Open Access
2006 Complex Structures in Electrodynamics
Stoil Donev, Maria Tashkova
J. Geom. Symmetry Phys. 7: 13-36 (2006). DOI: 10.7546/jgsp-7-2006-13-36


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.


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Stoil Donev. Maria Tashkova. "Complex Structures in Electrodynamics." J. Geom. Symmetry Phys. 7 13 - 36, 2006.


Published: 2006
First available in Project Euclid: 20 May 2017

zbMATH: 1125.78004
MathSciNet: MR2290123
Digital Object Identifier: 10.7546/jgsp-7-2006-13-36

Rights: Copyright © 2006 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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