Quantum Field Theory: Lecture Log

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QFT 1, Fall 2022 semester

August 23 (Tuesday):
August 26 (Thursday):
Syllabus and admin: course content, textbooks, prerequisites, homework, exams and grades, etc.
General introduction: reasons for QFT; field-particle duality.
Lagrangian mechanics: Lagrangian and action; least action principle; Euler–Lagrange equations; multiple dynamical variables; counting the degrees of freedom.
August 26 (Friday) [make-up lecture]:
Introduction to classical fields: Definition of a classical field; Lagrangian density; Euler–Lagrange equations for fields; Klein–Gordon example; multiple fields; complex fields; Landau–Ginzburg example; higher space derivatives and non-local Lagrangians for non-relativistic fields.
Relativistic fields: relativistic sign conventions; Einstein summation convention; relativistic ℒ and field equations; Klein–Gordon example; multiple scalar fields.
Started Relativistic electromagnetic fields: the 4–tensor Fμν=−Fνμ and the relativistic form of Maxwell equations.
August 30 (Tuesday):
Relativistic electromagnetic fields: 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.
Review of canonical quantization: Canonical quantization v. functional quantization; Hamiltonian formalism in classical mechanics; quantization, operators, and commutation relations; Poisson brackets and commutator brackets.
Introduction to quantum fields: Hamiltonian formalism for the classical fields; quantum fields; equal-time commutation relations.
September 1 (Thursday):
Finish Introduction to quantum fields: quantum Hamiltonian, Heisenberg equtions, and the quantum Klein–Gordon equation.
Quantum fields and particles: expanding free relativistic scalar fields into modes; creation and annihilation operators for a bunch of harmonic oscillators; eigenstates of the free quantum field's Hamiltonian; identifying the identical bosons; the Fock space.
Start General identical bosons: bosonic Fock space and its occupation number basis.
September 2 (Friday) [make-up lecture]:
General identical bosons: creation and annihilation operators; wave-function language vs. Fock-space language; one-body operators; two-body operators; non-relativistic quantum fields; “second quantization”.
September 6 (Tuesday):
Relativistic normalization of states and operators: Lorentz groups; momentum space geometry and Lorentz-invariant measure; relativistic normalization of states and operators.
Relativistic quantum fields: Expanding a free time-dependent scalar field into products of plane waves and creation/annihilation operators; massive vector field; charged scalar field and antiparticles; general free fields.
September 8 (Thursday):
Intro to relativistic causality: no superluminal particles or sigmals; signals in quantum mechanics: sending a signal requires [M̂(t2),Ŝ(t1)]≠0; QFT: local operators and relativistic causality.
Relativistic causality: local operator and fields; proof for free scalar fields; going forward and backward in time; causality for interacting fields.
Intro to Feynman propagators: why and how of time-ordering; defining the propagator; relation to D(x-y); scalar propagator is a Green's function of the Kelin–Gordon equation.
September 13 (Tuesday):
Feynman propagator and other Green's functions: Green's function in momentum space; regulating the integral over the poles; Feynman's choice; other types of Green's functions; Feynman propagators for vector, spinor, etc., fields.
Overview of symmetries of field theories: symmetries of the action; continuous and discrete symmetries; internal and spacetime symmetries; global and local symmetries; started continuous groups and their generators.
September 15 (Thursday):
Intro to Lie groups and Lie algebras: SO(3) example: generators and generator algebra; multiple commutator formula and finite rotations; representations of the algebra and of the group; general Lie groups and Lie algebras of their generators; representations of general groups and algebras.
Noether theorem: Global continuous symmetries and conserved currents; generators and currents for the SO(N) example; symmetry charges in the quantum theory; representation of the SO(N) symmetry in the Fock space; the phase symmetry and the net number of particles minus antiparticles; proof of the theorem; examples of Noether currents; translation symmetry and the stress-energy tensor; symmetrizing the Noether stress-energy tensor for non-scalar fields.
September 20 (Tuesday):
Local phase symmetry: local symmetry and covariant derivatives; gauge field and gauge transforms; algebra of covariant derivatives; coupling charged scalar fields to electromagnetism.
Covariant Schroedinger equation.
Aharonov–Bohm effect: Aharonov–Bohm effect; cohomology of the vector potential; charge quantization and the compactness of the U(1) phase symmetry.
