Almost all the mass of the electron is in the point particle, and the field has almost zero mass?

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.

Another heuristic proof of the Pauli exclusion principle: particles can only enter the same state through a spin-z = 0 state

Our blog post on March 7, 2021 left open several questions.

Our argument was that we have to be able to sum wave functions linearly. If ψ represents one particle, then 2 ψ would represent four particles, since the probability density is the wave function squared. That prohibits two particles from going into the same state.

If the requirement of a linear sum can be relaxed temporarily, then two particles could go to form the wave function

        sqrt(2) ψ,

which does conserve the number of particles, and is thus allowed.

We also claimed that a wave function must, in some sense, rotate if we want to conserve the number of particles. Rotation means a spin value which is not zero.

That suggests that if the spin-z of the particle can be 0, then the number of particles in that state is fuzzy, and we cannot demand linear summation of the wave function of each particle because there is no clear number of particles. Then linearity can be broken.

Two particles can then end up in the same state by going through the spin-z 0 state.

That explains why particles with a spin

        n + 1/2

can never end up in the same state, but particles with a spin

        n

can. Particles with a spin n possess a spin-z state 0, which allows linear summation of wave functions to be relaxed.

Electron-positron annihilation

If we sum the angular momenta of a spin-z 1/2 and spin-z -1/2 electrons, then apparently, the angular momentum is not zero. The angular momenta are not opposite.

But if we sum a spin-z 1/2 electron and a spin-z -1/2 positron, then the angular momentum is zero, we think. We have to check the spinor arithmetic from some source.

The positron is the time reverse of the electron. It makes sense that the angular momenta of an electron and a positron can be truly opposite.

In annihilation, an electron and a positron form a combination whose spin is zero. Annihilation changes the particle number. We suggest that a change in the particle number is possible when the total angular momentum of the system is zero.

There is a paradox in the Dirac equation: if it conserves the probability current, how can a particle be created at all? The solution is to use an interaction which breaks the linearity of the system. It may be that breaking the linearity requires that the total angular momentum is zero.

In the February 11, 2021 blog posting we introduced the hypothesis that a photon is a combination of a rotating electron and a positron. We conjectured that in pair annihilation, a zero energy photon reacts with the annihilating pair and helps to build the two outgoing photons. The concept of a "zero energy photon" is dubious, however. It assumes a particle in empty space. Maybe it is better to assume that the photons can be created from nothing, which means that pairs can be created from nothing, too.

The fact that the positron is a time reverse of the electron suggests that in annihilation, the wave function somehow forms a bridge between the two particles, just like in the Feynman diagram. Richard Feynman wrote that an electron is "scattered backward in time".

If the Dirac wave function stubbornly wants to conserve the number of particles, then changing the number may require turning the wave function backward in time. The backward branch is the positron.

Pauli exclusion says that the wave functions of electrons have to be orthogonal - particle-in-a-box

In the particle-in-a-box model known from textbooks, different stationary states (energy levels) have orthogonal wave functions. The integral (the inner product) over the product of two wave functions is zero.

Putting n particles to different states conserves the number of particles. We can sum the wave functions and get a description of the system state, without a need to construct a direct product of the wave functions.

The following two conditions are equivalent to the Pauli exclusion principle in this simple case.

1. We can sum the n individual wave functions and get the system wave function.

2. The particle number must be conserved.

Since the particles are indistinguishable, a direct product of wave functions does not sound right. If we cannot keep track of which particle is which, why and how to construct a product? How does that product react to spontaneus switching of particles?

However, if there is interaction between particles, then a direct product sounds the right solution. We have problems defining the interaction if we do not have separate coordinates for each particle.

In our blog postings in 2018 we remarked that there is no proof that an interacting system of n > 1 particles has any definite "quantum state" for each individual particle. It is an empirical observation that electrons in atoms seem to occupy states which are similar to the states of the hydrogen atom, and that the electrons seem to obey the exclusion principle.

