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

Annals of Physics, ISSN: 00034916, Vol: 351, Page: 382406
 Publication Year:
 2014
 Repository URL:
 http://philsciarchive.pitt.edu/id/eprint/11015
 DOI:
 10.1016/j.aop.2014.08.014
 Author(s):
 Publisher(s):
 Tags:
 Physics and Astronomy
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article description
In Dirac–Bergmann constrained dynamics, a firstclass constraint typically does not alone generate a gauge transformation. By direct calculation it is found that each firstclass constraint in Maxwell’s theory generates a change in the electric field E→ by an arbitrary gradient, spoiling Gauss’s law. The secondary firstclass 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 firstclass 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 firstclass 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 gaugeinvariant the relation q̇−δHδp=−Ei−pi=0 supports using the gauge generator and primary Hamiltonian rather than the separate firstclass constraints and the extended Hamiltonian.