Paradigms Lost


Every so often theoretical physics (TP) is visited by an idea that shakes its tail on a tree branch and causes a majority of TPers in that field to rush to the tree and yap excitedly at its base.

Bad dog! Heel!

So, yeah, great title, and great first sentence, but this is ground that I’ve covered in this blog – repeatedly. Indeed, it is ground upon which I first trod in my publications some 40 years ago. My original thought, for this blog installment, after deciding on that excellent title, Paradigms Lost, was to softly chide the pack of people known as theoretical physicists for chasing wildly after each and every brilliant new idea as though more of the same will surely lead to … to … to more good stuff. But chiding, whether soft or hard, has never, and will never, achieve its desired intent. You can’t stop a runaway locomotive with a stick.

My repetitive behavior reminds me of Joey in the movie Hackers.

• 00:43:51 Hi, my name’s Vicki and I’m an addict.
• 00:43:55 Hi. My name’s Hank and I’m an addict.
• 00:44:00 My name’s Joey, but…
• 00:44:03 I’m not an addict.
• 00:44:05 No, really. Listen.
• 00:44:06 I got in trouble with my computer.
• 00:44:08 My lawyer told the judge I’m an addict, but I’m not addicted to my computer.
• 00:44:12 No, really.
• 00:44:14 I’m not an addict.
• 00:44:15 I’m not.
• 00:44:18 Can I get some more coffee?

Grandpa’s here! Hide!

However, it’s about more than simple addiction; it’s also about aging. In principle growing old is not bad, although it will require help from a plethora of medical professionals for me to go very much further down this track. But getting older is not all cupcakes and buttercups. Well, it can be, if everyday you decide that today would be a great day to introduce cupcakes and buttercups into your life, largely forgetting that you did that same thing the day before, and the day before that. Not that cupcakes and buttercups are bad, per se, as an idée fixe – one could do worse – but one’s thoughts do tend to drift into repetition, as originality fades. I can see myself (d)evolving into that embarrassing grandparent who exhorts his grandchildren at each family gathering to pull his finger. Not that flatulence isn’t an eternally excellent source of humor; it is. And not that I actually have grandchildren; I don’t. I’ve consulted medical texts and it seems that a prerequisite for having grandchildren is to first have children. I have none of those either. But I digress; or do I?

You know, in 2014 my Seeable Matter; Unseeable Antimatter paper was published, in which it was demonstrated that division algebra mathematics, a la G M Dixon, implies that our universe is dominated by matter because antimatter has its own 4-d universe, and the two are linked by blah blah blah. But recently Researchgate notified me that a paper I published in 1989 (nine years after Quanta Magazine suggested I had given up on theoretical physics entirely) had reached 200 reads. A modest achievement, admittedly, but it prompted me to look at that paper. And what should I discover but that 31 years ago, at a minimum, and 25 years before my 2014 paper, I was already pointing out that … that … you know; what I said above. And although the 2014 paper is easily the most mathematically clean, still, I think I’d like to make a radical change and have cupcakes and buttercups today.

The cure

So, here I am bemoaning the repetitive nature of my blogging, and my life, and simultaneously … have I bemoaned the repetitive nature of my blogging in previous blogs? I think I have! Old published ideas tend to sink beneath the ocean of consciousness, then bubble up from the depths at some later time in the guise of originality.

The cure, as I see it, and have seen it, is to cease dwelling on the shortcomings inherent in the practice of theoretical physics, and to spend more time immersing myself, and both of my readers, in discussions of matters outside of science altogether. I see I’ve already done Star Wars, Game of Thrones, and Girls’ Last Tour … must-resist-desire-to-discuss-them-again … resistance is exhausting. I could do Paris; or flatulence; but I recently did a combination of Paris and flatulence, so that’s out. Think. Think!

I’ll get back to you. Not like I have any choice in the matter; writing has always been a compulsion of mine. I mean, after retiring I wrote five books. I’d have written more, but blogging … so much more immediate … and the past such a rich source of material.

Get off my lawn, ye young skallywags; and you too, ye blighted whippersnappers.



Here is a definition of “hebetude”: “The absence of mental alertness or physical sensitivity.” Next on the list is “hebetate”: “To blunt the sensitivity or keenness of.”

Let me explain. Decades ago, when giant beasts both strange and perilous roamed the earth, I began to collect words encountered while reading that I felt less than comfortable using in my own writing. I wrote these words down, along with definitions, in a notebook I dubbed Dixonary. I first encountered the word “hebetude” in The Chronicles of Thomas Covenant, a series of fantasy books. The author, Stephen R. Donaldson, was fond of sprinkling his prose with little used words, and I kept a dictionary at hand when reading these books.

So, yes, “hebetude” made it into the Dixonary, and as “hebetate” was nearby, it made it in too. Let’s use it in a sentence.

“The neutral and negative results of the LHC hebetated researches into theoretical particle physics worldwide, leading to a retrenchment of the field, but as custom reconciles us to everything, its practitioners have circled their wagons around stale – but beloved – ideas proven over the past 40 years to be ineffectual, but oh so comfortable.”

That was fun. In fact, however, I have never encountered the word “hebetate” in any of my reading, and even as I write it on my iPad, because it is unrecognized, it gets the dotted red underlining of death. Curiously, “hebetude” is recognized, and I actually did encounter it once. And only once.

