Hopf index

E = V x C. This implies an interesting interpretation of the Hopf index n, since that helicity is equal to the classical expression of the difference between the numbers of right-handed and left-handed photons contained in the field Nr — Nr (defined by substituting Fourier transform functions for creation and annihilation operators in the quantum expression). In other words, n = Nr — Nr- This establishes a relation between the wave and the particle understanding of the idea of helicity, that is, between the curling of the force lines to one another and the difference between right- and left-handed photons contained in the field. 

In this way, a complex function < )(r) can be interpreted as a map S3< S2. This is very important, since maps of this kind can be classified in homotopy classes labeled by a topological integer number called the Hopf index, so that the same topological property applies to any scalar field (provided that it is onevalued at infinity). 

Since F is closed in S3 whose second group of cohomology is trivial, it must also be exact or, in other words, there exists a 1-form A, well defined in S3 and such that F = dA. As was shown in 1947 by Whitehead [43], the integral of the form F through Tia which gives the Hopf index can be written as... 

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. 

Moreover, in order to maintain the orthogonality (70) through every time, it is necessary that the Hopf index of 4>0 and of 0q be equal ... 

The electromagnetic knot given in the previous subsections, a representative of the homotopy class C, can be easily generalized to classes C 2. To do that, we will need a property of the Hopf index. 

Consider a smooth map/ S3 —> S2. We have called the fiber of a point p C S1 to the inverse image f l p), which is generally a closed curve in S3. Now we define the multiplicity of the fiber / 1(p) to the number of connected components of / (//. Consider the map f1 S3 — S2, where n is an integer, for/" to be a good smooth map. The linking number of the closed curves that form the fibers of /" is equal to the linking number of the closed curves that form the fibers off (they are the same curves). However, the multiplicity of the fibers of/" is equal to n times the multiplicity of the fibers of/. Consequently, the Hopf index has the following property ... 

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

This was used to derive Eq. (15). A special case of equation (56) was previously used to classify the ways to build a box The Poincare index theorem was extended by Hopf to vector fields on arbitrary manifolds. For vector fields with m isolated hyperbolic critical points, the Poincare-Hopf index theorem is " ... 

Method 4. Index theory approach.. This method is based on the Poincare-Hopf index theorem found in differential topology, see, e.g., Gillemin and Pollack (1974). Similarly to the univalence mapping approach, it requires a certain sign from the Hessian, but this requirement need hold only at the equilibrium point. 

---