CONTEMPORARY PHYSICS2018, VOL. 59, NO. 4, 331–355https://doi.org/10.1080/00107514.2018.1527974

Dirac quantisation condition: a comprehensive review

Ricardo Heras

Department of Physics and Astronomy, University College London, London, UK

ABSTRACTIn most introductory courses on electrodynamics, one is taught the electric charge is quantised butno theoretical explanation related to this law of nature is offered. Such an explanation is postponedto graduate courses on electrodynamics, quantummechanics and quantum field theory, where thefamous Dirac quantisation condition is introduced, which states that a singlemagnetic monopole inthe Universe would explain the electric charge quantisation. Even when this condition assumes theexistence of a not-yet-detected magnetic monopole, it provides the most accepted explanation forthe observed quantisation of the electric charge. However, the usual derivation of the Dirac quanti-sation condition involves the subtle concept of an ‘unobservable’ semi-infinite magnetised line, theso-called ‘Dirac string,’whichmaybedifficult tograsp in a first viewof the subject. Thepurposeof thisreview is to survey the concepts underlying the Dirac quantisation condition, in a way that may beaccessible to advanced undergraduate and graduate students. Some of the discussed concepts aregauge invariance, singular potentials, single-valuedness of the wave function, undetectability of theDirac string andquantisation of the electromagnetic angularmomentum. Five quantum-mechanicaland three semi-classical derivations of the Dirac quantisation condition are reviewed. In addition, asimple derivation of this condition involving heuristic and formal arguments is presented.

ARTICLE HISTORYReceived 1 July 2018Accepted 17 September 2018

KEYWORDSMagnetic monopoles; chargequantisation; gaugeinvariance

1. Introduction

In the early months of 1931, Dirac was seeking for anexplanation of the observed fact that the electric charge isalways quantised [1]. In his quest for explaining thismys-terious charge quantisation, he incidentally came acrosswith the idea of magnetic monopoles, which turned outto be of vital importance for his ingenious explanationpresented in his 1931 paper [2]. In this seminal paper,Dirac envisioned hypothetical nodal lines to be semi-infinite magnetised lines with vanishing wave functionand having the same end point, which is the singularityof the magnetic field where the monopole is located (seeFigure 1). A quantum-mechanical argument on thesenodal lines led him to his celebrated quantisation con-dition: qg = n�c/2. Here, q and g denote electric andmagnetic charges, � is the reduced Planck’s constant, c isthe speed of light, n represents an integer number, and weare adopting Gaussian units. Dirac wrote [2]: ‘Thus at theend point [of nodal lines] there will be a magnetic pole ofstrength [g = n�c/(2q)].’ This is the original statementby which magnetic monopoles entered into the field ofquantum mechanics. In 1948, Dirac [3] presented a rel-ativistic extension of his theory of magnetic monopoles,in which he drew one of his most famous conclusions:

CONTACT Ricardo Heras ricardo.heras.13@ucl.ac.uk; ricardoherasosorno@gmail.com

‘Thus themere existence of one pole of strength [g] wouldrequire all electric charges to be quantised in units of[�c/(2g)]’.

For the modern reader, the Dirac argument for thequantisation of the electric charge involving the elusivemagnetic monopole is indeed ingenious. The basis of thisargument is the interaction of an electric charge with thevector potential of a magnetic monopole attached to aninfinitely long and infinitesimally thin solenoid, the so-called ‘Dirac string ’, which is shown to be undetectableby assuming the single-valuedness of the wave functionof the electric charge, and as a consequence the Diracquantisation condition qg = n�c/2 is required. Accord-ing to this condition, the existence of just one monopoleanywhere in the Universe would explain why the electriccharge is quantised. Indeed, if we identify the elemen-tary magnetic charge with g0, then q = n�c/(2g0). Nowfor n=1, we have the elementary electric charge e =�c/(2g0), which combines with q = n�c/(2g0) to give thelaw expressing the quantisation of the electric charge:q=ne. At the present time, the Dirac quantisation con-dition provides the most accepted explanation for theelectric charge quantisation even when it relies on theexistence of still undetected magnetic monopoles. It is

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332 R. HERAS

Figure 1. Nodal lines as envisioned by Dirac.

pertinent to note that there are excellent books [4–7]and reviews [8–17] on magnetic monopoles, which nec-essarily touch on the subject of the Dirac quantisationcondition and the Dirac string. So far, however, a reviewpaper dealing with the Dirac condition rather than withmagnetic monopoles seems not to appear in the standardliterature. The present review attempts to fill this gap forthe benefit of the non-specialist.

Typically, the Dirac condition is discussed in gradu-ate texts on electrodynamics [18–20], quantum mechan-ics [21] and quantum field theory [22–25]. The topic israrely discussed in undergraduate textbooks [26]. Thepurpose of this review is to survey the ideas underly-ing the Dirac quantisation condition, in a way that maybe accessible to advanced undergraduate as well as grad-uate students. After commenting on the status of theDirac quantisation condition, i.e. to discuss its past andpresent impact on theoretical physics, we find conve-nient to review the derivation of the Dirac conditiongiven in Jackson’s book [18]. We next present a heuristicderivation of the this condition in which we attempt tofollow Feynman’s teaching philosophy that if we cannotprovide an explanation for a topic at the undergradu-ate level then it means we do not really understand thistopic [27].We then review four quantum-mechanical andthree semi-classical derivations of the Dirac quantisationcondition. Some of the relevant calculations involved inthese derivations are detailed in Appendices. We thinkworthwhile to gather together the basic ideas underlyingthese derivations in a review, which may be accessible toadvanced undergraduate and graduate students.

2. Status of the Dirac quantisation condition:past and present

To appreciate the relevance of the method followed byDirac to introduce his quantisation condition, let us

briefly outline the historical context in which Diracderived this condition. As is well known, Maxwell builthis equations on the assumption that no free magneticcharges exist, which is formally expressed by the equation∇ · B = 0. With the advent of quantum mechanics,magnetic charges were virtually excluded because thecoupling of quantum mechanics with electrodynamicsrequired the inclusion of the vector potential A definedthrough B = ∇ × A. But it was clear that this equationprecluded magnetic monopoles because of the well-known identity ∇ · (∇ × A) ≡ 0. Before 1931, magneticmonopoles were irreconcilable within an electrodynam-ics involving the potential A, and hence with quantummechanics [28]. Furthermore, for quantum physicists ofthe early twentieth century, magnetic monopoles weremere speculations lacking physical content and weretherefore not of interest at all in quantum theory priorto 1931. This was the state of affairs when Dirac sug-gested in his 1931 paper [2] to reconsider the idea ofmagneticmonopoles. Using an innovativemethod, Diracwas able to reconcile the equations ∇ · B �= 0 and B =∇ × A, and therefore he was successful in showing thatthe interaction of an electron with a magnetic monopolewas an idea fully consistent in both classical and quantumphysics.

According to Dirac, the introduction of monopolesin quantum mechanics required magnetic charges to benecessarily quantised in terms of the electric charge andthat quantisation of the latter should be in terms of theformer. In his own words [2]: ‘Our theory thus allowsisolated magnetic poles [g], but the strength of suchpoles must be quantised, the quantum [g0] being con-nected with the electronic charge e by [g0 = �c/(2e)] . . .The theory also requires a quantisation of electric charge. . . ’. In his 1931 paper [2], Dirac seems to favour themonopole concept when he pointed out: ‘ . . . one wouldbe surprised if Nature hadmade no use of it ’. As Polchin-ski has noted [29]: ‘From the highly precise electriccharge quantisation that is seen in nature, it is then tempt-ing to infer that magnetic monopoles exist, and indeedDirac did so ’. However, Dirac was very aware that iso-lated magnetic monopoles were still undetected and heproposed a physical explanation for this fact.When inter-preting his result g0 = (137/2)e, he pointed out: ‘Thismeans that the attractive force between twoone-quantumpoles of opposite sign is 46921/4 times that betweenelectron and proton. This very large force may perhapsaccount for why poles of opposite sign have never yetbeen separated ’.

Let us emphasise that the true motivation of Dirac inhis 1931 paper was twofold; on one hand, he wanted toexplain the electric charge quantisation and on the other,to find the reason why the elementary electric charge

CONTEMPORARY PHYSICS 333

had its reported experimental value. Such motivationswere explicitly clarified by Dirac in 1978 [30]: ‘I was notsearching for anything like monopoles at the time. WhatI was concerned with was the fact that electric charge isalways observed in integral multiples of the electroniccharge e, and I wanted some explanation for it. Theremust be some fundamental reason in nature why thatshould be so, and also there must be some reason whythe charge e should have just the value that it does have. Ithas the value that makes [�c/e2] approximately 137. AndI was looking for some explanation of this 137’.

In his 1948 paper [3], Dirac stressed the idea that eachmagnetic monopole is attached at the end of an ‘unob-servable’ semi-infinite string (a refinement of the nodallines introduced in his 1931 paper [2]). In retrospec-tive, one can imagine that the idea of an unobservablestring might have seemed strange at that time, and ifadditionally the theory was based on the existence ofundetected magnetic monopoles, then it is not difficultto understand why this theory was received sceptically bysome of Dirac’s contemporaries. In a first view, Pauli dis-liked the idea of magnetic monopoles and sarcasticallyreferred to Dirac as ‘Monopoleon ’. But some years later,he reconsidered his opinion by saying that [31]: ‘This title[Monopoleon] shall indicate that I have a friendlier viewto his theory of “monopoles” than earlier: There is somemathematical beauty in this theory ’. On the other hand,Bohr, unlike Dirac, thought that one would be surprisedif Nature had made use of magnetic monopoles [32].

After Dirac’s 1931 seminal paper, Saha [33] presentedin 1936 a semi-classical derivation of the Dirac quanti-sation condition based on the quantisation of the elec-tromagnetic angular momentum associated to the staticconfiguration formed by an electric charge and a mag-netic charge separated by a finite distance, the so-calledThomson dipole ([34], see also [35]). This same deriva-tion was independently presented in 1949 by Wilson[36,37]. In 1944, Fierz [38] derived theDirac condition byquantising the electromagnetic angular momentum aris-ing from the classical interaction of a moving charge inthe field of a stationary magnetic monopole. Schwinger[39] in 1969 used a similar approach to derive a duality-invariant form of the Dirac condition by assuming theexistence of particles possessing both electric and mag-netic charges, the so-called dyons.

On the other hand, the Aharanov–Bohm effect [40]suggested in 1959 has been recurrently used to show theundetectability of the Dirac string [1,8–12,14–17,22,23,41,42], giving a reversible argument. If Dirac’s conditionholds then the string is undetectable, and vice versa, if thestring is undetectable then Dirac’s condition holds. Thepath-integral approach to quantummechanics, suggestedby Dirac in 1933 [43], formally started by Feynman

in his 1942 Ph.D. thesis [44] and completed by himin 1948 [45], has also been used to obtain the Diraccondition [22].

Several authors have criticised the Dirac argumentbecause of its unpleasant feature that it necessarilyinvolves singular gauge transformations [9]. A formalapproach presented by Wu and Yang [46] in 1975 avoidssuch annoying feature by considering non-singularpotentials, using the single-valuedness of the wave func-tion and then deriving the Dirac condition without usingthe Dirac string [4,8,9,11–13,16,24]. Other derivations ofthe Dirac condition have been presented over the years,including one by Goldhaber [47], Wilzcek [48,49] andJackiw [50–52].

