Category Archives: Yang-Mills

The Amplituhedron and Other Excellently Silly Words

Nima Arkani-Hamed recently gave a talk at the Simons Center on the topic of what he and Jaroslav Trnka are calling the Amplituhedron.

There’s an article on it in Quanta Magazine. The article starts out a bit hype-y for my taste (too much language of importance, essentially), but it has several very solid descriptions of the history of the situation. I particularly like how the author concisely describes the Feynman diagram picture in the space of a single paragraph, and I would recommend reading that part even if you don’t have time to read the whole article. In general it’s worth it to get a picture of what’s going on.

That said, I obviously think I can clear a few things up, otherwise I wouldn’t be writing about it, so here I go!

“The” Amplituhedron

Nima’s new construction, the Amplituhedron, encodes amplitudes (building blocks of probabilities in particle physics) in N=4 super Yang-Mills as the “area” of a multi-dimensional analog of a polyhedron (hence, Amplitu-hedron).

Now, I’m a big supporter of silly-sounding words with amplitu- at the beginning (amplitudeologist, anyone?), and this is no exception. Anyway, the word Amplitu-hedron isn’t what’s confusing people. What’s confusing people is the word the.

When the Quanta article says that Nima has found “the” Amplituhedron, it makes it sound like he has discovered one central formula that somehow contains the whole universe. If you read the comments, many readers went away with that impression.

In case you needed me to say it, that’s not what is going on. The problem is in the use of the word “the”.

Suppose it was 1886, and I told you that a fellow named Carl Benz had invented “the Automobile”, a marvelous machine that can get everyone to work on time (as well as become the dominant form of life on Long Island).

My use of “the” might make you imagine that Benz invented some single, giant machine that would roam across the country, picking people up and somehow transporting everyone to work. You’d be skeptical of this, of course, expecting that long queues to use this gigantic, wondrous machine would swiftly ruin any speed advantage it might possess…

The Automobile, here to take you to work.

Or, you could view “the” in another light, as indicating a type of thing.

Much like “the Automobile” is a concept, manifested in many different cars and trucks across the country, “the Amplituhedron” is a concept, manifested in many different amplituhedra, each corresponding to a particular calculation that we might attempt.


Each amplituhedron has to do with an amplitude involving a specific number of particles, with a particular number of internal loops. (The Quanta article has a pretty good explanation of loops, here’s mine if you’d rather read that). Based on the problem you’re trying to solve, there are a set of rules that you use to construct the particular amplituhedron you need. The “area” of this amplituhedron (in quotation marks because I mean the area in an abstract, mathematical sense) is the amplitude for the process, which lets you calculate the probability that whatever particle physics situation you’re describing will happen.

Now, we already have many methods to calculate these probabilities. The amplituhedron’s advantage is that it makes these calculations much simpler. What was once quite a laborious and complicated four-loop calculation, Nima claims can be done by hand using amplituhedra. I didn’t get a chance to ask whether the same efficiency improvement holds true at six loops, but Nima’s description made it sound like it would at least speed things up.

[Edit: Some of my fellow amplitudeologists have reminded me of two things. First, that paper I linked above paved the way to more modern methods for calculating these things, which also let you do the four-loop calculation by hand. (You need only six or so diagrams). Second, even back then the calculation wasn’t exactly “laborious”, there were some pretty slick tricks that sped things up. With that in mind, I’m not sure Nima’s method is faster per se. But it is a fast method that has the other advantages described below.]

The amplituhedron has another, more sociological advantage. By describing the amplitude in terms of a geometrical object rather than in terms of our usual terminology, we phrase things in a way that mathematicians are more likely to understand. By making things more accessible to mathematicians (and the more math-headed physicists), we invite them to help us solve our problems, so that together we can come up with more powerful methods of calculation.

