Perturbation Theory: String Theory’s Method of Approximation
The equations of string theory are incredibly complex, so they often can only be solved through a mathematical method of approximation called perturbation theory. This method is used in quantum mechanics and quantum field theory all the time and is a well-established mathematical process.
In this method, physicists arrive at a first-order approximation, which is then expanded with other terms that refine the approximation. The goal is that the subsequent terms will become so small so quickly that they’ll cease to matter. Adding even an infinite number of terms will result in converging onto a given value. In mathematical speak, converging means that you keep getting closer to the number without ever passing it.
Consider the following example of convergence: If you add a series of fractions, starting with 1/2 and doubling the denominator each time, and you added them all together (1/2 + 1/4 1/8 + . . . well, you get the idea), you’ll always get closer to a value of 1, but you’ll never quite reach 1.
The reason for this is that the numbers in the series get small very quickly and stay so small that you’re always just a little bit short of reaching 1.
However, if you add numbers that double (2 + 4 + 8 + . . . well, you get the idea), the series doesn’t converge at all. The solution keeps getting bigger as you add more terms. In this situation, the solution is said to diverge or become infinite.
The dual resonance model that Veneziano originally proposed — and which sparked all of string theory — was found to be only a first-order approximation of what later came to be known as string theory.
Work over the last 40 years has largely been focused on trying to find situations in which the theory built around this original first-order approximation can be absolutely proved to be finite (or convergent), and which also matches the physical details observed in our own universe.