The wave-particle duality of electricity: this explains the gyromagnetic ratio 2 of the electron?

Let us have a "generator" of electron-positron pairs. It sends electrons slowly upward and positrons slowly downward.

                                ^

                                | spin-z

                     electron cloud

                                ^

                                |

                         (======)    "generator"

                                |

                                v

                    positron cloud

                                | spin-z

                                v 

We imagine that the generator is analogous to a classical rotating electric dipole which produces a circularly polarized radio wave.

We first imagine that the electron-positron wave is a classical wave. The generator creates classical "clouds" of positive and negative charge.

The clouds have angular momentum, just like a rotating electric dipole gives up ħ of angular momentum per produced photon.

We assume that the z component of the angular momentum of the cloud is 1/2 ħ per produced elementary charge. The negative cloud and the positive cloud rotate to the opposite directions.

The wave-particle duality

This setup reminds us of the 19th century belief that electricity is some kind of a liquid which can flow. The particle model of electricity came later, when radioactivity was studied around the year 1900.

For electromagnetic waves, people recognize that a strong wave is a classical object whose quantum is the photon. We are very much aware of the wave-particle duality for electromagnetic waves.

On the other hand, we tend to conceive the electron purely as a particle. The classical aspect of an electron cloud is usually forgotten.

"Collapse" of a wave function

Absorption of photons shows that a weak classical electromagnetic wave somehow can "concentrate" its energy into a small spacetime 4-volume and excite, for example, a hydrogen atom.

An analogous process might be that we have a diffuse classical cloud of negative electricity, and the charge "collapses" into a small spacetime 4-volume, to be observed as an electron.

The gyromagnetic ratio 2 in the electron-positron generator

Our generator produces rotating clouds of electricity. The magnetic field of both clouds points to the same direction. Let the magnetic field of the rotating negative cloud be B.

Classically, we can measure a magnetic field strength of roughly 2 B, because the rotating negative and positive charge fields together form a "coil".

One may now ask that if the magnetic field "collapses" into a small spacetime 4-volume, what is its measured strength? Is it the field of a single electron, or double that field?

Empirically, it is double the field. The gyromagnetic ratio of the electron is 2.

Observing a single electron far from the generator

For photons, we have introduced in this blog the "teleportation" model where we conceptually move the rotating electric dipole very close to the absorbing antenna. That is the way to make the energy of the electromagnetic wave to "collapse" into a small spacetime 4-volume.

Similarly, we may imagine that when we measure the magnetic moment of the electron, we are doing it close to the electron-positron generator. That could explain the gyromagnetic ratio when we are observing an isolated electron.

Why we do not observe the electric charge of the positron when we measure an isolated electron?

The teleportation model has a hard time explaining why we measure -e as the charge of the electron, and do not see the positron charge +e around.

The explanation might be that the electric field has sources, while the magnetic field does not have. Let us measure the charge of a certain volume in the classical cloud of the electron-positron generator. We will classically get just the charge of the electrons there. On the other hand, the magnetic field classically is measured as 2 B.

The electron spin 1/2

In the electron-positron generator model, the pair is considered a single object. The summed absolute spin-z of the system is 1 ħ.

Thus, the wave of the pair does return to the same state after a single rotation. There is no problem of destructive interference which would happen if the system would have the spin-z 1/2 ħ.

If Nature has 1 ħ as the smallest possible absolute angular momentum to the z direction of an independent system, then the spin-z 1/2 ħ of the electron and the positron is the logical minimum possible spin. The spin of the photon is 1 ħ because the photon can be created independently from other particles. A single photon is an independent system.

The structure of the electron and the photon

Our new model claims that the "structure" of the electron is the imagined electron-positron generator and its classical electron and positron clouds very close to the generator. The clouds are rotating. That is the origin of the electron spin and angular momentum.

A collapse of the electron wave function teleports the generator very close to the observer.

This is similar to the photon. We claim that the "structure" of the photon is the rotating electric dipole which produces a classical electromagnetic wave. A collapse of the photon wave function teleports the rotating dipole very close to the absorbing system.

To explain the photon spin angular momentum, we do not imagine that the photon is a rotating little ball, or that it does some kind of zitterbewegung. Rather, we think that the photon, in an abstract way, possesses a part of the energy and the angular momentum of the classical electromagnetic wave.

