A Statistical Interpretation of Space and Classical-Quantum duality

By defining a prepotential function for the stationary Schrödinger equation we derive an inversion formula for the space variable $x$ as a function of the wave--function $\psi$. The resulting equation is a Legendre transform that relates $x$, the prepotential ${\cal F}$, and the probability density. We invert the Schrödinger equation to a third--order differential equation for ${\cal F}$ and observe that the inversion procedure implies a $x$--$\psi$ duality. This phenomenon is related to a modular symmetry due to the superposition of the solutions of the Schrödinger equation. We propose that in quantum mechanics the space coordinate can be interpreted as a macroscopic variable of a statistical system with $\hbar$ playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to $\tau=\partial_{\psi}^2{\cal F}$ is determined by the ``beta--function''. We propose that the quantization of the inversion formula is a natural way to quantize geometry.