Time Evolution in Canonical Quantum Gravity is Trivial

TL;DR

This paper shows that in a covariant canonical formulation of quantum gravity, the apparent time evolution is trivial and can be removed by gauge fixing, resolving longstanding issues with the Wheeler-DeWitt equation.

Contribution

It introduces a covariant canonical framework with embedding coordinates, demonstrating that time evolution in quantum gravity is trivial after gauge fixing.

Findings

Time evolution is equivalent to coordinate change in the wave function.

Gauge fixing removes the embedding coordinates and the time evolution equation.

The formalism provides a background independent quantum gravity description.

Abstract

The Wheeler-DeWitt (WdW) equation does not describe any explicit time evolution of the wave function, and somehow related to this issue, there is no natural way of defining an invariant inner product that provides a viable probability interpretation. We show that both of these difficulties are solved in a covariant canonical formulation of general relativity where the configuration space is extended by introducing the embedding coordinates as dynamical variables. The formalism describes the evolution of the wave function from one spacelike slice to another, but as in the case of spatial diffeomorphisms this is simply implemented by a coordinate change in the wave function. We demonstrate how the time evolution equation disappears after gauge fixing that removes the embedding coordinates. These findings indicate that the time evolution is trivial in a background independent formulation of quantum gravity.