Chemistry and Invariant Theory

In an alternative history of the world, perhaps quantum mechanics could have been discovered by chemists following up on the theories of two mathematicians from the late 1800s: Sylvester, and Gordan.

Both are famous for their work on invariant theory, which we would now call part of group representation theory. For example, we now use the Clebsch–Gordan coefficients to understand the funny way angular momentum ‘adds’ when we combine two quantum systems. This plays a significant role in physical chemistry, though Gordan never lived to see that.

But Sylvester already wanted to connect chemistry to invariant theory back in 1878! He published a paper on it, in the first issue of a journal he himself founded:

• James Joseph Sylvester, On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, with three appendices, American Journal of Mathematics 1 (1878), 64–104. (Available on JSTOR.)

The title suggests he is applying ideas from chemistry to invariant theory, rather than the other way around! I haven’t absorbed the paper, but this impression is somewhat confirmed by these passages:

To those unacquainted with the laws of atomicity I recommend Dr. Frankland’s Lecture Notes for Chemical Students, vols. 1 and 2, London (Van Voorst), a perfect storehouse of information on the subject arranged in the most handy order and put together and explained with true scientific accuracy and precision.

and then:

The more I study Dr. Frankland’s wonderfully beautiful little treatise the more deeply I become impressed with the harmony or homology (I might call it, rather than analogy) which exists between the chemical and algebraical theories. In travelling my eye up and down the illustrated pages of “the Notes,” I feel as Aladdin might have done in walking in the garden where every tree was laden with precious stones, or as Caspar Hauser when first brought out of his dark cellar to contemplate the glittering heavens on a starry night. There is an untold treasure of hoarded algebraical wealth potentially contained in the results achieved by the patient and long continued labor of our unconscious and unsuspected chemical fellow-workers.

So, he thinks the chemists may have found an ‘untold treasure of algebraical wealth’. What is this?

First he notes that you can use graphs to describe molecules: vertices represent atoms, and edges represent bonds.

This idea, utterly commonplace now, may have been only four years old when Sylvester published his work, since Wikipedia credits the use of graphs for describing molecules to this paper:

• Arthur Cayley, On the mathematical theory of isomers, Philosophical Magazine 47 (1874), 444–446.

Surely it’s not a complete coincidence that Sylvester was friends with Cayley, and that Sylvester was the first to use the term ‘graph’ to mean a bunch of vertices connected by edges!

But Sylvester noticed you can also use graphs to describe ways of building scalars from tensors: a vertex with n edges coming out is a tensor with n indices, and an edge between vertices means you sum over a repeated index, as in the ‘Einstein summation convention’. This idea is often attributed to Penrose, who explained it more clearly much later:

Penrose on spin networks.

Still later, Penrose’s spin networks and the theory of Feynman diagrams were unified via ‘string diagrams’ in category theory. I explain the story and the math here:

• John Baez and Aaron D. Lauda, A prehistory of n-categorical physics, in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson, Cambridge U. Press, Cambridge, 2011, pp. 13–128.

So, we can think of Sylvester’s chemistry-inspired work as another obscure chapter in the prehistory of n-categorical physics!

To be precise, in Sylvester’s setup a vertex with n edges out represents an atom with n bonds coming out, but also a binary quantic, meaning an element of V^{\otimes n} where V is a 2-dimensional vector space with an inner product on it. He notes that hydrogen, chlorine, bromine, and potassium have n = 1, oxygen, zinc, and magnesium have n = 2, and so on.

The inner product on V lets us raise or lower indices in tensors, so we don’t have to worry about which indices are superscripts and which are subscripts, which is usually a major aspect of the Einstein summation convention. In other words, it lets us identify V with its dual V^\ast, so we don’t have to worry about the difference between covariant and contravariant tensors.

It seems that by this method, Sylvester was able to see diagrams of molecules as recipes for building scalars from tensors! Here’s a nice page containing 45 separate figures explained in his paper:

But it wasn’t just Sylvester. Clifford, famous for inventing Clifford algebras, also thought about chemistry and invariant theory. In fact he wrote a letter about it to Sylvester! Sylvester published part of this along with his own work in the first issue of his journal:

• William Kingdon Clifford, Extract of a letter to Mr. Sylvester from Prof. Clifford of University College, London, American Journal of Mathematics 1 (1878), 126–128. (Available on JSTOR.)

Sylvester was so excited that he published this without Clifford’s permission, writing:

The subjoined matter is so exceedingly interesting and throws such a flood of light on the chemico-algebraical theory, that I have been unable to resist the temptation to insert it in the Journal, without waiting to obtain the writer’s permission to do so, for which there is not time available between the date of its receipt and my proximate departure for Europe. It is written from Gibraltar, whither Professor Clifford has been ordered to recruit his health, a treasure which he ought to feel bound to guard as a sacred trust for the benefit of the whole mathematical world.

