Dear Rick Lockyer,
Theta functions also describe tetrahedral-branes of eigen-rotational strings,, in that theory of elliptic function is applicable for the conformal mapping of eigen-rotational phase-transform, with holomorphic functions.
With best wishes,
Jayakar
Hi Rick,
I always enjoy reading your well-crafted and cogent arguments. I appreciate the beauty of reducing 480 multiplication tables to a manageable 64 and its identity with the 8 X 8 Hadamard matrix.
Why algebraic invariance over analytical covariance, however? Comparing your law of invariance with Lamport's Buridan's principle ("A discrete decision involving a continuous range of values cannot be made in a bounded length of time") I find that the choice of left and right, physically, is compelled by a continuous function, not an algebraic multiplicaiton rule. Nevertheless, I grok the utility of an axial-polar rotation relation between electric and magnetic fields -- as operations that result in a union of open and closed results in a 3-d coordinate system. (If you are interested, I recently posted a draft paper on my essay site that in fig. 6 pictorially shows a topological interpretation of the phenomenon, a closed external manifold connected to an open internal plane, with all external points mapped to all internal points.)
I think the models that fall out of your, Joy Christian's, Michael Goodband's and my research may converge in a deep sense. However:
"There is an assumption that it is impossible to define analycity within Octonion Algebra." Of course, you know that I am one of those who assume so. And I do agree with you that the tensor calculus is inadequate to the task of a closed logical judgment on wave propagation and electrodynamics in a 4-d continuum. The topological solution still most appeals to me, however -- not as a preference, but precisely because it eliminates the necessity for a preference; everything neatly follows from a free choice of topological initial condition.
Because we are working in the same 8-dimensional space, though, one can't help thinking that our 4-dimensional measure results all originate from a common source. All these mathematical methods may turn out to be dual.
Thanks for a great, forward-thinking essay, Rick! Best wishes in the contest.
Tom
Rick
"We must look for the connection between... and physical reality"
Indeed, any representational device must correspond with physical reality. So the question is, what underpins this mathematical system, and does that have proven correspondence with physical reality?
To put that question in context, I would suggest that physical reality does not have three dimensions, this is just the conceptual minimum. And the number of possible dimensions in physical reality is half (because dimension relates to a direction, either way) the number of possible directions that the smallest elementary particle could travel from a given spatial point. Dimension/size being an expression of 'physical presence' which can be conceptualised in terms of spatial footprint.
Paul
Hi Tom,
I do not at all put "algebraic invariance over analytic covariance". If you realize algebra and analysis are interlocking components, you see both from a more fundamental perspective. Algebraic invariance as I have defined it is a natural and simple principle that matches observation. All currents, forces, work, energy, energy flux, stresses and strains described in an Octonion framework are algebraic invariants. They are not simply such, they are the full complement of algebraic invariants available, complete. It is really difficult for me to think this is not a very loud statement the concept is a fundamental truth.
Without the application of a suitable analysis process, we have no connection to physical reality. Algebraic invariance can be understood without this connection as pure algebra, which I feel is important since it can be understood without the added complexity of physical reality and our current uncertain mathematical cover of it. After all, we are all hopeful we can improve the math side, whether we believe Octonion Algebra is they way or not. We are ahead in the game if we can separate out components, fully understand them, and then be able to better apply them to the greater whole.
The Ensemble Derivative is the interlocking of analytic and algebraic concepts. It works for the banal transformation of rectilinear coordinates to spherical-polar curvilinear coordinates in a Quaternion setting to the more interesting Lorentz covariance of Electrodynamics in the Octonion setting. Realizing both halves of the Electrodynamic field components transforming in an identical fashion is non-trivial. Algebraic invariance demands the algebraic basis element products for the transformed Electrodynamic field components exactly match those they rotate into. It is all there. So besides the fact that the Ensemble Derivative works, just what is your issue with it?
Thanks for your time and consideration on this Tom, and good luck in the competition. I read yours when it came out, but had no additional comments beyond interchanges we have already had. I still can't extrapolate as well as you can. I imagine I am missing things. I will try harder.
