Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes

0.18 MB·English

by  Jeronimo Cortez

Uniqueness ofthe Fockquantization offields withunitary dynamics innonstationary spacetimes
Jero´nimo Cortez
∗
DepartamentodeF´ısica,FacultaddeCiencias,UniversidadNacionalAuto´nomadeMe´xico,Me´xicoD.F.04510,Mexico.
Guillermo A. Mena Maruga´n and Javier Olmedo
† ‡
Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain.
Jose´ M. Velhinho
§
Departamento de F´ısica, Universidade da Beira Interior,
R. Marqueˆs D’A´vila e Bolama, 6201-001 Covilha˜, Portugal.
TheFockquantizationoffieldspropagatingincosmologicalspacetimesisnotuniquelydeterminedbecause
ofseveralreasons.Apartfromtheambiguityinthechoiceofthequantumrepresentationofthecanonicalcom-
1 mutationrelations,therealsoexistscertainfreedominthechoiceoffield:onecanscaleitarbitrarilyabsorbing
1 backgroundfunctions,whicharespatiallyhomogeneousbutdependontime. Eachnontrivialscalingturnsout
0 intoadifferentdynamicsand,ingeneral,intoaninequivalentquantumfieldtheory.Inthisworkweanalyzethis
2
freedomatthequantumlevelforascalarfieldinanonstationary, homogeneousspacetimewhosespatialsec-
n tionshaveS3topology. Ascalingoftheconfigurationvariableisintroducedaspartofalinear,timedependent
a canonicaltransformationinphasespace. Inthiscontext,weproveinfulldetailauniquenessresultaboutthe
J Fockquantizationrequiringthatthedynamicsbeunitaryandthespatialsymmetriesofthefieldequationshave

2 a natural unitary implementation. The main conclusion is that, with those requirements, only one particular
1 canonicaltransformationisallowed,andthusonlyonechoiceoffield-momentumpair(uptoirrelevantconstant
scalings). Thiscomplementsanotherpreviousuniquenessresultforscalarfieldswithatimevaryingmasson

] S3,whichselectsaspecificequivalenceclassofFockrepresentationsofthecanonicalcommutationrelations
c
undertheconditionsofaunitaryevolutionandtheinvarianceofthevacuumunderthebackgroundsymmetries.
q
Intotal,thecombinationofthesetwodifferentstatementsofuniquenesspicksupauniqueFockquantization
-
r forthesystem.Wealsoextendourproofofuniquenesstoothercompacttopologiesandspacetimedimensions.
g
[
PACSnumbers:03.70.+k,04.62.+v,98.80.Qc,04.60.-m

1
v
I. INTRODUCTION no general procedure to select a preferred quantum descrip-
7
9 tion. In this situation, physicalresultsdepend on the repre-
3 TheuniquecharacterofNatureisalludedinphysicsbythe sentation adopted, a fact that brings into question their sig-
2 nificance. It is then necessary to look for additional criteria
uniquenessofthetheoriesemployedtodescribeit. Inparticu-
. to warrant uniqueness and regain robustness in the quantum
1 lar,byimposingappropriatephysicalcriteria,thequantization
0 ofaclassicalsystemshouldyieldauniquequantumdescrip- predictions.
1
tion –up to unitary equivalence. Since the quantization pro- Theusualproceduretoselectapreferredrepresentationin
1
cess involveschoices that may lead to inequivalenttheories, fieldtheoryforagivensetofCCR’sistoexploittheclassical
:
v thespecificationofauniquedescriptionisanontrivialtask. symmetries. For instance, the invariance under the Poincare´
i
X Even in systems in which one already starts with a spe- groupisthecriterionimposedtoarriveatauniquerepresen-
cificchoiceofbasiccanonicalvariablesandanassociatedset tationinordinaryquantumfieldtheory. Thus,ifthefieldthe-
r
a of canonical commutation relations (CCR’s), there exists an orycorrespondstoascalarfield,Poincare´invariance,adapted
intrinsic ambiguity in the quantization process because the to the dynamics of the considered theory, selects a complex
CCR’scanberepresentedinnonequivalentways. Inthecase structure [2], which is the mathematical object that encodes
oflinearsystemswithafinitenumberofdegreesoffreedom, theambiguityinthequantizationanddeterminesthevacuum
theseambiguitiesareessentiallysuppressedbytheimposition state of the Fock representation. For stationary spacetimes,
ofcertainunitarityandcontinuityconditionsontherepresen- the time translation symmetry is exploited to formulate the
tationofthealgebraofobservables(asstatedintheStone-von so-calledenergycriterionandthenselectapreferredcomplex
Newmann theorem [1]), so that uniqueness follows without structure[3].Butwhenthesymmetriesareseverelyrestricted,
misadventures. Nonetheless,thesituationchangesdrastically asitisthecaseforgenericcurvedspacetimesorformanifestly
in the arena of field theory. These systems accept infinite nonstationarysystems,extrarequirementsmustbeimposedto
nonequivalent representations of the CCR’s [2] and there is complete the quantizationprocess. For example, in the case
ofdeSitterspacein1+1dimensions,itispossibletopickup
auniquedeSitterinvariantFockvacuumforafreescalarfield
by looking for an invariant Gaussian solution to a properly
Electronicaddress:[email protected] regulatedSchro¨dingerequation[4].
∗
Electronicaddress:[email protected]
†
Inthecontextofquantumcosmology,theextracriterionof
‡Electronicaddress:[email protected]
Electronicaddress:[email protected] a unitary implementationof the dynamicshas been success-
§
2
fullyemployedtospecifyaunique,preferredFockquantiza- thereforesimplifyinginpartthecorrespondingdynamics,al-
tion forthe Gowdyspacetimes. These are spacetimeswhich though there remain (or appear) time dependent potentials
possess two spacelike Killing isometries andspatial sections whichmanifestthatthescenarioisanonstationaryone.
of compact topology [5]. In the case of a three-torus topol-
Thequestionimmediatelyarisesofwhetheritisagainpos-
ogy and a content of linearly polarized gravitational waves,
sibletoinvokenaturalcriteriatoremove(atleastincertainsit-
thelocalgravitationaldegreesoffreedomcanbedescribedby
uations)theambiguitythatthisfreedominthechoiceoffield
a scalarfield witha specifictime dependentmassandwhich
introducesatthequantumlevel.Adetailedanalysisaboutthis
propagatesinanauxiliary,static backgroundwiththespatial
issue wasfirstcarriedoutforthe quantizationofthelinearly
topologyofthecircle[6].Forthischoiceofbasicfieldforthe
polarizedGowdymodelwiththree-torustopologyinRef. [9].
model, one is able to find a uniqueFock quantizationwhich
Thatworkstudiedafamilyoflinear,timedependentcanoni-
incorporatesthebackgroundsymmetriesassymmetriesofthe
caltransformationsthatinvolvea scaling ofthe field. Itwas
vacuumandimplementsthefielddynamicsasafamilyofuni-
proventhatthereactuallyexistsnofreedomleftinperforming
taryquantumtransformations[6–9].
a transformation of this kind, once the criteria of invariance
More recently, in a broadercontext, a unitary equivalence under the remaining spatial symmetries and the unitary im-
class of Fock representations has been specified for scalar plementation of the dynamics are imposed. More precisely,
fields withgenerictime varyingmass, defined on spheres in Ref. [9] shows that the considered transformations lead to
threeorlessdimensions[10,11]. Again,theprocedurecon- newdynamicssuchthatonecannotattaina unitaryquantum
sists in requiring a unitary dynamicsand the vacuuminvari- evolutioninaFockrepresentationwhilekeepingthesymme-
anceunderthesymmetriesofthefieldequation. Theparticu- try invariance of the vacuum.1 The requirements of unitary
larlyrelevantcaseofthethree-sphere,withthedimensionality evolutionand invariancethereforesuffice to select a specific
observedinouruniverse,wasconsideredinRef. [11]. scalingofthefieldandaprivilegedfamilyofequivalentFock
Apart from the inherent ambiguity in choosing the repre- quantizationsforit. Inotherwords,theuniquenessisguaran-
sentation of the CCR’s, the quantization of fields in curved teed bothfor the choice offundamentalfield (with its corre-
space-timesisaffectedbyanotherkindofambiguity.Itisdue spondingdynamics)andforthequantumrepresentationofthe
tothefreedominchoosingaspecificfieldparametrizationto correspondingCCR’s.
