Quantum Field Theory (I)

Supplementary Notes

Class of Dr. Kaplunovsky, Fall 2019 (unique 55155)

Notes used (or expected to be used) in class this Fall

• Relativistic electromagnetic fields.
• Notes on canonical quantization
  • Poisson brackets and commutator brackets.
  • Heisenberg equations for quantum fields.
• Fock space formalism.
• Operators in the Focks space and wave function languages.
• Expansion of relativistic fields into creation and annihilation operators.
• The saddle point method.
• Propagators and Green's functions.
• Local symmetries and gauge theories.
• Aharonov-Bohm effect and magnetic monopoles.
• Dirac spinor fields.
• Grassmann numbers (Wikipedia article).
• Bose–Einstein condensate and superfluidity.
• Fermionic algebra and Fock space; particles and holes.
• Finite multiplets of the Lorentz group [solutions to the exercises].
• Relativistic causality and Feynman propagator for the fermions.
• Spin-statistics theorem.
• Perturbation theory, Dyson series, and Feynman diagrams.
• Phase space factors.
• Mandelstam's variables s, t, and u.
• Dimensional analysis and allowed couplings.
• EM quantization and QED Feynman rules.
• Dirac trace techniques and muon pair production.
• Crossing Symmetry.
• Ward Identities and sums over photon polarizations.
• Annihilation and Compton Scattering.
• Resonances.
• QCD Feynman rules.
• Spontaneous symmetry breaking.
• The Higgs mechanism.
• Magnetic monopoles, Duality, and SUSY (1995 Trieste lectures by Jeffrey Harvey).
• Glashow–Weinberg–Salam theory of weak and EM interactions.
• Quarks and leptons in the Glashow–Weinberg–Salam theory.
• Kaons, CP violation, and CKM matrix (prof. Mark Thomson's lectures at Cambridge U, Fall 2009).
• Neutrino masses in the Standard Model.

Notes for the QFT (II) class

• Correlation functions of quantum fields.
• Lehmann–Symanzik–Zimmermann reduction formula.
• Dimensional Analysis and Renormalizability.
• QED Feynman rules in the counterterm perturbation theory.
• Renormalization of the EM field at one loop.
• Ward–Takahashi identities.
• Spontaneous symmetry breakdown.
• Form factors.
• QED vertex correction: the algebra, the anomalous magnetic moment, the electric form factor, the infrared divergence, the observed cross-sections, and the Appendix.
• Optical Theorem for Soft Photons.
• Gauge dependence of QED counterterms.
• Vacuum energy and effective potential.
• Higgs Mechanism.
• Renormalization scheme dependence and the Minimal Subtraction.
• Introduction to gauge theories.
• Path integrals in quantum mechanics.
• Path integral for the harmonic oscillator (in detail).
• Functional integration in quantum field theory.
• Functional quantuzation of fermion fields.
• Functional quantisation of the electromagnetism.
• Quantization of non-abelian gauge theories and the Faddeev–Popov ghosts.
• QCD Feynman rules and QCD Ward Identities.
• BRST symmetry.
• QCD beta function.

Recommended Reading

• Lie Agebras in Particle Physics: from Isospin to Unified Theories by Howard Georgi, 1999, Westview press, ISBN 9780813346113. (UT library ebook) [first semester].
• Monopoles, Instantons, and Confinement by Gerard 't Hooft, 1999 lectures at Saalburg, arXiv:hep-th/0010225 [second semester].