Poincaré Parasuperalgebra with Central Charges and Parasupersymmetric Wess–Zumino Model

Supersymmetry (SUSY), introduced in theoretical physics and mathematics, has a lot of interesting applications [1]. One of them consists in mixing fermionic and bosonic states. It has important consequences in the quantum field theory, namely, this property provides a mechanism for cancellation of the ultraviolet divergences. Moreover, supersymmetric quantum field theory (SSQT) allows to unify the space symmetries of the Poincaré group with internal symmetries [2]. It allows to overcome the“no-go” theorem of Coleman and Mandula. Supersymmetric quantum field theory (SSQFT) induced appearance of supersymmetric quantum mechanics (SSQM) [3]. SSQM stimulated deeper understanding of ordinary quantum mechanics and provided new ways for solving some problems [4]. SSQM has been generalized to the parasupersymmetric quantum mechanics (PSSQM) [5]. The latter deals with bosons and p = 2 parafermions having parastatistical properties. Here p is the so-called paraquantization order [6]. Soon an independent version of PSSQM yielding to positive defined Hamiltonians was proposed [7]. The crucial step in developing PSSQM was made by Beckers and Debergh [8] who required Poincaré invariance of the theory and formulated the group-theoretical foundations of the socalled parasupersymmetric quantum field theory (PSSQFT). This theory is a natural generalization of SSQFT, dealing with the Poincaré parasupergroup (or Poincaré parasuperalgebra (PPSA)) instead of the Poincaré supergroup (or Poincaré superalgebra (PSA)). Recently IRs of the PPSA for N = 1 have been described [9] and then IRs for arbitrary N and internal symmetry group have been found [10, 11]. The present paper consists of two main parts. In the first part we consider the Poincaré parasuperalgebra with central charges. The second part includes the physical model, which is invariant under the Poincaré parasuperalgebra.