Quantum Field Theory: Lecture Log

Navigation: Fall, Spring, Last regular lecture.

QFT 1, Fall 2020 semester

The Zoom sessions for all the regular lectures are at https://utexas.zoom.us/j/94371979747.
The sessions for the extra lectures are at a different URL, namely https://utexas.zoom.us/j/97782926154.

August 27 (Thursday):
Syllabus and admin: course content, textbooks, prerequisites, homework, exams and grades, etc.
General introduction: reasons for QFT; field-particle duality.
August 28 (Friday):
Review of classical mechanics: Lagrangian and action; least action principle; Euler–Lagrange equation; multiple dynamical variables; counting degrees of freedom.
Intro 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.
September 1 (Tuesday):
Non-relativistic and relativistic fields: Higher space derivatives and non-local Lagrangians for non-relativistic fields; relativistic sign conventions; Einstein summation convention; relativistic ℒ and field equations; Klein–Gordon example; multiple scalar fields.
Relativistic 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 3 (Thursday):
Finished Relativistic electromagnetic fields: 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.
September 4 (Friday):
Finished Review of canonical quantization: Poisson brackets and commutator brackets.
Introduction to quantum fields: Hamiltonian formalism for classical fields; quantum fields; equal-time commutation relations; quantum Klein–Gordon equation.
September 8 (Tuesday):
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.
September 10 (Thursday):
General identical bosons: Bosonic Fock space and its occupation number basis; creation and annihilation operators; wave-function language vs. Fock-space language; one-body operators; two-body operators; non-relativistic quantum fields; “second quantization”.
Regular lecture on September 11 (Friday):
Relativistic normalization of states and operators: Lorentz groups; momentum space geometry and Lorentz-invariant measure; relativistic normalization of states and operators.
Extra lecture on September 11 (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; BEC example (outline only).
September 15 (Tuesday):
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.
Intro to relativistic causality: superluminal particles in `relativistic' QM; signals in QM and in QFT; relativistic causality in QFT.
September 17 (Thursday):
Relativistic causality: local operator and fields; proof for free scalar fields; going forward and backward in time; causality for interacting fields.
Begin Feynman propagator for the scalar field: why and how of time-ordering; defining the propagator; relation to D(x-y).
September 18 (Friday):
Feynman propagator for the scalar field: 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 22 (Tuesday):
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.
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.
September 24 (Thursday):
Noether theorem: 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.
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 25 (Friday):
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.
Extra lecture on September 25 (Friday):
Bose–Einstein condensate and superfluidity: 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’; fluctuation spectrum in a moving condensate and superfluidity.
September 29 (Tuesday):
Magnetic monopoles: Heuristic picture; Dirac construction; charge quantization; gauge bundles.
Non-abelian local symmetries: Covariant derivatives and matrix-valued connections; non-abelian gauge transforms; Gell-Mann matrices and the component gauge fields.
October 1 (Thursday):
Non-abelian local symmetries: infinitesimal gauge transforms in components; non-abelian tensions fields; gauge transforms of the tension fields; the adjoint multiplet; Yang–Mills theory and normalization of the gauge fields; gauge theories with matter.
Overview of group theory: Lie groups, Lie algebras, representaions, multiplets.
October 2 (Friday):
Gauge symmetries: symmetry groups and multiplets of fields; general local symmetry groups and Lie-algebra-valued gauge fields; covariant derivatives for different multiplets types; multiple gauge groups; Standard Model example.
October 6 (Tuesday):
Finish gauge symmetries: classification of allowed gauge groups.
Lorentz symmetry: generators and multiplet types; unitary but infinite particle representations; little groups and Wigner theorem; massive particles have definite spins; massless particles have definite helicities; tachyons have nothing; maybe Wigner theorem in d≠4 dimensions.
October 8 (Thursday):
Tachyons: tachyons in QM; Wigner theorem for the tachyons; tachyon field and vacuum instability; interactions and scalar VEVs (vacuum expectation values).
Lorentz symmetry: finish Wigner theorem; Lorentz multiplets of fields; (j+,j) multiplets; Weyl spinors and Spin(3,1)≅SL(2,C); vectors and bispinors; tensors.
Regular lecture on October 9 (Friday):
Finish Lorentz multiplets of fields: vectors and bispinors; tensors.
Dirac spinors and spinor fields: Lorentz spinor multiplet; Dirac equation.
Extra lecture on October 9 (Friday):
Vortices: rotation and vortices in a superfluid; vortex energy; vortex in a superconductor; magnetic flux; cosmic strings.
Other types of topological defects: domain walls, monopoles, YM instantons; codimension.
October 13 (Tuesday):
Dirac spinor fields: covariance of the Dirac equation; Dirac conjugation; Dirac Lagrangian; Hamiltonian for the quantum Dirac field.
Grassmann numbers and classical limits of fermionic fields.
October 15 (Thursday):
Fermionic algebra and Fock space: Hilbert stace of one fermionic mode; multiple modes; Fermionic fock space; wave functions and operators.
Fermionic particles and holes: particles and holes; holes as quasiparticles; Fermi sea, extra particles and holes.
October 16 (Friday):
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 20 (Tuesday):
Charge conjugation symmetry: neutral particles and C-parity; Majorana fermions.
Dirac, Majorana, and Weyl fermions: counting degrees of freedom; relations between Majorana and Weyl fermions; Majorana mass term; massless and massive neutrinos.
October 22 (Thursday):
Parity and other discrete symmetries: G-parity (briefly); parity; CP; time reversal (briefly); CPT theorem; baryogenesys and Sakharov's criteria.
Chiral symmetry: vector, axial, and chiral symmetries; the U(N)L×U(N)R chiral symmetry; chiral symmetry in QCD; chiral gauge theories.
Regular lecture on October 23 (Friday):
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.
Extra lecture on October 23 (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 27 (Tuesday):
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.
Introduction to perturbation theory: interaction picture of QM; the Dyson series and the time-ordering.
Gave out the midterm exam.
October 29 (Thursday):
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 30 (Friday):
Perturbation theory in QFT and Feynman rules: momentum space Feynman rules; momentum conservation and connected diagrams; scattering amplidudes; summary of Feynman rules for the λΦ4 theory.
Phase space factors.
November 3 (Tuesday):
Loop counting: loop counting for the λΦ4 theory; adding cubic couplings; Mandelstam's s, t, and u; multiple fields.
November 5 (Thursday):
Dimensional analysis: dimensions of fields and couplings; trouble with δ<<0 couplings; types of Δ≥0 couplings in 4D; other dimensions.
Began Intro to Quantum Electro Dynamics (QED): quantizing EM fields; photon propagator in the Coulomb gauge..
Regular lecture on November 6 (Friday):
Finish Intro to QED: photon propagator in various gauges.
QED Feynman rules: propagators and vertices; external line factors; Dirac indexology; Gordon identities; sign rules.
Extra lecture on November 6 (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.
November 10 (Tuesday):
Coulomb scattering in QED: diagrams and amplitudes; non-relativistic limit; recovering the Coulomb potential; electron-electron vs. electron-positron Coulomb scattering.
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 muon pair production.
November 12 (Thursday):
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; analytically continuing the amplitudes; crossing symmetry in general; signs for crossed fermions; Compton vs. annihilation example.
November 13 (Friday):
Ward Identities: Ward identities for photons and gauge invariance of the amplitudes; sums over photon polarizations.
Electron-positron annihilation e^+e+→γ: tree diagrams and the amplitude; checking the Ward identities; summing over photon polarizations and averaging over fermions' spins.
November 17 (Tuesday):
Electron-positron annihilation: Dirac traceology; summary and annihilation kinematics; annihilation cross-section; crossing relation to Compton scattering.
Spontaneous symmetry breaking: symmetric Lagrangian/Hamiltonian but asymmetric vacuum; continuous families of degenerate vacua; massless particles; linear sigma model.
November 19 (Thursday):
Spontaneous symmetry breaking: Wigner and Goldstone modes of symmetries; Goldstone theorem.
