Quantum field theory 2

A.Y. 2020/2021
Overall hours
Learning objectives
Expand the core ideas of relativistic quantum field theory which have been introduced in Quantum Field Theory 1, specifically in what concerns analiticity, symmetry and invariance.
Expected learning outcomes
At the end of this course the student:
1. Will be able to use unitarity and the optical theorem to understand the analytic properties of amplitudes;
2. Derive the Ward identities for symmetres realized in Wigner-Weyl form;
3. Prove Glodstone's theorem for spontaneously broken symmetries, both at the classical and quantum level;
4. Construct and compute the effective potential;
5. Quantize a gauge theory and derive its Feynman rules with various gauge choices
6. Construct a gauge theory with massive field via the Higgs mechanism;
7. Renormalize quantum electrodymanics perturbatively;
8. Understand the quantum breaking of classical symmetries related to scale invariance (including chiral anomalies);
9. Write donw and solve the Callan-Symanzik equation (renormalization group equation);
10. Compute the operator-product (Wilson) expansion and the anomaloud dimensions of operators entering it.
Course syllabus and organization

Single session

Lesson period
First semester
Course syllabus
Vector fields. Yang-Mills theories and non-abelian gauge invariance. Faddev-Popov quantization. Lattice gauge theories. Perturbative renormalization: the case of the scalar field con with 4-bodies interaction. The methods of regularization. Dimensional regularization. Critical phenomena and divergencies. Scale invariance in quantum filed theories. Renormalization group according to Wilson. recursion relations for the case of the scalar field with one component. Fixed point and classification of scaling operators. Scaling of the correlation functions in the critical region. Anomalous dimension of the fields. Operator Product Expansion and Renormalization Group (RG). RG for the non-linear sigma model in two dimension. Asymptotic freedom. Universality of the first coefficients of the beta function. Lambda parameter. RG for the XY model and topological phase transitions. Generating functionals. Effective potentials. Spontaneously broken symmetries in quantum filed theories.
Prerequisites for admission
Knowledge of the basics of relativistic quantum field theory, special relativity and path integral methods
Teaching methods
The course consists of blackboard lectures in which the individual topic included in the syllabus are presented, fist introducing the basics and then discussing the main conceptual points and computational technique. Interaction with the students in class is very much encouraged, through questions and discussions.
Teaching Resources
Notes. The article by Wilson e Kogut, Phys. Rep. 12C, The Renormalization Group and the epsilon expansion
J. Cardy: Scaling and Renormalization Group in Statistical Physics.
J. Zinn-Justin: Quantum Field Theory and Critical Phenomena.
Assessment methods and Criteria
The exam is an oral test of about one hour, during which the student is asked to discuss one topic selected among those included in the syllabus of the course. During the exam, the student is asked a number of questions of variable complexity, which aim at ascertaining his basic understanding of the various topics covered in class, his ability to place them in the more general context of quantum field theory, and his ability to think critically and autonomously using these methods.
Lessons: 42 hours
Professor: Caracciolo Sergio