- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Wigner function shapelets: Symplectic representation of astronomical images

Shun Arai*

  • *Contact author: shunarai@kmi.nagoya-u.ac.jp
PDF

Phys. Rev. D 114, 023024 – Published 16 July, 2026

DOI: https://doi.org/10.1103/zgtf-m8y7

Abstract

We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing 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 Sp(4,R), which is quantized by a phase-space cell 2πƛ 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 SU(2) irreducible representations of the phase space with the Hopf tori. We introduce a scalar function Wkℓ(Q0,Q2) from the U(1)×U(1) covariant tori to a two-dimensional space of constants of motion (Q0,Q2)—the harmonic energy and axial angular momentum—thereby yielding a natural phase-space “band structure,” given a pair of winding numbers (k,ℓ)∈Z2. The WFS leverages key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way, (ii) its transport equation naturally evolves with a Liouville equation at ƛ→0, (iii) it admits positive/negative oscillatory patterns on the (Q0,Q2) plane that can be a 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 quantum information theory.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (100)

  1. A. Refregier, Mon. Not. R. Astron. Soc. 338, 35 (2003).
  2. A. Refregier and D. Bacon, Mon. Not. R. Astron. Soc. 338, 48 (2003).
  3. R. Massey and A. Refregier, Mon. Not. R. Astron. Soc. 363, 197 (2005).
  4. R. Massey, B. Rowe, A. Refregier, D. J. Bacon, and J. Bergé, Mon. Not. R. Astron. Soc. 380, 229 (2007).
  5. R. Mandelbaum, Annu. Rev. Astron. Astrophys. 56, 393 (2018).
  6. D. J. Bacon, D. M. Goldberg, B. T. P. Rowe, and A. N. Taylor, Mon. Not. R. Astron. Soc. 365, 414 (2006).
  7. X. Li, N. Katayama, M. Oguri, and S. More, Mon. Not. R. Astron. Soc. 481, 4445 (2018).
  8. X. Li, Y. Li, and R. Massey, Mon. Not. R. Astron. Soc. 511, 4850 (2022).
  9. J. P. Gardner et al., Space Sci. Rev. 123, 485 (2006).
  10. R. Laureijs et al., arXiv:1110.3193.
  11. LSST Science Collaboration, arXiv:0912.0201.
  12. R. Braun et al., arXiv:1912.12699.
  13. D. Spergel et al., arXiv:1503.03757.
  14. A. Tagore and C. Keeton, Mon. Not. R. Astron. Soc. 445, 694 (2014).
  15. N. Ephremidze, C. Chandrashekar, A. Çağan Şengül, and C. Dvorkin, Mon. Not. R. Astron. Soc. 542, 2610 (2025).
  16. S. Yatawatta, Astron. Astrophys. 692, A31 (2024).
  17. G. Nienhuis, Phys. Rev. A 48, 656 (1993).
  18. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, NJ, 1957), foundational text on angular momentum theory.
  19. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  20. E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Pure and Applied Physics Vol. 5 (Academic Press, New York, 1959), Expanded and improved edition; translated by J. J. Griffin.
  21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
  22. P. A. M. Dirac, The Principles of Quantum Mechanics, 1st ed. (Oxford University Press, Oxford, 1930), Landmark text in quantum mechanics.
  23. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1997), Standard graduate-level textbook; listed as core text in MIT OCW quantum mechanics syllabus (Fall 2018).
  24. E. P. Wigner, Phys. Rev. 40, 749 (1932).
  25. J. Ville, Câbles Transm. 2, 61 (1948) in French; Introduces the analytic signal formalism.
  26. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, New York, 2001).
  27. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, Cambridge, England, 1997).
  28. A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).
  29. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).
  30. H. J. Groenewold, Physica (Amsterdam) 12, 405 (1946).
  31. J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).
  32. R. Simon and G. S. Agarwal, Opt. Lett. 25, 1313 (2000).
  33. H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1950); English translation by H. P. Robertson of the German original: Z. Phys. 46, 1 (1927).
  34. R. L. Stratonovich, Sov. Phys. JETP 31, 1012 (1956).
  35. J. C. Várilly and J. M. Gracia-Bondía, Ann. Phys. (N.Y.) 190, 107 (1989).
  36. C. Brif and A. Mann, Phys. Rev. A 59, 971 (1999).
  37. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
  38. G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies Vol. 122 (Princeton University Press, Princeton, NJ, 1989), p. 288.
  39. H. Hopf, Math. Ann. 104, 637 (1931).
  40. H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).
  41. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
  42. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, England, 1999).
  43. D. J. Schroeder, Astronomical Optics, 2nd ed. (Academic Press, San Diego, 2000).
  44. E. Hecht, Optics, 4th ed. (Addison Wesley, San Francisco, 2002).
  45. D. Manzano, AIP Adv. 10, 025106 (2020).
