The sharp hammer and drum skin model for the electron static field is complicated. Let us simplify it further.
● swing frame
|
|
|
| kids'
| swing
-----
---->
impulses
at high
frequency
We push the swing with our hand at short intervals, so that the swing is permanently positioned to the right from the equilibrium position. The configuration is analogous to the static "pit" which the electron makes into the electric field.
Let us assume that the swing is moving back to the left. A single push of the hand can be decomposed in this way:
1. our hand absorbs the "reflection" of the previous impulse by stopping the movement of the swing to the left;
2. our hand emits a new impulse by pushing the swing again to the right.
If the resonant frequency of the swing is high (the rope in the swing is short), we must push the swing at very short time intervals to keep it positioned to the right. For slower resonant frequencies, we can push at longer time intervals.
The analogy between the drum skin and the swing
If we hit a drum skin with a sharp hammer, it produces a Dirac delta impulse, which in the Green's function contains all kinds of frequencies, high and low.
Let us fix a timestep. It might be 10^-20 s, for example. The hammer hits the drum skin at that interval. The swing analogy suggests that each hit of the hammer absorbs some spectrum of reflections of the Green's functions of previous hits. Low-frequency components take a longer time to reflect back and get absorbed.
The hit also produces a new impulse a new Green's function as a response.
Suppose then that the electron scatters from a nucleus and its velocity vector changes significantly.
virtual photon E, q
~~~~~~
/ \
e- --------------------------------------
| virtual
| photon
| p
Z+ --------------------------------------
A mixture of spectra from Green's functions from earlier hits is left "dangling". The spectra will not be absorbed completely by the electron because the electron changed its course. The dangling part is the virtual photon E, q in the Feynman diagram above. The electron will absorb some of the dangling part. This gives rise to the vertex correction. The rest is emitted as bremsstrahlung.
What is in the dangling part? It contains the output of many hits of a long-frequency component of the Green's function. The electron has many timesteps of time to try to absorb those long-frequency components.
The most important contribution to various phenomena obviously comes from the frequencies where the cycle time is of the same order as it takes the electron to scatter from the nucleus. We know that classical bremsstrahlung is concentrated to those frequencies. Let us call these mid-frequencies.
High-frequency components get absorbed quickly. The electron does not change its course significantly at a short time interval. We believe that the impact of high-frequency components is small on various phenomena.
For mid-frequency components, the Feynman diagram and the formula might be quite a realistic description of what happens. We may imagine that the dangling component of mid-frequencies was produced by a single hit by a large, non-sharp, hammer. Some of the component will be absorbed in the next hit.
What about low frequencies? If the dangling component was produced by, say, 10 hits of the large hammer, then some of it will be absorbed by the next 10 hits.
What is the impact of low-frequency components? The energy in the far field of the electron is small, which suggests that the impact is small.
Mid-frequencies are responsible for the reduction of the effective mass of the electron as it passes the nucleus. Mid-frequencies also produce most of classical bremsstrahlung.
Next we need to study carefully why the Feynman formula might describe the process correctly for mid-frequencies. We know that the Feynman formula does not work for high frequencies because regularization is needed for them.
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