*) Remark on the Lorenz-Convention falsely attributed to H.A. Lorentz

The Lorenz condition stems from the Danish mathematician and physicist Ludvig Valentin Lorenz (18. Januar 1829 - 9. Juni 1891):

Lorenz, L. "On the Identity of the Vibrations of Light with Electrical Currents." Philos. Mag. 34, 287-301, 1867.

Further References

Summary

We trace the derivation of the differential equations for the vector potential from the homogeneous Maxwell equations for the vacuum and back. However, due to an unclear formulation in [1] for the way back the authors erroneously use the version [1; (7)], where the Loren(t)z condition was applied, instead of the "Lorenz-free" version [R;(7a)]. Therefore just this term now appears additionally as "vacuum current density". A miraculous conclusion,which should have made the authors suspicious, that just the Lorenz term should yield a vacuum current. Surely, it's a pity that vacuum currents and vacuum energy have their origin merely in that wrong interpretation of Equ.(7) as "Lorenz-free", and all further speculations for a vacuum energy density are in vain.

Abstract of [2]: Great announcements ...
"It is shown that if the Loren(t)z condition is discarded, the Maxwell–Heaviside field equations become the Lehnert equations, indicating the presence of charge density and current density in the vacuum. The Lehnert equations are a subset of the O(3) Yang–Mills field equations. Charge and current density in the vacuum are defined straightforwardly in terms of the vector potential and scalar potential, and are conceptually similar to Maxwell's displacement current, which also occurs in the classical vacuum. A demonstration is made of the existence of a time dependent classical vacuum polarization which appears if the Loren(t)z condition is discarded. Vacuum charge and current appear phenomenologically in the Lehnert equations but fundamentally in the O(3) Yang–Mills theory of classical electrodynamics. The latter also allows for the possibility of the existence of vacuum topological magnetic charge density and topological magnetic current density. Both O(3) and Lehnert equations are superior to the Maxwell–Heaviside equations in being able to describe phenomena not amenable to the latter. In theory, devices can be made to extract the energy associated with vacuum charge and current."

Sketch of the authors' erroneous idea

In [1] B. Lehnert and S. Roy have recalled to their readers that the Maxwell equations can be solved by means of potentials A, Φ, which are solutions of the differential equations

[R;(7a)]                                 ¹/_c² A_tt− ΔA + Ñ (div A + ¹/_c² Φ_t) = μ_o j

and

[R;(7b)]                                 ¹/_c² Φ_tt − ΔΦ − (div A + ¹/_c² Φ_t) _t = ^ρ/_εo .

without any coupling between A and Φ to obtain the solutions B = curl A and E = − ÑΦ − A_t. However, in addition the well-known Lorenz condition

div A + ¹/_c² Φ_t = 0

can be imposed on the potentials to reduce the equations [R;(7a,b)] to inhomogeneous wave equations the solutions of which have well-known integral representations.

However, due to unclear formulations in [1] the authors believe the equations [R;(7a,b)] with applied Lorenz condition, the equations

[R;(7a')]                                 ¹/_c² A_tt− ΔA = μ_o j

and

[R;(7b')]                                 ¹/_c² Φ_tt − ΔΦ = ^ρ/_εo .

to be the "Lorenz-free" equations to be used. And so the forgotten Lorenz term gives rise to a "vacuum current density" just of the size of the forgotten term:

j_A = − ¹/_μo [grad (div A + ¹/_c² Φ_t)] .