Klein-Gordon equation within the real Hilbert space formalism

TL;DR

This work extends quantum mechanics to the real Hilbert space formalism ($\mathbb{R}$HS) and applies it to the Klein-Gordon equation (KGE) using both complex and quaternionic wave functions. By relaxing the Hermiticity constraint on momentum operators, it derives usual, generalized, and non-Hermitian formulations, each with its own continuity equation and real-valued observables, enabling non-conservative and non-stationary dynamics. The study reveals that the $\mathbb{R}$HS approach naturally accommodates negative energies, avoids the need for imaginary momentum in the Klein paradox, and yields mass-generation-like effects in the quaternionic sector via terms like $\mathscr A_{1\mu}\mathscr A_1^{\dagger\mu}$. Overall, the real Hilbert space framework provides a robust generalization of QM, extending to self-interacting and quaternionic extensions with potential applications to relativistic quantum problems and beyond.

Abstract

Within this article one finds the statement of the Klein-Gordon problem within the real Hilbert space formalism ($\mathbbm R$HS) in terms of complex wave functions, and in terms of quaternionic wave functions as well. The complex formulation comprises hermitian and non-hermitian cases, while the quaternionic solutions additionally set in motion self-interacting particles. The non-hermitian cases comprise non-conservative processes, while the self-interaction physically implies the increase of the effective mass of the particle, an effect that cannot be reproduced using a complex wave function. The obtained autonomous particle solutions, as well as the Klein problem agree to the previously discovered self-interacting non-relativistic particle, and thus reinforce $\mathbbm R$HS as viable and consistent way to explore open problems in quantum mechanics. Also important, the negative energy problem that plagues the usual formalism is eliminated within this approach.