Quantum Mechanics and the Generalized Uncertainty Principle

There is reason to believe in the existence of a minimal length that can in principle be measured. This viewpoint stems in part from the realization that if enough massenergy is confined to a small region of space, a black hole must form. For example if one increases the energy of colliding particles beyond the Planck energy, one expects the short distance effects to be hidden behind an event horizon. In fact, as the energy is increased, the size of this event horizon increases. One way an effective minimal length might arise is through the discretization of spacetime. One might also ask the question whether the generic appearance of a minimal length in a low-energy effective theory of quantum gravity survives in the full theory when its ultraviolet sector is completed. In a theory of quantum gravity a fundamental distance scale is expected to be of order the Planck length LP . The existence of a minimal length invites the possibility that there are corrections to the usual Heisenberg uncertainty principle such that it becomes what is known as a generalized uncertainty principle (GUP):