Quantum Field Theory

Supplementary Notes

Class of Dr. Kaplunovsky, Fall 2024 and Spring 2025

Notes for QFT (I)

• Why study QFT?
• Intro to classical field theory.
• Relativistic electromagnetic fields.
• Review of canonical quantization.
• Intro to quantum fields.
• Spectrum of the free scalar quantum field.
• Second quantization of bosons.
• Bose–Einstein condensate and superfluidity.
• Expansion of relativistic fields into creation and annihilation operators.
• The saddle point method.
• Relativistic causality.
• Propagators and Green's functions.
• Noether theorem.
• Local symmetries and gauge theories.
• Aharonov–Bohm effect.
• Electric-magnetic duality and magnetic monopoles.
• A note on multiplets and representations.
• Particle and field multiplets of Lorentz symmetry.
• Dirac spinor fields.
• Grassmann numbers (notes from CEA, France); see also Wikipedia article.
• Fermionic algebra and Fock space; particles and holes.
• Charge congugation, Majorana and Weyl fermions, parity and CP.
• Relativistic causality and Feynman propagator for the fermions.
• Spin-statistics theorem.
• Spinor fields in different dimensions.
• Perturbation theory, Dyson series, and Feynman diagrams.
• Fermi's Golden Rule and the phase space factors.
• Mandelstam's variables s, t, and u.
• Dimensional analysis and allowed couplings.
• Scattering in Quantum Mechanis.
• 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.
• Spontaneous symmetry breaking.
• The Higgs mechanism.
• Glashow–Weinberg–Salam theory of weak and EM interactions.
• Quarks and leptons in the Glashow–Weinberg–Salam theory.
• Cabibbo–Kobayashi–Maskawa matrix of flavor mising.
• Kaons, CP violation, and CKM matrix (prof. Mark Thomson's lectures at Cambridge U, Fall 2009); local copy.
• Neutrino masses in the Standard Model.

Notes for the QFT (II)

• Amputating the external leg bubbles.
• Introduction to caclulating loop diagrams.
• Overview of UV regularization schemes.
• Basics of dimensional regularization.
• Optical Theorem.
• Correlation functions of quantum fields.
• Lehmann–Symanzik–Zimmermann reduction formula.
• Dimensional Analysis and Renormalizability.
• QED: counterterms, Feynman rules, divergences, and renormalizability.
• Renormalization of the EM field at one loop.
• Ward–Takahashi identities:
  • Current conservation, formal proof, and Z₁=Z₂.
  • Diagrammatic proof.
• Vacuum energy and effective potential.
• Form factors.
• QED vertex correction.
• Optical Theorem for Soft Photons.
• Gauge dependence of QED counterterms.
• Introduction to the renormalization group.
• Plot of the running QCD coupling.
• Renormalization scheme dependence and the Minimal Subtraction.
• Path integrals in quantum mechanics.
• Functional integration in quantum field theory.
• Euclidean spacetime, discretization, and QFT ↔ StatMech analogy.
• 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.
• Renormalizability of non-abelian gauge theories.
• QCD beta function.
• Axial anomaly.
• Non-linear sigma models.

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].
• Group Theory for Unified Model Building by Richard Slansky, Physics Reports 79 (1981) pp. 1–128 (local copy) [technical aspects of group theory].
• Monopoles, Instantons, and Confinement by Gerard 't Hooft, 1999 lectures at Saalburg, arXiv:hep-th/0010225 [second semester].
• Magnetic monopoles, Duality, and SUSY (1995 Trieste lectures by Jeffrey Harvey).