Lectures on Geometric Quantization

Abstract

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.

Adiabatic deformations of quantum Hall droplets

“…However, we will steer clear of this terminology to avoid confusion with the actual Hamiltonian of a quantum system. The symplectic perspective is nevertheless useful thanks to its connection with geometric quantization [44,45,70,71], to which we now turn in order to define unitary deformations of wave functions.…”

“…3.3 and provides a self-contained review of the action of (certain) area-preserving diffeomorphisms on closed, oriented surfaces such as spheres or tori. In the context of geometric quantization (see [44,45,70,71]), the action that we consider coincides with automorphisms of prequantum line bundles, also known as prequantum operators or quantomorphisms [89]. A summary of the main points is as follows:…”

Shifting Operators in Geometric Quantization

“…In our earlier papers [9][10][11][12], we followed an algebraic analysis, similar to that used by Dirac [8], supplemented by heuristic guesses about the behaviour of the shifting operators at the points of singularity of the polarization. In particular, we assumed that a X ϑ vanishes on the states concentrated on a set of limit points of e t X ϑ (p) as t → ∞.…”

Shifting operators in geometric quantization

“…In a series of papers on Bohr-Sommerfeld-Heisenberg quantization of completely integrable systems [4], [5], [6], [11], we interpreted shifting operators as quantization of functions e ±iϑ j , where (I j , ϑ j ) are action angle coordinates. The aim of this paper is to show how these operators occur in prequantization, which is the first step of geometric quantization.…”