A. O. Barut and A. J. Bracken (1981) about zitterbewegung: an explanation for spin 1/2?

Our previous blog post conjectured that the electron spin and the magnetic moment exist because any wave equation which we form the general energy-momentum relation

       E²  =  p² + m²

is "ugly". (We have set c = 1 in the equation.)

Since the equation is ugly, also its solutions probably are ugly. The ugliness would give rise to the electron spin.

https://www.rle.mit.edu/cua_pub/8.421/Papers/Module%25202/Zitterbewegung%2520PhysRevD.23.2454.pdf

A. O. Barut and A. J. Bracken (1981) explain the Schrödinger argument (1930) of the zitterbewegung.

Positive and negative energy solutions: the complex value rotates either clockwise or counter-clockwise with time

The zitterbewegung seems to be associated with the fact that the Dirac equation admits solutions both with a positive E > 0 and a negative E < 0. Let us call these positive and negative frequency solutions.

Let us make a wave packet which contains an identical amount of positive and negative frequencies.

                   --->

                                  ___

              ____          /       \_____  positive

                      \___/

                                  ___

              ____          /       \_____  negative

                      \___/

                   --->

Let the wave packet move to the right. The wave at the front of the packet may be something like

        ψ(t) = exp(-i (E t - p x))  +  exp(-i (-E t + p x)).

The wave alternates between destructive interference and constructive interference. Does that mean that the expectation value of the position x of the particle moves back and forth? Probably yes.

Constructive interference happens when t = 0 and x = 0. Where is there destructive interference? When

       p x = π / 2,

that is 1/4 of the de Broglie wavelength

       λ = 2 π / p.

The expected location of the particle jumps back and forth the distance λ / 4.

We want the location to jump back and forth by a fixed distance that does not depend on p. An obvious solution is to "mix less" of the second term in the formula of ψ if |p| is small:

       ψ(t) = exp(-i (E t - p x))  +  C |p| * exp(-i (-E t + p x)).

There C is a (small) constant.

https://quantummechanics.ucsd.edu/ph130a/130_notes/node489.html

Above we have standard plane wave solutions for the Dirac equation (by Jim Branson, 2013).

Let us assume that p in the formula is non-zero only to the x direction. If we sum the ψ(1) in the upper left corner to the ψ(4) in the lower right corner, then the the first and the fourth component of the spinor wave function look somewhat like what we derived above. Maybe we found a simple model which describes the zitterbewegung?

Above ψ(1) is an electron with the spin-z up and ψ(4) is a positron with the spin-z down. That matches nicely pair production.

Is an "electron" actually a mix of electron and positron solutions?

Then the electron would not be the solution ψ(1) or ψ(2). It would be a mix of positive frequency (E > 0) and negative frequency (E < 0) waves. If the momentum |p| is small, then there is only little negative in the mix.

Why is the electron spin 1/2 and not 1?

The "mix" model above gives us a heuristic explanation. The length of the jump path, or the zitterbewegung is

        2 * 1/4 λ = 1/2 λ,

where λ is the de Broglie wavelength. For relatively large momenta |p|, the de Broglie wavelength is close to the electron Compton wavelength.

Thus, a nice value for the zitterbewegung pathlength is 1/2 of the Compton wavelength, which corresponds to the spin 1/2.

In this blog we have worked very hard trying to understand how the electron may return to its original state in zitterbewegung after moving just 1/2 of the Compton wavelength. 

The mix model explains this: the path is formed by the interference of two waves. The state of both of these waves only returns to the original after two constructive interference events.

This is probably the origin of the strange 720 degree rotation rules for the electron spin.

What is the origin of the gyromagnetic ratio 2?

Above we were able to make the interference pattern to move back and forth. How can we make it to follow a circular path?

Negative frequencies in a chirp

In this blog we have studied hypothetical Unruh and Hawking radiation. We learned that a "chirp" contains both positive and negative frequencies.

If we have an electron under an accelerating motion, then its wave function presumably is a chirp.

It seems to be so that an electron wave under an interaction always contains both positive and negative frequencies. It is not possible to restrict us to just positive frequencies.

Some people have claimed that the electron wave function should only contain positive frequencies, but that seems to be impossible to implement.

The zitterbewegung model of David Hestenes

http://geocalc.clas.asu.edu/pdf/ZBW_I_QM.pdf

David Hestenes (1990) suggests that the phase of the electron wave function determines its location in a circular motion. The circular motion is the electron spin.

The Hestenes model may be too bold.

How to make the Schrödinger equation more precise about the energy-momentum relation?

https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

If we try to improve the Schrödinger equation in such a way that it estimates the energy-momentum relation

       E = sqrt( p² + m² )

more precisely, then we have to add more terms.

