18
2 Relativistic Point Particle
such labeled trees has no boundary since any two trees in Fig. 2.2 smoothly cross
into each other when the affine parameter τ approaches 0. On the other hand, this
decomposition is not unique. For instance, we could require that the affine parameter
has to be larger than some value , say. This is equivalent to adding a piece of
propagator of length to each leg of the cubic vertex. This new cubic vertex defines
a product l
2 : V ⊗ V → V , related to ∗ through
l
2 (a, b) = e
H
(e
−H a) ∗ (e
−H b)
.
(2.28)
Note though that the composition of two binary products l
2 (l
2 (a, b), d) still
vanishes trivially due to the nilpotency of the ghost c. However, it is already clear
from Fig. 2.2 that we need to introduce a four-vertex in order to reproduce the correct
4-point amplitude. Indeed, the zero-length propagator s-, t-, and u-channel trees do
not touch anymore and we therefore need to add a quartic vertex
f
4 (ψ 4 , ψ 3 , ψ 2 , ψ 1 ) = =l
2 (ψ 4 , ψ 3 ), b e
−2 l
2 (ψ 1 , ψ 2 ) + (perm.) .
Using again the identity
ω(l
2 (a, b), c) = (−1)
|a|+|b|+1
2 (a, b), c
= (−1)
|a|+1
a, l
2 (b, c) = −ω(a, l
2 (b, c)) ,
we can rewrite the latter formula as
f 4 (ψ 4 , ψ 3 , ψ 2 , ψ 1 ) = ω(ψ 4 , l
2 (ψ 3 , b 0 e
H l
2 (ψ 2 , ψ 1 )))
(2.29)
+ω(ψ 4 , l
2 (ψ 1 , b 0 e
H l
2 (ψ 2 , ψ 3 )))
+ω(ψ 4 , l
2 (ψ 2 , b 0 e
H l
2 (ψ 3 , ψ 1 )))
≡ ω(ψ 4 , l 3 (ψ 3 , ψ 2 , ψ 1 )) .
The triple product l 3 is trivially Q-closed,
[Q, l 3 ] ≡ Q ◦ l 3 + l 3 ◦ (Q ⊗ 1 ⊗ 1) + l 3 ◦ (1 ⊗ Q ⊗ 1) + l 3 ◦ (1 ⊗ 1 ⊗ Q) = 0 ,
(2.30)
again due to the nilpotency of c. As a consequence, the BV action for three-vertices
with stubs is obtained simply by adding the quartic vertex to (2.16).
Of course, once a four-vertex is introduced this will imply a five-vertex upon
substitution into a tree with 5 legs and so forth so that eventually we will end up
with an infinite set of vertices, or maps {l
n }, with relations of the form given in
(2.30).
2 Relativistic Point Particle
such labeled trees has no boundary since any two trees in Fig. 2.2 smoothly cross
into each other when the affine parameter τ approaches 0. On the other hand, this
decomposition is not unique. For instance, we could require that the affine parameter
has to be larger than some value , say. This is equivalent to adding a piece of
propagator of length to each leg of the cubic vertex. This new cubic vertex defines
a product l
2 : V ⊗ V → V , related to ∗ through
l
2 (a, b) = e
H
(e
−H a) ∗ (e
−H b)
.
(2.28)
Note though that the composition of two binary products l
2 (l
2 (a, b), d) still
vanishes trivially due to the nilpotency of the ghost c. However, it is already clear
from Fig. 2.2 that we need to introduce a four-vertex in order to reproduce the correct
4-point amplitude. Indeed, the zero-length propagator s-, t-, and u-channel trees do
not touch anymore and we therefore need to add a quartic vertex
f
4 (ψ 4 , ψ 3 , ψ 2 , ψ 1 ) = =l
2 (ψ 4 , ψ 3 ), b e
−2 l
2 (ψ 1 , ψ 2 ) + (perm.) .
Using again the identity
ω(l
2 (a, b), c) = (−1)
|a|+|b|+1
2 (a, b), c
= (−1)
|a|+1
a, l
2 (b, c) = −ω(a, l
2 (b, c)) ,
we can rewrite the latter formula as
f 4 (ψ 4 , ψ 3 , ψ 2 , ψ 1 ) = ω(ψ 4 , l
2 (ψ 3 , b 0 e
H l
2 (ψ 2 , ψ 1 )))
(2.29)
+ω(ψ 4 , l
2 (ψ 1 , b 0 e
H l
2 (ψ 2 , ψ 3 )))
+ω(ψ 4 , l
2 (ψ 2 , b 0 e
H l
2 (ψ 3 , ψ 1 )))
≡ ω(ψ 4 , l 3 (ψ 3 , ψ 2 , ψ 1 )) .
The triple product l 3 is trivially Q-closed,
[Q, l 3 ] ≡ Q ◦ l 3 + l 3 ◦ (Q ⊗ 1 ⊗ 1) + l 3 ◦ (1 ⊗ Q ⊗ 1) + l 3 ◦ (1 ⊗ 1 ⊗ Q) = 0 ,
(2.30)
again due to the nilpotency of c. As a consequence, the BV action for three-vertices
with stubs is obtained simply by adding the quartic vertex to (2.16).
Of course, once a four-vertex is introduced this will imply a five-vertex upon
substitution into a tree with 5 legs and so forth so that eventually we will end up
with an infinite set of vertices, or maps {l
n }, with relations of the form given in
(2.30).