Parasupersymmetry in quantum mechanics

Abstract

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^pQ₁⁺+Q₁^p-1 Q₁⁺Q₁+...+Q₁Q₁⁺Q₁^p-1+Q₁⁺Q₁^p =2pQ₁^p-1H, [H,Q₁]=0; Q₁^p+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^pQ₁⁺+Q₁⁺Q₁^p= 2Q₁^p-1H. 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.