Tag Archives: quantum field theory

What’s an Amplitude? Just about everything.

I am an Amplitudeologist. In other words, I study scattering amplitudes. I’ve explained bits and pieces of what scattering amplitudes are in other posts, but I ought to give a short definition here so everyone’s on the same page:

A scattering amplitude is the formula used to calculate the probability that some collection of particles will “scatter”, emerging as some (possibly different) collection of particles.

Note that I’m using some weasel words here. The scattering amplitude is not a probability itself, but “the formula used to calculate the probability”. For those familiar with the mathematics of waves, the scattering amplitude gives the amplitude of a “probability wave” that must be squared to get the probability. (Those familiar with waves might also ask: “If this is the amplitude, what about the period?” The truth is that because scattering amplitudes are calculated using complex numbers, what we call the “amplitude” also contains information about the wave’s “period”. It may seem like an inconsistent way to name things from the perspective of a beginning student, but it is actually consistent with the terminology in a large chunk of physics.)

In some of the simplest scattering amplitudes particles literally “scatter”, with two particles “colliding” and emerging traveling in different directions.

A scattering amplitude can also describe a more complicated situation, though. At particle colliders like the Large Hadron Collider, two particles (a pair of protons for the LHC) are accelerated fast enough that when they collide they release a whole slew of new particles. Since it still fits the “some particles go in, some particles go out” template, this is still described by a scattering amplitude.

It goes even further than that, though, because “some particles” could also just be “one particle”. If you’re dealing with something unstable (the particle equivalent of radioactive, essentially) then one particle can decay into two or more particles. There’s a whole slew of questions that require that sort of calculation. For example, if unstable particles were produced in the early universe, how many of them would be left around today? If dark matter is unstable (and some possible candidates are), when it decays it might release particles we could detect. In general, this sort of scattering amplitude is often of interest to astrophysicists when they happen to get involved in particle physics.

You can even use scattering amplitudes to describe situations that, at first glance, don’t sound like collisions of particles at all. If you want to find the effect of a magnetic field on an electron to high accuracy, the calculation also involves a scattering amplitude. A magnetic field can be thought of in terms of photons, particles of light, because light is a vibration in the electro-magnetic field. This means that the effect of a magnetic field on an electron can be calculated by “scattering” an electron and a photon.

If this looks familiar, check the handbook section.

In fact, doing the calculation in this way leads to what is possibly the most accurately predicted number in all of science.

Scattering amplitudes show up all over the place, from particle physics at the Large Hadron Collider to astrophysics to delicate experiments on electrons in magnetic fields. That said, there are plenty of things people calculate in theoretical physics that don’t use scattering amplitudes, either because they involve questions that are difficult to answer from the scattering amplitude point of view, or because they invoke different formulas altogether. Still, scattering amplitudes are central to the work of a large number of physicists. They really do cover just about everything.

Am I a String Theorist?

5 Replies

Perimeter, like most institutes of theoretical physics, divides their researchers into semi-informal groups. At Perimeter, these are:

Condensed Matter
Cosmology
Mathematical Physics
Particle Physics
Quantum Fields and Strings
Quantum Foundations
Quantum Gravity
Quantum Information
Strong Gravity

I’m in the Quantum Fields and Strings group, which many people seem to refer to simply as the String Theory group. So for the past week or so, I’ve been introducing myself as a String Theorist. As I briefly mention in my Who Am I? post, this isn’t completely accurate.

Am I a String Theorist?

The theories that I study do derive from string theory. They were first framed by string theorists, and research into them is still deeply intertwined with string theory research. I’ve definitely had occasion to compare my results to those of string theorists, or to bring in calculations by string theorists to advance my work.

And if you’re the kind of person who views the world as a competition between string theory and its rivals (like Loop Quantum Gravity) then I suppose I’m on the string theory “side”. I’m optimistic, at least, that the reason why string theory research is so much more common than any other approach to quantum gravity is simply because string theory provides many more interesting and viable projects for researchers.

On the other hand, though, there’s the basic fact that the theories I work with are not, themselves, string theories. They’re quantum field theories, the broader class that encompasses the modern synthesis of quantum mechanics and special relativity. The theories I work with are often reasonably close to the well-tested theories of the real world, close enough that the calculations are more “particle physics” than the they are “string theory”.

