**Electromagnetism, Thermodynamics and Quantum Physics**

68 Trends in Electromagnetism – From Fundamentals to Applications

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*Spain*

**Quantization Rules**

<sup>2</sup>*Universidad Complutense de Madrid*

<sup>1</sup>*Universidad Rey Juan Carlos*

**Topological Electromagnetism: Knots and**

In this chapter, we revise the main features of a topological model of electromagnetism, also called the model of electromagnetic knots, that was presented in 1989 (Rañada, 1989) and has been developed in a number of references. Some of them are (Arrayás & Trueba, 2010; 2011; Irvine & Bouwmeester, 2008; Rañada, 1990; 1992; Rañada & Trueba, 1995; 1997; 2001; Rañada, 2003). One of the main characteristics of this model is that it allows to obtain interesting topological quantization rules for the electric charge (Rañada & Trueba, 1998) and the magnetic flux through a superconducting ring (Rañada & Trueba, 2006). We will pay

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 . This number is a measure of how much the force lines curl themselves the ones around the others. These lines coincide with the level curves of a pair of complex scalar fields *φ*(**r**, *t*), *θ*(**r**, *t*). In the model of electromagnetic knots, the physical space and the complex plane are compactified to *<sup>S</sup>*<sup>3</sup> and *<sup>S</sup>*2, so that the scalars can be interpreted as maps *<sup>S</sup>*<sup>3</sup> �→ *<sup>S</sup>*2, which are known to be classified in homotopy classes characterized by the integer value of the Hopf

The topological model of electromagnetism is locally equivalent to Maxwell's standard theory in the sense that the set of electromagnetic knots coincides locally with the set of the standard radiation fields (radiation fields are electromagnetis fields such that the magnetic field is orthogonal to the electric field at any point and at any instant of time). In other words, standard radiation fields can be understood as patched together electromagnetic knots. This can still be expressed as the statement that, in any bounded domain of space-time, any standard radiation fields can be approximated arbitrarily enough by electromagnetic knots. It is remarkable that the standard Maxwell's equations are the *exact linearization*, by change of variables *not by truncation*, of a set of nonlinear equations referring to the complex scalar fields *φ*(**r**, *t*) and *θ*(**r**, *t*). The fact that this change is not completely invertible has the surprising consequence that the linearity of the Maxwell's equations is compatible with the existence of topological constants of the motion which are nonlinear in the magnetic and electric fields. In

this chapter we will see how to find some of these topological constants.

**1. Introduction**

special attention to these features.

index *n*, which is related to the linking number .

Manuel Arrayás1, José L. Trueba1 and Antonio F. Rañada2
