# Thermodynamic Partition Function from Quantum Theory for Black Hole Horizons in Loop Quantum Gravity

Authors: Abhishek Majhi
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

We establish the link between the thermodynamics and the quantum theory of black hole horizons through the construction of the thermodynamic partition function, partly based on some physically plausible arguments, by beginning from the description of quantum states of the horizon, considering loop quantum gravity (LQG) as the underlying theory. Although the effective “thermalized” form of the partition function has been previously used in the literature to study the effect of thermal fluctuations of the black hole horizon, nonetheless the direct link to any existing quantum theory (which is here taken to be LQG), especially a derivation of the partition function from the quantum states of the horizon, appears to be hitherto absent. This work is an attempt to bridge this small, but essential, gap that appears to be present between the existing literature of quantum theory and thermodynamics of black holes. Furthermore, it may be emphasized that this work isonly concerned with themetric independent approaches to black hole thermodynamics.


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## Body

## 1. Introduction

The thermodynamic properties associated with spacetime geometry are intimately associated with the presence of a physical horizon. In the Euclidean quantum gravity approach, it had been actually proven that the entropy of the spacetime geometry vanishes in the absence of a horizon [1]. Similarly, using certain properties of Canonical quantum gravity, within certain approximations, it has been shown that the partition function of a spacetime reduces to that of the horizon only [2, 3]. The physical significance of the presence of the horizon looks even more compelling when we find that (i) the entropy of the black hole is given by its horizon area instead of the volume of the region behind the horizon [4, 5], (ii) the thermal radiation of the black hole is emitted at the temperature which is the one associated with the horizon [6, 7], and (iii) there are laws of thermodynamics associated with the horizon [8]. All these findings compel us to think that there may be some quantum theory of the horizon geometry itself which describes the independent quantum degrees of freedom of the horizon and then only we can have a proper understanding and logical reasoning behind the emergent thermodynamic properties of the horizon manifested in its own right. The knowledge of such quantum theory of the horizon will also enable us to formulate a spacetime-metric or back ground independent horizon thermodynamics. In fact, in recent literature there have been some serious attempts to study horizon thermodynamics without referring to the bulk spacetime by constructing the partition function for the horizon only and investigating the relevant thermodynamic properties.Thermal fluctuations of black hole horizons have been studied in various details by several authors in the recent past leading to interesting results regarding black hole thermodynamics [2, 3, 9–14]. In [2, 3, 14] thermal stability of black holes under thermal fluctuations was studied in canonical and grand canonical ensembles leading to a stability criterion for black holes which successfully explained the stabilities and instabilities of certain known black hole solutions. On the other hand, the issues of canonical and grand canonical entropy of black holes in presence of thermal fluctuations have been studied in [9–13]. The common and unique property of all these works is the use of the horizon partition function, whose construction is based on some heuristic modeling or assumptions or approximations regarding the underlying quantum structure of the horizon geometry, without referring to any classical background geometric structures, such as the metric and hence valid for arbitrary black hole horizons independent of the structure of the associated bulk spacetime. The thermodynamic properties associated with black hole horizons urge one to think of an inherent quantum description of the horizon geometry which will give rise to the notion of microstates so as to have a physical meaning of the associated entropy and hence thermality. This is the underlying physical essence of this particular statistical mechanical approach to horizon thermodynamics presented and investigated in [2, 3, 9–14].However, none of these works [2, 3, 9–14] presents a logical straightforward derivation of the partition function based on the fundamental quantum structure of black hole horizons, which would have made the link between the microscopic quantum theory of the horizon and the associated thermodynamics as had been urged by the works in [1, 4–8]. This crucial direct link still goes begging, even though there is a well defined quantum theory for the horizon geometry present in the literature for several years [15, 16]. Perhaps the clearest quantum description of the geometry of a black hole horizon is provided by loop quantum gravity [15, 16], where the quantum degrees of freedom of the horizon belong to the Hilbert space of a quantum Chern-Simons theory coupled to point like sources (to be explained in detali later).The theory, thus, provides a self-contained platform for the application of the statistical mechanical techniques so as to unravel the corresponding thermodynamic properties of the black hole horizon. This paper is aimed at providing a direct link between the quantum theory of horizon geometry and the associated thermodynamics through a derivation of the thermodynamic partition function associated with the horizon. This bridges the gap between the quantum geometric framework of equilibrium horizons laid down in [15, 16] and the thermodynamic aspects of the horizon studied in [2, 3, 9–14], which in turn answers the issues motivated by the works in [1, 4–8] as discussed earlier. This is precisely the important value addition of this work to the extant literature and hopefully may be of interest to the concerned readers. The structure of the paper can be debriefed as follows.


## 2. Equilibrium Horizon and Quantum Fluctuations: Microcanonical Ensemble

In modern day literature, a black hole horizon in gravitational and thermal equilibrium with ambient matter and radiation is described by the notion of an isolated horizon (IH) [17–19], which is a 2+1-dimensional null inner boundary of a 3+1-dimensional spacetime. An IH is a generalization of an event horizon of a stationary black hole spacetime to a more realistic scenario. As opposed to the global notion of the event horizon, an IH is defined locally, without any reference of the ambient bulk spacetime and hence can admit matter and radiation arbitrarily close to itself, whereas an event horizon cannot. The laws of black hole mechanics are completely realizable in the local framework of IH [19–21]. The mass and the surface gravity associated with the IH are also defined from a completely local perspective [19]. The strength of the IH framework is its local, background independent description in terms of the connection variables, which has led to rigorous analysis of the connection dynamics, symplectic structure, and so forth of the IH. The analysis reveals that the symplectic structure of the IH is that of an SU(2) Chern-Simons (CS) theory [17, 18] (a concerned reader may look into [22] where, taking the event horizon in the Schwarzschild spacetime as an example (which is a special case of an IH), it has been shown that there is indeed an SU(2) CS theory on the event horizon). In the quantum theory, the states of a QIH are given by that of the CS theory coupled to the edges of the bulk spin network which span the bulk quantum geometry and intersect the IH at specific points called punctures [15, 16].The Hilbert space of a quantum spacetime admitting QIH as an inner boundary is given byH=HV⊗HS modulo gauge transformations, where V denotes bulk and S denotes boundary (QIH) at a particular time slice [15, 16]. Mathematically, if the 4d spacetime (R×Σ) admits a 3d IH (Δ) as null inner boundary, then S≡Δ∩Σ denotes a cross section of the IH [17, 18]. Hence, a generic quantum state of the spatial geometry of such a spacetime can be written as Ψ=ΨV⊗ΨS, where ΨV is the wave function corresponding to the volume (V) or bulk states represented by an oriented graph, say Γ, consisting of edges and vertices [23] and ΨS denotes a generic quantum state of the QIH. ΨS∈HS≡ the Hilbert space of the CS theory coupled to the punctures P made by the bulk spin network Γ with the IH endowing them with the spin representations carried by the respective piercing edges which are solely responsible for all the relevant features of the QIH, the most important being the quantum area spectrum of the QIH. To be precise, for a given N number of punctures, with spins (j1,…,jN), the QIH Hilbert space is given by HS≡Inv(⨂i=1NHji) where “Inv” denotes the invariance under the local SU(2) gauge transformations on the QIH. Now, as it is seen that at the quantum level the full Hilbert space is the direct product space of the bulk and boundary Hilbert spaces, a generic quantum state of the QIH (boundary) can be written in terms of basis states on HS, independent of the bulk wave function. Hence, one should understand that a basis state of the QIH Hilbert space is actually a generic quantum state of the full Hilbert space, since the bulk part of the wave function is a linear combination of the basis states of the bulk geometry. In other words, a givenspin configuration on the QIH admits all possible graphs (Γ-s) in the bulk consistent with the given configuration. This spin configurations provide the area eigenstate basis, which is the all important material in the context of QIH entropy. Such a basis state of the QIH Hilbert space is denoted by the ket sj. This is an eigenstate of the area operator associated with the QIH, having the area eigenvalue given by A^Ssj=8πγlp2∑j=1/2k/2sjj(j+1)sj. Such a spin configuration (eigenstate) has a (N!/∏jsj!)-fold degeneracy due to the possible arrangement of the spins yielding the same area eigenvalue. Hence, a generic quantum state of the QIH can be written as (1)ΨS=∑sjcsjsj,where csj2=ωsj (say) is the probability that the QIH is found in the state sj.We would like to mention that the topological structures, that is, the punctures, arise only in the quantum theory and we can consider it to be a macroscopic parameter only if we deal with an ensemble of QIHs rather than an ensemble of IHs. Thus, the Hilbert space of a QIH for a givenN should be appropriately written with proper designation as HSk,N=Inv⨂l=1NHjl. But, the full Hilbert space of a classical IH designated by the corresponding CS level k takes into account all possible sets of punctures [15, 16] and is given by(2)HSk=⨁PInv⨂l=1NHjl,where(3)⨁P≡⨁N;1/2≤jl≤k/2∀l∈1,N∋∑l=1Njljl+1=k/2±O1/8πγand γ is the Barbero-Immirzi parameter. The CS level (k) is defined as k≡A/4πγlp2, where A is the classical area of the IH [15]. In the quantum theory, the CS coupling k is an integer which is a necessary prequantization condition for the quantization of the classical IH [16]. Since we shall begin with the application of quantum mechanical ensemble theory to the QIH, the macroscopic parameters will be considered to be k and N [24] and the microstates resulting in the quantum degeneracy for a given k and N will give rise to the statistical or microcanonical entropy as a function of k and N [24, 25].


