Course curriculum

  • 1

    Welcome to the course!

    • Introduction

    • Course Outline

    • Lecture Notes

    • Perusall
  • 2

    Week 1

    • Week 1 Guide

    • Optional: Classical mechanics of non-relativistic particle

    • Optional: Quantization of non-relativistic particle

    • Imaginary time

    • Optional: Path integrals

    • Observables in the Schroedinger and Heisenberg pictures

    • Time ordering

    • Euclidean time

    • Optional: Klein Gordon theory

    • Optional: Euclidean Klein Gordon theory

    • Functional integral quantization

    • Optional: IR regularization

    • Optional: Canonical quantization vs functional integral quantization

    • Vacuum expectation value of an operator at finite temperature

    • Euclidean propagator

    • Integral kernels and Green's functions

    • Properties of the propagator

    • Typical field configurations

    • Real time propagator

    • Quiz 1: Path, functional, and Gaussian integrals

    • Tutorial 1: Correlation functions and saddle-point approximation

    • Tutorial 1 Solutions

    • Comments or questions?
  • 3

    Week 2

    • Week 2 Guide

    • Introduction

    • Generating functionals

    • Klein Gordon generating functional

    • Wick's theorem

    • Optional: $$\phi^4$$ theory

    • Correlation functions for $$\phi^4$$ theory

    • Optional: Feynman diagrams in $$\phi^4$$ theory

    • Perturbation theory introduction

    • Optional: Feynman rules for $$\phi^4$$ theory

    • Optional: 0 and 2 point functions at leading order

    • Optional: Momentum space Green's functions

    • Optional: 4 point function at leading order

    • Structure of perturbation theory

    • Generating functional for Green's functions

    • Generating functional for connected diagrams

    • Introduction to loop expansion and effective action

    • Topology of Feynman diagrams

    • Loop expansion = semiclassical expansion

    • Background field

    • Definition of effective action

    • Properties of effective action

    • One-loop effective action

    • Proof of one-loop result

    • Quiz 2: Integral kernels, perturbation theory, generating functionals

    • Tutorial 2: Generating functionals for connected and irreducible diagrams

    • Solution 2

    • Homework 1: Perturbation theory

    • Comments or questions?
  • 4

    Week 3

    • Week 3 Guide

    • Quantum effective action recap

    • 1-loop effective action recap

    • Expanding the Tr log

    • First terms in position and momentum space

    • UV divergences and effective action

    • Optional: Regularization intro

    • Optional: Mass renormalization intro

    • Fermionic functional integral introduction

    • Generators of Grassmann algebra

    • Elements of Grassmann algebra

    • Conjugation

    • Example of multiplication and conjugation for $$N=1$$

    • $$N>1$$ and $$\mathbb{Z}_2$$ grading

    • Derivation

    • Integration

    • Gaussian integrals

    • Correlation functions

    • Dirac fields

    • Quiz 3: One-loop effective action and Berezin calculus

    • Tutorial 3: 1-loop effective action and Grassmann variables

    • Solution 3

    • Comments or questions?
  • 5

    Week 4

    • Week 4 Guide

    • Introduction and recap

    • Fermionic functional integrals

    • Correlation functions

    • Wick's theorem

    • Grassmann conjugate of $$\Psi$$

    • Optional: Non-relativisitic fermions

    • Lie groups review quiz

    • Introduction to non-abelian gauge theory

    • Fundamental scalar with global $$SU(2)$$

    • Fundamental fermion

    • Vector fields

    • Gauge transformations and covariant derivatives

    • Infinitessimal gauge transformations

    • Intro and recap

    • Covariant derivatives and gauge tranformations recap

    • Field strength

    • Yang-Mills action

    • Expanding the Lagrangian

    • Feynman rules for pure Yang-Mills

    • Coupling to matter

    • Quiz 4: Fermionic functional integrals and non-abelian gauge theory

    • Tutorial 4: Fundamental scalars and zero modes

    • Solution 4

    • Homework 2: Large N limit

    • Homework 2 PDF

    • Comments or questions?
  • 6

    Week 5

    • Week 5 Guide

    • Introduction to gauge fixing

    • Gauge fields recap

    • Functional integrals for gauge fields introduction

    • Examples of gauge fixing

    • Space of gauge field configurations

    • Optional: student questions

    • Physical configuration space

    • Gauge invariant measure introduction

    • Toy model for gauge fixing

    • Delta in group space

    • Gauge fixing in a functional integral

    • Introduction and recap

    • Inserting the identity

    • Jacobian

    • Gauge invariance and change of variables

    • Example of Faddeev Popov operator

    • Ghost action

    • Feynman rules

    • Corrections to the propagator

    • Quiz 5: Gauge fixing

    • Tutorial 5: Ghosts

    • Solution 5

    • Optional: Spontaneous symmetry breaking and the Higgs mechanism

    • Optional: Renormalization of non-abelian gauge theory

    • Comments or questions?
  • 7

    Lie group resources

    • From quantum mechanics to Lie groups, Representation of groups, Important Lie group examples by Gang Xu

    • Structure constants, Adjoint Representation of Lie algebra by Gang Xu

    • Gang Xu's videos

    • Lecture notes for week 1 of Quantum Theory

    • Lecture 1.1 - Groups, Lie groups and representation theory from Quantum Theory

    • Lecture 1.3 - Lie algebras and Lie groups from Quantum Theory