Texts and
Monographs
Series Editors: R.
Balian, Gif-sur-Yvette, France
W.
Beiglböck, Heidelberg, Germany
H.
Grosse, Wien, Austria E. H. Lieb, Princeton, NJ, USA N.
H.
Reshetikhin, Berkeley, CA, USA Spohn, München, Germany
W.
Thirring, Wien, Austria
in
Physics
Springer Berlin
Heidelberg New York
Hong Kong London Milan Paris
Tokyo
Physics and Astronomy
l ONLINE LIBRARY
http://www.springer.de/phys/
Ola Bratteli Derek W. Robinson
Operator Algebras and Quantum Statistical Mechanics 2 Equilibrium States. Models in Quantum Second Edition
Springer
Statistical Mechanics
Professor Ola Bratteli Universitetet i Oslo Matematisk Institutt Moltke Moes vei 3 1
0316 Oslo, Norway e-mail:
[email protected] Home page:
http://www.math.uio.no/~bratteli/
Professor Derek W. Robinson Australian National
University
School of Mathematical Sciences ACT 0200 Canberra, Australia e-mail:
[email protected] Home page:
http://wwwmaths.anu.edu.au/~derek/
Cataloging-in-Publication
Data
applied
for
published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic datais available in the Internet at . Bibliographie
information
Second Edition 1997. Second
Printing
2002
ISSN 0172-5998 ISBN 3-540-61443-5 2nd Edition ISBN 3-540-1038 1-3
Ist Edition
Springer- Verlag
Springer- Verlag
Berlin
Berlin
Heidelberg
Heidelberg
New York
New York
This work is
subject to Copyright. All rights are reserved, whether the whole or part of the material is conceraed, 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-Vedag. Violations are liable for prosecution under the German Copyright Law. Springer- Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business
Media GmbH
http://www.springer.de ©
Springer-Veriag Berlin Heidelberg 1981, Germany
1997
Printed in The
use
of
general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the specific Statement, that such names are exempt from the relevant probreak tective laws and regulations and free for general use.
absence of therefore Cover
a
design: dexign
Printed
on
&
production GmbH, Heidelberg
acid-free paper
SPIN 10885999
55/3141/ba
543210
To
Trygve Bratteli, Samuel Robinson, and Harald ROSS
Preface to the Second Edition
Fifteen years have and much has ments
passed since completion of the first edition of this book happened. Any attempt to do justice to the new develop-
would necessitate at least
edition of the current
one
new
volume rather than
a
second
Fortunately
other authors have taken up the challenge of describing these discoveries and our bibliography includes references to a variety of new books that have appeared or are about to one.
appear. We consequently decided to keep the format of this book äs a basic reference for the operator algebraic approach to quantum statistical mechanics and concentrated on correcting, improving, and the
updating
material of the first edition. This in itself has not been easy and changes occur throughout the text. The major changes are a corrected presentation of Bose-Einstein condensation in Theorem 5.2.30, insertion of a general result on the absence of symmetry breaking in Theorem 5. 3. 3 3 A, and an extended
in
The discussion of
in Sects. 6.2.6 and
description of the dynamics of the A^-Fmodel phase transitions in specific models, 6.2.7, has been expanded with the focus shifted from
model to
genuine quantum situations
such
äs
the
Example
6.2.14.
the classical
Heisenberg
Ising
and X-Y
models. In addition the Notes and Remarks to various subsections have been considerably augmented. Since
our
interest in the
subject of equilibrium states and models considerably in the last fifteen years
statistical mechanics has waned
of it
VIII
Preface to the Second Edition
impossible to prepare this second edition without the and encouragement of many of our friends and colleagues. We are Support indebted to Charles Batty, Michiel van den Berg, Tom ter Eist, particularly
would have been
Jürg Fröhlich, Taku Matsui, Andre Verbeure, helpful advice, and we apollatter. We the often for are especially grateful to Aernout ignoring ogize for Werner Reinhard and Enter van counselling us on recent developments and giving detailed suggestions for revisions.
Dai Evans, Mark Fannes,
and Marinus Winnink for information and
Oslo and Canberra 1996
Ola Bratteh Derek W. Robinson
Contents Volume 2
States in
Quantum Statistical Mechanics
5.1. Introduction
5.2. Continuous
3
Quantum Systems.
I
5.2.1. The CAR and CCR Relations
5.2.2. The CAR and CCR 5.2.3.
States and
Algebras Representations
6 6 15
23
5.2.4. The Ideal Fermi Gas
45
5.2.5. The Ideal Böse Gas
57
5.3. KMS-States
76
5.3.1. The KMS Condition
5.4.
l
76
5.3.2. The Set of KMS States
1 12
5.3.3. The Set of Ground States
131
Stability 5.4.1. 5.4.2.
and
Equilibrium
Stability Stability
144
of KMS States
144
and the KMS Condition
176
X
Contents Volume 2
5.4.3.
Gauge Groups and Systems
the Chemical Potential
197
5.4.4. Passive
211
Notes and Remarks
217
Models of
235
Quantum Statistical Mechanics
6.1. Introduction
237
6.2
Quantum Spin Systems
239
6.2.1. Kinematical and
239
Dynamical Descriptions Equilibrium The Maximum Entropy Principle Translationally Invariant States Uniqueness of KMS States Nonuniqueness of KMS States
6.2.2. The Gibbs Condition for
261
6.2.3.
266
6.2.4. 6.2.5. 6.2.6.
286 306 317
6.2.7. Ground States
6.3. Continuous
338
Quantum Systems. II
353
6.3.1. The Local Hamiltonians
355
6.3.2. The Wiener 6.3.3. The 6.3.4. The
Integral Thermodynamic Limit. Thermodynamic Limit.
366 I. The Reduced
Density
Matrices
II. States and Green's Functions
381 395
6.4. Conclusion
422
Notes and Remarks
424
References
463
Books and
Monographs
465
Articles
468
List of
487
Subject
Symbols Index
499
Contents Volume l
Introduction
l
Notes and Remarks
C*-Algebras 2.1.
and
16
von
Neumann
Algebras
C*-Algebras
19
2.1.1. Basic Definitions and Structure
19
2.2. Functional and 2.2.1.
Spectral Analysis
2.2.3.
Approximate
Representations 2.3.1.
25
Resolvents, Spectra, and Spectral Radius Identities and
32
Quotient Algebras
and States
Representations
39
42 42
2.3.2. States 2.3.3. Construction of
25
'
2.2.2. Positive Elements
2.3.
17
48
Representations
54
XII
Contents Volume l
2.3.4. Existence of
Representations
2.3.5. Commutative
2.4.
von
Neumann
2.4.1.
58
C*-Algebras
6l
Algebras
Topologies
on
65
^()
65
2.4.2. Definition and of
von
Elementary Properties Neumann Algebras
75
2.4.4.
79
Quasi-Equivalence
of
Representation
2.5. Tomita-Takesaki Modular of
von
2.5.1.
Neumann cr-Finite
Theory
and Standard Forms
Algebras Neumann
2.5.3.
Algebras Group Integration and Analytic Elements for One-Parameter Groups of Isometries on Banach Spaces
2.5.4.
Self-Dual Cones and Standard Forms
von
2.5.2. The Modular
2.6.
7l
2.4.3. Normal States and the Predual
83 84 86
97
102
Quasi-Local Algebras
118
2.6.1. Cluster
1 18
2.6.2.
129
2.6.3.
Properties Topological Properties Algebraic Properties
2.7. Miscellaneous Results and Structure 2.7.1.
Dynamical Systems and Crossed Products Operator Algebras Weights on Operator Algebras; Self-Dual Cones of General von Neumann Algebras; Duality and Classification of Factors; Classification of C -Algebras
2.7.2. Tensor Products of 2.7.3.
Notes and Remarks
Groups, Semigroups,
3.1. Banacb 3.1.1.
3.1.2.
3.1.3. 3.1.4.
3.1.5.
136 136 142
145
152
and Generators
Space Theory
Uniform
133
Continuity Strong, Weak, and Weak* Continuity Convergence Properties Perturbation Theory Approximation Theory
157
159 161
163 183
189 198
Contents Volume l
3.2.
XIII
205
Algebraic Theory 3.2.1. Positive Linear
205
3.2.2. General
228
3.2.3.
3.2.4. 3.2.5. 3.2.6.
Maps and Jordan Morphisms Properties of Derivations Spectral Theory and Bounded Derivations Derivations and Automorphism Groups Spatial Derivations and Invariant States Approximation Theory for Automorphism Groups
244
259 263 285
Notes and Remarks
298
Decomposition Theory
309
4.1. General
311
Theory
311
4.1.1. Introduction 4.1.2.
4.1.3.
