restriction of J3 to that subspace, a classical algebraic computation shows that σ(J 2 ) = {j(j + 1)|j ∈ N/2} and σ(J3 |Kj ) = {−j, −j + 1, · · · , j − 1, j}. (30) The effect of boundary conditions on the spectrum of operators can be quite dramatic, as the following comparison shows. The Hamiltonian (22) (with N = 1 for

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simplicity) with V ≡ 0, the so-called free Hamiltonian H0 = − 2m ∆ is unitarily equivalent by Fourier transform to a multiplication operator 2

H0 mult k 2 on L2 (Rd ). Its spectrum is then σ(H0 ) = R+ . The spectrum of the operator (26) is easily computed to be 2 2

2π 2 2 2 (n + n + · · · + n ) | n ∈ Z (31) σ(HΛ ) = j 2 d mL2 1 Another celebrated Hamiltonian is the harmonic oscillator, which will play a prominent role in the quantization of classical fields. In one dimension, this Schr¨odinger operator reads p2 γ + q 2 , where γ is a positive constant. 2m 2 Performing the (canonical) change of operators P and Q by P = (mγ)−1/4 p, Q = (mγ)1/4 q, so that [Q, P ] = i, the operator becomes Ho =

γ 1/2 ω 2 (P + Q2 ), with ω = . 2 m

(32)

The spectral analysis of (32) is essentially algebraic once one introduces the creation and annihilation operators by 1 1 a∗ = √ (Q − iP ), a = √ (Q + iP ), such that [a, a∗ ] = 1l . 2 2

(33)

The operator (32) takes the form 1 Ho = ω(a∗ a + ). 2 Then, defining the vacuum state state |0 as the (normalized) solution to the differential equation mω 1/4 mω 2 a|0 = 0 ⇐⇒ |0 = e− 2 q , π one sets by induction (a∗ )n |n = √ |0, n = 1, 2, · · · n! These vectors are normalized and take the form of a product of polynomials of degree n, known as Hermite polynomials, by the Gaussian |0. One sees easily, using the so-called canonical commutation relations (33), that √ √ 1 Ho |n = ω n + |n since a∗ |n = n + 1|n + 1 and a|n = n|n − 1. 2

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These eigenvectors are non-degenerate, such that ∆|n (p)∆|n (q) = (n+1/2), and span L2 (R). P4: The time evolution of the system is determined by its Hamiltonian H, the energy observable of the system. There are two equivalent standard descriptions: The Schr¨odinger picture, in which the state ψ evolves in time according to the timedependent Schr¨odinger equation in H i

d ψ(t) = Hψ(t), with ψ(0) = ψ. dt

(34)

The Heisenberg picture, in which the state is fixed, whereas the observables B evolve in time according to the Heisenberg equation in the space of self-adjoint operators on H d i B(t) = −[H, B(t)], with B(0) = B. (35) dt Introducing the unitary evolution group U (t) = e−itH/ (Spectral Theorem again), we get the relation between the Schr¨odinger and Heisenberg pictures through the identity Eψ(t) (B) = Eψ (B(t)), ∀t ∈ R, which follows from ψ(t) = U (t)ψ, and B(t) = U (t)∗ BU (t). As a consequence, ψ(t) remains normalized for all times and, since the Hamiltonian H commutes with the evolution group U (t) it generates, the observable energy is constant in time. This is also true for observables which are compatible with H, as (35) shows. The motivations behind the first order linear evolution equation (34) stem from physical observations leading to the so-called superposition principle implying linearity and from the fact that ψ at time 0 should determine completely the state at any later time t. This equation is the quantum equivalent of (3), whereas (35) is the quantum equivalent of (11). Applied to our example (2), the relation between (35) and (11) together with (9) and (23) are in keeping with the so-called correspondence principle stating that Poisson brackets are to be replaced by commutators in order to achieve formal quantization in that setting: [·, ·] . { · , · } !→ i This yields another motivation for (35). As an example, consider the case where the Hamiltonian has discrete nondegenerate spectrum {Ej }j∈N with associated eigenvectors {φj }j∈N , as is the case if the potential is confining. The time evolution of any initial state ψ reads

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ψ(t) =

cj e−itEj / φj , ∀t ∈ R, where cj = φj |ψ.

(36)

j∈N

Therefore, the probability of measuring the energy Ej0 ∈ σ(H) in the state ψ(t) is given by Pψ(t) (H ∈ {Ej0 }) = |φj0 φj0 |ψ(t) 2 = |φj0 |ψ(t)|2 = |cj0 e−itEj0 / |2 = |cj0 |2 and is constant. We used the convenient notation PH ({Ej }) = |φj φj |. Similarly, the probability to obtain an energy in a subset E = {Ej0 , Ejn , · · · , Ejn } of the spectrum of H is given by 2 n Pψ(t) (H ∈ E) = |φjk φjk |ψ(t) = |cjk |2 . (37) k

k

Note however, that the sole knowledge of the spectrum of H does not allow in general to get precise information about the evolution of states that are not eigenstates, due to the complicated interferences present in (36). A nice and sometimes useful exception to this rule is the case of coherent states for the harmonic oscillator. In the one-dimensional setting used in (32), these normalized states depend on a complex number α and are defined as |α = e− 2 |α| 1

2

∞ αn √ |n. n! n=0

They have the properties (which can be checked by means of (33) only) a|α = α|α, ∆|α (p)∆|α (q) = /2, |α|α=0 = |0. Their explicit expression as functions of L2 (R) reads as 1/2 mω 1/4 2mω 1 mω 2 ∗ q + |α = exp − α(α + α ) − αq . π 2 2 Then, using e−itH/ |n = e−iω(n+1/2)t |n, we get e−itH/ |α = e−iωt/2 |eiωt α. Finally, we note also here that in case the Hamiltonian is time-dependent, (34) gives rise under some regularity conditions to a two-parameter unitary evolution operator U (t, s) on H satisfying ∂ U (t, s) = H(t)U (t, s), with U (s, s) = 1l ∂t from which follows the relation

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U (t, r)U (r, s) = U (t, s), ∀r, s, t ∈ R. In such a case, the evolution operator is no longer an exponential and the future of initial wave functions is usually harder to describe. But again, in case the Hamiltonian is essentially a quadratic form in p and q, with time-dependent coefficients, explicit solutions to the above equation can be obtained, provided the initial conditions are of a coherent state type. 1.3 Fermions and Bosons So far we have considered systems which have no internal structure or degrees of freedom. Such internal degrees of freedom are introduced by taking a tensor product of the original Hilbert space H with K, another Hilbert space in which these degrees of freedom live, so that the system is now described by means of the new Hilbert space H ⊗ K. An important internal degree of freedom is the spin of a particle. It is a vector valued operator S in a finite dimensional space K Ck whose components also satisfy the commutation relations (25), and therefore displays the same spectral properties (30). A spin is half-integer or integer, depending whether it is true for the maximal quantum number s of S 2 . Consider now the state of a collection of N identical particles, that is sharing the same physical characteristics like masses, spins, charge, etc. In the framework of our example and slightly abusing notations, it is described by means of a wave function ψ(q1 , s1 ; q2 , s2 ; · · · ; qN , sN ) ∈ L2 (RdN ) ⊗ KN . The fact that the particles are identical is equivalent to saying that all observables B(q1 , p1 , s1 ; · · · ; qN , pN , sN ) applied to such states are invariant under permutations of their variables (qj , pj , sj ). An example of such observable is the kinetic energy part of (22), and it is also true of the potential part of this Hamiltonian due to (2). Therefore, if Pjk is the operator whose action is to permute the variables labeled j and k, one has 2 ∗ Pjk = 1, Pjk = Pjk , and [Pjk , B] = 0, ∀ B.

(38)

Hence, σ(Pjk ) = {+1, −1} and the observables and Pjk can be diagonalized simultaneously. Thus we can first diagonalize Pjk and then describe the observables restricted to the corresponding subspaces of Pjk . It is a law of nature that the eigenvalues of Pjk are either +1 for all pairs j, k or -1 for all pairs j, k. Therefore, identical particles divide themselves into two distinct sets of particles: those that are invariant under exchange of the variables of their wave function, and those that undergo a sign change under this operations. Particles belonging to the former set are called bosons whereas they are called fermions if they belong to the latter. As the properties (38) are true for all pairs jk of labels, they are also true for arbitrary permutations π ∈ PN in the group of permutations of N elements. In particular, if Pπ denotes the permutation of indices corresponding to π, then Pπ ψ = ψ for bosons and Pπ ψ = (−1)π ψ for fermions, where (−1)π is the signature of the permutation π. In other words, the above discussion shows that the physical Hilbert spaces for bosons and fermions are not the N -fold tensor product HN of the Hilbert space H of their one particle n n and H− , defined by descriptions, but rather H+

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Alain Joye n H± = S± Hn where S± =

1 (±1)π Pπ . N! π∈PN

The operators S± are easily checked to be orthogonal projectors onto the orthogonal n of Hn and vectors in these spaces are characterized by the propersubspace H± ties described above. These characteristics will have important consequences on the physical properties of collections of such particles. In particular, antisymmetry forbids two independent fermions to be in the same quantum state. Indeed, the sign change induced by exchange of these two particles in the antisymmetrization procedure makes the vector vanish. This goes under the name Pauli’s Principle. It turns out that the fermionic or bosonic nature of particles is linked to the properties of their spin. Indeed, it can be shown within the realm of relativistic quantum field theory that fundamental requirements of Physics as micro-causality and Lorentz’s invariance imply the so-called Spin-Statistic Theorem asserting that particles with half-integer spin are fermions, whereas those with integer spin are bosons. It is an experimental fact that no other statistics is present in nature. Electrons are thus fermions and photons are bosons (although the latter have an internal degree of freedom called helicity, instead of a spin).

2 Quantum Statistical Mechanics 2.1 Density Matrices We give here, in a particular setting, some heuristics behind the formal definition of state (or mixed state) in Quantum Statistical Mechanics which will be used later on. The first approach is based on a time dependent point of view supplemented by a postulate on the behavior of some phases, advocated in [3], for example. Let us start from (37) which says that in case the Hamiltonian H on H, of a system is time independent, discrete and non-degenerate, i.e. when the system is isolated from the rest of the world, its normalized wave function can be written as cj (t)φj , (39) ψ(t) = j∈N

with explicit time dependent complex valued coefficients cj (t). The basic postulates of Statistical Mechanics are formulated for isolated systems. However, in practice, one is often interested in a subsystem of the whole system only, so that, certain degrees of freedom are not observed and are incorporated in what we call the rest of the world. Thus Statistical Mechanics effectively deals with systems that interact with the external world, so that the truly isolated system is our initial system plus the rest of the world. The relevant Hilbert space to describe this new system is the tensor product R ⊗ H, where R is the Hilbert space of the rest of the world, and the corresponding scalar product is the product of the respective scalar products on R and H. The wave function of this larger isolated system can still be written as

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55

(39), with the proviso that the cj s are now time dependent elements of R such that j cj (t)|cj (t)R = 1, where the subscript specifies what scalar product is used. Now, if B is an observable acting on the original system only, technically of the form 1l ⊗ B as an operator on R ⊗ H, the instantaneous expectation value of a set of measurements of this observable on ψ(t) is given by cj (t)|ck (t)R φj |Bφk H . ψ(t)| 1l ⊗ Bψ(t)R⊗H = k,j

In an actual experiment, it is rather the time average of the above quantity that is measured, over a time that is large with respect to the ”molecular time scale”, but short with respect to the resolution of the measurement apparatus. Therefore, the measured quantity is actually B = cj (t)|ck (t)R φj |Bφk H , k,j

where the bar indicates time average. One postulate concerns the scalar products of the cj (t)’s about which we have minimal knowledge, since it deals with properties of the external world. It is postulated that these scalar products producing interferences are averaged to zero over the time scale on which we observe the system, this is the Random phases postulate: cj (t)|ck (t)R = 0, ∀ j = k. The consequence of this postulate is an effective description of the original system by means of the non-zero scalars λj = cj (t)|cj (t)R , such that λj = 1, (40) j

in the sense that the outcome of a measurement in that framework is given by B = λj φj |Bφj H . j

Therefore one says that the random phases postulate allows to regard the state of the system as an incoherent superposition of eigenstates of H or a mixed state. As a consequence, it is possible to represent the mixed state by a density matrix. Let ρ be the linear operator on H defined by ρ= λj |φj φj |. (41) j

It is a positive, trace class operator, of trace one, such that B = Tr (ρB).

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This operator contains all the information we have about the mixed state and allows to compute in a convenient way all expectation values by means of the trace operation. Note that a pure state χ as defined in the previous section corresponds to the density matrix ρχ = |χχ|. Actually, it is easy to see that a density matrix ρ corresponds to a pure state if and only if it is a rank one projector. Another approach of mixed states consists in noting that incomplete information about a system always leads to density matrices, without resorting to delicate properties of the time evolution. A first point of view consists in considering the system as an ensemble of true eigenstates, to be considered one at a time, where the relative proportion of the eigenstate φj is λj . The value λj ∈ [0, 1] is interpreted as the classical probability toget the pure eigenstate φj in the mixed state. This statistical interpretation ρ = j λj |φj φj | allows to avoid any consideration of effective coupling between the system and the ”external world” and makes no use of a priori knowledge about the time evolution. A second interpretation of incomplete knowledge about the system consists in splitting the total Hilbert space it lives in into R ⊗ H, where R concerns the degrees of freedom that are not known. Therefore, if {ϕj ⊗ ψk }j,k denotes an orthonormal basis of R ⊗ H made out of individual orthonormal bases of R and H and ρ is any density matrix on R ⊗ H, one introduces the corresponding reduced density matrix ρH on H by its matrix elements ϕk ⊗ ψi |ρ ϕk ⊗ ψj , ∀i, j. (ρH )ij = k

The matrix ρH is designed so that for any operator of the form 1l ⊗ B, one has Tr H (ρH B) = Tr (ρ 1l ⊗ B), where Tr H denotes the trace in H. This formula is in keeping with our ignorance of the degrees of freedom in R which are traced out. The point is that one can easily check that if ρ = |Ψ Ψ | for some Ψ ∈ R ⊗ H, then, in general, the corresponding reduced density matrix ρH characterizes a mixed state. Therefore we will adopt from now on the following definition: A mixed state (or simply state) in Quantum Statistical Mechanics is a positive trace class operator on H of trace 1. The time dependence of density matrices is governed by the following equation in the subset of density matrices in T (H), the linear space of trace class operators on H, d (42) i ρ = [H, ρ], ρ(0) = ρ0 . dt This equation stems from the cyclicity of the trace and the relation which must hold for all t and all observables B

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Tr (ρB(t)) = Tr(ρ(t)B). Note that (42) is in keeping with the evolution of the density matrix of a pure state.

3 Boltzmann Gibbs So far, we haven’t talked about equilibrium properties. The basic postulate in Quantum Statistical Mechanics describes the density matrix of an isolated system that has reached equilibrium. Assume the energy of the system is known to lie within the range [E, E + ∆], where ∆ << E. The Equal a priori Probability postulate, formalizing again our minimal knowledge of the total system, states within the framework where the φj ’s represent eigenstates of the Hamiltonian H, that at equilibrium, the λj ’s defining the density matrix ρeq are given by λ if E < Ej < E + ∆, λj = (43) 0 otherwise. The constant λ is normalized so that the trace of ρeq is one. Other ways of writing ρeq are E<Ej <E<∆ |φj φj | . (44) ρeq = PH (E, E + ∆)/ Tr (PH (E, E + ∆)) = #{j | Ej ∈]E, E + ∆[} Note that as ρeq is a function of the Hamiltonian H, it is constant in time due to (42), which is what we expect for the equilibrium density matrix. This above prescription for ρeq corresponds to the micro-canonical ensemble, where it is understood that the system under consideration has fixed energy and fixed number of particles. In this case, Boltzmann’s formula is used to define the entropy The entropy of a system at equilibrium in the microcanonical ensemble reads S(E) = k ln Γ (E), where Γ (E) = #{j | Ej ∈]E, E + ∆[}

(45)

and k 1, 38 × 10−23 J/K is Boltzmann’s constant. The quantity Γ (E), which is the denominator in (44), gives the number of quantum states that are accessible to the system. The entropy actually depends on other variables such as the volume V of the system, the number N of particles of the system, etc... that we omitted in the notation. The definition (45) makes the bridge between equilibrium Statistical Mechanics and thermodynamics, once the thermodynamic limit is taken. That is, once it is demonstrated that Boltzmann’s formula fulfills the following conditions:

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a. Extensivity, so that the thermodynamical limit as V → ∞, N → ∞ and E → ∞ exists i.e. 1 S(E, V, N ) → s(e, v), where E/N → e, V /N → v, S/N → s N where e, s and v are the densities of energy and entropy and v is the specific volume. b. The fact that if exterior parameters of the system initially at equilibrium are varied in such a way that the system can reach another equilibrium configuration, then the difference of entropies between these configurations is non negative. This is an expression of the second law of thermodynamics which implies that the equilibrium state maximizes the entropy. We’ll come back to this point shortly. Without discussing thermodynamics, we mention that assuming the existence of entropy and that properties a) and b) hold, all thermodynamical quantities can be computed from S(E, V, N ) through the definitions: ∂S 1 = defines the temperature, so that , T ∂E V,N U (S, V, N ) ≡ E(S, V, N ) the internal energy of the system exists and ∂U P =− defines the pressure whereas ∂V S,N ∂U defines the chemical potential. µ= ∂N S,V The above definitions yield the familiar differential dU (S, V, N ) = T dS − P dV + µdN,

(46)

motivating the physical interpretations of these derivatives. The extensivity property of U , i.e. homogeneity of degree one, associated with (46) implies U = T S − P V + µN.

(47)

In order to complete the picture, let us briefly recall that the first law of thermodynamics asserts that the differential dU = δQ − δW is exact, where δQ is amount of heat absorbed by the system and δW is the work done by the system in any transformation. A corollary of the second law of thermodynamics says that the differential δQ dS = is exact, T relating the experimental notion of heat to entropy. These two statements imply the existence of the functions S and U at equilibrium.

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It can be argued that the definition (45) satisfies requirement a), but we shall not provide the argument here. Let us consider b). This last property calls for a variational approach of the entropy. Hence we introduce a more general definition of the entropy of a state by The entropy of state ρ of a physical system is given by S(ρ) = −k Tr (ρ ln(ρ)),

(48)

where the function x !→ x ln(x) is defined to be zero at x = 0. If {λj }j∈N denotes the set of eigenvalues of the density matrix ρ= λj |φj φj |, j

the entropy is given by S(ρ) = −k

λj ln(λj ).

(49)

j

Therefore one sees that S(ρ) ≥ 0 with S(ρ) = 0 if and only if λj = δj,k for some k, i.e. ρ corresponds to a pure state. Also, the entropy (48) of the density matrix ρ describing two independent systems defined on H1 and H2 is the sum of the individual entropies, as expected. Indeed, in such a case, ρ = ρ1 ⊗ ρ2 on H = H1 ⊗ H2 , where ρj , j = 1, 2 are the density matrices of the individual systems. Using ln(ρ1 ⊗ ρ2 ) = ln(ρ1 ) ⊗ 1l + 1l ⊗ ln(ρ2 ), and taking partial traces, one gets S(ρ) = S(ρ1 ) + S(ρ2 ). More generally, it can be shown that for α ∈ [0, 1] and arbitrary density matrices ρ1 , ρ2 S(αρ1 + (1 − α)ρ2 ) ≥ αS(ρ1 ) + (1 − α)S(ρ2 ), which shows that S is concave as a function on the set of density matrices, i.e. mixing density matrices increases the entropy. And, on the other hand, the entropy is almost convex in the sense S(αρ1 + (1 − α)ρ2 ) ≤ αS(ρ1 ) + (1 − α)S(ρ2 ) − α ln α − (1 − α) ln(1 − α). More mathematical properties of the entropy are provided in [1] or [5], for example. Now, maximizing (49) over the probabilities λj ’s shows that the entropy is maximal when the eigenvalues are all constant. Thus, when the corresponding eigenvectors φj are those of the Hamiltonian, we get back both the equal a priori probability postulate and Boltzmann’s formula. The micro canonical ensemble is convenient to motivate definitions, but one often prefers to use the canonical ensemble for applications. In that setting, the system under consideration interacts with a thermal reservoir whose property is to remain at fixed temperature. Exchanges of energy are allowed between the reservoir and the

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system, under the constraint that the averaged energy of the system is kept fixed. The maximization of entropy in the canonical ensemble leads to Gibbs prescription for the density matrix, as we now (formally) argue. Consider the functional over the set of density matrices. F(ρ) = S(ρ) − kβHρ ,

(50)

where Hρ denotes the expectation value of the energy computed by means of the density matrix ρ, and β is a Lagrange multiplier associated with the energy constraint. We need to compute the first variation of F δF(ρ) =

d F(ρ + tη)|t=0 , dt

where the admissible variation η is any trace class operator of zero trace and Tr (ρ) = 1. In order to do so, we first justify the following intuitive relation: If A(t) is a t-dependent self-adjoint operator, such that A(t) = A(0) + tη, then d Tr (f (A(t))) = Tr (f (A(t))η). dt

(51)

Indeed, Hellman-Feynman formula applied to A(t) whose eigenvalues and normalized eigenvectors are denoted by (aj (t), ϕj (t)) reads aj (t) = ϕj (t)|A (t)ϕj (t).

(52)

Thus for any (reasonable) real valued function f , d Tr (f (A(t))) = f (aj (t))aj (t) = ϕj (t)|f (A(t))ϕj (t)aj (t). dt j j In the case under consideration, A (t) = η so that, by (52), aj (t) = ϕj (t)|ηϕj (t) and, using orthonormality of the ϕj ’s, the RHS of (53) equals ϕj (t)|f (A(t))ϕj (t)ϕj (t)|ηϕj (t) j

=

ϕj (t)|f (A(t))ϕk (t)ϕk (t)|ηϕj (t)

j,k

=

j

In our case, we get

ϕj (t)|f (A(t))ηϕj (t), which yields (51).

(53)

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δF(ρ) = −k Tr (η(ln(ρ) + 1l + βH)), which has to be zero for any admissible η if ρ extremalizes F. In particular, we can self adjoint choose η = j ηj |ϕj ϕj | where {ϕj } are the set of eigenvectors of the operator ln(ρ) + 1l + βH and {ηj } are a set of real numbers satisfying j ηj = 0. For that η, we have δF(ρ) = −k ηj vj , j

where the vj ’s denote the eigenvalues of ln(ρ) + 1l + βH. Choosing above η0 = −η1 = 0 and ηj = 0 if j ≥ 2, we get that δF(ρ) = 0 for any η0 implies v0 = v1 . Iterating, we get v0 = vj , for any j ∈ N. Hence δF(ρ) = 0 ∀η ⇐⇒ ln(ρ) + 1l + βH = v0 1l . The constant v0 will be determined by the normalization of the state. Therefore, exponentiating, we find as extremalizer the Gibbs distribution ρG =

e−βH . Tr (e−βH )

(54)

Explicit computations that we do not present here show that the second variation δ 2 F is negative. Hence we get that the Gibbs prescription yields a maximum of the functional (50). Here the parameter β used to insure constancy of the average energy of the system in that canonical ensemble setting will be identified with the inverse temperature given by the usual formula β=

1 . kT

(55)

The normalization of the Gibbs distribution Z := Tr (e−βH ) defines the (canonical) partition function. It is related to the internal energy of the system HρG by ∂ ln(Z) . HρG = − ∂β Again, in the thermodynamic limit, the partition function of Statistical Mechanics is directly linked to a thermodynamical quantity: the free energy F of the system, defined as F = U − T S (see below), where U is the internal energy HρG of the system. To substantiate this claim, let us formally compute by means of (54) (assuming the thermodynamic limit and extensivity holds), S = kβHρG + k ln(Z) F := −kT ln Z defines the free energy in Statistical Mechanics.

so that (56)

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Moreover, as a consequence of our variational approach, we get that the free energy of a system is minimized by the equilibrium state, which together with (56) are two familiar properties of the free energy. More precisely, in thermodynamics, F is a function of (T, V, N ), which is the result of its very definition: The thermodynamical free energy is the Legendre transform of the internal energy U (S, V, N ) with respect to the variable S F (T, V, N ) = (U − T S)(T, V, N ) where S(T, V, N ) is computed from ∂U ∂S V,N = T .

(57)

This operation allows to trade the entropy variable for the more natural temperature variable. Recall that the Legendre transform of a one variable function f : x !→ f (x) is the function a : p !→ a(p) defined by a(p) = f (x(p)) − px(p), where x(p) is obtained by inversion of d f (x) = p(x). dx There is no loss of information in the process as long as the inversion of f (x) is possible and when f is concave, respectively convex, its Legendre transform is convex, respectively concave. In case f depends on other variables y, one has the identities ∂ ∂ ∂ a(p, y) = −x(p, y), a(p, y) = f (x, y) ∂p ∂y ∂y allowing to recover all thermodynamic quantities from F via the relations ∂F ∂F ∂F = −S, = −P, = µ. ∂T V,N ∂V T,N ∂N T,V We can now provide a justification of the identification (55) as follows, assuming the thermodynamic limit is taken and extensivity holds. Indeed, by means of that identification, we get by explicit computation on (56) ∂ −βH Tr ( ∂β e ) ∂β ∂ HρG F = −k ln Z − T k = −k ln Z − = −S, ∂T Z ∂T T

which is identical to the first relation above. Let us present here the classical computation of partition functions associated with independent harmonic oscillators.

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Let H be the Hilbert space spanned by the eigenvectors {|n}n=0,··· ,∞ of the Hamiltonian Ho in (32) corresponding to the energies n = 12 + n, n = 0, · · · , ∞ (we assume ω = 1, without loss). Working in the canonical ensemble, the partition function of one harmonic oscillator reads −βHo

Z1 (β) = Tr (e

−β/2

)=e

∞

e−βn =

n=0

1 , eβ/2 − e−β/2

the internal energy U1 (β) = Ho ρG is given by U (β) = −

∂ ln(Z1 (β)) 1 = coth(β/2), ∂β 2

whereas the free energy F1 (β) reads F (β) = −kT ln(Z1 (β)) = kT ln(eβ/2 − e−β/2 ). In case we work with a d-dimensional harmonic oscillator, or, equivalently, with d independent oscillators with the same frequency, we denote by |n1 , n2 , · · · , nd ,

nj ∈ N, j = 1, · · · , d,

the eigenvector corresponding to the energy d2 + n1 + · · · + nd . Thus, the corresponding partition function reads e−β(n1 +···+nd ) = Z(β)d , Z(β, d) = e−βd/2 n1 ≥0,n2 ≥0,··· ,nd ≥0

so that the internal and free energies and are given by U (β, d) = dU (β), Fd (β, d) = dF (β).

(58)

Going from the micro canonical to the canonical ensemble, we have allowed energy exchanges between the system under consideration and a thermal reservoir. In a similar fashion, we can relax the condition that the number of particles in the system is fixed and allow particles exchanges with the reservoir as well, assuming the their average number only is fixed. This corresponds to working in the grand canonical ensemble. As we will see later on, allowing particles exchanges in Quantum Open Systems is essential, in the sense that the statistical properties of these particles, i.e. their bosonic or fermionic nature, have definite physical consequences. This calls for a precision about the Hilbert space suitable to describe such situations, the so-called second quantization formalism. The Hilbert space allowing variable numbers of particles is either the symmetrical or anti-symmetrical Fock space, depending on the statistics. These Hilbert spaces will be object of much mathematical care later on, so we will briefly and informally describe here the bosonic and fermionic Fock spaces F± (H). If H is the one-particle Hilbert space, the nn is the n-boson or n-fermion subfold properly symmetrized tensor product H± n , space. An element Ψ of F± (H) is a collection {ψ(n)}n∈N , where ψ(n) ∈ H±

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0 for all n > 0, ψ(0) ∈ C ≡ H± , with the obvious linear structure and norm 2 2 ψ(n) . Observables B on the Fock space can be constructed as Ψ ±= ± n B = n B(n), where the B(n)’s acting on the n-particle subspaces are given (with B(0) = 0). In particular, the number operator N defined by NΨ = {nψ(n)}n∈N has the form N = n n 1lHn± . Another case is that of one body operators. That is when n A = n A(n) with A(n) = j=1 Aj , where Aj = 1l ⊗ · · · 1l ⊗ A ⊗ 1l · · · 1l, and A acts on the j’th copy of H. With these preliminaries behind us, let us assume we are given a Hamiltonian H with the structure above. The equilibrium state in that framework is obtained by maximization of the entropy, under the constraints that both the average energy and average number of particles are fixed. This leads to the computation of the first and second variations of the functional G over the density matrices defined by

G(ρ) = S(ρ) − kβHρ + kβµNρ , where β and µ are Lagrange multipliers associated with the imposed constraints. They will be identified in the thermodynamic limit, with the inverse temperature and chemical potential, respectively. A maximizing procedure quite similar to the one performed above that we will not detail here yields the extremum ρGC =

e−β(H−µN) Z

where, due to the structure of H and N, the grand canonical partition function Z can be written as eβµn Tr Hn± (e−βH(n) ). Z= n βµ

The quantity z = e is also called the fugacity and with Zn the canonical partition function, we can rewrite z n Zn . Z= n

One can also verify that the maximal value of G is S(ρGC ) − kβHρGC + kβµNρGC = k ln Z.

(59)

To make the bridge with thermodynamics, consider the thermodynamical grand potential Φ defined by the Legendre transform of F with respect to N , i.e. Φ(T, V, µ) = (F − µN )(T, V, µ) ∂F = µ. One can then see by formal manipulations similar to those performed where ∂N above, assuming the thermodynamic limit and extensivity holds, that Φ is minimal at equilibrium. From (59) and (47), we get that this minimum is given by

Φ = −P V = −kT ln Z and we further have the thermodynamical relations

Introduction to Quantum Statistical Mechanics

∂Φ ∂T

= −S, V,µ

∂Φ ∂µ

65

= −N. T,V

The ensemble (microcanonical, canonical or grand canonical) chosen to describe a specific system is largely made according to convenience for the computations. Therefore it is comforting to know that the respective descriptions are all equivalent. This is the statement known as the equivalence of ensembles which says that in the thermodynamical limit, one can use either the microcanonical or the canonical ensemble to perform calculations of thermodynamical quantities because the results will agree. Instead of providing a justification of this statement here, we shall be content with the explicit verification of this fact for a system of independent harmonic oscillators considered in the microcanonical and canonical ensembles. In the microcanonical ensemble, we compute the entropy by means of (45). If we have N independent oscillators each of which has energy levels j + 1/2, j ∈ N, we get for N large, ⎫ ⎧ ⎬ E + N/2 ⎨ nj = E − N/2 , Γ (E) # nj ∈ N, j = 1, · · · , N, | ⎭ ⎩ N j using the combinatoric formula ⎧ ⎫ ⎨ ⎬ M + N − 1 # nj ∈ N, j = 1, · · · , N, | nj = M ∈ N = . ⎩ ⎭ N −1 j

Hence, by means of Stirling formula, we compute in term of the energy density e = E/N , 1 1 1 1 S(E, N ) N k (e + ) ln(e + ) − (e − ) ln(e − ) ≡ N s(e). 2 2 2 2 Therefore, the temperature is determined by ∂s(e) 1 = = k ln T ∂e

e+ e−

1 2 1 2

,

so that we get the following formula for the energy density 1 1 eβ + 1 = coth(β/2). e= 2 eβ − 1 2 For the same system considered in the canonical ensemble, we obtained in (58) with d = N, 1 U (β, N ) = N U (β) = N coth(β/2), 2 which yields the same energy density e = U/N .

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Let us consider now the computation of the grand canonical partition function, in the simple bosonic/fermionic context where particles do not interact with one another. Consider the normalized vector |n0 , n1 , n2 , · · · , nj , · · ·± ∈ F± (H) in the socalled occupation number representation relative to the eigenstates |n in H of some nondegenerate Hamiltonian H. This vector consists in a normalized, fully (anti)symmetrized tensor product of states |n ∈ H characterized by n0 factors |0, n1 factors |1, · · · nj factors |j, etc. The number of particles N in such a state is obviously given by N = k nk . In case of bosons, nj ∈ N without restriction, whereas in case of fermions, Pauli’s principle enforces nj ∈ {0, 1}, for any j. We’ll denote by N the set of allowed values of the nj ’s , depending on the statistics. The collection {|n0 , n1 , n2 , · · · , nj , · · ·± }n0 ∈N ,··· ,nj ∈N forms an orthonormal basis of with a fixed number of particles, one gets F± (H). If one considers only the states that the set {|n0 , n1 , n2 , · · · , nj , · · ·± | k nk = N } forms an orthonormal basis N . of the subspace H± Let n denote the eigenvalue of H corresponding to |n. Then the one body observable H in F± (H) constructed from H satisfies nk k |n0 , n1 , n2 , · · · , nj , · · ·± . H |n0 , n1 , n2 , · · · , nj , · · ·± = k

The corresponding physical system consists of a collection of independent fermions or bosons individually driven by the Hamiltonian H. Though quite simple, such systems allow to put forward the effect of the statistics. The canonical partition function ZN (β) of N independent fermions/bosons is −β nj j j ZN (β) = e , {nj |

j

nj =N }

where the restrictions on the nj ’s due to the statistics are implicit in the notation. Hence, with z = eβµ , −β nj j j Z(β, z) = zN e = (ze−β j )nj j N ≥0 N ≥0 {nj |

j

nj =N }

{nj |

⎧ −β j −1 ! ⎪ ) ⎪ (1 − ze ⎨ j −β j n = (ze ) = −β j ⎪ ) ⎪ n j ⎩ (1 + ze

j

nj =N }

for bosons, for fermions.

j

In particular, we compute

N ρGC

⎧ ze−β j ⎪ ⎪ ⎪ ⎨ 1 − ze−β j ∂ j ln(Z(β, z)) = =z ze−β j ⎪ ∂z ⎪ ⎪ ⎩ 1 + ze−β j j

for bosons, for fermions,

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67

which allows to determine z. Similarly, the average occupation numbers can be obtained as 1 ∂ ze−β j for bosons, ln(Z(β, z)) = nj ρGC = − for fermions. β ∂j 1 ∓ ze−β j Therefore, we get the expected relation N ρGC =

nj ρGC ,

j

where we clearly see the effects of the statistics and temperature on the the average occupation numbers.

References 1. O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Texts and Monographs in Physics, Springer, New York, Heidelberg, Berlin, 1981 2. J. Glimm, A. Jaffe, Quantum Physics, Springer, New York, Heidelberg, Berlin, 1981 3. K. Huang, Statistical Mechanics, J. Wiley & Sons, New York, London, Sydney, 1963 4. Ph. A. Martin, F. Rothen, Many -body Problems and Quantum Field Theory, Texts and Monographs in Physics, Springer, 2nd Edition, 2004 5. B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Series in Physics, Princeton New-Jersey, 1993

Elements of Operator Algebras and Modular Theory St´ephane Attal Institut Camille Jordan, Universit´e Claude Bernard Lyon 1, 21 av Claude Bernard, 69622 Villeurbanne Cedex, France, e-mail: [email protected] url: http://math.univ-lyon1.fr/˜ attal

1

2

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.1 1.2

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70 71

C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1 2.2 2.3 2.4 2.5

First definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representations and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commutative C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 73 79 83 84

von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Topologies on B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commutant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predual, normal states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86 89 90

Modular theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

3.1 3.2 3.3 4

4.1 4.2 4.3

The modular operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 The modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Self-dual cone and standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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1 Introduction 1.1 Discussion Techniques and tools coming from operator algebras, that is, C ∗ -algebras or von Neumann algebras, are central in all the approaches of open quantum system theory. They are intensively used in most of the courses of these three volumes. They are essential in the Hamiltonian approach for they provide the setup for describing equilibrium states in quantum statistical mechanics, the so-called K.M.S. states (see C.-A. Pillet’s course in this volume), for describing the observable algebras of free Bose and Fermi gas (see M. Merkli’s course in this volume), for computing the standard Liouvillian of quantum dynamical systems (see C.-A. Pillet’s course in this volume) which is the starting point of ergodic theory on these systems. In the Markovian approach, they are essential tools for proving the Stinespring and the Lindblad theorems, which are the mathematical foundation of the so-called quantum master equations (see R. Rebolledo’s course in the second volume), for developing the theory of quantum Markov processes, their dilations and their qualitative behavior (see F. Fagnola’s course in the second volume and Fagnola-Rebolledo’s course in the third volume). In the third volume of this series (“Recent developments”), there is no course which does not make a heavy use of all the techniques of von Neumann algebras, states, representations, modular theory. The aim of this course is to give a basic introduction to this theory. Writing such a course is a challenge, for these theories are difficult, deep and subtle. It suffices to have a look to the most well-known references ( [1], [2], [3], [4], [5], [6], [7], [8],) to understand that we here enter into a heavy theory. Hundreds of volumes have been written on operator algebras, it has been a life work of numerous famous mathematicians to try to understand them, to classify them. It is not our task here to enter into all these details, we just want to browse the most basic elements, the tools that appear everywhere. We try to give as many proof as possible, we only skip the very long and difficult ones (at least, we give the ideas). This course has to be considered as a very first step in the theory, a tool box for the other courses. Readers who are interested in more details or more developments are encouraged to enter into the classical literature that we quoted above. Among the huge literature on the subject, we have selected few references at the end of this course. The two volumes by Bratteli and Robinson,[1] and [2], are fundamental references for these three volumes, for they really develop the theory of operator algebras in perspective with applications in quantum statistical mechanics. They are the only references in our list which connect operator algebras with physics. Furthermore, their exposition is concise and pedagogical. The series of volumes by Kadison and Ringrose,[4], [5] and [6], are sorts of bibles on operator algebras. They provide a very complete exposition on all the old

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and modern theory of operator algebras. For example they completely treat the classification theory of von Neumann algebra, which we do not treat here. Sakai’s book [8] is a well-known reference on the basic elements of C ∗ and von Neumann algebras. It is a concise and dense book, it should maybe not considered as a beginner reference. It does not treat modular theory. Pedersen’s book [7] is very complete and more pedagogical. Dixmier’s book [3] is an older reference, far from the modern pedagogical and notational standards, but it has influenced a whole generation of mathematicians. 1.2 Notations All the vector spaces, Hilbert spaces, algebras, are here supposed to be defined on the field of complex numbers C. On a Hilbert space the scalar product < · , · > is linear in the right variable and antilinear in the left one. The notation for the scalar product does not refer to the underlying space, there should not be any possible confusion. In the same way, the norm of any normed space is denoted by || · ||, unless it is necessary for the comprehension to specify the underlying space. On any Hilbert space, the identity operator is denoted by I, without precising the associated space. The algebra of bounded operators on a Hilbert space H is denoted by B(H). In any normed space, B(x, r) denotes the closed ball with center x and radius r.

2 C ∗-algebras The two main operator algebras are C ∗ -algebras and von Neumann algebras. They can be represented as sub-algebras of bounded operator algebras B(H), with different topologies. But they also admit abstract definitions, without reference to any particular representation. This is with the abstract theory we start with, representation theorems come later. 2.1 First definitions A C ∗ -algebra is an algebra A equipped with an involution A !→ A∗ and a norm || · || satisfying, for all A, B ∈ A, all λ, µ ∈ C: i) A∗∗ = A ∗ ii) (λA + µB) = λA∗ + µB ∗ ∗ iii) (AB) = B ∗ A∗ i’) ||A|| is always positive and ||A|| = 0 if and only if A = 0 ii’) ||λA|| = |λ| ||A|| iii’) ||A + B|| ≤ ||A|| + ||B|| iv’) ||AB|| ≤ ||A|| ||B||

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i”) A is complete for || · || ii”) ||AA∗ || = ||A|| . An algebra with an involution as above satisfying i), ii) and iii) is called a ∗-algebra. An algebra satisfying all the conditions above but where ii”) is replaced by ii”) ||A∗ || = ||A|| is called a Banach algebra. 2

The basic examples of C ∗ -algebras are: 1) A = B(H), the algebra of bounded operators on a Hilbert space H. The involution is the usual adjoint mapping and the norm is the usual operator norm: ||A|| = sup ||Af || ||f ||=1

. 2) A = K(H), the algebra of compact operators on H. It is a sub-C ∗ -algebra of B(H). 3) A = C0 (X), the space of continuous functions vanishing at infinity on a locally compact space X. Recall that a function f is vanishing at infinity if for every ε > 0 there exists a compact K ⊂ X such that |f | < ε outside of K. The involution on A is the complex conjugation f and the norm is ||f || = sup |f (x)| . x∈X

We will see later that these examples are more than basic : every commutative C ∗ -algebra is of the form 3), and every C ∗ -algebra is a sub-algebra of a type 1) example. Proposition 2.1. On a C ∗ -algebra A we have ||A∗ || = ||A|| for all A ∈ A. Proof. We have ||A|| = ||A∗ A|| ≤ ||A∗ || ||A|| and thus ||A|| ≤ ||A∗ ||. Inverting the role of A and A∗ gives the result. 2

An element I of a C ∗ -algebra A is a unit if IA = AI = A for all A ∈ A. If a unit exists it is unique and norm 1 (unless A = {0}). But it may not always exists. Indeed, in the example K(H) there exists a unit if and only if H is finite dimensional. In the example C0 (X) there exists a unit if and only if X is compact. But if a C ∗ -algebra does not contain a unit one can easily add one as follows. Consider the vector space A = A ⊕ C and provide it with the product (A, λ)(B, µ) = (AB + λB + µA, λµ), with the involution

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73

∗

(A, λ) = (A∗ , λ) and with the norm ||(A, λ)|| = sup ||AB + λB|| . ||B||=1

∗

Equipped this way A is a C -algebra. It admits a unit (0, 1). The algebra A identifies to the subset of elements of the form (A, 0). The only delicate point is to check that ||(A, λ)|| = 0 if and only if A = 0 and λ = 0. One can assume that λ = 0 for if not we are in A. Hence one can assume that λ = 1. We have ||B − AB|| ≤ ||B|| ||(−A, 1)|| . Thus if ||(−A, 1)|| = 0 then B = AB for all B ∈ A. Applying the involution gives B = BA∗ for all B ∈ A. In particular A∗ = AA∗ = A and thus B = AB = BA. This means A is a unit and contradicts the assumption. Note that the above definition of the norm in A comes from the fact that in any ∗ C -algebra we have ||A|| = sup ||AB|| . ||B||=1

Indeed, there is obviously an inequality ≥ between the two terms above. The equality is obtained by considering B = A∗ / ||A||. 2.2 Spectral analysis Let A be a C ∗ -algebra with unit I. An element A of A is invertible if there exists an element A−1 of A such that A−1 A = AA−1 = I. One calls resolvent set of A the set ρ(A) = {λ ∈ C; λI − A is invertible}. We put σ(A) = C \ ρ(A) and call it the spectrum of A. If |λ| > ||A|| then the series 1 λ n

n A λ −1

is normally convergent and its sum is equal to (λI − A) . This implies that σ(A) is included in B(0, ||A||). Furthermore, if λ0 belongs to ρ(A) and if λ ∈ C is such that |λ − λ0 | < ||λ0 I − A||, then the series

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(λ0 I − A) normally converges to (λI − A) 1) the set ρ(A) is open

−1

2) the mapping λ !→ (λI − A) 3) the set σ(A) is compact.

−1

λ 0 − λ n λ0 I − A n

. In particular we have proved that:

−1

is analytic on ρ(A)

We define r(A) = sup{|λ| ; λ ∈ σ(A)}, the spectral radius of A. Theorem 2.2. We have for all A ∈ A 1/n

1/n

r(A) = lim ||An ||

= inf ||An ||

n

n

≤ ||A|| .

In particular the above limit always exists and σ(A) is never empty. 1/n

Proof. Let n be fixed and let |λ| > ||An || . Every integer m can be written m = pn + q with p, q integers and q < n. Thus we have A m A pn+q ||An || p ||A|| q = ≤ n λ λ |λ| |λ| m m m n−1 n p ||A|| ||A || ||A|| + ... + ≤ 1+ < ∞. n |λ| |λ| |λ| p Thus the series

1 λ m

m A λ −1

1/n

converges and its sum is equal to (λI − A) . This proves that r(A) ≤ ||An || 1/n and thus r(A) ≤ lim inf n ||An || . 1/n Let us now prove that r(A) ≥ lim supn ||An || . If we have 1/n

r(A) < lim sup ||An || n

then consider the open set 1/n

O = {λ ∈ C; r(A) < |λ| < lim sup ||An || n

}.

On O all the operators λI − A are invertible, thus so are the operators I − λ1 A. −1 n and its Taylor series n ( A The mapping λ !→ (I − λ1 A) is analytic on O λ) n n converges. But the convergence radius of the series n z A is exactly

Operators Algebras and Modular Theory 1/n −1

(lim sup ||An ||

)

75

.

n

This would mean

1 1/n −1 < (lim sup ||An || ) |λ| n

which contradicts the fact that λ ∈ O. We have proved the first part of the theorem. If r(A) > 0 then it is clear that σ(A) is not empty. It remains to consider the case r(A) = 0. But note that if 0 belongs to ρ(A) this means that A is invertible and 1/n 1/n 1 = ||An A−n || ≤ ||An || ||A−n ||. In particular, 1 ≤ ||An || ||A−n || . Passing to the limit, we get r(A) > 0. Thus if r(A) = 0 we must have 0 ∈ σ(A). In any case σ(A) is non empty. Corollary 2.3. A C ∗ -algebra A with unit and all of which elements, except 0, are invertible is isomorphic to C. Proof. If A ∈ A its spectrum σ(A) is non empty. Thus there exists a λ ∈ C such that λI − A is not invertible. This means λI − A = 0 and A = λI. All the above results made use of the fact that we considered a C ∗ -algebra with unit. If A is a C ∗ -algebra without unit and if A is its natural extension with unit, then the notion of spectrum and resolvent set are extended as follows. The spectrum of We extend the notion of resolvent set in A ∈ A is its spectrum as an element of A. the same way. An element A of a C ∗ -algebra A is normal if A∗ A = AA∗ , self-adjoint if A = A∗ . If A contains a unit, then an element A ∈ A is isometric if A∗ A = I, unitary if A∗ A = AA∗ = I. Theorem 2.4. Let A be a C ∗ -algebra with unit. a) If A is normal then r(A) = ||A||. b) If A is self-adjoint then σ(A) ⊂ [ − ||A|| , ||A|| ]. c) If A is isometric then r(A) = 1. d) If A is unitary then σ(A) ⊂ {λ ∈ C; |λ| = 1}. e) For all A ∈ A we have σ(A∗ ) = σ(A) and σ(A−1 ) = σ(A) f) For every polynomial function P we have σ(P (A)) = P (σ(A)). g) For any two A, B ∈ A we have σ(AB) ∪ {0} = σ(BA) ∪ {0}.

−1

.

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Proof. a) If A is normal then n 2 n 2 2n 2n−1 2n−1 2n (AA∗ ) A = A2 A∗ = (AA∗ ) = (AA∗ ) 2 2n−1 2n 2n+1 = (AA∗ ) . = . . . = ||AA∗ || = ||A|| It is now easy to conclude with Theorem 2.2 b) We only have to prove that the spectrum of any self-adjoint element of A is a subset of IR. Let λ = x + iy be an element of σ(A), with x, y real. We have x + i(y + t) ∈ σ(A + itI). But 2 2 ||A + itI|| = ||(A + itI)(A − itI)|| = A2 + t2 I ≤ ||A|| + t2 . This implies

2

2

2

|x + i(y + t)| = x2 + (y + t) ≤ ||A|| + t2 or else

2

2yt ≤ ||A|| − x2 − y 2 for all t. This means y = 0. c) If A is isometric then

2 ||An || = ||A∗ n An || = A∗ n−1 An−1 = . . . = ||A∗ A|| = ||I|| = 1.

d) Assume e) is proved. Then if A is unitary we have −1

σ(A) = σ(A∗ ) = σ(A−1 ) = σ(A)

.

This and c) imply that σ(A) is included in the unit circle. e) The property σ(A∗ ) = σ(A) is obvious. For the other identity we write λI − A = λA(A−1 − λ−1 I) and λ−1 I − A−1 = λ−1 A−1 (A − λI). f) Note that if B = A1 . . . An in A, where all the Ai are two by two commuting, we have that B is invertible if and only if each Ai is invertible. Now choose α and α1 , . . . αn in C such that P (x) − λ = α (x − αi ). i

In particular we have P (A) − λI = α

(A − αi I). i

As a consequence λ ∈ σ(P (A)) if and only if αi ∈ σ(A) for a i. But as P (αi ) = λ this exactly means that λ belongs to σ(P (A)) if and only if λ belongs to P (σ(A)). g) If λ belong to ρ(BA) then (λI − AB)(I + A(λI − BA)−1 B) = λI. This proves that λI −AB is invertible on the right, with possible exception of λ = 0. The invertibility on the left is obtained in a similar way. This proves one inclusion. The converse inclusion is obtained exchanging the role of A and B.

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Theorem 2.5. The norm which makes a ∗-algebra being a C ∗ -algebra, when it exists, is unique. Proof. By the above results we have ||A|| = ||AA∗ || = r(AA∗ ) 2

for AA∗ is always normal. But r(AA∗ ) depends only on the algebraic structure of A. Proposition 2.6. The set of invertible elements of a C ∗ -algebra A with unit is open and the mapping A !→ A−1 is continuous on this set. −1 then Proof. If A is invertible and if B is such that ||B − A|| < A−1 B = A(I − A−1 (A − B)) is invertible for r(A−1 (A − B)) ≤ A−1 (A − B) < 1 and thus I − A−1 (A − B) is invertible. The open character is proved. Let us now −1 show the continuity. If ||B − A|| < 1/2 A−1 then ∞ −1 −1 n −1 −1 −1 B − A = A (A − B) A − A n=0

∞ −1 A (A − B) n A−1 ≤ n=1

≤

−1 2 A ||A − B||

1 − ||A−1 (A − B)|| 2 ≤ 2 A−1 ||A − B|| . This proves the continuity.

In the following, we denote by 1l the constant function equal to 1 on C and by idE the function λ !→ λ on E ⊂ C. A ∗-algebra morphism is a linear mapping Π : A → B, between two ∗-algebras A and B, such that Π(A∗ B) = Π(A)∗ Π(B) for all A, B ∈ A. A C ∗ -algebra morphism is ∗-algebra morphism Π between two C ∗ -algebras A and B, such that ||Π(A)||B = ||A||A , for all A ∈ A. Theorem 2.7 (Functional calculus). Let A be a C ∗ -algebra with unit. Let A be a self-adjoint element in A. Let C(σ(A)) be the C ∗ -algebra of continuous functions on σ(A). Then there is a unique morphism of C ∗ -algebra C(σ(A)) −→ A f !−→ f (A) which sends the function 1l on I and the function idσ(A) on A. Furthermore we have σ(f (A)) = f (σ(A)) for all f ∈ C(σ(A)).

(1)

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Proof. When f is a polynomial function the application f !→ f (A) is well-defined and isometric for ||f (A)|| = sup{|λ| ; λ ∈ σ(f (A))} = sup{|λ| ; λ ∈ f (σ(A))} = ||f || . Thus it extends to an isometry on C(σ(A)) by Weierstrass theorem. The extension is easily seen to be a morphism also. The only delicate point to check is the identity (1). Let µ ∈ f (σ(A)), with µ = f (λ). Let (fn )n∈IN be a sequence of polynomial functions converging to f . The sequence (fn (λ)I − fn (A))n∈IN converges to µI − f (A). As none of the fn (λ)I − fn (A) is invertible then µI − f (A)is not either (Proposition 1.6). Thus f (σ(A)) ⊂ σ(f (A)). Finally, if µ ∈ C \ f (σ(A)) then let −1 −1 g(t) = (µ − f (t)) . Then g belongs to C(σ(A)) and g(A) = (µI − f (A)) . Thus µ belongs to C \ σ(f (A)). An element A of a C ∗ -algebra A is positive if it is self-adjoint and its spectrum is included in IR+ . Theorem 2.8. Let A be an element of A. The following assertions are equivalent. i) A is positive. ii) (if A contains a unit) A is self-adjoint and ||tI − A|| ≤ t for some t ≥ ||A||. iii) (if A contains a unit) A is self-adjoint and ||tI − A|| ≤ t for all t ≥ ||A||. iv) A = B ∗ B for a B ∈ A. v) A = C 2 for a self-adjoint C ∈ A. Proof. Let us first prove that i) implies iii). If i) is satisfied then tI − A is a normal operator and ||tI − A|| = sup{|λ| ; λ ∈ σ(tI − A)} = sup{|λ − t| ; λ ∈ σ(A)} ≤ t. This gives iii). Obviously iii) implies ii). Let us prove that ii) implies i). If ii) is satisfied and if λ ∈ σ(A) then t − λ ∈ σ(tI − A) and with the same computation as above |t − λ| ≤ ||tI − A|| ≤ t. But as λ ≤ t we must have λ ≥ 0. This proves i). We have proved that the first 3 assertions are equivalent. We have that √ v) implies iv) obviously. In order to show that i) implies v) it suffices to consider C = A (using the functional calculus of Theorem 2.7 and identity (1)). It remains to prove that iv) implies i). Let f+ (t) = t ∨ 0 and f− (t) = (−t) ∨ 0. Let A+ = f+ (A) and A− = f− (A) (note that when iv) holds true then A is automatically self-adjoint and thus accepts the functional calculus of Theorem 2.7). We have A = A+ −A− and the elements A+ and A− are positive by (1). Furthermore the identity f+ f− = 0 implies A+ A− = 0. We have (BA− )∗ (BA− ) = A− (A+ − A− )A− = −A3− . In particular −(BA− )∗ (BA− ) is positive.

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Writing BA− = S + iT with S and T self-adjoint gives (BA− )(BA− )∗ = −(BA− )∗ (BA− ) + 2(S 2 + T 2 ). In particular, as the equivalence established between i), ii) and iii) proves it easily, the set of positive elements of A is a cone, thus the element (BA− )(BA− )∗ is positive. As a consequence σ((BA− )(BA− )∗ ) ⊂ [0, ||B|| ||A− ||]. But by Theorem 1.4 g) we must also have σ((BA− )∗ (BA− )) ⊂ [0, ||B|| ||A − ||]. In 2 2 particular σ(−A3− ) ⊂ [0, ||B|| ||A− || ]. This implies σ(A3− ) = {0} and A3− = 3 0 = ||A− || . That is A− = 0. This notion of positivity defines an order on elements of A, by saying that U ≥ V in A if U − V is a positive element of A. Proposition 2.9. Let U, V be self-adjoint elements of A such that U ≥ V ≥ 0. Then i) W ∗ U W ≥ W ∗ V W ≥ 0 for all W ∈ A; −1

ii) (V + λI)

−1

≥ (U + λI)

for all λ ≥ 0.

Proof. i) is obvious from Theorem 2.8. ii) As we have U + λI ≥ V + λI, then by i) we have −1/2

(V + λI)

−1/2

(U + λI)V + λI)

≥ I.

Now, note that if W is self-adjoint and W ≥ I then σ(W ) ⊂ [1, +∞[ and σ(W −1 ) ⊂ [0, 1]. In particular W −1 ≤ I. This argument applied to the above inequality shows that 1/2

(V + λI)

−1

(U + λI) −1/2

Multiplying both sides by (V + λI)

1/2

V + λI)

gives the result.

≤ I.

2.3 Representations and states Note that a ∗-algebra morphism is always positive, that is, it maps positive elements to positive elements. Indeed we have Π(A∗ A) = Π(A)∗ Π(A). Theorem 2.10. If Π is a morphism between two C ∗ -algebras A and B then Π is continuous, with norm smaller than 1. Furthermore the range of Π is a sub-C ∗ algebra of B.

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Proof. If A is self-adjoint then so is Π(A) and thus ||Π(A)|| = sup{|λ| ; λ ∈ σ(Π(A))}. But it is easy to see that σ(Π(A)) is included in σ(A) and consequently ||Π(A)|| ≤ sup{|λ| ; λ ∈ σ(A)} = ||A|| . For a general A we have ||Π(A)|| = ||Π(A∗ A)|| ≤ ||A∗ A|| = ||A|| . 2

2

We have proved the first part of the theorem. For proving the second part we reduce the problem to the case where ker Π = {0}. If this is not the case, following the Appendix subsection 2.5, we consider the quotient of A by the two-sided closed ideal ker Π : AΠ = A/ ker Π which is a C ∗ algebra. We can thus assume ker Π = {0}. Let BΠ be the image of Π, it is sufficient to prove that it is closed. Consider the inverse morphism Π −1 from BΠ onto A. As previously, for A self-adjoint in A we have ||A|| = Π −1 (Π(A)) ≤ ||Π(A)|| ≤ ||A|| . Thus Π −1 and Π are isometric and one concludes easily.

A representation of a C ∗ -algebra A is a pair (H, Π) made of a Hilbert space H and a morphism Π from A to B(H). The representation is faithful if ker Π = {0}. Proposition 2.11. Let (H, Π) be a representation of a C ∗ -algebra A. Then the following assertions are equivalent. i) Π is faithful. ii) ||Π(A)|| = ||A|| for all A ∈ A. iii) Π(A) > 0 if A > 0. Proof. We have already seen that i) implies ii), in the proof above. Let us prove that ii) implies iii). If A > 0 then ||A|| > 0 and thus ||Π(A)|| > 0 and Π(A) = 0. As we already know that Π(A) ≥ 0, we conclude that Π(A) > 0. Finally, assume iii) is satisfied. If B belongs to ker Π and B = 0 then Π(B ∗ B) = 0. But ||B ∗ B|| = 2 ||B|| > 0 and thus B ∗ B > 0. Which is contradictory and ends the proof. Clearly we have not yet discussed the existence of representations for C ∗ algebras. The key tool for this existence theorem is the notion of state. A linear form ω on A is positive if ω(A∗ A) ≥ 0 for all A ∈ A. Note that for such positive linear form one can easily prove a Cauchy-Schwarz inequality: |ω(B ∗ A)| ≤ ω(B ∗ B) ω(A∗ A), 2

with the same proof as for the usual Cauchy-Schwarz inequality.

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Proposition 2.12. Let ω be a linear form on A, a C ∗ -algebra with unit. Then the following assertions are equivalent. i) ω is positive. ii) ω is continuous with ||ω|| = ω(I). Proof. By Theorem 2.8 ii), recall that a self-adjoint element A of A, with ||A|| = 1, is positive if and only if ||(I − A)|| ≤ 1. In particular, for any A ∈ A, we have that ||A∗ A|| I − A∗ A is positive. If i) is satisfied then ω(A∗ A) ≤ ||A∗ A|| ω(I). By Cauchy-Schwarz we have |ω(A)| ≤ ω(I)1/2 |ω(A∗ A)|

1/2

≤ ||A∗ A||

1/2

ω(I) = ||A|| ω(I).

(2)

This proves ii). Conversely, if ii) is satisfied. One can assume ω(I) = 1. Let A be a self-adjoint element of A. Write ω(A) = α + iβ for some α, β real. For every λ ∈ IR we have 2 2 ||A + iλI|| = A2 + λ2 I = ||A|| + λ2 . Thus we have 2 2 2 β 2 + 2λβ + λ2 ≤ α2 + i(β + λ) = |ω(A + iλI)| ≤ ||A|| + λ2 . This implies that β = 0 and ω(A) is real. Consider now A positive, with ||A|| = 1. We have |1 − ω(A)| = |ω(I − A)| ≤ ||I − A|| ≤ I. Thus ω(A) is positive.

When the C ∗ -algebra A does not contain a unit, the norm property ||ω|| = ω(I) above has to be replaced by ||ω|| = lim ω(Eα2 ) α

for an approximate unit (Eα ) in A (cf subsection 2.5), we do not develop the proof in this case. We call state any positive linear form on A such that ||ω|| = 1. We need an existence theorem for states. Theorem 2.13. Let A be any element of A. Then there exists a state ω on A such 2 that ω(A∗ A) = ||A|| . Proof. On the space B = {αI + βA∗ A; α, β ∈ C} we define the linear form f (αI + βA∗ A) = α + β||A|| . 2

One easily checks that ||f || = 1. By Hahn-Banach we extend f to the whole of A into a norm 1 continuous linear form ω. By the previous proposition ω is a state.

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We now turn to the construction of a representation which is going to be fundamental for us, the so called Gelfand-Naimark-Segal construction (G.N.S. construction). Indeed, note that if (H, Π) is a representation of a C ∗ -algebra A and if Ω is any norm 1 vector of H, then the mapping ω(A) = < Ω , Π(A)Ω > clearly defines a state on A. The G.N.S. construction proves that any C ∗ -algebra with a state can be represented this way. Theorem 2.14 (G.N.S. representation). Let A be a C ∗ -algebra with unit and ω be a state on A. Then there exists a Hilbert space Hω , a representation Πω of A in B(Hω ) and a unit vector Ωω of Hω such that ω(A) = < Ωω , Πω (A)Ωω > for all A and such that the space {Πω (A)Ωω ; A ∈ A} is dense in Hω . Such a representation is unique up to unitary isomorphism. Proof. Let Vω = {A ∈ A; ω(A∗ A) = 0}. The set Vω is a left ideal for if A ∈ Vω and B ∈ A then 0 ≤ ω((AB)∗ AB) ≤ ||B|| ω(A∗ A) = 0. 2

We consider the quotient space A/Vω . On A/Vω we define < [A] , [B] > = ω(B ∗ A) (We leave to the reader to check that this definition is consistent, in the sense that ω(B ∗ A) only depends on the equivalence classes of A and B). It is a positive sesquilinear form which makes A/Vω a pre-Hilbert space. Let Hω be the its closure. We put LA : A/Vω → A/Vω [B] !→ [AB]. We have

< LA [B] , LA [B] > = ω(B ∗ A∗ AB) ≤ ||A|| ω(B ∗ B) 2

for C !→ ω(B ∗ CB) is a positive linear form equal to ω(B ∗ B) on C = I. In partic2 ular < LA [B] , LA [B] > ≤ ||A|| < [B] , [B] >. One can extend LA into a bounded operator Πω (A) on Hω . If we put Ωω = [I] then the construction is finished. Let us check uniqueness. If (H , Π , Ω ) is another such triple, we have < Πω (B)Ωω , Πω (A)Ωω > = < Ωω , Πω (B ∗ A)Ωω > = ω(B ∗ A) = < Ω , Π (B ∗ A)Ω > = < Π (B)Ω , Π (A)Ω >. The unitary isomorphism is thus defined by U : Πω (A)Ωω !→ Π (A)Ω .

This G.N.S. representation theorem gives the fundamental representation theorem for C ∗ -algebras.

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Theorem 2.15. Let A be a C ∗ -algebra . Then A is isomorphic to a sub-C ∗ -algebra of B(H) for some Hilbert space H. Proof. For every state ω we have the G.N.S. representation (Hω , Πω , Ωω ). Put H = ⊕ω Hω and Π = ⊕ω Πω where the direct sums run over the set of all states on A. For every A ∈ A there exists a state ωA such that ||ΠωA (A)|| = ||A|| (Theorem 2.13). But we have ||Π(A)|| ≥ ||ΠωA (A)|| = ||A||. Thus we get ||Π(A)|| = ||A|| by Theorem 2.10. This means that Π is faithful by Proposition 2.11. In particular A is isomorphic to Π(A) which is, by Theorem 2.10 a sub-C ∗ -algebra of B(H). 2.4 Commutative C ∗ -algebras We have shown the very important characterization of C ∗ -algebras, namely they are exactly the closed ∗-sub-algebras of bounded operators on Hilbert space. We dedicate this last section to prove the (not very useful for us but) interesting characterization of commutative C ∗ -algebras . Let A be a commutative C ∗ -algebra . A character on A is a linear form χ on A satisfying χ(AB) = χ(A)χ(B) for all A, B ∈ A. On then calls spectrum of A the set σ(A) of all characters on A. Proposition 2.16. Every character is positive. Proof. If necessary, we extend the C ∗ -algebra A to A so that it contains a unit I. A character χ on A then extends to a character on A by χ(λI + A) = λ + χ(A). Thus we may assume that A contains a unit I. Let A ∈ A and λ ∈ σ(A). Then there exists B ∈ A such that (λI − A)B = I. Thus χ(λI − A)χ(B) = (λχ(I) − χ(A))χ(B) = χ(I) = 1. This implies in particular that λ = χ(A). We have proved that χ(A) always belong to σ(A). In particular χ(A∗ A) is always positive. As a corollary every character is a state and thus is continuous. The set σ(A) is a subset of A∗ , the dual of A. Theorem 2.17. Let A be a commutative C ∗ -algebra and X be the spectrum of A endowed with the ∗-weak topology of A∗ . Then X is a Hausdorff locally compact set; it is compact if and only if A admits a unit. Furthermore A is isomorphic to the C ∗ -algebra C0 (X) of continuous functions on X which vanish at infinity. Proof. Let ω0 ∈ X. Let A positive be such that ω0 (A) > 0. One can assume ω0 (A) > 1. Let K = {ω ∈ X; ω(A) > 1}. It is an open neighborhood of ω0 . Its closure K is included into {ω ∈ X; ω(A) ≥ 1}. The latest set is closed and included in the unit ball of A∗ which is compact. Thus X is locally compact. If A contains a unit I, then the same argument applied to A = 2I shows that X is compact.

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# # is a continuous complex Now, for all A ∈ A we put A(ω) = ω(A). Then A # is a morphism. Furthermore function and A !→ A 2 # # 2 ∗ 2 A(ω) = ||A|| = sup A$ A = sup A(ω) ω∈X

ω∈X

# is an isomorphism. for it exists an ω such that |ω(A∗ A)| = ||A||. Thus A !→ A # belong to The set Kε = {ω ∈ X; ω(A) > ε} is ∗-weakly compact and thus A # separates the points of X, thus by Stone-Weierstrass theorem, the C0 (X). Finally A # gives the whole of C0 (X). mapping A 2.5 Appendix This is an appendix of the C ∗ -algebra section, on Quotient algebras and approximate identities. It is not necessary at first reading. A subspace J of a C ∗ -algebra A is a left ideal if for all J ∈ J and all A ∈ A then JA belongs to J . In the same way one obviously defines right ideals and twosided ideals. If J is a two-sided, self-adjoint ideal of A, one can easily define the quotient algebra A/J by the usual rules: i)λ[X] + µ[Y ] = [λX + µY ], ii)[X] [Y ] = [XY ], iii)[X]∗ = [X ∗ ], where [X] = {X + J; J ∈ J } is the equivalence class of X ∈ A modulo J . We leave to the reader to check the consistency of the above definitions. We now define a norm on A/J by ||[X]|| = inf{||X + J|| ; J ∈ J }. The true difficulty is to check that the above norm is a C ∗ -algebra norm. For this aim we need the notion of approximate identity. If J is a left ideal of A then an approximate identity or approximate unit in J is a generalized sequence (eα )a of positive elements of J satisfying i) ||eα || ≤ 1, ii) α ≤ β implies eα ≤ eβ , iii) limα ||Xeα − X|| = 0 for all X ∈ J . Proposition 2.18. Every left ideal J of a C ∗ -algebra A possesses an approximate unit. Proof. Let J+ be the set of positive elements of J . For each J ∈ J+ put −1

eJ = J(I + J)

= I − (I + J)

−1

.

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It is a generalized sequence, it is increasing by Proposition 1.9 and ||eJ || ≤ 1. Let us now fix X ∈ J . For every n ∈ IN there exists a J ∈ J+ such that J ≥ nX ∗ X. Thus (X − XeJ )∗ (X − XeJ ) = (I − eJ )X ∗ X(I − eJ ) ≤

1 (I − eJ )J(I − eJ ) n

by Proposition 1.9. It suffices to prove that sup J(I − eJ )2 < ∞. J∈J+

But note that J(I −eJ )2 = J(I +J)−2 and using the functional calculus this reduces to the obvious remark that λ/(1 + λ2 ) is bounded on IR+ . We can now prove the main result of the appendix. Theorem 2.19. If J is a closed, self-adjoint, two-sided ideal of a C ∗ -algebra A, then the quotient algebra A/J , equipped with the quotient norm, is a C ∗ -algebra. Proof. Let us first show that ||[X]|| = lim ||eα X − X|| α

for all X ∈ J . By definition of the quotient we obviously have ||[X]|| ≤ lim ||eα X − X|| . α

As σ(eα ) ⊂ [0, 1] we have σ(I − eα ) ⊂ [0, 1] and ||I − eα || ≤ 1. This implies ||(X + eα X) + (Y + eα Y )|| = ||(I − eα )(X + Y )|| ≤ ||X + Y || . In particular lim supα ||(X + eα X)|| ≤ ||X + Y || for every Y ∈ J . This proves our claim. Now we have ||[X]|| = lim ||X − eα X|| = lim ||(X ∗ − X ∗ eα )(X − eα X)|| 2

2

α

α

∗

= lim ||(I − eα )(X X + Y ∗ )(I − eα )|| α

≤ ||X ∗ X + Y || for every Y ∈ J . This implies ||[X]|| ≤ ||[X]∗ [X]|| 2

and thus the result.

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3 von Neumann algebras 3.1 Topologies on B(H) As every C ∗ -algebra is a sub-∗-algebra of some B(H), closed for the operator norm topology (or uniform topology), then it inherits new topologies, which are weaker. On B(H) we define the strong topology to be the locally convex topology defined by the semi-norms Px (A) = ||Ax||, x ∈ H, A ∈ B(H). This is to say that a base of neighborhood is formed by the sets V (A; x1 , . . . , xn ; ε) = {B ∈ B(H); ||(B − A)xi || < ε, i = 1, . . . , n}. On B(H) we define the weak topology to be the locally convex topology defined by the semi-norms Px,y (A) = |< x , Ay >|, x, y ∈ H, A ∈ B(H). This is to say that a base of neighborhood is formed by the sets V (A; x1 , . . . , xn ; y1 , . . . , yn ; ε) = {B ∈ B(H); |< xi , (B − A)yj >| < ε, i, j = 1, . . . , n}. Proposition 3.1. i) The weak topology is weaker than the strong topology which is itself weaker than the uniform topology. Once H is infinite dimensional then these comparisons are strict. ii) A linear form on B(H) is strongly continuous if and only if it is weakly continuous. iii) The strong and the weak closure of any convex subset of B(H) coincide. Proof. i) All the comparisons are obvious in the large sense. To make the difference in infinite dimension assume that H is separable with orthonormal basis (en )n∈IN . Let Pn be the orthogonal projection onto the space generated by e1 , . . . , en . The sequence (Pn )n∈IN converges strongly to I but not uniformly. Furthermore, consider the unilateral shift S : ei !→ ei+1 . Then S k converges weakly to 0 when k tends to +∞ but not strongly. ii) Let Ψ : B(H) → C be a strongly continuous linear form. Then there exists x1 , . . . , xn ∈ H such that n ||Bxi || |Ψ (B)| ≤ i=1

for all B ∈ B(H) (classical result on locally convex topologies, not proved here). On B(H)n let P be the semi-norm defined by P (A1 , . . . , An ) =

n i=1

||Ai xi || .

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n On the diagonal of B(H) we define the linear form Ψ by Ψ (A, . . . , A) = Ψ (A). We then have Ψ (A, . . . , A) ≤ P (A, . . . , A). By Hahn-Banach, there exists a linear form Ψ on B(H)n which extends Ψ and such that

|Ψ (A1 , . . . , An )| ≤ P (A1 , . . . , An ). Let Ψk be the linear form on B(H) defined by Ψk (A) = Ψ (0, . . . , 0, A, 0, . . . , 0).

(A is at the k-th place)

Then |Ψk (A)| ≤ ||Axk || for every A. Every vector y ∈ H can be written as Axk for some A ∈ B(H). The linear form Axk !→ Ψk (A) is thus well-defined and continuous on H. By Riesz theorem there exists a yk ∈ H such that Ψk (A) = < yk , Axk >. We have proved that n < yk , Axk >. Ψ (A) = i=1

Thus Ψ is weakly continuous. iii) is an easy consequence of ii) and of the geometric form of Hahn-Banach theorem. Another topology is of importance for us, the σ-weak topology. It is the one determined by the semi-norms p(xn )n∈IN ,(yn )n∈IN (A) =

∞

|< xn , Ayn >|

n=0

where (xn )n∈IN and (yn )n∈IN run over all sequences in H such that n

2

||xn || < ∞

and

2

||yn || < ∞.

n

Let T (H) denote the Banach space of trace √ class operators on H, equipped with the trace norm ||H||1 = tr |H|, where |H| = H ∗ H. Theorem 3.2. The Banach space B(H) is the topological dual of T (H) thanks to the duality (A, T ) !→ tr(AT ), A ∈ B(H), T ∈ T (H). Furthermore the ∗-weak topology on B(H) associated to this duality is the σ-weak topology. Proof. The inequality |tr(AT )| ≤ ||A|| ||T ||1 proves that B(H) is included in the topological dual of T (H). Conversely, let ω be an element of the dual of T (H). Consider the rank one operators Eξ,ν = | ξ >< ν |. One easily checks that

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||Eξ,ν ||1 = ||ξ|| ||ν||. Thus |ω(Eξ,ν )| ≤ ||ω|| ||ξ|| ||ν||. By Riesz theorem there exists an operator A ∈ B(H) such that ω(Eξ,ν ) = < ν , Aξ >. The linear form tr(A ·) then coincides with ω on rank one projectors. One concludes that they coincide on T (H) by density of finite rank operators. This proves the announced duality. The ∗-weak topology associated to this duality is defined by the seminorms PT (A) = |tr(AT )| where T runs over T (H). But every trace class operator T writes T =

∞

λn | ξn >< νn |

n=0

for some orthonormal systems (νn )n∈IN , (ξn )n∈IN and some absolutely summable sequence of complex numbers (λn )n∈IN . Thus tr(AT ) =

∞

λn < νn , Aξn >

n=0

and the seminorms PT are equivalent to those defining the σ-weak topology.

Corollary 3.3. Every σ-weakly continuous linear form on B(H) is of the form A !→ tr(AT ) for some T ∈ T (H). We can now give the first definition of a von Neumann algebra. A von Neumann algebra is a C ∗ -algebra acting on H which contains the unit I of B(H) and which is weakly (strongly) closed. Of course the whole of B(H) is the first example of a von Neumann algebra. Another example, which is actually the archetype of commutative von Neumann algebra, is obtained when considering a locally compact measured space (X, µ), with a σ-finite measure µ. The ∗-algebra L∞ (X, µ) acts on H = L2 (X, µ) by multiplication. The C ∗ -algebra C0 (X) also acts on H. But every function f ∈ L∞ (X, µ) is almost sure limit of a sequence (fn )n∈IN in C0 (X). By dominated convergence, the space L∞ (X, µ) is included in the weak closure of C0 (X). But as L∞ (X, µ) is also equal to its weak closure, we have that L∞ (X, µ) is the weak closure of C0 (X). We have proved that L∞ (X, µ) is a von Neumann algebra and we have obtained it as the weak closure of some C ∗ -algebra.

