Abstract
We study classical and quantum dynamics on the Euclidean Taub-NUT geometry coupled to an abelian gauge field with self-dual curvature and show that, even though Taub-NUT has neither bounded orbits nor quantum bound states, the magnetic binding via the gauge field produces both. The conserved Runge-Lenz vector of Taub-NUT dynamics survives, in a modified form, in the gauged model and allows for an essentially algebraic computation of classical trajectories and energies of quantum bound states. We also compute scattering cross sections and find a surprising electric-magnetic duality. Finally, we exhibit the dynamical symmetry behind the conserved Runge-Lenz and angular momentum vectors in terms of a twistorial formulation of phase space.
Highlights
The Euclidean Taub–NUT (TN) geometry has been studied extensively and from several different points of view. It is interesting as a simple example of a gravitational instanton, it can be viewed as a Kaluza–Klein geometrisation of the Dirac monopole and it arises in the context of monopole moduli spaces, either directly, or, in its ‘negative mass’ form, as an asymptotic limit
We show that all the interesting algebraic features of ordinary TN dynamics carry over to the gauged case, and that, bounded motions and quantum bound states, neither of which are possible on TN alone, occur in the gauged dynamics
While there are no such ambiguities in the definition of the angular momentum operators, they do arise in defining a quantum version of the Runge–Lenz vector
Summary
The Euclidean Taub–NUT (TN) geometry has been studied extensively and from several different points of view It is interesting as a simple example of a gravitational instanton, it can be viewed as a Kaluza–Klein geometrisation of the Dirac monopole and it arises in the context of monopole moduli spaces, either directly, or, in its ‘negative mass’ form, as an asymptotic limit. In a limit where TN becomes flat Euclidean four-space, the four-dimensional magnetic field is constant, and the bound states that we find become ordinary Landau levels This picture of magnetic binding provides a qualitative explanation of the index found by Pope and of the form of the zero-modes discussed in [3].
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