Possible connection between probability, spacetime geometry and quantum mechanics

Following our discussion [E. Canessa, Physica A 375 (2007) 123] to associate an analogous probabilistic description with spacetime geometry in the Schwarzschild metric from the macro- to the micro-domain, we argue that there is a possible connection among normalized probabilities P, spacetime geometry (in the form of Schwarzschild radii rs) and quantum mechanics (in the form of complex wave functions ψ), namely Pθ,φ,t(n)≈Rs(n)/rs=|ψn(n)(X(n))|2/|ψn(x)|2. We show how this association along different (n)-nested surfaces—representing curve space due to an inhomogeneous density of matter—preserves the postulates of quantum mechanics at different geometrical scales.