describethe physicalsystem, namely, the freedomin declar- One may wonder whether the uniquenessin the choice of
ing a particular choice of field (together with its associated field description can also be guaranteed in other, more gen-
dynamics)asthefundamentalone. Letusconcentrateourat- eralsystemsthantheGowdymodel,andinparticularfornon-
tentiononthecaseofhomogeneousbutnonstationaryspace- stationarysettingswheretherealreadyexistresultsaboutthe
times, like those encountered as backgroundsin cosmology. uniqueness of the representation of the CCR’s. The case of
Inthesecircumstances,itismostnaturaltoconsiderfieldre- fields in 1+3 dimensional spacetimes with compact spatial
definitionswhichabsorbbackgroundfunctions. Thisleadsto topologyisspeciallyimportant,owingtoitsapplicationse.g.
ascalingofthefieldbyatimedependentfunction,suchthat to cosmology. A summaryofthe discussionforscalar fields
thelinearityofthefieldequationsandofthestructuresofthe propagatinginanonstationaryspacetimewithsectionsofS3
system are preserved. If this time dependence is nontrivial, topologywasalreadypresentedbyusinRef. [12],anticipat-
thetwofields(i.e.,thescaledandtheunscaledones)aregov- ingthattheanswertothequestionofuniquenessisintheaffir-
ernedbydifferentdynamics. Sinceachangeinthedynamics mative. Theaimofthepresentworkistoprovidefulldetails
typically calls for inequivalentrepresentations, the construc- ofthedemonstrationofthisresult.
tion of a quantumtheoryclearlydependson the selection of
We will consider a scaling of the field by a generic func-
aspecificfielddescriptionforthesystemamongallthosere-
tionoftime. Thisscalingcanalwaysbecompletedintoatime
latedbythesescalingtransformations.
dependentcanonicaltransformation. We demandsuchtrans-
As commentedabove, these considerationsare crucial for
formationtobecompatiblewithalllinearstructuresonphase
quantum matter fields propagating in inflationary or cosmo-
spaceandwiththesymmetriesofthefieldequations.Anyad-
logical backgrounds, which are spatially homogeneous but
missible canonicaltransformationisthen linearand, further-
not stationary. The discussion is also relevant for the quan-
more,can bedividedintotwo parts. Thefirst oneisa linear
tizationoflocalgravitationaldegreesoffreedom,incontrast
canonical transformation that is explicitly time independent
with the previous context of quantum matter fields in clas-
buttakesintoaccounttheinitialconditions,renderingsimple
sical spacetimes that are solutions to the gravitational field
ones for the remaining part, which incorporates then all the
equations. This latter class of systems includes, e.g., the
timedependence. We willdemonstratethatthereexistsonly
already mentioned Gowdy models and the case of gravita-
onepossiblechoiceofphasespacevariablessuchthatthere-
tional perturbations around cosmological backgrounds. For
sulting field theory admits a Fock quantization with unitary
thesegravitationalsystems,thereexistsagreatfreedominthe
dynamicsandanaturalimplementationofthesymmetriesof
choiceofparametrizationofthe metriccomponentsin terms
offields. Inallthesesituations, thechoiceofasuitablefield
parametrization involves a time dependentscaling related to
backgroundfunctionsandwhosespecificformdependsonthe
1Infact,inthisparticularlinearsystemonecanstillintroducearedefinition
particularsystemunderstudy.Thischoiceoftenleadstofields
ofthemomentumwhichimpliesnoscalingofthefield,butthisturnsoutto
whicheffectivelypropagateinanauxiliarystaticbackground, beirrelevantinasmuchasnonewnonequivalentFockquantizationarises.
3
the field equations. The unique choice which remains avail- severelyrestrictedbyourcriteriathatitturnsouttobefixed.
ableispreciselytheonewhichcorrespondstoatransformed Inaddition,wewillarguethattheanalysiscanbegeneralized
scalarfieldthatpropagatesinastaticspacetimewithS3 spa- tolowerdimensions,replacingthethree-spherewithS2orS1
tial topology,though in the presence of a time varyingmass (forthis last case, see Ref. [9]), as well as to other compact
term. Recall that, for this latter field, the uniqueness of the topologies.
representationoftheCCR’swasproveninRef. [11]. The content of the paper is organizedas follows. In Sec.
IIwesummarizetheresultsthatarealreadyknownaboutthe
Aswehavealreadymentioned,thelistofscenarioswhere
Fockquantizationofascalarfieldwithatimevaryingmassin
thisresultfindsdirectapplicationsincludesthecaseofinfla-
astaticspacetimewhosespatialsectionshavethetopologyof
tionarymodelswhereascalarfieldwithconstantmassprop-
S3. InSec. IIIweintroducethelinear,timedependentcanon-
agates in a Friedmann-Robertson-Walker (FRW) spacetime
icaltransformationwhichaccountsforthescalingofthefield
with compact spatial topology. In this case, one can check
and discuss its consequences at the quantum level. Sec. IV
thatalinear,timedependentcanonicaltransformationallows
contains the detailed proof that only one of these canonical
oneto rewritethe field equationas thatofa field in a space-
transformationsleadstoafielddynamicswhichiscompatible
time with identical spatial topology but static, whereas the
withourcriteriaofaquantumunitaryevolutionandthesym-
massbecomestimevarying.Anothertypeofsituationswhere
metry invariance of the vacuum. We discuss the results and
ourresulthasimplicationsisgivenbythequantizationof(in-
conclude in Sec. V. Finally, an appendix which deals with
homogeneous) perturbations around nonstationary homoge-
sometechnicalpartsofourdemonstrationisadded.
neoussolutionsoftheEinsteinequations,typicallycosmolog-
ical backgrounds[13–17]. Examplesare the gauge-invariant
energy density perturbationamplitude in an FRW spacetime
II. PRELIMINARIES:THESYSTEMANDTHEFOCK
withS3 spatialtopologyfilledwith aperfectfluid(whenthe
QUANTIZATIONOFREFERENCE
perturbations of the energy-momentum tensor are adiabatic
[14, 16]) or the matter perturbations around the same FRW
Letusstartbyreviewingsomeofthekeyaspectsandresults
spacetimeforamassivescalarfield[18].Withasuitablescal-
about the Fock quantization of a real scalar field φ subject
ing (and in an appropriate gauge in the case of the massive
to a time dependentpotential V(φ) = s(t)φ2/2, where s(t) is
field),thecorrespondingequationsofmotioncanberelatedto
in principle any regular function of time (conditions on this
thoseofascalarfieldinastaticspacetimewithatimedepen-
functionwillbeintroducedlateron).Thefieldpropagatesina
dent mass term (see Ref. [9] for additionaldetails). At this
staticbackgroundin1+3dimensionswhoseCauchysurfaces
point, it may be worth commenting that, although flat FRW
arethree-spheres,equippedwiththestandardroundmetric
universesreceiveaspecialattentionincosmologynowadays,
some recent works find reasons to prefer closed FRW mod- h dxadxb =dχ2+sin2(χ) dθ2+sin2(θ)dσ2 . (1)
els with S3 topology, for example from the pointof view of ab
h i
perturbation theory in relation with the choice of appropri- Here, χ and θ have a range of π, and σ S1. The time co-
ategaugeswhichembodyMach’sprinciple[19], orinanat- ordinate t runs over an interval I of the r∈eal line, so that the
tempttoaccountforalowmicrowavebackgroundquadrupole spacetimehasthetopologyofI S3. Itsmetricis
[20,21].Ontheotherhand,wewillarguelateronthatourre- ×
sultscanbegeneralizedtothecaseofflatbutcompactFRW ds2 = dt2+habdxadxb. (2)
−
universes.