In all the decades since, as the great reptiles slowly died out, I have never lost the feeling that if a word is encountered that rarely – once, or never – does that word lose legitimacy? I personally – despite their inclusion in my Dixonary – have always avoided the use of these words in my own writing, feeling certain that the vast majority of my readers would be as befuddled as I once was. (And now I’m wondering if the phrase “vast majority” is applicable if the number to which the phrase is being applied is ten or fewer.)

Eventually, because of my addiction to P.G. Wodehouse, I started adding meta-content to the Dixonary, viz., phrases. For example, let’s use a few:

“Theoretical particle physicists have been cut off from the exercise of making meaningful advances to their field because they labor under the misapprehension that old ways of thinking, shown to be inadequate, are preferable to giving rein to idle speculation that could imperil the cordial relations existing between mainstream theorists and the practitioners of science media.”

Anyway, it would be idle to deny that sharing my jaundiced view of theoretical physics could be hazardous. One doesn’t usually associate STEM folk with danger, but I recently read that a German mathematician was arrested for murder and cannibalism, than which nothing could be more calculated to give one the heebie jeebies. Deliquescence is preferable to ending up in a shallow grave with all my meaty bits gone into a mainstream stew.


Meanwhile, outside the mainstream, speculation continues to run rampant. L. Motl himself, as a kind of left handed compliment, suggested that Tony Smith and myself are (well, Tony has passed, and I have a medical death sentence, so maybe “were” is the mot juste) the archetypes for all modern day “crackpots”, of which there are many. However, although my work is founded on highly developed and unassailable pure mathematics, which is then interpreted as a fully fledged foundation for the Standard Model (unavoidably so), providing explanations for a multitude of puzzling aspects … sniff. I need a tissue. Where was I? Oh, yes, without exception, although my work is frequently cited (sort of), in not one instance does the work doing the citing build upon my work. Nay, my work is used as an excuse for the author’s own speculative ideas. You know, Gürsey and Günaydin used the octonion algebra mathematics as a way of explaining color SU(3). I cited them many times, and I built upon their results, at least those that did not try to connect to quantum theory. It never occurred to me to cite them, then ignore them, and carry on as though their seminal work was irrelevant, save as an excuse for me to do something similar. Sigh. So, yeah, Motl was right.

Curiously, the first six words in the Dixonary (which I just yesterday dug out of a musty pile of old creations) are: Abnegate; Abjure; Abeyance; Acrimony; Adumbrate; and Aegis. All rather fitting, n’est ce pas? The seventh word is Accolade, but the universe has stopped at number six. Further down the list is Apotheosis. Ooh, I like that one. Let’s do that!

Manga: read right to left.

Tu es le petit déjeuner de ma vie

After I retired, SWMBO (who is sometimes referred to by the pseudonym, Franscesca) and I spent several weeks each spring in Europe, each time lavishing Paris with our resplendent presence. I wrote several proleptically bestselling books during this time, the first of which covered my peripatetic years of being a young theoretical physicist on the go. The underlying theme of that tome was the statistically implausible frequency with which my travels were met by near disasters. Maybe this was just Nature’s way of punishing me for my highly efficacious theoretical effrontery. Nature has always seemed content with the dithering investigations of mainstream theorists into the arcana of strings, loops, it-from-qubit, and amplitudeology. These folks were never punished with floods, illness, terrorism, earthquakes, … the list goes on. No, Nature’s wrath was reserved for me, for I had actually uncovered one of her secrets. Bad Geoffrey! Bad!

Ok, settle down ego. So, anyway, I wrote two subsequent travelogues, each of which continued this theme of disasters narrowly avoided. The first is entitled, Paris: je dirais même plus, and it covers our first post retirement trip to Europe in 2016. This was the year that Paris, and much of the rest of Europe, flooded. I’d like to share a section of that book, and perhaps, in so doing, dispel any notion you may have that my character is entirely frivolous. Herewith, …

The Fart that Ate Paris

In extenuation I must explain that I did not know they were there. I’m not proud of what happened, and for a little while after I felt shame, but I got over it. Also, I’d been sick, and would continue to be sick for weeks.

Ok, so we set the scene: Franscesca and I are walking, more than flâner, but less than rushing to a destination. I feel a pressure build up in my abdomen as we walk up the incline of some street. I am almost positive I gave a quick look around making sure no one was in earshot of what I was about to deliver, and I had a feeling earshot in this case was going to be rather extensive. Satisfied that the coast was clear, I let loose.

So, as you have doubtless conjectured by now, someone was within earshot – someone other than my forgiving wife, Francesca. Where the hell did they come from? How did they get just behind us? Was this some sort of stealthy French ninja thing? Were they planning on picking our pockets? Why did I not even know they were there until Francesca informed me, and described their reaction to my gaucherie?

This is her description: the man stopped in his tracks, his arms came forward, and he bent over a bit, as though he’d just been gut shot by a sniper. The woman grabbed him and pulled him quickly across the street, giving Francesca a look that mingled unforgiving bile with wonder that she, Francesca, could spend any amount of time with this unmitigated boor – i.e., me. Well, I sometimes wonder that too, but so far so good.

