A central problem is what is an impulse (= source term in the inhomogeneous equation), and impulse response, in the Dirac equation. How do we interpret the impulse classically, when we think of the electron as a classical particle?
● field of the electron
|
| "spring"
|
e- ● --->
Z+ ●
Our example is classical Coulomb scattering of the electron from a nucleus. We believe that the vertex correction is a classical effect which is caused by the static electric field of the electron. But we have to prove that.
In quantum field theory, particles are created by hitting a field equation with a Dirac delta impulse. In Coulomb scattering, we do not really create any new electrons. But could it be that the Born approximation can be modeled as creating a small flux of new, scattered electrons? The basic idea of the Born approximation is that we treat the main solution of a field equation as constant, and calculate small scattered fluxes from small corrections to the equation.
Then classically, an impulse might be a force which affects the electron as it flies past the nucleus.
We think that the electron gives an impulse to its own electric field during the fly-by. Is there an opposite impulse to the electron? If yes, that would explain the Feynman diagram where the electron emits an off-shell photon, and becomes off-shell itself.
Note that if we model an electron as a free particle, but the electron is constantly under a force, then the electron wave does not satisfy the free Dirac equation, and we might say that the electron is off-shell. Suppose that the effect of the electric field of the electron is to reduce its mass. Then the electron will appear off-shell as long as its mass is reduced. Classically, the electron does not satisfy the energy-momentum relation
E² = p² + m²,
if its mass is reduced.
The role of a wave phase shift
If we calculate in the wave model, then reflection of waves and similar phenomena cause a phase shift. If we add such shifted wave to a wave in the original phase, there may be destructive interference.
significantly scattered flux
/
/
/
------- ●
---------------------------------- little scattered
---------------------------------- flux
If the Born approximation produces waves which are shifted by 180 degrees and those waves cancel some flux of significant scattering, then we assume that in the precisely modeled process, the canceled flux appears in the less scattered flux. The total flux has to be conserved in the precise model.
Let us check the phase of the vertex correction diagram. In the link (2015) we have the Feynman rules.
The simplest Coulomb scattering diagram has just one photon propagator line and two vertices. The phase factor in the Feynman formula is
-i * -i * -i = i.
The vertex correction has two photon propagator lines, two electron propagator lines and four vertices. The phase factor is
-i * -i * i * i * -i * -i * -i * -i = 1.
The vertex correction is shifted by 90 degrees from the simplest scattering diagram.
Why does the Born approximation work for the vertex correction?
Suppose that significant scattering only happens if the electron comes within a small minimum distance r of the nucleus. The cross section is small for large deflection angles.
Suppose then that there is some small disturbance which depends on r and, say, increases the minimum distance r by a small value dr for every electron. It is fundamentally a non-perturbative process because the entire flux of electrons shifts its course.
Let us try to use the Born approximation to calculate the effect of the disturbance as a perturbation. We have to assume:
1. Most electrons still go along their original path - the main wave stays as it is.
2. A small number of electrons change their path (4-momentum) substantially - we add a small "perturbation wave" (= wave "scattered" from the disturbance).
______
______/
_____/
-------------------------------->
0 r
Suppose that the original distribution of the electrons is such that very few have a small value of r. If we shift the distribution to the right by dr, that is almost equivalent to picking, say, 1/137 of the electrons and shifting them to the right by 137 dr. A small perturbation wave can be used to get the right spatial distribution with respect to r.
The general idea of the Born approximation. If there is a small disturbance, and we can approximate its effect in the final calculated result by imagining a large effect on a small number of electrons, and the rest of the electrons going their original route, then we can use the Born approximation.
Feynman diagrams use the Born approximation. They only work correctly if the thing we are trying to calculate can be approximated by keeping the original wave intact and adding a small perturbation wave. In earlier blog posts we remarked that Feynman diagrams do not model in the right way the bending of an electron beam in a constant electric field. The entire original wave is deflected to a constant angle. One cannot obtain such a result by keeping the original wave intact and adding a small perturbation wave.
The average 4-momentum of the total electron flux is sensible, but the 4-momentum change is grossly exaggerated for the perturbation wave.
The perturbation wave is "fictitious" from the point of view of momenta: there is nothing like the perturbation wave in the precise solution. If we try to solve a differential equation by a stepwise approximation method, then the Born approximation is only a single step. In some cases, using a single step does work.
Richard Feynman in his 1949 papers does not mention the problems of the Born approximation at all. Feynman right away assumes that using a small perturbation wave works. Feynman believed that the virtual particles, that is, the perturbation waves, "exist" in some sense. Our analysis above suggests that they are almost pure fiction.
Why does an electron perturbation wave absorb the photon sent originally by itself? Why not the photon of some other perturbation wave?
Let us again use the rubber plate model of the static electron electric field. The impulse makes a dent in the rubber plate. We can calculate the subsequent behavior of the rubber plate using Green's functions.
The Fourier decomposition of the impulse response is what the Feynman formula uses in calculation.
But classically, the electron does not care about the Fourier decomposition. It sees the classical wave.
Why does the Feynman formula work? Richard Feynman himself wrote that we must calculate all the ways that a "photon can propagate". What does that mean classically?
Feynman diagrams must conserve energy and momentum at the vertices. What is the associated classical phenomenon?
photon wave W-ph, E and q
\ | | /
\ /
\ /
e- | | | | | | | | | | | | | ||||||||||||||
electron wave W-e
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| momentum
| transfer p
Z+ | | | | | | | | | | |
nucleus wave
Let us use the wave representation of the classical electron and the classical nucleus. That is, we look at the path integral with a wavelength determined by the Planck constant (or any constant, as long as the constant is small enough).
Classically, every electron sends a certain photon wave. But let us use the Born approximation and assume that only a small number of electrons send a photon wave W-ph with certain E and q.
There is a small electron perturbation wave W-e with the energy E and momentum q subtracted from those of the incoming electron wave. When W-e passes close to the nucleus Z+, it absorbs momentum p.
Why does W-e absorb later its counterpart W-ph? Why not absorb some other photon wave?
The reason probably is the way that the Born approximation is built: we imagine a small perturbation wave which has a much exaggerated deflection in the 4-momentum. To cancel these exaggerated effects, we have to return back the original E and p which we borrowed from it to create the photon.
The mystery of allowing the electron to send ANY virtual photon in the vertex correction
E, q virtual photon
~~~~~~~
k / \
e- -----------------------------------------
|
| p virtual photon
|
Z+ -----------------------------------------
Suppose that we have calculated the scattering differential cross section using the simplest Feynman diagram. We should estimate the effect of the interaction of the electron with its own electric field. There is probably some effective mass reduction to the electron.
In the Born approximation we let most electrons follow their ordinary course. A small number of electrons get a large kick from emission of a large electromagnetic wave.
We know that classically, the electron which approaches the nucleus will make a rather smooth impulse to its own field. The smoothness of the impulse depends on p. A large value of p means a sharp turn and a sharp impulse.
The Feynman formula somehow gets the right result (after regularization and renormalization) by letting the electron make a Dirac delta impulse to its own field. The delta impulse makes a wave which contains all pairs E, q. How can it work without using p as an input?
We will return to this question in a subsequent blog post.
The anomalous magnetic moment of the electron
virtual photon E, q
~~~~~~~~
/ \
e- -------------------------------------
| virtual
| photon
| p
|
B magnetic field
(Matthew Schwartz, 2012)
The most famous result of the vertex correction (vertex function) is the calculation of the electron anomalous magnetic moment by Julian Schwinger in 1948.
In that calculation, the momentum p in the diagram above is allowed to go to zero. It is clear that the disturbance caused by the very weak p cannot be the reason for the correction. Our own calculation on March 2, 2021 suggested that the correction comes from the electron charge movement in zitterbewegung. The very rapid circling of the electron causes the rather substantial reduction in the effective mass and a larger magnetic moment.
Thus, the electron movement in the above diagram really is the light-speed zitterbewegung movement. The diagram is about coupling the field B to the spin of the electron. It is not the movement of the electron in the laboratory frame.
David Hestenes (2020) has a new paper about zitterbewegung. We have to check that.
There exists another vertex correction diagram for the electron movement in the laboratory frame. The mass reduction should be very small if the electron does a large circle at a speed slower than light.
We will write a new blog post about the anomalous magnetic moment. It opens new perspectives into zitterbewegung.
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