Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Coherent States, Wavelets, and Their Generalizations

Part of the book series: Theoretical and Mathematical Physics ((TMP))

Abstract

We start with a description of the canonical coherent states (CS) and some historical remarks on the evolution of the concept and its applications. Then we present in detail the organization of the book.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The acronym CS will be used throughout this book to mean “coherent state” or “coherent states,” depending on the context.

  2. 2.

    For convenience, we have split the bibliography into two separate lists, books and theses (denoted by letters) and articles (denoted by numbers).

References

  1. A. Arnéodo, F. Argoul, E. Bacry, J. Elezgaray, J.F. Muzy, Ondelettes, multifractales et turbulences – De l’ADN aux croissances cristallines (Diderot, Paris, 1995)

    Google Scholar 

  2. I. Bengtsson, K. Z̀yczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  3. M. Combescure, D. Robert, Shape Analysis and Classification: Theory and Practice (Springer, Dordrecht, Heidelberg, 2012)

    Google Scholar 

  4. V.V. Dodonov, V.I. Man’ko (eds.), Theory of Nonclassical States of Light (Taylor and Francis, London, New York, 2003)

    Google Scholar 

  5. D.H. Feng, J.R. Klauder, M. Strayer (eds.), Coherent States: Past, Present and Future (Proc. Oak Ridge 1993) (World Scientific, Singapore, 1994)

    Google Scholar 

  6. H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms. Lecture Notes in Mathematics, vol. 1863 (Springer, Berlin, Heidelberg, 2005)

    Google Scholar 

  7. J-P. Gazeau, Coherent States in Quantum Physics (Wiley-VCH, Berlin, 2009)

    Google Scholar 

  8. E.C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw Hill, New York, 1937)

    Google Scholar 

  9. J.R. Klauder, E.C.G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968)

    Google Scholar 

  10. J.R. Klauder, B.S. Skagerstam, Coherent States – Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985)

    Book  MATH  Google Scholar 

  11. J.R. Klauder, A Modern Approach to Functional Integration (Birkhäuser/Springer, New York, 2011)

    Book  MATH  Google Scholar 

  12. G.W. Mackey, Induced Representations of Groups and Quantum Mechanics (Benjamin, New York, 1968)

    MATH  Google Scholar 

  13. G.W. Mackey, Theory of Unitary Group Representations (University of Chicago Press, Chicago, 1976)

    MATH  Google Scholar 

  14. H. Meschkowsky, Hilbertsche Räume mit Kernfunktionen (Springer, Berlin, 1962)

    Book  Google Scholar 

  15. A.M. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986)

    Book  MATH  Google Scholar 

  16. S.T. Ali, Stochastic localisation, quantum mechanics on phase space and quantum space-time. Riv. Nuovo Cim. 8(11), 1–128 (1985)

    Article  Google Scholar 

  17. S.T. Ali, J-P. Antoine, Coherent states of 1+1 dimensional Poincaré group: Square integrability and a relativistic Weyl transform. Ann. Inst. H. Poincaré 51, 23–44 (1989)

    MathSciNet  MATH  Google Scholar 

  18. S.T. Ali, H.-D. Doebner, Ordering problem in quantum mechanics: Prime quantization and a physical interpretation. Phys. Rev. A 41, 1199–1210 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  19. S.T. Ali, E. Prugovečki, Systems of imprimitivity and representations of quantum mechanics on fuzzy phase spaces. J. Math. Phys. 18, 219–228 (1977)

    Article  ADS  MATH  Google Scholar 

  20. S.T. Ali, E. Prugovečki, Harmonic analysis and systems of covariance for phase space representation of the Poincaré group. Acta Appl. Math. 6, 47–62 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. S.T. Ali, J-P. Antoine, J-P. Gazeau, De Sitter to Poincaré contraction and relativistic coherent states. Ann. Inst. H. Poincaré 52, 83–111 (1990)

    MathSciNet  MATH  Google Scholar 

  22. S.T. Ali, J-P. Antoine, J-P. Gazeau, Relativistic quantum frames. Ann. Phys.(NY) 222, 38–88 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. S.T. Ali, J-P. Antoine, J-P. Gazeau, U.A. Mueller, Coherent states and their generalizations: A mathematical overview. Rev. Math. Phys. 7, 1013–1104 (1995)

    Google Scholar 

  24. S.T. Ali, J-P. Antoine, F. Bagarello, J-P. Gazeau (Guest Editors), Coherent states: A contemporary panorama, preface to a special issue on Coherent states: Mathematical and physical aspects. J. Phys. A: Math. Gen. 45(24) (2012)

    Google Scholar 

  25. P. Aniello, G. Cassinelli, E. De Vito, A. Levrero, Square-integrability of induced representations of semidirect products. Rev. Math. Phys. 10, 301–313 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Aniello, G. Cassinelli, E. De Vito, A. Levrero, Wavelet transforms and discrete frames associated to semidirect products. J. Math. Phys. 39, 3965–3973 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. N. Aronszajn, Theory of reproducing kernels. Trans. Amer. Math. Soc. 66, 337–404 (1950)

    Article  MathSciNet  Google Scholar 

  28. E.W. Aslaksen, J.R. Klauder, Unitary representations of the affine group. J. Math. Phys. 9, 206–211 (1968)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. E.W. Aslaksen, J.R. Klauder, Continuous representation theory using the affine group. J. Math. Phys. 10, 2267–2275 (1969)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. P.W. Atkins, J.C. Dobson, Angular momentum coherent states. Proc. Roy. Soc. London, A 321, 321–340 (1971)

    Google Scholar 

  31. H. Bacry, J-M. Lévy-Leblond, Possible kinematics, J. Math. Phys. 9, 1605–1614 (1968)

    Google Scholar 

  32. A.O. Barut, H. Kleinert, Transition probabilities of the hydrogen atom from noncompact dynamical groups. Phys. Rev. 156, 1541–1545 (1967)

    Article  ADS  Google Scholar 

  33. J. Ben Geloun, J. R. Klauder, Ladder operators and coherent states for continuous spectra. J. Phys. A: Math. Theor. 42, 375209 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  34. F.A. Berezin, General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. H. Bergeron, From classical to quantum mechanics: How to translate physical ideas into mathematical language. J. Math. Phys. 42, 3983–4019 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. H. Bergeron, J-P. Gazeau, P. Siegl, A. Youssef, Semi-classical behavior of Pöschl-Teller coherent states. Eur. Phys. Lett. 92, 60003 (2010)

    Google Scholar 

  37. D. Bernier, K.F. Taylor, Wavelets from square-integrable representations. SIAM J. Math. Anal. 27, 594–608 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. G. Bernuau, Wavelet bases associated to a self-similar quasicrystal. J. Math. Phys. 39, 4213–4225 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. I. Bogdanova, P. Vandergheynst, J-P. Antoine, L. Jacques, M. Morvidone, Stereographic wavelet frames on the sphere,. Appl. Comput. Harmon. Anal. 19, 223–252 (2005)

    Google Scholar 

  40. P.G. Casazza, G. Kutyniok, Frames of subspaces, in Wavelets, Frames and Operator Theory. Contemporary Mathematics, vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87–113

    Google Scholar 

  41. D.L. Donoho, Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data, in Different Perspectives on Wavelets. Proceedings of Symposia in Applied Mathematics, vol. 38, ed. by I. Daubechies (American Mathematical Society, Providence, RI, 1993), pp. 173–205

    Google Scholar 

  42. H. Führ, M. Mayer, Continuous wavelet transforms from semidirect products: Cyclic representations and Plancherel measure. J. Fourier Anal. Appl. 8, 375–396 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. D. Gabor, Theory of communication. J. Inst. Electr. Engrg.(London) 93, 429–457 (1946)