Suppose that there really is a meaningful "quantum state" for each electron in an atom. Then we can claim like in the particle-in-a-box case that putting two electrons into the same state would increase the number of particles by two. Pauli exclusion would follow from the requirement that the particle number must be conserved.

Conclusions

We have argued that Pauli exclusion follows from:

1. Wave functions can generally be summed to obtain a new wave function.

2. The number of particles must be conserved.

Assumed antisymmetry of a direct product of electron wave functions may have nothing to do with the Pauli exclusion principle.

We presented a hypothesis that bosons can enter the same state through the spin-z state 0, where the number of particles cannot be defined. For fermions, that route is not available.

Why is the value of the fine structure constant 1/137?

Short answer: there is no reason. It is just a random value.

----

The measured value of the fine structure constant is

       α = 1 / 137.035 999.

https://en.wikipedia.org/wiki/Fine-structure_constant

The fine structure constant is, among other things, the ratio

        electron classical radius
        -----------------------------------------------------
        radius of the electron charge spin loop

The charge spin loop means the imagined circle which the electron charge makes at the speed of light, and which produces the electron magnetic moment. The radius is

       r_m = λ_e / (2 π),

where λ_e is the electron Compton wavelength.

The electron mass spin loop radius is

       r_s = 1/2 r_m.

The mass spin loop is the imagined circle which the electron mass makes at the speed of light, and which produces the spin angular momentum.

The classical radius is determined by the energy density of the electric field and the mass of the electron.

In a sense, the fine structure constant is the ratio

       energy density of the electron electric field
       ---------------------------------------------------------
       Planck constant.

Does this ratio have a random value or is it determined through some physical mechanism? The numerical value of the ratio looks random.

The role of the fine structure constant in the photograph model of quantum field theory

Let us analyze the fine structure constant from the point of view of our photograph model.

The Planck constant determines the resolution of the photograph of classical processes. It determines how "sharply" we can see the deep down classical processes. Let us call our impaired vision the Planck microscope.

The electron electric field and its energy density are classical physical things. They are things which we look at through the Planck microscope.

Conceptually, the Planck constant is at a different level from the electric field. The Planck constant is much more fundamental. The electric field is just a random object which we look at through the Planck microscope.

It would be surprising if the Planck constant would be related to a random field in nature - the electric field. Thus, there probably is no deeper reason for the value of α.

Which things in the electron are classical and which are quantum (i.e., wavelike)?

Electromagnetism is a classical thing. The electric field of the electron is classical.

The existence of the spin is a quantum thing. It is required by the wave description of the electron, to conserve the number of particles.

The angular momentum in the spin is a classical thing. But we look at the angular momentum through the Planck microscope, which makes it behave in a quantum way.

The magnetic moment of the electron is a classical thing. The fact that the gyromagnetic factor is 2, is a quantum thing which arises from our impaired vision through the Planck microscope. We cannot see the mass spin loop because it has too small a radius.

The mass of the electron might be a quantum thing because the classical mass-energy of its static electric field is infinite. Another possibility is that the classical mass of the electron (if any) is in the point particle, and the "true" mass of its static field is zero. The field can store momentum, but it might be a result from retardation, and "tension" of electric field lines.

We will look at the mass problem when we analyze diverging Feynman diagrams.

A possible explanation for why the electron has the spin 1/2; a new proof of the Pauli exclusion principle

Suppose that an electron is confined in a finite volume. Classically, it bounces back and forth there.

How to define a standing wave which describes the movement and has a sensible probability density and a conserved probability current?

       ____         ____

     /         \___/        \   t = 0

     ________________  t = 1

     linearly polarized standing

     wave is identically

     zero at certain moments

If we try a linearly polarized wave, that is not going to succeed. A linearly polarized standing wave is at certain moments identically zero. That would spoil the probability density.

             

              _______

     ____/              \____

     jump rope rotates

      around the axis

A better idea is to make a rotating standing wave, like girls' jump rope. The value of the wave function stays non-zero at most places. It is a circularly polarized wave.

Where would that wave rotate? In some abstract space?

