The fundamental forces

By this point, we’ve talked quite a bit about what matter is made of, but we haven’t really said much about how it works. That is to say, we have no idea yet how the fundamental particles interact with each other. (Actually, that isn’t entirely true. We’ve talked about the fact that quarks combine to make hadrons as a result of “color”, and I’ll get back to this in a moment.) We currently have all the pieces, but not the rules of the game.

So what are the rules of the game? You likely already know the answer to that question, mostly because I wrote it in the title, but also because you know from last semester that essentially all of mechanics is using Newton’s second equation, F = dp/dt = ma, to determine what a system will do.

Alright, so forces are what we need to tell us how particles interact. At the moment, we believe there are four forces in nature*: the strong force, electromagnetism, the weak force, and gravity. I certainly hope you’ve heard of at least two of those, by the way. But is this really the whole list? What about the normal force or friction? Or the drag force or the spring force? For that matter, what about centrifugal force? (For those of you who don’t think it exists, I refer you to this xkcd comic: http://xkcd.com/123/ .)

The answer is that none of these other forces are truly fundamental; all of them are produced by one of the four forces on the list. In fact, in most cases, each of the forces I mentioned is just a product of electromagnetism. Is this surprising? Take the normal force for example: as I write this, I’m sitting on a chair on the third floor of a building at CERN. Gravity is pulling everything down, so why don’t I fall through the chair? For that matter, why doesn’t the chair fall through the floor? The reason is electromagnetism: the electrons in me repel the electrons in the chair, and the electrons in the chair repel the electrons in the electrons in the floor, and as a result everything stays right where it is.

Okay, so hopefully you believe me that every interaction is really the result of just four fundamental forces. This means that it’s time for another table:


There are a few aspects of this table I’d like to make special mention of. For one thing, each force is associated with a particular charge. This shouldn’t be surprising, especially if you’re in EM this semester, where you’ll have seen the equation relating the electric force to the field: F = qE. That is, if we want to know the effect (the force) of a field on a particular particle, we have to know how susceptible the particle is to that field; this is exactly what charge is: a highly charged particle will be affected quite a bit by a particular electric field, whereas an uncharged particle won’t be affected at all by the field.

But electric charge is only the most familiar kind of charge (the one associated with electromagnetism). Every other force has its own charge as well. As it turns out, the mysterious “color” quantum number from last blog is just the charge of the strong force. This is the “true” definition of color, and why it deserves to be believed in. (Actually, even this isn’t really enough, but showing that it fits into the framework is a good start.)

The charge of the weak force is often called “flavor”, for reasons passing understanding. As a matter of fact, you already know the flavors of the quarks and leptons: the names themselves are the flavors. Thus, quarks come in six flavors: up, down, charm, strange, top and bottom. Leptons are generally considered to come in three flavors: electron type, muon type, and tau type.

Finally, the charge of gravity is, naturally enough, mass. After all, the amount of mass an object has determines how strongly it’s affected by gravity, and even just comparing Newton’s law of gravitation to Coulomb’s law,

F_g = G_Nm₁m₂/r²

F_e = k_eq₁q₂/r²

suggests that mass relates to gravity the same way as electric charge relates to electromagnetism.

The third column of the table is actually a little bogus, because each force behaves quite differently as a function of distance, so “relative strength” depends a lot on how far back you decide to stand, so to speak. Still, there’s some useful information to be gleaned here, including the answer to the question I posed a few weeks ago: If protons repel each other electrically, then why do they get smashed together so tightly inside the nucleus? The answer is the strong force: protons and neutrons are made of quarks, which carry color, as we saw last week. Thus, they experience the strong force, much stronger than electromagnetism, which binds them together inside the nucleus.

While we’re still looking at the table, I’d like to talk about the “carrier particles” in the fourth column. Classically, the way forces operate was a bit mysterious, because they seemed to work from a distance. Magnets, for instance, attract other magnets even though there’s nothing physically connecting them; and the Sun attracts the Earth even through the vacuum of space. This really irritated Newton, who couldn’t figure out how objects could affect each other without some sort of tangible connection between them. Newton couldn’t figure a way out of the problem of “action-at-a-distance”, so he did exactly what I do when confronted with a tough homework question: he punted, and left the issue for someone else to solve.

The problem had to wait another two and a half centuries, but eventually the newly arrived “quantum field theory” proposed a good answer. Forces, according to field theory, in fact don’t cause action at a distance: they are instead communicated from one particle to another by a particle which “carries” the force. These particles are listed in the fourth column; in many ways they’re very similar, but they’re also just different enough to explain why the individual forces act the way they do.

To be fair, I should mention that this is where the story really starts becoming incomplete. First of all, we need a separate field theory to explain each of the different forces: electromagnetism, the “easiest” force, is described by “quantum electrodynamics” (QED); the strong force is described by “quantum chromodynamics” (QCD); and the weak force, which has already been merged with electromagnetism by a guy named Steve Weinberg (no relation), is described by the “electroweak model”. We’ll talk more about each of these in the next few weeks.

And what about gravity? What quantum field theory describes that force? Well, this is embarrassing, but it turns out we don’t have one. That’s right: the oldest and best-known of the fundamental forces, and we have no idea how it works at small distances. None. Some textbooks and Scientific American articles play down this problem, saying the theory isn’t “completely satisfactory”, or that “gravity doesn’t play a significant role in particle interactions”. (That last point is mostly true; a glance at the table shows us that gravity is a billion trillion trillion times weaker than even the weak force.) But the problem is actually much worse than that: it is in fact impossible to construct a valid quantum field theory for gravity. This is a big deal, and it means more than just that gravity needs to be left out: it means that our entire model, quantum field theory itself, must not be the final story.

I’ll let you mull this over until next week, but feel free to send me any comments or questions. Here are this week’s problems:


https://mywebspace.wisc.edu/mweinberg/web/fundamentalForces.pdf

*In a few weeks I’ll contradict myself: there are actually only three fundamental forces, because we have “unified” electromagnetism with the weak force. If we’re going to talk about these as two separate forces, we might as well split up electricity and magnetism and say there are five forces. Many people believe (and I agree) that there is only one fundamental force, and everything we see is a product of that.