Gravity: a "limp" rubber membrane is a model

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?

The perturbation 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.

***  WORK IN PROGRESS  ***

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.

Vacuum polarization in QED

UPDATE October 20, 2025: When two electrons approach each other, superlinear polarization reduces the energy of the electric field? Then the repulsion is weaker. We have to check if someone has measured the running of the coupling constant for Möller scattering. 

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

Wikipedia only mentions the LEP experiment, where the running was observed in Bhabha scattering e- e+ → e- e+. Also, Wikipedia states that vacuum polarization is not relevant in low-energy processes.

----

In our post on August 27, 2021 we claimed that the ultraviolet divergence of the Feynman vacuum polarization integral is canceled out by destructive interference.

Furthermore, we claimed that in the traditional analysis of vacuum polarization, in the integral there are two sign errors which cancel each other out. The Dirac sea is empty. This also solves the problem of the infinite energy density of the vacuum: the energy of the vacuum is zero, not infinite.

Let us analyze this in more detail. Our previous blog posts have taught us something about the ultraviolet divergence in the vertex correction. There, the diagrams with an ultraviolet divergence can be discarded altogether because they have a zero chance of happening. The loop will always send a real photon, which makes the loop integral to converge.

A semiclassical model of vacuum polarization

Suppose that an electron and a positron pass by each other at a very high speed. As the electric field strength grows, it "almost produces" a new electron positron pair. It is a "virtual pair" which electrically polarizes the vacuum.

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

The pair makes the vacuum between the electron and the positron to conduct electric lines better: it better "permits" electric lines of force:

                      e-

                      • -->

 

                      ° e+    virtual pair pulls on the

                      ° e-     electron and positron

      

                <-- •

                     e+

In the case an electron meeting an electron, the configuration is like this:

                        ° e-   virtual pair pulls on

                        ° e+  the upper electron

                    e- • -->

                  <--  • e-

   

                        ° e+   virtual pair pulls on

                        ° e-    the lower electron

In the above diagram, the field strength is the largest at the locations where the virtual pairs form.

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

If we have a medium where the electric polarization is superlinear in the electric field strength, then charges will behave just like above. We say that the electric susceptibility

        χ(E)

is superlinear in the field strength E.

We know that very high energy electrons and positrons will produce real pairs when they meet. It is natural to assume that the electric susceptibility is superlinear in E.

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

Classically, the extra polarization close to the meeting charges will always produce an electromagnetic wave. The virtual pair is a transient electric dipole which radiates away an electromagnetic wave.

The Feynman diagram above cannot happen. The virtual pair loop always emits a real photon.

The Feynman integral for the virtual pair loop has a logarithmic ultraviolet divergence at large 4-momenta, after using the Ward identity. Adding an emission of a real photon might make the integral to converge. But does that yield a result which matches the traditional renormalization technique? We have to check that.

In our August 27, 2021 post we suggested that destructive interference cancels out virtual pair loops with high 4-momenta. Does the emission of a real photon accomplish the same thing? The real photon makes the diagram asymmetric, which may complicate calculations greatly.

Toy model of superlinear polarization: flat plates of opposite charge

     E = 0        E = 2         E = 0

                       -->

                 |              |

                 |              |

                 |              |

                 |              |

                 |              |

                 +              -

We assume that the polarization of the air between the plates is linear in E when the electric field strength is

       |E|  ≤  1,

and the polarization is much stronger when

       |E|  >  1.

The plates, when alone, have the electric field |E| close to the plates. We bring the plates parallel, close to each other.

Then the field between the plates is E = 2, and there is a lot of extra polarization there. The energy of the electric field between the plates is reduced between the plates. We conclude that when we bring the negative plate close to the positive plate, the superlinear polarization increases the attractive force felt by the negative plate. The field has less energy => we were able to harvest more energy when we lowered the negative plate close to the positive plate.

Question. Is the electric attraction "asymptotically free"? If the polarization at a very strong electric field entirely erases the energy of the electric field, then opposite charges will not feel any force. This might happen when the distance of an electron and a positive charge e+ is the classical radius re = 2.8 * 10⁻¹⁵ m.

Since the Compton wavelength of the electron is much longer, 2.4 * 10⁻¹² m, the electron can only enjoy its asymptotic freedom for very short times. It is not a permanent.

The Uehling potential around an electron is steeper than the Coulomb potential. If we have a symmetric spherical shell of positive charge around an electron, and lower it close to the electron, then the entire energy of the electron electric field is exhausted earlier than for a Coulomb potential. The force felt by the shell is stronger than Coulomb, but the force should become zero earlier.

If the electron has a large kinetic energy, then LEP collider experiments show that the electric field does possess a force down to distances of 10⁻¹⁸ m or less. The "asymptotic freedom" would only hold for low-energy electrons.

A semiclassical model for an electron-positron pair

The rubber membrane model was able to clarify bremsstrahlung and the vertex correction. We hit the membrane twice, but the second hit is a little bit displaced. What escapes from the first Green's function is the bremsstrahlung. That part is not absorbed by the second hit. The missing part causes the infrared divergence of the elastic diagram.

