Readers' Choice: Taking on the 10-D Universe with 8-D Math

[deleted]

If Hydrogen doesnt fit - that would be the end of it. My rude and crude opinion is that if algebra does not make sense of Hydrogen ... it is junk. Physics is fabulous at accomodating facts, but not so great at explaining that miserable little muon, or why atoms fall. At least Hydrogen is a nice context in which to contemplate interaction - the main question in the lead-in article. Unfortunately Feynman's approach is painful for Hydrogen. Something is screwy in the model of fermions. If we knew how to properly define fermions there would not be any generation puzzle. The whole lot of them would click into place and make elementary algebraic sense and be consistent with space and time. I think octonions can bring it all together into a coherent model - not as a tool, because anything with space, motion and matter is more like the design of the world - which is not like any tools we are familiar with, which are usually subalgebras, eg, Pauli algebra. It seems to make more sense to start with (complex) octonions and look for stuff that looks like particles, and building up, rather than starting with strings and working one's way down. After all, octonions are excruciatingly elementary. The 'duality units' look like a promissing start - the parentheses in o(a(bc)) explicitly define what kind of duality is considered - a vertex and a face on an abstract tetrahedron

and the antisymmetric rules force consideration of all permutations and associations. All these things need do is oscillate and it begins to look like physics - which makes me wonder why physics ignores octonions. Then Hydrogen has to be a neutral quadruple of such oscillators. Of course, this is all head scratching and hand waving - but might be an interesting way to look at what octonions have to do with physics.

which brings up a question - does the energy-time uncertainty relation have anything to do with requiring associators ( or some mix of commutation and association ).

[deleted]

If the structure of spacetime is available from inspection then is it reasonable to expect quanta to be similarly available from inspection ?

[deleted]

Joel,

I think you have it wrong. Physics has not ignored Octonions, (almost all) physicists have. There is a difference.

You know of my website and have emailed me in te past. I think perhaps you have not looked at what I have presented, else you would not conclude that physics has ignored Octonions. Instead, you would conclude that Octonions sing the song of our physical world. The beautiful structure of the algebra gives us many clues as to how the physical world MUST be, if only we would listen to the melody.

But you must have faith in the structure to buy into the process, and you must not try to force other forms of faith into the discussion as a prerequisite for acceptance. This is where physicists fail Octonions, not where Octonions fail physicists.

Take non-associativity for instance. If it is your religion that physical reality must be associative, then you surely will have an issue with Octonions. But where does this notion come from, but outside the structure and the application of Octonions? It may have relavance within the confines of the non-octonion formalism, but it is a big leap of faith to believe the concept of associativity is a universal truth of physical reality and ANY mathematical method by which we model and try to better understand reality must abide by it.

Mathematics is robust, and the true value of any particular choice for application in physics is what the choice brings to the party. Octonions tell you up front, "hey you, multiplication order matters, get a clue!" Then it tells you, "hey you, its me again. I have more than one way of being defined. And those people saying 480 ways, forget about it."

480 does not come from the algebra itself, it comes from abstractionists whose dogma is defined outside the scope of Octonion Algebra, and whose faith is their religion is universally true and applicable to all. But I digress.

The structure of the algebra of Octonions tells us to look at the product histories of each product term we come up with when we multiply a full form differential operator on its target, be it an 8D potential function or previously differentiated form on it. When we sort the product terms out as to whether and how they may change if the definition of multiplication changes between rules that continue to yield an alternative composition algebras, we find recognizable differentiated forms from Electrodynamics for what we know are forces, expressions for work and energy density, energy density flux as a subset of all product terms that are exactly the same for ANY of the possible definitions of Octonion Algebra.

Physics ignoring the Octonions? I think not.

Physicists ignoring Octonions. I hardly have to itemize, the case is made.

What confounds me to this very day is how my own work has not been commented on by the likes of Drs Dray and Manogue, and especially John Baez. I sent all of them papers outlining my work before my website was up, and have pointed Baez in the direction of the website in his n-category blog. Can't imagine he has not looked at it. To have his long standing and keen interest in Octonions then being given the Octonion form for the structure of the field equations from potentials through the conservation of energy and momentum accurately describing known and trusted Electrodynamics and not illiciting a single comment, not even privately, is odd.

