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

Abstract

This is the first of two related papers analyzing and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic points of view. The QP arises in the quantum Hamiltonian under various procedures of quantization of natural systems, i.e., those whose Hamilton functions are positive-definite quadratic forms in momenta with coefficients depending on the coordinates in (n-dimensional) configurational space V _n thus endowed with a Riemannian structure. The result of quantization may be considered as quantum mechanics (QM) of a particle in V _n in the normal Gaussian coordinate system in the globally static space-time V _1,n . Contradiction of the QP to the General Covariance and Equivalence principles 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 a QP in the form that B. DeWitt found 26 years later. Efforts to avoid the QP or to 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 also shown that Feynman (path-integral) quantization of natural systems singles out two versions of the QP, which both determine two bi-scalar (independent of a choice of coordinates) propagators fixing two different algorithms of path integral calculation.

The accompanying paper under the same general title and the subtitle “The Relativistic Point of View” (published in the same issue of the journal and referred to as Paper II) considers a relation of the nonrelativistic QP to the quantum theory of a scalar field non-minimally coupled to the curved space-time metric.