One of the arguments often given for string theory is that it is somehow exceptionally “beautiful”. This has always mystified me, since that’s certainly not the way I would describe it. Over the years I’ve paid close attention whenever I see someone trying to explain exactly what it is about string theory that is so beautiful. Lubos Motl has just posted his own detailed answer to this question, something I read with interest.
As usual, Lubos is not exactly concise, so I won’t quote him extensively, but let me try and summarize his arguments for calling string theory beautiful, together with some of my own comments.
1. Symmetries are beautiful and just about every symmetry you can imagine gets used somewhere, somehow in string theory.
Even Lubos is not so sure of this argument, since he says ” I don’t really thing that we view symmetries as the most important reason why string theory is beautiful”. What is beautiful about symmetries is the way they constrain things. If your theory is based upon a simple symmetry principle (take for example gauge theory and the gauge symmetry principle), a huge amount of structure follows from a single, simple principle. String theory is not based on a simple symmetry principle, rather it is a complicated framework, into which you can fit all sorts of different symmetry principles. But because they are not fundamental, these symmetries don’t constrain the theory much if at all. This is very different than the standard model, where at a fundamental level the theory is built around a single symmetry principle, one that governs a large part of the structure of the theory and its physical predictions.
2. The way in which “miraculous” cancellations occur in string theory, constraining the theory by only allowing it to make sense for certain specific choices.
The most well known example of this is the way in which anomaly cancellation picks out 10 dimensions and SO(32) or E_8 times E_8 for the superstring. This was the main reason people got so excited back in 1984, when they thought that the anomaly cancellation principle would give them a nearly unique theory that could be used to make predictions. If the anomaly cancellation principle had picked out four dimensions and SU(3)xSU(2)xU(1), that certainly would have been a beautiful explanation of why the standard model is the way it is. In the standard model itself, anomaly cancellation for the chiral gauge symmetry does work in an impressive way. If you take just the leptons or just the quarks, you have an anomalous theory, but the anomalies of the one cancel those of the other.
In string theory, all anomaly cancellation does is pick out a much too large dimension of space-time and a much too large gauge group. You can certainly embed the standard model in this structure, but you could also embed just about anything you want in it because there is so much room. In the end you are stuck with some version of the “Landscape”, essentially an infinite number of different possibilities with no way to choose amongst them. The anomaly cancellation ends up providing very little constraint on what the structure of low energy physics looks like.
3. String theory is a unique theory that can predict everything about the physical world.
Lubos likes to go on about how unique and predictive string theory is. While I understand this is the dream of every string theorist, the reality of what they actually have is a long ways from what they hope is true. The vision of what they would like to be true may be beautiful, but the reality is something else. The reality is that there is no “unique” string theory that can reproduce the real world, just a dream that such a theory exists. And as for predictions of string theory, there are none. When Lubos says that “string theory predicts” things, what he really means is that if every thing he would like to be true actually were, then in principle you could predict things from string theory.
4. String theory manages to extend quantum field theory in a consistent way, something which is very non-trivial and the way this happens can be described as beautiful.
This seems to be Witten’s main argument these days for promoting the continued study of string theory and I have a certain amount of sympathy for it. There certainly is something of interest going on behind the complicated framework that people are studying under the name “string theory” and maybe it will someday lead to insight into something about physics, most likely the strong coupling behavior of gauge theories. But the fact that there is interesting structure you don’t understand doesn’t mean that this structure has anything to do with a fundamental unification principle for physics.
5. There are beautiful connections to new pure mathematical structures.
The relation of string theory to mathematics is a huge topic, and I’ll comment on it at length at some other time. In brief though, while I think string theory has been an utter disaster for theoretical physics during the past 20 years, it has lead to many interesting things in mathematics. However, most of these interesting things really come from 2d conformal QFT, and I would argue that it is QFT which is having a huge impact on mathematics, much more so than string theory. Witten’s Fields medal was for his work on the relation of QFT to math, not for anything he has done using string theory.