Partial revivals of wave packets are investigated by means of the action-angle Wigner function, which is formulated in terms of action quantum numbers and angle variables. In the eigenbasis of the Hamiltonian, the phase shifts of the expansion coefficients of the wave packet are approximated by a quadratic function of action quantum numbers. Under this approximation, the evolution of the action-angle Wigner function of the wave packet, similarly to the classical Liouville density, can be viewed as a result of each phase-space point moving along a classical orbit. The partial and full revivals of wave packets are direct consequences of two facts: (i) The action-angle Wigner function is distributed only on discrete tori of the phase space and (ii) phase-space points of nearby tori move against each other with a constant speed.Partial revivals of wave packets are investigated by means of the action-angle Wigner function, which is formulated in terms of action quantum numbers and angle variables. In the eigenbasis of the Hamiltonian, the phase shifts of the expansion coefficients of the wave packet are approximated by a quadratic function of action quantum numbers. Under this approximation, the evolution of the action-angle Wigner function of the wave packet, similarly to the classical Liouville density, can be viewed as a result of each phase-space point moving along a classical orbit. The partial and full revivals of wave packets are direct consequences of two facts: (i) The action-angle Wigner function is distributed only on discrete tori of the phase space and (ii) phase-space points of nearby tori move against each other with a constant speed.

Physical Review AMay 1, 199813 citations

Wigner-Type Theorems for Projections

Georges Chevalier

International Journal of Theoretical PhysicsMay 15, 20077 citations

The geometry of Wigner's function

Nandor L. Balazs

Lecture notes in physicsJan 1, 19871 citation

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.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:2602.01141Feb 3, 2026

\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.