A note on Gaussian integrals over para-Grassmann variables

Abstract

We discuss the generalization of the connection between the determinant of an operator entering a quadratic form and the associated Gaussian path-integral valid for Grassmann variables to the para-Grassmann case [θp+1 = 0 with p = 1 (p > 1) for Grassmann (para-Grassmann) variables]. We show that the q-deformed commutation relations of the para-Grassmann variables lead naturally to consider q-deformed quadratic forms related to multiparametric deformations of GL(n) and their corresponding q-determinants. We suggest a possible application to the study of disordered systems.