My All 2020 Mathematics A to Z: Renormalization


I have again Elke Stangl, author of elkemental Force, to thank for the subject this week. Again, Stangl’s is a blog of wide-ranging theme interests. And it’s got more poetry this week again, this time haikus about the Dirac delta function.

I also have Kerson Huang, of the Massachusetts Institute of Technology and of Nanyang Technological University, to thank for much insight into the week’s subject. Huang published this A Critical History of Renormalization, which gave me much to think about. It’s likely a paper that would help anyone hoping to know the history of the technique better.

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Renormalization.

There is a mathematical model, the Ising Model, for how magnets work. The model has the simplicity of a toy model given by a professor (Wilhelm Lenz) to his grad student (Ernst Ising). Suppose matter is a uniform, uniformly-spaced grid. At each point on the grid we have either a bit of magnetism pointed up (value +1) or down (value -1). It is a nearest-neighbor model. Each point interacts with its nearest neighbors and none of the other points. For a one-dimensional grid this is easy. It’s the stuff of thermodynamics homework for physics majors. They don’t understand it, because you need the hyperbolic trigonometric functions. But they could. For two dimensions … it’s hard. But doable. And interesting. It describes important things like phase changes. The way that you can take a perfectly good strong magnet and heat it up until it’s an iron goo, then cool it down to being a strong magnet again.

For such a simple model it works well. A lot of the solids we find interesting are crystals, or are almost crystals. These are molecules arranged in a grid. So that part of the model is fine. They do interact, foremost, with their nearest neighbors. But not exclusively. In principle, every molecule in a crystal interacts with every other molecule. Can we account for this? Can we make a better model?

Yes, many ways. Here’s one. It’s designed for a square grid, the kind you get by looking at the intersections on a normal piece of graph paper. Each point is in a row and a column. The rows are a distance ‘a’ apart. So are the columns.

Now draw a new grid, on top of the old. Do it by grouping together two-by-two blocks of the original. Draw new rows and columns through the centers of these new blocks. Put at the new intersections a bit of magnetism. Its value is the mean of whatever the four blocks around it are. So, could be 1, could be -1, could be 0, could be ½, could be -½. There’s more options. But look at what we have. It’s still an Ising-like model, with interactions between nearest-neighbors. There’s more choices for what value each point can have. And the grid spacing is now 2a instead of a. But it all looks pretty similar.

And now the great insight, that we can trace to Leo P Kadanoff in 1966. What if we relabel the distance between grid points? We called it 2a before. Call it a, now, again. What’s important that’s different from the Ising model we started with?

There’s the not-negligible point that there’s five different values a point can have, instead of two. But otherwise? In the operations we do, not much is different. How about in what it models? And there it’s interesting. Think of the original grid points. In the original scaling, they interacted only with units one original-row or one original-column away. Now? Their average interacts with the average of grid points that were as far as three original-rows or three original-columns away. It’s a small change. But it’s closer to reflecting the reality of every molecule interacting with every other molecule.

You know what happens when mathematicians get one good trick. We figure what happens if we do it again. Take the rescaled grid, the one that represents two-by-two blocks of the original. Rescale it again, making two-by-two blocks of these two-by-two blocks. Do the same rules about setting the center points as a new grid. And then re-scaling. What we have now are blocks that represent averages of four-by-four blocks of the original. And that, imperfectly, let a point interact with a point seven original-rows or original-columns away. (Or farther: seven original-rows down and three original-columns to the left, say. Have fun counting all the distances.) And again: we have eight-by-eight blocks and even more range. Again: sixteen-by-sixteen blocks and double the range again. Why not carry this on forever?

This is renormalization. It’s a specific sort, called the block-spin renormalization group. It comes from condensed matter physics, where we try to understand how molecules come together to form bulks of matter. Kenneth Wilson stretched this over to studying the Kondo Effect. This is a problem in how magnetic impurities affect electrical resistance. (It’s named for Jun Kondo.) It’s great work. It (in part) earned Wilson a Nobel Prize. But the idea is simple. We can understand complex interactions by making them simple ones. The interactions have a natural scale, cutting off at the nearest neighbor. But we redefine ‘nearest neighbor’, again and again, until it reaches infinitely far away.

This problem, and its solution, come from thermodynamics. Particularly, statistical mechanics. This is a bit ahistoric. Physicists first used renormalization in quantum mechanics. This is all right. As a general guideline, everything in statistical mechanics turns into something in quantum mechanics, and vice-versa. What quantum mechanics lacked, for a generation, was logical rigor for renormalization. This statistical mechanics approach provided that.

