Dirac-Kaehler fermion with noncommutative differential forms on a lattice

TL;DR

This paper introduces a noncommutative differential form framework on a lattice, defining a new Clifford product that naturally yields the Dirac-Kähler fermion, advancing lattice fermion formulations.

Contribution

It proposes a novel associative Clifford product on a lattice using noncommutative differential forms, enabling a natural realization of Dirac-Kähler fermions.

Findings

Defined a noncommutative differential form framework on a lattice.

Established a new associative Clifford product compatible with lattice structure.

Demonstrated the emergence of Dirac-Kähler fermions from this framework.

Abstract

Noncommutativity between a differential form and a function allows us to define differential operator satisfying Leibniz's rule on a lattice. We propose a new associative Clifford product defined on the lattice by introducing the noncommutative differential forms. We show that this Clifford product naturally leads to the Dirac-K\"ahler fermion on the lattice.