Theoretical Physics

INTERNATIONAL CENTRE FOR lcmini THEORETICAL PHYSICS

DYNAMICAL GROUPS AND SUPERSYMMETRY II. SUPERSYMMETRIC AHARONOV-BOHM AND ANYON SYSTEMS

INTERNATIONAL A.O. Barut ATOMIC ENERGY AGENCY and

P. Roy UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MtRAMARE-TRIESTE

T T TT

IC/92/227

1. INTRODUCTION International Atomic Energy Agency In this series of paper we study supersymmetric models in the more general framework and of dynamical groups. The latter aims to describe the totality of the states of a quantum system, United Nations Educational Scientific and Cultural Organization i.e. the complete Hilbeit space and the observables, from a representation of a dynamical algebra INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS including the Poincart algebra. The supersymmetry is a substructure of a large dynamical group structure which compares neighbouring (or next-to-near neighbouring) subsystems. In a given representation of the dynamical algebra certain subalgebras also close under both commutators and anticommutators thus giving rise to supersymmetric algebras.

DYNAMICAL GROUPS AND SUPERSYMMETRY II. In the firstpape r [1J we explored these relations in the examples of the Posch-Teller and SUPERSYMMETRIC AHARONOV-BOHM AND ANYON SYSTEMS related problems and of the radial Coulomb problem. In the present paper we discuss the supersymmetric version of the dynamical group of the Aharonov-Bohm system [2] and the supersymmetric version of the anyon system in two-dimensional space [3], and identify explicitly the representations of the supersymmetric algebras that occur in these problems. A.O.Barut* and P.Roy" International Centre for Theoretical Physics, Trieste, Italy. 2. TWO-BODY AN YON-SYSTEM ABSTRACT We consider a supersymmetric N = 2 quantum mechanical system in two dimensions. Let ir,( k), i = 1,2, k - 1,2, be operators depending on the (bosonic) coordinates and momenta The group structure of the supersymmetric version of the Aharonov-Bohm Hamilton ian x,pof the two particles (kinetic momenta) and bdk), spinors in some representations, in our and that of two-anyon system include the same superalgebra $pl(2,1) whose representation in case, Dirac matrices to be specified below. Such a system may be specified by the form of su- these problems are identified. percharges [7]

V2 (1) MIRAMARE - TRIESTE Here the t's anticommute September 1992 (2) and the IT'S have the commutation relations

(3)

The bosonic and fermionic parts, of course, commute.

(4)

* Permanent address: Department of Physics, University of Colorado, Boulder, CO 80309, USA. ** Permanent address: Electronics Unit, Indian Statistical Institute, Calcutta - 700 035, India. The supersymmetric Hamiltonian is characterized by the algebra of supercharges

T TT For the spinor parts we choose

=T°, 6i(2) (11) and with (5)

Or, with and introduce the combinations (6) we have

6f2) = j (12)

[H,Q] = (7) so that the number operator jV> is given by

The particular physical system we consider is a two-dimensional two-particle system (13) interacting via a vector potential with kinetic momenta which evaluated with (11) is ,0,1,1) ""- (? -in) we also have = "Pi! -

*i(l) = Py, +

-X2) (14) (8) which anticommute. with In polar coordinates with x + iy ~ e'$,X + iY - Reiv the supercharges can now be P* =(i, ~x2y written as The vector potential is thu=

*>-B^

-12) (9)

The corresponding magnetic field vanishes except at the origin

B = VXl x A = 0 . [ e-" (-idn - ~ 9,) (6?,, - (15) We can separate center-of-mass and relative coordinates by setting

1 , We can separate the relative and center-of-mass parts of the supercharges because the fermionic 1 = 11 - £2 parts of the center-of-mass and relative supercharges anticommute.

(10) cm l mt "< + Q ,Q*™' + Q-™} = {Q" ,Q } (16)

4 Thus we define In the polar coordinates the complex combinations

(22) With respect to the fermionic operators the state space is partitioned into sectors with definite quantum numbers n?. Q* increases np by one unit, and Q decreases np by one unit. can be written in terms of a± = {ax ± ins)/2 as One recognizes another supersymmetry by considering two neighbouring Hamiltonians Q = -^ e"* (-id, -X-dB + A,-i^f\o,~ with nj? = 0 and tif = 2, Tor example. Taking them to be the isospectral partners we have (23) 1 d1 (B+TTO2-1/4 * = j= «" (-«flt + \ dt + Ar + i 4± • ™ JT + ~~ '•"" ~- = which define the operators q and q*. 1 d2 I)2-1/4 (17) A convenient gauge choice for the vector potential is or, written jointly A = ^(-y,x,0) (24)

£ satisfying div A = 0, rot A = 0 {r y 0). Thus

Ar = 0, As = B = const. =B+m+- (18) Hence In the next section, we show that the same radial Hamiltonian occurs in the supersym- Q = ^= e -{& _ I a, _ i £) o = , metric version of the Aharonov-Bohm effect. t ff (25)

3. AHARONOV-BOHM SYSTEM The Hamiltonian can now be written in the two-dimensional space as

In this section we study the dynamical algebra of the supersymmetric version of the (26) Aharonov-Bohm effect.

