A plane wave of electrons
----------------- crests
----------------- of
----------------- waves
--------------------------> x
Let us start from the simplest case: a plane wave.
If the flux is electrons, then it is natural to set the equal weight for each path which starts from a specific coordinate x.
But now we face a dilemma: the real axis R is infinite. The density of the probability distribution cannot be defined. It cannot be 0 nor any ε > 0.
Using nonstandard arithmetic with infinitesimal numbers would be kind of a workaround in some cases, but that does not help if we need to pick a "random" real number.
Let us just cut off the infinity in the real line at some large M. That works in most cases. But it is unfortunate if we need to start regulararization and renormalization at this early stage!
What about using a localized wave packet? That does not help either because we can never make sharp edges to it. And the Fourier decomposition of the wave packets consists of plane waves.
Maybe the mathematically most beautiful solution would be to use infinitesimal numbers, distributions, and the like. But in practice, problems do not usually arise at this stage.
Path for a "photon"?
This is a hard problem. If we treat the photon as a particle, how does it enter the stage in the deep down classical world of particles?
The electron is in classical physics a particle, but the "photon" is a wave.
In a Feynman diagram, all particles, also photons, are plane waves. They do not have a "position" at all because they exist in the momentum space. But we want to calculate path integrals in the position space.
In an earlier blog posting we suggested treating photon absorption as reversed photon emission. In classical physics electrons create complex electromagnetic waves. We may do a Fourier decomposition of the emitted wave and imagine that each plane wave component is an incarnation of a photon.
We suggested that destructive interference wipes away all but one plane wave mode from the complex emitted electromagnetic wave.
How to reverse the process of emission? Many different classical world processes can emit the exact same plane wave, i.e., the photon. If we try to add to the path integral all these infinite number of processes, how do we assign weights to them?
The classical process behind the Feynman photon-electron scattering matrix
In a Feynman diagram, an electron can emit and absorb a photon. The diagram does not specify any classical path of events. It just assumes a certain scattering matrix for the input and the output.
The scattering matrix is derived from a semiclassical calculation where an electromagnetic wave moves an electron. Thus, there is a simple classical system behind the Feynman diagram, after all. Let us analyze what exactly is assumed in the classical process.
| | | | | beam of laser
| | | • | |
e-
We assume a classical plane wave, for example, a laser beam. We calculate how much it makes the electron oscillate, and calculate the emitted electromagnetic wave with a classical formula.
The input is a classical plane wave, and the output is a complex classical dipole wave.
The "processing system" is the electron, which is assumed to be initially static. What degrees of freedom does it have? Just its spatial location. In a path integral we simply assume that the location makes no difference. The issue of weights of paths does not come up.
The Lyman alpha absorption/emission
Let us assume that the input is a plane wave.
In this case the "processing system", the hydrogen atom, is very complex. Classically, it would be a classical electron particle orbiting a classical proton. There is an infinite number of possible paths.
We may approximate the atom as a harmonic oscillator. Or we can calculate using the known orbitals and Fermi's golden rule. In an earlier blog posting we showed that at the exact resonant frequency, the absorption cross section can be derived from classical angular momentum J conservation.
What are the weights that we put to different paths in these cases? It is not clear.
Pair production
The input in pair production is two colliding electromagnetic plane waves. The processing system is a pair which is born.
In a Feynman diagram, we assume that the photon-electron scattering process can be generalized to include scattering the electron to the past or the future. The bold generalization seems to work but why?
A more traditional calculation treats pair production as a process where a -511 keV electron is excited to a +511 keV state. This is analogous to the Lyman alpha case. A harmonic oscillator model probably gives a reasonable estimate. Why?
We in our blog have suggested using classical paths of the pair, but we have the problem how to assign weights to the various cases.
Pair annihilation
The Feynman diagram for pair annihilation is obtained by simply turning the pair production diagram upside down. That suggests that the process really is symmetric in the real world.
The photograph model may be able handle annihilation if we assume that the pair simply disappears when it has radiated away all its mass-energy.
Note that the static electric field loses its energy as the separation of the pair grows smaller. If we assume that all the mass-energy of the electron is in its static electric field, then annihilation can be understood as the disappearance of the static electric field.
A drum skin model might illuminate annihilation. One finger presses the skin upward and another presses the skin downward some distance away. The fingers have done work to deform the skin. If we suddenly remove the fingers, all the static deformation energy in the drum skin is converted to waves.
Let the pair collide head-on. Our analysis in an earlier blog post suggests that the pair sends a very sharp wave when they are at the distance 1.4 * 10^-15 m from each other.
It is probably a dipole wave. The Fourier decomposition of the wave contains extremely high frequencies because the wave is born from a very abrupt process. Hypothesis: the components which survive destructive interference have the distribution of a dipole wave to various directions.
If the pair does not collide head-on but have an impact parameter b, what is the produced wave like?
Zero angular momentum makes the pair to move along a straight line segment. A non-zero angular momentum means a hyperbolical orbit.
The orbit must take the pair close enough, so that the distance is ~ 1.4 * 10^-15 m. The radiation must carry away all the energy and the angular momentum.
The outgoing wave is probably a mixture of a dipole wave and a circularly polarized wave. Usually, two photons are born. They have to take away the angular momentum which means that they cannot originate from the exact same point.
We need to check from literature what is known about the angular distribution of annihilation photons and their polarization states. A dipole antenna outputs linearly polarized photons and a rotating dipole outputs circularly polarized photons.
The "size" of the degrees of freedom space somehow determines the cross section
Now that we understand the photon output of annihilation, how can we calculate the cross section of the reverse process and the directions of the output particles?
Gould and Schreder (1967) give the pair production cross section as
σ = β π r_0^2,
where the produced particle velocity is β c and r_0 is the classical electron radius.
Frank Rieger gave the annihilation cross section as
σ = 1 / β * π r_0^2,
where the velocity of the incoming particles is β c.
The formulas are remarkably similar. The production cross section is β^2 the annihilation cross section. As if we would have particles passing through a hole whose area is π r_0^2.
_____________
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____| |
| | |
| |
|____| |
|_____________|
pair,
size β^2 photons, size 1
hole,
size π r_0^2
The "vessel" on the pair side is β^2 the size of the vessel on the photon side. Incidentally, the momentum vector length for a pair is β times the vector for a photon, and the "area" of the available momentum space is β^2 the area of the one for the photons.
There is a rule that the "size" of the available degrees of freedom space determines the cross section in the fashion above. Obviously, this should follow from the steady state of maximum entropy. But we need to analyze this in detail.
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