Pitts, J. Brian (2017) Equivalent Theories Redefine Hamiltonian Observables to Exhibit Change in General Relativity. Classical and Quantum Gravity, 34. 055008. ISSN 13616382, Print ISSN 02649381

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Abstract
Change and local spatial variation are missing in canonical General Relativity's observables as usually defined, an aspect of the problem of time. Definitions can be tested using equivalent formulations of a theory, nongauge and gauge, because they must have equivalent observables and everything is observable in the nongauge formulation. Taking an observable from the nongauge formulation and finding the equivalent in the gauge formulation, one requires that the equivalent be an observable, thus constraining definitions. For massive photons, the de BroglieProca nongauge formulation observable A_{\mu} is equivalent to the StueckelbergUtiyama gauge formulation quantity A_{\mu} + \partial_{\mu} \phi, which must therefore be an observable. To achieve that result, observables must have 0 Poisson bracket not with each firstclass constraint, but with the RosenfeldAndersonBergmannCastellani gauge generator G, a tuned sum of firstclass constraints, in accord with the PonsSalisburySundermeyer definition of observables.
The definition for external gauge symmetries can be tested using massive gravity, where one can install gauge freedom by parametrization with clock fields X^A. The nongauge observable g^{\mu\nu} has the gauge equivalent X^A,_{\mu} g^{\mu\nu} X^B,_{\nu}. The Poisson bracket of X^A,_{\mu} g^{\mu\nu} X^B,_{\nu} with G turns out to be not 0 but a Lie derivative. This nonzero Poisson bracket refines and systematizes Kuchař's proposal to relax the 0 Poisson bracket condition with the Hamiltonian constraint. Thus observables need covariance, not invariance, in relation to external gauge symmetries.
The Lagrangian and Hamiltonian for massive gravity are those of General Relativity + \Lambda + 4 scalars, so the same definition of observables applies to General Relativity. Local fields such as g_{\mu\nu} are observables. Thus observables change. Requiring equivalent observables for equivalent theories also recovers HamiltonianLagrangian equivalence.
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