**Navigation:**
Fall,
Spring,
Last regular lecture.

The Zoom sessions for all the *regular* lectures are at
https://utexas.zoom.us/j/94371979747.

The sessions for the *extra* lectures are at a different URL, namely
https://utexas.zoom.us/j/97782926154.

- August 27 (Thursday):
- Syllabus and admin:
course content, textbooks, prerequisites, homework, exams and grades, etc.

General introduction: reasons for QFT; field-particle duality. - August 28 (Friday):
- Review of classical mechanics:
Lagrangian and action; least action principle; Euler–Lagrange equation;
multiple dynamical variables; counting degrees of freedom.

Intro to classical fields: Definition of a classical field; Lagrangian density; Euler–Lagrange equations for fields; Klein–Gordon example; multiple fields; complex fields; Landau–Ginzburg example;~~higher space derivatives and non-local Lagrangians for non-relativistic fields~~. - September 1 (Tuesday):
- Non-relativistic and relativistic fields:
Higher space derivatives and non-local Lagrangians for non-relativistic fields;
relativistic sign conventions; Einstein summation convention;
relativistic ℒ and field equations; Klein–Gordon example;
multiple scalar fields.

Relativistic 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 3 (Thursday):
- Finished Relativistic electromagnetic fields:
current conservation and gauge invariance of the action; counting the EM degrees of freedom.

Review of canonical quantization: Canonical quantization v. functional quantization; Hamiltonian formalism in classical mechanics; quantization, operators, and commutation relations;~~Poisson brackets and commutator brackets~~. - September 4 (Friday):
- Finished Review of canonical quantization:
Poisson brackets and commutator brackets.

Introduction to quantum fields: Hamiltonian formalism for classical fields; quantum fields; equal-time commutation relations; quantum Klein–Gordon equation. - September 8 (Tuesday):
- Quantum fields and particles: expanding free relativistic scalar fields into modes; creation and annihilation operators for a bunch of harmonic oscillators; eigenstates of the free quantum field's Hamiltonian; identifying the identical bosons; the Fock space.
- September 10 (Thursday):
- General identical bosons: Bosonic Fock space and its occupation number basis; creation and annihilation operators; wave-function language vs. Fock-space language; one-body operators; two-body operators; non-relativistic quantum fields; “second quantization”.
- Regular lecture on September 11 (Friday):
- Relativistic normalization of states and operators: Lorentz groups; momentum space geometry and Lorentz-invariant measure; relativistic normalization of states and operators.
- Extra lecture on September 11 (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; BEC example (outline only). - September 15 (Tuesday):
- Relativistic quantum fields:
Expanding a free time-dependent scalar field into products of plane waves and creation/annihilation operators;
massive vector field; charged scalar field and antiparticles; general free fields.

Intro to relativistic causality: superluminal particles in `relativistic' QM; signals in QM and in QFT; relativistic causality in QFT. - September 17 (Thursday):
- Relativistic causality:
local operator and fields; proof for free scalar fields; going forward and backward in time;
causality for interacting fields.

Begin Feynman propagator for the scalar field: why and how of time-ordering; defining the propagator; relation toD(x-y) . - September 18 (Friday):
- Feynman propagator for the scalar field:
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 22 (Tuesday):
- Feynman propagators for vector, spinor, etc. fields.

Overview of symmetries of field theories: symmetries of the action; continuous and discrete symmetries; internal and spacetime symmetries; global and local symmetries.

Noether theorem: Global continuous symmetries and conserved currents; generators and currents for the SO(N) example; symmetry charges in the quantum theory; representation of the SO(N) symmetry in the Fock space; the phase symmetry and the net number of particles minus antiparticles. - September 24 (Thursday):
- Noether theorem:
Proof of the theorem; examples of Noether currents;
translation symmetry and the stress-energy tensor;
symmetrizing the Noether stress-energy tensor for non-scalar fields.

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 25 (Friday):
- Covariant Schroedinger equation.

Aharonov–Bohm effect: Aharonov–Bohm effect; cohomology of the vector potential; charge quantization and the compactness of the U(1) phase symmetry. - Extra lecture on September 25 (Friday):
- Bose–Einstein condensate and superfluidity: 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’; fluctuation spectrum in a moving condensate and superfluidity.
- September 29 (Tuesday):
- Magnetic monopoles:
Heuristic picture; Dirac construction; charge quantization; gauge bundles.

