The idea is very much reminiscent of the corresponding Feynman diagrams:
e- ------>
|
/ \
\ / virtual e+ e-
|
e+ ------->
e- ------->
| _____ > e+
| /
\ _____ > e-
|
e+ ------->
The vertical line in the diagrams represents the (complicated) electromagnetic interaction between the electron and the positron. The line symbolizes a "quantum" of the interaction process. Feynman calls it a virtual photon. Let us call it in our representation a pseudoquantum to underline the fact the energy is probably not quantized as in a real photon of a definite frequency f, and the pseudoquantum itself symbolizes a complex process while the photon is the quantum of a simple planar wave.
Conversion of a pseudoquantum to a pair
The classical collision process of an electron and a positron, and most of the disturbance to the electromagnetic field, happens in a spacetime patch which is of the order 10^-15 m in diameter and 10^-23 s in duration, or less.
We already pointed out that if the collision energy would be in a single photon, its wavelength would be of the order 1,000 times longer than the collision area size. The collision happens in an almost pointlike area from the point of view of wave representation of particles.
What is the probability that our pseudoquantum tries to convert itself into a pair? Typically, the probability of tunneling out from a "bound state" grows linearly with the duration of the state. But the collision is not a bound state. Maybe the process is more like a particle bumping into a potential barrier? Then the pseudoquantum will try tunneling once. Feynman diagrams suggest that tunneling may be tried several times (= the number of virtual pair loops), but the probability will decrease exponentially if the coupling constant is < 1.
In the previous blog post, we introduced the model of an inelastic collision where a single particle runs up a hill within a groove. If the particle has a lot of kinetic energy, it will run up the hill so high that the neighboring groove has a lower potential. Tunneling into the neighbor groove corresponds to production of a real pair.
- - - - - - - - - - - - - - - - - - - - - -
e- e+ exist in this groove
- - - - - - - - - - - - - - - - - - - - - -
__________________________
/\ /\
/ \ / \------> o particle
___\/______________________
If the particle does not have enough energy to form a real pair, it will still try tunneling. We may interpret that the particle does enter the barrier but tunnels back to its original groove.
The Feynman diagram with several virtual loops would correspond to the particle bumping several times to the wall of its groove.
In the Feynman formulas, each vertex carries a factor of the coupling constant, square root of α (where alpha is the fine structure constant). Since a virtual pair loop contains two vertices, α measures the probability that the particle will bump once into the wall during the entire collision process. It will do two bumps at the probability α^2, and so on. The value of alpha is roughly 1/137. Several bumps are very unlikely.
A possibility we have been ignoring so far is that the collision contains enough energy to produce a real pair, but the produced pair meets and annihilates. The virtual pair loop in the first diagram above might actually be a real pair.
Classically, the produced electron and positron tend to fly to opposite directions. It is more likely that they will annihilate other particles than their original sibling.
In the previous blog post, we introduced the model of an inelastic collision where a single particle runs up a hill within a groove. If the particle has a lot of kinetic energy, it will run up the hill so high that the neighboring groove has a lower potential. Tunneling into the neighbor groove corresponds to production of a real pair.
- - - - - - - - - - - - - - - - - - - - - -
e- e+ exist in this groove
- - - - - - - - - - - - - - - - - - - - - -
__________________________
/\ /\
/ \ / \------> o particle
___\/______________________
If the particle does not have enough energy to form a real pair, it will still try tunneling. We may interpret that the particle does enter the barrier but tunnels back to its original groove.
The Feynman diagram with several virtual loops would correspond to the particle bumping several times to the wall of its groove.
In the Feynman formulas, each vertex carries a factor of the coupling constant, square root of α (where alpha is the fine structure constant). Since a virtual pair loop contains two vertices, α measures the probability that the particle will bump once into the wall during the entire collision process. It will do two bumps at the probability α^2, and so on. The value of alpha is roughly 1/137. Several bumps are very unlikely.
What if a real pair annihilates again?
A possibility we have been ignoring so far is that the collision contains enough energy to produce a real pair, but the produced pair meets and annihilates. The virtual pair loop in the first diagram above might actually be a real pair.
Classically, the produced electron and positron tend to fly to opposite directions. It is more likely that they will annihilate other particles than their original sibling.
We need to decipher the Feynman propagator of vacuum polarization
Feynman noticed in his 1949 papers that the Fourier decomposition of the electron propagator function K(2, 1) is simple. K(2, 1) denotes the probability amplitude of the electron starting from spacetime point x_1 and ending up at x_2.
The propagator is taken to be a function of x_2, we set x_1 = 0. The propagator function is then decomposed into Fourier modes.
For each Fourier mode, determined by the four-momentum p of the mode, the weight of the mode is
i p-slash + m
------ • ------------------
4π^2 p^2 - m^2
where m is the rest mass of the electron. The above formula is undefined if p^2 - m^2 = 0, that is, if the electron is on the mass shell, which means it is a real electron and not virtual. We choose a suitable limit procedure (Feynman contour integration) to make integrals over p well-defined.
