Operator algebras and quantum statistical mechanics 2; equilibrium states, models in quantum statistical mechanics 2nd edition

The firstapproach beginswith the specific descriptionof finiteSystems andtheir equilibrium states provided by quantum statistical mechanics.. One thenrephrases this description in an alg

Monographs Physics

Series Editors:

R.Balian, Gif-sur-Yvette,France

W.Beiglböck, Heidelberg, Germany

H Grosse, Wien,Austria

E H.Lieb,Princeton, NJ,USA

N Reshetikhin, Berkeley, CA,USA

H Spohn,München,Germany

W.Thirring,Wien,Austria

Home page:http://www.math.uio.no/~bratteli/

Professor DerekW Robinson

Australian NationalUniversity

School of Mathematical Sciences

ACT 0200Canberra,Australia

e-mail: Derek.Robinson@anu.edu.au

Home page:http://wwwmaths.anu.edu.au/~derek/

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SecondEdition 1997 SecondPrinting2002

ISSN0172-5998

ISBN3-540-61443-5 2ndEditionSpringer- VerlagBerlinHeidelbergNew York

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Trygve Bratteli,

SamuelRobinson,

andHarald ROSS

Fifteenyearshavepassed sincecompletion of the first edition of this bookand much has happened Any attempt to do justice to the new develop-ments would necessitate at least one new volume rather than a secondedition of the current one. Fortunately other authors have taken up thechallenge of describing these discoveries and our bibliography includesreferences to a variety ofnew books that have appeared or are about toappear Weconsequentlydecidedtokeep the format of this bookäs abasicreference for the operator algebraic approach to quantum statistical me-

chanics and concentrated on correcting, improving, and updating thematerial of the first edition This in itself has notbeen easy and changes

occurthroughoutthetext.Themajor changes are acorrectedpresentation

of Bose-Einstein condensation in Theorem 5.2.30, insertion ofa general

result on the absence ofsymmetry breaking in Theorem 5 3 3 3A, and an

extendeddescriptionof thedynamicsof the A^-Fmodel inExample6.2.14.The discussion ofphase transitions in specific models, in Sects 6.2.6 and

6.2.7, has been expanded with the focus shifted from the classical Isingmodel to genuine quantum situations such äs the Heisenberg and X-Ymodels In addition the Notes and Remarks to various subsections havebeen considerably augmented.

Since our interest in the subject of equilibrium states and models ofstatistical mechanics has waned considerably in the last fifteen years it

would have been impossible to prepare this second edition without theSupportandencouragementofmanyofourfriends andcolleagues. Weare

particularlyindebtedtoCharlesBatty,MichielvandenBerg,TomterEist,

Dai Evans, Mark Fannes, Jürg Fröhlich, Taku Matsui, Andre Verbeure,

and Marinus Winnink for information and helpful advice, and we ogize for often ignoring the latter We are especially grateful to Aernout

apol-vanEnterand ReinhardWernerforcounsellingus on recentdevelopmentsand giving detailed suggestions for revisions

Derek W Robinson

Statesin Quantum Statistical Mechanics l

5.4.3 Gauge Groupsand the Chemical Potential 197

6.2.1 Kinematical andDynamical Descriptions 239

6.3.3 TheThermodynamicLimit.I. The ReducedDensityMatrices 3816.3.4 TheThermodynamicLimit II States and Green's Functions 395

Introduction l

2.2.1 Resolvents, Spectra, andSpectralRadius 25

322.2.3 ApproximateIdentities andQuotient Algebras 39

2.3.4 Existence ofRepresentations 58

2.4.2 Definition andElementary Properties

2.5 Tomita-Takesaki Modular Theory and Standard Forms

2.5.3 IntegrationandAnalytic Elements

for One-ParameterGroups of Isometries

2.7.1 Dynamical Systems and Crossed Products 136

2.7.3 Weights onOperator Algebras; Self-Dual Cones

of GeneralvonNeumannAlgebras; Dualityand

Classification ofFactors; Classification of C -Algebras 145

3.2 Algebraic Theory 2053.2.1 Positive Linear Mapsand JordanMorphisms 205

3.2.3 Spectral Theoryand Bounded Derivations 244

3.2.5 Spatial Derivations and Invariant States 2633.2.6 ApproximationTheoryfor AutomorphismGroups 285

4.2 Extremal, Central, and Subcentral Decompositions 353

4.4.2 Spatial Decomposition andDecompositionof States 442

Quantum

In thischapter,and thefollowing one,weexamine various applicationsof algebras and their states to statistical mechanics Principally we analyze thestructuralproperties of theequilibriumstatesofquantum Systemsconsistingof

C*-a largenumber ofparticles. InChapter l wearguedthat this leads tothestudy

ofstates ofinfinite-particle Systemsäs an initialapproximation. Thereare twoapproaches to this studywhich are to a large extentcomplementary.

