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hopf index

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16 - The Hopfion Solution and the Hopf Map
from Part I - General Properties of Fields; Scalars and Gauge Fields
Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
Book:

Classical Field Theory

Published online:

04 March 2019

Print publication:

14 March 2019, pp 150-157
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Summary

In this chapter, we study the Hopfion solution of electromagnetism. It is a solution characterized by a topological number, the Hopf index, associated with the Hopf map from the 3-sphere to the 2-sphere (Hopf fibration). It is also characterized by nonzero “helicities,” which are defined as integrals of Chern–Simons forms on the spatial volume, as well as having a knot structure, where the electric and magnetic fields are linked, with nonzero linking number. Generalization of these electromagnetic knots are also given.