PARASUPERSYMMETRY IN QUANTUM-MECHANICS

The various aspects of parasupersymmetric quantum mechanics of one boson and one parafermion of (arbitrary) order p are discussed in some detail. In particular it is shown that the parasupersymmetry algebra is given by Q1(p)Q1+ + Q1(p-1)Q1+Q1+... + Q1Q1+Q1p-1 + Q1+Q1P = 2pQ1p-1 H, [H,Q1] = 0; Q1p+1 = 0 and the Hermitian conjugated relations where Q1 is the parasupercharge and H the Hamiltonian. It is also shown that such a system always possesses (p - 1 ) other conserved parasupercharges and p bosonic constants. Further, a special case is pointed out when the above algebra takes a very simple form as given by Q1Q1+Q1 = 2Q1H, Q1(p)Q1+ + Q1+Q1p = 2Q1p-1 H. A model of conformal parasupersymmetry of degree p is discussed and it is shown that in this case one has p supercharges, p conformal supercharges, and p bosonic constants which along with H, dilatation generator D, and conformal generator K form a closed algebra. A model of parasupersymmetry of degree p is discussed which is not conformal invariant and yet whose spectrum can be algebraically obtained by using the above conformally invariant supersymmetry model. Finally a model of parasupersymmetric quantum mechanics of one paraboson of (arbitrary) order q and one parafermion of order p is discussed.