Probability of photon to photon collision

Question

2 photons having sufficient energy can collide and form an electron positron pair (which then annihilate and form a new photon pair - with lower energy?). I assume this means that they can't collide (just pass through each other?) if not energetic enough? As you see I have many small subquestions. My main question, however, is: What is the probability of two photons colliding in a transparent vacuum container? I suspect it would be near zero when talking about sunlight but what about having two high energy laser beams (originating from the same laser) meet head on at a common focal point inside the container?

Answer 1

Photons are elementary particles. The probability of two photons to interact can be calculated using Feynman diagrams, here is the lowest order diagram for photon photon scattering :

A Feynman diagram (box diagram) for photon–photon scattering, one photon scatters from the transient vacuum charge fluctuations of the other

Each vertex contributes a factor of (1/137)^1/2, the electromagnetic coupling constant, and that has to be squared so the probability of scattering is very small and negligible for two light beams meeting. (1/137)^2

The crossection goes up with energy and at gamma ray energies there are proposals for gamma gamma colliders.

The interference patterns seen with light does not mean that the photons composing it are interacting. It is just the superposition of the quantum mechanical wave functions that build up the classical wave, and the patterns appear also with one photon at a time.

You state:

One more point: Even if a photon is represented by some probability cloud the "nearest neighbor" probability of two such clouds goes to zero proportional to the square of the distance so the number of photons per unit volume must be extremely high in order for two clouds to be within "touching" distance - assuming this distance is in the micrometer range or less.

Yes, distance at minimum approach is also a factor on when these two photon interactions can have measurable consequences and should be taken into account in designing collider experiments..

Answer 2

The amplitudes of such processes can be easily calculated using Field Theory. The Feynman diagram directly gives the M matrix whose amplitude square gives the probability density of the process. Also in the entire process the 4 momentum is conserved.