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 a measuring 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 that in a re-run of history, our ‘law’ for quantum systems really could have turned out very different, giving Schrödinger no reason to introduce his puzzling cat.
Whether our current physical laws are special and unique, or a different set would work just as well, there is something more to say about laws in general. And about their exceptions, and especially about transcending them. By that word we don’t mean that the laws are disobeyed. We mean that they become irrelevant because of a change in context, like the way a jumbo jet transcends gravity by using air-flow past its wings.
We’ll take Ohm’s law as an example, because it appears to be simple.
Matter is basically of two kinds with regard to electricity: either it’s an insulator, or it’s a conductor. If it’s a conductor, Ohm’s law applies: current equals voltage divided by resistance. So, for fixed resistance, a greater current requires a greater voltage. However, resistance need not be fixed, and this possibility lies behind some natural anomalies, like the way that lightning changes the insulating gas of the atmosphere into a conducting ionised path for the lightning strike, or ball lightning, which essentially folds up into the surface of a sphere. Being anomalies, these phenomena are automatically interesting. We can also play tricks with variable conductors of electricity, starting with thermionic valves (vacuum tubes) in the 1920s and continuing with semiconductors like transistors. The computer industry is built upon this trick.
The discovery of superconducting alloys – no electrical resistance – near zero Absolute temperature was a very interesting anomaly, which, as new alloys have been found that exhibit no resistance at higher and higher temperatures, promises to give us a whole new energy