Unfinished history and paradoxes of quantum potential. I. Non-relativistic origin, history and paradoxes

Abstract

This is the first of two related papers analising and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic point of view. QP arises in the quantum Hamiltonian, under various procedures of quantization of the natural systems, i.e. the Hamilton functions of which are the positive-definite quadratic forms in momenta with coefficients depending on the coordinates in ( n -dimensional) configurational space V n endowed so by a Riemannian structure. The result of quantization may be considered as quantum mecanics (QM) of a particle in V n in the normal Gaussian system of reference in the globally-static spacetime V 1,n . Contradiction of QP to the Principles of General Covariance and Equivalence is discussed.It is found that actually the historically first Hilbert space based quantization by E. Schrödinger (1926), after revision in the modern framework of QM, also leads to QP in the form that B. DeWitt had been found 26 years later. Efforts to avoid QP or reduce its drawbacks are discussed. The general conclusion is that some form of QP and a violation of the principles of general relativity which it induces are inevitable in the non-relativistic quantum Hamiltonian. It is shown also that Feynman (path integration) quantization of natural systems singles out two versions of QP, which both determine two bi-scalar (indepedendent on choice of coordinates) propagators fixing two different algorithms of path integral calculation.In the accompanying paper under the same general title and the subtitle "The Relativistic Point of View", relation of the non-relativistic QP to the quantum theory of the scalar field non-minimally coupled to the curved space-time metric is considered.

Quantum billiards in multidimensional models with branes

“…The Lapalace-Beltrami form of WDW equation was considered previously in [6,53,54,55,56]. Similar prescription appears in quantization of a point-like particle moving in a curved background, for a review see [57,58].…”