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3.2 Commutant Let M be a subset of B(H). We put M = {B ∈ B(H); BM = M B for all M ∈ M}. The space M is called the commutant of M. We also define M = (M ) , . . . , M(n) = (M(n−1) ) , . . . Proposition 3.4. For every subset M of B(H) we have i) M is weakly closed; ii) M = M = M(5) = . . . and M ⊂ M = M(4) = . . . Proof. i) If (An )n∈IN is a sequence in M which converges weakly to A in B(H) then for all B ∈ M and all x, y ∈ H we have | < x , (AB − BA)y > | ≤ n→∞

|< x , (A − An )By >| + |< x , B(A − An )y >| −−−−→ 0. Thus A belongs to M . ii) If B belongs to M and A belongs to M then AB = BA, thus A belongs to (M ) = M . This proves the inclusion M ⊂ M . But note that if M1 ⊂ M2 then clearly M2 ⊂ M1 . Applying this to the previous inclusion gives M ⊂ M . But as M is also equal to (M ) we should also have the converse inclusion to hold true. This means M = M . We now conclude easily. Proposition 3.5. Let M be a self-adjoint subset of B(H). Let E be a closed subspace of H and P be the orthogonal projector onto E. Then E is invariant under M (in the sense M E ⊂ E for all M ∈ M) if and only if P ∈ M . Proof. The space E is invariant under M ∈ M if and only M P = P M P . Thus if E is invariant under M we have M P = P M P for all M ∈ M. Applying the involution on this equality and using the fact that M is self-adjoint, gives P M = P M P for all M ∈ M. Finally P M = M P for all M ∈ M and P belongs to M . The converse is obvious. Theorem 3.6 (von Neumann density theorem). Let M be a sub-∗-algebra of B(H) which contains the identity I. Then M is weakly (strongly) dense in M . Proof. f Let B ∈ M . Let x1 , . . . , xn ∈ H. Let V = {A ∈ B(H); ||(A − B)xi || < ε, i = 1, . . . , n} be a strong neighborhood of B. It is sufficient to show that V intersects M. One can assume B to be self-adjoint as it can always be decomposed as a linear combination of two self-adjoint operators which also belong to M .

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= ⊕n H and π : B(H) → B(H) be given by π(A) = ⊕n A. Let Let H i=1 i=1 Let P be the orthogonal projection from H onto the closure x = (x1 , . . . , xn ) ∈ H. By Proposition 2.5 we have that P belongs of π(M)x = {π(A)x; A ∈ M} ⊂ H. to π(M) . to Mn (B(H)) it is easy to see that π(M) = Mn (M ) If one identifies B(H) and π(M ) ⊂ Mn (M ) (be aware that the prime symbols above are relative to different operator spaces!). This means that π(B) belong to π(M ) ⊂ Mn (M ) = π(M) . In particular B commutes with P ∈ π(M) . This means that the space π(M)x is invariant under π(B). In particular ⎞ ⎛ Bx1 ⎟ ⎜ π(B) (π(I)x) = ⎝ ... ⎠ Bxn belongs to π(M)x. This means that there exists a A ∈ M such that ||(B − A)xi || is small for all i = 1, . . . n. Thus A belongs to M ∩ V . As immediate corollary we have a characterization of von Neumann algebras. Corollary 3.7 (Bicommutant theorem). Let M be a sub-∗-algebra of B(H) which contains I. Then the following assertions are equivalent. i) M is weakly (strongly) closed. ii) M = M . As I always belong to M , we have that a C ∗ -algebra M ⊂ B(H) is a von Neumann algebra if and only if M = M . 3.3 Predual, normal states Let M be a von Neumann algebra. Put M1 = {M ∈ M; ||M || ≤ 1}. Note that the weak topology and the σ-weak topology coincide on M1 . Hence M1 is a weakly closed subset of the unit ball of B(H) which is weakly compact. Thus M1 is weakly compact. We denote by M∗ the space of weakly (σ-weakly) linear forms on M which are continuous on M1 . The space M∗ is called the predual of M, for a reason that will appear clear in next proposition. If Ψ belongs to M∗ then Ψ (M1 ) is compact in C, thus Ψ is norm continuous. Thus M∗ is a subspace of M∗ the topological dual of M. Proposition 3.8. i) M∗ is closed in M∗ , it is thus a Banach space. ii) M is the dual of M∗ .

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Proof. i) Let (fn )n∈IN be a sequence in M∗ which converges to a f in M∗ , that is sup |fn (A) − f (A)| −→n→∞ 0.

||A||=1

We want to show that f belongs to M∗ , that is f is weakly continuous on M1 . Let (An )n∈IN be a sequence in M1 which converges weakly to A ∈ M1 . Then |f (An ) − f (A)| ≤ |f (An ) − fm (An )| + |fm (A) − f (A)| + |fm (An ) − fm (A)| ≤ 2 sup |fm (B) − f (B)| + |fm (An ) − fm (A)| ||B||=1

→n→∞ 2 sup |fm (B) − f (B)| ||B||=1

→m→∞ 0. This proves i). ii) For a A ∈ M we put ||A||du =

|ω(A)|

sup ω∈M∗ ;||ω||=1

the norm of A for the duality announced in the statement of ii). Clearly we have ||A||du ≤ ||A|| . For x, y ∈ H we denote by ωx,y the linear form A !→ < y , Ax > on B(H) and ωx,y |M the restriction of ωx,y to M. We have ||A|| =

sup ||x||=||y||=1

|< y , Ax >| ≤

sup ω=ωx,y ;||ω||=1

|ω(A)| ≤ ||A||du . ∗

Thus M is indeed identified linearly and isometrically to a subspace of (M∗ ) . We just have to prove that this identification is onto. Let φ be a continuous linear form on M∗ . Let φ (x, y) = φ(ωx,y |M ). Then φ is a continuous sesquilinear form on H, it is thus of the form φ (x, y) = < y , Ax > for some A ∈ B(H). If T is a self-adjoint element of M then ωT x,y |M = ωx,T y |M and < AT x , y > = < T Ax , y > for all x, y ∈ H. Thus A belong to M = M. As ωx,y (A) = < y , Ax > = φ (x, y) = φ(ωx,y |M ) then the image of A in ∗ (M∗ ) coincides with φ at least on the ωx,y . Now, it remains to show that this is sufficient for A and φ to coincide everywhere. That is, we have to prove that an ∗ element a of (M∗ ) which vanishes on all the ωx,y is null. But all the elements of M∗ are linear forms ω of the form ω(A) = tr(ρA) for some trace class operator ρ. As every trace class operator ρ writes as λn |xn yn | ρ= n

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for some orthonormal basis (xn )n∈IN and (yn )n∈IN and some summable sequence (λn )n∈IN , we have that ω= λn ωxn ,yn n

where the series above is convergent in M∗ . One concludes easily. The two main examples of von Neumann algebra have well-known preduals. Indeed, if M = B(H) then M∗ = T (H) the space of trace class operators. If M = L∞ (X, µ) then M∗ = L1 (X, µ). Theorem 3.9 (Sakai theorem). A C ∗ -algebra is a von Neumann algebra if and only if it is the dual of some Banach space. Admitted.

A state on a von Neumann algebra M is called normal if it is σ-weakly continuous. The following characterization is now straightforward. Theorem 3.10. On a von Neumann algebra M, for a state ω on M, the following assertions are equivalent. i) The state ω is normal ii) There exists a positive, trace class operator ρ on H such that trρ = 1 and ω(A) = tr(ρA) for all A ∈ M.

4 Modular theory 4.1 The modular operators The starting point here is a pair (M, ω), where M is a von Neumann algebra acting on some Hilbert space, ω is a normal faithful state on M. Recall that ω is then of the form ω(A) = tr(ρA) for a positive nonsingular ρ, with trρ = 1. Let us consider the G.N.S. representation of (M, ω). That is, a triple (H, Π, Ω) such that i) Π is a morphism from M to B(H). ii) ω(A) = < Ω , Π(A)Ω > iii) Π(M)Ω is dense in H. From now on, we omit to mention the representation Π and identify M and M with Π(M) and Π(M ). We thus write ω(A) = < Ω , AΩ >.

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Proposition 4.1. The vector Ω is cyclic and separating for M and M . Proof. Ω is cyclic for M by iii) above. Let us see that it is separating for M. If A ∈ M is such that AΩ = 0 then ω(A∗ A) = 0, but as ω is faithful this implies A = 0. Let us now see that these properties of Ω on M imply the same ones on M . If A belongs to M and A Ω = 0 then A BΩ = BA Ω = 0 for all B ∈ M. Thus A vanishes on a dense subspace of H, it is thus the null operator. This proves that Ω is separating for M . Finally, let P be the orthogonal projector onto the space M Ω. As it is the projection onto a M -invariant space, it belongs to (M ) = M. But P Ω = Ω and thus (I − P )Ω = 0. As Ω is separating for M this implies I − P = 0 and Ω is cyclic for M . As a consequence the (anti-linear) operators S0 : MΩ −→ MΩ AΩ !−→ A∗ Ω F0 : M Ω −→ M Ω BΩ !−→ B ∗ Ω are well-defined (by the separability of Ω) on dense domains. Proposition 4.2. The operators S0 and F0 are closable and F 0 = S0∗ , S 0 = F0∗ . Proof. For all A ∈ M, B ∈ M we have < BΩ , S0 AΩ > = < BΩ , A∗ Ω > = < AΩ , B ∗ Ω > = < AΩ , F0 BΩ >. This proves that F0 ⊂ S0∗ and S0 ⊂ F0∗ . The operators S0 and F0 are thus closable. Let us show that F 0 = S0∗ . Actually it is sufficient to show that S0∗ ⊂ F 0 . Let x ∈ Dom S0∗ and y = S0∗ x. For any A ∈ M we have < AΩ , y > = < AΩ , S0∗ x > = < x , S0 AΩ > = < x , A∗ Ω >. If we define the operators Q0 and Q+ 0 by Q0 : AΩ − ! → Ax : AΩ ! → Ay − Q+ 0 we then have < BΩ , Q0 AΩ > = < BΩ , Ax > = < A∗ BΩ , x > = < y , B ∗ AΩ > = < By , AΩ > = < Q+ 0 BΩ , AΩ >.

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St´ephane Attal ∗ This proves that Q+ 0 ⊂ Q0 and Q0 is closable. Let Q = Q0 . Note that we have

Q0 ABΩ = ABx = AQ0 BΩ. This proves that Q0 A = AQ0 on Dom Q0 and thus AQ ⊂ QA for all A ∈ M. This means that Q is affiliated to M , that is, it fails from belonging to M only by the fact it is an unbounded operator; but every bounded function of Q is thus in M . In particular, if Q = U |Q| is the polar decomposition of Q then U belongs to M and the spectral projections of |Q| also belong to M . Let En = 1l[0,n] (|Q|). The operator Qn = U En |Q| thus belongs to M and Qn Ω = U En |Q| Ω = U En U ∗ U |Q| Ω = U En U ∗ Q0 Ω = U En U ∗ x. Furthermore we have Q∗n Ω = En |Q| U ∗ Ω = En Q+ 0 Ω = En y. This way U En U ∗ x belongs to Dom F0 and F0 (U En U ∗ x) = En y. But En tends to I and U U ∗ is the orthogonal projector onto Ran Q, which contains x. Finally, we have proved that x ∈ Dom F 0 and F 0 x = y = S0∗ x. That is, S0∗ ⊂ F 0. The other case is treated similarly. We now put S = S 0 and F = F 0 . Lemma 4.3. We have

S = S −1 .

Proof. Let z ∈ Dom S ∗ . We have < S0 AΩ , S ∗ z > = < A∗ Ω , S0∗ z > = < z , S0 A∗ Ω > = < z , AΩ >. Thus S ∗ z belongs to Dom S0∗ = S ∗ and (S ∗ )2 z = z. Let y ∈ Dom S and z ∈ Dom S ∗ , we have S ∗ z ∈ Dom S ∗ and < S ∗ z , Sy > = < y , (S ∗ )2 z > = < y , z >. This means that Sy belongs to Dom S ∗∗ = Dom S and S 2 y = S ∗∗ Sy = y. We have proved that Dom S 2 = Dom S and S 2 = I on Dom S. We had proved in Proposition 3.2 that F = S ∗ . Thus the operators F S and SF are (self-adjoint) positive. The operators F and S have their range equal to their domain, they are invertible and equal to their inverse. Let ∆ = F S = S ∗ S. Then ∆ is invertible, with inverse ∆−1 = SF = SS ∗ . As S, ∆ and thus ∆1/2 have a dense range then the partial anti-isometry J such that

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S = J(S ∗ S)1/2 (polar decomposition of S) is an anti-isometry from H to H. Furthermore S = J∆1/2 = (SS ∗ )1/2 J = ∆−1/2 J. Let x belong to Dom S. Then x = S 2 x = J∆1/2 ∆−1/2 Jx = J 2 x and thus J 2 = I. Note the following relations S = J∆1/2 F = S ∗ = ∆1/2 J ∆−1 = J∆J. The operator ∆ has a spectral measure (Eλ ). Thus the operator ∆−1 = J∆J has the spectral measure (JEλ J). Let f be a bounded Borel function, we have −1 < f (∆ )x , x > = f (λ) d< JEλ Jx , x > = f (λ) d< Jx , Eλ Jx > = f (λ) d< Eλ Jx , Jx > = < f (∆)Jx , Jx > = < Jx , f (∆)Jx > = < Jf (∆)Jx , x >. This proves

f (∆−1 ) = Jf (∆)J.

In particular ∆it = J∆it J ∆it J = J∆it . Finally note that SΩ = F Ω = Ω and thus ∆Ω = F SΩ = Ω which finally gives ∆1/2 Ω = Ω. Let us now summarize the situation we have already described. Theorem 4.4. There exists an anti-unitary operator J from H to H and an (unbounded) invertible, positive operator ∆ such that

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∆ = F S, ∆−1 = SF, J 2 = I S = J∆1/2 = ∆−1/2 J F = J∆−1/2 = ∆1/2 J J∆it = ∆it J JΩ =∆Ω = Ω. The operator ∆ is called the modular operator and J is the modular conjugation. It is interesting to note the following. If the state ω were tracial, that is, ω(AB) = ω(BA) for all A, B, we would have ||S0 AΩ|| = ||A∗ Ω|| = < A∗ Ω , A∗ Ω > = ω(AA∗ ) = ω(A∗ A) = ||AΩ|| . 2

2

2

Thus S0 would be an isometry and S =J = F ∆ = I. 4.2 The modular group Let A, B, C ∈ M. We have SASBCΩ = SAC ∗ B ∗ Ω = BCA∗ Ω = BSAC ∗ Ω = BSASCΩ. This proves that B and SAS commute. Thus SAS is affiliated to M . Let us assume for a moment that ∆ is bounded. In that case the operators ∆−1 = J∆J, S and F are also bounded. We have seen that SMS ⊂ M F M F ⊂ M. This way we have ∆M∆−1 = ∆1/2 JJ∆1/2 M∆−1/2 JJ∆−1/2 = F SMSF ⊂ F M F ⊂ M. We also have

∆n M∆−n ⊂ M

for all n ∈ IN . For any A ∈ M, A ∈ M , the function −2z

f (z) = ||∆||

< φ , [∆z A∆−z , A ]ψ >

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is analytic on C. It vanishes for z = 0, 1, 2, ... As ∆−1 = ||J∆J|| = ||∆|| we have −2 z | z| 2 |f (z)| = O ||∆|| (||∆|| ) = O(1) when %z > 0. By Carlson’s theorem we have f (z) = 0 for all z ∈ C. Thus ∆z M∆−z ⊂ M = M for all z ∈ C. But M = ∆z (∆−z M∆z )∆−z ⊂ ∆z M∆−z and finally

∆z M∆−z = M.

Furthermore JMJ = J∆1/2 M∆−1/2 J = SMS ⊂ M JM J = J∆−1/2 M∆1/2 J = F MF ⊂ M. We have proved

JMJ = M .

The results we have obtained here are fundamental and extend to the case when ∆ is unbounded. This is what the following theorem says. We do not prove it as it implies pages of difficult analytic considerations. We hope that the above computations make it credible. Theorem 4.5 (Tomita-Takesaki’s theorem). In any case we have JMJ = M ∆it M∆−it = M. Put

σt (A) = ∆it A∆−it .

This defines a one parameter group of automorphisms of M. Proposition 4.6. We have, for all A, B ∈ M ω(Aσt (B)) = ω(σt+i (B)A).

(1)

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Proof. < Ω , A∆it B∆−it Ω > = < ∆−it A∗ Ω , BΩ > = < ∆−it−1/2 A∗ Ω , ∆1/2 BΩ > = < ∆−it−1 ∆1/2 A∗ Ω , ∆1/2 BΩ > = < J∆−it+1 J∆1/2 A∗ Ω , ∆1/2 BΩ > = < J∆1/2 BΩ , ∆−it+1 J∆1/2 A∗ Ω > = = < Ω , B∆−i(t+i) AΩ > = < Ω , ∆i(t+i) B∆−i(t+i) AΩ > = ω(σt+i (B)A). It is interesting to relate the above equality with the following result. Proposition 4.7. Let ω be a state of the form ω(A) = tr(ρA) on B(K) for some trace-class positive ρ with trρ = 1. Let (σt ) be the following group of automorphisms of B(K): σt (A) = eitH Ae−itH for some self-adjoint operator H on K. Then the following assertions are equivalent. i) For all A, B ∈ B(K), all t ∈ IR and a fixed β ∈ IR we have ω(Aσt (B)) = ω(σt−βi (B)A). ii) ρ is given by ρ=

1 −βH e , Z

where Z = tr(exp(−βH)). Proof. ii) implies i): We compute directly 1 tr(e−βH AeitH Be−itH ) Z 1 = tr(AeitH Be(−it−β)H ) Z 1 = tr(Ae−βH e(it+β)H Be(−it−β)H ) Z 1 = tr(e−βH e(it+β)H Be(−it−β)H A) Z = ω(σt−βi (B)A).

ω(Aσt (B)) =

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i) implies ii): We have tr(ABρ) = tr(ρAB) = ω(AB) = ω(σ−βi (B)A) = tr(ρeβH Be−βH A) = tr(AρeβH Be−βH ). As this is valid for any A we conclude that Bρ = ρeβH Be−βH for all B. This means

B ρeβH = ρeβH B.

As this is valid for all B we conclude that ρ exp(βH) is a multiple of the identity. This gives ii). Another very interesting result to add to Proposition 3.6 is that the modular group is the only one to perform the relation (1). Theorem 4.8. σ· is the only automorphism group to satisfy (1) on M for the given state ω. Proof. Let τ. be another automorphism group on M which satisfies (1). Define the operators Ut by Ut AΩ = τt (A)Ω. Then Ut is unitary for ||Ut AΩ|| = < τt (A)Ω , τt (A)Ω > = < Ω , τt (A∗ A)Ω > = ω(τt (A∗ A)) = ω(τt+i (I)A∗ A) 2

= ω(A∗ A) = ||AΩ|| . 2

The family U· is clearly a group, it is thus of the form Ut = exp itM for a self-adjoint operator M . Note that Ut Ω = Ω and thus M Ω = 0. Let A, B be entire elements for τ· , then the relation ω(τi (B)A) = ω(AB) implies = < ∆1/2 B ∗ Ω , JJ∆1/2 AΩ > = < A∗ Ω , BΩ > = ω(AB) = ω(τi (B)A) = < Ω , e−M BeM AΩ > = . This means ∆ = eM and τ = σ.

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4.3 Self-dual cone and standard form We put P = {AJAJΩ; A ∈ M}. Proposition 4.9. i) P = ∆1/4 M+ Ω = ∆−1/4 M+ Ω and thus P is a convex cone. ii) ∆it P = P for all t. iii) If f is of positive type then f (log ∆)P ⊂ P. iv) If ξ ∈ P then Jξ = ξ. v) If A ∈ M then AJAJP ⊂ P. Proof. i) Let M0 be the ∗-algebra of elements of M which are entire for the modular group σ· (that is, t !→ σt (A) admits an analytic extension). We shall admit here that M0 is σ-weakly dense in M. For every A ∈ M0 we have ∆1/4 AA∗ Ω = σ−i/4 (A)σi/4 (A)∗ Ω = σ−i/4 (A)J∆1/2 σi/4 (A)Ω = σ−i/4 (A)Jσ−i/4 (A)JΩ = BJBJΩ where B = σ−i/4 (A). By σ−i/4 (M0 ) = M0 and by the density of M0 in M we have BJBJΩ ∈ ∆1/4 M+ Ω ⊂ ∆1/4 M+ Ω for all B ∈ M. Thus P ⊂ ∆1/4 M+ Ω ⊂ ∆1/4 M+ Ω. Conversely, M+ 0 Ω is dense in M+ Ω. Let ψ ∈ M+ Ω. There exists a sequence such that An Ω → ψ. We know by the above that ∆1/4 An Ω belongs (An ) ⊂ M+ 0 to P. But J∆1/2 An Ω = An Ω → ψ = J∆1/2 ψ and thus 2 1/4 ∆ (ψ − An Ω) = < ψ − An Ω , ∆1/2 (ψ − An Ω) > → 0. Thus ∆1/4 ψ belongs to P and ∆1/4 M+ Ω ⊂ P. This proves the first equality of i). The second one is treated exactly in the same way. ii) We have ∆it ∆1/4 M+ Ω = ∆1/4 ∆it M+ Ω = ∆1/4 σt (M+ )Ω = ∆1/4 M+ Ω.

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iii) If f is of positive type then f is the Fourier transform of some positive, finite, Borel measure µ on IR. In particular f (log ∆) = ∆it dµ(t). One concludes with ii) now. iv) JAJAJΩ = JAJAΩ = AJAJΩ. v) AJAJBJBJΩ = ABJAJJBJΩ = ABJABJΩ. Theorem 4.10. i) P is self-dual, that is P = P ∨ where P ∨ = {x ∈ H; < y , x > ≥ 0, ∀y ∈ P}. ii) P is pointed, that is,

P ∩ (−P) = {0}.

iii) If Jξ = ξ then ξ admits a unique decomposition as ξ = ξ1 − ξ2 with ξ1 , ξ2 ∈ P and ξ1 orthogonal to ξ2 . iv) The linear span of P is the whole of H. Proof. i) If A ∈ M+ and A ∈ M+ then < ∆1/4 AΩ , ∆−1/4 A Ω > = < AΩ , A Ω > = < Ω , A1/2 A A1/2 Ω > ≥ 0. Thus P is included in P ∨ . Conversely, if ξ ∈ P ∨ , that is < ξ , ν > ≥ 0 for all ν ∈ P, we put ξn = fn (log ∆)ξ where fn (x) = exp(−x2 /2n2 ). Then ξn belongs to ∩α∈C Dom ∆α and ξn converges to ξ. We know that fn (log ∆)ν belongs to P and thus < ξn , ν > = < ξ , fn (log ∆)ν > ≥ 0. Let A ∈ M+ then ∆1/4 AΩ belongs to P and < ∆1/4 ξn , AΩ > = < ξn , ∆1/4 AΩ > ≥ 0. ∨

Thus ∆1/4 ξn belongs to M+ Ω which coincides with M+ Ω (admitted). This finally gives that ξn belongs to ∆−1/4 M+ Ω ⊂ P. This proves i). ii) If ξ ∈ P ∩ (−P) = P ∩ (−P ∨ ) then < ξ , −ξ > ≥ 0 and ξ = 0. iii) If Jξ = ξ then, as P is convex and closed, there exists a unique ξ1 ∈ P such that ||ξ − ξ1 || = inf{||ξ − ν|| ; ν ∈ P}.

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We put ξ2 = ξ1 − ξ. Let ν ∈ P and λ > 0. Then ξ1 + λν belongs to P and 2

2

||ξ − ξ1 || ≤ ||ξ1 + λν − ξ|| . 2

2

2

That is ||ξ2 || ≤ ||ξ2 + λν|| , or else λ2 ||ν|| + 2λ%< ξ2 , ν > ≥ 0. This implies that %< ξ2 , ν > is positive. But as Jξ2 = ξ2 and Jν = ν then < ξ2 , ν > = < Jξ2 , Jν > = < ξ2 , ν >. That is < ξ2 , ν > ≥ 0 and ξ2 ∈ P ∨ = P. iv) If ξ is orthogonal to the linear span of P then ξ belongs to P ∨ = P. thus < ξ , ξ > = 0 and ξ = 0. Theorem 4.11 (Universality). 1) If ξ ∈ P then ξ is cyclic for M if and only if it is separating for M. 2) If ξ ∈ P is cyclic for M then Jξ , Pξ associated to (M, ξ) satisfy Jξ = J and Pξ = P. Proof. 1) If ξ is cyclic for M then Jξ is cyclic for M = JMJ and thus ξ = Jξ is separating for M. And conversely. 2) Define as before (the closed version of) Sξ : Aξ − ! → A∗ ξ Fξ : A ξ !−→ A∗ ξ. We have JFξ JAξ = JFξ JAJξ = J(JAJ)∗ ξ = A∗ ξ = Sξ Aξ. This proves that Sξ ⊂ JFξ J. By a symmetric argument Fξ ⊂ JSξ J and thus JSξ = Fξ J. Note that (JSξ )∗ = Sξ∗ J = Fξ J = JSξ . This means that JSξ is self-adjoint. Let us prove that it is positive. We have < Aξ , JSξ Aξ > = < Aξ , JA∗ ξ > = < ξ , A∗ JA∗ ξ > which is a positive quantity for ξ and A∗ JA∗ J belong to P. This proves the positivity of JSξ . We have 1/2 Sξ = Jξ ∆ξ = J(JSξ ). By uniqueness of the polar decomposition we must have J = Jξ . Finally, we have that Pξ is generated by the AJξ AJξ ξ = AJAJξ. But as ξ belongs to P we have that AJAJξ belongs to P and thus Pξ ⊂ P. Finally, P = P ∨ ⊂ Pξ∨ = Pξ and P = Pξ .

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The following theorem is very useful and powerful, but its proof is very long, tedious and cannot be summarized, thus we prefer not enter into it and give the result as it is (cf [B-R], p. 108-117). For every ξ ∈ P one can define a particular normal positive form ωξ (A) = < ξ , Aξ > on M. That is, ωξ ∈ M∗+ . Theorem 4.12. 1) For every ω ∈ M∗+ there exists a unique ξ ∈ P such that ω = ωξ . 2) The mapping ξ !−→ ωξ is an homeomorphism and 2

2

||ξ − ν|| ≤ ||ωξ − ων || ≤ ||ξ − ν|| ||ξ + ν|| . We denote by ω !−→ ξ(ω) the inverse mapping of ξ !−→ ωξ . Corollary 4.13. There exists a unique unitary representation α ∈ Aut(M) !−→ Uα of the group of ∗-automorphisms of M on H, such that i) Uα AUα∗ = α(A), for all A ∈ M, ii) Uα P ⊂ P and, moreover, ∗

Uα ξ(ω) = ξ(α−1 (ω)) for all ω ∈ M∗+ and where (α∗ ω)(A) = ω(α(A)). iii) [Uα , J] = 0. Proof. Let α ∈ Aut(M). Let ξ ∈ P be the representant of the state A !−→ < Ω , α−1 (A)Ω >. That is,

< ξ , Aξ > = < Ω , α−1 (A)Ω >.

In particular ξ is separating for M and hence cyclic. Define the operator U AΩ = α(A)ξ. We have ||U AΩ|| = < ξ , α(A∗ A)ξ > = < Ω , A∗ AΩ > = ||AΩ|| . 2

Thus U is unitary. In particular

2

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U ∗ Aξ = α−1 (A)Ω. Now, for A, B ∈ M we have U AU ∗ Bξ = U Aα−1 (B)Ω = α(Aα−1 (B))ξ = α(A)Bξ and

α(A) = U AU ∗ .

We have proved the existence of the unitary representation. Note that SU ∗ Aξ = Sα−1 (A)Ω = α−1 (A)∗ Ω = α−1 (A∗ )Ω = U ∗ A∗ ξ = U ∗ Sξ Aξ. Hence by closure J∆1/2 U ∗ = U ∗ Jξ ∆ξ

1/2

That is

= U ∗ J∆ξ . 1/2

U JU ∗ U ∆1/2 U ∗ = J∆ξ . 1/2

By uniqueness of the polar decomposition we must have U JU ∗ = J. This gives iii). For A ∈ M we have U AJAJΩ = α(A)Jα(A)Jξ. Since ξ belongs to P we deduce U P = P. If φ ∈ M∗+ we have = < ξ(φ) , U ∗ AU ξ(φ) > = < ξ(φ) , α−1 (A)ξ(φ) > = φ(α−1 (A)) ∗

= (α−1 (φ))(A) ∗

∗

= < ξ(α−1 (φ)) , Aξ(α−1 (φ)) >. By uniqueness of the representing vector in P ∗

U (α)ξ(φ) = ξ(α−1 (φ)). This gives ii) and also the uniqueness of the unitary representation.

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References 1. O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1. C*and W*-Algebras. Symmetry Groups. Decomposition of States, Texts and Monographs in Physics, 2nd ed. 1987, Springer Verlag. 2. O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2. Equilibrium States. Models in Quantum Statistical Mechanics, Texts and Monographs in Physics, 2nd ed. 1987, Springer Verlag. 3. J. Dixmier, Les C ∗ -alg`ebres et leurs repr´esentations, Gauthier-Villars, Paris, 1964. 4. R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras (I), Acad. Press, 1983. 5. R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras (II), Acad. Press, 1986. 6. R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras (III), Acad. Press, 1991. 7. G.K. Pedersen, C ∗ -algebras and their automorphism groups, London Mathematical Society Monographs, Academic Press, 1989. 8. S. Sakai, C ∗ -algebras and W ∗ -algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete 60, Springer Verlag.

Quantum Dynamical Systems Claude-Alain Pillet CPT-CNRS (UMR 6207), Universit´e du Sud, Toulon-Var, BP 20132, 83957 La Garde Cedex, France e-mail: [email protected]

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2

The State Space of a C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.1 2.2

3

Classical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.1 3.2

4

Basics of Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Classical Koopmanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

5

States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 The GNS Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

C ∗ -Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W ∗ -Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariant States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ergodic Properties of Quantum Dynamical Systems . . . . . . . . . . . . . Quantum Koopmanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132 139 141 142 147 153 161 165

KMS States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.1 5.2

Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Perturbation Theory of KMS States . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

1 Introduction Many problems of classical and quantum physics can be formulated in the mathematical framework of dynamical systems. Within this framework ergodic theory

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provides a probabilistic interpretation of dynamics which is suitable to study the statistical properties of the evolution of a mechanical system over large time scales. The conceptual foundation of ergodic theory is intimately related to the birth of statistical mechanics and goes back to Boltzmann and Gibbs. It started to develop as a mathematical theory with the pioneering works of von Neumann [48] and Birkhoff [10]. It is now a beautiful cross-disciplinary part of Mathematics with numerous connections to analysis and probability, geometry, algebra, number theory, combinatorics. . . The Koopman–von Neumann approach to ergodic theory [34] provides an effective way to translate ergodic properties of dynamical systems into spectral properties of some associated linear operator (which I shall call Liouvillean). The resulting spectral approach to dynamics is particularly well adapted to the study of open systems. During the last decade, this spectral approach has been successfully applied to the problem of return to equilibrium [27, 8, 16, 22, 21] and to the construction of non-equilibrium steady states for quantum open systems [29]. The reader should consult [30] for an introduction to these problems. The aim of this lecture is to provide a short introduction to quantum dynamical systems and their ergodic properties with particular emphasis on the quantum Koopman–von Neumann spectral theory. However, I shall not discuss the spectral analysis of the resulting Liouvillean operators. The interested reader should consult the above mentioned references. For other approaches based on scattering ideas see [40, 6, 26]. Ergodic theory also played an important role in the development of the algebraic approach to quantum field theory and quantum statistical mechanics, mainly in connection with the analysis of symmetries. Most of the results obtained in this framework rely on some kind of asymptotic abelianness hypothesis which is often inappropriate in the context of dynamical systems. The reader should consult Chapter 4 and in particular Sections 4.3 and the corresponding notes in [11] for an introduction to the results. I have assumed that the reader is familiar with the material covered in the first Lectures of this Volume [33, 32, 7]. Besides that, the notes are reasonably selfcontained and most of the proofs are given. Numerous examples should provide the reader with a minimal toolbox to construct basic models of quantum open systems. These notes are organized as follows. Section 2 is an extension of Subsection 3.3 in Lecture [7]. It consists of two parts. In Subsections 2.1, I review some topological properties of von Neumann algebras. I also introduce the notions of support and central support of a normal state. Subsection 2.2 explores some elementary consequences of the GNS construction. In Section 3, I briefly review some basic facts of the ergodic theory of classical dynamical systems. The discussion is centered around two simple properties: ergodicity and mixing. As a motivation for the transposition of these concepts to quantum mechanics I discuss the classical Koopman–von Neumann approach in Subsection 3.2. General references for Section 3 are [15], [35], [49] and [5]. The main part of these notes is contained in Section 4 which deals with the ergodic theory of quantum systems. The basic concepts of the algebraic theory of

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quantum dynamics – C ∗ - and W ∗ -dynamical systems and their invariant states – are introduced in Subsections 4.1–4.3. In Subsection 4.4, I define a more general notion of quantum dynamical system. The GNS construction provides an efficient way to bring such systems into normal form. This normal form plays an essential role in quantum ergodic theory. In particular it allows to define a Liouvillean which will be the central object of the quantum Koopman–von Neumann theory. In Subsection 4.5, I introduce the related notion of standard form of a quantum dynamical system. The ergodic properties – ergodicity and mixing – of quantum dynamical systems are defined and studied in Subsection 4.6. The quantum Koopman–von Neumann spectral theory is developed in Subsection 4.7. In many physical applications, and in particular in simple models of open systems, the dynamics is constructed by coupling elementary subsystems. Perturbation theory provides a powerful tool to analyze such models. In Subsection 4.8, I discuss a simple adaptation of the Dyson-Schwinger-Tomonaga time dependent perturbation theory – which played an important role in the early development of quantum electrodynamics – to C ∗ - and W ∗ -dynamical systems. General references for the material in Section 4 are Chapters 2.5, 3.1 and 3.2 in [11] as well as [47], [46] and [44]. More examples of dynamical systems can be found in [9]. Among the invariant states of a C ∗ - or W ∗ -dynamical system, the KMS states introduced in Section 5 form a distinguished class, from the physical as well as from the mathematical point of view. On the physical side, KMS states play the role of thermodynamical equilibrium states. As such, they are basic building blocks for an important class of models of open quantum systems where the reservoirs are at thermal equilibrium. On the mathematical side, KMS states appear naturally in the modular structure associated with faithful normal states (or more generally semi-finite weight) on von Neumann algebras. They are thus intimately connected with their mathematical structure. This tight relation between dynamics and the structure of the observable algebra is one of the magical feature of quantum mechanics which has no classical counterpart. In Subsection 5.2, I discuss the perturbation theory of KMS states. For bounded perturbations, the theory is due to Araki [2] and [4] (see also Section 5.4.1 in [12]). Extensions to unbounded perturbations have been developed in [43],[19] and [17]. This subject being very technical, I only give some plausibility argument and state the main results without proofs. Acknowledgments. I wish to thank Jan Derezi´nski and Vojkan Jakˇsi´c for fruitful discussions related to the material covered by these notes. I am particularly grateful to Stephan De Bi`evre for his constructive comments on an early version of the manuscript.

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2 The State Space of a C ∗-algebras This section is a complement to Lecture [7] and contains a few additions that are needed to develop the ergodic theory of quantum dynamical systems. It consists of two parts. In the first part I present the basic properties of normal states on a von Neumann algebra M. In particular I discuss the connection between the σ-weak topology on M and its algebraic structure. I also introduce the very useful concepts of support and central support of a normal state. The second part of the section deals with the GNS construction and its consequences on the structure of the state space of a C ∗ -algebra: enveloping von Neumann algebra and folium of a state, relative normality and orthogonality of states. I also discuss the special features of the GNS representation associated with a normal state. The material covered by this section is standard and the reader already familiar with the above concepts may skip it.

Warning: All C ∗ -algebras in this lecture have a unit I.

2.1 States A linear functional ω on a C ∗ -algebra A is positive if ω(A∗ A) ≥ 0 for all A ∈ A. Such a functional is automatically continuous and ω = ω(I) (see Proposition 5 in Lecture [7]). If µ, ν are two positive linear functionals such that ν − µ is a positive linear functional then we write µ ≤ ν. A state is a normalized ( ω = 1) positive linear functional. A state ω is faithful if ω(A∗ A) = 0 implies A = 0. Denote by A1 the unit ball of the Banach space dual A . By the Banach-Alaoglu theorem, A1 is compact in the weak- topology. The set of all states on A is given by E(A) = {ω ∈ A1 | ω(A∗ A) 0 for all A ∈ A}, and it immediately follows that it is a weak- compact, convex subset of A . Normal States Recall (Subsection 3.1 in Lecture [7]) that the σ-weak topology on a von Neumann algebra M ⊂ B(H) is the locally convex topology generated by the semi-norms A !→ |(ψn , Aφn )|, n∈N

for sequences ψn , φn ∈ H such that n ψn 2 < ∞ and n φn 2 < ∞. The σ-strong and σ-strong∗ topologies are defined similarly by the semi-norms

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A !→

n∈N

1/2 Aψn 2

,

A !→

n∈N

Aψn 2 +

111

1/2 A∗ ψn 2

,

n∈N

with n ψn 2 < ∞. Note that, except when H is finite dimensional, these topologies are not first countable. Therefore, the use of nets (directed sets) is mandatory. As a Banach space, the von Neumann algebra M is the dual of the space M of all σ-weakly continuous linear functionals on M. In particular, the predual M is a norm-closed subspace of the dual M = (M ) . Exercise 2.1. Show that the σ-strong∗ topology is stronger than the σ-strong topology which is stronger than the σ-weak topology. Show also that the σ-strong (resp. σ-weak) topology is stronger than the strong (resp. weak) topology and that these two topologies coincide on norm bounded subsets. Exercise 2.2. Adapt the proof of Proposition 8.ii in Lecture [7] to show that a linear functional ω on M is σ-weakly continuous if and only if it is σ-strongly continuous. Using Corollary 2 in Lecture [7] and the Hahn-Banach theorem show that ω is σweakly continuous if and only if there exists a trace class operator T on H such that ω(A) = Tr(T A) for all A ∈ M. The von Neumann density theorem (Theorem 13 in Lecture [7]) asserts that a ∗-subalgebra D ⊂ B(H) containing I is dense in D in the weak and strong topologies. In fact one can prove more (see for example Corollary 2.4.15 in [11]). Theorem 2.3. (Von Neumann density theorem) A ∗-subalgebra D ⊂ B(H) containing I is σ-strongly∗ dense in D . Thus, any element A of the von Neumann algebra generated by D is the σstrong∗ limit of a net Aα in D. By Exercise 2.1 the net Aα also approximates A in the σ-strong, σ-weak, strong and weak topologies. In particular, D coincides with the closure of D in all these topologies. It is often useful to approximate A by a bounded net in D. That this is also possible is a simple consequence of the following theorem (see Theorem 2.4.16 in [11]). Theorem 2.4. (Kaplansky density theorem) Let D ⊂ B(H) be a ∗-subalgebra and denote by D its weak closure. Then Dr ≡ {A ∈ D | A ≤ r} is σ-strongly∗ dense in Dr ≡ {A ∈ D | A ≤ r} for any r > 0. Recall also that a self-adjoint element A of a C ∗ -algebra A is positive if its spectrum is a subset of [0, ∞[ (see Theorem 5 in Lecture [7]). This definition induces a partial order on the set of self-adjoint elements of A: A ≤ B if and only if B − A is positive. Moreover, one writes A < B if A ≤ B and A = B. The relation ≤ is clearly a purely algebraic concept, i.e., it is independent on the action of A on some Hilbert space. However, if A acts on a Hilbert space H then its positive elements are characterized by the fact that (ψ, Aψ) ≥ 0 for all ψ in a dense subspace of H. In particular, A ≤ B if and only if (ψ, Aψ) ≤ (ψ, Bψ) for all ψ in such a subspace.

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Let Aα be a bounded increasing net of self-adjoint elements of B(H), i.e., such that Aα ≥ Aβ for α ( β and supα Aα < ∞. Then, for any ψ ∈ H, one has supα (ψ, Aα ψ) < ∞ and since a self-adjoint element A of B(H) is completely determined by its quadratic form ψ !→ (ψ, Aψ), there exists a unique A ∈ B(H) such that (ψ, Aψ) ≡ lim(ψ, Aα ψ) = sup(ψ, Aα ψ). α

α

It follows immediately from this definition that A = supα Aα . Since one has 0 ≤ A − Aα ≤ A, the estimate (A − Aα )ψ 2 = (ψ, (A − Aα )2 ψ) ≤ A − Aα (A − Aα )1/2 ψ 2 ≤ A (ψ, (A − Aα )ψ), further shows that Aα converges strongly to A. Moreover, since the net Aα is bounded, one also has limα Aα = A in the σ-strong and σ-weak topologies (Exercise 2.1). In particular, if M ⊂ B(H) is a von Neumann algebra and Aα ∈ M, then A ∈ M. Finally, we note that if B ∈ M then sup(B ∗ Aα B) = B ∗ (sup Aα )B. α

(1)

α

Definition 2.5. A positive linear functional ω on a von Neumann algebra M is called normal if, for all bounded increasing net Aα of self-adjoint elements of M, one has ω(sup Aα ) = sup ω(Aα ). α

α

In particular, a normal state is a normalized, normal, positive linear functional. Remark 2.6. This definition differs from the one given in Section 3.3 of Lecture [7]. However, Theorem 2.7 below shows that these two definitions are equivalent. Note that the concept of normality only depends on the partial order relation ≤ and hence on the algebraic structure of M. Since by Exercise 2.2 any σ-weakly continuous linear functional on M is of the form A !→ Tr(T A) for some trace class operator T , it is a finite linear combination of positive, σ-weakly continuous linear functionals (because T is a linear combination of 4 positive trace class operators). Thus, the following theorem characterizes the σ-weak topology on M in a purely algebraic way. Theorem 2.7. A positive linear functional on a von Neumann algebra is normal if and only if it is σ-weakly continuous. Proof. If ω is a σ-weakly continuous positive linear functional on the von Neumann algebra M and Aα a bounded increasing net of self-adjoint elements of M one has, in the σ-weak topology,

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ω(sup Aα ) = ω(lim Aα ) = lim ω(Aα ) = sup ω(Aα ). α

α

α

α

Hence, ω is normal. To prove the reverse statement let ω be a normal positive linear functional and consider the set A ≡ {A ∈ M | 0 ≤ A ≤ I, ωA ∈ M }, where ωA (X) ≡ ω(XA). If 0 ≤ B ≤ A ≤ I then the Cauchy-Schwarz inequality, |ωA (X) − ωB (X)|2 = |ω(X(A − B))|2 ≤ ω(X(A − B)X ∗ )ω(A − B) ≤ X 2 ω(A − B), yields that ωA − ωB 2 ≤ ω(A) − ω(B).

(2)

Let Aα be an increasing net in A and set A ≡ supα Aα . One clearly has 0 ≤ Aα ≤ A ≤ I and since ω is normal Equ. (2) shows that ωAα converges in norm to ωA . M being a norm-closed subspace of M one has ωA ∈ M and we conclude that A ∈ A. Thus, A is inductively ordered and by Zorn’s lemma there exists a maximal element N ∈ A. We set M ≡ I − N and note that if ω(M ) = 0 then Equ. (2) shows that ω = ωN ∈ M . To conclude the proof we assume that ω(M ) > 0 and show that this leads to a contradiction. Since M > 0 we can pick ψ ∈ H such that ω(M ) < (ψ, M ψ). Consider an increasing net Bα in the set B ≡ {B ∈ M | 0 ≤ B ≤ M, ω(B) ≥ (ψ, Bψ)}, and let B ≡ supα Bα . Then 0 ≤ Bα ≤ B ≤ M and since ω is normal ω(B) = sup ω(Bα ) ≥ sup(ψ, Bα ψ) = (ψ, Bψ), α

α

shows that B ∈ B. Hence B is inductively ordered. Let S be a maximal element of B. Remark that M ∈ B since ω(M ) < (ψ, M ψ). This means that T ≡ M − S > 0.

(3)

Next we note that if 0 ≤ B ≤ T and ω(B) ≥ (ψ, Bψ) then B + S ∈ B and the maximality of S yields that B = 0. It follows that 0 ≤ B ≤ T implies ω(B) ≤ (ψ, Bψ). Since for any B ∈ M such that B ≤ 1 one has T B ∗ BT ≤ T 2 ≤ T, we can conclude that ω(T B ∗ BT ) ≤ (ψ, T B ∗ BT ψ) = BT ψ 2 . By Cauchy-Schwarz inequality we further get

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|ωT (B)|2 = |ω(IBT )|2 ≤ ω(I)ω(T B ∗ BT ) ≤ BT ψ 2 . This inequality extends by homogeneity to all B ∈ M and shows that ωT is σstrongly continuous and hence, by Exercise 2.2, σ-weakly continuous. Finally, we note that ωN +T = ωN + ωT ∈ M and by Equ. (3), N < N + T = N + (M − S) = I − S ≤ I, a contradiction to the maximality of N .

Thus, the set of normal states on a von Neumann algebra M coincides with the set of σ-weakly continuous states and with the set of σ-strongly continuous states. It is given by N (M) = M ∩ E(M) ⊂ E(M), and is clearly a norm closed subset of E(M). If M acts on the Hilbert space H then, according to Exercise 2.2, a normal state ω on M is described by a density matrix, i.e., a non-negative trace class operator ρ on H such that Tr ρ = 1 and ω(A) = Tr(ρA). Lemma 2.8. Let M ⊂ B(H) be a von Neumann algebra. The set of vector states V (M) ≡ {(Ψ, ( · )Ψ ) | Ψ ∈ H, Ψ = 1} is total in N (M) i.e., finite convex linear combinations of elements of V (M) are norm dense in N (M). Proof. We first note that if µ, ν ∈ N (M) are given by density matrices ρ, σ then |µ(A) − ν(A)| = | Tr((ρ − σ)A)| ≤ ρ − σ 1 A , where T 1 ≡ Tr(T ∗ T )1/2 denotes the trace norm. Hence µ − ν ≤ ρ − σ 1 . Let µ ∈ N (M) and ρ a corresponding density matrix. Denote by pn ψn (ψn , · ), ρ= n

its spectral decomposition, i.e., (ψn , ψk ) = δn,k , 0 < pn ≤ 1, n pn = 1. From the trace norm estimate N −1 pn ψn (ψn , · ) = pn ψn (ψn , · ) = pn ≡ qN , ρ − n≥N n=1 n≥N 1 1

it follows that

N −1 pn µn + qN µN ≤ 2qN , µ − n=1

where µn = (ψn , ( · )ψn ) ∈ V (M). Since limN qN = 0 we conclude that finite convex linear combinations of vector states are norm dense in N (M).

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Exercise 2.9. (Complement to Lemma 2.8) Let D ⊂ H be a dense subspace. Show that the set of vector states VD (M) ≡ {(Ψ, ( · )Ψ ) | Ψ ∈ D, Ψ = 1} is total in N (M). Exercise 2.10. Show that a net Aα in a von Neumann algebra M converges σ-weakly (resp. σ-strongly) to 0 if and only if, for all ω ∈ N (M) one has limα ω(Aα ) = 0 (resp. limα ω(A∗α Aα ) = 0). Lemma 2.11. Let M, N be von Neumann algebras. A ∗-morphism φ : M → N is σ-weakly continuous if and only if it is σ-strongly continuous. Proof. Suppose that φ is σ-weakly continuous and that the net Aα converges σstrongly to 0. By Exercise 2.10, A∗α Aα converges σ-weakly to 0. It follows that φ(Aα )∗ φ(Aα ) = φ(A∗α Aα ) converges σ-weakly to zero and hence, by Exercise 2.10 again, that φ(Aα ) converges σ-strongly to 0. Suppose now that φ is σ-strongly continuous. Since any ω ∈ N (N) is σ-strongly continuous, so is the state ω ◦ φ. This means that ω ◦ φ is σ-weakly continuous for all ω ∈ N (N) and hence that φ itself is σ-weakly continuous. Corollary 2.12. A ∗-isomorphism between two von Neumann algebras is σ-weakly and σ-strongly continuous. Proof. Let M, N be von Neumann algebras and φ : M → N a ∗-isomorphism. If Aα is a bounded increasing net of self-adjoint elements in M then so is φ(Aα ) in N. Set A ≡ supα Aα . Since φ preserves positivity one has φ(Aα ) ≤ φ(A) and hence supα φ(Aα ) ≤ φ(A). Moreover, since φ is surjective there exists B ∈ M such that supα φ(Aα ) = φ(B) and φ(Aα ) ≤ φ(B) ≤ φ(A), holds for all α. These inequalities and the injectivity of φ further yield Aα ≤ B ≤ A, for all α. Thus, we conclude that B = A, that is, sup φ(Aα ) = φ(sup Aα ). α

(4)

α

By Theorem 2.7, any ω ∈ N (N) is normal and Equ. (4) yields sup ω(φ(Aα )) = ω(sup φ(Aα )) = ω(φ(sup Aα )), α

α

α

which shows that ω ◦ φ is normal and hence σ-weakly continuous. It follows that φ itself is σ-weakly continuous and, by Lemma 2.11, σ-strongly continuous.

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Functional Calculus Let A be a C ∗ -algebra and A ∈ A a self-adjoint element. By Theorem 4 in Lecture [7] there is a unique ∗-morphism πA : C(σ(A)) → A such that πA (f ) = A if f (x) = x. Accordingly, if f is continuous we write f (A) ≡ πA (f ). When dealing with a von Neumann algebra M ⊂ B(H), it is necessary to extend this morphism to a larger class of functions. This can be done with the help of Theorem 7 and Remark 10 in Lecture [32]. Let A ∈ B(H) be self-adjoint and denote by B(R) the ∗-algebra of bounded Borel functions on R. Then there exists a unique ∗-morphism ΠA : B(R) → B(H) such that (i) ΠA (f ) = f (A) if f ∈ C(σ(A)). (ii) If f, fn ∈ B(R) are such that limn fn (x) = f (x) for all x ∈ R and supn,x∈R |fn (x)| < ∞ then ΠA (fn ) → ΠA (f ) strongly. Again we write f (A) ≡ ΠA (f ) for f ∈ B(R). Thus, if A ∈ M is self-adjoint then f (A) ∈ M for any continuous function f . More generally, assume that f ∈ B(R) can be approximated by a sequence fn of continuous functions such that (ii) holds. Since M is strongly closed it follows that f (A) ∈ M. In particular, if χI denotes the characteristic function of an interval I ⊂ R then χI (A) is the spectral projection of A for the interval I and one has χI (A) ∈ M. The Support of a Normal State Exercise 2.13. Let Pα be an increasing net of orthogonal projections of the Hilbert space H. Denote by P the orthogonal projection on the smallest closed subspace of H containing all the subspaces Ran Pα . Show that s − lim Pα = sup Pα = P. α

α

Exercise 2.14. Let P and Q be two orthogonal projections on the Hilbert space H. Denote by P ∧ Q the orthogonal projection on Ran P ∩ Ran Q and by P ∨ Q the orthogonal projection on Ran P + Ran Q. i. Show that I − P ∨ Q = (I − P ) ∧ (I − Q). ii. Show that P ∨ Q ≤ P + Q ≤ I + P ∧ Q. iii. Set T ≡ P QP and show that Ran P ∩ Ran Q = Ker(I − T ). Mimic the proof of Theorem 3.13 to show that P ∧ Q = s − lim T n . n→∞

iv. Show that if M is a von Neumann algebra on H and if P, Q ∈ M then P ∧Q ∈ M and P ∨ Q ∈ M. v. Show that if ω ∈ E(M) then ω(P ∨ Q) = 0 if and only if ω(P ) = ω(Q) = 0 and ω(P ∧ Q) = 1 if and only if ω(P ) = ω(Q) = 1.

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Let ω be a normal state on the von Neumann algebra M. We denote by MP the set of orthogonal projections in M. Exercise 2.13 shows that the non-empty set Pω ≡ {P ∈ MP | ω(P ) = 0} is inductively ordered: any increasing net Pα in Pω has a least upper bound sup Pα = s − lim Pα ∈ Pω . α

α

By Zorn’s lemma, Pω has a maximal element P¯ω . For any P ∈ Pω one has P ∨P¯ω ≥ P¯ω and, by exercise 2.14, P ∨ P¯ω ∈ Pω . The maximality of P¯ω yields P ∨ P¯ω = P¯ω from which we can conclude that P ≤ P¯ω . Thus, one has P¯ω = sup Pω . The complementary projection sω ≡ I − P¯ω = inf{P ∈ MP | ω(P ) = 1}, is called the support of ω. For any normal state ω and any A ∈ M one has, by the Cauchy-Schwarz inequality |ω(A(I − sω ))| ≤ ω(AA∗ )1/2 ω(I − sω )1/2 = 0, from which we conclude that ω(A) = ω(Asω ) = ω(sω A). Exercise 2.15. Show that ω(A∗ A) = 0 if and only if Asω = 0. Conclude that the state ω is faithful if and only if sω = I. Hint: if Asω = 0 there exists > 0 and a non-zero P ∈ MP such that sA∗ As ≥ P and P = P sω . Exercise 2.16. Let M ⊂ B(H) be a von Neumann algebra. If K ⊂ H is a vector ¯ Use Proposition subspace, denote by [K] the orthogonal projection on its closure K. 10 in Lecture [7] to show that, for any subset M ⊂ H, [M M] ∈ M. Lemma 2.17. Let M ⊂ B(H) be a von Neumann algebra. The support of the state ω ∈ N (M) is given by sω = [M Ran ρ], where ρ is any density matrix on H such that ω(A) = Tr(ρA) for all A ∈ M. In particular, the support of the vector state ωΦ (A) = (Φ, AΦ) is given by sωΦ = [M Φ]. ωΦ is faithful if and only if Φ is cyclic for M . Proof. Set P = [M Ran ρ] and note that P ∈ M by Exercise 2.16. On the one hand I ∈ M implies Ran ρ ⊂ M Ran ρ and hence P Ran ρ = Ran ρ. Thus, P ρ = ρ and ω(P ) = Tr(P ρ) = Tr(ρ) = 1, from which we conclude that P ≥ sω . On the other hand, 0 = ω(I − sω ) = Tr(ρ(I − sω )) = Tr(ρ1/2 (I − sω )ρ1/2 ) = (I − sω )ρ1/2 22 , yields (I − sω )ρ = 0. It follows that sω Ran ρ = Ran ρ and sω M Ran ρ = M sω Ran ρ = M Ran ρ, implies sω P = P , that is P ≤ sω .

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The Central Support of a Normal State The center of a von Neumann algebra M is the Abelian von Neumann subalgebra Z(M) ≡ M∩M . One easily sees that Z(M) = (M∪M ) so that Z(M) = M∨M is the smallest von Neumann algebra containing M and M . The elementary proof of the following lemma is left to the reader. Lemma 2.18. Assume that M and N are two von Neumann algebras and let φ : M → N be a ∗-morphism. (i) If φ is surjective then φ(Z(M)) ⊂ Z(N). (ii) If φ is injective then φ−1 (Z(N)) ⊂ Z(M). (iii) If φ is bijective then φ(Z(M)) = Z(N). M is a factor if Z(M) = CI or equivalently M ∨ M = B(H). The central support of a normal state ω on M is the support of its restriction to the center of M, zω ≡ inf{P ∈ Z(M) ∩ MP | ω(P ) = 1}.

(5)

For any normal state ω one clearly has 0 < sω ≤ zω ≤ I and hence ω(A) = ω(Azω ) = ω(zω A) for all A ∈ M. The state ω is centrally faithful if zω = I. Lemma 2.17 shows that the central support of the vector state ωΦ is zωΦ = [M ∨ M Φ].

(6)

In particular, if Φ is cyclic for M or M or if M is a factor, then ωΦ is centrally faithful. More generally one has Lemma 2.19. Let ω be a normal state on the von Neumann algebra M ⊂ B(H). If sω is the support of ω, its central support is given by zω = [Msω H]. Proof. Set K ≡ Msω H and denote by P the orthogonal projection on K. We first claim that P ∈ Z(M). Using Proposition 10 in [7] and the relation Z(M) = (M ∪ M ) , this follows from the fact that Msω H and hence K are invariant under M and M . Next we note that Ran sω ⊂ K implies P ≥ sω and hence ω(P ) = 1. Finally, if Q ∈ MP ∩ Z(M) is such that ω(Q) = 1, Cauchy-Schwarz inequality yields that ω(A(I − Q)) = 0 for all A ∈ M. Exercise 2.15 further leads to (I − Q)Asω = A(I − Q)sω = 0 for all A ∈ M. This shows that (I − Q)P = 0, i.e., Q ≥ P and Equ. (5) yields that P = zω . By Corollary 2.12, a ∗-automorphism τ of a von Neumann algebra is automatically continuous in the σ-weak topology. In particular ω ◦ τ is a normal state for any normal state ω. It immediately follows from the definitions that sω◦τ = τ −1 (sω ) and zω◦τ = τ −1 (zω ).

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2.2 The GNS Representation Let A be a C ∗ -algebra and ω ∈ E(A). Throughout these notes I shall use the standard notation (Hω , πω , Ωω ) for the GNS representation of A associated to the state ω (Theorem 8 in Lecture [7]). Enveloping von Neumann Algebra and Folium of a State Since Ωω is cyclic for πω (A), ω ˆ (A) ≡ (Ωω , AΩω ), defines a centrally faithful normal state on the von Neumann algebra πω (A) . By the von Neumann density theorem, πω (A) is σ-weakly dense in πω (A) and hence we have a canonical injection N (πω (A) ) → E(A) ν˜ → πω (˜ ν ) = ν˜ ◦ πω . Thus, we can identify N (πω (A) ) with a subset N (A, ω) of E(A). Explicitly, ν ∈ N (A, ω) if and only if there exists a density matrix ρ on Hω and a corresponding normal state ν˜ on πω (A) such that ν(A) = ν˜ ◦ πω (A) = Tr(ρπω (A)). Definition 2.20. Let A be a C ∗ -algebra and ω ∈ E(A). (i) Aω ≡ πω (A) ⊂ B(Hω ) is the enveloping von Neumann algebra of A associated to ω. (ii) N (A, ω) ⊂ E(A) is the folium of the state ω. It is the image under πω of the set of states on πω (A) which have a unique normal extension to the enveloping algebra Aω . Its elements are said to be normal relative to ω, or simply ω-normal. Note that ω ˆ is the unique normal extension of the state πω (A) !→ ω(A) from ˆ is the πω (A) to its weak closure Aω . By a slight abuse of language I shall say that ω normal extension of ω to Aω . Similarly, if ν = ν˜ ◦ πω ∈ N (A, ω) I shall say that ν˜ is the normal extension of ν to Aω . I further denote by sν|ω the support of ν˜ and by zν|ω its central support. Abusing notation, I also set sω ≡ sω|ω = sωˆ . Definition 2.21. Let ω, ν be two states on the C ∗ -algebra A. (i) ν, ω are quasi-equivalent, written ν ≈ ω, if N (A, ν) = N (A, ω). (ii) They are orthogonal, written ν ⊥ ω, if λµ ≤ ν and λµ ≤ ω for some µ ∈ E(A) and λ ≥ 0 implies λ = 0. (iii) They are disjoint if N (A, ν) ∩ N (A, ω) = ∅.

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The GNS Representation of a Normal State In this subsection we study the special features of the GNS representation associated to a normal state ω on the von Neumann algebra M ⊂ B(H). The first result relates the central support zω of the state ω to the kernel of the ∗-morphism πω . Before stating this relation let me make the following remark. Remark 2.22. If P ∈ Z(M) is an orthogonal projection then Q ≡ I − P ∈ Z(M) and since MP H = P MH ⊂ P H and MQH = QMH ⊂ QH, any element of M can be written, according to the orthogonal decomposition H = P H ⊕ QH, as a 2 × 2-matrix B 0 A= , 0 C where B ∈ B(P H) and C ∈ B(QH). Using the injection B0 PM * → B ∈ B(P H), 0 0 we can identify P M with a von Neumann algebra on P H and similarly for QM. We then write M = P M ⊕ QM, and say that M is the direct sum of the von Neumann algebras P M and QM. Of course the same argument applies to the commutant and we also have M = P M ⊕ QM . It follows immediately that P M = (P M) and QM = (QM) as von Neumann algebras on P H and QH. With the same interpretation we can write Z(M) = P Z(M) ⊕ QZ(M) and Z(P M) = P Z(M), Z(QM) = QZ(M). Lemma 2.23. If ω ∈ N (M) then Ker(πω ) = (I −zω )M. In particular, πω is faithful (i.e., is a ∗-isomorphism from M onto πω (M)) if and only if ω is centrally faithful. More generally, the map π ˆω : Azω !→ πω (A), defines a ∗-isomorphism from the von Neumann algebra zω M onto πω (M) such that, ˆω−1 (B) = B and π ˆω−1 ◦ πω (A) = for all A ∈ M and all B ∈ πω (M) one has πω ◦ π zω A. Proof. For A, B ∈ M one has πω (A(I − zω ))πω (B)Ωω 2 = ω(B ∗ A∗ AB(I − zω )) = 0. Since πω (M)Ωω is dense in Hω one concludes that πω (A(I − zω )) = 0, i.e., M(I − zω ) ⊂ Ker(πω ). To prove the reverse inclusion note that πω (A) = 0 implies that ω(B ∗ A∗ AB) = πω (A)πω (B)Ωω 2 = 0, for all B ∈ M. Exercise 2.15 further gives ABsω = 0 for all B ∈ M and Lemma 2.19 yields Azω = 0, i.e., A = A(I − zω ). The proof of the last statement of the lemma is easy and left to the reader

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Corollary 2.24. Let M be a von Neumann algebra and ω, ν ∈ N (M). Then ν is ω-normal if and only if sν ≤ zω . Proof. Suppose that ν = νˆ ◦ πω for some νˆ ∈ N (Mω ). By Lemma 2.23 we have ν(I − zω ) = νˆ(πω (I − zω )) = νˆ(0) = 0. Thus, ν(zω ) = 1 from which we conclude that sν ≤ zω . ˆω−1 . Since π ˆω is a ∗-isomorphism νˆ Suppose now that sν ≤ zω and set νˆ ≡ ν ◦ π is normal. Moreover, from ν(A) = ν(zω A) we conclude that νˆ ◦ πω (A) = ν(ˆ πω−1 (πω (A))) = ν(zω A) = ν(A). The continuity properties of πω follow from the simple lemma: Lemma 2.25. The map πω is normal i.e., for any bounded increasing net Aα of selfadjoint elements of M one has sup πω (Aα ) = πω (sup Aα ). α

α

Proof. For any B ∈ M one has ω(B ∗ Aα B) = (πω (B)Ωω , πω (Aα )πω (B)Ωω ) and Equ. (1) allows us to write (πω (B)Ωω , sup πω (Aα )πω (B)Ωω ) = sup(πω (B)Ωω , πω (Aα )πω (B)Ωω ) α

α

= sup ω(B ∗ Aα B) α

= ω(B ∗ (sup Aα )B) α

= (πω (B)Ωω , πω (sup Aα )πω (B)Ωω ). α

Since πω (M)Ωω is dense in Hω the claim follows. Exercise 2.26. Prove the following lemma using Lemma 2.25 and following the proof of Corollary 2.12. Lemma 2.27. If ω ∈ N (M) then πω is σ-weakly and σ-strongly continuous. Lemma 2.28. If ω ∈ N (M) then πω (M) is a von Neumann algebra in B(Hω ), i.e., Mω = πω (M). ˆω (Mzω ) we can assume, without loss of generality, that Proof. Since πω (M) = π πω is faithful and hence isometric (Proposition 4 in Lecture [7]). Let B ∈ πω (M) . By the Kaplansky density theorem there exists a net Aα in M such that πω (Aα ) ≤ B and πω (Aα ) converges σ-weakly to B. Since Aα = πω (Aα ) ≤ B ,

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it follows from the Banach-Alaoglu theorem that there exists a subnet Aβ of the net Aα which converges σ-weakly to some A ∈ M. Since πω is σ-weakly continuous one has πω (A) = lim πω (Aβ ) = lim πω (Aα ) = B, β

and hence B ∈ πω (M).

α

Lemma 2.29. If ω ∈ N (M) then N (M, ω) = {ν ∈ N (M) | sν ≤ zω } ⊂ N (M). In particular, if ω is centrally faithful then N (M, ω) = N (M). Proof. By Lemma 2.27, if ω ∈ N (M) then πω is σ-weakly continuous. Hence N (M, ω) ⊂ N (M) and Corollary 2.24 apply. As an application of the above results let us prove the following characterization of the relative normality of two states on a C ∗ -algebra. Theorem 2.30. Let A be a C ∗ -algebra and ω, µ ∈ E(A). Denote the induced GNS representations by (Hω , πω , Ωω ), (Hµ , πµ , Ωµ ) and the corresponding enveloping von Neumann algebras by Aω , Aµ . Then µ ∈ N (A, ω) if and only if there exists a σ-weakly continuous ∗-morphism πµ|ω : Aω → Aµ such that πµ = πµ|ω ◦ πω . If this is the case then the following also hold. (i) N (A, µ) ⊂ N (A, ω). (ii) If ν ∈ N (A, µ) has the normal extension ν˜ to Aµ then ν˜ ◦ πµ|ω is its normal extension to Aω . (iii) πµ|ω is σ-strongly continuous. (iv) π ˆµ|ω : zµ|ω A !→ πµ|ω (A) is a ∗-isomorphism from zµ|ω Aω onto Aµ . (v) Ker πµ|ω = (I − zµ|ω )Aω (vi) πµ|ω (sµ|ω ) = sµ . (vii) If µ ˜ denotes the normal extension of µ to Aω then (Hµ , πµ|ω , Ωµ ) is the induced GNS representation of Aω . In particular Aµ = (Aω )µ˜ . Proof. Let µ ˆ(A) ≡ (Ωµ , AΩµ ) be the normal extension of µ to Aµ . If the morphism ˆ ◦ πµ = (ˆ µ ◦ πµ|ω ) ◦ πω . Since πµ|ω is σ-weakly πµ|ω exists then one has µ = µ continuous µ ˆ ◦ πµ|ω ∈ N (Aω ) and we conclude that µ ∈ N (A, ω). ˜ ∈ N (Aω ). Consider Assume now that µ ∈ N (A, ω), i.e., that µ = µ ˜ ◦ πω with µ ˜. By Lemma 2.27 φ is σ-weakly the GNS representation (K, φ, Ψ ) of Aω induced by µ and σ-strongly continuous. Since by the von Neumann density theorem πω (A) is σstrongly dense in Aω , we get K = φ(Aω )Ψ = φ(πω (A))Ψ . Finally, for any A ∈ A one has µ(A) = µ ˜(πω (A)) = (Ψ, φ(πω (A))Ψ ),

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and we can conclude that (K, φ ◦ πω , Ψ ) is a GNS representation of A induced by µ. By the unicity, up to unitary equivalence, of such representations there exists a unitary U : K → Hµ such that πµ (A) = U φ(πω (A))U ∗ and Ωµ = U Ψ . Set πµ|ω (A) ≡ U φ(A)U ∗ then (Hµ , πµ|ω , Ωµ ) is another GNS representation of Aω induced by µ ˜ and πµ = πµ|ω ◦ πω . Since πµ|ω is also σ-weakly and σ-strongly continuous Lemma 2.28 and the von Neumann density theorem yield (Aω )µ˜ = πµ|ω (Aω ) = πµ|ω (πω (A)) = πµ (A) = Aµ . This proves the existence of πµ|ω with Properties (iii) and (vii). To prove Properties (i) and (ii) let ν ∈ N (A, µ) and denote by ν˜ its normal extension to Aµ . One has ν = ν˜ ◦πµ = ν˜ ◦πµ|ω ◦πω and it follows that ν ∈ N (A, ω) and that its normal extension to Aω is ν˜ ◦ πµ|ω . Properties (iv) and (v) follow directly from Lemma 2.23. By Lemma 2.25, πµ|ω is normal and hence ˜(P ) = 1} πµ|ω (sµ|ω ) = πµ|ω inf{P ∈ AP ω |µ ˆ(πµ|ω (P )) = 1} = inf{πµ|ω (P ) | P ∈ AP ω,µ ˆ(Q) = 1} = inf{Q ∈ AP µ |µ = sµ , proves Property (vi).

3 Classical Systems 3.1 Basics of Ergodic Theory Let X be a measurable space, i.e., a set equipped with a σ-field F. A dynamics on X is a family of maps ϕt : X → X, indexed by a time t running in R, such that (x, t) !→ ϕt (x) is measurable and the group properties ϕ0 (x) = x,

ϕt ◦ ϕs = ϕt+s ,

hold. In particular, the map ϕt is one to one with inverse ϕ−t . Given x ∈ X, we set xt = ϕt (x) and call (xt )t∈R ⊂ X the orbit or trajectory starting at x. Observables are bounded measurable functions f : X → C. Instead of considering individual orbits, think of the initial configuration x as a random variable distributed according to a probability measure1 µ on X. Then math(xt )t∈R becomes a stochastic process and we denote by Eµ the corresponding ematical expectation. If f is an observable, Eµ [f (x0 )] = µ(f ) = f dµ is the expectation of f at time zero. Its expectation at time t is µt (f ) ≡ Eµ [f (xt )] = µ(f ◦ ϕt ) which defines the evolution µt of the measure µ. A measure µ is called invariant if 1

All measures in this section are probabilities, so I use the words measure and probability interchangeably.

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the corresponding process is stationary, i.e., if µt = µ for all t. Such an invariant measure describes a stationary regime of the system, or an equilibrium state. Invariant probabilities may fail to exist (see Exercise I.8.6 in [35]) but most physical systems of interest have a lot of them. For example, if x ∈ X is periodic of period T , then 1 T µx ≡ δxt dt, T 0 is invariant, supported by the orbit of x. Under additional topological assumptions, one can prove that there is at least one invariant measure. Exercise 3.1. Let M be a compact metric space and ϕt a continuous dynamics on M , i.e., assume that the map (t, x) !→ ϕt (x) is continuous. Recall the following facts: The set of continuous observables C(M ) is a separable Banach space. Its dual C(M ) is the set of Baire measures on M . The set of Baire probabilities is a compact, metrizable subset of C(M ) for the weak- topology. Every Baire measure has a unique regular Borel extension. Show that for any Baire probability µ on M , there exists a sequence Tn → +∞, such that the weak- limit Tn 1 µt dt, µ+ = w −lim n Tn 0 exists and defines an invariant probability for ϕt on M . Definition 3.2. A classical dynamical system is a triple (X, ϕt , µ) where X, the phase space of the system, is a measurable space, ϕt a dynamics on X and µ an invariant measure for ϕt . If (X, ϕt , µ) is a classical dynamical system, then U t f ≡ f ◦ ϕt defines a group of isometries on each Banach space Lp (X, dµ). Indeed, for any f ∈ Lp (X, dµ) we have U 0 f = f ◦ ϕ0 = f , U t U s f = U t (f ◦ ϕs ) = (f ◦ ϕs ) ◦ ϕt = f ◦ (ϕs ◦ ϕt ) = f ◦ ϕt+s = U t+s f, and U t f pp = µ(|f ◦ ϕt |p ) = µ(|f |p ◦ ϕt ) = µt (|f |p ) = µ(|f |p ) = f p . A function f ∈ Lp (X, dµ) is invariant if U t f = f for all t. A measurable set A ⊂ X is invariant modulo µ if its characteristic function χA is invariant, that is 0 = |χA − U t χA | = |χA − χϕ−1 (A) | = χA∆ϕ−1 (A) , t

t

which is equivalent to µ(A∆ϕ−1 t (A)) = 0. Ergodic theory deals with the study of invariant measures and their connections with the large time behavior of dynamical system. The cornerstone of ergodic theory is the following, so called Birkhoff or individual ergodic theorem.

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Theorem 3.3. Let (X, ϕt , µ) be a classical dynamical system. Then for any f ∈ L1 (X, dµ), the two limits 1 T →∞ T

T

f ◦ ϕ±t (x) dt,

(Pµ f )(x) = lim

0

exist and coincide for µ-almost all x ∈ X. They define a linear contraction Pµ on L1 (X, dµ) with the following properties: (i) Pµ f ≥ 0 if f ≥ 0, (ii) P2µ = Pµ , (iii) U t Pµ = Pµ U t = Pµ for all t, (iv) µ(f ) = µ(Pµ f ), (v) if g ∈ L1 (X, dµ) is invariant, then Pµ gf = gPµ f . In particular, Pµ g = g. Remark 3.4. Pµ is a conditional expectation with respect to the σ-field of invariant sets modulo µ. Even in the case of a smooth (continuous or analytic) dynamical system, the conditional expectation Pµ f (x) can display a weird dependence on the starting point x, reflecting the complexity of the orbits. Of special interest are the measures µ for which Pµ f (x) is µ-almost surely independent of x. Definition 3.5. The dynamical system (X, ϕt , µ) is called ergodic if, for all f ∈ L1 (X, dµ), one has Pµ f (x) = µ(f ) for µ-almost all x ∈ X. In this case, we also say that µ is an ergodic measure for ϕt , or that the dynamics ϕt is ergodic for µ. Proposition 3.6. The following propositions are equivalent. (i) (X, ϕt , µ) is ergodic. (ii) For any measurable set A invariant modulo µ, one has µ(A) ∈ {0, 1}. (iii) For any invariant f ∈ L1 (X, dµ) one has f = µ(f ). (iv) For any µ-absolutely continuous probability ρ one has 1 T →∞ T

lim

T

ρt (f ) dt = µ(f ), 0

for all f ∈ L∞ (X, dµ). Proof. Let us first show that (i), (ii) and (iii) are equivalent. (i)⇒(ii). If A is invariant modulo µ then its indicator function χA is invariant. Property (v) of Theorem 3.3 shows that χA = Pµ χA . If µ is ergodic we further have Pµ χA = µ(A). Therefore, χA = µ(A) and we conclude that µ(A) ∈ {0, 1}. (ii)⇒(iii). If f ∈ L1 (X, dµ) is invariant so are its real and imaginary parts. Without loss of generality we can assume that f is real valued. Then the set {x | f (x) > a} is invariant modulo µ. Therefore, the distribution function Ff (a) = µ({x | f (x) > a}) takes values in the set {0, 1} from which we conclude that f is constant µ-almost everywhere.

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(iii)⇒(i). If f ∈ L1 (X, dµ) then Pµ f is invariant by property (iii) of Theorem 3.3. Therefore, Pµ f = µ(Pµ f ) and by property (iv) of Theorem 3.3 this is equal to µ(f ). Now we consider property (iv). Denote by g ∈ L1 (X, dµ) the Radon-Nikodym derivative of ρ with respect to µ. Then, for any f ∈ L∞ (X, dµ), Fubini Theorem and the invariance of µ give 1 T 1 T ρt (f ) dt = g(x)f ◦ ϕt (x) dµ(x) dt T 0 T 0 1 T = g ◦ ϕ−t (x) dt f (x) dµ(x). T 0 Birkhoff Theorem together with Lebesgue dominated convergence Theorem further lead to 1 T ρt (f ) dt = µ((Pµ g)f ). ρ+ (f ) ≡ lim T →∞ T 0 Therefore, if µ is ergodic, we obtain ρ+ (f ) = µ(µ(g)f ) = µ(f ). Reciprocally, if ρ+ = µ for all µ-absolutely continuous probabilities ρ, we can conclude that for all g ∈ L1 (X, dµ) such that µ(g) = 1 one has Pµ g = 1 = µ(g). This clearly implies that µ is ergodic. The following ergodic decomposition theorem shows that ergodic measures are elementary building blocks of invariant measures. Theorem 3.7. Let M be a compact metric space and ϕt a continuous dynamics on M . Then the set Eϕ (M ) of invariant probabilities for ϕt is a non-empty, convex, weak− compact subset of the dual C(M ) . Denote by Xϕ (M ) the set of extremal points2 of Eϕ (M ): (i) µ ∈ Xϕ (M ) if and only if it is ergodic for ϕt . (ii) If µ, ν ∈ Xϕ (M ) and µ = ν, then µ and ν are mutually singular. (iii) For any µ ∈ Eϕ (M ) there exists a probability measure ρ on Xϕ (M ) such that µ = ν dρ(ν). The relevance of an ergodic measure µ in the study of the large time behavior of the system comes from the fact that the time average of an observable along a generic orbit is given by the ensemble average described by µ: 1 T lim f (xt ) dt = µ(f ), T →∞ T 0 2

µ is extremal if it can not be expressed as a non-trivial convex linear combination of two distinct measures: µ = αµ1 + (1 − α)µ2 with α ∈]0, 1[ and µi ∈ Eϕ (M ) implies µ1 = µ2 = µ.

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for µ-almost all x. Note however that, depending on the nature of µ, generic orbits may fill a very small portion of the phase space (as shown by the above example of a periodic orbit). Here small refers to some additional feature that the physical problem may induce on the space X. For example if X is a smooth manifold, small could mean of measure zero with respect to some (any) Riemannian volume on X (see [39] for a more detailed discussion of this important point). A more precise information on the large time asymptotics of the system is given by the following mixing condition. Definition 3.8. A dynamical system (X, ϕt , µ) is mixing if, for any µ-absolutely continuous measure ρ and all observables f ∈ L∞ (X, dµ), one has lim ρt (f ) = µ(f ).

t→∞

One also says that µ is mixing, or ϕt is mixing for µ. If we think of an invariant measure µ as describing an equilibrium state of the system then µ is mixing if all initial measures ρ which are not too far from equilibrium, i.e., which are absolutely continuous with respect to µ, converge to µ as t → ∞. For this reason, the mixing condition is often referred to as return to equilibrium. If µ is mixing, it is obviously ergodic from Property (iv) in Theorem 3.6. Here is another proof. If A is an invariant set modulo µ such that µ(A) > 0, then ρ(f ) = µ(f χA )/µ(A) defines a µ-absolutely continuous invariant measure with ρ(A) = 1. Therefore, if µ is mixing we have 1 = ρ(A) = ρt (A) = lim ρt (A) = µ(A). t→∞

From Theorem 3.6 we conclude that µ is ergodic. Thus mixing implies ergodicity, but the reverse is not true in general. Exercise 3.9. Let X = R/Z be the one dimensional torus, µ the measure induced on X by the Lebesgue measure on R and ϕt (x) = x + t (mod 1). Show that (X, ϕt , µ) is ergodic but not mixing. Exercise 3.10. Show that (X, ϕt , µ) is mixing if and only if, for all measurable subsets A, B ⊂ X one has lim µ(ϕ−1 t (A) ∩ B) = µ(A) µ(B).

t→∞

Ergodicity and mixing are only two elements of the so called ergodic hierarchy which contains many other properties like exponential mixing, K-system. . . The interested reader should consult the general references given in the Introduction. 3.2 Classical Koopmanism Ergodicity and mixing can be quite difficult to prove in concrete situations. One of the more powerful tools to do it is the Koopman–von Neumann spectral theory that I

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shall now introduce. The Koopman space of the dynamical system (X, ϕt , µ) is the Hilbert space H = L2 (X, dµ) on which the Koopman operators U t are defined by U t f ≡ f ◦ ϕt . In the following, I shall always assume that the Koopman space is separable. This is the case for example if X is a (locally) compact metric space equipped with its natural Borel structure. Lemma 3.11. (Koopman Lemma) If H is separable, U t is a strongly continuous group of unitary operators on H. Proof. We have already shown that U t is a group of isometries on H. Since U t U −t = I we have Ran U t = H and therefore U t is unitary. Finally, since t !→ (f, U t g) is measurable and H is separable, it follows from a well known result of von Neumann (Theorem VIII.9 in [41]) that the map t !→ U t is strongly continuous. Remark 3.12. The separability condition is satisfied if X is a finite dimensional manifold. In infinite dimensional cases it can often be replaced by a weak continuity assumption. Indeed, if the map t !→ f (x)f (ϕt (x)) dµ(x) is continuous for all f ∈ L2 (X, dµ), then U t is strongly continuous since U t f − f 2 = 2( f 2 − Re(f, U t f )).

(7)

By the Stone Theorem, there exists a self-adjoint operator L on H such that U t = e−itL . We call L the Liouvillean of the system. Note that 1 ∈ H and U t 1 = 1 for all t, from which we conclude that 1 ∈ D(L) and L1 = 0. In other words, 0 is always an eigenvalue of the Liouvillean, with the associated eigenfunction 1. The connection between Ker L and ergodic theory is the content of the following von Neumann or mean ergodic theorem. Theorem 3.13. Let U t = e−itA be a strongly continuous group of unitaries on a Hilbert space H and denote by P the orthogonal projection on Ker A. Then, for any f ∈ H, 1 T t U f dt = P f, lim T →∞ T 0 holds in H. Proof. Since U t is strongly continuous, f T =

1 T

T

U t f dt 0

is well defined as a Riemann integral. We first remark that for f ∈ Ran A, we have

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U t f = U t Ag = i∂t U t g for some g ∈ D(A). Hence an explicit integration gives i T (U − I)g = 0. T →∞ T

lim f T = lim

T →∞

Using the simple estimate uT − vT ≤ u − v , this result immediately extends to all f ∈ Ran A = Ker A⊥ . Since for f ∈ Ker A we have f T = f , we get for arbitrary f ∈ H lim f T = lim P f T + lim (I − P )f T = P f.

T →∞

T →∞

T →∞

Theorem 3.14. (Koopman Ergodicity Criterion) A dynamical system is ergodic if and only if 0 is a simple eigenvalue of its Liouvillean L. Proof. Let f ∈ Ker L, then f is an invariant function in L1 (X, dµ) and, by Theorem 3.6, if µ is ergodic we must have f = µ(f )1. Thus, Ker L is one dimensional. Assume now that Ker L is one dimensional. Let A be an invariant set modulo µ. It follows that χA ∈ H is invariant and hence belongs to Ker L. Thus, we have χA = µ(A)1, from which we may conclude that µ(A) ∈ {0, 1}. Ergodicity of µ follows from Theorem 3.6. Theorem 3.15. (Koopman Mixing Criterion) A dynamical system is mixing if and only if w − lim U t = (1, · )1. (8) t→∞

In particular, if the spectrum of the Liouvillean L is purely absolutely continuous on {1}⊥ , then the system is mixing. Proof. Assume first that the system is mixing, and set H1+ = {g ∈ H | g ≥ 0, µ(g) = 1}. Since any g ∈ H1+ is the Radon-Nikodym derivative of some µ-absolutely continuous probability ρ, we get for any f ∈ L∞ (X, dµ) (g, U t f ) = ρt (f ) → µ(f ) = (g, 1)(1, f ),

(9)

as t → ∞. Since any g ∈ H is a finite linear combination of elements of H1+ , Equ. (9) actually holds for all g ∈ H and f ∈ L∞ (X, dµ). Finally, since both side of Equ. (9) are H-continuous in f uniformly in t, Equ. (8) follows from the fact that L∞ (M, dµ) is dense in H. The reverse statement is proved in an analogous way. Suppose ρ is a µ-absolutely continuous probability and denote by g its Radon-Nikodym derivative. Assuming for a while that g ∈ H, we get from Equ. (8) that limt→∞ ρt (f ) = µ(f ) for all

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f ∈ L∞ (M, dµ). Since ρt (f ) is L1 -continuous in g, uniformly in t, the desired result follows from the fact that H is dense in L1 (M, dµ). To prove the last statement, we first remark that to obtain Equ. (8) it suffices to show that lim (f, U t f ) = 0, t→∞

⊥

for all f ∈ {1} (use the orthogonal decomposition H = C · 1 ⊕ {1}⊥ and the polarization identity). If L has purely absolutely continuous spectrum on {1}⊥ then its spectral measure associated to f ∈ {1}⊥ can be written as dνf (λ) = g(λ) dλ for some function g ∈ L1 (R). Therefore, t (f, U f ) = eitλ g(λ) dλ → 0, as t → ∞ by the Riemann-Lebesgue Lemma.

Note that (f, U t f ) is the Fourier transform of the spectral measure of L associated to f . It is clear from the above proof that a dynamical system is mixing if and only if the Fourier transform of the spectral measure of its Liouvillean associated to any vector in {1}⊥ vanishes at infinity. See Subsection 2.5 in Lecture [25]. Exercise 3.16. On the n-dimensional torus Tn = Rn /Zn , consider the dynamics ϕt (x) = x + tω (mod 1) where ω = (ω1 , · · · , ωn ) ∈ Rn . Show that the (Haar) measure dµ(x) = dx1 · · · dxn is invariant and compute the Liouvillean L of (Tn , ϕt , µ). Determine the spectrum of L and discuss the ergodic properties of the system.

4 Quantum Systems In the traditional description of quantum mechanics, a quantum system is completely determined by its Hilbert space H and its Hamiltonian H, a self-adjoint operator on H. The Hilbert space H determines both, the set of observables and the set of states of the system. The Hamiltonian H specifies its dynamics. An observable is a bounded linear operator on H. A states is specified by a “wave function”, a unit vector Ψ ∈ H, or more generally by a “density matrix”, a nonnegative, trace class operator ρ on H with Tr ρ = 1. The state associated with the density matrix ρ is the linear functional A !→ ρ(A) ≡ Tr(ρA),

(10)

on the set of observables. As a state, a unit vector Ψ is equivalent to the density matrix ρ = (Ψ, · )Ψ . Such density matrices are characterized by ρ2 = ρ and the corresponding states A !→ (Ψ, AΨ ), are called vector states. The role played by quantum mechanical states is somewhat similar to the role played by probability distributions in classical dynamical systems. However, since

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there is no quantum mechanical phase space, there is nothing like a trajectory and no state corresponding to the Dirac measures δx of classical dynamical systems. To any self-adjoint observable A, a state ρ associate a probability measure on the spectrum of A dρA (a) = ρ(EA (da)), where EA (·) is the projection valued spectral measure of A given by the spectral theorem. The measure ρA specifies the probability distribution for the outcome of a measure of the observable A. In particular, the expectation value of A in the state ρ is a dρA (a) = ρ(A). If two or more self-adjoint observables A, B, . . ., commute, they have a joint spectral measure and a similar formula gives the joint probability distribution for the simultaneous measurement of A, B, . . . However, if A and B do not commute, it is not possible to measure them simultaneously and there is no joint probability distribution for them (see also Lecture [33] in this Volume and [13] or any textbook on quantum mechanics for more details). By the spectral decomposition theorem for compact operators, a density matrix ρ can be written as pj (Ψj , · )Ψj , (11) ρ= j

where the Ψj are eigenvectors of ρ and form an orthonormal basis of H and the coefficients pj are the corresponding eigenvalues which satisfy pj = 1. 0 ≤ pj ≤ 1, j

Thus, a general state is a convex linear combination of vector states. pj is the probability for the system to be in the vector state associated to Ψj . In the physics literature, such a state is sometimes called statistical mixture (or incoherent superposition) of the states Ψj with the statistical weights pj . A wave function Ψ evolves according to the Schr¨odinger equation of motion i∂t Ψt = HΨt . Since H is self-adjoint, the solution of this equation, with initial value Ψ0 = Ψ , is given by Ψt = e−itH Ψ. According to the decomposition (11), a density matrix ρ evolves as ρt = e−itH ρ eitH , and thus, satisfies the quantum Liouville equation ∂t ρt = i[ρt , H].

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The expectation value of an observable A at time t is then ρt (A) = Tr(ρt A). This is the so-called Schr¨odinger picture of quantum dynamics. Since the cyclicity of the trace implies that Tr(ρt A) = Tr(e−itH ρ eitH A) = Tr(ρ eitH A e−itH ), we can alternatively keep the state fixed and let observables evolve according to At = eitH A e−itH . Such time evolved observables satisfy the Heisenberg equation of motion ∂t At = i[H, At ]. We obtain in this way the Heisenberg picture of quantum dynamics. Since ρt (A) = ρ(At ), the Schr¨odinger and the Heisenberg pictures are obviously equivalent. For systems with a finite number of degrees of freedom, this Hilbert space description of a quantum system is good enough because it is essentially unique (a precise form of this statement is the Stone–von Neumann uniqueness theorem, see Lecture [36]). However, for systems with an infinite number of degrees of freedom, (i.e., quantum fields) this is no more the case. An intrinsic description, centered around the C ∗ -algebra of observables becomes more convenient. 4.1 C ∗ -Dynamical Systems Definition 4.1. A C ∗ -dynamical system is a pair (A, τ t ) where A is a C ∗ -algebra with a unit and τ t a strongly continuous group of ∗-automorphisms of A. Since τ t ((z − A)−1 ) = (z − τ t (A))−1 , a ∗-automorphism τ t clearly preserves the spectrum. Hence, given the relation between the norm and the spectral radius (see the proof of Theorem 3 in Lecture [7]), it is isometric and in particular norm continuous. Strong continuity of τ t means that, for any A ∈ A, the map t !→ τ t (A) is continuous in the norm topology of A. From the general theory of strongly continuous semi-groups, there exists a densely defined, norm closed linear operator δ on A such that τ t = etδ . Since τ t (1) = 1, if follows that 1 ∈ D(δ) and δ(1) = 0. Differentiation of the identities τ t (AB) = τ t (A)τ t (B) and τ t (A∗ ) = τ t (A)∗ for A, B ∈ D(δ) further show that the generator δ is a ∗-derivation as defined by the following Definition 4.2. Let A be a ∗-algebra and D ⊂ A a subspace. A linear operator δ : D → A is called ∗-derivation if

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(i) D is a ∗-subalgebra of A. (ii) δ(AB) = δ(A)B + Aδ(B) for all A, B ∈ D. (iii) δ(A∗ ) = δ(A)∗ for all A ∈ D. Generators of strongly continuous groups of ∗-automorphisms are characterized by the following simple adaptation of the Hille-Yosida Theorem (see [11], Theorem 3.2.50). Proposition 4.3. Let A be a C ∗ -algebra with a unit. A densely defined, closed operator δ on A generates a strongly continuous group of ∗-automorphisms of A if and only if: (i) δ is a ∗-derivation. (ii) Ran(Id +λδ) = A for all λ ∈ R \ {0}. (iii) A + λδ(A) A for all λ ∈ R and A ∈ D(δ). If the C ∗ -algebra A acts on a Hilbert space H then a dynamical group τ t can be constructed from a group of unitary operators U t on H τ t (A) = U t A U t∗ . Such ∗-automorphisms are called spatial. They are particularly pleasant since it is possible to lift most of their analysis to the Hilbert space H itself. Example 4.4. (Finite quantum systems) Consider the quantum system with a finite number of degrees of freedom determined by the data H and H as described at the beginning of this section. On the C ∗ -algebra A = B(H) the dynamics is given by τ t (A) = eitH A e−itH . Exercise 4.5. Show that the group τ t is strongly continuous if and only if H is bounded. Thus, if τ t is strongly continuous it is automatically uniformly continuous and its generator is the bounded ∗-derivation δ(A) = i[H, A], on B(H). More specific examples are: 1. N -levels systems: H = CN and, without loss of generality, the Hamiltonian is a N × N diagonal matrix Hij = i δij . 2. Lattice quantum systems: H = l2 (Zd ) and the Hamiltonian is of tight-binding type 1 (Hψ)(x) = ψ(y) + V (x)ψ(x), 2d |x−y|=1

where |x| denotes the Euclidean norm of x = (x1 , · · · , xd ) ∈ Zd and V ∈ l∞ (Zd ). See Example 4 in Subsection 5.2 of Lecture [6] for a continuation of this example.

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Example 4.6. (The ideal Fermi gas) Let h be a Hilbert space and Γ− (h) the Fermionic Fock space over h. Recall that + (n) Γ− (h), Γ− (h) = n∈N (0)

(n)

where Γ− (h) = C and Γ− (h) = h ∧ h ∧ · · · ∧ h is the n-fold totally anti-symmetric tensor product of h. For f ∈ h, the action of the Fermionic creation operator a∗ (f ) (n) on a Slater determinant f1 ∧ f2 ∧ · · · ∧ fn ∈ Γ− (h) is defined by √ a∗ (f )f1 ∧ f2 ∧ · · · ∧ fn = n + 1 f1 ∧ f2 ∧ · · · ∧ fn ∧ f, (12) and is extended by linearity to the dense subspace Γﬁn − (h) =

k ,+

(n)

Γ− (h),

k∈N n=0

of Γ− (h). The annihilation operators are defined in a similar way starting from a(f )f1 ∧ f2 ∧ · · · ∧ fn =

n √ (f, fk )f1 ∧ · · · fk−1 ∧ fk+1 · · · ∧ fn . n k=1

A simple calculation shows that these operators satisfy a∗ (f )ψ = a(f )∗ ψ as well as the Canonical Anti-commutation Relations (CAR) {a(f ), a∗ (g)}ψ = (f, g)ψ, {a∗ (f ), a∗ (g)}ψ = 0, for all f, g ∈ h and ψ ∈ Γﬁn − (h). It immediately follows that a(f )ψ 2 + a∗ (f )ψ 2 = f 2 ψ 2 , from which we conclude that the closures of a(f ) and a∗ (f ) are bounded operators on Γ− (h). If we denote these extensions by the same symbols then a∗ (f ) = a(f )∗ and the canonical anti-commutation relations hold for all ψ ∈ Γ− (h). Exercise 4.7. Prove that a(f ) = a∗ (f ) = f (Hint: Use the CAR to compute (a∗ (f )a(f ))2 and the C ∗ -property of the norm). Compute the spectrum of a(f ), a∗ (f ) and a∗ (f )a(f ). Let u be a subspace of h, not necessarily closed, and denote by CAR(u) the C ∗ algebra generated by the family {a(f ) | f ∈ u}, i.e., the norm closure in B(Γ− (h)) of the linear span of monomials a∗ (f1 ) · · · a∗ (fn )a(fn+1 ) · · · a(fm ) with fj ∈ u. It follows from the CAR that this algebra has a unit. To any self-adjoint operator h on h we can associate the second quantization U t = Γ(eiht ) on the Fock space. By definition U t f1 ∧ · · · ∧ fn ≡ eith f1 ∧ · · · ∧ eith fn , and therefore U t is a strongly continuous unitary group on Γ− (h). Denote by τ t the corresponding group of spatial ∗- automorphisms of B(Γ− (h)).

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Exercise 4.8. Show that the generator H = dΓ(h) of U t is bounded if and only if h is trace class. Hence, by Exercise 4.5, if h is not trace class then τ t is not strongly continuous on B(Γ− (h)). Let me show that its restriction to CAR(u) is strongly continuous. It follows from Equ. (12) that τ t (a∗ (f )) = U t a∗ (f )U t∗ = a∗ (eith f ). Therefore, for any monomial A = a# (f1 ) · · · a# (fm ), where a# stands for either a or a∗ , we get a telescopic expansion τ t (A) − A m = a# (eith f1 ) · · · a# (eith fk−1 )a# (eith fk − fk )a# (fk+1 ) · · · a# (fm ), k=1

which, together with Exercise 4.7, leads to the estimate τ (A) − A m t

m−1 max fk max eith fk − fk .

1km

1km

We conclude that limt→0 τ t (A) − A = 0 for all such monomials and hence for arbitrary polynomials. Since these polynomials are norm dense in CAR(u) and τ t is isometric, the result follows from an ε/3-argument. The simplest non-trivial example of strongly continuous group of ∗-automorphisms of CAR(u) is the gauge group ϑt obtained by setting h = I, i.e., ϑt (a# (f )) = a# (eit f ).

(13)

The corresponding operator N ≡ dΓ(I) is the number operator. The one particle Hilbert space for an infinite d-dimensional Fermi gas is h = L2 (Rd , dx). The non-relativistic Hamiltonian is h = −∆. To any compact region Λ Rd one can associate the Hilbert space hΛ = L2 (Λ, dx) which is canonically embedded in h. Accordingly, the corresponding C ∗ -algebra CAR(hΛ ) can be identified with a subalgebra of CAR(h). Elements of , CAR(hΛ ), Aloc = ΛRd

are called local and we say that A ∈ CAR(hΛ ) is supported by Λ. Exercise 4.9. Show that Aloc is a dense ∗-subalgebra of CAR(h). From a physical point of view, local observables supported by disjoint subsets should be simultaneously measurable, i.e., they should commute. However, it follows from the fact that {AB, C } = A{B, C } − {A, C }B,

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that CAR(hΛ ) and CAR(hΛ ) anti-commute if Λ ∩ Λ = ∅ and will generally not commute. Thus the full CAR algebra is in a sense too big. To obtain an algebra fulfilling the above locality requirement, one introduces the so called even subalgebra CAR+ (h) generated by monomials of even degrees in a and a∗ . It follows from [AB, C D ] = {A, C }D B + A{B, C }D − C {D , A}B − AC {B, D }, that two elements of this subalgebra supported by disjoint subset commute. Another way to characterize the even subalgebra is to consider the ∗-morphism of CAR(h) defined by θ(a(f )) = −a(f ). Then one clearly has CAR+ (h) = {A ∈ CAR(h)|θ(A) = A}. Since τ t commutes with θ, it leaves CAR+ (h) invariant and (CAR+ (h), τ t ) is a C ∗ -dynamical system which describes an ideal Fermi gas. When the C ∗ -algebra A does not act naturally on a Hilbert space, the situation is more involved and Banach space techniques must be used. I will only mention a simple example based on the powerful technique of analytic vectors (see Chapter 3.1 in [11], Section 5 in [32] and Chapter 6.2 in [12] for a more systematic exposition of these techniques). Definition 4.10. Let T be an operator on a Banach space B. A vector x ∈ B is called analytic for T if x ∈ ∩n∈N D(T n ) and if there exists ρx > 0 such that the power series ∞ zn n T x n! n=0 defines an analytic function in {z ∈ C | |z| < ρx }. Note that the set of analytic vectors of an operator T is a subspace. If x is analytic for T one can define ∞ n t n T x, etT x ≡ n! n=0 for |t| < ρx . Moreover, if T is closed and t < ρx then it is easy to show that etT x is analytic for T and a simple manipulation of norm convergent series yields that esT etT x = e(s+t)T x as long as |s| + |t| < ρx . If the subspace of analytic vectors is dense in B and if one can prove that the linear operator etT defined in this way is bounded (this is usually done with the help of some dissipativity estimate), then it extends to all of B and using the group property its definition can be extended to all t ∈ R. Example 4.11. (Quantum spin systems) Let Γ be an infinite lattice (for example Γ = Zd , with d 1) and to each x ∈ Γ associate a copy hx of a finite dimensional Hilbert space h. For finite subsets Λ ⊂ Γ set hΛ ≡ ⊗x∈Λ hx and define the local C ∗ -algebra AΛ ≡ B(hΛ ).

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If Λ ⊂ Λ , the natural injection A !→ A ⊗ IhΛ \Λ allows to identify AΛ with a subalgebra of AΛ . Therefore, a C ∗ -norm can be defined on Aloc ≡ ∪Λ⊂Γ AΛ , the union being over finite subsets of Γ. Denote by A the C ∗ -algebra obtained as norm closure of Aloc . Each local algebra AΛ is identified with the corresponding subalgebra of A. C ∗ -algebras of this type are called uniformly hyperfinite (UHF). Interpreting h as the Hilbert space of a single spin, A describes the observables of a quantum spin system on the lattice Γ. An interaction is a map X !→ φ(X) which, to any finite subset X of the lattice Γ, associates a self-adjoint element φ(X) of AX describing the interaction energy of the degrees of freedom inside X. For example, in the spin interpretation, if X = {x} then φ(X) is the energy of the spin at x due to the coupling of its magnetic moment with an external magnetic field. If X = {x, y} then φ(X) is the coupling energy due to the pair interaction between the corresponding magnetic moments. Given an interaction φ, the local Hamiltonian for a finite region Λ ⊂ Γ is the self-adjoint element of AΛ given by HΛ ≡ φ(X), X⊂Λ

and a C ∗ -dynamical system is defined on A by τΛt (A) ≡ eitHΛ A e−itHΛ . Assume that the interaction φ has sufficiently short range, more precisely that φ σ ≡ sup φ(X) e2σ|X| < ∞, x∈Γ

Xx

for some σ > 0, where |X| denotes the cardinality of the subset X. We shall show that the limit τ t (A) ≡ lim τΛt (A), (14) Λ↑Γ

exists for all A ∈ A, t ∈ R and defines a strongly continuous group of ∗-morphisms. The generator of τΛt is the bounded derivation δΛ (A) = i[HΛ , A], hence we have a norm convergent expansion τΛt (A) =

∞ n t n δ (A). n! Λ n=0

For A ∈ AΛ0 , we can further write n (A) = i[φ(Xn ), i[φ(Xn−1 ), · · · , i[φ(X1 ), A] · · · ]]. δΛ X1 ,··· ,Xn ⊂Λ

(15)

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Since local algebras corresponding to disjoint subsets of Γ commute, this sum can be restricted by the condition Xj ∩ Λj−1 = ∅ with Λj−1 = Λ0 ∪ X1 · · · ∪ Xj−1 . n (A) We proceed to estimate the norm of δΛ n δΛ (A)

x1 ∈Λ0 X1 x1 x2 ∈Λ1 X2 x2

2n A |Λ0 | sup x1

2n A sup

x1 ,··· ,xn

|Λ1 | sup x2

X1 x1

···

X1 x1

···

2n A

xn ∈Λn−1 Xn xn

· · · |Λn−1 | sup

n

φ(Xi )

i=1 n

φ(Xi ) Xn xn i=1 n n |Xn |) φ(Xi ) . i=1

xn

X2 x2

(|Λ0 | + |X1 | + · · · +

Xn xn

Using the inequality e2σx (2σx)n /n! with x = |Λ0 |+|X1 |+· · ·+|Xn | we finally get the following uniform estimate in Λ n (A) ≤ δΛ

From this we conclude that

n (A) = δ (n) (A) = lim δΛ Λ

n! 2σ|Λ0 | e A φ nσ . σn

(16)

i[φ(Xn ), i[φ(Xn−1 ), · · · , i[φ(X1 ), A] · · · ]],

X1 ,··· ,Xn

exists and satisfies the same estimate (16). It is also clear from this argument that k lim δ (n) (δΛ (A)) = δ (n+k) (A). Λ

(17)

Introducing (16) into the expansion (15), we conclude that the limit τ t (A) = lim τΛt (A) = Λ

∞ n t (n) δ (A), n! n=0

(18)

exists for |t| < σ/ φ σ . Since τΛt is isometric from any local algebra AΛ0 into A, it follows that the limit t τ is norm continuous and extends by continuity to all of A. Furthermore, as a norm limit of ∗-morphisms, τ t is a ∗-morphism. For |s| + |t| < σ/ φ σ and A ∈ Aloc the continuity of τ t yields lim τ t (τΛs (A)) = τ t (τ s (A)), Λ

while Equ. (17) and the expansions (15)(18) lead, after a simple manipulation to lim τ t (τΛs (A)) = τ t+s (A). Λ

Thus τ t satisfies the local group property, and in particular is a ∗-isomorphism. Finally τ t can be extended to a group by setting τ nt0 +t = (τ t0 )n ◦ τ t and relation (14) then extends to all t ∈ R and A ∈ A.

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Example 4.12. (Continuous classical dynamical system) Let M be a compact metric space and ϕt a continuous dynamics on M . Then the space C(M ) of continuous functions on M is a commutative C ∗ -algebra with a unit on which the map τ t (f ) = f ◦ ϕt defines a strongly continuous group of ∗-automorphisms. 4.2 W ∗ -Dynamical Systems We have seen in Example 4.4 that even the simplest quantum mechanical system leads to a dynamics which is not strongly continuous when its Hamiltonian is unbounded. Thus, the notion of C ∗ -dynamical system is too restrictive for our purposes and we need to consider weaker topologies on the algebra of observables. Let M ⊂ B(H) be a von Neumann algebra. A group of ∗-automorphisms of M is σ-weakly continuous if for all A ∈ M the map t !→ τ t (A) is continuous in the σ-weak topology. This means that for any A ∈ M and any trace class operator T on H the map t !→ Tr(T τ t (A)) is continuous. Definition 4.13. A W ∗ -dynamical system is a pair (M, τ t ) where M is a von Neumann algebra acting on a Hilbert space H and τ t is a σ-weakly continuous group of ∗-automorphisms of M. Remark 4.14. More generally, M could be an abstract W ∗ -algebra, i.e., a C ∗ -algebra which is the dual Banach space of a Banach space M , and τ t a weak- continuous group of ∗-automorphisms. Since by Sakai theorem (see [42], Theorem 1.16.7) a W ∗ -algebra is ∗-isomorphic to a von Neumann subalgebra of B(H) for some Hilbert space H, I will only consider this particular situation. The predual M is then canonically identified with the quotient L1 (H)/M⊥ where L1 (H) is the Banach space of trace class operators on H and M⊥ = {T ∈ L1 (H) | Tr(T A) = 0 for all A ∈ M}, the annihilator of M. The weak- topology on M induced by M coincides with the σ-weak topology and elements of the predual corresponds to σ-weakly continuous linear functionals on M. We have seen in Subsection 2.1 (Corollary 2.12) that a ∗-automorphism of a von Neumann algebra is σ-weakly continuous. A σ-weakly continuous group of σweakly continuous linear operators is characterized by its generator, as in the strongly continuous case (see Chapter 3 in [11] for details). Thus, one can write τ t = etδ and there is a characterization, parallel to Proposition 4.3, of W ∗ -dynamical systems (see Theorem 3.2.51 in [11]). Proposition 4.15. Let M be a von Neumann algebra. A σ-weakly densely defined and closed linear operator δ on M generates a σ-weakly continuous group of ∗automorphisms of M if and only if: (i) δ is a ∗-derivation and 1 ∈ D(δ). (ii) Ran(Id +λδ) = M for all λ ∈ R \ {0}.

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(iii) A + λδ(A) A for all λ ∈ R and A ∈ D(δ). Example 4.16. (Finite quantum systems, continuation of Example 4.4) Consider now the case of an unbounded Hamiltonian H on the Hilbert space H. Clearly τ t (A) = eitH A e−itH defines a group of ∗-automorphisms of the von Neumann algebra M = B(H). For any unit vectors Φ, Ψ ∈ H the function t !→ (Φ, τ t (A)Ψ ) = (e−itH Φ, Ae−itH Ψ ) is continuous and uniformly bounded in t by A . If T is a trace class operator on H then one has T = n tn (Φn , · )Ψn with Ψn = Φn = 1 and n |tn | < ∞. It follows that tn (Φn , τ t (A)Ψn ), t !→ Tr(T τ t (A)) = n

is continuous (as a uniformly convergent series of continuous functions). Thus τ t is σ-weakly continuous and (M, τ t ) is a W ∗ -dynamical system. Example 4.17. (Ideal Bose gas) Let h be a Hilbert space and Γ+ (h) the Bosonic Fock space over h (see Section 2 in Lecture [36]). For f ∈ h, denote by 1 Φ(f ) = √ (a∗ (f ) + a(f )), 2 the self-adjoint Segal field operator. Since Φ(f ) is unbounded for f = 0, it is more convenient to use the unitary Weyl operators W (f ) = eiΦ(f ) . They define a projective representation of the additive group h on Γ+ (h) satisfying the Weyl relations W (f )W (g) = e−i Im(f,g)/2 W (f + g).

(19)

To any self-adjoint operator h on h we can, as in the Fermionic case, associate the second quantized strongly continuous unitary group U t = Γ(eith ) on Γ+ (h). The action of this group on Weyl operators is given by U t W (f )U t∗ = W (eith f ).

(20)

It follows from Equ. (19) that, for f = 0 and g = iθf / f 2 , W (g)∗ W (f )W (g) = e−iθ W (f ). This shows that the spectrum of W (f ) is the full unit circle and therefore W (f ) − W (g) = W (f − g) − e−i Im(f,g)/2 = 2, for f = g. This makes clear that in the Bosonic framework there is no chance to obtain a strongly continuous group from Equ. (20). On the other hand it follows from Exercise 4.16 that the von Neumann algebra W(h) = {W (f ) | f ∈ h} , together with the group τ t (A) = U t A U t∗ , ∗

form a W -dynamical system.

(21)

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Exercise 4.18. Show that the system {W (f ) | f ∈ h} is irreducible, i.e., that {W (f ) | f ∈ h} = CI. Conclude that W(h) = B(Γ+ (h)). (Hint: see Subsection 2.4 in Lecture [36]) 4.3 Invariant States Definition 4.19. If τ t is a group of ∗-automorphisms on a C ∗ -algebra A, a state µ on A is called τ t -invariant if µ ◦ τ t = µ for all t ∈ R. We denote by E(A, τ t ) ⊂ E(A) the set of τ t -invariant states. As in the case of classical dynamical systems, invariant states play a important role in the study of quantum dynamics. Theorem 4.20. Let τ t be a group of ∗-automorphisms of the C ∗ -algebra A. If there exists a state ω on A such that the function t !→ ω(τ t (A)) is continuous for all A ∈ A then E(A, τ t ) is a non-empty, convex and weak- compact subset of A . In particular, these conclusions hold if (A, τ t ) is a C ∗ - or W ∗ -dynamical system. Proof. To show that E(A, τ t ) is not empty, we follow the strategy of Exercise 3.1. For all A ∈ A consider the expression 1 T ωT (A) ≡ ω ◦ τ s (A) ds. T 0 Since the function s !→ ω ◦ τ s (A) is continuous, the integral is well defined and we clearly have ωT ∈ E(A) for all T > 0. Since E(A) is weak- compact, the net (ωT )T >0 has a weak- convergent subnet. The formula 1 t 1 T +t ωT (τ t (A)) = ωT (A) − ω ◦ τ t (A) ds + ω ◦ τ s (A) ds, T 0 T T leads to the estimate |ωT (τ t (A)) − ωT (A)| 2 A

|t| , T

from which it follows that the limit of any convergent subnet of (ωT )T >0 is τ t invariant. It is clear that the set of τ t -invariant states is convex and weak- closed. Definition 4.21. If τ t is a group of ∗-automorphisms of the von Neumann algebra M we denote by N (M, τ t ) ≡ E(M, τ t ) ∩ N (M) the set of normal τ t -invariant states. It immediately follows from the last paragraph in Subsection 2.1 that if ω ∈ N (M, τ t ) then τ t (sω ) = sω and τ t (zω ) = zω . We note that for a W ∗-dynamical system (M, τ t ) the compactness argument used in the proof of Theorem 4.20 breaks down if we replace E(M) by N (M). There is

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no general existence result for normal invariant states of W ∗ -dynamical systems. In fact, there exists W ∗-dynamical systems without normal invariant states (see Exercise 4.34). However, we shall see below that any invariant state of a C ∗ - or W ∗-dynamical system can be described as a normal invariant state of some associated W ∗-dynamical system. For this reason, normal invariant states play an important role in quantum dynamics. 4.4 Quantum Dynamical Systems Definition 4.22. If C is a C ∗ -algebra and τ t a group of ∗-automorphisms of C we define E(C, τ t ) ≡ {µ ∈ E(C, τ t ) | t !→ µ(A∗ τ t (A)) is continuous for all A ∈ C}. If µ ∈ E(C, τ ) we say that (C, τ t , µ) is a quantum dynamical system. Example 4.23. If (A, τ t ) is a C ∗ -dynamical system then E(A, τ ) = E(A, τ ) and (A, τ t , µ) is a quantum dynamical system for any τ t -invariant state µ. Example 4.24. If (M, τ t ) is a W ∗ -dynamical system then N (A, τ ) ⊂ E(A, τ ) and (M, τ t , µ) is a quantum dynamical system for any τ t -invariant normal state µ. Exercise 4.25. Show that E(C, τ t ) is a convex, norm closed subset of E(C, τ t ). In this subsection we shall study the GNS representation of a quantum dynamical system. We start with the following extension of the GNS construction. Lemma 4.26. Let (C, τ t , µ) be a quantum dynamical system and denote the GNS representation of C associated to µ by (Hµ , πµ , Ωµ ). Then there exists a unique selfadjoint operator Lµ on Hµ such that (i) πµ (τ t (A)) = eitLµ πµ (A)e−itLµ for all A ∈ C and t ∈ R. (ii) Lµ Ωµ = 0. Proof. For fixed t ∈ R one easily checks that (Hµ , πµ ◦ τ t , Ωµ ) is a GNS representation of C associated to µ. By unicity of the GNS construction there exists a unitary operator Uµt on Hµ such that, for any A ∈ C, one has Uµt πµ (A)Ωµ = πµ (τ t (A))Ωµ ,

(22)

Uµt Ωµ = Ωµ .

(23)

and in particular For t, s ∈ R we have Uµt Uµs πµ (A)Ωµ = Uµt πµ (τ s (A))Ωµ = πµ (τ t+s (A))Ωµ = Uµt+s πµ (A)Ωµ , and the cyclic property of Ωµ yields that Uµt is a unitary group on Hµ . Using Equ. (7) it follows from the continuity of the map

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t !→ (πµ (A)Ωµ , Uµt πµ (A)Ωµ ) = µ(A∗ τ t (A)), that Uµt is a strongly continuous. By Stone theorem Uµt = eitLµ for some self-adjoint operator Lµ and property (ii) follows from Equ. (23). Finally, for A, B ∈ C we get Uµt πµ (A)πµ (B)Ωµ = πµ (τ t (A))πµ (τ t (B))Ωµ = πµ (τ t (A))Uµt πµ (B)Ωµ , and property (i) follows from the cyclic property of Ωµ . To prove the uniqueness of Lµ note that Equ. (22) uniquely determines Uµt and that conditions (i) and (ii) imply that eitLµ satisfies (22). Recall from subsection 2.2 that µ ˆ(A) = (Ωµ , AΩµ ) defines a centrally faithful normal state on the enveloping von Neumann algebra Cµ = πµ (C) . Moreover, by property (i) of Lemma 4.26, the σ-weakly continuous group of ∗-automorphisms of B(Hµ ) defined by (24) τˆµt (A) ≡ eitLµ Ae−itLµ , leaves πµ (C) and hence its σ-weak closure Cµ invariant. Thus, (Cµ , τˆµt ) is a W ∗ dynamical system. By property (ii) of Lemma 4.26, µ ˆ is τˆµt -invariant. Definition 4.27. A quantum dynamical system (C, τ t , µ) is in normal form if (i) C is a von Neumann algebra on a Hilbert space H. (ii) τ t (A) = eitL Ae−itL for some self-adjoint operator L on H. (iii) µ(A) = (Ω, AΩ) for some unit vector Ω ∈ H. (iv) CΩ = H. (v) LΩ = 0. We denote by (C, H, L, Ω) such a system. The above considerations show that to any quantum dynamical system (C, τ t , µ) we can associate a quantum dynamical system in normal form (Cµ , Hµ , Lµ , Ωµ ). Definition 4.28. (πµ , Cµ , Hµ , Lµ , Ωµ ) is the normal form of the quantum dynamical system (C, τ t , µ). The operator Lµ is its µ-Liouvillean. The normal form of a quantum dynamical system is uniquely determined, up to unitary equivalence. Definition 4.29. Two quantum dynamical systems (C, τ t , µ), (D, σ t , ν) are isomorphic if there exists a ∗-isomorphism φ : C → D such that φ ◦ τ t = σ t ◦ φ for all t ∈ R and µ = ν ◦ φ. Exercise 4.30. Show that two isomorphic quantum dynamical systems share the same normal forms.

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Remark 4.31. If ω ∈ E(C, τ t ) and µ ∈ N (C, ω) ∩ E(C, τ t ) then µ = µ ˜ ◦ πω for some µ ˜ ∈ N (Cω ) and therefore µ(A∗ τ t (A)) = µ ˜(πω (A)∗ πω (τ t (A))) = µ ˜(πω (A)∗ eitLω πω (A)e−itLω ), is a continuous function of t ∈ R. It follows that µ ∈ E(C, τ t ). Denote by τˆωt the W ∗ -dynamics on Cω generated by the ω-Liouvillean Lω . Since µ is τ t -invariant one ˜(A) for all A ∈ πω (C) and by continuity µ ˜ is τˆωt -invariant. Let has µ ˜(ˆ τωt (A)) = µ πµ|ω : Cω → Cµ be the ∗-morphism of Theorem 2.30. From the identity τˆµt ◦ πµ|ω ◦ πω = τˆµt ◦ πµ = πµ ◦ τ t = πµ|ω ◦ πω ◦ τ t = πµ|ω ◦ τˆωt ◦ πω , it follows by σ-weak continuity that τˆµt ◦ πµ|ω = πµ|ω ◦ τˆωt ,

(25)

and since zµ|ω is invariant under τˆωt τˆµt ◦ π ˆµ|ω = π ˆµ|ω ◦ τˆωt .

(26)

We conclude that π ˆµ|ω is an isomorphism between the quantum dynamical systems ˜) and (Cµ , τˆµt , µ ˆ). (zµ|ω Cω , τˆωt , µ The normal form turns out to be a very useful tool in the study of quantum dynamics since it provides a unifying framework in which both C ∗ - and W ∗ -systems can be handled on an equal footing. Example 4.32. (Finite quantum systems) Let (M, τ t ) be the C ∗ - or W ∗ -dynamical system constructed from the Hilbert space H and the Hamiltonian H, i.e., M ≡ B(H) and τ t (A) ≡ eitH Ae−itH (Examples 4.4 and 4.16). A density matrix ρ on H such that e−itH ρ eitH = ρ defines a normal, τ t -invariant state µ(A) = Tr(ρA). If H has non-empty point spectrum, such states are easily obtained as mixtures of eigenstates of H. (M, τ t , µ) is a quantum dynamical system. Its normal form can be described in the following way. Set G ≡ Ran ρ, denote by ι : G → H the canonical injection and by L2 (G, H) the set of Hilbert-Schmidt operators from G to H. Then Hµ ≡ L2 (G, H), is a Hilbert space with inner product (X, Y ) ≡ Tr(X ∗ Y ). Since P ≡ ιι∗ is the orthogonal projection of H on G, a simple calculation shows that Ωµ ≡ ρ1/2 ι is a unit vector in Hµ . Setting πµ (A)X = AX, for all X ∈ Hµ , the map A !→ πµ (A) defines a faithful representation of M on Hµ . Since (X, πµ (A)Ωµ ) = Tr((ρ1/2 ιX ∗ )A), one immediately checks that πµ (M)Ωµ

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is dense in Hµ . Finally a simple calculation shows that for all A ∈ M one has µ(A) = (Ωµ , πµ (A)Ωµ ). Since µ is normal and centrally faithful3 the enveloping von Neumann algebra is πµ (M) and N (M, µ) = N (M). Let · denote an arbitrary complex conjugation on the Hilbert space G. Then ϕ ⊗ ψ¯ !→ ϕ(ψ, · ) extends to a unitary map U : H ⊗ G → Hµ such that πµ (A) = U (A ⊗ I)U ∗ . Thus the enveloping von Neumann algebra πµ (M) is unitarily equivalent to M ⊗ I. Exercise 4.33. Show that the µ-Liouvillean of the preceding example is given by

eitLµ X = eitH Xe−itH , where H is the restriction of H to G. What is the spectrum of Lµ ? Exercise 4.34. Show that if H has purely continuous spectrum, there is no trace class operator commuting with H, hence no normal invariant state. Example 4.35. (Ideal Bose gas, continuation of Example 4.17) Let D ⊂ h be a subspace and denote by CCR(D) the C ∗ -algebra generated by the Weyl system W (D) ≡ {W (f ) | f ∈ D}, i.e., the norm closure of the linear span of W (D). Since we can replace h by D we may assume, without loss of generality, that D is dense in h. Let H be a Hilbert space and π : W (D) → B(H) be such that π(W (f )) is unitary and (27) π(W (f ))π(W (g)) = e−i Im(f,g)/2 π(W (f + g)), for all f, g ∈ D. Then one can show (see Theorem 6 in Lecture [36]) that π has a unique extension to an injective ∗-morphism from CCR(D) into B(H). Thus, a representation of CCR(D) is completely determined by its restriction to W (D). A representation (H, π) of CCR(D) is regular if the map λ !→ π(W (λf )) is strongly continuous for all f ∈ D. Regular representations are physically appealing since by Stone theorem there exists a self-adjoint operator Φπ (f ) on H such that π(W (f )) = eiΦπ (f ) for all f ∈ D. The operator Φπ (f ) is the Segal field operator in the representation π. The corresponding creation and annihilation operators √ are obtained by linear combination, for example aπ (f ) = (Φπ (f ) + iΦπ (if ))/ 2. A state ω on CCR(D) is called regular if its GNS representation is regular. Since the finite linear combinations of elements of W (D) are norm dense in CCR(D), a state ω on this C ∗ -algebra is completely determined by its characteristic function D * f !→ Sω (f ) ≡ ω(W (f )). Clearly, if the function λ !→ Sω (λf ) is continuous for all f ∈ D, the state ω is regular and we denote by Φω (f ), aω (f ) and a∗ω (f ) the corresponding operators. The state ω is said to be C n , C ∞ respectively analytic if the function λ !→ Sω (λf ) has this smoothness near λ = 0. If ω is an analytic state, it is easy to see that Sω (λf ) 3

Because M is a factor.

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is actually analytic in an open strip around the real axis and that Ωω is an analytic vector for all field operators Φω (f ). The characteristic function Sω (f ) and therefore the state ω itself are then completely determined by the derivatives ∂λn Sω (λf )|λ=0 , or equivalently by the family of correlation functions Wm,n (g1 , · · · , gm ; f1 , · · · , fn ) = (Ωω , a∗ω (gm ) · · · a∗ω (g1 )aω (f1 ) · · · aω (fn )Ωω ). Characteristic functions of regular states on CCR(D) are characterized by the the following result (see Lecture [36] for a proof). Theorem 4.36. (Araki-Segal) A map S : D → C is the characteristic function of a regular state ω on CCR(D) if and only if (i) S(0) = 1. (ii) The function λ !→ S(λf ) is continuous for all f ∈ D. (iii) For all integer n 2, all f1 , · · · , fn ∈ D and all z1 , · · · , zn ∈ C one has n

S(fj − fk ) e−i Im(fj ,fk )/2 z¯j zk 0.

j,k=1

If the subspace D is invariant under the one-particle dynamics, i.e., if e−ith D ⊂ D for all t ∈ R, then it follows from Equ. (20) that the group τ t defined by Equ. (21) leaves W (D) and hence its closed linear span CCR(D) invariant. Thus, τ t is a group of ∗-automorphisms of CCR(D). Note that (CCR(D), τ t ) is neither a C ∗ dynamical (τ t is not strongly continuous by Example 4.17), nor a W ∗ -dynamical system (CCR(D) is not a von Neumann algebra). Suppose that the map S : D → C satisfies the conditions of Theorem 4.36 and (iv) S(eith f ) = S(f ) for all f ∈ D and t ∈ R. (v) limt→0 S(eith f − f ) = 1 for all f ∈ D. Then the corresponding regular state ω is τ t -invariant and ω(W (f )∗ τ t (W (f ))) = S(eith f − f ), is continuous at t = 0. By the Cauchy-Schwarz inequality the same is true of ω(W (g)∗ τ t (W (f ))) for any f, g ∈ D. Since the linear span of W (D) is norm dense in CCR(D) the group property of τ t allows to conclude that for all A ∈ CCR(D) the function t !→ ω(A∗ τ t (A)) is continuous. Therefore, ω ∈ E(CCR(D), τ t ) and (CCR(D), τ t , ω) is a quantum dynamical system. Definition 4.37. A state on CCR(D) is called quasi-free if its characteristic function takes the form 2 1 1 (28) S(f ) = e− 4 f − 2 ρ[f ] , where ρ is a closable non-negative quadratic form on D.

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We shall denote by ωρ the quasi-free state characterized by Equ. (28). We shall also use the symbol ρ to denote the non-negative self-adjoint operator associated with the quadratic form ρ. Thus, one has D ⊂ D(ρ1/2 ) and ρ[f ] = ρ1/2 f 2 for f ∈ D. The quasi-free state ωρ is clearly regular and analytic. If e−ith ρ eith = ρ for all t ∈ R then condition (iv) is satisfied and ωρ is τ t -invariant. Moreover, since ρ[eith f − f ] = 4 sin(th/2)ρ1/2 f 2 condition (v) is also satisfied and ωρ ∈ E(CCR(D), τ t ). To describe the normal form of (CCR(D), τ t , ωρ ) let g ≡ Ran ρ ⊂ h, denote by ι : g → h the canonical injection and set Hωρ ≡ L2 (Γ+ (g), Γ+ (h)), the set of Hilbert-Schmidt operators from Γ+ (g) to Γ+ (h) with scalar product (X, Y ) ≡ Tr(X ∗ Y ). For g ∈ g denote by W (g) the Weyl operator in Γ+ (g). For f ∈ D define πωρ (W (f )) : X !→ W ((I + ρ)1/2 f )XW (ι∗ ρ1/2 f )∗ . Using the CCR (19), one easily checks that πωρ (W (f )) is unitary and satisfies Equ. (27). Thus, πωρ has a unique extension to a representation of CCR(D) in Hωρ . Denote by Ω, Ω the Fock vacua in Γ+ (h), Γ+ (g) and set Ωωρ = Ω(Ω , · ). Exercise 4.38. Show that Ωωρ is cyclic for πωρ (C). (Hint: For X = Ψ (Φ, · ) with Ψ = a∗ (fn ) · · · a∗ (f1 )Ω and Φ = a∗ (gm ) · · · a∗ (g1 )Ω and f ∈ h compute ∂λ πωρ (W (λf ))X|λ=0 and ∂λ πωρ (W (iλf ))X|λ=0 .) A simple calculation shows that ωρ (W (f )) = (Ωωρ , πωρ (W (f ))Ωωρ ). Finally, the ωρ -Liouvillean is given by

eitLωρ X ≡ Γ(eith )XΓ(e−ith ), where h is the restriction of h to g. The representation of CCR(D) obtained in this way is called Araki-Woods representation [1]. We refer the reader to [36] for a more detailed discussion and to [18] for a thorough exposition of the representation theory of canonical commutation relations. 4.5 Standard Forms Recall that a subset C of a Hilbert space H is a cone if tψ ∈ C for all t ≥ 0 and ψ ∈ C. The dual of a cone C is the closed cone C# ≡ {ψ ∈ H | (φ, ψ) ≥ 0 for any φ ∈ C}. A cone C is self-dual if C# = C. A self-dual cone is automatically closed.

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Definition 4.39. A von Neumann algebra M ⊂ B(H) is said to be in standard form if there exist a anti-unitary involution J on H and a self-dual cone C ⊂ H such that: (i) JMJ = M . (ii) JΨ = Ψ for all Ψ ∈ C. (iii) AJAC ⊂ C for all A ∈ M. (iv) JAJ = A∗ for all A ∈ M ∩ M . We shall denote by (M, H, J, C) a von Neumann algebra in standard form. Theorem 4.40. Any von Neumann algebra M has a faithful representation (H, π) such that π(M) is in standard form. Moreover, this representation is unique up to unitary equivalence. Sketch of the proof. If M is separable, i.e., if any family of mutually orthogonal projections in M is finite or countable then M has a normal faithful state ω (Proposition 2.5.6 in [11]). The associated GNS construction provides a faithful representation (Hω , πω , Ωω ) of M and it follows from Tomita-Takesaki theory (Chapter 4 in Lecture [7]) that πω (M) is in standard form. The anti-unitary involution J is the modular conjugation and the self-dual cone C is the natural cone {AJAΩω | A ∈ Mω } (see Section 4.3 in Lecture [7]). This construction applies in particular to any von Neumann algebra over a separable Hilbert space which is the case most often encountered in physical applications. In the general case, the construction is similar to the above one, substituting faithful normal states with faithful normal semi-finite weights. The general theory of standard forms was developed by Haagerup [23] following the works of Araki [3] and Connes [14] (see also [45], where standard forms are called hyper-standard). Von Neumann algebras in standard form have two important properties which are of crucial importance in the study of quantum dynamical systems. The first one concerns normal states. Theorem 4.41. Let (M, H, J, C) be a von Neumann algebra in standard form and for any unit vector Ψ ∈ H denote by ωΨ ∈ N (M) the corresponding vector state ωΨ (A) = (Ψ, AΨ ). Then the map {Φ ∈ C | Φ = 1} → N (M) Ψ !→ ωΨ is an homeomorphism (for the norm topologies). In particular, for any normal state ν on M, there is a unique unit vector Ψν ∈ C such that ν = ωΨν . We call Ψν the standard vector representative of ν. Moreover, for any unit vectors Ψ, Φ ∈ C one has: (i) Ψ − Φ 2 ωΨ − ωΦ Ψ − Φ Ψ + Φ . (ii) Ψ is cyclic for M ⇐⇒ Ψ is cyclic for M ⇐⇒ ωΨ is faithful. (iii) More generally MΨ = JM Ψ .

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Proof. I shall not prove the fact that Ψ !→ ωΨ is an homeomorphism and the first inequality in (i) since this requires rather long and involved arguments. A proof can be found for example in [11]. The Second inequality in (i) follows from the polarization identity. The first equivalence in (ii) is a special case of (iii) which is a direct consequence of (i) and (ii) in Definition 4.39. The second equivalence in (ii) is a special case of the last statement of Lemma 2.17. The second important property of von Neumann algebras in standard form has to do with the unitary implementation of ∗-automorphisms. To formulate this property let me introduce the following definition. Definition 4.42. Let (M, H, J, C) be a von Neumann algebra in standard form. A unitary operator U on H is called standard if the following holds: (i) U C ⊂ C. (ii) U ∗ MU = M. Obviously, the set of standard unitaries form a subgroup of the unitary group of H. It is not hard to see that this subgroup is closed in the strong topology of B(H). The following is essentially a rewriting of Corollary 4 in Lecture [7]. Theorem 4.43. Let (M, H, J, C) be a von Neumann algebra in standard form. Denote by Us the group of standard unitaries of H equipped with the strong topology and by Aut(M) the group of ∗-automorphisms of M with the topology of pointwise σ-weak convergence. For any U ∈ Us denote by τU the corresponding spatial ∗-automorphism τU (A) = U AU ∗ . Then the map Us → Aut(M) U !→ τU is an homeomorphism. In particular, for any ∗-automorphism σ of M there is a unique standard unitary U such that σ = τU . We call U the standard implementation of σ. Moreover, for any U ∈ Us one has: (i) [U, J] = 0. (ii) U M U ∗ = M . (iii) U ∗ Ψω = Ψω◦τU for all ω ∈ N (M). Definition 4.44. A quantum dynamical system (C, τ t , µ) is in standard form if (i) C is a von Neumann algebra in standard form (C, H, J, C). (ii) (C, τ t ) is a W ∗ -dynamical system. (iii) µ ∈ N (C, τ t ). Suppose that (C, τ t , µ) is in standard form. By Theorem 4.43, τ t has a standard implementation U t for each t ∈ R. Since t !→ τ t (A) is σ-weakly continuous, U t is strongly continuous. Therefore, there exists a self-adjoint operator L on H such

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that U t = eitL . By Theorem 4.41, there exists a unique vector Φ ∈ C such that µ(A) = (Φ, AΦ). Moreover, it follows from properties (i)-(iii) of Theorem 4.43 that JL + LJ = 0

(29)

eitL C e−itL = C ,

(30)

LΨω = 0,

(31)

for all t ∈ R and all ω ∈ N (C, τ t ). Definition 4.45. We denote by (C, H, J, C, L, Φ) a quantum dynamical system in standard form and we call L its standard Liouvillean. As an immediate consequence of this definition we have the Proposition 4.46. If the quantum dynamical system (C, τ t , µ) is in standard form (C, H, J, C, L, Φ), its standard Liouvillean is the unique self-adjoint operator L on H such that, for all t ∈ R and all A ∈ C one has (i) e−itL C ⊂ C. (ii) eitL Ae−itL = τ t (A). Let (C, τ t , µ) be a quantum dynamical system, (πµ , Cµ , Hµ , Lµ , Ωµ ) its normal form and (H, π) a standard representation of Cµ . Then M ≡ π(Cµ ) is a von Neumann algebra in standard form (M, H, J, C). Note that since π is faithful it is σ-weakly continuous by Corollary 2.12. The same remark apply to its inverse π −1 : M → Cµ . Let η ≡ π ◦ πµ be the induced representation of C in H. Since πµ (C) is σ-weakly dense in Cµ we have M = η(C) . For A ∈ M we define σ t (A) ≡ π(eitLµ π −1 (A)e−itLµ ). It follows that for A ∈ C one has σ t (η(A)) = π(eitLµ π −1 (η(A))e−itLµ ) = π(eitLµ πµ (A)e−itLµ ) = π(πµ (τ t (A))) = η(τ t (A)). The group σ t defines a W ∗ -dynamical system on M. It has a standard implementation with standard Liouvillean L. Finally we remark that ω ≡ (Ωµ , π −1 ( · )Ωµ ) ∈ N (M, σ t ) satisfies ω ◦ η = µ. Denote by Φ ∈ C its standard vector representative. It follows that (M, σ t , ω) is in standard form. Definition 4.47. We say that (η, M, H, J, C, L, Φ) is the standard form of the quantum dynamical system (C, τ t , µ) and that L is its standard Liouvillean.

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If (π1 , D1 , K1 , M1 , Ω1 ) and (π2 , D2 , K2 , M2 , Ω2 ) are two normal forms of the quantum dynamical system (C, τ t , µ) then, by the unicity of the GNS representation, there exists a unitary U : K1 → K2 such that π2 = U π1 U ∗ , Ω2 = U Ω1 and M2 = U M1 U ∗ . Let (η1 , H1 ) and (η2 , H2 ) be standard representations of D1 and D2 and denote by (η1 ◦π1 , M1 , H1 , J1 , C1 , L1 , Φ1 ) and (η2 ◦π2 , M2 , H2 , J2 , C2 , L2 , Φ2 ) the corresponding standard forms of (C, τ t , µ). Since (η2 (U · U ∗ ), H2 ) is another standard representation of D1 , there exists a unitary V : H1 → H2 such that η2 (U AU ∗ ) = V η1 (A)V ∗ . It follows that η2 ◦π2 = V η1 (U ∗ π2 U )V ∗ = V η1 ◦π1 V ∗ . Thus, the standard form and in particular the standard Liouvillean of a quantum dynamical system are uniquely determined, up to unitary equivalence. The next proposition elucidate the relation between the µ-Liouvillean and the standard Liouvillean. It also shows that in many applications (see Lemma 5.11 below) the two coincide. Proposition 4.48. Let (C, τ t , µ) be a quantum dynamical system with normal form ˆ ≡ (Ωµ , ( · )Ωµ ) is faithful on Cµ then this (πµ , Cµ , Hµ , Lµ , Ωµ ). If the state µ von Neumann algebra is in standard form (Cµ , Hµ , J, C) and the standard form of (C, τ t , µ) is given by (πµ , Cµ , Hµ , J, C, Lµ , Ωµ ). In particular Lµ is its standard Liouvillean. Proof. If µ ˆ is faithful then, as mentioned in the sketch of the proof of Theorem 4.40, Cµ is in standard form. J is the modular conjugation associated to Ωµ and C is the associated natural cone. In particular, Ωµ is the standard vector representative of µ ˆ. If L is the standard Liouvillean then LΩµ = 0 by Equ. (31) hence L is the µ-Liouvillean. More generally, the relation between µ-Liouvillean and standard Liouvillean is given by the following result. Proposition 4.49. Let (C, τ t , µ) be a quantum dynamical system with standard form (η, M, H, J, C, L, Φ). Then the subspace K ≡ MΦ ⊂ H reduces the standard Liouvillean L and the µ-Liouvillean Lµ is (unitarily equivalent to) the restriction of L to K. In particular, one has σ(Lµ ) ⊂ σ(L). Proof. We reconstruct a normal form of (C, τ t , µ) out of its standard form. Recall that η : C → M is a representation such that M = η(C) , µ(A) = (Φ, η(A)Φ) and η(τ t (A)) = eitL η(A)e−itL . Denote by k : K → H the canonical injection. Since K is invariant under M, φ(A) ≡ k ∗ η(A)k defines a representation of C on K. By the von Neumann density theorem η(C) is σ-strongly dense in M and one has φ(C)Φ = k ∗ η(C)Φ = k ∗ MΦ = K. Finally µ(A) = (Φ, η(A)Φ) = (Φ, φ(A)Φ). Thus (K, φ, Φ) is the required GNS representation. Since eitL η(A)Φ = η(τ t (A))Φ the subspace K reduces L and the restriction of L to K is the µ-Liouvillean. If (C, τ t , ω) is a quantum dynamical system and µ is a τ t -invariant ω-normal state on C then (C, τ t , µ) is also a quantum dynamical system. The standard Liouvilleans of these two systems are not independent. Their relation is explicited in the next Theorem.

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Theorem 4.50. Let (C, τ t , ω) be a quantum dynamical system with standard form (η, M, H, J, C, L, Φ) and µ a τ t -invariant ω-normal state. Denote by P ∈ Z(M) the central support of the normal extension of µ to M. The standard Liouvillean of the quantum dynamical system (C, τ t , µ) is (unitarily equivalent to) the restriction of L to the subspace P H. In particular, its spectrum is contained in σ(L). Proof. We construct a standard form (ηµ , Mµ , Hµ , Jµ , Cµ , Lµ , Φµ ) of (C, τ t , µ). Since µ is ω-normal there exists µ ˜ ∈ N (Cω ) with central support zµ|ω such that µ = µ ˜ ◦ πω . There is also a ∗-isomorphism π : Cω → M such that π ◦ πω = η. ¯ ◦ η. We The normal extension of µ to M is µ ¯ ≡ µ ˜ ◦ π −1 so that µ = µ denote its standard vector representative by Φµ . It follows from Lemma 6 that P = [M ∨ M Φµ ] = π(zµ|ω ) and in particular that Φµ ∈ Hµ ≡ P H. By Remark 2.22 one has M = P M ⊕ (I − P )M and we can identify Mµ ≡ P M with a von Neumann algebra on Hµ . Clearly ηµ (A) ≡ P η(A) defines a representation of C in Hµ . By Property (iv) of Definition 4.39 one has JP J = P so that the subspace Hµ reduces J. It follows that the restriction Jµ of J to this subspace is an anti-unitary involution. Finally, Cµ ≡ P C is a cone in Hµ and Cˆµ = {ψ ∈ Hµ | (ψ, P φ) ≥ 0 for all φ ∈ C} = {ψ ∈ Hµ | (P ψ, φ) ≥ 0 for all φ ∈ C} = {ψ ∈ Hµ | (ψ, φ) ≥ 0 for all φ ∈ C} = Hµ ∩ C. Let me show that Hµ ∩C = Cµ . On the one hand P = JP J yields P = P 2 = P JP J and hence P C = P JP JC = P JP C ⊂ C by Properties (ii) and (iii) of Definition 4.39. But since P C ⊂ Hµ one concludes that P C ⊂ Hµ ∩ C. On the other hand if ψ ∈ Hµ ∩ C then ψ = P ψ ∈ C and hence ψ ∈ P C proving the claim and hence the fact that Cµ is self-dual. We have to show that Properties (i)-(iv) of Definition 4.39 are satisfied. By Remark 2.22, Mµ = P M and Property (i) follows from Jµ Mµ Jµ = JP MP J = P JMJP = P M P = Mµ . To prove Property (ii) note that for ψ ∈ Cµ ⊂ C we have Jµ ψ = Jψ = ψ. For A ∈ Mµ we further have A = AP = P A = P AP and hence AJµ ACµ = P (AP JAP C) ⊂ P C = Cµ which proves Property (iii). By Remark 2.22, Z(Mµ ) = P Z(M) so that if A ∈ Z(Mµ ) we have Jµ AJµ = JP AP J = (P AP )∗ = P A∗ P = A∗ and Property (iv) is verified. −1 ˆµ|ω By Theorem 2.30, π ˆµ|ω : zµ|ω Cω → Cµ is a ∗-isomorphism. Hence φ ≡ π ◦ π is a faithful representation of Cµ in Hµ such that φ(Cµ ) = π(zµ|ω Cω ) = P M = Mµ and (Mµ , Hµ , Jµ , Cµ ) is indeed a standard form of Cµ . To determine the standard Liouvillean note that since µ ¯ is an invariant state its central support P is also invariant eitL P e−itL = P. −1 ◦ It follows that P reduces the standard Liouvillean L. By Equ. (26), we have π ˆµ|ω −1 t t ˆµ|ω from which we derive τˆµ = τˆω ◦ π −1 φ ◦ τˆµt (A) = π ◦ τˆωt ◦ π ˆµ|ω (A) = eitL φ(A)e−itL .

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Finally, since eitL Cµ = eitL P C = P eitL C = P C = Cµ we conclude that the standard Liouvillean of τˆµt is the restriction of L to Hµ . Example 4.51. (Standard form of a finite quantum system) Consider the quantum dynamical system (M, τ t , µ) of Example 4.32 and suppose that the Hilbert space H is separable. To construct a standard form let ρ0 be a density matrix on H such that Ker ρ0 = {0}. It follows that Ran ρ0 = H and Lemma 2.17 yields that the normal state ω0 (A) ≡ Tr(ρ0 A) is faithful. The GNS representation of M corresponding to ω0 is given, according to Example 4.32, by Hω0 = L2 (H), πω0 (A)X = AX, 1/2

Ωω0 = ρ0 . By Lemma 2.23 this representation is faithful and by Lemma 2.28 the corresponding enveloping von Neumann algebra is Mω0 = πω0 (M). It follows that ω ˆ 0 is faithful on Mω0 and hence, by Proposition 4.48, that Mω0 is in standard form. Indeed, one easily checks that J : X !→ X ∗ and C ≡ {X ∈ L2 (H) | X ≥ 0} satisfy all the conditions of Definition 4.39. By Proposition 4.46, the standard Liouvillean L is given by eitL X = eitH Xe−itH , since this is a unitary implementation of the dynamics eitL πω0 (A)e−itL X = eitH (Ae−itH XeitH )e−itH = πω0 (τ t (A))X, which preserves the cone C. The standard vector representative of the invariant state µ(A) = Tr(ρA) is ρ1/2 ∈ C. One easily checks that the map X !→ X|G is isometric from Mω0 ρ1/2 into L2 (G, H) and has a dense range. It extends to a unitary map U between K ≡ Mω0 ρ1/2 and Hµ = L2 (G, H) such that, according to Proposition 4.49 U L|K U ∗ = Lµ . 4.6 Ergodic Properties of Quantum Dynamical Systems My aim in this subsection is to extend to quantum dynamics the definitions and characterizations of the ergodic properties introduced in Subsection 3.1. Since there is no obvious way to formulate a quantum individual ergodic theorem (Theorem 3.3), I shall use the characterization (iv) of Proposition 3.6 to define ergodicity. Mixing will then be defined using Definition 3.8. To do so I only need to extend the notion of absolute continuity of two measures to two states on a C ∗ -algebra. Example 4.52. Consider the classical C ∗ -dynamical system (C(M ), τ t ) of Example 4.12 and let µ be a τ t -invariant Baire probability measure on M . To construct the

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associated GNS representation denote by µ ˆ the regular Borel extension of µ and set µ). The C ∗ -algebra C(M ) is represented on this Hilbert space as Hµ ≡ L2 (M, dˆ πµ (f ) : ψ !→ f ψ. Since C(M ) is dense in L2 (M, dµ) the vector Ωµ = 1 is cyclic. Finally, one has µ(x) = µ(f ). (Ωµ , πµ (f )Ωµ ) = f (x) dˆ We note in passing that the corresponding enveloping von Neumann algebra is µ). πµ (C(M )) = L∞ (M, dˆ If ν is a µ-normal state on C(M ) then there exists a density matrix ρ on Hµ such that ν(f ) = Tr(ρπµ (f )). Let pn ψn (ψn , · ), ρ= n

be the spectral representation of ρ. It follows that F (x) ≡ pn |ψn (x)|2 ∈ L1 (M, dˆ µ), n

and hence ν(f ) =

F (x)f (x) dˆ µ(x).

Thus, if ν is µ-normal its regular Borel extension νˆ is absolutely continuous with respect to µ ˆ. dˆ ν 1/2 Reciprocally, if νˆ + µ ˆ and G = ( dˆ one has µ) ν(f ) =

f

dˆ ν dˆ µ = (G, πµ (f )G), dˆ µ

and we conclude that ν is µ-normal. This example shows that for Abelian C ∗ - or W ∗ -algebras absolute continuity is equivalent to relative normality. If A ⊂ M is a measurable set then its characteristic function χA , viewed as a multiplication operator on Hµ , is an orthogonal projection. In fact all orthogonal projections in πµ (C(M )) are easily seen to be of this form. Since the absolute continuity of the Borel measure νˆ with respect to µ ˆ means that µ ˆ(χA ) = 0 implies νˆ(χA ) = 0 we see that if ν is µ-normal then sν ≤ sµ . As the following example shows this is not necessarily true in the non-Abelian case. Example 4.53. Let M ≡ B(H) with dim H > 1 and µ ≡ (ψ, · ψ) for some unit vector ψ ∈ H. As already remarked in Example 4.32, any normal state on M is µnormal. In particular if ρ is a density matrix such that Ker ρ = {0} then ν ≡ Tr(ρ · ) is a faithful µ-normal state and hence I = sν > sµ = ψ(ψ, · ).

