ABSTRACT. A subtle form of nonlinearity is described. It is shown that it appears in a topological model of electromagnetism, where it explains why the electromagnetic helicity and the charge (both electric and magnetic) can be understood as topological constants of the motion.

1. Two ways of linearising

Nonlinear field equations are frequently linearised. This is usually performed in two different ways: by truncation or by change of variables. In the first method, the second and higher order terms are neglected in the Taylor expansion of the equation. In the second one, the old variables u^old_α (r,t) are changed to new variables v^new_β (r,t) by means of the equations

v^new_β = G_β(u^old_α ,∂_µu^old_α ). (1)

The first method is useful only when dealing with weak fields u^old _α, while the second one is possible, it has a much wider validity. In the latter case, the application to a particular nonlinear field equation would seem to suggest that the nonlinearity of the first equation is no more than an accident due to a particularly unfortunate election of the field variables, so that a completely linear theory is possible, with all the properties of the linearity.

This is certainly true if the following condition is verified: that the change of variables (1) is invertible so that the inverse change is well defined for any v^new_β as

u^old_α = H_α(v^new_β,∂_µv^new_β), (2)

which establishes a one to one correspondence between the solutions of the old and the new equations. In other words, if the application of (1) to the set of the solutions of the nonlinear equations gives the complete set of the solutions of the linear one, the converse being also true.

We will consider in this paper an example in which this property does not hold. More precisely, in which the change (1) does not give all the solutions of the linear equation while effectively linearising the nonlinear one. This implies that the theory can be formally presented in a linear language, but not all the linear combinations of solutions are themselves solutions. In other words, although the theory looks linear because all its solutions verify a linear equation, it retains some hidden form of nonlinearity.

This case is the Maxwell’s theory in empty space or with point charges. The electromagnetic field F_µν is writen in terms of a couple of scalar fields ф, θ which obey highly nonlinear equations, but there are linear combinations of solutions of the Maxwell’s equations that are not solutions of these equations. However, both equations, the linear and the nonlinear equations are locally equivalent, their difference being just on the behaviour at infinity.

2. Why a topological theory of electromagnetism

During a great part of the 19th century, electromagnetism was conceived in terms of lines of force, which were thought to be very real, not just a mathematical convenience. One reason for this understanding was, of course, the belief in the existence of the aether, which offered an appealing possibility to explain electromagnetic phenomena: the force lines were a manifestation of its streamlines and vorticity lines. It was expected, therefore, that the electromagnetism would be eventually understood thanks to the mechanics of fluids, in a model in which the force lines would coincide with lines of aether particles and would be therefore something real and tangible [1]. Maxwell himself was very much in favour of Kelvin’s suggestion in 1868 that atoms were knots or links of the vortex lines of the ether, a picture presented expressively in a paper called “On vortex atoms” [2, 3, 4]. He liked the idea, as it expressed for instance in his presentation of the term “Atomism” in the Encyclopaedia Britannica in 1875 [5, 6].

Kelvin had applied to his topological idea the then new Helmholtz’s theorems on fluid dynamics. He did not like the then widely held view of infinitely hard point atoms or, in his own words, “the monstrous assumption of infinitely strong and infinitely rigid pieces of matter” [7]. Kelvin was much impressed by the conservation of the strength of the vorticity tubes in an inviscid fluid according to Helmholtz’s theorems, thinking that this was an inalterable quality on which to base an atomic theory of matter without infinitely rigid entities. We know now that this is also a trait of topological models, in which some invariant numbers characterize configurations which are rigid and can deform, distort or warp. As he put it “Helmholtz has proved an absolutely unalterable quality in any motion of a perfect liquid... any portion [of it] has one recommendation of Lucretius’ atoms — infinitely perennial specific quality”.

Kelvin had the insight that such knots and links would be extremely stable, just as matter is. Furthermore, he thought that the variety of the properties of the chemical elements could be a consequence of the many different ways in which such curves can be linked or knotted. Two other important properties of matter, not known in his time [8], can also be understood. One is the ability of atoms to change into another kind in a nuclear reaction, which could be related to the breaking and reconnection of lines, as happens for instance to the magnetic lines in tokamak and astrophysical plasmas. The other is the discrete character of the spectrum, which is also a property of the nontrivial topological configurations of a vector field, as was shown by Moffatt [9].

The reception to Kelvin’s idea was good, but unfortunately it was soon forgotten. Ironically, this was mainly due to Maxwell’s monumental Treatise on Electromagnetism, after which, because of the successful developments of algebra and differential geometry, the line of force was relegated behind the concepts of electromagnetic tensor F_µν and electromagnetic vectors E_i, B_j, A_µ. It is usually now a secondary concept, always derived from F_µν as the integral lines of B and E.

It turns out that a topological theory of electromagnetism is possible. The rest of this paper contains a terse summary.

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