Adiabatic regularization for massive and massless spin-$1$ fields

TL;DR

This paper develops an adiabatic regularization approach for massive and massless spin-1 fields in FLRW cosmologies, addressing the challenges of the massless limit and relating the results to scalar field renormalization.

Contribution

It introduces a direct method for adiabatic regularization of spin-1 fields that handles the massless limit without changing degrees of freedom, linking it to scalar field results.

Findings

The renormalized stress-energy tensor for massless spin-1 fields relates to that of scalar fields.

A detailed construction of adiabatic subtractions for spin-1 fields is provided.

Comparison with BPHZ renormalization highlights methodological connections.

Abstract

The adiabatic regularization method is likely the most direct and intuitive renormalization scheme for FLRW cosmologies. The method requires one to start with a nonvanishing mass, but massless theories can be studied by taking the massless limit at the end of the calculations. For spin-$1$, however, this limit changes the number of degrees of freedom, so it cannot be performed directly. In this work, we show a direct approach that begins with the canonical quantization of the physical degrees of freedom of a massive Proca field. We give the details of the construction and show that, in the massless limit, the renormalized stress-energy tensor of the Proca field is closely related to that of a minimally coupled scalar field. For completeness and pedagogical purposes, we also include a brief orientation to the adiabatic method for scalar fields. The construction of the adiabatic subtractions in momentum space is compared with that of the BPHZ method for renormalizing Feynman loop integrals.