On Multidimensional Dirac–Hestenes Equation in Geometric Algebra

P. 323–335.

Rumiantseva S., Shirokov D.

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 present research outlines the 2d-dimensional Dirac–Hestenes equation. In the geometric algebra G_{1,3}, there is a lemma on the 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 G_{1,2d−1} is bigger than the dimension of the minimal left ideal for d>2. Hence, we consider the auxiliary real subalgebra of G_{1,2d−1} to prove a similar statement. We present the multidimensional Dirac–Hestenes equation for the case G_{1,2d−1}. 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.

Keywords: geometric algebra, Dirac-Hestenes equation, gauge invariance

In book

Advances in Computer Graphics: 41st Computer Graphics International Conference, CGI 2024, Geneva, Switzerland, July 1–5, 2024, Proceedings, Part III

Vol. 15340. , Springer, 2025.