What Are Observables in Hamiltonian Einstein–Maxwell Theory?
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Abstract
Is change missing in Hamiltonian Einstein-Maxwell theory? Given the most
common definition of observables (having weakly vanishing Poisson bracket with
each first-class constraint), observables are constants of the motion and
nonlocal. Unfortunately this definition also implies that the observables for
massive electromagnetism with gauge freedom (Stueckelberg) are inequivalent to
those of massive electromagnetism without gauge freedom (Proca). The
alternative Pons-Salisbury-Sundermeyer definition of observables, aiming for
Hamiltonian-Lagrangian equivalence, uses the gauge generator G, a tuned sum of
first-class constraints, rather than each first-class constraint separately,
and implies equivalent observables for equivalent massive electromagnetisms.
For General Relativity, G generates 4-dimensional Lie derivatives for
solutions. The Lie derivative compares different space-time points with the
same coordinate value in different coordinate systems, like 1 a.m. summer time
vs. 1 a.m. standard time, so a vanishing Lie derivative implies constancy
rather than covariance. Requiring equivalent observables for equivalent
formulations of massive gravity confirms that
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1572-9516