Science of Discworld III

sometimes several mathematically equivalent ways to describe something – different ways to say exactly the same thing in mathematical language. If one of them is true, they all are. But, their natural physical interpretations can be inconsistent.
    A good example arises in classical (non-quantum) mechanics. A moving particle can be described using (one of) Newton’s laws of motion: the particle’s acceleration is proportional to the forces that act on it. Alternatively, the motion can be described in terms of a ‘variational principle’: associated with each possible particle path there is a quantity called the ‘action’. The actual path that the particle follows is the one that makes the action as small as possible.
    The logical equivalence of Newton’s laws and the principle of least action is a mathematical theorem. You cannot accept one without accepting the other, on a mathematical level. Don’t worry what ‘action’ is. It doesn’t matter here. What matters is the difference between the natural interpretations of these two logically identical descriptions.
    Newton’s laws of motion are local rules. What the particle does next, here and now, is entirely determined by the forces that act on it, here and now. No foresight or intelligence is needed; just keep on obeying the local rules.
    The principle of least action has a different style: it is global. It tells us that in order to move from A to B, the particle must somehow contemplate the totality of all possible paths between those points. It must work out the action associated with each path, and find whichever one of them has the smallest action. This ‘computation’ is non-local, because it involves the entire path(s), and in some sense it has to be carried out before the particle knows where to go .
So in this natural interpretation of the mathematics, the particle appears to be endowed with miraculous foresight and the ability to choose, a rudimentary kind of intelligence.
    So which is it? A mindless lump of matter which obeys the local rules as it goes along? Or a quasi-intelligent entity with vast computational powers, which has the foresight to choose, among all the possible paths that it could have taken, precisely the unique one that minimises the action?
    We know which interpretation we’d choose.
    Interestingly, the principle of least action is a mechanical analogue of Feynman’s sum-over-histories method in optics. The two really are extremely close. Yes, you can formulate the mathematics of quantum mechanics in a way that seems to imply that light follows all possible paths and adds them up. But you are not obliged to buy that description as the real physics of the real world, even if the mathematics works.
    The many-worlds enthusiasts do buy that description: in fact, they take it much further. Not the history of a single photon bouncing off a mirror, but the history of the entire universe. That, too, is a sum of all possibilities – using the universe’s quantum wave function in place of the light intensity due to the photon – so by the same token, we can interpret the mathematics in a similarly dramatic way. Namely: the universe really does do all possible things. What we observe is what happens when you add all those possibilities up.
    Of course there’s also a less dramatic interpretation: the universe trundles along obeying the local laws of quantum mechanics, and does exactly one thing … which just happens, for purely mathematical reasons, to equal the sum of all the things that it might have done.
    Which interpretation do you buy?
    Mathematically, if one is ‘right’ then so is the other. Physically, though, they carry very different implications about how the world works. Our point is that, as for the classical particle, their mathematical equivalence does not require you to accept their physicaltruth as descriptions of reality. Any more than the equivalence of Newton’s laws with the principle of least action obliges you to believe in intelligent particles that can predict the future.
    The many-worlds interpretation of quantum mechanics, then, is resting on dodgy ground even though its mathematical foundations are impeccable. But the usual presentation of that interpretation goes further, by adding a hefty dose of narrativium. This is precisely what appeals to SF authors, but it’s a pity that it stretches the interpretation well past breaking-point.
    What we are usually told is this. At every instant of time,