Underdeterminacy and redundance in Maxwell’s equations. Origin of gauge freedom – Transversality of free electromagnetic waves – Gaugefree canonical treatment without constraints, EJTP 6

Abstract: Maxwell’s (1864) original equations are redundant in their description of charge conservation. In the nowadays used, ’rationalized ’ Maxwell equations, this redundancy is removed through omitting the continuity equation. Alternatively, one can Helmholtz decompose the original set and omit instead the longitudinal part of the flux law. This provides at once a natural description of the transversality of free electromagnetic waves and paves the way to eliminate the gauge freedom. Poynting’s inclusion of the longitudinal field components in his theorem represents an additional assumption to the Maxwell equations. Further, exploiting the concept of Newtonian and Laplacian vector fields, the role of the static longitudinal component of the vector potential being not determined by Maxwell’s equations, but important in quantum mechanics (Aharonov-Bohm effect) is elucidated. Finally, extending Messiah’s (1999) description of a gauge invariant canonical momentum, a manifest gauge invariant canonical formulation of Maxwell’s theory without imposing any contraints or auxiliary conditions will be proposed as input for Dirac’s (1949) approach to special-relativistic dynamics