Electron Self Energy Corrections

At this point we must take the unpleasant position that this (constant)
**infinite energy should just be subtracted** when we consider the overall zero of energy
(as we did for the field energy in the vacuum).
Electrons exist and don't carry infinite amount of energy baggage so we just subtract off the infinite constant.
Nevertheless, we will find that the **electron's self energy may change when it is a bound state** and that we
should account for this change in our energy level calculations.
This calculation will also give us the **opportunity to understand resonant behavior in scattering**.

We can **calculate the lowest order self energy corrections represented by the two Feynman diagrams** below.

This contains a term causing absorption of a photon and another term causing emission. We separate the terms for absorption and emission and pull out the time dependence.

The initial and final state is the same , and second order perturbation theory will involve a

We have dropped the subscript on specifying the photon emitted or absorbed leaving a reminder in the sum. Recall from earlier calculations that the creation and annihilation operators just give a factor of 1 when a photon is emitted or absorbed.

From time dependent perturbation theory, the rate of change of the amplitude to be in a state is given by

In this case, we want to use the equations for the the state we are studying, , and all intermediate states, plus a photon. Transitions can be made by emitting a photon from to an intermdiate state and transitions can be made back to the state from any intermediate state. We neglect transitions from one intermediate state to another as they are higher order. (The diagram is emit a photon from then reabsorb it.)

The **differential equations for the amplitudes** are then.

In the equations for , we

Our task is to **solve these coupled equations**.
Previously, we did this by integration, but needed the assumption that the amplitude to be in the initial state was 1.

Since we are attempting to **calculate an energy shift, let us make that assumption** and plug it into the equations to
verify the solution.

will be a complex number, the

**Substitute** this back into the differential equation for
to verify the solution
and **to find out what is**.
Note that the double sum over photons reduces to a single sum because we must absorb the same type of photon that was emitted.
(We have not explicitly carried along the photon state for economy.)

Since this a

We have something of the form

If we think of as a complex number, our integral goes along the real axis. In the upper half plane, just above the real axis, , the function goes to zero at infinity. In the lower half plane it blows up at infinity and on the axis, its not well defined. We will calculate our result in the upper half plane and take the limit as we approach the real axis.

This is well behaved everywhere except at . The second term goes to there. A little further analysis could show that the

Recalling that
,
the real part of
corresponds to an **energy shift in the state**
and the **imaginary part corresponds to a width**.

The right hand side of this equation is just what we previously derived for the decay rate of state , summed over all final states.

The

This also gives us the

In our calculation of the total decay rate summed over polarization and integrated over photon direction we computed the cosine of the angle between each polarization vector and the (vector) matrix element. Summing these two and integrating over photon direction we got a factor of and the polarization is eliminated from the matrix element. The

Note that we wish to use the

It is the **difference between the bound electron's self energy and that for a free electron** in which we are interested.
Therefore, we will **start with the free electron with a definite momentum**
.
The normalized wave function for the free electron is
.

It easy to see that this will go to negative infinity if the limit on the integral is infinite. It is quite reasonable to

If we were hoping for little dependence on the cut-off we should be disappointed. This self energy calculated is

For a **non-relativistic free electron** the energy
decreases as the mass of the electron increases,
so the **negative sign corresponds to a positive shift in the electron's mass**,
and hence an increase in the real energy of the electron.
Later, we will think of this as a **renormalization of the electron's mass**.
The electron starts off with some **bare mass**.
The self-energy due to the interaction of the electron's charge with its own radiation field
**increases the mass to what is observed**.

Note that the correction to the energy is a
**constant times , like the non-relativistic formula for the kinetic energy**.

If we

If we use the Dirac theory, then we will be justified to move the cut-off up to very high energy. It turns out that the relativistic correction diverges logarithmically (instead of linearly) and the difference between bound and free electrons is finite relativistically (while it diverges logarithmically for our non-relativistic calculation).

Note that the self-energy of the free electron depends on the momentum of the electron,
so we cannot simply subtract it from our bound state calculation.
(What
would we choose?)
Rather me **must account for the mass renormalization**.
We **used the observed electron mass in the calculation** of the Hydrogen bound state energies.
In so doing, we have already included some of the self energy correction and we must not double correct.
This is the subtraction we must make.

Its hard to keep all the minus signs straight in this calculation, particularly if we consider the bound and continuum electron states separately. The free particle correction to the electron mass is positive. Because we ignore the rest energy of the electron in our non-relativistic calculations, This makes a negative energy correction to both the bound ( ) and continuum ( ). Bound states and continuum states have the same fractional change in the energy. We need to add back in a positive term in to avoid double counting of the self-energy correction. Since the bound state and continuum state terms have the same fractional change, it is convenient to just use for all the corrections.

Because we are correcting for the mass used to calculate the base energy of the state , our correction is written in terms of the electron's momentum in that state.