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

Useful to know for computer hardware design

[goombah99]

Without having to understand the physics or worry if it's right or not there is an important fact to be gleaned for computer scientists here. Specifically, we won't have a strong need to ever build SIMD systems wider than 8 (well maybe 16). There might be advantages for parallelism beyond that but they are merely scaling advantages not representational advantages.

That is to say, we currently handle 4 wide floats efficiently in SIMD systems. That's not an accident. Systems like Silicon Graphics were specially designed for exactly the purpose of efficient 4x4 matrix multiplication to handle quaternion graphics. Four is the essential number needed to make the atomic unit of all those transactions be the quaternion size. It makes everything else easier if you are not having to do bookkeepping on the data representation of the 4-vectors.

One might have thought that well, make an 8 then someone will want a 16 then a 32. So there's nothing special about 8. But this says indeed there is something special about 8. It's the largest size you really need to worry about the bookkeeping on. It's the largest atomic unit most algebras will ever need to treat.

You could scale beyond that but you will want to make sure that the most efficient ops can work on 8-vectors in whatever designs you consider in the future. it's special.

And microcode desginers will also want to make 8-ops special as well. Page boundaries should be multiples of 32= (8*float) etc...

Maxwell's equations and quaternions

[dtmos]

The really amusing thing to me is that historically, James Clerk Maxwell’s mathematical theory of electromagnetism (published in 1865), which for the first time unified electricity and magnetism, was written in the form of quaternions. For this reason, it was viewed by the engineering world as obtuse and impenetrable – 20 equations in 20 unknowns! Little was done with it until Oliver Heaviside re-wrote the theory in 1884 using the curl and divergence concepts of vector calculus, replacing 12 of the 20 equations with four short differential equations. Ironically, these four equations are now taught to undergraduates as “Maxwell’s Equations,” even though Maxwell never saw them (he died in 1879).

I’ve never seen an electromagnetics textbook written after 1900 that uses the original quaternion description of electromagnetics – they all use Heaviside’s vector calculus approach. It would be supremely ironic if a distaste of quaternions set the search for Physics’ Unified Field Theory back 150 years.

Re: Maxwell's equations and quaternions

[Tomahawk]

It's also interesting that the equations could be changed that way. Maybe that works between all of levels.

Here's the Unified Theory of Everything in octonions using x formulae that explain everything.

Now that that's done, we can simplify them into the fewer equations in quaternions.

Now that that's done, we can simply again to fewer equations in complex numbers.

Now we have something that's much easier to understand, but to properly appreciate it and work with it and expand upon it you need to go back to the original octonion. Then resimplify.

(If simply is the correct word here)

Re:Maxwell's equations and quaternions

[JaxDefore]

after years of reading slashdot I was moved to create a login because of your post. It is the single most interesting thing I have read in ages. Thank you for it - this is why I slog through the posts every day - golden nuggets of fascinating insight. it reminds me of James Burke's Connections (which I hope comes across as a positive mention to you - that's how I meant it)

Re:quanternions for SR?

[BitterOak]

True, ordinary quaternions aren't that useful for describing spacetime but biquaternions give a very natural and elegant way to model the space-time of special relativity. In particular, Maxwell's equations can be written as one simple equation which is manifestly covariant. Lorentz transformations in this algebra have the matrix representation SL(2,C), the set of complex 2x2 matrices with determinant one which is the covering group of the 4x4 matrix algebra representing proper, orthochronous Lorentz transformations. In a sense, biquaternions are to Lorentz transformations in special relativity what quaternions are to rotations in three dimensional Euclidean space.

Simple answer

[Okian Warrior]

Can any of you smart mathematicians and physicists possibly down-translate this for the rest of us?

I'm sure I'm not alone in admitting I have not the slightest idea what the hell this is. OK, maybe I'm alone in admitting it, but I'm sure I'm not alone in having no idea what this is saying.

