Electromagnetism electromagnetic knots

D. A Family of Electromagnetic Knots with Hopf Indices n2... 

It is based on the idea of electromagnetic knot, introduced in 1990 [27-29] and developed later [30-32], An electromagnetic knot is defined as a standard electromagnetic field with the property that any pair of its magnetic lines, or any pair of its electric lines, is a link with linking number i (which is a measure of the extent to which the force lines curl themselves around one another, i.e., of the helicity of the field). These lines coincide with the level curves of a pair of complex scalar fields , 0. The physical space and the complex plane are compactified to Si and S2, so that the scalars can be... 

It will be shown in Section II.C that the two Hopf indices are equal, n , = ne = n, in the case of electromagnetic knots in empty space. 

Figure 1. Schematic aspect of several force lines (either magnetic or electric) of an electromagnetic knot. Any two of the six lines shown are linked once.

A very important property is that the magnetic and electric lines of an electromagnetic knot are the level curves of the scalar fields 4>(r, t) and 0(r, f), respectively. Another is that the magnetic and the electric helicities are topological constants of the motion, equal to the common Hopf index of the corresponding pair of dual maps constant with dimensions of action times velocity. 

In an electromagnetic knot, each line is labeled by a complex number. If there are m lines with the same label, we will say that m is the multiplicity. If all the pairs of line have the same linking number l, it turns out that the Hopf index is given as n = Im2. 

Here we summarize a program to find explicitly the Cauchy data of electromagnetic knots [25,27,30-32]. Let < 50, 00 —> S2 be two applications satisfying the following two conditions ... 

The level curves of < >0 must be orthogonal, in each point, to the level curves of 0o, since we know that electromagnetic knots are singular fields (E B = 0). This condition can be written as... 

Summarizing this subsection, the group-theoretic techniques allow us to obtain three maps S3 —>. S 2 whose velocity vectors are mutually orthogonal, and with the same linking number. Next, we have to build the Cauchy data of the electromagnetic knots based on these maps. 

Consequently, we have obtained the Cauchy data of an electromagnetic knot, a representative of the homotopy class C, for which, according to (63)... 

To find the electromagnetic knot, defined at every time, from the Cauchy data (91), we use the Fourier analysis. The magnetic and electric fields can be written as... 

For the electromagnetic knot with Cauchy data given by (91), we find... 

Introducing these vectors in (95), the expressions, for all the times, of one electromagnetic knot representative of the homotopy class C are... 

It is easy to show that we can also construct electromagnetic knots with Hopf index — n2 by means of the dual fields... 

The magnetic and electric fields of the electromagnetic knot are then... 

As a first step in constructing a topological model of the electromagnetic field, let us consider the set of electromagnetic knots defined by pairs of dual scalars (<)>, 0). If we try a theory based on these two scalars, the most natural election for the action integral is... 

Electromagnetic radiation fields—also called degenerate or singular by mathematicians—are defined by the condition det(F(lv) = 0 or, equivalently, by E B = 0, that is, by the orthogonality of the electric and magnetic vectors. As was stated above, the electromagnetic knots are of this type. This means that the model just described contains only radiation fields. 

The electromagnetic knots satisfy a very important property. In a precise way, the following proposition holds tme. 

This means that the difference between the set of the radiation solutions of the Maxwell equations and the set of the electromagnetic knots is not local but global. In other words Radiation fields and knots are locally equal. A proof is the following. 

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