Why do two beams of light pass through one another without interacting? Why is it that if I have two torches, each shining rays in perpendicular directions at perpendicular screens, such that the beams cross, then the images on the respective screens are independent of whether the alternate torch is switched on? In other words, why does light seem to pass through itself without interacting?
Apologies if this question is basic, and apologies if there is a similar question somewhere on the site but I haven't managed to find an answer. 
Thanks, 
A.
 A: Beams of light as we observe them are described classically by Maxwell's equations which have been validated over and over again. Optics is a very well studied subject. In this classical framework light is an electromagnetic wave that  moves in vacuum  with a constant velocity c in all frames and does not need a medium to propagate, in contrast to sound waves or water waves. Electromagnetic waves have been studied  to interfere with each other in the two slit experiment. The energy is distributed in an interference pattern.
In the above framework it makes no sense to have two beams interact in the sense of scattering off each other.
In the quantum mechanical frame, the underlying frame not only of light but of all nature, the electromagnetic wave is composed out of zillions of photons, and photons are elementary particles . There exists photon photon scattering but the probability of this to happen is very low. Thus  two light beams have no measurable interactions when crossing.
A: In other words, why does light seem to pass through itself without interacting?
Because it consists of waves. Anna mentioned photons, but see Wikipedia where you can read about E=hf or E=hc/λ, wherein f is frequency and λ is wavelength. Photons have a wave nature. Imagine minor ocean waves, intersecting. They ride over each other and keep on going. They did interact, but not in any permanent way. If however they were both high-energy waves they might ride up each other and turn into "rollers". The Breit-Wheeler process is something like this:
"However, in 1997, researchers at the Stanford Linear Accelerator Centre were able to conduct the so-called multi-photon Breit–Wheeler process using electrons to first create high-energy photons, which then underwent multiple collisions to produce electrons and positrons, all within same chamber."
See this report from Imperial, which relates to what CuriousOne was saying.  
