Bremsstrahlung: why a large value of the Planck constant makes it weak?

Let us continue our study of bremsstrahlung.

We think we have found the explanation why a big Planck constant h makes the cross section small for the case of the electron losing most of its kinetic energy in the emission of a large photon.

If we smoothen the 1 / r potential at the origin so that there is no deep well, then the Fourier decomposition of the new potential does not contain large momenta k. This shows that it is the deep well which contributes the large momentum k exchanges with the electron. This is just like in classical physics: the electron has to go very close to the nucleus for the electron to gain a large change in the momentum.

We see that it is the close encounters which produce large photons in the QED framework.

A close encounter happens quickly. The distortion in the electron wave function lasts for a very short time.

          __          __        

        /     \___/     \___/   photon

          __

        /     \   distortion

If the Planck constant h is small enough that the distortion "fits" in half a cycle of a large photon, then the "projection" of the distortion on the photon wave is large. There is a large possibility that the photon gets emitted in the process. Having a small h makes the process look like the classical limit. In the classical limit all 1 MeV electrons which come within 10^-14 m from the nucleus (a proton) will lose most of their kinetic energy in the encounter.

       ___________

     /                       \____________/  photon

                __

              /     \    distortion

But if the value of h is big, then the large photon has a long wavelength and a long cycle. The projection of the distortion on the photon wave is small: there is a low probability that a large photon gets emitted.

The wavelength of a 250 keV photon is 5 * 10^-12 m. To emit such a photon, the 1 MeV electron has to come within 10^-14 m of the proton. We conclude that in our universe the value of the Planck constant h is "big", and few large photons are emitted in bremsstrahlung.

This means that most close encounters are elastic or almost elastic. In a classical universe no almost elastic close encounters would happen.

The classical limit and what happens if we increase the value of the Planck constant

A way to analyze bremsstrahlung is first to look at it as a classical process.

If we set h to 1 / 100 of its usual value, then the de Broglie wavelength of a 1 MeV electron is roughly 10^-14 m. A close encounter to within 10^-14 m of a proton is "almost classical".

Increasing h back to its normal value suppresses the emission probability of a large photon by a factor 1 / 100.

What about our "rubber plate" model of the electron electric field? We explained the vertex correction with the classical model. Does our explanation work if we increase h?

It is hard to produce a large real photon of energy E in a small volume in a short time. That is what our analysis showed.

But there is no problem in exchanging a similar amount of momentum k in that small volume and short time. Coulomb scattering is not affected if we raise the value of h. As if pushing or pulling is not affected by a large h, but producing a vibration (a real photon) is hampered by a large h.

The effect of the rubber plate in the vertex correction is to push and pull on the electron. We conjecture that the rubber plate model works in the classical way even if h has a large value.

We need to check if the vertex function, or correction, in QED depends on the value of h.

The length scale problem is solved!

We have spent a lot of time wondering how the very sharp turn which the 1 MeV electron makes close to the proton can produce a relatively long wave photon, the wavelength ~ 5 * 10^-12 m and energy ~ 250 keV.

Now we have the solution. The sharp turn in rare (1 / 100) cases is projected into a long-wave photon. In other cases, the encounter is elastic or almost elastic.

If the Planck constant would be 100 times smaller, then the encounter would in most cases produce a short wavelength photon, ~ 5 * 10^-14 m. The energy of the photon would still be ~ 250 keV.

A long wavelength photon has problems going through a small hole in spacetime

This setup is clearly related to the analysis above. The bremsstrahlung photon whose wavelength is ~ 5 * 10^-12 m is "born" in the area of encounter, and that area is only ~ 10^-14 m in size and ~ 10^-14 m / c in time. The would-be photon has to make it through a small hole in spacetime, in order to break out into the external world.

Which scattering cross sections depend on the value of the Planck constant?

