On multidimensional Dirac-Hestenes equation
Sofia Rumyantseva -- HSE University
The four-dimensional Dirac--Hestenes equation is equivalent to the four-dimensional Dirac equation. One of the advantages to investigate the Dirac--Hestenes equation is that solutions of this equation are real. We present the multidimensional Dirac--Hestenes equation in the geometric algebra formalism. Since the matrix representation of the complexified geometric algebra $\mathbb{C}\otimes\operatorname{Cl}_{1,n}$ depends on a parity of $n$, we explore even and odd cases separately. We present a lemma about the unique decomposition of an element of the left ideal into the product of the idempotent and an element of the auxiliary real subalgebra of the geometric algebra. We use this subalgebra and properties of its elements to present the multidimensional Dirac--Hestenes equation. We prove that we might obtain a solution of the multidimensional Dirac--Hestenes equation using a solution of the multidimensional Dirac equation and conversely. We also get that the multidimensional Dirac--Hestenes equation has gauge invariance.