**Navigation:**
Fall,
Spring,
Last lecture.

- August 30 (Tuesday):
- Syllabus and admin: course content, textbooks, prerequisites, homework, exams and grades, etc.
(see the main web page for the class).

General introduction: reasons for QFT; field-particle duality.

Introduction to classical field theory, starting with a refresher of classical mechanics (the least action principle and the Euler-Lagrange equations). - September 1 (Thursday):
- Classical fields: definition, Euler–Lagrange equations; Klein–Gordon example;
coupled scalar fields; higher derivatives and counting of degrees of freedom;
relativistic notations.

The electromagnetic fields: the tension fields**E**and**B**and the potentials Φ and**A**; gauge transforms; the 4–vector A^{μ}and the 4–tensor F^{μν}; the homogeneous Maxwell equations as Jacobi identities. - September 6 (Tuesday):
- The electromagnetic fields: Maxwell equations in the relativistic form; the Lagrangian formalism;
current conservation and gauge invariance of the action; counting EM degrees of freedom.

Symmetries and conserved currents: conserved currents in field theory; current conservation and symmetries; symmetries of the action; groups; discrete vs continuous symmetries. - September 8 (Thursday):
- Symmetries: continuous symmetries and their generators; Lie groups and Lie algebras; global and local symmetries; Noether theorem and its proof; the SO(N) example.
- September 13 (Tuesday):
- Translational symmetry and the stress-energy tensor.

Local phase symmetry: complex fields and the phase symmetry; local symmetry and the covariant derivatives; relation to gauge transforms; multiple charged fields; math of covariant derivatives; non-commutativity. - September 14 (Wednesday) [supplementary lecture]:
- 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. - September 15 (Thursday):
- Aharonov–Bohm effect and magnetic monopoles:
the covariant Schroedinger equation; the Aharonov–Bohm effect; cohomology of the vector potential;
charge quantization and the compactness of the U(1) phase symmetry; magnetic monopoles;
Dirac quantization of the magnetic charge; gauge bundles.

Intoduction to quantization: canonical quantization vs. path integrals; Hamiltonian classical mechanics. - September 20 (Tuesday):
- Canonical quantization in mechanics:
canonical commutation relations; Schrödinger and Heisenberg pictures of quantum mechanics;
Poisson brackets and commutators.

Quantum scalar field: the canonical momentum field π(**x**,t) and the classical Hamiltonian; operator-valued quantum fields; the equal-time commutation relations; the Hamiltonian operator; the quantum Klein–Gordon equation.

Started quantum fields and particles: expanding free relativistic scalar field into harmonic oscillators. - September 22 (Thursday):
- Quantum fields and particles:
Eigenstates of the free quantum field's Hamiltonian; identifying the identical bosons;
the Fock space.

Going back: from identical bosons to harminic oscillators to quantum fields; the non-relativistic quantum fields; the second quantization.

Translating between the wave function and the Fock space languages for the operators (quick overview only). - September 27 (Tuesday):
- Relativistic quantum fields: Lorentz groups; relativistic normalization of particle states and operators; expanding a free time-dependent scalar field into products of plane waves and creation/annihilation operators; generalization to non-scalar fields.
- September 28 (Wednesday) [supplementary lecture]:
- Field theory of the superfluid: Non-relativistic QFT; Bose–Einstein condensation;
classical field theory of the condensate; superfluid velocity; irrorational flow; vortices.

Other kinds of topological `defects: domain walls, monopoles, YM instantons; co-dimension. - September 29 (Thursday):
- Relativistic causality: superluminal particles in `relativistic' QM; superluminal signals in QM and in QFT;
proof of causality for the free scalar field.

Feynman propagator for the scalar field: why and how of time-ordering; defining the propagator; Feynman propagator as a Green's function. - October 4 (Tuesday):
- Feynman propagator as 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.

Generators of the Lorentz symmetry.

