Science of Discworld III

ladders. At each roll of the dice, players move their counters from square to square on a board; traditionally there are a hundred squares. Some are linked by ladders, and if you land at the bottom you immediately rise to the top; others are linked by snakes, and if you land at the top you immediately fall to the bottom. Whoever reaches the final square first wins.
    To simplify the description, imagine someone playing solo snakes and ladders, so that there is only one counter on the board. Then at any instant, the ‘state’ of the game is determined by a single square: whichever one is currently occupied by the counter. In this analogy, the board itself becomes the phase space, our analogue of Platonia.The counter represents the entire universe. As the counter hops around, according to the rules of the game, the state of the ‘universe’ changes. The path that the counter follows – the list of squares that it successively occupies – is analogous to the world-line of the universe. In this interpretation, time does exist, because each successive move of the counter corresponds to one tick of the cosmic clock.
    Quantum snakes and ladders is very different. The board is the same, but now all that matters is the probability with which the counter occupies any given square – not just at one stage of the game, but overall. For instance, the probability of being on the first square, at some stage in the game, is 1, because you always start there. The probability of being on the second square is 1/6, because the only way to get there is to throw a 1 with the dice on your first throw. And so on. Once we have calculated all these probabilities, we can forget about the rules of the game and the concept of a ‘move’. Now only the probabilities remain. This is the quantum version of the game, and it has no explicit moves, only probabilities. Since there are no moves, there is no notion of the ‘next’ move, and no sensible concept of time.
    Our universe, Barbour tells us, is a quantum one, so it is like quantum snakes and ladders, and ‘time’ is a meaningless concept. So why do we naive humans imagine that time flows; that the universe (at least, the bit near us) passes through a linear sequence of changes?
    To Barbour, the apparent flow of time is an illusion. He suggests that Platonian configurations which have high probability must contain within them ‘an appearance of history’. They will look as though they had a past. It’s a bit like the philosophers’ old chestnut: maybe the universe is being created anew every instant (as in Thief of Time ), but at each moment, it is created along with apparent records of a lengthy past history. Such apparently historical clouds in Platonia are called time capsules. Now, among those high-probability configurations we find the arrangement of neurons in a conscious brain. In other words, the universe itself is timeless, but our brains are timecapsules, high-probability configurations, and these automatically come along with the illusion that they have had a past history.
    It’s a neat idea, if you like that sort of thing. But it hinges on Barbour’s claim that Platonia must be timeless because ‘there can only be once-and-for-all probabilities for each possible configuration’. This statement is remarkably reminiscent of one of Xeno’s – sorry, Zeno’s – paradoxes: the Arrow. Which, you recall, says that at each instant an arrow has a specific location, so it can’t be moving. Analogously, Barbour tells us that at each instant (if such a thing could exist) Platonia must have a specific probability mist, and deduces that this mist can’t change (so it doesn’t).
    What we have in mind as an alternative to Barbour’s timeless probability mist is not a mist that changes as time passes, however. That would fall foul of the non-Newtonian relation between space and time; different parts of the mist would correspond to different times depending on who observed them. No, we’re thinking of the mathematical resolution of the Arrow paradox, via Hamiltonian mechanics. Here, the state of a body is given by two quantities, position and momentum, instead of just position. Momentum is a ‘hidden variable’, observable only through its effect on subsequent positions, whereas position is something we can observe directly. We said: ‘a body in a given position with zero momentum is not moving at that instant, whereas one in the same position with non-zero momentum is