Abstract
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 assume these punctures to be indistinguishable, the microstate count for the horizon resembles that of Bose-Einstein counting formula for gas of particles. For the Bekenstein-Hawking area law to follow from the entropy calculation in the large area limit, the Barbero-Immirzi parameter (γ) approximately takes a constant value. As a by-product, we are able to speculate the state counting formula for the SU(2) quantum Chern-Simons theory coupled to indistinguishable sources in the weak coupling limit.
Highlights
One of the prime achievements of canonical quantum gravity, loop quantum gravity (LQG), is the provision of a description of microstates of equilibrium black hole horizon, modeled as quantum isolated horizon (IH) [1, 2], leading to an ab initio quantum statistical derivation of entropy from first principles
Unlike what occurs in case of quantum field theory one can develop a notion of particle number from the number operator and, by showing it as a conserved quantity, there is no such construction in existence for the number of punctures of quantum IH in LQG framework
With gj = 2j + 1, as the zeroth-order term. This formula is nothing but the state count for distinguishable punctures with configuration {sj}, obtained by ignoring the deeper details of the state counting associated with the internal symmetry of the quantum IH
Summary
One of the prime achievements of canonical quantum gravity, loop quantum gravity (LQG), is the provision of a description of microstates of equilibrium black hole horizon, modeled as quantum isolated horizon (IH) [1, 2], leading to an ab initio quantum statistical derivation of entropy from first principles. If these punctures are considered to be distinguishable, the microstate count for the quantum IH resembles that of Maxwell-Boltzmann (MB) counting for a gas of distinguishable particles [3,4,5,6] In this case, if the microcanonical entropy has to be given by Bekenstein-Hawking area law (BHAL) (i.e., one-fourth of the area of the horizon divided by Planck length squared), the Barbero-Immirzi parameter (γ) needs to take a certain fixed value. We explain that the resemblance between a quantum IH and a gas of particles follows if we view the quantum number j as denoting the area levels of an individual puncture, rather than “spin,” similar to the energy levels of an individual particle in a gas This provides an explanation of how our work differs from other instances in literature where BE statistics have been discussed in relation to quantum IH by considering the quantum number j as “spins” associated with the punctures (e.g., see [7]).
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