Comment on "Symmetries and Interaction Coefficients of Kelvin waves" [arXiv:1005.4575] by Lebedev and L'vov

TL;DR

This paper critiques a claim about Kelvin wave symmetries, clarifying that the local induction approximation inherently prohibits certain couplings due to symmetry constraints, and highlights a common mathematical misconception.

Contribution

It clarifies the role of symmetries in Kelvin wave interactions and corrects a misinterpretation related to the local induction approximation.

Findings

LIA prohibits Kelvin wave coupling due to symmetry constraints

The perceived coupling arises from a mathematical representation choice

Symmetry considerations are crucial for understanding Kelvin wave kinetics

Abstract

We comment on the claim by Lebedev and L'vov [arXiv:1005.4575] that the symmetry with respect to a tilt of a quantized vortex line does not yet prohibit coupling between Kelvin waves and the large-scale slope of the line. Ironically, the counterexample of an effective scattering vertex in the local induction approximation (LIA) attempted by Lebedev and L'vov invalidates their logic all by itself being a notoriously known example of how symmetries impose stringent constraints on kelvon kinetics---not only the coupling in question but the kinetics in general are absent within LIA. We further explain that the mistake arises from confusing symmetry properties of a specific mathematical representation in terms of the canonical vortex position field w(z) = x(z) + iy(z), which explicitly breaks the tilt symmetry due to an arbitrary choice of the z-axis, with those of the real physical system recovered in final expressions.