an:06714500
Zbl 1360.70030
Pitts, J. Brian
A first class constraint generates not a gauge transformation, but a bad physical change: The case of electromagnetism
EN
Ann. Phys. 351, 382-406 (2014).
00365635
2014
j
70H45 70S15 70S05 78A25 81S05
Dirac-Bergmann constrained dynamics; Gauge transformations; canonical quantization; observables; Hamiltonian methods; first-class constraints
Summary: In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not \textit{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 \(\overrightarrow{E}\) 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 \(\overrightarrow{E}\) 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 \textit{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 \textit{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.