September 22 (Thursday):
Magnetic monopoles: Heuristic picture; Dirac construction; charge quantization; gauge bundles; angular momentum in presence of a monopole; spin-statistics theorem for dyons.
Non-abelian local symmetries: Covariant derivatives and matrix-valued connections; non-abelian gauge transforms; Gell-Mann matrices and the component gauge fields; infinitesimal gauge transforms in components.
The extra lecture on September 23 (Friday):
Vortices and other kinds of topological defects: domain walls as topological defects; co-dimension; rotation and vorices in superfluid; vortex energy; vortex rings and the critical velocity of the superflow; vortices in superconductors and the magnetic flux they carry; cosmic strings; vortices as topological defects of co-dimension=2; magnetic monopoles as topological defects (co-dimension=3); (briefly) Yang–Mills instantons.
September 27 (Tuesday):
Non-abelian local symmetries: non-abelian tension fields; gauge transforms of the tension fields; the adjoint multiplet; Yang–Mills theory and normalization of the gauge fields; gauge theories with matter.
Group theory: simple and semi-simple Lie groups; compact and non-compact groups; the adjoint multiplet and the Killing form; multiplets and representations. General gauge theories: symmetry groups and multiplets of fields; general local symmetry groups and Lie-algebra-valued gauge fields; covariant derivatives for different multiplets types; the adjoint multiplet; multiple gauge groups; Standard Model example.
September 29 (Thursday):
General gauge theories: symmetry groups and multiplets of fields; general local symmetry groups and Lie-algebra-valued gauge fields; covariant derivatives for different multiplets types; the adjoint multiplet; multiple gauge groups; Standard Model example.
Lorentz symmetry: generators and representations; unitary but infinite particle representations; little groups and Wigner theorem; massive particles have definite spins; massless particles have definite helicities.
October 4 (Tuesday):
Tachyons: tachyons in QM; Wigner theorem for the tachyons; tachyon field and vacuum instability; interactions and scalar VEVs (vacuum expectation values).
More Lorentz symmetry: Wigner theorem in d≠4 dimensions; Lorentz multiplets of fields; (j+,j) multiplets; Weyl spinors and Spin(3,1)=SL(2,C); vectors and bispinors; tensors.
Started Dirac spinors and spinor fields.
October 6 (Thursday):
Dirac spinors and spinor fields: Dirac spinor representation of the Lorentz symmetry; Dirac equation and its covariance; Dirac conjugation and Dirac Lagrangian; Hamiltonian for the quantum Dirac field.
Grassmann numbers and classical limits of fermionic fields.
Started Fermionic algebra and Fock space: Hilbert stace of one fermionic mode; multiple modes.
The extra lecture on October 7 (Friday):
Conformal symmetry: definition; complex language in Euclidean 2D; conformal symmetry group and its generators; conformal algebra in d>2 dimensions, Euclidean or Minkowski.
Conformal field theories and their application: world-sheet QFT in string theory; condenced matter at a critical point; conformal window of QCD; AdS/CFT duality.
October 11 (Tuesday):
Fermionic Fock space: fermionic fock space; wave functions and operators; particles and holes; holes as quasiparticles; Fermi sea, extra particles 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); neutral particles and C–parity; Majorana fermions.
October 13 (Thursday):
Dirac, Majorana, and Weyl fermions: counting degrees of freedom; relations between Majorana and Weyl fermions; Majorana mass term; massless and massive neutrinos.
Parity and other discrete symmetries: parity; CP; time reversal (briefly); CPT theorem; baryogenesys and Sakharov's criteria.
Begin vector, axial, and chiral symmetries.
The extra lecture on October 14 (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).
October 18 (Tuesday):
Chiral symmetry: vector, axial, and chiral symmetries for Weyl fermions; U(N)L×U(N)R chiral symmetry; chiral gauge theories; electroweak example; chiral symmetry in QCD; Majorana and Dirac mass terms in chiral theories.
Relativistic causality for the fermions: commuting and anticommuting fields; checking anticommutativity of free Dirac fields at spacelike separation; spin-statistics theorem.
Feynman propagator for Dirac fermions.
Gave out the midterm exam.
