The n-Category Café

If I have time I’ll try to find the mistake in your paper.

But in fact, if a paper contains results contradicting a widely accepted theorem, the paper’s referees should request what you are requesting of me: that the author find a mistake in the proof.

In the case at hand, several proofs of Hurwitz’s theorem are sketched on Wikipedia, with links to more details. Discovering that any one of these proofs is wrong would instantly hurtle anyone to the status of a mathematical superstar.

Thanks, Theo!

Up to isomorphism, there are just two kinds of not-necessarily-associative unital 8-dimensional real algebra having a nondegenerate quadratic form $Q$ with

$$Q(x y) = Q(x) Q(y)$$

One is the octonions, where $Q$ is positive definite. The other is the split octonions, where $Q$ has signature $(4,4)$. Neither is associative.

This comes from the classification of composition algebras, and in particular the 8-dimensional ones, which are called octonion algebras.

I also skimmed the article, or rather the latest version on arxiv (https://arxiv.org/abs/1908.06172, note the section). One thing that stood out to me was in the appendix, in particular the definition of X and Y in (A.6). These certainly appear to be in $K^\lambda$, as they are each a linear combination of two of the basis elements, but the author does some trickery, gets a contradiction, and then goes on to argue that it’s not a problem because X and Y were not part of $K^\lambda$ in the first place.

Even amongst many proponents of geometric algebra in physics, Joy Christian is largely considered a pseudoscientific quack and a fraud whose ideas simply do not hold up to mathematical scrutiny whatsoever, so the field absolutely should not be judged upon the work of Joy Christian. It is an absolute shame the journal and whoever peer-reviewed this article failed to reject this article.

Geometric algebras as mathematical objects are typically defined as Clifford algebras over the real numbers, and quite frequently the terms ‘geometric algebra’ and ‘Clifford algebra’ are used as synonyms. The geometric/Clifford algebra approach to physics, and specifically the 3D geometric/Clifford algebra, is primarily pedagogical in nature in teaching undergraduate multivariate calculus, Euclidean geometry, and classical mechanics. It is an upgrade to using 3-dimensional vector spaces with a vector cross product, as is common in undergraduate physics, as it helps clarify the algebraic structure and conceptual ideas of 3-dimensional Euclidean geometry much better than the vector cross product algebra does, and integrates quaternions as used in 3D rotational mechanics into its framework as well, as the even-degree subalgebra of the 3D geometric/Clifford algebra. Non-relativistic quantum mechanics already uses geometric/Clifford algebras, they just represent the unit vectors as Pauli matrices instead of $i$,$j$,$k$ unit vectors. There also exists a 3+1 dimensional geometric/Clifford algebra for Minkowski spacetime/special relativity which simplifies Maxwell’s equations for classical electromagnetism into only one equation, but in relativistic quantum mechanics the geometric/Clifford algebra is usually represented by Dirac matrices instead.

My biggest criticism of the geometric/Clifford approach is the fact that implementing such a framework in undergraduate physics requires a complete overhaul of not only the physics courses but also the mathematics courses undergraduate physicists would have to study. Geometric/Clifford algebra would either be incorporated into an applied linear algebra class or as a separate successor course to said linear algebra course (probably as ‘applied Clifford algebras’ or something of that nature). The multivariate calculus class would have to be updated to include geometric calculus and Clifford analysis, and placed after a linear algebra course or the proposed geometric/Clifford algebra course. Entire undergraduate physics textbooks and curricula would have to be rewritten, and professors would have to learn and adopt the new approaches. Until that happens, the geometric/Clifford algebra approach to physics would not be adopted at all at the undergraduate level, and it is highly unlikely any undergraduate physics department is willing to take such drastic changes.

Because of that, the usage of geometric/Clifford algebras in physics outside of quantum mechanics would be akin to the usage of tau in mathematics and science as a replacement of pi, or the usage of ETCS + replacement axiom in formal mathematics as a replacement of ZFC, theoretically better than the commonly used alternative, but unlikely to be adopted by said community, because it was developed far later than the alternative (in the case of geometric/Clifford algebras, 80 years after the physics community adopted Gibbs’s vector cross product algebra), and community inertia would prevent any such change from happening in the first place.

Yes. This paper was brought to my attention after it was brought to his.

If I were you, I would check the facts before spreading false rumors. It is one thing when this is done by anonymous individuals on the Internet, and another thing when an established academic falls that low. Note that there is nothing wrong with the paper that was retracted by Annals of Physics. Despite my repeated requests, Annals of Physics never provided any evidence of mistake in it, even privately. In any case, the retracted paper is now republished in IEEE Access. And you claim that Florin Moldoveanu was a reviewer of my Royal Society paper, which remains published to date. What evidence do you have for this? His word? I put it to you that what Moldoveanu says on his blog is full of lies and that you are spreading those lies further.