Quantum mechanics on York slices

TL;DR

This paper develops a quantum theory of cosmological dynamics using York time in an anisotropic minisuperspace model, revealing a non-canonical structure and solving for the Hamiltonian's eigenspectrum.

Contribution

It introduces a novel quantisation approach for York time in anisotropic cosmology, handling non-canonical variables and deriving the Hamiltonian spectrum.

Findings

Quantum theory has no momentum representation, with a fundamental position basis.

Successfully solved for the Hamiltonian's eigenspectrum in the model.

Discussed potential extensions to non-homogeneous cases and perturbation theory.

Abstract

For some time the York time parameter has been identified as a candidate for a physically meaningful time in cosmology. An associated Hamiltonian may be found by solving the Hamiltonian constraint for the momentum conjugate to the York time variable, although an explicit solution can only be found in highly symmetric cases. The Poisson structure of the remaining variables is not canonical. Here we quantise this dynamics in an anisotropic minisuperspace model via a natural extension of canonical quantisation. The resulting quantum theory has no momentum representation. Instead the position basis takes a fundamental role. We illustrate how the quantum theory and the modified representation of its momentum operators lead to a consistent theory in the presence of the constraints that arose during the Hamiltonian reduction. We are able to solve for the eigenspectrum of the Hamiltonian. Finally we discuss how far the results of this model extend to the general non-homogeneous case, in particular perturbation theory with York time.