The Feynman checkerboard model — Feynman's discrete path-sum for the 1+1-dimensional Dirac electron — is, when its paths are confined to an absorbing strip 0<x≤L0<x\le L, exactly a six-vertex (ice-type) model with complex vertex weights built from the mass mm and the lattice step ε\varepsilon. For the model on the full line the ε→0\varepsilon\to0 continuum limit is known (convergence to the Dirac propagator), and in the strip the total reflection probability is known to converge to the classical thin-film (Airy) formula; but Ozhegov, Skopenkov and Ustinov (2024) state that no amplitude-level limit is known: an asymptotic formula for the strip amplitudes a±(x,t;τ)a_\pm(x,t;\tau) as ε↘0\varepsilon\searrow0 is explicitly posed as an open problem, the general ε↘0\varepsilon\searrow0 limit of the complex-weight six-vertex model is declared 'widely open', and even the eigenvalues and eigenfunctions of the governing transfer matrix are unknown.

Background

The Feynman checkerboard model is a discrete path-integral representation of the 1+1-dimensional Dirac equation, sketched by R. Feynman in 1965. A 'checker' moves on the lattice εZ2\varepsilon\mathbb{Z}^2 by diagonal steps (±ε,ε)(\pm\varepsilon,\varepsilon); every turn of the path multiplies the amplitude by the factor −imε-im\varepsilon, where m>0m>0 is the electron mass and ε>0\varepsilon>0 the lattice step, and summing (−imε)turns(p)/(1+m2ε2)ℓ(p)/2(-im\varepsilon)^{\mathrm{turns}(p)}/(1+m^2\varepsilon^2)^{\ell(p)/2} over all checker paths pp from (0,τ)(0,\tau) to (x,t)(x,t) gives the amplitudes a±(x,t;τ)a_\pm(x,t;\tau) of finding the right- or left-moving electron at xx at time tt in the more conventional normalization of equation (17) of the source (here turns(p)\mathrm{turns}(p) is the number of turns of the path and ℓ(p)\ell(p) its number of interior points — the same for all paths between fixed endpoints; the divisor keeps the single-step evolution unitary). The Problem Statement below instead uses the strip normalization of equations (15)–(16) of the source, in which the divisor is (1+imε)ℓ(p)(1+im\varepsilon)^{\ell(p)}; for fixed endpoints the two normalizations differ only by the overall phase factor e−iℓ(p)arctan⁡(mε)e^{-i\ell(p)\arctan(m\varepsilon)}, which cancels in probabilities but not in amplitudes. When the paths are confined to the strip 0<x≤L0<x\le L, the electron is absorbed at the lines x=0x=0 and x=L+εx=L+\varepsilon; this is the lattice model of thin-film reflection of light studied by Ozhegov–Skopenkov–Ustinov.

Painting the path segments red, the segments around each lattice point form one of the six allowed local configurations of the six-vertex (ice-type) model — the standard statistical-mechanical model on the square lattice in which every vertex has two incoming and two outgoing arrows (the 'ice rule') and carries one of three vertex weights a,b,ca,b,c according to the local arrow pattern. The checkerboard model is thus a six-vertex model whose vertex weights are complex: a vertex untouched by the path carries weight 11, a straight passage of the path through the vertex carries 1/1+m2ε21/\sqrt{1+m^2\varepsilon^2}, and a turn carries −imε/1+m2ε2-im\varepsilon/\sqrt{1+m^2\varepsilon^2} (Figure 5, middle row, of the source), so the path sum becomes a partition-function-like sum over all six-vertex-rule configurations. For real positive weights the six-vertex model is intensively studied — for example, its free energy is known exactly (Duminil-Copin–Kozlowski–Krachun–Manolescu–Tikhonovskaia 2022) — but the weights here are complex: they encode quantum amplitudes, not probabilities, and they carry the small parameter ε\varepsilon.

For the model without the strip, the continuum limit is understood: the checkerboard amplitudes converge as ε→0\varepsilon\to0 to the Dirac propagator — the fundamental solution of the Dirac equation (Feynman's 1965 consistency problem, solved rigorously in the survey of Skopenkov–Ustinov 2022), with extensions to an external electromagnetic field via mass renormalization (Ozhegov 2024) and to several interacting particles, where the model is likewise described as a six-vertex model with complex weights and the ε→0\varepsilon\to0 consistency of the expected charge density is proved via an exact hypergeometric solution (Skopenkov–Ustinov 2024). For the absorbing strip, exactly one limit is known, proved in the source paper itself: the total reflection probability P(ω,m,L,ε)P(\omega,m,L,\varepsilon) converges as ε↘0\varepsilon\searrow0 to the classical thin-film (Airy) formula (n2−1)2/((n2+1)2+4n2cot⁡2ωnL)(n^2-1)^2/((n^2+1)^2+4n^2\cot^2\omega nL) with refractive index n=1+2m/ωn=\sqrt{1+2m/\omega}. The proof goes through the spectral radius ρ(T)<1\rho(\mathbf T)<1 — the maximal modulus of the eigenvalues of T\mathbf T — of the finite-dimensional transfer matrix that advances the two-component amplitude distribution by one time step, together with Gelfand's formula ρ(T)=lim⁡n→∞∥Tn∥1/n\rho(\mathbf T)=\lim_{n\to\infty}\|\mathbf T^n\|^{1/n}, but no eigenvalues or eigenfunctions of T\mathbf T are known, and no limit at the level of individual amplitudes or of the complex-weight six-vertex configuration sums is known.

Problem Statement

Consider the Feynman checkerboard model of lattice step ε\varepsilon in the absorbing strip 0<x≤L0<x\le L — equivalently, the six-vertex (ice-type) model on the square lattice with the complex vertex weights induced by the checkerboard rules: a vertex untouched by the checker path carries weight 11, a straight passage of the path through a vertex carries 1/1+m2ε21/\sqrt{1+m^2\varepsilon^2}, and a turn carries −imε/1+m2ε2-im\varepsilon/\sqrt{1+m^2\varepsilon^2} (Figure 5 and equations (15)–(17) of Ozhegov–Skopenkov–Ustinov, arXiv:2407.03258). Establish the continuum limit of this discrete model at the amplitude level, as ε↘0\varepsilon\searrow0 with the other parameters fixed. Concretely:

  1. Find an asymptotic formula for the strip amplitudes a±(x,t;τ)  =  ∑s:(0,τ)⇝(x,t)(−imε)ℓ(s),a_\pm(x,t;\tau)\;=\;\sum_{s:(0,\tau)\rightsquigarrow(x,t)}(-im\varepsilon)^{\ell(s)}, where the sum runs over all light paths ss from (0,τ)⇝(x,t)(0,\tau)\rightsquigarrow(x,t) inside the strip 0<x≤L0<x\le L — a light path being a finite sequence of points of εZ2\varepsilon\mathbb{Z}^2 such that the vector from each point (except the last one) to the next one, called a step, is cooriented with (1,1)(1,1) or (−1,1)(-1,1), and each point (possibly, except the first and the last ones) belongs to the strip 0<x≤L0<x\le L, so that the starting point (0,τ)(0,\tau) may itself lie on the absorbing line x=0x=0 — a step can thus be (0,0)(0,0), (±ε,ε)(\pm\varepsilon,\varepsilon), (±2ε,2ε)(\pm2\varepsilon,2\varepsilon), etc., and the zero vector is viewed as cooriented with any vector, so that repeating points in ss (several scatterings at one lattice point) are allowed — and the last step is nonzero and cooriented with (ε,ε)(\varepsilon,\varepsilon) for a+a_+, (−ε,ε)(-\varepsilon,\varepsilon) for a−a_- — and ℓ(s)\ell(s) is the number of scatterings, i.e. the number of points of ss counted with repetitions, excluding the first and the last (equivalently, by equation (16) of the source, a sum over checker paths pp of (−imε)turns(p)/(1+imε)ℓ(p)(-im\varepsilon)^{\mathrm{turns}(p)}/(1+im\varepsilon)^{\ell(p)}), as ε↘0\varepsilon\searrow0 for fixed m,L,x,t,τm,L,x,t,\tau. This is the explicitly stated open problem of the source; the resulting limit is expected to be consistent with the already-proven limit of the total reflection probability, though the source does not pin the answer's form.

  2. More generally, establish the ε↘0\varepsilon\searrow0 limit theory for the six-vertex model with these complex weights — rigorous limiting expressions for its configuration sums and amplitude-level observables (the analogue, for complex weights, of the limiting free energy and correlation functions of the real-weight theory) — which the source declares, in general, a widely open problem.

  3. Find the eigenvalues and eigenfunctions of the transfer matrix T\mathbf T that advances the strip amplitudes by one time step, as functions of mm, ε\varepsilon, and LL.

The objective is the continuum limit of the discrete complex-weight model at the amplitude level, extending the known limit of the total reflection probability. The problem is not narrowed to any single observable, and no additional restrictions (geometry, parameter range, or answer form) beyond those stated in the source are imposed.

References

  1. Fedor Ozhegov, Mikhail Skopenkov, Alexey Ustinov, 'Feynman checkers: through the looking-glass', arXiv:2407.03258 (2024); published as 'Feynman Checkers: Through the Looking-Glass', The Mathematical Intelligencer 47(3) (2025) 210–221, DOI 10.1007/s00283-024-10401-x, https://arxiv.org/abs/2407.03258
  2. Mikhail Skopenkov, Alexey Ustinov, 'Feynman checkers: towards algorithmic quantum theory', Russian Mathematical Surveys 77:3 (2022), arXiv:2007.12879, DOI 10.1070/rm10025, https://arxiv.org/abs/2007.12879
  3. Mikhail Skopenkov, Alexey Ustinov, 'Feynman checkers: lattice quantum field theory with real time', Analysis and Mathematical Physics 14 (2024), arXiv:2208.14247, DOI 10.1007/s13324-024-00896-0, https://arxiv.org/abs/2208.14247
  4. Fedor Ozhegov, 'Feynman checkers: external electromagnetic field and asymptotic properties', Reviews in Mathematical Physics 36(7) (2024), arXiv:2209.00938, DOI 10.1142/S0129055X2450017X, https://arxiv.org/abs/2209.00938
  5. Mikhail Dmitriev, 'Feynman Checkers with Absorption', arXiv:2204.07861 (2022); published in Russian as 'Shashki Feinmana s pogloshcheniyem' ('Feynman checkers with absorption'), Sibirskie Elektronnye Matematicheskie Izvestiya 20 (2023) 626–637, DOI 10.33048/semi.2023.20.037, https://arxiv.org/abs/2204.07861
  6. Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, Tatiana Tikhonovskaia, 'On the Six-Vertex Model's Free Energy', Communications in Mathematical Physics 395 (2022) 1383–1430, DOI 10.1007/s00220-022-04459-x, https://doi.org/10.1007/s00220-022-04459-x