Lectures on String Theory

– 53 –
As we will discuss later on, one can perform an analytic continuation τ → −iτ under
which the exponent entering the vertex operator V − will transform as e iτ → e τ . Then
in the limit τ → −∞ we see that V − → 1 and in the Euclidean picture
lim V (k, 0)|0〉 =
τ→−∞ eik µx µ |0〉 ,
i.e. at τ → −∞ this vertex operator creates a particle with the momentum k µ .
Normal-ordering the product of exponents
Consider the normal product : e X :: e Y :, where X and Y are two operators with the
propagator 〈XY 〉. Clearly one gets
: e X :: e Y :=
∞∑
n,m=0
: X n : : Y m :
.
n! m!
To apply the Wick theorem we first calculate the number of ways we can pick up k
X’s from X n n!
, which is obviously . Analogously, the number of ways to pick
k!(n−k)!
up k Y ’s from Y m m!
is . Now we have to pair (i.e. to form propagators) k fields
k!(m−k)!
X with k fields Y
: X } .{{ . . X}
:: Y } .{{ . . Y}
: .
k
k
The are k! ways to pair all the terms in the last expression. Thus, application of the
Wick theorem gives
: e X :: e Y :=
=
∞∑
n,m=0
min(n,m)
∑
k=0
Thus, we find
∞∑
n,m=0
:
min(n,m)
∑
k=0
X n−k
: Xn−k
n!
Y m−k
(n − k)! (m − k)!
Y m−k
m!
:
:
〈XY 〉k
k!
n! m!
k!(n − k)! k!(m − k)! k! 〈XY 〉k =
=
∞∑ 〈XY 〉 k
k=0
: e X :: e Y :=: e 〈XY 〉+X+Y :
k!
∞∑
n,m=k
:
X n−k
Y m−k
(n − k)! (m − k)! :
It is now easy to see that the last formula can be generalized for the case of several
vertex operators as follows
∏
: e X i
:= e P i