Yes, the octonion algebra O is nonassociative. But for each division algebra (R,C,H,O) there are associated (and associative) algebras of actions of the algebra on itself. With R and C, because they are commutative and associative, this gives rise to nothing new. But H is noncommutative, so the algebra of all left multiplications of H on itself (HL elements do this: xH) is distinct from multiplications from the right (HR: Hx). Both HL and HR are isomorphic to H, but they are otherwise entirely distinct. And with respect to either the algebra H is a spinor space, and HL and HR are Clifford algebras (Cl(0,2)).

(My introduction to Clifford algebras was Ian Porteous's book Topological Geometry, which has a table of Clifford algebra isomorphisms involving R, C and H. This book also introduced me to the octonions.)

I am not the first person to point out that O can also be viewed as a spinor space (I think Conway and Sloane do so in Sphere Packings). The Clifford algebra in this case is OL, which consists of nested actions that look like this: x(y(...(zO)...)). It turns out you only need to nest to level 3, however. That is, OL consists of multiplicative actions of these forms: xO; x(yO); x(y(zO)). OL is trivially and necessarily associative, and it is isomorphic to the algebra of real 8x8 matrices. So, it it also isomorphic to the Clifford algebra, CL(0,6), which has an 8-dimensional spinor space, which in this context is O itself. And finally, a really cool consequence of nonassociativity is this: OL = OR (isomorphisc, but also the same algebra; this is different from the case of H, where we had isomorphism, but distinct algebras).

Anyway, I've got over 30 years of books and papers going into much deeper detail on all these things. The bottom line is this: the nonassociativity of O is a feature, not a bug, and an amazing feature at that, coming into play in just the perfect way.

O = spinor space; OL = Clifford algebra.

Hi Cristinel

Just downloaded your essay. Impressive. This contest will be long over before I could possibly dig into it deeply enough to say anything cogent.

You mentioned Furey's work using the Dixon algebra. I began working on that algebra some 40 years ago. I worked alone, and the work was far enough removed from mainstream thought that it was largely ignored. But it was, and is, right in its fundamentals.

Baez was the first to refer to RâŠ--CâŠ--HâŠ--O as the Dixon algebra, after noting that recent work exploiting this algebra largely ignored my decades of books and papers. Such is science, and there is no cure (as I point out in my book, A Fire in the Night).

Dear Geoffrey,

I greatly liked your essay!

I was struck in particular by the statement that Dirac even speculated that one day physics and mathematics will become one. I did not know Dirac had such a view, and I humbly mention that some considerations lead me to the same conclusion in my essay.

My thanks and regards,

Tejinder

Dear Geoffrey,

Thank you for the clear explanations, the properties you described are really great, in particular the dual action of quaternions (with which I am more familiar) and especially the double role of the octonion algebra. I definitely want to know more about your work and come back with more questions later. In the mean time, I hope this contest will give more visibility to your works and ideas.

Best wishes,

Cristi Stoica, Indra's net

I intend to work explicitely the quotient construction E8/SO(16). By saying E8/SO(16) = 128, as far as I know this means there is some vector space that is the 128-dim adjoint representation of --- well something. The thought has occurred to me it is U(8)xU(8), where each is 64 dimensions. The snag I can see with that is with Zamolodchikov representation of the golden quaternions, with magnitudes given by the golden mean, there would not likely be this even partition.

In fact I was going to work on this last May, but I had a death in the family. My great buddy and pal Umbriel, Umbra for sporadic group stuff and for the moon around neptune, was this big wonderful black pit bull that I adored. It might sound silly, but it took me a while to get over that.

I did a derivation quite some years ago on how Λ_{24} as derived by O^3. I used θ-functions. I think I have the derivation written in some notebook somewhere. The idea of a trilinear product is also something I have been kicking around. The Freudenthal diagonalization leads to three sets of eigenvalues. Behind this is the hyperdeterminant of Cayley and it seems to me there is a generalization of the 3-form that involves the associator. I have done a lot of work fairly recently on Morse theory with the Jordan algebra. It is work I did mostly about 2 years ago.

I have read on the higher sporadic groups, in particular Conway and SLoane Sphere Packing, lattices and groups. I have not though done any calculations on this. I figure if we can just get reasonable physics with O, E8, O^3 and J^3(O) then we might have something workable. Since these have automorphism properties on the FG monster this then will suggest a deeper layer of structure.

