TIME EVOLUTION AND OBSERVABLES IN CONSTRAINED SYSTEMS

The investigation of constrained systems is limited to first-class par ametrized systems, where the definition of time evolution and observab les is not trivial, and to finite-dimensional systems in order that te chnicalities do not obscure the conceptual framework. The existence of reasonable true, or physical, degrees of freedom is rigorously define d and called local reducibility. A proof is given that any locally red ucible system admits a complete set of perennials. For locally reducib le systems, the most general construction of time evolution in the Sch rodinger 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 H eisenberg type results: such observables can be identified with certai n classes of perennials and the structure of the classes depends on th e time evolution. The time evolution between two non-global transversa l surfaces is studied. The problem is posed and solved within the fram ework of ordinary quantum mechanics. The resulting non-unitarity is di fferent 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.