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Affine Quantization of Metric Variables

Permanent Link: http://ufdc.ufl.edu/UFE0022070/00001

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Title: Affine Quantization of Metric Variables
Physical Description: 1 online resource (82 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

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Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our study concerns a novel scheme for the quantization of positive-definite matrix degrees of freedom of the type associated with the spacial part of the metric tensor of general relativity. Standard canonical schemes for the quantization of such objects, wherein each matrix element is quantized independently, run into difficulties because the spectrum of the resulting 'quantum matrix' is not, in general, positive definite. We demonstrate that this situation can be overcome by abandoning canonical quantization in favor of 'affine quantization', that is, utilizing a quantum kinematical structure based the affine group, rather than the Heisenberg group. Our approach involves a generalization of the one-dimensional affine group, the group of translations and dilations of the real line. A multidimensional generalization of the one-dimensional affine Lie algebra possesses a representation in which, crucially, the spectrum of the matrix operator of interest is positive definite. The corresponding Hilbert space admits a group-defined coherent state representation, and we develop the formalism of this representation, including the formulation of coherent state overlap functions as suitably regularized path integrals. The coherent state label space, endowed with a suitable ray metric, may be viewed as a Riemannian manifold whose scalar curvature is shown to be a negative constant. This symmetry leads naturally, through the procedures of metrical quantization, to acceptable Hilbert spaces of high dimension. The incorporation of constraints into affine quantum mechanics represents an important problem, especially from the point of view of quantum gravity. We outline a relatively new technique, involving projection operators, that overcomes many of the difficulties associated with more standard approaches, and provide a simple example.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Klauder, John R.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-05-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022070:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022070/00001

Material Information

Title: Affine Quantization of Metric Variables
Physical Description: 1 online resource (82 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Our study concerns a novel scheme for the quantization of positive-definite matrix degrees of freedom of the type associated with the spacial part of the metric tensor of general relativity. Standard canonical schemes for the quantization of such objects, wherein each matrix element is quantized independently, run into difficulties because the spectrum of the resulting 'quantum matrix' is not, in general, positive definite. We demonstrate that this situation can be overcome by abandoning canonical quantization in favor of 'affine quantization', that is, utilizing a quantum kinematical structure based the affine group, rather than the Heisenberg group. Our approach involves a generalization of the one-dimensional affine group, the group of translations and dilations of the real line. A multidimensional generalization of the one-dimensional affine Lie algebra possesses a representation in which, crucially, the spectrum of the matrix operator of interest is positive definite. The corresponding Hilbert space admits a group-defined coherent state representation, and we develop the formalism of this representation, including the formulation of coherent state overlap functions as suitably regularized path integrals. The coherent state label space, endowed with a suitable ray metric, may be viewed as a Riemannian manifold whose scalar curvature is shown to be a negative constant. This symmetry leads naturally, through the procedures of metrical quantization, to acceptable Hilbert spaces of high dimension. The incorporation of constraints into affine quantum mechanics represents an important problem, especially from the point of view of quantum gravity. We outline a relatively new technique, involving projection operators, that overcomes many of the difficulties associated with more standard approaches, and provide a simple example.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Klauder, John R.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022070:00001


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Firstandforemost,IwouldliketothankmyadvisorJohnKlauderforteachingmemostofwhatlittleIknowaboutmathematicalphysics,andalsoforunreasonablygenerousamountsofpatienceovermanyyears.IwouldalsoliketothankBernardWhiting,forteachingmetothinkgeometrically,ZoranPop-Stojanovic,forteachingmetothinkmathematically,andPierreRamond,forteachingmetothinkaboutjustabouteverythingintermsofgroups.Gratitudeisalsoexpressedtothemembersofmysupervisorycommittee-StevenDetweiler,KhandkerMuttalib,PaulRobinsonandJohnYelton-fortheircontinuedinterestinthiswork.ThanksofadierentkindareduetoLarissa,Mum,Dad,andallthemembersofmyfamily,whoinsisted,inthefaceoferceresistance,thatIgobacktoschoolandnishthisproject. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 10 1.1QuantizationofPositiveDegreesofFreedom ................. 10 1.2CanonicalCoherentStates ........................... 11 1.3ReproducingKernelHilbertSpaces ...................... 13 1.4Group-denedCoherentStates ........................ 14 2ONE-DIMENSIONALAFFINECOHERENTSTATES .............. 16 2.1Overview .................................... 16 2.2RepresentationoftheOne-DimensionalAneAlgebra ........... 16 2.3One-DimensionalAneCoherentStates ................... 18 2.4One-DimensionalAneCoherentStatePathIntegral ............ 22 3GENERALIZEDAFFINECOHERENTSTATES ................. 25 3.1Overview .................................... 25 3.2TheGeneralizedAneAlgebra ........................ 25 3.3RepresentationoftheGeneralizedAneAlgebra .............. 26 3.4GeneralizedAneCoherentStates ...................... 30 3.5FactoringoutSO(n) .............................. 34 4EXPECTATIONVALUESANDTHEIRCLASSICALLIMIT .......... 40 4.1Overview .................................... 40 4.2ExpectationValues ............................... 40 4.3ExpansionoftheOverlapIntegral ....................... 44 4.4ClassicalLimit ................................. 47 5PATHINTEGRALREPRESENTATIONS ..................... 49 5.1Overview .................................... 49 5.2FormalPathIntegralforthePropagator ................... 49 5.3Continuous-timeRegularizationforthePropagator ............. 50 6GEOMETRYOFTHEPHASESPACE ...................... 52 6.1Overview .................................... 52 6.2ScalarCurvatureoftheExtendedPhaseSpace ................ 53 5

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............... 55 7CONSTRAINTS ................................... 57 7.1Overview .................................... 57 7.2ClassicalHamiltonianMechanics:aGeometricalViewpoint ......... 57 7.3GeometryoftheConstraintHypersurface ................... 59 7.3.1TheNon-DegenerateCase ....................... 60 7.3.2TheDegenerateCase .......................... 63 7.4FaddeevPathIntegralApproach ........................ 65 7.5ProjectionOperatorApproach ......................... 66 7.6AToyConstraint:detk=1 .......................... 67 7.6.1TheCasen=2 ............................. 68 7.6.2TheCasen=3 ............................. 69 7.6.3TheCasen>3 ............................. 71 7.7ConstrainedDynamics ............................. 72 8PRODUCTREPRESENTATIONS ......................... 74 8.1Overview .................................... 74 8.2FinitelyManyAneSystems ......................... 74 8.3InnitelyManyAneSystems ........................ 74 8.4InnitelyManyIdenticallyConstrainedAneSystems ........... 75 9CONCLUSION .................................... 77 REFERENCES ....................................... 79 BIOGRAPHICALSKETCH ................................ 82 6

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Figure page 7-1IntegrationregionRforI3,correspondingtothechoiceabc 70 7-2NumericalevaluationofI3(2t) ........................... 71 7

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Ourstudyconcernsanovelschemeforthequantizationofpositive-denitematrixdegreesoffreedomofthetypeassociatedwiththespacialpartofthemetrictensorofgeneralrelativity.Standardcanonicalschemesforthequantizationofsuchobjects,whereineachmatrixelementisquantizedindependently,runintodicultiesbecausethespectrumoftheresulting\quantummatrix"isnot,ingeneral,positivedenite.Wedemonstratethatthissituationcanbeovercomebyabandoningcanonicalquantizationinfavorof\anequantization",thatis,utilizingaquantumkinematicalstructurebasedtheanegroup,ratherthantheHeisenberggroup. Ourapproachinvolvesageneralizationoftheone-dimensionalanegroup,thegroupoftranslationsanddilationsoftherealline.Amultidimensionalgeneralizationoftheone-dimensionalaneLiealgebrapossessesarepresentationinwhich,crucially,thespectrumofthematrixoperatorofinterestispositivedenite.ThecorrespondingHilbertspaceadmitsagroup-denedcoherentstaterepresentation,andwedeveloptheformalismofthisrepresentation,includingtheformulationofcoherentstateoverlapfunctionsassuitablyregularizedpathintegrals. Thecoherentstatelabelspace,endowedwithasuitableraymetric,maybeviewedasaRiemannianmanifoldwhosescalarcurvatureisshowntobeanegativeconstant.Thissymmetryleadsnaturally,throughtheproceduresofmetricalquantization,toacceptableHilbertspacesofhighdimension. 8

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[Q;P]=i~I; involvesoperatorsQandPwhosespectraextendoverthefullrealline(;1).How,then,shouldweapproachthequantizationofamanifestlypositiveclassicaldegreeoffreedom?Therearevariousanswerstosuchaquestion.Onecan,forinstance,takesomelimitinwhichalargepotentialblocksothewavefunction'saccesstothenegativepartoftherealline.However,theredoesexistamorenaturalsolution,andthatistostartwithaHilbertspaceinvolvingfunctionsdenedonlyonthepositivehalf-line(0;1).SuchaconstructnecessarilyimpliestheabandonmentofthecanonicalmomentumPasaself-adjointoperator,sincethetranslationsitgeneratesnolongerrespectunitarity.Wemustinsteadturntothegeneratorofdilations,D,denedby tocompletethebasisforourLiealgebraofobservables.Multiplying( 1{1 )byQ,wearriveattheanecommutationrelation, [Q;D]=i~Q: Therelation( 1{3 )encapsulatestheLiealgebraoftheanegroup,whoseassociatedcoherentstatesarediscussedinchapter 2 Amoreinterestingquestion,withobviousrelevanceforquantumgravity,is,howshouldweapproachthequantizationofapositivedenitematrixdegreeoffreedom?Itturnsoutthatthereexistsanaturalmatrixgeneralizationof( 1{3 ),introducedbyKlauder 10

