Fractional linear maps in general relativity and quantum mechanics

TL;DR

This paper explores fractional linear transformations in general relativity and quantum mechanics, revealing new classifications and interpretations that could bridge the two theories through a mathematical framework involving Kleinian groups.

Contribution

It introduces a novel perspective on fractional linear maps, linking quantum Hamiltonians with BMS transformations and interpreting limit-point conditions via Kleinian groups.

Findings

Classification of limit-point conditions into loxodromic, hyperbolic, parabolic, and elliptic cases.

Establishment of a connection between second-order differential operators and Kleinian groups.

Construction of solutions for differential equations with singular behavior at both ends of the positive real line.

Abstract

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi-Metzner-Sachs transformations in general relativity. The analogy therefore arising, suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions.