14
2 Relativistic Point Particle
Fig. 2.1 Sketch of the
world-line of the first particle
(straight line) in the
background of the second
represented by circles in
analogy with to propagation
through a water wave
particle. Therefore, the correct generalization of the evolution kernel (2.11), linear
in ψ 2 , is obtained by replacing the right-hand side of (2.11) by
Δτ
ds
dc dφ f dφ i ¯
ψ 3 (φ f )
D[φ, b, c] e
iI [φ,b,c] δ(c(s)) ψ 2 (φ(s)) ψ 1 (φ i )
(2.15)
=
Δτ
ds
dc dφ f dφ i dφ ¯
ψ 3 (φ f ) K(τ f , s, φ f , φ) c(s) ψ 2 (φ)
× K(s, τ i , φ, φ i ) ψ 1 (φ i ) ,
where we used again that δ(c(s)) = c(s) in the second line and omitted the ghost c
from the arguments. The insertion of δ(c(s)) in the above expression is required to
ensure that the only reparametrizations that are gauge fixed are those which preserve
the interaction point. If we then strip off the free propagation which amount to letting
Δτ → 0 in the evolution kernel, this defines a product on V ,
∗ : V ⊗ V → V , ψ ⊗ ψ → cψ
2 ,
which is commutative, associative, and of degree one. It is not hard to see that (2.15)
can be reproduced upon adding a cubic term to (2.14), i.e.,
S[ψ] =
1
2
ψ, Q ψ +
1
3
ψ, ψ ∗ ψ .
(2.16)
The algebraic structure of the action (2.16) is that of a nilpotent abelian differential graded (dg) algebra (V , Q, ∗), together with a non-degenerate symplectic
form ω. The relations are
Q
2
= ∗
2
= Q∗ = ∗Q = 0 .
(2.17)
Finally, let us comment on the BV action corresponding to (2.16). Formally, this
action is invariant under ψ → ψ + Qλ where λ has degree −1. The idea of the
2 Relativistic Point Particle
Fig. 2.1 Sketch of the
world-line of the first particle
(straight line) in the
background of the second
represented by circles in
analogy with to propagation
through a water wave
particle. Therefore, the correct generalization of the evolution kernel (2.11), linear
in ψ 2 , is obtained by replacing the right-hand side of (2.11) by
Δτ
ds
dc dφ f dφ i ¯
ψ 3 (φ f )
D[φ, b, c] e
iI [φ,b,c] δ(c(s)) ψ 2 (φ(s)) ψ 1 (φ i )
(2.15)
=
Δτ
ds
dc dφ f dφ i dφ ¯
ψ 3 (φ f ) K(τ f , s, φ f , φ) c(s) ψ 2 (φ)
× K(s, τ i , φ, φ i ) ψ 1 (φ i ) ,
where we used again that δ(c(s)) = c(s) in the second line and omitted the ghost c
from the arguments. The insertion of δ(c(s)) in the above expression is required to
ensure that the only reparametrizations that are gauge fixed are those which preserve
the interaction point. If we then strip off the free propagation which amount to letting
Δτ → 0 in the evolution kernel, this defines a product on V ,
∗ : V ⊗ V → V , ψ ⊗ ψ → cψ
2 ,
which is commutative, associative, and of degree one. It is not hard to see that (2.15)
can be reproduced upon adding a cubic term to (2.14), i.e.,
S[ψ] =
1
2
ψ, Q ψ +
1
3
ψ, ψ ∗ ψ .
(2.16)
The algebraic structure of the action (2.16) is that of a nilpotent abelian differential graded (dg) algebra (V , Q, ∗), together with a non-degenerate symplectic
form ω. The relations are
Q
2
= ∗
2
= Q∗ = ∗Q = 0 .
(2.17)
Finally, let us comment on the BV action corresponding to (2.16). Formally, this
action is invariant under ψ → ψ + Qλ where λ has degree −1. The idea of the