Устинов Алексей Владимирович

Feynman Checkers: Through the Looking-Glass

2025 · ARTICLE · en

In his famous lectures, Richard Feynman gave an elementary introduction to quantum theory by discussing the reflection of light by glass. The purpose of this paper is to make his discussion mathematically rigorous while keeping it elementary. This leads us to accurate quantitative results and allows us to derive a well-known formula from optics, see Theorem 2 below. Feynman’s model is simple, and we may begin playing the following “game” immediately. Light entering a glass surface turns into a checker, just as Alice turns into a pawn in a well-known children’s novel; considering all possible checker paths, we compute what part of the light is reflected. But we prefer to begin with a summary of one of Feynman’s lectures to reflect on what it is all about. After that, we define a mathematical model and prove the desired formula. This model is not found in the literature, although it is essentially a particular case of models that are trendy now—quantum walks and the six-vertex model. Finally, along with the most patient readers, in the appendices we dive into previously postponed technicalities and generalizations. Yet some natural questions will remain open or be answered only in the references.

On q-Deformed Cubic Equations: The Quantum Heptagon and Nonagon

2025 · ARTICLE · en

The recent notion of $q$-deformed irrational numbers is characterized by the invariance with respect to the action of the modular group $\PSL(2,\Z)$, or equivalently under the Burau representation of the braid group~$B_3$. The theory of $q$-deformed quadratic irrationals and quadratic equations with integer coefficients is known and entirely based on this invariance. In this paper, we consider the case of cubic irrationals. We show that irreducible cubic equations with three distinct real roots and cyclic Galois group~$C_3$ (or $\Z/3\Z$) acting by a third order element of $\PSL(2,\Z)$, have a canonical $q$-deformation, that we describe. This class of cubic equations contains well-known examples including the equations that describe regular $7$- and $9$-gons.

К последней записи из математического дневника Гаусса

2025 · ARTICLE · ru

Первая часть заметки посвящена доказательству гипотезы из последней записи математического дневника Гаусса. Доказательство полностью элементарно и основано на свойствах сумм Якобшталя. Вторая часть заметки содержит элементарное доказательство одного результата Л. Гончаровой о распределении квадратичных вычетов по модулю простого числа. В недавней статье Кириченко, Цфасмана, Влэдуца, Захаревича этот результат был доказан методами алгебраической геометрии.

Зоногоны

2024 · ARTICLE · ru

В статье рассказывается о зоногонах и связанных с ними задачами геометрии и комбинаторики.

Feynman checkers: lattice quantum field theory with real time

2024 · ARTICLE · en

We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a line. It is a modification of the model known under the names Feynman checkers or one-dimensional quantum walk. It can be viewed as a six-vertex model with certain complex weights of the vertices. The discrete model is consistent with the continuum quantum field theory, namely, reproduces the known expected charge density as the lattice step tends to zero. It is exactly solvable in terms of hypergeometric functions. We introduce interaction resembling Fermi's theory and establish perturbation expansion.

О периодичности последовательностей Сомоса по модулю m

2024 · ARTICLE · ru

Для последовательностей Сомоса конечного ранга доказывается периодичность повторения остатков по произвольному модулю m. В качестве приложения доказывается периодичность остатков последовательности Сомос-(6).

Когда фигурные числа совпадают

2024 · ARTICLE · ru

В статье рассказано об известных случаях совпадения фигурных чисел.

On Periodicity of the Somos Sequences Modulo m.

2024 · ARTICLE · en

Для последовательностей Сомоса конечного ранга доказывается периодичность повторения остатков по произвольному модулю m𝑚. В качестве приложения доказывается периодичность остатков последовательности Сомос-(6).

Random walks and electrical circuits

2023 · CHAPTER · en

Волшебные пятерки

2023 · ARTICLE · ru

В статье рассказывается о неожиданном появлении числа пи в десятичной записи значений синуса специальных углов.