## 3. Quantum Fluctuations: The Microcanonical Ensemble

The number of microstates for a QIH with CS levelk and number of punctures N are given by the dimension of the Hilbert space HSk,N=Inv⨂l=1NHjl, where 1/2≤jl≤k/2∀l∈[1,N]. The formula for the number of microstates for an arbitrary sequence of spins (j1,…,jN) on N number of punctures on the QIH is given by [26](4)Ωj1,…,jN=2k+2∑a=1k+1sin⁡aπ2j1+1/k+2⋯sin⁡aπ2jN+1/k+2sin⁡aπ/k+2N-2.To obtain the total number of microstates of a QIH for given k and N, that is, the dimensionality of HSk,N, we must consider all possible values of spin from 1/2 to k/2 for each puncture. This is the precise argument which was provided in [24] in the other way round so as to render k and N to be macroscopic parameters designating a macroscopic state of a QIH that is taking the sum over all spins; the only parameters which are left bare are k and N. Hence we can write Ω(k,N)=∑j1,…,jNΩ(j1,…,jN), where each spin is summed from 1/2 to k/2. In principle and by definition, the microcanonical entropy of the QIH is given by SMC=log⁡Ω(k,N), where we have set the Boltzmann constant to unity. In practice [24], the calculation of Ω(k,N) is performed only approximately (the approximation is good enough in the limit k,N→∞ which is the appropriate for QIHs with large classical area and large number of punctures) by finding themost probable distribution through the application of the method of Lagrange undetermined multipliers. The calculation has been extensively carried out in [24] and is needless to repeat here. What we shall discuss here is the physical essence of the method alongside mentioning the crucial steps of the calculation. Firstly, one can express the sum over spin values as sum over spin configurations by the application of the multinomial theorem [24]; that is,(5)Ωk,N=∑j1,…,jNΩj1,…,jN=∑sjΩsj,where(6)Ωsj=N!∏jsj!2k+2∑a=1k+1sin2⁡aπk+2∏jsin⁡aπ2j+1/k+2sin⁡aπ/k+2sj.This puts forward a very clear picture of the physical scenario which manifests the quantum jumps of QIH state from one area eigenstate sj to another, since the spin configurations are the eigenstates of the area operator of the QIH as discussed earlier. The variation of spin values all over the punctures of the QIH is the quantum fluctuations of the system. Now, any puncture can take any spin value from 1/2 to k/2 with equal a priori probability and hence all the spin sequences are equally probable. But when we look at the spin configurations, the one allowing the maximum number of possible equi-probable microstates (spin sequences) that is maximally degenerate is the most probable spin configuration. Hence quantum state corresponding to the most probable configuration is the highest entropy state. At par with the basic postulates of equilibrium statistical mechanics, the degeneracy corresponding to this state overwhelmingly outnumbers the degeneracies corresponding to the other subdominant configurations which may be regarded as negligible quantum fluctuations. Thus, it is a good enough approximation to consider Ω(N,k)≃Ωsj⋆, where sj⋆ is the distribution corresponding to the most probable spin configuration and comes out to be(7)sj⋆≃N2j+1exp⁡-λjj+1-σ.It may be mentioned that the classical area (Acl⁡) is related to k as k=A/4πγlp2. It is straightforward to show that the microcanonical entropy comes out to be [24, 25](8)SMC=λk2+Nσ,where λ and σ are the Lagrange multipliers and we have