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

Wigner functions for curved spaces. II. On spheres

Miguel Angel Alonso, GEORGE S. POGOSYAN, Kurt Bernardo Wolf

Published Mar 20, 2003• 21 citations
Publisher

Abstract

The form of the Wigner distribution function for Hamiltonian systems in spaces of constant negative curvature (i.e., hyperboloids) proposed in M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids” [J. Math. Phys. 43, 5857 (2002)], is extended here to spaces whose curvature is constant and positive, i.e., spheres. An essential part of this construction is the use of the functions of Sherman and Volobuyev, which are an overcomplete set of plane-wave-like solutions of the Laplace–Beltrami equation for this space. Rotations that displace the poles transform these functions with a multiplier factor, and their momentum direction becomes formally complex; the covariance properties of the proposed Wigner function are understood in these terms. As an example for the one-dimensional case, we consider the energy eigenstates of the oscillator on the circle in a Pöschl–Teller potential. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.

DOI
10.1063/1.1559644

Related papers

Wigner functions for curved spaces. I. On hyperboloids

Miguel Angel Alonso, George S. Pogosyan, Kurt Bernardo Wolf

We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature, in this article on hyperboloids, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace–Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the one-dimensional case on a hyperbola branch, where this conic-oscillator is the Pöschl–Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.

arXiv:quant-ph/0205041•Journal of Mathematical Physics•Dec 1, 2002•36 citations

Wigner functions for curved spaces I: On hyperboloids

Miguel Angel Alonso, GEORGE S. POGOSYAN, Kurt Bernardo Wolf +2 more

We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature —in this paper on hyperboloids—, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace-Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the 1-dimensional case on a hyperbola branch, where this conic-oscillator is the Pöschl-Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.

•Jan 1, 2008•32 citations

Wigner function shapelets: Symplectic representation of astronomical images

Shun Arai

We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing \textit{Wigner function shapelets}(WFSs). Whereas conventional shapelets expand images separately in configuration or Fourier space using Hermite-Gaussian or Laguerre-Gaussian modes, WFSs represent images directly in the four-dimensional phase space with symplectic group $\mathrm{Sp}(4,\mathbb{R})$, which is quantized by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFSs consist of a bilinear form of the cross-Wigner function of the Laguerre-Gaussian modes as an orthogonal and complete basis for the Wigner function of an image, carrying out $\mathrm{SU}(2)$ irreducible representations of the phase space with the Hopf tori. We introduce a scalar function $\mathcal{W}_{k\ell} (Q_0,Q_2)$ from the $\mathrm{U}(1)\times \mathrm{U}(1)$ - covariant tori to a two-dimensional space of constants of motion $(Q_0,Q_2)$ -- the harmonic energy and axial angular momentum -- thereby yielding a natural phase-space ``band structure'', given a pair of winding number $(k,\ell) \in \mathbb{Z}^2$. % The WFSs leverage key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way; (ii) its trasport equation naturally involves with a Liouville equation at $\lambdabar \rightarrow 0$; (iii) it admits positive/negative oscillatory patterns on $(Q_0,Q_2)$ plane that can be sensitive spatial coherent structure of galaxy morphology and cosmological imprints; and (iv) systematics and noise can be manipulated as a quantum channel operation. This paper aims to bring together all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organizing them in terminologies of both astronomy and of quantum information theory.

astro-ph.COCross-listedarXiv:2602.01141•Feb 3, 2026

A study of the orthogonal polynomials associated with the quantum harmonic oscillator on constant curvature spaces

Christophe Vignat, Pedro W. Lamberti

Recently, Cariñena, et al. [Ann. Phys. 322, 434 (2007)] introduced a new family of orthogonal polynomials that appear in the wave functions of the quantum harmonic oscillator in two-dimensional constant curvature spaces. They are a generalization of the Hermite polynomials and will be called curved Hermite polynomials in the following. We show that these polynomials are naturally related to the relativistic Hermite polynomials introduced by Aldaya et al. [Phys. Lett. A 156, 381 (1991)], and thus are Jacobi polynomials. Moreover, we exhibit a natural bijection between the solutions of the quantum harmonic oscillator on negative curvature spaces and on positive curvature spaces. At last, we show a maximum entropy property for the ground states of these oscillators.

Journal of Mathematical Physics•Oct 1, 2009•2 citations

Wigner negativity and stellar rank for SU(1,1) states

A. B. Klimov, A. Muñoz, G. Leuchs +2 more

Quasiprobability distributions for systems endowed with SU(1,1) dynamical symmetry have received surprisingly little attention, despite the central role of this symmetry in two-photon physics, squeezed states, and nonlinear interferometry. Here, we fill this gap by constructing a full covariant family of $s$-ordered quasiprobability distributions defined on the two-sheeted hyperboloid, or equivalently, on the Poincaré unit disk via stereographic projection. A key result is that the Wigner function is strictly positive for all Perelomov SU(1,1) coherent states, in sharp contrast to the SU(2) case. This positivity endows Wigner negativity with an unambiguous operational meaning: any negative volume is a direct signature of genuinely quantum behavior. We further examine the stellar rank of SU(1,1) states, defined through the zeros of the Husimi $Q$-function, and show how it compares with Wigner negativity as a geometry-adapted witness of nonclassicality in this setting. We further introduce a hierarchy of generalized multipoles through a harmonic expansion of the density operator on the hyperboloid, providing a complementary framework for probing quantumness. This offers a comprehensive toolkit for characterizing and quantifying quantum resources in SU(1,1) systems.

