While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalizations have hitherto remained tangential. In this communication, 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 characterized by complex Hamiltonians invariant under spacetime reflection are shown to be equivalent to certain coquaternionic extensions of Hermitian quantum theories. One of the interesting consequences is that the spacetime dimension of these systems is six, not four, on account of the structures of coquaternionic quantum mechanics.While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalizations have hitherto remained tangential. In this communication, 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 characterized by complex Hamiltonians invariant under spacetime reflection are shown to be equivalent to certain coquaternionic extensions of Hermitian quantum theories. One of the interesting consequences is that the spacetime dimension of these systems is six, not four, on account of the structures of coquaternionic quantum mechanics.

arXiv:1012.0757Journal of Physics A Mathematical and TheoreticalJan 21, 201164 citations

Quantum Mechanics and Classical Physics

N. Rosen

Jan 1, 19900 citations

Classical Mechanics and Quantum Mechanics

Patrick A. Heelan

Springer eBooksJan 1, 19650 citations

Flux continuity and probability conservation in complexified Bohmian mechanics

Bill Poirier

Recent years have seen increased interest in complexified Bohmian mechanical trajectory calculations for quantum systems as both a pedagogical and computational tool. In the latter context, it is essential that trajectories satisfy probability conservation to ensure they are always guided to where they are most needed. We consider probability conservation for complexified Bohmian trajectories. The analysis relies on time-reversal symmetry considerations, leading to a generalized expression for the conjugation of wave functions of complexified variables. This in turn enables meaningful discussion of complexified flux continuity, which turns out not to be satisfied in general, though a related property is found to be true. The main conclusion, though, is that even under a weak interpretation, probability is not conserved along complex Bohmian trajectories.Recent years have seen increased interest in complexified Bohmian mechanical trajectory calculations for quantum systems as both a pedagogical and computational tool. In the latter context, it is essential that trajectories satisfy probability conservation to ensure they are always guided to where they are most needed. We consider probability conservation for complexified Bohmian trajectories. The analysis relies on time-reversal symmetry considerations, leading to a generalized expression for the conjugation of wave functions of complexified variables. This in turn enables meaningful discussion of complexified flux continuity, which turns out not to be satisfied in general, though a related property is found to be true. The main conclusion, though, is that even under a weak interpretation, probability is not conserved along complex Bohmian trajectories.

arXiv:0803.0193Physical Review AFeb 25, 200840 citations

Classical Mechanics and Quantum Mechanics

Erhard Scheibe

Fundamental theories of physicsJan 1, 20250 citations

Stable coherent states

Antonia Zipfel, Thomas Thiemann

We analyze the stability under time evolution of complexifier coherent states (CCS) in one-dimensional mechanical systems. A system of coherent states is called stable if it evolves into another coherent state. It turns out that a system can only possess stable CCS if the classical evolution of the variable $\mathbf{z}={e}^{\ensuremath{-}i{\mathcal{L}}_{{\ensuremath{\chi}}_{C}}}\mathbf{q}$ for a given complexifier $C$ depends only on $\mathbf{z}$ itself and not on its complex conjugate. This condition is very restrictive in general so that only a few systems exist that obey this condition. However, it is possible to access a wider class of models that in principle may allow for stable coherent states associated with certain regions in the phase space by introducing action-angle coordinates.We analyze the stability under time evolution of complexifier coherent states (CCS) in one-dimensional mechanical systems. A system of coherent states is called stable if it evolves into another coherent state. It turns out that a system can only possess stable CCS if the classical evolution of the variable $\mathbf{z}={e}^{\ensuremath{-}i{\mathcal{L}}_{{\ensuremath{\chi}}_{C}}}\mathbf{q}$ for a given complexifier $C$ depends only on $\mathbf{z}$ itself and not on its complex conjugate. This condition is very restrictive in general so that only a few systems exist that obey this condition. However, it is possible to access a wider class of models that in principle may allow for stable coherent states associated with certain regions in the phase space by introducing action-angle coordinates.

arXiv:1506.08613Physical review. D/Physical review. D.Apr 15, 20167 citations

C

depends only on

\mathbf{z}

itself and not on its complex conjugate. This condition is very restrictive in general so that only a few systems exist that obey this condition. However, it is possible to access a wider class of models that in principle may allow for stable coherent states associated with certain regions in the phase space by introducing action-angle coordinates.