Particles and states in string theory

Question

In QFT, put simply, we have some quantized field which lives on our spacetime and the excitations of this field correspond to particles. So some particular excitation would correspond to a particle at a given point on the spacetime, say $x^{\mu}$.

In string theory, we are essentially considering a $2$-dimensional cft which is itself a $2$-dimensional qft with fields $X^{\mu} : \Sigma \to M$ mapping from the two-dimensional world sheet to the $d$-dimensional target space. By the methods used in QFT, we can canonically quantize the theory creating a Fock space which defines the different types of particles that can exist. Then, as with any QFT, a particle would correspond to an excitation of the field $X^{\mu}$ and so some particular excitation of this field would correspond to a particle at a given point on the worldsheet, say at the point $(\tau, \sigma)$.

Is this the correct way to think about particles in string theory? This seems strange as I thought that a particular state corresponds to an excitations of the whole string, not simply some point on the string? What am I not seeing?

Answer 1

First, the one-particle state localized in the point, e.g. $\phi(x)|0\rangle$ is very singular. What behaves more nicely insteas is a smeared state $\int_V d^D x \eta(x) \phi(x)|0\rangle$. However if we look at this "particle at a point" we will see, $$ \phi(0)=\int \frac{d^{D-1}p}{(2\pi)^D\sqrt{2 E_p}} a^\dagger(\vec{p})|0\rangle $$ I.e. to produce ideally localized state we have to integrate over arbitrary high momentum.

If we translate to the string theory then it is important to distinguish between the worldsheet point of view and our spacetime point of view. Indeed the string is 2d QFT. However if we consider a single non-interacting (i.e. having a trivial topology) string it always describes what we would call a single particle. Vacuum on the string corresponds to a single particle that have the lowest mass. It is degenerate because this particle can have arbitrary momentum. It is even more degenerate in the GSO-projected superstring where it describes the massless supermultiplet of particles. Worldsheet multiparticle states still describe a single particle in spacetime with larger and larger mass.

However we also have particles with different momentum on the worldsheet. Becuase one of the coordinates of the string is restricted to a finite interval the spectrum of the worldsheet momentum is discrete. This is the origin of the infinite tower of the ladder operators $\alpha_n$. However the higher values of the worldsheet momentum lead to the higher spacetime mass. Thus single-particle states with higher momentum also look like a heavy single particle from a spacetime point of view.

And thus we return to our particle at point states. First you may notice that this ideally localized excitation describes a delta-like spike on an otherwise smooth string. Next we remember that to construct a local field operator we have to integrate over all momenta. In our case we have to sum over all discrete spectrum of worldsheet momentum. And this translates to this state from a spacetime point of view being a superposition of heavier and heavier particles over all the infinite tower of masses.