From path integrals to the Wheeler-DeWitt equation: Time evolution in spacetimes with a spatial boundary

TL;DR

This paper derives the Wheeler-DeWitt equation from the path integral in quantum gravity without explicit spacetime splitting, showing that boundary conditions at spatial boundaries govern time evolution.

Contribution

It presents a formal derivation of the Wheeler-DeWitt equation from path integrals using boundary variations, avoiding explicit 3+1 spacetime splitting.

Findings

Transition amplitudes depend on boundary conditions at spatial boundaries.

Time evolution is governed by boundary conditions on the gravitational field.

Boundary conditions at joints are necessary for nonsmooth boundaries.

Abstract

We reexamine the relationship between the path integral and canonical formulation of quantum general relativity. In particular, we present a formal derivation of the Wheeler-DeWitt equation from the path integral for quantum general relativity by way of boundary variations. One feature of this approach is that it does not require an explicit 3+1 splitting of spacetime in the bulk. For spacetimes with a spatial boundary, we show that the dependence of the transition amplitudes on spatial boundary conditions is determined by a Wheeler-DeWitt equation for the spatial boundary surface. We find that variations in the induced metric at the spatial boundary can be used to describe time evolution---time evolution in quantum general relativity is therefore governed by boundary conditions on the gravitational field at the spatial boundary. We then briefly describe a formalism for computing the dependence of transition amplitudes on spatial boundary conditions. Finally, we argue that for nonsmooth boundaries, meaningful transition amplitudes must depend on boundary conditions at the joint surfaces.