Quaternionic quantum mechanics in real Hilbert space

TL;DR

This paper develops a quaternionic quantum mechanics framework within a real Hilbert space, establishing foundational theorems and tools that support its consistency and potential for discovering new physics.

Contribution

It introduces a novel formulation of quaternionic quantum mechanics in real Hilbert spaces, including a spectral theorem and quaternionic Fourier series, demonstrating its consistency and utility.

Findings

Proves the spectral theorem for quaternionic quantum mechanics in real Hilbert space.

Derives a new quaternionic Fourier series applicable to this formalism.

Shows the formalism's consistency and potential for new physics exploration.

Abstract

A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After a brief discussion on unitary operators in this formalism, we conclude that this quantum theory is indeed consistent, and can be a valuable tool in the search for new physics.