Quantum gravity: vertex correction

In QED, the classical vertex correction is due to the fact that the energy of the far field of a charge does not have time to take part in the scattering process. This reduces the effective mass of the charged particle.

This is related to bremsstrahlung, in which the far field, and some of its energy, is "freed" from the charge.

In gravity, bremsstrahlung has a 16-fold energy compared to the analogous electromagnetic system.

What kind of an effective mass reduction, if any, could be happen in the vertex correction for gravity?

Does a mass reduction matter at all, if the inertia of the particle is reduced as much?

What is the energy density of the gravity field?

         ^  E  

         |

  

                      -----------  +       capacitor 

                      -----------  -        plates

If we have a static electric field E, we can extract energy from it locally by using capacitor plates which have opposite charges.

                        • m

                 -------------    scaffolding

                |              |

                       ●

                       M

To extract energy from a static gravity field, using a small mass m,  we must use a scaffolding which encloses the mass M. Then we can lower a small mass m and extract energy. Energy cannot be extracted locally.

This is the old problem about where is the energy of a gravity field located. Gravitational waves certainly carry energy. But is there energy in a static gravity field and what is the energy density, if not zero?

Electromagnetic/gravitational waves: a quantitative model

           F    M/Q                   M/Q    -F

         ------> ●                        ● <------

                              a                      d

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

                     accelerate     decelerate

                     time t             time t'

1.   First we accelerate to the right a particle of a charge Q or a mass M, with a constant force F for a time t. The particle moves a distance a.

2.   Then we decelerate it with a constant force -F. It stops after a distance d, after some time t'.

3.  The work lost,

       W  =  F (a  -  d)

escaped in radiation. The momentum

       p  =  F (t  -  t')

escaped in radiation.

Since gravitational waves carry 16X the energy of analogous electromagnetic waves, the difference a - d has to be 16X for gravity.

                \   rubber plate

                  \

                  ● ----> F

                  /

                /

Let us analyze the "effective inertia" of the particle in the process. Let us first look at a "rubber plate" model of the field of the particle. In phase 1, the far field of the particle does not have time to react. The effective inertia of the particle may be reduced because of that.

But it is also increased because the rubber plate in the diagram pulls the particle to the left.

                                \   rubber plate

                                  \

                     -F <----- ●

                                  /

                                /

In phase 2, the bending of the rubber plate actually helps -F to pull the particle to the left. The effective inertia is reduced. This explains why d < a.

How do we obtain a 16X energy to the radiated energy?

A.   If the rubber plate has a zero mass and resists stretching very much, then it carries away a lot of energy.

B.   If the rubber plate has mass, and is very elastic, then it carries away a lot of energy.

In the case of gravity, A is a more beautiful option. Assigning mass-energy to a static gravity field does not look nice.

The vertex correction

           particle 1

           • --------____

                                ------- >

                          ● particle 2

The classical vertex correction is due to the effective inertia being smaller in a rapid acceleration of the particle. Its field does not have time to react, and the particle is able to approach closer to another particle pulling it. Then more momentum can be exchanged.

If the gravity field is more elastic, then we can expect the vertex correction to be larger? But here is a problem: how do we know that the gravity charge of particle 1 does not become smaller, too?

Let us analyze the gravitational wave produced by an oscillating mass. For example, the quadrupole wave produced by a binary black hole. The fact that a gravitational wave is produced means that

       a > d

in the example of the previous section. That is, the effective inertia is smaller during the deceleration phase than the acceleration phase.

***  WORK IN PROGRESS  ***

Intuitive vacuum polarization model in QED – no vacuum polarization in gravity?

Let us, finally, try to construct an intuitive vacuum polarization model in QED, such that the model is not just hand waving.

                                             |  q momentum

                                             v      gained by e-

                         v ≈ c                 

                    e-  • --->

                                      R = minimum distance

                     /  |   \

                   |     |     |      E strong field,

                     \  |   /          dense energy

     

                         ● proton+

When e- passes close to the proton, the total energy of the electric field E drops, but the field between the two particles becomes stronger, its energy grows, and this growing energy creates a repulsion between the particles. The repulsion, of course, cannot cancel the overall attraction, but the repulsion is significant.

Vacuum polarization can reduce the energy of the field between the particles, and make the attraction stronger.

Vacuum polarization adds a new degree of freedom to the system. The field between the particles can use this freedom to reduce the total energy of the system. It is a good guess that if the energy W can escape to the Dirac field, then the total energy of the system electric field & Dirac field is reduced by W.

When the electron passes the proton at a distance R, the spatial size of the extra strong field E close to the proton is ~ R, and the time is ~ R / c. It is a smooth disturbance pulse ("bump") of those dimensions. The wavelength of the momentum exchange q is ~ 861 R.

Semiclassical analysis, with point particles e- and proton+

Let us first analyze this semiclassically in the zone of the size R. We Fourier decompose the bump. The most important components have the wavelength ~R.

The bump is the sum of the field of the electron and the proton. When they are close, the energy density of that part of the electric field E is double the sum of the individual fields.

    mildly relativistic

    e- •  --------------------------------------

                          | 

                          | q virtual photon

                          |

                       /     \  k + q  virtual electron-

                       \     /  positron pair e- e+

                          |  

                          | q virtual photon

                          |

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

  proton+

  static

The transient electric field E of the bump disturbs the Dirac field, through the coupling. We can construct the disturbance with Green's functions of the Dirac field.

We can imagine that q above designates the bump in E. The bump q hits the Dirac field through the coupling. The hit will create Fourier components k according to the propagator of the Dirac field. Because the bump is smooth, destructive interference will mostly cancel out |k| > |q|.

If q creates a virtual pair, then q will be absorbed to the pair, and the energy of the field E will weaken accordingly.

If the pair is long-lived, then the virtual pair will survive for the whole transit of the electron past the proton. The pair will reduce the energy of the field E for a long time. The pair will not escape as a real pair, because the electron e- is only mildly relativistic and cannot donate 1.022 MeV. The pair eventually must annihilate and give q back to E.

A good guess is that a long-lived virtual pair increases the attraction of the electron and the proton as if an extra momentum exchange q would happen between the electron and the proton.

A long-lived pair is called "almost-bremsstrahlung".

For q = 0, no almost-bremsstrahlung will happen. In the standard vacuum polarization calculation, this is represented by the integral

       Π₂(0).

If q ≠ 0, then there will be almost-bremsstrahlung:

       Π₂(0)  -  Π₂(q²).

In the vertex correction we saw that bremsstrahlung appeared as a missing part of the electric form factor F₁(q²) integral. The integral F₁(0) describes the process when the electric field of the electron can keep up with the movement of the electron. Bremsstrahlung is the field which "broke free".

