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

## TL;DR

This paper extends the Weyl-Heisenberg Lie algebra to quaternionic Hilbert spaces, defining momentum operators, analyzing the uncertainty principle, and introducing a quaternionic displacement operator with key properties.

## Contribution

It develops a quaternionic representation of the Weyl-Heisenberg algebra, including momentum operators, uncertainty relations, and a unitary displacement operator, expanding the algebraic framework to quaternions.

## Key findings

- Uncertainty principle saturates near the origin for the whole quaternion set.
- Uncertainty principle saturates on the entire slice within quaternions.
- Quaternionic displacement operator is square integrable, irreducible, and unitary.

## 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.