Equivalence of the Path Integral for Fermions in Cartesian and Spherical Coordinates

TL;DR

This paper demonstrates the equivalence of fermionic path integral formulations in Cartesian and spherical coordinates, and explores implications for black hole thermodynamics in curved spacetime.

Contribution

It explicitly establishes the equivalence of fermionic path integrals in different coordinate systems and connects flat spacetime calculations to black hole thermodynamics.

Findings

Path integral in spherical coordinates matches Cartesian results.

Near-horizon limit relates to black hole thermodynamics.

Provides a basis for studying fermions in curved spacetime.

Abstract

The path-integral calculation for the free energy of a spin-1/2 Dirac-fermion gas is performed in spherical polar coordinates for a flat spacetime geometry. Its equivalence with the Cartesian-coordinate representation is explicitly established. This evaluation involves a relevant limiting case of the fermionic path integral in a Schwarzschild background, whose near-horizon limit has been shown to be related to black hole thermodynamics.