If the energy of the field of the electron is infinite, then the electron plus the inner field has negative mass-energy

Let us write yet another post about the renormalization problem of a classical point charge.

Suppose that an electron collides with a positron. If the positron could survive very close to the electron, both would "see" each other having negative mass-energy. That is bad because negative mass-energies lead to strange phenomena.

https://www.mpi-hd.mpg.de/personalhomes/frieger/HEA10.pdf

Frank Rieger gives the cross sections for pair annihilation. For low-energy pairs, it is

       π r₀²  *  1 / β,

where r₀ is the classical electron radius and the velocity of the particles is β c. For high-energy pairs, the cross section is

      π r₀²  *  1 / γ  * (ln (2 γ) - 1),

where γ is the Lorentz factor.

For a low-energy electron, the field farther than r₀ / 2 has mass-energy 511 keV. Annihilation prevents the particles from coming that close to each other. We ignored here the focusing effect of Coulomb attraction, though. Anyway, annihilation is a way to prevent the particles from seeing each other with negative energy.

What about high-energy pairs? If the pair would come so close to each other that they would almost see each other with negative energy, the particles would scatter to large deflection angles, say, ~ 90 degrees. The particles roughly obey classical Coulomb scattering. If the pair would come within

       r₀ / γ

of each other, they would scatter to a large angle. Again, annihilation prevents the particles from seeing each other with negative energy.

The far field of an electron must have inertia of its own

In the previous blog post we speculated that the static field of the electron may have zero mass-energy. It might be so, but the far field must have inertia. Let us analyze the following thought experiment.

We push an electron with our hand for a time t, with some force. The pushing accelerates the electron and creates an electromagnetic wave of energy E. Besides giving the electron some kinetic energy, our hand does work against an additional force F(t) which produces the extra energy E. Let the velocity of our hand be v(t). Our hand does the work

       E = ∫ F(t) v(t) dt

and gives an impulse (momentum)

       p = ∫ F(t) dt.

A photon of energy E can only carry away a momentum

       p' = E / c.

The velocity v(t) is much less than c. Thus, the static far field of the charge must absorb almost all of the momentum p. That is only possible if the far field has inertia.

The total inertial mass of the electron is 511 keV / c². Some of that inertia must reside in the far field.

What happens if opposite charges cannot annihilate each other?

Suppose that an electron collides with an up quark whose charge is +2/3 e. What prevents the quark from seeing the electron with negative mass-energy?

Suppose that the electron is ~ 1 MeV. Then no pair production is possible. Classically, the electron may pass very close to the quark. The electron sends some classical bremsstrahlung as it passes the quark. The amount of classical bremsstrahlung would exceed 1 MeV if the electron would come much closer than r₀ from the quark. We conclude that we cannot model very close encounters with the classical model. Feynman diagrams calculate the amount of bremsstrahlung correctly, as far as we know.

If the electron has high energy, say, 100 MeV, then pair production may obfuscate the behavior in close encounters.

We also need to check if there is some electroweak mechanism which blocks negative energies in this case.

What prevents an electron having negative energy in the hydrogen atom? It is the uncertainty principle which forbids an orbit which is very close to the proton. Uncertainty blocks negative energies.

Conclusions

The far field of the electron must have inertia. That is the only way to explain how a radio transmitter works, and ensure momentum conservation.

In QED, annihilation hides the structure of the very near field from us. It hides a possible negative energy of the electron plus the inner field.

In the standard model, we cannot model a very close encounter of a quark and an electron classically.

Renormalization and regularization of electron self-energy

Our blog post on December 19, 2020 was about this problem. Let us look at it again.

The Feynman diagram for the electron self-energy is shown below.

                           virtual photon E, q

                               ~~~~

                            /               \

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

           momentum p

The electron emits a virtual photon and subsequently absorbs it.

Our sharp hammer model explains what this is about. The electron is a source in the massless Klein-Gordon equation which describes the electric field. It hammers the field at a very high frequency to make a pit into the field. The pit is the static Coulomb field of the electron.

https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qed/renorm-new.pdf

In the link there is a short exposition of renormalization in QED. It is assumed that the electron has a "bare mass". We calculate the propagator of the electron, including the additional effect of the self-energy diagram. The effect turns out to be infinite. We remove the infinity with regularization. The new propagator looks similar to the original electron propagator if we redefine the mass of the electron.

