In Gordon Kane's Supersymmetry and Beyond (p. 118), he states:

String theory has to be formulated in nine space dimensions or it is not a consistent mathematical theory. There doesn't seem to be a simple way to explain "Why nine?"

Which, rather unexpectedly, is followed immediately by this seemingly simple explanation:

What happens is that if theories to describe nature and to include gravity in the description are formulated in $d$ space dimensions, they lead to results that include terms that are infinite, but the terms that are infinite are multiplied by a factor $(d-9)$, and drop out only for a factor $d=9$.

This leads me to think

  1. that perhaps this explanation is not as simple as I perceive it to be, or
  2. that it perhaps misses out on subtleties and therefore perhaps is not quite correct, or
  3. it's not really considered an explanation by Kane because it doesn't explain why some terms are multiplied by $(d-9)$.

My guess is option 3, but I'd like to be sure. So, my main question is: Is the quoted "explanation" fully correct?


The answer in the book is almost correct albeit oversimplified. If you want to jot them down, then there are several reasons for requiring that $d=9+1$ in superstring theory. None of them however are in any way "simple" (compared to the kind of explanations that the book seems to give). Let me jot down some of the reasons that come to mind. (There may be more.)

Note: Here I take $d$ to be the number of space-time dimensions.

  1. In representation theory, massless particles must form representations of $SO(d-2)$. In superstring theory, this is only possible in $d=10$. In other dimensions, the quantization of the superstring is at odds with this requirement.

  2. A further requirement is the cancellation of the gravitational anomaly. Let me explain this first. Superstring theory relies heavily on conformal field theory. When put in an arbitrary gravitational background, the conformal symmetry is broken. (This breaking is called "anomaly".) This creates a huge problem in the formulation of strings. Fortunately, this anomaly is proportional to $d-10$ and so the anomaly vanishes (i.e., the symmetry is restored) in $d=10$.

  3. The Hilbert space of strings is plagued with negative norm states. Such negative norm states should not be present as it leads to negative probability and amplitudes which do not make sense in the quantum theory. It turns out, in $d=10$, these negative norm states are then elevated to states with zero norm. While these states are also difficult to handle, things settle down quite well when one identifies such zero norm states with pure gauge states. Thus existence of zero norm states points to gauge symmetries.

  4. Starting from a classical theory, there are often several ways to quantize a theory. If the theory is to be consistent, then one requires that all these different methods of quantization lead to the same physical Hilbert space of states. One also requires that all amplitudes be the same. This is called the no-ghost theorem. It turns out that this is only possible when $d=10$.

So there are several reasons to believe that $d=10$ within the framework of superstring theory. The above explanations are not independent of each other of course. You can simply pick whichever is your favourite and use that as your explanation for $d=10$.


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