Non-commuting coordinates in vortex dynamics and in the Hall effect related to “exotic” Galilean symmetry

Vortex dynamics in a thin superfluid 4He film as well as in a type II superconductor is described by the classical counterpart of the model advocated by Peierls, and used for deriving the ground states of the Fractional Quantum Hall Effect. The model has non-commuting coordinates, and is obtained by reduction from a particle associated with the “exotic” extension of the planar Galilei group. 1 Vortex dynamics and the Peierls substitution (Quantum) Mechanics with non-commuting coordinates[1, 2], {x, y} = θ, (1) has become the focus of recent research. Such a relation may appear rather puzzling. Below we argue, however, that it is inherent in a number of physical instances, and could indeed have been recognized many years ago. Our first example of non-commuting coordinates is provided by the effective dynamics of point-like flux lines in a thin film of superfluid He [3, 4]. For the sake of simplicity, we restrict ourselves to two vortices of identical vorticity. The center-of-vorticity coordinates are constants of the motion. For the relative coordinates x = x1 − x2 and y = y1 − y2, respectively, the equations of motion become[3, 4, 5] (ρLκ) ẋ = ∂yH, (ρLκ) ẏ = −∂xH, (2) where ρ and L are the density and the thickness of the film, respectively; κ is the (quantized) vorticity. The Hamiltonian reads