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
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* 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 The angular eigenfunctions were obtained in a closed form for any real value of v by Hurst [7] . Their behaviour neai' and on the string can be easily deduced from his formula. One finds , v i w "- lh*((-.)em " (7) ИЩА»*^í&^íWWt^ v - 5 where the h 's are regular at в - * and ar« different fraai жвго there. As а satter of fact Tame'в early paper [$} also contains this result. Indeed, although Таим pays attention only to the v-n case, his analysis of the be haviour of the angular eigenfunctions in the neighbourhood of the lines 9=w (the string) and ©"O is valid for non-integer values of v too. Notice by the way that his solution (34) for the radial wave functions is also valid for any real v . (7) shows that when v+n all the U's vanish on the string, the si's being Integers by definition. Therefore in the v+a case the electron cannot be found on the string. In principle this is an ob servable effect. It is easy to show (appendix 2) that in this севе A - describes an infinitely thin (i.e. idealized) Aharonov-Bohai solenoid Qlo] located along the string, the latter being observable through the A-B inter- ' ference effect too. Consider now the v*n case. One sees fron (7) that for the single value m->n the U's do not vanish on the string, in and due to the phase factor e а и takes on different values when one A reaches the string from different directions. Thus UR is singular (aulti- valued and discontinuous) on the string S — , being regular everywhere else. Let us now see in more detail how the string-type singularity turns up. For the v=n>0 case the angular eigenfunctions take a simple form (see also the note on page 24) A (в+f)1 l(s+f)«P , IO Q where 4* e*( ) is the well known dM0) function of Wigner (as given by Edmonds [lij) belonging to the 2 j+1 dimensional irreducible unitary representa tion of the SU(2) group. The v j - "• - 0, 1, 2, <•) в - -j, -j+1,..., j. * and m are given by the equations x " 3 (J+D / m = s + 5 Kotice that for even (odd) values of n, both j «ad в are integers (half-integers). a is of course always integer. The m»n case involves the dQ (e)»s, and they are indeed dif- 7* 2 ferent from sero when в»» . The particular case n»j-l gives Dirac's eigenfunction -Sb --siny e . (As a matter of fact la the notation of [l] and H the phase of ^ should be -e». The misprint in И is corrected in [5]-) In spite of the fact that these wave functions are eoasaon knowledge, there seems to be a widespread belief that on the string sll the wave func tions must vanish in the v-n case too. One of the reasons for this belief probably arises froa a misinterpretation of the famous "veto of Dirac", introduced by him in [ll]. However, the veto refers to the paths of a clas sical electron, and it does not imply that all the quantum mechanical .wave functions must vanish on the string. Another reason for this belief may be that in [l] only an example of a wave function with string-type singularity is given, but the general mechanism of the appeara.ice of this singularity is not discussed. Let us also remark that at the origin all the energy eigen- functions indeed vanish since all the radial wave functions R^(kr) do so, as one can easily see from eq. (34) of Tamm [5,|. To understand the physical significance of the peculiar behaviour of the energy eigenfunctions on the soring it is important to realize that the requirement that the electron has finite energy leads to such a behaviour for all the wave functions. To see this, consider some wave function Ф<й>(r) which in the neighbourhood of the string has the behaviour *l5i, (in must be integer, otherwise the wave function would be multivalued in some neighbourhood of the string). Now in H -ф -fc) a term p"2(ra-v)t appears. If v 4 n, • *-'(r) must vanish on the string for all m's to be able to keep the energy finite. However, when v - n, • - (£) may be non-zero on the string for the single value of m • n because m-v-n-n-O. One comes to the conclusion that in the v-n case all the wave functions of the electron either vanish on the string, or have the string- -type singularity there. The fact that regular (continuous and single-valued) functions such as e p are thereby discarded ceases to be surprising if one recalls that similar situations are often encountered when the Hamiltonian is singular. For example in the case of a perfectly repulsing core of radius R all the wave functions must vanish on the sphere, the physical reason for discarding e * |r> R| again being that it would require infinite energy to maintain the electron on the sphere. Of course, в p can be expanded in ^шлтттш^тгтщштвШШЯНШШШШ 7 - a serles of the energy eigenfunctions of the monopolé since the latter con stitute a basis in L(2), normalization to ő(E-E') being allowed. This follows from the considerations in [7] and can be verified without difficulty for any real v with the help of [13~J. However i - const •/dre"*lpH^)e",lp turns out to be infinite. On the other hand, it is obvious that the common phase eÍjéP drops out from all the absolute squares !• - \'r)| and leaves the scalar product (•i~» 1>2 ) of anv two wave functions uniquely determined. Since all the probability distributions can be expressed through absolute squares and scalar products, the string-type singularity is an allowed one. Let us recapitulate the result of the first step of the proof. For v # n all the wave functions vanish on the string. This makes the string observable; it is observable through the Aharonov-flohm effect too. Mhen v * n, there are wave functions which do not vanish on the string, and the A-B effect is null (see Appendix 2). Therefore when v » n there is a pos- sit-lity of the string being unobservable. The second step in the proof will show that when v f n the displa cement of the string does not correspond to a gauge transformation, whereas for v = n it does. Thus in the latter case the string is indeed unobserv able. 3 Displacement of the string Let S- •* s—, whore a and Ь are arbitrary unit vectors. Then A(-} * A*-* anc consequently H<ä) * Я^1. Since rot (A(-J - A ^ )«fi-H.