Stress-energy Tensor Correlators in N-dim Hot Flat Spaces via the Generalized Zeta-Function Method

TL;DR

This paper computes the stress-energy tensor correlators for a massless scalar field at finite temperature in N-dimensional flat spacetime using the generalized zeta-function method, providing insights into quantum fluctuations and stochastic gravity.

Contribution

It introduces a method to calculate stress-energy correlators at finite temperature in arbitrary dimensions, including their temperature dependence and singular behaviors.

Findings

Correlators exhibit T^N dependence at low temperatures.

At high temperatures, correlators scale as T^2 with exponential suppression.

Singular behavior analyzed in the coincident limit.

Abstract

We calculate the expectation values of the stress-energy bitensor defined at two different spacetime points $x, x'$ of a massless, minimally coupled scalar field with respect to a quantum state at finite temperature $T$ in a flat $N$-dimensional spacetime by means of the generalized zeta-function method. These correlators, also known as the noise kernels, give the fluctuations of energy and momentum density of a quantum field which are essential for the investigation of the physical effects of negative energy density in certain spacetimes or quantum states. They also act as the sources of the Einstein-Langevin equations in stochastic gravity which one can solve for the dynamics of metric fluctuations as in spacetime foams. In terms of constitutions these correlators are one rung above (in the sense of the correlation -- BBGKY or Schwinger-Dyson -- hierarchies) the mean (vacuum and thermal expectation) values of the stress-energy tensor which drive the semiclassical Einstein equation in semiclassical gravity. The low and the high temperature expansions of these correlators are also given here: At low temperatures, the leading order temperature dependence goes like $T^{N}$ while at high temperatures they have a $T^{2}$ dependence with the subleading terms exponentially suppressed by $e^{-T}$. We also discuss the singular behaviors of the correlators in the $x'\rightarrow x$ coincident limit as was done before for massless conformal quantum fields.