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, identified Cohl Furey hired a car and drove six hours south to Pennsylvania State University, enthusiastic start fucking talking to a physic prof there reputation Murat Gunaydin. Furey had figured out how to build on a discover of Gunaydin’s from 40 times earlier–a largely forgotten result that supported a potent hunch about fundamental physics and its relationship to pure math.

Quanta Magazine


Original story reprinted with dispensation from Quanta Magazine, an editorially independent book of the Simons Foundation whose mission is to enhance public understanding of science by include investigate developments and trends in mathematics and the physical and life sciences.

The suspicion, harbored by many physicists and mathematicians over the decades but rarely actively engaged, is that the peculiar panoply of forces and corpuscles that comprise actuality spring logically from the owneds of eight-dimensional numbers called “octonions.”

As quantities start, the familiar real numbers–those found on the count path, like 1, p and -8 3.777 — only get things started. Real quantities is also available paired up in a particular way to formation “complex numbers, ” firstly studied in 16 th-century Italy, that react like coordinates on a 2-D aircraft. Adding, subtracting, proliferating and partitioning is like translating and rotating points around the plane. Complex multitudes, appropriately paired, structure 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 solicitor friend of Hamilton’s, subsequently showed that pairs of quaternions realise octonions: numbers that define coordinates in an synopsi 8-D space.

John Graves, the Irish lawyer and mathematician who discovered the octonions in 1843.
MacTutor History of Mathematics

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 fractioned. The first three of these “division algebras” would soon lay the mathematical foot for 20 th-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 already led numerous researchers to wonder about the last and least-understood division algebra. Might the octonions support 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.

Gunaydin, the Penn State professor, was a graduate student at Yale in 1973 where reference is and his advisor Feza Gursey found a surprising link between the octonions and the strong force, which fastens quarks together inside atomic nucleu. An initial commotion of interest in the finding didn’t last-place. 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, feeble and electromagnetic forces( all the fundamental forces except seriousnes ). But rather than seek numerical answers to the Standard Model’s mysteries, most physicists residence their hopes in high-energy particle colliders and other experiments, expecting additional specks to show up and lead the way beyond the Standard Model to a deeper description of reality. They “imagined that the next chip of progress will come from some brand-new articles being fell onto the table,[ rather than] from thoughts harder about the parts we already have, ” said Latham Boyle, a theoretical physicist at the Perimeter Institute of Theoretical Physics in Waterloo, Canada.

Decades on, no particles beyond the interests of the Standard Model have been met. Meanwhile, the strange beauty of the octonions has continued to attract the occasional independent-minded researcher, including Furey, the Canadian grad student who seen Gunaydin four years ago. Looking like an interplanetary traveler, with choppy silver slams that taper to a stage between perforating blue-blooded seeings, Furey scrawled esoteric represents on a blackboard, trying to explain to Gunaydin that she had extended his and Gursey’s work by build an octonionic simulate of both the strong and electromagnetic forces.

“Communicating the details to him turned out to be a bit more of significant challenges than I had anticipated, as I struggled to get a word in edgewise, ” Furey remembered. Gunaydin had continued to study the octonions since the ’7 0s by way of their late connections to string theory, M-theory and supergravity–related ideologies that attempt to unify gravity with the other fundamental patrols. But his octonionic pursuings had always been outside the mainstream. He advised Furey to find another study activity for her PhD, since the octonions might close entrances for her, as he felt they had for him.

Susannah Ireland

But Furey didn’t–couldn’t–give up. Driven by a profound intuition that the octonions and other division algebras underlie nature’s laws, she told a colleague that if she didn’t find work in academia she planned to take her accordion to New Orleans and busk on the streets to support her physics habit. Instead, Furey property a postdoc at the University of Cambridge in the United Kingdom. She has since caused a number of results connecting the octonions to the Standard Model that experts are announcing plotting, strange, luxurious and novel. “She has taken significant steps toward solving some really deep physical perplexes, ” said Shadi Tahvildar-Zadeh, a mathematical physicist at Rutgers University who recently saw Furey in Cambridge after watching an online series of teach videos she made about her work.

Furey has yet to construct a simple octonionic simulate of all Standard Model particulate matter and thrusts in one go, and she hasn’t touched on seriousnes. She stresses that the numerical possibilities are many, and experts say it’s too soon to tell which method of combining the octonions and other disagreement algebras( if any) will lead to success.

“She has found some intriguing associations, ” said Michael Duff, a pioneering string theorist and prof at Imperial College London who has studied octonions’ role in cord speculation. “It’s certainly worth prosecute, in my view. Whether it will ultimately be the way the Standard Model is described, it’s hard to say. If “its been”, it would qualify for all the superlatives–revolutionary, and so on.”

Peculiar Numbers

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 exposed bruises from desegregated martial arts ), rolled-up jeans, socks with caricature immigrants on them and Vegetarian Shoes-brand sneakers, in person she was more Vancouverite than the otherworldly figure in her teach videos. We dawdled around the college lawns, ducking through medieval doorways in and out of the hot sunlight. On a different daytime I might have insured her make physics on a violet yoga mat on the grass.