Remarkably, in 1974 t’Hooft [53] and Polyakov [54]independently discovered monopole solutions for spon-taneously broken non-Abelian gauge theories. This orig-inated another way to understand why electric charge isquantised in grand unified theories, where monopolesare necessarily present. If the electromagneticU(1) gaugegroup is embedded into a non-Abelian gauge group,then charge quantisation is automatic, for considera-tions of group theory [4,11]. It is not surprising thenthat charge quantisation is now considered as an argu-ment in support of grand unified theories [4,29,55]. Inthe context of unified theories, Polchinski goes even fur-ther arguing that [29] ‘In any theoretical framework thatrequires charge to be quantised, there will exist magneticmonopoles ’. On the other hand, it has been noted that theinteger n in Dirac’s condition can be identified as a wind-ing number, which gives a topological interpretation ofthis condition [4,11,56]. Finally, it is pertinent to men-tion the recent claim that the Dirac condition also holdsin the Proca electrodynamics with non-zero photonmass[57], reflecting the general character of this quantisationcondition.

The preceding comments allowus to put in context thereview presented here on the basic ideas underpinningthe Dirac quantisation condition, such as gauge invari-ance, singular vector potentials, single-valuedness of thewave function, undetectibility of the Dirac string and thequantisation of the electromagnetic angular momentum.

The present review is organised as follows. InSection 3, we closely review Jackson’s treatment of theDirac quantisation condition. In Sections 3–6, we presenta new derivation of the Dirac condition based on heuris-tic and formal arguments, which does not consider theDirac string. The specific gauge function required in thisheuristic derivation is discussed. In Section 7, we exam-ine in detail the Dirac strings by explicitly identifyingtheir singular sources. In Section 8, we study the classicalinteraction of the electric chargewith theDirac string andconclude that this string has amathematical rather than a

334 R. HERAS

physicalmeaning. In Section 9,we examine the quantum-mechanical interaction of the electric charge with theDirac string and show that if the string is undetectablethen the Dirac quantisation condition holds. We reviewin Section 10, the Aharanov–Bohm effect and show howit can be used to derive theDirac condition. In Section 11,we outline Feynman’s path integral approach to quan-tum mechanics and show how it can be used to obtainthe Dirac condition. In Section 12, we briefly discuss theWu–Yang approach that allows us to derive the Diraccondition without the recourse of the Dirac string. InSection 13, we review three known semi-classical deriva-tions of the Dirac condition. The first one makes useof the Thomson dipole. The second one considers theinteraction between a moving charge and the field of astationary monopole, and the third one considers theinteraction between a moving dyon and the field of a sta-tionary dyon. In Section 14, we make some final remarkson the Dirac quantisation condition. In Section 15, wemake a final comment on the concept of nodal lines andin Section 16, we present our conclusions. In Appen-dices 1–5, we perform some calculations involved in thederivations of the Dirac condition.

3. Jackson’s treatment of the Dirac quantisationcondition

The first quantum-mechanical derivation of the Dirac con-dition we will review is that given in Jackson’s book[18]. The magnetic monopole is imagined either as oneparticle to be at the end of a line of dipoles or at theend of a tightly wound solenoid that stretches off toinfinity, as shown in Figure 2. Any of these equivalentconfigurations can be described by the vector poten-tial of a magnetic dipole A(x) = [m× (x − x′)]/|x −x′|3, where x is the field point, x′ is the source pointand m is the magnetic dipole moment. The line ofdipoles is a string formed by infinitesimal magneticdipole moments dm located at x′ whose vector potentialis dA(x) = −dm×∇(1/|x − x′|), where we have used∇(1/|x − x′|) = −(x − x′)/|x − x′|3. With the identifi-cation dm = gdl′, with g being the magnetic charge anddl′ a line element, the total vector potential for a string orsolenoid lying on the curve L reads

AL = −g∫Ldl′ ×∇

(1

|x− x′|). (1)

Using the result ∇ × (dl′/|x− x′|) = −dl′ ×∇(1/|x−x′|), we can write Equation (1) as

AL = g∇ ×∫L

dl′

|x− x′| . (2)

Figure 2. Representation of a magnetic monopole g as the endof a line of dipoles or as the end of a tightly wound solenoid thatstretches off to infinity.

Notice that this potential is already in the Coulombgauge: ∇ · AL = 0 because ∇ · [∇ × ( )] ≡ 0. InAppendix 1, we show that the curl of this potential gives

∇ × AL = gR2

R+ 4πg∫Lδ(x− x′) dl′, (3)

where δ(x− x′) is the Dirac delta function, R = |x−x′| and R = (x − x′)/R. To have a clearer meaning ofEquation (3), it is convenient to write this equation as

Bmon = ∇ × AL − Bstring, (4)

where

Bmon = gR2

R (5)

is the field of the magnetic monopole g located at thepoint x′ and

Bstring = 4πg∫Lδ(x − x′) dl′ (6)

is a singular magnetic field contribution along the curveL.

By taking the divergence to Bmon, it follows

∇ · Bmon = ∇ ·(

gR2

R)= 4πgδ(x − x′), (7)

where we have used ∇ · (R/R2) = 4πδ(x− x′). Simi-larly, if we take the divergence to Bstring, we obtain theresult

∇ · Bstring = ∇ ·(4πg

∫Lδ(x− x′) dl′

)

CONTEMPORARY PHYSICS 335

= −4πg∫L∇′δ(x− x′) · dl′

= −4πg δ(x − x′), (8)

where we have used ∇δ(x− x′) = −∇′δ(x− x′). WhenEquations (7) and (8) are used in the divergence ofEquation (3), we verify the expected result ∇ · (∇ ×AL) = 0. Expressed in an equivalent way, the fluxes of thefields Bmon and Bstring mutually cancel:∮

SBmon · da =

∫V

∇ · Bmon d3x = 4πg, (9)

∮SBstring · da =

∫V

∇ · Bstring d3x = −4πg, (10)

where da and d3x denote the differential elements of sur-face and volume, and the Gauss theorem has been used.As a particular application, let us consider the case inwhich the string lays along the negative z-axis and themagnetic monopole is at the origin. In this case dl′ =dz′z, and the corresponding potential is

AL = g∇ × z∫ 0

−∞dz′

|x− z′z| . (11)

In Appendix 1, we show that the curl of Equation (11)yields

∇ × AL = gr2r+ 4πgδ(x)δ(y)�(−z)z, (12)

where now r = |x|, r = x/r, and�(z) is the step functionwhich is undefined at z=0 but it is defined as �(z) = 0if z<0 and �(z) = 1 if z>0. The highly singular char-acter of the magnetic field of the string is clearly noted inthe second term on the right of Equation (12). It is inter-esting to note that in his original paper [2], Dirac wrotethe following solution for the vector potential in spheri-cal coordinatesAL = (g/r) tan(θ/2)φ and noted that thispotential gives the radial field gr/r2. He pointed out: ‘Thissolution is valid at all points except along the line θ = π ,where [AL] become infinite ’. The solution considered byDirac is equivalent to

AL = g1− cos θr sin θ

φ. (13)

This expression can be obtained by performing the inte-gration specified in Equation (11), which requires thecondition sin θ �= 0. This is shown in Appendix 2.

Clearly, the curl of Equation (13) subjected to sin θ �=0 gives only the field of the magnetic monopole ∇ ×AL = gr/r2 = Bmon. This is so because the singularityoriginated by sin θ = 0 is avoided in the differentiation

process. As far as the computation of the total magneticfield of the configuration formed by a string laying alongthe negative z-axis and amagneticmonopole at the originis concerned, it is simpler to take the curl to the implicitformof the potential defined byEquation (11) rather thantaking the curl of a regularised form of the potential inEquation (13) [see Appendix 4].

If an electric charge is interacting with the poten-tial given in Equation (2), then it is ultimately inter-acting with a magnetic monopole and a magnetisedstring. Dirac argued that the interaction must only bewith the magnetic monopole and therefore the charge qshould never ‘see’ the singular field Bstring defined byEquation (6). For this reason, he postulated that the wavefunction must vanish along the string. But this require-ment is certainly criticisable because it would mean thatthe string does not exist at all. This postulate is known asthe ‘Dirac veto ’, which in an alternative form states thatany interaction of the electric chargewith the string is for-bidden. In Dirac’s own words [30]: ‘You must have themonopoles and the electric charges occupying distinctregions of space. The strings, which come out from themonopoles, can be drawn anywhere subject to the condi-tion that theymust not pass through a region where thereis electric charge present ’.

The next step of the argument is to show thatEquation (4) does not depend on the location of thestring. To show this statement, consider two differentstrings L′ and L with their respective vector potentialsAL′ and AL. Evidently, the equivalence of these poten-tials will imply the equivalence of their respective stringsindicating that the location of the string is irrelevant. Thedifference of the potentials AL′ and AL can be obtainedfrom Equation (2) with the integration taken along theclosed curve C = L′ − L around the area S as shown inFigure 3. The result can be written as [18]

AL′ − AL = g∇ ×∮C

dl′

|x− x′| = ∇(g�C), (14)

where �C is the solid angle function subtended by thecurve C. The integral specified in Equation (14) is donein Appendix 3. The fact thatAL′ andAL are connected bythe gradient of a function reminds us of the gauge trans-formation A′ = A+∇�, where � is a gauge function.Without any loss of generality, we can then write A′ ≡AL′ ,A ≡ AL and� ≡ g�C. Notice thatAL′ andAL are inthe Coulomb gauge: ∇ · AL′ = 0 and ∇ · AL = 0. How-ever, this does not prevent these potentials from beingconnected by a further gauge transformation wheneverthe gauge function � is restricted to satisfy ∇2� = 0.We can verify that this is indeed the case by taking thedivergence to Equation (14) and obtaining ∇2� = 0,

336 R. HERAS

Figure 3. Representation of a magnetic monopole g as the endof a line of dipoles or as the end of a tightly wound solenoid thatstretches off to infinity. The solid angle �C is subtended by thecurve C = L− L′, which embeds the area S. The potentialsAL andAL′ correspond to the strings L and L′.

indicating that the potentials AL′ and AL are connectedby a restricted gauge transformation.

The remarkable point here is that different stringpositions correspond to different choices of gauge, or achange in string from L to L′ is equivalent to a gaugetransformation from AL to AL′ with the gauge function� = g�C. With the identification � = g�C, the asso-ciated phase transformation of the wave function ′ =eiq�/(�c) takes the form ′ = eiqg�C/(�c) . Now a cru-cial point of the argument. The solid angle �C under-goes a discontinuous variation of 4π as the observationpoint (or equivalently the charge q) crosses the sur-face S. This makes the gauge function � = g�C multi-valued which implies that eiqg�C is also multi-valued, i.e.eiqg�C �= eiqg(�C+4π). Thus the transformed wave func-tion of the charge qwill bemulti-valued when q crosses S,unless we impose the condition ei4πqg/(�c) = 1. But thiscondition and ei2πn = 1 with n being an integer, imply4πqg/(�c) = 2πn, and hence, the Dirac quantisationcondition qg = n�c/2 is obtained. Accordingly, the fieldof the monopole in Equation (4) does not depend on thelocation of the string. The price we must pay is the impo-sition of the Dirac condition. The lesson to be learnedhere is that gauge invariance and single-valuedness of thewave function are the basic pieces to ensemble the Diracquantisation condition.

The above derivation of the Dirac condition putsemphasis on the idea that the location of the string isirrelevant. But the argument might equally put empha-sis on the idea that the string is unobservable. In fact,consider the value �1 corresponding to one side of thesurface S and the value �2 corresponding to the other

side. They are related by �1 = �2 + 4π . It follows thateiqg�1/(�c) = eiqg(�2+4π)/(�c). This means that the wavefunction of the charge q differs by the quantity ei4πqg/(�c),and this would make the Dirac string observable as thecharge crosses the surface, unless we impose the con-dition ei4πqg/(�c) = 1, which is satisfied if qg = n�c/2holds, i.e. the price we must pay for the unobservabilityof the string is the imposition of the Dirac condition.

The standard derivation of theDirac quantisation con-dition explained in this section is appropriate to be pre-sented to graduate students. In Sections 4–9, we will sug-gest a presentation of the Dirac condition that encapsulesthe main ideas underlying this condition, which may besuitable for advanced undergraduate students.

