Earning a PhD by studying a theory that we know is wrong

I study a theory called N=4 super Yang-Mills.

When I say this to someone, I have a pretty good idea of how the conversation will go. First, the person will spend a few moments trying to pronounce the theory’s name. Giving up, they'll then try to bring things back to something they’ve heard of.

“N=4 super… umm… so, is that something they’re testing at the Large Hadron Collider?”

“Well, not really, no.”

“Is it astrophysics? Could you see it through a telescope?”

“No, nothing like that.”

“So… what sorts of experiments do you use to test it then?”

“None.”

There are no experiments that could test N=4 super Yang-Mills. Nor will there ever be, because N=4 super Yang-Mills doesn’t describe reality. In an everyday sense, N=4 super Yang-Mills is not “true.”

Yes, I study a theory that isn’t true.

Wait, what? How do you know...

First of all, N=4 super Yang-Mills involves supersymmetry. Some forms of supersymmetry are being searched for by the Large Hadron Collider. But those forms involve symmetries that are broken, which allow the particles to have distinctive characters.

In N=4 super Yang-Mills, supersymmetry is unbroken. Every particle has the same mass and the same charge. Furthermore, in N=4 super Yang-Mills that mass is equal to zero; like photons, the particles of N=4 super Yang-Mills would all travel at the speed of light.

There is no group of particles like that in the Standard Model. They can’t be undiscovered particles, either. Particles that travel at the speed of light are part of the everyday world if they have any interaction with normal matter whatsoever, so if the particles existed, we’d know about them. Since they don’t in N=4 super Yang-Mills, we know the theory isn't “true.”

Even with this knowledge, there is an even more certain way to know that N=4 super Yang-Mills isn't “true": it was never supposed to be true in the first place.

A theory by any other name

More than a few of you are probably objecting to my use of the word “theory” in the last few paragraphs. If N=4 super Yang-Mills isn't part of the real world, how could it possibly be a theory? After all, 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.”

That's courtesy of the American Association for the Advancement of Science. Confused? You must have been talking to the biologists again. Let’s explain.

In biology, a theory is indeed the confluence of multiple lines of real-world observations and evidence. That's precisely what scientists mean when they refer to evolution as a theory. And this is how it works in most other areas of science, from the germ theory of disease to the theory of plate tectonics to the big bang theory. But just because a term is used one way, that doesn't mean it isn't also frequently applied in another.

When something is called a theory, it is being compared to the other great theories of the past. In the case of something like the theory of evolution or the germ theory of disease, this comparison is saying that, like the theory of general relativity or evolution, a theory is so well-tested and so thoroughly incorporated into its field that it comes as close as science gets to final truth. Theoretical physics, on the other hand, often uses a different comparison; like general relativity, a theory in theoretical physics is a mathematical framework, a set of rules that describe the behavior of some system. Unlike general relativity, these systems don’t need to be grounded in experiment and they usually aren't even meant to describe the real world. N=4 super Yang-Mills isn't alone; check out Chern-Simons theory, Topological Quantum Field theories, or N=2 Superconformal Field theories.

What these theories do share is a certain level of rigor. Rather than being arbitrary, they involve precisely defined conditions that collectively give rise to interesting properties. While a theory in the theoretical physics sense isn't “true” in that it doesn’t describe the real world, it is “true” in that two researchers will agree on the theory’s properties. This allows interested parties to build off each other’s work.

While this sort of definition is perhaps most jarring in physics, other fields also define "theory" in a similar way. Essentially, every theory in mathematics is a theory in this sense (see Group theory and Category theory). The same is often true in closely related fields like computer science (Type theory, anyone?).

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.