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

- August 23 (Tuesday):
- Canceled.
- August 26 (Thursday):
- Syllabus and admin:
course content, textbooks, prerequisites, homework, exams and grades, etc.

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

Lagrangian mechanics: Lagrangian and action; least action principle; Euler–Lagrange equations; multiple dynamical variables; counting the degrees of freedom. - August 26 (Friday) [make-up lecture]:
- Introduction to classical fields:
Definition of a classical field; Lagrangian density; Euler–Lagrange equations for fields;
Klein–Gordon example; multiple fields; complex fields; Landau–Ginzburg example;
higher space derivatives and non-local Lagrangians for non-relativistic fields.

Relativistic fields: relativistic sign conventions; Einstein summation convention; relativistic ℒ and field equations; Klein–Gordon example; multiple scalar fields.

Started Relativistic electromagnetic fields: the 4–tensor F^{μν}=−F^{νμ}and the relativistic form of Maxwell equations. - August 30 (Tuesday):
- Relativistic electromagnetic fields:
the 4–vector potential A
^{μ}and the gauge transforms; the Lagrangian formulation; current conservation and gauge invariance of the action; counting the EM degrees of freedom.

Review of canonical quantization: Canonical quantization v. functional quantization; Hamiltonian formalism in classical mechanics; quantization, operators, and commutation relations; Poisson brackets and commutator brackets.

Introduction to quantum fields: Hamiltonian formalism for the classical fields; quantum fields; equal-time commutation relations. - September 1 (Thursday):
- Finish Introduction to quantum fields:
quantum Hamiltonian, Heisenberg equtions, and the quantum Klein–Gordon equation.

Quantum fields and particles: expanding free relativistic scalar fields into modes; creation and annihilation operators for a bunch of harmonic oscillators; eigenstates of the free quantum field's Hamiltonian; identifying the identical bosons; the Fock space.

Start General identical bosons: bosonic Fock space and its occupation number basis. - September 2 (Friday) [make-up lecture]:
- General identical bosons:
creation and annihilation operators; wave-function language vs. Fock-space language;
one-body operators; two-body operators; non-relativistic quantum fields; “second quantization”.

- September 6 (Tuesday):
- Relativistic normalization of states and operators:
Lorentz groups; momentum space geometry and Lorentz-invariant measure;
relativistic normalization of states and operators.

Relativistic quantum fields: Expanding a free time-dependent scalar field into products of plane waves and creation/annihilation operators; massive vector field; charged scalar field and antiparticles; general free fields. - September 8 (Thursday):
- Intro to relativistic causality:
no superluminal particles or
*sigmals*; signals in quantum mechanics: sending a signal requires [M̂(t_{2}),Ŝ(t_{1})]≠0; QFT: local operators and relativistic causality.

Relativistic causality: local operator and fields; proof for free scalar fields; going forward and backward in time; causality for interacting fields.

Intro to Feynman propagators: why and how of time-ordering; defining the propagator; relation to D(x-y); scalar propagator is a Green's function of the Kelin–Gordon equation. - September 13 (Tuesday):
- Feynman propagator and other Green's functions:
Green's function in momentum space; regulating the integral over the poles;
Feynman's choice; other types of Green's functions;
Feynman propagators for vector, spinor, etc., fields.

Overview of symmetries of field theories: symmetries of the action; continuous and discrete symmetries; internal and spacetime symmetries; global and local symmetries; started continuous groups and their generators. - September 15 (Thursday):
- Intro to Lie groups and Lie algebras:
SO(3) example: generators and generator algebra; multiple commutator formula and finite rotations;
representations of the algebra and of the group;
general Lie groups and Lie algebras of their generators; representations of general groups and algebras.

Noether theorem: Global continuous symmetries and conserved currents; generators and currents for the SO(N) example; symmetry charges in the quantum theory; representation of the SO(N) symmetry in the Fock space; the phase symmetry and the net number of particles minus antiparticles; proof of the theorem; examples of Noether currents; translation symmetry and the stress-energy tensor; symmetrizing the Noether stress-energy tensor for non-scalar fields. - September 20 (Tuesday):
- Local phase symmetry:
local symmetry and covariant derivatives; gauge field and gauge transforms;
algebra of covariant derivatives; coupling charged scalar fields to electromagnetism.