It is a mistake to think that the electron is a tiny classical particle which does zitterbewegung to produce the spin and the magnetic moment. Rather, the electron is a "collapse" of a classical cloud of charge. The properties of the electron reflect the properties of that classical cloud. In this thinking, the classical limit is reflected in the quantum world. One might claim that the classical limit is fundamental and the quantum, the electron, secondary.

Conclusions

Our new model has a similarity to the "pipe" model which we introduced a couple of years ago. The pipe was imagined to connect the electron to the corresponding positron of a pair. In the new model we explain the pipe functionality through the collapse of the wave function and the teleportation model.

We conjecture that if a particle carries a charge which must be conserved, then the particle must have the spin-z 1/2, and the "gyromagnetic" ratio for the corresponding field is 2. We need to check what is currently known about elementary particles.

We still do not understand why the Dirac equation gets the electron spin and the gyromagnetic ratio 2 right. Is it somehow aware of our electron-positron generator?

Does the new model solve the problem of the infinite energy of the electric field of a point particle? If the "particle" is just a quantum of a diffuse field, the problem may not arise.

Spin 1/2 rotation with a magnetic field: the phase change from the potential

In the Feynman lectures, an electron beam is turned through an angle. The usual way to do that probably is by applying a magnetic field.

                              ○        ○        ○

                              ○        ○        ○

     e- ●  --->

                              ○        ○        ○

                            magnetic field B

Let us use a magnetic field B whose lines of force are to the direction of the z axis. The z axis points up from the screen. We prepare a coherent electron beam whose spin-z points up. We want to turn the electron beam through 360 degrees using the magnetic field.

The magnetic field introduces a potential to the electrons, and the potential causes a phase shift to the electron wave function.

Let the electrons be non-relativistic, and their velocity v. Let the magnetic field strength be B.

The force is

       F = e v B.

We set F equal to the centrifugal "force":

       m v² / r = e v B

  <=>

       r = m v / (e B).

The length of the loop is 2 π r.

The magnetic moment of the electron to the direction of the z axis is approximately one Bohr magneton:

       μ = e ħ / (2 m).

The potential energy due to the magnetic moment is

       V = μ • B.

Let the kinetic energy of the electrons be much larger than V:

       E = p² / (2 m).

Because of the potential, the momentum p of the electrons in the magnetic field changes by a factor

       1 + 1/2 V / E

       = 1 + 1/2 e ħ B / p².

The de Broglie wavelength of the electron is

       λ = h / p.

The loop 2 π r contains

       n = 2 π p / (e B) * p / h

          = p² / (e ħ B)

wavelengths if we ignore the potential V.

Introducing the potential V causes a change of

       n * 1/2 V / E

       = p² / (e ħ B) * 1/2 e ħ B / p²

       = 1/2

wavelengths. That is, the wave function of the electron changes the sign.

We could explain the sign change of the electron wave function simply by the potential which the magnetic field imposes on the magnetic moment of the electron. There is no need to assume that the spinor is changed in the loop in any way.

"Rotation" around the y axis

Let us assume that we have prepared an electron to the spin-z = +1/2 state. The measuring apparatus points to the direction of the positive z axis.

We then rotate the measuring apparatus through an angle α around the y axis. We do another measurement of the spin.

We do not think it makes sense to speak of this experiment as a "rotation" of the spinor. We did not touch the electron at all and did not rotate its spinor.

Is it a rotation of coordinates? There is no need to rotate the coordinates if we turn the measuring apparatus.

It is a rotation of the measuring apparatus - not of the spinor or coordinates. We prepared the electron in the spin-z +1/2 state, and then measure an observable which does not commute with the prepared state. In particular, if α = 90 degrees, then we measure the spin-x, and have a 50% probability to get +1/2 and 50% for -1/2.

Conclusions

We need to find out if the spin 1/2 spinor rotation algebra in literature is a result of a confusion. People have not taken into account the magnetic potential which can explain a change of the phase in an experiment where a magnetic field is used to "rotate" a particle.

If the "rotation" would be made with an electric field, then the potential can be very small, and we might be able to change the path of an electron with a negligible change of the phase.

This reminds us of our analysis of the Aharonov-Bohm effect on December 21, 2021. Literature forgets that the potential from a magnetic field inside a solenoid does change the phase of an electron when the electron flies past the solenoid.

Classically, a spinning object tries to keep its axis, or precesses if there is a torque. What would the precession mean in quantum mechanics?

Rotation of a spinor: what does it mean?

UPDATE March 23, 2022: In the Pauli equation, the potential of the magnetic moment associated with the spin-z in a magnetic field B to the z direction is given by the σ • B term. The Pauli equation calculates the same phase shift in the wave function as we calculated below with our simplified "scalar electron" model.

https://en.wikipedia.org/wiki/Pauli_equation

The Feynman lectures do not mention at all how the electron beam is turned through an angle, and do not analyze what the turning force or potential does to the wave function. Feynman just assumes that the beam mysteriously turns.

----

Frame-dragging may offer an explanation for what a rotation of a spinor means.

Frame-dragging in a pool of water

                    --------

                /                \

              |                    |      

                \                /       

                    --------

                    <--- ω

Suppose that we have a round pool of water rotating with an angular velocity ω. Let there be a wave which circles the pool to the same direction, so that the angular velocity of the wave is 2 ω relative to a static observer. The pool of water is the frame which is being dragged.

The "rotation of a spinor" through an angle α means that the wave moves through the angle α relative to a static observer. The wave only moves an angle 1/2 α relative to the pool.

The angular position of the pool is considered relevant in the system. If we only specify the angle of the wave, we do not know the full state of the system. We have to know if the pool has been rotated less than 180 degrees or more than 180 degrees.

The rotation of a spinor through an angle α means that the wave has moved the angle α relative to a static observer.

The rotation is not a coordinate rotation. If it were, the system would return to its original state after a rotation of 360 degrees.

We can specify the state of the system by giving the angle  0 ≤ α < 360 of the wave and the angle 0 ≤ β < 360 of the pool.

The state is of the form

       (α, β).

Let us define that the rotation of both started from 0 degrees. Then

       β = α / 2,                                                 

   or                                                                    (1)

       β = α / 2 + 180.

What is the state if we rotate coordinates through an angle γ?

The new state is

       ( (α - γ) mod 360,  (β - γ) mod 360 ).

The rotations of the pool and the wave now started from -γ degrees. Formula (1) above no longer holds.

A rotation of coordinates is a different operation from watching the wave to move over an angle α.

Feynman lectures: a "rotation" of the spinor for an electron means that the electron moves along a physical loop

https://www.feynmanlectures.caltech.edu/III_06.html

Richard P. Feynman (1965) explains what we mean by a "rotation" of a spinor. He has several diagrams which show a beam of electrons turning, for example, 90 degrees to the left.

If the beam makes a full 360 degree turn, then it is claimed that the electron spinor changes its sign.

If the 360 degree loop adds an integer number of wavelengths to the path of the electron, the flip of the spinor sign will cause destructive interference with another beam which did not do the 360 degree loop.

Feynman writes that the "physics" does not change if the phase of the beam wave function changes. For example, the sign flip of the beam which made a 360 degree loop does not change anything, unless we let it interfere with a beam which did not do the loop.

It would be better to say that the phase shift does change the physics. In this blog we have claimed that a phase change of a wave function can only happen if it interacts with some external system. It is a "physical" interaction and it does change the "physics".

The spin-statistics "theorem"

https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem

In our blog post on March 5, 2021 we criticized the presented proof of the theorem in Wikipedia.

Let us analyze the Wikipedia proof again. There 

 

         R(π)

denotes a 180 degree "rotation" of the "spin polarization" of a particle annihilated by φ(x) at a location x. The text says that

      R(2 π) = -1.

Thus, R has to mean a physical rotation of the spinor, not of the coordinates.

But under the headline "Suggestive bogus argument", there is a coordinate rotation through 180 degrees. The Wikipedia author seems to confuse the two types of rotations:

1. A spinor rotation is a real physical process where, in our water pool example, the wave travels through an angle α as seen by a static observer. In the Feynman lectures, a beam of electrons is physically turned the angle α.

2. A coordinate rotation is just a change of coordinates. It is not a physical process. It a change of notation.

The Minkowski cosmological model and baryonic oscillations

The Milne model is not the right model for cosmology in

        the Minkowski & newtonian model

     +

        negative gravity charge particles.

That is because in the early universe pressure accelerates the expansion.

Let us call our new cosmological model the Minkowski cosmological model.

The name stresses the fact that after the initial pressure-dominated stage, the large-scale metric is Minkowski.

The Minkowski model is a bounce-back model. We can, in principle, trace the history of the universe to the time before the Big bang. There is no singularity. We expect matter and dark matter to bounce back if we calculate backward in time past the time of the Big bang.

Baryonic oscillations and the CMB

Baryonic oscillations are in ΛCDM the mechanism which is assumed to make perturbations in the cosmic microwave background (CMB) larger at feature sizes 1 degree, 0.4 degrees, and 0.25 degrees of arc.

In ΛCDM we are seeing the CMB from a radius which was only 30 million light-years when the photons were emitted. Features with a radius 0.01 radians are somewhat stronger than other sizes. The original radius of the feature in light-years was

       0.01 * 30 million light-years

       = 300,000 light-years.

The "current" radius is 330 million light-years, because the scale factor has grown by a factor 1,100.

In the Minkowski model we are seeing the CMB in which the photons were emitted some 14 billion years ago from the distance (in the Minkowski metric) of 14 billion light-years.

Large-scale distances have since then increased by a factor ~ 1,100. A cluster if galaxies may now have a radius of 330 million light-years. It was just 300,000 light-years when the CMB photons were emitted.

The radius 300,000 light-years is just 1 / 1,000th of a degree of arc viewed from 14 billion light-years away.

The resolution of the Planck probe map of the CMB is 0.07 degrees of arc. We cannot discern the seeds of the current clusters of galaxies in the map.

Can we in the Minkowski model explain why features whose size is 1 degree are strong in the CMB map? The radius 0.5 degrees corresponds to 140 million light-years when viewed 14 billlion light-years away.

In the Milne model, the age of the universe was

       14 billion / 1,100 years

       = 13 million years

when the CMB photons were emitted. The Milne model has a hard time explaining features of a size 140 million light-years.

In our Minkowski model, the expansion accelerated because of the pressure after the Big bang. It might be that the age of the universe was already 140 million years when the CMB photons were emitted.

We have to calculate the age, assuming some pressure in the early phase. If the age turns out to be less than 140 million years, then we have to explain features of the size 140 million light-years by something which happened before the Big bounce.

Conclusions

Let us assume that the Big bang actually was a Big bounce. We have to figure out what was the pressure in the early stage after the bounce, and how much time it took for the CMB photons to be freed.

Maybe some of the Big bounce proponents has already calculated this?

Can we explain the 1 / 100,000 uniformity of the CMB with a Big bounce model?

Is it certain that the Minkowski model, when calculated back in time, yields a Big bounce? The result might also be black holes of some kind.

The electron spin 1/2 comes from frame-dragging?

Aurelien Benoit-Levy and Gabriel Chardin (2012) in their paper mention an article by Brandon Carter (1968):

https://journals.aps.org/pr/abstract/10.1103/PhysRev.174.1559

The Kerr metric of a charged rotating black hole would explain the gyromagnetic ratio 2 of the electron.

https://www.researchgate.net/publication/350418929_Image_of_the_Electron_Suggested_by_Nonlinear_Electrodynamics_Coupled_to_Gravity

Irina Dymnikova (2021) has written a review article about the Schrödinger zitterbewegung approach to the "structure" of the electron, as well as the models inspired by the Kerr metric.

A couple of years ago in this blog we tried to explain the electron spin angular momentum

        1/2 ħ

and the gyromagnetic ratio 2 of the electron. We failed to find an explanation.

The magnetic moment of the electron suggests that it is orbiting at the speed of light along a circular path whose length is the Compton wavelength of the electron. This loop would be the zitterbewegung loop.

But the spin angular momentum says that the length of the loop is only 1/2 of the Compton wavelength.

The mystery:

How can the electron only do a half a wavelength in a loop? In an atom that would result in total destructive interference of the electron wave function. Such an orbit should be forbidden.

Frame-dragging is the solution?

The electron is a charged particle. We have explained in previous blog posts that it drags the frame of other charged particles.

Could it be that the electron drags its own frame? Then it could make a full circle from the point of view of an external observer, but in the dragged frame it only makes a half of a full circle. The wave function of the electron does not yet meet itself and destroy itself in the interference.

The magnetic field of the electron is not charged and is not affected by frame-dragging.

We conjecture that frame-dragging soļves the mystery. It would explain why a spin 1/2 particle has to be "rotated" 720 degrees for it to return to its original state. A rotation of 720 degrees in the global frame is a rotation of only 360 degrees in the dragged frame.

The Poynting vector: the inertia of the field of the electron

Much, or all, of the inertia of the electron comes from its field. Could it be that in the tight zitterbewegung orbit, the inner field of the electron does not contribute much to the inertia?

Or the electron has to be treated as a loop of electric current, and the current drags its frame along with it?

This could also solve the famous 4/3 problem. If the electron in a linear motion drags its frame, the momentum of its field may appear too large if calculated without taking into account the frame-dragging.

If we build the charge of a moving electron from small parts, then the movement of other parts may help a new part to move. This is analogous to the small charged shell which is expanding and contracting inside a larger charged shell. The collective movement of the charges helps to reduce energy shipping (= the Poynting vector) in the field.

This could also solve the renormalization mystery of the field of a point charge, like the electron. If frame-dragging is 100% in the inner field within the classical radius of the electron, then the inner field does not contribute at all to the inertia of the electron. The inertia of the electron is finite rather than infinite.

Let us recapitulate. The following open problems are all about the surprisingly low inertia of an electron in a movement:

1. The spin 1/2 and the gyromagnetic ratio 2 in the zitterbewegung loop.

2. The 4/3 problem in a linear motion.

3. The renormalization problem of the inertial mass of the electron. Why the inertia is not infinite in all kinds of motions?

We are used to thinking that a 1 kg mass has the same inertia in all kinds of movements. We argued in previous blog posts that its inertia would vary greatly depending on the movement, but a dark energy mechanism cancels the effect of distant galaxies.

In the case of the electron, its inertia seems to differ in the circular zitterbewegung motion from its inertia in a linear movement.

Think of an air-filled spherical balloon which is sunk into a water pool. If we move the balloon linearly, there is large inertia, which is caused by water flowing around the balloon. But we can rotate the balloon around its axis with minimal inertia. The zitterbewegung of the electron may be analogous.

Conclusions

The classical radius of the electron is only 2.8 * 10⁻¹⁵ m. The radius of the zitterbewegung loop is the reduced Compton wavelength 3.9 * 10⁻¹³ m, or 137 times larger. The electromagnetic field is strong enough to cause significant frame-dragging only at a distance which is at most a few times the classical radius. Could it be that the length contraction of the field in the light-speed loop makes the field strong enough to cause significant frame-dragging?

Let us assume that the frame-dragging is not electromagnetic. We conjecture that the frame-dragging is caused by the unknown force which makes the electron to circle in the zitterbewegung loop. Let us call the force the zitterbewegung force.

We will next study the properties of an arbitrary force which causes a particle to orbit in a tight loop. Let us have a frame which rotates with the particle. The "potential" of the centrifugal force is roughly -1/2 m c² at the distance where the electron moves at the speed of light. The potential of the zitterbewegung force is strong enough to cause significant frame-dragging.

We need to find a logical model for the motion of the electron. How does the wave function behave under such extreme conditions?

The Milne cosmological model follows from negative gravity charge particles

There is negligible extra inertia in a linear movement of a point test mass, compared to the inertia when a small shell of mass expands and contracts.

In our previous blog post we claimed that having negative gravity charge particles in dark matter and demanding that the total gravity charge of the universe is zero is the most beautiful way to explain the lack of extra inertia.

https://arxiv.org/abs/1110.3054

Aurelien Benoit-Levy and Gabriel Chardin (2012) have studied just this type of a cosmology.

Since gravity charges cancel each other at a large scale in the universe, the expansion has a constant velocity for a typical baryon. Gravity does not slow down the expansion. Neither is there a cosmological constant to speed up the expansion.

https://en.wikipedia.org/wiki/Milne_model

Edward Arthur Milne introduced the constant expansion velocity model in 1935.

Benoit-Levy and Chardin write that the uniformity of the universe can be explained without inflation in the Milne model.

Let us list advantages of the Milne model:

1. The Milne model explains why a point test mass in a linear movement has no extra inertia compared to an expanding and contracting small spherical shell of masses.

2. There is no ad hoc cosmological constant Λ.

3. No need for ad hoc inflation.

4. Gravity charges are like electric charges: charges cancel each other out at the large scale.

5. The Milne model is consistent with the ages of the oldest stars ~ 14 billion years.

6. The Milne model is simple. The metric for static observers is the Minkowski metric.

Disadvantages:

1. We have to assume ad hoc dark matter which has negative gravity charges.

2. The Milne model says that, in comoving coordinates with matter, the spatial metric has negative curvature. This may contradict some CMB data.

We need to check if there are any observations which might refute the Milne model.