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Is there any way to describe phsycially which each creation operator $a^{(i)+}_{n}$ in string theory does to the ground state string?

Here would be my guess (although it is likely to be totally wrong):

You can consider the ground state string as a string that is not moving. The creation operator $a^{(i)+}_{n}$ increases the oscillation harmonic in the direction $x^i$ by $n$. E.g. So that $a^{(1)+}_2$ acting on the ground state will do the following: enter image description here

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  • $\begingroup$ A creation operator stretches the ground state string. It increases the eigenvalue of an eigenvector. $\endgroup$ – Ernie Sep 17 '15 at 12:35
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The states of string theory are quantum states. They represent a "vibration" of the string in the same sense that a particle in standard QFT represents a "vibration" of the quantum field.

That is, they do not represent actual "physical" vibration at all. In particular, the states do not describe actual physical position, vibrations or whatever of the string. You may see the worldsheet as a propagator that turns states at one end into states at the other end - but these states live in Hilbert spaces associated to the strings, and do not describe anything about the string itself. Just like usual quantum field theory describes states in Hilbert spaces associated to spatial slices of spacetime, but that doesn't mean the spatial slice itself would behave/be formed in any particular way.

Furthermore, in some approaches to string theory, all you do is specify certain types of conformal field theories on the worldsheets, and the requirement is that their total central charge cancels to not have a Weyl anomaly. In this approach, you don't generally get the usual creation/annihilation operators, or it is at the very least not natural to interpret them as being related to any "vibration".

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