- 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 F^{μν}=−F^{νμ}; Maxwell equations in relativistic form; 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. - 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)→γ^{2}Ψ^{*}(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)*symmetry._{L}×U(N)_{R} - October 24 (Thursday):
- Chiral symmetry: vector, axial, and chiral symmetries;
the
*U(N)*chiral symmetry; chiral symmetry in QCD; chiral gauge theories._{L}×U(N)_{R}

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 λΦ
^{4}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^ the tree amplitude; the un-polarized scattering and the spin sums/averages; Dirac trace techniques; traces for the pair production;^{−}+μ^{+}:~~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=T*; masses of the vector fields and the Weinberg's mixing angle; charged and neutral weak currents.^{3}+Y - 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}↔K̅^{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.

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