Peter Bergmann on Observables in Hamiltonian General Relativity: A Historical-Critical Investigation

TL;DR

This paper critically examines Peter Bergmann's diverse ideas on observables in Hamiltonian general relativity, highlighting inconsistencies and proposing a revised, more coherent definition aligned with gauge covariance rather than invariance.

Contribution

It clarifies Bergmann's complex and sometimes conflicting views on observables and proposes a revised definition using the gauge generator G for better conceptual coherence.

Findings

Bergmann's original ideas on observables were inconsistent.

A new definition of observables based on the gauge generator G is proposed.

The revised approach aligns with the covariance of general relativity.

Abstract

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 Peter Bergmann, who invented observables. Whereas initially he 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: 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, plausible but not all consistent or remembered. At times he required observables to be locally defined (not changeless and global); at times he wanted them independent of the Hamiltonian formalism (not essentially involving separate first-class constraints). But typically he took observables to have vanishing Poisson bracket with each first-class constraint, purportedly justified by electrodynamics. He expected observables to be analogous to the transverse true degrees of freedom of electromagnetism. Hence there is no coherent concept of observables which he reliably endorsed. 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 $\int dt(p\dot{q}-H)$ by a boundary term. Bootstrapping from theory formulations with no first-class constraints, the "external" coordinate symmetry of GR calls for covariance ($4$-dimensional Lie derivative), not invariance ($0$ Poisson bracket), under $G$ (not each first-class constraint).