The Higgs Mechanism: SSB of a local U(1) symmetry; massive photon ‘eats&rdsqo; the would-be Goldstone boson; unitary gauge vs. gauge-invariant description.
Regular lecture on November 20 (Friday):
Non-Abelian Higgs Mechanism: SU(2) with a doublet; SU(2) with a real triplet; general case.
Extra lecture on November 20 (Friday):
SSB of QCD's chiral symmetry and sigma models: Chiral symmetry of QCD and its spontaneous breakdown (χSB); pions as pseudo–Goldstone bosons; linear sigma model of χSB non-linear sigma model; maybe general NLΣMs.
November 24 (Tuesday):
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 currents; Fermi's effective theory of weak interactions.
Fermion masses arising from scalar VEVs.
December 1 (Tuesday):
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; flavor-changing weak decays.
Began 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.
December 3 (Thursday):
Finished the CKM matrix: bases for the charged leptons and for the neutrinos; neutral weak currents: diagonal in the Standard Model, but non-diagonal (flavor-changing) in other models.
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.
Introduction to CP violation: 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.
December 4 (Friday):
Neutrino masses: neutrino masses and oscillations; Dirac vs. Majorana masses of neutrinos; seesaw mechanism.
Gave out the final exam.

QFT 2, Spring 2021 semester

The Zoom sessions for all lectures on Tuesdays, Thursdays, and Fridays are at https://utexas.zoom.us/j/97545121357.
The sessions for the Wednesday lectures — extra or make-up — are at a different URL, namely https://utexas.zoom.us/j/99561322180.

January 19 (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; UV divergence.
January 21 (Thursday):
UV cutoff, long-distance effective field theories, and renormalization: long-distance (low-energy) EFT and its independence on short-distance (high energy) details; cutting off the UV momenta; one-loop amplitude; bare and physical couplings; changing bare coupling to compensate for changing the UV cutoff; perturbative expansion in powers of the physical coupling.
January 22 (Friday):
Overview of UV regulators: Wilson's hard edge; Pauli–Villars; higher derivatives.
January 26 (Tuesday):
Finished overview of UV regulators: covariant higher derivatives; lattice (very briefly).
Dimensional regularization: basics; momentum integrals in non-integral dimensions; d→4 limit; (1/ε) as log(ΛUV).
Optical theorem: proof from unitarity of the S matrix.
Extra lecture on January 27 (Wednessday):
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).
January 28 (Thursday):
Optical theorem: application to Im M1 loop in λφ4 theory; mentioned cutting diagrams and putting cut propagators on-shell (details in homework).
Correlation functions of quantum fields: correlation functions; relation to the free fields and evolution operators; Feynman rules; connected correlation functions.
January 29 (Friday):
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. Began Perturbation theory for the two-point function: resumming the 1PI bubbles
February 2 (Tuesday):
Perturbation theory for the two-point function: Σ(p2) and the renormalization of the mass and of the field strength; mass renormalization in the λφ4 theory; fine tuning problem.
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).
February 4 (Thursday):
Finish field strength renormalization in the Yukawa theory: 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.
Began counting the divergences: superficial degree of divergence; graphs and subgraphs.
February 5 (Friday):
Counterterms and canceling the divergences: classifying divergent graphs, subgraphs, and amplitudes; canceling overall divergences; subgraph divergences and their cancelation in situ.
February 9 (Tuesday):
Finish divergence cancellation for λφ4: nested and overlapping divergences; BPHZ theorem.
Divergences and renormalizability: supeficial degree of divergence in the φk theories; super-renormalizable φ3 theory; 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.
Extra lecture on February 10 (Wednesday):
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.
February 11 (Thursday):
QED perturbation theory: the counterterms and the Feynman rules; divergent amplitudes and their momentum dependences; missing counterterms and Ward–Takahashi identities.
Dressed electron propagator.
February 12 (Friday):
Dressed photon propagator.
Σμν(k) at one loop order: calculation; checking the WT identity; the divergence and the δ3 counterterm.
No lectures on February 16, 18, and 19 (whole week):
Cancelled due to bad weather.
Extra lecture on February 23 (Tuesday):
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.
