We calculated that a collision of an electron and a positron happens in an area of 10^-15 m when the center of mass collision energy is a very modest 511 keV.
The Heisenberg uncertainty principle states
σ_x σ_p ≥ h / 4π,
where σ denotes the standard deviation of the position x or momentum p.
If the standard deviation of the position is 10^-15 m, then the same for the momentum p is 5 * 10^-35 / 10^-15 = 5 * 10^-20 kg m /s.
We may assume p = m c. Then the standard deviation of m is roughly 1.7 * 10^-28 kg, which is 200 electron masses.
Thus, if we have a particle whose mass is 1 electron mass, and we want to represent it as a wave packet which is in an area 10^-15 m in diameter, then we must in the wave packet use momentums p which correspond to 200 electron masses or more. This is probably the origin of off-shell particles in Feynman diagrams. To describe something which happens in a very small area, we must have short-lived particles whose momentum is much bigger than their energy would allow if they were real, eternal particles.
But is it necessarily so that we need wave packets of size 10^-15 m? We do not know the position of the colliding positron and electron at that precision. To clarify this question, let us assume that the electron collides with a very heavy static nucleus Z. Then the uncertainty relation allows us to know the position of Z precisely. To describe the collision, we do need to use wave packets of size 10^-15 m.
The Bohr radius of the hydrogen atom is 5 *10^-11 m. That is 50,000 times bigger than a collision area of 10^-15 m. When calculating the hydrogen atom, we do not need high-momentum wave packets. But in collisions, we do need.
The Feynman virtual pair loop revisited
e- ------->
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/\ virtual e- e+
\/
|
Z
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/\ virtual e- e+
\/
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Z
The above numerical calculations cast light on the problem which we uncovered in the previous blog post: why does the Feynman formula for the virtual pair loop ignore the fact that a local electric field tends to pull the virtual positron and electron apart.
If we represent the virtual positron and the electron as a wave packet whose size is 10^-15 m, then the momentums in the packet far outweigh the momentum produced by the electric pull. If we want to calculate the probability of the pair meeting again to annihilate, it is enough to work with the high momenta p in the wave packets.
The Feynman model seems to assume that the virtual photon in the collision lives in an area of 10^-15 m. If it tries to convert itself to a pair, that too lives in that same area. And to calculate the probability amplitude of the pair meeting again, we can concentrate on the momenta of the wave packets and forget about electric fields.
The divergence in the Feynman formula is the result of assuming that the virtual positron and electron are created in the exact same point of spacetime. Then the wave packets will contain arbitrary momenta and will cause the Feynman integral to diverge.
To remove the divergence, we have to find good grounds to claim that the created electron and the positron should be described by wave packets which are not pointlike. Our 10^-15 m is a good candidate for the wave packet size.
https://en.m.wikipedia.org/wiki/Causal_perturbation_theory
Feynman tries to calculate the probability amplitude of the pair meeting by using distributions, that is, Dirac delta functions. Laurent Schwartz showed that the product of two distributions cannot be defined consistently. If two point-like particles meet, it is like a product of distributions.
Generally, if we have two pointlike particles, their probability of meeting is zero almost always, and one cannot meaningfully calculate probabilities. Why do tree-like Feynman diagrams then work? Maybe we can in a tree-like diagram approximate a pointlike particle with a wave packet of a small size, and the limit of probability amplitudes exists when we let the size go to zero? In a loop, the limit does not exist.
Feynman assumes that we can model the virtual photon as a point particle. In our previous blog posts we showed that classically, the electromagnetic field of colliding particles behaves in a complex way, and that there probably does not exist an undivisible "quantum" of interaction. Maybe it does not make sense to model the interaction as a virtual point particle? And if the interaction tries to create an electron-positron pair, it does not make sense to treat them as point particles either?
We had the model where a particle who lives in a 4-dimensional timespace tries to transform into a pair which lives in a 7-dimensional timespace. If the original particle is a wave packet, can we model the pair in a natural way as a wave packet?
Heisenberg uncertainty principle of energy and time
If we have an electron wave packet which carries little energy but up to 200 electron masses worth of three-momentum, what is the lifetime of such packet, according to the Heisenberg uncertainty principle? The energy-time uncertainty is
ΔE Δt ≥ h / 4π.
If we set ΔE to 200 electron masses, then Δt = 5 * 10^-35 / (10^8 * 1.6 * 10^-19) = 3 * 10^-24 s. If the electron moves at the speed of light, then it moves a distance of 10^-15 m in that time.
Thus, the virtual electron of the Feynman virtual loop should only be able to move within the collision area, which is of the order 10^-15 m. Does the Feynman propagator take this into account? We have repeated this question several times in this blog, but have not yet dug up a definite answer.
Our calculation also shows that if we are studying the behavior in the immediate collision spacetime patch, then we can in some cases ignore the fact that the probability amplitude of an off-shell particle will decrease exponentially with time. In our collision example, if the "borrowed energy" in an off-shell particle is much less than 200 electron masses, then the decrease of amplitude is negligible in the short duration of the collision, 10^-23 s.
