- August 30 (Thursday):
- 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 principle of least action and the Euler-Lagrange equations).
- September 4 (Tuesday):
- Classical field theory: Euler-Lagrange field equations; relativistic notations; scalar field examples.
The electromagnetic fields: the
*A*potentials and^{μ}*F*tension fields; gauge transforms; the EM Lagrangian and the Maxwell equations in the 4D language.^{μν} - September 6 (Thursday):
- EM: current conservation, gauge invariance, and the number of degrees of freedom. Symmetries in field theories: examples and general rules; group theory; kinds of symmetry (global or local, discrete or continuous). Symmetries and conserved currents — Noether theorem.
- September 11 (Tuesday):
- Noether theorem: general proof; conserved current for the phase symmetry;
conserved currents for the SO(
*N*). Translations of spacetime and the stress-energy tensor*T*. Local phase symmetry: the gauge fields^{μν}*A*and the covariant derivatives._{μ} - September 13 (Thursday):
- Local phase symmetry: covariant derivatives; Aharonov–Bohm effect; magnetic monopoles and charge quantization.
- September 18 (Tuesday):
- Non-abelian local symmetries: covariant derivatives and the non-abelian gauge fields; non-abelian tensions; Yang–Mills theory; normalization of gauge fields.
- September 20 (Thursday):
- General gauge theory: Lie groups, Lie Algebras, and their representations; covariant derivatives for general multiplets; multiple gauge group factors.
- September 25 (Tuesday):
- Review of canonical quantization: Hamiltonian classical mechanics; classical variables and quantum operators;
Poisson brackets and commutatots; Heisenberg–Dirac equations.
Quantization of the scalar field: Hamiltonian formalism; quantum fields φ̂(
**x**,*t*) and π̂(**x**,*t*) and their commutation relations; Schroedinger and Heisenberg pictures of the quantum fields; quantum Klein–Gordon equation. - September 27 (Thursday):
- Quantum fields, multi-oscillator systems, and identical bosons:
expansion of the relativistic scalar field into creation and annihilation operators;
energy spectrum of the free field; the multi-oscillator system and the occupation numbers;
identical bosonic particles; the Fock space.

General identical bosons (relativistic or not): multi-boson Hilber spaces, the Fock space, and its interpretation as a multi-oscillator system. - October 2 (Tuesday):
- General identical bosons: creation and annihilation operators;
one-body and two-body operators in the Fock space;
non-relativistic quantum fields; «second quantization».

Liquid helium II and other Bose–Einstein condensates: non-relativistic QFT; the semiclassical limit and the Landau–Ginzburg theory for the consensate; gradient flow of the condensate. - October 4 (Thursday):
- Semi-classical behavior in QM and in QFT:
stationary states vs. moving wave packets; semiclassical behavior for highly excited states;
coherent states; coherent multi-particle states in QFT; emergence of semiclassical fields.

Bose–Einstein condensation: ground state with ‘φ̂’≠0; elementary excutations; Bogolyubov transform. - October 5 (Friday) [supplementary lecture]:
- Superfluidity: Hamiltonian of a moving superfluid; negative-energy excitation and dissipations; critical velocity and dissipationless flow; vortex rings and the critical velocity of macroscopic superflow.
- October 9 (Tuesday):
- Spontaneously broken symmetries: shifting scalar fields; why SSB does not happen in QM but happens in QFT; spontaneous breakdown of continuous symmetries; Nambu–Goldstone theorem; SO(N) example (the linear sigma model).
- October 11 (Thursday):
- Higgs mechanism: spontaneous breakdown of a local U(1); massive photon "eats" the would-be Goldstone boson; unitary gauge vs. gauge-invariant description; non-abelian higgs mechanism; SU(2) example; electroweak SU(2)⊗U(1).
- October 16 (Tuesday):
- Electroweak theory: the electric current and the weak currents; Fermi theory of (low-energy)
weak interactions.

Relativistic fields in spacetime: Relatifistic normalization of particle states and operators; free scalar field in the Heisenberg picture; charged fields, particles, and antiparticles; general free fields and their plane waves; decomposition of general fields into creation/annihilation operators. - October 18 (Thursday):
- Relativistic causality: signals and local operators;
physical local operators must commute at spacelike separations;
proof of causality for the free scalar field; problems with single-particle relativistic QM.

Feynman proparator for the scalar field: time-ordering of operators; calculating the propagator; the propagator as a Green's function of the Klein–Gordon equation. - October 19 (Friday) [supplementary lecture]:
- Superconductivity: Cooper pairs and the Landau–Ginzburg effective theory; the supercurrent; Meissner effect as a non-relativistic Higgs effect; quantization of magnetic fluxes; Josephson junctions and SQUIDs.
- October 23 (Tuesday):
- Green's functions: momentum-space formula;
regulating ∫ dk
_{0}by moving the poles off the real axis; evaluating ∫ dk_{0}by contour integration; Feynman's poles and the Feynman propagator; other Green's functions; Feynman propagators for non-scalar fields.

Lorentz symmetry: the generators and their commutation relations; field multiplets and particle multiplets; polarization states of particles and the little group of the momentum. - October 25 (Thursday):
- Lorentz multiplets of particles and the Wigner theorem:
Wigner theorem — massive particles have spins, massless particles have helicities,
tachyons have nothing;
massless photons and massless neutrinos; Wigner theorem in general dimensions.

Lorentz multiplets of fields: general form; plane waves and particle polarizations; physical plane waves vs. gauge artefacts.

Gave out the__midterm exam__. - October 30 (Tuesday):
- Dirac spinor fields:
Dirac spinor multiplet; Dirac spinor field and Dirac equation; Dirac Lagrangian.

Quantum Dirac fields: the anticommutation relations; Dirac Hamiltonian; expansion into modes; fermionic creation and annihilation operators. - November 1 (Thursday):
- Fermionic Fock space:
single fermionic mode; multiple modes and the Fermi statistics; the Fock space;
one-body and two-body additive operators for the fermions.

Particle-hole formalism: negative-energy modes and filled-up states; holes; creation and annihilation operators for the holes; Fermi sea; quasiparticles in the Fermi sea; applications to condensed matter, to atoms, and to nuclei.

Electrons and positrons: Dirac sea; postrons as holes; modern point of view.

Collected the__midterm exams__. - November 6 (Tuesday):
- Quantum Dirac field: expansion into electron and positron operators; charge conjugation symmetry; relativistic causality for fermions; Feynman propagator for the Dirac field.
- November 8 (Thursday):
- Feynman propagator for the Dirac field.

Chiral symmetries: vector and axial currents; chiral symmetry and Weyl fermions; chiral symmetry in QCD; gauged chiral symmetry and violation of**P**and**C**symmetries. - November 9 (Friday) [supplementary lecture]:
- Spin–Statistics Theorem: relation between the spin and the statistics; assumptions of the theorem; the proof; generalization to other dimensions.
- November 13 (Tuesday):
- Weak interactions of fermions:
the quarks, the leptons, and their
*chiral*SU(2)⊗U(1) quantum numbers; Yukawa couplings and the masses of quarks and leptons; weak currents;**P**and**C**violation; flavor changing in weak interactions and the Cabibbo–Kobayashi–Maskawa matrix;**CP**violation;~~neutrino masses and neutrino oscillations~~. - November 15 (Thursday):
- Perturbation theory and Feynman diagrams:
the interaction picture of QM and the Dyson series; the
**S**matrix and its elements; vacuum sandwiches, propagators, and the Feynman diagrams; combinatorics; Feynman rules in the coordinate space; vacuum bubbles and connected diagrams; Feynman rules in the momentum space. - November 16 (Friday) [supplementary lecture]:
- Grand Unified Theories: unifying strong, weak, and EM interactions in the same gauge group; multiplets of fermions; the SU(5) example; the gauge couplings and the Georgi-Quinn-Weinberg equations; baryon decays and other exotic processes.
- November 20 (Tuesday):
- Feynman rules:
connected diagrams and the scattering amplitudes;
loop counting in perturbation theory; Feynman rules for multiple fields and couplings.

Dimensional analysis: canonical dimensions of fields and couplings; UV problems with negative-dimension couplings; allowed couplings in 4D;~~allowed couplings in other dimensions~~. - November 27 (Tuesday):
- Renormalizable couplings in
*d*≠4.

Mandelstam variables*s*,*t*, and*u*.

Quantum EM fields: free**Ê**and**B̂**fields; gauge fixing the Â^{μ}potentials; photon propagator in the Coulomb gauge; photon propagator in general gauges; Landau and Feynman gauges.

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

- November 29 (Thursday):
- Coulomb scattering: tree diagrams and amplitudes; the non-relativistic limit;
reconstructing the Coulomb potential; scattering and potential for e
^{-}e^{+}.

Dirac traceology and muon pair production e^{-}e^{+}→μ^{-}μ^{+}: un-polarized amplitudes and Dirac traces; techniques for calculating the traces; muon pair production. - November 30 (Friday) [supplementary lecture]:
- Fermionic fields in different dimensions:
Dirac spinors in general
*d*; mass vs. parity in odd*d*; Weyl spinors in even*d*; conjugate Weyl spinors; charge conjugation and Majorana spinors in*d*=0,1,2,3,4 mod 8 only; Majorana–Weyl spinors in*d*=2 mod 8. - December 4 (Tuesday):
- Quark pair production and the R ratio.

Crossing symmetries: muon pair production vs. electron-muon elastic scattering; analytically continued amplitudes; general crossing symmetries; annihilation vs. Compton scattering.

Ward identities and sums over photons' polarizations. - December 6 (Thursday):
- Electron-positron annihilation:
the diagram and the amplitudes; checking the Ward identities;
summing over photon polarizations and averaging over fermion spins;
Dirac traceology; kinematics in the CM frame.

Compton scattering: crossing relation to the annihilation; kinematics in the lab frame; Klein–Nishina formula.

Gave out the__final exam__.

- January 15 (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. - January 17 (Thursday):
- Renormalization: bare coupling versus physical coupling; canceling the UV divergence;
finite amplutudes in terms of finite physical couplings; perturbative expansion beyond 1 loop.

UV regulators: hard-edge cutoff; Pauli–Villars regulator; higher-derivative regulator; comparing the cutoff scales Λ;~~dimensional regularization~~. - January 22 (Tuesday):
- Dimensional regularization.

Intro to renormalization group: high-energy scattering; re-summing of leading powers of log(E); effective coupling; renormalization group equation.

Optical theorem: partial waves, phase shifts, and optical theorem; formal proof from unitarity of the S matrix; application to scalar scattering; calculation of ℑℳ at one loop. - January 24 (Thursday):
- Correlation functions of quantum fields:
definition; perturbation theory; Feynman rules; the
*connected*correlation functions.

The two-field correlation function F_{2}*(x-y)*: Källén–Lehmann spectral representation; poles in momentum space and their relation to physical particles' masses and to the field strength. - January 29 (Tuesday):
- Analytic two-point function F
_{2}(*p*^{2}): poles and particles; branch cuts and the multi-particle continuum; physical and un-physical sheets; resonances.

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.

1-loop examples: mass renormalization in the φ^{4}theory; quadratic UV divergences and how to regulate them. - January 31 (Thursday):
- Field strength renormalization in the Yukawa theory:
the one-loop diagram; the trace; the denominator and the numerator;
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}

Lehmann–Symanzik–Zimmermann (LSZ) reduction formula: amputated cores and external leg bubbles; correlation functions F_{n}(*p*) and their poles; origin of the poles;_{1},…p_{n}*x*→±∞ limits and the asymptotic |in⟩ and ⟨out| states; the LSZ reduction formula; scattering amplitudes and the amputated diagrams.^{0}_{i} - February 5 (Tuesday):
- Counterterms perturbation theory:
ℒ
_{bare}=ℒ_{phys}+counterterms; 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*; nested and overlapping divergences; BPHZ theorem. - February 7 (Thursday):
- 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.

Counterterm perturbation theory for the QED: the counterterms and the Feynman rules. - February 12 (Tuesday):
- QED perturbation theory:
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 14 (Thursday):
- Renormalization of the electric charge unit
*e*: dressing up the photon propagator F^{μν}(*k*); quantum corrections to Coulomb scattering of heavy charged particles; the effective α_{QED}(E) at high energies; the Landau pole.

