A note on Gaussian integrals over paragrassmann variables

TL;DR

This paper extends the relationship between determinants and Gaussian integrals from grassmann variables to paragrassmann variables, introducing q-deformed algebraic structures and potential applications in disordered systems.

Contribution

It generalizes Gaussian integral connections to paragrassmann variables and explores q-deformed algebraic structures, opening new avenues for studying disordered systems.

Findings

q-deformed relations lead to q-determinants for paragrassmann variables

Establishment of a link between quadratic forms and multiparametric deformations of GL(n)

Potential application to disordered systems analysis

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 paragrassmann case [$\theta^{p+1}=0$ with $p=1$ ($p>1$) for grassmann (paragrassamann) variables]. We show that the q-deformed commutation relations of the paragrassmann 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.