The electrostatic field of the electron has zero energy, but it exerts forces on the electron

This idea may solve the mystery of the "inner field" of the electron, as well as the classical renormalization problem of the electron mass. The problem has been that if one calculates the mass-energy of the static electric (electrostatic) field of the electron, the mass-energy of the field exceeds the mass-energy 511 keV of the electron if we integrate closer than a half of the classical radius of the electron, that is, closer than 1.4 * 10⁻¹⁵ m.

That suggests that the "bare mass" of the electron is minus infinite, and it only becomes 511 keV once we add the infinite mass-energy of the field.

Having an object with negative mass-energy is problematic in classical physics. We would like all objects to have mass-energy which is > 0.

Furthermore, there is a "naturalness problem": if the bare mass is minus infinite, why does the field happen to raise the observed mass to a tiny value 511 keV? Why not 10¹⁰⁰ times larger? There has to exist incredible fine-tuning.

In a periodic movement, the effective inertial mass can appear negative

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

(The drawing is by KiraLakira. See https://commons.wikimedia.org/wiki/File:A_mechanical_model_giving_rise_to_the_negative_effective_mass_effect..jpg#mw-jump-to-license about licensing.)

Wikipedia has a drawing of a mechanical spring device which appears to have negative inertial mass if subjected to a periodic force of a certain frequency.

The idea is that a spring overcompensates the centrifugal "force" (the centrifugal "force" is a real force in a rotating, co-moving, coordinate system). We move a body with our hand along a circle, and feel then a centrifugal "force" from the inertia of the body. Suppose then that there is a spring which helps to keep the body in the orbit. We feel that the centrifugal "force" is weaker, as if the body would have less inertia. If the spring is very strong, we have to pull the body outward to keep it in its orbit. The inertial mass of the body appears to be negative.

The static electric field of the electron can be viewed as a spring system around the electron. If the electron does not radiate real photons, then the far field has to be static and helps to keep the electron in a periodic orbit, effectively reducing the inertia of the electron for that movement. In many quantum mechanical processes, no real photons are radiated.

The field only reduces the effective inertia for a periodic movement. It does not do that for a linear movement.

In this blog we have been claiming that the far field of the electron cannot keep up with the electron in sudden movements, and the effective inertial mass of the electron is thus reduced in such movements. We were thinking that the mass-energy of the far field is cut off from the mass of the electron. But another interpretation is that in a periodic movement, the far field of the electron exerts a force on the electron and helps to keep the electron in the orbit, reducing the effective inertia of the electron.

In this new interpretation, the mass-energy of the static electric field is zero. This solves the classical renormalization problem of the infinite energy of the electron electric field. The energy is not infinite - it is zero. The 511 keV mass of the electron is all in the particle, not in the field.

https://www.feynmanlectures.caltech.edu/II_28.html

The new interpretation also solves the famous "4/3" problem of the Poynting vector. Richard Feynman has written a lengthy discussion of that problem.

The self-energy of the electron in quantum electrodynamics is zero

                    virtual photon k

                         ~~~~~~~~

                       /                      \

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

Quantum electrodynamics calculates the "self-energy" of the electron from the Feynman diagram above. We integrate over all 4-momenta k. The integral is divergent. The mass of the electron is renormalized to straighten things up.

This bears a resemblance to the classical renormalization problem of the static field of the electron.

In this blog we have been claiming that the diagram above breaks conservation of the speed of the center of mass, and the diagram is thus prohibited. Imagine that the electron is initially static. The electron emits a photon with spatial momentum k to the left. The electron starts moving to the right. Then the electron absorbs the photon. The end result is that the electron moved to the right. The center of mass moved.

We have also been claiming that destructive interference wipes out the whole integral over all 4-momenta k. Imagine a static electron. There is no reason why it should emit the photon k at a time t rather than at a slightly later time t'. Summing over all the times produces total destructive interference.

Question. Should we claim that photons with pure spatial momentum k do exist? They do not "live in time" and carry no energy.

          t

          ^

          |                     |     /  crests of the waves of

          |                     |   /    photons k with

          |                     | /   /  energy != 0

          |                     |   /

          |                     | /

          |                     |

          |                     e-

           -----------------------------------------> x

In the diagram we have a static (= free) electron which emits photons. If the photon contains energy, the crests of the wave of the photon are oblique lines. There is no reason why the photon would be emitted at a time t rather than at a slightly later time t'. We conclude that destructive interference wipes out such waves.

However, a photon with pure spatial momentum k in the diagram has vertical crests of the waves. Why there should be destructive interference of such waves? Maybe they do exist.

Anyway, photons carrying energy:

1. seem to break conservation of the speed of the center of mass;

2. are totally wiped out by destructive interference for a free electron.

We have good reasons to claim that the field of a free electron contains no energy. The "self-energy" of the electron is zero. There is no need to renormalize the mass of the electron. Its "bare mass" is the same as the observed mass.

If we allow photons with pure spatial momentum k to exists, the Feynman self-energy integral is still zero because the set of such photons has a zero measure in the 4-momentum space ℝ⁴.

Vacuum polarization: the momentum q "disturbs" perfect destructive interference of the virtual pair?

We have been working hard to find a plausible interpretation for the vacuum polarization integral in QED.

http://users.physik.fu-berlin.de/~kleinert/kleiner_reb6/psfiles/index.html

The vacuum polarization integral above is from Section 12.16 of the book by Hagen Kleinert. The Ward identity has already been used in the integral to remove the badly divergent part. The integral above is only logarithmically divergent.

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

                                    |   virtual

                                    |   photon q

                                    |

                           -k    O    k + q virtual pair

                                    |

                                    |   virtual

                                    |   photon q

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

           proton

Above we have the relevant Feynman diagram.

Interpretation of the vacuum polarization integral.

Case of the free electron. Let the Feynman diagram describe an electron moving essentially freely, far away from the proton. We set q = 0. We claim that there is total destructive interference of the virtual pair wave function in the diagram. That is because there is no reason why the loop should be born at a position x of the electron rather than at a slightly later position x'.

Alternatively, if the electron is static, there is no reason why the loop should be born at a time t rather than at a slightly later time t'.

When we sum the probability amplitudes of all histories, there is total destructive interference of the virtual pair wave function.

We claim that the full integral

       Π_μν(0)

describes the virtual pair in the case that there is no momentum exchange between the electron and the proton. That integral is totally wiped out by destructive interference.

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

                             -k    O    k  virtual pair

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

          proton

The actual Feynman diagram is above. The virtual pair floats freely in space, undisturbed.

Case of momentum exchange q != 0. In this case, the free floating of the virtual pair is disturbed by the momentum exchange q between the electron and the proton.

