A few days ago we remarked that if a virtual pair absorbs a real photon and delays its progress (for half a wavelength), then conservation of the speed of the center of mass is broken.
The same problem seems to exist also with a virtual photon which carries just momentum in Coulomb scattering.
e- ----------------------------------------
|
O vacuum polarization
| p virtual photon
Z+ ----------------------------------------
In the diagram, the nucleus Z+ pushes a positron e+ in the vacuum polarization loop and gives it the momentum p. The positron continues its travel, pulls on the electron, and gives the momentum p to the electron.
After that, the positron annihilates with its pair, a virtual electron, and nothing is left of the pair: no energy, no charge, no momentum.
While the momentum p was piggybacking the positron, it moved the center of mass through the movement of the mass of the positron. But when the positron annihilates, the movement of the center of mass is lost.
Feynman's momentum space view enforces conservation of energy and momentum, but it does not enforce conservation of the center of mass.
In a Feynman diagram, the contribution of the virtual loop has a half a wave phase shift relative to the simplest diagram where the virtual photon carries p directly from Z+ to e- and there is no loop. The vacuum polarization contribution causes destructive interference and cancels part of the probability amplitude of the simplest diagram.
Can the phase shift save the center of mass speed?
The contribution of the simplest diagram might be seen as a wave packet where the packet is spread over one meter, or 3 nanoseconds for a relativistic speed. The uncertainty in the momentum of the input electron is 10^-33 kg m/s, which for the electron is one millimeter per second.
If we are monitoring one cubic meter, the flight of the scattered positron may take 3 nanoseconds to give the momentum p to our real electron.
The delay of 3 nanoseconds is significant: the destructive interference only cancels the later part of the electron wave packet. The end result is that in the sum, the electron wave packet has progressed too fast, compared to the simplest diagram, gor which we assume that the speed of the center of mass is conserved.
Obviously, if we have one history where center of mass is ok, and subtract another where it is not ok, then the center of mass is not ok in the sum.
We need to study the Feynman diagram machinery in more detail. Momentum space, which is used in calculations, ignores the location of particles and can lead to errors which break conservation of the speed of the center of mass.
The notion of "zero energy virtual pairs" which pop out of nothing, and affect what happens in the real world, and disappear again, is generally suspect. They easily break conservation laws of newtonian mechanics.
In section 7 of the 1949 Space-time paper, Feynman writes that closed loops must be allowed, so that a produced (almost real) pair can be annihilated again.
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