Quantum Walks, Weyl equation and the Lorentz group

TL;DR

This paper explores how quantum walks can model relativistic quantum fields, specifically the Weyl equation, by analyzing their symmetry groups and reconciling discreteness with Lorentz invariance.

Contribution

It characterizes the full symmetry group of the Weyl walk as a nonlinear realization of a semidirect product of the Poincaré group and dilations.

Findings

The Weyl walk recovers the Weyl equation in the small wave-vector limit.

The symmetry group of the Weyl walk is a nonlinear realization of a semidirect product of Poincaré and dilation groups.

The framework reconciles quantum automata discreteness with continuous Lorentz symmetry.

Abstract

Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata and walks starting from very general principles. The intrinsic discreteness of this framework is reconciled with the continuous Lorentz symmetry by reformulating the notion of inertial reference frame in terms of the constants of motion of the quantum walk dynamics. In particular, among the symmetries of the quantum walk which recovers the Weyl equation--the so called Weyl walk--one finds a non linear realisation of the Poincar\'e group, which recovers the usual linear representation in the small wave-vector limit. In this paper we characterise the full symmetry group of the Weyl walk which is shown to be a non linear realization of a group which is the semidirect product of the Poincar\'e group and the group of dilations.