Sym\'etrie et th\'eorie des groupes \`a travers la physique

TL;DR

This paper reviews how group theory links symmetry properties to physical phenomena across various fields like condensed matter, particle physics, and electromagnetism, highlighting its role as both a tool and a conceptual framework.

Contribution

It provides a comprehensive overview of the application of group theory to physics, emphasizing its insights into natural phenomena and its utility in analyzing symmetry-related properties.

Findings

Group theory explains forbidden properties in high-symmetry molecules.

Theorems like Noether and Goldstone relate symmetry to physical laws.

Applications span condensed matter, particle physics, quantum mechanics, and electromagnetism.

Abstract

Physical properties of matter are tightly related with the kind of symmetry of the medium. Group theory is a systematic tool, though not always easy to handle, to exploit symmetry properties, for instance to find the eigenvectors and eigenvalues of an operator. Certain properties (optical activity, piezoelectricity...) are forbidden in molecules or crystals of high symmetry. A few theorems (Noether, Goldstone...) establish general relations between physical properties and symmetry. Applications of group theory to condensed matter physics, elementary particle physics, quantum mechanics, electromagnetism are reviewed. Group theory is not only a tool, but also a beautiful construction which casts insight into natural phenomena. ----- Les propri\'et\'es de la mati\`ere sont li\'ees au type de sym\'etrie qui y r\`egne. La th\'eorie des groupes est un outil syst\'ematique, mais pas toujours commode, pour exploiter cette sym\'etrie, notamment quand il faut trouver les vecteurs propres et valeurs propres d'un op\'erateur. Certaines propri\'et\'es (pouvoir rotatoire optique, pi\'ezo\'electricit\'e...) sont interdites dans des cristaux ou mol\'ecules de haute sym\'etrie. Quelques th\'eor\`emes (Noether, Goldstone...) \'enoncent des relations g\'en\'erales entre les propri\'et\'es physiques et la sym\'etrie. On passe en revue quelques applications de la th\'eorie des groupes \`a la physique de la mati\`ere condens\'ee, \`a la physique des particules \'el\'ementaires, \`a la m\'ecanique quantique, \`a l'\'electromagn\'etisme. La th\'eorie des groupes n'est pas seulement un outil, mais aussi une belle construction qui permet de mieux comprendre la nature.