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In Search ofFundam entalD iscreteness in 2+ 1

D im ensionalQ uantum G ravity

T .G .B udd and R .Loll

Institute forTheoreticalPhysics,UtrechtUniversity,Postbus80195,3508 TD

Utrecht,The Netherlands

E-m ail:[email protected], [email protected]

A bstract. Inspired by previous work in 2+ 1 dim ensionalquantum gravity,which

found evidence fora discretization oftim e in the quantum theory,we reexam ine the

issueforthecaseofpureLorentzian gravity with vanishing cosm ologicalconstantand

spatially com pactuniversesofgenusg � 2. Taking asourstarting pointthe Chern-

Sim ons form ulation with Poincar�e gauge group,we identify a set oflength variables

corresponding to space-and tim elike distances along geodesics in three-dim ensional

M inkowskispace.TheseareDiracobservables,thatis,functionson thereduced phase

space,whosequantization isessentially unique.Forboth space-and tim elikedistance

operators,the spectrum iscontinuousand notbounded away from zero.

PACS num bers:04.60.K z,04.60.Pp,02.40.Tt

1. Introduction

It is not uncom m on to hear researchers of quantum gravity express the view that

spacetim eon Planckian distancescalesm ustpossessfundam entally discreteproperties.

Given the absence ofexperim entaland observationalevidence for or against such an

assertion,and our highly incom plete understanding ofquantized gravity,this points

perhapslessto a convergenceofdi�erentapproachesto theproblem ofnonperturbative

quantum gravity than a shared wish foran ultraviolet cut-o� to render �nite certain

calculations, for exam ple, of black-hole entropy.z Discussions in the context of

popular candidate theories of quantum gravity in four spacetim e dim ensions have

revealed num eroussubtletiesconcerning the nature and observability of\fundam ental

discreteness".Discreteaspectsofasym ptotically safequantum gravity derived from an

e�ectiveaverageaction and ofloop quantum gravity havebeen discussed recently in [3]

and [4,5],respectively. By contrast,quantum gravity derived from causaldynam ical

triangulations has so far not revealed any trace offundam entaldiscreteness (see,for

exam ple,[6]).W hetherornotPlanck-scalediscretenesscan even in principleberelated

z Fora reasoning along theselines,see[1]and referencestherein.Related argum entson theexistence

ofa m inim um length scalein quantum gravity can be found in G aray’sclassicreview [2].

http://arxiv.org/abs/0906.3547v1

In Search ofFundam entalDiscretenessin 2+1 Dim ensionalQuantum Gravity 2

to testable physicalphenom ena willhave to awaita deeperunderstanding ofquantum

gravity.

In thispaper,wewilladdressthem orespeci�c question ofthespectralproperties

of quantum operators associated with the length of curves in spacetim e, and will

concentrate on thesim pler,non-�eld theoretic setting ofpure quantum gravity in 2+1

spacetim edim ensions.W ewillidentifysuitablelength functionson theclassicalreduced

phasespaceand investigate whetherthespectra oftheirassociated quantum operators

arecontinuousordiscrete,and whetherthisproperty dependson thetim e-orspacelike

nature ofthe underlying curves. Indications ofa possible discrete nature oftim e in

2+1 dim ensionalquantum gravity com e from two distinctclassicalform ulationsofthe

theory. Firstly,in the so-called polygon approach [7],based on piecewise atCauchy

slicings ofspacetim e,the Ham iltonian takes the form ofa (com pact) angle variable,

suggestive ofa discrete conjugatetim e variablein thequantum theory.Unfortunately,

subtleties in the quantization [8]and the treatm ent of (residual) gauge sym m etries

[9,10]have so farprevented a rigorousconstruction ofan operatorim plem entation of

this m odel. Secondly,an analysis paralleling that of3+1 loop quantum gravity [11]

hasalso uncovered a discrete spectrum forthe tim elike length operator,albeitatthe

kinem aticallevel,that is,before im posing the quantum Ham iltonian constraint. By

contrast,thequantized lengthsofspacelike curvesarefound to becontinuous.

In line with these com m ents,it should be kept in m ind that there is as yet no

com pletequantization ofthree-dim ensionalLorentzian gravity forcom pactspatialslices

and in the generic case ofgenus g � 2,which would allow us to settle this question

de�nitively (see[12,13]forreviews).Thereisofcoursethe\frozen-tim e"Chern-Sim ons

form ulation thatleadsdirectly to the physicalphase space P,the cotangentbundle of

Teichm �uller space,to which a standard Schr�odinger quantization can be applied [14].

However,asem phasized early on by M oncrief[15],tryingtoanswerdynam icalquestions

willin generallead to algebraically com plicated,tim e-dependent expressions in term s

of the canonicalvariable pairs of this linear phase space, whose operator status in

thequantum theory isoften unclear.Theinvestigation oftim e-dependentquantitiesis

physicallym eaningfuland appropriate,sincesolutionstotheclassicalEinstein equations

in three dim ensions are known to possess initialor�nalsingularities and a nontrivial

tim edevelopm ent.Aswewillseebelow,thisissueisalsorelevantforthework presented

here.

Beyond the technical problem of identifying well-de�ned, self-adjoint quantum

operators, there is another layer of di�culty to do with their interpretation and

m easurability,which is rooted in the di�eom orphism sym m etry ofthe m odel,shared

with generalrelativity in four dim ensions. In a gauge theory,physically m easurable

quantities are usually those which are invariant under the action of the sym m etry

group.In thecanonicalform alism they arealsoknown asDiracobservables.Forgeneral

relativity they coincidewith thedi�eom orphism -invariantfunctionson phasespaceand

are necessarily nonlocal[16]. Since tim e translationsform partofthe di�eom orphism

group,gravitationalDiracobservableshavetheunusualpropertyofnotevolvingin tim e.


Theusualnotion ofatim eevolution can berecovered through partially gauge-�xingthe

di�eom orphism sym m etry,a procedure notwithoutitsown problem s,especially when

it com es to quantizing the theory. The disappearance oftim e,and the sim ultaneous

necessity to selectsom e kind ofevolution param eterto describe dynam icalprocesses{

which even classically ishighly non-unique{ form partoftheso-called problem oftim e

in (quantum )gravity [17].Theproblem sarem ostsevere in thequantum theory,since

di�erentwaysoftreating tim etypically giveriseto quantum -m echanically inequivalent

results,atleastin sim plem odelsystem swheresuch resultscan beobtained explicitly.

Dueto theircloseassociation with constantsofm otion,to �nd gravitationalDirac

observables one �rst has to solve the dynam ics,at least partially [18]. This is m ade

di�cultby thecom plexity ofthefullEinstein equations,and hardly any explicitDirac

observables are knownx. It is at this point that our 2+1 dim ensionaltoy m odelis

drastically sim plerthan the full,four-dim ensionaltheory ofgeneralrelativity: we can

solve its classicaldynam ics com pletely and explicitly write down the reduced phase

space,that is,the space ofsolutions m odulo di�eom orphism s. Any function on the

reduced phasespacecorrespondsto a Diracobservableand viceversa.

Having a com plete setofDiracobservablesisnotenough;onealso needsto know

whatphysicalobservablesthey representand { atleastforthecaseofa realistictheory

{ how they relate to actual, physical m easurem ents. Classically, this m ay not be

m uch ofa concern and at m ost lead to interpretationalsubtleties,without a�ecting

calculationalresults. However,during quantization one often hasto m ake a choice of

which observablesare to be represented faithfully asquantum operators,and di�erent

choices m ay welllead to di�erent conclusions,for exam ple,on the spectralnature of

geom etricquantum operators.

In this article we study the quantization of a distinguished set of geom etric

observables associated with physicallengths and tim e intervals. Unlike in previous

sim ilar investigations, they are genuine Dirac observables. The quantization of the

reduced phase space ofour m odelis straightforward and essentially unam biguous,in

contrastwith theloop quantum gravity approach to 2+1 dim ensionalquantum gravity

[11].W ethen presentan exactquantization ofboth space-and tim elikelength operators

and give a com plete analysis oftheir spectra. For the spacelike distances, we �nd

continuous operator spectra,which is perhaps less surprising. The behaviour ofthe

correspondingoperatorsfortim elikedistancesism oresubtle.Itdisplayscertain discrete

features,butthelengthspectrum isnotboundedawayfrom zero.Thissettlestheissueof

fundam entaldiscretenessin 2+1gravity,atleastfortheparticularsetoflengthoperators

underconsideration,in thenegative.Open questionsrem ain regardingthegenerality of

thisresultand itsrelation with physicalm easurem entsin theem pty quantum spacetim e

described by thetheory.

The rem ainder ofthe paper is organized as follows. In the following section we

review the theory ofgeneralrelativity in 2+1 dim ensions with vanishing cosm ological

x Fairly generalm ethodsforconstructing Diracobservableshavebeen putforward in [19,18,20,21].


constant,and the structure ofits reduced phase space with the standard sym plectic

structure. W e rem ind the reader that the physicalphase space for given spacetim e

topology can beidenti�ed with thetangentbundleto a Teichm �ullerspaceofhyperbolic

structures on a two-dim ensionalRiem ann surface ofgenus g. In Sec.3 we de�ne our

distinguished length observables.The�rstkind correspondstospacelikegeodesicsin the

locally M inkowskian spacetim e solutions.In orderto obtain also Dirac observablesfor

tim elike lengths,we then de�ne a second kind ofvariable which m easuresthe distance

between pairs ofsuch spacelike geodesics. Crucially,we are able to relate the length

variables to well-known functions on Teichm �uller space. W e show how the di�erent

characterofspace-and tim elikedistancesin M inkowskispacetranslatesinto particular

angleand length m easurem entsfrom theviewpointofhyperbolicgeom etry.In Sec.4we

quantize both space-and tim elike length observablesand analyze theirspectra,before

presenting our conclusions in Sec 5. In order to m ake the article m ore self-contained

and som e ofthe derivationsin the m ain textm ore explicit,we have collected various

m athem aticalresultsin fourappendices.They dealwith speci�c aspectsofLiegroups

and algebras,ofhyperbolic geom etry and the generalized W eil-Petersson sym plectic

structure.{ Throughoutthearticleweuseunitsin which c= 16�G = 1.In theseunits

thePlanck length isjustequalto ~.

2. G ravity in 2+ 1 dim ensions

Itiswellknown [22]thata Lorentzian m anifold M containing a Cauchy surface � has

theproducttopology M = R � �.M oreover,M adm itsafoliation by spacelikesurfaces

oftopology �. In the following we willassum e � to be com pact and orientable. As

a consequence the topology of�,and hence ofM ,iscom pletely characterized by the

genus g of�,thenum berofholes.

2.1.The phase space

Three-dim ensional\generalrelativity" on the m anifold M is de�ned by the standard

Einstein-Hilbertaction functionalofthem etricg,

S[g]=

Z

M

d3xp�g (R � 2�): (1)

W hen we take the cosm ologicalconstant � to be zero,the Euler-Lagrange equations

havethefam iliarform ofthevacuum Einstein equations

R �� = 0: (2)

Gravity in 2+1 dim ensions is relatively sim ple because the Riem ann tensor has no

additionaldegrees offreedom com pared to the Riccitensor [10],as is clear from the

algebraicrelation

R �� = g��R �� + g��R �� g��R �� g��R �� 1

2(g��g�� g��g��)R: (3)


Itfollowsthatsolutionsto theEinstein equationsare at:any sim ply connected region

in M is isom etric to a region in three-dim ensionalM inkowskispace. The dynam ics

residesin the transition functionsbetween sim ply connected M inkowski-like regionsin

a covering ofM .Aswewillseebelow,thisinform ation isneatly captured by so-called

holonom iesaround closed curvesin M .

Equivalently, we can consider the �rst-order form ulation of the theory. The

variables are given by two sets ofone-form s on M , the R3-valued triad ea and the

so(2;1)-valued spin-connection !a = �abc!bc.TheEinstein-Hilbertaction (1)with �= 0

now assum estheform

S[ea;!a]= �2

Z

M

ea ^ (d!a +

1

2�abc!

b^ !c): (4)

W e can com bine ea and !a into a single connection A taking valuesin the Lie algebra

iso(2;1)ofthe Poincar�e group (see Appendix A).In term softhe Poincar�e-connection

A theaction (4)up toboundary term stakestheform ofaChern-Sim onsaction [14,10],

nam ely,

S[A]= �

Z

M

TrB (A ^ dA +2

3A ^ A ^ A)

= �

Z

M

dx��B

�

A �;(dA)�� +2

3[A �;A �]

�

; (5)

whereB isthebilinearform on iso(2;1)de�ned in Appendix A.Denotingthecurvature

ofA by F(A)= dA + A ^ A,theequationsofm otion aresim ply given by F(A)= 0.

ThePoincar�eholonom y along a closed curve in M based ata pointx0 (together

with achosen basisofthetangentspaceatx0)isde�ned asthepath-ordered exponential

g ;x0 = P exp

Z

A 2 ISO (2;1) (6)

taking values in the Poincar�e group. The vanishing curvature ofA im plies thatg ;x0

isinvariantunderdeform ationsof ,up to conjugation.Asa consequence,fora given

connection A the holonom y isonly a function ofthe hom otopy class[ ]ofthe closed

curve . Solutions to the equations ofm otion are characterized by their holonom ies.

M ore precisely,M ess [23](see also [24]) has proved thatany suitable hom om orphism

from the fundam entalgroup �1 to ISO (2;1) corresponds to a unique m axim al at

spacetim ek,leading to theidenti�cation

P = Hom 0(�1;ISO (2;1))=ISO (2;1); (7)

forthephasespaceP.Thesubscript\0"indicatesarestriction tothosehom om orphism s

whose SO (2;1)-projections have a Fuchsian subgroup of SO (2;1) as im age (see

Appendix C). Note thatthe fundam entalgroup �1 ofM isequalto the fundam ental

group ofthespacelike surface�.

Now that we have learned how to assign a set ofPoincar�e holonom ies to a at

spacetim e,can wealso achieve theconverse,thatis,reconstructthe atspacetim e (by

k Foram axim al atspacetim eM anyisom etricim beddingin a atspacetim eN isnecessarilysurjective.


identifyingpointsin M inkowskispace)from agiven hom om orphism � :�1 ! ISO (2;1)?

Itwasproved in [23]thatthereexistsauniqueconvex open subsetU ofM inkowskispace

on which � actsproperly discontinuously,giving rise to a quotientspace ofU which is

a m axim alspacetim e,necessarily having therightholonom ies.Constructing thesubset

U isdi�cultforgeneral�,butcan beobtained in a constructiveway fora densesubset

ofphasespaceby them ethod ofgrafting [25,26,27].

A spaceclosely related to thephasespaceP isTeichm �ullerspace

T = Hom 0(�1(�);PSL(2;R))=PSL(2;R); (8)

describing the space of conform al or com plex structures on the surface �

(Appendix C). Identifying PSL(2;R) with the future-preserving Lorentz group

SO 0(2;1)(Appendix A),itisim m ediately clearthatwe obtain a canonicalprojection

�T ofP onto T by sim ply taking the SO 0(2;1)�= PSL(2;R)-partofthe ISO 0(2;1)-

holonom ies. Itturnsoutthat�T identi�esP with the tangentbundle ofTeichm �uller

space: given a path t! [�](t)in T ,�rsttaking the derivative with respect to tand

evaluating on a hom otopy class,and then reversing theordergivesa correspondence

TT = T(Hom 0(�1(�);PSL(2;R))=PSL(2;R))

�= Hom 0(�1(�);TPSL(2;R))=TPSL(2;R): (9)

Using the fact that T PSL(2;R) and ISO 0(2;1) are isom orphic (Appendix A), we

concludethat

P = T T : (10)

2.2.Sym plectic structure

To obtain the sym plectic structure on P we foliate (4) into constant-tim e slices and

identify thecanonicalm om enta,leading to thebasicPoisson brackets

feai(x);!bj(y)g= �

1

2�ij�

ab�(x;y); (11)

wheretheone-form sea and !a arerestricted to a constant-tim esurface�.In term sof

theconnection A wecan writethesym plectic structureasthetwo-form

=

Z

�

Tr(�A ^ �A) (12)

on the (in�nite-dim ensional)space ofconnectionsrestricted to �,which descendsto a

sym plecticstructure! on thespaceP of atconnections.Itcan beshown [28,29]that

forconnectionsin ageneralgaugegroup G this! correspondstoacanonicalsym plectic

structure [30]on Hom 0(�1(�);G)=G which isa generalization ofthe well-known W eil-

Petersson sym plectic structure!W P in thecaseofG = PSL(2;R)(seeAppendix C).

W e expect this generalized W eil-Petersson sym plectic structure (Appendix D)

corresponding to the tangent group TPSL(2;R) to be related to the standard W eil-

Petersson structure!W P on T .Indeed,itisstraightforward to associateto thetangent


bundleofasym plecticm anifold acanonicalsym plecticstructure,thetangentsym plectic

structure [31].To seethis,notethatthetwo-form !W P de�nesa linearm ap

~!W P :T T ! T�T (13)

by contraction. At the sam e tim e, the cotangent bundle T�T already possesses a

canonicalsym plectic structure !can,which we can pullback along ~!W P to obtain a

sym plectic form

! = ~!�

W P !can (14)

on T T .W eshow in Appendix D thatthiscoincideswith thegeneralized W eil-Petersson

sym plectic structureforthetangentgroup.

Therelation between ! and !W P ism osttransparentwhen welook atthePoisson

bracketsthey de�ne.Given a function f on T ,therearetwo functionson P = T T we

can naturally associatewith it.First,wecan justtakethetrivialextension f� �T off,

which wewillcontinuetodenotebyf.Second,wecan takethederivativedf :T T ! R,

which wecallthevariation off,and which in thefollowing wewilloften denoteby the

corresponding capitalletterF.Therelation between thetwo di�erentPoisson brackets

can besum m arized by [31]

ff1;f2gP = 0;

fdf1;f2gP = ff1;f2gT ; (15)

fdf1;df2gP = dff1;f2gT

forany pairf1 and f2 offunctionson Teichm �ullerspace.

Let us check explicitly that (15) yields the Poisson brackets fam iliar from the

literature. Following [10],de�ne the loop variable T 0[ ]:= 1

2Trg ,where g is the

SO (2;1)-holonom y around ,analogous to (6) above,and its variation by T1[ ]:=

dT0[ ].FortheirPoisson brackets,wederive[10]�T0[ 1];T

0[ 2]= 0

�T1[ 1];T

0[ 2]= �

1

2

X

i

�(pi)�T0[ 1 �i 2]� T

0[ 1 �i �1

2 ]�

(16)

�T1[ 1];T

1[ 2]= �

1

2

X

i

�(pi)�T1[ 1 �i 2]� T

1[ 1 �i �1

2 ]�;

where 1�i 2 denotesthepath obtained bycuttingopen 1 and 2 atthei’th intersection

pointpi and com posing them with the curve orientationsasindicated,and �(pi)= �1

depending on therelativeorientation ofthetwo tangentvectors.Clearly,(16)isofthe

form of(15)ifthePoisson bracketon Teichm �ullerspaceisgiven by

�T0[ 1];T

0[ 2]

T= �

1

2

X

i

�(pi)�T0[ 1 �i 2]� T

0[ 1 �i �1

2]�: (17)

However,accordingto[30]thisisprecisely thePoisson bracketwegetforthegeneralized

W eil-Petersson structure for the group SO (2;1). Due to the isom orphism between

SO (2;1) and PSL(2;R) it corresponds to the standard W eil-Petersson sym plectic


structure on Teichm �uller space. Finally,note that by construction the m ap ~!W P is

an isom orphism ofsym plectic m anifolds which identi�es the phase space P with the

cotangentbundle ofTeichm �ullerspace. Thiswillm ake the quantization ofthe theory

in Sec.4 straightforward.

