danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2024-68-1-18-27dan-1171Research ArticleФИЗИКАPHYSICSНерелятивистское приближение в теории Паули–Фирца для частицы со спином 3/2 в присутствии внешних полейNon-relativistic approximation in the Pauli–Fierz theory for a spin 3/2 particle in the presence of external fieldsИвашкевичА. В.IvashkevichA. V.

Ивашкевич Алина Валентиновна – аспирант.

Пр. Независимости, 68-2, 220072

Ivashkevich Alina V. – Postgraduate Student.

68-2, Nezavisimosti Ave., 220072, Minsk

ivashkevich.alina@yandex.byРедьковВ. М.Red’kovV. M.

Редьков Виктор Михайлович – д-р физ.-мат. наук, гл. науч. сотрудник.

Пр. Независимости, 68-2, 220072, Минск

Red’kov Viktor M. – D. Sc. (Physics and Mathematics), Chief Researcher.

68-2, Nezavisimosti Ave., 220072, Minsk

redkov@dragon.bas-net.byИшханянА. М.IshkhanyanA. M.

Ишханян Артур Михайлович – член-корреспондент Национальной академии наук Республики Армения, д-р физ.-мат. наук, профессор.

Гитаван, 15/2, Аштарак, 0203

Ishkhanyan Artur M. – Corresponding Member of the National Academy of Sciences of the Republic of Armenia, D. Sc. (Physics and Mathematics), Professor.

15/2 Gitavan, Ashtarak, 0203

aishkhanyan@gmail.comИнститут физики имени Б.И. Степанова Национальной академии наук БеларусиB.I. Stepanov Institute of Physics of the National Academy of Sciences of BelarusИнститут физических исследований Национальной академии наук Республики АрменияInstitute for Physical Research of the National Academy of Sciences of Republic of Armenia2024040320246811827Copyright © Ивашкевич А.В., Редьков В.М., Ишханян А.М., 20242024Ивашкевич А.В., Редьков В.М., Ишханян А.М.Ivashkevich A.V., Red’kov V.M., Ishkhanyan A.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1171

Исследуется нерелятивистское приближение для релятивистской системы из 16 уравнений в декартовых координатах для волновой функции частицы со спином 3/2 с трансформационными свойствами вектор-биспинора относительно группы Лоренца. При осуществлении нерелятивистского приближения для выделения в волновой функции больших и малых составляющих используется метод проективных операторов. Соответственно полная волновая функция представляется в виде суммы трех частей: зависящей от 6 переменных большой составляющей и двух малых составляющих, зависящих в совокупности от 14 переменных. Найдены два линейных ограничения на 6 больших компонент, и 2 ограничения на 14 малых. После выполнения процедуры нерелятивистского приближения выведено 6 уравнений с нерелятивистской структурой относительно 4 больших компонент. Показано, что только 4 уравнения из 6 являются независимыми. В результате найдено уравнение паулиевского типа для 4-компонентной волновой функции. Найден явный вид трех 4-мерных матриц, представляющих спин частицы. Анализ трасформационных свойств нерелятивистской волновой функции позволяет обобщить структуру найденного уравнения на случай искривленного трехмерного пространства.

In the paper, we examine the non-relativistic approximation in the relativistic system of equations in Cartesian coordinates for 16-component wave functions with transformation properties of the vector-bispinor under the Lorentz group. When performing the non-relativistic approximation, for separating large and small components in the complete wave function we apply the method of projective operators. Accordingly, the complete wave function is presented as a sum of three parts: the large part depends on 6 variables, and the small ones depend on 14 variables. We have found two linear constraints on large components and two constraints on the small ones. After performing the procedure of the non-relativistic approximation we have derived 6 equations with a needed non-relativistic structure, which include only 4 large components. It is proved that only 4 equations are independent, so we have arrived at the generalized Pauli-like equation for the 4-component wave function. The analysis of transformation properties of the non-relativistic wave function permits us to generalize the structure of the derived equation to an arbitrary curved 3-space.

частица со спином 3/2внешнее электромагнитное поледекартовые координатынерелятивистское приближениепроективные операторытетрадный формализмискривленное трехмерное пространствоspin 3/2 particleexternal electromagnetic fieldCartesian coordinatesnonrelativistic approximаtionprojective operatorscurved 3-spacetetrad formalismРабота выполнена при финансовой поддержке Комитета науки Армении (гранты № 21AG-1C064 и 21SC-BRFFR-1C021) и Белорусского республиканского фонда фундаментальных исследований (проект Ф21Арм-022)The work was carried out with financial support from the Armenian Science Committee (grant no. 21AG1C064 and 21SC-BRFFR-1C021) and the Belarusian Republican Foundation for Fundamental Research (project Ф21Арм-022)ReferencesPauli W., Fierz M. Über relativistische Feldleichungen von Teilchen mit beliebigem Spin im elektromagnetischen Feld. Helvetica Physica Acta, 1939, bd. 12, ss. 297–300 (in German).Pauli W., Fierz M. 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The authors declare that there are no conflicts of interest present.