On representations of the rotation group and magnetic monopoles

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<ul><li><p>Physics Letters A 324 (2004) 913</p><p>On roun</p><p>evolu</p><p>Febr</p><p>. Ho</p><p>Abstr</p><p>Rec nite-have a gene is th nbou 200</p><p>PACS:</p><p>Keywor</p><p>1. In</p><p>Thelectr</p><p>2=whereobtainmechthe wis basthe fothe op</p><p>J = r</p><p>* CoE-</p><p>fermin@</p><p>0375-9doi:10.14.80.Hv; 03.65.-w; 03.50.De; 05.30.Pr; 11.15.-q</p><p>ds: Monopole; Indefinite-metric Hilbert space; Nonassociativity; Infinite-dimensional representation</p><p>troduction</p><p>e Dirac quantization relation [1] between anic charge e and magnetic charge q ,</p><p>(1)n, n Z, = eq , and we set h = c = 1, has been</p><p>ed from various approaches based on quantumanics and quantum field theory [111]. One ofidely accepted proofs of the Dirac selection ruleed on group representation theory and consists inllowing: in the presence of magnetic monopoleerator of the total angular momentum</p><p>(2) (i eA)rr,</p><p>rresponding author.mail addresses: nesterov@cencar.udg.mx (A.I. Nesterov),</p><p>udgphys.intranets.com (F. Aceves de la Cruz).</p><p>has the same properties as a standard angular momen-tum and for any value of obeys the usual commuta-tion relations</p><p>(3)[Ji, Jj ] = iijkJk.The requirement that Ji generate a finite-dimensionalrepresentation of the rotation group yields 2 beinginteger and only values 2 = 0,1,2, . . . are al-lowed (for details see, for example, [3,5,79]).</p><p>Actually the charge quantization does not followfrom the quantum-mechanical consideration and ro-tation invariance alone. Any treatment uses some ad-ditional assumptions that may be not physically in-evitable.</p><p>Recently we have exploited this problem employ-ing bounded infinite-dimensional representations ofthe rotation group and nonassociative gauge transfor-mations. We argued that one can relax Diracs condi-tion and obtain the consistent monopole theory with</p><p>601/$ see front matter 2004 Elsevier B.V. All rights reserved.1016/j.physleta.2004.02.051representations of the rotation g</p><p>Alexander I. Nesterov , FermDepartamento de Fsica, CUCEI, Universidad de Guadalajara, Av. R</p><p>Received 27 January 2004; received in revised form 18</p><p>Communicated by P.R</p><p>act</p><p>ently [Phys. Lett. A 302 (2002) 253] employing bounded infirgued that one can obtain the consistent monopole theory withe weight of the Dirac string. Here we extend this proof to the u4 Elsevier B.V. All rights reserved.www.elsevier.com/locate/pla</p><p>p and magnetic monopolesAceves de la Cruzcin 1500, Guadalajara, CP 44420, Jalisco, Mexicouary 2004; accepted 23 February 2004</p><p>lland</p><p>dimensional representations of the rotation group weralized Dirac quantization condition, 2 Z, wherended infinite-dimensional representations.</p></li><li><p>10 A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 913</p><p>the generalized quantization condition, 2 Z, being the weight of the Dirac string [12,13]. In ourLetter we extend this proof to the unbounded infinite-dimen</p><p>2. M</p><p>Aspatiblmono</p><p>Diraccan w</p><p>B = rwhereby</p><p>hn =</p><p>The uSn pa</p><p>Fo</p><p>An =and th</p><p>ASW =the stvectofield,condip, p </p><p>Thweighinfinitpoten</p><p>An =and th</p><p>hn =Sincelence</p><p>Notice that two strings Sn and Sn are related by thegauge transformation n =</p><p>nd vn n antatioeighLe</p><p>f thetatioansf rn(r)</p><p>(r;g</p><p>herer </p><p>Thy n =n =herehis tan ber an</p><p>e m</p><p>. Reirac</p><p>Le2:</p><p>3</p><p>, b1 Je foEa</p><p>n eigsional representations of the rotation group.</p><p>agnetic monopole preliminaries</p><p>well known any vector potential A being com-e with a magnetic field B = qr/r3 of Diracpole must be singular on the string (the so-calledstring, further it will be denoted as Sn), and onerite</p><p>ot An + hn,hn is the magnetic field of the Dirac string given</p><p>(4)4qn</p><p>0</p><p>3(r n ) d.