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

Coherent states for the relativistic harmonic oscillator

V. Aldaya, Julio Guerrero

Published Jan 1, 1995• 0 citations
OpenAlex

Abstract

Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.

Related papers

Canonical coherent states for the relativistic harmonic oscillator

V. Aldaya, Julio Guerrero

In this paper there are constructed manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given SL(2,R) representation, once a change of variables z∈C→zD∈ unit disk is performed. Also introduced are higher-order, relativistic creation and annihilation operators, â,â°, with canonical commutation relation [â, â°]=1 rather than the covariant one [ẑ, ẑ°]≊ energy and naturally associated with the SL(2,R) group. The canonical (relativistic) coherent states are then defined as eigenstates of â. Finally, a canonical, minimal representation is constructed in configuration space by means of eigenstates of a canonical position operator.

arXiv:hep-th/9406080•Journal of Mathematical Physics•Jul 1, 1995•12 citations

Generalized Coherent States Associated with the Cλ-Extended Oscillator

C. Quesne

Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1

•Jan 1, 2001•18 citations

Coherent states for a generalization of the harmonic oscillator

David J. Fernández C., Véronique Hussin, Luis Miguel Nieto

Coherent states for a family of isospectral oscillator Hamiltonians are derived from a suitable choice of annihilation and creation operators. The Fock-Bargmann representation is also obtained.

•Jan 1, 1995•0 citations

SU (N) coherent states

Manu Mathur, Hamdi Mani

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

arXiv:quant-ph/0206005•Journal of Mathematical Physics•Oct 24, 2002•26 citations

Annihilation and creation operators in Lie algebra theory and physics

Kateřina JARKOVSKÁ

We show the use of the theory of Lie algebras, especially their oscillator realizations, in the context of quantum mechanics. One can construct oscillator realizations from matrix realizations. In the case of symplectic and special orthogonal algebra, we demonstrate an alternative method of obtaining oscillator realizations from symmetric or exterior power of a vector space of annihilation and creation bosonic or fermionic operators. We find Lie algebra of polynomials of degree at most two in phase space of a mechanical system, which form the semi-direct product of the Heisenberg algebra and symplectic algebra. It is shown that a classical system with Hamiltonian function in this algebra can be quantized by two equivalent representations - Schrödinger or Bargmann-Fock representation. The second mentioned representation generates the same operators of symplectic algebra as we got from their previous formal construction from symmetric power of a vector space of bosonic operators. Quantization is demonstrated on the bosonic harmonic oscillator. We use the similarities between bosonic and fermionic oscillator realizations to define the fermionic harmonic oscillator. Some properties of spinor representations of special orthogonal algebra are illustrated on its state space. Powered by TCPDF (www.tcpdf.org)

•Jan 1, 2016•0 citations

The relativistic Bargmann transform

V. Aldaya, Julio Guerrero

An integral transformation generalizing the Bargmann transform is established among two sets of wavefunctions associated with the configuration and Bargmann-Fock-Segal-like representations of a relativistic harmonic oscillator. The time-independent configuration-space wavefunctions are also studied and the lack of unitarity occurring when factorizing the time dependence of the wavefunction is solved by modifying the scalar product.

Journal of Physics A Mathematical and General•Nov 21, 1993•4 citations
Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.
In this paper there are constructed manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given SL(2,R) representation, once a change of variables z∈C→zD∈ unit disk is performed. Also introduced are higher-order, relativistic creation and annihilation operators, â,â°, with canonical commutation relation [â, â°]=1 rather than the covariant one [ẑ, ẑ°]≊ energy and naturally associated with the SL(2,R) group. The canonical (relativistic) coherent states are then defined as eigenstates of â. Finally, a canonical, minimal representation is constructed in configuration space by means of eigenstates of a canonical position operator.
Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.
We show the use of the theory of Lie algebras, especially their oscillator realizations, in the context of quantum mechanics. One can construct oscillator realizations from matrix realizations. In the case of symplectic and special orthogonal algebra, we demonstrate an alternative method of obtaining oscillator realizations from symmetric or exterior power of a vector space of annihilation and creation bosonic or fermionic operators. We find Lie algebra of polynomials of degree at most two in phase space of a mechanical system, which form the semi-direct product of the Heisenberg algebra and symplectic algebra. It is shown that a classical system with Hamiltonian function in this algebra can be quantized by two equivalent representations - Schrödinger or Bargmann-Fock representation. The second mentioned representation generates the same operators of symplectic algebra as we got from their previous formal construction from symmetric power of a vector space of bosonic operators. Quantization is demonstrated on the bosonic harmonic oscillator. We use the similarities between bosonic and fermionic oscillator realizations to define the fermionic harmonic oscillator. Some properties of spinor representations of special orthogonal algebra are illustrated on its state space. Powered by TCPDF (www.tcpdf.org)