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A number of arguments at the interplay of general relativity and quantum theory suggest an operational limit to spatial resolution, conventionally modeled as a generalized uncertainty principle (GUP). Recently, it has been demonstrated that the dynamics postulated as a part of these models are only loosely related to the existence of the minimal-length scale. In this paper, we intend to make a more informed choice on the Hamiltonian by demanding, among other properties, that the model be invariant under (possibly) deformed Galilean transformations in one dimension. In this vein, we study a two-particle system with general interaction potential under the condition that the addition of two wave numbers as well as the action of Galilean boosts on wave numbers be nonlinearly deformed so as to comply with the cutoff. We find that the customary GUP Hamiltonian does not allow for invariance under (any kind of) generalized Galilean transformations. Those Hamiltonians which allow for a deformed relativity principle have to be related to the ordinary Galilean ones by virtue of a momentum-space diffeomorphism, i.e., a canonical transformation. Far from being trivial, the resulting dynamics is deformed, as we show at the example of the harmonic interaction.
Bosso et al. (Mon,) studied this question.