Classical limit of the commutator is a derivative?

[pellman]

I just came across the following claim:

$$\lim_{\hbar\rightarrow 0}[\frac{1}{\hbar}(AB-BA)]$$

(which approaches the classical Poincare commutator) is a derivative with respect to $$\hbar$$. I know it looks like derivative, but is it really? Please elaborate.

 

[meopemuk]

I just came across the following claim:

$$\lim_{\hbar\rightarrow 0}[\frac{1}{\hbar}(AB-BA)]$$

(which approaches the classical Poincare commutator) is a derivative with respect to $$\hbar$$. I know it looks like derivative, but is it really? Please elaborate.

The commutator can be generally written as a series

$$[A,B] = i \hbarK_1 + i\hbar^2K_2 + i \hbar^3 K^3 + \ldots$$

where $$K_i$$ are Hermitian operators. This follows simply from the fact that [A,B] is antiHermitian and that it must tend to zero as $$\hbar \rightarrow 0$$. So the limit

$$-\frac{i}{\hbar}\lim_{\hbar\rightarrow 0}[A,B] = K_1$$

(which is the classical Poisson bracket of A and B) can be regarded as the derivative of [A,B] with respect to $$\hbar$$ at $$\hbar \rightarrow 0$$.

Eugene.

 

[pellman]

Very cool. Thanks, Eugene!