I value their opinions, wish I had them.

Rick Lockyer

[deleted]

Hi Rick -- sounds like more good reasons for another "octoshop" conference in 2010? :) The interest is certainly there! Regarding your work, from my end the challenge is connecting it to existing interests and terminology. Sure, you're doing good work on octonion multiplication tables and I learned from you. But your criticism of the "480 tables" is misleading, IMHO. They're not "wrong", but they're irrelevant in respect to the equivalence classes you require. The explanation that they're relevant for your "algebraic invariance sieve" then replaces a deeper explanation with another concept you're introducing. Don't get me wrong, I'm far from saying that your work is bad; to the contrary. But I do believe that the burden of connecting it to existing concepts and terminology is on your side IF you want responses. Here's my take: The left/right concept of yours reflects a double-cover, and the 8 tables each then reflect triality (as you can see by grouping them into pairs, and then pairs-of-pairs), both of which (double cover, triality) are outer automorphisms on SO(8). Those are well understood concepts and would explain your slicing of the 480 octonion tables (call it your equivalence classes, if you want). That is my hunch, which is enough for me to be interested. But it's far from rigorous; (dis)proving this would be something for you to contemplate. Then, your "algebraic invariance sieve" hints similarities with "left-invariant vector fields on k[S^7]" from Klim/Majid arxiv:0906.5026 , lemma 6.7 and after. Again, my question to you would be whether this is true (rigorously), or if not, why so. Nothing of this is simple, and nobody expects quick results. ... We need an octonion conference! Thanks, Jens

[deleted]

Hi Rick, your pdf was very inspiring to me.

Jens, an octoshop... wow! Even the complex numbers are difficult to grasp.

For instance...

We must leave the concept of the Argand plane and replace it with the concept of a piramid with 4 points. 6 ribs on piramid with 4 corners.

Complex number: 4 locations ( 1 , i , -1 , -i)

- 4 starting points (the locations 1 , i , -1 , -i)

- at a starting point there are 3 rotation directions

- at next point (location) there are only 2 rotation directions left.

so there are 3 * 2 = 6 rotations (3 rotations, 3 reversed rotations)

Those 6 rotations of the complex number are 6 locations of the imaginairy units of the quaternion. I didn't worked out the relation between locations and rotations yet but the complex number is foreshadowing the quaternion. and the quaternion must be foreshadowing the octonion.

Communicative:

a * b = b * a

(location a * rotation b) = (location b * rotation a)

Associative:

(a * b) * c = a * (b * c)

(location a * rotation b) * rotation c = location a * (location b * rotation c)

location ab * rotation c = location a * rotation bc

Peter

[deleted]

Rick - physics - as in the Standard Model - is innocent of any committment to octonions. Under your 'Calculus of H and O algebra' ... 'define a space of 4 or 8 and attach to this an algebraic struct ...". Well, I do not want to define a space, I want the algebra to do that automatically, with all the subspaces, and everything else needed to get a physically reasonable picture, especially anything relevant to why there are 3 generations of fermions. Thanks for putting in that ref to Max Zorn's article on automorphisms in your latest. One suspects there are multiple ways to look at products in octonion algebra rather than one 'true' way. Have you seen Buonocristiani's article on Maxwell in Octonions ?

Lawrence Crowell

The best approach to do physics with octonions is with the Jordan exceptional algebra. The J^3(O) contains J^2(O) with the octonion in a 2x2 matrix, and this is embedded in a matrix with an additional diagonal scalar and the off diagonal octonions which form the superpartners in O^2.

This J^3(O) system is 27-dimensional and under a lightcone gauge reduces to 26-dimensions. This has correspondences to the 26-dimensional bosonic string. It further is the starting structure for a 26-dimensional Lorentzian system that contains the Leech lattice. Both of these have lots of modular sturcture and are automorphisms over the Fischer-Griess group --- sometimes called the Monster group. This results in a matrix form of M-theory.

The approach taken all too often amounts to soldering octonions onto physics by imposing nonassociativity. The role of octonions are likely to be more subtle than this.

Lawrence B. Crowell

[deleted]

Hi Lawrence,

Do you know a good site at which I can learn more about jordan algebra in plane language?