Representations of groups and Lie algebras: matrix representations; generators; reducible and irreducible representations; QM: multiplets of quantum states and multiplets of operators; simple and non-simple algebras; compact and non-compact algebras; Lorentz multiplets: no nontrivial finite unitary representations; infinite unitary multiplets of particle states; finite non-unitary multiplets of fields; no finite unitary multiplets. - October 6 (Thursday):
- Particle representations of the Lorentz symmetry:
the little group G(p); Wigner theorem: massive particles have definite spins,
massless particles have definite helicities, tachyons have nothing;
generalization to
*d*≠3+1 dimensions.

Tachyons in QFT mean vacuum instabilities; scalar VEVs. - October 11 (Thursday):
- Lorentz multiplets of fields.

- Dirac spinor fields: Dirac matrices; Dirac spinor multiplet; Dirac equation; Dirac Lagrangian; Hamiltonian for the Dirac fields; classical limits of fermionic fields; Grassmann numbers.
- October 12 (Wednesday) [supplementary lecture]:
- Supefluidity: excitations of a supefluid; dispersion relation ω(
*k*) for the excitations; reasons for dissipationless flow; critical velocity. - October 13 (Thursday):
- Classical limits of fermionic fields: Grassmann numbers

Fermionic particles and holes: Fermionic Fock space; particle-hole formalism; Dirac electrons and positrons. - October 18 (Tuesday):
- Charge conjugation:
**C**:e^{−}↔e^{+};**C**:Ψ(x)→γ^{2}Ψ^{*}(x); Majorana fermions and other neutral particles; Majorana vs. Weyl.

Parity and CP symmetries. - October 19 (Wednesday) [supplementary lecture]:
- 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; Bott periodicity; Majorana–Weyl spinors in*d*≡2 (mod 8); general Bott periodicity for spinors of SO(*a,b*). - October 20 (Thursday):
- Chiral symmetry:
Vector and axial symmetries of a Dirac fermions; the currents;
chiral
*U(N)*symmetry._{L}×U(N)_{L}

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: the interaction picture of QM; the Dyson series and the time-ordering; the S matrix and its elements. - October 25 (Tuesday):
- Perturbation theory in QFT and Feynman diagrams: 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; connected diagrams and scattering amplitudes.
- October 27 (Thursday):
- Perturbation theory and Feynman rules:
Phase space factors;
loop counting for quartic and cubic couplings;
Mandelstam's s, t, and u.

Gave out the midterm exam. - November 1 (Tuesday):
- Feynman rules for multiple fields.

Good and bad interactions in perturbative QFT: dimensional analysis; trouble with Δ<0 couplings; types of Δ≥0 couplings in 4D; other dimensions.

Quantum Electro Dynamics (QED): quantizing EM fields; photon propagator and its gauge dependence. - November 3 (Thursday):
- Collect the the midterm exam.

QED Feynman rules: vertices, propagators, and external lines; Dirac indexology; Gordon identities; sign rules; Coulomb scattering example.

Muon pair production in QED,e →^{−}+e^{+}μ ^{−}+μ^{+}: the tree amplitude; un-polarized scattering and spin sums/averages. - November 8 (Tuesday):
Muon pair production in QED: Dirac trace techniques; the traces for the pair production; the cross-section; quark pair production e ^{−}+e^{+}→q+q̄ →hadrons.

Crossing symmetry: electron-muon scattering vs. pair production; analitically continuing the amplitudes; crossing symmetry in general; signs for crossed fermions; Compton vs. annihilation example.- November 9 (Wednesday) [supplementary lecture]:
- Resonances and unstable particles: Breit–Wigner resonance and its lifetime; propagators of unstable particles; making a resonance; cross-sections and branching ratios; J/ψ example.
- November 10 (Thursday):
- Ward Identities: Ward identities for photons and gauge invariance
of the amplitudes; sums over photon polarizations.

