Stochastic Force Due to a Quantum Scalar Field in Minkowski Spacetime

TL;DR

This paper develops a method to approximate the stochastic force tensor in stochastic gravity caused by quantum scalar field fluctuations in Minkowski spacetime, using a truncated Karhunen-Loeve transform and coarse graining.

Contribution

It introduces a novel approach to compute the stochastic force in stochastic gravity by applying a truncated Karhunen-Loeve transform and regularization techniques.

Findings

Derived approximate expressions for the stochastic force tensor.

Presented two realizations of the stochastic tensor in Minkowski spacetime.

Demonstrated the method's applicability to quantum scalar fields in vacuum state.

Abstract

A method is presented for computing approximate expressions for the stochastic force term $\xi_{ab}$ which appears in the Einstein-Langevin equation of stochastic gravity. Within this framework, $\xi_{ab}$ is a stochastic tensor field whose probability distribution mimics the probability distribution of the fluctuations of the quantum stress tensor operator; it is defined to be a random tensor field of zero mean whose correlation function is given by the expectation value of the symmetrized two point function of the stress energy fluctuation operator, called the noise kernel. Approximate expressions are obtained by means of a truncated Karhunen-Loeve transform defined on a random lattice of spacetime points. Due to the singular nature of the noise kernel, a coarse graining procedure is used to regulate divergences; as a result, the expressions obtained for $\xi_{ab}$ approximate values which might be seen by a probe measuring fluctuations in the stress energy using a sampling profile of finite width. Two realizations of $\xi_{ab}$ in Minkowski spacetime for the conformally invariant quantum scalar field in the Minkowski vacuum state are presented.