Around 1940 (IIRC), Eugene Wigner pointed out that symmetries in physics let is map physical theories to abstract groups, and this can place restrictions on what the correct equations have to be, in a way that lets us winnow down the possible theories to only those that satisfy the group topologies.

Suppose you have a square playing card nestled in a square indentation on a table (a regular playing card, except it's square instead of rectangular). How many ways are there to pick up the card and place it back down in the indentation?

The answer is 8 possible ways. If you paint one of the edges of the card, then there are 4 possible sides (of the hole) where the painted edge can go, and then you can have the card face-up or face-down. Each of these placements corresponds to a rotation or a flip of the card: Four rotations (including the identity rotation of 0 degrees), and four flips, along the vertical, horizontal, or two diagonal axes.

No matter how many rotations and flips you make, you always end up in one of the 8 basic positions. Thus, the operations form a group - called the "dihedral" group. The operations are closed: no matter how many flips and rotates you use, it ends up as the same as one of the original 8. Each operation has an inverse, and the 0 degree rotation acts as an identity element. (It's also associative, but that's difficult to show.)

Now imagine the card centered on the X-Y plane, and draw 4 vectors from the origin out to each of the four corners. You can define 8 matrices that flip the vectors in various ways, each matrix being associated with one of the flip or rotate operations.

Thus, the 8 matrices become a representation of the dihedral group. This puts some strong restrictions on the types of matrix you use: each matrix has to have length 1 (it can't change the length of the vectors), and you can't flip one edge over without flipping the opposite edge, because you can't "twist" the card. The matrix length can't be -1 because that would make the card a mirror image - the "J" of a Jack would curve to the right instead of the left.

You can now use matrix mathematics to prove things about your group.

For a different group, consider a vector going from the origin to the unit sphere. You can consider all matrices that rotate the vector in 3D without changing its length or moving its origin. This also forms a group (operations are closed, operations have inverses, and there's an identity operation), but it's an infinite group (a Lie group) and the sphere surface is "smooth". This means that you now can now use differential geometry to prove things about your group.

This group is called SU(3), the "Special Unitary group". It's "Unitary" because the rotations don't change the lengths of the vectors (the matrices are of length 1), and it's "Special" because it doesn't allow mirror-images: the determinant ("length") of the matrix cannot be -1, in the same way that we can't have a matrix of length -1 when rotating cards.

Now consider a physics experiment. We set up an apparatus, calculate the wave equation, and at the end we measure (for example) the energy. We measure energy by applying an operator to the wave equation that describes the experiment.

We can imagine rotating our point of view around the experiment, so that when we do the experiment we measure the energy looking from the other side of the apparatus.

We expect in that case to get the same value.

This means that the energy operator we apply to the wave eq

Re:cart before the horse?

[Pfhorrest]

Ontology does ask "what is?", but a possible answer to that question is "everything that could possibly be".

Even given that answer to that unqualified question, we can (and usually implicitly mean to) ask a more restricted version of it, like "What concretely exists?", "What actually exists?", or even "What presently exists?" In philosophy of time people argue about presentism vs eternalism, and one proposed resolution to that argument is just to note different senses of the word "exist", one in the present tense and one tenseless: only the present presently exists, now, but other times exist in a tenseless sense of the word "exist". Other possible worlds and other mathematical structures may likewise "exist" in increasingly broader sense of the term than the one that means "right now, in the actual configuration of this reality".

Re: quanternions for SR?

[Anonymous Coward]

Yep. Special Relativity only needs a standard 4-element vector s = [x,y,z,ct]. Quaternions multiply differently. Decades ago I used quaternions to represent rotations and orientations of 3D objects in 3-space. At one time they were popular for keeping track of a satellite's incremental changes in rotation over time. They are simpler to use and renormalize than 3x3 rotation matrices. They can also be represented as sparse 4x4 matrices, which eliminates all the weird quirks of three kinds of imaginary nests or whatever it is that's weird about quaternions.