Quantum gravity, minimum length and holography

Abstract

The Karolyhazy uncertainty relation states that if a device is used to measure a length l, there will be a minimum uncertainty \(\delta l\) in the measurement, given by \((\delta l)^3 \sim L_{\mathrm {P}}^2 l\). This is a consequence of combining the principles of quantum mechanics and general relativity. In this letter we show how this relation arises in our approach to quantum gravity, in a bottom-up fashion, from the matrix dynamics of atoms of space–time–matter. We use this relation to define a space–time–matter (STM) foam at the Planck scale, and to argue that our theory is holographic. By coarse graining over time-scales larger than Planck time, one obtains the laws of quantum gravity. Quantum gravity is not a Planck scale phenomenon; rather it comes into play whenever classical space–time background is not available to describe a quantum system. Space–time and classical general relativity arise from spontaneous localisation in a highly entangled quantum gravitational system. The Karolyhazy relation continues to hold in the emergent theory. An experimental confirmation of this relation will constitute a definitive test of the quantum nature of gravity.