An update about numerical results: crude numerical calculations tell us that a dipole particle model with a constant dipole moment gives in the low-momentum case double the correct value (Uehling potential - measured in the Lamb shift), but in the high-momentum case it only gives 1/5 of the correct value (measured in the LEP in CERN).
In the model, the dipole consists of an electron and a positron placed roughly 10⁻¹⁴ m apart.
One would expect the dipole moment to be larger in the high-momentum case. We need to find ways to calculate the dipole moment.
Analysis of electron-proton scattering, assuming a dipole particle
We again study scattering of an electron by a proton. This time we assume that it is the electron which creates the virtual pair and the extra polarization during the encounter.
e- ------------------------------------->
\ k | q / k
\ | /
~~~~~~~~~ dipole particle
| (= virtual pair)
| q
● proton
Above q is the exchanged spatial momentum, and k is arbitrary 4-momentum whose euclidean norm is
|k| < |q|.
Interactions k and q are all in a dipole potential which is of the form
-1 / r²,
and whose Fourier decomposition (= propagator) is of the form
1 / |k|.
The dipole particle is created by the first k and destroyed by the second k.
The 4-momentum k integration volume
When the electron arrives, it brings to the scene extra polarization besides what possible polarization the proton has created.
If we want to describe the extra polarization with a wave packet, that packet occupies a small volume of spacetime. The Fourier decomposition of the packet has to contain relatively high frequencies both in spatial directions and in the time direction.
Why would the upper limit for |k| be |q|? It cannot be larger because |q| tells us how "much" the electron can interact with a dipole particle which is that distance away.
An alternative explanation: the momentum exchange q tells us how "smoothly" the state of the motion of the electron changes in the process. If the electron tries to send out higher frequencies than the frequency associated with q, destructive interference wipes those high frequencies off almost completely.
q + k
q --------- O ----------
-k
In the ordinary Feynman vacuum polarization formula we assume mysterious 4-momentum k which circles the virtual pair loop. The momentum k seems to appear from nowhere and then disappear.
In Dirac's model, the electron in the loop is one of the negative energy electrons in the Dirac sea. The photon q excites the electron, which leaves a hole, and later the electron drops back to the hole, releasing the momentum q.
In Dirac's model the negative energy electron is like an independent existing particle which takes part in the process. We cannot prohibit the photon q from being temporarily absorbed by a negative energy electron with arbitrary 4-momentum k. The 4-momentum can be as large as we wish. This is the origin of the ultraviolet divergence.
In Dirac's model the process affects particles which we can see. Then it probably should also affect the negative energy electron somehow. This means that information flows from visible particles to the shadow world of negative energy electrons. Dirac's model becomes very complicated if the shadow world becomes rich in information.
In our model, there is no independent external particle which would take part in the process. That is why we can argue that high momenta k cannot appear. Our model is "causal": it does not assume a shadow world of negative energy electrons.
The propagator of the potential
The potential in the diagram is a dipole potential 1 / r².
Why do we take a Fourier decomposition of the potential? Because we know how to calculate the scattering effect of a sine wave potential on a particle.
The spectrum of the Fourier decomposition constitutes the propagator of the potential.
Question. Is it a coincidence that the propagator for the Coulomb potential is similar to the propagator of the massless Klein-Gordon equation? That coincidence allows us to call that propagator the "photon propagator".
The propagator of a dipole potential looks like the "square root" of the photon potential. What are the implications of that?
The propagator of a particle
The propagator of a particle is the Fourier decomposition of a Green's function for the homogeneous wave equation
K ( ψ(t, x) ) = 0
which governs the particle.
Question. What equation governs the "particle" which mediates the dipole potential?
A Green's function is the impulse response of the homogeneous wave equation to a Dirac delta impulse. It is like hitting the equation with a sharp hammer.
We interpret that an interaction term of a field A with a field B disturbs (perturbs) the field B. The interaction at one spacetime point is like a small Dirac delta, which we write as the source of the wave equation of B:
K ( ψ(t, x) ) = δ(t₀, x₀).
The basic idea of the Born approximation is that we can treat the solution of the above as a new wave.
We assume that K (ψ) is linear on ψ. Otherwise, we cannot sum solutions.
Question. Why we can in the calculation of the loop
electron - k - dipole particle - k - electron
in the diagram do the Wick rotation and integrate assuming that the energy and the mass are imaginary?
We already claimed that the dipole particle is tunneling, and that is why it "moves" in imaginary time. But we need to find a more detailed motivation for this. Why can we set the energy in the whole loop imaginary?
In Hagen Kleinert's book, similar integrals are calculated Wick-rotated.
The propagator must be set to 1 for "almost" on-shell particles
In our diagram, if the electron has high energy, and q is much smaller, then the electron stays almost on-shell during the whole process.
We believe that the electron propagator should be set to 1 in such cases. We should prove this somehow, if the proof is not in literature.
The classical analogue of vacuum polarization is a polarizable gas whose molecules weigh nothing
An aside: we asked a few days ago what kind of a classical material corresponds to vacuum polarization.
If the material is a gas whose molecules have infinitesimal mass, then the molecules cannot exert torque on our electron or proton. Only a molecule exactly on the line between the electron and the proton, just in the middle, can have an effect. Polarization always makes the Coulomb force stronger.
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