The firstapproach beginswith the specific descriptionof finiteSystems andtheir equilibrium states provided by quantum statistical mechanics One thenrephrases this description in an algebraic language which identifies the equilibrium states äs states over a quasi-local C*-algebra generated by subalgebras corresponding to the observables ofspatial Subsystems Finally, one attempts

tocalculateanapproximationof thesestatesby takingtheir limitästhe volume

of theSystemtendstoinfinity, the so-calledthermodynamiclimit The volume equilibrium states obtained in this manner provide the data for thecalculation of bulkpropertiesof thematterunder considerationäsfunctions ofthethermodynamicvariables By this we meanproperties suchäs the particle density, or specific heat, äs functions of thetemperature and chemicalpoten-

infinite-tial, etc. In fact,the infinite-volume data provides amuchmore detailed,even

microscopic, description of the equilibrium phenomena although one is only generallyinterested in the bulkproperties and their fluctuations Examination

of the thermodynamic limit also provides a test of the scope of the usualstatistical mechanical formalism If this formalism is rieh enough to describephase transitions, then at certain critical values of the thermodynamic para-metersthere should beamultiplicityof infinite-volume limitstatesarisingfromslight variations of the external interactions or boundary conditions Thesestates would correspond to various phases and mixtures of these phases. Insuch a Situation it should be possible to arrange the limits such that phase Separation takes place and then the equilibrium states would also provideinformation concerning interface phenomena suchäs surface tension

The secondapproach toalgebraicstatistical mechanics avoids discussion ofthethermodynamic limit and attempts to characterize andclassify theequilibriumstatesof the infiniteSystemäs states over anappropriate C*-algebra.Theelements of the C*-algebrarepresentkinematicobservables, i.e., observablesat

agiven time,and thestates describe the instantaneousstatesof theSystem For

a complete physical description it is necessary to specify the dynamical law

governing the change with time of the observables, or the states, and theequilibrium states are determined by their properties with respect to this dy-namics Thegeneralnatureof thedynamicallawcanbe inferred from the usual

quantum-mechanical formalism and it appears that there arevarious Uties Recall that for finite quantum Systems the dynamics is given by a one-

possibi-parameter group of*-automorphisms of thealgebra ofobservables,

general infinite Systems in which compHcated phenomena involving the localaccumulation of an infinite number of particles and energy can occur forcertain initialstates.Thus it is necessarytoexamine weaker forms of evolution.For example, one could assume the dynamics to be specified äs a group of

automorphisms of the von Neumann algebras corresponding to a subclass ofstates over the C*-algebra Alternatively one could adopt aninfinitesimal de-scription and assume that the evolution is determined by a derivation whichgenerates an automorphism group only in certain representations. Fach ofthesepossiblestructurescould inprinciplebe verified inaparticularmodelbya

thermodynamic limitingprocessand each suchstructureprovidesaframeworkfor characterizing equilibrium phenomena. To under stand the type of char-acterization which is possible it is useful to refer to the finite-volume descrip-tion ofequilibrium.

There arevariouspossible descriptions ofequilibriumstates, which allstemfrom theearlywork of Boltzmann and Gibbsonclassical statisticalmechanics,

and which differ only in their initial specification. The three most common

possibilities arethe microcanonical ensemble,the canonical ensemble, and thegrandcanonical ensemble Inthefirst, the energy andparticlenumberareheld

fixed; in the second, states of various energy are allowed for fixed particle

number; and in thethird, both the energyand the particlenumbervary. Fach

ofthese descriptions can be rephrased algebraically but the grand canonicaldescriptionis in several waysmore convenient Let be the Hubert space ofstatesfor allpossible energiesandparticlenumbers of the finiteSystem, andHand N, the selfadjoint Hamiltonian and number operators, respectively. TheGibbs grand canonical equihbrium state is defined äs a state over ^(), or

^^(), by

Tr^(.-/^^^)

^^-^(^)^Tr,M^) '

where K H iiN, // G [R, and it is assumed that e~^^ is a trace-class erator. Typically His lower semi-bounded and the trace-classpropertyis validfor all ß> Q Theparametersßand/icorrespondtothe inversetemperatureofthe System, in suitable units, and the chemical potential, respectively, andtherefore this description is well-suited to a given type of material at a fixedtemperature Now if the generalized evolution Tis defined by

op-A G^() ^ T,(^) -e^^^Ae-^^^ G^() ,

then the trace-class propertyofe~^^ allows one to deduce that the functions

f ^ cDß^^(Ait(B]]

areanalytic in the open strip0 < Im ^ < jS and continuous on the boundaries

of the Strip Moreover, thecyclicity of the trace gives

This is the KM S condition whichwebrieüy described inChapter l and whichwill play an important role throughout this chapter. One significance of thiscondition is that it uniquely determines the Gibbs state over ^^(), i.e., theonly State over^^() which satisfies the KMS condition with respectto T atthe valueßis the Gibbsgrandcanonicalequilibriumstate. Thiscanbeproved

by explicit calculation but it will in fact follow from the characterization ofextremal 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

Systemwhose kinematic observables formaC*-algebra ^ and whosedynamics

issupposedtobegiven byacontinuous group of*-automorphisms Tof^,it isnatural totake the KMS conditionäs anempiricaldefinition ofanequilibriumstate.

Priorto theanalysis of KMSstates weintroduce thespecific quasi-local algebras which provide the quantum-mechanical description of Systems ofpoint particles and examine various properties of their states and representa-tions In particular we discuss the equilibrium states ofSystems of non-inter-acting particles. This analysis illustrates the thermodynamic limiting process,utilizes the KMS condition äs a calculational device, and also provides a

C*-testing ground for the general formalism whichwesubsequently develop.

In the latter half of the chapter we discuss attempts to derive the KMScondition from firstprinciples.

