  Various theories of Quantum Gravity predict modifications of the Heisenberg
Uncertainty Principle near the Planck scale to a so-called Generalized
Uncertainty Principle (GUP). In some recent papers, we showed that the GUP
gives rise to corrections to the Schrodinger equation, which in turn affect all
quantum mechanical Hamiltonians. In particular, by applying it to a particle in
a one dimensional box, we showed that the box length must be quantized in terms
of a fundamental length (which could be the Planck length), which we
interpreted as a signal of fundamental discreteness of space itself. In this
paper, we extend the above results to a relativistic particle in a rectangular
as well as a spherical box, by solving the GUP-corrected Klein-Gordon and Dirac
equations, and for the latter, to two and three dimensions. We again arrive at
quantization of box length, area and volume and an indication of the
fundamentally grainy nature of space. We discuss possible implications.
