Space and time transformations with a minimal length

Phenomenological studies of quantum gravity have proposed a modification of the commutator between position and momentum in quantum mechanics to introduce a minimal uncertainty in position in quantum mechanics. In the present work, we show the influence of space and time transformations in shaping quantities such as momentum, energy, and their relations with the generators of transformations. Thus, such an influence determines, among other aspects, the time evolution of a quantum system. In the exemplary case of Galilean transformations, the Schrödinger equation is identical to the ordinary case.