Commutators involving $\Box$ and $\Box^{-1}$ How to determine the followings:
$$[\Box,\frac{1}{\Box}]\mathcal{O}=?$$
$$[\nabla,\frac{1}{\nabla}]\mathcal{O}=?$$ 
$$[\nabla^2,\frac{1}{\nabla^2}]\mathcal{O}=?$$
$$[\partial^{2}_{r},\frac{1}{\partial^{2}_{r}}]\mathcal{O}=?$$
Note: In the case of $\Box$ we know they do NOT commute. But is this also true for partial derivative case? We know in some very specific form of $\mathcal{O}$ they do commute, but generally it seems they do not? 
How one define $\Box^{-1}$,$\nabla^{-1}$ and etc in terms of integral? What would be the boundaries of the integral? 
and $\mathcal{O}$ is an operator in general (one can define between scalar, vector, tensor) (the easiest is scalar of course).
 A: The inverse operators are Green's functions, this is rather common notation for them. For example, $\square^{-1}$ is the Green's function for the Klein-Gordon operator $\square$. Of course, boundary conditions (or a pole prescription) must be imposed in order to find a unique inverse (Green's function), so such a choice is implicit in this notation.
Often a slightly different version of this notation appears with the symbols indexed by their argument. For example, $\square_x$ indicates that this operator is acting w.r.t. the x-argument, and $\square^{-1}_{xy}$ indicates that the Green's function depends on the two coordinates, $x$, and $y$ (recall that Green's functions depend on two arguments).
In order to answer your question, I think one would need to know what is meant by $\square^{-1} \mathcal{O}$. In my experience these equations are always written in an explicit representation. For example, in the position space representation, this would be written as $\square_{xy}^{-1}\varphi_y$, and the double contraction over the $y$ coordinate would indicate that an integral is being preformed, where $\varphi_y$ is a quantum field evaluated at the point $y$. Can you provide an example where you run into this notation?
