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Home , Dyon, John Michell, Magnetic monopole

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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 culmi nating in ’s hypothesis, we review the modern theory of magnetic monopoles beginning with Dirac’s 1931 paper. Saha’s procedure of mono ple quantisation, using the 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 -magnetic scattering in the lowest partial wave such as how acquires fractional , 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 to attract pieces of iron has been known since antiquity. The English word ‘’ 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 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 ’ 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 . He found in 1820 that an 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 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 . 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 only the phase difference of the 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 = § / B(:n) - d.§' = Q ( 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 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 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 , 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 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 , 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 in which large mass ratio of neutron to 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 ‘ = €>

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 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. Dirac’s quantisation condition has also to be generalised so that it ex hibits the duality invariance. We shall refer to a particle, carrying both electric charge e and magnetic monopole g, a (e, g). Consider two (e,,.,g,,,) and (e,,,g,,). The duality invariants are cj, + gfn, eg + gf, e,,,e,. — e,,g,,.. The last one replaces the angular momentum expression eg for (e,0) and (0, g) system. The Dirac condition now becomes‘

e,,,g,, — g,,,e,, : Nh/2 N : integral. OCR Output From duality invariance we can choose g = 0 for one of the , say in nature and define its charge as e. The general solution of the quantisation condition is then given by

en : Zn,1 C Zn,2 el gn : ézn,2 where Z,,_1 and Z,,_2 are integral. A corrolary of this result is that for all magnetically neutral systems, their electric charge is an integral multiple of electric charge. Further for magnetically non-neutral and electrically neutral systems, if they exist, e'/e should be a rational number.

5. Wu-Yang anproach

In all the previous work we have to work with a singular electromagnetic potential A"(z) to discuss a magnetic monopole. A nonsingular A"(a:) would lead to g = 0. Dirac string is therefore required. It was realised by Wu and Yang that this difficulty arises from our insistence to use a single A,,(z) for the whole configuration spacel To discuss an analogous situation the surface of the is perfectly smooth and yet it cannot be covered by any single two dimensional coordi nate system without introducing a singularity. The solution is to use more than one coordinate system which partially overlap and require nonsingular relationship in the overlap regions. Actually two coordinate patches suflice. Thus for a magnetic monopole g, located at origin, we have the magnetic field B : ga/1-’ Let us divide the space in two partly overlapping regions

Ra :r>0,§1r+6>0;0,21r>¢?_0

Rb :1·>0, 1r20>§1r—6, 27l’>¢20. OCR Output Their overlap region Rbb is given by

R,,b;¢>0, §+6>0>§—6, 21r>¢;0.

We define two nonsingular electromagnetic potentials, .#l'(“)(2:) in Rb, and A"(:z:) in Rb by _ (G) : (a) : Ap,) : g(1 — COS H) R° ` Ar A8 0, 4* rsin0 . (bi : an : an Z y(+l-—¤<>¤0) Rb ` A" A0 0’ A° 1·sin6 In the overlap region 12,,,, ; B`(r) : 6 X' A<¤> Z V X EW

A—l") z=lib) 95 $6* S : e2icg¢·

The wavefunction 1/:(:z:) of the electron moving in the magnetic field of the monopole g has to be generalised to wave sections ¢(“) in Rb and 1/> (") in Rb. In the overlap region we have ¢(¤) = g¢(*>)_

Since a wavesection should return to itself after going around the polar axis once the gauge transformation S has to be single valued. This leads to the Dirac quantisation condition

841ricg :

6. Eermion-Mononole sxsmrrrami £·aQi.¤_n___.1_...z.al ¢l<·>¤¢r‘°¢ char G

Dirac in his 1931 paper also studied electron wavefunctions in the field of a monopole using Schrodinger equation. The problem has been fur ther discussed in its various aspects by s. number of authors using both OCR Output Schrédingcr equation and for the electronw. Exact solution of Schrodinger equation in the field of a magnetic monopole plus Aharonov Bohm potential have also been obtainedl The scattering problem of a Dirac electron (mass = M) in the field of a magnetic monopole at origin, was Hrst discussed by Kazama, Yang and Goldhaber15·16. We have the hamiltonian H H Z a-(-N-eE)+;2M : az-1?+6M

We have the total angular momentum (q = eg)

L+§& ix (-iv - J) - gw, [1,11]

The eigenvalues of J2 are given by j( j -1- 1) where

i= lql — 1/2, lql + E. lql + 3/2,

The lowest partial wave j = |q| — 1/2 presents a number of unusual features. We therefore concentrate on this partial wave for the rest of this section. Removing the angular dependence of the wavefunction we obtain the radial wave equation

H,..dx(¢‘) = Ex(’r)

where M -1Zd/ dr 11,,,,; = d ` { ——Mdr a with the scalar product

(X1("),H,..4X2(")} — (HmaX1("l»X2(T)l = i lFf(0)Gz(0) _ F2(0)Gi(0)l· This has to vanish if Hm; is self adjoint. Further the boundary condition on X2(·r) (restricting the Domain of Hmd) must be such that this vanishing implies the same boundary condition on X1(r) (which restricts the Domain of the adjoint of Hm;). We thus have to impose the boundary conditions

F(0) :i tan + G(0) where 0 is a real parameter. We thus have a one parameter family of self adjoint extensions. Under CP conjugation X(r) —> X*(r). Thus ew —> e`“° under CP. Therefore for CP invariance we must have 0 = 0,1r and this is the choice of boundary conditions made by Kazama et al. If the mass M = 0, then it can be seen that a chiral rotation is equivalent to a shift in value of 0. Therefore there is no physical effect of the parameter 0 since it’s effect can be rotated away. If however the mass M is nonzero, the spectrum of H consists of oo > |E| 2 m, and a at E = Msin0 if cos0 < 0. Let H¢E(T) = E¢¤(T)» f`d°F¢L(F)¢E#(¢) = 6(E · E') The charge Q is then given by Q = —§/d”F(/M dEI¤!¤¤(F')|2 — dEl¢¤(F)|”+ |¢v(¢")I2

10 OCR Output where 1bB (1-) is the bound state wavefunction. We obtain

eM sin0 da: Q _ vr ju \/zz — M2(:z: + Mcos 0)(2eg) = —§-f;l for eg=1/2.

Thus monopole becomes a dyon and acquires an electric charge Q which is a fractional number in units of electronic charge. This agrees with Witten’s resultslg. This is consistent with Dirac quantisation condition for dyons given earlier. Another unusual feature of the fermion-monopole system in this lowest partial wave is the helicity leaking. The helicity h is represented by the operator E · if 0 0 3 · v?] and formally commutes with the Hamiltonian ie [H, h] = 0. In all the partial waves except the lowest the helicity is found to be conserved as expected. On the other hand for the lowest partial wave j = |q| — 1/2, one finds that the helicity oonservinga scattering amplitudes are zero while helicity changing amplitude is finite. The pathology arises from the fact that in this partial wave we have

h = pr (3 2) , where prf = —g%(rf and the radial momentum operator p, has no self adjoint extension for the configuration space 0 § r < oo. Helicity leaks through the magnetic monopole16·"~18. Fractional charge and helicity leakage occur together for configuration space 0 §_ r < oo. If the configuration space is 0 §·r $ R (R finite) then fractionally charged dyons can be nonleakingm. It is also possible to find helicity conserving regularisations of the hamiltonian in 0 3 r < oo but it is not clear as to what extra physics could lead to the like features at r = 0 found in this regularisationzl. An investigation of other physical consequences of the black hole monopole might be illuminating.

11 OCR Output When the internal structure of the monopoles of nonabelian theories, which we discuss in the next section, is taken into account then it is found that helicity is indeed conserved, but the charge may not be"·”. The effect of pair production however may change this picture2‘*. That these consideration could lead to number violating decays catalysed by magnetic monopoles was pointed out by Rubakovzs and Callanz

7. t’hc0ft-Polyakov Monopole

The magnetic monopoles which we have discussed so far are point ob jects. It may however be expected that monopoles in view of their strong coupling may well have structure. t’hooft—Polyakov found a classical static solution of Georgi-Glashow SO(3) gauge model with a triplet of Higgs field , which was of finite energy and represents a nonsingular model of mag netic monopole with a structure"; We have the lagrangian

L = ipe -1;,,.. +.. -1>#<1> 1 .D,,·1>.. .. - (i - <1» - am 4 2 4

With the ansatz

= j;H(§), A1= —¤»a;[1 — K(€)l 2 = 0 (f = aer)

the equations of motion reduce to

d2K 2 2

d2H 2 A 2 2 §2—E=2K H+EH(H —§

The finiteness of the monopole mass leads to the boundary conditions

H(€)~£ wd K(€)—>0 as E—·<><>, HSO(§) and K(§)—1 < 0(§) as §—» 0.

12 OCR Output The asymptotic behavior of the magnetic field for this solution leads to

y = -1/¢ and mass M, M : 24 fo./C). 2 The coupled equations for H (§ ) and K (tf ) have to be integrated numerically leading to a calculation of the unknown function f. It was realised by Prasad and Sommerfieldzl that in case A = 0, ie vanishing Higgs mass, we have the analytic solution

H(§) = fcothf — 1 K(£) = {/Sinhé and leads to f(0) = 1. The monopole mass in this case satisfies the Bogo molny bound on the monopole massz

M 2 41ra|g|.

This special case is known as Bogomolny-Prasad-Sommerfield (BPS) monopole. Note that for t’hooft-Polyakov monopole no Dirac string is needed as there is no singularity attached with it due to its internal structure. Monopole size is of the order of (1 /MW) where the mass of the massive gauge Mw = ae. The generalisation of t’hooft-Polyakov monopole to the dyon case was carried out by Julia and Zeez Magnetic monopole charges to be expected in a Yang-Mills theory with gauge group G and with a Higgs potential U (Q) can be obtained by topolog ical considerations without solving the dynamical equations. The potential U (Q) would have many degenerate minima. Let Q = Q0 be one of these. Let H be the subgroup of G that leaves QU invariant. Except "accidental” degeneracy, the manifold of the minima of U (Q) would be the coset space

G/H = {Q : Q : SlQ0,Q 6 G}.

We can associate a mapping from the two dimensional sphere S2 at spatial infinity into G /H . These mapping fall into topological equivalence classes

13 OCR Output which can be endowed with a group structure 1r2(G/H ), the second ho motopy group of G/H. 1r2(G/H) = n·1(H)/1r1(G). The group 1r1(H) is discrete and its elements give the possible values of the topological charges. Magnetic monopoles carry this topological chargeil The fundamental magnetic monopole solutions have been worked out for other gauge groups arising in grand unified and other modelsmm. In the BPS limit great progress has also been made in obtaining multimonopole so lutions by using Atiyah-Ward ansatz, ADHM Construction or using theoretic methods33. The role of grand unified monopoles has been also extensively discussed in astrophysics and cosmology32·34·°5

Acknowledgements

The author wishes to thank Prof. S.M. Roy for many discussions about magnetic monopoles. He also wishes to thank Prof. H. Banerjee for inviting him to write this review for the special Saha issue of this journal. This material was also presented as an invited talk at a Saha Centennial seminar organised by the Physics department of the Calcutta University on Oct. 8, 93.

14 OCR Output References

The hist0rica.l account in this section is based on E. Whittaker, (1960), A History of the theories of Aether and electricity, Harper Torchbooks. See also V. Singh, Science To-day (Jan. 1980) 19-23. P. Thuillier, (1991), From Philosophy to Electromagnetism: The Oersted Case, The World Scientist No. 38 60-68.

E. Amaldi and N. Cabibbo, (1972), On the Dirac Magnetic Poles in ‘Aspects of Quantum Thy ’ edited by A. Salam and E.P. Wigner (Cambridge, 1972).

G. Giacomelli, (1986), The experimental detection of magnetic monopoles, in Theory and Detection of Magnetic Monopoles in Gauge Theories, edited by N. Craigie, (Singapore, World Scientific). P.A.M. Dirac, (1931), Proc. Roy. Soc. (London) A133, 60; P.A.M. Dirac, (1948), Phys. Rev. B, 817 (1948). H. Poincaré, (1896), Compt. Rend. Acad. Sc. @, 530. J .J . Thomson (1990), Elements gf the Mathematical Theory of Electricit and Magnetism, (Mass. Cambridge; 1st ed); J .J . Thomson, (1904), Phil. Mag. 8, 331. M.N. Saha, (1936), Indian Jour. Physics 10, 141. See also H.A. Wilson, (1949), Phys. Rev. QQ, 308, M.N. Saha, (1949), Phys. Rev. Q, 1968. J. Schwinger, (1968), Phys. Rev. @, 1536; J. Schwinger, (1969), Science @, 757. A.O. Barut, (1971), in Topics in Modern Physics (E.U. Condon Volume), (Boulder, Univ. of Colorado Press).

I0. L. Silberstein, (1907), Ann. Phys. Chem. 22, 579; G.Y. Rainich, (1925), Trans. Ann. Math. Soc. Q, 106. 11. J. Schwinger, (1968), Phys. Rev. @, 1536. D. Zwanziger, (1968), Phys. Rev. @, 1489.

15 OCR Output 12. T.T. Wu and C.N. Yang, (1975), Phys. Rev. @, 3845; T.T. Wu and C.N. Yang, (1976), Nucl. Phys. B107, 365.

13. I. Tamm, (1931), Z. Phys. Z1, 141. M. Fierz, (1944), Helv. Physica Acta Q, 27. P.P. Banderet, (1946), Helv. Phys. Acta. §, 503. Harish Chandra, (1948), Phys. Rev. 74, 883. C.J. Eliezer and S. Roy, (1962), Proc. Camb. Phil. Soc. 58, II, 401. D. Zwanziger, (1968), Phys. Rev. QQ, 1480.

14. S.M. Roy and V. Singh, (1983), Phys. Rev. Letters Q, 2069.

15. Y. Kazama, C.N. Yang and A.S. Goldhaber, (1977), Phys. Rev. D15, 2287, 2300.

16. A.S. Goldhaber, (1977), Phys. Rev. E, 1815.

17. H. Yamagishi, (1983), Phys. Rev. D27, 2383.

18. B. Grossman, (1983), Phys. Rev. Lett. Q, 464.

19. E. Witten, (1979), Phys. Lett. §_6_l§, 283.

20. S.M. Roy and V. Singh, (1985), Pramana, 24, 611.

21. A.P. Balachandran, S.M. Roy and V. Singh, (1983), Phys. Rev. _D2_8, 2669.

22. C.G. Callan, (1982), Phys. Rev. D25, 2141; E, 2058, and Nucl. Phys. B212, 391.

23. R. Jackiw and C. Rebbi, (1976), Phys. Rev. .P£, 3398; C.G. Callan and S.R. Das, (1983), Phys. Rev. Lett. 51, 1155. W.J. Marciano and I..I. Muzinich, (1983), Phys. Rev. Lett. @, 1035. A.M. Din and S.M. Roy, (1983), Phys. Lett. 129B, 201.

24. J. Preskill, (1984), Ann. Rev. Nucl. Part. Sc. Q, 461. A. Sen, (1985), Nucl. Phys. B250, 1.

25. V. Rubakov, (1981), JETP Lett. Q, 644; V. Ruhakov, (1982), Nucl. Phys. B203, 311.

16 OCR Output 26. G. t’Hooft, (1974), Nucl. Phys. _l;7_9, 276; A.M. Polyakov, (1974), JETP Lett. @, 194; A.M. Polyakov, (1974), Sov. Phys. - JETP Q, 988.

27. M.K. Prasad and C.M. Sommerfield, (1975), Phys. Rev. Lett. Q, 760.

28. E.B. Bogomolny, (1976), Sov. J. Nucl. Phys. 24, 449.

29. B. Julia and A. Zee, (1975), Phys. Rev. l, 2227. For a Hamiltonian formulation see S. Wadia, (1977), Phys. Rev. R15, 3615.

30. Y.S. Tyupkin, V.A. Fateev and A.S. Schvarts, (1975), JETP Lett. L, 41. M.I. Monastyrskii and A.M. Perelemov, (1975), JETP Lett. 21, 43. S. Coleman, (1983), The Magnetic Monopole fifty years later in The Unity of Fundamental Interactions ed. by A. Zichichi (Plenum, New York).

31. P. Goddard and D.I. Olive, (1978), Rep. Prog. Phys. 41, 1357.

32. N.S. Craigie (ed.), (1986), Theory and Detection of Magnetic Monopoles in gauge theories, (Singapore, World Scientific) and Ref. 24.

33. N.S. Craigie, P. Goddard and W. Hahm (ed.), (1982), Monopoles in uantum Field Theory (Singapore, World Scientific).

34. V. Singh, (1983), Fortschritts der Physik Q, 569.

35. E.W. Kolb and M.S. Turner, (1990), The Early (Addison Wesley).

17 OCR Output