A coincidence in string theory: part 1

I recently noticed that 10+16=26. Now you’re asking, “what’s so significant about this? Isn’t this just basic math?” Ok yes, this is just basic math, but what’s important is where these numbers come from.

26: String theory

String theory is a theory that attempts to solve the infamous problem of combining general relativity and quantum mechanics. String theory states that what we see as particles are actually tiny strings. Just as particles move so as to minimize the length of their path in spacetime (a.k.a. Newton’s first law), strings move to minimize the area of their path in spacetime. However, since string theory is a quantum theory, strings are jiggling all the time, and there is the possibility of a string splitting.

It turns out that unless there happen to be 26 spacetime dimensions, various amplitudes (read: probabilities strings will do stuff) turn out infinite. Since 26 dimensions is quite a lot, physicists decided there must be a way to have string theory work in fewer dimensions. This way is supersymmetry.

10: Supersymmetry

Supersymmetry states that there exists a symmetry that relates fermions and bosons. In quantum mechanics, fermions are particles that carry half-integer spin, and bosons are particles that carry integer spin. Since these types of particles behave very differently, the idea of a symmetry relating them is shocking. However, nobody has been able to disprove supersymmetry, with the added caveat that in our world, supersymmetry is spontaneously broken. In practice, this means that the supersymmetry-predicted particles are very heavy, hence they haven’t been observed yet. Many supersymmetric theories are also believed to be self-consistent.

In the string-theoretic world, supersymmetry means introducing a family of spinorial strings. This means that if you rotate them through 360°, they don’t return to the original. You have to rotate through 720°. When these spinorial strings are added, instead of 26 dimensions, only 10 dimensions are required. But to you, the reader, the concept of spinors is weird. Let’s take a closer look.

16: Spinors

If you consider rotation through 360° as a single operation, you won’t see spinors. But in the real world, of course, you don’t rotate through 360° all at once. You rotate gradually. If you rotate gradually, then it turns out there’s a way to tell the difference between rotating 360° and rotating 720°.

Interlude: There’s an interesting real-life demonstration of spinor-like behavior called the belt trick. If you twist a belt 360°, you can’t untwist it while keeping the belt ends fixed, but you can if you twist it 720°. Do try this at home!

In the mathematical description of spinors, spinors look sort of like vectors, except instead of $d$ components, they have $2^{\lfloor\frac{d-1}{2}\rfloor}$ components. For example, in our familiar 3 dimensions, this works out to 2 components. In the case of the electron, these are known as “spin up” and “spin down”. (Note: the fact that quarks are also named up and down is not a coincidence. They are also spinors, but not in spacetime.)

Applying this to string theory in 10 dimensions, we find that the mysterious spinorial strings are described by 16-component vector-like objects.