Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
76/2008

Higher Asymptotics of Unitarity in ``Quantization Commutes with Reduction''

William Kirwin

Abstract

Let M be a compact Kaehler manifold equipped with a Hamiltonian action of a compact Lie group G. In [Invent. Math. 67 (1982), no. 3, 515--538], Guillemin and Sternberg showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M//G. This map, though, is not in general unitary, even to leading order in h-bar.

In [Comm. Math. Phys. 275 (2007), no. 2, 401--422], Hall and the author showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed h-bar, becomes unitary in the semiclassical limit as h-bar approaches 0. The unitarity of the classical Guillemin--Sternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M//G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as h-bar goes to 0.

Received:
Oct 30, 2008
Published:
Nov 3, 2008
MSC Codes:
53D50, 53D20, 41A60
Keywords:
geometric quantization, symplectic reduction, semiclassical limit

Related publications

Preprint
2008 Repository Open Access
William D. Kirwin

Higher asymptotics of unitarity in 'Quantization commutes with reduction'