Classical regularization and renormalization of the mass of a point charge

The infinite energy in the field of a point charge is a classical regularization problem. A way to remedy the classical problem is to assume an infinite negative energy for the bare electron. When we add the infinite energy of the field, we get the measured electron mass 511 keV - this is classical renormalization. Classical renormalization is ugly. The combined mass-energy of the electron and the field located at a radius < half of the classical electron radius becomes negative. Negative energies can have surprising consequences.

In the diagram, the electron has constant momentum p. If there is no external interaction, then it does not matter where the mass-energy of the electron is located. Renormalization is only relevant when the electron is under interaction - that is, it is the vertex correction where mass regularization and renormalization might have an effect on phenomena.

How does a negative mass move classically?

Let us investigate further the effect of negative mass-energies in classical physics.

                     spring                     ● charge P

            ● \/\/\/\/\/\/\/\/\/\ ●          

       mass M                  mass -m charge Q

Suppose that we have a smaller negative mass -m attached with a spring to a larger positive mass M. How does the system move?

Let us assume that the negative mass -m is charged. Let us assume that the charge P repels the charge Q.

Since -m is negative, Q starts to move to the wrong direction, toward the charge P. This is because of momentum conservation. Then -m pulls on M. The pulling moves both -m and M closer to P. This does not make sense.

Above we assumed a direct force between P and Q. Maybe we should assume that P and Q only interact through their electric fields?

Another option is to return to the December 19, 2020 hypothesis that a static electric field contains no mass-energy. All the mass-energy of the electron would be in the point particle itself.

Feynman diagrams do not restrict the mass and the charge of a particle. Therefore, they should be able to handle the classical renormalization problem, too. All the problems of the classical case should pop up in the Feynman diagrams, too.

A tentative solution to the classical renormalization problem

If the acceleration of the electron is not extremely high, we can assume that it is not a point particle. We can then assign a positive mass-energy both to the electron and its field. Calculations can be done using classical Maxwell equations.

If the acceleration is huge, then we have to assume that the electron is a point particle. To avoid negative mass-energies, we have to assume that the mass-energy of the static electric field of the electron is zero.

Question. Does it conflict with classical electrodynamics if we set the energy of a static field zero?

Especially, we are interested in how our model of reduced electron mass works in the anomalous magnetic moment of the electron, if we assume the static field to have zero mass-energy.

The fact that the electron in high-energy collisions (e.g. LEP 100 GeV) behaves like a point particle, suggests that the 511 keV mass-energy of the electron really resides in the particle itself, not in the field.

The classical limit of Feynman diagrams in QED : the limit seems to be broken in the electric vertex function

We want to find out which phenomena in QED depend on the Planck constant h, and which are classical in nature.

https://arxiv.org/abs/1009.2313

Stanley J. Brodsky and Paul Hoyer (2011) study which calculations with bound states are affected if we let the Planck constant h approach zero. The result depends on which parameters we take to depend on h.

https://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf

Tree diagrams are generally agreed to be classical. Feynman formulas for them do not depend on h. Robert C. Helling writes about their use in classical field equations.

The simplest Coulomb scattering diagram

      momentum p              momentum p'

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

                                   | virtual

                                   | photon

                                   | q

      Z+ ----------------------------------------------

Rather than letting h approach zero, we can increase the mass of the electron by a factor 2 and its charge by a factor sqrt(2). The probability distribution of the deflection angle of the electron should stay the same if the process is classical.

http://www.physics.mcgill.ca/~andrzej/Peskin/Chap5.pdf

Andrzej Pokraka has calculated Coulomb scattering for us. In his formula (3), the matrix element M is the scattered wave per solid angle. The value of

       u-bar(p') γ^μ u(p)  ~  m_e,

and

       A-tilde_μ(p' - p)  ~  e / m_e².

Thus,

        M  ~  e² m_e / m_e².

We see that M stays constant if we double m_e and multiply e by sqrt(2).

The electric vertex correction

                                   k

                            ~~~~~~~

               p        /                     \             p'

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

                                   | virtual

                                   | photon

                                   | q

          Z+ --------------------------------------

In the vertex correction we have three extra internal lines, as well as two extra vertices, compared to the simplest diagram.

https://web2.ph.utexas.edu/~vadim/Classes/2019f/vertex.pdf

https://www-thphys.physics.ox.ac.uk/people/FrancescoHautmann/antw-qcd/antw16noteRen.pdf

https://homepages.dias.ie/ydri/qft_chap_five.pdf

From the Internet we find several calculations of the electric form factor F₁(q²). The calculations are very complicated. Furthermore, regularization and renormalization is used. Renormalization is applied to make F₁(0) equal to 1.

