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## Inclusion of information about external particles to calculate scattering amplitudes in string theory

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.

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.

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I have initially asked this on physics SE but I am getting no answer there ... – Dilaton Jan 27 at 22:38
I have my doubts that this is a good question for MO as it is standard textbook material. String backgrounds determine a CFT, in this CFT there is a state-operator correspondence and the vertex operators used in string scattering computations are given by this correspondence in terms of the external scattering states one is interested in. This is discussed in Chapters 2,3 of volume I of Polchinski's book on string theory. Perhaps you want something else though as I find the mention of bound states and lifetimes and the integral over $d\tau$ confusing. Where did you get your schematic equation? – Jeff Harvey Jan 28 at 12:51
@Jeff Harvey I am sorry if this is inappropriate here, I just tried it here because on physics SE, the level has dramatically dropped and theoretical physics topics are disappearing, such that not even such slightly technical questions get answered anymore. I have this schematic equation from one of Lenny Susskinds video lectures. Please just tell me if I should delete the question, instead of closing and downvoting, and I will do it. I probably really should :-/ – Dilaton Jan 28 at 14:44