The Fourier mode for four-momentum p is
exp(-i p • x_2).
Four-momentum in Wikipedia is defined as a vector
(E / c, p_x, p_y, p_z),
where E is the total energy of a relativistic particle, including its rest mass, and the p_i are the relativistic momenta to each direction. The "momentum" to the time direction is defined as E / c.
Is there anything in the Feynman propagator which would cause the probability amplitude of an off-shell electron to decrease exponentially? Their lifetime is short.
(E / c, p_x, p_y, p_z),
where E is the total energy of a relativistic particle, including its rest mass, and the p_i are the relativistic momenta to each direction. The "momentum" to the time direction is defined as E / c.
Is there anything in the Feynman propagator which would cause the probability amplitude of an off-shell electron to decrease exponentially? Their lifetime is short.
As we mentioned earlier, a collision happens in a very small space compared to the Compton wavelength of the colliding electron. Maybe particles spend such a short time in the off-shell state, that we can ignore the exponential decrease of their wave function?
The diverging Feynman integral of the vacuum polarization loop is a convolution over the whole spacetime. The integral seems to include very long journeys by off-shell particles. Does Feynman take into account the short lifetime of off-shell particles?
The Feynman model looks like a simple billiard ball interaction model where conservation of energy and momentum prunes off most histories. Why does such a simple model predict some phenomena to 12 decimal places?
In the internal loops, the billiard ball model fails. Integrals diverge. The model is way too simple to handle loops correctly. Why did physicists in the 1950s resort to the black magic of renormalization rather than make the physical model more realistic?
Does Feynman take into account opposite momenta of virtual pairs?
If an electron shoots off a spacetime point with a three-momentum p and a positron from the same point with a momentum -p, then obviously it will be unlikely that they will ever meet if their energy is large enough to overcome the electric pull.
How does Feynman deal with this in his virtual pair loop integral?
Both the electron and the positron seem to start from a single spacetime point. The wave function of the electron is essentially the sum of all possible four-momenta, such that there is a constructive interference at that spacetime point. The same for the positron. One of the particles absorbs the virtual photon with four-momentum q. That is handled by adding q to p in each exp(i p • x) component of that particle.
What is the probability amplitude that they will meet at a future spacetime point? Let us represent the propagators in the Fourier-decomposed form. We need to calculate the integral of the product of the propagators over the whole future spacetime. The integral is dominated by the products of type
exp(i p • x) * exp(i * -p • x),
because the above product is equal to 1 and the integral of it over the whole spacetime infinite, whereas products where the values of p are not opposite numbers will have a phase which rotates and keeps the integral "small".
The integral is dominated by a term which looks like the particles having opposite four-momenta. If p = (E / c, p_x, p_y, p_z), then the other particle has a negative energy and opposite values of p_i. But that means they move in the same direction! No wonder they have a high probability of meeting.
That does not seem right. If a pair is born, both the electron and the positron have positive energies and an electric field will move them to opposite directions, not to the same direction. The model looks hopelessly flawed.
But why does a cutoff or a renormalization procedure heal the approximation method so that it produces accurate results?
Both the electron and the positron seem to start from a single spacetime point. The wave function of the electron is essentially the sum of all possible four-momenta, such that there is a constructive interference at that spacetime point. The same for the positron. One of the particles absorbs the virtual photon with four-momentum q. That is handled by adding q to p in each exp(i p • x) component of that particle.
What is the probability amplitude that they will meet at a future spacetime point? Let us represent the propagators in the Fourier-decomposed form. We need to calculate the integral of the product of the propagators over the whole future spacetime. The integral is dominated by the products of type
exp(i p • x) * exp(i * -p • x),
because the above product is equal to 1 and the integral of it over the whole spacetime infinite, whereas products where the values of p are not opposite numbers will have a phase which rotates and keeps the integral "small".
The integral is dominated by a term which looks like the particles having opposite four-momenta. If p = (E / c, p_x, p_y, p_z), then the other particle has a negative energy and opposite values of p_i. But that means they move in the same direction! No wonder they have a high probability of meeting.
That does not seem right. If a pair is born, both the electron and the positron have positive energies and an electric field will move them to opposite directions, not to the same direction. The model looks hopelessly flawed.
But why does a cutoff or a renormalization procedure heal the approximation method so that it produces accurate results?
The contribution of a Feynman virtual loop
If we put a largish cutoff to |p|, then the Feynman formula for the virtual loop describes a process where an electron and a positron are created in a smallish area C. One of them absorbs a small additional four-momentum q. Then we calculate the probability amplitude for them to meet at some future spacetime point.
The path integral contribution of the virtual photon is replaced by a product which contains contributions from the flight of the photon and the virtual pair contribution.
In our classical model, the pair is born in a small area of size 10^-15 m or less. Maybe that is the smallish area C?
We had a tunneling model where the pair tries to tunnel into a real pair but fails.
Why would the simple Feynman formula calculate the tunneling contribution correctly with our cutoff?
We still do not know the shape of the barrier which the pair in the tunneling model tries to tunnel through.
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