4.2.
315
Barycentric Decompositions Orthogonal Measures
333
4.1.4. Borel Structure of States
344
Extremal, Central, and Subcentral Decompositions
353
4.2.1. Extremal
353
4.2.2.
Decompositions Decompositions
362
Central and Subcentral
4.3. Invariant States 4.3.1.
4.3.2. 4.3.3
4.3.4
4.4.
367 367
Ergodic Decompositions Ergodic States Locally Compact Abelian Groups Broken Symmetry
386 400
416
432
Spatial Decomposition
433
4.4.1. General 4.4.2.
Theory Spatial Decomposition
and
Decomposition
of States
442
Notes and Remarks
451
References
459
Books and
Monographs
461
Articles
464
List of
481
Subject
Symbols Index
487
States in
Quantum Statistical Mechanics
5.1. Introduction
In this
chapter, and the following one, we examine various applications of C*algebras and their states to statistical mechanics. Principally we analyze the structural properties of the equilibrium states of quantum Systems consisting of a large number of particles. In Chapter l we argued that this leads to the study of states of infinite-particle Systems äs an initial approximation. There are two approaches to this study which are to a large extent complementary. The first approach begins with the specific description of finite Systems and their equilibrium states provided by quantum statistical mechanics. One then rephrases this description in an algebraic language which identifies the equili brium states äs states over a quasi-local C*-algebra generated by subalgebras corresponding to the observables of spatial Subsystems. Finally, one attempts to calculate an approximation of these states by taking their limit äs the volume of the System tends to infinity, the so-called thermodynamic limit. The infinitevolume equilibrium states obtained in this manner provide the data for the calculation of bulk properties of the matter under consideration äs functions of the thermodynamic variables. By this we mean properties such äs the particle density, or specific heat, äs functions of the temperature and chemical potential, etc. In fact, the infinite-volume data provides a much more detailed, even microscopic, description of the equilibrium phenomena although one is only generally interested in the bulk properties and their fluctuations. Examination of the thermodynamic limit also provides a test of the scope of the usual statistical mechanical formalism. If this formalism is rieh enough to describe phase transitions, then at certain critical values of the thermodynamic parameters there should be a multiplicity of infinite-volume limit states arising from slight variations of the external interactions or boundary conditions. These states would correspond to various phases and mixtures of these phases. In such a Situation it should be possible to arrange the limits such that phase Separation takes place and then the equilibrium states would also provide information concerning interface phenomena such äs surface tension. The second approach to algebraic statistical mechanics avoids discussion of the thermodynamic limit and attempts to characterize and classify the equili brium states of the infinite System äs states over an appropriate C*-algebra. The elements of the C*-algebra represent kinematic observables, i.e., observables at a given time, and the states describe the instantaneous states of the System. For a complete physical description it is necessary to specify the dynamical law
4
States in
Quantum Statistical Mechanics
governing the change with time of the observables, or the states, and the equilibrium states are determined by their properties with respect to this dynamics. The general nature of the dynamical law can be inferred from the usual quantum-mechanical formalism and it appears that there are various possibiUties. Recall that for finite quantum Systems the dynamics is given parameter group of *-automorphisms of the algebra of observables,
A^Tt(Ä)^e^^^Ae-^^^
by
a one-
,
where H is the
selfadjoint Hamiltonian operator of the System. Thus it appears dynamics of the infinite System should be determined by a continuous one-parameter group of *-automorphisms T of the C*-algebra of observables. This type of dynamics is certainly the simplest possible and it occurs in various specific models, e.g., the noninteracting Fermi gas, some of which we examine in the sequel. Nevertheless, it is not the general Situation. The difRculty is that a group of this kind automatically defines a continuous development of every state of the System. But this is not to be expected for general infinite Systems in which compHcated phenomena involving the local accumulation of an infinite number of particles and energy can occur for natural that the
certain initial states. Thus it is necessary to examine weaker forms of evolution. For example, one could assume the dynamics to be specified äs a group of of the von Neumann algebras corresponding to a subclass of C*-algebra. Alternatively one could adopt an infinitesimal description and assume that the evolution is determined by a derivation which generates an automorphism group only in certain representations. Fach of these possible structures could in principle be verified in a particular model by a thermodynamic limiting process and each such structure provides a framework for characterizing equilibrium phenomena. To understand the type of characterization which is possible it is useful to refer to the finite-volume description of equilibrium. There are various possible descriptions of equilibrium states, which all stem from the early work of Boltzmann and Gibbs on classical statistical mechanics, and which differ only in their initial specification. The three most common possibilities are the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble. In the first, the energy and particle number are held fixed; in the second, states of various energy are allowed for fixed particle number; and in the third, both the energy and the particle number vary. Fach of these descriptions can be rephrased algebraically but the grand canonical be the Hubert space of description is in several ways more convenient. Let states for all possible energies and particle numbers of the finite System, and H and N, the selfadjoint Hamiltonian and number operators, respectively. The Gibbs grand canonical equihbrium state is defined äs a state over ^(), or ^^(), by
automorphisms states
over
the
Tr^(.-/^^^)
^^-^(^)^Tr,M^)
'
Introduction
where K erator.
