Our previous post introduced the "photograph model" of quantum field theory. We claimed that deep down, particles behave like they do in classical physics.
A problem is that there is no photon in classical physics.
The photon might be less of a particle than the electron. An electron can stay at one place, like in an oil droplet in the Millikan and Fletcher experiment in the year 1909, and we can observe it through its electric field.
A photon always travels at the speed of light and we can only observe it when it excites some other system.
If we wave an electric charge, it produces electromagnetic waves. If the waves consist of photons, where exactly are those photons born? Do electrons emit the photons?
This is analogous to phonons. If we wave an elastic plate, where exactly are the phonons of vibration born?
In bremsstrahlung, in our photographic model, the plane wave output is the product of constructive interference of very complex waves. Its photons are not born in any specific place.
How can different observers see different photons in the "same" electromagnetic wave?
Suppose that we have a radio transmitter falling freely in a gravitational field. A comoving observer sees it to emit photons of a fixed frequency. But another observer far away in space sees its output as a chirp: the frequency is redshifting.
If the radio transmitter is sending right hand circularly polarized photons, the faraway observer may see some of the photons left hand polarized.
What are the photon "particles" in the output and how do they change into the ones seen by the faraway observer?
Does the same problem exist with electrons? What is a Dirac chirp wave like?
Why is the stable state of a hydrogen atom a standing wave?
In bremsstrahlung, an electron emits an electromagnetic wave, but the electron is not bound to the proton.
What about a bound electron?
Suppose that a proton and an electron enter the experiment. Let us assume that the proton is infinitely heavy and we can treat it as a classical particle. The electron will pass several times by the proton and lose energy to electromagnetic radiation.
The path integral approach says that we can calculate the outgoing radiation by summing all possible paths. The final state of the electron probably has to be a standing wave because then there is a total destructive interference of all outgoing electromagnetic waves. That is because in a standing wave an electron at a position x with a velocity v is exactly as probable as an electron with the velocity -v.
The electron cannot annihilate with the proton. It has to have some final state. The electron cannot come still close to the proton because the available energy does not allow that sharp a resolution of the final state.
The same argument implies that any stable bound state of charged particles has to be a standing wave.
Why can one hydrogen atom absorb the whole energy output of another hydrogen atom which decays to a lower state?
The photograph model explains this. We argued that destructive interference reduces the complex wave that is output in bremsstrahlung, into a very simple plane wave. Our photograph does not have enough resolution to depict anything more complex but a plane wave (which depends on the plane wave of the output electron).
We can argue similarly that the input wave which excites a hydrogen atom to a higher state, is a plane wave. The reason is the exact same: our photograph does not have enough resolution to specify the input in more detail.
If we would have no knowledge whatsoever about the location of a hydrogen atom A which decays, then we would be forced to describe the electromagnetic wave which it outputs as a plane wave. Then another hydrogen atom B could absorb the wave with a 100% probability if we would not know the location of B.
Why is that? Because the output of A always matches an input of B. A always outputs a plane wave, and B can always input the same plane wave.
But we can have knowledge of the location of A and B, and also of the location of the electron which outputs the wave. We know the location of the electron with a 10^-10 m accuracy.
The cross section that the wave that is sent by A is absorbed by B may be of the order the size of a hydrogen atom. We need to check what is the measured value.
The basic idea in this: the low resolution of our photograph forces us to describe the output of an atom A in a very simple way. The simple output easily matches a simple input of another atom B. This explains the miracle that another atom can absorb the entire energy of an electromagnetic wave output by A. The atom B is not really in any specific location. It is everywhere - that is why it can absorb the whole infinite plane wave which is produced by A.
Imagine a digital universe where the output of A is a binary number 101 or 100. If B accepts those outputs as its inputs, then B can absorb the entire output of A. The simpler is the description of the universe (the more "quantized" the universe is), the easier is absorption.
The universe of classical physics is so complex that complete absorption of an electromagnetic wave is impossible.
The concept of the Fock space contains the idea which we developed above. The distinct states of particles may be plane waves. The state of the Fock space is the photograph in our photographic model.
Do photons exist?
The ideas which we developed above do not mention the photon at all. The atom A outputs an electromagnetic wave which destructive interference reduces to a very simple wave, possibly a plane wave.
We could call this plane wave a photon. The photon exists in the higher, path integral description of events. It is not present in the classical low level description. The electron is a particle on the low level, as well as on the high level.
How do we work with photons in the underlying classical world?
In a path, we use classical particles with classical physics. If an electromagnetic wave is the output, the path integral reduces it to a plane wave.
We can handle an input electromagnetic wave by reversing time. For example, absorption is the reverse process to emission. The input wave in the path integral world is a plane wave.
How to restrict the cross section of an input?
The idea which we briefly sketched above claims that a plane wave is always absorbed by a suitable system which accepts that particular plane wave as an input. Since the cross section in the real world is usually very small, we need to find a way to restrict the cross section.
We mentioned that knowing the locations of the atoms A and B might reduce the cross section of absorption. We will develop that idea.
In a Feynman diagram, the cross section is restricted by the coupling constant as well as propagators for virtual particles.
Maybe we can restrict the cross section by arguing about the angular momentum that is contained in the movement of the two colliding particles. If the particles come very close, then the angular momentum has to be very small. And it is quite improbable that two particles, which we describe with plane waves, would possess so little angular momentum.
For example, if two photons collide and temporarily turn into virtual pair, then the angular momentum in the pair must match the angular momentum which the system of two photons received. If the electron and the positron are close to each other, then their mutual angular momentum necessarily is quite small. This means that such an event, or path, is quite rare.
This would restrict the contribution of very high momentum pairs, because they necessarily have to be deep in their mutual negative potential, and thus they are close to each other.
A classical case of reducing output/input complexity
^ finger
| movement
v second finger
____U_________U_____
drum skin
Suppose that we rhythmically press a drum skin with a finger. The finger produces a circular wave.
If we put another finger on the drum skin at some distance, that second finger will only absorb a small fraction of the energy output of the first finger.
| | another hand
/\\\\
-----------
| |
| |
-----------
\ ////
| | hand movement
^
|
v
Suppose then that we have a square elastic plate. We use one hand to wave one side of the square, to make plane waves to the square plate. If we have another hand holding the opposite side of the square, that second hand can, in principle, absorb all the energy which the first hand outputs to the plate.
A circular wave is more complex than a plane wave. The circular wave contains information of the location where the wave was born. That is why the second finger can only absorb a small fraction of the energy output.
A plane wave can be seen as a single "channel" of energy output. The finger sends energy to many channels, and the second finger can only absorb from one channel.
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