The problem seems to be the extra degree of freedom which the virtual pair loop introduces to the reaction.
The Dirac equation describes strictly one particle and conserves charge
Paul Dirac tried the massive Klein-Gordon equation to describe the electron. A problem was that the equation is about the second derivatives. To specify initial values, we would need to give the initial function and its first derivative. There is too much freedom in the equation.
Defining a probability density function for one particle does not succeed, nor defining the probability current.
Charge conservation requires that the electron is strictly one particle which travels around. It cannot be a fuzzy ensemble like a classical electromagnetic wave.
Dirac took a "square root" of the massive Klein-Gordon equation, and was able to define a Lorentz covariant equation where there is a probability density and a conserved current.
Pair production spoils the strictness of the Dirac equation
The Schrödinger equation is a "deterministic" equation in the sense that the number of particles does not change, and we can use path integrals or other methods to develop the wave function forward in time in a deterministic way.
Creating new pairs can spoil this determinism if the initial conditions do not dictate all the parameters of the new particles. Creating a virtual pair in photon-photon scattering is an example: besides an arbitrary direction of the spatial momentum vector p, also the length of |p| can be set arbitrarily large.
The space of directions in R^3 is finite and we can define a constant probability density function on it.
But if we allow |p| to be arbitrary, then we face the problem of defining a constant probability density function on the real line R. That is not possible.
Photon-photon scattering
q p k
~~~~~ -------------- ~~~~~
| | p - k
| | virtual
| | electron
~~~~~ -------------- ~~~~~
-q -p -k
Photon-photon scattering is a prime example of too much freedom. One can choose the 4-momentum |p| as large as one wants, and is only punished by the product of the four propagators, which is something like
~ 1 / |p|^4
for large |p|. The space of 4-momenta is R^4, which makes the Feynman integral to diverge.
The way forward may be to describe the virtual electron line as some kind of scattering from a 1 / r^2 potential. That way get a handle on what is happening at the deep down level, and can make the behavior more "deterministic".
The photon-photon scattering Feynman integral converges if we assume a maximum lifetime for a virtual electron which "borrowed" energy
If the euclidean length |p| is large in the diagram, then the virtual electron has to "borrow" a lot of energy and its lifetime t is at most
t ~ h / (4 π|p|).
The maximum "lifelength" of the virtual electron is
~ t c.
Why? Because if the virtual electron would move a large spatial distance x, then in another (moving) inertial frame the electron would appear to live a long time (as a tachyon).
If |p| is large, then the virtual electrons move almost at the light speed (or even faster). The emission of the photons at the vertices has to be very quick, because of the great speed of the electrons, which means that the spatial separation of the created photons is at most the "lifelength" apart.
This means that the cross section seen from the left, from the incoming photon side, is at most
~ 1 / |p_0|^2
for |p| > |p_0|.
The Feynman integral for the loop does converge if we use this additional constraint. The contribution of very large |p| is negligible because they require the photons to pass very close to each other.
Note that slow electrons can emit photons in such a way that the photons are "created" very far away from each other. A simple example is a slowly rotating dipole. If the speed of the electrons in the dipole is, say, 1 / 100 c, then the emitted photons are "created" at a separation which is 200 times the dipole length. This is seen by calculating the emitted angular momentum J.
What is the fundamental reason why we can get the Feynman integral to converge?
In the diagram, large |p| represent an infinite number of "channels" which can take energy from the photons on the left, and transmit the energy to the right. Our argumentation shows that those channels have to compete for a very small flux of energy which is provided to the channels when the photons pass very close to each other. Even if the channels can transmit all the energy in that flux, the effect is negligible.
Without our restriction, the integral thinks that it can tap the energy of photons which pass very far from each other, and use that energy to create a virtual electron with huge 4-momentum |p|. That sounds counter-intuitive.
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