Does a Second-Class Primary Constraint Generate a Gauge Transformation? Electromagnetisms and Gravities, Massless and Massive

Abstract

In constrained Hamiltonian dynamics there are two views regarding how first-class constraints generate gauge transformations: individually or only in a certain combination, the Rosenfeld-Anderson-Bergmann-Castellani gauge generator. This gauge generator $G$ preserves Hamilton's equations and changes the canonical action at most by a boundary term; Hamiltonian's equations are the Euler-Lagrange equations for the canonical action. Hence the canonical formalism is equivalent to the Lagrangian formalism, and indeed subsumed within it (in important examples) with many canonical momenta serving as auxiliary fields. $G$ generates transformations basically equivalent to the usual 4-dimensional Lagrangian expressions, such as a 4-gradient in electromagnetism or a space-time coordinate transformation (on-shell) in General Relativity. It has been shown recently that separate first-class constraints lead to inequivalent observables between Proca non-gauge and Stueckelberg gauge massive electromagnetism. There is, however, widespread agreement that second-class constraints do not generate gauge transformations. Here it is shown that in such a sense as the first-class primary constraint in Maxwell's theory generates a gauge transformation, the second-class primary constraint in Proca's massive electromagnetism also generates a gauge transformation. Likewise the second-class primary constraints in various massive spin $2$ relatives of General Relativity generate as much of a gauge transformation as do the corresponding first-class primary constraints in GR. Hence the view that first-class constraints typically generate gauge transformations _individually_ faces a puzzle not faced by the gauge generator view.