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I nt ernational Journol of Theoretical Plty sics,Vol. 14, No. 5 (1 97 5 ), pp. 327 -346 D i s t i n g u i s h a b l e -a n d I n d i s t i n g u i s h a b l e - P a r t i c l e D e s c r i p t i o n so f S y s t e m s o f I d e n t i c a l P a r t i c l e s W. M. DE MUYNCK Department of Ph1,sics,Eindhoven University of Technology, Eindhoten, the Netherlands R e c c i v e d : 9J u n e 1 9 7 5 Abstract The problem of distinguishability of identical particles is considered from both experimental and theoretical points ofview. It is argued that distinguishability has to be defined relative to a definite set of experiments and that the criterion by which the particies are distinguished should be specified. Faiiure to do so may cause mismatching between theory and experiment. On the theoretical level a distinction is made between indexed- and unindexed-particle theories, indices being unobserved intrinsic properties of the particles. A fieid theory of indexed particies is constructed and shown to be equivalent to the second quantization formalism, which is an unindexed-particle theory. l. Introduction It wasobserved by Mirman(1973) that the conceptsof "identicalparticles" and "distinguishability"or "indistinguishability" havebeenpoorly formulated in quantumtheory.In nrosttextbooksthesenotionsaretreatedin a rather vagueand intuitiveway, without a firm relationto experiment.More or less the sameremarksapply to the relationbetween(in)distinguishabilityand the symmetryof the wavefunctionof a many-particie system("statistics"). Let us startwith the notion of "identicalparticles."Most definitionsare rathertautological, like the one givenby Messiah(1965): Two particlesare identicaiif their physicalpropertiesareexactlythe same.Strictly speaking, this definitionwould infer that two particlesnevercanbe identical,unless positionis considered no longerasa physicalproperty.Thedistinctionmadeby Jauch(1966)betweenintrinsicproperties(i.e.,propertiesthat areindependent of the stateol the system)and extrinsicproperties(dependingon the stateof the system)seemsto be more promisingin this respect.Jauchdefinestwo (elementary) particlesto be identicalif they agreein ail theirintrinsicproperties. Position,asa dynamicalvariable,is an extrinsicproperty.So Messiah's problem d o e sn o t a r i s ea n y m o r e . @ 1976 Plenum Publishing Corporation. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, elecfronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher. 327 328 w. M. DE MUYNcK The distinction betweenintrinsic and extrinsicpropertiesalsointroducesthe possibilityof revealingthe dependenceof the identity of particleson the experimentaland theoreticalcontext. Thus, by enlargingthe classof experiments by which the identity of two particiesmay therearetwo causes considered, property may be discovered,showingthe difchange:Either a new intrinsic o f t w o p a r t i c l e tsh a t w e r et h o u g h tt o b e i d e n t i c a lo: r a n ( u p u n t i l n o w ) ference intrinsic property may show up dynamicalbehavior,thus making two different particlesidentical(when the other intrinsic propertiesare the same).We shall havereasonto changeJauch'sdefinition slightly in sucha way that this depen' denceof the identity of particleson the classof experimentsconsideredis more explicit. As this hasto do with (in)distinguishabilityof the particles,we considerthis next. As to (in)distinguishability,severaldifferent opinionscan be lound in the literature,e.g.,(a) in classicalaswell asin quantum mechanicsidentical particles (Messiah,1965), (b) identicalparticlesare distinguishable are indistinguishable quantum mechanically(Jauch,1966), (c) same classicallybut indistinguishable as (b) when the wavepacketsofthe particiesare overlapping;but identicai particlesare distinguishable alsoquantum mechanicallywhen they are far apart or when there is someconstantof the motion by which the particlescan be distinguished(Schiff, 1955),(d) classicallyaswell asquantum mechanically wheneverthe historiesof the particles identicalparticlesare distinguishable experimentallyallow a distinguishingcriterion (Mirman, 1973). Whereas Messiah(a) seemsto usethe conceptsof identity and indistinguish' ability of particlesin a tautologicalmanner,the othersgo beyondthis by trying to find a way to label or namethe identicalparticles,thus trying to give eachparticle a kind of individuality that is maintainedthroughout time. Cases (b), (c), and (d) havein common that the labelingis thought to be performed by a dynamicalvariableor extrinsicproperty, the intrinsic propertiesof identicalparticlesbeingthe same.In classicalmechanicsgenerallythe position alsohere, stemrning variablewill do, althoughMirman noticessomereservations from experimentalinability to follow the continuouspathsof particleswhen they the absence ofparticle trajectories arevery closetogether.In quantummechanics makesthe position variableinappropriatefor distinguishingpurposeswhenever the wavefunctionsof the two particlesare overlapping.This leadsJauchto the rigorousconclusionof indistinguishability,whereasSchiff points out some exceptionsto this rule.Accordingto the latter,two identicalparticlesare when the two-particleprobabilityamplitudea(\ ,2) of some distinguishable dynamicalvariableis different from zero only when the two particleshave their valuesin disioint rangesof the spectrumof the variable.Note, however, that this can neveroccur when the wave function is (anti)symmetric,for then a(2,1)= !o(1,2). In order to analyzethe experimentalmeaningof the conceptsof identical particlesand distinguishabilityMirman (1973) considerstwo electronsin different galaxies.A more common paradigmcould be providedby the situation of two identicalparticlescrossinga bubble chamberat the samemoment with- DISTINGUISHABLEANDINDISTINGUISHABLEPARTICLES 329 out colliding.Whenthe particle energiesarehigh enoughwe seetwo completely separatedand distinguishabletracks.With Mirman thesedata seemto be sufparticles:The particlesare distinguished ficient to speakoftwo distinguishable by their tracks.However,when the wavefunction is (anti)symmetric,this is clearlyin conflict with Schiff s view,becausein