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

Dogukan Bakircioglu; Pablo Arnault

REVIEW 4 major objections 3 minor 20 references

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

T0 review · 4 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The Dirac QCA's propagator genuinely has fermion doubling, and the paper gives its closed-form Green's function.

desk verdict The closed-form one-step GF and the FQCA GF are useful, but the advertised 'definitive proof' rests on an ill-defined contour integral—the poles lie on the contour—so the proof needs a prescription before it can be accepted. read the letter →

arxiv 2607.14874 v1 pith:NWYAGSIE submitted 2026-07-16 hep-lat math-phmath.MPquant-ph

Dogukan Bakircioglu , Pablo Arnault This is my paper

classification hep-latmath-phmath.MPquant-ph

keywords fermiondoublingDiracquantumcellularautomatondiscrete-timewalkGreen'sfunctionlatticegaugetheoryelectrodynamicsflavorstaggeringBrillouinzone

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the Dirac quantum cellular automaton—a unitary, chirality-dependent spacetime-lattice discretization of the Dirac equation used to build lattice QED models—really does suffer from fermion doubling, and not just in its dispersion relation. In 1+1 dimensions the authors prove the point by showing that every spurious pole of the Fourier-space Green's function contributes non-vanishingly to the one-time-step Green's function after contour integration. They also find that this one-step Green's function is a closed form built from Kronecker deltas and the sine and cosine of mε, much simpler than the Dirac equation's Green's function. For the flavor-staggered, doubling-free version of the model, they show that its Green's function is a parity-selected combination of the original model's chiral components. A sympathetic reader would care because the result identifies where spurious modes live in QCA-based QFT and provides exact two-point correlation functions to build on.

What carries the argument

The load-bearing objects are the Fourier-space equation-of-motion operator and its determinant D(E,p), whose zeros are the poles of the Green's function. The decisive move is the change of variable z=e^{-iεE}, which converts the energy integral over the Brillouin zone into a contour integral on the unit circle; the residue theorem then forces each pole to contribute with winding number -1, so the doublers cannot be discarded by a choice of contour. In direct space, the closed form of Eq. (25) is the exact one-step propagator, and it is what makes the doubling concrete. For the fixed model, the key machinery is the rhombus (diamond) Brillouin-zone ansatz with its factor of 2 and the decomposi

What would settle it

A direct numerical check: initialize the Dirac quantum walk on a single spatial site and measure the one-step Green's function matrix elements for all four chirality combinations; Eq. (25) predicts exact coefficients c_ε at x-ε and x+ε and -i s_ε at x. If any measured amplitude differs, the closed-form propagator is wrong. For the doubling claim, compute the full n-step Green's function and look at the residue at a spurious pole such as the one near (E,p)=(π/(2ε), -π/(2ε)); if that residue vanishes for some n or for some boundary conditions, the assertion that every zero contributes non-vanish

Watch

Extended reading notes

Core claim

The paper's central claim is that the free single-particle sector of the (1+1)D Dirac QCA—the Dirac discrete-time quantum walk with one-step unitary U_ε = [[cos(mε)S_ε, -i sin(mε)],[-i sin(mε), cos(mε)S_ε†]]—has a genuine fermion-doubling problem. The Fourier-space equation-of-motion determinant D(E,p) has extra zeros; writing the energy integral as a contour integral in z=e^{-iεE} and applying the residue theorem, the authors show that every such zero is a pole that contributes non-vanishingly to the direct-space one-time-step Green's function. That Green's function is the closed form G(t'+ε,x;t',x')=(1/ε)[[c_ε δ_{x',x-ε}, -i s_ε δ_{x',x}],[-i s_ε δ_{x',x}, c_ε δ_{x',x+ε}]]. For the flavore

Load-bearing premise

The proof abstracts from the interacting QFT to the free single-particle Dirac quantum walk, so the load-bearing assumption is that every spurious pole of the free one-particle Green's function—and nothing else—fully determines fermion doubling in the interacting QED-QCA; if the energy-contour witness misses or over-counts modes once gauge fields are present, the 'definitive proof' would not transfer to the full QFT.

