Positron annihilation in a strong magentic field.

The Dirac equation <strong>in</strong> a magnetic field
µ ( ) ν
⎡
⎣−γ ( i∂ µ
+ qAµ
) − m ⎤
0 ⎦ψ ± ( x ) = 0,
u
↑
n
ψ
E + m
( x) exp( iE t) u ( x)
( + )
( + ) n 0
( + ) ↑↓
= −
( + )
n n
2En
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎞
n−1
⎟
⎟
⎟
0 ⎟
⎟
p
⎟
( + ) ⎟
I
( + ) n−1
⎟
n
+ m ⎟
0 ⎟
⎟
2nb m
⎟
0 ( )
( ) I
+ ⎟
+ n ⎟
En
+ m
0
⎟
⎠
⎜
( x)
= ⎜
,
⎜ E
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−
I
= c = 1
µ
A ( x) = (0, A) = (0,0, xB,0)
( We adopted <strong>in</strong> many equations that follow)
Solution for electrons
u
0
⎛
⎞
⎜
⎟
⎜
⎟
⎜ ( + )
I
⎟
⎜ n ⎟
⎜
⎟
⎜
⎟
↓ ⎜ 2nb m0
( )
n
( x)
( ) I
+ ⎟
= ⎜−
+ n−1
⎟
⎜ En
+ m ⎟
⎜
0 ⎟
⎜
⎟
⎜ p
⎟
⎜−
I
( + ) ⎟
⎜ ( + ) n ⎟
⎜ En
+ m
0
⎟
⎝
⎠
n
2
i ⎡ b( x−a)
⎤ ⎛ x−a⎞
⎛ iayb⎞
In
= exp ⎢− H exp exp( )
1/4
2 ⎥ n
ipz
n
⎜ ⎟ ⎜−
2 ⎟
L( πλ / b) 2 n!
⎣ 2λC
⎦ ⎝ λ ⎠ ⎝ λC
⎠
C
Johonsn and Lippmann, PR, 76 (1949) 828,
Bhattacharya, arXiv:0705.4275v2 [hep-th]
7/37