Statistics, holography, and black hole entropy in loop quantum gravity

Amit GhoshSaha Institute of Nuclear Physics, 1/AF Bidhan Nagar, 700064 Kolkata, India

Karim NouiLaboratoire de Mathématique et Physique Théorique, 37200 Tours, France;

Fédération Denis Poisson Orléans-Tours, CNRS/UMR 6083;and Laboratoire APC–Astroparticule et Cosmologie, Paris 7,

75013 Paris, France

Alejandro PerezCentre de Physique Théorique, Aix Marseille Université, CNRS, CPT, UMR 7332,

13288 Marseille, France; Université de Toulon, CNRS, CPT, UMR 7332,83957 La Garde, France

(Received 23 December 2013; published 23 April 2014)

In loop quantum gravity the quantum states of a black hole horizon consist of pointlike discretequantum geometry excitations (or punctures) labeled by spin j. The excitations possibly carry otherinternal degrees of freedom, and the associated quantum states are eigenstates of the area A operator.The appropriately scaled area operator A=ð8πlÞ can also be interpreted as the physical Hamiltonianassociated with the quasilocal stationary observers located at a small distance l from the horizon. Thus,the local energy is entirely accounted for by the geometric operator A. Assuming that: Close to thehorizon the quantum state has a regular energy momentum tensor and hence the local temperaturemeasured by stationary observers is the Unruh temperature. Degeneracy of matter states is exponentialwith the area exp ðλA=l2

pÞ, which is supported by the well-established results of QFT in curvedspacetimes, which do not determine λ but assert an exponential behavior. The geometric excitations ofthe horizon (punctures) are indistinguishable. And finally that the semiclassical limit the area of theblack hole horizon is large in Planck units. It follows that: Up to quantum corrections, matter degrees offreedom saturate the holographic bound, viz., λ must be equal to 1

4. Up to quantum corrections, the

statistical black hole entropy coincides with Bekenstein-Hawking entropy S ¼ A=ð4l2pÞ. The number of

horizon punctures goes like N ∝ffiffiffiffiffiffiffiffiffiffiffiA=l2

p

q; i.e., the number of punctures N remains large in the

semiclassical limit. Fluctuations of the horizon area are small ΔA=A ∝ ðl2p=AÞ1=4, while fluctuations of

the area of an individual puncture are large (large spins dominate). A precise notion of local conformalinvariance of the thermal state is recovered in the A → ∞ limit where the near horizon geometrybecomes Rindler. We also show how the present model (constructed from loop quantum gravity)provides a regularization of (and gives a concrete meaning to) the formal Gibbons-Hawking Euclideanpath-integral treatment of the black hole system. These results offer a new scenario for semiclassicalconsistency of loop quantum gravity in the context of black hole physics, and suggest a concretedynamical mechanism for large spin domination leading simultaneously to semiclassicality andcontinuity.

DOI: 10.1103/PhysRevD.89.084069 PACS numbers: 04.60.Pp

I. INTRODUCTION

Recent developments in loop quantum gravity (LQG)have brought the problem of black hole physics back tothe central stage. Several pieces of evidence seem toindicate that a clear understanding of these emblematicsystems of quantum gravity is perhaps within the reach ofLQG. One important input for such new perspective is thequasilocal description of black hole mechanics in terms ofstationary observers at proper distance l from the blackhole horizon [1]. Such a treatment leads to an effective

notion of horizon energy (suitable for statistical mechani-cal considerations) in the form

E ¼ A8πl

: (1)

This has opened up the possibility of treating the statisticalmechanics of quantum isolated horizons [2] from the morenatural perspectives of the canonical and the grandcanonical ensembles [3]. Another important insight camefrom the application of spin foam tools combined with

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ideas of standard quantum field theory that has providedthe means of computing local temperature of quantumhorizons [4].A central input in the present work is the validity of a

weak form of holography for ‘matter’ fields1 close to thehorizon which is encoded in the assumption that thedegeneracy DðAÞ of matter states for a given horizon areaA behaves as

D ∝ expðλA=l2pÞ; (2)

where λ is an unspecified dimensionless constant, lp thePlanck length. The proportionality constant (or eventuallycorrections, as those considered in Appendix A) in (2)is unfixed and for simplicity we assume that D ¼expðλA=l2

pÞ in the paper. Several calculations in supportof such an assumption are available from studies ofquantum field theory in curved spacetimes, where thestatistical properties of matter degrees of freedom in asuitable vacuum state close to an event horizon are used inthe context of local stationary observers.2 The earliestmodel is the brick-wall paradigm [5,6]. It is followed bymany subsequent analysis attempting to explain the originof black hole entropy from entanglement of matter statesacross the horizon [7]. It is important to point out that,while these treatments are all physically well motivated,they only provide some qualitative answers because theresults are infected by UV divergences and other uncer-tainties (such as number of fields or species) are alsopresent (may be these are also related to the UV incom-pleteness of the treatments). The parameter λ is not fixedby semiclassical QFT considerations (see however, [8]).We will use this semiclassical result with an uncertaincoefficient λ and our analysis will show how theseambiguities disappear when quantum gravity effects aretaken into account.There are also some recent studies suggesting a relation-

ship between self-dual variables and holography [9,10]. Inthese work it is argued by means of an analytic continuationthat the dimension of the Chern-Simons Hilbert space—used

in modeling quantum isolated horizons [11,12]—growsexponentially with A=ð4l2

pÞ for large spin representations.A link between self-duality and thermal behavior atHawking temperature is also suggested by [13]. Even ifthese claims are quite striking in view of their potentialrelevance in the context of the present analysis, their validityneeds to be tested further as they rely on the existence of aquantum theory that is not completely under control atpresent, which is quantum gravity with complex variables.We will show that holography along with the assumption

