Just Ignore Me
Pascual Jordan developed Jordan algebras. He’s famous. So are von Neumann and Wigner. Together with Jordan, they were able to classify all finite dimensional Jordan algebras. They published their results in 1934.
This troika, in the years after Solvay, achieved a renown that elevated them to such a high celestial sphere that it is tantamount to heresy to question the worth of anything they produced. This includes Jordan algebras.
But what motivated Jordan? We get a good idea in this John Baez blog post. I quote:
“… in traditional quantum mechanics, self-adjoint n×n complex matrices count as observables. These aren’t closed under matrix multiplication. Instead, they’re closed under linear combinations and the commutative operation a∘︎b = ½(ab + ba).”
This Jordan product of observable operators forces the result of the product to in turn be an observable. So, yippee. We want that don’t we? Don’t we?
Well, I’ve never been anything approaching 100% sure, but I’m a maverick – not a pariah, maybe, but … Was I alone in thinking Jordan algebras were little more than mathematically pretty human constructs that Nature herself can do without? I did some googling.
I googled Jordan algebra critiques, and – quelle surpris – topping the list was a Luboš Motl blog post with which I found myself largely in agreement. Luboš – by way of providing context – is an aging (but aren’t we all) pit bull blogger on all things HEP and a great deal else. He falls short of being politically correct by a margin that dwarfs the Grand Canyon, but my reaction to some of his more outré pronouncements – and I haven’t read many – has generally been of the eye rolling sort. Sometimes he lacks tact. For example, he claimed most vehemently, and numerous times in the same pair of blog posts, that I (and many others) am a crackpot, and, more pointedly – because this same accusation was leveled at no one else – that I am not Lance Dixon. Well, he’s right about at least half of that. But the point – from which I have strayed by a goodly bit – is that he is not a fan of Jordan algebras. So I am not alone. Yay.
Decades of Hope
In Lie group theory there are infinite sequences of classical Lie groups associated with the division algebras R, C, and H. There is also a finite sequence of exceptional Lie algebras associated with the last division algebra, the octonions, O.
The same sort of thing occurs with finite dimensional real Jordan algebras: infinite sequences associated with R, C, and H; and a finite sequence associated with O. The last of these, usually called the exceptional Jordan algebra, is 𝔥₃(O), the set of 3×3 self-adjoint octonion matrices. This set is 27-dimensional, but if you take away multiples of the identity, which doesn’t really participate in this Jordan algebra, you’re left with 26 traceless elements of 𝔥₃(O). And in fact, the automorphism group of this Jordan algebra is the 52-dimensional Lie group, F₄, the 2nd of the 5 exceptional Lie groups, whose fundamental representation is 26-dimensional.
John Baez has a theorem: The subgroup of F₄ that preserves the splitting of each off-diagonal octonion into complex scalar and complex vector parts and preserves a copy of 𝔥₂(O) in 𝔥₃(O) is isomorphic to (SU(3)×SU(2)×U(1))/ℤ₆, which he calls “the true gauge group of the Standard Model”. (The ℤ₆ is a sort of anal flourish which makes rigorous mathematicians happy.)
This is the kind of result that has kept many people plugging away at this algebra for decades. It’s cool, but is it that cool? First, in a previous blog I mentioned the difference between what I called Ubjects and Mools. Ubjects (short for mathematical ur-objects) are things that in a sense exist without the need of any sentience to invent them. My partial list of Ubjects includes my favorite things: prime numbers; the parallelizable unit spheres in dimensions 1, 2, 4 and 8; and the exceptional laminated lattices in dimensions 1, 2, 8 and 24.
Mools are mathematical tools, like quantum field theory. QFT, while incredibly useful within our potentially limited lights, needed to be invented, as opposed to discovered. The same can be said of Jordan algebras, but in this case matters are worse, because they are not incredibly useful. Few people use them at all, although many people study them.
But what about the standard symmetry inside F₄? Surely that’s something. Well, I have a few things to say about this that explain why this leaves me less than excited. The restrictions on 𝔥₃(O) that lead to this subgroup are unmotivated and rather artificial. And F₄ mixes the 26 traceless elements of 𝔥₃(O), which are essentially 1-dimensional. Fermions cannot be represented in this way. You’ve got to attach Dirac spinors somehow, thereby increasing the artificiality. (Contrast this to my work on the tensored division algebras: the standard symmetry, when derived, automatically acts on Dirac spinors, and that’s because the theory begins with a spinor space, not some algebra or Lie group). And what about the whole algebra of observable thing that got the mathematical physics gods so excited back in the 1930s? Well, that’s been largely – or entirely – discarded, at which point 𝔥₃(O) is just some algebra someone happened upon that has properties that appear relevant to physics, and, hey, there’s that number 3, so maybe it could explain the 3 family generation problem. Who knows?
Peter Woit continues to blog, and his assessment of the future of HEP funding is very 2020. Basically, the science to which I have devoted most of my life (and it will always be most, as I have metastatic prostate cancer and will be dead within 5 years), but anyway, that science also appears to have an incurable disease, and researchers who desire funding are streaming to quantum computing and AI.
Peter, taking advantage of this chaotic decline, has decided to get daring, while few people are looking anymore, and develop ideas related to quantum gravity using Penrose’s twistors. Penrose, of course, is another mathematical physics god, whose thoughts and pronouncements, however outré, are generally considered beyond criticism.
Twistors live in a universe based upon the complex algebra, C, and if his one reference to my work is any indication, he was not at all interested in incorporating H and O. They are merely irrelevant oddities.
Peter is not so narrow minded, and is willing to consider some role for those two higher dimensional division algebras. Carlos Perelman (or Castro, depending on his mood) has already developed quantum gravity ideas based on T = C ⊗ H ⊗ O. I don’t believe these involve twistors (I confess myself unable to judge the ideas of either Peter or Carlos), but they do involve all the relevant division algebras, so my seal of approval is upon them. (I never carried my work to quantum gravity, but following the mathematics – and listening solely to the mathematics – did lead to a solution to the missing antimatter problem, which I consider excellent, like a multifaceted diamond.)
Twistors, by the way, are Mools, but I suspect that if you are a devotee they seem like Ubjects. I’m not withholding judgment on the matter, but am disinclined to debate it. My judgement is not firm enough.
So, there you are. Everything is changing: for me personally, having learned I have an expiration date; for the human population, beset by disease, political unrest, and economic distress, painfully giving birth, inevitably, to a new world order that is being resisted across the political spectrum; and let’s not forget the environment, transcending all of that, and looming ever larger as a threat to old ways of thinking, being, and doing.
Everything is changing, and all in this year. What a year.
I thought my last blog was to be my last. I was wrong. I hope to keep being wrong for as long as possible.