I have read that noncommutative geometry is a possible way to describe quantum gravity, and that it implies that space becomes discrete at the Planck scale, where the usual notions of distance and measurement break down. However, I do not understand how noncommutativity of space coordinates leads to this conclusion.

In ordinary quantum mechanics, noncommutativity of operators means that there is an uncertainty relation between their eigenvalues, and that they cannot be simultaneously diagonalized. For example, the position and momentum operators satisfy $[x,p] = i\hbar$, which implies the Heisenberg uncertainty principle. But this does not mean that position or momentum are discrete, only that they are incompatible observables.

Similarly, in noncommutative approach to quantum space-time, the space coordinates satisfy $[x^\mu,x^\nu] = i\theta^{\mu\nu}$, where \theta is a constant antisymmetric tensor. This means that there is a minimal uncertainty in the measurement of space coordinates, and that the usual notion of point-like events is replaced by smeared events. But i have also read that it implies that space is "discretized" into Planck cells, i.e. a spacetime point is replaced by a Planck cell. How the notions of "Planck cells" are derived ? Are they analogue of *commutative points *?

I would appreciate any references or explanations that could help me understand this topic better. Thank you.