### References & Citations

# General Relativity and Quantum Cosmology

# Title: Fractional linear maps in general relativity and quantum mechanics

(Submitted on 29 Mar 2021 (v1), last revised 22 Jul 2021 (this version, v4))

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.

## Submission history

From: Giampiero Esposito Dr. [view email]**[v1]**Mon, 29 Mar 2021 08:14:47 GMT (127kb)

**[v2]**Tue, 30 Mar 2021 12:51:50 GMT (129kb,D)

**[v3]**Sun, 13 Jun 2021 15:08:03 GMT (129kb,D)

**[v4]**Thu, 22 Jul 2021 09:42:55 GMT (125kb,D)

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