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• W1531487418 abstract "The irreducible unitary representations of the Banach Lie group $U_0(H)$ (which is the norm-closure of the inductive limit $cup_k U(k)$) of unitary operators on a separable Hilbert space $H$, which were found by Kirillov and Ol'shanskii, are reconstructed from quantization theory. Firstly, the coadjoint orbits of this group are realized as Marsden-Weinstein symplectic quotients in the setting of dual pairs. Secondly, these quotients are quantized on the basis of the author's earlier proposal to quantize a more general symplectic reduction procedure by means of Rieffel induction (a technique in the theory of operator algebras). As a warmup, the simplest such orbit, the projective Hilbert space, is first quantized using geometric quantization, and then again with Rieffel induction. Reduction and induction have to be performed with either $U(M)$ or $U(M,N)$. The former case is straightforward, unless the half-form correction to the (geometric) quantization of the unconstrained system is applied. The latter case, in which one induces from holomorphic discrete series representations, is problematic. For finite-dimensional $H=C^k$, the desired result is only obtained if one ignores half-forms, and induces from a representation, `half' of whose highest weight is shifted by $k$ (relative to the naive orbit correspondence). This presumably poses a problem for any theory of quantizing constrained systems." @default.
• W1531487418 created "2016-06-24" @default.
• W1531487418 creator A5060266645 @default.
• W1531487418 date "1994-11-23" @default.
• W1531487418 modified "2023-09-27" @default.
• W1531487418 title "The infinite unitary group, Howe dual pairs, and the quantization of constrained systems" @default.
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• W1531487418 hasPublicationYear "1994" @default.
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