I could preface this question with three pages of posts, but I'll just put it out there and see what shakes lose. Put two flashlights (torches, lasers, whatever) butt to butt. Then send out two light beams in opposite directions. How fast, realitive to each other, are those light beams traveling? If you picked one photon from Beam West and one photon Beam East and determined their relative velocity, what velocity would that be?
Undefined. You can't transform to a photon at rest, for any reason including to see how fast the other is going relative to it, because photons always move at the speed of light.
Yeah, there is no inertial frame of reference in which one can be at rest relative to a photon -- kind of a downer answer really as it was Einstein who posed the question about what would it be like to travel with a beam of light. Of course, if it weren't for the Higgs field, all particles would travel at c and we couldn't have this discussion.
I thought exactly the same thing at first, but on the other hand there is a powerful uniformity to the idea that c is an invariant.
I think you were quite correct and that c is an invariant. I've heard people state that time does not pass for a photon but it's kind of a meaningless statement as anthropomorphic experience and descriptions probably aren't applicable to such contexts. It's fascinating that the Higgs field allows the passage of time to be experienced by beings such as ourselves who are assemblages of hadrons and leptons. Discussion of what happened when before electroweak symmetry was broken seems like a misconception to me -- possibly nothing more than mathematical extrapolation. As for the origin of the arrow of time, I try not to think too much about it as it seems to lead to self-referential logic and inconsistency (see Loschmidt's paradox). After all, it led to tragic consequences for Boltzmann himself. Re the original question, a particle such as a pi-0 decaying to two gamma-ray photons would be a natural equivalent of the two butt-to-butt torches. The quantum state of the gammas would be entangled, of course, and that's a whole other kettle of weird fish that I probably can't get my head around.