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?