- August 26 (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 least action principle and the Euler-Lagrange equations).
- September 1 (Tuesday):
- Classical fields; Euler-Lagrange field equations; relativistic notations; Klein-Gordon equation.

Electromagnetic fields: the vector potential and the scalar potential; gauge transforms; the 4-vector field*A*; the tension fields^{μ}**E**and**B**and the 4--tensor*F*.^{μν} - September 3 (Thursday):
- Lagrangian formalism for the electomagnetic fields:
Jacobi identities; the EM Lagrangian density; the Euler-Lagrange equations;
conservation of the electric current and its relation gauge invariance; counding the EM degrees of freedom.

Conserved currents of fields; O(N) example; relation to symmetries. - September 8 (Tuesday):
- Symmetries: current conservation due to symmetry (example); O(N) and SO(N) symmetries;
infinitesimal symmetries and generators; Lie algebras of generators;
global vs local symmetries.

Noether theorem: proof; SO(N) and U(1) examples; the stress-energy tensor. - September 10 (Thursday):
- Local phase symmetry:
local symmetry and covariang derivatives; relation to gauge transforms;
multiple charge fields; properties of covariant derivatives; non-commutativity;
the covariant Schroedinger equation.

Aharonov–Bohm effect. - September 11 (Friday) [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 (Tuesday):
- Aharonov–Bohm effect and magnetic monopoles:
mathematical aspects of AB effect; charge quantization and the compactness of the U(1) phase symmetry;
magnetic monopoles; Dirac quantization condition of the magnetic charge; gauge bundles.

Overview of quantization: path integrals vs. canonical quantization; classical Hamiltonian and Hamilton equations; Poisson brackets and commutator brackets; canonical commutation relations in QM; Heisenberg–Dirac equation. - September 17 (Thursday):
- Finish canonical quantization for mechanics:
Schroedinger and Heizenberg pictures; equal-time commutation relations.

Hamiltonian formalism for a scalar field: the canonical momentum field π(**x**,t); the Hamiltonian; Hamilton equations.

The Quantum scalar fields: operator-values fields; equal-times canonical commutation relations; the Hamiltonian operator; quantum Klein–Gordon equation.

Quantum fields and particles: expanding free scalar fields into harmonic oscillators. - September 22 (Tuesday):
- Quantum fields and particles: reassembling harmonic oscillators into identical bosons; going back: from identical bosons to harmonic oscillators to quantum fields; non-relativistic quantum fields; second quantization.
- September 24 (Thursday):
- Relativistic quantum fields: 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 25 (Friday) [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. - September 29 (Tuesday):
- Relativistic causality:
superluminal particles in `relativistic' QM;
superluminal signals in QM and in QFT;
proof of causality for a free scalar field.

Feynman propagator for the scalar field: why and how of time-ordering; defining the propagator. - October 1 (Thursday):
- Feynman propagators and Green's functions:
Feynman propagator is a Green's function; Green's functions in momentum space;
regularizing the poles along the mass shells; Feynman's choice; other Green's functions;
propagators and Green's functions for non-scalar fields.

Introduction to symmetries: continuous (Lie) grouls and their generators; Lie algebras; representations and multiplets. - October 2 (Friday) [supplementary lecture]:
- Supefluidity:
excitations of a supefluid; dispersion relation ω(
*k*) for the excitations; reasons for dissipationless flow; critical velocity. - October 6 (Tuesday):
- Lorentz symmetry:
Lorentz generators; field multiplets and particle multiplets;
Wigner theorem: massive partcles have definite spins, massless particles have independent helicities;
tachyons; generalizations of Wigner theorem to
*d*≠3+1 dimensions. - October 8 (Thursday):
- Canceled lecture.
- October 13 (Tuesday):
- Dirac spinor fields: Dirac matrices, Dirac spinors, and their Lorentz transformations; Dirac equation and its covariance; Dirac Lagrangian and Hamiltonian; Quantum Dirac fields.
- October 15 (Thursday):
- Fermionic Fock space:
fermionic anticommutation relations;
Hilbert space of a single fermionic mode; multiple modes;
fermionic Fock space; 1-body and 2-body operators.

