Quantum Field Theory: Lecture Log


QFT1, Fall 2012 semester

August 30 (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. Introduction to classical field theory, starting with a refresher of classical mechanics (the principle of least action and the Euler-Lagrange equations).
September 4 (Tuesday):
Classical field theory: Euler-Lagrange field equations; relativistic notations; scalar field examples. The electromagnetic fields: the Aμ potentials and Fμν tension fields; gauge transforms; the EM Lagrangian and the Maxwell equations in the 4D language.
September 6 (Thursday):
EM: current conservation, gauge invariance, and the number of degrees of freedom. Symmetries in field theories: examples and general rules; group theory; kinds of symmetry (global or local, discrete or continuous). Symmetries and conserved currents — Noether theorem.
September 11 (Tuesday):
Noether theorem: general proof; conserved current for the phase symmetry; conserved currents for the SO(N). Translations of spacetime and the stress-energy tensor Tμν. Local phase symmetry: the gauge fields Aμ and the covariant derivatives.
September 13 (Thursday):
Local phase symmetry: covariant derivatives; Aharonov–Bohm effect; magnetic monopoles and charge quantization.
September 18 (Tuesday):
Non-abelian local symmetries: covariant derivatives and the non-abelian gauge fields; non-abelian tensions; Yang–Mills theory; normalization of gauge fields.
September 20 (Thursday):
General gauge theory: Lie groups, Lie Algebras, and their representations; covariant derivatives for general multiplets; multiple gauge group factors.
September 25 (Tuesday):
Review of canonical quantization: Hamiltonian classical mechanics; classical variables and quantum operators; Poisson brackets and commutatots; Heisenberg–Dirac equations. Quantization of the scalar field: Hamiltonian formalism; quantum fields φ̂(x,t) and π̂(x,t) and their commutation relations; Schroedinger and Heisenberg pictures of the quantum fields; quantum Klein–Gordon equation.
September 27 (Thursday):
Quantum fields, multi-oscillator systems, and identical bosons: expansion of the relativistic scalar field into creation and annihilation operators; energy spectrum of the free field; the multi-oscillator system and the occupation numbers; identical bosonic particles; the Fock space.
General identical bosons (relativistic or not): multi-boson Hilber spaces, the Fock space, and its interpretation as a multi-oscillator system.
October 2 (Tuesday):
General identical bosons: creation and annihilation operators; one-body and two-body operators in the Fock space; non-relativistic quantum fields; «second quantization».
Liquid helium II and other Bose–Einstein condensates: non-relativistic QFT; the semiclassical limit and the Landau–Ginzburg theory for the consensate; gradient flow of the condensate.
October 4 (Thursday):
Semi-classical behavior in QM and in QFT: stationary states vs. moving wave packets; semiclassical behavior for highly excited states; coherent states; coherent multi-particle states in QFT; emergence of semiclassical fields.
Bose–Einstein condensation: ground state with ‘φ̂’≠0; elementary excutations; Bogolyubov transform.
October 5 (Friday) [supplementary lecture]:
Superfluidity: Hamiltonian of a moving superfluid; negative-energy excitation and dissipations; critical velocity and dissipationless flow; vortex rings and the critical velocity of macroscopic superflow.
October 9 (Tuesday):
Spontaneously broken symmetries: shifting scalar fields; why SSB does not happen in QM but happens in QFT; spontaneous breakdown of continuous symmetries; Nambu–Goldstone theorem; SO(N) example (the linear sigma model).
October 11 (Thursday):
Higgs mechanism: spontaneous breakdown of a local U(1); massive photon "eats" the would-be Goldstone boson; unitary gauge vs. gauge-invariant description; non-abelian higgs mechanism; SU(2) example; electroweak SU(2)⊗U(1).
October 16 (Tuesday):
Electroweak theory: the electric current and the weak currents; Fermi theory of (low-energy) weak interactions.
Relativistic fields in spacetime: Relatifistic normalization of particle states and operators; free scalar field in the Heisenberg picture; charged fields, particles, and antiparticles; general free fields and their plane waves; decomposition of general fields into creation/annihilation operators.
October 18 (Thursday):
Relativistic causality: signals and local operators; physical local operators must commute at spacelike separations; proof of causality for the free scalar field; problems with single-particle relativistic QM.
Feynman proparator for the scalar field: time-ordering of operators; calculating the propagator; the propagator as a Green's function of the Klein–Gordon equation.