Furey, who is 39, said she was firstly drawn to physics at a specific moment in high school, in British Columbia. Her teacher told the class that only 4 fundamental troops underlie all the world’s complexity–and, furthermore, that physicists since the 1970 s had been trying to unify all of them within a single theoretical formation. “That was just the most beautiful thing I never heard, ” she “ve been told”, steely-eyed. She had a same feeling a few years later, as an undergraduate at Simon Fraser University in Vancouver, upon understand better the four schism algebras. One such quantity arrangement, or endlessly numerous, would seem reasonable. “But four? ” she echoes envisaging. “How peculiar.”

After interrupts from institution expend ski-bumming, bartending abroad and deeply studying as a mixed martial artist, Furey later gather the department algebras again in an advanced geometry course and learned just how peculiar they become in four strokes. When you double the dimensions with each step as you go from real numbers to complex numbers to quaternions to octonions, she excused, “in every step you lose a property.” Real numbers can be ordered from smallest to largest, for example, “whereas in the complex plane there’s no such concept.” Next, quaternions lose commutativity; for them, a x b doesn’t equal b x a. This builds gumption, since multiplying higher-dimensional numerals involves rotation, and when you switch the rules of rotations in more than two facets you end up in a different place. Much more bizarrely, the octonions are nonassociative, symbolizing( a x b) x c doesn’t equal a x( b x c ). “Nonassociative things are strongly detested by mathematicians, ” said John Baez, a scientific physicist at the University of California, Riverside, and a leading expert on the octonions. “Because while it’s very easy to imagine noncommutative situations–putting on shoes then socks is distinct from socks then shoes–it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.”

The octonions’ apparently unphysical nonassociativity has paralyzed many physicists’ efforts to exploit them, but Baez explained that their singular math has also always been their foreman appeal. Nature, with its four forces batting around a few dozen particles and anti-particles, is itself peculiar. The Standard Model is “quirky and idiosyncratic, ” he said.

In the Standard Model, elementary particles are manifestations of three “symmetry groups”–essentially, ways of interchanging subsets of the corpuscles that leave the equations unchanged. These three equality groups, SU( 3 ), SU( 2) and U( 1 ), correspond to the strong, strong and electromagnetic forces, respectively, and they “act” on six the different types of quarks, two different types of leptons, plus their anti-particles, with each type of particle coming in three reproduces, or “generations, ” that are identical except for their quantities.( The fourth fundamental thrust, gravitation, is described separately, and incompatibly, by Einstein’s general theory of relativity, which throws it as arches in the geometry of space-time .)

Sets of particles attest the equalities of the Standard Model in the same way that four angles of a square must exist in order to realize a equality of 90 -degree gyrations. The question is, why this symmetry group–SU( 3) x SU( 2) x U( 1 )? And why this particular particle representation, with the celebrated particles’ funny array of charges, strange handedness and three-generation redundancy? The conventional outlook toward such questions has been to treat the Standard Model as a burst article of some more complete theoretical design. But a competing predilection is to try to use the octonions and “get the weirdness from the laws of logic somehow, ” Baez said.

Furey began gravely prosecuting this possibility in grad school, when she learned that quaternions capture the direction particles translate and rotate in 4-D space-time. She wondered about particles’ internal belongings, like their charge. “I realized that the eight magnitudes of freedom of the octonions could correspond to one generation of corpuscles: one neutrino, one electron, three up quarks and three down quarks, ” she said–a bit of numerology that had raised eyebrows before. The co-occurrences had now been proliferated. “If this research project were a murder mystery, ” she said, “I would say that we are still in the process of obtaining clues.”

The Dixon Algebra

To reconstruct particle physics, Furey uses the product of the four department algebras, R [?] C [?] H [?] O( R for reals, C for complex numbers, H for quaternions and O for octonions )– sometimes called the Dixon algebra, after Geoffrey Dixon, a physicist who first took this tacking in the 1970 s and ’8 0s before failing to get a faculty job and leaving the field.( Dixon forwarded me a aisle from his memoirs: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”)

Whereas Dixon and others continued by mixing the divide algebras with extra scientific apparatu, Furey inhibits herself; in her scheme, the algebras “act on themselves.” Combined as R [?] C [?] H [?] O, the four digit methods chassis a 64 -dimensional abstract space. Within this space, in Furey’s model, particles are numerical “ideals”: elements of a subspace that, when multiplied by other parts, stay in that subspace, standing molecules to stay molecules even as they move, rotate, interact and alter. The theme is that these mathematical standards are the corpuscles of quality, and they reveal the equalities of R [?] C [?] H [?] O.

As Dixon knew, the algebra separates flawlessly into two parts: C [?] H and C [?] O, the products of complex numbers with quaternions and octonions, respectively( real numbers are insignificant ). In Furey’s model, the symmetries is connected with how molecules move and revolve in space-time, together known as the Lorentz group, arising during the quaternionic C [?] H part of the algebra. The equality radical SU( 3) x SU( 2) x U( 1 ), associated with particles’ internal belongings and mutual interactions via the strong, feeble and electromagnetic forces, comes here the octonionic persona, C [?] O.

