A conformal group approach to the Dirac-Kähler system on the lattice

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-Kähler type equation on the lattice. We make use of the Cayley transform $\phi (\mathbf{w})=\frac{1+\mathbf{w}}{1-\mathbf{w}}$ 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 $\phi {\left({\mathbf{z}}_j\right)}^{-\frac{x_j}{h}}$ ($x_j\in h\mathbb{Z}$) with membership in the paravector space $\mathbb{R}\oplus \mathbb{R}{\mathbf{e}}_j{\mathbf{e}}_{n+j}$.

14 pages

Cited by in corpus (7)

• Relativistic Wave Equations on the lattice: an operational perspective
• On fractional semidiscrete Dirac operators of Lévy-Leblond type
• Time-changed Dirac-Fokker-Planck equations on the lattice
• A discrete Dirac-Kähler equation using a geometric discretisation scheme
• A note on the discrete Cauchy-Kovalevskaya extension
• A discrete version of plane wave solutions of the Dirac equation in the Joyce form
• Chiral properties of discrete Joyce and Hestenes equations