Another way to view q ≈ 0: then R is large, and the transit of the electron past the proton lasts for a long time. Any virtual pair created by the bump will have plenty of time to annihilate, and they will not be able to reduce the average energy of E much.

The role of the coupling constants in the pair loop. We can understand why there is the (small) coupling constant e² / (4 π) in the first vertex which creates the virtual pair from q. But, since the pair necessarily has to annihilate as the electron leaves the proton, why is there the second coupling constant e² / (4 π) at the second vertex? Why is the coupling constant not 1?

One of the reasons might be time symmetry. The process must look the same if we reverse time.

Above we assumed natural units. In them, the coupling constant is e² / (4 π) ≈ 1/137.

The bump in E is at least crudely modeled by a single hit with a blunt hammer. The bump of the increasing E hits the Dirac field, and this hit, at least crudely, is like a hit with a blunt hammer. This partially explains the curious feature of Feynman diagrams that they seem to model complex processes with a single hammer strike. Why not 100 strikes?

Alternatively, the smooth orbit of the electron e- past the proton is a "blunt" disturbance, which can be modeled with a single hit of a blunt hammer.

Quantum magnification hypothesis

Quantum magnification hypothesis. In QED, the Feynman diagram of waves "imitates" the classical process with a

       861 X = 2 π / α

"magnification".

Above we analyzed the classical electric field E between the point particles e- and proton+. If the scattering angle of the electron is large, then

       R  ~  re  =  2.8 * 10⁻¹⁵ m.

That is 1/861 of the electron Compton wavelength λe = 2.4 * 10⁻¹² m.

The Fourier decomposition of the bump E has wavelengths which are ~ 1/861 of the electron Compton wavelength. Quantum phenomena only can have a resolution of ~ the Compton wavelength. Therefore, the wave representation of the process in a Feynman diagram must have waves ~ 861 times longer than the classical description. Is this a problem? Does a calculation with a 861 X magnification produce the same results as the classical calculation?

We know that the simplest Feynman diagram calculates elastic scattering probabilities which agree with classical formulae with point charges. At least in that basic case, the magnification works.

Bremsstrahlung. Let us have a mildly relativistic electron. Let us calculate from the Larmor formula the total energy W it loses if it passes at a distance R from a proton.

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

There,

       a  =  1 / (4 π ε₀)  * e² / R²  * 1 / me.

The time that the power P radiates is

       t  =  2 R / c.

The total radiated energy is

       W  =  4/3  *  1 / (4 π ε₀)³  *  e⁶ / c⁴  * 1 / R³

                        * 1 / me²

             ≈ 2 * 10⁻⁵⁷  *  1 / R³

joules. The mass-energy of an electron is 511 keV ≈ 10⁻¹³ J.

If R = re = 2.8 * 10⁻¹⁵ m, then

       W  ≈  10⁻¹³ J.

We conclude that the cross section of a mildly relativistic electron losing most of its kinetic energy in bremsstrahlung, classically, is very crudely π re², or

       0.3 * 10⁻²⁸ m²  =  0.3 barn.

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

The presumably correct empirical cross section is found from a Wikipedia article. It is something like 

       0.0008 barn.

We see that quantum mechanics efficiently prevents the electron from radiating when R << λe.

Quantum mechanics suppresses the production of high-energy real photons, but not virtual photons? The simple elastic scattering of an electron from a protons obeys the classical cross sections. But bremsstrahlung is severely suppressed when the minimum distance (impact factor) R is much less than the electron Compton wavelength λe.

Quantum mechanics allowed the momentum exchange q to be classical. The photon in q is virtual, it is pure spatial momentum.

Classical limit of the bump E in vacuum polarization

In the electron-proton scattering, let us make the charges N-fold and the masses N²-fold, where N is a large number. The new particle is a "heavy electron". We do not change the ordinary electron in any way. The heavy electron is a new particle, like a muon.

The heavy electron will still track the same classical path. This time, the bulge in the electric field E is classical. We can analyze the process in a localized spatial volume. We do not need to resort to the "momentum space" of Feynman diagrams.

In this semiclassical treatment, obviously, high 4-momenta |k| are strongly suppressed by destructive interference. The time-dependent electric field E disturbs the Dirac field. We can construct the impact response with Green's functions of the Dirac field. Destructive interference cancels high 4-momenta. It is like hitting a tense rubber membrane with a blunt hammer.

Feynman diagrams do not prohibit macroscopic masses or charges. In this classical limit, we know that destructive interference removes ultraviolet divergences.

The classical limit also demonstrates why Feynman diagrams can do with a single hit or the "sharp hammer", or a single application of a Green's function. When the heavy electron comes close to the proton, the change in the field E can be crudely approximated with a single hit of a blunt hammer. Generally, processes which last a limited, short time, often can be modeled with a single hammer hit.

Moving from the classical limit back to the quantum realm

                                             |  q momentum

                                             v      gained by e-

                         v ≈ c                 

                    e-  • --->

                                         R = minimum distance

                     /  |   \

                   |     |     |         E strong field,

                     \  |   /            the "bump"

     

                         ● proton+

The crucial question is what happens if we switch from the heavy electron back to the normal electron. Does the system still behave in a classical way?

When the mildly relativistic electron passes the proton, the process lasts a time

       t  =  R / c.

Let us calculate from the energy-time uncertainty relation how much energy W the system is allowed to "borrow" for such a short time.

       W t  =  h / (4 π)

   =>

       W  =  1 / (4 π)  *  h c / R.

The energy required to create a photon of  wavelength R is

       W'  =  h f  =  h c / R.

We see that we can "almost" borrow enough energy to describe a bump of a size R with a real photon, for the duration of the transit. This suggests that vacuum polarization, which is due to virtual pairs, could be replicated in the classical form, even though we have far too little energy to create real photons which would build the bump. We can borrow enough energy?

Peskin and Schroeder (1995) formula (7.91) has the vacuum polarization correction in QED. If we scale me and q by a factor N², the integral on the right does not change. But scaling the charge of the electron by N adds a factor N² to α = e² / (2 π) (in natural units).

In bremsstrahlung, the Planck constant h has a dramatic effect for the electron, because the electron cannot emit a wave such that h f is larger the kinetic energy of the electron. Apparently, no such dramatic effect is present in vacuum polarization.

Superlinear vacuum polarization as "almost-bremsstrahlung"

We can imagine that a pointlike electron constantly hits a tense rubber sheet with a sharp hammer. The formed pit is the electric field of the electron. Since the electric field interacts with the Dirac field, these hits indirectly also hit the Dirac field, forming some kind of (static) vacuum polarization.

If the electron is free and not under an acceleration, the system is static. The electron is surrounded by an electric field and, presumably, a nonzero Dirac field.