            #

            #======   EM field hits Dirac field

            v

      __       ___        membrane

          \__/

    e+ °     ° e-        Dirac wave

                              = created virtual pair

The electromagnetic "hit" to create the pair and the second "hit" to annihilate the pair may be somewhat similar? If there is a lot of energy available, then pair can become real and escape. That is like bremsstrahlung, this time consisting of pairs.

To create a real pair, we need at least 1.022 MeV. We get a strict upper bound on the wavelength of the escaping real "created pair bremsstrahlung". The Dirac equation does not have a solution where the pair possesses less energy than 1.022 MeV. Classically, no Dirac wave can escape if energy is missing.

                   spring            disturbance

            | /\/\/\/\/\/\/\/\/\      ----___----___----

     wall                                     <--- 

Our membrane model is not suitable for this. A better model is a spring whose resonant frequencies are high. A disturbance can squeeze the spring (virtual pair), but it can only make the spring to oscillate if the frequency of the disturbance is high enough (real pair).

The ultraviolet divergence in the vacuum polarization loop comes from the fact that we allow one of the particles to have an arbitrarily large negative energy (the absolute value is large). What is the semiclassical interpretation for this? Formally, the positron in the Dirac equation does possess a negative energy.

We observed in August 2021 that the pair, taken as a combination, is a boson: its components are fermions.

Also, we observed that a running coupling constant can break conservation of energy. If the force depends on the speed at which the particles meet, then the force may be non-conservative. This is not problem in Feynman diagrams, though, because conservation of energy is always enforced.

Negative energy particles are waves which rotate to the "opposite" direction, in a classical analogue

The basic formula for a particle wave is

       exp( i (-E t  +  p • r) )

in quantum mechanics. Let us, for a moment, forget that E designates energy. Then a "negative energy" wave is something which rotates to the opposite direction. There is nothing mystical about negative energy, if we think this way.

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

                               o  spider rotates string

                             //\\

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

In this blog we have earlier written about a "spider" which stands on a tense string and makes the right side to rotate clockwise and the left side to rotate counterclockwise. The tense string is like children's jump rope.

What is a Green's function like? Does the spider suddenly hit the string both on the left and on the right and produce sharp rotating waves? The waves rotate to opposite directions.

This model would explain why the Green's function in a Feynman diagram always must allow also negative energy particles. The complete set of waves cannot be built from just clockwise rotation.

In the case of pair production, is it so that the negative energy waves are positrons?

Real photons have a positive energy if they are circularly polarized whichever way. The rotation in negative/positive energy must happen in an abstract (complex plane) space. It is not about polarization of light.

The Green's function which creates the pair, and the ultraviolet divergence

    |

    |                e-      ___

    |        q            /        \  q

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

    |                e+  \____/

    |

    |                  virtual pair

    |

    e-

   ^  t

   |

Let us try to analyze the origin of the ultraviolet divergence in (virtual) pair production. The electron passes by a very massive charged particle X.

Let the photon q be pure spatial momentum, no energy. The Green's function hits the Dirac field with a double hammer, creating both clockwise and counterclockwise waves. The hit is very sharp: the Fourier decomposition contains waves with huge positive and negative energies E. The large energies come from the sharpness of the impulse, just like when a sharp hammer hits a rubber membrane.

The Feynman integral of the vacuum polarization diagram is infinite. But let us forget the infinity. If q = 0, then we can imagine that the second hammer strike absorbs everything from the first hammer strike.

If we let q differ from zero more, then the electric field pulls on the virtual electron and the positron, and disturbs the pair. The second hammer strike is not able to absorb "everything" from the first strike.

The escaped part is kind of "bremsstrahlung". It makes the absolute value of the integral smaller. The "bremsstrahlung" increases the cross section of the scattered electrons. The electric force appears stronger. This is the "running" of the coupling constant.

Real "bremsstrahlung" would be a real pair.

Peskin and Schroeder textbook (1995) about vacuum polarization

Peskin and Schroeder (An Introduction to Quantum Field Theory, 1995) calculate the effective coupling constant αeff for a momentum exchange q. The effective value of α is larger for large |q²|.

Dimensional regularization is used: we integrate in 4 - ε dimensions, where ε is a small positive real number. This makes the integral finite.

We then let ε approach 0. That makes the value of Π₂(0) "minus infinite". Peskin and Schroeder renormalize the infinite integral by declaring Π₂(0) to be the reference point, to which the integral with other values of q have to be compared:

The metric signature is (+ - - -), which means that q² < 0. The logarithm above has a negative value, and

       Π₂(q²)  -  Π₂(0)  >  0.

That is, for large |q|, Π₂(q²) is "less minus infinite" than Π₂(0).

In this blog we have claimed that if |q| ≈ 0, then the virtual pair should have very little effect on the propagation of the virtual photon q. But the Feynman integral claims that the effect is "infinite". If |q| is larger, then the integral claims that the effect is "somewhat less infinite". This sounds illogical. A large |q| means a strong electric field. Then, intuitively, the virtual pair should have a stronger effect on the virtual photon q. This is the sign error which we claim to happen in Feynman diagrams in this case. The renormalization makes the final value sensible, though.

