[Paper Review] Non-Abelian Tensor Gauge Fields. Enhanced Symmetries

George Savvidy|ArXiv.org|Apr 17, 2006

Black Holes and Theoretical Physics48 references3 citations

TL;DR

This paper introduces an extended non-Abelian gauge symmetry for tensor gauge fields of arbitrary integer spin, constructing generalized field strength tensors that yield two infinite series of gauge-invariant quadratic Lagrangians. By tuning coupling constants, the authors achieve enhanced gauge symmetry, leading to highly symmetric equations for rank-2 and rank-3 tensor fields, with the first term recovering the Yang-Mills Lagrangian and all interactions mediated by dimensionless couplings without higher derivatives.

ABSTRACT

We define a group of extended non-Abelian gauge transformations for tensor gauge fields. On this group one can define generalized field strength tensors, which are transforming homogeneously with respect to the extended gauge transformations. The generalized field strength tensors allow to construct two infinite series of gauge invariant quadratic forms. Each term of these infinite series is separately gauge invariant. The invariant Lagrangian is a linear sum of these forms and describes interaction of tensor gauge fields of arbitrarily large integer spins 1,2,.... It does not contain higher derivatives of the tensor gauge fields, and all interactions take place through three- and four-particle exchanges with dimensionless coupling constant. The first term in this sum is the Yang-Mills Lagrangian. The invariance with respect to the extended gauge transformations does not fix the coefficients - the coupling constants - in front of these forms. There is a freedom to vary them without breaking the extended gauge symmetry. We demonstrate that by an appropriate tuning of these coupling constants one can achieve an enhancement of the extended gauge symmetry. This leads to highly symmetric equations. We present the explicit form of the free equations for the rank-2 and rank-3 gauge fields. Their relation to the Schwinger free equation for the rank-3 gauge fields is discussed.

Motivation & Objective

• To extend the non-Abelian gauge principle to tensor gauge fields of arbitrary integer spin, beyond the standard Yang-Mills framework.
• To define generalized field strength tensors that transform homogeneously under extended gauge transformations.
• To construct two infinite series of gauge-invariant quadratic Lagrangians for tensor fields of all ranks.
• To investigate whether tuning the coupling constants in the Lagrangian can lead to enhanced gauge symmetry beyond the initial extended group.
• To derive explicit free field equations for rank-2 and rank-3 tensor gauge fields and relate them to known equations such as Schwinger's.

Proposed method

• Defining extended non-Abelian gauge transformations for rank-(s+1) tensor fields $ A^a_{\mu\lambda_1\dots\lambda_s} $, which are totally symmetric in the $ \lambda $ indices but not in $ \mu $.
• Introducing generalized field strength tensors $ G^a_{\mu\nu,\lambda_1\dots\lambda_s} $ that transform homogeneously under the extended gauge group.
• Constructing two infinite series of gauge-invariant quadratic forms $ \mathcal{L}_s $ and $ \mathcal{L}'_s $, each separately invariant under the extended gauge symmetry.
• Forming a full Lagrangian as a linear sum $ \mathcal{L} = \sum_s g_s \mathcal{L}_s + \sum_s g'_s \mathcal{L}'_s $, with $ \mathcal{L}_1 $ being the Yang-Mills Lagrangian.
• Demonstrating that the coupling constants $ g_s $ and $ g'_s $ are not fixed by gauge symmetry alone, allowing for tuning to enhance symmetry.
• Using explicit computation in momentum space to derive free field equations for rank-2 and rank-3 fields and identifying conditions under which enhanced symmetry arises.

Experimental results

Research questions

• RQ1 Can the non-Abelian gauge principle be consistently extended to tensor gauge fields of arbitrary integer spin, including spin > 1?
• RQ2 Do generalized field strength tensors exist that transform homogeneously under extended gauge transformations?
• RQ3 Is it possible to construct a gauge-invariant Lagrangian for all higher-rank tensor fields without introducing higher derivatives?
• RQ4 Can the coupling constants in the Lagrangian be tuned to achieve a symmetry enhancement beyond the initial extended gauge group?
• RQ5 What are the explicit free field equations for rank-2 and rank-3 tensor gauge fields, and how do they relate to known equations like Schwinger's?

Key findings

• The generalized field strength tensors $ G^a_{\mu\nu,\lambda_1\dots\lambda_s} $ are constructed and transform homogeneously under the extended gauge group, ensuring gauge invariance of the Lagrangian terms.
• Two infinite series of gauge-invariant quadratic forms $ \mathcal{L}_s $ and $ \mathcal{L}'_s $ are derived, each term separately invariant under the extended gauge symmetry.
• The full Lagrangian $ \mathcal{L} = \sum_s g_s \mathcal{L}_s + \sum_s g'_s \mathcal{L}'_s $ describes interacting tensor gauge fields of all integer spins without higher derivatives and with dimensionless couplings.
• For the rank-2 field, tuning the coupling ratio $ c_2 = g'_2 / g_2 = 1 $ enhances the gauge symmetry from the initial extended group to a larger group, indicating symmetry enhancement.
• The free field equations for rank-2 and rank-3 tensor fields are explicitly derived and shown to be consistent with the Schwinger equation for rank-3 fields under specific conditions.
• The analysis confirms that the coupling constants $ g_s $ and $ g'_s $ remain arbitrary under the initial gauge symmetry, providing a mechanism to achieve enhanced symmetry through fine-tuning.