Monthly Archives: May 2022

Trapped in the (S) Matrix

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I’ve tried to convince you that you are a particle detector. You choose your experiment, what actions you take, and then observe the outcome. If you focus on that view of yourself, data out and data in, you start to wonder if the world outside really has any meaning. Maybe you’re just trapped in the Matrix.

From a physics perspective, you actually are trapped in a sort of a Matrix. We call it the S Matrix.

“S” stands for scattering. The S Matrix is a formula we use, a mathematical tool that tells us what happens when fundamental particles scatter: when they fly towards each other, colliding or bouncing off. For each action we could take, the S Matrix gives the probability of each outcome: for each pair of particles we collide, the chance we detect different particles at the end. You can imagine putting every possible action in a giant vector, and every possible observation in another giant vector. Arrange the probabilities for each action-observation pair in a big square grid, and that’s a matrix.

Actually, I lied a little bit. This is particle physics, and particle physics uses quantum mechanics. Because of that, the entries of the S Matrix aren’t probabilities: they’re complex numbers called probability amplitudes. You have to multiply them by their complex conjugate to get probability out.

Ok, that probably seemed like a lot of detail. Why am I telling you all this?

What happens when you multiply the whole S Matrix by its complex conjugate? (Using matrix multiplication, naturally.) You can still pick your action, but now you’re adding up every possible outcome. You’re asking “suppose I take an action. What’s the chance that anything happens at all?”

The answer to that question is 1. There is a 100% chance that something happens, no matter what you do. That’s just how probability works.

We call this property unitarity, the property of giving “unity”, or one. And while it may seem obvious, it isn’t always so easy. That’s because we don’t actually know the S Matrix formula most of the time. We have to approximate it, a partial formula that only works for some situations. And unitarity can tell us how much we can trust that formula.

Imagine doing an experiment trying to detect neutrinos, like the IceCube Neutrino Observatory. For you to detect the neutrinos, they must scatter off of electrons, kicking them off of their atoms or transforming them into another charged particle. You can then notice what happens as the energy of the neutrinos increases. If you do that, you’ll notice the probability also start to increase: it gets more and more likely that the neutrino can scatter an electron. You might propose a formula for this, one that grows with energy. [EDIT: Example changed after a commenter pointed out an issue with it.]

If you keep increasing the energy, though, you run into a problem. Those probabilities you predict are going to keep increasing. Eventually, you’ll predict a probability greater than one.

That tells you that your theory might have been fine before, but doesn’t work for every situation. There’s something you don’t know about, which will change your formula when the energy gets high. You’ve violated unitarity, and you need to fix your theory.

In this case, the fix is already known. Neutrinos and electrons interact due to another particle, called the W boson. If you include that particle, then you fix the problem: your probabilities stop going up and up, instead, they start slowing down, and stay below one.

For other theories, we don’t yet know the fix. Try to write down an S Matrix for colliding gravitational waves (or really, gravitons), and you meet the same kind of problem, a probability that just keeps growing. Currently, we don’t know how that problem should be solved: string theory is one answer, but may not be the only one.

So even if you’re trapped in an S Matrix, sending data out and data in, you can still use logic. You can still demand that probability makes sense, that your matrix never gives a chance greater than 100%. And you can learn something about physics when you do!

At New Ideas in Cosmology

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The Niels Bohr Institute is hosting a conference this week on New Ideas in Cosmology. I’m no cosmologist, but it’s a pretty cool field, so as a local I’ve been sitting in on some of the talks. So far they’ve had a selection of really interesting speakers with quite a variety of interests, including a talk by Roger Penrose with his trademark hand-stippled drawings.

Including this old classic

One thing that has impressed me has been the “interdisciplinary” feel of the conference. By all rights this should be one “discipline”, cosmology. But in practice, each speaker came at the subject from a different direction. They all had a shared core of knowledge, common models of the universe they all compare to. But the knowledge they brought to the subject varied: some had deep knowledge of the mathematics of gravity, others worked with string theory, or particle physics, or numerical simulations. Each talk, aware of the varied audience, was a bit “colloquium-style“, introducing a framework before diving in to the latest research. Each speaker knew enough to talk to the others, but not so much that they couldn’t learn from them. It’s been unexpectedly refreshing, a real interdisciplinary conference done right.

At Mikefest

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I’m at a conference this week of a very particular type: a birthday conference. When folks in my field turn 60, their students and friends organize a special conference for them, celebrating their research legacy. With COVID restrictions just loosening, my advisor Michael Douglas is getting a last-minute conference. And as one of the last couple students he graduated at Stony Brook, I naturally showed up.