that geometric excitations of the horizon are indistinguish-able particles leads by itself to a remarkable agreementbetween the ‘fundamental’ loop quantum gravity descrip-tion and the low energy semiclassical treatments: in view ofour analysis, holography becomes a necessary condition forsemiclassical physics.In this paper we explore the statistical mechanical

properties of a certain type of noninteractive system thatnaturally arises in the description of quantum black holehorizons in loop quantum gravity. The basic inputs comefrom the generic results of quantum gravity and quantumfield theories in curved spacetimes. The results show aremarkable consistency of the semiclassical black holephysics with the naive continuum limit of loop quantumgravity. The lessons of this exercise may be far reaching.We hope it could provide new insights into the (elusive)way low energy and semiclassical limit of loop quantumgravity are obtained.The paper is organized as follows. The next section is

devoted to the computations of the black hole partitionfunction in both the canonical and the grand canonicalensembles. We assume the indistinguishability of thepunctures introducing first a Gibbs factor before consid-ering the exact quantum statistics. This allows us toestablish an equation of state for the black hole, to recoverthe A=4l2

p behavior for the black entropy in the semi-classical regime, and to compute other thermodynamicalquantities as area or energy fluctuations or heat capacity.Furthermore, we show that, when combined with theholographic principle, indistinguishability is essential torecover the expected classical behavior of the black hole. InSec. III, we show how the expression of partition functionswe found provide a regularization and gives a concretemeaning to the Gibbons-Hawking Euclidean path integral.We conclude in the last section by a discussion and someperspectives.

II. IMPLICATIONS OF INDISTINGUISHABILITY

Perhaps one of the most fundamental lessons of quantumstatistics is the indistinguishable nature of particles. Thisnature is intrinsic to all elementary particles and severalnonelementary composites such as atoms, molecules, etc.However, the situation in quantum gravity, especially in thecontext of LQG, remains uncertain so far. In the context ofblack holes, it was argued that their effective fundamental

1In this paper, by matter we mean all degrees of freedom thatdo not explicitly contribute to the area spectrum of the black hole(BH) horizon.

2On a Schwarzschild background a simple calculation leadingto (2) goes as follows: for massless fields the energy density atthermal equilibrium is u ¼ π2l−6

p T4=15 (Stefan-Boltzmann lawin geometric units and k ¼ 1) while matter entropy density issm ¼ u=T ¼ π2l−6

p T3 (as photons have zero chemical potential).Using the Unruh temperature T ¼ TU ¼ l2

p=ð2πxÞwhere x is theproper distance of stationary observers from the horizon, we get

D ¼ exp

�π2

15l6p

ZT3dV

�¼ exp

�A

240πl2pð1þOðl2

p=l2ÞÞ�;

(3)

where dV ¼ r2 sinðθÞdθdϕdx for x ∈ ½lPl;l�.

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excitations are described by punctures located at thehorizon. Although these punctures are quantum mechanicalexcitations of geometry and their locations are uncertain,they are often treated as distinguishable in their statisticalanalysis. In this section, we are trying to investigate thepossibility of regarding these punctures as indistinguish-able quantum excitations of gravity. The case of distin-guishable punctures is briefly discussed in Appendix Bwhere we show how it is inconsistent with semiclassicallity.

A. Modeling indistinguishability with the Gibbs factor

There has been some discussions in the past about theissue of statistics of the punctures that define the quantumblack hole states [14–16]. In this section, we explore theconsequences of assuming that punctures are indistinguish-able excitations of the quantum geometry of the horizon. Inour analysis, we also consider as an additional assumption,the validity of a weak form of holography principle, whichwe will precisely define below. Before specifying a con-crete statistics for these excitations we will model indis-tinguishability of punctures by introducing the well-knownGibbs factor. In Sec. II B, we will do a more concreteanalysis by committing to fermionic and/or bosonicstatistics. The results remain qualitatively the same forboth statistics. Similar results are expected for anyonicstatistics as well.

1. Canonical partition function

We are going to study the statistical mechanical proper-ties of quantum isolated horizons (IHs). From the frame-work of LQG, it turns out that the statistical properties of aquantum IH are very well described as a gas of itstopological defects, henceforth called “punctures.” Using(1), we take the appropriately scaled IH area spectrum [17]to be the energy spectrum of the gas

Hjj1; j2 � � �i ¼�γl2

p

l

Xp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijpðjp þ 1Þ

q �jj1; j2 � � �i; (4)

where γ is the Immirzi parameter, jp ∈ N=2 is the spinassociated with the pth puncture. A quantum IH contains afinite number N of punctures because each puncture carriesa minimal area associated with the lowest value for spinj ¼ 1=2. In this paper we are going to restrict ourselves to alarge number of punctures for which a statistical descrip-tion is well suited.We are going to treat these punctures as indistinguishable

quantum particles. The appropriate way to implement thisis to make a choice of statistics for these punctures which iswhat we will do in the following section. For simplicity, wefirst implement their indistinguishability by simply intro-ducing the Gibbs factor N! in the partition function ofa system of distinguishable particles obeying Maxwell-Boltzmann statistics. When compared with the Bose or

Fermi statistics, this reproduces the correct behavior inthe high temperature limit. Then, the canonical partitionfunction with the Gibbs correction factor is given by

Q½N; β� ¼ 1

N!

Xfsjg

D½fsjg�N!Qjsj!

