Monthly Archives: April 2022

Things Which Are Fluids

For overambitious apes like us, adding integers is the easiest thing in the world. Take one berry, add another, and you have two. Each remains separate, you can lay them in a row and count them one by one, each distinct thing adding up to a group of distinct things.

Other things in math are less like berries. Add two real numbers, like pi and the square root of two, and you get another real number, bigger than the first two, something you can write in an infinite messy decimal. You know in principle you can separate it out again (subtract pi, get the square root of two), but you can’t just stare at it and see the parts. This is less like adding berries, and more like adding fluids. Pour some water in to some other water, and you certainly have more water. You don’t have “two waters”, though, and you can’t tell which part started as which.

Some things in math look like berries, but are really like fluids. Take a polynomial, say $5 x^2 + 6 x + 8$ . It looks like three types of things, like three berries: five $x^2$ , six $x$ , and eight $1$ . Add another polynomial, and the illusion continues: add $x^2 + 3 x + 2$ and you get $6 x^2+9 x+10$ . You’ve just added more $x^2$ , more $x$ , more $1$ , like adding more strawberries, blueberries, and raspberries.

But those berries were a choice you made, and not the only one. You can rewrite that first polynomial, for example saying $5(x^2+2x+1) - 4 (x+1) + 7$ . That’s the same thing, you can check. But now it looks like five $x^2+2x+1$ , negative four $x+1$ , and seven $1$ . It’s different numbers of different things, blackberries or gooseberries or something. And you can do this in many ways, infinitely many in fact. The polynomial isn’t really a collection of berries, for all it looked like one. It’s much more like a fluid, a big sloshing mess you can pour into buckets of different sizes. (Technically, it’s a vector space. Your berries were a basis.)

Even smart, advanced students can get tripped up on this. You can be used to treating polynomials as a fluid, and forget that directions in space are a fluid, one you can rotate as you please. If you’re used to directions in space, you’ll get tripped up by something else. You’ll find that types of particles can be more fluid than berry, the question of which quark is which not as simple as how many strawberries and blueberries you have. The laws of physics themselves are much more like a fluid, which should make sense if you take a moment, because they are made of equations, and equations are like a fluid.

So my fellow overambitious apes, do be careful. Not many things are like berries in the end. A whole lot are like fluids.

W is for Why???

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Have you heard the news about the W boson?

The W boson is a fundamental particle, part of the Standard Model of particle physics. It is what we call a “force-carrying boson”, a particle related to the weak nuclear force in the same way photons are related to electromagnetism. Unlike photons, W bosons are “heavy”: they have a mass. We can’t usually predict masses of particles, but the W boson is a bit different, because its mass comes from the Higgs boson in a special way, one that ties it to the masses of other particles like the Z boson. The upshot is that if you know the mass of a few other particles, you can predict the mass of the W.

And according to a recent publication, that prediction is wrong. A team analyzed results from an old experiment called the Tevatron, the biggest predecessor of today’s Large Hadron Collider. They treated the data with groundbreaking care, mindbogglingly even taking into account the shape of the machine’s wires. And after all that analysis, they found that the W bosons detected by the Tevatron had a different mass than the mass predicted by the Standard Model.

How different? Here’s where precision comes in. In physics, we decide whether to trust a measurement with a statistical tool. We calculate how likely the measurement would be, if it was an accident. In this case: how likely it would be that, if the Standard Model was correct, the measurement would still come out this way? To discover a new particle, we require this chance to be about one in 3.5 million, or in our jargon, five sigma. That was the requirement for discovering the Higgs boson. This super-precise measurement of the W boson doesn’t have five sigma…it has seven sigma. That means, if we trust the analysis team, then a measurement like this could come accidentally out of the Standard Model only about one in a trillion times.

Ok, should we trust the analysis team?

If you want to know that, I’m the wrong physicist to ask. The right physicists are experimental particle physicists. They do analyses like that one, and they know what can go wrong. Everyone I’ve heard from in that field emphasized that this was a very careful group, who did a lot of things impressively right…but there is still room for mistakes. One pointed out that the new measurement isn’t just inconsistent with the Standard Model, but with many previous measurements too. Those measurements are less precise, but still precise enough that we should be a bit skeptical. Another went into more detail about specific clues as to what might have gone wrong.

If you can’t find an particle experimentalist, the next best choice is a particle phenomenologist. These are the people who try to make predictions for new experiments, who use theoretical physics to propose new models that future experiments can test. Here’s one giving a first impression, and discussing some ways to edit the Standard Model to agree with the new measurement. Here’s another discussing what to me is an even more interesting question: if we take these measurements seriously, both the new one and the old ones, then what do we believe?

I’m not an experimentalist or a phenomenologist. I’m an “amplitudeologist”. I work not on the data, or the predictions, but the calculational tools used to make those predictions, called “scattering amplitudes”. And that gives me a different view on the situation.