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1 ],andsincerevisitedbyanumberofauthors[ 2 ][ 3 ][ 4 ][ 5 ][ 6 ][ 7 ][ 8 ],whichisuptothetask.Adiscussionofthisgeneralizationformsthebackboneofourproject. AsisthecaseformostsituationsinvolvingtherepresentationofaLiegrouponaHilbertspace,thelanguageofgroup-denedcoherentstatesisaparticularlyelegantandconvenientone.Inthefollowingsectionswebrieydevelopthatlanguage,beginningwiththeparadigmaticexampleofcanonicalcoherentstates.Fullerandmoregeneraldiscussionsofcoherentstaterepresentationsappearin[ 9 ]and[ 10 ]. 11 ],andrstreferredtoassuchbyGlauber[ 12 ],arebasedonthecanonicalcommutationrelation( 1{1 ).TheroadtotheirdenitionbeginswiththetamingoftheunboundedoperatorsPandQviatheintroductionoftheunitaryWeyloperators, wheretherealparameters(p;q)2R2maybethoughtofasbeingcoordinatesonamanifoldweshallrefertoasthelabelspace.ThecompositionrulefortwoWeyloperators,oftenreferredtoastheWeylrelations,reads TheWeylrelations,representingan\exponentiated"formofthecanonicalcommutationrelation( 1{1 ),formthestartingpointforarigorousmathematicaltreatmentofquantummechanics.ThecelebratedStone-vonNeumanntheoremstatesthatallstronglycontinuousirreducibleunitaryrepresentationsof( 1{5 )havegeneratorsPandQthatareunitarilyequivalenttothefamiliarSchroedingeroperators[ 13 ]. CanonicalcoherentstatesresultfromtheactionofWeyloperatorsonagivenducialvectorj0iintheHilbertspace, 11

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(P+iwQ)j0i=0; forsomepositive\frequency"w. Variousremarksconcerningcanonicalcoherentstatesareinorder.Firstly,theyarestronglycontinuousinthelabelspandq,meaningthat limp0!p;q0!qjjjp0;q0ijp;qijj=0; wherewehaveusedtheusualnotationforthenormofavectorjVi, Itshouldbenotedthatthesmoothnessofthecoherentstatesimpliedby( 1{8 )isinstarkcontrastwiththesituationinvolvingtheformaleigenstatesofself-adjointoperatorssuchasPandQ.Secondly,thecanonicalcoherentstatesadmitaresolutionofunityoftheform whenintegratedoverthelabelspace. Appropriategeneralizationsof( 1{8 )and( 1{10 )havebeensuggestedastheminimumpropertiesageneralfamilyofstatesmustexhibitinordertoqualifyas\coherent".However,itisprobablyfairtosaythatapreciseandgenerallyagreed-upondenitiondoesnotexist. 12

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(;)=hji=Zhjp;qihp;qjidpdq Itisimportanttonotethatthisrepresentationcontainsonlyasubsetofallsquare-integrablefunctions,namely,thosefunctionsf(p;q)thatsatisfythereproducingrelation wherethereproducingkernelK(p0;q0;p;q)issimplythecoherentstateoverlapfunction, ThereproducingkernelcanthusbethoughtofasameanstoprojectfromL2ontotherepresentationspace. 7 ,weshallhaveoccasiontoperformtheprecedinganalysisbackwards.Thatis,weshallask,whatkindofHilbertspacecorrespondstoagivenreproducingkernel?ThejusticationforaskingsuchaquestionisprovidedbytheGNS(Gel'fand,NaimarkandSegal)Theorem[ 14 ],aconsequenceofwhichisthatanycontinuouskernelfunctionK(l0;l)satisfyingtheHermiticityrequirement andthepositivityrequirement 13

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(1{16) Theinnerproductbetweentwosuchvectorsisdenedby (;)=MXj=1NXk=1kjK(lk;lj); andthecompletionoftheHilbertspaceisachievedbyincludingthelimitpointsofCauchysequencesinthenorm,givenforvectorsinthedensesetby ThereproducingkernelK(l0;l)maythusbesaidtocompletelycharacterizeitsassociatedreproducingkernelHilbertspace.Notethattherescalingofthereproducingkernel resultsinadierentinnerproductonthesamefunctionalspace. 15 ]forexceptions),afamilyofcoherentstatesrestsonanunderlyingLiegroupstructure.Forinstance,inthecaseofcanonicalcoherentstates,thecompositionrule( 1{5 )representsthecombinationlawfortheHeisenberggroup,afactorrepresentationofthetwo-dimensionalAbeliangroup.Wenowaddresstheimportantquestionofhowtogoaboutconstructingaresolutionofunityforageneralgroup-denedfamilyofcoherentstates. 14

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whereglabelsanelementofaLiegroupG,andtheU(g)formastronglycontinuousirreducibleunitaryrepresentationofGonaHilbertspace.Considerthe\uniformdistribution"ofcoherentstateprojectionoperatorsdenedby =Zjgihgjd(g); whered(g)isaleft-invariantmeasureonG.ActingtotheleftwithanyU(g0),g02G,wendthat NowaccordingtoSchur'slemma,theonlyoperatorsatisfying( 1{23 )forallg02Gisamultipleoftheunitoperator.Ifthatmultipleisnotzeroorinnity(asitisforsomenoncompactgroups),thenisestablishedtobewithinafactorofaresolutionofunity. Theproblemofndingagivencoherentstateresolutionofunityisthusoftentantamounttotheproblemndingtheappropriateleft-invariantgroupmeasure.Thismaybeaccomplishedinpracticebyndingabasissetofleft-invariant1-formelds(knownastheMaurer-Cartanformsoftherepresentation[ 16 ]),andformingtheirwedgeproduct. 15

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1{3 ).Toemphasizethefactthatwearenolongerdealingwithacanonicalstructure,wenowabandonthenamesPandQ,replacingthemwithandrespectively.Setting~=1,weobtain [;]=i: Thecommutationrelation( 2{1 )isofthesamefundamentalsignicancetoanequantummechanicsthatthecommutationrelation( 1{1 )istocanonicalquantummechanics. TheLiealgebraimpliedby( 2{1 )isthatoftheso-called\ax+b"group,thegroupoftranslations(generatedby)anddilations(generatedby)oftherealline.Therearethreefaithful,inequivalent,irreducibleself-adjointrepresentationsof( 2{1 )[ 17 ][ 18 ].Thesemaybecharacterizedbythespectrumof,whichmaybestrictlypositive,strictlynegative,ornull.Wewilladopttherstoftheserepresentations,theothersbeing\unphysical"inthepresentcontext. IntheSchroedingerrepresentationof( 1{1 ),theoperatorsPandQpossessspectraextendingoverthewholerealline.Asdiscussedinchapter1,ifoneisattemptingtoquantizeamanifestlypositiveclassicaldegreeoffreedom,itmakessensetochooseananeframeworkratherthanacanonicalone. 2{1 )inwhich,aspromised,thespectrumofispositive.WechooseasarepresentationspacethespaceofL2functionsontheopeninterval(0;1),equippedwiththeinnerproductdenition 16

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Theoperatorsandmayberepresentedasfollows, 2(k+k); where dk: Itisimportanttonotethattheoperatorisnotself-adjoint(nordoesitpossessself-adjointextensions)inthisrepresentation.The\translations"itgeneratesarenotunitary,owingtothefactthatthedomainoftherepresentativefunctionsdoesnotincludek<0.Intheaneworld,then,cannotrepresentanobservable.Theoperator,ontheotherhand,generatesdilations, d(lnk)(k)=eB=2(eBk); whereitisstraightforwardtocheckthat Theoperatorsandserveasgeneratorsoftheunitarytransformations wherewehaveintroducedtherealparametersFandB.These(andothersimilar)transformationrelationsaremosteasilyobtainedbydierentiation.Forinstance, 17

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2{9 )gives whichuponintegrationyieldstherighthandsideof( 2{9 ). 2.2 bydeningafamilyofunitaryoperatorsU(F;B), Thecompositionrulefortheseoperatorsreads Theactionoftheunitaryoperatorsin( 2{12 )onsomespeciedducialvectorjithenyieldsafamilyofanecoherentstates, Aresolutionofunityforthecoherentstatesin( 2{14 )canbeestablished(uptoafactor)byintegratingthecoherentstateprojectionoperatorsjF;BihF;Bjagainsttheappropriateleft-invariantgroupmeasure.Thismeasurecanbewrittenasthewedgeproductofthetwoinvariant1-formseBdBanddF(theseformabasisfortheMaurer-Cartanformsoftherepresentation),andsotheresolutionofunitycanbewritten 18