There is still almost total destructive interference for large |k|. The Feynman integral in this case

       Π_μν(q)

still describes the part of the full integral which is wiped out by destructive interference. The difference

       Π_μν(0)  -  Π_μν(q)

is the part of the full integral which survives destructive interference. The rest of the full integral "transmits" the momentum q from the electron to the proton. Note that the "propagator" of transmitting q over the virtual loop does not depend on q. Why is that? In an earlier blog post we sketched the idea that the virtual loop is a "dipole particle" and the momentum q is transmitted over the dipole "simultaneously".

                                   |  spatial momentum q 

                                   |

                            ____|___   e+  4-momentum k

                          /        |       \

                          \___ ●___/  e-   4-momentum -k

                                   |

                                   |

                                   |  spatial momentum q

The diagram is above. At the dot ●, the momentum q flows simultaneously between the lower and upper momentum q lines. The flow does not affect the electron propagator.

Why is the formula Π_μν(q) the exact description of the part which is wiped out by destructive interference? Why not some other formula? The model of the next section would explain that.

Another interpretation of the full integral: the vacuum polarization effect on the electron is "bremsstrahlung" of virtual pairs

Let us first assume that the electron flies freely.

                                    k

                           ~~~ O ~~~

                  q     /        -k         \  q

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

In the diagram, the electron sends itself arbitrary 4-momentum q as a virtual photon. The virtual pair loop receives and sends q simultaneously. That is why q does not appear in the virtual pair 4-momenta k, -k or the propagators of the pair.

The full integral of the process above is Π_μν(0). The full integral is completely wiped out by destructive interference.

Suppose then that the flight of the electron is disturbed by a proton, between sending and receiving q. In that case, the virtual pair cannot receive and send q simultaneously. The 4-momentum q has to spend some time in the loop. Now q appears in the propagators of the loop.

The electron can still absorb most of the wave that it sent, especially the high frequencies. The integral for the absorbed wave is

        Π_μν(q).

The electron sent a wave described by Π_μν(0), but was only able to absorb Π_μν(q) back. What happened to the rest of the wave? It was absorbed by the proton. In a sense, the disturbance made part of the wave Π_μν(0) to get "detached" from the electron and be absorbed by the proton.

The sharp hammer model and the part of the wave which is not absorbed

Recall that our "sharp hammer" model of the static electric field of an electron explains bremsstrahlung as the part of the wave which gets "detached" from the electron in the wave created by hits of the sharp hammer (a hit is a Dirac delta source to the massless Klein-Gordon wave equation).

The static electric field is created by quickly repeating hits with a sharp hammer. If the electron changes its course, it cannot absorb the entire wave back. Part of the wave (some real photons) escapes as bremsstrahlung.

We have claimed that the vertex function is a classical electromagnetic process where part of the static electric field of the electron "lags behind". This sounds a lot like our ideas for bremsstrahlung. Part of the wave gets "detached" but is eventually absorbed by the electron itself.

In a forthcoming blog post we will analyze once again the QED vertex function (correction), and how we can explain it with our sharp hammer model.

The Ward identity in QED vacuum polarization: does it make sense?

Peskin and Schroeder in Section 7.4 in their book state the Ward identity in the following way:

Ward identity. If

        M(k) = ε_μ(k) M^μ(k)

is the probability amplitude of a QED process involving an external photon with 4-momentum k and the polarization vector ε_μ(k), then if the polarization vector of the photon would be "longitudinal", that is,

       ε_μ(k) = k_μ

for all coordinates μ, then the probability amplitude M(k) is zero. This is written using the Einstein notation as

       k_μ M^μ(k) = 0.

In this we assume that the probability amplitude of the process can be written as a sum of the amplitudes for each polarization vector component of the photon.

Suppose that the photon entering the process is a real photon. We know that its polarization has to be normal relative to its spatial momentum and 4-momentum. If the Ward identity would not hold, then by running the process backward we would be able to produce a longitudinally polarized real photon, which is a contradiction. Thus, we proved that the Ward identity must hold for real photons.

However, if the photon is an internal photon line which carries pure spatial momentum between an electron and a proton in a scattering experiment, then the "polarization" of the photon is not defined, or it is longitudinal. What does the Ward identity say in that case?

Maybe we should not talk about polarization but state the Ward identity as a rule about Feynman diagrams. In the book of Peskin and Schroeder, Section 7.4, the identity is proved by induction for Feynman rules.

The identity seems to be a consequence of the fact that the value of the electron propagator only depends on how "far" off-shell is the electron. For example, in the following diagrams, the electron in the internal line is "as much" off-shell in both diagrams.

                                  |  photon

                                  |  q

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

                         |  photon

                         |  k

                         |  photon

                         |  q

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

                                 |  photon

                                 |  k

The electron enters and leaves the diagrams on-shell.

The Ward identity in the vacuum polarization integral

https://www.southampton.ac.uk/~doug/qft/aqft3.pdf

The enclosed picture was copied from the link.

In the link, the Ward identity used in a vacuum polarization calculation is "proved" from the formula above. The integrals over all 4-momenta k on

       1 / (γ • (k - q) - m)

and

       1 / (γ • k - m)

are identical if one changes the variable k - q in the first formula to k. One can claim that the difference of the integrals is zero, which would prove the Ward identity.

Both integrals diverge badly. If we integrate over a volume which is symmetric with respect to 4-momenta 0 and q, then the integrals really have the same value.

If one uses an integration volume which is asymmetric, the difference of the integrals does not converge.

Question. Is it fair to say that the difference of the integrals is zero?

In this blog we have been claiming that the vacuum polarization integral really calculates the part which is wiped off by destructive interference. Is there some reason why we should use an integration volume which is symmetric with respect to 0 and q?

Green's functions: a "smooth" source cannot create high-frequency waves

We have been claiming that destructive interference wipes out high-frequency waves in natural phenomena. We can now present a more sophisticated argument for our claim.

Let us have a field. The field can be a drum skin, the Dirac field, or whatever. Let the wave equation for the undisturbed field be a homogeneous linear wave equation

       W(ψ(t, x)) = 0.

If the field is disturbed, we can calculate the approximate response of the field with the same linear wave equation, except that now we have a source I(t, x):

       W(ψ(t, x)) = I(t, x).

The disturbance to the field is typically an interaction with another field in which we have a wave packet. The letter "I" stands for interaction.

Let us assume that the Fourier transform of the source I(t, x) contains no high frequencies. The source I(t, x) is a smooth wave packet.

Since the wave equation is linear, we can build the response of the field by summing the responses to individual components

        S(t, x) = exp(-i (-E t + p x))

of the Fourier transform of the source I(t, x).

