Wightman axioms. Are test functions injectively mapped to operators In AQFT test functions f are mapped to operators $\phi(f)$. This operator is said to obey a Klein Gordon equation KG ($\phi(f)$) = 0 if $\phi(KG(f))$ = 0.
This means that if the map is injective it requires KG(f) = 0.
I do not find in the Wightman axioms anything about the properties of this map. Has $\phi$ to be injective?
Thanks.
 A: Referring to general Garding-Streater-Wightman axioms for a real scalar field, KG equation is not assumed, in particular because the description should be valid for interacting fields which do not obey KG equation. However, the linear map $f \mapsto \phi(f)$ is not required to be injective in general.
In Minkowski spacetime and referring to a free field, choosing the (unique) Gaussian Poincaré invariant state on the algebra of the fields and representing the algebra in terms of operators in the Fock-space representation,  the operators $\phi(f)$ satisfy all Wightman axioms together with further specific requirements, KG equation in particular, in the form you wrote. In this case the map $f \mapsto \phi(f)$ cannot be injective as I go to explain.
More generally, if you are dealing with KG fields in a globally hyperbolic curved spacetime $M$ (see, e.g., this recent review ) the generators of the unital $^*$-algebra of fields, $\phi(f)$, satisfy, in fact, $\phi(Kf)=0$ where $f \in C_0^\infty(M)$ and $K$ is the KG operator.
Now, since the map $f \mapsto \phi(f)$ is linear, you immediately have that it cannot be injective. In particular, it is possible to prove that, it holds
$\phi(f)=\phi(f')$ if and only if $f-f'=Kg$ for some $g \in C_0^\infty(M)$.
This result can be made even more precise with the following statement.
Proposition. If $\Sigma\subset M$ is a smooth spacelike Cauchy surface of the globally hyperbolic spacetime $M$ and $N \supset \Sigma$ is an open neighborhood of $\Sigma$, for every $\phi(f)$ there is $g$ whose support is included in $N$ such that $\phi(f)= \phi(g)$.
In in particular the subalgebra generated by a region arbitrarily concentrated around a Cauchy surface coincides with the whole $*$-algebra of fields. Notice that no preferred vacuum has been fixed here.
The result is a particular case of the so-called time slice axiom.
In Minkowski spacetime, referring to the general Wightman axioms (without KG equation), it holds into a  weaker form as a theorem. Explicitly referring to the Hilbert space representation of the algebra of the fields associated with the Poincaré invariant vacuum, every element of this algebra of densely defined operators can arbitrarily be approximated, in the weak operator topology, by the operator subalgebra generated by the smearing functions supported in any fixed neighborhood $N$ of any $t$-constant spacelike hypersurface.
