The Dirac Monopole and Induced Representations *
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1 The Drac Monopole and Induced Representatons * In ths note a mathematcally transparent treatment of the Drac monopole s gven from the pont of vew of nduced representatons Among other thngs the queston of bound states for the spnnng electron n the feld of a magnetc monopole s consdered In 948, as he was turnng from theoretcal physcs to mathematcs, Harsh-Chandra wrote one last paper [5] on a physcal topc, nvestgatng a queston nspred by Drac, and perhaps even proposed by hm Does an electron movng accordng to Drac s equaton n the feld of a magnetc monopole have a bound state? The technque nvolved, namely, separaton of varables, s of course elementary, and would reappear but at a much deeper level repeatedly n Harsh-Chandra s later work on harmonc analyss on semsmple groups However, ths tme, hs mnd on other matters, he handled t perfunctorly, and went astray wth the calculatons, concludng ncorrectly that there could be no bound states The matter has snce been dealt wth correctly [4], [6], [8] However, the dervaton of the radal equatons n [6] and [8] s not so effcent as t mght be In vew of the possble hstorcal nterest to students of Harsh-Chandra s later work, a bref, mathematcally transparent treatment does not seem out of place It s the purpose of ths note, whch does not touch on any questons of serous current nterest, to provde t In [2] Drac consders the wave functon ψ of a charged partcle wthout spn, observng that one can replace ψ by e β ψ wthout changng the dstrbuton ψ 2, whch s what counts The functon β depends on the coordnates g,x,x 2,x The substtuton replaces and but, snce do not appear alone but n the combnatons t by x j by t and t + A, * Appeared n Pac Jour Math, vol t + β t x j + β x j, x j x j + A j,
2 The Drac monoploe 2 ths effect can be countered by a change n the electromagnetc potentals A,A,A 2,A The upshot s that ψ can be regarded as a secton of a lne bundle wth metrc and wth a connecton defned by the electromagnetc potentals The pertnent lne bundle s on M, four-dmensonal space wth the lne x x 2 x removed Ths space s to be dentfed wth R R + H\G, where G s SU2 and H s the group of dagonal matrces As usual G s mapped to SO and thus acts on -space x,x 2,x y,y 2,y x,x 2,x Ag n such a way that the Paul matrces σ,σ 2,σ multpled by /2 correspond to nfntesmal rotatons through the x,x 2, and x -axes Thus y y y 2 y + y 2 y f x x x 2 x + x 2 x g Then r, g rag dentfes R + H\G wth -space mnus the orgn The lne bundle s defned by a one-dmensonal representaton ρ : e θ e θ e nθ of H, sectons beng complex functons f on R R + G satsfyng ft, r, hg ρhft, r, g The connecton, lke the bundle, s a product trval on the frst two factors Let H be the orthogonal complement of H n G and regard functons on H\G as H-nvarant functons on G We prescrbe that for X H the tangent vector X g at Hg defned by X g f d dt fexp tx g t acts on sectons by the same formula, and verfy that ths yelds a well-defned connecton The curvature of ths connecton can be computed on H\G and, snce [H, H ] H, s a twoform that takes the value 2 ρ[x,x 2 ] at X g X2 g Takng X σ /2,X 2 σ 2 /2 and recallng that when dvded by /2 the curvature on M yelds a two-form that gves the electrc and magnetc felds, we see that the assocated electrc feld s and the magnetc feld purely radal and equal to n 2r x,x 2,x
3 The Drac monoploe Observe that all constructons are nvarant under the acton of G on H\G, whch of course yelds the usual acton of G on M Consequently the magnetc feld s sphercally symmetrc and need only be evaluated at,, r Notce also that at a pont r, g where g, X r x 2,X2 r x However, the Drac equaton, wth whch [5] was concerned, s for electrons wth spn To ntroduce t we tensor the bundle wth the four-dmensonal trval bundle on M Ifσ s the representaton of G to tself, then the equatons are nvarant under the transformaton F F wth σgf t, xag F t, x So we replace F by f wth 2 ft, r, g σgf t, r,, Ag to obtan equatons nvarant under rght translatons In terms of f the Drac operator may be wrtten γ γ + γ + t r r X2 γ2 r X + r γ m The notaton s that of [] Ths need only be verfed at t, r, Usng equatons we see that the Drac operator may be wrtten as γ σg γ + t r X2 γ2 r X + γ σg m, r g beng set equal to after the dfferentaton Thus t s γ γ + γ + t r r X γ2 r X γ r σx2 + γ2 r σx m Snce and the equaton follows γ,γ j σ j σ j σx j 2 σ j σ j, j, 2,
