Comment on ‘Electromagnetic potentials without gauge transformations’

Abstract

In this comment it is demonstrated that contrary to the claim in the title, ‘Electromagnetic potentials without gauge transformations’, the potentials were actually calculated in the Coulomb gauge.

Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

“…Stewart showed that this representation of the time dependent vector potential satisfies Maxwell equations and even more, that it is equivalent to retarded representation obtained by Jackson [17]. However, its very easy to prove that (34) is in fact the potential (21) coming from the Helmholtz-Maxwell elliptic theorem. We have the following:…”

“…Corollary to theorem 5: the instantaneous Stewart representation (34) for the vector potential is equivalent to the retarded representation (21). Proof: Using standard vector identities and Green's functions properties we can write down…”

“…Now we put the decomposition in the Equations ( 17, (18)) to get, after identifying irrotational and solenoidal components-including an elliptic Helmholtz decomposition for the current density-the following equations: Comment II: theorem (5) is not as trivial as theorem (4), and some ambiguities appeared around its interpretation. A careful derivation of the Equations ((31)-(33)) appeared under a different interpretation in [20], but in [21] it was proved that the equations are really the Coulomb gauge equations. In [20] it was also proved that the potentials , Ф s A are gauge invariant quantities, so the Coulomb gauge is enough to fix the arbitrary degrees of freedom coming from gauge invariance of Equations ((17), (18)).…”

Reply to Comment on ‘Electromagnetic potentials without gauge transformation’

“…A result that has been given by Engelhardt and Onoochin using examples. For this reason we cannot believe that in any sense we are trying to refute statements in [2], instead we are offering a possible theoretical explanation of their results. (ii) The proof that the new potentials are gauge invariant quantities.…”

“…The foregoing is a summary of the conventional wisdom about the widespread method to solve Maxwell's equations by means of the g-potentials, and can be read in any textbook on electromagnetic theory (see, e.g., [1, pp 239-241]). In the critical comment by Engelhardt and Onoochin [2] on the paper [3] it is proved, in the right manner, that the field equations deduced for the introduced gauge invariant potentials ϕ g , A g are formally the same as those for the g-potentials in the Coulomb gauge. However, we believe that this is not a criticism because the authors overlooked the main achievement of [3] that for us relies on two independent points: (i) A general argument for the possibility of different solutions in different gauges unrelated by a gauge transformation.…”