4. How to construct a suitable quantisationcondition

The origin of the letter n appearing in the Diracquantisation condition qg = n�c/2 can be traced tothe trigonometric identity cos (2πn) = 1, where n =0± 1, ±2, ±3 . . . This trigonometric identity can beexpressed as

ei2πn = 1, (15)

which follows from Euler’s formula eiα = cosα + i sinα

with α = 2πn. Consider now spherical coordinates(r, θ ,φ) with their corresponding unit vectors (r, θ , φ).For fixed r and θ , the azimuthal angles φ and φ + 2π rep-resent the same point. This property allows us to define asingle-valued function of the azimuthal angle F(φ) as onethat satisfies F(φ) = F(φ + 2π). We note that the partic-ular function F(φ) = φ is not a single-valued functionbecause F(φ) = φ and F(φ + 2π) = φ + 2π take differ-ent values: F(φ) �= F(φ + 2π). We then say that F = φ isa multi-valued function.

The complex function F(φ) = ei2kφ with k being anarbitrary constant is not generally a single-valued func-tion because F(φ) = ei2kφ and F(φ + 2π) = ei2k(φ+2π)

can take different values: F(φ) �= F(φ + 2π). This is sobecause in general ei4πk �= 1 for arbitrary k. In this case,however, we can impose a condition on the arbitraryconstant k so that F = ei2kφ becomes a single-valuedfunction. By considering Equation (15), we can see thatei4πk = 1 holds when k is dimensionless and satisfies the‘ quantisation’ condition:

k = n2, n = 0,±1,±2,±3, . . . . (16)

Under this condition, F = ei2kφ becomes a single-valued function: F(φ) = F(φ + 2π). In short: the single-valuedness of F = ei2kφ requires the quantisation con-dition specified in Equation (16). Notice that a specific

CONTEMPORARY PHYSICS 337

value of k may be obtained in principle by consideringthe basic equations of a specific physical theory. We willsee that electrodynamics with magnetic monopoles andquantum mechanics conspire to yield the specific valueof k that leads to the Dirac quantisation condition.

5. Gauge invariance and the Dirac quantisationcondition

We will now to present a heuristic quantum-mechanicalderivation of the Dirac condition. The Schrödingerequation for a non-relativistic particle of mass m andelectric charge q coupled to a time-independent vectorpotential A(x) is given by

i�∂

∂t= 1

2m

(−i�∇ − q

cA)2

. (17)

This equation is invariant under the simultaneous appli-cation of the gauge transformation of the potential

A′ = A+∇�, (18)

and the local phase transformation of the wave function

′ = eiq�/(�c) , (19)

where�(x) is a time-independent gauge function. Equa-tions (17)–(19) are well known in textbooks.1

At first glance, Equations (17)–(19) do not seem to berelated to some quantisation condition. But a comparisonbetween the previously discussed function ei2kφ with thephase factor eiq�/(�c) appearing in Equation (19),

ei2kφ ←→ eiq�/(�c), (20)

suggests the possibility of constructing a specificquantisation condition connected with Equations (17)–(19). Consider first that k is an arbitrary constant. There-fore ei2kφ is not generally a single-valued function. Werecall that the gauge function � in the phase eiq�/(�c) ofthe transformation in Equation (19) is an arbitrary func-tion whichmay be single-valued ormulti-valued. In viewof the arbitrariness of k and �, we can make equal bothfunctions: eiq�/(�c) = ei2kφ , which implies

�q = 2k�cφ. (21)

This is the key equation to find a quantisation conditionthat leads to the electric charge quantisation. The genesisof this remarkable equation is the gauge invariance of theinteraction between the charge q and the potential A. Bydirect substitution, we can show that a particular solution

of Equation (21) is given by the relations

k = qg�c

, (22)

and

� = 2gφ, (23)

where the constant g is introduced here to make theconstant k dimensionless. The constant g has the dimen-sion of electric charge and its physical meaning isunknown at this stage. Notice that � in Equation (23)is a multi-valued gauge function. We require now thatthe phase eiq�/(�c) be single-valued. From eiq�/(�c) =ei2kφ , it follows that ei2kφ must be single-valued and thenk must satisfy the quantisation condition displayed inEquation (16). In other words, by demanding the single-valuedness of eiq�/(�c), Equations (16) and (22) yield thequantisation condition

qg = n2�c. (24)

If now the constant g is assumed to be the magneticcharge, then Equation (24) is the Dirac quantisation con-dition.

Notice that according to the heuristic approach fol-lowed here, the derivation of Equation (24) relies onthe existence of the gauge function � = 2gφ. In thefollowing section, we will discuss the feasibility of thisspecific gauge function and argue the identification of gwith the magnetic charge. For now we observe that theheuristic approach uses the same two fundamental piecesdiscussed in Section 3, namely, the single-valuedness ofthe wave function and gauge invariance. However, theheuristic approach makes use of these two pieces in asimpler way.

6. The gauge function� = 2gφ

It is convenient to assume first the existence of the gaugefunction � = 2gφ with the purpose of elucidating itsassociated gauge potentials. The gradient of � = 2gφ inspherical coordinates gives

∇� = 2gr sin θ

φ. (25)

Notice that this gradient is singular at r=0. This is areal singular point which is not problematic and we agreeit is allowed. However, this gradient is also singular atthose values of the polar coordinate θ satisfying sin θ =0, which represent lines of singularities involving non-trivial consequences, whichwill be discussed in Section 7.

338 R. HERAS

Presumably, there exist two vector potentials such that

A′ − A = 2gr sin θ

φ. (26)

Both potentials A′ and A must originate the samemagnetic field B, i.e. ∇ × A′ = ∇ × A = B. FromEquation (26), we can see that A′ and A may be of thegeneric form

A′ = g1− f (θ)

r sin θφ, A = −g 1+ f (θ)

r sin θφ, (27)

where f (θ) is an unspecified function such that it doesnot change the validity of Equation (26). Notice that A′and A have singularities originated by sin θ = 0. Thesewill not be considered for now.We observe thatA′ andAin Equation (27) are of the form A′ = [0, 0,A′φ(r, θ)] =A′φ(r, θ)φ andA = [0, 0,Aφ(r, θ)] = Aφ(r, θ)φ. The curlof a generic vector of the form F = F[0, 0, Fφ(r, θ)] inspherical coordinates reads

∇ × F = 1r sin θ

∂

∂θ

(sin θFφ

)r− 1

r∂

∂r(rFφ

)θ . (28)

When this definition is applied toA′ andA and sin θ �= 0is assumed, we obtain

∇ × A′ = ∇ × A = − gr2 sin θ

∂f∂θ

r, (29)

and therefore both potentials yield the same field

B = − gr2 sin θ

∂f∂θ

r. (30)

In the particular case f (θ) = cos θ , this field becomes

B = gr2r. (31)

The nature of the constant g is then revealed in thisparticular case. Equation (31) is the magnetic field pro-duced by a magnetic charge g located at the origin.In other words, the constant g introduced by hand inEquations (22) and (23) is naturally identified with themagnetic monopole!

The potentials A′ and A in Equation (27) are inthe Coulomb gauge. In fact, using the definition ofthe divergence of the generic vector F = F[0, 0, Fφ(r, θ)]in spherical coordinates ∇ · F = [1/(r sin θ)]∂Fφ/∂φ, itfollows that ∇ · A′ = 0 and ∇ · A = 0. Here, there isa point that requires to be clarified. At first glance,there seems to be some inconsistency when connect-ing A′ and A via a gauge transformation because bothpotentials are already in a specific gauge, namely, theCoulomb gauge. However, there is no inconsistence as

explained in Section 3, because even for potentials sat-isfying the Coulomb gauge there is arbitrariness. Evi-dently, the restricted gauge transformation A→ A′ =A+∇�, where∇2� = 0, preserves theCoulombgauge.The definition of the Laplacian of the generic scalarfunction f = f (φ) in spherical coordinates reads ∇2f =[1/(r sin θ)2]∂2f /∂φ2. Using this definition with f =� = 2gφ, it follows that∇2� = 0, indicating thatA′ andA are connected by a restricted gauge transformation.

Let us recapitulate. By assuming the existence of thegauge function� = 2gφ, we have inferred the potentials

A′ = g1− cos θr sin θ

φ, A = −g 1+ cos θr sin θ

φ. (32)

[these are A′ and A in Equation (27) with f (θ) = cos θ],which originate the same field given in Equation (31)whenever the condition sin θ �= 0 is assumed. This fieldis the Coloumbian field due to a magnetic monopole g.With the identification of g as the magnetic monopole,we can say that Equation (24) is the Dirac quantisa-tion condition. Evidently, we can reverse the argumentby introducing first the potentials A′ and A by meansof Equation (32) considering sin θ �= 0 and then provingthey yield the same magnetic field in Equation (31). Theexistence of these potentials guarantees the existence ofthe gauge function � = 2gφ.

Once the existence of the gauge function � = 2gφhas been justified with g being the magnetic monopole,the heuristic derivation of the Dirac quantisation con-dition has been completed. However, we should notethat this heuristic procedure involves an aspect thatcould be interpreted as an inconsistency. According tothe traditional interpretation, the existence of magneticmonopoles implies ∇ · B �= 0 and therefore we cannotwrite B = ∇ × A, at least not globally. This is so because∇ · (∇ × A) = 0. The origin of this apparent inconsis-tency deals with the singularity originated by the valuesin θ = 0 and its explanation will take us to one ofthe most interesting concepts in theoretical physics, theDirac string, which will be discussed in the followingsection.

7. Dirac strings

As previously pointed out, both potentials in Equation(32) yield the same magnetic field given in Equation (31)whenever sin θ �= 0 is assumed. The question naturallyarises: What does sin θ = 0 mean? The answer is simple:θ = 0 and θ = π . The first value represents the positivesemi-axis z, i.e. z>0,whereas the second value representsthe negative semi-axis z, i.e. z<0. Therefore, the condi-tion sin θ �= 0 means that the semi-axes z>0 and z<0

CONTEMPORARY PHYSICS 339

have been excluded in the heuristic treatment. Accord-ingly, when we took the curl to A′ and A, we obtainedthe magnetic field B = gr/r2 in all space except at r=0(which we agree it is allowed) and except along the neg-ative semi-axis in the case of A′, and also except alongthe positive semi-axis in the case of A. Expectably, if weadditionally consider the field contributions associatedto the Dirac strings located in the positive and negativesemi-axes, then we can reasonably assume the followingequations:

∇ × A′ = gr2r+ B′(along z < 0), (33)

∇ × A = gr2r+ B(along z > 0). (34)

Here B′(z < 0) and B(z > 0) represent magnetostaticfields produced by Dirac strings. The formal determina-tion of these fields is not an easy task because they arehighly singular objects. But, fortunately, heuristic consid-erations allow us to elucidate the explicit form of thesefields.We note that the semi-axis z<0 can be representedby the singular function−δ(x)δ(y)�(−z)z and the semi-axis z>0 by the singular function δ(x)δ(y)�(z)z. There-fore, the fields B′(z < 0) and B(z > 0) may be appropri-ately modelled by the singular functions

B′(z < 0) = −Kδ(x)δ(y)�(−z)z, (35)

B(z > 0) = Kδ(x)δ(y)�(z)z, (36)

where K is a constant to be determined. Using Equa-tions (33)–(36), we obtain

∇ × A′ = gr2r− Kδ(x)δ(y)�(−z)z, (37)

∇ × A = gr2r+ Kδ(x)δ(y)�(z)z. (38)

The divergence of Equation (37) gives

0 = 4πgδ(x)+ Kδ(x), (39)

where ∇ · (r/r2) = 4πδ(x) with δ(x) = δ(x)δ(y)δ(z)and ∂�(−z)/∂z = −δ(z) have been used. A similar cal-culation on Equation (38) gives Equation (39) again.From Equation (39), it follows that K = −4πg and thuswe get the final expressions

∇ × A′ = gr2r+ 4πgδ(x)δ(y)�(−z)z, (40)

∇ × A = gr2r− 4πgδ(x)δ(y)�(z)z. (41)

We should emphasise that simple heuristic argumentshave been used to infer Equations (40) and (41). We also

note that Equation (40) is the same as Equation (12),which was in turn derived by the more complicatedapproach outlined in Section 3. The advantage of theheuristic argument is that it has nothing to do with theidea of modelling a magnetic monopole either as theend of an infinite line of infinitesimal magnetic dipolesor as the end of a tightly wound solenoid that stretchesoff to infinity. Equation (40) is also formally derived inAppendix 1 by means of an integration process. Fur-thermore, Equation (40) can alternatively be obtained bydifferentiation, which is done in Appendix 4, where anappropriate regularisation of the potential A′ is required.

