We believe that we have corrected a fundamental error in the interpretation of regularization / renormalization of the vacuum polarization diagram in QED.
How a sign error gave rise to conceptual problems
The origin of the error was a wrong sign the Feynman integral which describes vacuum polarization. People believed that vacuum polarization weakens the Coulomb force, while the truth is that it strengthens the force.
The sign error was "corrected" by calculating the complement of the true physically relevant integral. This creates a further problem: the complement is formally infinite. If we believe that the complement physically "exists", how do we handle the conceptual problem that the physical entity is infinite?
The further problem was solved by two methods:
- regularization, which makes the infinite integral formally finite, and
- renormalization, which explains why the observed charge of the electron is finite while its "bare charge" is infinite deep down. The bare charge is screened by infinite polarization of the vacuum.
The machinery of regularization / renormalization can be viewed as a purely formal tool which does not imply or prove that any new physics is hiding from us at the Planck scale. However, people were tempted to think that there is a physical, real cutoff at the Planck scale, which would make the infinite integral physically finite.
Kenneth Wilson developed a conceptual framework where hidden physics at very short distances gives rise to the physics which we measure at distances of, say, 1 meter. The analogy is water, where complicated physics at the scale of molecules eventually explains the Navier-Stokes fluid behavior of water at the scale of 1 meter.
But there is a serious problem in the water analogy of the vacuum. If we do experiments with water, information from our experiments leaks into the hidden small-scale world of molecules. Molecules have to "exist" physically to carry information. They must contain mass-energy.
In the case of vacuum polarization, the role of water molecules is held by the negative energy electrons in the Dirac sea. If we do scattering experiments with electrons and protons, then information about the experiments leaks into the hidden world of the Dirac sea. Thus, the Dirac sea has to "exist" physically and must contain mass-energy.
Vacuum mass-energy brings two serious problems:
- How do we describe the information which leaks into the hidden world? What is the state of the vacuum?
- What is the mass-energy of the vacuum? Simple calculations give infinite energy per cubic meter. Why space does not collapse into a black hole?
Quite a mess was caused by a simple sign error in the vacuum polarization integral.
Consequences of correcting the interpretation of vacuum polarization: string theory and loop quantum gravity
Conceptual problems in regularization / renormalization have been a motivator in the development of string models ("string theory") and loop quantum gravity. This motivator is removed once we accept the correct interpretation of regularization.
However, there is another motivator for string models: trying to derive the masses and other properties of elementary particles. Are the masses really arbitrary or is there a hidden explanation for them?
Loop quantum gravity is a step toward discrete mathematical models of physics. Maybe the world at a very small scale is digital?
Renormalizability of theories
In our interpretation there is no renormalization. We do not need to "absorb" infinite quantities into the definition of parameters of the theory. The "true" values of the parameters, for example, the electron charge, are the values which we measure at the scale of long distances.
However, it might happen for some theories that integrals still diverge even if we take into account destructive interference. We have to find out if there are important theories which behave that badly.
Logarithmic ultraviolet divergences are probably benign. One can make them converge with only a moderate amount of destructive interference.
Quantum gravity
Gravity is not "renormalizable". One cannot absorb all its divergences into parameters of the theory.
In our interpretation, there is no need to absorb divergences. The non-renormalizability of gravity is a moot point.
However, a problem is that we do not know the behavior of gravity at very high energies. Do particles become black holes? If yes, how can we treat black holes in Feynman diagrams?
A simple solution is to cut off extremely high energies. Then gravity behaves well, or does it?
Gravity affects the geometry of spacetime. That brings problems for quantum field theory, which is traditionally done in the flat Minkowski metric.
There are many problems in formulating a quantum field theory of gravity. At low energies, we probably are fine with machinery similar to QED. But we are interested in phenomena in black holes, and there energies are potentially infinite.
The hierarchy problem and supersymmetry
One of the problems is why the Higgs boson mass is so tiny compared to the Planck mass. People think that the Higgs boson has a "bare mass" at the scale of very short distances. That bare mass has to be corrected with a self-energy calculation. If we set a bare mass at the Planck scale, the correction and the bare mass have to be incredibly fine-tuned to output the measured tiny mass of the Higgs boson.
We in our blog believe that it is the long-distance behavior of the Higgs boson and its couplings which gives rise to its short-distance behavior. Does that remove the need for fine-tuning?
This hierarchy problem is one of the motivators for supersymmetry.
The mass-energy and inertia of the static electric field of the electron is zero?
The electron self-energy involves the mystery of the electron "inner field". The mass-energy of the inner electric field of a point particle should classically be infinite. Why is the mass of the electron finite? We in this blog have not yet solved this mystery. The Higgs boson hierarchy problem is probably analogous to the electron inner field problem. We have to study this more.
virtual photon
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/ \
e- -------------------------------------
Our current view on the electron self-energy Feynman diagram is that the virtual photon cannot carry mass-energy, since that would break conservation of the speed of the center of mass. However, the virtual photon might carry spatial momentum.
We could ban the self-energy diagram altogether for a free electron. We can claim that destructive interference wipes out virtual photons of all frequencies. Virtual photons can then only exist if the electron is interacting with some other particle. The inertial mass of the electron would be a random parameter, and there would be no correction to it.
So far we have assumed that in a swift motion, the inertial mass of an electron is reduced since its far field does not have time to react. But it might be that the far field helps the movement of the electron in some of such situations. Then we do not have the problem of what happens in swift movements whose length is smaller than 1/2 of the classical radius of the electron. With the old model, the inertial mass of the electron itself would become negative, because the mass-energy of the field outside 1/2 of the classical radius is > 511 keV. That would be very ugly.
Our explanation of the Lamb shift relied on reduction of the inertial mass of the electron. Our various models of a rotating tight rope in the spring of 2021 explored the possibility that the rope becomes tighter and helps the end of the robe stay in a circular orbit.
We have been contemplating the possibility that the mass-energy and inertia of the static electric field of the electron is zero. That would solve the classical renormalization problem: why the mass of the electron is finite while the energy of the field of a point charge is infinite. It would also solve the classical 4/3 problem of the Poynting vector.
The goal: show that we can consistently set the mass-energy of the field zero, but still explain the Lamb shift and how a radio transmitter works. Our "rubber plate" model would then have the rubber plate having zero mass.
If this succeeds for the electron, it might succeed for the Higgs boson, too, and remove the hierarchy problem.
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