Most of us, however, suspect that a prerequisite for progress will be a worked-out theory that relates gravity to the micro-world. Back at the very beginning the entire universe could have been squeezed to the size of an elementary particle—quantum fluctuations could shake the entire universe, and there would be an essential link between cosmology and the micro-world. Of course, string theory and M-theory are the most ambitious and currently-fashionable attempts to do that. When we have that theory we at least ought to be able to formulate some physics for the very beginning of the universe. One question, of course, is whether we will find that space and time are so complicated and screwed that we can't really talk about a beginning in time. We've got to accept that we will have to jettison more and more of our commonsense concepts as we go to these extreme conditions.
The main stumbling block at the moment is that the mathematics involved in these theories is so difficult that it's not possible to relate the complexity of this 10- or 11-dimensional space to anything we can actually observe. In addition, although these theories may appear aesthetically attractive, and although they give us a natural interpretation of gravity, they don't yet tell us why our three dimensional world contains the types of particles that physicists study. We hope that one day this theory, which already deepens our insight into gravity, will gain credibility by explaining some of the features of the microworld that the current 'standard model' of particle physics does not..
Although Roger Penrose can probably manage four dimensions, I don't think any of these theorists can in any intuitive way imagine the extra dimensions. They can, however, envision them as mathematical constructs, and certainly the mathematics can be written down and studied. The one thing that is rather unusual about string theory from the viewpoint of the sociology and history of science—is that it's one of the few instances where physics has been held up by a lack of the relevant mathematics. In the past, physicists have generally taken fairly old-fashioned mathematics off the shelf. Einstein used 19th century non-Euclidean geometry, and the pioneers in quantum theory used group theory and differential equations that had essentially been worked out long beforehand. But string theory poses mathematical problems that aren't yet solved, and has actually brought math and physics closer together.
String theory is the dominant approach right now, and it has some successes already, but the question is whether it will develop to the stage where we can actually solve problems that can be tested observationally. If one can't bridge the gap between this ten-dimensional theory and anything that we can observe it will grind to a halt. In most versions of string theory the extra dimensions above the normal three are all wrapped up very tightly, so that each point in our ordinary space is like a tightly wrapped origami in six dimensions. We see just three dimensions: the rest are invisible to us because they are wrapped up very tightly. If you look at a needle it looks like a one-dimensional line from a long distance, but really it's three-dimensional. Likewise, the extra dimensions above our three could be seen if you looked at things very closely. Space on a very tiny scale is grainy and complicated—its smoothness is an illusion of the large scale. That's the conventional view in these string theories.
An extra idea which has become popular in the last two or three years is that not all the extra dimensions are wrapped up, but there might be at least one extra dimension which exists on a large scale. Lisa Randall and Raman Sundrum have developed this idea in their work on branes. According to their theory there could be other universes, perhaps separated from ours by just a microscopic distance. However, that distance is measured in some fourth spatial dimension of which we are not aware. Because we are imprisoned in our three dimensions we can't directly detect these other universes. It's rather like a whole lot of bugs crawling around on a big, two-dimensional sheet of paper, who would be unaware of another set of bugs that might be crawling around on another sheet of paper that could be only a short distance away in the third dimension. In a different way, this concept features in a rather neat model that Paul Steinhardt and Neil Turok have discussed, which allows a perpetual and cyclic universe, These ideas, again, may lead to new insights. They make some not-yet-testable predictions about the fluctuation of gravitational waves, but the key question is whether they have the ring of truth about them. We may know that when they've been developed in more detail.
Two of the potential explanations for the huge disparity in energy scales are supersymmetry and the physics of extra dimensions. Supersymmetry, until very recently, was thought to be the only way to explain physics at the TeV scale. It is a symmetry that relates the properties of bosons to those of their partner fermions (bosons and fermions being two types of particles distinguished by quantum mechanics). Bosons have integral spin and fermions have half-integral spin, where spin is an internal quantum number. Without supersymmetry, one would expect these two particle types to be unrelated. But given supersymmetry, properties like mass and the interaction strength between a particle and its supersymmetric partner are closely aligned. It would imply for an electron, for example, the existence of a corresponding superparticleŚcalled a selectron, in this caseŚwith the same mass and charge. There was and still is a big hope that we will find signatures of supersymmetry in the next generation of colliders. The discovery of supersymmetry would be a stunning achievement. It would be the first extension of symmetries associated with space and time since Einstein constructed his theory of general relativity in the early twentieth century. And if supersymmetry is right, it is likely to solve other mysteries, such as the existence of dark matter. String theories that have the potential to encompass the standard model seem to require supersymmetry, so the search for supersymmetry is also important to string theorists. Both for these theoretical reasons and for its potential experimental testability, supersymmetry is a very exciting theory.