Let us next attack QED vacuum polarization, based on the insights we got from the vertex correction. On November 4, 2025 we wrote a previous detailed analysis of vacuum polarization.
Is the Feynman rule, that a closed fermion loop adds -1, correct?
In this blog we have criticized that rule. Let us have the following Feynman diagram:
___
e- / \
photon ~~~~~ ~~~~~
e+ \_____/
The diagram, with the -1 rule, claims that the phase of the photon changes 180 degrees, even though it does not interact with any outside object. That cannot happen with classical waves. We do not believe it is possible with quantum waves, either.
The text in the link says that a fermion loop involves an odd number of fermion field swaps, and that the factor -1 comes from the fact that fermion fields anticommute.
In the link, Lubos Motl explains that an operator ψ₁↑ has to be transported over 3 other operators, anticommuting, and yielding -1³.
The subscripts 1 and 2 denote the two vertices of the vacuum polarization diagram. A propagator is the line between two vertices. But why should we transport ψ₁↑ to the end? This might suffice:
ψ₁ ψ₂↑ ψ₁↑ ψ₂.
The dagger version of the operator ψ can be interpreted either to annihilate an electron or to create a positron? Then the first two operators above create and annihilate an electron (= first propagator), and the two last create and annihilate a positron (= second propagator). The operator ψ₁↑ was only transported over two other operators. There is no sign change.
Classical processes suggest that there cannot be a factor -1 in a fermion loop
Classical radio antenna. Let us have a radio wave which meets a radio antenna.
reflected waves, 180º phase shift
/ \
/ \
radio wave ~~~~~~~
#
#
#
antenna
Radio waves are first absorbed by the antenna, and then emitted again (= reflected). The emitted wave has destructive interference with the wave which would continue directly through the antenna. The missing energy goes to the wave emitted to other directions. The emitted wave has a 180 degree phase shift relative to the original wave.
The antenna keeps some of the momentum of the incoming wave. Therefore, the antenna is able to re-emit the wave with a 180 degree phase shift. In vacuum polarization, the lonely electron-positron loop cannot absorb momentum. It is not plausible that vacuum polarization could do the same trick.
Classical metal body between the electron and the proton.
e- • --->
++ metal body is polarized and
-- increases the attraction
● proton+
A metal body acts as a radio antenna. What if we put that body between the electron abd the proton? The body is polarized and increases the attraction.
If we interpret the attraction as a virtual photon carrying only spatial momentum, the antenna does not cause any "destructive interference" to the virtual photon. Instead, it makes the virtual photon stronger. There is no phase shift.
Classical polarization:
1. Classical polarization makes a real photon to be reflected with a 180 degree phase shift. The polarized body must keep some spatial momentum.
2. Classical polarization increases the attraction between opposite charges. Here we assume that the charges are passing by each other, and that polarization increases in the strong electric field between them.
Classical analogy between a real photon and a photon only carrying spatial momentum. In the radio antenna example, the antenna absorbed some of the energy of the incoming radio wave. The antenna could not keep that energy. It had to send the energy to other directions. Let us investigate an equivalent process for the momentum exchange between the electron and the proton.
Fd
^
| • e+
e- • dipole
| • e-
| |
v v F' weak
F
● proton+
The proton pulls on the electron with a force F. We try to reduce the pull by putting a dipole near the electron. The dipole exerts an upward force Fd on the electron, and reduces the pull of the proton. The electron pushes the dipole down with the force -Fd.
We were able to divert some momentum transfer to the dipole, so that the electron does not receive it. The missing momentum transfer was absorbed by the dipole. Bu we now face a dilemma: to which body can the dipole "emit" that downward momentum it stole from the electron? The dipole should pull something else downward. It cannot pull the proton downward.
The only way for the dipole to get rid of its momentum is to collide with the proton. The dipole is a transient process created by the passing electron. It is highly inlikely that it could collide with the proton.
