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AlgebraicQuantum Gravity(AQG)IV.

Reduced PhaseSpaceQuantisation

of

Loop Quantum Gravity

K.Giesel1�,T.Thiemann1;2y

1 M PIf.Gravitationsphysik,Albert-Einstein-Institut,

Am M �uhlenberg 1,14476 Potsdam,Germany

2 PerimeterInstituteforTheoreticalPhysics,

31 CarolineStreetN,W aterloo,ON N2L 2Y5,Canada

PreprintAEI-2007-152

A bstract

W eperform acanonical,reducedphasespacequantisationofGeneralRelativitybyLoopQuantum

Gravity (LQG)m ethods.

The explicit construction ofthe reduced phase space is m ade possible by the com bination of

1. the Brown { Kucha�r m echanism in the presence ofpressure free dust �elds which allows to

deparam etrisethetheory and 2.Rovelli’srelationalform alism in theextended version developed by

Dittrich to constructthealgebra ofgauge invariantobservables.

Since the resulting algebra ofobservablesisvery sim ple,one can quantiseitusing the m ethods

of LQG.Basically, the kinem aticalHilbert space of non reduced LQG now becom es a physical

Hilbert space and the kinem aticalresults ofLQG such as discreteness ofspectra ofgeom etrical

operators now have physicalm eaning. The constraints have disappeared,however,the dynam ics

ofthe observables isdriven by a physicalHam iltonian which is related to the Ham iltonian ofthe

standard m odel(withoutdust)and which we quantisein thispaper.

1 Introduction

The objects ofultim ate interest in a �eld theory with gauge sym m etry are the gauge invariant

observables. There are two m ajor approaches to the canonicalquantisation ofsuch theories. In

theso called Diracapproach one�rstconstructsHilbertspacerepresentationsofgaugevariantnon

�gieskri@aei.m pg.deythiem ann@aei.m pg.de,tthiem ann@perim eterinstitute.ca

1

observablesand then im posesthevanishing ofthequantised version oftheclassicalgaugesym m etry

generators(constraints)asa selection principle forphysicalstates.The associated physicalHilbert

space then hopefully (ifthere are no anom alies)carriesa representation ofthe observable algebra.

In theso called reduced phasespaceapproach one�rstconstructstheclassicalobservablesand then

directly looksforrepresentationsofthatalgebra.

The advantage ofthe Dirac apporoach isthatthe unreduced phase space ofnon observablesis

typically a sm ooth (Banach)m anifold so thatthealgebra ofnon { observablesissu�ciently sim ple

and representations thereofare easy to construct. Its disadvantage is that one has to dealwith

spuriousdegreesoffreedom which isthe possible source ofam biguitiesand anom aliesin the gauge

sym m etry algebra. The advantage ofthe reduced phase space approach is that one never has to

careaboutkinem aticalHilbertspacerepresentations.However,itsdisadvantageisthatthereduced

phase space typically no longerisa sm ooth m anifold turning the induced algebra ofobservablesso

di�cultthatrepresentationsthereofarehard to �nd.

ThereducedphasespaceofGeneralRelativitywithstandardm atterishardtoconstructexplicitly.

However,on can com binetwo independentrecentdevelopm entsin orderto m akeprogress:

On the one hand,Brown & Kucha�r have shown in a sem inalpaper [1]that there is hope to

constructobservablesifoneaddspressurefreedusttothetheory.Thisisbecauseonecan then write

theconstraintsin deparam etrised form 1.

On the otherhand,there isRovelli’srelationalform alism [2]forconstructing observableswhich

we need in the extended form developed by Dittrich [3]. W ith this form alism one can write the

observables as an in�nite series Ff;T in term s ofpowers ofso called clock variables T and with

coe�cients involving m ultiple Poisson brackets between constraints C and non observables f such

thattheseriesis(form ally)gaugeinvariant2.Rem arkably [3,4],them ap FT : f 7! Ff;T isaPoisson

hom om orphism between thealgebraofnon observablesf and thealgebraofobservableswith respect

to a certain Diracbracket(which isuniquely determ ined by theconstraintsand thefunctionsT).

Now usually Dirac brackets m ake the Poisson structure so com plicated that one cannot �nd

representations thereof. However,as observed in [4],ifthe system deparam etrises,ifone uses as

clocksT thecon�guration variablesconjugateto them om enta P in C = P + H and ifoneconsiders

functionsf which donotdependonT;P 3 thenFT becom esaPoisson bracketisom orphism .M oreover,

the functionsH in C = P + H becom e physical,conserved Ham iltonian densities which drive the

physicalevolution oftheobservables.Thisim pliesthata reduced phasespacequantisation strategy

becom es available,since to �nd representations ofthe Ff;T is as easy as forthe f. The only non

trivialproblem leftisto �nd representationswhich supportthephysicalHam iltonian4.

In [5]these two independent observations were com bined and the algebra ofclassicalphysical

Observableswasconstructed explicitly by adding a generalscalar�eld Lagrangian withoutpotential

to theEinstein { Hilbertand standard m odelLagrangian.Itturnsoutthatam ong the,in principle,

in�nitenum berofphysicalObservablesthereisa unique,positiveHam iltonian selected.

In [6,7]thatfram ework wasfurtherim proved by using asspeci�c scalar�eld the pressure free

dustofBrown & Kucha�r.ThecorrespondingHam iltonian ispositive,reducestotheADM energy far

1G iven a system ofconstraintsCI on aphasespace,deparam etrisation m eansthatonecan � nd localcoordinatesin

theform oftwo m utually com m uting setsofcanonicalpairs(qa;pa);(TI;�I)such thattheconstraintscan bewritten

in the locally equivalentform CI = �I + H I wherethe H I only depend on the (qa;pa).

2Itism anifestly gaugeinvariantin an open neighbourhood ofthephasespaceiftheseriesconvergeswith non zero

convergenceradiuswhich hasto be checked.3ThisisnolossofgeneralitybecauseP can beelem inated in term softheotherdegreesoffreedom viatheconstraints

and T ispure gauge.4Thecaveatisthatthedeparam etrisation and thusthereduced phasespacequantisation isgenerically only locally

valid in phasespace.Thus,the globally valid Diracquantisation program m eshould be developed furtherin parallel.

2

awayfrom thesourcesand tothestandard m odelHam iltonian on atspace.Itgeneratesequationsof

m otionfortheobservablesassociatedtothenondustvariablesthatareinagreem entwiththeEinstein

equations for the system without dust,up to sm allcorrections which originate from the presence

ofthe dust. In particularone can develop a m anifestly gauge invariantcosm ologicalperturbation

theory to allorders which was shown to reproduce the linear order as developed by M ukhanov,

Feldm ann and Brandenberger[8]. The dustservesasa m aterialreference system which we couple

dynam ically as �elds rather than assum ing the usualtest observers in order to give the Einstein

equations(m odulogaugefreedom )theinterpretation ofevolution equationsofobservablequantities.

Thisleadsto in principle observable deviationsfrom the standard form alism which howeverdecay

during thecosm ologicalevolution.

In this paper we quantise the algebra ofobservables constructed in [6]. Actually there is not

m uch to do because that algebra is isom orphic to the Poisson algebra ofGeneralRelativity plus

the standard m odelon R � S where S is the dust space m anifold. Hence we can take over the

kinem aticalHilbertspace representation thatisused in Loop Quantum Gravity (LQG)[9,10].For

recentreviewson LQG see [11],forbookssee [12]. One m ay objectthatthisrepresentation isless

naturalhere than in usualDirac quantised LQG where itisuniquely selected on physicalgrounds

[13,14],nam ely one wantsto have a unitary representation ofthe spatialdi�eom orphism group of

the coordinate m anifold X which is a gauge group (passive di�eom orphism s) there. Since allour

observables are gauge invariant,we have no di�eom orphism gauge group any longer,hence that

physicalselection criterion isabsent.However,itisreplaced by a di�erentone:Itturnsoutthatthe

physicalHam iltonian hasthedi�eom orphism group ofthedustlabelspaceassym m etrygroup.These

di�eom orphism s change our observables,they are active di�eom orphism s since they m ap between

physically distinguishable dustspacelabels.Thuswem ay apply thesam eselection criterion.

Now the interesting rem aining question is whether that representation allows us to de�ne the

quantised version ofthe physicalHam iltonian. M aybe notsurprisingly,itturnsoutthatthe sam e

techniquesthatallowed to constructthe quantum Ham iltonian constraint[15]and the m astercon-

straint[16]in usualDiracquantised LQG can beused to de�nethequantised physicalHam iltonian.

This operatorispositive,hence sym m etric and upon taking itsnaturalFriedrich extension,itbe-

com es self{ adjoint. In orderto preserve itsclassical,active,spatialdi�eom orphism sym m etry it

turns out that one has to de�ne it in such a way that it preserves the graph ofa spin network

function thatitactson. The techniques developed in [17]can now be applied to show,using the

sem iclassicalstates introduced in [18],thatthe physicalHam iltonian has the correct sem iclassical

lim iton su�cinently �negraphs.In fact,in ordertogetrid ofthegraph dependenceonecan usethe

generalisation ofLQG to Algebraic Quantum Gravity [17]. Thiscastsquantum gravity com pletely

into thefram ework of(Ham iltonian)latticegaugetheory [19,20]with onecrucialdi�erence:There

isno continuum lim itto be taken because we are in a background independenttheory with active

di�eom orphism sassym m etries.

The attractive feature ofthis reduced phase space approach is that we no longer need to deal

with the constraints: No anom alies can arise,no m aster constraint needs to be constructed, no

physicalHilbert space needs to be derived by com plicated group averaging techniques. W e m ap

a conceptually com plicated gauge system to the conceptually safe realm ofan ordinary dynam ical

Ham iltonian system . The kinem aticalresults ofLQG such asdiscreteness ofspectra ofgeom etric

operators now becom e physicalpredictions. This is a concrete im plem entation ofthe program m e

outlined forthefulltheoryin [21]andgeneralisesthereduced phasespacetechniquesrecentlyadopted

forthe Loop Quantum Cosm ology (LQC)truncation ofLQG [22,23,24,25]which isa toy m odel

forthecosm ologicalsectorofLQG,to thefulltheory.

It\rem ains"toanalysethephysicalHam iltonian in detailsinceitencodesthecom pletedynam ics

3

ofGeneralRelativity coupled to thestandard m odel.Thefollowing tasksshould beaddressed in the

future:

1.Vacuum and spectralgap

Fora startwenoticethatthephysicalHam iltonian doesnotdepend explicitly on an external

tim e param eter. OurHam iltonian system which dynam ically couplesgeom etry and m atteris

a conservativesystem .Thisisin contrastto QFT on curved and in particulartim edependent

background spacetim em etricswhereonequantisesm atterpropagating on an externally given

background geom etry.TheHam iltonian ofthatQFT isnotpreserved and thuseven thenotion

ofaground stateorvacuum asalowestenergy eigenstatebecom estim edependentwhich leads

to constantparticlecreation problem setc.[26].In ourapproach thenotion ofa vaccum state

would notsu�erfrom thoseproblem s.Thisappearsasa conceptualim provem entalthough of

coursethelowesteigenvalueoftheHam iltonian could bevastly degenerate.Also,them inim um

ofthe spectrum ofthe Ham iltonian m ightnotlie in itsdiscrete (m ore precisely,pure point)

partso thatthe\ground state(s)" would notbenorm laisable.

