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

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

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.

Hello Joel,

you asked:

"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:

[math]

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

[/math]

Hello Joel,

In relation with the Dirac equation.

In the equations below: h = hbar.

[math]

\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[/math]

[math]

\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[/math]

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):

[math]\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[/math]

[math]\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[/math]

Peter van Gaalen

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.

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 nothing to distinguish one from another when the 16 rules are separated into the two structurally different groups of 8 which I have called Left and Right Octonion Algebra. They are structurally different because you can't put a Left Algebra into a Right Fano Plane representation without changing directions on some of the arrows, yet all 8 of each can easily be put into their own kind a multitude of ways without arrow direction changes.

When I thought about the different octonion representations, my first knee-jerk reaction was anything physical that could be described by octonions would have to be invariant between ALL representations, something I later called The Law of Octonion Algebraic Invariance. The bold step of calling it a "law" came about after I noticed each and every physical property I described with octonion algebra yielded exactly the same result independent of which of the 16 different rule sets I used. Beyond noticing things after the fact, I have also successfully taken my Octonion work-force Action Function to a different but totally equivalent form in which each term contains an outside differentiation by forming all possible invariant forms with the anticipated product histories. No small feat if you look at the pages of equations. But what this happily led to was the form for the Octonion Laws of Conservation of Energy and Conservation of Momentum. If you want to know what the Octonion Poynting Vector looks like, check out my website. The full structure of algebra demands these equations be exactly as I have shown, no more no less, nothing inserted and everything provided by the algebra itself. Of course, Electrodynamics from potentials through fields through work-force to conservation equations is complete, but is only a subset of the representation.

So I question if you are on a valid path if you must use one representation. What say you?

As for mapping the octonion elements to our seemingly 4D physical reality, I prefer a doubling up for the 8 octonion elements. Take one of the vector basis elements, does not matter which. Pair it with the scalar element as a C subalgebra fully analytic. Find the three permutations the scalar paired element lives in, cyclically rotate the terms until the chosen one is central. Observe the outside elements. For a "Left Octonion" set of 7 triplet juxtapositions, another triplet's elements will be on the left side, and right side if you had a "Right Octonion". This is a big clue also on how to fit the algebra into a Fano Plane representation, the central element is central in the Fano Plane, and the left or right side triplet are the triangle midpoints. Back on point, take the right or left triplet that is one-to-one with chosen element as a closed set multiplication rule for physical xyz. The other 3 triplets define open set multiplication rules over xyz. These rules duplicate the closet set vector element multiplication rules for 3D axial vectors, and the three permutations duplicate the open set element multipication rules between 3D polar and axial vectors. No surprise EM fits nicely

Look forward to your perspectives, check out my stuff at

www.octospace.com

Sorry for not making it a link, did not check how to do it first and fear losing my input here finding out now.

Rick Lockyer

Hi Rick,

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

[math]t^{2} - l^{2} b^{2} - f^{2} = E^{2} - p^{2} s^{2} - m^{2}[/math]

Written in the following form

[math]f^{2} - b^{2} - m^{2} - p^{2} = t^{2} - l^{2} - E^{2} - s^{2}

[/math]

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

[math]f^{2} l^{2} E^{2} s^{2} = t^{2} b^{2} m^{2} p^{2}

[/math]

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

Peter van Gaalen

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

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

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

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 of all 7 permutations == switch order in the permutation == change arrow direction in the Fano Plane representation.

Joel and LC,

The statement that there are 480 "different" octonion multiplication tables has to be qualified by a definition of "different" and a blind eye towards what is actually also different but not coming up with the desired number of 480. Proponents of 480 will not dispute that the multiplication rules for unlike octonion basis elements are described by a set of 7 permutation triplet rules where each basis element shows up in 3 of the 7. Once you have a set of 7 permutations, your only degree of freedom for change is negation of individual permutation rules, i.e. (e1 e2 e3) -> (e3 e2 e1). This would permit a maximum of 2^7=128 possible multiplication tables for a given set of 7 permutations. But alas, only 16 of these are alternative composition algebras, a requirement to be called octonion. These are the 16 (8 left and 8 right) I have mentioned.

So to get past 16, you have to say there is something significant with a specific subset of aliasing on the enumeration of the set of 7 vector basis elements. There are after all, 7! ways to alias, a number way bigger than 480. So once you open the door for aliasing, there are actually 7!*16 different tables. The abstractionist often states certain differences are unimportant, yet rarely carries the burden of proof that the assumption is still correct into further extensions of their logic.

When I am asked how many octonion tables there are, I provisionally state 2, since all left are isomorphic algebras and all right are isomorphic algebras. By this I mean they are structurally identical, one can be had from another by a one-to-one and onto re-enumeration, which is NOT possible between left and right. There is no algebraic significance to how we order or re-order the basis elements, we do this to keep them separate in our minds. So I see no fundamental claim of difference out of aliasing.

However, I have exploited the differences between all 16 for a non-aliased representation, so there is certainly physical significance to these differences. These operations I call extra-algebraic since one can only formulate mathematical expressions born out of a single pre-established rule for multiplication. My Sieve Process does break this rule, but only to demonstrate algebraic variance and invariance to sort out just where each product term fits in.

Jens,

Looking forward to your paper on non-associativity. How is it coming? I need to get smarter on your application of octonions and the issues you see from this application, not to mention non-associativity in general.

As you know from our previous dialog, I do not fret much over octonion non-associativity, since it is a central feature of the algebra I love, and I do not try to get outside the confines of the algebra with applications where associativity is assumed or at least required. I see octonion product chains as having both associative and non-associative components. To me this is a good thing, for if we see things in reality that are both, we should have an algebra that likewise has both. The vector triple product is not unknown in physics, so one can't say all of physical reality is associative. The possible lack of associativity is one of nature's way of telling us that product order has significance. This product history is central to the determination of octonion algebraic variance and invariance, and as I have shown, physical reality seems to be built on algebraic invariant forms.

Drs Dray and Monague, I really would like to know your impressions about what I and others have discussed here. Could you be so kind as to drop a line or two?

Thanks,

RL

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

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.

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.

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.

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

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

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 ?

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