Gauge theories a la Utiyama

Abstract

We review the construction of the gauge theory for semi-simple Lie groups by Utiyama in “Invariant Theoretical Interpretation of Interaction”^[1]. It is shown an auxiliary field $A_{\mu }^a\left(x\right)$ must be introduced in order to keep the system of fields ${\varphi }^A\left(x\right)$ invariant under a transformation group depending on $n$ parameters ${ϵ}^a\left(x\right)$. This auxiliary field interacts with $\varphi$ through the covariant derivative ${\nabla }_{\mu }{\varphi }^A$. We determine the transformation law for $A_{\mu }^a$ under the $x^{\mu }$-dependent Lie group and calculate the field strength $F_{\mu \nu }^a\left(x\right)$. Moreover, we specify the conserved current $J_a^{\mu }$ related to the invariance of the complete system. The paper ends with the application of the general theory to the cases of the charged particle in an electromagnetic field and of the Yang-Mills potential under isotopic spin space transformations; we briefly address the matter of the gravitational field as a gauge theory; finally, we comment on the extension of Utiyama's theory for ${\mathcal{L}}_A={\mathcal{L}}_A\left[A_{\mu }^a;{\partial }_{\nu }{A}_{\mu }^a;{\partial }_{\rho }{\partial }_{\nu }{A}_{\mu }^a\right]\left(x\right)$.

Keywords:
gauge theories; Utiyama method