A Simple Method for One-Loop Renormalization in Curved Space-Time

TL;DR

This paper introduces a straightforward method for one-loop renormalization in curved space-time, enabling explicit control over counterterms and applying it to a scalar field in an expanding universe, with comparisons to existing methods.

Contribution

The paper presents a new simple approach for deriving renormalization counterterms in curved space-time, including explicit finite parts and application to interacting scalar fields.

Findings

Method yields explicit counterterms with finite parts.

Agreement with Schwinger-DeWitt expansion results.

Disagreement with adiabatic subtraction in interacting theories.

Abstract

We present a simple method for deriving the renormalization counterterms from the components of the energy-momentum tensor in curved space-time. This method allows full control over the finite parts of the counterterms and provides explicit expressions for each term separately. As an example, the method is used for the self-interacting scalar field in a Friedmann-Robertson-Walker metric in the adiabatic approximation, where we calculate the renormalized equation of motion for the field and the renormalized components of the energy-momentum tensor to fourth adiabatic order while including interactions to one-loop order. Within this formalism the trace anomaly, including contributions from interactions, is shown to have a simple derivation. We compare our results to those obtained by two standard methods, finding agreement with the Schwinger-DeWitt expansion but disagreement with adiabatic subtractions for interacting theories.