Feynman checkers: towards algorithmic quantum theory

We survey and develop the most elementary model of electron motion introduced by R$\mathrm{.}$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$\mathrm{.}$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$\mathrm{.}$Narlikar from 1972 and by T$\mathrm{.}$Sunada-T$\mathrm{.}$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.