At first sight, this might seem to be a very oblique way of doing physics. It takes the stance that the way to understand the real world is to ignore it, and focus instead on all the possible alternative unreal worlds. Then we find some principle (in this case, minimum energy) to rule out nearly all of the unreal worlds, and see what's left. Wouldn't it be easier to start with the real world, and focus solely on that? No, it wouldn't. As we've just seen, the real world alone is too limited to offer a convincing explanation. What you get from the real world alone is 'the world is like it is, and there's nothing more to be said'. However, if you take the imaginative leap of considering unreal worlds, too, you can compare the real world with all of those unreal worlds, and maybe find a principle that picks out the real one from all the others. Then you have answered the question 'Why is the world the way it is, rather than something else?'
An excellent way to approach 'why' questions is to consider alternatives and rule them out. 'Why did you park the car round the corner down a side-street?' 'Because if I'd parked outside the front door on the double yellow lines, a traffic warden would have given me a parking ticket.' This particular 'why' question is a story, a piece of fiction: a hypothetical discussion of the likely consequences of an action that never occurred. Humans invented their own brand of narrativium as an aid to the exploration of I-space, the space of 'insteads'. Narrative provides I-space with a geography: if I did this instead of that, then what would happen would be ...
On Discworld, phase spaces are real. The fictitious alternatives to the one actual state exist, too, and you can get inside the phase space and roam over its landscape -provided you know the right spells, secret entrances and other magical paraphernalia. L-space is a case in point. On Roundworld, we can pretend that phase space exists, and we can imagine exploring its geography. This pretence has turned out to be extraordinarily insightful.
Associated with any physical system, then, is a phase space, a space of the possible. If you're studying the solar system, then the phase space comprises all possible ways to arrange one star, nine planets, a considerable number of moons and a gigantic number of asteroids in space. If you're studying a sand-pile, then the phase space comprises the number of possible ways to arrange several million grains of sand. If you're studying thermodynamics, then the phase space comprises all possible positions and velocities for a large number of gas molecules. Indeed, for each molecule there are three position coordinates and three velocity coordinates, because the molecule lives in three-dimensional space. So with N molecules there are 6 N coordinates altogether. If you're looking at games of chess, then the phase space consists of all possible positions of the pieces on the board. If you're thinking about all possible books, then the phase space is L-space. And if you're thinking about all possible universes, you're contemplating U- space. Each 'point' of U-space is an entire universe (and you have to invent the multiverse to hold them all ... )
When cosmologists think about varying the natural constants, as we described in Chapter 2 in connection with the carbon resonance in stars, they are thinking about one tiny and rather obvious piece of U-space, the part that can be derived from our universe by changing the fundamental constants but otherwise keeping the laws the same. There are infinitely many other ways to set up an alternative universe: they range from having 101 dimensions and totally different laws to being identical with our universe except for six atoms of dysprosium in the core of the star Procyon that change into iodine on Thursdays.
As this example suggests, the first thing to appreciate about phase spaces is that they are generally rather big. What the universe actually does is a tiny proportion of all the things it could have done instead. For instance, suppose that a car park has one hundred parking slots, and that cars are either red, blue, green, white, or black. When the car park is full, how many different patterns of colour are there? Ignore the make of car, ignore how well or badly it is parked; focus solely on the pattern of colours.
Mathematicians call this kind of question 'combinatorics', and they have devised all sorts of clever ways to find answers. Roughly speaking, combinatorics is the art of counting things without actually counting them. Many years ago a mathematical acquaintance of ours came across a university administrator counting light bulbs in the roof of a lecture hall. The lights were arranged in a perfect rectangular grid, 10 by 20. The administrator was staring at the ceiling, going '49, 50, 51 -'
'Two hundred,' said the mathematician.
'How do you know that?'
'Well, it's a 10 by 20 grid, and 10 times 20 is 200.' 'No, no,' replied the administrator. 'I want the exact number.'13 Back to those cars. There are five colours, and each slot can be filled by just one of them. So there are five ways to fill the first slot, five ways to fill the second, and so on. Any way to fill the first slot can be combined with any way to fill the second, so those two slots can be filled in 5 X 5 = 25 ways. Each of those can be combined with any of the five ways to fill the third slot, so now we have 25 x 5 = 125 possibilities. By the same reasoning, the total number of ways to fill the whole car park is 5 x 5 x 5 ... X 5, with a hundred fives. This is 5100, which is rather big. To be precise, it is
78886090522101180541172856528278622 96732064351090230047702789306640625
(we've broken the number in two so that it fits the page width) which has 70 digits. It took a computer algebra system about five seconds to work that out, by the way, and about 4.999 of those seconds were taken up with giving it the instructions. And most of the rest was used up printing the result to the screen. Anyway, you now see why combinatorics is the art of counting without actually counting; if you listed all the possibilities and counted them '1, 2, 3, 4 ...' you'd never finish. So it's a good job that the university administrator wasn't in charge of car parking.
How big is L-space? The Librarian said it is infinite, which is true if you used infinity to mean 'a much larger number than I can envisage' or if you don't place an upper limit on how big a book can be,14 or if you allow all possible alphabets, syllabaries, and pictograms. If stick to 'ordinarysized'
English books, we can reduce the estimate. A typical book is 100,000 words long, or about
600,000 characters (letters and spaces, we'll ignore punctuation marks). There are 26 letters in the English alphabet, plus a space, making 27 characters that can go into each of the 600,000 possible positions. The counting principle that we used to solve the car-parking problem now implies that the maximum number of books of this length is 27600,000, which is roughly 10860,000
(that is, an 860,000-digit number). Of course, most those 'books' make very little sense, because we've not yet insisted that the letters make sensible words. If we assume that the words are drawn from a list of 10,000 standard ones, and calculate the number of ways to arrange 100,000 words in order, then the figure changes. 10,000100,000 is equal to 10400,000, and this is quite a bit smaller ... but still enormous. Mind you, most of those books wouldn't make much sense either; they'd read something like 'Cabbage patronymic forgotten prohibit hostile quintessence'
continuing at book length.15 So maybe we ought to work with sentences ... At any rate, even if we cut the numbers down in that manner, it turns out that the universe is not big enough to contain that many physical books. So it's a good job that L-space is available, and now we know why there's never enough shelf space. We like to think that our major libraries, such as the British Library or the Library of Congress, are pretty big. But, in fact, the space of those books that actually exist is a tiny, tiny fraction of L-space, all the books that could have existed. In particular, we're never going to run out of new books to write.