                        v ≈ c

                  e- • ---->

                                               |   q momentum

                                               v       change

                      R

                      ●

                     proton+

When the electron flies past a proton, the hammer hits no longer form a static configuration. The electron gains a momentum q.

A new hammer hit fails to "absorb" everything from the previous hit. The failed part escapes as electromagnetic bremsstrahlung.

In the case of the Dirac field, the failed part is "almost-bremsstrahlung" of a virtual pair. This is the "superlinear polarization" we have been talking about. If the virtual pair does not possess 1.022 MeV, it cannot escape as a real pair – but it can linger for the duration of the electron transit.

The momentum gain q reflects the maximum loss W of the energy of the combined electric field E as the the electron flies by. These are roughly linear if we keep R constant:

       q  ~  W.

                        e-  • ---> 

                             |  q

                             |

                             O  virtual pair

                             |

                             |  q

                             ●

                             proton+

A formed virtual pair means that some electric field energy escaped to a new "degree of freedom". How much energy? Since the field energy fell by W as the electron came close to the proton, we can guess that the virtual pair drains the same energy W.

At the beginning if this blog post we noted that if an energy W can escape to a new degree of freedom, then the total energy of a system often falls by the same amount W. The falling total energy means that the attraction between the electron and the proton is stronger. The magnitude of the added attractive force comes from the relation q ~ W.

Almost-bremsstrahlung hypothesis. The increased attraction between the electron and the proton comes from the electric field energy escaping, for a while, to a virtual pair. This hypothesis explains why the correction makes the attraction stronger between charges of a different sign, and the correction is:

       Π₂(0)  -  Π₂(q²).

If the charges are of the same sign, the escaping energy makes the repulsion weaker.

Does the almost-bremsstrahlung interpretation solve all ultraviolet divergences in any vacuum polarization?

If in some quantum field theory,

       Π₂(0)  -  Π₂(q²)

ultraviolet diverges, then we have a problem. Classically, it would mean that an infinite amount of energy escapes, for a while, into a new degree of freedom, if the particle absorbs a momentum q. The energy in the almost-bremsstrahlung is infinite and the attractive force becomes infinite.

The same problem could appear as infinite bremsstrahlung if we just have a force field surrounding a particle. No vacuum polarization is required to get an ultraviolet divergence if we ignore the fact that the energy must come from somewhere.

These problems would arise from the fact that the energy of the force field of a point particle is, formally, infinite. The acceleration caused by q can then, potentially, extract an infinite energy.

An obvious solution to the problem is to claim that the force field of the particle cannot contain more mass-energy than the mass of the particle. Does that work?

Another possible solution is to appeal to destructive interference. But in the classic model, the field close to a point particle is a problem. Its Fourier decomposition contains very short wavelengths.

The true "gravity charge" is always positive in a Feynman diagram?

Classically, any disturbance in a field always carries a positive energy. Feynman diagrams allow arbitrary negative energies in a virtual particle, but is this "negative energy" just a formal tool, and does not represent genuine negative energy which would have a negative gravity charge?

A sharp hammer hit (Green's function) always puts positive energy into the disturbed field, even if a Fourier component would formally carry a negative energy.

A static field contains energy, even if the Fourier decomposition is time-independent, and formally has no energy.

In general relativity, people often require that the energy density in space is everywere ≥ zero. Otherwise, we could arrive at time paradoxes. This is called an energy condition.

In a Feynman diagram, the 4-momentum really tells us about a net transfer of 4-momentum during the entire process. The "gravity charge" of a line does not need to be the formal amount of energy specified in the line.

In QED, in the vacuum polarization loop, the electron may carry a positive energy W, and the positron a negative energy -W. But classically, polarization of a material always requires positive energy.

If the gravity charge is always positive, is vacuum polarization possible in gravity at all?

In QED, the electric field of colliding charges pumps energy into the Dirac field, creating electric polarization. What would be an analogous process in gravity? We would need energy to create a positive energy particle and pull it away from a negative energy particle? Opposite gravity charges, presumably, repulse each other. Why would we need energy to pull them apart?

Note that a collision of electric charges can produce a real electron and a real positron. Vacuum polarization can be seen as an "almost-production" of a pair. But we do not believe that the corresponding process is possible in gravity. If a particle has a negative energy and a negative gravity charge, the repulsion from masses could lift its energy close to zero, but not to a positive territory. And we not believe that a real particle can have a negative energy.

Hypothesis: no vacuum polarization in gravity. All particles, virtual or real, have a positive "true" energy and a positive gravity charge. There is no vacuum polarization in gravity.

If the hypothesis is true, it solves all problems with vacuum polarization in gravity. It also removes the awkward possibility that we could use quantum mechanics to create a zone of a negative gravity charge.

The hypothesis bans the existence of any virtual particles in empty space. In this blog, we have repeatedly claimed that there cannot exist virtual particles in empty space. Such particles would break unitarity of quantum mechanics, because they would introduce an unknown causal factor into quantum mechanical processes.

Always positive gravity charge and arbitrary loops in Feynman diagrams of gravity

Above we argued that the gravity charge always has to be positive for a virtual particle, even for a particle which formally has a negative energy in a Feynman diagram.

              ~~~~~~~~~~~~~~~~~~~  graviton

                            |         |

                            |         |  virtual gravitons

                            |         | 

              ~~~~~~~~~~~~~~~~~~~  graviton

       --> t

Loops with gravitons are known to ultraviolet diverge in quantum gravity. Let us analyze what our hypothesis would imply for them.

Is the diagram above sensible at all? If a graviton scatters from another graviton (denoted by the first vertical virtual graviton), how could it scatter a second time?

Conclusions

We found an intuitive model for QED vacuum polarization, and an explanation why the renormalized Feynman integral may calculate it correctly.

Vacuum polarization effects have been measured experimentally in high-energy electron-positron scattering experiments, like the LEP of CERN. The precision of the experiments probably is not very good. They do agree with QED predictions.

In low-energy experiments, vacuum polarization probably has a very small effect. We do not know if such experiments have tested vacuum polarization.

In nature, ultraviolet divergences are a concern only in connection with gravity. Our hypothesis of no vacuum polarization in gravity may solve the problems. But let us do more research. What is bremsstrahlung and the vertex correction like in gravity?

Gravity: a "limp" rubber membrane is a model – vacuum polarization

Now that we understand QED better, let us return to the problem of ultraviolet diverging loop integrals in quantum gravity.

What kind of a material has a Green's function which diverges very easily? The Fourier decomposition should have large terms for large |k|?

                  #

                  #========= sharp hammer

                  v

       _____     _____    "limp"

                 \/              rubber membrane

                

Let us have a "limp" rubber membrane whose elastic energy grows slower than quadratic. The force resisting stretching is sublinear.

Let us hit the membrane with a sharp hammer. That will produce a very deep pit. In the Fourier decomposition, there will be lots of waves with a large |k|.

