标题: On String Theory and SUSY
作者: sage
I am not going to touch the subject whether or why string theory will go downhill. It is very likely that interesting real stringy physics lies at very high energy scale. Therefore, it is probably to say it in really hard to test it directly or falsify it directly. Therefore, whether it will go up or down base on circumstantial evidences (and weak ones usually) is really a good subject in sociology. Since that is not my field, let me refrain from making that judgement.
It is usually said that string theory predicts supersymmetry. It is a statement based on the following logic (roughly). If you get bored about bosonic string theory, or motivated by hatred towards a tachyongic state, the next thing you could try is to introduce something called worldsheet supersymmetry by introducing worldsheet fermions which together with the worldsheet boson satisfy something called D=2 superconformal algebra (Notice this is not the same as the spacetime supersymmetry, and they don't really necessarily lead to each other). Then, you work out the spectrum. Too bad, you still have tachyons. However, there is a savior. It turns out that keeping all possible states around is not really consistent. For example, it will break some of the symmetries which keep the string theory consistent. And, fortunately, there is a discrete symmetry in theory (the so called worldsheet fermion number). Therefore, we could try to consistently keep only part of the states which is even or odd under this symmetry (You cannot if this is not a symmetry since even will go to odd and odd will go to even). Therefore, we could try to do this. There are several consistent choices (type I, Typy IIA, IIB). The procedure to go to these choices goes by the name GSO projections. Anyway, once you do that, miracle! you find spacetime supersymmetry (in 10 dimensions). These are good, well understood, so-called perturbative string vacua.
There are obvious caveats in this argument. First of all,tachyons may not be as bad as we thing. It usually just mean we are probably a wrong vacuum. It has to roll to some stable vacuum. Of course, it is usually not quite known whether the new theory will have the same degrees of freedom as the old one. However, it is conceivable one can just find theories with not much change, except the tachyon is stabilized. At least, it is not obvious at all why the stable theory will have supersymmetry. Moreover, we are usually talking about the so called critical string theories (means D=26 for bosonic string and D=10 for superstring), stem from the simpliest way one knows how to make a unitary string theory. However, it is not clear at all why these are the only possibilities. In fact, there are many works about non-critical string theories and there are many examples. They don't have tachyon, and usually don't really have supersymmetry. All these things are much less explored, not because they are not likely, but because they are hard. The couplings are usually not so small. Without supersymmetry, we don't usually know what to do in the case of large coupling. However, if string theory is realized in nature, it does not have to care whether we can compute or not. Proton exists, never mind the fact that we cannot compute its mass.
Even if we start from superstring theory, the supersymmetry it possese is usually not the supersymmetry we are talking about. What we usually talk about is the so-called N=1, D=4 supersymmetry with breaking terms on the order of TeV. This gives us a solution to the so-called hierarchy problem, gives us gauge unification, etc. It is an appealing feature. That's why many people, including me, is working on it.
However, to begin with, if the supersymmetry is unbroken in the string theory (compactify on T^6, for example), it translate into D=4, N=8 supersymmetry. This is not acceptable since it is not chiral. Therefore, typically, people pick manifolds (Calabi-Yaus, for example) which only preserves N=1 supersymmetry. Then break it again at TeV scale. You see, as nice as this picture maybe, it contains a lot of assumptions and/or wishes. It is always good to wish and dream in physics. However, it is not true that one should say go to hell if one dream is broken. For example, if one is breaking supersymmetry, naively, there is no reason why some of them should survive all the way down to the low energy scale. In fact, people tried very hard. It usually takes a lot of effort to break supersymmetry at the appropriate scale. Therefore, besides our good wishes, it is not unlikely that supersymmetry in string theory is just broken at very high scale. Again, this is a scenario for which little study has been done not only because it is a little disappointing, but also it is hard to do any calculations without supersymmetry. A symmetry which is broken pretty much at the cut-off of an effective field theory should not have any implications for the low energy physics at all.
Given all these caveats, let me emphasize again low energy supersymmetry is still very appealing idea. It is appealing independent of superstring theory since it solves a lot of problems and puzzles. The picture that it comes from string theory requires several assumptions. It is still an interesting and appealing possibility. However, one should always keep those caveats in mind.
To me, string theory is appealing not only because it offers the possibility of a theory of everything, but also that everything includes in particular gravity. However, string theory is still probably a little bit distance away from acheiving that goal. That's also why I think AdS-CFT is really a very deep theoretical break through since it does says some highly non-trivial about gravity.
Actually, I really like the one-liner proof mentioned by Polchinski. I think it gives a very good explanation to the finiteness of string theory by relating it to a unique feature of the string theory: the connection between UV and IR which could be easily demonstrateby by a simple string amplitude. This is sort of argument which will convince a physicist like me.
It is probably impossible to give a rigorous proof at this moment since we don't even know the full vacua of string theory yet. In many cases, we probably don't really know the appropriate degrees of freedom yet. However, I think we could probably safely say that all known string theories (maybe one has to say perturbative) are finite.
I am amazed that Lee is criticizing string theory for lack of mathematical rigor. Besides the fact the true mathematical rigor rarely buys us anything. They are almost always after thoughts, as Polchinski point out. There is no parallel at all between string theory and 'others', since things like loop quantum gravity does not even qualified to be called a theory.