Interaction of two moving photons at a distance :

Photons are quantum mechanical point particles in the standard model of particle physics.

Interactions of elementary particles are modeled by quantum field theory and can be calculated using feynman diagrams as a guide for the respective integrals. They always interact at a point, as all elementary particles, as far as our data show, are point particles.

There is no vertex for a photon-photon interaction, so the interactions have to occur in higher order diagram in the perturbative expansion.

Because the electromagnetic coupling is of order 1/137, the probability of such scatters is very much depressed, the photons' wavefunctions superpose but do not interact except very very seldom.

As a general rule, photons don't interact with each other, either at a distance or when they are right on top of each other. This is known as the superposition principle in electromagnetism, and it carries over to the quantum-mechanical photon-based view.

That said, photon-photon interaction can happen in two different settings:

• When the interaction is mediated by a medium which does not satisfy the superposition principle, i.e. in the regime of nonlinear optics.
• At extremely high intensities, above the so-called Schwinger limit, there are some quantum effects that introduce photon-photon interaction. This can happen with visible light, at intensities which we cannot yet reach (but for which there are light sources under development that should come online within the next decade or so), and it can also happen for high-photon-energy gamma rays (for which there is already some experimental evidence).

In both of these cases, the interaction typically happens when both photons are in the same location; non-local nonlinear optical processes are extremely rare (at best) but probably nonexistent.