Wigner function shapelets: Symplectic representation of astronomical images

Shun Arai

REVIEW 3 major objections 5 minor 105 references

Wigner function shapelets: Symplectic representation of astronomical images

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that cross-Wigner functions of Laguerre-Gaussian modes form an orthogonal, complete basis for the Wigner function of any galaxy image, and that the resulting phase-space 'band structure' separates morphology from rotation,

desk verdict A mathematically sound phase-space shapelet formalism whose central observable is not accessible from intensity-only images—worth refereeing as a formalism paper, but the authors must confront the phase-recovery problem before any astronomical estimator can be taken seriously. read the letter →

arxiv 2602.01141 v2 pith:WMTKX772 submitted 2026-02-01 astro-ph.CO astro-ph.GAquant-ph

Shun Arai This is my paper · ORCID

classification astro-ph.COastro-ph.GAquant-ph

keywords WignerfunctionshapeletsphasespaceLaguerre-GaussianmodesweakgravitationallensingcosmicshearparityviolationHopffibration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the familiar shapelet decomposition of galaxy images from separate position and Fourier spaces into a single four-dimensional phase space. It argues that the cross-Wigner functions of Laguerre-Gaussian modes form an orthogonal and complete basis for the Wigner function of a galaxy image, so every square-integrable image can be expanded in them, with coefficients equal to products of the usual Laguerre-Gaussian mode amplitudes. From this basis the paper builds a two-dimensional observable, W_kℓ(Q0,Q2), a kind of 'band structure' labelled by torus winding numbers that separates morphological charge from rotational charge and turns weak-lensing and parity-violating effects into explicit selection rules. If the formalism is correct, it would give galaxy image analysis, cosmic shear measurement, and parity tests on the sky a symmetry-preserving phase-space representation with rigorous completeness.

What carries the argument

The central object is the cross-Wigner function of two Laguerre-Gaussian modes, defined as W_ab(θ,p) = (2πλ)^{-2} ∫ d²ξ ψ_a(θ+ξ/2) ψ_b*(θ−ξ/2) e^{-ip·ξ/λ}; the collection {W^LG_{(j,s),(j',s')}} spans the Hilbert–Schmidt space of phase-space functions. The load-bearing identity is the Moyal–Plancherel relation, which converts mode orthogonality into orthogonality of the cross-Wigner basis. The Hopf spinor parametrisation (u,v) of phase space, with the level-set invariants Q0 (harmonic energy) and Q2 (axial angular momentum), organizes the phase space into Hopf tori; the torus harmonics χ_kℓ(φ_u,φ_v) and the projected observables W_kℓ(Q0,Q2) carry the gauge charges that separate morphology (J=

What would settle it

Simulate a galaxy image with a known complex field, apply a small shear γ, compute the Wigner function, expand it in WFS, and check whether the coefficients W_kℓ change exactly as Eq. (128) predicts; then repeat using only intensity data, attempting phase retrieval or forward-model fitting. If the intensity-only route cannot recover W_kℓ, the observable loses its observational handle regardless of the mathematical completeness of the basis.

Watch

Extended reading notes

Core claim

The paper establishes that the set of cross-Wigner functions of Laguerre-Gaussian modes, W^LG_{(j,s),(j',s')}, is an orthogonal and complete basis for the Hilbert–Schmidt space of Wigner functions in the four-dimensional phase space of an image. Therefore any quadratically integrable Wigner function can be expanded in this basis, and the expansion coefficients coincide with products of the LG-mode expansion coefficients of the image field. It further introduces Hopf-torus observables W_kℓ(Q0,Q2), which project the Wigner function onto torus harmonics labelled by winding numbers (k,ℓ), with selection rules: statistical isotropy forces k=ℓ, parity forces even k+ℓ, and chirality imposes W_kℓ(Q0

Load-bearing premise

The load-bearing premise is that the full complex image field ψ(θ), or equivalently its phase correlations, is accessible from observations, since the Wigner function is defined from ψ(θ) while ordinary detectors record only the intensity |ψ(θ)|².

