Quantum Field Theory: Lecture Log

January 23 (Wednesday):
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; physical vs. bare couplings.
January 28 (Monday):
Bare and physical couplings: The bare coupling and the cutoff; physical coupling in terms of a physical amplitude; resumming the perturbation theory in terms of physical coupling; cutoff independence.
The ultraviolet regulators: Wilson's hard-edge cutoff; Pauli–Villars; higher-derivative regulator; covariant higher derivative for gauge theories; mentioned the dimensional regularization and the lattice.
January 30 (Wednesday):
Dimensional regularization: D<4 as a UV regulator; Gaussian and non-Gaussian integrals in non-integer dimensions; taking the D→4 limit.
The optical theorem: origin in the unitary S matrix; one-loop example.
February 4 (Monday):
Correlation functions of quantum fields: correlation functions; relation to the free fields and evolution operators; Feynman rules; connected correlation functions.
The two-point correlation function: Källén–Lehmann spectral representation; features of the spectral density function; analytic two-point function F2(p2): poles and particles; pole mass = physical particle mass; branch cuts and the multi-particle continuum; physical and un-physical sheets of the Riemann surface; resonances. Begin perturbation theory for the F2: re-summing the 1PI bubbles; Σ(p2) and renormalization of the mass and of the field strength; mass renormalization in the φ4 theory; fine tuning problem.
February 6 (Wednesday):
Quadratic UV divergences: regulator dependence; dimensional regularization of quadratic divergences.
Field strength renormalization in the Yukawa theory: calculating the one-loop diagram: the trace, the denominator and the numerator, Wick rotation; the UV divergence structure: Σ(p2)=(div.constant)+(div.constant)×p2 +finite_f(p2); calculating the Σ(p2); dΣ/dp2 and the scalar field strength renormalization.
Counterterms perturbation theory: ℒbare=ℒphysical+counterterms; Feynman rules for the counterterms; adjusting δZ, δm, and δλ order by order in λ; one-loop examples.
Extra lecture on February 11 (Monday):
Lehmann–Symanzik–Zimmermann reduction formula: relating the correlation functions Fn(p1,…pn) to the scattering amplitudes: the amputated core and the external leg bubbles; the poles for the on-shell pi0→±E(pi) and their relations to the asymptotic x0i→±∞ limits; the asymptotic |in⟩ and ⟨out| states and the LSZ (Lehmann–Symanzik–Zimmermann) reduction formula; scattering amplitudes and the amputated diagrams.
Regular lecture on February 11 (Monday):
Counting the divergences: superficial degree of divergence; graphs and subgraphs; classifying the divergences; counterterms and canceling the divergences.
Subgraph divergences: in situ cancelation of subgraph divergences by counterterms; nested divergences; overlapping divergences; BPHZ theorem.
Divergences and renormalizability: supeficial degree of divergence in the φk theories; super-renormalizable, renormalizable, and non-renormalizable theories.
February 13 (Wednesday):
Troubles with non-renormalizable theories.
Dimensional analysis and renormalizability: canonical dimensions of fields and couplings; power-counting renormalizability; renormalizable theories in 4D; other dimensions.
QED perturbation theory: the counterterms and the Feynman rules; the divergent amplitudes and their momentum dependence; missing counterterms.
February 18 (Monday):
QED perturbation theory: missing counterterms and the Ward–Takahashi identities.
Σμν(k) at one loop order: calculation, checking the WT identity, the divergence and the δ3 counterterm.
Dressed propagators in QED: the electron's F2(); the photon's F2μν(k); equations for the finite parts of the δ2, δm, and δ3 counterterms.
February 20 (Wednesday):
Effective coupling at high energies: Loop corrections to Coulomb scattering; effective QED coupling αeff(E) and it's running with log(energy); running of other coupling types.
Ward–Takahashi identities: the identities; Lemma 1 (the tree-level two-electron amplitudes); Lemma 2 (the one-loop no-electron amplitudes); identities for the (bare) higher-loop amplitudes; Z1=Z2; generalizing to multiple charged fields.
Extra lecture on February 25 (Monday):
Spontaneous Symmetry breaking: scalar poantial, 〈Φ〉≠0, and spontaneous symmetry breaking; complex field example; linear sigma model; Wigner and Goldstone–Nambu modes of symmetries; Goldstone theorem; broken-symmetry currents in the linear sigma model; pion scattering; spontaneous symmetry breaking in d≠4 dimensions.
Regular lecture on February 25 (Monday):
Ward–Takahashi identities: δ12; in the counterterm perturbation theory; general Ward–Takahashi identities and their relation to the current conservation.
Form factors: probing nuclear and nucleon structure with electrons; the form factors; the on-shell form-factors F1(q2) and F2(q2); the gyromagnetic ratio.
Began the dressed QED vertex at one loop: the diagram and the denominator.
February 25 (Wednesday):
Dressed QED vertex at one loop: numerator algebra; calculating the F2 form factor and the anomalous magnetic moment; the experimental and the theoretical electron's and muon's magnetic moments at high precision.
