Spin and angular momentum operators and their conservation

TL;DR

This paper generalizes reciprocity relations in electromagnetism to include conservation laws for optical energy, spin, and momentum, extending Noether's theorem to electromagnetic wave symmetries.

Contribution

It introduces a new framework linking Maxwell's invariance transformations to conserved optical quantities using reciprocity relations and Hermitian operators.

Findings

Derived a free-space reciprocity relation generalizing momentum-energy conservation.

Defined operators for optical energy, spin, and angular momentum.

Extended Noether's theorem to electromagnetic wave symmetries.

Abstract

Lorentz's reciprocity lemma and Feld-Tai reciprocity theorem show the effect of interchanging the action and reaction in Maxwell's equations. We derive a free-space version of these reciprocity relations which generalizes the conservation of the momentum-energy tensor. This relation corresponds to the interference conservation of electromagnetic waves. We show that for any transformation or symmetry that leaves Maxwell's equations invariant, we can modify the reciprocity relation to introduce a conserving density, optical flux and stress tensor extending Noether's theorem to a different context. We apply this method to transformations that can be expressed as Hermitian operators and more specifically we define the operators associated with the optical energy, spin, linear and angular momentum.