Propagation in quantum walks and relativistic diffusions

TL;DR

This paper connects quantum walk propagation to Dirac fermions in electromagnetic fields and explores relativistic diffusions, providing new numerical insights and a geometric extension of Fick's law, with implications for quantum decoherence models.

Contribution

It demonstrates that 1D quantum walks can be described by Dirac equations with electromagnetic coupling and introduces a geometric generalization of Fick's law for relativistic diffusions.

Findings

Quantum walks relate to Dirac fermions in electromagnetic fields.

Numerical simulations of relativistic Ornstein-Uhlenbeck process show short-time propagation.

A geometric extension of Fick's law is proposed for relativistic diffusions.

Abstract

Propagation in quantum walks is revisited by showing that very general 1D discrete-time quantum walks with time- and space-dependent coefficients can be described, at the continuous limit, by Dirac fermions coupled to electromagnetic fields. Short-time propagation is also established for relativistic diffusions by presenting new numerical simulations of the Relativistic Ornstein-Uhlenbeck Process. A geometrical generalization of Fick's law is also obtained for this process. The results suggest that relativistic diffusions may be realistic models of decohering or random quantum walks. Links with general relativity and geometrical flows are also mentioned.