We do not know if the final results of these calculations have been tested empirically.

The final results do not stay constant if we double the mass m_e of the electron and multiply the elementary charge e by sqrt(2). That would mean that the final results break the classical limit. Is the physical model in the calculations wrong?

The Sudakov double logarithm

The leading terms in the electric form factor for large q are

       F₁(q²) = 1 - α / (4 π) ( log(-q² / m_e²) )²,

where the log squared is known as the Sudakov double logarithm.

A very large q requires that the electron has very large initial momentum, and passes extremely close to another charge. It might be a situation where the electron is detached from its static electric field. According to our previous blog post, the mass of the electron may appear as 1.022 MeV. If the kinetic energy is much larger than 1.022 MeV, the change in the electron mass should have a very small effect.

We have to check what empirical measurements say about such high-energy collisions.

https://physics.stackexchange.com/questions/257936/sudakov-double-logarithm

Sudakov double logarithms seem to cancel in many cases.

Conclusions

In this blog we have the hypothesis that Feynman diagrams are mostly a classical thing. Calculations for the electric vertex function seem to break this hypothesis. The calculations also seem to break the classical limit where we increase the electron mass and charge.

In previous blog posts we asked why a Feynman diagram (with regularization) calculates the vertex correction right. It might be that it does not calculate it right.

We will look at vacuum polarization loops in a subsequent blog post.

The charge of the electron is positive - its field is negative

We have the model where the electron is a small weight resting on a drum skin. It makes a pit into the skin. The pit is the static electric field (more precisely, its potential) of the electron.

The electric charge of the electron is positive

The hamiltonian contains the energy of the static electric field of the electron, as well as the potential of the positive charge of the electron in the potential pit of the electric field. The hamiltonian will get a smaller value if we make a potential pit into the electric field, and put the electron in that pit.

Yes, the electric charge of the electron is positive. That is why it creates a field which from the outside looks like the field of a negative charge.

If the electron would be a classical point particle, it would sink deeper and deeper in the electric potential. It would release an infinite amount of energy, and the energy of the field would become infinite, too.

       ---------          --------  electric potential

                   \ ●/ 

                     electron in potential pit

We conclude that a classical point charge cannot exist. We believe that zitterbewegung is the mechanism which rescues the quantum mechanical electron from the infinite negative potential.

Why is the electron mass 511 keV? Maybe it is just a random value. The muon mass is 105.7 MeV, which looks like a random value, too.

In the drum skin model, it is the small weight which creates the pit. Having a weight pressing the skin is equivalent to having a small hammer hitting the skin at the location of the electron with a very high frequency.

In the hammer model, the drum skin responds to each hit with a Green's function. It is the impulse response.

The spectrum of the Green's function behaves like this:

1. High frequencies are "reflected" back to the hammer quickly and the hammer absorbs them.

2. Low frequencies are reflected back slower. There is a buildup of low frequencies from many hits of the hammer in the skin.

We may imagine that the 1 / r² electric field of the electron is "built" from these frequencies.

The Feynman diagram virtual photon emitted by an electron

In a Feynman diagram, it is a single hit which creates the field of an electron. The spectrum is wrong, if we believe our own model. The Feynman model plays down low frequencies. There is no buildup of low frequencies from many hits. The Feynman model exaggerates the impact of high frequencies. The high-frequency error is corrected with regularization in QED.

The Feynman model is approximately right for mid-frequencies.

Question. The Feynman model is wrong for low frequencies. Why that does not show up as significant (e.g., ~ 10%) errors in Feynman calculations?

The simplest Feynman diagram - for an unknown reason - models in the right way classical relativistic Coulomb scattering. It might be that the same coincidence is the answer to the Question above.

The bare mass of the electron is 1.022 MeV?

Let us first have an electron without the interaction with the electromagnetic field. Its mass-energy might be 1.022 MeV. Once we couple it to the electromagnetic field, it sinks to a lower potential. It is a good guess that the electron sinks to a potential -1.022 MeV. The virial theorem suggests that the entire energy of the system, 511 keV, is then in the electric field of the electron.

