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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 22nd 2011
• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJul 21st 2016

While many applications of groups and their representations to quantum physics had more or less explicitly been observed before, Wigner stood out as making the mathematical formalism fully explicit.

More so than Weyl? Quantenmechanik und Gruppentheorie was published in 1928, after a paper of the same name in 1927.

There’s quite a lot of scholarship on this aspect of the early history of QM. Picking out some highlights of a talk:

Slide 143 here by Christophe Eckes ’Weyl and the mathematisation of Quantum Mechanics’:

Let us enumerate the protagonists who apply group theory (in a wide sense) to quantum mechanics during the period 1926-1931 : - Heisenberg (1926) - Wigner (1926-1931) - von Neumann (1927-1928) - Weyl (1927-1931) - van der Waerden (1929-1931) - Heitler (1927-1928) - London (1927-1928)

Another interesting slide 173:

Group theory is used in four areas during the period 1926-1931 (cf. M. Schneider) : (1) « Foundational questions » (Wigner, Weyl) - for instance, Wigner identifies the « conservation laws » of quantum systems by using group-theoretical methods. (2) « Quantum numbers, atomic and molecular spectra » (Wigner, von Neumann, Weyl) - for instance, Weyl and Wigner show that the determination of the quantum numbers ‘ and m are linked with the study of the representations of SO(3) and SO(2) (3) « Molecular bond » (Heitler, London, Weyl) (4) « Dirac wave equation », extension to relativistic quantum mechanics (Weyl, van der Waerden)

Eckes continues

Thanks to this classification, we can criticize two historical assumptions : (1) First assumption (due to Mackey) : « Weyl’s idea differed from that of Wigner in that he wanted to apply group representations to get a better understanding of the foundations of quantum mechanics in general and not so much to gain insight into particular problems ». « Wigner and Weyl not only introduced group representations into quantum mechanics in quite different ways with different goals but they reached this interaction between physics and mathematics from opposite directions. While Wigner was above all a theoretical physicist, Weyl was a pure mathematician ».

Although Wigner is a « theoretical physicist » and Weyl a « mathematician », their works on quantum mechanics are mathematically and physically very close : (1) Wigner also wants to « get a better understanding of the foundations of quantum mechanics » (cf. his article on conservation laws in quantum mechanics), (2) Weyl’s monograph on quantum mechanics is characterized by an « empirical turn » : Weyl pays great attention to the empirical data resulting from spectroscopy. Moreover, in 1927 von Neumann advises Wigner to read Weyl’s article on Lie groups — essentially the second part which is devoted to SO(n) and its representations. Wigner and von Neumann use this reference in a series of papers entitled « Zur Erklärung einiger Eigenschaften der Spektren aus der Quantenmechanik des Drehelektrons ».

189 Let us sum up our arguments on Weyl : - Weyl occupies a central position in the network of scientists using group-theoretical methods in quantum mechanics, - his monograph is a wide synthesis which contains various applications of group theory to quantum mechanics, - Contrary to van der Waerden, he is intransigent : this way of formalizing quantum mechanics can’t be replaced by group free methods, - he dogmatically advocates for group-theoretical methods in mathematical physics (in his monograph, in his articles and also in his talks and lecture courses during his stay in the United States (mainly Princeton and Berkeley, 1929)). The other protagonists belonging to this network are not so intransigent.

192 III.4. Reception of these group-theoretical methods among physicists There is no antagonism between « two camps » : defenders vs detractors of group-theoretical methods in quantum mechanics. More precisely, it would be misleading to focus on two extremes : - Weyl as the most intransigent advocate of these methods, - Born and Slater as the most virulent detractors of group theory when applied to quantum mechanics. Among scientists using group-theoretical methods in quantum mechanics, some of them are conciliatory : they admit the relevance of other approaches (cf. van der Waerden). Heitler is another interesting case. He becomes Born’s assistant at the end of the 20’s. Under Born’s influence, he doesn’t use anymore group-theoretical methods in the description of molecular bonds.

196 Conversely, theoretical physicists do not necessary agree with Born’s and Slater’s opinion, following which group-theoretical methods must be avoided in quantum mechanics because they are too technical. Three examples : (i) The expression « Gruppenpest » is due to Erhenfest in 1928. In fact, Erhenfest himself doesn’t reject categorically these methods. - He finds Weyl’s approach interesting. - On the other hand, he recognizes that he is not acquainted with this mathematical framework. (ii) In a 1929 letter to Weyl, Schrödinger claims that we must clarify first the physical foundations of quantum mechanics before using these methods (scepticism but not rejection). (iii) Heisenberg is clearly enthousiastic in his recension of Weyl’s monograph.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJul 21st 2016

And…

198 the reception of works using these methods in quantum mechanics is very complex and contrasted among physicists : - Rejection (Born, Slater), - Interest (Ehrenfest, Sommerfeld), - Support (Heisenberg, Uhlenbeck, Laporte, Casimir), - Scepticism (Hartree, Schrödinger). In particular, Schneider describes in detail the reception of Weyl’s monograph by physicists. Such a reception is crucial in order to determine this typology.

201 It seems clear that theoretical physicists are not massively opposed to these new methods. Moreover the three monographs due respectively to Weyl (1928, 1931), Wigner (1931) and van der Waerden (1932) have a certain audience among physicists. Conversely, we must not overestimate the impact of this new approach at the beginning of the 30’s. The theory of group representations will be considered as an essential tool in mathematical physics and theoretical physics only after the second world war. Scholz : « With the exception of such « heroic » but for a long time relatively isolated contributions, it needed a new generation of physicists and a diversification of problems and another problem shift in quantum physics, before group theory was stepwise integrated into the core of quantum physics ».

Erhard Scholz is generally very good, so I should take a look at WEYL ENTERING THE ’NEW’ QUANTUM MECHANICS DISCOURSE (pdf)

Sorry, all rather unprocessed.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJul 23rd 2016

One small addition at least. It’s clearly unfair to Weyl not to include him with Wigner as originators of group theory in QM, so I’ve joined them, and put in references by Scholz and Eckes.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeJul 23rd 2016

I wonder if Schrödinger’s scepticism had anything to do with Weyl’s affair with his wife, ’Mind and Nature’ (p. 5). But perhaps intellectuals were more liberal back then.

I also added a oft-quoted passage from John Slater’s autobiography.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 17th 2018
• (edited Feb 17th 2018)

Has anyone yet coined the term ’$\infty$-gruppenpest’?

• CommentRowNumber7.
• CommentAuthorAlex Shpilkin
• CommentTime7 days ago