UPDATE: See our post on October 13, 2018 on the Huygens principle. It gives a natural cutoff scale for all virtual loops in Feynman diagrams.
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Let us summarize our findings about regularization from the past three days. We believe we have solved the mystery of regularization/renormalization.
In Feynman diagrams, a virtual pair is born at a single point in spacetime. It has to climb up from an infinitely deep potential well of Coulomb attraction.
e+ e-
\ /
t \ /
^ \ /
| \ /
| \ /
| \/
|
--------------------------> x
The electron and the positron move at almost the speed of light within the light cone.
If one calculates the number of cycles which their wave function does during the climb, it is infinite. That is bad, because the point of the light cone is then a singularity, and singularities, like dividing with zero, tend to produce undefined or infinite results in calculations.
The obvious solution to the problem is to allow faster-than-light paths and forbid paths that go to the singularity.
e+ e-
\ /
\______/
The horizontal part in the path can be interpreted as faster-than-light travel. Equivalently, the particles are born with a small spatial distance. They do not start from the same point.
When the electron is in the superluminal phase of the path, there is nothing in its past light cone. It does not see the positron and does not feel any Coulomb force. It does not descend deeper in the potential well because there is no force. That explains why a pair can annihilate producing just 1.022 MeV of electromagnetic energy. This solves the mystery of the classical electron radius - why is the energy of the electric field of the electron just 511 kB rather than infinite as suggested by the 1 / r Coulomb potential.
All regularization methods cut off the singularity at the sharp point of the light cone. For example, imposing a cutoff on the momentum |p| makes the birthplace of each particle fuzzy. They are no longer born at the same point. The singularity is removed.
The Pauli-Villars regularization, as well as several others, involve fictitious particles or ghosts whose contribution is subtracted from the diverging Feynman integral. No one was able to give a physical explanation why such particles are around and why their contribution should be subtracted. The real physical explanation is that we have to cut off the rogue singularity.
An important thing is that we have to give up the notion that causality only works forward in time. In the light cone, the wider upper levels are the physically well-behaving ones. We need to cut off their past - the point of the light cone - to make the physical configuration well-behaved.
e- -------->
^
| e+ virtual
| e- pair
v
Z
Virtual pairs are born in collisions of particles, for instance. Our classical model showed that, to reduce the attraction, the virtual positron and electron have to move past the colliding particles. That is one of the reasons we are allowed to cut off the sharp point of the light cone. Pairs which for their whole lifetime remain in a very small spatial area, do not appreciably affect the electric attraction in the collision.
The Landau pole lurks at the distance of 10^-292 m in the light cone. We have to cut it completely off because it would involve rogue behavior like a collapse of the vacuum to a lower energy state.
Also a black hole lurks at the bottom of the light cone. The Coulomb potential is
E = -k e^2 / r.
The radius of a black hole of energy E is
r = 2G E / c^4.
We get
c^4 r / 2G = k e^2 / r
r^2 = k e^2 * 2G / c^4
r = 2 * 10^-36 m.
The associated energy is
E = k e^2 / r
= k e / r eV
= 10^27 eV.
The black hole dominates at energies of the order 10^27 eV, long before we get to a Landau pole. If we try to shoot an electron at a positron to make a Landau pole, a black hole will form rather than a Landau pole. That is fortunate because a black hole cannot cause a collapse of the vacuum.
The energy 10^27 eV is 1/10 of the Planck energy. Why?
E = sqrt(k e^2 * 2G / c^4) * c^4 / 2G
= sqrt(k / 2G) * e * c^2
E_P = sqrt(h c^5 / (2π G)).
Let us remove common factors:
sqrt(k) * e
sqrt(h c / π)
Is there a reason why these numbers are relatively close in the logarithmic scale? The upper number is about electricity, the lower about relativity and quantum mechanics. The ratio of those numbers is sqrt(α / 2). We rediscovered the fine structure constant.
We need to think about paths that venture at a distance of 2 * 10^-36 m. Time will freeze, but how does that affect the superluminal phase of the electron path?
Still one observation of the electron-positron annihilation. The electron gets reflected back in time. The attractive potential of the positron is a time mirror in timespace. Reflection of waves is caused by nonlinearity of the wave equation in a potential. An ordinary mirror only reflects waves back in space. It is probably the acceleration between the electron and the positron which makes the positron a time mirror. This may be relevant for the (non-existent?) Unruh radiation from accelerating mirrors.
If we have nonlinearity in the wave equation, its effects can be reduced by making the wavelength of the wave shorter. This is the reason why a high-energy electron can approach the positron closer in annihilation. It can move deeper in the light cone before reflecting back in time.
http://meta-phys-thoughts.blogspot.com/2018/09/the-optical-theory-of-electron.html
The optical theory of electron scattering stated that the vicinity of another charge is "optically dense" for the electron wave. The time mirror concept adds a further point of view: the field of a positron can scatter the electron wave back in time.
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