Of course, all of that could change. One of the great things about string theory is the way it connects lots of different interesting quantum field theories together. There’s a “string”, the “GKP string”, involved in the work of Basso, Sever, and Vieira, work that I will probably get involved with here at Perimeter. The (2,0) theory is a quantum field theory, but it’s much closer to string theory than to particle physics, so if I get more involved with the (2,0) theory would that make me a string theorist?

The fact is, these days string theory is so ubiquitous that the question “Am I a String Theorist?” doesn’t actually mean anything. String theory is there, lurking in the background, able to get involved at any time even if it’s not directly involved at present. Theoretical physicists don’t fall into neat categories.

I am a String Theorist. Also, I am not.

Hexagon Functions II: Lost in (super)Space

2 Replies

My new paper went up last night.

It’s on a very similar topic to my last paper, actually. That paper dealt with a specific process involving six particles in my favorite theory, N=4 super Yang-Mills. Two particles collide, and after the metaphorical dust settles four particles emerge. That means six “total” particles, if you add the two in with the four out, for a “hexagon” of variables. To understand situations like that, my collaborators and I created “hexagon functions”, formulas that depended on the states of the six particles.

One thing I didn’t emphasize then was that that calculation only applied to one specific choice of particles, one in which all of the particles are Yang-Mills bosons, particles (like photons) created by the fundamental forces. There are lots of other particles in N=4 super Yang-Mills, though. What happens when they collide?

That question is answered by my new paper. Though it may sound surprising, all of the other particles can be taken into account with a single formula. In order to explain why, I have to tell you about something called superspace.

A while back I complained about a blog post by George Musser about the (2,0) theory. One of the things that irked me about that post was his attempt to explain superspace:

Supersymmetry is the idea that spacetime, in addition to its usual dimensions of space and time, has an entirely different type of dimension—a quantum dimension, whose coordinates are not ordinary real numbers but a whole new class of number that can be thought of as the square roots of zero.

This is actually a great way to think about superspace…if you’re already a physicist. If you’re not, it’s not very informative. Here’s a better way to think about it:

As I’ve talked about before, supersymmetry is a relationship between different types of particles. Two particles related by supersymmetry have the same mass, and the same charge. While they can be very different in other ways (specifically, having different spin), supersymmetric particles are described by many of the same equations as each-other. Rather than writing out those equations multiple times, it’s often nicer to write them all in a unified way, and that’s where superspace comes in.

At its simplest, superspace is just a trick used to write equations in a simpler way. Instead of writing down a different equation for each particle we write one equation with an extra variable, representing a “dimension” of supersymmetry. Traveling in that dimension takes you from particle to particle, in the same way that “turning” the theory (as I phrase it here) does, but it does it within the space of a single equation.

That, essentially, is the trick that we use. With four “superspace dimensions”, we can include the four supersymmetries of N=4 super Yang-Mills, showing how the formulas vary when you go beyond the equation from our first paper.

So far, you may be wondering why I’m calling superspace a “dimension”, when it probably sounds like more of a label. I’ve mentioned before that, just because something is a variable, doesn’t mean it counts as a real dimension.

The key difference is that superspace dimensions are related to regular dimensions in a precise way. In a sense, they’re the square roots of regular dimensions. (Though independently, as George Musser described, they’re the square roots of zero: go in the same direction twice in supersymmetry, and you get back where you’re started, going zero distance.) The coexistence of these two seemingly contradictory statements isn’t some sort of quantum mystery, it’s just a consequence of the fact that, mathematically, I’m saying two very different things. I just can’t think of a way to explain them differently without math.

Superspace isn’t a real place…but it can often be useful to think of it that way. In theories with supersymmetry, it can unify the world, putting disparate particles together into a single equation.

Feeling Perturbed?

2 Replies

You might think of physics as the science of certainties and exact statements: action and reaction, F=ma, and all that. However, most calculations in physics aren’t exact, they’re approximations. This is especially true today, but it’s been true almost since the dawn of physics. In particular, approximations are performed via a method known as perturbation theory.

Perturbation theory is a trick used to solve problems that, for one reason or another, are too difficult to solve all in one go. It works by solving a simpler problem, then perturbing that solution, adjusting it closer to the target.

