What kind of operations in a Feynman diagram honor conservation of the speed of the center of mass?

NOTE December 14, 2020: We can allow negative masses in classical physics, and still honor conservation laws. A negative mass just moves in the opposite direction to the momentum vector p. The problem in the Feynman vacuum polarization formula seems to be that it allows the antiparticle "magically" make the particle to disappear, wherever the particle is.

An example of "negative mass" in classical physics is a balloon filled with air, submerged into a water pool. If we move the balloon to some direction, the system center of mass moves to the opposite direction.

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In an earlier post this week we remarked that if an electron in a vacuum polarization loop has positive energy and it carries some momentum q from a particle A to particle B at some distance, then conservation of center of mass is broken.

What is the root cause of the problem?

If the loop electron has zero energy, then it relays the momentum immediately forward, in zero time. Conservation is ok then.

The problem might be that we allow a negative energy particle, the loop positron. (We do not refer here to the formally negative E in the Dirac positron wave, but negative mass-energy as observed by an outsider.)

In classical mechanics we do not have negative mass particles. Also, a zero mass particle cannot absorb momentum for a time > 0. These recipes ensure that the speed of the center of mass is conserved.

What about particles which do not obey the energy-momentum relation

       E^2 = p^2 + m^2

(where c = 1)?

Can they break conservation laws in classical physics? A week ago we blogged about the analogy of a virtual particle in classical physics. If the particle is interacting, it may disobey the energy-momentum relation. Conservation laws are honored, though. Thus, at least some virtual particles do behave well.

Coulomb scattering

Let us consider the simplest interesting case: Coulomb scattering of an electron e- from a nucleus Z+.

No energy is exchanged. The electron receives a virtual photon from the nucleus. The photon carries pure momentum q.

            Z+ ●   

                            ^

                            |

                            |

                            |

                           e-

If a vacuum polarization electron has E > 0, then its wave travels "forward in time", while the positron wave travels "backward in time" since E < 0. How can the pair then annihilate each other? Time travel maybe?

Let us only allow pairs where both particles have E = 0. Then their waves only undulate in the spatial directions, not to the time direction. These particles are "frozen in time" like the virtual photon that only transfers momentum, no energy.

The virtual photon is a Fourier component of the static Coulomb potential. What kind of a time-independent curve do the frozen electrons represent?

The divergent vacuum polarization integral becomes very much different if we only allow E = 0. We need to calculate what it is then.