Dirac-Kaehler Fermion from Clifford Product with Noncommutative Differential Form on a Lattice

TL;DR

This paper introduces a novel formulation of Dirac-Kaehler fermions on a lattice using a noncommutative Clifford product, connecting it to existing fermion actions and enabling lattice QCD applications.

Contribution

It develops a new Clifford product framework for Dirac-Kaehler fermions on a lattice, deriving known fermion actions and extending to lattice QCD.

Findings

Derived Kogut-Susskind and staggered fermion actions from the new formulation.

Established a lattice QCD action with Dirac-Kaehler matter fermion.

Ensured Hermiticity through specific lattice structures.

Abstract

We formulate Dirac-Kaehler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac-Kaehler action requires to choose the lattice structure having both orientabilities on a link. The Kogut-Susskind fermion and the staggered fermion actions are derived directly from the Dirac-Kaehler fermion formulated by the Clifford product. The lattice QCD action with Dirac-Kaehler matter fermion is also derived via an inner product defined by the Clifford product.