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arXiv:2204.07861

Feynman Checkers with Absorption

Dmitriev, Mikhail

Published Apr 16, 2022• 1 citation
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Abstract

We give a new elementary proof of the theorem by Ambainis et al. that for a quantum walk, the probability amplitudes of absorption at the initial point after 4n steps are proportional to the Catalan numbers. We also calculate the absorption probabilities at points close to the initial one for the first time.
DOI
10.48550/arxiv.2204.07861

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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.
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In this paper, we consider limit probability distributions of the quantum walk recently introduced by Wang and Ye (C.S. Wang and X.J. Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Inf. Process. 15 (2016), no. 5, 1897-1908). We first establish several technical theorems, which themselves are also interesting. Then, by using these theorems, we prove that, for a wide range of choices of the initial state, the above-mentioned quantum walk has a limit probability distribution of standard Gauss type, which actually gives a new limit theorem for the walk.
We study the most elementary model of electron motion introduced by R.Feynman in 1965. It is a game, in which a checker moves on a checkerboard by simple rules, and we count the turnings. The model is also known as one-dimensional quantum walk. In his publication, R.Feynman introduces a discrete version of path integral and poses the problem of computing the limit of the model when the lattice step and the average velocity tend to zero and time tends to infinity. We get a nontrivial advance in the problem on the mathematical level of rigor in a simple particular case; even this case requires methods not known before. We also prove a conjecture by I.Gaidai-Turlov, T.Kovalev, and A.Lvov on the limit probability of direction reversal in the model generalizing a recent result by A.Ustinov.