Covariant Schroedinger equation.

Aharonov–Bohm effect: Aharonov–Bohm effect; cohomology of the vector potential; charge quantization and the compactness of the U(1) phase symmetry. - September 22 (Thursday):
- Magnetic monopoles:
Heuristic picture; Dirac construction; charge quantization; gauge bundles;
angular momentum in presence of a monopole; spin-statistics theorem for dyons.

Non-abelian local symmetries: Covariant derivatives and matrix-valued connections; non-abelian gauge transforms; Gell-Mann matrices and the component gauge fields; infinitesimal gauge transforms in components. - The extra lecture on September 23 (Friday):
- Vortices and other kinds of topological defects: domain walls as topological defects; co-dimension; rotation and vorices in superfluid; vortex energy; vortex rings and the critical velocity of the superflow; vortices in superconductors and the magnetic flux they carry; cosmic strings; vortices as topological defects of co-dimension=2; magnetic monopoles as topological defects (co-dimension=3); (briefly) Yang–Mills instantons.
- September 27 (Tuesday):
- Non-abelian local symmetries:
non-abelian tension fields; gauge transforms of the tension fields; the adjoint multiplet;
Yang–Mills theory and normalization of the gauge fields; gauge theories with matter.

Group theory: simple and semi-simple Lie groups; compact and non-compact groups; the adjoint multiplet and the Killing form; multiplets and representations. General gauge theories: symmetry groups and multiplets of fields; general local symmetry groups and Lie-algebra-valued gauge fields; covariant derivatives for different multiplets types; the adjoint multiplet; multiple gauge groups; Standard Model example. - September 29 (Thursday):
- General gauge theories:
symmetry groups and multiplets of fields;
general local symmetry groups and Lie-algebra-valued gauge fields;
covariant derivatives for different multiplets types; the adjoint multiplet;
multiple gauge groups; Standard Model example.

Lorentz symmetry: generators and representations; unitary but infinite particle representations; little groups and Wigner theorem; massive particles have definite spins; massless particles have definite helicities. - October 4 (Tuesday):
- Tachyons:
tachyons in QM; Wigner theorem for the tachyons;
tachyon field and vacuum instability; interactions and scalar VEVs (vacuum expectation values).

More Lorentz symmetry: Wigner theorem in*d*≠4 dimensions; Lorentz multiplets of fields; (j_{+},j_{−}) multiplets; Weyl spinors and Spin(3,1)=SL(2,**C**); vectors and bispinors; tensors.

Started Dirac spinors and spinor fields. - October 6 (Thursday):
- Dirac spinors and spinor fields:
Dirac spinor representation of the Lorentz symmetry; Dirac equation and its covariance;
Dirac conjugation and Dirac Lagrangian; Hamiltonian for the quantum Dirac field.

Grassmann numbers and classical limits of fermionic fields.

Started Fermionic algebra and Fock space: Hilbert stace of one fermionic mode; multiple modes. - The extra lecture on October 7 (Friday):
- Conformal symmetry:
definition; complex language in Euclidean 2D; conformal symmetry group and its generators;
conformal algebra in
*d*>2 dimensions, Euclidean or Minkowski.

Conformal field theories and their application: world-sheet QFT in string theory; condenced matter at a critical point; conformal window of QCD; AdS/CFT duality. - October 11 (Tuesday):
- Fermionic Fock space:
fermionic fock space; wave functions and operators;
particles and holes; holes as quasiparticles; Fermi sea, extra particles and holes.

Relativistic electrons and positrons: Naive diagonalization of the Dirac Hamiltonian; positrons as holes in the Dirac sea; expanding the Dirac fields into creation and annihilation operators.

charge conjugation symmetry:**C**:e^{−}↔e^{+};**C**:Φ(x)→Φ^{*}(x);**C**:Ψ(x)→γ^{2}Ψ^{*}(x); neutral particles and C–parity; Majorana fermions. - October 13 (Thursday):
- Dirac, Majorana, and Weyl fermions:
counting degrees of freedom; relations between Majorana and Weyl fermions;
Majorana mass term; massless and massive neutrinos.

Parity and other discrete symmetries: parity; CP; time reversal (briefly); CPT theorem; baryogenesys and Sakharov's criteria.

