A first class constraint generates not a gauge transformation, but a bad physical change: The case of electromagnetism
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Authors
Publication Date
2014-12Journal Title
Annals of Physics
ISSN
0003-4916
Publisher
Elsevier BV
Volume
351
Pages
382-406
Language
English
Type
Article
Metadata
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Pitts, B. (2014). A first class constraint generates not a gauge transformation, but a bad physical change: The case of electromagnetism. Annals of Physics, 351 382-406. https://doi.org/10.1016/j.aop.2014.08.014
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 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), 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.
Keywords
Dirac-Bergmann constrained dynamics, gauge transformations, canonical quantization, observables, Hamiltonian methods, first-class constraints, problem of time
Identifiers
External DOI: https://doi.org/10.1016/j.aop.2014.08.014
This record's URL: https://www.repository.cam.ac.uk/handle/1810/247013
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