Geoffrey,

I enjoyed your essay. This is because I read your https://arxiv.org/abs/1407.4818 paper some years ago. I do have a few questions. In particular is the 128 dimensional T^2 hyperspinor space the same as the E8/SO(16) = 128? The other is that I have written notes or a pre-paper on some work with the Jordan J^3(O). This is I think more general than the Leech lattice, or embeds the Leech lattice. I was wondering if you have done any work on this and automorphism of the FS "monster group."

I have pondered how it is that spin ½ leads to FD statistics. I have found myself thinking exactly what Feynman responded with, "I can't do it." It does seem plausible that because BE statistics integrates 1/(e^{-Eβ} - 1) into ζ-functions. The FD statistics 1/(e^{-Eβ} + 1) can be thought of as related to the BE with the general form 1/(e^{-Eβ} + e^{iθ}) for θ a phase angle. This is a bit like anionic statistics. It seems in a way this involves some deep relationship with the Riemann zeta function.

The motivation by mathematics can at times be compelling. I have some resonance with Dirac's call to seek beauty. It is though not clear to me whether mathematics is more fundamental than physics. There was a time when I thought this might be the case. Then as time goes on this seemed difficult to uphold, while on the flip side it appears to be a collapse of objectivity to just assume mathematics is a sort of game or human invention. I am at a stage where I have not the faintest idea what the deep relationship between mathematics and physics is.

Cheers LC

    Ah yes...

    The bananas must be thriving.

    Warm Regards,

    Jonathan

    Hello again Geoff,

    I wanted to thank you for the thoughtful 'play by play' review of my essay. It is amazing what one can learn seeing what you have written through someone else's eyes. I especially appreciate your catching me on the dodgy usage of the word 'likely' which has no place in academic writing, where the goal is to be crystal clear and mathematically precise.

    As for the Mandelbrot Set; it was the horse I rode in on, as it were. I had a few phone conversations with Ben Mandelbrot more than 30 years ago that greatly encouraged and shaped my learning. The last time we spoke; he called me out of the blue on an Easter Sunday morning. I had worked until midnight, the night before, and decided to sleep in rather than attending a church service - but I got lucky and talked to Ben instead.

    Since then; I've found out M doesn't stand alone, but connects with a number of other mathematical objects - so it was a good place to start me going in a worthwhile direction. I'm glad to have had your musings to refer to, as well, because there does appear to be a special significance to T. As I recall; the Sedenion sphere S15 only has three possible fibrations, S1, S3. & S7 - yielding the C, H, and O algebras. This would seem to indicate they are a foundational trio.

    I thought the comment above was priceless, and I'm very glad to see that your banana plantation is thriving!

    All the Best,

    Jonathan

    "In particular is the 128 dimensional T^2 hyperspinor space the same as the E8/SO(16) = 128?"

    I don't have any reason to believe it is or isn't. Tony Smith thinks so, but I am unwilling to lead the maths. I prefer to be led, even if led astray. At least it won't be my fault.

    "Jordan J^3(O). This is I think more general than the Leech lattice, or embeds the Leech lattice."

    Wilson, at U of London, and I have independently represented the Leech lattice over O3. Anyone with any interest in J^3(O) will naturally point out that there is a nice copy of O3 in J^3(O). I don't know if this is meaningful. If one could show that there was a bilinear of trilinear multiplication on the Leech lattice as a subset of J^3(O) that closed on this subset, then I personally would be hugely interested. But yes, one can stick it into J^3(O), but without some further interesting structure ...

    "I was wondering if you have done any work on this and automorphism of the FS "monster group.""

    No. I of course find the monster group enticing, because it is exceptional, but I haven't got around to looking into it.

    As to your paragraph 2, this leads into the quantum quagmire. If you can find two people who agree on much of anything in that quagmire, let me know. Meanwhile, the level of disagreement means to me that we are not ready to understand at a deep level.

    "... it appears to be a collapse of objectivity to just assume mathematics is a sort of game or human invention. I am at a stage where I have not the faintest idea what the deep relationship between mathematics and physics is."

    This is a hard one, because one has to define what one means by mathematics. I think of it as a thing that is there even in the absence of any intelligent life or any conception of consciousness. Many will likely give this idea a label named after a dead Greek and think they understand. Maybe they do. Anyway, in the presence of human intelligence we have this notion of Ur-mathematics, and we have the human symbols and formalism that is our lens onto this world. So, ignoring the invented formalism, I think of physics - I'm just making this up, but it sounds good to me - anyway, physics crystallizes on the exceptional, generative, and resonant bits of this Ur-maths, for only there is there a structure rich enough to nurture it. And to give rise to us, for what that's worth.

      I replied to this below, but may have failed to make it a reply, as opposed to a comment.

      And again, I check "I'm not a robot", but I likely am.

      Hello Geoffrey,

      I liked the way your essay is written, mainly the introductory part. I believe your essay makes much sense, no one can reach an agreement for fundamental. I also believe that mathematics is more fundamental than physics which I have written very descriptively on my essay.

      I gave a good rating to your essay because I find it unique and interesting and I hope you'd enjoy mine too.

      Kind Regards

      Ajay Pokharel

      Geoffrey Dixon

      Dear Geoffrey,

      I rarely feel so lucky and pleased to read something with which I agree almost completely. Your essay gave me this high satisfaction, so thank you. There may be two points where we slightly diverge, but only as a matter of preference. The first one is that I may be a bit biased towards geometry and consider complex, quaternionic, and octonionic structures as living on real vector spaces. The second one follows from the first one, since the compositions of transformations preserving structures is associative. I fully agree with the paramount role of spinors, but consequently I tend to see them as representations of Clifford algebras (a quite mainstream position among mathematicians). So my views about the Standard Model are shaped by this. Not that I would disagree with you, in fact I think you are right from another perspective. Another consequence of my view is that it kept me away from properly investing time in studying your work, although I I knew about the Dixon algebra and that you made a mathematically beautiful and physically insightful model for leptons and quarks. It's time to fix these lacunae I have and read carefully your writings. Your essay convinced me of this, even though you mentioned your work only incidentally, being focused on answering the question "what is 'fundamental'". I also just included in my paper about the Standard Model based on a Clifford algebra a mention of your model (fortunately my manuscript is still under review). I think your work deserves more attention. What I find intriguing is that, unlike Clifford algebras which are infinitely many, the Dixon algebra is one of a kind. Since I still am a bit biased against nonassociativity, I would like to ask you if you know some physical consequences of this feature of octonions. Congratulations for your excellent essay, and success!

      Best wishes,

      Cristi

        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?