Gauge fixing and field redefinitions of the Hartle--Hawking wavefunction path integral

TL;DR

This paper reviews recent advances in understanding the Hartle--Hawking wavefunction for the universe, focusing on gauge fixing, field redefinitions, and their implications for quantum cosmology within Einstein gravity with a positive cosmological constant.

Contribution

It introduces different prescriptions for the no-boundary wavefunction via field redefinitions, revealing an infinite set of solutions with equivalent semi-classical inner products.

Findings

Different field redefinitions lead to multiple Wheeler--DeWitt equations.

Inner products of the resulting Hilbert spaces are semi-classically equivalent.

Universal quantum predictions emerge despite ordering ambiguities.

Abstract

We review some recent results concerning the Hartle--Hawking wavefunction of the universe. We focus on pure Einstein theory of gravity in the presence of a positive cosmological constant. We carefully implement the gauge-fixing procedure for the minisuperspace path integral, by identifying the single modulus and by using diffeomorphism-invariant measures for the ghosts and the scale factor. Field redefinitions of the scale factor yield different prescriptions for computing the no-boundary ground-state wavefunction. They give rise to an infinite set of ground-state wavefunctions, each satisfying a different Wheeler--DeWitt equation, at the semi-classical level. The differences in the form of the Wheeler--DeWitt equations can be traced to ordering ambiguities in constructing the Hamiltonian upon canonical quantization. However, the inner products of the corresponding Hilbert spaces turn out to be equivalent, at least semi-classically. Thus, the model yields universal quantum predictions.