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Physics Letters A 324 (2004) 9–13

www.elsevier.com/locate/pla

On representations of the rotation group and magnetic monop

Alexander I. Nesterov∗, Fermín Aceves de la Cruz

Departamento de Física, CUCEI, Universidad de Guadalajara, Av. Revolución 1500, Guadalajara, CP 44420, Jalisco, Mexico

Received 27 January 2004; received in revised form 18 February 2004; accepted 23 February 2004

Communicated by P.R. Holland

Abstract

Recently [Phys. Lett. A 302 (2002) 253] employing bounded infinite-dimensional representations of the rotation grhave argued that one can obtain the consistent monopole theory with generalized Dirac quantization condition, 2κµ ∈ Z, whereκ is the weight of the Dirac string. Here we extend this proof to the unbounded infinite-dimensional representations. 2004 Elsevier B.V. All rights reserved.

PACS: 14.80.Hv; 03.65.-w; 03.50.De; 05.30.Pr; 11.15.-q

Keywords: Monopole; Indefinite-metric Hilbert space; Nonassociativity; Infinite-dimensional representation

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1. Introduction

The Dirac quantization relation [1] betweenelectric chargee and magnetic chargeq ,

(1)2µ= n, n ∈ Z,

where µ = eq , and we seth = c = 1, has beenobtained from various approaches based on quanmechanics and quantum field theory [1–11]. Onethe widely accepted proofs of the Dirac selection ris based on group representation theory and consisthe following: in the presence of magnetic monopthe operator of the total angular momentum

(2)J = r × (−i∇ − eA)−µrr,

* Corresponding author.E-mail addresses: nesterov@cencar.udg.mx (A.I. Nesterov)

fermin@udgphys.intranets.com (F. Aceves de la Cruz).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.02.051

has the same properties as a standard angular motum and for any value ofµ obeys the usual commutation relations

(3)[Ji, Jj ] = iεijkJk.

The requirement thatJi generate a finite-dimensionrepresentation of the rotation group yields 2µ beinginteger and only values 2µ = 0,±1,±2, . . . are al-lowed (for details see, for example, [3,5,7–9]).

Actually the charge quantization does not follofrom the quantum-mechanical consideration andtation invariance alone. Any treatment uses someditional assumptions that may be not physicallyevitable.

Recently we have exploited this problem emploing bounded infinite-dimensional representationsthe rotation group and nonassociative gauge transmations. We argued that one can relax Dirac’s contion and obtain the consistent monopole theory w

.

10 A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 9–13

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the generalized quantization condition, 2κµ ∈ Z, κbeing the weight of the Dirac string [12,13]. In oLetter we extend this proof to the unbounded infinidimensional representations of the rotation group.

2. Magnetic monopole preliminaries

As well known any vector potentialA being com-patible with a magnetic fieldB = qr/r3 of Diracmonopole must be singular on the string (the so-caDirac string, further it will be denoted asSn), and onecan write

B = rotAn + hn,

wherehn is the magnetic field of the Dirac string giveby

(4)hn = 4πqn

∞∫0

δ3(r − nτ ) dτ.

The unit vectorn determines the direction of a strinSn passing from the origin of coordinates to∞.

For instance, Dirac’s original vector potential rea

(5)An = qr × n

r(r − n · r),

and the Schwinger’s choice is

(6)ASW = 1

2(An + A−n),

the string being propagated from−∞ to ∞ [6]. Bothvector potentials yield the same magnetic monopfield, however, the quantization is different. The Dircondition is 2µ= p, while the Schwinger one isµ =p, p ∈ Z.

These two strings belong to a family{Sκn} ofweighted strings, κ being the weight of the seminfinite Dirac string [12,13]. The respective vectpotential is defined as

(7)Aκn = κAn + (1− κ)A−n,

and the magnetic field of the stringSκn is

(8)hκn = κhn + (1− κ)h−n.

SinceAκ−n = A1−κn , we obtain the following equiva

lence relation:Sκ−n � S1−κn .

Notice that two stringsSκn andSκn′ are related by thegauge transformation

(9)Aκ ′n′ =Aκ

n + dχ

and vice versa. Besides, an arbitrary transformaSκn → Sκ

′n′ can be realized as combination ofSκn →

Sκn′ and Sκn → Sκ′

n , where the first transformationrotation, and the second one results in changing ofweight stringκ → κ ′ without changing its orientation

Let denote byn′ = gn, g ∈ SO(3), the left actionof the rotation group induced bySκn → Sκn′ . Fromrotational symmetry of the theory it follows this gautransformationSκn → Sκn′ can be undone by rotatior → rg as follows

(10)Aκn′(r)= Aκ

n(r′)=Aκ

n(r)+ dα((r;g)),

(11)α(r;g)= e

r′∫r

Aκn(ξ ) · dξ , r′ = rg,

where the integration is performed along the geodrr′ ⊂ S2.

