In their lecture notes "Boson Quantum Field Models" (in "Mathematics of Contemporary Physics", R.Streater (ed.)), Glimm and Jaffe define an annihilation operator $a(k), k \in \mathbb{R}$ on a certain domain $\mathcal{D}$ in Fock space by \begin{equation} (a(k)\theta)_n(k_1, \ldots, k_n) = (n+1)^{1/2}\theta_{n+1}(k, k_1, \ldots, k_n) . \end{equation} (This is the equation between equations (4.4) and (4.5) in the paper.) They then make the point that the adjoint, $a^\ast(k)$, is not a well-defined operator on Fock space, although it is a quadratic form. However, in equation (4.9) on the next page, they state that \begin{equation} [a(k_1), a^\ast(k_2)] = \delta(k_1 - k_2) , \end{equation} and claim that this commutator ``can be verified directly''. I have no trouble verifying the commutator directly at the level of formal manipulation, but I am having trouble understanding how to make the calculation rigorous, or even how to make rigorous sense of the term $a(k_1) a^\ast(k_2)$ in the commutator, if $a^\ast(k_2)$ is only a quadratic form. Is there a standard way to represent $a^\ast(k_2)$ as an operator on some larger space in order to make sense of the commutator? Failing that, is there some other rigorous interpretation of the commutator?

(Glimm and Jaffe do not give an explicit formula for $a^\ast(k)$, but it is easy to derive, and can be found in many other references: \begin{equation} (a^\ast(k)\theta)_n(k_1, \ldots, k_n) = n^{-1/2} \sum_{l=1}^n \delta(k - k_l) \theta_{n-1} (k_1, \ldots, k_{l-1}, \hat{k_l}, k_{l+1}, \ldots k_n) , \end{equation} where $\hat{k_l}$ indicates that $k_l$ is omitted. I have not yet found a reference that explains a rigorous interpretation of the commutator, however.)