Wavefunction of the universe: Reparametrization invariance and field redefinitions of the minisuperspace path integral

TL;DR

This paper investigates the Hartle-Hawking wavefunction in a minisuperspace model, analyzing how field redefinitions affect the path integral measure and quantum predictions, ultimately finding a universal norm in the semi-classical approximation.

Contribution

It demonstrates that different field redefinitions lead to distinct wavefunctions but yield the same quantum predictions when the inner product is properly defined.

Findings

Path integral measures depend on field redefinitions.

Semi-classical wavefunctions differ with redefinitions.

Quantum predictions remain consistent across prescriptions.

Abstract

We consider the Hartle-Hawking wavefunction of the universe defined as a Euclidean path integral that satisfies the "no-boundary proposal." We focus on the simplest minisuperspace model that comprises a single scale factor degree of freedom and a positive cosmological constant. The model can be seen as a non-linear $\sigma$-model with a line-segment base. We reduce the path integral over the lapse function to an integral over the proper length of the base and use diffeomorphism-invariant measures for the ghosts and the scale factor. As a result, the gauge-fixed path integral is independent of the gauge. However, we point out that all field redefinitions of the scale factor degree of freedom yield different choices of gauge-invariant path-integral measures. For each prescription, we compute the wavefunction at the semi-classical level and find a different result. We resolve in each case the ambiguity in the form of the Wheeler-DeWitt equation at this level of approximation. By imposing that the Hamiltonians associated with these possibly distinct quantum theories are Hermitian, we determine the inner products of the corresponding Hilbert spaces and find that they lead to a universal norm, at least semi-classically. Quantum predictions are thus independent of the prescription at this level of approximation. Finally, all wavefunctions of the Hilbert spaces of the minisuperspace model we consider turn out to be non-normalizable, including the no-boundary states.