Ehrenfest's Principle and the Problem of Time in Quantum Gravity

TL;DR

This paper investigates a proposal linking Ehrenfest equations to the problem of time in quantum gravity, deriving strict consistency conditions that limit solutions to the Wheeler-DeWitt equation, and discusses implications for the proposal's viability.

Contribution

It provides a detailed analysis of the Ehrenfest-based approach to the problem of time, identifying restrictive conditions on solutions to the Wheeler-DeWitt equation.

Findings

Only a restricted class of solutions satisfy all consistency conditions.

The proposal may be abandoned or used as a boundary condition for physical solutions.

Derivation of exact consistency conditions from Ehrenfest equations.

Abstract

We elaborate on a proposal made by Greensite and others to solve the problem of time in quantum gravity. The proposal states that a viable concept of time and a sensible inner product can be found from the demand for the Ehrenfest equations to hold in quantum gravity. We derive and discuss in detail exact consistency conditions from both Ehrenfest equations as well as from the semiclassical approximation. We also discuss consistency conditions arising from the full field theory. We find that only a very restricted class of solutions to the Wheeler-DeWitt equation fulfills all consistency conditions. We conclude that therefore this proposal must either be abandoned as a means to solve the problem of time or, alternatively, be used as an additional boundary condition to select physical solutions from the Wheeler-DeWitt equation.