James Webb results agree with our Minkowski & newtonian cosmology?

UPDATE September 5, 2022: A major problem of the Milne model is how to explain the "first acoustic peak" in the cosmic microwave background (CMB) angular power spectrum. 

https://www.aanda.org/articles/aa/full_html/2012/01/aa16103-10/aa16103-10.html

A. Benoit-Levy and G. Chardin (2011) suggest that it might be associated with matter/antimatter zones in the universe, and not with baryon acoustic oscillations.

https://arxiv.org/abs/1906.11628

The Hubble constant value can be derived by looking at the angular diameter of these CMB acoustic baryon oscillations in the sky. Andrei Cuceu et al. (2020) quote a value of 67.4 km/s per megaparsec. That is smaller than the value 74 km/s per megaparsec which is measured from standard candles. Could it be that the first acoustic peak is not associated with baryon acoustic oscillations?

----

https://m.slashdot.org/story/403711

The first pictures from the James Webb telescope reveal surprisingly old and surprisingly well-developed galaxies.

"Galaxy formation models may now need a revision, as current ones hold that gas clouds should be far slower to coalesce into stars and galaxies than is suggested by Webb’s galaxy-rich images of the early universe, less than 500 million years after the big bang. “This is way outside the box of what models were predicting,” says Garth Illingworth of the University of California (UC), Santa Cruz."

https://www.science.org/content/article/webb-telescope-reveals-unpredicted-bounty-bright-galaxies-early-universe

http://meta-phys-thoughts.blogspot.com/2022/01/the-minkowski-newtonian-model-predicts.html

We wrote about our Minkowski & newtonian model on January 31, 2022. The expansion of the universe happens at a constant speed, just like in the Milne model.

In the Milne model the cosmic microwave background which we observe now was born much farther away from us than in the ΛCDM model. The same for the first galaxies which we observe now. The apparent angular diameter of the first galaxies is smaller in the Milne model than in the ΛCDM model.

         \                         /   now

           \                     /

             \                 /

               \             /

                 \         /   O  first galaxies

                   \     /

                     \ /   Big Bang

           Milne model

         \                         /  now

           \                     /

              \               /   O first galaxies

                 \         /

                     \ /  Big Bang

                  ΛCDM

                  

                  

In the Milne model the first galaxies have a larger redshift and their apparent angular diameter is smaller than in ΛCDM.

James Webb supports the Milne model.

The electron propagator in Thomson scattering is classical

Our analysis in the previous blog post brought up the problem what process, or wave, does the electron propagator really describe in a Feynman diagram.

We know that the photon propagator, for an unknown reason, models Coulomb scattering.

The electron attached to its electric field: the "rubber string" model

Classically, the electron possesses a static electric field. We may imagine that the lines of force are made of rubber.

If we shake the electron, waves will propagate along the rubber strings. The waves are electromagnetic wave.

We may interpret that the rubber mesh controls the movement of the pointlike particle electron. The electron is a kind of an oscillating mass attached to the rubber mesh.

Classical Thomson scattering

                                        electric

                                    lines of force

                                             |

              |    |    |               --- ● ---  

                                             |

           laser beam       electron e-

The laser beam shakes the electron. New electromagnetic waves propagate in the lines of force of the electron.

We may imagine that the laser beam gives small impulses to the electron at very short time intervals. The response of the electron and the rubber mesh is an impulse response.

Recall that a propagator is a Fourier component of the impulse response of a wave equation. For example, if we hit a drum skin with a sharp hammer, the Fourier decomposition of the wave is the set of propagators.

The impulses to the electron have a cycle determined by the frequency of the laser. Constructive interference strengthens the output at this frequency. Destructive interference wipes out other frequencies.

The electron will act as a radio transmitter and send an electromagnetic wave to many directions. It scatters the incoming laser light.

The propagator in this classical treatment tells us how effective the electron is in outputting energy to the scattered "channel", or scattered wave.

We can, in principle, calculate with a computer how the electromagnetic field behaves, and what is the value of the propagator.

The classical Feynman diagram looks like this:

        laser                              scattered wave

        ~~~~~~                       ~~~~~~~~~~~

                           \              /

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

                            virtual

                           electron 

It is essentially the same as the quantum electrodynamics (QED) Feynman diagram.

The "virtual electron" is the propagator. We may imagine that it represents the electron in the mesh after an impulse hit the electron.

QED Thomson scattering

As we wrote above, the Feynman diagram is essentially the same as in the classical process. Numerical results from Feynman formulae agree with the classical treatment.

Note the following thing: the propagator in the QED diagram is from the impulse response of the Dirac equation. We were able to connect the Dirac equation to a classical process.

The process in Thomson scattering is non-relativistic. The Dirac equation in that case is equivalent to the Pauli equation, or the Schrödinger equation. The propagator for the Schrödinger equation seems to be complicated.

What is a "virtual" electron and how does an electron "absorb" a photon

In the classical interpretation, a virtual electron means the system electron & its electric field where the system has been disturbed by an impulse.