To give an analogy: let’s say you want to find the area of a circle, but you only know how to draw straight lines. You could start by drawing a square: it’s easy to find the area, and you get close to the area of the circle. But you’re still a long ways away from the total you’re aiming for. So you add more straight lines, getting an octagon. Now it’s harder to find the area, but you’re closer to the full circle. You can keep adding lines, each step getting closer and closer.

And so on.

This, broadly speaking, is what’s going on when particle physicists talk about loops. The calculation with no loops (or “tree-level” result) is the easier problem to solve, omitting quantum effects. Each loop then is the next stage, more complicated but closer to the real total.

There are, as usual, holes in this analogy. One is that it leaves out an important aspect of perturbation theory, namely that it involves perturbing with a parameter. When that parameter is small, perturbation theory works, but as it gets larger the approximation gets worse and worse. In the case of particle physics, the parameter is the strength of the forces involves, with weaker forces (like the weak nuclear force, or electromagnetism) having better approximations than stronger forces (like the strong nuclear force). If you squint, this can still fit the analogy: different shapes might be harder to approximate than the circle, taking more sets of lines to get acceptably close.

Where the analogy fails completely, though, is when you start approaching infinity. Keep adding more lines, and you should be getting closer and closer to the circle each time. In quantum field theory, though, this frequently is not the case. As I’ve mentioned before, while lower loops keep getting closer to the true (and experimentally verified) results, going all the way out to infinite loops results not in the full circle, but in an infinite result instead. There’s an understanding of why this happens, but it does mean that perturbation theory can’t be thought of in the most intuitive way.

Almost every calculation in particle physics uses perturbation theory, which means almost always we are just approximating the real result, trying to draw a circle using straight lines. There are only a few theories where we can bypass this process and look at the full circle. These are known as integrable theories. N=4 super Yang-Mills may be among them, one of many reasons why studying it offers hope for a deeper understanding of particle physics.

Look what I made!

Gravity is Yang-Mills Squared

12 Replies

There’s a concept that I’ve wanted to present for quite some time. It’s one of the coolest accomplishments in my subfield, but I thought that explaining it would involve too much technical detail. However, the recent BICEP2 results have brought one aspect of it to the public eye, so I’ve decided that people are ready.

If you’ve been following the recent announcements by the BICEP2 telescope of their indirect observation of primordial gravitational waves, you’ve probably seen the phrases “E-mode polarization” and “B-mode polarization” thrown around. You may even have seen pictures, showing that light in the cosmic microwave background is polarized differently by quantum fluctuations in the inflaton field and by quantum fluctuations in gravity.

But why is there a difference? What’s unique about gravitational waves that makes them different from the other waves in nature?

As it turns out, the difference all boils down to one statement:

Gravity is Yang-Mills squared.

This is both a very simple claim and a very subtle one, and it comes up in many many places in physics.

Yang-Mills, for those who haven’t read my older posts, is a general category that contains most of the fundamental forces. Electromagnetism, the strong nuclear force, and the weak nuclear force are all variants of Yang-Mills forces.

Yang-Mills forces have “spin 1”. Another way to say this is that Yang-Mills forces are vector forces. If you remember vectors from math class, you might remember that a vector has a direction and a strength. This hopefully makes sense: forces point in a direction, and have a strength. You may also remember that vectors can also be described in terms of components. A vector in four space-time dimensions has four components: x, y, z, and time, like so:

$\left( \begin{array}{c} x \\ y \\ z \\ t \end{array} \right)$

Gravity has “spin 2”.

As I’ve talked about before, gravity bends space and time, which means that it modifies the way you calculate distances. In practice, that means it needs to be something that can couple two vectors together: a matrix, or more precisely, a tensor, like so:

$\left( \begin{array}{cccc} xx & xy & xz & xt\\ yx & yy & yz & yt\\ zx & zy & zz & zt\\ tx & ty & tz & tt\end{array} \right)$

So while a Yang-Mills force has four components, gravity has sixteen. Gravity is Yang-Mills squared.

(Technical note: gravity actually doesn’t use all sixteen components, because it’s traceless and symmetric. However, often when studying gravity’s quantum properties theorists often add on extra fields to “complete the square” and fill in the remaining components.)

