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September 23, 2020

New Normed Division Algebra Found!

Posted by John Baez

Hurwitz’s theorem says that there are only 4 normed division algebras over the real numbers, up to isomorphism: the real numbers, the complex numbers, the quaternions, and the octonions. The proof was published in 1923. It’s a famous result, and several other proofs are known. I’ve spent a lot of time studying them.

Thus you can imagine my surprise today when I learned Hurwitz’s theorem was false!

Abstract. We present an eight-dimensional even sub-algebra of the 2 4=162^4=16-dimensional associative Clifford algebra Cl 4,0\mathrm{Cl}_{4,0} and show that its eight-dimensional elements denoted as X\mathbf{X} and Y\mathbf{Y} respect the norm relation XY=XY\| \mathbf{X} \mathbf{Y}\| = \| \mathbf{X} \| \| \mathbf{Y} \|, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

Even more wonderful is that the author has discovered that the unit vectors in his normed division algebra form a 7-sphere that is not homeomorphic to the standard 7-sphere. Exotic 7-spheres are a dime a dozen, but those merely fail to be diffeomorphic to the standard 7-sphere.

Well, no: this article must be wrong.

Doesn’t Communications in Algebra have some mechanism where a few editors look at every paper — just to be sure they make sense?

Posted at September 23, 2020 5:54 PM UTC

23 Comments & 0 Trackbacks

Re: New Normed Division Algebra Found!


I cannot match your knowledge in mathematics as I am merely a physicist (you may even remember me from Perimeter Institute or from one of the meetings at Roger Penrose’s house in Oxford). But I believe that if you actually read my paper you might find that it is not all that wrong, misguided, or uninteresting. It might even give you a new perspective — a Geometric Algebra perspective. On the other hand, if you find a mistake in my calculations, then please do let me know. I am always eager to learn.


Posted by: Joy Christian on September 23, 2020 8:45 PM | Permalink

Re: New Normed Division Algebra Found!

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.

Posted by: John Baez on September 23, 2020 10:28 PM | Permalink

Re: New Normed Division Algebra Found!

I am not looking for stardom, so I leave that investigation in the capable hands of mathematicians. The reviewers for Communications in Algebra did not ask me to do what you are saying they should have. They simply checked my calculations and recommended the publication of my paper on that basis.

Posted by: Joy Christian on September 23, 2020 10:49 PM | Permalink

I guess I will not be submitting to Communications in Algebra.

I did actually skim the paper. This is not the first time I came across the language of “geometric algebra”. I was about to say something mean about it, and then decided that it’s best not to say anything, since, although this is not the first time, it remains a time when I see no reason to understand any benefit of “geometric algebra”.

In any case, the author at various times says that their algebra 𝒦 λ\mathcal{K}^\lambda is: (i) associative; (ii) isomorphic to the dual quaternions [ϵ]/(ϵ 2)\mathbb{H}[\epsilon]/(\epsilon^2), or perhaps to [ϵ]/(ϵ 21)\mathbb{H}[\epsilon]/(\epsilon^2 -1 ); (iii) has no zero divisors but does contain idempotent elements; (iv) enjoys XY=XY\|XY\| = \|X \|\|Y\| where \|-\| is calculated with “the fundamental geometric product” instead of the usual scalar product; (v) that i=0 7Z ie i= i=0 7Z i 2\|\sum_{i=0}^7 Z_i e_i\| = \sum_{i=0}^7 Z_i^2, where e ie_i is a basis of 𝒦 λ\mathcal{K}^\lambda over \mathbb{R}.

Based on skimming, I came away with the impression that: (i) is true, because 𝒦 λ\mathcal{K}^\lambda is defined as a subspace, closed under multiplication, of some Clifford algebra; (ii) wasn’t clear one way or the other; (iii) is self-contradictory, since if x 2=xx^2 = x, then x(1x)=0x(1-x) = 0; (iv) might well be true; and (v) is false. I’m guessing that the author’s “topologically distinct 7-sphere” is thus a 7-dimensional hyperboloid.

For the record, the notation λ\lambda refers to a choice of sign λ=±1\lambda = \pm 1. It matters only in the choice of basis for 𝒦 λ\mathcal{K}^\lambda.