When informed I’d just ruined a portion of the lives of two presumably innocent Parisians I felt contrite. After some time, as is my wont, I sought to mitigate my chagrin by finding some way of thinking about my behavior that would lessen my guilt, and put some of it on the Parisian couple. And so I thought back to a scene from Paris: The Luminous Years, describing the initial reaction of many Parisians to their first viewing of cubist art. A primitive cartoon illustrated this reaction, showing a chubby man walking past cubist paintings, getting more and more distressed with each canvas, so that when he reached the window at the end of the exhibit his only recourse was to jump through it to his death, thereby ending his misery. And in this way I turned my fart into a work of art, and the reaction of the people behind us into an example of hidebound criticism of my avant-garde masterpiece, unappreciated in its time, but surely leading to posthumous glory in some future decade. Meanwhile I am and shall remain a solitary unrecognized genius whose vision will shape a future I shall never see.

Cutting Edge and Exile

In an effort to maintain their reputations, and the concomitant privileges pertaining to same, some of those with any standing in the community of theoretical physics are surrendering their grasp on failed ideas, and jumping to the cutting edge with the embrace of decades old ideas (eg., Twistors; Schwinger’s stuff; …) founded by senescent – at best – eminences. Despite the reverence in which the founders of such ideas are held, the most important aspect of the ideas themselves is that they have languished; they never died, but neither did they ever catch fire, so insufficient attention has been paid to them over the decades to warrant finding irremediably fatal flaws, if such flaws exist. This makes them excellent intellectual hidey holes in which to sequester oneself during these End Days. Sigh. Heavy sigh. (As mentioned previously, most fugitive theorists have latched onto black holes and dark mattergy.)

Clearly I am bemoaning the deep backwater to which my own ideas have been relegated, but then, I am not an eminence. Nor do I display gravitas in my dealings with my fellow humans, so normal humans – those at all interested in theoretical physics, and who are even vaguely aware of me and my notions – find it difficult to imbue either me or my notions with any respect, so …

But my time is limited, so I am told, and I shall be denied senescence. Maybe that’s for the best. No one wants to witness me plum deeper into the depths of curmudgeonliness were I to grow too old, so let’s talk about something else. In particular, let’s talk about the Arts, paying close attention to those bits that I’ve found most enjoyable.

Although not as confirmed a reader as Tony Smith (see eulogy here), because I go out and play more than Tony ever did, or even wished to, I have read a fair bit in my life – although, as will become clear, much of my reading was in no way intended to advance my critical thinking, or to deepen my philosophical outlook, or even to help me be a better physicist. Much of it was frivolous, but – as I lack gravitas – this is to be expected.

Anyway, many moons ago a sister-in-law of mine, aware that I did read a fair bit, commented on that, and she asked the following question: “How many books have you read in your life? 50?”

To her 50 was a borderline inconceivably large number of books to have read in one lifetime. I did not tell her that I owned more than 70 books by a single author, many of which I’d read multiple times. But I did store this question in my hard drive. It made me aware of something to which I had hitherto given little thought: most people will read very few books in their lifetimes; some will read none. They may lead very contented lives, full and successful, but without books. It reminds me of a very funny refrigerator magnet I once saw. It depicted a young woman’s face, smiling whimsically, and thinking: “And yet another day I’ve gone without using calculus.”

So, if you haven’t guessed already, those 70 plus books were authored by P.G. Wodehouse. Were I to be exiled on the island of Elba, and told I could have the books of only a single author for all my years there, I would choose Wodehouse. This, even though it’s been a few years since I’ve read any. What’s important is choosing an author whose oeuvre is extensive, and whose books have already proven themselves worthy of repeated readings. Yes, I acknowledge that many serious thinkers consider his works frivolous, his stories contrived, their endings unrealistically contented, but all of that pleases my mildly autistic brain. However, the real reason for choosing Wodehouse is the language, the turns of phrase that leave one – if that one is me – basking blissfully in their linguistic refulgence. And it doesn’t hurt that all this brilliance illuminates goofy plots about goofy characters in a world free of cartels, serious thinkers, terrorists, and intellectual ossification. Replace each of those things with smiles, and you get the idea.

Meanwhile, the powers that exiled me on Elba were kind enough to give me a large screen TV and a DVD/Blu-ray player, but again, being of the persnickety punitive persuasion, they restrict me to discs associated with a single studio. Without hesitation I choose Ghibli. I mean, how many times can I watch Hulk smash, whereas Ghibli animations are transformative, transporting, and some other “t” word, meaning magical, to complete the alliteration. And unlike the majority of American films, Ghibli characters are not sharply divided into good and evil. Instead there is an ever fluctuating spectrum of behaviors manifesting self interest at times, and altruism at others, with occasional dollops of cynicism. The artwork and stories make me feel happy. Being myself concertedly non serious, that feeling pleases me. (Not that the stories aren’t at times very serious; the most serious of them all I have yet to view, and I shall likely never view it, for I am informed that in depicting humanity at its worst, it makes one feel the opposite of happy and contented. I refuse to spend my exile on Elba wallowing in humanity at its worst. Just leave me in peace.)

Art and music are more problematic. I’m quite fond of Cézanne, so I’d choose his works, but not with the same conviction as my literary and film choices. As to music, my tastes vary so rapidly that I’d simply ask for a guitar or piano instead, and failing that, I’d choose to sing or hum whatever struck my fancy. I’ve written a two chord piece of music for guitar that resonates so profoundly with my brain that I cannot conceive of it being bettered. When it comes to music, I am easily pleased, and just as easily annoyed. Best to leave me to my own devices.

I’ve been to Elba, by the way. It’s a pleasant place to be exiled. And I promise not to escape and attempt to be emperor. The responsibility of that would be just too much.

Battening down the hatches

“Never let anyone drive you crazy; it is nearby anyway and the walk is good for you.”

So, I’ve had an inspiration – nay, an epiphany. A revelation. Based on the weekly Boston area physics colloquia calendar – which includes Harvard, MIT, Boston University, Tufts, Brandeis … you know, a nontrivial and likely representative collection of first tier institutions – it is evident that theoretical particle physics is dead. Billions of dollars and euros and yuan and yen were spent on machinery intended to keep it alive, but the machinery failed to do so. Indeed, it hastened its demise. But that’s not the revelation. I’ve already covered the demise of particle physics.

Let’s recap. Big machines, intended to throw light on particle physics, as defined and envisioned by the mainstream, did nothing of the sort. On the contrary, the machines threw shade on their hopes and aspirations, a shade so deep that mainstream theorists wandered around quite blindly for a time, and then the survival instinct kicked in, and they rearranged their thinking. A lesson had been learned.

The lesson, sadly (here’s the revelation), is that evidence, if neutral or negative, is a bad thing, and given our inability to predict how evidence may turn out, we should migrate our intellectual efforts to areas that are largely immune to the vagaries of evidence, like black holes, dark matter, dark energy, and just about anything that involves the word “quantum” (even better if it also includes the word “interpretation”). That is safe ground indeed, and no amount of currency is likely sufficient to produce enough clear evidence to render these memes incontrovertibly pointless. Seven out of ten of the Boston area colloquia of this recent week were devoted to topics of this sort. The other three, by people resisting the poseur drift, were hard science topics that have half a chance or more of proving impactful some day.

“Would you tell me, please, which way I ought to go from here?”
“That depends a good deal on where you want to get to,” said the Cat.
“I don’t much care where—” said Alice.
“Then it doesn’t matter which way you go,” said the Cat.
“—so long as I get somewhere,” Alice added as an explanation.
“Oh, you’re sure to do that,” said the Cat, “if you only walk long enough.”

Concurrently, Nobel Prizes in physics are being strewn about, like garlands at a druidical rite, to work on black holes, thereby highlighting the notion that this research area is a safe harbor where one can wait out the experimental storm that sank the good ship Particle Physics, which roved too far out to sea in search of treasure (Nobel doubloons).

The theoretical black hole Nobel was awarded to 89 year old Roger Penrose for his Singularity Theorem. (And 87 year old Steven Weinberg also recently got a prize … ok, we’re all thinking the same thing, but let’s just not go there; it is what it is. I’m doomed never to even reach 80, so all the numerous prizes waiting in the wings for me will be awarded posthumously.) Singularities, like probabilities outside of the range 0 to 1, are mathematical hints that you’re doing something wrong. Penrose’s theorem, it would seem, proves that. Still, in the absence of a valid quantum theory of gravity with which to dispel geometric infinities, singularities are a juicy way for theoretical prima donnas to get those in control of the klieg lights to turn their brilliant illumination in their direction. Say cheese!

This reminds me. You know Hell? You know, it’s where you get dumped if you’ve been really really naughty during your life. Well, I now know what Hell is. It’s being locked in a room with two 10 year olds listening to them debate what happens when the unstoppable object runs into the immovable object … for all of eternity. Remind you of 2020 trends in theoretical physics? Huh? Does it?

But never mind all that. An academic helper of SWMBO recently confessed to her that – although he is as confirmed a Trekkie as you’re likely to encounter anywhere – if he had to spend the rest of his life rewatching just two TV series, he’d choose the superb animated series, Avatar: The Last Airbender, and Firefly, a series that was cancelled, but seamlessly concluded with the film Serenity. This young man is wise beyond his years. This is a brilliant choice. I say this with complete confidence, for I have already spent many contented hours watching and rewatching both, and will likely sneak in a few more viewings before I depart. At present, however, I am (re)working through the Ghibli catalogue – a safe harbor for a troubled mind.

I see dead physics

Pop physics stories on Flipboard have recently been dominated by conjectural fluff about black holes, wormholes, time travel, and finding planets in other galaxies. Consider that last one. Why is that a thing? Why wouldn’t there be planets in other galaxies? The probability that some nearby galaxy should lack planets is 0. The probability that there are scads of them is 1. The probability that one of those planets should be habitable is closer to 1 than 0. Should we discover that one is more than habitable, maybe even paradisiacal, the probability that we should ever be able to travel there and make it non paradisiacal, maybe even uninhabitable, is 0. Not that we couldn’t easily destroy its ecosystem; we just will never have the technology to make the trip. We will never have the technology to send humans to other stars in our own galaxy – or to wormholes, should these highly conjectural figments even exist, and even if in entering one you wouldn’t be crushed … So, no, we don’t live in a universe in which any sci fi plot device is possible, requiring only a little funding and effort. Looking for planets in other galaxies is like looking for sand on a beach. Take a plastic shovel and a bucket, et voilà.

