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arXiv:2607.14874

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

Dogukan Bakircioglu, Pablo Arnault

hep-latmath-phquant-phSubmitted Jul 16, 2026• Announced Jul 17, 2026• 0 citations• Revised Aug 12, 2026 (v4)
v1 · Jul 16, 2026v2 · Aug 8, 2026v3 · Aug 11, 2026v4 · Aug 12, 2026
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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) [arXiv:2505.0790], 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$\unicode{x2014}$which is nothing but the Dirac equation$\unicode{x2014}$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. Then, we compute the GF of the FD-fixed model put forward in the last cited reference, called Flavored Dirac QCA (FDQCA)$\unicode{x2014}$which staggers an extra, artificial flavor $only$, on a diamond spacetime lattice, and does not stagger chirality as staggered fermions in usual LGT. The structure of this FDQCA 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.

Comments
26 pages. 14 pages for the body of the article. 12 pages of appendices. 3 figures related to the body of the article. 2 figures in the appendices

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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) [arXiv:2505.0790], 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\unicodex2014\unicode{x2014}\unicodex2014which is nothing but the Dirac equation\unicodex2014\unicode{x2014}\unicodex2014even 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. Then, we compute the GF of the FD-fixed model put forward in the last cited reference, called Flavored Dirac QCA (FDQCA)\unicodex2014\unicode{x2014}\unicodex2014which staggers an extra, artificial flavor onlyonlyonly, on a diamond spacetime lattice, and does not stagger chirality as staggered fermions in usual LGT. The structure of this FDQCA 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.
Simulations of one quantum system by an other has an implication in realization of quantum machine that can imitate any quantum system and solve problems that are not accessible to classical computers. One of the approach to engineer quantum simulations is to discretize the space-time degree of freedom in quantum dynamics and define the quantum cellular automata (QCA), a local unitary update rule on a lattice. Different models of QCA are constructed using set of conditions which are not unique and are not always in implementable configuration on any other system. Dirac Cellular Automata (DCA) is one such model constructed for Dirac Hamiltonian (DH) in free quantum field theory. Here, starting from a split-step discrete-time quantum walk (QW) which is uniquely defined for experimental implementation, we recover the DCA along with all the fine oscillations in position space and bridge the missing connection between DH-DCA-QW. We will present the contribution of the parameters resulting in the fine oscillations on the Zitterbewegung frequency and entanglement. The tuneability of the evolution parameters demonstrated in experimental implementation of QW will establish it as an efficient tool to design quantum simulator and approach quantum field theory from principles of quantum information theory.
This study presents a unitary quantum cellular automaton (QCA) that, in the continuum limit, converges to the (1+1)-dimensional Generalized Dirac Equation (GDE). We outline the construction of the unitary, discrete-time evolution and derive the model's exact dispersion relation from its eigenvalue spectrum, showing it possesses the correct relativistic limit. We then explore its dynamics through numerical simulations. While the total probability density is shown to be insensitive to the chiral angle parameter (ρ)(ρ)(ρ) of the GDE, we demonstrate that this parameter has a profound and directly observable effect on the particle's local spin polarization. By measuring this quantity, we reveal the hidden internal dynamics of the spinor, providing a clear, visual confirmation of the physical consequences of the chiral mass term.
The Thirring Quantum Cellular Automaton (QCA) describes the discrete time dynamics of local fermionic modes that evolve according to one step of the Dirac cellular automaton, followed by the most general on-site number-preserving interaction, and serves as the QCA counterpart of the Thirring model in quantum field theory. In this work, we develop perturbative techniques for the QCA path sum approach, expanding both the number of interaction vertices and the mass parameter of the Thirring QCA. By classifying paths within the regimes of very light and very heavy particles, we computed the transition amplitudes in the two- and three-particle sectors to the first few orders. Our investigation into the properties of the Thirring QCA, addressing the combinatorial complexity of the problem, yielded some useful results applicable to the many-particle sector of any on-site number-preserving interactions in one spatial dimension.
We consider discrete spacetime models known as quantum walks, which can be used to simulate Dirac particles. In particular we look at fermion doubling in these models, in which high momentum states yield additional low energy solutions which behave like Dirac particles. The presence of doublers carries over to the `second quantised' version of the walks represented by quantum cellular automata, which may lead to spurious solutions when introducing interactions. Moreover, we also consider pseudo-doublers, which have high energy but behave like low energy Dirac particles, and cause potential problems regarding the stability of the vacuum. To address these issues, we propose a family of quantum walks, that are free of these doublers and pseudo-doublers, but still simulate the Dirac equation in the continuum limit. However, there remain a small number of additional low energy solutions which do not directly correspond to Dirac particles. While the conventional Dirac walk always has a zero probability for the walker staying at the same point after a single time step, we obtain the family of walks by allowing this probability to be non-zero.
Non-abelian quantum cellular automata: 1+11{+}11+1-dimensional SU(2)SU(2)SU(2) Yang--Mills with fermions
This work provides a digital quantum simulation scheme for 1+11{+}11+1-dimensional SU(2)SU(2)SU(2) Yang--Mills theory with Dirac fermions. It takes the form of a quantum circuit, infinitely repeating across space and time with Δt=Δx=εΔ_t=Δ_x=\varepsilonΔt​=Δx​=ε, whose wires follow lightlike propagation. The construction mirrors the logic of the standard quantum field theory approach, transposed to the discrete setting. Namely, we start from the Dirac quantum walk, restore SU(2)SU(2)SU(2) gauge symmetry by introducing the gauge field, lift the walk to a multi-particle QCA while preserving fermionic anticommutation, and equip the gauge field with its own dynamics. Rather than relying on Clebsch--Gordan decompositions, we use the pointwise-product structure of gauge-link updates, which yields self-contained proofs of unitarity and gauge covariance in quantum-computing notation. The construction provides an explicit algorithmic formulation of the theory, whose continuum limit we discuss.
We numerically compute renormalized expectation values of quadratic operators in a quantum field theory (QFT) of free Dirac fermions in curved two-dimensional (Lorentzian) spacetime. First, we use a staggered-fermion discretization to generate a sequence of lattice theories yielding the desired QFT in the continuum limit. Numerically-computed lattice correlators are then used to approximate, through extrapolation, those in the continuum. Finally, we use so-called point-splitting regularization and Hadamard renormalization to remove divergences, and thus obtain finite, renormalized expectation values of quadratic operators in the continuum. As illustrative applications, we show how to recover the Unruh effect in flat spacetime and how to compute renormalized expectation values in the Hawking-Hartle vacuum of a Schwarzschild black hole and in the Bunch-Davies vacuum of an expanding universe described by de Sitter spacetime. Although here we address a non-interacting QFT using free fermion techniques, the framework described in this paper lays the groundwork for a series of subsequent studies involving simulation of interacting QFTs in curved spacetime by tensor network techniques.