Editorial extensions

If this is right

The one-time-step Green's function of the Dirac QCA is known in closed form, so single-step propagation of any initial state on the lattice can be done exactly and cheaply.
Fermion doubling is a property of the propagator itself, so any interacting QED-QCA built on this Dirac QCA inherits the spurious modes unless the flavor-staggering fix is applied.
In ultrarelativistic regimes (small mε) the Dirac QCA approximates the continuum Dirac equation better than continuous-time naive lattice fermions, while the opposite holds in non-relativistic regimes (mε near π/2).
The flavored QCA's two-point function is fully determined by the four chiral components of the original model's Green's function via a simple parity rule, giving an exact and computationally tractable correlation function for the FD-fixed model.
The paper claims the Dirac QCA's doubling is less severe by a factor of three than that of discrete-time standard lattice gauge theories, because its extra modes are spatiotemporal rather than separate spatial and temporal doublers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the one-step Green's function is a closed form, the n-step Green's function likely satisfies a simple convolution recurrence; one could derive closed-form n-step propagators by repeated application of Eq. (25), which the paper leaves open.
The same contour-integral witness could be applied to the continuous-time LGT and to the 2+1 and 3+1 QCA models to settle whether their doublers also contribute non-vanishingly to their propagators, extending the proof beyond 1+1 dimensions.
The regime comparison suggests a practical guide for quantum simulators: choose the QCA realization for massless or very light fermions and the continuous-time LGT for massive, non-relativistic fermions; a hybrid scheme that switches in the intermediate regime might be worth testing, although the paper does not propose one.
The FQCA Green's function's parity selection rule is a sharp signature that an experiment on a two-flavor diamond-lattice walk could verify by preparing a single-site state and measuring the flavor-chirality correlations after an even or odd number of steps.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Good to get your take on this. I've now spent some time with it, and I largely agree with your conditional read—with one thing sharper: the stress-test note is right. The contour integral in App. B is not a well-defined object as written. After the change of variable z = e^{-iεE}, the roots z± = c_ε cos(pε) ± i sqrt(1 − c_ε² cos²(pε)) lie exactly on the unit circle, which is the integration contour. You can't assign an index to a point on the contour, and the residue theorem doesn't apply without a prescription. If you push the contour outward you get both poles' residues; if you push it inward you get zero. The paper just states Ind_C(z±) = −1 and applies the full residue formula. That's invalid. So the advertised 'definitive proof' that every determinant zero contributes non-vanishingly is not established by App. B.

That said, the main closed-form Eq. (25) is not in doubt. App. C derives the same thing directly from the quantum-walk propagator, and App. D does the Fourier integral. The paper's real value is that closed form, plus the FQCA Green's function in Eq. (65), which is a clean result. The regime comparison with LGT is also thoughtful, and the plots are honest. The FQCA structure proofs in Apps. J and K are solid given the rhombus-BZ ansatz from Ref. [3]; the factor of 2 in Eq. (45) is assumed, not derived, so the FQCA result inherits that assumption.

Other issues: the title says 'definitive proof' but the analysis is restricted to a free one-time-step sector; that's a paper-scope problem, not a fatal one. App. H has a typo in (H3d): the QW b term should be cos(εm) sin(εp), not sin(εm). And a couple of supporting claims are explicitly unpublished, which is fine but weakens the self-containedness.

Who is this for? People working on QCA-based QED and quantum simulation of Dirac dynamics will want the closed-form one-step GF and the fixed-model GF. The fermion-doubling argument needs to be cleaned up before anyone can rely on it as a proof, but the paper is useful and worth reviewing. I'd send it to a referee, with the instruction that the contour prescription must be addressed—either by an iη deformation or by reframing the 'pole contribution' argument in terms of the direct propagator. If the authors fix that, the paper becomes a solid contribution.

Referee Report

4 major / 3 minor

Summary. The paper studies the (1+1)-dimensional Dirac QCA of Arrighi et al. It derives a closed form for the one-time-step Green's function, Eq. (25), by energy contour integration, and interprets the poles of the Fourier-space determinant as fermion doublers. It compares the QW integrand with the continuum Dirac propagator and with continuous-time naive LGT in ultrarelativistic and nonrelativistic regimes. It then computes the Green's function of the flavored Dirac QCA of Ref. [3], obtaining Eq. (65), which expresses the FQCA Green's function in terms of the original QW Green's-function blocks, with a flavor operator I or σ_x depending on the parity of D+Δn.

