Wigner Function Shapelets I : formalism

TL;DR

The paper develops Wigner Function Shapelets (WFS) to analyze galaxy morphology directly in four-dimensional phase space, leveraging the symplectic group $ ext{Sp}(4,\mathbb{R})$ and Hopf fibration to reveal a natural phase-space band structure indexed by $(Q_0,Q_2)$ and torus quantum numbers $(k,\ell)$. It provides closed-form analytic Wigner-function representations for LG-based basis elements, connects WFS to BiPoSH and FPFS formalisms, and derives local cosmological responses to shear and flexion as well as parity-violation probes within this framework. The methodology yields a symmetry-preserving, information-rich description of images that can separate morphology from rotation, quantify systematic effects via quantum-channel language, and enable optimized recovery under PSF and noise. The work lays a baseline formalism with practical guidance on choosing scale parameters $(\sigma,\lambdabar)$ and discusses feasibility and future validation with simulations and data, potentially enabling novel cosmological inferences and robust handling of image systematics. It thus opens a cross-disciplinary bridge between quantum-information formalisms and astrophysical image analysis, with immediate implications for weak lensing, flexion, and parity-violating signatures.

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.

Figures (1)

• Figure 1: Analytic band matrix in the $(Q_0,Q_2)$ plane. Columns correspond to $(n_u,n_u')$ and rows to $(n_v,n_v')$. Each panel shows the radial product $R_{n_u n_u'}(\sqrt{Q_0+Q_2})\,R_{n_v n_v'}(\sqrt{Q_0-Q_2})$, masked to the physical wedge $|Q_2|\le Q_0$.