Fermionic particles and holes: filling up the negative-energy modes; Fermi sea as the ground state; holes as quasi-particle excitations; creation and annihilation operators for the holes; quantum numbers of the holes; particle-hole formalism in condenced matter, atomic physics, and nuclear physics.

Positrons: mode expansion of the electron fields; the Dirac sea; positrons. - October 16 (Friday) [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. - October 20 (Tuesday):
- Symmetries of fermions and other fields: Charge conjugation; parity; combined CP; time reversal T and CPT theorem; vector and axial symmetries of Dirac fermions; chiral symmetries; Majorana and Weil fermions, and their equivalence to each other.
- October 22 (Thursday):
- Finish fermions:
Relativistic causality for fermions; Feynman propagator for Dirac fermions.

Begin perturbation theory: the interaction picture of QM; the Dyson series; the S «matrix» and its elements. - October 23 (Friday) [supplementary lecture]:
- Fermions in different spacetime dimensions:
Dirac fermions in even vs odd dimensions; Weyl fermions;
Majorana fermions; Bott periodicity; when LH Weyl, RH Weyls, and Majorana fermions are equivalent
to each other and when they are not; Majorana–Weyl fermions in
*d*=2+8k; applications to string and M theory. - October 27 (Tuesday):
- Perturbation theory and Feynman diagrams:
matrix elements of field products; Feynman diagrams and Feynman rules;
getting rid of vacuum bubbles; Feynman rules in the momentum space;
connected diagrams; scattering amplitudes and cross-sections.

Gave out Mid-term exam. - October 29 (Thursday):
- Perturbation theory and Feynman rules:
Phase space factors;
loop counting for quartic and cubic couplings;
Mandelstam's
*s*,*t*, and*u*; Feynman rules for multiple scalar fields. - November 3 (Tuesday):
- Dimensional analysis of allowed couplings.

Quantum Electro-Dynamics: quantizing EM fields; gauge fixing; photon propagator; QED Feynman rules.

Collect mid-term exam. - November 5 (Thursday):
- Coulomb and Yukawa scattering.

Sums over spins, Dirac treaceology, and muon pair-production. - November 6 (Friday) [supplementary lecture]:
- Resonances: unstable particles and other resonances;
Breit–Wigner resonances; width and lifetime;
resonant scattering amplitudes, cross-sections, and branching ratios;
J/ψ and other vector resonances in e
^{−}e^{+}. - November 10 (Tuesday):
- Quark pair production and the R ratio.

Crossing symmetry.

Ward identities and sums over photon polarizations; checking Ward identities for the annihilation. - November 12 (Thursday):
- Annihilation and Compton scattering:
Σ|ℳ|
^{2}over spins and polarizations; Dirac traceology; annihilation kinematics; Compton scattering; kinematics in the lab frame; Compton formula; Klein–Nishina formula; Compton backscattering and photon colliders; Breit–Wheeler process; virtual photons and electromagnetic showers. - November 17 (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 Goldstone bosons.
- November 19 (Thursday): Higgs Mechanism: SSB of a local U(1) symmetry; massive photon "eats" the would-be Goldstone boson; unitary gauge vs. gauge-invariant description.
- November 20 (Friday) [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.
- November 24 (Tuesday):
- Non abelian gauge theories: non-abelian tension fields; gauge couplings and field normalization; Yang–Mills theory and QCD; perturbation theory and Feynman rules; general gauge groups and general multiplets; the SU(3)×SU(2)×U(1) Standard Model.
- December 1 (Tuesday):
- Non-abelian Higgs mechanism:
SU(2) with a doublet Higgs example; SU(2) with a real triplet Higgs example;
general formulae for the vector masses.

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

Quarks, leptons, and the origin of their masses. - Plan for December 3 (Thursday):
- Fermions in the GWS theory:
quarks, leptons, and their masses; charged and neutral weak currents;
flavor mixing and the CKM matrix; CP violation; neutrino masses.

Give out the final exam.

Non-abelian gauge symmetries: covariant derivatives and matrix-valued connections; non-abelian vector fields.

Last Modified: December 1, 2015. Vadim Kaplunovsky

vadim@physics.utexas.edu