The 1934 bremsstrahlung paper by Bethe and Heitler is freely readable at 

https://royalsocietypublishing.org/doi/10.1098/rspa.1934.0140

Bethe and Heitler seem to use the principle later known as Fermi's Golden Rule:

https://en.wikipedia.org/wiki/Fermi%27s_golden_rule

where the "density" of possible end states is counted.

      incoming         distorted      outgoing

 e- |   |   |   |   |     /    /    /    /     |        |        |

                  

                                                       ~~~~~~~~~~~

                                                       photon

                                      ● Z+ nucleus

Bethe and Heitler assume a plane wave of the incoming electron, which is perturbed (distorted) for a short time dt by the Coulomb potential of the nucleus Z+.

Then they take the projection of the distorted wave function to an off-shell electron plane wave. An off-shell electron plane wave is one where the energy-momentum relation

       E² = p² + m²

does not hold.

They further calculate the transition probability of this off-shell wave to an on-shell outgoing electron wave under a perturbation of the field of the emitted photon. If we look backward in time, it is the emitted photon which causes a state transition of the outgoing electron to an off-shell electron.

Besides the process described above, there is another - less intuitive - process where the electron emits the photon before the electron meets the nucleus.

                                          real photon

                                          ~~~~~~~~~~

                                        /

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

                             |  virtual

                             |  photon

                             |  k

     Z+ ---------------------------------------------

In the corresponding Feynman diagram, the electron absorbs a virtual photon which is sent by the nucleus Z+ and carries 4-momentum k. The electron goes off-shell. The electron then emits a real photon and returns to on-shell.

The probability of the electron receiving a virtual photon whose 4-momentum is k, is governed by the photon propagator. Richard Feynman in his famous paper derives the propagator from the Fourier decomposition of the Coulomb potential 1 / r. Another way to derive it is from the Green's function of the Klein-Gordon equation.

An aside: why is the Fourier decomposition of the Coulomb potential similar to the Green's function? Our "sharp hammer" hypothesis in this blog is a manifestation of this relationship.

Which processes depend on the value of the Planck constant h?

Coulomb scattering is equivalent in classical physics and quantum field theory. Cross sections for it do not depend on the value of h.

Cross sections in Compton scattering depend on the classical radius of the electron and on the mass-energy of the photon. They do not depend directly on the value of h.

But in bremsstrahlung, the cross section for emitting most of the kinetic energy in a photon is inversely proportional to h.

What is the difference in these processes? An obvious difference is that in bremsstrahlung the number of particles grows by one. Bethe and Heitler count the number of possible final states in a unit volume.

The ratio of the number of final states and the number of input states probably depends on h. If we make h smaller, the ratio will probably grow. This may explain why in bremsstrahlung making h smaller will increase the cross section.

In Coulomb scattering and Compton scattering the ratio probably does not depend on h because the process is symmetric.

Hypothesis. QED processes which conserve the number of particles happen like in classical physics - if one takes into account that the photon is a particle. Their cross sections do not depend on the Planck constant h. But if the number of particles grows in the process, then the cross section is reduced by quantum physics, and a smaller h increases the cross section.

Our hypothesis claims that the "mechanics" of a process is classical. The value of the Planck constant enters the stage when we calculate the number of possible states in a unit volume.

The hypothesis explains the coincidence that Coulomb scattering is equivalent in classical and quantum physics.

Our hypothesis smashes the hopes in our previous two blog posts: the fine structure constant is not determined by the geometry of the electron static electric field.

This is because calculating the number of possible states in a unit volume is not connected to the geometry of the electron electric field.

Or could it be? Are there "resonances" in vibrations of the static electric field of the electron? If the electron orbits a proton, could it be that these resonances enforce the Bohr orbits? We need to study this.

The fine structure constant: an update

In our previous blog post we claimed that in a classical model an electron might lose most of its kinetic energy if it passes the proton at a distance

       < 0.9 * 10^-15 m.

