Horizon entropy with loop quantum gravity methods

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Horizon

entropy

with

loop

quantum

gravity

methods

Daniele Pranzetti

∗

,

Hanno Sahlmann

InstituteforQuantumGravity,UniversityFriedrichAlexander,UniversityErlangen-Nürnberg(FAU),Staudtstrasse7/B2,91058Erlangen,Germany

a

r

t

i

c

l

e

i

n

f

o

a

b

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a

c

t

Articlehistory:

Received23January2015

Receivedinrevisedform29April2015 Accepted30April2015

Availableonline5May2015 Editor:M.Cvetiˇc

We show that the spherically symmetric isolated horizon can be described in terms of an SU(2) connectionandansu(2)-valued one-form,obeyingcertainconstraints.Thehorizonsymplecticstructure is precisely the one of 3dgravity in a ﬁrst order formulation.We quantize the horizon degrees of freedom inthe framework ofloop quantum gravity, withmethods recently developed for 3dgravity withnon-vanishingcosmologicalconstant.Bulkexcitationsendingonthehorizonactverysimilarlyto particlesin 3dgravity. The Bekenstein–Hawking lawis recoveredin the limitof imaginary Barbero– Immirziparameter.Alternativemethodsofquantizationarealsodiscussed.

1. Introduction

The standard black hole entropy calculation inloop quantum gravity (LQG) strongly relies on the interplay with the Chern– Simons theory describing the horizon degrees of freedom (dof). TherelevanceofTQFTinnon-perturbativequantumgravitywhen a boundaryof finite area is presentwas firstpointed out in [1]. Thecentral role ofChern–Simons theory was then further estab-lishedin [2]by meansofthe isolatedhorizon (IH)boundary con-ditions [3], providing a localdefinition of an isolated black hole moregeneralandphysicallyrelevantthanthenotionofevent hori-zon.TheU

(

1

)

gaugeﬁxingadoptedin

[2]

hasbeenmorerecently relaxedfor all physicallyrelevant kinds ofblack holes. This was systematically derived and developed in the sequence of papers

[4–6],providing afullySU

(

2

)

-invariant Chern–Simonsdescription of isolated horizons boundary theory. This analysis provided the theoreticalframework foranalytical

[7,8]

andnumerical[9] tech-niques developed for the counting of the number of boundary dof.Along thelinesoftheoriginal pointofviewof

[2]

,the lead-ingterm fortheIH entropyhasbeen shownto be inagreement withtheBekenstein–Hawkingsemiclassicalformula

[10]

foraﬁxed numerical value of the Barbero–Immirzi parameter

β

given by

β

0

=

0

.

274067

. . .

.See

[11]

forareviewoftheseresults.

This unexpected central role of the Barbero–Immirzi (BI) pa-rameter in recovering a semiclassical result of QFT on a ﬁxed geometry hasrecently motivated an alternative scenario. In [12]

*

Correspondingauthor.

E-mailaddresses:daniele.pranzetti@gravity.fau.de(D. Pranzetti),

hanno.sahlmann@gravity.fau.de(H. Sahlmann).

it hasbeen notedthat, by takingan analytic continuation ofthe dimension of the SU

(

2

)

Chern–Simons Hilbert space on a punc-tured2-sphere(modelingaquantumIH)toSL

(

2

,

C)

togetherwith some assumptions on the spin representations, the semiclassical resultcouldberecoveredwithoutthenumericalrestriction

β

= β

0. Such analytic continuation was interpreted as the passage to an imaginary BI parameter, and this choice is physically preferred due to the correct transformation of the Ashtekar self-dual con-nection

[13]

underspace–timediffeomorphisms

[14]

.Inparticular, in [15] it was shownhow local Lorentz invariance underlies the strict connection betweenthe analytic continuation to

β

=

i and thethermalityofthequantumIH.

However, regardless of the status of

β

, the fundamental role playedbyChern–Simonstheoryintheblackholeentropy calcula-tionisevident.InordertoclaimthisasafullsuccessoftheLQG approach, itwould be desirableto have aquantization ofthe IH boundarytheory completelywithinthe kinematicalframework of thetheoryandbeabletoperformthecountingwithoutrelyingon the Verlindeformulaforthe Chern–Simons Hilbertspace dimen-sion.Moreover,thestandardcouplingbetweenbulkandboundary theories,requiring identiﬁcationof certain structuresofLQG and Chern–Simonstheory,presentsanumberofambiguitieswhich af-fectstheentropycalculationandareatthecoreofsomeofthestill open issues. A more uniformtreatment uniquely in termsof the LQGformalism,besidesmakingthewholederivationmoresound, canhelptosolvethelatterandalsoprovidefurtherinsightonthe aspectsofthecalculationmentionedabove.

A ﬁrst attempt along this direction was madein [16], where some structuresofthe quantumdeformation SLq

(

2

)

of theSU

(

2

)

group (with q the deformation parameter), expected to be asso-ciated to the Chern–Simons theory, appeared; however, a clear

http://dx.doi.org/10.1016/j.physletb.2015.04.070

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Hilbert space structure was still lacking there. In this paper we proceedfurtheronthisroute.

More precisely, in Section 2 we show how the IH conserved presymplectic form can be re-expressed in terms of ﬁrst order gravityvariablesandlisttheboundaryconditionsthat thesehave to satisfy.In Section 3 we show how the Ashtekar–Barbero con-nection on the IH becomes non-commutative and we introduce a second new non-commutative connection in order to be able torely ontechniquesdevelopedin

[17,18]

inthecontext of2

+

1 gravitywithnon-vanishing cosmologicalconstant toquantizethe boundary theory using LQG techniques. The quantization is car-ried out inSection 4,where theIH quantum state is definedby regularizing pointpunctures withfiniteloops, asrequired bythe extendednatureoftheLQGconfigurationvariablesandinanalogy tothe proposalof [19]; we thendefine the physicalscalar prod-uct of the horizon theory,imposing the quantum version of the boundaryconditions.InSection5weusetheequivalence

[18]

be-tween the Chern–Simons observables expectationvalues andthe physicalamplitudesof2

+

1canonicalLQGtocomputethenumber ofIHdof by meansof thephysical scalarproduct previously de-fined.Wefindthatthedegeneracyoftheboundaryquantumstate satisfiestheBekensteinholographic boundfor

β

=

i, thus provid-ing furtherevidenceforthe newperspectiveadvocated above.In Section6analternativequantizationschemecloserinspirittothe approach of [16] is presented,by developing a comparison with the context of2

+

1 gravity coupled to point particles. Section 7

containsasummary ofourresults.Inthispaperwefocusour at-tentiononthesphericallysymmetriccase.

