Брст-бфв лагранжевы формулировки для полей высших спинов, подчиненных диаграммам Юнга с двумя столбцами Текст научной статьи по специальности «Математика»

Рассмотрены детали лагранжева описания неприводимых представлений высшего целого спина группы Пуанкаре с таблицей Юнга Y [ˆs1, sˆ2], имеющих 2 столбца для Бозе-частиц, распространяющихся в пространстве-времени Минковского произвольной размерности. Процедура основана, во-первых, на использовании вспомогательного пространства Фока, порожденного фермионными осцилляторами (антисимметричный базис), во-вторых, на построении модуля Верма и нахождении вспомогательной осцилляторной реализации для алгебры sl(2)⊕sl(2), которая кодирует подсистему связей второго рода в супералгебру симметрии высших спинов . Применение универсального БРСТ-БФВ подхода позволяет воспроизвести калибровочно-инвариантные лагранжианы с приводимыми калибровочными симметриями, описывающие свободную динамику как безмассовых, так и массивных смешанно-антисимметричных бозонных полей любого спина с подходяшим набором калибровочных и Штюкельберговых полей. Общая прескрипция обладает возможностью воспроизвести лагранжианы с БРСТ-инвариантными расширенными алгебрическими связями, которые описывают неприводимые представления группы Пуанкаре в терминах смешанно-антисимметричных тензорных полей с 2 группами индексов.

Похожие темы научных работ по математике , автор научной работы — Решетняк А. А.

BRST-BFV LAGRANGIAN FORMULATIONS FOR HS FIELDS SUBJECT TO TWO-COLUMN YOUNG TABLEAUX

The details of Lagrangian description of irreducible integer higher-spin representations of the Poincare group with an Young tableaux Y [ˆs1, sˆ2] having 2 columns are considered for Bose particles propagated on an arbitrary dimensional Minkowski space-time. The procedure is based, first, on using of an auxiliary Fock space generated by Fermi oscillators (antisymmetric basis), second, on construction of the Verma module and finding auxiliary oscillator realization for sl(2)⊕sl(2) algebra which encodes the second-class operator constraints subsystem in the HS symmetry superalgebra. Application of an universal BRST-BFV approach permits to reproduce gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive mixed-antisymmetric bosonic fields of any spin with appropriate number of gauge and Stukelberg fields. The general prescription possesses by the possibility to derive constrained Lagrangians with only BRST-invariant extended algebraic constraints which describes the Poincare group irreducible representations in terms of mixed-antisymmetric tensor fields with 2 group indices.

Текст научной работы на тему «Брст-бфв лагранжевы формулировки для полей высших спинов, подчиненных диаграммам Юнга с двумя столбцами»

BRST-BFV LAGRANGIAN FORMULATIONS FOR HS FIELDS SUBJECT TO TWO-COLUMN YOUNG

Department of Theoretical Physics, Tomsk State Pedagogical University, Kievskaya str., 60, 634061 Tomsk, Russia, Institute of Strength Physics and Material Science Siberian Branch of RAS, Akademicheskii av. 2/4, 634021 Tomsk,

E-mail: reshet@ispms.tsc.ru, reshet@tspu.edu.ru

The details of Lagrangian description of irreducible integer higher-spin representations of the Poincare group with an Young tableaux Y[si,«2] having 2 columns are considered for Bose particles propagated on an arbitrary dimensional Minkowski space-time. The procedure is based, first, on using of an auxiliary Fock space generated by Fermi oscillators (antisymmetric basis), second, on construction ofthe Verma module and finding auxiliary oscillator realization for sZ(2)©sZ(2) algebra which encodes the second-class operator constraints subsystem in the HS symmetry superalgebra. Application of an universal BRST-BFV approach permits to reproduce gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive mixed-antisymmetric bosonic fields of any spin with appropriate number of gauge and Stukelberg fields. The general prescription possesses by the possibility to derive constrained Lagrangians with only BRST-invariant extended algebraic constraints which describes the Poincare group irreducible representations in terms of mixed-antisymmetric tensor fields with 2 group indices.

Keywords: higher spins, BRST operator, Lagrangian formulation, Verma module, gauge invariance.

The belief to reconsider the problems of an unique description of variety of elementary particles and known interactions maybe resolved within higher-spin (HS) field theory whose revealing together with the proof of supersymmetry display, and finding a new insight on origin of Dark Matter remains by the aims in LHC experiment programm ( [1]). Because of the existence of so-called tensionless limit in the (super)string theory [2] which operates with an infinite tower of HS fields with integer and half-integer spins the HS field theory may be considered both of superstring theory part and as an method to study a superstring theory structure. On present state of HS field theory we recommend to know from reviews [3-6]. The paper considers the results of constructing Lagrangian formulations (LFs) for free integer both massless and massive mixed-antisymmetry tensor HS fields on flat R1,d-1-space-time subject to arbitrary Young tableaux (YT) with 2 columns Y[s1, s2] for s1 > s2 in Frons-dal metric-like formalism on a base of BFV-BRST approach [7], and precesses the results which appear soon in [8] (as well as for fermionic mixed-antisymmetric spin-tensor HS fields on R1,d-1-space-time subject to arbitrary Y[n1 + 1 ,n2 + 1 ] in [9]).

The irreducible Poincare or (Anti)-de-Sitter ((A)dS) group representations in the constant curvature space-times may be described both by mixed-symmetric HS fields subject to arbitrary YT with k rows, Y( s 1. sk), (case of symmetric basis) determined by more than one spin-like parameters s j [10,11]

and, equivalently, by mixed-antisymmetric tensor or spin-tensor fields subject to arbitrary YT now with I columns, Y[ s1. s ;], (case of antisymmetric basis) with integers or half-integers s1 > s 2 > . > s; having a spin-like interpretation [8,9]. Both mixed-symmetric and mixed-antisymmetric HS fields appear for d > 4 space-time dimensions, in addition to totally symmetric and antisymmetric irreducible representations of Poincare or (A)dS algebras. Whereas for the latter ones and mixed-symmetric HS fields case the LFs both for massless and massive free higher-spin fields is well enough developed [12-16] as well as on base of BFV-BRST approach, e.g. in [17-24], for the mixed-antisymmetric case the problem of their field-theoretic description has not yet solved except for constrained bosonic fields subject to Y[ s1,s 2] on the level of the equations of motion only [25] in so-called "frame-like" formulation.

We use, first, the conventions for the metric tensor = diag(+, -. —), with Lorentz indices = 0,1. d — 1, second, the notation e(A), gh(A) for the respective values of Grassmann parity and ghost number of a quantity A, and denote by [A, B> the super-commutator of quantities A, B, which for theirs definite values of Grassmann parities is given by [A, B> = AB — (—1)e(A)e(B)BA.

2 Derivation of integer HS symmetry superalgebra on R1,d-1

We consider a massless integer spin irreducible representation of Poincare group in a Minkowski

space R1^-1 which is described by a tensor field

„2 of rank s1 + S2 and

generalized spin s = (s1 , . sS2; sS2+i. sSl) = (2, 2. 2; 1. 1), (with omitting later a sign "*" under Sj and s1 > s2 > 0, s1 < [d/2]) subject to a YT with 2 columns of height s1, s2, respectively