Yj

e−βsjEj ; (5)

where, following Eq. (4), Ej ¼ γl2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðjþ 1Þp

=l is theenergy of the jth puncture, sj is the number of puncturescarrying spin value j and D½fsjg� is the number of statesassociated with the additional (matter or nongeometric[18]) degrees of freedom for a given configuration fsjg.The degeneracy factorD½fsjg� is an essential ingredient forsuccessfully implementing the quantum statistics for punc-tures. Before arguing why such a degeneracy factor isnecessarily present in LQG, we first show its effects in thestatistical mechanical properties of the system.For that purpose, we are going to assume (this is the only

assumption we will make) that the degeneracy D½fsjg� ismaximal in the holographic sense, namely it grows expo-nentially with A=l2

p. As already mentioned in the intro-duction, both the brick-wall paradigm [5,6] and all otherentanglement entropy calculations [7] imply that qualita-tively this must be the case. However, due to regularizationambiguities associated with UV divergencies one cannotfix the proportionality coefficient in front of A=l2

p in theseapproaches. Consequently, we shall leave this coefficientarbitrary and prove a posteriory that it is equal to 1=4 up toquantum corrections (this is one of the predictions of ouranalysis). Accordingly, we write the degeneracy as

D½fsjg� ¼ exp

�ð1 − δhÞ

A4l2

p

�¼Yj

expð1 − δhÞajsj

4l2p

;

(6)

where δh is a free parameter and aj ¼ 8πγl2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðjþ 1Þp

isthe area eigenvalue associated with a single spin j (seeAppendix A for the case where there are power lawcorrections to (6). As we shall see, (6) implies that thesystem is dominated by large spins. Essentially this meansthat the area spectrum linearizes,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðjþ 1Þ

p¼ jþ 1

2þOð1=jÞ; (7)

and Oð1=jÞ terms can be neglected for our purposes.Moreover, none of the results obtained in this paper woulddepend on the details of the area spectrum. In fact, as wewill see later, even in models where the area spectrum iscontinuous (such as in [19]), the conclusions of this paperwill remain valid. Using (6), Eq. (5) becomes

Q½N; β� ¼ 1

N!

Xfsjg

N!Yj

1

sj!e−ðβ−βUþδhβUÞsjEj ; (8)

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where βU ¼ 2πl=l2p is the inverse Unruh temperature,

which represents the local temperature measured by sta-tionary observers at proper distance l from the horizon. Sothe partition function takes the standard old form (exceptfor the additional Gibbs factor) without the degeneracy butwith a new effective inverse temperature β − βU þ δhβU.In [3] we have studied the statistical mechanical proper-

ties of the Boltzmann gas at the Unruh temperature. Fromthese studies we already know that at the semiclassicallimit, β is close to βU; so we can introduce a new smalldimensionless parameter δβ such that β ¼ βUð1þ δβÞwhere δβ must vanish in the limit lp → 0

3. As a result,the effective inverse temperature becomes

~β ¼ ðδβ þ δhÞβU: (9)

It will be convenient to use the notation δ≡ δβ þ δh. Wewill see below that, with the new ingredient of holographyand for macroscopic black holes (A=l2

p ≫ 1), the gas ofindistinguishable punctures attains an equilibrium close tothe Unruh temperature if and only if δ ¼ δβ þ δh ≪ 1; i.e.,δ is a small quantum correction.Since ~β ¼ δβU and δ becomes a multiplicative factor in

the area spectrum in (8), as we take δ → 0 we must havej → ∞ so that the combination δj remains finite. Thisjustifies why the area spectrum becomes effectively linearfor large black holes.With these inputs the partition function becomes simply

Q ¼ qN

N!; (10)

with q the partition function for a single puncture

q ¼X∞j¼1=2

expð−2πγδjÞ ¼ 1

expðπγδÞ − 1; (11)

which blows up as δ → 0, showing that there is no smoothlimit when δ approaches 0. This divergence will be crucialin what follows.

2. The grand canonical partition function

In the regime we consider here, we do not expect thenumber of punctures to be strictly conserved. Hence, it isbest to use the grand canonical ensemble. This will alsobecome obvious when we introduce suitable statistics forthe punctures in the next section.It is clear that in the limit of large j, when the LQG area

spectrum becomes linear, a single puncture can split intotwo punctures without changing the area. This means that,in the large area limit, an arbitrary number of punctures can

be created or destroyed for a given area. Since the numberof punctures cannot be conserved, the chemical potentialmust vanish. The situation is very much analogous to asystem of photons where the photon number is notconserved (in that case a system with a given energycan contain an arbitrary number of soft photons). Thisstatement is strictly true in the regime of linear spectrumand this is precisely the case we should consider when thetemperature is close to the Unruh temperature. For thatreason we set the fugacity z ¼ 1 in what follows.Thus the grand canonical partition function is given by

Z½β� ¼XN

qN

N!¼ expðqÞ: (12)

The mean energy U at the inverse temperature β ¼ βUð1þδβÞ is

U ¼ A8πl

¼ −∂β logðZÞ

¼ πγTUexp ðπγδÞ

½expðπγδÞ − 1�2 ; (13)

where δ ¼ δβ þ δh. This gives an equation of state relatingthe area (equivalently the energy) with the parameter δ,

A½δ�4l2

p¼ πγ expðπγδÞ

½expðπγδÞ − 1�2 : (14)

The equation of state gives a one-to-one relation betweenthe area and δ, which shows that for large black holesA=l2

p ≫ 1 we must consider δ ≪ 1. In fact, for a givenlarge area the equation of state can be solved for δ, giving

δ ¼ 1

πγln

1þ 2πγ

l2p

Aþ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπγl2

p

A

�πγl2

p

Aþ 1

�s !; (15)

which simplifies when A ≫ l2p according to

δ ¼ δβ þ δh ¼ffiffiffiffiffiffiffiffi4l2

p

πγA

sþO

�l2p

A

�: (16)

Although the above equation shows that only the combi-nation δ ¼ δh þ δβ is small in the large area limit and alsoin the classical limit ℏ → 0, we can also infer that in theselimits δh ≪ 1 because we already know that δβ is a smallquantum correction which also vanishes in these limits. Inother words, up to corrections that vanish when lp → 0,nongeometric degrees of freedom responsible for thedegeneracy of the LQG area spectrum saturate the holo-graphic bound in the sense of (6). This is a nontrivialprediction of LQG because the standard QFT calculationsare always infected by ultraviolet divergences and also by

3Such quantum corrections to Unruh temperature have beensuggested in [13].

