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GEOMETRICAL METHODS IN PHYSICS

Degree course: 
Corso di Second cycle degree in PHYSICS
Academic year when starting the degree: 
2022/2023
Year: 
1
Academic year in which the course will be held: 
2022/2023
Course type: 
Compulsory subjects, characteristic of the class
Language: 
English
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Basic knowledge of algebra, geometry, real analysis in several variables, mechanics, electromagnetism

Final Examination: 
Orale

Solution of written exercises proposed by the teacher + oral exam
At the end of the course, the docent will propose some exercises to be solved at home. It will be evaluated as first the ability to frame the problem in the right context, the correctness of the procedure, the correctness of the mathematical passages and the ability to provide an answer to the problem.
It follows an oral discussion in order to evaluate the general theoretical knowledge of the student.

Assessment: 
Voto Finale

Acquisition of operational capabilities in the use of geometric instruments in the physical field.
At the end of the course the student will be able to:
-recognize the role of geometry in different physical models;
-provide examples;
-apply techniques of differential geometry to problems in physics;
-analyze a physical model from the geometric viewpoint;
-compare different geometric models in order to select the most suitable one;
-create new models

1 Multilinear algebra
Tensors. Tensors in physics. The external algebra.
2 Representations of finite groups
General theory. Characters. Irreducible representations.
3 Tensors and symmetric group
Introduction. Schur functors. Examples.
4 Introduction to differential geometry
Differential varieties. Fiber bundles. Differential forms.
5 Lie Groups and Lie algebras
Lie groups. Simple Lie algebras. Representations of simple Lie algebras. Representations of sl(2).
6 Classification of simple Lie algebras
Cartan algebra and weights. Roots and Dynkin diagrams. Real simple algebras.
7 Representations of simple Lie algebras
Weights: integrality and symmetries. The highest weight. Irreducible representations of sl(n). The case n=3.
8 Orthogonal Groups
Bilinear and quadratic forms. Orthogonal groups. Quaternions. The group SO(2m). The group SO(2m+1).
9 Spin representations
Clifford algebras. The spin group. Spin representations of so(n).
10 Geometric structures
Fiber bundles. Geometry and Lie groups. Connections. Covariant derivatives. Curvature. Riemannian structure.
11 Dynamical theory of symmetries
Matter fields. Gauge fields. Yang-Mills theories. External symmetries, and relativity.
12 Scalar and spinoral fields
Particles and the Poincaré group. Klein-Gordon equation. Dirac equation. The SO(1,3) group. Spin 1 fields. Fields and gravity.
13 Geometry of the Standard Model of particles and GUT
The field content in the Standard Model. Constructing the Standard Model. Spontaneous symmetry breaking. The mass of neutrino. The seesaw mechanism. GUTs.

Convenzionale

Frontal lessons

Office hours: upon agreement with the teacher.

Professors

CACCIATORI SERGIO LUIGI

Borrowers