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2{16 )maybeusedtowritetheinnerproductoftwovectorsjiandjiintheform giving k: Itisclearthatnoteverychoiceof(k)givesawell-denedintegralforN.ThustheconditionN<1servesasanadmissibilitycriterionforcandidateducialvectors.Thismaybecontrastedtothecorrespondingsituationforcanonicalcoherentstates,whereanynormalizedvectorintherelevantHilbertspaceisttoserveastheducialvector. Wenowintroduceasubsetofadmissibleducialvectorsdescribedby whereandarerealpositiveparameters,andthefunctionC(;)ischosentobe inordertoobtainthenormalizationhji=1.Evaluationoftheintegralin( 2{18 )yieldsN=2=,verifyingtheadmissibilityofthisducialvectorfamily. Theducialvectorsof( 2{19 )havethedistinctionofbeingminimumuncertaintyvectors.Theyarethusinthissenseanalogsoftheconventionalchoiceofducialvectorforcanonicalcoherentstates,namely,thegroundstateofaharmonicoscillator.Theane 19

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where =hi; =hi: Theminimumuncertaintypropertyofourjicanbeseenbyrecallingthattheuncertaintyrelation,appliedtoageneralvectorji,canbederivedfromtheSchwarzinequality, bysetting Inorderfortheequalityin( 2{24 )tohold,werequirethevectorsjiandjitobecolinear,thatis, forsomecomplexnumber.Thisisindeedthecaseforthejiof( 2{19 ),forwhichitisstraightforwardtoshowthat 2=0; andhencethat( 2{27 )holdswith=i=. 20

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@FhF;Bji=@ @FhjeiBeiFji=hjeiB(i)eiBeiBeiFji=ieBhjeiBeiFji; @BhF;Bji=ihjeiBeiFji; andrearrange( 2{27 )slightlytogive 0ji=0: Substitutionof( 2{30 )and( 2{31 )in( 2{32 )thenrevealsthatthefunctionshF;Bjiintherepresentationinducethe(complex)polarizationofL2characterizedby @F1 @B0hF;Bji=0: Thecoherentstatesbasedontheducialvector( 2{19 )arerepresentedbythefunctions fromwhichwemayimmediatelyderivethecoherentstateoverlap, 21

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3 ,wenowintroducethenaturalvariableGdenedby Expressingtheresults( 2{16 )and( 2{35 )intermsofGratherthanB,wendthat and respectively.Noticethatthemeasureappearingintheintegralof( 2{37 )takesonits\canonical"formwhenexpressedintermsofthevariablesFandG1.Wehavethereforefoundanaturalcoordinatesysteminwhichtoformulatefunctionalintegralrepresentations. 2{38 )togoodusetondacoherentstatepathintegralrepresentationforageneralpropagator.Todoso,itisrstusefultodenetworelevantgeometricalobjectsonageneralHilbertspace-thesymplecticpotentialandtheraymetric. SupposethatasetofnormalizedvectorsfjligdenesacurveinaHilbertspace,thelabellbeingacontinuousrealparameteralongthecurve.Thentheoverlapoftwonearbyvectorsjldl=2iandjl+dl=2imaybeexpandedas whered,linearindl,isthe(real)symplecticpotential,andd2,quadraticindl,isthe(non-negative)raymetric. 22

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2{38 )impliesthatinthecaseofanecoherentstates,wemaywrite 2G1dF(+1)(2+1) 42G2(dF)22+1 8G2(dG)2+=expi2+1 2G1dF2+1 81G2(dF)2+G2(dG)2+: Thesymplecticpotentialandtheraymetricmaynowbereado, Wenowintroduceageneraltime-independentHamiltonianoperatorHwithuppersymbol,H(F;G),denedby andlowersymbol,h(F;G),denedimplicitlyby andoutlinethestandardconstructionofthecoherentstatepathintegralforthepropagator 23

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M+1HjFk;GkidFkd(G1k); where TheM!1limitisthentaken.Itiscustomary(thoughnotrigorouslyjustied)atthispointtointerchangetheorderofthelimitandtheintegrations,andtowritetheintegrandintheformitwouldtakeforcontinuousanddierentiablepaths.Oneisthenled,withtheaidofthesymplecticpotential( 2{41 ),tothestrictlyformalexpression whereMrepresentsasuitablenormalization. Analternativepathintegralrepresentationmaybegivenusingthetechniqueofcontinuoustimeregularization[ 19 ][ 20 ].ThisprocedureinvolvestheinsertionofanappropriateWienermeasureintotheintegral,andleadsto whereMisa(-dependent)normalizationfactor.Notethatinthisregularizationofthepathintegral,itisthelowersymbolfortheHamiltonianthatappears. 24

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2{1 )ismotivatedbythedenitionalpropertiesofthemetrictensorofclassicalgeneralrelativity.Thespacialpartofsuchatensor,atagivenpointinspacetime,appearsinagivencoordinatesystemasasymmetric33matrixfgijgthatisrequiredtobepositivedenite, forallnonzerovectorsVi.Variousalternativecharacterizationsofpositive-denitematricesareavailable[ 22 ],animportantonebeingprovidedbythefactthatamatrixispositivedeniteifandonlyifallitseigenvaluesarepositive.AnotheristhecomputationallyconvenientSylvester'scriterion:amatrixispositivedeniteifandonlyifallitsleadingprincipalminorsarepositive. Standardcanonicalquantizationproceduresrunintodicultieswhenappliedtoapositivedenitematrixdegreeoffreedom.Theusualprocedure,involvingtheintroductionofcanonicalconjugatesforthegij,failstoleadtotherequiredspectralproperties-asymptomofthenonlinearityoftheunderlyingphasespace.Ourgoalistoshowthatthesedicultiescanbeovercomebyinsteadadoptingananequantizationschemeinwhichthepositive-denitenatureofthematrixfgijgisalreadyfullyrespectedatthequantumkinematicallevel,inthesamewaythatthepositivenatureofisrespectedintheone-dimensionalanequantummechanicsofchapter 2 2{1 ),rstsuggestedbyKlauder[ 1 ].Itsconstructionproceedsviatheintroductionofasetofn2generatorsbaalongwiththeirn(n+1)=2symmetricaneconjugatesjk(=kj), 25

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2kabjbjka; 2bjak+bkaj; [ab;jk]=0; wherealltheindicestakeonvaluesfrom1ton.Foragivenn,thedimensionalityofthisLiealgebraisn2+n(n+1)=2=n(3n+1)=2.Weshallhoweverforthesakeofconveniencerefertoitasbeingthen-dimensionalanealgebra.Thecommutationrelation( 3{2 )isseentobethatofasetofmatricesfMbagwithelements(Mba)sr=i1 2arbs,sothatthefbagbythemselvesformabasisfortheLiealgebraofGL(n;R),thegenerallineargroupinndimensions.Thusthen-dimensionalanealgebracontainsgl(n;R)asasubalgebra. wheretheindicesrunfrom1ton,andthe+ontheintegralindicatesthatthedomainofintegrationextendsonlyoverthoseregionsinwhichthematrixkabissymmetricandpositivedenite.Wedenetheinnerproductoftwovectors(k)and(k)inthespacetobe 26

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2@(pb)k(pa)+k(pa)@(bp)=ik(ap)@(bp)+1 4(n+1)ba; with 2(kab+kba); 2@ @kab+@ @kba: WewilldenoteagenerallinearcombinationofthekjbyB,whereB=fBbagisamatrixofrealcoecients,and Similarly,wedenoteagenerallinearcombinationofthejkbyF,withF=fFabgasymmetricarrayofrealcoecients,and Theoperatorsbaandabaresubjecttothetransformationrules wherethe(invertible)matrixSisdenedby 27

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2 .Forinstance,inordertoverify( 3{14 ),letusset wheretisarealparameter.Thendierentiationwithrespecttotgives dtYba(t)=eiB[iBqppq;ba]eiB=1 2BqpeitB(bqpapabq)eitB=1 2Ybp(t)BpaBbpYpa(t): Inasimilarfashion,weset whereupondierentiationthistimeyields dtZba(t)=1 2Zbp(t)BpaBbpZpa(t): Comparisonof( 3{20 )and( 3{22 )revealsthatYba(t)andZba(t)satisfythesamedierentialequation.Furthermore,theyshareacommonboundarycondition,namely,thatYba(0)=Zba(0)=ba.Therefore and( 3{14 )holdsasaspecialcaseof( 3{23 )witht=1. Notethatin( 3{14 )and( 3{15 ),theoperatorstransformastensorsofthevalencesindicatedbyourchoiceofindexplacement. Anothertransformationofcentralimportanceisdescribedby 28

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(STkS)ab=SpakpqSqb; and(k)isafunctioninourrepresentationspace.Toverify( 3{24 ),werstnotethat Itthenremainstoshowthat Toverify( 3{27 ),wefollowanowfamiliarprocedure,introducingarealparametertanddemonstratingtheequalityoftheobjectsY(t)andZ(t)denedby and Variationof( 3{28 )withrespecttotyields whileasimilarvariationof( 3{29 )gives @kabZ(t)dt=(BT=2)k+k(B=2)ab@ @kabZ(t)dt=Bbak(bp)@(ap)Z(t)dt; 29

