Mathematics GR8250
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