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the growing number of matter species which together raisedoubts about the holographic bound. However, in LQG adetailed balance between the matter sector and geometryseems to play an important role so that neither the ultra-violet divergences nor the species problem appear tothreaten the holographic principle. Notice that we didnot impose the parameter δh to be small to begin with—that it turned out to be so is a definite LQG prediction.Now we can compute the entropy S ¼ βU þ logZ

which becomes4

SG½A� ¼A4l2

p

"1þ δβ þ 2

ffiffiffiffiffiffiffiffil2p

πγA

sþO

�l2p

A

��: (18)

One can also compute the area or energy fluctuations,ðΔUÞ2 ¼ −∂βU, which gives

ΔUU

¼ ΔAA

¼ exp

�−πγδ

2

�½expð2πγδÞ − 1�1=2

¼ffiffiffiffiffiffiffiffiffiffi2πγδ

pþOðδÞ: (19)

Therefore, these fluctuations are small for large black holes,more precisely ΔA

A ≃ αðl2p=AÞ1=4 where α is a numerical

multiplicative factor. We can also calculate the specificheat. From (14),

C ¼ −β2∂βU ¼ 2

πγδ3ð1þOðδÞÞ; (20)

which is positive and large. Note that we have always keptl fixed, be it while deriving the formula (1) in [1] or in allour subsequent calculations. Therefore, in contrast to othertreatments the quasilocal formulation of black hole thermo-dynamics is thermally stable. This can be compared withother stability conditions, like putting a box around thehorizon. We think that a fixed l is serving a similar purposein our case, while the specific heat C ¼ ΔE=ΔT → ∞because ΔT → 0 for fixed l.Another important thing to note is that the physical

quantities other than the quasilocal energy (1), such asentropy, energy fluctuations, and specific heat do notdepend on the length scale l of the local observers. Thisshows that l plays only a fiducial role in our treatment. Inthe expression of the energy, l plays a somewhat similar

role to that of a volume V for standard systems—l appearsin the energy spectrum just like V does. However, incontrast to standard thermodynamic systems, the scale ldoes not appear in the expression of the entropy, specificheat, and area fluctuations—we believe that this is areflection of the underlying background independence ofgravity.In the next section we will treat the system of

indistinguishable punctures by introducing some suitablestatistics. However, the expressions for the entropy andtemperature obtained here are not expected to changesubstantially as far as the leading order terms are con-cerned. Usually, for nongravitational systems differentstatistics give similar results in the limit of small β(high-temperature regime) because different energy levelsbecome equally likely and the precise forms of distributionsbecome somewhat irrelevant. However, in the present case,and as far as the statistics is concerned, the role of inversetemperature is played by ~β ¼ β − βU þ δβU. Therefore,high temperature behavior is achieved in the limit of small~β (which close to βU is equivalent to small δ). Thus, allenergy levels become equally likely when the physicaltemperature is close to the Unruh temperature. This occursbecause of holography: the exponential growth of degen-eracy is compensated by the exponential decay of theBoltzmann factor close to the Unruh temperature.Therefore, a naive analogy with nongravitational systems(as far as the leading order terms in physical quantities areconcerned) shows that the gas is as if at an infinitetemperature. This is why the use of exact quantum statisticsis not expected to affect the leading order terms and, as weshall see, it will not modify the qualitative behavior of thesubleading corrections either.

B. Quantum statistics

In the previous section we used the Gibbs factor tocorrect the Boltzmann distribution in order to deal with thegas of indistinguishable punctures. This is an inexpensiveshort-cut-method for dealing with the problem. A precisetreatment should instead use the actual quantum statistics,which is either bosonic or fermionic. From the frameworkof LQG, as it is not quite obvious what the actual statisticsof these punctures would be, we allow for both possibilitiesand investigate the cases when the punctures are bosonicand fermionic. In fact more complex possibilities may arisebecause the punctures lie on a two-dimensional surface. Wewould also like to discuss some of these other possibilitiesin this section.

1. Bosonic/fermionic punctures

We assume that all punctures are either bosons orfermions. Another natural choice is that punctures carryinghalf-integer spins are fermions while punctures carryinginteger spins are bosons. But this case can be workedout from the formulas given here. The generic formulas

4If one puts a nonvanishing fugacity in (12) then logZ ¼ zq ¼z∂z logZ ¼ N. The entropy formula in the grand canonicalensemble S ¼ βU − N log zþ logZ becomes

S ¼ βU þ ð1 − log zÞN: (17)

The choice of fugacity z ¼ e then kills theffiffiffiffiffiffiffiffiffiffiffiA=l2

p

qcorrections in

(18) which entirely comes from the term proportional to N. Thephysical relevance of this observation will be studied elsewhere.

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obtained in the case of either purely bosons or purelyfermions are also valid for the latter case.The canonical partition function takes the form

QN ¼Xfsjg

e−P

j~β sjEj ; (21)

where ~β ¼ β − βU þ δhβU andP

sj ¼ N, as in theBoltzmann case. The grand canonical partition functionat β ¼ βUð1þ δβÞ is

Z ¼XN

QN ¼Yj

½1� expð−2πγδjÞ��1; (22)

where δ ¼ δh þ δβ and the � signs describe fermions andbosons respectively. Again, contributions from the linear-ized spectrum dominates in the limit δ → 0. Equivalently,for fermions and bosons respectively

logZ ¼ �X∞j¼1=2

logð1� e−2πγδjÞ: (23)