We build the response from a Green's function of the wave equation for our field. A Green's function is the response of the field to a Dirac delta source. For example, we can hit a drum skin with a sharp hammer. That constitutes a Dirac delta source in the drum skin wave equation.

The Fourier transform of the Green's function typically contains arbitrarily high frequencies, because it is the response to a very "sharp" impulse.

Let us then look at the response to the source S(t, x). What happens to high frequency components of the Green's function? They are totally wiped out by destructive interference. The source S(t, x) spans the whole spacetime. The only surviving component of the Green's function is the one which "syncs" with the source S(t, x).

Note that when an electron "absorbs" a photon, it is just this syncing mechanism behind the process.

We have found the reason why in typical natural phenomena, low-frequency waves cannot give birth to high-frequency waves.

Are there processes where low frequencies can produce high frequencies? Yes. A gearbox can turn slow oscillation into fast oscillation. Such gearboxes do not exist in typical natural phenomena.

Coronal heating problem: a whiplash effect explains the high temperature in the corona of the Sun?

One of well-known open problems in physics is what heats the corona of the Sun to a temperature of over 1,000,000 kelvins while the visible surface of the Sun is only 6,000 K.

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

There are competing explanations:

1. Alfvén waves around magnetic lines of force;

2. magneto-acoustic waves;

3. magnetic reconnection theory.

Let us add a new hypothesis: the whiplash effect of a string which becomes thinner.

Magnetic field lines are in a constant movement inside the Sun. For example, in a prominence we see a magnetic field line rise up from the surface of the Sun and move farther into space.

A moving magnetic field line in plasma induces an electric current. Let us model the current as a very large point charge Q which moves inside the Sun.

                   space

        ----------------------------  surface of the Sun

                   ●  ---->

                  Q point charge

The movement of Q creates an electromagnetic wave. It is usually half a wave, not periodic.

Electromagnetic waves inside plasma obey the massive Klein-Gordon equation

        d²ψ / dt²  -  d²ψ / dx²  + m² ψ  = 0.

The velocity of the wave is slow in plasma. The effective mass m in the equation is large.

Plasma becomes thinner when we move out from the surface of the Sun, and farther into the corona. When the electromagnetic wave goes up, there is a whiplash effect. The effective mass m is much less up in the corona. It is like a wave traveling in a string which becomes a lot thinner.

                            wave ----->

            ===========--------------- . . . . . . .

            string

The amplitude of the wave grows at the thin end of the whip. This means that charged particles will move faster in the corona than inside the Sun. Collisions of fast electrons and protons then create a high temperature in the corona.

What are the consequences of the correct explanation of regularization / renormalization?

We believe that we have corrected a fundamental error in the interpretation of regularization / renormalization of the vacuum polarization diagram in QED.

How a sign error gave rise to conceptual problems

The origin of the error was a wrong sign the Feynman integral which describes vacuum polarization. People believed that vacuum polarization weakens the Coulomb force, while the truth is that it strengthens the force.

The sign error was "corrected" by calculating the complement of the true physically relevant integral. This creates a further problem: the complement is formally infinite. If we believe that the complement physically "exists", how do we handle the conceptual problem that the physical entity is infinite?

The further problem was solved by two methods:

- regularization, which makes the infinite integral formally finite, and

- renormalization, which explains why the observed charge of the electron is finite while its "bare charge" is infinite deep down. The bare charge is screened by infinite polarization of the vacuum.

The machinery of regularization / renormalization can be viewed as a purely formal tool which does not imply or prove that any new physics is hiding from us at the Planck scale. However, people were tempted to think that there is a physical, real cutoff at the Planck scale, which would make the infinite integral physically finite.

Kenneth Wilson developed a conceptual framework where hidden physics at very short distances gives rise to the physics which we measure at distances of, say, 1 meter. The analogy is water, where complicated physics at the scale of molecules eventually explains the Navier-Stokes fluid behavior of water at the scale of 1 meter.

But there is a serious problem in the water analogy of the vacuum. If we do experiments with water, information from our experiments leaks into the hidden small-scale world of molecules. Molecules have to "exist" physically to carry information. They must contain mass-energy.

In the case of vacuum polarization, the role of water molecules is held by the negative energy electrons in the Dirac sea. If we do scattering experiments with electrons and protons, then information about the experiments leaks into the hidden world of the Dirac sea. Thus, the Dirac sea has to "exist" physically and must contain mass-energy.

Vacuum mass-energy brings two serious problems:

- How do we describe the information which leaks into the hidden world? What is the state of the vacuum?

- What is the mass-energy of the vacuum? Simple calculations give infinite energy per cubic meter. Why space does not collapse into a black hole?

Quite a mess was caused by a simple sign error in the vacuum polarization integral.

Consequences of correcting the interpretation of vacuum polarization: string theory and loop quantum gravity

Conceptual problems in regularization / renormalization have been a motivator in the development of string models ("string theory") and loop quantum gravity. This motivator is removed once we accept the correct interpretation of regularization.

However, there is another motivator for string models: trying to derive the masses and other properties of elementary particles. Are the masses really arbitrary or is there a hidden explanation for them?

Loop quantum gravity is a step toward discrete mathematical models of physics. Maybe the world at a very small scale is digital?

Renormalizability of theories

In our interpretation there is no renormalization. We do not need to "absorb" infinite quantities into the definition of parameters of the theory. The "true" values of the parameters, for example, the electron charge, are the values which we measure at the scale of long distances.

However, it might happen for some theories that integrals still diverge even if we take into account destructive interference. We have to find out if there are important theories which behave that badly.

Logarithmic ultraviolet divergences are probably benign. One can make them converge with only a moderate amount of destructive interference.

Quantum gravity

Gravity is not "renormalizable". One cannot absorb all its divergences into parameters of the theory.

In our interpretation, there is no need to absorb divergences. The non-renormalizability of gravity is a moot point.

However, a problem is that we do not know the behavior of gravity at very high energies. Do particles become black holes? If yes, how can we treat black holes in Feynman diagrams?

A simple solution is to cut off extremely high energies. Then gravity behaves well, or does it?

Gravity affects the geometry of spacetime. That brings problems for quantum field theory, which is traditionally done in the flat Minkowski metric.

There are many problems in formulating a quantum field theory of gravity. At low energies, we probably are fine with machinery similar to QED. But we are interested in phenomena in black holes, and there energies are potentially infinite.

The hierarchy problem and supersymmetry

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

One of the problems is why the Higgs boson mass is so tiny compared to the Planck mass. People think that the Higgs boson has a "bare mass" at the scale of very short distances. That bare mass has to be corrected with a self-energy calculation. If we set a bare mass at the Planck scale, the correction and the bare mass have to be incredibly fine-tuned to output the measured tiny mass of the Higgs boson.

