Dave Farber
2018-08-03 01:48:19 UTC
Date: August 3, 2018 at 02:23:43 GMT+9
Subject: [Dewayne-Net] The Peculiar Math That Could Underlie the Laws of Nature
[Note: This item comes from friend Ed DeWath. DLH]
The Peculiar Math That Could Underlie the Laws of Nature
New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called âoctonions.â
By Natalie Wolchover
Jul 20 2018
<https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/>
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 GÃŒnaydin. Furey had figured out how to build on a finding of GÃŒnaydinâs from 40 years earlier â 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 âoctonions.â
As numbers go, the familiar real numbers â those found on the number line, like 1, Ï and -83.777 â just get things started. Real numbers can be paired up in a particular way to form âcomplex numbers,â 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 âquaternions,â discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublinâs Broome Bridge. John Graves, a lawyer friend of Hamiltonâs, 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 âdivision algebrasâ 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 Einsteinâs 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?
âOctonions are to physics what the Sirens were to Ulysses,â Pierre Ramond, a particle physicist and string theorist at the University of Florida, said in an email.
GÃŒnaydin, the Penn State professor, was a graduate student at Yale in 1973 when he and his advisor Feza GÃŒrsey 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 didnât last. Everyone at the time was puzzling over the Standard Model of particle physics â 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 Modelâs 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 âimagined 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,â 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 GÃŒnaydin 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 GÃŒnaydin that she had extended his and GÃŒrseyâs work by constructing an octonionic model of both the strong and electromagnetic forces.
âCommunicating 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,â Furey recalled. GÃŒnaydin had continued to study the octonions since the â70s by way of their deep connections to string theory, M-theory and supergravity â 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.
[snip]
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-------------------------------------------Subject: [Dewayne-Net] The Peculiar Math That Could Underlie the Laws of Nature
[Note: This item comes from friend Ed DeWath. DLH]
The Peculiar Math That Could Underlie the Laws of Nature
New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called âoctonions.â
By Natalie Wolchover
Jul 20 2018
<https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/>
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 GÃŒnaydin. Furey had figured out how to build on a finding of GÃŒnaydinâs from 40 years earlier â 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 âoctonions.â
As numbers go, the familiar real numbers â those found on the number line, like 1, Ï and -83.777 â just get things started. Real numbers can be paired up in a particular way to form âcomplex numbers,â 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 âquaternions,â discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublinâs Broome Bridge. John Graves, a lawyer friend of Hamiltonâs, 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 âdivision algebrasâ 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 Einsteinâs 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?
âOctonions are to physics what the Sirens were to Ulysses,â Pierre Ramond, a particle physicist and string theorist at the University of Florida, said in an email.
GÃŒnaydin, the Penn State professor, was a graduate student at Yale in 1973 when he and his advisor Feza GÃŒrsey 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 didnât last. Everyone at the time was puzzling over the Standard Model of particle physics â 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 Modelâs 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 âimagined 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,â 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 GÃŒnaydin 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 GÃŒnaydin that she had extended his and GÃŒrseyâs work by constructing an octonionic model of both the strong and electromagnetic forces.
âCommunicating 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,â Furey recalled. GÃŒnaydin had continued to study the octonions since the â70s by way of their deep connections to string theory, M-theory and supergravity â 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.
[snip]
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