nLab Gruppenpest

Contents

Context

Quantum systems

Group Theory

Idea
References

Idea

In the late 1920s, Eugene Wigner and Hermann Weyl highlighted the role that group theory and representation theory play in the analysis of quantum mechanics, for instance in the analysis of atomic spectra. While many applications of groups and their representations to quantum physics had more or less explicitly been observed before, Wigner and Weyl stood out as making the mathematical formalism fully explicit. This attitude was not well received by some of their colleagues, who felt that this formal mathematics had no place in physics. In particular Erwin Schrödinger is said (Wigner 1981) to have spoken of the Gruppenpest (German for “plague of group theory”) which ought to be abandoned.

In his autobiography John Slater, an MIT physicist, claimed:

It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their “Gruppenpest”: the pest of the group theory… The authors of the “Gruppenpest” wrote papers which were incomprehensible to those like me who had not studied group theory, in which they applied these theoretical results to the study of the many electron problem. The practical consequences appeared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. Yet there were no good texts from which one could learn group theory. It was a frustrating experience, worthy of the name of a pest.

I had what I can only describe as a feeling of outrage at the turn which the subject had taken…

As soon as this [Slater’s] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as “Slater has slain the ‘Gruppenpest’”. I believe that no other piece of work I have done was so universally popular.

Eventually this resistance vanished and turned into its opposite in theoretical fundamental physics: in the classification of fundamental particles by unitary representations of the Poincaré group introduced by Hermann Weyl, in the description of gauge theory in terms of associated bundles given by representations of gauge groups.

Today almost the first thing that one wants to know about any physical theory is its global symmetry-group and gauge group and the their relevant representations.

References

Historical texts:

Hermann Weyl, Quantenmechanik und Gruppentheorie, Zeitschrift für Physik 46 (1927) 1–46 [doi:10.1007/BF02055756]
Hermann Weyl, Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig, (1931), translated by H. P. Robertson: The Theory of Groups and Quantum Mechanics Dover (1950) [ISBN:0486602699, ark:/13960/t1kh1w36w]
Eugene P. Wigner: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Springer (1931) [doi:10.1007/978-3-663-02555-9, pdf]
Eugene P. Wigner: Group theory: And its application to the quantum mechanics of atomic spectra, 5, Academic

Press (1959) [doi:978-0-12-750550-3]

Transcript of an interview with Wigner where he mentions Schrödinger’s remark on the Gruppenpest:

Interview with Eugene Wigner, by Lillian Hoddeson, Gordon Baym and Frederick Seitz at the New Yorker Hotel January 24 (198) [Niels Bohr Library and Archives]

History and review:

Brian G. Wybourne, The “The Gruppen Pest” yesterday, today, and tomorrow, International Journal of Quantum Chemistry Symp. 7 (1973) 35-43 [doi:10.1002/qua.560070706]
Erhard Scholz, Introducing groups into quantum theory (1926-1930), Historia Mathematica 33 4 (2006) 440-490 [arXiv:math/0409571, doi:10.1016/j.hm.2005.11.007]
Erhard Scholz, Weyl entering the ‘new’ quantum mechanics discourse, in: C. Joas, C. Lehner, J. Renn (eds.), HQ-1: Conference on the History of Quantum Physics, MPI History of Science Berlin 350 II (2008) 253–271. (Berlin July 2–6, 2007) [pdfl]
Arianna Borrelli, Bretislav Friedrich, Eugene Wigner and the bliss of the “Gruppenpest” (2011) [pdf]
Christophe Eckes, Weyl and the mathematisation of Quantum Mechanics individual and collective perspectives (2020) [pdf]