October 20 (Thursday):
Perturbation theory in QFT and Feynman diagrams: the interaction picture of QM, the Dyson series, and the time-ordering; the 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 25 (Tuesday):
Golden Rule and the phase-space factors: Fermi's Golden Rule for transitions — derivation and an example; going beyond the first order; decay rate — the matrix element and the phase space factors; scattering of relativistic particles; calculating the phase space factors for the 2→2 scattering and 1→2 decays.
Summary of Feynman rules for the λΦ4 theory.
Phase-space factors.
Loop counting in perturbation theory: λΦ4 theory; adding a cubic coupling.
October 27 (Thursday):
Feynman rules for multiple scalar fields.
Mandelstam's s, t, u variables.
Dimensional analysis: dimensions of fields and couplings; trouble with δ<0 couplings; types of δ≥0 couplings in 4D; other dimensions.
Intro to Quantum Electro Dynamics (QED): quantizing the EM fields, need to fix a gauge.
Likbez lecture on October 28 (Friday):
Review of potential scattering in Quantum Mechanics: Scattering wave functions; Lippmann–Schwinger series; Born approximation; partial wave analysis.
November 1 (Tuesday):
Intro to Quantum Electro Dynamics (QED): photon propagator in the Coulomb gauge; other gauges.
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): Yukawa theory and its Feynman rules; non-relativistic fermion scattering; Yukawa potential.
Started Muon pair production in electron-positron collisions, e+e+→μ+: the tree amplitude; referral to homework#9 for the polarized cross-sections; intro to the unpolarized amplitudes.
November 3 (Thursday):
Dirac trace technology: Dirac traces for summing or averaging |M|2 over spins; techniques for calculating the Dirac traces.
Muon pair production in electron-positron collisions, e+e+→μ+: calculating the traces; summing over the Lorentz indices; the partial cross-section and its angular dependence; the total cross-section and its energy dependence.
Hadronic production e+e+→hadrons: quark pair production and jets; the R ratio; QCD corrections.
Quick review of traceology in homework#9.
November 8 (Tuesday):
crossing symmetry: muon pair production vs. electron-muon scattering; comparing spin-summed |M|2; comparing the ampitudes in the ultra-relativistic regime; analytically continuing the amplitudes; crossing symmetry in general; signs for crossed fermions; Compton vs. annihilation example.
Ward Identities: Ward identities for photons and gauge invariance of the amplitudes; sums over photon polarizations.
Electron-positron annihilation e+e+→γγ: tree diagrams and the amplitude; checking the Ward identities; summing over the photons' polarizations and averaging over the fermions' spins; Dirac traceology; summary and annihilation kinematics; annihilation cross-section; crossing relation to the Compton scattering.
November 10 (Thursday):
Compton scattering (briefly).
Spontaneous symmetry breaking: symmetric Lagrangian/Hamiltonian but asymmetric vacuum; continuous families of degenerate vacua; massless particles; linear sigma model; Wigner and Goldstone modes of symmetries; Goldstone–Nambu theorem and Goldstone bosons; spontaneous breakdown of approximate symmetries.
Extra lecture on November 11 (Friday):
Resonances and unstable particles: Breit–Wigner resonances in QM; resonances in QFT and unstable particles; making a resonance in a collision; cross-sections and branching ratios; quarkonia as resonaces in electron-positron collisions; calculating σ(e+e+→J/ψ).
November 15 (Tuesday):
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 mechanism: SU(2) with a doublet; SU(2) with a real triplet; general case.
Glashow–Weinberg–Salam theory: the bosonic fields and the Higgs mechanism; the unbroken electric charge Q=T3+Y; masses of the vector fields and the Weinberg's mixing angle; charged and neutral currents; Fermi's effective theory of weak interactions.
November 17 (Thursday):
Glashow–Weinberg–Salam theory: charged and neutral weak currents; Fermi's effective theory of weak interactions.
Fermions of the Glashow–Weinberg–Salam theory: Higgs origin of quark and lepton masses; charged and neutral weak currents of quarks and leptons.
Introduction to CKM matrix: Cabibbo mixing and Kaon decays; GIM mechanism and the charm quark; third family and the CKM matrix; the charged currents and the flavor-changing weak decays.