Such a classical system, of course, does not produce infinite waves or an infinite energy. Hammers are always blunt. Destructive interference very efficiently wipes out high |k|.

What is the analogy for gravity? We should have an attraction between gravitons, or waves in the limp rubber membrane.

If the elastic energy is quadratic, or the force resisting stretching is linear, then waves in the membrane do not interact. The wave equation is linear.

In a limp rubber membrane, waves can help each other to get to a lower energy state by interference. The energy of constructive interference is less than in the linear case. There is an attractive force between wave packets.

Let us place weights on the membrane. Together they can lower themselves to a lower vertical position than in the linear case. This is like the steepening gravity potential in general relativity.

We found a pretty good model for gravity, and for other fields which are prone to ultraviolet divergences. Let us study how this classical model avoids divergences.

We do not need to place weights on the rubber membrane? The limpness is enough

       ___        ____       ____   membrane

              \•/           \•/

         weight      weight

   

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

                  Earth

Actually, it might be that we do not need weights at all in this model. If particles are waves stretching the membrane, and a wave can get to a lower energy state by entering an area where the membrane is stretched more, then we have an attractive force. This makes the model much simpler and more beautiful.

The speed of waves is slower in a stretched part of the membrane, because it is "limper". That makes waves to turn toward the stretched parts, just like in gravity.

Avoiding divergences with classical fields

Energy and momentum must be conserved. Or the classical fields suffer from singularities. It is about the existence of solutions.

A real-world system, like a rubber membrane, can approximate a classical field. However, there is a cut-off at the scale of atoms. Energy conservation is guaranteed for these approximate solutions.

Ultraviolet divergent Feynman integral behave really badly: they break energy conservation even if we impose a relatively modest cutoff. Thus, the divergence in Feynman integrals is a crude error, which does not happen in a classical system.

Suppose that we allow point charges in the classical model. The classical field energy of the electron is then infinite. Maybe this can lead to infinities in classical scattering experiments?

A positron falling into an electron would generate an infinite energy. The energy would be radiated away as the positron spirals into the electron. We have to ban phenomena like this. Then the classical approximation (with a cut-off) will behave well and conserve energy. There are no ultraviolet divergences.

Graviton Feynman diagrams

      graviton   ~~~~~~~~~~~~~~

                                      |  graviton

      graviton   ~~~~~~~~~~~~~~

Does a Feynman diagram like the above make sense?

A graviton turns toward another because the speed of waves is slower there. Is it natural to model this as a force? Yes. The force in electromagnetism can be understood to come from the field energy. A force tries to reduce the field energy.

The electron and the positron behave like point charges in scattering experiments. It is likely that gravitons behave like point particles, too. But the wave description of a graviton or an electron is a wave, not a point particle.

A wave packet, which is far away from other wave packets, does behave like a particle.

The classical field model has to be augmented with point particles? Feynman diagrams do not contain particles, but plane waves.

Let us try to do with the wave model. In many cases, we can simulate a point particle with a wave packet.

The gravity attraction is steeper than the Coulomb force. The Green's function of the graviton must contain more high |k| than the photon Green's function. This agrees with our limp rubber membrane model.

              ~~~~~~~~~~~~~~~~~~~  graviton

                       |        |        |

                       |        |~~~|     gravitons

                       |        |        |

              ~~~~~~~~~~~~~~~~~~~  graviton

       --> t

According to literature, a Feynman integral with two graviton loops ultraviolet diverges. Above we have three loops. Let us try to interpret what the diagram is supposed to calculate.

It is a collision of two gravitons. The diagram tries to calculate a correction to the simplest tree diagram. Any 4-momentum can circulate in any loop of the diagram. The coupling constant at the vertices contains the mass-energy of one of the three gravitons, but which one?

The coupling in gravity

A brief Internet search says that graviton-graviton coupling is an "active field of research".

What is the problem?

In this blog we tentatively proved in May 2024 that general relativity does not have solutions for any "dynamic" system. But let us forget that for a while.

If we have two gravitational wave packets flying at a large distance from each other, then their coupling, presumably, is the newtonian gravity interaction.

In QED, we believe that photons can collide and produce electron-positron pairs. That is, electrons scatter like point particles.

If gravitons are like photons, then they behave like point particles. If we let two extremely large energy gravitons meet, they will in rare cases scatter to a large angle. Does general relativity predict this for classical wave packets? A packet has a much larger extension than a point particle.

https://physics.stackexchange.com/questions/721658/what-do-we-mean-when-we-say-gravitational-waves-are-non-linear-and-do-not-superp

Let f be a small perturbation of the flat Minkowski metric η. A perturbation to the linear wave equation might depend on the square of f, or the square of its (second) derivative.

Let us shoot two gravitons head-on, so that they will meet at a perpendicular area A. How does f change if we make A four times larger? The perturbation will be a half, and its square 1/4.

This fits the fact that the fuzzy gravitational wave packets attract each other. But does the perturbation reproduce the behavior of pointlike gravitons which attract each other?

General relativity already includes all the quantum effects? Probably not

The perturbation in the preceding section probably sends parts of waves to every direction. This fits the pointlike particle behavior. Does general relativity already include the point particle model?

Using Huygens's principle may resolve this? The Feynman path integral is a form of Huygens's principle. 

The "hammer" in Huygens's principle is sharp. It is like a point particle.

  

      gravitational wave packet

              ~~~  --->

                    <---  ~~~

                            gravitational wave packet

Let us first look at two gravitational wave packets passing each other at a large distance. The packets attract each other. If general relativity calculates this wave phenomenon correctly, the paths of the wave packets will bend according to laws of gravity.

If we would be looking at QED, and the wave packets would describe an electron and a positron, then we would need the electromagnetic field to model the attraction. But for gravity, the attraction must be handled by the field itself.

Electromagnetism adds nonlinearity to the Dirac field. Gravity itself is nonlinear.

                 --->          <---

                 ~~~          ~~~

      very strong gravitational

      wave packets

Let us shoot very strong gravitational wave packets head-on at each other. Both contain a huge number of gravitons. If general relativity is correct, it will calculate the scattering of the waves. Could quantum mechanics calculate a different result in this case?

If we, in the particle model, imagine the wave packets as swarms of N gravitons, the scattered flux to a large angle grows as

       ~ N².

The formula might be the same for the scattering of gravitational waves. We have to analyze it with Huygens's principle.

Electrodynamics describes the behavior of electromagnetic waves very well. The quantum description of the waves is identical to the classical field.

Absorbtion and emission of energy in units of

       E  =  h f

is a separate rule which we glue on top of the classical field.

In the past two months we saw that the vertex correction and vacuum polarization can be interpreted as classical effects.

