论文标题
严格量化多项式泊松结构
Strict quantization of polynomial Poisson structures
论文作者
论文摘要
我们展示了如何使用组合恒星产品来获得$ \ Mathbb r^d $对多项式泊松结构进行严格的变形量化，从而将已知结果概括为恒定和线性泊松结构的已知结果，以使其成为任意程度的多项式泊松结构。我们提供了几个非线性泊松结构的例子，并构建了明确的正式星星产品，它们的变形参数可以以$ \ hbar $的任何实际值进行评估，从而严格量化了$ \ mathbb r^d $的分析功能的空间，并具有无限的融合半径。我们还解决了进一步的问题，例如经典极限$ \ hbar \ to 0 $的连续性，与 *involutions的兼容性以及正线性函数的存在。后者可用于实现严格的量化为算子在希尔伯特空间上的 * - 代数，我们在具体的示例中证明了这一点。
We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of $\hbar$, giving strict quantizations on the space of analytic functions on $\mathbb R^d$ with infinite radius of convergence. We also address further questions such as continuity of the classical limit $\hbar \to 0$, compatibility with *-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as *-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.
