# Relativistic Wigner Function for Quantum Walks

## TL;DR

This paper introduces a relativistic Wigner function for 2D quantum walks on a lattice, deriving its transport equation and analyzing lattice-induced corrections, bridging discrete quantum models with continuous relativistic fermion descriptions.

## Contribution

It defines a relativistic Wigner function for discrete quantum walks and derives its transport equation, including lattice correction terms, connecting discrete models with continuous relativistic quantum mechanics.

## Key findings

- Derived the transport equation for the relativistic Wigner function.
- Identified the continuous limit matching 2D Dirac fermions.
- Computed first-order lattice corrections to the transport equation.

## Abstract

A relativistic Wigner function for free Discrete Time Quantum Walks (DTQWs) on the square $2D$ space-time lattice is defined. Useful concepts such as discrete derivatives and discrete distributions are also introduced. The transport equation obeyed by the relativistic Wigner function is obtained and degenerates at the continuous limit into the transport equation obeyed by the Wigner function of $2D$ Dirac fermions. The first corrections to the continuous equation induced by the discreteness of the lattice are also computed.