When a high-energy photon hits a nucleus, the primary scattering mechanism is production of real electron-positron pairs.
1. The coupling constant of the electromagnetic and the Dirac field determines how big a chance the photon has per one meter of its flight to try tunneling into an electron-positron pair.
2. The problem in tunneling is that the photon has too much momentum p compared to its energy E. If we try to convert the photon to an electron-positron pair which has combined momentum p, then the energy E does not suffice.
____ electron e- p' E'
/
p E photon ------------->
\____ positron e+ p'' E''
Above, p' + p'' should be p, but then necessarily E' + E'' > E.
Above, p' + p'' should be p, but then necessarily E' + E'' > E.
Since e- and e+ proceed at a speed less than light, there is no way to conserve both momentum and energy in the above process. That is the reason a photon cannot decay into a real pair in empty space.
3. What to do to make pair production to happen? Suppose that there is an atomic nucleus Z nearby. Let the positron "borrow energy" from empty space. Then we can have E' + E'' > E for a short period of time. Borrowing energy means that the positron becomes a "virtual particle" in the sense of our earlier blog post.
But the positron has to pay back the energy it borrowed. In tunneling, a particle may borrow energy to be able to climb up a potential wall, but it must pay back the energy when it slides down the wall on the other side.
But the positron has to pay back the energy it borrowed. In tunneling, a particle may borrow energy to be able to climb up a potential wall, but it must pay back the energy when it slides down the wall on the other side.
4. The positron can pay back the energy when it slides down the potential hill produced by the electric repulsion of the nucleus Z. Then it can become a "real particle", and we have witnessed a production of a real pair.
We may view pair production as a tunneling process where 1) a positron climbs up the electric attraction of the produced electron with borrowed energy and 2) the positron then slides down the electric repulsion of the nucleus Z and pays back the energy. Here we assume that the "bare mass" of the electron and the positron is zero. They acquire their mass-energy by climbing up the potential wall of their mutual electric attraction.
If the bare mass is > 0, then we need to imagine some other kind of a potential wall that the forming positron climbs up. Since an electron and a positron annihilate to pure electromagnetic energy - usually two photons - that suggests their bare mass really is zero.
What about the vacuum polarization diagram where the electron and the positron fail to tunnel through and come back to annihilate? If the energy of the photon is less than 1.022 MeV, a real pair cannot form.
Here our earlier blog post about "spike" wave functions is relevant. We may imagine that the electron and the positron start their journey from a very small spacetime patch. Then we have to model them with very sharp spike wave functions. The spike can be viewed as a superposition where the particle may have a very high momentum, and consequently very high energy.
Our blog post showed that calculating with spikes would require extreme precision to exclude solutions where a high-energy particle flies over the potential wall.
Furthermore, Feynman's integral formula seems to assume that the high-energy electron and positron magically find each other to annihilate. That does not happen if they behave anything like real high-energy particles.
e-
_____
/ \
photon --------> --------------> photon
\_____/
e+
Suppose that we send a flux of coherent low frequency laser light towards a static atomic nucleus. Think of the vacuum polarization diagram above.
For each short distance ds of the path of a flying photon, there is a small probability that it will convert itself to a real pair that tries to tunnel through a potential wall, as explained above. Here we are assuming that the bare masses of e- and e+ are zero. Then even a minuscule photon energy can produce a real pair.
For each short distance ds of the path of a flying photon, there is a small probability that it will convert itself to a real pair that tries to tunnel through a potential wall, as explained above. Here we are assuming that the bare masses of e- and e+ are zero. Then even a minuscule photon energy can produce a real pair.
What does the "flux" of created real electrons and positrons look like? The flux of these particles cannot tunnel through the potential wall. Rather, the flux is "reflected back" and annihilates to form a flux of new photons.
We may think of the pair as a single particle which moves in 6 spatial dimensions and one time dimension. Let us denote the electron spatial coordinates by x, y, z and positron coordinates x', y', z'. The particle starts from a "diagonal" subspace where x = x', y = y', and z = z'. Our 6-dimensional particle tries to tunnel through a potential wall that surrounds that subspace. The potential wall is the electric attraction of the electron and positron.
In the 6-dimensional space, our flux of real pairs looks like a flux of single particles. Thus, we may treat the process as a flux of real particles meeting a high potential wall. All particles will be reflected back from the wall because they do not have enough energy.
A question: after the reflection, do the particles immediately annihilate, or is it possible that they could move in a complex way in the 6-dimensional space? A positronium "atom" does annihilate if the electron and the positron come near each other. That suggests that the annihilation is immediate.