    Google Scholar 

  44. J-P. Gabardo, D. Han, Frames associated with measurable spaces. Adv. Comput. Math. 18, 127–147 (2003)

    Google Scholar 

  45. J-P. Gazeau, On the four Euclidean conformal group structure of the sturmian operator. Lett. Math. Phys. 3, 285–292 (1979)

    Google Scholar 

  46. J-P. Gazeau, A remarkable duality in one particle quantum mechanics between some confining potentials and \((R + L_{\epsilon }^{\infty })\) potentials. Phys. Lett. 75A, 159–163 (1980)

    Google Scholar 

  47. J-P. Gazeau, \(\mathrm{SL}(2, \mathbb{R})\)-coherent states and integrable systems in classical and quantum physics, in Quantization, Coherent States, and Complex Structures, ed. by J-P. Antoine, S.T. Ali, W. Lisiecki, I.M. Mladenov, A. Odzijewicz (Plenum Press, New York and London, 1995), pp. 147–158

    Google Scholar 

  48. J-P. Gazeau, P. Monceau, Generalized coherent states for arbitrary quantum systems, in Colloquium M. Flato (Dijon, Sept. 99), vol. II (Klüwer, Dordrecht, 2000), pp. 131–144

    Google Scholar 

  49. R. Gilmore, On properties of coherent states. Rev. Mex. Fis. 23, 143–187 (1974)

    Google Scholar 

  50. J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. R.J. Glauber, Coherent and incoherent states of radiation field. Phys. Rev. 131, 2766–2788 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  52. J. Glimm, Locally compact transformation groups. Trans. Amer. Math. Soc. 101, 124–138 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Grossmann, J. Morlet, Decomposition of functions into wavelets of constant shape, and related transforms, in Mathematics + Physics, Lectures on recent results. I, ed. by L. Streit (World Scientific, Singapore, 1985), pp. 135–166

    Chapter  Google Scholar 

  54. A. Grossmann, J. Morlet, T. Paul, Integral transforms associated to square integrable representations II: Examples. Ann. Inst. H. Poincaré 45, 293–309 (1986)

    MathSciNet  MATH  Google Scholar 

  55. A. Grossmann, R. Kronland-Martinet, J. Morlet, Reading and understanding the continuous wavelet transform, in Wavelets, Time-Frequency Methods and Phase Space (Proc. Marseille 1987), ed. by J.-M. Combes, A. Grossmann, P. Tchamitchian, 2nd edn. (Springer, Berlin, 1990), pp. 2–20

    Google Scholar 

  56. K. Hepp, E.H. Lieb, Equilibrium statistical mechanics of matter interacting with the quantized radiation field. Phys. Rev. A 8, 2517–2525 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  57. J.A. Hogan, J.D. Lakey, Extensions of the Heisenberg group by dilations and frames. Appl. Comput. Harmon. Anal. 2, 174–199 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  58. M. Holschneider, General inversion formulas for wavelet transforms. J. Math. Phys. 34, 4190–4198 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. J.R. Klauder, Continuous-representation theory II: Generalized relation between quantum and classical dynamics. J. Math. Phys. 4, 1058–1073 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  60. J.R. Klauder, Path integrals for affine variables, in Functional Integration, Theory and Applications, ed. by J-P. Antoine, E. Tirapegui (Plenum Press, New York and London, 1980), pp. 101–119

    Chapter  Google Scholar 

  61. J.R. Klauder, Are coherent states the natural language of quantum mechanics?, in Fundamental Aspects of Quantum Theory, ed. by V. Gorini, A. Frigerio. NATO ASI Series, vol. B 144 (Plenum Press, New York, 1986), pp. 1–12

    Google Scholar 

  62. J.R. Klauder, Quantization without quantization. Ann. Phys. (NY) 237, 147–160 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. J.R. Klauder, An affinity for affine quantum gravity. Proc. Steklov Inst. Math. 272, 169–176 (2011) and references therein

    Google Scholar 

  64. A.B. Klimov, C. Muñoz, Coherent, isotropic and squeezed states in a N-qubit system. Phys. Scr. 87, 038110 (2013)

    Article  ADS  Google Scholar 

  65. G. Lindblad, B. Nagel, Continuous bases for unitary irreducible representations of SU(1,1). Ann. Inst. H. Poincaré 13, 27–56 (1970)

