Lectures on QFT
I have been asked to contribute to a series of seminars and “informal discussions” about Quantum Field Theory (QFT). The group lead by Jingbo Wang from the Physics department of UWA is indeed interested in the connections between quantum random walks and the discretisation of the path integral formulation of QFT found in Lattice QCD. From the mathematical side, QFT is a fantastic playground for learning about the deep concepts of differential geometry, group theory and algebraic topology. So, the ambitious goal of these talks is to give some insight into the Standard Model of particles from the perspective of symmetry and explain the mathematics behind the formulation of a so-called “gauge theory” (Yang-Mills).
The modern method of constructing field theories is to first postulate a set of symmetries and then write down the most general Lagrangian that respects them. The advantage of abstracting the problem to an “algebraic” one (in the sense of Lie group theory) is that a lot can be understood from the structure of the symmetry groups (commutation relations, Noether charges, gauge fields as affine connections, etc..). In the case of the Standard Model, everything follows from space-time symmetry. The program is to revise a few notions of group representation theory (in particular Schur’s lemma), then study Noether’s theorem for fields under the Poincaré group (generators of Lorentz and momentum transformations), then build examples of gauge theories (scalar quantum electrodynamics). Ultimately, we will be in a position to understand symmetry breaking (Higgs mechanism) and the recent confirmation of the existence of the Higgs boson.
So far, we have covered:
Lecture 1: introduction to the path integral formulation of quantum mechanics, heuristics of Feynman propagator (as some sort of heat Kernel), free-particle example
Lecture 2-3: properties of the path integral in quantum mechanics, computing matrix elements of position operators, Ehrenfest theorem, partition function and analogy with statistical mechanics, introduction to functional methods (generating functions), perturbation theory with free-particle and harmonic oscillator examples (Green’s function), Feynman diagrams in quantum mechanics
Lecture 3-5: introduction to the path integral formulation of QFT, scattering events, recipe to compute expectation values with functional methods, example of single scalar field theory, Feynam diagrams in QFT, consequence of symmetry in the classical field theory (Lagrangian), brief presentation of the Standard Model Lagrangian of Fermions
I do not have time to write proper LaTeX notes, so for now the links above gives access to PDFs of my hand-written notes. Hopefully one day I will get around to producing something more user friendly.