Spinors and pre-metric electromagnetism

TL;DR

This paper explores a formulation of electromagnetism that does not rely on Minkowski spacetime structure, emphasizing bivectors, complex geometry, and spinor representations to generalize classical concepts.

Contribution

It introduces a pre-metric approach to electromagnetism, highlighting the role of bivectors and complex geometry, and discusses spinor representations within this framework.

Findings

Electromagnetic constitutive law replaces Minkowski metric

Complex geometry reveals Lorentz group structure

Spinor representations offer new insights into fields

Abstract

The basic concepts of the formulation of Maxwellian electromagnetism in the absence of a Minkowski scalar product on spacetime are summarized, with particular emphasis on the way that the electromagnetic constitutive law on the space of bivectors over spacetime supplants the role of the Minkowski scalar product on spacetime itself. The complex geometry of the space of bivectors is summarized, with the intent of showing how an isomorphic copy of the Lorentz group appears in that context. The use of complex 3-spinors to represent electromagnetic fields is then discussed, as well as the expansion of scope that the more general complex projective geometry of the space of bivectors suggests.