A momentum-space representation of Green's functions with modified dispersion relations on general backgrounds

TL;DR

This paper develops a momentum-space series representation for Green's functions of a massive scalar field with modified dispersion relations on curved backgrounds, incorporating geometric terms like Ricci tensor components.

Contribution

It introduces a new method to compute Green's functions using Fermi normal coordinates and series expansion, accounting for higher derivative spatial operators and background geometry.

Findings

Series representation involves geometric terms such as Ricci tensor

Method applies to general backgrounds with geodesic vector fields

Results align with previous specific-background calculations

Abstract

We consider the problem of calculating the Green's functions associated to a massive scalar field with modified dispersion relations. We analyze the case when dispersion is modified by higher derivative spatial operators acting on the field orthogonally to a preferred direction, determined by a unit time-like vector field. By assuming that the integral curves of the vector field are geodesics, we expand the modified Klein-Gordon equation in Fermi normal coordinates. By means of a Fourier transform, we find a series representation in momentum-space of the Green's functions. The coefficients of the series are geometrical terms containing combinations of the Ricci tensor and the vector field, as expected from previous calculations with different methods and for specific backgrounds.