The square root has the Taylor series:

       sqrt(1 + a) ≅ 1 + 1/2 a - 1/8 a² + 1/16 a³ ...

Let us assume that m = 1 and p² = a. Then we can use the series to approximate the energy-momentum relation.

How to add the term -1/8 a² to the Schrödinger equation? Could we use the fourth derivative

        d⁴ / dx⁴

to keep the equation linear?

The Scrödinger equation "codes" the value of p² into the second spatial derivative of the wave function Ψ. Can we code the value of p⁴ into the fourth spatial derivative? No, that does not work. We cannot make sure that the fourth derivative stays as the square of the second derivative.

What about adding a term

       (d² / dx² Ψ)² ?

That makes the equation nonlinear. We might try to solve the nonlinear equation by writing it as a linear equation plus a perturbation term. But a perturbation will scatter the wave. It is hard to maintain conservation of momentum, if the wave is scattered to various directions.

Make the electron to move at the speed of light and "bounce" in a pipe?

The energy-momentum relation is very simple for massless particles which move at the speed of light:

       E = | p |.

If we make the electron to be massless and move at the speed of light, then we maybe can keep the wave equation linear.

             pipe wall

        ------------------------------

         /\/\/\/\/\/\/\/\/\            bouncing electron

        ------------------------------

             pipe wall

The bouncing of the electron would be the zitterbewegung, and it would be responsible for the electron spin and the magnetic moment.

This is a method of simulating a massive particle with a massless particle. If we have a set of photons confined in a box, the photons, in a way, behave like a massive object.

We still have to find an explanation to why the spin-z of this bouncing has to be +- 1/2 ħ.

The electron path is curvy because the energy-momentum relation is ugly?

We may finally be approaching a solution of the electron spin after studying it for four years. If we write a wave equation using the energy-momentum relation as is, the wave equation is very ugly. An ugly equation does not allow beautiful sine wave solutions: the path of the electron must be ugly!

The spin of the electron would reflect a path which spirals very fast. The circular motion would be the origin of the spin and the magnetic moment.

The energy-momentum relation and the Klein-Gordon and Schrödinger equations

The energy-momentum relation of special relativity is

       E² = p² + m².

We assume just one spatial coordinate x and that c = 1 and ħ = 1.

Let us use the usual recipe to transform it into a wave equation.

The energy operator is 

       i d / dt

and the momentum operator is 

       -i d / dx.

The wave function Ψ is complex-valued. Furthermore, we use the metric signature (- + + +) to decide the sign of the square of an operator:

       d²/dt² Ψ = -d²/dx² Ψ + m² Ψ.

The equation is the massive Klein-Gordon equation.

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

As explained in the Wikipedia article about the Dirac equation, a second order wave equation has "too much freedom". It is hard to conserve the particle number. Charge conservation requires that the number of electrons must stay constant.

We want to reduce the equation to a first order equation. The problem is the square root in

       E  =  sqrt(p² + m²).

How to get rid of the ugly square root?

Erwin Schrödinger in 1925 devised a workaround in the case where p² << m²:

       E  ≅  p² / (2 m) + m,

       i d/dt Ψ  =  -1 / (2 m) * d²/dx² Ψ + m Ψ.

This is equivalent to the usual Schrödinger equation if we have the potential V(x, t) set to zero. The term m Ψ can be removed. It does not affect the physics.

The Schrödinger trick works if the possible momenta p have very small absolute values |p|.

Also, if we can switch coordinates to make |p| small, then we can solve the equation.

However, if the possible momenta p differ from each other a lot, what to do then?

The ugly energy-momentum wave equation may force curved paths on particles

We could try writing a wave equation like

       i dΨ/dt = sqrt( -d²/dx² Ψ + m² Ψ ),

but that is hard to solve.

A simple linear wave equation allows beautiful sine wave solutions:

       Ψ(t, x) = exp( -i (E t  -  p x) ).

An ugly equation like the one above does not allow them.

A beautiful sine wave corresponds to a particle moving along a straight path at a constant velocity.

An ugly wave equation may force the particle to move along a curved path!

That may be the origin of the spin of the electron.

The Dirac equation is somewhat ugly. Its solution for a general wave packet makes the electron to do the zitterbewegung. The electron does not move along a straight line. This probably comes from the ugly nature of the energy-momentum relation.

Hypothesis 1. The Dirac equation somehow simulates the general energy-momentum wave equation and is able to isolate the relevant features of the particle motion: a linear motion and a circular motion.

The underlying nature in various wave equations may be the path integral: they describe path integral values for a set of possible paths of a particle. The circular motion of the electron may be an interference pattern. The phases of various paths cause constructive interference in various locations in time and space.

Hypothesis 2. The spin and magnetic moment of the electron are not a result of its interaction with its own electric field. Rather, the field is dragged along the circular motion (zitterbewegung?) of the electron.

The interference pattern idea solves the question that has troubled us for a long time: what is the force which keeps the electron in the zitterbewegung loop? There is no force. The loop is just an interference pattern.

This is analogous to the double-slit experiment. What is the force which moves photons to the locations of constructive interference? There is no force.

The path integral aspect

In a path integral, a "lagrangian density" is integrated over "all" paths and then summed. The integral determines the phase at the endpoint of the path:

        exp(i S),

where S is the integral of the lagrangian density L over the path. The lagrangian density is typically the energy of the particle.

If the formula for the lagrangian density (energy) L is simple and beautiful, then we presumably end up with a beautiful wave equation which can be solved with a standard plane wave.

The lagrangian density which we get from the energy-momentum relation is ugly. Thus, the wave equation is ugly, and the solutions are ugly.

The zitterbewegung shows that the Dirac equation is incorrect?

In the Dirac equation, the spin-z of the electron is "hard-coded" in the components of the 4-component (spinor) wave function ψ.

If we make a standard wave packet and it exhibits the zitterbewegung, then we should see another spin-like motion in the electron.

No second spin has been observed. This suggests that the Dirac equation actually gives incorrect solutions in the case of a general wave packet.

The Dirac equation does seem to work in the restricted case where the electron is described as a single plane wave plus the spin-z value.

The energy-momentum wave equation is nonlinear. The success of the Dirac equation proves that one can estimate a solution with a linear motion of the electron plus the spin motion. The nonlinearity is not pathologically complex if one can make such an estimate.

What is a "free particle"?

We usually think that a free particle is something which is under no interaction and moves along a straight line.

In the double-slit experiment, a photon which has passed the slits is not under any interaction - but it does not make much sense to say that the photon moves along a straight line. Rather, the final position of the photon on a photographic plate depends on the interference pattern.

We conjecture that a "free" electron moves along a curved line. The curved path produces the spin and the magnetic moment of the electron.

Conclusions

This is by far the best idea that we have come up with to explain the electron spin and magnetic moment.

If our idea holds, then the "natural motion" of a particle in spacetime is generally not a linear motion. The natural motion depends on the wave equation of the particle. The natural motion is an interference pattern which arises from various paths that the particle may take.

If the Klein-Gordon equation describes a massless particle, then those particles naturally travel along a straight line. (However, from what does their integer spin come from?)

But a massive particle where the number of particles has to be conserved, naturally "moves", or its interference pattern moves, along a curved path. This is the origin of the spin for fermions.

The electron spin: the classical origin

We have previously introduced the rubber string model for the electron electric field. The field lines are rubber strings. The electron is "suspended" from rubber strings which repel each other.

                              |

                              | e-

                     ------- ● -------

                              |

                              |  field lines

Since the field lines cannot move faster than light, they might form a "wire cage" which tries to keep the electron static.

Suppose then that we could make the electron to move at the speed of light. The field lines will bend very tightly at the electron. The force against the electron might be able to keep it in a circular orbit.

The circular orbit would be the zitterbewegung orbit, and that would explain why the electron has a magnetic moment.

Quantum mechanics then would dictate why the orbit has a constant size: the spin and magnetic moment are constant. It is like the quantized orbits of the electron in the hydrogen atom.

However, we still do not understand why the electron spin is only 1/2 ħ and not ħ.

We can move the electron linearly at a speed less than light. There is no circular orbit in that case. The situation may be different if the electron moves at the speed of light.

The mass of the electron in this model is in the deformation of its electric field. Since the electron does the circle at the speed of light, its rest mass must be zero.

If the field lines somehow are able to "confine" the electron in a "box", then quantum mechanics says that the electron must move. Classical mechanics says that it is able to do a circle if the centripetal force is strong enough.

The mystery of the Dirac equation

The Dirac equation predicts the spin 1/2 and the magnetic moment of the electron. The equation probably does not know anything about the wire cage around the electron. How is the equation then able to predict these things?

The Dirac equation actually contains four fields: each component of the spinor is associated with field. These four fields interact in a very complex way in the Dirac equation.

Let us form a standard wave packet for the Dirac equation. Erwin Schrödinger showed that the expectation value of the electron position does circular motion at the speed of light. This is zitterbewegung.

Thus, the "natural motion" of a particle in the Dirac equation is zitterbewegung. In the Schrödinger equation the natural motion is a linear motion.

The Dirac equation does NOT describe a point particle which moves freely and independently in space.

The massless Klein-Gordon equation nicely and in a very simple way describes an independent particle which moves at the speed of light. The Schrödinger equation does the same for a massive particle which moves slower than light.

If we want an equation which describes a particle doing zitterbewegung, the equation probably must be more complex.

Why the complex equation should be one which we obtain by taking a "square root" of the Klein-Gordon equation? That is, the Dirac equation.

And why would that equation describe the behavior of the electron in its wire cage?

A clue: the Klein-Gordon equation describes the electromagnetic field nicely. And the electron is an electric charge. That probably is the connection. It remains to show that the electron in its wire cage must satisfy the Dirac equation.

A second clue: the Feynman propagator for the Dirac field calculates much the same thing as our classical model in the previous blog post about Compton scattering.

The static electric field of a charge as an optimization problem: the drum skin model

We have previously suggested that the static electric field around a spherical, non-pointlike, charge assumes a minimum energy configuration. The potential around the charge is reduced as long as the gain from the lower potential can cover the energy cost of creating the electric field.

This is analogous to putting a heavy metal sphere on a drum skin. It creates a pit and assumes a minimum energy configuration with the skin.

Can the sphere do a circular motion in the pit that it created? That is only possible if the sphere moves very fast compared to the movements of the skin. Otherwise, the pit follows the sphere.

Conclusions

In the hydrogen atom, the orbits of the electron have classical counterparts, as the Sommerfeld atom model proves.

Quantum mechanics restricts the permitted orbits to those which are stationary and do not self-destruct in destructive interference.

If the combined system electron plus its electric field is analogous to the hydrogen atom, then we should find a classical model where the electron does zitterbewegung inside its own electric field. Quantum mechanics would then somehow restrict the spin to 1/2.

Doppler effect in classical Compton scattering

The Doppler effect has surprising consequences in particle collision experiments. The most natural frame to work in is the center of mass frame.

If the laboratory frame is a different frame, we have to do a Lorentz transformation. The effects of the transformation can be called "Doppler effects".

The part which is not obvious is how a (dipole) wave behaves in a frame change. The natural frame for a wave is the one where the source of the wave is static. How does a moving observer see the intensity of the wave?

An example: Compton scattering

                 incoming beam

                 photon E

                   ~~~~>

       wave burst      source e-      wave burst

                 ( ( (                 ●                ) ) )

     <----- v ●         

          observer

                 <----------------->

                    distance d

Let us assume that an incoming photon of the energy E << 511 keV is linearly polarized in the vertical direction.

There, 511 keV is the electron mass-energy m c².

The photon hits a static electron and makes the electron to oscillate up and down. The electron emits a burst of waves.

Simultaneously, the photon pushes the electron and makes it to move.

We are interested in the intensity of the produced classical electromagnetic wave in the direction from which the photon arrived. We assume that the collision is almost head-on and that we observe a photon coming back from the process.

The electron receives an impulse

       p = 2 E / c

to the right.

We guess that the best frame to analyze the process classically is the one where the collision is half-way: the electron has received an impulse of p / 2 and is moving to the right at the velocity

       v = c E / 511 keV

          = α c,

where we denote by α the ratio E / 511 keV.

If we work in that moving frame, then the electron is static relative to the horizontal axis and is oscillating up and down, and the observer is moving to the left at the same velocity

       α c.

The observer sees a burst of waves arrive from the electron. How does he see the burst?

Let us consider a hypothetical (wrong) model where the electron stays at the same horizontal position throughout the collision process. Then the observer would also stay static. The observer would see a classical dipole wave emitted by the electron which oscillates up and down. The dipole wave would have the same frequency as the incoming beam. The intensity of the back scattered wave would not depend on α, as long as the intensity of the incoming beam stays constant.

Let us have a very naive observer in the laboratory frame. He uses the hypothetical (and wrong) model to calculate the frequency and the intensity of the back scattered wave. We compare the correct calculation to his naive calculation.

The Doppler effect on the wave emitted by the electron:

Doppler shift of the frequency. The observed frequency is smaller by the factor

       1 / (1 + α)

relative to the frequency emitted by the electron.

Doppler effect on the intensity. The intensity goes down as

        ~ 1 / d²

on the distance d.

We look at the spatial distance between the following events:

1. the collision event, and

2. the observation event at the location of the observer.

If the spatial distance is d in the laboratory frame, then it is

       (1 + α) d

in the moving frame.

