This is continuation to our post on March 18, 2021.
First mystery: why a simple photon propagator captures the complex classical Coulomb scattering?
There is a mystery in QED: why does the very simple photon propagator calculate correctly classical Coulomb scattering in 3 spatial dimensions? QED does not do it correctly if the dimension differs from 3. The classical process is quite complex. Why does such a simple propagator work? It is as if the propagator somehow would be equivalent to the equally simple Coulomb law of force in 3 dimensions.
Second mystery: why the simple vertex correction diagram captures the complex classical mass reduction?
Another mystery is why the simple vertex correction diagram of QED correctly computes the effect of reduced mass, after the Feynman integral is regularized and renormalized. This mystery may be easier to solve than the first one.
virtual photon
energy E, momentum q
~~~~~~
momentum k / \
e- ------------------------------------------
| virtual photon
| energy 0,
| momentum p
Z+ -------------------------------------------
Let us try to solve the classical problem through some kind of wave formalism. In wave formalism, particles are not restricted to their paths of newtonian mechanics. They can zigzag and take strange routes. It is like using the Huygens principle where any point in space can act as a new source of waves. The interference of waves eventually determines where the particle will end up.
A classical lagrangian and path integral resemble a Feynman diagram
Wave formalism is probably equivalent to a lagrangian method of calculation and a path integral.
● "field of the electron"
|
| spring
|
e- ●-------->
Z+ ●
Consider the following classical lagrangian problem. The electron is connected to a smallish extra mass through a spring. The extra mass plays the role of the mass of the field of the electron.
We are free to choose the path of the electron and the extra mass as we like, but energy and momentum must be conserved. Note that if we tune the path of the extra mass, we can steal momentum and energy from the electron.
We are studying a classical lagrangian. The interaction between the electron and the extra mass forms a loop in the classical path integral. The electron may give a packet of energy and momentum to the extra mass through the spring, and later take it back.
Is it possible that we get a divergence in the classical path integral? If we allow any energy and momentum to circulate in the loop, that very much resembles a divergent Feynman diagram. How to eliminate divergences in the classical case?
Is there a limit on how much momentum q can the extra mass exchange with the electron? Momentum and energy must be conserved. That puts some limits on E and q. We might approximate the process by assuming a "collision" which transfers some energy and momentum to the extra mass, and another "collision" later. A vertex in a Feynman diagram depicts such a collision.
In a Feynman diagram, the energy of a particle can be negative. In a classical system that is banned. How to reconcile these? If a classical particle moves back in time, then its energy is negative. Or maybe we can allow negative energies in a classical path integral, and that does not affect the final result? We are looking for extreme values of action. Allowing negative energies might not change those extremes? In tunneling there is negative kinetic energy. We have to allow negative energies. The importance of tunneling probably is minor in a typical scattering experiment.
The potential V in the spring complicates things. Let us define that the spring is part of the extra mass. Then the extra mass can flexibly receive various amounts of energy regardless of its momentum. The extra energy is stored in deformation of the spring.
Note that the electron does not satisfy the energy momentum relation
E² = p² + m_e²
in the classical process if the extra mass is not rigidly fixed to the electron. The effective mass m of the electron is less than m_e. The electron is "off-shell" in the classical process. Also the extra mass, the "photon", is off-shell because it does not move at the speed of light. This is an example of how particles in classical physics can be off-shell, or "virtual".
The reduced mass of the electron and the Larmor formula: classical versus quantum
In the blog post March 2, 2021 we calculated that the anomalous magnetic moment of the electron can be explained if the lagging field of the electron in the zitterwebegung orbit reduces the mass of the electron by the mass of the static field half a wavelength, or π "radians" away.
In the March 15, 2021 post we calculated that the Larmor formula of electromagnetic power dissipation is explained by the mass of the field more than 3/4 radians away using its inertia to pump the maximum power out of the oscillation. Classical bremsstrahlung is Larmor radiation in the case where the acceleration is not periodic.
The Larmor radiation seems to be a "stronger" process than mass reduction. Classically, both phenomena are, of course, always present if an electron is in accelerating motion.
In quantum mechanics, bremsstrahlung creates large quanta only in a small number of fly-bys of the electron. Most fly-bys are elastic, or only radiate soft photons. Classically, the situation is very different. Every fly-by creates also high-frequency radiation.
What about mass reduction? It is always present classically. Is it always present in quantum mechanics? At least in zitterbewegung mass reduction is always present.
The Feynman diagram: the classical interpretation as impulses
electric field
of the electron
|
| impulse
● e- --->
|
|
● Z+
Let us think of the electron as a particle and its static electric field as a field.
