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arXiv:0709.1198 · Symmetry Integrability and Geometry Methods and Applications

Complex Projection of Quasianti-Hermitian Quaternionic Hamiltonian Dynamics

G. Scolarici

Published Sep 8, 2007• 2 citations
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Abstract

We characterize the subclass of quasianti-Hermitian quaternionic Hamiltonian dynamics such that their complex projections are one-parameter semigroup dynamics in the space of complex quasi-Hermitian density matrices.As an example, the complex projection of a spin-1 2 system in a constant quasianti-Hermitian quaternionic potential is considered.

DOI
10.3842/sigma.2007.088

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We characterize the subclass of quasianti-Hermitian quaternionic Hamiltonian dynamics such that their complex projections are one-parameter semigroup dynamics in the space of complex quasi-Hermitian density matrices.As an example, the complex projection of a spin-1 2 system in a constant quasianti-Hermitian quaternionic potential is considered.
We characterize the quasianti-Hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra; moreover, we state a necessary and sufficient condition for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional (right) quaternionic Hilbert space. According to such results we obtain two alternative descriptions of a quantum optical physical system, in the realm of quaternionic quantum mechanics, while no alternative can exist in complex quantum mechanics, and we discuss some differences between them.