«0 everywhere except on the strings, the relation А(-}<£) -*&(*> - MCi--adx Thus, as far as the potential is concerned, S - * S -' induces a gauge transformation with a singularity on the strings. However, even if one disregards this singularity for a moment, one should remember that as a rule not all the classically possible x(r) 's lead to a quantum mechanical gauge transformation. This is because in quantum mechanics (10) must be completed with a corresponding transformation of the wave functionst в - l (E) é«a>(ö* *<ь)(£) . e ** ^(t) (и) and only such transformations are acoeptable under which the appropriate analytic properties of the wave functions, discussed in II. 1 are pre served. It will be shown below that this is the oase when v « n. Special attention will be paid to the role oS the singularity of (lO) . Let us nov find X(r). Making use of an integral representation of Л*-* (r) due to Jordan [14] who remarked that A - (r) can be viewed as the integral of the vector potentials of infinitesimal magnetic dipoles Vát' located along S- , one easily finds Here the open string formed by S*" and 5 For the special case b«-a the solid angle is ill-defined,because Fig. $ the plane fa,-a) is ill-defined. Namely, it is easy to see that - 9 - where <р is the azimuthal angle in a cylindrical coordinate system p>r «P . Z, with its -Z axis pointing in the direction a, and • is the „[-independent azimuthal angle in this system of the arbitrarily chosen plan« (c) in which S_ lies. Such r- independent azimuthal angles are clearly ir relevant both in (11) and in (12) and they will be ignored in what follows. The conclusion is that the r-dependence of X(r) - ft Let us now look at the analytic properties of the wave functions in (11). Sincce tth e properties of the Ф - (r)'s are well known from II.1 , let d . Then ... i» Q(d,b;r) ... ( * When a circulation along a closed path is made, fl(d,b;r) becomes ß(d,b;r) + 4uk with к = 0, + 1, -1,... depending on the path chosen. Therefore, when v + n, all the 4/ — s are multivalued everywhere. This is clearly unacceptable, i.e. in the v # n case we do not have a gauge trans formation, in agreement with the string S — being observable (see II.1). Of course, for -he new string S - a system of regular wave functions does exist, and the energy eigenfunctions of ti — can be found by a simple co ordinate transformation (but not by a gauge transformation!) of the solutions of Hurst [7]. They will obviously be different from the multivalued * 3's in (15) . Let us now come to the case v - n. The gauge factor exp[i 4°(d,b»r)J is then continuous and single-valued everywhere except along the strings. It can be easily seen either geometrically or from (5),(Ю) and-(14) that in the neighbourhood of the strings 1 £J(d,b;r) 4 - 2 ] ii(d,jj;r) % 2 It has been shown in II.1 that along S - the • -'s have the string-type singularity ein everywhere except possibly on the new string S — , in agreement with the fact that the Hamiltonian is now H - . Therefore when v » n the displa cement of the string amounts to a gauge transformation under which the string-type singularity follows the string*, i.e. remains an allowed sin gularity. Consequently all the probability distributions remain well defined and unchanged under this transformation, and the string is unobservable, q.e.d. It is easy to see (Appendix 1) that the proof goes through also if the new string is curvilinear, with the exception of those pathological strings which make infinite oscillations at infinity, like e.g. the string {y • x sinx, z -» оo) . In the latter case the solid angle ÍJlc'(r) for sj;c' cannot be defined. Since a rotated and a space reflected string is again a string, the rotations índ the space reflection of a string are unobservable for all values of n. It follows from this remark that rotation invariance and space reflection symmetry must hold and rotation and space reflection operators should exist for any n. In parts III and IV these operators will be found and their properties studied. III. ROTATION INVARIANCE 1. The rotation operator T _ Since the Hamiltonian H(-' is not explicitly rotation Invariant, the usual rotation operator R defined by the relation Кдф(а)(£) = ф<а>(д-1£) does not commute with H*— . Notice also that if ty^-'(r) has the string- -type singularity on S(—' as it should, R i|>— (£> will have it on the string s*9~ , and this is not allowed, the Hamiltonian being IT- . Therefore R cannot be the rotation operator in our case. (It should be stressed that while in part II both н'—' and ф1—' changed under a gauge transformation, now one has to rotate either the wave functions or the operators, but not. both. Here the former possibility has been chosen.) On the other hand, recalling that A<9§) _ A . „ gradi)(f,gd;r) ш Let us remark that the gauge factor never changes the absolute value of the wave functions. In particular, unlike the string-type singularity, the zero lines of the energy eigenfunctions ф -'(£) do no follow the string as one might chink. E -Il oné easily finds that the operator i3o(gd,d;r) T = e Rg (16) / J4 does commute with H —, since (-> R-i. »««-> - ,i5e(4'*,s) „(d).-1!"«»*'^ V B g Moreover, it follows from the results in part II that the ^-dependent phase factor in (16) brings back the eventual string-type singularity of R • — (r) to the right place. Notice also that similarly to R , т does not act on <\, g g the radial variable r. Thus T could be the rotation operator, but un- *\* % % fortunately it does \ot satisfy the composition law: T T #T_ As 9 9 9 92 1 2 1 ' a matter of fact one can show - making use of a geometrical approach - that the T 's form a ray representation, and even the r- Independent but g-de- pendent phase factor which transforms them into a representation T can be found geometrically (see Appendix 3) . However, T can also be found di rectly by making use of a simple relation (eq.{17) below) between the angular eigenfunctions U ( Turning to the derivation of eq.