Feynman Checkers: Number Theory in Quantum Electrodynamics

In the 40s, R. Feynman developed the simplest model of electron motion, now known asFeynman checkers. In this game, the checker moves around the chessboard according to simple rules and as a result describes the quantum-mechanical behavior of the electron. This model is also known as one-dimensional quantum walk or Ising model with imaginary temperature. We solve mathematically a problem by R. Feynman from 1965, which was to prove that the model reproduces the usual quantum-mechanical free-particle kernel for large time, small average velocity, and small lattice step. The main tools are the Fourier transform and the stationary phase method.

A more detailed description of the model and known results can be found in [1].

[1] M. Skopenkov, A. Ustinov, Feynman checkers: towards algorithmic quantum theory // arXiv:2007.12879 [math-ph] (2020).