2.Scattering theory

W ith a physicalHam iltonian H atourdisposalwe can in principle perform scattering theory,

that is,we can com pute m atrix elem ents ofthe tim e evolution operator U(�) = exp(i� H ).

Theanalyticalevaluation ofthosem atrix elem entsisofcoursetoo di�cultbutasin ordinary

QFT we m ay use Ferm i’sgolden rule and expand,forshorttim e intervals�,the exponential

asU(�)= 1H + i� H +O (�)2.Them atrix elem entsofH seem hopelessto com pute because it

involvessquare rootsofa positive selfadjointoperatorforwhose precise evluation we would

need theassociated projection valued m easurewhich ofcoursewedo nothave.However,since

in scattering theory initialand �nalstatesareexcitationsovera ground statewhich wedo not

know exactly butpresum ably can approxim ateby kinem aticalcoherentstates,onecan invoke

the technique developed in [17]to expand the square rootofthe operatoraround the square

rootofitsexectation value. W e willdo thisin a future project. Ofcourse there are issuesto

be resolved such asthose ofthe existence ofasym ptotic states[27]and how one im plem ents

them in ourform alism ,seee.g.[28]forsom ebasicideas.

3.Anom alies

Asalready m entioned,the Ham iltonian H hasa huge sym m etry group ofwhich Di�(S)isa

subgroup and itiseasy to im plem entthissym m etry atthe quantum level. However,there is

anotherin�nite classical,Abelian sym m etry group N which isgenerated by the Ham iltonian

density functions H (�) and in term s ofwhich the Ham iltonian reads H =R

Sd3� H (�).

Classically one hasfH (�);H (�0)g = 0 which ofcourse im pliesclassically thatfH (�);H g=0.

The Lie algebra ofthe totalclassicalsym m etry group thus consists ofin�nitesim alactive

di�eom orphism sand in�nitesim altransform ationsgenerated by theH (�).Thelatterform an

Abelian Poisson idealand thusN isan Abelian invariantsubgroup in thetotalsym m etry group

which hence isa sem idirectproductG = N o Di�(�).Presum ably,in the naive quantisation

ofH thatweconsiderasa prelim inary proposalin thispaper,thelattersym m etry isexplicitly

broken,oranom alousalthough sem iclassically itispreserved.In orderto reinstallit,onecan

try tom akeuseofrenorm alisation group techniquesassociated tosocalled im proved orperfect

actions[29].

4.Lattice num ericalm ethods

It transpires that within the fram ework proposed here m any ofthe conceptualproblem s of

4

canonicalquantum gravity have been solved and thetechnicaltaskshave been sim pli�ed and

reduced to a detailed analysis ofthe operator H ,ofcourse,at the price to have introduced

additional,albeit unobservable, m atter as a m aterialreference system and a possibly only

locally (in phase space)description.Since H isa com plicated operatorwhich isisform ulated

in term soflatticelikevariablesespecially in theAQG version,itisnaturalto useM onteCarlo

m ethodsin orderto study theoperatornum erically.

5.QFT on curved spacetim esand standard m odel

It is widely accepted that the fram ework ofQFT on curved spacetim es [26]should be an

excellent approxim ation to quantum gravity whenever the m etric uctuations are sm all. In

particular,when the background spacetim e is M inkowski,then the standard m odelm ust be

reproduced. Besidesthat,one would like to see whetherourbackground independentlattice

theory which is m anifestly UV �nite and non perturbative can explore the non perturbative

sectorofthestandard m odelsuch asQCD.Anotherinteresting question iswhetherourexplic-

itly geom etry { m atercoupled system can lead to an im proved understanding oftheHawking

e�ectdueto thepossibility to takecareofbackreaction e�ects.

6.E�ective action,universality,am biguities

Ourfram ework presents a canonicalquantisation ofthe �eld theory underlying the Einstein

Hilbertaction plusstandard m odelaction. Now com putationswithin perturbative QFT and

also string theory suggestthatthe e�ective action5 forgravity isan extension ofthe Einstein

{ HilbertLagrangian by higherderivative term s and an often asked question is whetherone

should notquantisethesem oregeneralactions.Thereareseveralrem arksin order:

A.The e�ective action isa com plicated,often even non local,action which takescare ofall

higherloop diagram m esobtained from asim plebareaction.Itlookslikeaclassicalaction

butitactually encodesallquantum uctuations.Thereforeitisinappropriatetoquantise

that classicalaction anew,it would not produce the sam e quantum theory as the bare

action.

B.Stillonecould justadd allpossiblehigherderivativeterm sfrom theoutset.W hileonecan

canonically quantise such theoires by the Ostrogradsky form alism ,this leads in general

to a drastic increase in the num berofdegrees offreedom [31]due to the appearance of

highertim ederivatives.

C.In the Euclidean form ulation ofQFT on M inkowskiasa path integralone entertains a

related (W ilson)notion ofe�ectiveaction astheaction thatoneobtainswhen integrating

out degrees offreedom labelled by (in Fourier space) m om enta above a certain energy

scale6.Thisalso producesvarioushigherderivative term satlowerenergiesascom pared

5Thereareseveralloosely equivalentde� nitionsforthee� ectiveaction.Thenotion wem ean hereisthefollowing:

Consider� rsta renorm alisabletheory.G iven a de� ning action with a � nite num berof� nite butunknown couplings

and m asses(param eters)one can perform perturbation theory and discovers,within a given regularisation schem e,

thattheparam etersareto bealtered by functionsofthedistancecuto� which divergein thelim itofvanishing cuto�

in orderto avoid singularitiesin loop diagram m es.Ifonedoesthisorderby orderthen oneendsup with theso called

bareaction which produces� nitehigherloop diagram m esto allorders.Thee� ectiveaction isa vehiclethatproduces

the sam escattering am plitudesorn�pointfunctionsasthe bareaction butofwhich oneonly needsto com pute tree

diagram m es(no loops). The de� nition for a non renorm alisable theory such as gravity is the sam e,just that then

num ber ofparam eters is in� nite. In renorm alisable theories a � nite num ber ofexperim ents is su� cient to � x the

unknown param eterswhile non renorm alisabletheorieshaveno predictivepower.6Thatenergy scalehasnothing to do with a perturbativecuto� ,wearetalking hereaboutan already wellde� ned

theory.

5

tothebareaction which isde�ned atin�niteenergy.Now thecouplingsofthebareaction

also also arein principle unknown,however,form any theoriesthatdoesnotm atterdue

toaphenom enon called universality:Thecouplingsofthehigherderivativeterm sdepend

on theenergy scaleand a coupling iscalled relevant,m arginalorirrelevantrespectively if

itgrows,rem ainsconstantordecreasesin thelow energy lim it.A universaltheory issuch

that allbut a �nite num ber ofthe couplings are irrelevant. One m ay ask whether one

can see universality also in the canonicalform alism ,however,there are severalobstacles

in answering thisquestion. Firstofall,the Euclidean form ulation usesa W ick rotation

which is only possible for background dependent theories where the background has a

presentation with an analyticdependenceon thetim ecoordinate.In quantum gravity the

m etric becom esan operator,hence W ick rotation and therefore a Euclidean form ulation

isnotpossible.Oneshould thereforede�netheW ilsonian e�ective action directly in the

Ham iltonian (Lorentzian)form ulation,however,thathasnotbeen doneso far.

It seem s to us that in order to m ake progress on this kind ofquestions one should �rst try

to de�ne a Ham iltonian notion ofe�ective action,see [33]fora possible direction. Then,if

thesym m etry argum entsm entioned under[3.]areinsu�cientin orderto �x thequantisation

(discretisation) am biguities in the de�nition ofH ,possibly universality studies m ay lead to

furtherunderstanding.

7.Singularity avoidance

In quantum gravity weexpectorwantto resolvetwo typesofsingularities:First,QFT kind of

shortdistancesingularitieswhich com efrom thefactthatin interacting�eld theoriesonehasto

dealwith productsofoperatorvalued distributions.Secondly,classicalGeneralRelativity kind

ofsingularitieswhich aresim ply a featureoftheEinstein equationsto predictthatgenerically

spacetim esaregeodesically incom plete.An analyticalm easureforsuch spacetim esingularities

aretypically divergencesofcurvatureinvariants.

Now as shown in [15],UV type ofsingularities are absent at the non gauge invariant level,

speci�cally,thequantum constarintsaredensely de�ned.In [21]itwasdiscovered,in thecon-

text ofusualLQG that expectation values ofnon gauge invariant curvature operators with

respectto non gaugeinvariantcoherentstatesthatarepeaked on a classically singular(FRW )

trajectory rem ain �nite asone reachesthe singularity,thusbacking up the m uch m ore spec-

tacularresultsof[22,23,24,25]which areatthelevelofthephysicalHilbertspacealbeitfor

a toy m odeland notthefulltheory.

W hile these are encouraging results,they are at the kinem aticallevelonly and thus are in-

conclusive.However,with thetechnology developed in thispaperwecan transferboth results

literally and with absolutely no changesto thephysicalHilbertspace.Asfarasthespacetim e

singularity resolution is concerned,this is stillnot enough because the coherent states that

we are using,while being now physicalcoherentstates,they are notadapted to the physical

Ham iltonian and thus m ay spread out under the quantum dynam ics generated by U(�). In

otherwords,given gaugeinvariantinitialdatam (0)and acoherentstate 0 thatweprepareat

� = 0 and which ispeaked on m (0),itm ay bethataftershorttim e� thestateU(�) 0 isvery

di�erentform the state � which ispeaked on the classicaltrajectory � 7! m (�). Therefore,

in orderto com eto conclusionsoneshould ratherstudy expectation valueswith respectto the

states U(�) 0 rather than �. In addition,one should try to construct dynam icalcoherent

statesforwhich such a spread doesnothappen. However,thisisa di�culttask already for

theanharm onicoscillator.

6

Theplan ofthepaperisasfollows:

In section two wereview theessentialsof[6,7]in ordertom akethisarticleself{contained.This

willlead to thereduced phasespaceand theclassicalphysicalHam iltonian.

In section three we quantise the reduced phase space using m ethodsfrom LQG and obtain the

physicalHilbertspacealm ostforfree.Then weim plem entthephysicalHam iltonian on thatHilbert

space.W edo thisboth forLQG and theAQG extension.

In section fourwesum m ariseand conclude.

2 R eview ofthe B row n { K ucha�r and relationalfram ew ork

2.1 B row n { K ucha�r Lagrangian

In [1]Brown and Kucha�radd thefollowing Lagrangian totheEinstein {Hilbertand standard m odel

Lagrangian on thespacetim em anifold M

SD = �1

2

Z

M

d4Xpjdet(g)j� [g�� U�U� + 1] (2.1)

wheretheoneform U isde�ned by U = �dT + W jdSj and theindex jtakesvalues1;2;3 while�;�

take values0;1;2;3.The action SD isa functionalofthe �elds�;g��;T;Sj;W j.Here T;S

j have

dim ension oflength,W j isdim ensionlessand thus� hasdim ension cm� 4.

Asshown in [6,7],in perform ing theLegendretransform ation of(2.1)according to the3+1 split

ofM �= R � X into tim eand spaceoneintroducesm om enta P;Pj;I;Ij conjugateto T;Sj;�;W j

respectively next to the m om enta P ab; p; pa conjugate to qab; n; na respectively one encounters

severalprim ary constraints. Here one has introduced a foliation ofM ,that is,a one param eter

fam ily ofem beddingst7! X t: X ! Xt whereXt aretheleavesofthefolitation and thecoordinates

on X are denoted by xa; a = 1;2;3. The vector�eld @tX�

t = nn� + naX�

t;a can be dexom psed in

com ponentsnorm aland tangentialtothelaveswheren� isthefutureoriented norm al.Thefunctions

n;na aretheusuallapseand shiftfunctionsand qab = g��X�;aX

�;b de�nesthethreem etricintrinsicto

X .Theaforem entioned prim ary constraintsare

Z =:I = 0;Z j := Ij = 0;Zj := Pj + PW j = 0;z:= p= 0;za := pa = 0 (2.2)

Thestability analysisoftheseconstraintswith respecttothecorrespondingprim ary Ham iltonian

leadsto thefollowing secondary constraints

ctot = c+ c

D;c

D =1

2[

P 2

�pdet(q)

+ �pdet(q)(1+ q

abUaUb)]

ctota = ca + c

Da ;c

Da = P[T;a � W jS

j;a]

~c =n

2[�

P 2

�2pdet(q)

+pdet(q)(1+ q

abUaUb)] (2.3)

and six m oreequationswhich can besolved fortheLagrangem ultiplierscorrespondingtoconstraints

Z j; Zj and which we do not display here. Here Ua = �T;a + W jSj;a = �cDa =P and c; cDa respec-

tively are the contributionsofgeom etry and standard m atterto the usualHam iltonian and spatial

di�eom orphism constraintrespectively.

7

The stability analysis of the secondary constraints with respect to the prim ary Ham iltonian

which isa linearcom bination ofthe constraints(2.2)and the �rsttwo constraintsin (2.3)reveals

thatthereareno tertiary constraints.M oreover,theclassi�cation ofthesetsofconstraintsinto �rst

and second classshowsthatthe constraintsz; za; ctot; ctota are �rstclasswhile,roughly speaking,

the pairs(Z;~c);(Zj;Zj)form second classconstraintswith non degenerate m atrix form ed by their

m utualPoisson brackets. Hence,to proceed,one passes to the corresponding Dirac bracket and

solvesthesecond classconstraintsexplicitly by setting

I := 0;IJ := 0;W j := �Pj

P; �

2 :=P 2

pdet(q)

[qabUaUb+ 1] (2.4)

Fortunately,theDiracbracketreduced to thegeom etry variablesqab;pab and therem aining m atter

variablesisidenticalto theoriginalPoisson bracket.

After using (2.4) and solving z = za by identifying lapse and shift as Lagrange m ultiplicator

functionsrespectively weareleftwith the�rstclassconstraints

ctot = c+ c

D;c

D = �

q

P 2 + qabcDa cDb

ctota = ca + c

Da ;c

Da = PT;a + PjS

j;a (2.5)

In principlewecould havechosen theothersign tosolvethequadraticequation for� in (2.4)butthe

detailed analysisin [6]revealsthattheotherchoice would produce theEinstein equationswith the

wrong sign in thelim itofvanishing dust�elds.In particularonem ustchoose�;P < 0 so thatthe

additionalm atterenterswith negativesign into theHam iltonain constraint.Thishastheim portant

consequence thatc> 0 thusenablescloseto atspacesolutions.

Asfarasthe physicalinterpretation oftheadditionalm atterisconcerned we justm ention that

itsEulerLagrangeequationsim ply thatthevector�eld U � = g��U� isageodesicin a�neparam etri-

sation,thatthe�eldsW j;Sj areconstantalongthegeodesicand thatthe�eld T de�nespropertim e

along each geodesic. ItfollowsthatSj = �j =const. labelsa geodesic while T = � =const. isan

a�ne param eteralong the geodesic. Furtherm ore,itsenergy m om entum tensoristhatofa perfect

uid with vanishing pressure and negative energy density7,hence itispressure free phantom dust.

Itserves asa dynam ical,m aterialreference system which also plays the role ofa phantom in the

literalsense because itisnotdirectly visible in the �nalpicture while leaving its�ngerprinton the

dynam ics.

2.2 B row n { K ucha�r M echanism

Theobservation ofBrown and Kucha�rwasthattheconstraints(2.5)can bewritten in deparam etrised

form .Thisholdsin m oregeneralcircum stances,nam ely wheneverwe considerscalar�eldswithout

potentialand m assterm saspointed outin [5]. The observation consistsin the factthatthe only

appearanceofT;Sj in ctot isin theform cDa .However,thism eansthatusing ctota = 0 wem ay write

(2.5)in theequivalentform

ctot = c+ c

D;c

D = �pP 2 + qabcacb

ctota = ca + c

Da ;c

Da = PT;a + PjS

j;a (2.6)

7W earenotviolatingany energy conditionsbecausewestillrequirethattheenergy m om entum tensorofobservable

(standard)m atterplusdustsatis� esthe energy conditions.In fact,itwould besu� cientifthe energy conditionsare

satis� ed by thestandard m atteralonebecausein the� nalanalysisthedustcom pletely disappearswhiletheequations

ofm otion forobservablem atterand geom etry assum etheirstandard form plussm allcorrections,see[6,7].

8

where equivalentm eansthat(2.5)and (2.6)de�ne thesam econstraintsurfaceand thesam egauge

invariantfunctions.

W e can now solve the �rst equation in (2.6) for P,rem em bering that P < 0 and the second

equation for Pj,m aking the assum ption that the m atrix Sj;a is everywhere non degenerate8 with

inverse Saj.Theresultis

~ctot = P + h;h = +pc2 � qabcacb

~ctotj = Pj + hj;hj = Saj[ca � hT;a] (2.7)

In solving (2.6)in term sofP we �nd atan interm ediate step thatP 2 = c2 � qabcacb.Hence,while

theargum entofthesquarerootin (2.7)isnotm anifestly positive,itisconstrained to be positive.

Notice that the function h is independent ofSj;T while hj stilldepends on both. Hence,we

have achieved only partialdeparam etrisation. However,thiswillbe su�cientforourpurposes. An

im portant consequence is thatthe constraints in the form (2.7)are m utually Poisson com m uting.

Thisfollowsim m ediately from an abstractargum ent9 [34],although onecan alsoverify thisby direct

com putation [1].Thisim pliesin particularthattheh(x)arem utually Poisson com m uting whilethe

h(x)do notPoisson com m utewith thehj(y)and neitherdo thehj(y)am ong each other.

2.3 R elationalfram ework

2.3.1 G eneraltheory

W e�rstconsidera generalsystem with �rstclassconstraintsC I with arbitrary index setI and later

specialiseto oursituation.

Considerany setoffunctionsTI on phasespacesuch thatthem atrix de�ned by thePoisson bracket

entriesM JI := fCI;T

Jg isinvertible.Considertheequivalentsetofconstraints

C0I :=

X

J

[M � 1]JI CJ (2.8)

such thatfC 0I;TJg � �JI where� m eans= m odulo term sthatvanish on theconstraintsurface.Let

X I betheHam iltonian vector�eld ofC0I and setforany setofrealnum bers�

I

X � :=X

I

�IX I (2.9)

Forany function f on phasespaceweset

��(f):= exp(X �)� f =

1X

n= 0

1

n!X

n� � f (2.10)

Now let�I beanothersetofrealnum bersand de�ne

O f(�):= [��(f)]��(T)= � (2.11)

8Thisisa classicalrestriction ofthe sam ekind asdet(q)> 0.9 The constraints(2.7)are � rstclass. Hence theirPoisson bracketsare linearcom binationsofconstraints. Since

the constraintsare linearin the m om enta P;Pj,their Poisson bracketsare independent ofP;Pj. Therefore we can

evaluate the linear com bination ofthe constraints that appear in the Poisson bracketcom putation in particular at

P = �h;P j = �h j.

9

where��(T)= � m eans��(TI)= �I forallI.Asonecan check,��(T

I)� TI + �I so that(2.11)is

weakly (i.e.on theconstraintsurface)equivalentto

O f(�):= [��(f)]�= �� T (2.12)

Noticethatafterequating� with � � T,thepreviously phasespaceindependentquantities� becom e

phase space dependent,therefore itisim portantin (2.12)to �rstcom pute the action ofX � with �

treated asphasespaceindependentand only then to setitequalto � � T.

Thesigni�canceof(2.12)liesin thefollowing facts:

1.ThefunctionsO f(�)areweak Diracobservableswith respectto theCI,thatis

fCI;O f(�)g � 0 (2.13)

Thisrem arkableproperty isdueto thekey observation thattheX I weakly com m ute[3,4].

2.Them ultiparam eterfam ilyofm apsO � : f 7! O f(�)isahom om orphism from thecom m utative

algebraoffunctionson phasespacetothecom m utativealgebraofweak Diracobservables,both

with pointwisem ultiplication,thatis

O f(�)+ Of0(�)= Of+ f0(�);Of(�)Of0(�)� Off0(�) (2.14)

Thelinearrelation isobvious,them ultiplicative onefollowsfrom thefactthat

��(ff0)= e

X � � ff0= e

X � � ff0e� X � � 1= [eX � � fe

� X �][eX �f0e� X �] (2.15)

whereweused theidentity

[eX � � fe� X �]=

1X

n= 0

1

n (2.16)

and whereX �;f respectively areconsidered asderivation and m ultiplication operatorsrespec-

tively on the algebra offunctions on phase space so that [X �;f]= X � � f. Here [X ;f](0) =

f;[X ;f](n+ 1) = [X ;[X ;f](n)].

3.Them ultiparam eterfam ily ofm apsO � : f 7! O f(�)isin facta Poisson hom om orphism with

respectto theDiracbracketf:;:g� de�ned by thesecond classsystem C I;TJ,thatis

fO f(�);Of0(�)g � fOf(�);Of0(�)g�� O ff;f0g�(�) (2.17)

wheretheDiracbracketisexplicitly given by

ff;f0g� = ff;f

0g� ff;CIg[M

� 1]IJfTJ;f

0g+ ff

0;CIg[M

� 1]IJfTJ;fg (2.18)

Here we have used in the �rststep thatboth O f(�); Of0(�)have weakly vanishing brackets

with the constraints. Relation (2.17) follows from the fact that the m ap �� is a Poisson

autom orphism on the algebra offunctions on phase space and the Poisson bracket m ust be

replaced by the Diracbracketbecause in evaluating fO f(�);Of0(�)g we m usttake care ofthe

factthat� = � � T isphasespacedependent.See[4]fortheexplicitproof.

The interpretation ofO f(�)isthatitisa relationalobservable,nam ely itisthe value off in the

gauge� = T � �.

