# GROUP THEORY

**PHYSICS 9012 (coorte 2019/2020)**- THEORETICAL PHYSICS 61842
- MATHEMATICAL METHODS IN PHYSICS 61843
- MATTER PHYSICS 2 61844
- NUCLEAR AND PARTICLE PHYSICS AND ASTROPHYSICS 2 61847

**PHYSICS 9012 (coorte 2020/2021)**- THEORETICAL PHYSICS 61842
- MATTER PHYSICS 2 61844
- NUCLEAR AND PARTICLE PHYSICS AND ASTROPHYSICS 2 61847

OVERVIEW

Group Theory (code 63662) has credit value 6 and it is taught in the second semester of the first and second year of the LM.

Lectures are usually given in Italian. Shuld it be requsted,, they can be given in English. This has been done in the past.

## AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to give the basic properties and constructions of finite dimensional representations of finite and compact groups and (some of) their applications In Quantum Mechanics, Another topic is the structure of Linear Lie Groups and their Lie Algebras

AIMS AND LEARNING OUTCOMES

- At the conclusion of the course the student should know the irreducible representations of the Symmetric Group, of SU(2) (and the Rotation Group) and SU(3)
- He should be able to understand the role played by these representations in Quantum Mechanics and in some elemantary particles models
- He should be able to use the theory of group representations in the solution of explicit problems.

PREREQUISITES

- Linear Algebra( finite dimensional), including the Spectral Theorem for an arbitrary set of commuting linear transformations on a finite dimensional vector space.
- Basic notions of Quantum Mechanics as given in the courses of LT

Teaching methods

Traditional: chalk and blackboard. The students are stongly invited to attend the lectures.

