Feynman checkers: through the looking-glass

TL;DR

The paper provides a rigorous, elementary lattice-based realization of Feynman’s thin-film reflection using a checkerboard quantum-walk framework. By defining path amplitudes $a_\pm$ and a reflection arrow $a(\omega,m,L,\varepsilon)$ on a 2D lattice, it derives a Dirac-type recurrence and solves it via a transfer-matrix approach, proving convergence and obtaining a closed-form optical reflection formula in the $\varepsilon\to0$ limit. The main result expresses the reflection probability as $\lim_{\varepsilon\searrow0} P(\omega,m,L,\varepsilon)= \frac{(n^2-1)^2}{(n^2+1)^2+4n^2\cot^2(n\omega L)}$ with $n=\sqrt{1+2m/\omega}$, connecting to the usual single-surface result and to wave-optics intuition. The work also links quantum walks and the six-vertex model, providing a combinatorial, particle-based perspective on optical phenomena and contributing to the broader program of exactly solvable lattice models in quantum theory.

Abstract

Feynman gave a famous elementary introduction to quantum theory by discussing the thin-film reflection of light. We make his discussion mathematically rigorous, keeping it elementary, using his other idea. The resulting model leads to accurate quantitative results and allows us to derive a well-known formula from optics. In the process, we get acquainted with mathematical tools such as Smirnov's fermionic observables, transfer matrices, and spectral radii. Quantum walks and the six-vertex model arise as the next step in this direction.

Figures (6)

• Figure 1: (Left) An experiment to measure the partial reflection of light by a single surface of glass. About $4\%$ of light is reflected and hits Detector A while the rest $96\%$ is transmitted and hits Detector B. (Right) A similar experiment with two reflective surfaces. Depending on the thickness of the glass, from $0\%$ to almost $16\%$ of light is reflected and hits Detector A. The additional surface can "turn off" or "amplify" reflection. This is a challenge for any reasonable theory.
• Figure 2: (Top) The percentage of light reflected by the glass, depending on its thickness. (Bottom) The theory explaining the plot is illustrated for three values of the thickness: the stopwatches for front and back reflection paths, front and back reflection arrows, and length squares of their sums are shown.
• Figure 3: (Left) Reflection as a result of scatterings. The left vertical line (red) depicts a monochromatic source that emits photons in a special predictable way. Their initial arrows at four moments are shown to the left from the source. The blue thin grid stands for the glass and red thick lines illustrate possible light paths. We use units such that the speed of light is $1$ so that the slope of thick lines is $\pm 1$. Black rhombi depict scatterings. The vertical gray line is detector A. (Right) A formalization of the left picture. Black points are the centers of the thin blue squares to the left. The bold number "$\mathbf{2}$" means two scatterings at the same point. In general, the number of scatterings can be arbitrary. We have added such a possibility to the original Feynman model to make it compatible with classical optics. See Definition \ref{['def-main']}.
• Figure 4: A light path $s$ contributing to the value $a_-(x,t)$ and the corresponding paths $s_0, s_1, s_2, \dots$ contributing to $a_-(x-\varepsilon,t+\varepsilon)$. Bold number "$\mathbf{2}$" means that the point is repeated twice in the path $s_2$. See the proof of Lemma \ref{['Dirac_1']}.
• Figure 5: (Top) Feynman checkers as a six-vertex model. The steps of a checker path are thick red segments; the remaining segments joining diagonal neighbors are thin light. Around each lattice point, they form one of the six configurations. The leftmost configuration does not appear but becomes possible in a more general model describing several moving electrons. We take the product of weights in the middle row over all lattice points. (Bottom) The corresponding configurations of ice molecules at a node of the square crystal lattice.

Theorems & Definitions (20)

• Definition 1
• Theorem 1: Thin-film reflection probability
• Example 1: Boundary conditions
• proof
• Lemma 1: Reccurence
• proof
• Lemma 2: Quasiperiodicity
• proof
• proof : Proof of Theorem \ref{['th-reflection']}
• Theorem 2: The wave function is well-defined