Magnetic monopoles

Like the curl condition is reminiscent of the Yang-Mills field, the quantization just mentioned is reminiscent of a study by Wu and Yang [76] for the quantization of Dirac s magnetic monopole [77-78]. As will be shown, the present quantization conditions just like the Wu and Yang conditions result from a phase factor, namely, the exponential of a phase and not just from a phase. 

XVIII. Technical Appendix C On the Absence of Magnetic Monopoles and Currents in 0(3) Electrodynamics References... 

In this section, the field equations (31) and (32) are considered in free space and reduced to a form suitable for computation to give the most general solutions for the vector potentials in the vacuum in 0(3) electrodynamics. This procedure shows that Eqs. (86) and (87) are true in general, and are not just particular solutions. On the 0(3) level, therefore, there exist no topological monopoles or magnetic charges. This is consistent with empirical data—no magnetic monopoles of any kind have been observed in nature. 

Yu. A. Simonov, Cluster Expansion, Non-Abelian Stokes Theorem and Magnetic Monopoles,... 

Helmholtz decomposition, defined by Eq. (40), can be readily applied to Maxwell s set of equations with magnetic monopoles, provided these equations are written as... 

The absence of magnetic monopole implies the conditions V = 0 and B = -Vx = 0, which are consistent with the relations given above, since B =VAA =0. However, as with the A vector potential, the equality Bx = = Vx / 0 enables us to define a scalar potential x that can be calculated on the basis of Biot-Savart law for a filiform (filament-shaped) circuit... 

These fields are regarded as independent functions in the method of functional variation. In developing their concept of a magnetic monopole, t Hooft and Polyakov identified cj> with a scalar component of the electromagnetic field, a component that they denoted F [46]. It is convenient for our purposes to identify <() with a scalar component B of the electromagnetic field in the vacuum. Therefore, there are two independent magnetic flux density components ... 

Another question is related to the symmetrization of Maxwell s equations. Dirac asked himself whether there existed magnetic monopoles, and proposed inclusion of a magnetic source to make Maxwell equations symmetric. 

Section III. F exhibits a different route to symmetrize the equations, without the introduction of magnetic monopoles. 

From the symmetric set, an extended set of Maxwell equations was exhibited in Section V.E. This set contains currents and sources for both fields E, B. The old conjecture of Dirac s is vindicated, but the origin of charge density is always electric (i.e., no magnetic monopole). Standard Maxwell s equations are a limiting case in far field. 

We consider now the Aharonov-Bohm effect as an example of a phenomenon understandable only from topological considerations. Beginning in 1959 Aharonov and Bohm [30] challenged the view that the classical vector potential produces no observable physical effects by proposing two experiments. The one that is most discussed is shown in Fig. 10. A beam of monoenergetic electrons exists from a source at X and is diffracted into two beams by the slits in a wall at Y1 and Y2. The two beams produce an interference pattern at III that is measured. Behind the wall is a solenoid, the B field of which points out of the paper. The absence of a free local magnetic monopole postulate in conventional... 

Since magnetic monopoles have never been found, the irreducible magnetic entity must be the magnetic dipole. In Ampere s law, we can define the magnetic flux density or magnetic induction B due to, or induced by, an electrical current j at a distance r, as... 

Even though magnetic monopoles do not exist, this magnetic dipole moment vector [ll can also be considered conceptually (Fig. 3.8a) as consisting of two magnetic monopoles, the "North pole" q, and the "South pole" —q linked by the vector 8 ... 

The interaction with a magnetic field may similarly be written in term of magnetic dipole, quadrupole etc. moments (there is no magnetic monopole, corresponding to electric charge). Since the magnetic interaction is substantially smaller in magnitude than the electric, only the dipole term is normally considered. ... 

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