Green functions and dimensional reduction of quantum fields on product manifolds

TL;DR

This paper analyzes Green functions on product manifolds, demonstrating that under certain conditions, the quantum field can be approximated by its zero mode, simplifying the analysis especially at large distances or high temperatures.

Contribution

It provides a rigorous approximation method for Euclidean Green functions on product manifolds, including estimates of the approximation's accuracy and implications for finite temperature quantum field theory.

Findings

Zero mode approximation is valid for compact manifolds M.

Approximation error diminishes at large distances on N.

High temperature limit reduces to a lower-dimensional theory.

Abstract

We discuss Euclidean Green functions on product manifolds P=NxM. We show that if M is compact then the Euclidean field on P can be approximated by its zero mode which is a Euclidean field on N. We estimate the remainder of this approximation. We show that for large distances on N the remainder is small. If P=R^{D-1}xS^{beta}, where S^{beta} is a circle of radius beta, then the result reduces to the well-known approximation of the D dimensional finite temperature quantum field theory to D-1 dimensional one in the high temperature limit. Analytic continuation of Euclidean fields is discussed briefly.