See in my field, precision is one of our biggest selling-points. If you want theoretical predictions to match precise experiments, you need our tricks to compute them. We believe (and argue to grant agencies) that this precision will be important: if a precise experiment and a precise prediction disagree, it could be the first clue to something truly new. New solid evidence of something beyond the Standard Model would revitalize all of particle physics, giving us a concrete goal and killing fruitless speculation.

This result shakes my faith in that a little. Probably, the analysis team got something wrong. Possibly, all previous analyses got something wrong. Either way, a lot of very careful smart people tried to estimate their precision, got very confident…and got it wrong.

(There’s one more alternative: maybe million-to-one chances really do crop up nine times out of ten.)

If some future analysis digs down deep in precision, and finds another deviation from the Standard Model, should we trust it? What if it’s measuring something new, and we don’t have the prior experiments to compare to?

(This would happen if we build a new even higher-energy collider. There are things the collider could measure, like the chance one Higgs boson splits into two, that we could not measure with any earlier machine. If we measured that, we couldn’t compare it to the Tevatron or the LHC, we’d have only the new collider to go on.)

Statistics are supposed to tell us whether to trust a result. Here, they’re not doing their job. And that creates the scary possibility that some anomaly shows up, some real deviation deep in the sigmas that hints at a whole new path for the field…and we just end up bickering about who screwed it up. Or the equally scary possibility that we find a seven-sigma signal of some amazing new physics, build decades of new theories on it…and it isn’t actually real.

We don’t just trust statistics. We also trust the things normal people trust. Do other teams find the same result? (I hope that they’re trying to get to this same precision here, and see what went wrong!) Does the result match other experiments? Does it make predictions, which then get tested in future experiments?

All of those are heuristics of course. Nothing can guarantee that we measure the truth. Each trick just corrects for some of our biases, some of the ways we make mistakes. We have to hope that’s good enough, that if there’s something to see we’ll see it, and if there’s nothing to see we won’t. Precision, my field’s raison d’être, can’t be enough to convince us by itself. But it can help.

The Undefinable

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If I can teach one lesson to all of you, it’s this: be precise. In physics, we try to state what we mean as precisely as we can. If we can’t state something precisely, that’s a clue: maybe what we’re trying to state doesn’t actually make sense.

Someone recently reached out to me with a question about black holes. He was confused about how they were described, about what would happen when you fall in to one versus what we could see from outside. Part of his confusion boiled down to a question: “is the center really an infinitely small point?”

I remembered a commenter a while back who had something interesting to say about this. Trying to remind myself of the details, I dug up this question on Physics Stack Exchange. user4552 has a detailed, well-referenced answer, with subtleties of General Relativity that go significantly beyond what I learned in grad school.

According to user4552, the reason this question is confusing is that the usual setup of general relativity cannot answer it. In general relativity, singularities like the singularity in the middle of a black hole aren’t treated as points, or collections of points: they’re not part of space-time at all. So you can’t count their dimensions, you can’t see whether they’re “really” infinitely small points, or surfaces, or lines…

This might surprise people (like me) who have experience with simpler equations for these things, like the Schwarzchild metric. The Schwarzchild metric describes space-time around a black hole, and in the usual coordinates it sure looks like the singularity is at a single point where r=0, just like the point where r=0 is a single point in polar coordinates in flat space. The thing is, though, that’s just one sort of coordinates. You can re-write a metric in many different sorts of coordinates, and the singularity in the center of a black hole might look very different in those coordinates. In general relativity, you need to stick to things you can say independent of coordinates.

Ok, you might say, so the usual mathematics can’t answer the question. Can we use more unusual mathematics? If our definition of dimensions doesn’t tell us whether the singularity is a point, maybe we just need a new definition!

According to user4552, people have tried this…and it only sort of works. There are several different ways you could define the dimension of a singularity. They all seem reasonable in one way or another. But they give different answers! Some say they’re points, some say they’re three-dimensional. And crucially, there’s no obvious reason why one definition is “right”. The question we started with, “is the center really an infinitely small point?”, looked like a perfectly reasonable question, but it actually wasn’t: the question wasn’t precise enough.

This is the real problem. The problem isn’t that our question was undefined, after all, we can always add new definitions. The problem was that our question didn’t specify well enough the definitions we needed. That is why the question doesn’t have an answer.

Once you understand the difference, you see these kinds of questions everywhere. If you’re baffled by how mass could have come out of the Big Bang, or how black holes could radiate particles in Hawking radiation, maybe you’ve heard a physicist say that energy isn’t always conserved. Energy conservation is a consequence of symmetry, specifically, symmetry in time. If your space-time itself isn’t symmetric (the expanding universe making the past different from the future, a collapsing star making a black hole), then you shouldn’t expect energy to be conserved.