Is there a cloud of virtual pairs in vacuum polarization?
e- ------>
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/\ /\ /\
\/ \/ \/
\ | /
Z
| | |
/\ /\ /\
\/ \/ \/
\ | /
Z
The Feynman model where the colliding particles exchange just one virtual photon is very rough. A more accurate model would have them exchange a large number of virtual particles.
If we have a whole swarm of virtual photons flying between the colliding particles, then each of these photons has roughly a 1/137 chance of producing a virtual pair. Then the virtual electrons and positrons from different pairs can come to annihilate each other. It is like a cloud of electrons and positrons all annihilating each other.
Since each additional photon involves a factor of 1/137 in the Feynman integral formula, the traditional way of calculating Feynman integrals ignores swarms of photons and their respective virtual loops. But is that the right way to calculate the process?
Feynman's original paper from 1949 explains his idea about how to model a flux of electrons flying in a constant electric field. He assumes that most electrons will fly unaffected by the field, but a few will absorb a virtual photon and change their course. That will turn the flux to the appropriate direction determined by the direction and the strength of the electric field. Why not let all electrons absorb a small virtual photon?
Our discussion in the previous blog posts suggested that there is really no "quantum" of electric attraction, like there is a quantum of energy in a plane electromagnetic wave. The attraction virtual photon can be divided in as many pieces as one likes.
The classical limit of electrons in a constant electric field has them flying along rather classical trajectories. It is not that just a few electrons are affected by the field, but all of them are. All electrons absorb several virtual photons.
To allow a cloud of virtual photons, we can change the Feynman rules in the way that we multiply each virtual photon line in a diagram to n copies without adding the 1/137 factor for each new line.
For example, a virtual photon carrying a four-momentum p can be split to two virtual photons that may carry, for example, p/2 each. A virtual photon contributes a factor 1/137 * exp(i p ∙ x) in the Feynman integral in the position space. We can as well put 1/137 * exp(i p/2 ∙ x) * exp(i p/2 ∙ x) instead.
Suppose that we have a cloud of virtual electrons and positrons. What is the probability amplitude for each electron meeting a positron to annihilate? The electron and the positron for a typical annihilating pair were born somewhere in the collision area, but not normally at the same point. This reminds us of calculating the annihilation probability amplitude for two wave packets of size 10^-15 m.
The idea of a cloud of virtual pairs has a serious flaw, though: even a single pair can screen the electric charge of the colliding electron. There is no room for many pairs.
If the pairs are very short-lived, then there is room for many virtual pairs. But can the pairs mix then in the annihilation?
The classical version of a collision does suggest that there should be many virtual pairs. A pair can form anywhere where the electric field is increasing. Why and how would nature restrict the number of virtual pairs to one?
Since each additional photon involves a factor of 1/137 in the Feynman integral formula, the traditional way of calculating Feynman integrals ignores swarms of photons and their respective virtual loops. But is that the right way to calculate the process?
Feynman's original paper from 1949 explains his idea about how to model a flux of electrons flying in a constant electric field. He assumes that most electrons will fly unaffected by the field, but a few will absorb a virtual photon and change their course. That will turn the flux to the appropriate direction determined by the direction and the strength of the electric field. Why not let all electrons absorb a small virtual photon?
Our discussion in the previous blog posts suggested that there is really no "quantum" of electric attraction, like there is a quantum of energy in a plane electromagnetic wave. The attraction virtual photon can be divided in as many pieces as one likes.
The classical limit of electrons in a constant electric field has them flying along rather classical trajectories. It is not that just a few electrons are affected by the field, but all of them are. All electrons absorb several virtual photons.
To allow a cloud of virtual photons, we can change the Feynman rules in the way that we multiply each virtual photon line in a diagram to n copies without adding the 1/137 factor for each new line.
For example, a virtual photon carrying a four-momentum p can be split to two virtual photons that may carry, for example, p/2 each. A virtual photon contributes a factor 1/137 * exp(i p ∙ x) in the Feynman integral in the position space. We can as well put 1/137 * exp(i p/2 ∙ x) * exp(i p/2 ∙ x) instead.
Suppose that we have a cloud of virtual electrons and positrons. What is the probability amplitude for each electron meeting a positron to annihilate? The electron and the positron for a typical annihilating pair were born somewhere in the collision area, but not normally at the same point. This reminds us of calculating the annihilation probability amplitude for two wave packets of size 10^-15 m.
The idea of a cloud of virtual pairs has a serious flaw, though: even a single pair can screen the electric charge of the colliding electron. There is no room for many pairs.
If the pairs are very short-lived, then there is room for many virtual pairs. But can the pairs mix then in the annihilation?
The classical version of a collision does suggest that there should be many virtual pairs. A pair can form anywhere where the electric field is increasing. Why and how would nature restrict the number of virtual pairs to one?
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