10

2.3.2 Specialisation to deparam etrised theories

For deparam etrised theories it is possible to �nd canonicalcoordinates consisting oftwo sets of

canonicalpairs (P I;TI) and (qa;pa) respectively (where the Poisson brackets between elem ents of

the �rstand second setsetvanish)such thatthe constraintsC I can be rewritten in the equivalent

form

CI = PI + hI(qa;pa) (2.19)

that is,they no longer depend on the variables TI. This is a very specialcase and m ost gauge

system s cannotbe written in thisform . Even with dustGeneralRelativity isa priorinotofthat

form ,however,wewillreduceitto thatform with an additionalm anipulation below.

Thesim pli�cationsthatoccurarenow thefollowing:

A.W eobviosly have

MJI = fCI;T

Jg = �

JI (2.20)

thereforeC 0I = CI and wedo nothaveto inverta com plicated m atrix.

B.By the sam e argum entasin the footnote after(2.7)we have fCI;CJg = 0 identically on the

fullphasespace,notonly on theconstraintsurfacewhich ofcourseim pliesthat[X I;X J]= 0,

the Ham iltonian vector�eldsofthe constraintsare m utually com m uting. Italso followsthat

fhI;hJg = 0 and thus fCI;hJg = 0 for allI;J which m eans that the hI are already Dirac

observables.

These sim pli�cationsm ean thatallthe previousweak equalitiesbecom e strong ones,i.e. identities

on thefullphasespace.TheDiracobservableassociated to TI

O TI(�)= [��(TI)]�� (T)= � = �

I (2.21)

is sim ply the constant (on phase space) function �I. The m om enta PI are already Dirac observ-

ables,howeverthey can beexpressed in term sofqa;pa via the constraints.M oreover,since O� isa

hom om orphism wehaveon theconstraintsurface

PI = O PI(�)= �OhI(�)= �hI(O qa(�);Opa(�))=:�HI (2.22)

In factwehavehI = H I becausehI isalready a Diracobservable.

Thereduced phasespace(wheretheconstraintshold and wherethegaugetransform ationshave

been factored out)isthereforecoordinatised by thefunctions

Qa(�)= Oqa(�); Pa(�)= Opa(�) (2.23)

and in whatfollowswe concentrate on functionsf which only depend on qa;pa.On such functions

the Dirac bracketreducesto the Poisson bracketsince fTI;fg = 0 forallI. Therefore the reduced

m ap O � : f 7! O f(�)isnow a m ulti{ param eterPoisson autom orphism with respectto thePoisson

bracket.In particularwenote

fPa(�);Qb(�)g = fOpa(�);Oqa(�)g = Ofpa;qbg(�)= O�ba(�)= �

ba (2.24)

which m eans thatthe reduced phase space hasa very sim ple sym plectic structure in term softhe

coordinates Pa := Pa(0); Qa := Q a(0) which in fact form a conjugate pair. It is this fact which

m akesreduced phasespacequantisation feasibleasobserved in [4].

11

Itseem sthatwehavetrivialised everything.However,thisisnotthecaseaswem ustinterprete

the� dependenceofourobservables.W enotice�rstofallthaton functionsf independentofTI;PIform ula (2.12)readsexplicitly

O f(�)= ��(f)= exp(X �)� f =

1X

n= 0

1

n!X

n� � f (2.25)

whereX � istheHam iltonian vector�eld ofthefunction H � = (�I � TI)H I.Herewehaveused that

theX I on f reduce to theHam iltonian vector�eld ofhI and since hI isindependentofPJ we m ay

writeO f(�)in theabovecom pactform .Itisnow a sim pleexercise to verify that[4]

@O f(�)=@�I = fH I;O f(�)g (2.26)

which m eansthatthestrongly Abelian group ofPoisson bracketautom orphism s�� isgenerated by

the\Ham iltonians" H I.Thus,ifwe interprettheTI asclocksthen wehave a m ulti{ �ngered tim e

evolution with Ham iltoniansH I.

In quantum theory then one would like to selecta suitable oneparam eterfam ily by prescribing

functions�I(s)in term sofa single param etersuch thatthe associated Ham iltonian ispositive and

haspreferred physicalproperties.

2.4 T he reduced phase space ofG eneralR elativity w ith dust

Now wespecialiseto oursituation which isa specialcaseofthegeneraltheory.Thishasbeen previ-

ously done in detail,including proofs,in [5]and wasalso reviewed in [6].Here we sum m arise those

results.

As previously m entioned,the Ham iltonian constraints in (2.7) are in deparam etrised form ,how-

ever,the spatialdi�eom orphism constraints are not. However,the idea is to exploitthe factthat

theconstraints(2.7)arem utually Poisson com m uting so thatonecan perform thereduction ofthe

phasespacein two steps:Firstwereducewith respectto thespatialdi�eom orphism constraintand

then with respectto theHam iltonian constraint.M oreprecisely,considerarbitrary functions�0;�j

on X and denoteby X � theHam iltonian vector�eld ofthefunction

ctot� :=

Z

X

d3� �

�(x)~ctot� (x) (2.27)

where we have de�ned ~ctot0 = ~ctot. Then forarbitrary functions�0(x)= �(x); �j(x):= �j(x)on X

thegeneralform ula reads

O f(�)= [��(f)]�� (T)= �; ��(f)= exp(X �)� f (2.28)

whereT0(x)= T(x);Tj(x)= Sj(x).W ereadily com putethat��(T�(x))= T�(x)+ ��(x)so that

O f(�)= [��(f)]�= �� T (2.29)

Now sinceSj(x)Poisson com m uteswith ~ctot(y)wem ay rewrite(2.29)in theform

O f(�)= [��0([�~�(f))]~�= ~�� ~S)]�0= �� T (2.30)

Itturnsoutthatonecan com putetheinnerargum entof(2.30)ratherexplicitly with an im m ediate

physicalinterpretation forjudiciouschoicesofthefunctions�j(x).Nam ely,forany scalarfunction f

12

builtfrom ofT;P;qab;pab and them atterofthestandard m odelone�ndsexplicitly thatforconstant

functions�j

[�~�(f(x))]~�= ~�� ~S = f(x)~S(x)= � (2.31)

In otherwords,whateverthe value ofx atwhich the function f isevaluated,(2.31)evaluatesitat

the pointxa� atwhich Sj(x)assum es the value �j. Since we have assum ed thatSj;a is everywhere

invertibleand thusde�nesa di�eom orphism between X and therangeofS j which isthedustspace

S,the value x� isunique. Form ula (2.31)isproved explicitly in [6]and willnotbe repeated here.

Thus,(2.31)takesa sim pleform ifwechooseasf oneofthefollowing functionson S

~T := T; ~P =P

J; ~qjk := qabS

ajS

bk; ~p

jk :=pabSj

;aSk;b

J(2.32)

where

J := det(@S=@x) (2.33)

aswellas

~aIj := aIbS

bj;~e

j

I :=eaIS

j;a

J; ~ �I := �I;

~� �I := � �I; ~�I := �I;~�I :=

�I

J(2.34)

forconnectionsaIb,electric�eldseaI,ferm ions �I;

� �I and Higgs�elds�I with conjugatem om entum

�I ofthe standard m odelwhere I labelsa basisin the Lie algebra ofthe appropriate gauge group,

see[15]forthecanonicalform ulation ofthestandard m odelcoupled togravity including appropriate

background independentHilbertspacerepresentations.

Itisclearthatthe evaluation ofthe functions (2.32)and (2.34)atx� isnothing else than the

pullback ofthecorresponding �eldsto S undertheinverse ofthedi�eom orphism S j : X ! S.W e

willdenotethecorresponding tensor�eldson S asin (2.32)and (2.34).Noticethatwhiletheseare

scalarson X they are tensordensitiesofthe sam e weighton S asthey have10 on X . In [1,6]itis

shown thatonecan arriveatthespatially di�eom orphism invariantfunctions(2.32)and (2.34)also

by sym plecticreduction with respecttothespatialdi�eom orphism constraintwhich isan alternative

proofofthe factthatcanonicalpairswithouttilde on X are m apped to canonicalpairson S. For

instance

f~pjk(�);~qm n(�0)g = ��

j

(m�kn)�(�;�

0) (2.35)

where � = 16�GN ewton. This also shows that it is su�cient to consider constant � j rather than

arbitrary functions.

Returning to (2.30)weseethatitrem ainsto com pute

O f(�;�):= [��0(f(�))]�0= �� T (2.36)

where f isnow an arbitrary function ofthespatially di�eom orphism invariantfunctions(2.32)and

(2.34).Now wecan usethesim pli�ed theory ofsection 2.3.2because~ctotiswritten in deparam etrised

10This statem ent sounds contradictory because of the following subtlety: W e have e.g. the three quantities

P (x); ~P (x) = P (x)=J(x); ~P(�) = ~P(x�). O n X ,P (x) is a scalar density while ~P (x) is a scalar. Pulling back

P (x)to S = S(X )by the di� eom orphism � 7! S�1 (�)resultsin ~P(�). Butpulling back ~P (x) back to S resultsin

the sam e quantity ~P (�).Since a di� eom orphism doesnotchangethe density weight,wewould getthe contradiction

that ~P (�) has both density weights zero and one on S. The resolution ofthe puzzle is that what determ ines the

density weightofP (x)on X isitstransform ation behaviourundercanonicaltransform ationsgenerated by the total

spatialdi� eom orphism constraintctota = cDa + ca where cDa ; ca are the dustand non dustcontributionsrespectively.

After the reduction ofctota ,whatdeterm ines the density weightof ~P (�)on S is its transform ation behaviourunder

([ca + P T;a]Saj=J)(x�)= ~cj(�)+ ~P (�)~T;j(�)and thisshowsthat ~P (�)hasdensity weightone.

13

form ,i.e. itdoesnotinvolve T;Sj any longer. Actually,form ula (2.36)would be awkward fornon

constantfunctions� becauseitdependson

ctot� =

Z

X

d3x (� � T)(x)~ctot(x) (2.37)

which isexpressed on the space X ratherthan dustspace S. However,forconstant� (2.37)isthe

integralofa density ofweightoneand can then bewritten in theform

ctot� =

Z

S

d3� (� � ~T)(�)[~P + ~h](�) (2.38)

where

~h(�) =

q

~c(�)2 � ~qjk(�)~cj(�)~ck)(�)

~c(�) =c

J(x�)

~cj(�) =caS

aj

J(x�) (2.39)

Noticethate.g.~cisjustthepullback ofcand thatonesim ply hasto replaceevery tensorwithout

tildeby theirpulled back im agewith tilde.Thusconstant� isuniquely selected by therequirem ent

thatctot� isspatially di�eom orphism invariant.

Itfollowsnow from section 2.3.2 that

O f(�;�)=

1X

n= 0

1

n!fH �;f(�)g(n); H � =

Z

S

d3� [� � T](�)~h(�) (2.40)

and thatd

d�O f(�;�)= fH ;Of(�;�)g; H =

Z

S

d3� ~h(�) (2.41)

Since the h(x) are m utually Poisson com uting it follows that also the ~h(�) are m utually Poisson

com m uting so that

H (�;�):= ��0(~h(�))�0= �� T =

~h(�)=:H (�) (2.42)

isindependentof� and already a Diracobservable.

