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.

Re:quanternions for SR?

[lgw]

nah, normally 4 vectors are used which are NOT quaternions.

You're thinking of General Relativity, not Quantum Mechanics. QM math is all about matrices of complex numbers..

Quaternions and and matrices are somewhat annoying, as they're not commutative. Octonions are neither commutative nor distributive, which makes them harder still to work with.

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.

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.

Re:quanternions for SR?

[MrMr]

I clicked on the link. Gave me the impression 'natural', 'elagant' and 'simple' may mean something different where you live.

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

Re:Is there a PhD in the house?

[Bob the Super Hamste]

I don't think the timecube guy was ever on slashdot.

Re:Feminist propaganda

[Pfhorrest]

Fuck women!

Not with that attitude you won't.

Another step in our understanding of Everything

[mykepredko]

Great article and illustrates how as we try to understand reality (for lack of a better word): we first find that our current level of physics can't explain what we observe so we need to go to the next level. That next level needs the appropriate mathematical tools which often end up being already invented and looking for a practical application.

From the perspective of using a branch of mathematics that is new to the field, there's a lot of similarity between this story and using mathematics to predict crime: https://science.slashdot.org/s...

I believe we need to promote and retell these stories to students so that they can look beyond the simple and search for mathematical analogues that allow them to understand and model the physical world in different ways.

Re:Clueless editor about singularity

[fisted]

It's almost as if the diagram targets scientifically curious laypeople, so your nerd rage about this irrelevant detail (given the context) is a bit over the top.

Re:cart before the horse?

[Pfhorrest]

So... you're telling me that reality is defined by an abstract algebra concept?
I thought we were using abstract algebras to *model reality*--not the other way around.

Yes. Reality will be defined by some mathematical structure or another. We can invent mathematical structures to describe any possible way that reality might be. Whatever way it turns out that reality is, whichever mathematical structure accurately describes it defines its properties.

One might even say (as Max Tegmark more or less does) that concrete existence, the kind of existence that applies to rocks and trees and such, is just a special case of abstract existence, the kind that applies to mathematical structures like numbers and triangles. All mathematical structures "exist" in that abstract sense, and the things that "exist" in a more concrete sense are just the things that are part of the same mathematical structure of which we are a part, i.e. of our physical reality.

Similar to how, as David Lewis puts it, "'actual' is indexical", i.e. in a multiverse of possible worlds (which, NB, would all be part of the concrete world we're talking about above), the "actual world" is just the one that we happen to be part of, and not ontologically different from any of the other possible worlds. We might likewise say that "'concrete' is indexical"; concrete reality is just the abstract structure of which we are a part, and not ontologically different from any other abstract structures.

It's still an empirical question to figure out which possible world (configuration) of which abstract structure we are a part of. But whatever the answer will turn out to be, there's some possible math that will describe it.

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...

Re:Useful to know for computer hardware design

[mikael]

There are the AVX, AVX-256 and AVX-512 extensions to the x64 instruction set. 128-bit width gives four floating point, 512-bit width registers gives sixteen floating point values or eight double precision values.

Re:Clueless editor about singularity

[dgatwood]

And yet nobody has trouble with square roots being defined, even though sqrt(4) is +/- 2.

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

[mykepredko]

I forgot about that. When I was in university and learning Maxwell's equations, it was explained to us that using Calculus was the "simpler" way - Maxwell's original method used an approach which lead to a dead end.

Thanx for the memories.

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: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

Re: Maxwell's equations and quaternions

[DCFusor]

You could hope for that. After all, a lot of problems in signal processing are solved more easily by flipping to a different domain (say, frequency instead of time), solving it there, and then converting back.
In fact, something similar is what Ed Witten was on about in joining the various (sub) string theories, so that a problem insoluble in one of them could be solved in another, then converted back.

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:Clueless editor about singularity

[burtosis]

And yet nobody has trouble with square roots being defined, even though sqrt(4) is +/- 2.

I think if the square root is negative you are imagining things.

Re: Clueless editor about singularity

[Tomahawk]

If the square is negative you're imagining things. A square root being negative is just reality.

