Peter Bergmann on observables in Hamiltonian General Relativity: A historical-critical investigation

The problem of observables and their supposed lack of change has been significant in Hamiltonian quantum gravity since the 1950s. This paper considers the unrecognized variety of ideas about observables in the thought of constrained Hamiltonian dynamics co-founder Peter Bergmann, who trained many students at Syracuse and invented observables. Whereas initially Bergmann required a constrained Hamiltonian formalism to be mathematically equivalent to the Lagrangian, in 1953 Bergmann and Schiller introduced a novel postulate, motivated by facilitating quantum gravity. This postulate held that observables were invariant under transformations generated by each individual first-class constraint. While modern works rely on Bergmann's authority and sometimes speak of "Bergmann observables," he had much to say about observables, generally interesting and plausible but not all mutually consistent and much of it neglected. On occasion he required observables to be locally defined (not changeless and global); at times he wanted observables to be independent of the Hamiltonian formalism (implicitly contrary to a definition involving separate first-class constraints). But typically he took observables to have vanishing Poisson bracket with each first-class constraint and took this result to be justified by the example of electrodynamics. He expected observables to be analogous to the transverse true degrees of freedom of electromagnetism. Given these premises, there is no coherent concept of observables which he reliably endorsed, much less established. A revised definition of observables that satisfies the requirement that equivalent theories should have equivalent observables using the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first-class constraints that changes the canonical action ∫dt(pq̇-H) by a boundary term. Bootstrapping from theory formulations with no first-class constraints, one finds that the "external" coordinate gauge symmetry of GR calls for covariance (a transformation rule and hence a 4-dimensional Lie derivative for the Poisson bracket), not invariance (0 Poisson bracket), under G (not each first-class constraint separately).