Does God Play Dice?

Ian Stewart

Dice and Determinism
In the next subsection I'm going to describe a new idea which gets round Bell's inequality in a very clever way. As preparation, let me first develop the argument that along with that mainstay of probability theory texts, the fair coin, the 'dice' metaphor is one of the most inappropriate ever invented. At least, unless we revise our idea of randomness.

I'm talking of an ideal die, a perfect inelastic cube, thrown on to a perfectly flat inelastic surface, subject to some precise law of friction, and obeying Newtonian mechanics. I have to do that to introduce the mathematics precisely. It seems to me that whatever makes a real die random ought to show up in this model too. Putting on Laplace's hat, however, it's clear that Vast Intellect could work out the final rest state of the die the moment it's thrown. With a video-camera and a supercomputer we ought, at least in principle, to be able to predict the outcome before the die does.

This isn't entirely a fantasy. J. Doyne Farmer, an American chaologist, developed a theory of the roulette wheel which improves considerably on pure chance. He's having trouble getting the casinos to let him play, though.
Anyway, if you can predict exactly what will happen, where does the randomness come from?

I can't do calculations for a die, but I'll do them for a simplified coin, close enough to show what's involved. The coin is a line segment of unit length, confined to a vertical plane. When it is tossed, starting at ground level, it's given a vertical velocity v and also a rotation rate of r turns per second. When it returns to ground level, it freezes: whichever side is then uppermost is considered to be the result of the toss.

If g is the acceleration due to gravity, then the coin takes 2v/g seconds to return to the horizontal, and so makes 2rv/g turns. The boundary between heads and tails occurs at exact half-turns, that is, when 2rv/g is half an integer. If this integer is is N,then the head/tail boundary is given by vr = gN/4.

Initial conditions for a spinning coin,  striped according to its eventual fate. Black = heads, white = tails.

If I could control the values of r and v exactly, then I'd be able to make the coin land whichever way up I want. However, in practice I can control these values only within limits. For example, suppose that I can keep v between 480 and 520 cm/sec, with r between 18 and 22 revolution per second. How does the outcome heads or tails - depend on v and r?

You can get the answer from the formula above. The rectangle of possible values of v and r divides into stripes: black for heads, white for tails (see diagram).
Any known values of the initial velocity and the rate of spin give a unique answer. Not only is the outcome deterministic I really can tell you, in advance, what it is.

But if all I know is that v and r lie within the given range, I can't prescribe the outcome. The best I can do is think of the rectangle as a kind of dartboard. Each coin-toss is like throwing a dart: if the dart hits a black stripe, I get a head, if white, I get a tail. If the darts are distributed uniformly over the rectangle, then the probability of a head is the proportion of the total area covered by black stripes.

In others words, the source of the randomness lies in the choice of initial conditions. Unless I can control them exactly, I can't make a precise prediction.
Here Laplacian determinism breaks down again - but in a subtly different way. The model coin isn't a chaotic system. It's a perfectly regular one.

What we see here is that, associated with any deterministic dynamical system, there is a probabilistic system that offers a kind of 'coarse-grained' representation. Instead of telling us exactly which point in phase space the system occupies at a given instant, it tells us just the probability that the point lies in a given region at some instant. The study of such probabilities, called invariant measures, goes back to the early days of statistical mechanics, when mathematicians and physicists were trying to understand gases as complex collections of molecules, bouncing madly off each other. Invariant measures explain why gases have well-defined average properties like density and pressure. You might say that before we understood the molecular basis of matter, the only things we knew about the dynamics of gases were probabilistic. Afterwards, we realized that the probabilities are derived from a deterministic - but incredibly complicated - underlying dynamic. So statistical mechanics does have a hidden variable theory, whose variables are the positions and velocities of the gas's component molecules.

Could quantum theory be similar? Our experience to date makes us think that it is irreducibly probabilistic, but where do the probabilities come from? Probabilities are patterns, of a kind, and it is actually rather bizarre to think of probability as a primary physical concept when in every case where we understand the deep structure, probabilities arise from a deterministic dynamic as invariant measures. Indeed the existence of well-defined statistical patterns is evidence for a kind of order that becomes apparent only when we average over long timescales. What makes the system's distant future resemble its past, even on average? If it's really random, why aren't they just different? If a radioactive atom decays in a manner that has we'll-defined statistical regularities, where do those regularities come from? To say they are fundamentally probabilistic, and leave it at that, is simply to postulate a pattern that ought to be explained.

For deterministic chaos, in contrast, there is a clear mathematical explanation of the associated probabilities and their statistical regularities. We know where they come from: they arise as invariant measures. Although we don't understand everything here that we'd like to - for example the existence of technically 'nice' invariant measures is widely conjectured but narrowly proved - we see very clearly that it is the determinism of the dynamics that makes the future look similar to the past. The reason is recurrence: deterministic systems keep returning close to their previous states. So, paradoxically, it is the underlying determinism of the system that makes probabilities applicable. In order for a coarse-grained model to fit, there has to be something fine-grained to coarsen.

It seems to me that a truly random system ought not to exhibit patterns at all, not even on average. I'm aware that this view is a minority one, and I'm also aware that in some ways it conflicts with Gregory Chaitin's beautiful 'algorithmic information theory', which defines randomness in a manner that necessarily implies statistical regularities. I'm aware that complexity theory (see Chapter 17) suggests that patterns need not 'come from' anywhere, and I'm inclined to agree. But I still think that there's a grain of truth in the idea. I think that the existence of statistical regularities in quantum level matter needs to be explained, not simply assumed; and some kind of chaotic hidden variable theory would fit the bill - if only it weren't for Bell's inequality.

However, there are more ways to Bell a cat than choking it with correlations. In principle, we can get round Bell's inequality. I don't know if this can be done while remaining consistent with every known experimental result about quantum mechanics that needs further research. However, despite Bell's inequality, we don't have to give up at the outset. Bell's inequality tells us that certain kinds of 'hidden variable' extension of conventional quantum theory can't possibly work, but it doesn't rule out every conceivable extension or alternative. It is a constraint that tells us a little bit about what kind of hidden variable model we can introduce.

I have a feeling that physicists may have been too easily impressed by a mathematical theorem. Mathematicians know that theorems have hypotheses, assumptions that you have to make before the theorem works. Indeed mathematicians spend a lot of time writing down their hypotheses very carefully, thereby infuriating the physicists, who prefer to be sloppy (they call it 'appeal to physical intuition') and leave the hypotheses tacit. This is fine for everyday nuts-and-bolts physics, but I think it can be horribly misleading when we get down to fundamental philosophical issues, where some careful logic-chopping is necessary.

A careful study of the requisite hypotheses reveals potential loopholes in the proof of Bell's inequality. There are several hidden assumptions, technical things like the uniqueness of the probability measure used to compute the correlation function and the convergence of various infinite series and integrals. These loopholes are probably pluggable, but very recently it has been shown that at least one is not.
Let's take a peep inside Pandora's quantum box.