that casewe cannot tell which of the two particlesis in which of the two tracks.Accordingto Schiff we could particleswhen there is a one-to-onecorrespononly speakof distinguishable dencebetweeneachof the tracksand one of the indicesI and2 of the two particles.It is clear from this that studyingthe experimentalmeaningof distinguishabilityamountsto studyingthe experimentalmeaningof the indices1 and 2 of the two particles.This can be done for classicaland quantum mechanics simultaneously,becausein the elementaryforms of both theoriestheseindices obtain. Sincean index is thoughtto be independentof the stateof the particle,it isto be considered asan intrinsicproperty.As a consequence of this, according to Jauch'sdefinition particlesthat havedifferent indicesare not identical.By ttris reasoningthe problem of distinguishabilityis definedaway,becausenonidentical particlesbeing distinguishable, this leadsto the conclusionthat the particlesdescribedby elementaryclassicaland quantum mechanicsare distinguishable.However,this statementfollows from purely theoreticalconSo it tellsmore about the form of the theoriesthanabout the siderations. experimentalmeaningof the indices.The only conclusionwe can draw is that the languages of elementaryclassicaland quantum mechanicsare those of distinguishableparticles.Suchlanguages are bound to be not very adequateto describeparticlesthat are indistinguishableexperimentally,which may be the reasonfor the different viewson distinguishabilityfound in literature.These languages would only be adequatewhen detectorswere existingthat are sensibleto the indices.Whenthis is not the caseit is preferableto uselanguages that areindependentof the indices,i.e.in which dynamicalvariableswithout experimentalmeaninglike r;, the positionvectorof particle7, do not showup. Suchlanguages do exist for quantum aswell as classicalmechanics:quantum field theory (secondquantization)and the classicallimit of the field theoretic approachto statisticalmechanics.Quantitiesthat do haveexperimentalmeanings for particleswithout indicesare, for instance,the probability p(r)dro of finding a particle in the volume elementdro nearthe spacepoint ro, or correlationfunctionslike p(ro, 16)correlatingthe probabilitiesof findingparticles in spacepoints ro and 16.It is essentialthat the indicesa andb do not refer to particleshere,but to detectorsat spacepointsro and 16.In this language the only relevantquestionis whethera detectorhascounteda particleor not, particleindicesplaying no role. Distinguishabilitynow reducesto Mirman's purelyexperimentalnotion (Mirman,1973),expressed by, e.g.,the mutual distanceof the two detectorpositionsro and16 and the conductol p(ro, 16) asa functionof theseparameters. This point of view seemsto be consistent with the methodLyuboshitz& Podgoretskii (1969, 1971)useto treat the interferenceof nonidenticalparticles. 330 W. M. DE MUYNCK 2. Experimental Relevanceol Indices Apart from particle creationand annihilation,elementaryquantum mechanics 1930).Also the and secondquantizationareequivalenttheories(Heisenberg, theoriesof particleswith and without indicesareequivalentwhen only classical that areindependentof the particleindices. resultsareconsidered Then, using asintrinsicproperties. As we saw,indicesareto be considered Jauch'sdeflnitionof identicalparticles,particlesthat areidenticalin the unindexedversionof the theory are differentin the indexedone.This is undesirof the sameset ablebecausethe theoriesdescribethe samesetof experiments of particles.To overcomethis difficulty we changethe definitionof identical particlesslightly,makinguseof the observation that apparentlythe index does not haveinfluenceon the dynamicalbehaviorofthe particles.So particiesare identicalin both theorieswhen we changethe definitionin the followingsense:1 Two particlesare identicalif they agreein all their dynamicallyrelevant intrinsicproperties. considered, the reievance of the set of experiments This definitionstresses intrinsicpropertiesthat aredynamicallyirrelevantwith respectto because may becomedynamicallyrelevanton extendingthis someset of experiments performedwith set.For instance,considerthe setof all collisionexperiments irrelevant two billiardballsthat differ onli, in color.Then color is a dynarnicall-v" intrinsicpropertyand assuchmay serveasan index.The two ballsareidentical with respectto the setof collisionexperiments. However,when we extendthe to includealsoabsorptionand reflectionof light, the two set of experiments ballsareno longeridentical,for they will showup differentdynamicalbel.ravior in absorbingand reflectinglight. of the notion of "identicalparticles"on the Besides showingthe dependence experimental setting.the examplerevealsanotheraspectpertainingto the identicalparticlesby indices:When experimental meaningof distinguishing particlesareidenticalwith respectto somesetof experirxents they canonl,'-be distinguished by an index when it is possibleto extendthe setof expcriments in sucha way that the index becomesdynarnicallyrelevantwith respecttr.r somenew property.Whensuchextensionis impossibleor unknorvnthe index the particles.ln in distinguishing doesnot haveany experimentalrelevance this caseonly Mirman'scriterionremains.This seemsto be the situationrn elementaryparticlephysics.However,this is not a reasonto deny all physical quanturn to the indicesofthe particlevariables obtainingin elenrentarv relevance by meansof indicesis basedon inrnechanics. As we saw,distinguishability of the descriptionof the dynamicalbehaviorof the particles. cornpleteness language is an indicationthat Then,the merepossibilityof an indexed-particle therernightbe somenew field of research connectedwith theseindices. I It rvill turn out to be essential that the definition also concerns particles that are not elementary. DISTINGUISHABLEANDINDISTINGUISHABLEPARTICLES 331 Someevidencein this directioncould be providedby the problemof a dynamicaldescriptionof the processof (anti)symmetrization of the wave function when two identicalparticlesapproacheachother from greatmutual distance(Mirman,1973).As Mirmanstipulates, when two identicalparticles are createdsimultaneouslyat greatdistancefrom eachother, immediate (anti)symmetryof