(17), a remark will be in order. when the rotation operator is R , half-integer j representations cannot occur for the spinless electron. However in our case T ф R , and such rep- resentations cannot be a priori excluded. Accordingly it is more appropriate to work with the SU(2) group than with the 0(3) group. Let us therefore introduce the SU(?) Euler angles a,ß,y parametrizing the elements of the SU(2) group. These angles correspond to rotations around the fixed Z, Y, Z axes, the rotation у being carried out first. The ranges are О < a < 2tr ; 0 The ranges of the polar angles c»(r°) = (p , ß(r°) - 0 , y(r°) - yQ where > can be chooson arbitrarily within the appropriate range. Although Y drops out from the final result, is would turn out to be inconvenient to оЬоояр Y - 0. 'o - 12 - Take now an arbitrary SU(2> rotation g and consider the group elemensntt r° = g r°. The Euler angles of r will be denoted through Ф, 6. Y, ct(r°) = Ф B(f°) =• 0 Y(r°) - yQ A glance at the composition rules of the Euler angles written out in Appen dix 4 reveals that 5, § (and yQ - yQ too) are Independent of у . Therefore Ф, § are the polar angles of the rotated coordinate vector g~ r_. Of course, any pair of SU(2) rotations with Euler angles a,6,0 < Y < 2w and «,6,Y + 2* lead to the same rotated vector g r. The desired relation obtains immediately if one notices that the composition law of the rotation matrices D3(g)= DMOBY) of the 2j + 1 di mensional irreducible unitary representation of the SU(2) group permits one to write for r° = g~ r° of n Taking into account that the d3 (ß)'s of Edmonds are real and making use of s' s the definition of the U's given in (8) one finds (see also eq.(38)) 1 S Mg"1, r°) , 3 г 3 3 3 17 e U n (Ф,б) - I D /a(J) U , П(Ф^9) ( > s'" I g'»-j s,s *,- 1 where ы is a function of two arbitrary SU(2) rotations g,h: o>(g,h) = «(h) + Y(h) - ct(g-1h) - у (g_1h) . (18) Especially, according to a remark made above, -1 ш(д , r°) « Ф + Y0 - 9 - Y0 in (17) depends only on Ф,0 and on the Euler angles of the rotation g but not on У . It is also independent of the radial variable r of course. One easily sees from (17i that the operator iSuMg-^r0) T = e Z R (19) g 9 - 13 - is the rotation operator on the linear space of the wave functions of our electron. Indeed, since R U-* _ ( Taking into account that for any fixed n » 2ue(t»c)~ the O's constitute a basis on the unit sphere and ththaa t T does not act on the radial coordinate r, one immediately concludes that T T - T , (21) g2 <зх g29i ( 2 (rg, H 4>3 - Qrg. j ] - o. g 0 g (21) can also be deduced using the obvious relation Rn»(h,r )Rrt • u(h,g r ) One then finds T T exp|l2-PlJJ | • T _ where 9j 9j Ч2ЯХ 1 1 1 1 1 £ swig" , r°) • ы(д1" , gj r°) - «(g* g^ , r°), and from the definition of <>> in (18) one immediately sees that С н 0, q.e.d. Let us now look at the properties of the phase factor in T . Prom "* (d) 9 the fact that both T and T commute with И - one deduces that their phase factors roust have the same £ -dependence. The precise relation 1 0 Mg- ^ ) - "(gd,d;r) -«(g) -Y(g) (22) obtains after a shor:-. calculation based on the composition rules for the Euler angles and on an analytic expression for the solid angle (Appendix 4). (22) shows that the operator T is discontinuous in r on the strings s — and s "- . Usually no discontinuity in r is tolerated in an operator which realizes a Lie group. In the present case however this singularity is needed to keep the eventual string-type singularity of T * - on the (d) " string SK-'. Let us consider now the g-dependence of T . To be a realization of the SU(2) group, T must be a single-valued continuous function on the group for any fixed value of Ф,9. This is indeed the case, as one can infer from the composition rule of the Euler angles and the definition (18) of » . Namely, one finds that w has a 4ir-discontinuity in g, which leaves, however, the phase-factor continuous for any n. The critical case 6(g) • * is easy to check. Tn this case fJ(-d,d,r.) is known tobe ill-defined. Clearly, the - 14 - arbitrary angle Ф in (13) where one now has to put a » d must be related to the Euler angles a(g) + у 2. Half-integer 1 representations (20) shows that for fixed values of n and j the U's are the basis of the 2j + 1 dimensional irreducible representation of the rotation group. From (9) we know that when n is odd, j mist be half-integer. Such representations could not be present among the wave functions of a spinless electron if the rotation operator were R . Indeed, for a 2* rotation - g R2lI round any axis 1 - (-)2j j 2j since D . (2w) = (-) в . . This condition would restrict the j'e to integer sfs s's values, i.e. the n's to even values. However, in T the phase factor is easily seen to be (-)n for a 2n rotation T, , and from (20) one finds (-)"= (-)2j in agreement with the spectrum condition (9) coming from the Schrödinger equation. Thus for odd values of n half-integer j representations (and only these) occur, in spite of the electron being spinless. i Let us remark th.it a 2TT rotation for n odd ( l T2Tf* ä) (r) = - }$> (r) has no observable effect since all the wave functions get the factor -1. Similarly to the usual spinor case the presence of the half-integer j rep resentations cpuld be tested only through such measurements as e.g. quantum state multiplicity and angular-momentum analysis. The theory of the Dirac monopolé is the only example known to us in which half-integer j representations of the rotation group appear with out the introduction of half-integer spin particles or fields at the beginning. - i5 - 3. The generators of the rotatloi group The rotation operators have been found above without making use of infinitesimal rotations. The generators can now be easily obtained since T is a continuous, different' Ъ1е function of g . The result coincides of course with that of Hurst \j] (23) In spherical coordinates, one obtains [б] +icp J ij e +1 cot0 x i y - " (± 4 h -1 ^ -i Яр " 1 • It is easy со check by direct calculation that these operators satisfy the usual commutation relations [Jx' Jy] i J , etc., (24) and act on the basis functions of the irreducible representations of the SU(2) group in the well known way 1 j s 1 1 j 0 (j + ÍJ ) u „ („,e) = Г№«)(1 i + )] " п(ч»' ) ' s,-— - s+1,- •=• J_ U|3J ( The latter equations show that for the v ь n case the -^ 's are well defined self-adjoint operators on the linear space of the wave functions of the