Significance. The closed-form one-step Green's function and the compact FQCA Green's-function expression are potentially useful results, and the comparison of lattice models in far-from-continuum regimes addresses a relevant quantum-simulation question. The combinatorial/inductive proof of Eq. (65) in Apps. J and K is a strength: it is self-contained and does not depend on the questionable Fourier normalization. The quantitative model comparison is also clearly presented. However, the 'definitive proof' of fermion doubling rests on a contour integration whose poles lie on the integration contour; as written, that part is not rigorous.

major comments (4)

[App. B, Eqs. (B10)-(B15)] The roots in Eq. (B14) satisfy |z_±|^2 = c_ε^2 cos^2(pε) + 1 - c_ε^2 cos^2(pε) = 1, so both poles lie on the unit circle C, which is the integration contour. The residue theorem with Ind_C(z_±) = -1 and full 2πi residues is therefore not applicable: a simple pole on the contour makes the integral ill-defined unless a principal value or an iη deformation is specified, and different prescriptions give different coefficients. This invalidates the Sec. IIIC claim that every zero of D(E,p) contributes non-vanishingly to the one-time-step GF. Please add an explicit regularization (e.g., replace e^{-iEε} by e^{-iEε-η} in L_B) and redo the computation, or prove the pole-contribution claim by the direct unitary-propagator route and state the distributional identity under that prescription.
[App. B, Eqs. (B7), (B15)] There is a sign/prefactor inconsistency between Eq. (B7) and the final result: combining dE = i dz/(εz) with the integrand in Eq. (B4) yields an overall sign different from that in Eq. (B15), and Eq. (B15) appears to give minus Eq. (22). Since Eq. (25) is independently confirmed by the propagator computation in App. C, this is not fatal, but the contour-integral derivation must be corrected to match.
[Sec. IV, Eq. (45)] The rhombus-Brillouin-zone Fourier ansatz, including the factor 2=(√2)^2, is assumed rather than derived. The final FQCA result Eq. (65) is nevertheless proven by the independent combinatorial and inductive arguments in Apps. J and K, so the main result survives; however, the Fourier-integral derivation in Sec. IVA3 is incomplete unless the integration measure and normalization are justified.
[Sec. I and Sec. V] The paper claims a definitive proof of fermion doubling for a QCA model used as a foundation for QED, but the explicit computation is limited to the free single-particle sector and to one time step. This is a scope limitation rather than a technical error—the resolvent pole structure is the standard free-field diagnostic—but the title/abstract should either restrict the claim accordingly or add a sentence explaining why the one-step free GF suffices for the interacting-QFT conclusion.

minor comments (3)

[App. H, Eq. (H3d); Sec. III.D, Eq. (41d)] The imaginary part of I^QW_11 is missing the sin(εp) factor and incorrectly contains cos(εm) sin(εm) instead of cos(εm) sin(εp). Eq. (41d) likewise has a spurious e^{iPD}; both should read (1/2π) cos(M) sin(P) for -Im{e^{-iPD}F^QW_11}.
[General] The manuscript contains many typos and grammatical errors (e.g., 'quantum cellular automatas', 'we use QCAs will be used', duplicated words). A careful proofread is needed.
[Ref. [3]] The FD-fixing model and the FQCA construction are taken from the authors' own preprint Ref. [3]; please ensure it is published or otherwise available, and clarify which definitions are being imported.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivations are self-contained and the FQCA Green's function is an explicit algebraic consequence of the model, not a relabeled input.

