"The context for this map S7 -> S3 that I consider is GR extended to 11-dimensions as it seems the scenario that makes most physical sense."
Makes sense to me too, Michael. Because S^3 has infinitely many copies in the Hopf Fibration, 8 dimensions is a sufficient formal framework to describe all physical phenomena, which after all are manifest and recorded on the S^2 manifold. Can this business be ultimately as simple as 8 3? Wow.
Tom
weak reasoning still and always the same repetitions.are you blocked in these 7 to 8 to 11 and the M Theory.
Mr Witten, forget these comics please. You are better than this strategy.
Hi Michael,
You asked me the question
"Can you explain how the algebraic structure of the octonions and quaternions is ultimately responsible for these homotopy group results? It obviously is, but I currently don't understand how."
Not sure how to answer this not knowing precisely what you mean by "ultimately responsible". The "obviously is" part is clear.
I hope you read my essay. If you did you will find my ideas on the fundamental structure of Octonion Algebra. O has a better automorphism group than G2 I call Iso() which is 8 dimensional and represented by compositions of columns in an 8x8 Hadamard matrix. This group structure appears in the signs of the products in any O multiplication table, in the 16 chiral choices for H subalgebras defining the full variability in O definition, and in the sieve process on the result space that permits separation of algebraic invariant product terms representing physical observables from algebraic variants that form 14 homogeneous equations of algebraic constraint. I suggest you also download the Hadamard document using the link in the References section of my essay for more detail. I asked you to take a look at this group structure in a post on your essay blog. You never answered. I had hoped for some enlightenment from your superior understanding you aptly demonstrated within your essay.
I think I have done a very good and undeniably accurate cover of O structure and just how H fits in that you will not find elsewhere. Maybe it will help you through understanding this responsible angle. But there is much more to my story than the algebraic and group structure manifested by O. I show this structure is fully compliant with our 4D understanding of relativistic Electrodynamics, where its O cover is only a subset of the presentation, and the remainder is fully provided by O, including Gravitation all without a split signature metric or intrinsic curvature. My personal opinion is everything else is there also, like QM without wave functions or "dice".
So I dance on the twin "third rails" of physical religious orthodoxy, GR and QT. The math is there whether or not the "faithful" wish to acknowledge it. I wish more people would just "get over it", and take an objective look at what I have presented.
I hope to continue a dialog with you Michael,
Rick
Hi Rick
I have experience with topological defects in quantum field theory, which is why my analysis is in terms of the homotopy groups of spheres and the small symmetry groups. The relation between the quaternions and spin, together with the sphere S3 in the quaternions being the group space of SU(2) means that they are within my physics experience. But the octonions are a different matter. The S7 in the octonions isn't a symmetry group space, but that of various symmetry group quotients - which isn't of much help to my physics intuition. My considerations involving S7 were just in terms of a separation into S3 fibre and S4 base-space because I could then just apply the homotopy groups and geometry. The octonion algebra itself I don't know in enough detail, which is why I was asking you.
The structure of the octonion algebra must provide more detail about the nature of the maps with the homotopy groups Pi7(S3) and Pi7(S4) that I have used. For example, there are 2 non-trivial maps S7 -> S3 (Pi7(S3) = Pi4(S3) = Z2) which have spatial chiralities Left and Right. This chirality is critical to the identification of the non-trivial map as being the chiral electroweak vacuum and it breaks up the S7, but without the symmetry principles with which I am familiar I am struggling to the see the structure.
Note that my S7 octonion part just covers the gauge symmetries of particles, the spatial part of GR has to be in addition - hence the spatial S3. A solely "octonion relativity" can't include both GR and the full gauge symmetries - not enough degrees of freedom.
Michael
Mr Witten,
these persons are not good for your credibility.Your works are relevant, their strategies no. Their methods also are not relevant and furthermore bad.
You cannot work with these kind of persons. It is not good for your works.Forget these businessmen, these false scientists.
Really, all will be easier.
Mr Tegmark, Mr Aguire,Mr Witten ...please sort your teams.
Hello Rick,
I agree the octonions are more an absolute endpoint or an ultimate starting point, rather than some obscure way station in a process of infinite doubling. It is far easier to make sense of things by asserting that octonions are the fundamental starting place from which the H, C, and R subalgebras are special cases, or steps in a sequential limiting of degrees of freedom.
Real numbers are the most common, and a lot of folks feel that the whole concept of number comes out of the natural or counting number system. First, of course; we must distinguish none or zero from one. Then there is the concept of many. So to imagine that the reals are a subset of an 8-dimensional number system is to some people rather far-fetched.