Renormalization in quantum mechanics we needed because of virtual particles. Quantum mechanics requires that particles can pop into existence, carrying momentum, and then pop back out again. This gives us electromagnetism, and the strong nuclear force (which holds particles together), and the weak nuclear force (which causes nuclear decay). Leave gravity over on the side. The more momentum in the virtual particle, the shorter a time it can exist. It’s actually the more energy, the shorter the particle lasts. In that guise you know it as the Uncertainty Principle. But it’s momentum that’s important here. This means short-range interactions transfer more momentum, and long-range ones transfer less. And here we had thought forces got stronger as the particles interacting got closer together.

In principle, there is no upper limit to how much momentum one of these virtual particles can have. And, worse, the original particle can interact with its virtual particle. This by exchanging another virtual particle. Which is even higher-energy and shorter-range. The virtual particle can also interact with the field that’s around the original particle. Pairs of virtual particles can exchange more virtual particles. And so on. What we get, when we add this all together, seems like it should be infinitely large. Every particle the center of an infinitely great bundle of energy.

Renormalization, the original renormalization, cuts that off. Sets an effective limit on the system. The limit is not “only particles this close will interact” exactly. It’s more “only virtual particles with less than this momentum will”. (Yes, there’s some overlap between these ideas.) This seems different to us mere dwellers in reality. But to a mathematical physicist, knowing that position and momentum are conjugate variables? Limiting one is the same work as limiting the other.

This, when developed, left physicists uneasy. It’s for good reasons. The cutoff is arbitrary. Its existence, sure, but we often deal with arbitrary cutoffs for things. When we calculate a weather satellite’s orbit we do not care that other star systems exist. We barely care that Jupiter exists. Still, where to put the cutoff? Quantum Electrodynamics, using this, could provide excellent predictions of physical properties. But shouldn’t we get different predictions with different cutoffs? How do we know we’re not picking a cutoff because it makes our test problem work right? That we’re not picking one that produces garbage for every other problem? Read the writing of a physicist of the time and — oh, why be coy? We all read Richard Feynman, his QED at least. We see him sulking about a technique he used to brilliant effect.

Wilson-style renormalization answered Feynman’s objections. (Though not to Feynman’s satisfaction, if I understand the history right.) The momentum cutoff serves as a scale. Or if you prefer, the scale of interactions we consider tells us the cutoff. Different scales give us different quantum mechanics. One scale, one cutoff, gives us the way molecules interact together, on the scale of condensed-matter physics. A different scale, with a different cutoff, describes the particles of Quantum Electrodynamics. Other scales describe something more recognizable as classical physics. Or the Yang-Mills gauge theory, as describes the Standard Model of subatomic particles, all those quarks and leptons.

Renormalization offers a capsule of much of mathematical physics, though. It started as an arbitrary trick to avoid calculation problems. In time, we found a rationale for the trick. But found it from looking at a problem that seemed unrelated. On learning the related trick well, though, we see they’re different aspects of the same problem. It’s a neat bit of work.


This and all the other 2020 A-to-Z essays should be at this link. Essays from every A-to-Z series should be gathered at this link. I am looking eagerly for topics for the letters S, T, and U, and am scouting ahead for V, W, and X topics also. Thanks for your thoughts, and thank you for reading.

My 2018 Mathematics A To Z: Nearest Neighbor Model


I had a free choice of topics for today! Nobody had a suggestion for the letter ‘N’, so, I’ll take one of my own. If you did put in a suggestion, I apologize; I somehow missed the comment in which you did. I’ll try to do better in future.

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Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Nearest Neighbor Model.

Why are restaurants noisy?

It’s one of those things I wondered while at a noisy restaurant. I have heard it is because restauranteurs believe patrons buy more, and more expensive stuff, in a noisy place. I don’t know that I have heard this correctly, nor that what I heard was correct. I’ll leave it to people who work that end of restaurants to say. But I wondered idly whether mathematics could answer why.

It’s easy to form a rough model. Suppose I want my brilliant words to be heard by the delightful people at my table. Then I have to be louder, to them, than the background noise is. Fine. I don’t like talking loudly. My normal voice is soft enough even I have a hard time making it out. And I’ll drop the ends of sentences when I feel like I’ve said all the interesting parts of them. But I can overcome my instinct if I must.

The trouble comes from other people thinking of themselves the way I think of myself. They want to be heard over how loud I have been. And there’s no convincing them they’re wrong. If there’s bunches of tables near one another, we’re going to have trouble. We’ll each by talking loud enough to drown one another out, until the whole place is a racket. If we’re close enough together, that is. If the tables around mine are empty, chances are my normal voice is enough for the cause. If they’re not, we might have trouble.

So this inspires a model. The restaurant is a space. The tables are set positions, points inside it. Each table is making some volume of noise. Each table is trying to be louder than the background noise. At least until the people at the table reach the limits of their screaming. Or decide they can’t talk, they’ll just eat and go somewhere pleasant.

Making calculations on this demands some more work. Some is obvious: how do you represent “quiet” and “loud”? Some is harder: how far do voices carry? Grant that a loud table is still loud if you’re near it. How far away before it doesn’t sound loud? How far away before you can’t hear it anyway? Imagine a dining room that’s 100 miles long. There’s no possible party at one end that could ever be heard at the other. Never mind that a 100-mile-long restaurant would be absurd. It shows that the limits of people’s voices are a thing we have to consider.

There are many ways to model this distance effect. A realistic one would fall off with distance, sure. But it would also allow for echoes and absorption by the walls, and by other patrons, and maybe by restaurant decor. This would take forever to get answers from, but if done right it would get very good answers. A simpler model would give answers less fitted to your actual restaurant. But the answers may be close enough, and let you understand the system. And may be simple enough that you can get answers quickly. Maybe even by hand.

And so I come to the “nearest neighbor model”. The common English meaning of the words suggest what it’s about. We get it from models, like my restaurant noise problem. It’s made of a bunch of points that have some value. For my problem, tables and their noise level. And that value affects stuff in some region around these points.

In the “nearest neighbor model”, each point directly affects only its nearest neighbors. Saying which is the nearest neighbor is easy if the points are arranged in some regular grid. If they’re evenly spaced points on a line, say. Or a square grid. Or a triangular grid. If the points are in some other pattern, you need to think about what the nearest neighbors are. This is why people working in neighbor-nearness problems get paid the big money.

Suppose I use a nearest neighbor model for my restaurant problem. In this, I pretend the only background noise at my table is that of the people the next table over, in each direction. Two tables over? Nope. I don’t hear them at my table. I do get an indirect effect. Two tables over affects the table that’s between mine and theirs. But vice-versa, too. The table that’s 100 miles away can’t affect me directly, but it can affect a table in-between it and me. And that in-between table can affect the next one closer to me, and so on. The effect is attenuated, yes. Shouldn’t it be, if we’re looking at something farther away?

This sort of model is easy to work with numerically. I’m inclined toward problems that work numerically. Analytically … well, it can be easy. It can be hard. There’s a one-dimensional version of this problem, a bunch of evenly-spaced sites on an infinitely long line. If each site is limited to one of exactly two values, the problem becomes easy enough that freshman physics majors can solve it exactly. They don’t, not the first time out. This is because it requires recognizing a trigonometry trick that they don’t realize would be relevant. But once they know the trick, they agree it’s easy, when they go back two years later and look at it again. It just takes familiarity.

This comes up in thermodynamics, because it makes a nice model for how ferromagnetism can work. More realistic problems, like, two-dimensional grids? … That’s harder to solve exactly. Can be done, though not by undergraduates. Three-dimensional can’t, last time I looked. Weirdly, four-dimensional can. You expect problems to only get harder with more dimensions of space, and then you get a surprise like that.

The nearest-neighbor-model is a first choice. It’s hardly the only one. If I told you there were a next-nearest-neighbor model, what would you suppose it was? Yeah, you’d be right. As long as you supposed it was “things are affected by the nearest and the next-nearest neighbors”. Mathematicians have heard of loopholes too, you know.

As for my restaurant model? … I never actually modelled it. I did think about the model. I concluded my model wasn’t different enough from ferromagnetism models to need me to study it more. I might be mistaken. There may be interesting weird effects caused by the facts of restaurants. That restaurants are pretty small things. That they can have echo-y walls and ceilings. That they can have sound-absorbing things like partial walls or plants. Perhaps I gave up too easily when I thought I knew the answer. Some of my idle thoughts end up too idle.


I should have my next Fall 2018 Mathematics A-To-Z post on Tuesday. It’ll be available at this link, as are the rest of these glossary posts.

My Mathematics Reading For The 13th of June


I’m working on the next Why Stuff Can Orbit post, this one to feature a special little surprise. In the meanwhile here’s some of the things I’ve read recently and liked.

