Mathematical Physics
// 2021
Symplectic Geometry
Mathematical Framework of Classical Mechanics & Geometric Quantization
Abstract
Advanced reading project on the mathematical foundations of Classical Mechanics. Incorporates Lie Algebra, Symplectic Manifolds, and Complex vector spaces exploring the analogy between the Wave-Corpuscular theory of light and Geometric Quantization.
Table of Contents / Outline
Mathematical Focus
Exploration of phase space as a smooth manifold endowed with a closed, non-degenerate 2-form (the symplectic form).
Covered Topics
- Symplectic Manifolds & Hamiltonian Vector Fields: Conservation of energy via Darboux’s theorem and Poisson brackets.
- Lie Groups & Moment Maps: Noether’s theorem formulated through momentum maps and symplectic reduction.
- Geometric Quantization: Bridging classical phase space to quantum Hilbert spaces via prequantization line bundles and polarizations.