2. Isolatedhorizonpresymplecticform

Inordertoexpresstheconservedpresymplecticforminterms ofBFvariables,1 letusrecallfirstsome usefulrelationsfollowing from the IH boundary conditions (see [11] for more details and definitions). The phase-space variables of gravity in the first or-der formalism are givenby the 2-formdensitized triad

i (with i

,

j

,

k

=

1

,

2

,

3 andI

,

J

=

0

,

i internalSL

(

2

,

C)

indices)deﬁnedas:

I J

≡

eI

∧

eJ

i

≡

ijk

jk (1) andthe1-formextrinsiccurvatureKi

₌

_ω

0i_,_where

_ω

I J _is_the_spin connection deﬁned by

ω

aI J

≡

eIb

∇

ae_bJ and related to the metric throughtherelationgab

=

eaIe

J

b

η

I J,where

η

I J

=

diag

(

−

1

,

1

,

1

,

1

)

. In terms of these phase-space variables we can write the presymplecticformforgravityas:

κ

(δ

1

, δ

2

)

=

M

δ

_[1

i

∧ δ

2]Ki

,

(2) where

κ

=

8

π

G, M is a Cauchy surface representing space and

δ

1

,

δ

2

∈

Tp

, i.e.theyarevectorsinthetangentspacetothe phase-space

at the point p.

is an inﬁnite-dimensional manifold whose points p are given by solutions to the Einstein equations andarelabeled byapairp

= (,

K

)

.

WenowwanttointroducetheAshtekar–Barberovariables de-ﬁnedthroughtheintroductionoftheconnection Ai

a:

A_ai

=

_ai

+ β

K_ai

,

(3)

1 _A_description_of_non-rotating_isolated_horizons_in_terms_of_symmetry_reduced

SO(1,1)BFtheorywasusedin[20].

where

i

= −

1₂

i jk

_ω

jk and

β

is the Barbero–Immirzi parameter. The connection Ai _is_still _conjugate _to

i _and_in_terms _of_it _the presymplecticform

(2)

takesthe form:

κ

(δ

1

, δ

2

)

=

1

β

M

δ

_[1

i

∧ δ

2]

β

Ki

+

1

β

M

δ

_[1

i

∧ δ

2]

i

−

1

β

M

δ

[1

i

∧ δ

2]

i

=

1

β

M

δ

_[1

i

∧ δ

2]Ai

−

1

β

∂M

δ

_[1ei

∧ δ

2]ei

,

(4) where

∂

M istheboundaryof M.Ifweassume

∂

M tocorrespond to a 2-sphere cross-section IH of M withan isolated horizon

, then the isolated horizon boundaryconditions [3]imply the fol-lowingrelationtoholdonthe2-sphere:

Fi ⇐

(

A+

)

= −

2

i ⇐ (5) fromwhich Fi

()

= −

Re

(

2

)

_⇐i

+

1 2

i jkK_⇐j

∧

K_⇐k

,

dKi ⇐

= −

Im

(

2

)

i ⇐ , (6) where Ai

+

=

i

+

i Ki,

2 is the only non-vanishing Weyl scalar, the curvature Fi

(

A

)

is givenby Fi

(

A

)

=

d Ai

+

1₂

i jkAj

∧

Ak; the double arrows denote the pull-back to 2-sphere IH and we will omit them fromnowon to lightenthe notation. Inparticular, in the spherically symmetric case (

2

=

2aπIH), the above conditions

imply

Fi

(

A

)

= −

π

aIH

(

1

− β

2

)

i

,

dKi

=

0

,

(7) whereaIH isthearea ofthe isolatedhorizon. In

[5]

,by meansof

aspecialgaugewherethetetrad

(

eI

)

issuch thate1 isnormalto IH and e2 _and_e3 _are _tangent _to _IH, _it _has_been _shown _that ₍₇₎ impliesK1

=

0,whichinturnshowsthat v

1

∧

K1

=

0,wherev isa vectorﬁeldtangent toIH.Since inthe chosengaugethepull backof

2 and

3 onthehorizoniszero,thenonehas

v

i

∧

Ki

=

0

.

(8)

Therefore, (8)beingtrue ina particular gauge istrue ingeneral, since itisagaugeinvariantrelation.Anotherusefulrelationvalid onIH is

[5]

Kj

∧

Kk

i jk

=

2

π

aIH

i

.

(9)

The IH boundary conditions also restrict the variations

δ

=

(δ,

δ

A

)

∈

Tp

()

such that for ﬁelds pulled back on the hori-zontheyaregivenbylinearcombinationsofSU

(

2

)

internalgauge transformationsanddiffeomorphismswhichpreservethepreferred foliationof

.

In

[5]

ithasbeenshownthattheIHboundaryconditionslisted above preservethe presymplecticform

(4)

,inthesense that itis independent of M.Weare nowgoingto showthat theboundary termin

(4)

canberewrittenintermsofﬁrstordergravityvariables onIH.

Proposition1.IntermsofAshtekar–Barberoconnectionandits

con-jugatemomentumvariablestheconservedpresymplecticstructureofa sphericallysymmetricIHtakestheform

κ

(δ

1

, δ

2

)

=

1

β

M

δ

_[1

i

∧ δ

2]Ai

+

1

β

2

aIH 2

π

IH

δ

_[1ei

∧ δ

2]Ai

.

