Multilevel systems and generalizations of the superalgebra

A new approach to constructing the various generalizations of the one-dimensional supersymmetric quantum mechanics is proposed, including the parasupersymmetric quantum mechanics constructed by Rubakov and Spiridonov as the special case. In particular, we derive the generalized superalgebra, which possesses the features both of the familiar superalgebra and of the parasuperalgebra. Namely, the generalized supercharges Q_i ^± and the Hamiltonian H forms the generalized superalgebra, where Q_i ^±2=0 (as for ordinary superalgebra), but the triple products of generalized supercharges obey the relations Q₁ ⁺Q_j ^–Q_j ⁺=Q_i ⁺H (i, j=1, 2) and Q_i ⁺Q_i ^–Q_j ⁺=(1/4)kQ_i ⁺, Q_i ⁺Q_i ^–Q_j ⁺=(1/4)kQ_i ⁺(i, j=1, 2; ij) (analogous to the parasuperalgebra). Furthermore, the generalized supercharges are conserved, i.e. H, Q_i ^±]=0.Presented at the International Workshop on Squeezed and Correlated States, Moscow, December 3–7, 1990.