Teaching and Schools

2023

Quantum Field Theory -1 (at the master-II of ICFP)
L1, Why QFT
L2-part1, Lagrangian and Hamiltonian for fields, Klein-Gordon as best guess at long distances

L2-part2, Canonical quantization free real scalar field, particle interpretation and bose statistic,
construction Lorentz generators

L3, Canonical quantization Charged scalar field, Wightman functions, Feynman's correlator,
Lorentz invariance and causality

2022

Quantum Field Theory -1 (at the master-II of ICFP, ENS Paris)

Lectures notes (for in-person student only, or email me)
L1, Why QFT
L2-part1, Lagrangian and Hamiltonian for fields, Klein-Gordon as best guess at long distances
L2-part2, Canonical quantization free real scalar field, particle interpretation and bose statistic,
construction Lorentz generators
L3, Canonical quantization Charged scalar field, Wightman functions, Feynman's correlator,
Lorentz invariance and causality
L4, Axioms of scalar QFT, Kallen-Lehmann spectral rep., analytic structure 2-point
correlators
and dispersion relations
L5, Perturbation theory and Dyson series, Causality vs Lorentz invariance
L6, Wick theorem, Feynman's diagrams in position and momentum space , momentum
conservation, tree vs loop diagrams
L7, Loops-expansion=coupling-expansion, quantum correction interpretation,
NDA, renormalization first encounter: the tadpole
L8, Functional generators Z[J] and W[J], Feynman's diagrams with charged fields
L9-part 1, Theory of Scattering and LSZ reduction
L9-part 2, LSZ reduction, Amputation and wavefunction renormalization, crossing symmetry
L10, Fields as Lorentz irreps, Weyl spinors, Sigma matrices, Dirac Spinors and gamma-matrices, Diracology
L11, Weyl, Dirac and Majorana actions and e.o.m., quantizing free Dirac,
spin-statistics, positivity of energy, causality with fermions, Wick theorem and Feynman diagrams with Dirac
L12, Massive vector field, eliminate spin-0 state, 3 polarizations and the little group SU(2), Hamiltonian and quantization,
propagator and the LSZ
L13 part -1, Massless vectors and the Photon: longitudinal polarization blowing up, current conservation,
gauge invariance as redundancy, Ward identity as consistency condition,
ISO(2) little group and triviality translations, gauge invariance from Lorentz-invariance and finite d.o.f., propagator
L13 part-2, QED as EFT of 2 oppositely charged Weyl fermions coupled to massless spin-1,
accidental P, C and T discrete symmetries of QED, transformations of fermion bilinears,
perturbative proof of CPT theorem

L14 parts_1,2 and 3: intro perturbative renormalization, Wick rotation, 1-loop corrections to the self-energy in phi^4,
1-loop corrections to the coupling in phi^4, locality of counter-terms and predictions on non-analytic terms,
checking the optical theorem at 1-loop, Feynman parametrization, the meaning of renormalization,
various regularisations, dimensional reg.

L15 effective action (skipped)

L16 part 1 and 2: -1-loop in phi^3 in D=6: self-energy and vertex renormalization, photon self-energy in QED,
photon transversity via charge conservation, 1PI resummation self energy in QED,
wave function renormalization, 1 loop QED self-energy in dim reg.,
renormalization of coulomb potential: short and long distance corrections,
vacuum polarization and hint to running coupling

L17 Renormalization group equation, beta functions, beta function in phi^4 in D=4, beta function in phi^3 in D=6,
resumming large logs, asymptotics of QFTs: beta>0, beta<0, beta=0, Landau pole and asymptotic freedom,
QED beta function via renormalised potential, QED beta function via renormalized self-energy

L18, part 1 and 2: Path integral in QM: from Euclidean to Minkowski, Gaussian Integration,
2pt-function harmonic oscillator via path integral , Path integral in QM for scalars,
2pt function for scalars via Gaussian integration, perturbative expansion and Dyson series via path int.,
Photon propagator via path integral and gauge fixing