Non-abelian local symmetries: Covariant derivatives and matrix-valued connections; non-abelian gauge transforms; Gell-Mann matrices and the component gauge fields. - October 1 (Thursday):
- Non-abelian local symmetries:
infinitesimal gauge transforms in components; non-abelian tensions fields;
gauge transforms of the tension fields; the adjoint multiplet;
Yang–Mills theory and normalization of the gauge fields; gauge theories with matter.

Overview of group theory: Lie groups, Lie algebras, representaions, multiplets. - October 2 (Friday):
- Gauge symmetries: symmetry groups and multiplets of fields; general local symmetry groups and Lie-algebra-valued gauge fields; covariant derivatives for different multiplets types; multiple gauge groups; Standard Model example.
- October 6 (Tuesday):
- Finish gauge symmetries:
classification of allowed gauge groups.

Lorentz symmetry: generators and multiplet types; unitary but infinite particle representations; little groups and Wigner theorem; massive particles have definite spins; massless particles have definite helicities;~~tachyons have nothing~~; maybe Wigner theorem in*d*≠4 dimensions. - October 8 (Thursday):
- Tachyons: tachyons in QM;
Wigner theorem for the tachyons;
tachyon field and vacuum instability;
interactions and scalar VEVs (vacuum expectation values).

Lorentz symmetry: finish Wigner theorem; Lorentz multiplets of fields; (j_{+},j_{−}) multiplets; Weyl spinors and Spin(3,1)≅SL(2,**C**);~~vectors and bispinors; tensors~~. - Regular lecture on October 9 (Friday):
- Finish Lorentz multiplets of fields:
vectors and bispinors; tensors.

Dirac spinors and spinor fields: Lorentz spinor multiplet; Dirac equation. - Extra lecture on October 9 (Friday):
- Vortices:
rotation and vortices in a superfluid; vortex energy; vortex in a superconductor; magnetic flux;
cosmic strings.

Other types of topological defects: domain walls, monopoles, YM instantons; codimension. - October 13 (Tuesday):
- Dirac spinor fields:
covariance of the Dirac equation; Dirac conjugation; Dirac Lagrangian;
Hamiltonian for the quantum Dirac field.

Grassmann numbers and classical limits of fermionic fields. - October 15 (Thursday):
- Fermionic algebra and Fock space:
Hilbert stace of one fermionic mode; multiple modes; Fermionic fock space; wave functions and operators.

Fermionic particles and holes: particles and holes; holes as quasiparticles; Fermi sea, extra particles and holes. - October 16 (Friday):
- 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 20 (Tuesday):
- Charge conjugation symmetry:
neutral particles and C-parity; Majorana fermions.

Dirac, Majorana, and Weyl fermions: counting degrees of freedom; relations between Majorana and Weyl fermions; Majorana mass term; massless and massive neutrinos. - October 22 (Thursday):
- Parity and other discrete symmetries:
G-parity (briefly); parity; CP; time reversal (briefly); CPT theorem;
baryogenesys and Sakharov's criteria.

Chiral symmetry: vector, axial, and chiral symmetries; the*U(N)*chiral symmetry; chiral symmetry in QCD; chiral gauge theories._{L}×U(N)_{R} - Regular lecture on October 23 (Friday):
- Chiral symmetry:
vector, axial, and chiral symmetries for Weyl fermions;
*U(N)*chiral symmetry; chiral gauge theories; electroweak example; chiral symmetry in QCD._{L}×U(N)_{R} - Extra lecture on October 23 (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 ofSO .^{+}(*a,b*) - October 27 (Tuesday):
- Relativistic causality for the fermions:
commuting and anticommuting fields;
checking anticommutativity of free Dirac fields at spacelike separation;
spin-statistics theorem.

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 (Thursday):
- 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 30 (Friday):
- Perturbation theory in QFT and Feynman rules:
momentum space Feynman rules; momentum conservation and connected diagrams;
scattering amplidudes;
summary of Feynman rules for the λΦ
^{4}theory.

