1999/01/01 by N.P. Landsman, Nicolaas P. Landsman
Mathematics · Physics and Astronomy · #(g,K)-module #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Canonical quantization #Geometric quantization #Hilbert space #Holomorphic function #Lie group #Mathematical analysis #Mathematics #Pure mathematics #Quantization (signal processing) #Quantum #Separable space #Symplectic geometry #Unitary group #Unitary state #math-ph #math.MP
paper · pdf · doi:10.2991/jnmp.1999.6.2.4
published as J. Nonlinear Math. Phys. 6 (1999), no. 2, 161-180
openalex publication_date 1999/01/01 · arxiv created 1999/04/01 · arxiv updated 2015/06/26 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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 -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 representation, 'half' of whose highest weight is shifted relative to the naive orbit correspondence. This is only possible when H is finite-dimensional.