electron. Therefore the Casimir operátor J2 » J2 + J2 + J2 — x у z 2 also axists and it is easy to see that it coincides with the J operator defined as the angular part of H — in (6). Thus one comes to the conclu sion that rotation invariance around the monopolé holds and the rotation operators and their generators exist for any integer value of v • n. The opposite conclusion is reached in [f\ for odd values of n. The argument is based on the statement that the operator r~ (r-J) has only integer eigenvalues. However, acting with this operator e.g. on Dirac's eigen- function Sa=cosj in [Í] , one easily finds the eigenvalue - j. - 16 - It should be noted that the J.'s satisfy the commutation rela- (d) -1 tions <24* and commute with H - even if an arbitrary v « 2Ue(hc) is substituted for n into (23). This might lead to the impression that rotation invariance holds for non-integer values of v too. However, it has been shown by Hurst [V] that the Lie algebra of the J.'s is not in- tegrable when v Ф n. This is in agreement with the fact that in the v f n case the string is observable, and thus rotation invariance cannot hold. Let us come back to the v = n case. The generators «i. are then self-adjoint operators and they commute with H — , therefore they should correspond to the angular momentum of the system. The first term in (23) does indeed have the canonical form of the angular momentum written in the Dirac gauge A = A — , however, an additional term also appears. This leads us to the problem of the conserved quantities and their relation to the generators of the corresponding symmetry group in the case of a system the potential of which is not explicitly invariant under that symmetry. This problem will be investigated in part V. Before coming to this, let us discuss the properties of the Dirac monopolé under space reflection. IV. SPACE REFLECTION The classical equation of motion M'*-![*""?f ] of the electron in the magnetic field (1) of the monopolé with magnetic charge u goes over under space reflection £ * ~ £ into the equation of motion of the electron in the field of a monopolé with charge -v. This is the content of the space reflection symmetry in the case of our electron- -monopole system. Accordingly, in quantum mechanics one should require that under space reflection P : £ * '" £ tne Hamiltonian К •* H_ , i.e. that the wave functions ty (r) -» ty (-r) = <|r ,(r). Notice also that -u implies -n * 2(-ye)(Tic) . In order to avoid confusion, in part IV the parameter у will be written out in those quantities which depend on it. It has been argued by Schwinger \£\ in the context of a field theoretical approach that space reflection symmetry holds for the monopolé only for even values of n. The conclusion reached below for the simple case of a spinless electron moving in the external field of a single monopolé is different. It will be shown that Schwinger's restrictive condition emerges only if one sticks to the usual definition "r "* - r" of the parity operator P. However, one can implement P with an appropriate r-dependent phase factor much in the same way as in the case of the rotation operator In part III, - 17 - Space reflection symmetry is then seen to hold for any integer value of Let us denote the potential with two strings 5 — and S{~-] in- troduced by Schwinger through A : д(п, ^(-n) A + -v The Hamiltonians containing the potentials A,1,- and A* will be denoted through Н1*У and н" respectively. ( _1 Since PA -* p К p"1 = »!y (25) p „^»p-1 - н^и> (26) (25) shows that under P in H merely the sign of is changed, i.e. that a space reflected initial state undergoes space reflected tine evolu tion if v •*• -u. This is Drecisely the content of space reflection symmetry in a magnetic field. Notice, however, that H is not invariant under space reflection, and parity selection rules should not be expected In the case of a single monopolé. If the monopolé is described by a potential with several strings, they should be all unobservable, i.e. Dirac's quantum condition « n. v, 1' it hold for each of them. Since the two potentials in A?s U are of equal strength, v, « v, == n. and one finds v » v.+ v_ = 2n. » n. Thus Schwinger's quantization rule is certainly sufficient to secure space reflection symmetry. Let us now look at (26) . It shows that under P in fT- not only M changes its sign, but also the string changes its position. However, it can be brought back to its initial position with the help of an r-dependent phase factor expift(n,-n;r). Taking into account (13), one finds '1пфп<^) „<-n> ^n^ (n) -u »% Therefore, introducing the modified parity operator -Infi (r) e - ~ P (27) V one arrives at the relatic W Hu [ u ' -H -C - 18 - in complete analogy with (25) but with no restriction on the value of n. The role of the phase factor in (27) can also be seen in the follow ing way. Let if1'- (r) be a state in the gauge A = A/,-* • If Ф*-* (£) has a string-type singularity, then P*^-* (r) = (И-* (-r)= •jjj* (£) will have it on the string S' —'. This is not allowed. The phase factor brings back the singularity to the right place and with the correct exponent -!•„(£)/ cor responding to -p . One also easily sees that with two strings '"§*—' and S — no supplementary phase transformation is necessary. However, then only even n values remain. Finally, notice that the new parity operator for the magnetic charge -M is of course р|? , not РЛ—' . Therefore the effect of two consecutive space reflections is (n) (n) inVE> -inV^ in»p(I) -in(«p (£)•») p-u Fl =e Pe - P - e - • e - P = (-) . In particular, for odd values of n the result is -1 as usual when the wave functions transform under rotation according to half-integer j repre sentations . V. SYMMETRIES AND CONSERVATION LAWS According to the theorem of Noether every continuous symmetry of the action Is connected with a quantity which is conserved during the motion. In particular, the conjerved quantity, belonging to the space translation symmetry, is the canonical momentum p_, while invariance of the Lagrangian under rotation implies conservation of the canonical momentum J = [rxg]. Let us consider the classical motion of a charged particle, in a given electromagnetic field. One may wonder whether