Polymerization, the Problem of Access to the Saddle Point Approximation, and Thermodynamics

TL;DR

This paper explores an alternative method using polymer quantization, inspired by loop quantum gravity, to address issues in applying the saddle point approximation to black hole thermodynamics.

Contribution

It introduces a polymer quantization approach to resolve divergences in the saddle point approximation for certain black hole models.

Findings

Polymer quantization can regularize the action in problematic models.

The method offers an alternative to counter-term addition.

Potential implications for quantum gravity and black hole thermodynamics.

Abstract

The saddle point approximation to the partition functions is an important way of deriving the thermodynamical properties of black holes. However, there are certain black hole models and some mathematically analog mechanical models for which this method can not be applied directly. This is due to the fact that their action evaluated on a classical solution is not finite and its first variation does not vanish for all consistent boundary conditions. These problems can be dealt with by adding a counter-term to the classical action, which is a solution of the corresponding Hamilton-Jacobi equation. In this work however, we seek an alternative solution to this problem via the polymer quantization which is motivated by the loop quantum gravity.