Relational time in generally covariant quantum systems: Four models

We analyze 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 analyze four cases. The first one is nonrelativistic 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. Second, 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 spacelike simultaneity surfaces. In order to mimic the structure of general relativity we study the $\mathrm{SL}\mathrm{}\left(2R\right)\mathrm{}$ 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.