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.

Not renormalizable theories in QFT: there is a confusion about overlapping probabilities

Wikipedia defines nonrenormalizable theories in this way:

"Not all theories lend themselves to renormalization in the manner described above, with a finite supply of counterterms and all quantities becoming cutoff-independent at the end of the calculation. If the Lagrangian contains combinations of field operators of high enough dimension in energy units, the counterterms required to cancel all divergences proliferate to infinite number"

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

Suppose that we have a process into which particles with a total energy E enter. Is it really a problem that we need an "infinite number" of counterterms?

    |                       k + q

    |                e-      ___

    |        q            /        \  q

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

    |                e+  \____/                       charge

    |                          -k

    |

    |                  virtual pair

    |

    e-

   ^  t

   |

Above we have the vacuum polarization diagram. The momentum q is the input which "disturbs" the processes in a pure vacuum polarization loop with no incoming or outgoing particles.

In the previous blog post we claimed that destructive interference cancels out the entire vacuum polarization loop integral, unless q ≠ 0 disturbs it. The momentum line q is the "input" to the process.

Hypothesis. Destructive interference cancels out all loop integrals if the "input" to them from external lines goes to zero.

Does the hypothesis solve the problem of an infinite number of counterterms?

A theory where the 4-momentum k interacts: divergences really are due to the input producing many Fourier components in another field?

In gravity, the mass-energy E interacts with other particles.

A Feynman diagram loop contains arbitrary values of the 4-momentum k. Can that cause a problem if we have many loops which are connected to each other?

Let us study the diagram below.

   particle    

     • ---------------------------------------

                              |  interaction

                              |

                              |

                              |  

                             __ q + k + j

                    ___/       \___  q + k

                  /       \___/       \

               /       q + k - j        \

     q ~~                                  ~~ q

                \_______________/

                                          q - k

We assume that interactions in the diagram can depend on the 4-momentum. There are nested loops with arbitrary 4-momenta k and j.

Destructive interference almost entirely cancels any 4-momentum

       |k| > |q|,

where | | denotes the euclidean norm of the 4-momentum:

       |(E, px, py, pz)|  =  sqrt(E² + px² + py² + pz²).

If q = 0, the cancellation is perfect. If q ≠ 0, then, in very rare cases, |k| can be huge. The nested, smaller loop will in those cases have a very large |j|. Very large |j| will have a huge interaction with the particle.

The integral over j can have a very large value. Is that a problem? It should not be. If the probability of a large |k| is infinitesimal, we do not need to care much of a huge integral. In QFT there seems to be a confusion about overlapping probabilities. As if a huge integral value in some extremely rare event with a probability

       P  <  ε  <<  1

would somehow make P large, or even infinite.

Hypothesis. The huge value of the integral means that the process produces many Fourier components of a wave at the same time. The probabilities overlap.

The confusion is the same as in bremmstrahlung in electron scattering. The large value of the integral describes a complex wave which contains many Fourier components. If |k| is allowed to be large, why would it produce just one Fourier component? More likely is that it will produce a large number of Fourier components, just as happens in bremsstrahlung.

The produced wave can still be seen as "one wave", but it just happens to have many Fourier components. In classical bremsstrahlung, the complex wave certainly contains many Fourier components.

Black holes. If j has a lot of energy, then the particle may interact with a black hole. But that is extremely improbable.

What if the input q is of the Planck scale? Then we will have black holes in the diagram, and it may be that Feynman diagrams do not work at all. However, particles with a Planck scale energy are rare, or nonexistent in nature.

It looks like nonrenormalizable theories work fine. We may have a cascade of loops where the energy rises to the Planck scale, but those have an infinitesimal probability of occurring, and we can ignore them.

Hypothesis. "Nonrenormalizability" of gravity is not a problem at all.

There are other problems in the quantum field theory of gravity, though. The interaction is nonlinear and complicated.

Assaf Shomer (2007): entropy of a black hole

https://arxiv.org/abs/0709.3555

Assaf Shomer (2007) argues that the entropy of a black hole is too large for it to be describable with a renormalizable quantum field theory. He claims that a renormalizable QFT is asymptotically a conformal field theory, and the entropy in such a system must be smaller than that of a black hole.

The entropy of a black hole is roughly the same as the entropy of the wavelength

       λ  =  16 π rs

radiation which we can use to feed and grow a black hole. There rs is the Schwarzschild radius. The reverse process would be the hypothetical Hawking radiation.

