ΔE Δt >= h / (4π),
where h is the Planck constant.
Let us calculate how many wavelengths does a virtual photon with a borrowed energy E = hf live:
h f Δt = h / (4π)
<=>
Δt = 1 / (4π f)
It lives for a length
L = c / f * 1 / (4π)
The wavelength λ = c / f. We see that a virtual photon which has borrowed all its energy lives for a meager 0.1 wavelengths.
In vacuum polarization, the pair borrows all its energy and momentum from the vacuum. The lifetime of the pair is extremely short, and we may speculate that its spatial extension is equally short, typically just 0.1 wavelengths.
The graphical appearance of the Feynman vacuum polarization diagram looks suspicious:
virtual photon
Z+ ~~~~~~~O~~~~~~~~ e-
loop of
virtual
e- and e+
How can a virtual pair which resides midway between a nucleus and an electron affect the Coulomb force felt by the electron?
The weakening of the Coulomb force is supposed to happen because the loop "reflects" the virtual photon and causes a half a wavelength phase shift to the photon. The phase shift does destructive interference to the probability amplitude calculated without the loop:
virtual photon
Z+ ~~~~~~~~~~~~~~~ e-
In an earlier blog post we wrote about the the phase shift which a vacuum polarization loop causes to a real photon flying in empty space. We argued that the phase shift would break conservation of the speed of the center of mass.
Additionally, destructive interference in a free real photon would break conservation of energy.
The concept of a photon bouncing from an "object with zero energy and zero momentum" does not sound right. In classical physics, that kind of a process would break conservation laws.
We suggest the following principle:
A photon flying "freely" cannot be affected by vacuum polarization loop pairs.
However, if the vacuum polarization loop is very close to the electron e-, then it can affect the behavior of the system.
______
/ \
Z+ e- e- e+
\_______/
virtual
pair
In the diagram, we have an example of classical polarization. The nucleus Z+ attracts the virtual positron e+ and repels the virtual electron e-. The effect is that the electron in the middle feels less Coulomb force.
We noted in an earlier blog post that an interacting classical particle can break the energy-momentum relation and be "virtual" in that special sense. The loop in the diagram has strong interactions present. It would be no suprise that the vacuum polarization effect is felt by the electron in the middle, even though it is not felt when the loop is farther away.
The divergence of the Feynman vacuum polarization integral can be seen as a result from the claim that virtual pairs anywhere in space would contribute to weakening of the Coulomb interaction. Our new rule makes this more sensible: only the pairs which classically could weaken the force are counted.
e- ------------> 3 * 10^-15 m <------------ e-
Suppose that we have two relativistic electrons which collide and exchange about 500 keV in momentum. Classically, they will come to within 3 * 10^-15 meters of each other (= the classical radius).
The Compton wavelength of 500 keV is 2 * 10^-12 m.
A virtual pair which has borrowed 500 keV will "live" for 0.1 wavelengths, or 2 * 10^-13 m.
We see that we cannot exclude virtual pairs whose 4-momentum is < 33 MeV. They can take part in vacuum polarization.
For larger 4-momenta (E, q), we can put an attenuating factor
C = 33 MeV / (|E| + |q|)
into the Feynman integral for the loop. Without the attenuating factor, the integral would diverge logarithmically, when we integrate spherical shells E^2 + q^2 = r^2 over r.
The attenuating factor makes the integral to converge.
We have argued that one should implement a smooth cutoff at 67 * the exchanged momentum of the electron collision. We need to calculate what kind of running does that hypothesis cause in the coupling constant.
Is there destructive interference in the vacuum polarization loop in a Feynman diagram?
Suppose that a virtual photon with a spatial momentum p approaches our electron. Suppose that at each point in its flight, it can produce a virtual electron whose 4-momentum is k, and the initial phase of the virtual electron is determined by the photon.
The phase of the virtual electron might at the creation be equal to the phase of the photon.
If the virtual electron has a very big |k| and it could live for many wavelengths, there would be significant destructive interference. But if the lifetime of the virtual electron is just 0.1 wavelenghts, then we can ignore destructive interference. This solves the question we have been pondering for a long time about destructive interference.
A model of real pair production from two photons: a model how a photon acts as a source of the Dirac equation
We can introduce a new model which explains how real pairs are produced.
A photon, virtual or not, always and everywhere disturbs the Dirac field and acts as a source for a new Dirac wave.
The electromagnetic field as a source for the Dirac field. A non-zero electromagnetic field constantly at every point in spacetime produces new virtual electrons using the Green's function for the Dirac equation. The electrons inherit their phase from the photon of the electromagnetic field.
The Green's function is the "impulse" response of the Dirac field: we kind of hit the Dirac field with a sharp hammer and look how it vibrates.
However, since the system the photon & the virtual electron and the virtual positron is off-shell, the "vibration" of the Dirac field dies off very quickly, after about 0.1 wavelenghts.
On-shell electrons can be seen as "resonant vibrations" of the Dirac field. They live forever. Off-shell vibrations die off very quickly.
Feynman diagrams apparently treat photons as sources of the Dirac field, just as we described above.
Real pair production can happen when the photon is colliding to another photon and is very close to it. Then there are strong interactions and a part of an off-shell electron wave can escape as an on-shell wave. The model qualitatively explains the cross-section of pair production from two photons.
