In CM, p is the generator of translations. On functions, exp[ i(-i) d/dx] is what translates. So the relationship [x,-id/dx]=i seems to just be some sort of group theoretic relation. Where did quantum mechanics come into this? Was it the fact that you represented the translation operator with the i in the exponential exp[ i(-i) d/dx] instead of writing it like exp[d/dx]? That is, a unitary operator operates on a state function f(x) to translate it to f(x+1)? I know in CM you can define an exponential map such that exp[operatator]x(0)=x(t), where x(t) is the particle trajectory. This looks suspiciously similar to exp[-iHt]f(0)=f(t) for the state function f(t), so I was wondering whether you can say that the difference between CM and QM has to do with the representation of the group elements that evolve time and also the objects on which the group elements act (in CM this object would be a function, and in QM this object would be a functional?), so the commutation relations are fundamental to the group but the different ways of representing the group distinguish CM from QM?