Monthly Archives: December 2012

Why I Am Not A Mathematician

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(No relation to Russel’s Why I Am Not A Christian. Well, not much.)

I am a theorist. I study theories. Not the well-supported theories of the AAAS definition, but simply potential lists of particles, and lists that, further, are almost certainly not “true”.

Most people find that disconcerting. Used to thinking of scientists as people who investigate the real world, people whose ideas are always tested in the fire of experiment, the idea of a scientist whose work has no direct connection to the real world is a major source of cognitive dissonance…for at least a few minutes. After that, a light dawns in most people’s heads, as they turn to me with a sigh of relief and say,

“Oh. So you’re a Mathematician.”

No.

No, I am not a Mathematician. There is a difference, subtle but vast, between what I do and a mathematician does.

An illustrative example: Quantum Electro-Dynamics, or QED, is the most successful theory in the entirety of science. Yes, I do mean the entirety of science. Quantum Electro-Dynamics, the theory of how electrons and light behave, agrees with experiments to ten decimal places. Ten digits of detail, predicted then observed. That’s more confirmed accuracy than anything else in physics, in science at all, has ever achieved.

And if you ask a mathematician who specializes in this sort of thing, they’ll tell you that QED probably doesn’t exist.

Now, by this they don’t mean that electrons don’t exist, or that light doesn’t exist. What they mean is that, if you follow the theory’s implications all the way, you get a contradiction. You can calculate each step of the way, getting reasonable results each time, results that keep agreeing perfectly with experiments…but if you were to go all the way, off to infinity, you get results that make your whole theory stop making any sort of reasonable sense.

But as physicists, we keep using it. Because before reaching infinity, for any real calculation, it works. Perfectly.

That’s the difference between a theoretical physicist and a mathematician: for a mathematician, everything must be completely rigorous, and every implication, out to infinity, has to be vetted. For a physicist, if a theory gives reasonable results, we don’t really care whether it is completely clear how it works mathematically. We use physical reasoning, using concepts that work in the physical world, even if we’re studying a theory that doesn’t actually exist in the physical world. And while that sounds like a poor way to study abstract ideas, it allows us to take risks mathematicians can’t, which sometimes means we can make discoveries that even the mathematicians find interesting.

N=4: Maximal Particles for Maximal Fun

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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!

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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

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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.