# Unfinished history and paradoxes of quantum potential. II. Relativistic point of view

*E.A. Tagirov*^{1}

(1) Joint Institute for Nuclear Research, Dubna, 141980, Russia

### Abstract

This is the second of the two related papers analyzing the origins and possible explanations of a paradoxical phenomenon of the quantum potential (QP). It arises in quantum mechanics (QM) of a particle in a Riemannian *n*-dimensional configurational space obtained by various procedures of quantization of non-relativistic natural Hamilton systems. Now, two questions are investigated: 1) Does a QP appear in non-relativistic QM generated by the quantum theory of a scalar field (QFT) non-minimally coupled to the space-time metric? 2) To which extent is it in accord with quantization of natural systems? To this end, the asymptotic non-relativistic equation for particle-interpretable wave functions and operators of canonical observables are obtained from the primary QFT objects. It is shown that, in globally static space-time, the Hamilton operators coincide at the origin of the quasi-Euclidean space coordinates in both alternative approaches for any non-minimality constant *ξ*, but a certain requirement of the Equivalence Principle to the quantum field propagator distinguishes the unique value *ξ* = 1/6. Just the same value had the constant *ζ* in the quantum Hamiltonians arising from the traditional quantizations of natural systems: the DeWitt canonical, Pauli-DeWitt quasiclassical, geometric and Feynman ones, as well as in the revised Schrodinger variational quantization. Thus, the QP generated by mechanics is tightly related to non-minimality of the quantum scalar field. Meanwhile, an essential discrepancy exists between the non-relativistic QMs derived from the two alternative approaches: QFT generates a scalar QP, whereas various quantizations of natural mechanics lead to QPs depending on the choice of space coordinates as physical observables and nonvanishing even in flat space if the coordinates are curvilinear.

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