Download Distinguishable- and Indistinguishable

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 .
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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.
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Haar, D. ter (1961). Elements of Statistical Mechanics. Holt, Rinehart and Winston, New
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Heisenberg,W. (1930). Die physikalischen Prinzipien der Quantentheorre. Verlag von S.
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Jauch, J. M. (1966). Foundations of QuantumMechanics, Chap. 15. Addison-Wesley,
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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.
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