Diffeomorphism Groups and Quantized Vortex Filaments

The quantum hydrodynamics of an incompressible inviscid fluid in $R^2$ or $R^3$ is described by means of representations of the group of volume-preserving diffeomorphisms. In $R^2$ we obtain the surprising result that within the framework of the Euler equations point vortices cannot be consistently quantized, and the encouraging outcome that vortex filaments can be. We also discuss analogs of such configurations in $R^3$.