Thermodynamic partition function from quantum theory for black hole horizons in loop quantum gravity

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\rq{} 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. Further, it may be emphasized that this work is {\it only} concerned with the {\it metric independent} approaches to black hole thermodynamics.

This paper presents an extensive work on the study of thermodynamics of black holes in LQG framework, namely quantum isolated horizons(QIH). Having reviewed the derivation of the microcanonical entropy of a QIH we proceed towards constructing the canonical and grand canonical partition functions for the QIH in the corresponding quantum mechanical ensembles. Some important issues regarding the conjugate parameter(µ) corresponding to the macroscopic variable N (number of punctures) are discussed in details, with possible explanations of the new physical consequences which can follow from its presence in the quantum theory and absence in the classical theory. The role of µ being dependent on the observer leads to interesting conclusions about the near horizon quantum phenomena, whereas the asymptotic physics remains unchanged. The extensive and detailed derivation of the canonical and grand canonical partition functions of the QIH lead to the effective 'thermalized' forms of the partition functions which had been previously used in the literature to study the effects of thermal fluctuations of black holes. A comparative study of the present derivation with those previous approaches is made. The previous procedures were based on some heuristic models and quite expectedly plagued with some technical caveats, leaving those approaches prone to doubts of having any sort of relation to black hole thermodynamics. The novelty of this work is to eliminate those shortcomings of the earlier approaches and put the formalism of statistical mechanical approach to black hole thermodynamics on a more sound basis than ever by beginning from the very fundamental structures of the quantum theory leading to the exact derivation of the horizon partition function, without having to make any sort of assumption or approximation regarding area spectrum, etc. Nowadays, a black hole horizon in gravitational and thermal equilibrium is described by an isolated horizon(IH) [1][2][3] which is a 2+1 dimensional null inner boundary of a 3+1 dimensional spacetime. An IH is a generalization of the concept of an event horizon of a stationary black hole spacetime to a more realistic scenario which can be briefly explained as follows. As opposed to the global notion of the event horizon, an IH is defined locally without referring to the situation in the bulk. Thus, an IH can admit matter and radiation arbitrarily close to itself, whereas an event horizon can not -any sort of dynamics is disallowed by the very need of Killing vector fields in the bulk. The local notion of an IH is devoid of the use of any particular metric, although it is still fully consistent with the Einstein general relativity. The laws of black hole mechanics are completely realizable in the local framework of IH [3][4][5]. The mass and the surface gravity associated with the IH are also defined from a completely local perspective [3]. 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, etc. of the IH. The analysis reveals that the symplectic structure of the IH is that of an SU (2) Chern-Simons(CS) theory [1,2] 1 . In the quantum theory, the states of a quantum IH (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 [7,8]. The theory of QIH provides a self-contained platform for the application of the statistical mechanical techniques so as to unravel the corresponding thermodynamic properties. This paper is aimed at providing a detailed and complete account on the thermodynamics of QIH in different quantum mechanical ensembles. The structure of the paper is explained as follows.
The quantum mechanical structure of a system lays the foundation for the application of statistical mechanics and QIH is not an exception. Hence, it is necessary to start with a discussion on the structure of QIH. In section(II), we begin with a short review of the Hilbert space structure of a QIH, namely the SU (2) CS theory coupled to point-like sources and the physical states of the QIH, which are the eigenstates of the area operator for the QIH. The macroscopic states of a QIH are designated by two integer parameters, namely, the CS level k and the number of punctures N . The number of microstates of a QIH for a given set of k and N represent the degree of quantum fluctuations underlying that particular macrostate. Calculation of the number of microstates yields the statistical entropy of the QIH as a function of k and N . The steps of the calculations have been reviewed in short in section(III). This is what we call the microcanonical entropy which arises from purely quantum fluctuations of the system and is a measure of the quantum uncertainly underlying the macroscopic structure of the system, which is here the QIH. Having calculated the microcanonical entropy we now turn on to study the quantum mechanical canonical and grand canonical ensemble scenario where the QIH is allowed to interact with the environment, the exterior spacetime and the macroscopic parameters are now allowed to fluctuate. These are the ensembles where the study of thermal fluctuations can actually be made. Beginning from the quantum mechanical partition functions, we arrive at the 'thermalized' forms of the partition function which are suitable for the study of thermal fluctuations of the system. This is the whole subject matter of section(IV). The subsections(IV A) and (IV B) deal with the canonical and grand canonical ensemble of QIHs where, beginning from the quantum mechanical ensembles we lay down the method to arrive at the form of the partition function from which we can study the effect of thermal fluctuations. Now, it is to be noted that the thermodynamics of QIH is quite different from that of IH because of the extra presence of the macroscopic variable N for QIH and the corresponding conjugate parameter(µ) analogous to the chemical potential of a gas of particles [11]. Section(V) deals with the issue of this 'chemical potential'(µ) for QIH. The role of this analog 'chemical potential' in case of QIH is quite subtle to explain and differs from the explanations in usual case of gas of particles. Some possible physical explanations of the role of the parameter are put forward. In subsection(V A), it is explained how a non-zero value of µ can give rise to the notion of interconversion of matter and geometry at the quantum level giving rise to quantum topological fluctuations of the QIH which will only be observable by a local observer close to the horizon. However, these quantum topological fluctuations won't be observable by an asymptotic observer to whom only the classical picture of the IH will be relevant. A detailed explanation of the difference between the local and asymptotic view of the QIH is given in subsection(V B). In subsection(V C), a very crucial observation is made which finally explains the difference between QIH and IH as far as the thermodynamical issues are concerned. Now, it is worth devoting a separate section to explain the novelty of the present work by making a comparative study with the prior background independent statistical mechanical approaches to black hole thermodynamics and understand how this present work overcomes the technical caveats which plague those earlier approaches. Section(VI) serves the purpose. Finally, we end with a discussion in section(VII).

