Lectures on String Theory

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It also follows from the scattering theory that appearance of a polynomial of degree
n as the residue of the amplitude signals that the exchanged particles of the mass
M 2 n carry spin up to the maximal value J max = n. Our later analysis of string in the
physical gauge will reveal that the exchanged particles delivering the resonances of
the Veneziano amplitude are those from the spectrum of open string!
Operator Product Expansion
Consideration of the Veneziano amplitude shows that the object of primary interest
in string perturbation theory is a correletation function of local operators
〈O i1 (x 1 )O i2 (x 2 ) . . . O in (x n )〉 ,
where O i (x) is a local operator. Here x ≡ (τ, σ) is a point on the two-dimensional
world-sheet. It is important to understand the behavior of the correlation function
when two operators are taken to approach each other. The technique to describe this
limit in known as the Operator product Expansion or OPE for short. The Operator
Product Expansion states that a product of two local operators can be approximated
to arbitrary accuracy by a sum of local operators
O i (x)O j (y) = ∑ k
C k ij(x − y, ∂ y )O k (y) .
Let us take as a local operator the stress tensor and try to work out the corresponding
OPE. Introduce the short-hand notation T ≡ T ++ and X µ ≡ X µ L . Consider
the component T −− of the stress tensor normalized as
T ≡ T −− = 1 α ′ : ∂ − X µ ∂ − X µ := 1 α ′ : ∂ − X ν L∂ − X Lν : .
We will also use the concise notation z = e i(τ−σ) and w = e i(τ ′ −σ ′) .
In what follows we consider the product of two stress tensors evaluated at two different
points
T (τ, σ)T (τ ′ , σ ′ ) = 1
α ′2 : ∂ −X µ (z)∂ + X µ (z) :: ∂ − X ν (w)∂ + X ν (w) :
and try to expand it over a basis of local operators. By using the Wick theorem, we get
T (τ, σ)T (τ ′ , σ ′ ) = 1
α ′2 : ∂ −X µ (z)∂ − X µ (z)∂ − X ν (w)∂ − X ν (w) :
+ 4
α ′2 〈∂ −X µ (z)∂ − X ν (w)〉 : ∂ − X µ (z)∂ − X ν (w) :
+ 2
α ′2 〈∂ −X µ (z)∂ − X ν (w)〉 〈∂ − X µ (z)∂ − X ν (w)〉 .