Gravitationally induced uncertainty relations in curved backgrounds

TL;DR

This paper explores how space-time curvature affects quantum uncertainty relations, deriving a second-order correction depending on Ricci scalar and shift vector, with applications in general relativity and extended gravity theories.

Contribution

It introduces a new formulation of uncertainty relations in curved backgrounds, incorporating geometric effects into quantum uncertainty measures.

Findings

Uncertainty relations depend on Ricci scalar and shift vector in curved spacetime.

Derived second-order corrections to quantum uncertainties in non-relativistic limit.

Applied results to examples in general relativity and extended gravity theories.

Abstract

This paper aims at investigating the influence of space-time curvature on the uncertainty relation. In particular, relying on previous findings, we assume the quantum wave function to be confined to a geodesic ball on a given space-like hypersurface whose radius is a measure of the position uncertainty. On the other hand, we concurrently work out a viable physical definition of the momentum operator and its standard deviation in the non-relativistic limit of the 3+1 formalism. Finally, we evaluate the uncertainty relation which to second order depends on the Ricci scalar of the effective 3-metric and the corresponding covariant derivative of the shift vector. For the sake of illustration, we apply our general result to a number of examples arising in the context of both general relativity and extended theories of gravity.