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Invariance Properties of the Dirac Monopole

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Invariance Properties of the Dirac Monopole fcs.- 0; /A }Ю-ц-^. KFKI-75-82 A. FRENKEL P. HRASKÓ INVARIANCE PROPERTIES OF THE DIRAC MONOPOLE Hungarian academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST L KFKI-75-S2 INVARIÄHCE PROPERTIES OF THE DIRAC MONOPOLE A. Frenkel and P. Hraskó High Energy Physics Department Central Research Institute for Physics, Budapest, Hungary ISBN 963 371 094 4 ABSTRACT The quantum mechanical motion of a spinless electron in the external field of a magnetic monopolé of magnetic charge v is investigated. Xt is shown that Dirac's quantum condition 2 ue(hc)'1 • n for the string being unobservable ensures rotation invarlance and correct space reflection proper­ ties for any integer value of n. /he rotation and space reflection operators are found and their group theoretical properties are discussed, A method for constructing conserved quantities In the case when the potential ie not explicitly invariant under the symmetry operation is also presented and applied to the discussion of the angular momentum of the electron-monopole system. АННОТАЦИЯ Рассматривается кваитоаомеханическое движение бесспинового элект­ рона во внешнем поле монополя с Магнитки« зарядом и. Доказывается» что кван­ товое условие Дирака 2ув(пс)-1 •> п обеспечивает не только немаолтваемость стру­ ны »но и инвариантность при вращении и правильные свойства при пространственных отражениях для любого целого п. Даются операторы времени* я отражения, и об­ суждаются их групповые свойства. Указан также метод построения интегралов движения в случае, когда потенциал не является язно инвариантным по отношению операции симметрии, и этот метод применяется при изучении углового момента сис­ темы электрон-монопопь. KIVONAT A spin nélküli elektron kvantummechanikai viselkedését vizsgáljuk u mágneses töltésű monopolus kttlsö terében. Megmutatjuk, hogy Dirac 2pe(tic)"l=n kvantumfeltétele a szál megfigyelhotetlenségével együtt a forgásinvarlanciát és a helyes tértükrözési tulajdonságokat is biztosítja bármely egész n érték­ re. Megadjuk a forgás és a tértükrözés operátorát és diszkutáljuk csoportel­ mélet! tulajdonságaikat. Bemutatunk továbbá egy módszert megmaradó mennyiségek megkeresésére olyan esetben, mikor a potenciál nem explicite invariáns a Vizs­ gált szimmetriatranszformációval szemben és e módszert alkalmazzuk az elektron- -monopol rendszer impulzusmomentumának vizsgálatakor. tm ЭД1г*^ШЗ¥!**г^;^ I. INTRODUCTION It has been stated by Dirac in his classical paper M on the mag­ netic monopolé that for any integer value n of the dimensionless quantity v в 2 MS Tic the singular line(hereafter the "string") of the vector potential A of the monopolé is unobservable. However the proof was not made explicit in [l] , and doubts have been expressed about the correctness of the statement. Schwinger [2] argued that the requirement of space reflection symmetry allows cnly for even values of n. Later Peres [з] came to the conclusion that rota­ tion invariance implies the same restriction, and recently it has been asked [4] whether the singular character of the gauge transformation connected with a displacement of the string does not invalidate Dirac's statement for all values of n. In the present note it is shown that Dirac's condition is sufficient (and, of course, necessary) for the string being unobservable, and that both rotation invariance and space reflection symmetry are guaranteed by this con­ dition. 'On the precise meaning of the expression "space reflection symmetry" in the present case see the beginning of part IV.) In part II the proof is given that an arbitrary change in the posi­ tion of the string amounts to a gauge transformation for any value of n. In particular, it is pointed out that some of the wave functions of the elec­ tron have a singularity on the string to be called "string-type singularity". For example the energy elgenfunction S.-ein» e ф written out in [Í] has such a singularity on the string, the latter being directed along the negative Z axis. It is then shown that under the transformation induced by a displacement of the string the string-type singularity follows the string. This is due precisely to the aforementioned singular character of the transformation along the old and the new strings. The conclusion that this transformation has no observable effects, i.e. that it is a bona fide gauge transformation easily follows. *•% j - 2 - The string being unobservable the symmetries of the Magnetic field В of the Monopole should be preserved, and rotation invariance and space reflection symmetry should hold for all values of n. Rotation lnvariance is discussed in part III. The rotation operator T representing a rotation g v is found. It coMMutes with the Haniltonlan and satisfies the coMposition law T9a T9i * T9a9i." Bowwr' lt