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

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Annals of Physics, ISSN: 0003-4916, Vol: 351, Page: 382-406

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Pitts, J. Brian
Elsevier BV
Physics and Astronomy
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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 pi,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 Aμ ), 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 q̇−δHδp=−Ei−pi=0 supports using the gauge generator and primary Hamiltonian rather than the separate first-class constraints and the extended Hamiltonian.