Editorial extensions

If this is right

Any galaxy image whose Wigner function is square-integrable can be represented exactly in the WFS basis, with coefficients inherited directly from the Laguerre-Gaussian expansion of the image field.
The (Q0,Q2) band map gives a sharp diagnostic: on a statistically isotropic, parity-symmetric ensemble only even-k diagonal modes W_kk survive, so off-diagonal or odd-k power flags systematics, anisotropy, or parity violation.
Cosmic shear and flexion act on the basis through explicit selection rules (Δs=±2 for shear; Δs=±1,±3 for flexion), so lensing responses can be computed locally in phase space rather than through global shape moments.
The parity-odd correlator ⟨W^E W^B*⟩ has zero disconnected contribution for parity-symmetric skies, making the connected four-point function a direct, contamination-free probe of parity-violating physics.
PSF convolution is represented as a quantum channel, and the entanglement-fidelity kernel M identifies which morphology and rotation charges can be recovered, guiding optimal deconvolution in a symmetry-adapted basis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same cross-Wigner machinery suggests a natural way to combine multi-band or polarised images, since mixed-index cross-Wigner functions could capture colour or polarisation coherence that single-band intensity analysis integrates out; the paper sketches but does not pursue this.
Editorial inference: the W_kℓ band map could serve as a compact morphological summary statistic for large surveys, analogous to power spectra; a testable extension would be to measure W_kℓ from image simulations and check whether it separates morphology classes more cleanly than current shapelet coefficients.
Editorial inference: because real detectors record intensity only, the formalism's observational reach depends on recovering phase information; a concrete practical test would be to reconstruct ψ(θ) from intensity-plus-speckle data and see whether the measured W_kℓ reproduces the predicted shear and flexion responses.
Editorial inference: framing PSF deconvolution as a quantum channel suggests that optimal image reconstruction could be posed as a quantum error-correction problem; the next step would be to construct the Petz–Wiener recovery map from the fidelity kernel and benchmark its shear recovery against existing methods in simulated surveys.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is best read as a pure formalism: it takes the known Wigner function of Laguerre–Gaussian modes, extends it to cross-Wigner functions, and shows that these form an orthogonal and complete basis for any square-integrable Wigner function. That claim is solid, following from the Moyal/Plancherel identity and the completeness of LG modes. The author is honest about the lineage, citing Simon & Agarwal and framing the contribution as a transposition to astronomical morphology rather than claiming the diagonal Wigner function as new. The derived shear response, the Hopf-torus observable W_kℓ, and the parity-violating trispectrum estimators are genuinely new in this context, and the quantum-channel treatment of PSF effects is a useful way to think about systematics.

The soft spots are real and one is load-bearing. The stress-test note is right: the Wigner function needs the complex amplitude ψ, but astronomical images give only I=|ψ|². The paper itself flags this in Sec. II.A and again in Sec. V.D, where it concedes that reconstruction 'few have addressed similar algorithms.' That is not a minor technical aside. For any ψ with the same intensity, ψ* gives a different Wigner function (mirrored in momentum), and the WFS coefficients transform non-trivially. So the central observable W_kℓ is not a function of the measured image. Without an explicit phase-retrieval step—or a restriction to interferometric or polarimetric data where phases are available—the shear and parity estimators in Sec. IV cannot be evaluated on standard CCD images. The paper neither provides that step nor shows that any measurable statistic is insensitive to the ambiguity.

Other concerns are proportionate. Some formulas, like the birefringence response Eq. (132), are asserted without derivation; the flexion response is at least derived in an appendix, so the reader’s complaint is partially addressed. There is no code or simulation, which is fine for a formalism paper but does mean practical confidence is limited. There are also minor typos (e.g., 'Swinger' oscillators, 'Correpspondence') that a referee would catch.

If the phase-recovery gap is resolved, this could become a genuinely useful framework for weak-lensing and morphology analyses. As it stands, the paper is a mathematically careful construction looking for an observational handle. I would send it to peer review, but with a strong request that the authors either supply a path from intensity data to the Wigner function, or explicitly limit the claims to data where phase information is available. The core math deserves referee time; the astronomical significance does not yet.

Referee Report

3 major / 5 minor

Summary. The paper introduces Wigner Function Shapelets (WFS), an orthogonal and complete basis for the Wigner function of a galaxy image, built from the cross-Wigner functions of Laguerre-Gaussian modes. It develops the Hopf-torus projection W_kℓ(Q0,Q2), derives closed forms for the basis, and connects the formalism to BiPoSHs and FPFS. It also derives linear weak-lensing shear and flexion responses, a cosmic-birefringence response, and parity-violating galaxy-shape correlators, and it frames PSF convolution and noise as quantum channels. The stated aim is to make quantum-information phase-space methods available for astronomical image analysis.

Significance. The mathematical core is a legitimate and explicit extension of Simon & Agarwal: orthogonality/completeness follows from the Moyal/Plancherel identity, and the closed forms in Eqs. (95) and (106) are valuable. The Sp(4)/SU(2)/Hopf-torus structure provides an elegant organizing principle for shape modes and lensing selection rules. However, the astrophysical impact is conditional on an unaddressed observational issue: the Wigner function is defined from a complex field, while CCD images deliver only intensity. The paper itself acknowledges this gap in Sec. V.D. If that gap can be closed, the framework could become a genuinely new phase-space morphology tool; as written, its estimators lack a demonstrated data connection.

major comments (3)