3. G eom etric observables

In the previous section we have established a fullcorrespondence between the phase

spaceP andthetangentbundletoTeichm �ullerspace.Thelatteriswellstudied andhasa

nicedescription in term sofhyperbolicgeom etryon Riem ann surfaces(seeAppendix C).

W ewillnow show how particularobservablesin our2+1 dim ensionalspacetim ecan be

interpreted asvariationsofgeom etricfunctionson Teichm �ullerspace.

Let us �rst exam ine how a Poincar�e holonom y acts on M inkowski space. W e

willrestrict ourselves to transform ations which describe boosts,since the Lorentzian

partsofthe nontrivialholonom ies in (7)are necessarily hyperbolic (Appendix C). In

the following we will often identify M inkowski space with the Lie algebra sl(2;R)

together with its inde�nite m etric B (as spelled out in Appendix A),and ISO (2;1)

with PSL(2;R)n sl(2;R). A holonom y (g;X ) 2 PSL(2;R)n sl(2;R) then acts on

M inkowskispaceby

Y ! Ad(g)Y + X : (18)

Ifg isnontrivial,Ad(g)willleave exactly one direction invariant,which according to

(B.4)isgiven by �l(g).{ For(g;X )toleaveageodesicin M inkowskispaceinvariant,the

latterm ustbealigned with theinvariantdirection �l(g),and thuscan beparam etrized

ast! Y + t�l(g).Itisinvariantifand only if

Ad(g)(Y + t�l(g))+ X = Y + (t+ L)�l(g) (19)

for som e L 2 R and allt. If we denote by P? the projection onto the subspace

�l(g)? � sl(2;R)perpendicularto �l(g),itfollowsthatwem usthave

(Ad(g)� 1)P? (Y )= �P? (X ): (20)

Since Ad(g)� 1 isa bijection when restricted to �l(g)? ,eq.(20)hasa unique solution

forY up to a shiftin thedirection �l(g).Itisnothard to seethatthissolves(19)when

wetakeL to be

L = B (�l(g);X ): (21)

Notethatby construction �l(g)isspacelike and ofunitnorm (c.f.Appendix B),which

im pliesthatL isa spacelike distance. W e conclude thatwe can describe a hyperbolic

Poincar�etransform ation asa translation by a distanceL along a geodesicfollowed by a

boostin theplaneperpendiculartothegeodesic(seetheleft-hand sideofFig.1).From

(B.8)itfollowsthat

B (Z;Ad(g)Z)= coshl(g) (22)

{ Here,�l(g)isthe variation ofthe hyperbolic length function l(g)on P SL(2;R)de�ned in (B.6).


lαlα

Lα

Figure 1. A Poincar�e transform ation (g�;X �) leaving a geodesic invariant can be

described asa translation by L� along thegeodesic,followed by a boostofrapidity l�

in theplaneperpendicularto thegeodesic(left).Atthesam etim e,l� can bethought

ofasthelength ofa uniqueclosed geodesicon an associated Riem ann surface(right).

fora unitvectorZ perpendicularto �l(g). W e deduce thatl(g)isprecisely the boost

param eter(orchangeofrapidity).

Supposenow wearegiven a spacetim e solution M 2 P.Forany closed curve� in

M wegeta Poincar�eholonom y (g�;X �)and two associated phasespacefunctions

l� = l(g�); (23)

L� = B (�l(g�);X �): (24)

From thede�nition (B.2)itisclearthatL� isjustthevariation ofl�,

L� = dl� :T T ! R: (25)

W hat is the interpretation ofthe observable L�? As we have m entioned earlier,the

spacetim e M can be reconstructed by taking the quotientofa subsetU ofM inkowski

space by the action ofallholonom ies. Thus,if the geodesic invariantunder(g�;X �)

would lieinsideU,itwould descend to a closed geodesicoflength L� in M hom otopic

to �. M oreover, it would be the path with m inim allength in the hom otopy class.

Unfortunately U is necessarily a convex subset and therefore cannot contain any

com plete geodesic. This m eans that when we try to m inim ize the length ofa path

in a hom otopy class,wewillnecessarily run into theinitialsingularity ofthespacetim e.

W e willnevertheless work with L� asa geom etric observable which probesthe length

scalesofthe spacetim e m anifold and som ewhatinaccurately referto itasthe \length

oftheclosed geodesic� in M ".W ewillreturn to thisissuein thediscussion section.

The function L� has already been studied in a slightly di�erent form in the

m athem aticsliterature,whereitisreferred toastheM argulisinvariant[32].In thework

ofM eusburger[33]l� andL� arecalled them assandspin of� andareusedasacom plete

setofobservableson phase space. In term sofhyperbolic geom etry (Appendix C)the

function l(g�)can beinterpreted asthehyperboliclength oftheuniqueclosed geodesic

hom otopicto � on theRiem ann surface(Fig.1,right).

Since the lengths L� only probe spacelike distances, we willnow de�ne a new

observable, the distance between two closed geodesics, which can be either space-

or tim elike. Let �1,�2 be two closed paths in M and denote their holonom ies by

(g1;X 1);(g2;X 2)2 PSL(2;R)n sl(2;R),with 1, 2 theassociated invariantgeodesics


ξl(g1)

ξl(g2)

h

Figure 2. The distance h between two geodesics in the unit hyperboloid H 1 is the

hyperbolic angle between the corresponding planes through the origin in M inkowski

space.

in M inkowskispace. W e are interested in the line-segm ent c connecting 1 and 2

at right angles. Since the directions ofthe geodesics are given by �l(g1) and �l(g2),

the direction ofc willbe their cross product,which in Lie algebra term s is just the

com m utator[�l(g1);�l(g2)].Fortwo pointsY1;Y2 2 sl(2;R)on thetwo geodesics 1 and

2,thesigned length ofcisequalto

D �1�2 =B (Y1 � Y2;[�l(g1);�l(g2)])

pjB ([�l(g1);�l(g2)];[�l(g1);�l(g2)])j

: (26)

For3-vectorsxa and ya wehavetheidentity

(x� y)� (x� y)= xayb�cabx

a0yb0�c0

a0b0�cc0 = (xaya)2 � (xaxb)(y

byb); (27)

which in ourcaseim plies

B ([�l(g1);�l(g2)];[�l(g1);�l(g2)])= B (�l(g1);�l(g2))2 � 1; (28)

wherewehaveused that�l(gi)isofunitnorm .Consequently,[�l(g1);�l(g2)]isspacelike

when B (�l(g1);�l(g2))> 1 and tim elikewhen B (�l(g1);�l(g2))< 1.

This raises the interesting question of how the two cases di�er at the level of

hyperbolic geom etry.Itturnsoutthatwhen the two closed geodesicson theRiem ann

surfacearenon-intersecting (Fig.3a),wehave

jB (�l(g1);�l(g2))j= coshh > 1; (29)

whereh isthe(shortest)hyperbolicdistancebetween thetwo.Ifthey do intersect(Fig.

3b),wehave

B (�l(g1);�l(g2))= cos� < 1; (30)

where� istheanglebetween thegeodesicsattheintersection point.A sim pleway tosee

thisisby considering thehyperboloid m odelH 1 (asdescribed in Appendix C).Thetwo


geodesicsde�netwoplanesthrough theorigin in M inkowskispacewith norm alsequalto

�l(g1)and �l(g2)(Fig.2),and intersection spanned by theouterproduct[�l(g1);�l(g2)].

The intersection willobviously only intersectH 1 ifthe two geodesicsintersect on H 1,

therefore[�l(g1);�l(g2)]istim elike ifand only ifthetwo geodesicsintersect.Now there

is a unique elem ent g 2 PSL(2;R) for which Ad(g) m aps �l(g1) to �l(g2) and leaves

[�l(g1);�l(g2)]invariant.

Ifthecom m utator[�l(g1);�l(g2)]isspacelike,thegroup elem entg ishyperbolicand

�l(g)isproportionalto [�l(g1);�l(g2)]. From (22)we deduce thatthe scalarproductof

the two vectorsisgiven by B (�l(g1);�l(g2))= coshl(g). The invariantgeodesic in H 1

corresponding to g istheintersection oftheplanespanned by �l(g1)and �l(g2)with H 1.

Ittherefore coincideswith the perpendicularly connecting geodesic,and the searched-

fordistance h isjustl(g). On the otherhand,if[�l(g1);�l(g2)]istim elike,the angle �

between the geodesics in H 1 is just the angle between the two planes,which satis�es

B (�l(g1);�l(g2))= cos�.

In ordertocalculatethevariation ofB (�l(g1);�l(g2))(thatis,ofh and �)toarriveat

theDiraclength observable,we�rstneed an identity forthederivativeof�l(g),nam ely,

d

dt

��t= 0

�l(exp(tX )g)= [Y;�l(g)]; (31)

whereY isapointon theinvariantgeodesic.Toprovethis,notethat�l(exp(t�l(g))g)=

�l(g)forallt,which m eansthatwecan replaceX by (Ad(g)� 1)Y according to (20),

d

dt

��t= 0

�l(exp(tX )g)=d

dt

��t= 0

�l(exp(t(Ad(g)� 1)Y )g)

=d

dt

��t= 0

�l(exp(tY )gexp(�tY ))

=d

dt

��t= 0

Ad(exp(tY ))�l(g)

=d

dt

��t= 0

exp(tad(Y ))�l(g)

= ad(Y )�l(g)= [Y;�l(g)]: (32)

Using thisresult,we�nd forthevariation ofB (�l(g1);�l(g2))

d

dt

��t= 0

B (�l(exp(tX 1)g1);�l(exp(tX 2)g2))

= B

�d

dt

��t= 0

�l(exp(tX 1)g1);�l(g2)

�

+B

�

�l(g1);d

dt

��t= 0

�l(exp(tX 2)g2)

�

= B ([Y1;�l(g1)];�l(g2))+ B (�l(g1);[Y2;�l(g2)])

= B (Y1 � Y2;[�l(g1);�l(g2)]); (33)

so that�nally

dh =dcoshh

pcosh

2h � 1

=B (Y1 � Y2;[�l(g1);�l(g2)])pB (�l(g1);�l(g2))

2 � 1; (34)


(a)

hα1α2

γ1

γ2

Dα1α2

c

(b)

θα1α2

γ2

γ1

Dα1α2

c

γ1

γ2

Figure 3. Distance D � 1� 2between (a) spacelike and (b) tim elike separated closed

geodesicsin M inkowskispaceand theirrelation to hyperbolicgeom etry.

and sim ilarly for�.W econcludethat

D �1�2 =

(

d��1�2 �1;�2 intersecton Riem ann surface

dh�1�2 otherwise; (35)

which isillustrated in Fig.3.