We found an analogy between a real photon and a momentum-only photon versus an "antenna". The difference is that the "antenna" cannot work in the momentum transfer case.
A real photon temporarily absorbed by an electron does change its phase by 180 degrees – what about the electron?
photon ~~~~~ ~~~~~~
\ /
e- ----------------------------------
This is just like the classical radio wave plus an antenna case. The electron is the antenna. The photon undergoes a 180 degree phase shift.
Does the electron undergo a 180 degree phase shift? It is, in a sense, reflected from the photon.
In the wave model, a photon can be understood as a "grid" which systematically disturbs the Dirac wave representing the electron. An electron wave absorbing the photon is scattered from the grid.
Under the Schrödinger equation, an electron scattered from the electric field of a proton does not undergo a phase shift. Light scattered from a grid is not phase-shifted.
Hypothesis: no phase shift unless a real particle is "absorbed". A phase shift is a major thing in the life of a real particle. If the real particle is, in some sense, absorbed, then there can be a conceptual "delay" in its re-emission, and a phase shift of 180 degrees is possible. The emitted particle is "created again" from scratch. A phase shift is not possible if a particle is not absorbed.
A real particle travels in time. A static observer sees the wave undulating up and down.
The hypothesis leaves open what happens with a virtual particle. Our reasoning in the previous section suggests that no phase shift can happen with virtual particles which only transport spatial momentum.
In the vacuum polarization diagram, we have a virtual photon which creates the pair into which it is "absorbed". If it were a real photon, it would be a miracle if this Baron Munchausen trick could change its phase. Also, the photon mostly transfers spatial momentum, which suggests that its phase cannot change.
Furry's theorem is almost correct
The classical limit of this theorem is very suspicious. It claims that if an electron passes a proton, and vacuum polarization occurs, then that vacuum polarization cannot interact with a second electron.
Why should that be? Vacuum polarization means that there are charges present in previously empty space. Those charges can be an "almost" on-shell electron-positron pair. That pair can, of course, interact with a foreign electron.
The pair is very short-lived. Therefore the probability of an interaction with a foreign electron is extremely low. Furry's theorem is almost correct.
Does vacuum polarization classically affect a foreign electron coming around?
• e-
• e- foreign
+ + polarization
- -
● proton+
The dipole field of the electron and the proton is reduced because of polarization. Polarization "conducts" field lines. Thus, vacuum polarization reduces the interaction of a foreign electron with the dipole.
The proof of Furry's theorem in Wikipedia goes like this: let us do a charge conjugation to the diagram, i.e., switch the signs of charges. Each of the three photon fields Aμ (lines) changes its sign. The method to calculate the diagram probability amplitude multiplies those photon fields. The calculation yields a factor (-1)³ = -1.
Most calculations in QED are unchanged under charge conjugation. If the Furry diagram changes its sign, and its value is nonzero, then the sum of the diagrams (= scattering amplitude) would change under charge conjugation. But charge conjugation must preserve the physics unchanged.
The conclusion in the proof is that the Furry diagram must have a zero probability amplitude.
Why does the calculation method cause a sign change in charge conjugation? A sign change means a 180 degree phase shift.
Does the sign change show that the calculation method gets the phase wrong? the previous sections of this blog post we showed that determining the phase is not trivial. QED has suspicious rules about the sign of the probability amplitude.
In the case of the Furry diagram, if the photons are pure momentum exchange, there should be no phase change, ever. Not after a charge conjugation, either.
We have to check the 1937 paper by W. H. Furry and determine what is the error there.
At the link we have two proofs of Furry's theorem. The link says that if we reverse the direction of the electron arrows in the Furry diagram, then the probability amplitude flips its sign. For a real physical process, we always have to sum both diagrams because we have no way of knowing which one happened. This sounds sensible. But why should the amplitude flip its sign if we change the direction of the electron arrow?
The vacuum polarization should be symmetric between the electron and the proton. How we formally draw the diagram should not matter.
*** WORK IN PROGRESS ***
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