Against the Wheeler-DeWitt equation

TL;DR

This paper proposes a covariant Hamiltonian quantization method for general relativity that avoids issues with the Wheeler-DeWitt equation, producing a wavefunction of the universe with different properties from traditional approaches.

Contribution

It introduces a mathematically rigorous covariant quantization technique using Rieffel induction, applied to a minisuperspace model, offering an alternative to the Wheeler-DeWitt equation.

Findings

Produces a wavefunction of the universe in a minisuperspace model

Differs from Vilenkin and Hartle-Hawking proposals for closed universes

Coincides with Hartle-Hawking in the open universe case

Abstract

The ADM approach to canonical general relativity combined with Dirac's method of quantizing constrained systems leads to the Wheeler-DeWitt equation. A number of mathematical as well as physical difficulties that arise in connection with this equation may be circumvented if one employs a covariant Hamiltonian method in conjunction with a recently developed, mathematically rigorous technique to quantize constrained systems using Rieffel induction. The classical constraints are cleanly separated into four components of a covariant momentum map coming from the diffeomorphism group of spacetime, each of which is linear in the canonical momenta, plus a single finite-dimensional quadratic constraint that arises in any theory, parametrized or not. The new quantization method is carried through in a minisuperspace example, and is found to produce a ``wavefunction of the universe". This differs from the proposals of both Vilenkin and Hartle-Hawking for a closed FRW universe, but happens to coincide with the latter in the open case.