Why does the Feynman propagator for a photon model correctly Coulomb's law?

https://physics.stackexchange.com/questions/44418/are-the-maxwells-equations-enough-to-derive-the-law-of-coulomb

The Feynman propagator for a photon is derived from the Klein-Gordon equation.

The Klein-Gordon equation is analogous to the wave equation (for the electric field E) which one can derive from Maxwell's equations.

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

One obtains the Feynman propagator from the following question: 

What is the "response" of a wave equation to a Dirac delta like source impulse at a point in space at a point in time?

https://en.wikipedia.org/wiki/Green%27s_function

The impulse response is called the Green's function.

https://physics.stackexchange.com/questions/279723/how-to-obtain-the-explicit-form-of-greens-function-of-the-klein-gordon-equation

We know that Feynman diagrams correctly model the Coulomb scattering of electrons and positrons. The scattering in classically governed by the Coulomb force. Why do Feynman diagrams work? They are derived from a wave equation, not from the Coulomb force equation.

The logic behind the renormalization group of a quantum field theory

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

We are currently studying renormalization groups, in order to understand why gravity is non-renormalizable.

https://arxiv.org/abs/0709.3555

Assaf Shomer has written a 10 page explanation of the non-renormalizability of gravity.

Let us calculate a Feynman path integral, using some large number Λ as a cutoff for momenta.

In QED, the integral over a (vacuum polarization) loop diverges badly, but by setting a cutoff we can calculate results which are empirically correct! Why is that? What is going on?

Shomer requires that the partition function (= the generator for all correlation functions) stays the same regardless of the cutoff. The correlation functions tell us the physical behavior of the system. Why would a relatively arbitrary cutoff Λ give the correct behavior and not another slightly different cutoff Λ'?

Maybe the right model is to adjust the values of coupling constants for various cutoff sizes, in a way that the integral which yields the partition function has the same value regardless of the cutoff size?

Shomer derives in his paper the equation (13), which determines the RG flow, that is, the dependency we must set on the coupling constants on the cutoff Λ, in order to have the partition function integral the same regardless of the cutoff.

How does this compare to the intuitive idea of scaling self-similar systems, as outlined in the Wikipedia article?

In what way does a higher Λ mean analyzing the system in more precise detail? Like we would analyze the block spin example of Leo P. Kadanoff in the Wikipedia article?

The analogy between a higher cutoff and more detail is not clear. We have to think more about this.

Bare charge and a dressed electron

The Wikipedia article contains the familiar claim that virtual electron-positron pairs around an electron can "screen" some of the  charge of the electron.

As if a "dressed" electron would appear to have a smaller charge.

This claim is misleading. Consider an electron in a classical polarized media. It is the polarization of the media close to the the observer which does screen some of the electron charge. That is, the molecules close to the observer are polarized, and cancel some of the electric field of the electron.

Suppose then that the observer moves closer to the electron. Does the electron charge appear larger then? That depends on the amount of polarization close to the electron. It is somewhat misleading to say, as in Wikipedia, that the observer "bypasses a screen of virtual particles" as he moves closer. The bypassing is not relevant but the magnitude of polarization closer to the charge.

Let us analyze a scattering experiment of an electron and a positron using Feynman diagrams. In the diagrams, there is a vacuum polarization loop which makes the interaction weaker. The integral for the loop diverges.

But in the diagram there is no cloud of virtual pairs which would screen the charge. Why would we invent an artificial "explanation" using imagined virtual pairs?

Quanta magazine claims that there is progress in the black hole information paradox

https://www.quantamagazine.org/the-black-hole-information-paradox-comes-to-an-end-20201029/ 

The Quanta article says that a group of researchers first considered black hole evaporation in the context of the conjectured AdS/CFT duality. Then they were able to eliminate the link to AdS/CFT using path integrals.

The Wikipedia page:

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

discusses the work of Penington et al.

The claims remind us of the announcement by Stephen Hawking in 2004 that he is able to recover the information which has fallen into a black hole, using Euclidean path integrals:

https://arxiv.org/abs/hep-th/0507171

Does Hawking radiation exist? Vladimir Belinski has claimed that the calculation by Hawking is erroneous. In this blog we have raised questions about conservation of momentum if we assume that the black hole horizon radiates photons. A photon carries away a momentum p. What, and how, could absorb the opposite momentum -p?

As far as we know, no one has refuted the criticism by Belinski, and no one has shown a mechanism which would conserve momentum.

What about the claims that we can use a path integral and show that the information falling toward the black hole horizon is preserved, after all?

Let us assume that a macroscopic black hole forms, and it devours and crushes a large part of the wave function, or, of the path integral.

In quantum mechanics, one cannot simply throw away a part of the wave function or a path integral. It is a strange claim that the remaining part would be equivalent to the entire original wave function.

