Non-abelian Gauge-Invariant Cellular Automata

Abstract

Gauge-invariance is a fundamental concept in Physics-known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of Cellular Automata. More precisely, the notions of gauge-invariance and gauge-equivalence in Cellular Automata are formalized. A step-by-step gauging procedure to enforce this symmetry upon a given Cellular Automaton is developed, and three examples of gauge-invariant Cellular Automata are examined.

A Discrete Relativistic Spacetime Formalism for 1+1-QED With Continuum Limits

“…The family (b − , b + ) forms an orthonormal basis of the two dimensional spin Hilbert space. Let us now rewrite equation (30) in this new orthonormal basis. A straightforward computation leads to:…”

“…Both drew inspirations from gauge-invariant Quantum Walks 17,26,27 . The question of extending these constructions to non-abelian gauge theories has been treated for Quantum Walks 28,29 and for Reversible CA 30 . Lifting this to obtain U(N) gauge-invariant QCA is no doubt one of the next steps lying ahead towards digital quantum simulation schemes for Yang Mills theories, such as QCD.…”