Phase space factors. - November 3 (Tuesday):
- Loop counting: loop counting for the λΦ
^{4}theory; adding cubic couplings; Mandelstam's s, t, and u; multiple fields. - November 5 (Thursday):
- Dimensional analysis:
dimensions of fields and couplings; trouble with δ<<0 couplings;
types of Δ≥0 couplings in 4D; other dimensions.

Began Intro to Quantum Electro Dynamics (QED): quantizing EM fields; photon propagator in the Coulomb gauge.. - Regular lecture on November 6 (Friday):
- Finish Intro to QED:
photon propagator in various gauges.

QED Feynman rules: propagators and vertices; external line factors; Dirac indexology; Gordon identities; sign rules. - Extra lecture on November 6 (Friday):
- Conformal symmetry:
definition; complex language in Euclidean 2D;
conformal symmetry group and its generators;
conformal algebra in
*d*>2 dimensions, Euclidean or Minkowski.

Conformal field theories and their application: world-sheet QFT in string theory; condenced matter at a critical point; conformal window of QCD; AdS/CFT duality. - November 10 (Tuesday):
- Coulomb scattering in QED:
diagrams and amplitudes; non-relativistic limit;
recovering the Coulomb potential;
electron-electron vs. electron-positron Coulomb scattering.

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 muon pair production.^{−}+μ^{+}: - November 12 (Thursday):
- 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; analytically continuing the amplitudes; crossing symmetry in general; signs for crossed fermions; Compton vs. annihilation example. - November 13 (Friday):
- Ward Identities:
Ward identities for photons and gauge invariance of the amplitudes;
sums over photon polarizations.

Electron-positron annihilatione^ : tree diagrams and the amplitude; checking the Ward identities; summing over photon polarizations and averaging over fermions' spins.^{−}+e^{+}→γ - November 17 (Tuesday):
- Electron-positron annihilation:
Dirac traceology; summary and annihilation kinematics; annihilation cross-section;
crossing relation to Compton scattering.

Spontaneous symmetry breaking: symmetric Lagrangian/Hamiltonian but asymmetric vacuum; continuous families of degenerate vacua; massless particles; linear sigma model. - November 19 (Thursday):
- Spontaneous symmetry breaking:
Wigner and Goldstone modes of symmetries; Goldstone theorem.

The Higgs Mechanism: SSB of a local U(1) symmetry; massive photon ‘eats&rdsqo; the would-be Goldstone boson; unitary gauge vs. gauge-invariant description. - Regular lecture on November 20 (Friday):
- Non-Abelian Higgs Mechanism: SU(2) with a doublet; SU(2) with a real triplet; general case.
- Extra lecture on November 20 (Friday):
- SSB of QCD's chiral symmetry and sigma models:
Chiral symmetry of QCD and its spontaneous breakdown (χSB);
pions as
*pseudo*–Goldstone bosons; linear sigma model of χSB non-linear sigma model; maybe general NLΣMs. - November 24 (Tuesday):
- 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 currents; Fermi's effective theory of weak interactions.

Fermion masses arising from scalar VEVs. - December 1 (Tuesday):
- Fermions of the Glashow–Weinberg–Salam theory:
Higgs origin of quark and lepton masses;
charged and neutral weak currents of quarks and leptons.

Introduction 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.

Began 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. - December 3 (Thursday):
- Finished the CKM matrix:
bases for the charged leptons and for the neutrinos;
neutral weak currents: diagonal in the Standard Model, but non-diagonal (flavor-changing) in other models.

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.

Introduction to CP violation: 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. - December 4 (Friday):
- Neutrino masses:
neutrino masses and oscillations; Dirac vs. Majorana masses of neutrinos; seesaw mechanism.

Gave out the final exam.

The Zoom sessions for all lectures on Tuesdays, Thursdays, and Fridays are at
https://utexas.zoom.us/j/97545121357.

The sessions for the Wednesday lectures — extra or make-up — are at a different URL, namely
https://utexas.zoom.us/j/99561322180.

- January 19 (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; UV divergence. - January 21 (Thursday):
- UV cutoff, long-distance effective field theories, and renormalization: long-distance (low-energy) EFT and its independence on short-distance (high energy) details; cutting off the UV momenta; one-loop amplitude; bare and physical couplings; changing bare coupling to compensate for changing the UV cutoff; perturbative expansion in powers of the physical coupling.
- January 22 (Friday):
- Overview of UV regulators: Wilson's hard edge; Pauli–Villars; higher derivatives.
- January 26 (Tuesday):
- Finished overview of UV regulators:
covariant higher derivatives; lattice (very briefly).

Dimensional regularization: basics; momentum integrals in non-integral dimensions;*d*→4 limit; (1/ε) as log(Λ_{UV}).

Optical theorem: proof from unitarity of the S matrix. - Extra lecture on January 27 (Wednessday):
- Resonances and unstable particles:
Breit–Wigner resonances in QM; resonances in QFT and unstable particles;
making a resonance in a collision; cross-sections and branching ratios;
quarkonia as resonaces in electron-positron collisions; calculating
σ(e
^{−}+e^{_}→J/ψ). - January 28 (Thursday):
- Optical theorem: application to
Im M
_{1 loop}in λφ^{4}theory; mentioned cutting diagrams and putting cut propagators on-shell (details in homework).

Correlation functions of quantum fields: correlation functions; relation to the free fields and evolution operators; Feynman rules; connected correlation functions. - January 29 (Friday):
- The two-point correlation function:
Källén–Lehmann spectral representation; features of the spectral density function;
analytic two-point function F
_{2}(*p*^{2}): poles and particles; pole mass = physical particle mass; branch cuts and the multi-particle continuum; physical and un-physical sheets of the Riemann surface; resonances. Began Perturbation theory for the two-point function: resumming the 1PI bubbles - February 2 (Tuesday):
- Perturbation theory for the two-point function:
Σ(
*p*^{2}) and the renormalization of the mass and of the field strength; mass renormalization in the λφ^{4}theory; fine tuning problem.

Quadratic UV divergences: regulator dependence; dimensional regularization of quadratic divergences.

Field strength renormalization in the Yukawa theory: calculating the one-loop diagram: the trace, the denominator and the numerator, Wick rotation; the UV divergence structure: Σ(*p*^{2})=(div.constant)+(div.constant)×*p*^{2}+finite_*f*(*p*^{2}). - February 4 (Thursday):
- Finish
field strength renormalization in the Yukawa theory:
calculating the Σ(
*p*^{2}); dΣ/d*p*^{2}and the scalar field strength renormalization.

Counterterms perturbation theory: ℒ_{bare}=ℒ_{physical}+counterterms; Feynman rules for the counterterms; adjusting δ^{Z}, δ^{m}, and δ^{λ}order by order in λ; one-loop examples.

Began counting the divergences: superficial degree of divergence; graphs and subgraphs. - February 5 (Friday):
- Counterterms and canceling the divergences:
classifying divergent graphs, subgraphs, and amplitudes; canceling overall divergences;
subgraph divergences and their cancelation
*in situ*. - February 9 (Tuesday):
- Finish divergence cancellation for λφ
^{4}: nested and overlapping divergences; BPHZ theorem.

Divergences and renormalizability: supeficial degree of divergence in the φ^{k}theories; super-renormalizable φ^{3}theory; super-renormalizable, renormalizable, and non-renormalizable theories; trouble with non-renormalizability.

Dimensional analysis and renormalizability: canonical dimensions of fields and couplings; power-counting renormalizability; renormalizable theories in 4D;~~other dimensions~~.

- Extra lecture on February 10 (Wednesday):
- Relating the correlation functions
F
_{n}(*p*) to the scattering amplitudes: the amputated core and the external leg bubbles; the poles for the on-shell_{1},…p_{n}*p*_{i}^{0}→±*E*(**p**_{i}) and their relations to the asymptotic*x*→±∞ limits; the asymptotic |in⟩ and ⟨out| states and the LSZ (Lehmann–Symanzik–Zimmermann) reduction formula; scattering amplitudes and the amputated diagrams.^{0}_{i} - February 11 (Thursday):
- QED perturbation theory:
the counterterms and the Feynman rules;
divergent amplitudes and their momentum dependences; missing counterterms and Ward–Takahashi identities.

Dressed electron propagator. - February 12 (Friday):
- Dressed photon propagator.

Σ^{μν}(*k*) at one loop order: calculation; checking the WT identity;~~the divergence and the δ~~._{3}counterterm - No lectures on February 16, 18, and 19 (whole week):
- Cancelled due to bad weather.
- Extra lecture on February 23 (Tuesday):
- 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.
- Make-up regular lecture on February 24 (Wednesday):
- Electric charge renormalization:
finish calculation of Π
^{1 loop}(*k*^{2}): the momentum integral, the δ_{3}counterterm, and the final result; loop corrections to Coulomb scattering and other high-momentum processes; effective QED coupling α_{eff}(*E*) and its running with log(energy). - February 25 (Thursday):
- Ward–Takahashi identities:
the identities; current conservation in quantum theories; contact terms;
formal proof of WT indentities; Z
_{1}=Z_{2}. - February 26 (Friday):
- Form factors:
probing nuclear and nucleon structure with electrons; the form factors;
on-shell form-factors
*F*_{1}(*q*^{2}) and*F*_{2}(*q*^{2}); the gyromagnetic ratio.

Begin the dressed QED vertex at one loop: the diagram and the denominator. - March 2 (Tuesday):
- Dressed QED vertex at one loop:
numerator algebra; calculating the
*F*_{2}form factor and the anomalous magnetic moment; the experimental and the theoretical electron's and muon's magnetic moments at high precision. The electric from factor*F*_{1}(*q*^{2}) at one loop: momentum integral; integral over Feynman parameters diverges; hints of IR divergence. - Make-up regular lecture on March 3 (Wednesday):
- Infrared divergence in QED:
IR divergence of the one-loop vertex correction; regulating the IR divergence with photon mass;
calculating the regulated
*F*_{1}(*q*^{2}); δ_{1}counterterm; momentum dependence of the IR divergence; Sudakov's douboe logarithms.

Begin Virtual and real soft photons: IR divergence of exclusive cross-sections due to virtual soft photons.~~soft-photon bremmsstrahlung and its IR divergence; finite inclusive cross-section~~. - March 4 (Thursday):
- Virtual and real soft photons:
soft-photon bremmsstrahlung and its IR divergence;
finite inclusive cross-sections (with or without soft photons);
detectable vs. undetectable photons, the observed cross-sections, and their finiteness;
briefly: higher loops and/or more soft photons.

Consequences of infrared divergence: Ill-defined Fock space in QED and other gauge theories; soft and collinear gluons in QCD; jets in theory and in experiment. - March 5 (Friday):
- Gauge dependence in QED:
gauge-dependent off-shell amplitudes and counterterms;
δ
_{1}(ξ)=δ_{2}(ξ).

Symmetries and counterterms: counterterms in general renormalizable QFT's; naturally small and unnaturally small couplings; Yukawa and QED examples.

Began intro to renormalization group: large log problem for*E≫m*; resumming leading logs in terms of runnibg λ(*E*). - March 9 (Tuesday):
- Intro to renormalization group:
large logarithms and running coupling λ(
*E*); off-shell renormalization schemes for couplings and counterterms.

Renormalization group basics: anomalous dimensions of quantum fields; running couplings and β functions.

Renormalization group equation for the λφ^{4}theory: solving the equation (in the one-loop approximation); no running below the mass threshold; boundary condition for the RGE and the threshold correction. - Make-up regular lecture on March 10 (Wednesday):
- Renormalization group for QED:
anomalous dimensions; β
_{e}to one-loop order; solving the RGE for QED; threshold correction and 2 loop correction.

Renormalization groups for general QFTs: β–functions for general couplings; Yukawa theory as an example; solving coupled RGEs. - March 11 (Thursday):
- Types of RG flows:
β>0, Landau poles, and UV incompleteness;
β<0, QCD example, and asymptotic freedom; Λ
_{QCD}; non-perturbative strong interactions at low energies.

Chromomagnetic monopole condensation and quark confinement. - March 12 (Friday):
- Fixed points β(
*g*^{*})=0 of RG flows: scale invariance and conformal symmetry; UV stability vs. IR stability; Banks–Zaks conformal window of QCD. - March 16–19:
- Spring break, no classes.
- March 23 (Tuesday):
- RG flows for multiple couplings:
Yukawa example; RG flows in the coupling space: fixed points and attractive lines.

