Interaction of non-Abelian tensor gauge fields

George Savvidy

doi:10.1016/j.physletb.2017.12.003

Abstract

The non-Abelian tensor gauge fields take value in extended Poincaré algebra. In order to define the invariant Lagrangian we introduce a vector variable in two alternative ways: through the transversal representation of the extended Poincaré algebra and through the path integral over the auxiliary vector field with the U(1) Abelian action. We demonstrate that this allows to fix the unitary gauge and derive scattering amplitudes in spinor representation.

Highlights

The recently proposed generalisation of Yang-Mills theory [26, 27, 28, 29, 34, 35, 36] is based on the extension of the Poincare algebra LG(P) with additional generators Lλa1...λn
Laeλ⊥1 ...eλ⊥n of the algebra and fulfils the equations: e2⊥ = −1, (n · e⊥) = 0, n2 = 0. These equations are reminiscent of the Abelian gauge field equations
We shall elevate this fact into a guiding principle postulating that the dynamics of the vector variables eλ(x) is governed by the Abelian U(1) action and the Lagrangian of the tensor gauge fields is

Summary

The generators have the form and the vacuum expectation values

The Killing metric contracts only the space-like components and guarantee the absence of negative norm time-like components of the tensor gauge fields in the Lagrangian L. 2. The non-Abelian tensor gauge fields Aaμλ1...λs(x), s = 0, 1, 2, ... It incorporates the Poincare and internal algebra LG with structure constants fabc. The algebra LG(P) has representation in terms of the differential operators:. The irreducible transversal representation is defined by invariant equations [30, 31, 32]: n2 = 0, (n · e⊥) = 0, e2⊥ = −1,. Where nλ is an arbitrary light-like vector. Where nλ is an arbitrary light-like vector2 These equations have the solution eμ⊥ = χnμ + eiφeμ+ + e−iφeμ−,.

The representation of the generators

The commutator of covariant derivatives