# Complex Equiangular Lines: The Unusual Shapes of Quantum Physics

How many intersecting lines can you draw such that the angle made by any pair is the same as the angle made by any other pair? What if you try in 3 dimensions, or 4, or 5? What if you let your coordinates become complex numbers? And what does all this have to do with quantum probability?!

As the kids say, “Like, quantum and subscribe!”

# Recollection

Mathematics was always my worst subject at school, right up until I went to college. I’ve heard a similar story from other physicists. I don’t know how useful my speculation about it will be to anybody else, but I think this is the reason why.

The kind of mistake I was prone to making, and the flaws in the way mathematics was taught, meshed perfectly. Carelessness cost more in math than in anything else, on the whole. If I was writing a history essay and I happened to swap Elba and St. Helena, I might only get docked a couple points out of a hundred, or perhaps none at all if the teacher had too many papers to grade. But if I wandered away from my pre-algebra homework, and upon my return my garishly awful handwriting had turned absolute-value bars into ordinary parentheses, my calculations would be completely off from that point onward. Nor did any of my teachers pick up on my problem — “Blake, you’ve got to be more careful!” — which makes me suspect that they weren’t much better at identifying what went wrong for other students, either.

In history and to a large extent in science, I was able to get by all through middle and high school with what I had learned out of books and documentaries on my own. (I was extraordinarily lucky to have a family that already had plenty of books around, and the means and the sense to provide me with more as I packed their contents into my brains.) I don’t think I had to learn anything in school that came across as wholly new. Everything was at most an elaboration of a topic I had already seen, something I’d grasped from a Larry Gonick cartoon guide, let’s say, done up with a few more details that might just have been included for the sake of having homework problems to assign. Algebra and geometry and trig and calculus, though, came closer to asking for a genuine production on my part.

Techniques of checking one’s work, which might have helped me to become a bit more generally competent, were either not taught or not motivated. “Plug your value of $x$ back in and check” might have been the last step of a few algebra exercises, but only because it was part of the rubric, devised to add another thing that could be graded.

The weird thing is that I had a sense of the importance of the mathematics, of the motivation for it. I knew why Kepler had cared about sines and cosines — to hear the music of the spheres, to turn comets from signs of dread into those of wonder. Exponentials tracked the explosion of populations and the decay of radioisotopes, each ominous in its own way. The subject offered wonders of pure thought and marvels of application. At the time, schoolwork merely seemed disconnected from those treasures which I saw in secondhand outline. Now, in retrospect, it appears almost a parody of them.

# From Gender to Gleason

… or, The Case of Adam Becker’s What Is Real? (2018).

It is easy to argue that the founders of quantum mechanics made statements which are opaque and confusing. It is fair to say that their philosophical takes on the subject are not infrequently unsatisfying. We can all use reminders that human flaws and passions are a part of physics. So, it would be nice to have a popular book on these themes, one that makes no vital omissions, represents its sources accurately and lives up to its own ideals.

# The Rise of Ironic Physics and/or Machine Physicists?

Over on his blog, Peter Woit quotes a scene from the imagination of John Horgan, whose The End of Science (1996) visualized physics falling into a twilight:

A few diehards dedicated to truth rather than practicality will practice physics in a nonempirical, ironic mode, plumbing the magical realm of superstrings and other esoterica and fret­ting about the meaning of quantum mechanics. The conferences of these ironic physicists, whose disputes cannot be experimentally resolved, will become more and more like those of that bastion of literary criticism, the Modern Language Association.

OK (*cracks knuckles*), a few points. First, “fretting about the meaning of quantum mechanics” has, historically, been damn important. A lot of quantum information theory came out of people doing exactly that, just with equations. The productive way of “fretting” involves plumbing the meaning of quantum mechanics by finding what new capabilities quantum mechanics can give you. Let’s take one of the least blue-sky applications of quantum information science: securing communications with quantum key distribution. Why trust the security of quantum key distribution? There’s a whole theory behind the idea, one which depends upon the quantum de Finetti theorem. Why is there a quantum de Finetti theorem in a form that physicists could understand and care about? Because Caves, Fuchs and Schack wanted to prove that the phrase “unknown quantum state” has a well-defined meaning for personalist Bayesians.

This example could be augmented with many others. (I selfishly picked one where I could cite my own collaborator.)

It’s illuminating to quote the passage from Horgan’s book just before the one that Woit did:

This is the fate of physics. The vast majority of physicists, those employed in industry and even academia, will continue to apply the knowledge they already have in hand—inventing more versatile lasers and superconductors and computing devices—without worrying about any underlying philosophical issues.

