The Peculiar Math That Could Underlie the Laws of Nature (quantamagazine.org)

xanthos writes: A fascinating article in Quanta magazine introduces us to Cohl Furey and the eight dimensional mathematics called octonions that she is using to model the interactions of strong and electromagnetic forces.

"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?"

"In her most recent published paper she consolidated several findings to construct the full Standard Model symmetry group for a single generation of particles, with the math producing the correct array of electric charges and other attributes for an electron, neutrino, three up quarks, three down quarks and their anti-particles. The math also suggests a reason why electric charge is quantized in discrete units -- essentially, because whole numbers are."

Comments

quanternions for SR?

[iggymanz]

nah, normally 4 vectors are used which are NOT quaternions. Not seeing what advantage their use would give over four-vectors since they wouldn't represent space-time but rather space and operations in space.

Clueless editor about singularity

[UnknownSoldier]

Who ever put together that diagram about "Four Special Number Systems" was completely clueless about Mathematical Singularities

When you add, subtract, multiply or divide the "real numbers" used in everyday life, you always get another real number

*facepalm*

NO, you do not. 0/0 is a singularity because it does NOT produce another real number. You get TWO numbers: +Infnity, and -Infinity and thus Mathematicians say the operation is "undefined".

Re:Clueless editor about singularity

[iggymanz]

+ and - infinity aren't numbers, and no they really don't solve the 0 / 0 problem. that quotient is just undefined for useful maths

Nearly every vector machine is SIMD 8

[Brannon]

You're assuming horizontal SIMD, and ignoring vertical SIMD. Horizontal SIMD places values in the SIMD lanes corresponding to dimensions 'x', 'y', 'z', etc. Vertical SIMD places values in lanes corresponding to the same dimension across different items: e.g. 'x0', 'x1', x2', ....

The former is arguably bounded to a small finite number, the latter isn't.

Re:Maxwell's equations and quaternions

[Anonymous Coward]

According to Michael J. Crowe, "A History of Vector Analysis", Maxwell developed his theory using component analysis in the 1860's. He began studying quaternions in 1870, and presented both component and quaternionic notation in his 1873 "Treatise on Electricity and Magnetism." A brief history can be found at http://fexpr.blogspot.com/2014/03/the-great-vectors-versus-quaternions.html

reminds me of O. Heaviside & Maxwell's equati

[volvox_voxel]

I'm reminded of Oliver Heaviside, who refactored Maxwell's equations into the useful and familiar vector notation that has adorned many tshirts of electrical engineering and physics students. Heaviside took an unwieldy set of twenty field equations, and reduced them to four. I do wonder what insights we can potentially learn if the model itself is refactored into an elegant form.

Her PhD thesis: https://arxiv.org/pdf/1611.091...

The mathematician John Baez has an engaging writing style, and gave an amusing account of octonian numbers (His blog is very interesting BTW): http://math.ucr.edu/home/baez/

"There are exactly four normed division algebras: the real numbers (R), complex numbers (C), quaternions (H), and octonions (O). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

http://math.ucr.edu/home/baez/...

Re:Can anybody dumb this down?

[Baloroth]

There are 4 consistent sets of numbers that allow you to construct a mathematical system with addition/subtraction and multiplication/division. These are the real numbers, the complex numbers, the quaternions, and the octonions. These systems have 1,2,4, and 8 units, respectively (and are therefore intrinsically 1,2,4, and 8 "dimensional" systems). Each one gains and looses some nice properties that are useful in various circumstances. The reals are useful for things like finance or sheep counting, the complex for quantum mechanics, and quaternions for 3D vectors (like CGI graphics). In principle you can always use the reals, but other systems have properties that naturally make it easier to do certain things.

Now, in physics there is something called the Standard Model (SM) that describes most of our understanding of particle physics: how they interact, how they're created and destroyed, and (almost all) their properties (there are a few exceptions, such as the neutrino mass, which is not included in the SM). The SM has been shown to be extremely accurate and predicts nearly every phenomenon that we see. There are a few things missing: notably, gravity is a completely separate model from the SM (not that they're opposed to each other, but no one's found a good way to integrate the two models together without running into mathematical absurdities).

Now, the SM uses regular old complex numbers, and adds a lot of very complicated and fancy math on top in order to make it's predictions. It all works, but the math could maybe be made simpler or more elegant: right now, it requires adding several sets of complex numbers together, because the dimensionality of the model is higher than the 2 that complex numbers alone can perform. Since octonions are 8 dimensional, it may be possible to re-write the math of the SM into a form that uses octonions with fewer groups of numbers. This would kinda be cool, and could maybe make the math easier (or reveal certain properties of the model that weren't obvious before), but wouldn't really change the physics (because that's already fairly well understood and, as mentioned above, very accurate). So far, Cohl Furey has managed to do that for one set ("generation") of particles in the SM. Note that the math for a single generation is far easier than for multiple generations, so it remains to be seen if it can be extended to include the entire SM, but from a purely mathematical standpoint, it's kinda cool.