Feynman checkers: towards algorithmic quantum theory

TL;DR

The paper addresses how a minimal lattice model of electron motion, Feynman checkers, can reproduce the continuum quantum propagator and what its large-time, small-lattice-step asymptotics look like. It develops a rigorous analysis of the lattice propagator, proving uniform asymptotics in a triple limit and establishing concentration of measure, thereby connecting discrete checker dynamics to continuum spin-1/2 propagation. Beyond the basic model, it introduces a second-quantized, gauge-coupled framework and outlines related interpretations via Ising, Young diagrams, and Jacobi polynomials, while aligning with the broader program of algorithmic quantum field theory. The results provide a foundation for precise, computable quantum-field-like evolutions on lattices and offer a pathway toward rigorous Minkowskian lattice QFT with potential computational applications.

Abstract

We survey and develop the most elementary model of electron motion introduced by R$.$Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a problem by R$.$Feynman from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results by J$.$Narlikar from 1972 and by T$.$Sunada-T$.$Tate from 2012. For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model.

Figures (6)

• Figure 1: The probability to find an electron in a small square around a given point (white depicts strong oscillations of the probability). Left: in the basic model from §\ref{['sec-basic']} (cf. Yepez-05). Middle: in the upgrade from §\ref{['sec-mass']} for smaller square side. Right: in continuum theory. For the latter, the relative probability density is depicted.
• Figure 2: Double-slit experiment
• Figure 3: The Feynman triple limit: $t\to+\infty$, $x/t\to 0$, $\varepsilon\to 0$
• Figure 4: Continuum limit: the point $(x,t)$ stays fixed while the lattice step $\varepsilon$ tends to zero
• Figure 5: The distribution of the electron position $x$ at time $t=100$ in natural units for the basic model from §\ref{['sec-basic']} (left, dots). Its normalized logarithm (middle, dots) and cumulative distribution function (right, dots). Their (weak) scaling limits as $t\to\infty$ (curves). The middle plot is also (minus the imaginary part of) the limiting free energy density in the Ising model. The non-analyticity of the curves reflects a phase transition.