Feynman Checkers: Number theory methods in quantum theory

In the 40's, R. Feynman invented a simple model of electron motion, which is now known as Feynman's checkers. This model is also known as the textit{one-dimensional quantum walk} or the imaginary temperature Ising model. In Feynman's checkers, a checker moves on a checkerboard by simple rules, and the result describes the quantum-mechanical behavior of an electron.

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. We compute the small-lattice-step and the large-time limits, justifying heuristic derivations by J. Narlikar from 1972 and by A. Ambainis et al. from 2001. The main tools are the Fourier transform and the stationary phase method. The talk is based on the joint paper with M. Skopenkov [1].

[1] M. Skopenkov, A. Ustinov, Feynman checkers: towards algorithmic quantum theory. (2020) https://arxiv.org/abs/2007.12879

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