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arXiv · 1704.02946

A Representation of Weyl-Heisenberg Lie Algebra in the Quaternionic Setting

B. Muraleetharan·K. Thirulogasanthar·I. Sabadini
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Abstract

Using a left multiplication defined on a right quaternionic Hilbert space, linear self-adjoint momentum operators on a right quaternionic Hilbert space are defined in complete analogy with their complex counterpart. With the aid of the so-obtained position and momentum operators, we study the Heisenberg uncertainty principle on the whole set of quaternions and on a quaternionic slice, namely on a copy of the complex plane inside the quaternions. For the quaternionic harmonic oscillator, the uncertainty relation is shown to saturate on a neighborhood of the origin in the case we consider the whole set of quaternions, while it is saturated on the whole slice in the case we take the slice-wise approach. In analogy with the complex Weyl-Heisenberg Lie algebra, Lie algebraic structures are developed for the quaternionic case. Finally, we introduce a quaternionic displacement operator which is square integrable, irreducible and unitary, and we study its properties.

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B. Muraleetharan, K. Thirulogasanthar, I. Sabadini. 2017-07-28. A Representation of Weyl-Heisenberg Lie Algebra in the Quaternionic Setting. https://doi.org/10.1016/j.aop.2017.07.014

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