Advertisement

Coherent States, Wavelets, and Their Generalizations

  • Syed Twareque Ali
  • Jean-Pierre Antoine
  • Jean-Pierre Gazeau

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

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 1-13
  3. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 15-36
  4. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 37-60
  5. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 61-103
  6. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 105-132
  7. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 133-164
  8. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 165-202
  9. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 203-243
  10. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 245-269
  11. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 271-304
  12. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 305-345
  13. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 347-377
  14. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 379-410
  15. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 411-456
  16. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 457-493
  17. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 495-513
  18. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 515-536
  19. Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau
    Pages 537-539
  20. Back Matter
    Pages 541-577

About this book

Introduction

This second edition is fully updated, covering in particular new types of coherent states (the so-called Gazeau-Klauder coherent states, nonlinear coherent states, squeezed states, as used now routinely in quantum optics) and various generalizations of wavelets (wavelets on manifolds, curvelets, shearlets, etc.). In addition, it contains a new chapter on coherent state quantization and the related probabilistic aspects. As a survey of the theory of coherent states, wavelets, and some of their generalizations, it emphasizes mathematical principles, subsuming the theories of both wavelets and coherent states into a single analytic structure. The approach allows the user to take a classical-like view of quantum states in physics.

 

Starting from the standard theory of coherent states over Lie groups, the authors generalize the formalism by associating coherent states to group representations that are square integrable over a homogeneous space; a further step allows one to dispense with the group context altogether. In this context, wavelets can be generated from coherent states of the affine group of the real line, and higher-dimensional wavelets arise from coherent states of other groups. The unified background makes transparent an entire range of properties of wavelets and coherent states. Many concrete examples, such as coherent states from semisimple Lie groups, Gazeau-Klauder coherent states, coherent states for the relativity groups, and several kinds of wavelets, are discussed in detail. The book concludes with a palette of potential applications, from the quantum physically oriented, like the quantum-classical transition or the construction of adequate states in quantum information, to the most innovative techniques to be used in data processing.

 

Intended as an introduction to current research for graduate students and others entering the field, the mathematical discussion is self-contained. With its extensive references to the research literature, the first edition of the book is already a proven compendium for physicists and mathematicians active in the field, and with full coverage of the latest theory and results the revised second edition is even more valuable.

Keywords

Canonical coherent states Coherent states Lie groups Probabilistic aspects Quantum measurement problem Quantum optics Relativity groups Reproducing kernels Square integrable nonlinear coherent states

Authors and affiliations

  • Syed Twareque Ali
    • 1
  • Jean-Pierre Antoine
    • 2
  • Jean-Pierre Gazeau
    • 3
  1. 1.Dept. of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Institut de Recherche en Mathématique et PhysiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Astroparticules et CosmologieUniversité Paris Diderot, Sorbonne Paris CitéParisFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-8535-3
  • Copyright Information Springer Science+Business Media New York 2014
  • Publisher Name Springer, New York, NY
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-1-4614-8534-6
  • Online ISBN 978-1-4614-8535-3
  • Series Print ISSN 1864-5879
  • Series Online ISSN 1864-5887
  • Buy this book on publisher's site