The electric from factor F1(q2) at one loop: calculating the integrals; the infrared divergence and its regulation; momentm dependence of the IR divergence; the δ1 counterterm.
Began IR finiteness of the measureable quantities: similar IR divergences of the vertex correction and of the soft-photon bremmsstrahlung.
March 4 (Monday):
IR finiteness of the measureable quantities: similar IR divergences of the vertex correction and of the soft-photon bremmsstrahlung; detectable vs. undetectable photons and the observed cross-sections; IR finiteness of the observed cross-sections.
Briefly: higher lops and/or more soft photons; optical theorem for the finite observed cross-sections; no Fock space for QED and other gauge theories; jets in QCD.
Gauge dependence in QED: δ1(ξ) and δ2(ξ).
Began symmetries and counterterms.
March 6 (Wednesday):
Finish symmetries and counterterms.
Intro to renormalization group: large log problem for Em; adjusting the counterterms to avoid large logs; off-shell renormalization schemes; running counterterms and running couplings.
Renormalization group basics: the anomalous dimension of a quantum field; the β function and the running coupling.
Extra lecture on March 11 (Monday):
Higgs mechanism: semiclassical abelian example; propagator of the massive photon; Higgs mechanism for general (abelian) SSB; non-abelian examples; general gauge theory.
Regular lecture on March 11 (Monday):
Renormalization group equations: RGE and its solution for the λφ4 theory; renormalization of QED: anomalous dimensions, βe to one-loop order, solving the RGE for QED; β–functions for general couplings; coupled RGE foe the Yukawa theory (briefly).
Begin types of RG flows: β>0, Landau poles, and UV incompleteness; β<0, QCD example, and asymptotic freedom; IR price of asymptotic freedom in QCD; confinement; chiral symmetry breaking.
March 13 (Wednesday):
Finish types of RG flows — fixed points: β(g*)=0; UV stability vs. IR stability; scale invariance and conformal symmetry; RG flows for multiple couplings; flow to UV versus flow to IR.
Renormalisation schemes: scheme dependence of the couplings and the β-functions; the minimal subtraction schemes MS and MS-bar; extracting β-functions from residues of the 1/ε poles.
March 18 and March 20:
Spring break, no classes.
March 25 (Monday):
Non-abelian gauge symmetries: review of the abelian gauge theory; non-abelian covariant derivatives, matrix-valued connections, and gauge transforms; non-abelian vector fields and tension tensors; Yang–Mills theory; QCD; tree-level QCD Feynman rules.
March 27 (Wednesday):
Non-abelian gauge symmetries: general gauge groups; matter fields in general multiplets; products of gauge symmetries; the Standard Model.
Introduction to path integrals: path integrals in QM; the Lagrangian and the Hamiltonian forms of the path integral; derivation of the Hamiltonian form.
Gave out the midterm exam.
April 1 (Monday):
Introduction to path integrals: Lagrangian path integrals — derivation and normalization; the partition function; harmonic oscillator example.
Functional integrals in QFT: “path” integrals for quantum fields; correlation functions; free fields and propagators; perturbation theory and Feynman rules; sources and the generating functionals.
Convergence of path integrals and imaginary time; Euclidean 4D for QFT path integrals.
April 3 (Wednesday):
Functional integrals in Euclidean spacetime: QFT↔StatMech corresponcence; Euclidean lattice as a UV cutoff; restoration of the SO(4)→Lorentz symmetry in the continuum limit; coupling↔temperature analogy.
Begin fermionic functional integrals: Grassmann numbers; Berezin integrals over fermionic variables.
Collect midterm exams.
Extra lecture on April 8 (Monday):
Glashow–Weinberg–Salam theory: Higgsing the SU(2)W×U(1)Y down to the U(1)EM; the photon and the massive W± and Z0 vectors; the mixing angle; the weak currents; the effective Fermi theory; ρ=1 for the neutral current.
Began Fermions in the GWS theory: Quantum numbers of the quarks and the leptons; the Yukawa couplings giving rise to the masses.
Regular lecture on April 8 (Monday):
Functional integrals for fermions: fermionic gaussian integrals; free Dirac field with sources; Dirac propagator; FI for fermions in EM background; Det(D̸+m) and the electron loops; 1/(D̸+m) and the tree diagrams.
Functional integral for the EM field: gauge transforms as redundancies; gauge-fixing conditions and the Faddeev–Popov determinants; Landau-gauge propagator from the functional integral; gauge-averaging and the Feynman gauge.
Briefly functuonal integrals at finite temperature: periodicity in Euclidean time, period=β; Det(−∂2+m2) over periodic fields; antiperiodic fermionic fields; Fadde'ev–Popov determinant and the number of photonic degrees of freedom.
April 10 (Wednesday):
Quantizing the Yang--Mills theory: fixing the non-abelian gauge symmetry; Faddeev–Popov ghost fields; gauge-fixed YM Lagrangian.
QCD: Quantum Lagrangian and the Feynman rules; generalization to other gauge theories; weakened Ward identities in QCD: the qq̄→gg example; ghost pair production and unitarity.