If the electron moves in the zitterbewegung loop, the mass-energy of the electron looks reduced because the far field does not have time to react. The acceleration in the zitterbewegung loop is moderate. What about high-energy collisions?

Suppose that we hit the electron with another particle. If the collision energy is ~ 1 MeV or larger, the electron can climb up from its potential pit. Does that have consequences which we could observe? The electron will move slower than an electron with a mass of 511 keV. After the collision, the electron will quickly sink back into a potential pit and emit 511 keV in radiation. The electron would for a short time be in an excited state - excited in its own field.

When the electron sinks back, the radiation which it emits can be confused with bremsstrahlung.

Excitation in the own field is somewhat fuzzy a concept. We might say that the field around the electron is excited, instead of the electron itself.

Kids' swing model of the static electric field of the electron

The sharp hammer and drum skin model for the electron static field is complicated. Let us simplify it further.

                ●  swing frame

                 |

                 |

                 |

                 |   kids'

                 |   swing

               -----

   ---->

   impulses

   at high

   frequency

We push the swing with our hand at short intervals, so that the swing is permanently positioned to the right from the equilibrium position. The configuration is analogous to the static "pit" which the electron makes into the electric field.

Let us assume that the swing is moving back to the left. A single push of the hand can be decomposed in this way:

1. our hand absorbs the "reflection" of the previous impulse by stopping the movement of the swing to the left;

2. our hand emits a new impulse by pushing the swing again to the right.

If the resonant frequency of the swing is high (the rope in the swing is short), we must push the swing at very short time intervals to keep it positioned to the right. For slower resonant frequencies, we can push at longer time intervals.

The analogy between the drum skin and the swing

If we hit a drum skin with a sharp hammer, it produces a Dirac delta impulse, which in the Green's function contains all kinds of frequencies, high and low.

Let us fix a timestep. It might be 10^-20 s, for example. The hammer hits the drum skin at that interval. The swing analogy suggests that each hit of the hammer absorbs some spectrum of reflections of the Green's functions of previous hits. Low-frequency components take a longer time to reflect back and get absorbed.

The hit also produces a new impulse a new Green's function as a response.

Suppose then that the electron scatters from a nucleus and its velocity vector changes significantly. 

                           virtual photon E, q

                           ~~~~~~

                         /                   \

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

                                  | virtual

                                  | photon

                                  | p

        Z+ --------------------------------------

A mixture of spectra from Green's functions from earlier hits is left "dangling". The spectra will not be absorbed completely by the electron because the electron changed its course. The dangling part is the virtual photon E, q in the Feynman diagram above. The electron will absorb some of the dangling part. This gives rise to the vertex correction. The rest is emitted as bremsstrahlung.

What is in the dangling part? It contains the output of many hits of a long-frequency component of the Green's function. The electron has many timesteps of time to try to absorb those long-frequency components.

The most important contribution to various phenomena obviously comes from the frequencies where the cycle time is of the same order as it takes the electron to scatter from the nucleus. We know that classical bremsstrahlung is concentrated to those frequencies. Let us call these mid-frequencies.

High-frequency components get absorbed quickly. The electron does not change its course significantly at a short time interval. We believe that the impact of high-frequency components is small on various phenomena.

For mid-frequency components, the Feynman diagram and the formula might be quite a realistic description of what happens. We may imagine that the dangling component of mid-frequencies was produced by a single hit by a large, non-sharp, hammer. Some of the component will be absorbed in the next hit.

What about low frequencies? If the dangling component was produced by, say, 10 hits of the large hammer, then some of it will be absorbed by the next 10 hits.

What is the impact of low-frequency components? The energy in the far field of the electron is small, which suggests that the impact is small.

Mid-frequencies are responsible for the reduction of the effective mass of the electron as it passes the nucleus. Mid-frequencies also produce most of classical bremsstrahlung.

Next we need to study carefully why the Feynman formula might describe the process correctly for mid-frequencies. We know that the Feynman formula does not work for high frequencies because regularization is needed for them.

The classical electron self-energy divergence problem

Our previous blog post mentioned the classical renormalization problem of a point charge.

Let us have a hamiltonian

        H = ∫ ε₀ E² dv + V + m c²

which contains the energy of the static electric field of a charge, the potential energy V of the charge in its own field, and the mass-energy of the charge. The hamiltonian is like for a point mass sitting on an elastic rubber sheet.

