Okay, this is not going to be a great explanation for three reasons. First: because my understanding isn't great. Second: because I'm going to try to write for a reader who hasn't taken quantum mechanics. And third: because the theory itself is young, only about forty years old, and there isn't a complete consensus among professional physicists yet on how to interpret it.
That said, the math predicts experimental results with incredible precision, and this description, unlike the quantum mechanics I've studied up until now, is consistent with our other knowledge of physics, with the theory of electromagnetism and relativity.
Here are the assumptions you have to make to get a working theory: 1. The particles of which everything is made behave, in many ways, like waves. 2. If you insist on thinking about particles instead of waves, you'll find some paradoxical properties, such as the idea that a particle which has a definite momentum does not have a definite location. Note: I didn't say that we can't know its location. I said it doesn't have one. 3. Time is a dimension almost exactly like length, width, and depth. It helps to think of "later" as "over there."
Now I'm going to explain these ideas further, but please understand that these are assumptions, not conclusions. It is not true that modern physics "proves" any of this. But if you make these assumptions, you can develop a theory of how particles interact which exactly matches, in every experiment to date (and there have been a lot of them, in those giant expensive particle accelerators) the way particles do interact. It is very possible that another description would match these results equally well, but we haven't found one yet (though some candidates -- incorporating large parts of Quantum Field Theory -- do exist.)
Okay, so what do I mean by that assumption number one? In what ways are particles like waves?
When waves encounter an obstacle, they spread out around it. (Imagine a box sitting in a pool, and ripples reflecting off of it in wide arcs.) Particles do something similar. If a beam of electrons encounters an obstacle, they too will spread out in different directions. Contrast this with billiard balls. If I send a bunch of billiard balls one at a time, from the same direction and the same speed toward some box, I expect them all to bounce off at the same angle. Electrons do no behave like billiard balls in these experiments. They also diffract around edges (billiard balls don't turn corners) and even show effects that are analogous to rainbows. Then too, like waves on Lake Michigan (I've watched them) and like light and sound waves, beams of electrons can interfere with each other, cancelling out in some places, and adding up in others (where electrons add up you get extra bright spots on your screen. Where water waves do, tall peaks and low troughs. Where sound waves do, loud noise, Etc.)
So why don't we just give up and talk about waves all the time, without ever mentioning particles? The fact is that both a beam of electrons and a beam of light are detected as a series of discrete collisions. Click, click, click, goes you photon detector or Geiger counter, one click at a time, whenever you shine your beam at it. (Photons are what we call the "particles" of light, the individual clicks on our devices.) You find charge and energy only in discrete lumps, whenever you go to measure it. So both electrons and light move like waves, but look like billiard balls, to our instruments.
The way physicists handle this is to say that the wave tells us something about the probability of detecting a particle. The bigger the waves at any given point in space and time, the more likely a detector located there is to go "click." So in the areas where the hypothetical waves add up to bigger waves, in these interference experiments, I detect lots of particles. That's my bright stripe of electrons or photons. In the places where the waves cancel out, I detect no particles.
Okay, now about that second assumption. That's called the "uncertainty principle." There's actually a couple of variations on it. The one I quoted, about a particle not having definite location and a definite momentum at the same time, is the most famous. It's experimentally testable -- just try measuring the positions of a beam of particles whose momentum you know, or the momenta of particles you've got trapped in a specific place. You'll find that under identical circumstances you get different results.
In fact, this is not nearly so confusing if you think about waves. A perfect wave repeats itself forever and ever at a uniform wavelength, so it doesn't really have a location. Everywhere looks just like everywhere else. By contrast if you have a single blip, where the water splashed up into one little peak, I can talk about the location, but not, really, about the wavelength. In traditional quantum mechanics, the wavelength tells you the momentum, so saying you can't know the momentum and the position at the same time, is just saying something obvious about waves.
Now picture a barrel full of water. Drop stones in the exact center of the barrel, and you may be able to create a circular wave pattern that holds its shape. The ripples spread out in circles, hit the circular wall of the barrel, reflect back as shrinking circles, and, if the wavelength is just right so that the reflections overlap the original waves, you get what's called a "standing wave pattern." It only works if the wavelength is just right, though, and the stones hit right in the center. This kind of pattern looks a little bit like the blip (it never spreads out past hte barrel) and a little like long train of incoming waves. I can talk about both its location and its wavelength, although both are a little ambiguous. And this kind of wave can persist, can hold its shape, but only if it's the right size to "fit" in the barrel. (For a simpler example of what I mean here, try drawing a box on your computer's paint program, and then drawing a wave with equally spaced crests inside it, that has a crest at both edges. You'll find that you have to get the spacing of the crests just right.)
It's waves that "fit" -- which look a little like blips and a little like wave-trains, which persist only if they have a certain shape and size -- which we actually see in real life. They're the ones that last long enough for us to see. The shape of these waves determines their energies, and only certain shapes, certain energies, are allowed.
When we try to measure the energies of electrons in real life situations, we don't get any old number. Over and over again, we find only the same discrete values -- the ones corresponding to those states that "fit" in our boundaries for the experiment. I'm talking experiment now, not theory. The same values over and over again. The theory is just our best guess as to why that would be. It all has to do with the fact that whether a wave-shape sticks around depends on its wavelength as well as where it start spreading out from.
All of this is ordinary quantum mechanics. It describes my electron (or whatever) as a probability wave, and tells me that I'm only going to get certain values for the energy.
What quantum field theory does, then, is try to explain why I should only get certain values for the number of electrons. In other words, if these thigns move like waves, why should my detector go click click click? Why do I never see half an electron? Why do I never see anything with half the electron's charge?
What if, says QFT, the wavefunction didn't describe that probability on a scale from zero to 100%. What if it described the strength of some field at that point in space and time. What if (for a specific wavelength) the bigger the field, the more particles I am likely to find at that point in space. In that case, for small energies, it still tells me the chaces of finding a particle here, but for large energies it tells me I might find more than one.
In other words, whether I detect a particle at some location does depends on the strength and shape of the field, but the field is more than just a probability function. For low energies, it doesn't matter; the effect is the same. But for high energies, I would seem to get particles popping out of nowhere. The idea here is that what actually exists is the field, not the particle. The particle is nothing but a click on my detector, the likelyhood of which depends on the field that exists.
But why should I get clicks at all? Wasn't that the question? If it's fields that are real, why do my instruments seem to see particles? Okay, and here's where it gets weird...
What if that field didn't have a specific value, the way a single blip on the water has a single location, or have a regular rate of change, the way a train of waves headed towards the beach has a specific wavelength... What if the charge, or whatever it is that my detector detects, depends on both so that only certain values "fit"? The way only certain shapes fit into my barrel. Those shapes didn't have a specific location or specific wavelength. They were spread out over a whole range of locations, with different wavelengths. In quantum field theory, my field doesn't have a specific value or rate of change, but a whole range of them.
So what I end up with, in effect, is like a probability function for a probability function. I look at one point in space and time, and I make a graph of how likely it is to have a field of a certain size, and that graph, of probability as a function of field strength, looks like some kind of wave. Only certain wave-shapes fit the boundaries that I draw, and those wave shapes correspond to the incidents where I detect a single particle at that location, or two particles, or three, or... Well, you get the idea.
It's a pretty big idea. When I discover an electron, the field doesn't have a single value but a range of possible values, a special wave-shaped range. If you like, you can picture a wave with a blurry edge, like the one below. That's what I'm picturing.
These are the things that are real, supposedly. At least, a theory describing these things predicts exactly what really happens, stuff that moves like waves but registers in our detectors like particles, predicts the results of experiments to an accuracy that is, as my professor likes to say, the same as giving the distance between New York and LA to within the width of a single human hair. (I didn't give him enough credit in my previous posts. He's a good professor, and he was trying. He just doesn't think about things in the same way as I like to.)
So that it...
Oh, but wait. You wanted to know about that "time is a direction" stuff. Well, actually that's not so much quantum field theory as relativity. But, as I mentioned at the beginning, quantum field theory (unlike old fashioned quantum theory) is consistent with relativity. Mainly in that it treats mass as a kind of energy (and even shows that mass can be "created" from energy, if you want to think about the probability of measuring more particles when the energy is bigger in that way) and treats time as a direction. So basically instead of having six directions to go (up down, forward backward, and left right) these waves actually have eight. When they scatter, there's a chance they will scatter "backwards in time." Now this sounds strange and spooky, but 1) mathematically it is mundane. A function of time is no different from a function of x or y or z, and I can graph them in just the same way and 2) the only implication it has for the real world of experiments, is that it means we should sometimes see particles which look just like our normal electrons and so on, but with the opposite charge, and a couple of other reversed properties. All we see in the lab is an "anti-particle" going forwards in time -- the two are equivalent. The only reason you might want to use this backwards-in-time description at all, is in describing what happens when a particle and an antiparticle occupy the same place at the same time. They "annhilate." At later points in time, neither one of them exists. They don't scatter off into any of our detectors. This spooky experimental fact actually seems less spooky of you think instead of a guy in a movie, getting into a time machine and going back. He vanishes, but at an earlier point in time there are two of him... Similarly you might think of someone from the future coming back to our time, which happens to be just before his own birth... In the future, there are two of him... (this is equivalent to the kind of particle creation I described earlier.) And if all of this seems far fetched, please remember that anti-particles are routinely observed in the lab (having been predicted by an early form of QFT) and that they experimentally can be created and annhilated. It's the universe that's far-fetched, and the theory faithfully describes it. If thinking about time travel gives you a headache, join the club. But take comfort in the fact that this mathematical description still says causes have to precede effects, and no paradoxes are possible (this is guarenteed by the fact that nothing can travel faster than the speed of light, but don't ask me how, exactly.)
You don't have to believe it. We'll probably come up with something that works even better, eventually. But considering the paradoxical experimental results that it had to explain, this one works amazingly well. Anyway, that's my understanding of quantum field theory, after a one quarter course on the subject. Considering how badly I did on the homework, you should take this with heaping spoonfuls of salt. Not only might the theory be wrong, but I might very easily be misunderstanding huge parts of the theory. But if you promise not to consider me an expert, I do like sharing my interpretations.