Notice that the physicalHam iltonian H is positive. It enjoys the following sym m etries: Since

it is an integralover a density ofweight one it is invariant under di�eom orphism s ofS. Notice

thatS isa labelspaceforgeodesicsand nota coordinatem anifold,hencein contrastto thepassive

di�eom orphism group Di�(X ),thegroup Di�(S)areactivedi�eom orphism s.In particular,itfollows

that

fH ;~cj(�)g= 0 (2.43)

which also isa consequenceofhaving chosen constant�,in which casethephysicalHam iltonian has

am axim alam ountofsym m etry.Had wenotchosen constant� then thephysicalHam iltonian would

notbe a Dirac observable.

Thisalso im pliesthat

fH ;Cj(�)g = 0; C(�;�):= ��0(~cj(�))�0= �� T =:Cj(�) (2.44)

14

isactually independentof�,although ~cj 6= Cj.Noticethat

H (�)=

q

C(�;�)2 � Q jk(�;�)Cj(�)Ck(�);C(�;�):= ��0(~c(�))�0= �� T (2.45)

Thesecond sym m etry ofH isofcoursethat

fH ;H (�)g= 0 (2.46)

Letuswriteforsom escalarand vectortestfunctionsf;uj respectively

H (f):=

Z

S

d3� f(�)H (�); C(u):=

Z

S

d3� u

j(�)Cj(�) (2.47)

then

fC(u);C(u0)g = ��C([u;u0])

fC(u);H (f0)g = ��H (u[f0])

fH (f);H (f0)g = 0 (2.48)

which showsthatthesym m etry generatorsgeneratean honestLiealgebra g in contrastto theDirac

algebra underlying GR as was pointed out already in [1]and further exam ined in [35]. That Lie

algebra hasa subalgebra generated by theC(u)and an Abelian idealgenerated by theH (f),hence

itisnotsem isim ple.ThecorrespondingLiegroupG = N o Di�(S)isthereforethesem idirectproduct

oftheAbelian invariantsubgroup N to which theH (f)exponentiateand theactivedi�eom orphism

group ofdustspace.

2.5 Physicalinterpretation and com parison w ith unreduced form alism

Thesym m etry algebra g and theassociated conservation lawsplay a crucialrolein showing [6]that

theequationsofm otion forthecanonicalpairsoftruedegreesoffreedom

(Q jk;Pjk);(A I

j;Ej

I);( �I; � �I);(�I;�

I) (2.49)

which aretheim agesofthecanonicalpairs

(~qjk; ~pjk);(~aIj;~e

j

I);(~ �I;

�~ �I);(~�I;~�I) (2.50)

under��0(:)�0= �� T at� = 0 assum ethestandard form thatthey havein GeneralRelativity without

dust [36],with two im portant m odi�cations: First,in usualGeneralRelativity without dust the

equationsofm otion generated by thecanonicalHam iltonian h(n;~n)= c(n)+ ~c(~n)which isa linear

com bination ofthe sm eared Ham iltonian constraint c(n) =R

Xd3xnc and spatialdi�eom orphism

constraint~c(~n)=R

Xd3xnaca,involve arbitrary lapse and shiftfunctions n;na on X which are in-

dependentofphase space. However,in ourform alism lapse and shiftfunctionsbecom e dynam ical

functions11 on S,nam ely N = C=H and N j = �Q jkCk=H . Secondly,without dust we stillhave

constraints c = ca = 0 while we have energy { m om entum conservation laws H = �; Cj = ��j

where�;�j arearbitrary functionson S independentof�.Thisturnsdynam icallapseand shiftinto

a function ofQ jk;�j=�. The functions�;�j express the in uence ofthe duston the othervariables

and arethepricetopay forhaving am anifestly gaugeinvariantform alism ratherthan assum ing non

dynam icaltestobserversthatturn geom etry and m atterinto observablequantities.

Thisconcludestheclassicalanalysisand thereview of[6].

11This issim ilarin spiritto [37]where one replaceslapse and shifttest� elds by hand by phase space dependent

functions,carefully chosen (via W itten spinor techniques that enter the proofofthe gravitationalpositive energy

theorem )so thatthe resulting Ham iltonian ispositive,atleaston shell.

15

3 R educed phase space quantisation ofG eneralR elativity

3.1 H ilbert space representation

Let us sum m arise the result of the previous section: By using the relationalform alism we can

explicitly com pute the reduced phase space ofGeneralRelativity with dust. It is identicalto the

unreduced phasespacewithoutdustwith properidenti�cation ofX with S and ofthegaugeinvariant

canonicalpairs(2.49)with thegaugevariantcanonicalpairs

(qab;pab);(aIb;e

bI);( �I;

� �I);(�I;�I) (3.1)

ofgeom etry and standard m atter.Theconstraintshavedisappeared,they havebeen solved and re-

duced.Instead ofalinearcom bination ofconstraintson thegaugevariantphasespacecoordinatised

by (3.1)which generatesgaugetransform ations,thereisa physicalHam iltonian (2.41)which gener-

atesphysicaltim eevolution on thegaugeinvariantphasecoordinatised by (2.49).From theclassical

pointofview one should now sim ply solve those equations in physically interesting situations. In

[6,7]we have done thisin the context ofcosm ologicalperturbation theory [6,7]which iswritten

in m anifestly gauge invariantform . Thisnotonly reproducesthe standard results[8]butalso will

allow usto investigate higherorderperturbation theory withoutrunning into problem swith gauge

invariance.

In the quantum theory we are looking for representations ofthe Poisson ��algebra generated

by (2.49)which supports a quantised version ofthe Ham iltonian H . The selection ofappropriate

representationswillbeguided by thesym m etry group G unveiled in theprevioussection.Firstofall,

since we considerferm ionic m atterwe are forced to work with tetradsratherfourm etrics. W e use

thesecond orderform alism asdisplayed in [15](thatis,wewritetheEinstein HilbertLagrangian in

term softhespin conection ofthetetrad which involvessecond orderderivativesratherthan usingthe

�rstorderPalatiniform alism )in orderto avoid torsion.Thism eansthatweform ulatethegeom etry

phase space in term sofsu(2)connectionsand canonically conjugate �elds(A Ij;E

j

I)ratherthan in

term softhe ADM variablesQ jk;Pjk where I isan su(2)index. Thiscaststhe geom etry sectorof

the phase space into a SU(2)Yang { M ills theory description. Th price to pay isthatthere isan

additionalGauss constraint on the phase space (which has been reduced only with respect to the

Ham iltonian and spatialdi�eom orphism constraint)given by

G I := @jEj

I + �IJK AJjE

j

K + ferm ion term s (3.2)

justasforthe m atterYang { M illsvariables(we assum e thatthe Cartan Killing m etric isalways

�JK by appropriatenorm alisation oftheLiealgebra basis).

The gauge �eld language suggests to form ulate the theory in term s ofholonom ies along one

dim ensionalpathsand electric uxesthrough two dim ensionalsurfaces,justasin unreduced LQG.

Thereonehasauniquenessresult[13,14]which saysthatcyclicrepresentationsoftheholonom y{ ux

algebrawhich im plem entaunitary representation ofthespatialdi�eom orphism gaugegroup Di�(X )

areuniqueand areunitarilyequivalenttotheAshtekar{Isham {Lewandowskirepresentation [9,10].

In ourcase we do nothave a di�eom orphism gauge group butrathera di�eom orphism sym m etry

group Di�(X )ofthe physicalHam iltonian H .Thisisphysicalinputenough to also insiston cyclic

Di�(S)covariantrepresentationsand correspondingly wecan copy theuniquenessresult.

Thuswesim ply choosethebackground independentand activedi�eom orphism covariantHilbert

spacerepresentation ofLQG used extensively in [15]and weaskwhetherthatrepresentation supports

a quantum operatorcorresponding to H .

16

3.2 Subtleties w ith the G auss constraints

Before we analyse this question in detail,we should m ention a subtlety: W hen one rewrites the

geom etry and standard m attercontributionsc;ca to the totalHam iltonian and and spatialdi�eo-

m orphism constraint in term s ofthe gauge theory variables,one can do this is G invariant form

(where G isthe com pactgauge group underlying the corresponding Yang M illstheory)only by in-

troducing term sproportionaltotheGaussconstraint,seee.g.[12].Forinstance,thecontribution to

thespatialdi�eom orphism constraintofa Yang M ills�eld on theunreduced phasespaceisgiven by

cY Ma = f

Iabe

bI � a

Iag

Y MI = ~cY Ma � a

Iag

Y MI (3.3)

where fIab = 2@[aaIb]+ �IJK a

Jaa

Kb is the curvature ofthe connection aIa and �IJK are the strcture

constants ofthe corresponding Lie algebra. The function (3.3) really generates Yang M ills gauge

transform ations,however,it is itselfofcourse not Yang { M ills gauge invariant due to the term

proportionalto theGaussconstraint

gY MI = @ae

aI + �IJK a

Jae

aK (3.4)

Likewise,thegeom etry contribution cgeo toccontainsaterm proportionaltoggeo

I [12](however,cY M

doesnot).Asfarasthede�nition ofthecom pleteconstraintsurfaceisconcerned,onecan drop the

variousGausslaw contributionstoc;ca sinceweim posetheGausslawsindependently anyway.This

givesan equivalentsetofconstraintswhich issuch thatc;ca arem anifestly invariantunderYang {

M illstype ofGausstransform ations. However,now the algebra ofthe ctot; ctota only closesup to a

term proportionalto thevariousGausslaws.

The question is now whether this spoils the argum ent that the constraints in the form (2.7)

are m utually Poisson com m uting. In fact,we only can conclude that their Poisson brackets are

proportionalto ~ctot; ~ctota and the various gY MI while they m ust not depend on the dust m om enta

P;Pj.Thism eansthattheirPoisson bracketsareproportionaltoaYangM illsgaugeinvariantlinear

com bination ofGaussconstraints. Hence,indeed the constraints ~ctot; ~ctota are Abelian only on the

constraintsurfaceoftheGaussconstraints.

Thisposesthequestion which consequencesthishasfortheform alism developed in theprevious

section.Firstofall,allrelationsthatwehavewritten thererem ain valid m odulo term sproportional

to the Gauss constraints. Secondly, the physicalHam iltonian is m anifestly Yang { M ills gauge

invariant,m anifestly Di�(X )invariantand invariantm odulo theGaussconstraintsunderN .

The strategy that we adopt is the following. In the presence ofgauge �elds we actually work

with the non Gauss invariant contributions to the spatialdi�eom orpphism constraints as in (3.3)

and with thenon Gaussinvariantcontribution to cgeo such thatalgebra ofHam iltonian and spatial

di�em orphism constraints closes without involvem ent ofthe Gauss constraints. This m akes the

analysis ofthe previous section go through without m odi�cations at the price that the physical

Ham iltonian isnotGaussinvariant.W hen we quantise itturnsoutthatonecan actually solve the

various Gauss constraints explicitly by Dirac constraint quantisation. That is,the Hilbert space

can be projected to the Gaussinvariantsubspace which hasan explicitly known orthonorm albasis

given by the Gauss invariant spin network functions (and their analog forthe gauge group ofthe

standard m odel).Therefore,on theGaussinvariantHilbertspaceonecan actually replacetheC Y Mj

by ~C Y Mj because the correction term proportionalto the Gauss constraint vanishes on the Gauss

invariantHilbertspace (upon appropriateordering ofthe Gaussconstraintoperatorto therightso

thatno com m utatorterm sarise). ThusCj isreplaced by itsGaussinvariantanalog and sim ilarly

onecan replaceC by itsGaussinvariantanalogso thatH and H becom em anifestly Gaussinvariant

operatorsand H should havethesym m etry group G aswell.