Nima and the Quanta article both make a big deal about how the amplituhedron gets rid of the principles of locality and unitarity, two foundational principles of quantum field theory. I’m a bit more impressed by this than Woit is. The fine distinction that needs to be made here is that the amplituhedron isn’t simply “throwing out” locality and unitarity. Rather, it’s written in such a way that it doesn’t need locality and unitarity to function. In the end, the formulas it computes still obey both principles. Nima’s hope is that, now that we are able to write amplitudes without needing locality and unitarity, if we end up having to throw out either of those principles to make a new theory we will be able to do so. That’s legitimately quite a handy advantage to have, it just doesn’t mean that locality and unitarity must be thrown out right now.

…and Disadvantages

It’s important to remember that this whole story is limited to N=4 super Yang-Mills. Nima doesn’t know how to apply it to other theories, and nobody else seems to have any good ideas either. In addition, this only applies to the planar part of the theory. I’m not going to explain what that term means here; for now just be aware that while there are tricks that let you “square” a calculation in super Yang-Mills to get a similar calculation in quantum gravity, those tricks rely on having non-planar data, or information beyond the planar part of the theory. So at this point, this doesn’t give us any new hints about quantum gravity. It’s conceivable that physicists will find ways around both of these limits, but for now this result, though impressive, is quite limited.

Nima hasn’t found some sort of singular “jewel at the heart of physics”. Rather, he’s found a very slick, very elegant, quite efficient way to make calculations within one particular theory. This is profound, because it expresses things in terms that mathematicians can address, and because it shows that we can write down formulas without relying on what are traditionally some of the most fundamental principles of quantum field theory. Only time will tell whether Nima or others can generalize this picture, taking it beyond planar N=4 super Yang-Mills and into the tougher theories that still await this sort of understanding.

Hexagon Functions – or, what is my new paper about?

I’ve got a new paper up on arXiv this week.

(For those of you unfamiliar with it, is a website where physicists, mathematicians, and researchers in related fields post their papers before submitting them to journals. It’s a cultural quirk of physics that probably requires a post in its own right at some point. Anyway…)

What’s it about? Well, the paper is titled Hexagon functions and the three-loop remainder function. Let’s go through that and figure out what it means.

When the paper refers to hexagon functions, it’s referring to functions used to describe situations with six particles involved. An important point to clarify here is that when counting the number of “particles involved”, we add together both the particles that go in and the particles that go out. So if three particles arrive somewhere, interact with each other in some complicated way, and then those three particles leave, that’s a six-particle process. Similarly, if two particles collide and four particles emerge, that’s also a six-particle process. (If you find the idea of more particles coming out than went in confusing, read this post.) Hexagon functions, then, can describe either of those processes.

What, specifically, are these functions being used for? Well, they’re being used to find the three-loop remainder function of N=4 super Yang-Mills.

N=4 super Yang-Mills is my favorite theory. If you haven’t read my posts on the subject, I encourage you to do so.

N=4 super Yang-Mills is so nice because it is so symmetric, and because it takes part in so many dualities. These two traits ended up being enough for Zvi Bern, Lance Dixon, and Vladimir Smirnov to propose an ansatz for all amplitudes in N=4 super Yang-Mills, called the BDS ansatz. (Amplitudes are how we calculate the probability of events occurring: for example, the probability of that “two particles going to four particles” situation I talked about earlier.)

Unfortunately, their formula was incomplete. While it was possible to prove that the formula was true for four-particle and five-particle processes, for six or more particles the formula failed. As it turned out though, it failed in a predictable way. All that was needed to fix it was to add something called the remainder function, the remaining part of the formula beyond the BDS ansatz.

The task, then, was to compute this remainder function.

I’ve talked before about how in quantum field theory, we calculate probabilities through increasingly complicated diagrams, keeping track of the complexity by counting the number of loops. The remainder function had already been computed up to two loops by working out these diagrams, but three looked to be considerably more difficult.

Luckily, we (myself, Lance Dixon, James Drummond, and Jeffrey Pennington) had a trick up our sleeves.