How exactly do the on-shell Dirac waves escape from the two colliding photons? We have no model of that. It is a strongly interacting system of the the electromagnetic and Dirac fields. It would be hard to calculate the exact behavior. It is like an oar making waves in water: close to the oar, water and the oar are interacting strongly. It is hard to calculate how water moves.
The converse process - annihilation of an electron and a positron - is equally hard to describe exactly. But we can use conservation of energy and momentum to determine which end results are possible. Dirac in 1930 was able to calculate the cross section using a semiclassical model.
The electromagnetic field as a source for the Dirac field
The lagrangian of QED describes how a classical electromagnetic field A and a non-zero Dirac field interact. Klein and Nishina in 1928 were able to calculate the behavior of an electron field under the classical electromagnetic field of a photon, and derived the formula for Thomson scattering.
But how does field A interact with a zero Dirac field? In pair production, A has to conjure up vibrations into the Dirac field, so that the real pair is produced. How does A do that?
Our hypothesis above says that A is constantly acting as a source for the Dirac field. Can we derive this from the lagrangian? We have to check if anyone has succeeded in that.
Production of a virtual pair can be viewed as "tunneling" of electromagnetic energy or momentum into the Dirac field. The lifetime of the pair is short. This brings up another question: in Schrödinger's equation, a tunneling particle has negative kinetic energy and an imaginary momentum. What effect does an imaginary momentum have on things?
The Dirac field as a source for the electromagnetic field
A classical electron has the Coulomb field around it. The Coulomb field can be modeled as "constant hammering" of the electric potential with a sharp hammer. That is, we can build the 1 / r potential by hitting the electromagnetic field with a Green's function at infinitesimal time intervals at the electron position.
t
^ hammer hits at short intervals
| :
| : # sharp hammer
| : #=========
| : #
| : v
| :
| :
---------------------------------------> x
e-
The time-independent components of the Green's function have constructive interference.
Time-dependent components would have complete destructive interference if their waves would reach to infinity. But the model we introduced above says that the waves only reach out some 0.1 wavelengths.
The time-independent components form the Fourier decomposition of the 1 / r potential.
Do the time-dependent components form the "dressed electron"? Is there any way to see the "dress"?
Anyway, here we have a model how a point particle can act as a source for the electromagnetic field A.
But how can the Dirac field act as a source?
If we have an electron with zero momentum somewhere in a large spatial volume, how do we create the Coulomb potential?
Maybe we should do just as we did in the previous section.
The Dirac field as a source for the electromagnetic field. A non-zero Dirac field constantly at every point in spacetime produces new virtual photons using the Green's function for the electromagnetic wave equation. The photons inherit their phase from the electron.
Feynman diagrams contain the above rule for the production of photons.
Created virtual pair as a "bound state"
We can view a virtual pair as a positronium atom which we have momentarily raised up from the zero energy, zero mometum state. Then it is a "bound state", and Feynman diagrams are known not to work for bound states.
If a virtual pair is created far away from a real electron, then in our model, it cannot pull or push on the real electron.
Why is that? Should we treat the electric field of the pair as the sum of the electron and the positron, and not separarely each field?
A hydrogen atom in the lowest energy state appears as electrically neutral to the outside world. At least in that case, we have to sum the electric field of the proton and the electron.
Or could it be that the short lifetime of the virtual pair prevents it from communicating with the electron far away?
Experimental measurements of the QED coupling constant at different energy scales
A quick Internet search reveals that the QED coupling constant α has been measured at the energy scale 0.6 GeV to 180 GeV. At 180 GeV, the coupling constant is some 7% stronger.
However, effects due to hadrons seem to dominate the change in α in the range from 0.6 GeV up. We cannot test our new hypothesis against that data.
We would need to know the constant at the 1 MeV scale, so that heavier particles do not affect vacuum polarization.
There is a claim that vacuum polarization in QED affects the Lamb shift by some 27 MHz. That contribution would come from low energies. We have to check that.
We can extend our new model to QCD and check if it affects vacuum polarization produced by hadrons.
Some back-of-the-envelope calculations suggest that α in our new model could differ by up to 1% from the value predicted by renormalization procedures, in the energy scale 10 MeV to 100 MeV. This is because in our model the cutoff depends on the energy scale of the collision experiment. In renormalization, the cutoff is often set at a constant large value Λ which does not depend on collision energy.
We need to check what our new model predicts about α at low energies, say, 1 eV to 1 keV. If we lower the cutoff to 30 eV, then the bare charge of the electron would become visible because the effect of vacuum polarization is very small?
The coupling constant α is experimentally a constant at hydrogen atom energies in the 10 eV scale. In our new model, should we put the cutoff at 30 eV or at 30 MeV in those phenomena?
What is the right way to calculate vacuum polarization effects?
We argued above that a Feynman diagram is an incorrect way to calculate the effect of vacuum polarization. Feynman diagrams do not work for bound states.
We were able to get the Feynman vacuum polarization integral to converge by removing those loops which cannot affect the scattering electron.
However, is there any reason why the corrected formula would model vacuum polarization right? The Feynman diagram might be a totally wrong model. Making the integral to converge would not make the formula any more correct.
We need to think about this. In an earlier blog post we introduced the principle that a photon field always acts as a source for the Dirac field. The coupling constant determines how efficient a source the photon field is.
If the photon field or the Coulomb field close to the electron produce virtual pairs, how do we calculate their effect on the electron?
The Feynman way of calculating the effect does look reasonable in the momentum space description of the process.
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