The infinite unitary group, Howe dual pairs, and the quantization of constrained systems
| dc.creator | Landsman, N. P. | |
| dc.date | 1994-11-23 | |
| dc.date.accessioned | 2026-07-07T04:20:48Z | |
| dc.date.available | 2026-07-07T04:20:48Z | |
| dc.description | 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. | |
| dc.description | LaTeX, 43 pages | |
| dc.identifier | https://arxiv.org/abs/hep-th/9411171 | |
| dc.identifier | http://arxiv.org/abs/hep-th/9411171 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/54075 | |
| dc.subject | High Energy Physics - Theory | |
| dc.title | The infinite unitary group, Howe dual pairs, and the quantization of constrained systems | |
| dc.type | text |