We consider, in this chapter, the violation of Lorentz symmetry. For this we make modifications in Compton scattering by amending the energy–momentum dispersion relation. This originates from a minimum spacetime cutoff à la in modern quantum gravity approaches. It maybe seen that two high-energy rays of different energies, for example, gamma rays, develop a time lag as, indeed, has been observed (with this modification). If the energies of the photons are \(\ge 1 \text {GeV}\) (this is near the so-called GZK cutoff) this separation becomes even more noticeable. In the same manner if gamma rays of cosmic origin are considered, we predict that such innumerable scattering processes occur before they reach us. This could explain the time-lag phenomena of gamma ray bursts (GRB)’s which have been observed. With the Snyder–Sidharth–Hamiltonian, it is possible to extend the GZK cutoff more than its usual limit, \(10^{20} eV\) . This is borne out by observations of ultra-high-energy cosmic rays thus extending the limits of special relativity.