Formulas in N=4 super Yang-Mills have a property called maximal transcendentality. I’ve talked about transcendentality before:  essentially, it’s a way of counting how many powers of pi and logarithms are in your equations. Maximal transcendentality means that every part of the formula has a fixed, maximum number for its degree of transcendentality. In the case of the remainder function, this is two times the number of loops. Thus, the two-loop remainder function has degree of transcendentality four, so it can have pi to the fourth power in it, while the three-loop remainder function (the one that we calculated) has degree of transcendentality six, so it can have pi to the sixth power.

Of course, it can have lots of other expressions as well, which brings us back to the hexagon functions. By classifying the sort of functions that can appear in these formulas at each level of transcendentality, we find the basic building blocks that can show up in the remainder function. All we have to do then is ask what combinations of building blocks are allowed: which ones make good physics sense, for example, or which ones allow our formula to agree with the predictions of other researchers.

As it turns out, once you apply all the restrictions there is only one possible way to put the building blocks together that gives you a functioning formula. By process of elimination, this formula must be the correct three-loop six-point remainder function. Every extra constraint then serves as a check that nothing went wrong and that the formula is sound. Without calculating a single Feynman diagram, we’ve gotten our result!

Just to give you an idea of how complicated this result is, in order to write the formula out fully would take 800 pages. We’ve got shorter ways to summarize it, but perhaps it would be better to give a picture. The formula depends on three variables, called u, v, and w. To show how the formula behaves when all three variables change, here’s a plot of the formula in the variables u and v, for a series of different values of w.


Without our various shortcuts to generate this formula, it would have taken an extraordinarily long amount of time. Luckily, N=4 super Yang-Mills’s nice properties save the day, and allow us to achieve what I hope you won’t mind me calling a truly impressive result.

N=4: Maximal Particles for Maximal Fun

Part Four of a Series on N=4 Super Yang-Mills Theory

This is the fourth in a series of articles that will explain N=4 super Yang-Mills theory. In this series I take that phrase apart bit by bit, explaining as I go. Because I’m perverse and out to confuse you, I started with the last bit here, and now I’ve reached the final part.

N=4 Super Yang-Mills Theory

Last time I explained supersymmetry as a relationship between two particles, one with spin X and the other with spin X-½. It’s actually a leeetle bit more complicated than that.

When a shape is symmetric, you can turn it around and it will look the same. When a theory is supersymmetric, you can “turn” it, moving from particles with spin X to particles of spin X-½, and the theory will look the same.

With a 2D shape, that’s the whole story. But if you have a symmetric 3D shape, you can turn it in two different directions, moving to different positions, and the shape will look the same either way. In supersymmetry, the number of different ways you can “turn” the theory and still have it look the same is called N.

N=1 symmetric shape

N=2 symmetric shape

Consider the example of super Yang-Mills. If we start out with a particle of spin 1 (a Yang-Mills field), N=1 supersymmetry says that there will also be a particle of spin ½, similar to the particles of everyday matter. But suppose that instead we had N=2 supersymmetry. You can move from the spin 1 particle to spin ½ in one direction, or in the other one, and just like regular symmetry moving in two different directions will get you to two different positions. That means you need two different spin ½ particles! Furthermore, you can also move in one direction, then in the other one: you go from spin 1 to spin ½, then down from spin ½ to spin 0. So our theory can’t just have spin 1 and spin ½, it has to have spin 0 particles as well!

You can keep increasing N, as long as you keep increasing the number and types of particles. Finally, at N=4, you’ve got the maximal set: one Yang-Mills field with spin 1, four different spin ½ particles, and six different spin 0 scalars. The diagram below shows how the particles are related: you start in the center with a Yang-Mills field, and then travel in one of four directions to the spin ½ particles. Picking two of those directions, you travel further, to a scalar in between two spin ½ particles. Applying more supersymmetry just takes you back down: first to spin ½, then all the way back to spin 1.

N=4 super Yang-Mills is where the magic happens. Its high degree of symmetry gives it conformal invariance and dual conformal invariance, it has been observed to have maximal transcendentality and it may even be integrable. Any one of those statements could easily take a full blog post to explain. For now, trust me when I tell you that while N=4 super Yang-Mills may seem complicated, its symmetry means that deep down it is one of the easiest theories to work with, and in fact it might be the simplest non-gravity quantum field theory possible. That makes it an immensely important stepping stone, the first link to take us to a full understanding of particle physics.