neglected the subleading terms in the entropy [24] that are irrelevant in the present analysis. The Lagrange multipliers are solvable in terms of k and N from the following equations [24, 25]:(9a)eσ=2λ21+32λe-3/2λ,(9b)kN=1+2λ+4λ3λ+2.It should be noted that the most probable spin configuration is determined by the given set of values of k and N.The physical essence of the microcanonical ensemble analysis is that the thermal fluctuations of the macroscopic variables are completely disallowed and what we deal with are purely quantum mechanical fluctuations of the system. It is easy to understand because we define the microcanonical ensemble by fixing the macroscopic variables, which are herek and N, and do not allow them to fluctuate. Thus the classically behaving macroscopic variables remain constant and the QIH jumps randomly from one quantum mechanical state to another which gives rise to the quantum uncertainty and hence the microcanonical entropy. In that sense, the resulting entropy is purely statistical. As we shall pass on to the grand canonical ensemble scenario, the effects of the thermal fluctuations will be taken into account because now the fixed macroscopic variables will be allowed to fluctuate.Note. At this point, having known the degeneracy corresponding to given k and N, we can write down the canonical partition function as(10)ZCβ,N=∑kΩk,Nexp⁡-βEk,Nand the grand canonical partition function as(11)ZGβ,μ=∑k,NΩk,Nexp⁡-βEk,N-μN,where E(k,N) is the mean energy or the statistical mean of the Hamiltonian operator for the QIH (although the classical energy associated with the classical IH [19] has not been quantized till date, there has been a proposal of the most general possible structure of the Hamiltonian operator associated with the QIH [27, 28], which satisfies the properties of the classical energy in the correspondence limit) with given k and N. This is like considering several different copies of microcanonical ensembles of QIHs designated by different sets of values of k and N and then dealing with the effective classical behaviour of the relevant quantities, the quantum fluctuations being averaged out [29]. So (10) and (11) can be regarded as the “thermalized” forms of the quantum mechanical canonical and grand canonical partition functions, respectively, since the many-body structure of the QIH is not manifested at all in these above forms and is glossed out by the classically behaving macroscopic variables.Similar forms of the partition functions have been extensively used to investigate the thermodynamics of black holes in the IH framework previously in literature [2, 3, 9, 11–14]. However, there are several technical caveats in common which plague the approaches which also curtains some important physical issues regarding the quantum statistics and thermodynamics of black holes. In the forthcoming sections, we shall arrive at the above “thermalized” forms of the partition functions by beginning from the treatment with quantum mechanical ensembles of QIHs to get an in depth understanding of the thermodynamical issues based on the underlying quantum structures. Remarkably, albeit expectedly, the technical caveats of the earlier approaches to black hole thermodynamics get removed automatically, alongside the fact that the associated physical understandings become far more transparent, which will be discussed in detail in a separate section.


