A representation of Weyl–Heisenberg Lie algebra in the quaternionic setting

Journal Article · Sun Oct 15 04:00:00 EDT 2017 · Annals of Physics

DOI:https://doi.org/10.1016/J.AOP.2017.07.014· OSTI ID:22701538

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.

• OSTI ID: 22701538
• Journal Information: Annals of Physics, Journal Name: Annals of Physics Vol. 385; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
HARMONIC OSCILLATORS
HILBERT SPACE
LIE GROUPS
POSITION OPERATORS
UNCERTAINTY PRINCIPLE