There is a regularization/renormalization problem in classical electromagnetism: if the electron is a point particle, then its static electric field has infinite mass-energy, assuming that the energy density is
~ E^2,
where E is the electric field strength.
We suggested in an earlier blog posting that maybe the energy of the static electric field is zero. Because of the finite speed of light, we can deform the field by accelerating the electron, and in that way store momentum and energy into the field, even though the field has zero "rest mass".
In our rubber plate model of the electric field this means that the mass of the plate is zero. Its apparent inertia is a result of the finite speed of light and the tension when the plate stretches.
In our electric field line model, electric field lines would be massless, but they still have tension, and can mimic the effects of inertia because they exert a force on the electron.
Electromagnetic waves propagate at the speed of light. If the field would have non-zero rest mass, then the waves would move slower. This suggests that the rest mass of a static electric field is zero. A static field is at rest: its mass has to be zero if the rest mass is zero.
Having a regularization problem in classical physics is ugly. In this blog we try to get rid of regularization in QED. We do not want the problem to persist in classical physics.
The famous 4/3 problem of classical electrodynamics would be solved by assuming zero mass-energy in the static electric field.
The anomalous magnetic moment of the electron
A week ago we presented a model where the effective mass of the electron is reduced when it circles the zitterbewegung loop. That would explain the anomalous magnetic moment, whose ratio is roughly
1 + α / (2 π) = 1 + 1 / (137 * 2 π),
where α is the fine structure constant.
Our argument assumed that the far field of the electron does not have time to react to the high-frequency circling motion of the electron, and that is why the electron has a lower effective mass.
In the previous section we suggested that the mass-energy of a static field actually is zero. How to save the anomalous magnetic moment?
There is no energy flow out from the system. The electron circles forever without losing energy.
We can prevent the energy flow by making the far field static: the field lines do not move at all despite the circling motion of the electron.
If the field lines cannot move far away, they must bend and stretch. That causes tension. The circling electron feels an extra centripetal force.
The centripetal force, which keeps the electron on its orbit, gets some help from the tension of the field lines. The net effect is the same as if we would make the electron lighter: in that case the centrifugal "force" would become weaker.
How much energy we can steal from an oscillator, using a small inertial mass?
Let us have a mass M doing circular motion at a velocity v with a radius r.
The acceleration
a = v^2 / r.
Suppose that we can with a small mass m somehow make a force
F = m a
on the big mass M and the force is always antiparallel to the motion of M. The drained power is
P = m v^3 / r.
The drained energy during one round is
E = m v^3 / r * 2 π r / v
= 2 π m v^2.
Suppose then that we do not want to drain any energy. We turn the force F = m a into a centripetal force for M. The net effect of that is to "reduce" the mass of M by m.
Example. Case A.
power P --->
___ ___
/ \___/ \__... tense rope
●
rotate
with hand
Consider the following example. We hold the end of a long tense rope in our hand. We rotate it to make a circularly polarized wave. The wave carries away some power P. In our hand we feel a tangential force F against which our hand has to do work. The tangential force is
F = T * 2 π r / λ,
where r is the radius of the rotation, T is the tension of the rope, and λ is the wavelength.
rotate
with hand
● ____
\
● attached to wall
Case B. We want to stop the power drain P. A simple way is to make the tense rope short and attach the other end to a wall. Let us rotate the rope. Now the tension and attachement to the wall adds an extra centripetal force to the rope. The force is felt in our hand if the distance to the wall is less than 1/4 of the wavelength.
Question. Quantum mechanics says that energy cannot be drained from an electron which is already in its lowest energy state. What does the field of the electron look like far away? Static, like we would have attached the rope to a wall?
We do not need to assume that there is any inertial mass m in the static electric field if we can produce the force F by some other means. The calculation stays the same.
The Edward M. Purcell derivation of the Larmor formula does not assume any mass m. It just calculates the extra energy in electric field lines which contain a "wrinkle" and are forced to become denser than in an undisturbed field.
Where does the mass of the electron reside?
If the mass-energy of the static field is zero, then the mass of a classical electron is in the point particle itself.
But the quantum electron does zitterbewegung, and its field has a magnetic moment. How does that affect the localization of mass-energy?
A static electric field which is in linear motion appears to include a magnetic field. We can say that the magnetic field is massless, too.
What about circular motion?
If we attach the other end of a tense massless rope to a wall, and rotate the rope in our hand, there certainly is energy in the motion of the rope.
How much energy there is in the field?
The rubber plate model of the electric field suggests that the near field travels faithfully with the electron. The resonant oscillation frequency of the near field is much higher than the frequency of zitterbewegung.
The radius of zitterbewegung is the Compton wavelength divided by 2 π, that is, 4 * 10^-13 m. The classical mass-energy of the field outside that radius is just 1 / 137 of the mass of the electron.
The anomalous magnetic moment suggests that the reduced mass of the electron in the zitterbewegung loop is
1 - 1 / (137 * 2 π)
times the measured mass of the electron. It is a good guess that the mass-energy of the rotating electric field is equal to this missing mass.
In collision experiments, the electron looks like a point particle. If the mass of the electron would reside in the field which is farther than 3 * 10^-15 m from the particle, that would probably show up in experiments. The electron would appear as a soft, large ball. This supports our conjecture that almost all of the mass of the electron is in the point particle itself.
But a fraction 1 / (137 * 2 π) of the mass resides in the rotating field of the electron, and quite far away, since 4 * 10^-13 m is a long distance in collisions. This fraction may be the reason for some first order corrections which are calculated from Feynman diagrams. We need to think about this.
What is the near electromagnetic field of the electron like?
The charge of the electron moves at the speed of light along a circle whose radius is 4 * 10^-13 m.
In classical physics, no charge can move at the speed of light. This makes it problematic to find out what the near field looks like.
Could it be that the speed of the charge is slightly less than the speed of light? Zitterbewegung is derived from the Dirac equation and it happens at the speed of light. Maybe QED corrections change the situation a little and drop the speed lower?
If we drop the speed by 1%, then the radius has to grow 1% since the wavelength grows by 1%. The current around the circle drops by 2%, and the area grows by 2%. The magnetic moment stays constant. It is not mandatory that the charge moves at the speed of light. We get the same effect from a slower speed.
The "correct" classical model for the electron may be one where it moves slower than the speed of light.
No comments:
Post a Comment