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"research has shown that the GZK cutoff limit results in the photon-pion production so the extra energy is carried off by the pion.",This is one explanation to GZK cutoff. Lorentz symmetry breaking is another explanation to GZK cutoff. In fact,to GZK cutoff,there lack experiments data.Experiments data is too little.So there don't exist accurate conclusions.So there exist muilti different understanding to GZK cutoff. My theory is only a mathematical model,a suppose,a idea. I think that my idea is very interesting.It's another new explanation to GZK cutoff.So I express it openly.

Thank you for your suggestion. English is not my native language.My english writing skills is very poor.My writing must rely on google translation. I just want to express my discovery,my thinking,my idea. I am not a career researchers.I am only a amateur on physics.Although I graduated from Physics Department,but my career as a programmer. My theory is a theoretical prediction only.In the current,there not exist supporting material,that does not matter.Perhaps will discover supporting material in future. To physics,propose theory at first,then discover supporting material.This is possible. Idea go ahead.

At first,we derive out energy-momentum equation,E=mc^2,from quantum mechanics. That is,Special Relativity Theory can be derived out from quantum mechanics. Special Relativity Theory is included in the quantum mechanics. Start from the new thinking,we derive out a new energy-momentum equation. In low energy region,the new energy-momentum equation can transition to E=mc^2. E=mc^2 is only approximate of new energy-momentum equation. The new energy-momentum equation can derive out GZK cut-off and Lorentz symmetry breaking. The new energy-momentum equation take another result.Energy have maximum value.Momentum have maximum value. From this thinking,the scattering problems in quantum filed theory maybe solved. If so,Renormalization is no longer needed. In the new energy-momentum equation,Lightspeed is no longer maximum speed value. If you have interesting to this,please download attach pdf file to read details. Welcome to discuss. My english writing skills is very poor.Please bear. My email is hidebrain@hotmail.com. Thanks. Please help me to distribute this paper to arxiv.org.My unique endorsement code is BFQMAO in arxiv.org. If you have this authority in arxiv.org,please recommend me in arxiv.org.If you can help,please click this url, https://arxiv.org/auth/endorse?x=BFQMAO Thank you very much. lixiaolin1.pdf