There’s much more to the connection than that, though. For one, it appears in the kinds of waves the two types of forces can create.

In order to create an electromagnetic wave you need a dipole, a negative charge and a positive charge at opposite ends of a line, and you need that dipole to change over time.

Change over time, of course, is a property of Gifs.

Gravity doesn’t have negative and positive charges, it just has one type of charge. Thus, to create gravitational waves you need not a dipole, but a quadrupole: instead of a line between two opposite charges, you have four gravitational charges (masses) arranged in a square. This creates a “breathing” sort of motion, instead of the back-and-forth motion of electromagnetic waves.

This is your brain on gravitational waves.

This is why gravitational waves have a different shape than electromagnetic waves, and why they have a unique effect on the cosmic microwave background, allowing them to be spotted by BICEP2. Gravity, once again, is Yang-Mills squared.

But wait there’s more!

So far, I’ve shown you that gravity is the square of Yang-Mills, but not in a very literal way. Yes, there are lots of similarities, but it’s not like you can just square a calculation in Yang-Mills and get a calculation in gravity, right?

Well actually…

In quantum field theory, calculations are traditionally done using tools called Feynman diagrams, organized by how many loops the diagram contains. The simplest diagrams have no loops, and are called tree diagrams.

Fascinatingly, for tree diagrams the message of this post is as literal as it can be. Using something called the Kawai-Lewellen-Tye relations, the result of a tree diagram calculation in gravity can be found just by taking a similar calculation in Yang-Mills and squaring it.

(Interestingly enough, these relations were originally discovered using string theory, but they don’t require string theory to work. It’s yet another example of how string theory functions as a laboratory to make discoveries about quantum field theory.)

Does this hold beyond tree diagrams? As it turns out, the answer is again yes!
The calculation involved is a little more complicated, but as discovered by Zvi Bern, John Joseph Carrasco, and Henrik Johansson, if you can get your calculation in Yang-Mills into the right format then all you need to do is square the right thing at the right step to get gravity, even for diagrams with loops!

zvi-bern-350

This trick, called BCJ duality after its discoverers, has allowed calculations in quantum gravity that far outpace what would be possible without it. In N=8 supergravity, the gravity analogue of N=4 super Yang-Mills, calculations have progressed up to four loops, and have revealed tantalizing hints that the uncontrolled infinities that usually plague gravity theories are absent in N=8 supergravity, even without adding in string theory. Results like these are why BCJ duality is viewed as one of the “foundational miracles” of the field for those of us who study scattering amplitudes.

Gravity is Yang-Mills squared, in more ways than one. And because gravity is Yang-Mills squared, gravity may just be tame-able after all.

A Wild Infinity Appears! Or, Renormalization

How (Not) to Sum the Natural Numbers: Zeta Function Regularization

16 Replies

$1+2+3+4+5+6+\ldots=-\frac{1}{12}$

If you follow Numberphile on YouTube or Bad Astronomy on Slate you’ve already seen this counter-intuitive sum written out. Similarly, if you follow those people or Sciencetopia’s Good Math, Bad Math, you’re aware that the way that sum was presented by Numberphile in that video was seriously flawed.

There is a real sense in which adding up all of the natural numbers (numbers 1, 2, 3…) really does give you minus twelve, despite all the reasons this should be impossible. However, there is also a real sense in which it does not, and cannot, do any such thing. To explain this, I’m going to introduce two concepts: complex analysis and regularization.

This discussion is not going to be mathematically rigorous, but it should give an authentic and accurate view of where these results come from. If you’re interested in the full mathematical details, a later discussion by Numberphile should help, and the mathematically confident should read Terence Tao’s treatment from back in 2010.

With that said, let’s talk about sums! Well, one sum in particular:

$\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+\frac{1}{6^s}+\ldots = \zeta(s)$

If s is greater than one, then each term in this infinite sum gets smaller and smaller fast enough that you can add them all up and get a number. That number is referred to as $\zeta(s)$ , the Riemann Zeta Function.

So what if s is smaller than one?

The infinite sum that I described doesn’t converge for s less than one. Add it up in any reasonable way, and it just approaches infinity. Put another way, the sum is not properly defined. But despite this, $\zeta(s)$ is not infinite for s less than one!

