In hypothetical Unruh radiation, empty space is claimed to warm an accelerating detector.
If space has zero energy, can it affect any physical process?
Let us analyze if an object with zero energy can affect anything in the physics of positive mass-energy objects.
Special relativity
We remarked in an earlier blog posting that if a zero mass object would absorb momentum for a non-zero time, that would break conservation of the speed of the center of mass (the center of mass theorem).
Of course, if there is some mechanism which restores the correct position of the center of mass, then there is no breach.
charge charge
● <-------------------> ●
Coulomb
force
In special relativity, a force between objects is an "object" which can transfer momentum.
In the center of mass coordinates the force transfers momentum in zero time. But if we look at the process in a moving frame, then it can appear that one object gains speed earlier than the other loses it. The concept of simultaneity is relative.
The moving observer can see a temporary breach of conservation of energy, momentum, and the speed of the center of mass. When the force no longer is in effect, conserved quantities are restored.
In our example, there is a frame where the force transfers momentum in zero time. If the separation of the events:
1. remove momentum p from the object A, and
2. add momentum p to the object B
is spacelike, then there exists a frame where the momentum transfer is in zero time.
We conclude that an "object" of zero energy can transfer momentum, as long as the transfer takes zero time in at least one frame. But if the momentum transfer is timelike, then conservation of the speed of the center of mass is broken.
Electron movement caused by zero-point fluctuations? the Lamb shift
The Wikipedia article contains the Lamb shift derivation by Theodore A. Welton (1948):
Welton assumes zero-point fluctuations which "smear" the position of the electron.
Hans Bethe in his 1947 papers offers a different explanation for the Lamb shift, using electron self-energy and mass renormalization.
Michael I. Eides et al. (2000) have written a very detailed QED treatment of the Lamb shift. The main contribution seems to come from vertex corrections.
We do not believe in the existence of zero-point fluctuations. A QED derivation does not assume them, but could the fluctuations be hidden in the regularization and renormalization of the diagrams?
Since the electron and the proton are both present, hypothetical zero-point fluctuations might act as a temporary "force" between them. Then conservation laws would be honored.
The rubber plate model of the electron electric field and the Lamb shift
In our rubber plate model of the electron electric field, the rubber plate lags behind in the accelerating motion of the electron toward the proton. In the receding phase, the rubber plate leads the electron and pulls the electron away from the proton. The QED vertex correction obviously is about this process.
In the rubber plate model, the effective electron mass is lower when it is close to the proton, because part of the rubber plate and its inertia do not have time to react. If the electron were less massive, its lowest energy according to the Sommerfeld fine structure expression would be higher.
Question. Does the rubber plate model explain the Lamb shift by making the electron effectively less massive when it is in large acceleration close to the proton?
Since the electron is "lighter" close to the proton, it moves past the proton quicker, and has less time to collect momentum. Consequently, the de Broglie wavelength close to the proton is longer than without this effect. A longer wavelength forces the orbital to become a little wider, which makes the energy level a little higher.
A non-relativistic formula for the momentum p close to the proton is
-V(r) = p^2 / (2 m_e)
which gives
p = sqrt(-2 V(r) m_e).
The effective mass of the electron is reduced roughly by V(r) / c^2 because the far electric field does not have time to react to the acceleration of the electron.
At the distance 3 * 10^-14 m, the effective mass is ~ 10% lower.
If the effective mass is 10% lower, then p might be ~ 5% lower. The de Broglie wavelength is ~5% larger.
We conclude that the reduction of the effective mass has a "significant" effect when r < 3 * 10^-14 m.
In Welton's calculation of the Lamb shift, the "smearing" has a significant effect on the effective Coulomb potential - and the de Broglie wavelength - when r < 3 * 10^-14 m.
This very rough calculation suggests that the rubber plate model might explain the Lamb shift.
The Darwin term
The Darwin term is 90 μeV while the Lamb shift is 4 μeV. The rubber plate model might be able to explain the Darwin term, too, though we doubt it.
The Darwin term is explained either by zitterbewegung or by the Foldy-Wouthuysen transformation.
When the electron is accelerating, its wave becomes a chirp: negative frequency waves (= the positron) are mixed with the normal positive frequency electron waves. That might explain zitterbewegung.
The Darwin term is related to the structure of the Dirac equation. We have not found a classical model for the spin and the magnetic moment of the electron.
Why the Welton method of zero-point fluctuations yields a numerically correct estimate for the Lamb shift?
If the rubber plate explains the Lamb shift, then zero-point fluctuations are not needed in the explanation. But this brings up another question: why zero-point fluctuations give a numerically correct result?
virtual photon q
~~~~~~
/ \
e- -------------------------------
| virtual
| photon
| p
Z+ -------------------------------
The reason might be that the vertex correction produces an effect which imitates hypothetical zero-point fluctuations. Various possible paths store some amount q of momentum to the electric field of the electron. The momentum is later returned to the electron. These effects might mimic random pushes caused by hypothetical zero-point fluctuations.
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