full rationale

The paper's main new computations are self-contained. The one-step Green's function, Eq. (25), is shown in App. C to equal the quantum-walk propagator, and in App. D to follow from the Fourier expression; this is an identity with the evolution operator, but it is derived explicitly rather than assumed, and the fermion-doubling witness argument does not rest on the 'astonishing simplicity' claim. The FD proof in Sec. IIIC and App. B is intended to show that every zero of the determinant D(E,p) contributes to the energy integral; this is a load-bearing step, but it is not circular in the sense of fitting inputs or defining the conclusion into the premises. The contour computation in App. B has a genuine mathematical gap: the roots z± in Eq. (B14) satisfy |z±|=1, so they lie on the integration contour, and the residue theorem as written is not directly applicable without an iη deformation or principal-value prescription. That is a rigor/correctness concern, not a circularity, because the flawed step does not reduce the theorem to its own assumptions by construction; moreover, Eq. (22)/Eq. (25) are independently reproduced in App. C as the propagator, so the Green's function itself is not merely an assumed input. The FQCA part relies on the rhombus-Brillouin-zone ansatz, Eq. (45), and on the model construction of the authors' Ref. [3]; this is a self-citation, but the paper's new claim is the explicit GF expression Eq. (65), which is proven in App. J by binomial expansion and in App. K by induction directly from the one-step FQCA operator and the original QW GF. No parameter is fitted and then called a prediction, no uniqueness theorem is imported from the authors, and no known result is merely renamed. The derivation chain is therefore not circular; the main risk is the mathematical validity of the contour argument, which is an error/omission issue rather than a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No free parameters are fitted to data. The central derivation relies on standard Fourier/residue mathematics plus two domain assumptions: free single-particle analysis is sufficient for a fermion-doubling diagnosis, and the rhombus-Brillouin-zone flavored construction is the correct fixing scheme. The only invented entity is the artificial flavor, with no independent evidence.

assumptions (5)

domain assumption The free, single-particle 'classical' Dirac field (Dirac quantum walk) is sufficient to diagnose and fix fermion doubling in the QCA/QFT models.
The paper explicitly restricts the analysis to the one-particle sector and the free model (Sec. I, Sec. II), although the motivating problem is interacting QED.
domain assumption Fermion doubling is witnessed by all zeros of the Fourier-space EOM determinant, and every such pole contributes via the residue theorem to the Green's function.
Sec. IIIC states this as the definitive detection criterion; App. B applies it to the one-time-step Green's function.
standard math Standard Fourier transform and contour-integration conventions, including the unit-circle change of variable z=e^{-iϵE} and the stated residue indices, are valid.
App. B uses the residue theorem for roots z± of Δ(z;p); the index computation is standard complex analysis.
ad hoc to paper The FQCA has a rhombus Brillouin zone, and the Fourier ansatz Eq. (45) with factor 2=(√2)^2 is the correct inverse transform.
Eqs. (45)-(51); this is the model-specific construction from Ref. [3] and is the basis for the FQCA Green's function derivation.
domain assumption An extra artificial flavor can be staggered on the diamond lattice without staggering chirality, yielding two independent Dirac fermions in the continuum.
Sec. IVA1 recaps this from Ref. [3]; the flavor is a model-building device, not an independently motivated physical degree of freedom.

invented entities (1)

Artificial flavor degree of freedom (flavor staggering)
purpose: Introduced to remove spurious fermion doublers by staggering an extra flavor only, not chirality, on a diamond spacetime lattice.
No independent experimental or falsifiable handle outside the construction is offered; it is adopted from the authors' earlier Ref. [3].

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Cite this review

Pith. "Pith review of Fermion-doubling problem in Chiral discretizations of Quantum field theory: Definitive proof, Fixing, and Computation of two-point correlation function." pith.science (2026). https://pith.science/paper/NWYAGSIE

@misc{pith2026260714874,
  author       = {Pith},
  title        = {Pith review of: Fermion-doubling problem in Chiral discretizations of Quantum field theory: Definitive proof, Fixing, and Computation of two-point correlation function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWYAGSIE}},
  note         = {Machine review of arXiv:2607.14874}
}

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.

Figures

Figures reproduced from arXiv: 2607.14874 by the authors.

