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Rob Knoops | CERN / University of Leuven | Belgium

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Can String Theory predict stuff?

What’s the deal with string theory? Why do people claim string theory is nonsense? Can we predict anything with it? As a theorist with too many experimental friends, these questions are thrown at me all the time. So today answering these will be my challenge.

Dislaimer: In the following I might have wiped too many ‘details’ under the doormat in order to keep everything readable. But feel free to post any comment if you would like me elaborate on specific parts.

Basically string theory says that the tiniest bits of matter are in fact little strings, in contrast to for example the Standard Model, where every particle is considered to be a point. This has a lot of interesting consequences, but I will only address the essential points that we will need along the way.
For this it is sufficient to believe that as soon one wants to quantize this string (with ‘to quantize’ I mean “to write a quantum mechanical theory of this string”), one quickly gets to the result that this theory we are talking about actually has to live in 10 dimensions (or 11 for M-theory, but let’s not talk about that).
I think the best way to look at this fact is to say ‘We need 10 dimensions to make the mathematics work out’. For people with a physics background: One way to look at it, is that we need 10 dimensions to make some anomalies vanish.

Anyway, we are now with a 10-dimensional theory. How should we look at these extra dimensions?
Mathematically it is simple: While you would usually work with (x,y,z) coordinates, we now instead work with (x1, x2, x3, … , x10 ) as coordinates. Physically it is more difficult, since the world as we observe it has only 4 dimensions (three spatial dimensions and we count ‘time’ as a dimension as well, adding up to 4). The question then arises: Where did the other six dimensions go? The theory should at least take into account why we can not see those six extra dimensions in our everyday live (or even in high energy experiments like the LHC at CERN until now).

The answer is that we think that these extra dimensions are ‘compactified’. So what does this mean?
Consider for example a circle. If I’d be walking along this circle long enough, I will end up at the same point again. Such a ‘rolled up’ direction we call a compactified dimension.
The reason why we can not see these extra dimensions in our everyday live and experiments, is that they are simply too small for us to notice.

Now, for our string theory, we need to compactify six dimensions. Stuff gets interesting when we are trying trying to compactify more than one dimension. For example, let’s roll up two dimensions. It is not hard to imagine that the surface of a sphere is an example of a compact two-dimensional object (we usually call these objects ‘manifolds’), but another option we have is to compactify the two dimensions in the form of a donut, or even a pretzel. It is already clear that we can make a huge amount of different objects, and the amount of choices we can make increases radically when we try to compactify six dimensions.
(The above is actually a bit oversimplified, string theorists for example like to compactify the extra dimensions on so-called ‘Calabi-Yau manifolds’)

Now the funny thing is, that for every different way you compactify the extra dimensions, the laws of Physics, as in the coupling constants, interactions and even the particle content in our four known dimensions will be different. Every such possibility we call a ‘vacuum’ of string theory.
The challenge is then, to compactify the extra dimensions in such a way that the theory we end up with would look like the Standard Model like we know it now ( + perhaps some extra particles that we have not discovered yet). People have actually found quite a lot of those configurations that look like our Standard Model.

Sidenote: One of the physical constants that varies from one vacuum to another, is the Cosmological constant, and I’m currently trying to find a way to make this one work out.

Now, the problem lies in the amount of vacua (or ‘different ways to compactify string theory down to four dimensions’) string theory has. Its exact number will depend on who you ask, but it is usually quoted as around 10500. That’s huge! This is more than there are particles in the universe. And every single one of those vacua will correspond to a different kind of universe, most of them that do not even look like the Standard Model at all! But then, what would it mean if we would find just one vacuum out of those 10500 that would correspond completely to the world and laws of nature we know? If we can just chose the one we like, how can the theory predict anything?

Honestly, at the moment, nobody knows. Some people have ideas: Perhaps there exists some dynamical principle that points to exactly the right vacuum of string theory in which we live, and the universe did not have a choice. Perhaps there is some anthropological principle going on: If the universe would have chosen to live in another vacuum of string theory, the laws of nature would be very different and would not allow for example for stars to be formed, so that we would not have been here to ask this question.

Instead of going all philosophic on this, let’s address the question: Why do you do string theory anyway? String theory has one big power, namely its mathematical machinery. In fact, string theory has taught us a lot about quantum field theories, like the Standard Model, in general. Also, it has shed light on the quantum mechanics (and entropy) of black holes. But perhaps the coolest thing that came out of string theory would be the AdS/CFT correspondence.

Very vaguely, the AdS/CFT correspondence states that there is a relation (‘duality’) between a theory of quantum gravity in some space, and a field theory (without gravity) that’s living only on the boundary of this same space. So, not only does it manage to relate a D-dimensional theory to a (D+1)-dimensional theory, but it relates also a theory with gravity to a theory without gravity. This idea, born in string theory, has been used to calculate stuff in a lot of other branches of physics, going from QCD in heavy ion collisions to superconducting materials. So even if it would turn out in the end that the elementary particles are not little strings, string theory already had its victories, purely by its mathematical machinery, what it has thought us about physics and how we can use this in other branches of physics.