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The Octonion Math That Could Underpin Physics

quantamagazine.org

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Re: The Octonion Math That Could Underpin Physics

#21

Post by davebryand »

fifnir wrote: post #14

A layman's quest to understand wtf this is... >In mathematics, the octonions are a normed division algebra over the real numbers . wtf is a normed division algebra?? >In mathematics, Hurwitz's theorem is a theorem [...] solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. ... right. I have the same problem when I try to understand any…

Here are a few that I love:

3 Blue 1 Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

VSauce: https://www.youtube.com/user/Vsauce

Numberphile: https://www.youtube.com/user/numberphile

Re: The Octonion Math That Could Underpin Physics

#22

Post by empath75 »

fifnir wrote: post #14

A layman's quest to understand wtf this is... >In mathematics, the octonions are a normed division algebra over the real numbers . wtf is a normed division algebra?? >In mathematics, Hurwitz's theorem is a theorem [...] solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. ... right. I have the same problem when I try to understand any…

Unfortunately there’s a lot of domain knowledge wrapped up in very short phrases. What you want to learn is abstract algebra, if you want to google for YouTube videos that break it down for you.

Re: The Octonion Math That Could Underpin Physics

#23

Post by throwawaymath »

sdenton4 wrote: post #13

I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "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…

For the curious, John Baez has written about octonions pretty extensively: http://math.ucr.edu/home/baez/octonions/

Re: The Octonion Math That Could Underpin Physics

#24

Post by madhadron »

_bxg1 wrote: post #15
ianai wrote: post #2

I think the following quote is the QED for academia ruining any chance at actual research: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”

It's my understanding that most of Einstein's theory was the product of intuition, backed up after-the-fact with mathematics and experimentation. Intuition isn't a bad compass, as long as you can set it aside if reality measurably contradicts it. In fairness, Einstein never accepted quantum mechanics because they flew in the face of his intuition, but it still got him pretty far.

This is very false. Einstein's reputation was built on explaining known phenomena such as the photoelectric effect and Brownian motion. Special relativity was heavily motivated by a pile of puzzling evidence and a bunch of existing mathematics.

Re: The Octonion Math That Could Underpin Physics

#25

Post by madhadron »

Ono-Sendai wrote: post #20

I have to disagree that quaternions underlie Special Relativity. Although special relativity does use 4-vectors, those aren't quaternions.

I remember going down that path when I was an undergraduate. Fortunately I had a very experienced theorist to hand who explained that, yes, people did try that, but stopped bothering because it doesn't generalize to general relativity, and there was no point keeping two mathematical toolboxes around when you could have one.

Re: The Octonion Math That Could Underpin Physics

#27

Post by jstanley »

nabla9 wrote: post #18
anaphylactic wrote: post #3

> 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. 1. I think they meant "the only kinds of numbers constructed in this way ". 2. Sedenions can still be added, multiplied, subtracted and divided. it's just that multiplication and division lose most of their useful properties. Wit…

Being composition algebras makes R,C,Q and O more interesting than others: N(xy) = N(x)N(y) - N is called norm. Without this property, you have zero divisors. edit: throwawaymath uses better notation: |xy| = |x| • |y|

What type of thing is N? What you wrote doesn't seem to make sense if N is just a constant, but I don't see how it makes sense if N is a function? Or maybe you didn't mean multiplication?

E.g. N = 3, x = 2, y = 3

N(xy) = 3(2 . 3) = 18

N(x)N(y) - N = 3(2) . 3(3) - 3 = 51, which is not 18

Re: The Octonion Math That Could Underpin Physics

#28

Post by throwawaymath »

davebryand wrote: post #21
fifnir wrote: post #14

A layman's quest to understand wtf this is... >In mathematics, the octonions are a normed division algebra over the real numbers . wtf is a normed division algebra?? >In mathematics, Hurwitz's theorem is a theorem [...] solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. ... right. I have the same problem when I try to understand any…

Here are a few that I love: 3 Blue 1 Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw VSauce: https://www.youtube.com/user/Vsauce Numberphile: https://www.youtube.com/user/numberphile

Those are good for supplementary exposition (especially 3Blue1Brown), but they're not suitable for learning on their own. They're also a bit of a hodge podge beyond calculus and linear algebra.

To understand what's going on here in any meaningful sense, the parent commenter should pick up an accessible undergrad textbook on abstract algebra. It doesn't have to be particularly advanced. Then they'll have a better foundation for understanding the algebraic features of various number systems.

Re: The Octonion Math That Could Underpin Physics

#29

Post by Micoloth »

fifnir wrote: post #14

A layman's quest to understand wtf this is... >In mathematics, the octonions are a normed division algebra over the real numbers . wtf is a normed division algebra?? >In mathematics, Hurwitz's theorem is a theorem [...] solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. ... right. I have the same problem when I try to understand any…

Well yeah, math books...

Jokes aside, i too think we need a new framework for divulgating math that actually tells what you need to know, without just handwaving at it, BUT without the amount of technical details of a mathematics class.

What do you know, i think this is Possible, too.. You can communicate a surprising amount of information if you use words properly.

Of course, since this has never been done except from basic maths, it would be quite a task to embark in, and one would only do it if it made economic sense..

Re: The Octonion Math That Could Underpin Physics

#30

Post by throwawaymath »

jstanley wrote: post #27
nabla9 wrote: post #18

Earlier quoted context omitted.

Being composition algebras makes R,C,Q and O more interesting than others: N(xy) = N(x)N(y) - N is called norm. Without this property, you have zero divisors. edit: throwawaymath uses better notation: |xy| = |x| • |y|

What type of thing is N? What you wrote doesn't seem to make sense if N is just a constant, but I don't see how it makes sense if N is a function? Or maybe you didn't mean multiplication? E.g. N = 3, x = 2, y = 3 N(xy) = 3(2 . 3) = 18 N(x)N(y) - N = 3(2) . 3(3) - 3 = 51, which is not 18

That's not a subtraction sign, it's a hyphen. N is the norm of x, denoted by |x|. Technically the norm is a scalar-valued function applied to a vector, hence the functional notation N(xy) = N(x) • N(y).

So to be explicit, they're saying |xy| = |x| • |y| implies you cannot have a 0 divisor.

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