[Paper Review] Non-Abelian tensor gauge fields and extended current algebra. Generalization of Yang-Mills theory

George Savvidy|arXiv (Cornell University)|Oct 30, 2005

Black Holes and Theoretical Physics53 references3 citations

TL;DR

This paper proposes a non-Abelian generalization of Yang-Mills theory by extending gauge symmetry to include higher-rank tensor gauge fields through an infinite-dimensional algebra based on extended current algebra. The framework introduces two gauge-invariant quadratic Lagrangians whose linear combination eliminates negative-norm states, yielding a consistent theory of massless helicity-two and helicity-zero charged tensor gauge bosons.

ABSTRACT

We suggest an infinite-dimensional extension of the gauge transformations which includes non-Abelian tensor gauge fields. Extended gauge transformations of non-Abelian tensor gauge fields form a new large group which has natural geometrical interpretation it terms of extended current algebra associated with compact Lie group. We shall demonstrate that one can construct two infinite series of gauge invariant quadratic forms, so that a linear combination of them comprises the general Lagrangian. The general Lagrangian exhibits enhanced local gauge invariants with double number of gauge parameters and allows to eliminate all negative norm states of the nonsymmetric second-rank tensor gauge field. Therefore it describes two polarizations of helicity-two and helicity-zero massless charged tensor gauge bosons.

Motivation & Objective

• To generalize Yang-Mills theory by extending local gauge symmetry to include non-Abelian tensor gauge fields of arbitrary integer spin.
• To resolve the problem of negative-norm states in nonsymmetric second-rank tensor gauge fields by enhancing gauge invariance.
• To construct a consistent gauge-invariant Lagrangian for higher-spin tensor fields using field strength tensors and extended current algebra.
• To unify vector, symmetric, and nonsymmetric tensor gauge fields under a single non-Abelian gauge structure with infinite-dimensional symmetry.
• To explore connections between the proposed field theory and tensionless string theory via shared symmetry structures.

Proposed method

• Introduces rank-(s+1) tensor gauge fields $ A^a_{ u au_1... au_s} $ that are symmetric in the last $ s $ indices and transform under extended non-Abelian gauge transformations.
• Defines extended gauge transformations using an infinite-dimensional algebra generated by $ L^a_{ au_1... au_s} $, isomorphic to the extended current algebra of the Lorentz group.
• Constructs field strength tensors $ G^a_{ u au, ho au} $ that transform homogeneously under the extended gauge group.
• Derives two independent gauge-invariant quadratic Lagrangians: $ ilde{rak{L}}_s $ and $ ilde{rak{L}}'_s $, both quadratic in field strengths.
• Combines the two Lagrangians into a total Lagrangian $ ilde{rak{L}}_s + c ilde{rak{L}}'_s $, where $ c $ is an arbitrary constant, to achieve enhanced gauge invariance.
• Demonstrates that the combined Lagrangian eliminates all negative-norm states of the nonsymmetric second-rank tensor field $ A^a_{ u au} $, allowing physical states of helicity $ \pm 2 $ and helicity $ 0 $.

Experimental results

Research questions

• RQ1 Can non-Abelian gauge invariance be consistently extended to higher-rank tensor gauge fields beyond the Yang-Mills vector case?
• RQ2 How can the negative-norm states in nonsymmetric second-rank tensor gauge fields be removed while preserving gauge invariance?
• RQ3 What is the structure of the extended gauge algebra that allows for the consistent coupling of higher-spin fields?
• RQ4 Can a unified Lagrangian be constructed from two gauge-invariant quadratic forms that describe both helicity-2 and helicity-0 states?
• RQ5 Is there a field-theoretic realization of the symmetry structure associated with tensionless strings?

Key findings

• The extended gauge transformations form a non-Abelian group with double the number of gauge parameters, enabling the removal of negative-norm states from the nonsymmetric second-rank tensor field.
• The field strength tensor $ G^a_{ u au, ho au} $ is constructed to transform homogeneously under the extended gauge group, ensuring gauge invariance.
• Two independent gauge-invariant Lagrangians are derived: one quadratic in $ G^a_{ u au, ho au} $, and another involving mixed contractions of field strengths.
• The linear combination $ ilde{rak{L}}_s + c ilde{rak{L}}'_s $, with arbitrary $ c $, leads to a fully gauge-invariant theory that describes two physical polarizations of a helicity-2 massless charged tensor gauge boson.
• The theory describes an additional helicity-zero state (axion-like) that arises from the symmetric part of the second-rank tensor field.
• The resulting Lagrangian for the third-rank tensor field $ A^a_{ u au heta} $ is explicitly constructed and shown to be gauge-invariant, with all terms canceling in the variation under the extended transformations.