Quantizing matter fields on curved spacetime makes a gravitational stress-energy tensor possible where classical GR proved none could exist

physicsphilosophy-of-physicsquantum-field-theorygeneral-relativitysemiclassical-gravitystress-energy-tensorrenormalizationtrace-anomalyhist-pharxiv

Curiel's 2019 non-existence theorem shows that classically, no tensor built from the metric and the Riemann curvature can satisfy all the conditions a gravitational stress-energy tensor would need — in particular, remaining non-zero wherever curvature is non-zero while also being covariantly divergence-free and dimensionally weight-zero under a constant rescaling of the metric. Ferreiro and Cavallaro (arXiv:2610.10588, 2026) argue this changes once matter fields are quantized on a fixed classical background — semiclassical gravity, short of full quantum gravity.

Renormalizing the quantum stress-energy operator (via point-splitting and Hadamard-parametrix subtraction) forces in a local, state-independent counterterm built purely from curvature. This term is not scale-invariant: under a constant rescaling of the metric it acquires a term proportional to the logarithm of the scale factor, because quantization breaks the classical conformal symmetry a massless field would otherwise have and replaces a vanishing trace with the trace (conformal) anomaly — a fixed multiple of the identity. Since Curiel's proof depended on the semiclassical source being dimensionally weight-zero, and the anomaly breaks exactly that condition, a candidate gravitational stress-energy tensor becomes available that classical GR had proved impossible.