Now as you might object, we only defined the Riemann Zeta Function for s greater than one. How do we know anything at all about it for s less than one?

That is where complex analysis comes in. Complex analysis sounds like a made-up term for something unreasonably complicated, but it’s quite a bit more approachable when you know what it means. Analysis is the type of mathematics that deals with functions, infinite series, and the basis of calculus. It’s often contrasted with Algebra, which usually considers mathematical concepts that are discrete rather than smooth (this definition is a huge simplification, but it’s not very relevant to this post). Complex means that complex analysis deals with functions, not of everyday real numbers, but of complex numbers, or numbers with an imaginary part.

So what does complex analysis say about the Riemann Zeta Function?

One of the most impressive results of complex analysis is the discovery that if a function of a complex number is sufficiently smooth (the technical term is analytic) then it is very highly constrained. In particular, if you know how the function behaves over an area (technical term: open set), then you know how it behaves everywhere else!

If you’re expecting me to explain why this is true, you’ll be disappointed. This is serious mathematics, and serious mathematics isn’t the sort of thing you can give the derivation for in a few lines. It takes as much effort and knowledge to replicate a mathematical result as it does to replicate many lab results in science.

What I can tell you is that this sort of approach crops up in many places, and is part of a general theme. There is a lot you can tell about a mathematical function just by looking at its behavior in some limited area, because mathematics is often much more constrained than it appears. It’s the same sort of principle behind the work I’ve been doing recently.

In the case of the Riemann Zeta Function, we have a definition for s greater than one. As it turns out, this definition still works if s is a complex number, as long as the real part of s is greater than one. Using this information, the value of the Riemann Zeta Function for a large area (half of the complex numbers), complex analysis tells us its value for every other number. In particular, it tells us this:

$\zeta(-1)= -\frac{1}{12}$

If the Riemann Zeta Function is consistently defined for every complex number, then it must have this value when s is minus one.

If we still trusted the sum definition for this value of s, we could plug in -1 and get

$1+2+3+4+5+6+\ldots=-\frac{1}{12}$

Does that make this statement true? Sort of. It all boils down to a concept from physics called regularization.

In physics, we know that in general there is no such thing as infinity. With a few exceptions, nothing in nature should be infinite, and finite evidence (without mathematical trickery) should never lead us to an infinite conclusion.

Despite this, occasionally calculations in physics will give infinite results. Almost always, this is evidence that we are doing something wrong: we are not thinking hard enough about what’s really going on, or there is something we don’t know or aren’t taking into account.

Doing physics research isn’t like taking a physics class: sometimes, nobody knows how to do the problem correctly! In many cases where we find infinities, we don’t know enough about “what’s really going on” to correct them. That’s where regularization comes in handy.

Regularization is the process by which an infinite result is replaced with a finite result (made “regular”), in a way so that it keeps the same properties. These finite results can then be used to do calculations and make predictions, and so long as the final predictions are regularization independent (that is, the same if you had done a different regularization trick instead) then they are legitimate.

In string theory, one way to compute the required dimensions of space and time ends up giving you an infinite sum, a sum that goes 1+2+3+4+5+…. In context, this result is obviously wrong, so we regularize it. In particular, we say that what we’re really calculating is the Riemann Zeta Function, which we happen to be evaluating at -1. Then we replace 1+2+3+4+5+… with -1/12.

Now remember when I said that getting infinities is a sign that you’re doing something wrong? These days, we have a more rigorous way to do this same calculation in string theory, one that never forces us to take an infinite sum. As expected, it gives the same result as the old method, showing that the old calculation was indeed regularization independent.

Sometimes we don’t have a better way of doing the calculation, and that’s when regularization techniques come in most handy. A particular family of tricks called renormalization is quite important, and I’ll almost certainly discuss it in a future post.

So can you really add up all the natural numbers and get -1/12? No. But if a calculation tells you to add up all the natural numbers, and it’s obvious that the result can’t be infinite, then it may secretly be asking you to calculate the Riemann Zeta Function at -1. And that, as we know from complex analysis, is indeed -1/12.