The horizon of a black hole is classically a one-way surface. Information can fall in, but can never come back.

Let us do a thought experiment: instead of a black hole, we have a horizon which leads to a wormhole, and the wormhole opens into a white hole in some other part of our universe. If we claim that the horizon necessarily returns back the information which has passed by, how do we explain that the same information ends up to another part of our universe? This is against the "no-copying" principle of quantum mechanics.

People who claim that a black hole horizon must necessarily give up the information it has devoured, kind of claim that the universe behind the horizon is "inferior" to our own universe. They think that the entropy should be calculated based on what is on our side of the horizon, and we should ignore what is behind the horizon. That does not sound like a reasonable assumption. Why would the other side be inferior to our side?

The Quanta magazine article points at the large number of assumptions and idealizations which Penington et al. use.  That is a weakness in the new work.

Does a system eventually radiate all its entropy out in an asymptotic Minkowski space?

Consider a block of glass. It is amorphic material and contains quite a lot of entropy. When a very long time passes, glass becomes crystallized, and its entropy dramatically decreases. After an immense time, the block of glass will assume a state of the lowest energy. That state might be one spherical crystal - spherical because of the gravitational attraction.

Hawking, Bekenstein, and others probably had this phenomenon in their mind when they conjectured that a black hole horizon must necessarily have a non-zero temperature since it encloses a lot of entropy inside.

But let us think about the wormhole example above. If we throw a block of glass through the horizon, the entropy of the glass will slowly seep out into the new part of the universe where the block of glass ends up. There is no obvious reason why the entropy should climb up the wormhole to the wrong direction and magically return back from the horizon.

If we take macroscopic one-way surfaces seriously, then the entropy will remain behind the surface. If it is a wormhole, then the entropy will pop out of a white hole. If the surface is a black hole horizon, then the entropy will remain confined behind the horizon.

Let us compare a block of glass to a black hole horizon. An observer can see that the atoms are in a disorder in the glass. He sees that there is a lot of entropy.

But if a black hole horizon rapidly becomes an ideal geometric object, and essentially black, too, then an outside observer does not see a lot of entropy there. He does know that the horizon devoured a lot of entropy, but he can no longer directly observe the disorder. This is in contrast to a block of glass. 

Do cosmological horizons somehow radiate back the entropy in the galaxies that they devoured?

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

In an accelerating expansion of an FLRW universe, galaxies eventually disappear behind a cosmological horizon.

If horizons generally would give back the entropy which they have devoured, would a cosmological horizon eventually return us all the information in the galaxies which it swallowed? That seems implausible.

The energy of a graviton has to be hf

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

Wikipedia states: "it is unclear which variables might determine graviton energy."

Let us assume that we have a mass M attached to a harmonic oscillator whose frequency is f. The harmonic oscillator device A is attached to the crust of Earth.

When the mass M swings in the oscillator, it produces a dipole gravitational wave.

Earth, in turn, produces an opposite dipole wave, which - far away - almost exactly cancels the dipole wave produced by M. This is the reason why observed gravitational waves are quadrupole, not dipole.

Let us then assume that we have another harmonic oscillator B of the frequency f close to our first oscillator A.

According to quantum mechanics, the oscillator A can only lose energy in units of hf, where h is the Planck constant.

If the gravitational interaction can transfer energy from A to B, it must happen in units hf. This strongly suggests that the energy of a single graviton is hf, just as it is for a single photon.

But does the oscillator A lose energy at all? Could it be that any energy state of A is stable under the gravitational interaction and cannot decay into a lower energy state?

If the mass M is huge, then we believe that gravitation behaves in a classical way. The oscillator B will certainly start to oscillate if A oscillates. This behavior might be measurable using a Cavendish torsion balance.

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

Thus, there is every reason to believe that the oscillator A can transfer energy packets of the size hf to B.

Conclusions

The energy of a graviton is most probably hf. It is the same energy as for a photon of the same frequency. The "reason" for the packet size is that a quantum harmonic oscillator can only gain or lose energy in packets of the size hf.

What does conservation of the ADM energy really mean?

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

The ADM formalism is supposed to prove conservation of the energy of a closed system when observed from "infinity".

But there is a conceptual problem in this. Suppose that we initially have a system S which is static, and we have a static solution of the Einstein equations. Let M be the ADM mass of S at infinity.

Let us then make some change to S. Some masses inside S move and produce gravitational waves. Conservation of energy requires that the total mass of S plus the energy of the waves stays the same.

Let us then calculate the new ADM mass as the limit at infinity. Since the speed of light is finite, no information about the change in S has yet reached infinity. The metric is the same as before, and the ADM mass is trivially the same old M!

What we would like to have is the conservation law:

(*) The total energy of the system S plus the energy of the emitted gravitational waves is conserved. We calculate the energy of the waves using some approximation method.

