The Functional Schrodinger Equation in the Semiclassical Limit of Quantum Gravity with a Gaussian Clock Field

TL;DR

This paper derives a functional Schrödinger equation for quantum fields in curved spacetime using a Gaussian dust clock, revealing how WKB time emerges in the semiclassical limit of quantum gravity.

Contribution

It introduces a derivation of the functional Schrödinger equation in quantum gravity with a Gaussian clock, clarifying the emergence of time in the semiclassical regime.

Findings

Functional Schrödinger equation includes a functional time derivative.

WKB expansion used to derive the equation in the semiclassical limit.

Standard Schrödinger evolution recovered in Minkowski spacetime.

Abstract

We derive the functional Schrodinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a WKB-type expansion of the wave functional in powers of the squared Planck mass. The functional Schrodinger equation that we obtain exhibits a functional time derivative that completes the usual definition of WKB time for curved spacetime, and the usual Schrodinger-type evolution is recovered in Minkowski spacetime.