Electromagnetic angular momentum and quantum mechanicsD. SingletonDepartment of Physics, Virginia Commonwealth University, Richmond, Virginia 23284-2000

~Received 22 September 1997; accepted 27 December 1997!

A quick way of arriving at the Dirac quantization condition between electric and magnetic chargesis to require that the electromagnetic field angular momentum of a Thomson dipole~a magneticmonopole and an electric charge! equal some integer multiple of the fundamental unit of quantummechanical angular momentum,\/2. Applying this same type of argument to the electromagneticfield angular momentum carried by a magnetic dipole–electric charge system leads to an infinitenumber of different quantization conditions, and an apparent incompatibility between quantummechanics and the dipole–charge system. However, a more careful analysis shows that the particleplus field angular momentum of this system does satisfy the standard angular momentumcommutation relationships and is therefore a good quantum mechanical angular momentum. Thisemphasizes that caution must be taken when applying such semiclassical quantization arguments.Finally, a possible connection between this dipole–charge field angular momentum and the nucleonspin crisis is given. ©1998 American Association of Physics Teachers.

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I. INTRODUCTION

One interesting aspect of including magnetic chargeselectromagnetism is that the system of a point electric cha1e and a point magnetic charge,1g, carry an angular mo-mentum in their electric and magnetic fields.1 By integratingthis field angular momentum density over all space the tfield angular momentum is2,3

Lem5eg

cr , ~1!

wherer is a unit vector which points from1e to 1g. Thisresult was used by several authors4 to arrive at the Diracquantization condition5 between electric and magnetcharge. Requiring that the field angular momentum in Eq.~1!come in some integer multiple of\/2 immediately leads tothe Dirac quantization condition ofeg/c5n(\/2). Thissemiclassical argument can be found in a numbertexts,2,6,7 and it has the virtue that one does not needdiscuss Dirac strings. It only uses the fact that the elecand magnetic fields of the particles are of Coulombic forA crucial point to this argument is that the field angumomentum in Eq.~1! depends on the charges,e andg, butnot on the distance between them. In this paper we willamine the less frequently discussed, but more physicallyalistic, system of an electric charge moving in the field omagnetic dipole. This system also carries a field angularmentum. Applying the semiclassical quantization argumto the dipole–charge case leads to an apparent conflictquantum mechanics. This indicates that the success of

697 Am. J. Phys.66 ~8!, August 1998

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argument for the Thomson dipole is somewhat of an ‘‘acdent,’’ and that caution should be taken when applying targument. By examining the dipole–charge system mcarefully it is found that the particle plus field angular mmentum is a proper quantum mechanical angular momtum. That a dipole–charge system carries a field angularmentum is remarked upon only occasionally, asFeynman’s lectures.8 This field angular momentum is usuallnot taken into account, although many real systems sucthe hydrogen atom have particles with both charge and mnetic dipoles. By making some rough order of magnitucalculations we will argue that for atomic size systems tangular momentum probably does not play any kind of snificant role. For nuclear size systems it may be importaand could play a part in explaining where the nucleonssome of their internal angular momentum, especially ifquantum chromodynamics~QCD! version of this field angu-lar momentum is considered.

II. CLASSICAL FIELD ANGULAR MOMENTUM

First we will find the classical field angular momentum fthe dipole–charge system. To do this we should integratefield angular momentum density over all space

Lem51

4pc E r3~E3B!d3x, ~2!

where E5er 8/r 83 is the Coulombic electric field of thecharged particle,1e, located atR ~so r 85r2R!, and B5(3r (r–M )/r 5)2(M /r 3) is the magnetic field of the dipole

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M , located at the origin. Instead of doing the integral in E~2! to find Lem directly we will use the result of Eq.~1! forthe field angular momentum from a Thomson dipole. Taktwo magnetic charges—one with charge1g at the originand another with charge2g at P—placing an electric charge1e at R, and adding up the field angular momentum of eamagnetic charge–electric charge set gives

Lem5e

c S 2gR

uRu1

g~R2P!

uR2Pu D . ~3!

