Analytical approximation for $< \phi^2 >$ of a quantized scalar field in ultrastatic asymptotically flat spacetimes

TL;DR

This paper develops analytical approximations for the vacuum expectation value of the squared scalar field in ultrastatic asymptotically flat spacetimes, covering both massive and massless cases with arbitrary curvature coupling, and distinguishes low- and high-frequency contributions.

Contribution

It introduces a novel analytical method to approximate <φ^2> in these spacetimes, including an expansion for high-frequency parts analogous to DeWitt-Schwinger, and applies it to perturbed flat backgrounds.

Findings

Derived high-frequency expansion similar to DeWitt-Schwinger

Calculated low-frequency contribution in perturbed flat spacetime

Discussed the applicability limits of the approximations

Abstract

Analytical approximations for $< \phi^2 >$ of a quantized scalar field in ultrastatic asymptotically flat spacetimes are obtained. The field is assumed to be both massive and massless, with an arbitrary coupling $\xi$ to the scalar curvature, and in a zero or nonzero temperature vacuum state. The expression for $< \phi^2 >$ is divided into low- and high-frequency parts. The expansion for the high-frequency contribution to this quantity is obtained. This expansion is analogous to the DeWitt-Schwinger one. As an example, the low-frequency contribution to $< \phi^2 >$ is calculated on the background of the small perturbed flat spacetime in a quantum state corresponding to the Minkowski vacuum at the asymptotic. The limits of the applicability of these approximations are discussed.