Wikipedia seems to claim that bremsstrahlung from a quantum process is much larger than the classical Larmor calculation would suggest. The Gaunt factor g_ff,Born, which measures the quantum/classical ratio (or does it?), may easily be 5 or larger.
In Wikipedia, if we multiply the electron mass and its charge by 10^6, then Wikipedia claims that quantum emission of, say, 500 eV photons is 10X larger than in the classical approximation. An electron of that mass should behave like a classical particle. The Wikipedia claim about the Gaunt factor has to be incorrect.
Electromagnetic radiation is a result of the reduced mass of the electron when the electron is under acceleration. If there is just a single jerk on the electron, instead of cyclic oscillation, then the reduced mass during the jerk is probably less than in regular oscillation. Therefore, the Larmor formula may underestimate the radiation in bremsstrahlung. But this can hardly explain a 5-fold difference.
Looking at literature reveals that the classical approximation describes bremsstrahlung quite well.
O. I. Obolensky and R. H. Pratt (2005) provide Figure 8 which shows good agreement, except for 2 - 30 eV.
George B. Rybicki and Alan P. Lightman (2004) explain the Gaunt factor. It turns out that the Gaunt factor in Wikipedia is determined by impact parameters b_min and b_max:
ln(b_max / b_min).
The impact parameter b_max is classical: to produce a wave of frequency f the bypass has to happen in time
~ 1 / f.
This is definitely a classical thing.
The parameter b_min is the larger of:
1. the value of b where the straight-line approximation breaks down;
2. the de Broglie wavelength of the electron.
The formula has a logarithmic divergence if we allow b_min to be arbitrarily small. In most cases the authors recommend setting b_min to the de Broglie wavelength.
Suppose that b_max is much larger than b_min.
Classically, the contribution of small b is negligible, which shows that the divergence is an artifact of a wrong formula. Thus, we can cut off at some small fixed b, for example at the de Broglie wavelength. Therefore, b_min is essentially a classical thing, too.
The Gaunt factor is mostly classical, not quantum.
Bremsstrahlung: classical or quantum?
A radio transmitter does many cycles and attains a steady state of power dissipation. Bremsstrahlung is an effect which is caused by an electron doing "half a cycle" as it scatters from a nucleus.
● Z+
e- ------>
Let an electron with 1 MeV kinetic energy pass a nucleus.
Two months ago we blogged about the "length scale problem". The electron may make a very sharp turn at a distance ~ 3 * 10^-15 m from the nucleus. Classically, the wave would then contain frequencies of up to 1 GeV energies. Does the length scale problem appear if the distance is large? No. The length scale problem only concerns distances which are less than the de Broglie wavelength of the electron. We cannot draw the path of the electron with a precision higher than the de Broglie wavelength.
We suggested that destructive interference of various paths in the path integral wipes away very large frequencies. Constructive interference strengthens frequencies which are compatible with the electron kinetic energy.
The rubber plate model (or any classical model) cannot explain the interference pattern. But we believe that the underlying process is classical.
If we increase the masses and the charges of the particles, we approach the classical limit. In the classical limit, the production of the bremsstrahlung wave is very probably a classical process.
We can appeal to the fact: quantum mechanics is affected by classical phenomena, unless those phenomena would break laws of quantum mechanics. In the hydrogen atom, the electron cannot crash into the nucleus, even though classical physics would suggest that happening. But classical corrections are visible in the hydrogen spectrum.
An electron passing a nucleus would classically radiate, and radiation does not break the rules of quantum mechanics. We conclude that the classical phenomenon probably is the "cause" of bremsstrahlung.
Thus, the rubber plate model probably explains bremsstrahlung at the low, classical level. The sharp turn of the electron at the nucleus makes the elastic electric field of the electron to vibrate. That vibration is bremsstrahlung.
Why does the Feynman diagram calculate bremsstrahlung correctly?
The Feynman diagram is
bremsstrahlung
~~~~~~~~~~ real photon q
/
e- ----------------------------------
| virtual
| photon
| p
Z+ ---------------------------------
and the other diagram where the real photon q is emitted after the encounter with the nucleus Z+.
Let us keep p almost fixed and small. Classically p tells the distance how close the electron came to the nucleus. Classically, this determines a complex electromagnetic wave pattern which the electron sends.
Since the acceleration is toward the nucleus, the classical electron sends a dipole wave where most of the energy is aimed to the direction of the electron velocity, and behind it.
The dipole wave is weak and it contains relatively high frequencies (the length scale problem).
Let us concentrate on the wave which the electron sends directly forward: q is parallel with the velocity v of the electron. If q is small, then p gets a small contribution which is at a right angle to p.
Note that the 4-momentum p contains no energy. It is just spatial momentum. The propagator of p is little changed by the contribution from q.
We get a result:
The spectrum of q is determined by the electron propagator.
The spectrum is not a classical phenomenon. The classical wave contains a small amount of energy for every passing electron, and is dominated by very high frequencies if the electron comes very close to the nucleus. The quantum output has no photon at all for most electrons, but a few electrons lose a large amount of energy to a single quantum.
Our photograph model of quantum field theory claims that the photograph tries to depict the classical process as well as it can, but is limited by the poor resolution of the photograph. The classical process drains a little bit of energy from every electron. The photograph model then decides to put the burden on a few electrons: they lose a full quantum of energy while most electrons lose nothing.
The electron propagator measures how much off-shell the electron is after the quantum of energy is confiscated from the electron. The more it is off-shell, the smaller the probability amplitude.
Thus, the electron propagator favors low-energy photons.
The quantum behavior is determined by the path integral. We cannot explain the quantum behavior from classical physics. It is the interference of various paths which generates the quantum behavior.
An infrared divergence in the low-frequency photons in bremsstrahlung
Since the electron propagator favors low-frequency photons, it overestimates their number from the process. If we sum over all low frequencies, we may get an infinite sum.
Amita Kuttner (2016) studies the reaction
e+ e- ----> μ+ μ- γ,
that is, a collision of a high-energy pair produces a muon pair and a photon.
Kuttner writes that the infrared divergence in bremsstrahlung is canceled by another infrared divergence in the vertex correction of the process
e+ e- ----> μ+ μ-.
The claim is strange. How can a correction to another process cancel the infinite energy of the photons in the first process?
The linked page claims that the infrared divergence in bremsstrahlung occurs in classical physics, too. The page suggests the same recipe as Amita Kuttner: let the infrared divergence in the vertex correction of elastic scattering cancel the infinite energy emitted in soft photons.
We need to investigate this. Is there really a divergence in classical bremsstrahlung?
Our photograph model would remove the quantum infrared divergence in this way: since the classical wave, which is produced by the passing electron, contains finite energy in very long wavelengths (actually, very little energy), then the quantum "photograph" of the process will contain at most that energy in very long wavelengths.
We need to find out the classical spectrum. Is there enough energy in very long wavelengths, so that every passing electron will send soft photons? Is there a classical infrared divergence? Is the classical calculation in Wikipedia correct?
The classical process in nonlinear. Usually it is impossible in mathematics to prove anything about nonlinear differential equations. We need to use heuristic arguments to show that the emitted energy in the classical process is finite.
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