Comment on ``Reduction of static field equation of Faddeev model to first order PDE'', arXiv:0707.2207

TL;DR

The paper critiques a proposed method for solving the Faddeev model by reducing it to first order PDEs, identifying an error that invalidates the original results.

Contribution

It analyzes and points out an error in the derivation of a method to solve the Faddeev model using first order PDE reduction.

Findings

The original method contains a critical derivation error.

The error invalidates the proposed solutions.

The critique clarifies the limitations of the method.

Abstract

The authors of the article Phys. Lett. B 652 (2007) 384, (arXiv:0707.2207), propose an interesting method to solve the Faddeev model by reducing it to a set of first order PDEs. They first construct a vectorial quantity $\bm \alpha $, depending on the original field and its first derivatives, in terms of which the field equations reduce to a linear first order equation. Then they find vectors $\bm \alpha_1$ and $\bm \alpha_2$ which identically obey this linear first order equation. The last step consists in the identification of the $\bm \alpha_i$ with the original $\bm \alpha$ as a function of the original field. Unfortunately, the derivation of this last step in the paper cited above contains an error which invalidates most of its results.