Particles, fluids and vortices

TL;DR

This paper explores the duality between particle mechanics and fluid flow on curved spaces, revealing how classical and quantum properties like vorticity and source strength relate through dual transformations.

Contribution

It introduces a duality framework connecting particle mechanics and fluid dynamics, including quantization conditions in quantum theories.

Findings

Duality transformation relates sources, sinks, and vortices in fluid models.

Quantization of vorticity and source strength are interconnected.

Reconstruction of particle mechanics counterpart from fluid dual theories.

Abstract

Classical particle mechanics on curved spaces is related to the flow of ideal fluids, by a dual interpretation of the Hamilton-Jacobi equation. As in second quantization, the procedure relates the description of a system with a finite number of degrees of freedom to one with infinitely many degrees of freedom. In some two-dimensional fluid mechanics models a duality transformation between the velocity potential and the stream function can be performed relating sources and sinks in one model to vortices in the other. The particle mechanics counterpart of the dual theory is reconstructed. In the quantum theory the strength of sources and sinks, as well as vorticity are quantized; for the duality between theories to be preserved these quantization conditions must be related.