In the canonical approach, the dynamics of the system are
In summary, the question that we are going to investigate governedbytheequations
is whetherthe criteria of unitarydynamicsandinvarianceof
1
thevacuumunderthesymmetriesofthefieldequationselecta P˙ = √h ∆φ s(t)φ , φ˙ = P , (3)
φ φ
uniqueFockquantizationamongallthosearisingfromdiffer- − √h
(cid:2) (cid:3)
enttimedependentscalingsofthefield. We willconcentrate whereP isthecanonicalmomentumofφ,h=sin2(θ)sin4(χ)
φ
ourdiscussiononthecasethataparticularscalingrendersthe
is the determinant of the metric (1), ∆ denotes the Laplace-
dynamics into that of a scalar field with time varying mass Beltrami operator on S3, and the dot stands for the time
propagating in a static spacetime, with inertial spatial sec-
derivative.
tionsthathavethetopologyofathree-sphere. Wewillstudy
Thecanonicalphasespaceofthetheoryisasymplecticlin-
at the quantum level the consequencesof local, time depen-
earspaceΓcoordinatizedbythefieldvariables(φ,P )(evalu-
φ
dentcanonicaltransformationswhichinvolveascalingofthe
atedonaparticularCauchysection,e.g. thesectiont = t for
0
field. These transformationsmust preservethe invarianceof
agivenvalueoftimet )andendowedwithasymplecticstruc-
0
thefieldequationunderthegroupofsymmetriesandthelin-
tureΩsuchthatthesevariablesformacanonicalpair,namely
earityofthespaceofsolutions. Transformationsofthiskind
theircorrespondingPoissonbracketis:
consistofascalingoftheconfigurationvariablebyafunction
oftime, theinversescalingofthe canonicalmomentum,and φ(x),Pφ(y) =δ(3)(x y), (4)
{ } −
possiblya contributionto themomentumthatislinearinthe
where the Dirac delta is defined on S3. The Eqs. of motion
configuration variable, the proportionality factor being time
(3)amounttothelinearwaveequation
dependent. Our main goal in this work is to provide a full
proof demonstrating that the canonical transformation is so φ¨ ∆φ+s(t)φ=0. (5)
−
4
SincetheLaplace-BeltramioperatoronS3 isinvariantunder character, because this is maintained if one removes a finite
therotationgroupSO(4),theaboveequationisclearlyinvari- number of modes.2 We will introduce the annihilation and
ant under this group as well. On the other hand, notice by creationlikevariables
comparisonwith the Klein-Gordonequation that a nonnega-
tivefunctions(t)canbeinterpretedasaneffectivenonnegative an = 1 ωn i qn . (8)
timedependentmassm(t)= s1/2(t). a∗n ! √2ωn ωn −i! pn !
Owing to the field character of the theory, the system ac-
Noticethatthesearepreciselythevariableswhichonewould
ceptsinfinitenonequivalentrepresentationsoftheCCR’s. Re-
naturallyadoptinthecaseofafreemasslessscalarfield.They
stricting one’s attention to representations of the Fock type,
formacompletesetinthephasespaceoftheinhomogeneous
this freedomis encodedin the complexstructure, which isa
sector. Givenasetofinitialdata a (t ),a (t ) atinitialtime
linear symplectic map j : Γ Γ, compatiblewith the sym- { n 0 ∗n 0 }
→ t ,itispossibletowritetheclassicalevolutiontoanarbitrary
plectic structure [in the sense that the bilinear map Ω(j, ) 0
is positive-definite], and such that j2 = 1 (see e.g. Re·fs·. timet Iintheform
∈
−
[2,22,23]). Differentchoicesofcomplexstructureselectdis-
a (t) α (t,t ) β (t,t ) a (t )
tinct,ingeneralnotunitarilyrelated,spacesofquantumstates n = n 0 n 0 n 0 . (9)
forthetheory;thus,physicalpredictionsdependonthechoice a∗n(t) ! β∗n(t,t0) α∗n(t,t0)! a∗n(t0) !
of j.
Let us call (t,t ) the linear evolution operator defined in
n 0
Actually,aswehavementioned,ithasbeenprovenrecently U
thisway.Sincethetimefunctionsα (t,t )andβ (t,t )provide
n 0 n 0
that,inthe(φ,P )description,thereexistsone(andonlyone)
φ asymplectomorphismonΓ,onehas
subfamilyofequivalentcomplexstructuressatisfyingthecri-
teriaofSO(4)invarianceandaunitaryimplementationofthe α (t,t )2 β (t,t )2 =1, (10)
n 0 n 0
dynamics. Letus sketch the main steps of the proofand ex- | | −| |
plain the correspondingquantization [11]. Given the invari- independently of the particular values of n, t , and t. Such
0
anceofthefieldequationunderSO(4),itisconvenienttoex- Bogoliubovcoefficientsα (t,t )andβ (t,t )ofthisevolution
n 0 n 0
pand the field in terms of (hyper-)sphericalharmonics Qnlm, map can be determined in the way explained in Ref. [11].
where the integern satisfies n 0, and the integersℓ andm For our present analysis, we only need to employ that their
≥
areconstrainedby0 ℓ nand m l[13,24,25]. Inthis asymptoticbehaviorwhenn isgivenby
basis,theLaplace-Be≤ltram≤ioperato|r|∆≤isdiagonalwitheigen- →∞
valuesequalto n(n+2).Althoughthe(hyper-)sphericalhar- 1 1
monicsarecom−plexfunctions,itis straightforwardto obtain αn(t,t0)=e−i(n+1)τ+O n!, βn(t,t0)=O n2!, (11)
a real basis from the real and imaginary parts of Q , with
nlm
whichonecandirectlyexpandtherealfieldφ(seeRef. [11] where τ = t t0 and the symbol O denotes the asymptotic
−
for details). The degrees of freedom are represented by the order. The derivationof this asymptotic behaviormakesuse
coefficientsq in this expansion, which can be understood ofthemildassumptionthatthefunction s(t) inEq. (5) must
nℓm
asadiscretesetofmodes. Thesearefunctionsoftimewhich be differentiable, with a derivativethat is integrablein every
satisfythelinearequation closedsubintervalofI. Tosimplifythenotation,wewillomit
inthefollowingthereferencetotheinitialtimet inthecoef-
0
q¨ + ω2+s(t) q =0, (6) ficientsoftheevolutionoperatorandintheinitialdata,called
nℓm n nℓm
h i now{an,a∗n}.
withω2 =n(n+2).Hence,themodesq aredecoupledfrom The SO(4) symmetry of the field equations is imposed at
n nℓm
eachother. Besides,togetherwiththeircanonicallyconjugate the quantum level by demanding that the complex structure
momentap =q˙ ,theyformacompletesetofvariablesin be invariant under this group. We call invariantthis class
nℓm nℓm
phasespace.Foreachfixedvalueofn,thereexistg =(n+1)2 of complex structures. By Schur’s lemma [26], any invari-
n
modeswiththesamedynamics,becausetheequationofmo- antcomplexstructurehastobeblockdiagonalwithrespectto
tionisindependentofthelabelsℓandm.Obviously,thequan- thedecompositionofthephasespaceasthedirectsumofthe
tity gn is justthedimensionofthecorrespondingeigenspace subspacesQn⊕Pn. Inotherwords,thecomplexstructurecan
oftheLaplace-Beltramioperator. Thecanonicalphasespace bedecomposedasadirectsum j= n jn,where jn isanin-
Γcanbesplitthenasadirectsum variantcomplexstructuredefinedonLthen-thsubspaceofthe
inhomogeneoussector of Γ. Moreover, a further application
Γ= , (7) of Schur’s lemma shows that each of the complexstructures
n n
Q ⊕P
Mn jn isagainblockdiagonalandindependentofthelabelsland
m, so that it can be characterized by a complex structure in
where and aretherespectiveconfigurationandmomen-
n n twodimensions,describinge.g.theactionontheannihilation
Q P
tumsubspacesforthemodeswithfixedn. Fromnowon,we
willomitthelabelsℓ andm,unlesstheyarenecessaryinthe
analysis.