(10)

(3)

Proof. weneedtoshow thatthephasespaceone-form

(δ)

de-ﬁnedby

(δ)

≡

IH ei

∧ δ

ei

+

1

β

aIH 2

π

IH ei

∧ δ

Ai (11)

isclosed,wheretheexteriorderivativeof

(δ)

isgivenby

d

0

(δ

1

, δ

2

)

= δ

1

(

0

(δ

2

))

− δ

2

(

0

(δ

1

)) .

Wesaw above that thegauge symmetry transformations allowed by the IH boundary conditions on IH are given by inﬁnitesimal SU

(

2

)

transformations anddiffeomorphisms tangent to the hori-zon.Thereforelet usconsidervariations ofthe form

δ

= δ

α

+ δ

v, where

α

:

IH

→

su

(

2

)

and v isavector ﬁeldtangent toIH.Under suchtransformationswehave

δ

αei

= [

α

,

e

]

i

, δ

αAi

= −

dA

α

i

,

δ

vei

= L

vei

=

v

dei

+

d

(

v

ei

)

= (δ

∗v

− δ

α(A,v)

)

ei

=

v

dAei

+

dA

(

v

ei

)

− [

v

A

,

e

]

i

,

δ

vAi

= L

vAi

= (δ

∗v

− δ

α(A,v)

)

Ai

=

v

Fi

(

A

)

+

dA

(

v

Ai

) ,

where

α

(

A

,

v

)

=

v

A and

δ

∗v isdeﬁnedas

δ

∗vAi

=

v

Fi

(

A

)

and

δ

∗vei

=

v

dAei

+

dA

(

v

ei

)

.

Letusalsoderiveausefulrelation,whichwillrepresentan ex-traboundaryconditionduetothedoublingoftheboundaryd.o.f. introducedwiththenewboundarytermin

(10)

,namely

dAei

=

dei

+ β

ijkKj

∧

ek

= β

i jkKj

∧

ek

= −β

2

π

aIH

i

,

(12)

where inthe second passage we have used the Cartan equation dei

₊

i

jk

j

∧

ek

=

0 andinthelastonetherelation

K_ai

= −

2

π

aIH

e_ai (13)

derived in

[5]

(fromwhich (9)follows). We alsorecall that on a 2-manifold A

∧

v

B

= −

v

A

∧

B forany2-formA and1-form B, whileA

∧

v

B

=

v

A

∧

B forany1-form A and2-form B.

Letusstartwiththegaugetransformations:

d(δ, δ

α

)

=

IH 2

δ

[ei

∧ δ

α]ei

+

1

β

aIH 2

π

IH

δ

[ei

∧ δ

α]Ai

− δ

[αei

∧ δ

]Ai

=

IH 4

δ

ei

∧

i jk

α

jek

−

2

β

aIH 2

π

IH

δ

ei

∧

dA

α

i

+

i jk

α

jek

∧ δ

Ai

= −

2

IH

δ(

i

+

1

β

aIH 2

π

dAe i

₎

_α

i

=

0

,

whereinthe lastpassage weused (12).Fordiffeomorphisms we have:

d(δ, δ

v

)

=

IH 4

δ

ei

∧ δ

vei

+

2

β

aIH 2

π

×

IH

δ

ei

∧ (δ

∗v

− δ

α(A,v)

)

Ai

)

− (δ

∗v

− δ

α(A,v)

)

ei

∧ δ

Ai

=

4

IH

δ

dei

∧

v

ei

+

2

β

aIH 2

π

IH

δ

ei

∧

v

Fi

(

A

)

+

2

β

aIH 2

π

IH

δ

dAeiv

Ai

= −

4

IH

δ(

ijk

j

∧

ek

∧

v

ei

)

−

(

1

− β

2

)

β

2

π

aIH

IH

δ

ei

∧

v

i

−

2

IH

δ

iv

Ai

= −β

IH

δ(

Ki

∧

v

i

)

=

0

,

where in the third linewe haveused the result ofthe previous calculationwith

α

=

v

A andtherelation

δ

dAAi

= δ

Fi

(

A

)

,inthe fourthCartan’sequation,andEq.(8)forthevanishingofei

∧

v

i inthelastline.

2

Hence, theIHconservedpresymplecticformcan be expressed in theform (10), which showshow theboundary theory can be parametrizedby thevariables

(

A

,

e

)

satisfyingthe boundary con-ditions Fi

(

A

)

= −

π

aIH

(

1

− β

2

)

i (14) dAei

= −β

2

π

aIH

i

.

(15) 3. Non-commutativeconnection

On the isolated horizon IH we have a 2

+

1 theory. In the previous section we have seen that, upon the standard 2

+

1 de-composition, the phase space ofthe theory can be parametrized bythepullbacktoIH oftheAshtekar–Barberoconnectionandthe triad. In local coordinates we can express them in terms of the 2-dimensionalconnection Ai

a andthedyadﬁeldeia wherea

=

1

,

2 are spacecoordinateindiceson IH and i

,

j

=

1

,

2

,

3 are

su(

2

)

in-dices. As already pointed put in [5], the boundary term in the presymplecticfrom(4)impliesthat thehorizondyadﬁeldsatisfy thePoissonbracket

{

e_ai

(

x

),

e_bj

(

y

)

} = −

κ

β

ab

δ

i j

δ

(2)

(

x

,

y

) ,

(16) where

ab is the 2d Levi-Civita tensor. At the same time, if we parametrizetheIHphasespaceintermsofﬁrstordergravity vari-ables

(

A

,

e

)

, the boundary term in the presymplectic from (10)

indicatesthatthePoissonbracketamongthemisgivenby

{

Ai_a

(

x

) ,

e

˜

_bj

(

y

)

} =

κ

β

ab

δ

i j

δ

(2)

(

x

,

y

)

(17) where

˜

e_ai

:=

1

β

aIH 2

π

e i a

.

(18)

The two Poissonbrackets (16)and (17)are consistent witheach otherassoonaswetakeintoaccounttherelation

(13)

holdingon thehorizon2-sphere.Inparticular,thisimpliesthattheAshtekar– Barbero boundary connection becomes non-commutative. This should not be surprisingifone wants, asstandardly done in the

(4)

literature,interprettheboundarycondition

(14)

astheeomofthe SU

(

2

)

Chern–Simons theory on a punctured 2-sphere, since the Chern–Simonsconnectioninthel.h.s.of

(14)

isnon-commutative. Despitesuchnon-commutativity,the theory

(17)

,

(14)

,(15)bears a strong resemblance with 2

+

1 gravity with cosmological con-stant.2 _Let _us _clarify _this _classical _set-up _so _that _it _will _then _be

straightforwardtoapplyLQGtechniquesdevelopedinthatcontext toquantizetheboundarytheory.Inordertodoso,weintroducea newconnection

˜

A_ai

=

Ai_a

+

α

(

aIH

)

e

˜

ai

,

(19)

where

α

(

aIH

)

isa functionoftheIHareaaIH to bedeterminedby

expressingthe condition (14)asaﬂatness condition forthe new connection.Moreprecisely,

Fi

( ˜

A

)

=

dA

˜

i

+

1 2

i lmA

˜

l

∧ ˜

Am

=

Fi

(

A

)

+ (

aIH 2

π

β

2

α

2 2

−

α

)

i

₌

₀

_,

where in he last passage we used the boundary condition (15). Therefore, the condition (14) is recovered once

α

_±

= β(β ±

1

)

2

π

/

aIH.TheIHboundaryconditionscanthusbere-expressed as

Fi

( ˜

A

)

=

0 (20) dAe

˜

i

= −

i

,

(21) where A_ai

=

_ai

+ β

K_ai

=

_ai

−

2

π

β

2 aIH

˜

ei_a

=

_ai

−

β

2

2P

(

1

− β

2

)

k

˜

e_ai

,

(22)

˜

A_ai

=

Ai_a

+

α

_±e

˜

_ai

=

_ai

±

2

π

β

aIH

˜

e_ai

=

_ai

±

1 2

2P

(

1

− β

2

)

k

˜

ei_a

,

(23)

andwehaveusedtherelation

[5]

k

=

aIH

4

π

2P

β(

1

− β

2

)

(24) betweentheChern–Simonslevelk andtheIHareaaIH.

The boundarycondition (20)imposes the ﬂatnessof the non-commutativeconnection

(23)

,inanalogyto thetreatmentof

[17,

18] for 2

+

1 gravity in presence of a non-vanishing cosmolog-ical constant. While (21) encodes a modiﬁcation of the Gauss constraint encoding singularities in the torsion of the Ashtekar– Barbero connection on the boundary in the form of punctures induced(inthequantumtheory)fromthebulkspinnetworklinks piercingIH. This isanalog to thecase of2

+

1 gravity coupledto pointparticles

[21]

.WecanthuscombinetheLQGtechniques de-velopedintheframeworkof2

+

1gravitytoquantizetheboundary theoryontheIH.

4. Quantization

WenowwanttoquantizetheIHboundarytheoryparametrized bytheBFvariables

(

Ai

,

e

˜

i

)

andsatisfyingtheconstraints

(20)

,

(21)

justrelyingonLQGtechniques.Then,onecanthinktoextendthe quantizationtechniquesofthebulktotheisolatedhorizon. There-fore,thebasickinematicalobservablesonthehorizonaregivenby theholonomyoftheconnectionandappropriately smeared func-tionals of the dyad ﬁeld e.

˜

Namely, one can ﬁnd an irreducible representationofthequantumcounterpartoftheseobservableson akinematicalHilbertspace

H

IH

kin whosestatesaregivenby func-tionals

[

A

]

ofthe(generalized) connection A whichare square-integrablewithrespecttoadiff-invariantmeasure.

2 _in_Section₆_we_will_present_an_alternative_point _of_view_where_the _horizon

theoryistreatedasgenuineBF2+1gravitycoupledtopointparticles.

Thenon-commutativityoftheAshtekar–Barberoconnectionon theIH2-spheredoesnotrepresentanobstacletotheconstruction oftheIHHilbertspace.Thisisthecasesincethenon-commutative holonomyacting ontheAshtekar–Lewandowski vacuum

[22]

still hasamultiplicativeaction.Moreover,theintersectionoftwosuch holonomies has been explicitly computed in [17] and shown to reproduce theKauffman’scrossing bracket

[23]

; inparticular, the actionofonenon-commutativeholonomyonanothercanagainbe recast in a multiplicative form. This allows us to apply standard LQG kinematicaltechniques tothe construction ofthe IHHilbert space.Therefore,holonomiesofA arequantizedasinthe3

+

1 the-oryontheHilbertspaceL2

(A

(2)

,

d

μ

AL

)

viamultiplication,3 while

˜

ei

(

η

)

=

η

˜

ei_a

η

˙

a (25)

istheanalogofthe3dﬂux,inthesensethatthequantity

(25)

rep-resentstheﬂux ofe acrosstheone-dimensional paths

η

a

₍

_t

₎

_∈

_IH, with

η

˙

a

₌

_d

_η

a

_/

_dt._It _is_quantized _analogously_such _that _the asso-ciated operator acts non-triviallyonly onholonomies hγ along a path

γ

∈

IH thataretransversalto

η

,namely

[ˆ˜

e

(

η

),

hγ

] =

ihκ

¯

β

p∈η∩γ

sign

(

ab

η

˙

a

γ

˙

b

(

p

))

hγ2(p)Jihγ1(p)

,

(26)

i.e.actsasthederivativeoperatore

ˆ˜

_ai

= −

ih

¯

κ

β

ab

δ

i_j

δ/δ

A_bj. Inordertoimposethecurvatureconstraint,wearegoingtouse its non-commutativeconnectionformulation

(20)

sotobeableto importtechniquesdevelopedin

[17,18]

.Butletusﬁrstconcentrate onthemodiﬁedGausslaw

(21)

.FromtheLQGquantizationofthe densitizedtriad

iinthebulkwehave

ab

ˆ

_abi

(

x

)