The mean energies for fermions and bosons are

U ¼ A8πl

¼ −∂β logZ

¼ γl2p

l

X∞j¼1=2

jexpð2πγδjÞ � 1

¼ γl2p

2l

X∞n¼1

nexpðπγδnÞ � 1

: (24)

where, in the last sum, the sum runs over integers and nothalf-integers. In the limit of small δ (or equivalently in thelimit of large area A) the sum can be interpreted as aRiemann sum and therefore, it can be approximated by anintegral according to

U ¼ γl2p

2lδ2δX∞n¼1

nδexpðπγδnÞ � 1

≃ γl2p

2lδ2

Z∞

0

xdxeπγx � 1

:

(25)

Changing the integration variable x ↦ x=ðπδÞ in theprevious integral, one sees immediately that all the relevantparameters can be scaled out and the area reduces to thefollowing simple expression:

A ¼ 4l2p

πγδ2

Z∞

0

xdxex � 1

¼ ϵ�πl2p

3γδ2(26)

where ϵ� ¼ 1; 2 for fermions and bosons respectively.Thus, for a fixed large area A we get the equation of stateδ ¼ lp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπϵ�=3γA

p.

To compute the entropy, one can proceed in a verysimilar way. One starts by approximating (23) with anintegral,

logZ ≃� 1

πγδ

Z∞

0

logð1� e−xÞdx ¼ ϵ�π12γδ

: (27)

Thus, the entropy S ¼ βU þ logZ is given by

S½A� ¼ A4l2

p

"1þ δβ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiπϵ�l2

p

3γA

sþO

�l2p

A

�#; (28)

where again δβ in the previous formula appears due to thefact that the equation of state only determines the value thecombination δ ¼ δβ þ δh.It is also possible that punctures labeled by half-integer

spins behave like fermions and punctures labeled by integerspins behave like bosons. A simple adjustment of equationsin the previous calculations would lead to a new value ofδ and entropy in that case. However, as expected the resultis qualitatively the same and no special feature occurs.So from the entropy it seems unlikely that one can settle theissue of statistics.For the case z ¼ 1, although there is no thermodynam-

ical meaning to the average number of punctures, it ismeaningful to calculate the average number of puncturescarrying spin-j, that is hsji. It has the standard form,

hsji ¼ ½expð2πγδjÞ � 1�−1; (29)

for Fermi and Bose cases, respectively. We can also viewhsji as a probability of occurring the spin value j in thesense that the larger the value of hsji is, the more abundantthe corresponding spin value j is. Thus we can introducethe probability distribution,

pj ¼hsjiPkhski

; (30)

where we omitted the index � in the notation p�j for

purposes of simplicity. Using the previous definition onecan compute the mean value of the spin contributing to thesurface states hji� ¼Pjjpj in the two different statistics.The mean value is a priori a complicated function but itsbehavior for small δ can be expressed easily using thefollowing results,

Xj

hsji ¼X∞n¼1

1

expðπγδnÞ � 1≈

1

πγδ

Z∞

πγδ

dxex � 1

(31)

≈

8<:

logð2Þπγδ ðfor Bose statisticsÞ

logðδÞπγδ ðfor Fermi statisticsÞ

; (32)

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where here the symbol ≈ means that we consider only theleading order term when δ is small. Furthermore, it isstraightforward to compute the behavior of hjpji for smallδ and to show that it is given by the formula,

hjpji ≈ϵ�

24ðγδÞ2 : (33)

As a consequence, the mean value hji� for Bose and Fermistatistics behaves as follows when δ is small:

hji− ≈ϵ−π

24γ logðδÞ1

δ∝

ffiffiffiffiffiffiffiffiffiffiffiA=l2

p

qlogðA=l2

pÞ(34)

hjiþ ≈ϵþπ

24γ logð2Þ1

δ∝

ffiffiffiffiffiffiffiffiffiffiffiA=l2

p

q: (35)

A similar calculation allows us to compute the fluctuationsΔj� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihj2i� − hji2�

pand to obtain their behavior for

small δ which are given by

Δj− ≈DþffiffiffiffiffiffiffiffiffiffiffiA=l2

p

q; Δjþ ≈D−

ffiffiffiffiffiffiffiffiffiffiffiA=l2

p

q; (36)

where D� are constants.Basically, as the area grows the distribution (30)

becomes flatter more slowly in the bosonic case than inthe case of fermions. That is the why the growth of hji witharea in the bosonic case is slower than in the case offermions. We also see that although the distribution (30) tellus that individually small spins are more likely, the meanvalue of the spin is large for large area. This is consistentwith the viewpoint that the true classical limit is to be takenfor hji → ∞, not some individual spin eigenvalue j → ∞.In other words, although the distribution (30) peaks atsmaller spin values, larger spin values contribute in adominant way—this fact is reflected in the large fluctua-tions Δj� around the mean value for large areas.

2. Anyonic statistics

An interesting question that often appears in the descrip-tion of quantum isolated horizons is whether the correctstatistics for punctures should be anyonic. This comes fromthe fact that the horizon degrees of freedom in 2þ 1dimensions are described by a Chern-Simons theory inwhich the braiding of sources occurs.5 As a result a newform of statistics appears, called anyonic statistics. Suchnontrivial statistics can be heuristically thought of as aresult of a (nonlocal) quantum interaction. Thus, it is hardto make analytic computations with anyons. Nevertheless,standard cluster expansion techniques (for standard short

range interactions these are excellent approximations in thehigh temperature and low density regime) have beenapplied to compute up to the second virial coefficients.These computations show explicitly that such coefficientsinterpolate between those for Bosons and Fermionsdepending on the value of a suitable deformation parameter.For black hole models such deformation parameter is thelevel of the Chern-Simons theory which grows with thehorizon area in Planck units. For macroscopic black holesone is really in the large level regime where anyons shouldbe well approximated by standard statistics. These pointssuggest that the qualitative structure of the results presentedhere should be preserved even in the case of anyonicstatistics.Notice, however, that it is not clear if anyons are

physically relevant for black hole physics in the contextof loop quantum gravity. The reason is that in LQG theeffective 2þ 1- dimensional quantum description of thehorizon should really be rooted in a 3þ 1 description.Therefore, it is by no means clear why braiding ofpunctures should play a fundamental role when in LQGwhen the notion of braiding is trivial from a four-dimensional perspective.