SYLLABUS/CONTENT

1. GENERAL PROPERTIES OF GROUPS

1.1. Review of definitions

1.2. Examples of finite grous

1.2.1. Cyclic group of order n

1.2.2. Symmetric group

1.2.3. Dihedral group

1.2.4. Other examples

1.3. Examples of non finite groups

1.4. Action of a group and conjugacy classes

2. REPRESENTATIONS OF FINITE GROUPS

2.1. Representations

2.1.1. General facts

2.1.2. Irreducible representations

2.1.3. Direct sum of representations

2.1.4. Intertwining operators and Schur lemmas

2.2. Characters and orthogonality relations

2.2.1. Functions on the group and matrix elements

2.2.2. Characters of representations and orthogonality relations

2.2.3. Applications to the decomposition of representations

2.3. The regular representation

2.3.1. Definition

2.3.2. The character of the regular representation

2.3.3. Isotypic decomposition

2.3.4. A base of the space of class function

2.4. Projection operators

2.5. Induced representations

2.5.1. Definition

2.5.2. Geometric interpretation

3. REPRESENTATIONS OF COMPACT GROUPS

3.1. Compact groups

3.2. The Haar measure

3.3. Representations of topological groups and Schur lemmas

3.3.1. General facts

3.3.2. Coefficients of a representation

3.3.3. Intertwining operators

3.3.4. Schur lemmas

3.4. Representations of compact groups

3.4.1. Complete decomposability

3.4.2. Orthogonality relations

4. LIE ALGEBRAS AND LINEAR LIE GROUPS

4.1. Lie algebras

4.1.1. Definition and examples

4.1.2. Morphisms

4.1.3. Commutation relations and structure constants

4.1.4. Complexification and real forms

4.1.5. Representations of Lie algebras

4.2. Review of the properties of the matrix exponential

4.3. One parameter subgroups of GL(n,K)

4.4. Linear Lie groups

4.5. The Lie algebra of a linear Lie group

4.6. Morphisms of Lie algebras and Groups

4.6.1. Differential of a morphism of Lie groups

4.6.2. Differential of a representation of a Lie group

4.6.3. The adjoint representation

5. THE GROUPS SU(2) AND SO(3)

5.1. The Lie algebras su(2) and so(3)

5.2. The covering morphism of SU(2) onto SO(3)

5.2.1. The Lie group SU(2)

5.2.2. The Lie group SO(3)

5.2.3. The projection of SU(2) onto SO(3)

6. THE REPRESENTATIONS OF SU(2) AND SO(3)

6.1. Irreducible representations of sl(2,C)

6.1.1. Representations

6.1.2. Casimir operator

6.1.3. The “ladder” operators

6.2. Irreducible representations of SU(2)

6.2.1. The representations

6.2.2. The characters of the representations

6.3. Irreducible representations of SO(3)

7. SPHERICAL HARMONICS

7.1. *Review of the space *

7.2. Harmonic polynomials

7.2.1. Groups representations in functions spaces

7.2.2. Spaces of harmonic polynomials

7.2.3. Representations of SO(3) in spaces of harmonic polynomials

7.3. Spherical harmonics

7.3.1. Representations of SO(3) in spaces of spherical harmonics

7.3.2. The Casimir Operator

7.3.3. Base in the space of spherical harmonics

7.3.4. Explicit formulas

8. REPRESENTATIONS OF SU(3)

8.1. Review of sl(n,C).Representations of sl(3,C)

8.1.1. Review of sl(n,C)

8.1.2. The example sl(3,C)

8.1.3. Cartan algebra

8.1.4. Representations of sl(3,C) and SU(3)

8.2. Adjoint representation and roots

8.3. The fundamental representation and its dual

8.3.1. The fundamental representation

8.3.2. The dual of the fundamental representation

8.4. Highest weight of a finite dimensional representation

8.4.1. The highest weight

8.4.2. Weights as linear combinations of fundamental weights

8.4.3. Finite dimensional representations and weights

8.4.4. Other examples the representations 6,8 and10

8.5. Tensor products of representations

8.6. The eightfold way

8.6.1. Barions

8.6.2. Mesons

8.6.3. Barionic resounances

8.7. Quarks and antiquarks

9. THE ROLE OF THE SYMMETRIC AND ROTATIONS GROUPS IN QUANTUM MECHANICS

9.1. The representations of the symmetric group

9.1.1. Conjugate classes of the symmetric group

9.1.2. Young frames and representations

9.1.3. Action of the symmetric group on tensor products

9.2. Representations of compact groups and perturbation theory

9.2.1. Structure of the multiplicity of an eigenvalue of the Hamiltonian

9.2.2. Splitting of a degenerate eigenvalue under a perturbation

9.3. One particle systems

9.3.1. Multiplicity of eigenvalue for a n electron in a central field

9.3.2. Spin orbit interaction and fine structure

9.3.3. Stark and Zeeman effects

9.4. Systems of identical particles

9.4.1. Pauli principle and N electrons atom

9.4.2. Eigenvalues of the Hamiltonian

9.4.3. Spectroscopic terms

9.4.4. Spin orbit interaction. Multiplets

RECOMMENDED READING/BIBLIOGRAPHY

- W. Miller, Symmetry groups and their applications, Academic Press 1972
- S. Sternberg, Group theory and physics, Cambridge University Press 1994
- B. Hall, Lie groups Lie algebras and representations, Springer 2004
- The teacher's notes are distributed to the students

## TEACHERS AND EXAM BOARD

Exam Board

STEFANO GIUSTO (President)

NICOLA MAGGIORE

SIMONE MARZANI

CAMILLO IMBIMBO (President Substitute)

## LESSONS

Teaching methods

Traditional: chalk and blackboard. The students are stongly invited to attend the lectures.

LESSONS START

Second semester

## EXAMS

Exam description

written exam

Assessment methods

The main aim of the course is to make the student able using the mathematical methods of the theory of representations of groups and Lie algebras. The role of the exam is to test that this aim has been achieved.

This is why the exam must be the solution of a non trivial problem. The solution of a really non trivial problem may require more time than that available for a class session.

With these motivations, the following is the structure of the exam. Each student is given the text of a problem (all different). The solutions are requested inside two/three days. In this way, it is possible to assign problems more structured that the immediate application of a formula seen during the course.

The final assessment is given after a discussion with the student on his solution,