Anyway, one thing I almost never see in pop physics stories anymore is any mention of high energy physics (HEP), and elementary particles. The vast majority of exceptions to this dearth are paeans of praise for the plethora of physics advances to which the 20th century gave birth. These stories extoll the virtues of the likes of Feynman, Dirac, and Heisenberg, virtually patting humanity on the back for the intellectual achievements of this ilk, while ignoring the what-have-you-done-lately meme. The science to which I devoted most of my research life is not dead, but certainly comatose. Well, it might be dead.

And speaking of the 20th century, 87 year old S Weinberg recently won a 3 million dollar Breakthrough Prize. Not that he doesn’t deserve prizes, but his breakthroughs occurred half a century ago. Although I may not live long enough to see it, when all the Standard Model Nobelists are gone, to whom will such prizes be given?

Arguably the last glory decade for HEP was the 1970s, and those involved in making it glorious are now more likely to appear in obituaries than in arxiv. Four ensuing decades of work on creating a viable theory of quantum gravity produced lots of interesting mathematics, but by and large fizzled as regards physics and reality. Wide-eyed youths who surrendered to the clarion call of this effort became lost in a swamp, and its viscous bogginess only relatively recently became apparent. They still dot the arxiv with matters supersymmetric, or stringy, especially as regards how these faded glories relate to black holes – but now far less often than earlier. Of course, the arxiv gatekeepers allow this dross into hep-th, only relegating to gen-phys ideas that do not relate to beloved failed dogmas. Sniff.

Still, no one cares anymore. Forbes main physics guy, Ethan Siegel, recently published an article entitled: Why Are Scientists So Cruel To New Ideas? This is filled with carefully crafted hurdles over which novel ideas must leap if they are to be taken seriously. But …

You know how in the olympics pole vaulters have to vault over ever higher bars, and with each raise of the bar vaulters get eliminated, and how sometimes the bar is set so high that no one can achieve victory? Well, the failure of the LHC vaulter to provide unequivocal evidence of new physics beyond the Standard Model has doomed the mainstream’s coterie of young theoretical vaulters to face a bar so inconceivably high it is invisible. And the slew of erstwhile enthusiastic pop sci onlookers have become disenchanted with this particular enterprise and mostly drifted away.

In Ethan’s article there is a picture of Bohr and Einstein lounging contentedly while they discuss and debate ideas that will be testable in their lifetimes. This picture exemplifies the notion that dead physicists are the best, as is dead physics. But it has utterly nothing to do with the milieu in which young physicists find themselves, and especially the mavericks amongst that group.

What to do? How do you avoid the slow decline of rigor into an almost spiritual acceptance of whatever notions fit your fancy, and evidence be damned? Because – let’s face it – evidence in any historically conventional sense is hard to find anymore. Is there a viable alternative to experimentally based progress?

Well, in my oh so humble and self-effacing opinion (IMosHaSEO) there is a way of assigning potential value to new ideas. It’s not the first time I’ve suggested this, but here is my notion of rigor based on mathematics:
1. Start with mathematics, and at least convince yourself that your chosen mathematics is unavoidable, resonant, and special;
2. Make your work on this mathematics unassailable;
3. Make the interpretation of this work as a contextual foundation of some part of theoretical physics unavoidable, or as much so as possible;
4. Ignore your own biases and let the mathematics speak – follow it – do not lead it or push it.

You know, as an example of applied mathematics that irks me, at the core of much of quantum mechanics is the mathematical notion of Hilbert space. There are infinitely many Hilbert spaces, but surely infinitely many of those are physically irrelevant. The problem is, Hilbert space is not really a space, nor even a mathematical object; it is an abstract collection of properties from which spaces and mathematical objects can be constructed. It is an entirely too flexible tool that suited the needs of 20th century theorists (now mostly dead) trying to make sense of things quantum. Again, IMosHaSEO Nature is not that nonspecific nor flexible. Nature is in fact not at all nonspecific. If you require yourself to use a Hilbert space, pick one, damn it. And make your mathematical application of it unassailable; and blah blah blah. And you might want to base it on the parallelizable spheres … if you want it to have anything to do with the universe we live in.

And by the way, Hilbert is dead too – beatified, sure, but … Even in science we are prone to thinking religiously.

Number theory for complete beach puppies

Not only optimal, but intended.

Caveat: I’m lazy, so grains of salt should be on hand while reading this blog. Due diligence and researching precedents are not my fortes. I depend on google for that, but I do not always avail myself of its services. If you’ve read my previous blogs, then you’re likely already prepared.

Let’s recap. These are a few of my favorite mathematical things:

Ⓐ Primes numbers;

➁︎ Parallelizable spheres in dimensions 1,2,4,8;

Ⅲ︎ Pretty pretty laminated lattices in dimensions 1,2,8,24.

In this episode I want to discuss prime numbers, and how I perceive my thinking about primes differs from – well, this is unclear. Let’s just say I’ve not encountered views similar to those I’m about to extoll, but then, this is not my field. I am merely an enthusiastic dilettante, like a puppy at the beach.

So, I recently happened upon an article suggesting that mathematicians will never stop finding novel ways to prove the prime number theorem. Which is what? Well, if you plot π(n) (the number of primes less than or equal to n) vs n, the plot rises in a manner that looks like it may relate to ln(n). This gave rise to the following approximation:

π(n) ~ n / ln(n).

