Syllabus

The main theme of this course is concepts and applications of Feynman path integrals, symmetries, and the renormalization group.

Lecture notes are available here.

Topics (tentative)

  • Symmetries and conservation laws. Noether’s Theorem. Lie groups and Lie algebras. Representation theory of Lorentz and Poincare group. Abelian gauge symmetry and QED. Nonabelian gauge symmetry. Yang-Mills lagrangian. Covariant derivatives. Field strengths. Bianchi identities. Chirality and the Standard Model. Gauging isospin and hypercharge.
  • Goldstone’s Theorem. Spontaneous symmetry breaking in the linear sigma model and Abelian Higgs model. Spontaneous breaking of electroweak symmetry and the Higgs mechanism. Higgs potential (“Mexican hat”). How W and Z bosons, and fermions, become massive. Unitary gauge. The Weinberg angle and Fermi decay constant.
  • Introducing the Feynman Path Integral and defining its measure for spin zero fields. Functional quantization for scalar fields. Correlation functions. Functional determinants. Free 2- and 4-point correlation functions. Introducing a source J; the generating functional Z[J]. n-point correlation functions. The Feynman contour.
  • The connection between the FPI and the partition function of statistical mechanics. The FPI’s role as generating functional and its relationship to Schwinger-Dyson equations. Z[J] for free vs interacting scalar field theories. The generating functional for connected diagrams, W[J], defined via Z[J]=exp(iW[J]). Legendre trees. The LSZ reduction formula. The Kallen-Lehmann spectral representation.
  • Path integral quantization for spin-half fermi fields. Grassmann integrals and derivatives.
  • Path integral quantization for spin-one fields. Gauge invariance. Abelian Fadeev-Popov ghosts. Path integral quantization for nonAbelian gauge fields and their Fadeev-Popov ghosts. Lorentz gauges. Feynman rules for Yang-Mills theories. BRST invariance.
  • Divergences in Feynman graphs. Dimensional regularization. Feynman parameters. Counterterms. One loop renormalization of quartic scalar field theory. Weinberg’s Theorem.
  • Wilsonian Renormalization Group (RG) and critical phenomena. UV cutoffs and counterterms. Callan-Symanzik equation. Fixed points.
  • One loop renormalization of QED. Power counting. Vacuum polarization / photon self-energy. Electron self-energy. QED vertex correction. Ward-Takahashi identities. Photon masslessness. Charge renormalization. Optical theorem. Cutkosky rules. Counterterms and the QED beta function.
  • Anomalies. The triangle anomaly. Anomaly cancellation in chiral field theories.