**Figure 1.** Comparison between the A M(P; M)/(2π)’s for the three models M = Dirac (blue), M = QW (red), and M = LGT (gold), as functions of P := ϵp, for 9 values of M := ϵm. Remember that these A M(P; M)/(2π)’s are the factors, within the integrands of the momentum integral yielding the one-time-step Green’s function, which are independent from the considered transition distance D = (x − x ′ )/ϵ ∈ Z. We have called the top row… view at source ↗

**Figure 2.** Comparison of the real parts R M 11(P; M, D = 1) of the 11 matrix components of the Fourier integrands of the onetime-step Green’s functions of the three models M = Dirac (blue), M = QW (red), and M = LGT (gold), respectively, for a transition of D = +1 spatial lattice step (in the direction of growing positions, hence the + in D = +1). The lecture order of the plots in the order of increasing rescaled masses M is,… view at source ↗

**Figure 3.** Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

**Figure 5.** Figure 5: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

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Reference graph

Works this paper leans on

20 extracted references · 4 linked inside Pith

[3]

GQW B;11(E, p)Iflav. 0 0G QW B;22(E, p)Iflav. # , (60a) GFQW BZ;n.d.(E, p):=

Finding the simplest expression for the direct-space FQW GF The difficulty here is the new, rhombus BZ. We will first perform some changes of variables that will make us able to compute the integral of Eq. (45) in half the situations we encounter. We rewrite Eq. (45) as GFQW(t, x;t′, x′) =I (1)(D,∆n) +I (2)(D,∆n) +I (3)(D,∆n), (53) 13 with I (1)(D,∆n) := ...
[1]

flavor staggering only

Recap on the flavored Dirac QCA In Ref. [3], we have proposed a solution to solve the FD problem of the Dirac QCA, problem which has been re- explained in detail in Sec. IIIC. We have called this so- lution “flavor staggering only”, because it is based on the staggering of extra flavor degrees of freedom that we in- troduced, but without staggering the ch...
[2]

Looking for an appropriate GF This rhombus shape we have for the BZ implies in par- ticular that a suitable ansatz form with which to look for the Green’s function of the Dirac FQCA is GFQW(t, x;t′, x′) = 2 ˆ π/ϵ −π/ϵ dp 2π eip(x−x′) ˆ π/ϵ−|p| −π/ϵ+|p| dE 2π e−iE(t−t′)GFQW BFQW(E, p), (45) instead of the ansatz of Eq. (17). The factor of2 = ( √ 2)2 in fro...
[4]

Flavouredlatticeschwingermodel with chiral anomaly,

DogukanBakircioglu,“Flavouredlatticeschwingermodel with chiral anomaly,” (2026), arXiv:2604.13146 [hep-lat]

arXiv 2026
[5]

A quantum cellu- lar automaton for one-dimensional QED,

P. Arrighi, C. Bény, and T. Farrelly, “A quantum cellu- lar automaton for one-dimensional QED,” Quantum Inf. Process.19, 88 (2020), arXiv:1903.07007

arXiv 2020
[6]

A relativistic discrete spacetime formulation of 3+1 QED,

N. Eon, G. Di Molfetta, G. Magnifico, and P. Arrighi, “A relativistic discrete spacetime formulation of 3+1 QED,” Quantum7, 1179 (2023), 2205.03148

arXiv 2023
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Fermion doubling in quantum cellular automata,

D. Bakircioglu, P. Arnault, and P. Arrighi, “Fermion doubling in quantum cellular automata,” arXiv:2505.07900 (2025)

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taking bothpϵgoing to zero on the one hand, andmϵ going to zero on the other hand

This implies that, if we write Eq. (29) as GM(t′ +ϵ, x;t′, x′) = ˆ BM −BM dp IM(p;ϵ,∆),(30) with IM(p;ϵ,∆) := 1 2π eip∆UM ϵ (p),(31) the previous announced focuses will be onIM 11(p;ϵ,∆ =ϵ) andI M 12(p;ϵ,∆ = 0) =I M 21(p;ϵ,∆ = 0). But before definitively turning to the aforeme...
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reproduces

Eleanor Chu,Discrete and Continuous Fourier Trans- forms: Analysis, Applications and Fast Algorithms (Chapman and Hall/CRC, 2008). 15 Appendix A:Naivecontinuum limit of the Dirac-QCA one-time-step Green’s function: matches the Dirac-equation Green’s function If one knows no ge...

