I am confused about the mode expansion in string theory. For instance, for a bosonic closed string, the field describing the string coordinates $X^\mu(\sigma,\tau)$ can be written as: (many thanks to @ACurousmind for writing it in an answer to another question)
$$X_\pm^\mu(\tau\pm\sigma) = \underbrace{\frac{x^\mu + c^\mu}{2}}_{\text{initial position}} + \underbrace{\frac{\pi\alpha'}{l}p^\mu(\tau\pm\sigma)}_{\text{center-of-mass motion}} + \underbrace{\sum_{n\in\mathbb{Z}-\{0\}} \frac{\alpha^{\pm\mu}_n}{n}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{l}n(\tau\pm\sigma)}}_{\text{vibrational modes}}$$
If I am not wrong, the first two terms do not appear in QFT because the fields permeate the entire space, so there is no such thing as an observer “outside the field”, and thus there are no terms for a center of mass position or momentum of the entire field itself. Is my understanding correct?