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3{31 )wehaveusedthesymmetrycondition( 3{5 ).SinceY(t)andZ(t)satisfythesamedierentialequationwiththesameboundarycondition(att=0),theyareidentical.Settingt=1in( 3{28 )and( 3{29 )thenestablishes( 3{27 ). Wemaynowverifyexplicitlythatourrepresentationisindeedunitary.Using( 3{24 ),weobtain whereinthesecondlineof( 3{32 )wehaveintroducedanewmatrixvariablek0denedby TheJacobianassociatedwiththechangeofvariablesfromktok0is(detS)n+1.Furthermoreandrathermoreobviously(using( 3{8 )), 2 ,wenowdenethefamilyofunitaryoperators Thecompositionrulefortheseoperatorsmaybedeterminedfromtheiractiononafunctionintherepresentationspace, 30

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(S0FS0T)ab=S0apFpqS0bq; (S0S)ba=S0bpSpa: Thecorrespondinggroup-denedcoherentstates,whichweshallrefertoasgeneralizedanecoherentstates,aredenedby wherejiisanasyetunspeciedducialvectorintherepresentationspace.Theyhavefunctionalrepresentatives Aresolutionofunityconstructedfromthesecoherentstatesmaybeestablishedintheusualway,namely,byintegratingthecoherentstateprojectionoperatorsjF;SihF;Sjagainsttheappropriateleft-invariantgroupmeasure, whereNisanormalizationconstanttobedetermined.Thevalidityof( 3{41 )maybecheckedbyevaluatingageneralinnerproduct, 31

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3{42 )isonlyapparentlydependentonk.Indeed,ifwedeneMtobeamatrixsuchthat thechangeofvariablesSba=MpabqS0qp,orequivalently, whoseassociatedJacobianis(detM)n,revealsthat with (STS)ab=XpSpaSpb: Thissimplicationarisescourtesyofthefactthat isaleft-invariantmeasureonGL(n;R).WethusarriveatanexpressionforthenormalizationconstantN, whichisonlywell-denediftheducialvectorsatisestheadmissibilitycriterion Ifwecanndaducialvectorsuchthat( 3{49 )holds,theexistenceofaresolutionofunityforthegeneralizedanecoherentstatesisassured.Tothisendwechooseanatural 32

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2 where>(n1)=4,>0,andthenormalizationconstantCn(;)ischosentobe with n(2+1)=Z+(detk)2etrkYabdkab; sothat Theintegralforn(2+1),amultidimensionalgeneralizationoftheintegralrepresentationoftheone-dimensionalgammafunction,doesinfactexist(thatis,itconverges).Weshallpostponeanexplicitvericationofthisfact,aswellasademonstrationoftheadmissibilityoftheducialvectorin( 3{50 ),untilthenextsection. Theoverlapoftwocoherentstatesbasedontheducialvectorin( 3{50 )maynowbeevaluated, wherethefrequentlyoccuringcombinationisdenedas 33

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andwehaveusedourusualshorthand NotethatXisguaranteedtobeinvertible,sinceSSTandS0S0T,beingpositive-denitematrices,havestrictlypositiveeigenvalues. TheX-dependencemaybeextractedfromtheintegralin( 3{54 )usingachangeofvariablesk!X1k, det(2X)]n(21)=det(S0S) det[(S0S0T+SST)=2+i(F0F)=2]: Itisevidentthattheoverlapfunction( 3{58 )onlydependsonthematrixSthroughthesymmetriccombinationSST.Ourrepresentationisthereforeinvariantundertransformationsofthetype wherehereM2SO(n).Itisappropriate,then,toeliminatetheredundantdegreesoffreedombyfactoringoutSO(n)fromtherepresentation.Tothisend,weintroducethesymmetricmatrixG=fGabganditsinverseG1=fGabg, 34

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InordertointegrateouttheSO(n)variablesfromtheresolutionofunity,itisnecessarytoexpressthemeasureappearingin( 3{41 )asaproductofSO(n)andnon-SO(n)parts.WebeginwithapolardecompositionofthematrixS, whereTisapositiven-dimensionaluppertriangularmatrixandM2SO(n).Forn=2,thisdecompositiontakestheform Itisstraightforwardtoshowthatinthiscase, wheretheJacobiandeterminantJisgivenby Togeneralizethisresultforn>2,weexpresstheSO(n)matrixMasaproductofn(n1)=2rotationmatrices.Denotingarotationabouttheij-axisthroughanangleij

PAGE 36

NowR12(12),whoseexplicitformis isagainresponsiblefortheintroductionofafactorT22intotheJacobian.Likewise,eachRij(ij)introducesafactorTjj.BuildinguptheentireJacobianinthisway,wendthat wheredistheinvariantmeasureonSO(n).HavingseparatedthematrixSintoits\radial"partTand\angular"partM,wearenowinapositiontomakethechangeofvariablefromTtoG(notethatboththesematricespossessn(n+1)=2degreesoffreedom), 36

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fromwhichitisstraightforwardtoshowthat Puttingtogether( 3{69 )and( 3{72 ),andnotingthat detS=detT=nYi=1Tii; wendthat Wemaynowusethemeasureintheform( 3{74 )tointegrateouttheSO(n)degreesoffreedomfrom( 3{41 ), 2nN1nZYjkdFjkZ+jF;GihF;Gj(detG)(n+1)YabdGab=11: HerenisthegroupvolumeofSO(n),whichmaybeexpressedasaproductofthesurfacevolumesofj-spheres[ 21 ], n=nYj=12j=2 37

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3{75 )resultswhenthemeasureisexpressedintermsoftheelementsofG1ratherthanthoseofG, 2nN1nZYjkdFjkZ+jF;GihF;GjYabdGab=11: Theatnatureofthemeasureintheintegralof( 3{77 )indicatesthatthecoordinatesFjkandGabarenormalcoordinatesonthefactorgroupinquestion. Aspromised,wenowreturntothequestionoftheconvergenceoftheintegralrepresentationforthegeneralizedgammafunctionof( 3{52 ).ACholeskyfactorizationofthepositive-denitematrixk, withQauniquelydeneduppertriangularmatrixwithpositivediagonalelements,yields n(2+1)=Z+(detQ)4ePj;k(Qkj)2Yabdkab: Integrating( 3{79 )overSO(n),andusinganexactanalogof( 3{74 ),weobtain n(2+1)=2n1nZdetQ>0(detQ)4+1ePj;k(Qkj)2Ya;bdQba; amanifestlyconvergentGaussianintegral. Wemayalsoputthemeasurein( 3{74 )toworkinordertoverifytheadmissibilityoftheducialvectorof( 3{50 ).Startingfrom( 3{48 ),wendthat n(2+1)<1; 38

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3{81 )isdependentonourrestrictionontheducialvectorparameter, 4: 39

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with requiresthenormalizationconstantCn(;)tobegivenby ItisclearthatCn(;)innowaydependsonthecoherentstatelabelsFandG.Itiscertainlytruethenthat @G(ab)hF;GjF;Gi=[Cn(;)]2@ @G(ab)Z+(G;k)Yrsdkrs=0; wherefornotationalconveniencewehaveintroduced 40

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4{4 )maybeperformedwiththehelpoftheidentities @G(ab)=@ @Gab+@ @Gba; and @Gab(detG)=Gab(detG); yielding [Cn(;)]2Z+[2Gab4k(ab)](G;k)Yrsdkrs=0: Wearethusledtotheabexpectationvalue, Thebaexpectationcannowbeevaluateddirectly, 4(n+1)bahkjF;GiYrsdkrs=Z+hF;Gjki(i)hk(ap)(iFbpGbp)+ wherewehavemadeuseof( 3{9 )and( 4{9 ),andintroduced 41

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4{1 )onemoretimegives @G(jk)@ @G(ab)hF;GjF;Gi=@ @G(jk)[Cn(;)]2Z+2G(ab)4k(ab)(G;k)Yrsdkrs=[Cn(;)]2Z+4Ga(jGk)b+(2Gab4k(ab))(2Gjk4k(jk))(G;k)Yrsdkrs=[Cn(;)]2Z+4Ga(jGk)b42GabGjk+162k(ab)k(jk)(G;k)Yrsdkrs=0; wherewehaveusedtheidentity leadingimmediatelyto 42

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2ba(i)k(jq)(iFkqGkq)+1 2kjhkjF;GiYrsdkrs=Z+ik(ap)(iFbpGbp)+1 2ba(i)k(jq)(iFkqGkq)+1 2kj(G;k)Yrsdkrs=2 Therealandimaginarypartsof( 4{15 )representtheexpectationsoftheHermitianoperator1 2(bakj+kjba)andtheantihermitianoperator1 2(bakjkjba)respectively, 2(bakj+kjba)jF;Gi=2 2(bakjkjba)jF;Gi=i Thelatterresultmayofcoursebeobtainedmoresimplybystartingwiththecommutationrelation( 3{2 )andusing( 4{10 ). Thetwopossiblefullcontractionsofbaji,namelyaabbandbaab,arescalarHermitianoperatorswhoseexpectationsare and 8: 43

