Texts and

Monographs

in

Physics

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

www.pdfgrip.com


Springer
Berlin

Heidelberg
New York

Hong Kong
London
Milan
Paris

Physics and Astronomy

Tokyo

l ONLINE LIBRARY

/>
www.pdfgrip.com


Ola Bratteli
Derek W. Robinson

Operator Algebras

and Quantum
Statistical Mechanics 2
Equilibrium States.
Models in Quantum

Statistical Mechanics

Second Edition

Springer

www.pdfgrip.com


Professor Ola Bratteli
Universitetet i Oslo
Matematisk Institutt
Moltke Moes vei 3 1

0316 Oslo, Norway
e-mail:
Home page:

/>
Professor Derek W. Robinson
Australian National

University

School of Mathematical Sciences

ACT 0200 Canberra, Australia
e-mail:
Home page:

/>
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



©

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

www.pdfgrip.com


To

Trygve Bratteli,
Samuel Robinson,
and
Harald ROSS

www.pdfgrip.com


www.pdfgrip.com


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


www.pdfgrip.com

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

Ola Bratteh

Oslo and Canberra 1996

Derek W. Robinson

www.pdfgrip.com


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

www.pdfgrip.com


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

www.pdfgrip.com



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

32

Identities and

Quotient Algebras

and States

39

42

Representations

42


2.3.2. States
2.3.3. Construction of

25

'

2.2.2. Positive Elements

2.3.

17

48

Representations

www.pdfgrip.com

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.

133

136
136
142

145

152


and Generators

Space Theory

Uniform

Continuity
Strong, Weak, and Weak* Continuity
Convergence Properties
Perturbation Theory
Approximation Theory

www.pdfgrip.com

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

461

Monographs

Articles

464

List of

481

Subject


Symbols
Index

487

www.pdfgrip.com


www.pdfgrip.com


States in

Quantum Statistical Mechanics

www.pdfgrip.com


www.pdfgrip.com


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

www.pdfgrip.com



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^)

www.pdfgrip.com

'


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

^()

T,(^)

^

then the trace-class property of
f

are

-


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.

www.pdfgrip.com


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.

www.pdfgrip.com


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

www.pdfgrip.com

<

+^|J

a


8

States in

Quantum Statistical Mechanics


for each

\l/ G D(N). It is evident that A^ is selfadjoint since it is already given in
spectral representation. Note that e^^^ leaves the subspaces 5.t(i^) invariant.
We will also use TVto denote the selfadjoint restrictions of the number
operator
to these subspaces.
The peculiar structure of Fock space allows the amplification of
operators
on I) to the whole spaces 5.^(1)) by a method
commonly referred to äs second
qiiantization. This is of particular interest for selfadjoint operators and uniits

taries.
If H is

selfadjoint operator

f),

on

one can

define Hn

on

I)'^ by setting HQ


=

Q

and

ffn(P(f\

^'"^

fn]]

=P\y^fl^f2^-'-^Hfi^-"^fn

for all

// G D(H), and then extending by continuity. The direct sum of the // is
essentially selfadjoint because (1) it is Symmetrie and hence closable, (2) it has a
dense set of analytic vectors formed by finite sums of (anti-) symmetrized
products of analytic vectors of H, The selfadjoint closure of this sum is called
the second quantization of H and is denoted by dY(H]. Thus

dY(H)

@Hn

=

.


;i>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-

=

www.pdfgrip.com

/

,


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| <'/2||/||||^(")||,
Thus

Quantum Systems.

a(f]

and

(+1)'/2|/||||^W||

<

have well-defined extensions to the domain

*(/)

.

D(N^f^]


of

N^f^ and

\\a*(fm<\\f\ ||(A^+l)^/Vll
for all

\l/ G D(N^^^), where a^(f}
adjoint relation

denotes either

a(f)

or

a*(/). Moreover,

one

has the

(a*(/)(p,iA)
for all

(p,iA

a(f]

and


D(N^^^).

G

<2^(/)

on

(/)

Finally,

we

=

define annihilation and creation operators

the Fock spaces
=

/'i

a(f)P^

((p,ß(/)iA)

5^(1)) by
a


,

; (/)

=

fl* (/)P

P

.