Let us compare the entropy of a black hole of an energy E to the entropy of a (classical) photon gas stored into a vessel of the same size, and having the same energy E. A photon gas, presumably, is governed by a conformal field theory.

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

Wikipedia says that the energy density of a black body radiation photon gas is

       dE / dV  ~  T⁴,

where T is the absolute temperature. The entropy density is

       dS / dV  ~  T³.

Note that there is an error in Wikipedia in the table for the entropy: the table claims that one can replace the volume V with the temperature T, and derive a strange formula S = 16 σ / (3 c) T⁴, which does not depend on V.

Let the Schwarzschild radius rs vary. The energy of a black hole is

       Ebh  =  C rs,

where C is a constant of nature. The entropy, according to Bekenstein and Hawking, is

       Sbh  ~  rs².

Let us then put photon gas worth Ebh = C rs to a vessel whose volume is

       V  ~  rs³.

The total energy inside the vessel is then

       ~ T⁴ rs³  ~  C rs,

which implies that the temperature

       T  ~  1 / sqrt(rs).

The entropy inside the vessel is

       S  ~  rs³  *  T³

           ~ rs^3/2.

We see that as rs grows, the entropy of a black hole grows quadratically, while the entropy of a photon gas vessel of the same energy only grows by an exponent 1.5.

Assaf Shomer has a calculation error in the paper. He claims:

Since d = 4, that would mean that S ~ rs^¾ Shomer confused the density of energy and entropy to the total energy and entropy in the vessel.

Anyway, Shomer's main argument still seems to stand: the entropy of a black hole grows faster than the entropy of a equivalent photon gas vessel, if we increase the total stored energy.

In this blog we have claimed that the ingoing matter "freezes" at the horizon of a black hole. The calculation above assumed that the vessel containing the photon gas has no freezing effect: time flows at the same rate everywhere in the vessel.

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

Assaf Shomer's argument suggests that a black hole cannot globally, in static spatial coordinates around a black hole, behave asymptotically like a conformal field theory.

But we are interested in local phenomena in freely falling coordinates, where particle energies are much less than the Planck energy. Thus, Shomer's argument does not prevent us from having a fruitful quantum field theory of gravity.

Destructive interference cancels large frequencies exponentially well

When an electron e- passes by a massive charge X+, the time variation of the electric field in comoving coordinates of the electron, or in the comoving coordinates of X+ is something like

       ΔE(t)  =  1 /  (1  +  (v t)²).

Let us assume that v = 1.

https://math.stackexchange.com/questions/1047977/find-the-fourier-transform-of-frac11t2

The "disturbance" ΔE(t) then has a Fourier transform

For large frequencies, or 4-momenta, the Fourier component is absolutely negligible. If f = 100, the component is ~ 10⁻⁶³⁰.

However, is this cancellation even too strong? Let us compare this to the renormalized vacuum polarization value. Is the contribution of |k| > 100 |q| absolutely negligible?

https://www.scribd.com/document/259722615/Quantum-Field-Theory-by-Hagen-Kleinert#page=892

Hagen Kleinert (2013) calculates the effect of q ≠ 0 by using the q² derivative of Π₂(q²):

The arbitrary 4-momentum is in his nomenclature p, not k as in our blog text. The contribution of large |q| seems to be a decreasing geometric series. It is not exponentially decreasing.

This is not paradoxical. Even if the "disturbance" is very much free of high frequencies, its "impact" may be less free.

Is it possible that for some loop, the impact of a disturbance diverges, too? What would an ultraviolet divergence mean in such a case? If there are overlapping probabilities, can it mean an infinite energy for the generated wave?

Quantum gravity

https://link.springer.com/article/10.12942/lrr-2002-5

Zvi Bern (2002) writes about divergences in quantum gravity. In Section 2.2 he states that gravity with matter typically diverges badly at one loop, and pure gravity at two loops.

Let us check if we can find a way to alleviate the problem.

There is another problem, too. The Feynman integral for just five loops of gravitons contains 10³⁰ terms! We have to find a simpler way to calculate the interaction. In our blog we hold the view that gravity is a combination of an attractive force and an increase in the inertia of a particle in a gravitational field. Could it be that calculations with inertia would be simpler than calculations with the metric tensor?

Ultraviolet divergence in a loop

In an infrared divergence, on September 24, 2025 we were able to explain the problem away by claiming that the produced wave contains an infinite number of low-energy bremsstrahlung photons. Does the same principle work for an ultraviolet divergence?

