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A Mathematical Introduction to Geometric Quantization

Kadri İlker Berktav, Burak Oğuz, Ömer Önder, Yunus Emre Sargut, Başar Deniz Sevinç, Deniz Nazif Taştan

TL;DR

This collection of notes introduces geometric quantization by building from symplectic geometry foundations (symplectic vector spaces, symplectic manifolds, canonical structures on $T^*M$) to Hamiltonian dynamics and Poisson algebra. It then develops the necessary Lie-theoretic machinery (Lie groups/algebras, adjoint/coadjoint actions, moment maps) and culminates with symplectic reduction and the Noether principle, before introducing principal bundles, associated bundles, and connections as the geometric backbone for quantization. The notes emphasize standard results and provide pathways to the Witten-inspired knot theory/topology application, while clarifying how classical phase-space structures inform quantum constructions. Overall, the work serves as an accessible, reference-style guide to the core geometric toolkit of geometric quantization, rather than presenting new research results.

Abstract

These notes are based on a series of lectures by Kadri İlker Berktav from May 2024 to November 2024, providing a detailed exposition of geometric quantization formalism and its essential components. They are organized into three parts: background in symplectic geometry, basics of geometric quantization formalism, and an application related to Edward Witten's work in knot theory and topology.

A Mathematical Introduction to Geometric Quantization

TL;DR

This collection of notes introduces geometric quantization by building from symplectic geometry foundations (symplectic vector spaces, symplectic manifolds, canonical structures on ) to Hamiltonian dynamics and Poisson algebra. It then develops the necessary Lie-theoretic machinery (Lie groups/algebras, adjoint/coadjoint actions, moment maps) and culminates with symplectic reduction and the Noether principle, before introducing principal bundles, associated bundles, and connections as the geometric backbone for quantization. The notes emphasize standard results and provide pathways to the Witten-inspired knot theory/topology application, while clarifying how classical phase-space structures inform quantum constructions. Overall, the work serves as an accessible, reference-style guide to the core geometric toolkit of geometric quantization, rather than presenting new research results.

Abstract

These notes are based on a series of lectures by Kadri İlker Berktav from May 2024 to November 2024, providing a detailed exposition of geometric quantization formalism and its essential components. They are organized into three parts: background in symplectic geometry, basics of geometric quantization formalism, and an application related to Edward Witten's work in knot theory and topology.

Paper Structure

This paper contains 49 sections, 28 theorems, 394 equations, 1 figure, 1 table.

Table of Contents

  1. Lectures on Symplectic Geometry
  2. References and our road map.
  3. Symplectic Vector Spaces
  4. Symplectic Manifolds
  5. Symplectic Form on the Cotangent Bundle
  6. Hamiltonian Functions and Vector Fields
  7. Some Lie Theory
  8. Lie Groups
  9. The matrix Lie groups.
  10. Lie Algebras
  11. The matrix Lie algebras.
  12. Poisson Algebra
  13. Group Actions on Manifolds
  14. Lie Group Actions
  15. Smooth $\mathbb R$-action on a smooth manifold:
  16. ...and 34 more sections

Key Result

theorem 1

Given a skew-symmetric bilinear map $\Omega$ on $V$, with $\dim V < \infty$, there exists a basis $\{u_1, \ldots, u_k, e_1, \dots, e_n, f_1, \dots, f_n\}$ of $V$ such that

Figures (1)

  • Figure :

Theorems & Definitions (171)

  • definition 1
  • theorem 1
  • proof 1
  • claim 1
  • proof 2
  • claim 2
  • proof 3
  • definition 2
  • definition 3
  • example 1
  • ...and 161 more