Electron-positron annihilatione tree diagrams and the amplitude; checking the Ward identities; summing over photon polarizations; averaging over fermions' spins; summary; kinematics.^{−}+e^{+}→2γ:

Maybe Compton scattering and other related processes. - November 15 (Tuesday):
- Spontaneous Symmetry breaking: multiple degenerate vacua;
continuous families of vacua and massless particles;
Wigner and Goldstone–Nambu modes of symmetries; Goldstone theorem;
linear-sigma-model example; scattering of Goldstone particles;
~~SSB of the chiral symmetry of QCD~~, pions as pseudo-Goldstone bosons. - November 16 (Wednesday) [supplementary lecture]:
- Spin-statistics theorem:
integer-spin particles are bosons, half-integer-spin particles are fermions;
assumptions of the theorem; plane waves, spin sums, and lemmas; proving the theorem;
generalization to
*d*≠3+1; proving the lemmas. - November 17 (Thursday):
- Non-abelian gauge symmetries: covariant derivatives, matrix-valued connections, and gauge transforms; non-abelian vector fields and tension tensors; Yang–Mills theory; QCD; maybe QCD Feynman rules.
- November 22 (Tuesday):
- General gauge theories: general gauge groups; matter fields in general multiplets;
products of gauge symmetries; the Standard Model.

The 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 examples: SU(2) with a doublet; SU(2) with a real triplet. General Case. - November 29 (Tuesday):
- The Higgs mechanism:
caveats about the unitary gauge; multiple Higgs fields;
SO(N)→SO(N−1)→SO(N−2) example; breaking both local and global symmetries.

Glashow–Weinberg–Salam theory: Higgsing theSU(2) down to the_{W}×U(1)_{Y}U(1) ; the photon and the massive W_{EM}^{±}and Z^{0}vectors; the mixing angle; the weak currents; the effective Fermi theory; ρ=1 for the neutral current.

Fermions in the GWS theory: Quantum numbers of the quarks and the leptons; the Yukawa couplings giving rise to the masses;~~the EM current and the weak currents, charged and neutral~~. - November 30 (Wednesday) [supplementary lecture]:
- Superconductivity: Cooper pairs, BCS, and the effective Landau–Ginsburg theory; non-relativistic Higgs mechanism and Meissner effect; vortices and magnetic field; types of superconductors; cosmic strings.
- December 1 (Thursday):
- Fermion mass due to Higgs VEV×Yukawa coupling; chiral symmetry breaking.

Quarks and leptons in the GWS theory: Quantum numbers of quarks and leptons; Yukawa couplings and masses; the EM current and the weak currents (charged and neutral) in terms of the fermion fields; flavor mixing and the Cabibbo–Kobayashi;–Maskawa matrix; CP violation in weak interactions; neutrino masses.

Give out the final exam.

- January 24 (Tuesday):
- Syllabus of the spring semester.

Loop diagrams: amputating the external leg bubbles.

Calculating a one-loop diagram: Feynman trick for denominators; Wick rotation to the Euclidean momentum space; cutting off the UV divergence; explaining the cutoff; physical vs. bare couplings. - January 26 (Thursday):
- Bare and physical couplings:
The bare coupling and the cutoff; physical coupling in terms of a physical amplitude;
resumming the perturbation theory in terms of physical coupling; cutoff independence.

The ultraviolet regulators: Wilson's hard-edge cutoff; Pauli–Villars; higher-derivative regulator; covariant higher derivative for gauge theories; dimensional regularization; the lattice. - January 31 (Tuesday):
- Dimensional regularization:
*D*<4 as a UV regulator; Gaussian and non-Gaussian integrals in non-integer dimensions; taking the*D*→4 limit.

The optical theorem: origin in the unitary S matrix; one-loop example. - February 2 (Thursday):
- 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 F_{2}(*p*^{2}): 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. - February 3 (Friday) [supplementary lecture]:
- Vacuum energy and effective potentials: zero-point energy and the Casimir effect; zero-point energy for fields with VEV-dependent masses; Feynman diagrams for the vacuum energy; one-loop calculation; general Coleman–Weinberg effective potential; Higgs mechanism induced by the Coleman–Weinberg potential.
- February 7 (Tuesday):
- Perturbation theory for the F
_{2}(*p*^{2}): re-summing the 1PI bubbles; Σ_{2}(*p*^{2}) and renormalization of the mass and of the field strength; mass renormalization in the φ^{4}theory; quadratic UV divergences and how to regulate them.

