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Mathematical methods in physics: geometry and group theory 1

A.Y. 2020/2021

Learning objectives

The course aims at providing students with competences in the areas of differential geometry and group theory, and the ability to use these methods to solve actual physics problems. In the first part of the course, starting from principles of tensor calculus and differential forms we arrive at extending differential operations to manifolds and curved spaces, with the emphasis on obtaining frame-invariant physical laws. In the second part, starting from group axioms we arrive at developing the theory of group representation and of characters for both discrete and continuous groups.

Expected learning outcomes

At the end of this course the student will master the following skills:

1) will be able to use and manipulate tensorial objects and the transformation laws thereof

2) will be able to formulate physical laws and classical field theories in covariant form, with application to electrodynamics and general relativity and will be able to derive covariant equations from invariant actions (e.g. the Einstein equations from the Einstein-Hilbert action)

3) will be able to use the language of differential forms and of differential manifolds along with the corresponding topological aspects

4) will be able to formalize physical symmetry operations in terms of groups and representations thereof

5) will be able to understand and utilize the algebraic properties of groups (conjugation classes, subgroups, cosets)

6) will be able to use discrete groups and the theory of representation and of characters thereof

7) will be able to understand and use the fundamental theorems of the theory of group representation (great orthogonality theorem, Schur lemmas and the demonstrations thereof)

8) will be able to understand and apply continuous groups such as SO(3), SU(2), Lorentz and Poincare' groups etc and the properties thereof to physical problems

9) will be able to apply group theory to problems of quantum mechanics (analysis of angular momentum, Clebsch-Gordan coefficients, Wigner small matrix etc) and problems of solid state physics (crystal symmetries, crystal field splitting, tensorial properties of crystals) and spectroscopy (level splitting)

1) will be able to use and manipulate tensorial objects and the transformation laws thereof

2) will be able to formulate physical laws and classical field theories in covariant form, with application to electrodynamics and general relativity and will be able to derive covariant equations from invariant actions (e.g. the Einstein equations from the Einstein-Hilbert action)

3) will be able to use the language of differential forms and of differential manifolds along with the corresponding topological aspects

4) will be able to formalize physical symmetry operations in terms of groups and representations thereof

5) will be able to understand and utilize the algebraic properties of groups (conjugation classes, subgroups, cosets)

6) will be able to use discrete groups and the theory of representation and of characters thereof

7) will be able to understand and use the fundamental theorems of the theory of group representation (great orthogonality theorem, Schur lemmas and the demonstrations thereof)

8) will be able to understand and apply continuous groups such as SO(3), SU(2), Lorentz and Poincare' groups etc and the properties thereof to physical problems

9) will be able to apply group theory to problems of quantum mechanics (analysis of angular momentum, Clebsch-Gordan coefficients, Wigner small matrix etc) and problems of solid state physics (crystal symmetries, crystal field splitting, tensorial properties of crystals) and spectroscopy (level splitting)

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

Lectures notes prepared by the lecturer, which include all steps in the mathematical derivations, will be made available to the student for each lecture. Explanatory videos of the learning material corresponding to each lecture, will be posted in Google Drive, and if possible also on Ariel, in asyncronous mode.

**Course syllabus**

This course aims to provide a solid background in differential geometry, tensor calculus and group theory for students that will subsequently attend Theoretical Physics 1, Theoretical Physics 2, and the majority of advanced courses in Structure of Matter, Nuclear Physics and Astrophysics.

- definitions of tensorial objects and the transformation laws thereof

-formulation of physical laws and classical field theories in covariant form

- extension of differential operations to curved spaces, derivation of Einstein equations via the Einstein-Hilbert action

- introduction to the language of differential forms and of differential manifolds along with the corresponding topological aspects

- formalization of physical symmetry operations in terms of groups and representations thereof

-algebraic properties of groups (conjugation classes, subgroups, cosets)

- Lie groups and Lie algebras, differential forms of generators

- discrete groups and the theory of representation and of characters thereof

- fundamental theorems of the theory of group representation (great orthogonality theorem, Schur lemmas and the demonstrations thereof)

- theory of representation and characters of continuous groups such as SO(3), SU(2), Lorentz and Poincare' groups etc and their properties, Weyl and Dirac representations and Pauli-Lubanski vector

- application of group theory to problems of quantum mechanics (analysis of angular momentum, Clebsch-Gordan coefficients, Wigner small matrix etc) and problems of solid state physics (crystal symmetries, crystal field splitting, tensorial properties of crystals)

- definitions of tensorial objects and the transformation laws thereof

-formulation of physical laws and classical field theories in covariant form

- extension of differential operations to curved spaces, derivation of Einstein equations via the Einstein-Hilbert action

- introduction to the language of differential forms and of differential manifolds along with the corresponding topological aspects

- formalization of physical symmetry operations in terms of groups and representations thereof

-algebraic properties of groups (conjugation classes, subgroups, cosets)

- Lie groups and Lie algebras, differential forms of generators

- discrete groups and the theory of representation and of characters thereof

- fundamental theorems of the theory of group representation (great orthogonality theorem, Schur lemmas and the demonstrations thereof)

- theory of representation and characters of continuous groups such as SO(3), SU(2), Lorentz and Poincare' groups etc and their properties, Weyl and Dirac representations and Pauli-Lubanski vector

- application of group theory to problems of quantum mechanics (analysis of angular momentum, Clebsch-Gordan coefficients, Wigner small matrix etc) and problems of solid state physics (crystal symmetries, crystal field splitting, tensorial properties of crystals)

**Prerequisites for admission**

Calculus, linear algebra, vector spaces, analytical geometry

**Teaching methods**

Traditional at the blackboard with full development of derivations and theorems

**Teaching Resources**

- K. Cahill "Physical Mathematics", Cambridge University Press

- S. M. Carrol "Spacetime and geometry", Cambridge University Press

- H. F. Jones "Groups, representations, and physics" Taylor & Francis

- A. Zee "Group theory in a nutshell", Princeton University Press

- S. M. Carrol "Spacetime and geometry", Cambridge University Press

- H. F. Jones "Groups, representations, and physics" Taylor & Francis

- A. Zee "Group theory in a nutshell", Princeton University Press

**Assessment methods and Criteria**

Oral exam

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6

Lessons: 42 hours

Professor:
Zaccone Alessio

Professor(s)