On representations of the rotation group and magnetic monopoles

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  • Physics Letters A 324 (2004) 913

    On roun

    evolu

    Febr

    . Ho

    Abstr

    Rec nite-have a gene is th nbou 200

    PACS:

    Keywor

    1. In

    Thelectr

    2=whereobtainmechthe wis basthe fothe op

    J = r

    * CoE-

    fermin@

    0375-9doi:10.14.80.Hv; 03.65.-w; 03.50.De; 05.30.Pr; 11.15.-q

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

    troduction

    e Dirac quantization relation [1] between anic charge e and magnetic charge q ,

    (1)n, n Z, = eq , and we set h = c = 1, has been

    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

    (2) (i eA)rr,

    rresponding author.mail addresses: nesterov@cencar.udg.mx (A.I. Nesterov),

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

    has the same properties as a standard angular momen-tum and for any value of obeys the usual commuta-tion relations

    (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]).

    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.

    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

    601/$ see front matter 2004 Elsevier B.V. All rights reserved.1016/j.physleta.2004.02.051representations of the rotation g

    Alexander I. Nesterov , FermDepartamento de Fsica, CUCEI, Universidad de Guadalajara, Av. R

    Received 27 January 2004; received in revised form 18

    Communicated by P.R

    act

    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 and magnetic monopolesAceves de la Cruzcin 1500, Guadalajara, CP 44420, Jalisco, Mexicouary 2004; accepted 23 February 2004

    lland

    dimensional representations of the rotation group weralized Dirac quantization condition, 2 Z, wherended infinite-dimensional representations.

  • 10 A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 913

    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

    2. M

    Aspatiblmono

    Diraccan w

    B = rwhereby

    hn =

    The uSn pa

    Fo

    An =and th

    ASW =the stvectofield,condip, p

    Thweighinfinitpoten

    An =and th

    hn =Sincelence

    Notice that two strings Sn and Sn are related by thegauge transformation n =

    nd vn n antatioeighLe

    f thetatioansf rn(r)

    (r;g

    herer

    Thy n =n =herehis tan ber an

    e m

    . Reirac

    Le2:

    3

    , b1 Je foEa

    n eigsional representations of the rotation group.

    agnetic monopole preliminaries

    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

    ot An + hn,hn is the magnetic field of the Dirac string given

    (4)4qn

    0

    3(r n ) d.

    nit vector n determines the direction of a stringssing from the origin of coordinates to .r instance, Diracs original vector potential reads

    (5)q r nr(r n r) ,

    e Schwingers choice is

    (6)12(An +An),

    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.

    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

    (7)An + (1 )An,e magnetic field of the string Sn is

    (8)hn + (1 )hn.An = A1n , we obtain the following equiva-relation: Sn S1n .

    A

    a

    S

    S

    ro

    w

    o

    ro

    trr

    A

    w

    r

    b

    A

    d

    w

    Tc

    th

    3D

    J

    J

    J

    th

    a(9)An + dice versa. Besides, an arbitrary transformationS

    n can be realized as combination of S

    n

    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

    rotation group induced by Sn Sn . Fromnal symmetry of the theory it follows this gauge

    ormation Sn Sn can be undone by rotationg as follows

    (10)= An(r)=An(r)+ d((r;g)),

    (11))= er

    r

    An( ) d , r = rg,

    the integration is performed along the geodesicS2.e transformation of the string Sn S n is given

    (12)An dn,(13)2q( ) (r n) dr

    r2 (n r)2 ,n is polar angle in the plane orthogonal to n.

    ype of gauge transformations being singular oneundone by combination of the inversion r

    d . In particular, if = 1 we obtainirror string: Sn Sn S1n .

    presentations of the rotation group ands quantization condition

    t be an eigenvector of the operators J3 and

    (14)= , J 2 = (+ 1) ,eing real numbers. Involving the operators J =

    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

  • A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 913 11

    of the operator J3. There are four distinct classes ofrepresentations [1719]:

    Rloin

    Ran

    Ran

    Rw

    y

    The nries ospectiThe rinfinitrepresare di

    Inyieldioperatationists thJ+|,aboveZ on

    the renally,whenJ|,and 2

    Inprobletheory

    Tasystem

    AN =AS =whereAN hon thecover

    related by the following gauge transformation:

    AS =AN 2q d.his iq. (1

    We

    = qhen

    =

    3 =

    2 =

    +ubstichr

    =e ha

    2Y (

    tartin

    = ehere

    quatiquati

    (1a =Tha pn

    lutio(, )n

    (, )n

    nd th

    n =epresentations unbounded from above and be-w, in this case neither + 0 nor 0 can betegers.epresentations bounded below, with +0 being

    integer, and 0 not equal to an integer.epresentations bounded above, with 0 being

    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+.

    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.

    king into account the spherical symmetry of the, the vector potential can be written as [10,11]

    q(1 cos) d,(15)q(1+ cos) d,

    (r, ,) are the spherical coordinates, and whileas singularity on the south pole of the sphere, AS

    north one. In the overlap of the neighborhoodsing the sphere S2 the potentials AN and AS are

    TE

    A

    T

    J

    J

    J

    SS

    H

    w

    JS

    Y

    w

    e

    e

    z

    totoso

    Y

    P

    a

    Cs the particular case of transformation given by2), when = 0 and = 1.start by choosing the vector potential as

    (1 cos) d.for the operators Ji s we have

    (16)ei(

    + i cot

    sin 1+ cos

    ),

    (17)i

    ,1

    sin

    (sin

    ) 1

    sin2 2

    2

    (18)2i1+ cos

    +2 1 cos

    1+ cos +2.

    tuting the wave function = R(r)Y (,) intodingers equation

    (19)E,ve for the angular part the following equation:

    (20), )= (+ 1)Y (,).g with J3Y =mY and assuming

    (21)i(m+)z(m+)/2(1 z)(m)/2F(z),z = (1 cos)/2, we obtain the resultant

    on in the standard form of the hypergeometricon,

    z)d2F

    dz2+ (c (a + b+ 1)z) dF

    dz abF = 0,

    (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

    n of Eq. (20) is of the form(23)(u)= Cn (1 u)/2(1+ u)/2P (, )n (u),

    (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)

    )1/2

    .

  • 12 A.I. Nesterov, F. Aceves de la Cruz / Physics Letters A 324 (2004) 913

    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].

    Ifm +represto cheintege2