Time evolution and observables in constrained systems

The investigation of constrained systems is limited to first-class parametrized systems, where the definition of time evolution and observables is not trivial, and to finite-dimensional systems in order that technicalities do not obscure the conceptual framework. The existence of reasonable true, or physical, degrees of freedom is rigorously defined and called local reducibility. A proof is given that any locally reducible system admits a complete set of perennials. For locally reducible systems, the most general construction of time evolution in the Schrodinger and Heisenberg form that uses only the geometry of the phase space is described. The time shifts are not required to be symmetries. A relation between perennials and observables of the Schrodinger or Heisenberg type results: such observables can be identified with certain classes of perennials and the structure of the classes depends on the time evolution. The time evolution between two non-global transversal surfaces is studied. The problem is posed and solved within the framework of ordinary quantum mechanics. The resulting non-unitarity is different from that known in field theory (Hawking effect): state norms need not be preserved so that the system can be lost during this kind of evolution.