From parabolic to loxodromic BMS transformations

TL;DR

This paper classifies BMS transformations using complex analysis, introduces four realizations of the asymptotic symmetry group, and links hyperbolic and loxodromic transformations to conditions in Sturm-Liouville problems, revealing a deep connection between spacetime symmetries and quantum mechanics.

Contribution

It proposes four realizations of BMS transformations based on their classification as parabolic, hyperbolic, elliptic, or loxodromic, and establishes a link between these transformations and Sturm-Liouville theory.

Findings

Four realizations of BMS transformations are proposed.

Hyperbolic and loxodromic transformations relate to limit-point conditions.

A connection between asymptotic spacetime symmetries and quantum mechanics is identified.

Abstract

Half of the Bondi-Metzner-Sachs (BMS) transformations consist of orientation-preserving conformal homeomorphisms of the extended complex plane known as fractional linear (or Mobius) transformations. These can be of 4 kinds, i.e. they are classified as being parabolic, or hyperbolic, or elliptic, or loxodromic, depending on the number of fixed points and on the value of the trace of the associated 2x2 matrix in the projective version of the SL(2,C) group. The resulting particular forms of SL(2,C) matrices affect also the other half of BMS transformations, and are used here to propose 4 realizations of the asymptotic symmetry group that we call, again, parabolic, or hyperbolic, or elliptic, or loxodromic. In the second part of the paper, we prove that a subset of hyperbolic and loxodromic transformations, those having trace that approaches infinity, correspond to the fulfillment of limit-point condition for singular Sturm-Liouville problems. Thus, a profound link may exist between the language for describing asymptotically flat space-times and the world of complex analysis and self-adjoint problems in ordinary quantum mechanics.