Nonunitarity of Simple 3+1-Dimensional Quantum Walk Models Corresponding to the Weyl and Dirac Equations

Abstract

It is well known that \(1+1\)-dimensional Weyl and Dirac equations in the relativistic quantum mechanics are related to some quantum walks. However, in \(3+1\)-dimensions, discrete-time quantum walks corresponding to the above two equations have not been obtained. We show that the \(3+1\)-dimensional Weyl and Dirac walkers cannot be obtained by generalizing the above well-known model to \(3+1\)-dimensional cases. Apparently, these \(3+1\)-dimensional extended models seem to be well defined, but they do not have unitary time evolution. Thus, the square of the magnitude of each spinor is not able to be any probability distribution. In addition, we present two Schrodinger walkers, and one of which has unitary time evolution. This unitary Schrodinger walker can move to all sites at each time step, unlike the existing models. Therefore, it is mentioned that adequate models of \(3+1\)-dimensional Weyl and Dirac walkers will also be given like this unitary Schrodinger walker.