Normal-product quantization of currents and renormalized Ward-Takahashi identities

Abstract

We discuss the renormalizable $\ensuremath{\sigma}$ model as the realization of Gell-Mann's current algebra. The Green's functions involving many currents are defined using the normal-product technique developed by Zimmermann and Lowenstein. The explicit definition of the so-called covariant $T$ product (${T}^{*}$) is given and reveals the "gauge-invariant" structure. This is compared with the renormalization of the corresponding gauge-field theory, and it is shown that the $\ensuremath{\sigma}$ model with currents acts like a classical approximation to the Higgs model.