October 19 (Friday) [supplementary lecture]:
Superconductivity: Cooper pairs and the Landau–Ginzburg effective theory; the supercurrent; Meissner effect as a non-relativistic Higgs effect; quantization of magnetic fluxes; Josephson junctions and SQUIDs.
October 23 (Tuesday):
Green's functions: momentum-space formula; regulating ∫ dk0 by moving the poles off the real axis; evaluating ∫ dk0 by contour integration; Feynman's poles and the Feynman propagator; other Green's functions; Feynman propagators for non-scalar fields.
Lorentz symmetry: the generators and their commutation relations; field multiplets and particle multiplets; polarization states of particles and the little group of the momentum.
October 25 (Thursday):
Lorentz multiplets of particles and the Wigner theorem: Wigner theorem — massive particles have spins, massless particles have helicities, tachyons have nothing; massless photons and massless neutrinos; Wigner theorem in general dimensions.
Lorentz multiplets of fields: general form; plane waves and particle polarizations; physical plane waves vs. gauge artefacts.
Gave out the midterm exam.
October 30 (Tuesday):
Dirac spinor fields: Dirac spinor multiplet; Dirac spinor field and Dirac equation; Dirac Lagrangian.
Quantum Dirac fields: the anticommutation relations; Dirac Hamiltonian; expansion into modes; fermionic creation and annihilation operators.
November 1 (Thursday):
Fermionic Fock space: single fermionic mode; multiple modes and the Fermi statistics; the Fock space; one-body and two-body additive operators for the fermions.
Particle-hole formalism: negative-energy modes and filled-up states; holes; creation and annihilation operators for the holes; Fermi sea; quasiparticles in the Fermi sea; applications to condensed matter, to atoms, and to nuclei.
Electrons and positrons: Dirac sea; postrons as holes; modern point of view.
Collected the midterm exams.
November 6 (Tuesday):
Quantum Dirac field: expansion into electron and positron operators; charge conjugation symmetry; relativistic causality for fermions; Feynman propagator for the Dirac field.
November 8 (Thursday):
Feynman propagator for the Dirac field.
Chiral symmetries: vector and axial currents; chiral symmetry and Weyl fermions; chiral symmetry in QCD; gauged chiral symmetry and violation of P and C symmetries.
November 9 (Friday) [supplementary lecture]:
Spin–Statistics Theorem: relation between the spin and the statistics; assumptions of the theorem; the proof; generalization to other dimensions.
November 13 (Tuesday):
Weak interactions of fermions: the quarks, the leptons, and their chiral SU(2)⊗U(1) quantum numbers; Yukawa couplings and the masses of quarks and leptons; weak currents; P and C violation; flavor changing in weak interactions and the Cabibbo–Kobayashi–Maskawa matrix; CP violation; neutrino masses and neutrino oscillations.
November 15 (Thursday):
Perturbation theory and Feynman diagrams: the interaction picture of QM and the Dyson series; the S matrix and its elements; vacuum sandwiches, propagators, and the Feynman diagrams; combinatorics; Feynman rules in the coordinate space; vacuum bubbles and connected diagrams; Feynman rules in the momentum space.
November 16 (Friday) [supplementary lecture]:
Grand Unified Theories: unifying strong, weak, and EM interactions in the same gauge group; multiplets of fermions; the SU(5) example; the gauge couplings and the Georgi-Quinn-Weinberg equations; baryon decays and other exotic processes.
November 20 (Tuesday):
Feynman rules: connected diagrams and the scattering amplitudes; loop counting in perturbation theory; Feynman rules for multiple fields and couplings.
Dimensional analysis: canonical dimensions of fields and couplings; UV problems with negative-dimension couplings; allowed couplings in 4D; allowed couplings in other dimensions.
November 27 (Tuesday):
Renormalizable couplings in d≠4.
Mandelstam variables s, t, and u.
Quantum EM fields: free and fields; gauge fixing the Âμ potentials; photon propagator in the Coulomb gauge; photon propagator in general gauges; Landau and Feynman gauges.
QED Feynman rules: propagators and vertices; external line factors; Dirac indexology; sign rules.
November 29 (Thursday):
Coulomb scattering: tree diagrams and amplitudes; the non-relativistic limit; reconstructing the Coulomb potential; scattering and potential for e-e+.