Ward–Takahashi identities: formulation of the basic WT identities and the outline of the proof. - February 19 (Tuesday):
- Proving the Ward–Takahashi identities.

General Ward–Takahashi identities and their relation to the electric current conservation. - February 21 (Thursday):
- Ward identity for the counterterms.

Infrared divergences of QED: one-shell IR divergences of the Σ(p̸) and Γ^{μ}(p',p); the soft-photon bremsstrahlung; cancelation of the IR divergence from the physical cross-sections. - February 26 (Tuesday):
- Electric and magnetic form factors of fermions.

One-loop QED vertex correction: vertex algebra; calculating the*F*_{2}form factor; the anomalous magnetic moment of the electron. - February 28 (Thursday):
- QED vertex correction: the electric form factor
*F*_{1}and its infrared divergence.

Optical theorem for the infrared divergences.

Gauge dependence of the counterterms and other off-shell quantities. - March 1 (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.
- March 5 (Tuesday):
- Symmetries and the counterterms:
δ
_{m}∝m in QED and the approximate axial symmetry; general rules for counterterms protected and un-protected by symmetries.

Introduction to the renormalization group: large logarithms at high energies; the `runnung' effective coupling λ(E); the off-shell renormalization conditions for the counterterms; the anomalous dimension; the β function and the renormalization group equation; solving the one-loop RGE. - March 7 (Thursday):
- Renormalization group for QED: anomalous dimensions for the
*A*^{μ}and Ψ fields; the beta finction; solving the RGE; initial condition for the RGE and the threshold corrections.

Renormalization group for the Yukawa theory: the anomalous dimensions; the beta functions β_{λ}and β_{g}; solving 2 coupled RG equations. - March 19 (Tuesday):
- Types of renormalization group flows:
β>0, Landau poles, and breakdown of perturbation theory at short distances;
flat β≡0 and scale invariance; conformal symmetry;
β<0 and asymptotic freedom; QCD example; dimensional transmutation;
non-perturbative behavior at long distances; confinement in QCD;
β(
*g*) changing sign at*g=g*and RG fixed points; UV-stable vs. IR-stable fixed points; Banks–Zaks example and the conformal window; RG flows for multiple couplings; UV vs IR starting points of the RG flow.^{*} - March 21 (Thursday):
- RG flow from UV towards IR: relevant, marginal, and irrelevant operators;
emergence of renormalizable effective theories in deep IR.

Renormalization schemes: scheme-dependence of couplings and beta functions; one-loop and two-loop βs are universal, higher orders are scheme-dependent.

Minimal subtraction: definition of the MS scheme; recursive formulae for the higher-order poles 1/ε^{n}in terms of the simple poles; formulae for the β functions; the MS-bar scheme.

Gave out the__midterm exam__. - March 26 (Tuesday):
- Path integrals in quantum mechanics:
path integrals for the evolution kernels;
deriving the phase-space path integrals from the ordinary QM;
the Lagrangian path integrals and their normalization;
the partition function Z; calculating Z for the harmonic oscillator.

Functional quantization in QFT: path integrals for the transition matrix elements; functional integrals for the correlation functions of quantum fields; the free field example; deriving Feynman rules from the functional integrals. - March 28 (Thursday):
- The generating functional for the connected correlation functions.

The Euclidean path integrals: analogies between QFT and Statistical Mechanics; convergence issues for path integrals; rotating the time axis*t→-ie*; Euclidean action; Feynman rules in Euclidean space._{e}

Field theories on the lattice: discretizing the Euclidean path integral; the discretized action for the scalar theory; the lattice as a UV regulator; SO(4) symmetry of the IR limit of the lattice theory; λ coupling acting as temperature.

Collected the__midterm exam__. - April 2 (Tuesday):
- Finite temperature and periodic Euclidean time.

Fermionic integrals: functions of odd Grassmann numbers; Berezin's integrals ∫dθ; Gaussian integrals for the fermions.

Path integrals over the fermionic fields: free Dirac fermions; fermionic path integral in QED; Det(iD̸-m) and the fermionic loops; fermionic sources and the open fermionic lines.

