Feynman checkers: towards algorithmic quantum theory
Abstract
We survey and develop the most elementary model of electron motion introduced by RFeynman. 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 RFeynman 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 JNarlikar from 1972 and by TSunada-TTate 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.
Keywords
Cite
@article{arxiv.2007.12879,
title = {Feynman checkers: towards algorithmic quantum theory},
author = {M. Skopenkov and A. Ustinov},
journal= {arXiv preprint arXiv:2007.12879},
year = {2025}
}
Comments
in English and in Russian; 54 pages, 17 figures. Changelog: An important reference to Sunada-Tate, 2012, has been added. This has not affected the main results. But some secondary ones have been replaced by the stronger ones from this reference. Minor correction of Example 4 has been performed, and a proof has been added in Section 12.5