I have not managed to understand Clifford’s ideas yet, but they may have been better than Sylvester’s—though unfortunately not developed, due to Clifford’s untimely death one year later in 1879. Olver and Shakiban write:

Although Sylvester envisioned his theory as the future of chemistry, it is Clifford’s graph theory that, with one slight but important modification, could have become a useful tool in computational invariant theory. The algebro-chemical theory reduces computations of invariants to methods of graph theory. Our thesis is that the correct framework for the subject is to use digraphs or “directed molecules” as the fundamental objects. One can ascribe both a graph theoretical as well as a chemical interpretation.

This is from here:

• Peter J. Olver and Cherzhad Shakiban, Graph theory and classical invariant theory, Advances in Mathematics 75 (1989), 212–245.

It sounds like what Clifford realized is that by using a directed graph we get a better theory that lets us drop the inner product on V. Having graphs with directed edges lets the graphical notation distinguish between covariant and contravariant tensors.

Now let’s jump forward a decade or two! At some point Paul Gordan read the work of Clifford and Sylvester and concluded that invariant theory could contribute to the understanding of chemical valence. But his own ideas were somewhat different. In 1900 he and his student W. Alexejeff wrote an article about this:

• Paul Gordan and W. Alexejeff, Übereinstimmung der Formeln der Chemie und der Invariantentheorie, Zeitschrift für Physikalische Chemie, 35 (1900), 610–633.

In 2006, Wormer and Paldus wrote:

The origins of the coupling problem for angular momenta can be traced back to the early—purely mathematical—work on invariant theory by (Rudolf Friedrich) Alfred Clebsch (1833–1872) and Paul (Albert) Gordan (1837–1912), see Section 2.5. Even before the birth of quantum mechanics the formal analogy between chemical valence theory and binary invariant theory was recognized by eminent mathematicians as Sylvester, Clifford, and Gordan and Alexejeff. The analogy, lacking a physical basis at the time, was criticised heavily by the mathematician E. Study and ignored completely by the chemistry community of the 1890s. After the advent of quantum mechanics it became clear, however, that chemical valences arise from electron–spin couplings … and that electron spin functions are, in fact, binary forms of the type studied by Gordan and Clebsch.

I learned of this quote from James Dolan, who happened to be studying the work of Eduard Study, who is mostly famous for his work on the so-called dual numbers, the free algebra on one generator that squares to zero. The paper by Wormer and Paldus is here:

• Paul E. S. Wormer and Josef Paldus, Angular momentum diagrams, Advances in Quantum Chemistry 51 (2006), 51–124.

Here’s another paper I should read:

• Karen Hunger Parshall, Chemistry through invariant theory? James Joseph Sylvester’s mathematization of the atomic theory, in Experiencing Nature: Proceedings of a Conference in Honor of Allen G. Debus, Springer, Berlin, 1997.

Sylvester was a colorful and fascinating character. For example, he entered University College London at the age of 14. But after just five months, he was accused of threatening a fellow student with a knife in the dining hall! His parents took him out of college and waited for him to grow up a bit more.

He began studies in Cambridge at 17. Despite being ill for 2 years, he came in second in the big math exam called the tripos. But he couldn’t get a degree… because he was Jewish.

In 1841, he was awarded a BA and an MA by Trinity College Dublin. In the same year he moved to the United States to become a professor of mathematics at the University of Virginia.

After just a few months, a student reading a newspaper in one of Sylvester’s lectures insulted him. Sylvester struck him with a sword stick. The student collapsed in shock. Sylvester thought he’d killed the guy! He fled to New York where one of his brothers was living.

Later he came back to Virginia. But according an online biography, “the abuse suffered by Sylvester from this student got worse after this”. Soon he quit his job.

He returned to England and took up a job at a life insurance company. He needed a law degree for this job, and in his studies he met another mathematician, five years younger, studying law: Cayley! They worked together on matrices and invariant theory.

Sylvester only got another math job in 1855, at the Royal Military Academy of Woolwich. He was 41. At age 55 they made him retire—that was the rule—but for some reason the school refused to pay his pension!

The Royal Military Academy only relented and paid Sylvester his pension after a prolonged public controversy, during which he took his case to the letters page of The Times.

When he was 58, Cambridge University finally gave him his BA and MA.

At age 62, Sylvester went back to the United States to become the first professor of mathematics at the newly founded Johns Hopkins University in Baltimore, Maryland. His salary was $5,000—quite generous for the time.

He demanded to be paid in gold.

They wouldn’t pay him in gold, but he took the job anyway. At age 64, he founded the American Journal of Mathematics. At 69, he was invited back to England to become a professor at Oxford. He worked there until his death at age 83.

One thing I’ve always liked about Sylvester is that he invented lots of terms for mathematical concepts. Some of them have caught on: matrix, discriminant, invariant, totient, and Jacobian! Others have not: cyclotheme, meicatecticizant, tamisage and dozens more.

But only now am I realizing how Sylvester’s fertile imagination, inspired by chemistry, connected graph theory and invariant theory in ways that would later become crucial for physics.

12 Responses to Chemistry and Invariant Theory

  1. Bill Bottenberg says:

    This was delightful John! We chemists have worked hard to build our material world only to be slighted by the lofty physicists and loftier mathematicians.
    (insert here)
    But seriously, adding Figure 28 and its 60deg rotated friend, properly normalized, gives you Kekule’s dream.