Hypothesis. Waves in nature must live in the familiar 3 + 1 -dimensional physical space and must hold energy and momentum. They have to be classical waves in that sense.

Electromagnetic waves satisfy the hypothesis.

We suggest that the rotating electron wave lives in the familiar 3 + 1 -dimensional space. It contains angular momentum. That angular momentum is the spin of the electron.

The Dirac equation solves the following problems in the massive Klein-Gordon equation:

1. There is too much freedom in specifying the initial values for the massive Klein-Gordon equation because it is a second order differential equation. We have to specify both the wave function value at a time t, and its temporal derivative.

2. One cannot define a probability density and a conserved probability current. This is probably associated with item 1.

The Dirac equation is a reasonably simple way to implement a probability density and a conserved probability current. Classically, the simple way is to use girls' jump rope, which inevitably includes a spin. This may be the underlying reason why there is a spin in the Dirac equation.

According to this reasoning, the existence of the spin follows from:

1. The wave equation has to be relativistic. It cannot be the Schrödinger equation.

2. There has to be a probability density and a conserved probability current. A spin is a simple way to implement that.

Why is the electron spin 1/2?

Why is the electron spin 1/2? Why not 1/4 or 1?

To have a conserved probability current, we have to avoid linearly polarized standing waves.

If the spin would be 1, then we could model the electron as a classical particle which moves at the speed of light along a circle whose length is one Compton wavelength. What if we would put another electron to circle to the opposite direction? The result might look like a linearly polarized standing wave. Particles with spin 1 (like the photon) can posses spin-z values 1, 0, -1. The value 0 is linearly polarized. 

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

The electron wave function in the Pauli equation is a two-component spinor wave function.

Spin-z states 1/2 and -1/2 are orthogonal in the spinor space of the Pauli equation. Suppose that we have two electrons with spin-z 1/2 and -1/2 confined into an "almost same" state. That is, the wave functions of the two electrons are of the form

       (0, ψ)   and   (ψ, 0),

where we have used spin-z as the basis.

The sum of the two wave functions is

       (ψ, ψ),

which is just right, so that the square

       ψ^2 + ψ^2

of the wave function corresponds to exactly two particles.

This did not yet explain why the electron spin has to be 1/2. The Dirac equation is kind of a "square root" of the massive Klein-Gordon equation. That might be associated with the value 1/2.

An electron spin 1 would be ok if we could ban the spin-z state 0.

A new proof of the Pauli exclusion principle, and a Pauli exclusion principle for photons

Note that if the spin-z for the two electrons were the same, then the sum of the two wave functions would be, e.g.,

       (2 ψ, 0).

The square of the wave function

       4 ψ^2

would correspond to four particles. That would break conservation of the probability current.

We get a new proof for the Pauli exclusion principle. If two electrons would end up in the exact same state, then the probability density of the summed wave function would be 4-fold, which would break conservation of the probability current.

The proof assumes:

1. We can sum two wave functions linearly.

2. The probability density of the summed wave function is obtained by squaring it.

In nature, many waves allow linear summation. The square of the wave function is often the energy of the wave. It makes sense to associate the energy with the number of particles.

More precisely, conservation of the probability current requires that electrons in a similar state must have their spinor vectors orthogonal:

Strong Pauli exclusion principle. If two electrons are in states

       (s, s') ψ   and   (r, r') ψ,

then the vectors (s, s') and (r, r') have to be orthogonal.

We get a "Pauli exclusion principle" for photons, too:

Pauli exclusion for photons. If two photons were created separately, they can never end up in the same state. The electric and the magnetic fields would be double in such a configuration, which would make the energy 4-fold. Conservation of energy prohibits such a state.

Photons can be created in the same state. A radio transmitter creates a huge number of coherent photons. Energy is conserved because the radio transmitter supplies the required energy.

There is a classical version of Pauli exclusion:

Classical Pauli exclusion principle. If two classical waves are created separately, they can never end up (without adding energy) in a configuration where constructive interference would dominate over destructive interference. That would break energy conservation.

It turns out that the Pauli exclusion principle is an energy conservation principle.

We cannot create electrons in the same state. If we could, we might be able to break the Pauli exclusion principle.

Question. Is the electric repulsion between electrons a result from the Pauli exclusion principle? Suppose that we have a small box and put more and more electrons there. They must occupy higher and higher energy states because of Pauli exclusion. Classically, we must win the electric repulsion. How do the energies compare?

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

Pauli exclusion causes electron degeneracy pressure.

At short distances, degeneracy pressure is stronger than the Coulomb repulsion. In the hydrogen atom, degeneracy pressure balances the Coulomb attraction.

We cannot derive the Coulomb force from degeneracy pressure.

Could it be that some kind of degeneracy anti-pressure is the mysterious attractive 1 / r^2 potential in pair annihilation?

How does the traditional antisymmetric wave function proof differ from our proof for Pauli exclusion?

In the traditional proof, the two electrons are distinct individual particles, and their combined wave function is some kind of a product of individual wave functions.

In our proof, the combined wave function of two electrons is the sum of wave functions. The electrons are kind of "bulk material" for the wave function. This is analogous to photons, for which we believe summing the wave functions is the correct procedure.

Which approach is right?

1. Electrons are indistinguishable particles. The traditional proof seems to assume that we have somehow marked them. That is dubious.

2. We do not know how to model the Coulomb potential between electrons if they are not particles. Simply summing the wave functions ignores this aspect.

Another question is how an electron-positron pair is created. If they are particles, where exactly these particles pop up in spacetime?

Photon wave functions can be summed, and must be summed to explain, for example, the interference pattern of two laser beams.

Let us check if coherent beams of electrons have been made.

https://www.nano.physik.uni-muenchen.de/nanophotonics/_assets/pdf/2015/PhysRevLett.114.227601.pdf

D. Ehberger et al. (2015) report about a highly coherent electron beam from a tungsten needle tip. It looks like electrons can behave like "bulk matter" just like photons. Then summing the wave functions is the right procedure.

Let us check the literature. Has anyone invented our simple proof for Pauli exclusion before us?

Proof of the spin-statistics theorem: can we really prove the Pauli exclusion principle?

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

Let us have two electrons, 1 and 2. We imagine that they are marked, so that we can distinguish them.

Let us prepare the two electrons in some way for our experiment, at positions which initially have spacelike separations. We calculate their wave function with a path integral. Let y be the coordinate of the electron 1 and z the coordinate of the electron 2. The coordinates may include also other parameters besides the R⁴ coordinates of the Minkowski space.

Let us prepare the electron 1 with a spinor s at an initial location -x.

Let us prepare 2 at the location x. The initial wave function of 2 is obtained by rotating the wave function of 1 through 180 degrees in a plane which includes time. The spinor s is transformed in the natural, smooth way to s'.

The wave function of the two particle system is initially

      R (φ) (x) (z) * φ(-x) (y),

where R (φ) performs the rotation to the spinor and the wave function, and where * denotes some kind of a "product" of the two wave functions.

Let us rotate the coordinates smoothly through 180 more degrees with R. The spinor s' is transformed in the smooth way to s''. The rotation also flips -x and x.

The wave function in the rotated frame is

       R (R (φ)) (-x) (z) * R (φ) (x) (y).

For a spin 1/2 particle, the spinor rotation through 360 degrees from s to s'' flips the sign:

       R (R (φ)) = - φ

We thus have

       - φ(-x) (z) * R (φ) (x) (y) = 

  

         R (φ) (x) (z) * φ(-x) (y).

The roles of y and z are switched on the two lines. That flips the sign. In this special case, the wave function Φ of the two particle system has to be antisymmetric - otherwise we would break the spinor algebra.

Analysis of the proof

The idea in the proof seems to be that we must find a configuration of two electrons where the states of the electrons seem to switch through a formal coordinate transformation. We do not need to touch the electrons physically at all. It is just a coordinate transformation to another frame. We rotate our coordinate system through 180 degrees, and if we transform the wave function in a natural, smooth way, then the sign of the wave function flips.

