Relational Evolution of Observables for Hamiltonian-Constrained Systems

TL;DR

This paper clarifies the structure of relational observables in Hamiltonian-constrained systems, establishing conditions for their evolution to be symplectomorphisms and correcting misconceptions in previous literature.

Contribution

It identifies the necessary and sufficient conditions for relational observable evolution to be a symplectomorphism in constrained systems, providing a clearer framework and correcting prior incomplete approaches.

Findings

Established conditions for symplectomorphism in relational evolution

Provided examples satisfying and violating these conditions

Clarified misconceptions in existing literature on relational observables

Abstract

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.