In section II.1.2 of Haag's Local Quantum Physics, he lists the Wightman axioms of QFT, in particular describing an axiom (F) called "completeness":
F. Completeness. By taking linear combinations of products of the operators $\Phi(f)$ one should be able to approximate any operator acting on $\mathcal H$. This may be expressed by saying that $\mathcal D$ contains no subspace which is invariant under all $\Phi(f)$ and whose closure is a proper subspace of $\mathcal H$. Alternatively one may say that there exists no bounded operator which commutes with all $\Phi(f)$ apart from the multiples of the identity operator (Schur's lemma).
$\mathcal D$ refers to a dense subspace of the Hilbert space $\mathcal H$ on which $\Phi$ reduces to an operator-valued function on spacetime, rather than an operator-valued distribution.
In other words, all operators on the Hilbert space are approximated by expressions of the form $a_1\phi(x_1)\cdots\phi(x_{n_1})+a_2\phi(x_1)\cdots\phi(x_{n_2})+\cdots$. I can see the appeal of such an axiom, but it seems to come out of nowhere. Is there a counterpart of this in "ordinary" (non-relativistic) quantum mechanics (NRQM)?
For example, considering a single non-relativistic free particle in one dimension without spin, can all operators on $L^2(\mathbb R)$ be approximated by linear combinations of products of $\hat x$ and $\hat p$? Is this a fact of a NRQM that has always been there, but just isn't important in that context? If it is new in QFT, is there a conceptual reason why it should hold there but not NRQM?
