Category Archives: General QFT

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.

A Theorist’s Theory

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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!