The problem of time in parametrized theories

TL;DR

This paper explores the problem of defining time in reparametrization invariant theories like quantum gravity, proposing a solution inspired by proper time formalism in relativistic quantum mechanics.

Contribution

It extends the proper time approach from relativistic quantum mechanics to quantum gravity, offering a new perspective on the problem of time.

Findings

Identifies the analogy between localization and problem of time.

Proposes a formalism to address the problem of time in quantum gravity.

Suggests a potential resolution based on proper time concepts.

Abstract

A common feature of reparametrization invariant theories is the difficulty involved in identifying an appropriate evolution parameter and in constructing a Hilbert space on states. Two well known examples of such theories are the relativistic point particle and the canonical formulation of quantum gravity. The strong analogy between them (specially for minisuperspace models) is considered in order to stress the correspondence between the ``localization problem'' and the ``problem of time,'' respectively. A possible solution for the first problem was given by the proper time formulation of relativistic quantum mechanics. Thus, we extrapolate the main outlines of such a formalism to the quantum gravity framework. As a consequence, a proposal to solve the problem of time arises.