Furthermore, in the following we restrict our study to the
inhomogeneoussector, namely, to modes with n , 0. This 2Therequirements onthequantization ofthezeromode, q0,mayleadto
extraconditionsonthefunctions(t). SeeRef. [11]andSec. Vforfurther
doesnotaffectthepropertiesofthesystemrelatedtoitsfield commentsonthisissue.
5
andcreationlikevariables(a ,a )foranyfixedmodelabelsl Here, the symbol denotes imaginary part. If one assumes
n ∗n ℑ
andm(seeRef. [11]formoredetails). that the evolution is unitary in the Fock quantization deter-
Ontheotherhand,letuscall j0thecomplexstructurewhich minedby j,sothatthesequences √gnβnj(t) areSQS t I,
inourbasisofvariables{an,a∗n}takesthediagonalform: one can prove that the sequence{{√gnλn} }must be S∀QS∈as
well [11]. But this summability is precisely the sufficient
i 0
j = . (12) andnecessaryconditionfortheunitaryimplementationofthe
0n 0 i!
− symplectictransformation intheFockrepresentationdeter-
K
A general complex structure j is related with j0 via a sym- minedby j0,whatamountstotheequivalenceofthetwocom-
plectictransformation j = j0 −1. Takingintoaccountthe plexstructures jand j0. Therefore,theSO(4)invarianceand
formoftheinvariantcompleKxstrKuctures,thesymplectictrans- the requirementof unitary dynamics select a unique equiva-
formation mustbealsoblockdiagonalandindependentof lenceclassofcomplexstructures,removingtheambiguityin
thedegeneKracylabelslandm. Wecall the2 2blockcor- thechoiceofrepresentationoftheCCR’s.
n
K ×
respondingtothen-thmode,forwhichweadoptthenotation:
= κn λn . (13) III. THEAMBIGUITYINTHECHOICEOFFIELDAND
Kn λ∗n κn∗ ! THEUNITARITYCRITERION
The symbol denotes again complex conjugation. Here,
κ 2 λ 2 =∗1 because isa symplectomorphism. Itfol- Although we have succeeded in selecting a preferredrep-
|lonw|s−in|pna|rticularthat κ Kn1 n N+. Notethatthereexist resentationoftheCCR’s forthe(φ,Pφ)variables,wecanal-
infiniteinvariantcompl|enx|s≥truc∀ture∈s.Actually,theyarenotall ways changefrom the (φ,Pφ) descriptionof the phase space
toanewcanonicaldescriptionbymeansofacanonicaltrans-
unitarilyequivalent,sothattheimpositionofSO(4)symmetry
formation. Since many canonical transformations fail to be
doesnoteliminatetheambiguityintheFockquantizationon
represented by unitary operators quantum mechanically, the
itsown.
classical equivalenceof these descriptionsmay be broken in
In order to select a class of equivalent invariant complex
the quantum arena, originating another type of ambiguity in
structuresweneedtoappealtoadditionalconditions. Auni-
the quantization. In our case, we are only interested in con-
taryimplementationoftheclassicaldynamicsatthequantum
sideringlinear,localcanonicaltransformations,whichrespect
level turns out to determine a preferred class, and hence a
thelinearityofthefieldequationsand,consequently,thelin-
unique Fock quantization up to equivalence. We recall that
ear nature of the structures of the system. As we have ex-
a symplectic transformationT is implementableas a unitary
plained in the Introduction, the class of canonical transfor-
transformation in the quantum theory for a given complex
mations that we want to analyze results in a time dependent
structure j if and onlyif j TjT 1 is a Hilbert-Schmidtop-
− − scalingofthefield. Itisthistimedependencewhatmakesthe
erator(ontheone-particleHilbertspacedefinedby j,seee.g.
transformationnontrivial;otherwise,anyadmissiblerepresen-
Refs. [3,23]). Inthecaseofthetimeevolutionoperator,and
tation of the original field would provide an admissible one
choosing the complexstructure j , this conditionis satisfied
0 forthetransformedfieldbylinearity. Butwhenthecanonical
ifandonlyif
transformationistimedependent,thefielddynamicschanges,
β (t)2 = g β (t)2 < t I, (14) affectingthepropertiesofthequantumtheory.
n n n
Xnℓm| | Xn | | ∞ ∀ ∈ Atimedependentscalingoftheconfigurationfieldvariable
canberegardedasacontacttransformation,whichcaneasily
i.e., if and only if the sequences √gnβn(t) are square be completedinto a canonicalone. Then, the canonicalmo-
{ }
summable(SQS)forallpossiblevaluesoftime. Then,since
mentummust experiencethe inversescaling and, optionally,
√gn = n+1, the asymptotic behavior(11) of the beta coef- maybemodifiedwiththeadditionofatermdependingonthe
ficientsguaranteesthedesiredsummability,ensuringthatthe
configurationfieldvariable,whichwerestricttobelinear(and
dynamics is implemented unitarily in the Fock quantization local),accordingtoourpreviouscomments.Thecoefficientin
picked up by j0, namely the complexstructure associated to thislineartermmayvaryintime,liketherestofcoefficients
the natural choice of annihilation and creationlike variables
inthelinearcanonicaltransformationunderconsideration.In
forthefreemasslesscase.
thisway,oneobtainsatransformationoftheform
Let us suppose now that we choose a different invariant
complexstructure j,whichcanbeobtainedfrom j0bymeans ϕ= F(t)φ, P = Pφ +G(t)√hφ. (16)
of a symplectic transformation , as we have commented. ϕ F(t)
K
Theunitaryimplementationoftheevolutionoperatorwithre-
We recall that the momentum variable is a scalar density of
specttothenewcomplexstructure jisequivalenttotheuni-
unitweight. Thisexplainsthesquarerootofthedeterminant
taryimplementationofatransformedevolutionoperatorwith
ofthe spatialmetric appearingin Eq. (16). In orderthatthe
respect to the complex structure j [11]. This transformed
0
transformation does not spoil the differential formulation of
evolution operator is obtained from the original one by the
actionof . Itsdiagonalblocksare (t) 1,withcorre- the field theory, nor produces singularities, F and G are re-
spondingKbetacoefficientsgivenby KnUn Kn− stricted to be two real and differentiable functions of time,
withF(t)differentfromzeroeverywhere.Noticealsothatthe
βj(t):=(κ )2β (t) λ2β (t)+2iκ λ [α (t)]. (15) homogeneityofFandGpreservestheSO(4)invarianceofthe
n n∗ n − n ∗n n∗ nℑ n
6
field dynamics. Inthe following,we will consideronlytime demonstratethatanysuchtransformation,excepttheidentity,
dependentcanonicaltransformationsoftheform(16). leadstoaclassicalevolutionwhichadmitsnounitaryimple-
Aswehavepointedout,differentchoicesofthebasicfield- mentationwithrespecttoanyoftheFockrepresentationsde-
likevariablestypicallyleadtodistinctdynamics.Forinstance, finedbyanSO(4)invariantcomplexstructure. Thus,ourcri-
acanonicaltransformationwithF(t)=1/a(t),whereaisaso- teria fix completely the choice of field description [up to a
lutionofthesecondorderdifferentialequation trivialtimeindependenttransformationofthetype(18)].
