书目名称 | Coherent States, Wavelets and Their Generalizations | 编辑 | Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre | 视频video | http://file.papertrans.cn/230/229225/229225.mp4 | 丛书名称 | Graduate Texts in Contemporary Physics | 图书封面 |  | 描述 | Nitya kaaler utshab taba Bishyer-i-dipaalika Aami shudhu tar-i-mateer pradeep Jaalao tahaar shikhaa 1 - Tagore Should authors feel compelled to justify the writing of yet another book? In an overpopulated world, should parents feel compelled to justify bringing forth yet another child? Perhaps not! But an act of creation is also an act of love, and a love story can always be happily shared. In writing this book, it has been our feeling that, in all of the wealth of material on coherent states and wavelets, there exists a lack of a discern able, unifying mathematical perspective. The use of wavelets in research and technology has witnessed explosive growth in recent years, while the use of coherent states in numerous areas of theoretical and experimental physics has been an established trend for decades. Yet it is not at all un common to find practitioners in either one of the two disciplines who are hardly aware of one discipline‘s links to the other. Currently, many books are on the market that treat the subject of wavelets from a wide range of perspectives and with windows on one or several areas of a large spectrum IThine is an eternal celebration ‘" A cosmic Festival of Light | 出版日期 | Textbook 20001st edition | 关键词 | Coherent states; Lie groups; group theory; harmonic analysis; quantum measurement problem; quantum-classi | 版次 | 1 | doi | https://doi.org/10.1007/978-1-4612-1258-4 | isbn_ebook | 978-1-4612-1258-4Series ISSN 0938-037X | issn_series | 0938-037X | copyright | Springer Science+Business Media New York 2000 |
1 |
Front Matter |
|
|
Abstract
|
2 |
,Introduction, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
The notion of coherent states (CS.) is rooted in quantum physics and its relationship to classical physics. The term ‘coherent’ itself originates in the current language of quantum optics (for instance, coherent radiation, sources emitting coherently, etc.). It was introduced in the 1960s by Glauber [145], . one of the founding fathers of the theory of CS, together with Klauder [189] and Sudarshan [Kl1, 272], in the context of a quantum optical description of coherent light beams emitted by lasers. Since then, coherent states have pervaded nearly all branches of quantum physics — quantum optics, of course, but also nuclear, atomic, and solid-state physics, quantum electrodynamics (the infrared problem), quantization and dequantization problems and path integrals, just to mention a few. It has been said, even convincingly [191], that “coherent states are the natural language of quantum theory!”
|
3 |
,Canonical Coherent States, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
This chapter is devoted to a fairly detailed examination of the quintessential example of coherent states — the .. It is fair to say that the entire subject of coherent states developed by analogy from this example. As mentioned in Chapter 1, this set of states, or rays in the Hilbert space of a quantum mechanical system, was originally discovered by Schrödinger transition from quantum to classical mechanics. They are endowed with a remarkable array of interesting properties, some of which we shall survey in this chapter. Apart from initiating the discussion, this will also help us in motivating the various mathematical directions in which one can try to generalize the notion of a CS.
|
4 |
,Positive Operator-Valued Measures and Frames, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
This chapter, and the three succeeding it, constitute a mathematical interlude, preparing the ground for the formal definition of a coherent state in Chapter 7 and the subsequent development of the general theory. As should be clear already, from a look at the last chapter, in order to define CS mathematically and obtain a synthetic overview of the different contexts in which they appear, it is necessary to understand a bit about positive operator-valued (POV) measures on Hilbert spaces and their close connection with certain types of group representations. In Chapter 2, we have also encountered examples of reproducing kernels and reproducing kernel Hilbert spaces, which in turn are intimately connected with the notion of POV measures and, hence, coherent states. In this chapter, we gather together the relevant mathematical concepts and results about POV measures. In the next chapter, we will do the same for the theory of groups and group representations. Chapters 5 and 6 will then be devoted to a study of reproducing kernel Hilbert spaces. The treatment is necessarily condensed, but we give ample reference to more exhaustive literature. Although the mathematically initiated reader
|
5 |
,Some Group Theory, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
In this chapter, we introduce a few concepts from the theory of groups, Lie algebras, transformation spaces, and group representations, presenting them in a form and notation adapted to the aims of this book. (A good source for more detailed information is, for example, [Bar].)
