quantum-field-theoryregularizationstring-theory
In other question, Ron Maimon says that he thinks string theory is the physical regulator. I did not know that string theory regularize divergencies.
So, Q1: How does string theory regularize the ultra-violet divergencies of the "low-energy" (Standard Model) fields? And Q2: Why it doesn't regularize its own divergencies.
By regularization I mean that the theory is ultra-violet finite before removing the regulator.
This is the proverbial sixty-four thousand dollar question for fundamental physics. It may be helpful to split it down into steps.
Once we have answered these questions the theoretical program to understand the foundations of physics is essentially complete and the rest is stamp collecting and experiment. That is not going to happen today but let's see where we are.
String Theory works very well as a perturbative theory of gravitons that appears to be finite at all orders, but there is no full proof that it is a complete theory of quantum gravity. It requires matter and gauge fields with supersymmetry to avoid anomalies. The size of gauge groups suggests that it could potentially include the standard model. It is too strong a claim to say that it does incorporate the standard model. A popular view is that it has a vast landscape of solutions which is sufficiently diverse to suggest that the standard model is covered, but crucial elements such as supersymmetry breaking and the cosmological constant problem are not yet resolved.
Supergravity theories are potentially alternative non-string theories that could provide a perturbative theory of quantum gravity. Indications are that they are finite up to about seven loops due to hidden E7 symmetry but they are likely to have problems at higher loops unless there are further hidden symmetries. These theories have multiplets of gauge groups and matter. The 4D theories do not have sufficiently large gauge groups for the standard model but compactified higher dimensional supergravity does. A more subtle problem is to include the right chiral structure and this may be possible only with the methods of M-theory.
It has long been the conventional wisdom that supergravity theories can only be made complete by adding strings. Recent work using twistor methods on 4D supergravity seems to support this idea (e.g. Skinner etc.)
Loop Quantum Gravity is an attempt to quantise gravity using the canonical formualism and it leads to a description of quantum gravity in terms of loops and spin network states which evolve in time. Although this is regarded as an alternative to string theory and supergravity it does not give a picture of a purtabative limit which would make it possible to compare with these approaches. It is possible that ST/SUGRA and LQG are looking at similar things from a different angle. In fact the recent progress on N=8 supergravity as a twistor string theory has some features that are similar to LQG. Both involve 2D worldsheet objects and network like objects.
The main distinctions are that LQG does not have supersymmetry and N=8 SUGRA does not use knots. Even then there has been some progress on a supersymmetric version of LQG and the Yangian symmetries used in N=8 SUGRA should be amenable to a q-deformation that brings in knots. It remains to be seen if these theories can be unified.
It is worth saying that all these approaches involve trying to quantise gravity in different ways. Although quantisation is not a completely unique procedure it is normal to expect that different ways of quantising the same thing should lead to related results, If something like supersymmetry or strings or knots are needed to get consistency in one approach the chances are that they will be needed in another.
I have not mentioned other approaches to quantum gravity such as spin foams, group field theory, random graphs, causal sets, shape dynamics, non-commutative geometry, ultra-violet fixed points etc. Some of these are related to the other main approaches but are less well developed. It should also be mentioned that there are always attempts to unify gravity and the standard model classically e.g. Garrett's E8 TOE, Weinstein's Geometric Unity etc. These may tell us something interesting or not, but it is only when you try to quantise gravity that strong constraints apply so there is no reason to think they should be related to the attempts to quantise gravity.
So in conclusion all approaches that have made any kind if real progress with quantising gravity look like they may be related. Much more has been revealed so far from this need to quantise consistently than from directly trying to unify gravity with the standard model. This may not be so surprising when you consider the enormous difference in energy scales between the two.
String theory includes every self-consistent conceivable quantum gravity situation, including 11 dimensional M-theory vacuum, and various compactifications with SUSY (and zero cosmological constant), and so on. It can't pick out the standard model uniquely, or uniquely predict the parameters of the standard model, anymore than Newtonian mechanics can predict the ratio of the orbit of Jupiter to that of Saturn. This doesn't make string theory a bad theory. Newtonian mechanics is still incredibly predictive for the solar system.
String theory is maximally predictive, it predicts as much as can be predicted, and no more. This should be enough to make severe testable predictions, even for experiments strictly at low energies--- because the theory has no adjustable parameters. Unless we are extremely unfortunate, and a bazillion standard model vacua exist, with the right dark-matter and cosmological constant, we should be able to discriminate between all the possibilities by just going through them conceptually until we find the right one, or rule them all out.
What "no adjustable parameters" means is that if you want to get the standard model out, you need to make a consistent geometrical or string-geometrical ansatz for how the universe looks at small distances, and then you get the standard model for certain geometries. If we could do extremely high energy experiments, like make Planckian black holes, we could explore this geometry directly, and then string theory would predict relations between the geometry and low-energy particle physics.
We can't explore the geometry directly, but we are lucky in that these geometries at short distances are not infinitely rich. They are tightly constrained, so you don't have infinite freedom. You can't stuff too much structure without making the size of the small dimensions wrong, you can't put arbitrary stuff, you are limited by constraints of forcing the low-energy stuff to be connected to high energy stuff.
Most phenomenological string work since the 1990s does not take any of these constraints into account, because they aren't present if you go to large extra dimensions.
You don't have infinitely many different vacua which are qualitatively like our universe, you only have a finite (very large) number, on the order of the number of sentences that fit on a napkin.
You can go through all the vacua, and find the one that fits our universe, or fail to find it. The vacua which are like our universe are not supersymmetric, and will not have any continuously adjustible parameters. You might say "it is hopeless to search through these possibilities", but consider that the number of possible solar systems is greater, and we only have data that is available from Earth.
There is no more way of predicting which compactification will come out of the big-bang than of predicting how a plate will smash (although you possibly can make statistics). But there are some constraints on how a plate smashes--- you can't get more pieces than the plate had originally: if you have a big piece, you have to have fewer small piece elsewhere. This procedure is most tightly constrained by the assumption of low-energy supersymmetry, which requires analytic manifolds of a type studied by mathematicians, the Calabi-Yaus, and so observation of low-energy SUSY would be a tremendous clue for the geometry.
Of course, the real world might not be supersymmetric until the quntum gravity scale, it might have a SUSY breaking which makes a non-SUSY low-energy spectrum. We know such vacua exist, but they generally have a big cosmological constant. But the example of SO(16) SO(16) heterotic strings shows that there are simple examples where you get a non-SUSY low energy vacuum without work.
If your intuition is from field theory, you think that you can just make up whatever you want. This is just not so in string theory. You can't make up anything without geoemtry, and you only have so much geometry to go around. The theory should be able to, from the qualitative structure of the standard model, plus the SUSY, plus say 2-decimal place data on 20 parameters (that's enough to discrimnate between 10^40 possibilities which are qualitatively identical to the SM), it should predict the rest of the decimal places with absolutely no adjustible anything. Further, finding the right vacuum will predict as much as can be predicted about every experiment you can perform.
This is the best we can do. The idea that we can predict the standard model uniquely was only suggested in string propaganda from the 1980s, which nobody in the field really took seriously, which claimed that the string vacuum will be unique and identical to ours. This was the 1980s fib that string theorists pushed, because they could tell people "We will predict the SM parameters". This is mostly true, but not by predicting them from scratch, but from the clues they give us to the microscopic geometry (which is certainly enough when the extra dimensions are small).
Best Answer
The answer to both questions is that string theory is completely free of any ultraviolet divergences. It follows that its effective low-energy descriptions such as the Standard Model automatically come with a regulator.
An important "technicality" to notice is that the formulae for amplitudes in string theory are not given by the same integrals over loop momenta as in quantum field theory. Instead, the Feynman diagrams in string theory are Riemann surfaces, world sheets, and one integrates over their possible conformal shapes (moduli).
Nevertheless, if one rewrites these integrals in a way that is convenient to extract the low-energy limit of string theory, one may see that the stringy diagrams boil down to the quantum field theory diagrams at low energies and the formulae are the same except for modifications that become large, $O(1)$, at energies of order $m_{\rm string}\sim \sqrt{T}$. The string scale is where perturbative string theory's corrections to quantum field theory become substantial and that's where the typical power-law increasing divergences in QFT are replaced by the exponentially decreasing, ultra-soft stringy behavior.
The reason/proof why/that string theory has no UV divergences has been known for decades. UV divergences would arise from extreme corners of the moduli space of Riemann surfaces in which the "length of various tubes" inside the degenerating Riemann surface would go to zero. But all such extreme diagrams are equivalent to diagrams with "extremely thin tubes" and may therefore be reinterpreted as IR divergences: it's the only right interpretation of these divergences and no "extra UV divergences" exist because it would be double-counting.
Bosonic string theory has infrared divergences due to the tachyon and dilaton and their long-range effects. However, in 10-dimensional superstring theory, one may prove that all the IR divergences – and there are just several a priori possible candidates that could be nonzero to start with – cancel, essentially due to supersymmetry. It follows that superstring theory is free of all divergences.