On complexified mechanics and coquaternions

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.

11 pages, version to appear in Journal of Physics A

References in corpus (9)

• Faster than Hermitian Quantum Mechanics
• Six-dimensional Methods for Four-dimensional Conformal Field Theories
• The Naimark dilated PT-symmetric brachistochrone
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Cited by in corpus (6)

• Edge states and topological phases in non-Hermitian systems
• Six-dimensional space-time from quaternionic quantum mechanics
• Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids
• Bifurcations and exceptional points in a PT-symmetric dipolar Bose-Einstein condensate
• Multicomplex solitons
• Extensions of real numbers using coset groups