|
6 |
,Hilbert Spaces with Reproducing Kernels and Coherent States, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
This chapter is somewhat technical in nature. On the other hand, the treatment of reproducing kernel Hilbert spaces given below is rather different from that normally found in the literature, and the level of generality adopted assures immediate applicability of the concept to the various geometric and functional analytic contexts in which they are later required. Unfortunately, the level of technicality apparent in our treatment of reproducing kernels is rather high, but it was deemed necessary for the development of the theory. We have tried to lighten the reading by including illustrative examples.
|
7 |
,Square Integrable and Holomorphic Kernels, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
In this chapter, we study two special types of reproducing kernel Hilbert spaces, which are probably the most widely occurring types in the physical literature. While the very general reproducing kernel Hilbert spaces, constructed in the last chapter, were spaces of vector-valued functions, they were not assumed to be Hilbert spaces of square integrable functions, with respect to any measure. Most reproducing kernel Hilbert spaces that arise in physics and in group representation theory do, on the other hand, turn out to be spaces of square integrable fuctions. Another widely occurring variety of reproducing kernel Hilbert spaces are spaces of holomorphic or square integrable holomorphic functions. We look at these two situations more closely in this chapter. Recall from the discussion in Chapter 2 that the family of canonical CS arise from a reproducing kernel Hilbert space of square integrable functions, and, indeed, they may also be associated to a space of analytic functions (the Bargmann space).
|
8 |
,Covariant Coherent States, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
By now, we have encountered several examples of coherent states — the canonical CS, associated to the Weyl-Heisenberg group, discussed in detail in Chapter 2; vector CS, introduced in Chapter 4, Section 4.2.1; CS associated to the discrete series representations of .(1,1), in Section 4.2.2 of the same chapter; and general coherent states, associated with any reproducing kernel Hilbert space, in Chapter 5, Section 5.3.2. At this point, certain common features are already seen to emerge, as follows.
|
9 |
,Coherent States from Square Integrable Representations, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
This chapter is devoted to a fairly detailed development of the theory of square integrable group representations. We have already seen examples of such representations. Indeed, the discrete series representations, . of .(1, 1), discussed in Chapter 4, Section 4.2.2, are square integrable.
|
10 |
,Some Examples and Generalizations, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
We have seen explicit examples of square integrable representations in the last chapter, related to the .(1, 1) and the connected affine groups. Here, we work out a few more examples to both illustrate the general theory of square integrable representations better and get a deeper understanding of the nature of the Duflo—Moore operator ., appearing in the orthogonality relations. We then move on to deriving a generalization of the notion of square integrability to accomodate CS of the Gilmore—Perelomov type and vector CS.
|
11 |
,CS of General Semidirect Product Groups, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
In Chapter 9, Section 9.1, we studied a class of semidirect product groups, the regular representations of which consisted entirely of a discrete sum of irreducible subrepresentations, all square integrable. These groups were of the general form . ℝ. ⋊ ., where . was an .-dimensional subgroup of .(.ℝ) and its action on the dual space gave rise to open free orbits. In this chapter, we generalize this setting and consider semidirect products of the type . ⋊ ., where . is an .-dimensional real vector space (. is assumed to be finite) and . is usually a subgroup of .(.) (the group of all nonsingular linear transformations of .). We shall again examine the action of . on the dual vector space .*, but now without the assumption that the orbits be open or free, and generally, the dimensions of these orbits could be lower than n, the dimension of the vector space . on which . acts. The analysis, however, will have features similar to those encountered in Section 9.1, although we shall no longer look for the sort of simple square integrability with respect to the entire group as was done there. An interesting interplay between the geometry of the orbits and the existence of CS, square integ
|
12 |
,CS of the Relativity Groups, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
In this chapter, we examine a few of the various relativity groups, that are of great importance in physics. The discussion will also illustrate the use of the results on semidirect products obtained in the last chapter.
|
13 |
,Wavelets, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
Wavelet analysis is a particular time-scale or space-scale representation of signals that has become popular in physics, mathematics, and engineering in the last few years. The genesis of the method is interesting for the present book, so we will spend a paragraph outlining it. After the empirical discovery by Jean Morlet (who was analyzing microseismic data in the context of oil exploration [152]), it was recognized from the very beginning by Grossmann, Morlet, and Paul [156]–[160] that wavelets are simply coherent states associated to the affine group of the line (dilations and translations). Thus, immediately the stage was set for a far-reaching generalization, using the formalism developed in Chapter 8 (it is revealing to note that two out of those three authors are mathematical physicists). But then the wind changed. Meyer [217] and Mallat [212] made the crucial discovery that orthonormal bases of regular wavelets could be built, and even with compact support, as shown by Daubechies [99], by changing the perspective (of course, the orthonormal basis of the Haar wavelets was known since the beginning of the century, but these are piecewise constant, discontinuous functions). Gr
|
14 |
,Discrete Wavelet Transforms, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
As we have seen in Section 12.5, the discretization of the CWT leads, among other things, to the theory of frames. For many practical purposes of signal processing, a tight frame is almost as good as an orthonormal basis. Actually, if one stays with the standard wavelets, as we have done so far, one cannot do better, since these wavelets do not generate any orthonormal basis (like the usual coherent states). There are cases, however, in which an orthonormal basis is really required. A typical example is data compression, which is performed (in the simplest case) by removing all wavelet expansion coefficients below a fixed threshhold. In order to not introduce any bias in this operation, the coefficients have to be as decorrelated as possible, and, of course, an orthonormal basis is ideal in this respect.
|
15 |
,Multidimensional Wavelets, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
Exactly as in one dimension, multidimensional wavelets may be derived from the similitude group of ℝ. (. > 1), consisting of dilations, rotations, and translations. Of course, the most interesting case for applications is . = 2, where wavelets have become a standard tool in image processing, including radar imaging [Me2, Me4]. Also, . = 3 may have a practical importance, since some important physical phenomena are intrinsically multiscale and 3-D. Typical examples may be found in fluid dynamics, for instance, the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in acoustics. In such cases, a 3-D wavelet analysis is likely to yield a deeper understanding [57]. The same comment is valid for the applications of the WT in quantum physics (quantum mechanics, atomic physics, solid-state physics, etc.). A good reference for the latter is the survey volume [Be2].
|
16 |
,Wavelets Related to Other Groups, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
Several applications exist in which data to be analyzed are defined on a sphere, in geophysics or astronomy, of course, but also in statistics and other instances (then spheres of dimension higher than two might occur). If one is interested only in very local features, one may ignore the curvature and work on the tangent plane, but when global aspects become important (description of plate tectonics on the Earth, for instance), one needs a genuine generalization of wavelet analysis to the sphere. Several authors have studied this problem, with various techniques, mostly discrete (see, for instance [269] for an efficient solution, based on second-generation wavelets). To preserve the rotational invariance of the sphere, however, a continuous approach is clearly necessary. A solution has been proposed in [176], with several ad hoc assumptions. It turns out that the general formalism developed in this book yields an elegant solution to the problem [39] and, in particular, allows one to derive all assumptions of [176].
|
17 |
,The Discretization Problem: Frames, Sampling, and All That, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
In the preceding chapters, we have encountered both continuous and discrete frames the latter are more traditional, but in fact the two are necessarily linked. Suppose one is dealing with a continuous frame, of rank one for simplicity. When it comes to numerical calculation, the integral has to be discretized, so that in effect one always restricts oneself to a . subset of ..
|
18 |
,Conclusion and Outlook, |
Syed Twareque Ali,Jean-Pierre Antoine,Jean-Pierre Gazeau |
|
Abstract
As must have become clear from the last few chapters of this book, research on wavelets — both theoritical and applied — has currently been gathering increasing momentum, perhaps more so because of its numerous applications in today’s cutting-edge technology. The use of coherent states, as an applied and theoritical tool in physical and mathematical research, is perhaps equally pervasive.
|
19 |
Back Matter |
|
|
Abstract
|
书目名称Coherent States, Wavelets and Their Generalizations影响因子(影响力) 
书目名称Coherent States, Wavelets and Their Generalizations影响因子(影响力)学科排名 
书目名称Coherent States, Wavelets and Their Generalizations网络公开度 
书目名称Coherent States, Wavelets and Their Generalizations网络公开度学科排名 
书目名称Coherent States, Wavelets and Their Generalizations被引频次 
书目名称Coherent States, Wavelets and Their Generalizations被引频次学科排名 
书目名称Coherent States, Wavelets and Their Generalizations年度引用 
书目名称Coherent States, Wavelets and Their Generalizations年度引用学科排名 
书目名称Coherent States, Wavelets and Their Generalizations读者反馈 
书目名称Coherent States, Wavelets and Their Generalizations读者反馈学科排名 
|
|
|