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

2013/10/31 by J. Brian Pitts · 3 citations
Physics and Astronomy · #gr-qc #hep-th #physics.hist-ph

paper · pdf · doi:10.1016/j.aop.2014.08.014

published as Annals of Physics 351 (2014), pp. 382-406

arxiv created 2014/10/03 · arxiv updated 2014/10/10

Abstract

In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not _alone_ generate a gauge transformation. 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 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 A), 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 q,t-dH/dp=-E-p=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. 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 electromagnetism help with GR.

Cited by

• Nomic Structure and Reduction
• The Limitations of the Notion of `Observable' in Diffeomorphism-Invariant Models
• Global Gauge Symmetry Breaking in the Abelian Higgs Mechanism

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• First-Class Constraints, Gauge Transformations, de-Ockhamization, and Triviality: Replies to Critics, Or, How (Not) to Get a Gauge Transformation from a Second-Class Primary Constraint
• Constraints and gauge transformations: Dirac's theorem is not always valid
• Strong limitations on allowable gauge transformations in electrodynamics
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