Begin Field strength renormalization in the Yukawa theory: the one-loop diagram; the trace; the denominator and the numerator. - February 9 (Thursday):
- Field strength renormalization in the Yukawa theory:
the UV divergence structure:
Σ(
*p*^{2})=(div.constant)+(div.constant)×*p*^{2}+finite_*f*(*p*^{2}); calculating the Σ(*p*^{2}); the imaginary part and the decay of the scalar into fermions;*dΣ/dp*and the scalar field strength renormalization.^{2}

The counterterm-based perturbation theory: ℒ_{bare}=ℒ_{phys}+counterterms; the counterterm vertices; canceling the infinities.

Counting the divergences: superficial degree of divergence; graphs and subgraphs; classifying the divergences; counterterms; cancelling sub-graph divergences*in situ*. - February 10 (Friday) [supplementary lecture]:
- Relating the correlation functions
F
_{n}(*p*) to the scattering amplitudes: the amputated core and the external leg bubbles; the poles for the on-shell_{1},…p_{n}*p*_{i}^{0}→±*E*(**p**_{i}) and their relations to the asymptotic*x*→±∞ limits; the asymptotic |in⟩ and ⟨out| states and the LSZ (Lehmann–Symanzik–Zimmermann) reduction formula; scattering amplitudes and the amputated diagrams.^{0}_{i} - February 14 (Tuesday):
- Subgraph divergences:
*in situ*cancelation of subgraph divergences by counterterms; nested divergences; overlapping divergences; BPHZ theorem.

Divergences and renormalizability: supeficial degree of divergence in the φ^{k}theories; super-renormalizable, renormalizable, and non-renormalizable theories; divergences in general quantum field theories and the energy dimensions of the couplings; super-renormalizable, renormalizable, and non-renormalizable couplings in*d*=4; other spacetime dimensions. - February 16 (Thursday):
- QED perturbation theory: the counterterms and the Feynman rules;
the divergent amplitudes and their momentum dependence;
the missing counterterms and the Ward–Takahashi identities.

Calculate the one-loop Σ^{μν}(*k*) and check the WT identity. - February 21 (Tuesday):
- Dressed propagators in QED:
the electron's F
_{2}(*p̸*); the photon's F_{2}^{μν}(*k*); equations for the finite parts of the δ^{2}, δ^{m}, and δ^{3}counterterms.

Effective coupling at high energies: Loop corrections to Coulomb scattering; effective QED coupling α_{eff}(*E*) and it's running with log(energy); running of other coupling types.

Begin Ward–Takahashi identities: the identities; Lemma 1 (the tree-level two-electron amplitudes). - February 23 (Thursday):
- Ward–Takahashi identities:
Lemma 2 (the one-loop no-electron amplitudes);
identities for the (bare) higher-loop amplitudes;
generalizing to multiple charged fields;
relation to the electric current conservation;
Ward identity δ
^{1}=δ^{2}for the counterterms; equation for the finite part of the δ^{1}. - February 24 (Friday) [supplementary lecture]:
- Grand Unification: unifying the EM, weak, and strong interactions in a single non-abelian gauge group; the SU(5) example; multiplets of fermions; the gauge couplings and the Georgi–Quinn:–Weinberg equations; the doublet-triplet problem; baryon decay and other exotic processes.
- February 28 (Tuesday):
- Form factors: probing nuclear and nucleon structure with electrons;
the form factors; the on-shell form-factors
*F*_{1}(*q*^{2}) and*F*_{2}(*q*^{2}); the gyromagnetic ratio.

The dressed QED vertex at one loop: the setup and the diagram; the denominator; the numerator algebra; calculating the*F*_{2}form factor and the anomalous magnetic moment; the experimantal and the theoretical electron's and muon's magnetic moments at high precision. - March 2 (Thursday):
- The electric from factor
*F*_{1}(*q*^{2}) at one loop: calculating the integrals; the infrared divergence and its regulation; momentm dependence of the IR divergence; the δ^{1}counterterm.

