A scalar product for computing fundamental quantities in matter

TL;DR

This paper presents a new scalar product method for calculating fundamental quantities like helicity and angular momentum in static matter, leveraging conformal symmetry invariance, with an application to magnetic Hopfions.

Contribution

It introduces a conformally invariant scalar product for static matter quantities, extending symmetry principles from Maxwell equations to static charge and magnetization distributions.

Findings

Derived explicit formulas for helicity and angular momentum in static matter.

Applied the method to compute quantities in a magnetic Hopfion.

Demonstrated invariance under the conformal group in static configurations.

Abstract

We introduce a systematic way to obtain expressions for computing the amount of fundamental quantities such as helicity and angular momentum contained in static matter, given its charge and magnetization densities. The method is based on a scalar product that we put forward, which is invariant under the ten-parameter conformal group in three-dimensional Euclidean space. Such group is obtained as the static restriction (frequency $\omega=0$) of the symmetry group of Maxwell equations: The fifteen-parameter conformal group in 3+1 Minkowski spacetime. In an exemplary application, we compute the helicity and angular momentum squared stored in a magnetic Hopfion.