Modified Dispersion Relations and trans-Planckian Physics

TL;DR

This paper explores how modified dispersion relations in quantum field theory on curved spacetime affect Green's functions, proposing a method to incorporate Lorentz invariance breaking via a preferred frame and deriving metric expansions.

Contribution

It introduces a framework for calculating Green's functions with modified dispersion relations in curved spacetime, including a four-derivative metric expansion and a connection to proper-time formalism.

Findings

Derived Green's function expansion up to four derivatives of the metric

Established a method to incorporate a preferred frame in ultra-static spacetimes

Connected the modified dispersion relations approach with the proper-time formalism

Abstract

We consider modified dispersion relations in quantum field theory on curved space-time. Such relations, despite breaking the local Lorentz invariance at high energy, are considered in several phenomenological approaches to quantum gravity. Their existence involves a modification of the formalism of quantum field theory, starting from the problem of finding the scalar Green's functions up to the renormalization of various quantum expectation values. In this work we consider a simple example of such modifications, in the case of ultra-static metric. We show how to overcome the lack of Lorentz invariance by introducing a preferred frame, with respect to which 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, and 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.