Rick
Well, now that you put it that way ... :-)
I could be persuaded. No matter -- I think we're on different pages of the same book with the same ending. Thanks again, Rick.
Tom
Hello again, Rick;
I just left a comment on Lawrence's page that bears mentioning here. He said that 'octonions are really a system of quaternions' (7 of them) which relates to a statement I made in the paper "In Defense of Octonions" with Ray Munroe. I wrote to Lawrence that while octonions could be represented that way, they do have to be resolved in an orderly fashion, and it's not the same as saying O is really H x 7.
In a paper I'm working on now; I suggest that working with octonion algebra is similar to assembling a watch. "Every layer or sub-assembly must mesh correctly, and then the layers must fit together in the correct relationship, for the watch to function. The same metaphor aptly describes what is required to do multiplication and division with octonions, as you must perform seven ordered groups of three operations in sequence."
Is this an apt characterization? My guess is that Lawrence's approach would treat the component quaternions in the same way that Physics folks normally treat grad, div, and curl - as independent or fundamental quantities, where in reality (or as you demonstrate) they are structured components of the quaternions. I suggested his statement might be made true if octonions are treated as an ordered or nested system of seven quaternions. Is this essentially correct? Do we also need an extra scalar value, for the Real component?
Enlighten us.
all the best,
Jonathan
Jonathan, Tom, Edwin,Brendan and friends, you are bad strategists meriting simply to ba analyzed by the laws. Sort your members Mr Tegmark and MR Aguire.
They have not theior place on this platfrom.They decrease the velocity of evolution of fQXi and its credibility must be universal.Sort these pseudos.
They have simply a strategy for the maney, they are just frusterated vanituious and envious. They must be sorted. They delete in correlation of their strategy of discriminations.In fact they fear that I arrive at New York, so they try with the discriminations. I have faith me, them no !Don't compare a thing which cannot be compared.
Mr Tegamrk and Mr Aguire, don't be troubled by their strategy and their words. These people have simply a heart without faith and reason. They are not scientists, but business man. And you ibm, forget also these persons !with the soa and its superimposings. Be rational and universal.
Steve
Jonathan,
The Octonions are at the end of the R to C to H to O chain, and their discovery order as well as the doubling process both follow simple to more complex direction. I prefer to think of the Octonions as the most fundamental since there is no higher dimension normed division algebra, and each of the more simpler division algebras are all O subalgebras, meaning their basis set is a subset of the O parent, and the subset forms a proper algebra all by itself. This means it abides by the three rules of algebraic element addition, multiplication by a scalar, and algebraic element multiplication closed for the subset of basis elements.
Geoffrey Dixon mentioned in his latest book his lack of enthusiasm over the doubling process, and I must say I fully agree with his position. This process has been the genesis of the "made from" mentality. As he mentions, you can double through the division algebras, but you can also double O to the Sedenions, which are not a division algebra, so the doubling does not conserve this very important characteristic. I further contend it has also led to a one O algebra mentality because many missed the fact doubling a commutative algebra (R to C or C to H) is not the same as doubling a non-commutative algebra as with H to O. The subalgebra perspective works end to end since R can have no subalgebra since it has only a single basis element.
Sedenion algebra defines 35 perfectly valid H subalgebras, and 15 Octonion-like subalgebras of which only 8 can be made into normed composition algebras. The latter is why Sedenions can't be made into a division algebra.
C has but one choice for definition of basis element multiplication, so no variability impact on the definition of H. But H does have 2 choices analogous to 3D right handed and left handed vector products. The subalgebra connection for H from O must accommodate these 2 choices, which is why there are 16 choices for proper O Algebra. The 16 choices, that is the full variability in O definition, is determined by handedness choices for the 7 H subalgebras that leave O a normed composition algebra. If as you question, O was simply Hx7, there would be 128 valid O for all possible H choices, but there are not.
The proper H selections are less important from the H perspective than they are from the O perspective, since things only get interesting when we make O algebraic element products. Then one "H" in a way multiples another "H", and the ins and outs of this sets the algebraic variance and invariance characteristics of fundamental importance.