Expanding 1/uR2Pu5(1/R)1(R–P/R3)1¯ and taking thelimit P→0, g→`, andgP→2M gives

Lem5eM

cR2

eR~M–R!

cR3 ~4!

for the field angular momentum in the dipole–charge systOne subtle point in this is that the dipole formed by takithe above limit of two magnetic point charges is differenear the origin from a magnetic dipole made from a currePhysically this comes about since a magnetic dipole pduced by a current loop, which has its magnetic mompointing along thez axis, will have a magnetic field whichnear the center of the loop also points along thez axis. For amagnetic dipole made from two point magnetic charges,magnetic field near the center between the two monopwill point along the negativez axis. Mathematically this istaken into account by adding a delta function term tostandard magnetic dipole field. One adds18pMd(r )/3 forthe magnetic dipole produced by currents and24pMd(r )/3for the magnetic dipole produced by monopoles. The liming procedure for obtaining Eq.~4! only produces the(3r (r–M )/r 5)2(M /r 3) part of the magnetic dipole fieldand not the delta function part. Thus for each of the tcases one should add to Eq.~4! the contribution of this deltafunction part of the magnetic dipole field. In both cashowever, the angular momentum density coming fromdelta function part of the magnetic field will be of the form

r3~E3Md~r !!. ~5!

When this is integrated over all space it will vanish becaof the delta function and the factor ofr in the integrand.Thus Eq.~4! is the expression for the field angular mometum for either a magnetic dipole produced by current ormonopoles.

It can be shown explicitly that the field angular mometum is required in order that the total angular momentumthe dipole–charge system be conserved. To have the anmomentum of the system be conserved the net torque onsystem should be zero. For the charged particle movingthe field of the dipole the torque is

tcharge5e

c~R3~V3B!!

5e

c~V~R–B!2B~R–V!!

5e

c S 2V~M–R!

R3 23R~M–R!~R–V!

R5 1M ~R–V!

R3 D ,

~6!

whereV is the velocity of the charged particle, and the forof the magnetic dipole field, at the location,R, of the

698 Am. J. Phys., Vol. 66, No. 8, August 1998

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charged particle has been used. The torque associatedthe field angular momentum is@using Eq.~4!#

tem5dLem

dt5

e

c S 2M ~R–V!

R3 2V~M–R!

R3 2R~M–V!

R3

13R~M–R!~R–V!

R5 D . ~7!

Finally, the moving charge creates a magnetic field atlocation of the magnetic dipole which puts a torque onThe magnetic field created by the moving charge isBc

5e(V3(2R))/cR3. The minus sign occurs because tmagnetic field of the moving charge is evaluated at the lotion of the dipole. This creates a torque on the dipole of

tdipole5M3Bc52e

c

~M3~V3R!!

R3

5e

c S R~M–V!

R3 2V~M–R!

R3 D . ~8!

Summing up the three torques from Eqs.~6!–~8! gives

ttotal5tcharge1tem1tdipole50. ~9!

Therefore the total angular momentum for this syst(charge1field1dipole) is conserved as is expected forclosed system. This shows that the field angular momenmust be taken into account if the angular momentum of tsystem is to be conserved.

III. QUANTUM MECHANICAL ANGULARMOMENTUM

For the Thomson dipole, several authors noticed4 that byrequiring the electromagnetic field angular momentumcome in some integer multiple of the quantum mechanunit of angular momentum~i.e., \/2!, the Dirac quantizationcondition was obtained in a quick and simple manner. Tmain aim of this paper is to show that this semiclassiargument must be used with caution, since for the magndipole–electric charge case it leads to apparent problwith quantum mechanics. Taking the dipoleM to be alignedalong thez axis, we find from Eq.~4! that the magnitude ofthe field angular momentum along thez axis is

uLzu5eM~12cos2 u!

cR, ~10!

whereu is the standard angle from thez axis. If thez com-ponent of the field angular momentum is required to eqsome integer multiple of\/2 a problem is encountered. If thelectric chargee is placed at a given position~so thatR andu are fixed!, then one gets a certain condition betweene andM by setting Eq.~10! equal ton\/2. If the electric charge isat some neighboring position, with slightly differentR andu,one gets a different quantization condition, which will usally be incompatible with the first quantization condition. Fevery possible position of the electric charge one will gedifferent quantization condition, which will be incompatibwith the quantization conditions from other positions.~Somepositions will yield the same quantization condition—for eample,u5p/4 andu53p/4 for the sameR will yield thesame condition. But in general different positions will yiedifferent conditions!. Unlike the case of the Thomson dipol

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where the field angular momentum is independent ofdistance between the constituents, in the dipole–chargethe field angular momentum depends on the distancetween the constituents. If the parts of the dipole–chargetem could only take up certain discrete positions relativeone another, in terms ofR and u, then this semiclassicamethod would work, but such a positional quantizationquirement is untenable for the dipole–charge system.