[Sec. II.A / V.D and Eqs. (2), (15), (69)] The Wigner function in Eq. (2) is defined from the complex amplitude ψ(θ), but astronomical images supply I(θ)=|ψ(θ)|², which is only the θ-marginal (Eq. 15). The WFS coefficients in Eq. (69), c_{js,j's'} = ψ_js ψ*_j's', are not functions of I alone: ψ and ψ* give identical intensity but W(θ,p) versus W(θ,−p), producing different W_kℓ and different shear responses. Section V.D concedes that image reconstruction in this framework is 'non-trivial' and that few practical algorithms exist. Without an explicit phase-retrieval or coherence-measurement prescription, the estimators in Secs. IV.A–C (Eqs. 128, 131, 132) have no observational handle. This is load-bearing for the paper's application claims.
[Sec. II.A, 'Parity' and 'Chirality' bullets] The text asserts as general properties that W is invariant under phase-space reflection and under (θ,p)→(θ,−p), with the justification that W is real. Both claims are false for generic complex ψ: realness does not imply W(θ,p)=W(θ,−p), and a Wigner function is not invariant under spatial reflection unless the underlying field has the corresponding symmetry. The later conditional selection rules in Sec. III.C are derived correctly as symmetry constraints, but they contradict the universal 'invariance' statements in Sec. II.A. The manuscript must correct these property statements, since they bear on the parity-violation discussion.
[Sec. IV.C, Eq. (132)] The cosmic-birefringence response W^I_kℓ − W^S_kℓ = ω(k−ℓ)ρ^S_kℓ is asserted without derivation. The symbol ρ^S_kℓ is undefined (presumably W^S_kℓ), and the physical mechanism is unclear: cosmic birefringence rotates the polarization angle, not the scalar intensity amplitude, so it is not obvious why the scalar Wigner coefficient responds as (k−ℓ)W^S_kℓ. Since this is one of the headline cosmological outputs, it needs a derivation or an explicit reference to a calculation.

minor comments (5)

[General] Typos: 'inrtoducing' (Sec. III.A), 'Correpspondence' (Contents and Sec. III.F), 'electrocmagnetic' (Sec. II.A), 'Swinger' (Appendix F title should be 'Schwinger'). Section II.A has 'funciton' and other small spelling errors.
[Sec. IV.C] Eq. (132) uses ρ^S_kℓ; define it and keep notation consistent with W_kℓ used elsewhere in the same section.
[Fig. 1 and Table I] Figure 1 is not referenced in the main text; the caption should explain the (Q0,Q2) normalization and the color scale. Table I is large; consider moving to an appendix or summarizing the count formula in the text.
[Sec. V.B] The variable λ is at times called a 'scale' and at others a dimensionless phase-space cell area. The derivation leading to λ≈1.37 is heuristic; clarify that this is an order-of-magnitude estimate, not a precision choice.
[Sec. II.B.1] The discussion of speckle statistics is difficult to follow and arguably tangential to static galaxy imaging. A brief remark distinguishing coherent speckle from incoherent galaxy-intensity statistics would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the WFS basis and its completeness follow from standard Moyal/Plancherel identities and an external LG-mode Wigner result; the paper's own phase-recovery caveat is a feasibility limitation, not a circular step.

full rationale

The central derivation is self-contained and non-circular. In Sec. III A, the orthogonality of the WFS basis (Eq. 65) is a direct consequence of the Moyal/Plancherel identity Eq. (21), which is proven in Appendix D from the definition of the cross-Wigner function and the LG inner products. The expansion coefficients Eq. (69), c_{(j,s),(j',s')} = ψ_js ψ*_j's', are obtained by substituting the LG-mode expansion of the field into the Wigner definition, not assumed as an input. The closed analytic form Eq. (95) explicitly extends the Wigner function of LG modes from Simon and Agarwal [32], an external reference, and the paper contains no self-citations. The scale parameters σ and λ are prescribed from image second moments, PSF width, and instrument bandwidth (Secs. V A and V B); they are not fitted to a target observable, and no fitted quantity is subsequently renamed as a prediction. The shear, flexion, and parity-violating response formulae in Sec. IV are derived by Lie transport and mode selection rules, not by tuning to a desired output. The paper does flag genuine limitations: Sec. II.A notes that ψ 'is often a real positive function as it loses phase information of electromagnetic fields, but we treat ψ as an arbitrary field that can be complex in general,' and Sec. V.D concedes that reconstructing galaxy images within the quantum-information framework is 'non-trivial' and 'few have addressed similar algorithms in practical image analysis in astronomy.' These statements identify an observational-feasibility gap between intensity-only data and the full complex field needed for the Wigner function; however, that is an identifiability or applicability limitation, not a circular reduction of the formalism to its own inputs. There is no load-bearing self-citation and no step in which a prediction is equivalent by construction to a fitted or assumed quantity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

