A Statistical Interpretation of Space and Classical-Quantum duality

Abstract

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 $\mathcal{F}$, and the probability density. We invert the Schrödinger equation to a third-order differential equation for $\mathcal{F}$ 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 $\hslash$ playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to $\tau ={\partial }_{\psi }^2\mathcal{F}$ is determined by the ``beta-function''. We propose that the quantization of the inversion formula is a natural way to quantize geometry. The formalism is extended to higher dimensions and to the Klein-Gordon equation.

11 pages. Standard Latex. Final version to appear in Physical Review Letters. Revised and extended version. The formalism is extended to higher dimensions and to the Klein-Gordon equation. A possible connection with string theory is considered. The $x-\psi$ duality is emphasized by a minor change in the title. The new title is: Duality of $x$ and $\psi$ and a statistical interpretation of space in quantum mechanics

References in corpus (1)

• Nonperturbative Renormalization Group Equation and Beta Function in N=2 SUSY Yang-Mills

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