Applying the semiclassical quantization method to tdipole–charge system seems to lead to the strange cosion that this system is incompatible with quantum mechacal angular momentum, despite the fact that there are a nber of real systems which possess both charges and magdipoles, and which are correctly treated using quantumchanics. We want to show that when this dipole–charge fiangular momentum is correctly considered, it does behava good quantum mechanical angular momentum. Wetake the commutation relationship

@Li ,L j #5 i\e i jkLk , ~11!

as the main property of a proper quantum mechanical anlar momentum. The angular momentum of the charged pticle plus the field angular momentum is given by

Li5e i lmRlDm1eMi

cR2

eRiRkMk

cR3 , ~12!

where the last two terms are the field angular momentumthe first term is the relative orbital angular momentum terbut with the ordinary momentum operator replaced bycovariant momentum,pm→Dm52 i\]m2eAm /c, with Ambeing the vector potential. By showing that the angular mmentum in Eq.~12! obeys the commutation relationshipEq. ~11! we will demonstrate that the angular momentumEq. ~12!—and also the total angular momentum which adthe spin of the dipole to Eq.~12!—is a proper quantum mechanical angular momentum. Inserting this into the commtator in Eq.~11! gives

@Li ,L j #5e i lme jpq@RlDm ,RpDq#

1e i lmRlFDm ,S eMj

cR2

eRjRhMh

cR3 D G2e jpqRpFDq ,S eMi

cR2

eRiRhMh

cR3 D G . ~13!

The first term in the first line in Eq.~13! can be written out as

@RlDm ,RpDq#5RlRp@Dm ,Dq#1Rl@Dm ,Rp#Dq

1Rp@Rl ,Dq#Dm

5 ie\

cemqkRlRpBk2 i\dmpRlDq

1 i\d lqRpDm , ~14!

where the relationships@Dm ,Rp#52 i\dmp and @Dm ,Dq#5 i (e\/c)emqkBk ~Ref. 9! have been used~Bk5kth compo-nent of the total magnetic field!. Putting thee i lme jpq factorback in the expression from Eq.~14!, and using the identity

e jpqemkq5d jmdpk2d jkdpm ~15!

leads to the following final form for this term:

699 Am. J. Phys., Vol. 66, No. 8, August 1998

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i\S e

ce i jkRkRpBp1RiD j2RjDi D

5 i\e i jk S eklmRlDm1e

cRkRpBpD , ~16!

where the antisymmetry property ofe i lm was used, Eq.~15!was used, and some dummy indices were renamed. Theond term in Eq.~13! can be written out as

e i lmRl S eMj

c FDm ,1

RG2eMh

c FDm ,RjRh

R3 G D5e i lmRl S ie\M jRm

cR3 2eMh

c S RjRhFDm ,1

R3G1Rj@Dm ,Rh#

1

R3 1@Dm ,Rj #Rh

R3D D5 i

e\

ce i lmRl S RjMm

R3 1RhMhdm j

R3 D , ~17!

where the formula@Dm , f (R)#52 i\]m( f (R)) has beenused several times. The third term in Eq.~13! can be ob-tained from Eq.~17! by letting i→ j . Defining the vectorsai5e i lmRlMm andbj5Rj /R3, one can write the sum of thesecond and third terms in Eq.~13! as

ie\

c~aibj2ajbi !22i

e\

ce i jkRk

MhRh

R3

5 ie\

ce i jkeklmalbm22i

e\

ce i jkRk

MhRh

R3

5 ie\

ce i jk S Mk

R2

3RkMlRl

R3 D , ~18!

where Eq. ~15! has been used several times and sodummy indices changed. Combining this with the first tefrom Eq. ~13! gives

@Li ,L j #5 i\e i jk S eklmRlDm1e

cRkRlBl D

1 ie\

ce i jk S Mk

R2

3RkMlRl

R3 D5 i\e i jk S eklmRlDm1

eMk

cR2

eRkMlRl

cR3 D5 i\e i jkLk , ~19!

