Wigner Function Shapelets (WFS)

• WFS is a phase-space formalism that uses cross-Wigner functions from Hermite-Gaussian and Laguerre-Gaussian modes to represent functions or images.
• It leverages the full four-dimensional phase space to maintain symplectic covariance, enabling rigorous asymptotic expansions and accurate expectation value computations.
• WFS facilitates efficient Monte Carlo sampling via positive spectrograms, improving high-frequency analysis in quantum mechanics and astronomical image processing.

Wigner Function Shapelets (WFS) constitute a formalism for expanding and analyzing functions or images directly in phase space using an orthogonal basis of cross-Wigner functions built from Hermite-Gaussian or Laguerre-Gaussian modes. Unlike conventional shapelet approaches, which are confined to configuration or Fourier space, WFS leverages the full four-dimensional phase space structure, maintaining explicit covariance under symmetry groups such as the full linear canonical (symplectic) group $\mathrm{Sp}(4,\mathbb{R})$. This framework is applicable both in quantum mechanics, where the Wigner function provides a quasi-probabilistic representation of quantum states, and in astronomical image analysis, where it encodes joint spatial and Fourier information with intrinsic symmetry preservation. The basis elements, or "shapelets," are positive spectrograms or cross-Wigner functions, permitting both rigorous asymptotic expansions and practical high-frequency sampling for expectation value computations (Keller, 2016, Arai, 1 Feb 2026).

1. Theoretical Foundations of Wigner Functions in Phase Space

The Wigner function for an $L^2$-normalized wavefunction $\psi\in L^2(\mathbb{R}^d)$ is defined as

$$W_\psi(q,p) = (2\pi)^{-d} \int_{\mathbb{R}^d} e^{ip\cdot y/\epsilon} \psi(q - y/2) \overline{\psi(q + y/2)}\,dy,$$

encoding the complete information of $\psi$ in the $2d$-dimensional phase space $(q,p)$. This quasi-probability distribution obeys correct marginals, with $\int W_\psi(q,p)\,dp = |\psi(q)|^2$ and $\int W_\psi(q,p)\,dq = |\hat{\psi}(p)|^2$ (where $\hat \psi$ is the Fourier transform), and enables exact computation of quantum expectations via

$$\langle \psi, Op^W(a)\psi \rangle = \int a(q,p) W_\psi(q,p)\,dq\,dp.$$

However, $W_\psi$ generally assumes negative values and thus is not a true probability density. In astronomical imaging, the analogous construction for a field $\psi(\vec\theta)$ enables the characterization of images in both position and spatial frequency, rendering all underlying symmetries manifest (Keller, 2016, Arai, 1 Feb 2026).

2. Shapelet Expansions: Hermite and Laguerre-Gaussian Bases

Conventional shapelet expansions employ Hermite-Gaussian (HG) or Laguerre-Gaussian (LG) modes as bases in configuration or Fourier space. In Wigner Function Shapelets, a countable orthogonal and complete basis for phase-space functions is built from cross-Wigner functions of such modes: $$W^{LG}_{(j,s),(j',s')}(q,p) \equiv W\big[\Psi^{LG}_{j,s}, \Psi^{LG}_{j',s'}\big](q,p),$$ with $\Psi^{LG}_{j,s}$ the normalized LG modes labeled by total order $j$ and spin $s$. These cross-Wigner functions constitute a basis in the Hilbert-Schmidt space of phase-space operators, satisfying the Moyal-Plancherel orthogonality relation: $\langle W_{ab} | W_{cd} \rangle_{HS} = \frac{1}{(2\pi\lambdabar)^2} \langle \psi_a|\psi_c\rangle \langle \psi_b|\psi_d\rangle.$ Any Wigner function can be expanded as

$$W(q,p) = \sum_{j,s, j', s'} c_{(j,s),(j',s')} W^{LG}_{(j,s),(j',s')}(q,p)$$

with coefficients determined by inner products in phase space (Arai, 1 Feb 2026, Keller, 2016).

3. Asymptotic Expansions and the Positive "Shapelet" Basis

For quantum expectation values, the Wigner function admits a finite-order asymptotic expansion in terms of Hermite spectrograms (positive, smooth probability densities). Specifically, for a given order $N$,

$$\mu_\psi^N(z) = \sum_{j=0}^{N-1} (-1)^j C_{N-1, j} \sum_{|k|=j} S_\psi^{\phi_k}(z),$$

where $S_\psi^{\phi_k}$ are Hermite spectrograms—i.e., the Wigner function smeared with the Wigner function of a Hermite window $\phi_k$—and $C_{N-1, j}$ are combinatorially defined coefficients. These spectrograms are genuine probability densities, and the expansion yields

$$W_\psi(q,p) \sim \sum_{j=0}^{N-1} (-1)^j C_{N-1, j}\sum_{|k|=j} S_\psi^{\phi_k}(q,p) + O(\epsilon^N).$$

All non-positivity in $W_\psi$ arises from cancellations among positive spectrogram basis elements, permitting an exact or arbitrarily precise representation of polynomial expectation values and systematic control of high-frequency error (Keller, 2016).

