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Complex Structures in Electrodynamics

Stoil Donev
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Abstract

In this paper we show that the basic external (i.e. not determined by the equations) object in Classical electrodynamics equations is a complex structure. In the 3-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 4-dimensional formulation the complex structure is extracted directly from the equations, it appears as a linear map $Φ$ in the space of 2-forms on $\mathbb{R}^4$. It is shown also that $Φ$ may appear through the equivariance properties of the new formulation of the theory. Further we show how this complex structure $Φ$ combines with the Poincare isomorphism $\mathfrak{P}$ between the 2-forms and 2-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.

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Stoil Donev. 2001-11-02. Complex Structures in Electrodynamics. https://arxiv.org/abs/math-ph/0106008

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