Search the web and you can probably get thousands of posts purporting to “explain” string theory. However, this post will explain string theory more in-depth. This post also presents string theory as a natural extension of normal physics.

## Newton’s first law and the action principle

Newton’s first law is the law of inertia. It states that if no forces act on a particle, then the particle travels in a straight line and constant speed. This very simple statement directly leads us to string theory. Yes, we don’t even need the second law.

Euler, Leibniz, and others realized that Newton’s first law can be written as an *action principle*. To do this, we introduce Minkowski and Einstein’s notion of spacetime. Basically, what you do is to treat time as another dimension. Then you can say that a particle traces out a *world-line* in spacetime. Then Newton’s first law states that this world-line is straight, provided no forces are acting on the object.

What other properties to straight lines have? They’re the shortest path between two points! This means we can state Newton’s first law as a “lazy particle” principle: a particle follows the shortest path between two points. We can say that a particle following a path takes some amount of effort or *action* that is the length of its world-line.

## Classical string theory: the Nambu-Goto action

Now suppose, for some reason, we’re interested in how strings behave. Then the action principle generalizes: a string follows a world-sheet in spacetime, and the action should be the area of the world-sheet. Also, the string is lazy, trying to minimize the action. Then oh wait, that means the string should collapse to zero length! So we add a new principle to classical string theory: the law of conservation of energy. The energy of a zero-length string is zero, so this prevents collapsing strings. Fortunately, this will turn out to be unnecessary.

## Explaining the action principle: the principle of stationary action

Now let’s ask a question: how do particles minimize their action? Think of going to a gas station. There’s usually several that are nearby. How do you get the best deal? Easy, if you find one of the nearby gas stations has a better price, you go to that one instead! So if a particle slightly modifies its path, the change in the action is not negative. Otherwise, the particle will modify its path! This implies the action can’t be changing much at all, since it turns out that if the action is increasing if the particle shifts in one direction, it will be decreasing at a similar rate if the particle shifts in the other direction. The only way both shifts can increase the action is if the rate of change of the action is very tiny. This is the stationary action principle.

## Explaining the action principle: quantum mechanics

I don’t believe that particles are always thinking, “can I get a better deal by slightly changing my path?” and neither do you. Fortunately, quantum mechanics naturally explains the action principle. Quantum mechanics says that the amplitude (basically the probability) a particle will travel along a path is $e^{iS}$ where $S$ is the action for that path. Or, $\cos S+i\sin S$. In this form, we can see that the amplitude is an oscillating function of the action.

Let’s look at the probability a particle travels close to a path minimizing the action. We’ve just argued that if a particle strays from such a path, the action changes only slightly. So the amplitude $e^{iS}$ also changes slowly, and so the amplitudes for all the nearby paths constructively add. So there’s a high probability of traveling close to such a path.

On the other hand, if the particle travels on a path where the action is changing quickly, the amplitudes oscillate rapidly and mostly cancel out. So the probability of traveling near such a path is very low. Thus, quantum mechanics provides a good explanation of the action principle. Note that the quantum action principle doesn’t need highly intelligent, precognitive, or omniscient particles.

## Bosonic string theory

So, combining the previous sections, we have a roughly complete description of what’s called bosonic string theory. The probability a string travels along a path is $e^{iA/A_p}$ where $A$ is the area of the world sheet and $A_p$ is the Planck area. We can get the probability a string goes from one configuration to the other by summing the probabilities for all the ways to get there. This simple model has some interesting points:

- The theory is only consistent in 26 dimensions.
- The theory contains Einstein gravity in 26 dimensions.
- So 22 of these dimensions must somehow be “compactified”
- It’s been known since 1919 that compactifying gravity in higher dimensions leads to so-called gauge fields. These gauge fields underlie all of known physics except gravity.
- Thus, bosonic string theory is a candidate for a theory of everything.
- If it weren’t for the fact that the electron cannot be described by bosonic string theory.

## Spinors and supersymmetry

Take a string. Rotate it 360 degrees in space. What did you get? It looks the same, obviously. Now rotate an electron 360 degrees in space. It does not look the same! Objects that behave like this are called *spinors*. Bosonic string theory is disproved by the existence of spinors like the electron.

It is possible to incorporate electrons into string theory. The key tool was developed long before string theory. The idea is the introduce a symmetry that related normal objects and spinors. Then, since we have normal objects in string theory, we will have spinors in *superstring theory*.

## Superstring theory

Unfortunately, superstring theory doesn’t have a nice intuitive explanation, unlike bosonic string theory. The only explanation of it I know of is that superstring theory is the simplest extension of bosonic string theory that includes supersymmetry. There are some interesting properties of superstring theory:

- The theory is only consistent in 10 dimensions.
- There are no less than five consistent theories of supertrings.
- There of the theories, types I, IIA, and IIB differ in what superstrings they contain.
- Type I superstring theory requires strings to have colored ends, where each end of the string can have one of 32 colors.
- There are two other theories, the
*heterotic string*theories. A heterotic string is a string where the left-moving waves on it are normal objects, while right-moving waves are spinors. - Spinorial strings only work in 10 dimensions, while bosonic strings only work in 26 dimensions. For a heterotic string theory, the 16 mismatched dimensions must be dealt with. There are two ways, the HO and HE theories.
- Funnily, and as a spoiler for things to come, the HO and type I superstring theories are exactly the same.

## Duality

We have five different consistent superstring theories: I, IIA, IIB, HO, and HE, or do we? We already know that I is the same as HO. There are other relationships between string theories. One is *T-duality*. T-duality means one theory is a circular dimension of small radius is the same as another theory with a circular dimension of large radius. Since we know that the 6 extra dimensions of superstring theory must be made into small circles, this is actually quite relevant. The other duality, *S-duality*, concerns the coupling constant. The coupling constant is the probability stings will interact. If two theories are S-dual, one theory with a weak coupling is the same as the other with a string coupling. So it seems all superstring theories are one…

## M-theory

Type IIA string theory has a very peculiar property. Specifically, it looks like a compactified 11-dimensional theory. The radius of the extra dimension is proportional to the coupling constant. So, if we let the coupling constant go to infinity, an extra dimension appears! The result of taking this limit is *M-theory*.

How do we reveal the true nature of M-theory? Start with a system of interest in type IIA string theory. Then let the coupling grow large, and launch it at an extremely high speed along the extra dimension. The only objects that survive this process are the *D0-branes*, the string endpoints. Funnily, these D0-branes look like point particles. In essence, M-theory is saying, “throw away the string! Only look at the endpoints!”

We started with the theory of point particles, and turned them into strings, only to find a theory of point particles! String theory has come full circle.