The absorption of a photon means that the system goes to a disturbed (excited) state.

Classically, the electron is a point particle. It cannot have excited states on its own. An excited state has to be the electron in an interaction with something else, in this case its own electric field.

When an excited electron emits a photon, that means that the oscillation of its electric field moves farther from the electron and starts a life of its own.

A propagator does not make much sense for a point particle. But a propagator for the system the electron & its field makes a lot of sense.

Question. How can we extend the classical propagator to an "electron field"? That is, we would not have a point particle but some kind of a field. This might show the connection between the Dirac equation and the classical model.

It is not clear if we can define an electron field in a reasonable way. If the field simply describes the position and the phase (in a path integral) of a single electron, then there is no obvious interaction between different parts of the field.

If the electron is either in the zone A or the zone B of space, there is no interaction between the zones A and B. This is very different from a drum skin where A and B always interact. In a drum skin, a wave equation is natural, but it is not natural for mutually exclusive histories.

The Dirac equation is used in Feynman diagrams to calculate the electron propagator. That is, a single electron is interacting with something else. Then it makes sense to consider the system the electron & its field.

The Dirac equation does have a conserved probability current and it does predict the magnetic moment of the electron. How can we explain these if the equation only describes an interacting electron?

Maybe the electron is a wave phenomenon, after all? The spin of an electromagnetic wave probably is a wave phenomenon.

But then we face the problem how to attach the electric field to the electron.

The Dirac equation describes the system the electron & its field?

The electron propagator in the Dirac field describes something which is off-shell, or not in its ground state.

If the QED propagator is able to calculate something similar as the classical propagator, then it is natural to assume that the QED propagator describes the combined system the electron and its field. We cannot remove the electric field from the electron. It makes sense that the Dirac equation describes the entire system.

However, it is not clear how the classical system gives rise to the Dirac equation. How do we explain zitterbewegung and the magnetic moment?

Conclusions

The analogue of a photon propagator is a sharp hammer hitting a drum skin (where the skin is actually the three-dimensional space).

The Fourier decomposition of the associated - 1 / r potential has the familiar formula

       ~ 1 / p²,

where p is the 4-momentum of the Fourier component.

              |

            _|_  

           |__| -->     ● e-

       hammer

The analogue of the electron propagator might be a hammer hitting the electron. The electric field of the electron moves a little bit. That is like adding a dipole where a positron e+ is put to the old position of the electron and a new electron is put to the new position.

The dipole potential is roughly

       ~ 1 / r²,

and the Fourier decomposition is roughly

       ~ 1 / |p|,

where p is the 4-momentum of the component.

The formula 1 / |p| is similar to the Feynman electron propagator if we set E = 0 and m = 0, where E is the energy of the electron and m is its mass.

Pair annihilation: the virtual electron is a carrier of a "force"

In our blog post February 16, 2021 we discussed the Feynman diagram of pair annihilation. It looks like scattering where the virtual electron carries (repulsive) momentum.

      momentum -p

             e+ ---------------- ~~~~~~~~~~  photon -k

                                     |

                                     |  virtual

                                     |  electron p - k

                                     |

              e-  --------------- ~~~~~~~~~~  photon k

      momentum p

Suppose that the positron and the electron approach from opposite directions at the same speed. The momenta are p and -p.

The momenta of the outgoing photons are k and -k.

The produced photons cannot carry away all the momentum of the particles. The electron gives the excess momentum p - k to the positron in the form of a virtual electron.

The virtual electron as a carrier of a "force"

The annihilation process looks like a scattering event where there is an approximate repulsive 1 / r² potential between the electron and the positron.

The Fourier transform of a 1 / r potential is

       ~ 1 / q².

The propagator of the photon is of this form.

https://mathematica.stackexchange.com/questions/239686/3d-fourier-transform-of-1-r2

The Fourier transform of a 1 / r² potential is of the form

       ~ 1 / |q|,

where q is the "momentum" of the Fourier component.

The propagator of the electron is very crudely of the form 1 / |q|.

Our blog post on February 16, 2021 shows that the potential in annihilation is actually even steeper than 1 / r², according to the cross sections calculated from Feynman diagrams.

We have been wrong in our attempts to explain annihilation by the electric attraction between the pair

The momentum transfer in the Feynman annihilation diagram does not happen through a photon.

Maybe it is not possible to make a (semi)classical model of annihilation using the electric force?

The sharp hammer model again: Huygens

We have explained the static electric field of an electric point charge with a sharp hammer which keeps hitting a "drum skin" at the charge, and makes a depression to the skin.

The Huygens principle is that a wave is absorbed by each point in space, and the point then acts as a new source for the wave.

In the case of the electron and its electric field, the Huygens principle might be something like a sharp hammer hitting a drum skin and creating the electron wave and its electric field.

The hit would produce a "point impulse" both in the Dirac field of the electron and the electromagnetic field.

