The title of this post is a famous question (posed, for example, by Joe Polchinski) which is modeled after an even more famous question by Ken Wilson, “What is Quantum Field Theory?”. I certainly can’t answer the first question, but Wilson’s question now does have a widely agreed upon answer (which is sadly not well presented in a popular literature that continues to repeat old myths about regularization) which I will mention a bit later
What I would mainly like to do, however, is to answer the much easier question, “What is string perturbation theory?” But before getting to that, let’s talk a bit about what perturbation theory is. Unfortunately, most of the equations in classical and quantum physics are not exactly solvable. Often, one can try to solve them on computers, but, particularly in the case of quantum field theory, the calculations can be long and difficult when they are even possible. Faced with this, most of physics (and other fields) is done by a variety of approximation techniques. The most prevalent of these is perturbation theory.
The idea behind perturbation theory is to start with a set of equations you can solve. You can then “perturb” these equations by adding terms with small coefficients. One expects that the solution to these perturbed equations should be close to the solutions of the unperturbed equations. (This is not always the case, and determining when one’s approximations are valid is an essential part of understanding the physics.) The solutions one obtains via perturbation theory are what are called formal power series — the “formal” just means that they don’t necessarily have to converge. By looking at the first few terms, even if the full series does not converge, one can usually obtain a good approximation to the exact solution. It’s not ideal — so called “non-perturbative” effects can be extremely important, and there’s often not a small parameter to let you even start your perturbation theory — but sometimes it’s the best you can do. For example, the determination of the “g-2” of the electron is mostly done using perturbation theory and is accurate to 12 or so digits.
Perturbation theory is the connection between quantum field theory and the idea of particles. As you might guess from it’s name, quantum field theory (QFT) is a theory of fields. At every point in spacetime, these fields can take their values, and we have to integrate over every possible way that can happen. Exact solutions are few and far between. So, we do perturbation theory. The easiest exact solutions are solutions to what are called free field theories. Free field theories have the property that the solutions can be expressed in terms of fundamental solutions which we can think of as particles. The difference between this and quantum mechanics is that quantum mechanics deals with a finite fixed number of particles, and free quantum field theory has as many as you want. This is why quantum field theory is sometimes considered as multibody quantum mechanics.
We can do perturbation theory around these free solutions. Because our solutions are close to the solution of the free theory, they can be thought of in terms of particles. The difference is that the particles can now interact with each other — an interacting theory is another name for a QFT which isn’t free. Feynman discovered an amazing and intuitive way of organizing the calculations involved in perturbation theory. Each term is a sum of graphs where the lines represent different types of particle in the free theory and the vertices correspond to the terms which describe the interaction. These are the famous Feynman diagrams:
These diagrams are remarkably useful as it’s very easy to picture them as particles interacting with each other. It’s always worth remembering, however, that they are tools for organizing an approximation. It’s unfortunate, however, that if you actually try to do the calculation the Feynman diagram is pictorially representing, you almost always get infinity. For a long time these infinities bedeviled theorists and were dealt with by a combination of black magic and handwaving. These days, however, we realize that these infinities are reflecting the fact that the theories we are working with are what are called “effective field theories”. This means that they are fundamentally incomplete, only capable of describing part of a world rather than all of it. The infinite answers that appear reflect the incompleteness of the theory, and we can use that insight to find the part of the answer that corresponds to the physics the theory does describe. The previous black magic was actually secretly doing this procedure, and that’s why it gave the correct answer.
Returning to perturbation theory, Feynman diagrams were far from Feynman’s only contribution to quantum field theory. He also showed how to take the fields out of free field theories. Instead of considering fields, one considers all possible ways a particle can move in spacetime. Mathematically, this corresponds to a theory that lives on a one dimensional line as opposed to on spacetime. This line is called the ‘worldline’ of the particle. The important field that lives on this one dimensional line is a map that embeds the line into spacetime (where we’d ordinarily be doing our physics). This values of this map give usual worldline of a particle as show here:
Because we’re doing quantum field theory on this one dimensional line, we integrate over every possible embedding. This gives the same answers as a free field theory in spacetime. It’s a weird inversion; instead of physics living in spacetime, we have that spacetime lives on this one dimensional line. But it works. We can extend this to perturbation theory by adding in the interactions by hand. We have graphs that embed into spacetime, and the same procedure reproduces the results of perturbation theory. You don’t get all of quantum field theory, but it’s still pretty cool.
String perturbation theory is a generalization of this final construction. All we do is replace our one dimensional line with a two dimensional surface called the worldsheet. Two dimensions is a lot more than one, so we have some freedom about the type of theory that can live there. This leads to the various types of string theories, ie, the bosonic string and the various superstring theories. Some amazing things begin to happen once you fix your theory, however. For one, you don’t have to put in interactions by hand anymore. If you look at every possible two dimensional shape to map into spacetime, you automatically get interactions. A sphere with three holes, for example, corresponds to two strings coming in and one string coming out in a diagram often called a “pair of pants”:
Another consequence is that not every spacetime you try to embed your string into ends up giving a sensible theory. Some of the classical theories (which always make sense) don’t have corresponding quantum theories (this is called an anomaly). In particular, the number of dimensions of spacetime is fixed in the simplest examples (there are other theories called non-critical strings which can have varying dimensions, but they tend to exhibit odd behavior.). It also turns out that the spacetime you map into is forced to obey the Einstein field equations. In other words, string perturbation theory only makes sense when you are doing perturbations around a spacetime that satisfies the equations of gravity.
This isn’t a surprise. When you look at how the vibrations of the string manifest themselves as particles in spacetime, one of them looks exactly like a graviton. On reasonably general grounds, any sensible theory that contains a graviton pretty much has to be Einstein’s theory. So, it seems that a minor miracle has occurred. Solely by generalizing Feynman’s description of perturbation theory from one dimensional objects to two dimensional objects, you automatically get gravity. It’s unavoidable. Still, you might object that this putative graviton could only look like a graviton — how do we know it actually is one? A graviton is supposed to be a small perturbation in the metric of spacetime. So, we can write down a theory on the two dimensional worldsheet mapping into a space with one of these perturbed metrics. By doing the usual tricks of perturbation theory, we can convert this into a state of the theory with the unperturbed metric. This state turns out to be precisely a bunch of our putative gravitons. Thus, we see that combining a ton of these gravitons precisely corresponds to changing the metric, and we can drop the “putative”.
That’s string perturbation theory. Just like in quantum field theory, the power series you get doesn’t converge. However, while it hasn’t been completely rigorously proven, there are very good arguments that, unlike in quantum field theory, every individual term you get is finite. This means that string theory doesn’t have to be an effective theory — it’s a possible theory of everything.
Well, it would be, except that it’s a perturbation theory. It’s only an approximation, and we don’t know to what. So, that’s the best I can do to answer the title of this post: string theory is the theory that has string perturbation theory as its perturbative approximation. Not very helpful, I realize, but in the last decade and a half of string theory, we’ve been able to learn a lot about the properties of this conjectural theory. We’ve even managed to give complete definitions in spacetimes that obey certain boundary conditions. But, it is still this puzzle that remains the fundamental question of string theory: we don’t know what it is.