EW2122

16 January
6 February
20 April (appello straordinario) - >> Students can sign up in this file excel writing their student ID number (matricola) to choose a date.

The exam consists of an oral session on the programme of the course (for students who attended the course with prof. Martinelli the exam will focus on the old programme)

Syllabus

23/02/2022 [2h]: Program and practical aspects of the course. Historical introduction to the problem of infinities in QFT [W1]
28/02/2022 [2h]: Origin of infinites as high-energy / small-distance effects and examples. Regularization procedures, dimensional regularization and example of the calculation of a divergent diagram [S], [C]
1/03/2022 [2h]: Expansion in d=4-eps of the divergent integral. Dimensionality of fields and couplings in d-dimensions. Power counting and superficial degree of divergence. (super)-renormalizable and non-renormalizable theories. [S], [PS]
2/03/2022 [2h]: Quick review of QED. Superficially divergent diagrams in QED. Renormalized perturbation theory. Beginning of vacuum polarization computation. [S], [PS]
7/03/2022 [2h]: Vacuum polarization (continued). Physics of the vacuum polarization at high and low q^2. Lamb shift and running coupling. General structure of the photon-electron vertex. [S], [PS]
8/03/2022 [2h]: Electron-photon vertex. Counterterm and anomalous magnetic moment. [S], [PS]
9/03/2022 [2h]: Electron self-energy. MSbar scheme. Pole mass and MSbar mass. Ward-Takahashi identity and universality of charge renormalization. [S], [PS]
14/03/2022 [2h]: Symmetries in classical mechanics (quick review): active and passive canonical transformations, point transformations, symmetries as canonical transformations, generator of a canonical transformation, generator as a constant of motion of a symmetry transformation. Symmetries in classical field theory (quick review): field transformations and symmetries, Noether theorem, conserved charge as the generator of the symmetry transformation. Symmetry transformations are groups.
15/03/2022 [2h]: Review on Lie groups: linear (matrix) Lie groups; Lie algebra of generators; connected, semi-simple, simple, compact Lie groups; classification of compact, simple linear Lie groups: classical and exceptional groups; from Lie algebra to Lie group: the exponential map. Global symmetries of a (classical) theory of N real or complex free scalar fields.
16/03/2022 [2h]: Symmetries in QFT: the Wigner-Weyl realization. Projective representations, conditions for a group to have intrinsic projective representations, examples of SO(3) and SO(3,1), universal covering. [W1 ch.2] Implications of Wigner-Weyl symmetries: action of the symmetry on the algebra of observables and fields, charges as generators of the symmetry group. [W1 sec. 7.3], [IZ]
21/03/2022 [2h]: Algebra of currents, Schwinger terms. Implications of Wigner-Weyl symmetries: invariance of the vacuum and Fabri-Picasso theorem; Coleman's theorem on current conservation; Ward identities. [IZ], [W1]
22/03/2022 [2h]: Approximate symmetries and spurion analysis: U(1) axial symmetry in QED and mass renormalization. Finite-dimensional representations of the Lorentz group: SL(2,C) as double cover of the Lorentz group; algebra so(3,1) of the Lorentz group from the complexification of the su(2) algebra; Weyl fermions as the smallest irreducible representations of the Lorentz group. [W1], [S]
23/03/2022 [2h]: Global symmetries of a theory of free Weyl fermions: Majorana mass term and its diagonalization, vectorial subgroup. Global symmetries of a theory of free Dirac fermions: mass term and its diagonalization, mass as a spurion of chiral symmetry, U(N) vs SU(N)xU(1). [S]
30/03/2022 [2h]: Example of spurion analysis: radiative corrections to fermion mass term in a theory with one Dirac fermion and one real scalar; technical naturalness and fine-tuning. Symmetries realized à la Nambu-Goldstone: introduction; real scalar theory with spontaneously broken Z2, analogy with quantum mechanics and the role of tunneling. [R], [S]
4/04/2022 [2h]: Spontaneous symmetry breaking as an emergent phenomenon in classical mechanics; role of the infinite volume limit in field theory. [W2] Example of the ferromagnet. Example of a theory of one complex scalar with spontaneously boken U(1): manifold of vacua, massless excitations, non-linear realization of U(1). O(N) linear sigma model: counting the massless fields. [R], [S], [PS], [A]
5/04/2022 [2h]: O(N) linear sigma model: broken currents interpolate NGBs, manifold of vacua, transformation law of NGBs; non-linear realization of the global symmetry. Spontaneously broken symmetries in quantum field theory: charge is ill defined; commutator of the charge with local fields and its properties. [IZ], [A]
6/04/2022 [2h]: Goldstone's theorem: two proofs. [IZ], [A]
