Dirac theory on a space with linear Lie type fuzziness

TL;DR

This paper develops a Dirac spinor theory on a noncommutative space with Lie algebraic structure, exploring lattice-like properties, fermion doubling issues, and quantum corrections in a novel noncommutative geometric framework.

Contribution

It introduces a spinor model on Lie type noncommutative space using Fourier space, addressing fermion doubling and deriving Feynman rules with quantum corrections.

Findings

Space exhibits lattice characteristics with discrete eigenvalues.

A projection reduces spinor degrees of freedom to match ordinary space.

One-loop quantum corrections are computed for the model.

Abstract

A spinor theory on a space with linear Lie type noncommutativity among spatial coordinates is presented. The model is based on the Fourier space corresponding to spatial coordinates, as this Fourier space is commutative. When the group is compact, the real space exhibits lattice characteristics (as the eigenvalues of space operators are discrete), and the similarity of such a \emph{lattice} with ordinary lattices is manifested, among other things, in a phenomenon resembling the famous \emph{fermion doubling} problem. A projection is introduced to make the dynamical number of spinors equal to that corresponding to the ordinary space. The actions for free and interacting spinors (with Fermi-like interactions) are presented. The Feynman rules are extracted and 1-loop corrections are investigated.