Direction of UV flow: IR to UV or UV to IR?

Relevant, irrelevant, and marginal operators; effective field theories. - Make-up regular lecture on March 24 (Wednesday):
- Renormalisation schemes: scheme dependence of the couplings and the β–functions; the minimal subtraction schemes MS and MS-bar; extracting β–functions from residues of the 1/ε poles.
- March 25 (Thursday):
- Introduction to path integrals:
path integrals in QM; the Lagrangian and the Hamiltonian forms of the path integral;
derivation of the Hamiltonian form;
Lagrangian path integrals — derivation and normalization;
partition function; harmonic oscillator example.

Gave out the midterm exam. - March 26 (Friday):
- Functional integrals in QFT:
“path” integrals for quantum fields; correlation functions;
free fields and propagators; perturbation theory and Feynman rules;
~~sources and generating functionals~~. - March 30 (Tuesday):
- Functional integrals in QFT:
sources and generating functionals.

Euclidean path integrals: convergence problems of path integrals; Euclidean time; discretization; harmonic oscillator example. - April 1 (Thursday):
- QFT and StatMech: Functional integrals in Euclidean spacetime; QFT↔StatMech analogy; coupling as temperature; QFT on a discrete lattice; lattice as a UV cutoff; recovering rotational / Lorentz symmetry in the continuum limit; custodial symmetries.
- April 2 (Friday):
- Fermions and Grassmann numbers:
Grassmann numbers; Berezin integrals;
Gaussian integrals over fermionic variables;
functional integrals over fermionic fields;
~~free Dirac field in Euclidean spacetime~~. - April 6 (Tuesday):
- Integrating over fermion fields in QED:
Dirac field in Euclidean spacetime;
functional integral in EM background: the determinant, and the source term;
Det(D̸+m) and the electron loops; 1/(D̸+m) and the tree diagrams.

Functional integral for the EM field: gauge transforms as redundancies; gauge-fixing conditions and the Faddeev–Popov determinants; Landau-gauge propagator from the functional integral; gauge-averaging, gauge-fixing terms, and the Feynman gauge. - Extra lecture on April 7 (Wednesday):
- Gauge theories on the lattice (abelian):
local U(1) symmetry on the lattice; gauge fields and link variables;
covariant lattice derivatives; plaquettes and tension fields; lattice EM action;
lattice ‘path’ integrals; compact QED.

Non-abelian lattice gauge theories: non-abelian gauge symmetries and link variables; covariant lattice symmetries; non-abelian plaquettes and tension fields; lattice YM action; integrals over link variables and the lattice ‘path’ integrals; brief history and applications of lattice QCD. - April 8 (Thursday):
- Quantizing the Yang–Mills theory:
fixing the non-abelian gauge symmetry; Faddeev–Popov ghost fields;
gauge-fixed YM Lagrangian.

QCD Feynman rules: physical, ghost, gauge-fixing, and counter- terms in the Lagrangian; propagators; physical vertices; counter-term vertices; handling the color indices of quarks.

Started QCD Ward identities: on-shell QCD Ward identities are weaker than in QED;*q+q̄→g+g*example: 3 tree diagrams and*k*_{μ}M^{μν}(1+2). - April 9 (Friday):
- QCD Ward identities:
*q+q̄→g+g*example: the third diagram; Ward identity holds for one longitudinal uark only; two longitudinal quarks are canceled by the ghost antighost pair.

Introduction to BRST symmetry: BRST transforms of QCD fields;~~nilpotency; BRST invariance of the net Lagrangian~~. - April 13 (Tuesday):
- BRST symmetry:
nilpotency; BRST invariance of the net Lagrangian;
physical and unphysical quanta in the QCD Fock space and BRST cohomology;
reducing 1-particle states to physical particles; multi-particle physical states and the S-matrix;
BRST symmetries of the amplitudes and cancellation of unphysical processes.

Started QCD renormalizability: renormalizability and the counterterm set; BRST and other manifest symmetries;~~allowed counterterms and renormalizability~~. - Extra lecture on April 14 (Wednesday):
- Wilson loops: Abelian and non-abelian Wilson loops; large loops and forces between probe particles; non-abelian probe particles; area law vs. perimeter law as test of confinement vs. deconfinement.
- April 15 (Thursday):
- QCD renormalizability:
modern approach; manifest symmetries; the bare Lagrangian;
Slavnov–Taylor identities for the QCD counterterms.

basic group theory — the Casimir and the index.