the syirmetries of the field lead to conservation laws through the Noether theorem or not. The answer is not direct since it is the symmetry of the action which is neces sary for a conservation law to exist. The action contains the potential rather than the field and the symmetries of the former do not conincide with those of the latter. This question has been raised and answered in [17,18,3 . Here ve follow a different line of argumentation. Let us show that every elementary symmetry group (space translation along a given direction; rotation around a given axis; time displacement) of the field leads through Noether's theoren to a corresponding conservation law. Let us assume that the electromagnetic field possesses a set of elementary symmetries. In the general case it is not possible to choose a - 19 - potential invariant under this whole set. But it can be shown in a simple way (see Appendix 5) that the totality of the equivalent potentials does possess the same set of symmetries as the field. In other words, one can always choose a gauge transformation which imbes the potential invariant under any of the elementary symmetries of the field. Let us now select one of the elementary symmetries, э.д. space translation »long the X axis. As mentioned above one can find a potential which is invariant under this translation. Let us denote this potential by A a ,• a and consider the quantity Pxa> =Px-i {x } S } = a'Pß - a0 ' Ч'РВ °- This quantity is gauge invariant since p - — A is a gauge in- variant combination and A a is a given function of the coordinates which are known to be gauge invariant. If we choose the gauge A • A then p a = p . But in this gauge the potential and the action are by definition invariant under space translation In the X direction and the corresponding constant of motion is just p . Therefore, p*a is thai: gauge invariant conserved quantity which, through Noether'4 theorem, belongs to the translation invariance of the field in the X direction. Rotation invariance can be treated in the same way. Assume, for example, that the field is invariant under rotation around the Z axis. Then the corresponding gauge invariant constant of motion is J C) (С) z = [»СВ-|(А-А 1)]Ж (29) where A (с) is a potential invariant under rotation around Z. In the gauge A = A (cT , J (c) Is с(ual to the Z-component of the canonical angular momentum М- As a simple example consider the homogeneous magnetic field. We choose the Z-axis along the field. The field is invariant under displacement in any direction but one cannot find a single potential with this same degree of symmetry. Potentials which are invariant under translation in the direc tion X or Y are given by the expressions л 'a> If » f. A р(Ь> _ £ A . «H • + eHx Г у Су C у c T c Invariance under a given elementary symmetry does not specify the potential uniquely. Therefore, for a given elementary symmetry of the field one can obtain a series of conserved quantities which, however, differ from each other merely by an additive constant. Let us consider, for example, the case of rotation around Z and let both A and A be invariant under this operation: (C (C 3A > 9A '> -55- = TS— - о <30> (c) (c') A , A represent any of the components in a cylindrical system of co- * (с) (с) ordinates . A and A differ from each other by a gauge transformation: A and, as a consequence of (30)grad -J- = 0. Therefore, const 32 Jz - Jz = [£х(А -A JJ ЭФ <- ' z as indicated. bet us now find the transformation law of the action (more precisely of Hamilton's principal function) г2 /1(гх, tx, r2, t2) •-: I dt L(r(t), £((-.)) under an eic.ientary symmetry g which transforms £ into g r. In the above expression the integration is performed alonq the actual path between the points r, = r(t.), r, - r(t2). Let A'(r) be a potential invariant under g. In this gauge the action л'(г. , £2) in which time coordinates has been suppressed is also invariant under g: 1 1 /i'(g" rj, g" r2) = A'(Li, £2). On the oilier hand, the action Л(г1( £2) written in an arbitrary gauge Л transforms under the gauge transformation Л » Л' ан - 21 - 4(£l, r2) = A'(rir £2) * X(r2) - X(£l). Therefore, _1 1 -1 _1 1 A(4~\V g r2) = лЧд- ^, g ^) + Hg r2) - Mg" ^) = 4 + a - (*i' L2^ (9:L2) - *(g? aj (33) where o(g;r) * X(g-1r) - X (r) . As stated above, gauge transformation which leads fron an arbitrary A to the invariant A' can always be found. There fore, (33) is the general transformation law of the action under a (yuanetry transformation of the field. As a consequence of this transformation law the Poissor-brackets of the conserved quantities may differ from the Lie-bracket relatione of the corresponding symmetry group by additive constants [l§. When, for example, the field is invariant under translations in X and Y directions, the corres ponding constants of motion satisfy the relation {PxU) ' РуШ} • conet- (34) This can be verified directly through (28) from which we get in a few steps a) (w + сь) b) a) The two rotors cancel since they are equal to the same magnetic field. The third term can be written as i„P-, where X is defined by the relation A = A + gradX. Since A and A are by definition invariant under the displacement in X and Y directions, respectively, we have (я) b) эА V , 7x Эу О. ,2. Hence graJ к ft - 0 and 1 ? " const which proves (34) . Let us turn now to the case of the monopolé. The field of the mono- pole is invariant under rotation around any axis going through the monopolé. Let the constants of motion belonging to the rotation symmetry around X Y Z a be Jx , J , J* ' respectively. Potentials invariant under one of these rotations are given by (3) with n - a, b, с antiparallel to the X, Y, Z axis respectively. The antiparallel choice ensures A £ =• A *'. Using (29) with A/c'- K^ and corresponding formulae for the X and у components, one gets for the conserved angular momenta the expressions - 22 - J a) (S) + x = t 4C)= [«to-* (* - *(£)>>] - [« (35) and (36) show that the conserved quantities 1 . j<»>_SU, i -jW-SE, т = j xx с ' уус' г г с are the components of the vector I - [r x (E - | A)] - »I | = m|r x rj - 2* f (37) and they satisfy the usual Poisson-bracket relations of the canonical angular momenta [i , I ] = I , etc. The first term in £ is the Hnetjc part of the angular momentum, while the second term has been shown [iSfj to be the angular momentum stored in the combined field of the monopolé and tl.e charged particle moving around it. The sum of these two terms, both having clear physical in terpretation, can be considered as the "natural" angular momentum. (c) The vector potential A'-' in (35) is symmetric under rotation around Z. As pointed out above, by choosing another potential of this same (c') (c) symmetry one can get a new angular momentum J, ' which differs from J' ' ic') (c') by a constant. Let us choose A,st ' in such a way as to make J' • equal to I . To this end we have to put into (31) a function n which satisfies the relation |^ = ^ , since in this case according to (32) J^0'* - J^c*- ^ = I . This can be achieved by taking n • p д (£.')