Exact path integrals on half-line in quantum cosmology with a fluid clock and aspects of operator ordering ambiguity

TL;DR

This paper performs exact path integral quantization of flat cosmological models with a fluid clock, addressing operator ordering ambiguities and their impact on symmetries and the uniqueness of quantum Hamiltonians.

Contribution

It introduces a specific operator ordering that preserves classical symmetries, leading to a unique quantum Hamiltonian for higher dimensions and analyzing symmetry loss in one dimension.

Findings

A particular operator ordering preserves lapse rescaling and covariance symmetries.

For D>2, the quantum Hamiltonian is uniquely determined by symmetry considerations.

In D=1, the lapse rescaling symmetry is lost, causing ambiguity in the quantum theory.

Abstract

We perform $\textit{exact}$ half-line path integral quantization of flat, homogeneous cosmological models containing a perfect fluid acting as an internal clock, in a $D+1$ dimensional minisuperspace setup. We also discuss certain classes of operator ordering ambiguity inherent in such quantization procedures and argue that a particular ordering prescription in the quantum theory can preserve two symmetries, namely arbitrary lapse rescalings and general covariance, which are already present at the classical level. As a result of this imposition, a large class of quantum Hamiltonians differing by operator ordering produces the same inner products between quantum states. This imposition of the two symmetries of the classical minisuperspace models leads to a unique prescription for writing the quantum Hamiltonian for minisuperspace dimension $D>2$. Interestingly, in the case of $D=1$, the lapse rescaling symmetry is lost in the quantum theory, leading to an essentially ambiguous description of the canonical theory. We provide general proof of this in the context of both canonical quantization and path integrals. We supply concrete examples to validate our findings further.