So here's what I've been doing for the past week or so, my excuse for neglecting my small-but-faithful audience: nothing.
And I've loved it. Nothing like -- riding in the car fifteen hours each way from Chicago to Denver and back reading out loud to keep Ken awake (The Sword in the Stone, which we finished, reading about the Boxing Day Boar Hunt on the day after Christmas) and watching the New Lemony Snicket movie and reading books seven, eight, nine, ten and eleven of the Series of Unfortunate Events.
The movie just about lives up to the books -- a high standard. All that's missing is the joy the books take in words themselves. In obscure vocabulary and playful definitions, in codes, puns, alliterations, anagrams, and literary allusions. I was extra-pleased by the mention of C.P. Snow's Corridors of Power, and I love all the T.S. Eliot. It's great that you can recognize villains by their fondness for Edgar Guest, while the well-read heroes prefer Herman Melville. I need to look-up some of the references which went over my head -- "Morel" Behavior in a Free Society, Archy and Mahitabel, Comyns, and Carl Van Vechten. I should also read the Gulag Archipelago, one of these days. But I think I got most of the references to Nietzche and Beverly Cleary.
Anyway, even some of that fun makes it into the movie, through Jude Law's pitch perfect narration. So it's worth the ticket price. We also rented "King Arthur" (which doesn't completement The Sword in the Stone as well as you'd think) and "Shaun of the Dead" (which does complement "Dawn of the Dead" -- Ken's favorite movie -- very well indeed.) I give it four stars, and I'm not the zombie-lover.
I picked out a wedding dress with my mom. Wedding plans are being revised, although I'm not making any more announcements yet.
Took Ken to Midnight Mass and to Casa Bonita. He's a good sport.
Christmas shopped up to the last possible moment, and then reaped an amazing haul on Christmas morning -- thanks for the cell-phone service especially, Dad and Mom.
And since we got back, I've spent a lot of time hanging out with Ken while he plays "Fable," which I got him for Christmas, after heavy hinting. I like watching video games. It makes more sense than watching sports -- at least I know the person I'm cheering for. And this game has a great story. I got totally caught up. The most hilarious part was when he took off his helmet and all of the game characters fell in love with him. His "renown" level was very high, as was his "attractiveness" score -- since his hair had grown back after his stint in prison.
There was a lot of recorded West Wing and Gilmore Girls to catch up on (thanks Becca) and more video games, movie rentals, books, and snow ball fights, and walking around on frozen ponds, somewhere in there...
It feels so good to do nothing, for a while.
Thursday, December 30, 2004
Wednesday, December 22, 2004
Even Though My Sister Hates the Poetry Posts...
The Journey of the Magi
by T.S. Eliot
'A cold coming we had of it,
Just the worst time of the year
For a journey, and such a long journey:
The ways deep and the weather sharp,
The very dead of winter.'
And the camels galled, sore-footed, refractory,
Lying down in the melting snow.
There were times we regretted
The summer palaces on slopes, the terraces,
And the silken girls bringing sherbet.
Then the camel men cursing and grumbling
And running away, and wanting their liquor and women,
And the night-fires going out, and the lack of shelters,
And the cities hostile and the towns unfriendly
And the villages dirty and charging high prices:
A hard time we had of it.
At the end we preferred to travel all night,
Sleeping in snatches,
With the voices singing in our ears, saying
That this was all folly.
Then at dawn we came down to a temperate valley,
Wet, below the snow line, smelling of vegetation;
With a running stream and a water-mill beating the darkness,
And three trees on the low sky,
And an old white horse galloped away in the meadow.
Then we came to a tavern with vine-leaves over the lintel,
Six hands at an open door dicing for pieces of silver,
And feet kicking the empty wine-skins.
But there was no information, and so we continued
And arrived at evening, not a moment too soon
Finding the place; it was (you may say) satisfactory.
All this was a long time ago, I remember,
And I would do it again, but set down
This set down
This: were we led all that way for
Birth or Death? There was a Birth, certainly,
We had evidence and no doubt. I had seen birth and death,
But had thought they were different; this Birth was
Hard and bitter agony for us, like Death, our death.
We returned to our places, these Kingdoms,
But no longer at ease here, in the old dispensation,
With an alien people clutching their gods.
I should be glad of another death.
by T.S. Eliot
'A cold coming we had of it,
Just the worst time of the year
For a journey, and such a long journey:
The ways deep and the weather sharp,
The very dead of winter.'
And the camels galled, sore-footed, refractory,
Lying down in the melting snow.
There were times we regretted
The summer palaces on slopes, the terraces,
And the silken girls bringing sherbet.
Then the camel men cursing and grumbling
And running away, and wanting their liquor and women,
And the night-fires going out, and the lack of shelters,
And the cities hostile and the towns unfriendly
And the villages dirty and charging high prices:
A hard time we had of it.
At the end we preferred to travel all night,
Sleeping in snatches,
With the voices singing in our ears, saying
That this was all folly.
Then at dawn we came down to a temperate valley,
Wet, below the snow line, smelling of vegetation;
With a running stream and a water-mill beating the darkness,
And three trees on the low sky,
And an old white horse galloped away in the meadow.
Then we came to a tavern with vine-leaves over the lintel,
Six hands at an open door dicing for pieces of silver,
And feet kicking the empty wine-skins.
But there was no information, and so we continued
And arrived at evening, not a moment too soon
Finding the place; it was (you may say) satisfactory.
All this was a long time ago, I remember,
And I would do it again, but set down
This set down
This: were we led all that way for
Birth or Death? There was a Birth, certainly,
We had evidence and no doubt. I had seen birth and death,
But had thought they were different; this Birth was
Hard and bitter agony for us, like Death, our death.
We returned to our places, these Kingdoms,
But no longer at ease here, in the old dispensation,
With an alien people clutching their gods.
I should be glad of another death.
Friday, December 17, 2004
Running Out of Classes
I feel ripped off.
I thought by this point in my education, I would know a lot more. I still look at papers, and feel like they're in a foreign language. I still look at the math, and feel like you'd have to be much smarter than me to understand it. (Including math that I've written.)
I do not feel like an expert. And this is bad, 'cause they're letting me help teach classes...
You know how they promise you, your whole life, "You'll understand when you're older"? They say, "Well, we can't explain this to you yet. You don't have enough math. You have to learn all this other stuff first."
But now I've done all that, I've sat through all their classes, almost. They haven't lived up to their side of the bargain. There's still so much I don't understand. And now I begin to suspect that they don't really understand either.
There are still lots of people above me, but they are not so far above me now. I'm getting to the point where it's not so much about people who know more than you, as people who know about different things than you, where we start to specialize.
So now, when I listen to physicists talk, I'm much less likely to believe they really know what they're talking about. I don't, and I've gotten this far, so why should I believe they do? And I feel the same way about people in other fields now, too. A doctorate is so much less impressive now that I think I actually might get one. Any club that would have me as a member, etc...
So running out of classes has made me feel a little cynical. Also smug, though. And old. (But I like feeling old.) Mainly, though, I'm just outraged about all the questions I now see I'm never going to get answers to. After so much work, and such a long time... Man! That sucks!
I thought by this point in my education, I would know a lot more. I still look at papers, and feel like they're in a foreign language. I still look at the math, and feel like you'd have to be much smarter than me to understand it. (Including math that I've written.)
I do not feel like an expert. And this is bad, 'cause they're letting me help teach classes...
You know how they promise you, your whole life, "You'll understand when you're older"? They say, "Well, we can't explain this to you yet. You don't have enough math. You have to learn all this other stuff first."
But now I've done all that, I've sat through all their classes, almost. They haven't lived up to their side of the bargain. There's still so much I don't understand. And now I begin to suspect that they don't really understand either.
There are still lots of people above me, but they are not so far above me now. I'm getting to the point where it's not so much about people who know more than you, as people who know about different things than you, where we start to specialize.
So now, when I listen to physicists talk, I'm much less likely to believe they really know what they're talking about. I don't, and I've gotten this far, so why should I believe they do? And I feel the same way about people in other fields now, too. A doctorate is so much less impressive now that I think I actually might get one. Any club that would have me as a member, etc...
So running out of classes has made me feel a little cynical. Also smug, though. And old. (But I like feeling old.) Mainly, though, I'm just outraged about all the questions I now see I'm never going to get answers to. After so much work, and such a long time... Man! That sucks!
Thursday, December 09, 2004
Everything I Know About Quantum Field Theory
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.
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.
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