Almost nothing is known about the existence of solutions for the Einstein equations. Therefore, (*) is an open problem.

We need to check the original papers about the ADM formalism. What do the authors state about conservation laws?

Quantization solves the existence problem of the solutions for the Einstein equations?

https://en.wikipedia.org/wiki/Exact_solutions_in_general_relativity#Existence_of_solutions

In 1993, Demetrios Christodoulou and Sergiu Klainerman were able to prove the stability of the Minkowski vacuum under small perturbations.

But the existence of solutions for the Einstein equations remains unproven for essentially all practical cases - that is, if we have a non-symmetric, non-uniform mass distribution.

The Navier-Stokes equations

It is notoriously hard to prove the existence of smooth solutions for non-linear differential equations. The most famous example is the Clay Millennium Problem about the smooth solutions of the Navier-Stokes equations.

Let us think about a real physical fluid, say, water. A milliliter of water contains some 3 * 10^22 water molecules H2O. The Navier-Stokes equations approximate a viscous flow of a very large number of water molecules. The equations are an idealized effective theory of a macroscopic amount of water.

A priori, there is no reason why the equations would make sense, or have smooth solutions, if we extend them to the case where a water molecule is infinitesimally small. The Clay Millennium problem may have little physical relevance. 

A water molecule size gives a natural cutoff scale for the Navier-Stokes equations. Approximate solutions of the equations are physically relevant provided that features whose size is of the order of a molecule do not affect the solution.

In the case of water, the quantum of water, a single molecule, saves us from the problem of the existence of smooth solutions.

Maxwell's equations

The electromagnetic field is another example of quantization. Maxwell's equations describe the behavior of a macroscopic classical field. We assume that a photon carries an energy hf, where h is the Planck constant and f is the frequency of classical (circularly polarized) macroscopic wave.

A very large number of coherent photons form a classical macroscopic wave. But Maxwell's equations do not describe the absorption of a single photon correctly. The equations are not aware of the quanta.

Maxwell's equations are an effective theory. Does it make sense to study the smoothness of the solutions for features whose size is much less than the photon wavelength? Probably not - a very short wavelength would involve a photon of a high energy. How could such a photon be produced? From where would the energy come from?

The Einstein equations

What about the existence and smoothness of solutions for the equations of General Relativity?

We do not know if the field is quantized. If it is, then we might get a cutoff scale which saves us from considering features smaller than a certain length.

We need to check if the problem in proving the stability of the Einstein equations involves very small features. If the problem is the appearance of singularities, then a cutoff scale would help.

In our earlier post today about a perpetuum mobile we again pointed at the possibility that the Einstein equations, combined to a lagrangian of an infinitely strong vessel, may have no sensible solutions at all. This potential problem is separate from the stability and smoothness of pure Einstein equations.

Conclusions

The smoothness and stability problems in various non-linear physical equations probably are not relevant, once we take into account the quantization of the field.

An improved perpetuum mobile for general relativity

The Schwarzschild interior solution has the peculiar property that the spatial metric is not affected by the pressure. Pressure only affects the temporal metric.

The radial spatial metric in the solution is stretched by mass-energy.

Suppose that we have an infinitely strong spherical vessel. If we place some mass-energy M at its center, then the volume of the vessel grows, while the area of the surface of the vessel remains constant.

Let us fill the vessel with incompressible fluid.

If we can somehow remove the mass M from the center of the vessel, then the pressure inside the vessel grows infinite, and we can, in principle, extract a large amount of energy from the system. We have a perpetuum mobile.

Last year, our attempts to construct a perpetuum mobile were thwarted by the gravitational field of the infinite pressure, if we try to remove M.

Let us try a new approach: we convert M instantaneously into light, and let the light escape from the vessel.

                 M

             _       _

            (_| -- |_)  vessel

Let us have a hollow tube which goes vertically through the vessel. We place the mass M inside the tube as an infinitely thin plate, like the dashes -- in the above diagram.

Let us convert M instantaneously into photons. Let us shoot the photons straight up and down from the plate.

Can the mass-energy M escape, leaving behind a vessel which has an infinite pressure?

A global observer perceives the speed of light to be "slower" close to the (large) mass M than inside the vessel. Or does he? If we have the energy M/2 moving as photons, what is the perceived speed by a global observer?

Could it be that the infinite pressure which is created inside the vessel, has enough time to affect the photons which carry M away, and can deflect almost all the photons back to the center of the vessel?

We do not know if the Einstein equations have any solution for the setup of the sketched perpetuum mobile. It might be that the equations do not give any prediction of the behavior.

In this blog we have conjectured that the Einstein equations are too "strict" and that no physically reasonable solution for them exists for many macroscopic setups. We have suggested that a switch to some kind of a rubber model would remove the excess strictness. How would a rubber model handle our perpetuum mobile?