Some remarks on the thermal and vacuum fluctuations of a massive scalar field

TL;DR

This paper clarifies the detailed calculations of thermal and vacuum fluctuations of a massive scalar field, highlighting subtle mathematical aspects and providing explicit derivations for the behavior of certain integrals in the context of accelerated observers.

Contribution

It provides detailed derivations and clarifications of the calculations related to thermal and vacuum fluctuations of a massive scalar field, addressing subtleties in integral behaviors.

Findings

Explicit derivations of Minkowski vacuum fluctuations for accelerated observers

Analysis of the leading order behavior of parameter-dependent integrals

Clarification of subtleties in fluctuation calculations using the Riemann-Lebesgue lemma

Abstract

Thermal fluctuations of a massive scalar field in the Rindler wedge have been recently obtained. As a by product, the Minkowski vacuum fluctuations seen by a uniformly accelerated observer have been determined and confronted with the corresponding Minkowski thermal fluctuations of the same field, seen by an inertial observer. Since some of the calculations of this previous work have not been detailed on it, and they present some important subtleties, they are explicitly done here. These subtleties have to do with the leading order behaviour of certain parameter dependent integrals. Some of the leading order expansions are derived using the Riemann-Lebesgue lemma.