17

Analternativeroutewouldbetoalsoreducethephasespacewith respecttotheGaussconstraints,

possibly using thefram ework of[38]and referencestherein.

3.3 Q uantum H am iltonian

3.3.1 Sign issues and strategy

Before we go into detailswe m ustworry aboutyetanotherissue: Aswe have seen in the classical

analysis,theexpression H 2 = C 2 � Q JK CJCK isconstrained to benon negative.Actually we have

seen thisonly forc2 � qabcacb butasweshowed

(C 2� Q

jkCjCk)(�)= ([c2 � q

abcacb]=J)(x�) (3.5)

and J > 0 bysassum ption (we have im posed J 6= 0 everywhere,hence eitherJ > 0 everywhere or

J < 0 everywhereby continuity and wechoosethe�rstoption).However,on thefull,reduced phase

space C 2 � Q jkCJCK m aybe inde�nite. In the quantum theory we therefore should derive,roughly

speaking,a selfadjointoperator(valued distribution)forH 2(�)and restrictthespectralresolution

ofthe Hilbertspace to the positive spectrum part. Thishas to be done forevery �. This m aybe

im possiblebecausethecorrespondingspectralconditionscould beincom patible.However,asalready

ponted outby Brown and Kucha�r[1],ifwe indeed m anage to quantise H 2(�)in such a way that

they arem utually com m uting12 then thecorresponding spectralprojectionscom m uteand theabove

requirem entisconsistent. Unfortunately,notonly m ay itbe hard to achieve com m utativity ofthe

operatorscorrespondingtothevariousH 2(�),m oreoveritwillbehard tocom putethecorresponding

projection valued m easures.

Therefore, as a �rst step, in this article we adopt the following strategy: Classically, in the

interesting partofthephasespace wehave C 2 � Q jkCjCk � 0.Thereforeon thispartofthephase

spacewehavetrivially C 2 � Q jkCjCk = jC 2 � Q jkCjCkj.Henceon thatpartofthephasespacewe

havetheidentity

H =

q

jC 2 � Q jkCjCkj=

r1

2([C 2 � Q jkCjCk]+ jC 2 � Q jkCjCkj) (3.6)

Thevirtueofthisrewriting isthatboth expressions,which areidenticalon thephysically interesting

pieceofthephasespace,can beextended tothefullphasespacewithoutbecom ingim aginary.In the

second version,the function actually vanisheson the unphysicalpartofthe phase space. In either

form ,the square rootnow m akessense in the quantum theory because itsargum entisnow a non

negativeexpression.

W e rem ark that a discussion ofsim ilar sign issues and whether one should allow states in the

quantum theory which violate the classicalpositivity ofH 2(�)which isenforced by a constraintof

the form P 2 � H 2 = 0 and where H 2 is not m anifestly positive while P 2 surely is,can be found

forinstance in [39]. There the authorsargue thatone should allow negative energy statesbecause

otherwise one would exclude the tunneling e�ects into the classically notallows regions which,as

weknow from quantum m echanicalexperim ents,do happen.W hathappensm athem atically isthat

in the operatorconstraintm ethod (Dirac approach)one quantisesboth P and H 2 asself{ adjoint

operatorson the kinem aticalHilbertspace and then solves the quantum constraint. The elem ents

12M ore precisely,one hasto dem and thatthe projection valued m easuresE � forthe H 2(�)m utually com m ute in

ordertoavoiddom ainquestions.NoticethatthePoissoncom m utativityoftheH (�)im pliesthePoissoncom m utativity

ofthe H 2(�)and vice versa.

18

ofthe corresponding physicalHilbertspace m ay have supportin the classically notallowed region

ofthecon�guration space(wherethey typically decay ratherthan oscillate)so thattheexpectation

valueofH 2 = P 2 becom esnegative.Thisispossibleonly becausetheoperatorcorresponding to P,

whilebeing a quantum Diracobservable,doesnotdescend to a selfadjointoperatoron thephysical

Hilbertspace. In a strictreduced phase space quantisation one would have to restrictthe physical

Hilbertspace to stateswhich have supportonly in theclassically allowed region ofthephase space

and thism ay wellbe the physically correctprocedure. However,forthe m om ent,aswe do notyet

have su�cient controloverthe spectrum ofH 2,we com ply with the conclusion of[39]and do not

m akeany restriction on thephysicalHilbertspace.

Thus,in thisarticlewethereforeproposetoquantisethe�rstversion of(3.6)which isaclassically

valid starting point13.W ethen adopta naivequantisation strategy and areableto constructa well

de�ned Ham iltonian operator.Thatquantisation notnecessarily hastheproperty thatthequantised

versionsoftheH 2(�)arem utually com m uting and thereforetheoperatorconstructed in thispaper

should only be considerd as a prelim inary step. However,that operator has the following three

properties:Itism anifestly Gaussinvariant,m anifestly Di�(S)covariantand hasthecorrectclassical

lim itin the sense ofexpectation valuesand uctuationswith respectto coherentstates. However,

itm aybe anom alouswith respectto the group N . In fact,the absence ofthatanom aly would be

m athem atically equivalentto showing thatthe Dirac algebra ofGeneralRelativity isim plem ented

non anom alously. W e stress,however,thatthe gauge sym m etries ofGeneralRelativity have been

exactly taken care ofin the reduced phase space approach. W e are talking here abouta sym m etry

group and not a gauge group. To break a localgauge group is usually physically inacceptable

especially in renorm alisabletheorieswherethecorresponding W ard identities�nd theirway into the

renorm alisation theorem s. However,itm ay orm ay notbe acceptable thata physicalsym m etry is

(spontaneouly,explicitly ...) broken. Forinstance,the explicitbreaking ofthe axialvectorcurrent

W ard identity in QED,also called theABJ anom aly,isexperim entally veri�ed.

In lack ofa physicaljusti�cation for why the N sym m etry should be broken, we view that

potentialanom aly asan indication thatthequantisation ofthepresentpaperhastobeim proved.In

fact,sincewearee�ectively working with abackground independentlatticegaugetheory,itisuseful

to adoptstrategiesfrom lattice gauge theory in orderto restore sym m etrieson the lattice thatare

broken in a naive quantisation. Itturnsoutthatin fortunate casesone can restore the sym m etry

by m aking theoperatorquasinon local.Thatis,in addition to nextneighbourinteractionsonehas

to considernextto nextneighbourinteractionsetc.which m akestheaction non local,howeverthe

coe�cients ofthose additionalinteractions decay exponentially with the lattice distance. See e.g.

[29]and referencestherein.

W e considerthe com pletion ofthisstep asa future research program m e. In the course ofthat

analysiswem ighteven beableto�xthequantisation (discretisation)am biguities,i.e.thecoe�cients

in frontofthevariousn-th neighbourcontributions.

W ith this cautionary rem arks out ofthe way, we can now consider a naive quantisation ofthe

Ham iltonian which isstrongly guided by analogoustechniques developed forthe Ham iltonian and

M aster constraint ofunreduced LQG [15,16]so that these constructions are also helpfulin the

presentreduced phasespaceapproach.

3.3.2 N aive Q uantisation

13A sim ilarstrategywasadopted forthequantisation ofthevolum ein LQ G :Classicallywehavedet(q)= det(E )> 0

butin orderto givem eaning topdet(q)in the quantum theory wem uststartfrom

pjdet(E )j.

19

3.3.2.1 C lassicalregularisation W ebegin with som eclassicalconsiderationsand wefocuson

the gravitationalcontributions to C;Cj and for C only on the Euclidean piece. For the m atter

contributions and the Lorenzian piece the necessary,com pletely analogous m anipulations can be

found in [17].Considera partition P ofS into cubes2 so that

H =X

22P

Z

2

d3�

q

jC 2 � Q jkCjCkj(�) (3.7)

LetV0(2)bethecoordinatevolum eof2 in any coordinatesystem and let�(2)besom ecoordinate

point inside 2 with respect to the sam e coordinate system . Then we can write (3.7) as lim it,in

which the partition becom es the continuum ,ofthe following Riem ann sum approxim ation ofthe

aboveintegral

H = limP ! S

X

22P

V0(2)

q

jC 2 � Q jkCjCkj(�(2)) (3.8)

Using theclassicalidentities

Qjk =

Ej

IEkJ�

IJ

det(Q);E

j

I=pdet(Q)e

j

I(3.9)

where I;J;::= 1;2;3 labela basis�I = �i�I (where �I denote the Paulim atrices)in su(2)and ej

I

denotesth triad itisnotdi�cultto verify that

C2 = [Tr(B )]2; Q

jkCjCk = [Tr(B �I)]

2=4=:C 2

I (3.10)

Here we have introduced the m agnetic �eld Bj

I= 1

2�jklF I

kl and have setBj = B

j

I�I;ej = eIj�I;B =

B jej whereeIj denotesthecotriad.W em ay furtherwrite

H = limP ! S

X

22S

pjC(2)2 � �IJCI(2)CJ(2)j (3.11)

where

C(2):=

Z

2

d3� C(�); CI(2):=

Z

2

d3� CI(�) (3.12)

Thestrategy isnow to quantisetheobjects(3.12)and to de�ne

bH := limP ! S

X

22S

q

jC(2)y C(2)� �IJCI(2)y CJ(2)j (3.13)

provided the lim itexists. ForC(2)thishasbeen done in the literature [15,16]and we follow the

sam estrategy here.In factwecan treatboth C;CI in a uni�ed way.W ehavewith �0 := 12

Z

2

d3� Tr(B ��)=

Z

2

Tr(F ^ e��)=1

�

Z

2

Tr(F ^ fV (2);Ag��) (3.14)

where

V (2)=

Z

2

d3�pdet(Q) (3.15)

isthephysicalvolum eof2.Actually thereisa sign ofdet(e)involved in (3.15)butthisiscancelled

in thesquaresthatappearin (3.11).

20

Thevirtueofwriting(3.14)in thisform isthat(3.15)can bequantised on theLQG Hilbertspace,

hence onereplacesthePoisson bracketby thecom m utatordivided by i~.Thusoneisleftwith the

quantisation oftheconnection A and itscurvatureF.Thisisthesourceofm any am biguitiesalready

in unreduced LQG because A;F do notexistasoperators,whatexistsare holonom iesalong paths

and loopsrespectively which can beused in orderto approxim ateA;F respectively.However,while

classically therearein�nitely m any waysto do thiswith thesam econtinuum lim it,in thequantum

theory each choiceleadstoadi�erentregularised operatorin unreduced LQG,see[15].In unreduced

LQG one can stillargue thatm ostofthe uncountably in�nite num berofchoicesare gauge related

underthespatialdi�eom orphism group and in factspatialdi�eom orphism invarianceisused in order

to carry outthelim itP ! X in a speci�coperatortopology [11,15].However,in reduced LQG the

spatialdi�eom orphism group isno longera gauge group,itisa sym m etry group ofthe dynam ics.