From relations (29) and (30) it is now straightforward to give a geom etric

interpretation ofthe functions ��1�2 and h�1�2 on P: h is the hyperbolic angle (or

boostparam eter)between 1 and 2 m easured alongtheconnecting geodesicc,and � is

theanglebetween 1 and 2 along c.W ith regard to ourquestforexpressing geom etric

quantitiesin M inkowskispace in term sof\Teichm �ullerdata",we can already see the

generalpictureem erging:spacelikegeodesicsin M inkowskispacearerelated togeodesics

on the Riem ann surface,and distances along them in M inkowskispace correspond to

variations ofhyperbolic distances. By contrast,tim elike geodesics relate to points in

theRiem ann surface,and tim elikedistancescorrespond to variationsofanglesatthose

points.

4. Q uantization

In section 2.2 we identi�ed the phase space P of2+1 dim ensionalgravity with the

cotangent bundle T�T to Teichm �uller space together with its canonical sym plectic

structure. Geom etric quantization ofthis phase space is straightforward. As Hilbert

space we take H = L2(T ;!3g�3

W P ), the space of square-integrable wave functions on

Teichm �ullerspacewith volum eform de�ned by theW eil-Petersson sym plecticstructure

!W P .A function f on Teichm �ullerspacebecom esa m ultiplication operator

f̂� = f � �; (36)

and itsvariation F a derivativeoperatoraccording to

F̂ � = i~ff;�gW P : (37)


τα

Figure 4.The Ham iltonian vector�eld H l� generatesa twistalong �.

Oneeasily checksthatthisyieldsan operatorrepresentation ofthePoisson algebra(15)

ofphasespacefunctionsatm ostlinearin thetranslationalpartoftheholonom ies.By

theStone-von Neum ann theorem thequantization ofthelatteralgebra isuniqueup to

unitary equivalence,because ourphase space can be broughtglobally to the canonical

form T�R6g�6 .+

The procedure for �nding the spectrum ofan operator F̂ corresponding to the

variation ofa function f on Teichm �uller space T is relatively straightforward. The

Ham iltonian vector�eld H f = ~!�1

W P (df)generatesthe Ham iltonian ow off on T . If

we take a wave function � with supporton a single orbitO ofthe ow,itwillbe an

eigenstateofF̂ with eigenvalueF ifitdescribesa wavein the ow param etert,thatis,

�jO (t)/ exp(�i

~Ft): (38)

W hetherthespectrum ofF̂ (restricted totheorbitO )iscontinuousordiscretedepends

on the dom ain oft. W henever the ow iswell-de�ned and injective fort2 R,F can

take any value in R. However,iftis restricted to take values in a bounded interval,

say,t2]0;r[,we can only have a discrete setofeigenstates with eigenvalues F which

are separated by a distance 2�~=r.The precise eigenvaluesdepend on the chosen self-

adjointextension ofF̂ or,equivalently,on thechosen boundary conditionsfor�.Toget

thefullspectrum ofF wem ustcom bineallspectra oftheindividualorbits,which need

notcoincide.

4.1.Spectra oflength observables

Recallthat the length ofa closed geodesic is given by the variation L� = dl� ofthe

hyperbolic length l�. A convenient globalcoordinate system for Teichm �uller space is

given by the Fenchel-Nielsen coordinates (li;�i),i= 1;:::;3g � 3 (see Appendix C),

corresponding toa pair-of-pantsdecom position which has� asoneofthecuts.In these

coordinatestheW eil-Petersson sym plectic form isgiven by (C.16),

!W P =X

i

dli^ d�i; (39)

+ By contrast,the so-called m odulispace M := T =M CG,obtained by taking a quotientwith respect

to them apping classgroup M CG of\largedi�eom orphism s" (generated by Dehn twists),isnotsim ply

connected. Som e ofthe di�cultieswhich arise when im plem enting M CG asa sym m etry group either

in the classicalorthe quantum theory aredescribed in [10].


(a)

l0l1

l2

θ

(b)

l2

2

τ1

2

l1

2

θ

l0

4

a

Figure 5. (a)Angle between two geodesicson the one-holed torus. (b)Cutting the

torusopen,one obtainsan octagon,which can be putinside the Poincar�edisc.

where the �i are the twistparam eters. The Ham iltonian ow ofl� issim ply the twist

ow along � (Fig.4). Since the twistparam etersascoordinateson Teichm �ullerspace

havedom ain equalto R,weconcludethatthespectrum ofL̂� istheentirereallineR.

Next,wewillinvestigate theoperator

D̂ �1�2 = i~f�;�gW P (40)

corresponding to a tim elikedistancebetween two geodesics.To startwith,considerthe

geom etric situation asdepicted in Fig.5,nam ely,a Riem ann surface ofgenus 1 with

a hole ofgeodesic boundary length l0.Fixing l0 m eansthatitshyperbolic geom etry is

described by atwo-dim ensionalTeichm �ullerspaceT .Oncewehavefound thespectrum

ofD̂ �1�2,we willargue thatthe resultholdsforany spatialtopology and forany two

sim pleclosed geodesics�1 and �2 with a singleintersection.

Oneway ofparam etrizingT (up toasign)isthrough thelengthsl1 and l2 of 1 and

2 asindicated in Fig.5.Cutting thesurfacealong 1 and along two shortestgeodesics

connecting the hole’sboundary to eitherside of 1,we obtain an eight-sided polygon

which wecan draw atthecentreofthePoincar�ediscasin Fig.5.Using thesym m etries

ofthesituation and applying thetrigonom etricidentitiesfrom Appendix C we�nd that

sinhl1

2sinh

l2

2� cosh

l0

4(41)

and

sin� =cosh l0

4

sinh l12sinh l2

2

: (42)

In view oftheexplicitform (40)oftheoperatorD̂ ,weareparticularly interested in the

range ofthe variable conjugate to �,thusin �nding a function � on Teichm �ullerspace

satisfying

f�;�gW P = 1: (43)

W e will for the m om ent restrict our attention to only half of Teichm �uller space,

corresponding to 0 < � < �

2, for which we can use l1 and l2 as coordinates (with


(a)1

2

3

4

5

1

2

3

4

5-20

0

20

1

2

3

4

5-20

0

20

ρ

l2 l1

(b)0 1 2 3 4 5

0

1

2

3

4

5

l1

l2

Figure 6. (a)Three-dim ensionalplotand (b)contourplotof� asfunction ofl1 and

l2 (with l0 = 1). The white line in (b)correspondsto sinh l1

2sinh l2

2= cosh l0

4. Note

the antisym m etry with respectto exchangeofl1 and l2.

dom ain given by (41)).W ecan �nd an explicitsolution for�(l1;l2)by solving a partial

di�erentialequation which weobtain from (42)using W olpert’sform ula (C.17),

1= f�;�gW P = cos�

�@�

@l1

@�

@l2�@�

@l2

@�

@l1

�

=@(sin�)

@l1

@�

@l2�@(sin�)

@l2

@�

@l1

=sin�

2

�

cothl2

2

@�

@l1� coth

l1

2

@�

@l2

�

: (44)

Thisequation can besolved using standard techniquesfor�rst-orderpartialdi�erential

equations.Thesolution willbeuniqueup toaddition ofafunction of�,which obviously

willPoisson-com m utewith �.To determ ine� uniquely (up to a constant)werequireit

to beantisym m etric in l1 and l2.An uninspiring calculation then leadsto

� =2

sin�sc�1

1

2

cosh l1

2

cosh l22

�cosh l2

2

cosh l12

! ��1�

sin2�

cosh2 l04

!

; (45)

where sc�1 isthe inverse Jacobielliptic function [34]. Forillustration,we show som e

M athem atica plotsof� asfunction ofl1 and l2 forsm allli in Fig.6.Itisnotdi�cult

to verify that�

(l1;l2)2 R2

> 0

��sinh

l1

2sinh

l2

2> cosh

l0

4

�

!

i

0;�

2

h

� R :(l1;l2)! (�;�) (46)

is sm ooth and injective. By allowing � to take values in ]0;�[, we obtain global

coordinateson Teichm �ullerspace.

To�ndthedom ainof� wenotethatx ! sc�1 (xjm )isabounded,strictlyincreasing

function for�xed m 2]� 1;1[.The asym ptotic valuesare �K (m )atx ! �1 ,where

K (m )isthe com plete elliptic integralofthe�rstkind [34].Hence,for�xed � we have

� 1

2�� l0(�)< � < 1

2�� l0(�),wherewehavede�ned

�� l0(�)=4

sin�K

1�sin2�

cosh2 l04

!

: (47)


(a)

Π

2ΠΘ

-4 Π

-2 Π

2 Π

4 Π

Ρ

(b)5 10 15 20 25

5

10

15

20

25

l0

2π

∆ρ(π/2)

Figure 7. (a)The dom ain of� and � (shaded area). The dotted curvescorrespond

to constantl1.(b)The value ofthe m inim um of��,asa function ofl0.

Thefunction �� l0(�)hasa m inim um at� = �=2,whereitassum esthevalue

�� l0

��

2

�

= 4 K

�

tanh2l0

4

�

: (48)

Com puting them inim um asa function ofl0 (Fig.7),oneobservesthatitstartsoutat

the value 2� atl0 = 0 and forincreasing l0 converges rapidly to l0 + c fora constant

c� 2:77.

W econcludethattheseparation oftheeigenvaluesofD̂ �1�2 dependson both � and

l0 and isgiven by

D �1�2 22�

�� l0(�)~Z; (49)

up to a constant which m ay depend on � and l0. For � near �=2 and l0 sm allthe

separation isapproxim ately equalto the Planck length ~. However,the discretization

disappearswhen � ! 0;� orl0 ! 1 .

