# Particles and states in string theory

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?

• In QFT in general the idea of localized particles - i.e., states where a particle is certainly at some point - is a subtle one. In that case one often works in a momentum representation, so instead of talking about "a particle at a given point" one talks about "a particle with a given momentum" or "a particle with such distribution of momentum" when talking about superpositions. I suggest reading section 6.5 ("Local fields, non-localizable particles!") of Duncan's "The conceptual framework of QFT" for more on this matter. – Gold Jan 20 '20 at 13:28
• Related/possible duplicate: physics.stackexchange.com/q/293947/50583 – ACuriousMind Jan 20 '20 at 17:20

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.
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.