Category Archives: General QFT

The Quantum Paths Not Traveled

Before this week’s post: a former colleague of mine from CEA Paris-Saclay, Sylvain Ribault, posted a dialogue last week presenting different perspectives on academic publishing. One of the highlights of my brief time at the CEA were the times I got to chat with Sylvain and others about the future forms academia might take. He showed me a draft of his dialogue a while ago, designed as a way to introduce newcomers to the debate about how, and whether, academics should do peer review. I’ve got a different topic this week so I won’t say much more about it, but I encourage you to take a look!

Matt Strassler has a nice post up about waves and particles. He’s writing to address a common confusion, between two concepts that sound very similar. On the other hand, there are the waves of quantum field theory, ripples in fundamental fields the smallest versions of which correspond to particles. (Strassler likes to call them “wavicles”, to emphasize their wavy role.) On the other hand, there are the wavefunctions of quantum mechanics, descriptions of the behavior of one or more interacting particles over time. To distinguish, he points out that wavicles can hurt you, while wavefunctions cannot. Wavicles are the things that collide and light up detectors, one by one, wavefunctions are the math that describes when and how that happens. Many types of wavicles can run into each other one by one, but their interactions can all be described together by a single wavefunction. It’s an important point, well stated.

(I do think he goes a bit too far in saying that the wavefunction is not “an object”, though. That smacks of metaphysics, and I think that’s not worth dabbling in for physicists.)

After reading his post, there’s something that might still confuse you. You’ve probably heard that in quantum mechanics, an electron is both a wave and a particle. Does the “wave” in that saying mean “wavicle”, or “wavefunction”?

The gif above shows data from a double-slit experiment, an important type of experiment from the early days of quantum mechanics. These experiments were first conducted before quantum field theory (and thus, before the ideas that Strassler summarizes with “wavicles”). In a double-slit experiment, particles are shot at a screen through two slits. The particles that hit the screen can travel through one slit or the other.

A double-slit experiment, in diagram form

Classically, you would expect particles shot randomly at the screen to form two piles on the other side, one in front of each slit. Instead, they bunch up into a rippling pattern, the same sort of pattern that was used a century earlier to argue that light was a wave. The peaks and troughs of the wave pass through both slits, and either line up or cancel out, leaving the distinctive pattern.

When it was discovered that electrons do this too, it led to the idea that electrons must be waves as well, despite also being particles. That insight led to the concept of the wavefunction. So the “wave” in the saying refers to wavefunctions.

But electrons can hurt you, and as Strassler points out, wavefunctions cannot. So how can the electron be a wavefunction?

To risk a bit of metaphysics myself, I’ll just say: it can’t. An electron can’t “be” a wavefunction.

The saying, that electrons are both particles and waves, is from the early days of quantum mechanics, when people were confused about what it all meant. We’re still confused, but we have some better ways to talk about it.

As a start, it’s worth noticing that, whenever you measure an electron, it’s a particle. Each electron that goes through the slits hits your screen as a particle, a single dot. If you see many electrons at once, you may get the feeling that they look like waves. But every actual electron you measure, every time you’re precise enough to notice, looks like a particle. And for each individual electron, you can extrapolate back the path it took, exactly as if it traveled like a particle the whole way through.

The same is true, though, of light! When you see light, photons enter your eyes, and each one that you see triggers a chemical change in a molecule called a photopigment. The same sort of thing happens for photographs, while an electrical signal gets triggered instead in a digital camera. Light may behave like a wave in some sense, but every time you actually observe it it looks like a particle.

But while you can model each individual electron, or photon, as a classical particle, you can’t model the distribution of multiple electrons that way.

That’s because in quantum mechanics, the “paths not taken” matter. A single electron will only go through one slit in the double-slit experiment. But the fact that it could have gone through both slits matters, and changes the chance that it goes through each particular path. The possible paths in the wavefunction interfere with each other, the same way different parts of classical waves do.