Duplicate of above

I intend to work explicitely the quotient construction E8/SO(16). By saying E8/SO(16) = 128, as far as I know this means there is some vector space that is the 128-dim adjoint representation of --- well something. The thought has occurred to me it is U(8)xU(8), where each is 64 dimensions. The snag I can see with that is with Zamolodchikov representation of the golden quaternions, with magnitude given by the golden mean, there would not likely be this even partition.

In fact I was going to work on this last May, but I had a death in the family. My great buddy and pal Umbriel, Umbra for sporadic group stuff and for the moon around neptune, was this big wonderful black pit bull that I adored. It might sound silly, but it took me a while to get over that.

I did a derivation quite some years ago on how Λ_{24} as derived by O^3. I used θ-functions. I think I have the derivation written in some notebook somewhere. The idea of a trilinear product is also something I have been kicking around. The Freudenthal diagonalization leads to three sets of eigenvalues. Behind this is the hyperdeterminant of Cayley and it seems to me there is a generalization of the 3-form that involves the associator. I have done a lot of work fairly recently on Morse theory with the Jordan algebra. It is work I did mostly about 2 years ago.

I have read on the higher sporadic groups, in particular Conway and SLoane Sphere Packing, lattices and groups. I have not though done any calculations on this. I figure if we can just get reasonable physics with O, E8, O^3 and J^3(O) then we might have something workable. Since these have automorphism properties on the FG monster this then will suggest a deeper layer of structure.

You said something above greatly of note Geoff..

Regarding the octonions, you stated "The bottom line is this: the nonassociativity of O is a feature, not a bug, and an amazing feature at that, coming into play in just the perfect way." I share your enthusiasm regarding this feature of the octonions, and I likewise exalt that it comes into play in a most amazing way.

Warm Regards,

Jonathan

Hello Geoffrey,

Good metaphor, thank you. Immediately we bump heads, tho, with C more fundamental than R. Given that the goal in our two essays is to have a satisfactory model of agency in the physical world at the level of the elementary particle spectrum, I'm of the view that R is more fundamental than C. This is the position taken by the geometric algebra community of the 'Hestenes school', as so simply and lucidly presented in his 1966 book, Spacetime Algebra, which resurrected Grassman and Clifford's original geometric intrepretation and introduced it to physics.

In the geometric view one can take the vacuum wavefunction to be comprised of the eight fundamental geometric objects of the 3D Pauli algebra - one scalar, three vectors (3D space), three bivectors, and one trivector. Endowing the geometric objects with topologically appropriate fields, this becomes an agent in the physical world.

Interaction of these agents/wavefunctions can be modeled by the nonlinear geometric product, which generates a 4D Dirac algebra of flat Minkowski. Time, the dynamics, emerges from the interactions, encoded in the 4D pseudoscalars. There is no need for complex numbers, for complex algebras, for this particular legacy of Euler.

re spinors, they are likewise understood as being comprised of a scalar plus bivector, can be visualized. Reinvention of Clifford algebra by Pauli and Dirac in the matrix representation has left the community stuck with something that is too abstract. Basis vectors of geometric algebra are equivalent of matrix representation....

I admire your knowledge of group theory, a knowledge that i sorely lack, hope that the above outline of the geometric wavefunction is helpful to you in applying that knowledge to the physics.

http://www.7stones.com/7_new/7_Pubs.html

What OS nd browser are you using? I just tried this, and it works on my old Mac using Firefox, and on my iPhone using Chrome.

https://arxiv.org/pdf/1407.4818.pdf

Likewise this link works fine on my Mac. Methinks the problem is closer to home. Please let me know if you figure it out.

both fine now, dunno what that was about. Glad to see what appear to me to be fairly strong connections between your work and what Michaele and I are doing. Sure wish we had your group theory expertise, tho.

I had the same issue...

Underline characters in the PDF links were turned into spaces, by acrobat or my browser, in the first example. And some part of the https:// got lost too. But I later got the links to work as well.

All the Best,

Jonathan

Geoffrey/Cristi/Jonathan/...

Looking at Geoffrey's comment

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Spinor wavefunction is scalar plus bivector if i understand correctly (please explain if wrong).

Handedness comes from the bivector, of which there are three in the Pauli algebra of 3D space. However it is the two component spinor that comprises wavefunction, not just the bivector (Bohr magneton).