Let us start from a configuration where the static field is zero. We can release energy by making a pit in the field and letting V become negative. The charge will fall into ever lower potential, and keeps releasing more energy.

Classically, the energy of the field of a point charge is infinite. To recover a sensible value for the total energy of the combined system of the charge and the field, we have to assume that the charge itself has negative infinite energy.  That is awkward.

A way to save us from infinities and negative energies is to make the electron into a ball whose radius is the classical radius of the electron. But particle collider experiments show that the electron radius cannot be that big. The radius of the charge has to be 10^-18 m or smaller.

Can zitterbewegung save the day?

If the electron moves at a speed of light in a small circle, that affects the geometry of its static field. Could it be that the infinite energy in the field of a point charge does not appear because of zitterbewegung?

The Darwin term of the hydrogen spectrum can be derived by assuming "smearing" of the charge in the zitterbewegung area, or more formally, using the Foldy-Wouthuysen transformation. It is probably not a coincidence that both approaches produce the correct result. If the charge is, in some sense, smeared, then there is no infinite energy in its field.

If a classical point electron moves at (almost) the speed of light, can we extract infinite energy by dropping an opposite charge into its field?

The classical renormalization problem is not really fundamental because the electron is a very small particle and it has to be treated quantum mechanically anyway.

We may have solved the mystery of regularization in QED

We think that we have solved the mystery of Feynman diagrams and regularization / renormalization.

                |  hits at a very high frequency

                v

                __

                |   |====== hammer

                \ /              

      -------         -------- drum skin

                \  /

                pit

Recall the drum skin & sharp hammer model where a sharp hammer at a very high frequency hits the drum skin at a specific location. Hitting the skin makes a pit in the skin. That pit is analogous to the static electric field of the electron.

        e- --------> 

            constant velocity v

If the electron moves at a constant velocity, the setup stays static. Destructive interference completely removes any dynamic waves made by the hitting hammer.

        e-  ---->

               v

                                ● Z+

But if the electron is scattered by a nucleus, then dynamic waves are created by the hitting hammer. The waves may be absorbed by the electron itself, or they may escape to infinity as bremsstrahlung.

If the waves are absorbed by the electron itself, then the phenomenon is the vertex correction.

In a Feynman diagram, an electron creates a photon from a Dirac delta impulse, as a Green's function. It is a single hit by the hammer. We need to figure out how this is related to our model with high-frequency hammer hits.

High-frequency components of the Green's function are wiped out by destructive interference in the fly-by of the electron. Only "medium-frequency" and low-frequency components remain.

Regularization is a method where we "subtract" the waves created in the constant velocity case (that is, the electron self-energy diagram) from the waves created in the scattering case. It is not a miracle that the method recovers a sensible and correct result in the scattering case. Regularization is a way to implement total destructive interference of high-frequency waves.

The drum skin model is classical. Consequently, the vertex correction is fundamentally a classical thing which is we look at through the microscope of quantum mechanics.

In our rubber plate model of the static electric field of the electron we never specified how the electron is attached to its rubber plate. Now we see that the attachment can be explained by the sharp hammer model. The static electric field is the response of the electromagnetic field to constant hammering by the electron charge.

      ------            ------   drum skin

                \●/       charge presses the skin

                pit

Alternatively, we may imagine that the charge is a small weight which makes a pit to the electromagnetic field. That is classically equivalent to the sharp hammer model.

A classical renormalization problem: the energy of the static field of a point charge is infinite

We will write a more detailed analysis of this new sharp hammer model.

A renormalization problem remains: a point charge makes an infinitely deep pit into the electric field. The energy of its field is infinite. We have to assume that the potential energy of the charge in the pit is infinitely negative, so that the total energy of the system is 511 keV. This is ugly.

Conclusions

The Feynman way of creating photons is explained by the sharp hammer model. The Dirac delta impulse of a Green's function is a single hit by the hammer.

Regularization works because it essentially implements the destructive interference in the classical model. The divergences in Feynman formulas are created because the formulas fail to consider destructive interference. The ad hoc method of Pauli-Villars regularization probably works because it somehow imitates the case where the electron moves at a constant velocity, and therefore implements the destructive interference.

If we are right, we have solved the 70-year-old mystery of regularization: why does it work?