Update on the Amplituhedron

2 Replies

Awhile back I wrote a post on the Amplituhedron, a type of mathematical object found by Nima Arkani-Hamed and Jaroslav Trnka that can be used to do calculations of scattering amplitudes in planar N=4 super Yang-Mills theory. (Scattering amplitudes are formulas used to calculate probabilities in particle physics, from the probability that an unstable particle will decay to the probability that a new particle could be produced by a collider.) Since then, they published two papers on the topic, the most recent of which came out the day before New Year’s Eve. These papers laid out the amplituhedron concept in some detail, and answered a few lingering questions. The latest paper focused on one particular formula, the probability that two particles bounce off each other. In discussing this case, the paper serves two purposes:

1. Demonstrating that Arkani-Hamed and Trnka did their homework.

2. Showing some advantages of the amplituhedron setup.

Let’s talk about them one at a time.

Doing their homework

There’s already a lot known about N=4 super Yang-Mills theory. In order to propose a new framework like the amplituhedron, Arkani-Hamed and Trnka need to show that the new framework can reproduce the old knowledge. Most of the paper is dedicated to doing just that. In several sections Arkani-Hamed and Trnka show that the amplituhedron reproduces known properties of the amplitude, like the behavior of its logarithm, its collinear limit (the situation when two momenta in the calculation become parallel), and, of course, unitarity.

What, you heard the amplituhedron “removes” unitarity? How did unitarity get back in here?

This is something that has confused several commenters, both here and on Ars Technica, so it bears some explanation.

Unitarity is the principle that enforces the laws of probability. In its simplest form, unitarity requires that all probabilities for all possible events add up to one. If this seems like a pretty basic and essential principle, it is! However, it and locality (the idea that there is no true “action at a distance”, that particles must meet to interact) can be problematic, causing paradoxes for some approaches to quantum gravity. Paradoxes like these inspired Arkani-Hamed to look for ways to calculate scattering amplitudes that don’t rely on locality and unitarity, and with the amplituhedron he succeeded.

However, just because the amplituhedron doesn’t rely on unitarity and locality, doesn’t mean it violates them. The amplituhedron, for all its novelty, still calculates quantities in N=4 super Yang-Mills. N=4 super Yang-Mills is well understood, it’s well-behaved and cuddly, and it obeys locality and unitarity.

This is why the amplituhedron is not nearly as exciting as a non-physicist might think. The amplituhedron, unlike most older methods, isn’t based on unitarity and locality. However, the final product still has to obey unitarity and locality, because it’s the same final product that others calculate through other means. So it’s not as if we’ve completely given up on basic principles of physics.

Not relying on unitarity and locality is valuable. For those who research scattering amplitudes, it has often been useful to try to “eliminate” one principle or another from our calculations. 20 years ago, avoiding Feynman diagrams was the key to finding dramatic simplifications. Now, many different approaches try to sidestep different principles. (For example, while the amplituhedron calculates an integrand and leaves a final integral to be done, I’m working on approaches that never employ an integrand.)

If we can avoid relying on some “basic” principle, that’s usually good evidence that the principle might be a consequence of something even more basic. By showing how unitarity can arise from the amplituhedron, Arkani-Hamed and Trnka have shown that a seemingly basic principle can come out of a theory that doesn’t impose it.

Advantages of the Amplituhedron

Not all of the paper compares to old results and principles, though. A few sections instead investigate novel territory, and in doing so show some of the advantages and disadvantages of the amplituhedron.

Last time I wrote on this topic, I was unclear on whether the amplituhedron was more efficient than existing methods. At this point, it appears that it is not. While the formula that the amplituhedron computes has been found by other methods up to seven loops, the amplituhedron itself can only get up to three loops or so in practical cases. (Loops are a way that calculations are classified in particle physics. More loops means a more complex calculation, and a more precise final result.)

The amplituhedron’s primary advantage is not in efficiency, but rather in the fact that its mathematical setup makes it straightforward to derive interesting properties for any number of loops desired. As Trnka occasionally puts it, the central accomplishment of the amplituhedron is to find “the question to which the amplitude is the answer”. By being able to phrase this “question” mathematically, one can be very general, which allows them to discover several properties that should hold no matter how complex the rest of the calculation becomes. It also has another implication: if this mathematical question has a complete mathematical answer, that answer could calculate the amplitude for any number of loops. So while the amplituhedron is not more efficient than other methods now, it has the potential to be dramatically more efficient if it can be fully understood.

