Phase space quantisation of interacting vortices in two dimensions

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Abstract
Studies the quantisation of point vortices in two dimensions by several alternative symmetrisation rules, and finds that only one rule yields a self-consistent quantum model. The symmetrisation rule is motivated by the attempt to construct an operator that represents the singular classical vorticity, and yields a quantum model which exhibits the effect of a finite vortex core at short distances. A generalisation of the Onsager-Feynman circulation theorem is obtained which reflects the idea that a vortex cannot be localised to within an accuracy greater than that allowed by the Heisenberg principle, and the combination of these two principles yields a vortex core size of the order of an inter-particle spacing. The model is also used to study the energy spectrum of a pair of interacting vortices. For vortices of equal circulation a discrete spectrum is obtained, reflecting the oscillator symmetry, while the energy of a pair with opposite circulation varies continuously with separation, reflecting translational symmetry in a fluid of infinite extent.