Non-local imprints of gravity on quantum theory

During the last two decades or so much effort has been devoted to the discussion of quantum mechanics (QM) that in some way incorporates the notion of a minimum length. This upsurge of research has been prompted by the modified uncertainty relation brought about in the framework of string theory. In general, the implementation of minimum length in QM can be done either by modification of position and momentum operators or by restriction of their domains. In the former case we have the so called soccer-ball problem when the naive classical limit appears to be drastically different from the usual one. Starting with the latter possibility, an alternative approach was suggested in the form of a band-limited QM. However, applying momentum cutoff to the wave-function, one faces the problem of incompatibility with the Schr\"{o}dinger equation. One can overcome this problem in a natural fashion by appropriately modifying Schr\"{o}dinger equation. But incompatibility takes place for boundary conditions as well. Such wave-function cannot have any more a finite support in the coordinate space as it simply follows from the Paley-Wiener theorem. Treating, for instance, the simplest quantum-mechanical problem of a particle in an infinite potential well, one can no longer impose box boundary conditions. In such cases, further modification of the theory is in order. We propose a non-local modification of QM, which has close ties to the band-limited QM, but does not require a hard momentum cutoff. In the framework of this model, one can easily work out the corrections to various processes and discuss further the semi-classical limit of the theory.