# Does "completeness" of operator fields in QFT have a counterpart in non-relativistic QM?

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?

It appears in ordinary QM for example when we use the Stone-von Neumann theorem to fix the representation of $$x$$ and $$p$$ as multiplication and differentiation on $$L^2(\mathbb{R})$$ - the statement of the theorem is that this is the unique irreducible representation of the CCR.