James Brian Pitts
Mohamed bin Zayed University of Artificial Intelligence and University of Cambridge
Corey Sargent Memorial Lecture Series in Astrophysics and Cosmology
Abstract:
Is change missing in Hamiltonian General Relativity and canonical quantum gravity? Given the most common definition of observables in canonical quantum gravity, they are constants of the motion and spatially nonlocal. This conclusion is due to a definition, that observables have 0 Poisson bracket with each first-class constraint, inferred from the doctrine that a first-class constraint generates a gauge transformation. Yet other definitions exist, such as one seeking Hamiltonian-Lagrangian equivalence due to Pons, Salisbury and Sundermeyer. They take gauge transformations to be generated by first-class constraints as a team, the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G; then observables have 0 Poisson bracket with G. For electromagnetism there are two first-class constraints but one arbitrary function; for GR there are eight first-class constraints but four arbitrary functions. Kucha艡 dropped the 0 Poisson bracket for the Hamiltonian constraint. The author has combined the G reform and a systematized non-zero Poisson bracket to define observables as invariant (0 Poisson bracket with G) under internal symmetries but covariant (having a space-time Lie derivative for the Poisson bracket with G on-shell) under external symmetries.
How can one test definitions? By using two formulations of a theory, one without gauge freedom and one with gauge freedom; being empirically equivalent, they must have equivalent observables. This platitude permits algorithmic testing of definitions using de Broglie-Proca and Stueckelberg massive electromagnetisms. The former has only second-class constraints and no gauge freedom, so everything is observable. The latter has Maxwell-like gauge freedom due to a new gauge compensation field (鈥淪tueckelberg trick鈥); setting it to 0 recovers de Broglie-Proca. Demanding equivalent observables permits a choice between separate constraints and G, favoring G over separate first-class constraints.
What is actually observable is quite indifferent to whether someone uses a canonical formalism or introduces any other auxiliary fields. Hence the correct notion of observables cannot be distinctively Hamiltonian, as Peter Bergmann (sometimes) agreed. Hence Hamiltonian-Lagrangian equivalence is obligatory, though common notions are distinctively Hamiltonian.
G for parametrized massive gravity yields a Lie derivative of the 4-metric (like GR). If observables must have 0 Poisson bracket with G, they are still spatio-temporally constraint. But non-gauge massive gravity has no first-class constraints, making everything observable and spatio-temporal variation manifest. So the 0 Poisson bracket is incorrect for external symmetries such as appear in gravity. Equivalence is achieved by requiring covariance under external symmetries (nonzero Lie derivative Poisson bracket): 4-dimensional tensor calculus all over again, at least on-shell.
The new definition handles mixed cases: Einstein-Maxwell theory combines internal electromagnetic and external coordinate symmetries. Calculating the Poisson brackets, one finds that the metric and the electromagnetic field strength are observables: coordinate-covariant and electromagnetically gauge-variant.
The gauge symmetry of General Relativity is so much like that of parametrized massive gravity (with a Lagrangian equivalent to GR plus 4 scalars) that the same definition of observables should apply. Consequently the metric and curvature, which transform with Lie derivatives under G, are observables. Observables are local, change, and are independent of the Hamiltonian formalism, as Bergmann (sometimes) intended. The non-zero Poisson bracket might suggest some analogous considerations in quantum gravity.
Finally, recently Berezhiani, Dvali and Sakhelashvili provided a quantization scheme providing the completion that the above classical account needed.