Quantum Systems. I

Therearetwo approachesto thealgebraicstructure associated withSystems ofpoint particlesin quantummechanics The first is quiteconcrete andphysical.One begins with the Hubert space ofvector states of the particles and subse-

quently introduces algebras of operators corresponding to certain particle

observables The second approachismoreabstract and consists ofpostulatingcertain structural features of a C*-algebra of observables and then proving uniqueness of the algebrạ One recovers the firstpoint of viewby passingto a

particular representation. We discuss the first concrete approach in this section and then in Section 5.2.2we examine the abstract formulation.The quantum-mechanical states ofn identical point particles in the config-uration space U^' are given by vectors of the Hubert space L~(R"^''). If thenumber ofparticlesisnotfixed, thestates aredescribedbyvectors of the direct

sub-sum space

77 >0

ịẹ, sequences\l/ = {jẤ'^}>o, where\l/^^^ G C, !/^^''^ GL'([R'^') for/? > l,and the

norm ofi/^ isgiven by

n>\'^

There is, however, a further restriction imposed by quantum statistics

Ifi/^ G 5is normalized, then

The first arises when the components \l/^"^ of each ij/ are Symmetrie underinterchangeof coordinates Particles whosestatestransform in thismanner are

called bosons andaresaid tosatisfyBöse(-Einstein)statistics The secondease

corresponds to anti-symmetry of the i/^^"^ under interchange of each pair ofcoordinates The associatedparticles arecalled/ermzö/?^andaresaidto satisfy

Fermi (-Dirac} statistics Thus to discuss these two types ofparticleone mustexamine the Hubert subspaces 5.^, of5, formed by the ij/ = {^ }n>Q whosecomponents are Symmetrie (the + sign) or anti-symmetric (the-sign). Thesesubspacesareusuallycalled Fockspacesbutwewill alsousetheterm formore

general direct sum spaces

To describeparticles which have internal structure,e.g.,anintrinsicangular

momentum, or spin, it is necessary to generalize the above construction ofFock space.

Assume that the states of eachparticleform a complex Hubertspace l) andlet l)'' = t)0I)(g) 0f) denote the /7-fold tensorproduct ofi^ with itself Further introduce the Fock space g(l)) by

5(1)) = © t)" ,

n>0

where if C Thus a vector \\j G 5(f)) is a sequence {iA''"''}/2>o ^^ vectors

i/^^"^ G t)'^ andl)"can be identifiedästhe closed subspaceofg(l)) formedbythevectors with all components exceptthe th equal to zero.

In Order to introduce the subspaces relevant to the description of bosonsand fermions we first define operators P on (5(^) by

for each\l/ GD(N).Itis evident thatA^isselfadjointsinceit isalready giveninitsspectral representation. Note thate^^^leaves thesubspaces 5.t(i^) invariant.

We will alsouseTVtodenote theselfadjointrestrictions of the numberoperator

to these subspaces.

The peculiar structure of Fock space allows the amplification ofoperators

on I) to the whole spaces 5.^(1)) by a methodcommonly referredto äs second

qiiantization. This is of particular interest for selfadjoint operators and taries

uni-IfHisselfadjointoperator onf), one candefineHn onI)'^ by settingHQ =Qand

ffn(P(f\ ^'"^fn]] =P\y^fl^f2^-'-^Hfi^-"^fn

for all// GD(H),andthenextending by continuity.The directsumof the// isessentially selfadjointbecause(1)itisSymmetrieand henceclosable, (2)it hasa

dense set of analytic vectors formed by finite sums of (anti-) symmetrized

products ofanalyticvectors of H, The selfadjointclosure of this sumis calledthe second quantization ofHand is denotedby dY(H]. Thus

If u is unitary, [/ is definedby L/o =H and by setting

Un(P(fl 0/20 ^fn)] =P(Ufl 0 ^/20 0Ufn)

andextending by continuity.The secondquantizationofUis denotedby r(f/),

where

r(u) = @u .

n>0

Notethatr(U)isunitary.The notation dY and P is chosen because ifUt= e'^^

is a stronglycontinuous one-parameter unitarygroup, then

r(t/,) -e'"^''(^) .

Next we wish to describe two C*-algebras of observables associated withbosons and fermions, respectively. Both algebras are defined with the aid ofparticle "annihilaüon" and "creation" operators which are introduced äs fol-lows For each/e1) wedefineoperators a(f),and*(/), on 5(1)) by initially setting a(/).A(0) =0,a*(/)^() = /, /e ^, and

a(/)(/i f2 -fn) = n^'^(f, /i)/2 /3 / ,

*(/)(/! /2 -/) = (+l)'^V

/l-Extension by linearity again yields two densely defined operators and if

i/^^"^ G l)\ one easilycalculates that

||a(/)^Wi| <'/2||/||||^(")||, ||a*(/),AW|| < (+1)'/2|/||||^W|| .

Thus a(f] and *(/) have well-defined extensions to the domain D(N^f^] of

N^f^ and

for all \l/ G D(N^^^), where a^(f} denotes eithera(f) ora*(/) Moreover, one

has the adjoint relation

(a*(/)(p,iA) = ((p,ß(/)iA)

for all (p,iA GD(N^^^) Finally, we define annihilation and creation operators

a(f] and<2^(/) on the Fock spaces 5^(1)) by

(/) =/'ia(f)P^ , a; (/) =P fl*(/)P .

The relations

(/)?, A) = (<?>,* (/W, iii(/)'/'ii < 11/11 ii(A^+i)'^Vii

follow from the correspondingrelations for a(f} and *(/) Moreover,

(/)-(/)^, a;(/)=Pfl*(/)

because a(f) leaves the subspaces g.j_([)) invariant Note that the maps

f^-^a(f] areanti-linear but the maps /i-^a(/) are linear

The physical interpretation of these operators is the following. Let

Q== (l, 0, 0,. .),then Qcorrespondstothezero-particlestate, thevacuum.Thevectors

A (/) = ;(/)"

identifywith elements of the one-particlespace 1^and hencea^ (/) "creates"a

particlein the state/. Thevectors

by anti-symmetry. Thus it is impossible to create two fermions in the same

state. This is the celebrated Pauliprinciple which is reflected by the operatorequation

a*_(/K_(/)=0 .

