Quantum Field Theory: Lecture Log

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QFT 1, Fall 2016 semester

August 30 (Tuesday):
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 (Thursday):
Classical fields: definition, Euler–Lagrange equations; Klein–Gordon example; coupled scalar fields; higher derivatives and counting of degrees of freedom; relativistic notations.
The electromagnetic fields: the tension fields E and B and the potentials Φ and A; gauge transforms; the 4–vector Aμ and the 4–tensor Fμν; the homogeneous Maxwell equations as Jacobi identities.
September 6 (Tuesday):
The electromagnetic fields: Maxwell equations in the relativistic form; the Lagrangian formalism; current conservation and gauge invariance of the action; counting EM degrees of freedom.
Symmetries and conserved currents: conserved currents in field theory; current conservation and symmetries; symmetries of the action; groups; discrete vs continuous symmetries.
September 8 (Thursday):
Symmetries: continuous symmetries and their generators; Lie groups and Lie algebras; global and local symmetries; Noether theorem and its proof; the SO(N) example.
September 13 (Tuesday):
Translational symmetry and the stress-energy tensor.
Local phase symmetry: complex fields and the phase symmetry; local symmetry and the covariant derivatives; relation to gauge transforms; multiple charged fields; math of covariant derivatives; non-commutativity.
September 14 (Wednesday) [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 (Thursday):
Aharonov–Bohm effect and magnetic monopoles: the covariant Schroedinger equation; the Aharonov–Bohm effect; cohomology of the vector potential; charge quantization and the compactness of the U(1) phase symmetry; magnetic monopoles; Dirac quantization of the magnetic charge; gauge bundles.
Intoduction to quantization: canonical quantization vs. path integrals; Hamiltonian classical mechanics.
September 20 (Tuesday):
Canonical quantization in mechanics: canonical commutation relations; Schrödinger and Heisenberg pictures of quantum mechanics; Poisson brackets and commutators.
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; the quantum Klein–Gordon equation.
Started quantum fields and particles: expanding free relativistic scalar field into harmonic oscillators.
September 22 (Thursday):
Quantum fields and particles: Eigenstates of the free quantum field's Hamiltonian; identifying the identical bosons; the Fock space.
Going back: from identical bosons to harminic oscillators to quantum fields; the non-relativistic quantum fields; the second quantization.
Translating between the wave function and the Fock space languages for the operators (quick overview only).
September 27 (Tuesday):
Relativistic quantum fields: Lorentz groups; 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 28 (Wednesday) [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; co-dimension.
September 29 (Thursday):
Relativistic causality: superluminal particles in `relativistic' QM; superluminal signals in QM and in QFT; proof of causality for the free scalar field.
Feynman propagator for the scalar field: why and how of time-ordering; defining the propagator; Feynman propagator as a Green's function.
October 4 (Tuesday):
Feynman propagator as 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.
Generators of the Lorentz symmetry.
Representations of groups and Lie algebras: matrix representations; generators; reducible and irreducible representations; QM: multiplets of quantum states and multiplets of operators; simple and non-simple algebras; compact and non-compact algebras; Lorentz multiplets: no nontrivial finite unitary representations; infinite unitary multiplets of particle states; finite non-unitary multiplets of fields; no finite unitary multiplets.
October 6 (Thursday):
Particle representations of the Lorentz symmetry: the little group G(p); Wigner theorem: massive particles have definite spins, massless particles have definite helicities, tachyons have nothing; generalization to d≠3+1 dimensions.
Tachyons in QFT mean vacuum instabilities; scalar VEVs.
October 11 (Thursday):
Lorentz multiplets of fields.
Dirac spinor fields: Dirac matrices; Dirac spinor multiplet; Dirac equation; Dirac Lagrangian; Hamiltonian for the Dirac fields; classical limits of fermionic fields; Grassmann numbers.
October 12 (Wednesday) [supplementary lecture]:
Supefluidity: excitations of a supefluid; dispersion relation ω(k) for the excitations; reasons for dissipationless flow; critical velocity.
October 13 (Thursday):
Classical limits of fermionic fields: Grassmann numbers
Fermionic particles and holes: Fermionic Fock space; particle-hole formalism; Dirac electrons and positrons.
October 18 (Tuesday):
Charge conjugation: C:e↔e+; C:Ψ(x)→γ2Ψ*(x); Majorana fermions and other neutral particles; Majorana vs. Weyl.
Parity and CP symmetries.
October 19 (Wednesday) [supplementary lecture]:
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; Bott periodicity; Majorana–Weyl spinors in d≡2 (mod 8); general Bott periodicity for spinors of SO(a,b).
October 20 (Thursday):
Chiral symmetry: Vector and axial symmetries of a Dirac fermions; the currents; chiral U(N)L×U(N)L symmetry.
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: the interaction picture of QM; the Dyson series and the time-ordering; the S matrix and its elements.
October 25 (Tuesday):
Perturbation theory in QFT and Feynman diagrams: 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; connected diagrams and scattering amplitudes.
October 27 (Thursday):
Perturbation theory and Feynman rules: Phase space factors; loop counting for quartic and cubic couplings; Mandelstam's s, t, and u.
Gave out the midterm exam.
November 1 (Tuesday):
Feynman rules for multiple fields.
Good and bad interactions in perturbative QFT: dimensional analysis; 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 3 (Thursday):
Collect the the midterm exam.
QED Feynman rules: vertices, propagators, and external lines; Dirac indexology; Gordon identities; sign rules; Coulomb scattering example.
Muon pair production in QED, e+e+μ+: the tree amplitude; un-polarized scattering and spin sums/averages.
November 8 (Tuesday):
Muon pair production in QED: Dirac trace techniques; the traces for the pair production; the cross-section; quark pair production e+e+q+q̄→hadrons.
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 9 (Wednesday) [supplementary lecture]:
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 10 (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; averaging over fermions' spins; summary; kinematics.
Maybe Compton scattering and other related processes.
November 15 (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 pseudo-Goldstone bosons.
November 16 (Wednesday) [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.
November 17 (Thursday):
Non-abelian gauge symmetries: covariant derivatives, matrix-valued connections, and gauge transforms; non-abelian vector fields and tension tensors; Yang–Mills theory; QCD; maybe QCD Feynman rules.
November 22 (Tuesday):
General gauge theories: general gauge groups; matter fields in general multiplets; products of gauge symmetries; the Standard Model.
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.
November 29 (Tuesday):
The Higgs mechanism: caveats about the unitary gauge; multiple Higgs fields; SO(N)→SO(N−1)→SO(N−2) example; breaking both local and global symmetries.
Glashow–Weinberg–Salam theory: Higgsing the SU(2)W×U(1)Y down to the 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.
Fermions in the GWS theory: Quantum numbers of the quarks and the leptons; the Yukawa couplings giving rise to the masses; the EM current and the weak currents, charged and neutral.
November 30 (Wednesday) [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.
December 1 (Thursday):
Fermion mass due to Higgs VEV×Yukawa coupling; chiral symmetry breaking.
Quarks and leptons in the GWS theory: Quantum numbers of quarks and leptons; Yukawa couplings and masses; the EM current and the weak currents (charged and neutral) in terms of the fermion fields; flavor mixing and the Cabibbo–Kobayashi;–Maskawa matrix; CP violation in weak interactions; neutrino masses.
Give out the final exam.

QFT2, Spring 2017 semester

January 24 (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; cutting off the UV divergence; explaining the cutoff; physical vs. bare couplings.
January 26 (Thursday):
Bare and physical couplings: The bare coupling and the cutoff; physical coupling in terms of a physical amplitude; resumming the perturbation theory in terms of physical coupling; cutoff independence.
The ultraviolet regulators: Wilson's hard-edge cutoff; Pauli–Villars; higher-derivative regulator; covariant higher derivative for gauge theories; dimensional regularization; the lattice.
January 31 (Tuesday):
Dimensional regularization: D<4 as a UV regulator; Gaussian and non-Gaussian integrals in non-integer dimensions; taking the D→4 limit.
The optical theorem: origin in the unitary S matrix; one-loop example.
February 2 (Thursday):
Correlation functions of quantum fields: correlation functions; relation to the free fields and evolution operators; Feynman rules; connected correlation functions.
The two-point correlation function: Källén–Lehmann spectral representation; features of the spectral density function; Analytic two-point function F2(p2): 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.
February 3 (Friday) [supplementary lecture]:
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.
February 7 (Tuesday):
Perturbation theory for the F2(p2): re-summing the 1PI bubbles; Σ2(p2) and renormalization of the mass and of the field strength; mass renormalization in the φ4 theory; quadratic UV divergences and how to regulate them.
Begin Field strength renormalization in the Yukawa theory: the one-loop diagram; the trace; the denominator and the numerator.
February 9 (Thursday):
Field strength renormalization in the Yukawa theory: the UV divergence structure: Σ(p2)=(div.constant)+(div.constant)×p2 +finite_f(p2); calculating the Σ(p2); the imaginary part and the decay of the scalar into fermions; dΣ/dp2 and the scalar field strength renormalization.
The counterterm-based perturbation theory: ℒbare=ℒphys+counterterms; the counterterm vertices; canceling the infinities.
Counting the divergences: superficial degree of divergence; graphs and subgraphs; classifying the divergences; counterterms; cancelling sub-graph divergences in situ.
February 10 (Friday) [supplementary lecture]:
Relating the correlation functions Fn(p1,…pn) to the scattering amplitudes: the amputated core and the external leg bubbles; the poles for the on-shell pi0→±E(pi) and their relations to the asymptotic x0i→±∞ limits; the asymptotic |in⟩ and ⟨out| states and the LSZ (Lehmann–Symanzik–Zimmermann) reduction formula; scattering amplitudes and the amputated diagrams.
February 14 (Tuesday):
Subgraph divergences: in situ cancelation of subgraph divergences by counterterms; nested divergences; overlapping divergences; BPHZ theorem.
Divergences and renormalizability: supeficial degree of divergence in the φk theories; super-renormalizable, renormalizable, and non-renormalizable theories; divergences in general quantum field theories and the energy dimensions of the couplings; super-renormalizable, renormalizable, and non-renormalizable couplings in d=4; other spacetime dimensions.
February 16 (Thursday):
QED perturbation theory: the counterterms and the Feynman rules; the divergent amplitudes and their momentum dependence; the missing counterterms and the Ward–Takahashi identities.
Calculate the one-loop Σμν(k) and check the WT identity.
February 21 (Tuesday):
Dressed propagators in QED: the electron's F2(); the photon's F2μν(k); equations for the finite parts of the δ2, δm, and δ3 counterterms.
Effective coupling at high energies: Loop corrections to Coulomb scattering; effective QED coupling αeff(E) and it's running with log(energy); running of other coupling types.
Begin Ward–Takahashi identities: the identities; Lemma 1 (the tree-level two-electron amplitudes).
February 23 (Thursday):
Ward–Takahashi identities: Lemma 2 (the one-loop no-electron amplitudes); identities for the (bare) higher-loop amplitudes; generalizing to multiple charged fields; relation to the electric current conservation; Ward identity δ12 for the counterterms; equation for the finite part of the δ1.
February 24 (Friday) [supplementary lecture]:
Grand Unification: unifying the EM, weak, and strong interactions in a single non-abelian gauge group; the SU(5) example; multiplets of fermions; the gauge couplings and the Georgi–Quinn:–Weinberg equations; the doublet-triplet problem; baryon decay and other exotic processes.
February 28 (Tuesday):
Form factors: probing nuclear and nucleon structure with electrons; the form factors; the on-shell form-factors F1(q2) and F2(q2); the gyromagnetic ratio.
The dressed QED vertex at one loop: the setup and the diagram; the denominator; the numerator algebra; calculating the F2 form factor and the anomalous magnetic moment; the experimantal and the theoretical electron's and muon's magnetic moments at high precision.
March 2 (Thursday):
The electric from factor F1(q2) at one loop: calculating the integrals; the infrared divergence and its regulation; momentm dependence of the IR divergence; the δ1 counterterm.
IR finiteness of the measureable quantities: similar IR divergences of the vertex correction and of the soft-photon bremmsstrahlung; detectable vs. undetectable photons and the observed cross-sections; IR finiteness of the observed cross-sections.
Briefly: higher lops and/or more soft photons; optical theorem for the finite observed cross-sections; no Fock space for QED and other gauge theories; jets in QCD.
March 7 (Tuesday):
Gauge dependence in QED; δ1(ξ) and δ2(ξ).
Symmetries and the counterterms: In QED, δmm because of chiral symmetry when m=0; in Yukawa theory, δmm because of discrete chiral symmetry, but δλ≠0 when λ=0; general rule: all counterterms of dim≤4 allowed by the symmetries of the physical Lagrangian.
Large logs and running couplings: large log problem for E≫m; adjusting the counterterms to avoid large logs; running counterterms and running couplings; renormalization schemes.
March 9 (Thursday):
Intro to the renormalization group: the off-shell renormalization schemes; the anomalous dimension of a quantum field; the β function and the running coupling; RGEs and their solutions; renormalization of QED; βe to one-loop order.
March 21 (Tuesday):
Renormalization of the Yukawa theory: the counterterms, the anomalous dimensions, and the β–functions; two coupled RGE equations.
Types of RG flow: flow to UV vs. flow to IR; β>0 and Landau poles; β<0 and the asymptotic freedom; IR price of asymptotic freedom; confinement and chiral symmetry breaking in QCD.
March 23 (Thursday):
Renormalization group flows: fixed points: RG flows for multiple couplings; flow to UV vs. flow to IR.
Relevant, irrelevant, and marginal operators; effective field theories.
Gave out the midterm exam.
March 28 (Tuesday):
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.
Introduction to path integrals: path integrals in QM; the Lagrangian and the Hamiltonian froms of the path integral; derivation and discretization; normalization of the path integral.
March 30 (Thursday):
Path Integrals: the partition function; the harmonic oscillator example; functional integrals in QFT; Feynman rules; sources and the generating functionals.
Collect the midterm exams.
March 31 (Friday) [supplementary lecture]:
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; a QFT at a β=0 point; exotic SCFTs in 5D and 6D and families of SCFTs in D=2,3,4; AdS/CFT duality.
April 4 (Tuesday):
Functional integrals in Euclidean spacetime: QFT–StatMech corresponcence; convergence and imaginary time; Euclidean action; Euclidean lattice as a UV cutoff; restoration of the SO(4)→Lorentz symmetry in the continuum limit; coupling–temperature analogy.
Fermionic functional integrals: Berezin integrals over fermionic variables; Gaussian fermionic integrals.
April 6 (Thursday):
Functional integrals for fermions: fermionic gaussian integrals; free Dirac field with sources; Dirac propagator; FI for fermions in EM background; 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 and the Feynman gauge.
April 7 (Friday) [supplementary lecture]:
Gauge theories on the lattice: local symmetry on the lattice; U(link) variables and the gauge fields; non-abelian symmetries and U(link) variables; U(plaquette) and the Fμν(x), abelian or non-abelian; action for the EM and the YM theories on the lattice. Briefly: computer simulations; strongly coupled compact U(1).
April 11 (Tuesday):
Gauge-averaging and the Feynman gauge in QED.
Quantiing the Yang--Mills theory: fixing the non-abelian gauge symmetry; Faddeev–Popov ghost fields; gauge-fixed YM Lagrangian.