One final note: you’re probably wondering why we stopped at N=4. At N=4 we have enough symmetry to go out from spin 1 to spin 0, and then back in to spin 1 again. Any more symmetry, and we need more space, which in this case means higher spin, which means we need to start talking about gravity. Supergravity takes us all the way up to N=8, and has its own delightful properties…but that’s a topic for another day.

Supersymmetry, to the Rescue!

Part Three of a Series on N=4 Super Yang-Mills Theory

This is the third in a series of articles that will explain N=4 super Yang-Mills theory. In this series I take that phrase apart bit by bit, explaining as I go. Because I’m perverse and out to confuse you, I started with the last bit here, and now I’m working my way up.

N=4 Super Yang-Mills Theory

Ah, supersymmetry…trendy, sexy, mysterious…an excuse to put “super” in front of words…it’s a grand subject.

If I’m going to manage to explain supersymmetry at all, then I need to explain spin. Luckily, you don’t need to know much about spin for this to work. While I could start telling you about how particles literally spin around like tops despite having a radius of zero, and how quantum mechanics restricts how fast they spin to a few particular values measured by Planck’s constant…all you really need to know is the following:

Spin is a way to categorize particles.

In particular, there are:
Spin 1: Yang-Mills fields are spin 1, carrying forces with a direction and strength.
Spin ½: This spin covers pretty much all of the particles you encounter in everyday matter: electrons, neutrons, and protons, as well as more exotic stuff like neutrinos. If you want to make large-scale, interesting structures like rocks or lifeforms you pretty much need spin ½ particles.
Spin 0: A spin zero field (also called a scalar) is a number, like a temperature, that can vary from place to place. The Higgs field is an example of a spin zero field, where the number is part of the mass of other particles, and the Higgs boson is a ripple in that field, like a cold snap would be for temperature.

While they aren’t important for this post, you can also have higher numbers for spin: gravity has spin 2, for example.

With this definition in hand, we can start talking about supersymmetry, which is also pretty straightforward if you ignore all of the actual details.

Supersymmetry is a relationship (or symmetry) between particles with spin X, and particles with spin X-½

For example, you could have a relationship between a spin 1 Yang-Mills field and a spin ½ matter particle, or between a spin ½ matter particle and a spin 0 scalar.

“Relationship” is a vague term here, much like it is in romance, and just like in romance you’d do well to clarify precisely what you mean by it. Here, it means something like the following: if you switch a particle for its “superpartner” (the other particle in the relationship) then the physics should remain the same. This has two important consequences: superpartners have the same mass as each-other and superpartners have the same interactions as each-other.

The second consequence means that if a particle has electric charge -1, its superpartner also has electric charge -1. If you’ve got gluons, each with a color and an anti-color, then their superpartners will also have both a color and an anti-color. Astute readers will have remembered that quarks just have a color or an anti-color, and realized the implication: quarks cannot be the superpartners of gluons.

Other, even more well-informed readers will be wondering about the first consequence. Such readers might have heard that the LHC is looking for superpartners, or that superpartners could explain dark matter, and that in either case superpartners have very high mass. How can this be if superpartners have to have the same mass as their partners among the regular particles?

The important point to make here is that our real world is not supersymmetric, even if superpartners are discovered at the LHC, because supersymmetry is broken. In physics, when a symmetry of any sort is broken it’s like a broken mirror: it no longer is the same on each side, but the two sides are still related in a systematic way. Broken supersymmetry means that particles that would be superpartners can have different masses, but they will still have the same interactions.

When people look for supersymmetry at the LHC, they’re looking for new particles with the same interactions as the old particles, but generally much higher mass. When I talk about supersymmetry, though, I’m talking about unbroken supersymmetry: pairs of particles with the same interactions and the same mass. And N=4 super Yang-Mills is full of them.

How full? N=4 full. And that’s next week’s topic.

Yang-Mills: Plays Well With Itself

Part Two of a Series on N=4 Super Yang-Mills Theory

This is the second in a series of articles that will explain N=4 super Yang-Mills theory. In this series I take that phrase apart bit by bit, explaining as I go. Because I’m perverse and out to confuse you, I started with the last bit here, and now I’m working my way up.

N=4 Super Yang-Mills Theory

So first these physicists expect us to accept a nonsense word like quark, and now they’re calling their theory Yang-Mills? What silly word are they going to foist on us next?

Umm…Yang and Mills are people.

Chen Ning Yang and Robert Mills were two physicists, famous for being very well treated by the Chinese government and for not being the father of nineteenth century Utilitarianism, respectively.

Has a wife 56 years younger than him

Did not design the Panopticon

In the 1950’s, Yang and Mills were faced with a problem: how to describe the strong nuclear force, the force that holds protons and neutrons in the nuclei of atoms together. At the time, the nature of this force was very mysterious. Nuclear experiments were uncovering new insight about the behavior of the strong force, but those experiments showed that the strong force didn’t behave like the well-understood force of electricity and magnetism. In particular, the strong force seemed to treat neutrons and protons in a related way, almost as if they were two sides of the same particle.

In 1954, Yang and Mills proposed a solution to this problem. In order to do so, they had to suggest something novel: a force that interacts with itself. To understand what that means and why that’s special, let’s discuss a bit about forces.

Each fundamental force can be thought of in terms of a field extending across space and time. The direction and strength of this field in each place determines which way the force pushes. When this field ripples, things that we observe as particles are created, the result of waves in the field. Particles of light, or photons, are waves in the field of the fundamental force of electricity and magnetism.

The electric force attracts charges with opposite sign, and repels charges when they have the same sign. Photons, however, have no charge, so they pass right through electric and magnetic fields. This is what I mean when I say that electricity and magnetism is a force that doesn’t interact with itself.

The strong force is different. Yang and Mills didn’t know this at the time, but we know now that the strong force acts on fundamental particles inside protons and neutrons called quarks, and that quarks come in three colors, unimaginatively named red, blue, and green, while their antiparticles are classified as antired, antiblue, or antigreen. Like all other forces, the strong force gives rise to a particle, in this case called a gluon. Unlike photons, gluons are not neutral! While they have no electric charge, they are affected by the strong force. Each gluon has a color and an anti-color: red/anti-green, blue/anti-red, etc. This means that while the strong force binds quarks together, it also binds itself together as well, keeping it from reaching outside of atoms and affecting the everyday world like electricity does.

Quarks and Gluons in a Proton

Yang and Mills’ description wasn’t perfect for the strong force (they had two types of charge rather than three) but it was fairly close to how the weak force worked, as other physicists realized in 1956. It was realized much later (in the 70’s) that a modification of Yang and Mills’ proposal worked for the strong force as well. In recognition of their insight, today the names Yang and Mills are attached to any force that interacts with itself.

A Yang-Mills theory, then, is a theory that contains a fundamental force that can interact with itself. This force generates particles (often called force-carrying bosons) which have something like charge or color with respect to the Yang-Mills force. If you remember the definition of a theory, you’ll see that we have everything we need: we have specified a particle (the force-carrying boson) and the ways in which it can interact (specifically, with itself).

Tune in next week when I explain the rest of the phrase, in a brief primer on the superheroic land of supersymmetry.

A Theorist’s Theory

Part One of a Series on N=4 Super Yang-Mills Theory

In my last post, I called Wikipedia’s explanation of N=4 super Yang-Mills theory only “half-decent”. It’s not particularly bad, though it could use more detail. What it isn’t, and what I wanted, was an explanation that would make sense to a general audience (i.e., you guys!).

Well, if you want something done right, you have to quote that cliché. Or, well, do it yourself.

This is the first in a series of articles that will explain N=4 super Yang-Mills theory. In this series I will take that phrase apart bit by bit, explaining as I go. And because I’m perverse and out to confuse you, I’ll start with the last bit and work my way up.

