Relational time in generally covariant quantum systems: four models

TL;DR

This paper investigates the conditions for defining consistent relational quantum evolution in generally covariant systems, analyzing four models to determine how clock choices affect self-adjointness and the structure of quantum observables.

Contribution

It provides a detailed analysis of how clock variables influence the self-adjointness of evolving constants in covariant quantum systems across four distinct models.

Findings

Standard time choice in non-relativistic case ensures self-adjointness.

Relativistic case yields Klein-Gordon representation with well-defined position.

Certain models allow consistent quantization with suitable clock variables.

Abstract

We analize the relational quantum evolution of generally covariant systems in terms of Rovelli's evolving constants of motion and the generalized Heisenberg picture. In order to have a well defined evolution, and a consistent quantum theory, evolving constants must be self-adjoint operators. We show that this condition imposes strong restrictions to the choices of the clock variables. We analize four cases. The first one is non- relativistic quantum mechanics in parametrized form. We show that, for the free particle case, the standard choice of time is the only one leading to self-adjoint evolving constants. Secondly, we study the relativistic case. We show that the resulting quantum theory is the free particle representation of the Klein Gordon equation in which the position is a perfectly well defined quantum observable. The admissible choices of clock variables are the ones leading to space-like simultaneity surfaces. In order to mimic the structure of General Relativity we study the SL(2R) model with two Hamiltonian constraints. The evolving constants depend in this case on three independent variables. We show that it is possible to find clock variables and inner products leading to a consistent quantum theory. Finally, we discuss the quantization of a constrained model having a compact constraint surface. All the models considered may be consistently quantized, although some of them do not admit any time choice such that the equal time surfaces are transversal to the orbits.