This is not a very good approximation. I don’t care if the step function π(n) and the smooth function [n / ln(n)] cross each other an infinite number of times (I don’t know that they do), if you look at the table here comparing these two functions, it is clear that Nature is largely uninterested in the correspondence. Still, the ratio of these two functions has been proven to converge to 1, and that is the prime number theorem, which, according to that aforementioned article, is a thing that mathematicians make a hobby of repeatedly proving. Never mind that they diverge in an arithmetic sense; but how about that ratio!

Still, better approximations abound. Maybe [1 / ln(x)] is the density of primes at x, in which case the integral of that from 2 to n, denoted Li(n), should be a good approximation to π(n). In fact, this is a better approximation, but while it is true that π(n) / Li(n) converges to 1, Li(n) does not at all behave like it is Nature’s intended smooth approximation to π(n).

Does Nature have intended smooth approximations to number theory step functions? Is this a thing? Yes, it is. Never mind how I know; just take my word for it.

Evidently Riemann found an exact form for π(n), which is discussed here. Riemann was there first for quite a bit of modern mathematics, and the zeros of his famous zeta function play a part in his exact form. As this involves the as yet unproven Riemann Hypothesis, I’m going to pretend Riemann never existed and carry on discussing π(n) from my own peculiar perspective.

Peculiar perspective

To begin with, there is a problem with the notion of just counting primes as they pop up. What you really want is a measure of prime occurrences. I mean, consider the integers from 1 to 10. There are 4 primes in that collection: 2, 3, 5, 7. But there are more occurrences of primes than just those primes. What I mean by that is this: the prime 2 occurs 3 times, because 2³ = 8 is less than 10. And the prime 3 occurs twice, because 3² = 9 is less than 10. But how do we measure that? In my puppy on a beach investigations I decided upon lcm(n) (least common multiple of the integers from 1 to n). So, for example, lcm(10) = 2³ x 3² x 5 x 7 = 2520.

If you plot lcm(n) it looks a lot like a step function version of en. Did you see? Therefore without a shadow of a doubt Nature’s intended smooth approximation to ln[lcm(n)] is a simple linear function:

ln[lcm(n)] ~ n-1.

Why subtract 1? Well, I did that to enforce equality at n = 1. Nature concurred.

Want another test? Well, if

lcm(n) ~ en-1,


[lcm(n)]1/(n-1) ~ e.

Plot that. Take your time.

Anyway, I will go to my grave (in the not too distant future, unfortunately) fully convinced that these smooth approximations of these number theory step functions are Nature’s intended approximations. And I just don’t understand why in all my beach puppy readings in number theory – back in my beach puppy days – I never encountered this sort of thing. I’m not the only person to have noticed this. If I ever write a book entitled Number Theory for Complete Beach Puppies, this lcm(n) stuff will form its core.

Well, so I used this Nature’s intended approximation to derive a really really good (Nature’s intended) approximation to π(n). And that’s my story. You’ll find that I will be sticking to it like a limpet on a rock.

Pollyanna at the Church of Jordan and Penrose

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 = CHO. 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.

May I have this Resonance

Ad Nauseam

Subjective resonance is that fuzzy feeling you get when an idea clicks. That is, prior to ever encountering said idea, you are psychologically inclined to react positively to it, maybe because you are on the verge of developing its concepts yourself, or possibly because it meshes well with ideas of which you’ve already formed very positive opinions. It resonates.

Subjective resonance requires a mind capable of making judgements.

Objective resonance is why the Tacoma Narrows Bridge collapsed. Yes, it was designed (poorly) and built by humans, but once built, even if all the humans magically disappeared shortly thereafter, it was doomed eventually to experience a catastrophic resonant response to wind energy, and collapse.

Again, many experiments designed to detect some phenomenon do so by measuring quantity Y as some other quantity X is varied, the hope being that around some value of X the value of Y will spike above the background, indicating that Y resonates there – the meaning of this depending upon the context. This resonant reaction of Y at said value of X exists even in the absence of whatever theory is being tested, and even in the absence of the experimental apparatuses designed and built by the helpful experimentalists.

In mathematics there are also resonances, and these too exist with or without sentient beings to discover them. The Euclidean dimensions 1, 2, 4 and 8 are resonant (this is our X). In support of this contention I would yet again tediously point out the following gases that bubble up (the Y) from these dimensions:
• The parallelizable spheres …
• The Hurwitz division algebras, R, C, H and O
• All the classical Lie groups …
• Jordan algebras …
• Clifford algebras …
• Oh, hell, the list goes on and on …
Clearly I find subjectively resonant the notion that these dimensions are objectively resonant. Let’s move on.

These dimensions are associated with different resonances in dimensions 1, 2, 8 and 24, pertaining to lattice theory. Given its conceptual simplicity, lattice theory is extraordinarily complicated and difficult, resisting almost every effort to apply induction to make predictions about lattices in dimension n+1 from those in dimensions 1 through n.

This devilish complexity, however, markedly diminishes in dimensions 1, 2, 8 and 24, and specifically regarding the laminated lattices in those dimensions (which can be nicely represented by points in R¹, C¹, H², and O³). These lattices are, Λ₁ (equally spaced dots), Λ₂ (also denoted A₂), Λ₈ (also denoted E₈), and Λ₂₄ (the Leech lattice).