The relations

(/)?, A)
follow from the

=

(<?>,* (/W,

corresponding

iii(/)'/'ii

relations for

(/)-(/)^,


a(f}

11/11 ii(A^

<

and

a;(/)=Pfl*(/)

a(f) leaves the subspaces g.j_([)) invariant.
f^-^a(f] are anti-linear but the maps /i-^a(/) are
The
==

physical interpretation of
.), then Q corresponds

(l, 0, 0,

.

i)'^Vii

*(/). Moreover,

because

Q


+

.

these
to the

Note that the maps
linear.

operators is

the

zero-particle state,

following.
the

vacuum.

Let

The

vectors

A
identify
particle


with elements of the
in the state

(/)

=

;(/)"

one-particle

space

1^ and

hence

a^ (/)

"creates"

a

The vectors

/.

iA(/i,...,/.)-^('^0~'/'<(/i)--<(/.)^
=


P(f\'--fn)

-particle states which arise from successive "creation" of particles in the
/,/_!,
/l Similarly the a (/) reduce the number of particles, i.e.,
they annihilate particles. Note that if fi=fj for some pair /, j with
are

States

l

.

.

.

,

.


iA_(/b...,/)-^-(/i^---^/)
by anti-symmetry.

Thus it is

=

o

impossible to create two fermions in the same
principle which is reflected by the operator

state. This is the celebrated Pauli

equation

a*_(/K_(/)=0

.

www.pdfgrip.com


10

States in

Quantum Statistical Mechanics

This last relation is the

simplest

of the commutation relations which link

case

the annihilation and creation operators.
One computes straightforwardly that

[

(/), a + (9)]=0

+

=

[<(/),<(ö)]

[fl+(/),a;((7)]=(/,ör)1

,

,

and

K(/),a-(^)}

0

=

K(/),al(ö)}

=

{.(/), al(./)}
where
are

we

have

used the notation

again

=

(/,^)1

{A, B}

called the canonical commutation relations

,

,

AB + BA. The first relations

(CCRs)

and the second the

ca-

nonical anti-commutation relations

(CARs).
Although there is a superficial similarity between these two sets of algebraic
rules, the properties of the respective operators are radically different. In applications to physics these differences are thought to be at the root of the
fundamentally disparate behaviors of Böse and Fermi Systems at low temperatures. In order to emphasize these differences we separate the subsequent
discussion of the CARs and CCRs but before the general analysis we give an
example of the creation and annihilation operators for point particles.
EXAMPLE 5.2.1.
functions of

tisymmetric (

n
-

If

l)

=

then

L~(U'),

5^(1))

{iA^"^}>o

consists of sequences

of

variables x/ G [R^' which are totally Symmetrie ( + sign) or totally ansign). The action of the annihilation and creation operators is given by

(a(/)A)'"'(^i,

(al(f]il,f\x,,.

.

.

.

.

.

.x)

,x]

=

=

(n

-'/2

l)'/'

+

fdxj{x)'^^"^'\x,x^,.

;^( l)'-'/(^,),A('-')(.x,,

.

.

.

.

.,x)

,.x,,

.

.

.

,

,x)

,

i=\

where i/ denotes that the i th variable is to be omitted. Note that

/->(/),
are

anti-linear and

a(f)

=

fl

(x),

and

a*_^ (x),

j dxW)^(^)

and then the action of these fields is

(a

W,A)W(^i,

(a;(;c)iA)("'(xi,

.

.

.

.

/^i(/)

linear, respectively,

tributions, i.e., fields

.

.

,x)

=

,x)

=

,

one

may introduce

operator-valued dis-

such that

<(f)

Jdxf(x)al(x)

=

,

given by

(n+

l)'/2 ,/.("+ ')(x,^,,

-'/2

^(

.

.

.

,x)

,

l)'-'5(.v -.x,OiA'"-"(-X|,

.

.

.

,x,,

/=!

In terms of these fields the number operator A^ is
A^-

the maps

äs

formally given by

[dxa-'^(x}a^(x)

www.pdfgrip.com

.

.

.

.

,x)

.