Representations of the Infinite Unitary Group from Constrained Quantization

DOI

10.2991/jnmp.1999.6.2.4How to use a DOI?

Abstract

We attempt to reconstruct the irreducible unitary representations of the Banach Lie group U0(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.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Download article (PDF)

TY  - JOUR
AU  - N.P. Landsman
PY  - 1999
DA  - 1999/05/01
TI  - Representations of the Infinite Unitary Group from Constrained Quantization
JO  - Journal of Nonlinear Mathematical Physics
SP  - 161
EP  - 180
VL  - 6
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.1999.6.2.4
DO  - 10.2991/jnmp.1999.6.2.4
ID  - Landsman1999
ER  -