the two-particlewavefunction wouid violate causality. This showsthe necessityfor studyingthis (anti)symmetrizationprocess,which cannotbe doneusingexistingquantumfield theories,because theseareconstructedas causaltheoriesthat do correspondto imnrediate(anti)symmetry of the wavefunction. Thereexistsat leastone physicalsituation in which different symmetriesof the wavefunctionsofinitial and final statesarenecessary to describeexperimentalevidence. It is not surprisingthat this situation,viz., that illustrating the famousGibbsparadox,hasbeenconnectedwith the probiemof distinguishability of identicalparticlesfor quite a long time (e.g.,Schrodinger, 1957). One considersan ideal gasof /y' identicalparticiesin voiume V, initially divided by an impermeable diaphragmin two parts,4 and.Bhavingthe sametemperaWhenthe diaphragmis removed,entropy,beingan extensive ture and pressure. property,shouldbehavein sucha way that the entropyof the combinedfinal stateshould equalthe sum of the entropiesof partsA andB in the initial state. From the quantummechanical definitionof entropy S=-kTrplnp (2.r) resultingin the Sackur-Tetrode expression (Schrodinger , 1951)for systemsin thermalequilibriurn,it canbe seenthat this is satisfiedwhen the densityoperator of the initial stateobeys P = PtPs (2.2) whereasthe final stateshouldbe totally (anti)symmetrized. The density operatorspa andp6 arethoughtto describethe subsystems,4 and.Band correspondto (anti)symmetrized statesof thesesubsystems separately. Whenthe particlesol A aredifferentfrom thoseof ,Ba diffusionprocessis startedby the removalof the diaphragm,resultingin a homogenous mixture of the two gases.Whenthe particlesin A andB are identicaithis diffusion processis often considered to be unnoticeable or evenunreal(Schrodinger, 1957).This paradoxicalsituationmight be resolvedwhen the particlesare indexedparticles.Thenthe diffusionprocessis asreal asin the caseof differentparticles.Moreover.whenthe (anti)symmetrization processis considered asa descriptionof this diffusionprocess, it is not evenunnoticeablesinceit clearlysubstantially influencesthe stateof the total system. The relationto distinguishability by meansof indicescomesinto play through equation(2.2). HereA andB areindices,not of particles,but of assemblies of particles(they correspond to particleswhenl and.Bcontainone particleeach). The possibilityof introducingdensityoperatorsof well-definedsubsystems,4 and.Bin the theoreticallanguage is clcarlyrelatedto the fact that the particles J3Z W. M. DE MUYNCK experimentallyfrom thoseof subsystem^B of subsystem,4are distinguishable aslong asthe impermeable diaphragmis present.So,4 andB aredistinguishing indices,correspondingto the regionsto which the subsystems,4and -Bare confined experimentallyin the initial state. In the indexedlanguage -T^ ^ rrBP, PA- -T^ ^ yBLLAP (2.3) whereTrl , T16 meanspartial tracingover the particlescontainedin A, B. The questionof distinguishabilityof the two subsystems by the indicesA andB after removalof the diaphragmis connectedwith the feasibilityof expressions (2.3)in this latter situation.It is often statedthat thereexistsa causalrelationship betweenindistinguishabilityand (anti)symmetryof the wave function. If this were true, the indicesA andB would loosetheir meaningin the processof (anti)symmetrization.However,we will seein the next sectionthat the causal relationshipmentionedabovedoesnot exist. This will remove(anti)symmetry of the wavefunction as an a priori obstacleto distinguishabilityof the two subsystems by meansof the indicesA andB, alsoin the final state. The experimentaldistinguishabilityof systems,4and,Bby the indices,4 andB in the initiai stateis basedon the one-to-onecorrespondence of some intrinsicproperty of the systemswith positionsof the particles.This correspondenceis lost in the final state.Becausein the diffusion processexperimental distinguishability in the senseof Mirmanby, e.g.,the positionsof the particles breaksdown, only the yet unknown property is left to distinguishA andB. lt is possibleto ignorethis property and to leaveit completelyoutsidethe theoreticallanguage.However,in acting this way we may neglectsomephenomena that seemto be not devoid of phvsicalrelevance. 3. Indistinguishability and "Statistics" As we sawin Sec.1 the ianguageof the elementarywavemechanicsof wave particles.Although functions29(*r,xz,. . .,xr) is a language of distinguishable it is not impossibleto stick to this descriptionwhen the indicesare thought to be physicallyirrelevant,someextra careis needed.Oneinstancewherethis becomesapparentis providedby the attemptto derive(anti)symmetryof the wavefunction ("statistics") from indistinguishability,madein sometextbooks (e.g.,l-andau& Lifshitz,1965;Blokhintsev,1964).Thereasoning is asfollows: C o n s i d etrh e e x p e c t a t i ovna l u e 3o f , e . g . at h e o b s e r v a b l e f ( x t , . . , x , , ) , V @ t , . . . , x n ) ) =) O * r . . . l d x " , f ( x y , . . . , x n ) l r l , @ r ,.. . , x , ) 1 " ( 3 1 ) Becasue of indistinguishability ol the particlesthe followingequalityis postulateda: (f(x r, . . ., xn))= (f(P(xr, . . ., x,))> (3.2) '^ *i = Xi, rry).17 being the spin variable of particle 7. ] IdxT stands for integration over the space variable and summation over the spin variable " The same should be done with functions of momentum or anv other one-oarlicle observable. DISTINGUISHABLEANDINDISTINGUISHABLEPARTICLES 333 P ( x r , . . . , x n ) b e i n ga n a r b i t r a r yp e r m u t a t i o n o f ( x 1 , . . . , x r ) . W h e n( 3 . 2 ) shouldhold for any functionl, the equalityinevitablyleadsto I / ( P ( x r , . . . , x ) ) 1 2 = l r l t ( xt , . . . , x n ) 1 2 (3.3) which impliessymmetry or antisymmetryof the wave function. But as the particlesare presupposedto be indistinguishable (at leastby way of indices)not everyobservable f(x r, . . ., xr) has physicalsignificance.Only thoseobservables that havea counterpartin the unindexedtheory shouldbe considered in (3.2).This beingpreciselythe classof totally symmetricfunctions for which (3.4) equation(3.2) is fulfilled trivially for any wave function. So indistinguishability by indicesdoesnot lead to any requirementon the form of the wavefunction. On the other hand, when the particlesare thought to be distinguishable by their indices,we haveno reasonto expectequality of, e.g.,the expectationvalues (r1) and (12)of the positionvariablesof two identicalparticlesI and 2 for all possiblestatesof the system.As we sawin Sec.2 it is possiblethen to prepare statesin which the particles1 and,2are in disjoint regionsof space.lt follows that in this casethe requirement(3.2) is not legitimate.So, alsofrom this point of view (anti)symmetrizationof the wave function is not invoked. A derivationoi (3.a) is givenby Schweber(1961),startingagainfrom (3.2), which is requiredby indistinguishabilityof the particles.It is interestingto note that (3.4) only follows irom (3.2) when this last equality shouldhold for c// functions ,!(x r, . . ., xr). Whenonly symmetricor antisymmetricwave functionswould play a role in the physicsol identicalparticles,it would be sufficientto consider(3.2) for statescorresponding to this classof functionsonly. But underthis circumstance (3.2) is satisfiedfor any observable/(x t,. . .,xn). So,in orderto arriveat (3.4) asa necessary conditionthe observables of a systemof identical particlesshould obey, it is essentialthat non- (anti)symmetric wavefunctionshaveexperimentalpertinence.But this impliesthat the whole reasoningbreaksdown because,as we sawin Sec.2, in this casethe indicesmay havean experimentallyreievantdistinguishingfunction. We arrive at the conclusionthat a fundamentalrelationshipbetween "statistics" and indistinguishabilityby meansof indicesof identicalparticles is not evident Note that this conclusionis not at variancewith the examplegivenin Sec.2, wheredeparturefrom "statistics"involveda kind of distinguishability. As a matter of fact, this distinguishability(the Schiif version)was arrivedat through position measurement,not through direct observationof somephysicalproperty correspondingto the index. The one-to-onecorrespondencebetweenposition and index obtaininghere is a very specialcircumstance,not liable to be a generalissue:Deviationfrom "statistics" is not bound to imply this distinguishabilityby indices.The intermediatestatesin the diffusion processdescribedin Section2 might furnish illustrationsto support this J-)+ W. M. DE MUYNCK view.providedthe statefunction(or densityoperator)ol the systemevolves in a continuousfashionfrom initial to final statealsoin this case. it shouldbe possible "statistics"not beingrelatedto (in)distinguishability, to describethis processusingeitheran indexed-or an unindexed-particle language an indexed-particle However,viewedasa diffusionprocess, language. unindexed-particle than an manner it in a conceptually simpler may describe language.This is the casebecausenow (anti)symmetryof the wave function of, e.g.,two identicalparticlescanbe interpretedin a simpleway by stating that, after the diffusionprocesshasreachedits stateof equilibriurn,the equality l i t ( x o . x o 1 1=2l Q ( x 6 .x o l l 2 expresses the equalityof the probabilityof findingparticles1 and 2 with coand the probabilityof findingthe particles ordinatesxo andx6, respectively, with their positionsand spinsexchanged. systems[e.g.. By the diffusionprocessan initial stateof two uncorrelated with densityoperator(2.2)] is turnedoverinto a final (equilibrium)statein which the systemsarecorrelated.It is preferableto describethis correlationin i.e.,by meansof the commutationrelationsof the the unindexedlanguage, it is not a correlationof the field operatorsof quantumfield theory,because indices(which seemto be distributedas randomlyaspossiblein the equilibriunr state),but ol the probabilitiesof findingparticlesin differentpoints of space (cf. Sec.I ). However,the theoriesbeingequivalent,the correlationshouldalso be expressible in the indexedlanguage. by A fundamentalrelationshipbetween"statistics"and indistinguishability meansof indicesnot beingevident,we haveto considerthe tendencl,of an assembiyof identicalparticlesto developits stateinto a stateshowinga specific or Fermi-Dirac"statistlcs").asan kind of correlation(eitherBose-Einstein independentand fundamentalpropertyof the particles.Approaching"statistics" the in statisticalmechanics in understanding in this way offersa big advantage factori/l by whichthe a priori probabilitiesshouldbe dividedin orderto get the (correct)Boltzmannweights(e.g.,$4.2 of ter Haar,1961).It is often effectbecauseonly statedthat this correctionis reallya quantummechanical of identicalparticleshavean experimental herewould indistinguishability the f.actor meaning(however,seeRushbrooke,1949).In quantummechanics of the restrictionto (anti)symmetricstates. ly'! turnsout to be a consequence i.e. to correlatedslates.But correlationis not a typically quantum nrechanical notion.We haveno reasonto supposethat this propertygetslost in the classical1imit.So we havethe possibilityof attainingthe factorIy'1alsoclassicalll,. when we postulatealso for classicalmechanicsthat identical particlesare necessarily conelated particles(at leastat equilibrium).In fact this is equrvalent to the replacement alreadyperformedby Gibbs,of "specificphases"b-v "genericphases."(Gibbs.1902).No allowancehasto be madefor indistinguishability of statesunderexchangeof particlesin phasespacewhen this "statistics" correiationis consideredalsoin classicalmechanicsasa fundamentalproperty ol systemsof identicalparticles.Apart from procuringthe factor,ly'!in classical mechanics, "statistics"correlationofthe particlesdoesnot seemto havecon- DISTINGUISHABLEANDINDISTINGUISHABLEPARTICLES 335 sequences. This is illustratedby the vanishingin the classicallimit of all interferencetermsbrought forth by (anti)symmetryof the wavefunctions,making expectionvaluescoincidewith thoseobtainedfrom unsymmetrized.wave functions.So it is possibleto continue usingthe conventionalclassicalmechanics particles,asthe "statistics"correlationmanifestsitself only on of uncorrelated the quantumlevel. 4. A Field Theory of Indexed Particles Elementaryquantum mechanicsis not unconditionallythe indexed-particle versionofquantum field theory,sinceparticlecreationand annihilationis not describedby the first theory. In this sectionwe will presenta field theory ol indexedparticles,i.e.,a theoryby which it is possibleto describecreationand annihiiationof indexedparticles.In orderto performthis extensionof quantum field theory, we start from the very clear formulation of this theory givenby Robertson(1973).In the unindexed-particle theory a systemofi/ bosonsor fermionsis represented by a statevectorin Fock spacelV17)whichis related to the (anti)symmetricwave function Vr("r, . . ., x,rr) through the formula I v , n , )(=w ) - t I 2J a xr ' ' J d x r v " t x , . . . . , x , , y ) 0 + ( x r ,0' )f'(' x' r ) l o ) (4.1) Uerer/r(x)[r/(x)] is the usualcreation(annihilation)operatorof an unindexed particleat x, obeyingthe comrnutationrelationsfor bosonsc.q. fermions [0(x),0(x')]+= 0 lU("),gt(x')l+ = 6(x- x') $.2a) (4.2b) , e f i n e db y l o ) i s t h e v a c u u ms t a t e d p ( x )l o )= 0 (4.3) It is shownby Robertson{1913) that the wavefunctionVr(xr, . . ., x1,.)is obtainedfrom the Fock state(4.1) through vr(x,, . . ., x,rr)= 11t.'.)-rlz <o\l)(xl). . . /(xr,,)l vru> @.4) We point out herethat (4.1) is not a uniqueexpression for the statevector lV,r,,).We may replacethe functionV"(xt, . . ., x,nl)in the integrandof (4.1) by an arbitrary function V(x, , . . ., x1,,)obeying V"(xr, . . .,x.nr)= (1///l) ; /v1r1x,, . . .,xry)) (4.s) the summationbeingperformedoverall permutationsof (x1, . . ., xry); for bosonseP = *1, for fermionseP = +1 or -1 when the permutationPis evenor odd. The result(4.4) is independentof the specificform of lV,rr) aslong as V(xr, . . ., x.nr)obeys(4.5). It is not possibleto constructa completedescriptionof indexedparticles usingjust a singlefield operator r/(x). Whenall particleshavedifferent indices, 336 W, M. T]E MUYNCK a different field operator p;(x) has to be associated with ear;hparticle.We will cali this operatoran indexed field operator.The vacuumstate,which is as usualdefinedby r f; ( x ) l o ) = 0 , V e r 9$) is the directproductofthe vacuumstateslor) ofall particlescontainedin the systemunder study (index set1). We will restrict our attention here to systemsin which the particlesare all characterized by different indices,althoughthis may turn out to be too restrictive, asis shownby the exampleof Sec.2, whereall particlesof subsystemA or B may be thought to havethe sameindex. However,field operators ,l,t@)19 a@)l describing the fieldsof the A(B) particlesarerelatedto the one-particlefield operator p(x) in the sameway as the unindexedfield operatorsof bosonand fermionfields.Sincein Sec.6 we shallfind the relation betweenindexedand unindexedfield operators,we do not needto study many-particlefield operatorslike rf7 (x), which moreoverseemto be less fundamentalfrom the point of view of distinguishability. We define the statevector correspondingto a systemof Iy' differently indexed particleswith wave function V"(x t , . . ., xry) by analogyof (4. 1) as 1 v r , . . . , r y ) = l o *J,axy,rvr(xr,. . .,xry){i,(x,nr)'. . r/f (x, ) |o) (4.7) Wealsorequirethat the wavefunctionis foundfrom lV1,. . .,1,,) through Vr(x r, . . ., x,rr)= b l t! tft r ) . . . {,nr(x,nr) lVr, . . .,ry) (4 8) In (4.1) the "statistics"correlationof the particlesis incorporatedin the statevectorin two ways:by the commutationrelationsof the field operators and by the symmetrycharacterof the functionV"(x1, . . ., xrr). We haveseen that the first one is sufficient to give the right "statistics." In the formulation we developin this sectionwe will choosethe secondmanner,i.e.,we introduce "statistics"in the statevector(4.7) by meansof the symmetrycharacterof the functionV"("r, . . ., x.rr).By doingso the possibilityis createdof a uniform description, independentof"statistics,"ofthe indexedparticles,be it bosons. fermions,particlesobeyingparastatistics, or evenparticleswith no correlation at ail. The commutation relationsof the field operatorsr/;(x), etc. may be the samein all cases. Sincefor uncorrelatedparticlesno symmetry requirementsare to be imposed on the function,Vr(x1,.. .,x,rr)[email protected]) and (4.8), it followsthat . gr ' ( x,^,,W hO*1. . . Vr l( y,) tol=fi s( x;- yi) (o l r!1 (x1. ). i= L (4.e) I N u DISTINGUISHABLEANDINDISTINGUISHABLEPARTICLES 337 Another aspectof applicabilityofthe theory to uncorrelatedparticlesis expressibleby the property that indexed field operatorswith different indices commute.Thus tgi(x),,1' i1)l _= [g;(x),pl fyn_= 0,i + j (4.10) It incidentallyfollows from (4.6) and (4.10) that { i ( x ) 1 V r . . . . . n ) =0 , i + { 1 , . . . , 1 / } (4.11) which is a most desirableresult. A third requirementfor the indexedfield operatorsis suggested by a relation that is equally valid for bosonsaswell as fermions,viz., , t , @ ) { t( y ) l o ) = 6 ( x - y ) l o ) As this clearlyis independentof "statistics", we expectthat it should hold also for our indexed field operators.So we require A { i { 1 ( l ) l o )= 6 ( x- y ) l o l (4.12) As the equalig (a.9) is a direct consequence of (a.10) and(4.12),we may omit the requirement(a.9). The one-and two-particleoperatorscorresponding to the configurationspaceoperators7';(x;) and V;i(xi, x), i * i are also definedby analogywith quantum field theory, as ri = (flri0'),tti(x) (4.13a) JaxP! Vii = | dx Iay,,i @Vi (y)Vii@,y),ltd y)'! t@), i + i (4.13b) Note that in quantum field theory, becauseof the presupposed indistinguishability of the particles,only operatorsareconsidered corresponding to observables that are symmetricin the particle indices,suchas t/ and ,Zrr'{*') N 2 Vii@i,xi) 'i,ji il Wth (4.13a) and (4.13b) it is quite easyto showthat (4.I 0) and (4.12) are sufficientfor the equivalence, underthe correspondence definedby @.7) ot the configurationspacerepresentation wave functions.Lr(xr, .. .,x,^r)] [by a n dt h e r e p r e s e n t a t i o bn y s t a t ev e c t o r sl V r . . . . . n ) . F o r a l l k i n d so f " s t a t i s t i c s " this equivalenceis expressedby the equalities '" ( @ r , . . . , . n r 1 V r , . . . , r=y ) lo*r f a x r y o f ( x r , . . . , x - n r ) V r ( x r.,. . , " , n r ) @.raa) ( @ r , . . . , , n r 1 4 1 v r , . .l .o,*. rn.r.). = i a x r o $ ( x , , . . . , x n r ) T i ( x i ) ,V. .".(,xxrN ) , ic_{i,. ,n} @.14b) 338 W. M. DE MUYNCK ( @ r.,. . , nl V i j l Vr , . . . , N > = tJf o r t " ' J d r " o f ( " , . . . , x y ) V ; 1 ( x i , x ; ) V " ( x r ,.. . , x N ) , i,j e {1,. . .,1/} @.1ac) F r o m( 4 . 1 1 )i t i s i m m e d i a t e lsye e nt h a t 7 " ; 1 V 1 , . . . , r ay n ) d V i 1 l 91 , . . . , 7 , / ) v a n i s h w h e n p a r t i c loeriT i s n o t p r e s e n t i n t h e s y s t e m d e s c r i b e1dVbry, . . . , 1 , , , ) . Also, the equivalenceof the Schrtjdingerequationof a systemofIy' indexed identicalparticles H ( l , . . . ,1 / ) l V r , . . , , v = ) r , . .. , N ) ) i h ( d l d tl V H ( 1 , .. ., AI)=.F10(1, . . .,il) + V(1,. . .,N N. 162 = > l axltitill ;;a* /Jvr i=lr +\ i I ;=l \ vutllr i!) \ ,{rr+ I ro,tf (ilv(, ,i,tt i,)tt {x) I a* |ay"1,! J (4.15a) J (4.1 sb) i=i with the usualnon-relativisticequationin configurationspace,is easilyderived. To concludethis sectionwe mention somefurther propertiesof the indexed field operators.From the physicalinterpretationof the operatorry';(x)asan operatordescribinga field of onTyone particle,it follows that ,lti$)rl,i(_y)= 0 (4.r6) In orderto study the commutatorof rf;(x) and r/lfv) we switchin the usual mannerto a discreterepresentation: 'lt{x)= \ af up@) (4.17) where{47.(x)}is somecompleteorthonormalsetof one-particle functions. Thenaf 1aff ) is the annihilation(creation)operatorof particie i in the state with wavefunctionu6(x). [email protected])into (4.10),(4.12),and (4.16)we get lof.oil = lo'f.oj l-=o. i + i ofo',rlo>= 6111o) ofoti=o ( a . 18 a ) (4.l 8b) (4.I 8c) Whenwe introduce for the one-particlestateof particlei with wave function .I,r(x) = uo(x) the notation lk1)= afT 1s7 we concludefrom (4.18b)thata! maybe represented by a! = lo)(kil (4.l e) (4.?.0) DISTINGUISHABLEANDINDISTINGUISHABLEPARTICLES 339 From (4.20) we easilyderivethat = 6(x* y) Ioi)(otl -,1t!0) loi) <oil v i@) @.2r) lf ;("),,l,lCyiland (4.22) I,t,{y), p,i("){i(")l-= 6(x * y)lt;(x) whichmaybe usedto find the equationof motionof r/;(x) in the Heisenberg picturewith theHamiltonian (4.15b)as do,(x) | I n2 ihry=l--: A+v@)ll)i(x) at l2M I nr' +4 ,t'i(x) t i-t Io, Vf*, y) + v(y, il Vl O)0iO) t (4.23) (i+i) It is alsoeasilyverified that the operator u,=lany!1n{i(x)= 1 o!'o! (4.24) commuteswith the Hamiltonian(4.15b),expressing conservation of the probability that particlei is presentin the systemdescribed by this Hamiltonian. The operatorcorrespondingto the number of particleswith statefunction up(x) is givenby ttk = ) afra! since N k a ! , ','I' o f i JO > =n r , a f l 't' ' o ! # ' l o , (4.2s) npbeingthe numberof timesk is represented in k1 ,. . .,kN. The operators Nk aredeflnedon the Hilbert spaceof indexedparticles.in which the vectors o ! , ' ' ' ' o f # l o ) . i / = O , 1 , 2 . . . , i p l , i 1* i y , c o n s t i t u t ea n o r t h o n o r m abl a s i s . The operatorcorrespondingto the total number of particlesnow follows from (4.24) and (4.25) as X=>N,=> wk=la!ra! (4.26) ikik For the Hamiltonian(4.15b)we arrivein the k representation at the expression N H ( r , . . . , 1 / ) =) i=l -+ - iy' l H f p ! 'a t i + l kl -+ .+ i,j=lklmn (i + i) rlnz I ufl,= | dxu[(x)| - n a + vg)lu{x\ "LLtrtl Vkt-n = ax at uf (xlufg)v(x, y)u^(,,)un@) ! ! 9.27) 344 w. M. DE MUYNCK Mirman (1973) strongly opposesagainstwhat is called"physical exchangeof particles,",inwhich two particleschangeplaces.Indeed,in a theory of indistinguishableparticlesthis notion makesno senseat all asit is then thought to be impossibleto distinguishexperimentallybetweenthe original stateand one in which the particleshavebeenexchanged.In a theory ofindistinguishableas particlesan interactionis neededto perform the exwell as distinguishable change.However,in an indexed-particletheory a new possibilityopensup for interpretingthe "physical exchangeof particles"that avoidsthe difficulties connectedwith the more or lessmysteriousway in which two materialparticles shouldbe able to exchangetheir places.For, ifnot the particlesthemselves, but just their indiceswould exchange,the final statecould not be distinguished experimentallyfrom a statethat resultsfrom a reai physicalexchangeofthe just the indices particles,includedindices.So an interactionthat exchanges could be interpretedas an exchangeinteraction. However,when index exchanginginteractionsare presentit is no longer possibleto usethis index for distinguishingpurposes.As a matter of fact preciselythe presenceof this kind of interactionwould givethe index the status particlesis possible of a dynamicalvariable.So a theory of distinguishable only when the interactionsare index preserving.The Hamiltonian(4. l5b) clearlyis of this kind as = o, x * y btgi@)'!i7)v(r,. .. ro,lft*w,,I(J/)to> 5. Indexed Boson Operators In order not to be botheredby phaseconventions,in the following we to bosons.Usingthe indexedparticleoperators restrictour considerations definedin (a.18) and (4.20) a normalizedsymmetricstateof y'y'indexed bosonsmay be represented by l i r " ' r r r r t k r " ' k l r ) s = { N r . I l 1 , n 1 , t . } - t l ' Z d : ' '" 4f,*'lo, (5.1) Againnyis the numberof timesthe singleparticlestatek occursamongk1, . . . , k u ' , f u r t h eP r k t , " ' , P k r y s t a n d fso r t h e p e r m u t a t i oPn( k t , " ' , k f f ) . I t i s possible(Jauch,1966)to definea projectionoperatoron the Fock spaceof symmetricstates,IIr, accordingto ++ I l r o f , " ' . . o ! # ' l o ) = Q { ) - r l 2 ( t l p n p l ) rl z l i r ' ' ' r r . rk; r ' ' ' k N )s $ . 2 ) This projection operatoris expressibleexplicitely in the indexed field operators followine < oIaflN' " o?,n' 4:' ' ' ' ' df,*' lo> ( s.3) In (5.3) the summationover i 1, . . ., iN is extendingover all possiblesetsof ly' different indices. 34I DISTINGUISHABLEANDINDISTINGUISHABLEPARTICLES The ooerators ,b! = tl"a!rl, ,nft = rt,oftn, (s.aa) (s.4b) propertiesthat are analogousto those of the are easilyverified to possess usualannihilationand creationoperatorsofunindexedbosons,viz., l i r " ' r , r rkt 1 ' ' ' k r ) s = ( N ) r l 2 $ t k n y l ) - t l 2 , b ! , , 1 ' ' ' , U f p f l o > ( 5 . 