Sarah Palin would hate Yang-Mills theory

I’m not a mathematician, however. I’m a physicist. I don’t study things merely because they are mathematically interesting. Given that, why do I (and many others) study theories that aren’t true?

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 know found that pretty ridiculous. After all, fruit flies are an iconic part of the popular image of biology, research animals that have been used for more than a century. And besides, hadn't we all grown up knowing about how they were used to discover HOX genes?

(Wait, you didn’t know about that? Evidently, you weren’t raised around biologists.)

HOX genes are how your body knows which body parts go where. When HOX genes are activated in an embryo, they control the fate of cells as they develop, telling them where the body’s arms and legs should go, what rib goes where, and so on.

Much of HOX genes’ power was first discovered in fruit flies. Because of the fruit flies' relatively simple genetics, scientists were able to manipulate the HOX genes, creating crazy frankenflies like Antennapedia. It was only later, as experimental techniques in biology got more sophisticated, that biologists began to track what HOX genes do in mammals (including humans). When they did, biologists made substantial progress in understanding debilitating mutations.

Thus, while “fruit fly research” might seem useless to Sarah Palin because it doesn’t study “important things” like human health, the chance to study a simpler system like the fruit fly allows scientists to learn a lot about properties generally applicable to many different organisms. In the end, this kind of research is a necessary first step to understanding the “real world” of humans.

In much the same way, N=4 super Yang-Mills is a simpler, more easily manipulated system that allows theoretical physicists to learn about the more complicated systems that describe the "real" physical world. Not only does it carry the advantage of every particle having the same mass and charge, it also has a property called "conformal symmetry."

In a theory with conformal symmetry, physics is the same no matter what scale you look at—whether you’re looking at events separated by light years over a period of centuries or only by nanometers and femtoseconds. If your theory is conformal, your predictions will be the same, regardless of scale. You can even use different scales in different places, warping your perspective, as long as you make sure that all angles remain the same. This means you can always write the answer to a problem in terms of angles with no mention of distances, letting you take powerful shortcuts in your calculations.

While N=4 super Yang-Mills shares the simplicity and ease of fruit fly manipulation, the analogy is a little more loose when it comes to connecting the theory to the real world. Fruit fly HOX genes tell us about human HOX genes because they are connected by our shared evolutionary heritage. Rather than one central principle like evolution, the links between N=4 super Yang-Mills and the rest of physics are many and varied. To illustrate this, let's consider a few examples.

Really helping reality

N=4 super Yang-Mills is linked to string theory in a number of ways; often this is a consequence of how the two theories are defined. As you are probably aware, string theory describes the world as a collection of ten dimensional string-like objects. Since the world that we are accustomed to has only four dimensions (three space and one time), the extra six dimensions must be curled up in some way so small that we can’t perceive them.

An important principle in string theory (inherited from general relativity) is the idea that space itself can be shaped in different ways, corresponding to different solutions of Einstein’s equations. Living on the surface of a sphere is very different from living on a flat sheet, no matter what coordinates you use. Similarly, living on a higher dimensional version of a sphere (called de Sitter space) is very different from living in ordinary “flat” space. Because of this principle, different ways to curl up the six extra dimensions result in different apparent physics in the four dimensions of the “normal world.” If you want that “normal world” to look like N=4 super Yang-Mills, some forms of string theory make your job quite simple. Just make each of the six extra dimensions a circle!

AdS/CFT is another way that string theory can give rise to N=4 super Yang-Mills, through what is called the holographic principle. In the phrase AdS/CFT, CFT stands for conformal field theory (a theory with conformal symmetry, described above). AdS is short for anti de Sitter space. While de Sitter space is like a higher dimensional sphere, anti de Sitter space is the “opposite.” Cross-sections of a sphere look like circles, but cross-sections of anti de Sitter space are hyperbolas. This tends to make the full space somewhat tricky to visualize. (I’ve heard it described as being like a saddle, or like a sideways black hole, but honestly I don’t pretend to be able to picture the space in my head either.)

If you look at the boundary of an anti de Sitter space in string theory, you end up finding conformal symmetry, which gives rise to a conformal field theory. In particular, if the anti de Sitter space is five-dimensional (and the remaining five dimensions of string theory are curled up into a sphere), the theory that you find on the boundary is N=4 super Yang-Mills theory in four dimensions.

This is where the “hologram” in “holographic principle” comes in. It turns out that, in these cases, the boundary—the four dimensional N=4 super Yang-Mills—has all the same information as the full, five dimensional space, just like a 2D hologram contains all the information for a 3D image.

Outside of string theory, similarities between formulas in N=4 super Yang-Mills and more realistic theories tend to show up unexpectedly. Often times they’re patterns that don’t yet have a clear explanation. This may seem surprising, but on a certain level this sort of thing is reasonable, as particle physics theories have very strict mathematical rules. There are only so many different formulas that can obey those rules, so we should expect nature to reuse them whenever possible.

Quantum Chromodynamics (QCD) is the theory (in the “theory of evolution” sense) of quarks and gluons, the particles that make up protons and neutrons. Calculations in QCD are much harder than comparable calculations in N=4 super Yang-Mills, but it turns out there is one part of both calculations that ends up exactly the same (the technical term for this part is the “leading transcendentality” piece). In a sense, this part is the most complex piece of a QCD formula, so understanding it sheds light on what might be called the “backbone” of the theory.

N=4 super Yang-Mills is also deeply connected to another theory called N=8 supergravity, an easier to manipulate form of gravity. (To put things into perspective, N=8 supergravity is related to gravity in the same way that N=4 super Yang-Mills is related to particle physics.) By arranging the results correctly, a calculation in N=4 super Yang-Mills can tell you the corresponding result in N=8 supergravity just by squaring the formula. Since calculations in gravity are generally much more complicated than in particle physics, this has greatly sped up progress in investigating N=8 supergravity. Further research has found that this relationship seems to apply to more real-world theories of gravity and particle physics as well.

Finally, the simplicity of N=4 super Yang-Mills makes it an ideal testing ground for some of the more ambitious and advanced methods of particle physics calculation. In quantum field theory, the precision with which we can predict something is described in units called loops, the values of which vary depending on the strength of the forces in question. The more loops you want, the harder the calculations become. The most accurate predictions in quantum field theory (and possibly in all of science) have been verified by experiments up to ten decimal places, and those predictions come from calculations done at four loops.

In N=4 super Yang-Mills it’s possible to do calculations up to six loops, and some specific results have been predicted up to an arbitrarily high number of loops. The techniques required to get these results can often be generalized to real-world particle physics. These techniques give us an idea of what methods might be needed when the harder calculations of the real world catch up to the precision available in N=4 super Yang-Mills. Methods range from guessing parts of the result through something called a symbol to constraining a solution by assembling pieces of it like a jigsaw puzzle.

The secret to good sushi: great rice

Got all that? Here's one more metaphor: according to Japanese tradition, an apprentice sushi chef spends the first five years without ever touching food. Once those five years are over, they are allowed to prepare rice. Only after they have mastered rice are they allowed to cut and clean fish.

The physics of the real world are more complicated than the most ornate sushi roll. We want to get things right, and for those who study N=4 super Yang-Mills, that means working with simpler theories. We're whetting our tools and sharpening our skills, making us better able to tackle those theories that are true. Like fruit flies in biology, N=4 super Yang-Mills allows physicists to do research that would be prohibitively difficult in more obviously relevant systems. In doing so, we can find the basic building blocks of particle physics, eventually advancing calculations in the whole field. And that is why I study a theory that isn't true.

This feature is based on material that originally appeared on the author's blog.