Our previous, extensive, post brought up the old problem about the wave-particle duality. When should we model particles as particles and when as waves, and what size should the wave packet be?
The Compton wavelength of the electron, 2 * 10^-12 m is way too large to model a close encounter with a positron, where the distance at 1 GeV energies can be as small as 10^-18 m. A modest 1 MeV energy corresponds to a distance of 10^-15 m.
A possible way to model tiny waves is to use sources. If we have a 30 cm long radio antenna, it can easily produce 300,000 km long waves if the charges in the antenna oscillate at 1 Hz.
The "dipole" model for photons, which we introduced in the previous blog post makes use of the fact that a tiny source in a wave equation can produce very long waves.
A teleportation model of the photon
We may imagine that a photon is a way to "teleport" the source of an electromagnetic wave very close to the receiving system, for example, a hydrogen atom.
When a hydrogen atom falls from the state 2s to 1s, it produces a photon whose wavelength is 122 nm. That is 1,220 times the diameter of the hydrogen atom, 0.1 nm.
When another hydrogen atom absorbs that photon, the receiving atom kind of sees all the energy in the photon concentrated to a tiny 0.1 nm volume.
If the atoms were adjacent, with a distance 0.1 nm, it would be easier to understand how all the energy is concentrated into a tiny volume.
The teleportation/dipole model makes a collision of two photons like a close encounter of two electrons and two positrons.
Pair production using the drum skin model with sources
The drum skin model of the previous blog post is an attempt to find a wave description for pair production. What happens if we use the dipole model to describe the colliding photons? Then we have sources to the Dirac equation, and the sources are separated only by ~10^-15 m.
A hit with a sharp hammer to a drum skin will produce all frequencies. A better model is to put the sharp hammer to touch the skin, and let the hammer move up and down smoothly. We can then produce sine waves of a desired length with a sharp, pointlike source.
How can we use a hamiltonian to justify the production of a pair? The electric field of the produced pair should reduce the potential energy of the oscillating electric field in the tiny, 10^-15 m volume. How do we model the Coulomb field of the produced pair?
As long as we treat the produced pair as particles, the Coulomb field can be modeled with the familiar 1 / r potential. But also in that case, we need to decide when exactly is the Coulomb field born, and how quickly it spreads to the environment.
What is the role of our massive Klein-Gordon drum skin then? The massive Klein-Gordon equation neatly explains why low-frequency photons cannot create pairs. In the particle model, the same restriction follows from conservation of energy.
e- e+
| | ^
| | | oscillation
| | v
e+ e-
dipole dipole
Let us then attempt a new hybrid model of a photon collision. We put two electron positron dipoles at a 10^-15 m distance from each other. The dipoles make one oscillation. At some point, the electric potential is so steep that a creation of a 1.022 MeV pair would reduce the total potential energy of the system.
Let us try to estimate how hard it is for a pair to tunnel into existence. There is a potential wall whose height is ~ 1 MeV and width is ~ 10^-15 m.
We have to "borrow" 1.6 * 10^-13 J for a time ~ 10^-23 s. The product 1.6 * 10^-36 is much less than the Planck constant. The energy-time uncertainty relation says that tunneling is very easy.
That is, if a new pair can reduce the potential energy of the electric field > 1.022 MeV, the pair will almost certainly be created.
In the massive Klein-Gordon equation, or the Dirac equation, we may create the new pair with two, almost pointlike, sources. The distance of the sources is ~ 10^-15 m.
How should we model the attraction between the newly created particles? It is hard to model that in the wave model. We should continue using the particle model until the distance of the pair is much more than 2 * 10^-12 m, i.e., the Compton wavelength. From then on, a sharp wave packet will work.
The classical limit
Let us imagine physics without gravity. Then point particles can be very heavy without being black holes. We can model a close encounter (10^-15 m) of two unit charges using a wholly classical description. The speed of the encounter can be much less than the speed of light. We can use newtonian physics.
The encounter will produce electron-positron pairs.
This suggests that the particle model actually is the right way to model close encounters. A wave model would be a wrong way.
Where do we need a wave model? To calculate interference patterns, for example. To calculate the electron orbitals of hydrogen. We could, in theory, calculate these with path integrals, which would be closer to a particle model.
An external observer does not see the sharp details of a particle collision
In the particle model, electrons can make sharp turns in a distance of 10^-15 m, much less than the Compton wavelength. But an external observer can never directly observe such detail. Whatever the collision will produce, will have the smallest detail size of 2 * 10^-12 m, the Compton wavelength.
One might speculate that the sharp detail does not exist at all - it is just some symbolic mechanism which we use to calculate probability amplitudes for particles coming out of the collision. Who knows.
The goal of our analysis of pair production has been to cast light on vacuum polarization. Let us look next at vacuum polarization, again.
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