Extra lecture on November 18 (Friday):
SSB of QCD's chiral symmetry and sigma models: Chiral symmetry of QCD and its spontaneous breakdown (χSB); pions as pseudo–Goldstone bosons; the linear sigma model of χSB; the non-linear sigma model; maybe general NLΣMs.
November 14–25 (whole week):
Fall break, no classes.
November 29 (Tuesday):
Quick review of quarks' and leptons' masses.
Origin of the CKM matrix: unitary charges of bases for each type of a fermion multiplet; 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; neutral and charged weak currents.
Neutral Kaons: GIM box and K0K0 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.
Introduction to CP violation: CPV in neutral kaon decays to pions; CPV in eigenstates and in decay rates; CPV in semi-leptonic decays.
December 1 (Thursday):
CKM origin of CP violation: 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 strong CP violation.
Neutrino masses: neutrino masses and oscillations; Dirac vs. Majorana masses of neutrinos; seesaw mechanism.
Give out the final exam.

QFT 2, Spring 2023 semester

January 10 and January 12 (first week):
January 17 (Tuesday):
Syllabus of the Spring semester (briefly).
Loop diagrams: amputating the external leg bubbles.
Calculating a one-loop amplitude: the diagrams; Feynman trick for denominators; Wick rotation to the Euclidean momentum space.
UV cutoff, long-distance effective field theories, and renormalization: UV divergence of the loop integral; effective long-distance / low-energy field theories; UV cutoff in condensed matter and in relativistic QFTs; bare and physical couplings; net one-loop amplitude in λΦ4 theory; bare and physical couplings at higher loop orders.
January 19 (Thursday):
Overview of UV regulators: Wilson's hard edge; Pauli–Villars; higher derivatives; covariant higher derivatives; lattice (very briefly).
Dimensional regularization: basics; momentum integrals in non-integral dimensions; d→4 limit; (1/ε) as log(ΛUV).
Quick introduction to partial wave analysis in QM and to optical theorem.
January 20 (Friday) [make-up lecture]:
Optical theorem: proof from unitarity of the S matrix; application to the Im M1 loop in the λφ4 theory; mentioned cutting diagrams and putting cut propagators on-shell (details in homework).
Started correlation functions of quantum fields: correlation functions; relation to the free fields and evolution operators.
January 24 (Tuesday):
Correlation functions of quantum fields: 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.
Perturbation theory for the two-point function: resumming the 1PI bubbles; Σ(p2) and the renormalization of the mass and of the field strength; mass renormalization in the λφ4 theory; fine tuning problem.
January 26 (Thursday):
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; optical theorem for the unstable particles.
Started the Lehmann–Symanzik–Zimmermann reduction formula: n-point correlators Fn(p1,…pn) and their poles at on-shell momenta; LSZ reduction formula for the common residue; the Feynman diagram explanation of the poles; the amputated diagrams and the scattering amplitudes.
January 27 (Friday) [make-up lecture]:
Deriving the LSZ reduction formula: The x0i→±∞ limits; in the coordinate space leading to the on-shell poles in the momentum space; residues and matrix elements of fewer fields; multiple poles and asymptotic |in⟩ and ⟨out| states; the physical S-matrix elements ⟨out|S|in⟩.
Started the counterterm perturbation theory: ℒbare=ℒphysical+counterterms; Feynman rules for the counterterms; adjusting δZ, δm, and δλ order by order in λ; one-loop examples; finite parts of the counterterms.
Regular lecture on January 31 (Tuesday):
Canceled due to bad weather.
Extra lecture on January 31 (Tuesday) [instead of the canceled regular 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; formal derivation of the Feynman diagrams.
February 2 (Thursday):
Counterterms and canceling the UV divergences: superficial degree of divergence; graphs and subgraphs; classifying divergent graphs, subgraphs, and amplitudes; canceling overall divergences; subgraph divergences and their cancelation in situ; nested and overlapping divergences; BPHZ theorem.
Divergences and renormalizability: divergences of φn theories; super-renormalizable φ3 theory; renormalizable φ4 theory.
February 3 (Friday) [make-up lecture]:
Divergences and renormalizability: non-renormalizable φn>4 theories; general super-renormalizable, renormalizable, and non-renormalizable theories; trouble with non-renormalizability.
Dimensional analysis and renormalizability: canonical dimensions of fields and couplings; power-counting renormalizability; renormalizable theories in 4D; other dimensions.
