2.2 Scattering Matrix and Minimal Model
17
The first and the last insertions of the evolution kernel K correspond to the free
propagation of the initial and final states. The standard procedure is to “amputate
these external lines” in order to isolate the scattering process. The amplitude (2.23)
then reduces to
∞
0
ds dφ 2 dφ 1
dc ( ¯
ψ 4 ∗ ψ 3 )(φ 2 ) K(s, φ 2 , φ 1 )
dc (ψ 2 ∗ ψ 1 )(φ 1 ) ,
(2.24)
where s = s 2 −s 1 . Here we used (2.12) as well as (2.15) and set τ i = 0 and τ f = ∞.
Remark 2.5 More precisely, the s-integral in (2.24) should be understood as
follows. Starting from (2.23) one should consider τ f of the form τ f = ae iθ with
θ ∈ (0,
π
2 ] and preform the limit a → ∞ in order to get (2.24). This will improve
the convergence of the resulting s-integral for an open subset of possible momenta
of the external states. The integrand is analytic in the first quadrant so that the result
does not depend on the value of θ . To continue, we take θ =
π
2 , which is equivalent
to the substitution τ → iτ , i.e., Wick rotation to imaginary time.
Returning to the canonical formalism, now with imaginary proper-time evolution
operator e −H τ , the expression (2.24) can be written equivalently as
4 ∗ ψ 3 , Q
−1 (ψ 1 ∗ ψ 2 ) = =ψ 4 , ψ 3 ∗ Q
−1 (ψ 1 ∗ ψ 2 ) ,
(2.25)
where we used the cyclicity ( ∗ b, c = (−1) |b| a, b ∗ c) and where Q −1 is the
propagator, the homotopy inverse of Q on the complement to H , |b| is the degree
of b. More precisely, let P be the projector to H , the cohomology of Q in V . Then
Q
−1 Q + QQ
−1
= 1 − P ,
Q
−1
=
b
H
(1 − P ) .
(2.26)
Note that due to (2.8) the action of the operator b is identified with
dc =
∂
∂c .
Remark 2.6 More precisely, in order to define Q −1 we first embed H in ker(Q)
ker(Q) = i(H ) ⊕ V T ,
(2.27)
where V T = Im(Q) in V . If V U is the linear complement to ker(Q) in V such that
ω| H and ω| V T ⊕V U are symplectic and ω| V T = ω| V U ≡ 0, then Q is an isomorphism
V U ∼ = V T . The homotopy inverse Q −1 is then defined as the inverse of Q on V T and
extended to be zero everywhere else.
The decomposition of any trivalent graph follows the same pattern. Thus, the pair
Q −1 and ∗ together with the symplectic form are sufficient to describe the tree-level
scattering of any number of spinless particles. We note that the moduli space of
17
The first and the last insertions of the evolution kernel K correspond to the free
propagation of the initial and final states. The standard procedure is to “amputate
these external lines” in order to isolate the scattering process. The amplitude (2.23)
then reduces to
∞
0
ds dφ 2 dφ 1
dc ( ¯
ψ 4 ∗ ψ 3 )(φ 2 ) K(s, φ 2 , φ 1 )
dc (ψ 2 ∗ ψ 1 )(φ 1 ) ,
(2.24)
where s = s 2 −s 1 . Here we used (2.12) as well as (2.15) and set τ i = 0 and τ f = ∞.
Remark 2.5 More precisely, the s-integral in (2.24) should be understood as
follows. Starting from (2.23) one should consider τ f of the form τ f = ae iθ with
θ ∈ (0,
π
2 ] and preform the limit a → ∞ in order to get (2.24). This will improve
the convergence of the resulting s-integral for an open subset of possible momenta
of the external states. The integrand is analytic in the first quadrant so that the result
does not depend on the value of θ . To continue, we take θ =
π
2 , which is equivalent
to the substitution τ → iτ , i.e., Wick rotation to imaginary time.
Returning to the canonical formalism, now with imaginary proper-time evolution
operator e −H τ , the expression (2.24) can be written equivalently as
4 ∗ ψ 3 , Q
−1 (ψ 1 ∗ ψ 2 ) = =ψ 4 , ψ 3 ∗ Q
−1 (ψ 1 ∗ ψ 2 ) ,
(2.25)
where we used the cyclicity ( ∗ b, c = (−1) |b| a, b ∗ c) and where Q −1 is the
propagator, the homotopy inverse of Q on the complement to H , |b| is the degree
of b. More precisely, let P be the projector to H , the cohomology of Q in V . Then
Q
−1 Q + QQ
−1
= 1 − P ,
Q
−1
=
b
H
(1 − P ) .
(2.26)
Note that due to (2.8) the action of the operator b is identified with
dc =
∂
∂c .
Remark 2.6 More precisely, in order to define Q −1 we first embed H in ker(Q)
ker(Q) = i(H ) ⊕ V T ,
(2.27)
where V T = Im(Q) in V . If V U is the linear complement to ker(Q) in V such that
ω| H and ω| V T ⊕V U are symplectic and ω| V T = ω| V U ≡ 0, then Q is an isomorphism
V U ∼ = V T . The homotopy inverse Q −1 is then defined as the inverse of Q on V T and
extended to be zero everywhere else.
The decomposition of any trivalent graph follows the same pattern. Thus, the pair
Q −1 and ∗ together with the symplectic form are sufficient to describe the tree-level
scattering of any number of spinless particles. We note that the moduli space of