Since the mid-1950s it has seemed that change is somehow missing in Hamiltonian General Relativity. How did this problem arise? How compelling are the axioms on which it rests? What of the 1980s+ reforming literature that has aimed to recover the mathematical Hamiltonian-Lagrangian equivalence that was given up in the mid-1950s, a reforming literature that is visible in journals but scarce in books? What should one mean by Hamiltonian gauge transformations and observables, and how can one decide? The absence of change in observables can be traced to (1) a pragmatic conjecture (initially by Peter Bergmann and his student Schiller and later by Dirac) that gauge transformations come not merely from a tuned sum of “first-class constraints” (the Rosenfeld-Anderson-Bergmann gauge generator), but also from each first-class constraint separately, and (2) an assumption that the internal gauge symmetry of electromagnetism is an adequate precedent for the external/space-time coordinate symmetry of General Relativity. Requiring that gauge transformations preserve Hamilton’s equations or that equivalent theory formulations yield equivalent observables shows that change is right where it should be in Hamiltonian General Relativity including observables, namely, essential time dependence (e.g., lack of a time-like Killing vector field) in coordinate-covariant quantities. A genuine problem of missing change might exist at the quantum level, however.
Philosophy of Physics Seminar Convenors for HT20: Oliver Pooley