
Abstract
There is a topological structure in the set of the electromagnetic radiation fields (with E·B = 0) in vacuum. A subset of them, called here the admissible fields are associated with maps S^{3} → S^{2} and can be classified in homotopy classes labelled by the value of the corresponding Hopf indexes, which are topological constants of the motion. Moreover, any radiation field can be obtained by patching together admissible fields and is therefore locally equal to one of them. There is, however, an important difference from the global point of view, since the admissible fields obey the topological quantum conditions that the magnetic and the electric helicities are equal to integer numbers n and m times an action constant α which must be introduced because of dimensional reasons, that is ∫A·Bd^{3}r = n α, ∫C·Ed^{3}r = m α, where B and E are the magnetic and electric fields and ∇ × A = B, ∇ × C = E. A topological mechanism for the quantization of the electric charge operates in the set of the admissible fields, in such a way that the electric flux through any closed surface around a point charge is always equal to √α times an integer number n', equal to the degree of a map S^{2} → S^{2}, corresponding to the existence of a fundamental charge with value q_{o} = √α/4 π. It is argued that results of this kind could help reaching a better understanding of quantum physics.
1. Introduction
There is little doubt that topology will in future play a major role in quantum physics. As Atiyah (1990) puts it, this is not surprising, since 'both topology and quantum physics go from the continuous to the discrete'. Topological ideas were already used in the study of the structure of matter more than a century ago by Lord Kelvin, who imagined in 1868 that the atoms could be knots or links of the vorticity lines of the aether, to which he applied the then new Helmholtz theorems on fluid dynamics (Kelvin 1868, Tait 1911, Archibald 1989). He understood, in a remarkable combination of geometrical insight and physical intuition, that such knots and links would be extremely stable, just as matter is. Furthermore, the many different ways in which curves can be linked or knotted offered an explanation for the remarkable variety of the properties of the chemical elements. From our modern perspective, we can add to stability and variety two other very important qualities of matter which were not known in Kelvin's time. One is transmutability, the ability of atoms to change into others of a different kind as an effect of nuclear reactions, which could be related to the breaking and reconnections of vorticity lines (as happens, for instance, to the magnetic lines in tokamaks during disruptions). The second is the discrete energy spectrum, which is also a characteristic of the nontrivial topological configurations of vector fields, as has been proved by Moffatt (1990a), making use of a theorem by Freedman (1988). At the time, Kelvin’s model had a good reception, being praised, for instance, by Maxwell. But neither topology nor atomic phenomenology were sufficiently developed to follow his very deep insight, which explains why it was later forgotten and remained unknown for a very long time.
Topology was also at the basis of one of Dirac’s (1931) most appealing and intriguing proposals, the monopole, which embodied a mechanism for the quantization of the electric charge, an idea developed later in other contexts (Polyakov 1974, ’t Hooft 1974). Since Aharonov and Bohm (1959) discovered the effect that bears their name, it has been known that the description of the electromagnetic field requires topological considerations. The sineGordon equation offers the simplest model with a conservation law of topological origin, based on the degree of a map S^{1}→ S^{1} of the circle on itself. Its extension to three dimensions allowed Skyrme (l961, 1988) to build a model with topological solitons and conserved current, the corresponding quantity taking only integer values equal to the degree of a map S^{3} → S^{3} between threedimensional spheres. As Skyrme explained he had three motives for proposing such a model: unification; renormalization; and what he called the fermion problem. His skyrmion, as his basic soliton became known, would be a fundamental boson from which he hoped to build all the particles and, because a topological theory must be nonlinear, the possibility of removing the infinities seemed a real and attainable aim.
In the past, the applications of topology to field theory concentrated on knots and links (Atiyah 1990), the classification of which had been attempted by Tait (1911) after discussions with Kelvin concerning his atomic models. He was then able to pose the problem and to formulate some conjectures. However, in spite of its interest and beauty, this branch of mathematics fell into oblivion for several decades, until the discovery in 1928 of the Alexander polynomials, which are invariants associated with each knot or link. In 1984, Jones found another set of polynomials, which were found to be very useful for classifying knots or links, which allowed proof of some of ’hit’s conjectures. Although these developments arose from pure mathematics, they turned out to be related to YangMills field theory, a very important physical application being the construction of a topological quantum field theory, as proposed by Witten (1988, 1989), which may open the way to a deeper understanding of quantum physics.
It will be shown in this paper that the set of the radiation solutions of Maxwell equations in vacuum (those which verify E·B = 0) has a subset, called here the admissible fields with a curious topological structure shown in the links formed by pairs of field strength lines. Any pair of maps ф, θ: S^{3} → S^{2} (with orthogonal level curves) is associated with an admissible electromagnetic field, such that the magnetic and electric lines are the level curves of ф and θ, respectively. As a consequence, the linking numbers of any pair of magnetic lines and of any pair of electric lies are two topological constants of the motion, taking only integer values n and m, equal to the Hopf index of the corresponding maps.
Journal of Physics A: Mathematical and General, 25 (1992) 16211641.