where we have inserted the magnetic dipole field@i.e., Bl

5(3Rl(RkMk)/R5)2(Ml /R3)# in going from the first to

second line andLk is the angular momentum given in Eq~12!. The charged particle’s angular momentum plus the fiangular momentum thus obey the correct commutation rtionships to be considered a quantum mechanical angmomentum. The total angular momentum for this systewhich would include the spin of the central dipole, alobeys the angular momentum commutation relationships.adding the spin of the dipole to the particle plus field angumomentum one immediately finds, in the usual way, that ttotal system angular momentum satisfies the commutatoEq. ~11!. It should be noted that the electromagnetic fieangular momentum given in Eq.~4! is not by itself a proper

699D. Singleton

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quantum mechanical angular momentum, since it doesobey the commutator in Eq.~11!. It is only the combinationin Eq. ~12! which satisfies the conditions for being a quatum mechanical angular momentum.

IV. PHYSICAL SYSTEMS WITH FIELD ANGULARMOMENTUM

After having shown that the field angular momentum odipole–charge system does obey the usual commutarules when considered correctly, one might ask why this fiangular momentum is not considered when discussing phcal systems which have both charges and magnetic dipoAs a first example we will consider the hydrogen atom inground state. The proton and electron each have a chargea magnetic dipole. This system is more complicated thanelectric charge–magnetic dipole system considered inprevious section. There are now two contributions tofield angular momentum: first from the interaction of tproton’s charge with the intrinsic magnetic dipole of telectron, and second from the interaction of the electrocharge with the intrinsic magnetic dipole of the proton. Hewe will simply give a rough estimate of the size of the clasical field angular momentum of this system, and comparesize to\/2. This is somewhat contrary to the previous sectwhere it was shown that such semiclassical argumentsnot to be trusted. However, given the more complex natof the systems now considered, a rigorous treatment worequire a much more intensive analysis. Taking the etron’s dipole moment to be along thez axis, Eq.~4! gives themagnitude of the field angular momentum along thez axis asuL zuem5(eM/cR)(12cos2 u). The maximum for this magni-tude occurs whenu5p/2 with uL zuem5(eM/cR). Now M5(ege\/4mec) and we will takeR58.031029 cm, whichis the expectation value forR for the hydrogen atom grounstate. We take the gyromagnetic ratio for the electron asge

52. Inserting numbers into this~we use the units of Ref. 2wheree54.8310210 esu and centimeters and grams arebe used! gives a value ofuL zuem'3.531025(\/2), which isfive orders of magnitude less than the smallest possibleof angular momentum of\/2. The contribution to the fieldangular momentum coming from the interaction of the eltron’s charge with the proton’s magnetic moment willsmaller than this by three orders of magnitude since the mof the proton is approximately 2000 times that of the elect~the gyromagnetic ratio for the proton is larger than for telectron, but they are still the same order of magnitude!. Inaddition the field angular momentum coming from the intaction of the electron’s charge with the proton’s dipole mment subtracts from the field angular momentum comfrom the interaction of the proton’s charge with the eletron’s dipole moment. Given the small size of field angumomentum, from the above order of magnitude calculaticompared to\/2, one could argue that the electromagnefield angular momentum does not play a role in the hydroatom or in any atomic size system in general.

Since the field angular momentum has a 1/r dependenceone expects that as the system gets smaller this field angmomentum should become progressively more importOne system where the dipole field angular momentum mplay a role is in quark bound states such as the protonneutron. There are many factors that in our simple analwould tend to change the magnitude of the field angu

700 Am. J. Phys., Vol. 66, No. 8, August 1998

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momentum of the quark bound states as compared toatomic bound states. We will make of list of these factoand discuss each in turn.

(a) Nuclear size—Since the nuclear size scale is five oders of magnitude smaller than the atomic size scale~i.e.^R&nuclear'10213 cm compared to R&atomic'1028 cm! thiswould tend to increase the field angular momentum byfactor of 105, making it on the order of\/2.

(b) Quark magnetic dipole moments—The quark dipolemoments can be estimated by fitting experimentally msured baryon dipole moments with the up, down, and straquark dipole moments.10 In this way one obtainsM p

51.85MN for the up quark moment andMd520.972MN

for the down quark moment.MN5e\/2mprotonc is thenucleon magneton. Since this involves the mass of the pton, this tends to reduce the field angular momentum bfactor of 103.

