The coefficients in the confluent hypergeometric equation specify
the Regge trajectories and the degeneracy of the angular momentum states. Bound
states are associated with real angular momenta while resonances are
characterized by complex angular momenta. With a centrifugal potential, the
half-plane is tessellated by crescents. The addition of an electrostatic
potential converts it into a hydrogen atom, and the crescents into triangles
which may have complex conjugate angles; the angle through which a rotation
takes place is accompanied by a stretching. Rather than studying the properties
of the wave functions themselves, we study their symmetry groups. A complex
angle indicates that the group contains loxodromic elements. Since the domain
of such groups is not the disc, hyperbolic plane geometry cannot be used.
Rather, the theory of the isometric circle is adapted since it treats all
groups symmetrically. The pairing of circles and their inverses is likened to
pairing particles with their antiparticles which then go on to produce nested
circles, or a proliferation of particles. A corollary to Laguerre’s
theorem, which states that the euclidean angle is represented by a pure
imaginary projective invariant, represents the imaginary angle in the form of a
real projective invariant.
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