QED and electrostatic attraction and repulsion

Factual Questions
1

I was watching a video of a lecture given by Richard Feynman on QED (and I read his QED book a number of years ago) and it does a good job of explaining the particle nature of photons and how they can appear to be waves. What I don’t understand is how any of this stuff explains why two point charges in empty space attract each other (or repel as the case may be). Does it explain it? Or does it merely explain the mechanics of it? I also don’t understand how it gives rise to the structure of an atom, nor how magnetism works, etc., but, in the words of Alton Brown, that’s another show.

Thanks,
Rob

2

Charged particles attract or repel via the exchange of virtual particles. Virtual particles are fairly weird animals and some would argue they aren’t even real as they’re off the mass shell. (They don’t obey Einstein’s equation that relates mass, energy and momentum) and they can’t ever be directly detected.

Pretending they’re real particles, I would guess you could see how they could cause two charged particles to repel each other via the exchange of momentum, but attraction is probably harder to visualize. About all I can say is that they have a strange property where their momentum vector can point in a direction opposite their direction of travel, and thus supply an attractive force to two oppositely charged particles.

3

To be clearer that should say virtual photons

4

You can derive all of Maxwell’s Equations from QED, which basically tells you everything there is to know about electromagnetism. Including the force laws.

5

Well, one could ask if there really is a difference between the two, but that’s probably not going to be helpful…

Anyway, the most quoted explanation is perhaps that two charges ‘exchange virtual particles’, thus ‘telling’ each other how to move. While that’s a perfectly appropriate picture, I’m not too sure it’s very illuminating. The problem is, the more illuminating picture is kinda hard to draw…

It’s perhaps easiest to just start out with a nondescript field of some sort. You might picture this as a kind of mattress-like lattice: points connected by springs that can oscillate around their rest position. Now, you can jump around on this mattress, creating all kinds of patterns of oscillation. It turns out that certain kinds of oscillations are favoured on the mattress – those that obey a certain relation between their frequency and wavelength. This relation, however, is simply a form of the well-known E = mc[sup]2[/sup] (or more accurately, E[sup]2[/sup] = p[sup]2[/sup]c[sup]2[/sup] + m[sup]2[/sup]c[sup]4[/sup], where p is the momentum), and thus, we call these oscillations ‘particles’. These particles are the quanta of the field; if we’re talking about the electromagnetic field, they’re photons.

Now, picture throwing some lumps of stuff on the mattress. These will cause disturbances in the field, corresponding, again, to particles. These need not, in principle, exactly obey the relation above; if they don’t, they’re called virtual particles, since they’re generally taken to be unobservable through direct means. Thus, the rocks do something to the field configuration, right? They couple to the field. Lumps coupling to the electromagnetic field are charges. So, now throw two lumps onto the mattress – what’s gonna happen? Well, this being quantum mechanics, they’re not just gonna sit there; rather, everything is subject to quantum fluctuations, and things are gonna jitter and jiggle around a bit.

The thing is, the presence of the lumps changes the amount of potential energy stored in the field. In the mattress analogy, just think about how pushing down requires more and more work, and how everything snaps right back into the original configuration if you release it. A similar thing happens with the quantum field, and here, the lumps can’t just ‘jump back up’ from the mattress. So, you have a certain amount of energy stored in the system of field + lumps. And, if you have, for instance, positive charges, it turns out that this energy gets smaller the farther apart both lumps are. Thus, with the quantum jiggling, the lumps will tend to get further separated – they experience a repulsive force. If you have opposite charges, and you keep track of the minus (always remember to make an even number of sign errors, not an odd one!), in the end, you come out with an attractive force.

And well, that’s that, or at least, that’s a picture of how the mechanics work that’s hopefully not too far off from the truth. In any case, it’s all I have time for right now…

6

Isn’t that a bit of a cheat, though? I mean, you basically put the Maxwell Lagrangian in to build QED, so it’s no great surprise that it comes out again…

7

It’s only a cheat if you try to pretend that Maxwell’s Equations weren’t an input. It’s still not completely trivial, though: There are plenty of things that take Maxwell as an input that don’t contain enough information to reproduce them.

8

Well, I certainly didn’t want to accuse you of doing that, sorry if it came across like I did.