We can use the Feynman diagrams to replicate the process of scattering with
strings. For particle interactions, we can do Feynman diagrams (for 
, see
this - just topologies). For strings, we can do the same; we call them "string
diagrams." A closed string forming two closed strings is depicted by changing
the point-particle by strings and word line by worldsheet.
The crossing line indicates (this one line is for collective dimensions.
However, there should be definitive for each one) that there is not one for
all Lorentz frame, unlike in point particle theory, but two. It can be
interpreted that the point-particle Feynman diagram is just a limiting case of
the string diagrams. Furthermore, one string diagram (with vertex function)
can be deformed to a few particle Feynman diagrams. That is one of the reasons
why there are not many string diagrams. Lorentz frames are also the reason for
the absence of ultraviolet divergence because of independent defined Lorentz
sites at interaction.
Similarly, one can do the one-loop of string diagrams as we do in point
particle. But, the convenience and Lorentz issue demand something better. We
do that by
conformally mapping the string diagrams. In this case, we map it to a topological disk (genus-0)
Among the advantages of conformal mapping, one being that there would not be
the h (associated with
) integrals in the matrix calculations. But what about the conservation of
quantum numbers after topological mapping? For that, we introduce vertex
operators. In the conformal image, cross-markers indicate the strings (the top
one shows the far past string, and the bottom two indicates the far future
newly born closed strings). The marked area is for the vertex operator. We can
introduce it with the symbol
, where m is for an m-type particle. This operation is effortless in a
1+1
system, which indeed we are following. The
is the operator for local absorption and emission of string states. We
can introduce another operator
, which is for the re-parametrization of the mapping. While W operators
account for Lorentz transformation, we must also take accounts of translation.
That is how we reach a well-known translation operator
The final operator for emission and absorption becomes
We need to fix the residual gauge invariance for the special linear group (when calculating the M-point functions). Conformal mapping for open strings (in this case, it should be on the boundary of the disk) is done in similar ways, however, they are different.
