Chemistry and Invariant Theory

28 March, 2023

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.

Four ‘Universes’

26 March, 2023

This chart made by Toby Ord shows four things:

• Everything we can observe now is the ‘observable universe’.

• Everything we can ever observe if we stay here is the ‘eventually observable universe’.

• Everything we can ever observe if we send spacecraft out in every direction at all speeds slower than light is the ‘ultimately observable universe’.

• Everything those spacecraft can ever affect is the ‘affectable universe’.

His chart is drawn in funny coordinates where a galaxy at rest moves straight up the page and light moves at 45° angles. The Big Bang is the horizontal line at the bottom, and the infinite future is the horizontal line at top. The expansion of the universe is hidden in these coordinates!

How big are these four things?

• When we observe distant galaxies we see what they were like long ago, when they were closer. Those galaxies now form a ball of radius 46 billion light years in diameter. So people say the radius of the observable universe is 46 billion light years. But beware: we can’t see what those galaxies look like now.

• The galaxies in the eventually observable universe now form a ball of radius 63 billion light years.

• The galaxies in the ultimately observable universe now form a ball of radius 80 billion light years.

• The galaxies in the affectable universe now form a ball of radius 16 billion light years.

These figures change with time. For example, shortly after the Big Bang the radius of the affectable universe was 63 billion light years. It has now shrunk to 16 billion light years. 90% of the galaxies we could in theory once reach—if we could have started right away—are lost to us now!

Of course, all these numbers are based on our current cosmology, which says that as the universe expands and ordinary matter thins out, the effect of dark energy becomes more important, and the universe starts expanding almost exponentially. If our theory of cosmology is wrong then these numbers are wrong!

You might wonder why the affectable universe has a finite radius even though the universe will last forever in our current theory. The reason is that because the universe is expanding faster and faster, it’s impossible to catch up with distant galaxies. So the only galaxies we can reach are those that are less than 16 billion light years away now.

For more, read Toby Ord’s paper:

• Toby Ord, The edges of our Universe.

and read my earlier blog post on this subject:

The expansion of the Universe.

The Galactic Center

24 March, 2023

You’ve probably heard there’s a supermassive black hole at the center of the Milky Way—and also that near the center of our galaxy there are a lot more stars. But did you ever think hard about what the Galactic Center is like?

I didn’t, until recently. As a kid I read about it in science fiction—like Asimov’s Foundation trilogy, where the capital of the Empire is near the Galactic Center on the world of Trantor, with a population of 40 billion. That shaped my impressions.

But now we know more. And it turns out the center of our galaxy is a wild and woolly place! Besides that black hole 4 million times the mass of our Sun, it’s full of young clusters of stars, supernova remnants, molecular clouds, weird filaments of gas, and more.

It’s in the constellation of Sagittarius, abbreviated ‘Sgr’. Let me go through the various features named above and explain them.

Sgr A contains the supermassive black hole called Sgr A*, which is worth a whole article of its own. Surrounding that is the Minispiral: a three-armed spiral of dust and gas falling into the black hole at speeds up to 1000 kilometers per second.

Also in Sgr A, surrounding the Minispiral, there is a torus of cooler molecular gas called the ‘Circumnuclear Disk’:

The inner radius of the Circumnuclear Disk is almost 5 light years. And inside this disk there are over 10 million stars. That’s a lot! Remember, the nearest stars to our Sun are 4 light years away.

Even weirder, among these stars there are lots of old red giants—but also many big, young stars that formed in a single event a few million years ago. These include about 100 OB stars, which are blue-hot, and Wolf-Rayet stars, which have blown off their outer atmosphere and are shining mainly in the ultraviolet.

Nobody knows how so many stars were able to form inside the Circumnuclear Disk espite the gravitational disruption of central black hole, and why so many are young. This is called the ‘paradox of youth’.

Stars don’t seem to be forming now in this region. But some predict that stars will form in the Circumnuclear Disk, perhaps causing a starburst in 200 million years, with many stars forming rapidly, and supernovae going off at a hundred times the current rate! As gas from these falls into the central black hole, life may get very exciting.

As if this weren’t enough, a region of Sgr A called Sgr A East contains a structure is approximately 25 light-years in width that looks like a supernova remnant, perhaps created between 35 and 100 thousand years ago. However, it would take 50 to 100 times more energy than a standard supernova explosion to create a structure of this size and energy. So, it’s a bit mysterious.

Moving further out, let’s turn to the Radio Arc, called simply ‘Arc’ in picture at the top of this article. This is the largest of a thousand mysterious filaments that emit radio waves. It’s obvious that the Galactic Center is wild, but these make it ‘woolly’. Nobody knows what causes them!

Here is the Radio Arc and some filaments:

Behind the Radio Arc is the Quintuplet Cluster, which contains one of the largest stars in the Galaxy—but more about that some other day.

Sgr B1 is a cloud of ionized gas. Nobody knows why it’s ionized. Like the filaments, perhaps it was heated up back when the black hole was eating more stars and emitting more radiation. Sgr B1 is connected to Sgr B2, a giant molecular cloud made of gas and dust, 3 million times the mass of the Sun.

The distance from Sgr A to Sgr B2 is 390 light years. That gives you a sense of the scale here! The whole picture spans a region in the sky 4 times the angular size of the Moon.

The two things called SNR are supernova remnants—hot gas shooting outwards from exploded stars. For example, in the top picture at lower right we see SNR 359.1-0.5, which looks like this close up:

The filament at right is called the Snake, while the Mouse at left is actually supposed to be a runaway pulsar. It looks like the Mouse is running away from the Snake! But that’s probably a coincidence.

Sgr D is another giant molecular cloud, and Sgr C is a group of molecular clouds.

So, a lot is going on in our galaxy’s center! Out here in the boondocks it’s more quiet.

Let me show you the first picture in all its glory without the labels. Click to enlarge:

It’s almost impossible to see the Galactic Center in visible light through all the dust, so this is an image in radio waves, made by the MeerKAT array of 64 radio dishes in South Africa. It was made by Ian Heywood with color processing by Juan Carlos Munoz-Mateos.

Here are two other versions of the same image, processed in different ways:

Click to enlarge!

Category Theory Outreach Panel

18 February, 2023

They just don’t quit! Besides their Joy of Abstraction book club, the Topos Institute also has another way for you to start learning category theory. It’s called the CT Outreach Panel, and it’s happening on March 16 at 17:00 UTC.

Some of the best explainers of category theory in the world—Emily Riehl, Eugenia Cheng, Tai-Danae Bradley, Paul Dancstep and Oliver Lugg—will explain their approaches to the subject and answer questions.

You can submit questions here: