A Wheeler-DeWitt Equation with Time

TL;DR

This paper introduces a method to incorporate explicit time dependence into the Wheeler-DeWitt equation by modifying the classical Hamilton-Jacobi-Einstein equation, demonstrated with a de Sitter minisuperspace example.

Contribution

It proposes a straightforward approach to include a meaningful notion of time in the Wheeler-DeWitt equation by making Hamilton's principal function explicitly time-dependent before quantization.

Findings

Derived a Wheeler-DeWitt equation with explicit time dependence.

Provided a working solution for the de Sitter minisuperspace.

Addressed the problem of time in canonical quantum gravity.

Abstract

The equation for canonical gravity produced by Wheeler and DeWitt in the late 1960s still presents difficulties both in terms of its mathematical solution and its physical interpretation. One of these issues is, notoriously, the absence of an explicit time. In this short note, we suggest one simple and straightforward way to avoid this occurrence. We go back to the classical equation that inspired Wheeler and DeWitt (namely, the Hamilton--Jacobi--Einstein equation) and make explicit, before quantization, the presence of a known, classically meaningful notion of time. We do this by allowing Hamilton's principal function to be explicitly dependent on this time locally. This choice results in a Wheeler--DeWitt equation with time. A working solution for the de Sitter minisuperspace is shown.