The transformation of the stringSκn → Sκ′

n is givenby

(12)Aκ ′n =Aκ

n − dχn,

(13)dχn = 2q(κ ′ − κ)(r × n) · drr2 − (n · r)2

,

whereχn is polar angle in the plane orthogonal ton.This type of gauge transformations being singularcan be undone by combination of the inversionr →−r andµ→ −µ. In particular, ifκ ′ = 1−κ we obtainthe mirror string:Sκn → Sκ−n � S1−κ

n .

3. Representations of the rotation group andDirac’s quantization condition

Let ψ�ν be an eigenvector of the operatorsJ3 and

J 2:

(14)J3ψλν = νψ�

ν , J 2ψ�ν = �(�+ 1)ψ�

ν ,

ν, � being real numbers. Involving the operatorsJ± =J1 ± J2 it is easy to show that the spectrum ofJ3 hasthe formν = ν0 + n, wheren= 0,±1,±2, . . . .

Each irreducible representation is characterizedan eigenvalue of Casimir operator and the spect

A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 9–13 11

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of the operatorJ3. There are four distinct classesrepresentations [17–19]:

• Representations unbounded from above and be-low, in this case neither�+ ν0 nor �− ν0 can beintegers.

• Representations bounded below, with �+ν0 beingan integer, and�− ν0 not equal to an integer.

• Representations bounded above, with �−ν0 beingan integer, and�+ ν0 not equal to an integer.

• Representations bounded from above and below,with � − ν0 and� + ν0 both being integers, thayields�= k/2, k ∈ Z+.

The nonequivalent representations in each of theries of irreducible representations are denoted,spectively, byD(�, ν0), D+(�), D−(�) andD(k/2).The representationsD(�, ν0), D+(�) andD−(�) areinfinite-dimensional;D(k/2) is (k + 1)-dimensionalrepresentation. The representationsD±(�) andD(�, ν0)

are discussed in detail in [14–18].In fact representationsD(�, ν) andD(−�− 1, ν),

yielding the same valueQ = �(�+ 1) of the Casimiroperator, are equivalent and the inequivalent repretations may be labeled asD(Q,ν) [19]. If there ex-ists the numberp0 ∈ Z such thatν + p0 = �, we haveJ+|�, �〉 = 0 and the representation becomes bounabove. In the similar manner if for a numberp1 ∈Z one hasν + p1 = −�, then J−|�,−�〉 = 0 andthe representation reduces to the bounded belownally, finite-dimensional unitary representation ariswhen there exist possibility of findingJ+|�, �〉 = 0 andJ−|�,−�〉 = 0. It is easy to see that in this case 2�,2mand 2ν all must be integers.

In what follows we will discuss the Dirac monopoproblem within the framework of the representatitheory outlined above.

Taking into account the spherical symmetry of tsystem, the vector potential can be written as [10,1

AN = q(1− cosθ) dϕ,

(15)AS = −q(1+ cosθ) dϕ,

where(r, θ,ϕ) are the spherical coordinates, and whAN has singularity on the south pole of the sphere,AS

on the north one. In the overlap of the neighborhocovering the sphereS2 the potentialsAN andAS are

related by the following gauge transformation:

AS =AN − 2q dϕ.

This is the particular case of transformation givenEq. (12), whenκ = 0 andκ ′ = 1.

We start by choosing the vector potential as

A= q(1− cosθ) dϕ.

Then for the operatorsJi ’s we have

(16)J± = e±iϕ

(± ∂

∂θ+ i cotθ

∂

∂ϕ− µsinθ

1+ cosθ

),

(17)J3 = −i∂

∂ϕ−µ,

J2 = − 1

sinθ

∂

∂θ

(sinθ

∂

∂θ

)− 1

sin2 θ

∂2

∂ϕ2

(18)+ 2iµ

1+ cosθ

∂

∂ϕ+µ2 1− cosθ

1+ cosθ+µ2.

Substituting the wave functionΨ = R(r)Y (θ,ϕ) intoSchrödinger’s equation

(19)HΨ =EΨ,

we have for the angular part the following equation

(20)J2Y (θ,ϕ)= �(�+ 1)Y (θ,ϕ).

Starting withJ3Y =mY and assuming

(21)Y = ei(m+µ)ϕz(m+µ)/2(1− z)(m−µ)/2F(z),

where z = (1 − cosθ)/2, we obtain the resultanequation in the standard form of the hypergeomeequation,

z(1− z)d2F

dz2 + (c− (a + b + 1)z

) dFdz

− abF = 0,

(22)a =m− �, b =m+ �+ 1, c=m+µ+ 1.