for all
H
5
// G [R, and it is assumed that e~^^ is a trace-class opH is lower semi-bounded and the trace-class property is valid
iiN,
Typically ß > Q. The parameters ß
and /i
to the inverse
correspond
temperature of
the System, in suitable units, and the chemical potential, respectively, and therefore this description is well-suited to a given type of material at a fixed
temperature. Now if the generalized evolution A G
^()
^
then the trace-class property of f
are
T,(^)
-
is defined
e^^^Ae-^^^
e~^^ allows ^
T
one
G
^()
by ,
to deduce that the functions
cDß^^(Ait(B]]
analytic in the open strip 0 < Im ^ < jS and continuous Strip. Moreover, the cyclicity of the trace gives
on
the boundaries
of the
^ß.M'^t(B]]\t^iß This is the KM S condition which will
role
=
o}ß^^(BA)
brieüy described throughout this chapter. we
.
in
Chapter l and which significance of this over ^^(), i.e., the
One
important uniquely determines the Gibbs state only State over ^^() which satisfies the KMS condition with respect to T at the value ß is the Gibbs grand canonical equilibrium state. This can be proved by explicit calculation but it will in fact follow from the characterization of extremal KMS states occurring in Section 5.3. It also follows under quite general conditions that the KMS condition is stable under limits. Thus for a System whose kinematic observables form a C*-algebra ^ and whose dynamics is supposed to be given by a continuous group of *-automorphisms T of ^, it is natural to take the KMS condition äs an empirical definition of an equilibrium play
an
condition is that it
state.
Prior to the analysis of KMS states we introduce the specific quasi-local C*algebras which provide the quantum-mechanical description of Systems of point particles and examine various properties of their states and representations. In particular we discuss the equilibrium states of Systems of non-interacting particles. This analysis illustrates the thermodynamic limiting process, utilizes the KMS condition äs a calculational device, and also provides a testing ground for the general formalism which we subsequently develop. In the latter half of the chapter we discuss attempts to derive the KMS condition from first principles.
5.2. Continuous
Quantum Systems.
I
5.2.1. The CAR and CCR Relations There are two approaches to the algebraic structure associated with Systems of point particles in quantum mechanics. The first is quite concrete and physical. One begins with the Hubert space of vector states of the particles and subsequently introduces algebras of operators corresponding to certain particle observables. The second approach is more abstract and consists of postulating certain structural features of a C*-algebra of observables and then proving uniqueness of the algebra. One recovers the first point of view by passing to a particular representation. We discuss the first concrete approach in this subsection and then in Section 5.2.2
we
examine the abstract formulation.
The
quantum-mechanical states of n identical point particles in the configuration space U^' are given by vectors of the Hubert space L~(R"^''). If the number of particles is not fixed, the states are described by vectors of the direct sum
space
^=@L\K"-) 77
i.e.,
\l/ {jA^''^}>o, where given by
sequences
of
norm
i/^
is
=
ll'Af
=
l'/'^'P
+
E
,
>0
\l/^^^
G
C,
!/^^''^
G
L'([R'^')
for
fdxr--dx\il^^"\x,,...,x)\^
/?
>
l, and the
.
n>\'^ There If
is, however, a further restriction imposed by quantum statistics. G 5 is normalized, then
i/^
dp(xi,...,Xn)
=
\il/^''\xi,...,Xn)\^dxi
-
dx^
is the quantum-mechanical probability density for \l/ to describe n particles at the infinitesimal neighborhood of the points ;ci x^ The normalization of \l/ , corresponds to the normalization of the total probability to unity. But in .
