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 it. 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?

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?