arXiv 2105.084002021-05-18quant-phmath.OA

整函数的Hilbert空间与欧氏平面的Toeplitz量子化

Hilbert Spaces of Entire Functions and Toeplitz Quantization of Euclidean Planes

Micho Durdevich, Stephen Bruce Sontz

备注：: 46 Classical Pages

摘要：

我们在前一篇论文中提出的Toeplitz量子化理论被扩展和进一步发展，以包括经典欧氏平面的各种有趣的非对易实现。这是使用整函数的希尔伯特空间来完成的，其中一个复变量中的多项式形成稠密子空间。复坐标自然地充当无界乘法算子，与其伴随算子一起生成高度非交换的算子 *-代数。然后，Toeplitz算子被几何地构造为来自该代数的特殊元素;它们与来自另一个二次非交换代数的符号相关联，该二次非交换代数可解释为要量化的平面上的多项式。这样一个概念框架促进了有趣的非平凡条件的初始标量积。这些都进行了详细的分析。计算了各种说明性示例。

英文摘要：

The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly non-commutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associated to the symbols from another quadratic non-commutative algebra, which is interpretable as polynomials over a plane to be quantized. Such a conceptual framework promotes interesting non-trivial conditions on the initial scalar product. These are analyzed in detail. Various illustrative examples are computed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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