2008
[16]

off shell

Dirac-equation Green’s function Let us start with the first part announced above. We consider the EOM operator of our Dirac QCA, Eq. (15b). Since by construction Tϵ ≡e iϵE ,(A1) withE :=i∂ t, and Sϵ ≡e −iϵPx ,(A2) withP x :=−i∂ x, then the expansion of Eq. (15b) at first order...
[17]

source field

Introduction In the context of mathematically classical fields—i.e., these fields are functions of spacetime that are number- valued, as opposed to operator-valued fields as in QFT for example—, there are essentially two principal types of situations in which GFs are useful—i....
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principle

Continuous-spacetime Schrödinger-like equations a. Cauchy problem Let us get into technicalities and into the actual solu- tions of these problems in the case of Schrödinger-like equations. We want to solve the following equation, Lu|t,x = 0,(E1) where L :=i∂ t − H,(E2) is a l...
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classical

Continuous-spacetime theories The aim of this subappendix is simple, and the follow- ing: set, or recall, the abstract-QM formalism (i.e., with position and momentum kets and bras) that is adapted to describe the single-particle sectors of usual QFTs, the latter being written ...
[20]

classical

Discrete-spacetime theories When one wants to spacetime discretize a continuum quantum-mechanical theory, or theory of “classical” fields—in the latter case, often with the perspective of 15 In the context of the most standard connections of QM to QFT, onesometimeshave(forinex...

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [3]

GQW B;11(E, p)Iflav. 0 0G QW B;22(E, p)Iflav. # , (60a) GFQW BZ;n.d.(E, p):=

Finding the simplest expression for the direct-space FQW GF The difficulty here is the new, rhombus BZ. We will first perform some changes of variables that will make us able to compute the integral of Eq. (45) in half the situations we encounter. We rewrite Eq. (45) as GFQW(t, x;t′, x′) =I (1)(D,∆n) +I (2)(D,∆n) +I (3)(D,∆n), (53) 13 with I (1)(D,∆n) := ...

[2] [1]

flavor staggering only

Recap on the flavored Dirac QCA In Ref. [3], we have proposed a solution to solve the FD problem of the Dirac QCA, problem which has been re- explained in detail in Sec. IIIC. We have called this so- lution “flavor staggering only”, because it is based on the staggering of extra flavor degrees of freedom that we in- troduced, but without staggering the ch...

[3] [2]

Looking for an appropriate GF This rhombus shape we have for the BZ implies in par- ticular that a suitable ansatz form with which to look for the Green’s function of the Dirac FQCA is GFQW(t, x;t′, x′) = 2 ˆ π/ϵ −π/ϵ dp 2π eip(x−x′) ˆ π/ϵ−|p| −π/ϵ+|p| dE 2π e−iE(t−t′)GFQW BFQW(E, p), (45) instead of the ansatz of Eq. (17). The factor of2 = ( √ 2)2 in fro...

[4] [4]

Flavouredlatticeschwingermodel with chiral anomaly,

DogukanBakircioglu,“Flavouredlatticeschwingermodel with chiral anomaly,” (2026), arXiv:2604.13146 [hep-lat]

arXiv 2026

[5] [5]

A quantum cellu- lar automaton for one-dimensional QED,

P. Arrighi, C. Bény, and T. Farrelly, “A quantum cellu- lar automaton for one-dimensional QED,” Quantum Inf. Process.19, 88 (2020), arXiv:1903.07007

arXiv 2020

[6] [6]

A relativistic discrete spacetime formulation of 3+1 QED,

N. Eon, G. Di Molfetta, G. Magnifico, and P. Arrighi, “A relativistic discrete spacetime formulation of 3+1 QED,” Quantum7, 1179 (2023), 2205.03148

arXiv 2023

[7] [7]

Fermion doubling in quantum cellular automata,

D. Bakircioglu, P. Arnault, and P. Arrighi, “Fermion doubling in quantum cellular automata,” arXiv:2505.07900 (2025)

arXiv 2025

[8] [8]

Green’s functions in quantum me- chanics,

Robert B. Littlejohn, “Green’s functions in quantum me- chanics,” Course given by Robert B. Littlejohn at Berke- ley University (2020)

2020