In ordertocom pleteourderivation,westillneed toshow that\isolatingahandle",

as we did above (c.f. Fig.5),does not constitute any loss ofgenerality. Let � be a

Riem ann surface ofany genusg � 2,and �1 and �2 two sim ple closed geodesicson �

with precisely oneintersection (an exam plewith g = 2isdepicted in Fig.8).Theunique

closed geodesic�0 in thehom otopy class[�1][�2][�1]�1 [�2]

�1 isnecessarily disjointfrom

�1 and �2.A pair-of-pantsdecom position containing�0 and �1 ascutswillthen contain

onepairofpantswhich hastheform ofaone-holed torus,asin ourpreviouscalculation,

the only di�erence being thatl0 isno longeran externalparam eter,buta function on

Teichm �ullerspace.Thesym plectic structureisgiven by

!W P = dl0 ^ d�0 + dl1 ^ d�1 +

3g�2X

i= 2

dli^ d�i; (50)

with l1 = l(�1),which can berewritten as

!W P = dl0 ^ d~�0 + d� ^ d� +

3g�2X

i= 2

dli^ d�i; (51)


α2

θα1

α0

Figure 8.A Riem ann surfacewith two sim pleclosed geodesics�1 and �2 intersecting

with angle�.

where ~�0 = �0 + �� 0(l0;l1;�1)and �� 0 isa function satisfying

@�� 0

@�1=

@�

@�1

@�

@l0�

@�

@l0

@�

@�1;

@�� 0

@l1=

@�

@l1

@�

@l0�

@�

@l0

@�

@l1: (52)

Onecan check thatthese di�erentialequationsareconsistent,thatis,@2�� 0=@l1@�1 =

@2�� 0=@�1@l1,and therefore can always be solved. W e conclude that �;�;l0;~�0;li;�i

form a new coordinate system for Teichm �uller space in which !W P is given by (51).

Thism eansthatthespectrum ofD̂ �1�2 wefound fortheone-holed torusisvalid in this

casetoo.

A sim ilarargum entcan be m ade in the case thatthe geodesics�1 and �2 do not

intersecton �.RecallthatthecorrespondingspacelikedistanceD �1�2 wasthevariation

ofthe hyperbolic length h�1�2 ofthe geodesic connecting them . Also in thiscase one

can always�nd apair-of-pantsdecom position having a particularpairofpantswith �1,

�2 and a third sim ple closed geodesic�0 asboundary com ponents,and which contains

the connecting geodesic. Since the geom etry ofa pairofpantsisfully determ ined by

the lengthsofitsboundary com ponentsl1,l2 and l0,the length h�1�2 asa function of

theFenchel-Nielsen coordinatesalso dependson l1,l2 and l0 only,and wecan write

D̂ �1�2 = i~

�@h

@l1

@

@�1+@h

@l2

@

@�2+@h

@l0

@

@�0

�

: (53)

Justasin thecaseofL̂�,theHam iltonian ow isa linear ow in thetwistparam eters

and thereforethespectrum ofD̂ �1�2,forspacelikedistancesD �1�2,isagain continuous.

5. D iscussion and conclusion

In thispaper,wehaveidenti�ed space-and tim elikelength variablesin 2+1dim ensional

gravitywithvanishingcosm ologicalconstant.Theyaregiveninterm soffunctionsonthe

reduced phasespaceofthetheory,obtained in aChern-Sim onsform ulation ofthethree-

dim ensionalPoincar�e group.Being linearin m om enta,the quantization ofthese Dirac

observables isessentially unique. A study oftheireigenvalues in the quantum theory

revealed continuous spectra spanning the entire realline forboth space-and tim elike

distanceoperators.� Asfarasweareaware,thisconstitutesthe�rstrigorousderivation

� The length eigenvaluescan haveeithersign becausethey cam efrom oriented lengths.


ofquantum spectra ofDiraclength observablesin Lorentzian three-dim ensionalgravity

forgenusg � 2.

Although ourresultsdonotcon�rm previousinvestigationsin [35,11],which found

evidenceforadiscretespectrum fortim elikedistances,wedid com eacrosssom ediscrete

aspectsin ourspectralanalysis.Although noneofthelength observablesweconsidered

were canonically conjugate to an angle,the tim elike distance D �1�2 was found to be

conjugate to a function with a �nite dom ain. However,the size ofthisdom ain isnot

bounded asa function on Teichm �ullerspace(thehabitatofthewave functions),which

im pliesthatthereisno \spectralgap" fortim elikedistances.]

Thediscrepancy with previousresultsm ay haveto do with thefactthatneitherof

them wasbased on acom pleteand consistentquantization ofthetheory on thereduced,

physicalphase space. The underlying form ulationsare su�ciently di�erentfrom ours

to m ake a directcom parison di�cult.Subtletieswith regard to theim plem entation of

the Ham iltonian constraint[9,4,5]m ay wellplay a role. They can be seen aspartof

a larger issue,present in allbut the sim plest system s with gauge sym m etry,nam ely,

to whatextentquantization and the im position ofconstraints com m ute [37,38]. Not

even forthecaseofgravity on a spatialtorus(g = 1),whosequantization hasreceived

a lot ofattention in the physics literature [10]has the question ofthe equivalence or

otherwiseofdi�erentquantizationsbeen settled com pletely.Partoftheproblem isthe

scarcity of\observables" which onewould liketo useto com parephysicalresults.

Thegeneric presence ofquantization am biguitieshighlightsthefactthattheissue

of\fundam entaldiscreteness" can be interpreted in m ore than one way,depending on

which quantization and operatorsoneappliesitto,and thereforem ay nothaveaunique

answer.In thepresentwork,wehavefocused on thewell-de�ned notion ofinvestigating

thespectraofDiracobservablesm easuringlengths,obtainedina\tim e-less"phasespace

reduction ofthree-dim ensionalquantum gravity. One could argue thatthis setting is

distinguished,becauseoftheabsenceofany choiceoftim e-slicing and thesim plicity of

theensuing (Schr�odinger)quantization.

The resultswe have been ableto derive com ewith som equali�cations.Firstly,as

already m entioned earlier,the lengths L� and D �1�2 are notinterpretable directly as

lengths ofcurves (orofdistances between such curves) inside the spacetim e m anifold

itself. Thishappensbecause there are no closed geodesicsin a nondegenerate solution

in theclassofgeom etrieswehavebeen considering (recallthateach solution isobtained

bym akingidenti�cationson a convexopen subsetof3d M inkowskispace).Nevertheless,

they constitutea com pletesetoflength variables\associated with a solution",in term s

ofwhich any otherobservable can be expressed. By the sam e token,we do notclaim

thatourlength variablesaredirectly m easurable.yy

There are related constructions which m ay yield length observables with a m ore

] This situation issom ewhatrem iniscentofthe num erically found propertiesofthe spectrum ofthe

volum eoperatoron higher-valencestatesin canonicalloop quantum gravity in 3+ 1 dim ensions[36].

yyO fcourse,physical\m easurability" isa som ewhatacadem icconceptin an unphysicaltoy m odellike

three-dim ensionalquantum gravity.


im m ediatephysicalinterpretation.Forexam ple,wecould considerthelength ofa path

in aparticularhom otopy classin thelim itasitapproachestheinitialsingularity,orthe

lengthsofclosed geodesicsin asurfaceofconstantcosm ologicaltim e[26].In eithercase

itisdi�culttocharacterizethecorrespondingfunctionson Teichm �ullerspaceexplicitly.

To quantizethem oneshould reform ulatethephasespaceentirely in term sofso-called

m easured lam inations.Thism ay befeasible,in thesensethatthesestructuresarewell

studied and a lotisknown abouttherelevantsym plectic structure[39,40,25,33].

Anotherpossibility ofconstructing physicalobservablesisby enlarging the phase

space slightly.Itisstraightforward to include pointparticlesinto the m odel,although

there are som e subtleties which prevent the na��ve use ofa quotient construction to

obtain thespacetim e.Theworld linesofm assiveparticlesde�netim elikegeodesicsand

onecould considerm easuring m inim aldistancesbetween them .An alternativem ethod

proposed recently by M eusburger[33]isto de�nedi�eom orphism -invariantobservables

corresponding to geodesics(in thiscase light-like),butparam etrized by the eigentim e

along theworldlineofan observer.They arean exam pleofRovelli’sevolving constants

ofm otion [41,19].

Note thatin our investigation we have only considered length spectra associated

with a subsetofcurves,nam ely,particulargeodesics(i.e. straightlines)in M inkowski

space. Ourconstruction doesnotallow foran easy generalization to arbitrary curves.

This,and thepeculiarbehaviourwefound when analyzing thespectrum ofthetim elike

distance between two spacelike geodesics in the previous section, nam ely, that the

discretization unitofthisdistancedependson therelativeanglebetween thegeodesics,

arean expression ofthe factthattheonly dynam icaldegreesoffreedom ofthetheory

areofaglobalnature,and arecaptured in acoupled and nonlocalway by variouslength

variables.Thisisnotafeatureonewould expecttobepresentin fourdim ensions,where

them etricdoespossesslocaldegreesoffreedom .

TheLorentziannatureofthespacetim ewascrucialforderivingtheresultspresented

here.Ifwereplaced thePoincar�egroup ISO (2;1)with theEuclidean group ISO (3),we

would obtain a theory closely related to theEuclidean latticegravity m odelofPonzano

and Regge [42],whose phase space can be identi�ed with the tangent bundle to (a

suitable subspace of) the space of at SU(2)-connections on �. One can repeat the

constructions ofSec.3 in term s ofinvariant geodesics to obtain the analogues ofthe

functionsL� and D �1�2.Thequantization iscom pletely analogous,with L̂� generating

a so-called generalized twist ow [30]on the SU(2)-equivalent ofTeichm �uller space.

However, it turns out that this twist ow is periodic with �xed period [43], which

im pliesthatthespectrum ofL̂� willbediscretized in unitsofa �xed m inim allength of

theorderofthePlanck length.Thisisin com plete agreem entwith resultsobtained in

theloop representation [44].

Finally, one m ay wonder whether any of the techniques we have used can be

extended to 3 + 1 dim ensions. An obvious starting point would be a generalization

to topological�eld theories with a di�erent gauge group. One such theory,perhaps

closest to generalrelativity in 3+ 1 dim ensions,is BF theory [45]with gauge group


SO (3;1). Since SO 0(3;1) is isom orphic to PSL(2;C),the isom etry group ofthree-

dim ensionalhyperbolicspace,oneshould beableto relatesom elength observablesin a

at3+ 1 dim ensionalspacetim eto functionsin three-dim ensionalhyperbolicgeom etry.

Further research is needed to determ ine whether this is feasible. Another connection

worthwhilepursuing m ay bethegeneralization to3+ 1dim ensionsof2+ 1gravity with

pointparticlesform ulated recently by ’tHooft[46].

A cknow ledgem ents. RL acknowledges support by ENRAGE (European Network

on Random Geom etry),a M arie Curie Research Training Network in the European

Com m unity’sSixth Fram ework Program m e,network contractM RTN-CT-2004-005616,

and by the NetherlandsOrganisation forScienti�c Research (NW O)undertheirVICI

program .

A ppendix A .T he gauge group

W edenotethefuture-preserving Lorentzgroup in 2+1 dim ensionsby SO 0(2;1),which

isprecisely the identity com ponentofSO (2;1). A basisforitsLie algebra so(2;1)is

given by

Jso0 =

0

B@

0 0 0

0 0 1

0 �1 0

1

CA Jso1 =

0

B@

0 0 �1

0 0 0

�1 0 0

1

CA Jso2 =

0

B@

0 �1 0

�1 0 0

0 0 0

1

CA : (A.1)

satisfying the com m utation relations[Ja;Jb]= �abcJc,where the totally antisym m etric

�-tensor is de�ned by �012 = ��012 = 1,and indices are raised and lowered with the

m etric�ab = diag(�1;1;1).ThegeneratorsJsoa form an orthonorm albasiswith respect

to theinde�nitebilinearform

Bso(X ;Y )=

1

2Tr(X Y ); B

so(Jsoa ;Jso

b )= �ab: (A.2)

W e willoften use the isom orphism SO 0(2;1) �= PSL(2;R) = SL(2;R)=fI;�Ig. To

m aketheisom orphism explicitwechoosethespeci�cbasis

Jsl0= 1

2

0 �1

1 0

!

Jsl1= 1

2

1 0

0 �1

!

Jsl2= 1

2

0 1

1 0

!

(A.3)

fortheLiealgebra sl(2;R).Thegeneratorssatisfy identicalcom m utation relationsand

areorthonorm alwith respectto thebilinearform

Bsl(X ;Y )= 2Tr(X Y ); B

sl(Jsla ;Jsl

b )= �ab: (A.4)

In fact,SO 0(2;1)em ergesastheadjointrepresentation ofPSL(2;R)on sl(2;R),when

written in thebasis(A.3).

Thegaugegroup of2+1gravity isthe2+1dim ensionalPoincar�egroup ISO 0(2;1),

which ism osteasilycharacterized asthesem i-directproductofSO 0(2;1)andtheabelian

group R 3,wheretheaction ofSO 0(2;1)on R3 isthefundam entalone,nam ely,

(g1;X 1)� (g2;X 2)= (g1g2;X 1 + g1X 2): (A.5)


Interm softhebasis(A.1),wecanidentifyso(2;1)withR 3,wheretheactionofSO 0(2;1)

now becom estheadjointaction,

(g1;X 1)� (g2;X 2)= (g1g2;X 1 + Ad(g1)X 2): (A.6)

In thisway weidentify ISO 0(2;1)with SO 0(2;1)n so(2;1),which isagain isom orphic

to PSL(2;R)n sl(2;R).

Furtherm ore,forany Liegroup G itssem i-directproductgroup G n gisisom orphic

to itstangentgroup TG,which isde�ned by taking asm ultiplication thetangentm ap

tothem ultiplication m ap G � G ! G.Explicitly,theisom orphism ,which elsewherewe

often useim plicitly,identi�estheelem entsoftheLiealgebra g with theright-invariant

vector�eldson G.{ To sum m arize,wehavethefollowing chain ofisom orphism s

ISO 0(2;1)�= SO 0(2;1)n so(2;1)�= PSL(2;R)n sl(2;R)

�= T SO 0(2;1)�= T PSL(2;R): (A.7)

Finally,a nondegeneratebilinearform on g givesriseto a naturalbilinearform on

theLiealgebraofTG by takingitsderivative.Forthecaseathand,weobtain anatural

nondegeneratebilinearform on iso(2;1)�= so(2;1)� so(2;1)by de�ning

B ((X 1;Y1);(X 2;Y2))= Bso(X 1;Y2)+ B

so(Y1;X 2): (A.8)

A ppendix B .Som e group theory

For a Lie group G and its associated Lie algebra g, we denote the left and right

m ultiplication m aps by lg;rg : G ! G. The conjugation m ap Cg : x ! gxg�1 is

an isom orphism ofG to itselfand itstangentm ap atthe origin Ad(g)= TeCg isthe

adjointrepresentation acting on g. Suppose B isa nondegenerate invariant(pseudo-)

m etricon G,i.e.B :g� g ! R isanondegeneratebilinearform invariantunderadjoint

transform ations.W edenoteby ~B theassociated m ap g ! g�.

Toadi�erentiablefunction f :G ! R wecan associateanaturalm ap �̂f :G ! g�,

which istherighttranslation ofthederivativeoff to theLiealgebra,

�̂f(g)= (Terg)�df(g): (B.1)

Using the m etric B ,we can de�ne the variation �f = ~B �1 � �̂f :G ! g off [30].

Equivalently,

B (�f(g);X )=d

dt

��t= 0

f(exp(tX )g) (B.2)

forX 2 g.Hereexp :g ! G isthestandard exponentialm ap forLiegroups.

From now on willwe assum e f to be a class function,i.e. a function which is

invariant under conjugation,C �

gf = f for allg 2 G. Using the well-known identity

Ch � exp = exp�Ad(h),we�nd that

�f(hgh�1 )= Ad(h)�f(g): (B.3)

Putting h = g weseethat�f(g)isinvariantunderAd(g),

Ad(g)�f(g)= �f(g): (B.4)


Consider now the speci�c case G = PSL(2;R)with the m etric as in (A.4). For

hyperbolicelem entsg2 G hyp = fgjTr(g)> 2g � G,wede�nethehyperboliclength l(g)

ofg by

Tr(g)= 2cosh(l(g)=2): (B.5)

Dueto thecyclicity ofthetrace,lisa classfunction,whose variation we can com pute

in a straightforward m anner. Applying relation (B.2),we �nd fordiagonalelem entsg

that�l(g)= diag(1=2;�1=2).Forgeneralelem entsg 2 G hyp which arediagonalized by

h 2 G,we have �l(g)= Ad(h)diag(1=2;�1=2).In particular,�l(g)isspacelike,ofunit

norm and thegroup elem entcan bewritten as[30]

g = exp(l(g)�l(g)): (B.6)

Ifdesired,thesetwo conditionscan betaken asthede�nition ofl(g)and �l(g),because

theexponentialm ap isa bijection from fX 2 sl(2;R)jB (X ;X )> 0g to G hyp.Onecan

writetheexponentialin (B.6)explicitly as

g = coshl(g)

21+

�

2sinhl(g)

2

�

�l(g) (B.7)

and

Ad(g)= coshl(g)1+ (1� coshl(g))�l(g)B (�l(g);�)+ sinhl(g)[�l(g);�]: (B.8)

Forelliptic elem ents we have a sim ilar construction. W e de�ne �(g)2 [0;2�[by

Tr(g)= 2cos(�=2).In thatcase,��(g)istim elike,ofunitnorm and thegroup elem ent

can bewritten as

g = exp(�(g)��(g)): (B.9)

A ppendix C .H yperbolic geom etry

In this appendix we willbrie y review som e notions ofhyperbolic geom etry, which

are used in the m ain text (see, for exam ple, [47]for details and proofs). For our

purposes,\hyperbolicgeom etry"willm ean thestudyofRiem ann surfacesandgeom etric

constructionson them . A Riem ann surface isa two-dim ensionalrealm anifold with a

com plex structure.In thisarticleweconsideronly com pactoriented Riem ann surfaces,

which aretopologically classi�ed by theirgenusg.W ewillbeconcerned only with the

caseg � 2.

Let� bea surface ofgenusg � 2.Itsfundam entalgroup isgenerated by a setof

2g hom otopy classesofclosed curvesfai;bigi= 1;:::;g satisfying therelation

gY

i= 1

aibia�1

i b�1

i = 1: (C.1)

The space ofinequivalentcom plex structureson � iscalled the m odulispace M g and

dependsonly on the genusg. In the following we willconsidera slightly largerspace,

the Teichm �uller space Tg, which is the universal covering of M g. One can de�ne


Teichm �uller space as the space of equivalence classes of m arked Riem ann surfaces.

\M arked"im pliesthatwepick acom plex structureandadistinguished setofgenerators

forthefundam entalgroup.Equivalently,weidentifytwocom plexstructureson �ifthey

arerelated by a biholom orphism hom otopicto theidentity m ap.Them odulispacecan

beobtained from Teichm �ullerspaceby taking thequotientwith respectto them apping

classgroup,

M g = Tg=M CG: (C.2)

Itiswellknown thatelem ents ofthe space ofcom plex structureson � are in one-to-

onecorrespondencewith conform ally inequivalentm etricson �.M oreover,forthecase

g � 2,every conform alequivalence class contains a unique hyperbolic m etric,i.e. a

m etric with constant curvature �1. Consequently,we can identify Teichm �uller space

with the space ofhyperbolic m etrics on � m odulo di�eom orphism s connected to the

identity.

