Pitts, J. Brian (2014) A First Class Constraint Generates Not a Gauge Transformation, But a Bad Physical Change: The Case of Electromagnetism. Annals of Physics.

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Abstract
In DiracBergmann constrained dynamics, a firstclass constraint typically does not \emph{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 (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 firstclass constraints, the AndersonBergmannCastellani gauge generator G, leaves E unchanged. Likewise only that combination leaves the canonical action invariantan 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 AndersonBergmann Lagrangian orientation of interesting canonical transformations.
The need to keep gaugeinvariant the relation $\dot{q} \frac{\delta H}{\delta p}= E_i p^i=0$ supports using the gauge generator and primary Hamiltonian rather than the separate firstclass constraints and the extended Hamiltonian.
Partly paralleling Pons's criticism, it is shown that
Dirac's proof that a firstclass 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 gaugegenerating property has been ascribed to the primaries alone, not the primarysecondary team G. Hence the Dirac conjecture about secondary firstclass constraints as generating gauge transformations rests upon a false presupposition about primary firstclass constraints. Clarity about Hamiltonian electromagnetism will be useful for an analogous treatment of GR.
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Item Type:  Published Article or Volume  

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Keywords:  DiracBergmann constrained dynamics; gauge transformations; canonical quantization; observables; Hamiltonian methods; firstclass constraints; 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:  07 Sep 2014 19:33  
Last Modified:  07 Sep 2014 19:33  
Item ID:  11015  
Journal or Publication Title:  Annals of Physics  
DOI or Unique Handle:  10.1016/j.aop.2014.08.014  
Subjects:  Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Fields and Particles Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances 

Date:  20 August 2014  
URI:  https://philsciarchive.pitt.edu/id/eprint/11015 
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