Ternary generalizations of Grassmann algebra

TL;DR

This paper introduces a ternary generalization of Grassmann algebra with ternary anti-commutativity, defining integrals and an analogue of the Pfaffian, expanding algebraic structures beyond classical fermionic systems.

Contribution

It develops a novel ternary Grassmann algebra framework, including integral definitions and explicit formulas for a cubic matrix Pfaffian analogue.

Findings

Defined integrals over ternary Grassmann algebras

Proved change of variables formula for these integrals

Calculated explicit form of the Pfaffian analogue for N=3

Abstract

We propose the ternary generalization of the classical anti-commutativity and study the algebras whose generators are ternary anti-commutative. The integral over an algebra with an arbitrary number of generators N is defined and the formula of a change of variables is proved. In analogy with the fermion integral we define an analogue of the Pfaffian for a cubic matrix by means of Gaussian type integral and calculate its explicit form in the case of N=3.