That role of the paths not taken, of the “what if”, is the heart and soul of quantum mechanics. No matter how you interpret its mysteries, “what if” matters. If you believe in a quantum multiverse, you think every “what if” happens somewhere in that infinity of worlds. If you think all that matters is observations, then “what if” shows the folly of modeling the world as anything else. If you are tempted to try to mend quantum mechanics with faster-than-light signals, then you have to declare one “what if” the true one. And if you want to double-down on determinism and replace quantum mechanics, you need to declare that certain “what if” questions are off-limits.

“What if matters” isn’t the same as a particle traveling every path at once, it’s its own weird thing with its own specific weird consequences. It’s a metaphor, because everything written in words is a metaphor. But it’s a better metaphor than thinking an electron is both a particle and a wave.

The Hidden Higgs

What Are Particles? The Gentle Introduction

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On this blog, I write about particle physics for the general public. I try to make things as simple as possible, but I do have to assume some things. In particular, I usually assume you know what particles are!

This time, I won’t do that. I know some people out there don’t know what a particle is, or what particle physicists do. If you’re a person like that, this post is for you! I’m going to give a gentle introduction to what particle physics is all about.

Let’s start with atoms.

Every object and substance around you, everything you can touch or lift or walk on, the water you drink and the air you breathe, all of these are made up of atoms. Some are simple: an iron bar is made of Iron atoms, aluminum foil is mostly Aluminum atoms. Some are made of combinations of atoms into molecules, like water’s famous H2O: each molecule has two Hydrogen atoms and one Oxygen atom. Some are made of more complicated mixtures: air is mostly pairs of Nitrogen atoms, with a healthy amount of pairs of Oxygen, some Carbon Dioxide (CO2), and many other things, while the concrete sidewalks you walk on have Calcium, Silicon, Aluminum, Iron, and Oxygen, all combined in various ways.

There is a dizzying array of different types of atoms, called chemical elements. Most occur in nature, but some are man-made, created by cutting-edge nuclear physics. They can all be organized in the periodic table of elements, which you’ve probably seen on a classroom wall.

The periodic table is called the periodic table because it repeats, periodically. Each element is different, but their properties resemble each other. Oxygen is a gas, Sulfur a yellow powder, Polonium an extremely radioactive metal…but just as you can find H2O, you can make H2S, and even H2Po. The elements get heavier as you go down the table, and more metal-like, but their chemical properties, the kinds of molecules you can make with them, repeat.

Around 1900, physicists started figuring out why the elements repeat. What they discovered is that each atom is made of smaller building-blocks, called sub-atomic particles. (“Sub-atomic” because they’re smaller than atoms!) Each atom has electrons on the outside, and on the inside has a nucleus made of protons and neutrons. Atoms of different elements have different numbers of protons and electrons, which explains their different properties.

Different atoms with different numbers of protons, neutrons, and electrons

Around the same time, other physicists studied electricity, magnetism, and light. These things aren’t made up of atoms, but it was discovered that they are all aspects of the same force, the electromagnetic force. And starting with Einstein, physicists figured out that this force has particles too. A beam of light is made up of another type of sub-atomic particle, called a photon.

For a little while then, it seemed that the universe was beautifully simple. All of matter was made of electrons, protons, and neutrons, while light was made of photons.

(There’s also gravity, of course. That’s more complicated, in this post I’ll leave it out.)

Soon, though, nuclear physicists started noticing stranger things. In the 1930’s, as they tried to understand the physics behind radioactivity and mapped out rays from outer space, they found particles that didn’t fit the recipe. Over the next forty years, theoretical physicists puzzled over their equations, while experimental physicists built machines to slam protons and electrons together, all trying to figure out how they work.

Finally, in the 1970’s, physicists had a theory they thought they could trust. They called this theory the Standard Model. It organized their discoveries, and gave them equations that could predict what future experiments would see.

In the Standard Model, there are two new forces, the weak nuclear force and the strong nuclear force. Just like photons for the electromagnetic force, each of these new forces has a particle. The general word for these particles is bosons, named after Satyendra Nath Bose, a collaborator of Einstein who figured out the right equations for this type of particle. The weak force has bosons called W and Z, while the strong force has bosons called gluons. A final type of boson, called the Higgs boson after a theorist who suggested it, rounds out the picture.

