Noneuclidean Tessellations and their relation to Reggie Trajectories

TL;DR

This paper explores the geometric and group-theoretic structures underlying Regge trajectories and quantum states in non-Euclidean tessellations, linking complex angles, symmetry groups, and particle-antiparticle relationships.

Contribution

It introduces a novel approach using non-Euclidean tessellations and isometric circle theory to analyze Regge trajectories and quantum symmetries without relying on hyperbolic geometry.

Findings

Complex angles indicate loxodromic elements in symmetry groups.

The theory of isometric circles is adapted for non-Euclidean tessellations.

Imaginary angles are represented as real projective invariants.

Abstract

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 one 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.