| q momentum
v gained by e-
v ≈ c
e- • --->
R = distance e- proton+
/ | \
| | | E strong field,
\ | / dense energy
● proton+
When e- passes close to the proton, the total energy of the electric field E drops, but the field between the two particles becomes stronger, its energy grows, and this growing energy creates a repulsion between the particles. The repulsion, of course, cannot cancel the overall attraction, but the repulsion is significant.
Vacuum polarization can reduce the energy of the field between the particles, and make the attraction stronger.
Vacuum polarization adds a new degree of freedom to the system. The field between the particles can use this freedom to reduce the total energy of the system. It is a good guess that if the energy W can escape to the Dirac field, then the total energy of the system is reduced by W.
When the electron passes the proton at a distance R, the spatial size of the extra strong field E close to the proton is ~ R, and the time is ~ R / c. It is a smooth disturbance pulse ("bump") of those dimensions. The wavelength of the momentum exchange q is ~ 861 R. But let us first analyze this semiclassically in the zone of the size R. We Fourier decompose the bump. The most important components have the wavelength ~R.
The bump is the sum of the field of the electron and the proton. When they are close, the energy density of that part of the electric field E is double the sum of the individual fields.
The Feynman diagram
mildly relativistic
e- • --------------------------------------
|
| q virtual photon
|
/ \ k virtual electron-positron
\ / pair
|
| q virtual photon
|
● ---------------------------------------
proton+
static
The transient electric field E of the bump disturbs the Dirac field, through the coupling. We can construct the disturbance with Green's functions of the Dirac field.
We can imagine that q above is the bump in E. It hits the Dirac field. The hit will create Fourier components k according to the propagator of the Dirac field. Because the bump is smooth, destructive interference will mostly cancel out |k| > |q|.
If q creates a virtual pair, then q will be absorbed to the pair, and the energy of the field E will weaken accordingly.
If the pair is long-lived, then the virtual pair will survive for the whole transit of the electron past the proton. The pair will reduce the energy of the field E for a long time. The pair will not escape as a real pair, because the electron e- is only mildly relativistic and cannot donate 1.022 MeV. The pair eventually must annihilate and give q back to E.
A good guess is that a long-lived virtual pair increases the attraction of the electron and the proton as if an extra momentum exchange q would happen between the electron and the proton.
A long-lived pair is called "almost-bremsstrahlung".
For q = 0, no almost-bremsstrahlung will happen. In the standard vacuum polarization calculation, this is represented by the integral
Π(0).
If q ≠ 0, then there will be almost-bremsstrahlung:
Π(0) - Π(q²).
In the vertex correction we saw that bremsstrahlung appeared as a missing part of the electric form factor F₁(q²) integral. The integral F₁(0) describes the process when the electric field of the electron can keep up with the movement of the electron. Bremsstrahlung is the field which "broke free".
Another way to view q ≈ 0: then R is large, and the transit of the electron past the proton lasts for a long time. Any virtual pair created by the bump will have plenty of time to annihilate, and they will not be able to reduce the average energy of E much.
The role of the coupling constants in the pair loop. We can understand why there is the (small) coupling constant e² / (4 π) in the first vertex which creates the virtual pair from q. But, since the pair necessarily has to annihilate as the electron leaves the proton, why is there the second coupling constant e² / (4 π) at the second vertex? Why is the coupling constant not 1?
One of the reasons might be time symmetry. The process must look the same if we reverse time.
Above we assumed natural units. In them, the coupling constant is e² / (4 π) ≈ 1/137.
The bump in E is at least crudely modeled by a single hit with a blunt hammer. The bump of the increasing E hits the Dirac field, and this hit, at least crudely, is like a hit with a blunt hammer. This partially explains the curious feature of Feynman diagrams that they seem to model complex processes with a single hammer strike. Why not 100 strikes?
Quantum magnification hypothesis
Above we analyzed the classical electric field E between the point particles e- and proton+. The Fourier decomposition of E has wavelengths which are ~ 1/861 of the electron Compton wavelength. Quantum phenomena only can have a resolution of ~ the Compton wavelength. Therefore, the wave representation of the process in a Feynman diagram must have waves ~ 861 times longer than the classical description. Is this a problem? Does calculating with a 861 X magnification produce the same results as the classical calculation?
We know that the simplest Feynman diagrams calculates elastic scattering probabilities which agree with classical formulae with point charges. At least in that basic case, the magnification works.
*** WORK IN PROGRESS ***
