Feynman Checkers: Through the Looking-Glass

Mathematical Intelligencer. 2025. Vol. 47. P. 210–221.

Fedor Ozhegov, Skopenkov M., Alexey Ustinov

In his famous lectures, Richard Feynman gave an elementary introduction to quantum theory by discussing
the reflection of light by glass. The purpose of this paper is to make his discussion mathematically rigorous while keeping it elementary. This leads us to accurate quantitative results and allows us to derive a well-known formula from optics, see Theorem 2 below. Feynman’s model is simple, and we may begin playing the following “game” immediately. Light entering a glass surface turns into a checker, just as Alice turns into a pawn in a
well-known children’s novel; considering all possible checker paths, we compute what part of the light is reflected. But
we prefer to begin with a summary of one of Feynman’s lectures to reflect on what it is all about. After that, we define a mathematical model and prove the desired formula. This model is not found in the literature, although it
is essentially a particular case of models that are trendy now—quantum walks and the six-vertex model. Finally, along with the most patient readers, in the appendices we dive into previously postponed technicalities and generalizations. Yet some natural questions will remain open or be answered only in the references.

Keywords: Feynman checkers, шашки Фейнмана

Publication based on the results of: Комбинаторные и числовые задачи на решетках (2023)