Quantum Field Theory: Lecture Log
- August 29 (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.
Refresher of classical mechanics
(the least action principle and the Euler-Lagrange equations).
- August 30 (Friday):
- Classical fields: definition; Euler–Lagrange equations; Klein–Gordon example;
coupled scalar fields; higher derivatives and counting of degrees of freedom.
- September 3 (Tuesday):
- Relativistic notations for field theory.
The electromagnetic fields:
the 4–tensor Fμν=−Fνμ;
Maxwell equations in relativistic form;
the 4–vector potential Aμ and the gauge transforms;
the Lagrangian formulation;
current conservation and gauge invariance of the action; counting the EM degrees of freedom.
- September 5 (Thursday):
- Intoduction to quantization: canonical quantization vs. path integrals; Hamiltonian classical mechanics;
canonical commutation relations; Schrödinger and Heisenberg pictures of quantum mechanics;
Poisson brackets and commutators.
- September 6 (Friday):
- 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;
quantum Klein–Gordon equation.
- September 10 (Tuesday):
- Quantum fields and particles:
expanding free relativistic scalar field into harmonic oscillators;
eigenstates of the free quantum field's Hamiltonian;
identifying the identical bosons; the Fock space.
Going back:
from identical bosons to the Fock space to harminic oscillators;
continuing to operators in the Fock space and hence to non-relativistic quantum fields.
- September 12 (Thursday):
- Identical bosons and non-relativistic fields:
wave-function and Fock-space formalisms;
one-body and two-body operators in the Fock space language;
examples; non-relativistic creation and annihilation fields; second quantization;
Landau–Ginzburg theory.
- Regular lecture on September 13 (Friday):
- Relativistic normalization of states and operators:
Lorentz groups; momentum space geometry and Lorentz-invariant measure;
normalization of states and operators.
- Extra lecture on September 13 (Friday):
- 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.
Intro to superfluids:
naive Bose–Einstein condensate as a coherent state;
classical and quantum fluctuation fields δφ(x);
Bogolyubov transform; ground state; fluctuation spectrum;
non-local force between helium atoms and the ‘rotons’.
- September 17 (Tuesday):
- Relativistic quantum fields:
expanding a free time-dependent scalar field into products of plane waves
and creation/annihilation operators; charged scalar field and antiparticles;
general free fields.
Relativistic causality: superluminal particles in ‘relativistic’ QM;
signals in QM and in QFT; local operators and fields; relativistic causality
in free and in interacting QFT.
- September 19 (Thursday):
- Relativistic causality: proof for the free fields.
Feynman propagator for the scalar field:
why and how of time-ordering; defining the propagator; relation to D(x-y).
- September 20 (Friday):
- Feynman propagator as a Green's function:
Checking that the propagator is 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.
- September 24 (Tuesday):
- Symmetries of field theory: symmetries of the action; global and local symmetries;
conserved currents and the Noether theorem; the SO(N) example;
phase symmetry of the quantum theory; SO(N) symmetry of the quantum theory;
proof of the Noether theorem; internal and spacetime symmetries.
- September 26 (hursday):
- Transpation symmetries and the stress-energy tensor.
Local phase symmetry:
local symmetry and covariant derivatives; gauge field and gauge transforms;
algebra of covariant derivatives; coupling charged scalar fields to electromagnetism.
- Regular lecture on September 27 (Friday):
- Aharonov–Bohm effect and magnetic monopoles:
covariant Schroedinger equation; Aharonov–Bohm effect; cohomology of the vector potential;
charge quantization and the compactness of the U(1) phase symmetry;
intro to magnetic monopoles and Dirac charge quantization.
- Extra lecture on September 27 (Friday):
- Field theory of superfluidity:
classical field for the Bose–Einstein condensate;
density and velocity of the superfluid; irrotational flow; vortices.
Other kinds of topological ‘defects’: domain walls, monopoles, YM instantons; co-dimension.
- October 1 (Tuesday):
- Finish magnetic monopoles:
Dirac construction; charge quantization; gauge bundles.
