Quantum Field Theory: Lecture Log

August 29 (Thursday):
Syllabus and admin: course content, textbooks, prerequisites, homework, exams and grades, etc. (see the main web page for the class).
General introduction: reasons for QFT; field-particle duality.
Refresher of classical mechanics (the least action principle and the Euler-Lagrange equations).
August 30 (Friday):
Classical fields: definition; Euler–Lagrange equations; Klein–Gordon example; coupled scalar fields; higher derivatives and counting of degrees of freedom.
September 3 (Tuesday):
Relativistic notations for field theory.
The electromagnetic fields: the 4–tensor Fμν=−Fνμ; Maxwell equations in relativistic form; the 4–vector potential Aμ and the gauge transforms; the Lagrangian formulation; current conservation and gauge invariance of the action; counting the EM degrees of freedom.
September 5 (Thursday):
Intoduction to quantization: canonical quantization vs. path integrals; Hamiltonian classical mechanics; canonical commutation relations; Schrödinger and Heisenberg pictures of quantum mechanics; Poisson brackets and commutators.
September 6 (Friday):
Quantum scalar field: the canonical momentum field π(x,t) and the classical Hamiltonian; operator-valued quantum fields; the equal-time commutation relations; the Hamiltonian operator; quantum Klein–Gordon equation.
September 10 (Tuesday):
Quantum fields and particles: expanding free relativistic scalar field into harmonic oscillators; eigenstates of the free quantum field's Hamiltonian; identifying the identical bosons; the Fock space.
Going back: from identical bosons to the Fock space to harminic oscillators; continuing to operators in the Fock space and hence to non-relativistic quantum fields.
September 12 (Thursday):
Identical bosons and non-relativistic fields: wave-function and Fock-space formalisms; one-body and two-body operators in the Fock space language; examples; non-relativistic creation and annihilation fields; second quantization; Landau–Ginzburg theory.
Regular lecture on September 13 (Friday):
Relativistic normalization of states and operators: Lorentz groups; momentum space geometry and Lorentz-invariant measure; normalization of states and operators.
Extra lecture on September 13 (Friday):
Seeing classical motion in quantum mechanics: stationary states smear motion; wave packets and their motion; coherent states of a harmonic oscillator.
Seeing classical fields in QFT: free fields are sums of harmonic oscillators; eigenstates show particles, coherent states show fields; perturbation theory.
Intro to superfluids: naive Bose–Einstein condensate as a coherent state; classical and quantum fluctuation fields δφ(x); Bogolyubov transform; ground state; fluctuation spectrum; non-local force between helium atoms and the ‘rotons’.
September 17 (Tuesday):
Relativistic quantum fields: expanding a free time-dependent scalar field into products of plane waves and creation/annihilation operators; charged scalar field and antiparticles; general free fields.
Relativistic causality: superluminal particles in ‘relativistic’ QM; signals in QM and in QFT; local operators and fields; relativistic causality in free and in interacting QFT.
September 19 (Thursday):
Relativistic causality: proof for the free fields.
Feynman propagator for the scalar field: why and how of time-ordering; defining the propagator; relation to D(x-y).
September 20 (Friday):
Feynman propagator as a Green's function: Checking that the propagator is A Green's function; Green's function in momentum space; regulating the integral over the poles; Feynman's choice; other types of Green's functions; Feynman propagators for vectors, spinor, etc., fields.
September 24 (Tuesday):
Symmetries of field theory: symmetries of the action; global and local symmetries; conserved currents and the Noether theorem; the SO(N) example; phase symmetry of the quantum theory; SO(N) symmetry of the quantum theory; proof of the Noether theorem; internal and spacetime symmetries.
September 26 (hursday):
Transpation symmetries and the stress-energy tensor.
Local phase symmetry: local symmetry and covariant derivatives; gauge field and gauge transforms; algebra of covariant derivatives; coupling charged scalar fields to electromagnetism.
Regular lecture on September 27 (Friday):
Aharonov–Bohm effect and magnetic monopoles: covariant Schroedinger equation; Aharonov–Bohm effect; cohomology of the vector potential; charge quantization and the compactness of the U(1) phase symmetry; intro to magnetic monopoles and Dirac charge quantization.
Extra lecture on September 27 (Friday):
Field theory of superfluidity: classical field for the Bose–Einstein condensate; density and velocity of the superfluid; irrotational flow; vortices.
Other kinds of topological ‘defects’: domain walls, monopoles, YM instantons; co-dimension.
October 1 (Tuesday):
Finish magnetic monopoles: Dirac construction; charge quantization; gauge bundles.
Non-abelian local symmetries: covariant derivatives, matrix-valued connections, and gauge transforms; non-abelian vector fields and tension tensors.
October 3 (Thursday):
Overview of group theory: Lie groups, Lie algebras, representaions, multiplets.
Non-abelian gauge theories: NA tension fields; Yang–Mills theory and gauge field normalization; gauge theories with matter; general gauge groups.
