Assume that, in addition to rotation, boost and parity symmetry, the Dirac equation also has a symmetry under
We wish to write the Dirac equation in a way that makes the symmetry between electron and positron clear.
Start from the Dirac equation and include the coupling to the EM field with the substitution that
The strategy is to try to write the charge conjugate of this equation then show that it is
equivalent to the Dirac equation with the right choice of charge conjugation operator for
First of all, the sign of
is expected to change in the charge conjugate equation.
(Assume the equation, including the constant
is the same but the sign of the EM field
Second assume, for now, that the Dirac spinor is transformed to its charge conjugate by the operation
where we are motivated by complex scalar field experience.
is a 4 by 4 matrix.
The charge conjugate equation then is
Take the complex conjugate carefully remembering that
will change signs.
Multiply from the left by
Compare this to the original Dirac equation,
The two equations will be the same if the matrix satisfies the conditions.
matrices in our representation,
are completely imaginary and will change sign upon complex conjugation,
are completely real and will not.
The solution in our representation (only) is
It anti-commutes with
producing a minus sign to cancel the one from complex conjugation.
It commutes with
giving the right + sign.
It anti-commutes with
giving the right - sign.
The charge conjugate of the Dirac spinor is given by.
Of course a second charge conjugation operation takes the state back to the original
Applying this to the plane wave solutions gives.
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 example
and its charge conjugate,
from the point of view that a 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.
The charge conjugate of the electron solution is an electron with the same charge
, opposite momentum
and spin opposite to the original state.
It satisfies the equation with the signs of the EM fields reversed and,
because the sign of the
term 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 about matter waves oscillating as
There is a symmetric set of solutions for the same particles moving ``backward in time'' oscillating as
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 at
, lets say in the state
, the electron interacts with the field and makes a transition to the state
which travels backward in time to
where it again interacts and makes a transition
From the other point of view, the electron starts out at
, then, at time
, the field causes the creation of
an electron positron pair both of which propagate forward in time.
, the positron and initial electron annihilate interacting with the field.
The electron produced at
propagates on into the future.
No reference to the ``negative energy'' sea is needed.
No change in the ``negative energy'' solutions is needed
although 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.