- Email: [email protected]
Supersymmetry of the Pauli equation in the presence of a magnetic monopole
Volume 137B, number 1,2
PHYSICS LETTERS
22 March 1984
SUPERSYMMETRY OF THE PAULI EQUATION IN THE PRESENCE OF A MAGNETIC MONOPOLE ~
Eric D'HOKER Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute o f Technology, Cambridge, MA 02139, USA
and Luc VINET Laboratoire de Physique NucHaire, Universit3 de MontrOal, Montrdal, Qudbec H3C3J7, Canada
Received 14 December 1983
It is shown that the Pauli hamiltonian for a spin 1/2 particle in the presence of a Dirac magnetic monopole possesses a dynamical conformal OSp(1,1) supersymmetry. Using this symmetry, the spectrum is constructed explicitly, and all but the lowest angular momentum states transform under irreducible representations of the supergroup. The lowest angular momentum states transform under irreducible representations of the SO (2,1) - subgroup of OSp (1,1) only, because the supercharges are not self-adjoint in that sector.
Several systems in quantum mechanics may be completely and explicitly solved because they exhibit a large symmetry group. For example, the Schr6dinger equation for the hydrogen atom possesses, in addition to its rotation symmetry, a "hidden" SO(3) symmetry [1,2]. Also, the Schr6dinger equations for a particle in the presence of a 1/r 2 potential [3] or a Dirac magnetic monopole [4,5 ], exhibit a "dynamical" SO(2,1) symmetry. The spectra o f such systems may be calculated explicitly by group theoretical techniques only. In the present paper, we shall show that the Pauli hamiltonian H = (l[2m)(p i - eAi) 2 - (e/2m)Bia i ,
i = 1,2,3
(1) for a spin ~1 particle in the field o f a Dirac magnetic monopole possesses a dynamical supersymmetry. Here
1 m is the mass of the spin i particle,Bi =gqi/q 3 is the monopole magnetic field strength and A i the corresponding vector potential that can be defined by patches so as to avoid string singularities [6]. The lagrangian formulation is convenient to establish transformation laws and to find the conserved Noether charges. The hamiltonian (1) can be derived from a lagrangian L --a~ m ~ 2 + ½i ~ i ~i + eAiqi" + ( e l m ) B i S i ,
with S i = - ~1 eijk ~ j ~ k •
72
(3)
Here, ~i are the real generators of a Grassmann algebra and describe the spin degrees of freedom o f a classical particle [7]. This lagrangian is invariant under spatial rotations. The expression for the angular momentum is well known and given by Ji = ei/kq/qk - egqi + Si "
¢~This work is supported in part through funds provided by the US Department of Energy (DOE) under contract DEAC02-76ER03069 and by the Natural Science and Engineering Research Council (NSERC) of Canada.
(2)
(4)
Furthermore, under the supertransformations 8Q and 6S, given by
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 137B, number 1,2 6Qqi=(ia/v~)t~i
,
PHYSICS LETTERS 6Q~i=-a~qi
6 s q i = (i[3/x/m)tdgi ,
,
~)S~i = --[3~¢/-m(tqi - qi),
(5a) (5b)
the lagrangian L changes by a total time derivative. In (5), ~ and/3 are anticommuting c-number parameters. In fact, the invariance u n d e r ~iQ and 6 S holds only if 3iBi = 0; consequently the origin (qi = O) should be excluded. We shall discuss the implications of this fact at the end of the paper. From the action o f 6Q and 6S, we deduce corresponding conserved supercharges using Noether's theorem: Q = vCmqi ~i
(6a)
'
Besides these supercharges, there also exist the conserved charges generating dilations (D) and special conformal transformations [4] (K): 1
4m{qi,qi}
(7a)
,
K = - t 2 H + 2tD + ~1m~2 .
