A unitary ‘quantization commutes with reduction’ map for the adjoint action of a compact Lie group

DOI10.1093/QMATH/HAY029zbMath1418.81047arXiv1709.08531OpenAlexW2963736367WikidataQ115270440 ScholiaQ115270440MaRDI QIDQ4968792

Brian C. Hall, Benjamin D. Lewis

Publication date: 9 July 2019

Published in: The Quarterly Journal of Mathematics (Search for Journal in Brave)

Abstract: Let $K$ be a simply connected compact Lie group and $T^{ast}(K)$ its cotangent bundle. We consider the problem of "quantization commutes with reduction" for the adjoint action of $K$ on $T^{ast}(K).$ We quantize both $T^{ast}(K)$ and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of $T^{ast}(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.

Full work available at URL: https://arxiv.org/abs/1709.08531

Mathematics Subject Classification ID

Applications of Lie (super)algebras to physics, etc. (17B81) Applications of Lie groups to the sciences; explicit representations (22E70) Geometry and quantization, symplectic methods (81S10) Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics (81S30) Geometric quantization (53D50)