A first class constraint generates not a gauge transformation, but a bad physical change: The case of electromagnetism

摘要

In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not alone generate a gauge transformation. By direct calculation it is found that 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 first-class 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(mu)), 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 the relation (q) - delta H/delta P = - E-i - p(i) = 0 supports using the gauge generator and primary Hamiltonian rather than the separate first-class constraints and the extended Hamiltonian.