Conserved charges for rational electromagnetic knots

Abstract

AbstractWe revisit a newfound construction of rational electromagnetic knots based on the conformal correspondence between Minkowski space and a finite $$S^3$$ 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 $$S^3$$ S 3 harmonics $$Y_{j;m,n}$$ 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 $$j=1$$ j = 1 . We demonstrate possible applications of our findings through some known previous results.