Relational evolution of observables for Hamiltonian-constrained systems

Evolution of systems in which Hamiltonians are generators of gauge transformations is a notion that requires more structure than the canonical theory provides. We identify and study this additional structure in the framework of relational observables (“partial observables”). We formulate necessary and sufficient conditions for the resulting evolution in the physical phase space to be a symplectomorphism. We give examples which satisfy those conditions and examples which do not. We point out that several classic positions in the literature on relational observables contain an incomplete approach to the issue of evolution and false statements. Our work provides useful clarification and opens the door to studying correctly formulated definitions.

Figure 1

The pictures illustrate the construction presented in the introduction. The picture (a) shows the phase space, the constraint surface, orbits of gauge transformations, the physical phase space, and the projection onto it. The picture (b) shows the physical phase space, the constraint surface, and the ($\tau$-dependent) sections which embed one into the other.