just short-lived pairs of virtual particles and antiparticles. And in fact one can often do something similar for networks. For even in the planar networks discussed on page 527 a great many different arrangements of connections can be viewed as being formed from different configurations of nearby pairs of non-planar persistent structures.

Talking about a random background affecting processes in the universe immediately tends to suggest certain definite relations between probabilities for different processes. Thus for example, if there are two different ways that some process can occur, it suggests that the total probability for the whole process should be just the sum of the probabilities for the process to occur in the two different ways.

But the standard formalism of quantum theory says that this is not correct, and that in fact one has to look at so-called probability amplitudes, not ordinary probabilities. At a mathematical level, such amplitudes are analogous to ones for things like waves, and are in effect just numbers with directions. And what quantum theory says is that the probability for a whole process can be obtained by linearly combining the amplitudes for the different ways the process can occur, then looking at the square of the magnitude of the result—or the analog of intensity for something like a wave.

So how might this kind of mathematical procedure emerge from the types of models I have discussed? The answer seems complicated. For even though the procedure itself may sound straightforward, the constructs on which it operates are actually far from easy to define just on the basis of an underlying network—and I have seen no easy way to unravel the various limits and idealizations that have to be made.

Nevertheless, a potentially important point is that it is in some ways misleading to think of particles in a network as just interacting according to some definite rule, and being perturbed by what is in essence a random background. For this suggests that there is in effect a unique history to every particle interaction—determined by the initial conditions and the configuration that exists in the random background.

But the true picture is more complicated. For the sequence of updates to the underlying network can be made in any order—yet each order in effect gives a different detailed history for the network. But if