II. QUANTUM GEOMETRY AND THE PHYSICAL STATES OF THE QIH
The Hilbert space of a quantum spacetime admitting QIH as an inner boundary is given by H = H V ⊗ H S modulo gauge transformations, where V denotes bulk and S denotes boundary(QIH) at a particular time slice [7,8]. Mathematically, if the 4d spacetime (R ⊗ Σ) admits a 3d IH (∆) as null inner boundary, then S ≡ ∆ ∩ Σ denotes a cross-section of the IH [1,2]. Hence, a generic quantum state of the spatial geometry of such a spacetime can be written as |Ψ = |Ψ V ⊗ |Ψ S , where |Ψ V is the wave function corresponding to the volume(V ) or bulk states represented by an oriented graph, say Γ, consisting of edges and vertices [9] and |Ψ S denotes a generic quantum state of the QIH. |Ψ S ∈ H S ≡ 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 (j 1 , · · · , j N ), the QIH Hilbert space is given by 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 H S , 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 given spin configuration on the QIH admit 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 |{s j } . This is an eigenstate of the area operator associated with the QIH, having the area eigenvalue given byÂ S |{s j } = 8πγℓ 2 Such a spin configuration (eigenstate) has a (N !/ j s j !)-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 is the probability that the QIH is found in the state | {s j } .
We would like to mention that the topological structures i.e. 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 given N should be appropriately written with proper designation as H k,N S = Inv N l=1 H j l . But, the full Hilbert space of a classical IH designated by the corresponding CS level k takes into account all possible sets of punctures [7,8] and is given by and γ is the Barbero-Immirzi parameter. The CS level(k) is defined as k ≡ A/4πγℓ 2 p where A is the classical area of the IH [7]. In the quantum theory, the CS coupling k is an integer which is a necessary prequantization condition for the quantization of the classical IH [8]. 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 [12] 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 [10,12].