4. Symplectic and Representation-Theoretic Structure

In WFS, the four-dimensional phase space $(\vec\theta,\vec p)$ is acted upon by the symplectic group $\mathrm{Sp}(4, \mathbb{R})$ via canonical transformations. The quantized cell size, $2\pi\lambdabar$, imposes a resolution limit corresponding to the phase space uncertainty principle. The basis of cross-Wigner functions can be organized using the irreducible representations of $\mathrm{SU}(2)$ on Hopf tori, which are parametrized by classical constants of motion: $$Q_0 = \tfrac{1}{2}(|u|^2 + |v|^2), \quad Q_2=\tfrac{1}{2}(|u|^2 - |v|^2),$$ where $(u, v)$ are dimensionless combinations of coordinates and momenta. Functions on fixed $(Q_0, Q_2)$ tori decompose into torus harmonics $$\chi_{k,\ell}(\varphi_u, \varphi_v)=e^{i(k\varphi_u+\ell\varphi_v)}$$, with winding numbers $(k,\ell)\in \mathbb{Z}^2$. This structure gives rise to a natural phase-space "band structure," in which image components with specific angular momentum and energy content are isolated in the $(Q_0, Q_2)$ plane (Arai, 1 Feb 2026).

5. Sampling, Quadrature, and Numerical Implementation

The expansion of Wigner functions in positive spectrograms underpins practical algorithms for computing quantum expectation values and for Monte Carlo sampling. Each spectrogram $S_\psi^{\phi_k}$ can be efficiently sampled by a Markov Chain Monte Carlo procedure using Metropolis-Hastings dynamics:

• Propose new phase-space points $z' = z_{n-1} + \sqrt{\epsilon}\xi$ with $\xi\sim N(0, I_{2d})$
• Accept according to the spectrogram value ratio
• Empirical averages converge at rate $O(n^{-1/2})$

Correspondingly, expectation values are approximated as linear combinations of averaged samples from these probability densities with explicitly calculated weights. For a wavefunction $\psi$, expansion order $N$, observable $a$, and $n$ samples,

$$\langle \psi, Op^W(a)\psi \rangle \approx \sum_{j=0}^{N-1} (-1)^j C_{N-1,j} \left(\frac{1}{n\,|K_j|} \sum_{k\in K_j}\sum_{m=1}^n a(z^{(j,k)}_m) \right),$$

where each $z^{(j,k)}_m$ is sampled from $S_\psi^{\phi_k}$ (Keller, 2016).

6. Applications in Image Analysis and Quantum Mechanics

WFS generalize radiative-transfer and weak lensing methods by enabling phase-space analysis that is symplectic and information-preserving. The explicit phase-space band structure admits sensitivity to spatially coherent galaxy morphology (e.g., spiral arms, bars, bulges) and enables the discrimination of features such as parity-violating signals. In quantum mechanics, the shapelet expansion allows precision quadrature of expectation values, recovering exact moments for polynomial observables of degree less than $2N$ and controlling errors in the high-frequency regime. Systematics and noise, including point-spread functions and stochasticities, can be modeled as quantum channels (completely positive trace-preserving maps) in Wigner space. A stationary point-spread function results in Gaussian convolution, while a stochastic PSF induces diffusion and damping of high-mode coefficients (Arai, 1 Feb 2026, Keller, 2016).

7. Limiting Behavior and Symmetry Considerations

In the classical limit $\lambdabar \to 0$, the von Neumann (Moyal) equation for time evolution reduces to the classical Liouville equation, and the Wigner function approaches a classical radiance on phase space. WFS are Sp(4,ℝ)-covariant, ensuring that all canonical transformations act naturally on the expansion coefficients, and SU(2) symmetry organizes basis elements according to constants of motion and torus charges. This symmetry structure facilitates the construction of expansions adapted to invariant quantities and supports efficient marginalization and analysis of morphological content. The use of Hopf tori and torus harmonics yields banded decompositions of Wigner functions, directly linking algebraic representation theory with concrete phase-space pattern recognition (Arai, 1 Feb 2026).

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Wigner Function Shapelets thus provide a unified, symmetry-preserving formalism for phase-space analysis, bridging quantum mechanical applications and advanced astronomical image processing, while enabling systematic computational schemes via expansions in positive-definite, sampling-friendly basis functions (Keller, 2016, Arai, 1 Feb 2026).