Lagrangian formulation for free $6D$ infinite spin field

TL;DR

This paper develops a Lagrangian formulation for free infinite spin fields in six-dimensional Minkowski space using the BRST approach, explicitly constructing the algebraic and representation-theoretic framework for such higher spin fields.

Contribution

It introduces a novel Lagrangian for 6D infinite spin fields based on BRST methods, explicitly realizing Poincaré generators and Casimir operators in this context.

Findings

Constructed a BRST-based Lagrangian for 6D infinite spin fields.

Explicit realization of Poincaré algebra generators in 6D.

Derived conditions for irreducible representations and confirmed their consistency.

Abstract

We construct a Lagrangian that describes the dynamics of a six-dimensional free infinite (continuous) spin field in $6D$ Minkowski space. The Lagrangian is formulated in the framework of the BRST approach to higher spin field theory and is based on a system of constraints defining an irreducible representation of the corresponding Poincar\'e group. The field realization of generators in the $6D$ Poincar\'e algebra and the second-, fourth-, and sixth-order Casimir operators are obtained in explicit form using additional spinor coordinates. Specific aspects of such a realization in six dimensions are discussed. We derive the conditions that determine the irreducible representation $6D$ infinite spin field and reformulate them as operators in the Fock space forming a first-class algebra in terms of commutators. These operators are used to construct the BRST charge and the corresponding Lagrangian. We prove that the conditions of the irreducible representation are reproduced as the consequence of the Lagrangian equations of motion, which finally provides the correctness of the results obtained.