[deleted]

I think that the most concise web source is John Baez "The Octonions." I attach it to this post. The section on the exceptional algebra is the most important for considering how to use octonions in physics.

Cheers LBAttachment #1: oct.pdf

[deleted]

... which closes the circle with the main article of this thread, the Dray/Manogue model of quaternionic spin over octonions (1999 and 1998), which one could understand embedded in J^2(O), an octonionic 2x2 matrix algebra; and the natural extension would lead to J^3(O), an octonionic 3x3 matrix algebra (e.g. Gillow-Wiles/Dray (2009)). All of this directly relates to the exceptional Lie algebra E6 (Wangberg (2007), a Ph.D. thesis supervised by Prof Dray), and frames nicely this enigmatic field of interest of the main article. Can this all be just incidental?

[deleted]

Jens,

Let me try to make my position more clear. Lets start at the beginning.

The multiplication of two unlike Octonion vector basis elements is a third different vector basis element. The three define a cyclic permutation multiplication rule. The complete set of multiplications between unlike vector basis elements requires seven of these cyclic permutation triplets. Each vector basis element will appear in three of the seven, and only once with each of the other six.

There are 30 ways to come up with seven sets of three different elements from a set of seven elements that satisfy the above requirements. The algebra of the Octonions may be FULLY DEFINED by picking any one of the 30 as a starting point. As I have said many times, in many places, once you have picked one of the 30, the other 29 represent ALIASES. They represent different basis name choices we may ARBITRARILY choose between in order to enumerate them.

Having picked one of the 30, we must now determine the order of elements in each of the seven permutations. We may pick either the alternative algebra rule or the composition algebra rule to validate our choices. Having done ALL POSSIBLE, I can say without a doubt there are only 16 ways to do this. The 16 ways can be shown to be 2 groups of 8, where the multiplication table for one in a group of 8 is nothing more than row/column swaps on the multiplication tables for any of the others in the group, but no such row/column swap is possible between members of different groups. Since row/column swaps do not fundamentally change the DEFINITION of the algebra, there are really only two structurally different Octonion Algebra representations.

With this, the Octonion Algebra is TOTALLY DEFINED.

480 multiplication tables come from 16 (actual) * 30 (29 ALIASES), so the number was not pulled out of the air. But there is ABSOLUTELY NOTHING of an algebraic definition value to any number past 16. It comes about by not putting any significance on the triplets, i.e. if a name change scheme comes up with the same set of permutations but in a different permutation order, that is assumed not new, thus the number is 30 and not 7!.

It is not for me to dismiss all possible fanciful applications of the ALIASING, it is for the proponents of 480 to come up with something physically significant requiring it besides enabling a GT square peg to be hammered into an Octonion round hole.

Lawrence and Joel,

While GT and the standard model are interesting in their own right, they do not DEFINE physics, nor do they set any absolute requirements on the application of Octonion Algebra to physics. Which approach will take our understanding the furthest is yet to be determined. Born out of associative algebras, GT related concepts may be limited to categorization and not full disclosure of detail.

Rick Lockyer

[deleted]

Hi Rick - thanks for your very concise summary! If I may comment cheekily, your work being interesting doesn't make others irrelevant :) ... Ok, seriously, it's very interesting how you slice octonions. Earlier I've thought that your equivalence class (that contains all "arbitrary" choices; the class that defines all "aliases" as you call it) was related to SO(8), but that's likely wrong; now I think it's related to O(7), also after your explanation. O(7) can rotate and also mirror 7 basis elements arbitrarily; it would reflect an arbitrary assignment of orthogonal axes to a 7 dimensional vector space. It's not the full answer yet, I'm just trying to inch in. But it's intriguing because O(7) is the smallest orthogonal group that contains G2, the (algebra) symmetry between all octonions ... and you're doing some interesting stuff on octonions on the vector-space level AND the algebra level. It's really fun! Thanks, Jens

PS: One remark on the "aliases"; they're not relevant with respect to the physical forces you're later modeling, but because your parameter space allows "aliasing", this will become part of conditions for specifying relativity, i.e., the conditions that describe equivalent frames of reference in nature, in respect to the force(s) you're trying to model. Therefore my continued interest here.

[deleted]

Within the Jordan exceptional algebra there is a natural way to define intervals, in general within 26-dimensional spacetime, as well as quantum amplitudes. So within this approach one can cast the octonions in a form that connects to C* algebras and then attempt to do some real physics.