Expressed differently, the potentials A′ and A appear-ing in Equations (40) and (41) produce respectively thefields B′ms = ∇ × A′ and Bms = ∇ × A, and so we canwrite

B′ms = Bmon + B′string, (42)

Bms = Bmon + Bstring, (43)

where the respective magnetic fields are defined as

Bmon = gr2r, (44)

B′string = 4πgδ(x)δ(y)�(−z)z, (45)

Bstring = −4πgδ(x)δ(y)�(z)z. (46)

Figures 4 and 5 show a pictorial representation of thefields appearing in Equations (42) and (43). It is con-ceptually important to identify the sources of the fieldsdescribed by Equations (42) and (43). The magnetic fieldBmon in Equation (44) satisfies

∇ · Bmon = 4πgδ(x), (47)

∇ × Bmon = 0. (48)

The magnetic field B′string in Equation (45) satisfies

∇ · B′string = −4πgδ(x), (49)

∇ × B′string = 4πg�(−z) [δ(x)δ′(y)x − δ′(x)δ(y)y],

(50)where δ′(x) = dδ(x)/dx and δ′(y) = dδ(y)/dy are deltafunction derivatives. The field Bstring in Equation (46)is shown to satisfy

∇ · Bstring = −4πgδ(x), (51)

340 R. HERAS

Figure 4. Pictorial representation of the monopole field Bmondefined by Equation (42). We have extracted the field of the stringB′string from the field B′ms to insolate the field Bmon of themagnetic monopole.

∇ × Bstring = −4πg�(z)[δ(x)δ′(y)x− δ′(x)δ(y)y

].

(52)Therefore, the fieldB′ms defined by Equation (42) satisfies

∇ · B′ms = 0, (53)

∇ × B′ms = 4πg�(−z) [δ(x)δ′(y)x− δ′(x)δ(y)y],(54)

and the field Bms defined by Equation (43) satisfies

∇ · Bms = 0, (55)

∇ × Bms = −4πg�(z)[δ(x)δ′(y)x− δ′(x)δ(y)y

].(56)

Let us return to the Schrodinger equation defined byEquation (17). According to this equation, the electriccharge q interacts with the potential A. From the gaugefunction � = 2gφ, we inferred the potentials A′ and Agiven in Equation (32). The curl of each of these poten-tials originates the field of the magnetic monopole plusthe field of the respective string as may be seen in Equa-tions (40) and (41). If any of these potentials is consideredin Equation (17), then a question naturally arises: Doesthe electric charge interact only with the monopole orwith the monopole and a Dirac string? In other words:Can the electric charge physically interact with a Diracstring? The answer is not as simple asmight appear at firstsight. The Dirac string is a subtle object whose physicalnature has originated controversy and debate.

Typically, the magnetic field of the Dirac string isdiscussed together with the Coulombian field of themag-netic monopole. But since we have identified the sourcesof themagnetic field of the string [those given on the rightof Equations (49) and (50) or also on the right of Equa-tions (51) and (52)], we can study themagnetic field of theDirac stringwith no reference to theCoulombian field. Inthe following section, we will discuss the interaction ofan electric charge with a Dirac string from classical andquantum-mechanical viewpoints.

Figure 5. Pictorial representation of the monopole field Bmondefined by Equation (43). We have added the field of the stringBstring to the fieldBms to insolate the fieldBmon of themagneticmonopole.

8. Classical interaction between the electriccharge and the Dirac string

In order to understand the possible meaning of the Diracstring, we should first study the sources of the magneto-static field produced by this string. Let us assume that thestring lies along the negative z -axis. From Equations (49)and (50), we can see that this string has the associatedcharge and current densities:

ρstring = −gδ(x), (57)

Jstring = cg�(−z) [δ(x)δ′(y)x− δ′(x)δ(y)y], (58)

which generate the magnetic field

B′string = 4πgδ(x)δ(y)�(−z)z. (59)

A regularised vector potential in cylindrical coordinatesfor the field B′string reads

Astring = 2g�(ρ − ε)�(−z)ρ

φ, (60)

where ε > 0 is an infinitesimal quantity. Notice that thepotential Astring for ρ > ε and z<0 is a pure gaugepotential, i.e. it can be expressed as the gradient of a scalarfield. To show that Astring generates B′string considerthe curl of the generic vector F = F[0, Fφ(ρ, z), 0] incylindrical coordinates

∇ × F = −∂Fφ

∂zρ + 1

ρ

∂

∂ρ

(ρFφ

)z. (61)

When this definition is applied to the potential Astring

defined by Equation (60), we obtain

∇ × Astring = 2�(ρ − ε)δ(z)ρ

ρ

+ 2gδ(ρ − ε)�(−z)ρ

z. (62)

CONTEMPORARY PHYSICS 341

Since we are only considering z<0, the first term van-ishes and then

∇ × Astring = 2gδ(ρ − ε)�(−z)ρ

z

= 4πgδ(x)δ(y)�(−z)z= B′string, (63)

where we have used the formula [58]:

δ(x)δ(y) = δ(ρ − ε)

2πρ, (64)

in which the limit ε→ 0 is understood.Having all the classical ingredients on the table, wewill

now proceed to interpret them from both mathematicaland physical point of views. These ingredients are highlysingular and therefore such interpretations are full of sub-tleties. Assuming the existence of magnetic monopoles,the classical interaction between amoving electric chargeq and the magnetic field B′string is given by the Lorentzforce F = q(v/c)× B′string. Expressing the velocity v ofthe charge in cylindrical coordinates v = (vρ , vφ , vz) andusing the regularised form of B′string = ∇ × Astring

defined in the first line of Equation (63), this force reads

F = −2qg�(−z)c

δ(ρ − ε)

ρ

[vφρ − vρφ

]. (65)

The singular character of this force becomes evident. Ifρ �= ε this force vanishes and then the charge q is insen-sitive to the string. If the charge q approaches too muchto the string, then ρ → ε, which implies ρ → 0 becauseε→ 0. In this case, we have

limρ→0

δ(ρ − ε)

ρ= 0, (66)

and again the force in Equation (65) vanishes indicat-ing that the charge q is also unaffected by the string inthis extreme case. However, from a mathematical pointof view, when ρ = ε the force in Equation (65) becomesinfinite (∞/0 = ∞), which is physically unacceptable.

Two results are then conclusive. On one hand, if theelectric charge q is outside the string, then q does not feelthe action of the magnetic field of the string. This is trueeven when the charge q is very close to the string. On theother hand, if ρ = ε, then the charge q feels an infiniteforce due to the magnetic field of the string. The idea ofan infinite force leads us to conclude that the Dirac stringlacks any physicalmeaning. Thus the common statementthat the Dirac string cannot be detected is meaningful inpurely classical considerations.

The interpretation of the potential in Equation (60) isalso somewhat subtle. There is no problem when ρ > ε

Figure 6. Geometry of the Dirac string and its associated vec-tor potential Astring. This potential satisfies ∇ × Astring =B′string.

because in this case Astring = 2g�(−z)φ/ρ exhibitsa regular behaviour which is drawn in Figure 6. Thereis also no problem when ρ < ε because in this caseAstring = 0. When ρ → ε, it follows ρ → 0 becauseε→ 0. In this case

limρ→0

�(ρ − ε)

ρ= 0, (67)

and againAstring vanishes. The problematic issue ariseswhen ρ = ε because in this caseAstring becomes unde-fined.

9. Quantum-mechanical interaction betweenthe electric charge and the Dirac string

The second quantum-mechanical derivation of the Diraccondition will now be reviewed. We have argued thatthe classical interaction of an electric charge with theDirac string is not physically admissible. Now we willconsider the possibility of a quantum-mechanical inter-action between the electric charge and the string. Dirac[2] noted that the interaction of an electric charge with avector potential is given by the phase in thewave function

= ei[q/(�c)]∫ x0 A(x′)·dl′0, (68)

where0 is the solution of the free Schrödinger equationand the line integral is taken along a path of the elec-tric charge from the origin to the point x. The quan-tum mechanical analogous to the classical Lorentz forceF = q(v/c)× B is given by the phase ei[q/(�c)]

∫ x0 A(x′)·dl′

appearing in Equation (68), which in turn representsthe solution of the Schrödinger equation given inEquation (17). This solution assumes that B = ∇ × A =

342 R. HERAS

0 holds in the considered region, otherwise the line inte-gral depends on the path. We note that the phase of thewave function can be discontinuous at some point but thewave function must be a continuous function.

Consider the particular case in which A = Astring,i.e. when the charge q interacts with the potentialAstring associated to the string L′. With this identifi-cation and using cylindrical coordinates, the Dirac con-dition can be implied by assuming (i) that the path is aclosed line surrounding the string

= ei[q/(�c)]∮C Astring·ρ dφφ 0, (69)

and (ii) that the phase change [q/(�c)]∮C Astring ·

ρ dφφ within Equation (69) satisfies the condition

q�c

∮CAstring · ρ dφφ = 2πn. (70)

Under these specific conditions, the possible quantum-mechanical effect of the string on the electric chargewill disappear because ei[q/(�c)]

∮C Astring·ρ dφφ = ei2πn =

1. Integration of the left-hand side of Equation (70) withthe potential defined by Equation (60) gives

q�c

∮CAstring · ρ dφφ = 2qg

�c�(ρ − ε)�(−z)

∫ 2π

0dφ

= 4πqg�c

�(ρ − ε)�(−z)

= 4πqg�c

, (71)

for ρ > ε and z<0. From Equations (70) and (71), wedirectly obtain the Dirac quantisation condition qg =n�c/2.We then conclude that fromquantum-mechanicalconsiderations the unobservability of the string (classi-cally well argued) implies the Dirac condition. The argu-ment can be reversed. If we start by imposing the Diraccondition, then the Dirac string turns out to be unde-tectable. The previous treatment to the Dirac string maybe seen as a complementary discussion to the heuristicapproach to theDirac condition. In the following section,we will review some of the well-known derivations of theDirac quantisation condition.

10. Aharonov–Bohm effect and the Diracquantisation condition

We will now review the third quantum-mechanicalderivation of the Dirac condition. According to theAharonov–Bohm (AB) effect [40], particles can beaffected by a vector potential even in regions where themagnetic field vanishes. We observe that this effect andthe derivation of theDirac quantisation condition require

Figure 7. The AB double slit experiment with the Dirac stringinserted between the slits. If we demand the string to be unde-tectable by the wave function, it follows that the Dirac quantisa-tion condition holds. Conversely, if the Dirac condition holds, thenthe string is undetectable.

similar objects: a long solenoid for the AB effect and asemi-infinite string for the Dirac condition. Therefore,we may think of the Dirac string as the AB solenoidand investigate as to whether the undetectability of theDirac string can be demonstrated via a hypothetical ABinterference experiment [4,8–12,14,15,17,22,23,41,42].

