130_notes.dvi

The matrix element is to be taken with the initialelectron at rest, ̄hk << mc, the final electron
(approximately) at rest, and hence the intermediate electron at rest, due to a delta function of
momentum conservation that comes out of the spatial integral.

Let the positive energy spinors be written as

u( 0 r)=

(

χ(r)

0

)

and the “negative energy” spinors as

u(r

′′)

0 =

(

0

χ(r

′′)

)

The matrixγ 4 γiconnect the positive and “negative energy” spinors so that the amplitude can be
written in terms of two component spinors and Pauli matrices.

γ 4 γi =

(

0 −iσi

−iσi 0

)

u(r

′′)†

0 iγ^4 γiu

(r)

0 =

(

0 ,χ(r

′′)†)

(

0 σi

σi 0

)(

χ(r)

0

)

=χ(r

′′)†

σiχ(r)

c(2)p~′,r′;k~′ǫˆ′(t) =

ie^2

4 mV

√

ω′ω

∑

r′′=3, 4

[〈 0 r′′|iγ 4 γnǫ′n| 0 r〉〈 0 r′|iγ 4 γnǫn| 0 r′′〉+〈 0 r′′|iγ 4 γnǫn| 0 r〉〈 0 r′|iγ 4 γnǫ′n| 0 r′′〉]

∫t

0

dt 2 ei(E

′−E+ ̄hω′− ̄hω)t 2 / ̄h

c(2)p~′,r′;k~′ǫˆ′(t) =

ie^2

4 mV

√

ω′ω

∑

r′′=3, 4

[

(χ(r

′′)†

~σ·ǫˆ′χ(r))(χ(r

′)†

~σ·ǫχˆ(r

′′)

) + (χ(r

′′)†

~σ·ˆǫχ(r))(χ(r

′)†

~σ·ǫˆ′χ(r

′′)

)

]

∫t

0

dt 2 ei(E

′−E+ ̄hω′− ̄hω)t 2 / ̄h

c

(2)

p~′,r′;k~′ǫˆ′(t) =

ie^2

4 mV

√

ω′ω

∑

r′′=3, 4

[

(χ(r

′)†

~σ·ǫχˆ(r

′′)

)(χ(r

′′)†

~σ·ˆǫ′χ(r)) + (χ(r

′)†

~σ·ǫˆ′χ(r

′′)

)(χ(r

′′)†

~σ·ˆǫχ(r))

]

∫t

0

dt 2 ei(E

′−E+ ̄hω′− ̄hω)t 2 / ̄h

c(2)~

p′,r′;k~′ǫˆ′

(t) =

ie^2

4 mV

√

ω′ω

χ(r

′)†[

(~σ·ǫˆ)(~σ·ǫˆ′) + (~σ·ˆǫ′)(~σ·ˆǫ)

]

χ(r)

∫t

0

dt 2 ei(E

′−E+ ̄hω′−hω ̄)t 2 / ̄h

c(2)p~′,r′;k~′ǫˆ′(t) =

ie^2

4 mV

√

ω′ω

χ(r

′)†

[σiσj+σjσi] ˆǫiǫˆ′jχ(r)

∫t

0

dt 2 ei(E

′−E+ ̄hω′− ̄hω)t 2 / ̄h

c

(2)

p~′,r′;k~′ǫˆ′(t) =

ie^2

2 mV

√

ω′ω

χ(r

′)†

ˆǫ·ǫˆ′χ(r)

∫t

0

dt 2 ei(E

′−E+ ̄hω′− ̄hω)t 2 / ̄h

c(2)p~′,r′;k~′ǫˆ′(t) =

ie^2

2 mV

√

ω′ω

ˆǫ·ǫˆ′δrr′

∫t

0

dt 2 ei(E

′−E+ ̄hω′− ̄hω)t 2 / ̄h