Nice lattices are more easily thought of as sphere packings, and even visualizable as such in dimensions 1, 2 and 3. A major lattice theory problem is finding the densest packing of spheres in any dimension. As it turns out, in dimensions 1, 2, 3, 8 and 24, we know that the densest sphere packings are associated with the laminated lattices in those dimensions (very little is known about all the other dimensions; for example, 49, and 343 … I could go on, but we don’t have time … ok, just one more: 117649). The dimension 3 is the outlier in the group, and the only reason anyone bothered proving this result (extremely difficult) is that we live in that dimension, and we have a vested interest in knowing the tightest way of packing oranges in a crate.

Of those other four dimensions, 8 and 24 are the most interesting, for they are relatively high dimensions, and knowledge of what goes on in neighboring dimensions is extremely limited, and likely to remain so. Proofs that the laminated lattices in dimensions 8 and 24 yield the tightest possible sphere packings can be found in The sphere packing problem in dimension 8, by Maryna S. Viazovska, and in The sphere packing problem in dimension 24, by Henry Cohn, et al.

Disclaimer 1: My own work in this area is much lower lying fruit than those two references, and my interest stems from my total conviction that our physical reality arises from resonant mathematics. Having demonstrated this in regard to dimensions 1, 2, 4 and 8, I spent some years dabbling with lattices in dimensions 1, 2, 8 and 24, finding a collection of relatively simple structures and results that had been missed in more highfalutin mathematics. I made a brave effort to connect this work to physics, but I’ll never know if I was on the right track.

Still, these resonant lattice dimensions crop up in physics from time to time (frequently as regards dimension 8), even in the mainstream. One example, and the reason I decided to write this blog, can be found in a recent article entitled, Sphere packing and quantum gravity.

Disclaimer 2: While I likely could, given the inclination to partake in months of study, reach a point where I came close to fully understanding Maryna S. Viazovska’s original 8-dimensional lattice proof, I have always found it difficult to go from interest, to inclination, to action. The initial hurdle is when I encounter modular forms, which are defined on the upper half of the complex plane. If any reason for this restriction has ever been proffered in any of the places I’ve encountered modular forms, I was too lazy to see it. If I dug deeper I’d figure it out. But I never have. Still, I gave a semi-educated skim of her entire paper, and was left with an overwhelming impression of mondo coolness.

However, I did not look over the paper extending Maryna’s proof to the Leech lattice in 24-dimensions (What would be the point? It’s clearly also mondo cool.), but I did look over the Sphere packing and quantum gravity paper, which looks at both the 8-d and 24-d lattice cases, and establishes “a precise relation between the modular bootstrap, used to constrain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry.” Keywords employed in that paper: “AdS-CFT Correspondence, Conformal Field Theory, Black Holes, Conformal and W Symmetry”.

Disclaimer 3: From early on in my so-called career as a theoretical physicist I avoided immersion in quantum field theory. I am aware that lots of attention has been paid to anti de Sitter (AdS) space and conformal field theory (CFT), not because it’s necessarily physically relevant (our universe is not anti de Sitter), but because it is more tractable than the de Sitter alternative, allowing the use of theoretical tools that people who study this kind of thing really like to use. In one of Peter W’s blogs the comments included much discussion of AdS/CFT, and I had to ask: “Am I correct in thinking that this surfeit of AdS/CFT talk is tantamount to looking for a nail in the dark under a lamppost because the light is better there, and anyway all you’ve got is a hammer, so the solution to your problem had better require a nail? Maybe two.” Peter replied: “Yes.”

So, anyway, the relevance of the Sphere packing and quantum gravity paper to reality is unimportant. What’s really cool is the remarkable correspondence between the modular bootstrap (don’t ask – it’s a thing used by a certain kind of theorist) and the sphere packing problem in 8 and 24 dimensions. It highlights something I said in Division Algebras, Lattices, Physics, Windmill Tilting: “… any theory, however bizarre, will find that things work best when the resonant dimensions, 1,2,4,8 and/or 24, are involved.” (Speaking of bizarre theories, the number of transverse dimensions in super string theory and bosonic string theory are 8 and 24, respectively. This is not unconnected to the quantum gravity paper cited above.)

So, that’s my take. You’ll find a nifty connection of the dimensions 8 and 24 to the division algebras and Clifford algebras in my last paper: Division Algebras, Clifford Algebras, Periodicity.


I think I may end my blogging here, as it can serve no further purpose (“further” … ha!). I’ve been spewing forth blather about mathematical resonance for decades now, but the notion has gained no traction. It’s stuck in the mud, a viscous, slippery stuff arising from a scientific culture that far more highly values the toolmaker Feynman than the architect Dirac.

Anyway, here’s a link to my presently favorite music video, the lyrics of which summarize my place in the world of theoretical physics. I want to be able to do that dance, but at my age I’d likely hurt myself.



I want to make a prediction, albeit one about which I harbor profound doubts that it will ever be verified by experiment or experience (it may simply never be recognized). Ok, focus (talking to myself here – I have a propensity to ramble … and there I go again).

So, right, here’s the deal. Let’s suppose some deity, or maybe the purported conscious universe communicating via a rock or burning bush or someat – or maybe just a super advanced bunch of aliens hovering in a spaceship above the earth, their scientific acumen and trustworthiness attested to by the fact that they’ve navigated to the earth from somewhere far far away … so, anyway, one of these things – or something else the opinions of which we should find it difficult to contest – it hands us on a silver platter a correct and complete TOE, blithely assuming we have the background to understand it, and are willing to give it a go.

So, right, here’s the deal – the real deal. I predict that by the vast majority of theoretical physicists this TOE would not be understood. Furthermore, while some few might have a notion of what it means, there is a 100% chance the TOE will contain concepts at odds with their cherished prejudices about how the universe should be described. The end result of all this generalized befuddlement and disaffection will be a tacit agreement to ignore this new TOE, to carry on tapping away at things that by mutual consent we label legitimate veins of truth.

Some might grab bits of the TOE from the silver platter on occasion and incorporate them into their own work, but without attribution, for the deity, or consciousness, or alien race, will have made the huge mistake of not publishing in a respected journal, and even worse, will not have the TOE listed in the arXiv, either because they didn’t bother trying, or because the arXiv gatekeepers saw their unusual and far too mathematical notions as unworthy and disruptive. (After all, the purported TOE doesn’t have a single Feynman diagram (all hail QFT!), so, crikey … there you are.)

Thanos was an idiot

So, let’s assume the TOE provider is an alien with the temperament of Thanos, and as punishment for our oblivious adhesion to orthodoxy decides to turn half of humanity to dust. The actual fictional Thanos did this to all life throughout the universe, and 5 years after having done so we saw an aerial view of a sports stadium – I think it was a baseball stadium in New York, so who cares … but anyway. The stadium looked unused, overgrown and rundown, for having lost half of humanity has caused … what did it cause?

1972. In the year 1972 the population of the earth was half what it is today. Was there no baseball in 1972? I seem to recall that there was baseball. I recall it. It was only 48 years ago. I was alive then. So, had the Avengers not mucked up his plans, Thanos’s big finger snap biocide would on the earth, within his lifetime, have been completely undone and the population back to where it is now. What an idiot. And anyway, he went off afterward to a seemingly uninhabited planet and became a contented farmer at peace with the universe. He could have done that without the whole finger snap incident. (And those 5 years later humanity was exceptionally mopey and depressed. Really? Think back to World War I and the 1918/19 flu epidemic that followed. Tens of millions died. Was all that horror followed by the Mopey 20s? Nay nay, it was followed by the Roaring 20s, a time of exuberance and invention. Anyway, just saying – human nature, right?)

In any case, if you’re handed a TOE on a silver platter by an alien – or even a deity – with the temperament of Thanos, act obsequious and grateful, and don’t go ignoring it until later, when the entity is well out of earshot.

All your hearts now seem so far from me
It hardly seems to matter now
And the nurse will tell you lies
Of a kingdom beyond the skies
But I am lost within this half-world
It hardly seems to matter now
Play me my song
Here it comes again
Play me my song
Here it comes again
Just a little bit
Just a little bit more time
Time left to live out my life
Play me my song
Here it comes again
Play me my song
Here it comes again

Discontented blather

Ethan Siegel, science guy for Forbes magazine, wrote an article entitled: No, The Universe Is Not Purely Mathematical In Nature. I couldn’t let this pass, so supplied the following comment everywhere I encountered the article, and comments were allowed:

“This by and large relates to the interface between reality and sentience – in this case, human. Delete all sentience from the universe and very little changes; the universe goes on as it had, despite there being nothing in the universe trying to understand. There are rules, and they are not based on magic. There are mathematical concepts that transcend any requirement of sentience to think them up. Prime numbers is a simple example. The universe isn’t going to quit working just because no sentient being is testing its validity, and mathematics underlies how it works, and even why it exists.”

I generally like Ethan’s posts, but he is mainstream, accustomed to thinking … well, here’s a quote from Sydney Coleman I encountered on LinkedIn:

“The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.”

The poster agreed with this, as did many of the commenters. The point is, Ethan’s view of what constitutes physics, and what constitutes mathematics, … ah, I see I’m about to go down a very well trodden path. So, uh, yeah … Thanos. What an idiot.

Here’s a quote from a recent Scientific American blog about scientific advances – especially the big ones:

“In fact, it may be necessary for pioneers to face the headwind of rejection for a while, or their idea might eventually be credited to the mainstream. An innovator has to persevere through an initial denial phase, as Weiss did, during which the mainstream rejects the idea so publicly that the proposer can later wear the rejection as a badge of honor. Under more typical circumstances, when a new idea is simply ignored, there is a real danger that mainstream proponents will claim it for themselves after introducing some cosmetic variations to its presentation.”

Over the last 20 years I’ve seen ominous signs that history is trending in this direction regarding my own work in theoretical physics. Is this hubris? Pshaw! My ideas are fundamentally correct, so no. (My PhD advisor used to say I had a hotline to god, so there you are. I have the silver platter to prove it.) People aware of my work are occasionally inspired, seemingly, not to build on its intractable ideas, but inspired by its Gestalt – by the idea that there could be an algebraic foundation to a TOE hitherto undiscovered and unincorporated – so they construct their own. It becomes a hobby among many fringe theorists – like being a ham radio enthusiast, or a computer hacker. Whatever. Cosmetic variations indeed.

How does one prepare oneself psychologically for this looming chunk of ineluctability? What are we even talking about? Who says things like “looming chunk of ineluctability”, and what does this refer to? I’m leaving. My Victory Garden isn’t going to grow itself.