# A First Class Constraint Generates Not a Gauge Transformation, But a Bad Physical Change: The Case of Electromagnetism

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 Dirac-Bergmann constrained dynamics, a first-class constraint typically does not \emph{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 $\dot{q}- \frac{\delta H}{\delta p}= -E_i -p^i=0$ supports using the gauge generator and primary Hamiltonian rather than the separate first-class constraints and the extended Hamiltonian.

Partly paralleling Pons's criticism, it is shown that
Dirac's proof that a first-class 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 gauge-generating property has been ascribed to the primaries alone, not the primary-secondary team G. Hence the Dirac conjecture about secondary first-class constraints as generating gauge transformations rests upon a false presupposition about primary first-class constraints. Clarity about Hamiltonian electromagnetism will be useful for an analogous treatment of GR.

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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Pitts, J. Brianjbp25@cam.ac.uk
Keywords: Dirac-Bergmann constrained dynamics; gauge transformations; canonical quantization; observables; Hamiltonian methods; first-class 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
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: http://philsci-archive.pitt.edu/id/eprint/11015

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