Posted by: Theo Johnson-Freyd on September 23, 2020 10:57 PM | Permalink


Thanks, Theo!

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

Q(xy)=Q(x)Q(y) Q(x y) = Q(x) Q(y)

One is the octonions, where QQ is positive definite. The other is the split octonions, where QQ has signature (4,4)(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.

Posted by: John Baez on September 23, 2020 11:24 PM | Permalink


I also skimmed the article, or rather the latest version on arxiv (, 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 λK^\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 λK^\lambda in the first place.

Posted by: Jake on September 24, 2020 1:46 AM | Permalink


That stood out to me as well.

Posted by: Blake Stacey on September 24, 2020 3:16 AM | Permalink


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 ii,jj,kk 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.

Posted by: Madeleine Birchfield on September 24, 2020 2:57 AM | Permalink

Dear Madeleine: it’s time for me to remind you and everyone here to be extremely polite — especially to other people reading this blog, but that could easily be anyone.

Posted by: John Baez on September 24, 2020 6:34 AM | Permalink


Geometric algebra experienced a rebirth in computer science, especially computer graphics and computer vision and robotics. Big business these days! See the book by Dorst er al.

There have been several valiant attempts to get something out of it in quantum information. But also, it has been a project of Joy Christian for about 13 years now. I wrote a paper about why these projects failed “Does Geometric Algebra provide a loophole to Bell’s Theorem? (with correction note)” published only recently after some further papers by Joy Christian were actually published (in RS Open Science, and in IEEE Access). Mine is: R.D. Gill, Entropy 2020, 22(1), 61 (21 pp.);

In fact, the present “Pure Maths”paper by Dr Christian is essentially one section lifted out of one of his earlier, published, papers. The problems with it have been known for years, also to Dr Christian himself though as he says here, he does not accept them, and they are mentioned on PubPeer and various internet blogs. I have spent many happy hours studying the first couple of sections of Johns big “Octonions” paper and I think I caught all the errors mentioned here. Unfortunately, as a mere statistician, my words do not carry as much weight as those of most contributors here.

I still hope to be able to use my knowledge of Octonions in e.g. Corona statistics, but the chance seems negligible.

Posted by: Richard Gill on September 24, 2020 10:00 AM | Permalink

Re: New Normed Division Algebra Found!

Has anyone written to the Editor-in-chief?

Posted by: David Roberts on September 24, 2020 5:45 AM | Permalink

Re: New Normed Division Algebra Found!

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

Posted by: John Baez on September 24, 2020 6:38 AM | Permalink

Re: New Normed Division Algebra Found!

The person who wrote to you and to the Editor-in-Chief before you was Richard D. Gill. He has taken this sort of actions against me, including repeatedly writing to my academic superiors, for the past ten years. For example, he previously tried to have three of my physics papers retracted. He succeeded in having one of them retracted by spooking an Editor-in-Chief, but there was nothing wrong with the paper and it is now republished in another journal of higher standing. Gill’s attempts to have that paper retracted again seems to have failed. Note that this is just one individual stalking me on every turn I take for the past ten years, with dire consequences for me personally and academically. As for the current math paper, either there is a mistake in my calculations or there isn’t. The reviewers checked my calculations and did not find a mistake in it, and therefore it is published.

Posted by: Joy Christian on September 24, 2020 9:11 AM | Permalink

Re: New Normed Division Algebra Found!

Of no scientific interest, but for the record.

Re: “Writing to academic superiors”. I have in the distant past pointed out to relevant “authorities” that Christian was faking his academic affiliation as their own. I also once complained (I think to a master of an Oxford college) that he was further attacking his academic opponents with obscene insults in public internet blog discussions. Finally, I complained to IEEE Access about plagiarism of my work on Pearle’s detection loophole model (it reproduces the singlet correlations in a local realistic way by data post-selection).

Sorry, all very childish and unnecessary.

Posted by: Richard Gill on September 24, 2020 10:43 AM | Permalink

Re: New Normed Division Algebra Found!