  46. P. H. van Cittert, Physica (Amsterdam) 1, 201 (1934).
  47. F. Zernike, Physica (Amsterdam) 5, 785 (1938).
  48. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  49. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).
  50. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, Greenwood Village, CO, 2005).
  51. R. Cerbino, Phys. Rev. A 75, 053815 (2007).
  52. A. J. F. Siegert, On the fluctuations in signals returned by many independently moving scatterers, Technical Report No. 465, MIT Radiation Laboratory, 1943.
  53. M. Lax, Rev. Mod. Phys. 32, 25 (1960).
  54. R. Hanbury Brown and R. Q. Twiss, Nature (London) 177, 27 (1956).
  55. R. Hanbury Brown and R. Q. Twiss, Proc. R. Soc. A 242, 300 (1957).
  56. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  57. N. Joshi, S. Jhingan, T. Souradeep, and A. Hajian, Phys. Rev. D 81, 083012 (2010).
  58. L. G. Book, M. Kamionkowski, and T. Souradeep, Phys. Rev. D 85, 023010 (2012).
  59. T. Kurita, D. Jamieson, E. Komatsu, and F. Schmidt, Phys. Rev. D 113, 063557 (2026).
  60. W. Heisenberg, Z. Phys. 43, 172 (1927).
  61. E. H. Kennard, Z. Phys. 44, 326 (1927).
  62. H. P. Robertson, Phys. Rev. 34, 163 (1929).
  63. P. Jordan, Z. Phys. 94, 531 (1935).
  64. J. S. Schwinger, On angular momentum, Technical Report No. NYO-3071, U.S. Atomic Energy Commission, 1952; Reprinted in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. Van Dam (Academic Press, New York, 1965); and in A Quantum Legacy, edited by K. A. Milton (World Scientific, Singapore, 2000).
  65. L. Poh-aun, S.-H. Ong, and H. Srivastava, Int. J. Comput. Math. 78, 303 (2001).
  66. E. Komatsu, Nat. Rev. Phys. 4, 452 (2022).
  67. O. H. E. Philcox, M. J. König, S. Alexander, and D. N. Spergel, Phys. Rev. D 109, 063541 (2024).
  68. J. Zhang, Mon. Not. R. Astron. Soc. 383, 113 (2008).
  69. D. J. Bacon, D. M. Goldberg, B. T. P. Rowe, and A. N. Taylor, Mon. Not. R. Astron. Soc. 365, 414 (2006).
  70. Y. Okura and T. Futamase, Mon. Not. R. Astron. Soc. 516, 668 (2022).
  71. Y. Minami and E. Komatsu, Astrophys. J. Lett. 902, L22 (2020).
  72. F. Naokawa, Phys. Rev. Lett. 136, 041004 (2026).
  73. Y. Minami and E. Komatsu, Phys. Rev. Lett. 125, 221301 (2020).
  74. C. Hirata and U. Seljak, Mon. Not. R. Astron. Soc. 343, 459 (2003).
  75. A. R. Thompson, J. M. Moran, and G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Springer, New York, 2017).
  76. J. E. Krist, R. N. Hook, and F. Stoehr, Publ. Astron. Soc. Pac. 117, 594 (2005).
  77. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, New York, 1998).
  78. J. R. Janesick, Scientific Charge-Coupled Devices, Monograph Volume PM83 (SPIE Press, 2001).
  79. D. Petz, Commun. Math. Phys. 105, 123 (1986).
  80. A. Mokeev, B. Saif, M. D. Lukin, and J. Borregaard, PRX Quantum 7, 010318 (2026).
  81. M. Kalash and M. V. Chekhova, Optica 10, 1142 (2023).
  82. N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, Nat. Commun. 10, 1865 (2019).
  83. G. Galilei, Sidereus Nuncius, or The Sidereal Messenger (University of Chicago Press, Chicago, 1989), translated by Albert Van Helden.
  84. C. Huygens, Systema Saturnium (Adrian Vlacq, The Hague, 1659), first detailed description of Saturn’s system; includes Mars sketches.
  85. G. D. Cassini, Découverte de la division de l’anneau de Saturne (Académie des Sciences, Paris, 1675). Memoir reporting discovery of the Cassini Division.
  86. W. Herschel, Scientific Papers of Sir William Herschel (Royal Society and Royal Astronomical Society, London, 1912), Vol. 1: Collected works, including nebula and cluster morphology.
  87. E. P. Hubble, Astrophys. J. 64, 321 (1926).
  88. E. P. Hubble, The Realm of the Nebulae (Yale University Press, New Haven, 1936).
  89. F. Zwicky, Morphological Astronomy (Springer, Berlin, 1957).
  90. G. de Vaucouleurs, Hand. Phys. 53, 275 (1959).
  91. M. Steinmetz and J. F. Navarro, New Astron. Rev. 46, 699 (2002).
  92. C. J. Conselice, Annu. Rev. Astron. Astrophys. 52, 291 (2014).
  93. R. J. Buta, Proc. Int. Astron. Union 6, 171 (2010).
  94. 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).
  95. K. G. Iyer, P. Behroozi, J. R. Primack, D. J. Wilkins, A. Dekel, and D. Ceverino, Astron. Astrophys. 684, A33 (2024).
  96. S. Bag, S. Bharadwaj, S. Sarkar, and S. K. Pandey, Mon. Not. R. Astron. Soc. 508, 2765 (2021).
  97. Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, L1 (2019).
  98. M. Bartelmann and P. Schneider, Phys. Rep. 340, 291 (2001).
  99. M. Kilbinger, Rep. Prog. Phys. 78, 086901 (2015).
  100. A. Lue, L. Wang, and M. Kamionkowski, Phys. Rev. Lett. 83, 1506 (1999).

Outline

Information

Phys. Rev. D 114, 023024– Published 16 July, 2026
  • Received 1 February 2026
  • Accepted 4 June 2026
Crossmark - Check for updates button

© 2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation