Quantum Field Theory: Lecture Log

This is the lecture log for the Quantum Field Theory classes PHY 396 K and PHY 396 L taught in 2024/25 by professor Vadim Kaplunovsky.

Most lectures should be video recorded and the records available on Canvas. For the few lectures that did not get recorded because of technical glitches, I shall scan the notes I have used in class and links the scans to this page.

Navigation: Fall, Spring, Last regular lecture.

QFT 1, Fall 2024 semester

August 26 (Monday):
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.
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.
August 28 (Wednesday):
Finish Intro to classical 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.
Relativistic electromagnetic fields: the 4–tensor Fμν=−Fνμ and the relativistic form of Maxwell equations; 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.
September 2 (Monday):
Labor Day holiday.
September 4 (Wednesday):
Introduction to quantum fields: Hamiltonian formalism for the classical fields; quantum fields; equal-time commutation relations; 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; Casimir effect (briefly).
Extra lecture on September 6 (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.
Bose–Einstent condensate: the naive BEC ground state v. the coherent state; classical Landau–Ginzburg theory and its ground state; quantum corrections and the fluctuation spectrum.
September 9 (Monday):
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”.
Groups of rotation and of Lorentz symmetries.
Started Normalization of states and operators: box and continuum normalizations; non-relativistic and relativistic normalization.
September 11 (Wednesday):
Relativistic normalization: Lorentz-invariant measure on the mass shell; 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.
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.
More Relativistic causality: local operator and fields; bosons, fermions, and measurable operators; proof of relativistic causality for the free scalar field.
September 16 (Monday):
Finished Relativistic causality: going forward and backward in time; causality for interacting fields.
Tachyons: tachyons in QM; tachyon field and vacuum instability; interactions and scalar VEVs (vacuum expectation values); no tachyons in the right vacuum state.
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; Green's functions in momentum space; regulating the integral over the poles: Feynman's choice and other types of Green's functions.
September 18 (Wednesday):
Feynman propagators as Green's functions: evaluating the ∫dk0 integrals for the Feynman propagator and for other kinds of Green's functions; Feynman propagators for vector, spinor, etc., fields.
Symmetries of classical field theories: symmetry groups; symmetries of field equations and symmetries of the action; continuous and discrete symmetries; internal and spacetime symmetries; global and local symmetries.
Symmetries in quantum mechanics: representation by unitary operators; Schrödinger and Heisenberg pictures; rorational symmetry and its generators; scalar and vector operators.
Extra lecture on September 20 (Friday):
Vortices and other kinds of topological defects: domain walls as topological defects; co-dimension; rotation and vorices in the superfluid; vortex energy; vortex rings and the critical velocity of the superflow; vortices as topological defects of co-dimension=2; magnetic monopoles as topological defects (co-dimension=3); (briefly) Yang–Mills instantons.
September 23 (Monday):
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.
Started Local phase symmetry: local symmetry and covariant derivatives; gauge field and gauge transforms; coupling charged scalar fields to electromagnetism.
September 25 (Wednesday):
Local phase symmetry: algebra of covariant derivatives; covariant field equations; [Dμ,Dν]=iQFμν.
Aharonov–Bohm effect: covariant Schrödinger equation for a charged quantum particle; gauge dependence of evolution kernels; Aharonov–Bohm effect; cohomology of the vector potential; charge quantization and the compactness of the U(1) phase symmetry.
Magnetic monopoles: electric-magnetic duality; magnetic monopoles; Dirac's charge quantization; Heuristic picture; Dirac construction; gauge bundles; electric-magnetic duality in QFT.
September 30 (Monday):
Finish magnetic monopoles: 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; 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.
Simple and compact Lie groups and their Lie algebras.
October 2 (Wednesday):
General gauge theories: Lie-algebra-valued gauge fields and gauge invariant Lagrangians; multiplets, representations, and matter fields. covariant derivatives for general multiplets; 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; tachyons have nothing.
Extra lecture on October 4 (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 7 (Monday):
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.
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.
October 9 (Wednesday):
Grassmann numbers and classical limits of fermionic fields.
Fermionic algebra and Fock space: Hilbert stace of one fermionic mode; multiple modes; fermionic fock space; wave functions and operators; particles and holes; holes as quasiparticles; 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.
October 14 (Monday):
Energy and charge of the Dirac sea.
Chiral symmetries: vector and axial symmetries of a massless fermion; Weyl fermions and chiral symmetries; U(N)L⊗U(N)R symmetry of N massless fermions; chiral symmetry in QCD; chiral gauge theories; electroweak example.
October 16 (Wednesday):
charge conjugation symmetry: C:e−↔e+; C:Φ(x)→Φ*(x); C:Ψ(x)→γ2Ψ*(x); 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.
Parity.
Gave out the midterm exam.
October 21 (Monday):
Parity and other discrete symmetries: parity; CP; time reversal (briefly); CPT theorem; baryogenesys and Sakharov's criteria.
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.
October 23 (Wednesday):
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.
Collected the midterm exam.
Extra lecture on October 25 (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 28 (Monday):
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; leading order cross-section in the λΦ4 theory.
Loop counting in perturbation theory: λΦ4 theory; adding a cubic coupling.
Mandelstam's S, T, and U parameters.
October 30 (Wednesday):
Feynman rules for multiple scalar fields.
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; photon propagator in the Coulomb gauge; propagators in other gauges; Landau and Feynman gauges.
Likbez lecture on November 1 (Friday):
Potential scattering in Quantum Mechanics: Scattering wave functions; Lippmann–Schwinger series; Born approximation; T(E) and S-matrix; phase shifts; calculating the phase shifts; small hard sphere example.
November 4 (Monday):
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.
Dirac Trace techniques: muon pair production example; polarized and un-polarized cross-sections; spin summed/averaged |M|2; relation to traces over Dirac indices; techniques for calculating the Dirac traces; traces for the muon pair production.
November 6 (Wednesday):
Muon pair production in electron-positron collisions: e−+e+→μ−+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.
Review traceology in homework#10.
Ward Identities: Ward identities for photons and gauge invariance of the amplitudes; sums over photon polarizations.
Begin annihilation, e−+e+→γ+γ: the tree-level amplitude; checking the Ward identities.
Extra lecture on November 8 (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 11 (Monday):
Annihilation, e−+e+→γ+γ: tree-level 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 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.
Compton scattering: crossing relation to the annihilation; kinematics in the lab frame; Compton formula; phase space in the lab frame; Klein–Nishina cross-section.
November 13 (Wednesday):
Finished Compton scattering: phase space in the lab frame.
Spontaneous symmetry breaking: symmetric Lagrangian/Hamiltonian but asymmetric vacuum state; tunneling vs. cluster expansion in QM and in QFT; dimension dependence.
Spontaneous breaking of continuous symmetries: continuous families of degenerate vacua and massless particles; complex field example; linear sigma model; Wigner and Goldstone modes of symmetries; Goldstone–Nambu theorem and Goldstone bosons; partial breaking of symmetry groups.
November 18 (Monday):
Pions in QCD: spontaneous breakdown of approximate symmetries; pseudo-Goldstone bosons; approximate chiral SU(2)L⊗SU(R)L symmetry of QCD and its spontaneous breakdown to the SU(2)V isospin; pions as pseudo-Goldstone bosons.
Abelian Higgs mechanism: SSB of a local U(1) symmetry; massive photon ‘eats’ the would-be Goldstone boson; unitary gauge vs. gauge-invariant description; Meissner effect in superconductors as non-relativistic Higgs Mechanism; massive photon's propagator in unitary and Rξ gauges.
Non-Abelian Higgs mechanism SU(2) with a Higgs doublet; SU(2) with a real Higgs triplet; started vector masses for general case.
November 20 (Wednesday):
More Higgs mechanism: general formulae for the vector masses; massive photon's propagator in unitary and Rξ gauges; mixed local/global symmetries.
Bosons of the 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 weak currents; Fermi's effective theory of weak interactions.
Extra lecture on November 22 (Friday):
SSB of QCD's chiral symmetry and sigma models: Approximate 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; general NLΣMs (briefly).
November 25–29 (whole week):
Fall break, no classes.
December 2 (Monday):
Fermions of the Glashow–Weinberg–Salam theory: SU(2)×U(1) quantum numbers and electric charges of quarks and leptons; Yukawa couplings to scalar VEVs and fermion masses; masses of quarks and leptons; 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.
December 4 (Wednesday):
Cabibbo–Kobayashi–Maskawa (CKM) matrix and its origin: third family and the CKM matrix; neutral and charged weak currents; flavor-changing weak decays; unitary basis redefinitions for each type of a fermion multiplet; matrices of Yukawa couplings; mass matrices for Weyl 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 Kaons: GIM box and K0↔K0 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.
Extra lecture on December 6 (Friday):
Making magnetic monopoles from non-abelian gauge and Higgs fields: Georgi–Glashow model; ‘hedgegog’ configuration of the Higgs and gauge fields; magnetic charge; multi-monopole solutions; Bogomol'nyj bound for the monopole mass; monopoles and the topology of the G/H vacuum space; no monopoles from the Glashow–Weinberg–Salam theory; monoploles from Grand unifications; magnetic and chromomagnetic charges of monopoles.
December 9 (Monday):
CKM origin of CP violation: CP action on W± and on leptonic charge currents; CP of hadronic charge currents and CKM↔CKM*; removable and unremovable CKM phases; Jarlskog invariant; calculating M(K0↔K‾0) and its imaginary part; maybe strong CP violation.
Neutrino masses: neutrino masses and oscillations; Dirac vs. Majorana masses of neutrinos; seesaw mechanism.
Gave out the final exam.

QFT 2, Spring 2024 semester

January 13 (Monday):
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.
Started Overview of UV regulators: Wilson's hard edge cutoff.
January 15 (Wednesday):
Overview of UV regulators: 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 overview of scattering in QM: partial wave analysis and optical theorem.
optical theorem in QFT: statement and proof from unitarity of the S matrix; application to the Im M1 loop in the λφ4 theory.
Extra lecture on January 17 (Friday):
Grand Unification (I): 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.
January 22 (Wednesday):
Optical theorem: Cutkosky's cutting rules.
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.
January 27 (Monday):
The two-point correlation function: 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; optical theorem for the unstable particles; 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); calculating the Σ(p2); dΣ/dp2 and the scalar field strength renormalization.
January 29 (Wednesday):
Finished Field strength renormalization in the Yukawa theory: calculating the dΣ&phi/dp2 and the Zφ.
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.
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⟩.
Counterterm perturbation theory: ℒbare=ℒphysical+counterterms; Feynman rules for the counterterms; adjusting δZ, δm, and δλ order by order in λ; theorem: all divergences are canceled by the δZ, δm, and δλ.
Started counting divergences: superficial degree of divergence; checking the subgraphs.
Extra lecture on January 31 (Friday):
Grand Unification (II): proton decay; fermion masses and GUT multiplet(s) for the SM Higgs; double-triplet problem; issues in SUSY GUTs.
February 3 (Monday):
Counterterms and canceling the UV divergences: 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: definitions of renormalizable, super-renormalizable, and non-renormalizable theories; divergences of Φn theories; super-renormalizable Φ3 theory; renormalizable Φ4 theory; started non-renormalizable Φn>4 theories.
February 5 (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.
4D theories from higher dimensions (brief overview): Kaluza–KLein, branes, string-theory context.
QED perturbation theory: the counterterms and the Feynman rules; divergent amplitudes and their momentum dependences; missing counterterms and Ward–Takahashi identities.
February 10 (Monday):
Finish QED perturbation theory: 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); 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 the WT indentities.
February 12 (Wednesday):
Ward–Takahashi identities: Z1=Z2 and generalization to multiple charged fields.
Diagramatic proof of Ward–Takahashi identities (briefly): 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).
Form factors: probing nuclear and nucleon structure with electrons; form factors; on-shell form-factors F1(q2) and F2(q2); the gyromagnetic ratio.
Started Dressed QED vertex at one loop: the dressed vertex and the form factors; the one-loop diagram and its denominator.
Extra lecture on February 14 (Friday):
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; formal derivation of these diagrams; one-loop calculation; general Coleman–Weinberg effective potential; Higgs mechanism induced by the Coleman–Weinberg potential.
February 17 (Monday):
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 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.
February 19 (Wednesday):
Bremsstrahlung: classical bremsstrahlung; flat frequency spectrum; implication for the quantum theory and for the IR divergences of QED; soft-photon bremsstrahlung in QED (tree-level); verifying I(v,v')=2fIR(q2).
Optical theorem for the soft photons: optical theorem for the μ-μ+→e-e+ 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.
Infrared problems in QCD: perturbative IR problems with soft an collinear gluons; non-perturbative confinement and hadronization; jets and their formation in e++e−→2 jets events; gluons and 3-jet events; jets in experiments and in perturbation theory.
gauge dependence in QED: gauge-dependent off-shell amplitudes and counterterms; δ1(ξ)=δ2(ξ).
February 24 (Monday):
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; Renormalization group with multiple couplings: general formulae for the β-functions; Yukawa theory example.
February 26 (Wednesday):
Renormalisation schemes: scheme dependence of the couplings and the β-functions; minimal subtraction schemes MS and MS-bar; extracting β-functions from residues of the 1/ε poles.
Types of RG flows: β>0, Landau poles, and UV incompleteness; β<0, QCD example, and asymptotic freedom; strong IR coupling and confinement; dimensional transmutation and ΛQCD.
Extra lecture on February 28 (Friday):
Confinement in QCD: Confinement in the Yang–Mills theory; triality; deconfining phase transition; effects of quarks and of the chiral symmetry breaking; phase transition v. crossover; (T,μ) phase diagram of QCD; nuclear matter and quark matter at high μ.
March 3 (Monday):
Fixed points β(g*)=0 of RG flows: scale invariance and conformal symmetry; UV stability vs. IR stability; Wilson's critical point in condensed matter; 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 RG flow, IR to UV or UV to IR?: Gell-Mann–Low equation and flow from the threshold to UV; in condensed matter at a critical point, flow from UV to IR; in modern HE theory: flow from UV to IR, check phenomenology, then flow back to UV.
March 5 (Wednesday):
Couplings and operators; relevant, irrelevant, and marginal operators; super-renormalizable, renormalizable, and non-renormalizable couplings; decoupling of irrelevant operators from long-distance effective field theories; caveats.
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.
Gave out the midterm exam.
March 10 (Monday):
Functional integrals in QFT: “path” integrals for quantum fields; correlation functions; 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.
March 12 (Wednesday):
Euclidean path integrals in QFT: sources, connected correlators, and Euclidean Feynman rules. QFT and Statistical Mechanics: QFT↔StatMech analogy; coupling as temperature; 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.
Started Integrating over fermion fields in QED: functional integral in EM background: the determinant, and the source term.
March 17–21 (whole week):
Spring break, no classes.
March 24 (Monday):
Integrating over fermion fields in QED: 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.
March 26 (Wednesday):
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.
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.
Extra lecture on March 28 (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 31 (Monday):
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.
Started 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.
April 2 (Wednesday):
QCD beta-function: 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).
