Is the Euclidean path integral always equal to the thermal partition function?

TL;DR

This work analyzes when the Euclidean path integral $Z^E$ matches the canonical thermal partition function $Z^C$ in curved static spacetimes. It shows that in non-compact spaces without Killing horizons the two quantities agree up to a vacuum-energy shift, while spacetimes bounded by horizons generally spoil the equality due to horizon-induced divergences and the mass-spectrum independence on $m$. The authors compare standard computation methods with a new regulator-based approach, proving that the latter yields the correct equality for both compact and non-compact spaces, including those with horizons. They also highlight multiple energy definitions in horizon spacetimes and demonstrate that derivatives of $\log Z^E$ with respect to $\beta$ do not universally reproduce any of these energies. The results have implications for covariant formulations of finite-temperature QFT in curved backgrounds and suggest a robust computational pathway via a regulator mass to ensure consistent thermodynamics in diverse geometries.

Abstract

The Euclidean path integral is compared to the thermal (canonical) partition function in curved static space-times. It is shown that if spatial sections are non-compact and there is no Killing horizon, the logarithms of these two quantities differ only by a term proportional to the inverse temperature, that arises from the vacuum energy. When spatial sections are bordered by Killing horizons the Euclidean path integral is not equal to the thermal partition function. It is shown that the expression for the Euclidean path integral depends on which integral is taken first: over coordinates or over momenta. In the first case the Euclidean path integral depends on the scattering phase shift of the mode and it is UV diverge. In the second case it is the total derivative and diverge on the horizon. Furthermore we demonstrate that there are three different definitions of the energy, and the derivative with respect to the inverse temperature of the Euclidean path integral does not give the value of any of these three types of energy. We also propose the new method of computation of the Euclidean path integral that gives the correct equality between the Euclidean path integral and thermal partition function for non-compact spaces with and without Killing horizon.