OVERVIEW

Advanced mathematical methods of physics (code 61843) has credit value 6 and it is taught in the first semester of the first or second year of the LM.

Il will introduce the tools of Group Theory to describe the symmetries of physical systems and their implementation 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

The course introduces group theory, the mathematical formalism used to describe symmetries, and its implementation in the path integral formulation of relativistic quantum theories.

It will provide the basics of Lie groups, Lie algebras and their representations. It will develop the functional formulation of quantum mechanics and quantum field theory and explain how the physical consequences of symmetries can be conveniently implemented in this formalism. 

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 methods of group theory to physical problems in quantum field theory.

PREREQUISITES

• Mathematical methods of Physics. Quantum mechanics. Special Relativity. 
• The path integral formulation of Quantum Field Theory will be introduced from scratch. Notions of relativistic quantum theories, introduced in the Theoretical Physics course, are not necessaries but they will be useful to fully appreciate the physical meaning of the methods developed in this course. 

TEACHING METHODS

Traditional: chalk and blackboard. Home assignements will be handed out weekly.

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. 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. Effective Lagrangians.

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
• Class notes will be made available to the students

TEACHERS AND EXAM BOARD

Exam Board

STEFANO GIUSTO (President)

PIERANTONIO ZANGHI'

NICODEMO MAGNOLI (President Substitute)

ANDREA AMORETTI (Substitute)

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 assignements. 

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 assignements during the oral exam. The exam also aims at assessing the knwoledge and comprehension of the results derived in class. 

Exam schedule