Feynman checkers: number-theoretic properties

TL;DR

This paper investigates the number-theoretic properties of Feynman checkers, a model of electron motion and quantum walks, revealing sign alternation patterns in the wave function and relating results to Young diagrams.

Contribution

It introduces new number-theoretic insights into Feynman checkers, connecting wave function properties to combinatorial structures like Young diagrams.

Findings

Sign alternation of real and imaginary parts of the wave function.

Comparison of Young diagrams with odd and even steps.

New number-theoretic results in quantum walk models.

Abstract

We study Feynman checkers, an elementary model of electron motion introduced by R. Feynman. In this model, a checker moves on a checkerboard, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk. We prove some new number-theoretic results in this model, for example, sign alternation of the real and imaginary parts of the electron wave function in a specific area. All our results can be stated in terms of Young diagrams, namely, we compare the number of Young diagrams with an odd and an even number of steps.