.
.
,
.
microscopic physics identical particles are indistinguishable and this is refiected by the symmetry of the probability density under interchange of the particle coordinates. This interchange defines a unitary representation of the permutation group and the symmetry is assured if the ij/ transform under a suitable subrepresentation. There are two cases of paramount importance.
Continuous
Quantum Systems.
I
7
\l/^"^
of each ij/ are Symmetrie under The first arises when the components Particles whose transform in this manner are of coordinates. states interchange called bosons and
said to
are
satisfy
Böse
statistics. The second
(-Einstein)
ease
anti-symmetry of the i/^^"^ under interchange of each pair of particles are called /ermzö/?^ and are said to satisfy Fermi (-Dirac} statistics. Thus to discuss these two types of particle one must examine the Hubert subspaces 5.^, of 5, formed by the ij/ {^ }n>Q whose sign). These components are Symmetrie (the + sign) or anti-symmetric (the subspaces are usually called Fock spaces but we will also use the term for more general direct sum spaces. To describe particles which have internal structure, e.g., an intrinsic angular momentum, or spin, it is necessary to generalize the above construction of corresponds
to
coordinates. The associated
=
-
Fock space. Assume that the states of each
particle form a complex Hubert space l) and f) denote the /7-fold tensor product of i^ with itself. Fur ther introduce the Fock space g(l)) by
let
l)''
=
t)
0
I)
0
(g)
5(1))
©
=
t)"
,
n>0
if
where
i/^^"^
C. Thus
a
\\j
vector
G
is
5(f))
a
{iA''"''}/2>o
sequence
^^ vectors
subspace of g(l)) formed by the vectors with all components except the th equal to zero. In Order to introduce the subspaces relevant to the description of bosons and fermions we first define operators P on (5(^) by G
t)'^ and l)"
can
^+(/i
for all
be identified
äs
/.)
=
P-(fl^f2^"'^ fn]
=
/2
/l, ...,/
(TII, 712,
,
7r)
G
^
^
f). The
sum
the closed
(n
!)"^ V /.,
(n !)
is
of the indices and n is
/.,
0
Z-^^
^
over
one
^ Bnfn, all
^
even
/.
TI;
(1.
,
fn.
permutations
if TT is
0
0
and minus
2,
one
Extension
...,)
F-^
if TC is odd.
l and by linearity yields two densely defined operators with ||P|| the P extend by continuity to bounded operators of norm one. The P+ and P_ restricted to i^", are the projections onto the subspaces of 1^" corresponding to the one-dimensional unitary representations n \-^ l and TC 8;^ of the permutation group ofn elements, respectively. The Bose-Fock space g^(t)) and the Fermi-Fock space (5_({)) are then defined by =
H-
S(^)=/'5(l)) and the
corresponding -particle subspaces I)'^ by I)^ on g(^) by
==
{)"
P
.
We also define
number operator N
D(N)
=
(lA; l
^
=
{^^"^}.>o, E^'ll^^'^^ll' '^>o
and
7v,A
=
{.AW}>o
0
The
simplest example
one
then has
of this second
quantization
jr(i)=7v If u is
and
unitary, [/
is defined
Un(P(fl
/2
0
0
extending by continuity.
by L/o
^
fn)]
=
The second
given by choosing
//
=
H ,
.
H and
P(Ufl
=
is
by setting 0
^/2
quantization
0
0
Ufn)
of U is denoted
by r(f/),
where
r(u)
@u
=
.
n>0
Note that r (U) is
is
a
strongly
unitary. The
notation dY and P is chosen because if Ut unitary group, then
r(t/,) Next
we
=
e'^^
continuous one-parameter
wish to describe two
bosons and fermions,
respectively.
-
e'"^''(^)
.
C*-algebras of observables associated with Both algebras are defined with the aid of
particle "annihilaüon" and "creation" operators which are introduced äs follows. For each / e 1) we define operators a(f), and *(/), on 5(1)) by initially setting a(/).A(0) 0,a*(/)^() /, / e ^, and =
a(/)(/i f2---fn)
*(/)(/! /2---/)
=
=
n^'^(f, /i)/2
/3
(+l)'^V
/l-
=
/
,
Continuous
Extension
i/^^"^
l)\
G
by linearity again yields easily calculates that
I
9
if
densely defined operators and
two
one
||a*(/),AW||
||a(/)^Wi|
