Evolution in totally constrained models: Schr\"odinger vs. Heisenberg pictures

TL;DR

This paper compares Schr"odinger and Heisenberg evolution pictures within totally constrained quantum theories, demonstrating their equivalence in specific models and discussing broader implications for quantum gravity approaches.

Contribution

It provides a detailed quantum-level comparison of two evolution frameworks based on Rovelli's evolving constants, highlighting their equivalence in certain models.

Findings

Schr"odinger and Heisenberg pictures are equivalent in the studied models

The approach clarifies the role of internal time in quantum constrained systems

Discussion on the generality and potential issues of the equivalence

Abstract

We study the relation between two evolution pictures that are currently considered for totally constrained theories. Both descriptions are based on Rovelli's evolving constants approach, where one identifies a (possibly local) degree of freedom of the system as an internal time. This method is well understood classically in several situations. The purpose of this paper is to further analyze this approach at the quantum level. Concretely, we will compare the (Schr\"odinger-like) picture where the physical states evolve in time with the (Heisenberg-like) picture in which one defines parametrized observables (or evolving constants of the motion). We will show that in the particular situations considered in this manuscript (the parametrized relativistic particle and a spatially flat homogeneous and isotropic spacetime coupled to a massless scalar field) both descriptions are equivalent. We will finally comment on possible issues and on the genericness of the equivalence between both pictures.