Is the operator identity $[X,F(P)]=[X,P]F'(P)$ always true?

Question

I came up with the operator identity in my QM textbook

$$ [X,F(P)]=[X,P]F'(P) $$

where $X,P$ are Hermitian operators whose commutator commutes with them: $$[X,[X,P]]=[P,[X,P]]=0.$$ $F(x)$ is some well-behaved function.

In the book, the identity is proved by verifying $$[X,P^n]=[X,P]nP^{n-1}$$ by induction and then expanding $F(x)$ into power series. However, the identity still works under conditions where this is quite impossible. For example, $$ [X,P^{-2}]=-2[X,P]P^{-3}\\ [X,\sqrt P]=\frac12[X,P]P^{-1/2} $$

is also true at least in some cases. (the second identity requires $P$ to be positive) As for the position and momentum case, it is beyond doubt because in the $p$-representation $X$ is $i\hbar\dfrac{d}{dp}$ and the identities are obvious. But difficulties occur when the commutator is not a constant.

Question Can we prove the identity is indeed true in such cases, or are they actually not true and there are some more restrictions on the function $F(x)$?

Answer 1

As it finally emerges from your graduated comments, you certainly ought to have indicated that your Hilbert space goes beyond the image of all functions of X and P! In the language of the field, this center Y is a "constant", to the extent X and P commutations don't affect it. You need not fuss with weird operators: Indeed, the identity holds for the simplest Heisenberg group, trivially. But, in physics, a "constant" is an element of the center of the Lie algebra!

In any case, your operators $$ X=e^{-i\phi} (-\partial_\theta + i \cot \theta ~ \partial_\phi), \qquad P=\cos\theta ,\\ \leadsto ~~~~ [X,P]\equiv Y =e^{-\phi} \sin\theta , \leadsto [Y,X]=[Y,P]=0, $$ do satisfy the identity, for F Laurent-expandible, avoiding the singularities inherent in the weird operators you chose (take θ small avoiding 0). Many operator identities of this form parallel similar Sylvester's formula identities for matrices.