Cheers LC

[deleted]

Hi Jens,

I was going to let your "cheeky" comment go, but think better of it now.

To be clear, in no way do I see the work of others as "irrelevant".

I have not (intentionally anyway) used any group theoretical concepts in my work, not because I feel they have no merit, but because they do not work for me personally as a method. That does not mean I believe GT concepts should not work for others, nor that my not needing such means nothing is to be learned from this perspective. My observation is that people work too hard on making a connection to GT sometimes. 480 Octonion multiplication tables is a good example, for the algebra in no way requires it.

I do take issue to any implication that my work needs a connection to today's popular thought in order to be legitimate, for mathematically it stands on its own. It does not need to be re-cast in a different terminology to be understood by anyone with reasonable math skills.

The position of there being 480 different Octonion multiplication tables has been adopted by many that do not really understand just what this means. There is on one hand an air of necessity, which only makes an already complicated algebra incorrectly orders of magnitude more complicated. This does a great disservice to the algebra. On the other hand, when the same people that support 480 are pressed on the issue, they state that, well actually all Octonion multiplication tables are equivalent, yet another incorrect opinion that does the algebra a disservice.

My goal is only to help others understand the algebra fundamentally, not to show up anyone, nor to prove they or their ideas are irrelevant. Not my style.

[deleted]

Hi Rick - thanks for writing! :) It's all good. And I'm beginning to believe that groups aren't sufficient for describing what you do, anyway. Connection to current (and past) thought enables acceptance by others, that's all; and terminology is a big human factor. Thanks for your patience, and for answering my questions. Jens

[deleted]

Jens you wrote (2 Oct) : "In contrast, the Dray/Manogue construction first projects out a single fermion generation, and always becomes the classical m^2 = E^2 - p^2 when squared. Projection into one fermion generation loses the nonassociative parts."

So Dray/Manoge uses the classical: [math]m^2 = E^2 - p^2[/math]

Below I want to settle my case for: [math]m^2 - s^2 = E^2 - p^2[/math]

Restmass:[math]m_0 = \sqrt{ \dfrac{E^2}{c^4} - \dfrac{p_x^2}{c^2}- \dfrac{p_y^2}{c^2} - \dfrac{p_z^2}{c^2} }

[/math]

Relativistic mass:[math]m_{rel} = \dfrac{m_0}{\sqrt{1 - \dfrac{v^2}{c^2}}}

[/math]

Velocity v is a vector quantity.

If we combine the above equations we get:

[math]\sqrt{m_{rel}^2 - \dfrac{m_{rel}^2v_x^2}{c^2} - \dfrac{m_{rel}^2v_y^2}{c^2} - \dfrac{m_{rel}^2v_z^2}{c^2} } = \sqrt{\dfrac{E^2}{c^4} - \dfrac{p_x^2}{c^2}- \dfrac{p_y^2}{c^2} - \dfrac{p_z^2}{c^2}}

[/math]

In my essay I wrote the above more correct as: [math]

m^2 - s_x^2c^2 - s_y^2c^2 - s_z^2c^2 = \dfrac{E^2}{c^4} - \dfrac{p_x^2}{c^2}- \dfrac{p_y^2}{c^2} - \dfrac{p_z^2}{c^2} [/math]

in which 's' is a new quantity. All quantities in the above equation are coordinate quantities.

My model has the advantage that each side in the equation can be decomposed into a quaternion. I made it a closed system. In my essay I combined this with spacetime (also composed of 8 dimensions) into an octonion model of gravity.

Jens, I hope I convinced you.

