Extension of the Poincar\'e Group and Non-Abelian Tensor Gauge Fields

TL;DR

This paper extends the Poincaré group to include non-Abelian tensor gauge fields, blending internal and space-time symmetries, and explores the algebraic structure and representations of this enlarged symmetry group.

Contribution

It introduces a novel extension of the Poincaré group incorporating non-Abelian tensor gauge fields and analyzes its irreducible representations and physical implications.

Findings

Extended Poincaré group with infinitely many generators

Identification of a vector variable with the derivative of the Pauli-Lubanski vector

Projection of non-Abelian tensor gauge fields onto the transversal plane

Abstract

In the recently proposed generalization of the Yang-Mills theory the group of gauge transformation gets essentially enlarged. This enlargement involves an elegant mixture of the internal and space-time symmetries. The resulting group is an extension of the Poincar\'e group with infinitely many generators which carry internal and space-time indices. This is similar to the super-symmetric extension of the Poincar\'e group, where instead of an anti-commuting spinor variable one should introduce a new vector variable. The construction of irreducible representations of the extended Poincar\'e algebra identifies a vector variable with the derivative of the Pauli-Lubanski vector over its length. As a result of this identification the generators of the gauge group have nonzero components only in the plane transversal to the momentum and are projecting out non-Abelian tensor gauge fields into the transversal plane, keeping only their positively definite space-like components.