The Peculiar Math That Could Underlie the Laws of Nature

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in 2014, a graduate student at the University of Waterloo, Canada, named Cohl Furey rented a car and drove six hours south to Pennsylvania State University, eager to talk to a physics professor there named Murat Günaydin

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 physicists placed their hopes in high-energy particle colliders and other experiments, expecting additional particles to show up and lead the way beyond the Standard Model to a deeper description of reality. They “imagined that the next bit 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.

. 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, harbored 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 chiseled 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.


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