The Standard Model also has new types of matter particles. Neutrinos interact with the weak nuclear force, and are so light and hard to catch that they pass through nearly everything. Quarks are inside protons and neutrons: a proton contains one one down quark and two up quarks, while a neutron contains two down quarks and one up quark. The quarks explained all of the other strange particles found in nuclear physics.

Finally, the Standard Model, like the periodic table, repeats. There are three generations of particles. The first, with electrons, up quarks, down quarks, and one type of neutrino, show up in ordinary matter. The other generations are heavier, and not usually found in nature except in extreme conditions. The second generation has muons (similar to electrons), strange quarks, charm quarks, and a new type of neutrino called a muon-neutrino. The third generation has tauons, bottom quarks, top quarks, and tau-neutrinos.

(You can call these last quarks “truth quarks” and “beauty quarks” instead, if you like.)

Physicists had the equations, but the equations still had some unknowns. They didn’t know how heavy the new particles were, for example. Finding those unknowns took more experiments, over the next forty years. Finally, in 2012, the last unknown was found when a massive machine called the Large Hadron Collider was used to measure the Higgs boson.

We think that these particles are all elementary particles. Unlike protons and neutrons, which are both made of up quarks and down quarks, we think that the particles of the Standard Model are not made up of anything else, that they really are elementary building-blocks of the universe.

We have the equations, and we’ve found all the unknowns, but there is still more to discover. We haven’t seen everything the Standard Model can do: to see some properties of the particles and check they match, we’d need a new machine, one even bigger than the Large Hadron Collider. We also know that the Standard Model is incomplete. There is at least one new particle, called dark matter, that can’t be any of the known particles. Mysteries involving the neutrinos imply another type of unknown particle. We’re also missing deeper things. There are patterns in the table, like the generations, that we can’t explain.

We don’t know if any one experiment will work, or if any one theory will prove true. So particle physicists keep working, trying to find new tricks and make new discoveries.

Neu-tree-no Detector

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I’ve written before about physicists’ ideas for gigantic particle accelerators, proposals for machines far bigger than the Large Hadron Collider or even plans for a Future Circular Collider. The ideas ranged from wacky but not obviously impossible (a particle collider under the ocean) to pure science fiction (a beam of neutrinos that can blow up nukes across the globe).

But what if you don’t want to accelerate particles? What if, instead, you want to detect particles from the depths of space? Can you still propose ridiculously huge things?

Neutrinos are extremely hard to detect. Immune to the strongest forces of nature, they only interact via the weak nuclear force and gravity. The weakness of these forces means they can pass through huge amounts of material without disturbing a single atom. The Sudbury Neutrino Observatory used a tank of 1000 tonnes of water in order to stop enough neutrinos to study them. The IceCube experiment is bigger yet, and getting even bigger: their planned expansion will fill eight cubic kilometers of Antarctic ice with neutrino detectors, letting them measure around a million neutrinos every year.

But if you want to detect the highest-energy neutrinos, you may have to get even bigger than that. With so few of them to study, you need to cover a huge area with antennas to spot a decent number of them.

Or, maybe you can just use trees.

That’s the proposal of Steven Prohira, a MacArthur Genius Grant winner who works as a professor at the University of Kansas. He suggests that, instead of setting up a giant array of antennas to detect high-energy neutrinos, trees could be used, with a coil of wire around the tree to measure electrical signals. Prohira even suggests that “A forest detector could also motivate the large-scale reforesting of land, to grow a neutrino detector for future generations”.

Despite sounding wacky, tree antennas have actually been used before. Militaries have looked into them as a way to set up antennas in remote locations, and later studies indicate they work surprisingly well. So the idea is not completely impossible, much like the “collider-under-the-sea”.

Like the “collider-under-the-sea”, though, some wackiness still remains. Prohira admits he hasn’t yet done all the work needed to test the idea’s feasibility, and comparing to mature experiments like IceCube makes it clear there is a lot more work to be done. Chatting with neutrino experts, one problem a few of them pointed out is that unlike devices sunk into Antarctic ice, trees are not uniformly spaced, and that might pose a problem if you want to measure neutrinos carefully.

What stands out to me, though, is that those questions are answerable. If the idea sounds promising, physicists can follow up. They can make more careful estimates, or do smaller-scale experiments. They won’t be stuck arguing over interpretations, or just building the full experiment and seeing if it works.

That’s the great benefit of a quantitative picture of the world. We can estimate some things very accurately, with theories that give very precise numbers for how neutrinos behave. Other things we can estimate less accurately, but still can work on: how tall trees are, how widely they are spaced, how much they vary. We have statistical tools and biological data. We can find numbers, and even better, we can know how uncertain we should be about those numbers. Because of that picture, we don’t need to argue fruitlessly about ideas like this. We can work out numbers, and check!

LHC Black Hole Reassurance: The Professional Version

If That Measures the Quantum Vacuum, Anything Does

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Sabine Hossenfelder has gradually transitioned from critical written content about physics to YouTube videos, mostly short science news clips with the occasional longer piece. Luckily for us in the unable-to-listen-to-podcasts demographic, the transcripts of these videos are occasionally published on her organization’s Substack.

Unluckily, it feels like the short news format is leading to some lazy metaphors. There are stories science journalists sometimes tell because they’re easy and familiar, even if they don’t really make sense. Scientists often tell them too, for the same reason. But the more careful voices avoid them.

Hossenfelder has been that careful before, but one of her recent pieces falls short. The piece is titled “This Experiment Will Measure Nothing, But Very Precisely”.

The “nothing” in the title is the oft-mythologized quantum vacuum. The story goes that in quantum theory, empty space isn’t really empty. It’s full of “virtual” particles, that pop in and out of existence, jostling things around.

This…is not a good way to think about it. Really, it’s not. If you want to understand what’s going on physically, it’s best to think about measurements, and measurements involve particles: you can’t measure anything in pure empty space, you don’t have anything to measure with. Instead, every story you can tell about the “quantum vacuum” and virtual particles, you can tell about interactions between particles that actually exist.

(That post I link above, by the way, was partially inspired by a more careful post by Hossenfelder. She does know this stuff. She just doesn’t always use it.)

Let me tell the story Hossenfelder’s piece is telling, in a less silly way:

In the earliest physics classes, you learn that light does not affect other light. Shine two flashlight beams across each other, and they’ll pass right through. You can trace the rays of each source, independently, keeping track of how they travel and bounce around the room.

In quantum theory, that’s not quite true. Light can interact with light, through subtle quantum effects. This effect is tiny, so tiny it hasn’t been measured before. But with ingenious tricks involving tuning three different lasers in exactly the right way, a team of physicists in Dresden has figured out how it could be done.

And see, that’s already cool, right? It’s cool when people figure out how to see things that have never been seen before, full stop.

But the way Hossenfelder presents it, the cool thing about this is that they are “measuring nothing”. That they’re measuring “the quantum vacuum”, really precisely.

And I mean, you can say that, I guess. But it’s equally true of every subtle quantum effect.

In classical physics, electrons should have a very specific behavior in a magnetic field, called their magnetic moment. Quantum theory changes this: electrons have a slightly different magnetic moment, an anomalous magnetic moment. And people have measured this subtle effect: it’s famously the most precisely confirmed prediction in all of science.

That effect can equally well be described as an effect of the quantum vacuum. You can draw the same pictures, if you really want to, with virtual particles popping in and out of the vacuum. One effect (light bouncing off light) doesn’t exist at all in classical physics, while the other (electrons moving in a magnetic field) exists, but is subtly different. But both, in exactly the same sense, are “measurements of nothing”.

So if you really want to stick on the idea that, whenever you measure any subtle quantum effect, you measure “the quantum vacuum”…then we’re already doing that, all the time. Using it to popularize some stuff (say, this experiment) and not other stuff (the LHC is also measuring the quantum vacuum) is just inconsistent.

Better, in my view, to skip the silly talk about nothing. Talk about what we actually measure. It’s cool enough that way.

Neutrinos and Guarantees

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The Higgs boson, or something like it, was pretty much guaranteed.

When physicists turned on the Large Hadron Collider, we didn’t know exactly what they would find. Instead of the Higgs boson, there might have been many strange new particles with different properties. But we knew they had to find something, because without the Higgs boson or a good substitute, the Standard Model is inconsistent. Try to calculate what would happen at the LHC using the Standard Model without the Higgs boson, and you get literal nonsense: chances of particles scattering that are greater than one, a mathematical impossibility. Without the Higgs boson, the Standard Model had to be wrong, and had to go wrong specifically when that machine was turned on. In effect, the LHC was guaranteed to give a Nobel prize.

The LHC also searches for other things, like supersymmetric partner particles. It, and a whole zoo of other experiments, also search for dark matter, narrowing down the possibilities. But unlike the Higgs, none of these searches for dark matter or supersymmetric partners is guaranteed to find something new.

We’re pretty certain that something like dark matter exists, and that it is in some sense “matter”. Galaxies rotate, and masses bend light, in a way that seems only consistent with something new in the universe we didn’t predict. Observations of the whole universe, like the cosmic microwave background, let us estimate the properties of this something new, finding it to behave much more like matter than like radio waves or X-rays. So we call it dark matter.

But none of that guarantees that any of these experiments will find dark matter. The dark matter particles could have many different masses. They might interact faintly with ordinary matter, or with themselves, or almost not at all. They might not technically be particles at all. Each experiment makes some assumption, but no experiment yet can cover the most pessimistic possibility, that dark matter simply doesn’t interact in any usefully detectable way aside from by gravity.

Neutrinos also hide something new. The Standard Model predicts that neutrinos shouldn’t have mass, since it would screw up the way they mess with the mirror symmetry of the universe. But they do, in fact, have mass. We know because they oscillate, because they change when traveling, from one type to another, and that means those types must be mixes of different masses.

It’s not hard to edit the Standard Model to give neutrinos masses. But there’s more than one way to do it. Every way adds new particles we haven’t yet seen. And none of them tell us what neutrino masses should be. So there are a number of experiments, another zoo, trying to find out. (Maybe this one’s an aquarium?)

Are those experiments guaranteed to work?

Not so much as the LHC was to find the Higgs, but more than the dark matter experiments.

We particle physicists have a kind of holy book, called the Particle Data Book. It summarizes everything we know about every particle, and explains why we know it. It has many pages with many sections, but if you turn to page 10 of this section, you’ll find a small table about neutrinos. The table gives a limit: the neutrino mass is less than 0.8 eV (a mysterious unit called an electron-volt, about ten-to-the-minus-sixteen grams). That limit comes from careful experiments, using E=mc^2 to find what the missing mass could be when an electron-neutrino shoots out in radioactive beta decay. The limit is an inequality, “less than” rather than “equal to”, because the experiments haven’t detected any missing mass yet. So far, they only can tell us what they haven’t seen.

As these experiments get more precise, you could imagine them getting close enough to see some missing mass, and find the mass of a neutrino. And this would be great, and a guaranteed discovery, except that the neutrino they’re measuring isn’t guaranteed to have a mass at all.

We know the neutrino types have different masses, because they oscillate as they travel between the types. But one of the types might have zero mass, and it could well be the electron-neutrino. If it does, then careful experiments on electron-neutrinos may never give us a mass.

Still, there’s a better guarantee than for dark matter. That’s because we can do other experiments, to test the other types of neutrino. These experiments are harder to do, and the bounds they get are less precise. But if the electron neutrino really is massless, then we could imagine getting better and better at these different experiments, until one of them measures something, detecting some missing mass.

(Cosmology helps too. Wiggles in the shape of the universe gives us an estimate of the total, the mass of all the neutrinos averaged together. Currently, it gives another upper bound, but it could give a lower bound as well, which could be used along with weaker versions of the other experiments to find the answer.)

So neutrinos aren’t quite the guarantee the Higgs was, but they’re close. As the experiments get better, key questions will start to be answerable. And another piece of beyond-the-standard-model physics will be understood.

Not Made of Photons Either

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If you know a bit about quantum physics, you might have heard that everything is made out of particles. Mass comes from Higgs particles, gravity from graviton particles, and light and electricity and magnetism from photon particles. The particles are the “quanta”, the smallest possible units of stuff.

This is not really how quantum physics works.

You might have heard (instead, or in addition), that light is both particle and wave. Maybe you’ve heard it said that it is both at the same time, or that it is one or the other, depending on how you look at it.

This is also not really how quantum physics works.

If you think that light is both a particle and a wave, you might get the impression there are only two options. This is better than thinking there is only one option, but still not really the truth. The truth is there are many options. It all depends on what you measure.

Suppose you have a particle collider, like the Large Hadron Collider at CERN. Sometimes, the particles you collide release photons. You surround the collision with particle detectors. When a photon hits them, these particle detectors amplify it, turning it into an electrical signal in a computer.

If you want to predict what those particle detectors see, you might put together a theory of photons. You’ll try to calculate the chance that you see some specific photon with some specific energy to some reasonable approximation…and you’ll get infinity.

You might think you’ve heard this story before. Maybe you’ve heard people talk about calculations in quantum field theory that give infinity, with buzzwords like divergences and renormalization. You may remember them saying that this is a sign that our theories are incomplete, that there are parameters we can’t predict or that the theory is just a low-energy approximation to a deeper theory.

This is not that story. That story is about “ultraviolet divergences”, infinities that come from high-energy particles. This story is about “infrared divergences” from low-energy particles. Infrared divergences don’t mean our theory is incomplete. Our theory is fine. We’re just using it wrong.

The problem is that I lied to you a little bit, earlier. I told you that your particle detectors can detect photons, so you might have imagined they can detect any photon you like. But that is impossible. A photon’s energy is determined by its wavelength: X-rays have more energy than UV light, which has more energy than IR light, which has more energy than microwaves. No matter how you build your particle detector, there will be some energy low enough that it cannot detect, a wavelength of photons that gives no response at all.

When you think you’re detecting just one photon, then, you’re not actually detecting just one photon. You’re detecting one photon, plus some huge number of undetectable photons that are too low-energy to see. We call these soft photons. You don’t know how many soft photons you generate, because you can’t detect them. Thus, as always in quantum mechanics, you have to add up every possibility.

That adding up is crucial, because it makes the infinite results go away. The different infinities pair up, negative and positive, at each order of approximation. Those pesky infrared divergences aren’t really a problem, provided you’re honest about what you’re actually detecting.

But while infrared divergences aren’t really a problem, they do say something about your model. You were modeling particles as single photons, and that made your calculations complicated, with a lot of un-physical infinite results. But you could, instead, have made another model. You could have modeled particles as dressed photons: one photon, plus a cloud of soft photons.

For a particle physicists, these dressed photons have advantages and disadvantages. They aren’t always the best tool, and can be complicated to use. But one thing they definitely do is avoid infinite results. You can interpret them a little more easily.

That ease, though, raises a question. You started out with a model in which each particle you detect was a photon. You could have imagined it as a model of reality, one in which every electromagnetic field was made up of photons.

But then you found another model, one which sometimes makes more sense. And in that model, instead, you model your particles as dressed photons. You could then once again imagine a model of reality, now with every electromagnetic field made up of dressed photons, not the ordinary ones.

So now it looks like you have three options. Are electromagnetic fields made out of waves, or particles…or dressed particles?

That’s a trick question. It was always a trick question, and will always be a trick question.

Ancient Greek philosophers argued about whether everything was made of water, or fire, or innumerable other things. Now, we teach children that science has found the answer: a world made of atoms, or protons, or quarks.

But scientists are actually answering a different, and much more important, question. “What is everything really made of?” is still a question for philosophers. We scientists want to know what we will observe. We want a model that makes predictions, that tells us what actions we can do and what results we should expect, that lets us develop technology and improve our lives.

And if we want to make those predictions, then our models can make different choices. We can arrange things in different ways, grouping the fluid possibilities of reality into different concrete “stuff”. We can choose what to measure, and how best to describe it. We don’t end up with one “what everything is made of”, but more than one, different stories for different contexts. As long as those models make the right predictions, we’ve done the only job we ever needed to do.

What Might Lie Beyond, and Why

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As the new year approaches, people think about the future. Me, I’m thinking about the future of fundamental physics, about what might lie beyond the Standard Model. Physicists search for many different things, with many different motivations. Some are clear missing pieces, places where the Standard Model fails and we know we’ll need to modify it. Others are based on experience, with no guarantees but an expectation that, whatever we find, it will be surprising. Finally, some are cool possibilities, ideas that would explain something or fill in a missing piece but aren’t strictly necessary.

The Almost-Sure Things

Science isn’t math, so nothing here is really a sure thing. We might yet discover a flaw in important principles like quantum mechanics and special relativity, and it might be that an experimental result we trust turns out to be flawed. But if we chose to trust those principles, and our best experiments, then these are places we know the Standard Model is incomplete:

Neutrino Masses: The original Standard Model’s neutrinos were massless. Eventually, physicists discovered this was wrong: neutrinos oscillate, switching between different types in a way they only could if they had different masses. This result is familiar enough that some think of it as already part of the Standard Model, not really beyond. But the masses of neutrinos involve unsolved mysteries: we don’t know what those masses are, but more, there are different ways neutrinos could have mass, and we don’t yet know which is present in nature. Neutrino masses also imply the existence of an undiscovered “sterile” neutrino, a particle that doesn’t interact with the strong, weak, or electromagnetic forces.
Dark Matter Phenomena (and possibly Dark Energy Phenomena): Astronomers first suggested dark matter when they observed galaxies moving at speeds inconsistent with the mass of their stars. Now, they have observed evidence for it in a wide variety of situations, evidence which seems decisively incompatible with ordinary gravity and ordinary matter. Some solve this by introducing dark matter, others by modifying gravity, but this is more of a technical difference than it sounds: in order to modify gravity, one must introduce new quantum fields, much the same way one does when introducing dark matter. The only debate is how “matter-like” those fields need to be, but either approach goes beyond the Standard Model.
Quantum Gravity: It isn’t as hard to unite quantum mechanics and gravity as you might think. Physicists have known for decades how to write down a naive theory of quantum gravity, one that follows the same steps one might use to derive the quantum theory of electricity and magnetism. The problem is, this theory is incomplete. It works at low energies, but as the energy increases it loses the ability to make predictions, eventually giving nonsensical answers like probabilities greater than one. We have candidate solutions to this problem, like string theory, but we might not know for a long time which solution is right.
Landau Poles: Here’s a more obscure one. In particle physics we can zoom in and out in our theories, using similar theories at different scales. What changes are the coupling constants, numbers that determine the strength of the different forces. You can think of this in a loosely reductionist way, with the theories at smaller scales determining the constants for theories at larger scales. This gives workable theories most of the time, but it fails for at least one part of the Standard Model. In electricity and magnetism, the coupling constant increases as you zoom in. Eventually, it becomes infinite, and what’s more, does so at a finite energy scale. It’s still not clear how we should think about this, but luckily we won’t have to very soon: this energy scale is vastly vastly higher than even the scale of quantum gravity.
Some Surprises Guarantee Others: The Standard Model is special in a way that gravity isn’t. Even if you dial up the energy, a Standard Model calculation will always “make sense”: you never get probabilities greater than one. This isn’t true for potential deviations from the Standard Model. If the Higgs boson turns out to interact differently than we expect, it wouldn’t just be a violation of the Standard Model on its own: it would guarantee mathematically that, at some higher energy, we’d have to find something new. That was precisely the kind of argument the LHC used to find the Higgs boson: without the Higgs, something new was guaranteed to happen within the energy range of the LHC to prevent impossible probability numbers.

The Argument from (Theoretical) Experience

Everything in this middle category rests on a particular sort of argument. It’s short of a guarantee, but stronger than a dream or a hunch. While the previous category was based on calculations in theories we already know how to write down, this category relies on our guesses about theories we don’t yet know how to write.

Suppose we had a deeper theory, one that could use fewer parameters to explain the many parameters of the Standard Model. For example, it might explain the Higgs mass, letting us predict it rather than just measuring it like we do now. We don’t have a theory like that yet, but what we do have are many toy model theories, theories that don’t describe the real world but do, in this case, have fewer parameters. We can observe how these theories work, and what kinds of discoveries scientists living in worlds described by them would make. By looking at this process, we can get a rough idea of what to expect, which things in our own world would be “explained” in other ways in these theories.

The Hierarchy Problem: This is also called the naturalness problem. Suppose we had a theory that explained the mass of the Higgs, one where it wasn’t just a free parameter. We don’t have such a theory for the real Higgs, but we do have many toy models with similar behavior, ones with a boson with its mass determined by something else. In these models, though, the mass of the boson is always close to the energy scale of other new particles, particles which have a role in determining its mass, or at least in postponing that determination. This was the core reason why people expected the LHC to find something besides the Higgs. Without such new particles, the large hierarchy between the mass of the Higgs and the mass of new particles becomes a mystery, one where it gets harder and harder to find a toy model with similar behavior that still predicts something like the Higgs mass.
The Strong CP Problem: The weak nuclear force does what must seem like a very weird thing, by violating parity symmetry: the laws that govern it are not the same when you flip the world in a mirror. This is also true when you flip all the charges as well, a combination called CP (charge plus parity). But while it may seem strange that the weak force violates this symmetry, physicists find it stranger that the strong force seems to obey it. Much like in the hierarchy problem, it is very hard to construct a toy model that both predicts a strong force that maintains CP (or almost maintains it) and doesn’t have new particles. The new particle in question, called the axion, is something some people also think may explain dark matter.
Matter-Antimatter Asymmetry: We don’t know the theory of quantum gravity. Even if we did, the candidate theories we have struggle to describe conditions close to the Big Bang. But while we can’t prove it, many physicists expect the quantum gravity conditions near the Big Bang to produce roughly equal amounts of matter and antimatter. Instead, matter dominates: we live in a world made almost entirely of matter, with no evidence of large antimatter areas even far out in space. This lingering mystery could be explained if some new physics was biased towards matter instead of antimatter.
Various Problems in Cosmology: Many open questions in cosmology fall in this category. The small value of the cosmological constant is mysterious for the same reasons the small value of the Higgs mass is, but at a much larger and harder to fix scale. The early universe surprises many cosmologists by its flatness and uniformity, which has led them to propose new physics. This surprise is not because such flatness and uniformity is mathematically impossible, but because it is not the behavior they would expect out of a theory of quantum gravity.

The Cool Possibilities

Some ideas for physics beyond the standard model aren’t required, either from experience or cold hard mathematics. Instead, they’re cool, and would be convenient. These ideas would explain things that look strange, or make for a simpler deeper theory, but they aren’t the only way to do so.

Grand Unified Theories: Not the same as a “theory of everything”, Grand Unified Theories unite the three “particle physics forces”: the strong nuclear force, the weak nuclear force, and electromagnetism. Under such a theory, the different parameters that determine the strengths of those forces could be predicted from one shared parameter, with the forces only seeming different at low energies. These theories often unite the different matter particles too, but they also introduce new particles and new forces. These forces would, among other things, make protons unstable, and so giant experiments have been constructed to try to detect a proton decaying into other particles. So far none has been seen.
Low-Energy Supersymmetry: String theory requires supersymmetry, a relationship where matter and force particles share many properties. That supersymmetry has to be “broken”, which means that while the matter and force particles have the same charges, they can have wildly different masses, so that the partner particles are all still undiscovered. Those masses may be extremely high, all the way up at the scale of quantum gravity, but they could also be low enough to test at the LHC. Physicists hoped to detect such particles, as they could have been a good solution to the hierarchy problem. Now that the LHC hasn’t found these supersymmetric particles, it is much harder to solve the problem this way, though some people are still working on it.
Large Extra Dimensions: String theory also involves extra dimensions, beyond our usual three space and one time. Those dimensions are by default very small, but some proposals have them substantially bigger, big enough that we could have seen evidence for them at the LHC. These proposals could explain why gravity is so much weaker than the other forces. Much like the previous members of this category though, no evidence for this has yet been found.

I think these categories are helpful, but experts may quibble about some of my choices. I also haven’t mentioned every possible thing that could be found beyond the Standard Model. If you’ve heard of something and want to know which category I’d put it in, let me know in the comments!

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!

4 gravitons

The trials and tribulations of four gravitons and a physicist