Stress Tensor Correlators in the Schwinger-Keldysh Formalism

TL;DR

The paper analyzes ultraviolet aspects of stress-tensor correlators for a massless scalar in flat space using the Schwinger-Keldysh formalism and position-space dimensional regularization. It demonstrates that the mixed $+-$ and $-+$ correlators are ultraviolet finite at one loop, while the diagonal $++$ and $--$ correlators contain divergences that reproduce the 't Hooft–Veltman counterterms associated with the graviton self-energy, specifically the $R^2$ and $R_{\mu\nu}R^{\mu\nu}$ structure in $D=4$. The diagonal divergences are isolated as $1/(D-4)$ poles, linking to local gravitational counterterms, whereas the finite, state-dependent parts of the mixed correlators take the form of derivatives acting on $\ln(\mu^2 \Delta x_{\pm\mp}^2)/\Delta x_{\pm\mp}^2$. This work clarifies the UV structure of stress-tensor fluctuations within the SK framework and connects to conventional graviton self-energy analyses, with implications for Casimir-type phenomena and fluctuations in curved spacetimes.

Abstract

We express stress tensor correlators using the Schwinger-Keldysh formalism. The absence of off-diagonal counterterms in this formalism ensures that the +- and -+ correlators are free of primitive divergences. We use dimensional regularization in position space to explicitly check this at one loop order for a massless scalar on a flat space background. We use the same procedure to show that the ++ correlator contains the divergences first computed by `t Hooft and Veltman for the scalar contribution to the graviton self-energy.