Hertzian electromagnetic potentials and associated gauge transformations

In spite of the wide use of Hertzian potentials in special problems there appears to be no account of the general theory which is completely satisfactory—especially with regard to (i) the arbitrariness of the potentials and the relation between different equivalent representations of the same electromagnetic field, (ii) the derivation of the Hertzian potentials for such equivalent representations from the physical sources, and (iii) electromagnetic fields not in vacuo. It is the purpose of this paper to fill this gap. It is shown that the Hertzian potentials may be subjected to a new type of gauge transformation which leaves invariant the electromagnetic field they represent. The particular integrals of the inhomogeneous Maxwell equations are generalized, so that they may be subjected to a related gauge transformation which leaves invariant the physical sources of the field; this leads to a treatment of (ii) above, which appears to be new. Examples, including the Whittaker and Debye—Bromwich two-scalar representations, are given. Finally, the theorem is established that, for any electromagnetic field in any stationary material medium, the particular integral of Maxwell equations may be so chosen that in general the complementary function can be expressed in terms of only two scalar functions (components of Hertzian potentials), previously only known to hold for source-free regions in vacuo .