dlffere £гои the »aual rotation operator P_ (R_"•<£)-•(9* £)) in an r-dependent phase factor, the role of which is clar­ ified. Particularly, it is shown that this phase factor Makes possible the odd values of n and accounts for the unusual feature that for these values of n half-integer j represen ttions of the SO(2) group occur in the case of the spinless electron. In part IV space reflection symmetry is considered. Unlike in the case of rotation invariance, it is possible to Maintain the usual parity operator P, but one then has to worK with two strings thereby constraining n to even values Qs]. However, if one introduces a new parity operator P which differes fro* p in an appropriately chosen r-dependent phase factor, one sees that all values of n are compatible with space reflection synMetry. The Monopole has a centrally symmetric Magnetic field and three quan­ tities of the type of angular momentum are known to be conserved during the I I Motion of the electron in the field of the Monopole.However, the potential and the action do not have that high degree of symmetry as does the Magnetic field and it is not straightforward to connect the conservation laws with the sym­ metry of the field through Noether's theorem. Part V is devoted to the gen­ eral discussion of this problem. II. GAUGE INVARIANCE 1. The vector potential of the monopolé It has been stressed by Dirac [l] that if one wishes to describe the magnetic field r H - M=» (1) ~ r of a monopolé of magnetic charge и (located at the origin o) with the help of a vector potential A(r) then the relation H - rot A (2) must be violated at least along one line going from the monopoie to infinity. This line, which may be curvilinear and not even planar, will be called the string S oi the potential. S**E-""^Щит/ШЮ**** э - For simplicity the discussion will be restricted in this not* to straight strings. The conclusions, however, remain valid for curvilinear strings too (see Appendix 1). The potential A*-* belonging to the straight string S - where n denotes the direction vector of the string can be written in the form On the string !* - I" " and (1) is violated! everywhere else it holds. The potential used in [l] belongs to the string S - lying on the negative Z axis. In spherical (г, в, <р) and cylindrical (p,4>, z) coordinates (Fig. l) only the Ф-component of A - is different from zero. It reads *!*> £ tg вf - J (1-COS6). (4) For later use notice that in the neigh­ bourhood of «Ф А (p % 0), Ф ^ P i.e. W) % 2y grad<p ; (p % 0). (5) Let us now consider the quantum mechanical motion of a spinless particle with electric charge e and mass M - the "electron" in what follows - in the field of force represented by the potential A — . The riamiltonian is then Suliig H<á> 5*M (-in* - !^)*. Unlike the magnetic field H in (1), A*-* and H^-) depend on the orientation of the string; in particular, they are singular (infinite) along the string. The proof that the string is nevertheless unobservable wher Dirac's condition v - n holds will be carried out in two steps. First, it will be pointed out that when van some of the well known elgenfunctions of 8'-' [JL,5,6,7,в] are singular on the string. Let us recall here that it has been stressed long ago by Neumann M that the wave functions may turn out to be singular at those places where the Hamiltonian is singular. The probability distributions of the observables must, however, be uniquely determined and the spatial proba- ! •^ЩШ^Ш^^' т"^^7!^т^т^тщт^т^ ^€Щт#^т:'*^щ^т^<'. i^>*m^«r^'W№>&>mmmewn 4 - bility distributions **»st be continuous everywhere including the singular places. It will be shown that the string-type singularity is of such an al­ lowed character. The second step of the proof consists in showing that when the po­ sition of the string S -' is changed and therefore the singular line in the Hariltonian is a.lro changed, the string-type singularity follows the string, i.e. it remains an allowed singularity. This is possible because the operator of the transformation from the old wave functions to the new ones has, itself,an appropriate singularity both on the old and the new strings. The rest of the proof reduces then to the usual gauge invariance argument which will not be repeated here. 2. String-type singularity The titie-independent Schrödinger equation e^V^w В v^ <*> written in spherical coordinates, separates for any real value of if one. puts 2 2 *g- (r) - RA(kr)U*<q»,0) ; ti k -2BM. The angular eigenfunctions satisfy the equation lT(y>,0) = ли„(ч>,0) 1' m m m=0,+l,+2,, where [б] , [7] -,2 - 1 -L( n0 A-) 1 Э si -j±-± + MW^). (6) 2 ilnO Э0 S0- siire 3<p 1+COS0 Эф 4 2 It should be stressed that J is defined here as the angular part of h -' for any real value of v and the notation merely anticipates but does not Imply that for v - n rotation invariance holds and that then J turns out to be the Casimir operator of the rotation group.
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