## 4. Thermalization: Other Quantum Mechanical Ensembles

Having reviewed the microcanonical ensemble results for a QIH, now we shall investigate the other quantum mechanical ensembles, namely, canonical and grand canonical. In the following paragraphs we shall explore in detail the construction of the canonical and grand canonical partition functions beginning from the very basic definitions and arrive at their respective “thermalized” forms.


### 4.1. Quantum Mechanical Canonical Partition Function

The Hilbert space structureHSk,N of a QIH with given k and N emulates that of a gas of particles [30] in the sense that every physical phenomenon concerning a QIH is a collective many-body effect, a puncture being an individual body. It is also evident from the quantum area of QIH that is 8πγlp2∑l=1Njl(jl+1) [15, 16] or the quantum energy of the QIH proposed in [27, 28]. Following this quantum “particle-like” structure of the QIH, the quantum mechanical canonical partition function can be written as(12)ZCβ,N=∑sjΩsjexp⁡-βEsj,where E[{sj}]=∑jϵjsj is the quantum energy of the QIH corresponding to the spin configuration {sj} and ϵj is the energy contribution from a single puncture carrying spin j and for any configuration, the total number of punctures must add up to N since it is kept fixed in the canonical ensemble.Now, from (7), considering that λ and σ are solvable in terms of k and N from (9a) and (9b), it is manifested that there is sj⋆ for every k and N. Thus, considering a fixed value of N, a different choice of k will give a different sj⋆. It is then completely physical to argue that the most probable distributions corresponding to k-1 and k+1 are the closest less probable distributions corresponding to the most probable distribution for k. Likewise, the other distributions in (12) are the most probable distributions for some k. Thus, for fixed N, in the quantum mechanical canonical ensemble, the sum over spin configurations can be replaced by sum over k; that is,(13)ZCβ,N=∑kΩk,Nexp⁡-βEk,N,where β is the temperature of the heat bath with which the QIH is considered to be in thermal contact. Since k is an integer and since we deal with the large k regime only the above expression can be well approximated by replacing with the summation by an integration as follows:(14)ZCβ,N≃∫dkΩk,Nexp⁡-βEk,N.Equation (14) is the expression for the canonical partition function for a QIH which is suitable for studying the thermal fluctuations. This may be regarded as the “thermalized” form of the quantum mechanical canonical partition function given by (12). The information about the many-body quantum structure of the QIH is glossed by the classical behaviour of the the macroscopic variables.


### 4.2. Quantum Mechanical Grand Canonical Partition Function

Now, we shall allow the macroscopic variableN to vary along side the energy. So, let us consider a grand canonical ensemble of QIHs where the number of punctures N is now allowed to vary along side the energy. The quantum mechanical grand canonical partition function can be written as(15)ZGβ,μ=∑N=1∞exp⁡βμNZCβ,N,where ZC is the partition function for a quantum mechanical canonical ensemble of QIHs. μ is some fictitious parameter conjugate to the macroscopic variable N, playing a roleanalogous to the chemical potential in case of a gas of particles. Let us rewrite (15) manifesting the many-particle structure of the QIH as(16)ZGβ,μ=∑sjΩsjexp⁡-βEsj+βμNsj.Now, we can argue in a similar way as in the case of the canonical ensemble. From (7), considering that λ and σ are solvable in terms of k and N from (9a) and (9b), it is manifested that there is sj⋆ for every k and N. Thus, different choices of k and N will give different sj⋆. Based on a physical argument similar to that of the quantum mechanical canonical ensemble, in the quantum mechanical grand canonical ensemble, the sum over spin configurations can be replaced by sum over k and N; that is,(17)ZGβ,μ=∑k,NΩk,Nexp⁡-βEk,N-μN.k,N are like “effective” quantum numbers for the energy and they are equispaced as they are integers. Since we are working in the large k and large N limits, the above discrete sum can be well approximated by replacing the summation by integration:(18)ZGβ,μ≃∫dkdNΩk,Nexp⁡-βEk,N-μN.Equation (18) is the expression for the canonical partition function for a QIH which is suitable for studying the thermal fluctuations. This may be regarded as the “thermalized” form of the quantum mechanical grand canonical partition function given by (15). The information about the many-body quantum structure of the QIH is glossed by the classical behaviour of the the macroscopic variables.