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2kj(G;k)Yrsdkrs=2 Therealandimaginarypartsofthisrelationaregivenby 2(abkj+kjab)jF;Gi=2 2(abkjkjab)jF;Gi=i Thelatterexpectationisagainseentobeconsistentwithasimplercalculationinvolving( 4{9 )andthistime( 3{3 ). 3{62 ).Considerthegeneratingfunction whereE=fEabgisanarbitraryrealsymmetricmatrix.Thepowerseriesexpansionfor(E)reads 2EabEjkhF;GjabjkjF;Gi+O(E3): 44

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4{23 )mayalsobewritten wherewehaveusedthecombinationrule( 3{36 ).Theoverlapformula( 3{62 )cannowbeutilizedtogive wherewehaveset Theexpressionfor(E)in( 4{26 )canbeexpandedasapowerseries, 2tr(Y2)1 22(trY)2+O(Y3)=1+i Comparisonoftherstandsecondordertermsin( 4{24 )and( 4{28 )yields and 45

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4{9 )and( 4{14 ). Asimilarprocedurecanbeusedtoevaluateexpectationsinvolvingtheji.Thistimeweconsiderthegeneratingfunction withAbaanarbitraryrealmatrix.Thepowerseriesexpansionfor(A)reads 2AbaAkjhF;SjabjkjF;Si+O(A3): Wemayalsowrite(A)as deth1 2G+eA=2G(eA=2)T+i wherewehaveusedtheoverlapformulain( 3{58 ).Comparisonoftherstandsecondordertermsin( 4{32 )and( 4{33 )yields and 2(bakj+kjba)jF;Gi=2 46

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4{10 )and( 4{16 ). Withthisinmind,wenowxtheducialvectorparametersandin( 3{50 ),setting andthereby inordertoobtain Higher-orderexpectationsaremorecomplicated(duetothepresenceofwhatmightbecalled\quantumcorrections"),butbecomesimplerinthelimit!1,whereforexample 47

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lim!1hF;GjjkabjF;Gi=GjkGab; lim!1hF;Gjaabbn=4jF;Gi=FaaFbb; lim!1hF;Gjbaabn(n+1)=8jF;Gi=FbaFab; lim!1hF;Gj1 2(abkj+kjab)jF;Gi=FkjGab: Thesecondtermsintheleft-handsidesof( 4{43 )and( 4{44 ),representingtheregularizationsrequiredtoobtainwell-denedlimits,arereminiscentofinnitegroundstateenergysubtractionsineldtheory,andcouldalternativelybeabsorbedintoareviseddenitionoftherespectiveexpectationvalues. Thelimits( 4{42 )-( 4{45 )maybethoughtofasbeing\classical"limits.Inordertorestore~toitstraditionalroleintheducialvectorof( 3{50 ),weset=a=~and=b=~,writing whereCn(;;~)isanormalizationfactor.Itisclearthatthelimits!1and~!0areequivalent. 48

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2 ,weendorseacontinuous-timeregularizationprocedure,whereaWienermeasure,basedontheraymetric,isusedtotameapurelyformalexpressionforthepropagator. 2.4 maybefollowedtobuildapathintegralrepresentationforthecoherentstatepropagatorJT(F00;G00;F0;G0),denedby whereHisaHamiltonianwithuppersymbolH(F;G)denedby andlowersymbolh(F;G)denedimplicitlyby Theoverlapformula( 3{62 )yields det(I+iG1dF=2)#=expi 49

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det(I+)=etrln(I+)=etr[2=2+O(3)]: Thesymplecticone-formdandtheraymetric(d)2mayimmediatelybereadothelastlineof( 5{4 ), (d)2=0 where,followingtheshorthandofchapter 2 ,wehavedened Thepropagatormaythenbewrittenas astrictlyformalexpressiontowhichtheremarksimmediatelypreceding( 2{49 )againapply. 50

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51

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5{7 ),asbeingthephasespace 5{7 )isclearlyofcrucialimportancetopathintegralregularizationsofthetypeappearingin( 5{10 )[ 23 ].Ofparticularinterestisthescalarcurvatureofthephasespace 2n(n+1)2: Thecalculationsleadingto( 6{1 )areoutlinedinthefollowingsections. Thephysicalconceptbehindthecontinuous-timeregularizationprocedureutilizedin( 5{10 )involvesthemotionofafreeparticleonacurvedspace(hereourphasespaceisregardedasthecongurationspace).QuantizationofsuchasystemleadstotheLaplace-Beltramioperatorplusapossibleadditionaltermoforder~2proportionaltothescalarcurvature[ 25 ].Asourscalarcurvatureisconstant,thistermrepresentsaharmlessfactorintheintegral( 5{10 )thatcanbeincludedintheoverallnormalization.Thiscrucialresultisobtaineddespitethefactthatthesectionalcurvature[ 26 ],denedschematicallyfortwovectorsUandVby 23 ]wehavereferredtothisspaceasgravitationalphasespace,forobviousreasons.2 24 ]. 52

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isnotconstant(atleastforn2).Theidenticationofthevariouscoordinatesaselementsofapositivedenitematriximpliesalackofisotropyonthemanifold. Inprinciple,itispossibletochooseaphasespacemetricforusein( 5{10 )\byhand"insteadoftheusingtheraymetricof( 5{7 ).However,inordertoobtaininterestingHilbertspaces(thatis,spacesofnon-trivialdimensionality),itisnecessarythatsuchmetricsshouldendowtheirphasespaceswithconstantscalarcurvature.Thisconditionimmediatelydisqualiesmanyapparentlyplausible-lookingmetricsfromseriousconsideration.Forinstance,thephasespacemetric[ 27 ]describedby (6{4) doesnotleadtoaphasespaceofconstantscalarcurvature,andthereforecannotgenerateviaananalogof( 5{10 )anythingotherthanaHilbertspaceoftrivialdimensionality. 6.1 ,itrstprovesconvenienttoreleasethesymmetryconditionsFjk=FkjandGab=Gba,thusdeningann2-dimensionalmanifold,whichweshallrefertoastheextendedphasespace.Weshalltakethemetriconthismanifoldtobe (d)2=1tr[(G1dF)2]+tr[(G1dG)2]=1GbcGdadFabdFcd+GbcGdadGabdGcd; whereforconveniencewehaveintroducedascalingparameterwhichplaysessentiallytheroleofin( 5{7 ). Theconnectioncoecientsassociatedwiththemetricin( 6{5 )maybecalculatedindirectlybyconsideringthedynamicsofaclassicalfreeparticlemovinginthegeometry 53

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2trh1(G1_F)2+(G1_G)2i=1 2h1GbcGda_Fab_Fcd+GbcGda_Gab_Gcdi: ExtremizationoftheactionintegralinvolvingtheLagrangianin( 6{6 )leadstotheequationsofmotionforFandG,whichmaybewrittenintheform Fjk+_Gab_FcdkdbcGja+_Gab_FcdjcadGbk=0; Gjk_GjaGab_Gbk2Gja_FabGbc_FcdGdk=0: Itwillbenoticedthatthesolutioncurvesof( 6{7 )and( 6{8 )representcurvesofextremallengthonthemanifold,namely,geodesics.Therefore( 6{7 )and( 6{8 )admitare-interpretationasgeodesicequations,beingoftheschematicform whereisthemetricconnection.Writingout( 6{9 )inanobviousnotationwhereagainrepresentstheconnectionassociatedwiththemetricof( 6{5 ),wehave Fjk+_Fab_FcdFjkFabFcd+2_Fab_GcdFjkFabGcd+_Gab_GcdFjkGabGcd=0; Gjk+_Fab_FcdGjkFabFcd+2_Fab_GcdGjkFabGcd+_Gab_GcdGjkGabGcd=0: Comparisonof( 6{10 )and( 6{11 )with( 6{7 )and( 6{8 )immediatelyyieldsexpressionsfortheconnectioncoecients, GjkFabFcd=(GjaGbcGdk+GjcGdaGbk)=2; FjkGabFcd=(kdbcGja+jcadGbk)=2; GjkGabGcd=(Gbcajdk+Gdacjbk)=2: 54

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28 ] Thesecondandthirdtermsontherighthandsideof(6{15)vanishasaconsequenceoftheconstantnatureofthedeterminantofthephasespacemetricin( 6{5 ).Evaluationoftheremainingtermsyields @GabGabFjkFlmGabFcdFjkFcdFlmGabFabGcdFjkGcdFlmFab=n2GmjGkl; @GabGabGjkGlmGabGcdGjkGcdGlmGabFabFcdGjkFcdGlmFab=nGmjGkl: Finally,contractionwiththeinversemetricthengivesthescalarcurvatureoftheextendedphasespacemanifold, ItisworthpointingoutthattheF-Gcrosstermsinvolving( 6{13 )playavitalroleinthecalculationabove,sincethescalarcurvatureofthecorrespondingn2-dimensionalmanifoldinvolvingonlytheelementsofthematrixG(andnotthoseofF)turnsouttoben(n21)=2,andnotsimplyn3=,asacursoryglanceat( 6{18 )mightsuggest. 6{5 )againapplies,subjectnowto 55

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Thescalarcurvatureofthesubmanifoldofsymmetricmatricesmayagainbeobtainedstartingfrom( 6{15 ).Thistimehoweveritisnecessarytoemploysymmetriccontractionsinordertolteroutthosecontributionspresentinthepreviouscalculationinvolvingskew-symmetricdirections.ThisprocessyieldstheRiccitensorelements 2G(m(jGk)l); andscalarcurvature Thetwo-dimensionalresult,R(n=1)=RE(n=1)=2=,willberecognizedastheconstantnegativescalarcurvatureofthePoincareplane.Noticethatthescalarcurvatureofthegeneralizedhigher-dimensionalphasespacemanifoldswehavedescribedremainsconstantforeachn. WeemphasizethatF-Gcrosstermsinvolving( 6{13 )areagainimportantinthederivationof( 6{21 ).ThescalarcurvatureofthecorrespondingsubmanifoldinvolvingonlythesymmetricmatrixGisn(n1)(n+2)=8. 56

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Traditionalapproachesfordealingwithquantumconstraintsare,forbetterorforworse,rmlyrootedinclassicalmechanics.Theclassicationofconstrainthypersurfacesinclassicalmechanicsisessentiallyageometricalone,andwethereforeundertakeabriefreviewofHamiltonianmechanicsfromageometricalperspective. Thesymplectic2-form!maybeusedtodeneaclassofvectoreldsonMknownasHamiltonianvectorelds.AHamiltonianvectoreldHisrequiredtosatisfy ~d[!(H)]=0; where~drepresentstheexteriorderivative.HamiltonianvectoreldsarethereforethosevectoreldsalongwhichtheLiederivativeof!vanishes.ThephysicalsignicanceofHamiltonianvectoreldsisobtainedbyassociatingtheirintegralcurveswithpossiblepathsofaphysicalsystemthroughphasespace,suitablyparameterizedbythetimet. 57

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ThefunctionhisknownastheHamiltonianofthephysicalsystemunderconsideration.Itis,uptoanadditiveconstant,uniquelydeterminedbyH. SupposethereexistsafunctionfonM.TherateofchangeoffalonganintegralcurveofaHamiltonianvectoreldHisgivenby dt=Hf=Hbf;b=h;a!abf;b: Theobjectin( 7{3 )iscalledthePoissonbracketofthefunctionsfandhandisconventionallywrittenff;hg. TheHamiltonianvectoreldcorrespondingtothePoissonbracketoftwofunctions,fa;bg,istheLiebracketofthetwocorrespondingvectorelds,[B;A].Thismaybeseenbyevaluatingtherateofchangeofanarbitraryfunctionfalongthevectoreld[B;A].DenotingtheLiederivativeby$, [B;A]f=~df;[B;A]=~df;$BA=$B~df;A$B~df;A=Bff;ag~d(Bf);A=fff;ag;bgfff;bg;ag=ff;fa;bgg; 58

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AnaturalandusefulcoordinatesystemonMisspeciedbyasetoffunctionsqaandpbwhosePoissonbracketssatisfyfqa;pbg=ab.Coordinatesthusconstructedarecalledcanonicalcoordinates.Theyarenotunique,andacoordinatetransformationbetweendierentsetsofcanonicalcoordinatesiscalledacanonicalcoordinatetransformation.Inasystemofcanonicalcoordinates,thePoissonbracketoftwoscalarfunctionsfandgassumesthefamiliarsimpleform @qa@g @pa@g @qa@f @pa: Associatedwitheachconstraintfunctionthereisaconstraintvector,withcomponentsb,suchthat TheconstraintvectormaybethoughtofastheHamiltonianvectoreldcorrespondingthetheHamiltonian.InwhatfollowsweassumethattheconstraintfunctionshavebeenchosensuchthattheircorrespondingconstraintvectorsconstitutealinearlyindependentsetateachpointoftheconstrainthypersurfaceM0. StatementsareoftenencounteredwhosetruthonlynecessarilyholdsonM0.Suchstatements,calledweakequalities,aretraditionallyindicatedwiththesymbol.Thestatementf0,forexample,indicatesonlythatthefunctionfvanishesonM0. 59

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Itisusefultodividetheconstraintfunctionsintotwoclasses.Aconstraintfunctionissaidtoberstclassifitcommutesweaklywithalltheotherconstraintfunctions, Aconstraintfunctionfailingtosatisfy( 7{9 )issaidtobesecondclass. ThePoissonbracketsofthevariousconstraintsmaybehousedconvenientlyinanmmantisymmetricmatrixC=f;g.Itisimportanttonotethatthepresenceofarstclassconstraintimmediatelyrendersthismatrixnon-invertible. InordertoimposeasymplecticstructureontheconstrainthypersurfaceM0,itisnecessarytoendowitwithasymplectic2-formofitsown.Themostnaturalwaytoaccomplishthisissimplytousetherestrictionof!toM0(sometimescalledthepre-symplectic2-form).Thisstrategymayimmediatelybesuccessful,anditspotentialsuccessisthesubjectofsubsection 7.3.1 .However,inmanycasesofphysicalinterest,therestricted2-formdoesnotdeneasymplecticstructureontheconstrainthypersurfacebecauseitfailstobeinvertiblethere.Thetraditionalwaytoproceedunderthesecircumstancesisoutlinedinsubsection 7.3.2 Theinvertibilityof!impliesthenon-existenceofanonzerovectorUtangenttoM0suchthat!0(U)=0.Inotherwords,thereisnononzerovectortangenttoM0whoseskewproductwitheveryothervectortangenttoM0vanishes. 60

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Itfollowsfrom( 7{7 )that foranyvectorVtangenttoM0.WethusdeducethatcannotbetangenttoM0,forsuchavectorwouldnecessarilysatisfy( 7{11 ).Hence,inthenon-degeneratecase,nolinearcombinationofconstraintvectorsistangenttotheconstrainthypersurface.Theconstraintvectorsthereforeformasub-basisspanningthedirectionsoM0. Associatedwiththevectoreld,thereisascalarfunction ThePoissonbracketofwithaconstraintfunction,evaluatedonM0,isgivenby SinceisnottangenttoM0,cannotcommutewithalltheconstraints.Therefore,inthenon-degeneratecase,allnonzerolinearcombinationsoftheconstraintfunctionsaresecondclass.ThusinturnistantamounttosayingthatthematrixCisinvertibleonM0.WedenoteitsinversebyC. ThePoissonbracketoftwofunctionsuandvonM0,constructedfromtherestrictedsymplectic2-form!0,iscalledtheDiracbracketofuandv,denotedfu;vgD.TorelatetheDiracbracketonM0totheoriginalPoissonbracketonM,wedecomposeageneralvector 61

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whereV0istangenttoM0andVisthecomponentofValongtheconstraintvector,obtainedbyoperatingonbothsidesof( 7{14 )with!(), Wendthat giving AnewandreducedHamiltoniansystembasedontherestricted2-form!0maybedenedontheconstrainthypersurface,inwhichtheconstraintsareseenasconstants, Conventionalwisdomdictatesthatonlyatthispointshouldanattempttoquantizethesystem,basedontheDiracbracketof( 7{17 ),bemade.Suchanapproachmaybecriticizedfromanumberofdierentpointsofview.Fromaphilosophicalstandpoint,itshouldbenotedthatthequantumsystemnevergetsachanceto\see"theconstraint,sinceitiseectivelyanentirelynew,unconstrainedclassicalsystemthatispresented 62

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AgivenHamiltonianfunctiononM0isthereforenotsucienttofullydeterminethedynamics,becauseitonlyspeciesaHamiltonianvectorelduptoanarbitrarylinearcombinationofnullvectors.Forsimplicity,weassumeinwhatfollowsthattherankof!0isconstantoverM0. Itfollowsfrom( 7{19 )thattheHamiltonianfunctionucorrespondingtoanullvectoreldUisconstantoverM0.Suchafunctionmay,inaneighborhoodofM0,beexpandedintermsoftheconstraintfunctions, Thefunctionusatises andisconsequentlyrstclass.Thedegeneratecaseisthereforecharacterizedbytheexistenceofatleastonerst-classlinearcombinationoftheconstraintfunctions. Itfollowsfrom( 7{4 )thattheLiebracketofanytworst-classconstraintvectorsisitselfarst-classconstraintvector.Thereforethesetofrst-classconstraintvectorsgeneratesaLiealgebrawhich,byvirtueofFrobenius'theorem,maybeusedtoconstruct 63

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InmanycasesofphysicalinterestitisconsideredinconvenienttoworkwiththequotientspacebecausesuchaspacemaynotpossesscertainusefulsymmetriesofthefullphasespaceM.UndersuchcircumstancesitisusualtoretainthefullconstrainthypersurfaceM0,withthephysicalinterpretationthatgaugemotionisunobservable.AnobservablefunctiononM0isdenedtobeafunctionwhosevalueisconstantovereachgaugeorbit. Dirac'sstrategyforthequantizationofrst-classconstraintsystemsistorstperformaconventionalcanonicalquantizationoftheoriginalunconstrainedsystem,andthentoimposeaphysicalityrequirementontheresultingHilbertSpace.Theconstraintfunctionsiarepromotedtoconstraintoperatorsi,andphysicallyallowedstatevectorsjiphysarerequiredtosatisfy ijiphys=0: CandidateHamiltonianoperatorsHarerequiredtogeneratetime-evolutionwithinthephysicalHilbertSpace,andmustthereforecommutewiththeiinthatspace, [H;i]jiphys=0: AlthoughtheDiracprocedurehasprovedtobeanextremelysuccessfultoolforthequantizationofmanygaugesystemsofphysicalinterest,itisclearthatitfailsingeneraltoprovidemeaningfulresults.Forinstance,ifthevaluezerodoesnotlieinthespectrumofagivenconstraintoperator,orevenifthevaluezeroliesinthecontinuousspectrumofagivenconstraintoperator,thenthephysicalHilbertspaceisempty,andthemethodreachesadeadend. 64

