Quantum Fields in Hyperbolic Space-Times with Finite Spatial Volume

TL;DR

This paper analyzes the one-loop effective action of a massive scalar field in a hyperbolic space-time with finite volume, using the Selberg trace formula, and explores divergences, regularizations, and finite temperature effects.

Contribution

It introduces a method to compute the effective action in hyperbolic spaces with finite volume using the Selberg trace formula, including divergence analysis and temperature effects.

Findings

The zeta function has a simple pole at s=0 for arbitrary gravitational coupling.

Explicit one-loop divergences are derived using proper-time regularization.

Finite temperature effects and high-temperature expansion are analyzed.

Abstract

The one-loop effective action for a massive self-interacting scalar field is investigated in $4$-dimensional ultrastatic space-time $ R \times H^3/\Gamma$, $H^3/\Gamma$ being a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the $\zeta$-function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that $\zeta(s)$ has a simple pole at $s=0$. The one-loop effective action is analysed by means of proper-time regularisations and the one-loop divergences are explicitly found. It is pointed out that, in this special case, also $\zeta$-function regularisation requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite temperature effects are studied and the high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like space-time is briefly discussed.