## 4.1. Quantum Mechanical Canonical Partition Function

The Hilbert space structureHSk,N of a QIH with given k and N emulates that of a gas of particles [30] in the sense that every physical phenomenon concerning a QIH is a collective many-body effect, a puncture being an individual body. It is also evident from the quantum area of QIH that is 8πγlp2∑l=1Njl(jl+1) [15, 16] or the quantum energy of the QIH proposed in [27, 28]. Following this quantum “particle-like” structure of the QIH, the quantum mechanical canonical partition function can be written as(12)ZCβ,N=∑sjΩsjexp⁡-βEsj,where E[{sj}]=∑jϵjsj is the quantum energy of the QIH corresponding to the spin configuration {sj} and ϵj is the energy contribution from a single puncture carrying spin j and for any configuration, the total number of punctures must add up to N since it is kept fixed in the canonical ensemble.Now, from (7), considering that λ and σ are solvable in terms of k and N from (9a) and (9b), it is manifested that there is sj⋆ for every k and N. Thus, considering a fixed value of N, a different choice of k will give a different sj⋆. It is then completely physical to argue that the most probable distributions corresponding to k-1 and k+1 are the closest less probable distributions corresponding to the most probable distribution for k. Likewise, the other distributions in (12) are the most probable distributions for some k. Thus, for fixed N, in the quantum mechanical canonical ensemble, the sum over spin configurations can be replaced by sum over k; that is,(13)ZCβ,N=∑kΩk,Nexp⁡-βEk,N,where β is the temperature of the heat bath with which the QIH is considered to be in thermal contact. Since k is an integer and since we deal with the large k regime only the above expression can be well approximated by replacing with the summation by an integration as follows:(14)ZCβ,N≃∫dkΩk,Nexp⁡-βEk,N.Equation (14) is the expression for the canonical partition function for a QIH which is suitable for studying the thermal fluctuations. This may be regarded as the “thermalized” form of the quantum mechanical canonical partition function given by (12). The information about the many-body quantum structure of the QIH is glossed by the classical behaviour of the the macroscopic variables.


## 4.2. Quantum Mechanical Grand Canonical Partition Function

Now, we shall allow the macroscopic variableN to vary along side the energy. So, let us consider a grand canonical ensemble of QIHs where the number of punctures N is now allowed to vary along side the energy. The quantum mechanical grand canonical partition function can be written as(15)ZGβ,μ=∑N=1∞exp⁡βμNZCβ,N,where ZC is the partition function for a quantum mechanical canonical ensemble of QIHs. μ is some fictitious parameter conjugate to the macroscopic variable N, playing a roleanalogous to the chemical potential in case of a gas of particles. Let us rewrite (15) manifesting the many-particle structure of the QIH as(16)ZGβ,μ=∑sjΩsjexp⁡-βEsj+βμNsj.Now, we can argue in a similar way as in the case of the canonical ensemble. From (7), considering that λ and σ are solvable in terms of k and N from (9a) and (9b), it is manifested that there is sj⋆ for every k and N. Thus, different choices of k and N will give different sj⋆. Based on a physical argument similar to that of the quantum mechanical canonical ensemble, in the quantum mechanical grand canonical ensemble, the sum over spin configurations can be replaced by sum over k and N; that is,(17)ZGβ,μ=∑k,NΩk,Nexp⁡-βEk,N-μN.k,N are like “effective” quantum numbers for the energy and they are equispaced as they are integers. Since we are working in the large k and large N limits, the above discrete sum can be well approximated by replacing the summation by integration:(18)ZGβ,μ≃∫dkdNΩk,Nexp⁡-βEk,N-μN.Equation (18) is the expression for the canonical partition function for a QIH which is suitable for studying the thermal fluctuations. This may be regarded as the “thermalized” form of the quantum mechanical grand canonical partition function given by (15). The information about the many-body quantum structure of the QIH is glossed by the classical behaviour of the the macroscopic variables.