But if the topological anatomy of the universe is something like what's being discussed by Joy Christian and Michael Goodband, then Octonions are far more fundamental than the so-called natural numbers, and they are indeed the algebra of everything.
Regards,
Jonathan
Michael,
I think both of us are looking outwards from our own perspective to the other's. I personally think I will benefit from thinking from an outside perspectives back into O algebraic structure, which is why I very much liked Joy Christian's work as well as your own. Perhaps you could benefit from looking at things from O rather than at O from your current perspective.
On your comment that "octonion relativity" has insufficient degrees of freedom to cover both GR and full gauge symmetries, perhaps you reach this conclusion because when you think of GR, you think of 20th century relativity, and when you think of the gauge symmetries it is in terms of the Standard Model. It just may be that 20th century relativity is a byproduct of insufficient degrees of freedom with its 4D framework, and the Standard Model actually does have too many knobs to twist. We all must be guided by our intuition, and mine is that O will do just fine.
Rick
Jonathan,
Happy to hear you agree with my premise. Sadly there are physical religion bigots that would not give the essay a look, much to their loss.
Algebra, analysis, topology and groups are interlocking parts. The most fundamental is the algebra, for it sets the tone for the remainder. For O, nobody has put it all together yet. My work is the easy part in many respects, for it is clear cut. Right is right and wrong can be demonstrated. I put a high value on the work of both Joy and Michael.
Rick
Rick,
The octonions are amazing. They are perhaps the Holy Grail of Mathematical Physics. But you have to admit they are ball busters dude. People who like algebra because they have memorized the rules of simplification hate the Octonions, because they thwart their best efforts at every turn. They are the epitome of difficulty, in that sense, because they are the most demanding of all the well-behaved algebras, in terms of keeping track of the order and/or syntax of mathematical statements and procedures. As I say in the above comment; it's like putting together a watch, to do proper algebra with octonion variables - at each stage, or with each cycle or operation. For some people, that takes the fun out of Math.
You and I are different, that way. The very thing that makes octonions demanding - their sequential or procedural ordering property - is what makes them fascinating to me. But when this is respected; algebraic invariance is preserved, and equivalently so are the physical symmetries such reversibility principles represent. In my mind; this makes octonions a kind of minimal starting place, as an octonion background space is what must be assumed if there are no added evolutive or limiting conditions. That is; when considering the question of what the minimum conditions are, to generate the universe of form we observe, the Octonions are likely as simple as you can get.
All the Best,
Jonathan
Rick,
I think you're right. Different disciplines in science have their own language and ways of thinking, such that there can be difficulties crossing between them. I have added a post (Sep 10) under my essay detailing how my work shows agreement with that of Joy Christian's, which would suggest that the particle gauge space is going to turn out to be S7. In which case, I may just be one of many with a particle physics background wanting the Octonion alegrbaic structure described in the language and perspective with which we are familiar. Joy's encountered 'difficulties' trying to explain the simpler topological structure of S3; trying to do the same for S7 is likely to be a lot more difficult. A translation exercise between the two perspectives may prove to be useful in the long term as ... well, it seems S7 is it. O does just fine for the paticle gauge space, physical space is extra.
As for the degrees of freedom count, that comes conserved charges:
1) 3 colour charges (red, green, blue): dependent upon the colour symmetry group either being SU(3) as it currently is, or the other possibility of SO(3) which is what I conclude it is. 3 colour charges in either case. Always have to find an answer to why are there 3 objects in a proton.
2) 3 isospin charges: dependent upon the W, W-, Z being gauge bosons of the weak force (that one looks settled)
3) 1 electric charge
4a) 3 spins for particles that are there own anti-particle
4b) 2 spins for particle/anti-particle pairs
Anyway you do it, that adds to 11 (remembering to count particle/anti-particle as a 'charge'). Since conserved charges are associated with continuous symmetries, that gives 11 dimensions in a pure extended Relativity. With fewer than 11 dimensions extra fields would have to be added. Having extra dimensions and extra fields could be viewed as being indecisive, and undermines the justifcation for extra dimensions in the first place.
Michael
I just wanted to second Jonathan on this one. The Octonions thwart efforts at simplification or principles to keep it together. But is this really the case? Especially for the decomposition into subspaces (S7->S3*S4->S3*S3*S1) of O that are likely to turn out to be relevant?
Michael
Hi Rick (and Michael),
You seem to be suggesting that there is a difference between S7 and O. Am I wrong?
In my view there is no fundamental difference between S7 and O.
S7 is simply a simply-connected set of all unit octonions, the algebra of which is O. This algebra gives powerful means to understand some aspects of the set, but so do the topological methods such as Hopf fibrations, Jones polynomials, and skein relations.
What am I missing?
