Quantum gravity, minimum length, and holography

Understand

The Karolyhazy uncertainty relation is the statement that if a device is used to measure a length $l$, there will be a minimum uncertainty $\delta l$ in the measurement, given by $(\delta l)^3 \sim L_P^2\; l$.

• This is a consequence of combining the principles of quantum mechanics and general relativity.
• In this note 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 foam at the Planck scale, and to argue that our theory is holographic.