Mathematics for quantum mechanics and general relativity [SMATB333]

Learning outcomes

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 tensor calculus. This lecture aims at giving students the intuitition of these theories through the mathematical formulation of their physical foundations. New mathematical tools, such as functional analysis and differential geometry, will be introduced on the way to the mathematical building of physical theories.

Emphasis is put on concepts, tools and their articulation with physical principles like quantum superposition or relativity.

Content

The course comprises two main parts:

- mathematical foundations of non-relativistic quantum mechanics ;

- introduction to classical field theory on space-time

The lecture will introduce mathematical formulation of some famous pillars of modern physics : Dirac' bra-ket formalism, Lorentz invariance, Schrödinger's and Heisenberg's vision of quantum mechanics ; spin and mass ; antimatter ; gauge theories ; symmetries in physics

Table of contents

Table of contents for 2023-2024 :

Part 1 : functional analysis and non-relativistic quantum mechanics

- Postulate of the state vector

Topological vector spaces ; Banach spaces ; separable complex Hilbert spaces ; Lebesgue space L^2

- Wave mechanics and/or matrix mechanics

Complete orthonormal sequences ; isomorphism between separable Hilbert spaces and spaces of square summable sequences ; equivalence of Schrodinger and Heisenberg representation of quantum mechanics

- Dirac's bra-ket formalism

Topological dual of Hilbert spaces, Riesz representation theorem ; distributions ; rigged Hilbert spaces

Part 2: introduction to classical field theory

- Minkowski space-time and symmetries

Relativistic invariance ; proper time and isometries ; external continuous and discrete symetries ; Lie groups ; Poincaré group (translations, rotations, boosts) ; group representation ; mass and spin : elementary particles

- Variational principles for field equations

Action and lagrangian density ; from particles to fields ; Lagrange chain model ; Noether theorem for fields

- Klein-Gordon equation and charged scalar field theory (spin 0)

- Dirac equation : free spinor field theory  (spin 1/2) ; charge conjugation and antimatter  ;

- Variational approach to covariant Maxwell equations (spin 1 ; potential 4-vector and Aharonov-Bohm effect)

- Relativistic gauge invariance and spontaneous U(1)-symmetry breaking (Meisner effect and supraconductivity, Brout-Englert-Higgs boson).