configuration is called the probability amplitude for that configuration. The square of the probability amplitude is the probability for observing that configuration.

More generally, to construct all possible states of two qubits, we add the four possibilities |uparrowuparrowrangle, |uparrowdownarrowrangle, |downarrowuparrowrangle, |downarrowdownarrowrangle, each multiplied by a separate number. That defines a four-dimensional space—you can step off distances in four different directions.

To describe the possible states of five quibits, we have up-or-down choices for each of them (e.g., |uparrowdownarrowuparrowuparrowdownarrowrangle or |uparrowuparrowdownarrowdownarrowuparrowrangle). There are 2 × 2 × 2 × 2 × 2 = 32 possibilities, and a general state has contributions from every one of them, each multiplied by a number. That's how we find ourselves with a thirty-two dimensional toy model on our hands. Some toy!

Quantum theory forces us to make much more room for physical reality. Traditional concepts of space are paltry by comparison.

Space is effervescent, substantial, weighty, and elastic. Each of these properties equates to specific, observable phenomena; they are not whimsical metaphors. Space has a life of its own, and exists independent of any matter that might occupy it. Indeed, in our most fundamental equations particles—the building blocks of matter—are described as disturbances in the activity of space-filling fields, or in other words of space itself.

While they might find details of these views of space surprising, and the concrete evidence for them startling and unfamiliar, Descartes and Newton—or even Aristotle and Lucretius—would have little difficulty understanding what they’re about. Philosophers and scientists have argued about similar questions, pro and con, for centuries. The progress is that now we have some answers.

Modern quantum physics brings in ideas of a different order. Quantum reality lives in spaces whose meaning, size, and structure transcends classical ideas about physical space. To get in tune with Nature, we must vastly expand our conceptual universe.

With new answers come new questions. The structure of space is encoded in the metric field. Like all fields the metric field is subject the laws of quantum mechanics. In particular, it is forever boiling with spontaneous fluctuations. When we calculate these fluctuations we find that they grow, as a fraction of the distance, for nearby points. Eventually, for distances below about 10-33 cm., the calculated fluctuations in distance become larger than the distance itself. Below this so-called “Planck length” our usual methods of calculation break down. Indeed, the whole concept of distance comes to look suspect. Now 10-33 cm. is a very small distance, far beyond practical access. Nevertheless this issue is of fundamental interest, not only in its own right, but also for cosmology. Indeed, our equations break down in describing extremely short time intervals (10-44 sec.), for similar reasons. Thus we aren’t able to describe the very earliest moments of the big bang. And so ultimate questions of origins remain up for grabs.

70