There’s an interesting new preprint by the historian of mathematics Erhard Scholz about the early history of the use of representation theory in quantum mechanics. Immediately after the beginnings of quantum mechanics in 1925, several people started to realize that the representation theory of the symmetric and rotation groups was a very powerful tool for getting at some of the implications of quantum mechanics for atomic spectra. One of the main figures in this was Eugene Wigner, who was trained as a chemical engineer, but worked on this topic with his fellow Hungarian, the well-known mathematician von Neumann.
Equally important was the role of the mathematician Hermann Weyl, who in 1925 had just completed his main work on the representation theory of compact groups, perhaps the most important mathematical work in a very illustrious career. Weyl was in close communication both with the group at Gottingen (Heisenberg, Born, Jordan) who were developing matrix mechanics, as well as Schrodinger who was working on wave mechanics. Weyl and Schrodinger both were professors in Zurich and knew each other well (Schrodinger’s first paper on quantum mechanics thanks Weyl for explaining to him some of the general properties of equations such as the Schrodinger equation). In 1927/8 Weyl gave a course on quantum mechanics and representation theory, which became the basis of his extremely influential book “The Theory of Groups and Quantum Mechanics”, first published in 1928.
Scholz has also posted another preprint about Weyl’s work, one that focuses on how his conception of the relation between matter and geometry evolved from 1915 to 1930. Weyl worked on general relativity and wrote an influential book about it (Space-Time-Matter, 1918). At that time he, Einstein and others believed that matter could somehow be described by a unified theory expressed in terms of some generalization of Riemannian geometry. Perhaps particles were some specific singularities or special solutions to the non-linear equations for the metric. The advent of quantum mechanics convinced Weyl (unlike Einstein), that this was a misguided notion, that matter should be described by a complex wave function. The right mathematics was not the geometry of a metric, but (in modern language) the geometry of gauge fields and of sections of a vector bundle with connection. The close connection between the basic ideas of representation theory and of quantum mechanics was quite clear to him, so, unlike Einstein, he enthusiastically adopted the new point of view of quantum physics.
One part of the close connection between Weyl and the history of quantum mechanics isn’t mentioned by Scholz. Weyl was not only a close friend of Schrodinger’s, he was Schrodinger’s wife’s lover. Schrodinger didn’t believe much in monogamy; it’s a well-known story that he discovered the Schrodinger equation while on holiday in the mountains with a girlfriend.