NOTE December 29, 2020: If the photon would be the whole time under a constant slowdown caused by vacuum polarization loops, then conservation of the speed of the center of mass would hold. Think about the photon entering a 1 meter interval. If it enters at the same speed as it moves during the interval, then the center of mass moves at a constant speed.
What would that imply? That a "bare" photon moves faster than light, but it is slowed down by virtual pairs.
However, this does not solve the loss of energy problem: if there is a half a wavelength phase shift in the photon which is reflected from a virtual pair, then destructive interference will eat up photon's energy.
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https://en.wikipedia.org/wiki/Center_of_mass
If no external forces act on a system, then the center of mass of the system will move at a constant velocity.
Consider the Feynman diagram for a real photon:
~~~~~~~~~~~~~
p ---->
and the vacuum polarization loop diagram:
~~~~~~O~~~~~~
p ---->
The Feynman rules seem (?) to imply that the photon in the second diagram gets a 180 degree phase shift. It has been "reflected" by the virtual electron-positron loop.
If there were no phase shift, then the second diagram would be equivalent to the first one, and vacuum polarization would have no effect at all.
photon
~~~~~~|~~~~|~~~~>
glass
pane
But a phase shift requires that some of the momentum of the particle was temporarily stored somewhere. For example, when a photon enters a glass pane, the photon moves slower. Some of its momentum has been transferred to the glass pane. When the photon exits the pane, it gets back its momentum.
The glass pane moved a little bit to the right in the diagram. The center of mass must move at a constant speed. The photon has not advanced as far as it would be without the pane - to make up, the pane had to move a little bit to the right.
Feynman diagrams enforce conservation of momentum, but they do not enforce conservation of the speed of the center of mass.
A vacuum polarization loop involves a zero-energy, zero-momentum system of a virtual electron and a positron. In the diagram, that system causes the phase of the photon to change. But since the system has zero energy, it cannot fix the discrepancy in the center of mass, like the glass pane did!
We conclude that a virtual pair cannot change the phase of the photon, and cannot have any effect on the physics of the system, unless the pair interacts with more than one particle.
In a particle collision, a virtual pair can interact with several particles. A virtual pair can even become "almost real" and only annihilate after some time. Thus, in a collision, vacuum polarization does have some effect.
We may have solved the problem of the divergence in vacuum polarization, in the case of a real photon! The diverging integral must be removed altogether.
In a particle collision, polarization effects must be calculated from more complex graphs where temporary pairs interact with several particles. We need to think what happens if the photon only transmits momentum, no energy.
In this blog we have stressed many times that conservation of momentum, and conservation of the speed of the center of mass have to be heeded.
Empty space cannot permanently absorb momentum, and it cannot permanently change the location of the center of mass.
A more detailed analysis of the phase shift when an electron temporarily absorbs a photon
A classical analogue is a string under tension, and a little extra weight attached to the string.
string extra weight
--------------------●------------------
wave --->
<--- --->
scattered
waves
The extra weight has inertia, and cannot keep up with the oscillation of the string. That is why the weight sends a scattered wave to both directions in the string. The scattered wave has a 180 degree phase shift relative to the original wave.
Does the weight delay the traveling of the wave energy?
We may model the string as a lattice of point masses joined by massless springs.
... /\/\/\/\ ● /\/\/\/\ ● /\/\/\/\ ...
The total energy of a string segment is the kinetic energy of the masses plus the elastic energy of the springs.
If one of the point masses is much larger than the others, its segment will, on the average, store more kinetic energy than the others, and most probably also the springs will be more stretched on the average, because they have to move the great mass.
Thus the segment will act as an energy reservoir in the motion of the string. Let us assume that each "energy packet" outside the extra weight segment moves at a constant (sound) speed v.
The large energy reservoir at the weight has to be filled with these energy packets. The reservoir will slow down the travel on many packets.
We do not know if the analogous thing happens in quantum mechanics with electromagnetic waves and electrons (which act as extra weights). But it is a reasonable assumption that any phase shift in a photon is associated with a delay in energy transport. An exception would be an infinite weight in our classical model. It would reflect all energy immediately, and store none. We need to think what this exception would be in quantum mechanics.
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