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Magnetic Monopoles i .L Y `·* M / ‘ *—~ OCR Output £wB%®3 tillnilnlllllliuiuiunlliilli Paooaoavi TIFR/TH/93-46 Magnetic Monopoles Virendra Singh Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005, India Abstract After a brief historical account of the classical electromagnetism culmi nating in Ampere’s hypothesis, we review the modern theory of magnetic monopoles beginning with Dirac’s 1931 paper. Saha’s procedure of mono ple charge quantisation, using the angular momentum considerations, is next described in its proper historical perspective. Some relevant consid eration arising out of natural symmetry of Maxwell’s equations between electric and magnetic fields and charges are emphasised. A brief descrip tion of Wu-Yang fibre bundle approach, using wave sections, is given. This approach avoids using singular electromagnetic potentials. We use this ap proach to describe unusual features which arise in Dirac fermion-magnetic monopole scattering in the lowest partial wave such as how magnetic monopole acquires fractional electric charge, and (ii) how usual conserva tion laws may be violated. Lastly we use grandunified theories to describe the internal structure of magnetic monopole following the work of t’hooft and Polyakov. The property of lodestone to attract pieces of iron has been known since antiquity. The English word ‘magnetism’ derives from the Greek word ‘magnetis lithos’ for the stone of magnesia. Pierre de Maricourt, a thirteenth century French crusader, carried out experiments in which he placed an iron needle on the surface of a spherical lodestone and marked the direction in which iron needle pointed. On join ing these direction he obtained curves on the surface of the lodestone which were closed. Further all these curves passed through two points on the sur face. The situation is analogous to the geographical situation of meridians of the longitude on the surface of the earth which are closed curves and pass through geographical north and south poles. He therefore named these two points on the surface of lodestone as magnetic north and south poles. William Gilbert, royal physician to the Queen Elizabeth of England, made a great advance in the study of magnetism when he published ‘De Magnete’ in 1600. He observed that the earth itself acts as a magnet and it’s two magnetic poles are located close to it’s geographical poles with south magnetic pole near the north geographical pole and vice versa. He also elucidated the law that unlike magnetic poles attract while one repel. John Michell published ‘A Treatise of Artificial Magnets’ in 1750. He realised that a magnet does not have to be spherical in order to have mag netic north and south poles. He noted "Wherever any magnetism is found, whether in the magnet itself or any piece of iron, etc. excited by the mag net there are always found two poles, which are generally called north and south, ........ ". He also enunciated the inverse square law of force between magnetic poles with north and south magnetic poles of a magnet having equal strength and opposite sign. The corresponding law for electric charges was only published by Charles A. Coulomb some thirty five years later in 1785. H. Oersted, motivated probably by ideas of "German romanticism",2 was the first person to demonstrate an undeniable relationship between magnetism and electricity. He found in 1820 that an electric current exerts a force on a magnetic needle placed parallel to it. J .P. Biot and F. Savart investigated the exact law of force between magnetic field and small electric current elements. OCR Output Ampere then experimentally investigated the forces exerted by two elec tric current carrying wires on each other. The brilliant mathematical anal ysis of the results of these experiments during 1822-27 led to the result that in its magnetic effects an electric current is equivalent to a magnetic shell. In view of this equivalence Ampere proposed his hypothesis that all observed magnetic phenomenon are due to small electric current loops present in magnetic materials. Ampere’s hypothesis has been a cornerstone of our present understand ing of all the electro-magnetic phenomenon observed so far in nature. All of them are explicable in terms of electric charges and their motions. No magnetic monopole charges are thus required and none have been seen so far3·‘*. As a result of this development the interest in magnetic monopoles declined. Magnetic monopoles were occasionally still used later in mag netostatics as a pedagogical device. Now and then they were sometimes discussed as something of purely theoretical interest. 