Erm, well sure ...

[cascadingstylesheet]

... I was gonna say that.

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?

[suutar]

The short form, I think, comes out to "we used somewhat complicated math to handle several things independently, but this even more complicated math handles it all together, which may help us predict more stuff."

summary inaccurate

[thePsychologist]

Says that the reals, complex numbers, quaternions, and octonions are the only kinds of numbers that can be added, subtracted, multiplied, and divided. This is obviously false, as that can be done in any division algebra (including any field, like finite fields, rational numbers, etc, and there are there are uncountably many fields).

What they meant to say is that those are the only normed division algebras - basically, algebras over the real numbers with a notion of distance and such that the distance is compatible with multiplication.

Geometric Algebra

[sfcat]

Octonions, quaternions and the like are algebras for dealing with dimensions represented by imaginary numbers. But they are special purpose, like most of the algebras used by physicists. They were 1 of 2 ways to represent vectors in math (the other being Vector Algebra). A way of uniting these two methods into 1 framework was discovered by Clifford about 50 years later but by that time there was a big split in math over VA vs Quaternions (which VA won). And most of the field ignored Clifford and his way to unite the two (VA and Quaternions). This is a big reason why its hard to unite multiple parts of physics as some still use Quaternions while most others use VA.

So in the 50s a mathematician named David Hestenes developed a new branch of math called Geometric Algebra (based upon Clifford Algebras) which could subsume all of the different algebras used by physicists (and many others too). Additionally, it can handle contravariance and covariance, any positive integer number of dimensions, and handle algebras over imaginary numbers. Quantum Loop Gravity uses Geometric Algebra for instance. The problem is that Geometric Algebra isn't taught yet except perhaps at a post-doc level to mathematicians. The first textbook covering GA for Computer Science was just published in 2017. There are hopes that reformulating physics in to GA will allow unifications that were either not possible or too difficult when each part of physics uses different types of algebras.

The problem with all of this? GA is really really really hard. There is even an extension to GA called Geometric Calculus that's even more difficult. Given how difficult most students find VA which is much easier than GA, I'm not sure when we can expect most physics to make new theories using GA instead of VA. But when we can climb that hill, we will likely be able to see new physics on the other side. There are also a great many CS applications of GA as well (which is what I do).

My take on TFA, is that this physicist is going down a wrong path because she was never taught GA. If she finds something, it will likely have to be converted into GA to unify it with other algebras used in other parts of physics. But I could be wrong, who knows but some of the greatest physicists in history have gone down this specific rabbit hole with nothing to show for it at the end. I wish her luck.

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: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.

Re:cart before the horse?

[Zorro]

Remember Mathematics only MODELS reality. It isn't reality itself. It is just a model of reality we can understand and manipulate.

Re:cart before the horse?

[Pfhorrest]

So is it like Bill Clinton said, "depends on what the meaning of "is" is?

Hah, I like making that joke/reference in this context myself. :-)

Another fun one: the field of mereology studies the relationships between parts and wholes, and the difference between continuous stuff that has no proper parts and is infinitely divisible, and discrete things that are made of other discrete things down to some atomic (indivisible) level. In that context, "stuff" and "things" are technical terms referring to those continuous and discrete kinds of beings. So what do mereologists study? Oh, you know... things, and stuff.

Re:cart before the horse?

[Pfhorrest]

The map is not the territory, true. Unless you have a perfectly detailed map at 1:1 scale, in which case you have just replicated the territory.

Mathematics models reality in that we don't know exactly what reality is like and we're trying to make a map of it. But whatever model it is that would perfectly map reality in every detail, would be identical to reality itself. We just don't know what model that is.

Cool name of author

[pchenden]

Her name sounds like a comic book super hero. Cold Fury?

Re:cart before the horse?

[wtrmute]

One might even say (as Max Tegmark more or less does) that concrete existence, the kind of existence that applies to rocks and trees and such, is just a special case of abstract existence, the kind that applies to mathematical structures like numbers and triangles. All mathematical structures "exist" in that abstract sense, and the things that "exist" in a more concrete sense are just the things that are part of the same mathematical structure of which we are a part, i.e. of our physical reality.

