Introducing Multidimensional Dirac-Hestenes Equation

TL;DR

This paper formulates a multidimensional Dirac-Hestenes equation within geometric algebra, explores its mathematical properties, and demonstrates its relation to the traditional Dirac equation, highlighting gauge invariance and solution correspondence.

Contribution

It extends the Dirac-Hestenes equation to multiple dimensions, analyzes algebraic decompositions, and establishes links to the standard Dirac equation in higher-dimensional settings.

Findings

Derived the multidimensional Dirac-Hestenes equation in Clifford algebra.

Proved the equivalence of solutions between multidimensional Dirac and Dirac-Hestenes equations.

Showed the gauge invariance of the multidimensional Dirac-Hestenes equation.

Abstract

It is easier to investigate phenomena in particle physics geometrically by exploring a real solution to the Dirac-Hestenes equation instead of a complex solution to the Dirac equation. The current research presents a formulation of the multidimensional Dirac-Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra $\mathbb{C}\otimes{C \kern -0.1em \ell}_{1,n}$ depends on the parity of $n$, we examine even and odd cases separately. In the geometric algebra ${C \kern -0.1em \ell}_{1,3}$, there is a lemma on a unique decomposition of an element of the minimal left ideal into the product of the idempotent and an element of the real even subalgebra. The lemma is used to construct the four-dimensional Dirac-Hestenes equation. The analogous lemma is not valid in the multidimensional case, since the dimension of the real even subalgebra of ${C \kern -0.1em \ell}_{1,n}$ is bigger than the dimension of the minimal left ideal for $n>4$. Hence, we consider the auxiliary real subalgebra of ${C \kern -0.1em \ell}_{1,n}$ to prove a similar statement. We present the multidimensional Dirac-Hestenes equation in ${C \kern -0.1em \ell}_{1,n}$. We prove that one might obtain a solution to the multidimensional Dirac-Hestenes equation using a solution to the multidimensional Dirac equation and vice versa. We also show that the multidimensional Dirac-Hestenes equation has gauge invariance.