We in our blog believe that it is the long-distance behavior of the Higgs boson and its couplings which gives rise to its short-distance behavior. Does that remove the need for fine-tuning?

This hierarchy problem is one of the motivators for supersymmetry.

The mass-energy and inertia of the static electric field of the electron is zero?

The electron self-energy involves the mystery of the electron "inner field". The mass-energy of the inner electric field of a point particle should classically be infinite. Why is the mass of the electron finite? We in this blog have not yet solved this mystery. The Higgs boson hierarchy problem is probably analogous to the electron inner field problem. We have to study this more.

                           virtual photon

                            ~~~~

                          /             \

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

Our current view on the electron self-energy Feynman diagram is that the virtual photon cannot carry mass-energy, since that would break conservation of the speed of the center of mass. However, the virtual photon might carry spatial momentum.

We could ban the self-energy diagram altogether for a free electron. We can claim that destructive interference wipes out virtual photons of all frequencies. Virtual photons can then only exist if the electron is interacting with some other particle. The inertial mass of the electron would be a random parameter, and there would be no correction to it.

So far we have assumed that in a swift motion, the inertial mass of an electron is reduced since its far field does not have time to react. But it might be that the far field helps the movement of the electron in some of such situations. Then we do not have the problem of what happens in swift movements whose length is smaller than 1/2 of the classical radius of the electron. With the old model, the inertial mass of the electron itself would become negative, because the mass-energy of the field outside 1/2 of the classical radius is > 511 keV. That would be very ugly.

Our explanation of the Lamb shift relied on reduction of the inertial mass of the electron. Our various models of a rotating tight rope in the spring of 2021 explored the possibility that the rope becomes tighter and helps the end of the robe stay in a circular orbit.

We have been contemplating the possibility that the mass-energy and inertia of the static electric field of the electron is zero. That would solve the classical renormalization problem: why the mass of the electron is finite while the energy of the field of a point charge is infinite. It would also solve the classical 4/3 problem of the Poynting vector.

The goal: show that we can consistently set the mass-energy of the field zero, but still explain the Lamb shift and how a radio transmitter works. Our "rubber plate" model would then have the rubber plate having zero mass.

If this succeeds for the electron, it might succeed for the Higgs boson, too, and remove the hierarchy problem.

The Casimir effect is not caused by "vacuum energy" - it is a van der Waals force

Our previous blog post suggests that vacuum energy is zero. Thus, vacuum energy cannot be responsible for the Casimir effect, which has been empirically measured at the precision of 1% already.

https://arxiv.org/abs/hep-th/0503158

Robert L. Jaffe (2005) from MIT writes about the derivation of the Casimir effect in QED.

Hendrik Casimir and Dirk Polder calculated in 1948 the van der Waals force between two polarizable molecules. 

                +     -                        +     -

          molecule 1             molecule 2

Configurations where the molecules are polarized and attract each other are energetically favored over configurations in which they are polarized and repel each other. This creates a weak attractive force. There is no reference to vacuum energy in this. It is common sense that an attractive force forms between the molecules. The force is a van der Waals force.

Casimir and Polder noticed that if we do not assume the van der Waals force between the molecules, we can still derive an attractive force by assuming "vacuum energy" and its pressure on the molecules. The derived force has the correct numerical value.

This observation created the myth that the Casimir effect somehow proves that there exists vacuum energy.

Our previous blog post suggests an explanation for the coincidence that the exact same force can be derived in two different ways. The vacuum energy derivation calculates the complement of the true physical effect. The true physical effect is the attractive van der Waals force.

Robert Jaffe explains in his paper that in the configuration with two metal plates, the fact that the plates are made of a polarizable material (metal) is obfuscated by establishing a boundary condition for assumed "vacuum fluctuations". The dipole explanation for the force is thus hidden from plain sight.

We probably solved the mystery of regularization in vacuum polarization

In this blog we have been claiming that destructive interference wipes off very high frequencies, and that there are no true divergences in Feynman integrals.

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

                            | virtual photon q

                            |

                    k     O     -k + q    virtual pair loop

                            |

                            | virtual photon q

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

     proton

            

In vacuum polarization, a sharp cutoff at the exchanged momentum |q| generates "almost right" results. We only integrate over those 4-momenta k for which the euclidean norm of k satisfies

       |k| < |q|.

It is immediately clear that a sharp cutoff at |q| does not describe destructive interference correctly. Wave phenomena are smooth. We should find a way to make a smooth cutoff.

Let us have a divergent Feynman integral K(q) where q is a parameter.

If we set q = 0, we have a "plain" version of the integral, K(0).

Let us then let q increase from zero. In the case of the vacuum polarization integral (12.448) in Hagen Kleinert's book, the absolute value of the integral decreases to the value K(q) (the integral is infinite, but assume a cutoff at some very large Λ).

We then interpret that the part

       K(q)

is the high frequency part of the integral which destructive interference wipes off.

The remaining part

       K(0) - K(q)

is the correct integral value. The part which was wiped off by destructive interference has been removed. Only the relevant part of the integral remains.

This interpretation has the following advantages:

1. We do not need to speculate about new physics at a high-energy cutoff Λ. The cutoff is purely a formal mathematical tool when we calculate the effect of increasing q.

2. The sign error in the Feynman vacuum polarization integral is removed. The integral makes the Coulomb force stronger, not weaker as literature claims.

3. This explains why various regularization methods do work, and generate the same numerical results. The role of regularization is just to make the integration volume big enough, but not infinite, so that we can calculate the effect of increasing q.

4. We do not need to assume the existence of the Dirac sea, or "vacuum fluctuations" which fill the vacuum. In some literature, it is assumed that the part K(q) in the integral is something real or physical. It could be the Dirac sea of negative energy electrons, or "vacuum energy" which fills the space. We only need to assume the existence of the real particles which enter the scattering experiment.

5. We do not need to renormalize the charge of the electron.

In the case of quantum gravity, the interpretation raises the following problem: since very high-energy quanta may be black holes, how do we calculate the effect of increasing q? Maybe we in gravity have to make a real, physical cutoff.

The interpretation above refutes the Wilsonian view that low-energy QED somehow arises from unknown physics at the Planck scale. In our model, the running of the QED coupling constant is a result of well-known QED physics at the low-energy scale.

A simple calculation example

Suppose that we have a divergent integral which, we believe, describes a physical process:

                   ∞

       K(q) = ∫  1 / (1 + |q|+ x) dx.

                 0

Let us assume that the correct integral I(q) for the physical process is obtained with a sharp integration cutoff at |q|, but with no q inside the integrand:

                  |q|

       I(q) = ∫  1 / (1 + x) dx.

                0

We have

       I(q) = K(0) - K(q),

where we assume that K is calculated with the integral cut off at some large Λ to make the integral finite, so that we can subtract the values.

