The End of Time: The Next Revolution in Physics

We no longer need many semiclassical solutions: one semiclassical solution is now sufficient to create one history. Nevertheless, at least one semiclassical solution remains – and will remain – the prerequisite for history. The core mathematical fact discovered by Hamilton keeps reappearing and being used in different ways. I feel sure that this is the true deep origin of history – we have already seen alpha-particle tracks form before our eyes. Watch a little longer, and even Henry VIII and his six wives will appear.

The second element in Bell’s account is collapse: crude, but effective. Little more needs to be said except that it is hard to believe that nature can behave so oddly. However, Bell’s down-to-earth account does show up the artificiality of the quantum measurement rules. These are formulated for individual observables, and insist that measurement invariably results in the finding of a single eigenvalue of a chosen observable. But in the case of the alpha particle ionizing an atom, no pure measurement results – there is simultaneous measurement of both position and momentum (both with imperfect accuracy, so that the uncertainty relation is not violated).

The third element in the creation of history is low entropy: the initial state of the system is highly special. The alpha particle, which could be anywhere, is inside the radioactive nucleus; the countless billions of cloud-chamber atoms, which could be in innumerable different excited states, are all in their ground states. The only reason we are not amazed by such order is our familiarity with the special. What we have known from childhood ceases to surprise us. But even the experiencing of coherent thoughts is most improbable. Among all possible worlds, the dull, disordered, incoherent states are overwhelmingly preponderant, while the ordered states form a miniscule fraction. But such states, sheer implausibility, must be presupposed if history is to be made manifest – at least it is in the normal view of things.

The initial ordered state creates history and a stable canvas on which it can be painted. The special position of the alpha particle gives rise to its semiclassical state. The thousand or so atoms it ionizes stand out as a vivid track on the un-ionized billions. Photographed before dispersal, the track becomes a record of history. If a large proportion of the atoms were already ionized, such a track could hardly form, let alone stand out. We might claim that history had unfolded, but there would be no evidence of it.

Records are all we have. We have seen one account of their creation. Except for quantum collapse, it does not seem outlandish. But Bell gives a second, fully quantum account in which the monstrously multidimensional configuration space of the cloud chamber is vital. This story of history is amazing. The next section prepares for it.

THE IMPROBABILITY OF HISTORY

The cloud chamber is treated schematically as a collection of hydrogen atoms, each consisting of a nucleus – a single proton – and an electron. We ignore the fact (here not an issue) that all protons are identical, and so are all electrons. It is also reasonable to assume that the protons are at fixed points, and to treat only the electrons and the alpha particle quantum mechanically. The coordinates of each electron can be three mutually perpendicular distances from its proton. A real cloud chamber may contain 10²⁷ atoms. It is daunting to contemplate a space with 3 × 10²⁷ (+ 3 for the alpha particle) dimensions, but we must do our best if we are to get a true feeling for what is going on in quantum mechanics.

The really important thing here is that each configuration point represents one totality of all electron positions in the chamber. If we keep all the electrons fixed except one, which we move, it explores just three of the dimensions. In a much more modest way, there is an analogy here with our existence on the Earth: we live in three dimensions, but are normally restricted to the Earth’s two-dimensional surface and do not normally move far in the third dimension. For the electron, the unexplored dimensions are not one but 3 × 10²⁷.