danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2022-66-6-574-586dan-1097Research ArticleФИЗИКАPHYSICSБезмассовая частица со спином 2: цилиндрическая симметрия, проективные операторы, калибровочные степени свободыMassless spin 2 particle: cylindrical symmetry, projective operators, gauge degrees of freedomИвашкевичА. В.IvashkevichA. I.

Ивашкевич Алина Валентиновна – аспирант.

пр. Независимости, 68-2, 220072, Минск

Alina V. Ivashkevich – Postgraduate Student, B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus.

68-2, Nezavisimosti Ave., 220072, Minsk

ivashkevich.alina@yandex.byБурыйА. В.BuryA. V.

Бурый Антон Васильевич – аспирант.

пр. Независимости, 68-2, 220072, Минск

Anton V. Bury – Postgraduate Student, B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus.

68-2, Nezavisimosti Ave., 220072, Minsk

anton.buryy.97@mail.ruОвсиюкЕ. М.OvsiyukE. M.

Овсиюк Елена Михайловна – кандидат физико-математических наук, доцент, заведующий кафедрой.

ул. Студенческая, 28, 247760, Мозырь, Гомельская обл.

Еlena М. Оvsiyuk – Ph. D. (Physics and Mathematics), Assistant Professor, Head of the Department, Mozyr State Pedagogical University named after I.P. Shamyakin.

28, Studencheskaya  Str.,  247760,  Mozyr

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

Кисель Василий Васильевич – кандидат физико-математических наук, доцент.

ул. П. Бровки, 6, 220013, Минск

Vasily V. Kisel – Ph. D. (Physics and Mathematics), Assistant Professor, Belarusian State University of Informatics and Radioelectronics.

6, P. Brovka Str., 220013, Minsk

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

Редьков Виктор Михайлович – доктор физико-математических наук, главный научный сотрудник.

пр. Независимости, 68-2, 220072, Минск

Viktor M. Red’kov – D. Sc. (Physics and Mathematics), Chief Researcher, B.I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus.

68-2, Nezavisimosti Ave., 220072, Minsk

redkov@dragon.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 Radioelectronics202203012023666574586Copyright © Ивашкевич А.В., Бурый А.В., Овсиюк Е.М., Кисель В.В., Редьков В.М., 20232023Ивашкевич А.В., Бурый А.В., Овсиюк Е.М., Кисель В.В., Редьков В.М.Ivashkevich A.I., Bury 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/1097

Исследуется безмассовая частица со спином 2, при этом применяется матричное уравнение в цилиндрической тетраде пространства Минковского. На решениях диагонализируются операторы энергии, третьей проекции полного момента и третьей проекции импульса; после разделения переменных получена система из 39 уравнений по полярной координате r. Для нахождения решений этих уравнений используется метод Федорова–Гронского, основанный на теории проективных операторов. В соответствии с этим 39 функций выражаются через пять основных функций от переменной r, строящихся в терминах функций Бесселя. Найден явный вид шести независимых решений. Чтобы исключить калибровочные степени свободы, используется явный вид четырех калибровочных решений, строящихся согласно теории Паули–Фирца на основе точных решений уравнения для безмассовой частицы со спином 1 в цилиндрических координатах. После исключения из шести решений четырех калибровочных найдены явные выражения для двух независимых решений, не содержащих калибровочных степеней свободы.

In the present paper, we have developed the theory of a massless spin 2 particle. We apply the matrix equation in Minkowski space-time, specifying it in cylindrical coordinates t, r, φ, z and tetrad. By diagonalizing energy operators, the third projection of total angular momentum, and the third projection of linear momentum, we derive the system of 39 differential equations in a polar coordinate r. In order to resolve this system, we apply the Fedorov–Gronskiy method based on the projective operator method. In accordance with this method, the dependence of all 39 functions is determined only by five different functions of the polar variable r that in the considered case are expressed in terms of Bessel functions. We find the explicit form of six independent solutions of the basic matrix equation. In order to eliminate gauge degrees of freedom, we use the general structure of gauge solutions according to the Pauli-Fierz approach, when the gauge solutions for the spin 2 field are constructed on the basis of the exact solution for a massless spin 1 field (in Bessel functions as well). In this way, we find the explicit form of two independent gauge solutions for the spin 2 field. In the end, we derive the explicit form of two gauge-free solutions for the massless spin 2 field, as should be expected by physical reason.

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