The electron magnetic moment comes from zitterbewegung?

Solutions of the Dirac equation are of the form

       exp(-i / h-bar (E t - p x)) * u,

where u is a 4-component Dirac spinor. We in this blog believe that the exp(...) part describes linear motion of the electron, while the spinor is a compact, "quantized" description of circling motion which may be called zitterbewegung.

Erwin Schrödinger calculated in 1930 that the Dirac electron position in a "typical" wave packet (such a packet always contains both a positive E and a negative E component) seems to move at the speed of light in a circle whose radius is

       λ_e / (4 π),

where λ_e is the Compton wavelength of the electron. The circling motion is called zitterbewegung. This movement, if it is classical motion, would explain the electron spin-z angular momentum

       L = 1/2 h / (2 π).

David Hestenes: the zitterbewegung interpretation of quantum mechanics

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

David Hestenes (1990) writes about zitterbewegung. Hestenes believes that the complex phase factor

        exp(-i / h-bar (E t - p x))

describes real spatial motion in the zitterbewegung circle. Hestenes calls this fundamental assumption the zitterbewegung interpretation of quantum mechanics.

Is it right to assign such a realistic role for a complex phase? Schrödinger's zitterbewegung has an angular velocity which is double of

        E / h-bar.

Which angular velocity describes the "real" zitterbewegung?

Suppose that a relativistic electron has its E doubled because it has lots of kinetic energy. Then the angular velocity is double in the above formula. But if the spin is classical motion, time dilation makes it spin at half the speed in the laboratory frame. We conclude that it is a bad idea to interpret the phase factor as classical circling motion.

The magnetic moment of the electron

The magnetic moment of the electron would be explained if the charge of the Dirac electron moves at the speed of light in a circle whose radius is

        λ_e / (2 π),

where λ_e is the Compton wavelength of the electron:

       λ_e = h / (m_e c).

That is, the charge does a loop whose radius is double the zitterbewegung radius.

Our March 7, 2021 post tried to explain why the electron gyromagnetic ratio is 2. Let us elaborate our explanation.

https://physics.stackexchange.com/questions/398401/total-angular-momentum-and-the-dirac-equation-commutator

The Dirac hamiltonian commutes with the total angular momentum operator

       J = L + 1/2 S,

not individually with 1/2 S. This suggests that 1/2 S is the spin classical angular momentum, and that has been confirmed experimentally, too.

Why would the magnetic moment then be determined by S, not 1/2 S? If we couple the Dirac equation with minimal coupling to a magnetic field B, we easily get the result that in the hamiltonian, the magnetic moment is determined by S.

Minimal coupling classically is something which works for linear movement of a charge. It also works in determining the magnetic moment of the Dirac electron. That is evidence for the hypothesis that the Dirac spinor does encode classical circling motion.

We conjectured in the March 7, 2021 blog post that in the path integral, the electron cannot draw a circle whose length is only 1/2 of its Compton wavelength. We claimed that instead, the path integral makes the electron to appear moving in a circle whose length is the Compton wavelength. How would the Dirac equation "know" about these intricacies of the path integral?

The Dirac equation "knows" about the spin 1/2 of the electron through the algebra of the gamma matrices.

Open problem. How does the Dirac equation "know" that the electron path integral has to have the circle length at least equal to the Compton wavelength, and knows to set the electron magnetic moment accordingly?

The anomalous magnetic moment of the electron

We were able to explain qualitatively the Schwinger formula for the anomalous magnetic moment by assuming that the static field of the electron has inertial mass, and that the far field cannot take part in the movement of the electron charge in the circle of the radius

        λ_e / (2 π).

That reduces the effective mass of the electron and increases its magnetic moment.

Our result strongly suggests that the electron magnetic moment is a result of real physical circling motion of the electron charge. That is, zitterbewegung is very real physical motion.

Why does the Born approximation work in Feynman diagrams?

Our blog post on March 23, 2021 tried to reduce the vertex correction Feynman diagram into classical physics.

A central problem is what is an impulse (= source term in the inhomogeneous equation), and impulse response, in the Dirac equation. How do we interpret the impulse classically, when we think of the electron as a classical particle?

           ● field of the electron

            |

            | "spring"

            |

       e- ● --->

                                 Z+ ●

Our example is classical Coulomb scattering of the electron from a nucleus. We believe that the vertex correction is a classical effect which is caused by the static electric field of the electron. But we have to prove that.