2021

Advanced QFT (10 Lectures at the EPFL)

2020

Topics in EFT (PhD course in Rome) (lecture notes here)

L1 Introduction to EFT via QM anharmonic oscillator

L2 Examples of 4D EFT (massless fermion and massless photon), matching and universality

L3 Scale transformations, irrelevant vs relevant perturbations, intro to Wilsonian RG-flow, phi^4 beta function

L4 Fast and slow modes, running of the mass term, anomalous dimensions, Wilson-Fisher fixed point

L5 UV sensitivity and naturalness, symmetries and spurions, technical vs natural

L6 Massless spin-1 and spin-2 particles, Little group, Locality+Lorentz+causality require gauge transformations

L7 Weinberg's Soft-theorems, electric charge conservation and equivalence principle from consistency of soft amplitudes, bootstrapping Yang-Mills theory

L8 Bootstrapping GR for the m=0 spin-2 self-interacting particle, using first-order formalism.

L9 Field Redefinitions and E.o.m., EFT for photons and gravitons, estimate strong coupling scales, leading higher-derivative corrections to Einstein-Maxwell EFT

L10 Causality and unitarity constraints in EFT

QFT-II (master ICFP at the ENS, Paris )

TD-1 Yang-Mills theory from consistency of Gauss law and 1st order formalism, Noether Theorem and conserved currents;

TD-2 Yang-Mills theory from consistency of Compton scattering via on-shell method (+ discussion homework)

TD-3+4 Ward identities, Global charges, spontaneous breaking and Goldstone theorem (+ discussion homework)

TD-5+6 Goldstone EFT, Higgs mechanism, and equivalence theorem, mostly for U(1) theory (+ discussion homework)

TD-7+8 cancelled

Homework: 1, 2, 3, 4, 5 (solutions: 1, 2, 3, 4, 5)

GGI School: Theory of fundamental Interactions: lectures, webpage

2019

QFT-II (master ICFP at the ENS, Paris )

TD-1 Non-abelian gauge theory from consistency of factorization in Compton scattering

TD-2 Yang Mills from consistency of Gauss law and 1st order formalism; Noether theorem, conserved currents and Ward-Takahashi identies

TD-3 Spontaneous breaking and Goldstone bosons

TD-4 solutions homework-3

TD-5 Higgs mechanism and Equivalence Theorem

TD- 6 Anomalies via path integral, anomaly cancellation in the SM, pi0 decay into gammas

Homework: 1, 2, 3, 4

GGI school Theory of Fundamental Interactions: website

2018

Aspects of (micro)causality in EFTs (at TUM)

causality and Lorentz, via Kallen-Lehmann, via Wick rotation, linear response theory, analyticity 2pt functions, analyticity refraction index, forward scattering, causality vs analyticity, dispersion relations, positivity bounds for scattering amplitudes

Advanced Topics in the Theory of Fundamental Interactions (at the University of Padova)
(lecture notes here)

table of contents

L1: dimensional analysis, intro to EFTs, Integrating out and tree-level matching

L2: Wilsonian effective action, decoupling, RG and beta-functions, Wilson-Fisher, anomalous dim

L3+L4: decoupling in QED, Landau pole, Banks-Zaks, UV sensitivity and naturalness, symmetries and spurions, chiral symmetry, estimating the cutoff and spurion analysis, technical vs natural, hbar counting, g-2

L5+L6: running in QCD, chiral symmetry breaking and Goldstone bosons, chiral effective theory, matching currents, Gell-Mann-Okubo, masses from quark mass splittings and electromagnetic interactions

L7+L8: CCWZ, cosetology, Goldeberger-Treiman, matching currents and neutron decay, unitarizations, rho-meson and photon mixing

L9: Higgs mechanism, restoring gauge invariance for massive spin-1 fields, equivalence theorem

2-component spinors