Quantum mechanics and General Relativity are two of the main pillars of modern physics. Besides of their wide range of applications in physics, they are build upon specific mathematics: functional analysis and differential geometry. This lecture aims at giving students the necessary mathematical background for their study of both theories.
Emphasis is put on concepts, tools and their articulation with physical principles like quantum superposition or relativity.
The course comprises two parts: first, riemanian differential geometry and Lie groups (including articulations with classical field theory) then functional analysis (including articulation with quantum mechanics).
Table of contents for 2020-2021 :
Part 1: riemanian differential geometry and Lie groups
Manifolds, tangent, cotangent vector and tensor fields, connexion and covariant derivative, torsion and curvature, riemanian and lorentzian metrics, Levi-Civita connexion, application to spacetimes
Introduction to Lie groups and representation theory with application to the important case of the Poincare group of spacetime symmetries ; classification of elementary particles in classical field theory
Part 2: functional analysis for quantum mechanics
Algebraic and topological aspects: Banach and Hilbert spaces, dual of Hilbert spaces and Riesz theorem
Analytical aspects : Lebesgue integral, L^p spaces, theory of distributions
Application: spectral triplet for the mathematisation of the axioms of quantum mechanics, Wigner theorem and classification of quantum states
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Standard | 0 | 4 | ||
| Standard | 3 | 4 |