Quantization, vector coherent states and wavelets on non-euclidean surfaces

在过去的六年里，我与比利时、法国、德国、意大利和墨西哥的几名博士生以及同事合作进行了三个方向的研究：(1)建立了理解小波变换和Wigner函数的统一群体理论框架，(2)在矩阵域上构造向量相干态，研究随之而来的Berezin量化问题，(3)在非欧几里德曲面上构造类小波变换和时频变换。这项工作对平方可积群表示和矩阵模型的理论以及非平坦几何上的信号分析的应用问题具有数学意义。在接下来的五年里，我打算继续朝着这三个方向努力。关于Wigner函数的工作将继续在一般对称空间上进行，作为早期结果的推广。这应该打开了为许多有趣的信号几何定义信号变换的可能性。从数学上讲，这将导致对相空间上定义的维格纳函数的更深层次的理解。在矩阵域上构造矢量相干态是我们提出的一个新概念，它推广了矢量相干态的标准理论。许多半单李群的齐次空间自然是矩阵域。我们在这里介绍的矢量相干态导致了关于矩阵变量的解析函数空间。目前，我正在研究Berezin型量子化，使用这种相干态及其相关的再生核。一个相关的问题是研究这种量子化理论的经典极限，特别是因为它们似乎触及了非对易几何的问题。我们打算深入探讨的另一个有趣的问题是为超对称哈密顿量构造矢量相干态的可能性。关于小波和时频变换的工作将建立在不同小组最近获得的一些结果的基础上。我们提出了一种群论和几何模型来发展这种变换。这项工作已经在研究界引起了广泛关注。

My research over the last six years, carried out in collaboration with several doctoral students and with co-workers in Belgium, France, Germany, Italy and Mexico has moved in three directions: (1) developing a unified group theoretical framework for understanding wavelet transforms and Wigner functions, (2) constructing vector coherent states over matrix domains, and studying the ensuing problem of Berezin quantization and (3) construction of wavelet-like and time-frequency transforms on non-Euclidean surfaces. The work is of mathematical interest in the theory of square-integrable group representations and matrix models, as well as to applied problems of signal analysis over non-flat geometries. In the next five years, it is my intention to continue working in all three directions. The work on Wigner functions will be continued over general symmetric spaces, as an extension of an earlier result. This, should open up the possibility of defining signal transforms for a number of interesting signal geometries. Mathematically, it would lead to a deeper understanding of the Wigner fuction, as defined on a phase space. The construction of vector coherent states over matrix domains was a new idea introduced by us, which extends the standard theory of vector coherent states. The homogeneous spaces of many semi-simple Lie groups are naturally matrix domains. The vector coherent states that we have introduced there lead to spaces of analytic functions, in terms of matrix variables. Currently, I am studying Berezin-type quantization, using such coherent states and their associated reproducing kernels. A related problem is to study the classical limits of such quantized theories, particularly since they seem to touch on questions of non-commutative geometry. Another interesting problem that we intend to explore in depth is the possibility of constructing vector coherent states for supersymmetric Hamiltonians. The work on wavelets and time-frequency transforms will build on some recent results obtained by different groups. We are proposing a group theoretical and a geometrical model for developing such transforms. The work has already attracted wide attention within the research community.