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Guillemin-Sternberg geometric quantization conjecture

Context

Geometric quantization

: Lagrangians and Action functionals + Geometric Quantization

Prerequisites

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Prequantum field theory

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    • prequantum 1-bundle = , regular, = lift of to

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Geometric quantization

Applications

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Physics

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Surveys, textbooks and lecture notes

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    • Axiomatizations

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        • -theorem

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    • Structural phenomena

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Symplectic geometry

Background

Basic concepts

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Classical mechanics and quantization

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Contents

Idea

The Guillemin-Sternberg conjecture states that, given a symplectic manifold equipped with a Hamiltonian action by some group, under suitable conditions the processes of geometric quantization and of symplectic reduction can be interchanged without changing the result (a representation of the group on the space of quantum states). This is usually abbreviated to:

  • quantization commutes with reduction.

Since symplectic reduction also means restricting a mechanical system to a constraint surface, one may also read this as saying:

  • quantization commutes with imposing constraints.

The original article (Guillemin-Sternberg 82) proved this statement for the case of compact Lie groups acting on compact Kähler manifolds and for geometric quantization via Kähler polarizations.

For the more powerful and more flexible geometric quantization by push-forward, the corresponding statement is widely known as the Guillemin-Sternberg conjecture. Since its formulation, this, too, has been proven in fair generality, see the References below.

Reductions of (-):

\downarrow \downarrow quotient by
\downarrow \downarrow quotient by

References

The original statement is due to

The first proof in geometric quantization by push-forward for compact groups acting on compact manifolds appears in

  • Eckhard Meinrenken, Symplectic surgery and the Spin cSpin^c-Dirac operator, Adv. Math. 134 (1998), 240-277.

A survey is in

  • Michèle Vergne, Quantification géométrique et r´éduction symplectique, Séminaire Bourbaki, Vol. 2000/2001. Astérisque No. 282 (2002), Exp. No. 888, viii, 249–278.

Discussion for Kähler polarizations of mildly singular phase spaces is in

The generalization to non-compact groups acting on non-compact spaces (but with compact quotients, hence “co-compactly”) by passing to KK-theory and defining the push-forward by the Baum-Connes analytic assembly map was proposed in

  • Klaas LandsmanFunctorial quantization and the Guillemin-Sternberg conjecture. In S. Ali, G. Emch, A. Odzijewicz, M. Schlichenmaier, and S. Woronowicz (eds.), Twenty years of Bialowieza: a mathematical anthology, pages 23–45, Singapore, 2005. World Scientific. (arXiv:math-ph/0307059)

and proven for various cases in

based on

  • Peter Hochs, Quantisation commutes with reduction for cocompact Hamiltonian group actions (pdf)

Discussion taking into account the metaplectic correction is in

  • Brian Hall, William Kirwin, Unitarity in “quantization commutes with reduction”, Comm. Math. Phys. 275 (2007), no. 3, pages 401 – 442 (arXiv:math/0610005)

The full statement for this general case was then proven in

Lecture notes on this are in

Further generalizations are discussed in

Similar discussion is in

based on

Discussion specifically for circle group actions is in

Last revised on August 29, 2015 at 03:26:19. See the history of this page for a list of all contributions to it.