 In classical physics, a field is is characterized as a scalar- (or vector- or tensor-) valued function \(\phi(x)\) on a domain that corresponds to some subset of spacetime points. In QFT, a field is characterized by means of an operator rather than a function. A field operator may be obtained from a classical field function by quantizing the function in the canonical manner – see Mandl (1959, pp. 1–17). For convenience, the field operator associated with \(\phi(x)\) is denoted below by the same expression (since the discussion below only concerns field operators). Field operators that are relevant for QFT are too singular to be regarded as realistic, so they are smoothed out over their respective domains using elements of a space of well-behaved functions known as test functions. There are many different test-functions spaces (Gelfand and Shilov 1977, Chapter 4). At first, the test-function space of choice for axiomatic QFT was the Schwartz space \(\Sigma\), the space of functions whose elements have partial derivatives of all orders at each point and such that each function and its derivatives decreases faster than \(x^{-n}\) for any \(n\in N\) as \(x\rightarrow \infty\). It was later determined that some realistic models require the use of other test-function spaces. The smoothed field operators \(\phi[f\)] for \(f \in \Sigma\) are known as quantum field operators, and they are defined as follows