A weak gravitational wave may contain just a single graviton. Its behavior would be like an electromagnetic wave of a single photon.

Nonlinearity of general relativity may come from vacuum polarization, or replace vacuum polarization. In QED, vacuum polarization makes electromagnetic waves nonlinear.

A brief Internet search reveals that the scattering of gravitational waves from each other is not known well in general relativity, unfortunately.

See below the section about dividing a gravitational wave into gravitons. It shows that general relativity probably does not include all quantum effects, e.g., Compton-like scattering.

What is the gravity field of matter in a quantum superposition?

This is a classical question. We can now present a solution. A weak gravitational wave packet is a single graviton in a quantum superposition. The energy in the graviton can be absorbed at different locations, with some probability density.

What is the gravity field of a gravitational wave packet like? Far away, it is like the gravity field of any matter located at the position of the wave. Close or inside the wave, the "gravity field" is the wave itself. The wave can "collapse". The square of the amplitude of the wave tells us the probability of it collapsing to a certain location, and releasing its energy.

The question is analogous to asking: what is the electromagnetic field of a single photon like? 

https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat

What is the gravitational field of Schrödinger's cat? The cat can stand, or lie flat on the floor of the box.

This is analogous to the question: what is the electric field of an electron like if the electron is in a superposition state? Far away, the electric field of the electron is essentially classical. But near the electron, the system is a quantum system, and its electric field is subject to how we measure it. There is no single classical field. The field is in a superposition, too.

In the case of the cat, the gravity field is macroscopic. The environment contains masses which "measure" the field continuously. We can assume that the field "collapses" immediately. The cat no longer is in a superposition state.

Certain authors have speculated that gravity mysteriously makes quantum mechanical wave functions to "collapse". Our view is that it does not do that. The electric field of the electron does not make the wave function of the electron to collapse.

But if the gravity field of an object is macroscopic, then we can treat the wave function as collapsed – just like we do for any macroscopic system.

Schrödinger's cat really is in a superposition state, but we can treat the system as if the wave function would have collapsed already.

Can we divide an energetic gravity wave into gravitons which scatter from each other and from matter? Probably yes

If we have a swarm of electrons colliding with a swarm of positrons, then the scattering reveals the "granularity" of the electron and positron fluxes. A small number will scatter to large angles. Electric charge allows us to count exactly the number of particles in each swarm.

But the number of photons in an energetic wave is not a clear concept. A wave packet contains various frequencies. We probably can "mine" the energy content of a wave and absorb various collections of photons from it. The collection of photons depends on the absorbing device.

If two electromagnetic wave packets collide, how do we determine their photon content?

If the waves are coherent, then we maybe can safely assume that they contain photons whose energy is h f?

The same questions concern gravitons. If we want to calculate the mutual scattering of two gravitational waves, how do we determine their graviton content? From the Fourier decomposition?

Suppose that we have a photon of a huge energy. Then we might treat it as a macroscopic particle. In the Wilson cloud chamber we, actually, see the tracks of almost macroscopic particles.

How do such macroscopic photons scatter from each other? They are very localized wave packets.

https://www.epj-conferences.org/articles/epjconf/pdf/2024/11/epjconf_snp2024_03003.pdf

Christian Schubert (2024) gives the low-energy cross section as:

and the high-energy cross section:

We see that high-energy photons should behave like very small particles. These formulae have not been verified experimentally.

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

But Compton scattering has been verified. We know that gamma photons collide with electrons like small billiard balls. If gravity behaves in a similar way, then general relativity is unlikely to describe this billiard ball effect.

Hypothesis. High-energy gravitons behave like high-energy photons. General relativity does not cover their billiard-ball-like behavior.

If a graviton has a mass of 10²⁶ kg, then its Schwarzschild radius is 15 cm, and we could define that its cross section for a "large deflection" of another graviton (> 0.1 radians) is a square meter.

A more realistic high-energy graviton has the mass of an electron, 10⁻³⁰ kg. The cross section is 10⁻¹¹² m².

If we have 1,000 kg of such gravitons, the cross section is

      10⁻⁷⁹ m².

That is, negligible!

There still remains a possibility that two gravitons colliding head-on would scatter significantly. If they colliden head-on, we cannot describe their behavior as wave packets which are separate and pass each other without overlapping.

Suppose that nonlinearity in gravity only shows up in very low potentials, close to a black hole, and nonlinearity is

       ~ |f|²,

where f is the metric perturbation. Then the only place where there could be significant scattering of gravitational waves from other gravitational waves is the place where strong waves are born: in binary black hole mergers. There the problem is to solve the nonlinear equation for the orbiting masses – it is not about scattering.

Scattering of classical gravitational waves is a problem of numerical relativity in black hole mergers. Numerical relativity, presumably, can calculate approximate solutions when large metric changes happen.

LIGO observations show that there is no huge scattering of gravitational waves when they travel 500 million light-years from a black hole merger. Very large scattering would garble the signal observed by LIGO.

If any scattering of gravitons is negligible, then Feynman diagrams describing scattering are a theoretical problem. We still have to explain why divergences do not break physics.

Head-on scattering of gravitons

Suppose that the wave packets describing to gravitons overlap. Then we cannot treat them as classical particles which pass each other.

This is the case where the Feynman diagrams above could become relevant.

We have to look at this.

Making the rubber membrane so limp that loops diverge badly – vacuum polarization in gravity

The limp rubber membrane is a model for an interaction for which the propagator allows a very large amount of high 4-momenta |k| – a "limp" interaction.

                                 k virtual boson

                          ~~~~~~

                        /                 \

                -------------------------------     particle 1

                                  |

                                  | q  virtual boson

                                  |

                -------------------------------     particle 2

In the diagram above, the virtual bosons represent the very limp interaction. In the virtual boson k loop, there might be a really bad ultraviolet divergence which cannot be renormalized. What would that mean?

Classically, particle 1 makes a very deep pit into the limp rubber. It meets the field of particle 2 and turns a little, absorbing the momentum q. There is no problem in this. No infinite energy is created.

     •  ------------------------------------  particle 1

                          | 

                       /     \  k   virtual electron-positron

                       \     /       pair

                          |  

                          | q virtual graviton

                          |

     ● -----------------------------------  heavy particle 2

Let us then study vacuum polarization in gravity. We assume that a very limp matter field exists, such that it can easily produce virtual particle pairs with large mass-energies

        E, -E.

Classically, we may have a medium, in which polarization is very easy, and grows at a fast pace for stronger fields. For gravity, this would mean that any mass M would spontaneously attract positive masses popping up in empty space, and repel negative masses. A black hole would form, with a zone of negative mass density surrounding it. That sounds nonsensical. At least, we know that it does not happen in nature.

Anyway, something like a diverging Feynman integral can happen in a classical model.