The hypergeometric functionF(a, b; c; z) reducesto a polynomial of degreen in z whena or b is equalto −n (n = 0,1,2, . . .) [20,21], and the respectivsolution of Eq. (20) is of the form

(23)Y(δ,γ )n (u)= Cn (1− u)δ/2(1+ u)γ/2P

(δ,γ )n (u),

P(δ,γ )n (u) being the Jacobi polynomials,u = cosθ ,

and the normalization constantC is given by

Cn =(∣∣∣∣2π 2δ+γ+15(n+ δ + 1)5(n+ γ + 1)

5(n+ 1)5(n+ δ + γ + 1)

∣∣∣∣)−1/2

.

12 A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 9–13

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The functionsY (δ,γ )n (u) form the basis of the repre

sentation bounded above or below. This case hasstudied in detail in [12,13].

If both of a and b are negative integers, thatm + � = −p, m + � = −k, p,k ∈ Z+, then therepresentation becomes finite-dimensional. It is eto check that in this casem + µ andm − µ must beintegers, that yields the Dirac quantization condit2µ ∈ Z.

In the rest of the Letter we will discuss threpresentationD(�,µ) unbounded above and beloWe are looking for the solutions of the Eq. (20) suthat being regular at the pointz = 0, in general,can have singularity atz = 1, where the Dirac stringcrosses the sphere. As a result we obtain the followrestrictions on the spectrum of the operatorJ3:

(24)m+µ= n, n= 0,±1,±2, . . . .

The respective solution is given by

Y(µ,n)� = C(�,µ,n)einϕzn/2(1− z)n/2−µF(a, b, c; z),

(25)a = n−µ− �, b = n−µ+ �+ 1, c = 1+ n,

whereC(�,µ,n) is a suitable normalization consta(for the details of the normalization procedure see [17,18]).

Consider now the other choice of the vector pottial

A= −q(1+ cosθ) dϕ,

which corresponds to the Dirac string crossingsphere at north pole (z = 0). In this case the solutioY(µ,n)� of the Eq. (20) being regular at the pointz = 1

takes the same form as in Eq. (25)

Y(µ,n)� = C(�,µ,n)einϕzn/2+µ(1− z)n/2

× F(a, b, c;1− z),

(26)a = n+µ− �, b = n+µ+ �+ 1, c = 1+ n.

The spectrum of operatorJ3 being different from (24)is found to be

(27)m−µ= n, n= 0,±1,±2, . . . .

Notice that the functionsY (µ,n)� can be obtained from

Y(µ,n)� by the change ofz �→ (1 − z) andµ �→ −µ,

that is agree with the gauge transformation

AS =AN − 2q dϕ

(see also Eqs. (12), (13)).The set of the functions{Y (µ,n)

� , Y(µ,n)� } form

the complete bi-orthonormal canonical basis ofrepresentationD(�,µ) in the indefinite-metric Hilberspace with the indefinite metric given by1

(28)ηmm′ = (−1)σ(m)δmm′ ,

where

(−1)σ(m) = sgn(5(�−m+ 1)5(�+m+ 1)

),

sgn(x) being the signum function. One can see tthe spectrum of the operatorJ3 is unbounded, doubledegenerate and discrete.

The general case of an arbitrary weighted stringSκncan be considered in the following way: form±µ= n

the weighted solutions of the Schrödinger equationgiven by

(29)Y(µ,n)κ,� = e−2iκµϕY

(µ,n)� , m= n−µ,

(30)Y(µ,n)κ,� = e−2iκµϕY

(µ,n)� , m= n+µ.

Since a Dirac string may be rotated by gauge transmation the widely accepted point of view is that tstring is unobservable. Thus, to avoid the appearaof an Aharonov–Bohm effect produced by a Dirstring, one has to impose the generalized Dirac qutization condition 2κµ ∈ Z. In particular casesκ = 1andκ = 1/2 it yields the Dirac and Schwinger seletional rules, respectively.

4. Concluding remarks

We have argued, by applying infinite-dimensionrepresentations of the rotation group, that the Diquantization condition can be relaxed and chanby 2κµ ∈ Z, where κ is the weight of the Diracstring. This selectional rule arises as natural condiof being consistent with an algebra of observaband ensures the absence of an Aharonov–Bohm eproduced by Dirac string. Moreover, since there

1 For discussion and details see Refs. [13,16–18].

A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 9–13 13

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no any restriction on the parameterκ , an arbitrarymagnetic charge is allowed.

It follows from our description that the spectruof the operatorJ3 is double-degenerate, discrete aunbounded,m = n ± µ. The physical interpretatioof this result is not clear yet. We believe that it cbe explained treating the charge-monopole systema free anyon with translational and spin degreesfreedom [22].

Acknowledgements

One of the authors, F.A., thanks Center for Theoical Physics of the Massachusetts Institute of Techogy where the part of this work has been done, forwarm hospitality. This work was supported by UdeGrant No. 5025.

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