Gauge Theory

One very important aspect of modern physics is something called gauge theory. Gauge theory explains how all four fundamental forces — gravity, electromagnetic, weak, and strong — work. However, it’s hard to find a nontechnical exposition of the principles of this theory. Here is the best attempt I’ve seen, although I still find it unsatisfying. Instead, I’m taking this absence as an opportunity for me to give my own explanation of gauge theory. [1]

Let me start with the strong force, which is the most straightforward example. This is the force between the quarks which makes them combine into protons and neutrons. Now, quarks have a property called “color”. A quark can have one of three colors, which physicists call red, blue, and green. However, these “colors” have nothing to do with the colors that you see. Each of these colors behaves the same way, and it’s only possible to tell what color a quark is is by comparing it with other quarks. A proton or neutron is made out of three quarks, one of each color, and similarly every particle which can be directly observed has a symmetrical combination of colors. So although we have names for these different colors, we can’t tell which is which.

This problem is a lot worse than it seems. You see, it’s actually fundamentally impossible to tell what color a quark is in an absolute sense. Let me explain: suppose some physicists in MIT decided to set a standard for quark colors. They start by isolating a quark[2] and making sure that it never changes color; they declare this to be the standard red quark. Comparing with this quark, they have a standard for when their own quarks are red. Now, the physicists in Harvard want to use this standard, so the physicists in MIT take another quark with the same color as their first quark, and send it to Harvard. You’d think that the quark in Harvard and the quark in MIT would have the same color. Now, just to verify this, two days later the physicists in MIT take another quark with the same color as the one they’re storing and send it to Harvard. Well, lo and behold, this second quark has a different color than the first quark. No matter how hard they try to prevent the quarks from changing color, the colors would still not be consistent.

Here’s the explanation: Color is not actually one universal concept. Every point in space and time has its own concept of quark color, and MIT September 5th color is something different from MIT September 7th color, which is different from Harvard September 7th color. It’s only possible to directly compare quark colors when they are in the same place and time. There has to be some relationship between these different concepts of color. After all, a quark which stays still between 7:00PM and 8:00PM must start with a color in the 7:00PM sense and end up with having color in the 8:00PM sense. Indeed, there’s something called a connection which links up these different sorts of color. The connection is the crucial component to this entire theory. For any journey a particle can take from one place to another, going in certain speeds in certain places in the middle, the connection describes a way to gradually transform a color for the beginning of the journey into a color for the end of the journey. When time passes for a quark and it needs to have another type of color, it always gets that color according to the way it went and the connection. So there is a correspondence between these color concepts, but it depends on the path you take from one to the other. That’s why the quarks in my imaginary experiment ended up with different colours; they took different paths.

Now, one of you might ask when are we going to get to strong force. Well, hypothetical reader, what I described is the strong force. The attraction of quarks with each other, their formation into protons and neutrons and other particles, everything about the strong force all comes as an indirect result of this connection. To give an example, you may have heard of gluons. The conventional explanation for the strong force is that it is generated by these gluon particles. What are gluons? Well, the connection only has an interesting role when multiple paths have inconsistent color changes. This inconsistency is called curvature. A gluon is a small area of space where this inconsistency is focused. The connection can always be described as a combination of many gluons, so in a sense, the strong force really is generated by gluons. A similar thing is true for all the other forces, so photons, gravitons, and the W and Z particles are all generated by the curvature of different connections.

The connection gives a theory for how quark color works, but how can it be force? How can it influence how a quark moves, rather than just what color it is? To answer that, I’ll move on to the second force I intend to discuss, the electromagnetic force.

The electromagnetic force is similar to the strong force. Here too, there is a connection, and here it influences every charged particle. One important thing is different, and it might seem a bit ridiculous: There’s only one color! So how can this connection do anything? Well, this whole time I have neglected the laws of quantum mechanics. However, they are important at everything that happens in this scale.

Let me summarize the rules for quantum mechanics. You would expect a particle to always be in certain place at any time, and for a quark to always have one specific color. Instead, particles are usually in something called a superposition. You can think of it being in a superposition as saying that the particle has different chances of being in any particular place. However, the superposition is more than that: it also gives something called a phase.

My description here of quantum mechanics is rather crude, but it will do for now. For better description, I reccommend this introduction, or for a more longer explanation these series of blog posts or the book QED: The Strange Theory of Light and Matter by Richard Feynmann.

Anyways, the crucial fact is that when a particle can be in many different positions, it is possible to compare the phases of these different possibilities. It is also possible to compare the phase between different times. Now, have you ever heard about how in quantum mechanics, a particle behaves like a wave? A wave is when the water is rising and falling in a regular pattern. In quantum mechanics, it is the phase that is changing in a particle which makes it act like a wave. Now, it turns out that how quickly the phase changes among different positions determines a particle’s momentum, and how quickly it changes over time determines it’s energy. Now, back to electromagnetism. You might have guessed by now what the connection does here: it influences phase. Well, just like in the case of colors, it doesn’t exactly change anything. More precisely, for charged particles phase is subjective and has different meanings in different places, and the connection gives a default for turning one kind of phase to another. This is just like the strong force.

Now it gets a bit subtle. When a particle is staying still in one spot, it will minimize energy, and so it will tend to not change its phase from the perspective of its own path. However, naturally it is not in one exact location, but in a superposition over a small region. As the phase tries to stay the same in each spot, the phases will be more and more inconsistent between different places. This means the particle will accumulate momentum, which will make the particle move. Put in a different way, the natural movement of phase acts as a sort of energy, which when it differs in different places generates a force.

So we know now how the connection influences the particle, but where does the connection come from? How did get to be in whatever state that it is in? Well, there are internal forces constraining the connection. Remember that the curvature of the connection measures how inconsistent it is in a small area. Well, the connection naturally is constrained so that it minimizes how much curvature it has. So in a vacuum, there is no curvature, and the connection doesn’t have any inconsistencies, except for small quantum fluctuations. However, when there is a charged particle, things are different. See, the particle has its own constraints, and it’s trying to move as smoothly as possible. How it moves is partially determined by the connection, and so when the particle is present, the connection naturally gives way. That is where the curvature of the connection comes from, and why there are forces between particles.

Now let’s get to gravity. With gravity what the connection changes is very interesting. First of all, here’s a question: Which way is up? There should be a fairly obvious answer. And yet, what you think is up, if travel far enough abroad, will turn out to be sideways are even down, with the real up. It can get even stranger: your watch is telling you the time is 6:00 AM, the locals insist that it’s 5:00 PM.

Of course, we all know that the Earth being round, and all of the strange consequences this implies, but that’s nothing profound. It’s just a strange convention. Although what you recognized as up is no longer called “up”, it’s still recognized as a direction. And it’s very easy for people from different locations to synchronize their watches. It’s still the same directions and times, just given different names.

At least, that’s what it seems. If you make very careful observations, you’ll find that direction, just like quark color, are not consistent in different places. Similarly, there is no absolute time which is consistent everywhere. These effects are all very subtle, except for one.

So now let’s really get to gravity. Throw something upwards, and afterwards it will fall. What makes it change direction? Nothing. That’s because it isn’t changing direction. So why does it hit the ground? Because the ground is moving, upwards and upwards until they collide.

More precisely, the connection also influences what it means to be move and what it means to stay still. That porcelain cup you threw is changing its path in the natural way with respect to the connection. The “move up” of one time is the same as the “stay still” of another which is the same as “move down” still further in the future. Our notion of staying still comes from comparing things with the ground, which accelerates upwards compared to the natural flow.

So why is the ground moving upwards? Well, for the small porcelain cup, following the connection is easy. It’s fairly self-consistent where the cup moves. But the ground is attached to the Earth, which is very big. And in spite of the curvature, the inconsistencies, all of the Earth has to move in unison. And it’s worse, since the curvature is generated by the Earth, and so it won’t ever move out of the way.

Now, finally, the weak force. Remember the rule with all the other forces: that a particle has an attribute that has different meanings in in different places, and so there can’t be a universal standard. Now let’s break these rules.

Back when people first studied the strong nuclear force, they found that it was very complicated. But in spite of the complexities, they did find one pattern: that protons and neutrons behave in the same way. The strong force between protons and protons and between protons and neutrons and between neutrons and neutrons is the same. Later on physicists discovered the pions. There are three types of pions, one positively charged, one negatively charged, and one neutral. They too all behave the same way with respect to the strong force, and all had approximately the same mass. Later more particles were discovered, and they always fitted into arrays of similar particles with the same strong interactions and about the same mass.

The whole thing reminded physicists of something else they’ve already seen, called spin. Spin is a property that every particle has. For example, electrons have spin 1/2. This means that each electron can be in one of two states: spin up or spin down. These are also called spin 1/2 and spin -1/2 (note the distinction: while the spin of a particular electron may be -1/2, the spin of electrons in general is always 1/2. Although the same word is used, these are different concepts). As you can guess from the name, flipping a spin up electron upside down makes a spin down electron, so spin is related to orientation. Excuse me while I ignore the question of what happens when you flip an electron sideways.

Not all particles are spin 1/2. Spin 0 particles are particles which only have one spin state, spin 0, while spin 1 particles can be in spin state -1, 0, or 1. In general, a particle has a set of consecutive spin states, more the higher the spin is. This is similar to the pattern described earlier for similar particles with different charges. Based on the analogy, this property of particles is called isospin.

So because of these similarities, it seems as though there’s some sort of virtual orientation[3] responsible for this isospin. Physicists considered the possibility that this virtual orientation has a connection to it, making yet another gauge theory. But to do that, the different virtual orientations must be indistinguishable. And yet, the proton and neutron, with isospin up and down, are clearly different. The virtual orientation with which the proton is isospin up is special, contradicting the principle that no virtual orientation is special.

What’s happening is something called symmetry breaking. Although fundamentally all of these virtual orientations are the same, there is a sort of pointer in each place in the universe which designates one virtual orientation as special. These pointers are called the Higgs field. Each pointer tends to align with the nearby pointer. However, this is impossible when the connection for the virtual is curved. So what happens is that any curvature of the connection is associated with movement of the Higgs field. This is called mixing. This mixture manifests as the Z and W particles. In addition, the Higgs field interacts with particles so that their behavior depends on how their virtual orientation aligns with it.

So there you have it. As I’ve just shown you, every fundamental force is based on the same idea: Some aspect of a particle is relative in place and time, and the connection mediates this aspect over different places and times to make it seem consistent locally. This is the basis of gauge theory. However, for each force this a different subtlety or tweak is added on. This makes each force behave in a unique way. But the underlying concept is always the same: gauge theory.

[1] While I was writing this article in my leisurely pace, Sean Carroll posted this article which is pretty much the sort of thing I was looking for, and so my justification for writing this is out of date. Oh well, these sorts of things happen in a 50 year old field.

[2] Actually, even getting this far is impossible. There is a phenomenon called quark confinement, which is that quarks never exist on their own, only in color-neutral groups. I hope you trust me when I tell you that this subtlety isn’t important, because I’m going to completely ignore it from now on.

[3] I will continue talking about this virtual orientation, but I want to note that it’s not a technical term. I don’t know if there is a technical term for it.

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