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

arXiv:1410.4678

$PT$ symmetry and a dynamical realization of the SU(1,1) algebra

Rabin Banerjee, Pradip Mukherjee

Published Oct 17, 2014• 0 citations
arXivHTMLPDF

Abstract

We show that the elementary modes of the planar harmonic oscillator can be quantised in the framework of quantum mechanics based on pseudo-hermitian hamiltonians. These quantised modes are demonstrated to act as dynamical structures behind a new Jordan - Schwinger realization of the SU(1,1) algebra. This analysis complements the conventional Jordan - Schwinger construction of the SU(2) algebra based on hermitian hamiltonians of a doublet of oscillators.

DOI
10.48550/arxiv.1410.4678

Related papers

PT symmetry and a dynamical realization of the SU(1, 1) algebra

Rabin Banerjee, Pradip Mukherjee

We show that the elementary modes of the planar harmonic oscillator can be quantized in the framework of quantum mechanics based on pseudo-hermitian Hamiltonians. These quantized modes are demonstrated to act as dynamical structures behind a new Jordan–Schwinger realization of the SU(1, 1) algebra. This analysis complements the conventional Jordan–Schwinger construction of the SU(2) algebra based on hermitian Hamiltonians of a doublet of oscillators.

Modern Physics Letters A•Jan 15, 2016•0 citations

The Harmonic Oscillator in the Plane and the Jordan-Schwinger Algebras

Rabin Banerjee, Pradip Mukherjee

The direct and indirect Lagrangian representations of the planar harmonic oscillator have been discussed. The reduction of these Lagrangians in their basic forms characterising either chiral, or pseudo - chiral oscillators have been given. A Hamiltonian analysis, showing its equivalence with the Lagrangian formalism has also been provided. Finally, we show that the chiral and pseudo - chiral modes act as dynamical structures behind the Jordan - Schwinger realizations of the SU(2) and SU(1,1) algebras. Also, the SU(1,1) construction found here is different from the standard Jordan - Schwinger form.

arXiv:quant-ph/0205143•May 23, 2002•1 citation

COHERENT STATES AND SCHWINGER MODELS FOR PSEUDO GENERALIZATION OF THE HEISENBERG ALGEBRA

Hossein Fakhri, B. Mojaveri, Alireza Dehghani

We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with [Formula: see text] and [Formula: see text] symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are [Formula: see text]-pseudo-Hermiticity and [Formula: see text]-anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.

Modern Physics Letters A•Aug 20, 2009•4 citations

The pseudo Hermitian invariant operator and time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry

Walid Koussa, Naima Mana, Oum Kaltoum Djeghiour +1 more

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.

Journal of Mathematical Physics•Jul 1, 2018•18 citations

A non-Hermitian oscillator Hamiltonian and su(1, 1): a way towards generalizations

C. Quesne

The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)

•Nov 21, 2012•12 citations

A non-Hermitian oscillator Hamiltonian and su(1,1): a way towards generalizations

C Quesne

The family of metric operators, constructed by Musumbu et al ( 2007 J. Phys. A: Math. Theor. 40 F75 ), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily -symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.

arXiv:0705.2868•Journal of Physics A Mathematical and Theoretical•Jul 12, 2007•34 citations
PTPTPT symmetry and a dynamical realization of the SU(1,1) algebra
We show that the elementary modes of the planar harmonic oscillator can be quantised in the framework of quantum mechanics based on pseudo-hermitian hamiltonians. These quantised modes are demonstrated to act as dynamical structures behind a new Jordan - Schwinger realization of the SU(1,1) algebra. This analysis complements the conventional Jordan - Schwinger construction of the SU(2) algebra based on hermitian hamiltonians of a doublet of oscillators.
The direct and indirect Lagrangian representations of the planar harmonic oscillator have been discussed. The reduction of these Lagrangians in their basic forms characterising either chiral, or pseudo - chiral oscillators have been given. A Hamiltonian analysis, showing its equivalence with the Lagrangian formalism has also been provided. Finally, we show that the chiral and pseudo - chiral modes act as dynamical structures behind the Jordan - Schwinger realizations of the SU(2) and SU(1,1) algebras. Also, the SU(1,1) construction found here is different from the standard Jordan - Schwinger form.
We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with [Formula: see text] and [Formula: see text] symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are [Formula: see text]-pseudo-Hermiticity and [Formula: see text]-anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.
We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)
The family of metric operators, constructed by Musumbu et al ( 2007 J. Phys. A: Math. Theor. 40 F75 ), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily -symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.