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5{9 ),hasbeenprovidedbyFaddeev[ 35 ].Theessenceofthemethodinvolvesanattempttowritedownaformalphasespacepathintegralinwhichtheonlypathswithsupportarethosewhichconnethemselvesatalltimestotheclassicalconstrainthypersurface.Thismaybeaccomplishedbymeansofameasureinvolvingdeltafunctionalsoftheconstraints.Suchaconstructionleads,ingeneral,toadivergentintegral,andsoitbecomesnecessaryto\xthegauge"withtheintroductionofdeltafunctionalsofthemutuallycommutingauxilliaryconditions Thisprescriptionresultsinapropagatoroftheform whereallthetermshavetheirstandardmeanings.TheFaddeev-Popovdeterminant,detfi;jg,ensuresformalinvarianceundercanonicaltransformations. Although( 7{25 )islocallyinvariantundergaugetransformations(thisresultisknownasFaddeev'stheorem),itfailsingeneraltobegloballyinvariant.Morespecically,twosetsofgauge-xingconditionscanonlybeguaranteedtoresultinthesameFaddeevpathintegraliftheydeneequivalentcoveringsofthespaceofconnectedgaugeorbits[ 36 ].TopologicalobstructionsbetweeninequivalentcoveringsleadtowhatareknownasGribovambiguities[ 37 ]. ItisimportanttonotethattheFaddeevapproachtorst-classconstrainedsystemsdiersphilosophicallyfromtheDiracapproachinthatquantizationfollowsreduction.Sincethesetwooperationsdonotingeneralcommute,thetwoapproachesmayleadtocompletelydierentresults. 65

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38 ]andothers.Thecentralphilosophyoftheapproachisthatconstraintsshouldbeinterpretednotasrelationsbetweenclassicalvariables,butratherasrestrictionsonthespectraofappropriatequantumoperators.Thetreatmentofrst-classsystemsinvolvesnogaugexing,andsothecorrespondingtopologicaldicultiesaresidestepped.Classicaleliminationofsecond-classconstraintsisnotrequired,andindeedrstandsecond-classconstraintsystemsaredealtwithonanequalfooting.Sincethemethodmakesnodirectreferencetopropertiesofanyclassicalconstrainthypersurface,itisparticularlywellsuitedtoadescriptionofconstraintsinanequantummechanics. Themethodbeginswiththespecicationofaprojectionoperatorappropriateforagivensetofconstraintoperatorsfag.First,anoperator isbuiltfromtheconstraints,whosespectrumisbydenitionnon-negative.Theprojectionoperatorofinterest,E(Z2),thenprojectsontothatpartofthespectrumofZlessthanorequalto2.Thestrategyisthen,roughlyspeaking,tomakeassmallaspossiblewithoutallowingEtovanish.OnceasuitableprojectionoperatorEhasbeenestablished,itsmatrixelementshF0;G0jEjF;Gican,inaccordancewiththeGNStheorem,beusedtodenethereproducingkernelofanewHilbertspace,tobeinterpretedasthephysicalHilbertspace. InthecasethattheconstraintoperatorsgeneratetheLiealgebraofacompactLiegroup, [a;b]=ifcabc; 66

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withd()theappropriateHaarmeasure. Ifoneormoreoftheconstraintoperatorspossessesacontinuousspectrum,thenacarefullimitingprocedureisnecessaryinordertoidentifythecorrectphysicalHilbertspace.Anexampleisprovidedinthefollowingsection. Toillustratetheprojectionoperatortechnique,weconsiderasingle(andthereforerst-class)constraintdetk=1,correspondingtotheconstraintoperator =det1: Sincetheconstraintrepresentsasinglepointininthecontinuousspectrumof,itisclearthat lim!0hF0;G0jE(<<)jF;Gi=0: Toovercomethissituation,weextractareproducingkernelappropriateforourquantumconstraintsubspacebyintroducingamagnicationfactor1=2, 2hF0;G0jE(<<)jF;Gi: Workinginthek-representation,andusingthecoherentstaterepresentativefunctions itisstraightforwardtoshowthat 67

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Therelation( 7{33 )denesthereproducingkernelonthequantumconstraintsubspace. WritingX=tY,witht=np TheintegralInisnowrevealedasonlybeingdependentonthematrixXthroughtheparticularcombinationdetX.Wediscussitsevaluationforvariousvaluesofthedimensionalityninthefollowingsubsections. Forthecasen=2,itispossibletoexpresstheintegralIninclosedform.Usingtheconvenientparametrization wendthat wherewehaveusedvariousintegralpropertiesofK0,thezero-ordermodiedBesselfunctionofthesecondkind[ 31 ][ 32 ]. 68

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Forthecasen=3,aparametrizationinthespiritof( 7{36 )becomesrathercumbersome,andweinsteadturntoachangeofvariablesbasedonthe\canonical"decompositionofthepositivedenitematrixk, withO2SO(3)andDthediagonalmatrixofeigenvaluesofk, arrangedsuchthatabc>0.Themeasureappearingintheintegral( 7{35 )canbewritten whered3istheinvariantmeasureonSO(3).Notethatatthosepointsink-spacewheretheeigenvaluesaredegenerate,themeasurein( 7{40 )vanishes.Intermsofthenewvariables,theintegralI3maybewritten whereRistheintegrationregiondepictedingure 7-1 ,and3isthegroupvolumeofSO(3).TheformofthefunctionI3,asafunctionof2t,isshowningure 7-2 .Acareful 69

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2(2t)3; andthatforlarget, Figure7-1. IntegrationregionRforI3,correspondingtothechoiceabc

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NumericalevaluationofI3(2t) Usingaparametrizationbasedon( 7{38 ),wherenowO2SO(n)andDisthediagonalmatrixofeigenvalues arrangedsuchthata1a2an>0,wendthat wherenisthegroupvolumeofSO(n),givenin( 3{76 ).Theintegral( 7{45 )doesnotlenditselftofurtherusefulsimplication,althoughitisclearthatthemainfeatures 71

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(2t)n(n1)=2; andthatforlarget, withnowt=np Givenaself-adjointHamiltonianoperatorHontheoriginalunconstrainedHilbertspace,dynamicsonthephysicalsubspacecanbedenedbyintroducingthepropagator limN!1hF0;G0jEeiHT=NEeiHT=NEEeiHT=NEjF;Gi=hF0;G0jEe(iEHET)EjF;Gi; whoseformmaybeinterpretedastheresultofinsertingaprojectionoperatorateveryinnitesimaltimesliceinordertocontinuouslyenforcethequantumconstraints.AsubtletyimmediatelyariseshereinthatalthoughEHEisHermitian,itmayfailtobeself-adjoint.However,inthecasethatthespectrumofHisboundedbelow,itwillalwaysadmitaself-adjointextension. Followinginthespiritofchapter 5 ,wemaygivethepropagatorin( 7{48 )theformalpathintegralrepresentation 72

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Itisclearthatanextraordinaryrangeofconstrainedsystemscaninprinciplebedealtwithusingtheformalismrepresentedby( 7{48 ).First-classsystemscanbetreatedwithouttheneedforgaugexing,whilesecond-classsystemsneednotsuertheindignityofbeingreducedbeforequantizationtakesplace.Thelatterfeatureisofparticularrelevancetoquantumgravity,wheretheconstraints,uponquantization,becomepartiallysecondclass. 73

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whereforthemomentweleavethechoiceofducialvectoropen,otherthanarequirementthatitbenormalized.Thereproducingkernelforthecompositesystememergesastheproductoftheindividualreproducingkernels, 8{2 )converges.Theconvergenceinquestionshouldexcludethecaseofdivergencetozero,thatis,whentheinniteproducthasthelimitingvaluezeroeventhoughnoneoftheindividualfactorsvanishes. 74

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Sincethecoherentstatesarenormalized,wemaydeducethatthetwoproductvectorsjF;GiandjF0;G0ipossessaconvergentinnerproductifandonlyiftheyhavethesame\tail",inthesensethat limN!11Xn=N1hFn;GnjFn;Gni=0; limN!11Xn=N1hFn;GnjF0n;G0ni=0; forsomexedproductvectorjF;Gi.Thus,inthelimitn!1,theoriginalHilbertspacefragmentsintoanuncountablenumberoforthogonalHilbertspaces,eachbuiltfromlinearcombinationsofproductvectorswiththesametail. Theoverlapexpansionin( 5{4 )maybeusedtoexplicitlycharacterizetheconvergencecondition( 8{3 )fortheparticularchoiceofducialvectorgivenin( 3{50 ),theresultbeingthathF0;G0jF;Giisconvergentifandonlyifthetwoconditions whereFn=F0nFnandGn=G0nGn,arebothmet. 7 ,involvesthereplacementoftheunconstrainedreproducingkernelshF0n;G0njFn;GniwiththeconstrainedreproducingkernelshF0n;G0njEjFn;Gni,whereEistheappropriateprojectionoperator[ 39 ].Suchareplacementleadstoa(candidate)expressionfortheproductsystemreproducingkernel, 75