Also, since the observer is moving away at the speed α c, the energy density of the wave appears to him by a ratio

       1 / (1 + α)

smaller than for an observer comoving with the electron.

The end result: the intensity is by a factor

       1 / (1 + α)³

less than what the very naive observer in the laboratory frame would calculate.

The Doppler effect on the incoming beam:

Doppler shift of the incoming beam (photon). Since the electron is moving away at the speed α c, the frequency of the incoming beam it sees is smaller by the factor

       1 / (1 + α)

relative to the frequency measured in the laboratory frame.

Doppler effect on the incoming beam (photon) intensity. The electron in the moving frame is moving also relative to the light source which produced the incoming beam. The intensity of the incoming beam that the electron sees is reduced by the factor

       1 / (1 + α)³

relative to what the very naive observer in the laboratory frame would calculate.

Above we assume that the incoming beam of light is produced by point sources. The analysis might be different if the incoming beam would be a true plane wave. Is it possible to create such a plane wave?

Conclusions:

1. The frequency of the observed photon is

       1 / (1 + α)²

of the incoming beam frequency.

2. The intensity of the back scattered wave is

       1 / (1 + α)⁶

times a constant, if we vary α but not the intensity of the incoming beam.

3. The cross section for back scattering is

       1 / (1 + α)⁴

times a constant if we vary α.

Comparison to Compton scattering in quantum mechanics

http://neo-classical-physics.info/electromagnetism.html

At the link, D. H. Delphenich has an English translation of a Walter Gordon 1926 paper where Gordon uses the Schrödinger equation to calculate Compton scattering. Gordon observed that the frequency and the intensity of the scattered wave is the geometric mean of the classically computed values at the start of the transition and at the end of the transition. This corresponds to our analysis above where we used a frame where the collision is "half-way".

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5709540/

Yuji Yazaki (2017) recounts the history the famous Klein-Nishina (1929) formula for Compton scattering.

Paul Dirac in 1926 used Heisenberg methods to calculate Compton scattering.

For the intensity of the scattered beam he obtained the formula

where I₀ is the incoming beam intensity, I is the scattered beam intensity, θ is the scattering angle, and φ is the angle between the electric field polarization and the propagating direction of the scattered wave.

If α = 0, then Dirac's formula is the classical result where the electron oscillates vertically and sends a dipole wave.

Above we analyzed the case where θ is roughly 180 degrees and φ is roughly 0. Our result agrees with that of Dirac.

https://en.wikipedia.org/wiki/Klein%E2%80%93Nishina_formula

This is the famous Klein-Nishina formula. The cross section is averaged over all incoming beam polarizations.

If the incoming beam is polarized, the formula is

If the energy of the incoming photon is << 511 keV, then the incoming wavelength λ' is quite close to the scattered wavelength λ, and

       2  ≌  λ / λ'   +   λ' / λ.

The cross section formula agrees with our analysis of back scattering above.

Conclusions

The Doppler effect on the intensity of the scattered beam is tricky to derive. We have to take into account the Doppler effect on the incoming beam, too.

We can approximate classically Compton scattering very precisely for photons << 511 keV, if we assume that the electron is "half-way" through the scattering process with the photon: the electron is already moving at half the speed it will eventually have.

Since the classical approximation is quite precise, then the corresponding Feynman diagram and the integral must describe an essentially classical process.

The classical process is this: if we disturb the electron by making it to oscillate, what happens in its electric field? It makes the field lines to oscillate.

Does the electron propagator describe the behavior of the electric field of the electron under a disturbance?

James Webb results agree with our Minkowski & newtonian cosmology?

UPDATE September 5, 2022: A major problem of the Milne model is how to explain the "first acoustic peak" in the cosmic microwave background (CMB) angular power spectrum. 

https://www.aanda.org/articles/aa/full_html/2012/01/aa16103-10/aa16103-10.html

A. Benoit-Levy and G. Chardin (2011) suggest that it might be associated with matter/antimatter zones in the universe, and not with baryon acoustic oscillations.

https://arxiv.org/abs/1906.11628

The Hubble constant value can be derived by looking at the angular diameter of these CMB acoustic baryon oscillations in the sky. Andrei Cuceu et al. (2020) quote a value of 67.4 km/s per megaparsec. That is smaller than the value 74 km/s per megaparsec which is measured from standard candles. Could it be that the first acoustic peak is not associated with baryon acoustic oscillations?

----

https://m.slashdot.org/story/403711

The first pictures from the James Webb telescope reveal surprisingly old and surprisingly well-developed galaxies.