4 The Drac monoploe 4 4 The egenfuncton equaton of the assocated Hamltonan s { σ σ r + σ X 2 σ r σ 2 X σ 2 r + } σ r σ f E m f E + m It cannot be nterpreted untl the formal expresson on the left has been completely defned as an operator H that wll be an extenson of the closure H of the obvously defned operator on functons wth compact support on R + G Snce H wll be seen not to be self-adjont f n, there wll be some freedom n the choce of H To analyze H and ts self-adjont extensons we can clearly consder the projectons onto the space of functons f transformng accordng to a gven representaton of G Take the representaton of dmenson d +actng on the polynomals of degree d n two varables wth orthonormal bass e j,k d j /2 x j y k,j + k d The entres of the column vector f wll then be matrx coeffcents of the representaton and we can take them all from one column, the several columns entalng a multplcty As a consequence of 2 the frst and the thrd entres wll be multples f /r and f /r of the matrx coeffcent on the row wth j k n +, and the second and fourth entres wll be multples f 2 /r and f 4 /r of the coeffcent n the row wth j k n The denomnator smplfes the nner product R + G f 2 r 2 drdg 4 f r 2 dr and, when we rewrte 4 n terms of the f, removes the last term on the left We have σ X 2 σ 2 X X 2 + X X 2 X and, n the Le algebra, X 2 + X 2 σ2 + σ X 2 X 2 σ2 σ, Moreover : e j,k jk + /2 e j,k+, : e j,k j +k /2 e j+,k
5 The Drac monoploe 5 So the left sde of 4 becomes 5 µ f r + µ f 2 r µ f µ f 4 and the rght sde 6 E m E m E + m E + m f f 2 f f 4 Here µ j +k /2 d + 2 n 2 /2,j+ k d, j k n In the specal case d + n, the coeffcents f 2 and f 4 are fcttous, µ, and 5 becomes 7 r f f If d +n then 5 becomes 8 r f2 f 4 Takng f f 2 f f 4 f ɛf g ɛg wth ɛ ±, we decouple the system gven by 5 and 6 nto the systems 9 { r + ɛµ r In the two exceptonal cases we have the system ± } f E m g E + m f E m f r g E + m g f g The operators appearng on the left of 9 and have stll to be defned They are to be selfadjont extensons of the closure G of the obvously defned operators on smooth functons of compact support on, We apply the theory of [], XIII 2, whch obvously extends to systems Denote the formal dfferental operator on the left of 9 or by τ
6 The Drac monoploe 6 of For equaton 9 the roots of the ndcal equaton of τ at are ±µ Snce µ, only one soluton τ f f ± g g s square-ntegrable on, ] and there are no boundary condtons [], XII42, XIII29, XIII22 For equaton the ndcal equaton has the multple root at So there are two ndependent boundary values For both equatons there s one square-ntegrable soluton of on [, and no boundary condton at We conclude that for 9 the operator T s already self-adjont It follows from Lemma XIII42 of [] and ntegraton by parts that for T F,F +F,T F fḡ + g f, wth F f,g t,f f,g t So the self-adjont extensons T of T are defned by f λg,λ R, org In all cases the dscrete spectrum of T + m m m m s obtaned by explctly solvng 9 or For a square ntegrable soluton E must be real For there s a square-ntegrable soluton of the equaton only f m 2 >E 2 and t s f e r m 2 E 2,g m E m + E e r m2 E 2 It satsfes the boundary condtons f and only f m + E/m E λ So there s a bound state f <λ< and none otherwse For 9 there could be a soluton square-ntegrable at only f m 2 >E 2 and then t would have to be a multple of the par wth f equal to the Whttaker functon W,ν αr [7] where α 2 m 2 E,ν µ 2,ɛ>,ν µ + 2,ɛ< Ths par cannot be square-ntegrable near We can sum the dscusson up wth a theorem cf [4]
7 The Drac monoploe 7 Theorem If n the operator H s self-adjont If n > then there s a -parameter famly of sphercally symmetrc self-adjont extensons H of H, parametrzed by λ R { } and no others The operator H + m m m m has a dscrete spectrum f and only f λ,, and then t conssts of the egenvalue E mλ 2 + λ 2 wth multplcty n References [] JD Bjorken and SD Drell, Relatvstc Quantum Mechancs, 964 [2] PAM Drac, Quantsed sngulartes n the electromagnetc feld, Roy Soc London, Proc, A 9 [] N Dunford and J Schwartz, Lnear Operators, Part II 96 [4] AS Goldhaber, Drac partcle n a magnetc feld: Symmetres and ther breakng by monopole sngulartes, Phys Rev D 6, No [5] Harsh-Chandra, Moton of an electron n the feld of a magnetc pole, Phys Rev, [6] Y Kazama, CN Yang, and AS Goldhaber, Scatterng of a Drac partcle wth charge ze by a fxed magnetc monopole, PhysRev D 5, No [7] ET Whttaker, An expresson of certan known functons as generalzed hypergeometrc functons, Bull Amer Math Soc, 9 [8] TT Wu and CN Yang, Drac monopole wthout strngs: Monopole harmoncs, Nucl Phys B, 7 976
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