„ л(с.) _ pgrad(p (c) instead of A -' in (35) one gets I directly - 23 - ^"} - l.rx( -|(&- A,£,)))J - [rxiir |A)J - ем г _ K er z ,|ц..1 I he potentials Л*-'' = л'-'- grady> , A(-'' = A*-' - gradq^ give 'x' 'у aspect Wei у. It is important to realize that J (с') is a gauge invariant expres sion in which the choice of the gauge is still open. It will be fixed if one (c') chooses a particular gauge for A in J . For example, if one puts Л = Aic' — ) , one gets J, (c')r- rxpn = I , that is, I takes its canonical — - - Z i*~ J 2 Z Z form in this particular gauge. One sees now that no gauge exists in which more than one component of the natural anyui at momentum can take its canonical form. This is because Л - »A — and A -°- arc different from each other and A can be equated at most to one of them at a time. For example in the gauge A = A'- , I = J*a a J do not h,»ve canonical for«. This xx IУ У conclusion is valid even in the classical theory of the monopolé. In the quantum mechanical case further restrictions arise. First of all, if one wishes to arrive at a rotation invariant situation, Dirac's condition у = 2[ie(hc) = n must be imposed. Indeed, even if one would not know that the Lie algebra of the 1 's cannot be integrated in the v ф n case [j} , one could come to the conclusion that the group they generate cannot be the invariance group of operators which realize rotations of the probability distributions |ф — (r)| -— \ty - (g~ £)| . Really, the general form of the unitary operators which carry out such rotations should be exp(iF(g,r,Э )} R . (d) ~ — " The conditions that they commute with H — and that their generators be the I.'s restrict F to the form F = (v /2^(n(gd,d ;r) +4(g>3 where n must be independent of r_ and of Э . Then single-valuedness of the rotated wave functions implies that in the v f n case rotation operators do not exist. Furthermore, as we have seen, not all the classically possible gauges arc allowed when v ••• r. . The gauge A = A1-' = A'— has been shown to lead : to correct wave funct. ort ф'-' (r_) . To go over to the gauge A = A — one плв to consider the transformed wave functions n ^-'Urj =el7V!*>(r). Clearly, they are acceptable only if n is even. Thus the conclusion is that in the caso of the quantized monopolé only one (n even), or none (n oc(d) of the three components of £he natural angular momentum can take its canonical form. Notice, finally, that in the gauge A = A -' usually adopted, none of th" components of the natural angular momentum t>J are canonical. In tlhis ri.iuqp I in (17) goes over (with v = n and p ' - ihV of course) into i ч - ' i he geniT-ifni - и of the rotation group given in (23). - 24 - ACKNOWLEDGEMENTS The authors are deeply indebted to P. Hasenfratz, H. Huszár, F. Niedermayer and Л. Patkós for their continuous interest in this work and for many valuable remarks. NOTE ON THE DEFINITION OF D3 AND U3 A few remarks are in order to explain our definition of D3, (g) 8,8 on page 12. This definition differs from that given in eq.(4.1.12) of Edmonds' book fll] by a change from a,8,Y, to -а, -В, -у. In other words, while in [ll] the rotation operator is defined in (4.1.$) to be *Ц * нен. u*"'*té-*£>>* we use instead _i«i .iß l -Ail. T Rg . e ** e T *> e * ** since this latter R (but not that of Edmoi.ds) corresponds to other commonly accepted definitions and notations we made use of. In particular, we define the action of R on a function f(h) of the SU(2) group element h to be R f(h)-f(g-1h) which leads to R_*(r)-*(g_1r), while Edmonds works with У у R g*(h)= f(hg), as can be seen e.g. from his formula (4.1.4). 2 Let us also notice that the eigenfunctions of J are written in (8) in a form wich is easily expressed through our D-"s: 4 4« i ?(Ф+У) 3 U-> n (ф,е) - D n (ф 8 y)e . (38) s,- J s,- ^ Indeed, according to [ll] (4.1.15) ,2O+B'+B I 0\ 2j-2o-s'-s therefore dj (-B) - (-)S'"S dj (B). .(cosßjsis 20+8'+S (sinfsjs ) Thus our eigenfunctions differ from those used in [1,5,6,7,8] only by a ph?se factor. - 25 - APPENDIX 1. Curvilinear strings. According to Jordan [14] , if the string of the potential A of the monopolé is curvilinear, then A can be written in the form л(£) • • I т^р- s where the integral has to be taken along the string 5 fron the point of location of the magnetic charge u to infinity. One easily verifies that on s |A|»«O, and everywhere else rot A » H. S*S*-' leads to (3). The transformation between two potentials A/ , A with strings S(a), S(b) now reads (compare with (12)) ib\r)- Mr) - v 6 fe"-)xd! - p grad ac(r) J " " " stc) |r-r'| TO where Яс(г) is the solid angle under which a surface bound by the strings is seen from r. The sign convention is the same as in the case of the straight jtrings. It is easy to see that apart from an r-independent additive constant nc(r) is uniquely determined for any pair of "non-pathological" (see page 14) strings. Introducing curvilinear cylindrical coordinate systems pa' фа'z a; pb' *b'z b witn tneir "z axls al°ng the strings 5***, S* ' respectively, all the results of part II concerning the string-type singular ity and gauge transformations easily obtain. In particular, for the case v = n any (non-pathological) curvilinear string can be transformed into the Dlrac string S - by a gauge transformation. This last remark makes trival the extension of the results of parts III and IV on rotation invariance space