Letusdiscussnowtheformofthenewdynamicsobtained
a¨
m2a2+s(t)=0, (17) with the transformation (19), and present the mathematical
a −
conditionnecessary for a unitary implementationof this dy-
leadsfromthe fieldequation(5)to the dynamicsofa Klein- namical evolution. We recall that the linear transformation
Gordon field with mass m propagating in the FRW back- (19)preservestheSO(4)invarianceofthefieldequationsand
groundds˜2 = a2(t)ds2 [see Eq. (2)]. Thisindicatesthatone thatwe demandthatthe (real)functions f(t) andg(t) bedif-
can extractinformationaboutthe dynamicsof differentfield ferentiable. Moreover, [like the functions F(t)] the function
theories by performinga time dependentcanonical transfor- f(t) is required to differ from zero everywhere. The sign of
mation of the above type. Let us emphasize, however, that the function f(t) is therefore constant and, since its initial
withthisprocedureoneisnottransformingagivenfieldthe- valuehasbeenfixedequaltotheunit,inwhatfollowswetake
oryinto anotherone, butratherconsideringdistinctfield de- f(t)>0 t I.
∀ ∈
scriptionsof a givenphysicalsystem, assuming that noneof Aswehavealreadycommented,sincethecanonicaltrans-
these descriptions is imposed from the start. In this kind of formation(19)dependsontime,theclassicalevolutionopera-
systems,onehastoaddresstheambiguityassociatedwiththe torthatdescribesthedynamicsofthepair(ϕ,P )differsfrom
ϕ
choiceoffieldparametrization(i.e.,withtheselectionoffun- thatcorrespondingtotheoriginalpair(φ,P ). Inordertode-
φ
damentalfield, togetherwith its associated dynamics). Then scribethenewdynamics,we willfollowthesameprocedure
itisnecessarytoinvokeadditional,physicallyacceptablecri- adoptedintheprevioussection. Namely,wefirstexpandthe
teriatopickupapreferredquantization;otherwise,thesignif- fieldϕanditsmomentumP in(hyper-)sphericalharmonics,
ϕ
icance of the predictionsof the quantumtheory would be in extractinginthiswaytheirspatialdependence,andthenintro-
question.Thecriteriathatwearegoingtoadoptareindeedthe duceannihilationandcreationlikevariables,definedinterms
samethatallowustoselectauniqueequivalenceclassofFock of the coefficientsof the expansionlike in Eq. (8). One can
representationsinthe(φ,P )representation,thatis,theSO(4) check [using the transformation (19) and the corresponding
φ
invarianceandtheunitaryimplementationoftheevolution. initialconditions]that,withthosevariables,theblocksofthe
Fortherestofouranalysis,itisconvenienttosplitthetime originalevolutionmatrix (t)arereplacedbynew2 2ma-
n
dependent canonical transformation (16) into two parts, one trices ˜ (t)= (t) (t)U,where3 ×
n n n
U T U
that takes care of the initial conditionson the functions F(t)
bWanyedFfiG0x(aton)n,dcaeGnad0n,tdrheefsoporetachltelivraentlhyia,nttihtciaearlirnrieeitfsieaarlellvnacthleueteitsmimFee(tt00d,)eaapnneddnddGeenn(toc0te)e.. Tn(t) :=  ff−+((tt))−+ii22ggωω((ttnn)) ff−+((tt))+−ii22ggωω((ttnn))  (20)
Then,anytransformationoftheform(16)canbeobtainedas and 2f (t) := f(t) 1/f(t). Finally, a straightforward com-
thecompositionofthecanonicaltransformation putatio±nallowsust±oobtaintheBogoliubovcoefficientsα˜ (t)
n
and β˜ (t) of the evolution matrices ˜ (t), which are of the
ϕ˜ = F ϕ, P = Pϕ +G √hϕ, (18) form n Un
0 ϕ˜ 0
F
0
g(t)
wmhatiicohndoofetshneottypvaer(y16in),time,withalinearcanonicaltransfor- α˜n(t) := f+(t)αn(t)+ f−(t)β∗n(t)+i2ωn[αn(t)+β∗n(t)],
g(t)
β˜ (t) := f (t)β (t)+ f (t)α (t)+i [α (t)+β (t)].(21)
ϕ= f(t)φ, P = Pφ +g(t)√hφ, (19) n + n − ∗n 2ωn ∗n n
ϕ
f(t)
OnecannowsimplyfollowtheprocedureexplainedinSec.
butsuchthatthefunctions f(t)andg(t)nowhavefixedinitial IIandwritedowntheconditionforaunitaryimplementation
values, namely f(t0) = 1 and g(t0) = 0. The transformation ofthedynamicsofthetransformedcanonicalpair(ϕ,Pϕ)with
(18) is just a time independentlinear one, with no impactin respecttoarepresentationoftheCCR’sdefinedbyanSO(4)
ourdiscussion,giventhelinearityoftheFockrepresentations invariantcomplexstructure. We againcall thesymplectic
of the CCR’s. In fact, if a quantization with SO(4) invari- transformationthatdeterminestheinvariantcKomplexstructure
anceandunitarydynamicsisachievedforthecanonicalpair junderconsiderationintermsofthecomplexstructureofref-
(ϕ,Pϕ),oneimmediatelyobtainsaquantizationwiththesame erence j0. Wealsoadoptthenotation(13)foritscoefficients,
propertiesforthetransformedpair(ϕ˜,Pϕ˜). Norealambiguity which do not depend on time. The new dynamics admits a
comes from this kind of transformations, since the quantum
representation for the original and the transformed fields is
actuallythesame(seeRef. [9]formoredetails).
Thus,weshallrestrictouranalysistothefamilyofcanoni- 3Whilethedependenceof n(t)ont0isnotshownexplicitlytosimplifythe
caltransformations(19)withfixedinitialconditions.Wewill notation,thematrix n(t)Uactuallydoesnotdependontheinitialtime.
T
7
unitaryimplementationwithrespecttotherepresentationde- Byourassumptions,thisexpressiontendstozeroonagiven
tetrmiIn,ewdhbeyrejifandonlyifthesequences{√gnβ˜nJ(t)}areSQS saunbdseq[uze]nctoezMer.oTohnenth,iws esugbesteqthuaetn|czen,|wmhuasttitmenpdliteostthheatutnhiet
n
∀ ∈ limitℑof [z ] 2isequaltoone.
β˜nj(t):=(κn∗)2β˜n(t)−λ2nβ˜∗n(t)+2iκn∗λnℑ[α˜n(t)]. (22) Suppo(cid:0)ℜse thne(cid:1)n that there really exists a particular subse-
quence M N+ such that the terms (26) tend to zero on it
⊂
forallpossiblevaluesoftime. Sincethefactor
IV. UNIQUENESSINTHECHOICEOFFIELD
DESCRIPTION f (t)cos[(n+1)τ], (28)
−
We will now present the detailed proof that the unitarity whichmultiplies [z2]inEq. (25),isboundedforeverypar-
ℑ n
condition introducedin the previoussection implies that the ticularvalueoft,itfollowsthat
transformation (19) must in fact be the identity transforma-
tion. 1+ℜ z2n f−(t)−2ℜ[zn] f+(t) sin[(n+1)τ] (29)
Letusassumethattheunitarityconditionissatisfied. Then, (cid:16)n h io (cid:17)
tthhee csoeqnusiednecreesd{in√tgernvβ˜anjl(tI).} IanrepSarQtiScufloarr,athllisvareluqeusireosftthimatethine mhyupsotthhaevsiesa1v−anℜishzi2nngtelinmdsittoonzeMro∀otn∈MI.asBweseildl,eos,nesifnucrethbeyr
terms √gnβ˜nj(t) of these sequences tend to zero in the limit obtainsthat h i
n . Sincebothg and κ aregreaterthan1, itmustbe
als→o t∞rue that β˜nj(t)/(κn∗n)2 ten|dns|to zero. Taking into account f−(t)−ℜ[zn] f+(t) sin[(n+1)τ] (30)
theasymptoticlimitsoftheBogoliubovcoefficientsαn(t)and musttendtoze(cid:0)roonMateachpo(cid:1)ssiblevalueofthetimet.
