Phys. Rev. Lett. 78, 163–166 (1997)Duality of x and ψ and a Statistical Interpretation of Space in Quantum MechanicsReceived 26 June 1996; revised 18 November 1996; published in the issue dated 13 January 1997 We introduce a “prepotential” F in quantum mechanics and show that the coordinate x is proportional to the Legendre transform of F with respect to the probability density. Inversion of the Schrödinger equation leads us to consider an x– ψ duality related to a modular symmetry. The scaling of x is determined by the “beta function,” suggesting that in quantum mechanics the space coordinate is a macroscopic variable of a statistical system with ħ playing the role of scale. The formalism is extended to higher dimensions and to the Klein-Gordon equation. © 1997 The American Physical Society URL:
http://link.aps.org/doi/10.1103/PhysRevLett.78.163
DOI:
10.1103/PhysRevLett.78.163
PACS:
03.65.Bz, 02.40.-k, 05.70.-a, 11.30.-j
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