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

This paper investigates the fermion-doubling problem within the framework of Dirac Quantum Cellular Automata (QCA) used for quantum simulation and the algorithmic construction of quantum field theory. While QCAs are praised for their native unitarity and local structure, the authors provide a rigorous proof that these models still suffer from fermion doubling—a phenomenon where spurious particles appear in the lattice discretization of fermionic fields. However, they demonstrate that this doubling is significantly less severe than in standard lattice gauge theories (LGTs) due to the use of chirality-dependent spatial shifts.

The authors perform their analysis in the (1+1)D Dirac QCA model. By examining the poles of the spacetime-Fourier-space Green's function, they confirm that the expected problematic poles indeed contribute to spurious doublers. A key contribution is the derivation of the one-time-step two-point correlation function (Green's function), which takes an remarkably simple closed form involving only Kronecker deltas. This contrasts sharply with the continuum Dirac equation, where the Green's function requires complex special functions like Bessel or Hankel functions.

Furthermore, the authors compare the Dirac QCA with continuous-time LGT spatial discretizations. They find that the Dirac QCA is superior for approximating the Dirac equation in ultrarelativistic regimes, whereas continuous-time LGT models perform better in non-relativistic regimes. Finally, the paper computes the Green's function for a previously proposed 'Flavored' Dirac QCA (FQCA), which fixes the doubling problem by introducing an artificial flavor on a diamond spacetime lattice.

Understanding the fermion-doubling problem is critical for researchers building quantum simulations of quantum field theories. Because experimental limitations often prevent researchers from reaching the true continuum limit, knowing which lattice discretization is most accurate in specific physical regimes (e.g., ultrarelativistic vs. non-relativistic) is essential for designing reliable simulations. The simplicity of the derived Green's functions provides a powerful, computationally efficient tool for analyzing these models without the overhead of complex analytical functions.