β (t), giveninEq. (11), andintroducingforconveniencethe
n Wenowmakeuseoftheresultprovenabovethat [z ] 2
notationzn =λn/κn∗,wearriveattheconclusionthat necessarilytendstotheunitonM. Then,thereexistsℜatlenast
(cid:0) (cid:1)
onesubsequenceM Msuchthat [z ]tendsto1orto 1
ei(n+1)τ−z2ne−i(n+1)τ f−(t)−2iznsin[(n+1)τ]f+(t) (23) onM′. Inanyofthe′s⊂ecases, givenℜthatnM′ isasubsequen−ce
h i ofM,andhenceexpression(30)musttendtozeroonM,we
must have a vanishing limit when n for all values of ′
t I. Recallthatτ=t t0. → ∞ conclude(usingthedefinitionof f±)thateither
∈ −
By considering separately the real and imaginary parts of sin[(n+1)τ]
theaboveexpression,wegetthatthetwosequencesgivenre- sin[(n+1)τ]f(t) or (31)
f(t)
spectivelyby
(orboth)haveavanishinglimitonthesubsequenceM N+
′
2ℑ[zn] f+(t)−ℑ z2n f−(t) sin[(n+1)τ] ∀t∈I. But,sincethefunction f(t)iscontinuousandvan⊂ishes
(cid:16) h i (cid:17) nowhere,thisimpliesthatsin[(n+1)τ]musttendto zeroon
+ 1 z2 f (t)cos[(n+1)τ] (24)
(cid:16) −ℜh ni(cid:17) − Mwh′efroer¯IailsltphoesdsiobmleavinaloubetsaionfetdimfreominIIa,fotrereaqushiviaftlebnytltyhe∀iτni∈tia¯Il,
and
timet .
0
1+ z2 f (t) 2 [z ] f (t) sin[(n+1)τ] Let us finally prove that this limiting behavior is not al-
ℜ n − − ℜ n + lowed. Take a positive number L such that [0,L] ¯I. We
(cid:16)n z2 fh (ti)ocos[(n+1)τ] (cid:17) (25) have, in particular, that sin2[(n + 1)τ] tends to zer⊂o on M
−ℑh ni − τ [0,L]. However, a simple applicationof the Lebesgue′
havetotendtozerowhenn t I. Here,thesymbol d∀om∈inatedconvergenceshowsthatthisstatementisfalse.The
→∞∀ ∈ ℜ
denotesrealpart. details are presented in the Appendix. Essentially, one can
Wecannowapplyargumentssimilartothosepresentedin see that the integral of sin2[(n+1)τ] over the interval [0,L]
Ref. [9]andshowthat,ifitistruethatthesequencesgivenin isboundedfrombelowbyastrictlypositivenumberforlarge
Eq. (25)tendtozeroforallvaluesoftime,thenitisimpossi- n,somethingwhichisincompatiblewithavanishinglimitfor
blethatthetwosequencesformedby this function in the entire interval. Therefore, one can ex-
cludethepossibilitythatthetwosequencesoftime indepen-
1 z2 and z2 (26) denttermsappearinginEq.(26)canbothconvergetozeroon
−ℜh ni ℑh ni asubsequenceM′ N+.
⊂
havesimultaneouslyavanishinglimitonany(infinite)subse- We willnowuse this factto demonstratethatthefunction
quenceofthepositiveintegersM N+(i.e.,forn M N+). f(t)isnecessarilyaconstantfunction. Letusstudyagainthe
⊂ ∈ ⊂
Letusseethisinmoredetail. realsequencesgivenbyEqs. (24)and(25)which,aswehave
We first note that [z ] 2 tends to the unit whenever the seen, must necessarily tend to zero in the limit n for
n
two terms in Eq. (26)ℜtend to zero. This can be checkedby allpossiblevaluesoftimet Iifthedynamicsofth→e(ϕ∞,P )
(cid:0) (cid:1) ϕ
∈
summingthesquareofthetwoterms(26),whichgives canonicalpairadmitsaunitaryimplementation.
Weconcentrateourattentiononaspecificsubsetofvalues
(1 z 2)2+4 [z ] 2. (27) oftheshiftedtimeτ,namely,allvaluesoftheformτ=2πq/p
n n
−| | ℑ
(cid:0) (cid:1)
8
where q and p can be any positive integers, except for the Theconditionofunitaritydemandsthat √gnβ˜nj(t)tendtozero
conditionthattheresultingvalueofτbelongstotheintervalof inthelimitn atallvaluesoftime,andthereforethesame
definitionofthisvariable,¯I. Foreachvalueof p,weconsider musthappent→ot∞hesequenceswithterms √gnβ˜nj(t)/κn∗2. Using
thesubsequenceofpositiveintegers thisconditionand takinginto accountthe knownasymptotic
behavior(11)ofα (t)andβ (t),aswellthatλ tendstozero
M := n=kp 1>0, k N+ . (32) n n n
p { − ∈ } and √gn/ωntendstotheunitforlargen,asimplecalculation
leadstotheresultthatthesequencesgivenby
Given p, the terms (24) and (25) tend to zero on the subse-
qquveanrcieesM. Tphwenh,ewnenr→eac∞htfhoercaollntchluesvioanlutehsaotfbτotrheachedwhen g(t)−4znωnsin[(n+1)τ]e−i(n+1)τ (37)
musthaveavanishinglimit t I. Wethenconsiderthereal
1 z2 f t + 2πq (33) andimaginarypartsofthese∀seq∈uences,namely
(cid:16) −ℜh kp−1i(cid:17) − 0 p ! g(t) 4z ω sin[(n+1)τ]cos[(n+1)τ δ ] (38)
n n n
− | | −
and
and
2πq
ℑhz2kp−1i f− t0+ p ! (34) 4|zn|ωnsin[(n+1)τ]sin[(n+1)τ−δn], (39)
wherewehavewrittenthecomplexnumbersz intermsofits
musttendtozeroask goestoinfinity. Thelimitmustvanish n
phaseandcomplexnorm:
foreverypossibleintegervalueof pandq. Notehoweverthat
the time independentfactors on the left of these expressions zn = zneiδn. (40)
are preciselythose givenin Eq. (26), whichwe haveproven | |
that cannot tend simultaneously to zero on any subsequence Althoughwealreadyknowthat z tendstozero,thelimitof
n
ofthepositiveintegers,e.g.thoseprovidedbyM foreachof theproduct z ω isstillundeterm| in|ed,becauseω growslike
p | n| n n
thevaluesof p. Therefore,theonlypossibilityleftisthatthe natinfinity.
function f (t +2πq/p)is equalto zeroatallthe considered Letussupposefirstthatthesequence z ω tendstozero.
