TITLE: Rigorous definition of the commutator $[a(k_1), a^\ast(k_2)]$ of creation and annihilation operators in boson quantum field models
QUESTION [7 upvotes]: 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.)

REPLY [6 votes]: I have the book right in front of me and from a quick glance at the relevant section it seems that GJ chose a somewhat non-standard presentation. I think the most commonly used approach is the one explained by yuggib or suggested by user1504 which involves smearing and understanding everything in some suitable "sense of distributions". However, as you rightly noticed, GJ are really talking about statements which are pointwise in the momentum variables. Your (formal) formula (I know that sounds bad) for $a^*(k)$ which contains a delta function immediately shows that there is a problem with the interpretation as an unbounded operator on $\mathscr{F}$. GJ are well aware of that and mention
on page 96 that the domain of this "operator" would have to be $\{0\}$. Then, they (adroitly!) pirouette this by saying that they do not define $a^*(k)$, for fixed $k$, as an operator but rather as a bilinear form $\mathscr{D}\times\mathscr{D}\rightarrow\mathbb{C}$. For $\theta_1,\theta_2\in \mathscr{D}$, the definition of this bilinear form
$$
\langle \theta_1, a^*(k)\theta_2\rangle
$$
is: compute à la physicist using the "formal formula" and then declare the outcome (which has no delta function) to be the definition.
Similarly, $a^*(k)a(k')$ is not the composition of two linear maps (with suitable domains) but a bilinear form defined by the same recipe.
I should add that what is really happening behind the scene here is multilinear algebra (the way for instance representation theorists use it, often with the help of diagrammatic algebra for duality pairings or tensor contractions) in infinite dimension. This is what Laurent Schwartz calls Volterra composition in the ultimate reference on the subject: volume 3 and volume 4 of his book on distributions. He also gave a quick summary in his ICM contribution "Théorie des noyaux".
I like to call this body of knowledge the Schwartz-Grothendieck Theory because of important input of Alexander Grothendieck needed to make it work, namely, the concept of nuclear space.