October 4 (Friday):
Finish gauge symmetry: general local symmetry groups; covariant derivatives for different multiplets types; multiple gauge groups; Standard Model example.
October 8 (Tuesday):
Lorentz symmetry: generators and commutation relations; finite field multiplets vs. unitary particle state multiplets; spinless particle representation; other particle representations and little groups G(p).
Wigner theorem: massive particles have spins; massive particles have helicities; tachyons have nothing.
October 10 (Thursday):
Finish Wigner theorem: tachyons are scalars; generalization to d≠3+1 dimensions.
Avoiding tachyons in QFT: vacuum instabilities and scalar VEVs.
Lorentz multiplets of fields.
Regular lecture on October 11 (Friday):
Dirac spinors and spinor fields: Lorentz spinor multiplet; Dirac equation; Dirac conjugation; Dirac Lagrangian.
Extra lecture on October 11 (Friday):
Bose–Einstein condensation and superfluidity: Bose–Einstein condensation; Bogolyubov transform; quasiparticles; perturbative corrections; superfluid liquid helium; quasiparticles in a moving fluid; dissipation vs. superfluidity.
October 15 (Tuesday):
Dirac spinor fields: Hamiltonian for the quantum fields; classical limits of fermionic fields, Grassmann numbers.
Fermionic algebra and Fock space: Hilbert stace of one fermionic mode; multiple modes; Fermionic fock space; wave functions and operators; particles and holes; holes as quasiparticles.
October 17 (Thursday):
Fermionic particles and holes: Fermi sea, extra fermions, and holes.
Relativistic electrons and positrons: Naive diagonalization of the Dirac Hamiltonian; positrons as holes in the Dirac sea; expanding the Dirac fields into creation and annihilation operators.
Charge conjugation symmetry: C:e↔e+; C:Φ(x)→Φ*(x); C:Ψ(x)→γ2Ψ*(x).
October 18 (Friday):
Neutral particles and C-parity.
Dirac, Majorana, and Weyl fermions: Majorana fermions; counting degrees of freedom; relations between Majorana and Weyl fermions; Majorana mass term; massless and massive neutrinos.
October 22 (Tuesday):
Parity and CP symmetries; (briefly) T symmetry; CPT theorem.
Vector and axial symmetries of a Dirac fermion. chiral U(N)L×U(N)R symmetry.
October 24 (Thursday):
Chiral symmetry: vector, axial, and chiral symmetries; the U(N)L×U(N)R chiral symmetry; chiral symmetry in QCD; chiral gauge theories.
Relativistic causality for the fermions: commuting and anticommuting fields; checking anticommutativity of free Dirac fields at spacelike separation; spin-statistics theorem.
October 25 (Friday):
Feynman propagator for Dirac fermions.
Introduction to perturbation theory: interaction picture of QM; the Dyson series and the time-ordering. Gave out the midterm exam.
October 29 (Tuesday):
Perturbation theory in QFT and Feynman diagrams: S matrix and its elements; vacuum sandwiches of field products; diagramatics; combinatorics of similar terms; coordinate space Feynman rules; vacuum bubbles and their cancellation; momentum space Feynman rules; momentum conservation and connected diagrams; scattering amplidudes.
October 31 (Thursday):
Perturbation theory in QFT and Feynam rules: summary of Feynman rules for the λΦ4 theory; phase space factors; loop counting. for quartic and cubic couplings; Mandelstam's variables s, t, and u.
Regular lecture on November 1 (Friday):
Collected the midterm exams.
More Feynman rules: cubic and quartic couplings; loop counting; multiple fields; Mandelstam's variables s, t, and u.
Begin Good and bad interactions in perturbative QFT: dimensional analysis and trouble with Δ<0 couplings.
Extra lecture on November 1 (Friday):
Fermionic fields in different spacetime dimensions: Dirac spinor fields; mass breaks parity in odd d; Weyl spinor fields in even d only; LH and RH Weyl spinors; Majorana spinor fields in d≡0,1,2,3,4 (mod 8) only; Majorana–Weyl spinors in d≡2 (mod 8); complex, real, and pseudoreal representation; Bott periodicity for spinors of SO(a,b).
November 5 (Tuesday):
Good and bad interactions in perturbative QFT: dimensions of fields and couplings; trouble with Δ<0 couplings; types of Δ≥0 couplings in 4D; other dimensions.
Quantum Electro Dynamics (QED): quantizing EM fields; photon propagator and its gauge dependence.
November 7 (Thursday):
QED Feynman rules: propagators and vertices; external line factors; Dirac indexology; Gordon identities; sign rules.
Coulomb scattering in QED: diagrams and amplitudes; non-relativistic limit; recovering the Coulomb potential; electron-electron vs. electron-positron Coulomb scattering.
Yukawa theory (briefly): Feynman rules; fermion-fermion and fermion-antifermion scattering; the non-relativistic limit and the Yukawa forces.