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Considering the special case of a W ∗ -dynamical system (M, τ t ) equipped with a normal invariant state µ, the following argument shows that the condition sν ≤ sµ is necessary if we wish to define the ergodicity of the quantum dynamical system (M, τ t , µ) by the condition (iv) of Proposition 3.6, i.e., 1 T →∞ T

T

ν ◦ τ t (A) dt = µ(A),

lim

(32)

0

for all A ∈ M. Indeed, since µ is τ t -invariant we have τ t (sµ ) = sµ and Equ. (32) leads to 1 T ν ◦ τ t (sµ ) dt = µ(sµ ) = 1, ν(sµ ) = lim T →∞ T 0 which means that sν ≤ sµ . These considerations motivate the following definition. Definition 4.54. If ν, µ ∈ E(C) are such that ν ∈ N (C, µ) and sν ≤ sµ then we say that ν is absolutely continuous with respect to µ and we write ν + µ. We also set S(C, µ) ≡ {ν ∈ E(C) | ν + µ}. When dealing with the set S(C, µ), the following lemma is often useful. Lemma 4.55. Let C be a C ∗ -algebra, µ ∈ E(C) and for λ > 0 set S λ (C, µ) ≡ {ν ∈ E(C) | ν ≤ λµ}. (i) S λ (C, µ) is weak- compact. (ii) The set S0 (C, µ) ≡

∞ ,

S n (C, µ),

(33)

n=1

is total in S(C, µ), i.e., the set of finite convex linear combinations of elements of S0 (C, µ) is a norm dense subset of S(C, µ). (iii) If (π, H) is a representation of C such that µ(A) = (Ψ, π(A)Ψ ) for some Ψ ∈ H then one has S0 (C, µ) = {(Φ, π( · )Φ) | Φ ∈ π(C) Ψ, Φ = 1}. Proof. We first prove (iii). Assume that ν ∈ S0 (C, µ) as defined in Equ. (33). Then there exists λ > 0 such that ν(A∗ A) ≤ λµ(A∗ A) = λ π(A)Ψ 2 for all A ∈ C. By the Cauchy-Schwarz inequality we further get |ν(A∗ B)| ≤ λ π(A)Ψ π(B)Ψ , for all A, B ∈ C. It follows that the map π(C)Ψ × π(C)Ψ → C (π(A)Ψ, π(B)Ψ ) !→ ν(A∗ B),

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is well defined. As a densely defined, bounded, non-negative sesquilinear form on K ≡ π(C)Ψ it defines a bounded non-negative self-adjoint operator M ∈ B(K), which we can extend by 0 on K⊥ and such that ν(A∗ B) = (π(A)Ψ, M π(B)Ψ ). Since ν(A∗ BC) = ν((B ∗ A)∗ C) we get (π(A)Ψ, M π(B)π(C)Ψ ) = (π(A)Ψ, π(B)M π(C)Ψ ), for all A, B, C ∈ C. Since M vanishes on K⊥ we can conclude that M ∈ π(C) . Set R ≡ M 1/2 , it follows that R ∈ π(C) , ν(A) = (RΨ, π(A)RΨ ), and RΨ = 1. Reciprocally, if ν is given by the above formula one has ν(A∗ A) = Rπ(A)Ψ ≤ R π(A)Ψ 2 = R µ(A∗ A) and thus ν ∈ S0 (C, µ). To prove (ii) let (Hµ , πµ , Ωµ ) be the GNS representation of C associated to µ. If ν ∈ S(C, µ) then there exists ν˜ ∈ N (Cµ ) such that ν = ν˜ ◦ πµ and ν˜(sµ ) = 1. Let ρ be a density matrix on Hµ such that ν˜(A) = Tr(ρA). From the condition Tr(ρsµ ) = 1 it follows that ρ = sµ ρsµ . Thus, ρ is a density matrix in the subspace Ran sµ = Cµ Ωµ and Exercise 2.9 shows that ν˜ can be approximated in norm by finite convex linear combinations of vector states ν˜n (A) = (ψn , Aψn ) with ψn ∈ Cµ Ωµ . (ii) now follows from (iii) and the fact that the map πµ : ν˜ !→ ν˜ ◦ πµ is norm continuous. Since E(C) is weak- compact in C (i) follows from the obvious fact that S λ (C, µ) is weak- closed. Definition 4.56. A quantum dynamical system (C, τ t , µ) is called: (i) Ergodic if, for any ν + µ and any A ∈ C, one has 1 T →∞ T

T

ν ◦ τ t (A) dt = µ(A).

lim

(34)

0

(ii) Mixing if, for any ν + µ and any A ∈ C, one has lim ν ◦ τ t (A) = µ(A).

t→∞

(35)

A state µ ∈ E(C) is called ergodic (resp. mixing) if it belongs to E(C, τ t ) and if the corresponding quantum dynamical system (C, τ t , µ) is ergodic (resp. mixing). These definitions are consistent with the notion of normal form of a quantum dynamical system. Proposition 4.57. The quantum dynamical system (C, τ t , µ) is ergodic (resp. mixing) if and only if its normal form (πµ , Cµ , Hµ , Lµ , Ωµ ) is ergodic (resp. mixing). Proof. We denote by µ ˆ the vector state associated to Ωµ and τˆµt the σ-weakly conˆ) tinuous group of ∗-automorphisms of Cµ generated by Lµ . Assume that (Cµ , τˆµt , µ is ergodic (resp. mixing) and let ν ∈ S(C, µ) and A ∈ C. Then there exists

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ν˜ ∈ N (Cµ ) such that ν˜ + µ ˆ and ν = ν˜ ◦ πµ . Since ν ◦ τ t (A) = ν˜ ◦ τˆµt (πµ (A)) and µ(A) = µ ˆ(πµ (A)) the convergence in Equ. (34) (resp. Equ. (35)) follows directly ˆ). from the corresponding statement for (Cµ , τˆµt , µ To prove the reverse statement assume that (C, τ t , µ) is ergodic (resp. mixing). Since τˆµt is isometric the map ν !→ ν ◦ τˆµt is norm continuous uniformly in t ∈ R. To ˆ) it is therefore prove convergence in Equ. (34) (resp. Equ. (35)) for all ν ∈ S(Cµ , µ ˆ). By Lemma 4.55, S0 (Cµ , µ ˆ) sufficient to prove it for all ν in a total subset of S(Cµ , µ ˆ) and is the union of the sets S n (Cµ , µ ˆ). Thus, it suffices to consider is total in S(Cµ , µ ˆ) for some n > 0. We set ν ∈ S n (Cµ , µ 1 t νt (A) ≡ ν ◦ τˆµs (A) ds, resp. νt (A) ≡ ν ◦ τˆµt (A) . t 0 It follows from the fact that µ ˆ is τˆµt -invariant that νt ∈ S n (Cµ , µ ˆ) for all t ∈ R. Since n ˆ) is weak- compact the set of cluster point of the net νt is non-empty and S (Cµ , µ ˆ) ⊂ N (Cµ ). If ν¯ is such a cluster point then it follows from contained in S n (Cµ , µ ˆ(πµ (A)). ν ◦ τˆµt (πµ (A)) = (ν ◦ πµ ) ◦ τ t (A) and ν ◦ πµ + µ that ν¯(πµ (A)) = µ Since ν¯ is normal we conclude that ν¯ = µ ˆ and hence that µ ˆ is the weak- limit of the net νt . Example 4.58. (Finite quantum systems, continuation of Example 4.32) If ψ is a normalized eigenvector of the Hamiltonian H then the state µ ≡ (ψ, ( · )ψ) is ergodic and even mixing. Indeed, since sµ = ψ(ψ, · ), the only state ν satisfying ν + µ is µ itself. Any normal invariant state which is a mixture of several such eigenstates is non-ergodic. In fact, the formula pj (ψj , Aψj ), ρ(A) = j

expresses the decomposition of the state ρ into ergodic components. Remark 4.59. Proposition 4.57 reduces the study of the ergodic properties of a quantum dynamical system to the special case of a W ∗ -dynamical systems (M, τ t , µ) with µ ∈ N (M, τ t ). For such a quantum dynamical system one has N (M, µ) = {ν ∈ N (M) | sν ≤ zµ }, by Lemma 2.29. Since sµ ≤ zµ we conclude that ν + µ if and only if ν ∈ N (M) and ν(sµ ) = 1, i.e., ν = ν(sµ ( · )sµ ). For ν ∈ N (M) one has either ν(sµ ) = 0 and thus ν(sµ ( · )sµ ) = 0 or ν(sµ ( · )sµ )/ν(sµ ) + µ. We conclude that {ν ∈ E(M) | ν + µ} = {ν(sµ ( · )sµ ) | ν ∈ M , ν ≥ 0, ν(sµ ) = 1}.

(36)

In particular, the set of linear combinations of states ν + µ is the set of linear functionals of the form ν(sµ ( · )sµ ) where ν ∈ M . As a quantum analogue of Proposition 3.6 we have the following characterizations of ergodicity which, using Proposition 4.57, can be applied to the normal form of a quantum dynamical system.

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Proposition 4.60. Let (M, τ t ) be a W ∗ -dynamical system and µ ∈ N (M, τ t ). Denote by Mτ,µ ≡ {A ∈ sµ Msµ | τ t (A) = A for all t ∈ R} the subalgebra of τ t -invariant observables modulo µ. The following propositions are equivalent. (i) (M, τ t , µ) is ergodic. (ii) If P ∈ Mτ,µ is an orthogonal projection then µ(P ) ∈ {0, 1}. (iii) If A ∈ Mτ,µ then A = µ(A)sµ . Proof. (i)⇒(ii). By Remark 4.59, the system is ergodic if and only if 1 T →∞ T

T

ν ◦ τ t (sµ Asµ ) dt = ν(sµ )µ(A),

lim

(37)

0

holds for all A ∈ M and ν ∈ M . Let P ∈ Mτ,µ be an orthogonal projection. Since P = sµ P sµ is τ t -invariant Equ. (37) yields ν(P ) = ν(sµ )µ(P ), i.e., ν(P − µ(P )sµ ) = 0 for any ν ∈ M . Thus, P = µ(P )sµ and from 0 = P (I − P ) = µ(P )sµ (I − µ(P )sµ ) = µ(P )(1 − µ(P ))sµ , we conclude that µ(P ) ∈ {0, 1}. (ii)⇒(iii). It clearly suffices to prove that A = µ(A)sµ for all self-adjoint elements of Mτ,µ . Let A ∈ Mτ,µ be self-adjoint and denote by Pa its spectral projection corresponding to the interval ] − ∞, a]. Note that if M ⊂ B(H) then we can interpret Mτ,µ as a von Neumann algebra on sµ H and the functional calculus yields Pa ∈ Mτ,µ . It follows from (ii) that for any a ∈ R either µ(Pa ) = 0 or µ(Pa ) = 1. In the first case we get Pa sµ = 0 and hence Pa = 0. In the second case we have Pa ≥ sµ and hence Pa = sµ . By the spectral theorem A is a multiple of sµ and (iii) follows. (iii)⇒(i). Let ν + µ and ν˜ ∈ E(M) be a weak- cluster point of the net 1 νT ≡ T

T

ν ◦ τ t dt. 0

Denote by Tα a net such that limα νTα (A) = ν˜(A) for all A ∈ M. For A ∈ M consider the corresponding net ATα ≡

1 Tα

0

Tα

τ t (A) dt ∈ M A ≡ {B ∈ M | B ≤ A }.

By the Banach-Alaoglu theorem M A is σ-weakly compact and there exists a subnet Tβ of the net Tα such that ATβ → A˜ σ-weakly. Since µ is σ-weakly con˜ = µ(A). By the usual argument (see the proof of tinuous and τ t -invariant µ(A) ˜ = A˜ for all t. Since µ is τ t -invariant we also Theorem 4.20) we have τ t (A) ˜ µ ∈ Mτ,µ . It follows from (iii) that sµ As ˜ µ = have τ t (sµ ) = sµ and hence sµ As ˜ ˜ µ(sµ Asµ )sµ = µ(A)sµ = µ(A)sµ . On the other hand, since ν is σ-weakly continuous and ν(sµ ) = 1, we have

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˜ µ ), ˜ = ν(sµ As ν˜(A) = lim νTα (A) = lim νTβ (A) = lim ν(ATβ ) = ν(A) α

β

β

and hence ν˜(A) = ν(µ(A)sµ ) = µ(A). Since this holds for any A ∈ M and any cluster point of the net νT we conclude that νT converges to µ in the weak- topology, i.e., that Equ. (34) holds. Note that if (M, τ t , µ) is ergodic and µ is faithful (sµ = I) it follows immediately that µ is the only normal invariant state of the W ∗ -dynamical system (M, τ t ). More generally, one has Proposition 4.61. If the quantum dynamical systems (C, τ t , µ1 ), (C, τ t , µ2 ) are ergodic then either µ1 = µ2 or µ1 ⊥ µ2 . Proof. Since µ1 , µ2 ∈ E(C, τ t ) we have ω ≡ (µ1 + µ2 )/2 ∈ E(C, τ t ) and hence ˆ ) be its normal form. From (C, τ t , ω) is a quantum dynamical system. Let (Cω , τˆωt , ω the fact that µi ≤ 2ω we conclude by Lemma 4.55 that µ1 and µ2 are ω-normal. Denote by µ ˜i the normal extension of µi to Cω , si ≡ sµi |ω its support, zi ≡ zµi |ω its central support and set Mi ≡ zi Cω . By Remark 4.31 the quantum dynamical system ˜i ) is isomorphic to the normal form (Cµi , τˆµt i , µ ˆi ) of (C, τ t , µi ) and is (Mi , τˆωt , µ therefore ergodic. Since the supports s1 , s2 are invariant under τˆωt so is P ≡ s1 ∧ s2 . P From si ≤ zi we conclude that P ≤ z1 ∧ z2 ≤ zi and hence P ∈ MP 1 ∩ M2 . ˜i (P ) ∈ {0, 1}. As in the proof of Since P = si P si , Proposition 4.60 yields that µ ˜2 (P )s2 and hence either P = 0 Proposition 4.60 this implies that P = µ ˜1 (P )s1 = µ or P = s1 = s2 . In the later case one has s1 ≤ s2 ≤ z2 and Corollary 2.24 yields µ1 + µ2 . It immediately follows from Equ. (34) that µ1 = µ2 . In the former case assume that for some λ > 0 and some state ν one has λν ≤ µi for i = 1, 2. Then λν ≤ ω and by Lemma 4.55, ν has a normal extension ν˜ to Cω which satisfies λ˜ ν≤µ ˜i . It follows that ν˜(I − s1 ) = ν˜(I − s2 ) = 0 and hence ν˜(s1 ) = ν˜(s2 ) = 1. By Exercise 2.14 we get ν˜(P ) = 1, a contradiction which shows that µ1 ⊥ µ2 . Proposition 4.62. A quantum dynamical system (C, τ t , µ) is ergodic if and only if the state µ is an extremal point of the set E(C, τ t ). Proof. Assume that (C, τ t , µ) is ergodic and that there exists µ1 , µ2 ∈ E(C, τ t ) and α1 , α2 ∈]0, 1[ such that α1 + α2 = 1 and µ = α1 µ1 + α2 µ2 . By Lemma 4.55 it ˜i its normal follows from the fact that αi µi ≤ µ that µi is µ-normal. Denote by µ ˜i ≤ µ ˆ and hence µ ˜i (I − sµ ) = 0, i.e., extension to Cµ . By continuity one has αi µ sµi ≤ sµ . Equ. (34) then yields µi = µ and we conclude that µ is extremal. Assume now that µ is extremal in E(C, τ t ) and let (Cµ , τˆµt , µ ˆ) be the corresponding normal form. Let (H, π) be a standard representation of Cµ and denote by (H, M, J, C) its standard form. Let L be the corresponding standard Liouvillean ˆ. The dynamics is implemented in M and Ψµ the standard vector representative of µ ¯(A) ≡ (Ψµ , AΨµ ) then the normal form is by σ t (A) = eitL Ae−itL and if we set µ ¯). isomorphic to the quantum dynamical system (M, σ t , µ If ν¯ ∈ N (M, σ t ) then ν¯ ◦ π ◦ πµ is µ-normal and invariant, hence it belongs to E(C, τ t ). In fact, since πµ (C) is σ-weakly dense in Cµ the map ν¯ !→ ν¯ ◦ π ◦ πµ is

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an injection from N (M, σ t ) into E(C, τ t ). It follows that µ ¯ is an extremal point of ¯1 , µ ¯2 ∈ N (M, σ t ) and α ∈]0, 1[ be such that µ ¯ = α¯ µ1 + N (M, σ t ). Indeed, let µ µ1 ◦ π ◦ πµ + (1 − α)¯ µ2 ◦ π ◦ πµ and since µ is (1 − α)¯ µ2 . Then we have µ = α¯ ¯2 . extremal this yields µ ¯1 = µ ¯(P ) ∈]0, 1[. Set P ≡ JP J ∈ M and Let P ∈ Mσ,¯µ and assume that α ≡ µ define the states µ ¯1 (A) ≡

(P Ψµ , AP Ψµ ) , P Ψµ 2

µ ¯2 (A) ≡

((I − P )Ψµ , A(I − P )Ψµ ) , (I − P )Ψµ 2

on M. Since P Ψµ 2 = (Ψµ , JP JΨµ ) = (P JΨµ , JΨµ ) = α, we have α¯ µ1 (A) + (1 − α)¯ µ2 (A) = (P Ψµ , AΨµ ) + ((I − P )Ψµ , AΨµ ) = (Ψµ , AΨµ ) =µ ¯(A), for any A ∈ M. Moreover, since e−itL P eitL = e−itL JP JeitL = Je−itL P eitL J = Jσ −t (P )J = JP J = P , and eitL Ψµ = Ψµ , one easily checks that µ ¯1 and µ ¯2 are invariant and hence belong ¯1 = µ ¯2 . Then µ ¯=µ ¯1 and for any A ∈ M we get to N (M, τ t ). Let us assume that µ (αΨµ , AΨµ ) = (P Ψµ , AΨµ ). It follows that 0 = [MΨµ ](P − α)Ψµ = [MΨµ ]J(P − α)JΨµ = J[M Ψµ ](P − α)Ψµ , from which we conclude that sµ (P − α)Ψµ = 0. Since P ∈ Mσ,¯µ it satisfies P = sµ P which yields P Ψµ = αΨµ . This is impossible since P is a projection and ¯2 provide a non-trivial deα ∈]0, 1[. This contradiction shows that the states µ ¯1 = µ composition of µ ¯, a contradiction to its extremality. We conclude that µ ¯(P ) ∈ {0, 1} ¯) is ergodic. Since for all P ∈ Mσ,¯µ and hence, by Proposition 4.60, that (M, σ t , µ this system is isomorphic to a normal form of (C, τ t , µ), the later is also ergodic by Proposition 4.57. The last result in this subsection is the following characterization of the mixing property. Proposition 4.63. Let (M, τ t ) be a W ∗ -dynamical system and µ ∈ N (M, τ t ). The quantum dynamical system (M, τ t , µ) is mixing if and only if, for all A, B ∈ M, one has (38) lim µ(Asµ τ t (B)) = µ(A)µ(B). t→∞

Proof. By Remark 4.59 the system is mixing if and only if lim ν ◦ τ t (sµ Bsµ ) = ν(sµ )µ(B),

t→∞

(39)

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holds for any ν ∈ M and B ∈ M. If the system is mixing then Equ. (38) follows from Equ. (39) with ν ≡ µ(A( · )) ∈ M . To prove the reverse statement suppose that Equ. (38) holds. By Proposition 4.57 we may also assume that the system is in normal form, i.e., that M ⊂ B(H), µ(B) = (Ωµ , BΩµ ) for some unit vector Ωµ ∈ H, MΩµ = H and sµ = [M Ωµ ]. Since τ t is isometric it suffices to show Equ. (39) for a norm total set of ν in S(M, µ). By Lemma 4.55, S0 (M, µ) is total in S(M, µ) and its elements are of the form ν(B) = (R∗ RΩµ , BΩµ ) with R ∈ M . But R∗ RΩµ can be approximated in norm by vectors of the type sµ A∗ Ωµ with A ∈ M and hence the set of linear functionals of the form ν(B) = (sµ A∗ Ωµ , BΩµ ) is also total in S(M, µ). For such ν, it does indeed follow from Equ. (38) that ν ◦ τ t (sµ Bsµ ) = µ(Asµ τ t (B)) → µ(A)µ(B) = ν(sµ )µ(B) as t → ∞.

4.7 Quantum Koopmanism To develop the spectral theory of quantum dynamical systems along the lines of Theorem 3.14 and 3.15 one may be tempted to use the µ-Liouvillean Lµ or the standard Liouvillean L. The next exercise shows that this is not possible. Exercise 4.64. In Examples 4.32, 4.58, assume that E is a degenerate eigenvalue of H and ψ a corresponding normalized eigenvector. Show that the µ-Liouvillean Lµ corresponding to the state µ(A) = (ψ, Aψ) is unitarily equivalent to H − E. Thus, even though µ is ergodic, 0 is not a simple eigenvalue of Lµ . Show also that if H has N eigenvalues (counting multiplicities) then 0 is an N -fold degenerate eigenvalue of the standard Liouvillean L. In fact a further reduction is necessary to obtain the analogue of the classical Liouvillean. Theorem 4.65. Let (C, τ t , µ) be a quantum dynamical system with normal form (πµ , Cµ , Hµ , Lµ , Ωµ ). Then the support sµ = [Cµ Ωµ ] reduces the µ-Liouvillean Lµ . We call reduced Liouvillean or simply “The Liouvillean” and denote by Lµ the restriction of Lµ to Ran sµ . (i) (C, τ t , µ) is ergodic if and only if Ker Lµ is one-dimensional. (ii) (C, τ t , µ) is mixing if and only if w − lim eitLµ = Ωµ (Ωµ , · ). t→∞

(iii) If the spectrum of Lµ on {Ωµ }⊥ is purely absolutely continuous then (C, τ t , µ) is mixing.

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Proof. Let τˆt (A) ≡ eitLµ Ae−itLµ . Since the vector state µ ˆ(A) ≡ (Ωµ , AΩµ ) is τˆt -invariant its support sµ satisfies eitLµ sµ e−itLµ = sµ , and hence L is reduced by Ran sµ . As in the proof of Proposition 4.57 we can replace S(C, µ) by S0 (C, µ) in Definition 4.56. For ν ∈ S0 (C, µ) we have ν ◦ τˆt (A) = (RΩµ , eitLµ Ae−itLµ RΩµ ) = (R∗ RΩµ , eitLµ AΩµ ). Since R∗ RΩµ = sµ R∗ RΩµ we can rewrite ν ◦ τˆt (A) = (sµ R∗ RΩµ , eitLµ sµ AΩµ ) = (sµ R∗ RΩµ , eitLµ sµ AΩµ ). From the mean ergodic theorem (Theorem 3.13) we get 1 T →∞ T

lim

T

ν ◦ τˆt (A) dt = (sµ R∗ RΩµ , P sµ AΩµ ),

0

where P is the orthogonal projection on Ker Lµ . Note that if P = Ωµ (Ωµ , · ) then the right hand side of the last formula reduces to ˆ(A). (sµ R∗ RΩµ , Ωµ )(Ωµ , sµ AΩµ ) = (RΩµ , RΩµ )(Ωµ , AΩµ ) = µ Since the systems {sµ R∗ RΩµ | R ∈ Cµ } and {sµ AΩµ | A ∈ Cµ } are both total in Ran sµ , we conclude that the system is ergodic if and only if P = Ωµ (Ωµ , · ). Since Ωµ always belongs to the kernel of Lµ , this is equivalent to the condition (i). The proof of (ii) is completely similar. Finally (iii) is proved as in the classical case (Theorem 3.15). In many applications the invariant state µ ˆ is faithful on Cµ . In this case the µLiouvillean and the standard Liouvillean coincide, sµ = I and the previous result reduces to the simpler Corollary 4.66. Let (C, τ t , µ) be a quantum dynamical system with normal form (πµ , Cµ , Hµ , Lµ , Ωµ ) and assume that sµ = [Cµ Ωµ ] = I. (i) (C, τ t , µ) is ergodic if and only if Ker Lµ is one-dimensional. (ii) (C, τ t , µ) is mixing if and only if w − lim eitLµ = Ωµ (Ωµ , · ). t→∞

(iii) If the spectrum of Lµ on {Ωµ }⊥ is purely absolutely continuous then (C, τ t , µ) is mixing. As in the classical case (see for example Theorem 9.12 in [5]), the spectrum of the Liouvillean of an ergodic quantum dynamical system is not arbitrary (see [24] and [31] for related results).

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Theorem 4.67. Let Lµ be the Liouvillean of the ergodic quantum dynamical system (C, τ t , µ). (i) The set of eigenvalues of Lµ is a subgroup Σ of the additive group R. (ii) The eigenvalues of Lµ are simple. (iii) The spectrum of Lµ is invariant under translations in Σ, that is σ(Lµ ) + Σ = σ(Lµ ). (iv) If Φ is a normalized eigenvector of Lµ then the corresponding vector state is the normal extension of µ to Cµ . (v) If (C, τ t , µ) is mixing then 0 is the only eigenvalue of Lµ . Proof. Let (πµ , Cµ , Hµ , Lµ , Ωµ ) be the normal form of (C, τ t , µ) and set µ ˆ(A) = (Ωµ , AΩµ ), τˆt (A) = eitLµ Ae−itLµ , sµ = [Cµ Ωµ ] and Kµ = sµ Hµ . Note that τˆt extends to B(Hµ ) and that τˆt (Cµ )Cµ = τˆt (Cµ τˆ−t (Cµ )) = τˆt (ˆ τ −t (Cµ )Cµ ) = Cµ τˆt (Cµ ), shows that τˆt (Cµ ) = Cµ . By Proposition 4.57, (Cµ , τˆt , µ ˆ) is ergodic. Denote by Σ the set of eigenvalues of Lµ . If Σ = {0} there is nothing to prove. Suppose that λ ∈ Σ \ {0} and let Φ ∈ Kµ be a corresponding normalized eigenvector. Since e−itLµ Φ = e−itLµ Φ = e−itλ Φ it follows that the state ν(A) ≡ (Φ, AΦ) is invariant. Moreover since (40) Cµ Φ ⊂ Cµ Kµ = Kµ , ˆ. Ergodicity yields that ν = µ ˆ and in particular one has sν ≤ sµ and hence ν + µ sν = sµ , i.e., Cµ Φ = Kµ . This proves (iv). ˆ(A∗ A) = AΩµ which shows For all A ∈ Cµ we have AΦ = ν(A∗ A) = µ that the linear map Tλ : Cµ Ωµ → Cµ Φ AΩµ !→ AΦ, is well defined, densely defined and isometric on Hµ . We also denote by Tλ its unique isometric extension to Hµ . For A, B ∈ Cµ we obtain (Tλ∗ BΦ, AΩµ ) = (BΦ, Tλ AΩµ ) = (BΦ, AΦ) = µ ˆ(B ∗ A) = (BΩµ , AΩµ ), from which we conclude that Tλ∗ BΦ = BΩµ .

(41)

It follows that Ran(Tλ∗ )⊥ = Ker Tλ = {0}, i.e., that Tλ is unitary. Since Tλ BAΩµ = BAΦ = BTλ AΩµ for all A, B ∈ Cµ we further get Tλ ∈ Cµ .

(42)

In particular, we have sµ Tλ = Tλ sµ , so that Tλ and Tλ∗ map Kµ into itself. Finally, for A ∈ Cµ , we have

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τˆt (Tλ )AΩµ = Aˆ τ t (Tλ )Ωµ = AeitLµ Tλ e−itLµ Ωµ = AeitLµ Tλ Ωµ = AeitLµ Φ = eitλ AΦ = eitλ Tλ AΩµ , which yields τˆt (Tλ ) = eitλ Tλ . itLµ

it(Lµ +λ)

(43) Tλ∗

= Tλ e which means that Lµ +λ The last relation can be rewritten as e is unitarily equivalent to Lµ . Property (iii) follows immediately. To prove (ii) suppose that λ is not a simple eigenvalue of Lµ . Then there exists a unit vector Ψ ∈ Kµ such that Ψ ⊥ Φ and Lµ Ψ = λΨ . Since (41) implies that Tλ∗ Φ = Ωµ it follows that Tλ∗ Ψ ∈ Kµ is a unit vector orthogonal to Ωµ . Moreover, Equ. (43) yields eitLµ Tλ∗ Ψ = eitLµ Tλ∗ Ψ = τˆt (Tλ )∗ eitLµ Ψ = (eitλ Tλ )∗ eitλ Ψ = Tλ∗ Ψ, from which we conclude that Tλ∗ Ψ is an eigenvector of Lµ to the eigenvalue 0, a contradiction to the simplicity of this eigenvalue. Thus, λ is simple. To prove (i) we first note that Equ. (43) yields that Φ ≡ Tλ∗ Ωµ ∈ Kµ is an eigenvector of Lµ to the eigenvalue −λ. Second, suppose that λ ∈ Σ \ {0} and let τ t (Tλ ) = Tλ be the corresponding unitary. We will then have τˆt (Tλ Tλ ) = τˆt (Tλ )ˆ it(λ+λ ) Tλ Tλ from which we can conclude again that Tλ Tλ Ωµ is an eigenvector e of Lµ to the eigenvalue λ + λ . To prove (v) suppose on the contrary that the system is mixing and that Lµ has a non-zero eigenvalue λ. Let Φ ∈ Kµ be a corresponding normalized eigenvector. It follows that Φ ⊥ Ωµ and (Φ, eitLµ Φ) = eitλ , does not converge to zero as t → ∞, a contradiction to the mixing criterion (ii) of Theorem 4.65. Using the relation between µ-Liouvillean and standard Liouvillean described in Proposition 4.49 it is easy to rephrase Theorem 4.65 in terms of the standard Liouvillean. Exercise 4.68. Let (C, τ t , µ) be a quantum dynamical system with standard form (η, M, H, J, C, L, Φ). Let sµ ≡ [M Φ] ∈ M be the support of the vector state µ ¯(A) ≡ (Φ, AΦ) and set sµ ≡ Jsµ J = [MΦ] ∈ M . (i) Show that Sµ ≡ sµ sµ is an orthogonal projection which reduces the standard Liouvillean i.e., eitL Sµ e−itL = Sµ . (ii) Show that the Liouvillean Lµ of Theorem 4.49 is unitarily equivalent to the restriction of the standard Liouvillean L to Ran Sµ . As we have seen, the µ-Liouvillean is good enough to study the basic ergodic properties of a quantum dynamical system. However, to get deeper results on the structure of the state space and in particular on the manifold of invariant states it is necessary to use the standard Liouvillean.

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Theorem 4.69. Let L be the standard Liouvillean of the W ∗ -dynamical system (M, τ t ). Then N (M, τ t ) = {ωΦ | Φ ∈ Ker L, Φ = 1}, and in particular (i) Ker L = {0} if and only if N (M, τ t ) = ∅. (ii) Ker L is one-dimensional if and only if (M, τ t ) has a unique normal invariant state. In this case, the corresponding quantum dynamical system is ergodic. Proof. If µ ∈ N (M, τ t ) then, by Proposition 4.41, µ has a standard vector representative Ψµ such that e−itL Ψµ = Ψµ◦τ t = Ψµ and hence Ψµ ∈ Ker L. Reciprocally, any unit vector Φ ∈ Ker L defines a normal τ t -invariant state ωΦ . 4.8 Perturbation Theory Let (A, τ0t ) be a C ∗ -dynamical system and denote by δ0 its generator. A local perturbation of the system is obtained by perturbing its generator with the bounded ∗-derivation associated with a self-adjoint element V of A: δV = δ + i[V, · ], with D(δV ) = D(δ). Using Theorem 4.3 it is easy to show that δV generates a t is an entire strongly continuous group of ∗-automorphisms τVt on A. In fact τλV analytic function of λ. An expansion in powers of λ is obtained by solving iteratively the integral equation (Duhamel formula) t τλV

(X) =

τ0t (X)

t s i[τ0t−s (V ), τλV (X)] ds.

+λ 0

The result is the Dyson-Robinson expansion t τλV (X) = τ0t (X) + ∞ N + λ N =1

0t1 ···tN t

(44) i[τ0t1 (V ), i[· · · , i[τ0tn (V ), τ0t (X)] · · · ]]dt1 · · · dtN ,

which is norm convergent for any λ ∈ C, t ∈ R and X ∈ A. Another useful representation of the locally perturbed dynamics is the interaction picture obtained through the Ansatz τVt (X) = ΓtV τ0t (X)Γt∗ (45) V . It leads to the differential equation ∂t ΓtV = iΓtV τ0t (V ), with the initial condition Γ0V = I. It follows that ΓtV is a unitary element of A which has the norm convergent Araki-Dyson expansion

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ΓtV

≡I+

∞

N

i

0t1 ···tN t

N =1

τ0t1 (V ) · · · τ0tN (V )dt1 · · · dtN .

(46)

Moreover, ΓtV satisfies the cocycle relations = ΓtV τ0t (ΓsV ) = τVt (ΓsV )ΓtV . Γt+s V

(47)

Note that the integrals in Equ. (44) and (46) are Riemann integrals of norm continuous A-valued functions. The interaction picture allows to obtain unitary implementations of τVt in an arbitrary representation (H, π) of A carrying a unitary implementation U0t of the unperturbed dynamics τ0t . Indeed, one has π(τVt (X)) = π(ΓtV τ0t (X)Γt∗ V ) t t = π(ΓV )π(τ0 (X))π(ΓtV )∗ = π(ΓtV )U0t π(X)U0t∗ π(ΓtV )∗ , from which we conclude that the unitary UVt = π(ΓtV )U0t implements τVt in H. The cocycle property (47) shows that UVt has the group property. From the expansion (46) we get norm convergent expansion (the integral are in the strong Riemann sense) UVt = U0t + ∞ iN + N =1

t −tN −1

0t1 ···tN t

U0t1 π(V )U0t2 −t1 · · · U0N

π(V )U0t−tN dt1 · · · dtN .

Let GV be the self-adjoint generator of UVt . Applying the last formula to a vector Φ ∈ D(G0 ) and differentiating at t = 0 we obtain Φ ∈ D(GV ) and GV = G0 + π(V ).

(48)

Note however that the unitary implementation of τVt in H is by no means unique. Indeed, eitK is another implementation if and only if eitGV π(X)e−itGV = eitK π(X)e−itK , for all X ∈ A and all t. Thus e−itK eitGV must be a unitary element of π(A) for all t. Assuming that D(K) = D(GV ), differentiation at t = 0 yields K = GV − W = G0 + π(V ) − W, itK −itGV e satisfies the for some self-adjoint element W ∈ π(A) . Then Γt W = e differential equation t t ∂t Γt W = −iΓW W , itGV = π(ΓtV )eitG0 with W t = eitGV W e−itGV and initial value Γ0 W = I. Since e itG0 −itG0 t itG0 −itG0 and e We ∈ π(A) for all t, we have W = e We and Γt W is given by the norm convergent expansion

Quantum Dynamical Systems

Γt W =I +

∞

167

(−i)N 0t1 ···tN t

N =1

W t1 · · · W tN dt1 · · · dtN .

Local perturbations of W ∗ -dynamical systems can be treated in a completely similar way, the only change is that the norm topology should be replaced with the σ-weak topology, i.e. the integrals in Equ. (44) and (46) have to be understood in the weak- sense. Unbounded perturbations of a W ∗ -dynamical systems (M, τ0t ) are common in applications. They require a slightly more sophisticated treatment. I shall only consider the case of a unitarily implemented unperturbed dynamics4 τ0t (X) = eitG0 Xe−itG0 . Let V be a self-adjoint operator affiliated to M, i.e., eitV ∈ M for all t ∈ R, and such that G0 + V is essentially self-adjoint on D(G0 ) ∩ D(V ). Denote by GV its self-adjoint extension and set τVt (X) = eitGV Xe−itGV . Defining the unitary

ΓtV = eitGV e−itG0 ,

the Trotter product formula (Theorem 15 in Lecture [32]) shows that t/n

(n−1)t/n

ΓtV = s − lim eitV /n τ0 (eitV /n ) · · · τ0 n→∞

(eitV /n ),

and since eitV /n ∈ M we get ΓtV ∈ M. By construction Equ. (45) and (47) remain valid. In particular τVt leaves M invariant and (M, τVt ) is a W ∗ -dynamical system. Another application of the Trotter formula further gives, for X ∈ M , n n eitGV X e−itGV = s − lim eitG0 /n eitV /n X e−itV /n e−itG0 /n , n→∞

which allows to conclude that eitGV X e−itGV = eitG0 X e−itG0 ,

(49)

for all X ∈ M . As in the C ∗ -case, other implementations of τVt can be obtained by choosing a self-adjoint W affiliated to M and such that G − W is essentially self-adjoint on D(G) ∩ D(W ). Denoting by GW its self-adjoint extension we set

itGW −itG e . Γt W =e By the Trotter formula argument we get Γt W ∈ M and Equ. (49) gives 4

This is not a real restriction since it is always possible to go to a standard representation where such an implementation always exists.

168

Claude-Alain Pillet t itG s isG itGV s −itGV i(t+s)GV (Γt )(Γs ) = Γt ΓW e e W ΓV e W ΓV e We itG s −itG i(t+s)GV = Γt ΓW e e We i(t+s)GV = Γt+s W e t+s i(t+s)G = Γt+s . W ΓV e

t itG Thus Γt = eitGW e−itG eitGV is a unitary group, and differentiation shows W ΓV e that its generator is given on D(G) ∩ D(V ) ∩ D(W ) by G + V − W . As an application let me now derive the perturbation formula for the standard Liouvillean.

Theorem 4.70. Let L0 be the standard Liouvillean of the W ∗ -dynamical system (M, τ t ) which we suppose to be in standard form (M, H, J, C). Let V be a selfadjoint operator affiliated to M and such that (i) L0 + V is essentially self-adjoint on D(L0 ) ∩ D(V ). (ii) L0 + V − JV J essentially self-adjoint on D(L0 ) ∩ D(V ) ∩ D(JV J). Denote by LV the self-adjoint extension of L0 + V − JV J. Then LV is the standard Liouvillean of the perturbed dynamical system (M, τVt ). Proof. W = JV J with D(W ) = JD(V ) is affiliated to M since eitW = Je−itV J ∈ M . Furthermore from Equ. (30) we get D(L0 ) = JD(L0 ) and L0 − W = −J(L0 + V )J, which is essentially self-adjoint on D(L0 ) ∩ D(W ) = J(D(L0 ) ∩ D(V )). Thus t Γt W = JΓV J,

and since LV is essentially self-adjoint on D(L0 ) ∩ D(V ) ∩ D(W ) we get from the above discussion eitLV = ΓtV JΓtV JeitL0 . The unitary group eitLV implements τVt and leaves the cone C invariant by property (iii) of Definition 4.39. This show that LV is the standard Liouvillean of the perturbed system.

5 KMS States 5.1 Definition and Basic Properties For a quantum system with an Hamiltonian H such that Tr e−βH < ∞, the GibbsBoltzmann prescription for the canonical thermal equilibrium ensemble at inverse temperature β = 1/kB T is the density matrix

Quantum Dynamical Systems

ρβ = Zβ−1 e−βH ,

169

(50)

where the normalization factor Zβ = Tr e−βH is the canonical partition function (see Section 3 in [33]). On the other hand the dynamics is given by the spatial automorphisms (51) τ t (A) = eitH Ae−itH . To avoid unnecessary technical problems, think of H as being bounded. The fact that the semi-group entering Equ. (50) reappears, after analytic continuation to the imaginary axis, in Equ. (51) expresses the very strong coupling that exists in quantum mechanics between dynamics and thermal equilibrium. To formalize this remark let us consider, for two observables A, B, the equilibrium correlation function ω(Aτ t (B)) = Zβ−1 Tr(e−βH AeitH Be−itH ) = Zβ−1 Tr(e−i(t−iβ)H AeitH B). Using the cyclicity of the trace, analytic continuation of this function to t + iβ gives ω(Aτ t+iβ (B)) = Zβ−1 Tr(e−itH Aei(t+iβ)H B) = ω(τ t (B)A). Thus there is a function Fβ (A, B; z), analytic in the strip Sβ ≡ {z ∈ C | 0 < Im z < β}, and taking boundary values Fβ (A, B; t) = ω(Aτ t (B)), t

Fβ (A, B; t + iβ) = ω(τ (B)A),

(52) (53)

on ∂Sβ . These are the so called KMS boundary conditions. In some sense, the analytic function Fβ (A, B; z) encodes the non-commutativity of the product AB in the state ρβ . A KMS state is a state for which such a function, satisfying the KMS boundary condition, exists for all observables A, B. Definition 5.1. Let (C, τ t ) be a C ∗ - or W ∗ -dynamical system. A state ω on C, assumed to be normal in the W ∗ -case, is said to be a (τ t , β)-KMS state for β > 0 if for any A, B ∈ C there exists a function Fβ (A, B; z) analytic in the strip Sβ , continuous on its closure and satisfying the KMS conditions (52) and (53) on its boundary. Remark 5.2. i. KMS states for negative values of β have no physical meaning (except for very special systems). However, for historical reasons, they are widely used in the mathematical literature. These states have the same definition, the strip Sβ being replaced by Sβ = {z ∈ C | β < Im z < 0}. ii. Our definition excludes the degenerate case β = 0. The KMS boundary condition then becomes ω(AB) = ω(BA) for all A, B ∈ C. Such a state is called a trace. iii. If ω is a (τ t , β)-KMS state then it is also (τ γt , β/γ)-KMS. Note however that there is no simple connection between KMS states at different temperatures for the same group τ t .

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If (C, τ t ) is a C ∗ - or W ∗ -dynamical system, an element A ∈ C is analytic for τ t if the function t !→ τ t (A), defined for t ∈ R, extends to an entire analytic function z !→ τ z (A) of z ∈ C. Let us denote by Cτ the set of analytic elements for τ t . It is easy to see that Cτ is a ∗-subalgebra of C. Indeed, τ z (A + B) = τ z (A) + τ z (B) and τ z (AB) = τ z (A)τ z (B) hold for z ∈ R and hence extend to all z ∈ C by analytic continuation. Moreover, if τ z (A) is entire analytic then so is τ z (A∗ ) ≡ τ z¯(A)∗ . For A ∈ C, set - ∞ 2 n τ t (A) e−nt dt, An ≡ π −∞ where the integral is understood in the Riemann sense in the C ∗ -case and in the weak- sense in the W ∗ -case. One has An ∈ C with An ≤ A and the formula - ∞ 2 n z τ t (A) e−n(t−z) dt, τ (An ) = π −∞ clearly shows that τ z (An ) is an entire analytic function of z such that 2

τ z (An ) ≤ en Im(z) A , and hence An ∈ Cτ . Finally, since ∞ √ 2 dt An − A = (τ t/ n (A) − A) e−t √ , π −∞ it follows from Lebesgue dominated convergence theorem that the sequence An converges towards A in norm in the C ∗ -case and σ-weakly in the W ∗ -case. This shows that the ∗-subalgebra Cτ is dense in C in the appropriate topology. Let ω be a (τ t , β)-KMS state on C, A ∈ C and B ∈ Cτ . Then the function G(z) ≡ ω(Aτ z (B)) is entire analytic and for t ∈ R one has G(t) = Fβ (A, B; t). Thus, the function G(z) − Fβ (A, B; z) is analytic on Sβ , continuous on Sβ ∪ R and vanishes on R. By the Schwarz reflection principle it extends to an analytic function on the strip {z | − β < Im z < β} which vanishes on R and therefore on the entire strip. By continuity Fβ (A, B; z) = ω(Aτ z (B)), holds for z ∈ Sβ and in particular one has ω(Aτ iβ (B)) = ω(BA).

(54)

As a first consequence of this fact, let us prove the most important property of KMS states. Theorem 5.3. If ω is a (τ t , β)-KMS state then it is τ t -invariant. Proof. For A ∈ Cτ , the function f (z) = ω(τ z (A)) is entire analytic. Moreover, Equ. (54) shows that it is iβ-periodic f (z + iβ) = ω(Iτ iβ (τ z (A))) = ω(τ z (A)I) = f (z).

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171

On the closed strip Sβ the estimate |f (t + iα)| τ t (τ iα (A)) = τ iα (A) sup τ iγ (A) < ∞, 0γβ

holds and therefore f is bounded on whole complex plane. By Liouville Theorem, f is constant. By continuity, this property extends to all observables. As shown by the next result, Property (54) characterizes KMS states in a way that is often more convenient than Definition 5.1 (See Proposition 5.3.7 in [12]). Theorem 5.4. A state ω on a C ∗ - (resp. W ∗ -) algebra C is (τ t , β)-KMS if and only it is normal in the W ∗ -case and there exists a norm dense (resp. σ-weakly dense), τ t -invariant ∗-subalgebra D of analytic elements for τ t such that ω(Aτ iβ (B)) = ω(BA), holds for all A, B ∈ D. Proof. It remains to prove sufficiency. We will only consider the C ∗ -case and refer the reader to the proof of Proposition 5.3.7 in [12] for the W ∗ -case. For A, B ∈ D, the function defined by Fβ (A, B; z) ≡ ω(Aτ z (B)), is analytic on the strip Sβ and continuous on its closure. Since D is invariant under τ t one has τ t (B) ∈ D and hence Fβ (A, B; t) = ω(Aτ t (B)), Fβ (A, B; t + iβ) = ω(Aτ iβ (τ t (B))) = ω(τ t (B)A). From the bound |ω(Aτ z (B))| ≤ A τ i Im z (B) we deduce that Fβ (A, B; z) is bounded on Sβ and Hadamard tree line theorem yields sup |Fβ (A, B; z)| ≤ A B .

(55)

z∈Sβ

Since D ⊂ Cτ is norm dense, any A, B ∈ C can be approximated by sequences An , Bn ∈ D. From the identity Fβ (An , Bn ; z) − Fβ (Am , Bm ; z) = Fβ (An − Am , Bn ; z) + Fβ (Am , Bn − Bm ; z), and the bound (55) we conclude that the sequence Fβ (An , Bn ; z) is uniformly Cauchy in Sβ . Its limit, which we denote by Fβ (A, B; z), is therefore analytic on Sβ and continuous on its closure where it satisfies (55). Finally, for t ∈ R one has Fβ (A, B; t) = lim ω(An τ t (Bn )) = ω(Aτ t (B)), n

Fβ (A, B; t + iβ) = lim ω(τ t (Bn )An ) = ω(τ t (B)A), n

which concludes the proof.

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Example 5.5. (Finite quantum systems, continuation of Example 4.16) Let H be a self-adjoint operator on the Hilbert space H and consider the induced W ∗ -dynamical system (B(H), τ t ). If ω is a (τ, β)-KMS state then there exists a density matrix ρ on H such that ω(A) = Tr(ρA) for all A ∈ B(H). Relation (54) with A ≡ φ (ψ, · ) yields (ψ, τ iβ (B)ρ φ) = (ψ, ρB φ), and hence τ iβ (B)ρ = ρB for all τ t -analytic elements B. If φ and ψ belong to the dense subspace of entire analytic vectors for eitH then B ≡ φ(ψ, · ) is analytic for τ t and τ iβ (B) = e−βH φ (eβH ψ, · ). For ψ = 0 we can rewrite the relation τ iβ (B)ρψ = ρBψ as (eβH ψ, ρψ) −βH e ρφ = φ, (ψ, ψ) from which we can conclude that ρ=

e−βH . Tr e−βH

Thus, the W ∗ -dynamical system (B(H), τ t ) admits a (τ t , β)-KMS state if and only if e−βH is trace class. Moreover, if such a state exists then it is unique. Example 5.6. (Ideal Fermi gas) Let ωβ be a (τ t , β)-KMS state for the C ∗ -dynamical system of Example 4.6. Then ωβ (a∗ (g)a(f )) is a sesquilinear form on h. Since |ωβ (a∗ (g)a(f )| a∗ (g) a(f ) g f , there exists a bounded self-adjoint T on h such that ωβ (a∗ (g)a(f )) = (f, T g).

(56)

Moreover, since ωβ (a∗ (f )a(f )) 0, T satisfies the inequalities 0 T I. For t ∈ R one has ωβ (a∗ (g)τ t (a(f ))) = (eith f, T g) and for f in the dense subspace of entire analytic vectors for h the analytic continuation of this function to t + iβ is given by (eβh eith f, T g) = (T eβh eith f, g). The CAR gives τ t (a(f ))a∗ (g) = −a∗ (g)τ t (a(f )) + (eith f, g) and hence ωβ (τ t (a(f ))a∗ (g)) = (eith f, g) − ωβ (a∗ (g)a(eith f )) = ((I − T )eith f, g), and the KMS boundary condition implies T eβh = I − T from which we conclude that 1 . T = 1 + eβh Consider now the expectation of an arbitrary even monomial (m + n even) Wm,n (g1 , . . . gm ; f1 , . . . fn ) = ωβ (a∗ (gm ) · · · a∗ (g1 )a(f1 ) · · · a(fn )). We first show that Wm,0 = W0,n = 0. For any f in the dense subspace of entire analytic vectors for h, the KMS condition and the CAR lead to

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173

W0,n (f, f2 , . . . fn ) = W0,n (f2 , . . . fn , eβh f ) = (−1)n−1 W0,n (eβh f, f2, . . . fn ), from which we get W0,n ((1 + eβh )f, f2 , . . . fn ) = 0. If g is an entire analytic vector for h, so is f = (1 + eβh )−1 g and we can conclude that W0,n (g, f2, · · · , fn ) = 0 for all g in a dense subspace and all f2 , . . . fn ∈ h and hence for all g, f2, . . . fn ∈ h. By conjugation, this implies that for all g1 , . . . gm ∈ h one also has Wm,0 (g1 , . . . gm ) = 0. We consider now the general case. Using the KMS boundary condition, we can write Wm,n (g1 , . . . ; f1 , . . .) = ωβ (a(e−βh fn )a∗ (gm ) · · · a∗ (g1 )a(f1 ) · · · a(fn−1 )). Using the CAR we can commute back the first factor through the others to bring it again in the last position. This leads to the formula Wm,n (g1 , . . . gm ; f1 , . . . fn ) = −Wm,n (g1 , . . . gm ; f1 , . . . e−βh fn )+ +

m

(−1)m−j (gj , e−βh fn )Wm−1,n−1 (g1 , . . . gj−1 , gj+1 , . . . gm ; f1 , . . . fn−1 ).

j=1

Replacing fn by (1 + e−βh )−1 fn and using Equ. (56), this can be rewritten as Wm,n (g1 , . . . gm ; f1 , . . . fn ) = m (−1)m−j W1,1 (gj ; fn ) Wm−1,n−1 (g1 , . . . gj−1 , gj+1 , . . . gm ; f1 , . . . fn−1 ). j=1

Iteration of this formula shows that Wm,n = 0 if m = n, and Wn,n can be expressed as sum of products of W1,1 (gj ; fk ) = (gj , T fk ). In fact, a closer look at this formula shows that it is nothing but the usual formula for the expansion of the determinant of the n × n matrix (W1,1 (gj ; fk ))jk , i.e., ωβ (a∗ (gm ) · · · a∗ (g1 )a(f1 ) · · · a(fn )) = δnm det{(fj , T gk )}. Definition 5.7. A state ω on CAR(h) or CAR+ (h) is called gauge-invariant if it is invariant under the gauge group ϑt (recall Equ. (13)). It is called gauge-invariant quasi-free if it satisfies ω(a∗ (gm ) · · · a∗ (g1 )a(f1 ) · · · a(fn )) = δnm det{(fj , T gk )},

(57)

for some self-adjoint operator T on h such that 0 ≤ T ≤ I, all integers n, m and all f1 , . . . , fn , g1 , . . . , gm ∈ h. Remark 5.8. It is easy to see that a gauge-invariant quasi-free state is indeed gaugeinvariant. One can show that, given an operator T on h such that 0 ≤ T ≤ I, there exists a unique gauge-invariant quasi-free state such that Equ. (57) holds (see the Notes to Section 2.5.3 in [12] for references).

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Thus, we have shown: Theorem 5.9. Let τ t (a(f )) = a(eith f ), then the unique (τ t , β)-KMS state of the C ∗ -dynamical system (CAR+ (h), τ t ) is the quasi-free gauge invariant state generated by T = (1 + eβh )−1 . Example 5.10. (Quantum spin system, continuation of Example 4.11) The reader should consult Chapter 6.2 in [12] as well as [38] for more complete discussions. For any finite subset Λ ⊂ Γ and any local observable A ∈ Aloc we set ωΛ (A) ≡

TrhΛ∪X (e−βHΛ A) , TrhΛ∪X (e−βHΛ )

whenever A ∈ AX . It is easy to see that ωΛ (A) does not depend on the choice of the finite subset X. By continuity, ωΛ extends to a state on A. We say that a state ω on A is a thermodynamic limit of the net ωΛ ,Λ ↑ Γ, if there exists a subnet Λα such that ω(A) = lim ωΛα (A), α

for all A ∈ A. Since E(A) is weak- compact the set of thermodynamic limits of the net ωΛ , is not empty. Let us now prove that any thermodynamic limit of ωΛ is a (τ, β)-KMS state. Following the discussion at the beginning of this subsection we remark that for any finite subset Λ ⊂ Γ and any A, B ∈ Aloc the function Fβ,Λ (A, B; z) ≡ ωΛ (AτΛz (B)), is entire analytic and satisfies the (τΛ , β)-KMS conditions (52), (53). Since for t, η ∈ R one has |ωΛ (AτΛt+iη (B))| ≤ A τ iη (B) , it is also bounded on any horizontal strip {t + iη | t ∈ R, a ≤ η ≤ b}. The boundary conditions (52), (53) and Hadamard three line theorem further yields the bound sup |Fβ,Λ (A, B; z)| ≤ A B .

(58)

z∈Sβ

Assume now that ω is the weak- limit of the net ωΛα and let A, B ∈ Aloc . Since for t ∈ R one has lim τΛt α (B) − τ t (B) = 0, α

it follows that lim Fβ,Λα (A, B; t) = ω(Aτ t (B)), α

lim Fβ,Λα (A, B; t + iβ) = ω(τ t (B)A). α

From the bound (58) and Montel theorem we conclude that some subsequence Fβ,Λαn (A, B; z) is locally uniformly convergent in the open strip Sβ . Let us denote its limit, which is analytic in Sβ , by Fβ (A, B; z).

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The estimate (16) and the expansion (18) show that, for B ∈ Aloc , lim τΛi (B) − B = 0,

→0

holds uniformly in Λ. From the identity Fβ (A, B; t + i) − ω(Aτ t (B)) = lim ωΛαn (A(τΛt+i (B) − τΛt αn (B))), αn n

and the bound (B) − τΛt αn (B)))| = |ωΛαn (τΛ−tαn (A)(τΛi αn (B) − B))| |ωΛαn (A(τΛt+i αn ≤ A τΛi αn (B) − B , we thus conclude that lim Fβ (A, B; t + i) = ω(Aτ t (B)).

↓0

Using the (τΛ , β)-KMS conditions (53) we prove in a completely similar way that lim Fβ (A, B; t + iβ − i) = ω(τ t (B)A),

↓0

and we conclude that Fβ (A, B; z) extends to a continuous function on the closed strip Sβ . Moreover, this function clearly satisfies the bound sup |Fβ (A, B; z)| ≤ A B ,

(59)

z∈Sβ

as well as the (τ, β)-KMS conditions (52) and (53). Finally, we note that any A, B ∈ A can be approximated by sequences An , Bn ∈ Aloc . As in the proof of Theorem 5.4 the bound (59) shows that the sequence Fβ (An , Bn ; z) converges uniformly on Sβ . Its limit, which we denote by Fβ (A, B; z) then satisfies all the requirements of Definition 5.1. I conclude this brief introduction to KMS states with the following complement to Proposition 4.48 which is very useful in many applications to open quantum systems. Proposition 5.11. Let (C, τ t , ω) be a quantum dynamical system with normal form (πω , Cω , Hω , Lω , Ωω ). Assume that there exists a ∗-subalgebra D ⊂ C with the following properties. (i) πω (D) is σ-weakly dense in Cω . (ii) For each A, B ∈ D there exists a function Fβ (A, B; z) analytic in the strip Sβ , continuous and bounded on its closure and satisfying the KMS boundary conditions (52), (53). Denote by ω ˆ = (Ωω , ( · )Ωω ) the normal extension of ω to the enveloping von Neumann algebra Cω and by τˆt (A) = eitLω Ae−itLω the W ∗ -dynamics induced by τ t on this algebra. Then the following hold.

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(i) ω ˆ is (ˆ τ t , β)-KMS and faithful. ˆ ) is in standard form. In particular the ω-Liouvillean Lω coincide (ii) (Cω , τˆt , ω with the standard Liouvillean L. (iii) The modular operator of the state ω ˆ is given by ∆ωˆ = e−βL . Proof. Let A, B ∈ Cω be such that A ≤ 1 and B ≤ 1. By the Kaplansky density theorem there exists nets Aα , Bα in πω (D) such that Aα ≤ 1, Bα ≤ 1 and Aα → A, Bα → B in the σ-strong∗ topology. In particular, if we set dα ≡ max( (Aα − A)Ωω , (A∗α − A∗ )Ωω , (Bα − B)Ωω , (Bα∗ − B ∗ )Ωω ), then limα dα = 0. By Hypothesis (ii) for each α there exists a function Fα (z) analytic in Sβ , continuous and bounded on its closure and such that Fα (t) = ˆ (ˆ τ t (Bα )Aα ) for t ∈ R. Hence, Fα (z) − Fα (z) is ω ˆ (Aα τˆt (Bα )) and Fα (t + iβ) = ω also analytic in Sβ , continuous and bounded on its closure. By the Cauchy-Schwarz inequality we have the bounds |Fα (t) − Fα (t)| ≤ 2(dα + dα ), |Fα (t + iβ) − Fα (t + iβ)| ≤ 2(dα + dα ), and the Hadamard three line theorem yields sup |Fα (z) − Fα (z)| ≤ 2(dα + dα ). z∈Sβ

Thus, Fα is a Cauchy net for the uniform topology. It follows that limα Fα = F exists, is analytic in Sβ and continuous on its closure. Moreover, we have F (t) = lim ω ˆ (Aα τˆt (Bα )) = lim(A∗α Ωα , eitLω Bα Ωω ) α

α

∗

= (A Ωω , e

itLω

BΩω ) = ω ˆ (Aˆ τ t (B))

and similarly F (t + iβ) = ω ˆ (ˆ τ t (B)A). This shows that ω ˆ is (ˆ τ t , β)-KMS. Let now A ∈ Cω be such that AΩω = 0. Then the function Fβ (A∗ , A; z) of Definition 5.1 satisfies Fβ (A∗ , A; t) = ω ˆ (A∗ τˆt (A)) = (AΩω , eitLω AΩω ) = 0. By the Schwarz reflection principle, this function extends to an analytic function on the strip {z ∈ C | − β < Im z < β} which vanishes on R. It is therefore identically ˆ (AA∗ ) = A∗ Ωω 2 = 0. For any B ∈ Cω , zero. In particular Fβ (A∗ , A; iβ) = ω ∗ the same argument shows that (BA) Ωω = A∗ B ∗ Ωω = 0 and since Ωω is cyclic ˆ is faithful and (ii) follows from we can conclude that A∗ = 0 and A = 0. Thus, ω Proposition 4.48. Note that by Remark 5.2 iii the state ω ˆ is (ˆ τ −βt , −1)-KMS. (iii) follows from t it of a faithful normal state is the the fact that the modular group σωˆ (A) = ∆ωˆ A∆−it ω ˆ unique W ∗ -dynamics for which ω is a β = −1 KMS state (Takesaki’s theorem, see Lecture [7], Theorem 18).

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Example 5.12. (Ideal Bose gas, continuation of Example 4.35) The characteristic function of the thermal equilibrium state of an ideal Bose gas can be obtained from the explicit calculation of the thermodynamic limit of the unique Gibbs state of a finite Bose gas (see Chapter 1 in [12], see also Subsection 4.4 in [36]). We shall get it by assuming that, as in the Fermionic case, the KMS state is quasi-free. Thus, we are looking for a non-negative operator ρ on h such that ω(W (f )) = e−(f,(I+2ρ)f )/4 ,

(60)

satisfies the (τ, β)-KMS condition. Using the CCR relation (19) we can compute ω(W (f )τ t (W (f ))) = ω(W (f + eith f ))e−i Im(f,e where

ith

f)

= e−(f,γ(t)f )/4 ,

γ(t) = (I + e−ith )(I + 2ρ)(I + eith ) + 2i sin th.

Assuming that f is an entire analytic vector for the group eith , analytic continuation to t = iβ yields γ(iβ) = (I + eβh )(I + 2ρ)(I + e−βh ) − 2 sh βh, and the KMS condition (54) requires that γ(iβ) = γ(0). This equation is easily solved for ρ and its solution is given by Planck’s black-body radiation law ρ=

1 . eβh − 1

(61)

It is clear from the singularity at h = 0 in Equ. (61) that if 0 ∈ σ(h) one can not hope to get a state on CCR(D) without further assumption on D. Assume that h > 0, i.e., that 0 is not an eigenvalue of h and that D = D(h−1/2 ). Then according to Definition 4.37, Equ. (60), (61) define a quasi-free, τ t -invariant state ω on CCR(D). Let (CCR(D), τ t , ω) be the associated quantum dynamical system and (πω , Mω , Hω , Lω , Ωω ) its normal form as described in Example 4.35. Note that since finite linear combinations of elements of W (D) are dense in CCR(D) one has Mω = πω (W (D)) and Mω = πω (W (D)) . D equipped with the scalar product (f, g)D ≡ (f, (I + 2ρ)g) is a Hilbert space. Since eith is a strongly continuous unitary group on D the subspace A ⊂ D of entire analytic vectors for eith is dense. It follows that any f ∈ D can be approximated in the norm of D by a sequence fn ∈ A. A simple calculation using the CCR shows that for f, f , g ∈ D, (πω (W (f )) − πω (W (f )))πω (W (g))Ωω 2 = 2(1 − e− f −f

2

D /4

cos θ),

where θ ≡ Im(f −f , g)+Im(f, f )/2. Since the elements of πω (W (D)) are unitary and Ωω is cyclic it follows that s − lim πω (W (fn )) = πω (W (f )), n

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whenever fn converges to f in D. We conclude that πω (W (A)) = πω (W (D)) = Mω , and hence πω (W (A)) = Mω . Denote by D ⊂ CCR(D) the linear span of W (A). By the von Neumann density theorem πω (D) is σ-weakly dense in Mω . Moreover, an explicit calculation shows that ω(W (f )τ t (W (g))) = e−φt (f,g)/4 where φt (f, g) = f 2D + g 2D + 2(g,

e−ith eith f ) + 2(f, g)D . D I + eβh I + e−βh

If f, g ∈ A then φt (f, g) is an entire analytic function of t and φt+iβ (f, g) = φt (g, f ). Thus, D fulfills the requirements of Proposition 5.11. It follows that ω ˆ (A) = (Ωω , AΩω ) is a β-KMS state for the W ∗ -dynamics generated by the Liouvillean L = Lω . Note that since Ker ρ = {0} the GNS representation associated to ω is given by Hω = L2 (Γ+ (h)), πω (W (f ))X = W ((I + ρ)1/2 f )XW (ρ1/2 f )∗ , Ωω = Ω(Ω, · ), where Ω is the Fock vacuum in Γ+ (h). The Liouvillean L is given by eitL X = Γ(eith )XΓ(e−ith ), Moreover, since ω ˆ is faithful we have [Mω Ωω ] = I and Corollary 4.66 applies. Exercise 5.13. Show that the standard form of Mω on Hω is specified by conjugation J : X !→ X ∗ and the cone C = {X ∈ L2 (Γ+ (h)) | X ≥ 0}. Exercise 5.14. Show that if h has purely absolutely continuous spectrum then 0 is a simple eigenvalue of L with eigenvector Ωω . Show that the spectrum of L on {Ωω }⊥ is purely absolutely continuous. Conclude that the quantum dynamical system (CCR(D), τ t , ω) is mixing. What happens if h has non-empty singular spectrum ? The thermodynamics of the ideal Bose gas is more complex than the above picture. In fact, due to the well known phenomenon of Bose-Einstein condensation, non-unique KMS states are possible. I refer to Lecture [36] and [12] for a detailed discussion. 5.2 Perturbation Theory of KMS States Consider the finite dimensional C ∗ -dynamical system defined by A = B(Cn ) and τ t (X) = eitH Xe−itH for some self-adjoint matrix H. Then

Quantum Dynamical Systems

ω(X) =

179

Tr(e−βH/2 Xe−βH/2 ) Tr(e−βH X) = , −βH Tr(e ) Tr(e−βH )

is the unique (τ t , β)-KMS state. If V is another self-adjoint matrix then the perturbed dynamics τVt as well as the perturbed KMS state ωV are obtained by replacing H by H + V . Note that, in the present situation, the definition (45) of the unitary cocycle ΓtV reads ΓtV = eit(H+V ) e−itH , which is obviously an entire function of t. Thus, we can express ωV in terms of ω as ωV (X) =

ω(XΓiβ V ) ω(Γiβ V )

iβ/2∗

=

ω(ΓV

iβ/2

XΓV

iβ/2∗ iβ/2 ω(ΓV ΓV )

)

.

(62)

Let (Hω , πω , Ωω ) be the cyclic representation of ω and L the Liouvillean (ωLiouvillean and standard Liouvillean coincide since ω is faithful). Then on has iβ/2

πω (ΓV

iβ/2

)Ωω = πω (ΓV

)e−βL/2 Ωω = e−β(L+πω (V ))/2 Ωω ,

(63)

by Equ. (48). Thus we can write Equ. (62) as ωV (X) =

(ΩωV , πω (X)ΩωV ) , (ΩωV , ΩωV )

where ΩωV = e−β(L+πω (V ))/2 Ωω . The cocycle property (47) further gives iβ/2

ΓV

iβ/4 iβ/4

= ΓV

τ

iβ/4

(ΓV

iβ/4 iβ/2

) = ΓV

τ

(τ −iβ/4 (ΓV

iβ/4

)),

−iβ/4

and τ −iβ/4 (ΓV ) = (ΓV )−1 . Since ΓzV¯∗ is analytic and equals (ΓzV )−1 for real z, they are equal for all z and iβ/4

iβ/2

ΓV

iβ/4 iβ/2

= ΓV

τ

iβ/4∗

(ΓV

).

We can rewrite the perturbed vector ΩωV as iβ/4

ΩωV = πω (ΓV

1/2

and since J∆ω

)e−βL/2 πω (ΓV

iβ/4 ∗

iβ/4

) Ωω = πω (ΓV

1/2 )e−βL/2 J∆ω πω (ΓV

iβ/4

)Ωω ,

= Je−βL/2 = eβL/2 J we conclude that iβ/4

ΩωV = πω (ΓV

iβ/4

)Jπω (ΓV

)Ωω ∈ C.

Thus ΩωV is, up to normalization, the unique standard vector representative of the perturbed KMS-state ωV . The main difficulty in extending this formula to more general situations is to show that Ωω ∈ D(e−β(L+πω (V ))/2 ). Indeed, even if V is bounded, the Liouvillean L is usually unbounded below and ordinary perturbation theory of quasi-bounded semi-groups fails. If V is such that τ t (V ) is entire analytic, this can be done using (63) since the cocycle ΓtV is then analytic, as the solution of a linear differential equation with analytic coefficients. It is possible to extend the result to general bounded perturbations using an approximation argument.

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Theorem 5.15. Let (C, τ t ) be a C ∗ - or W ∗ -dynamical system and V ∈ C a local perturbation. There exists a bijective map ω !→ ωV between the set of (τ t , β)-KMS states and the set of (τVt , β)-KMS states on C such that ωV ∈ N (C, ω) and (ωV1 )V2 = ωV1 +V2 . Let ω be a (τ t , β)-KMS state on C. Denote by L the (standard) Liouvillean of ˆ ) and by Ω ∈ Hω the standard vector the quantum dynamical system (Cω , τˆt , ω representative of ω ˆ . For any local perturbation V ∈ Cω one has: (i) Ω ∈ D(e−β(L+V )/2 ). (ii) Up to normalization, ΩV = e−β(L+V )/2 Ω is the standard vector representative of ω ˆV . (iii) ΩV is cyclic and separating for Cω . (iv) For any V1 , V2 ∈ Cω one has (ΩV1 )V2 = ΩV1 +V2 . (v) The Peierls-Bogoliubov inequality holds: e−β ωˆ (V ) ΩV 2 . ˆ (e−βV ). (vi) The Golden-Thompson inequality holds: ΩV 2 ω (vii) If Vn ∈ Cω strongly converges to V ∈ Cω then ΩVn converges in norm to ΩV and ωVn converges in norm to ωV . In the case of unbounded perturbations, one can use an approximation by bounded perturbations to obtain Theorem 5.16. Let (M, τ t , ω) be a quantum dynamical system and suppose that ω is a (τ t , β)-KMS state. Assume that the system is in standard form (M, H, J, C). Denote by L its standard Liouvillean and by Ω the standard vector representative of ω. Let V be a self-adjoint operator affiliated to M and such that the following conditions are satisfied: (i) L + V is essentially self-adjoint on D(L) ∩ D(V ). (ii) L + V − JV J is essentially self-adjoint on D(L) ∩ D(V ) ∩ D(JV J). (iii) Ω ∈ D(e−βV /2 ). Then the following conclusions hold: (i) Ω ∈ D(e−β(L+V )/2 ). (ii) Up to normalization, ΩV = e−β(L+V )/2 Ω is the standard vector representative of a (τVt , β)-KMS state ωV . (iii) ΩV is cyclic and separating for M. (iv) The Peierls-Bogoliubov inequality holds: e−β ωˆ (V ) ΩV 2 . ˆ (e−βV ). (v) The Golden-Thompson inequality holds: ΩV 2 ω (vi) For any λ ∈ [0, 1] the operator λV satisfies the hypotheses (i), (ii) and (iii) and one has limλ↓0 ΩλV − Ω = 0 and limλ↓0 ωλV − ω = 0.

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27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

Systems—QED”, held in Nordfjordeid, August 11—18 2004. To be published in Lecture Notes in Mathematics. Jakˇsi´c, V., Pillet, C.-A.: On a Model for Quantum Friction III. Ergodic Properties of the Spin-Boson System. Commun. Math. Phys. 178 (1996), 627. Jakˇsi´c, V., Pillet, C.-A.: Ergodic properties of classical dissipative systems I. Acta Math. 181 (1998), 245. Jakˇsi´c, V., Pillet, C.-A.: Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Commun. Math. Phys. 226 (2002), 131. Jakˇsi´c, V., Pillet, C.-A.: Mathematical theory of non-equilibrium quantum statistical mechanics. J. Stat. Phys. 108 (2002), 787. Jakˇsi´c, V., Pillet, C.-A.: A note on eigenvalues of Liouvilleans. J. Stat. Phys. 105 (2001), 937. Joye, A.: Introduction to the theory of linear operators. Part 1 in this volume. Joye, A.: Introduction to quantum statistical mechanics. Part 2 in this volume. Koopman, B.O.: Hamiltonian systems and transformations in Hilbert spaces. Proc. Nat. Acad. Sci. (U.S.A.) 17 (1931), 315. Man´e, R.: Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987. Merkli, M.: The ideal quantum gas. Part 5 in this volume. Rey-Bellet, L.: Classical open systems. Volume 2 in this series. Ruelle, D.:Statistical Mechanics. Rigorous Results. Benjamin, New York, 1969. Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95 (1999), 393. Ruelle, D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Phys. 98 (2000), 57. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York, 1972. Sakai, S.: C ∗ -Algebras and W ∗ -Algebras. Springer, Berlin, 1971. Sakai, S.: Perturbations of KMS states in C ∗ -dynamical systems. Contemp. Math. 62 (1987), 187. Sakai, S.: Operator Algebras in Dynamical Systems. Cambridge University Press, Cambridge, 1991. Stratila, S., Zsido, L.: Lectures on von Neumann Algebras. Abacus Press, Tunbridge Wells, 1979. Takesaki, M.: Theory of Operator Algebras II. Springer, Berlin, 2003. Thirring, W.: Quantum Mechanics of Large Systems. Springer, Berlin, 1980. von Neumann, J: Proof of the quasiergodic hypothesis. Proc. Nat. Acad. Sci. (U.S.A.) 17 (1932), 70. Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin, 1981.

The Ideal Quantum Gas Marco Merkli Institute of Theoretical Physics, ETH Z¨urich, CH-8093 Zurich, Switzerland e-mail: [email protected]

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

2

Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 2.1 2.2 2.3 2.4 2.5

3

185 188 191 194 197

The CCR and CAR algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.1 3.2 3.3 3.4 3.5

4

Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . Weyl operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The C ∗ -algebras CARF (H), CCRF (H) . . . . . . . . . . . . . . . . . . . . . . . Leaving Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The algebra CAR(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The algebra CCR(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schr¨odinger representation and Stone – von Neumann uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q–space representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium state and thermodynamic limit . . . . . . . . . . . . . . . . . . . .

199 200 203 207 209

Araki-Woods representation of the infinite free Boson gas . . . . . . . . . . 213 4.1 4.2 4.3 4.4 4.5

Generating functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ground state (condensate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical stability of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

214 217 222 224 228

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

present address: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada e-mail: [email protected]

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1 Introduction The goal of these lecture notes is to give an introduction to the mathematical description of a system of identical non-interacting quantum particles. An important characteristic of the systems considered is their “size”, which may refer to spatial extension or to the number of particles, or to a combination of both. Certain physical phenomena occur only for very large systems, say for systems which occupy an immense region of the universe or for a system the size of a laboratory, if the observed phenomenon takes place on a microscopic level. For the mathematical analysis it is often convenient to make an abstraction and to consider systems which are spatially infinitely extended (and which contain infinitely many particles). From a physical point of view, such a description can only be an approximation which is, however, justified by the fact that the mathematical models lead to correct answers to physical questions. An important part of these lectures is concerned with the description of infinite systems, or the passage of a finite system (a confined one, or one with only finitely many particles) to an infinite one. In some instances, this passage is called the thermodynamic limit. It is natural to consider first a system of finitely many (identical) quantum particles. States of such a system are described by vectors in Fock space, a Hilbert space that has a direct sum decomposition into subspaces, each of which describes a system with a fixed number n = 0, 1, 2, . . . of particles. The action of operators which are not reduced by this direct sum decomposition is interpreted as creation or annihilation of particles. So Fock space provides us already with a nice toolbox enabling the modeling of many physical processes. However, not all physical situations can be described by Fock space. Given any state in Fock space the probability of finding at least n particles in it decreases to zero, as n → ∞. Imagine a gas of particles which has a uniform nonzero density (say one particle per unit volume) and which is spatially infinitely extended. Such a state cannot be described by a vector in Fock space! How can we thus describe the infinitely extended system at positive density? The observable algebra (the one generated by the creation and annihilation operators on Fock space) has a certain structure determined by algebraic relations. Those are called the canonical commutation relations (CCR) or the canonical anticommutation relations (CAR) depending on whether one considers Bosons or Fermions. It can be viewed as an abstract algebra, merely determined by its relations, and not a priori represented as an operator algebra on a Hilbert space. Fock space emerges then just as one possible representation Hilbert space of the abstract algebra (called the CCR or the CAR algebra). A fundamental theorem regarding this setting is the von Neumann uniqueness theorem. It says that if we consider only finitely many particles then all the representations of the corresponding algebra are (spatially) equivalent. However, in the case of a system with infinitely many particles there are non-equivalent representations of the algebra! This is what happens in the case of the infinitely extended system with nonzero density; it is described by a vector in some Hilbert space which is not compatible with Fock space (the corresponding representations of the algebras are not spatially equivalent).

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It is one of the goals of these notes to calculate the representation Hilbert space of the infinitely extended gas for arbitrary densities. It may have become clear from this short introduction what kind of mathematics is involved in these notes. In the first chapter we will mainly deal with operators on Fock space (bounded and unbounded ones) and, in the second chapter, we move on to some aspects of the theory of C ∗ algebras in relation with the CCR and CAR algebras. The last chapter is devoted to the Araki-Woods representation, which gives the above mentioned representation of the infinitely extended free Bose gas for arbitrary momentum density distributions. These notes represent a composition of mostly well known concepts and results relevant to this collection of lecture notes, and they have, in the author’s view, an interest on their own. An effort has been made to render the material easy to understand for anybody with basic knowledge in functional analysis.

2 Fock space Fock space is the Hilbert space suitable to describe a system of arbitrarily many (identical) quantum particles. We start this section by introducing the Bosonic and Fermionic Fock spaces and the corresponding creation and annihilation operators. We will see that in the case of Bosons those operators are unbounded and it is thus convenient to “replace” them by Weyl operators. This leads us to the definition of the C ∗ algebras CCRF and CARF , for Bosons and Fermions, respectively. We discuss the “shortcomings” of these algebras in the last section, motivating the definition of the abstract CCR and CAR algebras. 2.1 Bosons and Fermions An ideal quantum gas is a system of identical (meaning indistinguishable), noninteracting quantum particles. A single particle is described by a complex Hilbert space H, i.e., a normalized ψ ∈ H is a (pure) state of the particle (ψ is also called the state vector). It is often useful to consider states which are determined by a linear (not necessarily closed) subspace D ⊆ H. (1) Typically, one may think of H = L2 (R3 , d3 x), then a normalized vector ψ ∈ H is called the wave function of the particle and has the following physical interpretation: |ψ(x)|2 is the probability density of finding the particle at location x ∈ R3 . An example for D is the set {f ∈ C0∞ (R3 ) | suppf ⊂ V } of all smooth functions with support in some compact set V ⊂ R3 ; D is called the test function space. We will see that the choice of the test function space often reflects physical properties of the system at hand, e.g., we may want to look only at particles confined to a region V in space. The Hilbert space of n distinguishable particles is given by the n-fold tensor product

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Hn = H ⊗ · · · ⊗ H.

