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Complex structures in electrodynamics. (English) Zbl 1125.78004

Considered are the Maxwell equations in vacuum for illustrating the relevance of a progressive level of abstraction in the characterization of the interplay between equations and solutions, so that eventually to one physical object corresponds one mathematical model object. In particular, starting from the linear Maxwell equations and the remarkable duality symmetry of their solutions (which in the simplest form is expressible by the invariance with respect to the substitutions \(E\to -B\) and \(B\to E\)), one casts the equations in a general covariant form involving the metric tensor and one then gets a coordinate free formulation in terms of external differential forms, on \(\mathbb{R}^2\) for the 3-dimensional formulation and on \(\mathbb{R}^4\) for the 4-dimensional one, respectively. The duality mentioned above then appears to be based on a linear transformations \(\phi\) on the corresponding form spaces which has the property \(\phi\circ\phi=-\text{identity}\), called accordingly a complex structure. The authors are then able to show that starting from the complex structures involved by the duality symmetry which leave the energy-momentum of the fields invariant, one can derive the relativistic pseudo-Euclidean metric structures of the Maxwell equations and their conformal symmetries.

MSC:

78A25 Electromagnetic theory (general)
35Q60 PDEs in connection with optics and electromagnetic theory
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