=

2

κ

β

p∈∩IH

δ(

x

,

xp

) ˆ

Ji

(

p

) ,

(27)

wheretheﬁxedgraph

⊂

M hasendpointsonIH denoted

∩

IH and the

ˆ

J s satisfy the

su(

2

)

algebra

[ ˆ

Ji

₍

_p

_),

ˆ

_Jj

₍

_p

₎

_]

₌

i j

k

ˆ

Jk

(

p

)

. Therefore, the bulk geometry induces conical singularities in the boundary torsion, which can be interpreted aspoint particles. It followsthat,inordertoremoveambiguityinthedeﬁnitionofthe boundary connection atthe location of thepoint particles, these havetobeblown-uptocircles

[21]

.Thesenewboundariesonthe horizontheninherit thespin- j irrepcarriedbythecorresponding bulklinkpiercingthehorizon.Inthisway,quantumIHstatescan thenberepresentedbyacollectionofsmallloops

i (i

=

1

,

. . . ,

N, N beingthetotalnumberofparticles)coloredwithSU

(

2

)

irreps ji, each surroundingonepunctureandconnectedbylinksforminga singleintertwiner4 _as_in

_{Fig. 1}

_.

With this regularization, the modified Gauss law (21) can be seen asa relationbetweentheflux ofthehorizonelectricfield

˜

e across a givencircle

i and the ﬂux of the bulk electric ﬁeld

acrossthesurface encircledby

i.The impositionofsuchrelation canthen beimplementedasin

[21]

by associatingan intertwiner

3 _As_explained_below,_the_restricted_set_of_boundary_observables_that_will_be

rel-evantfortheentropycalculationareformedonlybyloopsaroundeachpuncture whichdonotintersect eachothertogetherwithasetofholonomiesdeﬁnedon paths connecting each loopto asamesingle point.It isfor this secondsetof holonomiesthatoneshouldusetheKauffmanbrackettorepresenttheactionof the non-commutativeconnectioninamultiplicativeform.However,thephysical scalarproductoftheisolatedhorizontheorycanbedeﬁned(seebelow)suchthat thissetofholonomiesplaysnoeffectiveroleandtheboundaryobservablesbecome toallpurposescommutative.

4 _This_last_property_of_the_IH_states_is_a_consequence_of_a_global_constraint_that

followsfrom(14),namelythattheholonomyaroundacontractibleloopencircling allparticlesbetrivial.

(5)

Fig. 1. States of the quantum isolated horizon.

ι

i of SU

(

2

)

to each boundary

i; note however that, differently fromthecaseof

[21]

,thereisnofreemagneticnumberassociated toeach particlelink,sincethesearenowconnectedtotherestof thebulkgraph.Giventherestrictedstructureofthehorizonstates depictedin

Fig. 1

,each

ι

iisatrivialbivalentintertwiner.

Wenowanalyzetheimpositionofthecurvatureconstraint

(14)

. Weseethatawayfromtheparticles,the(non-commutative) con-nection Ai is ﬂat. Aroundeach particle,i.e. along each loop, the curvaturepicks up a contribution proportional to the ﬂux ofthe

iﬁeld.WesawinSection3thatthiscurvatureconstraintateach puncture can againbe expressed in terms ofthe ﬂatness condi-tionofanewnon-commutativeconnection

(23)

.Thisallowsusto usetheanalysisof

[18,24]

to deﬁnea projector intothephysical Hilbert space of the boundary theory. In fact, if we introduce a cellulardecomposition

IHofthehorizon2-sphereIH —with pla-quettes p

∈

IH of coordinatearea smaller or equalto

2 —the curvatureconstraintcanbewrittenas

C

[

N

] =

lim →0

p∈∪/ i tr

NpWp

(

A

)

+

lim →0

p∈∪i tr

NpWp

˜

A

=

0 (28) where Wp

=

1

+

2F

+

o

(

2

)

∈

SU

(

2

)

is the Wilson loop of the connection A, A in

˜

the spin-1/2 representation. It is immediate to see that the only non-vanishing contributions to the commu-tator of the constraint (28) with itself, when acting on a gauge invariant state, come from the commutator of anyof the terms p with itself, i.e. of the form

tr

NpWp

(

A

)

,

tr

MpWp

(

A

)

or

tr

NpWp

˜

A

,

tr

MpWp

˜

A

. In

[18]

,by meansof tech-niques developed in [17,25], it has been shown that such com-mutators are anomaly-free if and only if the inﬁnitesimal loop evaluatestothequantumdimension,namely

= (−)

2 j

_[

_{2 j}

₊

₁

_]

q

= (−)

2 j q2 j+1

−

q−(2 j+1) q

−

q−1

,

(29) wherenow q

=

⎧

⎨

⎩

e 2πiβ2 aIH κβ¯h 2

=

_e 2πi k β2 (1−β2)

_,

_{for p}

∈ ∪

_/

i e 2πiβ aIH κβ¯h 2

=

_e 2πi k β (1−β2₎

_,

_{for p}

_{∈ ∪}

i (30) asfollowsfromtheexpression (22)and(23).Therefore,the con-dition(29)implies that, ateach plaquette, therecoupling theory ofthe classical SU

(

2

)

group has to be replaced with the one of thequantumgroupUqSL

(

2

)

;however,whethertheplaquette sur-rounds a particle or not the deformation parameter q takes one ofthe two differentexpressions above. We thus have two quan-tum group recoupling theories entering the quantization of the curvatureconstrainton thehorizon. Thissuggeststhat, following theconstructionof

[18,24]

, thephysicalscalarproductforthe IH boundarytheorybetweentwohorizonstatess

,

scanbewritten as

s

,

s

phys

=

P

[

A

, ˜

A

]

s

,

s

,

(31) where P

[

A

, ˜

A

] =

lim →0

p∈∪/ i

δ(

Wp

(

A

))

p∈∪i

δ(

Wp

( ˜

A

))

=

lim →0

jp

p∈∪/ i

(

−)

2 j p

[

2 jp

+

1

]

q

χ

j p

(

Wp

(

A

))

×

p∈∪i

(

−)

2 j p

[

2 jp

+

1

]

q

χ

j p

(

Wp

( ˜

A

))

(32) istheprojector operatorintothephysicalHilbertspaceoftheIH boundarytheory.In thelast lineofthe expressionabove the de-formationparameterq ateachplaquette p takeseitheroneofthe two different values (30) according to the presence or not of a punctureinside p.

