Fractional linear maps in general relativity and quantum mechanics

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. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.