ar X iv : h ep - t h / 99 04 16 6 v 1 2 4 A pr 1 99 9 Degeneracy and Para - supersymmetry of Dirac Hamiltonian in ( 2 + 1 ) - Spacetime

The quantum mechanics of a spin 1 2 particle on a locally spatial constant curvature part of a (2 + 1)-spacetime in the presence of a constant magnetic field of a magnetic monopole has been investigated. It has been shown that these 2-dimensional Hamiltonians have the degeneracy group of SL(2, c), and para-supersymmetry of arbitrary order or shape invariance. Using this symmetry we have obtained its spectrum algebraically. The Dirac's quantization condition has been obtained from the repre-1 sentation theory. Also, it is shown that the presence of angular deficit suppresses both the degeneracy and shape invariance. Quantum theories, particularly quantum gravity in (2 + 1)-dimensions provide us with a useful field of investigation not only for theoretical and mathematical issues, but also in some cases, for actual physical problems [1, 2]. In the past decade many interesting physical problems in (2+1)-gravity have been solved, such as classical scattering, quantum scattering, 2 bound states of a scalar and spinor point particle both in the presence and absence of a magnetic monopole and also magnetic vortex [3, 4, 5, 6]. In reference [6] some interesting results have been obtained in studying quantum scattering and bound states of a scalar charged particle in the background metric corresponding to (2 + 1)-manifold both with local and global constant curvature in the presence of a magnetic monopole, which satisfy the coupled Einstein-Maxwell equations. Here in this article we investigate the quantum mechanics of a charged spin 1 2 point particle on a (2 + 1)-spacetime with spatial part of local constant curvature in the presence of a constant magnetic field of a magnetic monopole. We show that these 2-dimensional Hamiltonians have the degeneracy group of SL(2, c) type and para-supersymmetry of arbitrary order or shape invariance. Using these symmetries we have obtained their spectra algebraically. Also, the Dirac quantization follows naturally from the representation theory. In the case of local constancy of the curvature, the presence of angular deficit suppresses both the degeneracy and shape invariance. The paper is organised as follows: In section II we briefly describe the (2 + 1)-spacetime metric of reference [6] and assume that angular deficit is absent. Section III is devoted to the algorithm of the manipulation of the Dirac operator in these spacetimes. The Dirac operator has been given in terms of the generators of the sl(2, c) Lie algebra, which reduces the familiar Dirac operator on …