On the uncertainty principle in Rindler and Friedmann spacetimes

Thomas Schürmann

doi:10.1140/epjc/s10052-020-7694-3

Abstract

We revise the extended uncertainty relations for the Rindler and Friedmann spacetimes recently discussed by Dabrowski and Wagner in [9]. We reveal these results to be coordinate dependent expressions of the invariant uncertainty relations recently derived for general 3-dimensional spaces of constant curvature in [10]. Moreover, we show that the non-zero minimum standard deviations of the momentum in [9] are just artifacts caused by an unfavorable choice of coordinate systems which can be removed by standard arguments of geodesic completion.

Highlights

One of the open problems in contemporary physics is the unification of quantum mechanics and general relativity in the framework of quantum gravity
One of the difficulties for these systems is the position uncertainty measure for the particle. This is a consequence of the issue related to the choice of the operator for the azimuthal angle
For the 2-sphere the situation is even more complicated because of the absence of a self-adjoint momentum operator related to the azimuthal angle

Summary

Introduction

One of the open problems in contemporary physics is the unification of quantum mechanics and general relativity in the framework of quantum gravity. R should rather be interpreted as uncertainty and does not describe the standard deviation of position [10,11] Both the Rindler geometry and the foliations of the Friedmann cosmology at a given instant of time are spaces of constant curvature K. For k = 1, the space is isometric to the unit sphere and the ball with maximum position uncertainty is reached for r → πa(τ ), corresponding to the total domain of measure 2π 2a3 In this case, the right-hand side of (6) approaches zero such that the momentum dispersion can be arbitrary small the position uncertainty is still finite.

The uncertainty principle in Rindler space

The uncertainty principle in Friedmann cosmologies

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