130_notes.dvi

ψ

(3)

~p =

√

mc^2

|E|V

u

(3)

~p e

i(~p·~x+|E|t)/ ̄h →

√

mc^2

|E|V

u

(2)

−~pe

i(−~p·~x−|E|t)/ ̄h

ψ~p(4)=

√

mc^2

|E|V

u(4)~p ei(~p·~x+|E|t)/ ̄h → −

√

mc^2

|E|V

u(1)−~pei(−~p·~x−|E|t)/ ̄h

The charge conjugate of an electron state is the “negative energy” electron state, the
absence of which would produce a positron of the same energy,momentum, spin, and
velocity as the electron.That is, the conjugate is the hole needed to make a positron with the
same properties as the electron except that it has opposite charge.

Let us take one more look at a plane wave solution to the Dirac equation, for exampleψ(1)~p and its
charge conjugate, from the point of view thata positron is an electron moving backward in
time. Discard the idea of the “negative energy” sea. Assume that we have found a new solution to
the field equations that moves backward in time rather than forward.

ψ(1)~p =

√

mc^2

|E|V

u(1)~p ei(~p·~x−Et)/ ̄h → −

√

mc^2

|E|V

u(4)−~pei(−~p·~x+Et)/ ̄h≡

√

mc^2

|E|V

v~p(1)ei(−~p·~x+Et)/ ̄h

The charge conjugate of the electron solution is an electron with the same charge−e, opposite
momentum−~p, and spin opposite to the original state. It satisfies the equation with the signs of the
EM fields reversed and, because the sign of theEtterm in the exponential is reversed, it behaves
as a positive energy solution moving backward in time, with the right momentum and spin.

Our opinion of the “negative energy” solutions has been biased by living in a world of matter. We
know aboutmatter waves oscillating asei(~p·~x−Et)/ ̄h. There is asymmetric set of solutions for
the same particles moving “backward in time” oscillating asei(−~p·~x+Et)/ ̄h. These solutions
behave like antiparticles moving forward in time. Consider the following diagram (which
contributes to Thomson scattering)from two points of view. From one point of view, an electron

starts out att 1 , lets say in the stateψ(1)~p. At timet 3 , the electron interacts with the field and makes

a transition to the stateψ(4)p~′′ which travels backward in time tot 2 where it again interacts and makes

a transition toψp(1)~′. From the other point of view, the electron starts out att 1 , then, at timet^2 , the

field causes the creation of an electron positron pair both of which propagate forward in time. At
timet 3 , the positron and initial electron annihilate interacting with the field.The electron produced
att 2 propagates on into the future.