Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1996-06-12
Phys.Rev.Lett. 78 (1997) 163
Physics
High Energy Physics
High Energy Physics - Theory
11 pages. Standard Latex. Final version to appear in Physical Review Letters. Revised and extended version. The formalism is e
Scientific paper
10.1103/PhysRevLett.78.163
By defining a prepotential function for the stationary Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger 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. The formalism is extended to higher dimensions and to the Klein-Gordon equation.
Faraggi Alon E.
Matone Marco
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