The Theorem of the Day is just what the name offers. They’re fit onto single slides, so there’s not much text to read. I’ll grant some of them might be hard reading at once, though, if you’re not familiar with the lingo. Anyway, this particular theorem, the Lindemann-Weierstrass Theorem, is one of the famous ones. Also one of the best-named ones. Karl Weierstrass is one of those names you find all over analysis. Over the latter half of the 19th century he attacked the logical problems that had bugged calculus for the previous three centuries and beat them all. I’m lying, but not by much. Ferdinand von Lindemann’s name turns up less often, but he’s known in mathematics circles for proving that π is transcendental (and so, ultimately, that the circle can’t be squared by compass and straightedge). And he was David Hilbert’s thesis advisor.

The Lindemann-Weierstrass Theorem is one of those little utility theorems that’s neat on its own, yes, but is good for proving other stuff. This theorem says that if a given number is algebraic (ask about that some A To Z series) then e raised to that number has to be transcendental, and vice-versa. (The exception: e raised to 0 is equal to 1.) The page also mentions one of those fun things you run across when you have a scientific calculator and can repeat an operation on whatever the result of the last operation was.

I’ve mentioned Maths By A Girl before, but, it’s worth checking in again. This is a piece about Apéry’s Constant, which is one of those numbers mathematicians have heard of, and that we don’t know whether is transcendental or not. It’s hard proving numbers are transcendental. If you go out trying to build a transcendental number it’s easy, but otherwise, you have to hope you know your number is the exponential of an algebraic number.

I forget which Twitter feed brought this to my attention, but here’s a couple geometric theorems demonstrated and explained some by Dave Richeson. There’s something wonderful in a theorem that’s mostly a picture. It feels so supremely mathematical to me.

And last, Katherine Bourzac writing for Nature.com reports the creation of a two-dimensional magnet. This delights me since one of the classic problems in statistical mechanics is a thing called the Ising model. It’s a basic model for the mathematics of how magnets would work. The one-dimensional version is simple enough that you can give it to undergrads and have them work through the whole problem. The two-dimensional version is a lot harder to solve and I’m not sure I ever saw it laid out even in grad school. (Mind, I went to grad school for mathematics, not physics, and the subject is a lot more physics.) The four- and higher-dimensional model can be solved by a clever approach called mean field theory. The three-dimensional model .. I don’t think has any exact solution, which seems odd given how that’s the version you’d think was most useful.

That there’s a real two-dimensional magnet (well, a one-molecule-thick magnet) doesn’t really affect the model of two-dimensional magnets. The model is interesting enough for its mathematics, which teaches us about all kinds of phase transitions. And it’s close enough to the way certain aspects of real-world magnets behave to enlighten our understanding. The topic couldn’t avoid drawing my eye, is all.

How To Hear Drums


The @mathematicsprof tweet above links to a paper, by Carolyn Gordon and David Webb and published in American Scientist in 1996, that’s about one of those questions that’s both mathematically interesting and of obvious everyday interest. The question was originally put, in nice compact and real-world-relevant form, in 1966 by Mark Kac: can one hear the shape of a drum?

At first glance the answer may seem, “of course” — you can hear the difference between musical instruments by listening to them. You might need experience, but, after all, you’re not going to confuse a bass drum from a bongo even if you haven’t been in the music store much. At second glance, why would Kac bother asking the question if the answer were obvious? He didn’t need the attention. He had, among other things, his work in ferromagnetism to be proud of (and I should write about that some.) And could you tell one bass drum from another?

The question ties into what’s known as “spectral theory”: given a complicated bundle of information what can you say about the source? One metaphorical inspiration here is studying the spectrum of a burning compound: the wavelengths of light emitted by it give you information about what elements go into the compound, and what their relative abundances are.

The sound of a drum is going to be a potentially complicated set of sound waves produced by the drum’s membrane itself oscillating. That membrane oscillation is going to depend, among other things, on the shape of the membrane, and that’s why we might suppose that we could tell what the shape of the drum is by the sound it makes when struck. But then it might also be that multiple different shapes could produce the exact same sound.

It took to about 1990 to get a definite answer; Gordon and Webb, along with Scott Wolpert, showed that you can get different-shaped drums that sound the same, and very nicely showed an example. In the linked article, Gordon and Webb describe some of the history of the problem and how they worked out a solution. It does require some technical terms that maybe even re-reading several times won’t help you parse, but if you’re willing to just move on past a paragraph that looks like jargon to the rest I believe you’ll find some interesting stuff out, for example, whether you could at least hear the area of a drum, even if you can’t tell what the shape is.