2024/11/06 by Rumyantseva, S. V., Shirokov, D. S. · 1 citation
#15A66 #35Q41 #70S15 #81Q05 #81T13 #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2411.04214
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 ℂ⊗C \kern -0.1em ℓ1,n depends on the parity of n, we examine even and odd cases separately. In the geometric algebra C \kern -0.1em ℓ1,3, 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 ℓ1,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 ℓ1,n to prove a similar statement. We present the multidimensional Dirac-Hestenes equation in C \kern -0.1em ℓ1,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.