Wigner function shapelets: Symplectic representation of astronomical images

Abstract

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.

The paper asks whether astronomical-image shapelet analysis can be formulated jointly in position and spatial-frequency space, with image distortions and observational noise described using phase-space and quantum-information tools. It introduces Wigner function shapelets (WFSs), defined as cross-Wigner transforms of two-dimensional Laguerre–Gaussian modes, for expanding an image’s four-dimensional Wigner distribution. The construction uses the Wigner–Weyl correspondence and Moyal identity to establish orthogonality and completeness. For an image expanded with coefficients $\psi_{js}$, the WFS coefficients are $c_{(j,s),(j',s')}=\psi_{js}\psi^*_{j's'}$. A two-oscillator Schwinger representation gives closed expressions in associated Laguerre polynomials. Hopf coordinates separate radial variables $(Q_0,Q_2)$, satisfying $|Q_2|\leq Q_0$, from two angular phases. Fourier projection over these phases defines two-dimensional coefficient maps $\mathcal{W}_{k\ell}(Q_0,Q_2)$; retaining all integer sectors reconstructs the full distribution. The paper derives linear shear and flexion response formulas, relates WFS coefficients to Fourier-power shapelets and bipolar spherical harmonics, and proposes parity-sensitive correlations that are fourth order in the image field. It also formulates PSF effects and recovery using operator maps and entanglement fidelity, and discusses choosing image-scale and phase-space parameters from image moments and instrumental resolution. This is a formal development without image-simulation benchmarks or real-data tests. Practical shear-estimator construction, validation against existing image-processing methods, and demonstrations of advantageous reconstruction are deferred. Most calculations assume a static, single-band scalar image without polarization; broader extensions are outlined.