When we studied a collision of particles in the previous blog posts, we identified that the flow of energy within an electric, or rather, electromagnetic field, is a key concept.
Energy can flow between the field and the kinetic energy of charged particles, or energy can flow spatially. To be significant, a spatial flow has to happen regardless of the inertial coordinate system. An electron flying freely is static in the coordinates that move along with it - there is no energy flow.
An electromagnetic wave is clearly a classical phenomenon at least if the field is strong enough to be measured. What is not classical is that a quantum oscillator can absorb a whole quantum h f of energy in one shot from the field. A weak classical field would only give a minuscule energy to each oscillator. But a quantum field serves the energy in big portions, quanta.
It is as if the whole energy of the electromagnetic wave would be concentrated at a few tiny spots, the photons of the field.
One may imagine that the whole undulating wave is actually an image which these small dots draw like in an old cathode ray TV tube.
But when we have a non-periodic process like the energy flow in a collision, what is the quantum and how big is its energy?
If there is real pair production, then the quantums are the electron and the positron. We may consider the pair production as the excitation of a positronium atom from a virtual zero-energy state, to a state of > 1.022 MeV of energy. One would think that the energy quantum of the energy flow in the electromagnetic field has to be > 1.022 MeV to produce a real pair.
In a Feynman diagram, a virtual photon can carry any amount of energy and momentum which it can receive from a particle. In the simplest diagram, we typically assume that there is just one virtual photon which carries all the momentum and the energy which is exchanged in the process between two particles. But in a more complex diagram we may have two or more virtual photons doing the exchange.
The Feynman way of thinking is that we may assume that the whole energy exchange is one big virtual photon. It can transform to a pair if its energy is > 1.022 MeV.
The classical limit of the Feynman thinking would be one huge quantum which can be absorbed if two charged objects collide. Obviously, the energy of the quantum has to be restricted by the spacetime dimensions of the collision process.
When two electrons of combined kinetic energy 511 keV collide, they may approach to within 3 * 10^-15 m of each other. That is much less than the wavelength 2 * 10^-12 m of a 511 keV photon. Apparently, we may think that the whole collision process happens in an essentially pointlike zone compared to the wavelengths of the the particles emitted in the collision. This may be the reason why we can treat the whole collision energy as just one quantum.
If the collision process would occupy several wavelengths of the photon, then we might be forced to assume several quantums.
We have thought of the collision process as classical. The flow of energy happens in a tiny spot of the order 3 * 10^-15 m. Maybe it is a general principle that the quantum mechanical behavior of such tiny spot will allow all energy to be taken as a single quantum?
Classically, the collision process leads to an increase of the electric field in various zones in spacetime. A created pair would draw energy from such a zone. If there are many zones where the electric field increases, then the pair may be able to draw energy only from a tiny portion of the field. As an example, when two electrons approach each other, the electric field strength grows in most of space.
e- ----> <---- e-
A new created electron-positron pair can counteract the field strengthening only in a sector of space? Or, actually, can it counteract in the whole space?
e-
^
|
|
v
e- ----> e+ <---- e-
The positron will fly close to the colliding electrons, screening the charge of one electron. The created electron will fly away. The net result is that the two electron charges never came as close to each other as they would have come without the positron. The electric field never gained the energy it would have gained without the positron.
Thus, also classically, a pair can draw on much of the energy of the collision.
We do not need to speculate about the quantum of energy flow if the pair can also classically absorb most of the energy in the flow.
Another example is an electron flying close to a nucleus.
e- ----->
|
|
Z
The vertical line in the Feynman diagram symbolizes the energy flow in the combined electric field of the nucleus Z and the electron.
As the electron approaches the nucleus, it draws on the energy of the electric field. It comes to screen some of the positive charge in the nucleus, that is the fundamental reason for the attraction.
However the field in the zone between the nucleus and the electron grows. What if a pair is created in that zone? Then the electron of the pair can fly to screen the charge of the nucleus and the positron can fly to screen the charge of the original electron. The end result is that the original electron will draw less energy from the field to its kinetic energy.
e- ------>
| e+
| e-
Z
If we return to the rod model of electric attraction, pait production is like inelastic stretching of the rod. The rod transfers kinetic energy to the approaching electron. But some of the energy may go to inelastic (permanent) stretching of the rod. Then the electron will feel less pull and will gain less kinetic energy.
When the electron moves further from the nucleus, what if pair production happens in the more remote electric field which gains strength as the electron stops screening the nucleus? The created electron would fly to screen the nucleus and the positron would screen the original electron. The receding electron will pump less energy to the field and retain more kinetic energy. In the rod model, this corresponds to the rod stretching inelastically in the receding phase of the collision.
In the case of the nucleus and the electron it is not clear how much of the available energy can the pair production consume. Is it 50% or 1%?
A quantum of the energy flow is the produced pair? Mini-electrons?
It seems that there is no need to introduce a quantum to the energy flow in the electric field. The Dirac field, on the other hand, will absorb energy in produced pairs plus the kinetic energy of the pair.
If space has an infinite density of virtual pairs floating around, which virtual pair will absorb the energy and become real?
Maybe it is better to imagine that the energy flow produces a flux of electrons and positrons in a causal way. The pairs are quantums of this flux. Then we can discard the idea of virtual pairs floating around.
Suppose that there would exist mini-electrons who would have just 10^-9 of the charge and the energy of the electron. Then pair production would be less quantized and more "fluid". Can we write an "inverse" Maxwell equation for such charged fluid? A changing electric field produces a flux of charged fluid which tries to minimize the energy flow in the field?
The inverse Maxwell equation actually exists in a metal, where there is a lot of loose charges floating around. But in a metal, positive charges cannot move. Another analogy is a semiconductor or plasma, where we have post positive and negative carriers of charge.
Since the mass of an electron-positron pair is large, 1.022 MeV, real pairs cannot always form to reduce the energy flow.
Suppose that we had a light mini-electron whose mass would be smaller in proportion to the charge than in the electron. The classical radius of a mini-electron would be largish, otherwise its electric field would outweigh its rest mass.
A light mini-electron would pop up easily. We would observe charges appearing from empty space, reducing modest changes in an electric field. Such light pair production would be an everyday phenomenon. There would be less room for virtual pairs.
Is there some limit how small the charge of a mini-electron can be? Electromagnetic waves in the vacuum cannot be converted to any particle with a non-zero rest mass. Mini-electrons would not block electromagnetic waves. But they would make collisions of charged particles very inelastic and they would also draw energy from any process which produces electromagnetic waves.
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