Begin vector, axial, and chiral symmetries. - The extra lecture on October 14 (Friday):
- Fermionic fields in different spacetime dimensions:
Dirac spinor fields; mass breaks parity in odd
*d*; Weyl spinor fields in even*d*only; LH and RH Weyl spinors; Majorana spinor fields in*d*≡0,1,2,3,4 (mod 8) only; Majorana–Weyl spinors in*d*≡2 (mod 8); complex, real, and pseudoreal representation; Bott periodicity for spinors of SO^{+}(a,b). - October 18 (Tuesday):
- Chiral symmetry: vector, axial, and chiral symmetries for Weyl fermions;
U(N)
_{L}×U(N)_{R}chiral symmetry; chiral gauge theories; electroweak example; chiral symmetry in QCD; Majorana and Dirac mass terms in chiral theories.

Relativistic causality for the fermions: commuting and anticommuting fields; checking anticommutativity of free Dirac fields at spacelike separation; spin-statistics theorem.

Feynman propagator for Dirac fermions.

Gave out the**midterm exam**. - October 20 (Thursday):
- Perturbation theory in QFT and Feynman diagrams:
the interaction picture of QM, the Dyson series, and the time-ordering;
the
**S**matrix and its elements; vacuum sandwiches of field products; diagramatics; combinatorics of similar terms; coordinate space Feynman rules; vacuum bubbles and their cancellation; momentum space Feynman rules; momentum conservation and connected diagrams; scattering amplidudes. - October 25 (Tuesday):
- Golden Rule and the phase-space factors:
Fermi's Golden Rule for transitions — derivation and an example;
going beyond the first order; decay rate — the matrix element and the phase space factors;
scattering of relativistic particles; calculating the phase space factors for the
2→2 scattering and 1→2 decays.

- Summary of Feynman rules for the λΦ
^{4}theory.

Phase-space factors.

Loop counting in perturbation theory: λΦ^{4}theory; adding a cubic coupling. - October 27 (Thursday):
- Feynman rules for multiple scalar fields.

Mandelstam's*s, t, u*variables.

Dimensional analysis: dimensions of fields and couplings; trouble with δ<0 couplings; types of δ≥0 couplings in 4D; other dimensions.

Intro to Quantum Electro Dynamics (QED): quantizing the EM fields, need to fix a gauge. - Likbez lecture on October 28 (Friday):
- Review of potential scattering in Quantum Mechanics: Scattering wave functions; Lippmann–Schwinger series; Born approximation; partial wave analysis.
- November 1 (Tuesday):
- Intro to Quantum Electro Dynamics (QED):
photon propagator in the Coulomb gauge; other gauges.

QED Feynman rules: propagators and vertices; external line factors; Dirac indexology; Gordon identities; sign rules.

Coulomb scattering in QED: diagrams and amplitudes; non-relativistic limit; recovering the Coulomb potential; electron-electron vs. electron-positron Coulomb scattering.

Yukawa theory (briefly): Yukawa theory and its Feynman rules; non-relativistic fermion scattering; Yukawa potential.

Started Muon pair production in electron-positron collisions, e^{−}+e^{+}→μ^{−}+μ^{+}: the tree amplitude; referral to homework#9 for the polarized cross-sections; intro to the unpolarized amplitudes. - November 3 (Thursday):
- Dirac trace technology:
Dirac traces for summing or averaging |M|
^{2}over spins; techniques for calculating the Dirac traces.

Muon pair production in electron-positron collisions, e^{−}+e^{+}→μ^{−}+μ^{+}: calculating the traces; summing over the Lorentz indices; the partial cross-section and its angular dependence; the total cross-section and its energy dependence.

Hadronic production e^{−}+e^{+}→hadrons: quark pair production and jets; the*R*ratio; QCD corrections.

Quick review of traceology in homework#9. - November 8 (Tuesday):
- crossing symmetry:
muon pair production vs. electron-muon scattering;
comparing spin-summed |M|
^{2}; comparing the ampitudes in the ultra-relativistic regime; analytically continuing the amplitudes; crossing symmetry in general; signs for crossed fermions; Compton vs. annihilation example.

Ward Identities: Ward identities for photons and gauge invariance of the amplitudes; sums over photon polarizations.

Electron-positron annihilation e^{−}+e^{+}→γγ: tree diagrams and the amplitude; checking the Ward identities; summing over the photons' polarizations and averaging over the fermions' spins; Dirac traceology; summary and annihilation kinematics; annihilation cross-section; crossing relation to the Compton scattering. - November 10 (Thursday):
- Compton scattering (briefly).

Spontaneous symmetry breaking: symmetric Lagrangian/Hamiltonian but asymmetric vacuum; continuous families of degenerate vacua; massless particles; linear sigma model; Wigner and Goldstone modes of symmetries; Goldstone–Nambu theorem and Goldstone bosons; spontaneous breakdown of approximate symmetries. - Extra lecture on November 11 (Friday):
- Resonances and unstable particles:
Breit–Wigner resonances in QM; resonances in QFT and unstable particles;
making a resonance in a collision; cross-sections and branching ratios;
quarkonia as resonaces in electron-positron collisions;
calculating σ(e
^{−}+e^{+}→J/ψ). - November 15 (Tuesday):
- Higgs mechanism:
SSB of a local U(1) symmetry; massive photon ‘eats’ the would-be Goldstone boson;
unitary gauge vs. gauge-invariant description;
non-abelian Higgs mechanism: SU(2) with a doublet; SU(2) with a real triplet; general case.

Glashow–Weinberg–Salam theory: the bosonic fields and the Higgs mechanism; the unbroken electric charge Q=T^{3}+Y; masses of the vector fields and the Weinberg's mixing angle; charged and neutral currents;~~Fermi's effective theory of weak interactions~~. - November 17 (Thursday):
- Glashow–Weinberg–Salam theory:
charged and neutral weak currents; Fermi's effective theory of weak interactions.

Fermions of the Glashow–Weinberg–Salam theory: Higgs origin of quark and lepton masses; charged and neutral weak currents of quarks and leptons.

Introduction to CKM matrix: Cabibbo mixing and Kaon decays; GIM mechanism and the charm quark; third family and the CKM matrix; the charged currents and the flavor-changing weak decays. - Extra lecture on November 18 (Friday):
- SSB of QCD's chiral symmetry and sigma models: Chiral symmetry of QCD and its spontaneous breakdown (χSB); pions as pseudo–Goldstone bosons; the linear sigma model of χSB; the non-linear sigma model; maybe general NLΣMs.
- November 14–25 (whole week):
- Fall break, no classes.
- November 29 (Tuesday):
- Quick review of quarks' and leptons' masses.

Origin of the CKM matrix: unitary charges of bases for each type of a fermion multiplet; matrices of Yukawa couplings; mass matrices for Weyls fermions; diagonalizing the mass matrices and forming the Dirac fermions; basis mismatch for charge +2/3 and charge -1/3 quarks and the Cabibbo–Kobayashi–Maskawa (CKM) matrix; neutral and charged weak currents.

Neutral Kaons: GIM box and K^{0}↔K^{0}mixing; K-long and K-short; CP eigenstates K_{1}and K_{2}, and their decays to pions; K-short regeneration; semi-leptonic decays of neutral kaons; strangeness oscillations; the CPLEAR experiment.

Introduction to CP violation: CPV in neutral kaon decays to pions; CPV in eigenstates and in decay rates; CPV in semi-leptonic decays. - December 1 (Thursday):
- CKM origin of CP violation:
CP symmetry of chiral gauge theories; CP action on the W
^{±}and on the charged currents; CP: CKM↔CKM^{*}; quark phases and CKM phases; third family, Kobayashi–Maskawa, and CP violation;*briefly*strong CP violation.

Neutrino masses: neutrino masses and oscillations; Dirac vs. Majorana masses of neutrinos; seesaw mechanism.

Give out the**final exam**.

- January 10 and January 12 (first week):
- Canceled.
- January 17 (Tuesday):
- Syllabus of the Spring semester (briefly).

Loop diagrams: amputating the external leg bubbles.

Calculating a one-loop amplitude: the diagrams; Feynman trick for denominators; Wick rotation to the Euclidean momentum space.

