Unfinished History and Paradoxes of Quantum Potential. II. Relativistic Point of View

TL;DR

This paper investigates the origin of quantum potential in relativistic quantum field theory and compares it with non-relativistic quantum mechanics, revealing a deep connection to non-minimal coupling and highlighting discrepancies between approaches.

Contribution

It demonstrates the relationship between quantum potential and non-minimal scalar field coupling, and compares relativistic QFT derivations with traditional quantization methods.

Findings

Quantum potential relates to non-minimal coupling constant =1/6.

In static space-times, Hamiltonians from QFT and natural systems coincide at the origin.

Discrepancies exist between QFT-derived QP and coordinate-dependent PQs in non-relativistic quantum mechanics.

Abstract

This is the second of the two related papers analysing origins and possible explanations of a paradoxical phenomenon of the quantum potential (QP). It arises in quantum mechanics'(QM) of a particle in the Riemannian $n$-dimensional configurational space obtained by various procedures of quantization of the non-relativistic natural Hamilton systems. Now, the two questions are investigated: 1)Does QP appear in the non-relativistic QM generated by the quantum theory of scalar field (QFT) non-minimally coupled to the space-time metric? 2)To which extent is it in accord with quantization of the natural systems? To this end, the asymptotic non-relativistic equation for the particle-interpretable wave functions and operators of canonical observables are obtained from the primary QFT objects. It is shown that, in the globally-static space-time, the Hamilton operators coincide at the origin of the quasi-Euclidean space coordinates in the both altenative approaches for any constant of non-minimality $\tilde\xi$, but a certain requirement of the Principle of Equivalence to the quantum field propagator distinguishes the unique value $\tilde\xi = 1/6$. Just the same value had the constant $\xi$ in the quantum Hamiltonians arising from the traditional quantizations of the natural systems: the DeWitt canonical, Pauli-DeWitt quasiclassical, geometrical and Feynman ones, as well as in the revised Schr\"{o}dinger variational quantization. Thus, 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 altenative approaches: QFT generate a scalar QP, whereas various quantizations of natural mechanics, lead to PQs depending on choice of space coordinates as physical observables and non-vanishing even in the flat space if the coordinates are curvilinear.