III. QUANTUM FLUCTUATIONS : THE MICROCANONICAL ENSEMBLE
The number of microstates for a QIH with CS level k and number of punctures N is given by the dimension of the Hilbert space H k,N The formula for the number of microstates for an arbitrary sequence of spins (j 1 , ...., j N ) on N number of punctures on the QIH is given by the Kaul-Majumdar formula [28] written as To obtain the total number of microstates of a QIH for a given k and N i.e. the dimensionality of H k,N S , 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 [12] in the other way round so as to render k and N to be macroscopic parameters designating a macroscopic state of a QIH i.e. 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,··· ,jN Ω(j 1 , · · · , j N ) 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 S MC = log Ω(k, N ), where we have set the Boltzmann constant to unity. In practice [12], the calculation of Ω(k, N ) is performed only approximately 2 by finding the most probable distribution through the application of the method of Lagrange undetermined multipliers. The calculation has been extensively carried out in [12] 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 [12] i.e. where This puts forward a very crystal clear picture of the physical scenario which manifest the quantum jumps of QIH state from one area eigenstate |{s j } 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 are 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) i.e. 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, where s ⋆ j is the distribution corresponding to the most probable spin configuration. The expectation value of the area operator for the state |{s ⋆ j } is the closest to the classical area or the statistical mean area [12], which is related to k by the relation k = A/4πγℓ 2 p . It is straightforward to show that the microcanonical entropy comes out to be [10,12] where λ and σ are the Lagrange multipliers, which are solvable in terms of k and N from relevant equations involving λ, σ, k and N given by 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 here k 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 (III.6) and the grand canonical partition function as where E(k, N ) is the mean energy or the statistical mean of the Hamiltonian operator for the QIH 3 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 3 Although the classical energy associated with the classical IH [3] 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 [25,26], which satisfies the properties of the classical energy in the correspondence limit.
the relevant quantities, the quantum fluctuations being averaged out [17]. So eq.(III.6) and eq.(III.7) 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 are 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 [15,16,[18][19][20][21][22]. 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 details in a separate section.

IV. THERMALIZATION : OTHER QUANTUM MECHANICAL ENSEMBLES
Having reviewed the microcanonical ensemble results for a QIH, now it is time that we investigate the other quantum mechanical ensembles viz. canonical and grand canonical. In the following paragraphs we shall explore in details the construction of the canonical and grand canonical partition functions beginning from the very basic definitions and arrive at their respective 'thermalized' forms.

A. Quantum Mechanical Canonical Partition Function
The Hilbert space structure H k,N S of a QIH with given k and N emulates that of a gas of particles [11] 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 i.e. 8πγℓ 2 p N l=1 j l (j l + 1) [7,8] or the quantum energy of the QIH proposed in [25,26]. Following this quantum 'particle-like' structure of the QIH, the quantum mechanical canonical partition function can be written as where E[{s j }] = j ǫ j s j is the quantum energy of the QIH corresponding to the spin configuration {s j }, ǫ j 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, we have seen in the microcanonical ensemble analysis that there is a most probable spin distribution for every k and N . Thus, considering a fixed value of N , a different choice of k will give a different most probable distribution. 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. Thus, for fixed N , in the quantum mechanical canonical ensemble, the sum over spin configurations can be replaced by sum over k i.e.
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 Eq.(IV.3) 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 eq. (IV.1). The information about the many-body quantum structure of the QIH is glossed by the classical behaviour of the the macroscopic variables.

B. Quantum Mechanical Grand Canonical Partition Function
Now, we shall allow the macroscopic variable N 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 where Z C 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 role analogous to the chemical potential in case of a gas of particles. Let us rewrite eq.(IV.4) manifesting the many-particle structure of the QIH as where the notation N [{s j }] is used so as to manifest that we can define a number operator which when acts on the state |s j gives the corresponding number of punctures. Now we can argue in the similar way as in case of the canonical ensemble that we have seen in the microcanonical ensemble analysis that there is a most probable spin distribution for every k and N . Thus every different choice of k and N will give a different most probable distribution which is a less probable distribution from some other set of values of k and N . Hence, in the quantum mechanical grand canonical ensemble, the sum over spin configurations can be replaced by sum over k and N i.e.
k, N being integers are like 'effective' quantum numbers for the energy and it is trivial that k and N are equispaced. 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 Eq.(IV.7) 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 eq. (IV.4). The information about the many-body quantum structure of the QIH is glossed by the classical behaviour of the the macroscopic variables.

V. THE CHEMICAL POTENTIAL
The chemical potential for QIH is purely of quantum origin and is absent in the classical theory. This makes the role of the chemical potential quite subtle to understand and of course very different from our usual notion of chemical potential occurring in thermodynamics of ordinary systems. Let us explain the possible role of the chemical potential µ as far as the physical issues regarding the QIH are concerned and the possible new physics which may follow, in the following sections.

