Unfinished History and Paradoxes of Quantum Potential. I. Non-Relativistic Origin, History and Paradoxes

TL;DR

This paper explores the origins, manifestations, and paradoxes of the quantum potential in non-relativistic quantum mechanics, highlighting its unavoidable nature and conflicts with principles of general covariance and equivalence.

Contribution

It provides a detailed analysis of the quantum potential's origin, its relation to quantization procedures, and discusses its inevitable presence and implications in non-relativistic quantum mechanics.

Findings

Quantum potential arises from quantization of systems with quadratic Hamiltonians.

Schrödinger's original quantization also leads to quantum potential, revisited in modern frameworks.

Feynman quantization identifies two versions of quantum potential with different path integral algorithms.

Abstract

The first of the 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 systems whose Hamilton functions are the positive-definite quadratic forms in momenta with coefficients depending on the coordinates in (n-dimensional) configurational space (natural systems). Owing to the Riemannian structure thus introduced in the space, the result of quantization is considered as quantum mecanics (QM) of a particle. 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. Schr\"odinger (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 induced by it are apparently inevitable in the non-relativistic QM. It is shown also that Feynman quantization singles out two versions of QP, which both determine two bi-scalar propagators which fix 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.