    MathSciNet  MATH  Google Scholar 

  66. S.G. Mallat, Multifrequency channel decompositions of images and wavelet models. IEEE Trans. Acoust. Speech, Signal Proc. 37, 2091–2110 (1989)

    Google Scholar 

  67. M. Nauenberg, C. Stroud, J. Yeazell, The classical limit of an atom. Scient. Amer. 270, 24–29 (1994)

    Article  Google Scholar 

  68. H. Neumann, Transformation properties of observables. Helv. Phys. Acta 45, 811–819 (1972)

    MathSciNet  Google Scholar 

  69. M.M. Nieto, L.M. Simmons Jr., Coherent states for general potentials. Phys. Rev. Lett. 41, 207–210 (1987)

    Article  ADS  Google Scholar 

  70. A. Odzijewicz, On reproducing kernels and quantization of states. Commun. Math. Phys. 114, 577–597 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  71. D. Oriti, R. Pereira, L. Sindoni, Coherent states in quantum gravity: A construction based on the flux representation of loop quantum gravity. J. Phys. A: Math. Theor. 45, 244004 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  72. A.M. Perelomov, Coherent states and symmetric spaces. Commun. Math. Phys. 44, 197–210 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  73. E. Prugovec̆ki, Relativistic quantum kinematics on stochastic phase space for massive particles. J. Math.Phys. 19, 2261–2270 (1978)

    Google Scholar 

  74. C. Quesne, Coherent states of the real symplectic group in a complex analytic parametrization I, II. J. Math. Phys. 27, 428–441, 869–878 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  75. D. Robert, La cohérence dans tous ses états. SMF Gazette 132 (2012)

    Google Scholar 

  76. D.J. Rotenberg, Application of Sturmian functions to the Schroedinger three-body problem: Elastic e+–H scattering. Ann. Phys. (NY) 19, 262–278 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  77. E. Schrödinger, Der stetige Übergang von der Mikro- zur Makromechanik. Naturwiss. 14, 664–666 (1926)

    Article  ADS  MATH  Google Scholar 

  78. E.C.G. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277–279 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  79. A. Vourdas, Analytic representations in quantum mechanics. J. Phys. A: Math. Gen. 39, R65–R141 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  80. J.A. Yeazell, M. Mallalieu, C.R. Stroud Jr., Observation of the collapse and revival of a Rydberg electronic wave packet. Phys. Rev. Lett. 64, 2007–2010 (1990)

    Article  ADS  Google Scholar 

  81. J.A. Yeazell, C.R. Stroud Jr., Observation of fractional revivals in the evolution of a Rydberg atomic wave packet. Phys. Rev. A 43, 5153–5156 (1991)

    Article  ADS  Google Scholar 

  82. W.-M. Zhang, D.H. Feng, R. Gilmore, Coherent states: Theory and some applications. Rev. Mod. Phys. 26, 867–927 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  83. I. Zlatev, W.-M. Zhang, D.H. Feng, Possibility that Schrödinger’s conjecture for the hydrogen atom coherent states is not attainable. Phys. Rev. A 50, R1973–R1975 (1994)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada

    Syed Twareque Ali

  2. Institut de Recherche en Mathématique et Physique (IRMP), Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Jean-Pierre Antoine

  3. Astroparticules et Cosmologie (APC, UMR 7164), Université Paris Diderot, Sorbonne Paris Cité Paris, France

    Jean-Pierre Gazeau

  4. Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, Brasil

    Jean-Pierre Gazeau

Authors
  1. Syed Twareque Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Pierre Antoine
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Pierre Gazeau
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Ali, S.T., Antoine, JP., Gazeau, JP. (2014). Introduction. In: Coherent States, Wavelets, and Their Generalizations. Theoretical and Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8535-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-8535-3_1

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-8534-6

  • Online ISBN: 978-1-4614-8535-3

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation