Quantum Field Theory: Lecture Log
- January 23 (Wednesday):
- 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 28 (Monday):
- 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; mentioned the dimensional regularization and the lattice.
- January 30 (Wednesday):
- 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 4 (Monday):
- 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.
Begin perturbation theory for
the F2:
re-summing the 1PI bubbles; Σ(p2)
and renormalization of the mass and of the field strength;
mass renormalization in the φ4 theory; fine tuning problem.
- February 6 (Wednesday):
- 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(p2)=(div.constant)+(div.constant)×p2
+finite_f(p2);
calculating the Σ(p2);
dΣ/dp2 and the scalar field strength renormalization.
Counterterms perturbation theory:
ℒbare=ℒphysical+counterterms;
Feynman rules for the counterterms;
adjusting δZ, δm, and δλ
order by order in λ; one-loop examples.
- Extra lecture on February 11 (Monday):
- Lehmann–Symanzik–Zimmermann reduction formula:
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.
- Regular lecture on February 11 (Monday):
- Counting the divergences:
superficial degree of divergence; graphs and subgraphs; classifying the divergences;
counterterms and canceling the divergences.
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.
- February 13 (Wednesday):
- Troubles with non-renormalizable theories.
Dimensional analysis and renormalizability:
canonical dimensions of fields and couplings; power-counting renormalizability;
renormalizable theories in 4D; other dimensions.
QED perturbation theory:
the counterterms and the Feynman rules;
the divergent amplitudes and their momentum dependence; missing counterterms.
- February 18 (Monday):
- QED perturbation theory:
missing counterterms and the Ward–Takahashi identities.
Σμν(k) at one loop order:
calculation, checking the WT identity, the divergence and the δ3 counterterm.
Dressed propagators in QED:
the electron's F2(p̸);
the photon's F2μν(k);
equations for the finite parts of the δ2, δm,
and δ3 counterterms.
- February 20 (Wednesday):
- 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.
Ward–Takahashi identities:
the identities; Lemma 1 (the tree-level two-electron amplitudes);
Lemma 2 (the one-loop no-electron amplitudes);
identities for the (bare) higher-loop amplitudes; Z1=Z2;
generalizing to multiple charged fields.
- Extra lecture on February 25 (Monday):
- Spontaneous Symmetry breaking:
scalar poantial, 〈Φ〉≠0, and spontaneous symmetry breaking;
complex field example; linear sigma model; Wigner and Goldstone–Nambu modes of symmetries;
Goldstone theorem; broken-symmetry currents in the linear sigma model;
pion scattering; spontaneous symmetry breaking in d≠4 dimensions.
- Regular lecture on February 25 (Monday):
- Ward–Takahashi identities:
δ1=δ2; in the counterterm perturbation theory;
general Ward–Takahashi identities and their relation to the current conservation.
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.
Began the dressed QED vertex at one loop:
the diagram and the denominator.
- February 25 (Wednesday):
- Dressed QED vertex at one loop:
numerator algebra;
calculating the F2 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 F1(q2) at one loop:
calculating the integrals; the infrared divergence and its regulation;
momentm dependence of the IR divergence; the δ1 counterterm.
Began
IR finiteness of the measureable quantities:
similar IR divergences of the vertex correction and of the soft-photon bremmsstrahlung.
- March 4 (Monday):
- 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.
Gauge dependence in QED: δ1(ξ) and δ2(ξ).
Began symmetries and counterterms.
- March 6 (Wednesday):
- Finish symmetries and counterterms.
Intro to renormalization group: large log problem for E≫m;
adjusting the counterterms to avoid large logs; off-shell renormalization schemes;
running counterterms and running couplings.
Renormalization group basics: the anomalous dimension of a quantum field;
the β function and the running coupling.
- Extra lecture on March 11 (Monday):
- Higgs mechanism:
semiclassical abelian example; propagator of the massive photon; Higgs mechanism for general (abelian) SSB;
non-abelian examples; general gauge theory.
- Regular lecture on March 11 (Monday):
- Renormalization group equations:
RGE and its solution for the λφ4 theory;
renormalization of QED: anomalous dimensions, βe to one-loop order, solving the RGE for QED;
β–functions for general couplings; coupled RGE foe the Yukawa theory (briefly).
Begin types of RG flows:
β>0, Landau poles, and UV incompleteness; β<0, QCD example, and asymptotic freedom;
IR price of asymptotic freedom in QCD; confinement; chiral symmetry breaking.
- March 13 (Wednesday):
- Finish types of RG flows — fixed points:
β(g*)=0; UV stability vs. IR stability; scale invariance and conformal symmetry;
RG flows for multiple couplings; flow to UV versus flow to IR.
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 18 and March 20:
- Spring break, no classes.
- March 25 (Monday):
- Non-abelian gauge symmetries:
review of the abelian gauge theory;
non-abelian covariant derivatives, matrix-valued connections, and gauge transforms;
non-abelian vector fields and tension tensors; Yang–Mills theory; QCD;
tree-level QCD Feynman rules.
- March 27 (Wednesday):
- Non-abelian gauge symmetries:
general gauge groups; matter fields in general multiplets;
products of gauge symmetries; the Standard Model.
Introduction to path integrals:
path integrals in QM; the Lagrangian and the Hamiltonian forms of the path integral;
derivation of the Hamiltonian form.
Gave out the midterm exam.
- April 1 (Monday):
- Introduction to path integrals:
Lagrangian path integrals — derivation and normalization;
the partition function; harmonic oscillator example.
