Complex Structures in Electrodynamics

Abstract

In this paper we show that the basic external (i.e., not determined by the equations) object in Maxwell vacuum equations is a complex structure. In the three-dimensional standard form of Maxwell equations this complex structure $\mathcal{I}$ participates implicitly in the equations and its presence is responsible for the so called duality invariance. We give a new form of the equations showing explicitly the participation of $\mathcal{I}$. In the four-dimensional formulation the complex structure is extracted directly from the equations, it appears as a linear map $\Phi$ in the space of two-forms on $\mathbb{R}^4$. It is shown also that $\Phi$ may appear through the equivariance properties of the new formulation of the theory. Further we show how this complex structure $\Phi$ combines with the Poincaré isomorphism $\mathfrak{P}$ between the two-forms and two-tensors to generate all well known and used in the theory (pseudo)metric constructions on $\mathbb{R}^4$, and to define the conformal symmetry properties. The equations of Extended Electrodynamics (EED) do not also need these pseudometrics as beforehand necessary structures. A new formulation of the EED equations in terms of a generalized Lie derivative is given.