A. Quantum inter-conversion of geometry and matter
In case of a system of gas of particles, we have an easy understanding of the physical meaning to the chemical potential µ. It governs the physical exchange of particles between the system(the gas) and the ambient reservoir, both of which are 3+1 dimensional systems. Hence, most crucially, the particles retain their identity whether in the system or the reservoir and the scenario can be well visualized. But the scenario of QIHs is very much different. First of all the QIH is the quantum structure of the 2+1 dimensional IH, whereas the heat bath i.e. the exterior spacetime is 3+1 dimensional. The punctures, which are topological defects on the QIH are describable only as long as they are on the QIH. They are not defined elsewhere off the QIH. Thus there is no possible physical reservoir of punctures with which the QIH can physically exchange punctures. In this case the reservoir in contact with the QIH carries an abstract sense. Well, it can be argued that the punctures get detached to yield open ends in the bulk quantum geometry which are often considered as matter excitations [27], but still they are not punctures which are quantum geometric excitations. The physical process in either direction is a conversion between two different objects -i) detachment of punctures : conversion of horizon geometry to bulk matter ii) attachment of punctures : conversion of bulk matter to horizon geometry. Hence, the scenario of exchange of similar objects does not appear here. The very concept of 'exchange' gets replaced by the concept of 'conversion'. To be more precise and to make comparison(or rather distinction) between the two apparently similar scenarios we can say that the fluctuations of N imply 'exchange of particles' for a gas and 'mutual conversion of bulk matter and horizon geometry at the quantum level' in case of a QIH. It is quite clear that in case of QIHs the physical meaning of the parameter µ is drastically different from that in the case of a gas of particles. The situation urges us to understand the precise quantum dynamics which converts a geometrical excitation on the horizon into an excitation of matter field in the bulk.

B. Quantum Topology of the Isolated Horizon : Local vs Asymptotic View
What we have been dealing with in this work is pure gravity and there is no non-gravitational field in this scenario. In the LQG framework, the degrees of freedom of the QIH are completely described by CS theory coupled to the topological defects on the horizon which act as sources (topological charges) [7,8]. The macroscopic variables k and N are well defined as far as LQG is concerned and are pure gravity variables. Both the dynamical and the equilibrium states of the horizon are completely governed by purely quantum gravitational effects, devoid of any non-gravity fields in this scenario. Thus, as opposed to the 'tunable' potential corresponding to any non-gravity charges, the 'chemical potential' of a QIH corresponding to the topological charge N can not be tuned externally. What we can do is to observe the behaviour of the parameter and understand its consequences as far as the thermodynamics of a QIH is concerned. Hence, even though we define the grand canonical partition function for a QIH by fixing β and µ, but it may happen that a particular value µ is allowed for the equilibrium.
At thermodynamic equilibrium, the macroscopic variables of the QIH are related by an equation of state. No matter crosses the horizon. But, even in this thermodynamic equilibrium condition a non-zero 'chemical potential' i.e. µ = 0 will imply a tendency of fluctuation of N about the mean value N 0 is still dynamically present there. Since the horizon has attained equilibrium i.e. isolated, the area is now fixed, the fluctuations of N which can now take place are only those which can keep the area fixed. A bit of thinking will allow one to realize that these fluctuations of N are nothing but the changes in the quantum topology of the QIH (e.g. one puncture of area a get replaced by two punctures of area a/2), hence can be appropriately called quantum topological fluctuations. Thus, in spite of the fixed area at equilibrium, there will be a puncture dynamics going on for a nonzero chemical potential which gives rise to the visualization of the quantum description of radiation and accretion from the horizon [14].
However, tracing back to the original proposal of the existence of quantum hair N in [11], it should be remembered that these quantum topological fluctuations will only be observable for a local observer close to the horizon and the effects must vanish for an asymptotic observer who can only see the classical IH, the existence of punctures and corresponding dynamics being glossed out at infinity. This explanation is supported by the fact that the first law of IH thermodynamics [3] does not contain any µδN term as opposed to that of a QIH [11]. Thus what is observable at asymptopia is quantum topological equilibrium i.e. the smoothness of the topology of the horizon S 2 ⊗ R. Such a quantum topological equilibrium will be attained only for µ = 0.
But for a local observer very close to the horizon, the quantum topological fluctuations must be observable and in that case a positive chemical potential [10] must be there which will result in a quantum topological fluctuation going on for a fixed classical area of the IH. This is like quantum dynamics underlying a classical equilibrium. The situation gives rise to the scenario that an infalling observer just before hitting the horizon will see a region of quantum radiation. This radiation does not escape to infinity and not Hawking radiation, but must be present there due to the quantum uncertainties evoked by the quantum structure of the IH. This may be considered as an LQG description, albeit statistical in nature, of the near horizon concepts of energetic curtains [29] or firewalls [30].