If the matter field is the Dirac field, then the loop would have a propagator factor

        1 / |k|²,

and the couplings would add a factor |k|². The loop ultraviolet diverges extremely badly.

Can we somehow appeal to destructive interference which would establish a cut-off?

Let us assume that both particles are so heavy that we can treat them as classical, macroscopic particles. In a Feynman diagram, the momentum exchange q graviton is a time-independent wave of a form

       exp(i q • r).

We may assume that this wave fills, e.g., a cubic meter, into which particle 1 enters. Let us assume that particle 2 is static.

In the Feynman model, the wave of particle 1 scatters anywhere in the cubic meter, from the wave q. The scattering is not limited to the vicinity of particle 2. The scattering can happen at any time.

But in the classical model, particle 1 can only scatter with a momentum exchange q, in a narrow region close to particle 2. Furthermore, the momentum exchange q only spans a certain time period Δt. The classical model is dynamic, while the Feynman model is static.

The Feynman diagram depicts a process which can only happen for a short time when particle 1 is close.

We may interpret it as a pair forming "spontaneously", absorbing q from particle 2 and passing q to particle 1.

Is it possible that a pair with a very large |k| takes part in the process?

Let us claim that the approach of particle 1 causally produces the process seen in the Feynman diagram. Unitarity requires that the process is causal.

                               o  spider rotates string

                             //\\

            ------------------------------  tense string

Summing Green's functions for the Dirac field. Let us start from the idea of the "spider" on October 12, 2025, hammering the Dirac field. The spider makes the string to rotate into different directions on the left and the right. It accomplishes this by hitting with two special hammers.

The gravity field of particle 2 disturbs the Dirac field. The disturbance can (maybe) be calculated by summing the Green's functions of the Dirac field for each spatial point of the gravity field.

We can imagine a large number of little spiders hammering the Dirac field. In this model it is obvious that large 4-momenta |k| are canceled by destructive interference. The spider hits with two "sharp hammers" simultaneously, but otherwise this is analogous to one sharp hammer hitting a tense rubber membrane. A blunt hammer consists of a large number of sharp hammers. Its hit has destructive interference wiping off high |k|.

Only very close to particle 2 does the gravity field have a large spatial derivative, and we can expect large |k| to be significant there.

Earlier we calculated that the Fourier decomposition of

       1 / (x² + 1)

has large frequencies attenuated exponentially. A similar formula probably holds for the 1 / r² gravity field if r ≠ 0.

Exponential attenuation ensures that every Feynman loop integral ultraviolet converges rapidly. Note that an iteration of the process can still lead to a divergence. If one loop severely distorts the Dirac field, then the gravity field is updated a lot, and this update may generate even more distortion of the Dirac field. This would correspond to runaway classical polarization.

The field of particle 1 adds a time-dependent second gravity field to the system. This should not introduce large |k|.

Why does the Feynman diagram calculate an approximation of this (if we use an ultraviolet cut-off)? This is one of the great questions about Feynman diagrams. Why do they work at all?

The time-independent wave q describes the crude form of the gravity field at some radius R from particle 2. It is like one Fourier component of the 1 / r² gravity field.

Note that the wave q does not require that particle 1 passes at the distance R from particle 2. The wave q simulates the effect for particle 1 wherever it passes particle 2. It is in the "momentum space". We no longer need to care about the position of particle 1. Particle 1 will gain the momentum q at a small probability, wherever 1 passes. We could say that the wave q is a "kaleidoscope image" of the 1 / r² field.

The vacuum polarization loop is the hit with two hammers. It is the "typical" Green's function which disturbs the Dirac field at the distance R.

Only those vacuum polarization pairs which transfer momentum to particle 1 contribute to scattering of 1.

        wave q crudely describes this:

                 -    -    -             vacuum polarized

                 +   +   +            mass-energy

                      

                       • ---> v       particle 1

                      R

                      ●                 particle 2

                      +                vacuum polarized

                      -                 mass-energy

Why does the Feynman integral calculate something which approximates the impact of vacuum polarization?

The Green's function approach is like the sharp hammer constantly hitting the Dirac field, based on the wave q.

Vacuum polarization decreases the field energy of the combined field of particles 1 and 2. In the case of gravity, it makes gravity stronger.

Let us assume that the Compton wavelength of particle 1 is shorter than R, so that we can treat 1 as a somewhat classical particle.

                         v ≈ c

                         • --->            particle 1

                        R

                        ●                  particle 2

The disturbance which particle 1 causes to the Dirac field close to particle 2 is very crudely R wide and lasts a time R / c. Does this "blunt hammer hit" explain the Feynman integral?

A very crude calculation in QED with an electron passing a proton shows that the "wavelength" of q is

        ≈  861 R.

The gravity between an electron and a proton is a

        5 * 10⁻⁴⁰

times weaker force. Thus in gravity, the "wavelength" of q is

       ≈  2 * 10⁴² R.

On September 19, 2025 we introduced the "Quantum imitation principle". Let us add some magnification factors to it:

Quantum magnification hypothesis. In QED, the Feynman diagram of waves "imitates" the classical process with a

       861 X  =  2 π / α

"magnification". In the case of gravity, the magnification is 2 * 10⁴² X. 

In classical QED mildly relativistic electron scattering from the proton, the minimum distance has to be the classical radius

       ~  re  =  2.8 * 10⁻¹⁵ m

to obtain a momentum change q of the Compton wavelength

      ~  λe  =  2.4 * 10⁻¹² m.

The ratio re / λe is α / (2 π) ≈ 1/861. The "resolution" of quantum mechanics is bad. It has to imitate classical scattering with waves which are very long. In the case of gravity, the waves are hugely long.

Conclusions

We are working toward an intuitive model of vacuum polarization in QED and gravity. In this blog post, we introduced many ideas, like the limp rubber model, and the Quantum magnification hypothesis.

In our next blog post we will try to present the first intuitive model of vacuum polarization.

                                        |  q momentum

                                        v      gained by e-

                         v ≈ c                 

                  e-  • --->

                                     R = distance e- proton+

                     /  |  \

                   |    |     |      E strong field,

                     \  |  /          dense energy

     

                        ● proton+

The following idea might explain it. When an electron passes a proton, some of the transient and dense energy in the field E between e- and proton+ will repulse the two particles. But if that electric field E energy can escape for a long time to the Dirac field (i.e., a virtual pair), then the energy in E plus the Dirac field will repulse less. That, is the attraction is stronger.

"Long time" means that it is "almost bremsstrahlung", which is absorbed at as the electron e- recedes from the proton+. If |q| is small, then there is very little of this almost-bremsstrahlung.

The Feynman integral difference

       Π₂(0)  -  Π₂(q²)

calculates this almost-bremsstrahlung.

