  Various approaches to Quantum Gravity (such as String Theory and Doubly
Special Relativity), as well as black hole physics predict a minimum measurable
length, or a maximum observable momentum, and related modifications of the
Heisenberg Uncertainty Principle to a so-called Generalized Uncertainty
Principle (GUP). We propose a GUP consistent with String Theory, Doubly Special
Relativity and black hole physics, and show that this modifies all quantum
mechanical Hamiltonians. When applied to an elementary particle, it implies
that the space which confines it must be quantized. This suggests that space
itself is discrete, and that all measurable lengths are quantized in units of a
fundamental length (which can be the Planck length). On the one hand, this
signals the breakdown of the spacetime continuum picture near that scale, and
on the other hand, it can predict an upper bound on the quantum gravity
parameter in the GUP, from current observations. Furthermore, such fundamental
discreteness of space may have observable consequences at length scales much
larger than the Planck scale.
