POINCARE INVARIANCE AND QUANTUM PARASUPERFIELDS

Parasupersymmetries of arbitrary order p of paraquantization are seen in quantum field theory by requiring Poincare invariance in the N = 1, D = 4 context. We construct the corresponding minimal Lie parasuperalgebra as well as its (two) Casimir operators. Through its little parasuperalgebra in the timelike case, we characterize the irreducible (unitary) representations by constructing a generalized vacuum state with its partners. In the first nontrivial case, p = 2, we realize the operators in terms of para-Grassmannian variables and deduce parasupersymmetric equations leading to relativistic descriptions of spin 0, 1/2 and 1 (para)particles by considering parasuperfields and their para-Grassmannian components. In fact, we consider two nonequivalent realizations leading respectively to free and interacting relativistic descriptions, the latter corresponding to the extension of the Wess-Zumino model in this parasupercontext. The inclusion of symmetries and supersymmetries in the parasupersymmetries is enhanced.