"Galaxy formation models may now need a revision, as current ones hold that gas clouds should be far slower to coalesce into stars and galaxies than is suggested by Webb’s galaxy-rich images of the early universe, less than 500 million years after the big bang. “This is way outside the box of what models were predicting,” says Garth Illingworth of the University of California (UC), Santa Cruz."

https://www.science.org/content/article/webb-telescope-reveals-unpredicted-bounty-bright-galaxies-early-universe

http://meta-phys-thoughts.blogspot.com/2022/01/the-minkowski-newtonian-model-predicts.html

We wrote about our Minkowski & newtonian model on January 31, 2022. The expansion of the universe happens at a constant speed, just like in the Milne model.

In the Milne model the cosmic microwave background which we observe now was born much farther away from us than in the ΛCDM model. The same for the first galaxies which we observe now. The apparent angular diameter of the first galaxies is smaller in the Milne model than in the ΛCDM model.

         \                         /   now

           \                     /

             \                 /

               \             /

                 \         /   O  first galaxies

                   \     /

                     \ /   Big Bang

           Milne model

         \                         /  now

           \                     /

              \               /   O first galaxies

                 \         /

                     \ /  Big Bang

                  ΛCDM

                  

                  

In the Milne model the first galaxies have a larger redshift and their apparent angular diameter is smaller than in ΛCDM.

James Webb supports the Milne model.

The electron propagator in Thomson scattering is classical

Our analysis in the previous blog post brought up the problem what process, or wave, does the electron propagator really describe in a Feynman diagram.

We know that the photon propagator, for an unknown reason, models Coulomb scattering.

The electron attached to its electric field: the "rubber string" model

Classically, the electron possesses a static electric field. We may imagine that the lines of force are made of rubber.

If we shake the electron, waves will propagate along the rubber strings. The waves are electromagnetic wave.

We may interpret that the rubber mesh controls the movement of the pointlike particle electron. The electron is a kind of an oscillating mass attached to the rubber mesh.

Classical Thomson scattering

                                        electric

                                    lines of force

                                             |

              |    |    |               --- ● ---  

                                             |

           laser beam       electron e-

The laser beam shakes the electron. New electromagnetic waves propagate in the lines of force of the electron.

We may imagine that the laser beam gives small impulses to the electron at very short time intervals. The response of the electron and the rubber mesh is an impulse response.

Recall that a propagator is a Fourier component of the impulse response of a wave equation. For example, if we hit a drum skin with a sharp hammer, the Fourier decomposition of the wave is the set of propagators.

The impulses to the electron have a cycle determined by the frequency of the laser. Constructive interference strengthens the output at this frequency. Destructive interference wipes out other frequencies.

The electron will act as a radio transmitter and send an electromagnetic wave to many directions. It scatters the incoming laser light.

The propagator in this classical treatment tells us how effective the electron is in outputting energy to the scattered "channel", or scattered wave.

We can, in principle, calculate with a computer how the electromagnetic field behaves, and what is the value of the propagator.

The classical Feynman diagram looks like this:

        laser                              scattered wave

        ~~~~~~                       ~~~~~~~~~~~

                           \              /

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

                            virtual

                           electron 

It is essentially the same as the quantum electrodynamics (QED) Feynman diagram.

The "virtual electron" is the propagator. We may imagine that it represents the electron in the mesh after an impulse hit the electron.

QED Thomson scattering

As we wrote above, the Feynman diagram is essentially the same as in the classical process. Numerical results from Feynman formulae agree with the classical treatment.

Note the following thing: the propagator in the QED diagram is from the impulse response of the Dirac equation. We were able to connect the Dirac equation to a classical process.

The process in Thomson scattering is non-relativistic. The Dirac equation in that case is equivalent to the Pauli equation, or the Schrödinger equation. The propagator for the Schrödinger equation seems to be complicated.

What is a "virtual" electron and how does an electron "absorb" a photon

In the classical interpretation, a virtual electron means the system electron & its electric field where the system has been disturbed by an impulse.

The absorption of a photon means that the system goes to a disturbed (excited) state.

Classically, the electron is a point particle. It cannot have excited states on its own. An excited state has to be the electron in an interaction with something else, in this case its own electric field.

When an excited electron emits a photon, that means that the oscillation of its electric field moves farther from the electron and starts a life of its own.

A propagator does not make much sense for a point particle. But a propagator for the system the electron & its field makes a lot of sense.

Question. How can we extend the classical propagator to an "electron field"? That is, we would not have a point particle but some kind of a field. This might show the connection between the Dirac equation and the classical model.

It is not clear if we can define an electron field in a reasonable way. If the field simply describes the position and the phase (in a path integral) of a single electron, then there is no obvious interaction between different parts of the field.

