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

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.

💡 Research Summary

The paper investigates the relationship between the Euclidean path integral (Z_E) and the canonical thermal partition function (Z_C) for quantum fields on static curved space‑times. It begins by recalling the two standard approaches: the Euclidean (imaginary‑time) functional integral, originally used by Gibbons and Hawking, and the canonical ensemble defined by the trace over the normal‑ordered Hamiltonian. For ultrastatic manifolds ((g_{00}=1)) it is well known that the two quantities differ only by a temperature‑independent vacuum‑energy term, (\log Z_E = \log Z_C - \beta E_0), which does not affect thermodynamic observables.

The authors then examine more general static backgrounds where (g_{00}) varies spatially. They discuss three conventional ways of evaluating the functional determinant that appears in (Z_E): ζ‑function regularisation, heat‑kernel expansion, and the coincident‑point thermal Feynman propagator. All three are mathematically equivalent when the functional measure is taken to be covariant (no explicit factor of (g_{00})). However, when one rewrites the path integral in Hamiltonian form, the measure acquires a factor (g_{00}^{1/2}). This non‑covariant piece becomes crucial.

Two distinct orders of integration are considered:

1. Integrate over fields first (coordinate‑first) – the factor (g_{00}^{1/2}) remains in the measure and couples to the scattering phase shift of each mode. The resulting expression for (\log Z_E) contains ultraviolet (UV) divergences and a non‑linear dependence on the inverse temperature (\beta).

2. Integrate over canonical momenta first (momentum‑first) – the measure transforms to (g_{00}^{-1/2}) and the remaining integral becomes a total derivative. In space‑times with Killing horizons ((g_{00}\to0) on the horizon) this total derivative yields a term proportional to (\delta(0),\ln g_{00}), i.e. a divergence localized on the horizon.

Both procedures give results that differ from the canonical partition function, demonstrating that the naive Euclidean path integral is not universally equivalent to (Z_C).

The paper identifies three notions of energy:

• Vacuum energy (E_0) – the constant term that appears in the functional determinant regularisation.
• Thermal expectation value (\langle:\hat H:\rangle) – the usual canonical ensemble energy.
• Total energy including horizon surface contributions – relevant when a Killing horizon is present.

A key observation is that (-\partial_\beta \ln Z_E) does not coincide with any of these three definitions, contradicting the standard statistical‑mechanical identity that holds for (Z_C).

To resolve the discrepancy, the authors introduce a new computational scheme. They first perform a conformal (optical) transformation of the static metric, \