danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2021-65-6-668-679dan-1019Research ArticleФИЗИКАPHYSICSБезмассовое поле со спином 3/2: сферические решения и устранение калибровочных степеней свободыMassless spin 3/2 field, spherical solutions, eliminating of the gauge degrees of freedomИвашкевичА. В.IvashkevichA. V.

Ивашкевич Алина Валентиновна – аспирант

пр. Независимости, 68-2, 220072, Минск, Республика Беларусь

Ivashkevich Alina V. – Postgraduate student

68-2, Nezavisimosti Ave., 220072, Minsk, Republic of Belarus

ivashkevich.alina@yandex.byОвсиюкЕ. М.OvsiyukE. M.

Овсиюк Елена Михайловна – канд. физ.-мат. наук, доцент, заведующий кафедрой

ул. Студенческая, 28, 247760, Мозырь, Республика Беларусь

Оvsiyuk Еlena М. – Ph. D. (Physics and Mathematics), Assistant professor, Head of the Department

28, Studencheskaya Str., 247760, Mozyr, Republic of Belarus

e.ovsiyuk@mail.ruКисельВ. В.KiselV. V.

Кисель Василий Васильевич – канд. физ.-мат. наук, доцент

ул. П. Бровки, 6, 220013, Минск, Республика Беларусь

Kisel Vasily V. – Ph. D. (Physics and Mathematics), Assistant professor

6, P. Brovka Str., 220013, Minsk, Republic of Belarus

vasiliy_bspu@mail.ruРедьковВ. М.Red’kovV. M.

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

пр. Независимости, 68-2, 220072, Минск, Республика Беларусь

Red’kov Viktor M. – Ph. D. (Physics and Mathematics). Chief researcher

68-2, Nezavisimosti Ave., 220072, Minsk, Republic of Belarus

redkov@ifanbel.bas-net.byИнститут физики имени Б. И. Степанова Национальной академии наук БеларусиB. I. Stepanov Institute of Physics of the National Academy of Sciences of BelarusМозырский государственный педагогический университет имени И. П. ШамякинаMozyr State Pedagogical University named after I. P. ShamyakinБелорусский государственный университет информатики и радиоэлектроникиBelarusian State University of Informatics and Radioelectronics202126122021656668679Copyright © Ивашкевич А.В., Овсиюк Е.М., Кисель В.В., Редьков В.М., 20212021Ивашкевич А.В., Овсиюк Е.М., Кисель В.В., Редьков В.М.Ivashkevich A.V., Ovsiyuk E.M., Kisel V.V., Red’kov V.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/1019

Релятивистская система уравнений для вектор-биспинора, описывающего безмассовую частицу со спином 3/2, исследуется в сферической системе координат и соответствующей тетраде пространства Минковского. Представление волнового уравнения с использованием тензора Леви–Чивита выявляет существование калибровочных решений в виде 4-дивергенции от произвольного биспинора. Подстановка для 16-компонетной полевой функции основана на использовании функций Вигнера, она предполагает диагонализацию четырех операторов: энергии, квадрата и третьей проекции полного углового момента, а также оператора пространственного отражения. После разделения переменных выведена система из 8 радиальных уравнений. Детализируется общая структура калибровочных сферически симметричных решений, показывается, что эти радиальные функции обращают в тождества все 8 уравнений общей системы. Показывается, что общая система приводится к двум парам неоднородных дифференциальных уравнений второго порядка, их частные решения построены на основе использования калибровочных решений специального вида. Соответствующие однородные уравнения имеют одну и ту же структуру с тремя регулярными особыми точками и одной нерегулярной ранга 2. Построены их решения, исследована структура входящих в них степенных рядов с 4-членными рекуррентными соотношениями. Таким образом, построены два независимых класса решений с противоложными четностями, которые не содержат калибровочных компонент.

Relativistic system for a vector-bispinior describing a massless spin 3/2 field is studied in the spherical coordinates of Minkowski space. Presentation of the equation with the use of the covariant Levi-Civita tensor exhibits existence of the gauge solutions in the form of the covariant 4-gradient of an arbitrary bispinor. Substitution for 16-component field function is based on the use of Wigner functions, it assumes diagonalization of the operators of energy, square and third projection of the total angular momentum, and space reflection. We derive radial system for eight independent functions. General structure of the spherical gauge solutions is specified, and it is demonstrated that the gauge radial functions satisfy the derived system. It is proved that the general system reduces to two couples of independent 2-nd order and nonhomogeneous differential equations, their particular solutions may be found with the use of the gauge solutions. The corresponding homogeneous equations have one the same form, they have three regular singularities and one irregular of the rank 2. Frobenius types solutions for this equation have been constructed, and the structure of the involved power series with 4-term recurrent relations sre studied. Six remaining radial functions may be straightforwardly found by means of the simple algebraic relations. Thus, we have constructed two types of solutions with opposite parities which do not contain gauge constituents.

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The authors declare that there are no conflicts of interest present.