The Dirac-Hestenes Lagrangian

TL;DR

This paper develops a novel Lagrangian formulation for the Dirac-Hestenes equation within a Clifford algebra framework, utilizing a D-complex geometry to unify quantum mechanics and geometric algebra.

Contribution

It introduces a new variational principle for quantum mechanics based on Clifford algebra and D-complex geometry, deriving the Dirac-Hestenes Lagrangian in this setting.

Findings

Derived the Dirac-Hestenes Lagrangian within Clifford algebra framework

Mapped the Lagrangian onto a tensor product involving D-complex geometry

Provided a geometric algebraic formulation of quantum variational principles

Abstract

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