Correction to the quantum relation of photons in the Doppler effect based on a special Lorentz violation model

The possibility of the breaking of Lorentz symmetry has been discussed in many models of quantum gravity. In this paper we follow the Lorentz violation model in Ref. [14] (i.e., our previous work) to discuss the Doppler frequency shift of photons and the Compton scattering process between photons and electrons, pointing out that following the idea in Ref. [14] we have to modify the usual quantum relation of photons in the Doppler effect. But due to the current limited information and knowledge, we could not yet determine the specific expression for the correction coefficient in the modified quantum relation of photons. However, the phenomenon called spontaneous radiation in a cyclotron maser give us an opportunity to see what the expression for this correction coefficient might look like. Therefor, under some necessary constraints, we construct a very concise expression for this correction coefficient through the discussion of different cases. And then we use this expression to analyze the wavelength of radiation in the cyclotron maser, which tends to a limited value at v → c, rather than to 0 as predicted by the Lorentz model. And the inverse Compton scattering phenomenon is also discussed and we find that there is a limit to the maximum energy that can be obtained by photons in the collision between extremely relativistic particles and low-energy photons, which conclusion is also very different from that obtained from the Lorentz model, in which the energy that can be obtained by the photon tends to be infinite as the velocity of particle is close to c. This paper still follows the purpose in Ref. [14] that the energy and momentum of particles (i.e., any particles, including photons) cannot be infinite, otherwise it will make some physical scenarios invalid. When the parameter Q characterizing the degree of deviation from the Lorentz model is equal to 0, all the results and conclusions in this paper will return to the case as in the Lorentz model, so this paper also provides us with a possible experimental scheme to determine the value of Q in Ref. [14], although it still requires extremely high experimental energy.