Finite temperature quantum field theory in the heat kernel method

TL;DR

This paper derives a finite temperature free energy expression in quantum field theory using the heat kernel method, accounting for boundary effects and topological constraints in (D+1)-dimensional Euclidean spacetime.

Contribution

It introduces a novel approach to compute free energy at finite temperature using heat kernel traces in spacetimes with boundaries and topology.

Findings

Free energy contains volume and boundary contributions.

The method is valid for arbitrary inverse temperature values.

Absolute zero temperature is topologically forbidden.

Abstract

The trace of the heat kernel in a (D+1)-dimensional Euclidean spacetime (integer D > 1) is used to derive the free energy in finite temperature field theory. The spacetime presents a D-dimensional compact space (domain) with a (D-1)-dimensional boundary, and a closed dimension, whose volume is proportional the Planck's inverse temperature. The thermal sum appears due to topology of the closed Euclidean time. The obtained free energy in (3+1) and (2+1) dimensions contain two contributions defined by the volume of a domain and by the volume of the domain's boundary. This functional is finite and valid for arbitrary values of the Planck's inverse temperature. The absolute zero of thermodynamic temperature is forbidden topologically, and no universal low temperature asymptotics of the free energy can exist.