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
= ( π0
)
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)
β« π π0
π π π0( 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 β²β 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.
Basic derivations
In this later portion of the draft let us attempt to dig into the basic details and derivations that
lead us to the form of the scattering matrix (1).
We start with the field operator given for the scalar field (spin 0 boson)
(14.1)
πΜ( π₯ ) =
1
βπΏ3
β
ββ
β2π(πβ )
(π ππβ β π₯
π( πβ ) + πβ ππβ β π₯
πβ
( πβ ))
πβ
In here we shall be reminded that spatial momentum vector π is related to the wave number
vector πβ via de Broglieβs hypothesis, π = βπβ , where |πβ | = 2π/π. To make things momentarily
convenient, we resort to the Heaviside units (π = β = 1 ) so that π = πβ and reinsert the
appropriate values of the constants involved whenever required at the end of calculations
although I still retain β in some expressions like in the above. (So for loose convenience we refer
to πβ as spatial momentum.) Also we must take note that in my own personal convenient notation
I usually write the equivalence
(14.2)