A Statistical Interpretation of Space and Classical–Quantum duality
نویسندگان
چکیده
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 ψ. The resulting equation is a Legendre transform that relates x, the prepotential F , and the probability density. We invert the Schrödinger equation to a third–order differential equation for F and observe that the inversion procedure implies a x–ψ 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 h̄ playing the role of a scaling parameter. We show that the scaling property of the space coordinate with respect to τ = ∂2 ψF is determined by the “beta–function”. We propose that the quantization of the inversion formula is a natural way to
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