Electromagnetic knots defined

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... 

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... 

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... 

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. 

Note that the set of the electromagnetic knots contains some with very low energy, for which n is necessarily very small. Even if they can be defined as classical fields, the real system would have quantum behavior, since the action involved would be of the order of h. On the other hand, there are states with n small and even zero, which have, however, macroscopic energy. They are those for which Nr, Nr are large. When n is large, the photon contents are high and the energy is macroscopic. These are the states for which the classical approximation is valid. 

Helicity is defined as a qualitative measurement of how a topological configuration is linked, knotted, or twisted. If A is the vector potential of electromagnetism, the quantity then defines magnetic helicity. In order to obtain a nonnull value for K, the condition A (V A) /- 0 must be verified in the volume... 

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