^{3}.

In loop quantum gravity the quantum geometry of a black hole horizon consists of discrete nonperturbative quantum geometric excitations (or punctures) labeled by spins, which are responsible for the quantum area of the horizon. If these punctures are compared to a gas of particles, then the spins associated with the punctures can be viewed as single puncture area levels analogous to single particle energy levels. Consequently, if we

One of the prime achievements of canonical quantum gravity, more specifically

However, recently there has been a trend of doing black hole thermodynamics in the quantum IH framework considering the punctures to be

In this article, we shall revisit this issue of indistinguishability of punctures of quantum IH. We

An individual puncture of a quantum IH labeled with quantum number

On the other hand, let us consider a gas of particles with total energy

There is a manifest structural similarity between these two systems if we consider the following correspondence

As the underlying quantum theory of the gas of particles provides the details of

Now, for a quantum IH,

(

The counting details are available in standard textbooks of statistical mechanics (e.g., see [

This completes our effort to explain in what precise sense the punctures of a quantum IH obey BE statistics under the assumption of indistinguishability. However, we need to further clarify certain other issues regarding the analogy between a quantum IH and a gas of particles in order to differentiate our viewpoint from other instances in literature which discuss BE statistics in relation to quantum IH.

From the correspondence

The spins of particles do not explicitly appear in the microstate counting. It only implicitly affects the counting by imposing restriction on

In spite of all the above facts one can still wish to treat the quantum number

Now, we shall calculate the entropy of an IH with classical area

The next step is to choose

The allowed values of

The blue curves are the solutions of (

The view of the plot in Figure

In this section we discuss a few possible new ideas which germinate from this exercise of state counting and entropy calculation for quantum IH with indistinguishable punctures. To begin with, we briefly discuss the subject matter of [

Now, when we consider the punctures to be indistinguishable, we do not have a complete formula for the microstate count from which such a similar exercise can be performed. So, we try to go in the reverse direction in this case and look for the possible complete formula for the state count which takes into account the underlying symmetry of the IH and the associated Chern-Simons theory. Thus, the first step for the state counting for indistinguishable punctures is to ignore the minute details involved with the symmetries and so forth of the quantum IH and do the state counting for a configuration

Henceforth, taking cue from the case of distinguishable punctures, we speculate that the combinatorial form of formula (

Apart from this, the problem becomes more interesting due to the fact that it has a direct link with the counting of the conformal blocks of Wess-Zumino-Witten model [

What we have done here is the entropy calculation of the IH by assuming that the punctures of quantum IH are indistinguishable and we do not go into the debate whether this assumption is justified within this quantum IH framework. As far as the physical result is concerned, that is, the BHAL resulting from an

However, there is a very important issue which we are unable to address here in the context of the assumption of indistinguishability. The states of the quantum IH are actually given by that of the quantum SU

Here we have presented some magnified versions of the 3D plot in Figures

This is a magnified view of the plot in Figure

This is a magnified view of the plot in Figure

The author declares that there is no conflict of interests regarding the publication of this paper.

The author thanks Romesh Kaul for discussions regarding a few issues of this work. This work is funded by The Department of Atomic Energy, India.