It makes sense to say that K(q) is the "complement" of I(q).

Summary of divergences

Let us summarize our current view on cutoffs / regularization / renormalization.

Infrared divergences. The reason for infrared divergences is that Feynman integrals erroneously treat overlapping classical probabilities as separate cases.

At the low energy / low frequency / large distances scale, an electron looks like a classical particle whose position is reasonably well defined. In those frequencies, the electron sends classical electromagnetic radiation. That radiation contains many soft photons, for example, in bremsstrahlung. Thus, there is a large overlap in, say, emission of a soft photon A and a soft photon B. The emissions are not separate cases.

A cutoff at a moderate photon energy removes almost all overlapping in classical probabilities. That is why it works.

Ultraviolet divergences. The divergence is removed by destructive interference of high-frequency waves. The traditional regularization / renormalization process is a way to calculate what is left after destructive interference wipes off almost all of high frequencies.

There is a sign error in the Feynman vacuum polarization integral in literature. People have been thinking that the complement of the relevant integral value is the "physical, existing" thing. Paul Dirac thought that the infinite Dirac sea of negative energy electrons screens almost all of the (infinite) bare charge. But the complement is what really does not exist physically.

We have explained why the assumption that the complement exists, leads to problems with the model. Information flows from real particles to the shadow world of the complement. How are we supposed to describe the state of the complement? Why the negative energy electrons do not form "shadow" hydrogen atoms with real protons?

The problem of ultraviolet divergences exists also in various wave problems of classical physics. If we would use Green's functions and calculate over arbitrarily high frequencies, then many problems would diverge. In classical physics we instinctively assume that very high frequencies are cut off by some process (that is, destructive interference). In quantum field theory, this common sense was somehow lost by researchers, and they built the strange interpretation which involves a sign error, regularization, and renormalization.

The vacuum is totally empty in a particle model

Our interpretation probably removes the notion of "vacuum energy" from the calculation of the Casimir effect. There is an attractive dipole force between the metal plates. It is not that the pressure of "vacuum energy" is pushing them together. We need to look at literature, how the Casimir effect is calculated.

In this blog we have advocated a model where particles are fundamental, and wave phenomena arise from path integrals. In a particle model, the vacuum is totally empty. There are no particles in the vacuum. There is no shadow world of negative energy electrons nor vacuum fluctuations.

Textbooks of quantum field theory usually assume the existence of a lowest energy state which is designated as the "vacuum", even though we do not know what that state is, and if the state is unique. Particles are then created with a creation operator.

But in a particle model, the vacuum is totally empty, and real particles enter the experiment. Feynman diagrams are used to calculate the probability amplitude of various outputs. There is no need for a formal creation operator. A new particle is born at a vertex of the diagram. Though we could say that a creation operator creates it there.

Energy non-conservation in QED vacuum polarization: a solution which also solves the proton radius puzzle?

Michael E. Peskin and Daniel Y. Schroeder in their famous book calculate a more precise correction potential

        δV(r)

for the low-momentum vacuum polarization case, such that the Fourier transform of the potential is the vacuum polarization correction to the photon propagator (section 7.5, page 254).

https://www.amazon.com/Introduction-Quantum-Theory-Frontiers-Physics-ebook/dp/B07CYVPDQR

The potential goes to zero exponentially fast,

       δV(r)  ~  exp(-2 m r),

when r grows larger than 1/2 of the reduced Compton wavelength of the electron.

Energy non-conservation

We complained in our August 7, 2021 post that a running coupling constant breaks the classical limit and energy conservation. If a force depends on the exchanged momentum q, then the force depends on the state of the motion of the particle, for example, its speed v. Such a force usually is not conservative. Think about drag in water, for example.

A conservative force is often a result from a potential which only depends on the position of the particle.

The Fourier transform of most potentials does not describe the scattering cross sections of the potential

The vacuum polarization correction

       Π_μν(q)

to the photon propagator depends on the exchanged momentum q. The correction is used to calculate scattering cross sections.

But Peskin and Schroeder define a radial (Uehling) potential, which produces a conservative force field. What is going on?

The Fourier transform of the Coulomb potential 1 / r describes perfectly the classical scattering caused by the potential. The photon propagator gives the exact right classical cross sections. The 1 / r potential is a special case.

Other potentials do not necessarily behave that nicely. For example, the Fourier transform of a Dirac delta potential is a constant function. The Fourier transform does not describe the classical scattering caused by the deep and narrow potential well. The scattering cross section is infinitesimal, but that is not reflected by the Fourier transform.

According to the vacuum polarization matrix

       Π_μν(q),

the corrected photon propagator for small |q| is

       1 / q² + constant,

or

       1 / q² * (1 + q² * constant).

The correction may be significant even for quite small |q|.

However, the correction potential δV(r) goes to zero exponentially fast with r. For a small momentum exchange|q|, it is essentially zero. The corrected potential does not classically reproduce the scattering cross sections described by the corrected photon propagator.

The Schrödinger equation with the corrected potential does not reproduce the cross sections described by the corrected propagator, either.

Does the Feynman diagram calculate scattering cross sections or does it calculate the Fourier transform of a corrected potential?

We have been believing that a Feynman diagram calculates probability amplitudes for scattering.

But literature thinks that in the case of vacuum polarization, it calculates the Fourier transform of a corrected potential, which is a very different thing, as we explained above. Energy conservation and the classical limit are restored if we interpret that it calculates a potential, not scattering cross sections.

Energy conservation is very important. It makes sense to respect it.

Question. What is the corrected potential in the high-momentum q case? Does that potential reproduce (approximately) the scattering which is described by Π_μν(q)?

Question. How to interpret Feynman diagrams in the cases where they do not calculate probability amplitudes for scattering? Should we start using non-perturbative ideas? The momentum exchange q should be split into many smaller parts?

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

Question. If we correct the potential in such a way that it really reproduces the scattering amplitudes of the Feynman vacuum polarization diagram, does that solve the proton radius puzzle? The change might be around 1 / 1,000.

A possible solution to the energy conservation problem: the extra force is a dynamic process which cannot be reduced to a correction term in the Coulomb potential

If we enforce energy conservation separately for each encounter of the charged particles, then there is no energy conservation problem. Energy conservation is enforced in Feynman diagrams.

Then it is possible that the extra force does depend on |q| and not directly on the separation r of the particles. Then the muon in muonic hydrogen feels a different extra force at a distance r than the electron in ordinary hydrogen at the same distance.

In the standard calculation of the vacuum polarization contribution to the Lamb shift in ordinary hydrogen, it is assumed that the correction to the potential is a Dirac delta function. No assumption is made about the detailed form of the potential. It is just a pit very close to the proton.