Make-up regular lecture on February 24 (Wednesday):
Electric charge renormalization: finish calculation of Π1 loop(k2): the momentum integral, the δ3 counterterm, and the final result; loop corrections to Coulomb scattering and other high-momentum processes; effective QED coupling αeff(E) and its running with log(energy).
February 25 (Thursday):
Ward–Takahashi identities: the identities; current conservation in quantum theories; contact terms; formal proof of WT indentities; Z1=Z2.
February 26 (Friday):
Form factors: probing nuclear and nucleon structure with electrons; the form factors; on-shell form-factors F1(q2) and F2(q2); the gyromagnetic ratio.
Begin the dressed QED vertex at one loop: the diagram and the denominator.
March 2 (Tuesday):
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: momentum integral; integral over Feynman parameters diverges; hints of IR divergence.
Make-up regular lecture on March 3 (Wednesday):
Infrared divergence in QED: IR divergence of the one-loop vertex correction; regulating the IR divergence with photon mass; calculating the regulated F1(q2); δ1 counterterm; momentum dependence of the IR divergence; Sudakov's douboe logarithms.
Begin Virtual and real soft photons: IR divergence of exclusive cross-sections due to virtual soft photons. soft-photon bremmsstrahlung and its IR divergence; finite inclusive cross-section.
March 4 (Thursday):
Virtual and real soft photons: soft-photon bremmsstrahlung and its IR divergence; 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.
Consequences of infrared divergence: Ill-defined Fock space in QED and other gauge theories; soft and collinear gluons in QCD; jets in theory and in experiment.
March 5 (Friday):
Gauge dependence in QED: gauge-dependent off-shell amplitudes and counterterms; δ1(ξ)=δ2(ξ).
Symmetries and counterterms: counterterms in general renormalizable QFT's; naturally small and unnaturally small couplings; Yukawa and QED examples.
Began intro to renormalization group: large log problem for E≫m; resumming leading logs in terms of runnibg λ(E).
March 9 (Tuesday):
Intro to renormalization group: large logarithms and running coupling λ(E); off-shell renormalization schemes for couplings and counterterms.
Renormalization group basics: anomalous dimensions of quantum fields; running couplings and β functions.
Renormalization group equation for the λφ4 theory: solving the equation (in the one-loop approximation); no running below the mass threshold; boundary condition for the RGE and the threshold correction.
Make-up regular lecture on March 10 (Wednesday):
Renormalization group for QED: anomalous dimensions; βe to one-loop order; solving the RGE for QED; threshold correction and 2 loop correction.
Renormalization groups for general QFTs: β–functions for general couplings; Yukawa theory as an example; solving coupled RGEs.
March 11 (Thursday):
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.
March 12 (Friday):
Fixed points β(g*)=0 of RG flows: scale invariance and conformal symmetry; UV stability vs. IR stability; Banks–Zaks conformal window of QCD.
March 16–19:
Spring break, no classes.
March 23 (Tuesday):
RG flows for multiple couplings: Yukawa example; RG flows in the coupling space: fixed points and attractive lines.
Direction of UV flow: IR to UV or UV to IR?
Relevant, irrelevant, and marginal operators; effective field theories.
Make-up regular lecture on March 24 (Wednesday):
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 25 (Thursday):
Introduction to path integrals: path integrals in QM; the Lagrangian and the Hamiltonian forms of the path integral; derivation of the Hamiltonian form; Lagrangian path integrals — derivation and normalization; partition function; harmonic oscillator example.
Gave out the midterm exam.
March 26 (Friday):
Functional integrals in QFT: “path” integrals for quantum fields; correlation functions; free fields and propagators; perturbation theory and Feynman rules; sources and generating functionals.
March 30 (Tuesday):
Functional integrals in QFT: sources and generating functionals.
Euclidean path integrals: convergence problems of path integrals; Euclidean time; discretization; harmonic oscillator example.
April 1 (Thursday):
QFT and StatMech: Functional integrals in Euclidean spacetime; QFT↔StatMech analogy; coupling as temperature; QFT on a discrete lattice; lattice as a UV cutoff; recovering rotational / Lorentz symmetry in the continuum limit; custodial symmetries.