III. SEMICLASSICAL CONSISTENCY: GENERALRELATIVITY EMERGING FROM LQG

Our expression for the grand canonical partition functionin quantum statistics,

Z½β� ¼XN

Xfsjg

e−P

jðβ−βUþδβUÞsjEj

≈XN

Xfsjg

e−ðβ−βUÞ

Pjsj

aj

8πl2pl; (37)

where in the last line we have neglected the quantumcorrection δ in (9). In such regime of small δ (orequivalently to large horizon area) we may expect thepartition function (37) to arise from some semiclassicalconsiderations also. In the following we will see that this isexactly what happens.To explicitly see this, we first view the expression (37)

as a regularized version of some Euclidean functionalintegral where the partition function is obtained byintegrating over Euclidean metrics gð4Þ with some appro-priate weights that depend on the metric gð4Þ. Since in (37)we are summing over all punctures and spin configura-tions associated with a two-surface (the horizon), it isnatural to interpret the sum as a discretization of thecontinuum integral over the Euclidean metrics of thetwo-surface gð2Þ with the weight factor

Psjaj ¼ A as

area A½gð2Þ� of the two-surface associated with themetric gð2Þ

5See [20] for an argument leading to a nontrivial statistics ofpunctures within the LQG framework.

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Z½β� ¼Z

Dgð2Þ exp�−ðβ − 2πlÞ A½g

ð2Þ�8πl2

pl

�

≈ exp

�−ðβ − 2πlÞ A½g

ð2Þ�8πl2

pl

�: (38)

In fact, (37) provides a LQG definition for (38). Now itis well known that the expression (38) can be obtained froma functional integral over a Euclidean four-metric gð4Þ inthe leading semiclassical approximation (for a fixed back-ground metric gð4Þ0 )—this was explicitly shown in [21]for the case where one gives boundary conditions at infinity.In all generality the expression of the partition function is

Z½β� ¼Z

Dgð4Þβ exp

�−

1

16πl2p

ZMR½gð4Þ�

−1

8πl2p

Z∂M

ðK − K0Þ�; (39)

where the index β in gð4Þβ reminds us that in the semiclassicalEuclidean prescription one considers metrics with a conicaldeficit angle 2π − β=l at the horizon, R denotes the Ricciscalar, K is the extrinsic curvature at the boundary ∂M andK0 is a standard counter-term that is to be subtracted forconsistency.The expression (38) follows from (39) when one

evaluates the functional integral to a spacetime regioncontained between a stationary black hole horizon and theworld-sheet of stationary observers at proper distance lused in [1]. Then K is the extrinsic curvature of theboundary ∂M and the counter term K0 ¼ l−1 so that theresult is trivial in the A → ∞ limit where the spacetimeregion is isomorphic to (Euclidean) Rindler spacetime (seeFigure 1; in fact K0 is the value of the extrinsic curvature onthe boundary of that region in the Euclidean continuation).Under such conditions the boundary integral in (39) fallsoff like l3=ð ffiffiffiffi

Ap Þ and can be neglected. Therefore, (38)

entirely comes from the curvature term and the conicalsingularity at the horizon. For more details of the deriva-tion, see [22].Note that (39) does not depend on the detailed definition

of the path integral. It rather follows from the semiclassicalapproximation. Since (as argued before) in our case largespins dominate (analogue of the semiclassical approxima-tion), we expect that the semiclassical approximation of thespin foam transition amplitudes [23,24] would provide arigorous implementation of this last formal step. This isrelated to the ideas put forward in [4] when exploring thedynamics of a single plaquette (puncture).Nevertheless, this analysis shows that the holographic

behavior (6) along with the large area approximation isnecessary for the emergence of general relativity from LQGin the context of black holes. A similar point was alsodiscussed in [10]. It provides a semiclassical consistency ofLQG at least in this restricted context. Also note that sincethe statistical physics of a black hole (as described by a gasof punctures) is (in the A=l2

p → ∞ limit) the same asstatistical physics in flat spacetime as seen by Rindlerobservers, the arguments presented here apply to moregeneral cases than just black holes and thus provides newhints as to how to recover standard QFTs in flat spacephysics in the LQG formalism. Our study shows therelevance of (6) in LQG and its close connection to thesemiclassical consistency of LQG.

IV. DISCUSSION

We have shown that the naive introduction of holo-graphic degeneracy of matter states associated with aquantum horizon leads to a whole body of interestingphenomenology. This strongly suggests this degeneracy isa necessary ingredient for the continuum and semiclassicallimit of LQG. There are several independent quantum fieldtheoretic scenarios from which such holographic degener-acies (6) can be inferred. However, various issues attachedto this holography indicate that a complete and satisfactoryanalysis of degeneracy can only be made in a full quantumtheory of gravity. One famous example is the brick-wallmodel of ’t Hooft [5]. The computations of black holeentropy based on matter entanglement are closely related tothis example and produce results that are qualitatively same[7]. One limitation of these analysis is that the computa-tions are semiclassical (QFT on a classical backgroundspacetime) and thus the regularization ambiguities precludethe precise calculation of the proportionality constant infront of A=l2

p in (6). Moreover, the answer are typicallyproportional to the number of matter fields considered; thisis referred to as the “species problem.” The holographicdegeneracy (6) in all these cases is associated with matter ornongeometric degrees of freedom close to the horizon.Another example (which from a physical viewpoint remainsunclear at the moment) is the holographic degeneracy [10]found by analytic continuation from real to complex

FIG. 1. In the infinite area limit, one recovers flat spacetime inthe local formulation. The path integral is defined in the portionof spacetime between the horizon and the world sheet ofstationary observers at distance l.