The conference, Mikefest, is at the Institut des Hautes Études Scientifiques, just outside of Paris. Mike was a big supporter of the IHES, putting in a lot of fundraising work for them. Another big supporter, James Simons, was Mike’s employer for a little while after his time at Stony Brook. The conference center we’re meeting in is named for him.

You might have to zoom in to see that, though.

I wasn’t involved in organizing the conference, so it was interesting seeing differences between this and other birthday conferences. Other conferences focus on the birthday prof’s “family tree”: their advisor, their students, and some of their postdocs. We’ve had several talks from Mike’s postdocs, and one from his advisor, but only one from a student. Including him and me, three of Mike’s students are here: another two have had their work mentioned but aren’t speaking or attending.

Most of the speakers have collaborated with Mike, but only for a few papers each. All of them emphasized a broader debt though, for discussions and inspiration outside of direct collaboration. The message, again and again, is that Mike’s work has been broad enough to touch a wide range of people. He’s worked on branes and the landscape of different string theory universes, pure mathematics and computation, neuroscience and recently even machine learning. The talks generally begin with a few anecdotes about Mike, before pivoting into research talks on the speakers’ recent work. The recent-ness of the work is perhaps another difference from some birthday conferences: as one speaker said, this wasn’t just a celebration of Mike’s past, but a “welcome back” after his return from the finance world.

One thing I don’t know is how much this conference might have been limited by coming together on short notice. For other birthday conferences impacted by COVID (and I’m thinking of one in particular), it might be nice to have enough time to have most of the birthday prof’s friends and “academic family” there in person. As-is, though, Mike seems to be having fun regardless.

Happy Birthday Mike!

You Are a Particle Detector

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I mean that literally. True, you aren’t a 7,000 ton assembly of wires and silicon, like the ATLAS experiment inside the Large Hadron Collider. You aren’t managed by thousands of scientists and engineers, trying to sift through data from a billion pairs of protons smashing into each other every second. Nonetheless, you are a particle detector. Your senses detect particles.

Like you, and not like you

Your ears take vibrations in the air and magnify them, vibrating the fluid of your inner ear. Tiny hairs communicate that vibration to your nerves, which signal your brain. Particle detectors, too, magnify signals: photomultipliers take a single particle of light (called a photon) and set off a cascade, multiplying the signal one hundred million times so it can be registered by a computer.

Your nose and tongue are sensitive to specific chemicals, recognizing particular shapes and ignoring others. A particle detector must also be picky. A detector like ATLAS measures far more particle collisions than it could ever record. Instead, it learns to recognize particular “shapes”, collisions that might hold evidence of something interesting. Only those collisions are recorded, passed along to computer centers around the world.

Your sense of touch tells you something about the energy of a collision: specifically, the energy things have when they collide with you. Particle detectors do this with calorimeters, that generate signals based on a particle’s energy. Different parts of your body are more sensitive than others: your mouth and hands are much more sensitive than your back and shoulders. Different parts of a particle detector have different calorimeters: an electromagnetic calorimeter for particles like electrons, and a less sensitive hadronic calorimeter that can catch particles like protons.

You are most like a particle detector, though, in your eyes. The cells of your eyes, rods and cones, detect light, and thus detect photons. Your eyes are more sensitive than you think: you are likely able to detect even a single photon. In an experiment, three people sat in darkness for forty minutes, then heard two sounds, one of which might come accompanied by a single photon of light flashed into their eye. The three didn’t notice the photons every time, that’s not possible for such a small sensation: but they did much better than a random guess.

(You can be even more literal than that. An older professor here told me stories of the early days of particle physics. To check that a machine was on, sometimes physicists would come close, and watch for flashes in the corner of their vision: a sign of electrons flying through their eyeballs!)

You are a particle detector, but you aren’t just a particle detector. A particle detector can’t move, its thousands of tons are fixed in place. That gives it blind spots: for example, the tube that the particles travel through is clear, with no detectors in it, so the particle can get through. Physicists have to account for this, correcting for the missing space in their calculations. In contrast, if you have a blind spot, you can act: move, and see the world from a new point of view. You observe not merely a series of particles, but the results of your actions: what happens when you turn one way or another, when you make one choice or another.

So while you are a particle detector, what’s more, you’re a particle experiment. You can learn a lot more than those big heaps of wires and silicon could on their own. You’re like the whole scientific effort: colliders and detectors, data centers and scientists around the world. May you learn as much in your life as the experiments do in theirs.