e- ------>
|
/\ virtual
\/ pair
|
Z
Because we can interpret that the virtual photon is reflected from the pair it itself created. The pair has rest mass. Reflection from a denser material causes a 180 degree phase shift.
The 180 degree phase shift means that there is a destructive interference relative to the virtual photon which traveled without reflection.
In classical physics, reflection from a denser material involves the same 180 degree phase shift.
Classically, how is the virtual pair able to delay the effect of the push or pull by 180 degrees? That is a huge delay. We need to think about this. The impulse has to stay absorbed into the pair for almost the duration of the collision. It is like a shock absorber. How to implement the shock absorber in classical physics?
We have written about the screening of charges by the pair. The virtual electron will move close to the nucleus and the positron close to the passing original electron. This can be interpreted as the pull impulse being absorbed by the pair. The virtual positron will fly past the original electron and will pull it upwards in the diagram. That is, the virtual pair will counteract the original pull between the nucleus and the electron.
-----------------> e+
^ |
| |
e- |--------> |
e+ v
^ annihilation
e- |
Z | |
| |
v |
-----------------> e-
Diagram 1. Classical behavior of a virtual pair.
The vertical force on the electron without the virtual pair would be like this:
____ ____
\ /
\_____/
The virtual pair adds an effect like this to the above graph:
____
/ \
_______ / \______
\ /
\/
First there is no effect. When the pair has formed, the approaching virtual positron pulls the original electron down. When the positron passes, the virtual electron has screened some of the pull of the nucleus. The positron starts to pull the electron upwards. The end result is that the virtual pair screened some of the pull between the nucleus and the original electron.
Since the vertical force caused by the positron is a symmetric pull down / pull up, we can ignore it from the lower graph above and we get this graph:
____
_________/ \_____
We can interpret the graph above as the pull impulse being reflected from the virtual pair and having a 180 degree phase shift relative to the original pull impulse.
In a collision, the main wave is a longitudinal one, either push or pull. Furthermore, there is just half a cycle of that wave. For example, an elastic rod first contracts and then stretches.
It is always half a cycle - that may explain why the probability of forming a virtual pair involves the factor 1 / 137 regardless of the type or energy of a collision. There is a fixed probability per cycle that a wave will try to tunnel into a virtual pair.
Virtual pair creation means that the virtual photon itself creates a mirror from which it is reflected.
If we have a free photon of a plane wave, why it does not create a mirror and get phase-shifted by 180 degrees? Because reflection from a mirror always involves also a change of the path and requires some momentum. Momentum conservation in a plane wave prohibits the reflection.
Does this have any bearing on the divergence of the vacuum polarization Feynman integral? The first observation is that the traditional Feynman diagram does not depict the process in an intuitive way. The virtual pair loop should be very big like in Diagram 1, and the nucleus and the electron should go through the loop. The loop is a harness which counteracts the pull between the electron and the nucleus.
Annihilation cross section
What about the annihilation probability of the virtual pair? The pair is created in a very small patch of timespace, and the wave packets of the virtual electron and the positron must contain big momenta to make the packet small enough.
A fundamental problem in the Feynman vacuum polarization integral is that it tries to calculate the probability of two pointlike particles meeting at a point in spacetime. The first thought is that the probability is zero. Any attempt to "renormalize" it to something else than zero is on shaky grounds.
But we know that real electrons and positrons meet and annihilate. How do we calculate the cross section for such a process? Why not use that cross section to calculate the probability that a virtual pair meets to annihilate?
We have now isolated two fundamental problems in the vacuum polarization loop:
1. The pair is born at a spacetime point: we should use Dirac delta wave packets with arbitrarily high momenta. We discussed this problem earlier as "breaking into more degrees of freedom."
Possible solution: let the high p waves decay exponentially with time (tunneling). A drawback is the loss of Lorentz invariance. Theorem 2 of our previous blog post states that Lorentz invariance is broken in original Feynman formulas, too. Our tunneling formulas make the breach worse.
