Electromagnetism and magnetic monopoles

Abstract

Starting with the highly symmetric form of electromagnetism in tensor notation, the consideration of magnetic monopoles comes very natural. Following then the paper of J. M. Figueroa-O’Farrill (1998) we encounter the Dirac monopole and the 't Hooft-Polyakov monopole. The former, simpler -- at the cost of a singular Dirac string -- monopole already leading to the very important Dirac quantization condition, implying the quantization electric charge if magnetic monopoles exist. In particular the latter monopole, which is everywhere smooth and which has a purely topological charge, is found as a finite-energy, static solution of the dynamical equations in an $SO(3)$ gauge invariant Yang-Mills-Higgs system using a spherically symmetric ansatz of the fields. This monopole is equivalent to the Dirac monopole from far away but locally behaves differently because of massive fields, leading to a slightly different quantization condition of the charges. Then the mass of general finite-energy solutions of the Yang-Mills-Higgs system is considered. In particular a lower bound for the mass is found and in the previous ansatz a solution saturating the bound is shown to exist: The (very heavy) BPS monopole. Meanwhile particles with both electric and magnetic charge (dyons) are considered, leading to a relation between the quantization of the magnetic and electric charges of both dyons and, when assuming CP invariance, to an explicit quantization of electric charge. Finally, the $\mathbb{Z}_2$ duality of Maxwell's equations is extended. When $PSL(2,\mathbb{Z})$ duality is assumed for electric and magnetic monopoles satisfying the Bogomol'nyi mass bound a dyonic spectrum is found in the orbit of the electron.