</p><p>nit vector n determines the direction of a stringssing from the origin of coordinates to .r instance, Diracs original vector potential reads</p><p>(5)q r nr(r n r) ,</p><p>e Schwingers choice is</p><p>(6)12(An +An),</p><p>ring being propagated from to [6]. Bothr potentials yield the same magnetic monopolehowever, the quantization is different. The Diraction is 2= p, while the Schwinger one is =Z.</p><p>ese two strings belong to a family {Sn} ofted strings, being the weight of the semi-e Dirac string [12,13]. The respective vectortial is defined as</p><p>(7)An + (1 )An,e magnetic field of the string Sn is</p><p>(8)hn + (1 )hn.An = A1n , we obtain the following equiva-relation: Sn S1n .</p><p>A</p><p>a</p><p>S</p><p>S</p><p>ro</p><p>w</p><p>o</p><p>ro</p><p>trr</p><p>A</p><p>w</p><p>r</p><p>b</p><p>A</p><p>d</p><p>w</p><p>Tc</p><p>th</p><p>3D</p><p>J</p><p>J</p><p>J</p><p>th</p><p>a(9)An + dice versa. Besides, an arbitrary transformationS</p><p>n can be realized as combination of S</p><p>n </p><p>d Sn S n , where the first transformation isn, and the second one results in changing of thet string without changing its orientation.t denote by n = gn, g SO(3), the left action</p><p>rotation group induced by Sn Sn . Fromnal symmetry of the theory it follows this gauge</p><p>ormation Sn Sn can be undone by rotationg as follows</p><p>(10)= An(r)=An(r)+ d((r;g)),</p><p>(11))= er</p><p>r</p><p>An( ) d , r = rg,</p><p>the integration is performed along the geodesicS2.e transformation of the string Sn S n is given</p><p>(12)An dn,(13)2q( ) (r n) dr</p><p>r2 (n r)2 ,n is polar angle in the plane orthogonal to n.</p><p>ype of gauge transformations being singular oneundone by combination of the inversion r </p><p>d . In particular, if = 1 we obtainirror string: Sn Sn S1n .</p><p>presentations of the rotation group ands quantization condition</p><p>t be an eigenvector of the operators J3 and</p><p>(14)= , J 2 = (+ 1) ,eing real numbers. Involving the operators J =</p><p>2 it is easy to show that the spectrum of J3 hasrm = 0 + n, where n= 0,1,2, . . . .ch irreducible representation is characterized byenvalue of Casimir operator and the spectrum</p></li><li><p>A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 913 11</p><p>of the operator J3. There are four distinct classes ofrepresentations [1719]:</p><p> Rloin</p><p> Ran</p><p> Ran</p><p> Rw</p><p>y</p><p>The nries ospectiThe rinfinitrepresare di</p><p>Inyieldioperatationists thJ+|,aboveZ on</p><p>the renally,whenJ|,and 2</p><p>Inprobletheory</p><p>Tasystem</p><p>AN =AS =whereAN hon thecover</p><p>related by the following gauge transformation:</p><p>AS =AN 2q d.his iq. (1</p><p>We</p><p>= qhen</p><p> =</p><p>3 =</p><p>2 =</p><p>+ubstichr</p><p> =e ha</p><p>2Y (</p><p>tartin</p><p>= ehere</p><p>quatiquati</p><p>(1a =Tha pn</p><p>lutio(, )n</p><p>(, )n</p><p>nd th</p><p>n =epresentations unbounded from above and be-w, in this case neither + 0 nor 0 can betegers.epresentations bounded below, with +0 being</p><p>integer, and 0 not equal to an integer.epresentations bounded above, with 0 being</p><p>integer, and + 0 not equal to an integer.epresentations bounded from above and below,ith 0 and + 0 both being integers, thatields = k/2, k Z+.</p><p>onequivalent representations in each of the se-f irreducible representations are denoted, re-vely, by D(, 0), D+(), D() and D(k/2).epresentations D(, 0), D+() and D() aree-dimensional; D(k/2) is (k + 1)-dimensionalentation. The representationsD() andD(, 0)scussed in detail in [1418].fact representations D(, ) and D( 1, ),ng the same value Q = (+ 1) of the Casimirtor, are equivalent and the inequivalent represen-s may be labeled as D(Q,) [19]. If there ex-e number p0 Z such that + p0 = , we have = 0 and the representation becomes bounded. In the similar manner if for a number p1 e has + p1 = , then J|, = 0 andpresentation reduces to the bounded below. Fi-finite-dimensional unitary representation arisesthere exist possibility of finding J+|, = 0 and = 0. It is easy to see that in this case 2,2m all must be integers.what follows we will discuss the Dirac monopolem within the framework of the representationoutlined above.