Effective action for scalar fields and generalised zeta-function regularisation

TL;DR

This paper develops a generalized zeta-function regularisation method to compute the one-loop effective action for scalar fields in hyperbolic FRW spacetimes, addressing divergences and renormalization.

Contribution

It introduces a novel generalization of zeta-function regularisation applicable to hyperbolic manifolds, enabling analysis of quantum scalar fields in such curved spacetimes.

Findings

Identifies additional divergences at one-loop level in hyperbolic manifolds.

Derives one-loop renormalisation group equations using the new regularisation.

Discusses conditions for one-loop renormalisability of the model.

Abstract

Motivated by the study of quantum fields in a Friedman-Robertson-Walker (FRW) spacetime, the one-loop effective action for a scalar field defined in the ultrastatic manifold $R\times H^3/\Gamma$, $H^3/\Gamma$ being the finite volume, non-compact, hyperbolic spatial section, is investigated by a generalisation of zeta-function regularisation. It is shown that additional divergences may appear at one-loop level. The one-loop renormalisability of the model is discussed and making use of a generalisation of zeta-function regularisation, the one-loop renormalisation group equations are derived.