Now we come to the Feynman diagram principle that all "paths" which conserve energy and momentum are allowed, and their probability amplitudes must be summed.

The idea is that the when the electron wave arrives at a spacetime point x, the wave is "absorbed" and immediately recreated with an impulse which hits both the Dirac field and the electromagnetic field.

Our sharp hammer becomes more versatile: it hits two fields at once. All combinations of responses from the two fields are allowed.

The virtual electron in the annihilation diagram is one component of the impulse response. The photon flying away is another component.

If we hit a drum skin, only sine wave "on-shell" Fourier components can travel over a long distance. Other, "off-shell" components have a short range of the effect. This may explain why the electron and the positron must come close to each other in order to annihilate.

A particle model of electron-positron electric scattering

    

             <------  e+

                         |

                         |  electric attraction

                         |

                         e-  ------->

The particle model is very simple and intuitive. Both particles are treated as point charges with an attractive electric force. The paths are calculated with classical relativistic mechanics. This gives results which, according to literature, are (almost?) exactly the same as when calculated using the simplest Feynman diagram.

A wave model of the scattering

 

                   <------ e+

          |       |       |        |        |        overlapping

                |       |        |        |        | waves

                             e-  ------>

Let us then try to form an intuitive wave model from a Feynman diagram. Let us have an average of one electron and one positron in a cubic meter of space.

We do not know the positions of the particles, and represent them with standard plane wave solutions

       u(E, p) * exp(-i / ħ * (E t - p • x)),

where u is the spinor.

The Feynman diagram is

         

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

                                 |

                                 | virtual photon

                                 |

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

How can we relate this to the wave diagram above?

If we take seriously the wave interpretation, then the scattering of colliding beams of electrons and positrons is a nonlinear effect. If there were just one beam, then scattering would not happen. In a linear system we would be able to sum the solutions of the two beams to obtain a new solution: there would be no scattering.

Suppose that we try to add a "source" to the wave equation of the electron. Something like

       D(ψ) = f(φ, ψ),

where ψ is the electron wave function, φ is the positron wave function, and D(ψ) = 0 is the Dirac wave function of the free electron.

The source term f(φ, ψ) depends on both wave functions.

But if the wave functions φ and ψ are essentially constant in the cubic meter (save a phase factor), how can the source generate a wave which is significantly scattered, to an angle, say, 90 degrees?

To simulate the scattering of the point particles we should have scattered waves where the cross section is

       ~ 1 / α²

where α > 0 is the deflection angle. How to generate such waves without having a localized disturbance of the field?

That looks hard. In the hydrogen atom model, one particle, the proton is treated as a particle while the electron is treated as a wave.

Our view in this blog has been that particles are the "true" nature, and that any wave phenomena are due to path integrals.

An analysis of the electron-positron scattering Feynman diagram

We showed that a pure wave model of the scattering does not work. What does the Feynman diagram then really calculate?

         

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

                                 |

                                 | virtual photon

                                 |

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

Let us assume that the underlying process really is the classical Coulomb scattering of point particles. But we do not know the precise position of the particles. We have to calculate the path integral for very many possible paths which the particles can take. The path integral is a collective phenomenon of all the possible paths. The path integral is the "wave" associated with the process.

What is the photon in the Feynman diagram? It summarizes the interaction in these very many paths.

Why is the photon propagator

       ~ 1 / p²

the "right" way to calculate the effect of the interaction?

It is the Fourier component of the 1 / r potential, but there is no obvious reason why the component would correctly capture the cross section of classical scattering.

           \     \     \        scattered flux

              ________

             |              |

             |              |  cubic meter

             |_______|

                

                 \     \     \   scattered flux

                

Let us again have that cubic meter where electron and positron beams meet. The scattered fluxes come quite uniformly from the entire volume.

We can model the scattering by assuming that for each centimeter of the path of the electron, a small portion of the electron flux gets scattered to various deflection angles, and a corresponding part of the positron flux gets scattered to the opposite direction. It is like both fluxes would travel in a nonuniform medium.

The force which causes the scattering is the electric force, and the propagator for some reason happens to capture it correctly.

Since we cannot interpret the diagram purely with waves, we conclude that the Feynman diagram really does describe the encounter of two particles.

We may use a wave description for one of the particles, though.

What about using a wave packet description for both of the particles? We can make the packets to pass each other at some short distance. That might work reasonably well if the distance is larger than the wavelength.

Classically, we have two point charges, and their electromagnetic fields are kind of "waves".

The interpretation of the annihilation Feynman diagram

Our analysis of the Coulomb scattering diagram concluded that the photon propagator "for some reason" happens to work in that case.

In the annihilation diagram, the "interaction" is not by the photon propagator, but by the electron propagator. How do we analyze this?

Conclusions

A lot of questions but few answers. The key problem in this blog post is what does the electron propagator model in a Feynman diagram. Is it a wave? Is it a path integral of point-like particles?

We will analyze Thomson scattering in the next blog post. There we are able to connect the electron propagator to a classical process.