t
t
t
132No reference to the “negative energy” sea is needed. No change in the “negative
energy” solutions is neededalthough it will be helpful to relabel them with the properties of
the positron rather than the properties of the electron moving backward in time.
The charge conjugation operation is similar to parity. A parity operation changes the system to a
symmetric one that also satisfies the equations of motion but is different from the original system.
Both parity and charge conjugation are good symmetries of the Dirac equation and
of the electromagnetic interaction.The charge conjugate solution is that of an electron going
backward in time that can also be treated as a positron going forward in time.
36.19Quantization of the Dirac Field
The classicalfree field Lagrangian densityfor the Dirac electron field is.
L=−c ̄hψγ ̄μ∂
∂xμψ−mc^2 ψψ ̄Theindependent fieldsare considered to be the 4 components ofψand the four components ofψ ̄.
This Lagrange density is aLorentz scalarthat depends only on the fields. 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/∂x ̄ μ)= 0
∂L
∂ψ ̄= 0
−c ̄hγμ∂
∂xμψ−mc^2 ψ= 0