UV cutoff, long-distance effective field theories, and renormalization: UV divergence of the loop integral; effective long-distance / low-energy field theories; UV cutoff in condensed matter and in relativistic QFTs; bare and physical couplings; net one-loop amplitude in λΦ^{4}theory; bare and physical couplings at higher loop orders. - January 19 (Thursday):
- Overview of UV regulators:
Wilson's hard edge; Pauli–Villars; higher derivatives;
covariant higher derivatives; lattice (very briefly).

Dimensional regularization: basics; momentum integrals in non-integral dimensions;*d*→4 limit; (1/ε) as log(Λ_{UV}).

Quick introduction to partial wave analysis in QM and to optical theorem. - January 20 (Friday) [make-up lecture]:
- Optical theorem:
proof from unitarity of the S matrix; application to the
Im M
_{1 loop}in the λφ^{4}theory; mentioned cutting diagrams and putting cut propagators on-shell (details in homework).

Started correlation functions of quantum fields: correlation functions; relation to the free fields and evolution operators. - January 24 (Tuesday):
- Correlation functions of quantum fields:
Feynman rules; connected correlation functions.

The two-point correlation function: Källén–Lehmann spectral representation; features of the spectral density function; analytic two-point function 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.

Perturbation theory for the two-point function: resumming the 1PI bubbles; Σ(*p*^{2}) and the renormalization of the mass and of the field strength; mass renormalization in the λφ^{4}theory; fine tuning problem. - January 26 (Thursday):
- Quadratic UV divergences:
regulator dependence; dimensional regularization of quadratic divergences.

Field strength renormalization in the Yukawa theory: calculating the one-loop diagram: the trace, the denominator and the numerator, Wick rotation; the UV divergence structure: Σ(*p*^{2})=(div.constant)+(div.constant)×*p*^{2}+finite_*f*(*p*^{2}); calculating the Σ(*p*^{2}); dΣ/d*p*^{2}and the scalar field strength renormalization; optical theorem for the unstable particles.

Started the Lehmann–Symanzik–Zimmermann reduction formula:*n*-point correlators F_{n}(p_{1},…p_{n}) and their poles at on-shell momenta; LSZ reduction formula for the common residue; the Feynman diagram explanation of the poles; the amputated diagrams and the scattering amplitudes. - January 27 (Friday) [make-up lecture]:
- Deriving the LSZ reduction formula:
The
*x*→±∞ 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⟩.^{0}_{i}

Started the counterterm perturbation theory: ℒ_{bare}=ℒ_{physical}+counterterms; Feynman rules for the counterterms; adjusting δ^{Z}, δ^{m}, and δ^{λ}order by order in λ; one-loop examples; finite parts of the counterterms. - Regular lecture on January 31 (Tuesday):
- Canceled due to bad weather.
- Extra lecture on January 31 (Tuesday) [instead of the canceled regular lecture]:
- Vacuum energy and effective potentials: zero-point energy and the Casimir effect; zero-point energy for fields with VEV-dependent masses; Feynman diagrams for the vacuum energy; one-loop calculation; general Coleman–Weinberg effective potential; Higgs mechanism induced by the Coleman–Weinberg potential; formal derivation of the Feynman diagrams.
- February 2 (Thursday):
- Counterterms and canceling the UV divergences:
superficial degree of divergence; graphs and subgraphs;
classifying divergent graphs, subgraphs, and amplitudes; canceling overall divergences;
subgraph divergences and their cancelation in situ;
nested and overlapping divergences; BPHZ theorem.

Divergences and renormalizability: divergences of φ^{n}theories; super-renormalizable φ^{3}theory; renormalizable φ^{4}theory. - February 3 (Friday) [make-up lecture]:
- Divergences and renormalizability:
non-renormalizable φ
^{n>4}theories; general super-renormalizable, renormalizable, and non-renormalizable theories; trouble with non-renormalizability.

Dimensional analysis and renormalizability: canonical dimensions of fields and couplings; power-counting renormalizability; renormalizable theories in 4D; other dimensions.

- February 7 (Tuesday):
- QED perturbation theory:
the counterterms and the Feynman rules; divergent amplitudes and their momentum dependences;
missing counterterms and Ward–Takahashi identities; dressed electron propagator; dressed photon propagator;
finite parts of the counterterms.

