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I don't know much about string theory, apart from it being a theory of everything which brings QM, QED and nuclear forces and gravity under one single roof. I am curious to know from a mathematical view point, how the quantization of space-time is achieved in super string theory.

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string theory doesn't achieve quantization of space. string theory is 'background dependent', a description of how things happen IN space, taking the pre-existence of some kind of space as a given. much effort in string theory is spent arguing the shape and dimensions of that space. perhaps you are thinking of loop quantum gravity, a background independent view (like GR), which seems to take quantization of space itself as a starting point. –  JustJeff Apr 28 '11 at 22:26
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Dear Rajesh, the goal as well as main achievement of string theory is not to "achieve spacetime quantization" whatever this phrase is supposed to mean but to allow calculations about spacetime that are compatible with the postulates of quantum mechanics. It's not the same thing. The statement that "spacetime is quantized", whatever it means, is just a working hypothesis, not a holy grail that should be "achieved".

Instead, what is needed is to have a theory that is a quantum theory as a whole, i.e. one that agrees with the uncertainty principle, probabilistic character of predictions, observables' being expressed by linear operators on the Hilbert space, and so on.

On the other hand, the spacetime is an approximate concept in string theory. At distances much longer than the characteristic distance scale of string theory, the string scale (or the related gravitational Planck scale), classical general relativity is a good approximation - and some of its aspects remain exact at all distance scales. At distances comparable to the string scale (or Planck scale), entirely new set of physical phenomena take over.

It is not true that the only difference of these new phenomena from classical general relativity is that "spacetime is quantized". And in fact, it is not true in any sense and it cannot be true in a consistent theory of quantum gravity that spacetime becomes discrete. And in fact, geometric quantities - while remaining continuous whenever it makes sense to use them - may be shown in string theory to be invalid variables to describe physics at very short distances. To say the least, they're incomplete.

So if your question is How string theory confirms the prejudices and beliefs that the spacetime quantities survive as good variables up to arbitrarily short distance scales and/or that they become discrete, the answer is that string theory is a way to prove that both of these prejudices are dead wrong.

If your question is which quantum phenomena affecting spacetime are predicted by string theory, it is a valid question but it is too broad and a proper answer would require to review all of string theory - because all important insights of string theory, in some sense, show the consequences of the co-existence of quantum mechanics with a dynamical spacetime.

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