In pair production and annihilation, the initial state and the final state have the same dimension in the degrees of freedom. The pair side has the orientation of the momentum p, and the photon side has the orientation of the momentum q. The freedom is two-dimensional.
q p k
~~~~~ ---------- ~~~~~~~~~~ photon
| | virtual
| | electron
~~~~~ ---------- ~~~~~~~~~~ photon
-q -p -k
In the diagram, two photons that come from opposite directions are scattered by a virtual pair loop.
The sum of the spatial momenta
q - q = p - p = k - k = 0
is zero at each stage. Energy conservation requires that |q|= |k|.
Feynman rules would allow to add any 4-momentum to circulate in the virtual pair loop, which causes the Feynman integral to diverge logarithmically.
But we will now use the classical particle model of the virtual pair. Then |p| can have an arbitrarily high value which is compensated by making the electron-positron distance very short. The classical energy of the pair is set equal to the energy of the photons.
Thus, the momentum p may have an arbitrary direction, and |p| may be arbitrary. The freedom is 3-dimensional.
What are the separate "channels" of energy transmission through the diagram? If we set |p| to a constant value C, is that channel separate from the channel C' != C? If yes, then there is complete transmission because we have an infinite number of channels which carry energy to the other side.
The different number of dimensions of freedom is a fundamental problem. It is clearly connected to the divergence problem of Feynman integrals.
In a Feynman integral, the measure of the momentum space is the ordinary measure of R^3. Why should the measure be such? That is another fundamental question. Even if we through some trick get the integral to converge using the ordinary measure, it would anyway diverge if we choose some different measure over R^3.
In a deterministic universe, p would be uniquely determined by the input. The degrees of freedom of p should have the same dimension as those of q.
The universe has to be probabilistic if p has more degrees of freedom. But then we face the problem what weights should we put on various values of p, i.e., what measure to use in the space of the possible values of p. Why would p' have the same weight as p, if |p'| = 10^9 |p|?
There is a conceptual problem. This is not just about finding some trick, like regularization, which makes the Feynman integral to converge.
Restrict the cross section with the angular momentum?
If the combined energy of the photons is E << 1.022 MeV then the virtual pair has the maximum possible separation
r_0 / 2 = 1.4 * 10^-15 m,
where r_0 is the classical electron radius. Let us assume that the particles in the pair move at the speed of light.
The photons have the same angular momentum if they pass each other at the same distance r_0 / 2.
The maximum possible cross section would be 0.06 barn = 6 * 10^-30 m^2.
M. Bregant et al. (2008) report that the cross section for 532 nm photons scattering from each other is less than 3 * 10^-60 m^2. That would mean the photons passing each other at a distance < 10^-30 m. M. Bregant et al. say that the cross section calculated from Feynman diagrams is 10^-67 m^2.
It looks like angular momentum is not the way to restrict the cross section enough.
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