The vacuum polarization divergence in the Feynman integral happens when there is not enough energy available to form a real pair. The system tries to tunnel into a state where a pair screens the colliding charges, but because of the lack of energy, the attempt fails.
How could we classically model a tunneling attempt?
Let us first study a simpler tunneling process. A particle bumps into a potential barrier but does not have enough kinetic energy to climb over the barrier.
____
e- -----> | |
__________| |__________
A non-relativistic free particle has a wave function
exp(i (p x - E t))
where p is the momentum and E is the kinetic plus potential energy of the particle.
Inside the barrier, the momentum is imaginary and the absolute value of the wave function decreases exponentially the further we go inside the wall.
The Feynman integral formula describes a free particle moving in space. It does not describe a particle which is tunneling. A virtual pair is trying to tunnel into a real pair but fails.
Considering that the Feynman formula is not about tunneling, it is strange that it is of any use in dealing with virtual pairs. By assigning a suitable energy or momentum cutoff, the Feynman formula, however, does produce numerically accurate results. Why is that?
Classical inelastic collision
The classical analogue of a collision, with an option to pair production, is an inelastic collision, where the elastic rod or string can stretch or squeeze inelastically if there is enough energy.
---> O____________O <---
Above two colliding balls are attached with an elastic rod. The rod can buckle inelastically, absorbing some of the kinetic energy of the collision.
_
---> O____| |____O <---
Classically, if there is not enough energy available to cause the bending of the rod, then the collision is not affected by the fact that it could potentially bend. But in quantum mechanics, potential bending will affect the collision. The system will try to tunnel into the buckled state but fail.
A way to simulate buckling is to put the rightmost ball static against a hill and let the rod be perfectly elastic:
___ /
/ \___/
----> O---------------O/
x y
If the left ball comes close enough to the right one, it will push the right ball uphill into a pit which is higher in the potential.
Let x denote the position of the left ball and y the position of the right ball. If x is small, then the potential of the whole system as plotted against y looks like this:
__ /
/ \__/
\___/
y -->
But if x is large, the potential will look like this:
______ /
\__/
y -->
The push from the elastic rod will eventually roll y to the pit as x grows.
If the collision is high-energy, x will grow big and the system will roll to the "y in the pit" state. When x starts decreasing again, y will remain in the pit. It was an inelastic collision.
Quantum mechanical inelastic collision
If the collision is low-energy, then, classically, the pit will not matter. The collision will happen as if there were no pit in the potential graph of y. But in quantum mechanics, the pit does affect the behavior of the system even at a low energy. The particle will try to tunnel to the pit and that will affect the time evolution of the wave function.
We can simulate an inelastic collision with just one particle which moves in the potential as plotted against x and y above. We can calculate the quantum mechanical behavior through the ordinary Schrödinger equation. Then there are no divergences of Feynman integrals.
A problem remains, though. The inelastic collision model has to be realistic enough to be able to predict the numerical probabilities of various outcomes.
Electron-positron potential curve
If we use the potential method of the previous section, we do not know the potential function when the created electron and the positron are very close to each other. When the distance between the pair is greater and they screen the charges in the collision, we can calculate the potential. Thus, we know the curve of the potential in the pit of the previous section.
But we do not know the shape of the barrier which prevents the forming of the pair initially. If the barrier were very high, no pair production would occur. Since there is abundant empirical data about pair production, we can derive some knowledge about the barrier.
We calculated in a previous blog post that the electric potential of an elecron-positron pair is -511 keV when their distance is 3 * 10^15 m, that is, the classical electron radius.
The potential is -1.022 MeV when the distance is 1.4 * 10^-15 m.
Maybe the potential curve is flat at distances less than 1.4 * 10^-15 m? Note that the Compton wavelength of an electron is much larger, 2 * 10^-12 m.
If we would squeeze a particle into a box of size 10^-15 m, it would have of the order 100 MeV of kinetic energy. The electric potential -1.022 MeV is much too shallow to confine a particle. Maybe the forming pair initially lives in some extra dimensions where there is enough space for the pair to have a low kinetic energy? Or maybe a better explanation is that there initially is no pair, nothing in the 10^-15 m potential pit. It is a "quantum" of the electromagnetic interaction which tunnels there and becomes the pair. Feynman diagrams seem to have this way of thinking: a virtual photon transforms itself into a virtual pair. The coupling constant measures the barrier height in the original tunneling before the pair tries to tunnel through its own electrical attraction?
The process reverse to pair production is pair annihilation. Apparently, Feynman diagrams can be used to calculate accurate values for photon emission from annihilation.
However, a discrepancy of 0.1% from QED does exist in the 142 ns lifetime of orthopositronium, positronium where the spins are parallel.
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