Intro to BRST symmetry: BRST symmetry action on the fields; nilpotency of the BRST operator; BRTS invariance of the QCD Lagrangian.
Extra lecture on April 15 (Monday):
Quarks and leptons of the Standard Model: Yukawa couplings and masses; the EM current and the weak currents (charged and neutral) in terms of the fermion fields; flavor mixing and the Cabibbo–Kobayashi–Maskawa matrix.
Regular lecture on April 15 (Monday):
BRTS symmetry: the Fock space of QCD — the physical and the unphysical states; BRST cohomology and getting rid of the unphysical states; BRST symmetries of the amplitudes; cancellation of unphysical processes.
QCD renormalizability: renormalizability and the counterterm set; BRST and other manifest symmetries; allowed counterterms and renormalizability; Ward-like (but weaker) identities for the counterterms.
April 17 (Wednesday):
Basic group theory: the Casimir and the Index.
QCD β–function at the one-loop order: the beta function and the counterterms; calculating the δ2 for quarks; calculating the δ1 for quarks — the QED-like loop and the non-abelian loop; calculating the δ3 — the quark loop, the gluon loop, the sideways loop, and the ghost loop; assembling the β–function.
April 22 (Monday):
β–functions for QCD and other gauge theories: assembling QCD beta function; other gauge theories; allowing for Majorana and Weyl fermions; allowing for scalar fields; general formula.
Quark confinement in QCD: chromo-electric field lines and flux tubes; condensation of chromo-magnetic monopoles; flux tubes as hadronic strings; mesons and baryons; spinning hardonic string and the Regge trajectories.
April 24 (Wednesday):
Chiral symmetry: axial symmetry of a massless fermion; QED anomaly of the axial symmetry; SU(Nf)L×SU(Nf)R×U(1)V exact symmetry of QCD with massless quarks; approximate SU(2)L×SU(2)L and SU(3)L×SU(3)L of real-life QCD.
Spontaneous breaking of chiral symmetry: overview of SSB and of the Goldstone symmetry; pions as pseudo–Goldstone bosons; generalize to 3 light flavors.
Linear and non-linear sigma models: linear sigma model for the SU(2)×SU(2)→SU(2) SSB; taking the λ→∞ limit to the non-linear sigma model; general NLΣM's; the SU(2)–valued non-linear field and its symmetries; vector and axial currents of the NLΣM; pions as Goldstone bosons of the NLΣM; generalize to 3 flavors.
Extra lecture on April 29 (Monday):
CP Violation: weak interactions break C and P but might preserve CP; neutral K mesons and their mixing; GIM mechanism; decays of KS and KL; imaginary part of ΔM and the CP violation; CP violation for the neutral D and B mesons.
CP violation in the Standard Model: CP action on W±, on the charged currents, and on the CKM matrix; un-removable CKM phase for 3 families.
Regular lecture on April 29 (Monday):
Non-linear sigma models and chiral symmetry breakdown: NLΣMs in general; NLΣM for the SU(N)×SU(N)→SU(N) ΧSB; symmetry currents of the NLΣM; quark-qntiquark condensate in QCD and the NLΣM for the condensate; perturbation by the quark masses; U(3) NLΣM and η↔η′ mixing; constituent quark mass and the non-relativistic quark model.
Started axial anomaly in QED: Formal Ward-like identities for the axial current; hidden momentum shift and the regularization trouble.
May 1 (Wednesday):
Axial anomaly in QED: limiting the anomaly to the one-loop triangle diagrams; regulation analysis for the Pauli–Villars cutoff; ⟨∂μJμ5reg⟩= −2iMPV⟨η̄γ5η⟩; evaluating the triangle diagrams: relation to the PV regulator loops, numerator algebra, taking the integrals, summary; anomalies for QED with multiple fermions; decay of the neutral pion to 2 photons.
Extra lecture on May 6 (Monday):
Neutrino masses: neutrino oscillations; Dirac and Majorana masses for the neutrinos; SM origin of neutrino masses; seesaw mechanism.
Regular lecture on May 6 (Monday):
Anomalous decay of the neutral pion to 2 photons.
Axial anomaly in QCD: 2-gluon and 3-gluon anomalies; anomaly cancelation for the SU(Nf) currents and the η/π problem.
Chiral gauge theories: Weyl fermions and chiral currents; loops of Weyl fermions and the chiral anomaly; trace=0 conditions for anomaly cancelation; lepton and baryon anomalies in the Standard Model.
May 8 (Wednesday):
Gauge anomalies: triangle anomaly in chiral QED; anomalous gauge variance of log(det(̸D)); anomaly in non-abelian chiral theories; Wess–Zumino consistency conditions; anomaly coefficients Aabc and traces over chiral fermions; rules for anomaly cancellation; checking anomaly cancellation for the Standard Model; relations between quark and lepton charges; anomaly cancelation in general chiral theories; massive fermions do not affect the anomalies; cubic Casimirs and cubic anomaly indices for simple gauge groups; applications to Grand Unification.
Gave out the final exam.

Last Modified: May 8, 2019.
Vadim Kaplunovsky