Let us imagine a double-slit AB experiment witha Dirac string inserted between the slits as shown inFigure 7. Electric charges are emitted by a source at pointA, pass through two slits 1 and 2 of the screen locatedat point B, and finally are detected at point C. The wavefunction in a region of zero vector potential is simply = 1 +2, where 1 and 2 are the wave functionsof the charges passing through the slits 1 and 2. With-out the presence of the string, the wave function of thecharges combines coherently in such a way that the prob-ability density at C readsP = |1 +2|2. Since theDiracstring is inserted between the two slits, it is clear thateach of the wave functions 1 and 2 pick up a phasedue to the string potential Astring ≡ As. Thus the wavefunction of the charges is now given by

= e(iq/�c)∫1 As·ρ dφφ 1 + e(iq/�c)

∫2 As·ρ dφφ 2

=(1 + e(iq/�c)

∮C As·ρ dφφ 2

)e(iq/�c)

∫1 As·ρ dφφ

=(1 + ei4πqg/(�c) 2

)e(iq/�c)

∫1 As·ρ dφφ , (72)

where we have used the expression for Astring given inEquation (60) and written as∮

CAs · ρ dφφ =

∫2As · ρ dφφ −

∫1As · ρ dφφ. (73)

It follows now that the probability density at C reads

P = |1 + ei4πqg/(�c) 2|2. (74)

CONTEMPORARY PHYSICS 343

The effect of the Dirac string would be unobservable ifei4πqg/(�c) = 1 and this implies the Dirac quantisationcondition qg = n�c/2. Under this condition, the proba-bility density becomes P = |1 +2|2, meaning that nochange in the interference pattern would be observed dueto the Dirac string. In short: the Dirac string is unde-tectable if theDirac quantisation condition holds.We canreverse the argument: if theDirac quantisation conditionholds, then the Dirac string is unobservable.

11. Feynman’s path integral approach and theDirac quantisation condition

We will now discuss the fourth quantum-mechanicalderivation of the Dirac condition. The path-integralapproach to quantum mechanics, suggested by Dirac in1933 [43], formally started by Feynman in his 1942 Ph.D.thesis [44] and fully discussed by him in 1948 [45], pro-vides an elegant procedure to obtain the Dirac condi-tion, which is similar to a certain extent to that of theAharonov–Bohm effect.

Let us first briefly discuss the essence of the path-integral approach. Question [59]: If a particle is at aninitial position A, what is the probability that it will be atanother position B at the latter time? Schrödinger’s wavefunction tells us the probability for a particle to be in acertain point in time, but it does not tell us the transi-tion probability for a particle to be between two points atdifferent times.We need to introduce a quantity that gen-eralises the concept of wave function to include transitionprobabilities. According to Feynman, this concept is the‘transition probability amplitude’ (or amplitude for short)which relates the state of a wave function from the ini-tial position and time |(xi, ti)〉 to its final position andtime |(xf , tf )〉, and is given by the inner product K =〈(xf , tf )|(xi, ti)〉 , where we have used Dirac’s ‘bra-ket’ notation. It follows that the transition probability (orprobability for short) is defined as P = |K|2. Dirac [43]suggested that the amplitude for a given path is propor-tional to the exponent of the classical action associated tothe path e(i/�)S(x), where S(x) = ∫ L(x, x)dt, is the clas-sical action, with L being the Lagrangian. But a particlecan take any possible path from the initial to the finalpoint (there is no reason for the particle to take the short-est path). Therefore, to compute the amplitude, Feynmanproposed to sum over all the infinite paths that the parti-cle can take. More specifically, the transition probabilityamplitude K for a charged particle to propagate from aninitial point A to a final point B is given by the integralover all possible paths

K =∫

D(x) e(i/�)S(x), (75)

Figure 8. A Dirac string is encircled between two generic pathsγ1 and γ2 starting at A, ending at B, and forming the boundary ofthe surface S.

where∫ D(x) is a short hand to indicate a product of inte-

grals performed over all paths x(t) leading from A to Band S is the classical action associated to each path. Forexample, consider two generic paths γ1 and γ2 each ofwhich starts at A and ends at B. The amplitude is

K = K1 + K2 =∫

γ1

D(x) e(i/�)S(1)(x)

+∫

γ2

D(x) e(i/�)S(2)(x), (76)

where K1 is the amplitude associated to the integrationover all paths through γ1 and K2 is the amplitude associ-ated to the integration over all paths through γ2. Considerfirst the action for a free particle S0 =

∫mx2/2 dt. In

this case, there is not external interaction and thereforethe probability is simply P = |K1 + K2|2. Nothing reallyinteresting happens there. Consider now the case wherethe electric charge is affected by the potential due tothe magnetic monopole and the Dirac string given inEquation (2). Furthermore, suppose that the paths γ1 andγ2 pass on each side of the Dirac string and form theboundary of a surface S as seen in Figure 8.

The external vector potentialAL will affect the motionof the particle because the action acquires an interactionterm

S = S0 + qc

∫AL · dl. (77)

Thus the amplitude becomes

K =∫

γ1

D(x) e(i/�)(S(1)0 +(q/c)

∫(1) AL·dl)

+∫

γ2

D(x) e(i/�)(S(2)0 +(q/c)

∫(2) AL·dl)

=(K1 + e(iq/�c)

∮C AL·dl K2

)e(iq/�c)

∫(1) AL·dl, (78)

344 R. HERAS

where we have written∮CAL · dl =

∫(2)

AL · dl−∫

(1)AL · dl. (79)

Clearly, the contributions from γ1 and γ2 interfere, giv-ing the interference term e(iq/�c)

∮C AL·dl. Using Stoke’s

theorem and Equation (4), we can write the integral ofthis exponent as∮

CAL · dl =

∫S∇ × AL · da =

∫SBmon · da

+∫SBstring · da. (80)

Therefore, we may write the interference term as

e(iq/�c)∮C AL·dl = e(iq/�c)

∫S Bmon·da e(iq/�c)

∫S Bstring·da.

(81)The term e(iq/�c)

∫S Bmon·da is perfectly fine because the

charged particle should be influenced by the magneticmonopole. However, the second term must not con-tribute or otherwise the string would be observable.Therefore, we must demand e(iq/�c)

∫S Bstring·da = 1. But

the flux through the string is∫S Bstring · da = 4πg so

that ei4πqg/�c = 1, which implies the Dirac quantisationcondition qg = n�c/2.

As may be seen, the procedure to obtain the Diracquantisation condition based on Feynman’s path inte-gral approach is similar to the procedure based on theAharonov–Bohm effect. If one first teaches the latter pro-cedure in an advanced undergraduate course, then onemay teach the former procedure in a graduate course, fol-lowing Feynman’s opinion that [45]: ‘there is a pleasure inrecognising old things from a new point of view ’.

12. TheWu–Yang approach and the Diracquantisation condition

We will now examine the fifth quantum-mechanicalderivation of the Dirac condition. Let us rewrite Equa-tions (40) and (41) as follows:

B′ = ∇ × A′ = gr2r+ 4πgδ(x)δ(y)�(−z)z, (82)

B = ∇ × A = gr2r− 4πgδ(x)δ(y)�(z)z. (83)

A direct look at these equations reveals an unpleasantbut formal result: B′ �= B. This result follows from thedifference of the delta-field contributions of the respec-tive strings. Therefore, the potentials A′ and A are notequivalent. Strictly speaking they are not gauge poten-tials. However, it is possible to extend the gauge sym-metry to include contributions due to strings [8], but

this possibility will not be discussed here. Using theproperty �(−z) = 1−�(z), the difference of the mag-netic fields is given byB′ − B = 4πgδ(x)δ(y)z, where theright-hand side of this equation is a singular magneticfield attributable to an infinite string lying along the entirez -axis. The fact that B′ and B are different is not anunexpected result because the current densities produc-ing them are different as may be seen in Equations (50)and (52). However, we have argued that the Dirac stringsare unphysical and should therefore be unobservable.The question then arises: How should the potentials A′andA be interpreted? A rough answer will be thatA′ andA are equivalent because they produce the same mag-netic field [the first terms of Equations (82) and (83)]and because the field contributions of the strings [thelast terms of Equations (82) and (83)] can be physicallyignored. But we must recognise that this answer is notvery satisfactory from a formal point of view. In otherwords, A′ and A are physically but not mathematicallyequivalent.

Furthermore, it can be argued that the derivation ofthe Dirac condition involves some unpleasant featureslike singular gauge transformations and singular poten-tials [9]. Fortunately, a procedure due to Wu and Yang[46] avoids these unpleasant features and leads also to theDirac condition. The Wu–Yang method does not to dealwith singular potentials nor with singular gauge transfor-mations (except with the real singularity at the origin).The strategy of Wu and Yang was to use different vectorpotentials in different regions of space. Inmore colloquialwords, if the Dirac string is the cause of the difficultiesand subtleties, then the Wu–Yang approach provides asimple solution: to get rid of the Dirac string via a formalprocedure.

In the Wu–Yang method, the potentials A′ and A dis-played in Equation (32) are non-singular if we definethem in an appropriate domain:

A′ = g1− cos θr sin θ

φ, RN : 0 ≤ θ <π

2+ ε

2, (84)

A = −g 1+ cos θr sin θ

φ, RS :π

2− ε

2< θ ≤ π , (85)

where ε > 0 is an infinitesimal quantity. The potentialsA′ and A are in the Coulomb gauge: ∇ · A = 0 and∇ · A′ = 0. Furthermore, these potentials are non-globalfunctions since they are defined only on their respec-tive domains: RN and RS. The region RN , where A′ isdefined, excludes the string along the negative semi-axis(θ = π) and represents a North hemisphere. The regionRS, where A is defined, excludes the string along thepositive semi-axis (θ = 0) and represents a South hemi-sphere. The union of the hemispheres RN ∪ RS covers

CONTEMPORARY PHYSICS 345

Figure 9. The Wu–Yang configuration describing a magneticmonopole without the Dirac strings.

the whole space (except on the origin, where there is amagnetic monopole). In the intersection RN ∩ RS (the‘equator’) both hemispheres are slightly overlapped. Arepresentation of theWu–Yang configuration is shown inFigure 9.

Using Equation (28), the potentials A′ and A definedby Equations (84) and (85) yield the field of a mag-netic monopole: B = ∇ × A′ = ∇ × A = gr/r2. There-fore, the potentials A′ and A must be connected by agauge transformation in the overlapped region π/2−ε/2 < θ < π/2+ ε/2, where both potentials are welldefined. At first glance, A′ − A = 2gφ/(r sin θ). But inthe overlapped region, we have lim sin(π/2± ε/2) = 1as ε→ 0 and thus

A′ − A = 2gr

φ = ∇(2gφ) = ∇�, (86)

where� = 2gφ (the gauge function� satisfies∇2� = 0indicating that A′ and A are related by a restricted gaugetransformation). Suppose now that an electric charge isin the vicinity of the magnetic monopole. In this case,we require two wave functions to describe the electriccharge: ′ for RN and for RS. In the overlapped region,the wave functions ′ and must be related by the phasetransformation ′ = eiq�/(�c) , which is associated tothe gauge transformation given in Equation (86). Thisphase transformation with � = 2gφ reads

′ = ei2qgφ/(�c) . (87)

But the wave functions ′ and must be single-valued ( ′|φ = ′|φ+2π), which requires ei4πqg/(�c) =1, and this implies the Dirac quantisation conditionqg = n�c/2. Remarkably, Equations (84)–(87) do notinvolve unpleasant singularities. The approach suggestedby Wu and Yang constitutes a refinement of Dirac’s orig-inal approach. It is pertinent to say that the Wu–Yangapproach has become popular in many treatments of theDirac quantisation condition [4,8,9,11–13,16,24].