11/04/2022 [2h]: Particles as unitary representations of the Poincaré group, method of induced representations. Casimirs of the Poincaré group, classification of the irreducible representations. One-particle massive states, spin. One-particle massless states, ISO(2) as the little group. [W1], [WKT]
12/04/2022 [2h]: One-particle massless states (continued): helicity, examples of massless particles in nature, implication of P and CPT invariance on the physical spectrum. Massless vs massive neutrinos. Vacuum as the trivial representation of the Poincaré group. Transformation rules of creation and destruction operators from those of particles. [W1]
13/04/2022 [2h]: Constructing Lorentz covariant local fields: from the transformation rules of particles to the transformation rules of fields; massive case, Proca Lagrangian; massless case, gauge invariance from Lorentz invariance. [W1], [S]
20/04/2022 [2h]: Constructing gauge field theories: coupling to a matter field sector, covariant derivative. [W1] Non-abelian gauge theories: gauge transformation rule of gauge field, covariant derivative, field strength and Yang-Mills Lagrangian, equations of motion, self interactions of gauge bosons. [TG], [PS]
21/04/2022 [2h]: Gauge invariance as a redundancy of the description: invariance of observables and physical states, need of gauge fixing in a quantum theory. Routes to quantization of an abelian gauge theory: covariant and non-covariant gauges. [TQ] Quantization of non-abelian gauge theories, ghosts. Theta term and its importance in non-abelian gauge theories. Beta function and RG evolution for abelian and non-abelian gauge theories. [PS]
9/05/2022 [2h]: Beta-function and its calculation in QED using dimensional regularization and the MSbar scheme. [S] Phases of gauge theories (I): abelian and non-abelian Coulomb phase; Conformal phase, conformal window for a QCD-like theory, Banks-Zacks perturbative IR fixed point. UV behavior of gauge theories: asymptotic freedom; theories with unbounded couplings, Laundau poles, QED as an effective field theory; UV fixed points, asymptotic safety. [W2 sec. 18.3], [S sec. 23.2]
10/05/2022 [2h]: Phases of gauge theories (II): confinement, hadrons as asymptotic states, dimensional transmutation. [W2 sec. 18.7], [PS sec. 16.7, 17.1] Theory of a spin-1 particle as a gauge theory: rewriting the Proca Lagrangian using the Stueckelberg trick, xi-gauges, physical and unphysical poles in the propagators.
16/05/2022 [2h]: Equivalence between the theory of a massive spin-1 particle and an abelian gauge theory with spontaneously broken global symmetry (Brout-Englert-Higgs mechanism): theory with complex scalar field subject to a non-linear constraint; behavior of degrees of freedom in the high-energy limit. Power counting hbar factors.
17/05/2022 [2h]: BEH mechanism: theory with radial (Higgs) mode, Lagrangian in a generic xi gauge and in the unitary gauge. Validity of the Goldstone theorem in gauge theories with spontaneous symmetry breaking. Equivalence theorem and its application to the scattering of longitudinally-polarized vector bosons. [PS], [S]
23/05/2022 [2h]: Theories of self-interacting massive spin-1 vectors as spontaneously-broken gauge theories. Theories with U(N) -> U(N-1) spontaneous breaking: Lagrangian in the unitary gauge, physical spectrum. [CL], [PS], [R] Theory with SU(N)xSU(N)->SU(N) spontaneous breaking: Sigma field, unitary gauge, physical spectrum.
24/05/2022 [2h]: Theory with SU(2)xSU(2)->SU(2) spontaneous breaking: Lagrangian in the xi-gauge; scattering of longitudinal vectors bosons, loss of perturbative unitarity; recovering perturbative unitarity by adding the Higgs boson.
30/05/2022 [2h]: Contribution of the Higgs boson to the scattering amplitude of longitudinal vectors bosons; role of the Higgs boson in a spontaneously-broken gauge theory. Introduction to Effective Field Theories: locality and structure of an effective Lagrangian, effective coefficients and local operators, contribution of local operators to low-energy observables.
31/05/2022 [2h]: Two examples of EFTs: i) Fermi theory, and ii) heavy fermion coupled to a scalar and a gauge field. EFTs as full-fledged QFT, UV behavior of EFTs and their renormalizability. Sigma model as an EFT, sketch of 1-loop renormalization. Matching EFTs to known UV theories. Quantities that are calculable in an EFT.
6/6/2022 [2h]: Estimating the effective coefficients: example of the dipole operator in QED. Validity of an EFT: example of Fermi theory. Introduction to the Standard Model of particle physics: particles and fundamental interactions.
7/6/2022 [2h]: The Standard Model of particles physics as an effective gauge theory: field content and quantum numbers; renormalizable Lagrangian; spontaneous breaking of the electroweak symmetry, Weinberg angle; Lagrangian in the unitary gauge, masses and couplings; diagonalization of the Yukawa matrices, Cabibbo-Kobayashi-Maskawa matrix and its physical parameters. [PS], [S]