Electric charge renormalization: calculating the 1-loop Σ^{μν}(*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(*k*); the divergence and the δ^{2}_{3}counterterm; the finite result for the one-loop-order Π(*k*).^{2} - February 9 (Thursday):
- Electric charge renormalization:
loop corrections to Coulomb scattering and other high-momentum processes;
effective QED coupling α
_{eff}(E) and its running with log(energy).

Ward–Takahashi identities: the identities; current conservation in quantum theories; contact terms; formal proof of WT indentities;*Z*; multiple charged fields._{1}=Z_{2} - February 10 (Friday) [make-up lecture]:
- Diagramatic proof of Ward–Takahashi identities: WTI for tree-level 2-electron amplitudes; WTI for one-loop photonic amplitudes; WTIs for multi-loop diagrams (in the bare perturbation theory); WTIs in the counterterm perturbation theory (outline).
- February 14 (Tuesday):
- Form factors:
probing nuclear and nucleon structure with electrons; form factors;
on-shell form-factors
*F*_{1}(*q*^{2}) and*F*_{2}(*q*^{2}); the gyromagnetic ratio.

Dressed QED vertex at one loop: the dressed vertex and the form factors; the one-loop diagram and its denominator; numerator algebra; calculating the*F*_{2}form factor and the anomalous magnetic moment; the experimental and the theoretical electron's and muon's magnetic moments at high precision. - February 16 (Thursday):
- The electric from factor and the infrared divergence:
momentum integral for the
*F*_{1}(*q*^{2}); 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 δ_{1}counterterm; momentum dependence of the IR divergence; Sudakov's double logarithms.

Virtual and real soft photons: IR divergence of exclusive cross-sections due to virtual soft photons: IR divergence of the soft-photon bremmsstrahlung; finite inclusive cross-sections (with or without soft photons); detectable vs. undetectable photons, the observed cross-sections, and their finiteness; briefly: higher loops and/or more soft photons.

Bremsstrahlung: classical bremsstrahlung; flat frequency spectrum; implication for the quantum theory and fo r the IR divergences of QED; soft-photon bremsstrahlung in QED (tree-level); verifying*I(p,p′)*=2f_{IR}(*q*^{2}). - February 17 (Friday) [make-up lecture]:
- 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; soft and collinear gluons in QCD; jets in theory and in experiment.

Gauge dependence in QED: gauge-dependent off-shell amplitudes and counterterms; δ^{1}(ξ)=δ^{2}(ξ). - February 21 (Tuesday):
- Intro to renormalization group:
large log problem for
*E≫m*; running coupling λ(E); off-shell renormalization schemes for couplings and counterterms.

Renormalization group basics: anomalous dimensions of quantum fields; calculating γ_{φ}in the λφ^{4}theory; higher-loop corrections to the γ_{φ}; running couplings and β-functions; relating β-functions to the counterterms; β(λ) in the λφ^{4}theory; solving the renormalization group equation for the λ(E); anomalous dimensions and the β-function in QED; solving the renormalization group equation for QED. - February 23 (Thursday):
- Renormalization group with multiple couplings:
general formulae for the β-functions; Yukawa theory example.

Types of RG flows: β>0, Landau poles, and UV incompleteness; β<0, QCD example, and asymptotic freedom; Λ_{QCD}; non-perturbative strong interactions at low energies.

Chromomagnetic monopole condensation and quark confinement. - Extra lecture on February 24 (Friday):
- Grand Unification: unifying the EM, weak, and strong interactions in a single non-abelian gauge group; SU(5) example; multiplets of fermions; gauge couplings and Georgi–Quinn–Weinberg equations; the doublet-triplet problem; baryon decay and other exotic processes.
- February 28 (Tuesday):
- Fixed points β(
*g*^{*})=0 of RG flows: scale invariance and conformal symmetry; UV stability vs. IR stability; Banks–Zaks conformal window of QCD.

RG flows in spaces of multiple couplings: Yukawa example; RG flows in the coupling space; fixed points and attractive lines.

Direction of the flow: IR to UV or UV to IR? - March 2 (Thursday):
- Relevant, irrelevant, and marginal operators; effective field theories.

Renormalisation schemes: scheme dependence of the couplings and the β-functions; the minimal subtraction schemes MS and MS-bar; extracting β-functions from residues of the 1/ε poles.

