Feynman vacuum polarization and the "complement" problem

UPDATE August 29, 2021: Our method, when putting a sharp cutoff for |k| at |p|, gives for small |p| << M the correction term

      α p² / (16 π M²),

to the photon propagator, while in the correct formula 16 is replaced with 15.

http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-11-qed.pdf

See Formulae (12.446) and (12.465) in the book by Hagen Kleinert (2016) (there q takes the role of our p).

If we put a smooth cutoff, we might be able to get the exact right result. The calculation shows that we get a fine estimate for the Uehling potential with our method.

We need to study large |p| next.

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Our previous blog post was left at an intriguing problem.

        e- ----------------------------------------

                                |

                               O  vacuum polarization

                                |   virtual pair, 4-momenta

                                |   k, p - k

                                |

                                |  virtual photon, spatial

                                |  momentum p

        Z+ ----------------------------------------

There probably is a sign error in the Feynman vacuum polarization integral.

If the photon would scatter from a real pair, there might be a 180 degree phase shift, but we do not agree that a photon can change its phase by scattering from "empty space", or from a virtual pair which is created by the photon itself. That would be a Baron Munchausen trick. Literature skips over the phase determination discussion and immediately assumes that the vacuum polarization diagram should be subtracted from the plain photon propagator, not added to it.

We believe that large-4-momentum virtual pairs are wiped out by destructive interference. The natural cutoff is roughly  |p|, which is the momentum exchange of the electron and the proton. For larger 4-momenta, there is almost total destructive interference.

Or, we can set a smooth cutoff function

       |p| / |k|,

where k is the 4-momentum circling the virtual pair loop.

But why does the Feynman integral seem to calculate the "complement" of the effect of virtual pairs with small 4-momentum?

We want to calculate the value of the Feynman integral

       F(p²)

of the process if the 4-momentum in the integration is restricted to be smaller than a certain cutoff Λ.

Case A. Let us first set the cutoff Λ to a very large value.

The difference

       F(0) - F(p²) = D

is the correction that we have to add to the plain Coulomb scattering probability amplitude if we let the exchanged momentum be p. This is according to the usual Feynman calculus.

Case B. Let us set the cutoff Λ to |p|. We are interested in the value

       F(p²) = D'.

We have in this case fixed the Feynman integral sign error. We add D' to the plain Coulomb scattering probability amplitude to get the corrected amplitude. This is according to our new calculus.

If D = D', then the "complement" problem is solved. Both methods give the same numerical correction.

The value of D' corresponds to what effect is still left, D corresponds to what effect fell out. It would not be a big surprise if D = D'.

Calculations with Feynman integrals are cumbersome. We need to study detailed calculations to get numerical values out of our new cutoff procedure. Our new cutoff could be called "regularization", but the big difference is that our cutoff has a physical motivation in destructive interference, while regularization methods are ad hoc.