Equivalent Theories Redefine Hamiltonian Observables to Exhibit Change in General Relativity

Por • 23 sep, 2016 • Sección: Ciencia y tecnología

Brian Pitts

Abstract: Change and local spatial variation are missing in canonical GR’s observables as usually defined, part of the problem of time. Definitions can be tested using equivalent formulations, non-gauge and gauge, because they must have equivalent observables and everything is observable in the non-gauge formulation. Taking an observable from the non-gauge 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 observable A μ   is equivalent to the Stueckelberg-Utiyama quantity A μ +∂ μ ϕ.  Thus observables must have 0  Poisson bracket not with each first-class constraint, but with the 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 A .  The non-gauge observable g μν   has the gauge equivalent X A , μ g μν X B , ν .  The Poisson bracket of X A , μ g μν X B , ν   with G  turns out to be not 0  but a Lie derivative. This non-zero Poisson bracket refines and systematizes Kucha\v{r}’s proposal to relax the 0  Poisson bracket condition with the Hamiltonian constraint. Thus observables need covariance, not invariance, for external gauge symmetries. The Lagrangian and Hamiltonian for massive gravity are those of GR + Λ  + 4 scalars, so the same definition of observables applies to GR. Local fields such as g μν   are observables. Thus observables change. Requiring equivalent observables for equivalent theories also recovers Hamiltonian-Lagrangian equivalence.

arXiv:1609.04812v1 [gr-qc]

General Relativity and Quantum Cosmology (gr-qc); High Energy Physics – Theory (hep-th); History and Philosophy of Physics (physics.hist-ph)

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