This blog post returns to the problem of "bending" an electron beam under an electric field.
Suppose that we have a static nucleus and a uniform Dirac wave passes symmetrically around it on all sides.
------>
| | | |
| | | | ● Z+ nucleus
| | | |
e- electron wave
We can calculate the scattering using the simple Feynman diagram.
e- -------------------------
|
| photon with
| momentum p
Z+ -------------------------
The Fourier decomposition of the Coulomb electric potential 1 / r of Z+ consists of sine wave potentials for each p. The potential undulates in spatial dimensions and is constant in time.
potential for p component
----------------
e- ----->
----------------
----------------
In the diagram above, the lines mark the "crests" of the potential of the Fourier component
1/|p^2| sin(p • x)
of the Coulomb potential. In the formula, p is the momentum vector of the Fourier component and x is the spatial position vector.
The potential acts as a "diffraction grating" which scatters electrons up and down, giving them the momentum p up or down. The potential crests act as sources for the Dirac equation. These sources have constructive interference for electrons which have "absorbed" the momentum p up or down.
Note that the scattering is both up and down. That is ok because the electron wave arrives symmetric relative to Z+.
---->
| | | ● Z+ nucleus
| | |
| | |
e- electron wave
If we want to model the scattering for an electron wave which is not symmetric around Z+, then the Feynman diagram method does not work. It predicts scattering both up and down, but the real scattering is just up.
We can calculate the correct scattering pattern with the Schrödinger equation, or even with classical physics.
The Schrödinger calculation can be viewed as a non-perturbative calculation where the passing electron absorbs very many photons from the various Fourier components of the 1 / r potential. The sum of all these photons guides the electron to the correct path up.
The Feynman diagram method is a vastly simplified model where just one photon is exchanged between Z+ and e-. We could call it a minimal quanta method. If the method gives correct answers, it is, of course, much simpler to calculate than the Schrödinger method.
A big question is: why does the Feynman diagram work at all? Why does it work for a symmetric flyby? Is it a coincidence? We have not yet found an answer for that question.
An electron flies past a group of opposite charges
Suppose that we have a bound state of opposite charges. It could be a hydrogen atom, or just some ions attached together somehow. We have the charges static in space. An electron flies by.
The "minimal quanta" method of Feynman diagrams would assume that an electron normally absorbs just one photon, from one of the static charges.
● • Z+ e- hydrogen atom
e ----->
The one photon method would give a lot of scattering up and down. But we know that the atom appears electrically neutral to the outside world. There is no scattering.
The experiment has to be calculated using the Schrödinger method. We may imagine that the electron receives many photons from both of the opposite charges. The effects of these photons cancel each other out.
Feynman diagrams do not handle bound states. We have presented one example of that.
We could introduce a tentative rule:
Feynman diagrams are not the correct way to calculate a flyby of charges when the charges do not come "very close". In some (symmetric) cases, Feynman diagrams do work, though. Feynman diagrams do not work for bound states of opposite charges.
vacuum
polarization
loop
Z+ ~~~~~~~O~~~~~~~ e-
virtual
photon
A vacuum polarization loop very much looks like a bound state of opposite charges. Furthermore, the loop is usually "far away" from the electron flying by. There is no reason why a Feynman diagram would model the setup correctly. The divergence of the vacuum polarization integral is a symptom of an incorrect method.
Feynman diagrams do not work if we know something about the relative positions of the particles
The two examples of Feynman diagrams failing have one thing in common:
The particles in the experiment area are not at random positions. We know something about their relative positions.
In the first example we knew that the electrons pass just on one side of the nucleus.
If there is a bound state, then we know the relative positions of its component particles.
Feynman diagrams assume that free particles collide or fly by at random places in the experiment area. In that case, the single photon formula of Feynman diagrams produces (approximately) correct results.
If there is a collision of particles, say, two photons collide to produce a real pair, then we do know that the initial positions of the electron and positron are close to each other. Apparently, that is not a problem if the produced particles will be at random positions when they interact with other particles the next time.
A vacuum polarization loop contains two particles which are very close to each other for their whole lifetime. We cannot model the process as random collisions of free particles. That is why Feynman diagrams do not work for them.
The same problem as with vacuum polarization, exists for any diagram containing a loop. When the participants of the loop interact again, in most cases they are not at random positions relative to each other.
A correct way to handle bound states in a Feynman diagram might be to invoke the Schrödinger method at certain places. For example, if an electron interacts with a neutral hydrogen atom, we have to assume that the electron always interacts with both of the components of the atom. Or if we have a vacuum polarization loop, then a real electron always interacts with both the electron and the positron in the loop.
The Schrödinger equation calculates the electron wave function under all the potentials which are present. In a Feynman diagram, this is simplified to the one photon method in a special case.
It may be that the divergence in Feynman vacuum polarization is the result of applying the simplified method to a bound state, which is an error.
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