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FQXi FORUM
July 18, 2019

ARTICLE: Readers' Choice: Taking on the 10-D Universe with 8-D Math [back to article]

Tom wrote on Sep. 13, 2009 @ 22:29 GMT
"In 10 dimensions, octonions can be used to describe a particle’s momentum, but not its position. But after the description is collapsed down from 10 to four dimensions, particles can be described in both ways."

Consider: we cannot know both the position and momentum of a particle without 'observing' and affecting the situation. Before we observe the particle, it could exist in all 10 dimensions; HOWEVER when we view the particle from the perspective of a four dimensional setting, [and collect the position and momentum info] we inflict our forth dimensionality on the particle. If this was the case, it would work with the statement above. Specifically, by viewing the 10dim particle from a 4dim perspective, the octonian 'description' is collapsed into 4 dimensions, and the momentum and position can be known.

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mikelinpa wrote on Sep. 14, 2009 @ 01:32 GMT
Nice article, thanks. It is very interesting stuff. I wish I understood more of it. I love thinking about this stuff, but I haven't the math chops to do anything with it.

Here is a thought, (two actually,) that are mine. I thought them up myself. I do not thing they are probable, but interesting anyways. Tachyons are particles that are supposedly able to move backward and forward in time. What if there was only one tachyon and it was busy as hell being everywhere and everywhen? The same for electrons. Electrons are theoretically missing most of the time. What if there was only one electron, and it was busy as hell holding up every atom in the universe at once? (Would that make it the God particle? OK, that is a third thought...)

I do not believe these idea to be true. They are more like "What if..." If anyone would like to discuss them, or point me to more articles, please do.

Michael Lashinsky

mlashinsky@gmail.com

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Lawrence B. Crowell wrote on Sep. 14, 2009 @ 21:47 GMT
I didn't notice there was a new article. At the crux of my work is the Jordan matrix which is a 3x3 representation of octonions, where the off-diagonal elements are octonions. The three scalar diagonal elements obey a light cone condition and determine an elementary field. This is the "M2-brane," field in a sense. I am working through some eigenvalue problems with it to determine the AdS bounds on the dimension of the field which gives a quantum critical point.

I encountered years ago an octonion paper by Corinne Manogue, which I might still have somewhere either on HD archives or in my geological stack of papers in various boxes.

Cheers LC

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Lawrence B. Crowell wrote on Sep. 14, 2009 @ 21:53 GMT
The way I have worked in 11-dimensional supergravity with 8-dim octonion math is with the Jordan matrix. There are three octonions for a total of 24 dimensions (there is some S^3xSO(8) and SO(24) triality here) plus the 3 diagonal scalars. That is a total of 27 dimensions for the basic Jordan algebra. Now impose a light cone condition on the three scalars, which reduces a dimension to 26, corresponding to the 26 dimensional bosonic string. Now for there being (ignoring the constraint for the moment) three identical copies of the ocotonions that gives 8 + 3 = 11 dimensions, and the light cone constraint reduces that to 10.

Lawrence B. Crowell

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LuisM wrote on Sep. 17, 2009 @ 01:11 GMT
I wounder if this work could be combined with Garrett Lisi's work with E8? a 8 Dimentional system would work very well using 248 in base 10

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Jens Koeplinger wrote on Sep. 17, 2009 @ 05:37 GMT
What continues to intrigue me in your model is how well quaternionic spinors on octonionic background reproduce properties required for fermion generations, yet the role of nonassociativity in nature is obscure. How might one model "dynamics", i.e. the interplay of physical properties with the parameters they're modeled on?

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joel rice wrote on Sep. 17, 2009 @ 14:03 GMT
is there a recent article one can look at ?

If it turns out that Nature frowns upon supersymmetry

do octonions remain relevant ?

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Lawrence B. Crowell wrote on Sep. 17, 2009 @ 15:57 GMT
Jens,

I agree that the role of nonassociativity is not very clear. I will say that building things up from simple considerations might help. For instance with the Jordan algebra or matrix there are three copies of the octonions on the off diagonal. One might think of there being an additional color assigned to these octonion components, which amounts to introducng a G_2 action. This is similar to a QCD group action. From there physical aspects of nonassociativity might be developed.

