In the string theory the ladder operator were somewhat followed from the QFT. However, the definition of the $a_0\sim p^\mu$, defined from the zero mode expansion of $X^\mu$ directly, were not quite the same as the rising and the lowering operator in the QFT. There does not seem to be an obvious reason of why $a_0|0\rangle=0$.
In a related post Why does the QCD vacuum have zero momentum? this was explained from the Poincaré invariant. But in the string theory, the momentum operator $P = L_0-\bar L_0\sim a_0^2 -\bar a_0^2$, the annihilation condition was not constrained to $a_0$ or $\bar a_0$ individually.
Why does $a_0$ annihilate the vacuum states $|0\rangle$? Is it a convention or was there some constrains stated that $a_0|0\rangle$ must be true?
