Hello, I have some troubles understanding Hilbert representations for the standard free quantum particle On the one hand, we can represent Heisenberg algebra [Xi,Pj]= i delta ij on the space of square integrable functions on, say, R^3, with the X operator represented as multiplication and P operator as i times the gradient. On the other hand, we also represent the Galilean Lie group on the same Hilbert space So my question is : is that obvious that these two representations are "compatible"? Are they any constraints that one may derive by asking that the Hilbert must be the representation of both Lie Heisenberg Lie algebra + Galilean Lie algebra (in particular because P is both the momentum in [X,P] = i, and the generator of translations of the galilean group)? For instance, I could choose to represent QM on a finite interval; then it would break translation invariance. So which come first? Do we build the Hilbert as a representation of Heisenberg, and then impose some symmetry, or construct the Hilbert space as a representation of symmetry group and then define X via [X,P]= i ? Maybe this is not very clear, for I am indeed puzzled by the fact that P appears in both algebras. Thanks for any help!