Given two braid groups g and g' nonassocativity is a sort of map between them and a measure to what extent g'g^{-1} departs from unity. Nonassociativity is then a way in which nonunitay equivalence can be described according to a volume perserving or modular map.

Lawrence B. Crowell

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Lawrence B Crowell wrote on Sep. 17, 2009 @ 18:23 GMT
Joel,

The relationship between octonions and supersymmetry is a a bit of a thicket. The Jordan algebra is a graded algebra and is supersymmetric.

LC

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joel rice wrote on Sep. 17, 2009 @ 21:17 GMT
i worry about having more of a thicket than might be called for, figuring that if complexifying quaternions leads to such good things as Pauli Algebra and multivector algebra, then how much more do we get if we complexify octonions ?

After all it seems to lead to 4 dimensional Minkowski signatures, and not to the ++++, ---- or ++-- unless i am confused. And how does one know the correct and proper way to define a particle ?

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Lawrence B. Crowell wrote on Sep. 18, 2009 @ 00:33 GMT
You can have real-octonions, complex-octonions, quaternionic-octonions, and octo-octonions. These all involve generalizations of the Jordan matrix or algebra by extensions with exceptional algebras.

The signatures for Lorentzian structure is not something unique to octonions or extensions from the reals up the Cayley numbers. The fundamental basis for Lorentzian metric appears to be wrapped up in some properties of sporadic groups, in particular the Leech lattice, which is an automorphism with 26-dimensional Lorentzian structure on something called the Monster group. The mathematics of sporadic groups and the like is a very deep subject. It is worth noting that the Leech lattice has subgroups with three octonions, which is related to the Jordan algebra.

Cheers LC

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Peter van Gaalen wrote on Sep. 18, 2009 @ 19:14 GMT
I posted an essay about octonions, maybe you will find this essay interesting in regard to this discussion.

http://fqxi.org/community/forum/topic/509

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Lawrence B. Crowell wrote on Sep. 19, 2009 @ 00:16 GMT
I present some aspects of the Jordan exceptional algebra of octonions in:

http://www.fqxi.org/community/forum/topic/494

I have not looked at you paper yet. In fact I have not looked at most of them yet, and only voted on a couple. Usually I try to cover as many as I can after they are all submitted.

Fleshing out the physics of octonions is not easy. My paper really brushes on this issue rather than advance it in a central way.

Cheers LC

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joel rice wrote on Sep. 23, 2009 @ 13:12 GMT
does this mean that Alternativity is not an issue for OxH and OxO ?

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Jens Koeplinger wrote on Sep. 24, 2009 @ 01:27 GMT
Joel - OxH and OxO aren't alternative anymore in general. That's challenging, your suspicion is warranted ... in the Dray/Manogue approach, the dimensional reduction scheme reduces the formulation on nonassociative background to the traditional Dirac equation after fixing a (proposed) fermion generation, by means of quaternion spinors over an octonionc background. That's one way of dealing with nonassociativity, i.e., showing how traditional, associative formulations emerge in distinct cases - though it is still interesting to me what dynamics there may be in the general, nonassociative case. Are there physical principles that govern this territory?

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Lawrence B Crowell wrote on Sep. 25, 2009 @ 00:27 GMT
I have been discussing some of these issues on my essay site. In the last one I indicate how triplets or alternativity operates

http://www.fqxi.org/community/forum/topic/494

I think that nonassociativity is something which emerges from the G_2 action, which is the automorphism group on the octonions. This is related to the SU(3) holonomy on the Hanson-Iguchi metric and Ricci flat spaces of compactification.

Cheers LC

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Jens Koeplinger wrote on Sep. 25, 2009 @ 02:26 GMT
Dear Lawrence - thanks for responding. Let me just remark on this expression: "G2 action". That sounds like the path integral over some generalized action would be the underlying physical principle here; but on a nonassociative background, expressions like
$\int{L dx}$
quickly become ambiguous. Sure, G2 itself is associative; however, I don't see the physical principle that would drive nonassociativity as required (or desired). Without such a principle, nonassociativity appears to be more like a bolt-on feature (as opposed to naturally required). Using associative symmetries between nonassociative constructs seems like an excuse, something we have to do in the absense of better methods ... What would make it "natural"? JK

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Lawrence B Crowell wrote on Sep. 25, 2009 @ 17:43 GMT
Nonassociative operators probably have some subtle role that is different from what we currently understand. We might think of this as similar to noncommutative operators in quantum physics. Prior to the physical bais for quantum mechanics such structures had a limited role in classical mechanics. The only role they had was with angular momentum and related systems. It was with quantum mechanics the notion of [x, p] = iħ made physical sense. Then of course came spinors and then quaterionons and Clifford algebras. I think there is some underlying physics which is required to make sense of nonassociativity. The approach with G_2 I advocate here does not by itself impose nonassociativity onto physics, but is a way I think that S-matrix theory according to holography can be formulated.