When the electron approaches the nucleus, the nucleus pulls on the electron. The electron makes some kind of an impulse into its own static electric field. In the rubber plate model, the electron tugs the rubber plate as the electron accelerates toward the nucleus.
The impulse response in the electric (more precisely, electromagnetic) field can probably be, at least, approximated by a massless Klein-Gordon Green's function.
A Green's function is the response to a Dirac delta impulse. The delta impulse contains too much very high-frequency waves, in contrast to a realistic classical impulse. But if we do some kind of a cutoff of high frequencies, it might approximate the classical process.
The impulse response contains lots of off-shell waves. Typically the waves do not satisfy the energy-momentum relation
E² = p².
Classically, these are deformations which cannot escape to infinity as well-formed sine waves.
Bremsstrahlung is generated from the on-shell spectrum of the impulse response. Mass reduction is the result of the off-shell spectrum.
Let us think of the rubber plate model. When the electron accelerates, it stretches the rubber plate. When the electron decelerates, the rubber plate will exert various forces on the electron. The nearest parts of the plate will accelerate the electron, but very far parts are still gaining speed and their effect is to decelerate the electron.
If we assume that no waves can escape to infinity, then all the off-shell (and maybe also on-shell) waves have to be absorbed back by the electron. This is clearly the Feynman vertex correction diagram.
We now have a classical explanation for the virtual photon carrying E and q in the diagram. Various E and q combined (integrated) form the impulse response of the field when the electron is accelerated by the nucleus. A "virtual photon", with certain E and q, is a Fourier component of the true classical response.
We still need a classical explanation for the virtual electron and its propagator. We need to figure out a classical way to treat the electron as a wave.
How to model a classical electron as a wave - what is the classical limit of the Dirac equation?
Note the following: it makes sense to assume that all the inertial mass of the electron is in the particle itself. The mass of its static field is zero. The apparent reduced mass of the electron is a result of the forces which its far field exerts on the electron.
The assumption above would explain why the far field does not explicitly appear in the Feynman diagram. Only the impulse which the electron sends to its field is relevant.
k
e- --->
k
e- --->
k
e- --->
Let us have a random flux of classical electrons whose momentum is k. If they are free, then we can describe the classical process as a plane wave. The wavelength is determined by the Planck constant, but the wavelength is not relevant in this simple process of linear motion.
Classical scattering of electrons from the field of a nucleus looks like bending of waves in an optically dense material. It is not too big a surprise that the Schrödinger equation (slow electrons), or the Dirac equation (fast electrons), or a Feynman diagram calculates classical scattering correctly, at least in cases where tunneling is unimportant.
A classical wave representation for the electron must, of course, conserve the number of particles and keep the particle number >= 0 everywhere. It should be Lorentz covariant. The Dirac equation satisfies these.
Question. Is the Dirac equation the simplest wave equation which has a positive definite conserved probability density and is Lorentz covariant?
The Dirac equation is
i / c h-bar γ^0 dψ / dt
+ i h-bar γ^μ dψ / dμ
- I m_e c ψ
= 0,
where we sum over the values μ = 1, 2, 3 (meaning the x, y, z spatial coordinates), and I is the 4 × 4 identity matrix.
What is the classical limit of the Dirac equation? If we let m_e to grow large, or h-bar to become very small, then we have to shorten the wavelength of the solution. The spin of the electron, 1/2 h-bar, becomes insignificant in the classical limit.
Question. Has anyone proved that the limit of the Dirac equation or the Schrödinger equation really is classical physics?
Answer. The Ehrenfest theorem, to some extent, shows that the limit of the Schrödinger equation is classical mechanics:
Mario Bacelar Valente (2012) claims that QED is "upgraded" from classical physics:
If we in the Schrödinger equation let the wavelength to become shorter, that will reduce tunneling - as expected if the limit of the Schrödinger equation is classical physics.
Let us think about waves of visible light. The wavelength only matters when the wave meets an obstacle whose feature size is of the order of the wavelength or less. For example, if we have an acromatic lens, it bends all wavelengths in the same way if the wavelength is much smaller than the lens diameter.
In many phenomena, the "classical limit" of a very short wavelength is essentially the same as the phenomenon with substantially longer waves. This observation explains why electrons in many cases seem to follow classical paths.
What is an "impulse" to a classical electron wave representation?
We have no problems imagining in our mind what an impulse to a drum skin or to an electromagnetic field does and what the effect looks like.
But we are used to imagining the classical electron as a particle, not a wave. What would an impulse mean in this case?
Let us write a new blog post about this central problem.
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