# THEORETICAL PHYSICS

**PHYSICS 9012 (coorte 2019/2020)**- STATISTICAL PHYSICS 61867
- FIELDS THEORY 61876
- GROUP THEORY 63662
- GENERAL RELATIVITY (6 CFU) 61875
- LAB OF ANDVANCED THERMODYNAMICS 62424
- APPLIED ELECTRONICS 68873
- MESOSCOPIC SYSTEMS AND NANOSTRUCTURES 66800
- NANOSTRUCTURES 62744
- MATERIALS AND DEVICES FOR ELECTRONICS 62421
- FOUNDATIONS OF ASTROPHYSICS AND COSMOLOGY 61874
- PHYSICS OF ELEMENTARY PARTICLES 61872
- LABORATORIO DI FISICA DELLA MATERIA (6 CFU) 61862
- SOLID STATE PHYSICS 61861
- APPLIED NUCLEAR PHYSICS 61871
- PHYSICS OF THE OCEAN 68875
- PHYSICS OF ASTROPARTICLES 61873
- THEORY OF FUNDAMENTAL INTERACTIONS 62422
- LAB OF FUNDAMENTAL INTERACTIONS PHYSICS AND ASTROPHYSICS 61868
- LAB OF BIOPHYSICS 62739
- SOFT MATTER PHYSICS 61863

OVERVIEW

This course is the natural continuation of the course "Quantum Physics". The applications of interest of ordinaruy quantum mechanics require familiarity with specific techniques for the study of systems with many degrees of freedom. The main purpose of this course is poroviding the student with the basic ideas of these techniques and illustrating the relevant applications, both in a many-body, non relativistic context and in the context of the relativistic extension of quantum physics.

## AIMS AND CONTENT

LEARNING OUTCOMES

Providing the student with basis concepts in relativistic electrodynamics, and with quantum mechanics of many-body systems in the context of second quantization.

AIMS AND LEARNING OUTCOMES

The central aim of this course is learning techniques for the application of quantum mechanics to systems of actual physical interest: non-relativistic systems with many degrees of freedom, and relativistic systems.

Specifically, the student will be able to apply the formalism of second quantization to problems of interest. The idea of a canonical transformation will be introduced and developed, together with its main applications.

Finally, the basic ideas for the development of a covariant perturbation theory will be given.

PREREQUISITES

Classical physics: Foundations of analytical mechanics, statistical physics and classic electrodynamics.

Non-relativistic quantum mechanics: basic formalism, perturbation theory, scattering theory.

Special theory of relativity: foundations, four-vector formulation.

Teaching methods

The course is organized in six hours of lecture per week. Lectures are given at the blackboard, with no use of slides. About 30 to 40% of the total time is devoted to applications of the conceptual ideas, through exercises and problems.

SYLLABUS/CONTENT

1. Review of non-relativistic quantum mechanics

2. Systems of identical bosons

2.a The formalism of second quantization

2.b States and observables in second quantization

2.c Wick's theorem

2.d Density operator for mixed states

2.e Generalization of Wick's theorem to mixed states

3. Electromagnetic fields in empty space

3.a Normal modes

3.b Quantization in the radiation gauge

3.c Energy, momentum and spin of photons

3.d IGauge invariance and polarization

4. Relativistic fields

4.a Principle of least action and relativistic invariance

4.b Scalar field

4.c Symmetry-conservation theorem

4.d Quantization of the free real scalar field

4.e Quantization of the free complex scalar field

4.f U(1) symmetry; antiparticles

4.g Action for the free electromagnetic field

4.h Causality in field theory

5. Canonical transformations for bosonic systems

5.a Definition and general properties

5.b Coherent states

5.c Bogolubov transformations

5.d Coherent states of photons

6. Systems of identical fermions

6.a Formalism of second quantization

6.b States and observables in second quantization

6.c Wick's theorem

6.d Density operator for mixed states

6.e Generalization of Wick's theorem to mixed states

7. Canonical transformations for fermionic systems

7.a Bogolubov transformations

7.b Electron and holes

8. Spinor fields

8.a Spinor representations of the Lorentz group

8.b Weyl spinors

8.c Lagrangian density for right-handed Weyl spinors

8.d Quantization, energy and momentum of single particle states

8.e Parity inversion and left-handed spinors

8.f Dirac spinors

8.g Solutions of the free Dirac equation

9. Interactions

9.a Charged particles in an external electromagnetic field

9.b Gauge invariance in quantum mechanics

9.c Dirac equation in the presence of an external electromagnetic field

9.d Yukawa interaction

9.e Isotopic spin symmetry of strong interactions

9.f Interaction representation and induced interactions

10. Towards covariant perturbation theory

10.a Time-dependent perturbation theory

10.b Review of scattering theory

10.c Cross sections and decay rates

10.d Examples in a scalar theory

10.e Feynman rules for the scalar theory

10.f Feynman rules for quantum electrodynamics

10.g An axample: the Compton cross-section

RECOMMENDED READING/BIBLIOGRAPHY

Landau, Lifsitz 2 - Field theory

Landau, Lifsitz 3 - Quantum mechanics

Landau, Lifsitz 4 - Relativistic quantum theory

Gerry, Knight - Introductory Quantum Optics, Cambridge

Becchi, Ridolfi - An Introduction to relativistic processes and the standard model of electroweak interactions

Becchi - Appunti di fisica teorica (2018 version, reviewed by G. Ridolfi)

Maiani - Meccanica quantistica relativistica e introduzione alla teoria dei campi

Ridolfi - Notes on the Course of Theoretical Physics

## TEACHERS AND EXAM BOARD

**Ricevimento:** The students may contact the teacher at any time, for discussions, clarifications of the lectures, suggestions for the solution of problems. A preliminary contact via e-mail is suggested.

Exam Board

GIOVANNI RIDOLFI (President)

SIMONE MARZANI

NICODEMO MAGNOLI

NICOLA MAGGIORE

## LESSONS

Teaching methods

The course is organized in six hours of lecture per week. Lectures are given at the blackboard, with no use of slides. About 30 to 40% of the total time is devoted to applications of the conceptual ideas, through exercises and problems.

LESSONS START

September 24, 2019

## EXAMS

Exam description

The examination has a written test and an interview, usually a few days after the written test.

In the written test, the student will be asked to solve two problems in three hours. Students are allowed to look at books or notes.

The interview, usually 20 to 30 minutes, starts with a discussion of the results of the written test.

Assessment methods

The purpose of the written test is the assesment of the operational capabilities of the sudent. Two problems are usually proposed: one of them is about nonrelativistic applications of second quantization, while the seciond one is about relativistic quantum physics

During the interview, starting from a discussion of the written test, the student is asked to prove his understanding of the conceptual foundations at the basis of the study of many-body quantum systems.