Quantum phase origin of spacetime curvature

This paper presents an exhaustive derivation of the identity Gµν−iλΨ−1δΨ/δgµν = 0, which defines the semiclassical limit of quantum geometrodynamics. By utilizing the ADM (Arnowitt-Deser-Misner) decomposition of the Einstein-Hilbert action and the subsequent canonical quantization resulting in the Wheeler-DeWitt equation, we demonstrate that the Einstein tensor is the first-order momentum density of the gravitational field’s wavefunctional. We further extend the analysis to include second-order WKB corrections and the physical implications for the problem of time in quantum cosmology.