Functional integrals in QFT:
“path” integrals for quantum fields; correlation functions;
free fields and propagators; perturbation theory and Feynman rules;
sources and the generating functionals.
Convergence of path integrals and imaginary time; Euclidean 4D for QFT path integrals.
- April 3 (Wednesday):
- Functional integrals in Euclidean spacetime:
QFT↔StatMech corresponcence; Euclidean lattice as a UV cutoff;
restoration of the SO(4)→Lorentz symmetry in the continuum limit;
coupling↔temperature analogy.
Begin fermionic functional integrals:
Grassmann numbers; Berezin integrals over fermionic variables.
Collect midterm exams.
- Extra lecture on April 8 (Monday):
- 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.
Began Fermions in the GWS theory:
Quantum numbers of the quarks and the leptons; the Yukawa couplings giving rise to the masses.
- Regular lecture on April 8 (Monday):
- 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.
Briefly functuonal integrals at finite temperature:
periodicity in Euclidean time, period=β; Det(−∂2+m2)
over periodic fields; antiperiodic fermionic fields;
Fadde'ev–Popov determinant and the number of photonic degrees of freedom.
- April 10 (Wednesday):
- Quantizing 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 in QCD: the qq̄→gg example;
ghost pair production and unitarity.
Intro to BRST symmetry:
BRST symmetry action on the fields; nilpotency of the BRST operator;
BRTS invariance of the QCD Lagrangian.
- Extra lecture on April 15 (Monday):
- Quarks and leptons of the Standard Model:
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.
- Regular lecture on April 15 (Monday):
- BRTS symmetry:
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;
allowed counterterms and renormalizability;
Ward-like (but weaker) identities for the counterterms.
- April 17 (Wednesday):
- Basic group theory: the Casimir and the Index.
QCD β–function at the one-loop order:
the beta function and the counterterms; calculating the δ2 for quarks;
calculating the δ1 for quarks — the QED-like loop and the non-abelian loop;
calculating the δ3 — the quark loop, the gluon loop,
the sideways loop, and the ghost loop;
assembling the β–function.
- April 22 (Monday):
- β–functions for QCD and other gauge theories:
assembling QCD beta function; other gauge theories; allowing for Majorana and Weyl fermions;
allowing for scalar fields; general formula.
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 24 (Wednesday):
- Chiral symmetry:
axial symmetry of a massless fermion; QED anomaly of the axial symmetry;
SU(Nf)L×SU(Nf)R×U(1)V
exact symmetry of QCD with massless quarks;
approximate SU(2)L×SU(2)L and SU(3)L×SU(3)L
of real-life QCD.
Spontaneous breaking of chiral symmetry:
overview of SSB and of the Goldstone symmetry;
pions as pseudo–Goldstone bosons; generalize to 3 light flavors.
Linear and non-linear sigma models:
linear sigma model for the SU(2)×SU(2)→SU(2) SSB;
taking the λ→∞ limit to the non-linear sigma model;
general NLΣM's; the SU(2)–valued non-linear field and its symmetries;
vector and axial currents of the NLΣM; pions as Goldstone bosons of the NLΣM;
generalize to 3 flavors.
- Extra lecture on April 29 (Monday):
- CP Violation:
weak interactions break C and P but might preserve CP;
neutral K mesons and their mixing; GIM mechanism; decays of KS and KL;
imaginary part of ΔM and the CP violation; CP violation for the neutral D and B mesons.
CP violation in the Standard Model:
CP action on W±, on the charged currents, and on the CKM matrix;
un-removable CKM phase for 3 families.
- Regular lecture on April 29 (Monday):
- Non-linear sigma models and chiral symmetry breakdown:
NLΣMs in general; NLΣM for the SU(N)×SU(N)→SU(N) ΧSB;
symmetry currents of the NLΣM; quark-qntiquark condensate in QCD and the NLΣM for the condensate;
perturbation by the quark masses; U(3) NLΣM and η↔η′ mixing;
constituent quark mass and the non-relativistic quark model.
Started axial anomaly in QED:
Formal Ward-like identities for the axial current; hidden momentum shift and the regularization trouble.
- May 1 (Wednesday):
- Axial anomaly in QED:
limiting the anomaly to the one-loop triangle diagrams; regulation analysis for the Pauli–Villars cutoff;
⟨∂μJμ5reg⟩=
−2iMPV⟨η̄γ5η⟩;
evaluating the triangle diagrams: relation to the PV regulator loops, numerator algebra, taking the integrals,
summary; anomalies for QED with multiple fermions; decay of the neutral pion to 2 photons.
- Extra lecture on May 6 (Monday):
- Neutrino masses:
neutrino oscillations; Dirac and Majorana masses for the neutrinos;
SM origin of neutrino masses; seesaw mechanism.
- Regular lecture on May 6 (Monday):
- Anomalous 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;
trace=0 conditions for anomaly cancelation;
lepton and baryon anomalies in the Standard Model.
- May 8 (Wednesday):
- Gauge anomalies: triangle anomaly in chiral QED; anomalous gauge variance of log(det(̸D));
anomaly in non-abelian chiral theories; Wess–Zumino consistency conditions;
anomaly coefficients Aabc and traces over chiral fermions; rules for anomaly cancellation;
checking anomaly cancellation for the Standard Model; relations between quark and lepton charges;
anomaly cancelation in general chiral theories; massive fermions do not affect the anomalies;
cubic Casimirs and cubic anomaly indices for simple gauge groups;
applications to Grand Unification.
Gave out the final exam.
Last Modified: May 8, 2019.
Vadim Kaplunovsky
vadim@physics.utexas.edu