If you think of a photon as a particle, it’s easy to imagine that it could hit something. If the photon hits an object, that object will recoil. Which means that the photon has imparted momentum to the object. Which means that the photon must have had some momentum to begin with.
Also, if you imagine a particle-antiparticle pair annihilating to produce photons, the total momentum of the pair has to go somewhere. If the photons couldn’t carry momentum, the annihilation would be impossible to observe in any reference frame other than the center of mass. It would be even stranger if a particle interaction was impossible just because you were moving
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