arXiv math-ph/99040401999-04-01math-phmath.MP

无限酉群的约束量子化表示

Representations of the infinite unitary group from constrained quantization

Nicolaas P. Landsman

发表信息：: J. Nonlinear Math. Phys. 6 (1999), no. 2, 161-180
DOI：: 10.2991/jnmp.1999.6.2.4

摘要：

本文试图通过对可分Hilbert空间$\H$上的所有酉算子$U$的Banach李群$U_0(\H)$的余伴随轨道的约束量子化，重建出其不可约酉表示，其中对于可分Hilbert空间$\H$，$U-{\mathbb I}$是紧的，这是Kirillov和Ol'shanskii最初发现的.为此目的，余伴随轨道被实现为Marsden-Weinstein共轭。无约束的系统，作为一个Weinstein对偶对，是量化的相应的豪对偶对。然后进行约束量化，用Rieffel归纳法的$C^*$ -代数方法代替经典的辛约化过程。必须对$U(M)$（简单）或$U(M,N)$进行还原和诱导。在后一种情况下，我们从全纯离散级数表示中导出，如果忽略半形式，并从\rep中导出，则可以得到所需的结果，其中\rep的“一半”的最高权相对于朴素轨道对应移动。这仅在$\H$是有限维时才可能。

英文摘要：

We attempt to reconstruct the irreducible unitary representations of the Banach Lie group $U_0(\H)$ of all unitary operators $U$ on a separable Hilbert space $\H$ for which $U-{\mathbb I}$ is compact, originally found by Kirillov and Ol'shanskii, through constrained quantization of its coadjoint orbits. For this purpose the coadjoint orbits are realized as Marsden-Weinstein quotients. The unconstrained system, given as a Weinstein dual pair, is quantized by a corresponding Howe dual pair. Constrained quantization is then performed in replacing the classical procedure of symplectic reduction by the $C^*$-algebraic method of Rieffel induction. Reduction and induction have to be performed with respect to either $U(M)$, which is straightforward, or $U(M,N)$. In the latter case one induces from holomorphic discrete series representations, and the desired result is obtained if one ignores half-forms, and induces from a \rep, `half' of whose highest weight is shifted relative to the naive orbit correspondence. This is only possible when $\H$ is finite-dimensional.

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