UPDATE August 14, 2024: If a particle is born from a strong static field, there is no way to know the phase of the wave function of the particle. This breaks unitarity, and prevents the Sauter-Schwinger effect, or Hawking radiation from existing.
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https://en.wikipedia.org/wiki/Schwinger_effect
A typical idea is to assume that a pair of "virtual" particles can pop up. One of them has enough positive energy to escape as a real particle, while the negative energy particle "tunnels" down in the potential, and is somehow preserved. The preservation may be due to the particle gaining enough potential energy to become real, or through some kind of a "freezing" mechanism.
In our blog we have studied the Sauter-Schwinger effect in several posts in 2018, 2020, and 2021. Our general conclusion was that the effect probably does not exist.
An electron with a negative total energy in a potential
In the link it is claimed that a nucleus with at least 137 protons would allow a bound state where an electron has a zero energy, or even a negative energy. That is, the binding energy is more than m c² = 511 keV.
For some reason, nature seems to avoid such a strange state for the electron. There is no stable nucleus with 137 protons.
Why and how nature accomplishes this? We have to investigate that question.
The Sauter-Schwinger effect for a single laser beam
On September 24, 2018 we argued that an extremely strong laser beam cannot create Schwinger pairs, since the pair would carry less momentum than the photons of the same energy. Julian Schwinger came to the same conclusion in his 1951 paper.
For a strong static electric field, there is no problem with momentum conservation.
Pair production from a large gamma photon
-------------- e+
/
/
photon γ ------- ----------------- e-
| photon
nucleus Z --------------------------
----> time
Above we have the Feynman diagram for pair production which is empirically observed. A large-energy photon γ > 1.022 MeV comes close to a nucleus and produces a pair where the positron or the electron is real, and the other member of the pair gives the excess momentum p to the nucleus Z.
In our blog we explain the process like this:
1. the large photon γ is modeled as a wave in the electromagnetic (EM) field;
2. there is an interaction between the EM field and the Dirac field: a wave in the EM field acts as a "source" for the Dirac equation and causes "impulses" in the Dirac field;
3. the impulse response in the Dirac field is a Green's function;
4. if no component of the Green's function can get rid of the excess momentum p of the photon, then the disturbance in the Dirac field necessarily is "absorbed" back into the EM field,
5. but if a component can interact with the nucleus Z, and get rid of the excess momentum, then the disturbance of the Dirac field can live indefinitely: a pair is born.
In the case where a pair cannot be produced, the impulse to the Dirac field is like disturbing a solid object in a way which does not resonate with the object. A real particle is like a resonant wave which can carry energy away efficiently and last for a long time.
If we imagine the virtual pair as a single particle, the phase of the wave function of the pair is inherited from the incoming high-energy photon.
Can we make the photon γ to have zero energy in the above pair production diagram?
If γ would have zero energy, then the diagram above would describe "spontaneous" pair production. Let us first imagine that γ holds a small energy.
If the positron is produced as a real particle, then the electron has a large amount of negative energy. Can it "tunnel" close to the nucleus Z, so that it becomes a real electron?
The pair inherits its phase from the low-energy photon γ. The phase of the electron e- is probably deterministically determined by the phase that the virtual e- had before the tunneling process.
Let us now imagine that the process can happen with a zero energy photon γ. We may imagine that the phase of such a photon can be arbitrary. The virtual pair, and also the eventual real pair inherits its phase from γ. Since γ can have an arbitrary phase, there is total destructive interference of the wave function of the pair. The Schwinger process does not occur at all.
What about a low-energy photon γ? The probability of a pair produced with a phase which is inherited from γ is a little bit increased. But still, the most of the output from γ is destroyed in interference. Very few pairs are produced by a low-energy photon γ.
Ex nihilo nihil fit (= nothing comes from nothing)
In this blog we have several times claimed that the wave function has to develop deterministically with time. The phase of the output wave function must be determined by what is the input.
We wrote about this on September 28, 2018.
The hypothesis that particles can be born from a static field suffers from the problem: what would determine the phase of the wave function of the produced particles?
Hawking radiation
In Hawking radiation is the nondeterminism problem especially acute: if the particles are produced randomly from an essentially static field, how could they carry away the information which was poured into the black hole?
This comes on top of the problem of what determines the phase of the produced particles.
We found yet another reason why Hawking radiation cannot exist: the phase of the wave function would be nondeterministic.
Conclusions
The wave function phase problem seems to prevent the Sauter-Schwinger effect from existing.
The same argument says that Hawking radiation cannot exist.
Can an electron have a negative energy? As far as we know, we can set a static electric potential to > 511 kV. Under such conditions, an electron would have a negative total energy.
On the other hand, elementary particles and atoms do not allow bound states where an electron has a negative energy. Is there a deep reason for this?
If an electron and a positron would have a bound state at a very low potential, we could release more than 1.022 MeV from the pair. That would create an object whose mass is negative. Nature probably prohibits such an object.
We still do not know what is the inertia of an electron under different potentials. The spectrum of the hydrogen atom is not affected by the electric potential. At least in bound states, the inertia of an electron seems to be its rest mass. But if we move an electron freely around, then it would be strange if the energy flow in the electric field would not affect the inertia.
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