Joy
Hi Joy,
I see O as making S7 possible, so in this regard of an algebra as compared to suggested topology, they are not the same, but I will hazard a guess this is not your point.
I REALLY need to find some time to study your book. Tough when you have to earn a living doing something else. On the subject of work, I had better get going.
Good to hear from you Joy!
Rick
Good to see you back here too, Rick!
Thanks for your reply.
By the way, I have already rated yours (as well as Michael's) essay (as a member). You can venture a guess what marks I have given. I may be accused of bias; but, as you know, I have been accused of worse.
Happy working,
Joy
Thanks Michael, for the point of agreement;
It really depends how the subspaces relate to the octonionic background or bulk. Hans Van Leunen's 'Hilbert book model' uses quaternions to link us with the microscale and quaternions again to link us with the cosmic scale - and with other systems in general - through probability amplitude distributions. He calls them QPADs. But if in your theory; we are living in a quaternionic topological space - S3, that is embedded in or contiguous with octonionic space - S7, what does that imply conceptually? And how do things fit together? The linkage is all important.
My guess is that the orderliness of the octonions actually comes to the rescue there, or for helping reconcile your theoretical ideas with Joy Christian's, in terms of helping to sort out the way things link together and the order in which things must evolve.
I've been working for some time on a universal protocol for measurement or determination, and it turns out to have strong tie-ins with the octonion algebra's procedural aspect, as it requires precisely seven stages. I keep returning to points made by Connes, in papers from 12 years ago, regarding how we define smooth, topological, and measurable spaces. He spells out some of the precursors or prerequisites of measurability. But that's just a starting place. However; I think the octonions suggest and endpoint to that process.
More later,
Jonathan
Please note, Anonymous;
John Baez wrote a wonderful article on Octonions with John Huerta, and it was published in Scientific American - one of the sponsors of this contest. It seems ill mannered of you to invoke Baez' crackpot index here, for ideas John Baez might actually support.
Regards,
Jonathan
Dear "Anonymous" blog troll?
Care to elaborate and identify yourself?
I think it is high time someone produces
(1) a "stuck-in-the-box" index for all the dogmatic idiots of the scientific world
and
(2) computes the so-called "crackpot indices" of Kepler, Newton, and Einstein.
Jonathan and Rick
Just to clarify. In my model the S3 of space is *not* a subspace of S7 but the two S3 and S7 form a 10 dimensional product space. This can either be viewed as a torus - if we pick the S1 fibre from the S3 - with an S7 cross-section, or a sphere - if we pick out the S2 base-space of S3 - which has a S7 surface cross-section. The absolutely critical element of my model is that there is a twist in the S7 cross-section in going around the S3 (like a higher dimensional mobius strip or twisted torus) - this is the electroweak vacuum and it picks out the sub-spheres of S7, but in a way that mixes S7 with the S3 spin space. In my model, this gives the reason for why the electroweak charges of the particles are chiral. This is also why my results with a 10 dimensional product space are *not* actually in disagreement with the S7 of Joy's correlation results. Spin (space S3) is correlated with isospin (space S3) and hypercharge (space S1) which are sub-spaces of S7, but colour charges are independent and not correlated with the other observables.
So my model says that viewing S7 as all of space - as Joy and Rick do - will work for spins and electroweak charges - which are correlated in particle physics - but that leaves the colour space unaccounted for. Joy's correlation results imply the *independent* colour space would be S3 (as S7 isn't a group space). The twist in my model of a physical gauge space S7 in going around the space of a closed S3 universe has the peculiar effect of switching S3 spaces in the *measurement* space - the S3 spin space is swicthed for the S3 colour space. The linkage of the subspaces is all important - as is the distinction between physical spaces and measurement spaces - and just how 'the orderliness of the octonions' actually does the rescuing is the piece of the puzzle I'm looking for.
Michael
Ahahahah and they insist furthermore.You are not real christians. Me yes.Don't try with things that you do not understand.
How can you converge if you have not understood the reall message.It seems not possible. In fact your goodband is just a team without real innovations. A goodband, yes of course, a goodband of businessmen yes. And you try with bad strategies.Like poor thinkers frustrated and jealous and envious and full of hate. My universal love will eat your hate. Dead or alive, I will continue. even a gun on my head, I will continue.
You know ; kill me , it is better you know and more quick for your strategy.
Aahahah I am crazy, I have god with me , Jesus Christ also is with me and Buddah. You want really to try with the faith. Let me laugh band of bad men.
Irritating that I forgive you all no?You would in fact that I discriminate people and country. Let me laugh, I love more than you the USA. the United States of the Sphere you know. No? don't worry you shall understand on the entropical arrow of times. I love USA !
A goodband, yes of course.a good band for monney yes.
Regards