8.4
Problems
263
Problems
8.1 Consider a matrix element of the form
∫
∫
+ip f ·x ∂ μ A
μ −ipi ·x
M = d
3
x dt e
e
.
Assuming the integration is over all space–time and that
A
0
→ 0
as t → ±∞
and
|A| → 0
as |x| → ∞
use integration by parts to show
∫
∫
+ip f ·x ∂ 0 A
0 −ipi ·x
+ip f ·x A
0 −ipi ·x
(a)
dt e
e
= (−ip f0 ) dt e
e
∫
( ∫
)
+ip f ·x
∇ · Ae
−ipi ·x
+ip f ·x
Ae
−ipi ·x
(b)
d
3
x e
= +ip f ·
d
3
x e
.
Hence show that
∫
∫
d
3
+ip f ·x (∂ μ A
μ
x dt e
+ A
μ ∂ μ )e
−ipi ·x
∫
∫
d
3
+ip f ·x A
μ −ipi ·x
= −i(p f + p i ) μ
x dt e
e
.
8.2 Verify equation (8.27).
8.3 Evaluate (8.31) and interpret the result physically (i.e. compare it with
(8.27)).
(a) Using the u-spinors normalized as in (3.73), the φ
1,2 of (8.47), and
the result for σ · Aσ · B from problem 3.4(b), show that
(
)
k
′
iφ
1†
σ · k
′
· k
× kφ
1
′ ′
u
† (k , s = 1)u(k, s = 1) = (E+m) 1 +
+
.
(E + m) 2
(E + m) 2
(b) For any vector A = (A
1 , A
2 , A
3 ), show that φ
1†
σ · Aφ
1 = A
3 . Find
similar expressions for φ
1†
σ · Aφ
2 , φ
2†
σ · Aφ
1 , φ
2†
σ · Aφ
2 .
(c) Show that the S of (8.46) is equal to
( [
] 2
)
k
′
· k
(k
′
× k)
2
S = (E + m)
2
1 +
+
.
(E + m) 2
(E + m) 4