20
2 Relativistic Point Particle
which upon expanding ψ U in terms of ψ P using (2.32) produces an obstruction for
ψ P to give a solution to (2.32). This obstruction is of the form
k≥2
s k (ψ
P , . . . , ψ
P ) = 0 ,
(2.34)
which is a Maurer–Cartan equation for ψ P . On the other hand, upon inspecting the
first few terms in (2.33), that is,
s 2 (ψ
P , ψ
P ) = P (ψ
P
∗ ψ
P ) ,
s 3 (ψ
P , ψ
P , ψ
P ) = −P
Q
−1
ψ
P
∗ ψ
P
∗ ψ
P
+
ψ
P
∗ Q
−1
ψ
P
∗ ψ
P
,
and recalling that Q −1 is the propagator, we see that
P , s 2 (ψ
P , ψ
P )
is the scattering matrix for a three-particle scattering (which usually vanishes for the
kinematical reasons), while
P , s 3 (ψ
P , ψ
P , ψ
P )
is the scattering matrix for a four-particle scattering. It is not hard to see that this
identification holds for an arbitrary power of ψ P . To summarize, the maps s n which
are defined on the cohomology H can be interpreted as the scattering matrices for
an n+1 particle scatterings. Equation (2.32) 3 induces a nonlinear map F : H → V .
Explicitly, let SV be the symmetric algebra
SV =
∞
n=0
(V
⊗n ) sym ,
where V ⊗0 = C and similarly, SH the symmetric algebra on H . Then we can write
ψ = F (ψ
P ) = ψ
P
+ ψ
U
= π 1 (1 − Q
−1
∗)
−1 (e
ψ P − 1)
= ψ
P
− Q
−1 (ψ
P
∗ ψ
P ) + Q
−1 (Q
−1 (ψ
P
∗ ψ
P ) ∗ ψ
P ) + · · · ,
where π 1 : SV → V denotes the projection on one output. Similarly, we have
s n = P π 1 ∗ (1 − Q
−1
∗)
−1 i n ,
(2.35)
3 Here we are somewhat cavalier about the proper notion of embedding of ψ P in V .