(2)

If we restrict our attention to one-particle states in D then of course only the subspace D ⊗ · · · ⊗ D of Hn is relevant. To be able to describe processes involving creation and annihilation of particles, we build the direct sum Hilbert space + F(H) = Hn , (3) n≥0

where H0 = C. F(H) is called the Fock space over the Hilbert space H. The Hilbert space Hn identified as a subspace of Fock space is called the n-sector (or the n-th chaos, in quantum probability). The zero-sector is also called the vacuum sector. An element ψ of F(H) is a sequence ψ = {ψn }n≥0 with ψn ∈ Hn . We write sometimes the n-particle component ψn of ψ as [ψ]n . The scalar product on F(H) is given by ψn , φn Hn , (4) ψ, φ = n≥0

where ·, ·Hn is the scalar product of Hn , which we take to be antilinear in the first argument and linear in the second one. The direct sum in (3) is the decomposition of Fock space into spectral subspaces (eigenspaces) of the selfadjoint number operator, N , defined as follows. The domain of N is ⎫ ⎧ ⎬ ⎨ n2 ψn 2Hn < ∞ , (5) D(N ) = ψ ∈ F(H) ⎭ ⎩ n≥0

and the action of N is given, for ψ ∈ D(N ), by [N ψ]n = n[ψ]n .

(6)

Clearly, the spectrum of N is discrete and consists of all integers n ∈ N. The vector Ω ∈ F(H) given by [Ω]0 = 1 ∈ C, [Ω]n = 0 ∈ Hn , if n > 0,

(7)

is called the vacuum (vector). It spans the one-dimensional kernel of N . The degree of degeneracy of the eigenvalue n of N equals dim(Hn ) = (dim H)n . Let us now consider a system of indistinguishable particles. The indistinguishability is reflected in the symmetry of the state vector (wave function) under the exchange of particle labels. We are adopting in these notes the view that all quantum particles fall into two categories: either the state vectors are symmetric under permutations of indices, in which case the particles are called Bosons, or the state vectors are anti-symmetric under permutations of indices, in which case the particles are called Fermions. For example, let {fk }nk=1 ⊂ H be n state vectors of a single particle. The vector f1 ⊗ · · · ⊗ fn ∈ Hn is the state of an n-particle system where the particle labelled by k is in the state fk . The state describing n Bosons, one of which

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(but we cannot say which one, because they are indistinguishable) is in the state f1 , one of which is in the state f2 , and so on, is given by the symmetric state vector 1 fπ(1) ⊗ · · · ⊗ fπ(n) ∈ Hn , (8) n! π∈Sn

where Sn is the group of all permutations π of n objects. The corresponding vector describing n Fermions is given by 1 (π)fπ(1) ⊗ · · · ⊗ fπ(n) ∈ Hn , (9) n! π∈Sn

where (π) is the signature of the permutation π. 3 Let us introduce the symmetrization operator P+ and the anti-symmetrization operator P− on F(H). Set P± Ω = Ω and for {fk }nk=1 ⊂ H, n ≥ 1, set 1 P+ f1 ⊗ · · · ⊗ fk = fπ(1) ⊗ · · · ⊗ fπ(n) , (10) n! π∈Sn 1 P− f1 ⊗ · · · ⊗ fk = (π)fπ(1) ⊗ · · · ⊗ fπ(n) , (11) n! π∈Sn

and extend the action of P± by linearity to the sets .K / (k) (k) Dn = f1 ⊗ · · · ⊗ fn(k) K ∈ N, fl ∈ H ⊂ Hn , n ≥ 1.

(12)

k=1

1 It is clear that P± f1 ⊗· · ·⊗fn ≤ n! n∈Sn f1 · · · fn = f1 ⊗· · ·⊗fn , so P± is a contraction on Dn , P± ψ ≤ ψ for ψ ∈ Dn . Consequently the operators P± extend to all of Hn , for all n, and to F(H) by sector-wise action. 4 Of course P± are actually selfadjoint projections; i.e., P±2 = P± = P± ∗ and they satisfy P± = 1. We define the Bosonic Fock space, F+ (H), and the Fermionic Fock space, F− (H), to be the symmetric and anti-symmetric parts of F(H): + P± Hn . (13) F± (H) = P± F(H) = n≥0

The number operator (6) leaves F± (H) invariant. We will not distinguish in our notation between N and its restriction to those invariant subspaces. 3

Let us recall that every permutation π ∈ Sn is uniquely decomposed into a (commutative) product of cycles and that every cycle is a (non commutative, non unique) product of transpositions (a cycle of length two). The number of transpositions in the decomposition of each cycle is a constant modulo 2. One defines the signature of π to be (π) = (−1)#(transp) , where #(transp) is the number of transpositions in any decomposition of π. The permutation π is called even if (π) = 1 and odd if (π) = −1. Each cycle of length l(cycle) ≥ 2 is the product of l(cycle)−1 transpositions, so we have the relations #(transp in c)

4

(l(cycle)−1)

c:cycles = (−1)n−#(cycles) = (π) = (−1) c:cycles = (−1) (−1)n+#(cycles) , where we use c:cycles l(cycle) = n. 0 Formally this means that we consider n≥0 P± on F (H), which we denote simply again by P± .

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2.2 Creation and annihilation operators Given f ∈ H, we define the annihilation operator a(f ) in the following way: a(f ) : H0 → 0 ∈ F(H), a(f ) : Hn → Hn−1 , n ≥ 1, and for {fk }nk=1 ⊂ H, √ a(f )f1 ⊗ · · · ⊗ fn !→ n f, f1 f2 ⊗ · · · ⊗ fn , (14) where ·, · is the scalar product in H. Similarly, we define the creation operator a∗ (f ) : Hn → Hn+1 by √ a∗ (f )f1 ⊗ · · · ⊗ fn !→ n + 1 f ⊗ f1 ⊗ · · · ⊗ fn . (15) The map f !→ a(f ) is antilinear, while f !→ a∗ (f ) is linear. We extend the action of the creation and annihilation operators by linearity to Dn , see (12), for all n. We have the following relations, for ψn ∈ Dn and f ∈ H: √ a(f )ψn ≤ n f ψn , (16) √ ∗ a (f )ψn = n + 1 f ψn , (17) where the symbol · denotes the norm in the obvious spaces. The bound (16) follows from a(f )ψn =

sup φ∈Hn−1 , φ =1

|φ, a(f )ψn |

K 1 √ 21 2 (k) (k) (k) f, f1 φ, f2 ⊗ . . . ⊗ fn = sup n φ∈Hn−1 , φ =1 k=1 K 1 √ 2 (k) = sup f ⊗ φ, f1 ⊗ · · · ⊗ fn(k) n n−1 φ∈H , φ =1 k=1 K 1 2 √ (k) ≤ n f Φ, f1 ⊗ · · · ⊗ fn(k) sup n Φ∈H , Φ =1 k=1 √ = n f ψn . Equality (17) is shown as follows K √ (k) (k) a (f )ψn = n + 1 f ⊗ f1 ⊗ · · · ⊗ fn k=1 K (k) √ (k) = n + 1 f ⊗ f1 ⊗ · · · ⊗ fn k=1 √ = n + 1 f ψn . ∗

By continuity, the action of a(f ) and a∗ (f ) extends to Hn , for all n, and hence by component-wise action to the domain D(N 1/2 ) ⊂ F(H). We have

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a# (f )ψ ≤ f (N + 1)1/2 ψ ,

189

(18)

for ψ ∈ D(N 1/2 ), where a# stands for either a or a∗ . The bound (18) is easily obtained from a# (f )ψ 2 = n≥0 a# (f )ψn 2 , (16), (17) and the definition of the number operator N , (6). The appearance of the star in a∗ (f ) is not an arbitrary piece of notation, it signifies that a∗ (f ) is the adjoint operator a(f )∗ of a(f ). We show this now. For all ψ, φ ∈ D(N 1/2 ), f ∈ H, we have ψ, a(f )φ = a∗ (f )ψ, φ .

(19)

Relation (19) follows easily from f1 ⊗ · · · ⊗ fn−1 , a(f )g1 ⊗ · · · ⊗ gn = a∗ (f )f1 ⊗ · · · ⊗ fn−1 , g1 ⊗ · · · ⊗ gn , for any n, f, fj , gj ∈ H, which in turn follows directly from the definitions of a# (f ), see (14), (15). Equality (19) shows that a∗ (f ) ⊆ a(f )∗ (the adjoint of a(f ) is an extension of a∗ (f )), so a(f )∗ is densely defined and consequently a(f ) is closable (a closed extension of a(f ) is a(f )∗∗ ). Similarly, one sees that a∗ (f ) is a closable operator. We denote from now on by a# (f ) the closed creation and annihilation operators. To show that a∗ (f ) = a(f )∗ it remains to prove that a∗ (f ) ⊇ a(f )∗ . Let ψ ∈ D(a(f )∗ ) then ϕ !→ ψ, a(f )ϕ (20) is a bounded linear map on D(a(f )). Given ϕ ∈ D(a(f )) we choose ϕ(n) to be the vector in Fock space obtained by setting all components ϕk of ϕ equal to zero, for k > n. Due to the boundedness of the map (20) we have ψ, a(f )ϕ = lim

1

n→∞

n−1 2 ψ, a(f )ϕ(n) = lim ψk , a(f )ϕk+1 . n→∞

(21)

k=0

Equality (19) shows that for each fixed n we can move a(f ) to the left factor in the inner product, so ψ, a(f )ϕ = lim

n→∞

n−1

a∗ (f )ψk , ϕk+1 .

(22)

k=0

By the density of D(a(f )) the last equality extends to all vectors ϕ ∈ F(H) ∞ ∗ a (f )ψk 2 < ∞, so that and choosing ϕk+1 = a∗ (f )ψk shows that k=0 ∗ ∗ ∗ ψ ∈ D(a (f )). We conclude that D(a(f ) ) = D(a (f )). Since a∗ (f ) is closed we have a∗ (f )ψ = limn a∗ (f )ψ (n) , where ψ (n) is the truncation of ψ as explained above in the case of ϕ. Using this in (22) gives ψ, a(f )ϕ = a∗ (f )ψ, ϕ ,

(23)

for any ϕ in the dense set D(a(f )). Consequently, we have a(f )∗ ψ = a∗ (f )ψ which shows that a∗ (f ) ⊇ a(f )∗ . This finishes the proof of the statement a∗ (f ) = a(f )∗ .

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Notice that a(f )Ω = 0 for all f ∈ H and conversely, if ψ ∈ F(H) is s.t. a(f )ψ = 0 for all f ∈ H then ψ = zΩ, for some z ∈ C. The annihilation operators a(f ) leave the subspaces F± (H) invariant. This can be seen as follows. Let τi,j be the bounded linear operator on F(H) which interchanges indices i and j in the tensor product, e.g. τ1,2 is determined by τ1,2 f1 ⊗ f2 ⊗ f3 ⊗ · · · ⊗ fn = f2 ⊗ f1 ⊗ f3 ⊗ · · · ⊗ fn . An element ψn ∈ Hn is in the range of P± if and only if τi,j ψn = ±ψn , for all 1 ≤ i < j ≤ n. From the definition (14) of a(f ) we have for instance √ τ1,2 a(f )f1 ⊗ · · · ⊗ fn = n f, f1 f3 ⊗ f2 ⊗ · · · ⊗ fn = a(f )f1 ⊗ f3 ⊗ f2 ⊗ · · · ⊗ fn = a(f )τ2,3 f1 ⊗ · · · ⊗ fn , and in a similar fashion one sees that τi,j a(f ) = a(f )τi+1,j+1 . Consequently, if ψn is in the range of P± , then we have τi,j a(f )ψn = a(f )τi+1,j+1 ψn = ±a(f )ψn , so a(f )ψn is in the range of P± . We may write this also as P± a(f )P± = a(f )P± . The Bosonic (+) and Fermionic (−) creation and annihilation operators are defined to be the restrictions # a# ± (f ) = P± a (f )P± .

(24)

One then has a± (f ) = a(f )P± and a∗± (f ) = P± a∗ (f ). Using (14) and (15), it is not difficult to verify that a+ (g)a∗+ (f )f1 ⊗ · · · ⊗ fn =

n+1

g, fk P+ f1 ⊗ · · · ⊗ f#k ⊗ · · · ⊗ fn+1 ,

(25)

k=1

where the hat # means that the corresponding symbol is omitted, and where we have set fn+1 = f . Similarly, a∗+ (f )a+ (g)f1 ⊗ · · · ⊗ fn =

n

g, fk P+ f1 ⊗ · · · ⊗ f#k ⊗ · · · ⊗ fn+1 .

(26)

k=1

Bosonic creation and annihilation operators satisfy the canonical commutation relations (CCR): [a+ (g), a∗+ (f )] = g, f 1lF+ (H) , [a+ (f ), a+ (g)] = [a∗+ (f ), a∗+ (g)] = 0,

(27) (28)

for any f, g ∈ H, and where [x, y] = xy − yx is the commutator. Equations (27), (28) are understood in the strong sense on D(N ), on which products of two creation and annihilation operators are defined. Relation (27) follows directly from (25) and (26), and (28) can be established similarly. Fermionic creation and annihilation operators satisfy the canonical anti-commutation relations (CAR):

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{a− (g), a∗− (f )} = g, f 1lF− (H) ,

(29)

{a− (f ), a− (g)} = {a∗− (f ), a∗− (g)} = 0,

(30)

for any f, g ∈ H, and where {x, y} = xy + yx is the anti-commutator (a priori again understood in the strong sense on D(N ). However, it turns out that this relation extends to an equality of bounded operators, as we show now). Although the CCR and the CAR have a similar structure (just interchange commutators with anti-commutators), they impose very different properties on the respective creation and annihilation operators. For instance, it turns out that the Fermionic creation and annihilation operators extend to bounded operators, while this is not true in the Bosonic case. We see this by using the CAR to obtain 3 4 (31) a∗− (f )ψ 2 = ψ, a− (f )a∗− (f )ψ = − a− (f )ψ 2 + f 2 ψ 2 , for all ψ ∈ D(N ), from which it follows that a# − (f ) ≤ f . On the other hand, a∗− (f )Ω = f = f Ω , so a− (f )a∗− (f )Ω = f 2 = f a∗− (f )Ω , hence (32) a# − (f ) = f . Notice that this reasoning does not work for Bosons, because the minus sign on the r.h.s. of (31) would have to be replaced by a plus sign. The fact that a∗+ (f ) is an unbounded operator can be seen as follows. Let ψn ∈ F+ (H) be the normalized vector whose components are all zero except the n-particle component, which is f ⊗ f ⊗ · · · ⊗ f , for some f√ ∈ H, f = 1. Then we have √ a∗+ (f )ψn = n + 1ψn+1 , hence a∗+ (f )ψn = n + 1 → ∞, as n → ∞. This reasoning does not work for Fermions, because the vector ψn is not in the Fermionic Fock space. More generally, the Pauli principle says that it is impossible to have a state of several Fermions in which two among them are in the same one-particle state. This is expressed as (33) a∗− (f )a∗− (f ) = 0, for all f ∈ H, which follows immediately from (30). 2.3 Weyl operators On a mathematical level, dealing with unbounded operators is a delicate affair so from this point of view Fermionic creation and annihilation operators are more easily handled than the Bosonic ones. It is desirable to replace the set of Bosonic creation and annihilation operators by a set of bounded operators which are in a certain sense equivalent to the set of creation and annihilation operators. These bounded operators are called Weyl operators. We first form the (normalized) real and imaginary parts of a+ (f ) Φ(f ) =

a+ (f ) + a∗+ (f ) a+ (f ) − a∗+ (f ) √ √ , Π(f ) = , 2 2i

(34)

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defined as operators on D(N 1/2 ). We do not equip Φ and Π with an index + since we are going to use them only for Bosons (although one can do the same procedure for Fermions as well). We have Π(f ) = Φ(if ), so it suffices to consider the operators Φ(f ). Notice though that f !→ Φ(f ) is not a linear nor an antilinear map; it is only a real-linear map. Define the finite particle subspace of Fock space by

0 (H) = ψ = {ψn }n≥0 ∈ F+ (H) | all but finitely many ψn are zero . (35) F+ 0 Clearly, F+ (H) ⊂ D(N ν ) for any ν > 0. In particular, any polynomial in creation 0 (H). and annihilation operators is well defined as an operator on F+ 0 Proposition 2.1. 1. For any f ∈ H, Φ(f ) is essentially selfadjoint on F+ (H). If {fn } is a sequence in H converging to f ∈ H, i.e. fn − f → 0, then Φ(fn ) → Φ(f ) in the strong sense on D(N 1/2 ), i.e. (Φ(fn ) − Φ(f ))ψ → 0, for all ψ ∈ D(N 1/2 ). 0 (H), we have 2. On F+ eitN Φ(f )e−itN = Φ(eit f ), (36)

for any t ∈ R, f ∈ H. 3. For f, g ∈ H, we have the CCR [Φ(f ), Φ(g)] = iIm f, g ,

(37)

understood in the strong sense on D(N ). Proof. An elegant proof of essential selfadjointness can be given using the GlimmJaffe-Nelson commutator theorem, c.f. [13]. We opt for a more pedestrian proof involving analytic vectors, 5 because these are useful for concrete calculations. Nelson’s analytic vector theorem says that if the domain of a symmetric operator contains an invariant subspace C which itself contains a dense set (in Hilbert space) of analytic vectors, then the symmetric operator is essentially selfadjoint on C. (See e.g. [13], Theorem X.39). 0 is invariant under Φ(f ). We show that each Let f ∈ H be fixed. The dense set F+ 0 vector ψ ∈ F+ is analytic for Φ(f ). Because ψ is a finite sum of vectors ψn ∈ P+ Hn (for possibly varying n), it is enough to show that ψn is an analytic vector for Φ(f ), for any n. It is clear that ψn ∈ D(Φ(f )k ), for all k ≥ 0 and from √ √ √ Φ(f )k ψn ≤ 2 f (N + 1)1/2 Φ(f )k−1 Ψn ≤ 2 n + k f Φ(f )k−1 ψn it follows that Φ(f )k ψn ≤ 2k/2 This means that the series

tk k≥0

5

k!

5

(n + k)! f k ψn .

Φ(f )k ψn

Let A be a linear operator on a Hilbert space H. A vector ψ ∈ H is called analytic for A if ψ ∈ ∩k≥0 D(Ak ) and the complex power series k≥0 tk Ak ψ/k! has a nonzero radius of convergence. If the radius of convergence is infinite then ψ is said to be entire for A.

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converges for any t ∈ C, hence ψn is an analytic (even an entire) vector for Φ(f ). We now show the strong continuity property. Let ψ ∈ D(N 1/2 ) ∩ F+ (H). Then (Φ(fn ) − Φ(f ))ψ ≤ 2−1/2 a∗ (fn − f )ψ + 2−1/2 a(fn − f )ψ √ ≤ 2 fn − f (N + 1)1/2 ψ , and the result follows. To see 2., simply use the definition of the creation operator to obtain √ eitN a∗+ (f )e−itN P+ f1 ⊗ · · · fn = n + 1eit P+ f ⊗ f1 ⊗ · · · ⊗ fn = a∗+ (eit f )P+ f1 ⊗ · · · ⊗ fn , and similarly for annihilation operators. The proof of 3. is immediate from (27), (28).

From now on we denote by Φ(f ) the selfadjoint closure of (34). It generates a strongly continuous one-parameter group of unitaries on the Hilbert space F+ (H), R * t !→ eitΦ(f ) .

(38)

We define the Weyl operator W (f ), for f ∈ H, to be the unitary operator W (f ) = eiΦ(f ) .

(39)

We have encountered the CCR expressed in terms of creation and annihilation operators (see (27), (28)) and in terms of the operators Φ(f ) (see (37)). How are they expressed in terms of the Weyl operators? Taking into account (37), the BakerCampbell-Hausdorff formula gives (formally) W (f )W (g) = e− 2 Imf,g W (f + g) = e−iImf,g W (g)W (f ). 6 i

(40)

Relation (40) is called the Weyl form of the CCR. The following result is sometimes useful. Proposition 2.2. On the domain D(N ) of the number operator we have N W (f ) = W (f )N + W (f )(Φ(if ) + f 2 /2),

(41)

for any f ∈ H. This means in particular that the Weyl operators leave D(N ) invariant. It follows thus from (40) that any finite sum of products of Weyl operators leave D(N ) invariant. 6

The BCH formula is the non-commutative analogue of the formula ea eb = ea+b . Let A, B be bounded operators on a Hilbert space H. Then eA eB = exp{A + B + 12 [A, B] + 1 ([A, [A, B]]−[B, [A, B]])+· · · } (these are the first explicit terms in the BCH formula). 12 In case the commutator [A, B] is proportional to the identity the BCH formula simply 1 1 reduces to eA eB = eA+B+ 2 [A,B] = eA+B e 2 [A,B] . Formally (40) follows thus from (37). Recall though that the Φ(f ), Φ(g) are unbounded operators. It is correct to say that 0 (H), one has [Φ(f ), Φ(g)] = (40) implies (37); this can be seen by noticing that, on F+ 1 2 ∂ | (W (tf )W (sg) − W (sg)W (tf )), and then calculating the r.h.s. using (40). s=t=0 2 st i

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Proof. To show (41) we notice first that eitN W (f ) = W (eit f )eitN (which follows from (36)). Using ∂t |t=0 Φ(eit f ) = Φ(if ), ∂t |t=0 Φ(eit f )n = nΦ(f )n−1 Φ(if ) − i f 2

n(n − 1) Φ(f )n−2 , for n ≥ 2, 2

0 we obtain, in the strong sense on F+ ,

in Φ(eit f )n 1 1 ∂t |t=0 W (eit f )eitN = W (f )N + ∂t |t=0 i i n! n≥0 = W (f )N + W (f ) Φ(if ) + f 2 /2 , which extends to D(N ), giving (41).

We finish this section by examining the continuity properties of the map f !→ W (f ). Recall that for Fermionic creation and annihilation operators, f !→ a# − (f ) is a continuous map from H into the bounded operators (equipped with the operator-norm topology), see (32). As we show now only a weaker form of continuity holds for the map f !→ W (f ). This is a source of considerable trouble in many applications. Theorem 2.3. If fn → f in H, then W (fn ) → W (f ) in the strong sense on F+ (H), i.e., for any ψ ∈ F+ (H), (W (fn ) − W (f ))ψ → 0. However, for any f ∈ H, f = 0, we have W (f ) − 1l = 2. Let eith be a strongly continuous unitary group on H (h being its selfadjoint generator). Due to the theorem we have W (eith f ) − 1l = 2 (for f = 0), which implies that t !→ W (eith f ) is not norm continuous (the dynamics defined by eith is not continuous in the C ∗ algebra topology). Proof of Theorem 2.3. The previous proposition tells us that Φ(fn ) → Φ(f ), in 0 (H), which is a joint core for all the operators Φ(fn ) and the strong sense on F+ Φ(f ). Therefore, Φ(fn ) converges to Φ(f ) in the strong resolvent sense (see e.g. [13], Theorem VII.25]), from which it follows that eitΦ(fn ) converges to eitΦ(f ) in the strong sense, for all t ([13], Theorem VII.21). Let us show W (f ) − 1l = 2, for any f = 0. The CCR (40) give W (g)∗ W (f )W (g) = e−iImf,g W (f ), for any g ∈ H. Since W (g) is unitary, this tells us that the spectrum of W (f ) is invariant under rotations, hence it must be the whole unit circle. The assertion W (f ) − 1l = 2 follows now from the spectral theorem. 2.4 The C ∗ -algebras CARF (H), CCRF (H) The set of all Fermionic creation and annihilation operators generates a C ∗ -algebra of operators on F− (H), which we call CARF (H). The index F reminds us that the

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elements of this C ∗ -algebra are viewed as operators on Fock space F− (H). Similarly, the set of all Weyl operators generates a C ∗ -algebra of operators on F+ (H), which we shall call CCRF (H). Both algebras are unital C ∗ -algebras. For CARF (H) this follows from (29), and for CCRF (H) it follows from W (0) = 1l. Theorem 2.4. Let a∗ (f ) and a(f ) denote the Fermionic creation and annihilation operators, acting on F− (H). The linear span of vectors of the form a∗ (f1 ) · · · a∗ (fn )Ω, with fk ∈ H, n ≥ 0, is dense in F− (H). In particular, Ω is cyclic for CARF (H) in F− (H). 7 Moreover, CARF (H) acts irreducibly on F− (H). 8 Proof. The first statement follows from a∗ (f1 )a∗ (f2 ) · · · a∗ (fn )Ω =

√

n! P− f1 ⊗ f2 ⊗ · · · ⊗ fn .

To see irreducibility, we suppose that T is a bounded operator on F− (H) that commutes with all operators a# (f ), f ∈ H, and show that T = z1l, for some z ∈ C. We have a(f )T Ω = T a(f )Ω = 0, for all f ∈ H, so T Ω = zΩ, for some complex number z (see after (19)). It follows that T a∗ (f1 ) · · · a∗ (fn )Ω = a∗ (f1 ) · · · a∗ (fn )T Ω = za∗ (f1 ) · · · a∗ (fn )Ω, so by cyclicity of Ω, T ψ = zψ, for all ψ ∈ F− (H).

Theorem 2.5. The vacuum vector Ω ∈ F+ (H) is cyclic for CCRF (H) in F+ (H), and CCRF (H) acts irreducibly on F+ (H). Proof. As in the case of Fermions, it is clear that the span of {a∗ (f1 ) · · · a∗ (fn )Ω | fk ∈ H, n ≥ 0} is dense in F+ (H). But this is the same as the span of {Φ(f1 ) · · · Φ(fn )Ω | fk ∈ H, n ≥ 0}, so it is enough to prove that CCRF (H) is dense in that latter span. We show first that (42) (N + 1)k W (f )(N + 1)−k−1 is a bounded operator for all f ∈ H and all k ≥ 0. We proceed by induction in k. The statement is obvious for k = 0. Using (41) of Proposition 2.2 we get 7

8

Let ψ be a vector in a Hilbert space H and let M be a set of bounded operators on H, M ⊆ B(H). We say that ψ is cyclic for M in H if Mψ = {M ψ | M ∈ M} is dense in H. Let M be a set of bounded operators acting on a Hilbert space H. We say that M acts irreducibly if the only closed subspaces of H which are invariant under the action of M are the trivial subspaces {0} and H. M acts irreducibly on H if and only if its commutant is trivial, M = {T ∈ B(H) | T M = M T, ∀M ∈ M} = C1l.

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(N + 1)k (N + 1)W (f )(N + 1)−1 (N + 1)−k−1 = (N + 1)k W (f )(N + 1)−k−1 +(N + 1)k {W (f )(Φ(if ) + f /2)} (N + 1)−k−2 ,

(43) (44)

where we commuted N + 1 through W (f ) in the r.h.s. By the induction assumption, (43) is a bounded operator. The term with the field operator in (44) can be written as (N + 1)k W (f )(N + 1)−k−1 (N + 1)k+1 Φ(if )(N + 1)−k−2 , where the product of the first three operators is again bounded. It suffices thus to show that (45) (N + 1)k Φ(f )(N + 1)−k−1 is bounded, for all f ∈ H and k ≥ 0. Clearly we have a(f )N = (N + 1)a(f ), a∗ (f )N = (N − 1)a∗ (f ), and (45) follows easily. This finishes the proof of (42). Since the product of Weyl operators is again a Weyl operator (modulo a phase) we get a bounded operator also if we replace W (f ) in (42) by any sum of products of Weyl operators. Given any > 0 there exists a Tn () < ∞ such that Φ(f1 ) · · · Φ(fn )Ω = Φ(f1 ) · · · Φ(fn−1 )

W (tn fn ) − 1l Ω + O(), itn

provided tn ≤ Tn (), and where O() denotes a vector with norm less than . There exists a Tn−1 (, tn ) such that Φ(f1 ) · · · Φ(fn )Ω = Φ(f1 ) · · · Φ(fn−2 )

W (tn−1 fn−1 ) − 1l W (tn fn ) − 1l Ω + O(), itn−1 itn

provided tn−1 ≤ Tn−1 (, tn ). Continuing this process we see that there are numbers Tn (), Tk (, tn , . . . , tk+1 ), 1 ≤ k ≤ n − 1, such that if tk ≤ Tk (, tn , . . . , tk+1 ) and tn ≤ Tn () then Φ(f1 ) · · · Φ(fn )Ω W (tn−1 fn−1 ) − 1l W (tn fn ) − 1l W (t1 f1 ) − 1l ··· Ω + O(). = it1 itn−1 itn Since the operator acting on Ω in the above r.h.s. is an element of CCRF (H) cyclicity of Ω is shown. We finish the proof by showing irreducibility. Suppose T is a bounded operator on F+ (H) that commutes with all W (f ), f ∈ H. It follows that for any ψ ∈ D(Φ(f )), eitΦ(f ) − 1l eitΦ(f ) − 1l Tψ = T ψ −→ T Φ(f )ψ, it it as t → 0. This shows that T ψ ∈ D(Φ(f )) and that Φ(f )T ψ = T Φ(f )ψ, i.e. T leaves the domain of every Φ(f ) invariant and T commutes strongly with every Φ(f ). Since a(f ) = 2−1/2 (Φ(f ) + iΦ(if )), this means that T commutes with a(f ), in the strong sense, for all f ∈ H. Irreducibility is now shown exactly as in Theorem 2.4.

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2.5 Leaving Fock space We explain in this section why Fock space is not always the right Hilbert space to describe a physical system. As we have pointed out in Section 1.1, the very definition of Fock space gives the existence of a number operator, N , which is the operator of multiplication by n on the n-sector. Let ψ ∈ F(H) be a (pure) state of the quantum gas (the following reasoning applies equally well to mixed states given by density matrices, i.e., convex combinations of pure states). The probability of finding more than a fixed number n of particles in the state ψ is given by [ψ]k 2 , (46) ψ, P (N ≥ n)ψ = k≥n

where P (N ≥ n) is the spectral projection of N onto the set {n, n + 1, . . .}. The probability (46) vanishes in the limit n → ∞, simply because ψ is in Fock space (the series converges). This shows that, a priori, any state described by a vector (or a density matrix) in Fock space has only finitely many particles in the sense that the probability of finding n particles approaches zero as n increases to infinity. We will be interested in describing an ideal quantum gas which is extended in all of physical space R3 , and which has a nonzero density, say one particle per unit volume. Such a state cannot be described by a vector (or density matrix) in Fock space! We may describe such a state as a limit of states “living” in Fock space (i.e., given by a density matrix on Fock space), e.g. by saying that the system should first be confined to a finite box Λ0 ⊂ R3 , in which case it is described by a vector ψΛ0 ∈ F(L2 (Λ0 )) (of course, since the box is finite, and we specify a fixed density, there are only finitely many particles and Fock space can describe such a state). One then takes a sequence of nested boxes, Λ0 ⊂ Λ1 ⊂ · · · which increase to all of R3 , ∪k≥0 Λk = R3 , hence obtaining a sequence of states ψΛk ∈ F(L2 (Λk )). If one can show that ψΛk has a limit ψ∞ , in a suitable sense, and where the density or particles is fixed, as k → ∞, then ψ∞ can be regarded as being the infinitely extended state with nonzero density. This limit is called the thermodynamic limit. The limit state ψ∞ is naturally not a vector in Fock space any more. What kind of object is it? To answer this, we have to say in what sense we take the thermodynamic limit. To be specific, we carry out the following discussion for Bosons. It can be repeated for Fermions. For any finite box Λ, the vector ψΛ ∈ F+ (L2 (Λ)) gives rise to a positive, linear, normalized map on the von Neumann algebra of all bounded operators on F+ (L2 (Λ)) by the assignment B F+ (L2 (Λ)) * A !→ ωΛ (A) = ψΛ , AψΛ (47) (for a mixed state determined by the density matrix ρΛ , we set ωΛ (A) = tr(ρΛ A)). Since CCRF (L2 (Λ)) is irreducible (see Theorem 2.5), its weak closure is the set of all bounded operators (indeed, irreducibility implies that CCRF (H) = C1l, so CCRF (H) = B(H)). Without loss of generality, we may therefore consider (47)

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only for A ∈ CCRF (L2 (Λ)), i.e. we view ωΛ as a state on CCRF (L2 (Λ)), in the sense of the theory of C ∗ -algebras. 9 Consider the (so-called quasi-local) C ∗ -algebra A0 =

,

norm

CCRF (L2 (Λn ))

⊂ B F+ (L2 (R3 ))

n≥0

where norm means that we take the norm closure in the operator norm. Assume that the limit (48) ω∞ (A) = lim ωΛk (A) k→∞

exists, for any A ∈ CCRF (L (Λn )), any n. Then ω∞ defines a state on A0 . We point out once more that in general, ω∞ cannot be represented by a density matrix on Fock space F+ (L2 (R3 , d3 x)). One says that ω∞ is not normal with respect to the states ωΛk . 10 In the GNS representation (H∞ , π∞ , ψ∞ ) of (A0 , ω∞ ), the state ω∞ is represented as ω∞ (A) = ψ∞ , π∞ (A)ψ∞ . 2

In Section 4 we will discuss in detail the construction of the infinite-volume limit of a state describing a Bose gas with a given momentum density distribution and we will explicitly construct the corresponding GNS representation (the Araki-Woods representation). One may wonder about the dependence of the C ∗ -algebra CCRF (H) on its underlying Hilbert space, F+ (H). After all, we have just seen that density matrices on F+ (H) cannot describe certain states of physical interest. Therefore Fock space should not play a central role in the definition of a physical system. In an attempt to detach ourselves from Fock space we may define the CCR and CAR algebras as abstract C ∗ -algebras, without referring to a Hilbert space. Fock space is then just the GNS representation space of a certain state on the abstract algebras, represented by the Fock vacuum vector (recall that the Fock vacuum vector is cyclic for CCRF (H) and CARF (H), as we have shown in Theorems 2.4 and 2.5 above).

3 The CCR and CAR algebras In this section we introduce abstract CAR and CCR algebras and review some of their properties. Useful references are [4], [5] [14]. We remind the reader of the notion of the “test function space” D ⊆ H, introduced at the beginning of Section 2.1, see (1).

9

10

Let A be a (unital) C ∗ -algebra. A state ω on A is a positive linear functional ω : A → C which is normalized as ω(1l) = 1. Let ω1 and ω2 be two states on a C ∗ -algebra A. Then ω1 is called normal with respect to ω2 iff ω1 (A) = tr(ρπ2 (A)), where ρ is a trace class operator (density matrix) on H2 , and where (H2 , π2 , Ω2 ) is the GNS representation of (A, ω2 ).

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3.1 The algebra CAR(D) An (abstract) CAR algebra CAR(D) over D ⊆ H (where H is a Hilbert space) is defined to be a unital C ∗ -algebra generated by elements written as a(f ), f ∈ D, where the assignment f !→ a(f ) is an antilinear map, and where the following relations hold (49) {a(f ), a(g)} = 0, {a(f ), a∗ (g)} = f, g 1l. Here a∗ (f ) is the element in the C ∗ algebra obtained by applying the ∗operation to a(f ), and {a, b} = ab + ba is the anticommutator. We have already seen in the previous section that a C ∗ -algebra with these properties exists. Let us mention that the CAR (49) imply that a(f ) = f , (50) where · on the left hand side is the C ∗ norm and on the right hand side it is the norm of D induced by H. This follow since (a(f )a(f ) = 0 by the Pauli principle, see (33)) 2 ∗ a (f )a(f ) = a∗ (f ){a(f ), a∗ (f )}a(f ) = f 2 a∗ (f )a(f ), so that by the C ∗ norm property ( A∗ A = A 2 ), we have a(f ) 4 = f 2 a(f ) 2 . Alternatively, boundedness of the Fermionic creation and annihilation operators follows from the fact that π(a(f )) = f (51) in any representation π of the CAR, which is shown as in (32). Let fα be a net in D converging to f ∈ D (the closure of D ⊆ H). Then a(f )−a(fα ) = f −fα → 0, so a(f ) ∈ CAR(D) because CAR(D), being a C ∗ -algebra, is uniformly closed. This shows that (52) CAR(D) = CAR(D). The next result tells us that given D, the corresponding CAR algebra CAR(D) is unique. Theorem 3.1. (Uniqueness of the CAR algebra). Let D ⊆ H be a given test function space (see (1)), and let A1 , A2 be two CAR algebras over D (generated by a1 (f ) and a2 (f ), respectively, with f ∈ D). There is a unique ∗isomorphism α : A1 → A2 such that α(a1 (f )) = a2 (f ), for all f ∈ D. A proof can be found for instance in [5]. Once uniqueness is known in the sense above, one can easily prove the following result. Theorem 3.2. The C ∗ algebra CAR(D) is simple. 11 11

A C ∗ algebra A is called simple if it has no nontrivial closed two-sided ideals, i.e., if the only closed two-sided ideals are {0} and A. A subspace I ⊆ A is a two-sided ideal if A ∈ A and I ∈ I implies that IA and AI are in I.

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Proof. Let I = A1 = CAR(D) be a closed two-sided ideal of CAR(D). Define A2 = CAR(D)/I to be the C ∗ algebra generated by the equivalence classes a2 (f ) = [a(f )]. Theorem 3.1 tells us that the projection P : CAR(D) !→ CAR(D)/I is an isomorphism. Therefore the kernel of P , which is the span of I, must be zero: I = {0}. An interesting consequence of the simplicity is that every representation of CAR(D) is faithful (has trivial kernel). Indeed, let π be a (nonzero) representation of CAR(D). It is readily verified that ker π is a two-sided, closed ideal of CAR(D). Hence by Theorem 3.2, ker π = {0}. 3.2 The algebra CCR(D) An (abstract) Weyl algebra, or CCR algebra CCR(D) over a test function space D ⊆ H is defined to be the unital C ∗ algebra generated by elements W (f ), f ∈ D, satisfying the relations W (−f ) = W (f )∗ , W (f )W (g) = e− 2 Imf,g W (f + g). i

(53)

We have seen in the previous section that an algebra with these properties exists. The CCR (53) imply that f !→ W (f ) is not continuous (in the C ∗ norm topology). Indeed, the proof of Theorem 2.3 shows that we have W (f ) − 1l = 2, for any f = 0. Similarly to the CAR case, the Weyl algebra is unique. Theorem 3.3. (Uniqueness of the Weyl algebra). Let D ⊆ H be given and let W1 and W2 be two Weyl algebras over H (generated by W1 (f ) and W2 (f ), f ∈ D). There is a unique ∗isomorphism α : W1 → W2 such that α(W1 (f )) = W2 (f ), for all f ∈ D. A proof can be found in [5],[12]. As for the CAR algebra, simplicity of the CCR algebra follows from uniqueness. Theorem 3.4. The C ∗ algebra CCR(D) is simple. Due to the lack of continuity of the map f !→ W (f ) it is not true that the Weyl algebra over D is the same as the one over D if D = D. One can show that if D1 and D2 are two linear (not necessarily closed) subspaces of H then CCR(D1 ) = CCR(D1 ) ⇐⇒ D1 = D2 , see e.g. [5], Proposition 5.2.9. In particular, CCR(D) = CCR(D) if and only if D is closed. Another difficulty is generated by the lack of continuity of the map t !→ W (eith f ),

(54)

where t ∈ R and h is some selfadjoint operator on H (leaving D invariant). The assignment (54) is called a Bogoliubov transformation. It represents a dynamics of

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the system, where h is interpreted as the one-particle Hamiltonian. The lack of continuity prevents us from treating the dynamics with ease on an algebraic level; for instance, one cannot take the derivative (nor the integral) of the r.h.s. of (54) w.r.t. t – and these operations are important e.g. to define a perturbed dynamics. There are representations of the CCR for which weaker continuity properties hold; we look at them now. By a regular representation π of CCR(D) we understand one with the property that t !→ π(W (tf )) is continuous in the strong operator topology on the representation Hilbert space H, for all f ∈ D. A state ω on CCR(D) is called a regular state if its GNS representation is regular (see also Theorem 4.1). For a regular representation the map t !→ π(W (tf )) is a strongly continuous one-parameter group of unitaries on H. 12 The Stone-von Neumann theorem tells us that this group has a selfadjoint generator on H, which we denote by Φπ (f ), π(W (tf )) = eitΦπ (f ) . It is convenient to introduce annihilation and creation operators in the regular representation π by setting aπ (f ) =

Φπ (f ) + iΦπ (if ) Φπ (f ) − iΦπ (if ) √ √ , a∗π (f ) = . 2 2

(55)

Compare this with (34)! Definition (55) needs some explanation because Φπ (f ) and Φπ (if ) are both unbounded operators on H. Proposition 3.5. Let F = {f1 , . . . , fn } be a finite collection of elements in D. The operators {Φπ (fj ), Φπ (ifj )}N j=1 have a common set of analytic vectors which is dense in the representation Hilbert space H. This means that, for f ∈ D fixed, the domain Dπ,f := D(aπ (f )) := D(a∗π (f )) := D(Φπ (f )) ∩ D(Φπ (if ))

(56)

is dense in H. We understand the equalities (55) in the sense of operators on Dπ,f . Both aπ (f ) and a∗π (f ) are closed operators on Dπ,f . We have proved after equation (19) above that, for a# (f ) defined as in Section 2.2, the adjoint operator of a(f ) is a∗ (f ). This can be shown for any regular representation, i.e., we have (57) a∗π (f ) = aπ (f )∗ . A proof of (57) can be found in [5]. Proof of Proposition 3.5. The following “smoothing” is useful: let f ∈ D and consider the integral (understood in the strong sense on H) 12

The group properties follow from π(W (tf ))π(W (sf )) = π(W (tf )W (sf )) i = e 2 stImf,f π(W ((s + t)f )) = π(W ((s + t)f )) and π(W (f ))∗ = π(W (f )∗ ) = π(W (−f )) = π(W (f )−1 ) = π(W (f ))−1 .

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-

n π

R

ds e−ns Wπ (sf ), 2

(58)

where n > 0 and where we set π(W (f )) = Wπ (f ). The strong limit of (58), as n → ∞, is just the identity operator on H. We apply the operator Wπ (tf ) to the integral in (58) and obtain, after a change of variable, 2 −ns2 Wπ (tf ) ds e Wπ (sf ) = ds e−n(s−t) Wπ (sf ). (59) R

R

The r.h.s. of (59) has an analytic extension in t to the whole complex plane. Similarly, if fk is any element in F then the map ⎛ ⎞ N n N/2 2 2 t !→ Wπ (tfk ) ds1 · · · dsN e−n(s1 +···+sN ) Wπ ⎝ sj fj ⎠ (60) π R R j=1 is easily seen to have an analytic extension in t to all of C, and the r.h.s. of (60) converges in the strong sense to Wπ (tfk ), as n → ∞. This means that any vector of the form ⎛ ⎞ N n N/2 2 2 (61) ds1 · · · dsN e−n(s1 +···+sN ) Wπ ⎝ sj fj ⎠ ψ, π R R j=1 where ψ ∈ H is arbitrary, is an analytic (entire) vector for all operators in the set {Φπ (fj ), Φπ (ifj )}N j=1 . The set (61), where ψ varies over all of H, is dense in H, because (61) converges to ψ, as n → ∞. This shows the first part of the proposition. Let us now prove that the a# π (f ) are closed operators on Dπ,f , where f ∈ D is fixed. For any ψ ∈ Dπ,f we have, by (55), Φπ (f )ψ 2 + Φπ (if )ψ 2 = aπ (f )ψ 2 + a∗π (f )ψ 2 .

(62)

We use Wπ (sf )Wπ (itf ) = e−ist f Wπ (itf )Wπ (sf ) to get 2

1 2 ∂ |s=t=0 ψ, Wπ (sf )Wπ (itf )ψ = Φπ (f )ψ, Φπ (if )ψ i2 st = Φπ (if )ψ, Φπ (f )ψ + i f 2 ψ 2 , which implies that a∗π (f )ψ 2 = aπ (f )ψ 2 + f 2 ψ 2 .

(63)

Combining (62) and (63) yields the identity Φπ (f )ψ 2 + Φπ (if )ψ 2 = 2 aπ (f )ψ 2 + f 2 ψ 2 .

(64)

To show that aπ (f ) is a closed operator on Dπ,f assume that ψn ∈ Dπ,f is a sequence of vectors converging to some ψ ∈ H, such that aπ (f )ψn converges as n → ∞, i.e.,

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aπ (f )(ψn −ψm ) → 0, as n, m → ∞. It follows from (64) that both Φπ (f )(ψn − ψm ) and Φπ (if )(ψn −ψm ) converge to zero as n → ∞. Since Φπ (f ) and Φπ (if ) are closed operators (they are selfadjoint) we conclude that ψ ∈ D(Φπ (f )) and ψ ∈ D(Φπ (if )), i.e., ψ ∈ Dπ,f , and that Φπ (f )ψn → Φπ (f )ψ, Φπ (if )ψn → Φπ (if )ψ. Another application of (64) (with ψ replaced by ψn − ψ) shows that aπ (f )(ψn − ψ) 2 → 0 as n → ∞. Consequently aπ (f ) is a closed operator. In the same way one sees that a∗π (f ) is a closed operator. The Fock representation of CCR(D) is the regular representation defined by πF : CCR(D) → B(F+ (H)), πF (W (f )) = WF (f ),

(65)

where the operator on the r.h.s. is given by (39), and where the Bosonic Fock space F+ (H) was defined in (13). We mention another structural property of the Weyl algebra. Let D1 ⊆ H1 and D2 ⊆ H2 be two linear subspaces and let D1 ⊕ D2 ⊆ H1 ⊕ H2 be their direct sum (i.e., the not necessarily closed set of all f ⊕ g, f ∈ D1 , g ∈ D2 equipped with the usual direct sum operations). We have the relation CCR(D1 ⊕ D2 ) = CCR(D1 ) ⊗ CCR(D2 ).

(66)

This follows simply from the CCR (53), W (f1 ⊕ f2 ) = W (f1 ⊕ 0 + 0 ⊕ f2 ) = e 2 Imf1 ⊕0,0⊕f2 H⊕H W (f1 ⊕ 0)W (0 ⊕ f2 ), i

f1 ⊕ 0, 0 ⊕ f2 H⊕H = f1 , 0 + 0, f2 = 0 and the identifications W (f1 ⊕ 0) !→ W (f1 ) ⊗ W (0), W (0 ⊕ f2 ) !→ W (0) ⊗ W (f2 ). 3.3 Schr¨odinger representation and Stone – von Neumann uniqueness theorem Let us consider the easiest Weyl algebra CCR(C), where H = C is a one-dimensional Hilbert space. The Weyl operators are given by W (z), with z = s + it ∈ C, s, t ∈ R. They satisfy i

i

W (z) = W (s + it) = e 2 Ims,it W (s)W (it) = e 2 st W (s)W (it). Let us assume that we are in a regular representation of the CCR, i.e., τ !→ W (τ z) is a strongly continuous one parameter group (τ ∈ R) of unitaries on a (representation) Hilbert space. In particular, there are selfadjoint operators Φ, Π such that W (τ ) = eiτ Φ , W (iτ ) = eiτ Π . It is suggestive to write Φ = Φ(1) and Π = Φ(i), compare with (39). The generators satisfy the commutation relations [Φ, Π] = [Φ(1), Φ(i)] = iIm 1, i = i1l which can be seen by noticing that

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W (s)W (it)W (−s) = e−iIms,it W (it) = e−ist W (it),

(67)

which yields (by applying −i∂s |s=0 ) [Φ, W (it)] = −tW (it), and hence (by applying −i∂t |t=0 ) [Φ, Π] = i1l. These commutation relations remind us of [x, −i∂x ] = i, where x and −i∂x are selfadjoint operators on L2 (R, dx). We can define a regular representation πS of CCR(C) on L2 (R, dx) by i

πS (W (z)) = e 2 st U (s)V (t),

(68)

where z = s + it ∈ C, and U (s) and V (t) are the one-parameter (s, t ∈ R) unitary groups on L2 (R, dx) given by (U (s)ψ)(x) = eisx ψ(x), (V (t)ψ)(x) = ψ(x + t), with selfadjoint generators ΦS = x and ΠS = −i∂x . The representation (68) is called the Schr¨odinger representation of the CCR. Since this representation is regular we can introduce creation and annihilation operators (c.f. (34)) by 1 ΦS + iΠS √ = √ (x + ∂x ) 2 2 − iΠ 1 Φ S √ S = √ (x − ∂x ) . a∗S = 2 2

aS =

(69) (70)

Since both ΦS and ΠS are unbounded operators one has to take care in the exact definition of the unbounded (non-selfadjoint) operators a# S in (69), (70). This can be done by proceeding as in Proposition 3.5. These considerations show that L2 (R, dx) carries a Fock space structure, i.e., there are two (densely defined, closed, unbounded, non symmetric) operators aS and a∗S acting on L2 (R, dx) and satisfying the commutation relation [aS , a∗S ] = id. The commutator is understood in the strong sense on some dense set of vectors (e.g. the functions in C0∞ ). The vacuum vector ΩS ∈ L2 (R, dx) is given by the normalized solution of (x + ∂x )ΩS (x) = 0 (i.e. aS ΩS = 0), ΩS (x) = π −1/4 e−x

2

/2

.

We introduce a sequence of one-dimensional subspaces Hn ⊂ L2 (R, dx) spanned by (a∗S )n ΩS . Using the commutation relations for the creation and annihilation operator, one easily sees that the operator NS = a∗S aS =

1 2 −∂x + x2 − 1 2

(71)

leaves each Hn invariant, and that NS Hn = n id Hn . Notice that NS is just the Schr¨odinger operator (Hamiltonian) corresponding to a one-dimensional quantum

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205

harmonic oscillator (modulo the constant term −1/2). There are various ways to see that we have + Hn . (72) L2 (R, dx) = n≥0

For instance, one knows that the eigenvalues of NS are 0, 1, 2, 3, . . . and they are simple (harmonic oscillator!), so (72) is a consequence of the fact that the eigenvectors of NS span the entire space. The eigenvector ψn of NS with eigenvalue n ∈ N satisfies the equation NS ψn = nψn , which is equivalent to (c.f. (71)) (−∂x2 + x2 )ψn = (2n + 1)ψn ,

(73)

i.e. ψn is a harmonic oscillator eigenvector. The ψn are the Hermite functions, they have the form 2 1 2 1 1 (−1)n π −1/4 e 2 x (∂x )n e−x , ψn (x) = √ (a∗S )n ΩS (x) = √ n n! 2 n!

(74)

where √1n! is a normalization factor. The Schr¨odinger representation of CCR(Cn ) is defined as the n-fold tensor product representation of CCR(C), πS (W (z1 , . . . , zn )) =

n

i

e 2 sj tj Uj (sj )Vj (tj ),

(75)

j=1

acting on L2 (Rn , dn x) and where Uj (s), Vj (t) act on the variable xj in the obvious way. We may view Cn as C ⊕ · · · ⊕ C and compare (75) with (66). The above discussion shows that the Schr¨odinger and the Fock representations of CCR(C) are unitarily equivalent, the correspondence being (a∗S )n ΩS !→ (a∗F )n ΩF ,

(76)

where a∗F is the creation operator in the Fock representation, (65). This is not a coincidence, it can be viewed as a consequence of the following result. Theorem 3.6. (Stone – von Neumann uniqueness theorem). Let H be a finite dimensional Hilbert space. Any irreducible regular representation of CCR(H) is unitarily equivalent to the Fock representation of CCR(H). It is instructive to have a look at the mechanism behind the proof of the Stone – von Neumann uniqueness theorem. Outline of the proof of Theorem 3.6. Let {f1 , . . . , fn } be an orthonormal basis of H and define the non-negative operator (the number operator) Nπ =

n j=1

a∗π (fj )aπ (fj ),

(77)

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where the creation and annihilation operators are defined as in Proposition 3.5. One can show that Nπ is a non-negative selfadjoint operator on the representation Hilbert space which we shall call H. Using the CCR we find that, for any f ∈ H, Nπ aπ (f ) = aπ (f )(Nπ − 1).

(78)

Let n0 > −∞ be the infimum of the spectrum of Nπ and let P (Nπ ≤ n0 + 1/2) denote the spectral projection of Nπ associated with the interval [n0 , n0 + 1/2]. Take any normalized Ωπ ∈ Ran P (Nπ ≤ n0 + 1/2). Relation (78) tells us that P (Nπ ≤ x)aπ (f ) = aπ (f )P (Nπ ≤ x + 1) for any x, so we have aπ (f )Ωπ = 0, for any f ∈ H. Since π is irreducible the set Hπ = {Wπ (f )Ωπ | f ∈ H}, where Wπ (f ) = π(W (f )), is dense in H (the closure of Hπ is a closed subspace of H which is invariant under π(CCR(H)), and Hπ = {0} since 1l ∈ Hπ ). Proceeding as in the proof of Theorem 2.5 one shows that the closure of Hπ is the same as the closure of the set of vectors of the form a∗π (f1 ) · · · a∗π (fn )Ωπ , H = closure{a∗π (f1 ) · · · a∗π (fn )Ωπ | n ∈ N, f1 , . . . , fn ∈ H}.

(79)

Now we define the linear map U : H → F+ (H) by U a∗π (f1 ) · · · a∗π (fn )Ωπ = a∗F (f1 ) · · · a∗F (fn )ΩF .

(80)

It is easy to verify that U extends to a unitary map because the norms of a∗# (f1 ) · · · a∗# (fn )Ω# , # = π, F, can be calculated purely by using the fact that a# (fj )Ω# = 0 and the canonical commutation relations. This finishes the outline of the proof of the Stone – von Neumann uniqueness theorem. Since every representation can be decomposed into a direct sum of irreducible representations, Theorem 3.6 says that every regular representation of CCR(H), dim H < ∞, is a direct sum of Fock representations (in which case we say that the representation is quasi-equivalent to the Fock representation). If dim H = ∞ this is no longer true. In particular, the GNS representation corresponding to states of the infinitely extended free Bose gas with nonzero density which we will construct in Section 4 are not quasi-equivalent to the Fock representation. There is however a characterization of representations of CCR(H), where dim H = ∞, which are quasi-equivalent to the Fock representation. In view of the outline of the proof of the Stone – von Neumann uniqueness theorem this characterization is very natural, although its exact formulation is somewhat technical. The central object in the above proof of Theorem 3.6 is the number operator (77). It can be generalized by putting a∗π (fj )aπ (fj ), (81) Nπ = sup F

j

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207

where the supremum is over all finite-dimensional subspaces F of H, and the sum extends over an orthonormal basis {fj } of F . It is clear that a rigorous definition of (81) is not trivial. It can be given using quadratic forms rather than operators, see e.g. [5] Section 5.2.3. By proceeding as in the above outline of the proof of the Stone – von Neumann theorem one can show that a representation π of CCR(H) is a direct sum of Fock representations of CCR(H) if and only if the number operator (81) can be defined as a densely defined selfadjoint operator. This may be phrased as “π is quasi-equivalent to the Fock representation if and only if there is a number operator in the representation space of π”. A precise statement of this result can be found in [5], Theorem 5.2.14. 3.4 Q–space representation Our goal is to examine the unitary equivalence obtained from (76) when C is first replaced by Cn , and then n is taken to infinity. This will provide us with another representation of CCR(H), where H is a separable Hilbert space. The representation Hilbert space we construct is L2 (Q, dµ), where µ is a probability measure on Q, µ(Q) = 1. We give an explicit unitary equivalence between L2 (Q, dµ) and the bosonic Fock space F(H) (we write F instead of F+ ). The Q-space representation is particularly useful in the analysis of interacting fields, see e.g. [13]. The assignment a∗1 (f1 ) · · · a∗1 (fm )Ω1 ⊗ a∗2 (g1 ) · · · a∗2 (gn )Ω2 !→ a∗ (f1 ⊕ 0) · · · a∗ (fm ⊕ 0)a∗ (0 ⊕ g1 ) · · · a∗ (0 ⊕ gn )Ω, where fj ∈ H1 , gj ∈ H2 , establishes a unitary map between the Fock spaces F(H1 ) ⊗ F(H2 ) and F(H1 ⊕ H2 ) (compare with (66)). This means that F(Cn ) = F(C ⊕ · · · ⊕ C) ∼ = F(C) ⊗ · · · ⊗ F(C),

(82)

and taking into account the identification (76) we obtain F(Cn ) ∼ = L2 (R, dx) ⊗ · · · ⊗ L2 (R, dx) ∼ = L2 (Rn , dn x).

(83)

Let C be a conjugation on H, i.e., C is an antilinear isometry satisfying C 2 = 1. One may think of C as the operation of taking the complex conjugate of coordinates in a given basis of H. 13 Let {ej }∞ j=1 be an orthonormal basis of H such that each ej is invariant under C, Cej = ej . A consequence of introducing a basis of C invariant vectors is that if f = Cf and g = Cg then f, g = Cf , Cg = f, g, so the corresponding Weyl operators commute, W (f )W (g) = W (g)W (f ); c.f. (53) (similarly the field operators in a regular representation commute in the strong sense on a dense set of vectors). 13

If {fj } is any basis of H then define fj = fj + Cfj . The fj are invariant under C, Cfj = fj , and they span H. A Gram-Schmidt procedure yields an orthonormal basis {ej } of vectors satisfying Cej = ej . The action of C on coordinates w.r.t. the basis {ej } is complex conjugation.

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Let {f1 , . . . fn } be a finite collection of elements in {ej } and define Fn = closure{P (a∗ (f1 ), . . . , a∗ (fn ))Ω | P a polynomial } ⊂ F(H),

(84)

where Ω is the Fock vacuum and the a∗ are the creation operators in Fock representation, defined by (34) (we write a∗ instead of a∗+ ). Clearly, the map a∗ (f1 )k1 · · · a∗ (fn )kn Ω !→ a∗ (ζ1 )k1 · · · a∗ (ζn )kn ΩF (Cn ) ,

(85)

where ΩF (Cn ) is the vacuum vector in F(Cn ) and ζj ∈ Cn has zero components except for the j-th which equals one, extends to a unitary map between Fn and F(Cn ). The r.h.s. of (85) can be identified, via (83), (70), with the vector ζ1k1 · · · ζnkn

x1 − ∂x1 √ 2

k 1

···

xn − ∂xn √ 2 ⎛

where

Ωn = π −n/4 exp ⎝−

k n Ωn ∈ L2 (Rn , dn x),

n 1

2

(86)

⎞ x2j ⎠ .

(87)

j=1

We normalize Ωn to be the constant function by introducing the unitary map ⎞ ⎛ n 1 x2 ⎠ f (x) (T f )(x) = π n/4 exp ⎝ 2 j=1 j between L2 (Rn , dn x) and L2 (Rn , dµ1 × · · · × dµn ), where dµj = π −1/2 e−xj dxj . 2

Thus (85), (86) give a unitary map Un between Fn and L2 (Rn , dµ1 × · · · × dµn ) such that Un Ω = 1 (the constant function) and Un a∗ (fj )Un−1 = Un a(fj )Un−1 = √12 ∂xj 14 so that Un Φ(fj )Un−1 = Un

2xj −∂xj √ 2

,

a∗ (fj ) + a(fj ) −1 √ Un = xj . 2

Let Pj , j = 1, . . . , n be n polynomials in one variable. The unitarity of Un gives Ω, P1 (Φ(f1 )) · · · Pn (Φ(fn ))Ω = P1 (x1 ) · · · Pn (xn )dµ1 · · · dµn Rn

=

n

j=1

14

We have T ∂xj T −1 = ∂xj − xj

Ω, Pj (Φ(fj ))Ω .

(88)

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209

Let Q = ×∞ j=1 R be the set of sequences q = (q1 , q2 , . . .) equipped with the σ– algebra generated by countable products of measurable sets in R, and let µ = ⊗∞ j=1 µj . The pair (Q, µ) is a measure space (see e.g. Chapter VI of [9]), and the set of all polynomials P (q1 , . . . , qn ), n ∈ N, is dense in L2 (Q, dµ). The space Fn , (84), equals the closure of {P (Φ(f1 ), . . . , Φ(fn ))Ω}, where P ranges over all polynomials in n variables (see also the proof of Theorem 2.5). For any n ∈ N and any polynomial P in n variables, c(p1 , . . . , pn )xpj11 · · · xpjnn , P (xj1 , . . . , xjn ) = p1 ,...,pn

set U P (Φ(fj1 ), . . . , Φ(fjn ))Ω = P (qj1 , . . . , qjn ) ∈ L2 (Q, dµ).

(89)

Let us verify that U is norm preserving: P (Φ(fj1 ), . . . , Φ(fjn ))Ω 2 1 2 = c(p1 , . . . , pn )c(p1 , . . . , pn ) Ω, Φ(fj1 )p1 +p1 · · · Φ(fjn )pn +pn Ω p1 ,...,pn p1 ,...,pn

=

c(p1 , . . . , pn )c(p1 , . . . , pn )

p1 ,...,pn p1 ,...,pn

p +p1

Rn

qj11

p +pn

· · · qjnn

dµj1 · · · dµjn

|P (qj1 , . . . , qjn )|2 dµ.

=

(90)

Q

We use in the first step that the Φ’s commute, which is due to the fact that Cfj = fj and in the second step we make use of (88). Since the set of vectors P (Φ(fj1 ), . . . , Φ(fjn ))Ω, is dense in F(H) formula (90) shows that U extends to a unitary map from F(H) to L2 (Q, dµ), s.t. U Ω = 1 and U Φ(fj )U −1 = qj . 3.5 Equilibrium state and thermodynamic limit We focus in this subsection on Bosons and refer for more detail, as well as for the Fermionic case, to [5], Section 5.2.5. Let H be the one-particle Hamiltonian, acting on the one-particle Hilbert space H, and denote by dΓ (H) its second quantization acting on Bosonic Fock space F+ (H). dΓ (H) acts on the n sector as H ⊗ · · · ⊗ 1l + 1l ⊗ H ⊗ · · · ⊗ 1l + · · · + 1l ⊗ · · · ⊗ H. We set N = dΓ (1l), put Kµ = dΓ (H) − µN = dΓ (H − µ1l),

(91)

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Marco Merkli

where µ ∈ R is called the chemical potential, and assume that Zβ,µ = tre−βKµ

(92)

exists, for some inverse temperature β > 0. Here, tr denotes the trace on the Hilbert space F+ (H). It is not hard to show that (92) is finite if and only if tre−βH < ∞ and H − µ1l > 0,

(93)

see [5], Proposition 5.2.27; the trace here is of course over H. From the latter inequality it follows (H − µ1l has purely discrete spectrum) that there is a number η > 0 s.t. (94) dΓ (H − µ1l) = Kµ ≥ ηdΓ (1l) = ηN. The Gibbs (equilibrium) state on CCR(H) is defined by −1 tr e−βKµ A . ωβ,µ (A) = Zβ,µ

(95)

It depends on the inverse temperature β and the chemical potential µ. The Gibbs state satisfies the KMS relation (96) ωβ,µ (Aαt (B)) = ωβ,µ αt−iβ e−βµN BeβµN A , where αt (A) = eitdΓ (H) Ae−itdΓ (H) is the Heisenberg dynamics generated by the Hamiltonian H. Identity (96) makes sense for operators B s.t. eβ(dΓ (H)−µN ) Be−β(dΓ (H)−µN ) exists. If µ = 0, (96) reduces to the usual KMS relation ωβ (Aαt (B)) = ωβ (αt−iβ (B)A). In order to calculate (95) explicitly it is useful to extend the domain of definition of ωβ,µ to arbitrary (finite) products of creation and annihilation operators, i.e., to the polynomial ∗algebra P of unbounded operators on F+ (H), generated by {a# (f ) | f ∈ H}. This can be done in the following way. From (94) we see that (97) N k e−tKµ < ∞, for any t > 0 and for any k ≥ 0 15 . The operator e−βKµ /2 leaves the finite particle 0 invariant (see (35)). If Q ∈ P is any polynomial in creation and ansubspace F+ 0 and, by (97), extends to a nihilation operators then Qe−tKµ is well defined on F+ bounded operator on F+ , satisfying a# (f1 ) · · · a# (fk )e−tKµ ≤ C f1 · · · fk . Let µ > 0. For ψ ∈ F+ (H) we have 3 15 k −tK This follows from N e

µ

ψ ≤ N k ψ, e−2tηN N k ψ

41/2

= N k e−tηN ψ.

(98)

The Ideal Quantum Gas

1 2 ψ, e−βKµ /2 Qe−βKµ /2 ψ 1 2 = ψ, e−βKµ /2 eβµN/4 e−βµN/4 Qe−βµN/4 eβµN/4 e−βKµ /2 ψ 1 2 ≤ Qe−βµN/4 ψ, e−βKµ eβµN/2 ψ 3 4 = Qe−βµN/4 ψ, e−βKµ/2 ψ .

211

(99)

Since e−βKµ is trace class e−βKµ/2 is too (see (93)), so (99) shows that for any Q ∈ P, tr e−βKµ /2 Qe−βKµ /2 ≤ C, (100) where C < ∞ depends on Q, β and µ > 0. Therefore ωβ,µ can be extended to P, and we have (101) ωβ,µ a# (f1 ) · · · a# (fk ) ≤ C f1 · · · fk . Note that since e−βKµ commutes with the number operator, the l.h.s. of (101) is actually zero unless k is even and k/2 of the operators a# are creation operators. 0 We have in the strong sense on F+ e−βKµ /2 a∗ (f ) = a∗ e−β(H−µ)/2 f e−βKµ /2 , (102) and hence, using the cyclicity of the trace and the CCR, we obtain −1 ωβ,µ (a∗ (f )a(g)) = Zβ,µ tr a∗ (e−β(H−µ)/2 f ) e−βKµ a(e−β(H−µ)/2 g) = ωβ,µ a(e−β(H−µ)/2 g)a∗ (e−β(H−µ)/2 f ) 2 1 = g, e−β(H−µ) f +ωβ,µ a∗ (e−β(H−µ)/2 f )a(e−β(H−µ)/2 g) . Iterating this m times gives ωβ,µ (a∗ (f )a(g)) =

m 1 2 g, e−jβ(H−µ) f j=1

+ωβ,µ a∗ (e−mβ(H−µ)/2 f )a(e−mβ(H−µ)/2 g) . (103) In the limit m → ∞, the last term on the r.h.s. of (103) tends to zero, which follows from limm e−mβ(H−µ)/2 f = 0 (H − µ > 0 !) and the continuity of ωβ,µ , (101). The first term on the r.h.s. of (103) can be summed explicitly and we obtain 1 ωβ,µ (a∗ (f )a(g)) = g, β(H−µ) f . (104) e −1 Viewed as a function on H × H, (104) is called the two-point function of the state ωβ,µ . Similarly, one defines n-point functions for all n ≥ 1 by

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ωβ,µ (a∗ (f1 ) · · · a∗ (fn )a(g1 ) · · · a(gn )).

(105)

Notice that the average of a product of m creation operators and n annihilation operators in the state ωβ,µ vanishes unless m = n. A state with this property is called gauge invariant. The average of an arbitrary polynomial Q ∈ P is expressed in terms of the n-point functions by first normal ordering Q. This means that the CCR are used repeatedly to write Q as a sum of polynomials in a# , where in each polynomial all creation operators stand to the left of all annihilation operators. Proceeding in the same way as above, one can show that the n-point function (105) can be expressed as a sum of products of two-point functions. Consequently, (104) determines the state uniquely. Any state which is determined uniquely by its one- and two-point functions is called quasi-free. Using the quasi-free structure one can show that 1 eβ(H−µ) + 1 f ωβ,µ (W (f )) = exp − f, β(H−µ) 4 e −1 β(H − µ) 1 = exp − f, coth f . 4 2 So far, we have treated a general Hilbert space H and a Hamiltonian H with the property that H − µ1l > 0 is trace class. We consider now the case of the free Bose gas. The following discussion of the thermodynamic limit of the free Bose gas is summarized in [5], Proposition 5.2.29, see also [12]. 3 in R3 , s.t. 6 Let {Λk3}k≥0 ⊂ R be an increasing sequence of bounded regions 2 3 k Λk = R . Denote by −Hk the selfadjoint Laplace operator on L (Λk , d x) corresponding to a classical boundary condition. We choose µ s.t. there is a C > 0 satisfying (106) Hk − µ1l ≥ C1l, Λk denote the Gibbs state on CCR(L2 (Λk , d3 x)), see (95). uniformly in k. Let ωβ,µ The following results hold.

1. For any k and any A ∈ CCR(L2 (Λk , d3 x)), the limit Λ

k lim ωβ,µ (A) = ωβ,µ (A)

k →∞

(107)

exists and defines a state ωβ,µ on CCR(D), where D is the dense subspace of L2 (R3 , d3 x) given by , D= L2 (Λk , d3 x). (108) k≥0

The generating functional of ωβ,µ is given by β(H − µ) 1 ωβ,µ (W (f )) = exp − f, coth f , 4 2

(109)

for f ∈ D and where −H is the selfadjoint Laplace operator on L2 (R3 , d3 x). Note that due to (106) we can extend (109) to all f ∈ L2 (R3 , d3 x).

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2. The GNS representation (Hβ,µ , πβ,µ , Ωβ,µ ) of (CCR(D), ωβ,µ ) is regular. Let a# β,µ (f ), f ∈ D, denote the creation and annihilation operators in this representation. The state ωβ,µ can be extended to the polynomial algebra Pβ,µ generated by {a# β,µ (f ) | f ∈ D}. The extension is the gauge-invariant quasi-free state with two-point function ∗ 1 ωβ,µ aβ,µ (f )aβ,µ (g) = g, β(H−µ) f . (110) e −1 3. Let f ∈ L2 (R3 , d3 x) and let {fn } ⊂ D be a sequence approximating f , i.e., f − fn → 0. The strong limit lim πβ,µ (W (fn )) = Wβ,µ (f ) n

(111)

exists and defines a unitary operator Wβ,µ (f ) in the von Neumann algebra πβ,µ (A) ⊂ B(Hβ,µ ). The operators Wβ,µ (f ), for f ∈ L2 (R3 , d3 x), satisfy the Weyl CCR, (40). In other words, they define a representation of the algebra CCR(L2 (R3 , d3 x)). 4. The state ωβ,µ , viewed as a state on the von Neumann algebra πβ,µ (A) determined by the vector Ωβ,µ , is a (β, αt )-KMS state, where αt is the ∗automorphism group given by αt (Wβ,µ (f )) = Wβ,µ (e−iHt f ), for f ∈ L2 (R3 , d3 x). We point out that condition (106) gives a restriction on the possible values of µ. We must require µ < µ0 , where µ0 depends on the choice of the boundary condition. On the other hand, µ is related to the particle density of the system. It turns out that under condition (106), one cannot describe high particle densities – e.g. the situation where most particles are in the state of lowest energy. In order to describe this phenomenon, called Bose-Einstein condensation, one needs a more careful analysis of the thermodynamic limit. We refer for more detail to [5], Section 5.2.5.

4 Araki-Woods representation of the infinite free Boson gas The goal of this section is to find the GNS representation of states ω on CCR(D) which represent the infinitely extended ideal Bose gas in which the momentum density distribution of the particles is prescribed. Our approach is based on the original paper [3]. In a first step we show that the states of CCR(D) are in one-to-one correspondence with so-called generating functionals on D. Then we calculate explicitly the generating functional corresponding to the Bose gas in a box and with a prescribed momentum density distribution. We take the thermodynamic limit of the finite-volume generating functionals, where the box size tends to infinity and the momentum density distribution approaches a given limit. The infinite-limit generating functional corresponds to a unique state on CCR(D). We construct explicitly its GNS representation, which is commonly called the Araki-Woods representation.

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4.1 Generating functionals We consider in the remaining part of the notes the C ∗ algebra CCR(D), where D ⊆ H. Given a state ω on CCR(D), we may consider the (nonlinear) generating functional defined by E:D→C f !→ E(f ) = ω(W (f )).

(112)

The generating functional satisfies the following properties: 1. (normalization) E(0) = 1 2. (unitarity) E(f ) = E(−f ), f ∈ D 3. (positivity) for any K ≥ 1, zk ∈ C, fk ∈ D, k = 1 . . . K, K

zk zk e− 2 Imfk ,fk E(fk − fk ) ≥ 0. i

(113)

k,k =1

Properties 1. and 2. are obvious, and (113) is a consequence of the positivity of the state ω. Any positive element in a C ∗ algebra can be written as A∗ A, so in CCR(D), any positive element is approximated in C ∗ algebra norm by elements of the form

K k=1

∗ zk W (fk )

K

zk W (fk )

=

K

zk zk e− 2 Im−fk ,fk W (fk − fk ), i

k,k =1

k=1

for some zk ∈ C and fk ∈ D. Hence (113) is equivalent to ω(A∗ A) ≥ 0, for any A ∈ CCR(D). We now show that conversely, if a functional E with properties 1.-3. is given, then it determines uniquely a state on CCR(D), with respect to which it is the generating functional. Theorem 4.1. Suppose a map E : D → C satisfies 1.-3. above. For f ∈ D, set ω(W (f )) = E(f ) and extend ω by linearity to the linear span of the Weyl operators, K K zk W (fk ) = zk E(fk ). ω k=1

k=1

Then ω extends uniquely to a state on CCR(D). Moreover, E is continuous in the topology of D ⊆ H if and only if f !→ πω (W (f )) is a strongly continuous map from D into the bounded operators on the GNS Hilbert space associated to (CCR(D), ω). Remark. The statement “f !→ E(f ) is continuous in the topology of D” is equivalent to the statement “f !→ E(f + f0 ) is continuous in the topology of D, for any fixed f0 ∈ D”. We incorporate the proof of this remark in the proof of Theorem 4.1 given below.

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This theorem can be viewed as a non-commutative analog of the Bochner-Minlos theorem (see e.g. [8], Section 3.2 or [7], Sections 3.4 and A.6). Let S be the Schwartz space on Rn , 16 S its dual (the set of continuous linear functionals on S), and let SR be the set of real Schwartz distributions, i.e. the set of χ ∈ S satisfying χ; f = · ; · is the dual pairing. Let ν be a positive regular Borel χ; f , for all f ∈ S, where measure on SR , 17 s.t. S dν(χ) = 1. We define the Fourier transform of ν by R

E(f ) =

SR

e−iχ;f dν(χ),

f ∈ S. Then E satisfies 1’. E(0) = 1, 2’. f !→ E(f ) is continuous, 3’. for any K ≥ 1, zk ∈ C, fk ∈ S, k = 1 . . . K, we have K

zk zk E(fk − fk ) ≥ 0.

