so, is the commutator relation between two observables just a Lie bracket? And if so, I have two questions: I know from differential geometry that the Lie bracket of two vector fields gives me a third vector field. So, what do we mean when we say that [x,p] = i*hbar? In fact, is there at all a way to visualize (for better lack of word) these vector fields (if that is indeed what they are)? And do we always get a vector field back out (i.e. my example i*hbar). And also, I can think of a vector field as being the infinitesimal generator for some flow. If my Lie bracket vanishes (i.e. my vector fields commute), this implies that I can "flow along" the two corresponding flows in either direction (I can flow along flow 1 and then flow 2, or vice versa, and get the same thing either way). What does this mean physically? What is is saying about the universe? There's just one thing bothering me about all my reasoning: the operators, as I've understood them, represent linear transformations in a vector space. So, how is it that we can think of them as vector fields as well? Or can we take the Lie brackets of things besides vector fields? Is this kind of how we can represent any linear transformation as a matrix, then consider that matrix as an element of a matrix vector space?