Renormalization of the Quantum Stress Tensor Fluctuations and the Limits of Semiclassical Gravity

TL;DR

This work interrogates the validity of semiclassical gravity by renormalizing the stress-energy tensor and its fluctuations for quantum fields on curved spacetimes. Using Hadamard states and the operator product expansion, it provides a covariant framework to compute ⟨T_{ab}⟩_ren and ⟨T_{ab}T_{cd}⟩_ren, with explicit 2d results showing that squeezed vacua yield fluctuations comparable to the mean, thus violating semiclassicality. The authors derive a concrete, pointwise criterion for semiclassical gravity and demonstrate that even modest excitations away from the Minkowski vacuum can lead to large backreaction fluctuations, with profound implications for black hole evaporation and cosmology. Collectively, the findings suggest that fully quantum gravitational degrees of freedom are necessary to accurately describe backreaction in several physically important regimes.

Abstract

We analyze the expectation value of the energy-momentum tensor and its fluctuations in quantum field theory on curved spacetimes $\langle T_{ab} \rangle$. A necessary condition for the conceptual consistency of semiclassical gravity, where $\langle T_{ab} \rangle$ represent the sources of the Einstein equations, is that the fluctuations of the energy momentum tensor remain small compared to its expectation value. We study the renormalization of both the energy-momentum tensor $\langle T_{ab}(x)\rangle_{\rm ren}$ and the fluctuation tensor $\langle T_{ab}(x) T_{cd}(x) \rangle_{\rm ren}$ for suitable Hadamard states, using the operator product expansion for a free scalar field on a fixed curved background. We show that squeezed vacua -- arising naturally in black hole evaporation and in inflationary cosmology -- fail to satisfy the semiclassicality criterion.