</p><p>king into account the spherical symmetry of the, the vector potential can be written as [10,11]</p><p>q(1 cos) d,(15)q(1+ cos) d,</p><p>(r, ,) are the spherical coordinates, and whileas singularity on the south pole of the sphere, AS</p><p>north one. In the overlap of the neighborhoodsing the sphere S2 the potentials AN and AS are</p><p>TE</p><p>A</p><p>T</p><p>J</p><p>J</p><p>J</p><p>SS</p><p>H</p><p>w</p><p>JS</p><p>Y</p><p>w</p><p>e</p><p>e</p><p>z</p><p>totoso</p><p>Y</p><p>P</p><p>a</p><p>Cs the particular case of transformation given by2), when = 0 and = 1.start by choosing the vector potential as</p><p>(1 cos) d.for the operators Ji s we have</p><p>(16)ei( </p><p>+ i cot </p><p> sin 1+ cos</p><p>),</p><p>(17)i </p><p>,1</p><p>sin </p><p>(sin </p><p>) 1</p><p>sin2 2</p><p>2</p><p>(18)2i1+ cos</p><p>+2 1 cos </p><p>1+ cos +2.</p><p>tuting the wave function = R(r)Y (,) intodingers equation</p><p>(19)E,ve for the angular part the following equation:</p><p>(20), )= (+ 1)Y (,).g with J3Y =mY and assuming</p><p>(21)i(m+)z(m+)/2(1 z)(m)/2F(z),z = (1 cos)/2, we obtain the resultant</p><p>on in the standard form of the hypergeometricon,</p><p>z)d2F</p><p>dz2+ (c (a + b+ 1)z) dF</p><p>dz abF = 0,</p><p>(22)m , b =m+ + 1, c=m++ 1.e hypergeometric function F(a, b; c; z) reducesolynomial of degree n in z when a or b is equal(n = 0,1,2, . . .) [20,21], and the respective</p><p>n of Eq. (20) is of the form(23)(u)= Cn (1 u)/2(1+ u)/2P (, )n (u),</p><p>(u) being the Jacobi polynomials, u = cos ,e normalization constant C is given by(2 2++15(n+ + 1)5(n+ + 1)5(n+ 1)5(n+ + + 1)</p><p>)1/2</p><p>.</p></li><li><p>12 A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 913</p><p>The functions Y (, )n (u) form the basis of the repre-sentation bounded above or below. This case has beenstudied in detail in [12,13].</p><p>Ifm +represto cheintege2 </p><p>InrepresWe arthat bcan hcrosse</p><p>restric</p><p>m+The re</p><p>Y(,n</p><p>a =</p><p>where(for th17,18</p><p>Cotial</p><p>A=whichspherY(,n</p><p>takes</p><p>Y(,n</p><p>a =</p><p>The sis fou</p><p>mNoticY(,n</p><p>that is agree with the gauge transformation</p><p>AS =AN 2q dee a</p><p>The co</p><p>presace</p><p>mm =here</p><p>1)</p><p>n(x</p><p>e spegen</p><p>Than bee we</p><p>iven(,n),</p><p>(,n),</p><p>inceatioringf anringzationd onal</p><p>. Co</p><p>Wepres</p><p>uantiy 2ringf bend enrodu</p><p>1 Foboth of a and b are negative integers, that is = p, m + = k, p,k Z+, then theentation becomes finite-dimensional. It is easyck that in this case m+ and m must bers, that yields the Dirac quantization conditionZ.</p><p>the rest of the Letter we will discuss theentation D(,) unbounded above and below.e looking for the solutions of the Eq. (20) sucheing regular at the point z = 0, in general,</p><p>ave singularity at z = 1, where the Dirac strings the sphere. As a result we obtain the followingtions on the spectrum of the operator J3:</p><p>(24)= n, n= 0,1,2, . . . .spective solution is given by</p><p>) = C(,,n)einzn/2(1 z)n/2F(a, b, c; z),</p><p>(25)n , b= n+ + 1, c= 1+ n,</p><p>C(,,n) is a suitable normalization constante details of the normalization procedure see [13,]).nsider now the other choice of the vector poten-</p><p>q(1+ cos) d,corresponds to the Dirac string crossing the</p><p>e at north pole (z = 0). In this case the solution)</p><p>of the Eq. (20) being regular at the point z= 1the same form as in Eq. (25)) = C(,,n)einzn/2+(1 z)n/2</p><p> F(a, b, c;1 z),</p><p>(26)n+ , b= n++ + 1, c= 1+ n.