In this (schematic) equation to calculate the scattering amplitude A by integrating over all possible world sheets and lifetimes of the bound states

$$ A = \int\limits_{\rm{life time}} d\tau \int\limits_{\rm{surfaces}} \exp^{-iS} \Delta X^{\mu} (\sigma,\tau)$$

the information about the incoming and outgoing particles is still missing. It has to be inserted by hand by including additional multiplicative factors (vertex operators)

$$ \prod\limits_j e^{ik_{j_\mu} X^{\mu}(z_j)}$$

into the integrand as Lenny Susskind explains at about 1:18 min here. But he does not derive the fact that the information about the external particles can (or has to) be included by these additional multiplicative factors like this, he just writes it down.

Of course I see that these factors represent a particle with wave vector $k$, and $z$ is the location of injection (for example on the unit circle when conformally transforming the problem to the unit disk) over which has finally to be integrated too.

But I'd like to see a more detailed derivation of these vertex operators (do there exit other ones too that contain additional information about other conserved quantities apart from the energy and the momentum?) and how they go into the calculation of scattering amplitudes, than such "heuristic" arguments.