Conserved charges for rational electromagnetic knots

TL;DR

This paper constructs and analyzes electromagnetic knot solutions in Minkowski space using conformal mappings, computes their conserved charges, and explores their geometric and topological properties.

Contribution

It introduces a direct conformal approach to electromagnetic knots, computes all associated conserved charges, and provides explicit results up to spin j=1.

Findings

Conserved charges are either zero or proportional to energy.

Explicit charge calculations are provided for spins up to j=1.

The geometric structure of vector charges is elucidated.

Abstract

We revisit a newfound construction of rational electromagnetic knots based on the conformal correspondence between Minkowski space and a finite $S^3$-cylinder. We present here a more direct approach for this conformal correspondence based on Carter-Penrose transformation that avoids a detour to de Sitter space. The Maxwell equations can be analytically solved on the cylinder in terms of $S^3$ harmonics $Y_{j;m,n}$, which can then be transformed into Minkowski coordinates using the conformal map. The resultant "knot basis" electromagnetic field configurations have non-trivial topology in that their field lines form closed knots. We consider finite, complex linear combinations of these knot-basis solutions for a fixed spin $j$ and compute all the $15$ conserved Noether charges associated with the conformal group. We find that the scalar charges either vanish or are proportional to the energy. For the non-vanishing vector charges, we find a nice geometric structure that facilitates computation of their spherical components as well. We present analytic results for all charges for up to $j{=}1$. We demonstrate possible applications of our findings through some known previous results.