UPDATE September 10, 2021: We removed the erroneous claim that assigning q to just one side of the virtual pair loop is a restriction.
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We have been studying the calculations of Hagen Kleinert before Formula (12.454) at
Kleinert estimates the low-momentum behavior of the vacuum polarization diagram. He does a Wick rotation to get to use the euclidean metric.
The formula seems to describe the following process:
e- -----------------------------------------------
| virtual photon q
|
________
| | tunneling
k | | boson
| | k + q
proton ------------------------------------------------
The propagator of the boson is
1 / ( |p|² + m² ),
where | ... | is the square of the euclidean norm of the 4-momentum p, and m is the mass of the electron. That is,
|p|² = E² + s²,
where E is the energy and s the spatial momentum in the 4-momentum p.
The boson is the virtual pair. It is a boson because its spin is the sum of the electron and the positron half-integer spins.
The virtual pair cannot live long. It is presumably described by a wave of a type
ψ = exp(-i (i E t + s x)).
That is a typical wave for a tunneling particle. The energy i E in the wave is imaginary, which makes the wave to decay exponentially.
The massive Klein-Gordon equation is
d²ψ / dx² - d²ψ / dt² + m² ψ = 0.
Its propagator is
1 / (p² + m²).
Note that we use the east coast sign convention (- + + +) when we calculate the square of the 4-momentum p.
For low speeds,
E = m c² + s² / (2 m).
We see that if E is imaginary, we must make m, too, imaginary.
That is, a tunneling boson lives in a euclidean world. It is not in the Minkowski metric. It lives in imaginary time.
Question. Should we use 2 m as the pair mass?
If yes, then we can change the variable, make p = 2 p', and the propagator looks formally the same.
Question. Is the force in the diagram an attractive force?
The force in the diagram is attractive if it does not change the phase of the electron wave. Then the diagram will have constructive interference with the plain Coulomb scattering diagram. We do not see why the phase of the electron would change in the diagram.
For the original vacuum polarization diagram we argued that the photon cannot on its own change its phase. It has to be an attractive force.
How to fix the low momentum behavior?
The low-momentum q behavior of our diagram is not right. The product of the two massive boson propagators then becomes
1 / m⁴,
while it should be ~ 1 / m². Also, the diagram is missing a coefficient 1 / q².
The propagator for low momentum maybe has to be something like
1 / ( p² + m q ),
where q is the momentum exchange.
When a photon moves in a medium, its apparent "rest mass" depends on its energy and the refractive index. The mass is tiny for blue light, and even tinier for red light. It makes sense to let the effective mass in the propagator to depend on the momentum.
In the low-momentum regime no real pairs can be produced. Could that imply that we must use a model which is different from the high-momentum regime?
Why is the virtual pair a boson which is tunneling?
How do we motivate the claim that the virtual pair is a boson, and it is tunneling?
We wrote in an earlier blog post that the pair looks like a "bound state". Feynman diagrams do not handle bound states. The propagators for the pair in the Feynman diagram are for independent particles. A bound state from outside looks like a single particle. It makes sense to model it as a single particle.
The virtual pair clearly is doing tunneling, because its lifetime is short. In an earlier blog post we remarked that the Uehling potential looks like a force which is mediated by a massive particle. The force only reaches to a distance which is the reduced Compton wavelength of the electron.
We need to check if our new interpretation of the calculation of Hagen Kleinert allows us to keep the cutoff at its natural place |q|, and still reproduce his numerical results.
The proton can send massive virtual photons?
In the diagram above, we may interpret the boson as a photon with (rest) mass m. This opens up the possibility that a charged particle can send such particles, not just massless photons.
A massive photon is a virtual pair. It can couple to charges like a charged particle.
Above we referred to the index of refraction of a transparent material. A photon in such a material travels at less than the speed of light. This is explained by the fact that the photon causes polarization and acquires a positive mass. Since the photon makes electrons in the material to move, it obviously can couple to charged particles in the same way as a charged particle can.
How does the classical analogue behave?
If we embed the proton and the electron in a medium where polarization is superlinear on the electric field strength, we will see a classical process. The classical process is complicated: energy flows into polarization, and flows back when the electron recedes. This classical process must have a description with Feynman type diagrams. What are the diagrams like?
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