Bounds on QCA Lattice Spacing from Data on Lorentz Violation

TL;DR

The paper investigates whether spacetime discreteness, as realized by a quantum cellular automaton (QCA) corresponding to quantum electrodynamics (QED), can be constrained by observations. It shows that the QCA for massless bosons yields photon-like propagation in the long-wavelength limit but introduces small Lorentz-invariance violations (LIV) and directional anisotropies at higher energies, with a leading LIV correction that scales with the lattice spacing $\Delta x$. By mapping LIV and anisotropy predictions to experimental bounds from high-energy gamma-ray bursts and precision Michelson–Morley-type tests, the authors derive upper limits on $\Delta x$: $\Delta x \lesssim 5.8\times10^{-36}$ m from dispersion (approaching the Planck length) and $\Delta x \lesssim 6.5\times10^{-26}$ m from anisotropy. These results suggest that if spacetime is discrete, the lattice spacing must lie at or near the Planck scale, and they outline avenues for sharpening these bounds with next-generation observatories and quantum simulators.

Abstract

Recent work has demonstrated that discrete quantum walks, when extended to quantum cellular automata (QCA), can, in the continuum limit, reproduce relativistic wave equations and quantum field theories (QFTs), including free quantum electrodynamics (QED). This QCA/QFT correspondence bridges quantum information processing and high-energy physics, raising fundamental questions about the nature of spacetime: whether it is the continuum QFT or the discrete QCA that is fundamental. For while Lorentz invariance appears robust experimentally, it may only approximate a deeper discrete structure, particularly at Planck-scale energies. This high-energy Lorentz violation is potentially observable either through cumulative effects over cosmic distances or via small deviations at accessible energies. In this paper, we analyze the QCA corresponding to QED and show that it implies both a deviation from the speed of light and spatial anisotropies. Using current experimental and astrophysical constraints, we place upper bounds on the QCA lattice spacing, providing insight into the plausibility of a fundamentally discrete spacetime.