Abstract
We construct a duality between several simple physical systems by showing
that they are different aspects of the same quantum theory. Examples include
the free relativistic massless particle and the hydrogen atom in any number of
dimensions. The key is the gauging of the Sp(2) duality symmetry that treats
position and momentum (x,p) as a doublet in phase space. As a consequence of
the gauging, the Minkowski space-time vectors (x^\mu, p^\mu) get enlarged by
one additional space-like and one additional time-like dimensions to (x^M,p^M).
A manifest global symmetry SO(d,2) rotates (x^M,p^M) like d+2 dimensional
vectors. The SO(d,2) symmetry of the parent theory may be interpreted as the
familiar conformal symmetry of quantum field theory in Minkowski spacetime in
one gauge, or as the dynamical symmetry of a totally different physical system
in another gauge. Thanks to the gauge symmetry, the theory permits various
choices of ``time'' which correspond to different looking Hamiltonians, while
avoiding ghosts. Thus we demonstrate that there is a physical role for a
spacetime with two times when taken together with a gauged duality symmetry
that produces appropriate constraints.
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