A momentum-space representation of Green's functions with modified dispersion on ultra-static space-time

TL;DR

This paper develops a momentum-space method to represent Green's functions for scalar fields with modified dispersion on ultra-static spacetimes, overcoming limitations of traditional proper-time techniques.

Contribution

It introduces a novel momentum-space expansion of Green's functions in curved spacetime with modified dispersion, addressing higher-order derivatives and extending deWitt-Schwinger methods.

Findings

Green's functions expressed as frequency integrals in a preferred frame

Series expansion with geometric coefficients similar to deWitt-Schwinger

Expansion valid up to four derivatives of the metric tensor

Abstract

We consider the Green's functions associated to a scalar field propagating on a curved, ultra-static background, in the presence of modified dispersion relations. The usual proper-time deWitt-Schwinger procedure to obtain a series representation of the Green's functions is doomed to failure, because of higher order spatial derivatives in the Klein-Gordon operator. We show how to overcome this difficulty by considering a preferred frame, associated to a unit time-like vector. With respect to this frame, we can express the Green's functions as an integral over all frequencies of a space-dependent function. The latter can be expanded in momentum space, as a series with geometric coefficients similar to the deWitt-Schwinger's ones. By integrating over all frequencies, we finally find the expansion of the Green's function up to four derivatives of the metric tensor. The relation with the proper-time formalism is also discussed.