Gave out the midterm exam. - March 7 (Tuesday):
- Introduction to path integrals:
path integrals in QM; the Lagrangian and the Hamiltonian forms of the path integral;
derivation of the Hamiltonian form; the Lagrangian path integrals — derivation and normalization;
the partition function; harmonic oscillator example.

Functional integrals in QFT: “path” integrals for quantum fields; correlation functions; briefly free fields and propagators. - March 9 (Thursday):
- Functional integrals in QFT:
free fields and propagators; perturbation theory and Feynman rules; sources and generating functionals;
log Z[J] generates the connected correlation functions.

Euclidean path integrals: convergence problems of path integrals; Euclidean time; discretization; harmonic oscillator example.

QFT and StatMech: Functional integrals in Euclidean spacetime; QFT↔StatMech analogy; coupling as temperature. - March 11–19 (whole week):
- Spring break, no classes.
- March 21 (Tuesday):
- QFT on a lattice:
QFT↔StatMech analogy; discrete lattice as a UV cutoff;
recovering rotational / Lorentz symmetry in the continuum limit; custodial symmetries.

Fermionic functional integrals: Grassmann numbers; Berezin integrals; Gaussian integrals over fermionic variables; functional integrals over fermionic fields; free Dirac field in Euclidean spacetime.

Integrating over fermion fields in QED. functional integral in EM background: the determinant, and the source term; - March 23 (Thursday):
- Integrating over fermion fields in QED:
functional integral in EM background: the determinant, and the source term;
Det(D̸+m) and the electron loops; 1/(D̸+m) and the tree diagrams.

Functional integral for the EM field: gauge transforms as redundancies; gauge-fixing conditions and the Faddeev–Popov determinants; Landau-gauge propagator from the functional integral; gauge-averaging, gauge-fixing terms, and the Feynman gauge.

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

QCD Feynman rules: physical, ghost, gauge-fixing, and counter terms in the Lagrangian; propagators; physical vertices; counter-term vertices; handling the color indices of quarks. - Extra lecture on March 24 (Friday):
- Gauge theories on the lattice (abelian):
local U(1) symmetry on the lattice; gauge fields and link variables;
covariant lattice derivatives; plaquettes and tension fields; lattice EM action;
lattice ‘path’ integrals; compact QED.

Non-abelian lattice gauge theories: non-abelian gauge symmetries and link variables; covariant lattice symmetries; non-abelian plaquettes and tension fields; lattice YM action; integrals over link variables and the lattice ‘path’ integrals; brief history and applications of lattice QCD. - March 28 (Tuesday):
- QCD Ward identities:
on-shell QCD Ward identities are weaker than in QED;
*q+q̄→g+g*example: 3 tree diagrams and*k*_{μ}M^{μν}(1+2); the first 2 diagrams; the third diagram; Ward identity holds for one longitudinal quark only; production of longitudinal quarks is canceled by the ghost-antighost pair production.

Introduction to BRST symmetry: BRST transforms of QCD fields; nilpotency; BRST invariance of the net Lagrangian; physical and unphysical quanta in the QCD Fock space and BRST cohomology; reducing 1-particle states to physical particles; multi-particle physical states and the S-matrix; BRST symmetries of the amplitudes and cancellation of unphysical processes.

QCD renormalizability: modern approach; manifest symmetries; the bare Lagrangian; Slavnov–Taylor identities for the QCD counterterms. - March 30 (Thursday):
- QCD renormalizability:
modern approach; manifest symmetries; the bare Lagrangian; Slavnov–Taylor identities for the QCD counterterms.

QCD beta-function: relation to the counterterms; calculating the one-loop δ_{2}for the quarks; calculating the one-loop δ_{1}for the quarks: the QED-like loop and the non-abelian loop; calculating the one-loop δ_{3}for the gluons: the quark loop, the gluon loop, the sideways gluon loop, the ghost loop, the summary; β(QCD). - Extra lecture on March 31 (Friday):
- Wilson loops: Abelian and non-abelian Wilson loops; large loops and forces between probe particles; non-abelian probe particles; area law vs. perimeter law as test of confinement vs. deconfinement; examples.
- April 4 (Tuesday):
- Beta-function of general gauge theories:
β(QCD); generalizing to other gauge theories.

