OVERVIEW

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. 

AIMS AND CONTENT

LEARNING OUTCOMES

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.

AIMS AND LEARNING OUTCOMES

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. 

PREREQUISITES

• Mathematical methods of Physics. Quantum mechanics. Special Relativity. Theoretical Physics.
• The path integral formulation of Quantum Field Theory will be introduced from scratch. Notions of relativistic quantum theories, introduced in the Theoretical Physics course, will be useful to fully appreciate the physical meaning of the methods developed in this course. 

TEACHING METHODS

Traditional: chalk and blackboard. Home assignments will be handed out weekly and their solution will be verified during the final oral exam.

SYLLABUS/CONTENT

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. 

RECOMMENDED READING/BIBLIOGRAPHY

• H. Georgi, Lie Algebras in Particle Phyics, CRC Press 1999
• S. Weinberg, The Quantum Theory of Fields. Vol I, II, Cambridge University Press, 2005
• M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory, CRC Press
• Class notes will be made available to the students

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

Check the calendar at

 https://corsi.unige.it/corsi/9012/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral exam. During the exam the student will also be asked to discuss the solution of one of the home assignments. 

ASSESSMENT METHODS

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. 

FURTHER INFORMATION

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.