Space-time uncertainty relation and operational definition of dimension

TL;DR

This paper explores how quantum mechanics and general relativity imply an intrinsic imprecision in space-time, leading to a scale-dependent operational dimension that deviates from four near the Planck length, with current experiments still far from detecting this deviation.

Contribution

It introduces a scale-dependent operational definition of space-time dimension influenced by quantum and gravitational effects, highlighting deviations near the Planck scale.

Findings

Operational dimension slightly less than four at large scales

Significant deviation from four near the Planck length

Experimental bounds are currently far from theoretical predictions

Abstract

Operational definition of space-time in light of quantum mechanics and general relativity inevitably indicates an intrinsic imprecision in space-time structure which has to do with space-time dimension as well. The operational dimension of space-time turns out to be a scale dependent quantity slightly smaller than four at distances $\gg l_P$. Close to the Planck length the deviation of space-time dimension from four becomes appreciable. The experimental bounds on the deviation of space-time dimension from four coming from the electron ${\tt g} - 2$ factor, Lamb shift in hydrogen atom and the perihelion shift in the planetary motion are still far from the theoretical predictions.