(7b)
Next, we canonically quantize the system [7]. The m o m e n t u m conjugate to qi is given by
and the variables now satisfy the commutation relation (9)
{ ~i, gg] } = 6i]"
The only irreducible representation for the Clifford algebra generated by the ffi's is two dimensional and equivalent to that realized by the Pauli matrices: ffi = oi/~/2. Thus, we have S i = el~2 , and the hamiltonian takes the standard Pauli form given in (1). The conserved charges H, D, K, Q and S can be shown to verify [H,D] = ill,
G = O(3)rotations ® OSp(1,1)superconforma 1 •
[H,K] = 2iD,
[D,K] = iK,
(10a)
[H,Q] = [ K , S ] = 0 ,
(10b)
[H, S] = - i Q ,
(10c)
To determine the spectrum, we construct the irreducible representations of G. It is convenient to introduce a Caftan basis for the OSp(1,1) algebra: R = (1/2a 2) K + ½a2H,
(12a)
B+ = ( 1 / 2 a 2 ) K -
(12b)
[K, Q] = i S , ,
~a2H + iD , 1 •
(12c)
F+ = ( 1 / 2 a ) S ~ ~laQ ,
with B_+ = (F_+, F+ }. Here the parameter a is real and positive and has dimensions of the square root of length. The generator R is a compact operator with a discrete spectrum, whereas B± and/7_+ are raising (+) and lowering ( - ) operators, satisfying JR,B+] =+-B+_ ,
[B.,~_] =-2R,
1
[R,F+_] =+-~F+_ ,
(13a)
{F.,F }--R.
(13b)
As usual, the representations of the group G are completely specified by the eigenvalues of all Casimir operators of a canonical chain of subgroups of G. For the rotation group these Casimirs are j2 and J z . For the supergroup OSp(1,1), the canonical chain is OSp(1,1) D SO(2,1) D SO(2)
[D,S] = ~1. iS,
(10d)
(14)
and we shall call the corresponding Casimirs C, C O and R. R is given by (12a) and the Casimir C O is well known [4] and may be expressed in terms of the charges H , D and K, or R and B+ : C O -I{HK , }-DZ=R2~
aQ [D, QI = - ~~.
- ½(B+,B_}.
£Q,S} = -2D,
{S,S } =2K.
(lOe)
(15)
To construct the Casimir of OSp(1,1) we first compute C O in terms of the dynamical variables. We find CO = k (j2 _ Ji °i - egqi °i - eZg2) •
{Q,Q} = 2H,
(11)
(8)
Pi = ~L/~qi = m[li + eAi
[qi, P/l = ifi i ,
Again, these relations only hold when OiBi = 0. Also note that the commutation relations of these charges with H follow from the fact that they are time independent. Eqs. (10) constitute the structure equations for a superalgebra (or graded algebra) of rank 1, called OSp(1,1) [8] and satisfy a graded Jacobi identity. Furthermore, it is clear that all five charges H, D, K, Q and S commute with Ji" As a consequence, the full supersymmetry algebra of eq. (1) is
(6b)
S = - t Q + ~/mqi ~i •
O = tH-
22 March 1984
(16)
Then, we also compute the quantity A =i [Q,S] - ~x = - 2 [ F + , F _ ] -
1
(17)
73
V o l u m e 1 3 7 B , n u m b e r 1,2
PHYSICS LETTERS
J2lj, m,a,n)=j(/
in terms of dynamical variables, and we find ^ oi - ~x . A = oiJ i + egqi
22 M a r c h 1 9 8 4
(18)
(24a)
+ 1)lj, m , a , n ) ,
Jzlj,m,a,n)=mlj,
(24b)
m,a,n),
Clearly the combination
C=Co +kA +~ =&(j2 _ e2g2 +~_)
(19)
commutes with H, D, K, Q and S, since these charges are all rotation invariant, so that C is the quadratic Casimir o f OSp (1,1). The operator A itself is SO (2,1) invariant, and we shall henceforth choose to diagonalize A instead of C 0. We see from eq. (19), that in the magnetic monopole problem, the only representations of G we need are those for which the Casimirs j2 of the rotation group and C of the superconformal group are related. Finally, the Casimir eigenvalues of A (or equivalently of CO) are related to those of C. Indeed, upon expressing C O in terms of Q and S only, we find 1
C O = - g [Q,S] 2 - ~-i [Q,S] .
(20)
With the help of (17) and (19), we can express C in terms of A, and we find (21)
C=~A 2 .
From the above discussion, we see that the independent quantum numbers of the states of the theory are the eigenvalues o f J 2 , J z , sign (A) and R. The eigenvalues o f J 2 and Jz are denoted by ] ( / + 1) and m respectively. From standard arguments we have legl + ~1 . . . . .
j=legl-½,
(22)
The eigenvalue of C is computed from (19) and equals 41-d 2 , where d] = + [(/+ ~)2 _ e2g2]l/2 .