February 7 (Tuesday):
QED perturbation theory: the counterterms and the Feynman rules; divergent amplitudes and their momentum dependences; missing counterterms and Ward–Takahashi identities; dressed electron propagator; dressed photon propagator; finite parts of the counterterms.
Electric charge renormalization: calculating the 1-loop Σμν(k): the trace, the denominator, and the numerator; the good, the bad, and the odd; checking the Ward–Takahashi identity; the momentum integral for the Π(k2); the divergence and the δ3counterterm; the finite result for the one-loop-order Π(k2).
February 9 (Thursday):
Electric charge renormalization: loop corrections to Coulomb scattering and other high-momentum processes; effective QED coupling αeff(E) and its running with log(energy).
Ward–Takahashi identities: the identities; current conservation in quantum theories; contact terms; formal proof of WT indentities; Z1=Z2; multiple charged fields.
February 10 (Friday) [make-up lecture]:
Diagramatic proof of Ward–Takahashi identities: WTI for tree-level 2-electron amplitudes; WTI for one-loop photonic amplitudes; WTIs for multi-loop diagrams (in the bare perturbation theory); WTIs in the counterterm perturbation theory (outline).
February 14 (Tuesday):
Form factors: probing nuclear and nucleon structure with electrons; form factors; on-shell form-factors F1(q2) and F2(q2); the gyromagnetic ratio.
Dressed QED vertex at one loop: the dressed vertex and the form factors; the one-loop diagram and its denominator; 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.
February 16 (Thursday):
The electric from factor and the infrared divergence: momentum integral for the F1(q2); integral over Feynman parameters diverges for D≥4; the infrared divergence; tiny photon's mass as the IR regulator; (re)calculating the Feynman parameter integral; the δ1 counterterm; momentum dependence of the IR divergence; Sudakov's double logarithms.
Virtual and real soft photons: IR divergence of exclusive cross-sections due to virtual soft photons: IR divergence of the soft-photon bremmsstrahlung; finite inclusive cross-sections (with or without soft photons); detectable vs. undetectable photons, the observed cross-sections, and their finiteness; briefly: higher loops and/or more soft photons.
Bremsstrahlung: classical bremsstrahlung; flat frequency spectrum; implication for the quantum theory and fo r the IR divergences of QED; soft-photon bremsstrahlung in QED (tree-level); verifying I(p,p′)=2fIR(q2).
February 17 (Friday) [make-up lecture]:
Optical theorem for the soft photons: optical theorem for the μμ+→ee+ pair production; 3 two-loop diagrams for adding a photon, no net IR divergence; Cutkosky cuts; virtual and real soft photons; explaining cancelation of IR divergences.
Implications of the infrared divergence: Ill-defined Fock space in QED and other gauge theories; soft and collinear gluons in QCD; jets in theory and in experiment.
Gauge dependence in QED: gauge-dependent off-shell amplitudes and counterterms; δ1(ξ)=δ2(ξ).
February 21 (Tuesday):
Intro to renormalization group: large log problem for E≫m; running coupling λ(E); off-shell renormalization schemes for couplings and counterterms.
Renormalization group basics: anomalous dimensions of quantum fields; calculating γφ in the λφ4 theory; higher-loop corrections to the γφ; running couplings and β-functions; relating β-functions to the counterterms; β(λ) in the λφ4 theory; solving the renormalization group equation for the λ(E); anomalous dimensions and the β-function in QED; solving the renormalization group equation for QED.
February 23 (Thursday):
Renormalization group with multiple couplings: general formulae for the β-functions; Yukawa theory example.
Types of RG flows: β>0, Landau poles, and UV incompleteness; β<0, QCD example, and asymptotic freedom; ΛQCD; non-perturbative strong interactions at low energies.
Chromomagnetic monopole condensation and quark confinement.
Extra lecture on February 24 (Friday):
Grand Unification: unifying the EM, weak, and strong interactions in a single non-abelian gauge group; SU(5) example; multiplets of fermions; gauge couplings and Georgi–Quinn–Weinberg equations; the doublet-triplet problem; baryon decay and other exotic processes.
February 28 (Tuesday):
Fixed points β(g*)=0 of RG flows: scale invariance and conformal symmetry; UV stability vs. IR stability; Banks–Zaks conformal window of QCD.
