Fermion-doubling problem in Chiral discretizations of Quantum field theory: Definitive proof, Fixing, and Computation of two-point correlation function

Abstract: We give the definitive proof that the Dirac Quantum Cellular Automaton (QCA) used for both quantum simulation and algorithmic foundations of Quantum Field Theory (QFT), and especially of Quantum Electrodynamics (QED), as put forward in References https://doi.org/10.1007/s11128-019-2555-4 and https://doi.org/10.22331/q-2023-11-08-1179, does exhibit Fermion Doubling (FD), albeit thrice as less severe as discrete-time standard Lattice Gauge Theories (LGTs) -- as shown in Reference arXiv:2505.07900 -- , which are naive regarding the spacetime discretization of differential operators acting on fermionic fields. The proof is done for the $(1 + 1)$D Dirac-QCA model. We show that the (one-time-step) two-point correlation function, also called Green's function (GF), of the Dirac QCA, is of astonishing simplicity, which is in contrast with the GF of the Dirac equation. We also compare, both qualitatively and quantitatively, this Dirac QCA to the continuous-time-LGT spatial discretization of Dirac fermions regarding how well these two lattice models approximate their naive continuum limit -- which is nothing but the Dirac equation -- even when far away from that limit, a situation which must be considered because of experimental limitations in quantum simulation -- : the Dirac QCA is better for ultrarelativistic regimes, whereas continuous-time LGT is better for non-relativistic regimes. In a second part of this work, we compute the GF of the FD-fixed model put forward in the last cited reference, called Flavored Dirac QCA (FQCA) -- which staggers an extra, artificial flavor \emph{only}, on a diamond spacetime lattice, and does not stagger chirality as staggered fermions in usual LGT. The structure of this FQCA two-point correlation function is of extreme simplicity, and can be expressed in a very simple manner in terms of the four chiral components of the FD-suffering, original-model GF.