Next,considera surface� with a speci�c com plex structure.Asa consequence of

the well-known uniform ization theorem ,the universalcoverof� isthe com plex upper

half-plane H. Any elem ent ofthe fundam entalgroup of� therefore corresponds to

an autom orphism ofH. The autom orphism group ofH is easily seen to be equalto

PSL(2;R),acting according to

a b

c d

!

:z !az+ b

cz+ d: (C.3)

An elem ent g 2 PSL(2;R)which isnotequalto the identity issaid to be hyperbolic

ifjTr(g)j > 2,elliptic ifjTr(g)j< 2 and parabolic ifTr(g) = 2. Under the iden-

ti�cation PSL(2;R) �= SO 0(2;1) these correspond to boosts,rotations and lightlike

transform ationsrespectively.On H,they can becharacterized asthosetransform ations

which leave�xed two,no and onepointson theboundary respectively.

The set � ofautom orphism s corresponding to the elem ents ofthe fundam ental

group iscalled a Fuchsian m odelof�.W ecan write� asthequotient

�= H=�: (C.4)

Since � is a sm ooth m anifold, � m ust act properly discontinuously on H, which is

equivalent to � being a Fuchsian group, that is, a discrete subgroup ofPSL(2;R).

Such a group necessarily consists only ofhyperbolic elem ents (and the identity). It

turns out that any representation ofthe fundam entalgroup �1 as a Fuchsian group

in PSL(2;R)arises as the Fuchsian m odelofa com plex structure. W e therefore can

identify Teichm �ullerspaceas

Tg = Hom 0(�1(�);PSL(2;R))=PSL(2;R); (C.5)

wherethesubscript0 m eansthatwerestrictto injectivehom om orphism swhich havea

Fuchsian group asim age.

To establish the geom etric propertiesofa Riem ann surface itsu�cesto know the

geom etry ofitsuniversalcovering H.The hyperbolic m etricon H corresponding to its


(a) (b)

Figure C 1.Som egeodesicsin (a)thecom plexupper-halfplaneH and(b)thePoincar�e

disc�.

com plex structureisthePoincar�em etricds2H= jdzj2=Im 2

z.Thegeodesicswith respect

to thism etric are halfcirclescentered on the realaxis(see Fig.C1). Som etim es itis

convenienttousean equivalentm odelofthehyperbolicplane,thePoincar�edisc�with

thePoincar�em etric

ds2�=

4jdzj2

(1� jzj2)2: (C.6)

Now thegeodesicsarecirclearcsperpendicularto theboundary (seeFig.C1).

ξl(g) g

Figure C 2.The hyperboloid m odel.

Yet another representation ofthe hyperbolic plane,which m akes the relation to

2+1 dim ensionalspacetim e m osttransparent,isthehyperboloid m odel.Itisde�ned as

theunithyperboloid

H 1 = fX 2 R3jX � X = �1;X0 > 0g (C.7)

in three-dim ensionalM inkowskispace,asdepicted in Fig.C2,with theinduced m etric.

Thegeodesicsaregiven by theintersectionsofH 1 with two-dim ensionalplanesthrough

theorigin.Anelem entg 2 PSL(2;R)�= SO 0(2;1)actsonH 1 byLorentztransform ation

(the adjoint representation Ad(g)ofg,ifwe identify M inkowskispace with sl(2;R)).

Ifg ishyperbolic,itcorresponds to a boostin M inkowskispace. Such a boostleaves

a unique plane through the origin invariantand thusg determ ines a unique invariant


geodesic in H 1. Note that�l(g)isthe norm alto thisplane,since itisinvariantunder

Ad(g).

Trigonom etry can be developed in the hyperbolic plane in analogy with the

Euclidean case.W estateheresom etrigonom etricrelations[48]forhyperbolicpolygons,

which areused in them ain text.Referring to thenotation ofFig.C3,they are

a

b

c

a’

b’

c’

(a)

a

b

c

c’

Α

b’

(b)

a

b

c

Α

Β

Γ

(c)

Figure C 3.Exam plesofhyperbolicpolygons.Allunlabelled anglesarerightangles.

(a) Given any three num bers a;b;c 2 R > 0 there exists a unique convex right-angled

hexagon with alternating sidesoflength a,band c.Thelengthsofthesidessatisfy

therelations

sinha

sinha0=

sinhb

sinhb0=

sinhc

sinhc0; (C.8)

cosha0=coshbcoshc+ cosha

sinhbsinhc: (C.9)

Analogousrelationshold forb0and c0.

(b) A pentagon with fourrightanglesand a rem aining angle� satis�es

sinha

sin�=

coshb

sinhb0=coshc

sinhc0; (C.10)

cosha = sinhbsinhc� coshbcoshccos�: (C.11)

(c) A trianglewith arbitrary anglessatis�es

sinha

sin�=sinhb

sin�=sinhc

sin ; (C.12)

cosha = coshbcoshc� sinhbsinhccos�: (C.13)

W e saw above thata hyperbolic elem entg 2 PSL(2;R)leavesexactly two points

on the boundary ofthe com plex upper-halfplane H �xed,which im pliesthatthere is

a unique geodesic in H invariantunderg. The geodesic istranslated along itselfby a

hyperbolicdistance l(g),which isrelated to thetraceofg through eq.(B.5).Ifg isan

elem entoftheFuchsian m odelof� corresponding to a hom otopy class�,thisgeodesic

projectsto theuniqueclosed geodesicin � with length given by l(g).


Closed geodesics on the Riem ann surface are associated with a convenient set of

coordinates on Teichm �uller space,known as the Fenchel-Nielsen coordinates. Given

a Riem ann surface � ofgenus g � 2,one can always �nd a set of3g � 3 m utually

disconnected sim ple (that is,non-sel�ntersecting) closed geodesics f ig. Cutting the

surfacealong thesegeodesicsresultsin a decom position of� into 2g� 2 pairsofpants,

each oneagenus-0Riem ann surfacewith threegeodesicboundarycom ponents.Itisnot

hard to show (using the trigonom etric identity (a)above) thatthe com plex structure

on a pairofpantsiscom pletely determ ined by thelengthsofitsboundary com ponents.

In order to �x the com plex structure on � we therefore need to �x the lengths li of

the geodesics i and the way we re-glue the pairs ofpants. To quantify the latter

we introduce the so-called twist param eters �i,which m easure the distance between

particular distinguished points on i. It turns outthat both types ofvariables taken

together,

(li;�i)2 (R > 0 � R)3g�3 ; (C.14)

form a globalsetofcoordinateson Teichm �ullerspace.In particular,thisim pliesthat

dim Tg = 6g� 6: (C.15)

TheW eil-Petersson sym plectic structure(seeAppendix D)takeson a particularly

sim pleform in term softheFenchel-Nielsen coordinates[47],nam ely,

!W P =

3g�3X

i= 1

dli^ d�i: (C.16)

Fortwoclosedgeodesics� and�,thePoisson bracketoftheirassociatedlengthsvariables

isgiven by W olpert’sform ula,

fl�;l�gW P =X

p2�]�

cos�p; (C.17)

wherethesum runsovertheintersection pointsp and �p istheanglebetween � and �

atp. Note thatwe could have derived thisform ula by com bining (30)with (D.2)(see

also [30]).

A ppendix D .G eneralized W eil-Petersson sym plectic structure

Fora com pactoriented surface� ofgenusg > 1,we areinterested in hom om orphism s

from itsfundam entalgroup �1 to a Liegroup G.M orespeci�cally,wewantto consider

the space Hom (�1;G)=G,where G acts on Hom (�1;G) by overallconjugation. Fora

hom om orphism � :�1 ! G,wewilldenoteitsequivalenceclassby [�]2 Hom (�1;G)=G.

IfG possesses a (pseudo-)m etric,i.e. a nondegenerate bilinearform B on itsLie

algebra g,(a suitableopen subsetof)thespaceHom (�1;G)=G can begiven a canonical

sym plecticstructure!G ,known asthegeneralized W eil-Petersson sym plectic structure.

W ithoutgivingany detailsoftheconstruction,which involveshom ology,wewillsim ply


statethem ain resultof[49].Foraclassfunction f on G (seeAppendix B)and a closed

curve� in �,de�ne

f� :Hom (�1;G)=G ! R :[�]! f(�([�])): (D.1)

Given twosuch functions,f and f0,and two closed curves� and �0,thePoisson bracket

off� and f0

�0 turnsoutto be

ff�;f0

�0gG ([�])=X

p2�]� 0

"(p;�;�0)B (�f(�(�p));�f0(�(�0

p))); (D.2)

where�]�0isthesetofintersectionsin �.Thediscretevariable"(p;�;� 0)= �1depends

on theorientation oftheintersection,�p isjustthecurve� butwith basepointspeci�ed

to bep,and �f :G ! g isthevariation off asde�ned in Appendix B.

The above can also be applied to thetangentgroup TG,which wasintroduced in

Appendix A,togetherwith them etric

B TG ((X 1;Y1);(X 2;Y2))= B (X 1;Y2)+ B (Y1;X 2); (D.3)

which is essentially the derivative of B . W e will now show that the generalized

W eil-Petersson sym plectic form !TG forthe group TG isthe tangentsym plectic form

corresponding to !G . Instead ofTG we willuse the sem i-directproductG n g,which

isisom orphicto TG by righttranslation (seeAppendix A).