We have claimed that for q = 0, destructive interference cancels out all virtual pairs. Then the integral is zero. If |q| is larger, then destructive interference does not cancel out all virtual pairs: they make the electric susceptibility of the vacuum larger, and make the interaction stronger.

Can we somehow interpret that the vacuum polarization diagram makes the interaction stronger? Yes

If the renormalization calculates the interaction correctly, then, for some reason, the disturbance caused by q on the virtual pair makes the interaction stronger, so that the cross section of a q momentum exchange is larger. Can we find an intuitive explanation why this should be so?

      

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

                           | q

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

We assume that X+ is very heavy and has a positive charge. The tree level diagram makes most of the cross section.

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

                           |  q

                          O       vacuum polarization

                           |  q

          X+ ------------------------

The vacuum polarization loop maybe is an independent phenomenon from the tree level diagram? The loop denotes polarization between e- and X+. The polarization pulls both on e- and X+. Then the loop diagram would come on top of the tree level diagram, and increase the cross section. Classically, the loop would be an object which is polarized in the field of e- and X+.

We can interpret the vacuum polarization diagram this way: the field of X+ disturbs a virtual electron wave and the virtual electron wave absorbs the momentum q from the field of X+. That is, X+ pulled the virtual electron. The virtual positron pulls the real electron and absorbs the momentum -q from the real electron. Then the virtual pair annihilates. Since the pair was able to get rid of the extra momentum (q - q = 0) this process respects conservation laws and is allowed.

This all sounds very logical. But why does the Feynman method with renormalization calculate this right? We can imagine that if q = 0, then the Feynman integral calculates the "life" of a virtual pair in empty space. Having q ≠ 0 disturbs this peaceful life.

New kind of "bremsstrahlung": the virtual pair is a boson with a mass

Hypothesis of why a disturbed virtual pair contributes to the interaction.

1.   When the life of a virtual pair is peaceful, destructive interference, actually, cancels out the pair completely. It never existed and did not have any effect.

2.   If q ≠ 0, then not all virtual pairs are completely absorbed in the second "hammer strike". Non-absorbed pairs did not suffer a complete destructive interference. They came into existence.

3.   A virtual pair cannot escape as bremsstrahlung because it does not have enough energy.

4.   The pair must eventually annihilate and disappear. The pair did contribute to the interaction, transferring momentum between X+ and the real electron e-.

The hypothesis above explains why the "missing part" of the integral tells us the extra interaction between X+ and the real e-. In Feynman rules, this is implemented by flipping the sign of the integral, because of the fermion loop, and declaring the integral value for q = 0 the "benchmark", to which the integral with q ≠ 0 has to be compared.

The extra interaction between X+ and the real e- is kind of "bremsstrahlung", which has a real-world effect on the process.

Since the pair, if made real, has a mass 1.022 MeV, there is no infrared divergence in this new kind of bremsstrahlung. Recall that the infrared divergence of bremsstrahlung was removed by giving a small mass to the photon.

Note that the virtual pair is a boson. The virtual pair is analogous to a boson with a mass.

How does the new concept of bremsstrahlung affect our view of the anomalous magnetic moment?

Supersymmetry

We realized that the virtual pair can be seen as a massive photon. Does this have anything to do with supersymmetry?

A fundamental idea of supersymmetry is that the fermionic field and the bosonic field form one whole, similar to time and place being one whole in special relativity. If we have a polarizable material, then a photon can be defined as polarization in the material. Does this have anything to do with supersymmetry?

In this blog we have presented a hypothesis that a photon really is polarization of the vacuum. We will look at supersymmetry later.

Conclusions

We argued that the renormalization, or the counterterm, in the vacuum polarization integral really removes what "does not exist". The scale of things in the vacuum polarization process is determined by the momentum exchange q. There is a dynamic process as the electron passes close to the charge X+, and |q| determines the "scale", or the resolution, of the process. Momenta much larger than |q| are canceled out by destructive interference.

The Feynman vacuum polarization integral calculates how much of the Green's function is canceled out by destructive interference. The remaining part,

        Π₂(0)  -  Π₂(q²)

is the one which is not canceled, and pulls the charge X+ and the electron e- toward each other with the momentum exchange q.

In the correct interpretation, we do not need any "renormalization", meaning an obscure cancelation of an infinity. Rather, we simply remove the nonexistent part of the integral.

This blog post ends our analysis of QED divergences. We showed that they are not problematic:

1.   The infrared divergence of bremsstrahlung is the correct result: it describes the fact that an infinite number of low-energy real photons are always emitted.

2.   We can discard the vertex correction elastic scattering Feynman diagram which contains the other infrared divergence: the history can never happen because photons are always emitted.

3.   If we add a real photon emission in the vertex correction loop, then the Feynman integral converges.

4.   The "renormalization" in the vacuum polarization diagram just removes the part of the integral which is canceled by destructive interference. We, of course, should not integrate over something which does not happen at all. We do the logical, right thing, and there is no need to devise complicated explanations about why we can remove the infinity.

We will next look at theories which are not renormalizable. Our goal is to understand what problems exist in quantum theories of gravity.