Nonassociativity I think might emerge in the following way. The holographic principle or black hole complementarity exists on the basis that an asymptotic observer watches a string approach an event horizon in a way very differently from what an infalling observer records. The distant asymptotic observer records the string to time dilate and spread across the horizon. The transverse modes of the string slow down for a string with a tension T, and as a result appear to lengthen. Of course the infalling observer sees nothing like this as the string passes the event horizon. The holographic appearance of the string is worked on the tortoise coordinates

r* = r – 2m ln(|1 – 2m/r|)

which means the S-matrix theory (which is what string are) is on a domain of causal support appropriate for S-matrix theory. The infalling observer observes the string on a different causal domain. This means the two observers detect string physics on incommensurate bases of states.

The S-matrix theory is really a system of braids, knot theory, or Yang-Baxter equations. If there are two different causal domains with S-matrix theory over incommensurate states, this means there are two braid systems which are not related to each other by the Reidmeister operations of a braid group. So there is some sort of map m:g - ->g’, between two quantum groups (braid groups), which is such that g’g^{-1} is not a unit. So the map requires there to be an additional element A such that g’A(e)g^{-1} = 1, and the information preserving aspects of quantum theory are preserved. The symbol A(e) acts on an associator ~ g(eg^{-1}) – (ge)g^{-1} to give a unit. The braid g - - g ( or elements a - - b for a & b in g) is extended to an associator g - - g” - -g’, here g” = e, with a homotopy structure. This is a bit cryptic here, but the idea is that associative QM is a system which intertwines braids, or quantum groups.

The G_2 group that I am working with is a system of three forms on M^7, and this is a holonomy involved with brane wrappings and AdS/CFT. In this way I think that nonassociative structures might be shown to exist in quantum gravity.

Cheers LC

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joel rice wrote on Sep. 26, 2009 @ 19:23 GMT
Jens, i suppose one physical principle would be that 'construction is the association of building blocks', and that only certain associations make octonionic algebraic sense structurally, and ditto for building blocks.

If clifford is about the structure of space then a plausible guess is that octionion algebra is about the structure of matter. Perhaps there is something about Hydrogen, and fermion generations, that can not be said without Octonions - not necessarily dynamical ?

It is awfully curious that complex quaternions go off in a quantum direction with Pauli algebra, and in a geometrical direction with multivectors ... same algebra but different interpretations, and smells an awful lot like the puzzle of quantum gravity.

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Lawrence B. Crowell wrote on Sep. 27, 2009 @ 03:12 GMT
I think that Jens is saying, which I agree with, is that we can't just throw up some quantum-like operatprs amd start doing nonassociative calcuations. There must be some underlying physical principle nonassociativity reflects. Without that we really don't know for certain what we are doing.

Cheers LC

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joel rice wrote on Sep. 28, 2009 @ 00:37 GMT
LC I think a bunch of principles need to be examined because the SM relies on Dirac algebra. Last i looked octonions have 480 multiplication tables, so it is not clear what can be multiplied by what, in general, never mind operators and nonassociative calculations. I wish Graves had written 'Elementary Octonion Arithmetic' - and how it differs from ordinary and Clifford Arithmetic.

It might not be so mysterious - as to what physical principle nonassociativity reflects. The antisymmetric rules require paying attention to permutation and association.It might be right under our noses. Anything so fundamental has to be ubiquitous. Back in 11th grade (1965) a guy gave a talk saying that octonions ought to be about the structure of Hydrogen. Unfortunately, knowing only about complex numbers, it went in one ear and out the other. Maybe he was onto something. But it seems upside down from the 'Quaternionic Spin' approach.