2. The pair must meet at a spacetime point to annihilate. The probability of meeting at a point is zero. This is the time-reversed version of problem 1.
Possible solution: calculate the annihilation cross section for various momenta and trajectories, and use that as a criterion for annihilation.
Also tree-like Feynman diagrams involve problems 1 and 2. Real pairs are produced and real pairs meet to annihilate. We have to study how tree-like Feynman integrals deal with these problems.
Laurent Schwartz proved that the product of two distributions does not make sense. Problems 1 and 2 are connected to that.
The virtual photon lines in Feynman diagrams meet electrons/positrons at vertices. Since the photon symbolizes the interaction, we may take for granted that it meets the fermion at a spacetime point.
Feynman himself says in his 1949 papers that we are allowed to rotate the time/space coordinates by an arbitrary angle.
~~~~~~~~ ---------> e-
|
|
v e+
time ---->
The curly line is the arriving photon, the vertical line the leaving positron. Note that we drew the positron arrow to the opposite direction from Feynman. If we rotate the diagram slightly counterclockwise, it looks like pair production. A clockwise rotation makes it look like photon absorption by an electron.
Could rotation rid us of problem 1?
An electron-positron loop is a stationary state in spacetime?
Feynman thinks that the same electron travels first forward in time and then backward, returning to its original point in spacetime.
We have a problem in this: if we describe the electron with a wave, how can the wave return to the exact same form when the loop is completed?
An analogous phenomenon is the wave which describes the hydrogen atom. The spatial structure of the wave miraculously forms a loop where it returns to the exact same form after a rotation of space. The wave is in a stationary state.
When we describe the virtual pair as a wave, we could set the following axiom:
Axiom 1. A virtual pair must form a "stationary state" in the spacetime. The waveform must return to the exact same form after the loop is completed. An equation similar to the Schrödinger equation determines allowed stationary wave solutions.
The de Broglie wave is also in spacetime. We may now define the "Bohr atom model" of a virtual pair: the loop must contain an integer number of wavelengths.
Axiom 1 probably removes most of the need for high momentum p waves in the virtual loop and imposes a natural cutoff which will remove the divergence of the Feynman integral.
We need to think about the interaction of the virtual photon with the loop.
The loop is asymmetric: the wave function of the loops decays with the time in the laboratory frame. The asymmetry should be an indication of increasing entropy in the system.
e+ ___________
/ \
/ \
/ e- ----------------> \
\ /
\ Z /
e-_____________/
In the diagram, the hexagon represents the virtual pair loop.
The amplitude of the loop wave function grows smaller to the right. How can entropy increase in the setup? There certainly is a possibility of bremsstrahlung, a photon flying away. The electric force is in that sense nonconservative, and there cannot be time symmetry.
Could there be collisions where the particles arrive from the future? Then the wave function might decay to the past direction.
The connection between entropy and virtual pairs is an intriguing one.
The virtual pair loop imposes a condition on h, k e^2, c, m_e, where k is the Coulomb constant and m_e the electron rest mass. Could we use the condition to derive an exact mathematical value for the fine structure constant? That seems unlikely, but worth a thought.
The loop is asymmetric: the wave function of the loops decays with the time in the laboratory frame. The asymmetry should be an indication of increasing entropy in the system.
e+ ___________
/ \
/ \
/ e- ----------------> \
\ /
\ Z /
e-_____________/
In the diagram, the hexagon represents the virtual pair loop.
The amplitude of the loop wave function grows smaller to the right. How can entropy increase in the setup? There certainly is a possibility of bremsstrahlung, a photon flying away. The electric force is in that sense nonconservative, and there cannot be time symmetry.
Could there be collisions where the particles arrive from the future? Then the wave function might decay to the past direction.
The connection between entropy and virtual pairs is an intriguing one.
The virtual pair loop imposes a condition on h, k e^2, c, m_e, where k is the Coulomb constant and m_e the electron rest mass. Could we use the condition to derive an exact mathematical value for the fine structure constant? That seems unlikely, but worth a thought.
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