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Stephane Bronoff in The Standard Model of Particle Physics from Sedenions claims that studying the left-multiplication map of unit doubly-pure sedenions solves several mysteries of the standard model. First of all, the sedenion algebra is the next step in the Cayley-Dickson construction after the octonions (you lose the fact $ab=0 \rightarrow a=0\vee b=0$). A doubly-pure sedenion is one whose complex part is zero. I found Eigentheory of Cayley-Dickson Algebras by four Daniel's (referenced by the previous one) which is talking about the map that, given a sedenion $a$, left multiplies a sedenion by $\frac{a^2}{|a|^2}$. This is apparently a real-linear map and has the action of U(1), SU(3), and (approximately) SU(2)xSU(2) on some of its eigenspaces (I don't really understand how that could be the case since the eigenvectors of a real linear map only differ by a real factor after the map is applied to them).

Anyway, what I roughly understand about how Lie groups pop up in particle physics is what I understood from David Griffiths "Introduction to Elementary Particles" which I read (most of) a long time ago. If you have a Lagrangian for a fermion field where you can multiply the field by an element of a Lie group and not affect the Lagrangian, the profitable thing to do is instead multiply the field by an element of the Lie group that varies differentiably with respect to space and time and then throw in a boson field to cancel out derivatives and everything works. Maybe it's just going over my head, but I don't see how any of that is done in the paper, and it isn't obvious to me how it would all be formalized using the sedenions.

Aside from all that, the paper contains an off-putting number of typos for one that seems to be announcing such a tremendous result. Is it to be taken seriously?

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  • $\begingroup$ Is this published/peer-reviewed? $\endgroup$ – Qmechanic Jun 27 '18 at 13:00
  • $\begingroup$ @Qmechanic it's only on ResearchGate as far as I know . . . don't know if that's considered published or not. $\endgroup$ – Eben Cowley Jun 27 '18 at 20:06
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Amateur would-be physicists (and quite a few professional physicists) have discovered many mathematical formulas for the unexplained constants of the standard model, such as for the fine structure constant (approximately equal to 1/137). These will be expressions involving e, pi, natural numbers, square roots, and so on, in some combination. This is often called "numerology".

These formulas by themselves prove nothing, and indeed almost all of them must be coincidence, having nothing to do with physics. Such a formula is only meaningful if it can arise from a full theory that also has equations of motion, and the capacity to explain physical processes like particle decay, scattering events, interactions, bound states, etc.

There are, similarly, many papers which purport to motivate some of the algebra of the standard model. They propose some justification for the symmetry groups and the inventory of elementary particles. But as with works of physics numerology, these works of "algebrology" generally don't contain a theory in the sense that the standard model is a theory, i.e. something that makes dynamical predictions and can model actual physical processes.

Instead, a work in the algebrological genre just lists a number of algebraic facts, in which e.g. symmetry groups of the standard model do appear, but shorn of their physical context. Common omissions include any motivation for particle statistics or for the gauging of these groups.

As with the numerological propositions, the algebrological propositions would only be physically meaningful if they were part of a dynamical theory. At least Stephane Bronoff understands this, he wants his algebrology to follow from the opaque category-theoretic metaphysics that he describes in his earlier paper. But he certainly doesn't exhibit this deduction.

In the end you ask, "is this paper to be taken seriously?" Obviously not as a completed theory of everything. The odds are also massively against an individual work of algebrology being THE ONE which has chanced upon the right way to motivate the symmetry group of the standard model, for the same reason that almost all numerological formulas must be red herrings.

This paper belongs to the major subgenre of algebrology which tries to obtain the standard model gauge groups from hypercomplex numbers. Usually such efforts stop at the quaternions or octonions; though there is some prior work involving sedenions. It is a challenge to the plausibility of all such works, that in field theory and string theory, one may easily construct physical models with any gauge group you like, including gauge groups that have no special relation to division algebras or hypercomplex numbers. This might be a clue that this kind of algebrology is the wrong way to discover what's special about the standard model. (An example of an alternative approach to that task, which is better motivated although worse named, is the search for "swampland criteria" that will tell you which field theories cannot be realized in string theory.)

So - I always look at this stuff with interest. I actually like the quest for physics via hypercomplex numbers. It may eventually turn up something profound. (The "Musean hypernumbers", in particular, are crazy in an inspirational way.) But this paper is not physics. It contains nothing testable and no way to calculate anything physical. It is a work of algebrology, meaning, it contains algebraic propositions that are still in search of a physical theory that will give them meaning and life.

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