April 2 (Friday):
Fermions and Grassmann numbers: Grassmann numbers; Berezin integrals; Gaussian integrals over fermionic variables; functional integrals over fermionic fields; free Dirac field in Euclidean spacetime.
April 6 (Tuesday):
Integrating over fermion fields in QED: Dirac field in Euclidean spacetime; 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.
Extra lecture on April 7 (Wednesday):
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.
April 8 (Thursday):
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.
Started 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).
April 9 (Friday):
QCD Ward identities: q+q̄→g+g example: the third diagram; Ward identity holds for one longitudinal uark only; two longitudinal quarks are canceled by the ghost antighost pair.
Introduction to BRST symmetry: BRST transforms of QCD fields; nilpotency; BRST invariance of the net Lagrangian.
April 13 (Tuesday):
BRST symmetry: 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.
Started QCD renormalizability: renormalizability and the counterterm set; BRST and other manifest symmetries; allowed counterterms and renormalizability.
Extra lecture on April 14 (Wednesday):
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.
April 15 (Thursday):
QCD renormalizability: modern approach; manifest symmetries; the bare Lagrangian; Slavnov–Taylor identities for the QCD counterterms.
basic group theory — the Casimir and the index.
Renormalization of QCD: counterterms and the beta-function; calculating the one-loop δ2 for quarks; calculating the one-loop δ1 for quarks — the QED-like loop and the non-abelian loop (unfinished).
April 16 (Friday):
Renormalization of QCD — finish calculating one-loop δ1 for the quarks; calculate the one-loop δ3: the quark loop; the gluon loop; the sideways gluon loop; the ghost loop; summary.
April 20 (Tuesday):
Renormalization of gauge theories: finish calculating the one-loop δ3 counterterm; QCD beta function at one loop; generalizing to other gauge theories.
Introduction to axial anomaly: axial symmetry of massless electrons; anomaly and its origin in the path integral measure; would-be Ward identity and the hole in the argument.
Extra lecture on April 21 (Wednesday):
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.
April 22 (Thursday):
Axial Anomaly: the diagrams, the naive cancelation, and the regulation problem; Adler–Bardeen theorem; Pauli–Villars regulation of the anomaly.
April 23 (Friday):
Axial anomalies: Calculating the regulated triangle diagrams; axial anomaly in QCD.
April 27 (Tuesday):
More axial anomalies: anomaly of the measure of the fermionic functional integral; anomalies of generalized axial symmetries; η and π mesons in QCD.
Non-linear sigma models: non-linear field spaces; NLΣM of the chiral symmetry breaking. χSB in QCD context.
Extra lecture on April 28 (Wednesday):
Instantons: 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 29 (Thursday):
Non-linear sigma models of the chiral symmetry breaking: vector and axial currents; QCD context; weak decays of charged pions; quark masses as perturbations; η and η' mesons.
April 30 (Friday):
QED anomaly of the axial isospin and the neutral pion decay.
Chiral U(1) gauge theories: Weyl fermions and chiral currents; loops of Weyl fermions and the chiral anomaly; net anomalies of global symmetries.
May 4 (Tuesday):
Global symmetries of chiral gauge theories: trace formula for the anomaly; baryon and lepton number anomalies in the electroweak theory; instantons and sphalerons; baryogenesys by leptogenesys in early Universe; leptogenesys by our-of-equilibrium decays of sterile neutrinos.
Gauge anomalies: triangle anomaly in chiral QED and its effect on Ward identities.
Extra lecture on May 5 (Wednesday):
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.
May 6 (Thursday):
Gauge anomalies: 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.
Anomaly cancellation in chiral gauge theories: checking anomaly cancellation in the Standard Model; importance of tr(Qel)=0; in general gauge theories, massive fermions do not contribute to the anomaly.
May 7 (Friday):
Anomalies in general chiral gauge theories: cubic Casimirs and cubic anomaly indices for simple gauge groups; applications to Grand Unification; briefly anomalies in other dimensions.
Give out the final exam.

Last Modified: May 17, 2021.
Vadim Kaplunovsky