Relational evolution of the degrees of freedom of generally covariant quantum theories

TL;DR

This paper explores the classical and quantum dynamics of generally covariant theories with zero Hamiltonian, focusing on relational evolution, evolving constants of motion, and a generalized Heisenberg picture with multiple time variables.

Contribution

It introduces a method to find evolving constants of motion and extends the Heisenberg picture to multiple time variables in covariant quantum theories.

Findings

Derived the geometric meaning of relational evolution.

Proposed a new method for finding evolving constants.

Applied the framework to models mimicking general relativity.

Abstract

We study the classical and quantum dynamics of generally covariant theories with vanishing a Hamiltonian and with a finite number of degrees of freedom. In particular, the geometric meaning of the full solution of the relational evolution of the degrees of freedom is displayed, which means the determination of the total number of evolving constants of motion required. Also a method to find evolving constants is proposed. The generalized Heinsenberg picture needs M time variables, as opposed to the Heisenberg picture of standard quantum mechanics where one time variable t is enough. As an application, we study the parameterized harmonic oscillator and the SL(2,R) model with one physical degree of freedom that mimics the constraint structure of general relativity where a Schrodinger equation emerges in its quantum dynamics.