A First Class Constraint Generates Not a Gauge Transformation, But a Bad Physical Change: The Case of Electromagnetism

Abstract

In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not _alone_ generate a gauge transformation. Each first-class constraint in Maxwell's theory generates a change in the electric field E by an arbitrary gradient, spoiling Gauss's law. The secondary constraint p^i,_i=0 still holds, but being a function of derivatives of momenta (mere auxiliary fields), it is not directly about the observable electric field (a function of derivatives of A), which couples to charge. Only a special combination of the two first-class constraints, the Anderson-Bergmann-Castellani gauge generator G, leaves E unchanged. Likewise only that combination leaves the canonical action invariant---an argument independent of observables. If one uses a first-class constraint to generate instead a canonical transformation, one partly strips the canonical coordinates of physical meaning as electromagnetic potentials, vindicating the Anderson-Bergmann Lagrangian orientation of interesting canonical transformations. The need to keep gauge-invariant q,t-dH/dp=-E-p=0 supports using the gauge generator and primary Hamiltonian rather than the separate first-class constraints and the extended Hamiltonian. Partly paralleling Pons's criticism, it is shown that Dirac's proof that a first-class primary constraint generates a gauge transformation, by comparing evolutions from _identical_ initial data, cancels out and hence fails to detect the alterations made to the initial state. It also neglects the arbitrary coordinates multiplying the secondary constraints _inside_ the canonical Hamiltonian. Thus the gauge-generating property has been ascribed to the primaries alone. Hence the Dirac conjecture about secondary first-class constraints as generating gauge transformations rests upon a false presupposition about primary first-class constraints. Clarity about electromagnetism help with GR.