Analytical approximation for⟨φ2⟩of a quantized scalar field in ultrastatic asymptotically flat spacetimes

Abstract

Analytical approximations for $⟨{\ensuremath{\varphi}}^{2}⟩$ 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 $\ensuremath{\xi}$ to the scalar curvature, and in a zero or nonzero temperature vacuum state. The expression for $⟨{\ensuremath{\varphi}}^{2}⟩$ 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 $⟨{\ensuremath{\varphi}}^{2}⟩$ 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.