Syed Twareque Ali

Quantization, Coherent States, and Complex Structures

Quantization, Field Theory, and Representation Theory: On Quantum Mechanics in a Curved Spacetime with Absolute Time (D. Canarutto et al.). Massless Spinning Particles on the Antide Sitter Spacetime (S. De Bievre, S. Mehdi). A Family of Nonlinear Schrodinger Equations: Linearizing Transformations and Resulting Structure (H.D. Doebner et al.). Modular Structures in Geometric Quantization (G.G. Emch). Diffeomorphism Groups and Anyon Fields (G.A. Goldin, D.H. Sharp). On a Full Quanization of the Torus (M.J. Gotay). Differential Forms on the Skyrmion Bundle (C. Gross). Explicitly Covariant Algebraic Representations for Transitional Currents of Spin1/2 Particles (M.I. Krivoruchenko). The Quantum Su(2,2)Harmonic Oscillator (W. Mulak). GeometricStochastic Quantization and Quantum Geometry (E. Prugovecki). Prequantization (D.J. Simms). Classical Yang-Mills and Dirac Fields in the Minkowski Space and in a Bag (J. Sniatycki). Symplectic Induction, Unitary Induction and BRST Theory (Summary) (G.M. Tuynman). Coherent States, Complex and Poisson Structures: Spin Coherent States for the Poincare Group (S.T. Ali, J.P. Gazeau). Coherent States and Global Differential Geometry (S. Berceanu). Natural Transformations of Lagrangians into pforms on the Tangent Bundle (J. Debecki). SL(2,IR)Coherent States and Itegrable Systems in Classical and Quantum Physics (J.P. Gazeau). Symplectic and Lagrangian Realization of Poisson Manifolds (M. Giordano et al.). From the Poincare-Cartan Form to a Gerstehhaber Algebra of Poisson Brackets in Field Theory (I.V. Kanatchikov). Geometric Coherent States, Membranes, and Star Products (M. Karasev). Integral Representation of Eigenfunctions and Coherent States for the Zeeman Effect (M. Karasev, E. Novikova). QDeformations and Quantum Groups, Noncommutative Geometry: Quantum Coherent States and the Method of Orbits (B. Jurco, P.Stovicek). On the Deformation of Commutation Relations (W. Marcinek). The qdeformed Quantum Mechanics in the Coherent States Map Approach (V. Maximov, A. Odzijewicz). Quantization by Quadratic Polynomials in Creation and Annihilation Operators (W. Slowikowski). On Dirac Type Brackets (Yu.M. Vorobjev, R. Flores Espinoza). Quantum Trigonometry and Phasespace Propensity (K. Wodkiewicz, B.G. Englert). Noncommutative Space-Time Impled by Spin (S. Zadrzewski). Miscellaneous Problems of Quantum Dynamics: Spectrum of the Dirac Operator on the SU(2) Manifold as Energy Spectrum for the Polyaniline Macromolecule (H. Makaruk). On Geometric Methods in the Description of Quantum Fluids (R. Owczarek). Galactic Dynamics in the Siegel Halfplane (G. Rosensteel). Graded Contractions of so(4,2) (J. Tolar, P. Travnicek). The Berry Phase and the Geometry of Coset Spaces (E.A. Tolkachev, A.A. Tregubovich). Index.

Discrete Wavelet Transforms

This chapter is devoted to discrete wavelets. We start with the standard version, related to multiresolution analysis, and some of its generalizations. Next we extend the analysis to a group-theoretical approach to discrete wavelet transforms. Starting from wavelets on the finite field \(\mathbb{Z}_{p}\), we introduce pseudo-dilations and a group structure. Then we generalize this approach to wavelets on a discrete abelian group. Finally we discuss algebraic wavelets, by which we mean wavelets based on different numbers, replacing, for instance, the dilation factor 2 by the golden mean τ (we speak then of τ-wavelets) or arbitrary real numbers, which lead to Pisot wavelets.

Quantization and infinite-dimensional systems

Papers from the July 1993 workshop treat topics such as field theory, geometric quantization and symplectic geometry, coherent states methods, holomorphic representation theory, Poisson structures, non-commutative geometry, and supersymmetry. Annotation copyright Book News, Inc. Portland, Or.

Wavelets on Manifolds

In this chapter, we discuss the construction of wavelets related to other groups than similitude groups. The first, and most important, case is that of wavelets on the two-sphere \({\mathbb{S}}^{2}\). We start with the continuous approach, based on the use of stereographic dilations, i.e., dilations obtained by lifting to \({\mathbb{S}}^{2}\) ordinary dilations on a tangent plane by an inverse stereographic projection. Next we describe briefly a number of techniques for obtaining discrete wavelets on \({\mathbb{S}}^{2}\). Then we extend the analysis to wavelets on other manifolds, such as conic sections, a torus, general surfaces of revolution or graphs.

Quantum frames, quantization and dequantization

A continuous frame in a Hilbert space is a concept well adapted for constructing very general classes of coherent states, in particular those associated to group representations which are square integrable only on a homogeneous space. In addition, (quantum) frames provide a method of quantization which generalizes the coherent state approach and fits in neatly with the operational meaning of quantum measurements. We discuss this approach in detail, taking as our working example the case of the Poincare group in 1+1 space-time dimensions. We also compare this approach to the familiar geometric quantization method, which turns out to be less versatile than the new one.

Multidimensional Wavelets and Generalizations

The continuous wavelet transform can be extended to arbitrary dimensions and this is the topic of this chapter. We begin with the general mathematical analysis, with some emphasis on the distinction between isotropic and directional wavelets. Next we particularize to 2-D, the most important case for applications in image analysis, discussing its distinctive properties and some applications. Finally we describe in some detail a number of generalizations, such as multiselective wavelets, ridgelets, curvelets and shearlets.

Coherent States from Square Integrable Representations

This chapter is devoted to a detailed development of the theory of square integrable group representations, including the resulting orthogonality relations. Then we study a particular class of semidirect product groups, namely, groups of the form \(G = {\mathbb{R}}^{n} \rtimes H\), where H is an n-dimensional closed subgroup of GL\((n, \mathbb{R})\). Several concrete examples are presented. Finally we generalize the theory to representations that are only square integrable on a homogeneous space. This allows one to treat CS of the Gilmore-Perelomov type and, in particular, CS of the Galilei group.

The Discretization Problem: Frames, Sampling, and All That

This last chapter is devoted to the discretization problem: can one obtain discrete wavelets by sampling continuous ones? In particular, what happens to frames under that operation? We start with the Weyl–Heisenberg group underlying canonical CS and discuss Gabor frames. Next we describe discrete frames associated with affine semidirect product groups, such as the affine Weyl–Heisenberg group or the affine Poincare group. Finally we turn to groups without dilations, in particular, the Poincare groups in 1+1 and 1+3 dimensions.

Positive Operator-Valued Measures and Frames

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 may wish to skip these four mathematical chapters, the discussion of many of the topics here is sufficiently different from their treatment in standard texts to warrant at least a cursory glance at it.

Covariant Coherent States

This chapter returns to a group-theoretical context, namely, a systematic study of CS associated to group representations. After a general definition of these covariant CS, we describe the well-known Gilmore–Perelomov CS, as well as vector and matrix CS. Some generalizations are mentioned, including continuous semi-frames. We conclude the chapter by a thorough description of two interesting cases. First we treat CS on spheres constructed via heat kernels (such CS are not of the Gilmore–Perelomov type). Next we turn to CS on conformal classical domains, i.e., classical domains associated to the conformal group SO(n, 2).