1) A scalar particle travels from one spacetime region to another, carrying momentum k and scattering into momentum k'. This scattering process is described by a scattering matrix.
2) The scattering matrix involves second order derivatives of the vacuum-to-vacuum matrix element with respect to sources J(x) and J(x'). This vacuum-to-vacuum matrix has a factored exponential form.
3) The left-hand side of the scattering matrix gives the probability that a one-particle state with momentum k at initial time Tin will be found with momentum k' at later time Tout, and can be evaluated via path integration.
5. β«
π3
π₯β²
β(2π)3
πβππβ β²βπ₯β²
β«
π3
π₯
β(2π)3
π ππβ βπ₯
πΊ( π₯β² β π₯)
= β
1
(2π)2
β« ππ(2)
0
β« π3
πβ
(2) π
ππ(2)
0 ( π₯β² 0
β π₯0)
βπ(2)
2
(πβ (2)) + π2
+ ππ
(2π)3
Γ πΏ3
(πβ β² + πβ (2))
π₯β²
πΏ3
(πβ + πβ (2))
π₯
where it is specified that to the space region π₯ β² a vertex with the Dirac-delta function
(12.3)
πΏ3
(πβ β² + πβ (2))
π₯β²
and to the space region π₯ a vertex with its own Dirac-delta function
(12.4)
πΏ3
(πβ + πβ (2))
π₯
Clearly in (12.2) we initially see two vertices, one as already mentioned at the space region of π₯,
where we sum up two space momenta πβ and πβ (2) and the vertex at the space region of π₯ β² where
πβ β² and πβ (2) are summed up. Since it is indicated that we are to perform integration ove the πβ (2)
vec momentum variable, the vertex at the initial space region will disappear as there will be
picking of πβ (2) = β πβ . Given such picking, we have
(12.5)
π(2)
2
( πββ (2)) β π(2)
2
(βπββ ) = ( π(2)
0 )
2
β πββ β πββ
and then just relabel π(2)
0
to π0
so that
(12.6)
π(2)
2
(βπββ ) = π π π π = ( π
0
)
2
β πββ β πββ
Thus, (12.2) further results to
(12.7)
β«
π3
π₯β²
β(2π)3
πβππβ β²βπ₯β²
β«
π3
π₯
β(2π)3
π ππβ βπ₯
πΊ π₯β²
π₯
=
1
2(2π)4
β« π π0
π
ππ0 ( π₯β² 0
β π₯0)
( π0)2 β ( πβ β πβ + π2
+ ππ)
πΏ3
(πβ β² β πβ )
π₯β²
9. πΏ3
(πβ β²
β πβ β² (1))
π₯β² = β«
π3
π₯β²
(2π)3
πβ π( πβ β²
β πβ β²
(1)) β π₯β²
As a whole what does this 3rd term signify? Taking (13.5) and (13.7) altogether, there are two
separate propagations of the scalar field starting at two different initial spacetime points and
ending up to scatter at two different spatial points. The scalar field propagating in (13.5) starts at
the spacetime point y, then propagates towards the spatial point of π₯, carrying the spatial
momentum πβ (1) and then it scatters at this point carrying the spatial momentum πβ . For this said
scalar field, the spatial point of π₯ is where the scattering vertex is. At this scattering vertex
spatial momenta πβ and πβ (1) are summed up to zero and in turn implies a picking πβ (1) = βπβ
over the integration variable πβ (1). Meanwhile, the scalar field propagating in (13.7) starts at the
spacetime point yβ, then propagates towards the spatial point of π₯β²
, carrying the spatial
momentum πβ β²
(1) and then it scatters at this point carrying the spatial momentum πβ β²
. For this said
scalar field, the spatial point of π₯β²
is where the scattering vertex is. At this scattering vertex
spatial momenta πβ β²
and πβ β²
(1) are summed up to zero and in turn implies a picking πβ β²
(1) = πβ β²
over the integration variable πβ β²
(1). So in view of the 3rd term, there are two different scattering
vertices (processes), one at the spatial point of π₯ and the other one at π₯β²
and that the scatterings at
these vertices may not be simultaneous, one may happen earlier than the other. However, these
scatterings depend on the presence of their corresponding sources at two different initial
spacetime points.
References
[1]Baal, P., A COURSE IN FIELD THEORY
[2]Cardy, J., Introduction to Quantum Field Theory
[3]Gaberdiel, M., Gehrmann-De Ridder, A., Quantum Field Theory