In quantum field theory, particles are created by hitting a field equation with a Dirac delta impulse. In Coulomb scattering, we do not really create any new electrons. But could it be that the Born approximation can be modeled as creating a small flux of new, scattered electrons? The basic idea of the Born approximation is that we treat the main solution of a field equation as constant, and calculate small scattered fluxes from small corrections to the equation.

Then classically, an impulse might be a force which affects the electron as it flies past the nucleus.

We think that the electron gives an impulse to its own electric field during the fly-by. Is there an opposite impulse to the electron? If yes, that would explain the Feynman diagram where the electron emits an off-shell photon, and becomes off-shell itself.

Note that if we model an electron as a free particle, but the electron is constantly under a force, then the electron wave does not satisfy the free Dirac equation, and we might say that the electron is off-shell. Suppose that the effect of the electric field of the electron is to reduce its mass. Then the electron will appear off-shell as long as its mass is reduced. Classically, the electron does not satisfy the energy-momentum relation

       E² = p² + m²,

if its mass is reduced.

The role of a wave phase shift

If we calculate in the wave model, then reflection of waves and similar phenomena cause a phase shift. If we add such shifted wave to a wave in the original phase, there may be destructive interference.

                         significantly scattered flux

                       /

                     /

                   /

        -------        ●

        ---------------------------------- little scattered

        ---------------------------------- flux

If the Born approximation produces waves which are shifted by 180 degrees and those waves cancel some flux of significant scattering, then we assume that in the precisely modeled process, the canceled flux appears in the less scattered flux. The total flux has to be conserved in the precise model.

https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qed/feyn.pdf

Let us check the phase of the vertex correction diagram. In the link (2015) we have the Feynman rules.

The simplest Coulomb scattering diagram has just one photon propagator line and two vertices. The phase factor in the Feynman formula is

       -i * -i * -i = i.

The vertex correction has two photon propagator lines, two electron propagator lines and four vertices. The phase factor is

       -i * -i * i * i * -i * -i * -i * -i = 1.

The vertex correction is shifted by 90 degrees from the simplest scattering diagram.

Why does the Born approximation work for the vertex correction?

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

Suppose that significant scattering only happens if the electron comes within a small minimum distance r of the nucleus. The cross section is small for large deflection angles.

Suppose then that there is some small disturbance which depends on r and, say, increases the minimum distance r by a small value dr for every electron. It is fundamentally a non-perturbative process because the entire flux of electrons shifts its course.

Let us try to use the Born approximation to calculate the effect of the disturbance as a perturbation. We have to assume:

1. Most electrons still go along their original path - the main wave stays as it is.

2. A small number of electrons change their path (4-momentum) substantially - we add a small "perturbation wave" (= wave "scattered" from the disturbance).

                                   ______

                      ______/

          _____/

          -------------------------------->

         0                  r     

Suppose that the original distribution of the electrons is such that very few have a small value of r. If we shift the distribution to the right by dr, that is almost equivalent to picking, say, 1/137 of the electrons and shifting them to the right by 137 dr. A small perturbation wave can be used to get the right spatial distribution with respect to r.

The general idea of the Born approximation. If there is a small disturbance, and we can approximate its effect in the final calculated result by imagining a large effect on a small number of electrons, and the rest of the electrons going their original route, then we can use the Born approximation.

Feynman diagrams use the Born approximation. They only work correctly if the thing we are trying to calculate can be approximated by keeping the original wave intact and adding a small perturbation wave. In earlier blog posts we remarked that Feynman diagrams do not model in the right way the bending of an electron beam in a constant electric field. The entire original wave is deflected to a constant angle. One cannot obtain such a result by keeping the original wave intact and adding a small perturbation wave.

The average 4-momentum of the total electron flux is sensible, but the 4-momentum change is grossly exaggerated for the perturbation wave.

The perturbation wave is "fictitious" from the point of view of momenta: there is nothing like the perturbation wave in the precise solution. If we try to solve a differential equation by a stepwise approximation method, then the Born approximation is only a single step. In some cases, using a single step does work.

Richard Feynman in his 1949 papers does not mention the problems of the Born approximation at all. Feynman right away assumes that using a small perturbation wave works. Feynman believed that the virtual particles, that is, the perturbation waves, "exist" in some sense. Our analysis above suggests that they are almost pure fiction.

Why does an electron perturbation wave absorb the photon sent originally by itself? Why not the photon of some other perturbation wave?

Let us again use the rubber plate model of the static electron electric field. The impulse makes a dent in the rubber plate. We can calculate the subsequent behavior of the rubber plate using Green's functions.

The Fourier decomposition of the impulse response is what the Feynman formula uses in calculation.

But classically, the electron does not care about the Fourier decomposition. It sees the classical wave.

Why does the Feynman formula work? Richard Feynman himself wrote that we must calculate all the ways that a "photon can propagate". What does that mean classically?

Feynman diagrams must conserve energy and momentum at the vertices. What is the associated classical phenomenon?

             

                  photon wave W-ph, E and q

                            \     |     |     /

                       \                            /

                   \                                    /

   e-  | | | | | |     |     |     |     |   |   |   |    ||||||||||||||

                  electron wave W-e

                                        |

                                        |  momentum

                                        |  transfer p

     Z+ |   |   |   |   |   |   |   |        |        |        |

              nucleus wave

Let us use the wave representation of the classical electron and the classical nucleus. That is, we look at the path integral with a wavelength determined by the Planck constant (or any constant, as long as the constant is small enough).

Classically, every electron sends a certain photon wave. But let us use the Born approximation and assume that only a small number of electrons send a photon wave W-ph with certain E and q. 

There is a small electron perturbation wave W-e with the energy E and momentum q subtracted from those of the incoming electron wave. When W-e passes close to the nucleus Z+, it absorbs momentum p.

Why does W-e absorb later its counterpart W-ph? Why not absorb some other photon wave?

The reason probably is the way that the Born approximation is built: we imagine a small perturbation wave which has a much exaggerated deflection in the 4-momentum. To cancel these exaggerated effects, we have to return back the original E and p which we borrowed from it to create the photon.

The mystery of allowing the electron to send ANY virtual photon in the vertex correction

                         E, q virtual photon

                        ~~~~~~~

        k           /                      \

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

                                |

                                | p virtual photon

                                |

        Z+ -----------------------------------------

Suppose that we have calculated the scattering differential cross section using the simplest Feynman diagram. We should estimate the effect of the interaction of the electron with its own electric field. There is probably some effective mass reduction to the electron.

In the Born approximation we let most electrons follow their ordinary course. A small number of electrons get a large kick from emission of a large electromagnetic wave.

We know that classically, the electron which approaches the nucleus will make a rather smooth impulse to its own field. The smoothness of the impulse depends on p. A large value of p means a sharp turn and a sharp impulse.

The Feynman formula somehow gets the right result (after regularization and renormalization) by letting the electron make a Dirac delta impulse to its own field. The delta impulse makes a wave which contains all pairs E, q. How can it work without using p as an input?

We will return to this question in a subsequent blog post.

The anomalous magnetic moment of the electron

                       virtual photon E, q

                       ~~~~~~~~

                     /                        \

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

                                | virtual

                                | photon

                                | p

                                |

                                B   magnetic field

https://canvas.harvard.edu/files/936399/download%3Fdownload_frd%3D1%26verifier%3DDTWO5WywHiDDO9YoFWCbyxFi6dpxJBZlug5SQAvx

(Matthew Schwartz, 2012)

The most famous result of the vertex correction (vertex function) is the calculation of the electron anomalous magnetic moment by Julian Schwinger in 1948.

In that calculation, the momentum p in the diagram above is allowed to go to zero. It is clear that the disturbance caused by the very weak p cannot be the reason for the correction. Our own calculation on March 2, 2021 suggested that the correction comes from the electron charge movement in zitterbewegung. The very rapid circling of the electron causes the rather substantial reduction in the effective mass and a larger magnetic moment.

Thus, the electron movement in the above diagram really is the light-speed zitterbewegung movement. The diagram is about coupling the field B to the spin of the electron. It is not the movement of the electron in the laboratory frame.

https://arxiv.org/abs/1910.11085

David Hestenes (2020) has a new paper about zitterbewegung. We have to check that.

There exists another vertex correction diagram for the electron movement in the laboratory frame. The mass reduction should be very small if the electron does a large circle at a speed slower than light.

We will write a new blog post about the anomalous magnetic moment. It opens new perspectives into zitterbewegung.