13. Semi-classical derivations of the Diracquantisation condition

We will now discuss the first semi-classical derivation ofthe Dirac condition. In 1936, Saha wrote [33]: ‘If we takea point charge e at A and a magnetic pole μ at B, classi-cal electrodynamics tells us that the angular momentumof the system about the line AB is just eμ/c. Hence, fol-lowing the quantum logic, if we put this = h/(2π), thefundamental unit of angular momentum, we have μ =ch/(4πe)which is just the result obtained by Dirac ’. Thisrelatively simple semi-classical argument to arrive at theDirac condition [with n=1] remained almost ignoreduntil 1949 when Wilson [36,37] used the same argu-ment to obtain this condition [now with n integer]. Letus develop in more detail the derivation of Dirac’s con-dition suggested by Saha and also by Wilson. When theDirac condition is written as qg/c = n�/2, we can seethat the left-hand side has units of angular momentumbecause the constant � has these units. This suggests thepossibility that the quantity qg/c can be obtained from theelectromagnetic angular momentum:

LEM = 14πc

∫Vx× (E× B) d3x, (88)

with the idea that the field E is produced by the elec-tric charge q and the field B by the magnetic charge g,both charges at rest and separated by a finite distance.This configuration was considered by Thomson [34,35]in 1904, and is nowknown as the ‘Thomson dipole ’.Moreprecisely stated, the Thomson dipole is a static dipoleformed by an electric charge q and a magnetic charge gseparated by the distance a = |a|, where the vector a isdirected from the charge q to the charge g. For conve-nience, we place the charge q at x′ = −a/2 and the chargeg at x′ = a/2 as seen in Figure 10. Clearly, there is nomechanical momentum associated to this dipole becauseit is at rest. In Appendix 5, we show that the electromag-netic angular momentum due to the fields of the chargesq and g is given by

LEM = qgca, (89)

where a = a/a. This equation was derived by Thom-son [34,35]. Remarkably, the magnitude of LEM does notdepend on the distance between the charges. We notethat Equation (89) has been derived by several equivalentprocedures [60,61]. Notice also that this angularmomen-tum is conserved: dLEM/dt = 0. We now invoke a quan-tum mechanical argument: quantisation of the angularmomentum. As is well known in quantum mechanics,the total (conserved) angular momentum operator J ofa system reads [21]: J = L+ S , where L is the orbital

346 R. HERAS

Figure 10. Configuration of the Thomson dipole.

angular momentum operator and S is the spin angularmomentum operator. In order to obtain J for a givensystem, we first identify its corresponding classical coun-terpart. Evidently, Thomson dipole lacks of an orbitalangular momentum. We can therefore identify S withJ and make the substitution LEM→ J . If we measureJ along any of its three spatial components, say z, ittakes the discrete values Jz = n�/2 [21]. Therefore, if wechoose a = z in Equation (89), thenwe can quantise the zcomponent of this equation. Following this argument, weobtain Jz = qg/c = n�/2, which yields the Dirac condi-tion qg = n�c/2. We should emphasise that this methodis semiclassical in the sense that the angular momentumqg/c is first obtained from purely classical considerationsand then it is made equal to n�/2 by invoking a quantumargument.

We will now examine the second semi-classical deriva-tion of the Dirac condition. We can also arrive at the Diraccondition by another semiclassical method due to Fierz[38]. Consider an electric charge q moving with veloc-ity x in the field of a monopole g centred at the origin:B = gr/r2. This configuration is illustrated in Figure 11.The charge q experiences the Lorentz force

dpdt= q

(xc× B

), (90)

where p = mx is the mechanical momentum associatedto the charge q.

The field of the monopole is spherically symmet-ric and therefore one should expect the total angularmomentum of the system is conserved. To see this, wetake the cross product of Equation (90) with the positionvector x, use x× (dp/dt) = d(x× p)/dt, and obtain thecorresponding torque

d(x× p)

dt= q

c(x× (x× B)) = qg

c

(x× (x× x)

r3

)= d

dt

(qgcr), (91)

Figure 11. Dynamics of a moving electric charge in the field ofa magnetic monopole. In this configuration, the angular momen-tum r · J = −qg/c is constant. This means that the charge movesin a cone on the axis J, with the angle θ = cos−1(qg/Jc).

where we have used the identity

x× (x× x)r3

= drdt

. (92)

Clearly, the mechanical angular momentum x× p is notconserved d(x× p)/dt �= 0. This is an expected resultbecause there is an extra contribution attributed to theangular momentum of the electromagnetic field. FromEquation (91), it follows

ddt

(x× p− qg

cr)= 0. (93)

Hence, the total (conserved) angular momentum is

J = x× p− qgcr. (94)

This interesting result was observed by Poincaré [62] in1896, although it was already anticipated by Darbouxin 1878 [63]. From Equation (94), it follows that theradial component of this angular momentum is constantJ · r = −qg/c.With regard to the quantity qg/c, Fierz [38]pointed out: ‘ . . . the classic value qg/c, must be in quan-tum theory equal to an integer or half-integer multiple of�’. Following this argument, we can quantise the radialcomponent of the angular momentum in Equation (94):Jr = qg/c = n�/2 (the minus sign is absorbed by n) andthis yields the Dirac condition qg = n�c/2.

We will now review the third semi-classical derivationof the Dirac condition. Strictly speaking, we will reviewthe derivation of a generalised duality-invariant form ofthis condition due to Schwinger [39]. The approach fol-lowed by Schwinger is similar to that of Fierz but now

CONTEMPORARY PHYSICS 347

Figure 12. Dynamics of a moving dyon in the field of a station-ary dyon. In this configuration, the angular momentum r · J =−(q1g2 − q2g1)/c is constant. Thismeans that the dyonmoves ina coneon the axis J, with the angle θ = cos−1((q1g2 − q2g1)/Jc).

applied to the case of dyons, which are particles withboth electric and magnetic charge. The approach con-siders the interaction of a dyon of mass m carrying anelectric charge q1 and a magnetic charge g1, moving withvelocity x in the field of a stationary dyon with electriccharge q2 and magnetic charge g2 centred at the origin,as seen in Figure 12. The Lorentz force due to themovingdyon takes the duality-invariant form

dpdt= q1

(E+ x

c× B

)+ g1

(B− x

c× E

), (95)

where the electric and magnetic fields produced by thecharges q2 and g2 of the stationary dyon are

E = q2r2r, B = g2

r2r. (96)

Therefore, we may write Equation (95) as

dpdt= (q1q2 + g1g2

) rr2+ (q1g2 − q2g1

) x× xc r3

. (97)

To find the conserved angular momentum of the sys-tem, we take the cross product of Equation (97) with theposition vector x, use x× (dp/dt) = d(x× p)/dt, andobtain

d(x× p)

dt=(q1g2 − q2g1

)c

drdt

, (98)

where we have used Equation (92). The conserved angu-lar momentum is thus

J = x× p− (q1g2 − q2g1) rc, (99)

whose radial component J · r = −(q1g2 − q2g1)/c canbe quantised: Jr = (q1g2 − q2g1)/c = n�/2, yielding theSchwinger–Swanziger quantisation condition

q1g2 − q2g1 = n2�c. (100)

In contrast to the Dirac condition qg = n�c/2, whichfor a fixed value of n is not invariant under the dualchanges q→ g and g →−q, the Schwinger–Swanzigercondition is clearly invariant under these dual changes.Equation (100) was first obtained by Schwinger [64]and independently by Swanziger [65]. Interestingly,both of these authors argued that the quantisation inEquation (100) should take integer and not half-integervalues, i.e. Equation (100) should be written as q1g2 −q2g1 = n�c.

14. Final remarks on the Dirac quantisationcondition

The advent of the Dirac quantisation condition broughtus two news: one good and another bad. The good newsis that this condition allows us to explain the observedquantisation of the electric charge. The bad news is thatsuch an explanation is based on the existence of unob-served magnetic monopoles. One is left with the feelingthat the undetectability ofmagneticmonopoles spoils theDirac quantisation condition. Evidently, the fact that theDirac condition explains the electric charge quantisationcannot be considered as a proof of the existence of mag-netic monopoles. Although it has recently been arguedthat magnetic monopoles may exist, not as elementaryparticles, but as emergent particles (quasiparticles) inexotic condensed matter magnetic systems such as ‘spinice’ [66–68], there is still no direct experimental evidenceof Dirac monopoles. However, experimental searches formonopoles continue to be of great interest [69–73,76].It can be argued that the idea of undetected magneticmonopoles is too high a price to pay for explainingthe observed charge quantisation. But equally it can beargued that magnetic monopoles constitute an attractivetheoretical concept, which is not precluded by any funda-mental theory and has been extremely useful in moderngauge field theories [4,29].

In any case, magnetic monopoles are like the LochNess monster, much talked about but never seen.Although many theoretical physicists would say that theidea of magnetic monopoles is too attractive to set aside,we think it would be desirable to have a convincingexplanation for the electric charge quantisation withoutappealing to magnetic monopoles.

It is interesting to note that the introduction of mag-netic monopoles in Dirac’s 1931 paper [2] was not taken

348 R. HERAS

fondly by Dirac himself. He wrote: ‘The theory leads to aconnection, namely, [eg0 = �c/2], between the quantumofmagnetic pole and the electronic charge. It is rather dis-appointing to find this reciprocity between electricity andmagnetism, instead of a purely electronic quantum con-dition such as [�c/e2] ’. However, no satisfactory explana-tion for the charge quantisation was proposed between1931 and 1948 and this seemed to led him to reinforcehis idea about magnetic monopoles. In his 1948 paper,he wrote [3]: ‘The quantisation of electricity is one ofthe most fundamental and striking features of atomicphysics, and there seems to be no explanation for it apartfrom the theory of poles. This provides some grounds forbelieving in the existence of these poles ’.

The story of the Dirac quantisation condition may betraced to the story of a man [P. A. M. Dirac: the theo-rist of theorists!] who wanted to know why the electriccharge is quantised and why the electric charge of theelectron had just the numerical value that makes theinverse of the fine structure constant to acquire the valueα−1 = �c/e2 ≈ 137. Many years later, he expressed hisfrustration at not being able to find this magic number.He criticised his theory because it [30]: ‘··· did not leadto any value for this number [α−1 ≈ 137], and, for thatreason,my argument seemed to be a failure and I was dis-appointed with it ’. But the idea of explaining this numberseems to have been always important for him. With theconfidence of a master, Dirac wrote [30]: ‘The problem ofexplaining this number�c/e2 is still completely unsolved.Nearly 50 years have passed since then. I think it is per-haps the most fundamental unsolved problem of physicsat the present time, and I doubt very much whether anyreally big progress will be made in understanding thefundamentals of physics until it is solved ’.

Although Dirac was not successful in explaining whythe charge of the electron has its observed value, in thesearch for this ambitious goal, he envisioned a mag-netic monopole attached to a semi-infinite string, whichhe required to be unobservable by a quantum argu-ment, obtaining thus a condition that explains the elec-tric charge quantisation. This is indeed a brilliant ideanot attributable to an ordinary genius but rather to amagician, a person ‘whose inventions are so astounding,so counter to all the intuitions of their colleagues, thatit is hard to see how any human could have imaginedthem’ [74].

15. A final comment on nodal lines

Berry [77] has pointed out that the nodal lines introducedby Dirac in his 1931 paper [2] are an example of disloca-tions in the probabilitywaves of quantummechanics. Thehistory can be traced to 1974 when Nye and Berry [78]

observed that wavefronts can contain dislocation lines,closely analogous to those found in crystals. They definedthese dislocation lines as those lines on which the phaseof the complex wave function is undetermined, whichrequires the amplitude be zero, indicating that dislocationlines are lines of singularity (or lines of zeros). Remark-ably, the lines of singularity (also calledwave dislocations,nodal lines, phase singularities and wave vortices) aregeneric features of waves of all kinds, such as light waves,soundwaves and quantummechanical waves. These linesinvolve two essential properties: on these lines the phaseis singular (undetermined) and around these lines thephase changes by a multiple (typically ±1) of 2π . Eventhough the concept of the line of singularity has beenextensively discussed in the literature (see, for example,the collection of papers in the special issues mentionedin References [79–82]), its connection with the Diracstrings is not usually commented on. In his review on sin-gularities in waves [77], Berry has claimed: ‘He [Dirac]recognises that0 [appearing in Equation (68)] can havenodal lines around which the phase χ0 in the absence ofmagnetic field changes by 2nπ , i.e. he recognises the exis-tence of wavefront dislocations ’. However, it should beemphasised that the semi-infinite nodal lines introducedby Dirac are unobservable because of the Dirac quan-tisation condition. But in the general case, the lines ofsingularity are physical and can form closed loops, whichcan be linked and knotted [83].

