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

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.