Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

q deformations of the oscillator

R. J. Finkelstein

Published Dec 1, 1992• 3 citations
Publisher

Abstract

Generating functions that interpolate between the generating functions for Hermitian multilinears and the corresponding Grassmann multilinears are considered. The new functions that arise in this way may be related to state functions of q-deformed harmonic oscillators.
DOI
10.1063/1.529827

Related papers

A q-oscillator Green’s function

H. Ahmedov, I. H. Duru

By using the generating function formula for the product of two q-Hermite polynomials, q-deformation of the Feynman Green’s function for the harmonic oscillator is obtained.

arXiv:q-alg/9609028•Journal of Mathematical Physics•Aug 1, 1997•0 citations

q-deformed Hermite polynomials in q-quantum mechanics

Ralf Hinterding, Julius Wess, Sektion Physik Der Ludwig-maximilians-universität +1 more

The q-special functions appear naturally in q-deformed quantum mechanics and both sides profit from this fact. Here we study the relation between the q-deformed harmonic oscillator and the q-Hermite polynomials. We discuss: recursion formula, generating function, Christoffel-Darboux identity, orthogonality relations and the moment functional. 1 1

•Jan 1, 1998•14 citations

On the construction of generalized Grassmann representatives of state vectors

M. El Baz, Y. Hassouni

published in J. Phys. A37 (2004) 4361 Generalized Zk-graded Grassmann variables are used to label coherent states related to the nilpotent representation of the q-oscillator of Biedenharn and Macfarlane when the deformation parameter is a root of unity. These states are then used to construct generalized Grassmann representatives of state vectors. Recently in [1] we have constructed coherent states for k-fermions using Kerner’s [2] Z3-graded extension of the Grassmann variables [3]. These results were obtained in the case where the deformation parameter is a primitive cubic root of unity, i.e., k = 3. In order to obtain similar results in the generic case (by generic we mean here that q, the deformation parameter, is an arbitrary k th root of unity, i.e., for an arbitrary positive integer k) one should use Zk-graded generalizations of the Grassmann variables. Unfortunately, up to our knowledge, such structures have not been yet constructed in the spirit of Kerner’s variables. There exist however in the literature another point of view and other generalized Zk-graded Grassmann variables [4]. In this letter we investigate on the use of these latter variables for the description of k-fermions. Namely, we will construct k-fermionic coherent states labeled by Zk-graded Grassmann variables. The coherent states will be used to derive a space of (Zk-graded) Grassmann representatives in which state vectors are represented as ”holomorphic ” functions of the Grassmann variable. There exist many deformations of the harmonic oscillator which, for some values of the deformation parameter, give rise to k-fermions. In this letter we will illustrate the construction using the oscillator deformation of Biedenharn [5] and Macfarlane [6]. This q-oscillator is described by the operators {N, a, a +} with the following relations

•Sep 13, 2013•3 citations

Harmonic Oscillator Realization of the Deformed Bogoliubov (p,q)-Transformations without First Finite Fock States

M. H. Naderi, R. Roknizadeh, M. Soltanolkotabi

Considering a simple generalization of the (p,q)-deformed boson oscillator algebra, which leads to a two-parameter deformed bosonic algebra in an infinite dimensional subspace of the harmonic oscillator Hilbert space without first finite Fock states, we establish a new harmonic oscillator realization of the deformed boson operators based on the Bogoliubov (p,q)-transformations. We obtain exact expressions for the transformation coefficients and show that they depend on arbitrary functions of p and q which can be interpreted as the parameters of the (p,q)-deformed GL(2,C) group. We also examine the existence and structure of the corresponding deformed Fock-space representation for our problem.

Progress of Theoretical Physics•Nov 1, 2004•3 citations

Operators of the q-oscillator

Franciszek Hugon Szafraniec

We scrutinize the possibility of extending the result of [19] to the case of $q$-deformed oscillator for $q$ real; for this we exploit the whole range of the deformation parameter as much as possible. We split the case into two depending on whether a solu

arXiv:0711.3073•Banach Center Publications•Jan 1, 2007•11 citations

NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED

Jules Beckers, N. Debergh, José F. Cariñena +1 more

Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

arXiv:hep-th/0102004•Modern Physics Letters A•Jan 20, 2001•14 citations
published in J. Phys. A37 (2004) 4361 Generalized Zk-graded Grassmann variables are used to label coherent states related to the nilpotent representation of the q-oscillator of Biedenharn and Macfarlane when the deformation parameter is a root of unity. These states are then used to construct generalized Grassmann representatives of state vectors. Recently in [1] we have constructed coherent states for k-fermions using Kerner’s [2] Z3-graded extension of the Grassmann variables [3]. These results were obtained in the case where the deformation parameter is a primitive cubic root of unity, i.e., k = 3. In order to obtain similar results in the generic case (by generic we mean here that q, the deformation parameter, is an arbitrary k th root of unity, i.e., for an arbitrary positive integer k) one should use Zk-graded generalizations of the Grassmann variables. Unfortunately, up to our knowledge, such structures have not been yet constructed in the spirit of Kerner’s variables. There exist however in the literature another point of view and other generalized Zk-graded Grassmann variables [4]. In this letter we investigate on the use of these latter variables for the description of k-fermions. Namely, we will construct k-fermionic coherent states labeled by Zk-graded Grassmann variables. The coherent states will be used to derive a space of (Zk-graded) Grassmann representatives in which state vectors are represented as ”holomorphic ” functions of the Grassmann variable. There exist many deformations of the harmonic oscillator which, for some values of the deformation parameter, give rise to k-fermions. In this letter we will illustrate the construction using the oscillator deformation of Biedenharn [5] and Macfarlane [6]. This q-oscillator is described by the operators {N, a, a +} with the following relations
Considering a simple generalization of the (p,q)-deformed boson oscillator algebra, which leads to a two-parameter deformed bosonic algebra in an infinite dimensional subspace of the harmonic oscillator Hilbert space without first finite Fock states, we establish a new harmonic oscillator realization of the deformed boson operators based on the Bogoliubov (p,q)-transformations. We obtain exact expressions for the transformation coefficients and show that they depend on arbitrary functions of p and q which can be interpreted as the parameters of the (p,q)-deformed GL(2,C) group. We also examine the existence and structure of the corresponding deformed Fock-space representation for our problem.
We scrutinize the possibility of extending the result of [19] to the case of qqq-deformed oscillator for qqq real; for this we exploit the whole range of the deformation parameter as much as possible. We split the case into two depending on whether a solu