Thereforethesetwo argum entsareno longeravailableand thereforetheam biguity issueappearsto

bem uch worsein reduced LQG.Thisisthe�rstindication thatcallsfortheAQG generalisation.

In the next paragraph we willdiscuss to what extent those am biguities persist in reduced LQG,

in theparagraph afterthatweusetheAQG reform ulation.

3.3.2.2 R educed LQ G :Em bedded graphs W e want to de�ne the Ham iltonian operator bH

on the Gauss invariantHilbert space ofLQG which we willdenote by H . This Hilbertspace has

an orthonorm albasis consisting ofspin network functions T ;j;I where isa (sem ianalytic) graph

em bedded intoS,j= fjege2E ( )isacollection ofnon vanishing spin quantum num bers(oneforeach

edge)and I = fIvgv2V ( ) isa collection ofGaussinvariantintertwiners(oneforeach vertex).There

isa unitary action oftheactivedi�eom orphism son thisHilbertspacede�ned by

U(’)T ;j;I = T’( );j;I (3.16)

In unreduced LQG the di�eom orphism sare considered asgauge transform ationsand therefore the

states(3.16)areallgaugerelated.In thereduced form alism ofthispaperthestatesoftheform are

physically distinguishable. Therefore itdoes notm ake physicalsense to construct di�eom orphism

invariantdistributionswhich som etim esareused in theconstruction ofHam iltonian orm astercon-

straintoperatorsasalready pointed out.

This last point has crucialbearing on the quantisation strategy: Ifwe want to preserve the

classicalsym m etry oftheHam iltonian operatorunderdi�eom orphism s,then thisoperatorm ustbe

quantised in a graph non changing way [40]on H . By thisism eantthe following: LetH be the

closed linearspan ofspin network statesover .Then H isthedirectsum oftheH ,thatis

H = � H (3.17)

which showsthatthephysicalHilbertspaceH isnon separable.Thisisan im portantdi�erencewith

non reduced LQG wherethephysicalHilbertspacecan bem adeseparableifoneextendsthespatial

passivedi�eom orphism group beyond thedi�erentablecategory [41].Thisisasecond indication that

one should possibly leave the strictrealm of(reduced)LQG and passto anotherfram ework where

non separableHilbertspacescan beavoided.ThiscallsfortheAQG extension [17]which wediscuss

in thesubsequentparagraph.

In any case,graph non changing in the sense of[40]now m eans that the operator bH should

preserveeach H separately!Thisappearsasifwehad to assum ean in�nitenum berofconservation

lawsthatthe classicaltheory did nothave which isa second pointto worry aboutand presents a

third m otivation to switch to the AQG extension ofLQG.However,letussee how farwe can get

21

within theusualform alism .To thatend,weuse thenotion ofa m inim alloop originally introduced

in [28]and also used to som eextentin [16].

D e�nition 3.1.Given a graph ,considera vertexv 2 V ( )and a paire;e02 E ( )ofdistinctedges

incidentatv and with outgoing orientation. A loop � ;v;e;e0 in starting atv along e and ending

atv along (e0)� 1 is said to be m inim alprovided thatthere exists no other loop in with the sam e

propertiesand feweredgestraversed.The setofm inim alloopsin with data v;e;e0 willbe denoted

by L ;v;e;e0.

Noticethatthede�nition isbackground independentand di�eom orphism covariant.

Given a graph and a vertex v 2 V ( )wede�nefor� = 0;1;2;3

C�; ;v :=1

‘2PjTv( )j

X

(e1;e2;e3)2Tv( )

�IJK

1

jL ;v;eI;eJj

X

�2L ;v;eI;eJ

�Tr(��A(�)A(eK )[A(eK )� 1;V ;v]) (3.18)

whereTv( )isthesetofordered triples(i.e.orderm atters)ofdistinctedgesof incidentatv taken

with outgoing orientation,A(p)denotestheholonom y oftheconnection A along a path p and

V ;v = ‘3

P

s

j1

48

X

e1;e2;e32Tv( )

�(e1;e2;e3)�LM N X L

e1X M

e2X N

e3j (3.19)

isthe projection ofthe volum e operator[42]to14 H foran in�nitesim alneighbourhood ofv.Here

�v(e1;e2;e3)isthesign ofthedeterm inantofthem atrix form ed by thetangentsofthosethreeedges

at v and X e denotes the right invariant vector �eld on SU(2) associated with the copy ofSU(2)

coordinatised by A(e).

Finally weset

bH :=X

v2V ( )

r

jP [Cy ;vC ;v�

1

4Cy

I; ;vCI; ;v]P j (3.20)

whereP :H ! H denotestheorthogonalprojection and m akessurethat bH isnotgraph changing,

i.e. preservesH . The daggeroperation isthaton H forthe operatorde�ned in (3.18)using that

entriesofholonom iesm atricesarejustm ultiplication operatorsand that V ;v isselfadjoint.

Theoperator bH isnow sim plybH = �

bH (3.21)

Itiseasy to check thatitisdi�eom orphism invariant

U(’)bH U(’)� 1 = bH (3.22)

for all’. M oreover,it is m anifestly Gauss invariant. One m ay ask what happened to the lim it

P ! S.Theansweristhatwede�netheoperator bH asin (3.22)and justcheck thatitsexpectation

valueswith respectto suitable sem iclassicalstatesreproducesthe classicalfunction bH .Such states

in particularm ust use su�ciently large and �ne graphsin orderto �lloutS. W hatthe operator

doeson sm allgraphsisirrelevantfrom thepointofview oftheclassicallim it.

W ith them ethodsof[17]oneshould beableto verify thaton such graphsthesem iclassicallim it

oftheoperatoriscorrect.However,thatcalculation isofcoursegraph dependent.

14Thealternativevolum eoperator[43]wasruled outin [44]asinconsistentwith theclassicalPoisson bracketidentity

(3.14).In unreduced LQ G onecould stillsay thatthevolum eoperatorand thePoisson bracketidentity arerelations

am ong non observable objectsbutthisisno longertrue in reduced LQ G asconsiderd here and hence the objection

[44]m ustbe taken seriously.

22

3.3.2.3 R educed A Q G :A bstract (algebraic) graphs One ofthe m otivationsforthe AQG

extension ofLQG isthegraph dependenceofthesem iclassicalcalculations.Theotheristhenecessar-

ily graph perserving featureofdi�eom orphism invariantoperatorswhich appearsto say thatthereis

an uncountably in�nitenum berofconservation lawsthattheclassicaltheory doesnothave.Finally,

the non separability ofthe Hilbert space H even ifS is com pact without boundary is disturbing.

In a sense,to use allgraphsisa vastovercounting ofdegreesoffreedom ,atleastfrom theclassical

perspective.Toseethis,supposeforsim plicity thatS istopologically R 3 (oran open neighbourhood

thereof)and thuscan be covered by a single coordinate system . Considerpiecewise analytic paths

which consistofsegm entsalong the coordinate axes. Likewise,considerpiecewise analytic surfaces

which arecom posed outofsegm entsofcoordinateplanes.Itisclearthattheholonom iesalongthose

kind ofpaths and uxes through that kind ofsurfaces separates the points ofthe reduced phase

space.

Itistrue thatalso in canonicalQFT the qantum con�guration space isalwaysa distributional

enlargem entofthe classicalcon�guration space. However,there itisneverthe case thatthe label

setofthose �elds isuncountable. Forinstance,in free scalar�eld theory on M inkowskispace the

quantum con�guration space consists ofSchwarz distributions ratherthan sm ooth functions. The

labelsetofthe�eldsconsistsoftestfunctionsofrapid decreasewhich aredensein theHilbertspace

ofsquare integrable functionson R 3 and there existsa countable orthonorm albasisofthatHilbert

spaceconsisting ofSchwarzfunctions(e.g.Herm itefunctionstim esaGaussian).Thus,thequantum

�eldsare tested by a countable setoftestfunctionsand an orthonorm albasisin the QFT Hilbert

space is labelled by that countable set. In LQG on the other hand the quantum connections are

tested by allgraphswhich isan uncountablesetand statesoverdi�erentgraphsareorthogonal.So

thesituation iscom pletely di�erentwhich seem stobethepriceofhavingadi�eom orphism covariant

theory [13,14].

Onecould ofcourserestrictthelabelstothosem entioned abovebutthesewould notbepreserved

by di�eom orphism s.Itistruethatthedi�eom orphicim ageofa coordinatesegm entcan beapproxi-

m ated by coordinatesegm ents,however,thelength ofsay arotated segem entwhen approxim ated by

a staircase willdi�erlargely from theoriginallength.Thesam e happensforareasofsurfaces.The

only chance thatthisdoesnothappen isforobservables thatare integralsover three dim ensional

regionsaspointed outin [45].

To m akeprogresson thoseissueswethereforewillrestrictattention to operatorsthatcom efrom

integralsoverregionsofS such asthevolum eoperatorortheHam iltonian operator.Thisdoesnot

m ean that one cannot construct length and volum e operators,one just has to de�ne them in an

indirectway,see[46].In fact,wewillonly considerquantising functionswhich areDi�(S)invariant.

Them otivation fordoing thisisthatin physicswedo notspecify spatialregionsby considering a3D

subsetR ofS and de�ne,say,a Di�(X )invariantvolum e functions(i.e.a function invariantunder

passivedi�eom orphism starting from theunreduced form alism )by

V (R):=

Z

X

d3x �R (S(x))

pdet(q)(x) (3.23)

where�R denotesthecharacteristicfunction ofthesetR.Ratherweuseobservablem atterfordoing

this. To be sure,(3.23)isDi�(X )invariant,being the integralofa scalardensity overallofX .In

fact,wecan pullback thisexpression to S and obtain

V (R)=

Z

S

d3� �R (�)

pdet(~q)(�)=

Z

R

d3�pdet(~q)(�) (3.24)

where ~q = (S� 1)� q which would be a m athem atically naturalobject to consider in the reduced

theory (afterfurther applying ��0(:)�0= �� T). Itis,however,notDi�(S)invariant. However,from

23

thepointofview ofobservation onewould ratherliketo consideran objectoftheform

V (I):=

Z

X

d3x �I(�(x))

pdet(q)(x) (3.25)

whereI isa subsetoftherealaxisand � isa scalar�eld.Noticethat(3.25)isDi�(X )invariantbut

itisnota Diraconbservableyet.Itm easuresthevolum eofthesubsetofX in which � hasrangein

I.Now weapply them ap O � and obtain im m ediately

O V (I)(�;�):=

Z

S

d3� �I(�(�;�))

pdet(Q)(�;�) (3.26)

where �(�;�)= O�(x)(�;�)(forany x)isthe Dirac observable associated to �.Curiously,(3.26)is

a Dirac observable and itisDi�(S)invariant. Itm easures the physicalvolum e ofthe region in S

wherethephysicalscalar�eld � rangesin I.Theargum entshowsthatDi�(S)invariantobservables

naturally arisefrom thepointoftheunreduced theory and from operationalconsiderations.

