M-Theory with 10 space dimensions

Consider the apparent dimension of the universe.

According to M-theory, space-time has 10 space dimensions and 1 time dimension.

The idea is that 7 of the space dimensions are curled up so small that we don’t notice them, leaving us with the illusion that all that exist are the 3 remaining large dimensions we are familiar with.

One of the central open questions in M-theory is: Why, in our universe, aren’t there more large dimensions, and why are any dimensions curled up?

Many people would like to believe that there is some mechanism that causes all but 3 of the space dimensions to curl up spontaneously.

Alternatively, maybe all dimensions started small, but for some understandable reason three space dimensions expanded and the rest did not. It seems, however, that there is no dynamical reason for the universe to appear four-dimensional.

Instead, top-down cosmology predicts that the number of large space dimensions is not fixed by any principle of physics.

There will be a quantum probability amplitude for every number of large space dimensions from zero to ten. The Feynman sum allows for all of these, for every possible history for the universe, but the observation that our universe has three large space dimensions selects out the subclass of histories that have the property that is being observed.

In other words, the quantum probability that the universe has more or less than three large space dimensions is irrelevant because we have already determined that we are in a universe with three large space dimensions.

So as long as the probability amplitude for three large space dimensions is not exactly zero, it doesn’t matter how small it is compared with the probability amplitude for other numbers of dimensions.

It would be like asking for the probability amplitude that the present pope is Chinese. We know that he is German, even though the probability that he is Chinese is higher because there are more Chinese than there are Germans. Similarly, we know our universe exhibits 3 large space dimensions, and so even though other numbers of large space dimensions may have a greater probability amplitude, we are interested only in histories with three.

What about the curled-up dimensions? Recall that in M-theory the precise shape of the remaining curled-up dimensions, the internal space, determines both the values of physical quantities such as the charge on the electron and the nature of the interactions between elementary particles, that is, the forces of nature.

Things would have worked out neatly if M-theory had allowed just one shape for the curled dimensions, or perhaps a few, all but one of which might have been ruled out by some means, leaving us with just one possibility for the apparent laws of nature. Instead, there are probability amplitudes for perhaps as many as 10500 different internal spaces, each leading to different laws and values for the physical constants.

If one builds the history of the universe from the bottom up, there is no reason the universe should end up with the internal space for the particle interactions that we actually observe, the standard model (of elementary particle interactions). But in the top-down approach we accept that universes exist with all possible internal spaces. In some universes electrons have the weight of golf balls and the force of gravity is stronger than that of magnetism.

In ours, the standard model, with all its parameters, applies. One can calculate the probability amplitude for the internal space that leads to the standard model on the basis of the no-boundary condition.

As with the probability of there being a universe with three large space dimensions, it doesn’t matter how small this amplitude is relative to other possibilities because we already observed that the standard model describes our universe.

Our theory here is testable. In the prior examples, we emphasized that the relative probability amplitudes for radically different universes, such as those with a different number of large space dimensions, don’t matter. The relative probability amplitudes for neighboring (i.e., similar) universes, however, are important.

The no-boundary condition implies that the probability amplitude is highest for histories in which the universe starts out completely smooth. The amplitude is reduced for universes that are more irregular. This means that the early universe would have been almost smooth, but with small irregularities. As we’ve noted, we can observe these irregularities as small variations in the microwaves coming from different directions in the sky.

They have been found to agree exactly with the general demands of inflation theory; however, more precise measurements are needed to fully differentiate the top-down theory from others, and to either support or refute it.

These may well be carried out by satellites in the future. Hundreds of years ago people thought the earth was unique, and situated at the center of the universe. Today we know there are hundreds of billions of stars in our galaxy, a large percentage of them with planetary systems, and hundreds of billions of galaxies.

The results described in this chapter indicate that our universe itself is also one of many, and that its apparent laws are not uniquely determined. This must be disappointing for those who hoped that an ultimate theory, a theory of everything, would predict the nature of everyday physics. We cannot predict discrete features such as the number of large space dimensions or the internal space that determines the physical quantities we observe (e.g., the mass and charge of the electron and other elementary particles). Rather, we use those numbers to select which histories contribute to the Feynman sum.

We seem to be at a critical point in the history of science, in which we must alter our conception of goals and of what makes a physical theory acceptable. It appears that the fundamental numbers, and even the form, of the apparent laws of nature are not demanded by logic or physical principle.

The parameters are free to take on many values and the laws to take on any form that leads to a self-consistent mathematical theory, and they do take on different values and different forms in different universes. That may not satisfy our human desire to be special or to discover a neat package to contain all the laws of physics, but it does seem to be the way of nature.

There seems to be a vast landscape of possible universes. However, as we’ll see in the next chapter, universes in which life like us can exist are rare. We live in one in which life is possible, but if the universe were only slightly different, beings like us could not exist. What are we to make of this fine-tuning? Is it evidence that the universe, after all, was designed by a benevolent creator?

Or does science offer another explanation?