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8. Elementary Processes in Scalar and Spinor Electrodynamics
FIGURE 8.1
Coulomb scattering of s
+ .
In natural units (see appendices B and C) this has the value
α = e
2 /4π ≈
1
(8.3)
137
for the elementary charge e. α is called the fine structure constant. The smallness of α is the reason why a perturbation approach has been very successful
for QED.
2 A
2
To lowest order in α we can neglect the e
term and the perturbing
potential is then
V ˆ = ie(∂ μ A
μ + A
μ ∂ μ ).
(8.4)
For a scattering process we shall assume
1 the same formula for the transition
amplitude as in non-relativistic quantum mechanics (NRQM) time-dependent
perturbation theory (see appendix A, equations (A.23) and (A.24)):
∫
′ ∗ ˆ
A s + = −i d
4 x φ V φ
(8.5)
′
where φ and φ are the initial and final state free-particle solutions. The latter
are (recall equation (3.11))
−ip·x
φ = N e
(8.6)
′
N
′ −ip ·x
φ =
e
'
(8.7)
and we shall fix the normalization factors later. Inserting the expression for
ˆ
V into (8.5), and doing some integration by parts (problem 8.1), we obtain
∫
′ ∗
′ ∗
A s + = −i d
4 x {ie[φ (∂ μ φ) − (∂ μ φ )φ]}A
μ .
(8.8)
The expression inside the braces is very reminiscent of the probability current
expression (3.20). Indeed we can write (8.8) as
∫
μ
A s + = −i d
4 x j
(x)A μ (x)
(8.9)
em,s +
1 Justification may be found in chapter 9 of Bjorken and Drell (1964).