(23a)
From (21), the eigenvalue of A is seen to be od/., with c~= -+1. It remains to determine the eigenvalues o f R . From the properties of the SO (2,1) group, as applied to the magnetic monopole problem [4], we know that for a representation with Casimir eigenvalue C O = ~],c~(6],c~ - 1), the eigenvalues of R are r n = 6y,~ + n, with n integer. The value of 61,~ is readily computed from (19) and we find 1
1
1
6], e, = ~ - ~ a + ~ d] .
(23b)
Putting all together, we find the following expressions for the quantum numbers:
74
a,n) ,
(24c)
+n)lj,m,a,n).
(24d)
Alj,m, a,n)=odjlj,m, RIj,m,a,n)=(Sj,,~
It is straightforward to determine the action of the raising and lowering operators on these states. J+lj,m,a,n)
= [j(/'+ 1 ) - m ( m
+ 1)]l/21/,m + 1, a , n ) ,
(2Sa)
= [(6/,a + n)(61,a + n +- 1)
B+_lj, m , a , n )
- 8i, c~(6j, ~ - 1 ) ] l / 2 [ j , m , a , n
± 1) ,
(25b)
F+lj, m , c~,n)
=
[½(6j,
+n)+_
~_± ~1 ~djll/2 Ij, m , - a , n -
~1 a + l 5)
(25c) From (25c), we see that there is a unique state annihilated by F+: F+Ij, m, - 1 , - 1 ) = 0 ,
(26a)
and a unique state annihilated by F _ : F _ I1, m, 1,0) = 0 .
(26b)
Both (26a) and (26b) define irreducible representations X.~ o f G The first one, X;-, obtained by applyN/ " Y -I• i n g F to I j , m , - 1 , - 1 ) , the second one X;, obtained N by applying F~ to I j , m , 1,0). Since the operatorR is positive, only X7 is an admissible representation of the spectrum for the system at hand. The states (or weights) of X7 are presented in fig. la. Now that we have all the states, it is easy to construct the wave function. Starting with a basis II, lz ) ® 1½, Sz) for the angular dependence, we pass to a new basis IJ, m, a) in which j 2 , Jz and A are simultaneously diagonal. This change of basis is performed with the help of standard Clebsch-Jordan coefficients. In the latter basis, we determine the wave function for the case m = - j , and we find
Volume 137B, number 1,2
PHYSICS LETTERS
22 March 1984
Upon remarking that d2 -
ii
odj = (26],.
-
-
(32)
we recognize (31) as the radial equation for the harmonic oscillator, and we readily derive its solutions {bn (r) = N n (mr 2/a 2)6j, ~ - 3/4 X exp(-mr2/262)L28A~-l(Mr2/a2).
stote
-1
T
'0
1
o
b
(33)
Here N n is the normalization and L28j, a - 1 are generalized Laguerre polynomials. So far, we have worked in a basis in which the operator R is diagonal, but we may also choose to diagonalize the hamilton]an H. The states are now labelled by the quantum numbers], m, c~ and the energy E, so that
Fig. 1. The spectrum h/+ of the Paul] equation in the presence of a Dirac magnetic monopole is presented in (a) for the sector/ > legl + 1/2 and in (b) for the sector] = legl - 1/2. The dots represent states, the simple (respectively double) arrows represent the effect of raising operators F + (respectively B+).
HI]], m, a, E ) = El]], m , a, E ) .
(0, ¢, el]], - j , eL)
Also, the supercharge Q is now a raising (and lowering) operator for the quantum number a
![NY]+I/2,_]_
Q [//, m, a, E ) = ~ [ j ,
1/2(0, ~b)
='1~1~
Y]+ 1/2,-]+ 1/2( 0 , ~) +M]~ Y]- 1/2,-j+ 1/2( 0 ,~b~
(27) where o takes the values 1 or 2 according to whether the upper or lower component is chosen. Here Y are the standard monopole harmonics for a spinless particle [6], and N is a normalization constant and M~ is given by M / = N [ ( / + 1)/eg] [1 - a(j + ~)/di] .
(28)
The wave functions for states with m =~ -//are obtained by applying j]++m to the wave function (0, ~, o1//, - / , a). It is clear that the radial dependence of the wave function is factorized, so that (r, O, ~, ol//,m, a , n ) = {bn(r)( O, dp, ol//,m, a) .
(29)
The radial function {bn(r) is readily determined (see re]. [4]) from (24d), with the help o f the identity: 1 R = (1/2a 2) K + ( a 2 / 2 K ) ( C + D 2 - iD - 74 A - 1~6), (30) and we find that {bn (r) satisfies
( 1 1 6 2
1 m ) r +-(d12. - o d j ) - - r 2 {bn(r) 2m r dr 2 2mr 2 + 264
= (2/62)(8],~ + n) {bn(r).
