A Scalar Representation of Electromagnetic Fields

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Abstract
It is shown that in a region which is free of currents and charges, any electromagnetic field may be rigorously derived from a single, generally complex, scalar wave function V(x, t). In terms of this function the momentum density g(x, t) and the energy density w(x, t) of the field may be defined in such a way that they are represented by expressions analogous to the formulae for the probability current and the probability density in quantum mechanics; in a homogeneous isotropic medium g(x,t) = - (1/8πμ0c)[V*∇V + V ∇V*], w(x,t) = (1/8π)[(0/c2)VV* + (1/μ0)∇V.∇V*] The densities defined in this way differ from those given by the usual expressions, but the choice is justified since the differences disappear on integration over any arbitrary macroscopic domain. (The corresponding Lagrangian densities differ by a four divergence.) When V is of the form V0(x)e-iwt, g is found to form a solenoidal field which is orthogonal to the co-phasal surfaces arg V0 = constant.

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