Seems like both the non-commutative and the non-associative properties would come from the bivector, and don't quite understand how the scalar enters into it from consideration of those two broken properties (symmetries?). Is it nothing more than just the 'gauge', not playing any role as an additional topological object (the singularity) with the bivector in the wavefunction that is somehow involved in understanding what's going on?

coming back to my opening question

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Does this mean that the eight component Pauli wavefunction Michaele and I are working with has chiral symmetry breaking built in?

and ditto the 16 component Dirac algebra of the eight by eight geometric representation of the S-matrix generated by geometric products of Pauli wavefunctions?

    darn. another issue with formatting getting scrambled when working thru this interface. Trying again:

    Geoffrey/Cristi/Jonathan/...

    Looking at Geoffrey's comment nO = spinor space; OL = Clifford algebra.

    does this mean the left handed neutrino is built into OL?

    Spinor wavefunction is scalar plus bivector if i understand correctly (please explain if wrong). Handedness comes from the bivector, of which there are three in the Pauli algebra of 3D space. However it is the two component spinor that comprises wavefunction, not just the bivector (Bohr magneton).

    Seems like both the non-commutative and the non-associative properties would come from the bivector, and don't quite understand how the scalar enters into it from consideration of those two broken properties (symmetries?). Is it nothing more than just the 'gauge', not playing any role as an additional topological object (the singularity) with the bivector in the wavefunction that is somehow involved in understanding what's going on?

    coming back to my opening question

    O = spinor space; OL = Clifford algebra.

    does this mean the left handed neutrino is built into OL?

    Does this mean that the eight component Pauli wavefunction Michaele and I are working with has chiral symmetry breaking built in?

    and ditto the 16 component Dirac algebra of the eight by eight geometric representation of the S-matrix generated by geometric products of Pauli wavefunctions?

    ok giving up on this for the moment,

    gonna restart my computer,

    see if that helps.

    restart was no help, and i see Jonathan's post has the same issue

    Seeing that the character n now replaces all carriage returns, I think it is a system wide problem with the FQXi forum platform software. I think they may be trying to make returns appear as an en-dash for compactness but this is ridiculous.

    All the Best, JJD

    This is another test.

    The yellow banana grows on the sun. OL

    GD

    • [deleted]

    • Post #50

    Ok, so no carriage returns. My starting position is T = RâŠ--CâŠ--HâŠ--O. It's just a mathematical object. If it is an essential part of any viable mathematical model of reality, then I suggest it is required because fermion fields require parallelizable spheres in some manner as yet to be determined. Lacking a clear understanding of that I start with T. Like C, H and O, individually, and P =RâŠ--CâŠ--H, T is a spinor space. P is the spinor space of PL, which is essentially the Pauli algebra, so the associated geometry is 3-space. (Still just math.) It a very similar fashion T is the spinor space of the Clifford algebra TL, and its associated geometry is 9-space. In my first book I used the fact that a Dirac spinor is a pair of Pauli spinors (so P2, although this is actually an SU(2) doublet of Dirac spinors) to motivate basing my theory building on T2, which is an SU(2) doublet of Dirac spinors for a 1,9-spacetime. T itself is not only a spinor, it is also an algebra, and its identity can be decomposed into a set of orthogonal projection operators (a la Gürsey and company at Yale in the 1970s) with respect to which the bivectors of the 1,9-Clifford algebra, which is also the Lie algebra so(1,9), gets decomposed to so(1,3) x u(1) x su(3), and the spinor T2 is decomposable into a collection of su(2) doublet ordinary 1,3-Dirac spinors. Again, this is just pure mathematics, with some physics words thrown in because the maths has a kind of obvious interpretation in the physics context. Anyway, I think I won't try to duplicate all the details of two books and a number of papers in this comment, because I need breakfast. Since the neutrino was mentioned in the initial comment, I'll just add that interpreted as a basis for physics modelling, the neutrino that pops out of the maths is a Dirac neutrino, and it has the potential to have a Dirac mass. So, in conclusion: there is maths; there is physics (standard model); everything I've done is simply to show that one can use the maths as a skeleton onto which all of your preferred QFT-flesh can be attached. If the maths has something to say about the deeper quantum side of things, I am not competent to say what it is. If some other idea proves eventually to be the correct mathematical model of reality, T-maths will still have all these properties that look a lot like they ought to have something to do with our presently accepted theories of physics.