I sometimes hear people object to this. They ask, is it really true that energy isn’t conserved when space-time isn’t symmetric? Shouldn’t we just say that space-time itself contains energy?

And well yes, you can say that, if you want. It isn’t part of the usual definition, but you can make a new definition, one that gives energy to space-time. In fact, you can make more than one new definition…and like the situation with the singularity, these definitions don’t always agree! Once again, you asked a question you thought was sensible, but it wasn’t precise enough to have a definite answer.

Keep your eye out for these kinds of questions. If scientists seem to avoid answering the question you want, and keep answering a different question instead…it might be their question is the only one with a precise answer. You can define a method to answer your question, sure…but it won’t be the only way. You need to ask precise enough questions to get good answers.

Gateway Hobbies

Keeping It Colloquial

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In the corners of academia where I hang out, a colloquium is a special kind of talk. Most talks we give are part of weekly seminars for specific groups. For example, the theoretical particle physicists here have a seminar. Each week we invite a speaker, who gives a talk on their recent work. Since they expect an audience of theoretical particle physicists, they can go into more detail.

A colloquium isn’t like that. Colloquia are talks for the whole department: theorists and experimentalists, particle physicists and biophysicists. They’re more prestigious, for big famous professors (or sometimes, for professors interviewing for jobs…). The different audience, and different context, means that the talk plays by different rules.

Recently, I saw a conference full of “colloquium-style” talks, trying to play by these rules. Some succeeded, some didn’t…and I think I now have a better idea of how those rules work.

First, in a colloquium, you’re not just speaking for yourself. You’re an ambassador for your field. For some of the audience, this might be the first time they’ve heard a talk by someone who does your kind of research. You want to give them a good impression, not just about you, but about the whole topic. So while you definitely want to mention your own work, you want to tell a full story, one that gives more than a glimpse of what others are doing as well.

Second, you want to connect to something the audience already knows. With an audience of physicists, you can assume a certain baseline, but not much more than that. You need to make the beginning accessible and start with something familiar. For the conference I mentioned, a talk that did this well was the talk on exoplanets, which started with the familiar planets of the solar system, classifying them in order to show what you might expect exoplanets to look like. In contrast, t’Hooft’s talk did this poorly. His work is exploiting a loophole in a quantum-mechanical argument called Bell’s theorem, which most physicists have heard of. Instead of mentioning Bell’s theorem, he referred vaguely to “criteria from philosophers”, and only even mentioned that near the end of the talk, instead starting with properties of quantum mechanics his audience was much less familiar with.

Moving on, then, you want to present a mystery. So far, everything in the talk has made sense, and your audience feels like they understand. Now, you show them something that doesn’t fit, something their familiar model can’t accommodate. This activates your audience’s scientist instincts: they’re curious now, they want to know the answer. A good example from the conference was a talk on chemistry in space. The speaker emphasized that we can see evidence of complex molecules in space, but that space dust is so absurdly dilute that it seems impossible such molecules could form: two atoms could go a billion years without meeting each other.

You can’t just leave your audience mystified, though. You next have to solve the mystery. Ideally, your solution will be something smart, but simple: something your audience can intuitively understand. This has two benefits. First, it makes you look smart: you described a mysterious problem, and then you show how to solve it! Second, it makes the audience feel smart: they felt the problem was hard, but now they understand how to solve it too. The audience will have good feelings about you as a result, and good feelings about the topic: in some sense, you’ve tied a piece of their self-esteem to knowing the solution to your problem. This was well-done by the speaker discussing space chemistry, who explained that the solution was chemistry on surfaces: if two atoms are on the surface of a dust grain or meteorite, they’re much more likely to react. It was also well-done by a speaker discussing models of diseases like diabetes: he explained the challenge of controlling processes with cells, when cells replicate exponentially, and showed one way they could be controlled, when the immune system kills off any cells that replicate much faster than their neighbors. (He also played the guitar to immune system-themed songs…also a good colloquium strategy for those who can pull it off!)

Finally, a picture is worth a thousand words…as long as it’s a clear one. For an audience that won’t follow most of your equations, it’s crucial to show them something visual: graphics, puns, pictures of equipment or graphs. Crucially, though, your graphics should be something the audience can understand. If you put up a graph with a lot of incomprehensible detail: parameters you haven’t explained, or just set up in a way your audience doesn’t get, then your audience gets stuck. Much like an unfamiliar word, a mysterious graph will have members of the audience scratching their heads, trying to figure out what it means. They’ll be so busy trying, they’ll miss what you say next, and you’ll lose them! So yes, put in graphs, put in pictures: but make sure that the ones you use, you have time to explain.

4 gravitons

Stories about physics from someone who's been there