Topics in Representation Theory: Quantum Field Theory

Monday and Wednesday 2:40-3:55 pm

Mathematics 507

First meeting will be Wednesday, January 17.

This will be a course on quantum mechanics and quantum field theory for mathematicians, emphasizing a representation theory point of view on these topics. The course will be aimed towards a goal of explaining the details of a very specific quantum field theory: the Standard Model, which provides our best current mathematical model of fundamental physics.

There will be written course notes, updated here as the semester progresses.

Announcement: starting January 29, we'll be meeting in Math 507 instead of Math 622.

January 17: Classical mechanics, mainly Hamiltonian mechanics (chapter 2 of notes).

January 22: Introduction to quantization (chapter 3 of notes)

January 24: The Heisenberg group and its representations (chapter 4.1 of notes).

January 29: The symplectic group and the oscillator representation (chapter 4.2 of notes)

January 31: Polarizations and quantization

February 5: Pseudo-classical mechanics. Clifford algebras

February 7: The spinor representation

February 12: Free particles and the Dirac operator

February 14: Non-relativistic quantum field theory of free particles I

February 19: Non-relativistic quantum field theory of free particles II: Dynamics of quantum fields

February 21: Propagators and Euclidean quantum field theory

February 26: Path integrals. Gaussian integrals and perturbation theory

February 28: Geometry in four complex dimensions. Conformal symmetry

March 4: Geometry in four real dimensions

March 6: The Poincaré group and its representations

Spring Break

March 18: Relativistic scalar quantum fields

March 20: Spinor fields in four dimensions

March 25: Perturbation theory, diagrammatic methods

March 27: Renormalization

April 1: Connections and curvature, gauge fields.

April 3: Riemannian geometry and general relativity

April 8: No class due to Solar Eclipse.

April 10: Quantization of gauge fields: photons

April 15: Quantization of gauge fields: Yang-Mills theory

April 17: Quantum electrodynamics

April 22: Electroweak theory and the Higgs mechanism

April 24: The matter fields of the Standard Model

April 29: Quantum chromodynamics, lattice gauge theory