5. Entropy

In order to compute the microcanonical BH entropy we need to derive the dimension of the physical Hilbert space of the IH boundary theory and then take its logarithm, according to the standard relation S

=

log

(N )

, with

N

the number of horizon micro-statescompatiblewiththe givenmacroscopichorizonarea aIH.The quantity

N

can now be obtainedfrom the relation

be-tween the Chern–Simons partition function on a three manifold containingacollectionofunlinked,unknottedWilsonlinesandthe scalarproductbetweenstatesoftheassociatedHilbertspaceused byWitteninhisapproachtotheJonespolynomial

[26]

.

More precisely, given a three manifold M obtained from the connectedsumoftwothreemanifoldsM1 andM2 joinedalonga twosphereS2_and_containing_{N unlinked}_and_unknotted_circles_C

i withSU

(

2

)

irreps jiassociatedtothem,wedenotethe correspond-ingChern–Simonspartitionfunction(orFeynmanpathintegral)as Z

(

M

;

N_i₌₁Ci

)

;then Z

(

M

;

N

i=1 Ci

)

=

2

|

1

,

(33)

where

1 isthe vectordeterminedby theFeynmanpathintegral on M1 in the physical Hilbert space associated to the Riemann surface S2 and

2 the vector determined by the Feynman path integralonM2 inthedualHilbertspace;

·|·

indicatethephysical scalar product on this Hilbert space. The expression (33) corre-spond tothe unnormalized expectation value of thelink formed bythecollectionofcirclesCifromwhichJonesknotinvariantscan be derived [26].In thecase M

=

S2

_×

_S1_,_with _S1 _{corresponding} toacompacttimedirection,wehave

Z

(

S2

×

S1

;

N

i=1 Ci

)

=

dimHS2_;⊗ iji

,

(34)

wherether.h.s.correspondstothedimensionoftheHilbertspace onapuncturedtwosphere.TheReshetikhin–Turaev–Witten(RTW) invariant of a closed 3-manifold [27] provides a precise deﬁni-tion of the Chern–Simons path integral (33); at the same time, the Turaev–Viro (TV)invariant

[28]

,which representa state-sum modelfor3-dimensionalEuclideanquantumgravity withpositive cosmological constant

[29], has been shown to be related to the Witten’s Chern–Simons TQFT [26] by the theorem ZTV

(

M

)

=

|

ZWRT

(

M

)

|

2 [30]. From a LQG perspective, it has been shown in

[18] that the Turaev–Viro amplitudes can be recovered from the physical scalarproduct of the 2

+

1 theory with

>

0 using the same formalism introduced inthe previous section. Inparticular, with a proper relation between

andk (or aIH), the equivalent

(6)

ofthephysicalscalarproduct

(31)

,

(32)

providesanexplicit deﬁni-tionofther.h.s.of

(33)

,allowingustorecoverthelinkexpectation valuescomputedviatheChern–Simonspartitionfunction.

Therefore,wecan nowusetheseresultstogetherwiththe re-lation (34)to compute thenumber

N

of IHquantum states by meansofthephysicalscalarproduct

(31)

oftheboundaryHilbert space.Followingthislogic,wethushave

N

= <

P

∅,

>

=

lim →0

h dgh

i

χ

_ki

(

g_i

)

p∈∪/ i

jp

(

−)

2 j p

[

2 jp

+

1

]

q

×

χ

j p

(

Wp

(

A

))

×

p∈∪i

jp

(

−)

2 j p

[

2 jp

+

1

]

q

χ

j p

(

Wp

( ˜

A

))

(35) where g_i is the holonomyalong the loop going around the i-th particle in the IH state and dgh corresponds to the invariant

SU

(

2

)

-Haar measure. Inrelation tothe notation in (33), we have identiﬁed thestate

|

1

withthe isolated horizonstate depicted in Fig. 1 and

|

2

with the vacuum state; one can always ﬁnd a decompositionof M suchthat thisisthecaseandtheﬁnal re-sultisinsensitivetosuchchoice(thesameexpressionforther.h.s. of

(35)

would beobtained foranyother decomposition).We can graphicallyrepresentthephysicalscalarproduct

(35)

as

Inorder toproceed withthe evaluationofthe physicalscalar product, let us recall that, due to the (discrete) Bianchi identity, thereisa redundancyinthe intheproductofdeltadistributions entering theexpression oftheprojection operator andassociated totheplaquettesregulatingthetwospherehorizonsurface.A way to deal withsuch redundancy consists of eliminating the holon-omyWp

(

A

)

aroundanarbitraryplaquettep

∈ ∪

/

i.Bydoingso,it isimmediatetosee thatwhen weperform thegroup integration overtheedgesbelongingto p

∈ ∪

/

i theintertwinerstructure dis-appearsfromthescalarproduct(allthelinksconnectingtheloops

i are forcedintothe j

=

0 irrep). Thisishowthedisappearance oftheintertwinerstructurementioned abovetakesplaceandthe evaluationof (35)is considerablysimpliﬁed. As itis well known (see,e.g.,

[8]

), ignoring theintertwinerstructure affectsthe loga-rithmiccorrection totheentropy result,butdoesnot modifythe leading term. Thus, forthe purposes ofthispaper, such simpliﬁ-cationisirrelevantandourentropyresultshouldbecomparedto thelargek limitofthestandardcountingonecanﬁndinthe liter-ature.We,hence,endupwith

N

=

lim →0

h dgh

i

p

jp

(

−)

2 j p

[

2 jp

+

1

]

q

=

i

(

−)

2ki

[

2ki

+

1

]

q

=

i e2πiki

[

2ki

+

1

]

q

,

(36)

where,asshownin

[18]

,theq-boxintegrationhastobeperformed accordingtotherenormalizedskeinrelation

=

1

(

−)

2 j1

[

_{2 j}₁

+

₁

]

_q

δ

j1j2

Therefore,weseethatateachpuncture,besidestheusualterm givenbytheSU

(

2

)

irreduciblerepresentationdimensionassociated toit(inthelargek limit),a newdegeneracyfactorappearswhich reproducestheBekenstein’sholographicboundfor

β

=

i,namely exp

(

2

π

iki

)

=

exp

(

ai

/

4

2P

) ,

where ai

=

8

π

2

P

β

ki

.

