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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

#181

Post by romwell »

throwawaymath wrote: post #140
romwell wrote: post #136

Earlier quoted context omitted.

>Is there an easy explanation of what problems quaternions solve? Sure. Unit quaternions form a double-cover of SO(3). In other words, you can encode a rotation of a 3-dimensional object with a single unit quaternion. But wait, there's more! You could do the same with a matrix, or a triple of angles. Why not do that? Answer: interpolation. The "natural" way you want to go from one rotation to another corresponds to e…

In my opinion this is the best answer, because it neatly explains why quaternions are more useful than vectors for rotation. Vectors are "nicer" because they generalize to arbitrary dimensions. But quaternions handle 3-dimensional rotations in (essentially) a single step. The point about interpolation is really important, because all the machinery offered by vectors becomes a burden. It's also good to think about com…

Thanks!

On an unrelated note, thanks for that Feynman integral trick post.

Re: The Octonion Math That Could Underpin Physics

#182

Post by Retra »

PurpleBoxDragon wrote: post #118
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…

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a com…

One way to think of it is that numbers are mainly used to solve problems by studying the behavior of subsets of them that have specific properties. Like "prime numbers", "square-free numbers", and "solutions to this given equation", etc., and the interesting feature of numbers in general is that they can embed multiple such concepts at the same time. This is helpful because you can find numbers that share multiple properties in order to bridge between otherwise unrelated problems -- it makes it easier to compose solutions of simpler problems into solutions for larger, more complex problems.

From that point of view, quaternions and octonions solve problems in the same way more common numbers do, they just have different sets of properties and so help solving a different set of problems. Of course, they can only do this if we study them well enough to have a sufficiently large suite of concepts and relations in our toolbox.

Re: The Octonion Math That Could Underpin Physics

#183

Post by uryga »

uryga wrote: post #122
PurpleBoxDragon wrote: post #118

Earlier quoted context omitted.

Is there an easy explanation of what problems quaternion or octonions solve? Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers. But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a com…

Quaternions, in three words: "Rotations in 3D". I like this explanation: https://probablydance.com/2017/08/05/intuitive-quaternions/

One thing I'd like to add: they're not actually 4-Dimensional. (read the link for more on this)

Re: The Octonion Math That Could Underpin Physics

#184

Post by smaddox »

gjm11 wrote: post #83
smaddox wrote: post #77

Earlier quoted context omitted.

Not to mention operator calculus. Operators do not, in general, associate. Feynman developed his own notation for operator calculus.

How do operators in general fail to associate? Usually the "multiplication" operation for operators is function application, which is a paradigmatic example of a thing that is associative.

Operator calculus is much more general than multiplication. It can involve exponentiation, differentiation, integration, etc.

Re: The Octonion Math That Could Underpin Physics

#185

Post by fouc »

Why do I feel like the octonions are likely related to the spin networks of LQG (loop quantum gravity). The non-associative property is probably key to explaining how change in space is propagated through the spin networks. And thus why time is directional..

Re: The Octonion Math That Could Underpin Physics

#187

Post by swpdkfn »

romwell wrote: post #144
donbright wrote: post #137

Earlier quoted context omitted.

i did not understand quaternions until I read Hamilton's original works. Maybe i just have a 19th century brain or something. but i found them delightfully free of modern gobbldeygook. https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/

>Maybe i just have a 19th century brain or something. but i found them delightfully free of modern gobbldeygook. No, you can blame Bourbaki for that. People such as V. Arnold decried the way mathematics is now presented[1]. It was from Hamilton's book that I learned what the word vector means and why it's used. It simply means carrier (as in malaria vector that you might heard from biologists) - and carries the space…

I share the same sentiment. The state of modern mathematics exposition is well summarised IMO by "they like the logically most efficient path; that rarely coincides with the pedagogically most efficient one", to paraphrase.

I also have an anecdote similar to yours regarding Hamilton and vectors: I think it was in one of the "Analysis Infinitorum" (Euler) that I found the natural logarithm being called the "hyperbolic logarithm" (it was the English translation of course). When I was a kid I was perplexed by how everyone seemed to insist on using e as the base of their logarithms and exponentials -- why the hell? Reading Euler's treatment of the subject would have been very satisfying then.

Re: The Octonion Math That Could Underpin Physics

#188

Post by nessus42 »

nessus42 wrote: post #115
_bxg1 wrote: post #15

Earlier quoted context omitted.

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.

> Einstein never accepted quantum mechanics Wait? Since when did Einstein not accept Quantum Mechanics? He won the Nobel Prize for his work in discovering an important part of Quantum Mechanics. Einstein rejected the Copenhagen Interpretation of Quantum Mechanics. That's not the same thing as rejecting Quantum Mechanics. I've seen no claims that he would object to either the Everett or Bohm Interpretations of QM. (Bu…

I'm replying to myself here. I did a little research and discovered that Einstein was aware Bohm's interpretation, and Einstein seemed to think that it missed the mark. Einstein did die, however, two years before the publication of the Everett Interpretation, and I've seen no evidence that Einstein received any early information on that seminal Interpretation.

It would be very interesting to know how Einstein would have received the Everett Interpretation.

This is what Einstein had to say about Bohm's Interpretation, in a letter that Einstein wrote to Bohm about six months before he died:

"In the last few years several attempts have been made to complete quantum theory as you have also attempted. But it seems to me that we are still quite remote from a satisfactory solution to the problem. I myself have tried to approach this by generalising the law of gravitation. But I must confess that I was not able to find a way to explain the atomistic character of nature. My opinion is that if an objective description through the field as an elementary concept is not possible, than one has to find a possibility to avoid the continuum (together with space and time) altogether. But I have not the slightest idea what kind of elementary concepts could be used in such a theory."

From my reading of this, it seems that Einstein's greatest concern here is that GR and QM had not yet been unified.

There is a sense in which no competent scientist can fully accept either QM or GR, despite their incredible accuracy in making predictions, because we know that they are incompatible with each other, and consequently, both wrong or incomplete in some very important manner.

Re: The Octonion Math That Could Underpin Physics

#189

Post by dboreham »

XorNot wrote: post #143
scotty79 wrote: post #135

Earlier quoted context omitted.

Complex numbers tell you what happens if there is an i that i^2=-1. What happens is that you get cool way to express 2d rotations. Quaternions tell you what happens if there are three different i's that have this property. And it tells that it leads to nice algebra that expresses 3d rotations very well (and even 4d if you believe the article). Octonions tell what happens if there are seven such i's. And it leads to c…

I think the main curiosity stems from the fact that octonions are as far as you can go. In math infinity turns up all the time so having there be exactly a finite number of anything feels weird.

There are other "works only up to N, where N small +ve Integer" scenarios. e.g. :

https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem

Re: The Octonion Math That Could Underpin Physics

#190

Post by dboreham »

iainmerrick wrote: post #105
madhadron wrote: post #24

Earlier quoted context omitted.

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.

Special relativity was heavily motivated by a pile of puzzling evidence If you mean the Michelson–Morley experiment (apparent zero velocity of earth with respect to the ether), apparently Einstein himself claimed that wasn’t a motivation. I believe it was all about resolving the long-standing "action at a distance" question raised by Newtonian physics (Newton himself noted that this was philosophically disturbing but…

At the beginning of his paper he cites the relativistic nature of Maxwell's equations, and "unsuccessful attempts to discover any motion of the earth relatively to the 'light medium' ".

https://www.fourmilab.ch/etexts/einstein/specrel/www/

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