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!

15 thoughts on “Trapped in the (S) Matrix”

  1. Phil Cannata

    Can you make a comparison between an S Matrix and the matrices originally developed by Born, Jordan, and Heisenberg?

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    1. 4gravitons Post author

      Depends on the level of detail you’re looking for. I’ll give here the lowest level that makes sense given your question, if you want me to address something more specific just ask.

      The S Matrix is a “matrix” for the same reason the matrices from the early days of matrix mechanics are “matrices”: they’re operators on quantum states. You can work with it in the Born/Jordan/Heisenberg formalism, and it literally is a matrix in the same sense X and P are in that formalism. The S matrix in particular is closest in spirit to the time evolution operator.

      If there’s a difference from what you might be used to in the older formalism, it’s that the S Matrix is specifically connecting “in” and “out” states. So while it plays a similar role to the time evolution operator, here you’re not advancing from one finite time to another: you’re going from something you approximate as infinitely far in the past to something approximated as infinitely far in the future. (You’re allowed to do this because particle colliders are much much bigger than the reactions they study, and operate on much longer timescales, so from the point of view of the actual collision your initial and final states might as well be infinitely far away.)

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      1. Philip Cannata

        Great thanks.
        My understanding is that the matrices work when you can apply them to classical equations of motion, but that’s never been shown to be possible when trying to apply them to the General Relativity equations. Is that the crux of the problem here also?

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        1. 4gravitons Post author

          Not sure what you’re referring to by “the matrices work when you can apply them to classical equations of motion”, so I can’t say. Can you give a link to someone saying that so I can figure out what they mean?

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            1. Philip Cannata

              Regarding “matrices work when you can apply them to classical equations of motion”,

              - Heisenberg, in his first paper, re-interprets position and momentum but uses these new re-interpretations in standard classical equations of motion.

- Born and Jordan make this same assumption in their paper.

- And Dirac, in his paper "On the Theory of Quantum Mechanics" states this very clearly,

    “Heisenberg puts forward a new theory which suggests that it is not the equations of classical mechanics that are in any way at fault, but that the mathematical operations by which physical results are deduced from them require modification. All the information supplied by the classical theory can thus be made use of in the new theory.”

              Regarding “that’s never been shown to be possible when trying to apply them to the General Relativity equations”

              -  Sean Carrroll, in a recent podcast (https://www.quantamagazine.org/where-do-space-time-and-gravity-come-from-20220504/?mc_cid=cd1add88ff&mc_eid=bbefddb848) said,

    "But again, nevertheless, it has worked for electromagnetism, the nuclear forces and everything else. When you straightforwardly apply that quantization procedure to gravity — we have a classical theory, general relativity, we can quantize it. It just blows up. It just gives us infinite crazy answers."

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              1. 4gravitons Post author

                Ah ok. Yeah, so the first quote is describing quantization, a process by which you take a classical theory, interpret all the variables as quantum operators (matrices in the Heisenberg/Born/Jordan picture), and get a quantum theory out. This isn’t the only way to get a quantum theory, but it’s one of the most familiar ones. And Sean is indeed saying that if you do this with GR you get infinite answers.

                As to whether Sean/other people who say that are referring to the problem I discuss in this post, it varies. This is definitely one of those problems, but it’s perhaps not the most famous one: the issue I mention at the end of this post usually gets more emphasis. They’re distinct issues, and both of them are reasons why naively quantizing gravity can’t be the full story.

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  2. Andrew Larkoski

    The decay rate of neutrons is Lorentz invariant; it can’t (and indeed does not) depend on the velocity of the neutrons. I think what you mean instead is something like electron-neutrino scattering where you can experimentally dial in the center-of-mass collision energy. As you increase the collision energy within the four-Fermi theory, the cross section does grow with energy. Of course, the weak interactions unitarize it, corresponding to a new particle responsible for mediating the interaction.

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    1. 4gravitons Post author

      You’re right, the example I used of four-Fermi breaking unitarity doesn’t make sense, it needs to be an actual collision. I’ll have to think about what’s the best option to replace it (electron-neutrino scattering makes sense physically but has the drawback that it’s pretty hard to imagine an experimental setup for it, at least one that can be summarized in a few words). For now I’ll put a note in the post warning people the example is wrong.

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    2. 4gravitons Post author

      Ok, I figured out a way to phrase the example with electron-neutrino scattering. For what it’s worth it’s also possible to do with beta decay, but it requires saying something about the outgoing particles, and gets more complicated to phrase. You were quite right that “speeding up the neutron” by itself is irrelevant.

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  3. JollyJoker

    Do I remember correctly that I’ve heard probability amplitudes referred to as the square root of probabilities? Now that you say it’s the complex conjugate, I can of course see that squaring a complex number can’t work because of the -2ab term though.
    Just a verbal shortcut or am I thinking of something else or just mistaken?

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    1. 4gravitons Post author

      It’s just a verbal shortcut yeah. (If you want to get picky, the proper way to square a complex number is to multiply it by its complex conjugate, that’s the norm on the relevant space. 😉 )

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    1. 4gravitons Post author

      Hmm, I suspect someone’s been paying attention to recent amplitudes developments!

      I like to think of form factors as pieces of scattering amplitudes: if you have two form factors with matching off-shell legs, you can glue them to each other on those legs to get an amplitude, with the off-shell legs becoming an intermediate virtual particle.

      But that’s a very amplitudes-centric perspective. I think for other people, form factors are something beyond the S-matrix, thinking about QFT in terms of observables that don’t just involve on-shell states.

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