The role of time in relational quantum theories

TL;DR

This paper introduces a new approach to quantizing systems with a relational concept of time, addressing the problem of time in quantum gravity by capturing full classical dynamics and enabling evolution with an internal clock.

Contribution

It proposes a novel quantization strategy for systems with a single Hamiltonian constraint that preserves classical dynamics and introduces a minimal temporal structure for evolution.

Findings

Provides a consistent quantization method capturing full classical dynamics.

Applies the approach to Shape Dynamics, linking it to quantum gravity.

Results in a quantum theory similar to Dirac quantization of unimodular gravity.

Abstract

We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of Shape Dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity.