and Polyakov.the internal structure of magnetic monopole following the work of t’hoofttion laws may be violated. Lastly we use grandunified theories to describemonopole acquires fractional electric charge, and (ii) how usual conservamonopole scattering in the lowest partial wave such as how magneticproach to describe unusual features which arise in Dirac fermion-magneticapproach avoids using singular electromagnetic potentials. We use this aption of Wu-Yang fibre bundle approach, using wave sections, is given. Thiselectric and magnetic fields and charges are emphasised. A brief descriperation arising out of natural symmetry of Maxwell’s equations betweennext described in its proper historical perspective. Some relevant considple charge quantisation, using the angular momentum considerations, ismonopoles beginning with Dirac’s 1931 paper. Saha’s procedure of mononating in Ampere’s hypothesis, we review the modern theory of magnetic

After a brief historical account of the classical electromagnetism culmi

Abstract

Homi Bhabha Road, Bombay 400 005, India

Tata Institute of Fundamental Research

Virendra Singh

Magnetic Monopoles

Paooaoavi TIFR/TH/93-46tillnilnlllllliuiuiunlliilli

£wB%®3

M / ‘ *—~ OCR Outputi .L Y `·*

current elements. OCR Output

investigated the exact law of force between magnetic field and small electrica force on a magnetic needle placed parallel to it. J .P. Biot and F. Savartmagnetism and electricity. He found in 1820 that an electric current exertswas the first person to demonstrate an undeniable relationship between

H. Oersted, motivated probably by ideas of "German romanticism",21785.

was only published by Charles A. Coulomb some thirty five years later inequal strength and opposite sign. The corresponding law for electric chargesmagnetic poles with north and south magnetic poles of a magnet having

south, ........ ". He also enunciated the inverse square law of force betweennet there are always found two poles, which are generally called north andwhether in the magnet itself or any piece of iron, etc. excited by the magnetic north and south poles. He noted "Wherever any magnetism is found,realised that a magnet does not have to be spherical in order to have mag

John Michell published ‘A Treatise of Artificial Magnets’ in 1750. Healso elucidated the law that unlike magnetic poles attract while one repel.south magnetic pole near the north geographical pole and vice versa. Heit’s two magnetic poles are located close to it’s geographical poles withMagnete’ in 1600. He observed that the earth itself acts as a magnet andmade a great advance in the study of magnetism when he published ‘De

William Gilbert, royal physician to the Queen Elizabeth of England,points on the surface of lodestone as magnetic north and south poles.through geographical north and south poles. He therefore named these twoof the longitude on the surface of the earth which are closed curves and passface. The situation is analogous to the geographical situation of meridianswere closed. Further all these curves passed through two points on the suring these direction he obtained curves on the surface of the lodestone whichlodestone and marked the direction in which iron needle pointed. On joinexperiments in which he placed an iron needle on the surface of a spherical

Pierre de Maricourt, a thirteenth century French crusader, carried outword ‘magnetis lithos’ for the stone of magnesia.since antiquity. The English word ‘magnetism’ derives from the Greek

The property of lodestone to attract pieces of iron has been known

round any closed curve must be the same for all wavefunctions". This is OCR Outputpredictions, it was concluded that “the change in phase of a wavefunction

In order that this generalisation does not lead to ambiguity in physicalconnecting the same two points.connecting them. In general it could then be different for different pathsnot only depends on those two points but also depends on particular pathics in which the phase difference of the wavefunction between any two pointsDirac therefore investigated a generalisation of the usual quantum mechanmultiplied by an arbitrary constant phase factor without changing physics.any particular point, has a physical meaning since the wavefunction can befunction at two different points, and not the phase of the wavefunction atmore abstract. In quantum mechanics only the phase difference of the wavecal physics seemed to require its mathematical basis to become increasingly

Dirac was impressed by the fact that the continued progress of theoretito arise from quantum mechanical considerationssDirac in 1931 in which quantisation of magnetic pole strength was shown

The revival of interest in magnetic monopoles dates from a paper of

2. Dirac quantisation

discussed as something of purely theoretical interest.netostatics as a pedagogical device. Now and then they were sometimesdeclined. Magnetic monopoles were occasionally still used later in mag

far3·‘*. As a result of this development the interest in magnetic monopolesmagnetic monopole charges are thus required and none have been seen soof them are explicable in terms of electric charges and their motions. Noing of all the electro-magnetic phenomenon observed so far in nature. All

Ampere’s hypothesis has been a cornerstone of our present understandpresent in magnetic materials.

all observed magnetic phenomenon are due to small electric current loopsshell. In view of this equivalence Ampere proposed his hypothesis thatthat in its magnetic effects an electric current is equivalent to a magneticysis of the results of these experiments during 1822-27 led to the resulttric current carrying wires on each other. The brilliant mathematical anal

Ampere then experimentally investigated the forces exerted by two elec

Ig] Z h/2e. OCR Output

monopole. The least nonzero value of monopole is given byThis is the Dirac quantisation condition for the strength g of magnetic

:2 (n:i1,;r2,---)h

1.e.

41reg ·T*· = 21r1Lusing this principle,position to infinity. If the line of singularity goes through E then we get byassociated line of singularity in A(:n) must be running from the monopole

If the magnetic field B (az) is due to a monopole of strength g, then themay be different for different wavefunctions by arbitrary multiples of 27l'the physical prediction is that “the change in phase around a closed curve

Dirac now notes that what is rea.lly required for non ambiguousness ofin an electromagnetic field and nothing new has emerged.generalisation is equivalent to usual quantum theory of a particle movingloop G', the phase difference around a closed curve would be zero and theIn case there is no line of singularity passing through surface E enclosed by

B(:n) - d.§' = Q (magnetic flux through the surface 2 bounded loop O).§ /

which, by Stokes theorem, is given by (li = Curl

- Au) - asflyi

general then the phase difference around a closed curve C is given bywhere A(z) is the electromagnetic vector potentials for the system. In

Ao;) · asQ- Lby path P to be given byrequiring the nonintegrable phase difference between two points connectedsystem and not on its particular state. Dirac could implement these ideas bytum mechanics. Thus this change of phase must depend on the dynamicalnecessary in order to respect the principle of linear superposition in quan

which were suggested by Schwinger and othersg OCR Outputneutron, Saha’s model anticipated later models involving magnetic monopolesmonopole-antimonopole system. While this is not a tenable model for aratio of neutron to electron was attributed to neutron being a magnetic

Saha’s paper also contained a model of neutron in which large mass

quantisation conditionsJ along charge-monopole radial vector ei i.e. j· cl = eg leads to Dirac

o

Saha made the perceptive remark, in 1936, that the quantisation oftromagnetic angular momentum was a conserved quantity?also noted that the mechanical angular momentum together with the elecRemarkably it does not depend on the magnitude of the distance d. Hearated by a distance d along the direction d, and obtained a value egd.carried by the electromagnetic field for charge e-monopole g system, sep

d°FF><>< E’(f’)]*

i--/47r

He calculated in 1900, the angular momentum,portional to Poynting vector, is associated with an electromagnetic iield.

J .J . Thomson had discovered in 1893 that a momentum density, promomentum Jvof the systemccgi. He however did not identify this conserved quantity as total angularchanical angular momentum term and another radial contribution equal toa vectorial conserved integral of motion exists consisting of the usual me

Poincaré had also noted that for a charge-magnetic monopole system

3. Saha’s Derivation

string is essentially the same construction.nation of Birkeland’s experiments on motion of cathode ray beams°. Diracalence of a long thin straight magnet to a magnetic monopole in his expla

It is amusing to note that Poincaré in 1896 had already used the equivcondition ensures that the Dirac string is unobservable.called Dirac string, ending at the monopole position. The quantisationFor this case of magnetic monopoles A has a nodal line of singularity, now

e,,,g,, — g,,,e,, : Nh/2 N : integral. OCR Output

for (e,0) and (0, g) system. The Dirac condition now becomes‘e,,,e,. — e,,g,,.. The last one replaces the angular momentum expression egdyons (e,,.,g,,,) and (e,,,g,,). The duality invariants are cj, + gfn, eg + gf,electric charge e and magnetic monopole g, a dyon (e, g). Consider twohibits the duality invariance. We shall refer to a particle, carrying both

Dirac’s quantisation condition has also to be generalised so that it exso far are such that J; and Jf are proportional to each other.would be not that J; = 0 but rather all observed current densities observedIn view of this symmetry a more precise formulation of Ampere’s hypothesis

J—» Un JJ·» J-· Un J5 (JJE J5 (>(;)E ()the duality rotationsthan the Maxwell equations and Lorentwfoirce together are invariant under

cos 0 — sin 0 sin 0 cos 0

Let U (0) be the two dimensional rotation matrix

,’·

1i":J§E"+.Z,xB+J;B—.i;,xE.

The Lorentz-force F can be amended to

’

= J3, €·B=]_?.6 -13

Bt

.

= ——€XE—J.OB

Bt

‘

= €><B—JcBE

analogy to electron four—cu1·rent density Jf we havenetic fields (E,B).1° Introducing a magnetic f0ur—curreut density J; in

Ma.xwe1l’s equations have a natural symmetry between electric and mag

Rb :1·>0, 1r20>§1r—6, 27l’>¢20. OCR Output

Ra :r>0,§1r+6>0;0,21r>¢?_0

Let us divide the space in two partly overlapping regions

B : ga/1-’field

Thus for a magnetic monopole g, located at origin, we have the magneticrelationship in the overlap regions. Actually two coordinate patches suflice.than one coordinate system which partially overlap and require nonsingularnate system without introducing a singularity. The solution is to use moresmooth and yet it cannot be covered by any single two dimensional coordi

To discuss an analogous situation the surface of the sphere is perfectlyinsistence to use a single A,,(z) for the whole configuration spacel

It was realised by Wu and Yang that this difficulty arises from ourwould lead to g = 0. Dirac string is therefore required.potential A"(z) to discuss a magnetic monopole. A nonsingular A"(a:)

In all the previous work we have to work with a singular electromagnetic

5. Wu-Yang anproach

should be a rational number.

magnetically non-neutral and electrically neutral systems, if they exist, e'/etheir electric charge is an integral multiple of electric charge. Further for

A corrolary of this result is that for all magnetically neutral systems,where Z,,_1 and Z,,_2 are integral.

gn : ézn,2

en : Zn,1 C Zn,2 el

quantisation condition is then given byelectrons in nature and define its charge as e. The general solution of theFrom duality invariance we can choose g = 0 for one of the particles, say

ther discussed in its various aspects by s. number of authors using both OCR Outputof a monopole using Schrodinger equation. The problem has been fur

Dirac in his 1931 paper also studied electron wavefunctions in the field

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

841ricg :

Dirac quantisation condition

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

¢(¤) = g¢(*>)_

in Rb. In the overlap region we have

the monopole g has to be generalised to wave sections ¢(“) in Rb and 1/> (")The wavefunction 1/:(:z:) of the electron moving in the magnetic field of

S : e2icg¢·

A— z= 95l") lib) $6*

'

12,,,, ; B`(r) : 6 X A<¤> Z V X EW

In the overlap region

1·sin6. (bi : an : an Z y(+l-—¤<>¤0) Rb ` A" A0 0’ A°

rsin0_ (G) : (a) : Ap,) : g(1 — COS H) R° ` Ar A8 0, 4*

and A"(:z:) in Rb byWe define two nonsingular electromagnetic potentials, .#l'(“)(2:) in Rb,

R,,b;¢>0, §+6>0>§—6, 21r>¢;0.

Their overlap region Rbb is given by

<xi(¢).xz(¢)) = In dr xl(¢‘)xz(¢)OCR Output

with the scalar product

——M a11,,,,; = d `{ dr

M -1Zd/ drwhere

H,..dx(¢‘) = Ex(’r)

the radial wave equationsection. Removing the angular dependence of the wavefunction we obtainfeatures. We therefore concentrate on this partial wave for the rest of this

The lowest partial wave j = |q| — 1/2 presents a number of unusual

i= lql — 1/2, lql + E. lql + 3/2,

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

[1,11]

ix (-iv - J) - gw,

L+§&

We have the total angular momentum (q = eg)

H Z a-(-N-eE)+;2M : az-1?+6M

Goldhaber15·16. We have the hamiltonian Ha magnetic monopole at origin, was Hrst discussed by Kazama, Yang and

The scattering problem of a Dirac electron (mass = M) in the field ofBohm potential have also been obtainedlof Schrodinger equation in the field of a magnetic monopole plus AharonovSchrédingcr equation and Dirac equation for the electronw. Exact solution

10 OCR Output

Q = —§/d”F(/M dEI¤!¤¤(F')|2 — dEl¢¤(F)|”+ |¢v(¢")I2The vacuum charge Q is then given by

f`d°F¢L(F)¢E#(¢) = 6(E · E')

H¢E(T) = E¢¤(T)»

of oo > |E| 2 m, and a bound state at E = Msin0 if cos0 < 0. LetIf however the mass M is nonzero, the spectrum of H consists

0 since it’s effect can be rotated away.to a shift in value of 0. Therefore there is no physical effect of the parameter

If the mass M = 0, then it can be seen that a chiral rotation is equivalentand this is the choice of boundary conditions made by Kazama et al.

ew —> e`“° under CP. Therefore for CP invariance we must have 0 = 0,1r

adjoint extensions. Under CP conjugation X(r) —> X*(r). Thus

where 0 is a real parameter. We thus have a one parameter family of self

F(0) :i tan + G(0)

of the adjoint of Hm;). We thus have to impose the boundary conditionsimplies the same boundary condition on X1(r) (which restricts the Domainon X2(·r) (restricting the Domain of Hmd) must be such that this vanishingThis has to vanish if Hm; is self adjoint. Further the boundary condition

= i lFf(0)Gz(0) _ F2(0)Gi(0)l·

(X1("),H,..4X2(")} — (HmaX1("l»X2(T)l

become self adjoint. Let x(1·) = , then

tions at r = 0 so that the hamiltonian Hmd for the lowest partial waveYamagishiu and Grossmanls considered the allowed boundary condi

moment to the Hamiltonianling from the electron having an infinitesimally small anomalous magnetic

Kazama et al solved the difficulty by adding an interaction term aristhe difficulty earlierlthe boundary condition X(r) = 0 at r = 0. Harish Chandra had also notedIt is easy to see that there are no nonvanishing solutions if X(r) satisfies

11 OCR Output

consequences of the black hole monopole might be illuminating.at r = 0 found in this regularisationzl. An investigation of other physicalnot clear as to what extra physics could lead to the black hole like featureshelicity conserving regularisations of the hamiltonian in 0 3 r < oo but it isfractionally charged dyons can be nonleakingm. It is also possible to findspace 0 §_ r < oo. If the configuration space is 0 §·r $ R (R finite) then

Fractional charge and helicity leakage occur together for configuration

monopole16·"~18.the configuration space 0 § r < oo. Helicity leaks through the magneticand the radial momentum operator p, has no self adjoint extension for

h = pr , where prf = —g%(rf(3 2)that in this partial wave we havehelicity changing amplitude is finite. The pathology arises from the factone finds that the helicity oonservinga scattering amplitudes are zero whileexpected. On the other hand for the lowest partial wave j = |q| — 1/2,partial waves except the lowest the helicity is found to be conserved asand formally commutes with the Hamiltonian ie [H, h] = 0. In all the

E · if 0 0 3 · v?]operator

partial wave is the helicity leaking. The helicity h is represented by theAnother unusual feature of the fermion-monopole system in this lowest

given earlier.resultslg. This is consistent with Dirac quantisation condition for dyonsfractional number in units of electronic charge. This agrees with Witten’sThus monopole becomes a dyon and acquires an electric charge Q which is a

l

= —§-f; for eg=1/2.

Q _ vr ju \/zz — M2(:z: + Mcos 0)(2eg)da:eM sin0

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

12 OCR Output

HSO(§) and K(§)—1 < 0(§) as §—» 0.

H(€)~£ wd K(€)—>0 as E—·<><>,

The finiteness of the monopole mass leads to the boundary conditions

2 A 2 2 §2—E=2K H+EH(H —§d2H

2 2d2K

the equations of motion reduce to

(f = aer)2 = 0

<f> = j;H(§), A1= —¤»a;[1 — K(€)l

With the ansatz

4 2 4

mL = ipe -1;,, + -1>#<1> .D,,·1> - (<r> - <1» - a.. .. 1 .. .. i

We have the lagrangiannetic monopole with a structure";<I>, which was of finite energy and represents a nonsingular model of magsolution of Georgi-Glashow SO(3) gauge model with a triplet of Higgs fieldcoupling may well have structure. t’hooft—Polyakov found a classical staticjects. It may however be expected that monopoles in view of their strong

The magnetic monopoles which we have discussed so far are point ob

7. t’hc0ft-Polyakov Monopole

magnetic monopoles was pointed out by Rubakovzs and Callanzconsideration could lead to baryon number violating decays catalysed byeffect of pair production however may change this picture2‘*. That thesethat helicity is indeed conserved, but the charge may not be"·”. Thewhich we discuss in the next section, is taken into account then it is found

When the internal structure of the monopoles of nonabelian theories,

13 OCR Output

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

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

degeneracy, the manifold of the minima of U (Q) would be the coset spaceLet H be the subgroup of G that leaves QU invariant. Except "accidental”U (Q) would have many degenerate minima. Let Q = Q0 be one of these.ical considerations without solving the dynamical equations. The potentialgauge group G and with a Higgs potential U (Q) can be obtained by topolog

Magnetic monopole charges to be expected in a Yang-Mills theory withto the dyon case was carried out by Julia and Zeezgauge boson Mw = ae. The generalisation of t’hooft-Polyakov monopoleMonopole size is of the order of (1 /MW) where the mass of the massiveas there is no singularity attached with it due to its internal structure.

Note that for t’hooft-Polyakov monopole no Dirac string is neededThis special case is known as Bogomolny-Prasad-Sommerfield (BPS) monopole.

M 2 41ra|g|.

molny bound on the monopole masszand leads to f(0) = 1. The monopole mass in this case satisfies the Bogo

K(£) = {/Sinhé

H(§) = fcothf — 1

vanishing Higgs mass, we have the analytic solutionIt was realised by Prasad and Sommerfieldzl that in case A = 0, ie

leading to a calculation of the unknown function f.The coupled equations for H (§ ) and K (tf ) have to be integrated numerically

M : 2 fo./C).4 2and mass M,

y = -1/¢

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

14 OCR Output

93.

organised by the Physics department of the Calcutta University on Oct. 8,material was also presented as an invited talk at a Saha Centennial seminarhim to write this review for the special Saha issue of this journal. Thismagnetic monopoles. He also wishes to thank Prof. H. Banerjee for inviting

The author wishes to thank Prof. S.M. Roy for many discussions about

Acknowledgements

extensively discussed in astrophysics and cosmology32·34·°5theoretic methods33. The role of grand unified monopoles has been alsolutions by using Atiyah-Ward ansatz, ADHM Construction or using solitonBPS limit great progress has also been made in obtaining multimonopole sofor other gauge groups arising in grand unified and other modelsmm. In the

The fundamental magnetic monopole solutions have been worked outMagnetic monopoles carry this topological chargeildiscrete and its elements give the possible values of the topological charges.motopy group of G/H. 1r2(G/H) = n·1(H)/1r1(G). The group 1r1(H) iswhich can be endowed with a group structure 1r2(G/H ), the second ho

15 OCR Output

Phys. Rev. @, 1489.11. J. Schwinger, (1968), Phys. Rev. @, 1536. D. Zwanziger, (1968),

(1925), Trans. Ann. Math. Soc. Q, 106.

I0. L. Silberstein, (1907), Ann. Phys. Chem. 22, 579; G.Y. Rainich,

(E.U. Condon Volume), (Boulder, Univ. of Colorado Press).Science @, 757. A.O. Barut, (1971), in Topics in Modern PhysicsJ. Schwinger, (1968), Phys. Rev. @, 1536; J. Schwinger, (1969),

Q, 1968.

Wilson, (1949), Phys. Rev. QQ, 308, M.N. Saha, (1949), Phys. Rev.M.N. Saha, (1936), Indian Jour. Physics 10, 141. See also H.A.

Phil. Mag. 8, 331.and Magnetism, (Mass. Cambridge; 1st ed); J .J . Thomson, (1904),J .J . Thomson (1990), Elements gf the Mathematical Theory of Electricit

H. Poincaré, (1896), Compt. Rend. Acad. Sc. @, 530.

Dirac, (1948), Phys. Rev. B, 817 (1948).P.A.M. Dirac, (1931), Proc. Roy. Soc. (London) A133, 60; P.A.M.

edited by N. Craigie, (Singapore, World Scientific).in Theory and Detection of Magnetic Monopoles in Gauge Theories,G. Giacomelli, (1986), The experimental detection of magnetic monopoles,

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

Oersted Case, The World Scientist No. 38 60-68.P. Thuillier, (1991), From Philosophy to Electromagnetism: The

See also V. Singh, Science To-day (Jan. 1980) 19-23.A History of the theories of Aether and electricity, Harper Torchbooks.The hist0rica.l account in this section is based on E. Whittaker, (1960),

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16 OCR Output

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17 OCR Output

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