Parasupersymmetry and N-fold Supersymmetry in Quantum Many-Body Systems   II. Third Order
  Based on the general formalism of parafermionic algebra and parasupersymmetry proposed previously by us, we explicitly construct third-order parafermionic algebra and multiplication law, and then realize third-order parasupersymmetric quantum systems. We find some novel features in the third-order, namely, the emergence of a fermionic degree of freedom and of a generalized parastatistics. We show that for one-body cases the generalized Rubakov-Spiridonov model can be constructed also in our framework and find that it admits a generalized 3-fold superalgebra. We also find that a three-body system can have third-order parasupersymmetry where three independent supersymmetries are folded. In both cases, we also investigate the new concept of quasi-parasupersymmetry introduced by us and find that those of order (3,3) are indeed realized under less restrictive conditions than (ordinary) parasupersymmetric cases. 
Toshiaki Tanaka