(37)

The entropyresult

(36)

matches exactlytheone obtainedin

[19]

byexploitingthelocalCFTsymmetryintroducedateachpuncture byalsoblowingpointparticlestoinﬁnitesimal,butﬁniteloops.In both cases, such a regularization procedure plays a fundamental role.

Thepresenceofthenewdegeneracyfactor

(37)

hasbeen previ-ouslypostulatedin

[31]

inordertogetridofthequantumgravity correction totheentropyformulaassociatedtoachemical poten-tialtermfoundin

[32]

.A similaranalysisthenleadstotheentropy

S

=

2

π

i

i ki

+

o

(

√

aIH

)

=

aIH 4

2_P

+

o

(

√

aIH

) ,

(38)

inagreement withtheBekenstein–Hawkingformulaforan imag-inaryBIparameter.Ourresultprovidesyetanotherevidence,this time originatingentirelywithin theLQGformalism, insupportof the newperspective

[12,15]

intheLQGblackholeentropy calcu-lation allowing fortheremovalof thenumericalrestriction on

β

infavorofthephysicallybettermotivatedanalyticcontinuationto theAshtekarself-dualconnection.

Notice that, taking the limit

β

=

i, the deformation parame-ter (30) for plaquettes not containing any puncture reproduces the expression q

=

eπki _obtained _in _the _{Chern–Simons}

formula-tion of 2

+

1 gravity in presence ofa localpositive constant cur-vature (

>

0), in agreement withthedeformation parameter of thequantumgroupUqSL

(

2

)

enteringthedeﬁnitionoftheTuraev– Viro model that one would expect to appear due to the non-commutativity ofthe Ashtekar–Barberoconnectionon IH. Onthe other hand, forplaquettes around thepunctures the deformation parameter

(30)

becomesreal,againinagreementwiththeanalytic continuation from SU

(

2

)

to SL

(

2

,

C)

of the Verlinde formula for thedimensionoftheChern–SimonsHilbertspaceonapunctured 2-sphereperformedin

[12]

.

6. Alternativequantizationschemes

In thissection we wantto discusstherelationof thehorizon theoryanditsquantizationtocertainotherresultsintheliterature. Ontheonehand,thehorizonﬁelds

(

A

,

e

)

andtheboundary con-ditions

(14)

and

(15)

bearastrikingresemblancetotheﬁeldsand constraintsof3d

=

0 gravitycoupledtoparticlesinaﬁrstorder formulation.Ontheotherhand,in

[16]

itwassuggestedto imple-menttheboundaryconditionsonthehorizonﬁelddirectlyinthe LQG setting,thereby ignoring theboundary term ofthe presym-plecticstructure.Thehorizonﬁelds

(

A

,

e

)

seemtobeideallysuited forthisendeavor.Letusdiscussthesetwoperspectivesinturn.

(7)

6.1.Connectionto3dgravitywith

=

0

3d Euclidean gravity in ﬁrst order variables, has structure groupISU

(

2

)

.GeneratorsofthisLie-algebrawillbedenoted JI

,

Pi, i

=

1

,

2

,

3 with

[

Pi

,

Pj

] =

0

,

[

Pi

,

Jj

] =

i jkPk

,

[

Ji

,

Jj

] =

i jkJk

.

(39) Wewill closelyfollow

[24]

.The gravitationalphase spaceis em-beddedinthespaceofISU

(

2

)

connections

A

=

AiJi

+

eiPi (40)

equippedwithPoissonbracket

{

A_ai

(

x

) ,

e_bj

(

y

)

} =

ab

δ

i j

δ

(2)

(

x

,

y

) ,

(41) where e can be thought of asco-triad. Up to a prefactor thisis exactlythePoissonstructure comingfromthe boundary presym-plecticstructurein

(10)

.Couplingofparticlestogravityintroduces theﬁrstclassconstraints

ab

F

ab

(

0

)

= (

piJi

+

jiPi

)δ

2

(

x

−

x0

)

(42) withthe ISU

(

2

)

connection

F =

d(A)

A

and p

,

j are the particle dof.DecomposedintotranslationalandSU

(

2

)

components:

abFab

=

piJi

δ

2

(

x

−

x0

)

(43)

withtheSU

(

2

)

curvature F

=

d(A)A and

ab

(

dAe

)

ab

=

jiPi

δ

2

(

x

−

x0

) .

(44) ThePoissonbracket

(41)

canbequantizedinthestandardfashion on L2

(

A,

d

μ

AL

)

[24]. The particle dof obey some constraints on theirownandarequantizedontheHilbertspace

H

P

=

L2

(

SU

(

2

))

, fordetails see

[21]

.Whatis relevantforusare theconsequences forthegravitationaldof.

1. Theparticles transformundertheaction ofSU

(

2

)

.Constraint

(44) implies gauge invariance under the tensor product of gravitationalandparticleaction.Thismeansthatparticleand gravitationaldofhaveto becoupledby an intertwinertothe trivialrepresentation.

2. Constraint (43) determines the holonomy of loops: In the quantum theory, the connectionaround a loop istrivial ifit doesnotsurroundaparticle

hα

= I,

α

trivial (45)

andis givenbyan operator onthekinematicalHilbertspace oftheparticle,

hi

=

ie

miJ0

−1

i

,

(46)

wheremiisahalfintegerquantumnumberoftheparticle, J0 aﬁxedgeneratorofSU

(

2

)

and

iamultiplicationoperatoron theHilbertspaceoftheparticle.

