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.
  • Symmetry in Quantum Field Theory:   PDF File
    Noether's theorem. Ward identities in the path integral approach. Currents, charges and the operator picture. The Ward-Takahashi identity in QED. Symmetry and renormalization. Emergent symmetries.
  • 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
    Dimensional regularization. Counterterms and renormalization schemes. One loop renormalization of quartic scalar theory. One loop renormalization of QED. Vacuum polarization; loop calculations and physical interpretation. Decoupling in dimensional regularization.
  • 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. Faddeev-Popov ghosts and BRST symmetry. Feynman rules in Lorenz gauge. Large N. Lattice gauge theory. Topological terms and the vacuum angle. Perturbative renormalization of Yang-Mills. The beta function and asymptotic freedom. Confinement.

Problem Sheets


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.