Suppose that we want to push a boat away from a pier. We can use a rod and a virtual phonon, that is, the push of the rod, to accomplish the task. The boat recedes only slightly during the several seconds we push. We transfer a large amount of momentum to the boat but just a little bit of kinetic energy.
boat
____________
/ |
\____________|
|
| rod
o
/|\ man
/\
The man has also another way of transferring a large amount of momentum but just a little bit of energy. He can throw a heavy football at the boat. The ball bounces and the man catches it.
boat
_________
/\
/ \
/ \
O ball
The Feynman formula for the phase change in the wave function caused by a virtual photon is exactly like for a real photon. It depends on the transferred momentum p, not on the energy. The phase change is thus equivalent to the situation where two real photons are used to transfer the momentum. One can carry p / 2 and another -p / 2 to the other direction.
But a problem remains: the electric field which pulls the electron towards the nucleus is radial. How could we realistically treat it as a combination of two very high energy photons whose electric field is transverse? We need to find out what the combined electric field of the two photons looks like. Could it be a radial field?
We could use two very high energy longitudinal real phonons in our rod analogue of the virtual photon. But is their combined effect really equivalent to a single push of the rod?
In the boat and rod example, how can we implement a pull with a bouncing ball? There we must fall back to the rod and a pull.
We could use two very high energy longitudinal real phonons in our rod analogue of the virtual photon. But is their combined effect really equivalent to a single push of the rod?
In the boat and rod example, how can we implement a pull with a bouncing ball? There we must fall back to the rod and a pull.
Let us study the heavy electron flyby from the point of view of dipole radio transmitters. Let us choose a coordinate system where the nucleus moves right at a velocity v and the electron left at v.
Schematically, we have first:
-
+
and then:
-
+
This would be like two horizontal dipole antennas transmitting half a wave, if we would have static extra charges + and - placed in the system. The the initial position is:
+ -
+ -
and the later position is:
- +
- +
We see that the upper row is a dipole which sent half a wave, and the same for the lower row.
Let us analyze the system classically.
The charges in the flyby are either static or moving at a constant speed (constant because they are very heavy).
Is it possible that the system could radiate away real radio waves? A problem is if we can use a Fourier transform to decompose the electric field? A single static charge has a static electric field. Any Fourier decomposition of a static field would suggest that it radiates energy, which does not make sense. On the other hand, a Fourier decomposition of the changing dipole field of a radio transmitter might make sense.
We return to the old Larmor radiation formula problem.
Conjecture 1. To determine the radio waves sent by a configuration of moving charges, do a dipole decomposition. Take all pairs of charges. Calculate the dipole radiation of each pair. Sum up all the dipole radiation fields.
How to calculate the dipole radiation field? The key factor seems to be the acceleration of a charge relative to the other. Static charges do not radiate.
What about the case of the heavy electron passing by a heavy nucleus? We may treat this classically. Imagine a strong coherent laser wave that tries to push the charges apart when they are already receding from each other. Can the laser field give up some of its energy to the kinetic energy of the charges?
----
Let us return to the conundrum how transverse electromagnetic waves can create a pull or a push.
The electric field of transverse waves does not create pull or push, but the magnetic field would do the trick if the charge is moving with respect to the transverse wave.
In the electron-nucleus example, we could make the nucleus to send a transverse wave at the electron, and vice versa. In the dipole transmitter diagram above, the electric field of the dipole is horizontal and the magnetic field is a circle around the dipole. The magnetic field lines point up from the computer screen or down to the screen.
Now we see that the magnetic field can be used to generate the pull between the electron and the nucleus, since the electron is moving.
If the electron and the nucleus are static, then it is harder to explain the pull with a magnetic field. We should assume that the field is moving, which sounds strange.
Our goal is to reduce the pull into two real photons that are sent between the electron and the nucleus. Then the Feynman vacuum polarization diagram starts to make sense, and we can analyze the physical configuration further.
The magnetic field of a moving charge is proportional to the velocity of the charge. We may assume that the nucleus and the electron move at opposite directions at speeds v / 2. If the speed is much less than the speed of light, then the electric pull dominates the magnetic pull.
Note that the magnetic force goes as v^2 when v increases.
If we let the nucleus do short fast spurts but stay static at other times then it creates sharp magnetic pulses. If the electron does fast spurts just at the moment it receives these pulses, then the electron will receive strong pull impulses toward the nucleus. This might be the mechanism how the electron and the nucleus, by exchanging two real photons, achieve a strong pull impulse toward each other.
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