Lectures on String Theory

– 48 –
Thus, for τ > τ ′ one obtains the propagators 9
〈X R (τ, σ)X R (τ ′ , σ ′ )〉 = ηµν
8πT
〈X L (τ, σ)X L (τ ′ , σ ′ )〉 = ηµν
8πT
〈X R (τ, σ)X L (τ ′ , σ ′ )〉 = − ηµν
8πT ln z ,
where we made an identification
ln z −
ηµν
4πT ln(z − z′ ) ,
ln ¯z −
ηµν
4πT ln(¯z − ¯z′ ) ,
z = e i(τ−σ) , ¯z = e i(τ+σ) .
Computation for the case of open string is similar. For τ > τ ′ we have
〈X(τ, σ)X(τ ′ , σ ′ )〉 = −i ηµν
πT τ + 1
πT
Performing the sum one finds
[ (
〈X(τ, σ)X(τ ′ , σ ′ )〉 = − ηµν
log
4πT
+ log
∞∑
n=1
e iτ − e −i(σ−σ′) e iτ ′) + log
(
e iτ − e −i(σ+σ′) e iτ ′) + log
1
n e−inτ+inτ ′ cos nσ cos nσ ′ .
4.1.5 Vertex operators. Tachyon scattering amplitude
(e iτ − e i(σ−σ′) iτ ′)
e
(
e iτ − e i(σ+σ′) e iτ ′)] .
Here we approach for the first time the question about string interactions. It is
important to realize that the situation here is different to what one usually accounters
in QFT. The interaction of strings cannot be introduced by adding non-linear terms
to the string Lagrangian; in the latter case one would obtain non-linear interacting
theory but still of a single string.
Out
split string
on a mass−shell
Out
V
Insertion
of a local operator
emmiting a particle
In
In
Fig. 3. Open (closed) strings interact by means of joining and splitting.
Emission of a point particle on mass-shell is represented by insertion of a
local vertex operator.
9 Rigorous justification of these formulae requires an introduction of an IR regularization, because
correlation functions of the massless field in two-dimensions suffer from IR divergencies.