Feynman checkers: External electromagnetic field and asymptotic properties

Abstract

In this paper, we study Feynman checkers, one of the most elementary models of electron motion. It is also known as a one-dimensional quantum walk or an Ising model at an imaginary temperature. We add the simplest non-trivial electromagnetic field and find the limits of the resulting model for small lattice step and large time, analogous to the results by Narlikar from 1972 and Grimmet–Jason–Scudo from the 2000s. It turns out that the limits in the model with the added field are obtained from the ones without field by mass renormalization. Also, we find an exact solution of the resulting model.