quant-pharXiv:2607.22810•Jul 28, 2026•0 citations

The Wigner function of aq-deformed harmonic oscillator model

E I Jafarov, S Lievens, S M Nagiyev +1 more

The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.

arXiv:math-ph/0702082•Journal of Physics A Mathematical and Theoretical•Apr 30, 2007•13 citations
The form of the Wigner distribution function for Hamiltonian systems in spaces of constant negative curvature (i.e., hyperboloids) proposed in M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids” [J. Math. Phys. 43, 5857 (2002)], is extended here to spaces whose curvature is constant and positive, i.e., spheres. An essential part of this construction is the use of the functions of Sherman and Volobuyev, which are an overcomplete set of plane-wave-like solutions of the Laplace–Beltrami equation for this space. Rotations that displace the poles transform these functions with a multiplier factor, and their momentum direction becomes formally complex; the covariance properties of the proposed Wigner function are understood in these terms. As an example for the one-dimensional case, we consider the energy eigenstates of the oscillator on the circle in a Pöschl–Teller potential. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.
We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature, in this article on hyperboloids, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace–Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the one-dimensional case on a hyperbola branch, where this conic-oscillator is the Pöschl–Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.
We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature —in this paper on hyperboloids—, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace-Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the 1-dimensional case on a hyperbola branch, where this conic-oscillator is the Pöschl-Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.
We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing \textit{Wigner function shapelets}(WFSs). Whereas conventional shapelets expand images separately in configuration or Fourier space using Hermite-Gaussian or Laguerre-Gaussian modes, WFSs represent images directly in the four-dimensional phase space with symplectic group Sp(4,R)\mathrm{Sp}(4,\mathbb{R})Sp(4,R), which is quantized by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFSs consist of a bilinear form of the cross-Wigner function of the Laguerre-Gaussian modes as an orthogonal and complete basis for the Wigner function of an image, carrying out SU(2)\mathrm{SU}(2)SU(2) irreducible representations of the phase space with the Hopf tori. We introduce a scalar function Wkℓ(Q0,Q2)\mathcal{W}_{k\ell} (Q_0,Q_2)Wkℓ​(Q0​,Q2​) from the U(1)×U(1)\mathrm{U}(1)\times \mathrm{U}(1)U(1)×U(1) - covariant tori to a two-dimensional space of constants of motion (Q0,Q2)(Q_0,Q_2)(Q0​,Q2​) -- the harmonic energy and axial angular momentum -- thereby yielding a natural phase-space ``band structure'', given a pair of winding number (k,ℓ)∈Z2(k,\ell) \in \mathbb{Z}^2(k,ℓ)∈Z2. % The WFSs leverage key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way; (ii) its trasport equation naturally involves with a Liouville equation at $\lambdabar \rightarrow 0$; (iii) it admits positive/negative oscillatory patterns on (Q0,Q2)(Q_0,Q_2)(Q0​,Q2​) plane that can be sensitive spatial coherent structure of galaxy morphology and cosmological imprints; and (iv) systematics and noise can be manipulated as a quantum channel operation. This paper aims to bring together all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organizing them in terminologies of both astronomy and of quantum information theory.
Recently, Cariñena, et al. [Ann. Phys. 322, 434 (2007)] introduced a new family of orthogonal polynomials that appear in the wave functions of the quantum harmonic oscillator in two-dimensional constant curvature spaces. They are a generalization of the Hermite polynomials and will be called curved Hermite polynomials in the following. We show that these polynomials are naturally related to the relativistic Hermite polynomials introduced by Aldaya et al. [Phys. Lett. A 156, 381 (1991)], and thus are Jacobi polynomials. Moreover, we exhibit a natural bijection between the solutions of the quantum harmonic oscillator on negative curvature spaces and on positive curvature spaces. At last, we show a maximum entropy property for the ground states of these oscillators.
Quasiprobability distributions for systems endowed with SU(1,1) dynamical symmetry have received surprisingly little attention, despite the central role of this symmetry in two-photon physics, squeezed states, and nonlinear interferometry. Here, we fill this gap by constructing a full covariant family of sss-ordered quasiprobability distributions defined on the two-sheeted hyperboloid, or equivalently, on the Poincaré unit disk via stereographic projection. A key result is that the Wigner function is strictly positive for all Perelomov SU(1,1) coherent states, in sharp contrast to the SU(2) case. This positivity endows Wigner negativity with an unambiguous operational meaning: any negative volume is a direct signature of genuinely quantum behavior. We further examine the stellar rank of SU(1,1) states, defined through the zeros of the Husimi QQQ-function, and show how it compares with Wigner negativity as a geometry-adapted witness of nonclassicality in this setting. We further introduce a hierarchy of generalized multipoles through a harmonic expansion of the density operator on the hyperboloid, providing a complementary framework for probing quantumness. This offers a comprehensive toolkit for characterizing and quantifying quantum resources in SU(1,1) systems.
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.