Gunaydin and Gursey, in their early effort, already detected SU( 3) inside the octonions. Consider the basi create of octonions, 1, e 1 , e 2 , e 3 , e 4 , e 5 , e 6 and e 7 , which are unit distances in eight different orthogonal tendencies: They respect groupings of equalities announced G2, which happens to be one of the rare “exceptional groups” that can’t be mathematically classified into other existing symmetry-group categories. The octonions’ intimate connection to all the exceptional groups and other special numerical objects has compounded the notion in their significance, persuasion the famed Studies medalist and Abel Prize-winning mathematician Michael Atiyah, for example, that the final thought of quality must be octonionic. “The real possibility which we would like to get to, ” he said in 2010, “should include gravity with all these theories in a way that gravity is understood to be a consequence of the octonions and the exceptional groups.” He included, “It will be hard because we know the octonions are hard, but when you’ve discovered it, it should be a beautiful assumption, and it should be unique.”

Holding e 7 constant while transforming the other unit octonions shortens their symmetries to the group SU( 3 ). Gunaydin and Gursey utilized this fact to build an octonionic model of the strong force acting on a single generation of quarks.

Lucy Reading-Ikkanda/ Quanta Magazine

Furey has started further. In her most recent publicized newspaper, which appeared in May in The European Physical Journal C , she consolidated several determines to construct the full Standard Model symmetry radical, SU( 3) x SU( 2) x U( 1 ), for a single generation of particles, with the math creating the remedy array of electric charges and other aspects for an electron, neutrino, three up quarks, three down quarks and their anti-particles. The math too suggests a reason why electric charge is quantized in discrete units–essentially, because whole numbers are.

However, in that model’s way of formatting particles, it’s unclear how to naturally extend the model to cover the full three corpuscle generations that exist in nature. But in another new article that’s now flowing among experts and under review by Physical Letters B , Furey exploits C [?] O to construct the Standard Model’s two unbrokens equalities, SU( 3) and U( 1 ).( In sort, SU( 2) x U( 1) broken up into U( 1) by the Higgs mechanism, a process that steeps particles with mass .) In this case, the symmetries act on all three molecule generations and too allow for the existence of specks called infertile neutrinos–candidates for light subject that physicists are actively searching for now. “The three-generation model only has SU( 3) x U( 1 ), so it’s more rudimentary, ” Furey “ve been told”, pen poised at a whiteboard. “The question is, is there an obvious route to go from the one-generation picture to the three-generation picture? I think there is.”

This is the main question she’s after now. The numerical physicists Michel Dubois-Violette, Ivan Todorov and Svetla Drenska are also trying to model the three particle generations utilizing a organization that incorporates octonions called the exceptional Jordan algebra. After years of running solo, Furey is beginning to collaborate with researchers who take different approaches, but she prefers to stick with the product of the four schism algebras, R [?] C [?] H [?] O, behaving on itself. It’s complicated enough and offer flexible in the many access it can be chopped up. Furey’s goal is to find the framework that, in hindsight, feels inevitable and that includes mass, the Higgs mechanism, gravitation and space-time.

Already, there’s a sense of space-time in the math. She finds that all multiplicative series of the components of R [?] C [?] H [?] O can be generated by 10 matrices announced “generators.” Nine of the generators act like spatial facets, and the 10 th, which has the opposite sign, behaves like hour. String theory too predicts 10 space-time dimensions–and the octonions are involved there as well. Whether or how Furey’s work connects to string theory remains to be puzzled out.

So does her future. She’s looking for a faculty job now, but failing that, there’s always the ski slopes or the accordion. “Accordions are the octonions of the music macrocosm, ” she said–“tragically misunderstood.” She added, “Even if I prosecuted that, I would ever be working on this project.”

The Final Theory

Furey chiefly demurred on my more theoretical a matter of the relationship between physics and math, such as whether, deep down, they are one and the same. But she is taken with the riddle of why the property of division is so key. She also has a hunch, showing a common allergy to infinity, that R [?] C [?] H [?] O is actually an approximation that will be replaced, in the final hypothesi, with another, related scientific organisation that does not involve the infinite continuum of real numbers.

That’s only intuition talking. But with the Standard Model go tests to floundering perfection, and no enlightening brand-new molecules occurring at the Large Hadron Collider in Europe, a brand-new feeling is in the air, both unsettling and exciting, ushering a return to whiteboards and blackboards. There’s the burgeoning sense that “maybe we have not yet finished the process of fitting the current segments together, ” said Boyle, of the Perimeter Institute. He rates this possibility “more promising than numerous people realise, ” and said it “deserves more attention than it is currently getting, so I am very glad that some people like Cohl are seriously pursuing it.”

Boyle hasn’t himself written about the Standard Model’s possible relationship to the octonions. But like so many others, he acknowledges to hearing their siren call. “I share the hope, ” he said, “and even the doubt, that the octonions may end up playing a character, somehow, in fundamental physics, since they are very beautiful.”

Original story reprinted with dispensation from Quanta Magazine, an editorially independent book of the Simons Foundation whose mission is to enhance public understanding of science by plow research developments and trends in mathematics and the physical and life science .

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