130_notes.dvi

|c(2)p~′,r′;k~′ǫˆ′(t)|^2 =

(

e^2

2 mV

) 2

1

ω′ω

|ǫˆ·ǫˆ′|^2 δrr′ 2 πtδ(E′/ ̄h−E/ ̄h+ω′−ω)

Γ =

(

e^2

2 mV

) 2

2 π

ω′ω

∫

V k′^2 dk′dΩ

(2π)^3

|ˆǫ·ˆǫ′|^2 δrr′δ(ω′−ω)

Γ =

(

e^2

2 mcV

) 2

2 π

1

c

∫

V dΩ

(2π)^3

|ˆǫ·ǫˆ′|^2 δrr′

dσ

dΩ

=

(

e^2

2 mcV

) 2

2 π

V

c

1

c

V

(2π)^3

|ˆǫ·ǫˆ′|^2 δrr′

dσ

dΩ

=

(

e^2

4 πmc^2

) 2

|ˆǫ·ǫˆ′|^2 δrr′

This agrees with the other calculations and with experiment. The“negative energy” sea is
requiredto get the right answer in Dirac theory. There arealternativesto the “negative energy”
sea. Effectively we are allowing for the creation of electron positronpairs through the use of the
filled negative energy states. The same result could be obtained withthe possibility of pair creation,
again with the hypothesis that a positron is a “negative energy” electron going backward in time.

36.17Hole Theory and Charge Conjugation

Dirac postulated that the “negative energy” sea was entirely filled with electrons and that an anti-
electron would be formed if one of the “negative energy” electronswere elevated to a positive energy
state. This would yield a positive energy electron plus a hole in the “negative energy” sea. The hole
also has positive energy compared to the vacuum since it is lacking thenegative energy present in
the vacuum state. Therefore, both the electron and the positron would have positive energy. This
describes the process ofpair creation.

Similarly, any positive energy electron could make a transition to the now empty “negative energy”
state. Energy would be given off, for example by the emission of two photons, and the system would
return to the vacuum state. This is the process ofpair annihilation.

The tables belowcompare an electron and a positronwith the same momentum and spin.
For simplicity, the tables assume the momentum is along the z directionso that we can have spin
up and spin down eigenstates. The electron and positron have opposite charge so the standard
Electromagnetic currents are in opposite directions. The last row of the table shows the “negative
energy” electron state that must be unoccupied to produce the positron state shown in the row above
it. Ahole in the vacuum naturally produces a positron of oppositecharge, momentum,
and spin. Because the probability flux for the “negative energy” electron states is in the opposite
direction of the momentum, (and the charge and momentum are opposite the positron) the EM
current of the positron and of the “negative energy” state are inopposite directions, due the product
of three signs. This means thevelocities are in the same direction.