Cohl Furey, a mathematical physicist at the University of Cambridge, is finding links between the Standard Model of particle physics and the octonions, numbers whose multiplication rules are encoded in a triangular diagram called the Fano plane.
New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”
In 2014, a graduate student at the University of Waterloo, Canada, named Cohl Furey rented a car and drove six hours to Pennsylvania State University, eager to talk to a physics professor there named Murat Günaydin. Furey had figured out how to build on a finding of Günaydin’s from 40 years earlier -a largely forgotten result that supported a powerful suspicion about fundamental physics and its relationship to pure math.
The suspicion, harboured by many physicists and mathematicians over the decades but rarely actively pursued, is that the peculiar panoply of forces and particles that comprise reality spring logically from the properties of eight -dimensional numbers called “octonions.”
As numbers go, the familiar real numbers -those found on the number line, like 1, π and -83.777 – just get things started. Real numbers can be paired up in a particular way to form “complex numbers”, first studied in 16th -century Italy, that behave like coordinates on a 2-D plane. Adding, subtracting, multiplying and dividing is like translating and rotating positions around the plane. Complex numbers, suitably paired, form 4-D “quaternions,” discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiselled the formula into Dublin’s Broome Bridge. John Graves, a lawyer friend of Hamilton’s, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space.
There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. The first three of these “division algebras” would soon lay the mathematical foundation for 20th -century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein’s special theory of relativity. This has led many researchers to wonder about the last and least -understood division algebra. Might the octonions hold secrets of the universe?
“Octonions are to physics what the Sirens were to Ulysses,” Pierre Ramond, a particle physicist and string theorist at the University of Florida, said in an email.
Günaydin, the Penn State professor, was a graduate student at Yale in 1973 when he and his advisor Feza Gürsey found a surprising link between the octonions and the strong force, which binds quarks together inside atomic nuclei. An initial flurry of interest in the finding didn’t last. Everyone at the time was puzzling over the Standard Model of particle physics -the set of equations describing the known elementary particles and their interactions via the strong, weak and electromagnetic forces (all the fundamental forces except gravity). But rather than seek mathematical answers to the Standard Model’s mysteries, most physicist placed their hopes in high-energy particle colliders and other experiments, expeting additional particles to show up and lead the way beyond the Standard Model to a deeper description of reality. They “imagined that the next bid of progress will come from some new pieces being dropped onto the table, (rather than) from thinking harder about the pieces we already have,” said Latham Boyle, a theoretical physicist at the Perimeter Institute of Theoretical Physics in Waterloo, Canada.
Decades on, no particles beyond those of the Standard Model have been found. Meanwhile, the strange beauty of the octonions has continued to attract the occasional independent- minded researcher, including Furey, the Canadian grad student who visited Günaydin four years ago. Looking like an interplanetary traveler, with choppy silver bangs that taper to a point between piercing blue eyes, Furey scrawled esoteric symbols on a blackboard, trying to explain to Günaydin that she had extended his and Gürsey’s work by constructing an octonionic model of both the strong and electromagnetic forces.
“Communicating the details to him turned out to be a bit more of a challenge than I had anticipated, as I struggled to get a word in edgewise,” Furey recalled. Günaydin had continued to study the octonions since the ’70s by way of their deep connections to string theory, M-theory and supergravity – related theories that attempt to unify gravity with the other fundamental forces. But his octonionic pursuits had always been outside the mainstream. He advised Furey to find another research project for her Ph.D., since the octonions might close doors for her, as he felt they had for him.
Furey has yet to construct a simple octonion model of all Standard Model particles and foces in one go, and she hasn’t touched on gravity. She stresses that the mathematical possibilities are many, and experts say it’s too soon to tell which way of amalgamating the octonions and other division algebras (if any) will lead to success.
“She has found some intriguing links,” said Michael Duff, a pioneering string theorist and professor at Imperial College London who has studied octonions’ role in string theory. “It’s certainly worth pursuing, in my view. Whether it will ultimately be the way the Standard Model is described, it’s hard to say. If it were, it would qualify for all the superlatives -revolutionary, and so on.”
I met Furey in June, in the porter’s lodge through which one enters Trinity Hall on the bank of the River Cam. Petite, muscular, and wearing a sleeveless black T-shirt (that revealed bruises from mixed martial arts), rolled-up jeans, socks with cartoon aliens on them and Vegetarian Shoes–brand sneakers, in person she was more Vancouverite than the otherworldly figure in her lecture videos. We ambled around the college lawns, ducking through medieval doorways in and out of the hot sun. On a different day I might have seen her doing physics on a purple yoga mat on the grass.
Furey, who is 39, said she was first drawn to physics at a specific moment in high school, in British Columbia. Her teacher told the class that only four fundamental forces underlie all the world’s complexity — and, furthermore, that physicists since the 1970s had been trying to unify all of them within a single theoretical structure. “That was just the most beautiful thing I ever heard,” she told me, steely-eyed. She had a similar feeling a few years later, as an undergraduate at Simon Fraser University in Vancouver, upon learning about the four division algebras. One such number system, or infinitely many, would seem reasonable. “But four?” she recalls thinking. “How peculiar.”
As Dixon knew, the algebra splits cleanly into two parts: ℂ⊗ℍ and ℂ⊗𝕆, the products of complex numbers with quaternions and octonions, respectively (real numbers are trivial). In Furey’s model, the symmetries associated with how particles move and rotate in space-time, together known as the Lorentz group, arise from the quaternionic ℂ⊗ℍ part of the algebra. The symmetry group SU(3) × SU(2) × U(1), associated with particles’ internal properties and mutual interactions via the strong, weak and electromagnetic forces, come from the octonionic part, ℂ⊗𝕆.
Günaydin and Gürsey, in their early work, already found SU(3) inside the octonions. Consider the base set of octonions, 1, e1, e2, e3, e4, e5, e6 and e7, which are unit distances in eight different orthogonal directions: They respect a group of symmetries called G2, which happens to be one of the rare “exceptional groups” that can’t be mathematically classified into other existing symmetry-group families. The octonions’ intimate connection to all the exceptional groups and other special mathematical objects has compounded the belief in their importance, convincing the eminent Fields medalist and Abel Prize–winning mathematician Michael Atiyah, for example, that the final theory of nature must be octonionic. “The real theory which we would like to get to,” he said in 2010, “should include gravity with all these theories in such a way that gravity is seen to be a consequence of the octonions and the exceptional groups.” He added, “It will be hard because we know the octonions are hard, but when you’ve found it, it should be a beautiful theory, and it should be unique.”
Holding e7 constant while transforming the other unit octonions reduces their symmetries to the group SU(3). Günaydin and Gürsey used this fact to build an octonionic model of the strong force acting on a single generation of quarks.