Weak commutation relations of unbounded operators: nonlinear extensions

We continue our analysis of the consequences of the commutation relation $[S,T]=\text{Id}$, where $S$ and $T$ are two closable unbounded operators. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space \H where the operators act. {We also consider what we call, adopting a physical terminology}, a {\em nonlinear} extension of the above commutation relations.