In the Dirac theory, we have only one term in the **interaction Hamiltonian**,

Because it is linear in it can create a photon or annihilate a photon. Photon scattering is therefore second order (and proportional to ). The

The initial and final states are **definite momentum states**, as are the intermediate electron states.
We shall **first do the calculation assuming no electrons from the ``negative energy'' sea participate**,
other than to exclude transitions to those ``negative energy'' states.
The initial and final states are therefore the positive energy plane wave states
for
.
The intermediate states must also be positive energy states since the ``negative energy'' states are all filled.

The computation of the scattering cross section follows the same steps made in the development of the Krammers-Heisenberg
formula for photon scattering.
There is no
term so we are just computing the **two second order terms**.

As in the earlier calculation, the photon states have been eliminated from the equation since they give a factor of 1 with the initial state photon being annihilated and the final state photon being created in each term.

Now lets take a **look at one of the matrix elements**.
Assume the **initial state electron is at rest and that the photon momentum is small**.

For and , a delta function requires that . It turns out that , so that the cross section is zero in this limit.

This matrix only connects spinors to spinors because of its off diagonal nature. So, the calculation yields zero for a cross section in contradiction to the other two calculations. In fact, since the photon momentum is not quite zero, there is a small contribution, but far too small.

The above calculation misses some **important terms due to the ``negative energy'' sea**.
There are additional terms if we consider the possibility that the photon can elevate a ``negative energy'' electron
to have positive energy.

The matrix element is to be taken with the initial

Let the positive energy spinors be written as

and the ``negative energy'' spinors as

The matrix connect the positive and ``negative energy'' spinors so that the amplitude can be written in terms of two component spinors and Pauli matrices.

This agrees with the other calculations and with experiment. The

Jim Branson 2013-04-22