5 ) , b ! l t r ' ' ' i N ' , k r ' ' ' k r r ) , = p , n u r l 2 1 1 t - r l 2 V r i' ' ' ' ,bfr lir. i 1 i 1 ; k 1 k' ' ' kn), (s.6a) . . in,k1. . . kytg = 0 * p ) ( n k + l ) r l 2 ( N+ t ) - r | 2 f t r . . . i v r i ; k . . . . k N k > s (5.6b) where 1 1 ,i € { i r " ' r 1 1 } P;= | to,r+{ir' rl/} The equalities(5.5) and (5.6) lend supportto the interpretationoIthe operators and rbf+ asannihilationand creationoperatorsofan indexed "bf boson.Contrary to the creationand annilationoperatorsdefinedin Sec.4, they pertain to particlescorrelatedaccordingto "statistics." Yet they havea wider scopethen the creationand annihilationoperatorsof unindexedfield theory, asthe operators(5.4) are also definedoutsidethe Fock spaceof symmetricstates.However,sincethey are adaptedespeciallyto this space,it is questionablewhethera physicalmeaningcan be attributed to theseoperators outsidethe Fock spaceof symmetricstates.For this reasonwe will only study their behavioron this latter space.The commutation relationsof the operators ,b! andrbf* areeasilyderivedfrom (5.6) in this case.We get kbf , ,bjl_=1,bfr, ,bj1)_=o y,jn, (s.7) which relationsare evenvalid outsidethe Fock spaceof svmmetricstates.The commutationrelation [rDf , rDjf ]-is rather a bulky expressionwhich we shall not write down here. In orderto arriveat a dynamicaldescriptionofa systemofindexed bosons by meansof the indexedboson operators,we mention the equality allir " ' iyik1' ' 'k')" = p! lrr.. .iu;k.. '- kp)s (5.8) (note that there doesnot exist an analogousequality for the creationoperators). Replacingin(4.27) the operatorsaf etc.by their symmetrizedcounterparts, we may define the Hamiltonianoperator N _ .+ Z nhsbfsb!+, " 1 1 ( r . . . . . 4i= 42 = | kt t,,ii ktmn (5.9) 342 W. M. DE MUYNCK With (5.8) it is then easilyseenthat "' N;kt "' kNlfl(|, .,nDl1 "'1/;/r "'hl)s "(1 =s(I "' N;kt'.. kNlH(|, .,10t1...i/;/r .'./,17)s (5.10) So,on the Fock spaceof symmetricstateswe have (s.1 ) from which we concludethat-(5.9)is the Hamiltonianof a systemof ,A/indexed bosons.By the substitutiona! - ,bf alsothe numberoperatorsdefinedby (4.24), (4.25), and (4.26) are turned over into the correspondingnumber oDerators of indexedbosons. T h e o p e r a t orrb f t i s . u n l i k ea f * . n o t s i m p l yi n t e r p r e t a b laest h e c r e a t i o n operatorof a particle with index i in singleparticle statek. This is most clearly illustratedby the equality ,bfr,bjt = ,nft ,n,,1 (s.r2) which followsfrom the symmetryof the vectors(5.5). Equation(5.12) reflects the factsthat bosonsarecorrelatedparticlesand that the notion ofsingle particlestateshasa limited meaningfor suchparticles.For, althoughthe operator addsa particlewith index i and a single-particle statek to the "bi'because initial state, ofthe correlationin the final stateparticlei is not bound to be found in statek. 6. Relation to UnindexedBoson Theory In order to be able to comparethe indexedboson formalismwith the usual formalismof "indistinguishable," unindexedbosons(4.1)-(4.3)it is necessary to removethe restrictionin (5.9) on the summations of i andi to a specified numberof particles.We thereforereplace(5.9) by the Hamiltonian r11= : : HP,,b!',t',*i 7, rln (i+i) tkt vn,^n,u!',4t,bf ,bf 6.D wherethe summationsof i and7 now extend over the whole set1 of the particle indices.The Hamiltonian(6.1) hasthe desirableproperty that its matrir elements betweenstatevectorsnot belongingto a definitenumberof particles,of the form lV1;1)= Z M (.6.2) {"k} tziu-='xl are equalto those of the unindexedbosonHamiltonian H= > Hf1af,a1+|: kl klmn vy1^,al,at a^an (6 3) 343 D I S T l N G U I S H A B L E A N D INDISTINGUISHABLE PARTICLES betweenstatevectors l v )= c{nk}l {nk}) (6.4) providedcorresponding coefficientsin(6.2) and (6.4) areidentified,i.e., when we put c{nt}=r1tnn}'* (6.s) then = )( i L { i }l " t 1 l o 1 ; }) (Vl-F1lO (6.6) We note that the equality of the matrix elementsof with those of .F/is ",F1 independentof the choiceof the indicesin (6.2) aslong as(6.5) is satisfied. A secondremark amountsto the observationthat the statevectors(6.2) do not contain linear superpositionsof stateswith an equalnumber of particles but havingdifferent indexations.This restrictionis reasonable within the theory expoundedhere, sincethe indicesare consideredto be intrinsic propertiesof the particlesand so there shouldobtain a superselection rule regardingthese properties.This aspectis very important in tracingthe relationbetweenthe indexedand unindexedoperatorsa'f anda1. Ir-rdeed, on behalf of this superselectionrule an identificationof a6 with 2iaf asis suggested by (6.1) is ruled out. Apart from this argumentation,introduction of this identification in (6.3) givesrise to matrix elementsof which the potential energypart differs from that givenbV (6 !) by a factor of 2.It may alsobe verifiedthat the matrix elementsof 2 iaf do not equalthoseof the unindexedbosonoperatorcalculated in statesconnectedby (6.5), but that the matrix elementsof the operator 2 iaf lyrtz do (analogouslyfor the creationoperators).However,the physical meaningof this operatoris not clear. From the foregoingit follows that the resultsol "indistinguishable"unindexedbosontheory shouldbe recoveredby applicationof classicalensemble theory as far as the indicesare concerned.To this end we introducein the Hilbert spaceof symmetricindexedparticlevectors(6.2) equivalenceclasses of statevectorsin sucha way that statevectorsdiffering only by the indexation ofthe particlesbelongto the sameclass.Sucha classis then represented by the unindexedstatevector(6.4),c{nk} beinggivenby (6.5).A transitionbetween two unindexedparticlestateslV) and l@),specifiedby the transitionamplitude (O llli ) is now representedin the indexed-particlelormalismby a transition betweenthe correspondingequivalenceclasses, specifiedby the amplitude (O1; | V1; averagedover the different possibleindexationswithin the 1 1) equivalenceclasses. Because of the differentmeanings of the operatorsafandap it is not possible to recoverthe matrix elements(\L lar IO ) by simply averagingthe corresponding indexed-particlematrix elements.Indeed,in interpretingthe annihilationof w. M. DE MUYNCK 344 an unindexedparticlein terms of indexedparticleswhoseindicesremain unobserved,we should realizethat eachofthe particlesthat are presentin the initial statehas a probability of being annihilated.The sum of theseprobabilities shouldequalthe probability that a particle(no matter what