reflection symmetry to curvilinear strings. APPENDIX 2. The Aharonov-Bohrc solenoid In order to see in a simple way that in the v^n case the vector potential of the magnetic field of a magnetic charge у represents an infinitely thin (i.e. idealized) solenoid located along the strinq of the potential, let us move this charge and the end point of the Diri-c string r> - with it from the origin to infinity along the positive Z axis. The string will then occupy the whole Z axis, and the corresponding potential A obviously obtains if one puts в»* in (4): A = —u , А ж А = - 26- The magnetic flux Ф through any connected surface wnich has one point of intersection with the Z axis (the string) now is ф=ф A.dt = 4wp. Therefore A = (2»p) ф. This is the well Know Aharonov-Borin potential for an infinitely thin solenoid located along the Z axis [loj. The condition for a non-zero interference effect is n ф integer, where -a2e*(ch) = 2eu(cTi) 5 v. Thus this condition corresponds to our v^n case as expected. Let us now look at the behaviour of the energy eigenfunctione on the string. These functions are iimp J|m-v)(pk>e m = 0, + 1, + 2, (we merely substituted our notations Into eg. (3) of £iq]). When v#n they all vanish on the string (p = 0) as expected. One also sees that when v=n the functions with m=n do not vanish on the string and have the string-type singularity there: ln Thus when v=n one comes back to the monopolé case, but since now it has been moved to infinity, one has in fact a free field case." It is however obvious from the above considerations that if the magnetic charge is at the origin, the Dirac potential A - represents a monopolé when v=n and an Aharonov-Bohm solenoid located along the string .'> - when v#n. To observe the interference effect one must take of course an electron beam lying in the XY plane at large negative z value. It is hardly imaginable that a microobject corresponding to the observable string exists. Therefore in the u^n case the vector potential A - in (3) merely provides a somewhat^ idealized but convenient mathema tical description of a thin macroscopic solenoid with its axis along the string .' - . Only in the v-n case do we have a hypothetical microobject - the Dirac monopolé. к Л barrier along the axis of the solenoid would force all the wave functions to vanish there. Thus the remark on p. 448 in [in] concerning the irrele- vanco of such a barrier applies onlyito the v^n case (which was the case of interest in [in]), but not to the v~n case, - 27 - APPENDIX 3. Geometrical construction of Tg. Although the proof that T is the rotation operator is logically complete as given in the main text, it might be interesting to am« how can one find this operator by means of a purely geometric approach. To avoid problems connected with the double-valuedness of the half-integer j (odd n) representations of the geometrical rotation group and with the ambiguity of the geometrical definition of the solid angle Q(-d,d*r), let us restrict ourselves to cases when all the rotations involved have rotation angles smaller than 18C°. Of course at the end one would have to allow for all possible rotations and to make a link with the SU(2) group. This last step will however be omitted here because it is in fact contained in part.III. Let us show that the operators ^ i5 n(gd,d»r) 4 4 obey the composition law of ray representations * * = -15A<92'91) 1 4 *i «гЧ with expfi^r A)Independent of r. Recalling that R R • R _ and that V 2 ' - '2 91 9291 R ß(a,b;r)R~ - n(a,b;g~ r), one easily finds A 9 {1 <3 <3 + п + й ln (92' l) " ^2 1~' 2-''-^ ^2-'-''-^ (а»9291а»Е). obtaining this expression the obvious relation íl(q.d,á;g~ r) - ^(g.g.d^g.d;^) has been uaed. Thua A(g_,g.) is an algebraic sum of the three solid angles under which the (9 plane segments enclosed by pair« of the strings S*-*, $ 29lfl>f s\92§) are seen from r. It is easy to realize that A(g2,g.) is equal to the oriented area of the spherical triangle whose vertices on the unit sphere are the end points of the vectors d, g2g.d, g.d and this area is indeed independent of the choice of the point r, (Tig. 4). To be more precise, A(g_,g.) turns out to be independent of r only modulo 4ir. This is however sufficient to make expfi.- A)independent of £ for all values of n. •\, <\, л. One must also verify that the associativity rule T (T T ) « 9 3 92 9i = (T T )T is fulfilled. This will be the case if the equation g3 g2 gl Д(д3,д2) + Л(д3д2,д1) - Afa^g^) - Afa^) - 0 (3.1) - 28 - holds modulo 4*. But the S'-' first three terms in this expession combine to give the oriented area of the spheri- ,, vectors g3d, g^g^, g3g2d. On the other hand, this latter triangle obviously obtains through rotation g, of the triangle d, g-g.d, g2 area of which is A(g2,g..). Thus (3.1) holds. Let us now look for a function n(g) Independent of r such that the operators Pig. 4. i Ü n(g) i §(n(gd,djr) + n(g)) T = e ^ T=e ~ В g g g satisfy the composition law TT = т „ . This will be the case if the g2 9j Ч Ч relation 2 Х q = n g 9 n (3.2) \(42' 0 ( 2 l^ " <92) - nigj) holds modulo 4n. The fulfilment of the associativity rule for the T 's is then automatic. In order to construct n(g) geometrically it is convenient to introduce a rotating coordinate system K(XYZ) the starting position of which coincides with the fixed XYZ system. It is well known that for any rotation g one has Z я Y. x Z = Z' x 4' st Z . The rotations on the right should always be carried out first on both sides of this relation, and Y; Z' stand for the re dted positions of these axes. Consider now the rotations g2 and g29,- They carry К into K-=g2K and K_.r.g g К respectively. Then obviously ^21=g2g1^2 *2'Th e ^atter relation shows that the rotation K?>K can be carried out applying the rotation g. to К Keeping track of the Kuler angles of the rotations К »К,, К >K-. and - 29 - K-»K . one finds a * - T - !» T с - «21 - e2. 