TITLE: Can $AB-BA=I$ hold if both $A$ and $B$ are operators on an infinitely-dimensional vector space over $\mathbb C$?
QUESTION [7 upvotes]: Of course, it can't hold if operators are over finite-dimensional spaces, as is evident from trace considerations. Can it be true for infinite-dimensional spaces? I think not, but I don't see how we can argue in this case.

REPLY [2 votes]: The canonical commutation relations for position and momentum operators in quantum mechanics is $$[\hat x,\hat p]=i \hbar.$$
Here the vector space is the Hilbert space over $\mathbb C$.