## 5. The Chemical Potential: IH Thermodynamics from QIH Thermodynamics

It is worth mentioning that the most crucial consequence of all the above analysis, from the thermodynamic perspective, is that the grand canonical partition function of QIH withμ=0 is the canonical partition function for IH; that is, ZGQIH(β,0)=ZCIH(β). This is nothing surprising because looking at the structure of the full Hilbert space of an IH one would naturally consider the number of punctures to be unconserved for a given CS level (k). Thus, in the quantum mechanical canonical partition function of an IH, the sum over all possible quantum states underlying a classical IH implies sum overall possible spin sequences which give rise to area eigenvalues within O(lp2) of the classical area irrespective of the number of punctures. In fact the partition function written in [16], although in the area ensemble, takes a sum over number of punctures also. But the significance of QIH and quantum hair N remained unnoticed in [16]. In that sense it is more reasonable to consider N as a macroscopic variable and introduce μ on the first hand and then shift to the asymptotic view where the effect of the quantum hair disappears and μ is zero.However, one mustnot confuse the analysis and the arguments regarding the chemical potential (μ) of a QIH with that of black holes having Φ≠0 corresponding to any classical nongravitational charge, for example, Reissner-Nordstrom. One should remember that in this case there is an interplay between gravitational and nongravitational fields to attain the equilibrium. Hence, it is obvious that Φ need not vanish to attain the equilibrium from the asymptotic viewpoint. The most important point to be noted is that one can tune the parameter Φ from outside by controlling the nongravitational field, whereas μ cannot be controlled externally. The parameter Φ actually couples a nongravitational variable (Q) to the partition function but μ does so in the case of N which is purely (quantum) gravitational macroscopic variable. Although Φ and μ appear to be playing the same role in different cases, basically they are very different as far as their physical implications are concerned. Further, Φ has nothing to do with quantum gravity, whereas μ comes into play only when we consider the quantum gravity effects because the corresponding “charge” N is aquantum hair [24, 30].Finally, let us write down the IH partition function explicitly, which can be used directly to study the effects of Gaussian thermal fluctuations on the stability of a black hole [2]. The chemical potential μ is defined as(19)μ=-β-1∂SMC∂N=-σβ.Thus, from the asymptotic viewpoint, as discussed above, μ vanishes and consequently σ=0. Now, using σ=0 in (9a), one obtains(20)1=2λ21+32λe-3/2λ⟹λ=λ0.The above solution has been obtained graphically in [25] which gives the estimate λ0=1.2. Consequently, the microcanonical entropy is then given by(21)SMC=λ0k2=λ0Acl⁡8πγlp2usingk=Acl4πγlp2=Acl⁡4lp2by  choosingγ=λ02π.One may note that now we have k=Acl⁡/2λ0lp2. Also, as a parallel consequence, using λ=λ0 in (9b), one obtains(22)kN=1+2λ0+4λ03λ0+2=c0say.So, to calculate ZIH(β)=ZGQIH(β,0), it will not suffice to put μ=0 in (18), but we also have to impose the relation between k and N in (22) by using a Dirac delta function, namely, δ(k/c0,N). Thus, (18) now gets modified as(23)ZGQIHβ,0≃∫dkdNΩk,Nexp⁡-βEk,Nδkc0,N=1c0∫dkΩkexp⁡-βEk=12c0λ0∫dAcl⁡ΩAcl⁡exp⁡-βEAcl⁡,where we have set lp2=1 for convenience. Thus, we obtain the canonical partition function for the IH that is ZCIH(β) beginning from the fundamental quantum structures of the QIH. This is the same partition function which was studied in [2] and generalised to charged horizon in [3], to study the effect of Gaussian thermal fluctuations on the stability of black holes. But, as has been pointed out earlier, the derivations of the partition functions were based on some approximations and failed to establish a clear and direct link with the undelying quantum theory. One can see Appendix B for an elaborate discussion in this issue.