RG flows in spaces of multiple couplings: Yukawa example; RG flows in the coupling space; fixed points and attractive lines.
Direction of the flow: IR to UV or UV to IR?
March 2 (Thursday):
Relevant, irrelevant, and marginal operators; effective field theories.
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.
Gave out the midterm exam.
March 7 (Tuesday):
Introduction to path integrals: path integrals in QM; the Lagrangian and the Hamiltonian forms of the path integral; derivation of the Hamiltonian form; the Lagrangian path integrals — derivation and normalization; the partition function; harmonic oscillator example.
Functional integrals in QFT: “path” integrals for quantum fields; correlation functions; briefly free fields and propagators.
March 9 (Thursday):
Functional integrals in QFT: free fields and propagators; perturbation theory and Feynman rules; sources and generating functionals; log Z[J] generates the connected correlation functions.
Euclidean path integrals: convergence problems of path integrals; Euclidean time; discretization; harmonic oscillator example.
QFT and StatMech: Functional integrals in Euclidean spacetime; QFT↔StatMech analogy; coupling as temperature.
March 11–19 (whole week):
Spring break, no classes.
March 21 (Tuesday):
QFT on a lattice: QFT↔StatMech analogy; discrete lattice as a UV cutoff; recovering rotational / Lorentz symmetry in the continuum limit; custodial symmetries.
Fermionic functional integrals: Grassmann numbers; Berezin integrals; Gaussian integrals over fermionic variables; functional integrals over fermionic fields; free Dirac field in Euclidean spacetime.
Integrating over fermion fields in QED. functional integral in EM background: the determinant, and the source term;
March 23 (Thursday):
Integrating over fermion fields in QED: functional integral in EM background: the determinant, and the source term; 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, gauge-fixing terms, and the Feynman gauge.
Quantizing the Yang–Mills theory: fixing the non-abelian gauge symmetry; Faddeev–Popov ghost fields; gauge-fixed YM Lagrangian.
QCD Feynman rules: physical, ghost, gauge-fixing, and counter terms in the Lagrangian; propagators; physical vertices; counter-term vertices; handling the color indices of quarks.
Extra lecture on March 24 (Friday):
Gauge theories on the lattice (abelian): local U(1) symmetry on the lattice; gauge fields and link variables; covariant lattice derivatives; plaquettes and tension fields; lattice EM action; lattice ‘path’ integrals; compact QED.
Non-abelian lattice gauge theories: non-abelian gauge symmetries and link variables; covariant lattice symmetries; non-abelian plaquettes and tension fields; lattice YM action; integrals over link variables and the lattice ‘path’ integrals; brief history and applications of lattice QCD.
March 28 (Tuesday):
QCD Ward identities: on-shell QCD Ward identities are weaker than in QED; q+q̄→g+g example: 3 tree diagrams and kμMμν(1+2); the first 2 diagrams; the third diagram; Ward identity holds for one longitudinal quark only; production of longitudinal quarks is canceled by the ghost-antighost pair production.
Introduction to BRST symmetry: BRST transforms of QCD fields; nilpotency; BRST invariance of the net Lagrangian; physical and unphysical quanta in the QCD Fock space and BRST cohomology; reducing 1-particle states to physical particles; multi-particle physical states and the S-matrix; BRST symmetries of the amplitudes and cancellation of unphysical processes.
QCD renormalizability: modern approach; manifest symmetries; the bare Lagrangian; Slavnov–Taylor identities for the QCD counterterms.
March 30 (Thursday):
QCD renormalizability: modern approach; manifest symmetries; the bare Lagrangian; Slavnov–Taylor identities for the QCD counterterms.
QCD beta-function: relation to the counterterms; calculating the one-loop δ2 for the quarks; calculating the one-loop δ1 for the quarks: the QED-like loop and the non-abelian loop; calculating the one-loop δ3 for the gluons: the quark loop, the gluon loop, the sideways gluon loop, the ghost loop, the summary; β(QCD).
Extra lecture on March 31 (Friday):
Wilson loops: Abelian and non-abelian Wilson loops; large loops and forces between probe particles; non-abelian probe particles; area law vs. perimeter law as test of confinement vs. deconfinement; examples.
