The Quaternionic Dirac Lagrangian

TL;DR

This paper explores the formulation of the Dirac Lagrangian within quaternionic quantum mechanics, addressing the challenges posed by quaternion non-commutativity and deriving a complex-projected Lagrangian density for the two-component Dirac equation.

Contribution

It introduces a quaternionic approach to the Dirac Lagrangian, including the derivation of a complex-projected Lagrangian density necessary for quaternionic quantum theories.

Findings

Derived the quaternionic Dirac Lagrangian density

Addressed non-commutativity in quaternionic variational principles

Showed the necessity of complex projection in quaternionic Lagrangians

Abstract

We discuss the use of the variational principle within quaternionic quantum mechanics. This is non-trivial because of the non commutative nature of quaternions. We derive the Dirac Lagrangian density corresponding to the two-component Dirac equation. This Lagrangian is complex projected as anticipated in previous articles and this feature is necessary even for a classical real Lagrangian.