This last relation is the simplestcase of the commutation relations which linkthe annihilation and creation operators.

One computes straightforwardly that

[ +(/),a+(9)]=0 = [<(/),<(ö)] ,

and

K(/),a-(^)}= 0= K(/),al(ö)} ,{.(/), al(./)} = (/,^)1 ,

where we have again used the notation {A, B} AB + BA The first relations

arecalled the canonical commutation relations (CCRs) and the second the

ca-nonicalanti-commutation relations(CARs).

Although there isasuperficial similaritybetween these two setsofalgebraic rules, theproperties of the respective operators are radically different In ap-

plications to physics these differences are thought to be at the root of thefundamentally disparate behaviors of Böse and Fermi Systems at low tem-

peratures In orderto emphasize these differences we separatethe subsequentdiscussion of the CARs and CCRs but before thegeneral analysiswe give an

example of the creation and annihilation operators forpoint particles.

EXAMPLE 5.2.1 If l)=L~(U'), then 5^(1)) consists of sequences {iA^"^}>o offunctions ofnvariablesx/ G [R^' which aretotally Symmetrie (+ sign) ortotallyan-

tisymmetric (- sign).The action of the annihilation and creationoperatorsisgiven by

5.2.1.1 TheCAR Relations Wenextanalyzethepropertiesof the creation andannihilation operators obeying the CAR relations on the Fock space 5_(^).

We simplify Dotationby droppingthe suffix minus on theoperators

Proposition5.2.2 Letl) bea complexHubertspace, 5- (W ^^^FermiFockspace, and a(f) and a''(g] the corresponding annihilation and creation op-

erators on (5_(I)). Itfollows that

for all/G]^, and hence a(f] anda*(g) haveboundedextensions

(2) T/'Q= (1,0,0, .) and{/} isan orthonormal basis ofl), then

PROOF. (1) One has

(a\f)a(f)f=a^(f){a(f), a'(f)]a(f) = \\f\^a^(f)a(f)

and hence

ll(/)ll' =IIK(/)(/))'ll = il/ll'lk'(/M/)|| = ll/ll'i|a(/)f .

Asa(/) ^0 for/ ^0onecondudes that

To establish the lastequality oneconsiders the threecases n > m, n < m,andn =

m, separately.In the ßrstcasebothexpressionsare zerobecause thea(f) annihilate

moreparticles than thea*(g)create Inthe secondcase bothexpressions areagain

zero by complex conjugation.In the third case(/) a*(gß^)Q.isamultipleofQ

and the desiredequalityfollowsonce more.ThusT = (Q, rQ)1landirreducibilityisa

consequence ofProposition2.3.8

Quantum Statistical Mechanics

5.2.1.2 The CCR Relations The mainqualitativedifference between fermionsand bosons is the absence ofaPauli principle for the latterparticles. There is

no bound on the number ofparticles which can occupy a given statẹ This isquantitatively reflected by the unboundedness of the Böse annihilation andcreation operators If, for example, il/^^^ is the /7-fold tensor product of/G I)

withitself, then the annihilation operator satisfies

ă/) =2-'/2(<l.(/) +/n(/)), a*(/) = 2-'/2(<i,(/) _/n(/)) ,

and theăf} andß*(/) can berecuperatedfrom the <!>(/).Thus forfunctionalpurposes it suffices to examine the latter operators. Their basicproperties are

most easily examined on the subspace F(t)) c S+(t)) formed by the

finite-particle vectors, ịẹ, the ij/ = {^A }>o which have only a finite number ofnonvanishingcomponents.

Proposition 5.2.3 Let l) be a complex Hubertspace, 5_^(I)) the Böse Fockspace, andăf) andá'(g) annihilation andcreation operatorssatisfying thecanonicalcommutation relations Define^ by

0(/) = 2-i/2((/)+^*(/))

for all/G t) Itfollows that

(1) For each /Gl^,O(/) z^ essentially selfadjoint on F(bi) and if

\\f, -f\\ ->0, then ||<D(/J./'-<D(/)i/'|| -^ Qforall^ e)(7V'/2).

(2) If Q = (1,0,0, ) then the linear

span of the set

{$(/,).

-$(/)Q;/, e [), =0, l, ..} is dense in S+()).

(3) Foreach ^|/ e-D(Á') and,f,gel) onehas

(<D(/)(l)(ö-) -<D(ör)0(/)),A =^Im(/,6')iA

PROOF (1) The operatorO(/)isdenselydefined,andSymmetrie, hence closablẹ

To establish essentialselfadjointnessit suffieesto provethat^(f] hasadensesetofanalytie vectors. But if iẤ'^GV|, then i/^^"^ G D(<D(/)'") for all m and

^(/)V^"^ Gf)';+^ 0^1"^ The estimates

Thecontinuityfollows because

llW/a)-^(/))^|| <2-^/2||a(/, -fm+2-^/2||a*(/,-/)iA||

<2^/2||A-/l|||(7V+l)^/Vll

(2) The linear spans of {$(/i ) ^(fn]^\// eI),n >0} and {a* (/i ) a*(/)Q; //G I),W > 0} areidentical But the latter is densebythesameargumentsused inProposition 5.2.2

(3) This is immediate from the canonical commutation relations

Next weconsider the unitary groupsgenerated by the operators O(/), buthenceforthwe use thisSymbolto denote theselfadjointclosure of theprevious

>(/).

Proposition 5.2.4 Foreach /Gt) /e/O(/) denote theselfadjoint operator

<!)(/) =2-'/2(a(/)+fl*(/)).