April 4 (Tuesday):
Beta-function of general gauge theories: β(QCD); generalizing to other gauge theories.
Axial anomaly: axial symmetry of massless electrons; anomaly and its origin in the path integral measure; the diagrams, the naive cancelation, and the regulation problem; Adler–Bardeen theorem; Pauli–Villars regulation of the anomaly; calculating the loop of the PV compensator; net axial anomaly.
April 6 (Thursday):
Axial anomaly: anomaly of the measure of the fermionic functional integral; generalization to the non-abelian gauge theories; triangle and quadrangle anomalies in QCD; in QCD U(1)A is anomalous but SU(Nf)L×SU(Nf)R are anomaly-free; spontaneous axial symmetry breaking and the issue with the η and π mesons.
Non-linear sigma models: non-linear field spaces; NLΣM of the chiral symmetry breaking; vector and axial currents.
Extra lecture on April 7 (Friday):
Instantons: topological sectors and non-perturbative effects; topological sectors in the Yang–Mills theories; topological index I[Aμ] and its quantization; SE≥(8π2/g2)×∣I∣ and the topological sectors in the YM path integral; 't Hooft instantons and tunneling events; multiple instantons, cluster expansion, and the Θ angle.
April 11 (Tuesday):
Fπ and the decay of a charged pion.
Non-linear sigma models in QCD context: quark-antiquark condensation and the spontaneous chiral symmetry breaking; quark masses as perturbations; 2-flavor and 3-flavor models; axial anomaly as a perturbation, η and η′ mesons.
Electromagnetic anomalies: electromagnetic anomalies of quarks' symmetries; anomalous decays of the neytral pion π0→γγ.
Anomalies in chiral gauge theories: Weyl fermions and chiral currents; non-abelian chiral gauge theories; triangle and quadrangle diagrams for Weyl fermions; trace formula for the net chiral anomaly; QCD and SU(2)W examples; SU(2)W anomalies of baryon and lepton numbers
April 13 (Thursday):
Anomalies of baryon and lepton numbers: ΔB=ΔL=3×Index; instantons and sphalerons; leptogenesis.
Gauge anomalies: triangle anomaly in chiral QED and its effect on Ward identities; anomalous gauge variance of log(det(̸D)); anomaly in non-abelian chiral theories; triangle diagrams; quadrangle diagrams; net non-abelian anomaly; anomaly coefficients Aabc and traces over chiral fermions.
Anomaly cancellation in chiral gauge theories: anomaly in simple gauge groups; cubic Casimir and Cibic index; counting amonaly indices; SU(5) GUT example; anomalies in product gauge groups; 5 anomaly types in the Standard Model; checking anomaly cancellation in the Standard Model; importance of tr(Qel)=0.
Extra lecture on April 14 (Friday):
Instantons and fermions: instantons and axial anomaly; zero modes in instanton background; Atyah–Singer index theorem; zero modes in fermionic integrals chiral anomaly of the Θ angle; Θ=Θ+phase(det(quark mass matrix)); the strong CP problem; neutron's electric dipole; Peccei–Quinn symmetry.
April 18 (Tuesday):
Anomaly cancellation: checking anomaly cancellation in the Standard Model; importance of tr(Qel)=0; abelian and non-abelian anomalies; massive fermions do not contribute to the anomalies.
Quick overview of differential forms: forms and antisymmetric tensor fields; exterior derivative; closed and exact forms; gauge fields A and F as forms; nonabelian gauge fields as forms.
Differential forms for the anomalies: anomaly forms; Chern–Simons forms and their uses; descent equations; d+2 anomaly forms for the nonabelian anomalies in d dimensions.
Symmetric traces of degree (d+2)/2 for the anomalies; rules for anomaly cancellation in 2D; harder-to-satisfy rules for 6D and 10D.
April 20 (Thursday):
Wess–Zumino terms: flavor anomalies of the NLΣM; WZ term from 4D and 5D points of view; adding flavor gauge fields; flavor anomaly cancellation. 't Hooft's anomaly matching conditions: confiniement and χSB in chiral gauge theories; SU(5) example; surviving chiral symmetry and massless composite fermions; flavor anomaly matching.
Gave out the final exam.

Last Modified: April 22, 2023.
Vadim Kaplunovsky