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