On the interpretation of time-reparametrization-invariant quantum mechanics

TL;DR

This paper explores how time evolution in time-reparametrization-invariant quantum models can be understood through a proper time framework, providing an exact Heisenberg equation and examining the concept of a test clock.

Contribution

It demonstrates that time evolution can be described with respect to geometrical proper time, offering an exact Heisenberg equation, and introduces the concept of a viable test clock in semiclassical regimes.

Findings

Time evolution with respect to proper time is equivalent to standard quantum mechanics.

Exact Heisenberg equations are derived for these models.

A test clock can be used in the semiclassical limit to reveal time evolution.

Abstract

The classical and quantum dynamics of simple time-reparametrization- invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast to a similar proposal of Rovelli, evolution is with respect to the geometrical proper time, and the Heisenberg equation of motion is exact. The possibility of a ``test clock'', which would reveal time evolution while contributing negligibly to the Hamiltonian constraint is examined, and found to be viable in the semiclassical limit of large quantum numbers.