16. Conclusion

In this review paper, we have discussed five quantum-mechanical derivations, three semiclassical derivationsand a novel heuristic derivation of the Dirac quantisationcondition. They are briefly resumed as follows.

First quantum mechanical derivation. In this deriva-tion, the magnetic monopole is attached to an infiniteline of dipoles, the so-called Dirac string [18]. The vec-tor potential of this configuration yields the field of themagnetic monopole plus a singular magnetic field dueto the Dirac string. By assuming that the location of thestring must be irrelevant, it is shown that the two arbi-trary positions of the string are connected with two gaugepotentials, meaning that the change of a string to anotherstring is equivalent to a gauge transformation involvinga multi-valued gauge function. By demanding the wavefunction in the phase transformation be single-valued,the Dirac condition is required.

Heuristic derivation. (i) It starts with the relationei2kφ = eiq�/(�c), where k is an arbitrary constant, φ theazimuthal angle and � an unspecified gauge function;(ii) from this relation it follows the remarkable equation�q/(�c) = 2kφ. One solution of this equation is given

CONTEMPORARY PHYSICS 349

by k = qg/(�c) and � = 2gφ, where g is a constant tobe identified; (iii) if the phase eiq�/(�c) is required tobe single-valued, then ei2kφ must be also single-valuedand this implies the ‘ quantisation’ condition k = n/2with n being an integer; (iv) from this condition and k =qg/(�c), we get the relation qg = n�c/2; (v) the function� = 2gφwith g being themagnetic charge is proved to bea gauge function and this allows us to finally identify qg =n�c/2 with the Dirac quantisation condition; (vi) a weakpoint of this heuristic derivation is that the associatedDirac strings are excluded; (vii) classical considerationsindicate that the Dirac string lacks of physical mean-ing and is thus unobservable; (viii) quantum mechanicalconsiderations show that the undetectability of the Diracstring implies the Dirac condition.

Second quantummechanical derivation. The quantum-mechanical interaction of an electric charge q with thepotential A is given by the phase appearing in the wavefunction = ei[q/(�c)]

∫ x0 A(x′)·dl′0, where0 is the solu-

tion of the free Schrödinger equation and the line integralin the phase is taken along a path followed by q fromthe origin to the point x. If A = Astring = 2g�(ρ −ε)�(−z)φ/ρ and the path is a closed line surround-ing the string, we have [q/(�c)]

∮C Astring · ρ dφφ =

4πqg/(�c) for ρ > ε and z<0. If now we demand thisquantity to be equal to 2πn, then the effect of the string onthe charge q disappears because ei4πqg/(�c) = ei2πn = 1and this implies the Dirac condition.

Third quantum mechanical derivation. This deriva-tion is directly related to theAharonov–Bohmdouble-slitexperiment [40] with the Dirac string inserted betweenthe slits. Considering the vector potential of the string,it is shown that the corresponding probability density isP = |1 + ei4πqg/(�c) 2|2. The effect of the Dirac stringis unobservable if ei4πqg/(�c) = 1 and this implies theDirac condition. Vice versa, if this condition holds apriori, then the Dirac string is unobservable.

Fourth quantum mechanical derivation. According toFeynman’s path-integral approach to quantum mechan-ics [45], the amplitude of a particle reads K = ∫ D(x)e(i/�)S(x), where

∫ D(x) indicates a product of inte-grals performed over all paths x(t) going from A toB, and S is the classical action associated to eachpath. For two such generic paths in free space, γ1

and γ2, we have K = K1 + K2 =∫γ1

D(x) e(i/�)S(1)(x) +∫γ2

D(x) e(i/�)S(2)(x). Suppose that γ1 and γ2 pass on eachside of the Dirac string and form the boundary of a sur-face S. As a result, the action acquires an interactiontermS = S0 + (q/c)

∫AL · dl, whereS0 is the action for

the free path. Thus the amplitude becomes K = (K1 +e(iq/�c)

∮C AL·dlK2)e(iq/�c)

∫(1) AL·dl, and the interference

term is e(iq/�c)∮C AL·dl. Using the Stoke’s theorem and∇ ×

AL = Bmon + Bstring, the interference term becomese(iq/�c)

∮C AL·dl = e(iq/�c)

∫s Bmon·da e(iq/�c)

∫S Bstring·da. The

second exponential factor on the right should not con-tribute or otherwise the string would be observable.Thus we must demand e(iq/�c)

∫S Bstring·da = 1. But the

flux through the string is∫S Bstring · da = 4πg so that

ei4πqg/�c = 1, which implies Dirac’s condition.Fifth quantum mechanical derivation. This derivation

describes a magnetic monopole without Dirac strings[46] using two non-singular potentials which are definedin two different regions of space. In the intersectionregion, both potentials are connected by a non-singulargauge transformation with the gauge function � = 2gφ.The description of an electric charge in the vicinity of themagnetic monopole requires two wave functions ′ and , which are related by the phase transformation ′ =ei2qgφ/(�c) in the overlapped region. But ′ and

must be single-valued ( ′|φ = ′|φ+2π), which requiresei4πqg/(�c) = 1, and this implies Dirac’s condition.

First semi-classical derivation. This derivation consid-ers the Thomson dipole [34,35], which is a static dipoleformed by an electric charge q and a magnetic charge gseparated by the distance a = |a| [60,61]. The electro-magnetic angular momentum of this dipole is given byLEM = qga/c. By assuming that any of the spatial com-ponents of the angular momentum must be quantised ininter multiples of �/2, we obtain Dirac’s condition.

Second semi-classical derivation. This derivation con-siders an electric charge q moving with speed x in thefield of a monopole g [8,38]. The associated Lorentz forcedp/dt = q(x× B/c) is used to obtain total (conserved)angular momentum of this system J = x× p− qgr/c.The radial component J · r = −qg/c is then quantisedyielding Dirac’s condition.

Third semi-classical derivation. This derivation con-siders a dyon of mass m carrying an electric charge q1and a magnetic charge g1, moving with velocity x inthe field of a stationary dyon with charge q2 and g2located at the origin [39]. Using the duality-invariantform of the Lorentz force dp/dt = q1(E+ x× B/c)+g1(B− x× E/c), the total angular momentum of thissystem is found to be J = x× p− (q1g2 − q2g1)r/c.The radial component J · r = −(q1g2 − q2g1)/c is thenquantised yielding the Schwinger–Swanziger conditionq1g2 − q2g1 = n�c/2 which is a duality invariant form ofDirac’s condition.

Note

1. A derivation of Equations (17)–(19), which is more ped-agogical than that appearing in the standard graduatetextbooks (for example in Reference [21]), is available inthe author’s website: www.ricardoheras.com.

350 R. HERAS

Acknowledgements

I wish to thank Professor Michael V. Berry for bringing myattention to the important topic of wavefront dislocations andits connection with the Dirac strings.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes on contributor

Ricardo Heras is an undergraduate stu-dent in Astrophysics at University CollegeLondon. He has been inspired by Feyn-man’s teaching philosophy that if one can-not provide an explanation for a topic atthe undergraduate level, then it means onedoesn’t really understand this topic. Hisinterest in understanding physics has led

him to publish several papers inThe European Journal of Physicson the teaching of electromagnetism and special relativity. Hehas also authored research papers onmagneticmonopoles, pul-sar astrophysics, history of relativity and two essays in PhysicsToday. For Ricardo, the endeavour of publishing papers inphysics represents the first step towards becoming a physicistdriven by ‘The pleasure of finding things out’.

ORCID

Ricardo Heras https://orcid.org/0000-0003-1234-2481

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352 R. HERAS

Appendices

Appendix 1: Derivation of Equations (3) and (12)

The curl of Equation (2) gives

∇ × AL = ∇ ×(

∇ ×{∫

L

g dl′

|x − x′|})

= ∇(

∇ ·{∫

L

g dl′

|x− x′|})−∇2

{∫L

g dl′

|x − x′|}(A1)

= g∇∫L∇ ·

(dl′

|x − x′|)− g

∫L∇2

(1

|x − x′|)

dl′. (A2)

Using the result ∇ · (dl′/|x − x′|) = dl′ ·∇(1/|x− x′|), thefirst integral becomes∫

L∇ ·

(dl′

|x − x′|)=∫L∇(

1|x − x′|

)· dl′

= −∫L∇′(

1|x − x′|

)· dl′

= − 1|x − x′| . (A3)

Considering Equation (A3), the first term of Equation (A2)yields the field of the magnetic monopole

g∇∫L∇ ·

(dl′

|x − x′|)= g∇

(− 1|x − x′|

)= g

R2R, (A4)

where we have used∇(1/|x− x′|) = −R/R2. The second termof Equation (A2) yields the magnetic field of the Dirac string

− g∫L∇2

(1

|x − x′|)

dl′ = 4πg∫Lδ(x − x′) dl′, (A5)

where we have used ∇2(1/|x− x′|) = −4πδ(x − x′). TheAddition of Equations (A4) and (A5) yields Equation (3).

To derive Equation (12), we first take the curl of Equation(11),

∇ × AL = ∇ ×(

∇ ×{z∫ 0

−∞g dz′

|x − z′z|})

= ∇(

∇ ·{z∫ 0

−∞g dz′

|x − z′z|})−∇2

{z∫ 0

−∞g dz′

|x− z′z|}

= g∇∫ 0

−∞∂

∂z

(dz′

|x − z′z|)− g z

∫ 0

−∞∇2

(dz′

|x − z′z|).

(A6)

To simplify the first term, we may write

∂

∂z

(1

|x − z′z|)= − z − z′(

x2 + y2 + (z − z′)2)3/2 , (A7)

so that∫ 0

−∞∂

∂z

(dz′

|x − z′z|)= −

∫ 0

−∞z − z′(

x2 + y2 + (z − z′)2)3/2 dz′.

(A8)Consider the substitution u(z′) = x2 + y2 + (z − z′)2. Hence,du = −2(z − z′) dz′, and the right-hand side of the integral in

Equation (A8) takes the form

12

limβ→∞

∫ u(z′=0)

u(z′=−β)

duu3/2= lim

β→∞−1√u

∣∣∣∣u(z′=0)u(z′=−β)

= − 1|x| + lim

z′→−∞1

|x − z′z| = −1r. (A9)

Using this result in the first term in Equation (A6), we obtainthe monopole field

g∇∫ 0

−∞∂

∂z

(dz′

|x − z′z|)= g∇

(−1r

)= g

r2r. (A10)

To simplify the second term in Equation (A6), consider

∇2(

1|x − z′z|

)= −4πδ(x− z′z)

= −4πδ(x)δ(y)δ(z − z′). (A11)

Using this equation in the second term of Equation (A6), weobtain the string field

− gz∫ 0

−∞∇2

(dz′

|x − z′z|)= 4πgδ(x)δ(y)

×{∫ 0

−∞δ(z − z′) dz′

}z = 4πgδ(x)δ(y)�(−z)z, (A12)

where in the last step we have used the integral represen-tation of the step function �(ξ − α) = ∫ ξ

−∞ δ(τ − α)dτ toidentify the quantity within the brackets { } in Equation (A12)as �(−z) = ∫ 0

−∞ δ(z − z′) dz′. Addition of Equations (A10)and (A12) yields Equation (12).

