The Science of Discworld IV

alternatives are often dismissed out of hand. There is even a piece of mathematics, Bell’s theorem, which allegedly proves that quantum mechanics cannot be embedded within a broader deterministic local model, one that does not allow instant communication between widely separated entities.
    All of the above notwithstanding,
Pan narrans
has problems with quantum indeterminacy. How does nature
know what to do
? This is the thinking behind Einstein’s famous remark about a (non-) dicing deity. Generations of physicists have become accustomed to the problem – the mathematics says ‘it’s just like that’, and there’s no need to worry about interpretations. But it’s not quite that simple, because working out the implications of the mathematics requires some extra bolt-on assumptions. ‘What it’s like’ could be a consequence of those assumptions, not of the mathematics itself.
    It’s curious that we and Einstein use, as our icon for chance, the image of dice. A die (singular) is a cube, and when it is thrown, and bounces, it obeys the deterministic laws of mechanics. In principle, you ought to be able to predict the outcome as soon as the die leaves the hand. Of course there are modelling issues here, but that statement ought at least to be true of the idealised model. However, it’s not, and the reason is that the corners of the die amplify tiny errors of description. This is a form of chaos, related to the butterfly effect but technically different.
    Mathematically, the probabilities of the die landing on its various faces derive from the dynamical equations as a so-called invariant measure. One chance in six for each face. There is a sense in which the invariant measure is like a quantum wavefunction. You can calculate it from the dynamical equations and use it to predict statistical behaviour, but you can’t observe it directly. You infer it from a repeated series of experiments. There is also a sense in which an observation (of the final state of the die) ‘collapses’ this wavefunction. The table,and friction, force the die into an equilibrium state, which might be any of the six possibilities. What determines the observed value of the wavefunction is the secret dynamics of a rolling die, bouncing off a table. That’s not encoded in the wavefunction at all. It involves new ‘hidden variables’.
    You can’t help wondering whether something similar is happening in the quantum world. The quantum wavefunction may not be the whole story.
    When quantum mechanics was introduced, chaos theory didn’t exist. But the whole development might have been different if it had existed, because chaos theory tells us that deterministic dynamics can mimic randomness
exactly
. If you ignore some very fine detail of the deterministic system, what you see looks like random coin tosses. Now, if you don’t realise that determinism can mimic randomness, you can’t see any hope of connecting the apparent randomness observed in quantum systems with any deterministic law. Bell’s theorem knocks the whole idea on the head anyway. Except – it doesn’t. There are chaotic systems that closely resemble quantum ones, generate apparent randomness deterministically and, crucially, do not conflict with Bell’s theorem in any way.
    These models would need a lot more work to turn them into a genuine competitor for conventional quantum theory, even if that’s possible. The Rolls-Royce problem raises its head: if the test of a new design of car is that it has to outperform a Roller, innovation becomes impossible. No newcomer can hope to displace what is already firmly established. But we can’t help but wonder what would have happened if chaos theory had appeared
before
the early work in quantum mechanics did. Working within a very different background, one in which deterministic models were not seen to conflict with apparent randomness, would physicists still have ended up with the current theory?
    Maybe – but some aspects of the standard theory don’t make a lot of sense. In particular, an observation is represented mathematically as simple, crystalline process, whereas a real observation requires ameasuring device whose detailed quantum-mechanical formulation is far too complicated ever to be tractable. Most of the paradoxical features of quantum theory stem from this mismatch between an
ad hoc
add-on to the Schrödinger equation, and the actual process of observing, not from the equations as such. So we can speculate