Universal gauge-invariant cellular automata

TL;DR

This paper formalizes the concept of gauge extension in cellular automata, showing how to create gauge-invariant CA that can mediate interactions, bridging ideas from physics and computer science.

Contribution

It introduces a formal framework for gauge extension in cellular automata and demonstrates the universality of gauge-invariant CA with interaction capabilities.

Findings

Gauge-invariant CA are exactly the globally symmetric (colour-blind) CA.

Any CA admits a non-relative gauge extension.

Constructed gauge-invariant CA can mediate interactions within the initial CA.

Abstract

Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs `matter', and features a global symmetry. One then extends the theory so as make the global symmetry into a local one (a.k.a gauge-invariance). We formalise a discrete counterpart of this process, known as gauge extension, within the Computer Science framework of Cellular Automata (CA). We prove that the CA which admit a relative gauge extension are exactly the globally symmetric ones (a.k.a the colour-blind). We prove that any CA admits a non-relative gauge extension. Both constructions yield universal gauge-invariant CA, but the latter allows for a first example where the gauge extension mediates interactions within the initial CA.