Beta functions of general gauge theories.
Started introduction to axial anomaly. axial symmetry of massless electrons; anomaly and its origin in the path integral measure; the diagrams and the regulation problem (briefly).
April 7 (Monday):
Adler–Bell–Jackiw anomaly: 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.
Formal analysis of the axial anomaly: anomaly of the measure of the fermionic functional integral; generalization to the non-abelian gauge theories; triangle and quadrangle anomalies in QCD; flavor symmetries of QCD: the U(1)A is anomalous but the SU(Nf)L×SU(Nf)R are anomaly-free; spontaneous chiral symmetry breaking and the issue with the η and π mesons.
April 9 (Wednesday):
Non-linear sigma models: non-linear field spaces; NLΣM of the chiral symmetry breaking; vector and axial currents; 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.
Fπ and the decay of a charged pion.
Electromagnetic anomalies: electromagnetic anomalies of quarks' symmetries; anomalous decay of the neutral 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 example.
Extra lecture on April 11 (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 14 (Monday):
Anomalies in chiral gauge theories: QCD and SU(2)W examples; SU(2)W anomalies of baryon and lepton numbers; leptogenesis.
Instantons: topological index of non-abelian gauge fields; disjoint sectors of configuration spaces; self-dual and anti-self-dual fields; 't Hooft's one-instanton solution; multi-instanton solutions; cluster expansion and the instanton angle Θ; the Θtr(εFF) term in the Lagrangian.
April 16 (Wednesday):
Instantons: cluster expansion and the instanton angle Θ; the Θtr(εFF) term in the Lagrangian.
Instantons and fermions: instantons and anomaly; fermionic zero modes; Atiyah–Singer theorem; zero modes and amplitudes; non-conservation of anomalous charges.
Strong CP violation: chiral transforms of quark mass matrices and of the instanton angle; the Θ͞ invariant; Θ͞ and strong CP violation; neutron electric dipole; Peccei–Quinn symmetry.
Anomalies of gauge currents: abelian 3-photon anomaly; non-abelian 3-gluon anomaly; non-abelian 4-gluon anomaly; intro to anomaly cancellation.
April 21 (Monday):
Anomaly cancellation: anomaly in simple gauge groups; cubic Casimir and cubic index; counting anomaly indices; SU(5) GUT example; anomalies in product gauge groups such as the Standard Model; checking anomaly cancellation in the Standard Model; importance of tr(Qel)=0; massive fermions do not contribute to anomalies.
Confining chiral gauge theories: can chiral flavor symmetries survive confinement?; massless composite fermions and 't Hooft's anomaly matching conditions; SU(5) example; self–Higgsing; Higgs→confinement duality.
April 23 (Wednesday):
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; descent from a 6-form to the non-abelian anomaly in d=4 dimensions; other even dimensions d: nonabelian anomalies descent from (d+2) forms.
Anomaly cancellation in different dimensions: symmetric traces, Casimirs, and Indices of degree n=(d/2)+1; index counting for anomaly cancelation in 2D versus 4D; harder-to-satisfy rules for 6D and 10D.
Gravitational anomalies in 4D.
extra lecture on April 25 (Friday):
Green–Schwarz mechanism for anomaly cancellation: trace anomaly in 4D; gravitational and mixed anomalies in higher dimensions; antisymmetric tensor field Bμν and its couplings; 4D anomalies this field may cancel; generalization to higher dimensions; applications to 10D superstring theory.
April 28 (Monday):
Wess–Zumino terms: matching low-energy and high-energy anomalies in QCD; anomalous Wess–Zumino terms for the nonlinear sigma model; Wess–Zumino–Witten terms from 4D and 5D points of view; gauging WZW terms; flavor anomaly matching; WZW terms in other even dimensions.
Gave out the final exam.
Last Modified: April 28, 2025.
Vadim Kaplunovsky
vadim@physics.utexas.edu