0 n n
valuesof p−andq. Usingthefactthat f(t) > 0 t I,thelast In this case, recalling that the sequence{|s g|ive}n in Eq. (38)
resultamountstotheequality ∀ ∈ should tend to zero t I, it follows immediately that g(t)
must be the zero fun∀ctio∈n on I, as we wanted to prove. Fi-
2πq
f t + =1 p,q. (35) nally,letusdemonstratethatthealternatepossibility,i.e. the
0
p ! ∀ hypothesisthat z ω doesnottendtozero,leadstoa con-
n n
{| | }
tradiction.Wemakeuseofthefactthatthesequencesformed
Realizingthatthesubsetoftimevalues t +2πq/p isdense
in I ⊂ R andthat the function f(t) is co{n0tinuous,w}e are led zbeyroth,ethteerremms(u3s9t)exteinstdstaoszuebrsoeq∀ut.enIfce{|zMn|ωonf}thdeoepsonsiottivteenidnteto-
totheconclusionthat f(t)mustequaltheunitfunctiononits
gerssuchthatthe(positive)sequence z ω isboundedfrom
entiredomain. belowonM. Thus,onthatsubsequen{c|en,| n}
It remainsto be proventhat the functiong(t) in the trans-
formation (19) necessarily vanishes, under the condition of sin[(n+1)τ]sin[(n+1)τ δ ] (41)
n
unitary dynamics. Note first that the identity f(t) = 1 that −
wehavejustdemonstratedimpliesthatz tendstozerowhen mustnecessarilyhaveazerolimit τ I. But,asshownalso
n ∀ ∈
n . In fact, after introducing this identity in Eq. (23), intheAppendix,thislaststatementcanneverbetrue. Again
on→ese∞esthatthesequences z sin[(n+1)τ] musttendtozero thecrucialargumentinvolvestheapplicationoftheLebesgue
n
τ ¯I. Therefore, in orde{r to avoid again}the false conclu- dominatedconvergence.
∀sion∈thatsin2[(n+1)τ]tendstozeroonsomesubsequenceof As a result, the only function g(t) that is allowed by the
thepositiveintegersforallvaluesofτina compactinterval, condition of unitarity is the zero function. In total, we have
it is necessary that the complex sequence z has a vanishing demonstrated that the only canonical transformation of the
n
limit. Takingintoaccountthat κ 2 = λ 2+1,itisstraight- type(19)whichispermittedonceoneacceptstheunitaritycri-
n n
forwardtocheckthatthesequen|ce| form| ed| bythecoefficients terionisthetrivialone,i.e.theidentitytransformation.Inthis
λ must tend to zero, and that 1/κ 2 (and κ 2) approaches way,thechoiceofafieldparametrizationforthesystemturns
n n n
theunitinthelimitoflargen,wh|ati|mplies|in|particularthat outtobecompletelyfixed(uptoirrelevantconstantscalings)
thesequencegivenbyκ isbounded. bytherequirementsofinvarianceunderthe symmetrygroup
n
Tocompletetheproofthatg(t)vanishes,weconsideragain ofthefieldequations,SO(4),andtheunitaryimplementation
thesequences √gnβ˜nj(t) ,particularizednowtotheonlyvalue ofthedynamics. Theambiguityintheselectionofafieldde-
{ } scriptionistotallyremoved.
allowed for the function f(t), namely the identity, so that
f (t)=1and f (t)=0. Employingthedefinitionofthecoef-
+
ficientβj(t),giv−eninEq. (15),onecancheckthattheleading
n V. CONCLUSIONSANDDISCUSSION
termsinβ˜j(t)are
n
g(t) Inthiswork,wehavebegunouranalysisbyreviewingthe
β˜j(t)(cid:27)βj(t)+i (κ )2α (t)+λ2α (t) . (36)
n n 2ω n∗ ∗n n n Fockquantizationofascalarfieldwithatimevaryingmassin
n h i
9
astaticbackground,inwhichtheinertialspatialsectionshave fraredproblemappearsandchangesthescenariodrastically.4
S3 topology. For this particular scenario, we have seen that On the other hand, the ultraviolet divergences are absent in
the criteria of: i) invariance of the vacuum under the SO(4) thesystempreciselybecauseweareusinganappropriaterep-
symmetry of the field equations; and ii) unitary implemen- resentationoftheCCR’s. Thisrepresentationturnsouttobe
tation of the field dynamics, are sufficient to select a unique theonenaturallyassociatedwithafreemasslessscalarfield.
FockrepresentationoftheCCR’s. Thereasonisthat,intheasymptoticlimitoflargewavenum-
bers,whicharetherelevantmodesfortheultravioletregime,
Anadditionalquestionconcernsthepossibilityofchanging thebehaviorofthesystem(whenthefieldisproperlyscaled)
thefielddescription,ifoneallowsforascalingofthefieldby approachessufficiently fast the behavior of a massless field.
timedependentfunctions.Thisisasituationfrequentlyfound Only Fockquantizations(with the desiredinvariance)which
incosmology,whereitiscommontointroducescalingsofthe areequivalenttotheonethatwehavechosenkeepthisgood
fields in order to absorb part of the time dependence of the ultravioletproperty.Inthisway,oneobtainsasinglefamilyof
cosmological background. The prototypical example is that unitarilyequivalentFockquantizationswhichincorporatethe
offieldsinanFRWspacetimewithcompacttopology(S3for symmetries of the field equation and respect the unitarity in
our discussion), or the closely related scenario of field per- theevolution. Concerningthe choiceof fielddescription,let
turbationsaroundanFRWbackgroundofthatkind. Thereis usalsonotethatnonstationaryspacetimesgiverisetodamp-
thereforeanextraambiguityaffectingthequantizationofsuch ingterms(firstordertimederivativesofthescalarfield)inthe
systems, namely the choice of the field description, which equationsofmotion.Inrelationwithourpreviouscomments,
necessarilyaffectsthedynamics. suchcontributionsspoiltheunitaryimplementationofthedy-
namicsatthequantumlevel.Fortunately,asuitablescalingof
thefieldrelegatesalltheinformationaboutthenonstationarity
In the above mentioned systems, it is generally the case
ofthesystemtothe(effective)massterm.
that a time dependent scaling of the field renders the field
equationsinto a formdescribingthe effective propagationin Letusseethislastpointinsomemoredetail. Aswehave
explained,inordertoquantizeascalarfieldinanonstationary
astatic backgroundwitha timevaryingmass, i.e. themodel
settingalongthelinespresentedinthispaper,onegenerically
that we considered initially. We have demonstratedhere the
performsacanonicaltransformationwhichinvolvesatimede-
resultthatweanticipatedinRef. [12],namely,thatourcrite-
pendentscalingofthefield,sothatthetransformedfieldeffec-
ria of symmetry invarianceand unitary evolution allow only
tivelypropagatesinastaticbackground.Letuscallϕands˜(t),
forone admissiblefield descriptionamongallthose thatcan
respectively,thefieldanditstimedependentmasspreviousto
be reached by means of time dependent canonical transfor-
thediscussedtransformation. Ascommentedabove,thecor-
mations that include a time dependent scaling of the field.
respondingfield equationcontainsa dampingterm, which is
The analyzed canonical transformations are linear, in order
linear in ϕ˙. We call r(t) the function multiplying ϕ˙ in this
to maintainthe linearityof all the structureson phasespace,
dampingcontribution. We now want to give the explicitex-
andpreservethesymmetryofthefieldequations.
pressions of the time dependent scaling factor, F(t) [see Eq
(16)], and ofthe mass function s(t) forthe field φ = ϕ/F(t).
Toarriveatthisuniquenessresult,verymildrequirements
Astraightforwardcalculationshowsthat
havebeenimposedonthemassfunctions(t)appearinginthe
field equation (5). Specifically, the only condition that has t r(τ) [r2(t)+2r˙(t)]
been assumed is that the mass function has a first derivative F(t)= F0exp dτ , s(t)= s˜(t) .