</p><p>pectrum of operator J3 being different from (24)nd to be</p><p>(27)= n, n= 0,1,2, . . . .e that the functions Y (,n) can be obtained from) by the change of z (1 z) and ,</p><p>(s</p><p>thre</p><p>sp</p><p>w</p><p>(sgthd</p><p>c</p><p>thg</p><p>Y</p><p>Y</p><p>Sm</p><p>sto</p><p>sttia</p><p>ti</p><p>4</p><p>re</p><p>qbsto</p><p>a</p><p>plso Eqs. (12), (13)).e set of the functions {Y (,n) , Y (,n) } formmplete bi-orthonormal canonical basis of theentation D(,) in the indefinite-metric Hilbertwith the indefinite metric given by1</p><p>(28)(1)(m)mm ,</p><p>(m) = sgn(5(m+ 1)5(+m+ 1)),) being the signum function. One can see thatectrum of the operator J3 is unbounded, double-erate and discrete.e general case of an arbitrary weighted string Snconsidered in the following way: for m= nighted solutions of the Schrdinger equation areby</p><p>(29)= e2iY (,n) , m= n,(30)= e2iY (,n) , m= n+.</p><p>a Dirac string may be rotated by gauge transfor-n the widely accepted point of view is that theis unobservable. Thus, to avoid the appearanceAharonovBohm effect produced by a Dirac</p><p>, one has to impose the generalized Dirac quan-n condition 2 Z. In particular cases = 1= 1/2 it yields the Dirac and Schwinger selec-rules, respectively.</p><p>ncluding remarks</p><p>have argued, by applying infinite-dimensionalentations of the rotation group, that the Diraczation condition can be relaxed and changed Z, where is the weight of the Dirac</p><p>. This selectional rule arises as natural conditioning consistent with an algebra of observablessures the absence of an AharonovBohm effect</p><p>ced by Dirac string. Moreover, since there is</p><p>r discussion and details see Refs. [13,1618].</p></li><li><p>A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 913 13</p><p>no any restriction on the parameter , an arbitrarymagnetic charge is allowed.</p><p>It follows from our description that the spectrumof theunbouof thibe exa freefreedo</p><p>Ackn</p><p>Onical Pogy wwarm</p><p>Grant</p><p>Refer</p><p>[1] P.[2] J.</p><p>[3] M. Fierz, Helv. Phys. Acta 17 (1944) 27.[4] A.S. Goldhaber, Phys. Rev. B 140 (1965) 1407.[5] A. Peres, Phys. Rev. 167 (1968) 1443.6] J.7] A.8] H.</p><p>209] D.0] T.1] T.2] A.</p><p>253] A.</p><p>pu4] M5] E.6] S.7] S.8] S.9] B.</p><p>Yo0] M</p><p>Fu1] G.</p><p>br2] A.</p><p>Fioperator J3 is double-degenerate, discrete andnded, m = n . The physical interpretations result is not clear yet. We believe that it canplained treating the charge-monopole system as</p><p>anyon with translational and spin degrees ofm [22].</p><p>owledgements</p><p>e of the authors, F.A., thanks Center for Theoret-hysics of the Massachusetts Institute of Technol-here the part of this work has been done, for thehospitality. This work was supported by UdeG,No. 5025.</p><p>ences</p><p>A.M. Dirac, Proc. R. Soc. A 133 (1931) 60.Tamm, Z. Phys. 71 (1931) 141.</p><p>[[[</p><p>[[1[1[1</p><p>[1</p><p>[1[1[1[1[1[1</p><p>[2</p><p>[2</p><p>[2Schwinger, Phys. Rev. 144 (1966) 1087.Hurst, Ann. Phys. 50 (1968) 51.</p><p>J. Lipkin, W.I. Weisberg, M. Peskin, Ann. Phys. 53 (1969)3.Zwanziger, Phys. Rev. D 3 (1971) 880.</p><p>T. Wu, C.N. Yang, Phys. Rev. D 12 (1975) 3845.T. Wu, C.N. Yang, Nucl. Phys. B 107 (1976) 365.I. Nesterov, F. Aceves de la Cruz, Phys. Lett. A 302 (2002)3.I. Nesterov, F. Aceves de la Cruz, JHEP, submitted forblication.. Andrews, J. Gunson, J. Math. Phys. 5 (1964) 1391.G. Beltrami, G. Luzatto, Nuovo Cimento 29 (1963) 1003.S. Sannikov, Sov. J. Nucl. Phys. 3 (1966) 407.S. Sannikov, Sov. J. Nucl. Phys. 6 (1968) 788.S. Sannikov, Sov. J. Nucl. Phys. 6 (1968) 939.G. Wybourne, Classical Groups for Physicists, Wiley, Newrk, 1974.. Abramowitz, I.A. Stegun, Handbook of Mathematicalnctions, Dover, New York, 1965.E. Andrews, R. Askey, R. Roy, Special Functions, Cam-idge Univ. Press, New York, 1999.I. Nesterov, F. Aceves de la Cruz, Rev. Mexicanas. 49 (Suppl. 2) (2003) 134, hep-th/0209007.</p><p>On representations of the rotation group and magnetic monopolesIntroductionMagnetic monopole preliminariesRepresentations of the rotation group and Dirac's quantization conditionConcluding remarksAcknowledgementsReferences</p></li></ul>