## 6. Discussion

The prime motive of the paper was the exact derivation of the horizon partition function using the fundamental quantum theory of the horizon geometry, which, we hope, has been elaborately worked out here up to a satisfactory level. The fact that the entropy of spacetime geometry vanishes in the absence of a horizon [1] strongly motivates one to think of independent quantum degrees of freedom of the horizon giving rise to the thermodynamic properties uniquely associated with the horizon; and the derivation of the thermodynamic partition function right from the scratch, which has been presented in this work, satifactorily quenches the urge by establishing the all needed direct link between the quantum and thermodynamic aspects of the horizon. Hence, the major acquisition one can have by studying this work is the logical step by step derivation of the theromodynamic partition functions of the black hole horizon based on the underlying quantum geometric framework and a clear understanding about the role of quantum and thermal fluctuations in the corresponding thermodynamics. Following this, the “thermalized” form of the partition function derived here, which was previously used in the literature to study the thermal fluctuations of the black hole horizons, now has a clear meaning and stands on the firm ground provided by the quantum geometric description of black hole horizon [15, 16]. This can be considered as an important value addition to the literature of black hole horizon thermodynamics. Apart from this, an important clarification made in this paper is that the QIH thermodynamics is delicately different from IH thermodynamics due to the role of the quantum variable N which is present in the quantum scenario, but absent in the classical thermodynamic picture. In fact, by beginning from IH framework one completely misses out the essence of the underlying quantum structure, resulting in the incorporation of ad hoc assumptions and unnecessary approximations in the procedure. As has been pointed out earlier, the identification of N as a macroscopic variable for QIH is the difference. Now, looking at the full Hilbert space of an IH which allows arbitrary number of punctures for a given k, one may be tempted to ask that why at all shall we consider N as a macroscopic variable a priori. The answer to this question may be that the Hamiltonian operator [27, 28] commutes with the number operator for punctures which renders N to be a constant of motion or, in other words, apparently there is no dynamics that that can change the number of punctures. On the other hand, there is no role of N at the classical level which is manifested by the absence of any μδN term in the first law derived from the classical theory of IH [19]. Hence, it is most appropriate to begin from the study of quantum mechanical ensembles of QIHs by considering N as a macroscopic variable and take the limit μ=0 to arrive at the IH scenario, as shown in this paper. However, all the arguments regarding the role of μ in the local and asymptotic views remain at the qualitative level, the only understanding being the redshift factor as the cause of the behaviour of μ. A quantitative understanding of how μ vanishes at the asymptopia, by an LQG calculation, may bring forth some new insights. Finally, it may be pointed out that although we restrict our analysis to purely gravitational case and stop at canonical partition function of the IH, one can trivially consider nongravitational charge and extend the analysis to the grand canonical partition function of charged IHs. One can look into [3] for this extension to charged horizons. But, one must be alert that there will be tricky issues involved if one tries to generalize the analysis to rotating horizons, because we still do not have a proper quantum theory of rotating IH and this may be considered as a challenging research problem with potential consequences.



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