In quantum mechanics, we are allowed to multiply the wave function by any complex number α, where |α| = 1, before starting the experiment. We are not allowed to multiply it in the middle of the experiment.

Are we allowed to flip the sign in a coordinate transformation? Then we could declare that the sign of the wave function did not flip in the proof above, after all.

Let us just ban extra sign flips as ugly.

Could there be configurations where we can switch the electrons, like above, but the wave function sign does not change?

Rotations through 180 degrees seem to flip the sign because the two spinors are rotated a total of 360 degrees.

Does the proof show that in the general case, the wave function of two electrons is antisymmetric?

First we have to decide what "switching the electrons" exactly means.

If we have two electrons, let us move to their center of mass frame.

The spinors point at arbitrary directions. Generally, we cannot switch the electrons through a 180 degree rotation.

Let us look into literature. Is there a general proof?

The Pauli exclusion principle

Our goal is to prove the Pauli exclusion principle: two electrons cannot end up in the same quantum state.

Let us assume that we prepared the two electrons like in the first section, and their wave function starts as antisymmetric.

Does the wave function stay antisymmetric? We do not know, but let us forget that problem for a while.

    

                    t

                    ^

                    |

                    |

                    |

   ------●-------------●-------> x

          e-                e-

The electrons start their life in the diagram above. In our coordinates they travel forward in time, but in the rotated coordinates, backward in time.

Let us assume that the electrons end up in some stationary states in our coordinates:

      |b > |c >  (our coordinates).

In our coordinates, that configuration has a wave function value A_bc.

In the rotated coordinates, the state is

       |c > |b >  (rotated),

and the wave function value -A_bc.

What about the state

       |c > |c >  (our coordinates) ?

If that state has the wave function value C in our coordinates, and -C in the rotated coordinates, is C necessarily zero?

We do not see a reason why it should be.

If we could show that in our coordinates, the wave function values for

       |b > |c >  (our coordinates)

and

       |c > |b >  (our coordinates)

have the sign flipped, then |c > |c > would have a zero wave function value in our coordinates.

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

The proof of the Pauli exclusion principle in Wikipedia seems to assume that the particles are switched without any coordinate transformation.

Generally, a two electron wave function

       Φ(y, z)

is not antisymmetric in our coordinates. The electron at y may permanently reside in a location where z never comes. Then it is not a symmetric or antisymmetric function.

Let us check the literature. Has anyone come up with a solution which proves the Pauli exclusion principle from antisymmetry?

Wikipedia says about the spin-statistics theorem:

"An elementary explanation for the spin-statistics theorem cannot be given despite the fact that the theorem is so simple to state."

That sounds ominous.

We think that the theorem is not "simple to state", because it is unclear what switching the electrons really means.

https://www.mdpi.com/2073-8994/13/1/21/pdf

Ilya G. Kaplan (2021) writes that correctness of various proofs of the spin-statistics theorem has been under debate for the past 80 years. Wolfgang Pauli held the opinion that the Pauli exclusion principle is an empirical result and no genuine proof for it exists.

http://meta-phys-thoughts.blogspot.com/2018/11/the-spin-statistics-theorem.html

In 2018, we analyzed the correctness of various proofs of the spin-statistics theorem.

We also analyzed the conceptual difficulty in what an exclusion principle could mean. It is not clear if an interacting system has separate "states" for individual particles.

http://meta-phys-thoughts.blogspot.com/2019/01/is-there-quantum-state-of-individual.html

We suggested that something like the Pauli exclusion principle may follow from the extremely strong repulsion between the magnetic fields of two electrons when the spin direction is the same.

http://meta-phys-thoughts.blogspot.com/2018/11/a-heuristic-derivation-of-pauli.html

If magnetic repulsion is the reason for the empirical Pauli exclusion principle, then it has nothing to do with switching the particles.

The rubber plate model may explain the anomalous magnetic moment of the electron; the photograph model may explain spin 1/2

The first vertex correction diagram increases the magnetic moment of the electron by the factor

       1 + r_e / λ_e = 1 + α / (2π),

where

       r_e = 2.82 * 10^-15 m

is the classical radius of the electron,

       λ_e = 2.43 * 10^-12 m

is the Compton wavelength of the electron, and

       α = 1 / 137

is the fine structure constant.