Let f : G ! R be a class function on G (see Appendix B). W e associate to

f two class functions on G n g, its trivialextension f̂ = f � �G and its variation

F : (g;X ) ! B (�f(g);X ). To com pute the Poisson brackets (D.2) we will need

the variations of both f̂ and F. The variation of f̂ is easily seen to be given by

�f̂(g;X ) = (0;�f(g)) 2 g � g. The variation ofF is a bit trickier. From de�nition

(B.1)wehave

B TG (�F (g;X );(Y;Z))= dF((g;X );Ter(g;X )(Y;Z))

=d

ds

��s= 0

F (exp(sY )g;X + sZ + s[Y;X ]))

=d

dt

��t= 0

d

ds

��s= 0

f(exp(tX + tsZ + ts[Y;X ])exp(sY )g)

=d

dt

��t= 0

d

ds

��s= 0

f(exp(sY )exp(tX + tsZ)g)

= B (�f(g);Z)+d

dt

��t= 0

B (�f(exp(tX )g);Y): (D.4)

Hence

�F (g;X )=

�

�f(g);d

dt

��t= 0

�f(exp(tX )g)

�

: (D.5)

Applying thevariationsto two classfunctionsf and f0weget

B TG (�f̂(g;X );�f̂0(h;Y ))= 0

B TG (�F (g;X );�f̂0(h;Y ))= B (�f(g);�f0(h)) (D.6)

B TG (�F (g;X );�F 0(h;Y ))=d

dt

��t= 0

B (�f(exp(tX )g);�f0(exp(tY )h)):


Itnow followsfrom (D.2)that

ff̂�;f̂0

�0gTG = 0;

fF�;f̂0

�0gTG = ff�;f0

�0gG � �G ; (D.7)

fF�;F0

�0gTG = dff�;f0

�0gG ;

which isprecisely the structure (15)we expectforthe Poisson bracketscorresponding

to thetangentsym plectic structure.

R eferences

[1]SorkinR 2005Ten thesesonblackholeentropyStud.Hist.Philos.M od.Phys.36 291{301(Preprint

hep-th/0504037)

[2]G aray L 1995 Q uantum gravity and m inim um length Int.J.M od.Phys.A 10 145{166 (Preprint

gr-qc/9403008)

[3]ReuterM and SchwindtJ M 2006 A m inim allength from thecuto� m odesin asym ptotically safe

quantum gravity JHEP 01 070 (Preprint hep-th/0511021)

[4]Dittrich B and Thiem ann T 2009 Are the spectra ofgeom etricaloperators in Loop Q uantum

G ravity really discrete? J.M ath.Phys.50 012503 (Preprint0708.1721)

[5]RovelliC 2007 Com m enton ’Are the spectra ofgeom etricaloperatorsin Loop Q uantum G ravity

really discrete?’by B.Dittrich and T.Thiem ann (Preprint 0708.2481)

[6]Am bj�rn J,G �orlich A,Jurkiewicz J and LollR 2008 The nonperturbative quantum de Sitter

universePhys.Rev.D 78 063544 (Preprint 0807.4481)

[7]’tHooftG 1993 Canonicalquantization ofgravitating pointparticlesin (2+ 1)-dim ensionsClass.

Quant.Grav.10 1653{1664 (Preprintgr-qc/9305008)

[8]Eldering J 2006 The polygon m odelfor 2+ 1D gravity: The constraint algebra and problem s of

quantization (Preprint gr-qc/0606132)

[9]W aelbroeck H and Zapata J 1996 2+ 1 Covariantlattice theory and ’tHooft’sform ulation Class.

Quant.Grav.13 1761{1768 (Preprintgr-qc/9601011)

[10]Carlip S 2003 Quantum gravity in 2+1 dim ensions (Cam bridge:Cam bridgeUniversity Press)

[11]FreidelL,Livine E R and RovelliC 2003 Spectra oflength and area in (2+ 1) Lorentzian loop

quantum gravity Class.Quant.Grav.20 1463{1478 (Preprintgr-qc/0212077)

[12]Loll R 1995 Q uantum aspects of (2+ 1) gravity J. M ath. Phys. 36 6494{6509 (Preprint

gr-qc/9503051)

[13]Carlip S 2005 Q uantum gravity in 2+ 1 dim ensions:Thecaseofa closed universeLiving Rev.Rel.

8 1 (Preprint gr-qc/0409039)

[14]W itten E 1988 (2+ 1)-Dim ensionalG ravity asan Exactly Soluble System Nucl.Phys.B 311 46

[15]M oncriefV 1990How solvableis(2+ 1)-dim ensionalEinstein gravity? J.M ath.Phys.31 2978{2982

[16]TorreC 1993G ravitationalobservablesand localsym m etriesPhys.Rev.D 48 2373{2376(Preprint

gr-qc/9306030)

[17]Isham C J 1995 Structuralissuesin quantum gravity (Preprintgr-qc/9510063)

[18]Dittrich B 2006 Partialand com plete observables for canonicalgeneralrelativity Class.Quant.

Grav.23 6155{6184 (Preprintgr-qc/0507106)

[19]RovelliC 2004 Quantum Gravity (Cam bridge:Cam bridgeUniversity Press)

[20]Thiem ann T 2006 Reduced phase space quantization and Dirac observablesClass.Quant.Grav.

23 1163{1180 (Preprintgr-qc/0411031)

[21]PonsJ,Salisbury D and Sunderm eyerK 2009Revisitingobservablesin generallycovarianttheories

in the lightofgauge�xing m ethods(Preprint 0905.4564)

[22]W ald R M 1984 Generalrelativity (Chicago:University ofChicago Press)

[23]M ess G 2007 Lorentz spacetim es of constant curvature Geom . Dedicata 126 3{45 (Preprint

0706.1570)

hep-th/0504037

gr-qc/9403008

hep-th/0511021

0708.1721

0708.2481

0807.4481

gr-qc/9305008

gr-qc/0606132

gr-qc/9601011

gr-qc/0212077

gr-qc/9503051

gr-qc/0409039

gr-qc/9306030

gr-qc/9510063

gr-qc/0507106

gr-qc/0411031

0905.4564

0706.1570


[24]Andersson L,Barbot T,BenedettiR,Bonsante F,G oldm an W ,Labourie F,ScannellK and

SchlenkerJ 2007 Noteson a paperofM essGeom .Dedicata 126 47{70 (Preprint0706.0640)

[25]M cM ullen C 1998 Com plex earthquakesand Teichm �ullertheory J.Am .M ath.Soc.11 283{320

[26]BenedettiR and G uadagniniE 2001 The cosm ologicaltim e in (2+ 1)-gravity Nucl.Phys.B 613

330{352 (Preprintgr-qc/0003055)

[27]M eusburgerC 2006 G rafting and Poisson structure in (2+ 1)-gravity with vanishing cosm ological

constantCom m un.M ath.Phys.266 735{775 (Preprintgr-qc/0508004)

[28]Alekseev A Y and M alkin A Z 1995 Sym plectic structure ofthe m odulispace of atconnection

on a Riem ann surfaceCom m un.M ath.Phys.169 99{120 (Preprint hep-th/9312004)

[29]Atiyah M F and Bott R 1983 The Yang-M ills equations over Riem ann surfaces Phil.Trans.R.

Soc.London.Ser.A 308 523{615

[30]G oldm an W M 1986 Invariant functions on Lie groups and Ham iltonian ows ofsurface group

representationsInv.M ath.85 263{302

[31]G rabowskiJ and UrbanskiP 1995 TangentliftsofPoisson and related structuresJ.Phys.A 28

6743{6777 (Preprintmath/0701076)

[32]G oldm an W 2002 TheM argulisinvariantofisom etricactionson M inkowski(2+ 1)-spaceRigidity

in Dynam icsand Geom etry (Berlin:Springer-Verlag)pp 187{198

[33]M eusburger C 2009 Cosm ological m easurem ents, tim e and observables in (2+ 1)-dim ensional

gravity Class.Quant.Grav.26 055006 (Preprint0811.4155)

[34]Abram owitzM and Stegun I1965Handbook ofm athem aticalfunctions (New York:CourierDover

Publication)

[35]’t Hooft G 1996 Q uantization of point particles in 2+ 1 dim ensional gravity and space-tim e

discretenessClass.Quant.Grav.13 1023{1040 (Preprintgr-qc/9601014)

[36]Brunnem ann J and RideoutD 2008 Propertiesofthevolum eoperatorin loop quantum gravity I:

ResultsClass.Quant.Grav.25 065001 (Preprint0706.0469)

[37]LollR 1990 Noncom m utativity ofconstraining and quantizing: A U(1) gauge m odelPhys.Rev.

D 41 3785{3791

[38]Carlip S 2001 Q uantum gravity: a progress report Rept. Prog. Phys. 64 885 (Preprint

gr-qc/0108040)

[39]BenedettiR and Bonsante F 2005 CanonicalW ick rotations in 3-dim ensionalgravity (Preprint

math/0508485)

[40]BonsanteF 2005 Linearstructureson m easured geodesiclam inations(Preprintmath/0505180)

[41]RovelliC 1990 Q uantum m echanicswithouttim e:A m odelPhys.Rev.D 42 2638{2646

[42]O oguriH and SasakuraN 1991Discreteand continuum approachesto three-dim ensionalquantum

gravity M od.Phys.Lett.A 6 3591{3600 (Preprinthep-th/9108006)

[43]Je�rey L C and W eitsm an J 1992 Bohr-Som m erfeld orbitsin them odulispaceof atconnections

and the Verlinde dim ension form ula Com m un.M ath.Phys.150 593{630

[44]RovelliC 1993 The Basis ofthe Ponzano-Regge-Turaev-Viro-O oguriquantum gravity m odelin

the loop representation basisPhys.Rev.D 48 2702{2707 (Preprinthep-th/9304164)

[45]BaezJC 1996Four-dim ensionalBF theory asatopologicalquantum �eld theory Lett.M ath.Phys.

38 129{143 (Preprintq-alg/9507006)

[46]’tHooftG 2008 A locally �nitem odelforgravity Found.Phys.38 733{757(Preprint 0804.0328)

[47]Im ayoshiY and TaniguchiM 1992An introduction to Teichm �ullerspaces 1sted (Tokyo:Springer-

Verlag)

[48]FenchelW 1989 Elem entary geom etry in hyperbolic space (Berlin:De G ruyter)

[49]G oldm an W 1984Thesym plecticnatureoffundam entalgroupsofsurfacesAdv.M ath.54 200{225

0706.0640

gr-qc/0003055

gr-qc/0508004

hep-th/9312004

math/0701076

0811.4155

gr-qc/9601014

0706.0469

gr-qc/0108040

math/0508485

math/0505180

hep-th/9108006

hep-th/9304164

q-alg/9507006

0804.0328

insearchoffundamentaldiscretenessin2+1 dimensionalquantum...

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