Appendix 2: Derivation of Equation (13)

Using Equation (11), we obtain

AL = g∇ × z∫ 0

−∞dz′

|x − z′z| = g(

∂

∂yx − ∂

∂xy)

×∫ 0

−∞dz′

|x − z′z|

= g∫ 0

−∞

{∂

∂y

(x

|x− z′z|)+ ∂

∂x

(y

|x − z′z|)}

dz′.

(A13)

Now,

∂

∂y

(1

|x − z′z|)= − y

(x2 + y2 + (z − z′)2)3/2, (A14)

∂

∂x

(1

|x − z′z|)= x

(x2 + y2 + (z − z′)2)3/2. (A15)

Inserting these equations in Equation (A13), we obtain

AL = g(−yx + xy

) ∫ 0

−∞dz′

(x2 + y2 + (z − z′)2)3/2. (A16)

The integral can be solved by a variable change and an appro-priate substitution. We can write (z − z′)2 = (z′ − z)2. Nowwe let u(z′) = z′ − z so that du = dz′. Hence, the integral in

CONTEMPORARY PHYSICS 353

Equation (A16) may be written as

limβ→∞

∫ u(z′=0)

u(z′=−β)

du(x2 + y2 + u2)3/2

. (A17)

An appropriate substitution for solving this integral is u(v) =√x2 + y2 tan(v), where v = tan−1(u/

√x2 + y2). This relation

assumes√x2 + y2 �= 0, indicating that the negative z-axis asso-

ciated to the Dirac string has been avoided. It follows that du =sec2(v)dv and then the integral in Equation (A17) becomes

limβ→∞

∫ v(u(z′=0))

v(u(z′=−β))

√x2 + y2 sec2(v)(

(x2 + y2)(tan2(v)+ 1))3/2 dv. (A18)

Using the identity sec2(v) = tan2(v)+ 1, the denominator inEquation (A18) simplifies to (x2 + y2)3/2 sec3(v). It follows

1x2 + y2

limβ→∞

∫ v(u(z′=0))

v(u(z′=−β))

dvsec(v)

= 1x2 + y2

limβ→∞

×∫ v(u(z′=0))

v(u(z′=−β))

cos(v) dv

= limβ→∞

sin(v)

x2 + y2

∣∣∣∣v(u(z′=0))

v(u(z′=−β))

,

(A19)

where cos(v) = 1/ sec(v) has been used. Considering theidentity sin(tan−1(α)) = α/

√α2 + 1, we can easily evaluate

Equation (A19)

limβ→∞

sin(v)

x2 + y2

∣∣∣∣v(u(z′=0))

v(u(z′=−β))

=(

1x2 + y2

)lim

β→∞u√

x2 + y2√

u2x2+y2 + 1

∣∣∣∣u(z′=0)u(z′=−β)

=(

1x2 + y2

)lim

β→∞z′ − z√

x2 + y2 + (z − z′)2

∣∣∣∣z′=0z′=−β

= 1x2 + y2

(1− z√

x2 + y2 + z2

). (A20)

From Equation (A20) in Equation (A16), we obtain

AL = g(−yx + xy

)x2 + y2

(1− z√

x2 + y2 + z2

). (A21)

Considering spherical coordinates r = √x2 + y2 + z2, r sin θ

= √x2 + y2, r cos θ = z and φ = (−yx + xy)/(√x2 + y2),

Equation (A21) takes the formAL = g[(1− cos θ)/(r sin θ)]φ,which is Equation (13).

Appendix 3: Derivation of Equation (14)

Consider the first equality in Equation (14)

AL′ − AL = g∇ ×∮C

dl′

|x − x′| . (A22)

Using Stoke’s theorem and ∇(1/|x − x′|) = −∇′(1/|x − x′|),Equation (A22) becomes

AL′ − AL = −g∇ ×∫S∇′(

1|x − x′|

)× da′

= ∇ ×(

∇ ×{∫

S

g da′

|x − x′|})

= ∇(

∇ ·{∫

S

g da′

|x − x′|})−∇2

{∫S

g da′

|x − x′|}.

(A23)

Making use of ∇ · (da′/|x − x′|) = da′ ·∇(1/|x − x′|),Equation (A23) reads

AL′ − AL = g∇∫S∇(

1|x − x′|

)· da′

− g∫S∇2

(1

|x − x′|)

da′

= g∇∫S

(x′ − x) · da′|x − x′|3 + 4πg

∫Sδ(x − x′) da′,

(A24)

where we have used ∇(1/|x − x′|) = −(x − x′)/|x− x′|3 and∇2(1/|x− x′|) = −4πδ(x − x′). The integral in the first termof Equation (A24) is the solid angle [75]

�(x) =∫S

(x′ − x) · da′|x − x′|3 , (A25)

and therefore

AL′ − AL = g∇�+ 4πg∫Sδ(x− x′) da′. (A26)

The delta integral contribution vanishes at any point x noton the surface S and can therefore be dropped [7]. Thus weobtain AL′ − AL = g∇�, which is Equation (14). Discussionson Equation (A26) can be found in References [7,12,84].

Appendix 4: Derivation of Equation (40)

Consider the first vector potential given in Equation (32),namely A′ = [g(1− cos θ)/(r sin θ)]φ which is valid for z <

0. For convenience, we express this potential in cylindricalcoordinates

A′ = gρ

(1− z√

ρ2 + z2

)φ, (A27)

where we have used cos θ = z/√

ρ2 + z2, and r sin θ = ρ, withρ = √x2 + y2. A regularised form of this potential can beobtained by making the replacements [58]: 1/ρ → �(ρ −ε)/ρ and z/

√ρ2 + z2 → z/

√ρ2 + z2 + ε2, where � is the

step function and ε > 0 is an infinitesimal quantity. It follows

A′ε =g �(ρ − ε)

ρ

(1− z√

ρ2 + z2 + ε2

)φ. (A28)

Clearly, in the limit ε→ 0 we recover Equation (A27). Con-sider now the definition of the curl of the generic vec-tor F = F[0, Fφ(ρ, z), 0] in cylindrical coordinates given inEquation (61). Using this definition in Equation (A28), we

354 R. HERAS

obtain

∇ × A′ε = −g�(ρ − ε)

ρ

(ρ2 + ε2

(ρ2 + z2 + ε2)3/2

)ρ

+ g�(ρ − ε)

ρ

(z

(ρ2 + z2 + ε2)3/2

)z

+{gδ(ρ − ε)

ρ− gzδ(ρ − ε)

ρ√

ρ2 + z2 + ε2

}z

= g�(ρ − ε)

(ρ2 + z2 + ε2)

(ρρ + zz√

ρ2 + z2 + ε2

)

− ε2 g�(ρ − ε)ρ

ρ(ρ2 + z2 + ε2)3/2

+{gδ(ρ − ε)

ρ− gzδ(ρ − ε)

ρ√

ρ2 + z2 + ε2

}z. (A29)

In the last term enclosedwithin the brackets {}, we add the exactzero quantity [gδ(ρ − ε)/ρ − gδ(ρ − ε)/ρ]z ≡ 0, and obtain

∇ × A′ε =g �(ρ − ε)

(ρ2 + z2 + ε2)

(ρρ + zz√

ρ2 + z2 + ε2

)+ 2g δ(ρ − ε)z

ρ

− ε2 g�(ρ − ε)ρ

ρ(ρ2 + z2 + ε2)3/2− g δ(ρ − ε)

ρ

×(√

ρ2 + z2 + ε2 + z√ρ2 + z2 + ε2

)z. (A30)

This is a regularised form of the magnetic field produced bythe potential A′ε . The first two terms of Equation (A30) arethe only non-vanishing terms in the limit ε → 0. The thirdterm is shown to vanish easily because there is a term ε2 in thenumerator. However, it is not clear why the last term shouldvanish. Let us analyse this term. Consider an arbitrary point z0on the negative z-axis. For small ε, we can make the replace-ment [22]:

√ρ2 + z2 + ε2 + z→ (ρ2 + ε2)/(2z0). With this

replacement, the last term in Equation (A30) becomes(gδ(ρ − ε)ρ

2z0√

ρ2 + z2 + ε2+ gδ(ρ − ε) ε2

2ρz0√

ρ2 + z2 + ε2

)z. (A31)

In the limit ε→ 0, it follows that Equation (A31) vanishesbecause ε2 → 0 and δ(ρ)ρ = 0. Hence,

limε→0

∇ × A′ε = limε→0

{g�(ρ − ε)

(ρ2 + z2 + ε2)

(ρρ + zz√

ρ2 + z2 + ε2

)

+2gδ(ρ − ε)zρ

}= g

rr2+ 4πgδ(x)δ(y)�(−z)z, (A32)

where we have used r = (ρρ + zz)/(√

ρ2 + z2), and inserted�(−z) = 1 to specify that this expression is valid onlyfor z < 0.

Appendix 5: Derivation of Equation (89)

Consider the electromagnetic angular momentum of theThomson dipole whose configuration is shown in Figure 10.

The electric and magnetic fields of this dipole are

E = q(x+ a/2)|x + a/2|3 , B = g

(x− a/2)|x − a/2|3 . (A33)

These fields satisfy

∇ · E = 4πqδ(x+ a/2), ∇ × E = 0, (A34)

∇ · B = 4πgδ(x − a/2), ∇ × B = 0. (A35)

In particular, the electric field can be expressed as the gradientof the electric potential E = −∇�, where

�(x) = q|x + a/2| . (A36)

Using E = −∇�, we write E× B = −∇�× B, which com-bines with ∇ × (�B) = �∇ × B+∇�× B to obtain E×B = −∇ × (�B). If we define the vector W = �B, thenE× B = −∇ ×W. Using this expression in the integrand ofEquation (81), we obtain

x× (E× B) = −x× (∇ ×W). (A37)

To write Equation (A37) in an appropriate form, we can use thefollowing identity expressed in index notation [61]:

[x× (∇ ×W)]i = −∂j(xjWi − 2Wjxi

)+ ∂ i(xjWj)

− 2xi∂jWj. (A38)

Here summation convention on repeated indices is adoptedand εijk is the Levi-Civita symbol with ε123 = 1 and δij is theKronecker delta. Equation (A38) can be readily verified. Firstwe write

[x× (∇ ×W)]i = εijkxj (∇ ×W)k

= εijkxjεklm∂ lWm

= (δilδjm − δ

jlδ

im) xj∂ lWm

= xm∂ iWm − (xm∂m)Wi, (A39)

where we have used the identity εijkεklm = δilδjm − δ

jlδ

im. Now,

consider the identically zero quantities

2(∂mWmxi − ∂mWmxi

) ≡ 0, (A40)(∂ ixmWm + 2Wm∂mxi − ∂mxmWi) ≡ 0. (A41)

Adding Equations (A40) and (A41) to Equation (A39), weobtain Equation (A38). When Equation (A38) is integratedover a volume, the first two terms of the right-hand side can betransformed into surface integrals which are shown to vanishfor a large r. Therefore,∫

V[x× (E× B)]i d3x = 2

∫Vxi∂jWj d3x

= 2∫Vxi(∂j�Bj +�∂jBj) d3x

= −2∫Vxi(EjBj) d3x+ 2

∫Vxi�(∂jBj) d3x. (A42)

Using Equation (A42) in Equation (88), we obtain

LEM = − 12πc

∫Vx (E · B) d3x+ 1

2πc

∫Vx�(∇ · B) d3x

= 12πc

∫Vx�(∇ · B) d3x, (A43)

CONTEMPORARY PHYSICS 355

where the integral in the first term has vanished because inte-grand is an odd function of x for the chosen origin. UsingEquations (A35) and (A36), we substitute ∇ · B = 4πgδ(x−a/2) and � = q/|x + a/2| into the second integral, obtainingthe expected result

LEM = 2qgc

∫V

δ(x − a/2)(

x|x + a/2|

)d3x

= 2qgc

x|x + a/2|

∣∣∣∣x=a/2

= qgca. (A44)