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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.

• Ronald Cook

“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.”

Or perhaps the tiniest bits of matter are not in fact little strings.

• Jack

If you wanted to model every particle at every point throughout the history of the universe at the planck scale, that would be (using overestimates) 10^100 particle wave functions with four dimensions of 10^64 points each, making a grand total of 10^356 data points. Even assuming billions of potential states we don’t know about, that might takes us to 10^400. You can model the entire history of the universe to the planck scale with massively fewer data points than the number of universes predicted by String theory.

I continue to fail to understand the enthusiasm for saying “we don’t understand how to create the standard model, so we’ll increase the degrees of freedom by an infinite amount (going from a point to a string) and hope that the solution lies in there somewhere”. Of course it will, it’s the same as any set of data points and trying to fit curve to it – if you use y=SUM(x^n) where x can go to infinity, you’ll fit the points many times – but with absolutely no predictive power.

• Charlie

I’m not qualified to comment on this, but is 10^500 models really comparable to 10^400 different states of the universe? What if 10^499 of those models predict a universe with only one state? Or more reasonably, 10^499 models predict universes with states that don’t resemble the kind of states we see in ours.

Of course, if you had to test each model individually, that would take a while. But if you could devise tests to exclude 90% of the models at each step, that would be only 500 steps. (I’m just a biology prof so my questions are in pure ignorance and innocence. I would have a hard time convincing NIH to fund me if I told them I needed to do 500 experiments to test my hypothesis.)

• Xezlec

Aren’t you overstating things just a bit? “No predictive power” is a pretty strong thing to say.

Doesn’t string theory predict supersymmetry, extra dimensions, fuzzy black holes, and so forth? And hasn’t it already demonstrated some sort of analytical (though perhaps not “predictive” in the sense you mean) power via the AdS-CFT correspondence and the holographic principle? I mean, it’s one thing to say that its predictions are practically untestable, but quite another to say it makes no predictions.

Besides which, how do you know those 10^500 vacua have no structure? How can you be so sure that they can represent absolutely anything anybody wants? And even if they can, how can you be sure that’s a problem necessarily? One could say the same thing about field theory (I bet I could set up a field that would do just about anything, if you let me write down enough terms), but that doesn’t make field theory useless. We just have to use experiment to find the right fields.

Similarly, string theory provides a mathematical tool that (perhaps) lets you combine general relativity and quantum mechanics sensibly. Now that you have a powerful enough tool, you can start looking for the shape of the problem to use it on, but that’s a separate step. It might not make sense to expect the tool itself to tell you that without any experimental guidance, any more than just knowing how to vary a field theory lagrangian can be expected to tell you what the fields of the standard model are.

• Jack

Indeed not comparable, I was comparing apples and oil tankers. Big numbers, nothing much else to compare.

As you say, it’s only a matter of reducing the solution space a bit more. String theory has managed to reduce the potential solutions from infinite to only 10^500 which is an infinite reduction in solution space size, so getting down from 10^500 to a few dozen should be easy in comparison. I look forward to the results.

• Jack

My concept of predictive power is indeed something testable. Otherwise we’re not talking science, and while discussing metaphysics is always entertaining, I took the subject of the blog to refer to physics, not philosophy

• Hi Charlie,
You are actually exactly touching the point I was trying to make come across in this blog post, but let me correct your confusion:

At this stage there is a huge amount of different vacua for string theory, most of which are not even liveable (for example they can have a too small electric charge, protons would decay or even no chiral fermions are available in this model), so we are quite sure we do not live in them and throw them away. All this of course under the assumption that string theory is the right theory, which of course we don’t know either, to reply to Ronald Cook’s comment.

If we would try to answer the question, ‘why’ can we throw them away, I’m afraid this would be philosophy at this point. Perhaps we’ll understand string theory better one day to answer this, but I’m not going to try it.

Then, of course, this huge amount of vacua is too big to take a look inside every vacuum to see what’s happening in there. Therefore, for the big majority of vacua, we don’t know in detail the laws of physics that lie inside.
However, people did find quite a few vacua that ‘look like’ the standard model (or supersymmetric extensions thereof).

Now if there would be only one vacuum that looked like the standard model, we could investigate it, calculate some measurable quantities and put it to some tests.
The problem however, is that there are a lot, too many in fact. So we would not be able to give experimenters a predicted quantity they can measure.

On the other hand, some vacua with large extra dimensions have some common predictions that could possibly be found at the LHC (nothing yet).
A lot of vacua also contain so-called Kaluza-Klein modes, and if they are light enough, they could in principle be seen with the LHC as well. But here again, since you don’t know in which vacuum we live, we can not tell the experimenters an exact mass range or properties of these particles.

@Jack:
I wouldn’t consider modelling any particle in the universe as physics. Our goal is not to find a potential for every particle out there and to describe its equations of motion.
The goal of Physics is to understand what’s beneath, to answer more fundamental questions like “What are we made of?”, “Why does stuff behave like this?”…

And sometimes nature seems to be so weird, that one has to come up with the weirdest theories when you try to explain things.

• Jack

“I wouldn’t consider modelling any particle in the universe as physics”. A lot of physicists specialising in ballistic mechanics might disagree with you there. The goal of physics is to use the results of experiments to formulate scientific laws which can then be used to predict other phenomena. While there have been a few extraordinary individuals (Einstein’s approach to GR springs to mind, or Dirac with relativistic QM) who started with concepts and ended up with a theory that predicted unexpected new phenomena, the vast majority of physicists do it the other way round taking experimental results, initially modelling what they see, and trying to come up with a pattern that can be expressed as a set of mathematical rules. Hopefully, you’re one of those extraordinary individuals and you’ll produce something useful from String theory that makes a testable prediction. Best of luck

• string theorists…like to compactify the extra dimensions on so-called ‘Calabi-Yau manifolds’” that are achiral (“mirror-symmetric” for starters). Assuming photon vacuum symmetries for matter causes parity violations (Weak interaction), chiral anomalies (hadron mass), and symmetry breakings (empirically invalid SUSY).

Opposite shoes embed in trace chiral vacuum (mount a left foot) with different energies. They vacuum free fall along trace unlike minimum action trajectories (Equivalence Principle violation). Confront spacetime geometry with test mass geometry in an Eotvos experiment comparing chemically and visibly identical, single crystal test masses in enantiomorphic space groups: P3(1)21 vs. P3(2)21 alpha-quartz or P3(1) vs. P3(2) gamma-glycine.

When theory drips excuses, real world look.

• Doug Little

Are there some rules that the vacua obey when it comes to the particles and interactions that they predict? Is there a geometrical link between a certain configuration of 1 or more of the extra dimensions and a certain particle so that degrees of freedom can be eliminated along with their resulting vacua?

• Robin

I was thinking precisely this.

• Simon

Interesting article, thanks to broaden my knowledge!

I really lack experience so I want to ask some stupid questions:
I wonder if it is foreseen that one can test gravity unification theories in the future. Are there any constant/variables rolling out of the theory that can be used for this (maybe the Cosmological constant) ?

Also at the moment just before the big bang, time and space were not defined/existant. How does the string theory deal with the “non existent space”, is there a nice workaround that would lead to a broader knowledge of what was there before the big bang ?

• Craig

I guess I have read and understand that string theory describes all 4 forces (therefore ‘everything’), but what predictions does it make? What philosophical implications?
How does it reconcile singularities with infinite mass in a black hole?
How does it reconcile the double slit quantum eraser?
How does it reconcile Shrodinger’s cat?
What other philosophical implications does it bring to the table?

Vibrating strings. Got it. “Theory of Everything”? Show me.

• Tom

Physicist Rudolf Haag: ‘In many popularised presentations the starting point of string theory is explained as the replacement of the fundamental notion of “particles” with its classical picture of a point in space or a world line in space-time by a string in space respectively a two-dimensional worldsheet in space-time. This, I think, is a misunderstanding of existing wisdom. First of all, paraphrasing Heisenberg, one may say “Particles are the roof of the theory, not its foundation”.’