The central derivation rests on standard mathematical results (LG completeness, Wigner/Plancherel identities) which are not circular. The main free parameters σ and λ are physical scales chosen from image/instrument properties, not fitted to a target result. The most consequential assumption is the accessibility of the complex field ψ(θ) (phase information) from astronomical images, which the paper acknowledges but does not solve. The WFS basis and W_kℓ are new mathematical constructs without independent experimental evidence.

free parameters (3)

σ (image scale) = data-dependent, e.g., σ_g from second moments or argmax of f00(σ)
Determines the spatial scale of the LG modes; chosen to match the galaxy size, not a universal constant.
λ (phase-space cell scale) = λ = σ_θ σ_p, O(1)
Sets the discretisation scale of phase space; chosen from instrument resolution and bandwidth.
α_cut (top-hat scale cut) = 3–5
Multiplicative factor for the scale-cut radius r_cut = α_cut σ; chosen by hand to isolate the galaxy from surroundings.

assumptions (6)

standard math Laguerre-Gaussian modes form a complete orthonormal basis of L²(R²)
Invoked in Sec. III.A to define the WFS basis and in Eq. (65) for orthogonality.
standard math Moyal/Plancherel identity for cross-Wigner functions (Eq. 21)
Used to derive the Hilbert-Schmidt orthogonality of WFS and the coefficient relation c = ψ ψ* in Sec. III.A.
domain assumption The image field ψ(θ) is a complex amplitude accessible in phase space
The Wigner function Eq. (2) uses the full complex field; astronomical images record intensity only. Acknowledged in Sec. II.A and Sec. V.D but not resolved.
standard math Stratonovich-Weyl correspondence uniquely fixes the Wigner transform
Cited in Sec. II.A (Eq. 23) to justify the displaced-parity form of the Wigner kernel.
domain assumption Paraxial propagation and symplectic phase-space model of image plane
The whole framework assumes the image can be described by paraxial optics with phase space (θ,p) ∈ Sp(4,R), introduced in Sec. II.A.
domain assumption Galaxy image ensemble can be represented as a density matrix and processed by CPTP channels
Used in Sec. II.C and V.C to describe PSF and noise as quantum channels; this is a modeling choice specific to the paper's quantum-information framing.

invented entities (2)

WFS basis W^{LG}_{(j,s),(j',s')}
purpose: Complete orthogonal basis for decomposing the Wigner function of galaxy images in phase space.
A new mathematical construction; no independent falsifiable handle outside the paper yet.
Wigner-torus observable W_kℓ(Q0,Q2)
purpose: Provides a 'band structure' observable to characterize galaxy morphology and parity-violating correlations.
Predicted to be sensitive to coherent structure and parity violation, but no data or simulation yet demonstrates this sensitivity.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Wigner function shapelets: Symplectic representation of astronomical images." pith.science (2026). https://pith.science/paper/WMTKX772

@misc{pith2026260201141,
  author       = {Pith},
  title        = {Pith review of: Wigner function shapelets: Symplectic representation of astronomical images},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMTKX772}},
  note         = {Machine review of arXiv:2602.01141}
}

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 $2\pi\lambdabar$ 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.

Figures

Figures reproduced from arXiv: 2602.01141 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

105 extracted references · 22 linked inside Pith

[1]

Charge preserva- tion is therefore controlled by commutators with ˆK

Deterministic PSF For a deterministic PSF acting coherently on the com- plex field amplitude, the forward map is a single-operator channel ˆρO =E coh(ˆρI ) = ˆKˆρI ˆK †,(110) where ˆK = KPSF( ˆp) is the optical transfer function (OTF) acting multiplicatively in Fourier space, equivalently, a function of the momentum operator ˆp. Charge preserva- tion is t...
[2]

gauge charges

Stochastic PSF We describe the other extreme case that a galaxy image is blurred by a non-stationary PSF in position space, re- sulting in stochastic displacements of the image structure by a certain probability ΠP SF(∆). In this case, we obtain the PSF operation as EPSF(ˆρ) = Z d2∆ ΠPSF(∆) ˆT(∆)ˆρˆT †(∆),(61) where ˆT denotes the translation operator. Th...

