The Stormy Onset of Group Theory in the New Quantum Mechanics

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.”

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65 Responses to The Stormy Onset of Group Theory in the New Quantum Mechanics

  1. Juan R. says:

    I agree in that in general new physics implies often new math.

    The term new math would be taken on a broad sense, e.g. new applications in physics of old mathematical stuff. A example is the formulation of GR by Einstein and based in math done by mathematicians.

    For science, the rule is first physics after math. This rule is violated in string M theory, where there is some advance in pure math but few or no advance in physics. This indicates, at least to me, that the entire endeavour is completely wrong. It is not the problem of finding some correction term or some new magical concept solving all problems, we may simply ignore the approach and focus in another theory/-ies.


    There are problems now intractable (but that wiil be solve in a future) and other may be totally intractable to practical effects forever.

    One of that intractable forever problems is the formulation of a TOE. It does not exist.

    I don’t think that “intractability” is the source of time arrow, without a concept of entropy.

    Time arrow is based in certain topological effects linked to intimate structure of spacetime. Entropy plays a fundamental role and it is the source for the well-known link between thermodynamics and gravitation.

  2. Anonymous says:

    It’s easy to let mathematical formalism bury physical insight (or to use it to disguise a lack of physical insight).

  3. Chris W. says:

    (That’s Dawson and Nielsen. Sorry, Michael.)

  4. Chris W. says:

    This may stimulate some discussion:

    “I have been impressed by numerous instances of mathematical theories that are really about particular algorithms; these theories are typically formulated in mathematical terms that are much more cumbersome and less natural than the equivalent formulation today’s computer scientists would use.” — Donald E. Knuth

    [from D. E. Knuth. Computer science and its relation to mathematics. Amer. Math. Month., 81(4), April 1974.]

    This quote begins the introduction of The Solovay-Kitaev algorithm (Nielson and Dawson, quant-ph/0505030, 6 May 2005).

  5. Robert says:


    There is significant contact between number theory, the Reimann hypothesis and quantum chaos, described in Chapter 11 of Marcus du Sautoy’s ‘Music of the Primes’. Perhaps it is not so strange that Bombieri’s April 1st 1997 announcement of a proof of the RH described a physically motivated analysis that drew on insights from supersymmetric fermionic-bosonic systems – a near absolute zero ensemble of a mixture of anyons and morons.

  6. JC says:

    A lot of the math used in string theory before the 1984 Schwarz anomaly cancellation paper, didn’t appear to be much more complicated than the sort of math one comes across in quantum field theory and general relativity. A lot of the “new math” seems to have surfaced quite quickly after Witten started to publish a lot of string papers, especially complex algebraic geometry related stuff like Calabi-Yau manifolds. (Some of it looks like it was carried straight over from supergravity compactification type of problems).

  7. quantoken says:

    Peter said:
    “I think most people are resistant to learning a new abstract formalism unless there is good evidence that it really does something useful. One has a limited amount of time and energy, and learning a new formalism can be time-consuming.”

    Usefulness is really in the eyes of the beholder. Each individual math field is certainly considered some what useful at least to some people, otherwise there would be no one studying them. But out of all of possible math fields, which in principle could be an infinity, those applicable or useful to physics, which must be a finity since we are talking about a finited universe, such math applicable to physics must be a very small portion.

    For example number theory is very useful. But it does not seem to be related to physics. What does it do with physics whether all prime numbers lies on a straight line in the Liemann Hyperthesis? Nothing. Similarly the P and NP problem is unrelated to physics, too. Whether there is an efficient way of cracking the RSA encryption would not tell us howto unify gravity and QM.

    In mathematics you can surely imagine a 11 dimentional world, and derive tons of seemingly interesting mathematics out of it. You could also imagine what if the world is two dimentional. But it’s really not relevant at all. The world is 3+1 D as we know it and there hasn’t been any evidence it could be otherwise.

