Pitts, J. Brian (2013) A First Class Constraint Generates Not a Gauge Transformation, But a Bad Physical Change: The Case of Electromagnetism. [Preprint]

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Abstract
In DiracBergmann 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 p^i,_i=0 still holds, but being a function of derivatives of momenta, it is not directly about E (a function of derivatives of A_mu). Only a special combination of the two firstclass constraints, the AndersonBergmann (1951)Castellani gauge generator G, leaves E unchanged. This problem is avoided if one uses a firstclass constraint as the generator of a _canonical transformation_; but that partly strips the canonical coordinates of physical meaning as electromagnetic potentials and makes the electric field depend on the smearing function, bad behavior illustrating the wisdom of the AndersonBergmann (1951) Lagrangian orientation of interesting canonical transformations.
The need to keep gaugeinvariant the relation dot{q} dH/dp= E_i p^i=0 supports using the primary Hamiltonian rather than the extended Hamiltonian. The results extend the Lagrangianoriented reforms of Castellani, Sugano, Pons, Salisbury, Shepley, _etc._ by showing the inequivalence of the extended Hamiltonian to the primary Hamiltonian (and hence the Lagrangian) even for _observables_, properly construed in the sense implying empirical equivalence.
Dirac and others have noticed the arbitrary velocities multiplying the primary constraints outside the canonical Hamiltonian while apparently overlooking the corresponding arbitrary coordinates multiplying the secondary constraints _inside_ the canonical Hamiltonian, and so wrongly ascribed the gauge quality to the primaries alone, not the primarysecondary team G. Hence the Dirac conjecture about secondary firstclass constraints rests upon a false presupposition. The usual concept of Dirac observables should also be modified to employ the gauge generator G, not the firstclass constraints separately, so that the Hamiltonian observables become equivalent to the Lagrangian ones such as the electromagnetic field F.
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Item Type:  Preprint  

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Keywords:  empirical equivalence, gauge freedom, quantum gravity, problem of time  
Subjects:  Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Fields and Particles Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances 

Depositing User:  Dr. Dr. J. Brian Pitts  
Date Deposited:  06 Oct 2013 14:34  
Last Modified:  06 Oct 2013 14:34  
Item ID:  10038  
Subjects:  Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Fields and Particles Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances 

Date:  5 October 2013  
URI:  http://philsciarchive.pitt.edu/id/eprint/10038 
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