Since the energy of the pair is less than 1.022 MeV, they will surely meet again. That might justify that in the Feynman integral formula, the electron and the positron find each other without trouble. Question: how does the wave function phase angle develop in the Feynman formula versus our interpretation?
We may think of the pair as a single particle which moves in 6 spatial dimensions and one time dimension. Let us denote the electron spatial coordinates by x, y, z and positron coordinates x', y', z'. The particle starts from a "diagonal" subspace where x = x', y = y', and z = z'. Our 6-dimensional particle tries to tunnel through a potential wall that surrounds that subspace. The potential wall is the electric attraction of the electron and positron.
In the 6-dimensional space, our flux of real pairs looks like a flux of single particles. Thus, we may treat the process as a flux of real particles meeting a high potential wall. All particles will be reflected back from the wall because they do not have enough energy.
A question: after the reflection, do the particles immediately annihilate, or is it possible that they could move in a complex way in the 6-dimensional space? A positronium "atom" does annihilate if the electron and the positron come near each other. That suggests that the annihilation is immediate.
Since the energy of the pair is less than 1.022 MeV, they will surely meet again. That might justify that in the Feynman integral formula, the electron and the positron find each other without trouble. Question: how does the wave function phase angle develop in the Feynman formula versus our interpretation?
The wavelength of the 6-dimensional particle flux is obviously roughly the same as the wavelength of the incoming laser light. This is because the pairs are formed "causally" from the laser light. The production of pairs depends smoothly on the electromagnetic field in the laser beam. There is no reason why the field of produced particles would have "detail" much smaller than the laser light wavelength. And the energy of a pair is roughly the energy of a photon. That is why the wavelength of particles is roughly the laser wavelength. Therefore we can calculate the (failed) tunneling process by setting an energy cutoff.
Conjecture 1. When calculating vacuum polarization, we can set the energy cutoff roughly at the energy scale of the scattering experiment. The wavelength of waves in the Dirac field cannot be much shorter than this scale.
We can, in a sense, appeal to the classical limit of the system when we claim that the virtual pairs are produced "causally" from the laser light. If the laser beam is strong, it can be seen as a classical electromagnetic field. Then the waves in the Dirac field have to be determined by the laser field, and this is deterministic. There is no room for "vacuum fluctuations" to produce indeterminism in the system. The outgoing flux of light depends somewhat on the Dirac field waves, and the outgoing flux has to be deterministic in the classical limit.
Conjecture 2. It makes no sense to talk about "vacuum fluctuations" or vacuum energy. All fluxes in physical processes are deterministic and depend only on input fluxes. A hypothetical "structure of the vacuum" does not affect the outcome in any way.
If the vacuum energy is zero, that explains why the vacuum does not collapse under its own gravitational pull, an observation which would solve the famous puzzle about vacuum energy and gravitation.
What about the energy of a hypothetical cosmic inflaton field? Is there any sense in talking about the energy if that does not manifest itself in processes involving real particles? Or could we interpret the inflation of space as such a process?
If the vacuum energy is zero, that explains why the vacuum does not collapse under its own gravitational pull, an observation which would solve the famous puzzle about vacuum energy and gravitation.
What about the energy of a hypothetical cosmic inflaton field? Is there any sense in talking about the energy if that does not manifest itself in processes involving real particles? Or could we interpret the inflation of space as such a process?
Conjecture 3. There is no sense in speculating about the polarization of the vacuum around a static charge. That would require us to assume something about the structure of the vacuum. Vacuum polarization is a process that only has consequences when energy is available in a dynamic process that involves real particles.
According to Conjecture 3, the name "vacuum polarization" is misleading. It is not a vacuum when real particles are in a dynamic process in the area. The polarization is the result of a photon trying to tunnel into a real electron-positron pair. It is not about the photon affecting the routes of some hypothetical virtual pairs that exist in a totally empty vacuum.
Question 4. Can we formulate the above tunneling argument in the traditional path integral approach? Can we get rid of spike wave functions in the path integral by "bundling" together many spikes to form a smooth curve? How does the above argument solve divergences in toy models of quantum field theory, for example, in the phi cubed model of Mark Srednicki's book?
In our tunneling argument we were thinking of a real photon entering a scattering experiment. In a Feynman diagram, a virtual photon with a momentum p mediates the electric attraction/repulsion. What exactly is this virtual photon? What is the electric field in it like? How does the flux of electron-positron pairs form in this case?
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