130_notes.dvi

The Dirac equation is invariant under charge conjugation, defined as changing electron states into
the opposite charged positron states with the same momentum andspin (and changing the sign of
external fields). To do this the Dirac spinor is transformed according to.

ψ′=γ 2 ψ∗

Of course a second charge conjugation operation takes the state back to the originalψ. Applying
this to the plane wave solutions gives

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(1)~p ei(−~p·~x+Et)/ ̄h

ψ(2)~p =

√

mc^2

|E|V

u(2)~p ei(~p·~x−Et)/ ̄h →

√

mc^2

|E|V

u(3)−~pei(−~p·~x+Et)/h ̄≡

√

mc^2

|E|V

v(2)~p ei(−~p·~x+Et)/h ̄

ψ~p(3)=

√

mc^2

|E|V

u(3)~p ei(~p·~x+|E|t)/ ̄h →

√

mc^2

|E|V

u(2)−~pei(−~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

which defines new positron spinorsv(1)~p andv(2)~p that are charge conjugates ofu(1)~p andu(2)~p.

1.43 The Dirac Equation

To proceed toward a field theory for electrons and quantization ofthe Dirac field we wish to find
a scalar Lagrangian that yields the Dirac equation. From the study of Lorentz covariants we know
thatψψ ̄ is a scalar and that we can form a scalar from the dot product of two4-vectors as in the
Lagrangian below. The Lagrangian cannot depend explicitly on the coordinates.

L=−c ̄hψγ ̄μ

∂

∂xμ

ψ−mc^2 ψψ ̄

(We could also add a tensor term but it is not needed to get the Dirac equation.) Theindependent
fieldsare considered to be the 4 components ofψand the four components ofψ ̄. TheEuler-
Lagrange equationusing theψ ̄independent fields is simple since there is no derivative ofψ ̄in the
Lagrangian.

∂

∂xμ

(

∂L

∂(∂ψ/∂x ̄ μ)

)

−

∂L

∂ψ ̄

= 0

∂L

∂ψ ̄

= 0

−c ̄hγμ

∂

∂xμ

ψ−mc^2 ψ= 0

(

γμ

∂

∂xμ

+

mc

̄h

)

ψ= 0

Thisgives us the Dirac equationindicating that this Lagrangian is the right one. The Euler-
Lagrange equation derived using the fieldsψis theDirac adjoint equation,