Abstract
Maxwell equations have two types of asymmetries between the electric and magnetic fields. The first asymmetry is the inhomogeneity induced by the absence of magnetic charge sources. The second asymmetry is due to parity. We show how both asymmetries are naturally resolved under an alternative formulation of Maxwell equations for fields or potentials that uses a compact complex vector operator representation. The developed complex symmetric operator formalism can be easily applied to performing the continuity equation, the field wave equations, the Maxwell equations for potentials, the gauge transformations, and the 4-momentum representation; in general, the developed formalism constitutes a simple way of unfolding the Maxwell theory. Finally, we provide insights for extending the presented analysis within the context of (i) bicomplex numbers and tessarine algebra; and (ii) Lp-spaces in nonlinear Maxwell equations.
Highlights
There are various representations of Maxwell equations
The presented formulation of Maxwell equations constitutes a much simpler and compact way of unfolding the Maxwell theory compared to previous complex formulations
The presented formulation of Maxwell equations constitutes a much simpler way of unfolding the Maxwell theory compared with previous complex formulations
Summary
There are various representations of Maxwell equations. Some examples are the following: standard complex representation [1,2], spinor form [3], Silberstein–Bateman–Majorana form [4,5,6], Kemmer–Duffin–Petiau form ( known as the meson algebra) [4,7], matrix representation [8], Dirac form [9,10,11], Poincaré algebra [12], Debye sources [13,14], Penrose’s transformation presented in terms of integral geometry [15,16], integral representation [17], and multipolar presentation [18]. This paper uses the complex vector representation of Maxwell equations in order to develop the presented complex operator formalism. This developed formalism: (i) emerges naturally from the symmetry between electric and magnetic fields; and (ii) exhibits a compact set of equations for the fields and their potentials. The presented formulation of Maxwell equations constitutes a much simpler and compact way of unfolding the Maxwell theory compared to previous complex formulations (e.g., continuity equation, wave equations, Maxwell equations for potentials, gauge symmetry). The presented formulation of Maxwell equations constitutes a much simpler way of unfolding the Maxwell theory compared with previous complex formulations (e.g., continuity equation, wave equations, Maxwell equations for potentials, gauge symmetry).
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