Moreover let W(f) denote the unitary operatorexp{/<I)(/)}. Itfollows that

(1) For eachpair f.g^l), W(f)D(^(g)) =D(^(g)} and

PROOF (1) Each \l/^eF(I)) is analyticforO(/) and one candefine ^(g]W(fY

on\l/y^ by power-seriesexpansion.This expansion yieldstheidentity

ButF(I))isa coreforO(6f).Thus ifi//G)(>(öf)), one may choose theil/,^such that

'Aa^^2.nd^(ö')'Aa -^^(^)iA-Itimmediatelyfollows that^(g]W(fY\l/^ convergesand therefore r(/)Ve/)(O(^))and ^(^)r(/)*iAcc -^<3[>(^)^^(/)*'A- ThusD(O(^))

isinvariant under each W(f) and

'^(g)w(fr = H^(/)*(<i>(3) -im(/; 9)1)

on_D(<1)(6()).

(2) For\l/ F(I)) one canexploit the invariance ofD((^(g)]under W(f+g]',

etc., and the closedness of each<!>(/) to derive the identity

W(f)W(g)W(f^gY=e-^^^^^'^^^/'~

which is equivalentto the stated result

(3) If r is a bounded operator which commutes with each lV(f), then

n)(<D(/)) CD(^(f)) and^(f)Til/= T^(f]ilJfor each\jj D(^(f]]because<!>(/)isthe infinitesimal generator oft G ^^W(tf]. But a(f) =2-^/-(O(/)+i^(if)) and

so r commuteswitha(f)in thesame mannerandirreducibilityfollowsbythesame

calculation used for the CARs inProposition 5.2.2

(4) F(I)) is a core of (D(/) and ||($(/a) -^(f}W\ ->0 for all i//GF(!)) byProposition 5.2.3 Therefore, it follows from Theorem 3.1.28 that W(f^] convergesstronglyto W(f].

(5) Itfollows from part(1) that

w(itf]^(f]w(itfr =^(/)-^ii/f1for allt G [R Thus the spectrum of^(f) mustbe the whole real line Now considertheunitarygroup W(tf} and its spectral representation

W(tf) = f

dE(l)e'^^-.

J

For eachi//G (?+(f)) onehas

IhAf- II ^(/)^-n~ = 2 /^OA, ^W^A)(i -cos/) .

Thus if'([7r+ ,7i-])i// =i/^withO< e < 71/2

4-||^(/)-l)'AllVlhAf <2|l-cos|

andhence ||^(/)-1]|| =2

The operators W(f) introduced in the above propositionareusually calledWeyloperators and the commutation relations

W(f}W(g) = e-^^^^f^^^'^W(f-^g] -e-^^^^^^^^W(g]W(f]

arecalled the Weylform ofthe canonicalcommutation relations

In the last subsection we derived the canonical commutation and mutation relations and constructed the Weyl operators associated with theformer To completethis discussion we next examine the abstract C*-algebras generated byelements satisfying the CARs orthe Weyl form of the CCRs Itwill result that thesealgebrasareuniquelydeterminedbytheappropriateform

anti-com-of commutation relation

Again wedivide the discussion into two separate parts

5.2.2.1 The CARAlgebrạ TheforegoingFock space construction establishedthe existence of boundedoperators satisfyingthe CARs The nextresult char-acterizes the abstractpropertiesof theC*-algebra generated bytheseoperators.Theorem5.2.5.LetI)beapre-Hilbertspacewith closure5and let^/, i = 1,2,

be two C*-algebras generated by the identity D and elements ai(f], /GI),

satisfying

(1) /->fl/(/) is antilinear,

(2) {/(/), /(^)}-0 ,

(3) {,(/), a,(gr} = (/, g]^

forallf,gệi= L2

Itfollows that thereexists aunique ^-isomorphistn a : ^iF-^^2such that

ăa,(f])=â(f]

for all /G^. Thus there exists a unique, up to *-isomorphism, C*-algebra

^ 2X(]^) = ^(5) generated by elementsăf], satisfying the canonicalcommutationrelations over l).

anti-Furthermore

(1) Mf)\\ = \\f\\ forallf&\^.

(2) Ifl) is n dimensional, where n<-\-OQ, then ^(1^) is isomorphic withtheC*-algebra of2" x 2" complexmatrices

(3) ^(f)) isseparable if, andonly if, f) isseparablẹ

then there existsa uniqiie *-automorphism y of ^(I)) such that

and we use the Standard notation *(/) ==<^(fT- I^ follows from the CARs that

{ejy }.y^i 2 ^^^^^ families ofmutually commuting2x2matrixunits,

The separabilityStatement(3)is immediate from the construction of^il(I)) above,and thesimplicity follows fromCorollary2.6.19or can be provedäs follows: since

{(/), ß*(/)} = ll/fD, it follows that 7r(fl(/)) ^0 for all representations of TT of

^21(1)), and henceTT is faithful by the uniqueness of^l(l)). Hence *:ll(l)) has no non

trivial closed two-sided ideals Statement(5) followsbyapplyingthe first Statement

of theProposition ona\(f} =(/), a^d] =a(Uf] +fl*(F/) .

Note that fromStatement(2)of the theorem and the construction of^(I)) itfollows that this algebra is a UHF algebra (see Example 2.6.12).

The transformations described in part (5) of the theorem are often called

Bogoliiibov transformations.

We next investigate localstructure of the CAR algebra ^21(1)) over a Hubertspace I).

Proposition5.2.6 Let^(t)) be the CARalgebra over aHubertspace I),andlet Ibea netofclosednonemptysubspaces ofl), orderedbyinclusion such that:

(1) IfM^I, there existsanNeI such thatM I.N

(2) IfM ^N andM ^K, there exists an LeI such that M _L L and

N.KCL.