Havingm otivated toconsideronly Di�(S)invariantobservableswearenow ready toconsiderthe

AQG fram ework.Sinceforsuch observablesthecoordinatesystem playsno rolewegeneralise from

em bedded tonon em bdeed graphsand theaboveargum entshowsthatin�nitecubicalgebraicgraphs

should be su�cientalthough a generalisation to arbitary countable algebraic graphsassketched in

[17]would bedesirable15.In thispaperwewilljustconsiderthecubicgraph forsim plicity.

Atthealgebraiclevelthenotion ofDi�(S)and even ofS itselfism eaningless.Noticethatin AQG

thein�nitealgebraicgraphisafundam entalobject.Thisfundam entalgraphdoesnochchange.W hat

doeschangeunderthedynam icsaresubgraphsofthealgebraicgraph.In otherwords,subgraphsof

thefundam entalalgebraicgraph arenotpreserved underthequantum dynam ics16.Thede�nition ofbH in AQG ism uch sim plerand no longerinvolvestheprojection operatorsP ,so wedonothavethe

awkward conservation lawsany longer. In fact,there isno dependence on any algebraic subgraph

whatsoever.In com pleteanalogy17 to [17]itisgiven by thefollowing listofform ulae

C�;v :=1

24‘2P

X

s1;s2;s3= � 1

s1s2s3�I1I2I3

�Tr(��A(�v;I1s1;I2s2)A(ev;I3s3)[A(ev;I3s3)� 1;Vv]) (3.27)

whereev;Is istheedgebeginningatv in positive(s= 1)ornegative(s= �1)I direction and �v;Is;Js0

isthe unique m inim alloop in the cubic algebraic graph with data v;ev;Is;ev;Js0. Form ula (3.27)is

actually thespecialisation of(3.18).Theoperator Vv isthealgebraicvolum eoperator

Vv = ‘3

P

s

j1

48

X

s1;s2;s3= � 1

s1s2s3�IJK �LM N X L

ev;Is1X M

ev;J s2X N

ev;K s3

j (3.28)

15The idea would be to consider the m ost generalsuch graph which is the m axim alalgebraic graph. This is an

algebraic graph with a countably in� nite num berofverticesand with a countably in� nite num berofedgesbetween

each pair ofvertices including loops. This generalises the notion ofa com plete graph which is a graph in which a

singleedgeconnectseach pairofvertices.16This bearssom e resem blance with the m odels forem ergentgravity considered in [47]although the dynam icsof

thosem odelsnotobviously m odelsthe dynam icsofH .17In [17] we considered the extended M aster constraint which involves, in the language of this paper, [C 2 +

Q jkCjCk]=pdet(Q ) rather than

pjC 2 �Q jkCjCkj. Hence apart from the sign in front ofQ jkCjCk we only need

to changethe powerofthe volum eoperatorfrom V1=2v in [17]to Vv here.

24

Finally

bH :=X

v

r

jCy

0;vC0;v �1

4Cy

I;vCI;v]j (3.29)

where the sum is over allofthe in�nite num ber ofvertices ofthe algebraic graph. The operator

(3.29)ism anifestly Gaussinvariant.

TheHilbertspaceofAQG isthein�nitetensorproduct(ITP)ofHilbertspacesL2(SU(2);d�H ),

one foreach edge ofthe graph (this can be generalised to de�ning di�erent ITP’s thatcom e into

play when constructing Gaussinvariantstates).ThisHilbertspaceisnotseparablebutitisa direct

sum ofseparable Hilbert spaces which assum e a Fock like structure and which are preserved18 bybH . Asfarasthe sym m etry group G isconcerned,atthe algebraic levelforinstance we no longer

havespatialdi�eom orphism s.However,wehaveitsalgebraicversion which consistsin thefollowing:

Considerthem asterconstraintlikefunctional

M :=

Z

S

d3�aH 2 + bQ jkCjCk

pdet(Q)

(3.30)

where b > a > 0 are any realnum bers. Then a classicalfunction F is invariant with respect to

transform ationsgenerated by H ;Cj respectively ifand only iffF;fF;M ggM = 0 = 0.Thefunctional

(3.30)can bequantised on theAQG Hilbertspaceby literally thesam etechniquesasin [17].Thus,

wehavethepossibility to analysetheanom aly issuewith respectto G atthealgebraiclevelaswell.

Finally,wecan com putetheexpectation valueofbH with respectto sem iclassicalstatesasin [17]

and to zeroth order in ~ we should �nd that the classicalvalue is reproduced with sm all uctua-

tions.Asforthem asterconstraint,thereally astonishing factisthat bH isa �niteoperatorwithout

renorm alisation thanksto ourm anifestly background independentform ulation.Nam ely,atthefun-

dam entalquantum leveltheoperatoralgebra islabelled by asingle,countably in�niteabstract,that

isnon em bedded,graph �. There isno such thing asa lattice distance which would need a back-

ground m etric.However,thesem iclassicalstatesdepend on a di�erentialm anifold X ,an em bedding

Y ofthealgebraicgraph � into X ,a cellcom plex Y (�)� dualto Y (�)aswellasa point(A 0;E 0)in

theclassicalreduced phasespace.Thus,thesem iclassicalstatesm akecontacttotheusual(reduced)

LQG form ulation which in particularusesan atleasttopologicalm anifold X .HenceAQG describes

alltopologiessim ultaneously. The pointisnow that,since � isan in�nite graph,theem bedding of

� can be as�ne aswe wish,with respectto the spatialgeom etry described by E 0 even ifX isnot

com pact19.Theexpectation valuesofouroperatorssuch as bH willnow give,to zeroth orderin ~,a

Riem ann sum approxim ation ofthedesired continuum integralbH asin (3.14)in term sofholonom ies

along edges ofthe em bedded graph and the volum e ofthe cubes in the dualcellcom plex. That

Riem ann sum willapproxim ate the integralthe better,the �nerthe em bedding. Itisin thissense

thatin (3.29)no continuum lim ithasto beperform ed.

18W e do notknow whether bH isdensely de� ned on allofthe ITP.However,ifitisde� ned on a single vectorin a

given separablesectorthen itisdensely de� ned on theentiresector.Now each separablesectoroftheITP islabelled

by a cyclic vector which isexplicitly known. Now bH isde� ned on a given ifand only ifitisdensely de� ned on

the corresponding sector.Hence,foreach wejusthaveto perform thistestand wesim ply rem ovethe sectorsfrom

the ITP on which bH itisnotdensely de� ned,ifany,since they areunphysical.bH iscertainly densely de� ned on the

sectorsbuiltfrom sem iclassical ,hence the surviving partofthe ITP certainly includesallthe sem iclassicalstates.19In the com pactcase the em bedding necessarily hasaccum ulation pointsbutwe can choose ourstatesnotto be

excited on edgesthatare m apped underthe em bedding into a suitably sm allneighbourhood ofevery accum ulation

point.

25

4 Sum m ary and O utlook

As com pared to the M aster Constraint Program m e [16]the present fram ework has the advantage

thatthe M asterconstraintand itssolutionsare notneeded. W e directly considera representation

ofthegaugeinvariantphase spaceand itsHam iltonian.Celebrated resultsofunreduced LQG such

asthe discretenessofthe spectrum ofkinem aticalgeom etric operators[42,43,48,49]which isnot

granted to survive when passing to thephysicalHilbertspace [50,51]in theusualDiracconstraint

quantisation now becom esaphyscialprediction ifthecurves,surfacesand regionsthatonem easures

length,areaand volum eofarelabelled by dustspace.TheGaussinvariant[52]kinem aticalcoherent

states[18]ofunreduced LQG now becom ephysicalcoherentstates.

However,thephysicalHilbertspaceofreduced LQG isnon separablewhich appearsto bea vast

overcountingofquantum degreesoffreedom .PassingtoAQG m eanstoswitch from em bedded tonon

em bedded graphsand thusrem ovesthe overcounting. Since forspatially di�eom orphism invariant

operators(ondustspace)such astheHam iltonian bH oranyotheroperationallyinterestingobservable

(which doesnotreferdirectly to the dustlabelspace)the em bedding ofa graph isim m aterial,we

can considerthe AQG reform ulation asan econom ic description ofreduced LQG in the sense that

di�eom orphism related em beddings would lead to isom orphic sectorssuperselected by thiskind of

observables.TheadditionaladvantageofAQG isthatitdoesnotrequireatopologicalm anifold and

thatitisfreefrom com plicationsthathaveto do with graph preservation.

Thechallengeofthepresentfram ework isto im plem entthe(algebraicversion of)thesym m etry

group G in the de�nition of bH which willrequire toolsfrom lattice gauge theory. The �nalAQG

version ofthe reduced phase space isin any case very sim ilarto Ham iltonian lattice gauge theory

with the im portantdi�erence thatno continuum lim ithasto be taken which iswhy the theory is

UV �nite. Anotherim portantquestion ishow one can understand from the com plicated,non per-

turbative Ham iltonian bH the signi�cance ofthe standard m odelHam iltonian on M inkowskispace.

Theanswerto thisquestion m ustliein theconstruction ofa m inim um energy eigenstateofbH which

issim ultaneously a m inim aluncertainy state forallthe observablesand which ispeaked around

atvacuum (no excitationsofobservable m atter)spacetim e. Presum ably,ifone studiesm atterex-

citationsof and considersm atrix elem entsof bH in such statesthen theresulting m atrix elem ents

can be considered as the m atrix elem ents ofan e�ective m atter Ham iltonian on M inkowskispace

which should beclosetheHam iltonian ofthestandard m odelon M inkowskispace.Thisexpectation

issupported by theanalysisof[6,7]which showsthattheequationsofm otion ofthegaugeinvariant

geom etry and m atter degrees offreedom perturbed around a hom ogeneous and isotropic (FRW )

solution isdescribed e�ectively by theusualHam iltonian on aFRW background expanded tosecond

order in the perturbations. Ofcourse,this is only a classicalargum ent. See [28]form ore details

aboutthe quantum aspectsofthisidea. W e leave thisand the research projectsm entioned in the

introduction forfutureanalysis.

Acknowledgem ents

The authors thank the KITP in Santa Barbara for hospitality during the workshop \The Quan-

tum NatureofSpacetim eSingularities" held in January 2007during which partsofthisprojectwere

com pleted.T.T.wassupported there in partby the NationalScience Foundation underGrantNo.

PHY99-07949.

W e thank Abhay Ashtekar,M artin Bojowald,Steven Giddings,Jim Hartle,Gary Horowitz,Ted

26

Jacobson,Jurek Lewandowski,Don M arolf,Rob M yers,Herm ann Nicolai,Joe Polchinski,Stephen

Shenker,Eva Silverstein,LukaszSzulcand Erik Verlindeforinspiring discussions.

K.G.isgratefulto thePerim eterInstitute forTheoreticalPhysicsforhospitality and �nancialsup-

portwherepartsofthepresentwork werecarried out.Research perform ed atthePerim eterInstitute

forTheoreticalPhysicsissupported in partby theGovernm entofCanada through NSERC and by

theProvinceofOntario through M RI.

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