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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?

Week 2 Guide

Introduction

Generating functionals

Klein Gordon generating functional

Wick's theorem

Optional: $${\varphi }^4$$ theory

Correlation functions for $${\varphi }^4$$ theory

Optional: Feynman diagrams in $${\varphi }^4$$ theory

Perturbation theory introduction

Optional: Feynman rules for $${\varphi }^4$$ 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?

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?

Week 4 Guide

Introduction and recap

Fermionic functional integrals

Correlation functions

Wick's theorem

Grassmann conjugate of $$\mathrm{\Psi }$$

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?

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?

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