Axial anomaly: axial symmetry of massless electrons; anomaly and its origin in the path integral measure; the diagrams, the naive cancelation, and the regulation problem; Adler–Bardeen theorem; Pauli–Villars regulation of the anomaly; calculating the loop of the PV compensator; net axial anomaly. - April 6 (Thursday):
- Axial anomaly:
anomaly of the measure of the fermionic functional integral;
generalization to the non-abelian gauge theories;
triangle and quadrangle anomalies in QCD;
in QCD U(1)
_{A}is anomalous but SU(N_{f})_{L}×SU(N_{f})_{R}are anomaly-free; spontaneous axial symmetry breaking and the issue with the η and π mesons.

Non-linear sigma models: non-linear field spaces; NLΣM of the chiral symmetry breaking; vector and axial currents. - Extra lecture on April 7 (Friday):
- Instantons:
topological sectors and non-perturbative effects;
topological sectors in the Yang–Mills theories;
topological index I[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; multiple instantons, cluster expansion, and the Θ angle. - April 11 (Tuesday):
- F
_{π}and the decay of a charged pion.

Non-linear sigma models in QCD context: quark-antiquark condensation and the spontaneous chiral symmetry breaking; quark masses as perturbations; 2-flavor and 3-flavor models; axial anomaly as a perturbation, η and η′ mesons.

Electromagnetic anomalies: electromagnetic anomalies of quarks' symmetries; anomalous decays of the neytral pion π^{0}→γγ.

Anomalies in chiral gauge theories: Weyl fermions and chiral currents; non-abelian chiral gauge theories; triangle and quadrangle diagrams for Weyl fermions; trace formula for the net chiral anomaly; QCD and SU(2)_{W}examples; SU(2)_{W}anomalies of baryon and lepton numbers - April 13 (Thursday):
- Anomalies of baryon and lepton numbers:
ΔB=ΔL=3×Index;
instantons and sphalerons; leptogenesis.

Gauge anomalies: triangle anomaly in chiral QED and its effect on Ward identities; anomalous gauge variance of log(det(̸D)); anomaly in non-abelian chiral theories; triangle diagrams; quadrangle diagrams; net non-abelian anomaly; anomaly coefficients A^{abc}and traces over chiral fermions.

Anomaly cancellation in chiral gauge theories: anomaly in simple gauge groups; cubic Casimir and Cibic index; counting amonaly indices; SU(5) GUT example; anomalies in product gauge groups; 5 anomaly types in the Standard Model;~~checking anomaly cancellation in the Standard Model; importance of tr(~~.*Q*_{el})=0 - Extra lecture on April 14 (Friday):
- Instantons and fermions: instantons and axial anomaly; zero modes in instanton background; Atyah–Singer index theorem; zero modes in fermionic integrals chiral anomaly of the Θ angle; Θ=Θ+phase(det(quark mass matrix)); the strong CP problem; neutron's electric dipole; Peccei–Quinn symmetry.
- April 18 (Tuesday):
- Anomaly cancellation:
checking anomaly cancellation in the Standard Model; importance of tr(
*Q*_{el})=0; abelian and non-abelian anomalies; massive fermions do not contribute to the anomalies.

Quick overview of differential forms: forms and antisymmetric tensor fields; exterior derivative; closed and exact forms; gauge fields A and F as forms; nonabelian gauge fields as forms.

Differential forms for the anomalies: anomaly forms; Chern–Simons forms and their uses; descent equations;*d+2*anomaly forms for the nonabelian anomalies in*d*dimensions.

Symmetric traces of degree*(d+2)/2*for the anomalies; rules for anomaly cancellation in 2D; harder-to-satisfy rules for 6D and 10D. - April 20 (Thursday):
- Wess–Zumino terms:
flavor anomalies of the NLΣM;
WZ term from 4D and 5D points of view;
adding flavor gauge fields; flavor anomaly cancellation.
't Hooft's anomaly matching conditions:
confiniement and χSB in chiral gauge theories; SU(5) example;
surviving chiral symmetry and massless composite fermions;
flavor anomaly matching.

Gave out the final exam.

Last Modified: April 22, 2023. Vadim Kaplunovsky

vadim@physics.utexas.edu