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This is a thought experiment. Consider two photons traveling at right angles and intersecting at some point. As seen from Earth these photons will not interact with each other because their energy is too low for particle creation. Now consider a rocket traveling at high relativistic speed towards the intersection point. People on that rocket will "see" these photons extremely blue shifted. They would appear as gamma rays. Now from the rocket point of view there is a higher probability that particle creation would occur, because these photons appear so energetic. Any produced particles like electrons or positrons would be captured by the rocket and brought back to Earth as a proof. The experiment could be repeated to get good statistics. How many particles would that rocket bring to the Earth. None? Or maybe the amount predicted by observers on the rocket?

Disclaimer. I am not a physicist. Probably this "paradox" has a very simple resolution just like other relativity paradoxes. I probably made some wrong assumptions.

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Well, all the cross sections in quantum field theory are calculated as relativistic scalars, so that they don't depend on the reference frame.

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Well, that was easy. I guess i have to do some reading and learn how this scalar is actually calculated. –  Eiver Sep 5 '13 at 13:15
    
If you are interested, just go through some decent book on special relativity. The complications introduced by quantum mechanics and then quantum field theory are not relevant to this question. –  Enucatl Sep 5 '13 at 13:24
    
@Eiver Alternatively you can note that if the proposed reaction can not satisfy both energy and momentum conservation in one frame then it can not satisfy them in any frame. This is required by the relativity principle and enforced by the transformation properties of the energy and momentum. Writing the limit in terms of explicit scalars is a worthwhile exercise, because it makes this fact crystal clear. –  dmckee Sep 5 '13 at 22:20

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