Schwinger uses renormalization to remove a logarithmic divergence in the constant field case. Maybe the divergence is a result of an underlying assumption that empty space contains an infinite density of virtual pairs which may tunnel into real pairs? That infinite density is the origin of "non-causality" in physical processes and we aim to make physics causal in this blog.
----
The Schwinger limit is the electric field strength above which pair production of real electrons and positrons becomes significant. Consequently, the Maxwell equations of electromagnetic waves become significantly nonlinear.
https://en.wikipedia.org/wiki/Schwinger_limit
Julian Schwinger calculated the limit to be around 1.32 * 10^18 V / m.
https://journals.aps.org/pr/abstract/10.1103/PhysRev.82.664
Julian Schwinger in his 1951 paper estimated the pair production rate for a constant electric field.
Static field of a capacitor
________|________
^ ^ ^ ^
| | | |
_________________
|
The arrow lines in the diagram represent the electric field. We should observe real electrons and positrons materializing between the plates from time to time, if the Schwinger effect exists for a static electric field.
The materialization process can be seen as a tunneling process where a virtual electron-positron pair is created and the particles move in the electric field far enough to gain the combined energy 1.022 MeV and become real.
If we reverse the direction of time, then we see a real electron and a real positron tunnel against the electric field and meet to annihilate each other, in such a way that the energy is zero, and no photon is created.
How can we calculate the electron-positron production rate of the Schwinger process? That is a problem because the pair seems to appear "non-causally" from the space between the plates. If we assume that the space between the plates contains virtual electron-positron pairs that try to tunnel into real pairs, how do we know the number of virtual pairs per cubic meter? Equivalently, if we assume that between the plates there is a Dirac sea filled with electrons in a negative energy state, how do we know the density of these negative energy electrons? The more there are such electrons, the larger the flux of tunneling pairs should be.
We need to study the Schwinger 1951 paper and check how he restricts the number of negative energy electrons or virtual pairs.
If there are electrons in one plate, then a single electron can tunnel through the potential wall which is holding it in the plate. The electron tunnels and appears in the space between the plates. One might interpret this process as a virtual pair forming and the positron flying to annihilate the electron in the plate. The tunneling process is causal because it is a process of a real electron in the plate.
A very slowly changing electric field of a laser beam
We may create an almost static electric field if we have a very strong laser which sends radio waves of an immense wavelength. If the cycle of waves is longer than, say, 1 second, then we might treat the electric field created by the laser beam at some point of space as essentially constant, or static.
The laser beam consists of real photons. Conservation of momentum and energy does not allow the conversion of photons to real electron-positron pairs.
Lorentz invariance, too, implies that no pairs can be produced, since if we switch to a suitable reference frame which moves to the direction of the photons, we can make the power flux per square meter of the laser beam appear to be arbitrarily low, and consequently, it cannot create real pairs. This Lorentz invariance argument is mentioned in Wikipedia.
How would Julian Schwinger explain why the slowly varying electric field of a laser does not produce pairs, but a constant electric field does?
How would Julian Schwinger explain why the slowly varying electric field of a laser does not produce pairs, but a constant electric field does?
Zitterbewegung
There is a hypothesis that the strange zitterbewegung - trembling motion - of most solutions of the Dirac equation is really due to a process where the static electric field of the electron creates a virtual pair, and the electron annihilates with the virtual positron, making the virtual electron real.
If the above explanation is correct, then we might consider zitterbewegung a Schwinger process in the static field of a single electron.
Conservation of momentum and conservation of the center of mass is a problem in zitterbewegung. If we would be able to measure zitterbewegung, how could we ensure that conservation laws are obeyed? Classically, a point particle cannot jump around unless other particles are involved.
In our capacitor thought experiment, if a virtual pair is created between the plates, and the virtual positron annihilates with an electron on a plate, then we might see this as zitterbewegung where the electron jumped a macroscopic distance to the position where the virtual electron was.
Schwinger process only occurs with dynamic fields?
Our thought experiment of a laser beam suggests that there really is no Schwinger process with static electric fields. The tunneling of electrons from one plate is a real phenomenon, but it is not the Schwinger process where a pair appears non-causally between the plates.
We do witness creation of real pairs all the time, but that happens in dynamic situations, in particle colliders and in nature.
According to Wikipedia, people are trying to confirm the Schwinger process by shooting two laser beams at each other. Then the creation of pairs may be seen as photons colliding. The situation is very dynamic.
Conjecture 1. There is no Schwinger process with static or very slowly changing electric fields. Thus, there is no non-causal production of pairs. In dynamic situations, real pairs are created causally.
A general goal in this blog is to prove that there are no vacuum fluctuations. All that we observe is a causal consequence of real particles. If there were a Schwinger process in a static electric field, that would break the causality principle.
An observer flying in a tube of an alternating electric field
We need to study more carefully what exactly is different between a static electric field and a laser beam with a very low frequency.
Let us consider another thought experiment. Suppose that we have an observer flying at almost the speed of light in a tube where we have a static electric field pointing alternately up and down.
^ ^ ^ ^ ^ ^
| | | | | | | | | | |
v v v v v
-------> observer
Let the observer carry a vertical dipole antenna. He will observe the charges in the antenna to start oscillating. He may interpret this in the way that he is static in a beam of a low-frequency laser and the "quasi-photons" in the laser beam excite electrons and nuclei in the antenna and make the antenna to oscillate. He sees the photons as quasi-photons in the sense that he can measure the speed of the electromagnetic waves and conclude that they propagate at a speed slightly less than light in the vacuum.
A static observer, on the other hand, interprets that the kinetic energy of the observer is converted to the oscillation of the antenna.
Here we have an interesting dichotomy: the moving observer sees the antenna absorbing quasi-photons, while the static observer sees the charges in the antenna start moving because of their own kinetic energy.
What about pair production? The moving observer sees the quasi-photons move at a speed slightly less than light. He may interpret that they have a rest mass > 0. Since they have a non-zero rest mass, one can conserve energy and momentum and convert them into a real pair with a non-zero rest mass. But can the pair have a rest mass = 1.022 MeV?
Light propagates in an optically dense medium at a speed less than light in the vacuum. We may think of photons in a dense medium as quasi-photons with a non-zero rest mass. We know that such photons will create real pairs if the energy of the photon is > 1.022 MeV.
What if we make our alternating tube such that the quasi-photons have a very low energy? It just requires that the alternating sections if the tube are very long.
If there is a Schwinger effect in a static electric field, then our observer will encounter those pairs moving at almost the speed of light. How does he interpret what he sees? He thinks that he is in a flux of very low-energy quasi-photons. How can that flux create 1.022 MeV real pairs?
This must have something to do with the fact that the virtual photons in Feynman diagrams are not quantized: we can always split a virtual photon carrying a momentum p to two virtual photons each carrying a momentum p / 2. If we have an observer moving relative to these virtual photons, then his hypothetical collisions with virtual photons are not "quantized".
The Schwinger effect would create pairs which are very much quantized: they are point particles with a 511 keV rest mass. If there is really no quantum of a static electric field, then how can it give rise to quantums of the Dirac field?
Light propagating in a polarizable tube
Suppose that we send an electromagnetic wave down an electrically polarizable tube. Then the tube at any instant of time will look like the tube in the diagram of the previous section. There will be alternating sections of tube where the electric field points up and down.
If we have a static observer inside the tube, he will see roughly the same things as the moving observer in the previous section of our blog post.
The electromagnetic wave propagating through the tube is like light propagating in an optically dense medium.
Suppose that in an optically dense medium, we have an observer who moves as fast as light moves in that medium. For him, a beam of light moving to the same direction looks like he would be static in a static electric field. If the Schwinger effect for a static field would exist, he should see electron-positron pairs materializing.
The Schwinger effect requires the static electric field to be immensely strong. A static observer would see a very intense laser beam in a medium. The density of individual photons is large. Can these photons somehow create electron-positron pairs if the energy of a single photon is low?
A Feynman diagram which builds a 1.022 MeV pair from a collision of atoms and 1 eV photons would contain over a million vertexes. It must be extremely unlikely that pairs are created if the photons are low energy.
In this last thought experiment, we found two ways of looking at the experiment:
1. a fast moving observer who sees the electromagnetic wave standing still; if Schwinger is right he should also see pairs being created;
2. a static observer who sees low-frequency light propagating in a medium; no pairs should be produced because the photon is low-energy.
Who is right, 1 or 2?
In this last thought experiment, we found two ways of looking at the experiment:
1. a fast moving observer who sees the electromagnetic wave standing still; if Schwinger is right he should also see pairs being created;
2. a static observer who sees low-frequency light propagating in a medium; no pairs should be produced because the photon is low-energy.
Who is right, 1 or 2?
A photon in a medium has a "rest mass"?
Our observation about light propagating in an optically dense medium opens some interesting points of view.
Since the speed of the photon is less than the speed of light in the vacuum, we can interpret the photon as a non-zero rest mass particle whose energy is split between the rest mass and the kinetic energy.
If the medium is optically denser, then the rest mass of the photon is larger. The photon makes the charges in the medium to vibrate and in that way "adopts" some of the rest mass of the charges.
In our example of a polarizable tube, the heavier are the polarizable charges are in the tube, the slower the electromagnetic wave propagates.
In our blog we have raised the hypothesis that electromagnetic waves are mechanical waves of "virtual electron-positron pairs" of a zero rest mass, where the coupling between the charges is the Coulomb force. In the tube example, if we make the charges to have a zero rest mass, then the speed of electromagnetic waves is probably the speed of light in the vacuum. What is the density of these "virtual pairs" per cubic meter? Does the density matter?
When a photon comes close to a nucleus, it can "adopt" some of the rest mass of the nucleus. Because the photon now has a rest mass, it can convert itself to a real pair and conserve energy and momentum. This is one way of viewing pair production.
When a photon comes close to an electron, it can adopt some of the rest mass of the electron, but it cannot get a rest mass high enough to convert itself into a pair.
No comments:
Post a Comment