Kind of a long answer, but hopefully you will find it satisfactory.
Rick
Weak reasoning ! :)
Edwin, Lawrence, Jonathan,Joy,Tom,Rick and friends.
It is weak, even the strategies are weak, it is easy to find the holes. 0 really.And me alone, integre, transparent, without tool,just with rational sciences. Between us ,it is ironical no? In all case me I laugh. Because even like that I have teached you so many things ahahah incredible no?
And they insist furthermore with the compactification and the geomatrical algebras. If you you understand the 0, the 1, the infinity, the numbers, complexs, naturals, reals, R C O H or this or that.If you you understand the finite groups and the real infinity. Me really I am the queen of England you know. If you pondered intresting between us, ok, but no, even like that you insist on your stupidities for I don't know me.Probably a problem of vanity or a kind of play just due to your unconsciousness.In fact,you are not really skillings.Because If I learn here on fqxi , it is not with your team you know.
In fact you are not general, here is the probelm.And even your details are not good.So you imagine my pity , you can delete, betrween us, you understand, isn't it ? your hate increases, logic, your strategies are just a simple bad play of a kind of super team , but in fact it is a team who makes pity.really. You are even ready for all in fact. You are bad persons simply. When I see all this story since the begining. It is incredible in fact with your false universalism and false patriotism, I am a better american that you furthermore because me I am a real universalist loving his fellowman. A real christian. And you have made all this just because you are vanitous , envious and full of hate against people who critics universally. It is a sad team with bad tools, and bad strategies. You are not universal. Fqxi merits more than this kind of comportment. The integrity, universal is essential.
How can you have this kind of comportment in fact.What is your heart to make that. I pray for you, I have pity. All my pc is checked.All my platforms where I discuss.Linkedin,xing,fqxi also,facebook,....it is really bizare.
I forgive always. It is sad this story in fact. How people can make that ? the world is sick, if already the imrpotant systems of foundamental sciences are bizare.Where are we going? It is bizare simply.
Hi Rick
I think that Tom is definitely right and our work is related at a deep level. My considerations have just been at the level of the homotopoy groups of spheres, initially for a map from a particle gauge space of S7 to a closed spatial universe of S3 - the underlying structure in question is obviously that of the octonions and quaternions.
Consider a non-trivial map S7 -> S3 and out falls an electroweak vacuum with a Weinberg angle given geometrically by sin2 = 5/21, which is smack in the experimental range (derivation is in the technical notes of my essay and my paper. I feel like the only one in stunned surprise at this result ... but this isn't the end of it!
The map S7 -> S3 is from a S3 subspace of the S4 base-space of S7 to the other S3 space, and this picking out of the S3 from S7 gives an effective sphere decomposition that is locally S3*S3*S1. In the context of field theory, this amounts to a symmetry breaking from the space S7 to a space containing S1, which by homotopy group relations means that there *must* arise topological monopoles in the 3-space of the spatial universe. The spectrum of these topological monopoles will be given by the number of ways of picking out the S4 base-space from the S7, which is given by the homotopy group for the map S7 -> S4 and gives a 3 by 4 table of possibilities. The charge eigenvalues for this 3 by 4 table is given by identifying the spheres S3*S3*S1 with group spaces, which for the map S7 -> S3 must locally be SO(3)*SU(2)*U(1) and this gives a 3 by 4 spectrum of topological monopoles which the same charge eigenvalues as the particles - specifically including 1/3 electric charges.
Topological monopoles in field theories have generally been thought of as only bearing magnetic charges, but the underlying structure says not - there are two distinct topological maps for these monopoles, one would be expected to have magnetic charges, but the other would give electric charges. So one spectrum of these 12 topological monopoles would bear electric charges, and so look just like the 12 fundamental particles. Am I the only one in stunned surprise?
These homotopy group results obviously come from the underlying quaternion and octonion spaces in which the spheres S3 and S7 reside. I am amazed that the simple consideration of a map from one to the other yields the correct electroweak vacuum and the correct spectrum of particle charges and NO more. 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.
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.
Michael
"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