When I first started studying quantum mechanics I read quite a bit about the remarkable history of the subject, especially about the brief period from 1925-27 when the subject grew dramatically out of the incoherent ideas of the old quantum theory to the full quantum mechanical formalism that is still taught today. This was the work of a small group of physicists: especially Heisenberg, Born and Jordan in Göttingen, Schrödinger in Zurich, Dirac in Cambridge, and Pauli in Hamburg. Recently I’ve been reading again about some of this history, but paying attention especially to the interactions of mathematics and physics during these years. An excellent very recent article that covers some of this is by Luisa Bonolis, entitled “From the Rise of the Group Concept to the Stormy Onset of Group Theory in the New Quantum Mechanics”. (It seems that this link is inaccessible unless you’re at a university site that has a subscription. The article should also be available at most physics research libraries as vol 27, numbers 4-5 of the 2004 issue of Rivista del Nuovo Cimento.)
I’ve written a bit about this history before, especially about the mathematician Hermann Weyl’s role, but quite a few other mathematicians were closely involved, including Hilbert, von Neumann, Emmy Noether, and van der Waerden. Much of the interaction between mathematicians and physicists took place at Göttingen, where Hilbert was the leading mathematical figure, and Weyl was sometimes a visitor, with both of them lecturing on quantum mechanics. This period was very much a high point of the interaction of mathematics and physics, interactions of a sort that were not seen again until the 1980s. Heisenberg and his collaborators learned about matrices from Hilbert and the other mathematicians at Göttingen, and Weyl was responsible for educating physicists about group representation theory and turning it into an important tool in quantum mechanics.
The Bonolis article has some amusing quotes from physicists who were having trouble absorbing what the mathematicians were telling them. Heisenberg wrote to Jordan “Now the learned Göttingen mathematicians talk so much about Hermitian matrices, but I do not even know what a matrix is,” and to Pauli “Göttingen is divided into two camps, those who, like Hilbert (or also Weyl, in a letter to Jordan), talk about the great success which has been scored by the introduction of matrix calculus into physics; the others, like Franck, who say that one will never be able to understand matrices.” Pauli was scornful about this new, unphysical, mathematical formalism of matrices, drawing a testy response from Heisenberg: “When you reproach us that we are such big donkeys that we have never produced anything new physically, it well may be true. But then, you are also an equally big jackass because you have not accomplished it either.”
Immediately after having to get used to matrices, physicists were confronted by Weyl with high-powered group representation theory, which they found even harder to understand than matrices. Famously, Pauli referred to the group theory that mathematicians were talking about as the “Gruppenpest”, but the late twenties saw a very fruitful exchange of ideas between mathematicians and physicists around this topic. Weyl’s proof of the Peter-Weyl theorem and von Neumann’s work on representation theory grew out of quantum mechanics, and the Brauer-Weyl theory of spinor representations was inspired by Dirac’s work on the Dirac equation.
It’s also interesting to note how in the years just preceding this period, much interaction between math and physics had grown out of general relativity. Noether’s work on what is now known as the Noether theorem came about because she was asked questions by Einstein and Hilbert who were trying to sort out conservation laws in GR. Weyl took up representation theory as a result of his work on the symmetries of the curvature tensor.
An amusing story I hadn’t heard before that is in the Bonolis article was one told by Edward Condon about Hilbert. He claims that when Born and Heisenberg went to Hilbert to get help with matrices, he told them that “the only times that he had ever had anything to do with matrices was when they came up as a sort of by-product of the eigenvalues of the boundary-value problem of a differential equation. So if you look for the differential equation which has these matrices you can probably do more with that. They had thought it was a goofy idea and that Hilbert did not know what he was talking about. So he was having a lot of fun pointing out to them that they could have discovered Schrödinger’s wave mechanics six months earlier if they had paid a little more attention to him.”