Uncertainty principle in quantum mechanics with Newton's gravity

TL;DR

This paper derives a generalized position-momentum uncertainty relation incorporating Newtonian gravity, showing how gravitational effects influence quantum uncertainties and minimum measurable quantities.

Contribution

It introduces a new derivation of the uncertainty relation that explicitly includes gravitational interactions in a quantum framework.

Findings

Minimum length depends on particles' relative energy.

Gravity modifies the standard uncertainty bounds.

The relation is expressed in a new compact form.

Abstract

A new derivation is given of the known generalized position-momentum uncertainty relation, which takes into account gravity. The problem of two massive particles, the relative motion of which is described by the Schroedinger equation, is considered. The potential energy is defined as a sum of `standard' non-gravitational term and the second one, which corresponds to gravitational attraction of particles as in Newton's theory of gravity. The Green's function method is applied to solve the Schroedinger equation. It is assumed that the solution of the problem in the case, when the gravitational interaction is turned off, is known. Gravity is taken into account in linear approximation with respect to the gravitational coupling constant made dimensionless. Dimensional coefficients at additional squares of mean-square deviations of position and momentum are written explicitly. The minimum length, determined as minimal admissible distance between two quantum particles, and the minimum momentum appear to be depending on the energy of particles' relative motion. The theory allows one to present the generalized position-momentum uncertainty relation in a new compact form.