 The mathematical aspect of the problem is that a field at a point, \(\phi (x)\), is not an operator on a Hilbert space. The physical counterpart of the problem is that it would require an infinite amount of energy to measure a field at a point of space-time. One way to handle this situation—and one of the starting points for axiomatic reformulations of QFT—is not to consider fields at a point but instead fields which are smeared out in the vicinity of that point using certain functions, so-called test functions. The result is a smeared field \(\phi (f) = \int \phi (x) f(x) dx\) with supp\((f)\subset \mathcal{O}\), where supp(\(f\)) is the support of the test function \(f\) and \(\mathcal{O}\) is a bounded open region in Minkowski space-time. In Wightman’s field axiomatics from the early 1950s, the basic entities are then polynomial algebras \(P(\mathcal{O})\) of smeared fields, i.e., sums of products of smeared fields in finite space-time regions \(\mathcal{O}\).  Thus it replaces the mapping \(x \rightarrow \phi (x)\) in the conventional formulation of QFT by \(\mathcal{O} \rightarrow P(\mathcal{O})\).