(3)

l)-UMe/^-Leta^(M) C 91(1))be the sub-C''-algebra generated by {(/); /GM} foreachMG/ Then (^(f)), {^(M)}^^^) is a quasi-local algebra in the sense of Definition 2.6.3, mth o(a(f]] = -a(f] for allfei).

PROOF First, ifMI CMj,weevidentlyhave 5l(Mi) C2I(M2), and, second, äs any

A G^(f)) canbe approximated by finite polynomials in a(f) anda*(^), it followsfrom the relation ||fl(/)|| = \\f\\ andassumption (3)that5I(^) =UMe/^(^)-Third,the ^(M) have a common identity by definition Finally, let er be the unique

*-automorphismof ^ such thata(a(f]]= (/) for all/G Thisautomorphismexists by applying Theorem 5.2,5 part (5) io U = l and F =0 Then (7~ = i and

(j(5I(M)) =^(M) for all MG/ Each element A G^(M) can be uniformly ap

proximated bya sequenceofpolynomialsP in thea(f)and0^(0)with/, ö^ G M.But

if^iseven,it follows that the(P +ö-(P))/2 alsoconvergeto ^1 =(.4+G(A)]/2.But

(P +(j(P,i})/2isan even polynomialin thea(f) anda*(g). Byasimilarreasoningodd elementscanbeapproximated byoddpolynomials.Hence it sufficesto provethe

commutation relations forpolynomials ButnotingthatA~ andB commuteif^ and

Banti-commute, these follow directlyfrom the CARs

EXAMPLE 5.2.7 Let E)=L^(U'')and, for each bounded openset A C [R^define

^IA ästhe C*-subalgebra generated by {a(f) : /GL^(A)}. It follows that the CARalgebra^(l)) isaquasi-local algebrawithrespecttothisgeneratingnet.Inparticular

A

We conclude by mentioning an equivalent way ofdescribing the CAR al

gebra which is analogous to the description of the CCRs in terms of theoperators {O(/);/ G1^}.

One defines a family of elementsB(f)] /G f) by

{B(f],B(g]}=s(f,g]

for allf.g^H,where/fisarealvectorspaceand^isarealpositive symraetric

bilinear form over H If in this context /isany operator such that

s(Jf, g) = -s(f, Jg), ß = -l

,

one can introduce annihilation, and creation operatorsaj(f) and aj(g) by

aj(f) =2-'/2(5(/) +iB(Jf)), a}(g) = 2-^/^(B(g) -iB(Jg)]

and one has

{aj(f),a:,(g)}=s(f,g]-ris(f,Jcj] ,

etc.Thus incomparisonwith thepreviousdiscussions(f, g] correspondstothereal partof(f,g) ands(f,Jg) corresponds to the imaginary part

Althoughthis latterdescriptionseems moregeneralweremark that if//isa

real Hubert space and ifs is the nondegenerate inner producton H, then a /with the above properties exists if, and only ifH has even (or infinite) di-mension Inthiscase ffisacomplexHubertspace ands is the realpart of theinner product. One has the identification

(ll+Ü2)(^-;.LC^+A2/^

for all A/ G [R and (^ G//, and

(f.g]=s(f,g)-ris(f,Jg] .

The / can be constructed by first choosing an orthonormal basis

{^b n\i ^2-> ^27 } of^ ^^d then definingJ by

We will sHghtly generalize the Situation described in Section 5.2.1.2 andexamine afamilyofWeyloperators W(f] defined forelements/ofa real linearspace Hequippedwitha nondegenerate symplectic bilinear form fj, i.e., eris a

map fromH xH into [R such that

^(/, g) = -ö-(6^, /)

for all f, g e H, and if

ff(/, g]=0for all/e//, theng (^ Forexample, onecould takeT/tobeacomplexpre-Hilbert space and a to be given by

a(f, g) = Im(/, g]

and then one recuperates the CCR relations describedpreviously.

In fact, the generalization is very slight because if er is a nondegenerate symplectic bilinear form on H and there exists an operatorJ on H with theproperties

a(Jf,g) = ~a(f,Jg), ^' = -l ,then H is a pre-Hilbert space with scalar multiplication and inner productdefined by

(^1 +Ü2)/-AI/-f^2Jf, l/ G ff^, /G// ,

(f.g} = ^(L JQ] +i^(f 9) /,^G// ,and clearly (T(f, g) =Im(/, g).

Note that ifHissequentially complete withrespectto thetopologydefined

by (T,a/with the aboveproperties existsif, andonly if,Hdoesnothave finiteodd dimension Toconstruct/onefirstuses aproceduresimilartothe Gram-Schmidtorthogonalization proceduretofind elements {^/, rjj} inHspanninga

densesubspace and such that

(j(^t.^j) ^(^(^n^j}=^ ,(T(^i, r]j) =öij ,

Then one defines/by J^i =

r]^ andJrj^ = c/ and extension by continuity.

In thefollowing, whentalkingabout the CCR algebraover acomplexHilbert space H, it is always understood that er is the imaginarypart of theinner product.

pre-Theorem 5.2.8 Let Hbe a real linearspace equippedwith anondegenerate

symplecticbilinearforma and let 2l/,z = l, 2, be two C*-algebras generated

by nonzero elements Wi(f),fGH, satisfying

(1) Wi(-f) = w,(fr,

(2) Wi(f)Wi(g) =e (/'9)/2f^.(/+g) ßrallf,geH.