Non-abelian local symmetries:
covariant derivatives, matrix-valued connections, and gauge transforms;
non-abelian vector fields and tension tensors.
- October 3 (Thursday):
- Overview of group theory: Lie groups, Lie algebras, representaions, multiplets.
Non-abelian gauge theories:
NA tension fields; Yang–Mills theory and gauge field normalization; gauge theories with matter;
general gauge groups.
- October 4 (Friday):
- Finish gauge symmetry:
general local symmetry groups; covariant derivatives for different multiplets types;
multiple gauge groups; Standard Model example.
- October 8 (Tuesday):
- Lorentz symmetry:
generators and commutation relations; finite field multiplets vs. unitary particle state multiplets;
spinless particle representation; other particle representations and little groups G(p).
Wigner theorem:
massive particles have spins; massive particles have helicities; tachyons have nothing.
- October 10 (Thursday):
- Finish Wigner theorem:
tachyons are scalars; generalization to d≠3+1 dimensions.
Avoiding tachyons in QFT: vacuum instabilities and scalar VEVs.
Lorentz multiplets of fields.
- Regular lecture on October 11 (Friday):
- Dirac spinors and spinor fields:
Lorentz spinor multiplet; Dirac equation; Dirac conjugation; Dirac Lagrangian.
- Extra lecture on October 11 (Friday):
- Bose–Einstein condensation and superfluidity:
Bose–Einstein condensation; Bogolyubov transform; quasiparticles;
perturbative corrections; superfluid liquid helium; quasiparticles in a moving fluid;
dissipation vs. superfluidity.
- October 15 (Tuesday):
- Dirac spinor fields:
Hamiltonian for the quantum fields; classical limits of fermionic fields,
Grassmann numbers.
Fermionic algebra and Fock space:
Hilbert stace of one fermionic mode; multiple modes;
Fermionic fock space; wave functions and operators;
particles and holes; holes as quasiparticles.
- October 17 (Thursday):
- Fermionic particles and holes:
Fermi sea, extra fermions, 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)→γ2Ψ*(x).
- October 18 (Friday):
- Neutral particles and C-parity.
Dirac, Majorana, and Weyl fermions:
Majorana fermions; counting degrees of freedom; relations between Majorana and Weyl fermions;
Majorana mass term; massless and massive neutrinos.
- October 22 (Tuesday):
- Parity and CP symmetries; (briefly) T symmetry; CPT theorem.
Vector and axial symmetries of a Dirac fermion.
chiral U(N)L×U(N)R symmetry.
- October 24 (Thursday):
- Chiral symmetry: vector, axial, and chiral symmetries;
the U(N)L×U(N)R chiral symmetry;
chiral symmetry in QCD; chiral gauge theories.
Relativistic causality for the fermions:
commuting and anticommuting fields; checking anticommutativity of free Dirac fields at spacelike separation;
spin-statistics theorem.
- October 25 (Friday):
- Feynman propagator for Dirac fermions.
Introduction to perturbation theory:
interaction picture of QM; the Dyson series and the time-ordering.
Gave out the midterm exam.
- October 29 (Tuesday):
- Perturbation theory in QFT and Feynman diagrams:
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 31 (Thursday):
- Perturbation theory in QFT and Feynam rules:
summary of Feynman rules for the λΦ4 theory;
phase space factors;
loop counting.
for quartic and cubic couplings; Mandelstam's variables s, t, and u.
- Regular lecture on November 1 (Friday):
- Collected the midterm exams.
More Feynman rules: cubic and quartic couplings; loop counting; multiple fields;
Mandelstam's variables s, t, and u.
Begin Good and bad interactions in perturbative QFT:
dimensional analysis and trouble with Δ<0 couplings.
- Extra lecture on November 1 (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).
- November 5 (Tuesday):
- Good and bad interactions in perturbative QFT:
dimensions of fields and couplings; 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 7 (Thursday):
- 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):
Feynman rules; fermion-fermion and fermion-antifermion scattering;
the non-relativistic limit and the Yukawa forces.
