9.2 Lorentz Violation Effects in Ultra-High-Energy Cosmic Rays

Our theory is based on discrete lattice graph $\Lambda$. Although Lorentz symmetry is perfect emergent result at low energies, when approaching lattice scale (Planck scale $l_P$), this continuous symmetry must be broken by discrete structure.

9.2.1 Dispersion Relation of Light Speed

Wave packets propagating on lattices no longer have perfect dispersion relation $E=pc$, but have lattice correction terms:

$$E^2\approx p^2{c}^2+\eta \frac{p^4}{M_P^2}$$

where $M_P$ is Planck mass, $\eta$ is coefficient depending on lattice geometry.

This means ultra-high-energy photons (or neutrinos) will have velocity weakly dependent on their energy.

• If $\eta <0$, high-energy photons are slower than low-energy photons (subluminal).

• If $\eta >0$, high-energy photons are faster than low-energy photons (superluminal).

9.2.2 Observational Opportunity

Universe provides us natural super accelerators—Gamma-Ray Bursts (GRB).

When a GRB billions of light-years away bursts, it simultaneously emits photons observable across all wavelengths.

If QCA theory is correct, then after billions of years of flight, even extremely tiny velocity differences $\delta v\sim (E/M_P{)}^2$ will accumulate into observable time delays $\Delta t$.

9.2.3 Prediction

Instruments like Fermi Gamma-ray Space Telescope (Fermi LAT) should observe: High-energy photons from same burst source arrive at Earth systematically earlier (or later) than low-energy photons.

Although current observational data hasn’t given conclusive evidence, there are already some puzzling “outlier photons” suggesting this possibility. As detection precision improves, this is not only a test of QCA, but direct measurement of spacetime discreteness.