2000
[3]

Stochastic PSF A translationally invariant PSF corresponds to a ran- dom displacement channel, K(ˆρ) = Z d2∆K(∆) ˆT(∆)ˆρˆT †(∆) ˆT(∆) = exp − i λ ∆ ˆπ† + ∆∗ ˆπ ,(113) with ˆπ= i√ 2 (ˆa† u −ˆav). In the Heisenberg picture, K†( ˆO) = Z d2∆K(∆) ˆT †(∆) ˆO ˆT(∆).(114) Applying this map to the morphology generator ˆT0, one finds K†( ˆT0) = ˆT0 + ⟨|∆|2⟩PSF 2λ2 ...
[4]

Massey, B

R. Massey, B. Rowe, A. Refregier, D. J. Bacon, and J. Berg´ e, Mon. Not. R. Astron. Soc.380, 229 (2007), arXiv:astro- ph/0609795 [astro-ph]

arXiv 2007
[5]

endpoint

and flexions [ 6], cosmic birefringence[ 66], and higher- moment of galaxy shapes [ 59, 67] in the language of the Wigner function and WFS. Throughout this section, we will argue that the response function is well-defined at the local level of the image plane, implying a possibility to handle image systematics and noise at a local level. Note that we repr...
[6]

Refregier, Mon

A. Refregier, Mon. Not. R. Astron. Soc.338, 35 (2003)

2003
[7]

Refregier and D

A. Refregier and D. Bacon, Mon. Not. R. Astron. Soc.338, 48 (2003)

2003
[8]

Massey and A

R. Massey and A. Refregier, Mon. Not. R. Astron. Soc.363, 197 (2005)

2005

Show all 105 references

[9]

Mandelbaum, Ann

R. Mandelbaum, Ann. Rev. Astron. Astrophys.56, 393 (2018), arXiv:1710.03235 [astro-ph.CO]

2018 arXiv
[11]

X. Li, N. Katayama, M. Oguri, and S. More, Mon. Not. R. Astron. Soc.481, 4445 (2018), arXiv:1805.08514 [astro-ph.CO]

2018 arXiv
[12]

X. Li, Y. Li, and R. Massey, Mon. Not. R. Astron. Soc.511, 4850 (2022), arXiv:2110.01214 [astro-ph.CO]

2022 arXiv
[13]

J. P. Gardner, J. C. Mather, M. Clampin, R. Doyon, M. A. Greenhouse, H. B. Hammel, J. B. Hutchings, P. Jakobsen, S. J. Lilly, K. S. Long, J. I. Lunine, M. J. McCaughrean, M. Mountain, J. Nella, G. H. Rieke, M. J. Rieke, H.-W. Rix, E. P. Smith, G. Sonneborn, M. Stiavelli, H. S....

2006
[14]

Laureijs et al., arXiv e-prints (2011), arXiv:1110.3193 [astro-ph.CO]

R. Laureijs et al., arXiv e-prints (2011), arXiv:1110.3193 [astro-ph.CO]

2011 arXiv
[15]

LSST Science Collaboration, LSST Science Book, Version 2.0 (arXiv e-prints, 2009) arXiv:0912.0201 [astro-ph.IM]

2009 arXiv
[16]

Braun et al., arXiv e-prints (2019), arXiv:1912.12699 [astro-ph.IM]

R. Braun et al., arXiv e-prints (2019), arXiv:1912.12699 [astro-ph.IM]

2019 arXiv
[17]

Spergel, N

D. Spergel, N. Gehrels, C. Baltay, D. Bennett, J. Breckinridge, M. Donahue, A. Dressler, B. S. Gaudi, T. Greene, O. Guyon, C. Hirata, J. Kalirai, N. J. Kasdin, W. Moos, S. Perlmutter, M. Postman, B. Rauscher, J. Rhodes, Y. Wang, D. Weinberg, D. Benford, M. Hudson, W. Jeong, Y....

2015 arXiv
[18]

Tagore and C

A. Tagore and C. Keeton, Mon. Not. Roy. Astron. Soc.445, 694 (2014), arXiv:1408.6297 [astro-ph.CO]

2014 arXiv
[19]

Ephremidze, C

N. Ephremidze, C. Chandrashekar, A. C ¸a˘ ganS ¸eng¨ ul, and C. Dvorkin, arXiv e-prints , arXiv:2502.18571 (2025), arXiv:2502.18571 [astro-ph.CO]

2025 arXiv
[20]

Yatawatta, Astron

S. Yatawatta, Astron. Astrophys. (2024), open access

2024
[21]

Nienhuis, Phys

G. Nienhuis, Phys. Rev. A48, 656 (1993)

1993
[22]

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1957) foundational text on angular momentum theory; see also review by H. C. Bolton (1959)

1957
[23]

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988)

1988
[24]

E. P. Wigner, Group Theory and its Application to theQuantum Mechanics of Atomic Spectra , Pure and Applied Physics, Vol. 5 (Academic Press, New York, 1959) expanded and improved edition; translated by J. J. Griffin

1959
[25]

Allen, M

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A45, 8185 (1992)

1992
[26]

P. A. M. Dirac, The Principles of Quantum Mechanics , 1st ed. (Oxford University Press, Oxford, 1930) landmark text in quantum mechanics; reviewed in Lennard-Jones (1931) Mathematical Gazette

1930
[27]