In muonic hydrogen, the exact form of the potential is relevant because the Bohr radius of muonic hydrogen is only 2.6 * 10⁻¹³ m, which is just over half of the reduced Compton wavelength of the electron.

Thus, it might be that the Feynman diagram does calculate scattering probability amplitudes, and it is an error to interpret it as a way to calculate the Fourier transform of a static corrected Coulomb potential which is the same for all different cases and particles. It is a dynamic process when an electron or a muon encounters a proton. The process does conserve energy, but it does not define a single static force field in space.

We solved the proton radius puzzle?

The observation of the previous section may solve the proton radius puzzle. The vacuum polarization contribution in muonic hydrogen was incorrectly calculated. That is the reason why it gives a wrong value for the proton radius.

The correction potential, which goes exponentially to zero for large r, is a wrong approximation for large r. It should be replaced with a potential which approaches zero slower for large r.

We still need to check if other methods of measuring the proton radius assume anything about a fixed, corrected Coulomb potential, or do they estimate the radius using scattering amplitudes.

A very crude calculation shows that using a potential which better reflects scattering amplitudes may affect the vacuum polarization correction up to 10%. We only need to explain a difference of 0.15%.

http://hyperphysics.phy-astr.gsu.edu/hbase/hydwf.html

In the link is the radial probability density for the 2s and 2p orbitals. If we just fix the potential at > 5 Bohr radii to reflect scattering amplitudes, then the effect is to increase the Lamb shift by about

       1,000 eV * 1 / 100,000 * 0.05 = 0.5 meV,

while we should increase it by 0.3 meV to explain the measurement by Pohl et al. 

The classical limit problem

If we keep an electron close to a proton for a long time, then the exchanged momentum q may be very large over a long period of time. We do not believe that the correction for the Coulomb force should depend on q in such a case. Should we split q in smaller pieces, and if yes, how?

A Feynman diagram calculates the dynamics of a short encounter, a collision of particles. How should we calculate long lasting or static events?

General observations about QED vacuum polarization

We have been thinking hard in the past days how to make the dipole particle model of vacuum polarization more concrete. Meanwhile, we have made some general observations about the vacuum polarization calculation in Hagen Kleinert's book and other sources.

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

                             | virtual photon q

                             | 

                       k   O   -k + q   virtual loop

                             |

                             | virtual photon q

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

      proton

Above q is spatial momentum and k is arbitrary 4-momentum.

What is the sign in the vertex of the loop and the virtual photon? The vertex of the photon and the electron naturally carries a coefficient -e, and the vertex of the photon and the proton +e. But what is the coefficient when the photon couples to the loop? Does it couple to the electron (like in the Dirac sea model), or to the positron? In the dipole model, the force between the dipole and the charge is always attractive.

The Feynman parametrization is a relevant procedure in the calculation. The loop integral diverges quadratically. The Feynman parametrization affects the terms in the integral. Then terms which have an odd power of the arbitrary 4-momentum k, are dropped. That is, the terms in the new formula reveal symmetric integrals which are zero if the integration area is symmetric. Mathematically, this is dubious. Why the symmetries in this particular written form of the integral are relevant? Why not some other form? 

The Ward identity is strange. In the integral, Hagen Kleinert claims (Formula (12.447)) that a particular part of the integral (which actually seems to diverge) "should be" zero based on a Ward identity. The identity claims that the loop matrix

      Π_μν(q)

represents the "polarization" of the photon q, and it must be normal to q because the polarization of a (real) photon is always normal to the momentum of the photon.

We do not understand this. The virtual photon q is a direct momentum exchange from the Coulomb force. It does not have any defined polarization. Moreover, if we define the polarization as the direction of the electric field, the photon q is longitudinally polarized.

A semiclassical "bump into" model. Think about a model where the electron "bumps into" a virtual pair and gives it the momentum q. Then it may happen that the pair bumps into the proton and the proton absorbs the momentum q. Does this process make the Coulomb force appear stronger? If the electron would get the momentum -q toward the proton, then classically, the pair will move away from the proton, and it is improbable that it will bump into the proton. Momentum conservation bans a process where the pair does not give up its momentum. Thus, classically, the process makes the attraction appear weaker. The momentum q points toward the proton.

But what about an electron and an antiproton? In this case, the process would make the repulsion appear stronger. The model is not consistent with vacuum polarization.

What about phase shifts? What would the phase be for the electron which bumped into the virtual pair? Suppose that a mildly relativistic electron scatters to a large angle from the proton. The phase shift when the electron is roughly at the distance of the electron classical radius from the proton is around 1/1,000 cycles. The total phase shift, if the electron is sent from a distance 1 meter, might be around 15/1,000 cycles. The phase shift from bumping into a virtual pair might be of a similar magnitude. We conclude that phase shifts are almost negligible.

Cross sections in this classical model would be larger than for plain Coulomb scattering. The electron could get the momentum exchange q in two ways: through the Coulomb force or through bumping.

What would the propagator be? Maybe something like

        1 / q² * 1 / q² * 1 / m²

for small q. But that becomes infinite for very small q. We believe that the lifetime of the virtual pair is very short. The formula fails to take that into account and is nonsensical. For large q, the value is too small.

Why does the traditional calculation of vacuum polarization give empirically correct values even though there are so many ad hoc tricks in the calculation? Above we mentioned some fuzzy parts of the calculation. On top of that there is regularization. Even if the motivation for the traditional calculation is shaky, it does give the correct vacuum polarization contribution to the Lamb shift at least with a precision 1 / 1,000.

For high-momentum scattering, measurements at the LEP have an accuracy 10% or worse. Thus, we do not know if the traditional calculation is very accurate in the high-momentum case.

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

The measurement of the Lamb shift of muonic hydrogen (Randolf Pohl et al., 2010) gives a proton charge radius which is 4% less than the radius measured from electron scattering. This suggests that there might be something wrong with traditional vacuum polarization calculations when the momentum exchange is of an intermediate size. The measured Lamb shift in muonic hydrogen is 206.3 meV, and the discrepancy, which gives the strange proton radius, is 0.3 meV.

The 4-momentum in the virtual pair loop comes from the colliding particles: the Coulomb force is weaker, after all?

We have been claiming that the vacuum polarization diagram can only make the Coulomb force stronger - not weaker.

                          q + k

              q ~~~~~ O ~~~~~

                            -k

We argued that the photon cannot change its phase by colliding with "nothing", that is, with a virtual pair which pops up from empty space. We called such a phase change a Baron Munchausen trick - pulling oneself from a swamp by one's own ponytail.

              q ~~~~~~~~~~~

If the phase of the photon q does not change, then the vacuum diagram adds to the probability amplitude of the plain diagram above. That is, there is more scattering, and the Coulomb force must be stronger.

