### The mathematical structures for symmetries:

How elementary particles are classified and interact?

M. Rausch de Traubenberg

IPHC-DRS, Unistra, CNRS, IN2P3; 23 rue du Loess, Strasbourg, 67037 Cedex, France

**Abstract**

The concept of symmetry together with its associated mathematical structure have a crucial importance in physics. The study of the symmetries of a system could reveal a better understanding of the system itself, and, in particular group theoretical methods have illuminated much of modern physics. The purpose if these series of lectures is to describe from an elementary point of view some of the basic mathematical notions associated to the description of symmetries. All concepts are gradually introduced and illustrated through many examples. Few notions are needed to attend these lectures. Applications in physics, in particular for the description of elementary particles will be given but can be easily extend in condense matter or whatever. All the material of these lectures are taken from the recent book [1].

The first part of these lectures firrstly review the underlying algebraic notions associated to symmetries. Many examples relevant in physics will be given. In particular, we will focus on two important structures: Lie algebras (and their associated Lie groups) and Lie superalgebras (and their associated Lie supergroups). It will then be shown that the principles of Quantum Mechanics lead naturally to the mathematical structure associated for the description of symmetries. As a preamble to a more general study, important matrix Lie groups and matrix Lie algebras, and their differential or oscillators realisations, will be considered. The important notion of representation is also introduced.

The second part of these lectures will be devoted to the study of Lie groups and Lie algebras, mainly the simple complex and simple real Lie algebras together with their representations. Instead of following a formal presentation, it will be shown that these structures are natural generalisations of the well know groups SO(3) (rotations in three dimensions) and SU(3) (unitary transformations in three complex dimensions). The very important concepts of roots, Dynkin diagrams and Cartan matrices will be introduced together with the important notions of weights central to classify representations (the l ∈ ½ℕ of Quantum Mechanics!). A dierential realisation of some Lie algebras which enables to have all unitary representations in an ease manner will be given and illustrated mostly in the context of SU(3). Two applications of Lie algebras will be considered. Firstly, we will identify the Lie algebra(s) associated to the description of the symmetry of the space-time. Thus, as a consequence of space-time symmetries, an elementary particle will be characterised by two quantum numbers: its mass and its spin. Secondly, it will be established that fundamental interactions are associated to internal symmetries. Time permitting we will show how all fundamental interactions could be unify.

The last part of these lectures will be concerned by the Lie superalgebras and the Lie supergroups.

We will identify how these algebras could play a r^ole in physics, in particular to implement a symmetry

between fermions and bosons (called supersymmetry). Two Lie superalgebras will be studied. Their relationship to supersymmetry will be investigated.

The lectures will take place from 16:00 to 18:00 in

**IPHC, Bât 20 room 115C (or Bât 26 room 130): 30 March**, 1; 6; 8; 14; 15; 27; 29 April 1.

References

[1] R. Campoamor-Stursberg et M. Rausch de Traubenberg, Group Theory in Physics: A Practitioner’s Guide, World Scientic (2018), ISBN: 978-981-3273-60-3,

www.worldscientic.com/worldscibooks/10.1142/11081.

____________________________

1The two rooms are closed to each other, one is bigger than the other the room for the lecture will depend on the number

of persons who attend to it.

## Cours ED182 2019-2020

## The mathematical structures for symmetries:

How elementary particles are classified and interact?

M. Rausch de Traubenberg

IPHC-DRS, Unistra, CNRS, IN2P3; 23 rue du Loess, Strasbourg, 67037 Cedex, France

AbstractThe concept of symmetry together with its associated mathematical structure have a crucial importance in physics. The study of the symmetries of a system could reveal a better understanding of the system itself, and, in particular group theoretical methods have illuminated much of modern physics. The purpose if these series of lectures is to describe from an elementary point of view some of the basic mathematical notions associated to the description of symmetries. All concepts are gradually introduced and illustrated through many examples. Few notions are needed to attend these lectures. Applications in physics, in particular for the description of elementary particles will be given but can be easily extend in condense matter or whatever. All the material of these lectures are taken from the recent book [1].

The first part of these lectures firrstly review the underlying algebraic notions associated to symmetries. Many examples relevant in physics will be given. In particular, we will focus on two important structures: Lie algebras (and their associated Lie groups) and Lie superalgebras (and their associated Lie supergroups). It will then be shown that the principles of Quantum Mechanics lead naturally to the mathematical structure associated for the description of symmetries. As a preamble to a more general study, important matrix Lie groups and matrix Lie algebras, and their differential or oscillators realisations, will be considered. The important notion of representation is also introduced.

The second part of these lectures will be devoted to the study of Lie groups and Lie algebras, mainly the simple complex and simple real Lie algebras together with their representations. Instead of following a formal presentation, it will be shown that these structures are natural generalisations of the well know groups SO(3) (rotations in three dimensions) and SU(3) (unitary transformations in three complex dimensions). The very important concepts of roots, Dynkin diagrams and Cartan matrices will be introduced together with the important notions of weights central to classify representations (the l ∈ ½ℕ of Quantum Mechanics!). A dierential realisation of some Lie algebras which enables to have all unitary representations in an ease manner will be given and illustrated mostly in the context of SU(3). Two applications of Lie algebras will be considered. Firstly, we will identify the Lie algebra(s) associated to the description of the symmetry of the space-time. Thus, as a consequence of space-time symmetries, an elementary particle will be characterised by two quantum numbers: its mass and its spin. Secondly, it will be established that fundamental interactions are associated to internal symmetries. Time permitting we will show how all fundamental interactions could be unify.

The last part of these lectures will be concerned by the Lie superalgebras and the Lie supergroups.

We will identify how these algebras could play a r^ole in physics, in particular to implement a symmetry

between fermions and bosons (called supersymmetry). Two Lie superalgebras will be studied. Their relationship to supersymmetry will be investigated.

The lectures will take place from 16:00 to 18:00 in

IPHC, Bât 20 room 115C (or Bât 26 room 130): 30 March, 1; 6; 8; 14; 15; 27; 29 April 1.References

[1] R. Campoamor-Stursberg et M. Rausch de Traubenberg, Group Theory in Physics: A Practitioner’s Guide, World Scientic (2018), ISBN: 978-981-3273-60-3,

www.worldscientic.com/worldscibooks/10.1142/11081.

____________________________

1The two rooms are closed to each other, one is bigger than the other the room for the lecture will depend on the number

of persons who attend to it.