N=4 Super Yang-Mills Theory

Now as a relatively well-educated person, you may be grumbling at this point. “I know what a theory is!”

“A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment.”

Ah. It appears you’ve been talking to the biologists again. This is exactly why we needed this post. Let’s have a chat.

To be clear, when a biologist says that something (evolution, say, or germ theory) is a theory, this is exactly what they mean. They are describing an idea that has been repeatedly tested and that actually describes the real world. Most other scientists work the same way: geologists (plate tectonics theory), chemists (molecular orbital theory), even most physicists (big bang theory). But this isn’t what theoretical physicists mean when they say theory. In contrast, most things that theorists call theories have no experimental evidence, and usually aren’t even meant to describe the real world.

Unlike the AAAS definition above, theoretical physicists don’t have a formal definition of their usage of theory. If we did, it might go something like this:

“A theory (in theoretical physics) consists of a list of quantum fields, their properties, and how they interact. These fields do not need to be ones that exist in the natural world, but they do have to be (relatively) mathematically consistent. To study a theory is then to consider the interactions of a specific list of quantum fields, without taking into account any other fields that might otherwise interfere.”

Note that there are ways to get around parts of this definition. The (2,0) theory is famously mysterious because we don’t know how to write down the interactions between its fields, but even there we have an implicit definition of how the fields interact built into the theory’s definition, and the challenge is to make that definition explicit. Other theories stretch the definition of a quantum field, or cover a range of different properties. Still, all of them fit the basic template: define some mathematical entities, and describe how they interact.

With that definition in hand, some of you are already asking the next question: “What are the quantum fields of N=4 super Yang-Mills? How do they interact?”

Tune in to the next installment to find out!

Why I Study a Theory That Isn’t “True”

I study a theory called N=4 super Yang-Mills. (There’s a half-decent explanation of the theory here. For now, just know that it involves a concept called supersymmetry, where forces and matter are very closely related.) When I mention this to people, sometimes they ask me if I’m expecting to see evidence for N=4 super Yang-Mills at the Large Hadron Collider. And if not there, when can we expect a test of the theory?


Never? Yep. N=4 super Yang-Mills will never be tested, because N=4 super Yang-Mills (sYM for short) is not “true”.

We know it’s not “true”, because it contains particles that don’t exist. Not just particles we might not have found yet, but particles that would make the universe a completely different and possibly unknowable place.

So if it isn’t true, why do I study it?

Let me give you an analogy. Remember back in 2008 when Sarah Palin made fun of funding “fruit fly research in France”?

Most people I talked to found that pretty ridiculous. After all, fruit flies are one of the most stereotypical research animals, second only to mice. And besides, hadn’t we all grown up knowing about how they were used to research HOX genes?

Wait, you didn’t know about that? Evidently, you weren’t raised by a biologist.

HOX genes are how your body knows what limbs go where. When HOX genes activate in an embryo, they send signals, telling cells where to grow arms and legs.

Much of HOX genes’ power was first discovered in fruit flies. With their relatively simple genetics, scientists were able to manipulate the HOX genes, creating crazy frankenflies like Antennapedia (literally: antenna-feet) here.

A fruity fly’s HOX genes, and the body parts they correspond to.

Old antenna-feet. Ain’t he a beauty?

It was only later, as the science got more sophisticated, that biologists began to track what HOX genes do in humans, making substantial progress in understanding debilitating mutations.

How is this related to N=4 super Yang-Mills? Well, just as fruit flies are simpler to study than humans, sYM is simpler to study than the whole mess of unconnected particles that exist in the real world. We can do calculations with sYM that would be out of reach in normal particle physics. As we do these calculations, we discover new patterns and new techniques. The hope is that, just like HOX genes, we will discover traits that still hold in the more complicated situation of the real world. We’re not quite there yet, but it’s getting close.


By the way, make sure to watch Big Bang Theory on Thursday (11/29, 8/7c on CBS). Turns out, Sheldon is working on this stuff too, and for those who have read arXiv:1210.7709, his diagrams should look quite familiar…