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arXiv:2602.01141

Wigner Function Shapelets I : formalism

Shun Arai

astro-ph.COastro-ph.GAquant-phSubmitted Feb 1, 2026• Announced Feb 3, 2026
v1 · Feb 1, 2026v2 · Jul 7, 2026
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Abstract

We extend shapelets for the analysis of galaxy images to be available in a phase space, introducing \textit{Wigner Function Shapelets (WFS)}. Whereas conventional shapelets expand images separately in configuration or Fourier space using Hermite-Gaussian or Laguerre-Gaussian modes, WFS represents images directly in the four-dimensional phase space with symplectic group $\mathrm{Sp}(4,\mathbb{R})$, which is quantised by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFS consists 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 $(k,\ell) \in \mathbb{Z}^2$. % WFS 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 all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organising in both terminologies of astronomy and of quantum information theory.

DOI
10.1103/zgtf-m8y7
Comments
29 pages, 1 figure

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We show that the cross Wigner function can be written in the form [Formula: see text] where [Formula: see text] is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.
Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces ${\mathbb{S}}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$, i.e., for the canonical pair angle and orbital angular momentum, was published [H. A. Kastrup, Phys. Rev. A 94, 062113 (2016)] in which the main properties of these functions are derived and discussed and their usefulness is illustrated with examples. The present paper is a continuation which compares the properties of the Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions for planar phase spaces in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase-space functions is discussed. The $\ensuremath{\star}$ product formalism is shown to be completely implementable. In addition, basic dynamical laws for Wigner and Moyal functions are derived as generalized Liouville and energy equations. They are very similar to those in the planar case but also show characteristic differences.
The Wigner distribution function (WDF) is a quasi-probability distribution that provides a phase-space representation of quantum states. It offers insights into the state dynamics and contains statistical information of it, such as the marginal probabilities. This thesis investigates a protocol for estimating the WDF of quantum states using symplectic transformations, which are linear canonical transformations that preserve phase-space structure and can be implemented experimentally using the technique of Gaussian optics. These transformations are studied because the time evolution of the WDF can be described by means of these transformations in cases where the Hamiltonian is a second-order (quadratic) polynomial in the phase-space variables. Furthermore, it was found that the Fourier transform of the marginal probability (measured at the laboratory) is related to the characteristic function (Fourier transform of the Wigner function). This relationship is given by Equation 2.46, which states that, with each intensity data obtained by applying a given symplectic transformation, the value of the characteristic function (Fourier transform of the WDF) at a given point of the phase space can be obtained. Based on this, an optical setup was implemented to perform symplectic transformations, such as free-space propagation and passing through a lens. The study focuses on the WDF of a one-dimensional electric field of two Gaussian beams, separated by a distance 2d, where d = 1.388 ± 0.004mm. The two beams are passed through the transformation system (propagation-lens-propagation) and then the marginal probability (intensity) is measured with a CCD camera. After this, the Fourier transform was computationally applied to the intensity data and compared with the theoretical plots of the characteristic function (Equation 2.2). In doing so, it was found that it is necessary to multiply the data obtained by an additional scaling factor to improve them to better fit the theoretical results. Finally, the protocol to be followed to reconstruct the Wigner function of this state with the obtained data was described.