Polarization of the conformal vacuum in the Milne universe

TL;DR

This paper investigates the vacuum expectation values of a massless scalar field in the Milne universe, revealing that the Minkowski vacuum appears thermal relative to the conformal vacuum, with temperature characteristics depending on spatial dimensions.

Contribution

It provides an integral representation for the difference of Wightman functions and analyzes the thermal nature of the Minkowski vacuum in various dimensions.

Findings

Minkowski vacuum appears as a thermal state relative to the conformal vacuum.

Thermal factor is Bose-Einstein in odd dimensions.

Thermal factor is Fermi-Dirac in even dimensions.

Abstract

The vacuum expectation values (VEVs) of the field squared and energy-momentum tensor for a massless scalar field are investigated in the Milne universe with general number of spatial dimensions. The vacuum state depends on the choice of the mode functions in the canonical quantization procedure and we assume that the field is prepared in the conformal vacuum. As the first step an integral representation for the difference of the Wightman functions corresponding to the conformal and Minkowski vacua is derived. The mean field squared and energy-momentum tensor are obtained in the coincidence limit. It is shown that the Minkowski vacuum state is interpreted as a thermal one with respect to the conformal vacuum. The thermal factor is of the Bose-Einstein type in odd dimensional space and of the Fermi-Dirac type in even number of spatial dimensions.