If the electron is either in the zone A or the zone B of space, there is no interaction between the zones A and B. This is very different from a drum skin where A and B always interact. In a drum skin, a wave equation is natural, but it is not natural for mutually exclusive histories.

The Dirac equation is used in Feynman diagrams to calculate the electron propagator. That is, a single electron is interacting with something else. Then it makes sense to consider the system the electron & its field.

The Dirac equation does have a conserved probability current and it does predict the magnetic moment of the electron. How can we explain these if the equation only describes an interacting electron?

Maybe the electron is a wave phenomenon, after all? The spin of an electromagnetic wave probably is a wave phenomenon.

But then we face the problem how to attach the electric field to the electron.

The Dirac equation describes the system the electron & its field?

The electron propagator in the Dirac field describes something which is off-shell, or not in its ground state.

If the QED propagator is able to calculate something similar as the classical propagator, then it is natural to assume that the QED propagator describes the combined system the electron and its field. We cannot remove the electric field from the electron. It makes sense that the Dirac equation describes the entire system.

However, it is not clear how the classical system gives rise to the Dirac equation. How do we explain zitterbewegung and the magnetic moment?

Conclusions

The analogue of a photon propagator is a sharp hammer hitting a drum skin (where the skin is actually the three-dimensional space).

The Fourier decomposition of the associated - 1 / r potential has the familiar formula

       ~ 1 / p²,

where p is the 4-momentum of the Fourier component.

              |

            _|_  

           |__| -->     ● e-

       hammer

The analogue of the electron propagator might be a hammer hitting the electron. The electric field of the electron moves a little bit. That is like adding a dipole where a positron e+ is put to the old position of the electron and a new electron is put to the new position.

The dipole potential is roughly

       ~ 1 / r²,

and the Fourier decomposition is roughly

       ~ 1 / |p|,

where p is the 4-momentum of the component.

The formula 1 / |p| is similar to the Feynman electron propagator if we set E = 0 and m = 0, where E is the energy of the electron and m is its mass.

Pair annihilation: the virtual electron is a carrier of a "force"

In our blog post February 16, 2021 we discussed the Feynman diagram of pair annihilation. It looks like scattering where the virtual electron carries (repulsive) momentum.

      momentum -p

             e+ ---------------- ~~~~~~~~~~  photon -k

                                     |

                                     |  virtual

                                     |  electron p - k

                                     |

              e-  --------------- ~~~~~~~~~~  photon k

      momentum p

Suppose that the positron and the electron approach from opposite directions at the same speed. The momenta are p and -p.

The momenta of the outgoing photons are k and -k.

The produced photons cannot carry away all the momentum of the particles. The electron gives the excess momentum p - k to the positron in the form of a virtual electron.

The virtual electron as a carrier of a "force"

The annihilation process looks like a scattering event where there is an approximate repulsive 1 / r² potential between the electron and the positron.

The Fourier transform of a 1 / r potential is

       ~ 1 / q².

The propagator of the photon is of this form.

https://mathematica.stackexchange.com/questions/239686/3d-fourier-transform-of-1-r2

The Fourier transform of a 1 / r² potential is of the form

       ~ 1 / |q|,

where q is the "momentum" of the Fourier component.

The propagator of the electron is very crudely of the form 1 / |q|.

Our blog post on February 16, 2021 shows that the potential in annihilation is actually even steeper than 1 / r², according to the cross sections calculated from Feynman diagrams.

We have been wrong in our attempts to explain annihilation by the electric attraction between the pair

The momentum transfer in the Feynman annihilation diagram does not happen through a photon.

Maybe it is not possible to make a (semi)classical model of annihilation using the electric force?

The sharp hammer model again: Huygens

We have explained the static electric field of an electric point charge with a sharp hammer which keeps hitting a "drum skin" at the charge, and makes a depression to the skin.

The Huygens principle is that a wave is absorbed by each point in space, and the point then acts as a new source for the wave.

In the case of the electron and its electric field, the Huygens principle might be something like a sharp hammer hitting a drum skin and creating the electron wave and its electric field.

The hit would produce a "point impulse" both in the Dirac field of the electron and the electromagnetic field.

Now we come to the Feynman diagram principle that all "paths" which conserve energy and momentum are allowed, and their probability amplitudes must be summed.

The idea is that the when the electron wave arrives at a spacetime point x, the wave is "absorbed" and immediately recreated with an impulse which hits both the Dirac field and the electromagnetic field.

Our sharp hammer becomes more versatile: it hits two fields at once. All combinations of responses from the two fields are allowed.

The virtual electron in the annihilation diagram is one component of the impulse response. The photon flying away is another component.