But there just isn’t a clean dividing line between “underlying philosophical issues” and “more versatile computing devices”! In fact, the foundational question of what the nature of “quantum states” really are overlaps with the question of which quantum computations can be emulated on a classical computer, and how some preparations are better resources for quantum computers than others. Flagrantly disregarding attempts to draw a boundary line between “foundations” and “applications” is my day job now, but quantum information was already getting going in earnest during the mid-1990s, so this isn’t a matter of hindsight. (Feynman wasn’t the first to talk about quantum computing, but he was certainly influential, and the motivations he spelled out were pretty explicitly foundational. Benioff, who preceded Feynman, was also interested in foundational matters, and even said as much while building quantum Hamiltonians for Turing machines.) And since Woit’s post was about judging whether a prediction held up or not, I feel pretty OK applying a present-day standard anyway.

In short: Meaning matters.

But then, Horgan’s book gets the Einstein–Podolsky—Rosen thought-experiment completely wrong, and I should know better than to engage with what any book like that on the subject of what quantum mechanics might mean.

To be honest, Horgan is unfair to the Modern Language Association. Their convention program for January 2019 indicates a community that is actively engaged in the world, with sessions about the changing role of journalism, how the Internet has enabled a new kind of “public intellectuals”, how to bring African-American literature into summer reading, the dynamics of organized fandoms, etc. In addition, they plainly advertise sessions as open to the public, which I can only barely imagine a physics conference doing more than a nominal jab at. Their public sessions include a film screening of a documentary about the South African writer and activist Peter Abrahams, as well as workshops on practical skills like how to cite sources. That’s not just valuable training, but also a topic that is actively evolving: How do you cite a tweet, or an archived version of a Wikipedia page, or a post on a decentralized social network like Mastodon?

Dragging the sciences for supposedly resembling the humanities has not grown more endearing since 1996.
Continue reading The Rise of Ironic Physics and/or Machine Physicists?

# New Paper Dance (-ing With Myself)

B. C. Stacey, “Misreading EPR: Variations on an Incorrect Theme” [arXiv:1809.01751].

Notwithstanding its great influence in modern physics, the EPR thought-experiment has been explained incorrectly a surprising number of times.

22 pages; an unknown number of bridges burned.

# Two Recent Items Concerning Wikipedia

A few years ago, I found a sentence in a Wikipedia page that irritated me so much, I wrote a 25-page article about it. Eventually, I got that article published in the Philosophical Transactions of the Royal Society. On account of all this, friends and colleagues sometimes send me news about Wikipedia, or point me to strange things they’ve found there. A couple such items have recently led me to Have Thoughts, which I share below.

This op-ed on the incomprehensibility of Wikipedia science articles puts a finger on a real problem, but its attempt at explanation assumes malice rather than incompetence. Yes, Virginia, the science and mathematics articles are often baffling and opaque. The Vice essay argues that the writers of Wikipedia’s science articles use the incomprehensibility of their prose as a shield to keep out the riffraff and maintain the “elite” status of their subject. I don’t buy it. In my opinion, this hypothesis does not account for the intrinsic difficulty of explaining science, nor for the incentive structures at work. Wikipedia pages grow by bricolage, small pieces of cruft accumulating over time. “Oh, this thing says [citation needed]. I’ll go find a citation to fill it in, while my coffee is brewing.” This is not conducive to clean pedagogy, or to a smooth transition from general-audience to specialist interest.

Have no doubt that a great many scientists are terrible at communication, but we can also imagine a world in which Wikipedia would attract the scientists that actually are good at communication.

There’s communication, and then there’s communication. (We scientists usually get formal training in neither.) I know quite a few scientists who are good at outreach. They work hard at it, because they believe it matters and they know that’s what it takes. Almost none of them have ever mentioned editing Wikipedia (even the one who used his science blog in his tenure portfolio). Thanks to the pressures of academia, the calculation always favors a mode of outreach where it’s easier to point to what you did, so you can get appropriate credit for it.

Thus, there might be a momentary impulse to make small-scale improvements, but there’s almost no incentive to effect changes that are structured on a larger scale — paragraphs, sections, organization among articles. This is a good incentive system for filling articles with technical minutiae, like jelly babies into a bag, but it’s not a way to plan a curriculum.

The piece in Vice says of a certain physics article,

I have no idea who the article exists for because I’m not sure that person actually exists: someone with enough knowledge to comprehend dense physics formulations that doesn’t also already understand the electroweak interaction or that doesn’t already have, like, access to a textbook about it.

You’d be surprised. It’s fairly common to remember the broad strokes of a subject but need a reference for the fiddly little details.

Writers don’t just dip in, produce some Wikipedia copy, and bounce.

I’m pretty sure this is … actually not borne out by the data? Like, many contributors just add little bits when they are strongly motivated, while the smaller active core of persistent editors clean up the content, get involved in article-improvement drives, wrangle behind the scenes, etc.

[EDIT TO ADD (24 November): To say it another way, both the distribution of edits per article and edits per editor are “fat tailed, which implies that even editors and articles with small numbers of edits should not be neglected.” Furthermore, most edits do not change an article’s length, or change it by only a small amount. The seeming tendency for “fewer editors gaining an ever more dominant role” is a real concern, but I doubt the opacity of technical articles is itself a tool of oligarchy. Indeed, I suspect that other factors contribute to the “core editor” group becoming more insular, one being the ease with which policies originally devised for good reasons can be weaponized.]

If you want “elitism,” you shouldn’t look in the technical prose on the project’s front end. Instead, you should go into the backroom. From what I’ve seen and heard, it’s very easy to run afoul of an editor who wants to lord over their tiny domain, and who will sling around policies and abbreviations and local jargon to get their way. Any transgression, or perceived transgression, is an excuse to revert.

Just take a look at “WP:PROF” — the “notability guideline” for evaluating whether a scholar merits a Wikipedia page. It’s almost 3500 words, laying out criteria and then expounding upon their curlicues. And if you create an article and someone else decides it should be deleted, you had better be familiar with the Guide to deletion (roughly 6700 words), which overlaps with the Deletion process documentation (another 4700 words). More than enough regulations for anyone to petulantly sling around until they get their way!

And on the subject of deletion, over on Mastodon the other day I got into a chat about the story of Günter Bechly, a paleontologist who went creationist and whose Wikipedia page was recently toasted. The incident was described by Haaretz thusly:

If Bechly’s article was originally introduced due to his scientific work, it was deleted due to his having become a poster child for the creationist movement.

I strongly suspect that it would have been deleted if it had been brought to anyone’s attention for any other reason, even if Bechly hadn’t gone creationist. His scientific work just doesn’t add up to what Wikipedia considers “notability,” the standard codified by the WP:PROF rulebook mentioned above. Nor were there adequate sources to write about his career in Wikipedia’s regulation flat, footnoted way. The project is clearly willing to have articles on creationists, if the claims in them can be sourced to their standards of propriety: Just look at their category of creationists! Bechly’s problem was that he was only mentioned in passing or written up in niche sources that were deemed unreliable.

If you poke around that deletion discussion for Bechly’s page, you’ll find it links to a rolling list of such discussions for “Academics and educators,” many of whom seem to be using Wikipedia as a LinkedIn substitute. It’s a mundane occurrence for the project.

And another thing about the Haaretz article. It mentions sockpuppets arriving to speak up in support of keeping Bechly’s page:

These one-time editors’ lack of experience became clear when they began voting in favor of keeping the article on Wikipedia – a practice not employed in the English version of Wikipedia since 2016, when editors voted to exchange the way articles are deleted for a process of consensus-based decision through discussion.

Uh, that’s been the rule since 2005 at least. Not the most impressive example of Journalisming.

# To Thems That Have

Occasionally, I think of burning my opportunities of advancing in the physics profession — or, more likely, just burning my bridges with Geek Culture(TM) — by writing a paper entitled, “Richard Feynman’s Greatest Mistake”.

I did start drafting an essay I call “To Thems That Have, Shall Be Given More”. There are a sizable number of examples where Feynman gets credit for an idea that somebody else discovered first. It’s the rich-get-richer of science.
Continue reading To Thems That Have

# 17 Equations that Clogged My Social-Media Timeline

An image burbled up in my social-media feed the other day, purporting to be a list of “17 Equations that Changed the World.” It’s actually been circulating for a while (since early 2014), and purports to summarize the book by that name written by Ian Stewart. This list is typo-ridden, historically inaccurate and generally indicative of a lousy knowledge-distribution process that lets us down at every stage, from background research to fact-checking to copy-editing.
Continue reading 17 Equations that Clogged My Social-Media Timeline

# A Potent Quotable

Here is physicist John Archibald Wheeler, interviewed in the documentary The Creation of the Universe (1985):

There’s nothing deader than an equation. You write that down in a square on a tile floor. And on another tile on the floor you write down another equation, which you think might be a better description of the Universe. And you keep on writing down equations hoping to get a better and better equation for what the Universe is and does.

And then, when you’ve worked your way out to the end of the room and have to step out, you wave your wand and tell the equations to fly.

And not one of them will put on wings and fly.

Yet the Universe flies!

It has a life to it that no equation has, and that life to it is a life with which we are also tied up.

I saw that documentary as a kid, and that little speech was one part that stuck with me ever after. For the story of how Wheeler made this point to his physics classes, see arXiv:1405.2390, page 292.

# Concerning “Great Books”

Shimer College: the worst school in America?

Subhead: This tiny, eccentric institution in Chicago was just voted the worst place to study in America. But does Shimer, which shuns lectures and has no societies or clubs, deserve such an accolade? Jon Ronson went there to investigate.