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Ashtekar variables. The validity of such an analytic con-tinuation depends on the possibility of defining the quantumtheory in terms of complex self-dual variables which at thisstage remains an open issue.Our analysis shows that thermal equilibrium at Hawking

temperature (Unruh temperature in our local analysis)implies that as soon as quantum geometry effects arebrought in (such as the area spectrum), degeneracy ofthe area spectrum due to nongeometric degrees of freedom(that for simplicity are here referred to as matter and includeany other excitation that does not affect the horizonarea eigenvalues) grows exponentially saturating the holo-graphic bound up to quantum gravity corrections. Ourtreatment removes the two central problems mentionedabove—one, the UV divergences of QFTs (because of theunderlying discreteness of LQG) and two, the ambiguitiesof the proportionality factor, also known as the speciesproblem. These results are obtained here in an indirect waythrough the inputs (2). However, the key ingredient thatproduces the UV regularization is the area gap, and the factthat quanta of the area operator are pointlike (hence,discrete) excitations. For instance, our conclusions alsoapply for an area spectrum as the one obtained in [10](where the factor

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðjþ 1Þp

in the area spectrum is

replaced byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ 1=4

pwhere s ∈ R is a continuous label).

Notice that according to our results Unruh temperaturefor the local observers (Hawking temperature for stationaryobservers at infinity) is analogous to a Hagedorn temper-ature in quantum gravity [25]. We hope that a detailedanalysis of the nongeometric degrees of freedom in theframework of LQG would reproduce this result. We expectthis to arise from the dynamical restrictions on nongeo-metric degrees of freedom imposed by the Hamiltonianconstraint in combination with the requirement that thesuitable quantum gravity physical state describing the BHsystem approximates the near horizon geometry of astationary black hole in the semiclassical regime.

A. Indistinguishability or not?

In this work we have strongly used indistinguishabilityof LQG excitations of the horizon geometry, i.e., punctures.We strongly believe that this is the correct way to treatpunctures at the horizon. We base our assumption on one ofdeepest implications of quantum mechanics about thenature of particles. In a relativistic quantum field theory,particles are excitations of an underlying field (like waveson the surface of a lake). Such excitations by their verynature are indistinguishable. Then consistency (in thestandard quantum field theoretic context) requires thatthese excitations are either fermionic of bosonic (or moreexotic possibility like anyonic in lower dimensions). Butindistinguishability is a more fundamental issue. From theanalysis of the present paper we see that semiclassicalconsistency in the context of black hole physics implies the

necessity of indistinguishability in a strong way (see alsoAppendix B).This seems to contradict some results of the past where

punctures were taken to be distinguishable [26]. We believethat even though the statements made in [26] are correctin their context, their validity is only a peculiarity of theeffective models where such results were derived.Moreover, in such models where no holographic feature(6) is present among other things because matter degrees offreedom are completely ignored, BH entropy will not growlinearly with A=l2

p if punctures are taken to be indistin-guishable [15,16]. It is only when one includes the effect ofmatter through (6) that indistinguishability leads to thecorrect leading-order entropy and the correct leading ordertemperature.

B. Matter vs geometry

For quite some time there has been a tension in the field asto what could be the correct source for the huge number ofdegrees of freedom that lead to black hole entropy fornonextremal black holes. Some argue that it is of a purelygeometric origin, while others argue that it entirely comesfrom matter entanglement close to the horizon. Examples ofthe first are either the heuristic derivations based on theEuclidean path integral approaches or the more precise butolder entropy calculations in LQG that completely neglectmatter fields. Examples of the second type are the brick wallmodels and entanglement entropy calculations. In thescenario presented here, both sources contribute. The geo-metric component coming from the quantization of the areain LQG and the nongeometric component that produces thedegeneracy (6) of the area spectrum. In this paper the twoaspects are seamlessly combined which produces results thatare compatible with the semiclassical regime.

C. Continuum and semiclassical limit of LQG

From a broader perspective the results presented heremight also clarify the difficult issue of how one is expectedto recover the continuum and/or semiclassical limit of LQGfrom a fundamentally combinatorial structure of the theory.In this respect also there have been different views. On onehand, there is an idea that the continuum limit of LQG is tobe obtained through physical states constructed out of thevery fine grained structures, i.e., spin network graphs withmany edges and nodes per unit volume. According to thisview the geometric quantum numbers (the spin labels)coloring spin networks seem to be small. On the other hand,investigations in the context of spin foams [27] indicate thatthe basic building blocks of quantum geometry admit aclassical geometric interpretation when colored by largespins. Moreover, it is in the large spin limit where spin foamamplitudes can be related to general relativity (via a Reggeregularization) in the semiclassical regime [23,24]. Fromthe present analysis of Sec. II B 1—Eqs. (34) and (35)—wefind

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hNi− ∝ffiffiffiffiffiffiffiffiffiffiffiA=l2

p

qlogðA=l2

pÞ; hNiþ ∝ffiffiffiffiffiffiffiffiffiffiffiA=l2

p

q(40)

for Bose and Fermi statistics, respectively. These numbersas well as (34) and (35) are all large in the macroscopicregime and thus, the tension evoked above disappears.Thus, the system is dynamically driven to the configura-tions one would expect from naive semiclassical andcontinuum limit considerations. Notice that the thermalstate of the system respects the hierarchy of scales,

l2p ≪ a ¼ l2

phji ≪ A; (41)

that is found to correspond to the semiclassical and lowenergy limit in [28] and coincides with the analog criterionfound in the context of spin foams [23]. Perhaps notsurprisingly, this is complemented by the clear-cut linkwith the semiclassical Euclidean semiclassical formulationas described in Sec. III.

