arXiv:hep-th/9304011 v1 Apr 5 1993

13.4. Tree-Level Collective Field Theory S-Matrix
From the classical scattering matrix, we may derive the tree-level quantum S-matrix
by interpreting the left- and right-moving fields as incoming and outgoing quantum fields:
ψ ± → − √ π 1 µ (∂ t ± ∂ τ )χ ±
χ + = i
χ − = i
∫ ∞
−∞
∫ ∞
−∞
dξ
√
4πξ
α + (ξ) e iξ(t+τ)
dξ
√
4πξ
α − (ξ) e iξ(t−τ)
[α ± (ξ), α ± (ξ ′ )] = −ξ δ(ξ + ξ ′ ) .
(13.24)
Now following Polchinski, we interpret the relation (13.23) as a relation between incoming
and outgoing Fourier modes:
α ± (η) = ∑ ( 1 ∫
Γ(1 ∓ iη) 1 ∞
µ )p−1 d p ξ δ(η − ∑ ξ i ) : α ∓ (ξ 1 ) · · · α ∓ (ξ p ): . (13.25)
Γ(2 ∓ iη − p) p!
p≥1
−∞
Quantum mechanically, the Fourier modes in (13.24) are creation and annihilation operators
for left- and right-moving particles. Let us consider the S-matrix element for one
incoming left-mover of energy ω to decay to m outgoing particles of energies ω = ∑ ω i :
S c (ω →
m∑
m∏
ω i ) = 〈0|α − (−ω) α + (ω j )|0〉 c , (13.26)
i=1
j=1
where the vacuum is defined by α + (−ω)|0〉 = 0 for ω > 0. From (13.25) we may read off
without further calculation the result:
Sc
CF (ω →
m∑
ω i ) = −i( 1 µ )m−1 ω
i=1
The corresponding Euclidean S-matrix is
m∏
k=1
ω k
Γ(−iω)
Γ(2 − m − iω) . (13.27)
µ |q| R m+1 (q 1 , . . . q m , q) = ( 1 µ )m−1 i m |q| ∏ |q i |( ∂
∂µ )m−2 µ |q|−1 , (13.28)
a formula we will obtain in the next chapter via continuum methods.
Other S-matrix amplitudes can be derived analogously [138]. The S-matrix is not
analytic in the energies ω i and does not satisfy crossing symmetry. In general we
must divide momentum space into kinematic regions. These are defined as follows. (It
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