Local momentum space: Scalar field and gravity

TL;DR

This paper develops a local momentum space method to derive propagator expansions for scalar fields and gravitons in curved spacetime, enabling the calculation of one-loop divergences in quantum gravity theories.

Contribution

It introduces a novel application of local momentum space techniques to compute propagators and divergences in scalar-gravity systems, highlighting differences in effective actions with and without gravity.

Findings

Derived propagator expansions up to first order in curvature.

Computed one-loop divergences for non-minimally coupled scalar fields.

Showed that limits of effective actions depend on the order of calculations.

Abstract

We use the local momentum space technique to obtain an expansion of the Feynman propagators for scalar field and graviton up to first order in the background curvature. The expressions for the propagators are cross-checked with the past literature as well as with the expressions for the traced heat kernel coefficients. The propagators so obtained are used to compute one-loop divergences in the Vilkovisky-Dewitt's effective action for a scalar field non-minimally coupled with gravity for an arbitrary spacetime metric background. The Vilkovisky-DeWitt effective action is then compared with the standard effective action in the limit $\kappa =0$, where $\kappa = 2/M_P$ in terms of the Planck mass. The comparison yields the important result that taking the limit $\kappa=0$ after computing the Vikovisky-DeWitt effective action is not equivalent to computing the Vikovisky-DeWitt effective action for the same theory in the absence of gravity.