String theory is a perturbation theory of quantum gravity starting with perfectly linear Regge trajectories self-interacting in a consistent bootstrap. Bootstrap means that the interaction of the trajectories is only by exchange of other trajectories, so that the system is self-consistent, or, in 1960s terminology, that it pulls itself up by its own bootstraps.
The best way to learn what string theory is, is to get a copy of Gribov's "The Theory of Complex Angular Momentum", and learn the basic principles of Regge theory. You don't have to learn the Reggeon calculus covered later (although it is interesting), just the basic principles. The point of this theory is to understand spectral properties --- S-matrix states, not detailed microscopic field theory, which breaks down at the Planck scale. The S-matrix is valid at any scale, it is the fundamental observable object in relativistic quantum mechanics, when you don't have point probes.
In QCD, you can make little black holes and use them as point probes. You can even use electrons as point probes, without going to the trouble of making a black hole. This shows that the scale of QCD is not the appropriate scale for string theory, but nevertheless, string theory was discovered by trying to find a consistent bootstrap at this scale. This is very fortunate historically, and required vision and persistance.
Linear Regge trajectories can be understood as string-like excitations. They are the quantized states of an extended relativistic string described by a Nambu Goto action. But the string action by itself doesn't tell you anything about interactions. The interactions of these objects is only by exchanging states found in their own spectrum, with the condition that S-channel exchange is dual to T-channel exchange. They have no other interactions. This defines a dual string theory, the kind people study.
States and observables
String theory is a quantum mechanical S-matrix theory, so the state variables (in an asymptotically flat space) are the following:
- A classical configuration of fields at infinity, which defines the background.
- A finite number of incoming particles in a quantum superpostion of plane waves, which define the in-state at minus infinity.
The dynamical law is to produce the out-state, given these ingredients. There is only one observable,
- The S-matrix between in and out states.
That's it. Every other observable has to be extracted from this one, by some trick. This is extraordinarily difficult, but in practice, there are some simplifications.
- The classical field configuration is a background which must be consistent with the string theory itself. This gives classical equations which the background field must satisfy. These come from the condition that the string is still conformally invariant in the background, so that the $\beta$ function is zero. These equations define the classical allowed backgrounds, and from this you can extract the classical dynamics at infinity, which is what you use almost all the time, or the approximate quantum field theory description, which is what you use almost all the rest of the time.
- You can take a field theory limit, and start telling quantum field stories of particle propagation. Almost all the work in the 1980s was based on low-energy supergravity approximations, sometimes with higher order effective action corrections. Such a description can be thought of as an approximation scheme to string theory, by adding higher order string corrections to a quantum field theory, to get a string-corrected effective action. This way of dealing with strings is philosophically least challenging, but it doesn't seem to me to be a very convergent process. The string theory is not a quantum field theory, after all.
More dynamical formulations
There are more dynamical formulations than the S-matrix theory, and more honest formulations than the "effective-action" string-corrected quantum field theory. These honest formulations are due to Mandelstam, Kaku-Kikkawa, Banks-Fischler-Susskind-Shenker, and Maldacena-Witten-Gubser-Klebanov-Polyakov.
- When you absolutely need to use string theory itself, instead of quantum field theory with effective action corrections, you can move to a dynamical picture where the string tells a story which is local in a version of spacetime. In such a picture, the string theory can be thought of as a normal quantum theory, not an S-matrix theory. The states are defined by superpositions of configurations, jsut like any other quantum theory. The Mandelstam description of strings is one such picture, and because it exists, one could go to a second quantized string field theory, by defining creation and annihilation operators for the string states. So string field theory de-S-matrixes the S-matrix theory. But it is defined on a light cone, and it is technically complicated. But in such a picture, the basic states of string theory are quantum superpositions of light-cone configurations of strings.
- In 11-dimensional matrix theory, you have a point-particle description in which you can define the state space and evolution again like any other quantum field theory--- as superpositions of noncommutative matrix model configurations, with a normal quantum dynamics in 0+1 dimensions. This is the easiest state space to formulate.
- In asymptotically AdS backgrounds, instead of incoming/outgoing particles and an S-matrix, you have a full honest to goodness quantum field theory's worth of information at the boundary of space-time. This quantum field theory maps in a not-completely-understood way to the interior description, but the state-space and dynamics are obvious--- they are just like any other quantum field theory.
Notice that the descriptions of the state space is entirely different in the different formulations! This is important, both because their mutual self consistency is an insanely stringent consistency constraint, which has zero chance of being satisfied unless there is a consistent gravitational theory behind it, and also it gives completely different types of observables in different asymptotic backgrounds.
There is no unified way of defining the state space on all backgrounds at once. Each type of background has its own formulation. It requires physical intuition to move between the pictures, and there is no way to communicate the results to a mathematician without communicating the physical picture, because the theory isn't 100% complete.
Literature and Misconceptions
The best review for me was Mandelstam review from 1974. It is very important to learn these old-fasioned ideas, because otherwise you will have all sorts of nonsense in your head about what you can do to strings.
- Dual strings do not describe statistical polymer properties. It doesn't work, they aren't those types of strings. Polymers interact by self-intersection, strings don't.
- Dual strings don't work to describe vortex lines in a quantum field theory, although this idea was one of the ways in which they were discovered, by Nielsson. The vortex line picture was simultaneous with the flux-line picture, but the flux line picture is now known to be correct, while the vortex line picture, as far as I know, has no precise version beyond that the string is extended in 1d, like a vortex line. If you make an effective theory of vortex lines in a scalar/gauge field theory, they will interact in crazy non-string ways. In gauge theories with a gravitational dual, like N=4 gauge theory, you probably can make the string be a vortex line, so Neilsson's idea is not altogether wrong (I think), but then some duality will have to link up the vortex and flux line.
- Dual strings have no deformations: you can't make dual strings interact at collision points, you can't make them attach or detach other non-string objects. They are either a theory of everything or a theory of nothing.
- Dual strings only allow S-matrix probing. You can't calculate off-shell behavior, meaning you can't describe their detailed dynamics in space-time. You can formulate their world-sheet behavior in space and time. There is string field theory, which attempts to take strings off shell, but now we know the right way to do this is AdS/CFT, although string field theory is still very important.
- The only way to touch strings to classical objects is to fiddle with asymptotic values of fields. You can't probe them using local quantum fields, because they generate their own local fields. You can make them move in a classical background, but their dynamics determines the quantum properties of all the backgrounds.
- You can also make them interact with branes, but this is a surprise, and it only works because the branes are weak dual black holes, where strings can partially fall through. It doesn't work for arbitrary surfaces, and there are strong constraints on which branes are allowed.
If you learn the old-fasioned string theory of the 1960s and 1970s, you can understand the rest of the stuff. If you don't, you can't.
You suggest that we can use a nonrenormalizible theory (NR) at energies greater than the cutoff, by meausuring sufficiently many coefficients at any energy.
However, a general expansion of an amplitude for a NR that breaks down at a scale $M$ reads
$$
A(E) = A^0(E) \sum c_n \left (\frac{E}{M}\right)^n
$$
I assumed that the amplitude was characterized by a single energy scale $E $. Thus at any energy $E\ge M$, we cannot calculate amplitudes from a finite subset of the unknown coefficients.
On the other hand, we could have an infinite stack of (NR) effective theories (EFTs). The new fields introduced in each EFT could successively raise the cutoff. In practice, however, this is nothing other than discovering new physics at higher energies and describing it with QFT. That's what we've been doing at colliders for decades.
Best Answer
Theories like QED, where one has a finite number of relevant operators are very rare. Many important predictions are performed by means of effective theories (Please see the following reviews by: Aneesh V. Manohar and Scherer and Schindler). It is sometimes said that that since these theories are nonrenormalizable, then any loop correction beyond the tree level is not useful because we can actually have an infinite numbers of free parameters that can fit the theory to any data that we have.
However, Consider for example chiral perturbation theory which describes the low energy degrees of freedom of QCD (with 3 flavors):
$$ \mathcal{L} = \frac{1}{4} f_{\pi}^2\partial_{\mu}U^{\dagger}\partial_{\mu}U + ...$$
($U \in SU(3)$ is generated by the meson fields). Here, one loop corrections give rise to 8 counterterms whose coefficients can be estimated from various processes. There is evidence for improvement in the precision with respect to the tree level computation.
The improved predictions can be explained as follows: Even though the theory is not renormalizable in the usual sense, but if we restrict the Lagrangian to terms with less than a given number of derivatives, and a given number of loops, then, there is a finite number of counterterms. The example mentioned above corresponds to terms with up to 4 derivatives. The same terms of order 4 also serve as counterterms needed to renormalize the one loop contribution of the terms with up to 2 derivatives. Thus, if we limit ourselves low energy processes, we need only a finite number of counterterms. In other words up to a given energy scale, we have control on the counterterms
In the example of chiral perturbation theory, we know that it is a low energy effective theory, thus we know that we should stop at some level of the number of derivatives (or momenta).
This procedure is known as approximate renormalizability, in contrast to the "exact" renormalizability present in QED, where any energy scale can be reached. Actually, this exact renormalizability is not of much practical use, since QED itself is not valid to very high energies (other interactions become important).
Thus given that we want to work up to some energy scale, we can treat the effective field theory as a renormalizable theory and perform loop expansions which generate counterterms with no higher scale.
The question is how can we know where to stop. The answer lies in our knowledge of the degrees of freedom outside the theory. For example, the Fermi theory of weak interactions (which includes an effective four fermion term) gives good predictions for beta decays up to energies of the order of magnitude of the mass of the $W$-boson which is integrated out in the Fermi theory.
This "relaxation" of the renormalizability requirements does not mean that we have an infinite number of effective theories at our disposal. Extra structures are needed to create the property of approximate renormalizability. For example, chiral perturbation theory stems from the origin of pions as the Goldstone bosons of the chiral symmetry breaking.
A major factor which can spoil the approximate renormalizability are anomalies. If we try to gauge an anomalous symmetry, then we loose control on the number of derivatives in the counterterms. Moreover, if we gauge only anomaly free subgroups, then the counterterms will be gauge invariant up to total derivatives in the Lagrangian, and we have Ward identities to each scale.