photon e+
~~~~~ --------- ~~~~~~ photon
| |
| |
~~~~~ --------- ~~~~~~ photon
photon e-
A concrete example: let us assume that two vertically polarized 0.500 MeV photons collide from opposite directions.
They cannot create a real pair, but can create an "almost real" pair, which classically can move to a separation 7 * 10^-14 m from each other if all the 1.000 MeV energy is spent to the potential energy of the pair.
Feynman rules allow the electron and the positron in the diagram to have arbitrary 4-momenta, as long as the sum of energies is 1.000 MeV and the sum of momenta is 0.
For example, the electron might have 1 GeV energy and the positron -999 MeV energy.
Or the electron would have 1 GeV / c momentum to the x direction, and the positron -1 GeV / c momentum. Here c is the speed of light.
The virtual pair as the mediator particle of a "force" between the photons; a Yukawa potential?
We may interpret the diagram above that two photons scatter from each other through some kind of a force.
The Feynman propagator of the electron resembles the propagator of the massive Klein-Gordon equation. The Fourier decomposition of the Yukawa potential, in turn, looks like that propagator.
The Yukawa potential describes a force which has a "finite" range, in contrast to the Coulomb force which is said to have an "infinite" range:
V_Yukawa (r) = -g^2 exp(-α m r) / r.
The Coulomb potential is recovered by setting the "mass of the mediating particle" m to zero. Alpha is a coefficient which should be set to a suitable value, so that the range of the force is as measured.
Calculations in literature claim that the cross section of photon-photon scattering is ~ 1 microbarn. We argued in a blog post that the cross section should be much larger, at least ~ 1 millibarn.
The force between photons is attractive. We can deduce that through the following reasoning. If we have two wave packets coming from opposite directions and they graze each other, then virtual pair production somehow disturbs the progress of the waves. Since waves cannot move faster than light, a disturbance probably slows them down. Then the wavelength is shorter in the grazing area, which causes the wave packets to turn toward each other.
If we assume that virtual electrons cannot travel very far, then the potential might be Yukawa.
A particle model based on classical particles
Let us assume that the electron and the positron in the virtual pair are very ordinary classical particles. It just happens that their mutual potential energy V is so much negative that the total energy of the pair is < 1.022 MeV.
If the particles in the just created pair have moderate momenta, they are at a distance whose order of magnitude is 1.4 * 10^-15 m.
If the just created particles have very high momenta, then in our classical model they must be very close to each other. The pair has very large negative potential energy V, which is compensated by very large positive kinetic energy.
Does creation of a pair which has very high momenta require that the colliding photons are very close to each other?
Let us assume that the answer is yes.
If the particles have a lot of kinetic energy they move at almost the speed of light. Then the spatial momentum p and the kinetic energy E of the particles roughly satisfies
|p| = E / c.
The cross section for creating a pair whose particles have very large spatial momentum > |p| is then ~ 1 / |p|^2.
What about creating a pair where one particle has very large negative energy and the other very large positive energy? In our classical model, one might assign the large negative potential V in an arbitrary way to the particles. Then one would have large negative energy and the other large positive energy. But such arbitrary assignment does not change the actual physical system in any way. The most natural assignment is such that the particles have equal energies.
How to conserve angular momentum?
Let us consider the angular momentum J produced by the momenta p and -p of the colliding photons, as well as their relative spatial separation r. If the collision is very close, then J is very small. When the pair annihilates, the new photons have to leave with a very small spatial separation.
But now there is a problem with high momenta pairs: classically, according to the Larmor formula, they radiate their energy away very quickly, because the acceleration is huge. How can we conserve J? Obviously, the created pair must possess the same J as the pair of photons. Since the electron and the positron move at a speed < c, the separation of the pair must be slightly larger than the separation of the photons was. In annihilation, how can they produce photons with the correct (small) separation?
In classical physics, conservation of angular momentum is not a problem because we can assume that the colliding laser beams possess no angular momentum, and the scattered beams have zero angular momentum, too.
Maybe we can solve the angular momentum problem in the following way: we calculate the sum of the waves produced by various possible collisions and spacetime points. The sum is essentially the classical picture. Then we require that the observed photons conserve angular momentum.
A position space analysis of the Feynman model
In our classical model, rogue pairs are no problem. They contribute little to the cross section. Why are they a problem in the Feynman integral? Why does the integral diverge? What is the difference of the classical model to the Feynman model?
photon
~~~~~ -------------------- e+ 4-momentum q
| virtual
| electron
~~~~~ -------------------- e- 4-momentum p
photon
Let us consider the above diagram where we allow the outgoing particles be virtual, that is, have any 4-momentum, even off-shell.
laser laser
t -----> <-----
^
| / \ / \
| / \ / \
| / \ / \
| / \ / \
-----------------------------------------> x
2 * 10^-14 m
oblique lines are the crests of the
waves of two colliding laser beams
Let us try to visualize the situation in position space where we have a cube whose side is, say, 4 * 10^-14 m, and two laser beams of energy 0.500 MeV collide. We may assume that both beams have one photon in our cube.