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

                             |   virtual photon

                             |   spatial momentum q

                             |

                          /    \ 

                        /        \    massive boson

                      /  k        \  q + k   4-momentum

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

   proton

http://users.physik.fu-berlin.de/~kleinert/kleiner_reb6/psfiles/index.html

But on September 5, 2021 we wrote that Formula (12.448) in Hagen Kleinert's book looks like the diagram above. In the diagram, the proton sends a massive Klein-Gordon boson whose propagator is something like

       1 / (p² + m²),

where p is the 4-momentum of the particle.

The particle can be interpreted as a photon which has the rest mass m. In the diagram, q is spatial momentum and k is arbitrary 4-momentum.

Note that we in this blog use the east coast sign convention (- + + +) in the Minkowski metric. That is why m² has the plus sign.

For the diagram above, we no longer can claim outright that it has to make the Coulomb force stronger. It could be that it makes the force weaker.

What is different in the new diagram? The 4-momentum k no longer pops up from empty space. It comes from the proton. The new diagram makes a lot of sense: it is the proton which is generating polarization around itself. The proton should provide the resources for the process. It should not rely on something coming out from empty space.

Earlier we have suggested that the natural cutoff for |k| is |q|. Could the natural cutoff for |k| in the diagram be the proton mass 1 GeV? For the Lamb shift, that would be high enough to act as an "ultraviolet" cutoff.

If we assume that the new diagram makes the Coulomb force weaker, and put the cutoff at 1 GeV, we will get the exact same numerical results as Hagen Kleinert in Formula (12.543) for the vacuum polarization contribution to the Lamb shift.

The high-momentum case

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

                               | virtual photon

                               | spatial momemtum q

                               |

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

Let us then consider electron-positron scattering (Bhabha scattering) which was measured in the LEP at CERN.

In the case of the LEP, the energy of the collision, say, 80 GeV, is the natural cutoff. Is that high enough?

In our new model, a charge gets more polarization around it if the charged particle is heavier, or is a member in a tightly bound system, like the quarks inside the proton.

Question. Why does the hydrogen atom then appear neutral to the outside world? The proton is much heavier. The charge of the proton should be screened by larger polarization. This, of course, depends on the measuring apparatus, too. What is the mass of the system which is used in the measurement?

Could it be that the massive boson cannot reach far? No. Then the charge would appear to be smaller at a short distance.

In a collision, the effective measured charge will depend on both the mass-energy of the charge and the exchanged momentum q.

We need to check the results from the LEP and other tests of the running coupling constant. Do they measure the effective charge as a function of q only?

What do the propagators and integration volumes really mean in the vacuum polarization diagram in QED?

An update about numerical results: crude numerical calculations tell us that a dipole particle model with a constant dipole moment gives in the low-momentum case double the correct value (Uehling potential - measured in the Lamb shift), but in the high-momentum case it only gives 1/5 of the correct value (measured in the LEP in CERN).

In the model, the dipole consists of an electron and a positron placed roughly 10⁻¹⁴ m apart.

One would expect the dipole moment to be larger in the high-momentum case. We need to find ways to calculate the dipole moment.

Analysis of electron-proton scattering, assuming a dipole particle

Let us analyze from a semiclassical point of view. We want to make some general observations about the machinery of Feynman diagrams.

We again study scattering of an electron by a proton. This time we assume that it is the electron which creates the virtual pair and the extra polarization during the encounter.

     e- ------------------------------------->

              \ k          | q          / k

                 \          |           /     

                  ~~~~~~~~~         dipole particle

                            |                       (= virtual pair)

                            | q

                          ●  proton

Above q is the exchanged spatial momentum, and k is arbitrary 4-momentum whose euclidean norm is

       |k| < |q|.

Interactions k and q are all in a dipole potential which is of the form

       -1 / r²,

and whose Fourier decomposition (= propagator) is of the form

       1 / |k|.

The dipole particle is created by the first k and destroyed by the second k.

The 4-momentum k integration volume

When the electron arrives, it brings to the scene extra polarization besides what possible polarization the proton has created.

If we want to describe the extra polarization with a wave packet, that packet occupies a small volume of spacetime. The Fourier decomposition of the packet has to contain relatively high frequencies both in spatial directions and in the time direction.

Why would the upper limit for |k| be |q|? It cannot be larger because |q| tells us how "much" the electron can interact with a dipole particle which is that distance away.

An alternative explanation: the momentum exchange q tells us how "smoothly" the state of the motion of the electron changes in the process. If the electron tries to send out higher frequencies than the frequency associated with q, destructive interference wipes those high frequencies off almost completely.

                       q + k

         q  --------- O ----------

                        -k

In the ordinary Feynman vacuum polarization formula we assume mysterious 4-momentum k which circles the virtual pair loop. The momentum k seems to appear from nowhere and then disappear.

In Dirac's model, the electron in the loop is one of the negative energy electrons in the Dirac sea. The photon q excites the electron, which leaves a hole, and later the electron drops back to the hole, releasing the momentum q.

In Dirac's model the negative energy electron is like an independent existing particle which takes part in the process. We cannot prohibit the photon q from being temporarily absorbed by a negative energy electron with arbitrary 4-momentum k. The 4-momentum can be as large as we wish. This is the origin of the ultraviolet divergence.

In Dirac's model the process affects particles which we can see. Then it probably should also affect the negative energy electron somehow. This means that information flows from visible particles to the shadow world of negative energy electrons. Dirac's model becomes very complicated if the shadow world becomes rich in information.

In our model, there is no independent external particle which would take part in the process. That is why we can argue that high momenta k cannot appear. Our model is "causal": it does not assume a shadow world of negative energy electrons.

The propagator of the potential

The potential in the diagram is a dipole potential 1 / r².

Why do we take a Fourier decomposition of the potential? Because we know how to calculate the scattering effect of a sine wave potential on a particle.

The spectrum of the Fourier decomposition constitutes the propagator of the potential.

Question. Is it a coincidence that the propagator for the Coulomb potential is similar to the propagator of the massless Klein-Gordon equation? That coincidence allows us to call that propagator the "photon propagator".

The propagator of a dipole potential looks like the "square root" of the photon potential. What are the implications of that?

The propagator of a particle

The propagator of a particle is the Fourier decomposition of a Green's function for the homogeneous wave equation

       K ( ψ(t, x) ) = 0

which governs the particle.

Question. What equation governs the "particle" which mediates the dipole potential?

A Green's function is the impulse response of the homogeneous wave equation to a Dirac delta impulse. It is like hitting the equation with a sharp hammer.

We interpret that an interaction term of a field A with a field B disturbs (perturbs) the field B. The interaction at one spacetime point is like a small Dirac delta, which we write as the source of the wave equation of B:

       K ( ψ(t, x) ) = δ(t₀, x₀).