IR finiteness of the*measureable*quantities: similar IR divergences of the vertex correction and of the soft-photon bremmsstrahlung; detectable vs. undetectable photons and the*observed*cross-sections; IR finiteness of the observed cross-sections.

Briefly: higher lops and/or more soft photons; optical theorem for the finite observed cross-sections; no Fock space for QED and other gauge theories; jets in QCD. - March 7 (Tuesday):
- Gauge dependence in QED; δ
^{1}(ξ) and δ^{2}(ξ).

Symmetries and the counterterms: In QED, δ^{m}∝*m*because of chiral symmetry when*m*=0; in Yukawa theory, δ^{m}∝*m*because of discrete chiral symmetry, but δ^{λ}≠0 when λ=0; general rule:*all*counterterms of dim≤4 allowed by the symmetries of the physical Lagrangian.

Large logs and running couplings: large log problem for*E≫m*; adjusting the counterterms to avoid large logs; running counterterms and running couplings;~~renormalization schemes~~. - March 9 (Thursday):
- Intro to the renormalization group:
the off-shell renormalization schemes;
the anomalous dimension of a quantum field; the β function and the running coupling;
RGEs and their solutions; renormalization of QED; β
_{e}to one-loop order. - March 21 (Tuesday):
- Renormalization of the Yukawa theory:
the counterterms, the anomalous dimensions, and the β–functions;
two coupled RGE equations.

Types of RG flow: flow to UV vs. flow to IR; β>0 and Landau poles; β<0 and the asymptotic freedom; IR price of asymptotic freedom; confinement and chiral symmetry breaking in QCD. - March 23 (Thursday):
- Renormalization group flows: fixed points: RG flows for multiple couplings;
flow to UV vs. flow to IR.

Relevant, irrelevant, and marginal operators; effective field theories.

Gave out the midterm exam. - March 28 (Tuesday):
- 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.

Introduction to path integrals: path integrals in QM; the Lagrangian and the Hamiltonian froms of the path integral; derivation and discretization; normalization of the path integral. - March 30 (Thursday):
- Path Integrals: the partition function; the harmonic oscillator example;
functional integrals in QFT; Feynman rules; sources and the generating functionals.

Collect the midterm exams. - March 31 (Friday) [supplementary lecture]:
- 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; a QFT at a β=0 point; exotic SCFTs in 5D and 6D and families of SCFTs in D=2,3,4; AdS/CFT duality. - April 4 (Tuesday):
- Functional integrals in Euclidean spacetime:
QFT–StatMech corresponcence; convergence and imaginary time; Euclidean action;
Euclidean lattice as a UV cutoff;
restoration of the SO(4)→Lorentz symmetry in the continuum limit;
coupling–temperature analogy.

Fermionic functional integrals: Berezin integrals over fermionic variables; Gaussian fermionic integrals. - April 6 (Thursday):
- Functional integrals for fermions:
fermionic gaussian integrals; free Dirac field with sources; Dirac propagator;
FI for fermions in EM background; Det(D̸+m) and the electron loops;
1/(D̸+m) and the tree diagrams.

Functional integral for the EM field: gauge transforms as redundancies; gauge-fixing conditions and the Faddeev–Popov determinants; Landau-gauge propagator from the functional integral;~~gauge-averaging and the Feynman gauge~~. - April 7 (Friday) [supplementary lecture]:
- Gauge theories on the lattice:
local symmetry on the lattice; U(link) variables and the gauge fields;
non-abelian symmetries and U(link) variables;
U(plaquette) and the F
^{μν}(x), abelian or non-abelian; action for the EM and the YM theories on the lattice. Briefly: computer simulations; strongly coupled compact U(1). - April 11 (Tuesday):
- Gauge-averaging and the Feynman gauge in QED.

Quantiing the Yang--Mills theory: fixing the non-abelian gauge symmetry; Faddeev–Popov ghost fields; gauge-fixed YM Lagrangian.