Peter

[deleted]

some preliminary thoughts on the two Arxiv papers on Octonions, E6 and Particle Physics. Is it possible that Dirac is saying more than is needed to determine the ontological issues regarding how algebra necessitates the existence of particles in space - with generational structure. One would like to have a theory where all the particles emerge with all their measurable properties, but the gist of why particles are needed in the first place might be a much simpler question in octonion algebra. Is there anything tricky about what Dirac was doing ? Looking at Schweber and Lindsay-Margenau and Kramers, one does see various qualms which seem to have been fixed by Feynman having antiparticles go backwards in time. It makes me wonder if there comes a time when we might have to throw Dirac under the bus, and go with the simplest octonion algebra that makes sense of why these particles exist. This goes against the grain of sticking to what is observable. Feynman can deal with the muon by changing a constant, but it sheds no light on why the muon, or any other particle must exist. There seems a logical possibility that Dirac is sufficient for physics but Dirac algebra is not the only way to deal with relativity, and it might be that two wrongs make a right - as far as physics goes, but it might create a headache at the ontological level. If there is a different explanation of antimatter then there is a serious problem in going with Dirac-Feynman, for example if antimatter is a signature-reversal issue, instead of going forward and backward in time. This is not to argue that QM would be at that more elementary level, only that Euclid might have had the right idea about tetrahedra fitting with three dimensional space, but he did not know that Octonions would require the tetrahedral thingy to be oriented, associated and have a special vertex, and there are lots of them.

[deleted]

Is physics upside down with respect to octonion algebra ?

physics explains association in terms of forces, but in algebra

association is a defining relation. If the idea is association of

particles then octionions should be about the construction of atoms.

It ought to look more like structural chemistry rather than looking

like the standard model - algebra treats matter in a kind of top-down

manner. If so it would explain why Hamilton did not see any use for

octonions in physics - in 1844 - well, of course -they didnt know

about atoms or particles, never mind particles associating in space.

[deleted]

Joel,

I think you have it upside down.

The concept of algebra is a first principle notion for mathematics. It's rules must be established before meaningful mathematical expressions can be formulated.

Physics is the application of mathematics to explain the workings of our physical world. Therefore, the mathematics must be established before meaningful discussions on Physics can be had.

So, the proper top down order is

Reality

Physics

Mathematics

Algebra

Octonion algebra does not "look like the standard model", nor should it. Octonion algebra is simply a round peg of suitable size as to not prevent the inclined from forcefully hammering it into the square hole of the model in question. The connection is contrived, it does not flow naturally from the algebra itself.

Algebra defines the operations of addition, subtraction, multiplication by a scalar, and when applicable multiplication and division between members. To think it could or should look like "structural chemistry" let alone the standard model is over the top. It is many steps removed from either of these applications of the more fundamental notions of the applied mathematics and the algebra on top of which the mathematics is constructed.

I seriously doubt Hamilton had any bottom up - top down questions that led him to any conclusions on the Octonions. He had paternal closeness to his quaternions, and wanted to promote them. The sensibilities of his age had issues with four dimenions, issues that also hindered Einstein, Lorentz, Minkowski et al some 60 years later. Eight dimensions in his time would have been as well received as the concept that the earth is not the center of the universe was in an earlier time. We all know what that led to.

But, that was then. This is now.

[deleted]

Rick - it has been an enduring mystery to me that physics can merrily ignore octonions with seemingly no ill effects. How long that can continue is anyone's guess. My feeling is that octonion arithmetic and ordinary arithmetic are almost opposites. we do not multiply two electrons to yield another electron. The parentheses are not so much about expression evaluation as indicating that an association exists - an association that physicists describe with forces, which might explain why physics thinks it does not need octonions. An octonionic physics would be philosophically different because the assumptions are so different from Dirac algebra - one would expect an octonionic layer underneath the Standard Model, in which the generation structure of fermions would make sense - it would look complementary to physics, rather than looking like a useful tool for physics as we now see it. That would address Streater's complaint that octonions look like a lost cause -not by making octonions look more like physics but by arguing that if you inject signature antisymmetry and antiassociativity you have no choice but to acknowledge that it is a whole new ballgame. Also - regarding the 8 dimensional stuff - Graves letter to Hamilton shows an octonion as 1 a b c ab bc ca abc , which is more three dimensional than 8 dimensional. There is more perspicuity in the combinatorial approach. Append another letter to get complex octonions and the Minkowski structure pops out automatically, with no need for dimensional reduction. So, i agree that it does look upside down, and by saying so I probably cause one's crackpot detector to burst into flames, but as long as octonions yield Peano's axioms of ordinary arithmetic, which physics depends on, my faith in octonions must remain undiminished. Hence i am driven to contemplate that it is actually physics that is upside down. Who knows - maybe the LHC will exhibit patterns relevant to G2 some day.

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