Time and Time Functions in Parametrized Non-Relativistic Quantum Mechanics

TL;DR

This paper examines the use of evolving constants to define time in parametrized non-relativistic quantum mechanics, highlighting interpretational issues and the necessary restrictions for consistency with standard quantum theory.

Contribution

It investigates the range of time functions in evolving constants and identifies the restrictions needed for consistency with usual quantum mechanics.

Findings

Allowed time functions must relate to spacetime geometry.

Restrictions are necessary to prevent inconsistent predictions.

Suitable restrictions are unlikely to extend to reparametrization invariant theories like gravity.

Abstract

The ``evolving constants'' method of defining the quantum dynamics of time-reparametrization-invariant theories is investigated for a particular implementation of parametrized non-relativistic quantum mechanics (PNRQM). The wide range of time functions that are available to define evolving constants raises issues of interpretation, consistency, and the degree to which the resulting quantum theory coincides with, or generalizes, the usual non-relativistic theory. The allowed time functions must be restricted for the predictions of PNRQM to coincide with those of usual quantum theory. They must be restricted to have a notion of quantum evolution in a time-parameter connected to spacetime geometry. They must be restricted to prevent the theory from making inconsistent predictions for the probabilities of histories. Suitable restrictions can be introduced in PNRQM but these seem unlikely to apply to a reparametrization invariant theory like general relativity.