D. IR vs UV Newtonian constants

For simplicity of presentations here we have not explic-itly considered the possibility that the value of Newton’sconstant G appearing in the fundamental (or UV) areaspectrum in units of lp could be different from its lowenergy or large scale value. As in any quantum field theory,G is expected to flow with a suitably defined phenomeno-logical scale. Even though the explicit way such a flowwould occur cannot be addressed in the framework of thepresent calculation, one can explore the possible modifi-cation of our results in the case where we assume thatGUV ≠ G, where GUV and G are the fundamental high-energy and low-energy values of the gravitational con-stants, respectively. This would translate into two differentvalues of the Planck lengths lUV

p ≠ lp. In particular,lUVp ¼ ffiffiffiffiffiffiffiffiffiffiffiffi

ℏGUVp

will appear in the fundamental area spec-trum while G will appear in the form of the effectiveHamiltonian. Explicitly, Eq. (4) becomes

Hjj1; j2 � � �i ¼�ℏγlGUV

G

Xp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijpðjp þ 1Þ

q �jj1; j2 � � �i:

(42)

Notice that the change with respect to (4) can be hidden bythe shifting the Immirzi parameter γ → γG=GUV. By justlooking at the dependence of our results on the Immirziparameter one concludes that the flow of Newtons constantdoes not affect the leading order result of our calculations.In particular, the low energy Newton’s constant G appearsin the leading term of the black hole entropy and onegenuinely recovers the approapriate form of Bekentein-Hawking entropy formula suggested by the classicalfirst law.

E. Conformal invariance at Unruh temperature

Notice that in the limiting case A=l2p → ∞, δ → 0 and

the spacetime close to the horizon becomes flat (this is theRindler limit, the limit of zero curvature). The probabilitydistribution (30) becomes independent of the value of j.This is like a local scale invariance, local because the abovestatement holds at each puncture independently and scaleindependent because the spacetime close to the horizonpossesses no intrinsic scale. Is this the origin of anunderlying conformal invariance advocated in someapproaches [29,30]? At first sight this seems to be differentbecause the conformal invariance that emerges in suchtreatments is associated with the r − t plane, while here thescale invariance appearing in our context is on the two-surface of the horizon. Nevertheless, the mechanism thatmakes the results of these treatments generic might be alsorelevant here for similar geometric reasons. This is animportant issue that is to be clarified by further analysis.

ACKNOWLEDGMENTS

We are grateful to E. Frodden, M. Geiller, M. Han, andD. Pranzetti for exchanges and discussions contributing tothis work. A. P. thanks the quantum gravity group atMarseille for input and discussions during an informalpresentation of this work, especially C. Rovelli for dis-cussions and support.

APPENDIX A: LOGARITHMIC CORRECTIONS

In (6) we have assumed only the leading asymptoticgrowth of the density of matter states. The leading term isoften accompanied by a power law suppression leading toan expression for the degeneracy of the form

D½fsjg� ≈ ðA=l2pÞα expðð1 − δÞA=ð4l2

pÞÞ

¼�X

j

ajsj

�αY

j

expð1 − δÞajsj

4l2p

; (A1)

where α is in general an arbitrary real number (usuallynegative). For simplicity let us choose α ¼ k where k is apositive integer. In such case the partition function becomeshigher derivative of (12),

Z½β� ¼ ωð−∂ ~βÞk expðqÞ ¼ ωðTUÞkð−∂δÞk expðqÞ; (A2)

where in the following discussion the exact form of theprefactor ω will not be needed. From (12) and consideringsmall δ or equivalently large q, ð−∂δÞeq ≈ ðπγÞq2eq.Therefore, to the leading order,

logZ½δ� ¼ qþ 2k log qþ oð1=qÞ

≈1

πγδ− 2k logðδÞ þ oðδÞ: (A3)

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From the above formula, it is easy to see that themean energyis not affected up to the orders we are considering now,

U ¼ A8πl

¼ −∂β logZ

¼ l2p

2π2lγδ2½1þ oðδÞ�: (A4)

Therefore, the relation (16) between δ and the area isnot modified to leading orders. However, the entropyS ¼ βU þ logZ receives a logarithmic correction,

SG½A� ¼A4l2

p

"1þ δβ þ 4

ffiffiffiffiffiffiffiffil2p

πγA

s #

þ k log

�Al2p

�þ oð1Þ: (A5)

Thus, the logarithmic corrections to the entropy areassociated to the power-law suppression of theexponential growth of states (6). The entropy formula(A5) can be extended to some negative real numberα by analytic extensions. This shows that the mechanismleading to the logarithmic corrections by loop correctionsinvestigated in [31] will also lead to the very samecontributions in our analysis. This is due to the fact thatsuch logarithmic corrections would modify (6) accordingto (A1).

APPENDIX B: DISTINGUISHABLE PUNCTURES

It is easy to see that the distinguishable natureof the punctures is not consistent with holography(in the precise sense we have implemented holographyin this paper). Close to the Unruh temperature thegrand canonical partition function is given by ZMB ¼P

NqN . Its convergence requires q < 1, which means

δ > log 2=ðπγÞ; hence, a deviation from the holographicbound is expected.One could try to ignore this and analytically continue the

series from its value for jqj < 1 and write

ZMB ¼XN

qN ≔expðπγδÞ − 1

expðπγδÞ − 2: (B1)

However, the above partition function becomes negative forsmall δ and cannot be used to construct a real thermody-namical potential logZMB.Distinguishability of punctures seems to be incompatible

also with holography in the sense of having a factor close to1=4 in (6). But it would be compatible with a prefactorclose to ðπγ − logð2ÞÞ=ð4πγÞ in this case. However, asimple calculation shows that if that were the case, theentropy would not be close to Hawking entropy and thecorrespondence between the statistical mechanical partitionfunction and the Euclidean path integral semiclassicalpartition function would be lost.

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