264
8. Elementary Processes in Scalar and Spinor Electrodynamics
(d) Using cos θ = k · k
′ /(|k||k
′
|), |k| = |k
′
| and v = |k|/E, show that
S = (2E)
2 (1 − v
2 sin
2 θ/2).
8.5 Verify equation (8.55).
8.6 Check that γ
0 γ
μ† γ
0 = γ
μ .
8.7 Verify equation (8.79) for the lepton tensor L
μν .
8.8 Evaluate L
00 as in equation (8.80).
8.9 Verify equation (8.87).
+
→
− +
8.10 Verify equation (8.96) for the e
− s
e s amplitude to O(e
2 ).
8.11 Check that both the scalar and the spinor current matrix elements (8.27)
and (8.55), satisfy ∂ μ j
μ (x) = 0.
8.12 Verify equation (8.120).
8.13 Verify equation (8.136) for the Fourier transform of ρ(x) given by (8.135).
2
Show that the mean square radius of the distribution (8.135) is 12a .
8.14 Check the gauge invariance of M γe − given by (8.162), by showing that
if ∈ μ is replaced by k μ , or ∈
∗ by k ν
′ , the result is zero.
ν
8.15
(a) The spin-averaged squared amplitude for lowest-order electron Compton scattering contains the interference term
∑
(s)
(u)∗
M γe − M γe −
λ,λ ' ,s,s '
where (s) and (u) refer to the s- and u-channel processes of figure 8.14(a) and (b) respectively. Obtain an expression analogous
to (8.172) for this term, and prove that it is, in fact, zero. [Hint :
work in the massless limit, and use relations (J.4) and (J.5).]
(b) Explain why the term
∑
(u)
(u)∗
M M
γe −
γe −
λ,λ ' ,s,s '
is given by (8.177) with s and u interchanged.
8.16 Recalculate the interference term of problem 8.16(a) for the case k
2 =
2
′ 2
−Q
2 (but with k
′ 2 = p = p = 0), and hence verify (8.181).