Let δ > 0 be a fixed small real number. In the Feynman diagram, for any 4-momentum p_0, we are allowed to assume the existence of a "standard flux" of virtual or real electrons whose 4-momentum p satisfies
|p - p_0| < δ,
where | | denotes the euclidean length of the 4-momentum vector. We may assume that the cube contains one such electron.
We should calculate the flux of electrons which scatter from both photons, and become virtual positrons.
The Feynman formula reduces the flux of scattered electrons with the electron propagator, which is ~ 1 / |p| for very large |p|.
What is the contribution of electrons whose
r < |p| < r + dr?
It is ~ r^3 dr / r, or
~ r^2 dr.
If we try to integrate over r, it diverges very badly.
What is the problem? The input of photons to the cube is a finite flux. How can they cause an infinite flux of scattered electrons?
The problem is the approximation which pretends that the flux of photons is not reduced at all by scattering. A single photon can cause an infinite number of electrons to scatter.
A "first order perturbation" approximation makes the assumption: the input flux is not affected by a small scattered flux. But in this case, the assumption is very much wrong. The scattered flux is not small - it is infinite.
The Feynman formula reduces the scattered flux by a coefficient 1 / |p|. Is that sensible?
We have argued that the "lifelength" of a virtual particle is ~ 1 / |p|. If that is the case, the virtual electron has to meet the other photon in a volume ~ 1 / |p|^3. That coefficient would cut off high momenta very efficiently. But that does not sound right. The coefficient 1 / |p|^3 corresponds to a meetup of three particles. We believe that the pair is created by a collision of two photons.
The classical model which we introduced in a section above suggests that the coefficient should be ~ 1 / |p|^2. It is like the propagator for the photon. The Yukawa potential model for photon interaction, which we introduced in a section above, suggests a similar coefficient.
The coefficient 1 / |p|^2 applies to the collision of the two photons. The virtual pair will meet and annihilate with a 100% probability if the combined energy of the photons is < 1.022 MeV. What is the behavior if the combined energy is > 1.022 MeV? If the created particles are very close, they classically radiate away all their energy very quickly, and must annihilate. It might be that the probability of annihilation is slowly reduced when we increase the energy to several MeV.
Destructive interference in the position space model
Let us again return to the difficult problem of destructive interference. What is the phase of the virtual particles which are produced in a photon collision? Should we calculate the effect of destructive interference on them?
In Feynman rules, destructive interference is ignored, except in the final result.
In the double-slit experiment, we certainly have to take into account the interference pattern.
If the phase of the created virtual particles depends on the phases of the colliding photons in some simple and natural way, then we expect that the phase of the created high-momentum virtual particles is almost constant in a spatial area whose size is
λ = h / |p|,
where h is the Planck constant and p is the spatial momentum of the virtual particle. There λ is the de Broglie wavelength of the virtual particle.
The phase is almost constant also during a time interval λ / c at a constant spatial location x.
The phase of the real photons changes quite slowly relative to our high-momentum virtual particles. Therefore, we expect the virtual particles in such a small area be born with a relatively constant phase.
If the lifelength of the virtual particles is just 0.1 λ, then we can ignore interference effects. Waves born far away never reach the spatial position x we are interested in.
Note that this is a general rule with oscillating physical systems: if we have a field F which disturbs another field G, and the disturbance does not "resonate" with G, then the disturbance in G moves a very short distance and is retained in G for a very short time. If the "wavelength" of the disturbance in G is short relative to waves in F, then we can ignore interference effects.
A virtual particle is a disturbance which does not "resonate" with the field.
We conclude that we can ignore interference effects if the virtual particles are a lot off-shell. If |p| is large, then they are a lot off-shell.
Why do we need to care about very small features at all, like high |p|? Classically, an oscillating system where wavelengths are long does not care much about small detail. The answer: quanta can be understood as particles, and particles care about small detail; for instance, about other particles.
We will look at this fundamental question of the wave-particle duality in the next blog post.
Conclusions
Our classical model suggests that high-4-momentum virtual pairs do contribute to scattering of photons, but their contribution is not large, something like ~ 1 / |p|^2.
The Feynman loop integral diverges for the following reasons:
1. To work, a first order perturbation approximation requires that the scattered flux is small compared to the input flux. In the case of a Feynman loop, the scattered flux is infinite, which makes the approximation nonsensical.
2. The electron propagator, which is ~ 1 / |p| for large momenta, overestimates the chance that photons come very close. The coefficient should rather be ~ 1 / |p|^2. The process is not about an electron bouncing from two photons, but about two photons colliding.
3. Feynman allows arbitrary energy E to circulate in the loop. The classical model suggests that energy must be divided evenly between the virtual positron and the virtual electron.
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