C. 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 with µ = 0, is the canonical partition function for IH i.e. Z QIH G (β, 0) = Z IH C (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 over all possible spin sequences which give rise to area eigenvalues within O(ℓ 2 p ) of the classical area irrespective of the number of punctures. In fact the partition function written in [8], 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 [8]. 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 must not 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 non-gravitational charge e.g. Reissner-Nordstrom, etc. One should remember that in this case there is an interplay between gravitational and non-gravitational 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 non-gravitational field, whereas, µ can not 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 a quantum hair [11,12].

VI. A COMPARATIVE STUDY OF BACKGROUND INDEPENDENT STATISTICAL MECHANICAL APPROACHES TO BLACK HOLE THERMODYNAMICS
Now, let us discuss the different background independent statistical mechanical approaches to black hole horizon thermodynamics that has gradually evolved step by step in literature, which will finally give us an understanding about the shortcomings of these earlier approaches and how the related technical caveats get eliminated by the present work based on the QIH framework. As far as our knowledge is concerned, a generic statistical mechanical approach without prior use of any background metric was used to study the canonical ensemble scenario with the aim to investigate the effects of thermal fluctuations for black holes in [18,19,23]. The canonical partition function was written as for a black hole(assuming in the hindsight). This is tantamount to model the quantum theory of the black hole as that of a single particle, alike the harmonic oscillator, whose energy spectrum is characterized by a single quantum number and also equispaced. Without the equispaced energy spectrum, the discrete sum can not be approximated as an integration in any limit. The usual thermodynamic results following from the effective single particle model were applied to selected black holes and relevant conclusions were drawn. The above partition function, if not stated to be only a heuristic model, has hardly any relationship with actual black hole quantum states originating from the theory of QIH. The above model got improved a bit and some properties of QIH were incorporated in [20]. The canonical partition function was written as considering that the energy is a function of the area of the horizon and then, assuming that all the punctures on the horizon carry spin-half (neglecting the effects of other spins), the area spectrum of the horizon is made equispaced i.e. A ∝ N . The energy states are now characterized by a single quantum number but the energy spectrum is no more equispaced. Thus, the effective single particle model for black holes has got more generalized and for large horizons with sufficiently densely packed quantum states the discrete sum over the quantum number is replaced by an integration over a continuous variable which represents the quantum number in the continuum limit. It was only in [15] that a logical explanation of the partition function was given starting from the IH framework, albeit in a superficial approach using the tools of LQG. The partition function was no more any model but a true partition function derived from the fundamental theory of IH. But still the technical assumptions and approximations regarding the linearized area spectrum, etc. plagued the calculations resulting in the effective one particle picture.
Similar to the canonical ensemble study of the effective one-particle model, the fluctuations of non-gravity charges were studied in the grand canonical ensemble in [19,21], considering that the effective one-particle carries a quantized charge representing the non-gravity charge of the black hole. The partition function is written as using similar assumptions and approximations in the appropriate limits as discussed above for the canonical case. In [16] the partition function was derived in the IH framework, albeit in a superficial approach using the tools of LQG. The explanation of the Hamiltonian associated with the horizon was improved and the structure of the formalism was put on a stronger ground than ever. The partition function was a generic one for charged IH. But still the technical assumptions and approximations regarding the linearized area spectrum, etc. plagued the calculations resulting in the effective one particle picture. It is worth explaining that the assumption of the linearized quantum area spectrum is tantamount to disregard the role of N as an independent macroscopic parameter for QIH. Hence, the existence of the macroscopic variable at the quantum level continued to be opaqued by the assumption of the linearized quantum area spectrum. The relevance of the many-particle nature of the QIH remained unnoticed and hence, the canonical ensemble of QIHs and IHs have not been differentiated. It was in [11] that the crucial observation was made and was justified with simple reasonings in [12], that the number of punctures on the QIH should be considered to be an independent macroscopic variable which characterizes a macrostate of a QIH. Hence, it is worth investigating the crucial difference between the canonical ensemble thermodynamics of QIH and classical IH. The canonical partition function is defined for a fixed N for QIH, whereas for classical IH there is no concept of N . That is why the canonical partition function in eq.(IV.3) is tagged by the parameter N , where as there is no such tag for the canonical partition function of IH in [15]. In case of IH the canonical partition function is defined for fixed non-gravitational charges [15] which are allowed to fluctuate in the corresponding grand canonical ensemble [16]. But for the QIH, the many-body quantum structure compels us to treat the complete thermodynamics in the grand canonical ensemble even in the absence of non-gravitational charges.
Further there are some very important consequences of the study of the thermodynamics of the black holes beginning from the fundamental quantum structure of QIH rather than from the classical IH framework. For a canonical ensemble of IH [15] the energy is considered to be a function of area which is again assumed to be equispaced so that the summation can be replaced by integration. This assumption is severe and requires that only similar spins can be considered to be there at each puncture, preferably spin half, which is not true, although statistically spin half is the majority as found from the most probable distribution in the microcanonical ensemble. In the usual background independent approach to study the horizon thermodynamics in [15,16,[20][21][22] it is considered that 'the energy is a function of area'. In a precise sense, this is a very confusing statement. The statement should be made very precise, otherwise the underlying essence is not captured. Suppose one says the classical energy of an IH is the function of its classical area(or equivalently the CS level), which is fine. But this is not any sort of energy spectrum, rather the mean energy from the perspective of the QIH scenario [25,26]. Thus the statement 'equispaced area spectrum' does not have a clear meaning in the context of classical IH and hence approximations and assumptions had to be incorporated in [15,16,[20][21][22].
In this work, as we begin from the very fundamental quantum structure of the IH i.e. the QIH framework, the physical and technical aspects of the calculations are crystal clear. It is very much transparent that how the mean energy shows up in the partition function even though we begin from the quantum mechanical ensemble using the Hamiltonian operator for the QIH and hence the true energy spectrum of the same. Moreover, without having to make any assumption, we have the mean energy as the function of k and N , which are integers and hence equispaced. Thus the thermalized forms of the partition functions automatically reduce to the convenient forms which can further be approximated by replacing summation by integration.