Or, as Aristotle says, the substance of a thing like a rock or a tree has a prior existence to the concrete instances of it: "The essence, i.e. the substantial reality, no one has expressed distinctly. It is hinted at chiefly by those who believe in the Forms; (...) they furnish the Forms as the essence of every other thing, and the One as the essence of the Forms." (Metaphysics, Book I, part 7)

Similar to how, as David Lewis puts it, "'actual' is indexical", i.e. in a multiverse of possible worlds (which, NB, would all be part of the concrete world we're talking about above), the "actual world" is just the one that we happen to be part of, and not ontologically different from any of the other possible worlds. We might likewise say that "'concrete' is indexical"; concrete reality is just the abstract structure of which we are a part, and not ontologically different from any other abstract structures.

Aristotle again? "Actuality, then, is the existence of a thing not in the way which we express by 'potentially'; we say that potentially, for instance, a statue of Hermes is in the block of wood and the half-line is in the whole, because it might be separated out, and we call even the man who is not studying a man of science, if he is capable of studying; the thing that stands in contrast to each of these exists actually."

Re:cart before the horse?

[ffkom]

Yes. Reality will be defined by some mathematical structure or another. We can invent mathematical structures to describe any possible way that reality might be. Whatever way it turns out that reality is, whichever mathematical structure accurately describes it defines its properties.

You are turning math into a religion, in a very unconvincing way.

Following the same line of argumentation, I could state "The God of my religion X has created the Universe and defined the one true reality. It is all written in our holy book. Whatever observation you might come up with that contradicts the colorful stories in our holy book, we will just adjust the book to make it fit! So no matter how many convoluted additions we will need to make, we can invent them!".

Math is a utility. A useful one to describe a model of reality as we know it. But it does not define or create reality.

Just because some solutions exists for formulas used in general relativity that could be interpreted as "Tachyons" does not mean such "Tachyons" exist in reality. It might be worth looking for them, but the result can be there are none. Whatever mathmatical model you can invent, it will certainly allow for possibilities that do not exist in reality. And it might still miss to model real things that do not become known to any intelligent lifeform before those ceases to exist.

Re:cart before the horse?

[Pfhorrest]

Your religious analogy is flawed in that you're talking about people claiming that some particular book defines reality, and then freely and unabashedly modifying that book to fit reality; whereas what I'm talking about is more like saying "there is some possible book that could be written that would perfectly describe the rules of reality. Whatever rules would be written in that book, those rules define reality." It's pretty much a tautology.

There is some rigorous formal (i.e. mathematical) system that would be a perfect description of reality. Whatever the rules of that system are, those are the rules of reality, because that system is defined as whichever one has the rules of reality as its rules.

What the author of the paper in question here is saying that if this math regarding octonions is part of the mathematical system that perfectly describes reality, then no further explanation for the discrete values of electrical charges is needed, because that phenomenon is just an automatic consequence of integers having discrete values, in such a system.

It's like saying that if the geometric structures called ellipses describe the motion of the planets, and the Earth is a planet whose motion is described by that structure, then no further explanation for the apparent retrograde movement of the other planets in the sky is necessary, because that relative apparent motion is just an automatic consequence of the geometry of elliptical motion.

Re:Meh

[kenai_alpenglow]

And, BTW, what the heck does this have to do with 8 dimensions anyway?

Re:cart before the horse?

[sjames]

Yes, but when you don't yet know the answer, it helps if you can at least confine your search within what is possible. Math in general defines what is possible. We can further confine the search based of what math in particular can describe what we already know about reality.

What's really interesting is that the algebra of Octonions seems to have the physical properties of the standard model naturally fall out as a result of the properties of the underlying algebra rather than having to be "bolted on" piece by piece. In general, that's a sign that you're on to something. Compare elliptical orbital mechanics controlled by one equation and a few simple parameters vs an infinite series of epicycles each hand tuned to fit observation.