Lectures on geometric quantization. (English) Zbl 1345.53001

Śniatycki, Jędrzej

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 17th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 5–10, 2015. Sofia: Avangard Prima. 95-129 (2016).

Summary: These lectures notes are meant as an introduction to geometric quantization. In Section 1, I begin with presentation of the historical background of quantum mechanics. I continue with discoveries in the theory of representations of Lie groups, which lead to emergence of geometric quantization as a part of pure mathematics. This presentation is very subjective, flavored by my own understanding of the role of geometric quantization in quantum mechanics and representation theory. Section 2 is devoted to a review of geometry of Hamiltonian systems. Geometric quantization is discussed in the next two sections: prequantization in Section 3 and polarization in Section 4. In particular, I discuss geometric quantization with respect to polarizations given by Kähler structure, cotangent bundle projection and completely integrable system. More advanced topics, like metaplectic structure, pairing of polarizations, and commutation of quantization and reduction, are not included.
For the entire collection see [Zbl 1330.53003].

MSC:

53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53D50	Geometric quantization
81S10	Geometry and quantization, symplectic methods

Keywords:

Bohr-Sommerfeld quantization; co-adjoint orbits; geometric quantization; polarization; prequantization; reduction of symmetries; representations of Lie groups; symplectic geometry