Qualitative Quantum Mechanics ----------------------------- ASPECTS AND SQUIGGLES If we wish, we can take the single holistic state of the universe, and cut it up conceptually into aspects, so that it can be described by picking one value on each of the following lines: Electron A has x coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3.... Electron A has y coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3.... Electron A has z coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3.... Electron A has spin: up down Electron B has x coordinate: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3.... Electron B has y coordinate integer part: ... -5 -4 -3 -2 -1 0 1 2 3 4 ... Electron B has y coordinate fractional part: 0....0.2....0.35....0.64....0.79....0.99 Electron B has z momentum: ....-5....-4.7....-3.2....-1.7.....0.2.....2.3....7.3.... Electron B has spin: left right ... Then we can imagine a vertical squiggle going down through the lines above, whose position at each line indicates what the value of that aspect is -- such a squiggle yields a state of the universe. There are obviously very many possible such squiggles. Physicists call these squiggles "base states". Each is a possible state of the universe. But there are more possibilities than just the squiggles! Typically, say for an electron in a hydrogen atom, the electron is not at a specific point, but rather at a cloud of points, all around the nucleus. Electron A might be at such a cloud, meaning that the current state of the universe must somehow be a combination of many different squiggles, going through different possibilities for A's coordinates. So squiggles are possible states of the universe, but combinations of squiggles are also possible states (and are much more common!). (Physicists call such combinations of squiggles "superpositions of base states".) Above, we listed B's z momentum rather than its z coordinate. Why not list both? Because they are two different ways of talking about the same thing. It turns out that the momentum and the position are the Fourier transforms of each other, meaning that a specific value (or more likely the "value cloud") for one lets you work out the value cloud for the other, so you can't just arbitrarily specify both -- you only get to arbitrarily pick one of them. But clearly we have some freedom when picking what aspects to list, for example we got to choose between position and momentum, and clearly we could have picked a different xyz coordinate system for any of the electrons. Not so clearly, but just as truly, for measuring the electron's spin, you can pick any pair of opposite directions that you want. For example, if you tell me how much an electron is spinning "up" and "down", I can tell you how much it's spinning "left" and "right". (Don't worry about this too much, since nobody has a very good picture of what the "spin" means anyway!) COMBINATIONS If electron B is also at a cloud (like A is) rather than a single point, then the current state is the combination of a whole lot of squiggles -- for each possible path through A's aspects and each possible path through B's aspects, there is a contributing squiggle! If it takes N squiggles to represent a cloud, then it takes N^2 squiggles to represent two clouds. (Of course, if you believe in continuity, N is infinite.) Now, let's consider a peculiar state of the universe: Suppose electron A is in its hydrogen atom cloud, and electron B is at the "opposite" point in the cloud, so if A is one Angstrom to the right of the nucleus, then B is one Angstrom to the left of the nucleus, and so on. (Does this even make any sense? I mean, A's cloud is itself symmetric ball-shaped cloud around the nucleus, so isn't the "opposite" cloud just the same thing? Let's see.) Well, let's see if we can represent this idea with squiggles. For each squiggle contributing to this state, if it specifies that A's x coordinate is 1.3, then it should specify that B's x coordinate be -1.3, and so on. That was easy! We see how both A and B can be at the "same" cloud, yet they are opposite each other. We could also easily imagine how they could be at the same cloud, at the same place as each other (instead of opposite each other). Or, they could be "either at the same place or opposite each other" -- this would consist of both "same place" squiggles and "opposite place" squiggles -- this too is one possible state of the universe. In a way, these peculiar relationships between A and B are even simpler than when there is no relationship. When there was no relationship, it took N^2 squiggles to represent the state of the universe in terms of our chosen aspects, but when they are related in one of these peculiar ways, it only takes N squiggles (or 2N for the "same or opposite" one) to represent the state. Of course, if we had picked different aspects to build up our base states, then the number of needed squiggles would be different, so there's not much physical meaning to an analysis like this. RELATED & UNRELATED So anyway, the universe's state may be such that A and B are related, or it may be such that they are not related. If they are related, then the way a squiggle goes through B's aspects will depend on how it went through A's aspects. But if they are not related, then we can independently specify A's cloud and B's cloud, and this is enough to tell us what squiggles are contributing. For example, A and B could both have the same cloud around a nucleus, but be unrelated. This would mean that knowing how a squiggle picked A's aspects doesn't tell you anything about how that squiggle picks B's aspects. Look at how we described B's y coordinate above. We used two different aspects: the integer part, and the fractional part. Suppose that B's cloud has sharp edges, and goes from y=-2.1 to y=2.1. Then if a squiggle picks B's y coordinate's integer part to be 2, then it has to pick a value between 0 and .1 for the fractional part. So in this case, the integer part and fractional part are related aspects. It seems like these two aspects almost have to be related! Can we imagine a situation where they are not related? Sure, say the y coordinate is known to be exactly an integer -- the "y coordinate cloud" is all bunched up at a bunch of separate points corresponding to the integers on a number line. Then the fractional part is always 0, so there is no relation between the fractional part and the integer part. Put another way, we can give a "cloud" for the integer aspect and another cloud for the fractional aspect (this cloud is all bunched up at 0). As another example, say the y coordinate cloud just goes from 3.4 to 3.7. Then the integer part must always be 3, so again there is no relation between the integer and fractional aspects. Put another way, it makes sense to talk about separate clouds for the integer aspect (all bunched up at 3) and for the fractional aspect (going from .4 to .7). But of course, usually, these two aspects are related, and one cannot in general cut up the "y coordinate cloud" into a separate cloud for the integer aspect and an unrelated cloud for the fractional aspect. (Similarly, one usually cannot even separate B's y coordinate cloud out from its whole 3D cloud. But it is quite normal to be able to separate B's cloud from A's cloud.) Mathematicians, who like to keep things simple, would say that two unrelated things are "independent". Physicists, who like to keep things confused, would say that two related things are "entangled". Both of these terms are somewhat unfortunate. Conceptually, when aspects are related, we think of them as really describing just one single thing. For example, we would usually think of the "integer" and "fractional" aspects as really just describing the y coordinate. And we would probably think of A's 3 coordinates as together describing its position. Instead of the three coordinate aspects, we could have just listed one aspect, the "3D position", with each squiggle picking a value (a point in space) for that aspect. When aspects are unrelated, we think of them as describing two separate things. For example, if A's cloud and B's cloud are not related, then we can think of two separate clouds (perhaps at the same place), one for A and one for B. In real life, A and B will only be unrelated if they are not near each other. This is why we think of them as two separate particles in the first place. If they are near each other, then they will become related to each other. (Chemical bonding works like this.) TIME We keep talking about "states of the universe". A state of the universe is what the universe is like at a particular time. If we want to make any progress, we'll need to talk about how the universe evolves with time. That is, how does the state of the universe change over time? In this section, we're going to use some mathematical notation. Maybe I'll come back and rewrite it after I understand things better, but for now, we're going to have to use complicated numbers. (Mathematicians call them "complex numbers".) A complicated number is like an ordinary number, except that it has a direction (like north, or east, or south-by-southwest). So -3 north is the same thing as 3 south. If you add 12 north plus 12 east, you get 17 northeast. For 0, the direction doesn't matter. Pretty simple, really -- instead of a number line, we have a number area. Now we have been saying that a state of the universe is a combination of contributing squiggles, without discussing exactly how squiggles are actually combined. Well, combining them is easy -- you just list them, and each one indicates its contribution with a complicated number. Squiggles that don't contribute indicate this with the number zero. Squiggles that contribute more have larger numbers. And the directions? Well, we'll talk about them in a minute. Anyway, over time, a squiggle "leaks" into "neighboring" squiggles according to various laws of physics. "Leaking" means that it adds a little bit of its complicated number to its neighbors' complicated numbers (perhaps with a little twist prescribed by the physics). But what are its neighbors? Its neighbors are squiggles that are "close" in a qualitative sense, in that the various aspects are what we would call close to each other. For example, if two squiggles are the same except that one picks A's x coordinate to be 3.72 while the other picks it to be 3.73, then those two squiggles are pretty darn close. For aspects like spin with just a couple of possible values, those values are considered close. "Close" can be considered the word we use to describe squiggles that we feel to be qualitatively "near" each other. So what does near mean? It means they're "almost the same". What does that mean? Well, in the end it just means that the physics treats them very similarly, and they leak into each other. This leaking over time is how the universe progresses. If an electron is moving to the right, that means that the complicated numbers for the parts of its cloud are set up in such a way that when they leak, the net result will be a rightward leakage, i.e. a rightward movement of the cloud as a whole. MEASUREMENTS Sometimes, when we do an "experiment", we take a "measurement". The universe doesn't know we're doing this, and doesn't care. It doesn't have to watch a young experimenter and "judge" whether what they did was really a valid experiment or not. The universe just chugs along, doing what it always does. If we have chosen our aspects so that we regard the universe's state as the combination of two squiggles, then we might be able to set things up so that one of the two squiggles evolves very differently from the other. For example, maybe a photon is either "over here" or "over there", and we have placed a delicate detector at one of the positions. What is a detector? It's a device where, when a photon enters its detection receptacle, the photon interacts with some other particles, which interact with more particles, and the chain reaction eventually consists of human- made gears cranking about and light bulbs being turned on and off, i.e. large-scale high level things happening. Now, whereas our two original squiggles may have been able to become neighbors again and interact somehow (say if mirrors were to recombine the two possible positions for the photon so that they would "interfere" with each other), the detector really changes this, so after the detector has cranked its gears about (following only one of the squiggles), the two squiggles' offspring are hopelessly different -- so many aspects are completely different between the two clans that it would be practically impossible for any further offspring to ever coincide. Maybe if the whole experiment falls into a black hole... But what if we watch the whole same procedure but using a different set of aspects, so that at the beginning we saw just one squiggle instead of two? Well, the physicists haven't figured everything out yet -- in fact they have figured out very little, and they only know how to analyze or predict the leakage formulas for certain limited situations. In particular, they are only able to work with certain sets of aspects, but they will hasten to point out that this doesn't matter, since the universe will do the same thing no matter what aspects you choose to view it in terms of, so it really doesn't matter what aspects you pick. Well, this leaves us not really knowing how to think about the measurement except for the two-squiggle, two-clan point of view. Oh well. Anyway, at some point we observe the measurement -- we look at the detector's lights, and jot something down in our notebook. At this point, one of the two clans is representing us jotting down a "no", while the other clan is representing us jotting down a "yes". Now we are quite familiar with jotting things down, and we know that the experience is of only doing one thing, not of doing both. But this makes sense, since the two clans are nowhere near each other, and thus they aren't interacting at all, and each clan is in effect totally unaware of the other clan. Now some people (e.g. Everett) say that there is a you that jots down "no" and a you that jots down "yes", and that the two of you just never know about each other. Most people, though, say that the you you don't know about doesn't exist. As we are limited to observing what we can experience, the debate is purely philosophical. Anyway, if you do the experiment a large number of times, you can compare how many times you jotted down "no" with how many times you jotted down "yes", and you can ask the statistical question, "What is the chance that the next one will be a yes?" The answer is that the chance of becoming purely in the clan of descendents of a certain squiggle turns out to be exactly proportional to the size of its complicated number! (Actually, to the size of a circle centered at zero and containing the complicated number -- but this only matters when doing exact calculations, which only physicists know how to do.) And nobody has a good explanation for why this statistical observation is what it is. It is just a fact of nature, according to the physicists. LOGIC I'd like to get an intuitive feel for "quantum logic"... I still have several papers to read on that front yet though. Obviously it will work along the lines of having quantum boolean variables: v1: true false v2: true false v3: true false ... and then if the variables start out being unrelated, then each one can start out as either true, false, or wishy-washy (combination of true and false), and we'll naturally start out with 2^n squiggles where n is the number of "wishy washy" variables. This lets us do an exponential number of calculations at once, in effect trying all possibilities at the same time. This could sure speed up a lot of searches! But there are some outstanding issues: How do we generate output? How do we keep the variables in line during the calculation (normally they would slowly leak between their two values)? RELATIVITY I'd really like to get an understanding of how relativity fits into all this. For one thing, we'll have to abandon the notion of "a state of the universe", and replace it with something very local. But I haven't been able to find any qualitative descriptions of how these fit together. Maybe someday...All heckling, questions, insights, and other comments can be sent to Matthew Cook.