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prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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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:
Since symplectic reduction also means restricting a mechanical system to a constraint surface, one may also read this as saying:
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.
symplectic reduction, presymplectic reduction (gauge reduction), Poisson reduction, BV-BRST formalism
Reductions of (pre-)symplectic manifolds:
symplectic geometry | physics |
---|---|
presymplectic manifold | covariant phase space |
$\downarrow$ gauge reduction | $\downarrow$ quotient by gauge symmetry |
symplectic manifold | reduced phase space |
$\downarrow$ symplectic reduction | $\downarrow$ quotient by global symmetry |
symplectic manifold | reduced phase space |
The original statement is due to
The first proof in geometric quantization by push-forward for compact groups acting on compact manifolds appears in
A survey is in
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
and proven for various cases in
based on
Discussion taking into account the metaplectic correction is in
The full statement for this general case was then proven in
Volume 225, Issue 3, 20 October 2010, Pages 1224–1247 (arXiv:0806.3138)
Lecture notes on this:
Further generalizations are discussed in
Peter Hochs, Varghese Mathai, Quantizing tame actions (arXiv:1309.6760)
Jord Boeijink, Klaas Landsman, Walter van Suijlekom?, Quantization commutes with singular reduction: cotangent bundles of compact Lie groups (arXiv:1508.06763)
Similar discussion is in
based on
Discussion specifically for circle group actions is in
Last revised on March 27, 2021 at 05:19:15. See the history of this page for a list of all contributions to it.