A classical model where the electron moves at the speed of light along a circular orbit of the length λ_e would explain the Dirac equation magnetic moment of the electron.

                          λ_e      Compton wavelength

                    O --------

            e- orbits

        along the letter O

In the rubber plate model of the electron electric field, the field which is far from the electron "lags behind".

Let us guess that the electric field farther than λ_e / 2 does not contribute to the effective mass of the electron in its circulating motion. Why is the guess reasonable? In a standing wave, the first node is at λ / 4 from the central oscillation. Since the mass at the node does not move, it does not contribute to the effective mass. It is a reasonable guess that the total reducing effect might be equivalent to the entire mass farther than λ / 2 away. (In the future, we need to calculate an estimate somehow.)

The effective mass of the electron is reduced by the factor

        1 - r_e / λ_e.

The effective Compton wavelength grows by the factor

        1 + r_e / λ_e.

The electron must then circulate a path which is a little bit longer. The magnetic moment of the path grows by the factor

       1 + r_e / λ_e,

which is the right value.

The photograph model may explain the spin 1 / 2

The electron angular momentum in the above model is

        L = h-bar,

where h-bar is the Planck constant divided by 2 π. That is, the spin of the electron would be 1.

But the measured spin of the electron is 1/2.

We suggest the following model, which is inspired by our photograph model of quantum field theory.

The classical electron deep down does circulate at the speed of light along a path whose length is only

        λ_e / 2.

The spin of that path is 1/2.

But the path integral determines what size of features in the motion we can "see". There would be total destructive interference if we would try to put the electron into a path which is that short.

The smallest circular orbit, which the path integral allows, has the length of one Compton wavelength. That may be the reason why the electric field of the electron appears to circulate at the speed of light along a path of that length when we measure the magnetic moment of the electron. That is the shortest path we can "see".

In zitterbewegung, the electron moves at the speed of light along a path whose length is one Compton wavelength.

If we see the electric field in that way, why we do not see the angular momentum of the electron in the same way? That is a problem of our model. Maybe we can only measure the angular momentum through direct interaction with the classical electron deep down?

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

The model may explain the strange nature of the spinor: the spinor has to be "rotated" twice through 360 degrees to get it to the original configuration. It is like walking around a Möbius strip.

In our model, the classical electron makes two rounds when we outside observers only see one round.

        ____

       |O O|

       |O O|

        ------

Imagine four rotating discs placed side by side like in the diagram above. Let us put a tight rubber band loop around the discs. The lines in the diagram depict the rubber band.

If we rotate the discs two full rounds, then the rubber band makes approximately one full round.

The path integral may behave like the rubber band. Two rounds of classical rotation may appear as one rotation in the path integral.

How does the vertex function know how to calculate the rubber plate effect?

               virtual photon p

                      ~~~~~~~

                    /                   \

     e-  --------------------------------------------

                             /

                           /

         ~~~~~~

    virtual photon q

    from the magnetic field B

In the vertex correction diagram, the spatial momentum from the external magnetic field B, q, can be arbitrarily small.

The virtual photon which the electron sends to itself may contain an arbitrary 4-momentum p. Its magnitude does not depend on q at all. It is kind of an internal process of the electron.

In our classical model of the electron, the reduction in the effective mass of the electron is an "internal process". It depends only on the frequency of the circulation of the electron in the zitterbewegung orbit. It does not depend on q at all.

How does the vertex correction "know" about the zitterbewegung orbit, so that it can calculate the effect correctly?

The answer has to lie in the way how the magnetic moment in the Dirac equation is coupled to the external magnetic field B. The coupling might be somewhat similar to a classical electron in the zitterbewegung orbit. The coupling then "knows" something about the orbit.

Let us imagine an unknown force which makes a "scalar" pseudoelectron to move in the zitterbewegung orbit. The effective mass of the electron would be reduced just like in our above model.