The idea was that all of the inertia of the electron is located in its static electric field farther than

       r₀ / 2 = 1.4 * 10^-15 m

from the pointlike electron. There

       r₀ = 2.8 * 10^-15 m

is the classical radius of the electron.

A more detailed analysis reveals that it is enough that a half of the inertia of the electron "lags behind" in the movement. Imagine that a half of the mass of the electron is attached to it with an elastic rubber band. The rest of the mass is rigidly fixed to the pointlike electron.

              ● half of the electron mass

               |

               |  rubber band

               |

              ● e- electron

              v  ------->

Let the initial velocity vector of the system be v.

If the pointlike electron suddenly bounces back in the field of a proton, so that its velocity vector becomes -v, then all of the original kinetic energy of the system will go to stretching the rubber band. That is, the kinetic energy is totally converted to vibration or electromagnetic waves.

Where is the inertia of the electron located? We again meet the mystery of the "inner field" of the electron. We do not know the inertia distribution close to the classical radius of the electron. The Larmor formula suggests that in the far field of the electron, the inertia is in the mass-energy of the static electric field.

In our previous blog post, the cross section, which we calculated to be 25 millibarn, could be off by a factor 100, depending on the distribution of the inertia of the electron. That is bad news for our claim that the fine structure constant is determined by the geometry of the electron electric field.

The fine structure constant is determined by a semiclassical model?

In our July 14, 2021 blog post, we noticed that the bremsstrahlung approximation that is used in astrophysics can be explained by a semiclassical model where the electron comes very close to the nucleus, so that most of the kinetic energy of the electron is converted into vibration of its electromagnetic field.

Let the nucleus be a single proton.

Recall our "rubber plate" model of the static electric field of the electron. The nucleus suddenly pulls on the electron, which makes the "rubber plate" to vibrate. We have used the rubber plate model to explain qualitatively the birth of electromagnetic waves.

The model is not completely classical because the vibration frequency in the rubber plate would be much higher than the the frequency of the emitted bremsstrahlung photon. This is the "length scale problem" which we have discussed in many posts during this year.

In the astrophysical bremsstrahlung approximation, the classical Larmor formula is used to calculate the radiation, but the closest approaches to the nucleus are brutally cut off at the distance b, where b is the de Broglie wavelength of the electron.

How close must the electron come to the nucleus, so that most of the kinetic energy goes into the vibration of the rubber plate?

The mass of the electron resides in the energy of its static electric field at a distance

       > r₀ / 2,

from the pointlike electron, where

       r₀ = 2.8 * 10^-15 m

is the electron classical radius.

Let us assume that the electron is mildly relativistic, its total energy 1 MeV.

If the electron orbits about 90 degrees around the nucleus, traveling a distance 1.4 * 10^-15 m, then the complete field of the electron "lags behind" in the movement, and it might be that most of the kinetic energy is converted into vibrational energy. The radius of that orbit is 0.9 * 10^-15 m.

The cross section for emitting a large photon (containing most of the kinetic energy) is then something like

       A = π * (0.9 * 10^-15 m)^2

           = 2.5 * 10^-30 m^2

           = 25 millibarn.

We calculated earlier from the astrophysical approximation that the cross section for emitting a large photon is roughly

       A' = 10^-7 * π * (2.4 * 10^-12)^2

           = 2 * 10^-30 m^2

           = 20 millibarn.

Is it a coincidence that the semiclassical model roughly reproduces the right cross section?

The quantum mechanical way to calculate the cross section is the Bethe-Heitler formula. The cross section in the formula linearly depends on the fine structure constant, which is approximately 1 / 137.

If our semiclassical model is the "right" way to explain bremsstrahlung, then our semiclassical model fixes the value of the fine structure constant.

The fine structure constant depends on the Planck constant. If our model is "right", it fixes the value of the Planck constant.

The semiclassical model is about the "geometry" and mass-energy of the electric field. The Planck constant does not appear in the semiclassical model.

We need to study in more detail bremsstrahlung. Can we show that the semiclassical model qualitatively works for different energies and photon deflection angles, and different numbers of protons in the nucleus?

What would it mean if the Planck constant were determined by the semiclassical model? We need to investigate that.

The mass-energy of the electric field depends on vacuum permittivity ε₀. The Planck constant in our semiclassical model is determined by ε₀?

Comparison of classical and quantum bremsstrahlung

On May 28, 2021 we wrote about classical bremsstrahlung. If we imagine that the Planck constant goes to zero, we expect quantum bremsstrahlung to approach the corresponding calculation in classical electrodynamics.

Astrophysicists calculate bremsstrahlung with the classical approximation, cutting off impact parameters b where b is smaller than the de Broglie wavelength of the electron.

They motivate their cutoff approximation by claiming that for smaller b, "quantum effects" spoil the classical approximation.

Their motivation is very ad hoc. Why should we ignore the classical contribution of small b? Classically, most of bremsstrahlung energy is emitted with very small b.

Let us try to find a sensible motivation for the cutoff.

The radiated energy in the classical approximation is very small for b > the Compton wavelength

In the earlier blog post we calculated that classically, a "moderately relativistic" electron (1 MeV) which passes a proton at the distance 2.4 * 10^-12 m (the Compton wavelength) will only radiate ~ 10^-7 times of its kinetic energy away, roughly 0.05 eV. Classically, this energy is evenly spread over photons whose energy is up to 500 keV.

If we take the cutoff of astrophysicists literally, we have to assume that through some mysterious mechanism, a classical electron can produce a 500 keV photon even though the electron passes a proton far away, at a distance > 2.4 * 10^-12 m. This does not make much sense.

                                     500 keV photon

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

                    /

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

                          |

                          | virtual photon

                          |

      p -------------------------------------------------

Let us then think about the Feynman diagram of the process. The momentum transfer in the virtual photon is very large.

Semiclassically, for the moderately relativistic electron to lose a lot of its spatial momentum to the proton, it has to pass the proton at a distance < 2.8 * 10^-15 m (the classical radius of the electron).

The cross section of such a very close encounter is something like

       ~ 10^-7

times the cross section of an encounter at a distance < 2.4 * 10^-12 m.

We see that very close encounters can explain the flux of 500 keV photons which in the astrophysical cutoff approximation were the result of encounters roughly 2.4 * 10^-12 m away.

Now we have a semiclassical approximation which qualitatively explains why the astrophysical cutoff approximation might work for large-energy photons.

What about lower energy photons? A similar argument works for them.

Is there some deep underlying reason why the Feynman diagram method (qualitatively?) reproduces the cutoff approximation of astrophysicists?

How to conserve energy if the number of photons changes?

Let us analyze further the photon number change effect which we discovered in the previous blog post.

Let us have an obstacle made of a polarizable material. Let the obstacle be much smaller than the wavelength of a coherent source, e.g., a radio transmitter or a laser. The obstacle makes a very sharp area of deformation in the electromagnetic wave. An observer who moves through this deformation will in the temporal Fourier decomposition of the wave see high frequencies.

Does the effect really exist? It almost certainly does. A coherent laser beam, or a coherent beam from a radio transmitter, is a macroscopic object. It would be very surprising if the measuring device would not see high frequencies in Fourier decompositions of complex waveforms.

Thus, we must conclude that even if a laser only emits photons of energy E, a measuring device which moves at a relativistic constant velocity v may sometimes detect photons which have much higher energy E'.

Classically, energy conservation is not a problem. But how to get conservation in quantum mechanics?

Could it be that in quantum mechanics, several photons can sometimes merge into a photon with higher energy?

Maybe we should abolish the notion that the laser beam consists of photons? The beam is created by atoms decaying into a lower energy state. That is, the beam is created as kind of "photons". But maybe the beam itself does not "consist of photons"? Photons are absorbed from the beam. The beam is destroyed in units of a "photon".

In our earlier blog posts we have suggested that one should assume the minimum amount of information in the course of a quantum mechanical process. The prime example of this is that a photon does not have a definite path. It is a step further if we claim that a laser beam does not have definite "photons" either.

A "photon" would only exist when we perform a measurement. We may observe that a certain atom has decayed into a lower energy state, or has been excited to a higher energy state.

We can call this framework "minimalist quantum mechanics". Quantum mechanics moves closer to classical mechanics if we do not assume the existence of quanta in intermediate states of a physical process. Only measurements will observe definite quanta.

Conservation of energy

How is energy conserved if the number of quanta is not defined?

Maybe the quanta which are eventually absorbed from the wave will always have energy less or equal to the energy that went to create the wave?

How would nature do the bookkeeping to ensure conservation of energy?

An accelerating observer: Unruh and Hawking radiation

The derivation of hypothetical Unruh or Hawking radiation uses a Fourier decomposition of a wave as seen by an accelerating observer. Unruh and Hawking believed that negative frequencies in the decomposition represent photons popping out of nothing.

In this blog we hold the opinion that Unruh and Hawking made an error: a negative frequency is simply a classical wave which has counter-clockwise circular polarization, if we define a positive frequency wave to have clockwise polarization.

The problem of a changing number photons is prominent for an accelerating observer. What kind of quanta will he observe in his detectors if he moves within a coherent electromagnetic wave?

Classically the answer is clear: his detectors will absorb photons of various frequencies. Again we face the problem: how to conserve energy?

Does some extra energy come from the kinetic energy of the measuring device?

Suppose that we have static negative electric charges ordered in a line at equal distances. Let an antenna move along the line at a constant velocity v. The antenna will observe a fluctuating electric field and can extract energy from it.

The energy must come from the kinetic energy of the antenna. We may imagine that the antenna collides with a virtual photon sent by a charge, loses momentum, and kinetic energy is freed.

However, if the antenna moves in a coherent electromagnetic wave, then some of the energy comes from the wave, and it might be that none of the energy comes from the kinetic energy of the antenna.

The number of photons may change between inertial frames: a new "Unruh effect"

Let us work in semiclassical physics. Let us have a monochromatic laser beam which is reflected or is refracted by a small object (is scattered by the object). Let us assume that the object is smaller than the wavelength.

        laser beam

        ~~~~~~~~~~~~~~~~~~~

        ~~~~~~~~~~~~  ● object

        ~~~~~~~~~~~~~~~~~~~

                                          <o> measuring device

What kind of photons an observer may measure in the scattered light?

If the system and the measuring device are fixed in the laboratory frame, then the observer will see all oscillation of fields having the same frequency f as the laser. He will see photons whose energy matches the photons in the laser beam.

But let us then make the measuring device to move at a constant speed v relative to the laboratory frame.

The waveform close to the object has a complex form. The Fourier decomposition of the signal received by the measuring device will have many frequencies, some of them very high.

The observer may see a photon which has much higher energy than the photons emitted by the laser.

How do we interpret this? If the laser sends, say, one photon of energy E in a second, how can the observer see a photon with a much higher energy E'? Is energy conserved?

Seeing a high-energy photon has an extremely low probability, though.

We have discussed the analogous problem in the context of an accelerating laser, or an accelerating observer. There is no obvious way to match individual photons in emission to photons in absorption. We have called this problem the "real Unruh effect".

In purely classical physics there is no problem. Energy in a wave of frequency f can be transformed to energy in a wave of much higher frequency f'. It is quantization which poses the problem here.

The length scale problem is ubiquitous

We have been studying bremsstrahlung in the past weeks. The "length scale" problem is that an electron passing very close to a proton should classically emit photons of very high energy. What wipes away these photons? Our new observation in this blog post shows that a similar problem exists in very mundane scattering of laser light.