It is also full of lies. Gill repeatedly wrote to my academic superiors, such as to the President of my College at Oxford University, because he wanted to discredit me academically and personally. Why? Because he disagreed with my research for dogmatic reasons. He tried to have four (and now fifth) of my published papers retracted because he wants to discredit me academically and personally. Ten years of obsession with one individual can only be described as personal. Unfortunately, there is no law against this kind of stalking in academia. How dare he claims I am faking affiliations? It is pure nonsense. And it is Gill who has been insulting me all over the Internet (not to mention in private emails) as anyone can easily find out by googling. And talking of plagiarism, it is Gill who has stolen my work on Pearle’s detection loophole model reproducing the singlet correlations in a local realistic way. My work was published on the arXiv a whole year before Gill claimed it to be his own and posted it on the arXiv. Even in discussions on public forums I have priority on that work and I can prove it date by date. He has quite a way of turning everything around and throwing it back at me in a manner a certain politician in the US does.

Posted by: Joy Christian on September 24, 2020 11:45 AM | Permalink

Re: New Normed Division Algebra Found!

For context, it may be interesting to know that the author already has a history of writing papers with strongly overlapping content, and then getting some of them published in journals where this ends up causing upstir and discussion about how this could possibly have happened and what to do about it.

The first case I know of is a retracted paper at Annals of Physics in 2016. (See also Not Even Wrong.) Another case is a 2018 paper published at Royal Society Open Science, with a long debate between the author and Richard Gill in the comments at the bottom. There is a very recent open letter written by one of the referees who had recommended rejection(!), and co-signed by other physicists.

It’s saddening that this story now seems to be repeating itself with a mathematics journal. (Coincidentally one in which I’ve just had my own paper appear…)

Posted by: Tobias Fritz on September 24, 2020 9:05 AM | Permalink

Re: New Normed Division Algebra Found!

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.

Posted by: Joy Christian on September 24, 2020 9:31 AM | Permalink

Re: New Normed Division Algebra Found!

I must agree, the behaviour of the editors of “Annals of Physics” was despicable. And totally incompetent.

I have been studying the works of many Bell-denialists for many years. Yes, this makes me a kind of bottom feeder, I admit it, but I see it as important “Science Outreach”. It is admittedly highly entertaining, too. And it led to serious joint work with really interesting people like Nicolas Gisin and Anton Zeilinger. While fighting 20 years ago with Luigi Accardi, and later with Karl Hess and Walter Philipp (who published a huge paper with deeply hidden mathematical mistake in PNAS), I introduced martingale methods into the standard literature on Bell experiments, and my methods were used and cited by all four big loop-hole free Bell experiments of 2015 (Delft, Munich, NIST, Vienna).

Posted by: Richard Gill on September 24, 2020 10:19 AM | Permalink

Re: New Normed Division Algebra Found!

None of that undoes your stalking of me and my work for the past ten years, with the kind of obsession with me that is bewildering, especially for an academic.

Posted by: Joy Christian on September 24, 2020 10:29 AM | Permalink

Re: New Normed Division Algebra Found!

I was for a while bewildered by my own obsession. One thing is, I can’t stand to see intelligent people with many admirable qualities ruining their own scientific lives by their own stupidity. And being a somewhat autistic pure mathematician type I get obsessed when smart people don’t understand simple things and cause damage to themselves and others by their mistakes.

Of course, this attutude is not appreciated. Older and wiser, I now try to leave this kind of work to the next generations of scientists.

But if people publish scientific work which is evidently badly wrong in a field in which I am ideally situated to make critical comments, I raise the issue first with the author and later if necessary with a responsible editor. In this case Dr Christian heard every single mathematical issue raised on this forum earlier from me, in the context of his other papers, but chose to ignore me. At least, I feel I passed my exam on Clifford algebras and the Octonions with flying colours.

Posted by: Richard Gill on September 24, 2020 1:29 PM | Permalink

Re: New Normed Division Algebra Found!

Gill continues to lie about things in public thinking that no one will notice. For example, Gill claims above that I ignored his arguments. How did I ignore them? By repeatedly pointing out his extremely elementary mathematical mistakes in the following detailed refutations of all of his arguments?

Posted by: Joy Christian on September 24, 2020 1:41 PM | Permalink

Re: New Normed Division Algebra Found!

I think we have enough information here, or actually quite a bit more than enough, so I’m going to close down comments on this thread.

Posted by: John Baez on September 24, 2020 4:29 PM | Permalink