    One math branch that interests me is the problem of tractability. Some math problems are seeming intractable. Are those truely intractable by nature, or are they merely due to our shallow knowledge of math in our era? If intractability can be proven as a natural occurance, then apply it to quantum computing, it could explain the emergence of the time arrow, without the entropy. In another word, our world could be constructed using a series of quantum one way hash functions, so it could only move forward in time but never backwards.


  8. Alejandro says:

    I dissent. String or not string, most theoretical physicists are fond of using new mathematics. It is only that their discovery path does not coincide with the one used by mathematicians; so the new math they use come mainly from other theoretical physicists.

  9. ksh95 says:

    JC asked Peter:
    “…Excluding the string theory crowd, why are some physicists resistant to using “new mathematics”?..”

    Ksh95 will answer:

    People are resistant to learning new mathematics for the same reasons they are resistant to sticking shards of glass in their eyes. Very painfull, lots of screaming, plenty of cursing…

  10. Peter Woit says:

    Hi JC,
    I think most people are resistant to learning a new abstract formalism unless there is good evidence that it really does something useful. One has a limited amount of time and energy, and learning a new formalism can be time-consuming.

    Heisenberg et. al. had some good reasons to be dubious about thinking of p and q operators in terms of matrices. It wasn’t so clear how useful this was, and in the end Schrodinger ended up showing that representing these operators as differential operators was much more useful than thinking of them as matrices.

    For more than twenty years, string theorists have been pushing a long list of proposed abstract formalisms, none of which have gone anywhere in terms of giving any insight into unification. By now, most everyone is pretty dubious whenever they hear about another such proposal.

  11. Juan R. says:

    Peter, since I am new on this blog I read now your previous historic post.

    The usual presentation of history by physicists is usually wrong and omit important detailed well-proved. Perhaps the most radical manipulation was those of Newton, when recent research has demonstrated that his chemical career was omited…

    There are several example of rewritings of history by physicists. This is not so strange for understanding. Think during one instant in string theory and the manipulation of mass media, the neglect of other schools (many laymen still think that string theory is the only approach to QG), and the rewriting of string theory history.

    This is also true of usual history for group theory. For a more realistic view I recommend

    Foundations of Chemistry 2001, 3, 55–78.

    This paper traces the origins of Eugene Wigner’s pioneering application of group theory to quantum physics to his early work in chemistry and crystallography. In the early 1920s, crystallography was the only discipline in which symmetry groups were routinely used. Wigner’s early training in chemistry, and his work in crystallography with Herman Mark and Karl Weissenberg at the Kaiser Wilhelm institute for fiber research in Berlin exposed him to conceptual tools which were absent from the pedagogy available to physicists for many years to come. This both enabled and pushed him to apply the group theoretic approach to quantum physics. It took many years for the approach first introduced by Wigner in the 1920s – and whose reception by the physicists was initially problematical – to assume the pivotal place it now holds in physical theory and education. This is but one example that attests to the historic contribution made by the periphery in initiating new types of thought-perspectives and scientific careers.

    More data

    When Abraham Pais asked Wigner whether the vastly increased complexity of the calculations involved in the transition from three to four particles (in the Schrödinger equation) marked his first full awareness of the power of group theory, Wigner replied that his first awareness of the power of group theory in facilitating calculations arose out of his work on the lattice structure of rhombic sulfur.

  12. Anonymous says:

    Doesn’t this article require a subscription?

  13. JC says:


    Excluding the string theory crowd, why are some physicists resistant to using “new mathematics”?

    I can perhaps understand why an experimentalist would be resistant to “new mathematics”, when most “new math” doesn’t really help them much in their day to day research work. I’ve noticed quite a number of particle phenomenology folks and even some string theorists who are particularly resistant to “new mathematics”, unless the “new math” is “forced” upon them by the “experts” in the field (ie. like a Gell-Mann or a Witten).

  14. Anonymous says:

    “What is a matrix?”
    -Werner Heisenberg, 1925

    “What is the matrix?”
    -Keanu Reeves, 1999

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