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

Re:Meh

[Anonymous Coward]

It is left as an exercise for the reader to decide what Trump does when he sees something he doesn't understand.

Denies it happened, deems it fake.

I'm sure you could fill volumes with stuff Trump doesn't understand.

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.

Re:Meh

[Anonymous Coward]

Yes, economic growth at the expense of ethics, personal freedoms, and national dignity. You carry on believing that.

Re: Clueless editor about singularity

[Tomahawk]

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

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.

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]

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.