Hermitian structures defined by linear electromagnetic constitutive laws

TL;DR

This paper explores how linear electromagnetic laws induce Hermitian structures on the bundle of 2-forms in four-dimensional manifolds, linking geometric structures to electromagnetic oscillators and symplectic forms.

Contribution

It demonstrates that electromagnetic constitutive laws naturally define Hermitian and symplectic structures on 2-form bundles, connecting geometry with electromagnetic oscillator models.

Findings

Hermitian structures arise from electromagnetic constitutive laws.

Real and imaginary parts correspond to Hamiltonian and symplectic forms.

Complex oscillator equations are derived from geometric structures.

Abstract

It is demonstrated that when the bundle of 2-forms on a four-dimensional manifold M admits an almost-complex structure any choice of "real + imaginary" subspace decomposition of the bundle defines a conjugation map, as well as a Hermitian structure for the bundle. When the almost-complex structure comes from a linear electromagnetic constitutive law, the real and imaginary parts of the Hermitian structure are then shown to represent the Hamiltonian for an anisotropic three-dimensional electromagnetic oscillator at each point of M and a symplectic structure for each fiber. The complex form of the oscillator equations is also definable in terms of the geometric structures that were introduced.