In Weinberg's The Quantum Theory of Fields,volume I, the author quotes a theorem that left me a bit mystified. He states
Any operator $O: \mathscr{H} \rightarrow \mathscr{H}$ may be written
$$O=\sum_{i,j} \int C_{ij}(q,q')\, a^{\dagger}(q_1)\cdot \cdot\, \cdot a^{\dagger}(q_i)a(q_1)\cdot \cdot\, \cdot a(q_j) dq dq'$$
where the $a$'s are annihilation operators the sums range over (possibly a subset of) $\mathbb{N}_0^2$, and $dqdq'=dq_1 dq_2 ... dq_i dq_1' dq_2' ... dq_j'$.
I am not sure as to what conditions are supposed to be met for this to apply. The Hilbert space $\mathscr{H}$ is supposedly separable, but as to the operator, I don't know if it is supposed to be bounded or not.
EDIT: the Hilbert space and the creation an annihilation operators TLDR
Following jd27's comments, indeed the Hilbert space $\mathscr{H}$ is not arbitrary but a Fock space. In a Fock space, creation and annihilation operators are defined as follows.
We first recall what physicists call the occupancy basis, where we decompose the (boson, for simplicity, but for fermions we just replace the symmetry by antisymmetry) Fock space in $n$-particle states $$|n_i \rangle:=N \sum_{\sigma \in S_n} e_{i_{\sigma(1)}} \otimes ... \otimes e_{i_{\sigma(n)}}$$ where the $e_i$ are a basis of the single-particle Hilbert space, $n_i$ represents a sequence of integers such that $\sum_i n_i =n$ and $N$ is a normalization constant.
In this basis, we first define the linear maps \begin{align*} &A_n^{\dagger}(e_k) |n_1,n_2,...,n_k,... \rangle=\sqrt{n_k+1} |n_1,n_2,...,n_k+1,... \rangle \\ &A_n(e_k) |n_1,n_2,...,n_k,... \rangle=\sqrt{n_k} |n_1,n_2,...,n_k-1,... \rangle\end{align*} And thus, the creation and annihilation operators are defined for every element of the Fock space, if we put $A(f)[g]:=A_n(f)[g]$, if $f \in \mathscr{H}$ and $g \in F_n, n\geq 1$ ($F= \oplus_n F_n$, $F_0= \mathbb{C}$, $F_1=\mathbb{H}$), the same for $A^{\dagger}$, and if $g \in F_0$, $A^{\dagger}(f)[g]=g\,f$, $A(f)[g]=0$.
Now, in the important case where $\mathscr{H}= L^2(\mathbb{R})$, the creation and annihilation operators have become operator-valued distributions and as such they have the representation \begin{align*} A(f)=\int a(x) f^*(x) dx \hspace{1cm} A^{\dagger}(f)=\int a^{\dagger}(x) f(x) dx\end{align*}
The properties of the "operator distributions" inside the integral may be inferred from applying the definitions and thinking "in the sense of distributions".
Now, physicists like to treat the symbollic integrand representation of a distribution as acting on the Hilbert space itself. In that perspective, Weinberg for example writes $a^{\dagger}(q) \Phi_{q_1 \,... q_N}=\Phi_{q\, q_1...}$ where the labels $q_i$ include the four-momenta of the particle, to signify that the $\Phi$'s are states of definite momenta and other observables of interest. Of course all this is purely formal and it only attains meaning when one integrates the expressions on both sides of the equality against test functions.
The meaning of the expression on the right side of $O$ is then that, to act on a general state, you must bring it inside the integral and let the creation and annihilation operators "act on it" symbolically, if needed by "expanding the state in the 'continuous basis' of the 'states' of definite four-momenta". This is of course very formal and notational but is merely a prescription of how to distributionally think of the "little a" operators. For example, the calculation that I will mention in the next paragraph would look something like this:
$$(\Phi_0,O\Phi_0)=(\Phi_0,\sum_{i,j} \int C_{ij}(q,q')\, a^{\dagger}(q_1)\cdot \cdot\, \cdot a^{\dagger}(q_i)a(q_1)\cdot \cdot\, \cdot a(q_j) dq dq' \Phi_0)=(\Phi_0,\sum_{i,j} \int C_{ij}(q,q')\, a^{\dagger}(q_1)\cdot \cdot\, \cdot a^{\dagger}(q_i)a(q_1)\cdot \cdot\, \cdot a(q_j) \Phi_0 \, dq dq' )=(\Phi_0,C_{00} \Phi_0)= C_{00} $$ since $\Phi_0$ is a normalized "vacuum state", belonging to $F_0$ which is annihilated by the "little a's".
Returning to the main discussion, he proves this claim by induction in the annihilation-creation basis. Obviously, $O_{00}=(\phi_0,O\phi_0)=C_{00}$ so you just have to choose the first coeficient to be $O_{00} $. Then if you assume this works for $O_{MN}$, for $N<L, M \leq K$, and do the same for $O_{LK}$, you get $O_{LK}=K!L!C_{LK}+ A(C_{NM})$ where $A$ is the expression related to the products up until $q_M,q_N$. Clearly the $C_{LK}$ can be chosen to satisfy this equation.
I can't see where this proof wouldn't work, although I'm really wary of all those products of distributions. What is going on here, though? Is this precise? Why does this work? Is this a special property of the creation operators? If so, what exactly is that? If not, is this a general trend where you have a set of operators that generate a given operator algebra? What conditions then must the operators in that set verify?
I have a lot of questions but maybe describing my focus will help in providing a concise answer. In most approaches to QFT, there is a "procedure" called canonical quantization in which classical functions on phase space (which represent observable quantities) are directly replaced by operators in some Hilbert space satisfying canonical commutation relations. For example, when classical scalar fields are quantized, the approach of canonical quantization is to expand them out in Fourier modes and then replace the amplitude functions by creation operators and their adjoints. However, canonical quantization gets a bad rep. The "quantization correspondence map" that does the replacement of the functions by operators cannot satisfy all requirements of canonical quantization for instance.
Weinberg's approach is instead to say we don't need this procedure, and that the form of quantum fields (the operator distributions occurring in QFT) as possessing creation operators and their adjoints can be inferred from two principles:
- The theorem presented in this post
- A locality principle imposed on the quantum fields' interaction (S-matrix) called the cluster decomposition principle
This is why I'm so interested in knowing if these results are actually precise and what properties the $a$'s possess that are actually critical for this to work. If there is a more general and robust theory of which this example is just a specific instance, I would also like to know that.
