Changing Observables in Canonical General Relativity from Hamiltonian-Lagrangian Equivalence
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J. Brian Pitts (Cambridge) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "Changing Observables in Canonical General Relativity from Hamiltonian-Lagrangian Equivalence". Abstract: Is change missing in classical canonical General Relativity? If one insists on Hamiltonian-Lagrangian equivalence, then there is Hamiltonian change just when there is no time-like Killing vector field. Change has seemed missing partly due to Dirac’s belief that a first-class constraint, especially a primary, generates a gauge transformation. Pons showed that Dirac’s argument stops too soon: working to second order in time brings in first-class secondaries and hence the gauge generator G, a tuned sum of first-class constraints used by Anderson and Bergmann (1951) and recovered by Mukunda, Castellani et al. from the 1980s. I observe that trouble happens immediately: a first-class primary constraint generates an illegal change of initial data in GR, Maxwell and Yang-Mills. Dirac’s subtractive derivation misses it by cancellation; confusion between the electric field E(dA) and canonical momenta p (auxiliary fields in the canonical action \int dt (p \dot{q}-H) also obscures the problem. Dirac’s conjecture that a first-class secondary constraint generates a gauge transformation rests on a false assumption. Looking for gauge symmetries of the canonical action, one finds that the gauge generator G changes the action by at most a boundary term, but an isolated first-class constraint does not. The gauge generator G generates spatio-temporal coordinate transformations (not just spatial ones) for the space-time metric (not just the spatial metric). But are there locally varying observables in canonical General Relativity? Hamiltonian-Lagrangian equivalence guarantees that Hamiltonian observables are equivalent on-shell to Lagrangian observables. (Historically, Lagrangian-inequivalent observables may have arisen within Bergmann’s school due to novel postulation in Bergmann-Schiller 1953.) With first-class constraints exposed as not generating gauge transformations, observables’ Poisson brackets should be taken with the gauge generator G, as noted by Pons, Salisbury and Sundermeyer. Heeding Einstein’s point-coincidence argument excludes primitive point individuation and thus active diffeomorphisms in favor of (4-d) tensor calculus. Kuchař’s unsystematic waiver of the vanishing Poisson brackets condition to permit change has a more principled extension: observables should be internally gauge invariant (0 Poisson bracket with G for Maxwell, Yang-Mills, etc.) but externally gauge covariant. Hence the Poisson bracket with the coordinate-changing G should be the Lie derivative, indeed the Lie derivative of a geometric object (on-shell). For GR with no matter gauge group, observables are (on-shell) space-time geometric objects (components in coordinates with a transformation law). Hence the space-time metric and its concomitants (connection, curvature, etc.) are locally varying observables. Questions regarding Legendre projectability when an internal gauge group is also present and regarding the mixed supergravity transformations are noted. Velocity-dependent gauge transformations call for phase space extended by time---“phase space-time”; GR’s Lie derivative is an example. Vacuum GR’s phase space-time has 20 infinity^3 + 1 dimensions and 8 infinity^3 first-class constraints; one should not have expected a reduced phase space description of a theory with many-fingered time. Classical clarity might be of some use in quantization.