Dirac traceology and muon pair production e-e+→μ-μ+: un-polarized amplitudes and Dirac traces; techniques for calculating the traces; muon pair production.
November 30 (Friday) [supplementary lecture]:
Fermionic fields in different dimensions: Dirac spinors in general d; mass vs. parity in odd d; Weyl spinors in even d; conjugate Weyl spinors; charge conjugation and Majorana spinors in d=0,1,2,3,4 mod 8 only; Majorana–Weyl spinors in d=2 mod 8.
December 4 (Tuesday):
Quark pair production and the R ratio.
Crossing symmetries: muon pair production vs. electron-muon elastic scattering; analytically continued amplitudes; general crossing symmetries; annihilation vs. Compton scattering.
Ward identities and sums over photons' polarizations.
December 6 (Thursday):
Electron-positron annihilation: the diagram and the amplitudes; checking the Ward identities; summing over photon polarizations and averaging over fermion spins; Dirac traceology; kinematics in the CM frame.
Compton scattering: crossing relation to the annihilation; kinematics in the lab frame; Klein–Nishina formula.
Gave out the final exam.

QFT2, Fall 2013 semester

January 15 (Tuesday):
Syllabus of the spring semester.
Loop diagrams: amputating the external leg bubbles.
Calculating a one-loop diagram: Feynman trick for denominators; Wick rotation to the Euclidean momentum space; cutting off the UV divergence; explaining the cutoff.
January 17 (Thursday):
Renormalization: bare coupling versus physical coupling; canceling the UV divergence; finite amplutudes in terms of finite physical couplings; perturbative expansion beyond 1 loop.
UV regulators: hard-edge cutoff; Pauli–Villars regulator; higher-derivative regulator; comparing the cutoff scales Λ; dimensional regularization.
January 22 (Tuesday):
Dimensional regularization.
Intro to renormalization group: high-energy scattering; re-summing of leading powers of log(E); effective coupling; renormalization group equation.
Optical theorem: partial waves, phase shifts, and optical theorem; formal proof from unitarity of the S matrix; application to scalar scattering; calculation of ℑℳ at one loop.
January 24 (Thursday):
Correlation functions of quantum fields: definition; perturbation theory; Feynman rules; the connected correlation functions.
The two-field correlation function F2(x-y): Källén–Lehmann spectral representation; poles in momentum space and their relation to physical particles' masses and to the field strength.
January 29 (Tuesday):
Analytic two-point function F2(p2): poles and particles; branch cuts and the multi-particle continuum; physical and un-physical sheets; resonances.
Perturbation theory for the F2(p2): re-summing the 1PI bubbles; Σ2(p2) and renormalization of the mass and of the field strength.
1-loop examples: mass renormalization in the φ4 theory; quadratic UV divergences and how to regulate them.
January 31 (Thursday):
Field strength renormalization in the Yukawa theory: the one-loop diagram; the trace; the denominator and the numerator; UV divergence structure: Σ(p2)=(div.constant)+(div.constant)×p2 +finite_f(p2); calculating the Σ(p2); the imaginary part and the decay of the scalar into fermions; dΣ/dp2 and the scalar field strength renormalization.
Lehmann–Symanzik–Zimmermann (LSZ) reduction formula: amputated cores and external leg bubbles; correlation functions Fn(p1,…pn) and their poles; origin of the poles; x0i→±∞ limits and the asymptotic |in⟩ and ⟨out| states; the LSZ reduction formula; scattering amplitudes and the amputated diagrams.
February 5 (Tuesday):
Counterterms perturbation theory: ℒbare=ℒphys+counterterms; counterterm vertices; canceling the infinities.
Counting the divergences: superficial degree of divergence; graphs and subgraphs; classifying the divergences; counterterms; cancelling sub-graph divergences in situ; nested and overlapping divergences; BPHZ theorem.
February 7 (Thursday):
Divergences and renormalizability: supeficial degree of divergence in the φk theories; super-renormalizable, renormalizable, and non-renormalizable theories; divergences in general quantum field theories and the energy dimensions of the couplings; super-renormalizable, renormalizable, and non-renormalizable couplings in d=4; other spacetime dimensions.
Counterterm perturbation theory for the QED: the counterterms and the Feynman rules.
February 12 (Tuesday):
QED perturbation theory: the divergent amplitudes and their momentum dependence; the missing counterterms and the Ward–Takahashi identities.
Calculate the one-loop Σμν(k) and check the WT identity.
