Coherent States, Wavelets, and Their Generalizations by Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau

By Syed Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau

This moment version is absolutely up to date, overlaying particularly new different types of coherent states (the so-called Gazeau-Klauder coherent states, nonlinear coherent states, squeezed states, as used now in many instances in quantum optics) and numerous generalizations of wavelets (wavelets on manifolds, curvelets, shearlets, etc.). furthermore, it includes a new bankruptcy on coherent country quantization and the comparable probabilistic elements. As a survey of the idea of coherent states, wavelets, and a few in their generalizations, it emphasizes mathematical ideas, subsuming the theories of either wavelets and coherent states right into a unmarried analytic constitution. The technique permits the consumer to take a classical-like view of quantum states in physics.

Starting from the normal concept of coherent states over Lie teams, the authors generalize the formalism by means of associating coherent states to team representations which are sq. integrable over a homogeneous area; one other step permits one to dispense with the gang context altogether. during this context, wavelets might be generated from coherent states of the affine team of the true line, and higher-dimensional wavelets come up from coherent states of different teams. The unified heritage makes obvious a whole diversity of houses of wavelets and coherent states. Many concrete examples, reminiscent of coherent states from semisimple Lie teams, Gazeau-Klauder coherent states, coherent states for the relativity teams, and a number of other types of wavelets, are mentioned intimately. The booklet concludes with a palette of strength purposes, from the quantum bodily orientated, just like the quantum-classical transition or the development of sufficient states in quantum details, to the main leading edge recommendations for use in facts processing.

Intended as an creation to present examine for graduate scholars and others getting into the sphere, the mathematical dialogue is self-contained. With its broad references to the study literature, the 1st version of the booklet is already a confirmed compendium for physicists and mathematicians energetic within the box, and with complete insurance of the newest idea and effects the revised moment version is much more valuable.

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Additional info for Coherent States, Wavelets, and Their Generalizations

Example text

A more detailed treatment may be found in, for example, [Ber66]. Note that similar considerations on the relationship between POV measures1 and frames have appeared in a recent paper by Moran et al. [472]. Let X be a metrizable, locally compact space. (All the group spaces and parameter spaces for defining CS will be of this type. Metrizability is a technical assumption, which entails that Baire subsets and Borel subsets of X coincide). The Borel sets B(X) of X consist of elements of the σ -algebra formed by its open sets.

10. The problem is that these groups have no square integrable representations in the usual sense; hence the preceding theory is not applicable as it stands. There exist, however, families of vectors in certain representation spaces of these groups, which strongly resemble coherent states, and appear in an essential way in localization and quantization problems, but are not obtainable by the Gilmore–Perelomov method [22–25, 511, 512]. A possible way to incorporate all of these families of states within a unified framework of generalized coherent states was suggested in [13] and later developed in a series of papers [15, 17, 26–30].

This book is about mathematical physics; hence, its audience consists of both mathematicians and physicists. Of course, it is impossible to please everyone at the same time. So, we have on some occasions dwelled on mathematics longer than expected, following the internal dynamics of the topic at hand. Conversely, we have, at times, indulged in a thorough discussion of certain physical examples or applications. We apologize to both communities for these various wanderings, our sole excuse being our desire to illustrate the general theory and make the text livelier.

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