Some topological issues for ferromagnets and fluids

TL;DR

This paper investigates the topological and canonical structure of continuum ferromagnet models, revealing phase space complexities and their connections to fluid mechanics, with implications for quantum theory in two dimensions.

Contribution

It clarifies the canonical structure of ferromagnet models, addresses phase space connectivity issues, and links these to fluid mechanics and quantum theory.

Findings

Moments of momentum density related to volume-preserving transformations can be defined.

A phase space enlargement is needed for a nonsingular definition of other moments.

The phase space connectivity affects quantum formulations, especially in two dimensions.

Abstract

We analyze the canonical structure of a continuum model of ferromagnets and clarify known difficulties in defining a momentum density. The moments of the momentum density corresponding to volume-preserving coordinate transformations can be defined, but a nonsingular definition of the other moments requires an enlargement of the phase space which illuminates a close relation to fluid mechanics. We also discuss the nontrivial connectivity of the phase space for two and three dimensions and show how this feature can be incorporated in the quantum theory, working out the two-dimensional case in some detail.