Topics in Representation Theory: Quantum Field Theory

Monday and Wednesday 2:40-3:55 pm

Mathematics 622

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.

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: More relativistic scalar quantum fields, interactions

March 25: Spinor fields in four dimensions

March 27: Weyl spinor fields in four dimensions (Minkowski and Euclidean)

April 1: Principal G bundles: connections and curvature

April 3: Frame bundles and Riemannian geometry. Associated vector bundles

April 8: No class due to Solar Eclipse.

April 10: Quantization of gauge fields: photons

April 15: Quantization of gauge fields: covariant gauges, BRST, Yang-Mills

April 17: Helicity and duality for gauge fields

April 22: The Anderson-Higgs mechanism

April 24: The electroweak theory and QCD

April 29: The matter fields of the Standard Model and their masses