In an the January 22, 2021 blog post we claimed that the Feynman diagram gives too low a probability amplitude for photon-photon scattering.
It turns out that Richard Feynman maybe was right and we were wrong.
Let the combined energy of the photons be < 1.022 MeV.
photon photon
~~~~ ----------- ~~~~~
| | virtual
| | pair
~~~~ ----------- ~~~~~
photon photon
There are four electron propagators in the loop.
We argued that once the virtual electron and positron are "born", they will annihilate at a probability 100%. Therefore, the Feynman diagram underestimates the probability amplitude of annihilation, since it keeps annihilation behind 3 extra electron propagators after the pair is born.
The annihilation is certain - that is true - but the events have too high a "resolution" because the pair is short-lived. A high resolution in the time direction requires high energy and a large palette of frequencies, which are not available.
Let us try to draw the events from the birth of the pair to its annihilation. Drawing the detailed classical paths would require extremely high frequencies and momenta, and a large palette.
They would be available if the particles were macroscopic and we would have measured their position and momentum very accurately. We then could describe the particles with sharp wave packets. Those packets contain a very wide palette of frequencies.
But such a palette is not available. We have to simplify the picture to one plane wave being born and annihilated. The simple plane wave describes the "system" of the virtual pair in the simplest possible way. The sharpest detail in the simplified picture is the short lifetime of the pair.
We assume that our photons are very precisely tuned to a certain energy. The wave packet has to be at least 10 million cycles long to ensure that the uncertainty in energy is less than one millionth.
input ------> | virtual pair| -----> output
precisely scattered
1,000,000 eV photons
of energy
The input to the virtual pair picture essentially has just a single frequency. We cannot really draw the virtual pair in the photograph in a meaningful way.
But probably the "channel" in that precise frequency (1,000,000 eV) still relays some energy to the output.
Drawing a short-lived virtual pair is like making a wave packet whose time dimension is short. The packet has a spectrum of frequencies (= energies). Of that spectrum, the narrow band which matches the input frequency can relay energy to the output.
The lifetime of a virtual pair as classical particles versus the oscillator model
The classical model of a virtual pair tells us that it is very short-lived if the energy is even 1 eV below 2 m_e.
The "lifetime" of forced oscillation in a harmonic oscillator is
E / (E - E') cycles,
if E is the resonance frequency of the oscillator and E' is the frequency of the driving sinusoidal force.
If h f = 1 MeV, then one cycle is
1 / f = h / 1.6 * 10^-13 J
= 4 * 10^-21 s.
Let E' < E. What is the lifetime of a virtual pair whose energy is
2 m_e E' / E?
Let the distance of the pair be r, and largish. The classical acceleration is
a = k e^2 / (m_e r^2).
We get a rough approximation for the fall-down time t by setting
r = 1/2 a t^2.
Then
t = sqrt(2 m_e r^3 / (k e^2)).
Let us calculate some examples. If r = 2 * 10^-12 m, then the acceleration is
a = 5 * 10^25 m/s^2,
and
t = 3 * 10^-19 s,
or 75 cycles. The oscillator formula gives a longer lifetime, 1300 cycles.
What if we let the virtual pair orbit each other and calculate the lifetime with the Larmor formula?
The dissipated power is
P = e^2 a^2 / (6 π ε_0 c^3)
= 10^-2 W.
The Larmor lifetime is
t = 1.6 * 10^-13 J / 10^-2 W
= 1.6 * 10^-11 s.
Very long! But if we assume that the virtual pair is created from a close collision of photons, maybe the orbiting pair has way too much angular momentum? Also, the orbiting pair would send very long wave radiation. We assume that the pair was created from two large photons.
If the distance of the virtual pair is very short, say, r = 2 * 10^-15 m, then our classical model gives an extremely short lifetime t ~ r / c = 10^-23 s for the pair. The oscillator model gives a lifetime ~ 10^-20 s. Which one is right?
The oscillator model would simplify all temporary states to behave in the same way: the lifetime of the state would only depend on how many percent of energy we are lacking. That sounds implausible. The classical model allows temporary states to have versatile behavior. We conclude that the oscillator model is too simplified. But we do not know yet if the classical model is the right way.
If the particles are very heavy, then we are working in the classical limit, and then classical physics is certainly the right way to calculate the lifetime of a temporary state. Annihilation of macroscopic particles will send a huge number of photons. The converse reaction is practically impossible then.
How does the path integral cause virtual pair paths to have an almost complete destructive interference?
We have claimed that the wave-like behavior in the photograph model of quantum field theory is a result of path integrals.
What are the "paths" in our example and how does the destructive interference occur? We need to study this.
What is the effect of high-momentum virtual pairs?
We have shown that classical virtual pairs have an extremely short lifetime if they have high momentum. An extremely short lifetime means that drawing the picture would require a huge palette of energies. That, in turn, implies an almost complete destructive interference for the process.
But there is an uncountable number of high-momentum states of virtual pairs. How do we assign "weights" to them?
The Sommerfeld atom model
Let us consider classical orbits of the electron and the positron. The energy E and the angular momentum J determine the shape of the ellipse.
This sounds a lot like the Sommerfeld atom model (1915) with elliptical orbits. His model explains most of the spectrum of hydrogen - this fact supports our claim that deep down particles are classical, and wave behavior is the result of path integrals.
We may model the pair as a single particle which orbits under the mutual potential.
The particle is under huge acceleration and radiates its energy away very quickly. The polarization of the produced photons reveals the current direction of the velocity vector v of the particle.
But what determines |v|? It affects the sharpness of the wrinkle in the electromagnetic wave produced by the particle. However, destructive interference wipes away the information about the sharpness in the final wave output.
Should we assume that the virtual pair decides |v| at "random"? Random according to what measure?
The problem is: if the "channels", which drain energy from a system, have an extra degree of freedom, what happens?
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