The main objective of this course is to teach the concepts of continuous and discrete symmetries in the framework of the quantum and classical laws of physics, making use of the techniques of the group theory. The purpose is to provide the basic mathematical tools of group theory for the understanding of the theory of fundamental interactions among elementary particles and their phenomenological implications.

In particular:
D1. Knowledge and ability to
understanding: basic knowledge of the principles of discrete and continuous symmetries, and of group theory applied to quantum mechanics and its applications in elementary particle physics; ability to understand the scientific literature on the topic.

D2. Applied knowledge and understanding: knowledge and ability to use group theory and their representations useful for solving fundamental physical problems in quantum mechanics and elementary particle physics.

D3. Making judgements: ability to critically understand textbooks as well as scientific works in the field of quantum mechanics and elementary particle physics that make use of group theory applied to the resolution of associated physical problems.

D4. Communication skills: ability to explain and critically discuss general topics related to the principles of symmetry in physics and their applications through the use of group theory.

D5. Ability to learn: ability to use the knowledge acquired to solve physical problems in quantum mechanics that require the use of group theory.

basic knowledge of classical analytical mechanics, non-relativistic quantum mechanics, and special relativity.

This course consists in a series of lectures on the role of the continuous and discrete symmetries of space-time and of internal symmetries in quantum physics and their implications for the properties of the fundamental interactions among elementary particles. In particular this includes: the study of the continuous Lie groups, abelian and non-abelian, concept of generators of a non-abelian Lie group and its associated algebras, and their implications for the study of the fundamental interactions at quantum level. Aspects of symmetries of the strong interactions under the action of the internal groups SU(2) and SU(3), predictions for the mass spectrum of hadronic multiplets and other phenomenological implications. Role of the parity symmetry breaking in the weak interactions and its phenomenological implications. Introduction to the properties of the Lorentz and Poincare` groups in relativistic physics, algebra of associated generators and their main representations.

1) Landau-Lifsits, vol. IV, Relativistic Quantum Theory, Edition Mir, second edition 1991.

2) W. Greiner - B. Muller, Quantum Mechanics symmetries, second edition, Springer Verlag 1994

3) Relativita' - teoria di Dirac, N. Cabibbo lectures 2003 , in italian, (provided by the teacher).

-Symmetries and conservation laws in classical mechanics: invariance of physical systems for spatial translations and rotations and temporal translations, Noether's theorem, conservation laws in the external field.

-Symmetries in quantum mechanics: space-time translations, rotations.

-Introduction to group theory: continuous groups and the concept of generators of a continuous group. SO(3) rotation group, properties and representations. Generators of the rotation group in classical and quantum mechanics.

- Irreducible representations of the rotation group: vector, spin 1/2 and spin-1; Pauli matrix algebra; matrix representation of the angular momentum operator. Wigner D-matrices.

- Introduction to Lie groups: properties of simple and semi-simple Lie groups. Algebra of generators of a non-abelian Lie group, definition of structure constants, Cartan subalgebra, rank and roots of an algebra. Casimir operators, Racah theorem. SO(3) and SU(2) groups.

- Symmetry groups of the Hamiltonian and its implications: invariance for translations, rotations, degenerate energy eigenstates. Algebra of the generators of spatial translations and rotations; generalization to rotations in an N-dimensional Euclidean space.

- Discrete symmetries: parity transformations, time inversion and charge conjugation, and associated operators.

- Dynamic symmetries: Coulomb degeneracy of the hydrogen atom;

- Non-compact Lie groups, generators of rotations in spaces with non-positive metric.

- Lorentz group and its representations, and extension to the Poincare` group

-Internal symmetries and properties of strong interactions: SU(2) group of strong interactions; concept of isotopic spin, mass degeneracy of proton and neutron states; conservation of the baryon number; adjoint representation of the SU(2) group and its properties. SU(2) invariance test of strong interactions: G-parity, selection rules.

- SU(3) symmetry group of strong interactions: strangeness and hypercharge quantum number Y; classification into degenerate SU(3) multiplets for hadron and meson states; Gell-Mann matrices; properties of the SU(3) group, algebra of generators.

- Fundamental representations of the SU(3) group: concept of quarks and anti-quarks; representations of SU(3) multiplets starting from products of fundamental representations and its applications to the hadronic spectrum.

- Color of quarks: phenomenological motivations, SU(3) color group for quarks and its physical implications for strong interactions.

- SU(3) and SU(2) symmetry breaking and predictions of the quark model, Okubo-Gell-Mann relation for mass splitting in baryon and meson multiplets.

- Weak charged interactions: parity violation, beta decay, phenomenon of oscillations in neutral K mesons.

Lectures at the board or in alternative, preregistered online lectures. Notes of the lecturer (in English) and text books.

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By means of questions in class during lessons. Written final exam, method used:15 multi-answer questions. Oral examination if necessary. The final examination may be conducted in either Italian or English, at the student's discretion.