The idea of using Lie groups in physics can be easily understood intuitively, but what are the origins of the use of representation theory of Lie groups and Lie algebras in physics?

We mathematicians use representation theory as a tool (among others, such as cohomology) to get information about groups and other algebraic structures that could not be gathered otherwise. But I guess physicists do not have the same use of it?

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    $\begingroup$ Look up the work of Eugene Wigner. $\endgroup$ – KCd May 15 at 12:31
  • $\begingroup$ Ok thanks, maybe could you specify some book(s) ? $\endgroup$ – huurd May 15 at 12:33

Explicit applications of group representation to physics start with

E. Noether, Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 1918: 235–257.

where the celebrated Noether's Theorem was proved.

Applications to quantum mechanics were systematically developed by H. Weyl and E. Wigner:

H. Weyl, The theory of groups in quantum mechanics, first edition 1928


E. Wigner, Group theory and its application to the quantum mechanics of atomic spectra, first edition 1931.

There are many good modern books with "group theory" and "physics" or "quantum mechanics" in the title, my favorite is S. Sternberg, Group theory and physics, Cambridge, 1994.

Remark. The word "explicit" in the beginning of my answer is important. Considerations based on symmetry are as old as physics itself: they were used already by Aristotle. A major 19th century application was crystallography.

  • $\begingroup$ It seems that nowadays physicists use the word "symmetry" in a very broad sense of "automorphism of some structure", including homeophisms for example (which are not precisely the kind of intuitive idea of symmetry one has in mind). $\endgroup$ – huurd May 15 at 15:20
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    $\begingroup$ @huurd could you give an example of a homeomorphism being called a symmetry? The books in this answer, if I recall correctly, are very concrete in applying representation theory to discrete symmetry groups and simple lie groups (with the usual additions such as lorentz group), which are the things physicists usually think when talking about symmetries $\endgroup$ – cesaruliana May 15 at 18:47
  • $\begingroup$ @cesaruliana I heard some physicist saying that some groups of homeomorphisms of space-time are also considered as "symmetries". $\endgroup$ – huurd May 16 at 10:36
  • $\begingroup$ @huurd: the key word here is "group", not "homeomorphism". $\endgroup$ – Alexandre Eremenko May 16 at 12:57
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    $\begingroup$ "Group" is a mathematical notion which corresponds to the notion of "symmetry" from the "real world". I do not understand why you call this "abusive". $\endgroup$ – Alexandre Eremenko May 16 at 13:46

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