A conformal group approach to the Dirac-K\"ahler system on the lattice

TL;DR

This paper introduces a novel approach using conformal group representations to derive solutions for the Dirac-Kähler system on a lattice, connecting Lorentz pseudo-sphere geometry with group theory.

Contribution

It develops a method based on conformal group representations and Cayley transform to construct solutions to the Dirac-Kähler equation on the lattice, linking geometric and algebraic frameworks.

Findings

Derived a class of solutions using conformal group representations

Connected Lorentz pseudo-sphere geometry with the general linear group framework

Described solutions as a commutative n-ary product involving quasi-monomials

Abstract

Starting from the representation of the $(n-1)+n-$dimensional Lorentz pseudo-sphere on the projective space $\mathbb{P}\mathbb{R}^{n,n}$, we propose a method to derive a class of solutions underlying to a Dirac-K\"ahler type equation on the lattice. We make use of the Cayley transform $\varphi({\bf w})=\dfrac{1+{\bf w}}{1-{\bf w}}$ to show that the resulting group representation arise from the same mathematical framework as the conformal group representation in terms of the {\it general linear group} $GL\left(2,\Gamma(n-1,n-1)\cup\{ 0\}\right)$. That allows us to describe such class of solutions as a commutative $n-$ary product, involving the quasi-monomials $\varphi\left({\bf z}_j\right)^{-\frac{x_j}{h}}$ ($x_j \in h\mathbb{Z}$) with membership in the paravector space $\mathbb{R}\oplus \mathbb{R}{\bf e}_j{\bf e}_{n+j}$.