The basic idea of the Born approximation is that we can treat the solution of the above as a new wave.

We assume that K (ψ) is linear on ψ. Otherwise, we cannot sum solutions.

Question. Why we can in the calculation of the loop

       electron - k - dipole particle - k - electron

in the diagram do the Wick rotation and integrate assuming that the energy and the mass are imaginary?

We already claimed that the dipole particle is tunneling, and that is why it "moves" in imaginary time. But we need to find a more detailed motivation for this. Why can we set the energy in the whole loop imaginary?

In Hagen Kleinert's book, similar integrals are calculated Wick-rotated.

The propagator must be set to 1 for "almost" on-shell particles

In our diagram, if the electron has high energy, and q is much smaller, then the electron stays almost on-shell during the whole process.

We believe that the electron propagator should be set to 1 in such cases. We should prove this somehow, if the proof is not in literature.

The classical analogue of vacuum polarization is a polarizable gas whose molecules weigh nothing

An aside: we asked a few days ago what kind of a classical material corresponds to vacuum polarization.

If the material is a gas whose molecules have infinitesimal mass, then the molecules cannot exert torque on our electron or proton. Only a molecule exactly on the line between the electron and the proton, just in the middle, can have an effect. Polarization always makes the Coulomb force stronger.

Classical "vacuum" polarization

In the previous blog posts we have suggested that the classical analogue of vacuum polarization is superlinear electric polarization of a material.

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

Nonlinear optics studies phenomena in such materials. The Wikipedia article says that above the Schwinger limit

       1.3 * 10¹⁸ V / m

"the vacuum itself is expected to become nonlinear."

Let us calculate the electric field strength of a proton at the distance of the electron reduced Compton wavelength

       4 * 10⁻¹³ m.

It is

      k e / r² = 10¹⁶ V / m,

where k is the Coulomb constant and e is the electron charge. The Uehling potential shows that there is slight nonlinearity already at that distance scale. If r is 1/10 of the reduced Compton wavelength, then the Schwinger limit is reached.

What material is analogous to vacuum polarization?

Our material should be a gas, because we believe that virtual pairs can move around in the vacuum just like molecules or atoms in a gas. Though the movement may not be relevant because the collision of, e.g., the electron and the proton happens quickly. Let us try to visualize the polarization process.

                e-  ------->

              

                              +

                              -

                             ● proton

One effect is that superlinearity causes an extra electric field to appear between the proton and the electron. Superlinear polarization, in a sense, brings some of the charge of the electron closer to the proton, which results in larger attraction.

There is also an opposite effect. The electric field strength is ~ 1 / r². But the area of a 3D sphere is ~ r². Therefore, polarization which depends linearly on the field strength, hides some fixed amount of the proton charge when viewed by the electron at any distance. If polarization is superlinear, then the closer the electron gets to the proton, the more of the proton charge is hidden.

Which of the two effects wins? Does the Coulomb force appear stronger when we go closer?

It seems to depend on the formula of superlinearity. If there is 1% extra polarization after the first doubling of the electric field, and more than 2% after the second doubling, then the attractive force might win. That is, if superlinearity grows faster than

        ln(E),

where E is the electric field strength. In practice, polarization grows very steeply once we approach the ionization threshold.

Visible light inside glass

Let us consider visible light inside glass. The speed of light is slower there. Let the refractive index be, for instance, n = 1.1. If v = 0.9 c, then the familiar γ of special relativity is

       γ = 1 / sqrt(1 - v² / c²)

          = 2.3.

That is, the effective "rest mass" of a photon is roughly half of its total energy.

The photon probably satisfies the massive Klein-Gordon equation.

What happens if we raise photon energy so much that it can ionize the material?  What is the effective mass of the photon then? We could not find any empirical data. Apparently, the material becomes opaque when photon energy is high enough.

For low photon energies the refractive index of various materials is fairly constant.

Photon propagator in a polarizable material

Maybe the on-shell photon propagator in a polarizable material should be something like

       1 / ( p² - m² ),

where m is the effective photon "mass" at the "scale" of the momentum p? The scale of the momentum should be taken in the euclidean norm.

The Coulomb force in a linearly polarizable material is weaker than in the vacuum. The propagator for a pure momentum photon is

        1 / (p² n),

where n >= 1 is the refractive index.

The propagators are very different. It looks like we cannot define a single photon propagator in a polarizable medium.

Empirical values for the Coulomb force strength

Empirical measurements of the Uehling potential (through the Lamb) shift show that the Coulomb force is ~ 1 / 30,000 stronger at the distance of the electron reduced Compton wavelength

 

       4 * 10⁻¹³ m

from the proton.

In the collision of a 100 GeV electron and a positron, significant deflection requires that they come within 1 / 100,000 of the electron reduced Compton wavelength, that is,

       4 * 10⁻¹⁸ m.

At that distance, the Coulomb force appears roughly 7% stronger.

Vacuum polarization always strengthens the Coulomb force?

In our previous blog post we claimed that a virtual pair cannot absorb any angular momentum because it could not give it to the proton. That dictates that the momentum q transfers to the virtual pair have to be simultaneous.

The proton, the electron, and the virtual pair apparently have to be on the same line. Can the virtual pair exert a repulsive force on the electron?

                    •  e+ virtual positron

              e-  ●  real electron

                    •  e- virtual electron

                   ●  proton

There is something strange in the diagram. How can the virtual positron end up "behind" the real electron, even though the summed electric field there pulls it toward the real electron?

If the virtual pair is between the proton and the electron, it makes attraction stronger.

Let us assume that the electron alone produces some polarization on the line of the diagram. The polarization pulls it equally up and down. There is no net force.

When we add the proton, it makes polarization stronger below the electron, and weaker up from the electron. In both cases the new polarization causes a downward force on the electron. This suggests that vacuum polarization always strengthens the Coulomb force. It cannot make the force weaker.

What about a solid? We know that polarization of a solid makes the Coulomb force weaker.

                            • e+ virtual positron

               ● e-

                         • e- virtual electron

               ● proton

In the diagram above, the virtual pair denotes polarization which is caused by the proton. It is hard to make a diagram where the torque on the virtual pair is zero, unless all the particles are on the same line. In a solid the torque is no problem. The solid can absorb the torque.

Hypothesis. Vacuum polarization always makes the Coulomb force stronger, while polarization of a solid in most cases makes the force weaker.

Our hypothesis would explain why the Feynman diagram of vacuum polarization makes the Coulomb force stronger. In the Feynman diagram we have corrected the sign error in the effect of polarization - an error which has lingered in literature ever since the first paper by Dirac in 1934.