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

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 soobtained 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.

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

“…Now, we take the real coefficient of the imaginary unit e 1 in (33), and we add the equation with equation (32), and observing that…”

Canonical, Squeezed and Fermionic Coherent States in a Right Quaternionic Hilbert Space with a Left Multiplication on It

“…The operators can be taken as a † = q (multiplication by q) and a = ∂ s (left slice regular derivative), see [17,14]. It is also not difficult to see that (a † ) † = a, [a, a † ] = I H B r and aη q = q · η q (see also [15]). In the same way canonical CS can also be defined on a left quaternion Hilbert space [18].…”

“…With the right multiplication on a right quaternion Hilbert space a displacement operator similar to the harmonic oscillator displacement operator cannot be defined as a representation of the Fock space [2,18]. However, in [15] we have shown that, with the aid of a left multiplication defined on a right quaternionic Hilbert space, an appropriate harmonic oscillator displacement operator can be defined. We have proved that this operator is square integrable, irreducible and a unitary representation and also it satisfies most of the properties of its complex counterpart.…”

“…For various attempts and their drawbacks we refer the reader to [1]. However, in [15], we have given a through discussion on the possibility of defining various momentum operators. In fact we have shown that, by using the notion of left multiplication in a right quaternionic Hilbert space it is possible to define a linear self-adjoint momentum operator in complete analogy with the complex case.…”

Fredholm operators and essential S-spectrum in the quaternionic setting

“…These restrictions can severely prevent the generalization to the quaternionic case of results valid in the complex setting. Even though most of the linear spaces are one-sided, it is possible to introduce a notion of multiplication on both sides by fixing an arbitrary Hilbert basis of H. This fact allows to have a linear structure on the set of linear operators, which is a minimal requirement to develop a full theory [23,22]. However, in this manuscript we develop the theory on V R H without introducing a left multiplication on it.…”