Time evolution and observables in constrained systems

TL;DR

This paper investigates the definition of time evolution and observables in constrained, finite-dimensional systems, providing a rigorous framework and exploring non-unitary evolution phenomena within quantum mechanics.

Contribution

It introduces the concept of local reducibility, proves the existence of complete perennials, and constructs a general geometric approach to time evolution in constrained systems.

Findings

Existence of complete sets of perennials in locally reducible systems

A geometric construction of time evolution that does not require symmetry

Non-unitary evolution allowing state loss during system evolution

Abstract

The discussion 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 {\em 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 Schroedinger and Heisenberg form that uses only geometry of the phase space is described. The time shifts are not required to be 1symmetries. A relation between perennials and observables of the Schroedinger 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 the ordinary quantum mechanics. The resulting non-unitarity is different from that known in the field theory (Hawking effect): state norms need not be preserved so that the system can be lost during the evolution of this kind.