Pitts, J. Brian (2016) What Are Observables in Hamiltonian Theories? Testing Definitions with Empirical Equivalence. In: UNSPECIFIED.

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Abstract
Change seems missing in Hamiltonian General Relativity's observables. The typical definition takes observables to have $0$ Poisson bracket with \emph{each} firstclass constraint. Another definition aims to recover Lagrangianequivalence: observables have $0$ Poisson bracket with the gauge generator $G$, a \emph{tuned sum} of firstclass constraints.
Empirically equivalent theories have equivalent observables. That platitude provides a test of definitions using de Broglie's massive electromagnetism. The nongauge ``Proca'' formulation has no firstclass constraints, so everything is observable. The gauge ``Stueckelberg'' formulation has firstclass constraints, so observables vary with the definition. Which satisfies the platitude? The team definition does; the individual definition does not.
Subsequent work using the gravitational analog has shown that observables have not a 0 Poisson bracket, but a Lie derivative for the Poisson bracket with the gauge generator $G$. The same should hold for General Relativity, so observables change locally and correspond to 4dimensional tensor calculus. Thus requiring equivalent observables for empirically equivalent formulations helps to address the problem of time.
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