A relationship between scalar Green functions on hyperbolic and Euclidean Rindler spaces

TL;DR

This paper establishes a mathematical link between Green functions in hyperbolic and Euclidean Rindler spaces, facilitating calculations in quantum field theory at finite temperatures and on curved manifolds.

Contribution

It derives a simple, dimension-independent formula connecting Green functions on Euclidean Rindler and hyperbolic spaces, with applications to various space-time geometries.

Findings

Green functions are equal at specific momenta in both spaces.

The relation simplifies in momentum space with an additive relation between mass and momenta.

Applications include finite temperature Green functions and Green functions on cones and Milne space.

Abstract

We derive a formula connecting in any dimension the Green function on the D+1 dimensional Euclidean Rindler space and the one for a minimally coupled scalar field with a mass m in the D dimensional hyperbolic space. The relation takes a simple form in the momentum space where the Green functions are equal at the momenta (p_0,\bf p) for Rindler and (m,\bf p) for hyperbolic space with a simple additive relation between the squares of the mass and the momenta. The formula has applications to finite temperature Green functions, Green functions on the cone and on the 9compactified) Milne space-time. Analytic continuations and interacting quantum fields are briefly discussed.