QCD: Quantum Lagrangian and the Feynman rules; generalization to other gauge theories; weakened Ward identities; the qq̄→gg example.
April 13 (Thursday):
Weakeaned Ward identities for QCD: the qq̄→gg example; relations between longitudinal gluons and ghosts.
BRTS symmetry: the BRST generator and its nilpotency; invariance of ℒQCD; the Fock space of QCD — the physical and the unphysical states; BRST cohomology and getting rid of the unphysical states; BRST symmetries of the amplitudes; cancellation of unphysical processes.
QCD renormalizability: renormalizability and the counterterm set; BRST and other manifest symmetries; weakened Ward identities for the counterterms;
April 18 (Tuesday):
QCD renormalizability: renormalizability and the counterterm set; BRST and other manifest symmetries; weakened Ward identities for the counterterms.
QCD β–function at the one-loop order: the beta function and the counterterms; calculating the δ2; calculating the δ2 — the QED-like loop and the non-abelian loop; began calculating the δ2 &mdash just the quark loop.
April 20 (Thursday):
Finish calculating the QCD β–function: the gluon loop, the sideways loop, and the ghost loop; assembling the β– function and generalizing to other non-abelian gauge theories.
Quark confinement in QCD: chromo-electric field lines and flux tubes; condensation of chromo-magnetic monopoles; flux tubes as hadronic strings; mesons and baryons; spinning hardonic string and the Regge trajectories.
April 21 (Friday) [supplementary lecture]:
Instantons: topological index I[Aμ] and its quantization; SE≥(8π2/g2)×|I| and the topological sectors in the YM path integral; 't Hooft instantons and tunneling events.
April 25 (Tuesday):
Overview of comfinement and chiral symmetry breaking in QCD: confinement, flux tubes, mesons, baryons, the hadronic string; chiral symmetry breaking and its order paramater ⟨Ψ̄Ψ⟩; deconfinement and chiral symmetry restoration at high temperatures; the phase transition and its type.
Chiral symmetry breaking in detail: U(Nf)L×U(Nf)R and its currents; anomalous breaking of the axial U(1); flavor structure of the ⟨Ψ̄Ψ⟩ condensate; the linear and the non-linear sigma models of the SU(2)×SU(2)→SU(2) chiral symmetry breaking; NLΣM for the SU(3)×SU(3)→SU(3) breaking; the NLΣM in QCD context, its perturbation by the quark masses, and the masses for the pseudoscalar mesons.
April 26 (Wednesday) [extra supplementary lecture at 3:30 to 5 PM, in RLM 9.304]:
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 27 (Thursday):
Axial Anomaly: The η meson's mass problem in QCD and the anomaly; the massless QED example; the naive Ward identities for the axial Jμ5 current and what's wrong with them; the triangle graphs and the problems with their UV regulators; calculating the anomaly using the Pauli–Villars regulator.
April 28 (Friday) [supplementary lecture]:
Instantons: multi-instanton confugurations; the Θ angle.
Instantons and fermions: zero modes in instanton background; Atyah–Singer index theorem; relation to the axial anomaly; chiral anomaly of the Θ angle and Θ̅=Θ+phase(det(quark mass matrix)); the strong CP problem.
May 2 (Tuesday):
Anomalies in QED: multiple fermions; decay of the neutral pion to 2 photons.
Axial anomaly in QCD: 2-gluon and 3-gluon anomalies; anomaly cancelation for the SU(Nf) currents and the η/π problem.
Chiral gauge theories: Weyl fermions and chiral currents; loops of Weyl fermions and the chiral anomaly; the trace=0 conditions for anomaly cancelation.
May 4 (Thursday):
Anomaly cancelation and miscancelation: gauge anomaly cancelation in the Standard Model; gauge anomaly cancelation in GUTS; SU(2)W anomalies of the baryon and lepton numbers; baryon decay and baryogenesys.
Give out the final exam.

Last Modified: May 4, 2017.
Vadim Kaplunovsky