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Professore Associato
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Fisica

Insegnamento: Advanced Quantum Field Theory

Physics - Fisica (D.M.270/04) (Offerta formativa 2022)

Obiettivi formativi

Knowledge and understanding :
This is a follow up course on Quantum Field Theory (QFT). In the first course we aimed to give students the main tools of the discipline and applied them to some introductory topics such as scalar field theory and spinor field theory. In the present course instead we are going to study more advanced materials such as the quantization of theories based on local (gauge) symmetry, i.e. gauge theories. These theories are indeed the main ingredients of the Standard Model of particle physics that describes with amazing precision the interactions of elementary particles.

Applying knowledge and understanding
At the end of the course the student will develop the skills to apply the knowledges acquired to an arbitrary QFT.

Making judgement
At the end of the course students should be able to choose by themselves the suitable mathematical approach to face the various problems arisen during the lessons.

Communications skills:
At the end of the course students should be able to describe with the appropriate technical language and the correct mathematical formalism the subject presented in the course.

Learning skills
At the end of the course students should have developed the skill to deepen by themselves the various general approached treated during the lessons and extend them to arbitrary QFT’s.


Knowledge of introductory topics of quantum field theory is strongly suggested. Specifically, introductory knowledge of functional methods in scalar field theories and spinor field theories, and their application to the perturbative computation of scattering processes is suggested. The necessary level is the one provided by a first-year Master Degree course of QFT.

Programma del corso

1) Introduction
Review of scalar, spinor QFT and abelian gauge theory: terminology, formalism, conventions, goals and main tools, Feynman rules and scattering amplitudes, LSZ reduction formulas. The Noether's theorem in QFT. Ward identities in spinor Quantum Electrodynamics and their consequences. 1 CFU

2) Nonabelian quantum field theory.
Hints of representation theory for non abelian groups. Nonabelian gauge field theory: generalities, nonabelian global and local symmetries, covariant derivatives, field strengths, Yang Mills lagrangian, representations. Path integrals, gauge-fixing, Fadeev-Popov trick. Feynman rules in pure Yang Mills theory, and spinor and scalar Quantum Chromo-Dynamics (QCD). Tree level and one-loop amplitudes in QCD. One loop divergences in spinor QCD and one loop beta functions. BRST quantization for a relativistic particle and for (pure) Yang Mills and its physical interpretation. BRST Ward identities and Slavon-Taylor operator in Yang Mills theory. Chiral theories, and problem of masses, spontaneous symmetry breaking of global and gauge symmetries. 2 CFU

3) Hints of Standard Model Physics.
Particle content, global and gauge symmetries, spontaneous symmetry breaking and Higgs sector. Leptonic and quark sectors. Flavor problem and CKM matrix. 1 CFU

4) Introduction to integrability (in QFT)
What is an integrable model? Historical remarks. Examples of integrable models. Hamiltonian mechanics. Integrals of motion. Liouville integrability. Phase space structure. Chaos vs. integrability.
Lax pair. Classical R-matrix. Spectral parameter. Spectral curve. Dynamical divisor. Examples of Integrable Classical Field Theory models. Kortevegde Vries equation. Solitons and factorized scattering. Integrability structures. Lax monodromy and Lax scattering. Inverse scattering method. Spectral curves. Heisenberg magnet, Riemann surface of monodromy matrix, quasi-momentum, periods and moduli, finite-gap construction. Heisenberg spin chain. Periodic and open boundary conditions. Coordinate Bethe ansatz: magnon states, scattering factor, factorized scattering, solution of the infinite chain. Bethe equations for spin chains with boundary conditions (open or periodic). Heisenberg XXX model with higher spin. Bethe ansatz for higher-rank algebras. Scattering matrix and nested Bethe ansatz. 2 CFU

Metodi didattici

Blackboard lectures. On some specific subjects workgroup may also be proposed.

Testi di riferimento

M. Srednicki, Quantum Field Theory, Cambridge University Press 2007.
L. H. Ryder, Quantum Field Theory, Cambridge University Press 1996.
G. Sterman, An Introduction to Quantum Field Theory, Cambridge University Press 1993.
M. E. Peskin, D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus Book Publishing 1995.

Verifica dell'apprendimento

Informal homework assignments will be handed out in each class. Two noncompulsory midterm take-home written exams based on the homework problems will be handed out and a final oral exam, of about an hour, will be given, where the students knowledge of all the subjects of the course will be established. Although midterm exams are not compulsory, they may provide a verification on the ability of students in solving problems. If a student decides not to take them, problem solving skills will be verified during the oral exam.

Risultati attesi

Knowledge and understanding:
At the end of the course, students will have acquired all the necessary (standard) tools of quantum field theory (QFT).

Applying knowledge and understanding:
The knowledge will enable them to perform the computations of the arbitrary scattering processes, involving scalar particles, spinor particles and vector particles.

Making judgments:
Thanks to the variety of examples provided at the end of the course students will be able to recognize autonomously descriptive approaches and calculation methods appropriate to different types of problems of QFT.

Communicating skills:
Class discussions with teachers and peers will allow students to properly discuss standard arguments of quantum field theory, and some advanced ones.

Learning skills:
The knowledge acquired will allow them to independently study and learn other advanced topics of QFT and related subjects.