General Lagrangian Formulation for Higher Spin Fields with Arbitrary Index Symmetry. I. Bosonic fields

TL;DR

This paper develops a universal Lagrangian framework for describing arbitrary higher-spin bosonic fields with mixed symmetry, using BRST methods and auxiliary representations, applicable to massless and massive cases.

Contribution

It introduces a novel universal procedure for constructing unconstrained gauge-invariant Lagrangians for higher-spin bosonic fields with arbitrary Young tableaux.

Findings

Constructed explicit Lagrangian for a massless rank-4 tensor field with (2,1,1) symmetry.

Developed auxiliary representations of the constraint algebra isomorphic to sp(2k).

Demonstrated the method's applicability to fields with three-row Young tableaux.

Abstract

We construct a Lagrangian description of irreducible integer higher-spin representations of the Poincare group with an arbitrary Young tableaux having k rows, on a basis of the universal BRST approach. Starting with a description of bosonic mixed-symmetry higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space associated with special Poincare module, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a first-class constraint system. For this purpose, we find, for the first time, auxiliary representations of the constraint subalgebra, to be isomorphic due to Howe duality to sp(2k) algebra, and containing the subsystem of second-class constraints in terms of new oscillator variables. We propose a universal procedure of constructing unconstrained gauge-invariant Lagrangians with reducible gauge symmetries describing the dynamics of both massless and massive bosonic fields of any spin. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by the constraints corresponding to an irreducible Poincare-group representation. As examples of the general procedure, we formulate the method of Lagrangian construction for bosonic fields subject to arbitrary Young tableaux having 3 rows and derive the gauge-invariant Lagrangian for new model of massless rank-4 tensor field with spin $(2,1,1)$ and second-stage reducible gauge symmetries.