2.2 Scattering Matrix and Minimal Model
19
Remark 2.7 It turns out that this procedure of adding stubs to the vertices has a
familiar physical interpretation in terms of a tree-level Wilsonian effective action.
Indeed, adding an infinitesimal stub of length to the cubic vertex is the same
as inserting a momentum cut-off in the propagators since the values of p 2 =
g −1 (p, p) > >
1
are exponentially suppressed. Thus, the contribution from
large momenta to the scattering amplitudes are contained in the higher vertices
{l
n , n = 3, 4, . . .}. At order the effective action so obtained contains a four-vertex.
At order 2 a quartic as well as a quintic vertex is produced. We can repeat this
procedure until all momenta of the internal propagators are integrated out. The
resulting effective action has no propagators left and therefore has the physical
interpretation of a generating function for scattering amplitudes. We denote the
corresponding maps by {s n }.
The algebraic counterpart of this procedure, the minimal model theorem, which
for the model at hand states that, given a dg-algebra (V , ∗, Q), there exists a quasiisomorphism 2 F from (H, {s n }) to (V , ∗, Q), which induces an isomorphism on the
physical subspace contained in the cohomology H . Furthermore the maps {s n } are
just the matrix elements of the scattering matrix of physical states.
There is a simple and intuitive procedure to obtain the minimal model map by
constructing a perturbative solution to the equation of motion derived from the
action (2.16). To begin with we split the field ψ into a “physical” state P ψ = ψ P ∈
H and its complement ψ U + ψ T in V U ⊕ V T . The subspace V U corresponding to
the projection map P U = −Q −1 Q represents the nonphysical states, i.e., the states
not annihilated by Q, and the subspace V T represents the space of trivial states, i.e.,
Q-exact states. We can then consistently set ψ T = 0 and solve for ψ U starting from
the equation of motion for ψ. Since Qψ P = 0 by construction, we have
Qψ
U
= −ψ ∗ ψ .
(2.31)
Acting on this equation with Q −1 we get
ψ
U
= −Q
−1
(ψ
P
+ ψ
U ) ∗ (ψ
P
+ ψ
U )
(2.32)
= −Q
−1
ψ
P
∗ ψ
P
+ Q
−1
Q
−1
ψ
P
∗ ψ
P
∗ ψ
P
+Q
−1
ψ
P
∗ Q
−1
ψ
P
∗ ψ
P
,
up to terms of order four and higher. This gives a perturbative expression for ψ U
in terms of ψ P which satisfies (2.31) up to an element in the kernel of Q −1 .
Substituting the right-hand side of (2.32) into (2.31) and using (2.26) as well as
Q 2 = 0 we find
P
(ψ
P
+ ψ
U ) ∗ (ψ
P
+ ψ
U )
= 0 ,
(2.33)
2 Cf. Remark 2.10 in Sect. 2.3 concerning the use of the term minimal model in the world of cyclic
homotopy algebras.
19
Remark 2.7 It turns out that this procedure of adding stubs to the vertices has a
familiar physical interpretation in terms of a tree-level Wilsonian effective action.
Indeed, adding an infinitesimal stub of length to the cubic vertex is the same
as inserting a momentum cut-off in the propagators since the values of p 2 =
g −1 (p, p) > >
1
are exponentially suppressed. Thus, the contribution from
large momenta to the scattering amplitudes are contained in the higher vertices
{l
n , n = 3, 4, . . .}. At order the effective action so obtained contains a four-vertex.
At order 2 a quartic as well as a quintic vertex is produced. We can repeat this
procedure until all momenta of the internal propagators are integrated out. The
resulting effective action has no propagators left and therefore has the physical
interpretation of a generating function for scattering amplitudes. We denote the
corresponding maps by {s n }.
The algebraic counterpart of this procedure, the minimal model theorem, which
for the model at hand states that, given a dg-algebra (V , ∗, Q), there exists a quasiisomorphism 2 F from (H, {s n }) to (V , ∗, Q), which induces an isomorphism on the
physical subspace contained in the cohomology H . Furthermore the maps {s n } are
just the matrix elements of the scattering matrix of physical states.
There is a simple and intuitive procedure to obtain the minimal model map by
constructing a perturbative solution to the equation of motion derived from the
action (2.16). To begin with we split the field ψ into a “physical” state P ψ = ψ P ∈
H and its complement ψ U + ψ T in V U ⊕ V T . The subspace V U corresponding to
the projection map P U = −Q −1 Q represents the nonphysical states, i.e., the states
not annihilated by Q, and the subspace V T represents the space of trivial states, i.e.,
Q-exact states. We can then consistently set ψ T = 0 and solve for ψ U starting from
the equation of motion for ψ. Since Qψ P = 0 by construction, we have
Qψ
U
= −ψ ∗ ψ .
(2.31)
Acting on this equation with Q −1 we get
ψ
U
= −Q
−1
(ψ
P
+ ψ
U ) ∗ (ψ
P
+ ψ
U )
(2.32)
= −Q
−1
ψ
P
∗ ψ
P
+ Q
−1
Q
−1
ψ
P
∗ ψ
P
∗ ψ
P
+Q
−1
ψ
P
∗ Q
−1
ψ
P
∗ ψ
P
,
up to terms of order four and higher. This gives a perturbative expression for ψ U
in terms of ψ P which satisfies (2.31) up to an element in the kernel of Q −1 .
Substituting the right-hand side of (2.32) into (2.31) and using (2.26) as well as
Q 2 = 0 we find
P
(ψ
P
+ ψ
U ) ∗ (ψ
P
+ ψ
U )
= 0 ,
(2.33)
2 Cf. Remark 2.10 in Sect. 2.3 concerning the use of the term minimal model in the world of cyclic
homotopy algebras.