# David Skinner: Quantum Field Theory II

These are the lecture notes for the second Quantum Field Theory course offered to Part III students. They discuss Path Integrals, Wilsonian Effective Theory, the Renormalization Group, and non-Abelian Gauge Theories.

# Contents

**Introduction: PDF File**

Choosing a QFT. Space-time, fields, actions. Path integrals, the partition function and correlators.**QFT in Zero Dimensions: PDF File**

Review of the path integral. Free theory and Wick's theorem. Perturbation theory, asymptotic expansions and Feynman diagrams. Supersymmetry and localization. Effective theory of a coupled system.**QFT in One Dimension: PDF File**

The path integral approach to Quantum Mechanics and its relation to the operator approach. Brownian motion and the path integral measure. Effective Quantum Mechanics. 1d Quantum Gravity as the worldline approach to QFT.**The Renormalization Group: PDF File**

Wilson's approach to renormalization. Renormalization group flow. Beta functions, anomalous dimensions and Callan-Symanzik equations. Renormalization group trajectories. Counterterms and the continuum limit. Polchinski's equation. The local potential approximation. Gaussian and Wilson-Fisher fixed points in scalar theory. Zamolodchikov's c-theorem.**Perturbative Renormalization: PDF File**

One loop renormalization of quartic scalar theory. Mass renormalization. The on-shell renormalization scheme. Renormalization of the quartic coupling. Irrelevant interactions and the quantum effective action. Dimensional regularization and the MS-bar scheme. The beta function and triviality. One loop renormalization of QED. Vacuum polarization. Counterterms and the beta function of QED. The Euler-Heisenberg effective action.**Symmetries in QFT: PDF File**

Symmetries and conserved charges in classical field theory. Symmetries of the quantum effective action. Ward-Takahashi identities. Current conservation in QFT. Emergent symmetries. Low energy effective field theory. Charges, quantum states and representations.**Classical Yang-Mills Theory: PDF File**

Principal bundles and vector bundles. Connections, curvature and holonomy. The Yang-Mills action and Yang-Mills equations. Matter and minimal coupling. The Yang-Mills path integral. Gauge transformations are redundancies, not symmetries.**Perturbative Non-Abelian Gauge Theory: PDF File**

Faddeev-Popov ghosts and gauge fixing. BRST transformations and their Ward-Takahashi identities. BRST cohomology and the physical Hilbert space. Feynman rules in Lorenz gauge. Vacuum polarization diagrams in Yang-Mills theory. Background field method. The beta function and asymptotic freedom. Topological terms and the vacuum angle.

# Problem Sheets

**Problem Sheet 1: PDF File**

**Problem Sheet 2: PDF File**

**Problem Sheet 3: PDF File**

**Problem Sheet 4: PDF File**

If you're enrolled on the Part III/ MASt course in either Maths, Physics or Astrophysics you can sign up for problem classes here.

# Recommended Books

Any one of Schwartz *Quantum Field Theory and the Standard Model*,
Srednicki *Quantum Field Theory* or Peskin & Schroeder *An Introduction to Quantum Field Theory* will be useful books for the course, although our approach will initially be rather different. Many further suggestions for further reading are given in the first chapter of the Lecture Notes.