Cohen-Tannoudji, B

C. Cohen-Tannoudji, B. Diu, and F. Lalo¨ e,Quantum Mechanics (Wiley, New York, 1997) standard graduate-level textbook; listed as core text in MIT OCW quantum mechanics syllabus (Fall 2018)

1997
[28]

E. P. Wigner, Phys. Rev.40, 749 (1932)

1932
[29]

Ville, Cˆ ables et Transmissions2, 61 (1948), in French; introduces the analytic signal formalism

J. Ville, Cˆ ables et Transmissions2, 61 (1948), in French; introduces the analytic signal formalism

1948
[30]

W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001)

2001
[31]

Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, 1997)

U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, 1997)

1997
[32]

A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys.81, 299 (2009)

2009
[33]

Weedbrook, S

C. Weedbrook, S. Pirandola, R. Garc ´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys.84, 621 (2012)

2012
[34]

H. J. Groenewold, Physica12, 405 (1946)

1946
[35]

J. E. Moyal, Proc. Cambridge Philos. Soc.45, 99 (1949)

1949
[36]

Simon and G

R. Simon and G. S. Agarwal, Optics Letters25, 1313 (2000)

2000
[37]

Quantenmechanik und Gruppentheorie,

H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1950) english translation by H. P. Robertson of the German original: H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1 (1927)

1950
[38]

R. L. Stratonovich, Sov. Phys. JETP31, 1012 (1956)

1956
[39]

J. C. V´ arilly and J. M. Gracia-Bond ´ ıa, Ann. Phys.190, 107 (1989)

1989
[40]

Brif and A

C. Brif and A. Mann, Phys. Rev. A59, 971 (1999)

1999
[41]

Hillery, R

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep.106, 121 (1984)

1984
[42]

G. B. Folland, Harmonic Analysis in Phase Space , Annals of Mathematics Studies, Vol. 122 (Princeton University Press, Princeton, NJ, 1989) p. 288, published Mar 21, 1989

1989
[43]

Hopf, Math

H. Hopf, Math. Ann.104, 637 (1931)

1931
[44]

Kogelnik and T

H. Kogelnik and T. Li, Appl. Opt.5, 1550 (1966)

1966
[45]

A. E. Siegman, Lasers (University Science Books, 1986)

1986
[46]

Born and E

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999)

1999
[47]

D. J. Schroeder, Astronomical Optics, 2nd ed. (Academic Press, San Diego, 2000)

2000
[48]

Hecht, Optics, 4th ed

E. Hecht, Optics, 4th ed. (Addison Wesley, San Francisco, 2002)

2002
[49]

Manzano, AIP Advances10, 025106 (2020), arXiv:1906.04478 [quant-ph]

D. Manzano, AIP Advances10, 025106 (2020), arXiv:1906.04478 [quant-ph]

2020 arXiv
[50]

P. H. van Cittert, Physica1, 201 (1934)

1934
[51]

Zernike, Physica5, 785 (1938)

F. Zernike, Physica5, 785 (1938)

1938
[52]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985)

1985
[53]

J. W. Goodman, Journal of the Optical Society of America66, 1145 (1976)

1976
[54]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, Greenwood Village, CO, 2005)

2005
[55]

Cerbino, Physical Review A75, 053815 (2007)

R. Cerbino, Physical Review A75, 053815 (2007)

2007
[56]

A. J. F. Siegert, On the Fluctuations in Signals Returned by Many Independently Moving Scatterers , Tech. Rep. 465 (MIT Radiation Laboratory, 1943)

1943
[57]

Lax, Reviews of Modern Physics32, 25 (1960)

M. Lax, Reviews of Modern Physics32, 25 (1960)

1960
[58]

Hanbury Brown and R

R. Hanbury Brown and R. Q. Twiss, Nature177, 27 (1956)

1956
[59]

Hanbury Brown and R

R. Hanbury Brown and R. Q. Twiss, Proceedings of the Royal Society of London Series A242, 300 (1957)

1957
[60]

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (1988)

1988
[61]

Joshi, S

N. Joshi, S. Jhingan, T. Souradeep, and A. Hajian, Phys. Rev. D81, 083012 (2010), arXiv:0912.3217 [astro-ph.CO]

2010 arXiv
[62]

L. G. Book, M. Kamionkowski, and T. Souradeep, Phys. Rev. D85, 023010 (2012), arXiv:1109.2910 [astro-ph.CO]

2012 arXiv
[63]

Kurita, D

T. Kurita, D. Jamieson, E. Komatsu, and F. Schmidt, arXiv e-prints , arXiv:2509.08787 (2025), arXiv:2509.08787 [astro-ph.CO]

2025 arXiv
[64]

Heisenberg, Zeitschrift fur Physik43, 172 (1927)

W. Heisenberg, Zeitschrift fur Physik43, 172 (1927)

1927
[65]

E. H. Kennard, Zeitschrift fur Physik44, 326 (1927)

1927
[66]

H. P. Robertson, Physical Review34, 163 (1929)

1929
[67]

Jordan, Zeitschrift f¨ ur Physik94, 531 (1935)

P. Jordan, Zeitschrift f¨ ur Physik94, 531 (1935)

1935
[68]

J. S. Schwinger, On Angular Momentum, Tech. Rep. NYO-3071 (U.S. Atomic Energy Commission, 1952) technical Report; 29 reprinted in L.C.Biedenharn and H.Van Dam (eds.), *Quantum Theory of Angular Momentum* (Academic Press, 1965) and in K.A.Milton (ed.), *A Quantum Legacy* (World...