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8{8 )denesanacceptablereproducingkernelfortheconstrainedproductsystem. Intheeventthatthetaildoesnotlieentirelywithinthephysicalsubspace,itispossibletorescueanonvanishingreproducingkernelfortheproductsystembymeansofarescalingprocedure.Forconvenience,werstxthetail,taking withtheducialvectorjichosensuchthathjEji6=0(thatis,jiischosentobecompatiblewiththegivenconstraint).Anacceptablereproducingkernelfortheproductsystemisthenobtainedbysimplyrescalingeachoftheindividualreproducingkernels, hjEji: 76

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Inthepreviouschapters,wehaveshownhowaquantumframeworkappropriatetoapositive-denitematrixdegreeoffreedommaybebuilt.Ourconstruction,basedonageneralizationoftheane(\ax+b")group,fullyrespectsthenonlinearityoftheunderlyingphasespace,andleadstothecorrectspectralproperties. OurcentralresultisthattheresultingHilbertspacemaybecharacterizedbyareproducingkerneloftheform Wehavediscussedatlengththecorrespondingcoherentstaterepresentation,establishingtheconnectionbetweenthecoherentstatelabelspaceandtheclassicalphasespace,andshowingthatforanappropriatechoiceofducialvector, Wehaveshownhowvariouscoherentstateoverlapfunctionscanberealizedasphasespacepathintegrals,pointingouttheavailabilityofaWienermeasureregularizationprocedureasanalternativeandintuitivelyattractivewaytogivemeaningtootherwisepurelyformalexpressions. Ourstudythroughouthasbeenmotivatedbycertainfeaturesofclassicalgeneralrelativity.Oneimportantaspectofthistheoryisitsconstraintstructure.Therearefourclassicalgravitationalconstraintfunctions(threedieomorphismandoneHamiltonian),collectivelyexhibitinganopenrst-classPoissonbracketstructure.However,uponquantization,thisconstraintsystembecomespartiallysecond-class.Thisfeaturepresentsdicultiesforstandardmethodologies,whichgenerallyspeakingarefocusedondealingwithpurelyrst-classconstraintsystems.Theprojectionoperatormethodwehave 77

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7 ,however,hasthevirtueoftreatingrstandsecond-classconstraintsonanequalfooting,andwethereforestronglyendorseitsuseinthecontextofquantumgravity. Thesimplestexampleofaeldtheory(thatis,atheorywithaninnitenumberofdegreesoffreedom)isprovidedbyaproductsystem.Wehavediscussedthekinematicsofaneproductsystems,withspecialregardtotheenforcementofconstraints. Theroadtowardafullaneeldtheorybeginswiththepromotionoftheoperatorsin( 3{2 )-( 3{4 )toeldoperators, 2kabj(x)bjka(x)(xy); 2bjgak(x)+bkgaj(x)(xy); [gab(x);gjk(y)]=0; wheregabcorrespondstothespacialpartofthemetrictensoreld,andkjtoananeversionoftheADMcanonicalmomentumeld,dubbedthe\momentic"tensoreldbyKlauder.Anoverviewofaproposedanequantumgravityprogram,startingfrom( 9{4 )-( 9{6 ),maybefoundin[ 40 ]and[ 41 ]. Inclosingthen,itishopedthatinadditiontothrowingsomelightontheinterestingstructureofanequantummechanics,wehaveinthisworkmanagedtoprovidesometoolsthatmaybeusefulinaddressingatleastsomeoftheproblemsassociatedwiththeconstructionofafullquantumtheoryofgravity. 78

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[1] KlauderJR1970inRelativityedMSCarmeli,SIFicklerandLWitten(PlenumPress)p1 [2] IshamCJandKakasAC1984Agrouptheoreticalapproachtothecanonicalquantisationofgravity.I.ConstructionofthecanonicalgroupClass.QuantumGrav.1621 [3] IshamCJandKakasAC1984Agrouptheoreticalapproachtothecanonicalquantisationofgravity.II.UnitaryrepresentationsofthecanonicalgroupClass.QuantumGrav.1633 [4] PilatiM1983Strong-couplingquantumgravityI.SolutioninaparticulargaugePhys.Rev.D262645 [5] PilatiM1983Strong-couplingquantumgravity.II.SolutionwithoutgaugexingPhys.Rev.D28729 [6] FranciscoGandPilatiM1985Strong-couplingquantumgravity.III.QuasiclassicalapproximationPhys.Rev.D31241 [7] WatsonGandKlauderJR2000Generalizedanecoherentstates:anaturalframeworkforthequantizationofmetric-likevariablesJ.Math.Phys418072 [8] KlauderJR2002TheanequantumgravityprogrammeClass.Quant.Grav.19817 [9] KlauderJRandSkagerstamB-S1985CoherentStates(WorldScientic,Singapore) [10] PerelomovAM1986GeneralizedCoherentStatesandTheirApplications(Springer-Verlag) [11] SchroedingerE1926Naturwissenschalften14664 [12] GlauberRJ1963PhotonCorrelationsPhys.Rev.1312766 [13] ReedMandSimonB1972FunctionalAnalysis(AcademicPress) [14] EmchGG1972AlgebraicMethodsinStatisticalMechanicsandQuantumFieldTheory(Wiley,NewYork) [15] KlauderJR1993Coherentstateswithoutgroups:quantizationonnonhomogeneousmanifoldsModernPhys.Lett.A8 NakaharaM2003Geometry,TopologyandPhysics(IOPPublishing) [17] Gel'fandIMandNaimarkMA1947UnitaryrepresentationsofthegroupofanetransformationsofthestraightlineDokl.Akad.NaukSSSR55570 79

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KlauderJRandAslaksenEW1968UnitaryrepresentationsoftheanegroupJ.Math.Phys.2206 [19] DaubechiesI,KlauderJRandPaulT1987WienermeasuresforpathintegralswithanekinematicvariablesJ.Math.Phys.2885 [20] KlauderJR1999inQuantumFutureEds.P.BlanchardandA.Jadczyk(Springer-Verlag,Berlin)129 [21] GilmoreR1974Liegroups,Liealgebras,andsomeoftheirapplications(Wiley,NewYork) [22] BhatiaR2007Positivedenitematrices(PrincetonSeriesinAppliedMathematics) [23] WatsonGandKlauderJR2002MetricandcurvatureingravitationalphasespaceClass.QuantumGrav193617 [24] WeinbergS1972GravitationandCosmology(Wiley,NewYork,1972)142 [25] DestriC,MaranerPandOnofriE1994OnthedenitionofaquantumfreeparticleoncurvedmanifoldsNuovoCimA107,237 [26] Kay,D1988TheoryandProblemsofTensorCalculus(McGraw-Hill) [27] KlauderJR1990inProbabilisticMethodsinQuantumFieldTheoryandQuantumGravityEds.P.H.Damgaard,H.HuelandA.Rosenblum(North-Holland,NewYork)73 [28] SchutzB1980GeometricalMethodsofMathematicalPhysics(CambridgeUniversityPress) [29] KlauderJR,McKennaJandWoodsEJ1966Direct-productrepresentationsofthecanonicalcommutationrelationsJ.Math.Phys.5822 [30] ArnowittR,DeserSandMisnerC1962inGravitation:AnIntroductiontoCurrentResearchEd.L.Witten(Wiley&Sons,NewYork)227 [31] MorsePMandFeshbackH1953MethodsofMathematicalPhysics(McGraw-Hill,NewYork)volume21323 [32] GradshteynISandRyzhikIM1965TableofIntegrals,SeriesandProducts(AcademicPress)736 [33] HenneauxMandTeitemboimC1992QuantizationofGaugeSystems(PrincetonUniversityPress) [34] DiracPAM1967LecturesonQuantumMechanics(AcademicPress) [35] FaddeevLD1970Theor.Math.Phys.11 80

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GovaertsJ1991HamiltonianQuantisationandConstrainedDynamics(LeuvenUniversityPress) [37] GribovVN1978Nucl.Phys.B139 KlauderJR1997CoherentstatequantizationofconstraintsystemsAnnalsPhys.254419 [39] KlauderJR1999ProductrepresentationsandthequantizationofconstrainedsystemsProc.V.A.SteklovInst.Math.226212 [40] KlauderJR1999Noncanonicalquantizationofgravity.I.FoundationsofanequantumgravityJ.Math.Phys405860 [41] KlauderJR2001Noncanonicalquantizationofgravity.II.ConstraintsandthephysicalHilbertspaceJ.Math.Phys424440 81

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TheauthorwasborninLondon,England,in1968,andearnedaBachelorofArtsdegreefromOxfordUniversityin1989. 82