If we hit a drum skin, only sine wave "on-shell" Fourier components can travel over a long distance. Other, "off-shell" components have a short range of the effect. This may explain why the electron and the positron must come close to each other in order to annihilate.

A particle model of electron-positron electric scattering

    

             <------  e+

                         |

                         |  electric attraction

                         |

                         e-  ------->

The particle model is very simple and intuitive. Both particles are treated as point charges with an attractive electric force. The paths are calculated with classical relativistic mechanics. This gives results which, according to literature, are (almost?) exactly the same as when calculated using the simplest Feynman diagram.

A wave model of the scattering

 

                   <------ e+

          |       |       |        |        |        overlapping

                |       |        |        |        | waves

                             e-  ------>

Let us then try to form an intuitive wave model from a Feynman diagram. Let us have an average of one electron and one positron in a cubic meter of space.

We do not know the positions of the particles, and represent them with standard plane wave solutions

       u(E, p) * exp(-i / ħ * (E t - p • x)),

where u is the spinor.

The Feynman diagram is

         

         e+ ----------------------------------------

                                 |

                                 | virtual photon

                                 |

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

How can we relate this to the wave diagram above?

If we take seriously the wave interpretation, then the scattering of colliding beams of electrons and positrons is a nonlinear effect. If there were just one beam, then scattering would not happen. In a linear system we would be able to sum the solutions of the two beams to obtain a new solution: there would be no scattering.

Suppose that we try to add a "source" to the wave equation of the electron. Something like

       D(ψ) = f(φ, ψ),

where ψ is the electron wave function, φ is the positron wave function, and D(ψ) = 0 is the Dirac wave function of the free electron.

The source term f(φ, ψ) depends on both wave functions.

But if the wave functions φ and ψ are essentially constant in the cubic meter (save a phase factor), how can the source generate a wave which is significantly scattered, to an angle, say, 90 degrees?

To simulate the scattering of the point particles we should have scattered waves where the cross section is

       ~ 1 / α²

where α > 0 is the deflection angle. How to generate such waves without having a localized disturbance of the field?

That looks hard. In the hydrogen atom model, one particle, the proton is treated as a particle while the electron is treated as a wave.

Our view in this blog has been that particles are the "true" nature, and that any wave phenomena are due to path integrals.

An analysis of the electron-positron scattering Feynman diagram

We showed that a pure wave model of the scattering does not work. What does the Feynman diagram then really calculate?

         

         e+ ----------------------------------------

                                 |

                                 | virtual photon

                                 |

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

Let us assume that the underlying process really is the classical Coulomb scattering of point particles. But we do not know the precise position of the particles. We have to calculate the path integral for very many possible paths which the particles can take. The path integral is a collective phenomenon of all the possible paths. The path integral is the "wave" associated with the process.

What is the photon in the Feynman diagram? It summarizes the interaction in these very many paths.

Why is the photon propagator

       ~ 1 / p²

the "right" way to calculate the effect of the interaction?

It is the Fourier component of the 1 / r potential, but there is no obvious reason why the component would correctly capture the cross section of classical scattering.

           \     \     \        scattered flux

              ________

             |              |

             |              |  cubic meter

             |_______|

                

                 \     \     \   scattered flux

                

Let us again have that cubic meter where electron and positron beams meet. The scattered fluxes come quite uniformly from the entire volume.

We can model the scattering by assuming that for each centimeter of the path of the electron, a small portion of the electron flux gets scattered to various deflection angles, and a corresponding part of the positron flux gets scattered to the opposite direction. It is like both fluxes would travel in a nonuniform medium.

The force which causes the scattering is the electric force, and the propagator for some reason happens to capture it correctly.

Since we cannot interpret the diagram purely with waves, we conclude that the Feynman diagram really does describe the encounter of two particles.

We may use a wave description for one of the particles, though.

What about using a wave packet description for both of the particles? We can make the packets to pass each other at some short distance. That might work reasonably well if the distance is larger than the wavelength.

Classically, we have two point charges, and their electromagnetic fields are kind of "waves".

The interpretation of the annihilation Feynman diagram

Our analysis of the Coulomb scattering diagram concluded that the photon propagator "for some reason" happens to work in that case.

In the annihilation diagram, the "interaction" is not by the photon propagator, but by the electron propagator. How do we analyze this?

Conclusions

A lot of questions but few answers. The key problem in this blog post is what does the electron propagator model in a Feynman diagram. Is it a wave? Is it a path integral of point-like particles?

We will analyze Thomson scattering in the next blog post. There we are able to connect the electron propagator to a classical process.