The Feynman checkerboard: discrete quantum mechanics

Joint work with A. Ustinov.
We study the most elementary model of electron motion introduced by R.Feynman. It is a game, in which a checker moves on a checkerboard by simple rules, and we count the turnings. We give a first rigorous proof that the continuum limit of the model reproduces the retarded Green function for the $(1+1)$-dimensional Dirac equation, and provide an explicit estimate for the convergence rate. This justifies a heuristic derivation by J.Narlikar from 1972. In a sense, this is also a continuum limit of a 1-dimensional Ising model at imaginary temperature (H.Gersch, 1981), and a new approach to making quantum field theory rigorous and algorithmic. For the model, we also show an exact charge conservation and a coupling to lattice gauge theory, and state visual open problems.
Most of the talk is accessible to undergraduate students; no knowledge of physics is assumed.
The work was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018-2019 (grant N18-01-0023) and by the Russian Academic Excellence Project “5-100”.