The following is an excerpt from Chapter 3 of the book PCT, Spin and Statistics, and All That by Streater and Wightman right after introducing some of the Wightman axioms.
The preceding assumptions define what we mean by a field in a relativistic quantum theory; however they do not yet characterize a field theory. For example, in any relativistic quantum theory the scalar field $\varphi(x) = c1$, $c$ a constant, satisfies our assumptions so any relativistic quantum theory has at least a one-parameter family of fields, albeit trivial ones. To be a field theory, a relativistic quantum theory must have enough fields so its states can be uniquely characterized using fields and functions of fields.
In old-fashioned field theory, this requirement was often met by assuming that the fields provided an irreducible set of operators satisfying the canonical commutation relations at a given time: $$[\varphi_k(\mathbf{x},t), \pi_l(\mathbf{y},t)] = i\delta(\mathbf{x}-\mathbf{y})\delta_{jk} \tag{3-7}$$ (Here $j$ runs over a suitable subset of the indices and the $\pi_j$ are some combinations of the fields and their space-time derivatives). However, (3-7) requires that the fields make sense as operators when smeared in $\mathbf{x}$ only, and this is an additional strong assumption which goes beyond our axioms. Furthermore there are hints from examples that, in general, $[\varphi(\mathbf{x}, t), (\partial\varphi/\partial t')(\mathbf{y}, t')]$ has singularities at $t - t' = 0$, even after smeared in $\mathbf{x}$ and $\mathbf{y}$. In this case it is difficult to give (3-7) a meaning. Thus, one is reluctant to accept canonical commutation relations as an indispensable requirement on a field theory.
The text suggests that the difference between a quantum field and a quantum field theory is that a quantum field acts on a state and satisfies a few axioms whereas a quantum field theory is one in which the Hilbert space of states has a basis given by the action of the quantum fields (e.g. a basis can be written by acting on the vacuum by the fields or functions of the fields). Why is this an important property for a quantum field theory to have and why would one expect that the CCR would ensure that it is true?
I was surprised reading the second half of the passage since I have always thought the CCR are fundamental in defining a field theory. How are we able to dispense of the CCR without any issues?
