On complexified mechanics and coquaternions

TL;DR

This paper introduces quaternionic and coquaternionic extensions of Hamiltonian mechanics, unifying complexified classical and quantum theories, and reveals that coquaternionic quantum mechanics naturally resides in six-dimensional space-time.

Contribution

It presents a novel framework using coquaternions for complexified mechanics, linking classical and quantum theories in a unified manner.

Findings

Coquaternionic quantum mechanics implies six-dimensional space-time.

Complex Hamiltonians invariant under space-time reflection relate to coquaternionic extensions.

Provides a unifying mathematical structure for complexified classical and quantum mechanics.

Abstract

While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and coquaternionic (split-signature analogue of quaternions) extensions of Hamiltonian mechanics are introduced, and are shown to offer a unifying framework for complexified classical and quantum mechanics. In particular, quantum theories characterised by complex Hamiltonians invariant under space-time reflection are shown to be equivalent to certain coquaternionic extensions of Hermitian quantum theories. One of the interesting consequences is that the space-time dimension of these systems is six, not four, on account of the structures of coquaternionic quantum mechanics.