(114)

k,k =1

Inequality (114) holds because the l.h.s. is just is the Bochner–Minlos theorem:

SR

2 K k=1 zk e−iχ;fk dν(χ). Here

Theorem 4.2. Suppose a map E : S → C satisfies 1’.-3’. above. Then there exists a unique normalized positive regular Borel measure ν on the real Schwartz distribution space SR such that E is the Fourier transform of ν. Proof of Theorem 4.1. Parts of our proof are inspired by [1]. Let S be the ∗algebra generated by the Weyl operators W (f ), i.e., S is the ∗algebra of finite linear combinations of products of elements W (f ) ∈ CCR(D). S is a subalgebra of CCR(D) and inherits the notion of positivity induced by CCR(D). Inequality (113) implies that ω is a positive linear map on S. Positivity implies boundedness as follows: let first A ∈ S be a selfadjoint element satisfying A < 1. Then we have S * 1l − A > 0, so ω(1l) − ω(A) = ω(1l − A) ≥ 0, and consequently ω(A) ≤ ω(1l) = E(0) = 1. Next consider any A ∈ S s.t. A < 1. From A∗ A ≤ A∗ A 1l ≤ A 2 1l < 1l and the Cauchy-Schwarz inequality, |ω(A∗ B)|2 ≤ ω(A∗ A)ω(B ∗ B), which is valid for A, B ∈ S (note that we only need S to be a ∗algebra here, it does not have to be a Banach algebra), we obtain the estimate 16

17

the set of f ∈ C ∞ (Rn ) s.t. all seminorms f k,l = supx∈Rn xk ∂ l f (x) are finite, for any multi-indices k, l ∈ Nn . The topology of S is the one induced by these seminorms. A Borel measure µ on Q is called regular if for any Borel subset E of Q we have µ(E) = inf{µ(U ) | U ⊃ E, U open } and µ(E) = sup{µ(K) | K ⊂ E, K compact }. The Borel σ-algebra of S is generated by the open sets of the weak∗ topology on S . A base for this topology is given by the collection of all cylinder sets. Cylinder sets are of the form {χ ∈ S | ( χ; f1 , . . . , χ; fn ) ∈ B ⊂ Cn }, where f1 , . . . , fn ∈ S and B is open. An open set in the weak∗ topology is a union of cylinder sets.

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|ω(A)|2 = |ω(1lA)|2 ≤ ω(1l)ω(A∗ A) ≤ A 2 ω(1l)2 = A 2 . This shows that |ω(A)| ≤ A , for A ∈ S. Thus ω extends to a state on CCR(D). Next we show that if f !→ E(f ) is continuous then ω is a regular state. Let (Hω , πω , Ωω ) be the GNS representation of (CCR(D), ω). Suppose fn → f is a convergent sequence in D. Define a family of unitary operators Un = πω (W (fn )) and U = πω (W (f )) on the Hilbert space Hω . We show that lim (Un − U )ψ = 0,

n→∞

(115)

for any ψ ∈ Hω . Due to unitarity it suffices to show weak convergence, i.e. (115) is equivalent to (116) lim φ, (Un − U )ψ = 0, n→∞

for all φ, ψ ∈ Hω . Because Ωω is cyclic, we that for any > 0, there are vectors have K L φ = k=1 zk πω (W (gk ))Ωω and ψ = l=1 ζl πω (W (hl ))Ωω , s.t. φ − φ < and ψ − ψ < . Now |φ, (Un − U )ψ − φ , (Un − U )ψ | ≤ φ − φ (Un − U ) ψ +2 φ (Un − U ) ψ − ψ ≤ 4( φ + ψ ), uniformly in n (we use here that φ < 2 φ , for small ). Thus it is enough to prove that (117) lim πω (W (g))Ωω , (Un − U )πω (W (h))Ωω = 0, n→∞

for any g, h ∈ D. This scalar product is just Ωω , πω (W (−g)(W (fn ) − W (f ))W (h))Ωω = e− 2 Im(−g,fn +−g+fn ,h ) E(−g + fn + h) i

−e− 2 Im(−g,f +−g+f ,h ) E(−g + f + h), i

which converges to zero as n → ∞. Next we show that if ω is a regular state then f !→ E(f ) is continuous. Let fn be a sequence in D converging to f ∈ D. We have E(fn ) − E(f ) = ω(W (fn ) − W (f )) = ω (W (fn )W (−f ) − 1l)W (f ) i = ω e 2 Imfn ,f W (fn − f ) − 1l W (f ) 2 i 1 = Ωω , e 2 Imfn ,f πω (W (fn − f )) − 1l πω (W (f ))Ωω . (118) Since πω (W (fn − f )) converges strongly to zero as n → ∞ we have the desired continuity of E. Finally we prove the assertion of the remark after Theorem 4.1. Suppose that fn → 0 and that E(fn ) → 1. Our goal is to show that E(fn + f0 ) → E(f0 ), where

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217

f0 ∈ H is fixed. The above considerations leading to (115) show that πω (W (fn )) → 1l in the strong sense on Hω . Then we write, as in (118), E(fn + f0 ) − E(f0 ) 2 1 i = Ωω , e 2 Imfn ,f0 πω (W (fn )) − 1l πω (W (f0 ))Ωω . The r.h.s. converges to zero as n → ∞.

Suppose that Ek , k = 1, 2, . . . is a sequence of generating functionals, each satisfying conditions 1.-3. above. If Ek has a limit in the sense that there is a map E : D → C s.t. E(f ) = limk→∞ Ek (f ), for any f ∈ D, then it is clear that E satisfies conditions 1.-3. as well. In the next section we use this fact to construct the generating functional and the GNS representation of the free Bose gas extended to all of physical space in a state determined by a given momentum density distribution. We close this section with the calculation of EF (f ) = Ω, W (f )Ω, the Fock generating functional, corresponding to the vacuum state on CCRF (H). Using the series expansion of the Weyl operator in Fock space, we can write EF (f ) =

i2n 3 4 Ω, Φ(f )2n Ω , (2n)!

(119)

n≥0

where we have used that all odd powers of Φ(f ) have a vanishing vacuum expectation value. We use the commutation relations (27), (28), and the fact that a(f )Ω = 0 to get 3 4 4 2n − 1 4 1 3 f 2 Ω, Φ(f )2n−2 Ω . Ω, Φ(f )2n Ω = √ Ω, a(f )Φ(f )2n−1 Ω = 2 2 3 4 2 n (2n)! By induction, we arrive at Ω, Φ(f )2n Ω = f2

2n n! , which we use in (119) to obtain 2 (120) EF (f ) = e− f /4 . 3

4.2 Ground state (condensate) We construct in this section the representation of the CCR describing the infinitely extended Bose gas in its ground state where all particles are in the same state. The ground state is an example of a condensate (macroscopic occupation of a particular one-particle state of an infinitely extended system), it is parametrized by the particle density ρ ≥ 0. Consider first the free non-relativistic Bose gas confined to a finite box V = ⊂ R3 of volume |V |. We will let the volume and the number of particles, n, tend to infinity while keeping the density ρ = n/|V | fixed. For any finite n, the Bose gas is described using Fock space F(L2 (V, d3 x)), it is just a system 1 1/3 , |V |1/3 ]3 8 [−|V |

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Marco Merkli

of n non-interacting particles whose symmetric wave function ψ ∈ L2 (V n , d3n x) evolves according to the Schr¨odinger equation i∂t ψ(x1 , . . . , xn ) = (Hψ)(x1 , . . . , xn ), where H=

n

−∆j

(121)

(122)

j=1

is the selfadjoint Hamiltonian operator on L2 (V n , d3n x) with periodic boundary conditions (of course, ∆j is the Laplacian with respect to the variable xj ). The system is in its ground state ΨV (the one having the lowest energy) if each of the n particles is in the state fV of minimal energy (relative to −∆j ), given by fV (x) = |V |−1/2 , x ∈ V,

(123)

which we have normalized as fV L2 (V,d3 x) = 1. Consequently, 1 ΨV = √ a∗F (fV )n ΩF , n!

(124)

where √1n! is a normalization factor, a∗F is the Bosonic creation operator on Fock space, and ΩF is the Fock vacuum. The generating functional corresponding to ΨV is EV (f ) = ΨV , W (f )ΨV =

1 ΩF , aF (fV )n WF (f )a∗F (fV )n ΩF , n!

(125)

where WF = eiΦF (f ) is a Weyl operator in the Fock representation, 1 ΦF = √ (a∗F (f ) + aF (f )). 2 Our plan is to calculate the right hand side of (125) explicitly and take the limit n → ∞, keeping ρ fixed. This provides us with a generating functional EGS (depending on the number ρ) which we interpret as the generating functional of the ground state of the infinite system. Knowing EGS , we explicitly construct the GNS representation of the ground state of the infinite system (it will not be the Fock representation – i.e., there is no vector (or density matrix) on Fock space representing the ground state of the infinite system – we have already discussed this in Section 1.5.). In order to calculate the r.h.s. of (125) we “pull” (or commute) the annihilation operators to the right, through WF (f ) and through the creation operators, by using the canonical commutation relations. Whenever an annihilation operator is completely pulled through, it hits the vacuum ΩF yielding zero. The value of the r.h.s. of (125) is given by all extra terms (contractions) one generates, using the CCR, in this procedure. Let us first show how to pull the annihilation operators through the Weyl operator. Using the series expansion of WF (f ) = eiΦF (f ) and that

The Ideal Quantum Gas

[aF (f ), ΦF (g)k ] = 2−1/2 k f, g ΦF (g)k−1 ,

219

(126)

which follows easily from the CCR (27), (28), one verifies without difficulty that [aF (f ), WF (g)] = 2−1/2 i f, g WF (g).

(127)

All these relations can be understood in the strong sense on the finite particle subspace. We view the pulling through procedure as follows. Consider aF (fV )n WF (f ). Among the n annihilation operators, k (= 0, 1, . . . , n) are commuted through WF (f ) to the right while n − k have undergone a contraction of the form (127). For each fixed value of k, there are nk ways of choosing which annihilation operators are safely pulled through the Weyl operator. We obtain n−k n n i fV , f n √ aF (fV ) WF (f ) = WF (f )aF (fV )k . (128) k 2 k=0 One can of course prove (128) as well by induction, which is an easy task. The generating functional can thus be written as n−k n 3 4 i fV , f 1 n √ ΩF , WF (f )aF (fV )k a∗F (fV )n ΩF . EV (f ) = k n! 2 k=0 A similar pull through argument as above, plus the facts that aF (fV )ΩF = 0 and fV = 1 yields aF (fV )k a∗F (fV )n ΩF = n(n − 1) · · · (n − k + 1) fV , fV a∗F (fV )n−k ΩF n! a∗ (fV )n−k ΩF , = (n − k)! F k

from which we get n−k n 3 4 n! i fV , f 1 n √ ΩF , WF (f )a∗F (fV )n−k ΩF . k (n − k)! n! 2 k=0 (129) We pull the n − k creation operators to the left through the Weyl operator by using the adjoint relation to (128) (recall also that WF (f )∗ = WF (−f )) EV (f ) =

WF (f )a∗F (fV )n−k

=

n−k l=0

n−k l

−i fV , −f √ 2

n−k−l a∗F (fV )l WF (f ).

Clearly, only the term l = 0, where no creation annihilator arrives safely to the left of WF (f ) will give a non-zero contribution to expression (129) (because aF (fV )ΩF = 0, once again), so n−k n n! 1 n 1 2 EV (f ) = ΩF , WF (f )ΩF . − | fV , f | k (n − k)! n! 2 k=0

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We denote the Fock vacuum generating functional by EF (f ) = Ω, WF (f )Ω (see (119)) and observe that for f with compact support and large enough |V |, we have ρ 1/2 # −1/2 f (x)d3 x = (2π)3 f (0), fV , f = |V | n V where f#(k) =

1 (2π)3/2

d3 k e−ikx f (x)

(130)

R3

is the Fourier transform. Consequently we have n n−k n! ρ 1 n EV (f ) = EF (f ) −(2π)3 |f#(0)|2 . k (n − k)! n! 2n

(131)

k=0

We recall that the Laguerre polynomials are defined by n n! dn 1 1 n Ln (z) = ez n (e−z z n ) = (−z)n−k k (n − k)! n! dz n!

(132)

k=0

for n = 0, 1, . . . and z ∈ C. Next, it is known that (see e.g. [2], formula 22.15.2) √ lim Ln (z/n) = J0 (2 z), (133) n→∞

where the Bessel function J0 satisfies π 5 dθ −i(α cos θ+β sin θ) e = J0 α2 + β 2 , −π 2π

(134)

for any α, β ∈ R (see e.g. [11], formula (5.3.66)). In conclusion, we have calculated the infinite volume generating functional to be 5 EGS (f ) = EF (f )J0 (2π)3/2 2ρ |f#(0)| . (135) This generating functional defines a state on CCR(D) which we view as being the ground state of the infinite Bose gas. The test function space is given by 7 8 D = f ∈ L2 (R3 , d3 x) | f#(0) exists . (136) Any function in Schwarz space is contained in D, so D is dense in L2 (R3 , d3 x). Let us now construct the GNS representation of the infinite Bose gas. Consider the Hilbert space HGS = F(L2 (R3 , d3 x)) ⊗ L2 S 1 , dσ , (137) where the left factor is the Bosonic Fock space over L2 (R3 , d3 x) and the right one is the Hilbert space of all square integrable functions on the unit circle with uniform measure. It is convenient to parametrize the circle by the angle θ ∈ [−π, π]. Set

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221

ΩGS = ΩF ⊗ 1, (138) where 1 ∈ L2 S 1 , dσ is the constant function. We define the representation map πGS : CCR(D) → B(HGS ) as √ −i(2π)3/2 2ρ Ref#(0) cos θ+Imf#(0) sin θ . (139) πGS : W (f ) !→ WF (f ) ⊗ e Using relation (134) it is easily seen that for any f ∈ D we have ΩGS , πGS (W (f ))ΩGS = EGS (f )

(140)

so the representation gives the correct generating functional. To show that the GNS Hilbert space of (CCR(D), ωGS ), where the state ωGS is represented by ΩGS in HGS , is actually the entire HGS , we need to verify that ΩGS is cyclic for πGS . To show this, define the family of functions e−s|x| π sz 2 , fz,s (x) = 2 x +1 where x ∈ R3 , z ∈ C and s > 0. Clearly, fz,s L2 (R3 ,d3 x) → 0 as s → 0+ , while f#z,s (0) → z for s → 0+ . Due to the strong continuity of the Weyl operators in Fock space, we have for any z ∈ C πGS (W (fz,s )) → 1l ⊗ e−i(2π)

3/2 √

2ρ (Rez cos θ+Imz sin θ)

,

in the strong sense on HGS , as s → 0+ . Since the constant function 1 is cyclic in L2 S 1 , dσ for the set of multiplication operators {ei(a cos θ+b sin θ) | a, b ∈ R} 18 , we see that ΩF ⊗ L2 S 1 , dσ is contained in the closure of πGS (CCR(D)) ΩGS . Because ΩF is cyclic for {WF (f ) | f ∈ D} in F(L2 (R3 , d3 x)) (we use here that D is dense in L2 (R3 , d3 x)) we can approximate arbitrarily well any given ψ ∈ F(L2 (R3 , d3 x)) by some WF (f )ΩF , f ∈ D. It follows that for an appropriate choice of z ∈ C and for s small enough the vector ψ ⊗ 1 ∈ HGS is approximated arbitrarily well by πGS (W (f )W (fz,s ))ΩGS . This shows that ΩGS is cyclic for πGS in HGS . The representation (HGS , πGS , ΩGS ) is a regular. The creation and annihilation operators are given by √ a∗GS (f ) = a∗F (f ) ⊗ 1 − (2π)3/2 ρ f#(0) 1l ⊗ e−iθ , √ aGS (f ) = aF (f ) ⊗ 1 − (2π)3/2 ρ f#(0) 1l ⊗ eiθ . At zero density, ρ = 0, the ground state representation of the infinite Bose gas coincides with (is isomorphic to) the Fock representation. 18

Any function f ∈ L2 (S 1 , dσ) has a Fourier series expansion.

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4.3 Excited states Our goal for this section is to extend the above method to construct the generating functional and the GNS representation corresponding to an (infinite volume) state of the CCR with a continuous momentum distribution ρ(k). Consider first the situation where, in our box V (as in the last section), we have nj particles with momentum kj , i.e., with wave function fVj (x) = |V |−1/2 eikj x , where j = 1, . . . , p, and where |kj |2 are (discrete) eigenvalues of the Laplacian (in j the box V ⊂ R3 with periodic boundary conditions). The 2 eigenfunctions of 1 fV are the Laplacian and satisfy the orthonormality condition fVj , fVl

= δjl (Kronecker

symbol). We will let the box tend to all of R with the result that in the limit, the values of k can range continuously throughout R3 (this reflects the fact that −∆ on L2 (R3 , d3 x) has purely absolutely continuous spectrum). The state of the gas in the box with densities ρj = nj /|V | of particles with momenta kj , j = 1, . . . , p, is given by 3

ΨV = 5

1 a∗F (fV1 )n1 · · · a∗F (fVp )np ΩF , n1 ! · · · np !

and the corresponding generating functional EV (f ) = ΨV , WF (f )ΨV can be calculated just as in the previous section. It is an easy exercise to obtain the expression ρ1 # ρp # EV (f ) = EF (f )Ln1 (2π)3 |f (k1 )|2 · · · Lnp (2π)3 |f (kp )|2 , 2n1 2np where Ln (z) are the Laguerre polynomials defined in (132). We have used that for any f with compact support, −1/2

fj , f = |V |

−ikj x

e V

3

f (x)d x =

ρj nj

1/2

(2π)3/2 f#(kj ),

for |V | big enough and where f# is the Fourier transform (130). Using (133), we take the limits nj → ∞, j = 1, . . . , p, while leaving ρj fixed. The infinite volume generating functional is 5 5 E(f ) = EF (f )J0 (2π)3/2 2ρ1 |f#(k1 )| · · · J0 (2π)3/2 2ρp |f#(kp )| . (141) Our next task is to let the discrete distribution ρj , j = 1, . . . , p, tend to a continuous distribution ρ(k). We do this for simplicity first in the one-dimensional case, k ∈ R, and we will deduce the general formula afterwards. Let thus R * k !→ ρ(k) ∈ R+ be a given momentum density and consider an interval [−L, L]. We partition [−L, L]

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223

into small intervals with endpoints kj = −L + 2Lj/p, j = 0, . . . , p, each of length 2L/p, and we will let p → ∞. The density ρj of particles having momenta in the interval with left endpoint kj is given by ρj = 2Lρ(kj )/p. Our goal is to calculate 9 p E(f ) 4Lρ(kj ) # 1/2 |f (kj )| . (142) lim log log J0 (2π) = lim p→∞ p→∞ EF (f ) p j=0 Notice that the power of 2π is now 1/2, in one dimension. A simple Taylor expansion 2 shows that the leading term of log J0 () for small is − 4 , so that we have E(f ) lim log p→∞ EF (f ) L p 2Lρ(kj ) # 1 1 2 |f (kj )| = − 2π dk ρ(k)|f#(k)|2 . = − 2π lim 2 p→∞ j=0 p 2 −L √ If we take f such that ρf# is square integrable then we can take L → ∞ and obtain for the generating functional of the infinite Bose gas in three dimensions and with momentum density ρ

(2π)3 d3 k ρ(k)|f#(k)|2 . Eρ (f ) = EF (f ) exp − (143) 2 R3 The test function space consists of functions s.t. the integral in (143) exists, 7 8 √ D = f ∈ L2 (R3 , d3 x) | ρf# ∈ L2 (R3 , d3 k) ,

(144)

it depends on the function ρ. It is convenient to pass to a representation of the one-particle Hilbert space where the energy operator is diagonal; in the case of the Laplacian this means that we consider L2 (R3 , d3 k), the Fourier-transformed position space L2 (R3 , d3 x). The Fourier transform is an isometric isomorphism between L2 (R3 , d3 x) and L2 (R3 , d3 k) which induces a C ∗ algebra isomorphism between CCR(D ) and CCR(D), where

√ D = f ∈ L2 (R3 , d3 k) | ρf ∈ L2 (R3 , d3 k) , (145) and the corresponding generating functional is given by

(2π)3 Eρ (f ) = EF (f ) exp − d3 k ρ(k)|f (k)|2 , f ∈ D, 2 R3 where EF (f ) = e− f

2

, for f ∈ D (see (120)). Hence

4 13 3 Eρ (f ) = exp − f, (1 + 16π ρ)f , f ∈ D. 4

(146)

/4

(147)

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One can carry out the construction for a general selfadjoint Hamiltonian H (not necessarily of the form (122)) and one arrives at (143) where f# stands for the eigenfunction expansion of f corresponding to H. Formula (147) gives a generating functional which defines a state ωρ on CCR(D), according to Theorem 4.1. We give now the GNS representation of (CCR(D), ωρ ) for densities ρ(k) such that k !→ ρ(k) is continuous, ρ(k) > 0 a.e., d3 k ρ(k) < ∞. (148) R3

The representation Hilbert space Hρ and the cyclic vector Ωρ are Hρ = F(L2 (R3 , d3 k)) ⊗ F(L2 (R3 , d3 k))

(149)

Ωρ = Ω F ⊗ Ω F ,

(150)

where F(L (R , d k)) is the Bosonic Fock space over L (R , d k) and ΩF is the vacuum therein. The representation map πρ : CCR(D) → B(Hρ ) is given by 5 √ 1 + µ f ⊗ WF µf , (151) πρ (W (f )) = WF 2

3

3

2

µ(k) = 8π 3 ρ(k).

3

3

(152)

Notice that in the Weyl operator on the right factor there appears the complex conjugate of f . Using expression (120) it is an easy matter to verify that Ωρ , πρ (W (f ))Ωρ = Eρ (f ),

(153)

where Eρ (f ) is given by (147). πρ is a regular representation and the creation and annihilation operators are given by 5 √ a∗ρ (f ) = a∗F 1 + µ f ⊗ 1l + 1l ⊗ aF µf , (154) 5 √ aρ (f ) = aF 1 + µf ⊗ 1l + 1l ⊗ a∗F µf . (155) Since the a# ρ (f ) are obtained from the represented Weyl operators by strong differentiation it follows that Ωρ is cyclic for πρ if Ωρ is cyclic for the polynomial algebra P generated by all creation and annihilation operators a# ρ (f ), f ∈ D. The set √ { µf | f ∈ D} is dense in L2 (R3 , d3 k) due to condition (148). Since ΩF is cyclic for the Fock creation operators it follows from (155) that ΩF ⊗ F(L2 (R3 , d3 k)) lies in the closure of PΩρ . Similarly (154) shows that F(L2 (R3 , d3 k)) ⊗ ΩF is in that closure. Hence Ωρ is cyclic for πρ . If ρ(k) = 0 then µ(k) = 0 and the representation (151) reduces to the Fock representation. 4.4 Equilibrium states The results of the previous two sections can be combined to describe the infinitely extended free Bose gas with a momentum density distribution which has some condensate part characterized by the density ρ0 ∈ R+ and some continuous part given by ρ(k). The corresponding generating functional is

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Eρ0 ,ρ (f )

(156)

3 5 (2π) = EF (f ) exp − d3 k ρ(k)|f (k)|2 J0 (2π)3/2 2ρ0 |f (0)| , 2 3 R

compare with (135) and (146), for functions f in the test function space

√ D = f ∈ L2 (R3 , d3 k) | ρf ∈ L2 (R3 , d3 k), |f (0)| < ∞ .

(157)

The GNS representation Hilbert space Hρ0 ,ρ and the cyclic vector Ωρ0 ,ρ associated to (CCR(D), ωρ0 ,ρ ), where ωρ0 ,ρ is the state defined by (156), are given by Hρ0 ,ρ = F(L2 (R3 , d3 k)) ⊗ F(L2 (R3 , d3 k)) ⊗ L2 S 1 , dσ , (158) Ωρ0 ,ρ = ΩF ⊗ ΩF ⊗ 1, and the representation map is 5 √ πρ0 ,ρ (W (f )) = WF ( 1 + µ f ) ⊗ WF ( µ f ) ⊗ e−iΦ(f,θ) , with µ(k) = 8π 3 ρ(k), and where we introduce the phase 5 Φ(f, θ) = (2π)3/2 2ρ0 (Ref (0) cos θ + Imf (0) sin θ) .

(159)

(160)

Note that Φ is real linear in the first argument. The creation and annihilation operators in this regular representation are not difficult to calculate: 5 √ a∗ρ0 ,ρ (f ) = a∗F 1 + µ f ⊗ 1l ⊗ 1 + 1l ⊗ aF µf ⊗ 1 √ −(2π)3/2 ρ0 f (0) 1l ⊗ 1l ⊗ e−iθ , (161) 5 ∗ √ 1 + µf ⊗ 1l ⊗ 1 + 1l ⊗ aF µf ⊗ 1 aρ0 ,ρ (f ) = aF √ (162) −(2π)3/2 ρ0 f (0) 1l ⊗ 1l ⊗ eiθ . The dynamics on CCR(D) generated by the Hamiltonian (122) is given by W (f ) !→ αt (W (f )) = W (eitω f ),

(163)

where ω(k) = |k|2 . It is clear from (156) that Eρ0 ,ρ (eitω f ) = Eρ0 ,ρ (f ), for all t ∈ R. Consequently ωρ0 ,ρ is a time translation invariant i.e. stationary state, for any choice of ρ0 , ρ(k). We wish to examine which particular momentum density distributions correspond to equilibrium states of the system. We have (164) WF (eitω f ) = eitH WF (f )e−itH , where H is the free field Hamiltonian (in Fock space) given by the second quantization of the multiplication by ω(k). It is easy to see from (159) and (164) that the dynamics (163) is unitarily implemented as πρ0 ,ρ (αt (W (f )) = eitL πρ0 ,ρ (W (f ))e−itL ,

(165)

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where L is the so-called Liouvillian, given by L = H ⊗ 1l ⊗ 1 − 1l ⊗ H ⊗ 1.

(166)

An equilibrium state ω is a state that satisfies the KMS condition ω(Aαt (B)) = ω(αt−iβ (B)A),

(167)

see also (96). We assume here that B is such that αz (B) exists for values of z in a strip around the real axis. Since an equilibrium state is necessarily αt –invariant, (167) is equivalent to ω(AB) = ω(α−iβ (B)A). It is evident from the explicit form of Ωρ0 ,ρ that ωρ0 ,ρ can be extended to the polynomial algebra generated by the creation and annihilation operators (161), (162), giving a gauge-invariant quasifree state (see after (105)). To see which densities give an equilibrium state it is thus necessary and sufficient to solve the equation 4 3 4 3 Ωρ0 ,ρ , a∗ρ0 ,ρ (f )aρ0 ,ρ (g)Ωρ0 ,ρ = Ωρ0 ,ρ , eβL aρ0 ,ρ (g)e−βL a∗ρ0 ,ρ (f )Ωρ0 ,ρ , (168) which should hold for all f, g, for ρ0 and ρ(k). We calculate 3 4 Ωρ0 ,ρ , a∗ρ0 ,ρ (f )aρ0 ,ρ (g)Ωρ0 ,ρ = g, µf + (2π)3 ρ0 f (0)g(0) and 3 4 3 4 Ωρ0 ,ρ , eβL aρ0 ,ρ (g)e−βL a∗ρ0 ,ρ (f )Ωρ0 ,ρ = g, e−βω (1 + µ)f +(2π)3 ρ0 f (0)g(0). Consequently ρ0 ≥ 0 can be arbitrary and µ must satisfy µ = e−βω (1 + µ), i.e., 1 . The density is thus given by (see (152)) µ(k) = eβω(k) −1 ρ(k) = (2π)−3

1 eβω(k)

−1

,

(169)

which is the Planck distribution of black body radiation. Let us focus on massless relativistic Bosons, where ω(k) = |k|. Other dispersion relations are discussed in an analogous way. The total density of particles in the state of equilibrium is d3 k 1 c 3 d kρ(k) = ρ0 + 3 (170) = ρ0 + 3 , ρtot = ρ0 + β|k| 8π β e − 1 R3 R3 ∞ 2 where c = 2π1 2 0 ess−1 ds is a fixed constant. We can deduce from (170) the following qualitative behavior of the system. Suppose ρtot is fixed and suppose we decrease the temperature of the system (β → ∞). Then ρ0 tends to ρtot which means that for low temperatures the system likes to form a condensate. If we fix an inverse temperature β and increase the total density ρtot of the system then ρ0 increases as well. These considerations show that we are likely to observe a condensate if either the temperature is low or the density is high (this, of course, is also an experimental fact).

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We close this section with a result about the thermodynamic limit of Gibbs states which is due to Cannon, [6]. Fix an inverse temperature 0 < β < ∞ and define the critical density by d3 k , (171) ρcrit (β) = (2π)−3 βω −1 R3 e which coincides with the total density (170) in the equilibrium state for ρ0 = 0. Let V be the box defined by −L/2 ≤ xj ≤ L/2 (j = 1, 2, 3) and define the canonical state at inverse temperature β and density ρtot by c

Aβ,ρtot ,V =

trAPρtot V e−βHV , trPρtot V e−βHV

(172)

where the trace is over Fock space over L2 (V, d3 x), Pρtot V is the projection onto the subspace of Fock space with ρtot V particles (if ρtot V is not an integer take a convex combination of canonical states with integer values ρ1 V and ρ2 V extrapolating ρtot V ). The Hamiltonian HV is negative the Laplacian with periodic boundary conditions. The observable A in (172) belongs to the Weyl algebra over the test function space C0∞ , realized as a C ∗ algebra acting on Fock space. Cannon shows that for any β, ρtot > 0 and f ∈ C0∞ , . 3 4 z∞ 1 − 14 f 2 − 2 f, eβω −z∞ f c e , ρtot ≤ ρcrit (β) (173) W (f )β,ρtot ,V −→ e ρtot ≥ ρcrit (β) Eρ0 ,ρ (f ), for any sequence L → ∞. Here, z∞ ∈ [0, 1] is such that for subcritical density, the momentum density distribution of the gas is given by ρ(k) = (2π)−3 so that z∞ is the solution of −3

ρtot = (2π)

z∞ , eβω − z∞

(174)

z d3 k. eβω − z

(175)

The generating functional Eρ0 ,ρ in (173) is the one obtained by Araki and Woods, (156), where ρ is the continuous momentum density distribution prescribed by Planck’s law of black body radiation (169), and where ρ0 = ρtot − ρcrit .

(176)

This gives the following picture for equilibrium states: if the system has density ρtot ≤ ρcrit then the particle momentum distribution of the equilibrium state is purely continuous, meaning that below critical density there is no condensate. As ρtot increases and surpasses the critical value, ρtot > ρcrit , the “excess” particles form a condensate which is immersed in a gas of particles radiating according to Planck’s law. Finally we mention the work [10] which treats the thermodynamic limit for the grand-canonical ensemble.

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4.5 Dynamical stability of equilibria Take the infinitely extended Bose gas initially in a state which differs from the equilibrium state at a given temperature only inside a bounded region of space. As time goes on we expect the local perturbation to spread out and propagate off to spatial infinity. This property, sometimes called the property of return to equilibrium, is a priori not built into the definition of equilibrium states, i.e., the KMS condition, but it has to be verified “by hand”. In this section we investigate the large time limit of initial states which are local perturbations of an equilibrium state. Let us first describe sates which are local perturbations of a given state ω of the infinitely extended Bose gas. Let f ∈ D ⊂ L2 (R3 , d3 x) be a test function which is supported in a compact region Λ0 ⊂ R3 . If g is supported in the complement R3 \Λ0 then we have W (f )W (g) = e−iImf,g W (g)W (f ) = W (g)W (f ).

(177)

Consequently the state A !→ ω (A) := ω(W (f )∗ AW (f )) does not differ from A = the n state ω on observables supported away from Λ0 (i.e., on observables z W (f ), where the f are supported away from Λ ). The state ω is a loj j j 0 j=1 cal perturbation of ω. More generally, if B is an observable (an element of the Weyl algebra) we say the state ω(B ∗ · B) (178) ωB (·) := ω(B ∗ B) is a local perturbation of ω. The set of all local perturbations of ω is defined to be the set of all convex combinations of states of the form (178). The dynamical stability of an equilibrium state ωβ (w.r.t. the dynamics αt ) is expressed as lim ωB (αt (A)) = ωβ (A),

t→∞

(179)

for all observables A, B. We start our investigation of return to equilibrium by some purely algebraic considerations. Let A be an element in the Weyl algebra CCR(D). Given any there are complex numbers zj and test functions fj ∈ D s.t. n n iωt A − (180) zj W (fj ) = αt (A) − zj W (e fj ) < , j=1 j=1 where we use the fact that αt is an isometry. Let g be fixed. We have ⎛ ⎞ n iωt W (g)∗ ⎝αt (A) − ⎠ z W (e f ) W (g) j j j=1 n iωt α = (A) − z W (e f ) j j < , t j=1

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and the l.h.s. equals n iωt iωt i − 2 Im[−g,e fj +e fj ,g ] iωt W (g)∗ αt (A)W (g) − zj e W (e fj ) < . j=1 (181) 3 4 3 4 Since limt→∞ −g, eiωt fj + eiωt fj , g = 0 (this follows from the RiemannLebesgue Lemma) there is a number T0 () < ∞ s.t. if t > T0 then n n − 2i Im[−g,eiωt fj +eiωt fj ,g ] iωt z e W (e f ) − z α (W (f )) j j j t j < . (182) j=1 j=1 It follows from (180), (181) and (182) that W (g)∗ αt (A)W (g) − αt (A) < 2, for t > T0 (), and consequently lim W (g)∗ αt (A)W (g) − αt (A) = 0,

t→∞

(183)

for all observables A and all test functions g. Relation (183), which merely involves observables and the dynamics (and no reference to any state is made), is a form of asymptotic abelianness w.r.t. αt . In fact, it follows easily from (183) and (180) that lim Bαt (A) − αt (A)B = 0,

(184)

t→∞

for any observables A, B ∈ CCR(D). If ω is any αt –invariant state then (183) shows that lim ω(W (g)∗ αt (A)W (g)) = ω(A).

t→∞

(185)

To prove that (185) holds if ω is an equilibrium state, and for W (g) replaced by any observable B, i.e. to show return to equilibrium as in (179), we need to use special properties of equilibrium states. The property of asymptotic abelianness, (184), does not suffice. Let ωβ denote an equilibrium state of the free Bose gas with a continuous density (169) and with a fixed condensate density ρ0 ≥ 0. The expectation functional is given by (156). We have ωβ (W (g)W (eiωt f )W (h)) = e− 2 Im[g,e i

iωt

f +g+eiωt f ,h]

ωβ (W (g + eiωt f + h))

and (using again the Riemann-Lebesgue Lemma) lim ωβ (W (g)W (eiωt f )W (h))

i (2π)3 √ ρ(g + h) 2 = e− 2 Img,h EF (g + h) exp − 2

3 (2π) √ ρf 2 ×EF (f ) exp − 2 5 ×J0 (2π)3/2 2ρ0 |g(0) + f (0) + h(0)| .

t→∞

(186)

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In the absence of a condensate, ρ0 = 0, J0 (0) = 1, equation (186) is just lim ωβ (W (g)W (eiωt f )W (h)) = ωβ (W (g)W (h)) ωβ (W (f )).

t→∞

(187)

Using an easy approximation argument (as in (180)), this yields the property of return to equilibrium (ρ0 = 0), lim ωβ (Bαt (A)C) = ωβ (BC)ωβ (A),

t→∞

(188)

for any A, B, C ∈ CCR(D). What happens in presence of a condensate, ρ0 > 0? Formula (188) is not valid in this case, because the Bessel function in (186) does not split into a product. However, the integrand in the representation (134) of J0 does split into a product according to 5 J0 (2π)3/2 2ρ0 |g(0) + f (0) + h(0)| π 5 dθ exp −i(2π)3/2 2ρ0 [Re(g(0) + h(0)) cos θ + Im(g(0) + h(0)) sin θ] = 2π −π 5 × exp −i(2π)3/2 2ρ0 [Re(f (0)) cos θ + Im(f (0)) sin θ]. (189) This suggests that for an equilibrium state with a condensate and a fixed value of θ, the property of return to equilibrium holds. To cast this into a precise form we write π dθ θ ωβ (W (f )), ωβ (W (f )) = (190) −π 2π where

ωβθ (W (f )) = e−iΦ(f,θ) Ωρ , πρ (W (f ))Ωρ ,

(191)

with Ωρ given in (150) and πρ defined in (151), (152). This decomposition is in accordance with the decomposition of the Hilbert space into a direct integral, Hρ0 ,ρ =

⊕

dσ Hρ ,

(192)

S1

see (149), (158). The GNS representation of (CCR(D), ωβθ ) is given by (Hρ , πβθ , Ωρ ), where

πβθ (W (f )) = e−iΦ(f,θ) πρ (W (f )).

(193)

The representation map πβ associated to the state ωβ is the direct integral of the fibers πβθ , and the representation vector Ωβ of ωβ is the direct integral with constant fiber Ωρ : ⊕ ⊕ dθ θ dθ πβ , Ω β = Ωρ . πβ = (194) 2π 2π [−π,π] [−π,π]

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It is clear from (191) that for each θ fixed, ωβθ is a (αt , β)-KMS state. The (αt , β)KMS state ωβ is a uniform superposition of the ωβθ , θ ∈ S 1 . We can form other equilibrium states by taking different superpositions of the ωβθ : Given any probability measure dµ on [−π, π], π ωµ (W (f )) := dµ(θ) ωβθ (W (f )) (195) −π

is an (αt , β)-KMS state. As follows directly from (188) and (191), for each fixed θ we have limt→∞ ωβθ (Bαt (A)C) = ωβθ (BC)ωβθ (A), so lim ωµ (Bαt (A)C) =

t→∞

π

−π

dµ(θ) ωβθ (BC) ωβθ (A).

(196)

In general, the r.h.s. of (196) does not equal ωµ , so the time-asymptotic state depends on the initial state. If the perturbation of the state ωµ is given by B, C s.t. ωβθ (BC) = 1 for all θ then we have return to ωµ in the usual sense. What is special about the equilibrium states ωβθ ? They are factor19 equilibrium states. The fact that each ωβθ is a factor state follows from M := πωθ (A) = β

B(F(L2 (R3 , d3 k))) ⊗ 1l, M = 1l ⊗ B(F(L2 (R3 , d3 k))), hence M ∩ M = C1l ⊗ 1l. On the other hand, it is clear that ωβ , (190), is not a factor state since 1l ⊗ 1l ⊗ M (denoting the multiplication operators on L2 (S 1 , dσ)) belongs to the center of its von Neumann algebra, see (159). The decomposition (192)–(194) is called a factor decomposition of the state ωβ , or a decomposition into extremal states. Let us now see how the emergence of a multitude of equilibrium states for a fixed inverse temperature β can be viewed as a consequence of spontaneous symmetry breaking – here, the gauge group symmetry is broken. The general scheme is this: suppose a dynamics αt on a C ∗ -algebra A has a symmetry group σs , i.e. σs is a group of automorphisms of A satisfying σs ◦ αt = αt ◦ σs , for all s, t (s may belong to a discrete or continuous set, t ∈ R). If ω is any (β, αt )-KMS state then ωs := ω ◦ σs is a (β, αt )-KMS state as well: ωs (Aαt (B)) = ω(σs (A)αt (σs (B))) = ω(αt−iβ (σs (B))σs (A)) = ωs (αt−iβ (B)A).

(197)

(We implicitly assume that αt−iβ (σs (B)) is well defined.) If there is a (β, αt )-KMS state which is not invariant under σs for some s, i.e., ω ◦ σs = ω, then we say the symmetry σs is spontaneously broken, because there are equilibrium states which 19

A state ω on a C ∗ –algebra A is a factor state iff the von Neumann algebra πω (A) is a factor. (Here, πω is the GNS representation map associated to (A, ω).) A von Neumann algebra M ⊂ B(H) is a factor iff its center is trivial, Z := M ∩ M = C1l. We point out that it follows from general considerations that an equilibrium state is a factor state iff it is extremal (see [5], Theorem 5.3.30). A state ω is called extremal iff the relation {ω = λω1 + (1 − λ)ω2 , for some 0 < λ < 1 and some states ω1 , ω2 } implies that ω1 = ω2 = ω.

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“have less symmetry” than the dynamics. This gives rise to the existence of several equilibrium states at the same temperature. Consider the continuous symmetry group σs on CCR(D) given by σs (W (f )) = W (eis f ), s ∈ R, f ∈ D,

(198)

called the gauge group. Clearly we have αt ◦ σs = σs ◦ αt (where αt is given in (163)). Using (191) we obtain ωβθ (σs (W (f ))) = e−iΦ(e

is

f,θ)

3

4 Ωρ , πρ (W (eis f ))Ωρ ,

(199)

3 4 and (147), (153) show that Ωρ , πρ (W (eis f ))Ωρ = Ωρ , πρ (W (f ))Ωρ , while (160) gives 5 Φ(eis f, θ) = (2π)3/2 2ρ0 Re(eis f (0)) cos θ + Im(eis f (0)) sin θ 5 = (2π)3/2 2ρ0 (Re(f (0)) cos(θ − s) + Im(f (0)) sin(θ − s)) = Φ(f, θ − s). (200) This shows that the equilibrium states ωβθ break the gauge group symmetry, hence giving rise to an S 1 -multitude of equilibrium states ((200) shows that we get the whole family ωβθ by varying s in any interval of length 2π). Let us finally examine the mixing properties of the equilibrium states with respect to space translations. Given a vector a ∈ R3 we define τa (W (f )) := W (fa ),

(201)

where fa (x) := f (x − a) is the translate of f by a. τa defines a (three parameter) group of automorphisms on CCR(D). We say that a state ω on CCR(D) has the property of strong mixing w.r.t. space translations if lim ω(W (f )τa (W (g))) = ω(W (f ))ω(W (g)),

|a|→∞

(202)

for any f, g ∈ D. This means that if two observables (W (f ) and W (g)) are spatially separated far from each other then the expectation of the product of the observables is close to the product of the expectations (independence of random variables). Intuitively, this means that the state ω has a certain property of locality in space: what happens far out in space does not influence events taking place, say, around the origin. Condition (202) is also called a cluster property. It is easy to calculate explicitly the l.h.s. of (202) for ω = ωβ , the equilibrium state of the free Bose gas with a continuous density (169) and with a fixed condensate density ρ0 ≥ 0 (whose expectation functional is given by (156)): lim ωβ W (f )τa (W (g)) |a|→∞ = ωβ (W (f ))ωβ (W (g)) exp −8π 3 ρ0 Re(f (0)g(0)) . (203)

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Consequently, the equilibrium state is strongly mixing w.r.t. space translations if and only if ρ0 = 0, i.e., if and only if there is no condensation. In presence of a condensate, the system exhibits long range correlations (what happens far out does influence what happens say at the origin). On the other hand, it is easily verified that each state ωβθ is strongly mixing. We may understand that limit states (limt→∞ of states of the form (195)) depend on the initial state as a consequence of the long-range correlations in presence of a condensate. Even as time reaches its asymptotic value the system “remembers” the initial state.

References 1. Araki, H.: Hamiltonian Formalism and the Canonical Commutation Relations in Quantum Field Theory. J. Math. Phys., 1, 492-504 (1960) 2. Abramowitz, A., Stegun, I. A.: Handbook of mathematical functions. Dover Publications, Inc., New York, Ninth Printing, 1970 3. Araki, H., Woods, E.: Representations of the canonical commutation relations describing a non-relativistic infinite free Bose gas. J. Math. Phys. 4, 637-662 (1963) 4. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, Berlin, second edition, 1987 5. Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Berlin, second edition, 1997 6. Cannon, J.T.: Infinite volume limits of the canonical free Bose gas states on the Weyl algebra. Commun. Math. Phys., 29, 89-104 (1973) 7. Glimm, J., Jaffe, A.: Quantum Physics. A functional integral point of view. Springer, New York, second edition, 1981 8. Hida, T.: Brownian Motion. Springer-Verlag, Berlin, 1980 9. Jacobs, K.: Measure and Integral. Academic Press, New York, 1978 10. Lewis, J.T., Pul`e, J.V.: The equilibrium states of the free Boson gas. Commun. Math. Phys. 36, 1-18 (1974) 11. Morse, P. M., Feshbach, H.: Methods of Theoretical Physics, Part I. McGraw-Hill Book Company, Inc., New York, Toronto, London, 1953 12. Petz, D.: An invitation to the Algebra of Canonical Commutation Relations. Leuven Notes in Mathematical and Theoretical Physics, Volume 2, Series A: Mathematical Physics, Leuven University Press, 1990 13. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness. Academic Press, London, 1975. 14. Takesaki, M. :Theory of Operator Algebra I. Springer, Berlin, second edition, 2002

Topics in Spectral Theory Vojkan Jakˇsi´c Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada e-mail: [email protected]

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

2

Preliminaries: measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 2.1 2.2 2.3 2.4 2.5 2.6 2.7

3

238 238 240 240 241 242 247

Preliminaries: harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

4

Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riesz representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lebesgue-Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier transform of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differentiation of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson transforms and Radon-Nikodym derivatives . . . . . . . . . . . . . Local Lp norms, 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local version of the Wiener theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson representation of harmonic functions . . . . . . . . . . . . . . . . . . The Hardy class H ∞ (C+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Borel transform of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 253 253 254 255 256 258 261 263

Self-adjoint operators, spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digression: The notions of analyticity . . . . . . . . . . . . . . . . . . . . . . . . . Elementary properties of self-adjoint operators . . . . . . . . . . . . . . . . . Direct sums and invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic spaces and the decomposition theorem . . . . . . . . . . . . . . . . . . The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of the spectral theorem—the cyclic case . . . . . . . . . . . . . . . . . . Proof of the spectral theorem—the general case . . . . . . . . . . . . . . . .

267 269 269 272 273 273 274 277

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4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 5

Harmonic analysis and spectral theory . . . . . . . . . . . . . . . . . . . . . . . . Spectral measure for A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The essential support of the ac spectrum . . . . . . . . . . . . . . . . . . . . . . . The functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Weyl criteria and the RAGE theorem . . . . . . . . . . . . . . . . . . . . . . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering theory and stability of ac spectra . . . . . . . . . . . . . . . . . . . . Notions of measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 280 281 281 283 285 286 287 290 291

Spectral theory of rank one perturbations . . . . . . . . . . . . . . . . . . . . . . . . 295 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Aronszajn-Donoghue theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simon-Wolff theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some remarks on spectral instability . . . . . . . . . . . . . . . . . . . . . . . . . . Boole’s equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poltoratskii’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. & M. Riesz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

296 298 299 300 301 302 304 308 309

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

1 Introduction These lecture notes are an expanded version of the lectures I gave in the Summer School on Open Quantum Systems, held in Grenoble June 16—July 4, 2003. Shortly afterwards, I also lectured in the Summer School on Large Coulomb Systems—QED, held in Nordfjordeid August 11—18, 2003 [13]. The Nordfjordeid lectures were a natural continuation of the material covered in Grenoble, and [13] can be read as Section 6 of these lecture notes. The subject of these lecture notes is spectral theory of self-adjoint operators and some of its applications to mathematical physics. This topic has been covered in many places in the literature, and in particular in [4, 23, 24, 25, 26, 17]. Given the clarity and precision of these references, there appears to be little need for additional lecture notes on the subject. On the other hand, the point of view adopted in these lecture notes, which has its roots in the developments in mathematical physics which primarily happened over the last decade, makes the notes different from most expositions and I hope that the reader will learn something new from them. The main theme of the lecture notes is the interplay between spectral theory of self-adjoint operators and classical harmonic analysis. In a nutshell, this interplay can be described as follows. Consider a self-adjoint operator A on a Hilbert space H and a vector ϕ ∈ H. The function

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F (z) = (ϕ|(A − z)−1 ϕ) is analytic in the upper half-plane Im z > 0 and satisfies the bound |F (z)| ≤ ϕ 2 /Im z. By a well-known result in harmonic analysis (see Theorem 3.11) there exists a positive Borel measure µϕ on R such that for Im z > 0, dµϕ (x) . F (z) = R x−z The measure µϕ is the spectral measure for A and ϕ. Starting with this definition we will develop the spectral theory of A. In particular, we will see that many properties of the spectral measure can be characterized by the boundary values limy↓0 F (x+iy) of the corresponding function F . The resulting theory is mathematically beautiful and has found many important applications in mathematical physics. In Section 5 we will discuss a simple and important application to the spectral theory of rank one perturbations. A related application concerns the spectral theory of the WignerWeisskopf atom and is discussed in the lecture notes [13]. Although we are mainly interested in applications of harmonic analysis to spectral theory, it is sometimes possible to turn things around and use the spectral theory to prove results in harmonic analysis. To illustrate this point, in Section 5 we will prove Boole’s equality and the celebrated Poltoratskii theorem using spectral theory of rank one perturbations. The lecture notes are organized as follows. In Section 2 we will review the results of the measure theory we will need. The proofs of less standard results are given in detail. In particular, we present detailed discussion of the differentiation of measures based on the Besicovitch covering lemma. The results of harmonic analysis we will need are discussed in Section 3. They primarily concern Poisson and Borel transforms of measures and most of them can be found in the classical references [18, 19]. However, these references are not concerned with applications of harmonic analysis to spectral theory, and the reader would often need to go through a substantial body of material to extract the needed results. To aid the reader, we have provided proofs of all results discussed in Section 3. Spectral theory of self-adjoint operators is discussed in Section 4. Although this section is essentially self-contained, many proofs are omitted and the reader with no previous exposition to spectral theory would benefit by reading it in parallel with Chapters VII-VIII of [23] and Chapters I-II of [4]. Spectral theory of rank one perturbations is discussed in Section 5. Concerning the prerequisites, it is assumed that the reader is familiar with basic notions of real, functional and complex analysis. Familiarity with [23] or the first ten Chapters of [27] should suffice. Acknowledgment. I wish to thank St´ephane Attal, Alain Joye and Claude-Alain Pillet for the invitation to lecture in the Grenoble summer school. I am also grateful

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to Jonathan Breuer, Eugene Kritchevski, and in particular Philippe Poulin for many useful comments about the lecture notes. The material in the lecture notes is based upon work supported by NSERC.

2 Preliminaries: measure theory 2.1 Basic notions Let M be a set and F a σ-algebra in M . The pair (M, F) is called a measure space. We denote by χA the characteristic function of a subset A ⊂ M . Let µ be a measure on (M, F). We say that µ is concentrated on the set A ∈ F if µ(E) = µ(E ∩ A) for all E ∈ F. If µ(M ) = 1, then µ is called a probability measure. Assume that M is a metric space. The minimal σ-algebra in M that contains all open sets is called the Borel σ-algebra. A measure on the Borel σ-algebra is called a Borel measure. If µ is Borel, the complement of the largest open set O such that µ(O) = 0 is called the support of µ and is denoted by supp µ. Assume that M is a locally compact metric space. A Borel measure µ is called regular if for every E ∈ F, µ(E) = inf{µ(V ) : E ⊂ V, V open} = sup{µ(K) : K ⊂ E, K compact}. Theorem 2.1. Let M be a locally compact metric space in which every open set is σ-compact (that is, a countably union of compact sets). Let µ be a Borel measure finite on compact sets. Then µ is regular. The measure space we will most often encounter is R with the usual Borel σalgebra. Throughout the lecture notes we will denote by 1l the constant function 1l(x) = 1 ∀x ∈ R. 2.2 Complex measures Let (M, F) be a measure space. Let E ∈ F. A countable collection of sets {Ei } in F is called a partition of E if Ei ∩ Ej = ∅ for i = j and E = ∪j Ej . A complex measure on (M, F) is a function µ : F → C such that µ(E) =

∞

µ(Ei ),

(1)

i=1

for every E ∈ F and every partition {Ei } of E. In particular, the series (1) is absolutely convergent. Note that complex measures take only finite values. The usual positive measures, however, are allowed to take the value ∞. In the sequel, the term positive measure

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will refer to the standard definition of a measure on a σ-algebra which takes values in [0, ∞]. The set function |µ| on F defined by |µ(Ei )|, |µ|(E) = sup i

where the supremum is taken over all partitions {Ei } of E, is called the total variation of the measure µ. Theorem 2.2. Let µ be a complex measure. Then: (1) The total variation |µ| is a positive measure on (M, F). (2) |µ|(M ) < ∞. (3) There exists a measurable function h : M → C such that |h(x)| = 1 for all x ∈ M and h(x)d|µ|(x), µ(E) = E

for all E ∈ F. The last relation is abbreviated dµ = hd|µ|. A complex measure µ is called regular if |µ| is a regular measure. Note that if ν is a positive measure, f ∈ L1 (M, dν) and µ(E) = f dν, E

then

|f |dν.

|µ|(E) = E

The integral with respect to a complex measure is defined in the obvious way, f dµ = f hd|µ|. Notation. Let µ be a complex or positive measure and f ∈ L1 (M, d|µ|). In the sequel we will often denote by f µ the complex measure (f µ)(E) = f dµ. E

Note that |f µ| = |f ||µ|. Every complex measure can be written as a linear combination of four finite positive measures. Let h1 (x) = Re h(x), h2 (x) = Im h(x), h+ i (x) = max(hi (x), 0), ± ± (x) = − min(h (x), 0), and µ = h |µ|, i = 1, 2. Then h− i i i i − + − µ = (µ+ 1 − µ1 ) + i(µ2 − µ2 ).

A complex measure µ which takes values in R is called a signed measure. Such a measure can be decomposed as

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µ = µ+ − µ− , where µ+ = (|µ| + µ)/2, µ− = (|µ| − µ)/2. If A = {x ∈ M : h(x) = 1}, B = {x ∈ M : h(x) = −1}, then for E ∈ F, µ+ (E) = µ(E ∩ A),

µ− (E) = −µ(E ∩ B).

This fact is known as the Hahn decomposition theorem. 2.3 Riesz representation theorem In this subsection we assume that M is a locally compact metric space. A continuous function f : M → C vanishes at infinity if ∀ > 0 there exists a compact set K such that |f (x)| < for x ∈ K . Let C0 (M ) be the vector space of all continuous functions that vanish at infinity, endowed with the supremum norm f = supx∈M |f (x)|. C0 (M ) is a Banach space and we denote by C0 (M )∗ its dual. The following result is known as the Riesz representation theorem. Theorem 2.3. Let φ ∈ C0 (M )∗ . Then there exists a unique regular complex Borel measure µ such that f dµ, φ(f ) = M

for all f ∈ C0 (M ). Moreover, φ = |µ|(M ). 2.4 Lebesgue-Radon-Nikodym theorem Let (M, F) be a measure space. Let ν1 and ν2 be complex measures concentrated on disjoint sets. Then we say that ν1 and ν2 are mutually singular (or orthogonal), and write ν1 ⊥ ν2 . If ν1 ⊥ ν2 , then |ν1 | ⊥ |ν2 |. Let ν be a complex measure and µ a positive measure. We say that ν is absolutely continuous w.r.t. µ, and write ν + µ, if µ(E) = 0 ⇒ ν(E) = 0. The following result is known as the Lebesgue-Radon-Nikodym theorem. Theorem 2.4. Let ν be a complex measure and µ a positive σ-finite measure on (M, F). Then there exists a unique pair of complex measures νa and νs such that νa ⊥ νs , νa + µ, νs ⊥ µ, and ν = νa + ν s . Moreover, there exists a unique f ∈ L1 (R, dµ) such that ∀E ∈ F, νa (E) = f dµ. E

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The Radon-Nikodym decomposition is abbreviated as ν = f µ + νs (or dν = f dµ + dνs ). If M = R and µ is the Lebesgue measure, we will use special symbols for the Radon-Nikodym decomposition. We will denote by νac the part of ν which is absolutely continuous (abbreviated ac) w.r.t. the Lebesgue measure and by νsing the part which is singular with respect to the Lebesgue measure. A point x ∈ R is called an atom of ν if ν({x}) = 0. Let Aν be the set of all atoms of ν. The set Aν is countable and x∈Aν |ν({x})| < ∞. The pure point part of ν is defined by

νpp (E) =

ν({x}).

x∈E∩Aν

The measure νsc = νsing − νpp is called the singular continuous part of ν. 2.5 Fourier transform of measures Let µ be a complex Borel measure on R. Its Fourier transform is defined by µ ˆ(t) = e−itx dµ(x). R

µ ˆ(t) is also called the characteristic function of the measure µ. Note that −ihx e |ˆ µ(t + h) − µ ˆ(t)| ≤ − 1 d|µ|, R

and so the function R * t !→ µ ˆ(t) ∈ C is uniformly continuous. The following result is known as the Riemann-Lebesgue lemma. Theorem 2.5. Assume that µ is absolutely continuous w.r.t. the Lebesgue measure. Then lim |ˆ µ(t)| = 0. (2) |t|→∞

The relation (2) may hold even if µ is singular w.r.t. the Lebesgue measure. The measures for which (2) holds are called Rajchman measures. A geometric characterization of such measures can be found in [20]. Recall that Aµ denotes the set of atoms of µ. In this subsection we will prove the Wiener theorem: Theorem 2.6. Let µ be a signed Borel measure. Then 1 T →∞ T

T

|ˆ µ(t)|2 dt =

lim

0

x∈Aµ

µ({x})2 .

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Proof: Note first that

ˆ(t)ˆ µ(t) = |ˆ µ(t)|2 = µ Let 1 KT (x, y) = T

e

−it(x−y)

0

Then 1 T Obviously,

T

e−it(x−y) dµ(x)dµ(y).

R2

. (1 − e−iT (x−y) )/(iT (x − y)) dt = 1

if x = y, if x = y.

T

|ˆ µ(t)| dt = 2

0

R2

KT (x, y)dµ(x)dµ(y).

. 0 if x = y, lim KT (x, y) = T →∞ 1 if x = y.

Since |KT (x, y)| ≤ 1, by the dominated convergence theorem we have that for all x, lim KT (x, y)dµ(y) = µ({x}). T →∞

R

By Fubini’s theorem, ; : KT (x, y)dµ(x)dµ(y) = KT (x, y)dµ(y) dµ(x), R2

R

R

and by the dominated convergence theorem, 1 lim T →∞ T

T

|ˆ µ(t)| dt = 2

0

=

µ({x})dµ(x) R

µ({x})2 .

x∈Aν

2.6 Differentiation of measures We will discuss only the differentiation of Borel measures on R. The differentiation of Borel measures on Rn is discussed in the problem set. We start by collecting some preliminary results. The first result we will need is the Besicovitch covering lemma. Theorem 2.7. Let A be a bounded set in R and, for each x ∈ A, let Ix be an open interval with center at x.

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(1) There is a countable subcollection {Ij } of {Ix }x∈A such that A ⊂ ∪Ij and that each point in R belongs to at most two intervals in {Ij }, i.e. ∀y ∈ R, χIj (y) ≤ 2. j

(2) There is a countable subcollection {Ii,j }, i = 1, 2, of {Ix }x∈A such that A ⊂ ∪Ii,j and Ii,j ∩ Ii,k = ∅ if j = k. In the sequel we will refer to {Ii } and {Ii,j } as the Besicovitch subcollections. Proof. |I| denotes the length of the interval I. We will use the shorthand I(x, r) = (x − r, x + r). Setting rx = |Ix |/2, we have Ix = I(x, rx ). Let d1 = sup{rx : x ∈ A}. Choose I1 = I(x1 , r1 ) from the family {Ix }x∈A such that r1 > 3d1 /4. Assume that I1 , . . . , Ij−1 are chosen for j ≥ 1 and that Aj = A \ ∪j−1 i=1 Ii is non-empty. Let dj = sup{rx : x ∈ Aj }. Then choose Ij = I(xj , rj ) from the family {Ix }x∈Aj such that rj > 3dj /4. In this way be obtain a countable (possibly finite) subcollection Ij = I(xj , rj ) of {Ix }x∈A . Suppose that j > i. Then xj ∈ Ai and ri ≥

3 3rj sup{rx : x ∈ Ai } ≥ . 4 4

(3)

This observation yields that the intervals I(xj , rj /3) are disjoint. Indeed, if j > i, then xj ∈ I(xi , ri ), and (3) yields |xi − xj | > ri =

2ri ri rj ri + > + . 3 3 3 3

Since A is a bounded set and xj ∈ A, the disjointness of Ij = I(xj , rj /3) implies that if the family {Ij } is infinite, then lim rj = 0.

j→∞

(4)

The relation (4) yields that A ⊂ ∪j I(xj , rj ). Indeed, this is obvious if there are only finitely many Ij ’s. Assume that there are infinitely many Ij ’s and let x ∈ A. By (4), there is j such that rj < 3rx /4, and by the definition of rj , x ∈ ∪j−1 i=1 Ii . Notice that if three intervals in R have a common point, then one of the intervals is contained in the union of the other two. Hence, by dropping superfluous intervals from the collection {Ij }, we derive that A ⊂ ∪j Ij and that each point in R belongs to no more than two intervals Ij . This proves (1). To prove (2), we enumerate Ij ’s as follows. To I1 is associated the number 0. The intervals to right of I1 are enumerated in succession by positive integers, and the intervals to the left by negative integers. (The ”succession” is well-defined, since no point belongs simultaneously to three intervals). The intervals associated to even

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integers are mutually disjoint, and so are the intervals associated to odd integers. Finally, denote the interval associated to 2n by I1,n , and the interval associated to 2n + 1 by I2,n . This construction yields (2). Let µ be a positive Borel measure on R finite on compact sets and let ν be a complex measure. The corresponding maximal function is defined by Mν,µ (x) = sup r>0

|ν|(I(x, r)) , µ(I(x, r))

x ∈ supp µ.

If x ∈ supp µ we set Mν,µ (x) = ∞. It is not hard (Problem 1) to show that the function R * x !→ Mν,µ (x) ∈ [0, ∞] is Borel measurable. Theorem 2.8. For any t > 0, µ {x : Mν,µ (x) > t} ≤

2 |ν|(R). t

Proof. Let [a, b] be a bounded interval and Et = [a, b] ∩ {x : Mν,µ (x) > t}. Every point x in Et is the center of an open interval Ix such that |ν|(Ix ) ≥ tµ(Ix ). Let Ii,j be the Besicovitch subcollection of {Ix }. Then, Et ⊂ ∪Ii,j , and µ(Et ) ≤

µ(Ii,j ) ≤

i,j

1 |ν|(Ii,j ) t i,j

1 2 = |ν|(∪j Ii,j ) ≤ |ν|(R). t i=1 t 2

The statement follows by taking a → −∞ and b → +∞. In Problem 3 the reader is asked to prove: Proposition 2.9. Let A be a bounded Borel set. Then for any 0 < p < 1, Mν,µ (x)p dµ(x) < ∞. A

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We will also need: Proposition 2.10. Let νj be a sequence of Borel complex measures such that lim |νj |(R) = 0.

j→∞

Then there is a subsequence νjk such that lim Mνjk ,µ (x) = 0

k→∞

for µ − a.e. x.

Proof. By Theorem 2.8, for each k = 1, 2, . . . , we can find jk so that 8 7 µ x : Mνjk ,µ (x) > 2−k ≤ 2−k . Hence,

∞ 8 7 µ x : Mνjk ,µ (x) > 2−k < ∞, k=1

and so for µ-a.e. x, there is kx such that for k > kx , Mνjk ,µ (x) ≤ 2−k . This yields the statement. We are now ready to prove the main theorem of this subsection. Theorem 2.11. Let ν be a complex Borel measure and µ a positive Borel measure finite on compact sets. Let ν = f µ + νs be the Radon-Nikodym decomposition. Then: (1) ν(I(x, r)) = f (x), for µ − a.e. x. lim r↓0 µ(I(x, r)) In particular, ν ⊥ µ iff lim r↓0

ν(I(x, r)) = 0, µ(I(x, r))

for µ − a.e. x.

(2) Let in addition ν be positive. Then lim r↓0

ν(I(x, r)) = ∞, µ(I(x, r))

for νs − a.e. x.

Proof. (1) We will split the proof into two steps. Step 1. Assume that ν + µ, namely that ν = f µ. Let gn be a continuous function with compact support such that R |f − gn |dµ < 1/n. Set hn = f − gn . Then, for x ∈ supp µ and r > 0, |hn |µ(I(x, r)) gn µ(I(x, r)) f µ(I(x, r)) µ(I(x, r)) − f (x) ≤ µ(I(x, r)) + µ(I(x, r)) − gn (x) + |gn (x) − f (x)|.

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Since gn is continuous, we obviously have gn µ(I(x, r)) − gn (x) = 0, lim r↓0 µ(I(x, r)) and so for all n and x ∈ supp µ, f µ(I(x, r)) lim sup − f (x) ≤ Mhn µ,µ (x) + |gn (x) − f (x)|. µ(I(x, r)) r↓0 Let nj be a subsequence such that gnj → f (x) for µ-a.e. x. Since |hnj |dµ → 0 as j → ∞, Proposition 2.10 yields that there is a subsequence of nj (which we denote by the same letter) such that Mhnj µ,µ (x) → 0 for µ-a.e. x. Hence, for µ-a.e. x, f µ(I(x, r)) − f (x) = 0, lim sup µ(I(x, r)) r↓0 and (1) holds if ν + µ. Step 2. To finish the proof of (1), it suffices to show that if ν is a complex measure such that ν ⊥ µ, then lim r↓0

|ν|(I(x, r)) = 0, µ(I(x, r))

for µ − a.e. x.

(5)

Let S be a Borel set such that µ(S) = 0 and |ν|(R \ S) = 0. Then |ν|(I(x, r)) χS (|ν| + µ)(I(x, r)) = . (µ + |ν|)(I(x, r)) (µ + |ν|)(I(x, r))

(6)

By Step 1, lim r↓0

χS (|ν| + µ)(I(x, r)) = χS (x), (|ν| + µ)(I(x, r))

for |ν| + µ − a.e. x.

(7)

Since χS (x) = 0 for µ-a.e. x, (6) and (7) yield (5). (2) Since ν is positive, ν(I(x, r)) ≥ νs (I(x, r)), and we may assume that ν ⊥ µ. By (6) and (7), lim r↓0

ν(I(x, r)) = 1, (ν + µ)(I(x, r))

and so lim r↓0

µ(I(x, r)) = 0, ν(I(x, r))

for ν − a.e. x,

for ν − a.e. x.

This yields (2). We finish this subsection with several remarks. If µ is the Lebesgue measure, then the results of this section reduce to the standard differentiation results discussed, for

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example, in Chapter 7 of [27]. The arguments in [27] are based on the Vitali covering lemma which is specific to the Lebesgue measure. The proofs of this subsection are based on the Besicovitch covering lemma and they apply to an arbitrary positive measure µ. In fact, the proofs directly extend to Rn (one only needs to replace the intervals I(x, r) with the balls B(x, r) centered at x and of radius r) if one uses the following version of the Besicovitch covering lemma. Theorem 2.12. Let A be a bounded set in Rn and, for each x ∈ A, let Bx be an open ball with center at x. Then there is an integer N , which depends only on n, such that: (1) There is a countable subcollection {Bj } of {Bx }x∈A such that A ⊂ ∪Bj and each point in Rn belongs to at most N balls in {Bj }, i.e. ∀y ∈ R, χBj (y) ≤ N. j

(2) There is a countable subcollection {Bi,j }, i = 1, · · · , N , of {Bx }x∈A such that A ⊂ ∪Bi,j and Bi,j ∩ Bi,k = ∅ if j = k. Unfortunately, unlike the proof of the Vitali covering lemma, the proof of Theorem 2.12 is somewhat long and complicated. 2.7 Problems [1] Prove that the maximal function Mν,µ (x) is Borel measurable. [2] Let µ be a positive σ-finite measure on (M, F) and let f be a measurable function. Let mf (t) = µ{x : |f (x)| > t}. Prove that for p ≥ 1,

∞

|f (x)|p dµ(x) = p M

tp−1 mf (t)dt.

0

This result can be generalized as follows. Let α : [0, ∞] !→ [0, ∞] be monotonic and absolutely continuous on [0, T ] for every T < ∞. Assume that α(0) = 0 and α(∞) = ∞. Prove that ∞ (α ◦ f )(x)dµ(x) = α (t)mf (t)dt. M

0

Hint: See Theorem 8.16 in [27]. [3] Prove Proposition 2.9. Hint: Use Problem 2. [4] Prove the Riemann-Lebesgue lemma (Theorem 2.5).

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[5] Let µ be a complex Borel measure on R. Prove that |µsing | = |µ|sing . [6] Let µ be a positive measure on (M, F). A sequence of measurable functions fn converges in measure to zero if lim µ({x : |fn (x)| > }) = 0,

n→∞

for all > 0. The sequence fn converges almost uniformly to zero if for all > 0 there is a set M ∈ F, such that µ(M ) < and fn converges uniformly to zero on M \ M . Prove that if fn converges to zero in measure, then there is a subsequence fnj which converges to zero almost uniformly. [7] Prove Theorem 2.12. (The proof can be found in [9]). [8] State and prove the analog of Theorem 2.11 in Rn . [9] Let µ be a positive Borel measure on R. Assume that µ is finite on compact sets and let f ∈ L1 (R, dµ). Prove that 1 lim |f (t) − f (x)|dµ(t) = 0, for µ − a.e. x. r↓0 µ(I(x, r)) I(x,r) Hint: You may follow the proof of Theorem 7.7 in [27]. [10] Let p ≥ 1 and f ∈ Lp (R, dx). The maximal function of f , Mf , is defined by 1 |f (t)|dt. Mf (x) = sup r>0 2r I(x,r) (1) If p > 1, prove that Mf ∈ Lp (R, dx). Hint: See Theorem 8.18 in [27]. (2) Prove that if f and Mf are in L1 (R, dx), then f = 0. [11] Denote by Bb (R) the algebra of the bounded Borel functions on R. Prove that Bb (R) is the smallest algebra of functions which includes C0 (R) and is closed under pointwise limits of uniformly bounded sequences.

3 Preliminaries: harmonic analysis In this section we will deal only with Borel measures on R. We will use the shorthand C+ = {z : Im z > 0}. We denote the Lebesgue measure by m and write dm = dx. Let µ be a complex measure or a positive measure such that dµ(t) < ∞. 1 + |t| R

Topics in Spectral Theory

The Borel transform of µ is defined by dµ(t) Fµ (z) = , R t−z

z ∈ C+ .

(8)

The function Fµ (z) is analytic in C+ . Let µ be a complex measure or positive measure such that dµ(t) < ∞. 1 + t2 R The Poisson transform of µ is defined by dµ(t) Pµ (x + iy) = y , 2 2 R (x − t) + y

249

(9)

y > 0.

The function Pµ (z) is harmonic in C+ . If µ is the Lebesgue measure, then Pµ (z) = π for all z ∈ C+ . Note that Fµ and Pµ are linear functions of µ, i.e. for λ1 , λ2 ∈ C, Fλ1 µ1 +λ2 µ2 = λ1 Fµ1 + λ2 Fµ2 , Pλ1 µ1 +λ2 µ2 = λ1 Pµ1 + λ2 Pµ2 . If µ is a positive or signed measure, then Im Fµ = Pµ . Our goal in this section is to study the boundary values of Pµ (x+iy) and Fµ (x+ iy) as y ↓ 0. More precisely, we wish to study how these boundary values reflect the properties of the measure µ. Although we will restrict ourselves to the radial limits, all the results discussed in this section hold for the non-tangential limits (see the problem set). The nontangential limits will not be needed for our applications. 3.1 Poisson transforms and Radon-Nikodym derivatives This subsection is based on [14]. Recall that I(x, r) = (x − r, x + r). Lemma 3.1. Let µ be a positive measure. Then for all x ∈ R and y > 0, 1 Pµ (x + iy) = y

1/y 2

µ(I(x,

u−1 − y 2 ))du.

0

Proof. Note that 1/y2 5 µ(I(x, u−1 − y 2 ))du = 0

1/y 2

= R

: R

0

Since

5

0

1/y 2

; χI(x,√u−1 −y2 ) (t)dµ(t) du ! χI(x,√u−1 −y2 ) (t)du dµ(t).

(10)

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|x − t| < we have

5

u−1 − y 2

0 ≤ u < ((x − t)2 + y 2 )−1 ,

⇐⇒

χI(x,√u−1 −y2 ) (t) = χ[0,((x−t)2 +y2 )−1 ) (u),

and

1/y 2

0

χI(x,√u−1 −y2 ) (t)du = ((x − t)2 + y 2 ))−1 .

Hence, the result follows from (10). Lemma 3.2. Let ν be a complex and µ a positive measure. Then for all x ∈ R and y > 0, |Pν (x + iy)| ≤ Mν,µ (x). Pµ (x + iy) Proof. Since |Pν | ≤ P|ν| , w.l.o.g. we may assume that ν is positive. Also, we may assume that x ∈ supp µ (otherwise Mν,µ (x) = ∞ and there is nothing to prove). Set 5 Ix,y (u) = I(x, u−1 − y 2 ). Since

1/y 2

ν(Ix,y (u))du = 0

0

1/y 2

ν(Ix,y (u)) µ(Ix,y (u))du µ(Ix,y (u))

1/y 2

≤ Mν,µ (x)

µ(Ix,y (u))du, 0

the result follows from Lemma 3.1. Lemma 3.3. Let µ be a positive measure. Then for µ-a.e. x, dµ(t) = ∞. 2 R (x − t)

(11)

The proof of this lemma is left for the problem set. Lemma 3.4. Let µ be a positive measure and f ∈ C0 (R). Then for µ-a.e. x, lim y↓0

Pf µ (x + iy) = f (x). Pµ (x + iy)

(12)

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251

Remark. The relation (12) holds for all x for which (11) holds. For example, if µ is the Lebesgue measure, then (12) holds for all x. Proof. Note that P|f −f (x)|µ (x + iy) Pf µ (x + iy) ≤ − f (x) . Pµ (x + iy) Pµ (x + iy) Fix > 0 and let δ > 0 be such that |x − t| < δ ⇒ |f (x) − f (t)| < . Let M = sup |f (t)| and dµ(t) C = 2M . 2 |x−t|≥δ (x − t) Then P|f −f (x)|µ (x + iy) ≤ Pµ (x + iy) + Cy, Pf µ (x + iy) Cy ≤+ − f (x) . Pµ (x + iy) Pµ (x + iy)

and

Let x be such that (11) holds. The monotone convergence theorem yields that y lim = y↓0 Pµ (x + iy)

dµ(t) (x − t)2

−1 =0

and so for all > 0, Pf µ (x + iy) − f (x) ≤ . lim sup Pµ (x + iy) y↓0 This yields the statement. The main result of this subsection is: Theorem 3.5. Let ν be a complex measure and µ a positive measure. Let ν = f µ+νs be the Radon-Nikodym decomposition. Then: (1) Pν (x + iy) = f (x), for µ − a.e. x. lim y↓0 Pµ (x + iy) In particular, ν ⊥ µ iff lim y↓0

Pν (x + iy) = 0, Pµ (x + iy)

for µ − a.e. x.