Since our Magnification hypothesis ties the Feynman diagram to the classical electric field E between the electron e- and the proton+, we can study the process also localized in space, semiclassically.

Localized in space, we can appeal to the fact that the Green's functions in the Dirac field created by E, have almost all their high |k| destroyed by destructive interference. This would solve all ultraviolet divergence problems in quantum field theory.

Vertex correction and vacuum polarization in QED: a more thorough analysis

Our September 24 and 29, and October 12, 2025 posts contained many ideas, and the analysis was left superficial.

Let us continue the study of an electron e- scattering from a massive charge X+.

Overlapping probabilities in Feynman diagrams: we cannot assume no overlap

Consider the tree level elastic scattering diagram, and the tree level bremsstrahlung diagram.

It is clear that these cannot describe non-overlapping probabilities.

Elastic scattering, actually, never happens in the real world. The classical limit shows thst an infinite number of low-energy photons is always emitted, at least when |q| is small enough, so that we can describe the electron as a wave packet, passing X+ at a considerable distance.

Rule: Feynman diagrams do not necessarily describe non-overlapping probabilities. The infrared divergence in the bremsstrahlung diagram means that the electron emits an infinite number of photons. The probabilities for photons of various 4-momenta p overlap. Also, the probabilities in the tree level elastic scattering diagram overlap with those of the bremsstrahlung diagram.

In each individual case, we have to analyze which probabilities are disjoint, and which overlap.

The vertex function F1(q²): it is useless and should be ignored? No, there exists a classical vertex function

                                 k

                            ~~~~~

                p       /                \

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

                                | q

          X+ ---------------------------------

We can imagine that the electron hits the electromagnetic field with a sharp hammer when it arrives close to X+. The Green's function creates virtual and real photons of various 4-momenta k.

If q would be 0, then the electron would absorb everything which it sent in the Green's function. But q disturbs this. A part of the wave from the hammer hit escapes as real photons, bremsstrahlung. That part is seen in the Feynman integral as the (negative) infrared divergence. Something is "missing" from the integral when q ≠ 0.

            ∫ d⁴k f(q²)    -    ∫ d⁴k f(0).

       near k₀               near k₀

For 4-momenta close to some k₀, we define the "missing part" as the difference of the integral for q ≠ 0 from q = 0.

Large real photons cannot escape since the electron does not have enough kinetic energy to create them (though, classically, they would be able to escape). But the integral does have a "missing part" for them. The crucial question is how we should interpret the missing part?

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

                                | q

          X+ ---------------------------------

Plane wave analysis of elastic scattering. Above we have the tree level diagram for elastic scattering. Let us analyze the tree diagram and the loopy diagram from the plane wave point of view.

1.   We imagine that the plane wave describing the electron enters a cubic meter m³ where there is a time-independent electromagnetic wave q created by X+. An electron wave which is scattered by q absorbs the spatial momentum q.

2.   A certain flux of the electron plane wave "absorbs" the momentum q and is scattered, according to the tree level diagram.

3.   The loopy diagram means that a certain flux φ of the electron plane wave is scattered by a virtual photon k (which the electron itself sent).

4.   That flux φ can be scattered by the q wave. Later, the flux is again scattered by k, this time absorbing back k. A part of the flux φ absorbed the momentum q.

5.   The probability of the original electron plane wave scattering from q is the same as of the flux φ scattering from q.

6.    It does not matter for the electron wave if it was the original planar wave, or the flux φ. The probability of absorbing q was the same.

7.   We conclude that the probabilities described by the tree level diagram and the loopy diagram overlap completely. The loopy diagram does not contribute anything to the scattering probability.

8.   Classically, reducing the electron mass makes the scattering probability larger, but that involves at least two momentum exchanges between the electron and X+. A Feynman diagram with just a simple q line should not be aware of this.

Empirical evidence. So far, we have not found data about scattering experients which would be accurate enough to reveal the numerical value of the vertex loop correction. The CERN LEP experiment probed vacuum polarization.

The electric vertex function F1(q²) is extremely small for small q². The paper at the link

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

claims that for q² << me², the electric form factor is

In the hydrogen atom, the kinetic energy of the electron is

       Ekin  =  p² / (2 me), 

and we can assume that q ≈ p. Then

       q² / me²  ≈  2 Ekin / me 

                       ≈ 20 eV / 511 keV

                       ≈ 4 * 10⁻⁵,

and

       α / (3 π)  *  q² / me²  ≈  3 * 10⁻⁸.

The scattering probability changes very little from the (claimed) electric form factor. It is unlikely that such tiny changes can be measured.

The classical limit: is it nonsensical for the vertex function F₁(q²)? The fine structure constant is defined

       α  =  1 / (4 π ε₀)  *  e² / (ħ c)

            ≈ 1/137.

However, in natural units, the fine structure constant is simply e².

Let us increase the charge of the electron by some large factor N, and its mass by a factor N², so that it becomes a macroscopic particle. Then we can track its path in a classical fashion.

We assume that the electron passes X+ at some fixed distance R. In the vertex correction,

       α q² / m²

stays constant, since α grows by a factor N², q by N, and m by N².

The formula for the form factor F₁(q²) would claim that the apparent charge of a macroscopic particle would significantly (about 0.1% * q² / me²) depend on the momentum q it absorbs from a large, macroscopic charge X+. Is this nonsensical? Classically, the far electric field of the electron does not have time to react as the electron passes X+. The electron will have somewhat reduced mass, which causes it to go closer to X+ and receive more momentum. But is the effect so large that it could be 0.1% * q² / m²?

If we increase the mass of the electron, then its Compton wavelength will become smaller than its classical radius. Maybe there is a law of nature which prohibits this?

If we take the classical limit by decreasing h, then the vertex correction claims that the apparent charge of an electron varies very much depending on the momentum q it absorbs. Actually, h is set 1 in the formulae. We cannot change it.

The assumption of a fixed distance R when we grow the mass and charge may not be realistic. If the distance grows with the mass, then the correction does go to zero. This could be called a classical limit.

                           k

                       ~~~~~

                    /                \

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

                           |      \

                           |        ~~~~~~   real photon

                           | q

      X+   -----------------------------------

Convergence when the loop radiates bremsstrahlung. On September 29, 2025 we remarked that the loop integral probably does not have an ultraviolet divergence if the electron inside the loop radiates a real photon. That is because the product gains one more electron propagator. Our analysis above suggests that the probability of this diagram overlaps with the tree level diagram, and we should not add the probability to the scattering of the electron. That is, we should ignore this diagram if we just look at the electron scattering.

If we are interested in bremsstrahlung, then we must analyze if the diagram calculates correctly the effect of the electron mass reduction.

The classical vertex correction

Let us calculate an order of magnitude estimate for the electron in the hydrogen atom. The frequency of the orbit is

       6.6 * 10¹⁵ Hz.

