Feynman checkers: towards algorithmic quantum theory

TL;DR

This paper analyzes Feynman's elementary checker-based model of electron motion, demonstrating its consistency with continuum quantum theory, studying its asymptotic behavior, and introducing second quantization.

Contribution

It mathematically proves the consistency of Feynman's discrete model with continuum quantum mechanics and explores its asymptotic properties and measure concentration.

Findings

Proved the model's consistency with continuum quantum theory for large time and small lattice step.

Improved previous asymptotic results by Narlikar and Sunada-Tate.

First to observe and prove measure concentration in the small-lattice-step limit.

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.