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

If we hit a rubber membrane with an infinitely sharp hammer, then it, presumably, creates the Coulomb field of a point charge, which has an infinite energy. That is why the hammer is never allowed to be infinitely sharp: destructive interference must cancel the infinite energy. The hammer must be blunt.

   --> t

Above we have a Feynman diagram. Let the lines represent gravitons and the vertices their gravity interaction. Two gravitons enter from the left, create new virtual gravitons, interact, and exit on the right.

The Feynman integral calculates the "4-point function", or the probability amplitude for the process to happen, assuming that the input flux from the left has a certain value.

The input gravitons coming from the left have some modest probability amplitudes. If the Feynman integral is infinite, that would mean that the output flux of the gravitons on the right is infinite. That is, we have created an infinite energy. This is clearly nonsensical. What is the problem?

The process has a classical limit. The input gravitons could be wave packets, and the output gravitons are wave packets. General relativity is supposed to conserve energy. We conclude that the ultraviolet divergent Feynman integral miscalculates the process, and badly.

Let us imagine that the graviton q interacts with the electron and the positron gravitationally. In that case, the interaction is proportional to 

       |k|.

Maybe the Feynman integral diverges, even after renormalization? What does that mean, physically?

The contribution to the probability amplitude of the process might be such that each interval

         n  ≤  |k|  <  n + 1

contributes an equal amount. The sum is infinite. Then the classical probabilities must overlap. That would mean that the process at the same time launches many pairs with various momenta k. We have suggested that the pair, as a whole, is a boson. Many bosons with various k are launched at the same time. The right side of the diagram then would describe a simultaneous absorption of all these bosons. In this interpretation, the various k are not separate classical probabilities. They overlap. Then there is no divergence in the classical probability.

An analogy: a sharp hammer hit to a rubber membrane produces many Fourier components with a large |k|. The next hit absorbs them all.

Suppose that q produces a wave packet of a pair which carries the momentum q. The Fourier decomposition of the wave packet contains many different momenta k. Why would the sum of the classical probabilities for each k be less than 1, or even finite?

Destructive interference should almost completely cancel high 4-momenta. For classical waves, it probably establishes a cutoff.

Why does a Feynman integral diverge? It may be due to the following:

1.   the integral does not take into account destructive interference, or

2.   the integral does not understand that classical probabilities for various 4-momenta k in a loop overlap.

Classically, the infrared divergence in bremsstrahlung is easy to understand. But having a large amount of very high 4-momenta k would be strange. When an electron passes a charge X+ inside a material, the polarization of the material behaves varies smoothly. The Fourier decomposition of the polarization will not contain high frequencies.

Classically, a hit with an infinitely sharp hammer will contain very high frequencies. The energy of the hit probably is infinite. If the Green's function is not completely absorbed, then it may "leak" lots of very high frequencies, and we might get a classical divergence. An infinite amount of energy is created.

We already discussed one type of a classical ultraviolet divergence. If an electron receives an instantaneous impulse, and its acceleration is infinite, it will radiate an infinite energy.

Why do Feynman diagrams operate with a single hit of the hammer? The method does work in some cases, but in the general setting, it is prone to create an infinite energy – fail miserably.

Question. Is destructive interference the correct method to regularize and renormalize all Feynman integrals? It is suspicious that integrals of the type

        ∫ d⁴k  * 1 / kⁿ

only converge slowly, or not at all. Exponential convergence might be more realistic?

In quantum mechanics, one natural "convergence mechanism" is the uncertainty relation which says that a particle can "borrow" an energy E at most for a time

       t  <  ħ / (2 E).

The "convergence" in it is quite slow.

Conclusions

Let us close this long post. We will continue the study of ultraviolet divergences in an upcoming post.

We have to analyze more thoroughly what does an ultraviolet divergent Feynman integral really mean. It can remain divergent also after the "renormalization", if the input (like q ≠ 0 in vacuum polarization) "disturbs" the integral value enough.

Classically, an ultraviolet divergent approximation formula is a very bad approximation, since it generates an infinite energy from a finite input. Feynman integrals may in some cases calculate the processes wrong, regardless of regularization and renormalization procedures.

In quantum gravity, ultraviolet divergences seem to happen inevitably at two loops. Furthermore Einstein formulae are too complicated, so that a simple Feynman integral may contain 10³⁰ terms. We have to find a simpler method to calculate interaction processes.