On the zero helicity condition for quantum vortex defects

In this note we provide an analytical proof of the zero helicity condition for systems governed by the Gross-Pitaevskii equation (GPE). The proof is based on the hydrodynamic interpretation of the GPE, and the direct use of Noether's theorem by applying Kleinert's multi-valued gauge theory. As a by-product we also demonstrate the conservation and quantization of the circulation for the GPE.