(2) Assume in addition that ν is positive. Then lim y↓0

Pν (x + iy) = ∞, Pµ (x + iy)

for νs − a.e. x.

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Proof. The proof is very similar to the proof of Theorem 2.11 in Section 2. (1) We will split the proof into two steps. Step 1. Assume that ν + µ, namely that ν = f µ. Let gn be a continuous function with compact support such that R |f − gn |dµ < 1/n. Set hn = f − gn . Then, P|hn |µ (x + iy) Pgn µ (x + iy) Pf µ (x + iy) Pµ (x + iy) − f (x) ≤ Pµ (x + iy) + Pµ (x + iy) − gn (x) + |gn (x) − f (x)|. It follows from Lemmas 3.2 and 3.4 that for µ-a.e. x, Pf µ (x + iy) − f (x) ≤ M|hn |µ,µ (x) + |gn (x) − f (x)|. lim sup P (x + iy)) µ y↓0 As in the proof of Theorem 2.11, there is a subsequence nj → ∞ such that gnj (x) → f (x) and M|hn |µ,µ (x) → 0 for µ-a.e. x, and (1) holds if ν + µ. Step 2. To finish the proof of (1), it suffices to show that if ν is a finite positive measure such that ν ⊥ µ, then lim y↓0

Pν (x + iy) = 0, Pµ (x + iy)

for µ − a.e. x.

(13)

Let S be a Borel set such that µ(S) = 0 and ν(R \ S) = 0. Then PχS (ν+µ) (x + iy) Pν (x + iy) = . Pν (x + iy) + Pµ (x + iy) Pν+µ (x + iy)

(14)

By Step 1, lim y↓0

PχS (ν+µ) (x + iy) = χS (x), Pν+µ (x + iy)

for ν + µ − a.e. x.

(15)

Since χS (x) = 0 for µ-a.e. x, lim y↓0

Pν (x + iy) = 0, Pν (x + iy) + Pµ (x + iy)

for µ − a.e. x,

and (13) follows. (2) Since ν is positive, ν(I(x, r)) ≥ νs (I(x, r)), and we may assume that ν ⊥ µ. By (14) and (15), lim y↓0

Pν (x + iy) = 1, Pν (x + iy) + Pµ (x + iy)

and so lim y↓0

This yields part (2).

Pµ (x + iy) = 0, Pν (x + iy)

for ν − a.e. x,

for ν − a.e. x.

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253

3.2 Local Lp norms, 0 < p < 1. In this subsection we prove Theorem 3.1 of [28]. ν is a complex measure, µ is a positive measure and ν = f µ + νs is the Radon-Nikodym decomposition. Theorem 3.6. Let A be a bounded Borel set and 0 < p < 1. Then Pν (x + iy) p lim dµ(x) = |f (x)|p dµ(x). y↓0 A Pµ (x + iy) A (Both sides are allowed to be ∞). In particular, ν A ⊥ µ A iff for some p ∈ (0, 1), Pν (x + iy) p dµ(x) = 0. lim y↓0 A Pµ (x + iy) Proof. By Theorem 3.5, Pν (x + iy) p = |f (x)|p lim y↓0 Pµ (x + iy) By Lemma 3.2,

for µ − a.e. x.

Pν (x + iy) p p Pµ (x + iy) ≤ Mν,µ (x) .

Hence, Proposition 2.9 and the dominated convergence theorem yield the statement. 3.3 Weak convergence Let ν be a complex or positive measure and dνy (x) =

1 Pν (x + iy)dx. π

(16)

Theorem 3.7. For any f ∈ Cc (R) (continuous functions of compact support), lim f (x)dνy (x) = f (x)dν(x). (17) y↓0

R

R

In particular, Pν1 = Pν2 ⇒ ν1 = ν2 . Proof. Note that R

and so

: f (x)dνy (x) =

R

y π

R

; f (x)dx dν(t), (x − t)2 + y 2

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Vojkan Jakˇsi´c

|Ly (t)|d|ν|(t) f (x)dνy (x) − f (x)dν(x) ≤ , 1 + t2 R R R

where

y Ly (t) = (1 + t ) f (t) − π

2

R

f (x)dx (x − t)2 + y 2

(18)

.

Clearly, supy>0,t∈R |Ly (t)| < ∞. By Lemma 3.4 and Remark after it, lim Ly (t) = 0, y↓0

for all t ∈ R (see also Problem 2). Hence, the statement follows from the estimate (18) and the dominated convergence theorem. 3.4 Local Lp -norms, p > 1 In this subsection we will prove Theorem 2.1 of [28]. Let ν be a complex or positive measure and let ν = f m + νsing be its RadonNikodym decomposition w.r.t. the Lebesgue measure. Theorem 3.8. Let A ⊂ R be open, p > 1, and assume that sup |Pν (x + iy)|p dx < ∞. 0
A

Then: (1) νsing A = 0. (2) A |f (x)|p dx < ∞. (3) For any [a, b] ⊂ A, π −1 Pν (x + iy) → f (x) in Lp ([a, b], dx) as y ↓ 0. Proof. We will prove (1) and (2); (3) is left to the problems. Let g be a continuous function with compact support contained in A and let q be the index dual to p, p−1 + q −1 = 1. Then, by Theorem 3.7, −1 gdν = lim π g(x)Pν (x + iy)dx, A

y↓0

A

and 1/q 1/p q p g(x)Pν (x + iy)dx ≤ |g(x)| dx |P (x + iy)| dx ν A

A

A

≤C

1/q |g(x)| dx . q

A

Hence, the map g !→ A g(x)dν(x) is a continuous linear functional on Lq (A, dx), and there is a function f˜ ∈ Lp (A, dµ) such that

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255

g(x)f˜(x)dx.

g(x)dν(x) = A

A

This relation implies that ν A is absolutely continuous w.r.t. the Lebesgue measure and that f (x) = f˜(x) for Lebesgue a.e. x. (1) and (2) follow. Theorem 3.8 has a partial converse which we will discuss in the problem set. 3.5 Local version of the Wiener theorem In this subsection we prove Theorem 2.2 of [28]. Theorem 3.9. Let ν be a signed measure and Aν be the set of atoms of ν. Then for any finite interval [a, b], ⎛ ⎞ b 2 2 ν({a}) ν({b}) π + + Pν (x + iy)2 dx = ⎝ ν({x})2 ⎠ . lim y y↓0 2 2 2 a x∈(a,b)∩Aν

Proof. 2

Pν (x + iy) = y and

2 R2

dν(t)dν(t ) , ((x − t)2 + y 2 )((x − t )2 + y 2 )

b

Pν (x + iy)2 dx =

y a

where gy (t, t ) =

b

R2

gy (t, t )dν(t)dν(t ),

y 3 dx . + y 2 )((x − t )2 + y 2 )

a ((x − Notice now that: (1) 0 ≤ gy (t, t ) ≤ π. (2) limy↓0 gy (t, t ) = 0 if t = t , or t ∈ [a, b], or t ∈ [a, b]. (3) If t = t ∈ (a, b), then dx π 3 lim gy (t, t) = lim y = , 2 + y 2 )2 y↓0 y↓0 (x 2 R

t)2

(compute the integral using the residue calculus). (4) If t = t = a or t = t = b, then ∞ dx π lim gy (t, t) = lim y 3 = . 2 + y 2 )2 y↓0 y↓0 (x 4 0 The result follows from these observations and the dominated convergence theorem. Corollary 3.10. A signed measure ν has no atoms in [a, b] iff b lim y Pν (x + iy)2 dx = 0. y↓0

a

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3.6 Poisson representation of harmonic functions Theorem 3.11. Let V (z) be a positive harmonic function in C+ . Then there is a constant c ≥ 0 and a positive measure µ on R such that V (x + iy) = cy + Pµ (x + iy). The c and µ are uniquely determined by V . Remark 1. The constant c is unique since c = limy→∞ V (iy)/y. By Theorem 3.7, µ is also unique. Remark 2. Theorems 3.5 and 3.11 yield that if V is a positive harmonic function in C+ and dµ = f (x)dx + µsing is the associated measure, then for Lebesgue a.e. x, lim π −1 V (x + iy) = f (x). y↓0

Let D = {z : |z| < 1} and Γ = {z : |z| = 1}. For z ∈ D and w ∈ Γ let pz (w) = Re

1 − |z|2 w+z = . w−z |w − z|2

We shall first prove: Theorem 3.12. Let U be a positive harmonic function in D. Then there exists a finite positive Borel measure ν on Γ such that for all z ∈ D, pz (w)dν(w). U (z) = Γ

Proof. By the mean value property of harmonic functions, for any 0 < r < 1, π 1 U (0) = U (reiθ )dθ. 2π −π In particular, 1 sup 2π 0

π

−π

U (reiθ )dθ = U (0) < ∞.

For f ∈ C(Γ) set Φr (f ) =

1 2π

π

U (reiθ )f (eiθ )dθ. −π

Each Φr is a continuous linear functional on the Banach space C(Γ) and Φr = U (0). The standard diagonal argument yields that there is a sequence rj → 1 and a bounded linear functional Φ on C(Γ) such that Φrj → Φ weakly, that is, for all f ∈ C(Γ), Φrj (f ) → Φ(f ). Obviously, Φ = U (0). By the Riesz representation theorem there exists a complex measure ν on Γ such that |ν|(Γ) = U (0) and

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257

Φ(f ) =

f (w)dν(w). Γ

Since Φrj (f ) ≥ 0 if f ≥ 0, the measure ν is positive. Finally, let z ∈ D. If rj > |z|, then π 1 U (zrj ) = U (rj eiθ )pz (eiθ )dθ = Φrj (pz ), (19) 2π −π (the proof of this relation is left for the problems—see Theorem 11.8 in [27]). Taking j → ∞ we derive U (z) = Φ(pz ) =

pz (w)dν(w). Γ

Before proving Theorem 3.11, I would like to make a remark about change of variables in measure theory. Let (M1 , F1 ) and (M2 , F2 ) be measure spaces. A map T : M1 → M2 is called a measurable transformation if for all F ∈ F2 , T −1 (F ) ∈ F1 . Let µ be a positive measure on (M1 , F1 ), and let µT be a positive measure on (M2 , F2 ) defined by µT (F ) = µ(T −1 (F )). If f is a measurable function on (M2 , F2 ), then fT = f ◦ T is a measurable function on (M1 , F1 ). Moreover, f ∈ L1 (M2 , dµT ) iff fT ∈ L1 (M1 , dµ), and in this case f dµT = fT dµ. M2

M1

This relation is easy to check if f is a characteristic function. The general case follows by the usual approximation argument through simple functions. If T is a bijection, then g ∈ L1 (M1 , dµ) iff gT −1 ∈ L1 (M2 , dµT ), and in this case gdµ = gT −1 dµT . M1

M2

Proof of Theorem 3.11. We define a map S : C+ → D by S(z) =

i−z . i+z

(20)

This is the well-known conformal map between the upper half-plane and the unit disc. The map S extends to a homeomorphism S : C+ → D \ {−1}. Note that S(R) = Γ \ {−1}. If Kz (t) =

y , (x − t)2 + y 2

z = x + iy ∈ C+ ,

then (1 + t2 )Kz (t) = pS(z) (S(t)). Let T = S −1 . Explicitly, T (ξ) =

ξ−1 . iξ + i

(21)

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Vojkan Jakˇsi´c

Let U (ξ) = V (T (ξ)). Then there exists a positive finite Borel measure ν on Γ such that pξ (w)dν(w). U (ξ) = Γ

The map T : Γ \ {−1} → R is a homeomorphism. Let νT be the induced Borel measure on R. By the previous change of variables, pξ (w)dν(w) = (1 + t2 )KT (ξ) (t)dνT (t). Γ\{−1}

Hence, for z ∈ C+ , V (z) = pS(z) (−1)ν({−1}) +

R

(1 + t2 )Kz (t)dνT (t).

Since pS(z) (−1) = y, setting c = ν({−1}) and dµ(t) = (1 + t2 )dνT (t), we derive the statement. 3.7 The Hardy class H ∞ (C+ ) The Hardy class H ∞ (C+ ) is the vector space of all functions V analytic in C+ such that (22) V = sup |V (z)| < ∞. z∈C+

H ∞ (C+ ) with norm (22) is a Banach space. In this subsection we will prove two basic properties of H ∞ (C+ ). Theorem 3.13. Let V ∈ H ∞ (C+ ). Then for Lebesgue a.e. x ∈ R, the limit V (x) = lim V (x + iy) y↓0

(23)

exists. Obviously, V ∈ L∞ (R, dx). Theorem 3.14. Let V ∈ H ∞ (C+ ), V ≡ 0, and let V (x) be given by (23). Then | log |V (x)|| dx < ∞. 1 + x2 R In particular, if α ∈ C, then either V (z) ≡ α or the set {x ∈ R : V (x) = α} has zero Lebesgue measure. A simple and important consequence of Theorems 3.13 and 3.14 is: Theorem 3.15. Let F be an analytic function on C+ with positive imaginary part. Then: (1) For Lebesgue a.e. x ∈ R the limit

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259

F (x) = lim F (x + iy), y↓0

exists and is finite. (2) If α ∈ C, then either F (z) ≡ α or the set {x ∈ R : F (x) = α} has zero Lebesgue measure. Proof. To prove (1), apply Theorem 3.13 to the function (F (z) + i)−1 . To prove (2), apply Theorem 3.14 to the function (F (z) + i)−1 − (α + i)−1 . Proof of Theorem 3.13. Let dνy (t) = V (t + iy)dt. Then, for f ∈ L1 (R, dt), f (t)dνy (t) ≤ V |f (t)|dt. R

R

The map Φy (f ) = R f dνy is a linear functional on L1 (R, dt) and Φy ≤ V . By the Banach-Alaoglu theorem, there a bounded linear functional Φ and a sequence yn ↓ 0 such that for all f ∈ L1 (R, dt), lim Φyn (f ) = Φ(f ).

n→∞

Let V ∈ L∞ (R, dt) be such that Φ(f ) =

R

V (t)f (t)dt. Let

f (t) = π −1 y((x − t)2 + y 2 )−1 . A simple residue calculation yields V (t + iyn )dt Φyn (f ) = π −1 y = V (x + i(y + yn )). (x − t)2 + y 2 R Taking n → ∞, we get V (x + iy) = π −1 y

R

V (t)dt , (x − t)2 + y 2

(24)

and Theorem 3.5 yields the statement. Remark 1. Theorem 3.13 can be also proven using Theorem 3.11. The above argument has the advantage that it extends to any Hardy class H p (C+ ). Remark 2. In the proof we have also established the Poisson representation of V (the relation (24)). The next proposition is known as Jensen’s formula. Proposition 3.16. Assume that U (z) is analytic for |z| < 1 and that U (0) = 0. Let r ∈ (0, 1) and assume that U has no zeros on the circle |z| = r. Let α1 , α2 , . . . , αn be the zeros of U (z) in the region |z| < r, listed with multiplicities. Then

π n 1 r = exp log |U (reit )|dt . (25) |U (0)| |αj | 2π −π j=1

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Remark. The Jensen formula holds even if U has zeros on |z| = r. We will only need the above elementary version. Proof. Set n r 2 − αj z . V (z) = U (z) r(αj − z) j=1 Then for some > 0 V (z) has no zeros in the disk |z| < r + and the function log |V (z)| is harmonic in the same disk (see Theorem 13.12 in [27]). By the mean value theorem for harmonic functions, π 1 log |V (0)| = log |V (reiθ )|dθ. 2π −π The substitution yields the statement. Proof of Theorem 3.14. Setting U (eit ) = V (tan(t/2)), we have that | log |V (x)|| 1 π dx = | log |U (eit )||dt. 1 + x2 2 −π R

(26)

Hence, it suffices to show that the integral on the r.h.s. is finite. In the rest of the proof we will use the same notation as in the proof of Theorem 3.11. Recall that S and T are defined by (20) and (21). Let U (z) = V (T (z)). Then, U is holomorphic in D and supz∈D |U (z)| < ∞. Moreover, a change of variables and the formula (24) yield that π 1 − r2 1 U (eit )dt. U (reiθ ) = 2π −π 1 + r2 − 2r cos(θ − t) (The change of variables exercise is done in detail in [19], pages 106-107.) The analog of Theorem 3.5 for the circle yields that for Lebesgue a.e. θ lim U (reiθ ) = U (eiθ ).

r→1

(27)

The proof is outlined in the problem set. We will now make use of the Jensen formula. If U (0) = 0, let m be such Um (z) = z −m U (z) satisfies Um (0) = 0 (if U (0) = 0, then m = 0). Let rj → 1 be a sequence such that U has no zeros on |z| = rj . Set π π 1 1 it log |Um (rj e )|dt = −m log rj + log |U (rj eit )|dt. Jrj = 2π −π 2π −π The Jensen formula (applied to Um ) yields that Jri ≤ Jrj if ri ≤ rj . Write log+ x = max(log x, 0), log− x = − min(log x, 0). Note that sup log+ |U (eit )| ≤ sup |U (eit )| < ∞. t

t

Fatou’s lemma, the dominated convergence theorem and (27) yield that

Topics in Spectral Theory

Jr1 +

1 2π

π

−π

log− |U (eit )|dt ≤

Hence,

1 2π

261

π

−π

log+ |U (eit )|dt < ∞.

π

−π

| log |U (eit )||dt < ∞,

and the identity (26) yields the statement. 3.8 The Borel transform of measures Recall that the Borel transform Fµ (z) is defined by (8). Theorem 3.17. Let µ be a complex or positive measure. Then: (1) For Lebesgue a.e. x the limit Fµ (x) = lim Fµ (x + iy), y↓0

exists and is finite. (2) If Fµ ≡ 0, then

R

| log |Fµ (x)|| dx < ∞. 1 + x2

(28)

(3) If Fµ ≡ 0, then for any complex number α the set {x ∈ R : Fµ (x) = α},

(29)

has zero Lebesgue measure. Remark. It is possible that µ = 0

and

Fµ ≡ 0.

(30)

For example, this is the case if dµ = (x − 2i)−1 (x − i)−1 dx. By the theorem of F. & M. Riesz (see, e.g., [19]), if (30) holds, then dµ = h(x)dx, where h(x) = 0 for Lebesgue a.e. x. We will prove the F. & M. Riesz theorem in Section 5. Proof. We will first show that Fµ (z) =

R(z) , G(z)

where R, G ∈ H ∞ (C+ ) and G has no zeros in C+ . If µ is positive, set G(z) =

1 . i + Fµ (z)

Then, G(z) is holomorphic in C+ , |G(z)| ≤ 1 (since Im Fµ (z) ≥ 0), and

(31)

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Fµ (z) =

1 − iG(z) . G(z)

If µ is a complex measure, we first decompose µ = (µ1 − µ2 ) + i(µ3 − µ4 ), where the µi ’s are positive measures, and then decompose Fµ (z) = (Fµ1 (z) − Fµ2 (z)) + i(Fµ3 (z) − Fµ4 (z)). Hence, (31) follows from the corresponding result for positive measures. Proof of (1): By Theorems 3.13 and 3.14, the limits R(x) = limy↓0 R(x + iy) and G(x) = limy↓0 G(x+iy) exist and G(x) is non-zero for Lebesgue a.e. x. Hence, for Lebesgue a.e. x, Fµ (x) = lim Fµ (x + iy) = y↓0

R(x) . G(x)

Proof of (2): Fµ (x) is zero on a set of positive measure iff R(x) is, and if this is the case, R ≡ 0 and then Fµ ≡ 0. Hence, if Fµ ≡ 0, then R ≡ 0. Obviously, | log |Fµ (x)|| ≤ | log |R(x)|| + | log |G(x)||, and (28) follows from Theorem 3.14. Proof of (3): The sets {x : Fµ (x) = α} and {x : R(x) − αG(x) = 0} have the same Lebesgue measure. If the second set has positive Lebesgue measure, then by Theorem 3.14, R(z) = αG(z) for all z ∈ C+ , and Fµ (z) ≡ α. Since limy→∞ |Fµ (x + iy)| = 0, α = 0, and so Fµ ≡ 0. Hence, if the set {x : Fµ (x) = α} has positive Lebesgue measure, then α = 0 and µ = 0. The final result we would like to mention is the theorem of Poltoratskii [21]. Theorem 3.18. Let ν be a complex and µ a positive measure. Let ν = f µ + νs be the Radon-Nikodym decomposition. Let µsing be the part of µ singular with respect to the Lebesgue measure. Then lim y↓0

Fν (x + iy) = f (x), Fµ (x + iy)

for µsing − a.e. x.

(32)

This theorem has played an important role in the recent study of the spectral structure of Anderson type Hamiltonians [15, 16]. Poltoratskii’s proof of Theorem 3.18 is somewhat complicated, partly since it is done in the framework of a theory that is also concerned with other questions. A relatively simple proof of Poltoratskii’s theorem has recently been found in [14]. This new proof is based on the spectral theorem for self-adjoint operators and rank one perturbation theory, and will be discussed in Section 5.

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3.9 Problems [1] (1) Prove Lemma 3.3. (2) Assume that µ satisfies (9). Prove that the set of x for which (11) holds is Gδ (countable intersection of open sets) in supp µ. [2] (1) Let C0 (R) be the usual Banach space of continuous function on R vanishing at infinity with norm f = sup |f (x)|. For f ∈ C0 (R) let f (t) y fy (x) = dt. π R (x − t)2 + y 2 Prove that limy↓0 fy − f = 0. (2) Prove that the linear span of the set of functions {((x − a)2 + b2 )−1 : a ∈ R, b > 0}, is dense in C0 (R). (3) Prove that the linear span of the set of functions {(x − z)−1 : z ∈ C \ R} is dense in C0 (R). Hint: To prove (1), you may argue as follows. Fix > 0. Let δ > 0 be such that |t − x| < δ ⇒ |f (t) − f (x)| < . The estimates |f (t) − f (x)| y |fy (x) − f (x)| ≤ dt π R (x − t)2 + y 2 ≤ + 2 f

y π

|t−x|>δ

1 dt (x − t)2 + y 2

≤ + 4π −1 f y/δ, yield that lim supy↓0 fy − f ≤ . Since is arbitrary, (1) follows. Approximating fy by Riemann sums deduce that (1) ⇒ (2). Obviously, (2) ⇒ (3). [3] Prove Part (3) of Theorem 3.8. [4] Prove the following converse of Theorem 3.8: If (1) and (2) hold, then for [a, b] ⊂ A, b

|Pν (x + iy)|p dx < ∞.

sup 0
a

[5] The following extension of Theorem 3.9 holds. Let ν be a finite positive measure. Then for any p > 1, ⎛ ⎞ b p p ν({b}) ν({a}) + + lim y p−1 Pν (x + iy)p dx = Cp ⎝ ν({x})p ⎠ . y↓0 2 2 a x∈(a,b)

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Prove this and compute Cp in terms of gamma functions. Hint: See Remark 1 after Theorem 2.2 in [28]. [6] Prove the formula (19). [7] Let µ be a complex measure. Prove that Fµ ≡ 0 ⇒ µ = 0 if either one of the following holds: (a) µ is real-valued. (b) |µ|(S) = 0 for some open set S. (c) R exp(p|x|)d|µ| < ∞ for some p > 0. [8] Let µ be a complex or positive measure on R and (x − t)dµ(t) 1 −1 Hµ (z) = π (iPµ (z) − Fµ (z)) = . π R (x − t)2 + y 2 By Theorems 3.5 and 3.17, for Lebesgue a.e. x the limit Hµ (x) = lim Hµ (x + iy), y↓0

exists and is finite. If dµ = f dx, we will denote Hµ (z) and Hµ (x) by Hf (z) and Hf (x). The function Hµ (x) is called the Hilbert transform of the measure µ (Hf is called the Hilbert transform of the function f ). (1) Prove that for Lebesgue a.e. x the limit 1 dµ(t) , lim

→0 π |t−x|> x − t exists and is equal to Hµ (x). (2) Assume that f ∈ Lp (R, dx) for some 1 0

R

and deduce that Hf ∈ Lp (R, dx). (3) If f ∈ L2 (R, dx), prove that HHf = −f and deduce that 2 |Hf (x)| dx = |f (x)|2 dx. R

R

[9] Let 1 ≤ p < ∞. The Hardy class H p (C+ ) is the vector space of all analytic functions f on C+ such that f pp = sup |f (x + iy)|p dx < ∞. y>0

R

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265

(1) Prove that · p is a norm and that H p (C+ ) is a Banach space. (2) Let f ∈ H p (C+ ). Prove that the limit f (x) = lim f (x + iy), y↓0

exists for Lebesgue a.e. x and that f ∈ Lp (R, dx). Prove that f (t) y dt. f (x + iy) = π R (x − t)2 + y 2 (3) Prove that H 2 (C+ ) is a Hilbert space and that sup |f (x + iy)|2 dx = |f (x)|2 dx. y>0

R

R

Hence, H 2 (C+ ) can be identified with a subspace of L2 (R, dx) which we denote by 2 the same letter. Let H (C+ ) = {f ∈ L2 (R, dx) : f ∈ H 2 (C+ )}. Prove that 2

L2 (R, dx) = H 2 (C+ ) ⊕ H (C+ ). [10] In this problem we will study the Poisson transform on the circle. Let Γ = {z : |z| = 1} and let µ be a complex measure on Γ. The Poisson transform of the measure µ is 1 − |z|2 dµ(w). Pµ (z) = 2 Γ |z − w| If we parametrize Γ by w = eit , t ∈ (−π, π] and denote the induced complex measure by µ(t), then π 1 − r2 dµ(t). Pµ (reiθ ) = 2 −π 1 + r − 2r cos(θ − t) Note also that if dµ(t) = dt, then Pµ (z) = 2π. For w ∈ Γ we denote by I(w, r) the arc of length 2r centered at w. Let ν be a complex measure and µ a finite positive measure on Γ. The corresponding maximal function is defined by Mν,µ (w) = sup r>0

|ν|(I(w, r)) , µ(I(w, r))

if x ∈ supp µ, otherwise Mν,µ (w) = ∞. (1) Formulate and prove the Besicovitch covering lemma for the circle. (2) Prove the following bound: For all r ∈ [0, 1) and θ ∈ (−π, π], |Pν (reiθ )| ≤ Mν,µ (eiθ ). Pµ (reiθ )

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You may either mimic the proof of Lemma 3.2, or follow the proof of Theorem 11.20 in [27]. (3) State and prove the analog of Theorem 3.5 for the circle. (4) State and prove the analogs of Theorems 3.7 and 3.13 for the circle. [11] In Part (4) of the previous problem you were asked to prove the relation (27). This relation could be also proved like follows. Let pr,θ (t) =

1+

r2

1 − r2 . − 2r cos(θ − t)

Show first that 1 lim sup |U (re ) − U (e )| ≤ lim sup r→1 r→1 2π iθ

π

iθ

1 ≤ lim sup 2

↓0

−π

pr,θ (t)|U (eit ) − U (eiθ )|dt

|U (eit ) − U (eiθ )|dt, I(θ, )

and then use Problem 9 of Section 2. [12] The goal of this problem is to extend all the results of this section to nontangential limits. Our description of non-tangential limits follows [21]. Let again Γ = {z : |z| = 1} and D = {z : |z| < 1}. Let w ∈ Γ. We say that z tends to w non-tangentially, and write z→w ∠ if z tends to w inside the region ∆ϕ w = {z ∈ D : |Arg(1 − zw)| < ϕ}, for all ϕ ∈ (0, π/2). Arg(z) is the principal branch of the argument with values in (−π, π]. In the sector ∆ϕ w inscribe a circle centered at the origin (we denote it by Γϕ ). The two points on Γϕ ∩ {z : Arg(1 − zw) = ±ϕ} divide the circle into two arcs. The open region bounded by the shorter arc and the rays Arg(1 − zw) = ±ϕ is denoted Cϕ w . Let ν and µ be as in Problem 10. (1) Let ϕ ∈ (0, π/2) be given. Then there is a constant C such that sup z∈Cϕ w

|Pν (z)| ≤ CMν,µ (w), Pµ (z)

for µ − a.e. w.

(33)

This is the key result which extends the radial estimate of Part (2) of Problem 10. The passage from the radial estimate to (33) is similar to the proof of Harnack’s lemma. Write the detailed proof following Lemma 1.2 of [21]. (2) Let ν = f µ + νs be the Radon-Nikodym decomposition. Prove that Pν (z) = f (w), lim z → w Pµ (z) ∠

for µ − a.e.w.

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267

If ν is a positive measure, prove that Pν (z) = ∞, lim z → w Pµ (z)

for νs − a.e. w.

∠

(3) Extend Parts (3) and (4) of Problem 10 to non-tangential limits. (4) Consider now C+ . We say that z tends to x non-tangentially if for all ϕ ∈ (0, π/2) z tends to x inside the cone {z : |Arg(z − x) − π/2| < ϕ}. Let T be the conformal mapping (21). Prove that z → w non-tangentially in D iff T (z) → T (w) non-tangentially in C+ . Using this observation extend all the results of this section to non-tangential limits.

4 Self-adjoint operators, spectral theory 4.1 Basic notions Let H be a Hilbert space. We denote the inner product by (·|·) (the inner product is linear w.r.t. the second variable). A linear operator on H is a pair (A, Dom (A)), where Dom (A) ⊂ H is a vector subspace and A : Dom (A) → H is a linear map. We set Ker A = {ψ ∈ Dom (A) : Aψ = 0},

Ran A = {Aψ : ψ ∈ Dom (A)}.

An operator A is densely defined if Dom (A) is dense in H. If A and B are linear operators, then A + B is defined on Dom (A + B) = Dom (A) ∩ Dom (B) in the obvious way. For any z ∈ C we denote by A + z the operator A + z1, where 1 is the identity operator. Similarly, Dom (AB) = {ψ : ψ ∈ Dom (B), Bψ ∈ Dom (A)}, and (AB)ψ = A(Bψ). A = B if Dom (A) = Dom (B) and Aψ = Bψ. The operator B is called an extension of A if Dom (A) ⊂ Dom (B) and Aψ = Bψ for ψ ∈ Dom (A). If B is an extension of A one writes A ⊂ B. The operator A is called bounded if Dom (A) = H and A = sup Aψ < ∞.

(34)

ψ =1

We denote by B(H) the vector space of all bounded operators on H. B(H) with the norm (34) is a Banach space. If A is densely defined and there is a constant C such that for all ψ ∈ Dom (A), Aψ ≤ C ψ , then A has a unique extension to a bounded operator on H. An operator P ∈ B(H) is called a projection if P 2 = P . An operator U ∈ B(H) is called unitary if U is onto and (U φ|U ψ) = (φ|ψ) for all φ, ψ ∈ H. The graph of a linear operator A is defined by

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Γ(A) = {(ψ, Aψ) : ψ ∈ Dom (A)} ⊂ H ⊕ H. Note that A ⊂ B if Γ(A) ⊂ Γ(B). A linear operator A is called closed if Γ(A) is a closed subset of H ⊕ H. A is called closable if it has a closed extension. If A is closable, its smallest closed extension is called the closure of A and is denoted by A. It is not difficult to show that A is closable iff Γ(A) is the graph of a linear operator and in this case Γ(A) = Γ(A). Let A be closed. A subset D ⊂ Dom (A) is called a core for A if A D = A. Let A be a densely defined linear operator. Its adjoint, A∗ , is defined as follows. Dom (A∗ ) is the set of all φ ∈ H for which there exists a ψ ∈ H such that (Aϕ|φ) = (ϕ|ψ),

for all ϕ ∈ Dom (A).

Obviously, such ψ is unique and Dom (A∗ ) is a vector subspace. We set A∗ φ = ψ. It may happen that Dom (A∗ ) = {0}. If Dom (A∗ ) is dense, then A∗∗ = (A∗ )∗ , etc. Theorem 4.1. Let A be a densely defined linear operator. Then: (1) A∗ is closed. (2) A is closable iff Dom (A∗ ) is dense, and in this case A = A∗∗ . ∗ (3) If A is closable, then A = A∗ . Let A be a closed densely defined operator. We denote by ρ(A) the set of all z ∈ C such that A − z : Dom (A) → H is a bijection. By the closed graph theorem, if z ∈ ρ(A), then (A − z)−1 ∈ B(H). The set ρ(A) is called the resolvent set of A. The spectrum of A, sp(A), is defined by sp(A) = C \ ρ(A). A point z ∈ C is called an eigenvalue of A if there is a ψ ∈ Dom (A), ψ = 0, such that Aψ = zψ. The set of all eigenvalues is called the point spectrum of A and is denoted by spp (A). Obviously, spp (A) ⊂ sp(A). It is possible that sp(A) = spp (A) = C. It is also possible that sp(A) = ∅. (For simple examples see [23], Example 5 in Chapter VIII). Theorem 4.2. Assume that ρ(A) is non-empty. Then ρ(A) is an open subset of C and the map ρ(A) * z !→ (A − z)−1 ∈ B(H), is (norm) analytic. Moreover, if z1 , z2 ∈ ρ(A), then (A − z1 )−1 − (A − z2 )−1 = (z1 − z2 )(A − z1 )−1 (A − z2 )−1 . The last relation is called the resolvent identity.

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4.2 Digression: The notions of analyticity Let Ω ⊂ C be an open set and X a Banach space. A function f : Ω → X is called norm analytic if for all z ∈ Ω the limit lim

w→z

f (w) − f (z) , w−z

exists in the norm of X. f is called weakly analytic if x∗ ◦ f : Ω → C is analytic for all x∗ ∈ X ∗ . Obviously, if f is norm analytic, then f is weakly analytic. The converse also holds and we have: Theorem 4.3. f is norm analytic iff f is weakly analytic. For the proof, see [23]. The mathematical theory of Banach space valued analytic functions parallels the classical theory of analytic functions. For example, if γ is a closed path in a simply connected domain Ω, then < f (z)dz = 0. (35) γ

(The integral is defined in the usual way by the norm convergent Riemann sums.) To prove (35), note that for x∗ ∈ X ∗ , < < ∗ x f (z)dz = x∗ (f (z))dz = 0. γ

γ

∗

Since X separates points in X, (35) holds. Starting with (35) one obtains in the usual way the Cauchy integral formula, < f (w) 1 dw = f (z). 2πi |w−z|=r w − z Starting with the Cauchy integral formula one proves that for w ∈ Ω, f (z) =

∞

an (z − w)n ,

(36)

n=0

where an ∈ X. The power series converges and the representation (36) holds in the largest open disk centered at w and contained in Ω, etc. 4.3 Elementary properties of self-adjoint operators Let A be a densely defined operator on a Hilbert space H. A is called symmetric if ∀φ, ψ ∈ Dom (A), (Aφ|ψ) = (φ|Aψ). In other words, A is symmetric if A ⊂ A∗ . Obviously, any symmetric operator is closable. A densely defined operator A is called self-adjoint if A = A∗ . A is self-adjoint iff A is symmetric and Dom (A) = Dom (A∗ ).

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Theorem 4.4. Let A be a symmetric operator on H. Then the following statements are equivalent: (1) A is self-adjoint. (2) A is closed and Ker (A∗ ± i) = {0}. (3) Ran (A ± i) = H. A symmetric operator A is called essentially self-adjoint if A is self-adjoint. Theorem 4.5. Let A be a symmetric operator on H. Then the following statements are equivalent: (1) A is essentially self-adjoint. (2) Ker (A∗ ± i) = {0}. (3) Ran (A ± i) are dense in H. Remark. In Parts (2) and (3) of Theorems 4.4 and 4.5 ±i can be replaced by z, z, for any z ∈ C \ R. Theorem 4.6. Let A be self-adjoint. Then: (1) If z = x + iy, then for ψ ∈ Dom (A), (A − z)ψ 2 = (A − x)ψ 2 + y 2 ψ 2 . (2) sp(A) ⊂ R and for z ∈ C \ R, (A − z)−1 ≤ |Im z|−1 . (3) For any x ∈ R and ψ ∈ H, lim iy(A − x − iy)−1 ψ = −ψ.

y→∞

(4) If λ1 , λ2 ∈ spp (A), λ1 = λ2 , and ψ1 , ψ2 are corresponding eigenvectors, then ψ1 ⊥ ψ2 . Proof. (1) follows from a simple computation: (A − x − iy)ψ 2 = ((A − x − iy)ψ|(A − x − iy)ψ) = (A − x)ψ 2 + y 2 ψ 2 + iy((A − x)ψ|ψ) − iy((A − x)ψ|ψ) = (A − x)ψ 2 + y 2 ψ 2 . (2) Let z ∈ C \ R. By (1), if (A − z)ψ = 0, then ψ = 0, and so A − z : Dom (A) → H, is one-one. Ran (A − z) = H by Theorem 4.4. Let us prove this fact directly. We will show first that Ran (A − z) is dense. Let ψ ∈ H such that ((A − z)φ|ψ) = 0 for all φ ∈ Dom (A). Then ψ ∈ Dom (A) and (ψ|Aψ) = z ψ 2 . Since

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271

(ψ|Aψ) ∈ R, and Im z = 0, ψ = 0. Hence, Ran (A − z) is dense. Let ψn = (A − z)φn be a Cauchy sequence. Then, by (1), φn is also a Cauchy sequence, and since A is closed, Ran (A − z) is closed. Hence, Ran (A − z) = H and z ∈ ρ(A). Finally, the estimate (A − z)−1 ≤ |Im z|−1 is an immediate consequence of (1). (3) By replacing A with A − x, w.l.o.g. we may assume that x = 0. We consider first the case ψ ∈ Dom (A). The identity iy(A − iy)−1 ψ + ψ = (A − iy)−1 Aψ, and (2) yield that iy(A − iy)−1 ψ + ψ ≤ Aψ /y, and so (3) holds. If ψ ∈ Dom (A), let ψn ∈ Dom (A) be a sequence such that ψn − ψ ≤ 1/n. We estimate iy(A − iy)−1 ψ + ψ ≤ iy(A − iy)−1 (ψ − ψn ) + ψ − ψn + (iy(A − iy)−1 ψn + ψn ≤ 2 ψ − ψn + (iy(A − iy)−1 ψn + ψn ≤ 2/n + Aψn /y. Hence,

lim sup iy(A − iy)−1 ψ + ψ ≤ 2/n. y→∞

Since n is arbitrary, (3) follows. (4) Note that λ1 (ψ1 |ψ2 ) = (Aψ1 |ψ2 ) = (ψ1 |Aψ2 ) = λ2 (ψ1 |ψ2 ). Since λ1 = λ2 , (ψ1 |ψ2 ) = 0. A self-adjoint operator A is called positive if (ψ|Aψ) ≥ 0, for all ψ ∈ Dom (A). If A and B are bounded and self-adjoint, then obviously A±B are also self-adjoint; we write A ≥ B if A − B ≥ 0. A self-adjoint projection P is called an orthogonal projection. In this case H = Ker P ⊕ Ran P . We write dim P = dim Ran P . Let A be a bounded operator on H. The real and the imaginary part of A are defined by 1 1 ImA = (A − A∗ ). ReA = (A + A∗ ), 2 2i Clearly, ReA and ImA are self-adjoint operators and A = ReA + iImA.

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4.4 Direct sums and invariant subspaces Let H1 , H2 be Hilbert spaces and A1 , A2 self-adjoint operators on H1 , H2 . Then, the operator A = A1 ⊕ A2 with the domain Dom (A) = Dom (A1 ) ⊕ Dom (A2 ), is self-adjoint. Obviously, (A − z)−1 = (A1 − z)−1 ⊕ (A2 − z)−1 . This elementary construction has a partial converse. Let A be a self-adjoint operator on a Hilbert space H and let H1 be a closed subspace of H. The subspace H1 is invariant under A if for all z ∈ C \ R, (A − z)−1 H1 ⊂ H1 . Obviously, if H1 is invariant under A, so is H2 = H1⊥ . Set Dom (Ai ) = Dom (A) ∩ Hi ,

Ai ψ = Aψ,

i = 1, 2,

Ai is a self-adjoint operator on Hi and A = A1 ⊕ A2 . We will call A1 the restriction of A to the invariant subspace H1 and write A1 = A H1 . Let Γ be a countable set and Hn , n ∈ Γ, a collection of Hilbert spaces. The direct sum of this collection, + Hn , H= n

is the set of all sequences {ψn }n∈Γ such that ψn ∈ Hn and ψn 2Hn < ∞. n∈Γ

H is a Hilbert space with the inner product (φ|ψ) = (φn |ψn )Hn . n∈Γ

Let Bn ∈ B(Hn ) and assume that supn Bn < ∞. Then B{ψn }n∈Γ = {Bn ψn }n∈Γ , is a bounded operator on H and B = supn Bn . Proposition 4.7. Let An be self-adjoint operators on Hn . Set Dom (A) = {ψ = {ψn } ∈ H : ψn ∈ Dom (An ), An ψn 2Hn < ∞}, n

Aψ = {An ψn }. Then A is a self-adjoint operator on H. We write + An . A= n∈Γ

Moreover: (1) For z ∈ C \ R, (A − z)−1 = ⊕n (An − z)−1 . (2) sp(A) = ∪n sp(An ). The proof of Proposition 4.7 is easy and is left to the problems.

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4.5 Cyclic spaces and the decomposition theorem Let H be a separable Hilbert space and A a self-adjoint operator on H. A collection of vectors C = {ψn }n∈Γ , where Γ is a countable set, is called cyclic for A if the closure of the linear span of the set of vectors {(A − z)−1 ψn : n ∈ Γ, z ∈ C \ R}, is equal to H. A cyclic set for A always exists (take an orthonormal basis for H). If C = {ψ}, then ψ is called a cyclic vector for A. Theorem 4.8. (The decomposition theorem) Let H be a separable Hilbert space and A a self-adjoint operator on H. Then there exists a countable set Γ, a collection of mutually orthogonal closed subspaces {Hn }n∈Γ of H (Hn ⊥ Hm if n = m), and self-adjoint operators An on Hn such that: (1) For all n ∈ Γ there is a ψn ∈ Hn cyclic for An . (2) H = ⊕n Hn and A = ⊕n An . Proof. Let {φn : n = 1, 2, · · · } be a given cyclic set for A. Set ψ1 = φ1 and let H1 be the cyclic space generated by A and ψ1 (H1 is the closure of the linear span of the set of vectors {(A − z)−1 ψ1 : z ∈ C \ R}). By Theorem 4.6, ψ1 ∈ H1 . Obviously, H1 is invariant under A and we set A1 = A H1 . We define ψn , Hn and An inductively as follows. If H1 = H, let φn2 be the first element of the sequence {φ2 , φ3 , · · · } which is not in H1 . Decompose φn2 = φn2 + φn2 , where φn2 ∈ H1 and φn2 ∈ H1⊥ . Set ψ2 = φn2 and let H2 be the cyclic space generated by A and ψ2 . It follows from the resolvent identity that H1 ⊥ H2 . Set A2 = A H2 . In this way we inductively define ψn , Hn , An , n ∈ Γ, where Γ is a finite set {1, . . . , N } or Γ = N. By the construction, {φn }n∈Γ ⊂ ∪n∈Γ Hn . Hence, (1) holds and H = ⊕n Hn . To prove the second part of (2), note first that by the construction of An , + (A − z)−1 = (An − z)−1 . n

If A˜ = ⊕An , then by Proposition 4.7, A˜ is self-adjoint and + (A˜ − z)−1 = (An − z)−1 . n

˜ Hence A = A. 4.6 The spectral theorem We start with:

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Theorem 4.9. Let (M, F) be a measure space and µ a finite positive measure on (M, F). Let f : M → R be a measurable function and let Af be a linear operator on L2 (M, dµ) defined by Dom (Af ) = {ψ ∈ L2 (M, dµ) : f ψ ∈ L2 (M, dµ)},

Af ψ = f ψ.

Then: (1) Af is self-adjoint. (2) Af is bounded iff f ∈ L∞ (M, dµ), and in this case Af = f ∞ . (3) sp(Af ) is equal to the essential range of f : sp(Af ) = {λ ∈ R : µ(f −1 (λ − , λ + )) > 0 for all > 0}.

The proof of this theorem is left to the problems. The content of the spectral theorem for self-adjoint operators is that any selfadjoint operator is unitarily equivalent to Af for some f . Let H1 and H2 be two Hilbert spaces. A linear bijection U : H1 → H2 is called unitary if for all φ, ψ ∈ H1 , (U φ|U ψ)H2 = (φ|ψ)H1 . Let A1 , A2 be linear operators on H1 , H2 . The operators A1 , A2 are unitarily equivalent if there exists a unitary U : H1 → H2 such that U Dom (A1 ) = Dom (A2 ) and U A1 U −1 = A2 . Theorem 4.10. (Spectral theorem for self-adjoint operators) Let A be a selfadjoint operator on a Hilbert space H. Then there is a measure space (M, F), a finite positive measure µ and measurable function f : M → R such that A is unitarily equivalent to the operator Af on L2 (M, dµ). We will prove the spectral theorem only for separable Hilbert spaces. In the literature one can find many different proofs of Theorem 4.10. The proof below is constructive and allows to explicitly identify M and f while the measure µ is directly related to (A − z)−1 . 4.7 Proof of the spectral theorem—the cyclic case Let A be a self-adjoint operator on a Hilbert space H and ψ ∈ H. Theorem 4.11. There exists a unique finite positive Borel measure µψ on R such that µψ (R) = ψ 2 and dµψ (t) , z ∈ C \ R. (37) (ψ|(A − z)−1 ψ) = R t−z The measure µψ is called the spectral measure for A and ψ. Proof. Since (A − z)−1 = (A − z)−1∗ , we need only to consider z ∈ C+ . Set U (z) = (ψ|(A−z)−1 ψ) and V (z) = Im U (z), z ∈ C+ . It follows from the resolvent identity that

Topics in Spectral Theory

V (x + iy) = y (A − x − iy)−1 ψ 2 ,

275

(38)

and so V is harmonic and strictly positive in C+ . Theorem 3.11 yields that there is a constant c ≥ 0 and a unique positive Borel measure µψ on R such that for y > 0, dµψ (t) V (x + iy) = cy + Pµψ (x + iy) = cy + y . (39) (x − t)2 + y 2 R By Theorem 4.6, V (x + iy) ≤ ψ 2 /y

and

lim yV (x + iy) = ψ 2 .

y→∞

The first relation yields that c = 0. The second relation and the dominated convergence theorem yield that µψ (R) = ψ 2 . The functions dµψ (t) , Fµψ (z) = R t−z and U (z) are analytic in C+ and have equal imaginary parts. The Cauchy-Riemann equations imply that Fµψ (z) − U (z) is a constant. Since Fµψ (z) and U (z) vanish as Im z → ∞, Fµψ (z) = U (z) for z ∈ C+ and (37) holds. Corollary 4.12. Let ϕ, ψ ∈ H. Then there exists a unique complex measure µϕ,ψ on R such that dµϕ,ψ (t) , z ∈ C \ R. (40) (ϕ|(A − z)−1 ψ) = t−z R Proof. The uniqueness is obvious (the set of functions {(x − z)−1 : z ∈ C \ R} is dense in C0 (R)). The existence follows from the polarization identity: 4(ϕ|(A − z)−1 ψ) = (ϕ + ψ|(A − z)−1 (ϕ + ψ)) − (ϕ − ψ|(A − z)−1 (ϕ − ψ)) + i(ϕ − iψ|(A − z)−1 (ϕ − iψ)) − i(ϕ + iψ|(A − z)−1 (ϕ + iψ)). In particular, µϕ,ψ =

1 (µϕ+ψ − µϕ−ψ + i(µϕ−iψ − µϕ+iψ )) . 4

(41)

The main result of this subsection is: Theorem 4.13. Assume that ψ is a cyclic vector for A. Then A is unitarily equivalent to the operator of multiplication by x on L2 (R, dµψ ). In particular, sp(A) = supp µψ .

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Proof. Clearly, we may assume that ψ = 0. Note that (A − z)−1 ψ = (A − w)−1 ψ, iff z = w. For z ∈ C \ R we set rz (x) = (x − z)−1 . rz ∈ L2 (R, dµψ ) and the linear span of {rz }z∈C\R is dense in L2 (R, dµψ ). Set U (A − z)−1 ψ = rz . If z = w, then (rz |rw )L2 (R,dµψ )

1 = rz rw dµψ = z − w R =

(42)

R

(rz − rw )dµψ

> 1 = (ψ|(A − z)−1 ψ) − (ψ|(A − w)−1 ψ) z−w

= ((A − z)−1 ψ|(A − w)−1 ψ). By a limiting argument, the relation (rz |rw )L2 (R,dµψ ) = ((A − z)−1 ψ|(A − w)−1 ψ), holds for all z, w ∈ C \ R. Hence, the map (42) extends to a unitary U : H → L2 (R, dµψ ). Since U (A − z)−1 (A − w)−1 ψ = rz (x)rw (x) = rz (x)U (A − w)−1 ψ, (A − z)−1 is unitarily equivalent to the operator of multiplication by (x − z)−1 on L2 (R, dµψ ). For any φ ∈ H, U A(A − z)−1 φ = U φ + zU (A − z)−1 φ = (1 + z(x − z)−1 )U φ = x(x − z)−1 U φ = xU (A − z)−1 φ, and so A is unitarily equivalent to the operator of multiplication by x. We finish this subsection with the following remark. Assume that ψ is a cyclic vector for A and let Ax be the operator of multiplication by x on L2 (R, dµψ ). Then, by Theorem 4.13, there exists a unitary U : H → L2 (R, dµψ ) such that U AU −1 = Ax .

(43)

However, a unitary satisfying (43) is not unique. If U is such a unitary, then U ψ is a cyclic vector for Ax . On the other hand, if f ∈ L2 (R, dµψ ) is a cyclic vector for Ax , then there is a unique unitary U : H → L2 (R, dµψ ) such that (43) holds and U ψ = f ψ / f . The unitary constructed in the proof of Theorem 4.13 satisfies U ψ = 1l.

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4.8 Proof of the spectral theorem—the general case Let A be a self-adjoint operator on a separable Hilbert space H. Let Hn , An , ψn , n ∈ Γ be as in the decomposition theorem (Theorem 4.8). Let Un : Hn → L2 (R, dµψn ), be unitary such that An is unitarily equivalent to the operator of multiplication by x. We denote this last operator by A˜n . Let U = ⊕n Un . An immediate consequence of the decomposition theorem and Theorem 4.13 is 0 Theorem 4.14. The map U : H → n∈Γ L2 (R, dµψn ) is unitary and A is unitarily 0 equivalent to the operator n∈Γ A˜n . In particular, sp(A) =

,

supp µψn .

n∈Γ

Note that if φ ∈ H and U φ = {φn }n∈Γ , then µφ =

n∈Γ

µφn .

Theorem 4.10 is a reformulation of Theorem 4.14. To see that, choose cyclic vectors ψn so that n∈Γ ψn 2 < ∞. For each n ∈ Γ, let Rn be a copy of R and , M= Rn . n∈Γ

You may visualize M as follows: enumerate Γ so that Γ = {1, . . . , N } or Γ = N and set Rn = {(n, x) : x ∈ R} ⊂ R2 . Hence, M is just a collection of lines in R2 parallel to the y-axis and going through the points (n, 0), n ∈ Γ. Let F be the collection of all sets F ⊂ M such that F ∩ Rn is Borel for all n. Then F is a σ-algebra and µψn (F ∩ Rn ), µ(F ) = n∈Γ

is a finite measure on M (µ(M ) = n∈Γ ψn 2 < ∞). Let f : M → R be the identity function (f (n, x) = x). Then + + A˜n , L2 (M, dµ) = L2 (R, dµψn ), Af = n∈Γ

and Theorem 4.10 follows. Set µac (F ) =

n∈Γ

µψn ,ac (F ∩ Rn ),

n∈Γ

µsc (F ) =

µψn ,sc (F ∩ Rn ),

n∈Γ

µpp (F ) =

n∈Γ

µψn ,pp (F ∩ Rn ).

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Then L2 (M, dµac ), L2 (M, dµsc ), and L2 (M, dµpp ) are closed subspaces of the space L2 (M, dµ) invariant under Af and L2 (M, dµ) = L2 (M, dµac ) ⊕ L2 (M, dµsc ) ⊕ L2 (M, dµpp ). Set Hac = U −1 L2 (M, dµac ), Hsc = U −1 L2 (M, dµsc ), Hpp = U −1 L2 (M, dµpp ). These subspaces are invariant under A. Moreover, ψ ∈ Hac iff the spectral measure µψ is absolutely continuous w.r.t. the Lebesgue measure, ψ ∈ Hsc iff µψ is singular continuous and ψ ∈ Hpp iff µψ is pure point. Obviously, H = Hac ⊕ Hsc ⊕ Hpp . The spectra spac (A) = sp(A Hac ) =

,

supp µψn ,ac ,

n∈Γ

spsc (A) = sp(A Hsc ) =

,

supp µψn ,sc ,

n∈Γ

sppp (A) = sp(A Hpp ) =

,

supp µψn ,pp

n∈Γ

are called, respectively, the absolutely continuous, the singular continuous, and the pure point spectrum of A. Note that sp(A) = spac (A) ∪ spsc (A) ∪ sppp (A), and spp (A) = sppp (A). The singular and the continuous spectrum of A are defined by spsing (A) = spsc (A) ∪ sppp (A),

spcont (A) = spac (A) ∪ spsc (A).

The subspaces Hac , Hsc , Hpp are called the spectral subspaces associated, respectively, to the absolutely continuous, singular continuous, and pure point spectrum. The projections on these subspaces are denoted by 1ac (A), 1sc (A), 1pp (A). The spectral subspaces associated to the singular and the continuous spectrum are Hsing = Hsc ⊕ Hpp and Hcont = Hac ⊕ Hsc . The corresponding projections are 1sing (A) = 1sc (A) + 1pp (A) and 1cont (A) = 1ac (A) + 1sc (A). When we wish to indicate the dependence of the spectral subspaces on the operator A, we will write Hac (A), etc.

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4.9 Harmonic analysis and spectral theory Let A be a self-adjoint operator on a Hilbert space H, ψ ∈ H, and µψ the spectral measure for A and ψ. Let Fµψ and Pµψ be the Borel and the Poisson transform of µψ . The formulas (ψ|(A − z)−1 ψ) = Fµψ (z), Im (ψ|(A − z)−1 ψ) = Pµψ (z), provide the key link between the harmonic analysis (the results of Section 3) and the spectral theory. Recall that µψ,sing = µψ,sc + µψ,pp . Theorem 4.15. (1) For Lebesgue a.e. x ∈ R the limit (ψ|(A − x − i0)−1 ψ) = lim(ψ|(A − x − iy)−1 ψ), y↓0

exists and is finite and non-zero. (2) dµψ,ac = π −1 Im(ψ|(A − x − i0)−1 ψ)dx. (3) µψ,sing is concentrated on the set {x : lim Im (ψ|(A − x − iy)−1 ψ) = ∞}. y↓0

Theorem 4.15 is an immediate consequence of Theorems 3.5 and 3.17. Similarly, Theorems 3.6, 3.8 and Corollary 3.10 yield: Theorem 4.16. Let [a, b] be a finite interval. (1) µψ,ac ([a, b]) = 0 iff for some p ∈ (0, 1)

b

lim y↓0

= >p Im (ψ|(A − x − iy)−1 ψ) dx = 0.

a

(2) Assume that for some p > 1

b

= >p Im (ψ|(A − x − iy)−1 ψ) dx < ∞.

b

= >2 Im (ψ|(A − x − iy)−1 ψ) dx = 0.

sup 0
a

Then µψ,sing ([a, b]) = 0. (3) µψ,pp ([a, b]) = 0 iff lim y y↓0

a

Let Hψ be the cyclic subspace spanned by A and ψ. W.l.o.g. we may assume that ψ = 1. By Theorem 4.13 there exists a (unique) unitary Uψ : Hψ → L2 (R, dµψ )

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such that Uψ ψ = 1l and Uψ AUψ−1 is the operator of multiplication by x on ⊥ . Recall that L2 (R, dµψ ). We extend Uψ to H by setting Uψ φ = 0 for φ ∈ Hψ Im (A − z)−1 =

1 ((A − z)−1 − (A − z)−1 ). 2i

The interplay between spectral theory and harmonic analysis is particularly clearly captured in the following result. Theorem 4.17. Let φ ∈ H. Then: (1) (Uψ 1ac φ)(x) = lim y↓0

(ψ|Im (A − x − iy)−1 φ) , Im (ψ|(A − x − iy)−1 ψ)

for µψ,ac − a.e. x.

(2) (Uψ 1sing φ)(x) = lim y↓0

(ψ|(A − x − iy)−1 φ) , (ψ|(A − x − iy)−1 ψ)

for µψ,sing − a.e. x.

Proof. Since P(Uψ φ)µψ (x + iy) (ψ|Im (A − x − iy)−1 φ) = , Im (ψ|(A − x − iy)−1 ψ) Pµψ (x + iy) (1) follows from Theorem 3.5. Similarly, since F(Uψ φ)µψ (x + iy) (ψ|(A − x − iy)−1 φ) = , −1 (ψ|(A − x − iy) ψ) Fµψ (x + iy) (2) follows from the Poltoratskii theorem (Theorem 3.18). 4.10 Spectral measure for A Let A be a self-adjoint operator on a separable Hilbert space H and let {φn }n∈Γ be a cyclic set for A. Let {an }n∈Γ be a sequence such that an > 0 and an φn 2 < ∞. n∈Γ

The spectral measure for A, µA , is a Borel measure on R defined by µA = an µφn . n∈Γ

Obviously, µA depends on the choice of {φn } and an . Two positive Borel measures ν1 and ν2 on R are called equivalent (we write ν1 ∼ ν2 ) iff ν1 and ν2 have the same sets of measure zero.

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Theorem 4.18. Let µA and νA be two spectral measures for A. Then µA ∼ νA . Moreover, µA,ac ∼ νA,ac , µA,sc ∼ νA,sc , and µA,pp ∼ νA,pp . Theorem 4.19. Let µA be a spectral measure for A. Then supp µA,ac = spac (A),

supp µA,sc = spsc (A),

supp µA,pp = sppp (A).

The proofs of these two theorems are left to the problems. 4.11 The essential support of the ac spectrum Let B1 and B2 be two Borel sets in R. Let B1 ∼ B2 iff the Lebesgue measure of the symmetric difference (B1 \ B2 ) ∪ (B2 \ B1 ) is zero. ∼ is an equivalence relation. Let µA be a spectral measure of a self-adjoint operator A and f (x) its Radon-Nikodym derivative w.r.t. the Lebesgue measure (dµA,ac = f (x)dx). The equivalence class associated to {x : f (x) > 0} is called the essential support of the absolutely coness (A). With a slight abuse of terminology we tinuous spectrum and is denoted by Σac ess (A) as an essential support of the ac will also refer to a particular element of Σac ess (A)). For example, the set spectrum (and denote it by the same symbol Σac

x : 0 < lim(2r)−1 µA (I(x, r)) < ∞ , r↓0

is an essential support of the absolutely continuous spectrum. Note that the essential support of the ac spectrum is independent on the choice of µA . ess (A) be an essential support of the absolutely continuous Theorem 4.20. Let Σac ess spectrum. Then cl(Σac (A) ∩ spac (A)) = spac (A).

The proof is left to the problems. ess (A) ⊂ spac (A) equals spac (A), Although the closure of an essential support Σac ess Σac (A) could be substantially ”smaller” than spac (A); see Problem 6. 4.12 The functional calculus Let A be a self-adjoint operator on a separable Hilbert space H. Let U : H → L2 (M, dµ), f , and Af be as in the spectral theorem. Let Bb (R) be the vector space of all bounded Borel functions on R. For h ∈ Bb (R), consider the operator Ah◦f . This operator is bounded and Ah◦f ≤ sup h(x). Set h(A) = U −1 Ah◦f U.

(44)

Let Φ : Bb (R) → B(H) be given by Φ(h) = h(A). Recall that rz (x) = (x − z)−1 .

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Theorem 4.21. (1) The map Φ is an algebraic ∗-homomorphism. (2) Φ(h) ≤ max |h(x)|. (3) Φ(rz ) = (A − z)−1 for all z ∈ C \ R. (4) If hn (x) → h(x) for all x, and supn,x |hn (x)| < ∞, then hn (A)ψ → h(A)ψ for all ψ. The map Φ is uniquely specified by (1)-(4). Moreover, it has the following additional properties: (5) If Aψ = λψ, then Φ(h)ψ = h(λ)ψ. (6) If h ≥ 0, then Φ(h) ≥ 0. We remark that the uniqueness of the functional calculus is an immediate consequence of Problem 11 in Section 2. Let K ⊂ H be a closed subspace. If K is invariant under A, then for all h ∈ Bb (R), h(A)K ⊂ K. For any Borel function h : R → C we define h(A) by (44). Of course, h(A) could be an unbounded operator. Note that h1 (A)h2 (A) ⊂ h1 ◦ h2 (A), h1 (A) + h2 (A) ⊂ (h1 + h2 )(A). Also, h(A)∗ = h(A) and h(A) is self-adjoint iff h(x) ∈ R for µA -a.e. x ∈ M . In fact,√ to define h(A), we only need that Ran f ⊂ Dom h. Hence, if A ≥ 0, we can define A, if A > 0 we can define ln A, etc. The two classes of functions, characteristic functions and exponentials, play a particularly important role. Let F be a Borel set in R and χF its characteristic function. The operator χF (A) is an orthogonal projection, called the spectral projection on the set F . In these notes we will use the notation 1F (A) = χF (A) and 1{e} (A) = 1e (A). Note that 1e (A) = 0 iff e ∈ spp (A). By definition, the multiplicity of the eigenvalue e is dim 1e (A). The subspace Ran 1F (A) is invariant under A and cl(int(F ) ∩ sp(A)) ⊂ sp(A Ran 1F (A)) ⊂ sp(A) ∩ cl(F ).

(45)

Note in particular that e ∈ sp(A) iff for all > 0 Ran 1(e− ,e+ ) (A) = {0}. The proof of (45) is left to the problems. Theorem 4.22. (Stone’s formula) For ψ ∈ H, lim y↓0

y π

b

Im(A − x − iy)−1 ψdx =

a

> 1= 1[a,b] (A)ψ + 1(a,b) (A)ψ . 2

Proof. Since y lim y↓0 π

a

b

⎧ ⎪ ⎨0 1 dx = 1/2 ⎪ (t − x)2 + y 2 ⎩ 1

the Stone formula follows from Theorem 4.21.

if t ∈ [a, b], if t = a or t = b, if t ∈ (a, b),

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Another important class of functions are exponentials. For t ∈ R, set U (t) = exp(itA). Then U (t) is a group of unitary operators on H. The group U (t) is strongly continuous, i.e. for all ψ ∈ H, lim U (s)ψ = U (t)ψ.

s→t

For ψ ∈ Dom (A) the function R * t !→ U (t)ψ is strongly differentiable and lim

t→0

U (t)ψ − ψ = iAψ. t

(46)

On the other hand, if the limit on the l.h.s. exists for some ψ, then ψ ∈ Dom (A), and (46) holds. The above results have a converse: Theorem 4.23. (Stone’s theorem) Let U (t) be a strongly continuous group on a separable Hilbert space H. Then there is a self-adjoint operator A such that U (t) = exp(itA). 4.13 The Weyl criteria and the RAGE theorem Let A be a self-adjoint operator on a separable Hilbert space H. Theorem 4.24. (Weyl criterion 1) e ∈ sp(A) iff there exists a sequence of unit vectors ψn ∈ Dom (A) such that lim (A − e)ψn = 0.

n→∞

(47)

Remark. A sequence of unit vectors for which (47) holds is called a Weyl sequence. Proof. Recall that e ∈ sp(A) iff 1(e− ,e+ ) (A) = 0 for all > 0. Assume that e ∈ sp(A). Let ψn ∈ Ran 1(e−1/n,e+1/n) (A) be unit vectors. Then, by the functional calculus, (A − e)ψn ≤

sup

|x − e| ≤ 1/n.

x∈(e−1/n,e+1/n)

On the other hand, assume that there is a sequence ψn such that (47) holds and that e ∈ sp(A). Then ψn = (A − e)−1 (A − e)ψn ≤ C (A − e)ψn , and so 1 = ψn → 0, contradiction. The discrete spectrum of A, denoted spdisc (A), is the set of all isolated eigenvalues of finite multiplicity. Hence e ∈ spdisc (A) iff

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1 ≤ dim 1(e− ,e+ ) (A) < ∞, for all small enough. The essential spectrum of A is defined by spess (A) = sp(A) \ spdisc (A). Hence, e ∈ spess (A) iff for all > 0 dim 1(e− ,e+ ) (A) = ∞. Obviously, spess (A) is a closed subset of R. Theorem 4.25. (Weyl criterion 2) e ∈ spess (A) iff there exists an orthonormal sequence ψn ∈ Dom (A) ( ψn = 1, (ψn |ψm ) = 0 if n = m), such that lim (A − e)ψn = 0.

n→∞

(48)

Proof. Assume that e ∈ spess (A). Then dim 1(e−1/n,e+1/n) (A) = ∞ for all n, and we can choose an orthonormal sequence ψn such that ψn ∈ Ran 1(e−1/n,e+1/n) (A). Clearly, (A − e)ψn ≤ 1/n, and (48) holds. On the other hand, assume that there exists an orthonormal sequence ψn such that (48) holds and that e ∈ spdisc (A). Choose > 0 such that dim 1(e− ,e+ ) (A) < ∞. Then, limn→∞ 1(e− ,e+ ) (A)ψn = 0 and lim (A − e)1R\(e− ,e+ ) (A)ψn = 0.

n→∞

Since (A − e) Ran 1R\(e− ,e+ ) (A) is invertible and the norm of its inverse is ≤ 1/, we have that 1R\(e− ,e+ ) (A)ψn ≤ −1 (A − e)1R\(e− ,e+ ) (A)ψn , and so limn→∞ 1R\(e− ,e+ ) (A)ψn = 0. Hence 1 = ψn → 0, contradiction. Theorem 4.26. (RAGE) (1) Let K be a compact operator. Then for all ψ ∈ Hcont , 1 T Ke−itA ψ 2 dt = 0. (49) lim T →∞ T 0 (2) The same result holds if K is bounded and K(A + i)−1 is compact. Proof. (1) First, recall that any compact operator is a norm limit of finite n rank operators. In other words, there exist vectors φn , ϕn ∈ H such that Kn = j=1 (φj |·)ϕj satisfies limn→∞ K − Kn = 0. Hence, it suffices to prove the statement for the finite rank operators Kn . By induction and the triangle inequality, it suffices to prove

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the statement for the rank one operator K = (φ|·)ϕ. Thus, it suffices to show that for φ ∈ H and ψ ∈ Hcont , 1 T lim |(φ|e−itA ψ)|2 dt = 0. T →∞ T 0 Moreover, since (φ|e−itA ψ) = (φ|e−itA 1cont (A)ψ) = (1cont (A)φ|e−itA ψ), w.l.o.g we may assume that φ ∈ Hcont . Finally, by the polarization identity, we may assume that ϕ = ψ. Since for ψ ∈ Hcont the spectral measure µψ has no atoms, the result follows from the Wiener theorem (Theorem 2.6 in Section 2). (2) Since Dom (A) ∩ Hcont is dense in Hcont , it suffices to prove the statement for ψ ∈ Dom (A) ∩ Hcont . Write Ke−itA ψ = K(A + i)−1 e−itA (A + i)ψ , and use (1). 4.14 Stability We will first discuss stability of self-adjointness—if A and B are self-adjoint operators, we wish to discuss conditions under which A + B is self-adjoint on Dom (A) ∩ Dom (B). One obvious sufficient condition is that either A or B is bounded. A more refined result involves the notion of relative boundedness. Let A and B be densely defined linear operators on a separable Hilbert space H. The operator B is called A-bounded if Dom (A) ⊂ Dom (B) and for some positive constants a and b and all ψ ∈ Dom (A), Bψ ≤ a Aψ + b ψ .

(50)

The number a is called a relative bound of B w.r.t. A. Theorem 4.27. (Kato-Rellich) Suppose that A is self-adjoint, B is symmetric, and B is A-bounded with a relative bound a < 1. Then: (1) A + B is self-adjoint on Dom (A). (2) A + B is essentially self-adjoint on any core of A. (3) If A is bounded from below, then A + B is also bounded from below. Proof. We will prove (1) and (2); (3) is left to the problems. In the proof we will use the following elementary fact: if V is a bounded operator and V < 1, then 0 ∈ sp(1 V ). This is easily proven by checking that the inverse of 1 + V is given + ∞ by 1 + k=1 (−1)k V k . By Theorem 4.4 (and the Remark after Theorem 4.5), to prove (1) it suffices to show that there exists y > 0 such that Ran(A + B ± iy) = H. In what follows y = (1 + b)/(1 − a). The relation (50) yields

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B(A ± iy)−1 ≤ a A(A ± iy)−1 + b (A ± iy)−1 ≤ a + by −1 < 1, and so 1 + B(A ± iy)−1 : H → H are bijections. Since A ± iy : Dom (A) → H are also bijections, the identity A + B ± iy = (1 + B(A ± iy)−1 )(A ± iy) yields Ran(A + B ± iy) = H. The proof of (2) is based on Theorem 4.5. Let D be a core for A. Then the sets (A ± iy)(D) = {(A ± iy)ψ : ψ ∈ D} are dense in H, and since 1 + B(A ± iy)−1 are bijections, (A + B ± iy)(D) = (1 + B(A ± iy)−1 )(A ± iy)(D) are also dense in H. We now turn to stability of the essential spectrum. The simplest result in this direction is: Theorem 4.28. (Weyl) Let A and B be self-adjoint and B compact. Then spess (A) = spess (A + B). Proof. By symmetry, it suffices to prove that spess (A + B) ⊂ spess (A). Let e ∈ spess (A + B) and let ψn be an orthonormal sequence such that lim (A + B − e)ψn = 0.

n→∞

Since ψn converges weakly to zero and B is compact, Bψn → 0. Hence, (A − e)ψn → 0 and e ∈ spess (A). Section XIII.4 of [RS4] deals with various extensions and generalizations of Theorem 4.28. 4.15 Scattering theory and stability of ac spectra Let A and B be self-adjoint operators on a Hilbert space H. Assume that for all ψ ∈ Ran 1ac (A) the limits Ω ± (A, B)ψ = lim eitA e−itB ψ, t→±∞

exist. The operators Ω ± (A, B) : Ran 1ac (A) → H are called wave operators. Proposition 4.29. Assume that the wave operators exist. Then (1) (Ω ± (A, B)φ|Ω ± (A, B)ψ) = (φ|ψ). (2) For any f ∈ Bb (R), Ω ± (A, B)f (A) = f (B)Ω ± (A, B). (3) Ran Ω ± (A, B) ⊂ Ran 1ac (B). The wave operators Ω ± (A, B) are called complete if Ran Ω ± (A, B) = Ran 1ac (B); (4) Wave operators Ω ± (A, B) are complete iff Ω ± (B, A) exist.

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The proof of this proposition is simple and is left to the problems (see also [25]). Let H be a separable Hilbert space and {ψn } an orthonormal basis. A bounded positive self-adjoint operator A is called trace class if (ψn |Aψn ) < ∞. Tr(A) = n

The number Tr(A) does not depend on the choice of orthonormal basis. More generally, a bounded self-adjoint operator A is called trace class if Tr(|A|) < ∞. A trace class operator is compact. Concerning stability of the ac spectrum, the basic result is: Theorem 4.30. (Kato-Rosenblum) Let A and B be self-adjoint and B trace class. Then the wave operators Ω ± (A + B, A) exist and are complete. In particular, ess ess (A + B) = Σac (A). spac (A + B) = spac (A) and Σac For the proof of Kato-Rosenblum theorem see [25], Theorem XI.7. The singular and the pure point spectra are in general unstable under perturbations — they may appear or disappear under the influence of a rank one perturbation. We will discuss in Section 5 criteria for ”generic” stability of the singular and the pure point spectra. 4.16 Notions of measurability In mathematical physics one often encounters self-adjoint operators indexed by elements of some measure space (M, F), namely one deals with functions M * ω !→ Aω , where Aω is a self-adjoint operator on some fixed separable Hilbert space H. In this subsection we address some issues concerning measurability of such functions. Let (M, F) be a measure space and X a topological space. A function f : M → X is called measurable if the inverse image of every open set is in F. Let H be a separable Hilbert space and B(H) the vector space of all bounded operators on H. We distinguish three topologies in B(H), the uniform topology, the strong topology, and the weak topology. The uniform topology is induced by the operator norm on B(H). The strong topology is the minimal topology w.r.t. which the functions B(H) * A !→ Aψ ∈ H are continuous for all ψ ∈ H. The weak topology is the minimal topology w.r.t. which the functions B(H) * A !→ (φ|Aψ) ∈ C are continuous for all φ, ψ ∈ H. The weak topology is weaker than the strong topology, and the strong topology is weaker than the uniform topology. A function f : M → B(H) is called measurable if it is measurable with respect to the weak topology. In other words, f is measurable iff the function M * ω !→ (φ|f (ω)ψ) ∈ C is measurable for all φ, ψ ∈ H. Let ω !→ Aω be a function with values in (possibly unbounded) self-adjoint operators on H. We say that Aω is measurable if for all z ∈ C \ R the function ω !→ (Aω − z)−1 ∈ B(H), is measurable.

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Until the end of this subsection ω !→ Aω is a given measurable function with values in self-adjoint operators. Theorem 4.31. The function ω !→ h(Aω ) is measurable for all h ∈ Bb (R). Proof. Let T ⊂ Bb (R) be the class of functions such that ω !→ h(Aω ) is measurable By definition, (x − z)−1 ∈ T for all z ∈ C \ R. Since the linear span of {(x − z)−1 : z ∈ C \ R} is dense in the Banach space C0 (R), C0 (R) ⊂ T . Note also that if hn ∈ T , supn,x |hn (x)| < ∞, and hn (x) → h(x) for all x, then h ∈ T . Hence, by Problem 11 in Section 2, T = Bb (R). From this theorem it follows that the functions ω !→ 1B (Aω ) (B Borel) and ω !→ exp(itAω ) are measurable. One can also easily show that if h : R !→ R is an arbitrary Borel measurable real valued function, then ω !→ h(Aω ) is measurable. We now turn to the measurability of projections and spectral measures. Proposition 4.32. The function ω !→ 1cont (Aω ) is measurable. Proof. Let {φn }n∈N be an orthonormal basis of H and let Pn be the orthogonal projection on the subspace spanned by {φk }k≥n . The RAGE theorem yields that for ϕ, ψ ∈ H, 1 n→∞ T →∞ T

T

(ϕ|1cont (Aω )ψ) = lim lim

(ϕ|eitAω Pn e−itAω ψ)dt,

(52)

0

(the proof of (52) is left to the problems), and the statement follows. Proposition 4.33. The function ω !→ 1ac (Aω ) is measurable. Proof. By Theorem 3.6, for all ψ ∈ H, (ψ|1ac (Aω )ψ) = lim lim lim M →∞ p↑1 ↓0

1 πp

M

−M

=

>p Im (ψ|(Aω − x − i)−1 ψ) dx,

and so ω !→ (ψ|1ac (Aω )ψ) is measurable. The polarization identity yields the statement. Proposition 4.34. The functions ω !→ 1sc (Aω ) and ω !→ 1pp (Aω ) are measurable. Proof. 1sc (Aω ) = 1cont (Aω ) − 1ac (Aω ) and 1pp (Aω ) = 1 − 1cont (Aω ). Let M (R) be the Banach space of all complex Borel measures on R (the dual of C0 (R)). A map ω !→ µω ∈ M (R) is called measurable iff for all f ∈ Bb (R) the map ω !→ µω (f ) is measurable. We denote by µω ψ the spectral measure for Aω and ψ. ω ω Proposition 4.35. The functions ω !→ µω ψ,ac , ω !→ µψ,sc , ω !→ µψ,pp are measurable.