References

[W1] S. Weinberg, The Quantum Theory of Fields, Volume 1, Cambridge Univ. Press
[W2] S. Weinberg, The Quantum Theory of Fields, Volume 2, Cambridge Univ. Press
[S] M. Schwartz, Quantum Field Theory and the Standard Model, Cambridge Univ. Press
[PS] M. Peskin, D. Schroeder, An introduction to Quantum Field Theory, Perseus Books
[R] L. H. Ryder, Quantum Field Theory, Cambridge Univ. Press
[C] J. Collins, Renormalization, Cambridge Univ. Press
[IZ] C. Itzykson, J-B. Zuber, Quantum Field Theory, Dover
[A] I.J. Aitchison, An informal introduction to gauge field theories, Cambridge Univ. Press
[CL] T-P. Cheng, L-F. Li, Gauge Theory of elementary particle physics, Oxford
[D] J. Donoghue, E. Golowich, B. Holstein, Dynamics of the SM (second edition), Cambridge Univ. Press
[WKT] Wu-Ki Tung, Group Theory in Physics, World Scientific
[TG] D. Tong, Gauge Theory, link to lecture notes
[TQ] D. Tong, Quantum Field Theory, link to lecture notes

For a review on symmetries in classical mechanics and classical field theory, see for example:

R. Shankar, Principles of Quantum Mechanics, chapter 2.

[IZ ] sections 1-1-1, 1-2-3

For an introduction to group theory, see for example:

H. Georgi, Lie Algebras in Particle Physics, Westview press.

A. Zee, Group Theory in a Nutshell for Physicists, Princeton Univ. Press

Wu-Ki Tung, Group Theory in Physics, World Scientific

Notes on Young Tableaux

For a review on the quantization of abelian gauge theories (Coulomb gauge and Lorenz gauge), see for example:

D. Tong, Quantum Field Theory, chapter 6.

For an introduction to Effective Field Theories, see for example:

H. Georgi, Effective Field Theory, Ann. Rev. Nucl. Part. Sci. 43 (1993) 209.

D.B. Kaplan, Lectures at the 17th National Nuclear Physics Summer School, Berkely, arXiv: nucl-th/0510023

Lectures at the GGI School on Fundamental Interactions, 2016

A. Manohar, Lectures at Les Houches 2017 Summer School, arXiv:1804.05863

On electron and muon (g-2):

S. Laporta and E. Remiddi, Nucl.Phys.B 181 (2008) 10
A. Hoecker and W.J. Marciano, Muon Anomalous Magnetic Moment, review for the PDG
Notes by P. Agrawal prepared for The Allure of Ultrasensitive Experiments, Fermilab Academic Lectures 2013-2014.

On Coleman's theorem:

Coleman, J. Math. Phys. 7 (1966) 787

Useful Material

Notes:

On SU(2) and SO(3)

Useful Resources

Quantum Field Theory Lectures by Sidney R. Coleman (videos)
David Tong: Lectures on Theoretical Physics
Leonard Susskind: The Theoretical Minimum
Gerard 't Hooft: How to become a GOOD Theoretical Physicist
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