Coherent states and Hermite polynomials on quaternionic Hilbert spaces

Abstract

Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117-26) it is proved that these coherent states cannot be realized as a group action on a quaternionic Hilbert space. As an extension of the complex Hermite polynomials, quaternionic Hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions.

Coherent states on Hilbert modules

“…Recently they have also been obtained in [16]. Here we stress their similarity with our general construction over C * -modules.…”

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

“…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]. In the following we shall briefly see the Heisenberg uncertainty relation.…”

Squeezed States in the Quaternionic Setting

“…Coherent states on right quaternionic Hilbert spaces. The main content of this section is extracted from [19] as needed here. For an enhanced explanation we refer the reader to [19].…”

“…The main content of this section is extracted from [19] as needed here. For an enhanced explanation we refer the reader to [19]. In [19] the authors have defined coherent states on V R H and V L H , and also established the normalization and resolution of the identities for each of them.…”