November 8 (Friday):
Muon pair production in QED, e^+e+μ^+: the tree amplitude; the un-polarized scattering and the spin sums/averages; Dirac trace techniques; traces for the pair production; partial and total cross-sections.
November 12 (Tuesday):
Pair production in electron-positron collisions: partial and total cross-sections for the muon pair production; quark pair production and jets; hadronic production e^+e+q+q̄→hadrons and the R ratio.
Crossing symmetry: electron-muon scattering vs. pair production; analitically continuing the amplitudes; crossing symmetry in general; signs for crossed fermions; Compton vs. annihilation example.
November 14 (Thursday):
Ward Identities: Ward identities for photons and gauge invariance of the amplitudes; sums over photon polarizations.
Electron-positron annihilation e+e+→2γ: tree diagrams and the amplitude; checking the Ward identities; summing over photon polarizations and averaging over fermions' spins; Dirac traceology (part 1).
Regular lecture on November 15 (Friday):
Annihilation and Compton scattering: Dirac traceology (part 2); summary and annihilation kinematics; annihilation cross-section; crossing to Compton scattering; lab frame kinematics; Klein—Nishina formula.
Extra lecture on November 15 (Friday):
Resonances and unstable particles: Breit—Wigner resonance and its lifetime; propagators of unstable particles; making a resonance; cross-sections and branching ratios; J/ψ example.
November 19 (Tuesday):
Spontaneous symmetry breaking: continuous families of degenerate vacua; massless particles; linear sigma model; Wigner and Goldstone modes of symmetries; Goldstone theorem; approximate symmetries and pseudo-Goldstone bosons; chiral symmetry of QCD and the pions.
November 21 (Thursday):
The Higgs Mechanism: SSB of a local U(1) symmetry; massive photon “eats” the would-be Goldstone boson; unitary gauge vs. gauge-invariant description; non-abelian Higgs examples: SU(2) with a doublet; SU(2) with a real triplet; general case.
Regular lecture on November 22:
The Higgs Mechanism: vector masses for general gauge groups and Higgs multiplets.
Glashow–Weinberg–Salam theory: bosonic fields and the Higgs mechanism; unbroken electric charge Q=T3+Y; masses of the vector fields and the Weinberg's mixing angle; charged and neutral weak currents.
Extra lecture on November 22:
Making magnetic monopoles: topology of the Higgs field in the Georgi–Glashow model; the “hedgehog” solution and its magnetic field; multimonopole solutions and their magnetic charges; monopoles in general spontaneously broken gauge groups; monopoles in Grand Unified Theories.
November 26 (Tuesday):
Fermi effective theory of weak interactions.
Fermions of the Glashow–Weinberg–Salam theory: quarks' and lepton's masses arising from their Yukawa couplings to the Higgs scalar; charged and neutral weak currents of quarks and leptons.
December 3 (Tuesday):
History of hadronic symmetries: isospin; strangeness; SU(3)flavor; quarks; non-relativistic quark model; color; QCD; confinement; SU(3)×SU(3)→SU(3) chiral symmetry breaking and the π, K, and η pseudo-Goldstone bosons.
Intro to CKM matrix: Cabibbo mixing and Kaon decays; GIM mechanism and the charm quark; third family and the CKM matrix; the charged currents; flavor-changing weak decays.
Origin of the CKM matrix: SM fermions come in sets of 3 for each multiplet type; unitary charges of bases; matrices of Yukawa couplings; mass matrices for Weyls fermions; diagonalizing the mass matrices and forming the Dirac fermions; basis mismatch for charge +2/3 and charge -1/3 quarks and the Cabibbo–Kobayashi–Maskawa (CKM) matrix; flavor changing charged currents; bases for the charged leptons and for the neutrinos; netral weak currents: diagonal in the Standard Model, but non-diagonal (flavor-changing) in other models.
December 5 (Thursday):
Neutral kaons: GIM box and K^0↔K̅0 mixing; K-long and K-short; CP eigenstates K1 and K2, and their decays to pions; K-short regeneration; semi-leptonic decays of neutral kaons; strangeness oscillations; the CPLEAR experiment.
Intro to CP violation (CPV): CPV in neutral kaon decays to pions; CPV in eigenstates and in decay rates; CPV in semi-leptonic decays; CPV and the CKM matrix.
CP symmetry and its violation by weak interactions: CP symmetry of chiral gauge theories; CP action on the W± and on the charged currents; CP: CKM↔CKM*; quark phases and CKM phases; third family, Kobayashi, Maskawa, and CP violation.
Briefly other aspects of CP violation: Possibility of CPV by strong interactions; CPV and baryogenesis in the early Universe.
Plan for December 6 (Friday):
Neutrino masses: neutrino masses and oscillations; Dirac vs. Majorana masses of neutrinos; seesaw mechanism.
Give out the final exam.

Last Modified: December 5, 2019.
Vadim Kaplunovsky