(c) Number of constituents—Compared to the hydrogeatom system which had two contributions to the field angumomentum~proton’s charge–electron’s dipole moment, aelectron’s charge–proton’s dipole moment! the nucleonbound state will have six contributions to the field angumomentum~the charge of each quark with the dipole mment of each of the other two quarks!. Unlike the hydrogenatom system, where the contribution to the field angular mmentum from the electron’s charge interacting with the pton’s dipole moment made essentially no contribution, innucleon system all the contributions would be importasince the up and down quarks have should have magnmoments that have a similar magnitude. Whether thecreased number of constituents would increase or decrthe field angular momentum would depend on whetherquark spins~and therefore their magnetic moments! werearranged so that the field angular momenta of each quaquark subset added to or subtracted from the field angmomenta of the other quark–quark subsets.

(d) Quark charges—The quark charges are either twothirds or one-third of the electron’s charge. This would teto slightly decrease the field angular momentum as compato the hydrogen atom system.

These rough estimates indicate that, relative to atoscale systems, the decreased size of the nuclear systemcreases the field angular momentum by five orders of mnitude. However, the estimates of the quarks’ dipole mments in point ~b! would decrease the field angulamomentum by two or three orders of magnitude. Thus,though for nuclear scale systems the field angular momtum should be larger than for atomic scale systems, it issomewhat unlikely that it would play a significant role evin nuclear scale systems. Nevertheless, one might postuthat this field angular momentum may be a possible expnation for the nucleon spin crisis. In 1987, experimentsCERN11 ~see Ref. 12 for a review! indicated that a considerable fraction of the proton’s or neutron’s spin does ncome from the spins of the up and down valence quaThis is contrary to the naive picture where the nucleon’s sof 1/2 comes from two of the valence quarks having spins11/2 while the other quark has a spin of21/2. Possibleexplanations for this involve placing the extra internal anglar momentum with the nonvalence quark–antiquark paor with the gluon spin/orbital angular momentum. The altnative put forth in this paper~that the extra angular momentum may reside in the electromagnetic field angular momtum! is equivalent to saying that the extra internal angu

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momentum of the nucleon comes from photons rather tgluons. Actually the present argument could more realically be applied to the gluons as well. Just as the quacarry ordinary electric charges they also carry color chargIf one took an approximation where radiative correctiowere essentially ignored, then one would get a strong foversion of the Breit–Fermi potential.13,10This would give aninteraction between the color charge and color magneticpole in analogy with the electromagnetic interaction, whwould lead to a field angular momentum coming from tcolor charges and dipoles. In Ref. 13 this procedure was uto explain the mass splittings between hadrons in termthe color hyperfine interaction between quarks. Although tapproach lacks a firm theoretical foundation~i.e., why is onejustified in making a one-gluon exchange approximationget this QCD Breit–Fermi potential10?!, it nevertheless givesresults that are qualitatively and also quantitatively in agrment with the experimental hadron masses. Using the sapproximation in the present case would indicate that cocharges and color dipoles should give rise to a field angmomentum as in the electromagnetic case. If the cocharges interacting with the color magnetic dipole momedid result in a field angular momentum, then this internangular momentum would certainly play a role in the nspin of the nucleon, since the color charges are larger ttheir electromagnetic counterpart~i.e., the strong fine structure constant,as , is larger than the electromagnetic finstructure constant,aem!. Assuming that this field angular momentum is the source of the nucleon spin puzzle, one wadistinguish between whether the electromagnetic or strinteraction is the source of the extra angular momentum icompare the proton and neutron spin structure. If the elecmagnetic field is the source then one would expect a difence between the proton and neutron since they have dient charge distributions, while if the color field is the sourthen the proton and neutron should be roughly similar wrespect to this field angular momentum.

