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From the early days of electromagnetism, it was noticed that the sources of electric and magnetic fields are distinct. The sources of the electric field are point charges (such as the electron whose charge e is a fundamental constant of nature). The electric field E of a point charge is isotropic. Field lines point outward of a positive charge as in Fig. (a). When positive and negative charges are set close together, field lines diverging from the positive charge converge toward the negative charge, thereby forming an electric dipole as shown at Fig. (b). The sources of the magnetic field B are magnetic dipoles, with north and south poles as in Fig. (c), whose similarity to Fig. (b) is evident. But so far magnetic point charges (so called magnetic monopoles) are not found in Nature. Thus, while positive and negative electric charges are ubiquitous [the upper part of Fig. (d)], magnetic monopoles [the lower part of Fig. (d)] are absent. As children, some of us probably tried to get magnetic monopoles by breaking a magnetic bar. The result was a complete failure, see Fig. (e).
It was shown by Dirac about 86 years ago that if there is a magnetic charge (monopole) of strength g then there is a quantization rule 2eg = n~c ( n = 0,1,2… is an integer, ~ is Planck constant and c is the speed of light). This amazing quantum mechanical constraint stimulated an enormous research activity in numerous areas of physics (including fruitless search attempts). One of the earliest endeavors was to solve the hydrogen-like problem of electron in the field of magnetic monopole. Here I approach this problem from a ”condensed matter point of view” using a tight binding model. When the sites upon which the electron resides and hops are the vertices of a highly symmetric object, the Dirac quantization comes out naturally and the underlying physics employs beautiful mathematical tools such as gauge theory, spherical geometry, perfect polytopes, graph theory and point groups. At the end I will suggest possible experimental realizations.