Holographic Quantum Statistics from Dual Thermodynamics

TL;DR

This paper introduces a dual thermodynamic framework for black holes, mapping black hole properties to quantum systems, satisfying all thermodynamic laws, and providing a basis for quantum statistical models consistent with holography.

Contribution

It proposes a dual thermodynamics for black holes that satisfies all thermodynamic laws and constructs toy quantum models illustrating the duality and quantum corrections.

Findings

Dual variables satisfy all thermodynamic laws including the third law.

Constructed toy models from quantum gases replicate black hole thermodynamics.

Obtained quantum corrections to entropy, such as logarithmic terms.

Abstract

We propose dual thermodynamics corresponding to black hole mechanics with the identifications E' -> A/4, S' -> M, and T' -> 1/T in Planck units. Here A, M and T are the horizon area, mass and Hawking temperature of a black hole and E', S' and T' are the energy, entropy and temperature of a corresponding dual quantum system. We show that, for a Schwarzschild black hole, the dual variables formally satisfy all three laws of thermodynamics, including the Planck-Nernst form of the third law requiring that the entropy tend to zero at low temperature. This is in contrast with traditional black hole thermodynamics, where the entropy is singular. Once the third law is satisfied, it is straightforward to construct simple (dual) quantum systems representing black hole mechanics. As an example, we construct toy models from one dimensional (Fermi or Bose) quantum gases with N ~ M in a Planck scale box. In addition to recovering black hole mechanics, we obtain quantum corrections to the entropy, including the logarithmic correction obtained by previous papers. The energy-entropy duality transforms a strongly interacting gravitational system (black hole) into a weakly interacting quantum system (quantum gas) and thus provides a natural framework for the quantum statistics underlying the holographic conjecture.