2 21 21 - *1 where a,. s a(g-g,), etc. and c, e2, e21 denote the angles of the spherical triangle т of the unite sphere the vertices of which are the pe,inta of intersection of the S, *2 and *21 axes with this sphere (Fig. 5)* Recalling now the definiton of afa,^) , and that the string St^) is antiparallel to I, on* iaawdlately realises that -A(g2#g1) is equal to the oriented area c-i-c21+c2-ii of the spherical triangle т. Taking the case when -A(g2,g.) > О one obtains -Mg^gj) - e + c21 + c2 - w - +т -(e21 • У21)- («2 2^ " ("i^j). Comparison with (3.2) shows that n(g) - -a(g)- y(g), Fig. 5. as expected. This concludes the geometrical construction of the operator T . APPENDIX 4. Derivation of eq.(22) The composition rules for the Euler angles corresponding to the SU(2) relation g, * g2g, are given e.g. in [20] . In our notations they read: совв, * cose, совв, - sine, sine. coe(a. + У2)> i((i3-t«2) . e = -g^'ng' (sinejcoeej + coeejSinejCOsOij+Yj) + leine^einicij+Yj))* " "а2*Ъ'*1 Y Y 3 V 2 V 2\ 1 / в в1 * в 1 1 9 в l coe ein e - sin ^ 1 - -V - -<- 28i n 1 ' -V ) C09-в- : V •/ T -в- в 1 n -я- е / . (4.1) - 30 - Notice also that the relations between the Euler angles of the rotation g and g are a(g-1) - 2*p - y(g); efg"1) - 6(g); Y(g_1) = 4nq-2»p-a(g) where the integers p,q are uniquely fixed by the requirement that all the angles be in their appropriate ranges. Let us now indicate how to derive eq. (22). From (4.1) one finds a tT Y в в + t-g 3~°2,3- l _ с°4<У1 1> .tg 1—5Ъ — cos|(e2+e1) One has then to recall a formula [2\\ for the solid angle ft(e.,e~i ) measured by the area of a spherical triangle on the unit sphere with sides е., e_ and enclosed engle 6: в в níe^e^) 2sin -£ sin -± tg ж •— = r 1 г .. cos= (9^2 ) tg|+cos| (e2+e1)cot| g and Starting with r° • g~ r instead of g, = 92 l noticing thatft(e,6(g) ,a(g)- APPENDIX 5. A theorem on elementary symmetry. Let us call "an elementary symmetry" a./ the invariance under the translations along a given direction, b./ the rotations around a given axis, and c/ the translations in time. Theorem. Let the electromagnetic field E, H possess an elementary symmetry. The potential А, ф can always be chosen to possess the same symmetry. Before going to the proof, two remarks are in order: 1./ The potentials cannot have elementary symmetries other than those possessessessed by the field. If, e.g., Э H Ф 0 then ЪJb » 0 would contradict the relation H = rot A 2.1 if the field has several elementary symmetries, then in general there is no such potential which would have all these symmetries. As an - 31 - example, consider a homogeneous magnetic field of magnitude H in the +S direction. The elementary symmetries are the translations along any given axis, and the rotations around any axis parallel to the t axis. The poten tials Invariant under (i) translations in the X and Z directions (ii) transla tions in the У ar.d Z directions (ill) translations in the Z direction and rotations around the Z axis are (1) Ax - -Hy, Ay « Az - 0, (ii) A^ - Hx, M -A s - °' (ill) h"x - - | hy, Ay - \ Hx, h"x - 0. (Ф-О in all three cases.) The gauge transformations between these potentials read " 1 A' • A + H grad xy, A - A + |H grad xy. There is no such potential which would possess all the elementary symmetries of H. Proof of the theorem a./ Let the elementary symmetry of the field be the translations along a given direction, and the Z axis be parallel to this direction. Then by hypothesis TT"0' <5Л> ЭЕ TT"0 (5.2) where a refers to Descartes components. Suppose that А, ф are potentials belonging to the field, i.e. H - rot A, (5.3) , ЭА 5 * ' с Ti "gra d ф# (5,4) One has to show that one can find new potentials A' = A + gradx, (5.5) - 32 - •'-•-S?É (5.6) ЗА' such that the equations -5^ = O, -Л— = ű hold. From (5.1) and (S.3) one finds that i)A / »A \ SA rot — = 0 where I ^~ J = -«-2 . Hence there exists a function f(r,t) such that ЭА g§ - - grad f. (5.7) f is determined up to an additive function n(t). To fix it, notice that (5.2) and (5.4) give 1 a2* эл rad с ш£ * 9 a if - ° and with (5.7) one obtains grad (-£ - — -rr) = 0, i.e. (г Z С or Эф 1 3f ... n(t) can be ctiosen su~h that a(t) = 0 holds, (if this is not the case, go over to a new f = f - - / u(t')dti) Thus without loss of generality one can write It is now eis'- to set that the function ,(x,y,z,t) = Í dz'f(x,y,zjt) is such that the potentials А' Ф' possess the required elementary symmetry. Ind(iot) - 33 - ЗА' »А Г7 - TT * *rad sí " - 9"d f • grad f - О Ьг u a a due to (5.7) and (5.9). Furthermore Д11 „ |i_ lifi- ЛЛ".о Эг Эг с 3t32 3z с « due to (5.9), (5.B) q.e.d. b./ Let the elementary symmetry of the field be the rotations around a given axis, identified with the Z axis. Then 3E a Эф (5.Ю) ЭН a (5.11) Зф where a refers now to cylindrical coordinates (p, ЗА where \ Эф/о ~ Зф Thus again a function f(r,t) exists such that ЗА (5.12) with an arbitrary additive n(t) in f. Since "grad" is also free from the Ф coordinate, (5.4) and (5.10) give - -п^т+ grad -'•* = о С ЭфН 3 (X Эф and (5.12) leads to 34 1 3f _ ,.s _1 -- _ -, a(t). Let us choose n(t) such that rt(t) = 0 holds and then define the function - 34 - X(p, It is easy to verify that the potentials A' •' in (5.5), (5.6) with this X are independent of Ф q.e.d. c.l For time translation the theorem holds since a time independent field can always be represented by time independent potentials. REFERENCES ~l] P.A.M.Dirac, Proc.Roy.Soc, Л133, 60, 1931. [2] J.Schwinger, Phys.Rev. Ъ|4, 1087, 1966. И A.Peres, Phys.Rev. 16J, 1449, 1968, [4J B.M.Bolotovskij and Yu.A.Usachev, pp. 17-18 in Monopol Diraka, Edition Mir, Moscow, 1970. (in russian) И i.E.Tamm, Ztschr. Phys. 71, 141, 1931. И M.Fierz, Helv. Phys. Acta, 17, 27, 1944. [7] C.A.Hurst, Annals of Phys. SO, 51, 19b8. [8] A.Goldhaber, Phys.Rev. 140B 1407, 1965. [9] J. von Neumann, Göttinger Nachrichten, Math. Phys. 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[21] R.S.Burimjton, Handbook of Mathematical Tables and Formulas p.23, Handbook Publisher,Inc. Sandusky, Ohio 1958. - L 35 - FI6URE CAPTIONS Fig. 1 Notations for coordinates y Fig. 2 The closed string Sm ' Fig. Э The solid angle il(a,b»r) Fig. 4 The spherical triangle t(g2,g1). d2 stands for g2d, etc. Fig. 5 The spherical triangle т \ I - L i l Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Pintér György, a KFKI Részecske- és Magfizikai Tudományos Tanácsának szekció elnöke Szakmai lektor: Huszár Miklós Nyelvi lektor: H. Shenker Példányszám: 370 Törzsszám: 75-1389 Készült a KFKI sokszorosító üzemében Budapest, 1976. január hó