All that said, it’s important to remember that the amplituhedron is still limited in scope. Currently, it applies to a particular theory, one that doesn’t (and isn’t meant to) describe the real world. It’s still too early to tell whether similar concepts can be defined for more realistic theories. If they can, though, it won’t depend on supersymmetry or string theory. One of the most powerful techniques for making predictions for the Large Hadron Collider, the technique of generalized unitarity, was first applied to N=4 super Yang-Mills. While the amplituhedron is limited now, I would not be surprised if it (and its competitors) give rise to practical techniques ten or twenty years down the line. It’s happened before, after all.

The Parke-Taylor Amplitudes: Why Quantum Field Theory Might Not Be So Hard, After All

2 Replies

If you’ve been following my blog for a while, you know that Quantum Field Theory is hard work. To calculate anything, you have to draw an ever-increasing number of diagrams, translate them into formulas involving the momentum and energy of your particles, and add all those formulas up to get your final result, the amplitude of the process you’re interested in.

As I said in that post, my area of research involves trying to find patterns in the results of these calculations, patterns that make doing the calculation simpler. With that in mind, you might wonder why we expect to find any patterns in the first place. If Quantum Field Theory is so complicated, what insurance do we have that it can be made simpler? Where does the motivation come from?

Our motivation comes from a series of discoveries that show that things really do simplify, often in unexpected ways. I won’t go through all of these discoveries here, but I want to tell you about one of the first discoveries that showed amplitudes researchers that they were on the right track.

Let’s try to calculate a comparatively simple process. Say that we’ve got two gluons (force carrying bosons for the strong force, an example of a Yang-Mills field). Suppose the two gluons collide, and some number of gluons emerge. It could be two again, or it could be three, or more.

For now, let’s just think about diagrams at tree level, that is, diagrams with no loops. The particles can travel from place to place in the diagram, but they can’t form closed loops on the inside.

Gluons have two types of interactions, places where particle lines can come together. You can either have three lines meeting at one point, or four.

If two gluons come in and two come out, we have four possible diagrams:

4ptMHV

Note that while the last diagram looks like it has a loop in it (in the form of the triangle in the middle), actually that triangle just represents that two particles are passing each other without colliding, so that their lines cross.

The number of diagrams increases substantially as you increase the number of outgoing particles. With two particles going to three particles, you get fifteen diagrams. Here are three examples:

5ptMHV

Since the number of diagrams just keeps increasing, you’d expect the final amplitude to become more and more complicated as well. However, Steven Parke and Tomasz Taylor found in 1986 that for a particular arrangement of the spins of the particles (for technical people: this is the Maximally Helicity Violating configuration, or two particles with negative helicity and all the rest with positive helicity) the answer simplifies dramatically. In the sort of variables we use these days, the result can be expressed in an incredibly simple form:

$\frac{\langle 1 | 2 \rangle^4}{ \langle 1 | 2 \rangle\langle 2 | 3 \rangle\langle 3 | 4 \rangle \ldots \langle n-1 | n \rangle\langle n | 1 \rangle}$

Here the angle brackets represent momenta of the incoming (for 1 and 2) and outgoing (all the other numbers) particles, with n being the total number of particles (two going in, and however many going out). (Technically, these are spinor-helicity variables, and those interested in the technical details should check out chapter 3 of this or chapter 2 of this.)

Nowadays, we know why this amplitude looks so simple, in terms of something called B C F W recursion. At the time though, it was quite extraordinary.

This is the sort of simplification we keep running into when studying amplitudes. Almost always, it means that there is some deeper principle that we don’t yet understand, something that would let us do our calculations much faster and more efficiently. It indicates that Quantum Field Theory might not be so hard after all.

4 gravitons

Stories about physics from someone who's been there

Tag Archives: quantum field theory

What’s an Amplitude? Just about everything.

Am I a String Theorist?

Hexagon Functions II: Lost in (super)Space

Feeling Perturbed?

Look what I made!

Gravity is Yang-Mills Squared

A Wild Infinity Appears! Or, Renormalization

How (Not) to Sum the Natural Numbers: Zeta Function Regularization

Update on the Amplituhedron

The Parke-Taylor Amplitudes: Why Quantum Field Theory Might Not Be So Hard, After All