Relativistic Wigner Function for Quantum Walks

Abstract

A relativistic Wigner function for free Discrete Time Quantum Walks (DTQWs) on the square 2 D space-time lattice is defined. The transport equation obeyed by the relativistic Wigner function is derived in terms of discrete derivatives and degenerates in the continuum limit into the transport equation obeyed by the Wigner function of 2 D Dirac fermions. The first corrections to the continuous equation induced by the discreteness of the lattice are also determined and the similarities with non-quantum relativistic transport in phase space are discussed further.