(31)
(34)
m, - a , E ) ,
(35)
and we see that states again fall into irreducible, two dimensional representations at any given non-zero energy. The radial equation for the energy eigenfunctions is obtained with the help of an identity similar to (28) and we find (
1 1 d2 I ) . . . . r + (d2 - oed]) {bE(r ) =E{bE(r ) 2m r dr 2 2mr 2 (36) The solution to (36) are Bessel functions {bE(r) = N E r - 1 / 2 J28i, ~ - 1 ( ~
r) .
(37)
So far, we have assumed that OiBi = 0 in constructing the super-algebra and its representations. For the magnetic monopole, this relation is of course violated and we have instead ~iBi = 4rrg83 (r). As a consequence, the supersymmetry algebra is realized only on those states whose wave]unction vanishes at the origin, thus missing the contributions of the Dirac 8-function. This 1 condition is fulfilled when//~> legl + ~, but fails when 1 1 //= legl - ~ where d! = 0. So the//= legl - ~ states cannot fall into representations of the full OSp (1,1) supergroup. It must be remarked that the identity OiBi = 0 has been used only in the derivation of the supercharges Q and S. Thus the operators H, D and K are still symmetry generators and span an SO(2,1) algebra. In fact, one can deduce the lack of supersymmetry in the//= 75
Volume 137B, number 1,2
PHYSICS LETTERS
1
[egl - ~ mode also directly from the supercharges. Remember that Q is nothing but the helicity operator, which for the Dirac magnetic monopole in the j = legl - ½ sector is not self-adjoint [9]. Worse, its deficiency indices [10] are n+ = 2 and n _ = 0, so that it can never admit any self-adjoint extension. In other words, Q is not a legitimate quantum operator. The same holds for S, as evolution is proportional to Q. It is also known [9] that the hamiltonian is not selfadjoint in the j = legl - ~ sector. However, its deficiency indices are equal to n_+ = 1 and it admits the well known one parameter family of self-adjoint extensions [10], and it can be made into a legitimate quantum operator. The same holds true for the operators D and K. Thus, the states are now labelled by only three quantum numbers/', m and n, and the single tower of states is depicted in fig. lb.
We are grateful to Professor Roman Jackiw for several stimulating conversations and for making unpublished notes by S. Fubini on the supersymmetric 1/r 2 potential available to us. We also thank Dr. Baha Balantekin and Professor Edward Farhi, Professor Daniel Freedman and Professor Francesco Iachello for helpful discussions. One of us (L.V.) would like to thank MIT for the warm hospitality extended to him during the course of this work.
76
22 March 1984
References
[1] v. Fock, Z. Phys. 98 (1935) 145; V. Bargmann, Z. Phys. 99 (1936) 576. [2] M. Bander and C. Itzykson, Rev. Mod. Phys. 38 (1966) 330,346; M.J. Englefield, Group theory and the Coulomb problem (Wiley-Interscience, New York, 1972). [3] V. De Alfaro, S. Fubini and G. Furlan, Nuovo Cimento A34 (1976) 569. [4] R. Jackiw, Ann. Phys. (NY) 129 (1980) 183. [5 ] J. Beckers, J. Harnad, M. Perroud and P. Winternitz, J. Math. Phys. 19 (1978) 2126. [6] T.T. Wu and C.N. Yang, Nucl. Phys. B107 (1976) 365. [7] F.A. Berezin and M.S. Marinov, JETP Lett. 21 (1975) 30; Ann. Phys. (NY) 104 (1977) 336; R. Casalbuoni, Phys. Lett. 62B (1976) 49; Nuovo Cimento A33 (1976) 389; B. Zumino, in: Fundamentals of quark models, Proc. 17th Scottish Universities Summer School in Physics, 1976, eds. I.M. Baboni and A.T. Davies. [8] E. D'Hoker, Phys. Rev. D28 (1983) 1346. [9] Y. Kazama, C.N. Yang and A.S. Goldhaber, Phys. Rev. D15 (1977) 2287; A.S. Goldhaber, Phys. Rev. D16 (1977) 1815; C. Callias, Phys. Rev. D16 (1977) 3168. [10] N. Akhiezer and I. Glazman, Theory of linear operators in Hilbert space, Vol. II (F. Ungar, New York, 1978).