VII. DISCUSSION
Let us end with a discussion on some delicate issues regarding this work and also investigating some intriguing possibilities of some future work that can follow from here on.
First of all, it is mention worthy that the major acquisition one can have by studying this work is a very logical step by step derivation of the 'thermalized' partition functions of the QIH based on the quantum geometric framework and a clear cut understanding about the role of quantum and thermal fluctuations in the corresponding thermodynamics; the 'thermalized' forms of the partition functions pave the way to study the consequences of the thermal fluctuations. But the most interesting observation made in this paper is that the QIH thermodynamics is quite different from IH thermodynamics. 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 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 [25,26] commutes with the number operator for punctures which renders N to be a constant of motion 4 . 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. 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.
Beginning from the quantum mechanical ensembles, the key to the passage to the 'thermalized' forms of the partition functions is the microcanonical ensemble result which gives us the number of microstates for given k and N . Now, at asymptopia where µ = 0, which in turn implies that σ = 0, gives a unique value of λ from eq.(III.5) (determined graphically in [10]). As a result k and N are linearly related as manifested by eq.(III.5), which in turn implies that the macroscopic or classical area of the IH is linearly discretized by N . This has a profound implication as far as the issue of black hole radiation is concerned. If the IH decay from one equilibrium state to the next one by a dynamical process, this will result in a discretized emission (or absorption) spectrum and a quantized back reaction. The issue needs to be looked at with further rigor and may result in a future publication [31].