V. CONCLUSIONS

The semiclassical method4 of restricting the field angulamomentum of a magnetic charge—electric charge systemsome integer multiple of\/2 is often used in textbooks2,6,7

and articles to arrive at the Dirac quantization condition.applying the same argument to the more realistic systeman electric charge and a magnetic dipole we run intoproblem that there are an infinite number of incompatiquantization conditions for different positions that the dipoand charge can take up relative to one another. This coabout since the field angular momentum for the dipocharge depends on the distance,R, and on the relative ori-entation,u, between the charge and dipole. This indica

701 Am. J. Phys., Vol. 66, No. 8, August 1998

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that it may be somewhat of an accident that this semiclacal method works for the charge–monopole system, andany case shows that caution should be used when applthis method to other systems. It is shown that even thothe dipole–charge system has trouble with the semiclassquantization method, the charge plus field angular momtum of the system~as well as the total system angular mmentum! does obey the standard commutation rule,@Li ,L j #5 i\e i jkLk , so that it is a proper quantum mechanical anglar momentum. Finally some physical systems are discuswhere this dipole field angular momentum could play a roFrom rough estimates it is argued that for atomic size stems, such as the hydrogen atom, one would not expectfield angular momentum to be significant, while for nuclesystems, such as the proton, it could play a role. This mprovide a possible explanation for the nucleon spin puzespecially if one considers a QCD version of the field anlar momentum.

ACKNOWLEDGMENTS

I would like to thank Chris Ashman and Herbert Schmfor useful suggestions and comments.

1J. J. Thomson,Elements of the Mathematical Theory of Electricity anMagnetism~Cambridge U.P., Cambridge, 1904!, 3rd ed., Sec. 284.

2J. D. Jackson,Classical Electrodynamics~Wiley, New York, 1975!, 2nded., p. 251.

3E. F. Carter and H. A. Cohen, ‘‘The Classical Problem of Charge aPole,’’ Am. J. Phys.41 ~8!, 994–1006~1973!.

4M. N. Saha, ‘‘On the Origin of Mass in Neutrons and Protons,’’ Ind.Phys.10, 145–155~1936!; ‘‘Note on Dirac’s Theory of Magnetic Poles,’’Phys. Rev.75, 1968 ~1949!; H. A. Wilson, ‘‘Note on Dirac’s Theory ofMagnetic Poles,’’ibid. 75, 309 ~1949!.

5P. A. M. Dirac, ‘‘Quantised Singularities in the Electromagnetic FieldProc. R. Soc. London, Ser. A133, 60–72~1931!; ‘‘Theory of MagneticPoles,’’ Phys. Rev.74, 817–830~1948!.

6T. P. Cheng and L. F. Li,Gauge Theory of Elementary Particle Physic~Clarendon, Oxford, 1984!, p. 456.

7B. R. Holstein, Topics in Advanced Quantum Mechanics~Addison–Wesley, Reading, MA, 1992!, pp. 122.

8R. P. Feynman, R. B. Leighton, and M. Sands,The Feynman Lectures onPhysics~Addison–Wesley, Reading, MA, 1977!, Vol. II, pp. 27–28.

9H. J. Lipkin, W. I. Weisberger, and M. Peskin, ‘‘Magnetic Charge Quatization and Angular Momentum,’’ Ann. Phys.53, 203–214~1969!.

10J. F. Donoghue, E. Golowich, and B. R. Holstein,Dynamics of the Stan-dard Model~Cambridge U.P., Cambridge, 1992!, pp. 322, 365–370, and284.

11J. Ashmanet al., ‘‘A Measurement of the Spin Asymmetry and Determnation of the Structure FunctionG(1) in Deep Inelastic Muon-ProtonScattering,’’ Phys. Lett. B206, 364–370~1988!; ‘‘An Investigation of theSpin Structure of the Proton in Deep Inelastic Scattering of PolariMuons on Polarized Protons,’’ Nucl. Phys. B328, 1–35~1989!.

12Internal Spin Structure of the Nucleon, edited by C. Cavata and V. Hughe~World Scientific, Singapore, 1995!.

13A. DeRujula, H. Georgi, and S. L. Glashow, ‘‘Hadron Masses in a GauTheory,’’ Phys. Rev. D12, 147–162~1975!.

TEACHERS’ MISTAKES

Nothing is more bracing for students than to discover the fallibility of their exalted teachers.Students, God knows, are brimming with their own human weaknesses, and if their great mentorscan make mistakes, well then, anything might happen.

Alan Lightman,Dance for Two~Pantheon Books, New York, 1996!, p. 37.

701D. Singleton