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What has string theory really reached in quantum gravity?
#1
As a researcher in a completely different direction, I have completely ignored string theory. This is my personal bet. It may have been a bad choice, but even in this case it would be too late for me to switch to string theory. So, about string theory I'm in the position of a layman, and in fact only interested in understanding what they have really reached. In this thread, the question is what has been reached in the domain of quantum gravity.  

For comparison, quantum gravity is not really a problem in ether theory. It reintroduces a fixed Newtonian background into the theory. This background gives translational invariance. There is also a non-degenerated Lagrange formalism, and, as a consequence, Noether conservation laws and a non-degenerated Hamilton formalism with local energy and momentum conservation laws. So, all the conceptual problems related with the failure to create all this in a background-free theory disappear. Even more important is that the gravitational field follows usual condensed matter equations like the continuity and Euler equations. With a conserved density and a well-defined velocity field it is easy to introduce a comoving lattice as a regularization. And this lattice regularization, which gets rid of the field-theoretic infinities, is in no way obliged to have a well-defined limit for arbitrary small distances, because it has an own right to exist, as a fundamental atomic ether theory. In other words, we know how to quantized classical condensed matter theories, so that in quantum ether theory all we have to do is to follow known prescriptions, and not a problem at all.

The situation with quantization of general relativity is, instead, quite horrible. There is no background, which in itself leads to a quantum variant of the hole problem, and it is not even clear how to define GR as a quantum field theory. The topology is not fixed, and for the quantum domain one cannot exclude topologies which cannot appear as solutions of the Einstein equations, which requires to consider a topological foam. Einstein causality, defined by the gravitational field, becomes uncertain, thus, essentially undefined and undefinable. Because of diffeomorphism invariance, the Hamilton formalism can be defined only in a degenerate, problematic form, which leads to the "problem of time". Given that all the things which cause the problems, in particular, diffeomorphism invariance, are highly valued as deep insights of GR, it is almost anathema to abandon them, so that quantum GR researchers appear trapped by the rigorous requirements of GR philosophy.  

The situation is much easier in string theory, where we have a fixed Minkowski background. Even if this background is some 10- or 11-dimensional one, it may be sufficient to solve at least most of the conceptual problems caused in GR by its background freedom.

What remains is, of course, the field-theoretic part of the problem, the non-renormalizability of GR. Here, the problem is more serious than in ether theory, because one cannot simply consider a lattice regularization as an in itself satisfactory theory - the very point of string theory is that it will be true for arbitrary small distances. So, one really needs a finite theory for arbitrary small distances here. On the other hand, it is quite plausible that the very idea of strings instead of point particles allows to get rid of some infinities. Nonetheless, it does not mean that a really finite theory for arbitrary small distances has been actually reached. So, this is the first question I would be interested to learn more about, in particular about the actual state: Is there a satisfactory field theory of gravity, without any infinities, based on string theory or not?
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#2
Some answer to the question is given in Woit, Not even wrong - the failure of string theory, Jonathan Cape, London, 2006.  Here, a longer extract from p.183 ff:
Quote:The situation in superstring theory is that for any physical process, what the theory gives one is a method for assigning numbers to possible two-dimensional world-sheets swept out by moving strings. These world-sheets can be organised topologically by counting the number of holes each one has. A superstring theory calculation gives one a number for zero holes, another number for one hole, yet another for two holes, etc. The fundamental conjecture of superstring theory is that this infinite sequence of numbers is some kind of perturbation expansion for some unknown and well-defined theory. This conjectural underlying theory used to be referred to as non-perturbative superstring theory, but now is really what is meant by the term M-theory. No one knows what it is.

One might hope that expansion in the number of holes is actually a convergent expansion and thus good enough to calculate anything one wants to know to arbitrary precision, or that at least it is a good asymptotic series. There are strong arguments that the expansion is not convergent and that as one calculates more and more terms one gets a result that becomes infinite. In principle, the expansion could be a useful asymptotic series, but this is not quite what superstring theorists actually want to happen. There are some features of the terms in the superstring hole expansion that they like and they want the expansion to be good for calculating those. There are other features that are very problematic and they would like the expansion to fail completely when one tries to use it to calculate them.

The main features of the calculation that superstring theorists would like to keep are ones showing that, in the low energy limit, the theory looks like a theory of Yang-Mills fields and gravitons, since this was the main original motivation for superstring theory. There are quite a few features of the calculation that they would like to disown as things that should disappear in the true underlying M-theory. One of these features is the supersymmetry of the theory's vacuum. In the hole expansion, each term is exactly supersymmetric, with no spontaneous  supersymmetry breaking. Somehow, whatever M-theory is, it is supposed to contain an explanation of where spontaneous supersymmetry breaking comes from. It should also allow calculation from first principles of the 105 extra parameters of the minimal supersymmetric standard model. Finally, it should solve all the problems of supersymmetric quantum field theories explained in the last section. There is no evidence at all for the existence of an M-theory that actually does this.

The other feature of the hole expansion that superstring theorists would like M-theory to get rid of is the so-called vacuum degeneracy. Recall that superstring theory makes sense only in ten dimensions. In the hole expansion there is an infinite list of possible ten-dimensional spaces in which the superstring could be moving. Superstring theory is a background-dependent theory, meaning that to define it one has to choose a ten-dimensional space as background in which the superstring moves. There are an infinite number of consistent choices of how to do this, only some of which have four large space-time dimensions and six small dimensions wrapped up in a Calabi-Yau space. There may or may not be an infinite number of possible Calabi-Yau spaces that would work, but no matter what, if one chooses one it will be characterised by a large number of parameters that govern its size and shape. The vacuum degeneracy problem is that any Calabi-Yau of any size and shape is equally good as far as the superstring hole expansion is concerned. In recent years possible mechanisms have been found for fixing these sizes and shapes by adding some new structures into the problem, but these lead to a vast number of possibilities, and the implications of this will be examined in a later chapter. What superstring theorists would like M-theory to do is to somehow pick out a Calabi-Yau of a specific size and shape, but again there is no evidence at all for the existence of an M-theory that does this.

So, this answer to my question is a clearly negative one:
  • The theory of gravity based on string theory is not shown to be finite;
  • String theories where one can hope for showing that it may be finite are exactly supersymmetric, and, therefore, not viable;
Of, course, this is the state of 2006, as presented by a person critical of string theory.  But if this is true, string theory would be clearly inferior to ether theory even in the domain of quantum gravity.
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