1952
[69]

Poh-aun, S.-H

L. Poh-aun, S.-H. Ong, and H. Srivastava, International Journal of Computer Mathematics78, 303 (2001)

2001
[70]

Komatsu, Nature Reviews Physics4, 452 (2022), arXiv:2202.13919 [astro-ph.CO]

E. Komatsu, Nature Reviews Physics4, 452 (2022), arXiv:2202.13919 [astro-ph.CO]

2022 arXiv
[71]

O. H. E. Philcox, M. J. K¨ onig, S. Alexander, and D. N. Spergel, Phys. Rev. D109, 063541 (2024), arXiv:2309.08653 [astro-ph.CO]

2024 arXiv
[72]

Zhang, Mon

J. Zhang, Mon. Not. R. Astron. Soc.383, 113 (2008), arXiv:astro-ph/0612146 [astro-ph]

2008 arXiv
[73]

D. J. Bacon, D. M. Goldberg, B. T. P. Rowe, and A. N. Taylor, Mon. Not. R. Astron. Soc.365, 414 (2006), arXiv:astro- ph/0504478 [astro-ph]

2006
[74]

Okura and T

Y. Okura and T. Futamase, Mon. Not. R. Astron. Soc.516, 668 (2022), arXiv:2109.00155 [astro-ph.CO]

2022 arXiv
[75]

Minami and E

Y. Minami and E. Komatsu, Astrophys. J. Lett.902, L22 (2020), apJ Lett. 902, L22 (2020)

2020
[76]

Naokawa, arXiv e-prints , arXiv:2504.06709 (2025), arXiv:2504.06709 [astro-ph.CO]

F. Naokawa, arXiv e-prints , arXiv:2504.06709 (2025), arXiv:2504.06709 [astro-ph.CO]

2025 arXiv
[77]

Minami and E

Y. Minami and E. Komatsu, Phys. Rev. Lett.125, 221301 (2020), arXiv:2011.11254 [astro-ph.CO]

2020 arXiv
[78]

Hirata and U

C. Hirata and U. Seljak, Mon. Not. R. Astron. Soc.343, 459 (2003), arXiv:astro-ph/0301054 [astro-ph]

2003 arXiv
[79]

A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Springer, 2017)

2017
[80]

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005)

2005
[81]

J. E. Krist, R. N. Hook, and F. Stoehr, PASP117, 594 (2005)

2005
[82]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998)

1998
[83]

J. R. Janesick, Scientific Charge-Coupled Devices (SPIE Press, 2001)

2001
[84]

Petz, Communications in Mathematical Physics105, 123 (1986)

D. Petz, Communications in Mathematical Physics105, 123 (1986)

1986
[85]

Mokeev, B

A. Mokeev, B. Saif, M. D. Lukin, and J. Borregaard, arXiv e-prints , arXiv:2509.09465 (2025), arXiv:2509.09465 [quant-ph]

2025
[86]

Kalash and M

M. Kalash and M. V. Chekhova, Optica10, 1142 (2023), arXiv:2207.10030 [quant-ph]

2023 arXiv
[87]

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, Nature Communications10, 1865 (2019), arXiv:1803.04126 [physics.optics]

2019 arXiv
[88]

Galilei, Sidereus Nuncius, or The Sidereal Messenger (University of Chicago Press, Chicago, 1989) translated by Albert Van Helden

G. Galilei, Sidereus Nuncius, or The Sidereal Messenger (University of Chicago Press, Chicago, 1989) translated by Albert Van Helden

1989
[89]

Huygens, Systema Saturnium (Adrian Vlacq, The Hague, 1659) first detailed description of Saturn’s system; includes Mars sketches

C. Huygens, Systema Saturnium (Adrian Vlacq, The Hague, 1659) first detailed description of Saturn’s system; includes Mars sketches
[90]

G. D. Cassini, D´ ecouverte de la division de l’anneau de Saturne(Acad´ emie des Sciences, Paris, 1675) memoir reporting discovery of the Cassini Division
[91]

Herschel, Scientific Papers of Sir William Herschel , Vol

W. Herschel, Scientific Papers of Sir William Herschel , Vol. 1 (Royal Society and Royal Astronomical Society, London,
[92]

