The discussion revolves around a problem from Ballentine concerning the trace of the commutator of two operators, specifically ##[Q, P]##, and the implications of this trace in the context of quantum mechanics. Participants explore the mathematical intricacies of finite and infinite-dimensional matrices, the concept of trace-class operators, and the paradox that arises when applying these concepts to quantum operators.
Participants express differing views on the implications of the trace computation and the nature of the operators involved. There is no consensus on the resolution of the paradox or the interpretation of the trace in the context of infinite-dimensional spaces.
Participants highlight limitations regarding the definitions of operators in finite versus infinite-dimensional spaces and the challenges posed by infinite sums in trace calculations. The discussion remains open-ended with unresolved mathematical steps and assumptions.
No, it's not. An infinite sum like ##(\sum_{n=0}^\infty 1)## is divergent.PeterDonis said:[...] taking the trace (and seeing that it gives an infinite sum of terms each of which is ##i \hbar##) is simple.
Yes, I know that. I'm just saying that that infinite sum is what formally taking the trace of the infinite identity matrix gives you. I think something like that is what Ballentine intends to illustrate with the problem under discussion.strangerep said:An infinite sum like ##(\sum_{n=0}^\infty 1)## is divergent.