  Quantum gravitational effects suggest a minimal length, or spacetime
interval, of order the Planck length. This in turn suggests that Hilbert space
itself may be discrete rather than continuous. One implication is that quantum
states with norm below some very small threshold do not exist. The exclusion of
what Everett referred to as maverick branches is necessary for the emergence of
the Born Rule in no collapse quantum mechanics. We discuss this in the context
of quantum gravity, showing that discrete models (such as simplicial or lattice
quantum gravity) indeed suggest a discrete Hilbert space with minimum norm.
These considerations are related to the ultimate level of fine-graining found
in decoherent histories (of spacetime geometry plus matter fields) produced by
quantum gravity.