Renormalization of QCD: counterterms and the beta-function; calculating the one-loop δ_{2}for quarks; calculating the one-loop δ_{1}for quarks — the QED-like loop and the non-abelian loop (unfinished). - April 16 (Friday):
- Renormalization of QCD —
finish calculating one-loop δ
_{1}for the quarks; calculate the one-loop δ_{3}: the quark loop; the gluon loop; the sideways gluon loop; the ghost loop;~~summary~~. - April 20 (Tuesday):
- Renormalization of gauge theories:
finish calculating the one-loop δ
_{3}counterterm; QCD beta function at one loop; generalizing to other gauge theories.

Introduction to axial anomaly: axial symmetry of massless electrons; anomaly and its origin in the path integral measure; would-be Ward identity and the hole in the argument. - Extra lecture on April 21 (Wednesday):
- Grand Unification: unifying the EM, weak, and strong interactions in a single non-abelian gauge group; SU(5) example; multiplets of fermions; gauge couplings and Georgi–Quinn–Weinberg equations; the doublet-triplet problem; baryon decay and other exotic processes.
- April 22 (Thursday):
- Axial Anomaly: the diagrams, the naive cancelation, and the regulation problem; Adler–Bardeen theorem; Pauli–Villars regulation of the anomaly.
- April 23 (Friday):
- Axial anomalies: Calculating the regulated triangle diagrams; axial anomaly in QCD.
- April 27 (Tuesday):
- More axial anomalies:
anomaly of the measure of the fermionic functional integral;
anomalies of generalized axial symmetries; η and π mesons in QCD.

Non-linear sigma models: non-linear field spaces; NLΣM of the chiral symmetry breaking. χSB in QCD context. - Extra lecture on April 28 (Wednesday):
- Instantons:
topological index I[A
^{μ}] and its quantization; S_{E}≥(8π^{2}/g^{2})×∣I∣ and the topological sectors in the YM path integral; 't Hooft instantons and tunneling events; multiple instantons, cluster expansion, and the Θ angle. - April 29 (Thursday):
- Non-linear sigma models of the chiral symmetry breaking: vector and axial currents; QCD context; weak decays of charged pions; quark masses as perturbations; η and η' mesons.
- April 30 (Friday):
- QED anomaly of the axial isospin and the neutral pion decay.

Chiral U(1) gauge theories: Weyl fermions and chiral currents; loops of Weyl fermions and the chiral anomaly; net anomalies of global symmetries. - May 4 (Tuesday):
- Global symmetries of chiral gauge theories:
trace formula for the anomaly; baryon and lepton number anomalies in the electroweak theory;
instantons and sphalerons; baryogenesys by leptogenesys in early Universe;
leptogenesys by our-of-equilibrium decays of sterile neutrinos.

Gauge anomalies: triangle anomaly in chiral QED and its effect on Ward identities. - Extra lecture on May 5 (Wednesday):
- Instantons and fermions: instantons and axial anomaly; zero modes in instanton background; Atyah–Singer index theorem; zero modes in fermionic integrals chiral anomaly of the Θ angle; Θ=Θ+phase(det(quark mass matrix)); the strong CP problem; neutron's electric dipole; Peccei–Quinn symmetry.
- May 6 (Thursday):
- Gauge anomalies: anomalous gauge variance of log(det(̸D));
anomaly in non-abelian chiral theories; Wess–Zumino consistency conditions;
anomaly coefficients A
^{abc}and traces over chiral fermions.

Anomaly cancellation in chiral gauge theories: checking anomaly cancellation in the Standard Model; importance of tr(*Q*_{el})=0; in general gauge theories, massive fermions do not contribute to the anomaly. - May 7 (Friday):
- Anomalies in general chiral gauge theories:
cubic Casimirs and cubic anomaly indices for simple gauge groups;
applications to Grand Unification;
briefly anomalies in other dimensions.

Give out the final exam.

Last Modified: May 17, 2021. Vadim Kaplunovsky

vadim@physics.utexas.edu