170
6. Quantum Field Theory II: Interacting Scalar Fields
k
2
2
where in the last step we have used ω
2 =
+ m and written ‘i∈’ for ‘2i∈ω k ’
k
C
since what matters is just the sign of the small imaginary part (note that ω k is
defined as the positive square root). In this final form, the Lorentz invariance
of the scalar propagator is indeed manifest.
We shall have more to say about this propagator (Green function) in section 6.3.3. For the moment we simply note two points: first, it is the Fourier
transform of i/k
2
− m
2 + i∈, as stated in appendix G, where k
2 = k 0
2
− k
2 ;
C
and second, it is a function of the coordinate difference x 1 − x 2 , as it has to
be since we do not expect physics to depend on the choice of origin. This
second point gives us a clue as to how best to perform the x 1 − x 2 integral
in (6.91). Let us introduce the new variables x = x 1 − x 2 , X = (x 1 + x 2 )/2.
Then (problem 6.6) (6.91) reduces to
∫
∫ d
4 k
i
′
′
iq·x
−ik·x
(−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B ) d
4 x e
e
k 2 − 2
(2π) 4
m + i∈
C
(6.99)
′
′
= (−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B )
i
(6.100)
2
q 2 − m + i∈
C
′
′
where q = p A − p = p − p B is the 4-momentum transfer carried by the
B
A
exchanged C-quantum in figure 6.4, and we have used the four-dimensional
version of (E.26). We associate this single expression, which includes the
two coordinate space processes of figure 6.2, with the single momentum–space
Feynman diagram of figure 6.4. The arrows refer merely to the flow of 4momentum, which is conserved at each ‘vertex’ (i.e. meeting of three lines).
Thus although the arrow on the exchanged C-line is drawn as indicated, this
has nothing to do with any presumed order of emission/absorption of the
exchanged quantum. It cannot do so, after all, since in this diagram the states
all have definite 4-momentum and hence are totally delocalized in space–time;
equivalently, we recall from (6.91) that the amplitude in fact involves integrals
over all space–time.
A similar analysis (problem 6.7) shows that the contribution of the contractions (6.88) to the S-matrix element (6.74) is
′
′
(−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B )
i
(6.101)
2
(p A + p B ) 2 − m + i∈
C
which is represented by the momentum–space Feynman diagram of figure 6.5.
At this point we may start to write down the Feynman rules for the ABC
theory, which enable us to associate a precise mathematical expression for an
amplitude with a Feynman diagram such as figure 6.4 or figure 6.5. It is clear
′
′
that we will always have a factor (2π)
4 δ
4 (p A + p B − p − p ) for all ‘connected’
A
B
diagrams, following from the flow of the conserved 4-momentum through the
diagrams. It is conventional to extract this factor, and to define the invariant
amplitude M fi via
S fi = δ fi + i(2π)
4 δ
4 (p f − p i )M fi
(6.102)
6. Quantum Field Theory II: Interacting Scalar Fields
k
2
2
where in the last step we have used ω
2 =
+ m and written ‘i∈’ for ‘2i∈ω k ’
k
C
since what matters is just the sign of the small imaginary part (note that ω k is
defined as the positive square root). In this final form, the Lorentz invariance
of the scalar propagator is indeed manifest.
We shall have more to say about this propagator (Green function) in section 6.3.3. For the moment we simply note two points: first, it is the Fourier
transform of i/k
2
− m
2 + i∈, as stated in appendix G, where k
2 = k 0
2
− k
2 ;
C
and second, it is a function of the coordinate difference x 1 − x 2 , as it has to
be since we do not expect physics to depend on the choice of origin. This
second point gives us a clue as to how best to perform the x 1 − x 2 integral
in (6.91). Let us introduce the new variables x = x 1 − x 2 , X = (x 1 + x 2 )/2.
Then (problem 6.6) (6.91) reduces to
∫
∫ d
4 k
i
′
′
iq·x
−ik·x
(−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B ) d
4 x e
e
k 2 − 2
(2π) 4
m + i∈
C
(6.99)
′
′
= (−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B )
i
(6.100)
2
q 2 − m + i∈
C
′
′
where q = p A − p = p − p B is the 4-momentum transfer carried by the
B
A
exchanged C-quantum in figure 6.4, and we have used the four-dimensional
version of (E.26). We associate this single expression, which includes the
two coordinate space processes of figure 6.2, with the single momentum–space
Feynman diagram of figure 6.4. The arrows refer merely to the flow of 4momentum, which is conserved at each ‘vertex’ (i.e. meeting of three lines).
Thus although the arrow on the exchanged C-line is drawn as indicated, this
has nothing to do with any presumed order of emission/absorption of the
exchanged quantum. It cannot do so, after all, since in this diagram the states
all have definite 4-momentum and hence are totally delocalized in space–time;
equivalently, we recall from (6.91) that the amplitude in fact involves integrals
over all space–time.
A similar analysis (problem 6.7) shows that the contribution of the contractions (6.88) to the S-matrix element (6.74) is
′
′
(−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B )
i
(6.101)
2
(p A + p B ) 2 − m + i∈
C
which is represented by the momentum–space Feynman diagram of figure 6.5.
At this point we may start to write down the Feynman rules for the ABC
theory, which enable us to associate a precise mathematical expression for an
amplitude with a Feynman diagram such as figure 6.4 or figure 6.5. It is clear
′
′
that we will always have a factor (2π)
4 δ
4 (p A + p B − p − p ) for all ‘connected’
A
B
diagrams, following from the flow of the conserved 4-momentum through the
diagrams. It is conventional to extract this factor, and to define the invariant
amplitude M fi via
S fi = δ fi + i(2π)
4 δ
4 (p f − p i )M fi
(6.102)