Equivalent theories redefine Hamiltonian observables to exhibit change in general relativity

Citation data:

Classical and Quantum Gravity, ISSN: 1361-6382, Vol: 34, Issue: 5, Page: 055008

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Repository URL:
http://arxiv.org/abs/1609.04812; http://philsci-archive.pitt.edu/id/eprint/12870
Pitts, J. Brian
IOP Publishing
Physics and Astronomy; General Relativity and Quantum Cosmology; High Energy Physics - Theory; Physics - History and Philosophy of Physics
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article description
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, non-gauge and gauge, because they must have equivalent observables and everything is observable in the non-gauge 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 Broglie-Proca non-gauge formulation observable Ais equivalent to the Stueckelberg-Utiyama gauge formulation quantity A+ ∂φ, which must therefore be an observable. To achieve that result, observables must have 0 Poisson bracket not with each first-class constraint, but with the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first-class constraints, in accord with the Pons-Salisbury- Sundermeyer 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. The non-gauge observable ghas the gauge equivalent X,gX,The Poisson bracket of X,gX,with G turns out to be not 0 but a Lie derivative. This non-zero 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 + Λ + 4 scalars, so the same definition of observables applies to General Relativity. Local fields such as gare observables. Thus observables change. Requiring equivalent observables for equivalent theories also recovers Hamiltonian-Lagrangian equivalence.