Itfollows that thereexists aunique *-isomorphism a;^11-^^2 such that

a(fF,(/)) = ff2(/)

for all /G// Thus there exists a unique, up to *-isomorphism, C*-algebra

^= ^(H) generated by Weyl operators W(f).

Furthermore

(1) ^(0) =

D, W(f), is unitaryforall/G//,and \\W(f) -^=2for

allnonzero /G//

forallf, g eH, then thereexistsauniqiie'""

-aiitomorphism y of'^(H)suchthat

Next let US regard the linearspace // äs a discrete additive abelian group The

W(f] give a unitary representation ofH up to a phase, or multipHer b(f, g] =

exp{-/ö-(/, g] /l}. It is subsequently of importance that f^^Xg(f) =b(f, g] is a

character of//,e.g.,

b(fi+f2.ö]=b(f,,g]b(f2^g] .

OUT aim is to prove that the C*-algebras %,/ =1,2, are*-isomorphic.

Consider the two representationsR^, andR ofHdefinedon l^(H) by

(Rh(g}F}(f)=b(f,g)F(f+g)

and

(R(g}F)(f)=F(f+g) .

One calculates that^/,is a unitary representationup to themultiplier b andÄ is a

unitary representation in the usual sense. (Note that W(g) =Rij(g) defines a re

presentationof the CCRs and hence a CCRalgebra exists.)

Wemayassumethat ^li and% arefaithfullyrepresentedon Hubertspaces ,and2and definenewmultiplierrepresentations Wfx R on/"(//; /) =/0/^(//).

An element ij/G /"(//; /) isafunction over Hwith values ing,-, andwe set

Ui(WiXR)(g)U^ =^i^Rb(g) ,

where H/ is the identity on y Let ^/ be the C*-algebra generated by

{(WiX R) (g)]g G//}. Then, by the above identity, there exists a *-isomorphism T

from 93l onto ^2such that

for all/l/e C, // e//,and72 > 1 Therepresentation W xRis,via Fourier transform

on /^(//), unitary equivalent tothe representation W x R on l~(H;9)]definedby

((WxR](g]^|^](^]= W(g]l(g]^|^(y:), i^Hand hence

(1) At thebeginningof theproofweestablished that each W(f) isunitaryand

^(0)=t.The CCRsnowgive

W(g)W(f)W(gY=e-^-^^^^^W(f)

and hence thespectrumofW(f)is invariant under rotations Hence thespectrumisequal tothe circle and ||^(/)-11|| = 2bythespectral radius formula

(2) Assume that ^(//) is separable and let {^/J>o be acountable dense

se-quence of elements of^(//).Thus for eacht e ^and each/G //theremustbean rit

such that

(3) Let 71 be arepresentation of5I(//), then by the first Statementof the

the-oremthere existsa*-isomorphismafrom'^l(H} onton(^l(H)) such thata(PF(/)) =

n(W(f}) for all /G //. But then (x = n because ^l(//) is generated by the W(f).

Hence the kernel ofTIiszero.Butn wasarbitraryandtherefore^(H)mustbesimple.(4) This follows by applying the uniqueness Statement to W\(/)= W(f) and

^2(/) - W(Tf].

As in the CARcase,the transformations describedinpart(4)of the theorem

are often calledBogoliubov transformations,

There is an important difference between this theorem and the

corre-sponding result Theorem 5.2.5, for the CARs There is no Statement gousto ^l(I)) = '21(5) ^nd, and in fact, one has thefollowing Situation

analo-Proposition5.2.9 Let ^(//) be the CCRalgebra over the real vectorspace

Hand let ^(M) be the C'-suba!gebra generated by {W(f}', /GM} whereM

is a siibspace ofH

Itfollows that

^(M) = ^(H)

if, andonly if, M=H

PROOF IfM7^//, consider the representationof^l(H) onl^(H) defined by

(W(g}F)(f)=b(f,cj)F(f^g),i.e.,therepresentationused in theproofofTheorem 5.2.8 Assumeg eH\Mand let

andhence lV(g)i ^(M) .

We now considerquasi-local structure on the CCRalgebra.

Proposition 5.2.10 Let Hbealinearspace mthanondegenerate symplectic

bilinearform a, IfNandMaresubspaces ofH, defineN ^- M ifandonly if

(j(f^g) =Qfor allf^ Nandg^M Let Ibea netofnonempty subspaces of

H, orderedbyinclusion, such that:

(1) IfMel, there existsanN ^Isuch thatM I.N

(2) IfM A^N andM\.K, there exists an L eI such that M L and

N.KCL.

(3) // =

UMe/^-Let^(H} be the CCRalgebraoverH,and lett((M) be thesub-C^-algebras

generated by {W(f]\ fGM} for each MG/

777^/7 (^(H], {^(M]}^^f) is aquasi-local algebra in the sense of Defini

tion 2.6.3, mth a l.

PROOF This is immediate from the relation

W(f]W(g) =e-'^^^^'^^W(g)W(f)

for/,g G //.

Although this Statement on quasi-local structure is similar to Proposition

5.2.6, there is a distinction which arises from the phenomena described inProposition 5.2.9 and which is illustrated by thefollowing example.

EXAMPLE 5.2.11. LetHbe the subspaceofL-([R') formed bythe functions withcompact Support Moreover, let

a(J] g) =Im(/, g) .