PHYSICS LETTERS
22 March 1984
SUPERSYMMETRY OF THE PAULI EQUATION IN THE PRESENCE OF A MAGNETIC MONOPOLE ~
Eric D'HOKER Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute o f Technology, Cambridge, MA 02139, USA
and Luc VINET Laboratoire de Physique NucHaire, Universit3 de MontrOal, Montrdal, Qudbec H3C3J7, Canada
Received 14 December 1983
It is shown that the Pauli hamiltonian for a spin 1/2 particle in the presence of a Dirac magnetic monopole possesses a dynamical conformal OSp(1,1) supersymmetry. Using this symmetry, the spectrum is constructed explicitly, and all but the lowest angular momentum states transform under irreducible representations of the supergroup. The lowest angular momentum states transform under irreducible representations of the SO (2,1) - subgroup of OSp (1,1) only, because the supercharges are not self-adjoint in that sector.
Several systems in quantum mechanics may be completely and explicitly solved because they exhibit a large symmetry group. For example, the Schr6dinger equation for the hydrogen atom possesses, in addition to its rotation symmetry, a "hidden" SO(3) symmetry [1,2]. Also, the Schr6dinger equations for a particle in the presence of a 1/r 2 potential [3] or a Dirac magnetic monopole [4,5 ], exhibit a "dynamical" SO(2,1) symmetry. The spectra o f such systems may be calculated explicitly by group theoretical techniques only. In the present paper, we shall show that the Pauli hamiltonian H = (l[2m)(p i - eAi) 2 - (e/2m)Bia i ,
i = 1,2,3
(1) for a spin ~1 particle in the field o f a Dirac magnetic monopole possesses a dynamical supersymmetry. Here
1 m is the mass of the spin i particle,Bi =gqi/q 3 is the monopole magnetic field strength and A i the corresponding vector potential that can be defined by patches so as to avoid string singularities [6]. The lagrangian formulation is convenient to establish transformation laws and to find the conserved Noether charges. The hamiltonian (1) can be derived from a lagrangian L --a~ m ~ 2 + ½i ~ i ~i + eAiqi" + ( e l m ) B i S i ,
with S i = - ~1 eijk ~ j ~ k •
72
(3)
Here, ~i are the real generators of a Grassmann algebra and describe the spin degrees of freedom o f a classical particle [7]. This lagrangian is invariant under spatial rotations. The expression for the angular momentum is well known and given by Ji = ei/kq/qk - egqi + Si "
¢~This work is supported in part through funds provided by the US Department of Energy (DOE) under contract DEAC02-76ER03069 and by the Natural Science and Engineering Research Council (NSERC) of Canada.
(2)
(4)
Furthermore, under the supertransformations 8Q and 6S, given by
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 137B, number 1,2 6Qqi=(ia/v~)t~i
,
PHYSICS LETTERS 6Q~i=-a~qi
6 s q i = (i[3/x/m)tdgi ,
,
~)S~i = --[3~¢/-m(tqi - qi),
(5a) (5b)
the lagrangian L changes by a total time derivative. In (5), ~ and/3 are anticommuting c-number parameters. In fact, the invariance u n d e r ~iQ and 6 S holds only if 3iBi = 0; consequently the origin (qi = O) should be excluded. We shall discuss the implications of this fact at the end of the paper. From the action o f 6Q and 6S, we deduce corresponding conserved supercharges using Noether's theorem: Q = vCmqi ~i
(6a)
'
Besides these supercharges, there also exist the conserved charges generating dilations (D) and special conformal transformations [4] (K): 1
4m{qi,qi}
(7a)
,
K = - t 2 H + 2tD + ~1m~2 .
(7b)
Next, we canonically quantize the system [7]. The m o m e n t u m conjugate to qi is given by
and the variables now satisfy the commutation relation (9)
{ ~i, gg] } = 6i]"
The only irreducible representation for the Clifford algebra generated by the ffi's is two dimensional and equivalent to that realized by the Pauli matrices: ffi = oi/~/2. Thus, we have S i = el~2 , and the hamiltonian takes the standard Pauli form given in (1). The conserved charges H, D, K, Q and S can be shown to verify [H,D] = ill,
G = O(3)rotations ® OSp(1,1)superconforma 1 •
[H,K] = 2iD,
[D,K] = iK,
(10a)
[H,Q] = [ K , S ] = 0 ,
(10b)
[H, S] = - i Q ,
(10c)
To determine the spectrum, we construct the irreducible representations of G. It is convenient to introduce a Caftan basis for the OSp(1,1) algebra: R = (1/2a 2) K + ½a2H,
(12a)
B+ = ( 1 / 2 a 2 ) K -
(12b)
[K, Q] = i S , ,
~a2H + iD , 1 •
(12c)
F+ = ( 1 / 2 a ) S ~ ~laQ ,
with B_+ = (F_+, F+ }. Here the parameter a is real and positive and has dimensions of the square root of length. The generator R is a compact operator with a discrete spectrum, whereas B± and/7_+ are raising (+) and lowering ( - ) operators, satisfying JR,B+] =+-B+_ ,
[B.,~_] =-2R,
1
[R,F+_] =+-~F+_ ,
(13a)
{F.,F }--R.
(13b)
As usual, the representations of the group G are completely specified by the eigenvalues of all Casimir operators of a canonical chain of subgroups of G. For the rotation group these Casimirs are j2 and J z . For the supergroup OSp(1,1), the canonical chain is OSp(1,1) D SO(2,1) D SO(2)
[D,S] = ~1. iS,
(10d)
(14)
and we shall call the corresponding Casimirs C, C O and R. R is given by (12a) and the Casimir C O is well known [4] and may be expressed in terms of the charges H , D and K, or R and B+ : C O -I{HK , }-DZ=R2~
aQ [D, QI = - ~~.
- ½(B+,B_}.
£Q,S} = -2D,
{S,S } =2K.
(lOe)
(15)
To construct the Casimir of OSp(1,1) we first compute C O in terms of the dynamical variables. We find CO = k (j2 _ Ji °i - egqi °i - eZg2) •
{Q,Q} = 2H,
(11)
(8)
Pi = ~L/~qi = m[li + eAi
[qi, P/l = ifi i ,
Again, these relations only hold when OiBi = 0. Also note that the commutation relations of these charges with H follow from the fact that they are time independent. Eqs. (10) constitute the structure equations for a superalgebra (or graded algebra) of rank 1, called OSp(1,1) [8] and satisfy a graded Jacobi identity. Furthermore, it is clear that all five charges H, D, K, Q and S commute with Ji" As a consequence, the full supersymmetry algebra of eq. (1) is
(6b)
S = - t Q + ~/mqi ~i •
O = tH-
22 March 1984
(16)
Then, we also compute the quantity A =i [Q,S] - ~x = - 2 [ F + , F _ ] -
1
(17)
73
V o l u m e 1 3 7 B , n u m b e r 1,2
PHYSICS LETTERS
J2lj, m,a,n)=j(/
in terms of dynamical variables, and we find ^ oi - ~x . A = oiJ i + egqi
22 M a r c h 1 9 8 4
(18)
(24a)
+ 1)lj, m , a , n ) ,
Jzlj,m,a,n)=mlj,
(24b)
m,a,n),
Clearly the combination
C=Co +kA +~ =&(j2 _ e2g2 +~_)
(19)
commutes with H, D, K, Q and S, since these charges are all rotation invariant, so that C is the quadratic Casimir o f OSp (1,1). The operator A itself is SO (2,1) invariant, and we shall henceforth choose to diagonalize A instead of C 0. We see from eq. (19), that in the magnetic monopole problem, the only representations of G we need are those for which the Casimirs j2 of the rotation group and C of the superconformal group are related. Finally, the Casimir eigenvalues of A (or equivalently of CO) are related to those of C. Indeed, upon expressing C O in terms of Q and S only, we find 1
C O = - g [Q,S] 2 - ~-i [Q,S] .
(20)
With the help of (17) and (19), we can express C in terms of A, and we find (21)
C=~A 2 .
From the above discussion, we see that the independent quantum numbers of the states of the theory are the eigenvalues o f J 2 , J z , sign (A) and R. The eigenvalues o f J 2 and Jz are denoted by ] ( / + 1) and m respectively. From standard arguments we have legl + ~1 . . . . .
j=legl-½,
(22)
The eigenvalue of C is computed from (19) and equals 41-d 2 , where d] = + [(/+ ~)2 _ e2g2]l/2 .
