Generalized imaginary units in quantum mechanics

TL;DR

This work investigates whether the imaginary unit $i$ in quantum mechanics can be generalized beyond its standard role. It contrasts a complex-number generalization $i \to i e^{i theta}$, which yields non-stationary, non-conservative dynamics and nontrivial continuity-equation terms, with a quaternionic generalization via a unit eta in a real-Hilbert-space framework, leading to a quaternionic Schrödinger equation $\hbar \partial_t \Psi \eta = \widehat{\mathcal{H}} \Psi$ and a Hamiltonian with quaternionic potentials. The complex approach deforms quantum operators and non-conservative probabilities without a full operator-generalization, whereas the quaternionic formulation provides a consistent generalization of the imaginary unit and reveals new time-evolution structures, including finite-time evanescence and stationary states. Overall, the paper argues that $\mathbb{H}$QM offers a more natural and fruitful extension of quantum dynamics, with potential implications for non-Hermitian physics, gauge-like phases, and generalized uncertainty relations.

Abstract

The generalization of the imaginary unit is examined within the instances of the complex quantum mechanics ($\mathbb C$QM), and of the quaternionic quantum mechanics ($\mathbb H$QM) as well. Whereas the complex theory describes non-stationary quantum processes, the quaternionic theory does not admit such an interpretation, and associates the generalized imaginary unit to a novel time evolution function. Various possibilities are opened as future directions for future research.