The far electric field which does not have time to take part in the scattering (= orbit) is

       1.5 * 10⁻¹⁶ s  *  c

       = 4.5 * 10⁻⁸ m

       = r

away. The ratio

       Δ  =  re / r  ≈  5 * 10⁻⁸

tells us how much the mass of the electron is reduced.

       

                                 ^

                           ●  /      proton

              e- • --------

Let us denote by R = 1 the Bohr radius. As the electron passes past the proton, it receives an impulse which accelerates it the distance ~ 1 up in the diagram.

If the mass of the electron is reduced by some small fraction Δ, then the electron passes slightly closer to the proton, say,

       Δ / 4.

The cross section of the scattering grows because of this, by a factor Δ / 2:

       Δ / 2  ~  2.5 * 10⁻⁸.

The order of magnitude is the same as in the QED vertex correction.

Let us study how the classical vertex correction depends on q. If we make R = 2, then q is halved. The mass reduction Δ is halved because the time to go past the proton is double.

The upward force in the diagram is 1/4 but the time to go past the proton is double. The acceleration upward still moves the electron the distance 1 upward.

The effect on the scattering is 1/4, because Δ is halved and R is doubled. This agrees with the QED vertex correction.

Hypothesis. The "missing part" of the vertex correction integral, which cannot escape as bremsstrahlung, is "detached" from the electron during the scattering, and reduces the effective mass of the electron. This, in the classical way, increases the scattering amplitude of the electron.

Question. How can this classical effect in the Feynman integral depend on the Planck constant h?

If literature always sets h = 1 in the calculations, then the formulae above contain a hidden factor h. Then the value does not depend on h, after all.

The QED vacuum polarization for small |q|has a roughly similar magnitude as the vertex correction. Why?

Hypothesis 2. The QED vertex correction is the classical effect. We have been suspecting this in our blog for many years.

If Hypothesis 2 is true, then the renormalization in the vertex correction is what is needed to make the calculation correct and classical. It is not ad hoc, but is mandatory.

Vacuum polarization

    |

    |                e-      ___

    |        q            /        \  q

    |    ~~~~~~              ~~~ ● X+ massive charge

    |                e+  \____/

    |

    |                  virtual pair

    |

    e-

   ^  t

   |

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

Polarization P reduces the energy of the electric field, and thus makes the Coulomb force weaker between charges. Another way to measure polarization is the electric displacement D.

https://physics.stackexchange.com/questions/724130/how-to-make-sense-of-the-coulombs-law

The relative permittivity εr ≥ 1. The Coulomb force is weaker in a medium because the electric field energy is reduced by polarization. Polarization happens because it takes the system to a lower energy state. Thus, it is trivial that polarization reduces field energy. By linear polarization we mean that εr is constant regardless of the electric field.

We define superlinear polarization as the case in which

       εr(E)

the relative permittivity grows when the electric field |E| grows. Superlinear polarization further reduces the energy in the electric field. This makes the Coulomb force between opposite charges stronger. That is because we can further reduce the field energy by taking the charges closer.

Between charges of the same sign, superlinear polarization reduces the Coulomb force because it reduces the field energy when we take the charges closer to each other.

Superlinear polarization differs from the traditional interpretation of QED. In the traditional thinking, taking charges of the same sign closer to each other would increase the Coulomb repulsion because they would "see" the bare charge of each other.

Which is right: vacuum polarization increases the repulsive force or decreases it? If the electron and the positron were very massive, there would be no vacuum polarization. If we make them light, we increase the "freedom" of the system. Increasing the freedom should take the system to a lower energy state, which decreases Coulomb repulsion. It is very surprising if the traditional QED interpretation is right.

The classical limit. Peskin and Schroeder (1995) give:

Recall that in the metric signature (+ - - -), q² < 0. Let us then grow e by a large factor N and m by a factor N². For small |q|,

       Π₂(q²)  ~  e² q² / m²,

which is

                    ~ e⁴ / m²

if the electron passes at some fixed distance R from X+. The value of Π₂ does not change. The correction will stay reasonably large.

For small momenta |q|, the vacuum polarization correction is equivalent to the Coulomb potential correction term above. The term is called the Uehling potential, and it makes the potential pit deeper.

For large momenta |q|, the coupling constant grows by the formula above. There, A = exp(5/3).

If |q²| << m², then the integral for Π₂(q²) - Π₂(0) looks much like the vertex correction, and probably does not contain h as a factor. That is, vacuum polarization might be a "classical" effect for small |q|. But what classical effect is it?

If the electric field tries to hit the Dirac field in order to create a pair and reduce the energy of the electric field, this does not need to depend on the Planck constant h. The energy to create the pair is 2 me c², which does not contain the Planck constant.

The Planck constant is involved in the energy and wavelength of real particles. A transient hit to a field does not create real particles, and it might be that we do not need to bother about the value of the Planck constant.

In our favorite model, the rubber membrane and the sharp hammer model, the hit produces various transient waves. If some of them would escape, then in the quantum description, we would need to worry about the fact that the energy is h f. But transient waves may have a lot of freedom to be whatever they like. That would explain the absence of h.

In this blog we have remarked that momentum transfers are not quantized. It may be that most transient phenomena are not quantized.

Hypothesis 3. Vacuum polarization for small |q| is a phenomenon of the "classical" Dirac field and the electromagnetic field.

Conclusions

We once again stressed that Feynman diagrams may calculate overlapping classical probabilities. The infrared divergence of bremsstrahlung is a prime case: the electron always sends an infinite number of real photons.

We observed that the classical vertex correction, which is due to the far field of the electron not following instantaneously the electron, may be the correct electric form factor in QED. The mass of the electron is reduced because the far field does not follow it.

In QED we see various formulae for the electric form factor F₁(q²), but they always depend on the "photon mass", which is used to cut off the infrared divergence. Thus, we do not know what researchers suggest that F₁(q²) should be. Anyway, the classical vertex correction is the best bet, and satisfies the classical limit.

In vacuum polarization, we have strong evidence for our claim that it makes the force between charges of a different sign stronger, but the force between charges of the same sign becomes weaker. Thus, it is not about high-energy electrons "seeing" the bare charge behind a "cloud" of polarization. The intuitive picture in literature is wrong.

Vacuum polarization is analogous to a medium where the polarization is superlinear in the electric field E.

The Feynman integral formula for vacuum polarization for small |q| does not depend on the Planck constant h. In this sense, vacuum polarizations is "classical". If we treat the Dirac field as a classical field, we will probably obtain the same (or almost the same) vacuum polarization formula.

It may be so that anything which we calculate in QED, which does not depend on h, is a "classical" phenomenon.

Now that we understand QED better, we will in the next blog post study ultraviolet divergences in gravity, and in other problematic field equations.