(23a)
From (21), the eigenvalue of A is seen to be od/., with c~= -+1. It remains to determine the eigenvalues o f R . From the properties of the SO (2,1) group, as applied to the magnetic monopole problem [4], we know that for a representation with Casimir eigenvalue C O = ~],c~(6],c~ - 1), the eigenvalues of R are r n = 6y,~ + n, with n integer. The value of 61,~ is readily computed from (19) and we find 1
1
1
6], e, = ~ - ~ a + ~ d] .
(23b)
Putting all together, we find the following expressions for the quantum numbers:
74
a,n) ,
(24c)
+n)lj,m,a,n).
(24d)
Alj,m, a,n)=odjlj,m, RIj,m,a,n)=(Sj,,~
It is straightforward to determine the action of the raising and lowering operators on these states. J+lj,m,a,n)
= [j(/'+ 1 ) - m ( m
+ 1)]l/21/,m + 1, a , n ) ,
(2Sa)
= [(6/,a + n)(61,a + n +- 1)
B+_lj, m , a , n )
- 8i, c~(6j, ~ - 1 ) ] l / 2 [ j , m , a , n
± 1) ,
(25b)
F+lj, m , c~,n)
=
[½(6j,
+n)+_
~_± ~1 ~djll/2 Ij, m , - a , n -
~1 a + l 5)
(25c) From (25c), we see that there is a unique state annihilated by F+: F+Ij, m, - 1 , - 1 ) = 0 ,
(26a)
and a unique state annihilated by F _ : F _ I1, m, 1,0) = 0 .
(26b)
Both (26a) and (26b) define irreducible representations X.~ o f G The first one, X;-, obtained by applyN/ " Y -I• i n g F to I j , m , - 1 , - 1 ) , the second one X;, obtained N by applying F~ to I j , m , 1,0). Since the operatorR is positive, only X7 is an admissible representation of the spectrum for the system at hand. The states (or weights) of X7 are presented in fig. la. Now that we have all the states, it is easy to construct the wave function. Starting with a basis II, lz ) ® 1½, Sz) for the angular dependence, we pass to a new basis IJ, m, a) in which j 2 , Jz and A are simultaneously diagonal. This change of basis is performed with the help of standard Clebsch-Jordan coefficients. In the latter basis, we determine the wave function for the case m = - j , and we find
Volume 137B, number 1,2
PHYSICS LETTERS
22 March 1984
Upon remarking that d2 -
ii
odj = (26],.
-
-
(32)
we recognize (31) as the radial equation for the harmonic oscillator, and we readily derive its solutions {bn (r) = N n (mr 2/a 2)6j, ~ - 3/4 X exp(-mr2/262)L28A~-l(Mr2/a2).
stote
-1
T
'0
1
o
b
(33)
Here N n is the normalization and L28j, a - 1 are generalized Laguerre polynomials. So far, we have worked in a basis in which the operator R is diagonal, but we may also choose to diagonalize the hamilton]an H. The states are now labelled by the quantum numbers], m, c~ and the energy E, so that
Fig. 1. The spectrum h/+ of the Paul] equation in the presence of a Dirac magnetic monopole is presented in (a) for the sector/ > legl + 1/2 and in (b) for the sector] = legl - 1/2. The dots represent states, the simple (respectively double) arrows represent the effect of raising operators F + (respectively B+).
HI]], m, a, E ) = El]], m , a, E ) .
(0, ¢, el]], - j , eL)
Also, the supercharge Q is now a raising (and lowering) operator for the quantum number a
![NY]+I/2,_]_
Q [//, m, a, E ) = ~ [ j ,
1/2(0, ~b)
='1~1~
Y]+ 1/2,-]+ 1/2( 0 , ~) +M]~ Y]- 1/2,-j+ 1/2( 0 ,~b~
(27) where o takes the values 1 or 2 according to whether the upper or lower component is chosen. Here Y are the standard monopole harmonics for a spinless particle [6], and N is a normalization constant and M~ is given by M / = N [ ( / + 1)/eg] [1 - a(j + ~)/di] .
(28)
The wave functions for states with m =~ -//are obtained by applying j]++m to the wave function (0, ~, o1//, - / , a). It is clear that the radial dependence of the wave function is factorized, so that (r, O, ~, ol//,m, a , n ) = {bn(r)( O, dp, ol//,m, a) .
(29)
The radial function {bn(r) is readily determined (see re]. [4]) from (24d), with the help o f the identity: 1 R = (1/2a 2) K + ( a 2 / 2 K ) ( C + D 2 - iD - 74 A - 1~6), (30) and we find that {bn (r) satisfies
( 1 1 6 2
1 m ) r +-(d12. - o d j ) - - r 2 {bn(r) 2m r dr 2 2mr 2 + 264
= (2/62)(8],~ + n) {bn(r).
(31)
(34)
m, - a , E ) ,
(35)
and we see that states again fall into irreducible, two dimensional representations at any given non-zero energy. The radial equation for the energy eigenfunctions is obtained with the help of an identity similar to (28) and we find (
1 1 d2 I ) . . . . r + (d2 - oed]) {bE(r ) =E{bE(r ) 2m r dr 2 2mr 2 (36) The solution to (36) are Bessel functions {bE(r) = N E r - 1 / 2 J28i, ~ - 1 ( ~
r) .
(37)
So far, we have assumed that OiBi = 0 in constructing the super-algebra and its representations. For the magnetic monopole, this relation is of course violated and we have instead ~iBi = 4rrg83 (r). As a consequence, the supersymmetry algebra is realized only on those states whose wave]unction vanishes at the origin, thus missing the contributions of the Dirac 8-function. This 1 condition is fulfilled when//~> legl + ~, but fails when 1 1 //= legl - ~ where d! = 0. So the//= legl - ~ states cannot fall into representations of the full OSp (1,1) supergroup. It must be remarked that the identity OiBi = 0 has been used only in the derivation of the supercharges Q and S. Thus the operators H, D and K are still symmetry generators and span an SO(2,1) algebra. In fact, one can deduce the lack of supersymmetry in the//= 75
Volume 137B, number 1,2
PHYSICS LETTERS
1
[egl - ~ mode also directly from the supercharges. Remember that Q is nothing but the helicity operator, which for the Dirac magnetic monopole in the j = legl - ½ sector is not self-adjoint [9]. Worse, its deficiency indices [10] are n+ = 2 and n _ = 0, so that it can never admit any self-adjoint extension. In other words, Q is not a legitimate quantum operator. The same holds for S, as evolution is proportional to Q. It is also known [9] that the hamiltonian is not selfadjoint in the j = legl - ~ sector. However, its deficiency indices are equal to n_+ = 1 and it admits the well known one parameter family of self-adjoint extensions [10], and it can be made into a legitimate quantum operator. The same holds true for the operators D and K. Thus, the states are now labelled by only three quantum numbers/', m and n, and the single tower of states is depicted in fig. lb.
We are grateful to Professor Roman Jackiw for several stimulating conversations and for making unpublished notes by S. Fubini on the supersymmetric 1/r 2 potential available to us. We also thank Dr. Baha Balantekin and Professor Edward Farhi, Professor Daniel Freedman and Professor Francesco Iachello for helpful discussions. One of us (L.V.) would like to thank MIT for the warm hospitality extended to him during the course of this work.
76
22 March 1984
References
[1] v. Fock, Z. Phys. 98 (1935) 145; V. Bargmann, Z. Phys. 99 (1936) 576. [2] M. Bander and C. Itzykson, Rev. Mod. Phys. 38 (1966) 330,346; M.J. Englefield, Group theory and the Coulomb problem (Wiley-Interscience, New York, 1972). [3] V. De Alfaro, S. Fubini and G. Furlan, Nuovo Cimento A34 (1976) 569. [4] R. Jackiw, Ann. Phys. (NY) 129 (1980) 183. [5 ] J. Beckers, J. Harnad, M. Perroud and P. Winternitz, J. Math. Phys. 19 (1978) 2126. [6] T.T. Wu and C.N. Yang, Nucl. Phys. B107 (1976) 365. [7] F.A. Berezin and M.S. Marinov, JETP Lett. 21 (1975) 30; Ann. Phys. (NY) 104 (1977) 336; R. Casalbuoni, Phys. Lett. 62B (1976) 49; Nuovo Cimento A33 (1976) 389; B. Zumino, in: Fundamentals of quark models, Proc. 17th Scottish Universities Summer School in Physics, 1976, eds. I.M. Baboni and A.T. Davies. [8] E. D'Hoker, Phys. Rev. D28 (1983) 1346. [9] Y. Kazama, C.N. Yang and A.S. Goldhaber, Phys. Rev. D15 (1977) 2287; A.S. Goldhaber, Phys. Rev. D16 (1977) 1815; C. Callias, Phys. Rev. D16 (1977) 3168. [10] N. Akhiezer and I. Glazman, Theory of linear operators in Hilbert space, Vol. II (F. Ungar, New York, 1978).