Steinmetz and J

M. Steinmetz and J. F. Navarro, New Astron. Rev.46, 699 (2002)

2002
[93]

E. P. Hubble, Astrophys. J.64, 321 (1926)

1926
[94]

E. P. Hubble, The Realm of the Nebulae (Yale University Press, New Haven, 1936)

1936
[95]

Zwicky, Morphological Astronomy (Springer, Berlin, 1957)

F. Zwicky, Morphological Astronomy (Springer, Berlin, 1957)

1957
[96]

de Vaucouleurs, Handbuch der Physik53, 275 (1959)

G. de Vaucouleurs, Handbuch der Physik53, 275 (1959)

1959
[97]

S. Bag, S. Bharadwaj, S. Sarkar, and S. K. Pandey, Mon. Not. R. Astron. Soc.508, 2765 (2021)

2021
[98]

C. J. Conselice, Ann. Rev. Astron. Astrophys.52, 291 (2014)

2014
[99]

R. J. Buta, Proc. Int. Astron. Union6, 171 (2010)

2010
[100]

C. J. Lintott, K. Schawinski, A. Slosar, K. Land, S. Bamford, D. Thomas, M. J. Raddick, R. C. Nichol, A. Szalay, D. Andreescu, P. Murray, and J. Vandenberg, Mon. Not. R. Astron. Soc.389, 1179 (2008)

2008
[101]

K. G. Iyer, P. Behroozi, J. R. Primack, D. J. Wilkins, A. Dekel, and D. Ceverino, Astron. Astrophys.684, A33 (2024)

2024
[103]

Event Horizon Telescope Collaboration, Astrophys. J. Lett.875, L1 (2019)

2019
[104]

Bartelmann and P

M. Bartelmann and P. Schneider, Phys. Rep.340, 291 (2001)

2001
[105]

Kilbinger, Rep

M. Kilbinger, Rep. Prog. Phys.78, 086901 (2015)

2015
[106]

A. Lue, L. Wang, and M. Kamionkowski, Phys. Rev. Lett.83, 1506 (1999)

1999
[1912]

collected works, including nebula and cluster morphology

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Charge preserva- tion is therefore controlled by commutators with ˆK

Deterministic PSF For a deterministic PSF acting coherently on the com- plex field amplitude, the forward map is a single-operator channel ˆρO =E coh(ˆρI ) = ˆKˆρI ˆK †,(110) where ˆK = KPSF( ˆp) is the optical transfer function (OTF) acting multiplicatively in Fourier space, equivalently, a function of the momentum operator ˆp. Charge preserva- tion is t...

[2] [2]

gauge charges

Stochastic PSF We describe the other extreme case that a galaxy image is blurred by a non-stationary PSF in position space, re- sulting in stochastic displacements of the image structure by a certain probability ΠP SF(∆). In this case, we obtain the PSF operation as EPSF(ˆρ) = Z d2∆ ΠPSF(∆) ˆT(∆)ˆρˆT †(∆),(61) where ˆT denotes the translation operator. Th...

2000

[3] [3]

Stochastic PSF A translationally invariant PSF corresponds to a ran- dom displacement channel, K(ˆρ) = Z d2∆K(∆) ˆT(∆)ˆρˆT †(∆) ˆT(∆) = exp − i λ ∆ ˆπ† + ∆∗ ˆπ ,(113) with ˆπ= i√ 2 (ˆa† u −ˆav). In the Heisenberg picture, K†( ˆO) = Z d2∆K(∆) ˆT †(∆) ˆO ˆT(∆).(114) Applying this map to the morphology generator ˆT0, one finds K†( ˆT0) = ˆT0 + ⟨|∆|2⟩PSF 2λ2 ...

[4] [4]

Massey, B

R. Massey, B. Rowe, A. Refregier, D. J. Bacon, and J. Berg´ e, Mon. Not. R. Astron. Soc.380, 229 (2007), arXiv:astro- ph/0609795 [astro-ph]

arXiv 2007

[5] [5]

endpoint

and flexions [ 6], cosmic birefringence[ 66], and higher- moment of galaxy shapes [ 59, 67] in the language of the Wigner function and WFS. Throughout this section, we will argue that the response function is well-defined at the local level of the image plane, implying a possibility to handle image systematics and noise at a local level. Note that we repr...

[6] [6]

Refregier, Mon

A. Refregier, Mon. Not. R. Astron. Soc.338, 35 (2003)

2003

[7] [7]

Refregier and D

A. Refregier and D. Bacon, Mon. Not. R. Astron. Soc.338, 48 (2003)

2003

[8] [8]

Massey and A

R. Massey and A. Refregier, Mon. Not. R. Astron. Soc.363, 197 (2005)

2005