An 8-dimensional realization of the Clifford algebra in the 5-dimensional Galilean space-time

TL;DR

This paper constructs an 8-dimensional Clifford algebra representation in 5D Galilean space-time via dimensional reduction from 6D Minkowski space, and derives solutions to a Dirac-type equation in this setting.

Contribution

It introduces a novel 8-dimensional Clifford algebra realization in 5D Galilean space-time through dimensional reduction from 6D Minkowski space.

Findings

Explicit 8x8 gamma matrices for the algebra

Solutions to the Dirac-type equation in 5D Galilean space-time

Diagonalization of the Klein-Gordon divisor using Galilean boost

Abstract

We give an 8-dimensional realization of the Clifford algebra in the 5-dimensional Galilean space-time by using a dimensional reduction from the $(5+1)$ Minkowski space-time to the $(4+1)$ Minkowski space-time which encompasses the Galilean space-time. A set of solutions of the Dirac-type equation in the 5-dimensional Galilean space-time is obtained, based on the Pauli representation of $8\times 8$ gamma matrices. In order to find an explicit solution, we diagonalize the Klein-Gordon divisor by using the Galilean boost.