Quaternionic Wavefunction

Abstract

We argue that quaternions form a natural language for the description of quantum-mechanical wavefunctions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the non-relativistic limit to find the Schr\"odinger's equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the non-interacting non-relativistic limit the quaternionic description effectively reduces to the regular complex wavefunction language. We provide an easy to use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell's equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.