The Dirac-Hestenes Lagrangian

Abstract

We discuss the variational principle within Quantum Mechanics in terms of the noncommutative even Space Time sub-Algebra, the Clifford $\text{Ra}$-algebra $C{l}_{1,3}^{+}$. A fundamental ingredient, in our multivectorial algebraic formulation, is the adoption of a $\text{D}$-complex geometry, $\text{D}\equiv spa{n}_{\text{RR}}\{1,{\gamma }_{21}\}$, ${\gamma }_{21}\in C{l}_{1,3}^{+}$. We derive the Lagrangian for the Dirac-Hestenes equation and show that such Lagrangian must be mapped on $\text{D}\otimes \mathcal{F}$, where $\mathcal{F}$ denotes an $\text{Ra}$-algebra of functions.

Comments: 14 pages, RevTex. To appear in Int. J. Theor. Phys