Appendix B: Dispersion Relation and Lorentz Violation in Dirac-QCA

This appendix demonstrates how Lorentz invariance emerges as a low-energy approximation when we discretize continuous spacetime into a Quantum Cellular Automaton (QCA), and the origin of higher-order correction terms ($O(p^4)$).

1. Discrete Evolution Operator

Consider a one-dimensional lattice with lattice constant $a$ and time step $\Delta t$. Define the single-step evolution operator $U$ as the product of translation operator $S$ and coin operator (internal rotation) $C$:

$$U=S(p)\cdot C(\theta )=\left(\begin{array}{cc}{e}^{ipa}& 0\\ 0& e^{-ipa}\end{array}\right)\left(\begin{array}{cc}\cos \theta & -i\sin \theta \\ -i\sin \theta & \cos \theta \end{array}\right)$$

where $\theta$ is related to mass, and $p$ is momentum.

2. Derivation of Dispersion Relation

Solving the eigenvalue equation $\det (U-e^{-i\omega \Delta t}I)=0$, we obtain the strict discrete dispersion relation:

$$\cos (\omega \Delta t)=\cos (pa)\cos \theta$$

3. Continuous Limit and Corrections

In the low-energy limit ($p\ll 1/a$, $\omega \ll 1/\Delta t$), we perform Taylor expansion of the cosine functions:

$$1-\frac{(\omega \Delta t{)}^2}{2}\approx (1-\frac{(pa{)}^2}{2})(1-\frac{{\theta }^2}{2})$$

Retaining second-order terms, we recover the Dirac equation $E^2=c^2{p}^2+m^2{c}^4$ (where $c=a/\Delta t$).

But if we retain higher-order terms, we discover Lorentz violation terms:

$$E^2\approx c^2{p}^2+m^2{c}^4-\alpha \frac{p^4}{M_{scale}^2}\text{(B.1)}$$

The $p^4$ term represents the geometric saturation effect brought by lattice structure. Since this term is fourth power in momentum, it is extremely difficult to observe at low energies, explaining why the macroscopic universe appears so smooth.