The Uncertainty Principle

In a finite dimensional space, if the commutator of two operators is a multiple of the identity, then the commutation relation is zero. We are saying that we have good reason to believe that the hilber space of our observable part of the universe is finite dimensional. It's not an absolute statement that we know the answer to a. But as you might guess, if you have a finite dimensional hilbert space bthat dimensionality is really, really like ten to the ten to the hundred and 20 dimensions, that's pretty close to infinity for a lot of local purposes.