- November 8 (Friday):
- Muon pair production in QED,
e^−+e+→μ^−+μ+:
the tree amplitude; the un-polarized scattering and the spin sums/averages;
Dirac trace techniques; traces for the pair production;
partial and total cross-sections.
- November 12 (Tuesday):
- Pair production in electron-positron collisions:
partial and total cross-sections for the muon pair production;
quark pair production and jets; hadronic production
e^−+e+→q+q̄→hadrons
and the R ratio.
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 14 (Thursday):
- Ward Identities:
Ward identities for photons and gauge invariance of the amplitudes; sums over photon polarizations.
Electron-positron annihilation e−+e+→2γ:
tree diagrams and the amplitude; checking the Ward identities;
summing over photon polarizations and averaging over fermions' spins;
Dirac traceology (part 1).
- Regular lecture on November 15 (Friday):
- Annihilation and Compton scattering:
Dirac traceology (part 2); summary and annihilation kinematics; annihilation cross-section;
crossing to Compton scattering; lab frame kinematics; Klein—Nishina formula.
- Extra lecture on November 15 (Friday):
- 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 19 (Tuesday):
- Spontaneous symmetry breaking:
continuous families of degenerate vacua; massless particles; linear sigma model;
Wigner and Goldstone modes of symmetries; Goldstone theorem;
approximate symmetries and pseudo-Goldstone bosons;
chiral symmetry of QCD and the pions.
- November 21 (Thursday):
- 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.
- Regular lecture on November 22:
- The Higgs Mechanism: vector masses for general gauge groups and Higgs multiplets.
Glashow–Weinberg–Salam theory:
bosonic fields and the Higgs mechanism; unbroken electric charge Q=T3+Y;
masses of the vector fields and the Weinberg's mixing angle; charged and neutral weak currents.
- Extra lecture on November 22:
- Making magnetic monopoles:
topology of the Higgs field in the Georgi–Glashow model; the “hedgehog” solution
and its magnetic field; multimonopole solutions and their magnetic charges;
monopoles in general spontaneously broken gauge groups; monopoles in Grand Unified Theories.
- November 26 (Tuesday):
- Fermi effective theory of weak interactions.
Fermions of the Glashow–Weinberg–Salam theory:
quarks' and lepton's masses arising from their Yukawa couplings to the Higgs scalar;
charged and neutral weak currents of quarks and leptons.
- December 3 (Tuesday):
- History of hadronic symmetries:
isospin; strangeness; SU(3)flavor; quarks; non-relativistic quark model;
color; QCD; confinement; SU(3)×SU(3)→SU(3) chiral symmetry breaking
and the π, K, and η pseudo-Goldstone bosons.
Intro
to CKM matrix:
Cabibbo mixing and Kaon decays; GIM mechanism and the charm quark;
third family and the CKM matrix; the charged currents;
flavor-changing weak decays.
Origin of the CKM matrix:
SM fermions come in sets of 3 for each multiplet type; unitary charges of bases;
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;
flavor changing charged currents;
bases for the charged leptons and for the neutrinos;
netral weak currents: diagonal in the Standard Model, but non-diagonal (flavor-changing)
in other models.
- December 5 (Thursday):
- Neutral kaons:
GIM box and K^0↔K̅0 mixing; K-long and K-short;
CP eigenstates K1 and K2, and their decays to pions;
K-short regeneration;
semi-leptonic decays of neutral kaons; strangeness oscillations; the CPLEAR experiment.
Intro
to CP violation (CPV):
CPV in neutral kaon decays to pions; CPV in eigenstates and in decay rates; CPV in semi-leptonic decays;
CPV and the CKM matrix.
CP symmetry and its violation by weak interactions:
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 other aspects of CP violation:
Possibility of CPV by strong interactions; CPV and baryogenesis in the early Universe.
- Plan for December 6 (Friday):
- Neutrino masses:
neutrino masses and oscillations; Dirac vs. Majorana masses of neutrinos;
seesaw mechanism.
Give out the final exam.
Last Modified: December 5, 2019.
Vadim Kaplunovsky
vadim@physics.utexas.edu