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

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

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

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

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

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?

is there a recent article one can look at ?

If it turns out that Nature frowns upon supersymmetry

do octonions remain relevant ?

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

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

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 ?

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

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

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

does this mean that Alternativity is not an issue for OxH and OxO ?

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?

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

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 [math]\int{L dx}[/math] 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

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

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.

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