Description of the center of mass of a string in string theory

Question

For a bosonic closed string, the field describing the string coordinates $X^\mu(\sigma,\tau)$ can be written as: (ethernal thanks to @ACuriousmind for writing it in an answer to another question)

$$X_\pm^\mu(\tau\pm\sigma) = \underbrace{\frac{x^\mu + c^\mu}{2}}_{\text{initial position}} + \underbrace{\frac{\pi\alpha'}{l}p^\mu(\tau\pm\sigma)}_{\text{center-of-mass motion}} + \underbrace{\sum_{n\in\mathbb{Z}-\{0\}} \frac{\alpha^{\pm\mu}_n}{n}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{l}n(\tau\pm\sigma)}}_{\text{vibrational modes}}$$

So the general string state is the tensor product of a single-particle-like relativistic wavefunction representing the center of mass, and acted by operators like $x^\mu$ and $p^\mu$, tensor producted with the fock space of the internal string state (which deals with the string internal vibrational modes). The wavefuction seems to be treated not as a field but as a the wavefunction of a relativistic single particle.

When we learn relativistic QM we learn about the KG and the Dirac equations, and all the problems that they have, and we are told all these issues dissapear in QFT when we interpret the wavefunctions as fields.

Question: Why is it ok to have a single particle wave function when describing the CM of a bosonic string? Why is this no longer a problem as it was in relativistic QM, is it supossed to be dealt in a future version of string field theory, or what? I just started learning ST, so I dont know what is coming next, but nothing was mentioned in the text so far about this being an issue.

Answer 1

Note that the operator $X^\mu(\tau, \sigma)$ is parametrised by the worldsheet variables $\tau$ and $\sigma$. Thus each individual $X^\mu$ defines a scalar field on the two-dimensional worldsheet. It is only when the set of $d(=26)$ such fields are considered together that one gets a spacetime vector which describes the position of the string. In other words, instead of trying to describe the wave function directly on a $d$-dimensional spacetime, we define a QFT given by a set of scalar fields on a 2D surface, and let each field describe a single coordinate of the string.