"−Z 2 # − 4
which is integrable in all closed subintervals of the domain t0
(42)
of definition. In addition, if one wants that the zero mode
Wealsonotethattheconditionimposedons(t)forthevalidity
ofthescalarfield(thehomogeneoussector)canbequantized
ofouruniquenessresultismet,forinstance,ifs˜(t)satisfiesthe
consistently in the standard Schro¨dinger representation with
sameconditionandr(t)hasasecondderivativewhichisinte-
the Lebesgue measure (on R), an extra condition has to be grableinallcompactsubintervalsofthetimedomainI.Onthe
added: the mass s(t) has to be nonnegative for all possible
otherhand,thepositivityofthemassfunction(forastandard
valuesoftime.
quantizationofthehomogeneoussector)amountsjustto
Letuscommentonsomekeypointsunderlyingourunique- [r2(t)+2r˙(t)]
s˜(t) t I. (43)
nessresult. Afundamentalquestionistounderstandwhyone ≥ 4 ∀ ∈
canreachunitarityinthequantumevolutionandhowthisuni-
Letusnowaddresspossiblegeneralizationsofourresults,
tarity selects a unique field description as well as a unique
startingwiththecaseofscalarfieldsindifferentcompactspa-
equivalenceclassof complexstructuresforit, amongthe set
tial manifolds. The analysis carried out here, together with
ofallsymmetryinvariantcomplexstructures. Inthisrespect,
wefirstnoticethatinfrareddivergencesarenotanissuetobe-
ginwith,owingtothefactthatthespatialsectionshavecom-
pacttopology(leadinginparticulartoadiscretespectrumfor
4Forinstance,thewellknowninequivalenceofthequantumrepresentations
theLaplace-Beltramioperator). Likeformanyotherconsid-
correspondingtofreescalarfieldsofdifferentmassesinMinkowskispace-
erationsincosmology,thecompactnessofthespatialsections
timeispreciselyduetothelongrangebehaviorofthequantumfields.See
isessential. Whenthespatialtopologyisnotcompact,thein- Ref.[27]foranaccount.
10
thedimensionalargumentsexplainedinRef. [11]inrelation focusedonscalarfields,theredoesnotseemtoexistanytech-
to the uniqueness of the representation of the CCR’s for the nical or conceptual obstacle to extend the analysis to other
fielddescriptionselectedbyourcriteria,stronglyindicatethat kindoffields,applyingtothemourcriteriainordertopickup
the resultsthatwe haveachievedforthe three-spherecan be auniqueFockquantization. Forinstance,aninterestingcase
extendedtoothercompactspatialmanifoldsprovidedthatthe isprovidedbythetracelessanddivergencelesstensorpertur-
spatial dimension d is equal or smaller than three. Suppose bationsofthemetricaroundanFRWspacetimewithcompact
that,inthesecases,therepresentationofthesymmetrygroup spatial topology. These tensor perturbations describe gravi-
ofthefieldequationisirreducibleineachoftheeigenspaces tationalwaves. Theprimordialgravitationalwavesgenerated
oftheLaplace-Beltramioperator[likeithappensforSO(4)in in the early universe can also contribute to the power spec-
the case of the three-sphere]. This propertyis actuallysuffi- trumofthe CMB, intheformoftensormodes. Infact, with
cient(thoughnotnecessary)tocharacterizetheinvariantcom- aconvenientscalingandinconformaltime,thesetensorper-
plex structures in a block diagonal form similar to that dis- turbations satisfy again equations of motion like those for a
cussed in this work. One can then follow the same kind of freefieldwithatimedependentquadraticpotentialinastatic
steps that have allowed us to complete the proof of unique- spacetime when the perturbations of the energy-momentum
ness,reachinganalogousconclusions. tensor are isotropic (see Ref. [14]). Let us mention also
Inallthecaseswithd 3,ourargumentsthereforesupport the case of fermionic fields. The study of the perturbations
theexpectationthat,whe≤noneadoptsthescalarfielddescrip- aroundaclosedFRWspacetimeproducedbyfermionsofcon-
tionwithpropagationinastaticbackground,thefreemassless stantmasswascarriedoutinRef. [28],whereaquantization
representationprovidestheunique(equivalenceclassof)Fock wasachievedafterexpandingtheperturbationsinspinorhar-
quantizationthatsatisfiesourcriteriaofsymmetryinvariance monicsonthethree-sphere. Preliminarycalculationsindicate
andunitarydynamics[11]. Besides,ourcriteriaareexpected that the kind of techniques employed here can be extended
tofixagainthefunction f(t)inthecanonicaltransformations to dealas well with the uniquenessof the Fock quantization
of the type (19). This ensures that there is no ambiguity in forfermions. Itisworthemphasizingthatthecriteriaforthis
the scaling of the field, either. The only freedom remaining uniquenessarethenaturalimplementationofthesymmetries
in the canonicaltransformationis givenbythe functiong(t). of the field equationsand the unitarity of the evolution. For
Itisnotdifficulttorealize,repeatingtheargumentsdiscussed cases other than the scalar field (and gravitational waves, as
here,thatwhetherornotthefunctiong(t)isfixedtovanishde- noticedabove), these criteria may notnecessarily imply that
pendson the square summabilityof the sequence √gn/ωn . theselectedfielddescriptioncorrespondstoafieldpropagat-
IfthesequenceisnotSQS,asithappensfortheca{sesofth}e inginastationarybackground.
two-sphere and the three-sphere, the function g(t) must van-
TheFockquantizationoffieldsinthecontextofmodernap-
ish. However, if the sequence is SQS, there exists an arbi-
proachestoquantumcosmologyisanotherinterestingframe-
trarinessandourcriteriadonotdeterminethedefinitionofthe
workwhereourresultscanhaveapplications.Oneofthemost
momentum P completely. For instance, this is the case of
ϕ promisingapproachesiswhatnowadaysiscalledLoopQuan-
the circle S1 [9]. It is worth pointing out that, nevertheless,
tum Cosmology [29–31]. LQC employs the techniques of
this freedom has nothing to do with the scaling of the field,
LoopQuantumGravity (LQG)[32–34] in thestudyof mod-
leaving intact the time evolution. If a choice of momentum
els of interest in cosmology, obtained from General Relativ-
and of invariant complex structure permits a unitary imple-
ity by the imposition of certain symmetries. In the specific
mentationofthedynamics,thesamecomplexstructureleads
caseofanFRWspacetimecoupledtoascalarfield(seeRefs.
to a unitary evolutionfor any other admissible choice of the
[35–37]),wherehomogeneityandisotropyareimposed,LQC
momentumcanonicallyconjugatetothefield. Inotherwords,
predictsthattheclassicalBigBangsingularityisreplacedby
this freedomto change the momentumby addinga time de-
a Big Bounce, which connects the observed branch of the
pendentcontributionlinearintheconfigurationfieldvariable,
universe with a previous branch in the evolution. For semi-
whenavailable,doesnotallowonetoreachanewrepresenta-
classical states with certain properties [38], the evolution is
tionsatisfyingourcriteria.
peakedarounda trajectorywhich showsa behaviordifferent
As we have explained in the Introduction, a framework from the classical one in Einstein’s theory. Then, one could
whereourresultsfindanaturalapplicationisinthequantiza- usesuchatrajectorytodefineaneffective,quantumcorrected
tionof(inhomogeneous)perturbationsarounda closedFRW background. If inhomogeneousmatter fields are introduced,
spacetime. In this context, the simplest system is a scalar their scaling by background functions would then provide a
field coupled to a homogeneousand isotropic universe with different time dependent scaling with respect to the conven-
compactspatialsections. Thissystemisspeciallyrelevantin tional case in General Relativity. At the quantum level, the
cosmology. On the one hand, the considered perturbations combination of the use of loop techniques for the homoge-
providetheseedsforstructureformation. Ontheotherhand, neous background with a standard Fock quantization of the
those perturbationsexplain the anisotropies imprinted in the inhomogeneousfields,whichpropagateinit,isknowninthe
powerspectrumofthecosmicmicrowavebackground(CMB). literature as hybridquantization[39–42]. This quantization
Ourcriteriatoeliminatethequantizationambiguitiescannow procedureassumes thatthe mostrelevantquantumgeometry
beappliedintheirquantumtreatmentandthesubsequentanal- effects (characteristic of LQG) are those that affect the ho-
ysisofthepowerspectrum. mogeneous degrees of freedom of the gravitational field. A
Althoughthediscussionthatwehavecarriedouthasbeen familyofsystemsinwhichtheapplicationofsuchaquantiza-