If A isaboundedopensetof U^' and^IAis theC*-subalgebraof5l(//) generated by

{^(/);/ e^"(A)} then (^(//), {^Aj^^^^v) isa quasi-local algebra in thesense ofDefinition 2.6.3 However, ^(//) isnot equal to the CCRalgebra overL-([R'') be-

cause ofProposition 5.2.9

5.2.3 States and Representations

We continue the analysis of the GAR and CCR algebras witha discussion ofvariouspropertiesofstatesandrepresentationsof thesealgebras.Mostof these

properties are related to the existence of creation and annihilation operatorsand thiscauses adistinction between the CARs and CCRs The CAR algebra

contains creation and annihilation operators but these operators can only beaffiliated with special representations of the CCR algebra. We begin with a

discussion of these representations and associated states. We concentrate on

the CCRalgebraover apre-Hilbertspacel)and eschew thecaseofarealvector

space with symplectic form Aswehave explainedin Section 5.2.2.2 this latter

case is barelymore general.

There isa certain arbitrariness in the definition of the CCRalgebrafrom thegenerators ^(/) in Proposition 5.2.4, i.e., one could define this algebra by taking other functions of>(/) than /l -^e^^^\ In fact the C*-subalgebraof theCCR algebra generated by {W(tf)\t e IR} where / ef) is a fixed nonzero

element is isomorphic with the set of almost periodic functions on

[R=cr(>(/)), and there is no inherent reason not to operate with the set ofcontinuoLis functions on [R vanishing at infinity, or any other subalgebra of

Ciy(U)whichseparatespoints ofIR A consequence of thisanalysisis that oneisnot too interested in general representations of the CCR algebra, but only

representationswhere thegeneratorsof^^ ^(^/)exist Thesearethe so-called

regulär representations.

Arepresentation (, TI) of the CCRalgebra 51(1)), overthepre-Hilbertspace1), is said to be regulär if the unitary groups t G U\-^n(W(tf)) are stronglycontinuous for all /e f). If TT is regulär, then one can introduce, on , theselfadjoint infinitesimal generators ^T^(f),fGf) of the groups t^^n(W(tf]),

/GI), and thenuse these to define annihilation and creation operators.Similarly a state co over ^(l)) is said to be regulär if the associated cyclic

representation (^^, n.o-, ^co) is regulär. Note that

\\(n,,(W(tf)]-^]7L,,(W(cj))^,,\\^

=2 -e-^^^'^^^^^^^(D(W(tf]] -e^^^^^^^^^^a}(W(-tf))

and hence it easily follows that co is regulär if, and only if,

^ G [R^G)(W(tf)] G C is continuous for all /G i). We usethe notation^,o(f]

todenote the infinitesimalgeneratorof theunitarygroupTi^,:,(W(tf])associatedwith a regulär state.

The simplest example ofa regulär state is the Fock state cDp defined by thevacuum, or no-particle state, Q= (1,0,0, ) G e?_^(f)). One easily calculatesthat

cof,(W(f)) = (Q, W(f)ü) =^-ll/ll'/4

.

Even if co is a regulär state, there are certain technical domain problemswhich complicate the introduction of annihilation and creation operators.These arehandled, however, by the following quite straightforward result

Lemma 5.2.12 Let UI(f)) be the CCR algebra over the pre-Hilbert space

l)andCD a regulär state over 5l(^). For each /G l) denote the infinitesimal

generator ofthe unitary group t\-^7i(,j(W(tf}} by )f,j(/).

It follows that the operators {)c^(/), 0f^(//),/G M} have a common

denseset of analytic vectorsfor everyfinite-dimensional subspace MCI).

Moreover the annihilation andcreation operatorsdefined for eachfG f) by

/)(a,,(/)) =Z)((D,X/))n/)(a),,(//)) -D(/))

and

a^(f) =2-'/^($4/) +/(!)(//)), <(/) = 2-1/2 ($4/) _ .^_^(.y-))

aredensely defined, closed, flco(/)* =^w(/)' ^'^^

ll<I>(/)<Pll' +ll<l>('/)'?'ll' =2|K(/)'?'lP+ ll/ll'll'Pll'

for all (p er>(a(/)).

PROOF WemayassumethatMis acomplexsubspace. Let {fj-J= l,2. ,m} b&

anorthonormal basis ofM and definean operator R ong^ by

Rn= O'"/'^" ^'1 ^^'"'^^''^""''^'' ^""''"^'"''^(^(E(^^-+'''>)/>

It follows fromrotational invariance thatRn isindependentof the choice of basis.Moreover, if for i//ef^ one defines ij/^^ by i//,, /?j/^ then ||i/^,j-i/^|| -^0 by a si-milarargument tothat usedto proveProposition 2.5.22 Butusing

W(tfj)W((sj+itj]fj]=e-'"'I^W((sj+ s +itj)fj)

andachangeof variable in themultiple integraldefiningRn onecalculates thati// isanalyticfor eachO^^(/)).Hencei/^ isanalyticfor eachoperatorOfj(/),with/6 M.

It immediately follows that ciio(f] ^ndß*j(/) are densely defined on

D(0,,(/))nD(0,,(z/)). Moreover, (/)* D<(/) Thus a,,(f] has a densely defined adjoint which means that it is closable Similarly a^(/) is closable Now if

l|1?X/)('A-i/'JII-0 and ||(D4,/)(^-^J||^0. But <!.(/) and (D^//) are

selfadjoint, and in particular closed, and hence i/^ejD(Oa;(/))n)(<I)f,j(//))

=D(aco(f))- Another application of the identity gives \\aco(f)(^n~^}\\ ~^^ ^^^thereforeflfo(/) is closed A similarargumentis valid fora*j(/).

It remainsto provethat flfo(/)* =^lj(f)-This will be deduced äs acorollaryof

Corollary5.2.15

Ournext aim isto establish criteria fora state to be normal withrespect tothe Fock representation, i.e., thedefining representation of 9t(i^) on the Fock