225
8.1. Coulomb scattering of charged spin-0 particles
Note that the field A μ is not quantized: it is being treated as an ‘external’
classical potential. The expansion for the field φ ˆ is given in (7.16). As in
(6.48), the lowest-order amplitude is
∫
+
′
+
ˆ
= −i ′
| d
4 x H (x)|s , p>
(8.24)
A s +
s
where (cf (6.49))
√
+
|s , p> = 2Ea ˆ
† (p)|0>.
(8.25)
We are, of course, anticipating in our notation that (8.24) will indeed be the
same as (8.12). The required amplitude is then
∫
+
+
A s + = −i d
4 x ′
| ˆ j
μ
(x)|s , p>A μ (x).
(8.26)
em,s
Using the expansion (7.16), the definition (8.25) and the vacuum conditions
(7.30), and following the method of section 6.3.1, it is a good exercise to check
that the value of the matrix element in (8.26) is (problem 8.2)
+
+
−i(p−p )·x
′
| ˆ j
μ
(x)|s , p> = e(p + p
′ )
μ e
'
.
(8.27)
em,s
This is exactly the same as the expression we obtained in (8.11) for the wave
mechanical transition current in this case, using the normalization N = N
′ =
1, which is consistent with the field-theoretic normalization in (8.25). Thus
our wave mechanical transition current is indeed the matrix element of the
field-theoretical electromagnetic current operator :
μ
+
+
j
(x) = ′
| ˆ j
μ
(x)|s , p>.
(8.28)
em,s +
em,s
Combining all these results, we have therefore connected the ‘wavefunction’
amplitude and the ‘field-theory’ amplitude via
∫
μ
A s + = −i d
4 x j
(x)A μ (x)
em,s +
∫
+
+
d
4
′
| ˆ j
μ
= −i
x A μ (x).
(8.29)
We note that because of the static nature of the potential, and the noncovariant choice of A
μ (only A
0 = 0), our answer in either case cannot be
/
expected to yield a Lorentz invariant amplitude.
8.1.3 Coulomb scattering of s −
The physical process is (figure 8.2(a))
s
− (p) → s
− (p
′ )
(8.30)