Fully renormalized stress tensor correlator in flat space

TL;DR

This paper develops a comprehensive method for renormalizing the stress tensor two-point function in flat spacetime, providing explicit formulas and extending the approach to various states and potential curved backgrounds.

Contribution

It introduces a detailed renormalization procedure for the stress tensor correlator in position space, applicable to massive fields and general states, with potential extensions to curved spacetimes.

Findings

Explicit counterterms and finite parts for the massive scalar case

Agreement with previous massless results and momentum space calculations

Potential applicability to inflationary spacetimes

Abstract

We present a general procedure to renormalize the stress tensor two-point correlation function on a Minkowski background in position space. The method is shown in detail for the case of a free massive scalar field in the standard Minkowski vacuum state, and explicit expressions are given for the counterterms and finite parts, which are in full accordance with earlier results for the massless case. For the general case in position space, only regularized --- but not renormalized --- results have been obtained previously. After a Fourier transformation to momentum space, we also check agreement with a previous calculation there. We generalize our results to general Hadamard states. Furthermore, the proposed procedure can presumably be generalized to the important case of an inflationary spacetime background, where the transition to momentum space is in general not possible.