Group Theory: Mathematical Expression of Symmetry in Physics
The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-07-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/8/1354 |
| _version_ | 1797521987842605056 |
|---|---|
| author | Jean-Pierre Antoine |
| author_facet | Jean-Pierre Antoine |
| author_sort | Jean-Pierre Antoine |
| collection | DOAJ |
| description | The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group <i>G</i> is based on a unitary representation of <i>G</i>. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups. |
| first_indexed | 2024-03-10T08:20:15Z |
| format | Article |
| id | doaj.art-294ce5332a7a415a989295ca71a9b6bd |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T08:20:15Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-294ce5332a7a415a989295ca71a9b6bd2023-11-22T09:59:55ZengMDPI AGSymmetry2073-89942021-07-01138135410.3390/sym13081354Group Theory: Mathematical Expression of Symmetry in PhysicsJean-Pierre Antoine0Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, B-1348 Louvain-la-Neuve, BelgiumThe present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group <i>G</i> is based on a unitary representation of <i>G</i>. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.https://www.mdpi.com/2073-8994/13/8/1354group theoryLie groupsymmetryrepresentationsquantum physicselementary particles |
| spellingShingle | Jean-Pierre Antoine Group Theory: Mathematical Expression of Symmetry in Physics Symmetry group theory Lie group symmetry representations quantum physics elementary particles |
| title | Group Theory: Mathematical Expression of Symmetry in Physics |
| title_full | Group Theory: Mathematical Expression of Symmetry in Physics |
| title_fullStr | Group Theory: Mathematical Expression of Symmetry in Physics |
| title_full_unstemmed | Group Theory: Mathematical Expression of Symmetry in Physics |
| title_short | Group Theory: Mathematical Expression of Symmetry in Physics |
| title_sort | group theory mathematical expression of symmetry in physics |
| topic | group theory Lie group symmetry representations quantum physics elementary particles |
| url | https://www.mdpi.com/2073-8994/13/8/1354 |
| work_keys_str_mv | AT jeanpierreantoine grouptheorymathematicalexpressionofsymmetryinphysics |