It is sufficient to consider the two-dimensional motion of a particle in the vector potential Denoting its eigenstates by ( *' J we have A of the solenoid. The dynamical variables of the supersymmetric model are p", x and the Pauli qq* matrices. The supercharges for this so-called N = 2 supersymmetric system are formed from (27) tensor products of these variables we separate the radial equations by setting l Q = -j= [(-Pv + Av)az + (px + Ax)ay]

2 (28) Q = -j= + AX)

r ~ T T" These radial equations themselves can be written in the 2-component form, thus exhibit- ing another "supersymmetry" [Q*,Q-] = : lQi,Q±] = ±Q± ±Q± (30) with supersymmetric generators

[Q±,V±] =

(31) so that we have aa* 0 (32)

To determine the superalgebra structure, we define the following set of operators

[K,^±] = -T (34)

The commutation and anticommutation relations in (34) are those of the Lie superalgebra Spl( 2,1) [8] whose even part Lie algebra is sl( 2) xg£(l) generated by Q+,Q_,Q3 and Y and whose odd part is generated by sl{2) spinors V±, W± carrying hypercharges Y - \ and Y = — j. The superalgebra Spl( 2,1) hap two Casimir operators Ki satisfying the relations

--*&$ --«• 1 = 1,2,3, m= 1,2,3, a=l,2,3,4 . (35)

/2 The explicit forms of the Casimir operators are given by [8]

v; = ~= 2 K2 =Q* -Y + ~UCU (33) K-i - YKi + ^rY UCU + i UQE TCU +-^-Ue fCUQ (36) 4 0 12 where a± = (ai ± ia2) /2, a, being the usual Pauli matrices and Q± = y (Q ] ± 1Q2) • where in matrix notation Now after a straightforward calculation one can verify the following commutation and anticommutation relations: -1 M • (37) 1 -1 Now using the generators (given in (34)) it can be shown that for our problem the Casimir REFERENCES operators both vanish identically

[1] A.O. Barut and P. Roy, in M. MosMnsky Festschrift (World Scientific, 1992).

(38) [2] W. Ehrenberg and R.E. Siday, Proc. Phys. Soc. London B62 (1949) 8; M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect (Springer-Verlag, Berlin, so that the systems considered here realize the simplest possible representations. 1989); L. O'Raifeartaigh, N. Straumann and A. Wipf, Comments Nucl. Part. Phys. 20 (1991) 15. 4. CONCLUSION [3] J. Leinass and J. Myrhcim, II Nuovo Cimento 37 (1977) 1; In this paper we have studied the SUSY two anyon and the Aharonov-Bohm systems. In R. Mackenzie and F. Wilczek, Int. J. Mod. Phys. A3 (1988) 2827; particular the dynamical superalgebra underlying these systems have been identified. Clearly the P.S. Gerbert. Int. J. Mod. Phys. A6 (1991) 173; solutions to these problems are free field solutions with shifted angular momentum. It is interesting F. Wilczek, Phys. Rev. Lett. 48 (1982) 114; ibid, 49 (1982) 957; to note that the configuration space of the two anyon system is non simply connected and does not F. Haldane, Phys. Rev. Lett. 51 (1983) 605; contain the origin i.e. two anyons cannot overlap [3]. Thus all wave functions are required to The Quantum Hall Effect, eds. R. Prange and S. Girvin (Springer-Verlag, 1987), vanish at the origin. However, in the SUSY Aharonov-Bohm case (which is a one particle system) [4] Z. Hlousek and D. Spector, Nucl. Phys. B344 (1990) 763. there is apriori no reason to exclude the origin and it can be shown [5, 9] that square integrable singular solutions are allowed in this case. We shall deal elsewhere with the question of whether [5] P. Roy and R. Tarrach, Phys. Lett. B274 (1992) 59; the symmetry superalgebra can be realized over such states. A.O. Barut and R. Wilson, Ann. Phys. 164 (1985) 223. [6] R. Jackiw, Ann. Phys. 201 (1990) 83. [7] M. de Crombrugghe and V. Rittenburg, Ann. Phys. 151 (1983) 99. [8] M. Scheunert, W. Nahm and V. Rittenburg, J. Math. Phys. 18 (1977) 155.

[9] C. Manuel and R. Tarrach, Phys. Lett. B268 (1991) 222.

Acknowledgments

One of the authors (P.R.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

r r TT