February 14 (Thursday):
Renormalization of the electric charge unit e: dressing up the photon propagator Fμν(k); quantum corrections to Coulomb scattering of heavy charged particles; the effective αQED(E) at high energies; the Landau pole.
Ward–Takahashi identities: formulation of the basic WT identities and the outline of the proof.
February 19 (Tuesday):
Proving the Ward–Takahashi identities.
General Ward–Takahashi identities and their relation to the electric current conservation.
February 21 (Thursday):
Ward identity for the counterterms.
Infrared divergences of QED: one-shell IR divergences of the Σ(p̸) and Γμ(p',p); the soft-photon bremsstrahlung; cancelation of the IR divergence from the physical cross-sections.
February 26 (Tuesday):
Electric and magnetic form factors of fermions.
One-loop QED vertex correction: vertex algebra; calculating the F2 form factor; the anomalous magnetic moment of the electron.
February 28 (Thursday):
QED vertex correction: the electric form factor F1 and its infrared divergence.
Optical theorem for the infrared divergences.
Gauge dependence of the counterterms and other off-shell quantities.
March 1 (Friday) — supplementary lecture:
Vacuum energy and effective potentials: zero-point energy and the Casimir effect; zero-point energy for fields with VEV-dependent masses; Feynman diagrams for the vacuum energy; one-loop calculation; general Coleman–Weinberg effective potential; Higgs mechanism induced by the Coleman–Weinberg potential.
March 5 (Tuesday):
Symmetries and the counterterms: δm∝m in QED and the approximate axial symmetry; general rules for counterterms protected and un-protected by symmetries.
Introduction to the renormalization group: large logarithms at high energies; the `runnung' effective coupling λ(E); the off-shell renormalization conditions for the counterterms; the anomalous dimension; the β function and the renormalization group equation; solving the one-loop RGE.
March 7 (Thursday):
Renormalization group for QED: anomalous dimensions for the Aμ and Ψ fields; the beta finction; solving the RGE; initial condition for the RGE and the threshold corrections.
Renormalization group for the Yukawa theory: the anomalous dimensions; the beta functions βλ and βg; solving 2 coupled RG equations.
March 19 (Tuesday):
Types of renormalization group flows: β>0, Landau poles, and breakdown of perturbation theory at short distances; flat β≡0 and scale invariance; conformal symmetry; β<0 and asymptotic freedom; QCD example; dimensional transmutation; non-perturbative behavior at long distances; confinement in QCD; β(g) changing sign at g=g* and RG fixed points; UV-stable vs. IR-stable fixed points; Banks–Zaks example and the conformal window; RG flows for multiple couplings; UV vs IR starting points of the RG flow.
March 21 (Thursday):
RG flow from UV towards IR: relevant, marginal, and irrelevant operators; emergence of renormalizable effective theories in deep IR.
Renormalization schemes: scheme-dependence of couplings and beta functions; one-loop and two-loop βs are universal, higher orders are scheme-dependent.
Minimal subtraction: definition of the MS scheme; recursive formulae for the higher-order poles 1/εn in terms of the simple poles; formulae for the β functions; the MS-bar scheme.
Gave out the midterm exam.
March 26 (Tuesday):
Path integrals in quantum mechanics: path integrals for the evolution kernels; deriving the phase-space path integrals from the ordinary QM; the Lagrangian path integrals and their normalization; the partition function Z; calculating Z for the harmonic oscillator.
Functional quantization in QFT: path integrals for the transition matrix elements; functional integrals for the correlation functions of quantum fields; the free field example; deriving Feynman rules from the functional integrals.
March 28 (Thursday):
The generating functional for the connected correlation functions.
The Euclidean path integrals: analogies between QFT and Statistical Mechanics; convergence issues for path integrals; rotating the time axis t→-iee; Euclidean action; Feynman rules in Euclidean space.
Field theories on the lattice: discretizing the Euclidean path integral; the discretized action for the scalar theory; the lattice as a UV regulator; SO(4) symmetry of the IR limit of the lattice theory; λ coupling acting as temperature.
Collected the midterm exam.
April 2 (Tuesday):
Finite temperature and periodic Euclidean time.
Fermionic integrals: functions of odd Grassmann numbers; Berezin's integrals ∫dθ; Gaussian integrals for the fermions.
Path integrals over the fermionic fields: free Dirac fermions; fermionic path integral in QED; Det(iD̸-m) and the fermionic loops; fermionic sources and the open fermionic lines.
Path integrals over the gauge fields: trouble with the photon propagator; gauge fixing; the Landau gauge; constraints and Jacobians; the Faddeev–Popov determinant; path integrals over the gauge-fixed Aμ(x).
April 4 (Thursday):
Gauge fixing and photon propagators; the Landau gauge; gauge averaging and the Feynman gauge.
Non-abelian gauge theories: fixing the non-abelian gauge symmetry; the Faddeev–Popov ghost fields; the gauge-fixed QCD Lagrangian; QCD Feynman rules; generalization to other gauge theories.
April 5 (Friday) — supplementary lecture:
Gauge theories on the lattice: local symmetry on the lattice; U(link) variables and the Aμ(x) fields; plaquettes, U(□) and the Fμν(x); lattice EM action; non-abelian local symmetries on the lattice; non-abelian plaquettes; lattice Yang--Mills theory; computer simulations.
April 9 (Tuesday):
Finish QCD Feynman rules.
Weakeaned Ward identities for QCD; the qq̄→gg example; relations between longitudinal gluons and ghosts.
BRTS symmetry: the BRST generator and its nilpotency; invariance of ℒQCD; the Fock space of QCD — the physical and the unphysical states; BRST cohomology and getting rid of the unphysical states; BRST symmtries of the amplitudes; cancellation of unphysical processes.
April 11 (Thursday):
QCD renormalization: symmetries of the gauge-fixed theory; allowed counterterms and the BRST relations between them; the β function and the δ1, δ2, and δ3 counterterms; Lie–algebra factors in loop diagrams; Casimirs and indexes of group multiplets; calculating the δ1 and δ2 counterterms.
April 16 (Tuesday):
Calculating the δ3 counterterm and the β function for QCD and other non-abelian gauge theories.
Asymptotic freedom, dimensional transmutation, and ΛQCD; chromo-electric field lines, flux tubes and confinement; mesons, baryons, and glueballs; flux tubes as hadronic strings; flux tubes from condensation of chromo-magnetic monopoles.
April 16 (Thursday):
Chiral symmetry in QCD: spontaneous chiral symmetry breaking by the ⟨Ψ̄Ψ⟩ condensate; the non-linear sigma model; the quark masses and the pseudoscalar meson masses; the eta-meson's mass and the anomaly of the axial U(1) symmetry.
April 19 (Friday) — supplementary lecture:
Wilson loops: abelian and non-abelian Wilson loops; dependence on charge / multiplet type; large loops and signs of confinement; Wilson loops and forces between probe particles; non-abelian probe particles.
April 23 (Tuesday):
Axial Anomaly: massless QED example; naive Ward identities for the axial Jμ; the triangle graphs and the problems with their UV regulators; calculating the anomaly using the Pauli–Villars regulator; axial anomalies of multiple charge fermions; decay of the π0 meson to 2 photons.
April 25 (Thursday):
Announced how to prepare to my SUSY class in the Fall 2013.
Axial anomalies in QCD; Chern--Simons form.
Axial anomaly of the fermionic path integral: formal analysis, regulation, and calculation.
Chiral gauge theories: formulation; loop diagrams; ‘flavor’ anomalies; gauge anomalies.
April 26 (Friday) — supplementary lecture:
Strong coupling expansion in lattice gauge theories: basic idea; expansion of the Wilson loop in compact QED; quark Wilson loop in QED; Wilson loop for an adjoint probe; relation to confinement.
April 30 (Tuesday):
Chiral anomaly cancellation: complex and real multiplets; massive fermions do not contribute to the anomaly; anomaly cancellation in the Standard Model; anomaly cancelation in GUTs; anomalies of global symmetries; non-conservation of baryon and lepton numbers.
Topological sectors of Yang–Mills theories: topological index I[Aμ] and the Chern--Simons form; integer I for configurations with finite Euclidean action; splitting the path integral into sectors.
May 2 (Thursday) — last lecture:
Instantons: self-dual gauge fields; 't Hooft's instanton; multi-instanton configurations; the vacuum angle Θ.
Instantons and fermions: chiral anomaly of the Θ angle; Θ̅=Θ+phase(det(mass matrix)); the strong CP problem; fermionic zero modes in instanton background; the index theorem; relation to the axial anomaly.
Gave out the final exam.

Last Modified: May 2, 2013.
Vadim Kaplunovsky