2. Dirac quantisation The revival of interest in magnetic monopoles dates from a paper of Dirac in 1931 in which quantisation of magnetic pole strength was shown to arise from quantum mechanical considerationss Dirac was impressed by the fact that the continued progress of theoreti cal physics seemed to require its mathematical basis to become increasingly more abstract. In quantum mechanics only the phase difference of the wave function at two different points, and not the phase of the wavefunction at any particular point, has a physical meaning since the wavefunction can be multiplied by an arbitrary constant phase factor without changing physics. Dirac therefore investigated a generalisation of the usual quantum mechan ics in which the phase difference of the wavefunction between any two points not only depends on those two points but also depends on particular path connecting them. In general it could then be different for different paths connecting the same two points. In order that this generalisation does not lead to ambiguity in physical predictions, it was concluded that “the change in phase of a wavefunction round any closed curve must be the same for all wavefunctions". This is OCR Output necessary in order to respect the principle of linear superposition in quan tum mechanics. Thus this change of phase must depend on the dynamical system and not on its particular state. Dirac could implement these ideas by requiring the nonintegrable phase difference between two points connected by path P to be given by Q- L Ao;) · as where A(z) is the electromagnetic vector potentials for the system. In general then the phase difference around a closed curve C is given by flyi- Au) - as which, by Stokes theorem, is given by (li = Curl § / B(:n) - d.§' = Q (magnetic flux through the surface 2 bounded loop O). In case there is no line of singularity passing through surface E enclosed by loop G', the phase difference around a closed curve would be zero and the generalisation is equivalent to usual quantum theory of a particle moving in an electromagnetic field and nothing new has emerged. Dirac now notes that what is rea.lly required for non ambiguousness of the physical prediction is that “the change in phase around a closed curve may be different for different wavefunctions by arbitrary multiples of 27l' If the magnetic field B (az) is due to a monopole of strength g, then the associated line of singularity in A(:n) must be running from the monopole position to infinity. If the line of singularity goes through E then we get by using this principle, 41reg ·T*· = 21r1L 1.e. :2 h(n:i1,;r2,---) This is the Dirac quantisation condition for the strength g of magnetic monopole. The least nonzero value of monopole is given by Ig] Z h/2e. OCR Output For this case of magnetic monopoles A has a nodal line of singularity, now called Dirac string, ending at the monopole position. The quantisation condition ensures that the Dirac string is unobservable. It is amusing to note that Poincaré in 1896 had already used the equiv alence of a long thin straight magnet to a magnetic monopole in his expla nation of Birkeland’s experiments on motion of cathode ray beams°. Dirac string is essentially the same construction. 3. Saha’s Derivation Poincaré had also noted that for a charge-magnetic monopole system a vectorial conserved integral of motion exists consisting of the usual me chanical angular momentum term and another radial contribution equal to cgi. He however did not identify this conserved quantity as total angular momentum Jvof the systemc J .J . Thomson had discovered in 1893 that a momentum density, pro portional to Poynting vector, is associated with an electromagnetic iield. He calculated in 1900, the angular momentum, i--/47r d°FF><>< E’(f’)]* carried by the electromagnetic field for charge e-monopole g system, sep arated by a distance d along the direction d, and obtained a value egd. Remarkably it does not depend on the magnitude of the distance d. He also noted that the mechanical angular momentum together with the elec tromagnetic angular momentum was a conserved quantity? Saha made the perceptive remark, in 1936, that the quantisation of J along charge-monopole radial vector ei i.e. j·o cl = eg leads to Dirac quantisation conditions Saha’s paper also contained a model of neutron in which large mass ratio of neutron to electron was attributed to neutron being a magnetic monopole-antimonopole system. While this is not a tenable model for a neutron, Saha’s model anticipated later models involving magnetic monopoles which were suggested by Schwinger and othersg OCR Output Ma.xwe1l’s equations have a natural symmetry between electric and mag netic fields (E,B).1° Introducing a magnetic f0ur—curreut density J; in analogy to electron four—cu1·rent density Jf we have BE ‘ = €><B—Jc Bt OB . = ——€XE—J. Bt 6 -13 = J3, €·B=]_?.’ The Lorentz-force F can be amended to 1i":J§E"+.Z,xB+J;B—.i;,xE.,’· Let U (0) be the two dimensional rotation matrix cos 0 — sin 0 sin 0 cos 0 than the Maxwell equations and Lorentwfoirce together are invariant under the duality rotations E J—»() Un EJJ·» J5 (>(;) J-· Un J5 (JJ In view of this symmetry a more precise formulation of Ampere’s hypothesis would be not that J; = 0 but rather all observed current densities observed so far are such that J; and Jf are proportional to each other.
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