Quantum Field Theory: Lecture Log

QFT1, Fall 2015 semester

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

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 Z⁰ 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.