The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write:
Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the associated Fock space over $\mathscr{H}$ with domain of essential self-adjointness $D$. Corresponding to such $A$ we can define and operator $d\Gamma(A)$ on $\mathcal{F}(\mathscr{H})$ as follows. Let $A^{(n)} = A\otimes I \otimes \cdots \otimes I + I \otimes A \otimes \cdots \otimes I+ \cdots + I\otimes I \otimes \cdots \otimes A$ on $\bigotimes_{k=1}^{n}D$. Let $D_{A} \subset \mathcal{F}(\mathscr{H})$ be the set of $\psi =\{\psi_{0},\psi_{1},...\}$ such that $\psi_{n} = 0$ for $n$ large enough and $\psi_{n} \in \bigotimes_{k=1}^{n}D$ for each $n$. $D_{A}$ is dense in $\mathcal{F}(\mathscr{H})$ because $D$ is dense in $\mathscr{H}$. Define $A^{(0)} = 0$ and $d\Gamma(A) = \sum_{n=0}^{\infty}A^{(n)}$.
My problem is that they use the same notation for different things when it comes to tensor products and densely-defined operators. So, I would like to know:
- Should $\bigotimes_{k=1}^{n}D$ be taken to be the algebraic tensor product of $D$, that is, the vector subspace of all finite linear combinations of elements of $\psi_{1}\otimes \cdots \otimes \psi_{n}$, $\psi_{i} \in D$ for every $i = 1,...,n$? Or should it be taken to be its closure, that is, the "actual" tensor product?
- What do they exactly mean by $A^{(n)} = A\otimes I \otimes \cdots + I \otimes A \otimes \cdots \otimes I+ \cdots + I\otimes I \otimes \cdots \otimes A$? Is it the linear operator $A^{(n)} (\psi_{1}\otimes \cdots \psi_{n}) = A\psi_{1}\otimes \psi_{2}\otimes \cdots \otimes \psi_{n} + \cdots +\psi_{1}\otimes \cdots \otimes A\psi_{n}$ defined on $\bigotimes_{k=1}^{n}D$ or its closure?