QCD: Quantum Lagrangian and the Feynman rules; generalization to other gauge theories; weakened Ward identities; the qq̄→gg example. - April 13 (Thursday):
- Weakeaned Ward identities for QCD:
the qq̄→gg example; relations between longitudinal gluons and ghosts.

BRTS symmetry: the BRST generator and its nilpotency; invariance of ℒ_{QCD}; the Fock space of QCD — the physical and the unphysical states; BRST cohomology and getting rid of the unphysical states; BRST symmetries of the amplitudes; cancellation of unphysical processes.

QCD renormalizability: renormalizability and the counterterm set; BRST and other manifest symmetries; weakened Ward identities for the counterterms; - April 18 (Tuesday):
- QCD renormalizability: renormalizability and the counterterm set;
BRST and other manifest symmetries; weakened Ward identities for the counterterms.

QCD β–function at the one-loop order: the beta function and the counterterms; calculating the δ_{2}; calculating the δ_{2}— the QED-like loop and the non-abelian loop; began calculating the δ_{2}&mdash just the quark loop. - April 20 (Thursday):
- Finish calculating the QCD β–function:
the gluon loop, the sideways loop, and the ghost loop; assembling the β– function
and generalizing to other non-abelian gauge theories.

Quark*confinement*in QCD: chromo-electric field lines and flux tubes; condensation of chromo-magnetic monopoles; flux tubes as hadronic strings; mesons and baryons; spinning hardonic string and the Regge trajectories. - April 21 (Friday) [supplementary lecture]:
- Instantons: topological index I[A
^{μ}] and its quantization; S_{E}≥(8π^{2}/g^{2})×|I| and the topological sectors in the YM path integral; 't Hooft instantons and tunneling events. - April 25 (Tuesday):
- Overview of comfinement and chiral symmetry breaking in QCD:
confinement, flux tubes, mesons, baryons, the hadronic string;
chiral symmetry breaking and its order paramater ⟨Ψ̄Ψ⟩;
deconfinement and chiral symmetry restoration at high temperatures; the phase transition and its type.

Chiral symmetry breaking in detail: U(N_{f})_{L}×U(N_{f})_{R}and its currents; anomalous breaking of the axial U(1); flavor structure of the ⟨Ψ̄Ψ⟩ condensate; the linear and the non-linear sigma models of the SU(2)×SU(2)→SU(2) chiral symmetry breaking; NLΣM for the SU(3)×SU(3)→SU(3) breaking; the NLΣM in QCD context, its perturbation by the quark masses, and the masses for the pseudoscalar mesons. - April 26 (Wednesday) [extra supplementary lecture at 3:30 to 5 PM, in RLM 9.304]:
- 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 27 (Thursday):
- Axial Anomaly:
The η meson's mass problem in QCD and the anomaly;
the massless QED example;
the
*naive*Ward identities for the axial J^{μ5}current and what's wrong with them; the triangle graphs and the problems with their UV regulators; calculating the anomaly using the Pauli–Villars regulator. - April 28 (Friday) [supplementary lecture]:
- Instantons: multi-instanton confugurations; the Θ angle.

Instantons and fermions: zero modes in instanton background; Atyah–Singer index theorem; relation to the axial anomaly; chiral anomaly of the Θ angle and Θ̅=Θ+phase(det(quark mass matrix)); the strong CP problem. - May 2 (Tuesday):
- Anomalies in QED: multiple fermions; decay of the neutral pion to 2 photons.

Axial anomaly in QCD: 2-gluon and 3-gluon anomalies; anomaly cancelation for the SU(N_{f}) currents and the η/π problem.

Chiral gauge theories: Weyl fermions and chiral currents; loops of Weyl fermions and the chiral anomaly; the trace=0 conditions for anomaly cancelation. - May 4 (Thursday):
- Anomaly cancelation and miscancelation:
gauge anomaly cancelation in the Standard Model;
gauge anomaly cancelation in GUTS;
SU(2)
_{W}anomalies of the baryon and lepton numbers; baryon decay and baryogenesys.

Give out the final exam.

Last Modified: May 4, 2017. Vadim Kaplunovsky

vadim@physics.utexas.edu