Line geometry and electromagnetism IV: electromagnetic fields as infinitesimal Lorentz transformations

TL;DR

This paper explores the algebraic and geometric structures underlying electromagnetic fields, linking bivectors, 2-forms, and Lorentz transformations with line geometry and complex structures, revealing new mathematical representations.

Contribution

It establishes a novel connection between electromagnetic field representations and line geometry using bivectors, 2-forms, and Lie algebra isomorphisms, incorporating complex structures.

Findings

Electromagnetic fields can be represented by motors like screws and wrenches.

The Cartan-Killing form on so(1, 3) is isometric to the scalar product on bivectors.

Complex structures enable new algebraic representations of electromagnetic fields.

Abstract

It is first shown that the scalar product on any orthogonal space (V, g) allows one to define linear isomorphisms of the vector spaces of bivectors and 2-forms on V with the underlying vector spaces of the Lie algebra so(p, q) and its dual, respectively. When those isomorphisms are applied to the electromagnetic excitation bivector and field strength 2-form, resp., one can associate various algebraic constructions that pertain to them as bivector fields and 2-forms with corresponding constructions in terms of so(1, 3) and its dual. The subsequent association with corresponding things in line geometry will then become straightforward. In particular, the fields can be represented by motors, such as screws and wrenches, while the Cartan-Killing form on so(1, 3) is isometric to the scalar product on bivectors that gives the Klein quadric. When the space of bivectors (and therefore the space of 2-forms) is given an almost-complex structure (and therefore, a complex structure), one can also represent most of the constructions on the former and its dual in terms of so(3; C) and its dual.