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Proof. Since for any Borel set B, (1B (Aω )ψ|1ac (Aω )ψ) = µω ψ,ac (B), (1B (Aω )ψ|1sc (Aω )ψ) = µω ψ,sc (B), (1B (Aω )ψ|1pp (Aω )ψ) = µω ψ,pp (B), the statement follows from Propositions 4.33 and 4.34. Let {ψn } be a cyclic set for Aω and let an > 0 be such that an ψn 2 < ∞. n

We denote by µω =

an µω ψn ,

n

the corresponding spectral measure for Aω . Proposition 4.35 yields ω ω Proposition 4.36. The functions ω !→ µω ac , ω !→ µsc , ω !→ µpp are measurable. ess,ω Let Σac be the essential support of the ac spectrum of Aω . The map 1 ess,ω (x) ∈ L (R, dx), ω !→ (1 + x2 )−1 χΣac ess,ω does not depend on the choice of representative in Σac .

Proposition 4.37. The function ess,ω (x)dx, ω !→ h(x)(1 + x2 )−1 χΣac R

is measurable for all h ∈ L∞ (R, dx). Proof. Clearly, it suffices to consider h(x) = (1 + x)2 χB (x), where B is a bounded interval. Let µω be a spectral measure for Aω . By the dominated convergence theorem, Pµωac (x + iδ) ess,ω dx, (53) χΣac (x)dx = 2 lim lim ω

↓0 δ↓0 P (x + i) + Pµωac (x + iδ) B B µac and the statement follows.

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4.17 Non-relativistic quantum mechanics According to the usual axiomatization of quantum mechanics, a physical system is described by a Hilbert space H. Its observables are described by bounded self-adjoint operators on H. Its states are described by density matrices on H, i.e. positive trace class operators with trace 1. If the system is in a state ρ, then the expected value of the measurement of an observable A is Aρ = Tr(ρA). The variance of the measurement is Dρ (A) = (A − Aρ )2 ρ = A2 ρ − A2ρ . The Cauchy-Schwarz inequality yields the Heisenberg principle: For self-adjoint A, B ∈ B(H), |Tr(ρi[A, B])| ≤ Dρ (A)1/2 Dρ (B)1/2 . Of particular importance are the so called pure states ρ = (ϕ|·)ϕ. In this case, for a self-adjoint A, Aρ = Tr(ρA) = (ϕ|Aϕ) = xdµϕ (x), R

Dρ (A) =

x dµϕ − 2

R

R

2 xdµϕ

,

where µϕ is the spectral measure for A and ϕ. If the system is in a pure state described by a vector ϕ, the possible results R of the measurement of A are numbers in sp(A) randomly distributed according to Prob(R ∈ [a, b]) = dµϕ , [a,b]

(recall that µϕ is supported on sp(A)). Obviously, in this case Aρ and Dρ (A) are the usual expectation and variance of the random variable R. The dynamics is described by a strongly continuous unitary group U (t) on H. In the Heisenberg picture, one evolves observables and keeps states fixed. Hence, if the system is initially in a state ρ, then the expected value of A at time t is Tr(ρ[U (t)AU (t)∗ ]). In the Schr¨odinger picture, one keeps observables fixed and evolves states—the expected value of A at time t is Tr([U (t)∗ ρU (t)]A). Note that if ρ = |ϕ)(ϕ|, then Tr([U (t)∗ ρU (t)]A) = AU (t)ϕ 2 .

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The generator of U (t), H, is called the Hamiltonian of the system. The spectrum of H describes the possible energies of the system. The discrete spectrum of H describes energy levels of bound states (the eigenvectors of H are often called bound states). Note that if ϕ is an eigenvector of H, then AU (t)ϕ 2 = Aϕ 2 is independent of t. By the RAGE theorem, if ϕ ∈ Hcont (H) and A is compact, then 1 T AU (t)ϕ 2 dt = 0. (54) lim T →∞ T 0 Compact operators describe what one might call sharply localized observables. The states associated to Hcont (H) move to infinity in the sense that after a sufficiently long time the sharply localized observables are not seen by these states. On the other hand, if ϕ ∈ Hpp (H), then for any bounded A, 1 lim T →∞ T

T

AU (t)ϕ 2 dt = 0

1e (H)A1e (H)ϕ 2 .

e∈sppp (H)

The mathematical formalism sketched above is commonly used for a description of non-relativistic quantum systems with finitely many degrees of freedom. It can be used, for example, to describe non-relativistic matter—a finite assembly of interacting non-relativistic atoms and molecules. In this case H is the usual N -body Schr¨odinger operator. This formalism, however, is not suitable for a description of quantum systems with infinitely many degrees of freedom like non-relativistic QED, an infinite electron gas, quantum spin-systems, etc. 4.18 Problems [1] Prove Proposition 4.7 [2] Prove Theorem 4.9. [3] Prove Theorem 4.18. [4] Prove Theorem 4.19. [5] Prove Theorem 4.20. [6] Let 0 < < 1. Construct an example of a self-adjoint operator A such that ess spac (A) = [0, 1] and that the Lebesgue measure of Σac (A) is equal to . [7] Prove that ψ ∈ Hcont iff (49) holds for all compact K. [8] Prove that A ≥ 0 iff sp(A) ⊂ [0, ∞). [9] Prove Relation (45).

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[10] Prove Part (3) of Theorem 4.27. [11] Prove Proposition 4.29. [12] Let M * ω !→ Aω be a function with values in self-adjoint operators on H. Prove that the following statements are equivalent: (1) ω !→ (Aω − z)−1 is measurable for all z ∈ C \ R. (2) ω !→ exp(itAω ) is measurable for all t ∈ R. (3) ω !→ 1B (Aω ) is measurable for all Borel sets B. [13] Prove Relation (52). [14] Let ω !→ Aω be a measurable function with values in self-adjoint operators. (1) In the literature, the proof of the measurability of the function ω !→ 1sc (Aω ) is usually based on Carmona’s lemma. Let µ be a finite, positive Borel measure on R, and let I be the set of finite unions of open intervals, each of which has rational endpoints. Then, for any Borel set B, µsing (B) = lim

sup

n→∞ I∈I,|I|<1/n

µ(B ∩ I).

Prove Carmona’s lemma and using this result show that ω !→ 1sc (Aω ) is a measurable function. Hint: See [3] or Section 9.1 in [2]. (2) Prove that ω !→ 1pp (Aω ) is a measurable function by using Simon’s local Wiener theorem (Theorem 3.9). The next set of problems deals with spectral theory of a closed operator A on a Hilbert space H. [15] Let F ⊂ sp(A) be an isolated, bounded subset of sp(A). Let γ be a closed simple path in the complex plane that separates F from sp(A) \ F . Set < 1 (z − A)−1 dz. 1F (A) = 2πi γ (1) Prove that 1F (A) is a (not necessarily orthogonal) projection. (2) Prove that Ran 1F (A) and Ker 1F (A) are complementary (not necessarily orthogonal) subspaces: Ran 1F (A) + Ker 1F (A) = H and Ran 1F (A) ∩ Ker 1F (A) = {0}. (3) Prove that Ran 1F (A) ⊂ Dom (A) and that A : Ran 1F (A) → Ran 1F (A). Prove that A Ran 1F (A) is a bounded operator and that its spectrum is F . (4) Prove that Ker 1F (A) ∩ Dom (A) is dense and that A (Ker 1F (A) ∩ Dom (A)) → Ker 1F (A).

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Prove that the operator (55) is closed and that its spectrum is sp(A) \ F . (5) Assume that F = {z0 } and that Ran 1z0 (A) is finite dimensional. Prove that if ψ ∈ Ran 1z0 (A), then (A − z0 )n ψ = 0 for some n. Hint: Consult Theorem XII.5 in [26]. Remark. The set of z0 ∈ sp(A) which satisfy (5) is called the discrete spectrum of the closed operator operator A and is denoted spdisc (A). The essential spectrum is defined by spess (A) = sp(A) \ spdisc (A) [16] Prove that spess (A) is a closed subset of C. [17] Prove that z !→ (A − z)−1 is a meromorphic function on C \ spess (A) with singularities at points z0 ∈ spdisc (A). Prove that the negative coefficents of the Laurent expansion at z0 ∈ spdisc (A) are finite rank operators. Hint: See Lemma 1 in [26], Section XIII.4. [18] The numerical range of A is defined by N (A) = {(ψ|Aψ) : ψ ∈ Dom (A)}. In general, N (A) is neither open nor closed subset of C. Prove that if Dom (A) = Dom (A∗ ), then sp(A) ⊂ N (A). For additional information about numerical range, the reader may consult [11]. [19] Let z ∈ sp(A). A sequence ψn ∈ Dom (A) is called a Weyl sequence if ψn = 1 and (A − z)ψn → 0. If A is not self-adjoint, then a Weyl sequence may not exist for some z ∈ sp(A). Prove that a Weyl sequence exists for every z on the topological boundary of sp(A). Hint: See Section XIII.4 of [26] or [32]. [20] Let A and B be densely defined linear operators. Assume that B is A-bounded with a relative bound a < 1. Prove that A + B is closable iff A is closable, and that in this case the closures of A and A + B have the same domain. Deduce that A + B is closed iff A is closed. [21] Let A and B be densely defined linear operators. Assume that A is closed and that B is A-bounded with constants a and b. If A is invertible (that is, 0 ∈ sp(A)), and if a and b satisfy a + b A−1 < 1, prove that A + B is closed, invertible, and that (A + B)−1 ≤ (A + B)−1 − A−1 ≤

A−1 , 1 − a − b A−1 A−1 (a + b A−1 ) . 1 − a − b A−1

Hint: See Theorem IV.1.16 in [17]. [22] In this problem we will discuss the regular perturbation theory for closed operators. Let A be a closed operator and let B be A-bounded with constants a and

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b. For λ ∈ C we set Aλ = A + λB. If |λ|a < 1, then Aλ is a closed operator and Dom (Aλ ) = Dom (A). Let F be an isolated, bounded subset of A and γ a simple closed path that separates F and sp(H) \ F . (1) Prove that for z ∈ γ, B(A − z)−1 ≤ a + (a|z| + b) (A − z)−1 . (2) Prove that if A is self-adjoint and z ∈ sp(A), then (A − z)−1 = 1/dist{z, sp(A)}. (3) Assume that Dom (A) = Dom (A∗ ) and let N (A) be the numerical range of A. Prove that for all z ∈ N (A), (A − z)−1 ≤ 1/dist{z, N (A)}. >−1 = and assume that |λ| < Λ. (4) Let Λ = a + supz∈γ (a|z|) + b) (A − z)−1 Prove that sp(Aλ ) ∩ γ = ∅ and that for z ∈ γ, (z − Aλ )−1 =

∞

= >n λn (z − A)−1 B(z − A)−1 .

n=0

Hint: Start with z − Aλ = (1 − λB(z − A)−1 )(z − A). (5) Let Fλ be the spectrum of Aλ inside γ (so F0 = F ). For |λ| < Λ the projection of Aλ onto Fλ is given by < 1 (z − Aλ )−1 dz. Pλ = 1Fλ (Aλ ) = 2πi γ Prove that the projection-valued function λ !→ Pλ is analytic for |λ| < Λ. (6) Prove that the differential equation Uλ = [Pλ , Pλ ]Uλ , U0 = 1, (the derivatives are w.r.t. λ and [A, B] = AB − BA) has a unique solution for |λ| < Λ, and that Uλ is an analytic family of bounded invertible operators such that Uλ P0 Uλ−1 = Pλ . Hint: See [26], Section XII.2. (7) Set A˜λ = Uλ−1 Aλ Uλ and Σλ = P0 A˜λ P0 . Σλ is a bounded operator on the Hilbert space Ran P0 . Prove that sp(Σλ ) = Fλ and that the operator-valued function λ !→ Σλ is analytic for |λ| < Λ. Compute the first three terms in the expansion Σλ =

∞

λn Σn .

(56)

n=0

The term Σ1 is sometimes called the Feynman-Hellman term. The term Σ2 , often called the Level Shift Operator (LSO), plays an important role in quantum mechanics and quantum field theory. For example, the formal computations in physics involving Fermi’s Golden Rule are often best understood and most easily proved with the help of LSO. (8) Assume that dim P0 = dim Ran P0 < ∞. Prove that dim Pλ = dim P0 for

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|λ| < Λ and conclude that the spectrum of Aλ inside γ is discrete and consists of at most dim P0 distinct eigenvalues. Prove that the eigenvalues of Aλ inside γ are all the branches of one or more multi-valued analytic functions with at worst algebraic singularities. (9) Assume that F0 = {z0 } and dim P0 = 1 (namely that the spectrum of A inside γ is a semisimple eigenvalue z0 ). In this case Σλ = z(λ) is an analytic function for |λ| < Λ. Compute the first five terms in the expansion z(λ) =

∞

λ n zn .

n=0

5 Spectral theory of rank one perturbations The Hamiltonians which arise in non-relativistic quantum mechanics typically have the form (57) HV = H0 + V, where H0 and V are two self-adjoint operators on a Hilbert space H. H0 is the ”free” or ”reference” Hamiltonian and V is the ”perturbation”. For example, the Hamiltonian of a free non-relativistic quantum particle of mass m moving in R3 is 1 ∆, where ∆ is the Laplacian in L2 (R3 ). If the particle is subject to an external − 2m potential field V (x), then the Hamiltonian describing the motion of the particle is HV = −

1 ∆ + V, 2m

(58)

where V denotes the operator of multiplication by the function V (x). Operators of this form are called Schr¨odinger operators. We will not study in this section the spectral theory of Schr¨odinger operators. Instead, we will keep H0 general and focus on simplest non-trivial perturbations V . More precisely, let H be a Hilbert space, H0 a self-adjoint operator on H and ψ ∈ H a given unit vector. We will study spectral theory of the family of operators Hλ = H0 + λ(ψ| · )ψ,

λ ∈ R.

(59)

This simple model is of profound importance in mathematical physics. The classical reference for the spectral theory of rank one perturbations is [29]. The cyclic subspace generated by Hλ and ψ does not depend on λ and is equal to the cyclic subspace generated by H0 and ψ which we denote Hψ (this fact is an immediate consequence of the formulas (61) below). Let µλ be the spectral measure for Hλ and ψ. This measure encodes the spectral properties of Hλ Hψ . Note that ⊥ ⊥ = H0 Hψ . In this section we will always assume that H = Hψ , namely Hλ Hψ that ψ is a cyclic vector for H0 . The identities

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(Hλ − z)−1 − (H0 − z)−1 = (Hλ − z)−1 (H0 − Hλ )(H0 − z)−1 = (H0 − z)−1 (H0 − Hλ )(Hλ − z)−1 ,

(60)

yield (Hλ − z)−1 ψ = (H0 − z)−1 ψ − λ(ψ|(H0 − z)−1 ψ)(Hλ − z)−1 ψ, (H0 − z)−1 ψ = (Hλ − z)−1 ψ + λ(ψ|(Hλ − z)−1 ψ)(H0 − z)−1 ψ. Let Fλ (z) = (ψ|(Hλ − z)−1 ψ) =

R

(61)

dµλ (t) . t−z

Note that if z ∈ C+ , then Fλ (z) is the Borel transform and Im Fλ (z) is the Poisson transform of µλ . The second identity in (61) yields F0 (z) = Fλ (z)(1 + λF0 (z)), and so Fλ (z) = Im Fλ (z) =

F0 (z) , 1 + λF0 (z)

(62)

Im F0 (z) . |1 + λF0 (z)|2

(63)

These elementary identities will play a key role in our study. The function dµ0 (t) , G(x) = 2 R (x − t) will also play an important role. Recall that G(x) = ∞ for µ0 -a.e. x (Lemma 3.3). In this section we will occasionally denote by |B| the Lebesgue measure of a Borel set B. 5.1 Aronszajn-Donoghue theorem Recall that the limit Fλ (x) = lim Fλ (x + iy), y↓0

exists and is finite and non-zero for Lebesgue a.e. x. For λ = 0 define Sλ = {x ∈ R : F0 (x) = −λ−1 , G(x) = ∞}, Tλ = {x ∈ R : F0 (x) = −λ−1 , G(x) < ∞}, L = {x ∈ R : Im F0 (x) > 0}.

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In words, Sλ is the set of all x ∈ R such that limy↓0 F0 (x + iy) exists and is equal to −λ−1 , etc. Any two sets in the collection {Sλ , Tλ , L}λ=0 are disjoint. By Theorem 4.11, |Sλ | = |Tλ | = 0. As usual, δ(y) denotes the delta-measure of y ∈ R; δ(y)(f ) = f (y). Theorem 5.1. (1) Tλ is the set of eigenvalues of Hλ . Moreover,

µλpp =

xn ∈Tλ

1 δ(xn ). λ2 G(xn )

(2) µλsc is concentrated on Sλ . (3) For all λ, L is the essential support of the ac spectrum of Hλ and spac (Hλ ) = spac (H0 ). (4) The measures {µλsing }λ∈R are mutually singular. In other words, if λ1 = λ2 , then 1 2 and µλsing are concentrated on disjoint sets. the measures µλsing Proof. (1) The eigenvalues of Hλ are precisely the atoms of µλ . Let T˜λ = {x ∈ R : µλ ({x}) = 0}. Since µλ ({x}) = lim yIm Fλ (x + iy) = lim y↓0

y↓0

yIm F0 (x + iy) , |1 + λF0 (x + iy)|2

(64)

T˜λ ⊂ {x : F0 (x) = −λ−1 }. The relation (64) yields µλ ({x}) ≤ λ−2 lim y↓0

1 y = 2 , Im F0 (x + iy) λ G(x)

and so T˜λ ⊂ {x : F0 (x) = −λ−1 , G(x) < ∞} = Tλ . On the other hand, if F0 (x) = −λ−1 and G(x) < ∞, then lim y↓0

F0 (x + iy) − F0 (x) = G(x), iy

(the proof of this relation is left to the problems). Hence, if x ∈ Tλ , then F0 (x + iy) = iyG(x) − λ−1 + o(y), and µλ ({x}) = lim y↓0

yIm F0 (x + iy) 1 > 0. = 2 |1 + λF0 (x + iy)|2 λ G(x)

Hence Tλ = T˜λ , and for x ∈ T˜λ , µλ ({x}) = 1/λ2 G(x). This yields (1). (2) By Theorem 3.5, µλsing is concentrated on the set {x : lim Im Fλ (x + iy) = ∞}. y↓0

The formula (63) yields that µλsing is concentrated on the set

(65)

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{x : F0 (x) = −λ−1 }.

(66)

If F0 (x) = −λ−1 and G(x) < ∞, then by (1) x is an atom of µλ . Hence, µλsc is concentrated on the set {x : F0 (x) = −λ−1 , G(x) = ∞} = Sλ . (3) By Theorem 3.5, dµλac (x) = π −1 Im Fλ (x)dx. On the other hand, by the formula (63), the sets {x : Im F0 (x) > 0} and {x : Im Fλ (x) > 0} coincide up to a set of Lebesgue measure zero. Hence, L is the essential support of the ac spectrum of Hλ for all λ. Since µ0ac and µλac are equivalent measures, spac (H0 ) = spac (Hλ ). (4) For λ = 0, µλsing is concentrated on the set (66). By Theorem 3.5, µ0sing is concentrated on {x : Im F0 (x) = ∞}. This yields the statement. 5.2 The spectral theorem By Theorem 4.13, for all λ there exists a unique unitary Uλ : Hψ → L2 (R, dµλ ) such that Uλ ψ = 1l and Uλ Hλ Uλ−1 is the operator of multiplication by x on L2 (R, dµλ ). In this subsection we describe Uλ . For φ ∈ H and z ∈ C \ R let Mφ (z) = (ψ|(H0 − z)−1 φ), and

Mφ (x ± i0) = lim(ψ|(H0 − x ∓ iy)−1 φ), y↓0

whenever the limits exist. By Theorem 3.17 the limits exist and are finite for Lebesgue a.e. x. For consistency, in this subsection we write F0 (x + i0) = limy↓0 F0 (x + iy). Theorem 5.2. Let φ ∈ H. (1) For all λ and for µλ,ac -a.e. x, (Uλ 1ac φ)(x) =

+

1 Mφ (x + i0) − Mφ (x − i0) − λMφ (x + i0) 2i Im F0 (x + i0) λ (Mφ (x + i0) − Mφ (x − i0))F0 (x + i0) . 2i Im F0 (x + i0)

(2) Let λ = 0. Then for µλ,sing -a.e. x the limit Mφ (x + i0) exists and (Uλ 1sing φ)(x) = −λMφ (x + i0).

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Proof. The identities (60) yield (ψ|(Hλ − z)−1 φ) =

Mφ (z) . 1 + λF0 (z)

Combining this relation with (62) and (63) we derive 1 Mφ (z) − Mφ (z) (ψ|Im (Hλ − z)−1 φ) = Im (ψ|(Hλ − z)−1 ψ) 2i Im F0 (z) λ F0 (z)Mφ (z) − F0 (z)Mφ (z) . + 2i Im F0 (z) Similarly,

(ψ|(Hλ − z)−1 φ) Mφ (z) = . −1 (ψ|(Hλ − z) ψ) F0 (z)

(67)

(68)

(1) follows from the identity (67) and Part 1 of Theorem 4.17. Since µλ,sing is concentrated on the set {x : limy↓0 F0 (x + i0) = −λ−1 }, the identity (68) and Part 2 of Theorem 4.17 yield (2). Note that Part 2 of Theorem 5.2 yields that for every eigenvalue x of Hλ (i.e. for all x ∈ Tλ ), (69) (Uλ 1pp φ)(x) = −λMφ (x + i0). This special case of Theorem 5.2 (which can be easily proven directly) has been used in the proofs of dynamical localization in the Anderson model; see [1, 6]. The extension of (69) to singular continuous spectrum depends critically on the full strength of the Poltoratskii theorem. For some applications of this result see [16]. 5.3 Spectral averaging In the sequel we will freely use the measurability results established in Subsection 4.16. Let µ(B) = µλ (B)dλ, R

where B ⊂ R is a Borel set. Obviously, µ is a Borel measure on R. The following (somewhat surprising) result is often called spectral averaging: Theorem 5.3. The measure µ is equal to the Lebesgue measure and for all f ∈ L1 (R, dx), ; : f (x)dx = f (x)dµλ (x) dλ. R

R

R

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Proof. For any positive Borel function f , ; : λ f (t)dµ(t) = f (t)dµ (t) dλ, R

R

R

(both sides are allowed to be infinity). Let f (t) =

y , (t − x)2 + y 2

where y > 0. Then f (t)dµλ (t) = Im Fλ (x + iy) = R

Im F0 (x + iy) . |1 + λF0 (x + iy)|2

By the residue calculus, R

Im F0 (x + iy) dλ = π, |1 + λF0 (x + iy)|2

(70)

and so the Poisson transform of µ exists and is identically equal to π, the Poisson transform of the Lebesgue measure. By Theorem 3.7, µ is equal to the Lebesgue measure. Spectral averaging is a mathematical gem which has been rediscovered by many authors. A detailed list of references can be found in [30]. 5.4 Simon-Wolff theorems Theorem 5.4. Let B ⊂ R be a Borel set. Then the following statements are equivalent: (1) G(x) < ∞ for Lebesgue a.e. x ∈ B. (2) µλcont (B) = 0 for Lebesgue a.e. λ. Proof. (1) ⇒ (2). If G(x) < ∞ for Lebesgue a.e. x ∈ B, then Im F0 (x) = 0 for Lebesgue a.e. x ∈ B. Hence, for all λ, Im Fλ (x) = 0 for Lebesgue a.e. x ∈ B, and Im Fλ (x)dx = 0. µλac (B) = π −1 B

By Theorem 5.1, the measure B is concentrated on the set A = {x ∈ B : G(x) = ∞}. Since A has Lebesgue measure zero, by spectral averaging, µλsc (A)dλ ≤ µλ (A)dλ = |A| = 0. µλsc

R

R

Hence, µλsc (A) = 0 for Lebesgue a.e. λ ∈ R, and so µλsc (B) = 0 for Lebesgue a.e. λ.

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(2) ⇒ (1). Assume that the set A = {x ∈ B : G(x) = ∞} has positive Lebesgue measure. By Theorem 5.1, µλpp (A) = 0 for all λ = 0. By spectral averaging, λ µcont (A)dλ = µλ (A)dλ = |A| > 0. R

R

Hence, for a set of λ of positive Lebesgue measure, µλcont (B) > 0. Theorem 5.5. Let B be a Borel set. Then the following statements are equivalent: (1) Im F0 (x) > 0 for Lebesgue a.e. x ∈ B. (2) µλsing (B) = 0 for Lebesgue a.e. λ. Proof. (1) ⇒ (2). By Theorem 5.1, for λ = 0 the measure µλsing B is concentrated on the set A = {x ∈ B : Im F0 (x) = 0}. Since A has Lebesgue measure zero, by spectral averaging, R

µλsing (A)dλ ≤

µλ (A)dλ = 0. R

Hence, for Lebesgue a.e. λ, µλsing (B) = 0. (2) ⇒ (1). Assume that the set A = {x ∈ B : Im F0 (x) = 0} has positive Lebesgue measure. Clearly, µλac (A) = 0 for all λ, and by spectral averaging, µλsing (A)dλ = µλ (A)dλ = |A| > 0. R

R

Hence, for a set of λ of positive Lebesgue measure, µλsing (B) > 0. Theorem 5.6. Let B be a Borel set. Then the following statements are equivalent: (1) Im F0 (x) = 0 and G(x) = ∞ for Lebesgue a.e. x ∈ B. (2) µλac (B) + µλpp (B) = 0 for Lebesgue a.e. λ. The proof of Theorem 5.6 is left to the problems. Theorem 5.4 is the celebrated result of Simon-Wolff [31]. Although Theorems 5.5 and 5.6 are well known to the workers in the field, I am not aware of a convenient reference. 5.5 Some remarks on spectral instability By the Kato-Rosenblum theorem, the absolutely continuous spectrum is stable under trace class perturbations, and in particular under rank one perturbations. In the rank one case this result is also an immediate consequence of Theorem 5.1. The situation is more complicated in the case of the singular continuous spectrum. There are examples where sc spectrum is stable, namely when Hλ has purely singular continuous spectrum in (a, b) for all λ ∈ R. There are also examples where H0 has purely sc spectrum in (a, b), but Hλ has pure point spectrum for all λ = 0. A. Gordon [10] and del Rio-Makarov-Simon [7] have proven that pp spectrum is always unstable for generic λ.

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Theorem 5.7. The set {λ : Hλ has no eigenvalues in sp(H0 )}, is dense Gδ in R. Assume that (a, b) ⊂ sp(H0 ), and that G(x) < ∞ for Lebesgue a.e. x ∈ (a, b). Then the spectrum of Hλ in (a, b) is pure point for Lebesgue a.e. λ. However, by Theorem 5.7, there is a dense Gδ set of λ’s such that Hλ has purely singular continuous spectrum in (a, b) (of course, Hλ has no ac spectrum in (a, b) for all λ). 5.6 Boole’s equality So far we have used the rank one perturbation theory and harmonic analysis to study spectral theory. In the last three subsections we will turn things around and use rank one perturbation theory and spectral theory to reprove some well known results in harmonic analysis. This subsection deals with Boole’s equality and is based on [6] and [22]. Let ν be a finite positive Borel measure on R and Fν (z) its Borel transform. As usual, we denote Fν (x) = lim Fν (x + iy). y↓0

The following result is known as Boole’s equality: Proposition 5.8. Assume that ν is a pure point measure with finitely many atoms. Then for all t > 0 |{x : Fν (x) > t}| = |{x : Fν (x) < −t}| =

ν(R) . t

Proof. We will prove that |{x : Fν (x) > t}| = ν(R)/t. Let {xj }1≤j≤n ,

x1 < · · · < xn ,

be the support of ν and αj = ν({xj }) the atoms of ν. W.l.o.g. we may assume that ν(R) = j αj = 1. Clearly, Fν (x) =

n j=1

αj . xj − x

Set x0 = −∞, xn+1 = ∞. Since Fν (x) > 0 for x = xj , the function Fν (x) is strictly increasing on (xj , xj+1 ), with vertical asymptotes at xj , 1 ≤ j ≤ n. Let r1 < · · · < rn be the solutions of the equation Fν (x) = t. Then

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|{x : Fν (x) > t}| =

n

303

(xj − rj ).

j=1

On the other hand, the equation Fν (x) = t is equivalent to n k=1

or

n

αk

(xj − x) = t

n

(xj − x),

j=1

j=k

(xj − x) − t

−1

j=1

n k=1

αk

(xj − x) = 0.

j=k

Since {rj } are all the roots of the polynomial on the l.h.s., n

rj = −t−1 +

j=1

n

xj

j=1

and this yields the statement. Proposition 5.8 was first proven by G. Boole in 1867. The Boole equality is another gem that has been rediscovered by many authors; see [22] for the references. The rank one perturbation theory allows for a simple proof of the optimal version of the Boole equality. Theorem 5.9. Assume that ν is a purely singular measure. Then for all t > 0 |{x : Fν (x) > t}| = |{x : Fν (x) < −t}| =

ν(R) . t

Proof. W.l.o.g. we may assume that ν(R) = 1. Let H0 be the operator of multiplication by x on L2 (R, dν) and ψ ≡ 1. Let Hλ = H0 + λ(ψ| · )ψ and let µλ be the spectral measure for Hλ and ψ. Obviously, µ0 = ν and F0 = Fν . Since ν is a singular measure, µλ is singular for all λ ∈ R. By Theorem 5.1, for λ = 0, the measure µλ is concentrated on the set {x : F0 (x) = −λ−1 }. Let Γt = {x : F0 (x) > t}. Then for λ = 0,

. 1 if −t−1 < λ < 0, µλ (Γt ) = 0 if λ ≤ −t−1 or λ > 0.

By the spectral averaging,

|Γt | =

R

µλ (Γt )dλ = t−1 .

A similar argument yields that |{x : Fν (x) < −t}| = t−1 . The Boole equality fails if ν is not a singular measure. However, in general we have

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Theorem 5.10. Let ν be a finite positive Borel measure on R. Then lim t |{x : |Fν (x)| > t}| = 2νsing (R).

t→∞

Theorem 5.10 is due to Vinogradov-Hruschev. Its proof (and much additional information) can be found in the paper of Poltoratskii [22]. 5.7 Poltoratskii’s theorem This subsection is devoted to the proof of Theorem 3.18. We follow [14]. We first consider the case νs = 0, µ compactly supported, f ∈ L2 (R, dµ) real valued. W.l.o.g. we may assume that µ(R) = 1. Consider the Hilbert space L2 (R, dµ) and let H0 be the operator of multiplication by x. Note that Fµ (z) = ( 1l |(H0 − z)−1 1l ), For λ ∈ R, let

Ff µ (z) = ( 1l |(H0 − z)−1 f ).

Hλ = H0 + λ( 1l | · )1l,

and let µλ be the spectral measure for Hλ and 1l. To simplify the notation, we write Fλ (z) = ( 1l |(Hλ − z)−1 1l ) = Fµλ (z). Note that with this notation, F0 = Fµ ! By Theorem 5.1, the measures {µλsing }λ∈R are mutually singular. By Theorem 3.5, the measure µsing = µ0sing is concentrated on the set {x ∈ R : lim Im F0 (x + iy) = ∞}. y↓0

We also recall the identity Fλ (z) =

F0 (z) . 1 + λF0 (z)

(71)

By the spectral theorem, there exists a unitary Uλ : L2 (R, dµ) → L2 (R, dµλ ), such that Uλ 1l = 1l and Uλ Hλ Uλ−1 is the operator of multiplication by x on L2 (R, dµλ ). Hence (Uλ f )(x) λ dµ (x) = F(Uλ f )µλ (z). ( 1l |(Hλ − z)−1 f ) = x−z R In what follows we set λ = 1 and write U = U1 .

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For a ∈ R and b > 0 let hab (x) = 2b((x − a)2 + b2 )−1 , w = a + ib, and rw (x) = (x − w)−1 (hence hab = i−1 (rw − rw )). The relation U hab = hab + λi−1 (F0 (w)rw − F0 (w)rw ),

(72)

yields that U hab is a real-valued function. The proof of (72) is simple and is left to the problems. Since the linear span of {hab : a ∈ R, b > 0} is dense in C0 (R), U takes real-valued functions to real-valued functions. In particular, U f is a real-valued function. The identity ( 1l |(H0 − z)−1 f ) = (1 + ( 1l |(H0 − z)−1 1l ))( 1l |(H1 − z)−1 f ), can be rewritten as ( 1l |(H0 − z)−1 f ) = (1 + F0 (z))F(U f )µ1 (z).

(73)

Im ( 1l |(H0 − z)−1 f ) = Re F(U f )µ1 (z) + L(z), Im F0 (z)

(74)

It follows that

where L(z) =

Re (1 + F0 (z)) Im F(U f )µ1 (z). Im F0 (z)

We proceed to prove that for µsing − a.e. x,

lim Im F(U f )µ1 (x + iy) = 0, y↓0

lim L(x + iy) = 0, y↓0

for µsing − a.e. x.

(75) (76)

We start with (75). Using first that U f is real-valued and then the CauchySchwarz inequality, we derive ? ? Im F(U f )µ1 (x + iy) = P(U f )µ1 (x + iy) ≤ Pµ1 (x + iy) P(U f )2 µ1 (x + iy). Since the measures (U f )2 µ1sing and µsing are mutually singular, lim y↓0

P(U f )2 µ1 (x + iy) = 0, Pµ (x + iy)

for µsing − a.e. x,

(see Problem 4). Hence, lim 5 y↓0

Since

Im F(U f )µ1 (x + iy) 5 = 0, Pµ1 (x + iy) Pµ (x + iy)

for µsing − a.e. x.

(77)

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Pµ1 (x + iy)Pµ (x + iy) = Im F1 (x + iy)Im F0 (x + iy) =

(Im F0 (x + iy))2 ≤1 |1 + F0 (x + iy)|2

for all x ∈ R, (77) yields (75). To prove (76), note that |L(x + iy)| = 5

≤5

Im F(U f )µ1 (x + iy) |Re (1 + F0 (x + iy))| Im F0 (x + iy) 5 Im F0 (x + iy) |1 + F0 (x + iy)| Pµ1 (x + iy) Pµ (x + iy) Im F(U f )µ1 (x + iy) 5 . Pµ1 (x + iy) Pµ (x + iy)

Hence, (77) yields (76). Rewrite (74) as F(U f )µ1 (z) =

Im ( 1l |(H0 − z)−1 f ) + Im F(U f )µ1 (z) − L(z). Im F0 (z)

(78)

By Theorem 3.5, lim y↓0

Im ( 1l |(H0 − x − iy)−1 f ) Pf µ (x + iy) = lim = f (x), y↓0 Pµ (x + iy) Im F0 (x + iy)

for µ − a.e. x.

Hence, (78), (75), and (76) yield that for µsing − a.e. x.

(79)

1 + 1 F(U f )µ1 (x + iy). F0 (x + iy)

(80)

lim F(U f )µ1 (x + iy) = f (x), y↓0

Rewrite (73) as Ff µ (x + iy) = Fµ (x + iy)

Since |F0 (x + iy)| → ∞ as y ↓ 0 for µsing -a.e. x, (79) and (80) yield lim y↓0

Ff µ (x + iy) = f (x), Fµ (x + iy)

for µsing − a.e. x.

This proves the Poltoratskii theorem in the special case where νs = 0, µ is compactly supported, and f ∈ L2 (R, dµ) is real-valued. We now remove the assumptions f ∈ L2 (R, dµ) and that f is real valued (we still assume that µ is compactly supported and that νs = 0). Assume that f ∈ L1 (R, dµ) and that f is positive. Set g = 1/(1 + f ) and ρ = (1 + f )µ. Then lim y↓0

1 Fµ (x + iy) Fgρ (x + iy) = lim = , y↓0 Fρ (x + iy) F(1+f )µ (x + iy) 1 + f (x)

for µsing -a.e. x. By the linearity of the Borel transform,

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lim y↓0

307

F(1+f )µ (x + iy) Ff µ (x + iy) = lim − 1 = f (x), y↓0 Fµ (x + iy) Fµ (x + iy)

for µsing -a.e. x. Since every f ∈ L1 (R, dµ) is a linear combination of four positive functions in L1 (R, dµ), the linearity of the Borel transform implies the statement for all f ∈ L1 (R, dµ). Assume that µ is not compactly supported (we still assume νs = 0) and let [a, b] be a finite interval. Decompose µ = µ1 + µ2 , where µ1 = µ [a, b], µ2 = µ R \ [a, b]. Since Ff µ1 (z) + Ff µ2 (z) Ff µ (z) = , Fµ (z) Fµ1 (z)(1 + Fµ2 (z)/Fµ1 (z)) and limy↓0 |Fµ1 (x + iy)| → ∞ for µ1,sing -a.e. x ∈ [a, b], lim y↓0

Ff µ (x + iy) = f (x), Fµ (x + iy)

for µsing -a.e. x ∈ (a, b).

Since [a, b] is arbitrary, we have removed the assumption that µ is compactly supported. Finally, to finish the proof we need to show that if ν ⊥ µ, then lim y↓0

Fν (x + iy) = 0, Fµ (x + iy)

(81)

for µsing -a.e. x. Since ν can be written as a linear combination of four positive measures each of which is singular w.r.t. µ, w.l.o.g. me may assume that ν is positive. Let S be a Borel set such that µ(S) = 0 and that ν is concentrated on S. Then lim y↓0

FχS (µ+ν) (x + iy) = χS (x), Fµ+ν (x + iy)

for µsing + νsing -a.e. x. Hence, lim y↓0

Fν (x + iy) = 0, Fµ (x + iy) + Fν (x + iy)

for µsing - a.e. x, and this yields (81). The proof of the Poltoratskii theorem is complete. The Poltoratskii theorem also holds for complex measures µ: Theorem 5.11. Let ν and µ be complex Borel measures and ν = f µ + νs be the Radon-Nikodym decomposition. Let |µ|sing be the part of |µ| singular with respect to the Lebesgue measure. Then lim y↓0

Fν (x + iy) = f (x), Fµ (x + iy)

for |µ|sing − a.e. x.

Theorem 5.11 follows easily from Theorem 3.18.

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5.8 F. & M. Riesz theorem The celebrated theorem of F. & M. Riesz states: Theorem 5.12. Let µ = 0 be a complex measure and Fµ (z) its Borel transform. If Fµ (z) = 0 for all z ∈ C+ , then |µ| is equivalent to the Lebesgue measure. In the literature one can find many different proofs of this theorem (for example, three different proofs are given in [19]). However, it has been only recently noticed that F. & M. Riesz theorem is an easy consequence of the Poltoratskii theorem. The proof below follows [16]. Proof. For z ∈ C \ R we set dµ(t) , Fµ (z) = R t−z and write Fµ (x ± i0) = lim Fµ (x ± iy). y↓0

By Theorem 3.17 (and its obvious analog for the lower half-plane), Fµ (x ± i0) exists and is finite for Lebesgue a.e. x. Write µ = h|µ|, where |h(x)| = 1 for all x. By the Poltoratskii theorem, lim y↓0

|Fµ (x + iy)| = |h(x)| = 1, |F|µ| (x + iy)|

for |µ|sing -a.e. x. Since by Theorem 3.5, limy↓0 |F|µ| (x + iy)| = ∞ for |µ|sing -a.e. x, we must have limy↓0 |Fµ (x + iy)| = ∞ for |µ|sing -a.e. x. Hence, if |µ|sing = 0, then Fµ (z) cannot vanish on C+ . It remains to prove that |µ| is equivalent to the Lebesgue measure. By Theorem 3.5, d|µ| = π −1 Im F|µ| (x + i0)dx, so we need to show that Im F|µ| (x + i0) > 0, for Lebesgue a.e. x. Assume that Im F|µ| (x + i0) = 0 for x ∈ S, where S has positive Lebesgue measure. The formula (t − x)dµ(t) ydµ(t) Fµ (x + iy) = + i , 2 2 2 2 R (t − x) + y R (t − x) + y and the bound

ydµ(t) (t − x)2 + y 2 ≤ Im F|µ| (x + iy), R

yield that for x ∈ S, lim Fµ (x + iy) = lim

y→0

y→0

R

(t − x)dµ(t) . (t − x)2 + y 2

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Hence, Fµ (x − i0) = Fµ (x + i0) = 0,

for Lebesgue a.e. x ∈ S.

(82)

Since Fµ vanishes on C+ , Fµ does not vanish on C− (otherwise, since the linear span of the set of functions {(x − z)−1 : z ∈ C \ R} is dense in C0 (R), we would have µ = 0). Then, by Theorem 3.17 (i.e., its obvious analog for the lower half-plane), Fµ (x − i0) = 0 for Lebesgue a.e x ∈ R. This contradicts (82). 5.9 Problems and comments [1] Prove Relation (65). Hint: See Theorem I.2 in [29]. [2] Prove Theorem 5.6. [3] Prove Relation (72). [4] Let ν and µ be positive measures such that νsing ⊥ µsing . Prove that for µsing -a.e. x Pν (x + iy) = 0. lim y↓0 Pµ (x + iy) Hint: Write

Pν (z) Pνac (z) + sing Pµsing (z) Pµsing (z) Pν (z) = , Pµac (z) Pµ (z) +1 Pµsing (z)

and use Theorem 3.5. [5] Prove the Poltoratskii theorem in the case where ν and µ are positive pure point measures. [6] In the Poltoratskii theorem one cannot replace µsing by µ. Find an example justifying this claim. The next set of problems deals with various examples involving rank one perturbations. Note that the model (59) is completely determined by a choice of a Borel probability measure µ0 on R. Setting H = L2 (R, dµ0 ), H0 = operator of multiplication by x, ψ ≡ 1, we obtain a class of Hamiltonians Hλ = H0 + λ(ψ| · )ψ of the form (59). On the other hand, by the spectral theorem, any family Hamiltonians (59), when restricted to the cyclic subspace Hψ , is unitarily equivalent to such a class. [7] Let µC be the standard Cantor measure (see Example 3 in Section I.4 of [23]) and dµ0 = (dx [0, 1] + dµC )/2. The ac spectrum of H0 is [0, 1]. The singular continuous part of µ0 is concentrated on the Cantor set C. Since C is closed,

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spsing (H0 ) = C. Prove that for λ = 0 the spectrum of Hλ in [0, 1] is purely absolutely continuous. Hint: See the last example in Section XIII.7 of [26]. [8] Assume that µ0 = µC . Prove that for all λ = 0, Hλ has only pure point spectrum. Compute the spectrum of Hλ . Hint: This is Example 1 in [31]. See also Example 3 in Section II.5 of [29]. [9] Let n

−n

µn = 2

2

δ(j/2n ),

j=1

and µ = n an µn , where an > 0, n an = 1, n 2n an = ∞. The spectrum of H0 is pure point and equal to [0, 1]. Prove that the spectrum of Hλ in [0, 1] is purely singular continuous for all λ = 0. Hint: This is Example 2 in [31]. See also Example 4 in Section II.5 of [29].

[10] Let νj,n (A) = µC (A + j/2n ) and 0

µ = cχ[0,1]

∞ n=1

n

−2

n

2

νj,n ,

j=1

where c is the normalization constant. Prove that the spectrum of Hλ on [0, 1] is purely singular continuous for all λ. Hint: This is Example 5 in Section II.5 of [29]. [11] Find µ0 such that: (1) The spectrum of H0 is purely absolutely continuous and equal to [0, 1]. (2) For a set of λ’s of positive Lebesgue measure, Hλ has embedded point spectrum in [0, 1]. Hint: See [8] and Example 7 in Section II.5 of [29]. [12] Find µ0 such that: (1) The spectrum of H0 is purely absolutely continuous and equal to [0, 1]. (2) For a set of λ’s of positive Lebesgue measure, Hλ has embedded singular continuous spectrum in [0, 1]. Hint: See [8] and Example 8 in Section II.5 of [29]. [13] del Rio and Simon [8] have shown that there exists µ0 such that: (1) For all λ, spac (Hλ ) = [0, 1]. (2) For a set of λ’s of positive Lebesgue measure, Hλ has embedded point spectrum in [0, 1]. (3) For a set of λ’s of positive Lebesgue measure, Hλ has embedded singular continuous spectrum in [0, 1]. [14] del Rio-Fuentes-Poltoratskii [5] have shown that there exists µ0 such that: (1) For all λ spac (Hλ ) = [0, 1]. Moreover, for all λ ∈ [0, 1], the spectrum of Hλ is

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purely absolutely continuous. (2) For all λ ∈ [0, 1], [0, 1] ⊂ spsing (Hλ ). [15] Let µ0 be a pure point measure with atoms µ0 ({xn }) = an , n ∈ N, where xn ∈ [0, 1]. Clearly, ∞ an . G(x) = (x − xn )2 n=1 √ (1) Prove that if n an < ∞, then G(x) < ∞ for Lebesgue a.e. x ∈ [0, 1]. uniformly distrib(2) Assume that xn = xn (ω) are independent random variables √ uted on [0, 1] (we keep an deterministic). Assume that n an = ∞. Prove that for a.e. ω, G(x) = ∞ for Lebesgue a.e. x ∈ [0, 1]. (3) What can you say about the spectrum of Hλ in the cases (1) and (2)? Hint: (1) and (2) are proven in [12].

References 1. Aizenman M.: Localization at weak disorder: Some elementary bounds. Rev. Math. Phys. 6 (1994), 1163. 2. Cycon H.L., Froese R.G., Kirsch W., Simon B.: Schr¨odinger Operators, SpringerVerlag, Berlin, 1987. 3. Carmona R., Lacroix J.: Spectral Theory of Random Schr¨odinger Operators, Birkhauser, Boston, 1990. 4. Davies E.B.: Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1995. 5. del Rio R., Fuentes S., Poltoratskii A.G.: Coexistence of spectra in rank-one perturbation problems. Bol. Soc. Mat. Mexicana 8 (2002), 49. 6. del Rio R., Jitomirskaya S., Last Y., Simon B.: Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization. J. d’Analyse Math. 69 (1996), 153. 7. del Rio R., Makarov N., Simon B.: Operators with singular continuous spectrum: II. Rank one operators. Commun. Math. Phys. 165 (1994), 59. 8. del Rio R., Simon B.: Point spectrum and mixed spectral types for rank one perturbations. Proc. Amer. Math. Soc. 125 (1997), 3593. 9. Evans L., Gariepy G.: Measure Theory and Fine Properties of Functions, CRC Press, 1992. 10. Gordon A.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys. 164 (1994), 489. 11. Gustafson K., Rao D.K.M.: Numerical Range, Springer-Verlag, Berlin, 1996. 12. Howland J.: Perturbation theory of dense point spectra. J. Func. Anal. 74 (1987), 52. 13. Jakˇsi´c V., Kritchevski E., Pillet C.-A.: Spectral theory of the Wigner-Weisskopf atom. Nordfjordeid Lecture Notes. 14. Jakˇsi´c V., Last Y.: A new proof of Poltoratskii’s theorem. J. Func. Anal. 215 (2004), 103. 15. Jakˇsi´c V., Last Y.: Spectral structure of Anderson type Hamiltonians. Invent. Math. 141 (2000), 561.

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Vojkan Jakˇsi´c 16. Jakˇsi´c V., Last Y.: Simplicity of singular spectrum in Anderson type Hamiltonians. Submitted. 17. Kato T.: Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976. 18. Katznelson A.: An Introduction to Harmonic Analysis, Dover, New York, 1976. 19. Koosis P.: Introduction to Hp Spaces, second edition, Cambridge University Press, New York, 1998. 20. Lyons R.: Seventy years of Rajchman measures. J. Fourier Anal. Appl. Publ. Res., Kahane Special Issue (1995), 363. 21. Poltoratskii A.G.: The boundary behavior of pseudocontinuable functions. St. Petersburgh Math. J. 5 (1994), 389. 22. Poltoratskii A.G.: On the distribution of the boundary values of Cauchy integrals. Proc. Amer. Math. Soc. 124 (1996), 2455. 23. Reed M., Simon B.: Methods of Modern Mathematical Physics, I. Functional Analysis, Academic Press, London, 1980. 24. Reed M., Simon B.: Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness, Academic Press, London, 1975. 25. Reed M., Simon B.: Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, London, 1978. 26. Reed M., Simon B.: Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, London, 1978. 27. Rudin W.: Real and Complex Analysis, 3rd edition, McGraw-Hill, Boston, 1987. 28. Simon B.: Lp norms of the Borel transforms and the decomposition of measures. Proc. Amer. Math. Soc. 123 (1995), 3749 29. Simon B.: Spectral analysis of rank one perturbations and applications. CRM Lecture Notes Vol. 8, pp. 109-149, AMS, Providence, RI, 1995. 30. Simon B.: Spectral averaging and the Krein spectral shift. Proc. Amer. Math. Soc. 126 (1998), 1409. 31. Simon B., Wolff T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Communications in Pure and Applied Mathematics 49 (1986), 75. 32. Vock E., Hunziker W.: Stability of Schr¨odinger eigenvalue problems. Commun. Math. Phys. 83 (1982), 281.

Index of Volume I

∗-algebra, 72 morphism, 77 C ∗ -algebra, 71 morphism, 77 C0 semigroup, 35 W ∗ -algebra, 139 ∗-derivation, 132 µ-Liouvillean, 143 Adjoint, 6 Algebra ∗, 72 C ∗ , 71 Banach, 72 von Neumann, 88 Analytic vector, 32, 136, 202 Approximate identity, 84 Aronszajn-Donoghue theorem, 297 Asymptotic abelianness, 229 Baker-Campbell-Hausdorff formula, 193 Banach algebra, 72 Birkhoff ergodic theorem, 125 Bogoliubov transformation, 200 Boltzmann’s constant, 57 Boole’s equality, 302 Borel transform, 249, 261 Bose gas, 140, 145, 177 Boson, 53, 186 Canonical anti-commutation relations, 190 Canonical commutation relations, 50, 190, 192 Canonical transformation, 43

Cantor set, 310 CAR, CCR algebra CARF (h), 195 CCRF (h), 195 quasi-local, 198 simplicity, 199, 200 uniqueness, 199, 200 CAR-algebra, 134, 172 Cayley transform, 8 CCR-algebra, 140, 145, 177 Center, 118 Central support, 118 Chaos, 186 Character, 83 Chemical potential, 58 Commutant, 89 Condensate, 217 Configuration space, 42 Conjugation, 10 Contraction semigroup, 37 Critical density, 227 Cyclic subspace, 22, 295 vector, 22, 195, 273 Deficiency indices, 8 Density matrix, 55, 114, 290 Dynamical system C ∗ , 132 W ∗ , 139 classical, 124 ergodic, 125, 156 mixing, 127, 156

314

Index of Volume I

quantum, 142 Ensemble canonical, 60 grand canonical, 63 microcanonical, 57 Entropy Boltzmann, 57 Enveloping von Neumann algebra, 119 Essential support, 281 Evolution group, 29 Exponential law, 203 Factor, 118 Faithful representation, 80 Fermi gas, 134, 172 Fermion, 53, 186 Finite particle subspace, 192 Finite quantum system, 133 Fock space, 186 Folium, 119 Free energy, 61 Functional calculus, 16, 25, 281 G.N.S. representation, 82 Hahn decomposition theorem, 240 Hamiltonian, 290 Hamiltonian system, 43 Hardy class, 258 Harmonic oscillator, 50, 205 Heisenberg picture, 51 Heisenberg uncertainty principle, 49, 290 Helffer-Sj¨ostrand formula, 17 Hille-Yosida theorem, 37 Ideal left, 84 right, 84 two-sided, 84 Ideal gas, 185 Indistinguishable, 186 Individual ergodic theorem, 125 Infinitesimal generator, 35 Internal energy, 58 Invariant subspace, 22, 272 Invertible, 73 Isometric element, 75

Jensen’s formula, 259 Kaplansky density theorem, 111 Kato-Rellich theorem, 285 Kato-Rosenblum theorem, 287 Koopman ergodicity criterion, 129 Koopman lemma, 128 Koopman mixing criterion, 129 Koopman operator, 128 Lebesgue-Radon-Nikodym theorem, 240 Legendre transform, 62 Liouville equation, 43 Liouville’s theorem, 43 Liouvillean, 128, 143, 150, 161, 168 Lummer Phillips theorem, 38 Mean ergodic theorem, 32, 128 Measure absolutely continuous, 240 complex, 239 regular Borel, 238 signed, 239 space, 238 spectral, 274, 280, 295 support, 238 Measurement, 48 simultaneous, 49 Measures equivalent, 280 mutually singular, 240 Modular conjugation, 96 operator, 96 Morphism ∗-algebra, 77 C ∗ -algebra, 77 Nelson’s analytic vector theorem, 32 Norm resolvent convergence, 27 Normal element, 75 Normal form, 143 Observable, 42, 46, 123, 290 Operator (anti-)symmetrization, 187 closable, 5, 268 closed, 2, 268 core, 31, 268 creation, annihilation, 50, 188 dissipative, 37

Index of Volume I domain, 2 essentially self-adjoint, 7 extension, 2 field, 192 graph, 3, 268 linear, 2 multiplication, 14, 273 number, 186 positive, 271 relatively bounded, 12, 285 Schr¨odinger, 47 self-adjoint, 7 symmetric, 5 trace class, 286 Weyl, 193 Partition function, 61, 64 Pauli’s principle, 54, 191 Perturbation theory rank one, 295 Phase space, 42 Planck law, 226 Poisson bracket, 44 Poisson representation, 256 Poisson transform, 249 Poltoratskii’s theorem, 262, 304 Positive element, 78 linear form, 80 Predual, 90 Pressure, 58 Quantum dynamical system, 142 Quasi-analytic extension, 16 RAGE theorem, 284, 290 Reduced Liouvillean, 161 Representation, 80 Q-space (CCR), 207 Araki-Woods, 224 faithful, 80 Fock, 203 GNS, 120 GNS (ground state of Bose gas), 221 Quasi-equivalent, 206 regular (of CCR), 201 ¨ Schrdinger, 204 Resolvent, 3 first identity, 4, 268

norm convergence, 27 set, 3, 268 strong convergence, 194 Resolvent set, 73 Return to equilibrium, 127, 230 Riemann-Lebesgue lemma, 241 Riesz representation theorem, 240 Schr¨odinger picture, 51 Sector, 186 Self-adjoint element, 75 Simon-Wolff theorems, 300 Spatial automorphism, 133 Spectral averaging, 299 Spectral radius, 74 Spectral theorem, 23, 274, 298 Spectrum, 3, 73, 83, 268 absolutely continuous, 278 continuous, 278 essential, 284 point, 268 pure point, 278 singular, 278 singular continuous, 278 stability, 301 Spin, 53 Standard form, 148 Standard Liouvillean, 150, 168 Standard unitary, 149 State, 81, 198 absolutely continuous, 155 centrally faithful, 118 coherent, 52 disjoint, 119 equilibrium, 124 extremal, 159 factor, 231 faithful, 110, 117 gauge invariant, 173, 212 generating functional, 214 Gibbs, 210 ground (Bose gas), 220 invariant, 141 KMS, 169, 210 local perturbation, 228 mixed, 54 mixing, 232 normal, 92, 112 orthogonal, 119

315

316

Index of Volume I

pure, 46, 56 quasi-equivalent, 119 quasi-free, 147, 173, 212 relatively normal, 119, 198 tracial, 96 Stone’s formula, 282 Stone’s theorem, 30 Stone-von Neumann uniqueness theorem, 205 Strong resolvent convergence, 194 Support, 117 Temperature, 58, 61 Thermodynamic first law, 58 limit, 184, 197 second law, 58 Topology σ-strong, 111 σ-weak, 87, 111 strong, 86

uniform, 86 weak, 86 weak- , 139 Trotter product formula, 33 Unit, 72 approximate, 84 Unitary element, 75 Vacuum, 186 Von Neumann density theorem, 111 Von Neumann ergodic theorem, 33, 128 Wave operators, 286 complete, 286 Weyl (CCR) relations, 193 Weyl commutation relations, 140 Weyl quantization, 47 Weyl’s criterion, 283 Weyl’s theorem, 286 Wiener theorem, 241, 255

Information about the other two volumes

Contents of Volume II

Ergodic Properties of Markov Processes Luc Rey-Bellet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Markov Processes and Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Transition probabilities and generators . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stationary Markov processes and Ergodic Theory . . . . . . . . . . . . . . . . 4 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Control Theory and Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Hypoellipticity and Strong-Feller Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Liapunov Functions and Ergodic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 4 7 12 14 24 26 28 39

Open Classical Systems Luc Rey-Bellet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Derivation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 How to make a heat reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Markovian Gaussian stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 2.3 How to make a Markovian reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ergodic properties: the chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Strong Feller Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Liapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Heat Flow and Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Positivity of entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fluctuation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Kubo Formula and Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 44 44 48 50 52 56 57 58 66 69 71 75 77

Contents of Volume II

319

Quantum Noises St´ephane Attal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2 Discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.1 Repeated quantum interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.2 The Toy Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.3 Higher multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3 Itˆo calculus on Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.1 The continuous version of the spin chain: heuristics . . . . . . . . . . . . . . 93 3.2 The Guichardet space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3 Abstract Itˆo calculus on Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.4 Probabilistic interpretations of Fock space . . . . . . . . . . . . . . . . . . . . . . 105 4 Quantum stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.1 An heuristic approach to quantum noise . . . . . . . . . . . . . . . . . . . . . . . . 110 4.2 Quantum stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.3 Back to probabilistic interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5 The algebra of regular quantum semimartingales . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Everywhere defined quantum stochastic integrals . . . . . . . . . . . . . . . . 124 5.2 The algebra of regular quantum semimartingales . . . . . . . . . . . . . . . . . 127 6 Approximation by the toy Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.1 Embedding the toy Fock space into the Fock space . . . . . . . . . . . . . . . 130 6.2 Projections on the toy Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.3 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.4 Probabilistic interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.5 The Itˆo tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7 Back to repeated interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.1 Unitary dilations of completely positive semigroups . . . . . . . . . . . . . . 140 7.2 Convergence to Quantum Stochastic Differential Equations . . . . . . . . 142 8 Bibliographical comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Complete Positivity and the Markov structure of Open Quantum Systems Rolando Rebolledo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 1 Introduction: a preview of open systems in Classical Mechanics . . . . . . . . . 149 1.1 Introducing probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 1.2 An algebraic view on Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 2 Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3 Completely bounded maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4 Dilations of CP and CB maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5 Quantum Dynamical Semigroups and Markov Flows . . . . . . . . . . . . . . . . . . 168 6 Dilations of quantum Markov semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.1 A view on classical dilations of QMS . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.2 Towards quantum dilations of QMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

320

Contents of Volume II

Quantum Stochastic Differential Equations and Dilation of Completely Positive Semigroups Franco Fagnola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 2 Fock space notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 3 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4 Unitary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5 Emergence of H-P equations in physical applications . . . . . . . . . . . . . . . . . . 193 6 Cocycle property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8 The left equation: unbounded Gα β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9 Dilation of quantum Markov semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 10 The left equation with unbounded Gα β : isometry . . . . . . . . . . . . . . . . . . . . . . 213 11 The right equation with unbounded Fβα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Index of Volume-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Information about the other two volumes Contents of Volume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Volume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents of Volume III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Volume III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

224 228 232 236

Index of Volume II

Adapted domain, 114 Algebra Banach, 156 von Neumann, 157 Algebraic probability space, 154 Banach algebra, 156 Brownian interpretation, 107 Brownian motion, 13 canonical, 14 Chaotic expansion, 106 representation property, 106 space, 106 Classical probabilistic dilations, 174 Coherent vector, 96 Completely bounded map, 162 Completely positive map, 158 Conditional expectation, 173 Conditionally CP map, 170 Control, 24 Dilation, 208 Dilations of QDS, 173 Dynkin’s formula, 21

Fock space toy, 84 multiplicity n, 90 Gaussian process, 13 Generator, 7 Gibbs measure, 45 H¨ormander condition, 27 Independent increments, 13 Initial distribution, 5 Integral representation, 85 Itˆo integrable process, 99 integral, 15, 99 process, 16 Lyapunov function, 21 Markov process, 4 Martingale normal, 105 Measure preserving, 8 Mild solution, 217 Mixing, 9 Modification, 13

Elliptic operator, 27 Ergodic, 8

Normal martingale, 105

Feller semigroup strong, 7 weak, 7 First fundamental formula, 186

Obtuse system, 90 Operator process, 185 Operator system, 156

322

Index of Volume II

Poisson interpretation, 107 Predictable representation property, 105 Probabilistic interpretation, 87, 107 p-, 88 Probability space algebraic, 154 Process, 2 distribution, 3 Gaussian, 13 Itˆo integrable, 99 Ito, 16 Markov, 4 strong, 20 modification, 13 operator, 185 adapted, 111 path, 3 stationary, 7 Product p-, 89 Poisson, 108 Wiener, 108 Quantum dynamical semigroup, 170, 208 minimal, 210 Quantum Markov semigroup, 170, 208 Quantum noises, 111 Quantum probabilistic dilations, 174, 180 Regular quantum semimartingales, 128

Sesqui-symmetric tensor, 91 Spectral function, 48 State normal, 155 Stationary increments, 13 Stinespring representation, 164 Stochastic integral, 15 quantum, 115 Stochastically integrable, 185 Stopping time, 21 Strong Markov process, 20 Structure equation, 107 Tensor sesqui-symmetric, 91 Topology uniform, 156 Total variation norm, 12 Totalizing set, 205 Toy Fock space, 84 multiplicity n, 90 Transition probability, 5 Uniform topology, 156 Uniformly continuous QMS, 170 Vacuum, 96 von Neumann algebra, 157

Contents of Volume III

Topics in Non-Equilibrium Quantum Statistical Mechanics Walter Aschbacher, Vojkan Jakˇsi´c, Yan Pautrat, and Claude-Alain Pillet . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Conceptual framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mathematical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Non-equilibrium steady states (NESS) and entropy production . . . . . 3.3 Structural properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 C ∗ -scattering and NESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 C ∗ -scattering for open quantum systems . . . . . . . . . . . . . . . . . . . . . . . 4.3 The first and second law of thermodynamics . . . . . . . . . . . . . . . . . . . . 4.4 Linear response theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fermi Golden Rule (FGR) thermodynamics . . . . . . . . . . . . . . . . . . . . . 5 Free Fermi gas reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The simple electronic black-box (SEBB) model . . . . . . . . . . . . . . . . . . . . . . 6.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The equivalent free Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Thermodynamics of the SEBB model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Non-equilibrium steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Hilbert-Schmidt condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The heat and charge fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Entropy production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Equilibrium correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Onsager relations. Kubo formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 5 5 8 10 11 14 14 15 17 18 22 26 26 30 34 34 36 37 40 43 43 44 45 46 47 49

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8

FGR thermodynamics of the SEBB model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Historical digression—Einstein’s derivation of the Planck law . . . . . 8.3 FGR fluxes, entropy production and Kubo formulas . . . . . . . . . . . . . . 8.4 From microscopic to FGR thermodynamics . . . . . . . . . . . . . . . . . . . . . 9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Structural theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Hilbert-Schmidt condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi Golden Rule and Open Quantum Systems Jan Derezi´nski and Rafał Fr¨uboes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fermi Golden Rule and Level Shift Operator in an abstract setting . . 1.2 Applications of the Fermi Golden Rule to open quantum systems . . . 2 Fermi Golden Rule in an abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Level Shift Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 LSO for C0∗ -dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 LSO for W ∗ -dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 LSO in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The choice of the projection P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Three kinds of the Fermi Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . 3 Weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Stationary and time-dependent weak coupling limit . . . . . . . . . . . . . . 3.2 Proof of the stationary weak coupling limit . . . . . . . . . . . . . . . . . . . . . 3.3 Spectral averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Second order asymptotics of evolution with the first order term . . . . . 3.5 Proof of time dependent weak coupling limit . . . . . . . . . . . . . . . . . . . . 3.6 Proof of the coincidence of Mst and Mdyn with the LSO . . . . . . . . . . 4 Completely positive semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Completely positive maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stinespring representation of a completely positive map . . . . . . . . . . . 4.3 Completely positive semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Standard Detailed Balance Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Detailed Balance Condition in the sense of Alicki-Frigerio-GoriniKossakowski-Verri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Small quantum system interacting with reservoir . . . . . . . . . . . . . . . . . . . . . 5.1 W ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Algebraic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Semistandard representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Standard representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Two applications of the Fermi Golden Rule to open quantum systems . . . . 6.1 LSO for the reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 LSO for the Liouvillean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 50 53 54 56 58 58 60 63 67 68 68 69 71 71 72 73 74 74 75 75 77 77 80 83 85 87 88 88 89 89 90 91 93 93 94 95 95 96 97 97 99

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Relationship between the Davies generator and the LSO for the Liouvillean in thermal case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Explicit formula for the Davies generator . . . . . . . . . . . . . . . . . . . . . . . 6.5 Explicit formulas for LSO for the Liouvillean . . . . . . . . . . . . . . . . . . . 6.6 Identities using the fibered representation . . . . . . . . . . . . . . . . . . . . . . . 7 Fermi Golden Rule for a composite reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 LSO for a sum of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Multiple reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 LSO for the reduced dynamics in the case of a composite reservoir . 7.4 LSO for the Liovillean in the case of a composite reservoir . . . . . . . . A Appendix – one-parameter semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.3

100 103 104 106 108 108 109 110 111 112 115

Decoherence as Irreversible Dynamical Process in Open Quantum Systems Philippe Blanchard and Robert Olkiewicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Notes on the Qualitative Behaviour of Quantum Markov Semigroups Franco Fagnola and Rolando Rebolledo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 2 Ergodic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3 The minimal quantum dynamical semigroup . . . . . . . . . . . . . . . . . . . . . . . . . 167 4 The existence of Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.1 A general result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.2 Conditions on the generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.4 A multimode Dicke laser model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.5 A quantum model of absorption and stimulated emission . . . . . . . . . . 181 4.6 The Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5 Faithful Stationary States and Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.1 The support of an invariant state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.2 Subharmonic projections. The case M = L(h) . . . . . . . . . . . . . . . . . . 185 5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6 The convergence towards the equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7 Recurrence and Transience of Quantum Markov Semigroups . . . . . . . . . . . 193 7.1 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.2 Defining recurrence and transience . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.3 The behavior of a d-harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . 200 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

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Continual Measurements in Quantum Mechanics and Quantum Stochastic Calculus Alberto Barchielli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 1.1 Three approaches to continual measurements . . . . . . . . . . . . . . . . . . . . 207 1.2 Quantum stochastic calculus and quantum optics . . . . . . . . . . . . . . . . 207 1.3 Some notations: operator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 2 Unitary evolution and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 2.1 Quantum stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 2.2 The unitary system–field evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 2.3 The system–field state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 2.4 The reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 2.5 Physical basis of the use of QSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 3 Continual measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 3.1 Indirect measurements on SH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 3.2 Characteristic functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 3.3 The reduced description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.4 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 3.5 Optical heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 3.6 Physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 4 A three–level atom and the shelving effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 4.1 The atom–field dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 4.2 The detection process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 4.3 Bright and dark periods: the V-configuration . . . . . . . . . . . . . . . . . . . . 263 4.4 Bright and dark periods: the Λ-configuration . . . . . . . . . . . . . . . . . . . . 266 5 A two–level atom and the spectrum of the fluorescence light . . . . . . . . . . . . 268 5.1 The dynamical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 5.2 The master equation and the equilibrium state . . . . . . . . . . . . . . . . . . . 273 5.3 The detection scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 5.4 The fluorescence spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Index of Volume III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Information about the other two volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents of Volume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Volume I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents of Volume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Volume II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

296 297 301 305 308

Index of Volume III

T fixed points set, 166 Λ configuration, 258 ω-continuous, 123 σ-finite von Neumann algebra, 163 σ-weakly continuous groups, 113 Absorption, 271 Adapted process, 213 regular, 213 stochastically integrable, 214 unitary, 218 Adjoint pair, 216 Affinities, 19, 49, 56 Annihilation, creation and conservation processes, 212 Antibunching, 263 Araki’s perturbation theory, 11 Bose fields, 212 Broad–band approximation, 228 CAR algebra, 26 even, 28 CCR, 148, 210 Central limit theorem, 20 Characteristic functional, 234 operator, 234 Classes of bounded elements left, 127 right, 127 Classical quantum states, 126 Cocycle property, 219 Coherent vectors, 209

Completely positive map, 89 semigroup, 90 Conditional expectation, 122, 166 ψ-compatible, 122 Continual measurements, 229 Correlation function, 20, 47 Counting process, 263 quanta, 229 Current charge, 36, 45, 55 heat, 17, 36, 45, 55 output, 247 Dark state, 266 Davies generator, 97 Decoherence –induced spin algebra, 142 environmental, 119 time, 125 Demixture, 224 Density operator, 27 Detailed Balance Condition, 92 AFGKV, 93 Detuning parameter, 260 shifted, 275 Direct detection, 246 Dynamical system C∗, 5 W ∗ , 94 weakly asymptotically Abelian, 11

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Index of Volume III

Effective dipole operator, 279 Emission, 271 Entropy production, 9, 18, 19, 24, 39, 44, 46, 55 relative, 9 Ergodic generator, 114 globally, 114 Exclusive probability densities, 250 Experimental resolution , 139 Exponential domain, 209 vectors, 209 Exponential law, 29, 37 Fermi algebra, 26 Fermi Golden Rule, 54, 76 analytic, 76 dynamical, 76 spectral, 76 Fermi-Dirac distribution, 27 Field quadratures, 212, 230 Fluctuation algebra, 21 Fluorescence spectrum, 279 Flux charge, 36, 45, 55 heat, 17, 36, 45, 55 Fock space, 26, 209 vacuum, 209 Form-potential, 194 Friedrichs model, 40 Gauge group, 26 Gorini-Kossakowski-Sudershan-Lindblad generator, 140 Harmonic operator, 184 Heterodyne detection, 251 balanced, 253 Indirect measurement, 229 Infinitely divisible law, 238 Input fields, 231 Instrument, 243 Interaction picture, 221 Isometric-sweeping decomposition, 133 Ito table, 214 Jacobs-deLeeuw-Glicksberg splitting, 136 Jordan-Wigner transformation, 31

Junction, 15, 35 Kinetic coefficients, 19, 25 Kubo formula, 20, 25, 50, 56 Laser intensity, 275 Level Shift Operator, 73 Lindblad generator, 90 Linear response, 18, 25 Liouville operator, 226 Liouvillean -Lp , 8 -ω, 8, 28 semi–, 96 perturbation of, 11 standard, 8 Localization properties, 236 Møller morphism, 11, 15 Mandel Q-parameter, 248 Markov map, 89 Master equation, 226 Modular conjugation, 28 group, 39 operator, 28 Narrow topology, 165 NESS, 8, 43 Nominal carrier frequencies, 227 Onsager reciprocity relations, 20, 25, 50, 56 Open system, 14 Operator number, 26 Output characteristic operator, 239 Output fields, 231 Pauli matrices, 30 Pauli’s principle, 29 Perturbed convolution semigroup, 151 of promeasures, 151 Poisson, 152 Phase diffusion model, 224, 272 Photoelectron counter, 247 Photon scattering, 271 Picture Heisenberg, 89 Schr¨odinger, 89 Standard, 89

Index of Volume III Pointer states, 123 continuous, 139 Poissonian statistics sub–, 249 super–, 249 Polarization, 227 Potential operator, 194 Power spectrum, 278 Predual space, 163 Promeasure, 149 Fourier transform of, 150 Quantum Brownian motion, 155 Quantum dynamical semigroup, 91, 122 on CCR algebras, 153 minimal, 167 Quantum Markovian semigroup, 22, 52 irreducible, 185 Quantum stochastic equation, 225 Quasi–monochromatic fields, 227 Rabi frequencies, 260 Reduced characteristic operator, 240 dynamics, 243 evolution, 122 Markovian dynamics, 122 Representation Araki-Wyss, 28 GNS, 5, 27 semistandard, 95 standard, 94 universal, 6 Reservoir, 14 Response function, 247 Rotating wave approximation, 227 Scattering matrix, 41 Semifinite weight, 121 Semigroup C0 -, 112 C0∗ -, 113 one-parameter, 112 recurrent, 199 transient, 199 Sesquilinear form, 164 Shelving effect, 256 electron, 257 Shot noise, 278 Singular coupling limit, 140 Spectral Averaging, 83 Spin system, 31, 138 State, 5

chaotic, 14 decomposition, 6 ergodic, 5, 28 factor, 8 factor or primary, 27 faithful, 163 invariant, 5, 27, 163 KMS, 8, 19, 27, 28 mixing, 5, 28 modular, 8, 27 non-equilibrium steady, 8, 43 normal, 163 primary, 8 quasi-free gauge-invariant, 27, 31, 35 reference, 3, 9 relatively normal, 5 time reversal invariant, 15 States classical, 224 disjoint, 6 mutually singular, 6 orthogonal, 6 quantum, 224 quasi-equivalent, 7, 27, 44 unitarily equivalent, 7, 27, 44 Subharmonic operator, 184 Superharmonic operator, 184 Test functions, 234 Thermodynamic FGR, 24, 56 first law, 17, 24, 37 second law, 18, 24 Tightness, 165 Time reversal, 15, 42 TRI, 15 Two-positive operator, 127 Unitary decomposition, 130 V configuration, 257 Van Hove limit, 22, 50 Von Neumann algebra enveloping, 5 universal enveloping, 6 Wave operator, 41 Weak Coupling Limit, 22, 50, 77 dynamical, 76 stationary, 76 Weyl operator, 210 Wigner-Weisskopf atom, 40

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