Actually, when an electron orbits a proton in the hydrogen atom, the electron has a slightly reduced mass because its far electric field lags behind. The orbital frequency is

        f = 6.6 * 10^15 / s.

Light moves

        s = c / f = 4.5 * 10^-8 m

in that time.

The effective mass reduction factor is

       1 - r_e / s = 1  -  6 * 10^-8,

or 0.03 eV.

Let us check if this effective mass reduction is known in the literature.

Regularization and renormalization of the anomalous magnetic moment vertex correction

In quantum field theory, the infrared and the ultraviolet divergences in the vertex correction are cut off in an ad hoc fashion.

"Too large" and "too small" 4-momenta p are banned in the virtual photon which the electron sends to itself. The cutoff procedure has been chosen in a "natural" way, for example, adding a little mass for the photon, and imagined heavy particles in Pauli-Villars regularization. The cutoff, for an unknown reason, yields exactly the measured results.

How the rubber plate model removes ultraviolet and infrared divergences

In our rubber plate model, we do not need any ad hoc cutoffs of infinite values.

We do cut off the far electric field which lags behind and does not follow the movement of the electron, but that is a finite cutoff and has an intuitive physical explanation.

Suppose that there is an impulse on 1 square meter of the rubber plate, and the impulse lasts for 1 second. We use Green's functions to calculate the response. A Green's function calculates the response to a Dirac delta impulse. We have to sum such "spike" impulses over the 1 square meter and the 1 second.

Very high frequencies are removed by destructive interference. What removes very slow frequencies? Apparently, it is the small extent of the impulse which removes very slow frequencies. Waves of length, say, 1 kilometer would require either a long lasting impulse or an impulse which is 1 kilometer wide.

The impulse itself is kind of a wave packet. The packet contains a negligible spectrum of very high and very low frequencies. When we use Green's functions to calculate the response, very high and very low frequencies cannot contribute much to the response.

The problem of divergences arises from new degrees of freedom which are introduced by loops in Feynman diagrams. One can put an arbitrary 4-momentum p to circulate in the loop, and one gets a valid Feynman diagram.

The way out of too much freedom may be to take constraints from the deep down classical level of processes. In an earlier posting, we already removed the divergence from photon-photon scattering by appealing to classical virtual pairs.

The rubber plate model may explain the cancelation of bremsstrahlung against infrared divergence in vertex correction

http://scipp.ucsc.edu/~haber/ph218/Soft_Photons_Amita.pdf

According to Amita Kuttner (2016), the infrared divergence in the vertex correction cancels against a similar bremsstrahlung divergence.

In the rubber plate model, bremsstrahlung is born when the far electric field of the electron lags behind and pumps energy out from the vibration of the near field.

In the rubber plate model, vertex correction is the result of the far field lagging behind, and reducing the effective mass of the electron.

These processes may be related, which might explain why their infrared divergences cancel each other.

Bremsstrahlung is banned in an elastic collision. How to ban it? One may introduce an imagined wave which makes the far field to stand still. That might be equivalent to reducing the effective mass of the electron by cutting off the far field.

Why the vertex function affects the electron anomalous magnetic moment but not its charge? SOLVED!

On February 28, 2021, we asked the question in the title of this blog post.

https://www.fzu.cz/~kupco/QCD/2017/qcd06_anomalni_magneticky_moment.pdf

https://cluster28-files.instructure.com/files/1875~936399/download%3Fdownload_frd%3D1%26verifier%3DDTWO5WywHiDDO9YoFWCbyxFi6dpxJBZlug5SQAvx

Alexander Kupco (2017) and Matthew Schwartz (2012) have written derivations of the anomalous magnetic moment.

They use the Gordon decomposition, or identity, to divide the interaction into the Coulomb interaction and the magnetic interaction. It turns out that the vertex function only affects the magnetic part. That is, the vertex function has no effect when we measure the electron charge in the Millikan-Fletcher experiment, but it does have an effect when we measure the magnetic moment in a Penning trap.