Advanced mathematical methods of physics (code 61843) has credit value 6 and it is taught in the second semester of the first year of the LM.
It will introduce the basic concepts and techniques of the functional formalism for Quantum Mechanics and Quantum Field Theory and the fundamental principles of Group Theory as a tool to describe the symmetries of physical systems. It will be shown how the consequences of symmetries can be efficiently implemented within the functional formalism of Quantum Field Theory.
Lectures are given in Italian.
The calculus of variations is a general method to derive differential and partial differential equations
used in physics.
We will show how to solve these equations using the theory of distributions and the Green's function method.
This teaching unit introduces the functional formalism for quantum theories, starting from non-relativistic Quantum Mechanics and continuing to Quantum Field Theory. It will also provide a concise introduction to Group Theory, the mathematical formalism used to describe symmetries, with particular emphasis to Lie Groups, that describe continuos simmetries. Finally, it will explian how the physical consequences of symmetries can be conveniently implemented in the path integral formulation of relativistic quantum theories.
Emphasis will be given to fundamental concepts and computational tools, rather than to generality and mathematical rigour.
At the end of the course students should be able to apply the tools of functional integration and the methods of Group Theory to the study of Quantum Field Theories and their symmetries.
Traditional: chalk and blackboard. Home assignments will be handed out weekly and their solution will be verified during the final oral exam.
1) Path integrals in quantum mechanics and in relativistic quantum field theories. The bosonic and fermionic path integral. Correlation functions and their euclidean continuation. Generating functionals of connected and 1PI correlation functions and the effective action. Correlators of composite operators.
2) General properties of groups and their representations. Lie groups and Lie algebras. A brief introduction to the roots and weights of a Lie algebra.
3) Symmetries in classical field theories: Noether's theorem. Symmetries in quantum field theories: the operatorial and the functional approaches. Implementation of symmetries in the functional formalism: Schwinger-Dyson and Ward–Takahashi identities.
4) Spontaneously broken global symmetries. Goldstone's theorem, coset manifolds.
Ricevimento: Students can request an appointment by email: stefano.giusto@ge.infn.it
STEFANO GIUSTO (President)
PIERANTONIO ZANGHI'
NICODEMO MAGNOLI (President Substitute)
ANDREA AMORETTI (Substitute)
Check the calendar at
https://corsi.unige.it/corsi/9012/studenti-orario
Oral exam. During the exam the student will also be asked to discuss the solution of one of the home assignments.
A list of problems will be handed out weekly. To verify that students are able to apply the techniques of group theory and the functional formalism to problem solving, students will be asked to present the solution of one of the home assignments during the oral exam. The exam also aims at assessing the knowledge and comprehension of the results derived in class.
Students who have valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the instructor and with Professor Sergio Di Domizio (sergio.didomizio@unige.it), the Department’s disability liaison.
