态的对称约化II：CPn的一个非对易正态分布
Symmetry Reduction of States II: A non-commutative Positivstellensatz for CPn
摘要：
我们给出了CP^n的一个非对易的Positivstellenum：的实代数集CP^n上的多项式的（交换）*-代数与点态积可以通过相空间约化实现为C^{1+n}上的U（1）-不变多项式，仅限于C^{1 +n}内的实（2n+1）-球面，Schmüdgen的Positivstellenum给出了CP^n上的实值U（1）-不变多项式在球面上严格逐点正的代数描述。类似于这种交换情形，我们考虑C^{1+n}上多项式的非交换 *-代数，即Weyl代数，并给出了CP^n上线丛的全纯截面的希尔伯特空间上的实值U（1）-不变多项式在某些 *-表示中为正的代数描述。特别值得注意的是，非对易的结果不仅适用于严格正的元素，而且适用于所有正的元素。作为应用，确定了CP^n上多项式 *-代数的量化的所有 *-表示，例如通过相空间约化或Berezin-Toeplitz量化获得的。
英文摘要：
We give a non-commutative Positivstellensatz for CP^n: The (commutative) *-algebra of polynomials on the real algebraic set CP^n with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C^{1+n}, restricted to the real (2n+1)-sphere inside C^{1+n}, and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on CP^n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative *-algebra of polynomials on C^{1+n}, the Weyl algebra, and give an algebraic description of the real-valued U(1)-invariant polynomials that are positive in certain *-representations on Hilbert spaces of holomorphic sections of line bundles over CP^n. It is especially noteworthy that the non-commutative result applies not only to strictly positive, but to all positive elements. As an application, all *-representations of the quantization of the polynomial *-algebra on CP^n, obtained e.g. through phase space reduction or Berezin--Toeplitz quantization, are determined.
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