String Theory Fundamentals

From a point particle to a string

A relativistic point particle traces a one-dimensional worldline through spacetime. Its classical action is proportional to the length of that worldline. A string sweeps out a two-dimensional surface, called the worldsheet, as it moves. The natural classical action is therefore proportional to the area of that surface.

The replacement matters because an interaction is spread over a finite length rather than concentrated at a point. At ordinary energies the string looks pointlike, but at energies near the string scale its extended nature changes the ultraviolet behavior of scattering amplitudes.

Vibrational modes are particles

A quantum string has infinitely many oscillator modes. Each allowed mode has a mass, spin, and set of charges determined by the boundary conditions and the compactification. A single string can therefore produce a tower of particles rather than one species. The lowest modes are the most important at energies far below the string scale; higher modes are heavier and usually inaccessible.

Closed strings have a left-moving and a right-moving sector. Among their massless modes is a symmetric tensor that behaves like a spin-2 graviton, an antisymmetric two-form field, and a scalar called the dilaton. Open strings have endpoints and can carry gauge degrees of freedom. Their massless vector modes behave like gauge bosons.

Supersymmetry and consistency

The simplest bosonic string has serious problems: it contains a tachyon and requires 26 spacetime dimensions. Superstring theories add worldsheet fermions and supersymmetry, removing the tachyon in their stable spectra and reducing the critical dimension to 10. Supersymmetry pairs bosons with fermions, although any supersymmetry in nature must be broken at accessible energies because no superpartners have been observed.

The five consistent ten-dimensional superstring theories are type I, type IIA, type IIB, heterotic SO(32), and heterotic E8 x E8. They are not five unrelated final answers. Strong-weak and geometric dualities connect them, leading to the broader idea of M-theory in eleven dimensions.

What quantization does

Quantization turns the classical modes into creation and annihilation operators and imposes constraints from worldsheet gauge symmetries. Those constraints remove unphysical states and force relations among the dimension, background fields, and allowed interactions. The result is why string theory is more than simply drawing a tiny rubber band: consistency of the two-dimensional theory controls the possible spacetime physics.

Read the worldsheet description next.