A key problem in superstring theory is figuring out, whether there is a finite or infinite number of “universes” that it can describe. In a recent article, hep-th / 0606212, an attempt is made to prove that this number is finite.

Superstring theory, whose language and objects may seem like something completely different from our world, recently faced an unexpected problem: there are suspicions that it can describe any world, and therefore cannot predict anything. Now physicists are trying to figure out whether this theory is so “omnipotent” (image from the site pyweb.swan.ac.uk)

Superstring theory is one of the main candidates for a complete description of all interactions of elementary particles, including gravity, at an ultrahigh concentration of energy. The demands of mathematical self-consistency and correspondence to the real world have led physicists to one the only possible all-encompassing superstring theory, to the only possible fundamental “law of order” of our world – the so-called M-theory. (Of course, if we abandon the particle-string hypothesis, then other possibilities of description appear.)

After the discovery of M-theory, physicists hoped that soon the properties of the universe around us would be fully explained: that is, the world at low energy. But in the years that followed, those hopes began to dwindle and eventually led to a crisis in string theory. However, after a period of despair, physicists got down to business again, and gradually the possible ways out of the crisis began to become clear. The recent article by B. S. Acharya, M. R. Douglas, hep-th / 0606212, in which an attempt is made to answer the key question – is there a finite number of those variants of the structure of our world that the theory of superstrings gives?

The crux of the crisis in superstring theory is as follows. M-theory describes the “life” of extended objects in 11-dimensional space-time at very high temperatures. 11-dimensional space is not a whim, but the only way to satisfy all imposed conditions at once. If we want to get the properties of our world from this theory, then we must gradually lower the temperature and see what happens to this 11-dimensional space and objects flying in it.

It so happens that 7 of these 11 dimensions become unstable and spontaneously fold into small self-closed configurations, leaving three spatial dimensions plus time “large” – that is, our Universe. The details of this mechanism are not yet fully understood, and today it seems that a huge number of different configurations of folded space are possible in superstring theory. Each such configuration will lead to a “finite universe” with its own characteristics: force of interactions, masses of particles, etc. This entire set of finite universes, which can be obtained from a single theory by means of different “convolutions”, physicists called the “landscape” of the theory.

The trouble with superstring theory is that it cannot (yet) predict which exactly the convolution is realized in reality, which means that it cannot predict what kind of finite universe the M-theory will turn into with a decrease in temperature. Many fear that superstring theory can be obtained at all any the final state of our world; in other words, the landscape of superstring theory is infinite. In the worst case, this would mean that such a theory cannot be refuted at all:
any result of any experiment can be explained in terms of superstring theory.

However, superstringers hope that a careful study of the issue will still reveal a mechanism that dictates exactly how space should be folded. Finding such a mechanism is a very difficult mathematical problem, and therefore many researchers prefer to approach the problem from the other side – to study the properties of the “landscape”, to find out how many and what universes can be obtained after various folds of unnecessary dimensions.

It is clear that before arguing whether there are many or few such options, it is necessary to prove that there are finite number… Article hep-th / 0606212 is just about trying to prove that the number of options, consistent with observational data, Certainly.

Where does the infinite number of options come from in this theory? First of all, due to the diverse topologies
collapsing unnecessary dimensions. To illustrate, imagine how many different ways you can tie knots on a rope. Obviously, there are infinitely many such possibilities, because the imposition of new and new nodes will lead to a new configuration. However, another thing is immediately clear: if the thickness of the rope is not less than a given number and the length is not more than a certain limit, then only the final number of nodes. Knots can look and intertwine in different ways, but in the end it turns out that from any given rope you can get only a finite number of types of knotting.

Very similar requirements are used by the authors of the article. A rope that is too thin corresponds to too much vacuum energy density, and too much folded space will inevitably lead to a large number of new ultralight particles. Neither one nor the other is observed in our world. Therefore, in principle, there can be infinitely many options for convolution, but only finite number does not contradict the experiment.

Having reformulated the physical requirements in a rigorous mathematical language, the authors noticed that this condition exactly coincides with Cheeger’s finiteness theorem from Riemannian geometry. There is, however, one “but”: this theorem is valid only for smooth convolutions, without kinks, and in string theory convolutions with kinks are also allowed. For a complete proof, it will be necessary to generalize the theorem for such situations, and the authors have already outlined the ways of proof.

However, this will only be half the battle. Even with the same folding of space gravity device on it can be very different, and it is necessary to prove that there are also a finite number of such options. The authors have shown that for this it will be sufficient to prove two statements. The first is that the space of all possible devices of gravity is limited, and the second is that too close points of this space (that is, too similar realizations of gravity) do not differ
from the point of view of physics… Roughly speaking, universes that differ noticeably, not the hundredth decimal place in any parameter.

The authors found out that some not yet proven mathematical hypotheses, after “translation” into the required language, are just suitable for resolving this issue. Once the proofs of these statements are obtained, it will be possible to combine the two ideas – a finite number of convolutions and a finite number of solutions for each convolution – and the finiteness of physically meaningful solutions in string theory will be proved.

However, even if this approach is successful, it will still not be able to give even an approximate answer to the question of how many solutions are possible in superstring theory. New ideas will be needed to resolve this issue and overcome the crisis.

Igor Ivanov

Source: www.elementy.ru