  2. Raphael says:

    I wonder how this is related to “contemporary” chemical graph theory where we look at the eigenvalues of the “adjacency matrix” of the molecular graphs because they represent (scaled) orbital energies.

  3. Wolfgang says:

    From Sylvester’s graphs it seems he was also close to have invented Lewis formulas.

  4. Toby Bartels says:

    It’s too bad that I don’t live in the alternate universe where representation theory is called chemical algebra.

  5. Greg Egan says:

    After the advent of quantum mechanics it became clear, however, that chemical valences arise from electron–spin couplings … and that electron spin functions are, in fact, binary forms of the type studied by Gordan and Clebsch.

    I don’t understand the sense in which valence is supposed to be connected to electron spin per se. Isn’t the primary determinant of valence the way the atomic number is related to the number of electrons in each orbital and shell? For example, Lithium has a valence of 1 because it has two electrons in the 1s orbital, and one in the higher-energy 2s orbital, so it can be energetically favourable for it to donate that higher-energy electron to another atom with an unfilled shell where it can raise the energy by a smaller amount.

    How does that relate to contracting tensor products of the Hilbert space for single electrons?

    • John Baez says:

      That’s a good point, Greg. I should have said that I don’t see how Gordan and Alexejeff’s scheme could actually succeed in explaining chemical bonding, despite Wormer and Paldus’ optimistic-sounding assessment. I’m just imagining that their approach based on group representation theory might eventually, with lots of trial and error, have morphed into something like quantum chemistry as we know it.

      I have a feeling that while Wormer and Paldus say ‘electron spin functions’ it would be more forgiving to interpret all this early work in terms of SU(2) representation theory using the fact that the symmetry group of the bound states of a single-electron atom is SU(2) × SU(2), with one SU(2) for angular momentum and another for the Runge–Lenz vector.

      For example, we can imagine an alternative history where someone noticed that a large chunk of the periodic table can be explained by the representation theory of SU(2) × SU(2), or in a more ad hoc way using just SU(2). Using this, the valences of atoms can be understood, at least partially, using the Madelung rules—until we hit problems with heavier elements, like the scandal of scandium.

      But I didn’t feel like rendering a clear opinion on this because I have not yet actually managed to read Gordan and Alexejeff’s paper. It’s paywalled and in German, and the latter obstacle has lessened my motivation to circumvent the former. I should still try to read it!

      There is however something interesting about Sylvester’s earlier attempts: though phrased in terms of ‘binary quantics’, i.e. elements of \mathbb{C}^2 \otimes \cdots \mathbb{C}^2, and their invariants, i.e. elements invariant under \mathrm{GL}(2,\mathbb{C}), these have some interesting relation to SU(2) representation theory, and it’s interesting to try to figure out what they has to do with anything chemical.

    • Greg Egan says:

      Thanks for clarifying that, John. I guess I was wondering if there really was enough of the relevant representation theory going on here for anyone pre-QM to figure out the numbers of electrons in (what we now call) the n=1,2,3… shells, and I couldn’t see how this would get them there.

      • Raphael says:

        On top the valence and the number of valence electrons is not the same in general. E. g for nitrogen the common valence is 3 while it has 5 valence electrons (=in the uppermost shell).

        On a side note: for a large section of chemistry spin plays the role of “pairing” the electrons. And that’s mostly it. Especially bonding theory is concerned usually overwhelmingly with what we call “closed shell systems”.

        Not sure if it can contribute to a clarification, but that would be the situation from a Chemists perspective.

      • John Baez says:

        I don’t think the most interesting question is whether there are things that Sylvester and Gordan’s approaches can’t get right about chemistry—there are clearly lots—but whether there is anything that they can get right.

        I can’t comment on Gordan’s ideas yet, but here’s one thing that Sylvester’s approach gets right: linear combinations of allowed states are allowed states. There are probably others, but this alone, if pursued diligently and repeatedly corrected, might have given people a leg up on quantum mechanics.

        (But it didn’t.)

  6. Chuck Jackson says:

    What an interesting and enjoyable post! Thank you.

  7. “Meicatalecticizant would more completely express the meaning of that which, for the sake of brevity, I denominate the catalecticant.”

    You’re not going to bring up his poetry?

    “On a celebrated occasion, at a public reading of the Rosalind poem [400+ lines which all rhyme with Rosalind, or at least, with Rose-aligned] he decided to begin by going through the footnotes, and it was an hour-and-a-half before he got to the poem itself”

    • John Baez says:

      I like this:

      After retiring in 1870 he lived in London, spending most days at the Athenaeum Club (he had been elected a member of the Club in 1856). At first it looked as though he might give up mathematics since he published his only book at this time and it was on poetry. Clearly Sylvester was proud of this work, entitled The Laws of Verse, since after this he sometimes signed himself “J J Sylvester, author of The Laws of Verse“.

      Somehow his spending days at the Athenaeum Club reminds me of Mycroft Holmes at the Diogenes Club, though I guess it was fairly common for British gentlemen to spend time in such club.

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