Letuscompare thisto the quantum theory of the isolated hori-zon. To bring out the analogy, we proceed as in the 3d gravity case,byregardingtheboundaryconditions

(14)

,

(15)

constraintsto beimplementedlater.NotethatduetodAe

= β

K

∧

e therelation

(13)followsimmediatelyfrom

(15)

.Thus theconnection A would be commutative initially, as in the BF-formulation of 3d gravity. Hencethekinematicalquantizationofthegravitydofonthe hori-zonwouldbe thesame.Instead oftheparticleHilbert space, we wouldhaveapieceofthebulkHilbertspaceatthepuncture

Fig. 2. Operatorsofthebulkholonomy-ﬂuxalgebragiveoperatorsinthesurface theory.

H

p

=

j=0,1/2,1,...

H

j

,

(47)

whichwouldplaytheroleoftheparticleHilbertspace.

H

j isthe spin j irrepofSU

(

2

)

.Theconsequencesof

(14)

,

(15)

inthe quan-tumtheoryareasfollows.Awayfromthepunctures,thequantized versionof

(15)

wouldenforcegaugeinvarianceofthestatesonthe surface.Atthepuncture,itwouldenforcegaugeinvarianceofthe tensorproduct ofbulkandboundarystate,resulting in kinemati-calsurface statesofexactlythesamenatureasin3dgravitywith particles.We integrate (15)over thedisc D bounded bya loop

surroundingapuncture:

D h−_x01F

(

x

)

hx0d2x

=

c

D h−_x01

(

x

)

hx0d2x

.

(48)

Hereh areholonomiesfromsomeﬁxedpoint 0 on

tothepoint x and c is a constant.The quantization ofthe right-hand side is givenbyh−_p01LiJihp0,where L isanangularmomentumoperator on

H

p,anditcanbeshownthatthereisabasissuchthatLiJi is

λ

J0 forsomeﬁxedgenerator J0 ofSU

(

2

)

andsuitablenumbers

λ

, i.e.

D

h−_x01F

(

x

)

hx0d2x

=

c

λ

h−_p01J0hp0

.

(49) Ontheotherhand,expanding

(46)

uptoﬁrstorder,wehave

I +

D

h−_x01F

(

x

)

hx0d2x

= I +

m J0

−1

.

(50) Comparing

(49)

and

(50)

weseeastrongsimilarityinthequantum theory.Thissuggeststhattheboundarytheorymightalternatively bequantizedasEuclidean

=

0 3dgravitywithparticles.A quan-tization of the exponential of (14) has been suggested in [33], leadingtotheexpectationvalue

tr h

=

q(2 j+1)

−

q−(2 j+1)

q

−

q−1 (51)

for a givenquantum number j of the puncture (including j

=

0 forthe caseof aloop not enclosinganypuncture). Thisisof the sameformas

(29)

(uptothesignofq),however,q isnowgivenby q

=

eπki_,_with_the_level_{k from}

(24)

_._That_{q is}_different_is_no

contra-dictionto

(29)

,astheholonomiesbelongtodifferentconnections. Moreover,aswepointedoutattheendofSection5,oncewetake

β

=

i thedeformationparameter

(30)

reproducestheusual expres-sionaboveobtainedintheliterature.

Letusconcludewithanobservationonapossibledescriptionin termsofbulkoperators.In

[16]

itwassuggestedthattheboundary quantumtheorycouldbetakenasarestrictiontotheboundaryof the bulk quantumtheory.Indeed, thiscould be carriedout here. Holonomiesentirelyinthesurfacerepresenttheconnectiononthe surface, holonomiesthat endonthesurface contributedefects or particles, see Fig. 2. The ﬂux operators in the bulk, which have a transversal intersection with the surface, give, when restricted

(8)

to thisintersection, an operator that has the same commutation relations withholonomiesasthe two-dimensional ﬂux (25). This is remarkable, because they do correspond to different classical quantities. There is no contradiction, however, because the op-erators are the results of quantizing two different presymplectic spaces,thebulkandthesurface one,respectively.Constraint (14)

hasbeen investigatedin thiscontext

[33]

,but itis not yet clear whether solutionsareproperstatesontheholonomyﬂuxalgebra. The constraint(15)hasnot yetbeen investigatedinthiscontext. Presumably,itisagainlinkedtogaugeinvarianceonthehorizon.

7. Summary

As a ﬁrst step of our analysis, we have re-expressed the IH conservedpresymplecticforminterms ofﬁrstorder2

+

1gravity variables. That this is possible is very satisfying, as these vari-ables havean immediategeometric interpretation. Then we have quantized the boundary theory uniquely in terms of LQG tech-niques,withoutrelyingonstructuresoftheChern–Simonstheory onapunctured2-sphere.Wehaveshownhowthephysicalscalar product of canonical LQG can be used to compute the quantum IHstatedegeneracy,leading toan entropyinagreementwiththe Bekenstein–Hawkingformulafor

β

=

i.

Our analysis avoids several ambiguities present in the usual couplingofthebulkLQGandtheboundaryChern–SimonsHilbert spaces performed in previous literature, thus presenting a more coherent and sound picture of black hole entropy calculation in LQG, based on a nice interplay between the three and the four-dimensional theories.

We would like to point out again how a key ingredient for theentropyresult,namelythenewdegeneracyfactor

(37)

,is rep-resented by the punctures regularization via ﬁnite circles,which appearsnaturallywhenemployingLQGstructures.Itwouldbe in-terestingtostudythealgebraofobservablesthatonecoulddeﬁne living on these new boundaries, in analogy withthe analysisof

[19],andinvestigateifaVirasorostructuremightemergejustfrom withintheLQGformalism.

Finally,wenote thatthe horizonpunctures behaveverymuch likeparticles inthe2

+

1 quantumgravitydescribing thehorizon. Thisbeautifulpictureunderscoresagainthatthehorizon thermo-dynamicsisthethermodynamicsofthesepunctures.

References

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