Anomaly in canonical semiclassical gravity

TL;DR

The paper investigates whether a canonical semiclassical gravity framework—where gravity is classical and matter is quantum—can be self-consistent when matter contributions are replaced by their quantum expectation values. It builds effective constraints ${\cal H}^{\rm eff}$ and ${\cal C}^{\rm eff}$ and evolves the gravitational field using these, while the matter state evolves via a functional Schrödinger equation. A central result is that the Poisson brackets of these effective constraints do not close in general; anomalous terms appear that depend on the metric $q_{ab}$ and the matter state, signaling inconsistency for inhomogeneous spacetimes. This non-closure provides indirect evidence that gravity must be quantized and cannot be consistently treated purely semiclassically in the canonical framework; time-gauge fixes do not resolve the anomaly, and the paper discusses alternative semiclassical schemes and their limitations.

Abstract

We show that the canonical formulation of the semiclassical Einstein equation, where the matter terms in the constraints are replaced by expectation values of the corresponding operators in quantum states, is inconsistent due to the non-closure of the resulting constraint algebra.