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Lawrence B. Crowell wrote on Sep. 28, 2009 @ 02:39 GMT
You are right there are 480 nultiplication tables. I was thinking there were 168 of them, but that comes from the Fano plane. The projective Fano plane PSL(2,7) has 168 elements, and SL(2,7) = Z_2xPSL(2,7) has 336 elements. The 480 multiplications are then determined by Z^7_2x(Z^7/Z_2)xPSL(2,7).

The physics invovles some sort of underlying structure to quantum mechanics which we currently don't understand. Take a look at Grgin's paper in the essay contest. There quantions are discussed which have a difference between complex conjugate norm and distance. I have a rather large post on his blog site on what this might mean with respect to quantum gravity.

Cheers LC

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joel rice wrote on Sep. 29, 2009 @ 15:24 GMT
Having a look at Florin's paper on quantions, and the Fernandez-Rodriquez on

"Gravitation as a plastic distortion of Lorentz Vacuum". A lot on the plate.

Perhaps all these multiplication tables can be turned to advantage to define a collection of particle-oscillators. Since one wants to get Dirac algebra, a cheap way is to take the direct product of quaternion subalgebras, for a pairwise relation. But the object of interest is Hydrogen e(uud) and also

v(udd). Dirac did not deal with that. Is Hydrogen an algebraically natural object in its own right, apart from an explanation in terms of forces ?

Building Blocks seem almost out of place in the context of Mark Ronan's

considerations - 'Symmetry and the Monster' - but I dont think Hydrogen

is going to become a non-issue.

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Peter van Gaalen wrote on Sep. 29, 2009 @ 18:59 GMT
Hello Joel,

"If clifford is about the structure of space then a plausible guess is that octionion algebra is about the structure of matter. Perhaps there is something about Hydrogen, and fermion generations, that can not be said without Octonions - not necessarily dynamical ?"

Here are the two octonions responsible for the generalization of spacetime. (and they encompass general relativity.) i, j, k and L are imaginairy units. c = speed of light, G = gravitational constant. t = time, l = length, f = gm-flux, b = burst, E = energy, p = momentum, m = mass, s = string:

[equation]O_{f} = f + i \ c l_x + j \ c l_y + k \ c l_z + L \ \dfrac{G}{c^3} \ E + iL \ G s_{x} + jL \ G s_{y} + kL \ G s_{z} \\ \\

O_{t} = -c^2t - i \ \dfrac{b_x}{c} - j \ \dfrac{b_y}{c} - k \ \dfrac{b_z}{c} -L \ \dfrac{G}{c} \ m - iL \ \dfrac{G}{c^2} \ p_x - jL \ \dfrac{G}{c^2} \ p_y - kL \ \dfrac{G}{c^2} \ p_z[/equation]

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Anonymous wrote on Sep. 30, 2009 @ 05:29 GMT
Hello Joel,

In relation with the Dirac equation.

In the equations below: h = hbar.

$\Xi^2 = - \left( \dfrac{h \delta}{\delta t} \right)^2 - \left( \dfrac{h \delta}{\delta b_x} \right)^2 - \left( \dfrac{h \delta}{\delta b_y} \right)^2 - \left( \dfrac{h \delta}{\delta b_z} \right)^2$

$\Theta^2 = - \left( \dfrac{h \delta}{\delta f} \right)^2 - \left( \dfrac{h \delta}{\delta l_x} \right)^2 - \left( \dfrac{h \delta}{\delta l_y} \right)^2 - \left( \dfrac{h \delta}{\delta l_z} \right)^2$

And in the case of Dirac, we don't need the Dirac matrices but we can use the imaginairy elements of the octonions (h = hbar):

[equation]\Xi^2 = \left(

L\dfrac{h \delta}{\delta t} + iL\dfrac{h \delta}{\delta b_x} + jL\dfrac{h \delta}{\delta b_y} + kL\dfrac{h \delta}{\delta b_z} \right)^2[/equation]

$\Theta^2 = \left(-L\dfrac{h \delta}{\delta f} -iL\dfrac{h \delta}{\delta l_x} -jL\dfrac{h \delta}{\delta l_y} -kL\dfrac{h \delta}{\delta l_z} \right)^2$

Peter van Gaalen

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joel rice wrote on Sep. 30, 2009 @ 14:40 GMT
LC: first pass on quantions - I like minimalist philosophy, and am wary of putting R,C,H,O in a 2x2 matrix - preferring plain ol complex octonions, and

the automorphism groups contain SU(3) anyway - see Georgi Lie Algebras - and

does not need 10d spacetime, but appears quite happy with +--- and -+++. Perhaps I am being naive but expect that if Hydrogen is algebraically natural then the physics ought to flow naturally, without any extra dimensions and unwanted particles.

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Rick Lockyer wrote on Oct. 1, 2009 @ 02:31 GMT
Hi,

I am curious about your statement on using only one of the octonion representations. I presume you are talking about the 16 different ways to define octonion multiplication once you settle on the seven sets of triplets, coming down to an 2 up order choice defining the cyclic permutation rules in each of the seven.

This seems somewhat counter-intuitive, since there really is...

view entire post

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Anonymous wrote on Oct. 1, 2009 @ 05:41 GMT
Hi Rick,

I start with the metric I found (I leave away the c and G):

$t^{2} - l^{2} + b^{2} - f^{2} = E^{2} - p^{2} + s^{2} - m^{2}$

Written in the following form

$f^{2} - b^{2} - m^{2} - p^{2} = t^{2} - l^{2} - E^{2} - s^{2}$

this metric can be decomposed into two 'hyperbolic octonions' (i^2 = 1). (Minkoskwi used the hyperbolic quaternion in understanding spacetime)

Written in the following form

$f^{2} + l^{2} + E^{2} + s^{2} = t^{2} + b^{2} + m^{2} + p^{2}$

this metric can be decomposed into two normal octonions (i^2 = -1)

Peter van Gaalen

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Jens Koeplinger wrote on Oct. 1, 2009 @ 19:55 GMT
Hmm - Peter, maybe I'll need to read your contribution in the current essay contest more closely; but on first sight, if you take Alexander MacFarlane's hyperbolic quaternions, which are already nonassociative and their 2-form isn't multiplicative, then it is not clear to me how this could be grown into a 'hyperbolic octonion' of the kind you're writing about. What would you want to preserve in the dimensional doubling from 4 to 8? Similarly, if you take a direct sum of two octonions (if orthogonal), I'm not sure how you could recover a multiplicative 2-form ... Well, coming from the other side, if you would not require the sum of squares in your post/paper to be multiplicative, I would ask what quantities are preserved between equivalent frames of reference, to warrant relativity.

Thanks, Jens

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Peter van Gaalen wrote on Oct. 1, 2009 @ 21:27 GMT
Hi Jens.

Maybe I use the wrong name. With hyperbolic quaternion I don't mean the MacFarlane quaternion. What I mean with hyperbolic quaternion is reversing all arrows on the imaginairy quaternion sphere. With hyperbolic octonion I mean the phano plane with all arrows reversed and aditional i^2 = 1.

A reason why I came up with the idea of more dimensions: If we take m^2 + p^2 = E^2 and we translate m^2 + p^2 as the product of the quaternion and conjugate then the energy would be the norm. But that makes no sense. Energy is just another quantity that differs from momentum like momentum differs from mass. m^2 + p^2 = E^2 + s^2 makes more sense. (but you have to understand both the difference between quantities and proportional quantities and the concept of periodicity.) In case of t^2 - l^2 = S^2 the only thing about the invariant of spacetime is that it's invariant. It doesn't say how many quantities it's composed of. That's why it's more illuminating to write it like: t^2 - l^2 = f^2 - b^2.

Peter

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Jens Koeplinger wrote on Oct. 2, 2009 @ 00:26 GMT
Unusual ... and interesting! You're right, I was indeed eluding to some concept of conjugation, norm, or a directionless magnitude similar to proper time or mass in its classical understanding.

Two remarks: Similar to what Rick writes above, if you reverse all arrows on the quaternion sphere, or in the Fano plane, you're not leaving the algebra; you're just creating a different quaternion or octonion multiplication table. Call it left-/right-handed, or of different chirality - your choice. And second, from your description of the quaternion construction, it sounds you're indeed describing MacFarlane's hyperbolic quaternion.

The invariance condition you're proposing, for equivalent frames of reference, then is not given by a single magnitude, but an algebraic equality that may in general contain terms from every octonion basis element, which when multiplied become a sum of squares. Fun! 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.

I'll have a look at your essay submission. Thanks for writing!

Jens

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Rick Lockyer wrote on Oct. 2, 2009 @ 01:46 GMT
Hi Peter,

If you reverse all of the arrows in a Fano Plane representation of an octonion multiplication rule set, you get another octonion rule set of the opposite handedness, but still an octonion.

There are 8 "Right Octonion" rules and 8 "Left Octonion" rules for a given set of 7 permutation triplets. For every right(left) there is a left(right) representation that is the negation...

view entire post

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Lawrence B. Crowell wrote on Oct. 2, 2009 @ 03:12 GMT
It appears there has been discussions going on here I have missed. The issues of nonassociativity in physics is the real problem here. I think it has to do with a departure in the definition of an operator norm in the sense of the Born identity, and the distance as measured geometrically --- such as spacetime. Nonassociativity is a very subtle aspect of how quantum mechanics embeds the two notions of a distance into one structure.

Lawrence B. Crowell

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Peter van Gaalen wrote on Oct. 2, 2009 @ 08:32 GMT
Hi Jens,

Wiki: "The Proceedings of the Royal Society at Edinburgh published "Hyperbolic Quaternions" in 1900, a paper in which Macfarlane regains associativity for multiplication by reverting to complexified quaternions."

That's why I don't mean the MacFarlane quaternion. The MacFarlane quaternion regains associativity. The way this 'regaining' is done looks artificial to me. The same for the Pauli matrices. I am not interested in regaining associativity.

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joel rice wrote on Oct. 2, 2009 @ 16:11 GMT
jens, i thought the Quaternionic Spin paper gets 3 generations in section 6

But i am still perplexed at why one needs a 2x2 hermitian with RCHO entries

to get Minkowski signatures (with supersymmetry). It is not like supersymmetry

is an established fact that must be dealt with. It creates the impression that we ought to go with 10d spacetime and it is somehow inevitable, or too nice to be wrong. The reason I turned to Octonions was that Clifford algebra has +--- in one algebra and -+++ in a different algebra, but both are in Complex Octonions - it just pops out for free, and to stick with alternativity we do not want to generalize it. I have not looked at the algebra isomorphisms of the ten dimensional apparatus but fear a problem similar to Clif - if Dirac over R can be ++++, ----, +--- then how is Nature to decide in favor of Minkowski ?

You might get the same problem in 10d. If Complex Octionions rule then Nature does not have any choice - it is always Minkowski - in plain old 4d. We do not have to do anything to constrain it further. Of course, to have any serious physics content it needs more than just spacetime structure, and the SU(3) in G2 indicates particle content. There are complaints that exceptional groups do not make much geometrical sense, but why should they if its about particles ?

BTW - if Nature has no choice then one automatically concludes that space must be three dimensional and spacetime 4d Minkowski - and something has to be exercizing the SU(3) content, somehow. My fear is Platonism run amok.

Does it not tickle the imagination that if you complexify octonions you get Minkowski - without asking for it, and without any extra dimensions. It kinda makes one wonder about assumptions.

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joel rice wrote on Oct. 3, 2009 @ 21:25 GMT
LC one aspect of nonassociativity might go a step further than Heisenberg's commutator and involve associators and anti-associators, but it seems like an inscrutable problem. Another kind of question is - do we want to ask what is the mystery with nonCommutativity, or go with Hamilton explaining how to compute finite rotations in space, so the ab = -ba is a detail, and the star of the show is the quaternion being a relation of two vectors in space. How does one extend that to Octonions, or if quaternions are the even subalgebra of complex quaternions, and Octonions, then what distinguishes them in the different contexts. One expects a relationship of 3 objects. The only elementary objects that fit the bill are quarks, and one is then in despair for an intuitive classical model, unless one might get into color vision and cone cells in the retina. If distance is a relation of 2 points, i suspect that nonassociativity requires 3.

But there is something curious about the whole thing - physics manages to do quite nicely without countenancing Octonions. This begs for an explanation as to how they get away with that. People were aware of rotation long before Gauss and Hamilton formalized it, so perhaps physics is speaking octonionic prose already, without making a conscious point of it.

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Jens Koeplinger wrote on Oct. 3, 2009 @ 22:55 GMT
Joel - The post you just wrote (Oct. 3, 2009 @ 21:25 GMT) is a nice inspirational, I enjoyed following this stream-of-consciousness.

Hi Rick - it's good to hear from you. Of course I'm writing on a paper on nonassociativity (what else? hehe!) and will bring up the very questions from above. "Deadline" (I dread this word) is 21 October. Earlier this year, it was exciting to be exposed to experts in the field in Denver, including Prof. Dray, and I claim that I learned a lot - I am certainly thankful for all the *attempts* to teach me something! Fermion generations from quaternionic spinors are intriguing. In your work at octospace.com, what you call "algebraic invariance sieve" continues to be interesting to me, I'm still working on understanding the terminology that you've developed (in isolation, I understand that of course). There is a lot of good work going on with octonions, and I've got to stop rambling in online forums and better get back to producing results! :) Just kidding, it's all good fun.

Thanks, Jens

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Lawrence B. Crowell wrote on Oct. 4, 2009 @ 01:20 GMT
I think the role of octonions is in the way in which quantum mechanics and spacetime physics treats observables. Quantum mechanics gives a modulus square as the summation of components which determines a unity, or under a partial sum a probability. In general relativity the distance is the invariant interval. This too is a summation of terms, though finite as opposed to a possible infinite sum in QM, which gives the invariant of GR. The departure of course comes from the fact that in GR the interval can be zero, so there are null intervals. This is the projective geometry of the theory. In QM there is a projective Hilbert space as well, with a Fubini-Study metric or fibration What if these two measures are related to each other by nonassociativity?

Noncommutative systems are braid groups, or quantum groups. Yet we know that for quantum fields in curved spacetime there exist unitary inequivalence. The existence of an event horizon introduces two inequivalent ways in which fields can be expanded. For black holes the occurrence of an horizon is extended to infinity in tortoise coordinates. This is the basis for the S-matrix to an external observer. This observer will witness a string, an S-matrix object evolve in certain ways. For the observer who comoves with the infalling string will witness something very different. In these cases there are two ineqivalent braid groups G and G', so if we take g and g' in these respectively we find that there does not exist an inverse element so that g^{-1)g' = 1. The two braid groups in not mutually unital. So we introduce an element h so that we can associate the element of G and G' with the element h so that (gh)g' - g(hg') = [g, h, g']. This is the higher level term or associahedron on operators. The associator is then a map between different braids, or basis elements for the S-matrix.

Lawrence B. Crowell

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joel rice wrote on Oct. 5, 2009 @ 15:31 GMT
If GR and QFT are effective field theories then perhaps Penrose is right that we can expect foundational problems with both. Does a Hydrogen atom actually have a stress-energy that gravity can couple to ?

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Lawrence B. Crowell wrote on Oct. 5, 2009 @ 22:10 GMT
I am not sure how the hydrogen atom fits into this. The quantum Kepler problem for the H-atom has a group theoretic SO(4) structure. This is similar to the Lorentz group which is SO(3,1). However, this is hyperbolic, and that results in a number of departures.

LC

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joel rice wrote on Oct. 7, 2009 @ 13:41 GMT
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 ).

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joel rice wrote on Oct. 10, 2009 @ 13:32 GMT
If the structure of spacetime is available from inspection then is it reasonable to expect quanta to be similarly available from inspection ?

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Rick Lockyer wrote on Oct. 10, 2009 @ 19:06 GMT
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...

view entire post

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Jens Koeplinger wrote on Oct. 10, 2009 @ 22:19 GMT
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

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Peter van Gaalen wrote on Oct. 12, 2009 @ 06:26 GMT
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

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joel rice wrote on Oct. 12, 2009 @ 18:17 GMT
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 ?

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Lawrence B Crowell wrote on Oct. 13, 2009 @ 00:05 GMT
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

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Peter van Gaalen wrote on Oct. 13, 2009 @ 05:18 GMT
Hi Lawrence,

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

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Lawrence B. Crowell wrote on Oct. 13, 2009 @ 12:06 GMT
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 LB

attachments: oct.pdf

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Jens Koeplinger wrote on Oct. 13, 2009 @ 13:10 GMT
... 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?

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Rick Lockyer wrote on Oct. 13, 2009 @ 14:03 GMT
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

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Jens Koeplinger wrote on Oct. 13, 2009 @ 15:35 GMT
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.

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Lawrence B. Crowell wrote on Oct. 14, 2009 @ 00:11 GMT
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

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Rick Lockyer wrote on Oct. 17, 2009 @ 04:59 GMT
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.

RL

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Jens Koeplinger wrote on Oct. 17, 2009 @ 10:02 GMT
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

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Peter van Gaalen wrote on Oct. 31, 2009 @ 18:59 GMT
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:
$m^2 = E^2 - p^2$

Below I want to settle my case for:
$m^2 - s^2 = E^2 - p^2$

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

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

Velocity v is a vector quantity.

If we combine the above equations we get:

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

In my essay I wrote the above more correct as:
$m^2 - s_x^2c^2 - s_y^2c^2 - s_z^2c^2 = \dfrac{E^2}{c^4} - \dfrac{p_x^2}{c^2}- \dfrac{p_y^2}{c^2} - \dfrac{p_z^2}{c^2}$

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

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joel rice wrote on Nov. 14, 2009 @ 17:05 GMT
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.

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joel rice wrote on Dec. 3, 2009 @ 03:34 GMT
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.

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Rick Lockyer wrote on Dec. 4, 2009 @ 07:03 GMT
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.

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joel rice wrote on Dec. 4, 2009 @ 18:40 GMT
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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joel rice wrote on Dec. 16, 2009 @ 12:20 GMT
Has anyone noticed that Feynman did for complex arithmetic what Hamilton did for quaternion arithmetic. Complex arithmetic is more difficult to make sense of because it makes no spatial sense. But both Feynman and Hamilton suffer from an overloading problem. If you use the quaternion division algebra then you are stuck with polar and axial conventions or rules, which should be made explicit by using complex quaternions. Complex arithmetic needs complex octonions to make explicit what is crammed into the imaginary unit, which can be expressed as

any permutation or association of o(a(bc)) . So in his "QED strange theory" he shows how to generalize the Principle of Least Time by analyzing a mirror by considering possible paths and doing complex arithmetic - now connected to the best tested theory in Physics. It should extend to complex octonions, which should have direct physical meaning.

Maybe physicists have been making Octonionic Sense all along without explicitly noticing it, and that octonions are so ordinary that we just take it for granted.

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joel rice wrote on Dec. 20, 2009 @ 18:06 GMT
Given that the 'unreasonable effectiveness of math in physics' seems

to depend on the good behavior of algebra, somehow, is it possible to

decide whether Alternativity is a criterion of good behavior ? Would

it imply, say, an inability to get Classical Mechanics from Quantum Mechanics ?

Or would a lack of alternativity not be a problem for the system as a whole ?

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Bashir Yusuf wrote on Dec. 13, 2010 @ 23:38 GMT
The core Idea we postulate it is that the nature has same fundamentals. In this scientific article we

will explore the broad area in physical science in different aspect and compare to existing known

integrates and interprets to a better Unified theory.

Gravity is the basic interaction and the Photon is the ultimate elementary particle that every

thing is made of. Sphere is dominating shape of the Nature. Multi-Dimensions are important

issue for sphere Geometry. Dark matter which has two faces is also another face of the

universe’s matter.

We focus particle theory in both astrophysical and subatomic particles including WIMPs MACHOs,

Quarks, and Leptons. The Natures Elementary charge’s characteristic is significant in the charged

subatomic particles , such as Proton, Electrons, while it is trivial in the other Neutral particles, such

as Neutron, Neutrino, and Neutron star, this phenomena is seemingly based on quantum of what

may called ultimate elementary particles. It is about ODD and EVEN numbers, DISORDERED and

ORDERED systems. Our conclusion of particle system hierarchy is that there are two main

categories due to quantity of Photons

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Bashir Yusuf wrote on Dec. 14, 2010 @ 01:45 GMT
dimensions and the shape seems to be a key of understanding of the natures fundamentals

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Steve wrote on Feb. 26, 2015 @ 17:52 GMT
2^0 = Q (Real), 2^1 = C(Complex)I, 2^2 = H(quaternions)IJK, 2^3 = O(octonions )

Is there a next level?

2^4 = Hextonian? 16 dimensions? ijklmnopqrstuvw

15-Point Projective Space?

http://demonstrations.wolfram.com/15PointProjectiveSpa
ce/

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