Thus, no renormalization is needed if the scattering amplitudes are calculated in the correct way.
Our analysis of of the QED lagrangian brought this question up again.
Scattering of two electrons with low energy
Let us consider the scattering of two electrons which possess much less than 1.022 MeV of kinetic energy. Let us assume the bounce is symmetric.
^ ^
\ /
\ /
|~~~| virtual photon
/ \
/ \
e- e-
There is electric repulsion between the two electrons which makes them to bounce off each other. The virtual photon marks the repulsion. The virtual photon is not a particle in any way.
An analogue for the repulsion is a spring:
e- ---|\/\/\/\/\/\/\/\/|--- e-
The electrons push each other with a rod containing a spring.
What is analogous to the virtual electron-positron loop in the Feynman diagram for the virtual photon?
~~~~O~~~~
loop
An analogue is that the spring can give way not just in the spring /\/\/\ part but also in the straight rod parts ---.
If we want to calculate the bounce very precisely, we need to take into account all other degrees of freedom where the kinetic energy of electrons can be stored temporarily, not just the electric field.
In the case of the spring, the straight rods could store a little energy. It is similar for a virtual electron-positron loop: it can make the repulsive potential between the two electrons a little less steep. The loop will store some energy for the time when the electrons pass by. The loop will return the energy back to the kinetic energy of the electrons when they start to recede.
The virtual loop, which is also called vacuum polarization, might create a temporary charge distribution like this:
- e- ++ e- -
The positive charge density between the electrons makes the repulsion a little weaker in the bounce.
But the running coupling constant makes the electron repulsion stronger at short distances. What can cause that?
We remarked in our earlier post about a radio transmitter that if an EM wave is created by a disturbance of the EM field, and the wave is guaranteed to get absorbed soon again by another disturbance, then the wave may store a lot of momentum relative to the energy. A free plane wave in an EM field stores less momentum per energy.
The same is probably true for a short-lived wave in the electron field. We would need a correct QED lagrangian to analyze this in detail. The temporary field is born by interaction from the rapidly changing electric field between the bouncing electrons. We should show from the correct QED lagrangian that the electron field, indeed, is disturbed by the rapidly changing electric field, and stores some energy and momentum for a short time.
Let us assume that a temporary wave in the electron field is able store a little energy and considerable momentum. The temporary field does not need the 1.022 MeV of energy which would be needed for a real pair.
Is our analysis of the waves fully classical? Where does quantum mechanics enter the picture?
Note that our qualitative analysis did not refer to quanta anywhere except that the colliding electrons in the pictures were assumed to be particles.
A more precise analysis would assume that the electrons are waves obeying the Dirac equation when moving free of interactions.
The waves are smooth. The fuzziness of the waves in space it at least of the order of the Compton wavelength of the electrons.
If we solve the waves fully classically, where does quantum mechanics enter the picture? Maybe only at the measuring device. It will measure particles whose probability distribution can be derived from the classical wave solution.
What is the divergence in the Feynman loop diagram?
The well-known problem in the Feynman diagram formulas is the divergence of the calculation of the loop contribution. If we integrate over all possible momenta carried by the loop, the integral diverges.
How does that divergence show up in our classical analysis?
It does not because when two classical waves of a wavelength λ meet, we do not need to consider detail much smaller than λ in the reaction of the electron field. We can use a cutoff at λ, and intuitively know that finer detail has very little effect.
Using large momenta in the electron field would involve fine spatial detail in the electron field.
A Feynman diagram calculates all possible paths of the bouncing electrons. Electrons are point particles in the Feynman diagram and can come to a very close distance from each other. Very fine detail in the reaction of the electron field to that close encounter does have an effect on the Feynman calculation formula. If we try to calculate an intermediate result after the loop, the result may well diverge.
However, that intermediate result is not what we measure from the experiment. It makes no sense to calculate such.
If we only calculate the end results of the experiment, and if our intuition that fine detail has a vanishing effect is right, then the end results will not diverge.
If we are right, the divergence in Feynman diagrams is just an artifact from a wrong integration order. We must not calculate the diverging intermediate result.
The general problem of divergences in partial differential equations
A Millennium problem is to prove the existence and the smoothness of solutions for the Navier-Stokes equation. The problem is in turbulence. Does its infinitely fine detail have a large effect on the solution?
The QED classical wave equations might have a similar problem. We need to prove that no turbulence-like phenomenon can appear. If we cannot prove the existence and smoothness, then the divergence of Feynman intermediate results is a symptom of a real mathematical problem.
If we can prove the smoothness and existence of QED wave solutions, then the divergence is just an artifact.
We have in this blog post shown the connection between the existence of smooth solutions for a classical partial differential equation and the need to renormalize a Feynman diagram calculation.
To recapitulate:
1. If the classical equation has smooth solutions, then renormalization is really not needed. The apparent need for renormalization is a result of trying to calculate nonsensical intermediate results.
2. If the classical equation does not have smooth solutions, then it is not a proper physical model. We need to modify it, for example, by introducing a cutoff, which is equivalent to renormalization with a cutoff.
We need to look at the literature about the existence of smooth solutions for various partial differential equations. Coupled field equations are probably nonlinear in most cases. Little is known about the existence of solutions for nonlinear equations.
Linearized general relativity is not renormalizable. How does that show up when we try to solve the classical Einstein equations?
Conclusions
Renormalization may be unnecessary in QED.
The need for renormalization, or new physics at the Landau pole scale or the Planck scale, is connected to the existence of solutions for the corresponding classical field equations.
If smooth solutions exist, no renormalization nor any new physics is required. In such a case, the concept of an effective field theory is unnecessary.
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