is its index) is annihilated.This picture is in agreementwith the equality l ( { n p } ', n x - T l a s l{ n p } ) 1 7= n k N ( n r l l V ) =:|,(tr "'+"' i y ; k 1 " ' k " ' k u l a f l i r " ' r r , 'k; 1 " ' ' ky),12 (u.r) A.similarargumentobtainsfor creationoperatorsand for arbitrary products p[alrj, {a;}) oi creationand annihilationoperators. Thus for instancefor k * I | ({ny}', nr, - l, n 1 - | la1ra 7l {n1rDl} = tlkltt =1i(1/ - \ -#L; l) 1(// ' ' i u ; k t ' " ' t ' " ' i 1 y ' , k 1''' k p s ) r l 2 ' + ' = I. lr{i, " " j' k " t ,lafajlil'' ''t (6 8) Here1(1/ - l) is the numberof different waysin which two indexedparticles may be annihilatedfrom a systemof ly' indexedparticles.Note that the order in which particlesi andj arc annihilatedis relevant,givingriseto two different transition processes. It doesnot seempossibleto relatedirectly the amplitudesof the indexedtheoriesin a way consistentwith the interpretation and the unindexed-particle givenabove,because ofthe appearance ofthe factorsly'-1/2expoundedin (5.6). Sinceamplitudesdo not havephysicalrelevance,this doesnot haveconsequences in the casesstudiedabove.However,when we try to generalizethe foregoingto matrix elementsbetweenarbitrary statevectors,we arriveat the conclusionthat this can only be done when one of the vectorshas a definite number of particles.Thus l(vlr l{{al}, {a;})lo)12 tr).tsj (69) Whenboth vectorshavean indefinite number of particlesthe indexedparticleprobabilitiesdiffer from the unindexed-particle onesas may be seen from the following simpleexample.With l i D )= a 1| { n / r } )+ b r l { m y } ) , l * l = a z l { n r } ' ,n r - l ) + b z l { m r } ' ,m p - 1 ) N =2r* 1M=)my kk we get l(@larlV)12= lala2tfnp+ nlb 21/mrl2 (6.10) t n DISTINGUISHABLE AND INDISTINGUISHABLE PARTICLES Dut ) l(o1i] lof l,I,{r})|'?= Nlafa2t/(n*llg+ b|b2\/fuklI},t)12 I + (M * Nlblb/./(mxlruDlt (6.11) which is different irom (6. I 0) whenM * N, the differencebeing proportional to (NlIt4)t12- 1. theoriesfound The discrepancybetweenthe indexed-and unindexed-particle here could be usedto test the applicabilityof the theories.For this it is esential to considertransitionsin which both initial and final statesare coherent superpositionsof stateswith different numbersoI particles. 7. Comparisonwith an Earlier Theory By Marx (19'72)creationand annihilationoperatorshavebeenintroduced of "numbered" bosonsand fermions.The creationoperatornumberedTis definedin sucha way that the (anti)symmetricstatevector in (generalized) F o c k s p a c ec o n t a i n i n g p a r t i c lneus m b e r e d1 , 2 , . . . , i - 1 , i + 1 , . . . , I y ' i s turned over into a statevector of the samesymmetry characterwith particles numberedl, 2, . . ., /y'.Analogouslythe annihilationoperatornumbered7 destroysa particle with this samenumber,maintainingthe symmetry character ofthe statevector. In view of(5.6) the creationand annihilationoperatorsof numberedparticlesshow at first sight someresemblance to the indexed-particle operatorsdefinedby (5.4). The numbered.p*article operatorsof Marx refer to ' correlatedparticles,as do the operators and sb! . Apart from this similarity "Df the two kinds of operatorsare quite different becauseof the different meanings of "numbering" and "indexation" of particles.Thesedifferencesare for instance reflectedby the commutation relationsof the operators,which show quite a different behavior. In Section2 we definedthe particle indicesto be intrinsic propertiesof the particles..As we sawat the end of Sec.5, we then cannotinterpretthe ,-t operator,bi' as the creationoperatorof a particlewith indexi in single particlestatek. As Marx defineshis numberedparticle operatorsprecisely accordingto this interpretation,we may concludethat his numberscannot be intrinsic propertiesof the particles.This conclusionis confirmedby the commutation relationsthe numberedparticle operatorsobey fMarx, 1972, Equations(2.6)-(2.10)].Theserelationsshowa dependence of the numbering ofthe particleson the order in which two particlesare createdor destroyed successively in two different single-particle states.This could neverbe the caseif the numbersrepresentedintrinsic propertiesof the particles.In fact, the resultsof the numbered-particle theory of Marx are compatiblewith an interpretationof the numberingasa numberingof occupiedsingle-particle states,rather than as an indexationof the particles.This meansthat "particle T"refersto the particle that is attributed to theTth occupiedsingle-particle 346 w. M. DE MUyNCK state.Thus, when a particle is addedin somesingle'particlestate,all particles that were alreadypresentin stateswith a higher ordinal number should be -+j+ 1. This clearlyshowsthe differencebetweenparticle renumberedT numberingand indexation. References Blokhintsev, D. I. (1964). Quontum Mechanics, Chap. 19. D. Reidel, Dordrecht, Holland. Gibbs, J. W. (1902) Elementary Pinciples in Statistical Mechanics, Chap. 15. Yale University Press, New Haven or The Collected l,lorks of J. W. Gibbs, Vol. II' Yale University Press, New Haven,1948). Haar, D. ter (1961). Elements of Statistical Mechanics. Holt, Rinehart and Winston, New York. Heisenberg,W. (1930). Die physikalischen Prinzipien der Quantentheorre. Verlag von S. Hkze| Leipzig. Jauch, J. M. (1966). Foundations of QuantumMechanics, Chap. 15. Addison-Wesley, Reading, Massachusetts. landau, L. D. and Lifshitz, E. M. (1959). Quantum Mechanics, Chap. 9. Pergamon Press, London. Lyuboshitz, V. L. and Podgoretskii, M. I. (1969). Soviet Physics JETP, 28,469. Lyuboshitz, V. L. and Podgoretskii, M. I. (1971). Soviet Physics JETP, 33,5. Marx, E. (1972). Internotional Joumal of Theoretical Physrcs,6, 359. Messiah,A. (1965). Quantum Mechanics, Chap. 14. North-Holland, Amstetdam. Mirman, R. (1973). Nuovo Cimento l8B, 110. Robertson, B. (l 9'l 3). A merican J ournal of P hy sics, 41, 67 8. Rushbrooke, G. S. (1949). Introduction of Statistical Mechanics, Chap. 3. Oxford. Schiff, L. I. (1955). Quantum Mechanics, Chap. 9. McCraw-Hill Inc., New York. Schrodinger, E. (1957). Statistical Thermodynamrcs, Chap. 8. Cambridge. Schweber, S. S. (1961). An Introduction to Relativistic Quantum Field Theory, Chap. 6.b. Row, Peterson and Cy., Evanston, Illinois.