Path integrals over the gauge fields: trouble with the photon propagator; gauge fixing; the Landau gauge; constraints and Jacobians; the Faddeev–Popov determinant; path integrals over the gauge-fixed A^{μ}(x). - April 4 (Thursday):
- Gauge fixing and photon propagators; the Landau gauge; gauge averaging and the Feynman gauge.

Non-abelian gauge theories: fixing the non-abelian gauge symmetry; the Faddeev–Popov ghost fields; the gauge-fixed QCD Lagrangian; QCD Feynman rules; generalization to other gauge theories.

- April 5 (Friday) — supplementary lecture:
- Gauge theories on the lattice:
local symmetry on the lattice; U(link) variables and the A
^{μ}(x) fields; plaquettes, U(□) and the F^{μν}(x); lattice EM action; non-abelian local symmetries on the lattice; non-abelian plaquettes; lattice Yang--Mills theory;~~computer simulations~~. - April 9 (Tuesday):
- Finish QCD Feynman rules.

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 symmtries of the amplitudes; cancellation of unphysical processes. - April 11 (Thursday):
- QCD renormalization: symmetries of the gauge-fixed theory;
allowed counterterms and the BRST relations between them;
the β function and the δ
_{1}, δ_{2}, and δ_{3}counterterms; Lie–algebra factors in loop diagrams; Casimirs and indexes of group multiplets; calculating the δ_{1}and δ_{2}counterterms. - April 16 (Tuesday):
- Calculating the δ
_{3}counterterm and the β function for QCD and other non-abelian gauge theories.

Asymptotic freedom, dimensional transmutation, and Λ_{QCD}; chromo-electric field lines, flux tubes and confinement; mesons, baryons, and glueballs; flux tubes as*hadronic strings*; flux tubes from condensation of chromo-magnetic monopoles. - April 16 (Thursday):
- Chiral symmetry in QCD: spontaneous chiral symmetry breaking by the ⟨Ψ̄Ψ⟩ condensate; the non-linear sigma model; the quark masses and the pseudoscalar meson masses; the eta-meson's mass and the anomaly of the axial U(1) symmetry.
- April 19 (Friday) — supplementary lecture:
- Wilson loops: abelian and non-abelian Wilson loops; dependence on charge / multiplet type; large loops and signs of confinement; Wilson loops and forces between probe particles; non-abelian probe particles.
- April 23 (Tuesday):
- Axial Anomaly:
massless QED example;
*naive*Ward identities for the axial J^{μ}; the triangle graphs and the problems with their UV regulators; calculating the anomaly using the Pauli–Villars regulator; axial anomalies of multiple charge fermions; decay of the π^{0}meson to 2 photons.

- April 25 (Thursday):
- Announced how to prepare to my SUSY class in the Fall 2013.

Axial anomalies in QCD; Chern--Simons form.

Axial anomaly of the fermionic path integral: formal analysis, regulation, and calculation.

Chiral gauge theories: formulation; loop diagrams; ‘flavor’ anomalies; gauge anomalies.

- April 26 (Friday) — supplementary lecture:
- Strong coupling expansion in lattice gauge theories: basic idea; expansion of the Wilson loop in compact QED; quark Wilson loop in QED; Wilson loop for an adjoint probe; relation to confinement.
- April 30 (Tuesday):
- Chiral anomaly cancellation:
complex and real multiplets; massive fermions do not contribute to the anomaly;
anomaly cancellation in the Standard Model; anomaly cancelation in GUTs;
anomalies of global symmetries; non-conservation of baryon and lepton numbers.

Topological sectors of Yang–Mills theories: topological index I[A^{μ}] and the Chern--Simons form; integer I for configurations with finite Euclidean action; splitting the path integral into sectors. - May 2 (Thursday) — last lecture:
- Instantons:
self-dual gauge fields; 't Hooft's instanton; multi-instanton configurations;
the vacuum angle Θ.

Instantons and fermions: chiral anomaly of the Θ angle; Θ̅=Θ+phase(det(mass matrix)); the strong CP problem; fermionic zero modes in instanton background; the index theorem; relation to the axial anomaly.

Gave out the__final exam__.

Last Modified: May 2, 2013. Vadim Kaplunovsky

vadim@physics.utexas.edu