In the body, we have a bit more detail:

This is a ‘great books’ college. The great books of the western tradition, not the professors, are the teachers: Da Vinci’s Notebooks and Aristotle’s Poetics and Homer’s Odyssey and de Beauvoir’s Ethics of Ambiguity and Kafka and Derrida and Nietzsche and Freud and Marx and Machiavelli and Shakespeare and the Bible.

# Reflections

Prompted by this review of Colin McGinn’s Basic Structures of Reality (2011), I read a chapter, courtesy the uni library. It was endumbening. To the extent that he ever has a point, he says in many words what others have said more clearly in few. He confuses the pedagogy of a particular introductory book with the mature understanding of a subject, displays total ignorance of deeper treatments of his chosen topic, blunders into fallacies, and generally leaves one with the impression that he has never done a calculation in all the time he spent “studying physics”. Truly an amazing achievement.

A few years ago, I might have blogged my way through the whole darn book. I must be getting old (“REALLY? NO WAY!” declares my weak knee). But is it a healthy and mature sense of priorities, or a senescent academic crustiness? Have I become one of those people, concerned with my vita to the exclusion of all else? Dark thoughts for this cold autumn evening, dark as our current season of superhero movies—Fimbulwinter 3: Flame of Despair….

You know what I’d like to see? I’d like to have all the course materials necessary for a good, solid undergraduate physics degree available online, free to access and licensed in a way which permits reuse and remixing. I’d like it all in one place, curated, with paths through it mapped out to define a curriculum. When I say all the course materials, I mean that this webzone should have online textbooks; copies of, or at least pointers to, relevant primary literature; video lectures; simulation codes; sample datasets on which to practice analysis; homework and exam problems with worked-out solutions; interactive quizzes, so we can be trendy; and ways to order affordable experimental equipment where that is possible, e.g., yes on diffraction gratings, but probably no on radioactive sources. I’m talking about physics, because that’s what I nominally know about, but I’d like this to encompass the topics which I got sent to other departments to learn about, like the Mathematics Department’s courses in single- and multivariable calculus, differential equations, linear algebra, group theory, etc.

One way to think about it is this: suppose you had to teach a physics class to first- or second-year undergraduates. Could you get all the textual materials you need from Open-Access sources on the Web? Would you know where to look?

What with Wikipedia, OpenCourseWare, review articles on the arXiv, science blogs, the Khaaaaaan! Academy and so forth, we probably already have a fair portion of this in various places. But the operative word there is various. I, at least, would like it gathered together so we can know what’s yet to be done. With a project like, say, Wikipedia, stuff gets filled in based on what people feel like writing about in their free time. So articles grow by the cumulative addition of small bits, and “boring” content — parts of the curriculum which need to be covered, but are seldom if ever “topical” — doesn’t get much attention.

I honestly don’t know how close we are to this ideal. And, I don’t know what would be the best infrastructure for bringing it about and maintaining it. Idle fantasies and pipe dreams!

I’d like to have this kind of resource, not just for the obvious practical reasons, but also because it would soothe my conscience. I’d like to be able to tell people, “Yes, physics and mathematics are difficult, technical subjects. The stuff we say often sounds like mystical arcana. But, if you want to know what we know, all we ask is time and thinking — we’ve removed every obstacle to your understanding which we possibly can.”

I don’t think this would really impact the physics cranks and crackpots that much, but that’s not the problem I’m aiming to (dreaming that we will) solve. Disdain for mathematics is one warning sign of a fractured ceramic, yes: I’ve lost count of the number of times I’ve seen websites claiming to debunk Einstein “using only high-school algebra!” We could make learning the mathematical meat of physics easier, but that won’t significantly affect the people whose crankishness is due to personality and temperament. Free calculus lessons, no matter how engaging, won’t help those who’ve dedicated themselves to fighting under the banner of Douche Physik.

Alchemists work for the people. —Edward Elric

# “More Decimal Digits”

On occasion, somebody voices the idea that in year $N$, physicists thought they had everything basically figured out, and that all they had to do was compute more decimal digits. I won’t pretend to know whether this is actually true for any values of $N$ — when did one old man’s grumpiness become the definitive statement about a scientific age? — but it’s interesting that not every physicist with an interest in history has supported the claim.

One classic illustration of how the old guys with the beards knew their understanding of physics was incomplete involves the specific heats of gases. How much does a gas warm up when a given amount of energy is poured into it? The physics of the 1890s was unable to resolve this problem. The solution, achieved in the next century, required quantum mechanics, but the problem was far from unknown in the years before 1900. Quoting Richard Feynman’s Lectures on Physics (1964), volume 1, chapter 40, with hyperlinks added by me: